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MIXING LIGHT AND MATTER WAVES: PRINCIPLES AND APPLICATIONS
By
YUPING HUANG

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy
2009

 

 

ABSTRACT
MIXING LIGHT AND MATTER WAVES: PRINCIPLES AND APPLICATIONS
By
YUPING HUANG

The work of this dissertation is committed to theoretically explore rich physics
involving quantum-mechanical mixing of light and matter waves, while specifically
seeking applications in the fields of quantum interferometry, quantum information
processing, and testing fundamental quantum mechanics. Towards this goal, the
present research is guided by two lines. The first line is to study and manipulate col-
lective behaviors of multi-atom systems at quantum—degenerate temperature, where
the wave nature of atoms is maximized. Specifically, a variety of phase-coherent mix—
ing processes of two macroscopic matter-waves, in the form of gaseous Bose-Einstein
condensate (BEC), are investigated and engineered via i) tuning atomic collisional
interaction and/or inter-wave tunneling rate; ii) mixing with optical waves of phase-
locked lasers. By these means, a series of novel applications are proposed for gen—
erating highly nonclassical states, Heisenberg-uncertainty phase measurements and
ultra-fast quantum state mapping between light and matter waves. The second line
is to coherently mix single atoms with light beams in free space. It is well known that
the free-space atom-photon interactions are weak, usually dominated by incoherent
dissipation via spontaneous emission. Usable couplings between atoms and photons
are routinely realized by confining them in high—finesse optical cavities in the strong
CCupling regime. The goal of the present work is to use ultrahigh—sensitivity quantum
interferometry and the quantum Zeno effect to overcome the weak free-space atom-
PhOton coupling, thus leading to implementations of quantum information processing
in free Space.

Along the first line of this dissertation, chapter II describes a dynamical approach

 

to create many-particle Schrodinger cat states, created in a Bose-Einstein conden-
sate trapped in a double—well potential, via the technique of Feshbach resonance. A
detection scheme for cat states is proposed via revivial of the initial state. Also par-
ticularly studied is the environmental decoherence due to laser—induced interaction
between BEC atoms and the electromagnetic vacuum. Chapter III then presents
an optimized Mach-Zender interferometry scheme with gaussian number-difference
squeezed input states for sub—shot-noise phase resolution over a large phase—interval,
also implemented in the double—well BEC system. We find utilizing adaptive measure—
ment schemes allows any phase to be measured at Heisenberg-scaling precision within
only a few measurements. The scheme can be readily implemented in a double-well
BEC system. Chapter IV focuses on searching for the optimal operational strategy in
a matter-wave amplification process, in order to facilitate potential applications in the
fields of matter-wave interferometry and quantum information processing. In partic-
ular, the laser propagation effect inside two spatially overlapping spinor condensates
is studied via the semiclassical multi—mode theory.

Along the second line, chapter V proposes to generate entanglement between
single-atom qubits via a common photonic channel within the framework of quantum
interferometry at ultrahigh phase sensitivities. Via this new approach, a scalable
circuit element for quantum information processing is demonstrated at close-to—unity
success probability and fidelity. Chapter VI employs environmental dissipation to
implement on—demand interaction- and measurement-free quantum logic gates for
hybrid quantum bits (qubits) of single atoms and single photons. By forcing qubits
to stay in a decoherence-free subspace via the quantum Zeno effect, the problem of
qubit decoherence, otherwise constituting a major challenge in implementing quantum
COmputation and teleportation, is overcome counter-intuitively with the help of fast

environmental dissipation.

 

 

Once described, it is modified

-by Laozi, ~ 580 BC.

iv

 

 

ACKNOWLEDGMENTS

First, I would like to take this chance to thank my Ph.D advisor, Michael Moore,
for guiding me through the wonderland of quantum optics. He is a very “quantum”
scientist, having his own unique viewpoints on quantum physics, most of which are
simply brilliant and shining (perhaps except the everybody-being-a—robot idea). Hav-
ing the fortune of being his first Ph. D student, I learned from him every operator,
Hamiltonian and trick in quantum optics and beyond. To me, he has always been
more than an advisor. I shall never forget. that he let me stay with his family (Julie
and Brian) for more than half a year, treated me innumerable dinners, and helped us
whenever needed. I respect his personality, knowledgability, creativity and wisdom.
I also appreciate his patience and understanding on me.

Second, I want to thank my wife for endless supports through my PhD study.
While almost hating physics courses, she has been understanding and encouraging
my devotion in this field. Literally, she married a poor physics student without an
engagement ring, or even a wedding. I always understand that for me, she gave up
too much of her own opportunities. I love her.

Third, I want to thank my mom for giving me the chance to experience this
transient, but wonderful journey of life in this world, although not much has been
achieved by her son yet. I also want to thank my and my wife’s family members,
especially our two grandmas who are now living elsewhere, for their loves on me. ,

Finally, I would like to thank all professors and friends who have helped me in
various ways through my learning and researching. A few of them are L. Zhu, X. Liu,
Z. Yuan, M. Bottcher, C. Piermarocchi, S. ltiahanti, J. W. Liu, H. Lin, T. Yang and
J. Kost.

 

 

Table of Contents

List of Figures ................................. viii
1 Introduction ............................... 1
1.1 Bose-Einstein condensation ..................... 1
1.2 Quantum information processing .................. 19

2 Creation, detection and decoherence of Schrfidinger cat states

in double-well Bose-Einstein condensates .............. 35
2.1 Introduction .............................. 36
2.2 The model ............................... 41
2.3 Dynamical generation of cat states ................. 44
2.4 Detection of cat state ......................... 50
2.5 Laser-induced decoherence and losses ................ 53
2.6 Dephasing induced by atomic collisions ............... 74
2.7 Conclusion ............................... 76
2.8 Derivations and further discussions ................. 79

3 Optimized Double—well interferometry with Gaussian-squeezed

states ................................... 97
3.1 Introduction .............................. 98
3.2 Double-well MZ interferometer .................... 103
3.3 Bayesian analysis of phase uncertainties ............... 110
3.4 Optimized Double-well interferometer ................ 114
3.5 Conclusion ............................... 129
3.6 Extension: Bayesian analysis of PS and QDFT interferometers . . 130
4 Resonant Matter Wave Amplification in Mean Field Theory . . 138
4. 1 Introduction .............................. 139
4.2 The Model ............................... 142
4.3 Quantum Coupled-Mode theory .................. 145
4.4 Semiclassical Multi-mode Theory .................. 159
4.5 Conclusion ............................... 177
4.6 Derivations and further discussions ................. 178

5 On-demand generation of entanglement of atomic qubits via op-
tical interferometry ........................... 202

5.1 Introduction .............................. 203
5.2 The model ............................... 206
5.3 Implementations ........................... 208
5.4 Examples of applications ....................... 223
5.5 Conclusion ............................... 225
5.6 Further discussions .......................... 226
6 Interaction- and measurement-free Quantum Zeno gates on-
demand for single-atom and single-photon qubits ......... 239
6.1 Introduction and Overview ...................... 241
6.2 Quantum Zeno Gate: General theory ................ 245
6.3 Implementations for single atomic and photonic qubits ...... 252
64 Success Probability and Fidelity upon heralded success ...... 261
6.5 Zeno phase gate for two quantum-dots ............... 266
6.6 Conclusion ............................... 268
6.7 Derivations and further discussions ................. 271
7 Summary and Outlook ......................... 282
Bibliography .................................. 288

vii

2.1

2.2

2.3

2.4

2.6

2.7

List of Figures

Evolution of the probability distribution P(n) = [on]? versus n, with
N = 100. At t = 0, the initial state is taken as the ground state
withg = T/(0.9N) Then, the two body interaction is suddenly switched
to the negative value 9 = —T/(O.9N). The wavepacket then evolves,
where it shows a collapse and revival process. It is seen that, at a
time of t 2: 14.5/7', a well peaked cat state is obtained with two well
separated wavepackets centers at around 71. = 25, 75, respectively.

Time evolution of the expectation value of the projector Peat with sys-
tem parameters chosen as in Fig. (2.1) and a filtering window of width
no = 15. A well peaked cat state is created at t = 14.5/7, 24/T, 34/1...
Note that Peat reaches a minimum at t a: 10/T,20/T,30/T..., which
corresponds to the revival of the initial coherent state. ........
Phase distribution of the created Schr6dinger cat state in Fock space.
The y axis is the phase 6n = Arg(cn). .................
The measurement Peat versus interaction strength variation, with g 2:
0797-7V(1+69)' Other system parameters and projection operators other
identical to Fig.(2.2). It is shown that in the vicinity of 69 = 0, the
cat state is created. Note that the projection is measured at different
times for each 69, so that the value of Pcat is maximized. ......
Revival dynamics from the cat state back to the coherent state. This
figure shows the continued dynamical evolution following that shown
in Fig.2.1 with g = —0.01. At t = 14.5/7‘, the quantum system is in a

cat state, then at a time of t z 21, the initial coherent state is revived.

Dynamical evolution of a partially incoherent cat state. The parame-
ters are chosen the same as Fig.2.5. At t = 14.5 / T, the quantum system
is in a mixed state with no coherence between the two wavepackets.
Then as it evolves, the system exhibits oscillations, but will not evolve
back to the initial state. .........................
Dynamic evolution of completely decoherent cat state. At t = 14.5 / T,
we let the system collapse onto a statistical mixture of Fock-states. It
is seen that the dynamic is completely disordered. ..........

viii

46

47

49

5O

51

51

2.8 The decoherence parameter )u(s) as a function of I95 and s/AL. From
the diagram, Ms) is approximately 1 when the separation of two modes
3 is much smaller than the laser wavelength ’\L- As 3 increases, p(s)
decreases at a speed of 1/3 and eventually approaches zero as .3 goes
much larger than ’\L- Note that ,u(s) is also dependent on 65, where

at large 5, 11(3) ~ sin2 63. ........................

2.9 Magnitude of the density matrix elements for the initial coherent state.

2.10 Magnitude of the density matrix elements of the dynamical created cat
state at t = teat with no decoherence ...................
2.11 Magnitude of density matrix elements of the resulting state under weak
decoherence (a) and strong decoherence (b). System parameters are
chosen the same as in Fig.(2.1), and the resulting states (a) (b) are
measured at a time of teat = 14.5/7. ..................
2.12 Probability distribution P(n) of the resulting states corresponding to
different coherence times. Parameters are choose as in Fig. (2.11),
while from line (1) to (7), the coherent time tcoh is chosen as oo, 3tcat,
3tcat / 2, teat, 3tcat/4, 3tcat / 5, teat / 2, respectively. ..........
2.13 Effects of decoherence on the revival of the initial coherent state.
Shown is the magnitude of the density matrix elements of the system
at time t = teat + thold + trev under no decoherence (a), the decoher-
ent resulting state with thold = tcoh (b), and thold = NQtCOh/4 (c).
System parameters are chosen the same as in Fig. 2.1. ........
2.14 The evolution of the projector Peat under decoherence due to inelastic
atom loss. The curves correspond to the dynamically-created cat state
(Blue solid line), the perfect cat state (Red dashed line) and less ex-
treme cat state (2.32) (Green dashed-dotted line). The figure plots the
estimated probability to remain in a cat-like state versus time. Start-
ing from 1002atoms, it is estimated that 63 atoms are lost by the final
- _ A
time t — l—Q—lzf. ..............................
2.15 Magnitude of the resulting density matrix elements corresponding to

Fig. 2.14 at time t = Gal—$2? .....................

2.16 Probability distributions P(n.) of the six leading eigenmodes {IKI/i)} of
the density matrix shown in Fig. (2.15), where P,- is the eigenvalue for
eigenmode |\II,-) .............................

2.17 Dephasing process of cat state with nonzero atomic interaction. The
collapse is measured by Pcol = tr{pcp(t)}, where pc and p(t) are the
densities of the initial cat state and dephased state after a hold time
thold respectively. Parameters are given the same as in Fig. 2.1 while
thold has units of 1/|g|. .........................

ix

 

60
62

62

63

67

70

71

72

 

2.18

2.19

2.20

2.21

2.22

2.23

2.24

3.1
3.2

3.3

3.4

Dynamic evolution of dephased cat state after tuning on the tunneling.
The parameters are chosen as in Fig. 2.1 with a collapse time of tool =
1.5/7. In the first picture (a), the hold time is set to be thold = tcol,
while for (b), we set thold = 5tcol° It is shown that for (a), the system
still revives to the initial state, while in (b), the initial state can not
be restored. ................................
The energy gap Egap as a function of ratio g/T. Here, Egap = E1 —E0,
with E0, E1 being the eigen energies of the ground and first excited
state respectively. ............................
The wave-packet c(n) of the ground state (Blue solid) and first excited
sate (Green dashed) with (a) 9/7 = 0.01, (b) 9/7 = —0.01 and (c)
9/7 = —0.02 respectively. ........................
The probability distribution P(n.) at time t = teat is plotted versus
the level of random noise in the initial state or. We see that even at
or = 0.05 the output state is still a good cat-state. This corresponds
to random fluctuations in the initial en’s at about 5% of their exact
ground state values ............................
The phase distribution 6n = arg(cn) at t = tent is plotted for the
case of random noise in the initial state with or = 0.05. We see that
the phase distribution agrees well with Fig. 2.3 up to a non-physical
overall phase-shift. The disagreement occurs only in the regions where
the amplitude of on is small ........................
The effect of imprecise control over the atomic interaction strencth
g. The noise of g is introduced by adding a real random number,
where or is the width of the corresponding Gaussian distribution, to
the value of g in Hamiltonian (2.4). The cat state is measured at a
time of t = 14.5/ T and plotted as a function of or. It is shown that
uncertanties in 9 up to 3% can be tolerated. ..............
Dependence of the inelastic loss parameter g on the spatial size of
condensate, which is indicated by the dimensionless parameter a as
shown in Eq. (2.80). It is seen that 5 is approximately unity when the
size of condensate is much less than the laser wavelength, i.e. 0 << 1,
which means there is negligible atom loss. As the condensate size
approaches half AL, é decreases to around 0, corresponding to the
maximum loss rate .............................

The schematic graph of a MZ interferometer. .............
State engineering for MZ interferometry in the Bloch-sphere picture,
where (a) plots the interferometer input of number squeezed state, (b)
shows the state after the first BS. ....................
State engineering for M2 interferometry in the Bloch-sphere picture,
where (a) shows the state after phase acquisition, and (b) plots the
final output state obtained after the second BS. ............
A comparison of the exact ground state of a double-well BBC and the
Gaussian envelope (3.13). ........................

 

77

81

81

83

86

103

107

108

 

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

Probability packet dynamics of the double—well BEC system undergo-
ing MZ interferometric measurement with N = 800 and a phase shift
of 6 = 7r/4. Figure (a) and (b) are shown for an moderately number-
squeezed and the PS input state, respectively. In figure, the process of
phase imprinting, occurring at t = 7rT/4, is not shown. ........
Bloch-sphere analysis of the MZI with a GS input state: (a) A typical
GS state at input and output stages; (b) output states for optimized
input states with 0 = 0, iffy, 1%, i2}, 21:15, :53; (c) (continued on the
next page) geometric origin of the Jr input noise, AJ}; (d) (contin-
ued on the next page) numerical results plotting AJé/N versus a on
a log-log scale for three different N —values, validating the functional
form of AJ; derived geometrically from (c). The dashed vertical lines

correspond to 0 = 34;, where AJ}; drops to zero. ..........
A continuation of Fig.3.6. ........................
The probability distribution of optimized double-well interferometer
output for 9 = 60 = 7r/ 4, where figure (a) and (b) are for N = 100 and
1000, respectively. In both figures, scatters are exact results from nu-
meric calculation, while solid lines represent the Gaussian distribution
(3.24). ...................................
The phase density of optimized double—well interferometer, where figure
(a) and (b) are for (9 = 0 and 7r/4, respectively. In both figures the
dashed and solid lines are each for N = 100 and N = 1000. .....
Figure (a): Optimal width 0min versus N for different 0; (b): corre-
sponding minimized phase uncertainty ABmz-n. In both figures, from
(i) to (iv) the interferometer phases are 0 = 0, 0.01, 0.1 and 1. The
data points represent numerical results from strict Bayesian analysis
using the exact ground states of a double-well BEC, while the straight
lines represent the asymptotical forms of Eq.(3.26) and Eq.(3.28), re-
spectively. ................................
Monte-Carlo simulation results showing the percentage of runs which
achieved the maximum precision of 3.5 / N after (1M + 1) measurements,
plotted versus (M + 1). .........................
Pseudo Monte-Carlor simulation results of 104 runs. The true initial
phase is 7r/6, and the a-priori phase interval is [—7r/ 3,7r/ 3]. Results
are shown for the TF, PS, and GS states. In the inset, we see that the
formula from Eq. (3.31) (with one added for the final measurement
to go from 10/N to 3.5/N) agrees fairly well with pseudo simulation
results. ..................................
Phase density distribution P(cb|6) for Mach-Zehnder interferometer
with the PS input. Here P(¢|6 = 0) is a single peak located at ()5 = 0,
whereas, for 0 = 01,02, 0.3, P(d>|6) is split into twin peaks at around
45 = :l:0.1, :l:0.2, i0.3, respectively. For each 9, solid lines and dashed
lines are corresponding to N = 1000, 100, respectively. ........

xi

111

117
118

120

121

123

126

128

3.14

3.15

3.16

3.17

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Phase uncertainty Ad) versus N for several 6. Here, because of the
symmetric nature of the phase density, Adi is determined by 68% con-
fidence interval of the absolute phase distribution of P(|¢| l6).

The phase uncertainty versus true phase in the limit N ——> 00, as de-
termined by exact Bayesian analysis for large, but finite N. Results
are shown for TF and PS states based on a single measurement. Ap-
proximate fitting functions are also shown. ..............
Phase distribution P(cf>|6) for different 6s in QDFT-based interferom-
eter. From left to right, the peaks correspond to 6 = 0, 7r/8, 7r/4, 37r/8,
where for each 6, solid lines and dashed lines are corresponding to
N = 1000, 100, respectively. .......................
Plot of 01 = N Ad) versus N for QDFT-interferometry, with N from
100 to 1000. Here figure (b) is a close view of figure (a) with N E
[960,1000]. The solid, dashed, pointed and dashed—pointed lines in
both figures correspond to 6 = 0, 7r/ 8, 7r/ 4, 37r/ 8, respectively.

A MWA system configured on Raman transitions. Figure (a) plots
a toy mode showing the A-level scheme. Figure (b) shows a physical
implementation using Cesium’s Dl-line transition. Figure (c) draws
the schematic setup using copropagating beams. ...........
Figures (a), (b) and (c) show the dynamics of differential population
fraction in the off-resonance, overdamping and Rabi regimes. In all
figures, solid lines are numerical results, while dashed lines represent
formulas (4.17), (4.23) and (4.28), respectively. ............
Figure (a) shows the transfer rate R as a function of Rabi-frequency
|Q| of a resonant driving field, where F3 is fixed, given the same as in
4.2. Figure (b) shows R as a function of F3 for a fixed IQI = 1057. In
both figures, solid points are numeric results obtained by solving the
meanfield equations (4.6), while solid lines are plotted for the back—of-
the—envelope formula (4.15). .......................
Figures (a), (b) and (c) show the evolution of relative phase 612(t) in
the off-resonance, overdamping and Rabi regimes. Parameters are the
same with corresponding figures in Fig. 4.2. ..............
(Color online) Figures (a)-(c) are the surface plots of P2(E, f) in the off-
resonance, overdamping and Rabi dynamical regimes. In each figure,
the evolution time has been rescaled by the corresponding transfer time
7' obtained via QCM model, which for (a), (b) and (c) is 8.96F;1',
436.8I‘ ;1 and 1800I‘;1, respectively. ..................
Figures (a)-(c) are the surface plots of |CL(E, 5)] at 13,3) = 0 for corre-
sponding regimes in figure 4.5. The evolution time is similarly rescaled
according to the transfer time. Notice the distinct scales of color bar
in each figure. ..............................
Figures (a)—(c) are similar surface plots but for ICS(E, f)|. ......

xii

133

134

135

137

144

154

156

158

162

4.8

4.9

Dynamical evolution of the differential population fraction n(t), where
(a), (b) and (c) are plotted for the off-resonance, overdamping and Rabi
regimes, respectively. All parameters are the same as for corresponding
figures in Fig 4.5, with the evolution time similarly rescaled. .....
The same as figure 4.8 but with 1/ = 1. For (a), (b) and (c), t is rescaled

with T = 8.96r§1,436.8r§1 and 1800P§1, respectively. .......

4.10 Seff as a function of evolution time, which is rescaled with the transfer

4.11

4.12

4.13

4.14

4.15

4.16
5.1

5.2

5.3

time 7' for each dynamical regime. Parameters are given the same as
in figure 4.5. ...............................
Figures (a)-(c) are the surface plots of 612(5, f) at f, 3) = 0 for dynam—
ical regimes in figure 4.5 (a)-(c). Initially, 612(2, 0) = 0. .......
The cooperative parameter f for Thomas-Fermi condensates, where
figure (a) and (b) show the real and imaginary parts of f R and f]. In
both figures, discrete points are numerical results for Thomas-Fermi
profile, while lines are plotted for formula (4.72) and (4.76), respec-
tively. ...................................
The evolution of atom loss fraction 61083, where figures (a), (b) and (c)
show off-resonance, overdamping and Rabi regimes, respectively. In all
figures, solid lines are numerical results, while dashed lines represent
the analytical formula (4.109). Parameters for each dynamical regime
are the same with figure 4.2. ......................
The loss fraction per decibel gain 7) as a function of rescaled time for
u = 9,99 and 999. Here, N = 3 x 106 is fixed, A = 10, and all other
relevant parameters are given the same as in section 4.3.1. ......
The atom loss fraction 6103 3(7) as a function of rescaled time, where (a),
(b) and (c) are for the off-resonance, overdamping and Rabi regimes.
In all figures, solid and dashed lines are the numerical results via SMS
and QCM, respectively. Parameters for each regime are the same as
given in corresponding figures 4.5. ...................
The same as figure 4.15, but with u = 1. ................

Schematic setup of entanglement generation with optical interferom—
eters. Figure (a) shows the setup with the M2 interferometer which
consists of two linear 50/50 beamsplitters (BS). Figure (b) shows the
setup with the NOON-state interferometer consisted of only one non-
linear beamsplitter (NBS). .......................
(color online) An example of intrinsic error due to interferometer sensi-
tivity. Errors for MZ interferometer with coherent 5 (solid), TF input
17 (dashed), and N OON-state interferometer K (dashed-dotted) are plot
as functions of N6 (with N = 103), respectively. Note while 5 is de-
pendent on N 62 both 17 and K are dependent on N6. ........
Comparison of 17 033 and 17. Note that both depend only on the product
N 6. ....................................

xiii

169

196

198

199
201

208

211

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8
6.9

6.10

6.11

Level scheme for a two—state system with coherent coupling at Rabi
frequency 9 and upper-state decay at rate I‘. ............. 248
Scheme for the direct two-qubit quantum Zeno gate via a single IF M.
Each state is of the form |q1q2), where ql and q2 are the probe and
object qubits. The coherent Rabi oscillation is applied to the probe
qubit as indicated by the blue arrows, while the green arrow indicates
spontaneous decay. ............................ 249
Scheme for a two-qubit phase gate with an ancillary qubit. Each state
is of the form lqlqga), where ql and q2 are the logical qubits and a is
the ancillary qubit. Note that the coherent Rabi coupling, indicated
by the blue arrows, is applied to the ancillary qubit. ......... 252
Level scheme for the elementary IFPG and IFRG gates. The states
|V0) and |H 1) are resonantly coupled to the excited state Ie) with
coupling strength 9, while the state le) spontaneously decays at rate '7. 253
Figure (a): a schematic illustration of the physical implementation of

IFPG; (b): its graphic representation. ................. 253
Figure (a): a schematic illustration of the physical implementation of
IFRG; (b): its graphic representation. ................. 254

Graphical logic circuits for atom-atom (a) and photon-photon (b) CNOT
gate. In both figures, we use ‘0’ for the control and ‘T’ for the target
qubit. ................................... 259
Graphical logic circuits for the atom-photon phase gate. ....... 259
Scheme of the dynamics of the system under different initial states.

The blue (gray) arrow represents the electron (hole) spins. The energy
levels for QDl are the empty dot (lower) and the first exciton level
(upper). The energy levels in QD2 and QD3 are charged dot ground
states (lower) and trion levels (upper). (a)-(c) If either electron in

QD2 and QD3 is spin down, the exciton in QDl can decay into the
neighboring dots and be detected (detectors QD2 and / or QD3 switched

on). Therefore the QZE prevents the Rabi Oscillation in QDI, no phase

shift is present after a 7r pulse. ((1) Only if both detectors are switched

off, 3 7r Rabi Oscillation can be performed in QDl. .......... 267
Decay parameters A0 and A1 versus the atom-photon scattering pa-
rameters £0 and 61. Figure (a) shows A0 versus 60 for N = 102 (solid

line and circles), N = 103 (dashed line and squares), and N = 104
(dahsed-pointed line and triangles). All lines are the analytic approxi-
mations given in equation (6.26), while the points correspond to exact
numerical calculations. In figure (b) we similarly plot A1 versus N 61,
where because the analytical approximation of A1 (6.27) depends only

on the product N 61, three lines for different N s are overlapped. Both
figures show excellent agreements between analytic approximations and
numerical calculations. ......................... 270
Graphical state-transfer circuits for atom-to—photon (a) and photon-to-
atom(b), where |<I>) and |\II) denote the arbitrary atomic and photonic
states, respectively. ........................... 276

xiv

 

6.12 Schematic setup for long—distance entanglement generation with built-
in error correction. A photon in |H) polarization state is passed
through a MZ interferometer with three 7r-IFPGS, and is then mea-
sured by two detectors placed at the two interferometer outputs. The
MZ interferometer consists of two 50 / 50 linear beamsplitters (BS) with
a 7r/ 2 phase imprinted on the reflected photons. In figure, we label the
ancillary and the two logical qubits as a-, :r- and y-qubit, respectively.

7.1 The flow chart of this dissertation. ...................

XV

281

Chapter 1

Introduction

1.1 Bose-Einstein condensation

Bose-Einstein condensate (BBC) in dilute gases is a new form of matter that has
revolutionized the subject of many-body physics at low temperature. This part of the
introduction briefly reviews this relatively new subject and, particularly, discuss the
rich connections with various subjects in many-body physics. We first address i) what
a BBC is, ii) why gaseous BEC’s are so unique and important, and iii) recent experi—
mental progresses. Then, we focus on reviewing three important applications directly
related to chapters 11, III, and IV of this dissertation, including BBC in a double—
well trapping potential, matter-wave interferometry and matter-wave amplification in

dilute BEC gases.

1.1.1 The conception of Bose-Einstein condensation

Although the wave behavior of massless photons was observed by Young and
Fresnel in the early 1800’s, the wave nature of massive particles was not demon-
strated until the year of 1927, when for the first time Davisson and Germer observed

diffraction patterns of electrons analogous to that of photons [1]. The reason, now

 

well understood in quantum mechanics, is that the de Broglie wavelengthes of mas-
sive particles are normally much shorter than that of photons. Thus any observable
diffractive phenomenon of massive particles would require much narrower slits. Ac—
cording to the de Broglie relation, a massive particle of momentum p has a wavelength
AdB given by [2, 3]

h
)‘dB — 1; (H)

where h is the Planck constant. For a noninteracting free gas in thermal equilibrium
at temperature T, the average momentum p, according to Boltzmann distribution, is
determined by the relation

[)2 3

=—k T, 1.2
2m 2 B ' ( )

with m. the particle mass and k B the Boltzman constant. This immediately leads to

(1.3)

At room temperature (300 kelvin), the de Broglie wavelength for typical atoms of
10“26 Kg is ~ 10—11m, which is of the order of the Bohr radius. To observe the
wave behavior of room-temperature atoms in Young’s double—slit experiment, it would
require extremely narrow, almost inaccessible, slits of width close to atom size. They
can thus be safely treated as microscopic billiard balls under most circumstances,
exhibiting only the particle nature. If, however, we decrease the temperature to 1
mk, )‘d B will increase to 10 nm, providing more wave-like atomic behaviors. At the
temperature of T = 1 pk, AdB is ~ 1 pm, which becomes comparable to a typical
photon’s wavelength in the infrared regime. By further cooling, the atomic de Broglie
wavelength can eventually be made longer than the wavelength of visible light. For
example, AdB is obtained at ~ 10 pm if further lowering the temperature to T = 1

nk.

 

 

As they get cooled, massive particles behave more and more wave-like. For
bosonic atoms, when )‘d B reaches the inter-atom separation, a quantum phase tran-
sition occurs as an atom can no longer be distinguished from its neighbors due to
wave-packet overlapping. For an ideal gas of bosonic atoms (with integer spins), a
new type of matter forms at a critical temperature for which 70% B = 2.612, with n
the peak gas density. This new matter, characterized by a major portion of atoms
occupying the same quantum-mechanical state, was predicted by S. N. Bose and A.
Einstein in 19208 [4, 5], and has now come to be known as BBC.

The underlying physics of BBC is the quantum-mechanical phase transition in
quantum statistics of bosonic particles. Consider an ideal gas of N noninteracting
bosons confined in volume V, the number of particles in the quantum state of energy

6 is given by Bose—Einstein statistics, as [6]

1
a . .T = ,
"Bldg” ) exp[(e—,u)/kBT] —— 1'

 

(1.4)

where ,u is the chemical potential. In coordination with the fact 6 Z 0, where e = 0
simply corresponds to the zero momentum state, it must be less or equal to zero to
keep 12. B E positive. In practice, )4 is determined by the number-sum condition, where

the occupation sum of all energy levels is equal to the total particle number,

00

N = N0(T) + V/() nBE(e,,u,T)g(e)de, (1.5)

where 9(6) or \/E is the unit-volume density of states at energy 6 in the three—dimension
space. N0(T) S N is the atom number in the zero momentum state with e = 0. At
high temperature, the number-sum condition (1.5), together with the Bose-Einstein
distribution (1.4), requires a to be a large negative number and N0 << N. When
cooling the gas while keeping the volume fixed, )1 increases (decreases in magnitude)

while N0 stays small. Eventually, the phase transition occurs when T reaches a

 

critical point Tc where

00
N :2 V/ RBE(€,/L z 0,Tc)g(e)de, (1.6)
0

the condition of which leads to the critical parameter of 71A: B = 2.612. This corre-
sponds to the point when the excited states can not accommodate most atoms any
more. Further decreasing T to below Tc results in rapid building up of the population
fraction in the ground state, while remaining atoms continue to occupy higher-energy
levels according to the Bose-Einstein distribution for e > 0. Using equation (1.4) and

taking )1 = 0, it is straightforward to obtain that

T 3/2
1V0(T < Tc) = IV [I — (f) (1.7)

 

Clearly, in the limit of absolute zero temperature T = 0, then N0 = N, and a pure
BBC is obtained with all atoms occupying a single quantum mode, giving macroscopi-
cal quantum-degeneracy of translational states. It is noted that the above discussions
have assumed non-interacting atoms. In practice, however, particles may be interact-
ing with each other, in which case T c will be shifted up or down, depending on the
interaction being repulsive or attractive [7]. Further, in the above derivation, only
statistical properties of the system are used, so that the phase transition condition
of n/\(3i B = :1: (:1: is a constant determined by the atom-atom interaction) applies to
arbitrary systems consisting of spin-integer particles or quasi-particles. In fact, the
most recognized phenomena associated with BBC, before the realization in dilute
gases, was the superfiuid Helium consisting of bosonic 4He isotopes [8]. In these
two examples, the critical temperatures are 2.17 kelvin for Helium and up to ~ 100
kelvin for superconductors, both much higher than that for gaseous BEC, which is

sub microkelvin.

 

1.1.2 Why Bose-Einstein condensates in dilute gases

The question is then why a BEC in dilute gases, if after all what we obtain is

just another realization of Bose-Einstein condensation? There are multiple answers
to this question, but perhaps the most convincing one is that compared to the liquid
or solid systems, gaseous BEC’s yield much lower densities, thus substantially re-
ducin g the inter-particle interaction. In practice, such interactions have been shown
to Si gnificantly modify the nature of the BEC phase transition in liquid Helium and
solid superconductors. In fact, due to strong interactions, only ~ 10% 4He atoms in
liquid Helium can be condensated. Yet, atoms in a gaseous BEC interact far more
strongly than photons in a laser beam, the latter of which are generally considered
as noninteracting unless mediated by nonlinear medias. These, together with the
fact that the gases can be stably trapped and well isolated, make a gaseous BEC an
eXt~I“c‘I.J0rdinarily pure and easily accessible platform for rich many-body physics.

From the prospective of fundamental quantum physics, BEC’s in dilute gases
provide a unique bridge connecting quantum theory and experiments. Perhaps the
“1081: distinctive feature of a gaseous BBC is that with hundreds of thousands of
atoms occupying a single quantum state, a macroscopic matter wave is formed as a
magnification of a single-atom wavefunction by a factor of the square root of the atom
nurfiber. Imaging the BEC cloud by ordinary absorption or fluorescence method can
direCtly visualize the shape of the single-particle wavefunction. Further, by overlap-
ping two condensates, the relative phase between the two matter waves can be read
Out by simply imaging the resulting interference pattern [9]. In contrast, a similar
phase is inferred by Josephson oscillations in superconductors [10], or through the
Inotion of quantized vortices in the system of superfluid helium [11].

A second unique and very useful feature of gaseous BEC’s is that the inter-atom
interaction can be easily tuned by the technique of Feshbach resonance [12, 13]. This,

l 1‘ ' . . . . . . .
p 0Vlded With remarkable senSitIVIty of the system dynamics to such interactions, has

 

 

opened a door to a series of new quantum phenomena. A well justified example is
the realization of BCS-BEC crossover in dilute gases of Fermions. The BCS-BEC
crossover is important not only for studying superconductors, but also for testing
many-body quantum theory, for it constituting one of few non-perturbative prob-
lems of N-body systems that are presently solvable. Previous studies on BCS-BEC
crossover in superconductors have been hindered by strong interactions in such sys-
tems- In contrast, with tunable interactions via Feshbach resonance, gaseous BEC‘s
can serve as a good platform for testing the fundamental many-body physics. Re-
cently, the BCS to BBC crossover has been observed in a two-component gas of
Fermionic atoms [14, 15].
An initial gaseous BEC can also serve primarily as a well-defined starting point
for the generation of more exotic highly-correlated and/or entangled many-body
States. Important applications include implementing the phase transition from su-
Perfluid to Mott-insulator in an optical lattice and the crossover to a ‘fermionized’
TC>11ks-Girardeau gas under quasi-one—dimensional confinement [16, 17, 18, 19, 20].
Also appealing is the crossover from a vortex-lattice to a nontrivial entangled many—
body state in the ground state of a rapidly rotating BEC [21, 22]. In experiments, a
rev’Bl‘sible change from a superfluid to a Mott-insulator state was observed when cross—
ing from tunneling-dominated to the collision-dominated regime, for atoms trapped
in a. lattice potential with repulsive atom-atom interactions. The analogous transi-
tion for the case of attractive interactions involves the crossover from a superfluid
into a state often described as a ‘Schrédinger cat state’ where the atomic popula-
tion collapses into a single lattice site, with the true ground state being a symmetric
Su1T>erposition over all possible lattice sites as the final occupied site. In addition to
providing fundamental insights into the nature of the transition from the quantum to
the Classical descriptions of reality, such states may have important applications in

preciSion measurement [23, 24, 25, 26, 27] and quantum information processing.

 

Another notable application is to use a disordered BEC system as a powerful
“quantum emulator” for simulating complex behaviors in many-body physics. Dis-
order plays a critical role in condensed-matter physics as most materials inevitably
contain defects, which, in solid systems, are often difficult to control, thus hindering
our understandings. The disorder in BBC is usually provided by random potentials,
and can be remarkably controlled by adjusting trapping as well as atomic scatter-
ing parameters. This, aided by the intrinsic cleanliness of the quantum-degenerate
systems, offers unique opportunities to study disordered quantum behaviors [28, 29].
Lastly, with its well—established large degree of spatial and temporal coherence,

the system of gaseous BEC has emerged as a viable candidate for matter-wave inter-
ferometry [30, 31, 32, 33, 34, 35, 36, 37]. Compared to optical systems, a BEC-based
interferometer has the unique advantages of sensing inertial forces, measuring gravity
gradient, and allowing measurements at the single-particle level with high—efficiency
detectors [38, 39]. Furthermore, the feature of tunable interaction serves a natural
SOUI‘Ce of dynamical nonlinearity which can lead to nonclassical many-particle states
and eventually improved measurement precision [40]. Another remarkable applica-
tion of gaseous BEC is the implementation of matter-wave amplification, which is a
phElse-coherent wave mixing processing between two matter-waves and two optical-
WaVes. Further reviews on matter—wave interferometry and matter-wave amplification

Will be given in the later context.

1 - 1 .3 Overview of experimental progresses

The first gaseous BEC was created by the group of Cornell and VVieman at J ILA
in 1995, where a dilute vapor of about 2000 87Rb atoms was cooled to below 170nk
[41l- Then, about four months later, the group of Ketterle at MIT independently
Created a second BEC of 23Na atoms [42], the size of which was about a hundred

tuneS greater than the one at JILA. Shortly following the seminal work at JILA

 

 

and 1\/IIT, gaseous BBC has been realized in many other atomic systems. By now,
documented BEC gases include 7Li produced in 1995 [43, 44], Hydrogen produced in
1998 [45], 85Rb in 2000 [46], 41K in 2001[47], metastable He* in 2001 [48, 49], 133Cs
in 2003 [50], 174Yb in 2003 [51], and 52Cr in 2005 [52].

Typical experimental procedures for creating gaseous BEC include two major

stages, the laser cooling and evaporative cooling. In the laser cooling stage, a trapped
thermal vapor is first cooled to Doppler temperature, typically ~ 100pk, by the
means of Doppler cooling [53, 54]. The vapor is then further cooled to sub-Doppler
temperature of ~ 113k via a new type of cooling mechanism called Sisyphus cooling
[~55] - Then in the second stage, the cooling laser is turned off, and the trapping
Potelltial is lowered to further cool the vapor to below the BEC critical temperature.
This cooling mechanism, proposed by Hess in 1986 [56], is called evaporative cooling,
developed in an analogy to the cooling of a hot liquid by evaporation of energetic
part icles of higher momenta. It is noted the evaporative cooling is not the only cooling
method to obtain a BEC. A demonstrated alternative approach is the sympathetic
CO01ing, where a vapor is cooled by an ancillary BEC via mutual interactions [57, 47].
FiIlally, the existence of BEC is verified by time—of—fiight measurements of the atoms’
VelOCity distribution. That is, after the BBC is formed, atoms are released from the
trapping potential and undergo free expansion for a preset time. The spatial density
of the expanded vapor is then imaged by an absorption or phase-contrast method,
from which the velocity distribution is reconstructed. A sharp peak in the velocity
distribution announces the existence of a BEC.

A remarkable achievement in the field of experimental BEC is the development
of atom laser. Just as optical lasers have led to revolutionary advancements in optical
Science, the realization of the atom laser has the potential to substantially impulse
the SUbject of atom optics. The first atom laser in BBC was realized in 1997 by

t . . . .
he KGtterle group [58]. In the experiment, a BEC vapor in, say, a spin-up state Is

generated in a magnetic trapping potential. The magnetic trap is attractive to spin-up
atoms, and repulsive to spin-down ones. By applying short pulses at radio-frequency
(rf) , the condensate is evolved into a two-state coherent state of spin up and spin
down. The spin-down atoms are then ejected from the trap, and fall down due to
gravity. In this way, an output coupler similar to that for optical lasers is realized.
Since there is a matter-wave ejection for every beat of the rf pulse, a sequence of
coherent matter pulses, or an atom laser, is produced. The pulse length and intensity
can be controlled by tuning the rf pulse and the trapping potential. Such an atom
laser serves as a prospective starting point for many applications in the field of atom
optics [59].
Another notable experimental progress is loading, cooling and trapping BEC
813398 on microfabricated atom chips (for a review, see [60]). An atom chip is a
I'nicrostructure consisting of ultrathin conductors electroplated on a substrate. The
typical dimension of atom chips is of the order of millimeters or even smaller, and
the typical conductor width is only several to tens of micrometers. These conductors
allow high current densities of ~ 106 A/cm2, and can thus produce strong magnetic
IniCI‘O-traps near the surface of atom chips [61]. Such traps are compared to con-
verltional magneto-optic traps (MOT’S) employing large-scale coils of table size. By
a'rranging conductors, magnetic traps of arbitrary geometry can be produced, includ-
ing Strongly-confining ultra-long wave guides which are difficult with MOT’s due to
diffraction of the optical trapping beams. By varying applied currents, the trap size
and potential barriers can be fine tuned, providing the feasibility to implement, for
e"CaIIIpIe, superfluid to Mott-insulator phase transition in on-chip lattices. All these
give rise to the realizations of integrated atom-optics devices on miniaturized chip-
Cells, including portable and assemblable atom interferometers, quantum information
p r OCessing units, Tonks—Girardeau gas cells and so on. In experiments, the first BEC

gas 011 an atom chip was produced in 2001 by Ott et al [62] and by Hansel et al [63].

In 2005, matter-wave interferometry of BBC gases was demonstrated on atom chips
[64, 65, 37].

Other important experimental developments not elaborated here include the
observations of vortices in a spinning BEC, superfluid to Mott-insulator transition
in an Optical lattice and the crossover to a ‘fermionized’ Tonks—Girardeau in 1D
[16, 1 7, 18, 19, 20]. All these progresses give rise to implementations of rich many-body
physics at ultra-low temperature and applications in the filed of quantum interfer—
ometry and quantum information processing. In the following, we will review three
important applications related to the work of this dissertation, including double—well

BEC, BEC-based matter-wave interferometry and matter-wave amplification.

1. 1 .4 Double—well Bose-Einstein condensates

For various applications, a gaseous BEC system consisting of two well-isolated
yet. weakly-coupled quantum components, generally referred to as a bimodal BBC,
is reCOgnized as a well-justified platform [66, 67, 68, 69, 70, 71]. Its power is from
its exceptional simplicity and high controllability compared to competing systems
SUCh as Josephson—junction of superconductors. The two modes can be two internal
hyperfine states or two external spatial modes. In the first case, the modes are
Spatially overlapped, and are connected by coupling lasers [72, 73]. In the second
Case, the two modes are spatially separated, each in one well of a double-well potential
[74: 75] . Connection between the two modes is provided by weak inter-well tunneling.
The main difference between the two models is the presence of inter-mode cross-phase
modulation in the hyperfine system, as a consequence of mode overlapping. Its effect
is nonetheless physically equivalent to an addition of self-phase modulation possibly
together with a relative shift in chemical potentials of the two modes, and is thus
unimportant for most applications. For concreteness, this dissertation is focused on

t} .
le blmodal BEC system implemented in a double-well trapping potential, or the

10

 

 

so—called “double-well BEC”.
At zero temperature, the system is conveniently described in the two-mode ap-
proximation, in which the annihilating field operator ‘Il(r) for atoms at position r is

delocalized into orthogonal localized left— and rights modes,

A

an t) = eutwur) + 6120mm. (1.8)

where 6- L and 5R are bosonic annihilation operators for the left and right modes,
respectively. The localized modes correspond to the wavefunctions of atoms in left-
and right- well, formally defined via a linear transformation of the ground and first-
excited states. In this approximation, the system’s dynamics are governed by a two-
mode version of the Bose-Hubbard Hamiltonian, as [76]

‘ . c - . . 2 -2 .[2.2

H = —h'r(cchR + CTRCL) + 59(CE CL + CR CR), (1.9)
where ’T describes the inter-mode tunneling, and g is the strength of nonlinear intra—
mode interaction. The quantum state of this bimodal system is written in a super—
position of number-difference states In) [70]

N/2

It) = 2 cm), (1.10)
n=—N/2

Where
(CE)1V/2+72(C1;2)N/2—n

 

N N
['TL)E -2-+n,—+n =

2 '[vacuum). (1.11)

\/(N/2 + 72.)!(N/2 — 72.).
Her e, N is the total number of atoms in the condensate which, for convenience, is
taken as a even number, and n denotes half number difference between the two modes.
In praCtice, 7' 'can be easily adjusted by varying the trapping potential, while 9 can be

preCiSely tuned via the technique of Feshbach resonance [12, 13]. This extra freedom

11

 

of controllability has allowed a variety of important applications.

The first application is to implement a full range of Josephson-junction dynamics
in a neutral-atom system, in an analogy to charged superconducting junctions. This
includes the well-known dc, ac and Shapiro tunnelling phenomena, and most remark-
ably, the macroscopic quantum self-trapping (MQST) [67, 68]. MQST is a striking
quantum—equilibrium phenomenon where the tunneling between the two modes is
self-frozen with more atoms in one well than the other. It occurs when nonlinear
atom interaction is sufficiently strong to shift the tunneling out of resonance. Thus
by tuning 9 via Feshbach resonance, the system can be switched between Joseph—
son oscillation and MQST regimes. All these effects give rise to new apparatuses in
atom Optics with interacting matter—waves, which have not been possible with con—
ventional optics of noninteracting photons. Experimentally, the Josephson oscillation
and self-trapping was first observed by the Oberthaler group in 2005 [71].

A second important application is to generate a variety of highly nonclassical
states, including number squeezing, phase squeezing, and the macroscopic Schréidinger
cat States. In the absence of intra—mode interaction, the system’s ground state under
Hamiltonian (1.9) is a two-mode coherent state, corresponding to a Gaussian distri-
bution with on or exp(—n2/02), with the distribution width 0 = \/'1\7. In presence of
some repulsive interaction with g > 0, the ground state becomes number squeezed,
corresponding to a Gaussian with a reduced width of a < W. This is because in
or der to minimize the interaction energy, atoms tend to be localized in each well. In
the limit of large 9, the system’s ground state is a twin—Fock state with on = ",0.
This State is characterized by an equal and definite number of atoms in two quantum
InodeS. On the other hand, in the presence of attractive interaction with g < 0,
the ground state becomes phase squeezed, characterized by a spreading in number
distribution with a > \/]_V_. In the limit of large attractive interaction, the ground

St , . . . . .
ate tllrns out to be a superposmon of all atoms in the right and left well, which

12

\

 

 

is an N—atom realization of the famous Schrodinger—cat state [77]. Starting with a
double-well BEC system in the ground state without interaction, a whole series of
nonclassical states, ranging from number squeezing to phase squeezing, can be pro-
duced by adiabatically turning on interactions via Feshbach resonance. Such states
are proven to be very useful in testing fundamental quantum mechanics [78, 79] and
for applications in the fields of quantum interferometry and quantum information
processing [37, 80, 81, 82, 83]. It is noted that in practice, Schrodinger cat states
cannot be adiabatically generated by tuning g, hindered by the energy degeneracy of
8r011nd and first-excited states. Instead, it can be produced via a dynamical scheme,
as Will be discussed in Chapter II in this dissertation.
Furthermore, another well- j ustified application is to implement. ult.rahigh-sensitivity
mat ter—wave interferometry in tunneling multi-mode BEC systems, as will be dis-

cussed in the next s1.1bsection.

1 - 1 - 5 Matter-wave interferometry

An interferometer is a measurement device which exploits the wave nature of pho—
tons and/or massive particles to explore inhomogeneities in probing environments.
The first interferometry on file, which dated back to the 19th century, was imple-
mented with interfering optical rays [84, 85, 86]. From then on, optical interferom-
eters have been serving as primary tools for precise measurements in various fields.
Only much later was the first matter-wave interferometry demonstrated, including
that Of employing electrons in 1952 [87], and neutrons in 1962 [88]. Today, matter-
Wave interferometers using electrons, neutrons and atoms as probes, have been widely
recognized as invaluable tools for testing fundamental physics, precise measurements
of physical constants, as well as a variety of important applications in astronomy,

geo . . . . . .
thetry and in the emerging field of quantum information processmg (for rev1ews,

Se _
Q [89, 90, 91,331).

13

Compared to optical interferometry, interferometers built on matter waves have
several immense advantages. In the first place, atoms are massive particles subjected
to inertial-frame acceleration forces, such as gravitation. This makes matter-wave in-

terferometers sensitive detectors for probing inertial effects, for instance, gravity gra-
dients, since phase shifts induced by inertial forces increase with the mass of probing
particles [92]. In practice, this feature has remarkably led to developments of atom-
based accelerometers, gradiometers, rotation meters and gravimeters [93, 94]. Second,
atoms are sensitive to the external electromagnetic (EM) force, making atom interfer-
ometers powerful devices for precisely measuring local fluctuations of EM fields. This
is contrary to optical interferometers employing photons, which basically non-interact
With external EM fields. A notable application is the measurement of the Aharonov-
Casher phase via matter-wave interferometry in the middle of 1990’s [95, 96]. Third
and perhaps most importantly, matter-wave interferometers employing atoms have
the most unique advantage that interfering atoms are rich in internal varieties such
as h}’I)erfine structure, excited levels, isotopes and so on. This provides the possibil-
ity of differential interactions of probing beams with local environments, providing
Selec’tive measurements of external quantities. Using this feature, in practice, several
iInI)OI‘tant fundamental experiments were carried out, including testing the equiv-
alenCe principle in general relativity [97] and measuring the Aharonov-Bohm effect
[98] ,
This far, a variety of atom interferometers have been proposed, all of which rely
on Creating matter-wave gratings. For thermal atoms with short de Broglie wave-
leIlgthes, the gratings are formed either by a tight set of tiny apertures (e.g., narrow
slits) in a solid membrane, or a tight standing-wave grating formed by interfering
aserS- For ultra-cold atoms, such as BEC gases, the de Broglie wavelengths become
11'1th longer, allowing the use of a set of new, less-tight gratings at optical dimen-

Sio . . . . .
118: Such as tunneling optical lattices or micro-traps on an atom chip [35]. More

14

importantly, the contrast of the interference signal, which determines the measure-
ment sensitivity, increases with the brightness of the source of probing particles, i.e.,
the number of particles per quantum mode. With macroscopic occupation of a single
quantum mode, the advent of BBC gases holds promise to promote ultra-sensitive
matter-wave interferometry, in a way that the invention of laser has revolutionized
optical interferometry [30, 9, 32, 31, 33, 34, 35, 36, 37]. Another remarkable feature is
that , the atom-atom interaction in gaseous BEC is adequately tunable by the method
of Feshbach resonance. This provides the feasibility to implement some beneficial non-
linear interferometric procedures, as well generate number or phase squeezed input
states - All these give rise to significant improvements on interferometry performances.
An archetypical, perhaps most notable BEC interferometer is implemented in a
d0111:)le-well system, the configuration of which was discussed in the above section.
To implement a double-well interferometer, two phase-coherent condensates are pre—
pared , usually by coherently splitting one condensate into two through adiabatically
changing the shape of the trapping potential. The two condensates then separately
interact with each’s local environment, acquiring independent phase shifts. Finally,
they are recombined to form interference pattern. The relative phase information is
then extracted by measuring the interference pattern. In practice, a first proof-0f-
priIlCiple experiment was carried out by the Ketterle group in 1997, where two BEC’S
Wer e prepared independently in a double-well potential [9]. The potential was then
turned off, allowing the two gases to freely expand. As they overlapped, high-contrast
interference fringes were observed. If, on the other hand, spatially-inhomogeneous dc
a1'ICI/ Or rf magnetic fields are applied to disturb the condensate phases during free-
e3(138118ion, the fringe contrast is observed to be significantly reduced. This confirmed
that fringes were determined by the relative phase of the overlapping matter-waves.
Later in 2004, a whole procedure of double—well interferometric measurement was

de
ITlOIlstrated by the same group [31]. In the experiment, a condensate was split

15

into two phaselocked condensates by deforming a single-well potential into two sep-
arated wells, thus realizing a coherent matter-well beamsplitter analogous to optical
beamsplitters. The two separated condensates were then applied separately with ac
Stark shifts, followed by recombination via free expansion. Phase information is then
extracted from interference patterns, where good agreements are found between ap-
plied and measured phases. Recently, matter-wave interferometry was demonstrated
on atom chips [64, 65, 37]. This holds the promise to build miniaturized, portable

measurement devices for astronomy, geology and military uses.

1. 1 - 6 Matter-wave Amplification in BEC

hrlatter-wave amplification (MWA) in BBC is a stimulated four-wave mixing
(FVVIV'I) process in which two coherent matter waves, a pump mode and a signal
mode, are mixed with a single pump laser [99, 100, 101, 102, 103, 104, 105, 106, 107,
108, 109, 110, 111, 112, 113, 114, 115, 116, 117]. The pump and signal waves are two
BBC ’S in different momentum or hyperfine states, with a well-defined relative phase.
During the MWA process, a photon from the pump laser is scattered by a pump
atom , imparting a momentum kick of photon recoil. Due to the quantum stimulation
Effect, the kicked atom preferentially scatters into the populated signal wave. Con-
SeQUQntly, a fourth coherent wave, namely the idler wave, is generated in the form
of a. light beam emitted along the phase-matching angle. Generated in a coherent
proceSs, the phase of the optical idler beam is locked to be the relative phase between
the Signal and pump matter-waves. As the amplification process preserves the rela-
tive phase between the two matter waves, MWA has been viewed as an active device
with potential applications in matter-wave interferometry [118, 37, 81, 119, 83, 40].
This point of view, however, is limited by the fact that the signal matter-wave is
Eirrlplified at the expense of the pump wave, whose depletion ultimately suppresses

th
e IVIVVA gain mechanism. In addition, since the idler light carries the relative

16

phase information between the two matter waves, MWA can in principle allow for the
’mapping’ of an arbitrary quantum state from a two-mode atomic field onto a two—
mode optical field [101, 100, 112]. This has the potential to revolutionize the field
of matter-wave interferometry and quantum information processing. For example,
the application of matter-wave interferometry is largely hindered by the fact that the
relative phase between two matter-waves is read via the interference pattern between
the two, upon spatially overlapping, which primarily restricts such measurements to
be Within short spatial range. However, by using a pair of phase-locked lasers to si—
mul taneously drive two distant MWA systems, and then collecting and interfering the
emitted idler lights, the relative phase information of the two matter-wave systems
can be read out, without physically bringing them together. This leads to elimi—
nating the short—range restriction in the matter-wave interferometry. Furthermore,
the reverse process of MWA, in which an optical signal is amplified along with the
generation of an atomic idler field, works equally well via the same mechanism [108].
ThUS quantum information generated in a matter—wave field could be transferred to
an Optical field for detection or long-distance transfer to a second bimodal atomic
System [120]. In some sense, MWA can be viewed as a dynamical counterpart to adi—
abattic electromagnetically-induced-transparancy (EIT) based quantum storage and
retrieval techniques (for a review, see [121, 122]).
There are two theoretical models for the MWA systems. The first one, pioneered
by the work of Moore and Meystre [103], is based on quantum treatment of light
and matter coupling in Markovian approximation [123]. In this theory, the scattered
light field is eliminated to form a closed set of dynamical equations for atomic fields
only, which are then further simplified by assuming time—invariant momentum side
modes [105, 124, 125]. We henceforth refer to it as the quantum coupled mode
(QCIVI) model. Due to the elimination of scattered light, the MWA manifests itself

-n atomic stimulation process, in which decaying into a. particular Side mode is

17

enhanced by a factor of the mode occupation number. In the second semiclassical
picture, however, the MWA is viewed as an ordinary Bragg scattering process, in
which atoms are diffracted by optical standing waves formed by the laser and scattered
beam [126, 127, 128, 129, 130, 131, 132, 133, 117]. The superradiant enhancement is
due to the diffraction efficiency being proportional to the square of the grating depth,
and thus the atom number in the signal wave. Unlike QCM, the semiclassical model
is capable of incorporating multi—mode treatment of the light-matter coupling in a
more computationally accessible way, via the Maxwell—Schrodinger equations. We
thus call it the semiclassical Maxwell-Schrodinger (SMS) model.

While the two show fair agreements in the weak driving regime, the QCM model
is demonstrated to fail in interpreting several important observations in the strong
driving regime [134, 108]. Such include the asymmetry between the forward and
baCkWard superradiance mode, and the subexponential growth of the scattered field
[1 29, 131], both of which are, however, well explained in the SMS theory. It turns
011‘: that the invalidity of the QCM model is due to employing time-invariant spatial
modes , thus excluding any spatial dynamics. Consequences of spatial mode evolution,
usl-lir‘tlly referred to as spatial (or propagation) effects, are shown to be important
in the strong driving regime when atomic motions can be neglected [127]. In this

CliSS€31‘tation, we will explore the differences and connections between the QCM and
SKIS model in chapter IV.

In a close examination of the quantum approach employing master equations,
\ve nOnetheless find the spatial effect naturally arises as a consequence of the retarda-
tion Effect of the collective atomic correlation via light medium, when appropriately
treating the second-order perturbation. Physically, the retardation and thus the spa-
tial effect is intuitively understood in a picture where along any light propagation
direction, an atom in an ensemble only “sees” other atoms sitting behind it, but

no . . . . .
t t1lose in front. Thus for light traveling along 2, the strongest two—body nonlin-

18

ear effect occurs at far +2, as having been observed [108, 135]. A master equation
incorporating the spatial effect can then be viewed as a “quantum” version of the
semiclassical Maxwell-Schrodinger equation, with the extra capability of describing
the initialization of superradiance triggered by vacuum fluctuations, as well as the
induced dipole-dipole interaction due atomic exchange of virtual photons. We note
the dipole-dipole interactions would potentially spoil the MWA, by distorting the
seed—mode phase via cross-phase modulation. In practice, however, this quantum
approach is much more computationally complicated, when solving for the dynam—
ics beyond few-mode approximation. In contrast, the Maxwell-Schrodinger method
iS more computationally accessible, especially within the slow-varying envelope ap—
PrOXimation, and has been employed to successfully explain the. asymmetric behavior

observed in recent MWA experiments.

1 - 2 Quantum information processing

The technology of quantum information has emerged as a promising candidate
for the next-generation information processing, having the potential to substantially
a'(lV’Etnce our modern society. In this part of the introduction, I will give an overview of
this emerging field, presenting i) the conception of quantum information processing; ii)
the advantages over classical information processes; iii) implementations of quantum
in fOrInation processes; and iv) important recent experimental progresses.

1 ‘ 2.1 The idea and advantages of quantum information pro-

cessing

The technology of information processing has proven to be the key to recent
mense advancements in c1v1hzation, in various aspects ranging from seience, eco-

no - . .
mlcs to industry. The past decades have Witnessed tremendous developments of

19

information technology, mainly attributed to the exponential miniaturization of elec-
tronic elements, whose speed follows the famous Moore’s law which sets the objective
that the number of transistors on a chip doubles every one and a half year. To date,
the advanced computational devices are now employing microchips with microscopic
transistors linked with strip-conductors of ~ 10 nm in width. This corresponds to a
width of ~ 1000 parallel atoms, providing the validity to describe the system with
classical physics. If Moore’s law persists, soon in ten years from now, the strip con-
ductors will be miniaturized to a width of only tens of atoms. Under this scale,
however, the classical laws break down, and quantum effects would unavoidably oc-
cur- In modern technologies, information is digitally coded in the form of sequential
electric currents or light pulses. Entering the few-particle quantum regime would
then lead to loss of information precision, due to inherent fluctuations caused by the
Heisenberg uncertainty principle, as well as the probability nature of quantum mea-
surements. The quantum effects thus put an ultimate limit on the maximal number of
electronic elements on a unit microchip, and thus the achievable performance of cor—
responding information processing devices. Such a constraint is from physical laws,
and thus cannot be conquered by technological advancements. This gives rise to the
supposition that it is impossible to advance information technology soon as further
miniaturization is prohibited by quantum effects.

80, is this the whole story? While quantum effects are unavoidable, why not
just GXploit them for information processing, possibly with a novel coding system
other than the classical digitalization? If this is the case, then the miniaturization
can be pushed to the minimal single-particle level, potentially improving information
‘prOCQSsing capabilities by many orders of magnitude [136, 137]. Even more appealing,
can (Illantum effects expediently reduce the computational complexity associated with
110"Vaday computers, for example, in searching large databases [138]?

These thoughts regarding processing information in quantum-mechanical systems

20

have led to a new information science, namely quantum information processing (QIP).
Although the first ideas of QIP were proposed in the early 19803 [139, 140], only
much later was it recognized as the next-generation information technique, thanks to
several milestone experimental progresses (see [136] for a review). In the QIP, unit
information, namely a quantum-bit (qubit) analogous to binary bits, is encoded in
the quantum state of a quantum-mechanical object. Such an object can be a single
particle, such as an isolated atom, nuclei or ion (approximately) occupying by two
electronic or translational states, or a polarized photon traveling at the speed of light.
01' it can be a mesoscopic object in collective states, such as an ensemble of atoms,
01‘ a light beam containing over thousands of photons. In any physical form, a qubit
is described by a quantum superposition of two logic bases, namely [0) and [1), in a

ge Iieral density form of
1

 

I I2 I |2(col0>+61|1>) (1.12)
(:0 +61

Where (:0 and c1 are arbitrary complex numbers. This is in contrast to a classical bit,
WhEre the state is either [0) or [1), but not the superposition of the two.
One of the most striking applications of the superposition principle is the quan-
tum parallel computation, also widely referred to as quantum parallelism. In this
al gorithm, each basis state is treated as a single argument to a function and an op-
eration on the quantum register is then simultaneously performed on all arguments.
This allows generation of parallel outputs within only a single operation. In classi-
Cal processing, in contrast, each argument must be addressed individually, so that
a SirIlilar task would require an exponential number of steps. For example, suppose
our goal is to compute all values of a function g(x), where x is a binary vector of
163118th N. In classical computation, this calculation requires either one copy of the
CEIleulation circuit and 2N time steps, each with a different input. Or one time step

b
u t With 2N copies of the circuit and 2N different inputs. In either way, the computa-

21

tional complexity grows exponentially with N. In quantum parallelism, however, one
can simply construct an operation defined by [:r) +—> g(x)|:z:). Preparing an N -qubit
quantum register of an equally weighted superposition state

[\rm): |00~-0>+|10-~0)+---|11---1), (1.13)

I
—(
‘/2N
with the state basis [ij - - - k) : [-1')1[j)2 - - - [M3, and performing the operation g(x)

once, the register will evolve into an output state of

ppm“) .—_ \/——-:=N(g(00~-0)[(l0~-0) +g(10...0)[10...0> +-~g(lI---1)[ll-~1).

(1.14)
In t: 11 is way, all 2N values of g(xi) are obtained using a single quantum circuit and in a
Single time step. This literally corresponds to a dramatic reduction of computational
complexity by a factor of 2N from the classical algorithm.

The catch, or more precisely, the seeming catch, of this quantum parallelism is
that While multiple outputs can be generated simultaneously, it is impossible to read
tllem all out. In fact, according to the quantum measurement postulate, only the
amplitude of g(xi) can be statistically determined by the means of many repeated
E“3‘:I>el‘imental runs. This supposedly makes quantum parallelism useless. What saves
it is that in many cases only the aggregate properties of the g(x,)’s are of interest.
Rather than reading out each individual g(xi), one can instead use quantum paral-

leliSm as the intermediate calculation step, while not performing measurement until
the end. Along this line of thought, there are now several important algorithms via
quantUm parallel computation, which are demonstrated to be much more efficient
than via classical algorithms. The first one is the Deutseh algorithm proposed by

avid Deutseh in 1985 [140], where it was shown that the task to determine whether
an unknown function is balanced or constant can be performed in a quantum com-

D11 . . . . . .
1191‘ in one call, while a classrcal computer requires two. The g1st of the Deutseh

22

algorithm is to apply a single-qubit operation after the quantum parallel operation,
a-priori to measurement. The net effect is to change the measurement basis from
{[O) , [1)} to {([0) :l: [1))/\/§}. In this way, it is the relative phase between [0) and
[1) after the parallel operation determines the measurement result, thus becoming
an observable. Later in 1992, the Deutseh algorithm was generalized to an N -qubit
version, namely the Deutseh-Jozsa algorithm [141], where the balance-constant prob-
lem , requiring ~ 2N evaluations in classical solution, is solved in a single evaluation
with the quantum algorithm. This literally reduces the computational complexity
by a factor of 2N. Later in 1995, Peter Shor devised a landmark for efficiently find—
ing the periodicity of a set of integer numbers using the discrete quantum Fourier
transformation (DQF T), which is now widely known as Shor’s algorithm [142]. The
Shor’s algorithm is particularly useful in factorizing large numbers, for example, into
tVVO prime numbers, for it reducing the exponential complexity in classical algorithm
to Only polynomial scaling. In practice, a proof-of-principle experiment was demon-
Strated with a NMR—based quantum computer in 2001 [143]. Another revolutionary
quantum algorithm is the Grover algorithm proposed by Lov Grover in 1996, which
ConceI‘ns the problem of efficient searching in an unstructured and unsorted database
of size N >> 1 [144]. In classical algorithms, this task requires typically ~ N / 2 oper-
ationS. In the Grove algorithm, however, only ~ WV Operations are required, giving
an improvement factor of ~ 103 for moderate N = 106.

A second remarkable application of QIP is quantum cryptography, also referred
to as quantum secret-key distribution. The purpose of cryptography is to prevent
information leakage t0 an unauthorized third party during long-distance communi—
Cations. Quantum cryptography is a novel cryptographic protocol utilizing quantum
propel‘ties to radically eliminate the possibility of eavesdropping by physical laws
[1 36] - This is in contrast to classical cryptography approaches, where however pro-

O‘r . .
(3 am med, secret codes are always subjected to the risk of eavesdropping. To explain

23

quantum cryptography, we first briefly review the principles of classical cryptography
[137]. There are two major methods for classical cryptography. One is to use public-
key encryption, the security of which relies on the fact that certain mathematical
functions are extremely difficult to invert, for example, the factorization of a product
of two large prime numbers. The problem is, however, there is no mathematical proof
that an efficient algorithm does not exist; in fact, the aforementioned Shor’s algorithm
is designed for this exact purpose (which explains why Shor’s algorithm has attracted
so much military attentions). The second option is to use the so-called Vernam ci-
pher [145], where the information sender and receiver share a random, unique pair of
binary sequences, the secret key, for encoding each message. The Vernam cipher is
absolutely secure because there are no patterns for cryptanalysts to recognize. The
difficulty with this scheme is how to transfer these secret keys absolutely securely
between the sender and receiver.

This question remained unanswered until 1984, when Bennett and Brassard pro-
posed to use the quantum no-clone principle to detect the presence of eavesdropping
[139] - In the scheme, the sender transmits a sequence of randomized keys in quan-
tum Sliperposition states to the receiver. If eavesdropping occurred, keys received
by the receiver will be different from the original keys, due to the fact that 1) a
SuperPosition state can not be duplicated and 2) any measurement will immediately
COllapse it to one of the eigenstates according to measurement bases. By randomized
rneasuring on the received keys and comparing some measurement results with the
Sender, the receiver and sender can tell if eavesdropping has occurred or not. If yes,
the procedure is repeated. If no, then a secret key can be generated from the re—
Irlaining key sequence. Another quantum cryptography method is built on quantum
entanglement, using Bell’s inequality to detect eavesdropping [146]. In this scheme,

a Sequence of entangled qubits are generated. For each pair, one is sent to the sender

an
d the other to the receiver. The sender and receiver then each measure the qubits

24

they received on randomly chosen bases. Afterwards, they communicate the mea-
surement bases through public keys. Then, they compare the measurement results
via public keys for those measurements performed under orthogonal bases, based on
which Bell-inequality analysis is performed. If Bell’s inequality is violated, then the
eavesdropping did not occur with certainty, so the secret keys are secure. In this case,
measurement results performed on the same bases are used as secret keys. Other-
wise, secret keys are considered eavesdropped, and the whole process is repeated. The
theme of this protocol is that any measurement on the entangled pairs will inevitably

destroy the entanglement, in which case the Bell inequality can not be violated.

1. 2- 2 How to process quantum information

Having seen its power, in this section, we present required elementary operations
t0 execute QIP. To proceed, it is illustrative to recall that in classical information pro-
cessing, there are two basic protocols, namely information computation and network.
The information computation is operated with a series of basic logic gates, such as
One-bit NOT and two—bit CNOT gates. The information network is accomplished by
firSt Converting the solid bits, e.g., coded in directed ferromagnetic materials, into
propagating bits, e.g., in the form of sequential electronic or light pulses. The propa—
gating bits are then transported via wires, optical fibers or free space. The processes
of quflntum information are very analogous, except that with qubits, the computation
and network are executed with quantum logic circuits and quantum teleportation.

A quantum logic circuit consists of single-qubit and two—qubit gates. There are
three kinds of widely-used single-qubit gates, the NOT gate, Z gate and Hadamard

gate [147], which in the state basis {[0), [1)} are defined by matrices of

0 1 1 0
NOT : , Z : , Hadamard :

1
— (1.15)
1 0 0 —1 fl 1 —1

25

In the Bloch-sphere representation of qubits [148], they each correspond to a 7r-
rotation along the zit-axis, 7r—rotation along the y-axis, and a 7r-rotation along 2. fol-
lowed by a 7r / 2—rotation along y axis. It’s easy to verify that any single—bit operation

can be decomposed into these three basic gates. Widely used two-qubit gates in-
clude the conditional phase gate, XOR gate and controlled-NOT (CNOT) gate, all
of which are, however, inter—transformable with the aid of single-qubit operations.

Arnon g them, the CNOT gate corresponds to a transformation matrix of

(10 0)

0
0 1 0 0
CNOT : (1.16)
0 0 0 1
0 1

 

 

lo 0/

in the state basis of {[00), [01), [10), [11)}. The CNOT operation thus switches the
Coefficients of the [10) and [11) state. It is noted that any gate operation can be
conStructed from the combination of three basic single-qubit gates and one two-qubit
gate [136].

In quantum teleportation, the quantum states of information carriers (qubits),
While not the physical carriers themselves, are transported from one computational
device to another. Unlike in a classical information network, a qubit cannot trans-
ferred by measurement-reconstruction procedures due to the quantum measurement
p OStUIate. Instead, quantum teleportation is performed by making use of quantum
entanglement and joint-state measurements. The first and also widely—used telepor—
tation scheme was proposed by Bennett et al. back in 1993 [149]. In the scheme, the

goal iS to transfer a source qubit held by Alice to the target qubit held by Bob. The

26

source qubit is in an arbitrary state of

M. = 1 (cama +c1I1>a (1.17)

[Col2 +|P1|2

 

To perform transportation, a pair of particles (usually photons), namely particle 2'

and b, are generated in a Bell state of

[em = Jfiuommb +I1>,;u>a (1.18)

The particle i is then sent to Alice, while particle b is sent to Bob. The quantum state
Of the three qubits is then [\Il)a[<1>+) ib- Alice then performs Bell-state measurement

(BSIVI) on the joint state a and '17, projecting it onto one of the four Bell bases,

lei)...- = finale»: i mall)». (1.19)
mi) -— '1_([0)a[1>-ii[1>a[0>i)- (1.20)

at " fl

After BSM, the final state of the b qubit [\II) b is obtained by projecting the measured
Bell basis onto the three-qubit state. It is straightforward to show that depending

on the measurement result being [<1>+), [<D_), [\II+) or [\Il—), [\Il)b is collapsed to

 

x 1
\/l 2 2(Col0>b + Cllllb)a ———2l——2'(C0[0>b — Cilllb), —-1——(01|0>b +
Col +|61| Icol +|61| lco|2+|01|2
CO] 1) b) 01‘ —-———1———(c0[1) b —- c1[0)b), respectively. Alice then sends the measure-
|00|2+l61|2

“lent result via a classical communication channel to Bob, who becomes aware of
I‘1')b- The last step is to apply appropriate single-qubit operations to recover the
SourCe qubit, thereby completing the transportation. It is worth noticing that the
teleportation is not accomplished until Bob receives the measurement result from Al-
Qe’ SO that no information is transferred faster than speed light. Besides, upon the

(To
Inpletion of teleportation, the original qubits are destroyed, so that the quantum

27

no—cloning theorem is not violated. Finally, while the aforementioned Bennett et al.
scheme is now widely accepted as the standard approach, there exist other quantum
teleportation protocols [150, 151, 40]. In this dissertation, we describe a. new scheme

via quantum interferometry.

1 - 2 - 3 Overcoming decoherence

Experimental implementation of QIP relies on the ability to precisely manipulate
an isolated quantum object, which can be a single atom, ion, photon, or a collective
atomic or photonic ensemble. For quantum computation and memory, it is of the
general interest to use the atomic or quasi-atom qubits as stationary qubits, as they
Yield long coherence times and are easy manipulated by optical means. For quantum

C0111 munication, it is optimal to use photons as information messengers, as they are
rC)}31~lst carriers of quantum information that travel at the speed of light. The diffi-
culties in implementing hybrid atom-photon QIP include preparing and storing an
atomic qubit in a noisy background, creating strong atom-photon coupling, execut-
ing Single— or two—qubit logic gates with high precision, transporting optical qubits
Ovel‘ lossy channels, and so on, all of which are experimentally challenging. These
di fiiCrulties are not only from the routine technical challenges in quantum control over
":1 iQI‘Oscopic objects, but also from decoherence of qubits due to interaction with sur-
I‘()11rlding environments. The underlying physics of the decoherence is the differential
0011IDIing of the qubits’ basis states to orthogonal environmental degrees of freedom.
The couplings are irreversible and dissipative, resulting in decoherence and thus the
In f()I‘Ination loss. It is worth noting that the coupling may or may not be associated
w i t 11 energy dissipation, although a major decoherence source is through spontaneous
decay of photons or phonons. A common decoherence source without energy dissipa-
tion is the Rayleigh or Raman scattering of lasers by atomic qubits, widely occurring

(1 1 1 r i . . . , _
11g optical manipulations of atomic qubits.

28

 

In the presence of qubit decoherence, a successful quantum information process

must be executed in a time interval shorter than the decoherence time scale. In
practice, while the operation time is physically limited, great efforts are made to
reduce the decoherence rate. As given by Fermi’s Golden rule, the decoherence rate
is proportional to the density of states in the bath at the transition energy. By
providing a stronger isolation / confinement, the state density of the surrounding bath,
and thus the decoherence rate, can be reduced. For example, due to coupling to
the phonon bath, the lifetime 7' of the exciton in an (In,As)Ga/GaAs quantum dot,
usually of the order of 1 ns, can be significantly extended by embedding the system
into an optical cavity, thus reducing the density of photon state it couples to [152].
I 11 th is way, a significantly extended lifetime of 7' ~ 10 ns has been demonstrated in a
recent experiment [153]. Implementations of QIP in cavity-QED systems are operated
on a similar principle. In free space, the atom—photon interaction is weak, so that
generating useful atom-photon entanglement in this manner will generally fail due to
inCOherent Rayleigh/Raman scattering. By putting them in a high—finesse cavity in
the Strong—coupling regime, however, the interaction is enhanced due to the reduction
ill the photons’ quantization volume. In the meanwhile, the spontaneous emission
rate can be decreased by reducing the state density of the bath, using the Purcell
effect [154]. In this way, the atom-photon coupling can overcome the spontaneous

emission, allowing useful QIP in such sytems [155, 156].

While the cavity-QED approach is a well-justified method to create useful atom-
at, 0111 and atom-photon entanglement at a single-particle level, it relies on confining
each atomic qubit in a closed space, giving rise to the difficulty to manipulate these
qubits by external electromagnetic fields. If, however, the spontaneous emission can
S'()Irlehow be overcome in free space, then the qubits can be easily manipulated by
lasers and magnetic pulses. In principle, there are two ways to reduce the spontaneous

9111 ‘
‘ lSSlon rate of atomic qubits in free space.

29

The first one is to tune the interacting photon(s) to be far-off resonance with
respective to atomic transitions. Let g be the bare atom-photon coupling constant in
free space, I‘ be the natural linewidth of the atomic transition, and A be the detuning
of the photon from resonance, for A >> 9, the effective atom-photon coupling strength
is given by [gl2 / 2A, which scales as 1 / A. The spontaneous emission rate of a single
atom is [gl2F/4A2, scaling as 1/A2. The ratio between the two is 2A / I‘, which can
be 1113118 > 1 by making the detunning A large. Thus in principle, the spontaneous
emission can be overcome simply using far-off resonant photons. The problem is
that in the far-off resonant regime, the effective atom-photon interaction is so weak
that no useful entanglement can be created. One approach is to use many atoms,
inStead of a single atom, as an ensemble qubit. In this way, the interaction between
the eensemble qubit and a photonic qubit is enhanced by a factor of the number
of Eltcoms in the ensemble. The inherent disadvantage in using ensemble qubits is,
h()VVeV'er, that they are subjected to collisional dephasing effects due to atom—atom
interaction, thus a relative short quantum memory time. Instead, if one uses photonic-
eIISeInble qubits of laser beams, this collisional dephasing problem can be eliminated,
as photons are non—interacting. The difficulty is that, as well-known, while laser beams
O f many photons can strongly drive single-atom transitions, the back-action of a single
atom onto a laser beam, in the form of phase shift, is very weak, so that generating
usef 111 entanglement between them will generally fail due to spontaneous emission. To
Overcome this difficulty, in this dissertation, we propose to use the extreme sensitivity
of Sllb—shot-noise interferometers [157, 158, 119, 159, 160, 24, 25, 26, 27] to detect the
We ak phase imprinted on the forward scattered light in the regime where spontaneous
emiSSion is negligible, thus generating useful entanglements between isolated atomic
qubits via a common photonic channel. This work is presented in chapter V.
In the second approach, it is observed that even if photons are resonant with

a-t: . . . . . .
onllc eXCItation, the spontaneous emiSSion can still be suppressed by the quantum

30

Zeno effect. The quantum Zeno effect (QZE) occurs when a rapid sequence of mea-
surements is performed on a slowly evolving quantum system, with the result that
the system is frozen in its initial state [161]. Consider the coupling of a single atom

and photon qubits, for A = 0 and 9 << I‘, the effective spontaneous emission rate of
a. sin gle—atom qubit is given by 7eff = lgl2/I‘ << I‘. Counter-intuitively, c/eff scales
at 1 / T, meaning that the stronger natural linewidth, the slower the atomic qubit will
lecay. Unlike the off-resonance case, however, here the interaction is purely dissi-
‘dtive, in a sense that no phase-shift, thus no entanglement, is created between the
t 0111 ic and photonic qubit. A possible protocol to generate entanglement out of this
111‘er dissipative interaction is to introduce slow state-evolution of the atom or the
hoton using a weak driving laser or an optical rotator. The resonant atom-photon
Lt€Breaction is then used as a probe to monitor each other’s quantum state during the
"01 ution. If the driving is sufficiently weak, then the two qubits are forced to stay in
(leecherence-free subspace (DF S) [162, 163, 164], due to the quantum Zeno effect.
8 a result, the outcome state of one qubit is dependent on the other, leading to
1‘: anglement between them. VVhat’s appealing in this approach is that there is no
fr ect interaction between the two qubits. Instead, the potential that such an inter-
Ct’ion will lead to decoherence prevents the interaction to actually happen. In this
Ense, the entanglement is generated in an “interaction-free” manner. This helps to
3duee the failure possibility due to decoherence when executing the QIP. In chapter
r1 Of this dissertation, a series of universal, scalable QIP elements are proposed by
l a-1‘Eing use of the quantum Zeno effect. The work of chapter V and VI can potentially
3.3(1 to practical implementations of QIP in free space.

‘

‘ 2.4 Recent experimental progresses

The past decade has witnessed steady experimental progress in the realization

Cl Ilantum information processing protocols [136], including those for quantum com-

31

putation [165, 166, 167, 168, 169, 170, 171, 172, 173, 174], quantum communication
[1 75, 176, 177], quantum cryptography [178], quantum dense coding [179] and optimal
phase estimation [180]. The systems used in these experiments have involved purely
photonic qubits [181], mixtures of atomic and photonic qubits [182, 183, 184, 185],
or purely atomic qubits [186, 187, 188], where ‘atomic’ here refers to any massive
particle or ensemble of massive particles.
From the quantum logical gate prospective, one-qubit gates can now be com-
111 on I)! realized by applying sequences of Rabi pulses. Implementing two-qubit gates
are, however, much more difficult, mainly because they in general require strong
quhit-qubit interaction. Several important proof-of-principle two-qubit gates are ex-
Pel‘imentally demonstrated in systems of 1) all-optical in 2000 [189]; 2) cavity-QED
[182 , 183, 184, 185]; 3) trapped ions in 2004 [186, 187]; 4) nuclear spin resonance
of molecules in 1997 [190, 191]; 5) solid-state in 2009 [192]; and Rydberg atoms in
2009 [193]. A major difficulty in these schemes are, however, to maintain a strong
interaction while avoiding decoherence. Such primarily limits the existing gate suc-
CQSS probability to be only slightly above the threshold value of 66.7%, which is
t’ he Upper-boundary of classical means. In order to overcome this difficulty, another
erllel‘ging line of quantum computation is to use the quantum Zeno effect to replace
the qubit-qubit interaction. In this scheme, the Zeno effect forces the system to
relTlain in a decoherence free subspace (DFS) [162, 163, 164], and thus generating en-
t anglement between two qubits. Several schemes for quantum entanglement manip-
111 at ion and /or gate operation via a quantum Zeno gate have been recently proposed
[ 1 94 , 195, 196, 197, 198], including the work of chapter V in this dissertation.
Quantum teleportation, on the other hand, relies on the capability to generate
Cor 1‘ elated photon pairs and perform (Bell state measurement) BSM, both of which
are considerably challenging due to the difficulty to pair photons and low-efficiency

in . . . . . .
I)hoton detection. In the first series of experiments by the group of A. Zeilinger in

32

 

1997 [175] and Boschi et al in 1998 [199], qubits are encoded in the polarization state

of photons. The ancillary photon pairs are generated from the so—called EPR source
of type II parametric down-conversion device. The BSM is performed by first mixing
the measuring photons with polarized beamsplitter (PBS) and then detecting the out—
come photons. A second type of photonic quantum teleportation was demonstrated in
2003 by the Gisin group, where instead of polarization qubits, quantum information
3 coded in the form of a timed pulse sequence, thus improving sustainability against
)(’)larization relaxation in optical fibers [181]. Further, instead of qubit teleportation,
II C? so—called entanglement teleportation, in which entanglement is swapped from Al-
f6? with Bob to Alice with Charles, was also demonstrated in a 2001 experiment
Sing four photon entanglement [200]. However, all these schemes are probabilistic
[1C1 post-selection, where successful teleportation events are selected after comple-
On Of the experiment, by searching for a subset of experiments matching a preset
PCOnstruction operation. They are thus of little usability for practical QIP, which
31.168 on unconditional teleportation of qubits. In contrast, deterministic quantum
aleportation was demonstrated in 2004 by the Blatt group in Austria [186] and the
vfilleland group in NIST [187]. In these schemes, the quantum state of a trapped ion
leit is teleported to a target ion in a neighboring trap with the aid of an ancillary

)n qubit. The working principle is similar to the Zeilinger experiment [175]. That

‘

3

3 in the first step, the target and ancillary ions are prepared in Bell state by the
fleans of Raman gates. Then, the BSM is performed on the joint state of ancillary
-11(1 target qubits employing controlled phase gates. In both experiments, a fidelity of
H 750/0 are achieved, thus demonstrating deterministic (non-post-selection) quantum
eleportation with a fidelity exceeding the classical limit by ~ 10%. The problem is,
l()VVever, the state transfer is restricted to within neighboring ion traps, correspond-
rlg to a length scale of ~ 0.1mm, and is thus not precisely “teleportation”. A true

t'}1()ligh highly probabilistic teleportation over a 1m distance was (leiiionstrated by

33

the Monroe group in JQI in Jan 2009 [201]. In their scheme, an arbitrary teleporting
qubit is encoded in the ground hyperfine states of a trapped ion. The target qubit
of another ion separated by 1m is prepared in a known superposition state. A laser
pulse then excites each ion, leading to spontaneous emission. The spontaneous pho-
tons are then collected along certain directions and then interfere at a beamsplitter
(BS) - Conditioned on coincident detection of BS outcome, the entanglement of the
two ion qubits are announced, upon which teleportation is accomplished with the
addition of single-qubit operation. This scheme, While showing a heralded fidelity
0f ~ 90%, relies on coincide spontaneous emission of the two ions into two preset
direc tion (with a probability of ~ 10—8), and is thus highly probabilistic.

To conclude, while steady experimental progresses have been demonstrated in
VariOus systems, realizing usable QIP remains a long-term effort. To date, the highest
fiClelity among all reported experiments is S, 90%, while a practical implementation
of N-qubit QIP would require deterministic operations with a fidelity of >> 1/N ~
0 r 999 The gap between current achievements and practical requirements is thus
considerably large. Committed to narrow this gap, two new implementations of QIP

are proposed in chapter V and VI in this dissertation.

34

Chapter 2

Creation, detection and
decoherence of Schrodinger cat

St ates in double—well Bose-Einstein

condensates

In this chapter, we study the possibility to create many-particle Schredinger eat—
like States by using a Feshbach resonance to reverse the sign of the scattering length of
a. Bose—Einstein condensate trapped in a double-well potential. To address the issue
Of eD'Eperimental verification of coherence in the cat-like state, we study the revival
Of the initial condensate state in the presence of environmentally-induced decoher-
enee- As a source of decoherence, we consider the interaction between the atoms
:hq the electromagnetic vacuum, due to the polarization induced by an incident laser

e 1 C1 ~ We find that the resulting decoherence is directly related to the rate at which

SD
th aneously scattered photons carry away sufficient information to distinguish be—

tw-
.1 Gag!) the two atom-distributions which make-up the cat state. We Show that for

I) erfect’ cat-state, a single scattered photon will bring about a collapse of the su-

 

perposition, while a less-than-perfect cat-like state can survive multiple scatterings
before collapse occurs. In addition, we study the dephasing effect of atom-atom colli-
sions 0n the cat-like states. The major results of this chapter have been published in
“Creation, detection, and decoherence of macroscopic quantum superposition states
in double-well Bose-Einstein condensates”, Y. P. Huang and M. G. Moore, in Phys.
Rexr. A 73, 023606 (2006).

The organization of this chapter is as follows. In section 2.1, we give an introduc-
tion on the background of Schrodinger cat states and review previous implementations
in BEC systems. In Section 2.2, the model of bimodal BBC in Double-well system is
introduced and the cat states in such a system are defined; in Section 2.3, We propose
generating a cat-like state through dynamic evolution following a sudden flipping of
the Sign of the atomic interaction, accomplished via Feshbach resonance. In Section
2 ~41 3 the method of detecting the cat-state via revival of the initial state is discussed
in details. In Section 2.5, we investigate the decoherence of cat-states, via a master

equ‘c‘tt ion derived for laser-induced decoherence. In addition, the lifetime of cat-states
in the presence of laser fields is determined. In Section 2.6, the dephasing effects due
to t-11e nonlinear two-body interaction is studied. And finally, we briefly summarize

o
111' results in Section 2.7.

2-1

Introduction

Interest in Schrodinger cat-like states, loosely defined as quantum superpositions

Of I:F‘lacroscopically distinguishable many-body states, goes back to the earliest days of
qu ahtum mechanics. In 1935, in oder to demonstrate the limitations inherent in using
: 11 ahtum mechanics to describe everyday phenomena, Erwin Schrodinger proposed a
3:13 eriment in the macroscopic state of a cat is entangled with that of an unstable

11 L1
Q1 ei, allowing the cat to enter a coherent superposition of being dead and alive [77].

36

Such states were thought at the time to be logically untenable, although today we
might consider only that they would not be observed due to rapid environmentally-
induced decoherence [202]. To this day, this famous gedanken experiment illustrates
the fundamental difficulties inherent in postulating a well—defined boundary between
a. ’classical’ level of reality and an underlying level governed by quantum mechanics,
one of the fundamental open questions in quantum theory.

IVIany of the early gedanken experiments, whose consideration led to the earli-
est 11 nderstanding of quantum mechanics, are now being transformed into laboratory
realities. One such effort has been the ongoing quest for Schredinger-cat-like states,
With the most striking experimental results to date being the observation of a su-
perposition between clock-wise and counter—clock—wise currents in a superconducting
QUantum interference device [203] and the observation of double—slit diffraction for
l’IlElSSix/e C750 molecules [204]. Additional efforts include work in trapped ions at low
temperature [205, 206], nanoscale magnets[207], and superconductors [208, 209, 210].
This progress is generating deeper insight into the meaning of quantum theory, as well
8‘3 potential applications in the fields of precision measurement and quantum informa—
t ion processing. Even with the current rapid rate of progress, the quest to obtain and
utilize large cat-like states remains a serious experimental challenge, due primarily
to tlee presumed scaling of fragility (with respect to environmentally-induced deco-
- erelice) with number of particles. For most appliciations, the utility of the cat-state

Increases with particle number [78, 79], so there is a strong need to find systems in
which such maximally-entangled states of large numbers of particles can be generated
Wi t hout rapid collapse due to decoherence.

An atomic BEC [211, 212, 213] may be such a system, as its large degree of
SI3.3‘tial and temporal coherence is well established b ' t [214] Th b'modal
B y experimen . e i
- EC , which is constructed by isolating a pair of weakly-coupled atomic-field modes,

1% t
11 e of the simplest quantum systems used in the study of entanglement physics and

37

has therefore been the subject of extensive theoretical study [69, 70]. In this system,
[V bosons are restricted to occupy one of two modes, and entanglement is established
via two-body interactions between the bosons. For the case of an atomic BBC, the
two modes can either be two potential-wells spatially separated by a potential barrier
(so—called Double—well BEC’S) [74, 75] or two spatially-overlapping modes in different
hyperfine states [72, 73]. In addition, a recent paper proposes an analogous system in
which a double-well ‘potential’ in momentum-space, formed from the band-structure
in a lattice potential, can be used to create cat-states involving two momentum-states
[2 1 5] -

The quantum ground-state of the bosonic double-well system has been studied for
the case of repulsive interactions in the context of the superfluid to Mott-insulator
transition [216], as well as for the case of attractive interactions. For the latter,
it has been shown under certain conditions that a Schodinger cat-like macroscopic
S‘~113€1‘1‘position state, can be generated [217, 218]. In this cat-like many-body state,
a. large number of atoms are collectively either in one mode or in another. The
(lynamics of the bimodal system have also been investigated, both via a meanfield
approach and a full many-body treatment within the two-mode approximation. In the
IIleaII-field treatment, the dynamic evolution is treated as a Bose Josephson Junction,
which is complementary to that of a superconductor tunnel junction [219, 220]. This
t. reatment predicts a degenerate ground-state, corresponding to all amplitude in one
Well or the other, but is not capable of predicting the dynamical evolution of cat-
llke states, as these states lie outside of the domain of mean—field theory. In the
twg‘ hiode many-body treatment, the many-particle cat-like state is predicted to arise
quri . . . . . . . ,

_ 11g the dynamical evolution of an initial minimally-entangled condensate state
111 the f 1" ° t t' [72 75 ' ' '
presence 0 repu sive in erac ions , ], but With the Sign of the tunneling

fi‘icient reversed by imprinting a 7r-phase shift onto one well. In this chapter, we

Dr

Q.I3ose that such cat states can be realized for the case of an atomic BEC trapped

38

 

in a double-well potential by employing a magnetic Feshbach resonance to tune the

atomic interaction from repulsive to attractive [12, 13]. It is found that for suitable
parameters, the initial state will evolve into a cat—like state after a certain period of
time, and then it will revive back to the initial state.

Despite these theoretical advances, there remain many challenges, some of which

vve address in the present manuscript. One challenge is the problem of experimental
verification of the coherence between macroscopically distinguishable states, which
is related to the challenge of understanding the role played by environmentally-
induced decoherence in collapsing the cat-state onto a statistical mixture of the two
Inacroscopically-distinguishable possibilities. In this chapter we focus in detail on the
decoherence which will occur if laser-light is used for trapping and/or probing of the
cat states. In addition the influence of atom-atom interactions on the proposed de-
tection scheme is studied in some detail. We show the revival of the initial state can
})e llSed as an unambiguous signal that the supposed cat-state is indeed a coherent
macrosc0pic superposition, as opposed to an incoherent mixture. We also demon-
strate, however, that collisional de—phasing can mask the revival without a true loss
of COherence.

The effects of decoherence on these cat—like states have primarily been discussed
for the case of coupling to the thermal-cloud of non—condensate atoms. There is
disagreement in the literature regarding whether or not this effect is significant
[22 1 a 222, 72]. We note, however, that these authors treat the non-condensate frac-
t io .

11 as a Markovian reservoir, which implies an infinitely-short memory (relative
t0 QOndensate evolution times) for correlations between the condensate and non-
Qghdensate terms. We question this assumption, primarily due to the fact that the
“(311‘ . . .

q - Condensate fraction typically has a very low temperature and denSity. In ad-
1 t i911, the non-condensate atoms are trapped inside the condensate volume by the

t. 1‘ ,
I3t3ing potential, thus there is no irreversible loss of information due to propagation

39

of the entangled atoms outside of the system volume. Hence, we suspect that a non-
Markovian treatment is necessary to accurately evaluate the thermal-cloud induced
decoherence. However, modeling this is a difficult task that we do not address at
resent. Instead, we assume that this effect is ne ‘li 'ible and concentrate rimarily
p 8 8 P
on an analysis of a new source of decoherence: that due to spontaneous scattering
of far-off-resonant photons from laser fields which we presume are used in either the
tra.pping and/or probing of the system. In this case, the correlation time of the
reservoir is governed by the irreversible loss of the scattered photons from the sys-
ten1 volume at the extremely fast rate of L / c where L is the atomic mode size and
C is the speed of light. Hence, the i\-‘Iarkov approximation works extremely well for
describing the physics of photon-scattering. As is well known, this approach leads
to effective interactions as well as decoherence, effects which include induced dipole-
C1iIDOle interactions and collective spontaneous emission (superradiance) in the atomic
el”ISel'nble.
VVhen analyzing the effects of environmentally-induced decoherence, this cha)-
l
ter is devoted to answer two basic questions: (1) What limitations are imposed on
t . . . . . .
he ability to create cat-states, and rev1ve the initial condensate states by the pres-
e .
nee of decoherence, and (2) how long can one ‘hold’ the cat—state by decreasmg the
el-coupling to zero, Without collapse of the cat-state onto a mixed state. The
question is raised in antiCipation of applications in preCISion measurement and
11 ahtum information processmg where the cat state is operated on and / or held ready
fo
r fLiture operations during a complicated protocol. We therefore need to determine
t h Q
lifetime of the cat states in the presence of decoherence, as well as the effect of

(16
'QQherence on the dynamical evolution from the initial state into the cat-state.

40

 

 

2.2 The model

Our proposed scheme involves loading a BBC into a. double-well potential and

t hen employing a Feschbach resonance to vary the atomic scattering length in order
to produce a Schrodinger cat-like state. For the present we consider only the case
of a Spatial double-well potential. However, an analogous system can be formed by
using a Raman—scheme to couple two hyperfine states in a spinor BBC [72, 73]. The
primary difference between this model and the double-well system lies in the presence
of collisional interactions between the two modes, due to the spatial overlap of the
hyperfine modes. Our model neglects these collisions, as they are negligible for the
Case of well-separated potential minima. Most of our results should apply to both

S}-’Stems.
“7e begin our analysis from the usual many—body Hamitonian describing atomic

BEC,
A 2
H = [d3r®*(r>[—;—nv2+vtmp<r>ixi1<r>+%/d3r~il*<r)\ifi<r>®<r>xil<r>. (2.1)

where m is the atomic mass, Vtmp is the trapping potential and \II is the annihilating
eld operator for the atoms in Heisenberg picture. The two-body interaction strength

is gi 2 . . .
Ven by U0 = 47rah / m, With a denoting the s—wave scattering length. In our

ITIQCIel, the BBC is assumed to occupy a symmetric double-well potential.

The single-atom ground state and first excited state for this system are denoted
as 1258“) and i/Ja(r), respectively, where 2123(r) is symmetric with respect to the double-
we] 1 symmetry and z/Ja(r) is antisymmetric. We denote the energy gap between these
St E34363 as hr, which is typically much smaller than the gap between the fiTSt and
SQQth excited states. Because of this gap, we assume throughout that the atomic

13%
I3’111ation is restricted to these two modes alone. It is convenient to introduce two

41

1ocalized states w L", d) R [223],

wL(I‘) = %[‘l§j’s(r)+¢a(r)la
war) = —\1—5[¢S(r>-wa<r>i (2.2)

EX I)anding the field operator \IJ(r) in forms of these two modes gives

A

‘I/(n t) = antiwar) + mama). (2.3)

‘Vllere & L and 6R are bosonic atom—aiiniliilation operators for the left and right modes,
respectively. Inserting this expansion back into the many-body Hamiltonian (2.1), one

recovers the two-mode version of the Bose-Hubbard model

f1 = -hr(alLa-R + éLéL) + age-225% + 633.5%). (2.4)

111 deriving this expression, we have implicitly assumed that. atomic collisions in the
O“"el‘lapping region of the two modes are negligible, valid under the assumption that
the two modes are well separated. The intra—well two-body interaction strength is
iIlcliCJated by 9, where g = U0/2hf d3r|wL,R(r)|4. Note that in deriving the bimodal
Hamiltonian (2.4), the wavefunction for each mode is assumed to be independent
of particle number in the trap, and terms proportional to the total atom number-
C)I)el‘ator have been dropped, as atom number is assumed to be a conserved quantity.
The N-particle quantum state of this bimodal system can be expressed as a

3
ll I3Qrposition of FOCk-states In) [70]
N /2

nil) = 2 ml”), (2.5)
n=—N/2

42

vvhere l T
— (CL)N/2+n(CR)N/2—n

I") _ \/(N/2 + n)!(N/2 — n)! '0)' (2'6)

 

 

Here, N is the total number of atoms in the condensate which for convenience we
take as an even number, and n denotes half the number difference between the two

nuod es. The two corresponding operators are

N = alLeL + aka-R,

- 1 . . - .
'n = -2-(CTLCL — CECR). (2.7)

II]. t. l’liS representation, the conventional Schrodinger cat state is defined as

l , .
lcat) = 72f(|]\//2)+|—1\//2)), (2.8)

“'llere the N particles have equal probability to be all in the left or all in the right
mode. This cat—state is an ideal maximally-entangled state, however it will be difficult
to CIil‘eate such a state in this bimodal system. Instead, a more experimental accessible
Catrlike state is one containing two-well-separated wave-packets in the distribution of

C71 - For example, a typical cat-like state can be of the form

)2

n—n 2 n+n
cn o< 63:33)— + 6%, . (2.9)
“’i th no > a.

In the bimodal system we discuss below, the class of cat-like states is more gen-
eral than the Gaussian form (2.9). A useful eat like state only needs to satisfy two
COnditions: (i) the probability distribution in Fock—space should be approximately
S3r‘r11111etric; and (ii) the two wave-packets should be well separated in order to corre-

erd to macroscopically distinguishable states. In the present model, condition (i) is

43

satisfied provided the initial state is symmetric, due to the symmetry of Hamiltonian

(2-4). As we will see is Section 2.5, decoherence may break this symmetry. Nonethe—
less, to distinguish cat-like states from non-cat state, it is convenient to introduce a
projection operator f’cat

. N/2
a... = Z <In><n| + I — ‘ri><—'ri|)- (2.10)

n = n (-

I 'h is Operator acts as a filter which picks up cat—states with a minimum wave-packet
separation of 2710, where up is preset parameter. The expectation value (Peat) can
be a efficient way to determine whether a state is cat-like. It is straightforward to

see that for a cat-like state we will have (150,”) z 1, whereas for a non-cat state the

tra.ce will be noticeably less than unity.

2 - 3 Dynamical generation of cat states

As shown both theoretically and experimentally , the s—wave scattering length
of C=<>1d atoms can be modulated by applying a varying magnetic field. In the vicinity

of a Feshbach resonance, it takes the form [12]

—), (2.11)

vVllel‘e a0 is the background value, BO is the resonant value of the magnetic field, and

A B is the width of the resonance. By tuning the magnetic field B, the scattering

length a can be efficiently tuned in an adequate range based on the Feshbach reso—

nanee, as demonstrated in a recent experiment [13]. Thus, the two body interaction

term g in Hamiltonian (2.4), which is determined by scattering length a, is implicitly
S311 Ined to be an arbitrary adjustable number.

Before discussing the methods of generating cat-states, we note that attractive

44

BEC’S can collapse due to instability, as been demonstrated in experiments [224].
A trapped BEC, however, can be stable against collapse under the condition of low
atomic density [225]. In terms of the BEC atom number N, the attractive interaction
strength 9 and the energy gap between the first-excited state and ground state Aw,

the stability condition can be written as

N < A“: (2.12)
lgl
In the present discussion one can increase the energy gap between the ground and
excited state by adjusting the trapping potential, and in this way, we can ensure
our system will not collapse when the sign of the scattering length is switched from
pOSitive to negative.

There were some efforts to generate Schrodinger cat states via adiabatic manip-
lllat ion of the many-body ground state [217, 218], achieved by a continuous variation
of tl’le interaction strength 9 from a positive to a negative value. This approach, how-
ever a is extremely difficult to implement, due to the energy—degeneracy in the ground
and first-excited state, as explained in appendix 2.8.1. An alternate approach to a
Schrédinger cat state, based on dynamical evolution, has been studied for some time
[72] - In this subsection, we show that it is feasible to dynamically create the cat
State from a coherent state by using a Feshbach resonance to make a sudden change
in tlie scattering length. Our proposed scheme is as follows. First, our system is
I)I‘EIDEJed in the ground state with repulsive interactions and strong tunnelling be-
twee . . . . .

11 wells. This could be accomplished, e.g. by preparing a BBC in a Single well,

a
nd then adiabatically raising a barrier to divide the well into two equal parts. Note
here

the

i

, the quantum state of this system is coherent in the sense that each atom in
Condensate is independently in a superposition of the two localized states. The

tQI‘action parameters in this initial stage are chosen to satisfy 9 = Fig—N as this

45

 

Figllre 2.1: Evolution of the probability distribution P(n) = Icnl2 versus 71., with N =
100- At t = 0, the initial state is taken as the ground state withg = T/(0.9N). Then,
the two body interaction is suddenly switched to the negative value 9 = —7'/ (0.9N).

he wavepacket then evolves, where it shows a collapse and revival process. It is
Seen that, at a time of t = 14.5 /’r, a well peaked cat state is obtained with two well
Separated wavepackets centers at around n = 25,75, respectively.

will lead to the optimal cat-like state attainable in this scheme. Once this state is
%tablished, the Feshbach resonance is used to achieve a sudden switch of the sign of
the scattering length, i.e. we go from g = 067V to g = 7373],. In order to avoid the
collapse of the condensate when scattering length (1 becomes negative, the amplitude
of atomic interaction 9 should be small. At this point, with tunneling still on, the
thtial state is no longer an eigenstate, thus it will start a dynamic evolution under
thQ new Hamilton. The system dynamics is then calculated by numerically solving

1:
he underlying Schrodinger equations.

46

 

n' l

(16*

cat

0.4,

0.2 U ‘

00 2'0 4'0 60
t[1/T]

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.2: Time evolution of the expectation value of the projector Pcat with system
I3"31‘35I-Ineters chosen as in Fig. (2.1) and a filtering window of width no = 15. A well
peaked cat state is created at t = 14.5/7‘, 24/7, 34/1... Note that Peat reaches a

:nihimum at t z 10/T,20/T,30/T..., which corresponds to the revival of the initial
Oherent state.

47

The evolution of the initial coherent state after the change in the sign of g is shown
in Figure 2.1. It is seen that cat-like states are formed periodically, and between two
consequent cat states the system approximately revives back to the initial coherent
state. We have determined that the best cat state appears at time t = 14.5/7', which
corresponds to the final state shown in Fig. 2.1. Once the cat state is formed, one
needs to freeze the dynamic evolution at a time when the system is in the cat state.
Th is can be accomplished by suddenly raising the barrier between the two modes to
recluce the tunneling coeffient, T, to zero.
Since the present proposal is based on a dynamic procedure, predicting the time
VVhen the cat state is produced teat is of critical importance. Due to the imposed
Constraint 7' = 0.9Ng, which is necessary to obtain an optimal cat-state, teat can be

expressed solely in terms of the tunneling rate 7' as

tc at = .7: (2.13)
‘Vllel‘e n is determined from our simulations to be r; x 14.5.

In Figure 2.2, we show the dynamical evolution of projection operator Peat-
Again, it is shown that the optimum cat-like state forms at around t = tcat = 14.7/7'
and reappears again at intervals of t = 10/7’. It is also noticed that between each cat
State, there are valleys of Peat: which correspond to revivals of the initial coherent
St ate.

It is also of interest to study the relative phases between the Fock states in the
Cat‘like state generated in this manner. The symmetry between left and right modes
IS alltomatically guaranteed in our model since our system Hamiltonian (2.4) and our
Init ial state are both symmetric. In Figure 2.3, we see that the phase of created cat
St ate is exactly symmetric around 72 = 0. In addition, we see that the phase flattens

8
(>111 ewhat in the Vicinity of n = :tQ-S, where the peaks in the probability are located.

48

 

 

 

 

 

1:‘iignlre 2.3: Phase distribution of the created Schrodinger cat state in Fock space.
The y axis is the phase 9n. = Arg(cn).

\Ve have claimed that imposing the condition T = 0.9Ng is necessary to produce
the optimal cat state. It is worth pointing out that with other choices, left-right
S3"11‘ltnetric states can be created, but these are neither well-separated nor double
I3€3€il<ted This is shown in Figure 2.4, where we plot the expectation value of the
I)rC-).le(:tor Peat versus (59, where we have taken 9 = U—gjvfl + rig). In this figure, we
See the cat-like state is only created in the vicinity of 69 = 0 with an uncertainty of
approximately :l:5%. That means to create a cat state in the present system, a precise
CCI’lltrol of the atomic interaction 9 or tunneling rate 7' is required. It is important
t0 Ilote that in generating Figure 2.4 we have varied the evolution time tcat for each
I>a’rameter choice, in order to maximize the projection Pcat- Thus Figure 2.4 can be
111terpreted as verifying the conditions 9 = (REV and tcat = 14.7/7.

Finally, this method of generating a cat state is robust against noise in the initial
8 t ate, which could be due to a finite temperature. On the other hand, it does require

c

I) recise knowledge of atomic interaction strength 9 to be within a deviation of about

49

 

 

 

 

 

-0.04 0 0.04 0.08

6
9

F lgure 2.4: The measurement Peat versus interaction strength variation, with g =

fiv (1 + 69). Other system parameters and projection operators other identical to

Fig- (2.2). It is shown that in the vicinity of 69 = 0, the cat state is created. Note
that the projection is measured at different times for each (lg. so that the value of
peat is maximized.

2 /O - Detalled discussmns are glven 1n appendlx 2.8.2.

2 - 4 Detection of cat state

Presuming a cat—like state can be created in the above manner, it remains a
cl1‘3-11enge to experimentally verify the coherence between the macroscopically distin-
gIliShable states. Neverthless, a possible method to detect Schrodinger cat states in
this bimodal BEC’S lies in detecting the revival of the initial coherent state. In Fig.
2 ‘ 5 Vve show the evolution of the cat state starting from t = tcat with the system
I)Ia’rameters still satisfying 9 = 067V This shows that under the condition of no
Cl9"Qoherence, the system will evolve back to a coherent state after a certain period
of time, which we denote as trey. The revival time trey again is estimated from our
Si tr‘llllations, yielding trey = 7/T.

In the presence of decoherence, however, the system may not exhibit a distinct

P(n)

0.08 7,. W7
UL,” ’1’ ".fl/ '1 /
0.04 , ,/ Vt @1f 1 /W{//// 0
0 ////-/ // /',/ ”7.3,; 25
15 / “f“‘f - ' 0
17 /-25 n
t[l/Tl 19 -50

Figure 2.5: Revival dynamics from the cat state back to the coherent state. This
figure shows the continued dynamical evolution following that shown in Fig.2.1 with
g = —0.01. At t = 14.5/7, the quantum system is in a cat state, then at a time of
t z 21, the initial coherent state is revived.

  

/ ./ fl //
‘1 , ///////

   

a07/114

  

t [1m 19 -50

Figure 2.6: Dynamical evolution of a partially incoherent cat state. The parameters
are chosen the same as Fig.2.5. At t = 14.5/7, the quantum system is in a mixed
state with no coherence between the two wavepackets. Then as it evolves, the system
exhibits oscillations, but will not evolve back to the initial state.

51

revival of the coherent state. To illustrate this we first consider two states which
we define as the partially incoherent state and the totally incoherent state. These
states are defined by the introduction of a coherence ‘length’ in Fock-Space, Icoh,
so that coherent superpositions between Fock-states In) and Im) are destroyed for
In—ml >> I coh, while superpositions satisfying ln—ml S, I CO}, are relatively unaffected
by decoherence. The partially incoherent state is then defined in the sense that the
coherence length I 00}, is small compared to the distance between the two separated
peaks of the cat state, but larger than the width of either wavepacket, i.e., ICU}, <
n0 yet ICC}, > a in Eq. (2.9). For the completely incoherent state, on the other
hand, the cat state is taken to have collapsed to a single Fock state, i.e. a mixed
state maintaining the probability distribution of the cat-state but with no coherence
between Fock-states.

The evolution of those two states in time is shown in Fig.(2.6) and Fig.2.7,
respectively. Unlike the cat state, the incoherent states will not exhibit a revival of
the initial coherent state at t = trey. The partially incoherent state evolves instead
into a disinct three-peaked state, whereas the completely incoherent state exhibits
small oscillations with no discernable peaks after a very short collapse time.

Based on these considerations, one prospective method to verify the coherence
of the cat state would involve: i) first, measuring the probability distribution, P(n),
which is accomplished by repeated measurements of the atom-number in each mode;
ii) then after forming the cat-state, the state should be held for a time t = thold
and then with tunneling restored, the revival of the initial coherent state should be
observed after a time trey; iii) a known source of decoherence should be added while
holding the system in the cat state at 7' = 0 for a duration tholdr after which a lack of
revival should be observed. To add decoherence, one could simply add a laser field to
the system. This will introduce decoherence due to spontaneous photon scattering,

as described in Section 2.5. If the revival of the initial state can be observed in the

52

 

Figure 2.7: Dynamic evolution of completely decoherent cat state. At t = 14.5/7', we
let the system collapse onto a statistical mixture of Fock-states. It is seen that the
dynamic is completely disordered.

case of no additional decoherence, but disappears when the decoherence is added in a
controlled way, than one should have a strong claim for the observation of coherence

between macrscopically distinguishable states.

2.5 Laser-induced decoherence and losses

While trapped ultracold atoms are typically well-isolated from the environment,
the presence of lasers for trapping and / or detection of atoms will polarize the atoms
and thus couple them to the vacuum modes of the electromagnetic field. This coupling
allows for spontaneous Rayleigh scattering and /or diffraction of laser photons, which
can lead to losses and / or decoherence of the atomic system. If the scattered light field
acquires sufficient information to determine the atom-number difference between the
left and right wells, then this spontaneous scattering will lead to a dynamical collapse

from the cat-state into a mixture of Fock states, an example of environmentally-

53

induced decoherence.
In order to model these effects, we derive the master-equation which governs the

density operator of the atomic field ps, where

Mr»?
A

p5(t) = —2'H5p3(t) —i/d3-r \Ill(r)\Il(r)

— / (137“d3'r’Q(r)££rA——Efl§2*(r’)\lll(r)\I'(r)\Ill(r’)\I/(r’)ps(t) (2.14)

A

+ / d37‘d3r'Q*(r)-lfl:lfl9(r’)@l(r)\II(r)p3(t)\Ill(r,)\ll(r’) + H.c..

A detailed derivation of this master equation is given in appendix 2.8.3. Here, \Il(r)
is the annihilation operators for atoms. A = we — w L is the detunning of laser
field from resonant excitation. The atom-laser interaction is governed by the Rabi-
frequency Q(r) L(r — r’) is the nonlinear two-body correlation coefficient defined in
appendix 2.8.3, equation (2.60).

Underlying the dynamics governed by master equation (2.14) is the exchange of
photons between atoms and between an atom and the EM vacuum. As such photon
exchanges can involve significant momentum exchanges due to photon recoil, it is
useful to first define elastic and inelastic regimes with respect to the initial atomic
center-of-mass state. If the spatial size of the initial atomic mode is small compared to
the laser wavelength, then one would be in the elastic regime, as photon-recoil would
not be sufficient to carry the atom outside of the momentum distribution of the initial
state. In the inelastic regime, on the other hand, the initial mode is large compared to
the laser wavelength, which implies that it is narrow in momentum space compared to
the photon momentum, so that the recoil from a single photon is sufficient to remove
an atom from its initial mode and place it in an orthogonal mode.

In the strongly elastic regime, there will be negligible loss of atoms from their
initial modes. As we shall demonstrate, the master equation (2.14) may still lead

to decoherence/collapse of the two-mode cat-states. In the inelastic regime, on the

other hand, the predominant effect of the atom—field interaction will be scattering of
atoms out of the initial modes and into a quasi-continuum of modes. This loss will be
accompanied by a collapse of state of the remaining atoms into a mixture of states,
which may or may not be cat-like in themselves. A quantitative discussion of atom

loss via master equation is given in appendix 2.8.4.

2.5.1 Decoherence in elastic regime

In the present scheme, N atoms are trapped in a. double-well potential with
a finite spatial separation between the wells. Coupling of this system to the EM
vacuum modes will result in decoherence only when scattered photons carry away
sufficient information about the number of atoms in each well. It thus follows that
if the separation between the two condensates, s, is less than the laser wavelength
AL, no decoherence should occur, as the information contained in the photons is
diffraction-limited to a resolution of no better than a wavelength. On the other hand,
if s is larger than AL, the locations of the atoms can in-principle be determined from
the phase-information contained in the scattered photons, this irreversible transfer
of information to the environment will be reflected in decoherence of fock-space—
superpositions of different atom-number distributions.

In the deeply elastic regime, the size of the left and right modes is the smallest
length scale in the problem, so that we can replace the atomic density distributions

with delta-functions, or equivalently we take
A - S A
\I!(r) = 6(r — —)éL + c)(r + -2-)CR, (2.15)

where c L and 6 R are the annihilation operators for the left and right mode respec-

tively. Inserting this expansion into the master equation (2.14) and assuming again

for simplicity 9(r) = Qeka'r, gives

 

PS:

IQI2F .1- At. .1. t
— 2A2 [CLCLCLCLp-9 + CRCRCRCR’OS

+ZCOS(kL - S);r(s)élLéLé}2€°Rp3 — éEéLpSéiéL
—élRéRpsé};éR — cos(kL - s)p(s)éEéLPsé)?éR

— cos(kL -s)/1(s)élRéRp.gé'EéL)] + H.c (2.16)

where again the imaginary parts are absorbed via a transformation to the appropriate
rotating frame. The resulting decoherence is governed by the parameter ,u.(s), defined

as

2%[L(s)]/I‘
sin C

5

MS)

+(1—3c05263)(5%— S$333)]. (2.17)

 

2 Shin2 95

where C = 27rs/ /\ L and 65 is the angle between the induced polarization direction d
and the relative coordinate s.

From this master equation (2.16), we derive the equation of motion for the fock-
space matrix elements pm” = (nlpslm), yielding

. [m2 2
pn'm = — F02 — 7") (1 — COS(kL ‘ S)IU(S))p-n7n. (2.18)

This shows that the diagonal matrix elements (mzn) do not decay, reflecting the
absence of condensate losses in the elastic regime. The off-diagonal elements, on the
other hand, decay as at a rate proportional to (1 — cos(kL - s)p.(s))(n — m)2.

The dependence of g(s) on the mode separation is shown in Fig. 2.8, which plots
g(s) versus 3 / A L and 63 = arg s. We see that under the condition of 3 << A L we have
cos(k - s)p(s) a: 1, and therefore no decoherence. As .9 increases, ,u(s) will decrease

and the decoherence rate will increase. In the limit of 3 >> AL, we have ”(3) z 0,

56

Q 2
A
note that g(s) is angle-dependent, as seen in Fig.2.8.

and the decoherence converges to the rate 771m = F(n — m)2. It is interesting to

To understand the reason for this decoherence rate(2.18) we will consider the
limiting cases .3 >> A L and 5 << AL, corresponding to a mode separation much larger
than or much smaller than the laser wavelength, respectively. For the case where
the mode separation is large compared to the laser wavelength, we have g(s) ——> 0
so that the decoherence rate is 771m = I—Z—g—F (n — m)2. Our goal is thus to explain
the peculiar (72. ~— m)2 dependence in this expression. we begin by noting that in
this regime the phase information in the scattered photon wavefronts can in principle
determine whether the photon scattered from the L or R mode. Thus the question
becomes: on what timescale, T, does the radiation field scattered by L or R acquire
sufiicient information to distinguish the state In) from the state I-m). This timescale
will then determine the decay rate for the off-diagonal elements pnm, which give the
degree of coherence between the states In) and m). The rate at which photons will
scatter from the left mode is lggrnfi where n L is the number of atoms in the left
mode. The reason for the n% dependence, rather than the usual n L: is due to Bose
stimulation, as the atoms remain in their initial mode after elastic scattering.

The actual scattering of photons is a random process, so that the number of
L scattered photons np over a time interval t is drawn from a distribution func-
tion PL(np, n L, t). It is reasonable to assume that PL(np, n L: t) is at least approxi-
mately Poissonian, with a center at np(nL, t) = Lglf—I‘n%t, and a width Anp(nL, t) =
W = Jg-Im n L‘ Based on this assumption, the criterion for distinguish-
ing state In) from state Im) is that the distributions p L(np,n L2 t) and pL(np, m L,t)
should be distinguishable, so that the scattered photon number can be attributed to
one distribution or the other. In this expression n L is the number of atoms in mode

L for state In). This requires minimal overlap between the two distributions, which

57

can be approximately formulated as

'fzp(nL,T) + Anp(nL, T) = np(mL,T) — Anp(mL, T), (2.19)

where we have temporarily assumed m L > "L' Solving this equation for the deco-

hel'ence rate 711777, : l/fd yields

 

2
IQI2 I‘ [172% — niI

~ , , = . . 2.20)
17?"? A2 [."IL + ”L12 (
From eq. (2.7) it follows that
N
'n L = 3 — n
N
"R = —2- + n. (2.21)

which also hold for m L and m R with a simple substitution n —+ m. Inserting this

into (2.20) yields
1m?

Emil — m)2. (2.22)

’l’n m =

We note that the result for mode R instead of L can be found via n <—+ —n and
m H —m, while the result for m L < n. L can be found via n +—> m, all of which leave
the result (2.22) unchanged. Thus we have verified our interpretation that the decay
of the pnm coherence is governed by the timescale on which the information carried
by the scattered light becomes sufficient to distinguish between the states In) and
Im).

If we consider the opposite case 3 << /\ L we can no longer assume that scattered
photons carry information concerning which mode they scattered from, due to the
standard diffraction limit. In this case one might be tempted to assume that the

total scattering rate is proportional to 71% + 77?2 = N 2 / 2 + 272.2, as photons must still

58

scatter from one mode or the other and each mode will experience Bose-stimulation
of the recoiling atom. This would imply that the total scattering rate would be
different for the states In) and Im), so that given sufficient time the environment
could learn which state is scattering the light and thus destroy the coherence. This
effect, however, is compensated for by superradiance [226] so that the scattering rate
is proportional simply to N 2, which is the same for both In) and Im). Superradiance
is a many-body quanturn-interference effect whereby atoms within a distance less
than one Optical wavelength of emitted photons experience enhanced decay, identical
to Bose stimulation, even for distinguishable atoms.

To verify this interpretation one can calculate the scattered light intensity and
see precisely how it scales with n L and n R- This can be accomplished by using Eqs
(2.51) and (2.52) to obtain the Heisenberg equation of motion for ékA:

(I . . .
217““ = ‘lwkaka

— igg;,e—'ith /(137‘(%i(kL_k)'r\IJI(r)\I'(r), (2.23)

where we have substituted the adiabatic solution 2,6(r) = %\Il(r) . Substituting the
mode expansion (2.15) and noting that a L and 71R are constants of motion in the

absence of tunneling gives upon time integration

 

(310(15): dkA(O)€—lwkt _ [ii-591:”: 2(wk+wL)t/2
(wk — WLl/2

From this expression we can compute the mean number of scattered photons via

"190") = kaiéi‘iékAl, yielding

Q 2
np(t) = l—Z-S—IQ—I‘ In% + "f? + 2cos(kL - s),u.(s) nLnRI t (2.25)

      
 

I’M/Ia
v’t/I/ .4

Figure 2.8: The decoherence parameter g(s) as a function of 95 and s/AL. From the
diagram, a(s) is approximately 1 when the separation of two modes 3 is much smaller
than the laser wavelength ’\L- As 3 increases, “(8) decreases at a speed of 1/3 and
eventually approaches zero as 3 goes much larger than )‘L- Note that [1(8) is also
dependent on 03, where at large 5, ,u(s) ~ sin2 93.

where we have converted the sum to an integral in the limit of infinite quantization
volume and made the approximation sin2(a:t)/:c2 z 7rt 6(x) when integrating over It.
This expression reveals an additional component to the scattering rate proportional
to cos(k L - s),u(s)nLnR. This term is only significant when a(s) ~ 1, which requires
that the mode separation be comparable to or less than the optical wavelength. In
addition it depends on the exact spacing and orientation of the modes with respect to
the polarization of the lasers, via the implied dependence of p(s) on s. We interpret
this additional scattering as superradiance, as it arises from the imaginary part of the
two—body dipole-dipole interaction. We see that in the limit as s / /\ L —» 0 we have
cos(k L - s),u(s) —+ 1 so that the total scattering rate is proportional to N 2. As stated
previously, this means that the radiation field never acquires sufficient information to

distinguish the states In) and Im), therefore no decoherence occurs.

60

We now focus on the tunneling dynamics in presence of elastic decoherence,
where the total atom number can be treated as a constant of motion. In addition,
we will assume that the separation of condensates are much larger than the laser
wavelength, so that the dynamics is determined Eq. (2.18) but with as = 0. Under
those assumptions, it is straightforward to guess the coherence time tcoh as

4A2

t = —,
COh 4~r2I9I2P

(2.26)

by observing that the peaks of the dynamically created cat-state are separated by
no 2 iN/4. This is the time-scale on which coherence between states separated
by N/ 4 should decay based on the decoherence rate (2.22). During the generation
of the cat state, it is expected that if the coherence time tcoh is large compared to
teat: the cat state can still be produced. Otherwise, the system should evolve into a
mixed-state rather than a pure cat-state.

The effects of decoherence on the dynamical evolution of cat—states is shown in
Figures 2.9-2.11. In Fig. 2.9 we Show the magnitude of the density-matrix elements
for the initial coherent state at t = 0. Under the condition of no decoherence, this
state evolves into a cat-state at t = teat, which is shown in Fig. 2.10. In Figures
2.11 (a) and (b) we show the resulting state for the cases of weak and strong deco-
herence, respectively. In Fig. 2.11(a) the decoherence strength is chosen such that
tcoh = 5tcat7 while for the strong decoherence of Fig. 2.11 (b), we have tcoh = tcat-
It is seen that with tcoh = 5tcat we can still create cat-like states, while as the de—
coherence strength increases up to tcoh = tcat, the outcome state is a mixed state
with no coherence between macroscopically distinguishable distributions. Therefore,
to successfully produce cat state, the coherence time tcoh should be much longer than
the time need to produce cat state teat-

It is also quite interesting to observe from Fig. 2.11 that not only the off diagonal,

61

 

~50 —50

Figure 2.9: Magnitude of the density matrix elements for the initial coherent state.

0.04
0.02 '

50

 

-50 -50

Figure 2.10: Magnitude of the density matrix elements of the dynamical created cat
state at t = teat with no decoherence.

62

 

   

0.04 V 2‘ fiég/Z/w ‘1 : 14' I,
///

  

////// ”ff/ff}?

   
 

(b)

Figure 2.11: Magnitude of density matrix elements of the resulting state under weak
decoherence (a) and strong decoherence (b). System parameters are chosen the same
as in Fig.(2.1), and the resulting states (a) (b) are measured at a time of teat = 145/7.

63

but also the diagonal of density matrix of the final states are affected by decoherence
effect. In Figure 2.12, we plot the probability distribution P(n) E pnn of several
resulting states corresponding to different coherence times. In the figure we see that
the probability distribution of the mixed states formed under strong decoherence is
significantly different from that of the pure cat state. This is because under strong
decoherence, the coherences between different number basis states are continuously
destroyed during the dynamics, so that the population dynamics are modified. In
other words, if decoherence dominates the system dynamics, it will not yield the
same P(n). Thus the probability distribution alone supplies a direct way to verify
the existence of cat state at time teat, without the need to detect revivals. We note
that we have so far only established this result for the elastic regime. We suspect
that the elastic regime will be much more difficult to achieve experimentally than the
inelastic regime, due to the necessity of sub-wavelength confinement.

In Figure 2.13 we examine the effects of decoherence on the revival process, under
the assumption that teat < tech: but assuming that the system is held at T = 0 for
a time thold’ which may be longer than the coherence time. These figures show the
magnitude of the density-matrix elements at time t = teat + thold + trey, calculated
by direct numerical solution of the master equation (2.16). In Figure 2.13 (a) we show
the magnitude of the density-matrix elements for the case t hold = 0 and for tcoh = 00,
i.e. no decoherence is present. Note that this figure corresponds to t = 21 in Fig. 2.5.
In Figure 2.13 (b) we plot the magnitude of the resulting density matrix for the case
thold = tcoh = 10tcat- This choice gives sufficient time to destroy coherence between
fock states where n differs by N / 2, corresponding to the distance between peaks in the
initial cat state. This shows that the revival is effectively destroyed by decoherence
on the the timescale governed by the coherence time (2.26). This figure corresponds
to the ‘partially incoherent state’ shown in Fig. 2.6 at time t = 21, in the sense that

the choice thold = tcoh implies the condition a < [m], < "0- Lastly, Figure 2.13

64

 

0.04

0.03
P(n)
0.02

 

0.01

 

 

 

 

3%

Figure 2.12: Probability distribution P(n) of the resulting states corresponding to
different coherence times. Parameters are choose as in Fig. (2.11), while from line (1)
to (7), the coherent time tech is chosen as oo, 3tcat, 3tcat/2, tcata Stem/4, 3tcat / 5,
teat / 2, respectively.

65

2
(c) shows the case thold = thcoh with t = 10tcat which gives sufficient time

coh
to destroy coherence between any fock states. We see this case that the system has
collapsed onto an almost pure mixture of fock-states, and it is clear that no revival

occurs. This figure corresponds to the totally incoherent state of Fig. 2.7.

2.5.2 Loss and Decoherence in the inelastic regime

In this subsection we turn to the inelastic regime, which takes into account the
finite size of the atomic modes, so that the scattering of laser photons is not entirely
elastic due to photon recoil effects. The resulting losses from the two initially occupied
modes should necessarily be accompanied by decoherence, which may or may not be
sufficient to ’collapse’ the two-mode cat state. We focus solely on the case where the
mode separation is large compared to the wavelength, so that we can safely ignore
the cross terms in Eq. (2.16). The master equation in this situation is found by

substituting the two-mode expansion \Il(r) = q5 L(r)c L + ¢R(r)éR, yielding

_M
2A2
+ €[ans(t)nL + an.e(t)"Rll + H-Cw (2'27)

ac) [Nps(t) + ani + n}, _ Nipsm

with 6 defined in Eq. (2.79). Deriving the equation of motion for the density matrix

elements is then straightforward, giving

1912

1"?“ _ §)N + €(n — meme). (2.28)

anft) = ‘F

Under the condition of spatially broad (relative to AL) condensates, g z 0, as seen if

Figure 2.24, and this equation is reduced to

)2
_,.ISI

prnmU’): (2‘29)

anft) =

66

(a)

 

(b)

  
       
 
  
 

 
    
  
   

/% » /
%¢/////z//%////////cav 27%,
' ///%2; :'

/ I ’

         

-50 —50 (C)

Figure 2.13: Effects of decoherence on the revival of the initial coherent state. Shown
is the magnitude of the density matrix elements of the system at time t = teat+thold+
trey under no decoherence (a), the decoherent resulting state with thold = tcoh. (b),
and thold = N 2tcoh /4 (c). System parameters are chosen the same as in Fig. 2.1.

67

which shows that the decoherence rate is exactly equal to the atomic loss rate (2.78).
The reason for this is that the loss of an atom takes us from the manifold with N
atoms to that with N — 1 atoms. This is equivalent to n —> n :l: 1 where the choice
of i depends on whether the atom was lost from the L or R mode (— for L and + for
R). Thus one might suspect that a careful treatment of the scattering of the atom
into the quasi-continuum of recoil modes would reveal that the loss of coherence on
the matrix element pnm could be accompanied by a corresponding appearance of
coherence on the matrix element pni1.m:l:1 which lies in the N - 1 manifold.

A complete quantum description of this inelastic process should include both the
initial cat-state modes as well as a quasi-continuum of recoil modes.We can therefore
describe the initial unscattered systems with a density of product pg. = pN ® pr,
where pg}. is the full density operator having m atoms in the initial modes and N — m,

—m

atoms in the uasi-continuum, and m and {Y are the corres ondin reduced
C1 P .0 P g

density matrices for the system and reservoir. After one photon is scattered, the full

density matrix goes from pill. to pg.—

1, due to the transfer of one atom from the initial
modes into the quasi-continuum. As the modes are well-separated, the scattered
photon carries sufficient phase information into the environment to determine whether
the atom recoils from the L or the R mode. Assuming that atoms from L and R modes
scatter into different manifolds of recoil modes, the reduced density operator of the
two-mode system, obtained by tracing out the quasi-continuum modes, becomes
N—l 1 N N

p = WW” c}, + cm) ck) (2.30)
with N the normalization factor. In other words the system collapse onto an inco-
herent mixture of the state with one atom lost from mode L and the state with one

atom lost from mode R.

68

As an example, we now consider a N particle cat state of
PN = I’IIC><’IICI (2-31)

and to distinguish cat states with different total atom numbers, here we use a new

notation

I‘I’cl = %(

where In L,n R) stands for a Fock basis with n L atoms in left mode and ”R in the

 

N/4, 3N/4) +

 

3N/4, N/1)). (2.32)

right. This cat state is collapsed after one photon—detection into a statistical mixture

of two N — 1 particle states,

pm=§<Iw1><w11+1w2><wzn (2.33)

where
(11,) = —\;—§(IN/4 — 1, 3N/4) + I3N/4 — 1, N/4)), (2.34)
(112) = lair/4, 3N/4 — 1) + I3N/4, N/4 — 1)). (2.35)

,5

The important point here is that each of these states is still a good cat-like state, so
that while we don’t know which one the system has collapsed into, we do know that
the system remains in a good cat-like state. The means that the effect of scattering
a photon and losing an atom may not have a significant detrimental effect on cat-like
states. Similarly, after a second photon is scattered, the system will collapse into a
statistical mixture of four N — 2 particle cat states, and so on. We note, however, that
for the ’perfect’ cat state [IN 0) + ION )I /\/2 a single scattering results in a mixture
of the states |(N — 1)0) and I0(N — 1)), neither of which is itself a cat-state. This

means that while the ideal cat state is so fragile that a single scattered photon is

69

 

0.8
0.6
cat

0.4

0.2

--—I——--I-‘irq—————.l—w

 

 

 

 

’0 0:2 0T4 06 0.8 w 1
2 2
t [A “FIQI )l

Figure 2.14: The evolution of the projector Peat under decoherence due to inelastic
atom loss. The curves correspond to the dynamically—created cat state (Blue solid
line), the perfect cat state (Red dashed line) and less extreme cat state (2.32) (Green
dashed-dotted line). The figure plots the estimated probability to remain in a cat-like
state versus time. Starting from 100 atoms, it is estimated that 63 atoms are lost by

the final time t = IQAI 1"

sufficient to collapse that cat onto all-left or all-right states, the cat-like states we are
considering should be significantly more robust, giving them a significant advantage
provided they are still suitable for whatever application is desired.

To show the dynamics under decoherence induced by these inelastic condensate
losses, in Fig. 2.14, we plot the time evolution of the expectation value of the pro-
jector Peat, defined in (2.10). Three initial states are shown, the extreme cat state
IIN,0) + ION)]/\/2, the less extreme cat state, II%—i4-M) + |3j"l,i}’[)I /\/2 and the
wave-packet cat-states produced via dynamical evolution as in Section 2.3.

Figure 2.14 is obtained by assuming a single photon is detected in each time
interval At = W, where N (t) is the number of remaining atoms at time t.

After each scattering a new density matrix is obtained by applying Eq. (2.30). We

70

 

Figure 2.15: Magnitude of the resulting density matrix elements corresponding to
. . __ A
Fig. 2.14 at time t — 0.3IQI I"

note that the use of (2.30) automatically weighs the probabilities that the scattering
occurs from the L or R mode due to the action of the annihilation operators being
proportional to the square root of the mode occupation. The resulting density matrix

is then diagonalized and written
p = Z elm-mt (2.36)
i

where I\II,;) denotes the ith eigenmode, with P,- its statistical weights over the mixture
state. The resulting eigenmodes I\II,-)’s are then analyzed to determine whether or
not they are sufficiently cat-like. This is accomplished by first assigning a value of
zero to any state where Zn>0 ICnI2 51$ Zn<0 IcnI2, where on is again the probability
amplitudes for the state In). The states which survive this test are then assigned a
value equal to the expectation value of the projector Pcat- The assigned values for
each eigenmode of the density matrix at time t are then averaged together and the

results are plotted versus time.

71'

 

 

 

0.08

 

 

 

 

 

 

 

 

 

 

 

 

' P,='0.4T36 ' P,='0.4464 ' P,=0.0374
P(n)
0
0.08 n P4T0'021n9 ' 50.0018 I PEROOOI
P(n) II

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,LWI LJ U

-36 0 336-36’ 0 ’36 -36 0 36
n 11 11

Figure 2.16: Probability distributions P(n) of the six leading eigenmodes {IQ/2)} of
the density matrix shown in Fig. (2.15), where P,- is the eigenvalue for eigenmode
Ni)-

72

From the figure, we verify that the perfect cat state is completely collapsed
immediately after one photon scattering. In contrast, the less extreme cat states has
a high probability to remain in a cat-like state even after many scatterings, and, the
dynamically created cat-states are similarly robust against decoherence. We note that
in the figure out of an initial population of N = 100 atoms, N = 37 remain at the

. A2
end-time t = IQI I"

In Fig. 2.15, we plot the magnitude of the density matrix after a time of
thold = 0.36:]; as in Fig. (2.14) (Blue solid line). In this figure, an initial cat
state is formed at time teat via dynamic evolution with no decoherence. The state
is then held with g = 0 and T = 0 for time thold with photon-scattering acting as
a gradual source of decoherence. The figure shows a collapse of the reduced density
matrix onto a statistical mixture of left-centered and right-centered peaks, and thus
the collapse of cat state. This conclusion is, however, in seeming contradiction to
the result via numerical diagonalization of this matrix, which reveals that that the
system is in fact in a mixture of different cat states, as shown in Fig. 2.16, we plot
the major eigenmodes Iklli) of this density matrix. From the picture, some of these
resulting eigenmodes, such as I‘lli)’s with i = 1, 2, 4, 5 are still cat-like, in a sense that
separation between two peaks are larger than the width of either peak. While for
some others, like I\113) and I\II6), the separation between peaks are not larger than
the width of each peak, and thus they are not cat-like. It is noted I‘I’ll and I‘Ilg)
are symmetric and antisymmetric states around n = 0, respectively. From here, one
might think that such a collapsed state, being a mixture of cat-like states, can still
exhibit the macroscopic property of a pure cat state. The flaw in this argument is
that the two major eigenmodes I\Ill) and I‘IIQ) are nearly degenerate, and would then
simultaneously response to any measurements or operations, instead of individually.

As a consequence, they always act as an incoherence mixture of left-centered and

right-centered Fock-states. This is understood in the simplest case of a single qubit

73

in a mixture state of

pqubit. = ,1, (10> <01 + 11><11> . (2.37)

Due to the degeneracy of IO) and I1), this density can be rewritten in the form of

(|+><+| + |-)<-|), (2-38)

{\Dli—|

pqubit =

where I+) and I—) are superposition states of

1 .

I+) = $00) +I1))1 (2-39)
1

I‘) = 772-00) '— IU)- (240)

An arbitrary operation on state basis I0) and I1) would unavoidably act on both I+)
and I—). The system then responses as a mixture of IO) and I1), without exhibiting

coherence between the two.

2.6 Dephasing induced by atomic collisions

In this section we consider the effects of atom-atom collisions on a cat-state which
is ’frozen’ by reducing the tunneling strength, T, to zero. If during this hold state the
atomic interaction strength, 9, is nonzero, the system will undergo a dephasing process
[227, 216], whose dynamics is determined by Hamiltonian (2.44). The main effect of
this dephasing process is that, while it will not collapse the coherence of the cat-state,
once the state is dephased, it cannot evolve back to the initial coherent state. In this
way, a detection scheme based on revivals will incorrectly create the appearance of
collape. Therefore, in general, the dephasing effect needs to be suppressed by making
the atomic interaction strength 9, and/ or the hold time small.

In studying the dephasing effect, first, we note that at each period of 71/ g, the

74

 

0.75 -

0.5“

col

0.25 r

 

 

 

 

0 0.01 0.02 0.03 0.04 0.05
tl1/Igll

Figure 2.17: Dephasing process of cat state with nonzero atomic interaction. The
collapse is measured by P601 = tr{pcp(t)}, where pc and p(t) are the densities of the
initial cat state and dephased state after a hold time thold respectively. Parameters
are given the same as in Fig. 2.1 while thold has units of 1/IgI.

relative phase of each on is exactly restored and thus the many-body states is revived.
While between two consequent revivals, nonzero relative phases are introduced to Fock
basis, and the cat state will collapse. In Fig. 2.17, we show such a rapid dephasing
process, where the cat-state decays with a collapse time around tcol = 0.02/IgI,
determined from numerical simulations.

To obtain an analytic estimate of the collapse time scale, we assume that our
system is prepared in a Gaussian—like cat state (2.9), which is a good approximation of
the realistic cat state in our system. And then, we associate the collapse and revival
of our bimodal system with the evolution of one-body two—mode correlation function
A.

A = (catIéEc’iRIcat). (2.41)

At short time scale, the correlation function (2.41) is obtained, under the condition of

75

that the Gaussians are well peaked around n = ing, to a good approximation I227],
-(90'1t)2
A(t) = A(0)e (2.42)
And therefore, the collapse time tcol can be estimated by
(2.43)

As discuss above, the collisional dephasing will affect the revival process. Gen-
erally speaking, if the holding time thold is less than collapse time tcol, the system
can still evolve back to the initial state as shown in Fig. 2.18 (a). But under the
condition of thold > tool, the cat state is dephased, and the coherent state can not be
restored. Rather it will evolve to a disordered state as shown Fig. 2.18 (b). There-
fore, for the success of the detection scheme, one should make sure that the dephasing
collapse time is longer than the hold time and the revival process is only affected by

decoherence.

2.7 Conclusion

We have considered the double-well BEC system and prOposed that cat-like states
can be formed by switching the sign of the interaction strength from positive to
negative using a F eshbach resonance. We examined the possibility of either adiabatic
or sudden switching with the following results. In the adiabatic case, the ground
state evolves into a macroscopic superposition state with all of the atoms collectively
occupying either the left or right well. However, by examining the degeneracy of the
ground state, we conclude that the cat state is unstable against small perturbations
and may readily evolve into a localized state. For the case of a sudden change in

the scattering length, we find that an initial condensate state evolves dynamically

76

p(n) thold

0.05 w”: W W
/_

      
     

o //
15 [////// o
(80
Pm) thold
0.1 /%
0.05 ////// /r’/ C
0 ”1a 5.
, t1.
til/Tl 24 _50 ‘25 n (b)

Figure 2.18: Dynamic evolution of dephased cat state after tuning on the tunneling.
The parameters are chosen as in Fig. 2.1 with a collapse time of tool = 1.5 /T. In
the first picture (a), the hold time is set to be thold 2 tool? while for (b), we set
thold = 5tcol- It is shown that for (a), the system still revives to the initial state,
while in (b), the initial state can not be restored.

77

into a double-peaked superposition of number states, with one peak corresponding
to the majority of atoms being in one well, and the other peak corresponding to the
majority being in the opposite well.

We have shown that this process is stable against perturbations in the initial
state, as well as in the control parameters. For this reason we believe that the
dynamical evolution scheme is more likely to be demonstrated experimentally. In
addition, we have demonstrated that continuing the evolution after the cat-state
formation results in a nearly complete revival of the initial state. However, this
revival does not occur if the cat-state has collapsed into an incoherent mixed state,
so that this revival can be taken as proof of coherence of the initial cat—state. We
note that the states formed via adiabatic switching are maximum ‘all or nothing’ cat
states, while those formed dynamically are less distinct cat-like states corresponding
to a double—peaked distribution in Fock—space.

The effects of decoherence due to spontaneous scattering of laser photons have
also been studied in detail. We have determined that decoherence is strongest for the
case where the single-photon momentum recoil is not sufficient to remove the atom
from its initial state, due to Bose-stimulation as the atom remains in an strongly-
occupied atomic-field mode. In the opposite regime, where the scattering rate is
smaller by a factor of N, N being the atom number, we find that the maximally
entangled ‘all or nothing’ cat states are destroyed by the scattering of a single pho-
ton, while the dynamically-formed less-extreme cat-like states can survive multiple
scatterings without leaving the subspace of cat-like states, in analogy to the effect of
atomic collision losses studied in [72]. Because of this enhanced survivability, the less-
extreme cat-states may be more useful for applications, providing they are suitable
for the desired protocol.

In this context of ‘freezing’ the cat states for a prolonged time, we have also

considered the de—phasing effects of atom—atom interactions during this stage, and

78

conclude that collisions can mimic the effects of decoherence with regard to a revival-
type measurement, however with regard to applications, they may or may not be
detrimental, depending on the type of interaction and / or measurements involved.
We note that in the present proposal, there are three different time scales which
we have established: the time need to create cat state teat: the coherence time tcoh and
the collisional dephasing time tcol' First, to generate the cat state, the coherence time
scale tcoh must be longer than teat- While later, in order to hold the cat state for later
use in an application, a hold time thold shorter than tcoh is required to maintain the
macroscopic superposition state. And finally, another condition tcol > thold should
be satisfied to avoid the dephasing quasi-collapse, if such dephasing is detrimental

to the desired application. The coherence time, t is tunable by adjusting the

cob»
laser intensity and detuning, while the dephasing time @0me can be adjusted if

the interaction strength 9 is modified during the hold stage.

2.8 Derivations and further discussions

2.8.1 Adiabatic generation of cat states: why impractical

In this subsection, we explain the difficulty to generate cat-states via adiabatic
manipulation of the system’s ground state. In order to better understand the adia-
batic evolution of the ground state, we first examine the ground state under certain
parameter values for which it can be determined exactly. First, in presence of zero

tunneling, the bimodal Hamiltonian (2.4) is reduced to
H = 2119712. (2.44)

The ground state of this Hamiltonian is determined solely by the parameter 9. For g >

0, corresponding to repulsive interactions, the ground state will be I0) in our number-

79

difference representation (2.6), i.e., an insulator state. While in the case of 9 <0,
the ground state is doubly degenerate, corresponding to any arbitrary superposition
of the two extreme states IN / 2) and I — N / 2), and its orthogonal counterpart. These

states can be written as

———,1——(crIN/2) + (3) — N/2), (2.45)
with 01,13 being arbitrary complex numbers. Note when a = ,8, one recovers the
Shréidinger cat state (2.8).

In the other extreme case, if there is no atomic interaction but only the tunneling
term, the ground state of Hamiltonian (2.4) becomes the two-site analogue of the

superfluid phase [70]

N/2 N
19> = 2””2 Z In). (2.46)
—N/2 n

which is a binomial distribution of F ock states, and thus approximately Gaussian when
N is a large number. Henceforth we shall refer to this class of states as a ‘coherent
state’, meaning that each atom is in a coherent superposition of both modes and there
are no atom-atom correlations or many-body entanglement.

The more general ground state of our model, with the presence of both collisions
and tunneling, can be understood as a smooth cross-over between these two extremes.
First, under the condition of a repulsive atomic interaction, the ground states of this
system can be well approximated by a single Gaussian-like distribution of on peaked
at n = 0

en oc e—n2/(202), (2.47)

where the relative atom number dispersion a is obtained by minimizing by the mean-
energy of Hamiltonian. Second, under the condition of attractive two-body interac—

tion, the ground state of this system will be split into two separated wave-packets.

80

 

gap

 

 

 

—0902 —0.01 0 0.01
g/t

Figure 2.19: The energy gap Egap as a function of ratio g/ T. Here, Egap 2 E1 — E0,
with E0, E1 being the eigen energies of the ground and first excited state respectively.

0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. ' 1a)
c(n)0-- ’
—0.2
0.2 - , - _. _ .(b)
c(n)0 " V-
—0.2 .
0.2 k T 1%
c(n)0- ,
’04750 —25 9 25 50

Figure 2.20: The wave-packet c(n) of the ground state (Blue solid) and first excited

sate (Green dashed) with (a) g/T = 0.01, (b) g/T = —0.01 and (c) g/T = —0.02
respectively.

81

Thus instead of (2.47), a good approximation of this state turns out to be a superpo-
sition of two Gaussian distributions (2.9) [218], where again the minima center no and
spread width 0 are obtained by minimizing the mean-energy. For repulsive atomic
interactions, the energy minima is found at no = 0, which indicates that the atom
number distribution does not split. For attractive interactions, n0 3% 0, and under
the condition of no > a, the wavepacket of ground state will split and become a
superposition of two well-separated components, i.e. it becomes a cat-like state (2.9).
Note for this state, the relative phase for any Fock basis pair In) and I — n) is zero.
Based on these considerations, it is of theoretical interest to determine whether
or not the Shréidinger cat state in this bimodal BEC system can be generated by an
adiabatic process [218, 217]. As a basic scheme, the system would be first prepared
in the many-body ground state with positive 9, corresponding to a coherent state
centered at n = 0. Then by slowly tuning g to be negative, the distribution will split
and finally end up with a Shrodinger eat like state under the condition of no > 0.
There is a significant difficulty with this pr0posal, however, in that when the two
wavepackets in Fock space are well separated, the ground state, which is the desired
cat state, is nearly degenerate with the first excited state, as shown in Fig.(2.19).
It is observed that when 9 is positive, there is a relatively large energy gap between
the ground and first excited states. Then as 9 decreases, the energy gap decreases
so that at some point the energy gap becomes exponentially small. In Fig. 2.20, we
show the corresponding wavefunctions of the ground and first excited states c(n) with
different ratios g/ T. It is clear that the ground state does not evolve into cat state
until the bimodal system enters the degenerate regime. In the degenerate regime,
any small perturbation can mix the two states. As the first excited state is nearly
identical with the ground state, but with a 7r phase-shift between the left and right
wave-packets, an arbitrary superposition of these two states could easily correspond

to a localized state, which is equivalent to a collapse of the Schrc'idinger cat wave-

82

 

Figure 2.21: The probability distribution P(n) at time t = teat is plotted versus the
level of random noise in the initial state or. We see that even at or = 0.05 the output
state is still a good cat-state. This corresponds to random fluctuations in the initial
en’s at about 5% of their exact ground state values

function. Strictly speaking, this degeneracy implies that to adiabatically create the
cat state, the process requires an exponentially long time. Thus experimental real-
ization appears impractical, particularly when considering the inevitable exposure of

the system to noise.

2.8.2 Experimental Feasibility

In section 2.3, we have only considered the ideal situation where the initial state
is exactly prepared in the ground state and the atomic scattering length can be tuned
in a precise way. This may not necessarily hold true under experimental conditions,
where random fluctuations in system parameters may affect the formation of the cat-
state. Here, to test the susceptibility of the present proposal to such fluctuations, we

study the dependence of final cat—state on the precision of the initial state by adding

83

random noise to the exact ground state. Thus, the initial state becomes on, + (ST. The
random noise, (Sr, is a complex with an amplitude drawn from a Gaussian distribution,
and a completely random phase. The probability distributions for the amplitude and

phase are therefore

P(6) = —. (2.48)

Figure 2.21 shows the resulting cat-state as a function of 07-. We. see that a well
peaked cat state can be produced up to 0r a: 0.05, which is about c0(t = 0) / 6, i.e,
the peak of initial coherent state we make the cat state from. Thus this method of
generating a cat state appears robust against noise in the initial state, which could
be do to a finite temperature.

Another possible source of error in creating the cat state is from the inaccuracy
in control of the atomic interaction strength 9 and/or the tunneling rate 7'. This
effect is simulated by adding a Gaussian-distributed random number to negative 9.
We note that this is not equivalent to time-dependent fluctuations in g and T, but
rather represents imprecision in the control over the values of the parameters. The
result is shown in Fig. 2.23, where it is seen that the evolution system is sensitive to
the noise of the atomic interaction g, with only about i2% deviation being tolerable.
However, it does not necessarily mean that one can not create a cat-state without
the well defined 9. Instead, when 9 is shifted, the time need to create cat state tcat
is also shifted, and one can still obtain the cat-state on another time. The primary
difficulty here is that if the variation in g is unknown for a given run, then the precise

time to stop the dynamics, tcat is not known a—priori.

84

 

271' I I I

l

5% -251 o 25 50
n

 

 

 

Figure 2.22: The phase distribution 9n = arg(cn) at t = teat is plotted for the case of
random noise in the initial state with or = 0.05. We see that the phase distribution
agrees well with Fig. 2.3 up to a non-physical overall phase-shift. The disagreement
occurs only in the regions where the amplitude of en is small.

 

 

Figure 2.23: The eflect of imprecise control over the atomic interaction strencth g.
The noise of g is introduced by adding a real random number, where or is the width
of the corresponding Gaussian distribution, to the value of g in Hamiltonian (2.4).
The cat state is measured at a time of t = 14.5 / 7' and plotted as a function of or. It
is shown that uncertanties in 9 up to 3% can be tolerated.

86

2.8.3 Derivation of Master equation

In the presence of laser coupling to a BEC system, the total Hamiltonian for
the system, which includes the Hamiltonian for the atoms in condensate HS, the
reservoir of vacuum electromagnetic field HT and the interaction between them Vsr

can be written as

H = HS + Hr + Vtgr, (2.49)
_ 3_ All A .3, All A r
H,» — d 1A\Ile(r)\Ile(r) + d rQ(r)\Ile(r)\Ilg(r), (2.00)
Hr = Zwkd.L)\flk/\, (2.51)
k)‘
Vsr = Z / (137“ gkyeik'rcwlzt\Ill:(r)\llg(r)dk/\ + H.c., (2.52)
kA

where ‘Ile(r) and \Ilg(r) are the annihilation operators for atoms in the excited and
ground state respectively, and dkA is the photon annihilation operator for momentum
hk and polarization state A. In addition, we, w L and wk denote the frequency of
atomic excited state, laser field, and the reservoir photons respectively, while A = we—
w L is the detunning of laser field from the excited state. The atom-laser interaction is
governed by the Rabi-frequency Q(r), while the atom-reservoir interaction is governed
by 91:). = 275ng - sky, with (1 being the atomic dipole moment in mks units. We
note that for the purpose of deriving the master equation it is not necessary to include
terms in H s governing the free evolution of the ground state, as this involves very slow
time scales relative to the dynamics of the excited-state/EM-vacuum system. These
terms can be added to the system Hamiltonian at the end of the calculation, as any
energy-shifts they introduce will be negligible compared to the natural line-width of
the excited state.

The quantum state of the system+reservoir is described by the density opera-

tor psr. An effective equation of motion for the reduced system density operator,

87

p3 = trr{p3r}, can be derived via second-order perturbation theory by following the
standard approach (see for example [59]). For the case of a zero-temperature bath

this leads to the master equation of

p.9(t) = _2Hflgpg(t )— i/0T dT tr,{VI(T )P9,(t)} — (2.53)

ll
é/OT dT [07,, dT” t17{l1(T)V1( P97() VI(T,)ng-(t)VI(T )} + H.c.,
where P3,. is the system-reservoir density [237. in interaction picture,

1981‘“) = €i(H8+HT)tpsr€_i(HS+Hr)t. (2.54)

The interaction picture system—reservoir interaction, V1 is defined as

V1(r) = ei(H3+H7')Tl/e—i<H3+HrlT (2.55)

Z/d3TJAAP —i(wk-wL)t\Ill(r r)\Ilg(r)ak)\ + HHC (2.56)

where we assume that T is sufficiently small that exp[iH3T] 5:: 1, yet large compared
to the relaxation time of the reservoir so that p3r(t — T) m p3(t) (8 pkg, Q] where p[el esq]
the density matrix for the equilibrium state of the reservoir. For a thermal reservoir
we would have plea] = e—BHT/Zr, where ,6 is the inverse temperature and Zr is the
partition function. In the tracing over reservoir of vacuum modes, we assume that
the all modes are unoccupied, i.e. zero temperature, so that
(5110‘) = (filly) = 0,
may,» = claw =0,
(abakw) = 0,

(ékA51L/AI) = 6kk’6A/\” (2-57)

88

Invoking the Markoff approximation for the radiation field of the atoms,

I
1 7' t ll I ” , '
;/0 2.2/0 .1. e—M-wwt—t >W(w,,_..L)+_zl’_ (2.58)

wk—wL’

The resulting master—equation is then

(i.e(t) = -iH.s-ps(f)
/ d3r(l37" [L(r — r')\Ill:(r)\Ilg(r)\IIL(r’)\I/e(r’)ps(f)
— L*(r — r’)\i:;‘,(r)xiz..(r)p..(r)xizl(r’)xi:g(r’)] + H.c., (2.59)
where
L(r — r’> = :2; lgkil2€ik'(r‘r’){7r5(wk — m + $471.} (260)

with 79 indicating a principal value. In the limit of infinite quantization volume, the
summation becomes an integral, which can be done analytically [228], yielding

L(r — r’) = —I‘e”iC [sin2 Oé- + (1 — 3cos2 6) [é — Ely—H (2.61)

where F = widQ/(Bweohc3) is the single-atom spontaneous emission rate, C = kLlr —

r’ I and 0 is the angle between the dipole moment (1 and r — r’

. The real part of
L(r — r’) will contribute to decoherence in the master equation (2.59), while the
imaginary part contributes an energy shift due to photon exchange between atoms.

In the limit |r — r’l —+ 0 we find
+ 2'6, (2.62)

where the imaginary part, 6 is an infinite quantity, reflecting the fact that the standard
two-level atom-field interaction model is not properly renormalized. Physically, this

term is a Lamb-type shift in the energy of the excited-state due to interaction with

89

the vacuum modes of the EM field. In our system we can always choose a rotating
frame to absorb this shift, so that we can take 6 = 0 without loss of generality.

For most optical traps, the laser is detuned very far from the atomic resonance
frequency, so that the excited state occupation number is << 1. In this regime it is
useful to perform an adiabatic elimination of the excited-state operators. To eliminate
the excited state in the master equation (2.59), we note that for a far-off—resonant
laser field, the excitation rate in BEC is extremely low, which enables us to expand

the density matrix to a good approximation as

P 3 me + 001 + 910 + 911, (263)

where p00 and p11 stand for the density matrix of zero-excitation and one excitation
state respectively, while p01 and p10 describe coherence between these two manifolds.
In fact, it is convenient to define a projection operator Pj, which is a projection into

a subspace with j atomic excitations of the BEC system, such that
p — .pP (2.64)
W ith this expansion, one obtains

P00 = -i/d37‘9(r)‘i’;(r)‘i’e(r)001+

[d3rd3r'L(r—r’)\i1},(r)xi: er()p11\II:£(r’ mtg-(r ’r)+H.c., (2.65)

pm = -i/d3 TQ*(I‘ )‘I’l( )‘1’g( )poo
+i/d37“9* ()1011‘1’12 (r)‘1’g( )
—/d d3’rd3r L(r —r>w*< > antic N! < )p10 — zApm. <2 66>

90

(301 = p10, (26?)

and

bio —z' f d3rs*( Nineteen/+01
_ / d3rd3r' L(r — r’)\II:(r)\Ilg(r)\I/;(r,)\I/e(r,)p11

+H.(:.. (2.68)
In the assumption of low excitation rate, one can adiabatically eliminate p01 and pm
by demanding p01 z 0 and p10 z 0 respectively, yielding

— /d3r 0A0.) ‘i’l(r)‘i’g(r)p00 +

 

.010

9* ~ ~
d3. (”pus/Haas)

 

_ I A A A A
2: / d3rd3-r' fig—53:29:.(r)\Ilg(r)\I';(r’)‘Ile(r’)/)10 (2.69)

Solving this equation perturbatively up to the order 512, one obtains

 

m = — / (13+ 9 “’ ‘i’l(r)‘1’g(r)P00 + + [.13. Q—gflpmfilmar) (2.70)

 

 

A
I I/L_’§2* ”. - . - . II. II
—i/d3rd3r d3 (r :32 (r )el(r)eg(r)eg(r’)we(r’)wl(r )xpg(r )p00
3 gig/(L(r— r')n*<r”)A( . ., ,. , .,. n
[d rrd d 212 warmarwarmarmim-war>.

and p01 = p31. Inserting this result for p01,p10 back into Eq. (2.68), we find

A

T611 = i / 61371137" I mm:(r)‘i’g(r)poo‘i’;(r')‘1’e(r')

—i/d3rd3r ’M®l(r)

)

A A I II ”I
\pg(r)\pL(r’)\pe(r’)p11+/d3rd3r d3r d31-

91

L “ ‘1');er ’wlrguil (1)/900x

®L(r”’>w (r ”’wle’ )th ) L(r ”me ”>

/d3 d3 ’d3 ”d3 ”’ L*(I‘ "- r')5:"‘(I‘)Q(r
A

 

III

A

)\I’;(r)\llg(r)\II;r,(rm)\Ile(rm) x

 

A I A I A A
male war )wl<r”)we<r”)

+i/d3rd3r'ur—r’)+i/I.(r)+1/,(r )qfl‘u’ )\Ilg(r')p11 +H.c.. (2.71)

Next, we apply a similar adiabatic elimination process eliminate the excited state
matrix p11. To do that, first we normal order the ground state operators in above

equation, and then by setting p11 z 0, obtains

p11 = i/d37‘d37" %L)Wl(r r)\IIg(r r)

><()00‘ji’;(1")\1’e(r')

[d3 m3 “gig—“let )‘1’;(r'r)‘i’g(r)r‘i’e(r')p11 (2.72)

fir A A
/ d3 '22:} wi<r>1le<r>pu

 

Lr — ’ . . . .
— [d3rd3r —(—,,-,.—r—)\IIZ<r)\I/I,(r’)wg(r)we(r’>pu

I
3 3 ’ 3,”L*(r”-r’)9*(r)9(r)*‘r *
+/d Td r d 7 2A2F \I’e(r)\Ilg(r) x

A1. I A I A1- I/ A ll
POO‘I’g(r )‘I’g(r )‘I’g(r )‘I’e(r )
III ///*”—’* .
_/d3rd3r d3, d3, L (r rm (r)Q(r by:
2A2I‘

x11 < r'”)/)11‘1’lz(rl r>wg<r r'>\ill<r”)\ile(r”>

 

 

+H.c.,

92

Again, we solve this equation perturbatively up to order K12, and arrive at

 

=1: I >1: ’
p11: /d3Td3T d3T ” L (r” -r )9 “>90. )¢I:(r)‘llg(r)

 

 

223%
1- , A1- II A II
><pOO‘I’g (r ljl‘i’glr )‘I’gfr )‘I’e(1‘ ) (273)
, _ / * III/_ m =1: II ’”
_/d3rd3r’d3,,”d3r”d3rmzL(r r)L (r 1‘ )Q (r )Q(r ) X
2A2r2
A A A III A A
swim.warm.)¢1u~>ammo+1<rm Mr )wL(r””>\I+e<r””>
II
_fd3"13,!d3,”d3,,’”d3r/mL(r-1")L*(r””-r'”)9*(r’”)9(r ) X
2A2I‘2

A A . A A A A III A III A A
«(Res/L(r)warm(r’>wl<r’”’>w§5<r””)wt(r >wg<r )/)00‘I’L(I‘")‘Pe(r")

+H.c.,

This expression for p11 can be simplified by normal ordering. Finally, we insert

the normal ordered p01, p01, p11 back into Eq. (2.65), and keep only terms up to

X12. Meanwhile, since the number density of atoms in BBC is small, we drop any

3~body and above terms. With these two assumptions, we obtain the resulting master
equation (2.14),

1912

page) = —iH.ps<t)——A—2 [d3r®L(r)@g(r)ps<t>—

r r—r * ’ . , ~ ,
[d3rd3r’m W A2 )9 ( key. )1; ®;(r)wg(r)wg(r)ps(t)+

/ d3Td3T’Q MWA} r )9“lagent/9(a)),(t)xi1;(r’)\ixg(r’) + H.c..

 

 

where we have written p3 E p00.

93

2.8.4 Atom loss dynamics: elastic and inelastic regime

To derive the atom decay rate of a single atomic-field mode, we will need to make

use of the commutation relation
We. W (was) = —¢(r>ci\il<r> -— crew" (me. (274)

where c is the annihilation operator for atoms in mode o(r). Then, from the master
equation (2.14), it is straight forward to obtain the evolution of the mean mode

occupation no = (clc) as

so = _ / (137213719?—[¢*(r)53(r—r’)(ait(r’))

where G(r, r’) = Q(r)L(r —— r’)§2*(r’). Making a single mode. approximation,
to) = (15(r)é, (2.76)
one obtains
7’10 = ——A2—2 /d3rd3r'§R [0(r,r’)] |¢(r)|2 x [53(r—r’) — |¢(r’)|2] 730(277)

To better understand the physics described by this equation, we make the simpli-
. I
fying assumption f2(r) z QeZI‘L'(r_r ), where k L is the laser wave—vector, satisfying
kL - d = 0. This leads to
1913

7'70 = ‘37 ( — {)no, (2-78)

94

 

0.6 1
0.4 -

0.2 }

 

 

 

 

(5‘ 0:2 0:4 0:64 0:8 1
0'

Figure 2.24: Dependence of the inelastic loss parameter 6 on the spatial size of con-
densate, which is indicated by the dimensionless parameter a as shown in Eq. (2.80).
It is seen that 5 is approximately unity when the size of condensate is much less than
the laser wavelength, i.e. 0 << 1, which means there is negligible atom loss. As the
condensate size approaches half )1 L, 45 decreases to around 0, corresponding to the
maximum loss rate.

where

5: % / d3rd3r’ae’33L‘3—r3ur - r')ll¢g(r)l2|¢g(r')|2- (2.79)

In the elastic regime we have |<f>(r)|2 % 6(r) relative to L(r — r’), which leads to g = 1
so that there is no atomic loss, as expected. As the size of condensate grows, é will

decrease, as shown in Fig. 2.24, which plots 6 versus condensate size for the case of

a spherically symmetric Gaussian condensate wavefuction

 

1 —7'2/(20'2A%)

¢<r> = (wees/48 .

(2.80)

where a is the ratio of the condensate size to the laser wavelength, ’\L- From the
figure, we see that the decay rate of the condensate can be neglected under the
condition 0 << 1, while for a mode whose size is comparable to or larger than the

laser wavelength, we have 6 ——> 0 so that the population will decay at the expected rate

95

|Q|2F/A2, which is simply the electronically-excited state fraction times the excited

state lifetime.

96

Chapter 3

Optimized Double-well
interferometry with

Gaussian—squeezed states

In this chapter, we Show that a Mach-Zender interferometer with a gaussian
number-difference squeezed input state can exhibit a sub—shot-noise phase resolution
over a large phase-interval. We obtain the optimal level of squeezing for a given phase-
interval A60 and particle number N, with the resulting phase-estimation uncertainty
smoothly approaching 3.5 / N as A60 approaches 10 / N , achieved with highly squeezed
input state near Fock regime. We then analyze an adaptive measurement scheme
which allows any phase on (-—7r/ 2,7r/ 2) to be measured with a precision of 3.5/N
requiring only a few measurements, even for very large N. We obtain an asymptotic
scaling law of A9 z (2.1+3.2ln(ln(Nt0t tan A00)))/Nt0t, resulting in a final precision
of z 10/Nt0t, with Ntot the total particle number used in adaptive measurements.
This scheme can be readily implemented in a double-well Bose-Einstein condensate
system, as the optimal input states can be obtained by adiabatic manipulation of the

double-well ground state. The major results of this chapter have been published in

97

“Optimized Double-W’ell Quantum Interferometry with Gaussian Squeezed States”,
Y. P. Huang and M. G. Moore in Phys. Rev. Lett. 100, 250406 (2008).

The organization of this chapter is as follows. In section 3.1, we provide a back-
ground on quantum interferometry and briefly review previous apparatus to achieve
measurement precisions beyond the classical limit. In section 3.2, we present the
model of Mach-Zehnder (I‘V‘IZ) interferometer and a realization in a double-well BEC
system. In section 3.3, we introduce the Bayesian analysis as a reliable method to
evaluate the interferometers" performance. The major result of this work is presented
in section 3.4, where we show that ultrahigh-sensitivity phase measurements can be
realized by feeding a Mach-Zehnder interferometer with optimally number—squeezed
input states. Such states are obtained in a double-well BEC system as the ground
state in presence of repulsive atomic interaction, providing experimental feasibility in
present or near-future laboratories. We then briefly conclude this chapter in 3.5. For
a comparison, in appendix 3.6, using the Bayesian analysis, we analyze the perfor-
mances of two existing interferometers, one employing number-squeezed input states

while the other based on quantum discrete Fourier transformation.

3. 1 Introduction

The goal of quantum interferometry is to overcome inherent limitations of classi-
cal interferometers, thereby significantly improving the phase measurement precision
provided with similar amounts of resources [229]. Due to the nonexistence of a well-
defined Hermitian phase operator [230], a phase eigenstate in general does not exist, so
that the standard quantum measurement postulate does not apply. Instead, a phase
shift 0 can only be estimated by measuring some phase-dependent observables. The
accuracy of this estimation, measured by a phase uncertainty A6, is then a critical

parameter in evaluating the interferometers performance. Ultimately, the minimal

98

obtainable A0 is constrained by the inherent quantum fluctuation in the interfer-
ometer output, which, unlike classical noises, can not be suppressed by improving
measurement precision. In a Mach-Zehnder (MZ) interferometer with coherent input
states [231], the participating particles are uncorrelated with each other. The output
fluctuation is then limited by the shot noise, corresponding to a m uncertainty
in the number difference between the two output ports of the interferometer (Ntot
is the total particle number). The phase uncertainty, determined by the signal to
noise ratio, is then A6 ~ M/Ntot = 1/ \flVt—o—f This precision is widely recog—
nized as the standard quantum limit (SQL), also referred to as “shot noise”. If the
particles are correlated, the output fluctuation can be suppressed, leading to a re-
duced phase uncertainty from SQL [232, 233, 157]. The minimal phase uncertainty
allowed by quantum-mechanics, given in a analogy to the Heisenberg-uncertainty, is
Admin = l/Ntot- Such is the so—called Heisenberg limit. Interferometric devices
yielding phase uncertainties at or near the Heisenberg limit, with A0 = fi/Ntot and
,8 Z, 1, are considered efficient, generally referred to as Heisenberg scaling (HS) inter-
ferometers. Given the same amount of resources Ntot: a HS interferometer tremen-
dously reduces the phase uncertainty from SQL by an immense factor of \[Nfo—t-
The improvement from SQL to HS turns out to be very useful, mainly due to
two reasons. The first is to overcome the physical limitation on resources available
for interferometry uses. For example, the maximally attainable Ntot for an optical
interferometry is limited by the achievable laser intensity. Eventually, this limita-
tion can be overcome as technology develops. The second, perhaps more crucial
reason is that in various applications requiring non-demolition measurements, HS
interferometers are the only feasible candidates, as is usually the case in the field
of quantum information processing. For example, in a MZ-interferometric approach
to generate entanglement between isolated single—atom qubits, the allowed photon

number Ntot is primarily restricted by the need to avoid the spontaneous emission

99

of atomic qubits. It turns out that under such a constraint, a HS phase-sensitivity
is a required, unless optical resonators are used [234]. All these give rise to the
importance of HS interferometry, as reflected by the continuous interdisciplinary ef-
forts to construct HS interferometers in the past decades. Thus far, particular efforts
have been made in elevating the performance of MZ interferometry with nonclassical
squeezed/entangled input states [232, 157, 158, 30, 235, 236, 237, 238, 239]. Re—
cently, a range of non-MZ interferometries employing nonlinear beamsplitters and / or
novel measurement protocols have been proposed [26, 240, 27, 180, 241, 242, 243],
among which the maximally-entangled N-particle NOON state, in a superposition of
—\}——2—(|N, 0) + [0, N )) has attracted intensive attentions [24, 25, 26, 27].

According to their usages, interferometers can be generally categorized into two
classes, namely phase switches and phase meters. A phase switch is built for the
purpose of detecting the presence of a phase imbalance induced by environmental in-
homogeneities, for example, the gravity gradient. A phase meter, on the other hand,
is used to quantitatively read out an unknown phase shift. In general, constructing a
phase meter at a sub-SQL precision is much more challenging than building a phase
switch at the same precision. This is because the quantum fluctuation in an interfer-
ometer output is usually dependent on the value of phase shift 6, and thus cannot be
congruously suppressed with the a-priori information on 0 [229]. For example, for a
MZ interferometer, at a small phase imbalance, the quantum fluctuation at the inter-
ferometer output can be efficiently suppressed by number-squeezing the input state.
This mechanism has allowed a series of phase switches operating at HS sensitivity
[230, 232, 157, 158, 24, 235, 236, 119]. For not-so—small phases, however, a strong
number-squeezing would adversely increase the output fluctuation. As a matter of
fact, a MZ interferometer with maximally-squeezed input states has been shown to
yield a worse-than—SQL phase resolution for [6] >> l/Ntot [244].

To evaluate the performance of an interferometer, or precisely, to determine the

100

has

the)

Olly?

film)"
? - ,
Lllt‘ t
Fl

JEAN

.)- 1
ul,[

P
‘Llil
it 1
. ["1

.21“,
Y]. I

' ‘3 1
“Mil

phase uncertainty A6, there exist two major statistical methods. The first is the
linearized error propagation analysis, where assuming 72(6) is a 6—dependent output
observable, A6 is linearly related to the quantum fluctuation An in 71(6) by A6 =
(g%)_1An. This method, being extremely intuitive and easy to apply, has been very
popular among scientists in this field. Only recently was it found to be valid only if
the probability distribution of 72(6) is a singly-peaked Gaussian, which is, however,
seldom the case in the quantum interferometry [245, 244, 246, 119]. A second, more
general method is the Bayesian analysis applying to quantum statistics [247]. In this
approach, the conditional probability of estimating a physical phase 6 to be (1), namely
P(gbl6), is evaluated utilizing the classical Bayesian theorem. The phase uncertainty
A6 is estimated at the confidence interval, which can be 68%, 95% or 99%. By this
analysis, quite a few of existing interferometers, which were previously predicted by
the error propagation method to be HS, turn out to yield a near or worse-than SQL
phase resolution when measuring nonzero phases [230, 232, 157, 24, 235, 236, 26, 240,
27, 180, 241, 242, 243].

This far, it has been shown in theory that the Twin-Fock (TF) state and related
Pezze-Smerzi (PS) state can achieve HS precision when measuring a zero phase, thus
implementing HS phase switches [248, 244, 119, 131]. The TF state is a maximally
number-squeezed state of [N / 2,N / 2), i.e., having an equal and definite number of
particles in the two input modes, whereas the PS state is a two-mode superposition
of —\}3|N/2 + 1, N/2 — 1) + |N/2 - 1, N/2 +1). We find, however, that for |9| > 1/N
the phase precisions of both TF and PS interferometry quickly degrade to worse-
than-SQL. Eventually in the limit of large N, the phase uncertainty saturates at a
constant, N -independent value. This saturation behavior for the TF state was first
observed by Kim et al in 1998, where they found the phase uncertainty A6 ~ 1 /Ntot
for 6 < l/N, and growing rapidly thereafter [248]. The work of this chapter similarly

finds that TF and PS states become worse than shot-noise for [6] >> I/N. On the

101

other hand, we verify that a truly HS phase meter, i.e., A6 or 1/N for any 6, can be
implemented with quantum discrete fourier transformation (QDFT). The drawback
is, however, that implementing QDFT in practice requires the extremely-challenging
capability to perform quantum computation over a large number of qubits, making
it experimentally unfeasible in the present or even near-future laboratories.
Recently, with the well-established large degree of spatial and temporal coher—
ence, the system of Bose-Einstein condensate (BBC) has emerged as a viable can-
didate for matter-wave interferometry [30, 31, 32, 33, 34, 35, 36, 37]. Compared to
optical systems, a BEC—based interferometer has the unique advantages of sensing in—
ertial forces, measuring gravity gradient, and allowing measurements at single-particle
level with high-efficiency detectors [38, 39]. Furthermore, unlike photons, atoms are
strongly interacting particles. This feature serves a natural source of nonlinearity
which can lead to nonclassical many-particle states [249]. In particular, in BEC, the
strength of inter-atom interaction via s—wave scattering is conveniently tunable by
Feshbach resonance technique, providing the ease to engineer the system’s quantum
state [31, 118]. A most accessible, archetypical BEC interferometer is realized in a
double-well BEC system. The beamsplitter, in an analogue to MZ interferometers,
is implemented by manipulating the potential barrier between the two trapping wells
[37, 80, 81, 82, 83]. For this system, the squeezed-vacuum protocols are not appli-
cable, while the NOON state is not suited to determine an unknown phase due to
the periodicity of the phase—distribution [250, 244]. This leaves number-difference
squeezed states as a viable candidate, although to date there has been no system-
atic study of how to measure arbitrary phases at HS precision in the large-N limit.
In this chapter, we perform such an analysis and show that an asymptotic scaling of
(ln(ln Ntot))/Ntot can be achieved via multiple adaptive measurements with Gaussian

number-squeezed states, which can be readily created in a. double—well BEC.

102

 

 

 

 

 

Figure 3.1: The schematic graph of a. MZ interferometer.

3.2 Double-well MZ interferometer

3.2. 1 MZ interferometer

A MZ interferometer is a balanced homodyne measurement device consisting of
two 50/50 beamsplitters (BS), as shown in figure 3.1. It has two input and output
ports. During a phase measurement, the input beam, entering the interferometer
via one or two input ports, passes through the first 50/50 BS, thereby coherently
bifurcating into two beams traveling in the upper and lower arms. As it propagates,
each beam independently acquires local phase shift of the arm it travels in. The two
beams are then recombined at the second 50/ 50 BS to give the interferometer output
in the form of interference pattern. The output state is determined by the relative
phase shift of the two arms, which is inferred by measuring the number difference of

particles coming from the two output ports.

Describing the upper and lower arm with bosonic annihilators (lo and [21, it is

103

convenient to introduce the following momentum operators [157]

A 1 A . . A
Jr 2 5(a5a1 + a]a0), (3.1)
A 1 A A A A
Jy = Elahal — cling), (3.2)
A 1 A A A A
.13 = §( gag—a]a1), (3.3)
and the particle number operator
M—aia +3)“ 34

It is straightforward to verify that Jr, J? , .Iz constitute a SU(2) group. The total spin

A N N
J2=— —
2(2+1) (35)

J satisfies

The function of 50/ 50 BS is then represented by a J1: rotation for a 77/ 2 angle, with
the propagator
UBS = exp (13.1213) . (3.6)

The phase shift is a. 6-rotation along J2 axis, described by the propagator

A

Uphasew) = exp (i6Jz) . (3.7)

In the above spin-rotation formalism, the interferometer input and output state are

connected by a passage propagator 0, given by
Ilka-at) = 0(0)l\pin> (3-8)

where

A A

11(9) = UBsUphaseWWBs- (3-9)

104

During the transformation, the total spin of the system J is conserved. Using the
commutation relations (3.1)-(3.3) and that eAeB = eBeAelA’Bl, the propagator (I (6)
is reduced to

A

U(6) = exp(—2'6Jy) exp('i.7rJ_T). (3.10)

Thus for a symmetric interferometer input, the output state is merely a —6 rotation
along the Jy axis.

Due to the conservation of total spin, the state transformation of the MZ inter-
ferometry can be visualized in the Bloch sphere picture [237]. In this picture, a state

[‘11) corresponds to a quasi-probability on Bloch sphere of radius V J 2 z N/ 2, defined

 

as
PM. at) = I<N/2Ie“y(”/ 333332311013, (3.11)
where the Fock state [N / 2) = 1m((13)N|vacuum). Note any N-atoms state for
1 .

which all atoms are in the same single-particle state can be written in the form
e—‘ij 26be—ij 9(7r/ 2_‘9)|N/ 2), which has eigenvalue N/ 2 with respect to the projection
of J onto the axis defined by 6,¢. In this picture, the two-mode coherent state
corresponds to a round of radius \/N / 2 on the sphere centered at (J3) = (Jy) =
0. Number squeezing corresponds to compressing the round along the Jz-direction,
resulting in an ellipse, as shown in figure 3.2 (a). Phase squeezing, on the hand,
corresponds to compressing along the J9; direction. Specifically, a maximally-squeezed
TF state is a thin equatorial ring on the sphere.

In the Bloch-sphere picture, the process of MZ interferometric measurement is
intuitively described by a set of rotations, as shown in figure 3.2. For a number-
squeezed input state, the first BS rotate the state along J3; axis for an angle of
7r/ 2, transforming it to a phase-squeezed state, as shown in figure (b). The phase
acquisition corresponds to a J2 rotation for an angle of 6, as seen in figure (c). The

second BS, performing another J1: rotation, transforms the state into the output state

shown in figure (d). Clearly from the figure, the total effect of rotations corresponds
to a Jy rotation for an angle of —6. Phase information is obtained by measuring the
number difference between the two interferometer modes, which projects the output
state onto a Jz eigenstate. Quantum fluctuations in this measurement are governed by
the projection of the output distribution onto the Jz-axis. Due to the rigid rotation,
the width of the projection will be determined by a 6-dependent combination of the

J; and Jr noise of the input distribution.

3.2.2 Double-well BEC as MZ interferometer

hilatter-wave interferometers have emerged as powerful tools for ultra-precise
detection of gravitational waves and inertial forces [232, 251]. Among them, the
double-well BEC system has manifested itself as a promising candidate, with its
established large degree of spatial and temporal coherence and the recent progress
towards single-atom counting. As discussed in chapter one, a double-well BEC can be
consisted of atoms in two internal spin states, or in two spatially-separated trapping
potentials. In either case, the system can be approximately described on the state
basis of two localized modes. The modes are mapped onto the upper and lower
channel in a MZ interferometer as shown in figure 3.1, and are similarly described by
annihilation Operators of [1.0 and £11. In this two-mode approximation, the double-well

BEC system is described by the Hamiltonian,

A

H(X) = -271, + 6sz + U12, (3.12)

where T is the inter-well tunneling rate, U is the atom-atom interaction strength, and 6
is the asymmetric tilt of the double-well, presumably due to the external perturbation

being measured. For repulsive atom-atom interaction and 6 a: 0, the ground state is

106

NI2 \ 4.,

N/4~-

 

 

-N/4+

 

-NI2>

 

>;_/

\ "’0’
N/4N\’2/fi

Figure 3.2: State engineering for MZ interferometry in the Bloch—sphere picture.
where (a) plots the interferometer input of number squeezed state, (b) shows the

state after the first BS.

107

NI2~ AZ:

N/4 ~

 

 

-N/4-+ 3

 

 

-NI2 A

 

Figure 3.3: State engineering for MZ interferometry in the Bloch—sphere picture,
where (a) shows the state after phase acquisition, and (b) plots the final output state
obtained after the second BS.

108

very close to a Gaussian squeezed (GS) state of the form

N/2 2 2
[0) cc 2 e—" /40 [72), (3.13)
72.2—N/2
where [71) is a number-difference eigenstate satisfying Jz|n) = Tiln). The width 0,

depends on the parameter u = U / T, and is given by a2 = N/4x/1—m [227]. This
formula turns out to be a good approximation to exact for a wide range of system
parameters, as shown in figure 3.4. We note that u. can be tuned by varying U / T
via a Feshbach resonance and/ or changing the shape of the double-well potential,
and allows 0 to be varied between 0 and \/N / 2, corresponding to maximal number-
difference squeezing and no squeezing, respectively.

To implement a MZ interferometer, we set U = 0 and allow tunneling for t =
7r / 4T duration, thus realizing a linear 50 / 50 beamsplitter, described by the propagator
eigj‘”. This is followed by a sudden raising of the potential barrier to turn off
tunneling and allow phase acquisition due to the small but non-vanishing 6. Holding
the system for a measurement time T, a phase shift of 6 2 —(5T will be acquired,
described by the propagator 826.73. The barrier is then lowered again to implement
a second beamsplitter. The evolution of the system is shown in figure 3.5, where we
plot the probability packet dynamics for a phase shift of 6 = 7r / 4, with a moderately
number-squeezed state and the PS input state in figure (a) and (b), respectively.
In both figures, the probability-packet of the system, initially centered at n = 0, is
shifted from the center at the end of measurement by a distance determined by 6.
Measuring the number difference of output state then provides information on 6. In
figure 3.5 (a) with an input of some number—squeezing, the final probability-packet
of the system exhibits a sharp peak, suggesting a high signal to noise ratio, thus

a small phase uncertainty. In contrast, using the highly-squeezed PS input state

as shown in figure (b), the system evolves into two peaks, each peak yielding large

109

 

 

 

 

 

 

 

 

0.6 f . .
—— N=100 Exact Ground state with u=N/16-1/N
_ - - - N=100 Gaussian Envelope state with 0:1
0'5 ~- . -‘ N=10000 Exact Ground state with u=N/16-1/N 3
N=10000 Gaussian Envelope state with o=1
0.4L -
0.3 - _
i=2
0- 0.2 - 1
0.1 ~ -
-10 -5 0 5 10

Figure 3.4: A comparison of the exact ground state of a double-well BBC and the
Gaussian enve10pe (3.13).

oscillatory tails. This indicates a strong quantum fluctuation, which gives rise to a
large phase uncertainty. From here, we can preliminarily conclude that too much

number squeezing acts adversely in minimizing phase uncertainty.

3.3 Bayesian analysis of phase uncertainties

As discussed earlier, a phase shift 6 can only be estimated indirectly by measuring
some 6-dependent observables [230]. The estimation accuracy, measured by the phase
uncertainty A6, is commonly estimated either by the linearized error propagation

method or by the Bayesian analysis [252, 232, 253, 82, 158, 244, 119]. In the error

110

 

 

 

     

   

 

@005v'62’
O-t :
/ . ’ 400
0’ ..__._- 200
' 1 w] '200 n
t/T 1.5 -400

(b)

Figure 3.5: Probability packet dynamics of the double—well BEC system undergoing
MZ interferometric measurement with N = 800 and a phase shift of 6 = 7r / 4. Figure
(a) and (b) are shown for an moderately number-squeezed and the PS input state,
respectively. In figure, the process of phase imprinting, occurring at t 2 TT / 4, is not
shown.

111

propagation method, with a 6—dependent observable [1(6), A6 is estimated to be [252]
d 6 —1

Aq’) = I—I‘QI Art, (3.14)

where ”(6) = (M6)) is the expected value of 6(6), and All is the variance, given by

Au = (II )- #2- (3-15)

This method, while widely used in the field of quantum interferometry, turns out
to be valid only if the output distribution is a single Gaussian [244, 246, 119]. In
more general cases where the output is not a Gaussian, the error propagation method
gives imprecise, sometimes incorrect, predictions. For example, the NOON-state
interferometer, employing the maximally-entangled two-mode state of fiflN, 0)) +
IN, 0)), is predicted to yield a HS phase resolution for any 6 by the error propagation
method [24, 26, 254, 240, 241, 180, 242]. Yet as lately revealed [250, 244], for such
a interferometer, phase shifts of 6 and 6 + 2k7r/N (I: is a nonzero integer) always
lead to identical experimental outcomes, due to the existence of N -periods in the
interference pattern. Thus, phase shifts of this class are then indistinguishable, giving
rise to a phase uncertainty of 2n. An arbitrary phase 6 can be determined within an
uncertainty A6 ~ l/N only if 6 is already known to be within a. interval of period
27r/N, an evident paradox.

A improved measure of phase uncertainty, applying to arbitrary interferometer
outputs, is provide by the Bayesian analysis [247]. In this approach, we seek to
determine the conditional phase probability P(qblui), for a given measurement result

I32“ P(gbmi) is indirectly obtained using Bayes theorem, via

P(/1't|¢)P(¢)

P(elm) = 130%) ,

 

(3.16)

112

where P(pilgb) is the probability distribution of interferometer output in presence of
a phase shift ab, and can be calculated directly using input-output formalism (3.8).
P(pi) and P(qfi) represent priori information on the distributions of p,- and (p, be-
fore the phase measurement. Restricting the unknown phase to be in [—7r / 2, 7r/ 2], in
the case of no priori information, P(qb) = H (n/ 2 — [6]), where H (:r) is the Heav-
iside step function for 2:. P(ni) is then determined by normalization to satisfy

jib/32 P(¢llti)P(/li)d¢ = 1, resulting in

P(#t|¢)P(¢)
f P(MWWWWW

 

P(cfil/tt‘) = (3.17)

With P(¢l#i)+ an experimenter can then perform phase estimation upon a measured
result [2,. For example, the unknown phase 6 can be estimated to be the most likely
value fibmax, for which P(g’omi) is maximized. The phase uncertainty A6 is then given
by confidence interval. Or if P(¢>Ilti) is singly peaked, 6 can also be estimated by the

mean value of 6, where ()3 = f P(gblpiwdqb. The uncertainty can then be given by the

 

RMS variance, with A6 = \/f P((blpi)q§2dqb — (132.

In order to theoretically evaluate the performance of an interferometer, however,
we must take into account the fact that the measurement result, pi, is a random
outcome obeying the quantum probability P(pil6). This requires to average over all
possible outcomes to obtain a statistical mean phase distribution. For a phase 6, such

a distribution, namely the phase density, is given by

Pele) ZPquM (MW)

 

13032]wa (¢)P(.uil6)
Z I P (Hil¢’)P( ¢>’)d¢’ (3'18)

Clearly, this phase density characterizes the probability of estimating a true phase

6 to be (,1). From here, the phase uncertainty can be estimated by the confidence

113

interval,

Min(7r/2,6+A6)
/ P(¢|6)dqb = t, (3.19)
Max(0,6—A6)
where e = 0.68, 0.95, 0.99 correspond to a phase uncertainty at the confidence level of
68%, 95% and 99%, respectively.

In appendix 3.6, we use Bayesian analysis to evaluate two existing interferometric
apparatuses emerging as prospective HS interferometers. One is the MZ interferom-
eter with the PS input state. The other is a. non-MZ interferometer built on QDFT.
We find for the PS-state interferometer, while exhibiting HS sensitivity for a zero
phase, the phase uncertainty blows up to worse than SQL when measuring a nonzero
phase. Besides, such a state, in a superposition of two next-neighbor TF states, shall
be difficult to be produced in experiments. In contrast, the QDFT-interferometer
yields a phase uncertainty _<_ 7.47/N for any 6, and hence justifies itself as a HS
phase meter. Yet, experimental implementations would require performing quantum

computation over a large number of qubits, which is extremely challenging at present

or even in a foreseeable future.

3.4 Optimized Double-well interferometer

In observation of the lack of an efficient phase meter to measure arbitrarily un-
known phases, in this section, we propose to optimize the level of number-squeezing
level of input states for MZ interferometers, thus significantly elevating the measure-
ment resolution to near HS for any nonzero phase. For experimental obtainability,
the input states are restricted to be the ground state of the double—well BEC system,

given in the form of GS state (3.13).

114

3.4.1 Squeezing optimization

As discussed in section 3.2.1, in the Bloch-sphere picture, a CS input state cor-
responds to an ellipse located at J3 = Jy = 0. The output state is obtained by a
—6 rotation along Jy. In figure 3.6 (a), we give the quasi-probability distributions
of a typical GS input and output state. Before we present numerical results from
a rigorous Bayesian analysis, we first use linearized error propagation to provide an
approximate analytical description of the interferometer performance. Such analysis
shall be approximately valid given only moderate squeezing. We note that an ana-
lytical result is important for predicting the behavior at large N, where a numerical
result is inaccessible. In the case of M2 interferometry, the phase uncertainty is esti-
mated by evaluating A6 = [6(J3)/06]—1AJ2 at. the interferometer output. For input

states symmetric around 72 = 0, the expectation values at the output are related to

 

those at the input via (J3) = sin 6(J].) and AJZ = \/0082 6AJ;2 + Sin2 6AJ332. For
GS states, (J12) z N/2, and AJ; = a, which immediately leads to (I2) = N(sin6)/2.

 

In figure 3.6 (c) we see that AJ; z N/2 -— \/N2/4 — Aijz. Since GS state is a mini—
mum uncertainty state with AJéAJg; = (Jfr) / 2, we see that AJ; = N / 40. This leads
to AJ; = GIN/02, with a z 0.06. Exact numerical calculations verify this analytic
form for 1 << 0 << W/2, as shown in figure 1(d), but with a = 0.09. Inserting these

results into the error-propagation formula, we find

 

2
20 1+ [0.09Ntan6] . (3.20)

A6 ~ —
N a3
The TF and PS states roughly correspond to a fixed 0 S, 1, resulting in liin1\/_,OO A6 =
.18 tan 6/02, which quickly becomes saturated to an N—independent constant for
6 >> I/N, a result we have verified numerically with exact Bayesian calculations

in above section. On the other hand, if holding u fixed so that 0 ~ N1/4, the phase

uncertainty scales as A6 ~ 1/N3/4 for 6 = 0, as discussed in [255], and would even-

115

tually saturate to ~ 1/\/I—\I for 6 75 0. Rather than holding a or u fixed, we propose
varying u and thus a with N in order to minimize the phase variance. By setting

dA6/d0 = 0 we find

22

em,-,,(9, N) .503(1v tan (6|)1/ 3, (3.21)

A9,,,,;,,(9, N) A: 1.23(tan [6])” 3/N2/ 3 (3.22)
From self-consistency, these expressions are valid only when
10/N 5, [6| 5, tan—1(.137\/N) z 7r/2. (3.23)

In figure 3.6 (b), we show the output states for optimized input states with various
phases 6. From the figure, as 6 increases in amplitude, less squeezing is optimal.

We now employ rigorous Bayesian analysis to quantify the phase uncertainty and
validate our approximate analytic results, again assuming that [6] is not too close to
7r/ 2. The error-propagation result (3.22) is close to the 68% confidence interval of
P(qbln) because the underlying number distribution of the optimized GS output state

is approximated by the gaussian distribution

._ in 2
P(n]6)z 1 exp (_(n. NS 6/2) ), (3.24)

 

 

 

\/27rAn 2An2
where
tan2 6
An 2 \/1+ mUTnin(6a, N), (3.25)

with 6 being the unknown phase, and 6,; being the assumed phase used for Opti-
mization. This ansatz roughly agrees with the exact numerics, as shown in figure
3.8. Provided that |6| ~ [60,], the dependence on 6 is weak, and we have An a:
Mammme). This shows that the most-probable outcome is it = N sin 6/ 2,

which is sensitive to the sign of 6. Because An is only weakly dependent on 6, the

116

 

 

 

 

 

 

 

Figure 3.6: Bloch-sphere analysis of the MZI with a GS input state: (a) A typical
GS state at input and output stages; (b) output states for optimized input states
with 6 = 0,ifi,i%,i§,i§,i%; (c) (continued on the next page) geometric
origin of the Jr input noise, AJg; ((1) (continued on the next page) numerical results
plotting AJg/N versus a on a log-log scale for three different N -values, validating
the functional form of AJ; derived geometrically from (c). The dashed vertical lines

correspond to a = 34;, where AJg drops to zero.

117

 

 

 

 

 

 

 

 

 

_ (3 _ 4
10 4_ N—1o5 :
A N=10 :

10‘3 —o.09ts'2 :

  
 

 

1 ”5110

Figure 3.7: A continuation of Fig.3.6.

118

 

inverted distribution P(q'9|'n.) will also be close to Gaussian in the small—angle regime,
with width given by (3.22).

Using the Bayesian-analysis approach, together with the exact double-well ground-
state, we numerically find umm, the value of U / T which minimizes the phase uncer-
tainty. In figure 3.9 (a) and (b), we plot phase densities of optimized double-well
interferometer for 6 = 0 and TT / 4. In both cases, the phase density exhibits a narrow
and sharp peak, indicating a small phase uncertainty. Moreover, as N increases from
100 to 1000, the peak is significantly narrowed, leading to a significant reduction in
the phase uncertainty. This is in contrast to the MZ interferometer with a. PS in-
put showing in figure 3.13 in the appendix, where the phase density almost keeps
unchanged as N increases, which eventually gives rise to a constant, N-independent
phase uncertainty. In figure 3.10 (a), we plot the corresponding 0mm = U(IIn,.,-,.,) as
a function of N for several 6s. Also shown is a. least-squares fit to the N > 103 data

(including many data points not shown explicitly) to the analytic form (3.21), giving

0mm(9. N) = (3.26)
0.45(Ntan|6|)1/3, )9] > IO/N,

in good agreement with our analytical result. Inverting Eq. (3.26) leads to

. . 9 v _ 11%-,[p |6|<10/N; 3,27
umm< 2* )— 1.52 _ 1 l9] >10/N. ( - )
(tan|9|)4/3N1/3 7‘"

 

In Fig. 3.10 (b) we plot the corresponding minimized A6mz-n versus N for several

phases, achieved by setting a = 0m,n(6, N). Again fitting the N > 103 data to the

119

 

I f I W

 

0.25“ (a) I exact numerical result ‘1
Gaussian ansantz

 

 

 

 

 

 

 

 

 

 

 

 

 

I ' l ' l ' l
(b) I exact numerical result
- - - Gaussian ansantz
0.10 "' - -1
I n
I I
A
5 - .:
D. I q
0.05 - n! 4
I J,
I
i- "
I I‘,
I I
, '-‘

 

 

T

300 320 340 330 330 ' 460

Figure 3.8: The probability distribution of optimized double-well interferometer out-
put for 6 = 6a 2 7r/4, where figure (a) and (b) are for N = 100 and 1000, respectively.
In both figures, scatters are exact results from numeric calculation, while solid lines
represent the Gaussian distribution (3.24).

120

 

120- (a) I ' ' .,

 

 

 

 

100‘ - - - N=100 ‘

' ‘—N=1000 l

a 8°: .
_e.

v _ .i

0. 6°. ,

 

 

 

 

 

-0.10 -0.05 0.00 0.05 0.10

 

 

 

 

 

 

 

 

 

25- (b) -
- i

20- ---N=1oo ,
‘ ~—N=1ooo

§_ 15. .

3 ,

0- 10- -
5- .
0-#"’ ~~--=I'L¥

0.6 1.0

Figure 3.9: The phase density of optimized double-well interferometer, where figure
(a) and (b) are for 6 = 0 and 7r/4, respectively. In both figures the dashed and solid
lines are each for N = 100 and N = 1000.

121

analytic form of (3.22), we find

3.50/N, |6| < 10/N;
A6nrin(6: N) = 3 3 (3-28)
1.63(tan |0|)1/' /N2/‘, |9| > IO/N.

The difference between the prefactor here and (3.22) is primarily due to a factor of
approximately fl which comes from the definition of P(qbld).

In practice, 6 is not known a—priori, hence it. is not clear what value for (9 to
use in determining arm-”(0, N) via Eq. (3.26). If we assume prior knowledge of the
form P(H) oc exp[—(6 — 00)2/2A6(2)], we should first remove 60 by adding 60/T to
the tilt 6 during phase acquisition, and then use amm(A60, N) After obtaining a
measurement result n1, the estimated uncertainty A91 is then be determined via
fgglleAggll d6’P(6’ I721), with P(6’ |n1) being given by Bayes theorem. This will result
in A61 ~ Admin(A60,N). Based on Eq. (3.28), this uncertainty appears to scale

2/3, only a slight improvement of N1/6 over the SQL. However, in many

only as N "
applications requiring high precision, the phases are very small, in which case the
phase uncertainty can be reduced considerably due to the explicit phase dependence in
(3.28). This is in contrast to a shot-noise-limited interferometer, where A0 = 1/ W
for all BS not too close to :l:7r/ 2. The explicit theta-dependence in the optimized
scheme is due to the fact that stronger number—squeezing can be tolerated at smaller
angles before the 1% noise becomes detrimental. For example, if the phase is knOwn
to be smaller than 1/\/i_’\7, we have A6,,”-n S 1.63/N5/6, which is now an N“3

improvement over the SQL. As can be seen from Eq. (3.28), a maximum sensitivity

of 3.5/N can be achieved for |9| < 10/N, which is true Heisenberg scaling.

3.4.2 Adaptive measurements

In fact, almost any phase between —7r/ 2 and 77/ 2 can be measured at the max-

imum precision of 3.5/N if the present scheme is combined with multiple adaptive

122

 

 

 

 

 

 

   
  

 

 

 

1 (a) (”Mr """"""
10 r ..... A """ i
_ “.A ,,,, (H -
P """ lll _
r- |‘.A“ ".
~ ..... A “““ ‘,..n"’ j
- ....A “““ .. " .. i
.E - ,A """ ,Ir" (n)
E """""" 9’ x'“
b t ’n”’ Q’T" _,
.v‘l" O’T"
100L‘ “"' ,9" (i)
8 v ”3’ ° ° ° ° ‘i:
' l l l LLLIII I l I l llIl I
2 3 4
1O 10 10
I I I I III! b I I I I IIII I
-1
10 ,,,,, ( )
,,,,,, A 4
-2
.E 10 E” ......
(DE E " . .
< - 2i
-3
10 r
-4
1O
2 3 4
10 1O 10

N

Figure 3.10: Figure (a): Optimal width 0min versus N for different 6; (b): corre-
sponding minimized phase uncertainty A6min. In both figures, from (i) to (iv) the
interferometer phases are 6 = 0, 0.01, 0.1 and 1. The data points represent numer-
ical results from strict Bayesian analysis using the exact ground states of a double-
well BEC, while the straight lines represent the asymptotical forms of Eq.(3.26) and

Eq.(3.28), respectively.

123

measurements [157, 238]. After the first measurement as described above, we can
again rebalance the interferometer by adding 61/T to the tilt, followed by a second
measurement with 02 = 0min(A61, N), with result 722. The Bayesian distribution

for 62 will then be approximately
P(92IN2, 711) 0< eXvi-((92 - 92)2/2y§l, (329)

where 63' = sin—1(2n j /N) and l/yJQ- = Zizo 1 /A612;' Since A6j is much smaller
than A6j_1 we can say yj z A6J- ~ A6mm(A6J-_1, N). In other words, since the
distribution after a. measurement is much narrower than the previous distribution,
multiplying the distributions has little effect, so that the final uncertainty is effectively

determined by the resolution of the final measurement alone. After M iterations, with

oj : 0min(A0j—191V) z 0.57(Ntan A60/2.1)3_J, we find

3—AI

2.1 Ntan A6
MM ~ IT, (Til) . (3.30)

While the above expressions are good estimates of the expected behavior, in practice
each 0j and A6j would be computed exactly by applying Bayes theorem after each
measurement. This procedure should be repeated only until A6 M S 10/N, after
which an addition measurement will push the phase uncertainty to 3.5/N. The final
measurement is then made using the GS state with a = 1, which lies at the edge
of the maximally-squeezed Fock regime defined by 0 << 1. Thus an arbitrary phase
can be measured at 3.5/N precision with M + 1 measurements in total. Setting

A6 M = 10/N, and solving for M gives

.M 2 O.91n(ln(N tan A60/2.1)) — 0.4, (3.31)

124

1L

where again this is just an estimate subject to run-to—run fluctuations. For 67- : 7r/ 3
and N = 104, M = 1.6, i.e. only 2 or 3 total measurements will be required. For N =
1012, M = 2.6, requiring 3 or 4 measurements. Even for A60 extremely close to 7r / 2,
M remains small, for example, A60 = 7r/2 — 10—10 gives M x 2.7 for N = 104, and
M as 3.1 for N = 1012. Hence, for arbitrary phases in (—7r/ 2, 7r/ 2), our interferometer
converges quickly to the 3.5 / N precision within a few measurements, regardless of N.
The final experimental value for the initial unknown phase is then 6 2 2311:163-
with a quantum-limited uncertainty of A6 = 3.5 / N . The total number of atoms used
to obtain this precision is Ntot = (M + 1)N. For large enough N, we can approximate
ln(NtOt/2(M +1)) z ln(Nt0t), so that .M + 1 % 0.91n(ln(Nt0t tan A60» + 0.6, which

leads to the asymptotic scaling law
A0 % (2.1+ 3-21n(ln(-Nt0t tan A60)))/Nt0t. (3.32)

That the scaling law should depend on the initial phase interval has been previously
pointed out [256]. As our final approximation effectively overestimates Ntot, the
uncertainty approaches (3.32) from below as N increases.

In order to verify the accuracy of Eq. (3.31), as well as determine the magnitude
of the run-to—run fluctuations, we have carried out exact Monte-Carlo simulations of
many measurements of the phase 7r / 6, with an initial uncertainty A60 = 7r / 3. During
each simulation run, the measurement outcome was randomly selected according to
the output distribution, and the phase information was determined numerically via
Bayes theorem. The prescribed measure-rebalance process was iterated until the
estimated phase uncertainty reaches 3.5 / N . Figure 3.11 shows the percentage of runs
which attain the desired resolution on the (M + 1)th iteration, for two different N
values, with 104 runs each. The averages are MTI = 2.2 for N = 500, and 2.5

for 5000. Equation (3.31) gives 2.2 and 2.5 as well. The corresponding variances

125

 

 

 

 

 

 

100_ .32 - .
8 C;3N=5oo
oi :lN=5000‘

so- an _
o\
o .

a, '5

0:60— fi .,\° -

(U x

a 8.

a: E o

O V

540- k -

&’ sf?

‘ i5!
20- .wiissss ss-
183 83 88
I H idea 00 oo
0 i l._i—1
o 5 e

Reqdired Nifmber 0:; Measfirements (M+1)

Figure 3.11: Monte-Carlo simulation results showing the percentage of runs which
achieved the maximum precision of 3.5/N after (M + 1) measurements, plotted versus
(M + 1).

A(M + 1) are 0.4 and 0.6. We note that as the average approaches a half-integer
value, the minimum possible variance approaches 0.5. because M is constrained to
integer values. Thus the fluctuations are close to minimum allowed values.

For TF and PS states, adaptive measurement schemes are extremely inefficient.
This is due in part to their inability to distinguish positive from negative phases,
which makes rebalancing impossible. But even if this were overcome, the primary
difficulty is that the phase—uncertainty is N -independent for large phases, so that
~ N 2 measurements are required to obtain l/N precision. This results in 1 / NSC/,3
scaling, worse than SQL. To verify this, we perform pseudo Monte—Carlo simulation
of a repeated adaptive measurements with the TF and PS state, with the following
adaptive algorithm: repeated measurements are made at the phase 6, as well as 6 +
7r / 4, with results for each measurement randomly drawn from a gaussian distribution
with A6 L 3(6) given by the fitting curves shown in figure 3.15 in the appendix. For

each measurement, a phase distribution was generated, given by a gaussian centered at

126

the measurement with width A6fit(6). The phase distributions for the 6 and 6 + 7r/ 4
measurements are separately multiplied to generate two final distributions. Sufficient
measurements are performed so that these distributions are separated by a distance
larger than their widths. The order of the two peaks determines the sign, and the
phase can be determined from the center of the 6 distribution. This resulting phase
is subtracted from the tilt, and the process is then repeated until a phase uncertainty
of 3.5/N is obtained. We note that errors in determining the sign due to bad luck are
self-corrected by the algorithm. This happens frequently and increases the number
of measurements required. This error can be reduced by taking more measurements
at each step to further separate the 6 and 6 + 7r/4 peaks, with the result that the
prefactor changes, but the scaling with N stays the same. The results are shown in
figure 3.12, where we plot the average number of measurements versus particle number
N, based on 104 simulations for each N —value. A least-squares fitting results in a
scaling as N 2 for the TF state, N 1'94 for the PS state. For a comparison, a similar
pseudo Monte Carlo simulation for the GS state is also shown, using instead the
alternative algorithm described in last paragraph, as a CS interferometer is capable
of distinguishing positive and negative phases. The agreement with Eq. (3.31) is
shown to be fairly good. Exact simulations are shown for N = 500 and N = 5000,

showing even better agreement with Eq. (3.31).

3.4.3 Experimental stability

The above discussions have assumed that the input state is optimally squeezed
to width 0min- A realistic input state, however, may deviate from 0mm, due to
imprecise control over u and/ or imprecise knowledge of N. A straightforward error
analysis shows that our scheme is extremely robust against such uncertainties. The

increase in the single-measurement phase uncertainty 5A6 due to fluctuations in u

127

 

' I I rfi'III' U r UUIIUI' I I UI'UI'I

 

 

 

 

 

 

  

 

 

 

 

 

 

0 GS State A TF State I PS State I”
1 09 0.91 In In(N tan(n/3)/2.08)—O.41+‘l .
---0.84 N200 0.29 N194 .v"
I 3‘
.9 x‘
5 I» ,-
g ,x f
5 6 [I l.’
(010 /k .
"5 I,” I
8 , .,.I i - a - f q, ,1
.U .
E 3 2.5 .
g 10 I' .
D
2'01 l 0 Exact GS Simulation ]‘
1o2 103 104 105
10LT2 . .......,3 . .......,4 . ........5
10 10 N 10 10

Figure 3.12: Pseudo Monte-Carlor simulation results of 104 runs. The true initial
phase is 7r/6, and the a—priori phase interval is [—7r/ 3,7r/ 3]. Results are shown for
the TF, PS, and GS states. In the inset, we see that the formula from Eq. (3.31)
(with one added for the final measurement to go from 10/N to 3.5/N) agrees fairly
well with pseudo simulation results.

128

and N, using formula (3.20), is found to be

aznonv, u)
(92a

0A6(N, 21)

(SN
6N +

 

6A6 = ' $51.2 (3.33)

 

 

 

evaluated at u = 21mm. The scaling with 6212 reflects the fact that u = “min is a

local minimum with respect to the phase uncertainty. This gives

 

— —-—— +
A9,m,;.,, 3 N 8

 

6A6 2m 1 5' 2
_ ( u ) , (3.34)

”min

so that a 10% variation in N leads to a 7% variation in the phase uncertainty, while
even a 100% uncertainty in u only results in a 13% variation. For our purposes,
these increases are essentially negligible, and are independent of the values of N or
6. Of course there are many other potential sources of error, e.g. the precision with
which the tilt can be rebalanced, and the precision with which the scattering length
can be set to zero during interferometer operation. Reaching the Heisenberg limit in
a double-well BEC interferometer will clearly require major technological advances
in many areas. Assuming that a level of precision significantly below the SQL is
eventually obtained, the scheme we have developed will be the optimal method to

obtain this precision, whether or not it is close to the Heisenberg limit.

3.5 Conclusion

To conclude this chapter, we have used the Bayesian analysis to evaluate the per-
formance of important existing interferometers for measuring non-zero phases. The
MZ interferometer employ highly number-squeezed TF or PS state is shown to be
worse than SQL, due to large quantum fluctuations in the interferometer output.
The QDFT-based interferometer, on the other hand, can measure arbitrary phases at

Heisenberg scaling. Yet the experimental implementation is extremely challenging.

129

Instead, we propose to use an adaptive GS-state scheme, which has three advantages
over previously discussed MZ interferometry schemes. It (1) can readily be imple-
mented in a double-well BEC system, (2) can achieve a resolution well beyond the
SQL for a wide range of phases with a single measurement, and (3) quickly converges
to a final precision z 10/Nt0t within only a few adaptive measurements. In this way,

we have realized a Heisenberg-scaling interferometer for measuring arbitrary phase,

which could be realized in nowaday or near-future laboratories.

3.6 Extension: Bayesian analysis of PS and QDFT
interferometers

3.6- 1 MZ interferometer with PS input

Using Bayesian analysis, Hradil etal first examined the sensitivity of a MZ in-

terferomter with twin Fock state IN / 2, N / 2) as input [244]. Surprisingly, such inter-
ferOInetry turned out to have a phase sensitivity that is even worse than the. shot
noise. The reason is that while the phase density distribution P(cpl6) does have a
narrow, N -sensitive central peak,the probability is majorly distributed among broad,
N‘insensitive tails outside the central peak region. As a consequence, its phase un-
certainty, as determined by confidence interval (3.19), will exceed that of a SQL
interferometer with coherent input, which yields a wider central peak but no such
tails- To address this problem, Pezzé et.al proposed to use an input state of the
form (IN/2 +13|N/2 —1)+|N/2 — 1, |N/2 +1))/\/2, in order to destructively can-
Cel Out those tails [119]. Indeed, the tails are successfully eliminated at 6 = 0 and
a I‘Ieisenberg-limited phase switch is discovered. For a single measurement, such is
a"ITICDIIg the best sensitivities of a MZ interferometer having been reported so far.

The problem with PS-state interferometry is, in the first place, that it is ex-

130

 

20

P(<l>|9)

 

 

 

 

 

 

Figure 3.13: Phase density distribution P(q6|6) for Mach-Zehnder interferometer with
the PS input. Here P(gbl6 = 0) is a single peak located at (:5 = 0, whereas, for
9 = O.1,0.2,0.3, P(gbl6) is split into twin peaks at around (1) = :l:0.1,:l:0.2,:l:0.3,
respectively. For each 6, solid lines and dashed lines are corresponding to N =
1000, 1 00, respectively.

Del‘imentally challenging to create such a PS input state. More importantly, even
Provided that the PS state can be prepared, it turns out that such a interferome-
ter, While proven to be a HS phase switch, is in fact a worse-than-SQL phase meter.
That’s, as soon as 6 deviates from zero, its phase density P(qbl6) behaves significantly
differently from that of 6 = 0, as shown in Fig. 3.13. In figure, the phase density dis-
tribut ion at 6 = 0 is sharply peaked and N -sensitive, i.e., it dramatically shrinks as N
increases. For nonzero 6, however, P(qb|6) is split into two symmetric packets around
9 a 0 , each of which consists of dominating background bases. The symmetric pattern
makes it unable to distinguish between positive and negative phases. The dominating
bases are N -insensitive, resisting large even as N increases. Consequently, the phase
uncertainty for nonzero phases, determined from confidence interval of P(¢|6), is ex-
p ected to be bonded by a fixed, N -independent lower limit. This is shown clearly in
Fig- 3. 14, where we plot Act calculated from the absolute phase density distribution
P09“ I6) In figure, at 6 = 0 the log-log show of Ad) v.s. N is a exact straight line,

W
her e we recover the previous result A6 = 267/N reported in [119]. For nonzero

Dha‘SQS, the behavior is significantly different. For instance, for small N s, Ad) as of

131

9 2:: 0.01 coincides with that of 6 = 0. Whereas as soon as N exceeds a critical
value Ncr = 100, it deviates from the straight line and saturates to be flat. Similar
phase-transition behaviors are found for Aqfis at 6 = 0.1, 0.001, but at Ncr = 10, 1000
correspondingly. This phase transition is understood as follows. For N < Ncr, P(¢|6)
is broad and dominated by peaks, resulting in a Art that is N-sensitive. While as
A7 exceeds Ncr, although the peaks continue to shrink, the background bases remain
constant and hence become dominating. The N —insensitivity of the bases gives rise
to the saturation of Ad).

In Fig.3.15 we show the asymptotical value A6 L B as N goes infinity for various
values of 6. At 6 = 0, A6LB = 0, due to that A6) is infinitesimal as N increase. For

9 5* O, a good fit of A6LB is found to be
A6LB = —0.45 + exp(—0.8(6 —1)2), (3.35)

Which in the small 6 limit approaches 0.726. Also shown in figure is the asymptotical

phase uncertainty for the TF state, which constantly lies above the PS curve.
Hence, for the PS interferometer the highest possible signal-to—noise ratio for

a norlZero 6 is given by A6LB/6 = 0.72, regardless of N. In practice, in order to

distinguish between two phases 61, 62, it requires in general
91 — 92 > A61 + A62, (336)

whel‘ e we assume 61 > 62 > 0. For any N, this gives 61 > 6.162. That’s, a phase shift
can at the best be distinguished from ones whose amplitudes are at least six times

greater. Evidently, such interferometer is unsuitable to measure an arbitrary phase.

132

 

 

0 I v 1 VY‘I’I 1 v 1 1 ffiT‘r' l v v f YTjI' v v
q
\ ‘
‘
I

A
or

   
 

 

—0=0

10 ---0=0.001

I ------ 9:0.01

' -e=0.1

101 102 103
N

v.
..
rrrrr
’l

 

 

 

 

 

 

Figlu‘e 3.14: Phase uncertainty A96 versus N for several 6. Here, because of the
SYIIlInetric nature of the phase density, Act is determined by 68% confidence interval
0f the absolute phase distribution of P(le‘il [6).

3.6-2 QDFT-based interferometer

To the author’s best knowledge, the QDFT-interferometer is so far the only truly
HS phase meter under Bayesian analysis. To proceed with describing this interferom-
eter, it is recalled that the action of a QDFT is a unitary mapping between two, say,

N + 1~dimension Hilbert spaces of basis {|k)} and {IO}, with the projection defined

by <€|k>=7,—,—1,—=

+le—z'27rlk/(N+1). That’s, a QDFT operates on Ik) via

N
QDFT: |k) Ze—iZ‘W’k/(ler). (3.37)
z 0

1
l-——-) —
\/N + 1 _
A reverse QDFT, denoted as QDFTI, is defined in a conjugate manner. During a mea-
s
ure"lent, a single-Fock input state of [0) is first transformed to a superposition state

of 1 N . . . . . .
m 21:0 |€) Via QDFT. Then, the measuring phase 6 is acquired, resulting 111

133

0.0
0.0 0.2 0.4 0.6 0.81.0 1.2 1.4

Flgure 3.15: The phase uncertainty versus true phase in the limit N —> 00, as deter-
:llned by exact Bayesian analysis for large, but finite N. Results are shown for TF
11d PS states based on a single measurement. Approximate fitting functions are also

ShOWn_

 

 

 

 

 

 

o
pj’i‘l‘\
\.\\
TF
PS

0.7[1-Exp(-89)]
- - - -0.45+Exp[-0-8(9-1)21

 

 

 

 

I

0

134

 

30

I

I

20
P((ble)

10

 

I

 

 

 

 

   

Figure 3.16: Phase distribution P(cb|6) for different 68 in QDFT—based interferometer.
From left to right, the peaks correspond to 6 = 0, 7r/ 8, 7r/4, 37r/8, where for each 6,
solid lines and dashed lines are corresponding to N = 1000, 100, respectively.

\/ AI—Jt— 1 211:0 G—imflf‘), Finally, a QDFTIL is applied, upon which the interferometer
output is obtained as
AI

I‘Pf) = 2: email»). (338)
1:20

Where
1 1 _ e—i(N+1)6

cue) = m, _ ezi2nk/(N+1)-9I

The Output probability distribution is then P(kl6) = |ck(6)|2. By measuring the

 

(3.39)

Output state on {|k)} basis, the phase density P(cbl6) is calculated via Bayesian ap-
pr 030h. In Fig.3.16, we plot the obtained P(cbl6) for several 65. Compared to the PS
interferometry in Fig. 3.13, here P((bl6) is singly peaked for arbitrary 6, so that posi-
tive and negative phase shifts are now distinguishable. Further, all background bases
are eliminated, and as N increases, the distribution width is dramatically reduced.
This is a clear sign of HS phase precision-

To show that the phase uncertainty indeed yields 1/N scaling, in Fig. 3.17, we

plot 0 = Nch) versus N. For any 6, for zero phase, a = 2.61. While for nonzero

135

phases, (1 yields a periodical N -dependent value oscillating between a = 2.61 and
7-47, as shown in figure (a). Such is a characteristic HS behavior. It is recalled that
the PS interferometer yields a very close sensitivity of Aqb = 267/N. To explain
the oscillatory behavior of a, it is noticed that the probability distribution P(kl6)
in Eq.(3.38) is a periodical function of N, with a period of T = 27r/6. For N at

N1 = 2k07r/6 — I (assume both N1 and k0 are integers), P(kl6) is reduced to
P(kl6) = at, (340)

corresponding to a maximally sharped distribution. In contrast, at N2 2 (2kg +

1)7r/9 — I, the probability distribution,

2 ‘T—k —- 2/
OCH“ 9 I/ )7],
N +1 (N +1)

 

 

P(kl6) = (3-41)

is Illaximally broadened. As a consequence, the phase density distribution is sharply
Peaked at N = N1, but much wider at N = N2. This leads to the periodicity of a
as a. function of N, with a period of T = 27r/ 6. This periodicity is clearly shown in a
close View of figure (b), where for a nonzero phase 6, a exhibits a Sine-like behavior
in N, with a period of T = 16,8,16/3 for 6 = 7r/8,7r/4,37r/8, as predicted by the

formula (3.41).

136

 

 

 

 

(f 00 200 500 700 900 1000
N

 

 

 

 

 

u ’ t ‘ ' . I: I e ‘ > ' (b), a
= I I, :I ‘ i. i ' j I, . . ‘\ z: " if I,
2, ~\_: : 5 g‘ _ -
(I. 5 7 '- e 1,:— e i R . ‘- .‘: ’1- "‘ I. I
I I ‘ :' ' " ‘ = I
I ,‘ \ I I , ‘\ I,
360 970 980 990 1 000
N

Figure 3.17: Plot of a = N A05 versus N for QDFT-interferometry, with N from
100 to 1000. Here figure (b) is a close view of figure (a) with N E [960,1000].
The solid, dashed, pointed and dashed-pointed lines in both figures correspond to
0 h 0,7r/8,7r/4,37r/8, respectively.

137

Chapter 4

Resonant Matter Wave
Amplification in Mean Field

Theory

In this chapter, we study the four-wave mixing process involving two matter-
Waves of gaseous BEC’s and two phase-locked lasers, widely recognized as the matter-
Wave amplification (MWA). We focus on searching for an Optimal operational strategy
for MWA, motivated by the goal to facilitate applications in the fields of matter-wave
interferometry and quantum information processing. Specifically, we compare three
different dynamical regimes, namely the off-resonance, overdamping and Rabi regime.
0111‘ results show that for a given laser intensity, the fastest amplification process is
achieved when tuning the driving laser to be on-resonance, whereas previous studies
are focused on the far-off resonance regime. In the meanwhile, the total atom loss is
nearly identical in all dynamical regimes. Furthermore, due to the phase distortion
by the ac-Stark shift, it turns out an off-resonance three-level system is unsuitable for

State mapping between light and matter waves, while a resonant system is not subject

t . . . . .
O thls problem. All these give rise to superiorities of resonant MWA processes. We

138

identify the laser depletion and spatial effect by comparing the quantum coupled-
mode and semiclassical multi-mode theory.

The organization of this chapter is as follows. After an overview of the present
work in section 4.1, we present the level scheme of the three-level Raman MWA system
in section 4.2, . In section 4.3, we use the quantum few-mode theory to give a pellucid
picture of the MWA dynamics. We then employ the semiclassical multi-mode theory
in section 4.4, to develop a more ‘physical’ model incorporating the laser depletion

and spatial effect. We finally conclude in section 4.5.

4 . 1 Introduction

To employ MW’A for coherent amplifiers or as quantum information protocols, a
maj Or challenge, commonly existing with any quantum devices, is to overcome fast
dephasing and/or decoherence whose rates usually scale as the square of matter-
Wave sizes. This naturally requires processing MWA in a shortest time window. In a
PrOtOtypical three-level system, for N1 and N2 being atom numbers in pump 1 and the
Sign a1 wave, the instant amplification rate is P€F3N1N2° Here, F3 is the superradiance

emiSSion rate into the signal mode, and P6 is the fraction of pump atoms in the
1012

2|o|2+r§+4A2’

fr eQUency and detuning of the driving laser pump 2. In most MWA or superradiance

 

eKeited state, approximately given by Pe = With 9 and A the Rabi
experiments carried out with BECs, the driven laser is tuned far off-resonance with

2
A >> IQII‘S, so that the amplification rate is oc {FA-7P3 [102, 101, 108, 112, 135]. If

~ 2
lustead going on resonance, the rate then becomes oc 2|9I5+P F3, which is always
3

higher than that the off-resonance case for any driving strength 9. This suggests to

u

eIIIli>loy resonant, rather than off-resonance, driving laser for a fast MWA process.
D eQifically, in case of IQI < PS, the system is in the overdamping dynamical regime

W‘ . . 2 . .
1th Pe << 1, for an amplification rate of o< #4:. The ultimately mammal rate

139

of CC §Fs is obtained in the Rabi regime with IQI > PS, where the excited state is
macroscopically populated.

Ultimately, the source of the nonlinearity in the MWA dynamics is the atom-
field dipole interaction, which becomes a quadratic nonlinearity in terms of atomic
operators upon second-quantization of the matter-wave field. In the far-off resonance
or overdamping regime, the electronically excited atomic state can be adiabatically

eliminated, resulting in an effective cubic nonlinearity whereby MWA behaves as a
standard four-wave mixing process. In the Rabi regime, however, the excited state is
macroscopically populated, so that a fifth wave of the excited atomic state is required
in Order to describe the system dynamics. Thus asides from practical point of view,
it is also of interest of fundamental nonlinear optics to study the collective response
Of at ()mic ensembles to a resonant driving. Physically, treating the excited level dy-
Ila'rfliCxally is equivalent to extending nonlinear susceptibility to infinite orders. In
pr act, ice, the increased nonlinearity will potentially lead to nontrivial profiles of both
light and matter waves, as having been observed in a recent Rayleigh superradiance
63(1) eriment under near-resonant driving [117]. Besides, the scattered light in the Rabi
Fegi me yield a characteristic spectrum with two side bands, whose linewidth depends
only on the optical depth of the samples [257]. On the other side, due to a larger am—
I)lifBlC1ation rate, the MWA under resonant driving can be processed in a time window
8110I‘ter than any time scales related to atomic motion, environmental decoherence,
or 111eanfield dephasing. This leaves the resonant MWA a relatively “pure” quantum
system whose dynamics is solely determined by the light and matter coupling, thus
prgviding the ease to engineer it as a useful quantum device. Theoretically, the prob-
lem . . . .
Of forward-light scattering of a BEC illuminated by resonant and short pulses
hag been addressed, under single-mode and undeleted pump approximations [258].

T11
§ Work of this chapter is to go beyond these approximations and study important

C1 \r 11
amical consequences.

140

 

Following these considerations, in this chapter, we study the MWA process under

various dynamical circumstance ranging from the off-resonance, overdamping to Rabi
regimes. For purposes of illustrating important underlying physics, we will focus on

a simplest, prototypical three-level Raman system. For comparison, we employ both
the quantum and semiclassical theories, the former of which adopts a common time-
invariant spatial mode of the matter waves, while the later uses multi-mode treatment.
Our results show that the two models predict similar population transfer dynamics
in the beginning MWA stage, but then become divergent as the matter-wave modes,
evaluated via semiclassical Maxwell-Schrodinger (SMS) equations, grow out of their
initial shapes. The divergence turns out to be due to a combination of the laser
depletion and spatial effect, both not included in the quantum coupled-mode (QCM)
n10 del. Interestingly, both the spatial mode evolution and the population transfer
dynamics are found to be qualitatively similar in the off-resonance and Rabi regimes,
only for the later occurring on a much short time scale. The population dynamics
in both regimes is shown to agree well with QCM results before the time when half
of pump 1 atoms are depleted, but undergoes a sudden slowdown afterwards. This
S1:"g4gests identical spatial effects on the two regimes. In the overdamping regime,
hOWever, the sudden slowdown does not appear, and the population dynamics is
constantly following the quantum results. We attribute this behavior as a consequence
of St rong laser depletion effect. In addition, the relative phase between between pump
1 and the signal wave turns out to undergo ac-Stark shift and 7r-switching in the off-
33638 Qnance and Rabi regime, but is preserved in the overdamping regime. This result
provides the feasibility of using the present MWA system in the overdamping or Rabi
I‘Qgilnes, but not in the off-resonance regime, as a quantum state-mapping device.
illally, the atom loss dynamics is shown to be inversely proportional to the optical

C1
eh 13h, and is nearly independent of dynamical regimes.

141

4-2 The Model

A MWA process can be implemented with Raman or Rayleigh transitions, corre-

Sp onding to the pump 1 and signal matter waves in different electronic or translational
momentum states. Comparatively, while functioning on similar principles, the former
requires spatial overlap of two spinor condensates in different hyperfine states, while
the latter relies on forming a density grating of single-spin condensates. Thus a MWA
employing Raman transitions is the only scheme that can generate an idler wave co-
PI‘Opagating with pump 2. Besides, in a Rayleigh MWA process, multiple atomic
recoil modes, aside from the seeded pump 1 and signal modes, would unavoidably
app ear due to a higher-order process involving the reabsorption of idler beam and
Stimulated re-emission into pump 2 [108, 127, 129]. In a Raman MWA, however, such
prOCesses can in general be efficiently suppressed by using appropriate level configu-
r at ion [124, 131]. In this way, the Raman system can serve as a simplest, prototypical
implementation of MWA, with clear physical interpretations of experimental observa-
tions- In practice, due to a higher controllability, a Raman MWA system is a better
candidate for potential quantum information processes using light and matter waves.
BaSed on the above considerations, in this chapter we will focus our discussions on

the Raman MWA system.

The ‘A’ level diagram and schematic setup of a typical Raman MWA system is
dr awn in Fig. 4.1. Figure (a) shows a toy-model of the A level scheme, where spatially—
Qverlapped pump 1 and signal waves, in ground ll) and I2) states, are coupled to the
excited [3) level by pump 2 and idler light beams, respectively. Pump 2 can be off-
Ifes()nance or on-resonance, depending on whether or not the detuning A is greater
than both its Rabi-frequency Q and the linewidth of I3) level. Both pump 1 and

t e Signal wave are initially populated. During the MWA process, atoms in ll) are
firgt driven to I3) by pump 2. Due to the quantum stimulation effect, l3) atoms will

Dr
gferentially decay to the seeded I2) state, resulting in phase—coherence amplification

142

of the signal wave. This process is imparted by emitting an idler light beam along
the phase matching direction, carrying phase information of the system.

A realistic example of the A-level scheme configured on the D1 transition in

137Cs atoms is shown in Fig. 4.1 (b). The states I1) and I2) correspond to the ground
hy'perfine levels of IF = 3,mF = —3) and IF = 4,mF = —3), with a level spacing of
9-2 GHz between them. The excited I3) level corresponds to the IF = 4, m F = —4)
hyperfine level. Both pump 2 and the idler beam are 0‘ polarized, and are different in
frequency by the ground hyperfine splitting of 9.2 GHz. Two possible side transitions
in this level scheme, including exciting I1) atoms by the idler beam or exciting I2)
atoms by pump 2, are plotted in figure (b) with dashed lines. They are off-resonance
by amounts of 9.2 — A GHz and 9.2 + A GHz, and can thus be strongly suppressed,
le"=i-‘rillg an effective A MWA system. Within Alkaline atoms, a feasible configuration
is Siiltznilarly found in Rubidium, but not in Sodium due a smaller ground hyperfine
SE)1i13ting.

The schematic setup employing 2-copropagating pump 2 and idler beams is shown
in figure 4.1 (c). Both beams are a—-polarized along the 715-6: — z'y) direction. They
are distinguishable in that their frequency difference is greater than each’s linewidth,
as is the case for Cs and Rb atoms. This co-propagation arrangement is different
from the commonly used end-fire configuration where pump 2 and idler beams are
along perpendicular propagating directions [102, 108, 112]. Comparatively, thelaser
de:Dletion effect is thus expected to be stronger in our scheme. On the other hand,
here the recoil momentum of I2) atoms, which is of the order of 10 m—l, is effectively
zero in that it is much smaller than the typical momentum dispersion of ~ 104 m‘4
for condensates. This unique feature allows performing MWA with stationary matter

W ‘
a‘\’€S, which may benefit applications such as state mapping between light and matter

143

 

 

 

(a) (b)
_—E_L_A 5P 112
Q
681/2 a
2 ‘-‘
I I1) l > |1):mF=-3

 

 

 

 

 

 

 

 

Ji
W

Input E L Spinor Condensates Output E L

Scattered E s

   

 

 

F
gg‘ure 4.1: A MWA system configured on Raman transitions. Figure (a) plots a toy
Q Qde showing the A—level scheme. Figure (b) shows a physical implementation using

i11 isium’ 3 D1- line transition. Figure (c) draws the schematic setup using copropagat-
11% beams.

144

4.3 Quantum Coupled-Mode theory

4-3.1 The meanfield dynamics

We now build the mean-field model of atomic dynamics via the master-equation
approach where the light-matter coupling is treated in the Markovian approximation.
A detailed discussion on this approach is given in appendix 4.6.1. From the master

equation 4.45, the mean-field dynamics for each level is determined by

 

firm) = —z'”*,(r)¢3<r) (4.1)
+ / d3r’L’i‘(r — r’)w§(r’)w1(r’)w3(r)
gm) = [d3r’L;(r-r'wgu’wzu’wgu) (4.2)
Egan) = —z9§—’)wl(r>—(§+m)¢3(r> (4.3)
+ [43,1 Z Ljo-—r’)w}(r’>w3(r’>wj(r>. (4.4)
j=1,2

Here the Schrodinger fields are defined as 215 (r) = tr{‘II j (r) p} tracing over the whole
a't‘CDIilic Hilbert space, with \Ilj (r) (j = 1,2, 3) the field annihilate operator for level
lj > - A is the detuning of pump 2 from resonance, 9 the Rabi frequency, and 7 is
the Spontaneous relaxation rate of I3) level. Lj(r — r’) (j = 1,2) is the nonlinear
two‘body cooperative coefficient for transitions I3) -+ | j), defined in equation (4.47)
in appendix 4.6.1. In getting this result, a mean-field factorization of field operators
has been applied. While directly solving the meanfield dynamics (4.1)-(4.3) is in
131\ilileiple feasible, in practice, however, is extremely hard due to the presence of
l / | 1‘ ~— r’ I and 1/ Ir — r'I2 terms in Lj(r — r’). Instead, we expand the Schrodinger

fi
eIds onto momentum quasi-modes,

21,-(m) = Wag-(Mr), j=1,2 (4.5)

145

2123(r,t) = \/1—V_03(t)¢(r)eikL'r,

Where q5(r) is the time-invariant spatial profile, and k L the wavevector of pump 2.
In this expansion, both pump 1 and signal wave are assumed to have zero momenta.
This is the case for the Cesium example shown in Fig. 4.1, where the momentum

mismatch between pump 1 and signal wave due to the hyperfine splitting is negligi-
ble compared to the Fourier-transferred momentum dispersion of condensates. The
atomic pOpulation in the I j) state (3' = 1,2,3) is then given by N, = N chI2.

In the above expansion, we have assumed a single momentum mode for each
iuterrial atomic state. In practice, however, the system is simultaneously coupled to
Other modes, via spontaneous or superradiance emission. To the leading dynamical
or der, however, coupling to other modes does not interfere with the MWA process,
Since they deal with orthogonal electromagnetic modes. The net effect is the atom
1038 from the c1, c2, C3 manifold. Thus the single-mode expansion (4.5) is valid as long

as the atom loss fraction is small. Taking the coupling to other modes into account,
the total atom loss can be described by an effective relaxation rate of x37. Here x3
is the superradiance enhancement factor on atom loss, which is obtained by solving
a. toy multi-mode system, as in section 4.6.4. 7 is spontaneous emission rate of the
e3":C3i‘ted level I3). Using moderately elongated condensates, we find x3 z 1, so that the
enhancement effect is not significant and the atom loss is dominated by spontaneous
decay,

Inserting the mode expansion (4.5) and incorporating the atom loss enhancement,

1:
he equation set (4.1) is now reduced to

d 19* 1 ,, 2

_ = __ _ 4.6
dtCl 2 C3+2P1f1Nl63| 01 ( )
d 1 a: 2

_ = — 40
(1,02 2P2f2N|03| C2 ( 7)

146

J’J

Dl

d _ X37 - i9 1 2
EC3 — —— (——2— + 2A) C3 — ?c1 -— 5 .21; PJfJNIcJI 03, (4.8)
.7: a

with the cooperative parameter fj ( j = 1, 2) given by
2 - I
fj = r7 / d3rd3r’l¢<r>¢(r'>I2LJ-(r — r’>e—2kL'<r—r> (4.9)
.7

With 13 the spontaneous relaxation rate from level I3) to I j). An evaluation of fj for
condensates of Thomas—Fermi profile is given in appendix 4.6.2.
The real part of fj is often referred to as optical depth for an atomic cloud, while
the imaginary part contributes to the atomic mean-field shift due to dipole-dipole
interaction. For simplicity, we take I‘l = F2 = 'y/ 2 in the following discussions. In
Practice, for the level scheme configured on 03’s Dl-line transitions shown in figure
4- 1 (b), we find F1 = 7/12, F2 = 77/12, while for a similar level scheme on Rb’s
D 1—lime, F1 = 7 / 6, F2 = 7 / 2. Generalizing present conclusions and discussions to
Sp e(tific level schemes shall be straightforward. Finally, because of the employment
of SiSingle-mode expansion, any spatial dynamics of mode functions is excluded in this
no del. Thus the QCM theory is valid only if variations of the mode functions remain
insignificant. As we will show in the second part of the chapter, this is the case
through the point where half of pump 1 atoms are depleted.
We now investigate the atomic dynamics of the Raman MWA system by solving
the QCM equations (4.6)-(4.8). To characterize population transfer behaviors, we

(1
efine a differential population fraction n, as

N2—N1-N3

. 4.10
2(N1 + N2 + N3) ( )

 

n:

T h ‘ . .
I 18 quant1ty measures the relative populatlon distribution between the |1)&I3) and
2
> II13.nifold, for atoms remaining in the condensates. For negligible atom loss, it is

re
duced to n z %(Ic2l2 - Ic1I2 — IC3I2). To measure the speed of MWA, we intro-

147

duce the transfer time 7', defined as the evolution duration for which the differential
population is inverted, i.e., 72(7) = —n(0). We note ’7" is meaningful only if the initial
size of pump 1 is much greater than the signal wave. The transfer rate R can then

be defined as the inverse of transfer time, with R = %.
For a back-of-the-envelope estimation of 7', we assume that the fraction of excited

population, P6 = N3 / (N1 + N3), is constant during the dynamics, given by

W

= , 4.11
2M2 + (x37 + I‘s)2 + 4A2 ( )

 

Pe

Here R; = F2 f RN is the superradiant emission rate. The validity of this formula is
in that it provides correct asymptotical results for whichever parameter much lager
than the rest.

Under extreme considerations, the system can be classified into off-resonance,
overdamping and Rabi regime, depending on pump 2. For A >> IQIa X37 + P3, the
Syst elm is in the off—resonance regime, with Pg = 111%; For A smaller than IQI
or X .97 + 1" 3, the system is (effectively) in the resonant regime. Depending on |Q|

smaller or greater than x37 + I}, the system undergoes overdamping dynamics with
P Q 2

8 §
(Xs‘H'Fs)

atom loss, we have N = N2 + 11;? and thus n = 17%? - 2’ SO that

, or Rabi oscillations with the time-averaging P8 = %. Neglecting

d 1
an = -Per (n2 — a) - (412)

'1‘ -
his equation of motion can be analytically solved to give

 

1 1 _1 1- V
n(t) — 2 coth (533F313 + coth (1 + u)) . (4.13)

W
here V = 1;;3) is the initial population ratio. The gain factor of the signal wave,

148

 

 

 

defined as 0Q: N —2((——N0 )), is then
_ 1 + l/ 1 _1 1 — V
G'Q — 2 [coth (2PeF3t + coth (1 + u)) +1] . (4.14)
F‘llrther, the transfer time T is given by
— 2 my (4 15)
7'1 — Peps , .

Clearly, for any given 9 and PS, the least transfer time is always achieved at A = 0,
i- e - , when using a resonant driving laser. The ultimate shortest transfer time of
7' = 1:1; lnu is achieved when [9| 2, F3, A, for which Pe reaches its maximum of 1/2.

In the following subsections, we analyze in detail the three dynamical regimes, us-
ing adiabatical elimination or coarse-graining method. We then compare our analysis

to exact numerics.

03— resonance Regime

In this regime, A > |Q|, X37+F3, so that state l3) can be adiabatically eliminated

to give C3 = —Q%c1. Neglecting atom loss, the equations of motion (4.6) are reduced

to

d I__QI261_I___QI2

* 2
— z 4.
d '9‘2 * 2
$02 ~ 835sz Nlcll C2

In
the equation of motion for cl, the first term is the ac-Stark shift induced by

th
e far—off resonance driving. Since f = f R + 2' f], the second term contains real

d imaginary part, respectively corresponding to population decay and cross-phase

149

modulation by c2. Analytically solving these equation finds

 

1 |Q|2 _1 1— u
t = — — -
n0( ) 2 coth (8A2Pst + coth (1+ 11)) , (4 17)
and a transfer time of
7' — jfi lnu (418)
° _ M21“. ' '

These results are asymptotes of the back-of—the-envelope formula (4.13) and (4.15) in
the limit of A > |Q|,X3'y + P3.

'10 study the dynamics of relative phase between c1 and 02, we write

'6-
cj = , mjez 3 (4.19)

for j = 1,2,3. The relative phase between Cl and c2, defined as 612 = (91 — 62, is
then given by

d _ IQI2 |Q|2
E912 - E+meINn,
Inn2
M. (4.20)

This . o 2 o 2 o 2 . .
i ls because E121“ f IN 77. << 41111-21371. << JZA—’ so that the phase dynamics 18 dom-
118.111: 1y attributed to ac-Stark shift of pump 1 atoms, driven by pump 2. Assuming

i - -
mtlally 912 = 0, upon population inverting, the accumulated relative phase is

2A
012(70) = ———1/ >> 1. (4.21)
Ps
lea-3313', the relative phase between pump 1 and signal wave is not preserved during the
M
WA dynamics. Consequently, the phase of emitted idler beam is distorted, in which
8
91136 an off-

resonant MWA can not be used as a hybrid state-mapping device between

1i
ght and matter waves. On the other hand, the phase distortion of the signal wave,

150

due to cross-phase modulation c2, is < if}?! << 1 for typical elongated condensates.

Hence, while not suitable for quantum state mapping between matter and light waves,

an off-resonance MWA system can be used to phase-coherently amplify a signal matter

wave -

Overdamping Regime

In this regime, A = 0 and |fl| << F3, so that the excited level is only weakly

pumped due to the quantum Zeno effect. Neglecting atom loss and adiabatically

eliminating C3 lead to

Q 2
2|I'\f|‘N '62l2cli
|0|2
2PfN

 

~

 

ic
dt 1
d 2

d—t62 |01| C2-

Again, these equations can be analytically solved to give

1 [9'2 _1 1— V
= _ — h _—
nw(t) 2 coth (2F3t + cot (1 11)),

811d

_ 2r,

’Tw —|—$-2|—21nl/.

(4.22)

(4.23)

(4.24)

2 2
Here We have used the fact that f R > f I so that % % 1%]; Again, these results

are consistent with the back-of-the-envelop formula (4.13) and (4.15) in the resonant

W 2
eak driving limit, in which P8 = 1%}.

S
The motion of relative phase 912 is given by

2
d6 — '9' iI—n

dt12 — if}; .

151

(4.25)

rv—v

«‘1‘.

The magnitude of accumulated phase shift upon population inversion is then
|612(Tw)| < LI- ln V, (4.26)
fR

which is negligibly small given f I E, 0.01fR, if using a typical cigar-shape condensate.
Unlike in the off-resonance regime, here the relative phase between pump 1 and signal
wave is preserved, due to the elimination of ac Stark shift using resonant driving. This
provides the feasibility to use a overdamped MWA as a quantum device for state

mapping between light and matter waves, in addition to a matter wave amplifier.

Rabi Regime

For a strong and resonant driving with IQI >> P3, the dynamics is saturated
in the Rabi regime, characterized by rapid Rabi oscillations between II) and B)
Averaging over multiple Rabi oscillations, the mean atom number occupying II) and
'3) are approximately equal. To solve the mean-field dynamics 4.6, we separate two
diStinct time scales of Rabi oscillation and population transfer, using coarse-graining

approach. This leads to

 

d P3 2
_ = __ _ , 4.2
dtn 8 (4n 1) ( 7)
With a solution of
1 I‘fRN —1 1 — V
= _ __ . 4.28
ns(t) 2coth( 4 t+coth (1+u)) ( )
a.
11d hence a transfer time of
7-2 = :1— ln V. (4-29)
F3

E
0th match with (4.13) and (4.15) in strong driving regime where P8 = %. Similarly,
t
he phase (32 nearly does not change in time, so that the evolution of relative phase

12 is due to the Rabi oscillation of C1. Via a straightforward iterative analysis, the

152

dynamic solution for c1 is

C1(t) = V g — 1:2 cos 92—1: (4.30)

This gives rise to the relative phase

012a) = g(l — Sign [cos (933)] ). (4.31)

The relative phase between pump 1 and signal wave is thus an oscillatory square
function in time with an amplitude of 7r / 2 and a period of 21r/Q. The MWA in this
regime is therefore not a phase-preserving process, so that it is in general unsuit-
able for applications of state mapping. Nonetheless, in contrast to complete phase
randomization as in the off-resonance regime, here 012 is switching between 912(0)
and 612(0) + 7r during the dynamics, so does the phase of emitted idler beam. In an
interferometry measurement, a phase 0 can not be distinguished from its 7r-conjugate
Of 9 + 77. In this way, the initial phase 012(0) can still be read out via interferometrical
measurements of idler beams, thus providing the feasibility of using Rabi MWA for

the St ate mapping purposes.

Numerical Results

To verify the preceding analysis, we numerically solve the mean-field equations
(4‘6)~(4.8), using the following parameters: I‘l = F2 = 7/2, N = 3 x 106, W =
1()()/ k L and A = 10. The optical depth N f R, given in (4.72), is then 105. We choose
the initial occupation fraction to be C1(O) = m, 02(0) = m and 03(0) = 0-

In figure 4.2, we plot the dynamics of differential population fraction n(t), where
(a) , (b) and (c) are showing for A = 101‘s &, Q = I‘s, A = O & Q = 0.11} and A =
0 &C Q = 10l‘3. They are then each corresponding to the off-resonance, overdamping

and Rabi regime. In all regimes, the analytical formula are found to agree well with

153

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OAT
0.2-
n(t) ‘
0.0-
.02. Numefifl
l - - - - Formula
-0.4 . .
0 1000 2000 3000
0.4. (b)
0.2- r
"(0 ‘
0.01
' — Numerica
'0'2‘ - - - - Formula
-0.4 , . .
0 200 400 600 800
4 ' I f I ' I
0.4- (c)
0.2-
"(0 ‘
0.0« .
-o 2 ' —- Numerioa
' ' - - - - Formula
‘0.4 I I é ' I f
0 4 12
t /l" S

1‘

Fi . . . . .
f g13lI‘e 4.2: Figures (a), (b) and (0) show the dynanucs of differential population
tion in the off-resonance, overdamping and Rabi regimes. In all figures, solid lines

r e Ilumerical results, while dashed lines represent formulas (4.17), (4.23) and (4.28),

Dectively.

154

exact numerical results, thus validating our analysis. From figure 4.2, the transfer
times for each case are determined to be 18OOI‘S_1, 436.8I‘;1 and 8.96I‘3—1, compared
to analytical results of 1757I‘3—1, 439.4(I‘3)_1 and 8.79I‘3—1, respectively given by
formula (4.18), (4.24) and (4.29) in each dynamical regime.
As discussed before, the least transfer time and thus the maximum transfer rate
is obtained when the dynamics is saturated in Rabi regime with strong resonant
driving. To study the saturation behavior, in Fig. 4.3 we plot the transfer rate R, as
a function of |Q| for A = O. The analytical formula (4.15) is shown to agree well with
numerical results. For IQI smaller than I‘s, the transfer rate R grows quadratically
as | (2| increases. This is because in this overdamping regime, R or PC or IQI2, due
to quantum Zeno effect. As |Q| grows to be comparable with 1‘3, R is saturated
to R oc F3/4, as Fe m 1/2 in this Rabi regime. The phase transition between
overdamping and Rabi regimes occurs when |Q| a: F3. In practice, it is thus optimal
to Set |§2| ~ F3 for a fastest amplification process.
On the other hand, the transfer rate as a function of optical thickness P3 for
a fiXed |Q| is shown in figure 4.3 (b). Again, formula (4.15) agrees well with the
nu IIIerical data. For I}; < IQI, the system is in the Rabi regime, so that R increases
11136me with F3. The maximum R is achieved at the phase transition point with
F8 "*5 IQI. As I‘s continues to increase, R decreases linearly in I‘s—l, as the system
enters the overdamping regime. The fact that R reaches its peak value when F3
and l Ql are comparable indicates that given a finite laser intensity, the fastest MWA
prOCeSs is obtained by matching 1‘3 to IQI, via manipulating the trapping potential.
We Ilote for F3 << IQI, the formula result is divergent from numerics. Such is because
In that region, the atom loss becomes significant, so that the formula is invalid.
The relative phase evolution 012(t) for the three dynamical regimes is shown in
ig‘ 4.4 (a),(b) and (c). In the off-resonance regime, 012(t) oscillating at a period of
fig 3 as expected. In the weak resonant regime, 612(t) increases only slowly. The ac-

155

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-1.'I l f "I I I
10
-2:
10
'7” I
a -3
a. 10 a
_4 .
Formula 1
103 0 Numerical :
-5 l
10 ......... m...
0.01 0'1” 1 10 100 1000
I‘ll [1‘s]
10-1; (b) . Formula -
' 0 Numerical ,
E1023 ‘
a: 3 .
-3
10 ‘

 

 

 

1's Ilflll

Eggul‘e 4.3: Figure (a) shows the transfer rate R as a function of Rabi-frequency |Q|

Sh resonant driving field, where F3 is fixed, given the same as in 4. 2. Figure (b)

1111 0W8 R as a function of [‘3 for a fixed |Q|- — 1057. In both figures, solid points are

8.1-; II‘lllel‘ic results obtained by solving the meanfield equations (4. 6), while solid lines
e plotted for the back-of-the-envelope formula (4.15).

156

cumulated phase drift upon population inverting is about ~ 0.24, which is marginally
consistent with the estimated value of 0.13. In practice, further suppression of the
phase drift is easily achievable by increasing the condensate width W or reducing the

aspect ratio A. In the Rabi regime, 612 flips between 0 and 1r at a period of 27r/IQI,

as predicted.

4 - 3 -2 Further Discussions

In this section, we have developed the QCM theory for the three-level Raman
MWA system, whose dynamics is treated with single-mode approximation. We stud-
ied the dynamical behaviors according to the off-resonance, overdamping and Rabi
regimes, respectively driven by off—resonance, weak resonant, and strong resonant
lasers. For any given laser intensity, the largest amplification rate is achieved with
resonant driving. The ultimate maximum rate is obtained in the Rabi regime, when
the Rabi frequency of driving laser is comparable to or greater than the superradi-
ance emission rate. The atom loss fraction upon completing population transfer, as
diSCussed in details in appendix 4.6.4, is a universal quality independent of dynamical
regime the system is in.

In all regimes, the phase of signal wave is preserved, except for a small meanfield
Shift by cross phase modulation, which can be suppressed by adjusting the condensate
g63(3111etry, for example, changing the aspect ratio of the condensate. Thus MWA in
all regimes is qualified as a matter-wave amplification device. In the off-resonance
regime, due to the ac-Stark shift of pump 1 atoms by the driving, the relative phase
between pump 1 and signal wave is completed distorted, making it unsuitable for the
application of state mapping between light and matter waves. In Rabi regime, the
rela-tive phase switches between its initial value 912(0) and 012(0) + 7r. While this
I‘a‘tlclornizes the relative phase, the initial phase value 612(0) can still be read out via

in
telTferometrically measurement of the emitted idler beam. Thus to some extent, a

157

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

2.

912“) o -
4 ¢
.4 . , . , .
o 1000 2000 3000
0.3 . , . , . . .

(b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-o.1 .
40 200 400 600 800
. a
S'TTFTFFHT'IHHTHF
912(92:
1.
o ' s T 1'0 _1' 1'5 ' 20
tug

1Figure 4.4: Figures (a), (b) and (c) show the evolution of relative phase 012(t) in A
e off-resonance, overdamping and Rabi regimes Parameters are the same with
QC)l'responding figures in Fig. 4.2.

158

Rabi MWA system can be still used as a state mapping device, for the benefit of a
short est operation time. In contrast, the phase distortion is strongly suppressed in the
overdamping regime, thus establishing a viable candidate for quantum information
purposes.

On the other hand, the phase distortion can alternatively be overcome in a four-
level Rayleigh MWA system in the so—called ‘weak pulse’ regime [100, 108, 129]. It
is characterized by PeI‘s << am, with the atom recoil frequency w R = $1372]? (m is
the atomic mass). Such is imposed in order to inhibit further coupling of the matter
Waves to other modes by phase mismatching. Otherwise, the MWA process will be
8130in because coupling to other modes is turned out to occur on a similar time
Scale with MWA, since they are stimulated by the emitted idler light field. In this
f011r—level Rayleigh system, pump 1 and the signal wave, corresponding to identical
electronic states, are simultaneously driven by pump 2. They therefore experience
Similar phase shifts, so that the phase distortion is eliminated. Typically, I‘s ~ 109
Hz and w R ~ 105 Hz, so that it requires Pg < 10—4. This requires the use of off-
resonance driving with IQI S, 10—2A or resonant driving with IQI ,S 10’2I‘3. Under
Similar conditions, however, the transfer rate is orders of magnitude smaller than that

a”llc""v'ed in the present Raman system, in overdamping or Rabi regimes.

4~4 Semiclassical Multi-mode Theory

In the previous part, we have developed quantum coupled-mode theory at the
Illeanfield level for the three-level Raman MWA system, employing time-invariant
Spatial modes. Such a theory, while extremely intuitive and easy to access, does

not, take into account any effects accommodating propagation of light fields inside
t*he condensates. In practice, such effects have been shown to account for several

important experimental observations [113, 127 , 129, 131]. For a more physically

159

accurate model, we now develop the semiclassical Maxwell-Schrodinger (SMS) theory,
via semiclassical treatments of dipole coupling between light and matter waves via

Maxwell-Schriidinger (MS) equations.

4 - 4 - 1 Spatial-Mode Evolution

The semiclassical MS equations for this system are derived from a generic set of
nonlinear coupling equations for lights propagating inside a polarized atomic medium,

as given in appendix 4.6.3. In terms of the following dimensionless units and variables

f = Pst,
f‘ = kLl‘,
that) = k’3/2 2/2-(r ,t), i=1,2,3
2d12
CL,s(r,t) = —Eli+5)‘(rh1‘st)
the equation of motion can be is found to be
d ~ ~ 2 * ~ “' ~ “'
Emmi) = §<L(r.t>t3<r.t> (4,32)
d "' ~ ' * ~ ~ ~ "
d—{¢2(r.t") = icsu t) (r t) (4,33)
(1 ~ ,, 2A+
—=¢3(r.f) = «firxflau (rt)+- ;[< Le. t>t31(r.t )(+Csr <rt‘>t2(r (t'>]<4.>34
6__CL( (135) _ 3_77 "*~ " ~
T2 — szz/j1(r,£)w3(r>£) (4'35)
0_C—~__S(r (Li) _ .3_7r ~* ~ ~ "' ~
T2 — sz¢2(r,t)1/J3(r,0 (4'36)

In deriving these equations, we have neglected translational degrees of freedom, using
the fact that the MWA is processed within a time scale of 1 ~ 103ns, much shorter
than any time scales related to atomic motions. Also, we have neglected the retar-

dation effect associated with light propagating inside the condensate, which yields a

160

time scale of 5 10-123, much shorter than atomic dynamics time scales of ~ 1.0—98.

These equations are then numerically solved via Runge—kutta method to simulate the
multi—mode MWA dynamics.

First, we investigate the dynamical evolution of the spatial modes for both matter

and light waves. We take similar parameters as used in section 4.3.1, where F1 =

F2 = ’y/ 2, N = 3 x 106 and an initial population ratio of 1/ = 9. Initially, the matter

waves are in Thomas-Fermi profile (4.54) with W = 100/kL, L = 10W. In Fig. 4.5,

we plot the signal wave density integrated over the transverse dimension, defined as

‘ P2(2.t) = / did37|152(1~‘,0|2r (4.37)

The driving parameters are A = 101}, and Q = F3 for figure (a), A = 0 and f2 =
0~ 1F3 for figure (b), and A = 0, S2 = 10Fs for figure (c). Here 0 = 2h‘1d1EL(—oo,t)
is the Rabi frequency of the applied driving laser. Thus each of the figures are
respectively corresponding to the off-resonance, overdamping and Rabi regimes in
the QCM model discussed in section 4.3.1.

In figure 4.5 (a) for the off-resonance regime, the signal wave is seen to be am-
plified firstly in the back with respect to the beam propagation direction, i.e., the
+3 region. In the front, signal-wave amplification becomes significant only when
t > T/ 2. Thus initially in spatially symmetric, the mode function of the signal wave

firstly evolves to an asymmetric profile weighted in the back. Without reabsorption
of the idler beam, an atom transferred to the signal wave stays there, so that the sym-
metry is gradually restored as the front is lately amplified. A very similar dynamical
behavior is seen in the Rabi regime in figure (0), but on the much shorter time scale.
In contrast, in figure (0) for the overdamping regime, the dynamical asymmetry is
reversed, where the amplification firstly occurs in the front and then subsequently

extends to the back.

161

 

0 0.5 1 1 .5
Dimensionless Time tl-r

Figure 4.5: (Color online) Figures (a)-(c) are the surface plots of P2(2, f) in the off-
resonance, overdamping and Rabi dynamical regimes. In each figure, the evolution
time has been rescaled by the corresponding transfer time T obtained via QCM model,
which for (a), (b) and (c) is 8.961‘;1,436.81“;1 and 180011.71, respectively.

162

The dynamical asymmetry is due to two competing factors of laser depletion of
pump 2 and spatial effect of the idler beam [127, 129, 131, 117]. The depletion of pump
2 is due to photon absorption by pump 1 atoms, resulting in intensity attenuation as
it travels inside the medium. Atoms in the signal wave, however, are not coupled with

the pump 2 due to a large detunning, and are thus effectively ‘dark’ to the driving.
Hence, the depletion effect is strongest in the early stage, becoming weaker as atoms
are transferred to the signal wave, and is eventually gone upon depletion of pump 1.
A rough method to measure the significance of the laser depletion is to compare the
photon number in driving pump Np and atom number N in the medium. This method
is very intuitive: because an atom can at most absorb one photon, the depletion is
Significant only if there are more atoms than photons. For a time duration of At, it

is straightforward to find

eow2cAtEg(—oo)
th ’
oQNAt

= 6 47rf‘3 (4.38)

 

 

as'Suming a beam focal area of W2. Taking At to be the transfer time 7' given in
(415), this gives
Np _ 292 + I“? + 4A2
N 327d“?

 

ln V. (4.39)

Thus, for |Q|,A >> F3, Np > N, so that the laser depletion effect is expected to
Small in off-resonance and Rabi regimes. For IQI < 1",}, on the other hand, Np < N,
indicating a strong depletion effect in overdamping regime. For parameters for figure
4.5 (a), (b) and (c), we have {12? = 8.8, 0.02 and 4.3, so that the laser depletion is
Significant only for figure (b). This analysis agrees with the exact numerics shown in
figure 4.6, where the amplitude of pump 2 across the medium at is = g] = O is plotted
as a function of time. In figure (a) and (c) for the off-resonance and Rabi regimes,

pump 2 is almost constantly uniform in amplitude across the condensate. While in

163

figure (b) for the overdamping regime, there appears two distinct front and back
regions, with a tiny laser amplitude in the back, exhibiting a strong laser depletion.
In the early stage, pump 2 is only able to penetrate into the shallow of the atomic
medium. The penetration becomes deeper as atoms are subsequently transferred out
of pump 10.

The second factor is attributed to the so—called spatial effect, a characteristic
phenomenon where the simulation effect for the scattering process is stronger in the
back region than in the front [108, 127, 129, 131]. In the picture of Bragger scattering,
SuCh is due to a higher intensity of idler beam in the back. As seen in equation (4.36),
prOpagating along +2, the idler beam at Z is proportional to the integration of 11;;2/33
from --00 to 2. Given that 152 and 153 are spatially homogeneous in phases, its
iInKansity monotonically builds up in the 2 direction. According to equation (4.33),

the amplification rate of the signal wave is higher in the back, i.e., the area of of large
idler field. As an example, in figure 4.11, we plot the amplitude of idler beam |C3|
as a. function of 2 and f at i‘, 3'] = O. In figure (a) for the off-resonance regime, lel
is Shown to be much greater in the front than in the back, which is a clear sign of
the spatial effect. The idler field at the +2 edge exhibits a characteristic ‘ringing’
behavior, where a second peak appears after the main pulse [129, 131]. The ‘ringing’
is due to reabsorption of the idler beam in the back, in which atoms in the signal
Wave are excited by the idler field, and then repumped to pump 1 by stimulated
emission back to the laser. Due to the reabsorption, a small outcome idler field does
DOt necessarily imply a equally small field inside the sample. In fact, at around t = 7',
the density of idler beam is peaked close to the center, not the edge. Ultimately, the
reabsorption is due to the relative phase between pump 1 and the signal wave evolving
into mismatched front and back patterns, as we will plot later in figure 4.11 (a). On

the other hand, the reabsorption effect is nonetheless insignificant in the present

system, in that only a small portion of signal-wave atoms are repumped. This is

164

 

Dimensionless Time t]?

Figure 4.6: Figures (a)-(c) are the surface plots of [CL(2,t)| at 2,37 = O for corre-
sponding regimes in figure 4.5. The evolution time is similarly rescaled according to
the transfer time. Notice the distinct scales of color bar in each figure.

165

clearly seen in figure 4.5 (a), where the density of signal wave is nearly monotonically
built up in all areas. Shown in figure (c), similar envelope behaviors are found for the
Rabi regime, with the addition of fast Rabi oscillations. Such is expected, since both
are exposed to similar spatial effects.

Interestingly, in figure (0) for the overdamping regime, the idler beam is shown
to yield uniform intensity from a time-variant edge to the back. The edge is initially
at the front of condensates and then gradually shrinks towards back. This seemingly
counterintuitive behavior is in fact due to the strong laser depletion effect in this
regime, because of which the MWA occurs firstly in the front. The uniform distribu-
tion from edge to back reflects negligible reabsorption of idler beam. Physically, this
is due to the preservation of phase homogeneity in the area of strong idler field, as
we will show later in figure 4.11 (a).

To this end, we can now explain the dynamical asymmetry seen in figure 4.5 in
context of laser depletion and spatial effect. In the off-resonance and Rabi regime, the
laser depletion effect is negligible, and pump 2 is capable of propagating through the
condensate and simultaneously driving population transfer. Due to the spatial effect,
atoms in the back experience a stronger simulation effect by the idler beam, resulting
in a larger amplification rate of the signal wave [127, 129, 131]. In the overdamping
regime, however, due to a strong laser depletion effect, the MWA is driven firstly in
the front. As atoms in the front are transferred to the signal wave, the driving laser
pump 2 is allowed it to penetrate further and drive MWA in the back. We found
that for A = 0, fl = F3, the competing effects of propagation and laser depletion are

somehow neutralized to give a more symmetric mode profile during the dynamic.

4.4.2 Population Transfer

In this section, we study the population transfer dynamic via the SMS model,

taking into account the laser depletion and spatial effect. The population fraction in

166

 

o 0.5 1 1 .5
Dimensionless Time t/T

Figure 4.7: Figures (a)-(c) are similar surface plots but for |(S(E,fi|.

167

the each matter-wave mode is given by nj = f d3? gj(i'), for j = 1, 2, 3. In figure 4.8
(a)-(c), we plot the evolution of differential population fraction n, defined in (4.10), in
the three dynamical regimes, using the same parameters as for figure 4.5. From figure
(a) and (c), although on distinct time scales, qualitatively similar behaviors are found
for the off-resonance and overdamping regime. For t < T / 2, both closely follow the
QCM dynamics. Soon as t reaches T / 2, corresponding to about half of pump 1 atoms
being depleted, n(t) obviates from the QCM results, downgrading to a slower growth
rate. The population inverting occurs at t = 5.47- for both cases, which is about five
times longer than the QCM transfer time in corresponding regimes. The behavior
is dramatically different in the overdamping regime, as shown in figure (b), where
n(t) roughly follows the QCM curve throughout the whole dynamics. Surprisingly,
for t < T/ 2, the population transfer is even slightly faster than via the QCM model.
Such a behavior is counterintuitive, since both the laser depletion and spatial effect
are expected to slow down the population transfer.

To explain these behaviors, we define a relative collectivity parameter gaff with

respect to f R, as

67r
f R”

 

 

2 00 ~ 00 ~ ~ ~ ~ ~ ~
/ dz / dp px/ 92(1), 709360. Z)
77.3 —00 0

~

Z ~I
x / dz (Mama/3.20. (4.40)

gaff“)

 

—00

Here, the angular integration has been performed analytically to give 27r, by using the
azimuthal symmetry. For condensates initially prepared in azimuth-symmetric modes,
such as the Thomas-Fermi profile (4.54), the azimuthal symmetry is preserved during
the dynamics since atoms are uncoupled in transverse dimensions, as a consequence
of single k-vector approximation for the idler beam. We note that because the mode

functions evolve in time, gaff is a time-variant quantity. The equation of motion for

168

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.4. g .
. .’ a
0.2- . ( ) -
n(t) 1 I,
0.0- .
-o.2. - - - QCM -
—SMS
’0-4 I T T I f I f
o 2 4 6 6 1o
0.4-l (b) -i
. i
0.2- , ’ .
n(t) . [I
0.0. ,I _
I, 1
_02, , - - - QCM -
i , ’ —sms ,
.004 I ’ l I fir r I
0.0 04 0.8 12 16
. ,_--1-_-----: ........ ' .........
04- ,
. .' (c)
0.2. .' -
n(t) . .'
0.0- .
-o.2- - - - QCM .
‘ ——SMS
'0-4 r F l l '
o 3 6 9 12
tit]

Figure 4.8: Dynamical evolution of the differential population fraction n(t), where
(a), (b) and (c) are plotted for the off-resonance, overdamping and Rabi regimes,
respectively. All parameters are the same as for corresponding figures in Fig 4.5, with
the evolution time similarly rescaled.

169

n2 can then be written in a compact form of

d
d_t~n2 = {effn2n3. (4.41)

assuming spatially homogeneous phases for matter waves, i.e., Oj (f') = 0j(r’), for
j = 1, 2, 3. In the QCM model, 55712 = n2n3, so that the deviation of {eff from 1 is
a measure of the significance of the spatial effect.

In figure 4.10, we plot the time evolution of éeff for the three dynamical regimes
shown in figure 4.5. In all regimes, gaff = 1 at t = 0, indicating the physical
equivalence of the semiclassical Maxwell-Schrodinger equation and quantum master
equation. In the off-resonance regime, éeff smoothly downgrades from 1 to around
0.5, as t increases from O to T/ 2. It then slowly oscillates around 0.5 in a small
amplitude. A similar envelop function of {eff is found for the Rabi regime, but with
the additions of sharp oscillations. Because the laser depletion effect is weak in these
two regimes, the slowdown of population transfer at the half-depletion time is mainly
due to the downgrade of gaff' Essentially, the descending of “gaff is a consequence
of the spatial effect, due to which the mode functions of the matter waves evolve to
asymmetric profiles. This is seen in equation (4.40), where gaff is determined by
the overlapping of 92 and 93, or equivalently the densities of pump 1 and the signal
waves. In the early stage, pump 1 and the signal wave are spatially matched. Due to
the spatial effect, the depletion of pump 1 occurs firstly in the back, imparting with
the amplification of the signal wave also in the back. This leads to a spatial mismatch
of 92 and 93, thus a reduced relative collectivity gaff-

Comparatively, in the overdamping regime, Seff sharply decays to 0.2, and then
quickly regains to around 0.5. After t = 27', it gradually decreases to 0. Due to
quantum Zeno effect, the transfer rate in this regime is inversely proportional to

the optical thickness, and thus Seff' That’s, the spatial effect, while resulting in a

170

 

reduced optical thickness, in fact leads to a enhanced transfer rate. This exact reason
accounts for the counterintuitive phenomenon seen in figure 4.8 (b), where population
transfer is slightly faster in the SMS model. Of course, in the overdamping regime, the
laser depletion effect is strong, acting adversely on the population transfer. The two
competing factors somehow annihilate with each other, resulting in roughly agreeing
SMS and QCM population dynamics.

As discussed above, the downgrade of éeff is due to the matter waves evolving
into spatially mismatched modes, and is not directly related to the population dis-
tribution among different modes. Thus the spatial effect would similarly affect the
population dynamics for various initial population ratio V. To verify this, in figure
4.9 we present the population dynamics for u = 1, i.e., with evenly feeded pump 1
and signal waves. All other parameters are the same as for figure 4.8. In figure (a)
and (c) for the off—resonance and overdamping regimes, the SMS dynamics coincides
with the QCM results for n < 0.25, before pump 1 is half depleted. Afterwards, the
transfer rate is then significantly reduced, exhibiting similar divergent behaviors with
figure 4.8 (a) and (c) where V = 9. In figure (b) for the overdamping regime, however,
n(t) only moderately follows the QCM curve. The transfer speed in the SMS model

is shown to be slower than that via QCM.

4.4.3 Phase dynamics

In section 4.3. 1, we studied the phase coherence property of the three-level Raman
MWA system within the QCM model. A strong phase distortion effect is found in
the off-resonance regime, due to the ac-Stark shift by the pump 2 laser. In the Rabi
regime, the relative phase between pump 1 and signal wave is switched between its
initial value 612(0) and 012(0) + 7r rapidly. The only phase preserving process is,
however, realized in the overdamping regime, yet with a small drift due to cross

phase modulation via atomic dipole interaction.

171

 

0.4«

 

 

 

 

 

 

 

 

 

 

 

 

n(t)
0.2- .
0.0 . . . e . .
o 2 4 6 8 1o
, , ’ ' ’ (b)
0.4. , , -
n(t) ‘ I I
0.2. ,' .
, .’ - - - QCM
,, —SMS
60 ' . . . .
0.0 o 4 0.8 1 2 1 6

 

 

 

 

 

 

 

 

 

 

Figure 4.9: The same as figure 4.8 but with u = 1. For (a), (b) and (c), t is rescaled
with 7' = 8.961‘g1, 436.8I‘g1 and 1800F§1, respectively.

172

 

f I I T

 

- - - Overdamping
- Rabi

1 '2 d — Off-resonance

 

 

 

:
1
8
I
I
I
O

I

. O
.....

 

I,
HI“, "II|I' _.
I v I
I "Ulrl’luif'
‘ ‘H‘IHIIfl'.

 

 

 

t[T]

Figure 4. 10: gef as a function of evolution time, which IS rescaled with the transfer
time T for each ynamical regime. Parameters are given the same as in figure 4. 5.

Unlike in the QCM model, the atomic dipole interaction, essentially a virtual
photon exchange effect, is not included in the SMS model since it only accounts for
‘real’ photons. Thus the SMS model is valid only if the dipole interaction is negligible.
In practice, this requires k LW >> A, as discussed in section 4.6.2. For kLW = 100
and A = 10, we have seen in 4.3.1 that the phase shift due to dipole interaction
stays small throughout the dynamics, thus validating the SMS model using these
parameters.

In the SMS model, in both off-resonance and Rabi regimes, the laser depletion
effect is negligible, allowing pump 2 to homogenously excite atoms across the con-
densate. Thus all atoms contemporarily experience ac-Stark shift or phase switching
as in the QCM dynamics, with spatial homogeneity preserved. In the weak damping
regime, however, the laser depletion effect is strong, and amplification of the signal

wave is sequentially driven from front to back. Yet due to the elimination of ac-Stark

173

shift and that the phase of signal wave is preserved during MWA, the relative phase
between pump 1 and the signal wave is homogeneously conserved at all time. Con-
sequently, the generated idler beam constantly possesses the initial phase of 1912(0),
making an overdamped MWA a good candidate for state mapping.

To verify the above analysis, in figure 4.11 we plot the phase 612(2, 5) at 513, 37 = 0.
In figure (a) for the off-resonance regime, 012(t) is shifted linearly in time in a 27r
interval, as expected. Its spatial homogeneity is preserved through t = T/ 2, after
which the rear +2 region grows out of phase with the front —2 region. While this
indicates the emergence of phase inhomogeneity, the fact is that in the out-of—phase
region, pump 1 is (nearly) depleted, so that its phase relative to the signal wave
become meaningless. The spatial homogeneity is preserved in a sense that 012 stays
uniform across the region where both pump 1 and signal wave are macroscopically
populated. In figure (b) for the overdamping regime, 612(t) is found to be at its initial
value of zero through the dynamics in regions where pump 1 atoms are undepleted,
thus preserving the spatial homogeneity. In figure (0) for the Rabi regime, 012 is
switched between 0 and 7r at the Rabi-frequency of pump 2, agreeing with the QCM
result. Similarly to the off-resonance regime in figure (a), here the phase homogeneity

is as well preserved.

4.4.4 Further discussions

In this part of the chapter, we have studied the MWA dynamics via the semi-
classical full-mode theory, taking into account the laser depletion and spatial effects.
In both off-resonance and Rabi regimes, the laser depletion is negligible. Because
of the spatial effect, the signal wave is firstly amplified in the back along the beam
propagation direction. In the two regimes, the population transfer is fast before half
of pump 1 atoms are depleted, and agrees well with the QCM theory. Then, as the

spatial modes evolve into mismatching distributions, the population transfer is sig-

174

 

KLz
O

-1000

-2000

 

tl'rl

Figure 4.11: Figures (a)-(c) are the surface plots of 012(2, f) at 17:, 37 = 0 for dynamical
regimes in figure 4.5 (a)-(c). Initially, 012(2, 0) = 0.

175

nificantly slowed down, becoming divergent from the QCM results. In all dynamical
regimes, the phase of each matter wave is found to be practically homogeneous across
the condensates. Asides from small mean-field shift due to atomic dipole interaction,
the phase of signal wave is preserved during the MWA dynamics. In contrast, the
relative phase between pump 1 and the signal wave respectively experiences ac-Stark
shift and 7r-switching in the off-resonance and Rabi regimes, but is conserved in the
overdamping regime.

Additionally, as shown in appendix 4.6.4, the atom loss dynamics similarly agrees
with QCM until the half-depletion point, and then diverges. The ultimate atom loss
fraction upon completion of population transfer is enhanced by a factor of 3 ~ 4.
In the overdamping region, however, due to a dominating laser depletion effect, the
signal wave is sequentially amplified from front to back. The population transfer yield
smooth dynamics, which moderately agrees with the QCM results all the time. The
atom loss fraction, however, is constantly a factor of z 4 greater than that via QCM.

Hence, for purpose of phase-coherently amplifying a matter wave, it is feasible
to operate MWA in any dynamical regimes. For a given laser intensity, however,
the amplification rate is always higher if using resonant driving, with which Pe is
maximized, as discussed in section 4.3.1. For applications of state mapping between
light and matter waves, the off-resonance MWA is found to be unsuitable, due to
the strong phase distortion effect via ac-Stark shift. While as well undergoing phase
distortion, the MWA in the Rabi regime can nonetheless be used for state mapping
in that the phase information can be read out via interferometric methods. In the
overdamping regime, the phase distortion can be strongly suppressed, also allowing
state mapping.

In the resonant regime, due to the sudden slowdown of population transfer be-
yond the half-depletion point, for efficiency it is the optimal strategy is to stop the

dynamics at the half-depletion time. In this domain, the SMS dynamics turns out to

176

agree well with the QCM predictions. Thus we can approximate the SMS dynamics
by the analytical formulas derived in the QCM model. Using equation (4.13), the

half-depletion time Th, for which n(Th) = %n(0) + i, is given by

2
7,, = '13; In (3” + 7” + 2) . (4.42)

 

V2+3V

The atom loss fraction at Th for V < A2 is then

’7 1
€l088(Th) = f; In (1 + 51/) , (4.43)

as from equation (4.112) in the appendix.

' In the overdamping regime, both the dynamics of differential population fraction
n(t) and atom loss fraction 51038 (t) is more smooth through the dynamics, following
the QCM dynamics. It is thus equally efficient to evolve the system until the depletion
of pump 1. The transfer dynamics can then be approximated by equation (4.13). The
final atom loss fraction is, however, about four times greater than that of QCM, given
by

,7
9033(7) = 411—3 ln(1 + V). (4.44)

4.5 Conclusion

To conclude, focusing on a prototypical three-level Raman system, we have ex-
tended the study of matter-wave amplification to the resonant driving regime, in
order to find the optimal scheme for applications of phase-coherently amplifying a
matter-wave, as well as quantum state mapping between light and matter waves.
We developed the quantum single-mode as well the semiclassical multi-mode the-
ory. Comparatively, the former yields more intuitive and computationally accessible

dynamics associated with clear physical interpretations, while the latter takes into

177

account the laser depletion and spatial effects and is thus more ‘physical’.
Depending on driving laser, the system dynamics can be classified into off-
resonance, overdamping and Rabi dynamical regimes. For population dynamics, the
single-mode theory predicts correct behaviors in the overdamping regime, but is valid
only through half depletion of pump 1 atoms in the off-resonance and Rabi regimes,
after which the population transfer becomes inefficient. We find that for a given laser
intensity, the largest amplification rate is obtained using resonant driving, whereas
the ultimate maximum rate is achieved in the Rabi regime. On the other hand, the
atom loss fraction upon completing population transfer is a universal quality inde-
pendent of dynamical regimes. In all dynamical regimes, the phase of each matter
wave is effectively homogeneous across the condensates. The relative phase between
pump 1 and the signal wave respectively experiences ac-Stark shift and 7r-switching
in the off-resonance and Rabi regimes, but is preserved in the overdamping regime.
In short, it is the optimal strategy to use resonant driving, instead of the com-
monly used off-resonance one, to perform MWA for the purposes of amplifying a
matter wave and / or mapping quantum states between light and matter waves. If in
the Rabi regime, it is most efficient to stop the dynamics once half of pump 1 atoms

are transferred to the signal wave.

4.6 Derivations and further discussions

4.6.1 Master-equation formalism

In this section, we develop the master-equation formalism for the A—MWA system.
Let III j (r) ( j = 1, 2, 3) be the field annihilate operator for level lj), the standard master
equation for the Raman MWA system depicted in figure 4.1 (a) reads [59] (A detailed

178

derivation of this kind of master equation is given in the appendix of chapter II)

p(t) = —i[HO,p(t)] — [413711371 x (4.45)
2 [Ljo- — Mimi- (r011,- (r) ®3<r’)p(t>
j=1,2

_L;(r _ r3111}(r)\t3(r)p(t)ig(r’)\i,(r’) + no.) .
Here, the one-body Hamiltonian H0 is given by

so = / d3r(A—z'%)\Ill;(r)\I13(r)+

+/d3r9%i;(r)xi1(r)+ H.C', (4.46)

with A the detuning of pump 2 from resonance and Q the Rabi frequency. '7 is
the spontaneous relaxation rate of I3) level. The nonlinear two-body cooperative

coefficient is given by [123]

2
L-(r—r’) = —djc—/d3k k(1— |d~ i1?)
3 2(27r)3€0 J

 

(4.47)

x eik-(r—r’) [7rd(ck — ’Uj + A) +z' P
c — v-

1: 3+4:

for j = 1,2. Here, hvj is the level-spacing between l3) and |j), while dj and cf]-
are amplitude and unit vector of dipole moment for corresponding transitions. The
symbol ’P stands for taking principle value when performing integration. This integral

can be carried out analytically, giving

3F- _- . ' 1 '
LJ- (r — r') =—Je K3 sin2 01+ (1 — 3cos2 6’) —2 — i3 , (4.48)
4 Sj C]. (j

179

with 9' = (123' — A)|r — r’I/c, and 0 being the angle between dj and r — r'. P, =
(vj -— A)3d§/(37r60hc3) is the spontaneous decay rate from the A-detuned l3) to
ground I j ) level.

In the master equation (4.45), the real part of Lj (r — r’), related to the optical
depth of collective emission, characterizes the population transfer between level I3)
and L7). The imaginary part contributes an energy shift due to (virtual) photon
exchange between atoms, the laser-induced dipole-dipole interaction. In the limit
Ir—r’l —>0we find

um=g+m as)

where the imaginary part, 6, is an infinite quantity, reflecting the fact that the stan-
dard two-level atom-field interaction model is not properly renomalized. Physically,
this term is a Lamb-type shift in the energy of the excited-state due to interaction with
the vacuum modes of the EM field. In our system we can always choose a rotating

frame to absorb this shift, so that we can take 6 = 0 without loss of generality.

4.6.2 Evaluating cooperative parameter

In the strong interaction regime, the spatial profile of the condensate is well

approximated by the Thomas-Fermi distribution, as [6]

 

m m
sips)? = 4.1-.2. (4 — -2- (4in + 4322)) (4.50)

with a is the s-wave scattering length, m is atomic mass, and

 

2 2 2/5

41/2—

is the chemical potential determined by wavefunction normalization. Here, we have

assumed the external potential to be an azimuthal-symmetry harmonic trap, with “CL

180

and wz the transverse and longitude trapping frequency. Such a density distribution

corresponds a RMS width of

1/5
1 15am?)
W = — (— (4.52)
2 3 ’
\/7 m (.u_L
and length of
1 lswinz ”5
L = — —— . (4.53)
1/7 ( m2w§ )

With such defined W and L, the Thomas-Fermi distribution (4.50) can then be sim-
plified to

 

2 2
|<I>:I"F(r)|2 = 56¢;sz [I — % (%5 + fifl . (4.54)

Depending on W smaller or greater than L, the condensate is in a ’cigar’ or ’disk’
shape. Inserting this profile back to (4.9), the collective parameter fj can be nu-
merically evaluated. The evaluation is nonetheless difficulty, since it involves a six-
dimension integral with poles. In the following, we show a method to reduce the
dimension of this integral from six to three, providing numerical accessibility.

To proceed, the collectivity parameter is given by
f = / d3'rd3r’L(Ar)l¢(r)|2l¢(r’)|2e’k0'A’ (4.55)
with Ar = r — r’ and

L(Ar) = £32-£e_iC [sin2 9% + (1 — 3cos2 6) (32 — 225)] (4.56)

where c = kL|Ar|, 6 is the angle between A the dipole moment and Ar. Defining

dimension-less coordinates

x_a:+a:' _y+y’ Z_:z_-l_-_z:
C 2fiw’ yc zfiw C 2\/7L

181

 

 

 

 

y
CC = , = Z — 4.57
7' WW yr WW 7' fiL ( )
we have
225
f = 64? d3Tcd3TrL($riyrizr)
2 7‘2 2 T2
1— rc+Zr+rC-r7~ 1— rc+745—rc-rr (4.58)
with
L(rr,y7-,z7~) = —e sm 6— + (1 — 3008 6) — — — . (4.59)
2
Here, c- —kL\/7W2p%+7L2z,g and cos 219 = 79L? with p,- = rg+y12n We

Pr'l'zr
calculate for k0 = kLz. First, we do the d3'rc integral, where we let rr = rré.

- W d TrL($riyr,Zr)€
Min{(1—1'2 —r 2/4)/rcr7-,1} 2 2
x/ c r dcosd 1— r§+r—" —rgr3coszl9
—Min{(1—rc—r,2./4)/rcr7-,1} 4

22 . 1—rr/2
= ——5 d37‘7-L(xr,y7~, zr)e‘kL‘/7Lz"27r {/0 drc 7%

1/1— 4
f- 225 3 ZkaLZrzW/ Tr/

647r2

1 21—rT/4dr
/l((>> w) 4,.
(l—rg—rg/4V'Fc7‘r 7‘2
2 d 1 — 2 — 2 2 2
Tc ./;(1—rc—r72~/4)/Tc7‘r 'UJ ( (TC + 4 Try) —Tcrr# c}

225

= —2' d37‘rL(1‘ri yr: ZrleikLfiLzr
64%2

X 27r(—2 + Tr

_ 15
— 71687r

)4 32 + Tr(64 'I' 3Tr(8 + Tr))
3360

d3r7~L(:z:7-, yr, haiku/IL” (—2 + m4 [32 + rr(64 + 3M8 + rr)l

 

182

Note here e‘kL fiLZ’" can be replaced by cos(k L \/7 L27), due to the symmetry on zr.
This integral is hard to be solved, because it is oscillatory. To overcome this difficulty,

we change the integrand to

15
7168f737rkiw2L
xeiz0(-2 + r1)4 [32 + r1(64 + 311 (8 + 71)] (4.60)

 

/ d3T0L($o. yo. 20)

where

 

2 2 2
7.1 = $0+y0 + Z0
7(kLW)2 7(kLL)2

r3 sin2 6 + r3 cos2 6
7(kLW)2 7061.162
_ T0 2 2
_ 1 — 1 — 1 A 6, 4.61
WHWV ( /)ms < )

 

 

 

The boundary condition of r1 _<_ 2 is then cos2 6 2 (1 — 28kiW2 #3) / (1 — 1/A2) E
122, valid for the aspect ratio A > 1. The two-body nonlinearity is scaled to give
L(O) = 1 +16

2'

L(TO) 9, (25) = 36—2-71) [(1 — sin2 6sin2 (”é + (1 — 3sin2 6sin2 (b) (7% — :—

0
000

(4.62)

 

where r = r2 + y2 + 22. The integral is then
0 0 0 0 _

15 2fikLWA 2 —,i 1
f = / dro r0 / dcos6+/ dcos6
7168fi37rkgw2L o —1 u

. ' 27r
erO COS9(—2 + 11)4 [32 + TI (64 + 3r1(8 + r1)]/ d¢L(r0, 6, 45) (4.63)
0

 

 

15 /2~/7kLWA
o

—u
2
dro r0 2 / d cos 6 cos(ro cos 6)
7168f737rkgw2L —1

183

27r
x (—2 + r1)4 [32 + r1(64 + 3r1(8 + 71)] f0 dqu(r0, 6, (25) (4.64)

 

where p = \/(1 — 28k%W2/r8)/(1 — 1/A2) or 0 if 28k%W2/r(2) > 1. In this way, we
have successfully reduced the six-dimension integral to only 3, providing the feasibility
to be numerically solved by mathematica codes.

For an analytical expression of fj, we approximate the Thomas-Fermi profile

(4.54) by a Gaussian distribution of

I‘D (r)|2= 1 exp __i__zz_ (4 65)
GS (2«)3/2w2L 2W2 2L2 ’ '

 

which yields to the same RMS width and length with (4.54). Using formula (4.47)

for Lj (r — r’), the collective parameter fj is then explicitly given by

 

 

3 3 . 2 2 . P
fj 87%}, [d k k(1-|dj ~kl )Xj(k) [r662 — kL) +zk_ 1%], (4.66)
with
Xj(k) = / d3rd3r’l¢(r)¢(r')lze"(k"‘L)‘(r—"). (4.67)

Here we have assumed ”01,2 — A = ckL, i.e., the [1) and [2) level are degenerate. For
dl = dz, we then have X1(k) = X2(k) E X(k), and therefore f1 = f2 E f. Using
straightforward center-of—mass and relative coordinate transformation, Xj (k) can be

analytically evaluated to give
x(k) = exp (—k2 sin2 6W2 — (k cos6 — kL)2L2) , (4.68)

for k L = kLZ. Inserting this back to (4.66), it is clear that f is dominantly contributed

by integration in the regime near 6 = 0, due to the exponentially decrease of Xj (k)
in 62.

184

To proceed calculating this integral, we introduce the following dimensionless
parameters, 10 = kLW, 6 = kLL. From equation (4.66), for dj = —\}—§(:i: :l: 2'33), the

integrand is azimuth-symmetric, so that

 

 

3 0° 31+cosz6 . ’P
f — 471.ki/_oodkdcos6ls: _T—X3(k) 7r6(k—kL)+zk_kL:[ ,(4.69)

Introducing a small-valued parameter V = 1 — cos 6, the above integral is rewritten

in the form of

 

oo 2 2
f = 1 dis: / dV k3(1— V + 5—) exp[—w2k2(2V — V2) —— 62(117 — 1 — kV)2]
47r —oo 0 2
x [was — 1) +ik1—D1[ , (4.70)

where we have substituted [1: —+ k/kL. It is noticed that for practical parameters of
10,6 >> 1, this integral is no-vanishing only at V << 1. To calculate the real part f R,

using 2V > V2 and 1 — V w 1, and performing the 6(k — 1) integral give us

2 2
fR x 2/0 exp(—2w2V — €2V2)dV = 3—23/5: exp(w4/€2)erfc(w7). (4.71)

It turns out that this function can be well approximated by a much more evident

form of
fix/77

fR = 8(64/3 + (\/7_rw2)4/3)3/4.

 

(4.72)

To evaluate the imaginary principle-value part, we introduce a: = k — 1, so that

the corresponding integral becomes

3 0° 2 (1+:r)3 V2
f1 — Eflmdx/O dV—T—(l—V+-2—)

exp[—w2(:r +1)2(2V — V2) — 62(51: — V — 33V)2], (4.73)

185

 

To the leading order in V and :c, the integral is simply

3 OO

2
f1 = d2: [0 dV (% + 3) exp[—2w2V — 62(23 — V)2[, (4.74)

47—00

(4.75)

The expression for imaginary part is then obtained in the limit of large and small

Fresnel number .77 by

 

3 (21n(€/w2) — ’76. f >> 1;

 

8 6

f1 W “3 2 (4.76)

3 6 +112 ,7: << 1

8fiw 6 ’ '
(4.77)

Here 76 = 0.577216 is Euler’s constant, and the Fresnel number
16sz

.F — 27r L (4.78)

is a measure of the number of superradiant endfire modes within the end-fire angle
63 = 1%. When .7: >> 1, there are many competing end-fire modes, while for .7: S 1,
there are just two end-fire modes, one for each direction along the z-axis.

To show the validity of these formulas, in figure 4.12, we compare (4.72) and
(4.76) to the exact results by numerically evaluating f for the Thomas-Fermi profiles
(4.54). For all range of width W and aspect ratio A = L/ W, good agreements are
found for the real part of f. For the imaginary part, however, formula (4.76) is valid
only for W 2, 26L: especially for large aspect ratio A. In practice, this parameter re-
gion nonetheless matches that of typically prepared condensates in undergoing MWA
experiments [102, 101, 108, 112].

Finally, for typical MWA experiments performed with ‘cigar’-shape condensates,
kLW >> A [108], so that fI/fR z A/(kLW) << 1. This indicates that mean-

186

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

'I'"'

W

Figure 4.12: The cooperative parameter f for Thomas-Fermi condensates, where
figure (a) and (b) show the real and imaginary parts of f R and f]. In both figures,
discrete points are numerical results for Thomas—Fermi profile, while lines are plotted
for formula (4.72) and (4.76), respectively.

187

 

field dephasing due to dipole-dipole interaction is negligible compared to population
transfer, thus allowing amplifying the signal wave without introducing significant
phase distortion. Henceforth, all analysis and discussions are based on f R > f I- We
note asides from this laser-induced dipole interaction, other dephasing channels, such
as s-wave collision, can also be safely neglected due to the short time scale (~ ns) of

the MWA process under considerations.

4.6.3 Derivation of Maxwell-Schrtidinger equation

Let \Ilj (r) (j=1,2,3) be the Schrédinger field of state [7), and E2“ and ESP) be
the positive-frequency optical fields of the laser pump 2 and scattered idler field, the

generic Maxwell-Schréidinger equation reads [257]

71—6 ‘11 - —h2 V2 V ‘1’ d E(_)\Il 4 79

2 at 1(r) — —2m + trap“) 1(r,t) “ 1' L 3(r,t), ( - l
8 F 1‘12 - (_ “)xII

iii—5 W2(r, t) = ——2 V2 + vt’rapo.) 4120‘, t)— d2- ES (r, t), (4.80)

253-W3“, t)

h2

2m

 

V2+Vtrap(r )+h(w w3—Z'X_2"')

\II3(r, t)

 

 

— d* EE+)\111(r,t t)+d* E‘SW ‘1’2r( ,t)], (4.81)
2
_E(i) _ 22(i)13(i)
Here, E ( LS)’— — E( LHS). are the negative-frequency electric fields, 12.03 is bare energy

of level [3), and we have assumed |1) and |2) to be degenerate. The non-Hermitian
term of —ihX3'y, with X3 the time-dependent superradiance enhancement factor given
in Eq. (4.106) in appendix, describes the atom loss from the matter-waves under
consideration via spontaneous and superradiant decays of the [3) level. The trapping
potentials for [1), I2) and |3) states are Vtmp, Vt’mp and thr’up’ respectively. d1,2 =

1 )3) and )2) H

72:6: — iy)d1,2 are the dipole moments for the II) H [3) transitions.

188

P]? are the macroscopic polarizations of the atomic medium, defined as

1355’s) = "‘1\/‘-2’(i"i5’)d1,2‘1’i2(r)‘1’3(r)- (4.83)

As seen in the single-mode model, in present dynamical regimes of interests,
the MWA is processed within a time scale of 1 ~ 103ns, much shorter than any
time scales related to atomic motions. Thus we can neglect translational degrees of
freedom, leaving only the light-matter coupling terms in equation (4.79)-(4.82). For
2 being the common propagation direction of pump 2 and idler beams, we introduce

the following slowly varying envelope functions,

6:604) = “if“2'?)E]f§(r,t)e’(kLz‘“’L’), (4.84)

with w L = ck L laser frequency. The normalized slowly-varying Schréidinger fields are

introduced in a similar manner, where

‘1’1,2(r. t) = W $1,203 0

‘1'3(r,t) = Ww3(r,t)ei(kLZ—WL0

with N the total atom number.

In the rotation frame of laser frequency 02L, the Maxwell-Schriidinger equation

(4.79)-(4.82) is then rewritten as

6 'd *
52416.6 = 1,165) <r.t>43<r,t>, (4.85)
6 'd *
52420.0 = 5,265+) (66436.6, (4.86)
a s 'd* 4*
#30“ t) = — (2'4 + £21) 436, t) + 351-47226, t)¢1(r, t) + %E§+’(r. 0426,87)
<+>
3 6E (r,t) 'd N *
EE[+)(r,t) = —c_L.52_—+z_1_2“%6_¢1(r,t)¢3(r,t), (4.88)

189

<+>
6 8E ,t d N
#2?“an = 43—45%. l—O-g’f 436 t)¢3(r t) (4.89)

with A = (.03 — 9L- In obtained these equations, we have applied the standard

slowly-varying envelope approximation, where

6522.6 ,t)

at < wL]E£+)(r, t)]

 

VE(+ )(r,t t) < kL]E2]g(r,t)]

 

 

[gt-(M2243) << %]¢i,2¢3]-

8 0

—1 37: +c383 = 235, reducing

Defining new variables 19 = t+c-lz, q = t— c z, we have

the equation of motion for light fields to

 

 

(+)
6'EL (1', t) _ idleN*
—6p _ 1pi‘r( ,r,t)1p3( t), (4.90)
(+ )
6E5 (1‘ 1‘, t) _ idzigN’l/fk
————ap — 116t(,nt)¢3( t.) (4.91)
Usingt %(p + q), and z— — g(p— q), we further rewrite
E‘+S)<r t>= _ 1322(53 ——cp q) -<p+q)) (4.92)
and
* _. C 1
4,- (r, t)¢3(r, t) = 0,3 (p. ,6 - 4). 5(1) + 4)) , (4.93)

for j = 1, 2. In these new forms, equation (4.90) and (4.91) converted into integration

forms of
E2” (5: §<p— q), $64.4) = (4.94)

190

 

,dleN/p _,C I 1 / I
_ __ .. d
z 460 _ 013 p.200 q).2(p+q) 19

1
ES) (5, go) — q). 507 + q)) = EL(—oo) (4.95)
d N p 1
+ i-%- [_00023 (5.5-(10' - q). -2-(p’ + (1)) 6129’-

Here E L(—oo) is the applied electric field of pump 2 outside the medium. Defining

z’ = g(p’ — q), and recognizing %(p + q) = t, g(p — q) = z, the above equations are

simplified to

 

 

 

d w N 7' z — z’
E(+) -' t = L _, I _ I 4.
L (p,z, )) z 2ce0 _00 013 p,z ,t c dz, ( 96)
N z - ’
5359+) (,5, z, t) = EL(—OO) + id2wL / 023 (,5, z,,t — z z ) dz’.(4.97)
2ce0 _00 c

These expressions have a clear physical interpretation: a 2—propagating electric field
at position (,5', z) and time t is determined by the summation of dipole moments at

I
Z

an anterior position 2’ S 2 and a prior time t - 3;

C . The retardation in time of

52—2: is the time it takes for the light to travel from 2’ to .2. In practice, the typical
condensate sizes are of sub-millimeters. Thus the retardation within condensates is
,3 10—128, which is much shorter than atomic dynamics time scales of ~ 10‘93, and
can thus be neglected. Physically, neglecting retardation corresponds to Markovian
approximation, in which electric fields adiabatically follow atomic dynamics [123].

Converting back to differential forms, we have

 

 

(+)
8E (r,t) .d N *
Laz _ = z 12:) 1,01(r,t)7,b3(r,t), (4.98)
(+)
8E (rat) (1 Cd N
S _ .- 2 L at:
__(92 — z 2060 ¢2(r,t)2/}3(r,t). (4.99)

Compared to the original equation (4.88) and (4.89), here time partial-derivative

terms are dropped. The resulting equation directly leads to equation (4.32)—(4.33).

191

 

 

Finally, it is noticed that the dynamics is uncoupled along it and y directions, a
consequence from that light beams are assumed to propagate along the 2 direction.
This enables us to separately solve for the dynamics at different locations in the 2: — y

plane, thus greatly reducing computational complexity.

4.6.4 Atom loss: QCM and SMS theory
Atom loss via QCM model

As previously discussed, atom loss in this scheme is due to |3) coupling to other
quasi modes, a process initialized by spontaneous emission, and potentially enhanced
by the superradiance effect. To study the superradiance enhancement effect on atom
loss, we consider a many-mode system where I3) is simultaneously coupled to multiple
quasi modes besides that of c1 and 02. Let Nike: be the atom number in the side
mode of |j,k3) ( j is the electronic level and h(k L — k3) is momentum), the rate

equation is approximately

(1
EN1 = —FPeN1fRN2 - PeNl Z ijj,kijaks - 7P8N1’
j’ks
d
aN2 = rPeleng + 1), (4.100)
d

ENJ'ks = r,- PeN 1fj.ks (Nj,ks +1)-

Here I‘j is the spontaneous emission emission rate from the excited |3) to the j-th
level, satisfying 29' I‘j = '7. fj,k3 is the collectivity parameter for the | j, k3) mode,

which for condensates of Thomas-Fermi profile (4.54) is

f. ~ 3(1 — “A33 ' (ijlz)
.Lks 8k%W\/W2 0032 03 + L2 Sin2 68.

 

(4.101)

 

192

Here 03 is angle between kg and 2?, dj is the unit vector along the dipole direction for
the corresponding transition. Note here for each electronic level, st fj,k3 = 1.

As 03 increases from 0 to 7r/ 2, fj,ks sharply decreases from f R to 21] f R for ai-
polarized or 0 for 7r-polarized dipole moment. For an analytical solution of equation

(4.100), we approximate fj,k3 with a step function

fRa 68 S 06;
fj,ks = 1 (4.102)
ZfRa (93 > 6’8-

Here Ge = ¥ is the superradiance-emission angle. Due to A > 1, fj,k3 is much
smaller when 03 > 03. Thus, the quasi modes can be classified to end-fire (EF) and
non-end-fire (NEF) modes, corresponding to yielding a 63 less or greater than (96.
Due to a much smaller collective parameter, the population in each N EF mode stays
well below one during the dynamics, as a consequence of mode competition with EF
ones. To verify this, we iteratively solve equation (4.100) assuming small atom loss
fraction. To leading order, we find upon depletion of pump 1 that

Nj,k3(oo) = (1+ V)“(j’k3) — 1. ' (4.103)
Here u( j, ks) = {fl—3'31, and V = 1%1%%% is the initial ratio for populations in pump

R 2

1 and the signal wave. For I‘j = I‘ (j = 1,2,3-n), the occupation numbers in
EF and NEF modes are respectively V and %1n(1 + V). For typical parameters of
A, V = 10 ~ 100, while EF modes are mesoscopically populated, the N EF mode
remains nearly empty, with a occupation significantly less than one. Therefore, su-
perradiance emission occurs only for the EF modes, and decaying into NEF modes
is effectively a spontaneous process. This iterative approach has been validated by

verifying Ziks N j,k 3(00) < N for self-consistence.

193

Following the above arguments, we simplify the equation (4.100) to

d

EiNl = -PeI‘fRN1 N2 + MEFZij - ’YPeNl,
J'

d

3N2 = PePfRN1(N2 + 1),

d

EiNj’O = PePfRNflNjfi +1).

Here we have used I‘j = l", and replaced st with the number of EF mode M E F,
given by

2k%W2
MEF = A2 . (4.104)

 

These equations can be iteratively solved, where for F3 >> 7,

 

1 + V
N- z 1 — . .1
”0 V ( V + exp(Pe1‘st)) (4 O5)

Defining a superradiance enhancement factor

 

3V 1 + V
t = 1 -—-— 1 — 4.106
XS( ) + 4A2 ( V + exp(Pel"3t)) ( )

the equation of motion for N1, N2 can be rewritten in a compact form of

d

d
Et-Ng = FPeleR(N2 +1). (4.108)

At t = 0, X8 = 1, so that there is no superradiance enhancement effect, as the EF
modes are initially vacuum. Then, as the EF modes are gradually populated, x3
increases monochromatically, and eventually saturates to an asymptotical value of

1 + 13f? For negligible superradiance enhancement on atom loss, it then requires

194

 

 

V << A2, so that Xs z 1. For the parameters A = 10,V = 9 as in section 4.3.1,
X3 S 1.07, indicating only weak enhancement effect.

To quantify the atom loss, we define the atom loss fraction, as Eloss (t) = 1 -—
7%; (N 1 +N2+N3). An analytical formula for 51038 is obtained by solving the dynamics

making use of equation (4.13), which for V < A2 is approximately

 

(4.109)

In V+eP€P3t —ln 1+V
Eloss“) z m’)’ (Pet — ( P8) ( )> .

As shown in figure 4.13, this formula agrees well with the exact numerical results in
all dynamical regimes. Upon completion of population transfer, the total loss fraction
is

3V )7 ln(1 + V)
8A2 P3

 

5,0380») = (1+ . (4.110)

This result is independent of P6, and is thus a universal property for different dynami-
cal regimes. The universality origins in that the loss fraction is essentially determined
by the branch ratio of MWA and atom loss, which is independent of pump 2. Phys-
ically, while the instant atom loss rate is proportional to occupation fraction P8 in
the excited state, the transfer time is proportional to the inverse of P6. The final
loss fraction, obtained by integrating loss rate over transfer time, is then independent
of P6, or pump 2. For parameters used in section 4.3.1, it gives eloss(oo) = 0.045,
compared to the numerical result of 0.043, as seen in figure 4.13.

Under short evolving time of t < T, the loss fraction 51033 w fimflgyt,

while the gain factor G'q z 1 + fiPeI‘St. Thus atom loss per decibel gain 17 is then

gloss X37
= = V._ 4.111
77 In G PS i ( )

For V < A2, ,/x3 x 1, so that n z ’7/I‘3, which is proportional to the inverse of

optical depth. A plot of 77 as a function of time for V = 9, 99 and 999, corresponding

195

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.04- (a) -
gloss(t)0_02_ ,
— Numerics
- - - - Formula
0.00 . 1
O 1000 2000 3000
0.04- (b) -
gloss(t)0 02‘ _
— Numerics ,
- - - - Formula
0.00 r I I l
0 200 400 600 800
0.04- 4
gloss“)
0.024 -
— Numerics
- - - - Formula
0.00 . . 4 - .
O 4 8 12
t/rS

Figure 4.13: The evolution of atom loss fraction Gloss, where figures (a), (b) and
(0) show off-resonance, overdamping and Rabi regimes, respectively. In all figures,
solid lines are numerical results, while dashed lines represent the analytical formula
(4.109). Parameters for each dynamical regime are the same with figure 4.2.

196

 

to 742(0) = 0.1, 0.01 and 0.001, is shown in figure 4.14. For t < 0.257', 17 is shown to lie
closely with 'y/I‘s, regardless of V. As evolving, 77 only slightly grows for V = 9 and 99,
but sharply increases to about twice larger than 'y/I‘s for V = 999. Upon population
converting when t = 7', respectively for V = 9, 99 and 999, 17 reaches 0.0194, 0.0205
and 0.040, and then eventually saturates to 0.020, 0.022 and 0.040. Thus to a good
approximation, 1) is given by 'y/I‘s throughout the dynamics for both V = 9 and 99.

The loss fraction can then be fairly estimated by

7
5,0330) = F—s 1n age), (4.112)

valid for V < A2. Notice that at t = 00, Gq as 1 + V, sloss(oo) = (7/F3)1n(1 + V),
thus recovering formula (4.110). For V = 999, n is much greater, because in this case,

V > A2 so that X3 is no longer a small quantity.

Atom loss via SMS model

We now study the laser depletion and spatial effects on atom loss. In figure 4.15,
we plot the dynamics of atom loss fraction eloss for the three dynamical regimes in
figure 4.8 (a)-(c). In figure (a) for the off-resonance regime, 51083 closely follows the
QCM curve through t z T/ 2, the half-depletion time of pump 1. Then, instead of
saturating to 0.451 at t = 7' as in the QCM model, it continues to grow and eventually
approaches an asymptotical value of 0.145. A qualitatively similar behavior is shown
for the Rabi regime in figure (b), but on a much shorter time scale. The agreements
of SMS and QCM in the two regimes before half-depletion time is because the mean
occupation of the excited level 72.3, and thus the loss rate, is the same in the two
models, since the laser depletion is negligible. For t > 7', the majority of atoms
are transferred to the signal mode in the QCM model. Whereas in the SMS model,

due to a reduced effective optical thickness, the population transfer is greatly slowed

197

 

 

 

r l ' I ' l

0'04+ IIIIIIIIIIIIIIIIII1

 

 

 

 

 

O v=9 1
I A v=99
77 0.03- l I v=999 -
Y/rs
' 4

“A“‘AAAAAA(

0.02- l-' II”MPH-i...-nun-.1

 

 

 

I ' l’ r l

0.0 ' 0.5 1 0 1.5

tm '

Figure 4.14: The loss fraction per decibel gain 17 as a function of rescaled time for
V = 9, 99 and 999. Here, N = 3 x 106 is fixed, A = 10, and all other relevant
parameters are given the same as in section 4.3.1.

198

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.15~ (a) ‘
0.10— 1
gloss“) 1 I —SMS ‘
- - - QCM
0.05- _____________________________ :
0.00 v I ' I ' I ' I
0 2 4 6 8 10
0.15. (b) 4
0.104 ‘
a (t) . N—SMS .
loss - - - QCMI
0.05- _______________ -
000 "'. . . , . . . r
0 0 0 4 0.8 1 2 1 6
0.15- (C) .
l
0.10- ‘
a (t) , —s~us .
loss _ , _ QCMI
0.05- ______________________________ ~
0.00 . . . r
0 4 8 12
UP S
Figure 4.15: The atom loss fraction 6108 3(7') as a function of rescaled time, where (a),
(b) and (c) are for the off-resonance, overdamping and Rabi regimes. In all figures,

solid and dashed lines are the numerical results via SMS and QCM, respectively.
Parameters for each regime are the same as given in corresponding figures 4.5.

199

down, leaving roughly half atoms in pump mode at the time of t = 7'. The atom
loss thus continues through the depletion of pump 1, which occurs at a much later
time. In contrast, in the overdamping regime, the loss dynamics in SMS model is
about four times larger than that the QCM result, throughout the dynamics. The
ultimate loss fraction is 17%. The boost of 51033 in SMS dynamics is attributed to a
larger occupation in the excited level, due to a reduced optical thickness and thus a
weaker Zeno suppression effect. As discussed in section 4.6.4, the ultimate atom loss
fraction is inversely proportional to the optical thickness. In all regimes, the spatial
effect reduces the optical thickness by about a half, so that the atom loss is expected
to be doubled. This is compared to an enhancement factor of 3 ~ 4 from the numeric
results shown in figure 4.15. For V = 1, very similar divergent behavior is observed,
as seen in figure 4.16. Again, this reflects that the spatial effects depends little on

the population distribution between pump and the signal wave.

200

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.06 . . 4 . . . . .
(a)
0.04- 4
gloss“)
—QCM
0.02- sms -
0.00 . , - l - I . I
0 2 4 6 8 10
0.06 v I ' I ' I ' T
, (b)
0.04— 4
810880) . QCM .
0.... 1|
0.00 ” . . . . . . . .
0.0 0.4 0.8 12 16
0.06 - . . .
. (c)
0.04- -
gloss“) QCM *
0.02- SMS _
0.00 4 . . . A
0 4 UP 8 12
S

 

Figure 4.16: The same as figure 4.15, but with V = 1.

201

Chapter 5

On—demand generation of
entanglement of atomic qubits via

optical interferometry

In an effort to realize practical quantum information processing (QIP), in this
chapter, we examine the feasibility to create entanglement between distant single-
atom qubits via a common photonic channel in an optical interferometer. As well
understood, when a coherent state is used as the input of a Mach-Zehnder interfer-
ometer, a high-finesse optical cavity is required to overcome sensitivity to spontaneous
emission. However, we find that a number-squeezed light field in a twin-Fock (TF)
state can in principle create useful entanglement without cavity-enhancement. Both
approaches require single photon counting detectors, and best results are obtained
by combining cavity-feedback with twin-Fock inputs. Such an approach may allow
a fidelity of .99 using a two-photon input and currently available mirror and de-
tector technology. If using maximally entangled NOON state (fiflN, 0) + IO, N )))
and nonlinear beamsplitter, similar performance can be achieved without the need to

count single photons. The problem is, however, the NOON state is subject to fast

202

 

 

decoherence. To overcome this difficulty, we propose to use less-extreme schr6dinger
state, which can provide similar performance yet is more robust against decoherence.
The presently proposed approaches are scalable, and can be conveniently extended to
generate multi-site entanglement and entanglement swapping, both of which are ne-
cessities in quantum networks. The major results of this chapter have been published
in “On-demand generation of entanglement of atomic qubits via optical interferome-
try”, by Y. P. Huang and M. G. Moore in Phys. Rev. A. 77, 032349 (2008).

This chapter is organized as follows. In section 5.1, we briefly review the exist-
ing implementations of quantum information processing employing isolated atomic
qubits, and then give an overview on our work. In Sec. 5.2, we present a basic model
of the interferometrical generation of entanglement between two atomic qubits. In
section 5.3.1, we study the MZ—interferometrical approach using the TF input light
field. Then in Sec.5.3.2, we investigate an alternative approach employing NOON
states and nonlinear beamsplitters. In Sec. 5.4, as examples, we briefly show how
the present scheme can be applied to realize deterministic teleportation, multi-site
entanglement, and entanglement swapping. This is followed by a short discussion and

conclusion in Sec. 5.5

5. 1 Introduction

Future quantum information processing, if can ever reach practical level, will
rely on deterministic computational gates and communication protocols that operate
at high fidelities [136, 149, 175]. As quantum entanglement lies in the heart of QIP,
this necessarily requires realtime generation of entanglement amongst arbitrary qubits
performed at near-unit success probability and fidelity. For atom-type qubits, this
entanglement can be generated either via a photonic channel, utilizing entangled

photon-pairs [175, 181] or cavity-decay photons [182, 183, 184, 185], or an atomic

203

 

 

channel as in recent trapped-ion experiments [186, 187]. For high-speed quantum
computation and / or long-distance communication, a photonic quantum channel is
clearly ideal, as photons are robust carriers of quantum information that travel at the
speed of light. Since isolated trapped-atomic qubits have long coherence times and are
easily manipulated with electromagnetic fields, it is of general interest to consider the
problem of creating entanglement between two isolated atomic qubits via their mutual
interaction with a single photonic channel. The primary obstacle to such a protocol
lies in the problem of eliminating spontaneous emission while obtaining a sufficiently
strong atom-photon interaction. Recent attempts to overcome this difficulty have
primarily relied on the use of collective-state qubits in atomic ensembles to enhance
the dipole moment of the qubit [150, 259, 260]. This enhancement effect has allowed
Duan, Cirac, Zoller and Polzik to implement a quantum teleportation scheme between
two atomic samples, where a coherent beam is passed successively through and the
entanglement is generated by measuring its final Faraday-rotation angle [150]. Very
recently, a probabilistic scheme to entangle two distant quantum dots using cavity
enhancement has been proposed using bright coherent light via homodyne detection
and post-selection [261].

In this chapter, we investigate an approach in which single-atom qubits are de-
terministically entangled by use of an optical interferometer, thus avoiding collisional
decoherence mechanisms inherent in atomic ensembles. It is well-known that the back-
action of a single atom onto a focused laser pulse is very weak, so that generating
useful atom-photon entanglement in this manner will generally fail due to spontaneous
emission [262]. Our goal, however, is to overcome this difficulty by using the extreme
sensitivity of sub-shot-noise interferometers [157, 158, 119, 159, 160, 24, 25, 26, 27] to
detect the weak phase imprinted on the forward scattered light in the regime where
spontaneous emission is negligible. In addition, we also consider the more generic ap-

proach of using high-finesse optical resonators [155, 156] to enhance the atom-photon

204

 

 

interaction.

Our interferometry apparatus follows the Faraday-rotation scheme of Duan, Cirac,
Zoller, and Polzik [150], with the collective atomic ensembles replaced by single
trapped atoms, and with the coherent light replaced by a highly non-classical many-
photon state. As we will show, an interferometer employing coherent light input
is limited to shot-noise sensitivity, and will thus not work when the ensembles are
replaced by single atoms without the introduction of extremely high-finesse optical
resonators. If the coherent state input is replaced with a twin-Fock (TF) input state,
however, we find that a cavity is in principle no longer required. Cavity feedback
may still provide additional improvement in performance. For example, f = .99 can
be achieved if we use the TF state with 4 x 104 photons and no cavity, or only two
photons and cavities with M = 2 x 104. The later requires a single photon-on—demand
[263] injected into each interferometer input, with an accurate measurement of the
two—photon output state, which appears within the realm of experimental feasibility.
Both MZ-interferometer—based approaches require detectors with single-photon res-
olution [264]. This requirement, however, can be overcome by employing a non-MZ
interferometer based on NOON states and nonlinear beamsplitters. Such an interfer-
ometer yields a sensitivity close to the TF state in detecting phase imbalance, and
thus can achieve similar performance without counting single photons. While the TF
and NOON states have recently been shown as unable to measure any phase below
shot-noise in a single measurement due to large quantum fluctuations [244, 250, 119],
our present work shows that single-measurements with these states can still be highly
useful as ‘quantum switches’ with Heisenberg-limited sensitivity.

Our proposed interferometry approach to entangle atomic-qubits can be per-
formed on-demand and is scalable. We envision generalizing such a device to a com-
plete set of quantum information processing protocols whereby stationary single-atom

qubits are held in isolated traps, with arbitrary single-atom and multi-atom operations

205

 

 

achieved via sequences of light pulses guided amongst the atoms and into detectors
by fast optical switching. The goal of this chapter is to perform a theoretical analysis
of interferometrical generation of entanglement between two arbitrary qubits, and to

determine the fundamental limitations imposed by quantum mechanics.

5.2 The model

In our scheme, a single pulse of light is passed through an optical interferome-
ter, with the different ‘arms’ of the interferometer corresponding to different photon
polarization states. The beam passes through two atomic qubits, i.e. trapped ions,
neutral atoms and / or quantum dots, such that each polarization state interacts with
a different internal atomic state. This can be achieved using an ‘X’-type scheme, as
described in [150], in which the Zeeman sublevels of an F = 1 / 2 ground state form the
qubit, or in a A-type level scheme, with the m = :l:1 states of an F = 1 ground state
forming the qubits. In both cases, the ‘arms’ of the interferometer would correspond
to orthogonal circular polarization states. The interferometer output is determined
by a state—dependent phase-shift acquired via the atom-photon interaction. This re-
quires a large detuning from the atomic resonance, as there is no phase acquired on
resonance. Measurement of a phase imbalance at the interferometer output cannot
determine which qubit contributed the phase-shift, resulting in entanglement between
them.

We consider atomic qubits based on two degenerate hyperfine states, arbitrar-
ily labeled as [0) and [1). For a general consideration, our goal is to entangle two

uncorrelated qubits, labeled :1: and y, which are initially in states of [(033), Iwy), where

Ida.) = X600)“ + x‘l’llm (5-1)
and )u 6 {33,31}. The qubits are placed inside an optical interferometer with the setup

206

 

 

depicted in Fig. 5.1, where the states [0)a; and l0)y interact with photons in the
upper arm of the interferometer, while |1)g; and [1);] interact with the lower. Such

interaction is represented by the qubit-photon interaction propagator,

0,, = exp[—i0(&g&OéLOéflo + flasks/“)1, (5.2)
where 6pm is the annihilation operator for an atom at location 14 E {53, y} in internal
state m E {0, 1}. This interaction operator is valid in the far-off-resonance regime,
where the electronically excited state can be adiabatically eliminated. The interaction
is governed by the phase—shift

14800127

6 = 7?, (5.3)

where 7' is the atom-photon interaction time, A is the detuning between the laser and
atomic resonance frequencies, dis the electric dipole moment and E (w) = hw / (280 V)
is the ‘electric field per photon’ for laser frequency w and mode-volume V. Introduc-
ing the spontaneous emission rate F = d2w3/(37r50hc3), taking the photon mode as
having length L and width W (at the location of the atom), and taking the interaction
time as 7' = L / c, we arrive at the single-atom phase-shift
2
9 = 3:37; [$] 2%, (5.4)
where A is the laser wavelength. This is the phase-shift acquired by an off-resonant
photon forward-scattered by a single atom, and is independent of the pulse length.
The interferometer output is then determined by the phase-shift acquired via
the atom-photon interaction. Introducing the qubit-pair basis |2'j) E [2);]; ® Ij)y with
2', j = 0,1, the states [01) and |10) both correspond to a balanced interferometer with

zero net phase-shift, and thus constitute a ‘balanced’ qubit-pair subspace. In contrast,

the states [00) and [11) have equal and opposite non-zero phase-shifts, and thus

207

 

 

Upper Channel

“0 “0
Input BS BS Output
owerChanne
(3)

Upper Channel
00 00
Input NBS Output

  

ower Channe

(b)

Figure 5.1: Schematic setup of entanglement generation with optical interferome-
ters. Figure (a) shows the setup with the MZ interferometer which consists of two
linear 50/ 50 beamsplitters (BS). Figure (b) shows the setup with the NOON-state
interferometer consisted of only one nonlinear beamsplitter (NBS).

constitute an ‘imbalanced’ subspace. Measuring the photon number distribution at
the interferometer output distinguishes between zero and nonzero magnitudes of the
phase-shifts, and thus collapses the qubits onto the balanced or imbalanced subspaces,

based on which entanglement between the two is established.

5.3 Implementations

In this section, we present two feasible implementations of the prescribed pro-

tocol. The first employs MZ—interferometry with highly-number squeezed TF states

208

 

 

 

as input. The second uses a N OON-state interferometer configured on nonclassical

beamsplitters.

5.3.1 MZ interferometry with Twin-Fock input states

The basic set-up for entanglement generation using MZ interferometer is shown
in Fig.5.] (a). The MZ interferometer consists of two 50/ 50 linear beamsplitters. The
input light field is bifurcated at the first beamsplitter, guided to interact sequently
with the qubits, and then recombined at the second beamsplitter. Passage of photons

through the MZ interferometer can be described by the propagator,
U = UBSUyUzUBSa (5-5)
where U B S is the 50/ 50 beamsplitter propagator,
UBS = exp[—z'(a[,al + dam/4]. (5.6)

Without specifying the input light field, the initial states of the system can be written

in a general form

14.) = «1445.400 4 (44> e 14.». (5.7)

where [0) is electromagnetic vacuum state and @(65, 0]) defines the light field. The

state of the system at the interferometer output is then given by

liq) = UN.) (58)

= 004301. 0410010 4 04> a 14.).

209

 

 

Introducing dual-qubit spin operator

1 . . .. .
0'z = 2 Z (CLOCMO — Ideal), (5-9)
u=xm
we find that
051501 = iei6[sin(602)d[) + cos(602)ci[], (5.10)
0&[01 = iew[cos(602)d[) — sin(602)€z[]. (5.11)

The final state can now be rewritten as

1’17): 2: XiXXqu’az'j» @123) (512)
iJ=01

where |<I>(6.,-j» is the output light field in the presence of qubits-dependent interfer-
ometer phase 6,]- = 0 x (1 — i — j). It is now evident that the interferometer output
is determined by the joint states of the qubits. The states |01) and |10) result in
zero phase-shifts with 601 = 610 = 0, while [00) and |11) result in equal and opposite
phases with 000 = —611 = 6. If the interferometer is incapable of distinguishing
positive and negative phases, a measurement of the output light field will therefore
collapse the qubits onto either balanced or imbalanced subspaces (resulting in zero
or nonzero phase-shifts), and in this way generate entanglement between them.

In the framework of optical MZ interferometry, a first thought might be to use
coherent input states, as they are most attainable in practice. As explained in details
in appendix 5.6.1, however, this method is prohibited by fundamental physics laws,
originated in the well-known fact that the atom-photon interaction in free space is
dominated by the spontaneous emission. To overcome this difficulty would require
to put the atomic qubits in optical resonators. In appendix 5.6.1, we found that a

fidelity of .99 can be achieved using ring cavities which. cycle photons for 6.6 x 105

210

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 I fiE-T f -.
\\ ’ .
r‘ i ‘_ .‘
0 8 r 1 - ,‘ \’ Tl !_
' | ‘_ ,‘ \ ----- x ’
l \ ' f
‘ “ ~I “ — 8 -,
54 0.6 t 1 " [I \ .’ -
g. \ ‘ I \. I
- ‘ I ‘ ! J
m 0.4 b “ "‘ I ‘ .I
\ ‘\ I ‘— I
‘ I
0 2 1' ‘ \ " “\ I ‘
‘ " I Is -
\ , I I‘ \ ’ ~ \ I
~ \ ' -
0 12):. . ~_¢’ 2?-)c” ‘~
0 1 2 3N0 4 5 6

Figure 5.2: (color online) An example of intrinsic error due to interferometer sensi-
tivity. Errors for MZ interferometer with coherent 8 (solid), TF input 7) (dashed),
and N OON-state interferometer K. (dashed-dotted) are plot as functions of N 0 (with

N = 103), respectively. Note while 5 is dependent on N192, both 77 and 14 are depen-
dent on N 0.

times, with about 4 photons needing to be measured accurately at one output.

To achieve a higher fidelity, and / or to eliminate the need for a high-finesse res-
onator, we now consider using sub-shot-noise interferometers to overcome the spon-
taneous emission to phase sensitivity. In this section, we investigate the fundamental
limits when a twin-Fock (TF) photon input state is used to increase the phase sensi-
tivity of the MZ interferometer. In this setup, the photon number-differenCe between
the outputs must be measured. A result of zero number-difference constitutes a null
result, projecting the qubits onto balanced subspace. Otherwise, the qubits are pro-
jected to the imbalanced subspace.

To formulate the process, the input state is given in the form of |N, N), with the
dual-Fock basis defined as

lie, 1) = (ag)k(a[)l|0)/ k! u. (5.13)

211

Following equation (5.12), the output state is given by

 

N
I‘I’f) = ZXfXinJ') ‘3 Z €m(92'j)|N+ m,N - m). (5-14)
z'j m=—N
where
min{N,N—m}
€m(9z’j) = Z (—1)m+‘ m +1 I x

l=max{0,—m} N N

\/(N + m)!(N — m)!

 

N! (sin Oij)m+2l(cos 19,-j)2N_m'—2l.
The desired two-qubit entangled state is then created by measuring the photon num-
ber difference between the upper and lower outputs. It is seen from Eq. (5.14) that

the probability of detecting a difference of 2m is given by

P(2m) = Mm), + (1 — A)g,2,,(0), (5.15)

I2 [2. The probability to detect zero photon number

where again A = [xglef + lx‘fx’é

difference (or a null result) is thus
P(O) = A(1- 77) + 77, (5.16)

where 77 = {8 (9) is the probability of a false null result. On detecting the null result,

the qubit state will collapse onto

1
[‘23) = \/A(1 _ 7]) + 'r] X (5.17)

[X8X’fl01)+ xfxg|10>+ Jflxfix‘élOO) + xfxlflllfl] ,

 

 

212

 

 

with the corresponding fidelity

f _ A
W” _ A+(1—A)77'

 

(5.18)

The remaining time, a photon number difference m 75 0 is detected, with the qubits

collapsed to

mm = X8x3|00>+(-1)mx‘fx’1’|11>- (5.19)

Here, similar to the coherent state, the exact photon number difference must be
measured in order to successfully disentangle the qubits. The overall fidelity in this

entanglement generation is then

favg = l- (1 —A)€

IV
H
l
m

(5.20)

The TF input yields a intrinsic error due to interferometer sensitivity given by

2
n. In contrast, for the coherent input, the intrinsic error is e = e—N‘9

, as derived in
appendix 5.6.1. A comparison plot of 17 and 5 is shown in Fig. 5.2, where it is seen that
17 decreases with N much faster than 6. In fact, for N 0 < 1, 17 z e—N292, which is
characteristic of a Heisenberg-limited phase sensitivity. This means that significantly
fewer photons are required to obtain equal fidelity, with a corresponding reduction in
spontaneous emission. In Fig. 5.2, we see that the false-null probability 77 is exactly
zero for a periodic set of values of N 6. The first such zero occurs at N 6 = 1.196 E 231.
Thus if one can precisely control N 0, it is possible to achieve teleportation without

intrinsic error due to false-null results. In this case, the success of teleportation is

governed only by spontaneous emission probability Psp = 2NOF/A = 23:1I‘ / A. The

213

 

 

 

condition N6 = :51, together with (5.4), means that F/A = (8131/N) (W/A)2, so that

16:1:2 W 2

For the case of a tightly focussed beam, we can take W/A z 3 this gives Psp = 206/N.
The theoretical limit to fidelity therefore scales as ~ 1 — 200/N, thus a fidelity of
f = .99 would require N = 2 x 104 (or a total of 4 x 104 photons), while a fidelity
of f = .999 could be achieved with N = 2 x 105. An extremely high fidelity of
f = .999999 would therefore require N = 2 x 108. The addition of a Q—switched cavity
with M cycles replaces N with the effective photon number MN, resulting in the
spontaneous emission probability results in P3? = 206/ (NM), which for M = 2 x 104,
would reduce the photon numbers to N = 1 for f = .99, N = 10 for f = .999, and
N = 104 for f = .999999.

The exactly elimination of false-null-induced reduction in fidelity requires the
precise control of single-particle phase shift 6, as well as the particle number N.
Imprecise controls of either will lead to 17 79 0, and thus a reduction in overall fidelity.
To estimate this effect, we let N 6’ = $1 + 6, with 6 resulted from the displacement of

0 and/or N. Expanding 77(131 + 6) near 17(xl) = 0 gives
17(231 + 6) z 1.3 62, (5.22)

For a fidelity of f = 1 — 200/N (with 17 = 200/N), it requires 6 < 12.4/\/N. This
then requires 61 - 9.5 6['5 < 0 < 01 + 9.5 61'5, where 61 = zl/N is the desired
per—atom phase shift. This allows a relatively flexible control of 9.

Lastly, we note that for the TF input and the present parameter choice of N 6 =
1:1, a single photon loss will immediately reduce the fidelity, with a worst-case result
of f = 0.73 and thus disrupt the on-demand entanglement generation scheme. This is

because a lost photon will lead to a rapid degradation of the interferometer sensitivity.

214

 

 

To see this, for the TF input state, one photon lost from the q-th path during qubit-

to-qubit prOpagation will result in the final state

 

1. . - . -
I‘P'f) = V NUBSUyaqUCEUBSI‘I’ila (523)
Use the identity
. .. . . - «I .
UBSUyaqUJULS = ezaqucy<1(aq — 151_q)/\/2, (5.24)
The identity
. . . . - .7 .
UBSUyaqUJU],S = 6961/104qu — 1&1_q)/\/2, (5.25)
leads to
1 1 105], (Egg . .. «
[‘17) = W 6 q (aq-wl—q) Ul‘I’z'), (5-26)
1 .. N
= VfifongqlijW Z §m(91'j) X
11 m=-N

[(—i)q\/N+m|N+m — 1,N -m) +

(—i)1_q\/N — m|N + m, N — m — 1)],

with the projector R, defined as

A

. .T -
Pq = 62.0ququ

 

ewlq>y<ql + 11 — q>y<1— ql- (5.27)

A lost photon will therefore result in odd number differences of photons measured

in the two output ports. Since without photon loss, a TF state will always result in

215

 

even number differences, it is in this way possible to determine the loss of a single
photon (while without knowing which path it is lost from). Seemingly, this makes it
possible to detect the phase imbalance if we accordingly redefine a null result as the

measured photon number difference being 1. The false-null rate as given by

17’0” = W) + 151(9)\/1 + 1/NI2. (5.28)

is, however, no long a small quantity. A comparison of 17‘033 and 17 is plotted in
Fig. 5.3, where it is shown 171033 behaves as the envelope of 17 without the zero-value
points. In particular, with the present choice of N 0 = 131, it is found ”loss z 0.27, in
contrast to the corresponding rate 17 = 0 without the loss. Depending on the qubits’
states, a photon loss will thus immediately degrade the fidelity to f 2 0.73.

On the other hand, 171033 is yet much smaller than the corresponding false-null
rate 8 for the coherent state. Hence, if we presume one photon will be lost and set
the value of N 0 accordingly, we may still generate entanglement without the cavity

enhancement. In fact, as shown in Fig. 5.3, a least-square fit finds

7033 0.33
= _. 5'29
77 N 0 ( )
Letting nloss = Psp and using equation (5.4) gives

2.6

Psp = W1 (5-30)

for W//\ = 3. The limit to fidelity thus scales as 1 — 2.6/N1/3, and a fidelity of
f = .99 will require N = 1.8 x 107, compared to N = 2 x 104 without photon loss.
Entanglement can in this sense still be generated without the need of cavities, while

the single-photon loss can be compensated by using more photons. Lastly, we note

that f ~ 1 — 2.6/N 1/ 3 is also the lower limit on the fidelity achievable when the

216

 

 

 

 

 

 

 

 

 

 

 

 

0.2 u!
1"! _ _ _ 1,|loss
0.15- ”1‘, —n 1
g ‘1, ----- 0.33/N0
2: 0.1 ’
£-
0.05-
00

 

loss

Figure 5.3: Comparison of 17 and 17. Note that both depend only on the product

N6.

phase shift can not be tuned such that 17(Nl9) = 0. This is simply because 171033 is
the envelop of 17, and for any N and 6, 17 _<_ 1710‘”. -

To sum up, in this section we have shown that for a MZ interferometer with the
TF input, atom-atom entanglement can be generated with much higher fidelity, and
the need for high-finesse optical resonators can in principle be eliminated. Particu-
larly, we found a fidelity of .99 is quantum-mechanically allowed with 20, 000 photons,
or more intriguingly with only 2 photons, provided ring cavities which cycle photons
2 x 104 times are additionally incorporated. The 2-photon TF state could be gen-
erated with a pair of single-photon-on-demand sources (one for each input) and a
precise photon detector to measure the two-photon output state, technologies that
are rapidly advancing at present. Finally, we have shown the present scheme is rela-
tively insensitive to deviations in the per-atom phase shift, yet is highly sensitive to
loss of a single photon. This is somewhat mitigated by the fact that for the 2-photon
state, the loss of a photon could be readily detected, so that success is heralded by

the detection of both photons.

217

 

 

5.3.2 NOON-state interferometry with nonlinear beamsplit-

ters

In above sections, we have discussed generating entanglement between atomic
qubits using an optical MZ interferometer with TF input states and linear beamsplit-
ters. While both are shown to be able to achieve a close-to-unit fidelity in the presence
of intrinsic quantum errors, they require precise measurement of output light field at
the single-photon level. In this section, we show this requirement can be overcome by
using a non-MZ interferometer based on NOON states and nonlinear beamsplitters
[242].

A NOON state is a Shriidinger cat state that corresponds to an equally-weighted
superposition of all-upper—channel and all—lower—channel states [24, 265, 266, 267, 180]

1

.11
fi(|N,0) + |0,N)), (5.31)

|NOON)=

where <73 is the relative phase which we take for zero for simplicity. The nonlinear
beamsplitter can either be a four-wave mixer [26] or a quantum circuit constructed
from CNOT gates [136]. The action of such beamsplitters is presented by a projecting

operator 0' N B S of the general form

0.4135 = -\/1-§Z(e’¢ls.0>+l0.s>)<0,sl+e"'°(e"’ls.0)—|0,s>)<0,sl. (5.32)

where for simplicity we let the relative phases (,0 = (p = 0. Note for the four-wave
mixer, this projector is valid only for even-number s. A realization of this nonlinear
beamsplitter in nonlinear Kerr medium is presented in the appendix. Using this
nonlinear beamsplitter, a NOON state can be generated from a single Fock state
[N , 0), i.e., by injecting N photons in its upper input channel.

The set-up of the NOON-state interferometer differs from a MZ interferometer in

218

 

 

 

that now the first beamsplitter is dropped (or more precisely, it is formally replaced
by the assumption of a NOON input state) while the second one is replaced by the
nonlinear beamsplitter, as shown in Fig.5.1 (b). For measurement, a photo detector
is placed in the lower (or equivalently, the upper) output port to detect the presence
of outcoming photons, while without counting them. Because the output light field
consists of [N , 0) and IO, N) states, corresponding to all photons coming out from
upper or lower port, a photon detector with a resolution of < N / 2 would be sufficient
to distinguish them. This exhibits an essentially improvement from the previous MZ
interferometry schemes, where the detector resolution must be less than one. A null
result, meaning no photon is detected at the lower channel, will collapse the qubits

onto

X0X1|01) + xfxoyl10>+ln>. (5.33)

with In) the intrinsic state error due to a false-null result. In contrast, if the detector

has detected a photon, the qubits will collapse into the imbalanced subspace

xfixglom + x1 x1 31.111) (534)

Here, we emphasize that this state is independent on the exact number difference 11
between the upper and lower outputs. This is essentially different from the corre-
sponding ones in (5.71) with the coherent state and in (5.19) with the TF state, both
of which are dependent on 12.

To derive these results, we follow the previous approach and find the final state

of system as

lwf> = UNBSUyU. [£0540 + IO. N» 2 14> 3 14.)]

= Z xix] ylij) ®[—cosN6,;]-[N,0) isinNOi j,]0 N)],
1,1: —01

219

 

 

 

where we have dropped an irrelevant global-phase term in the last step. The proba-

bility of a null result is thus
Pnull = A(1 — K.) + I‘E, (5.35)

with K. = cos2 N6 is the intrinsic error rate. On detecting this null result, the qubits

will collapse onto

1

 

 

 

11:: $y01+$y10+n$y00+$y11
| B) \/A(1—Is)+n [xoxll > X1X0| > \/—(X0X0| > xlxll D]
(5.36)
where re = cos2 N6 is the false null rate. The corresponding fidelity is then
A
fnull — 11+ (1 _ A)”. (5.37)

The remaining time, a not-null result is detected, projecting the qubits onto the

imbalanced subspace,

1
1_

 

mm = (x8x3l00> + xfxi’lm) . (5.38)

>

The overall fidelity averaged over the null and not-null results is given by

favg = 1 — (1 — 1))/€- (5.39)

The fidelity in this entanglement generation is thus similar to the MZ interferom-
etry cases but with the intrinsic error rate given by n. A comparison of e, 17, K. is shown
in Fig. 5.2, where it is shown that for N6 < 1, K. is close to 17, exhibiting a Heisenberg-
limited phase sensitivity. Furthermore, 11: is exactly zero at N6 = 1r / 2, compared to
at 1.196 for 17. This means the N OON-state interferometer can achieve similar per-

formances with the TF state. The fidelity after taking into account the possibility of

220

 

 

spontaneous emission therefore scales as f z 1 — 350/N. A fidelity of f = .99 and
f = .999 will then require N = 3.5 x 104 and N = 3.5 x 105 photons, respectively.
Since there is no requirement on exactly counting output photons, N can in principle
be made large, allowing an arbitrary close-to—unit fidelity, at least quantum mechani-
cally. Finally, similar to TF state, in order to suppress false-null rate, for a fidelity of
z 1 — 350/N, 6 must be tuned within an interval of (61 — 8.16]'5, 61 + 8.16['5) with
61 = 1r / 2N .

The present entanglement generation using NOON states will be completely dis-
rupted by a single-photon loss. This is due to the fact that a randomly lost photon
will immediately collapse the NOON state to a statistical mixture of all-upper-channel

and all-lower—channel states, whose reduced density is given by
1
p10“ = §(|N — 1,0)(N — 1,0| + W —1,0)(N —1,0|). (5.40)

This mixture state is apparently incapable of detecting phase imbalances. This prob-
lem, however, might be overcome by using a class of less-extreme cat-like states
[268, 249]. Such states correspond to a symmetric superposition of two well-separated
wavepackets in number-difference space. In the case of small photon losses, instead
of being completely destroyed, they will decay into a mixture of smaller-sized cat-like
states, which are still suitable for the purpose of detecting phase imbalance. Hence,
asides from a reduction in fidelity due to phase randomization, faithful entanglement
might be generated despite of photon loss. This study is presented in appendix 5.6.3.

Finally, we note that the present N OON-state interferometer relies on highly
nonclassical light source with definite photon number. This requirement is, however,
not a necessity. For example, our scheme can be directly extended to use more

’classical’ cat input states that correspond to the superposition of coherent states,

|a>0 ® |0>1 + |0>0 ®|a>1- (5.41)

221

 

 

Such a state can be rewritten as,

Zf(m)(lm. 0) + I0. m». (5.42)
m
where f (m) is the coefficient of the coherent state and can be approximated by
f(m) = i—e—(m—NPMN. I (5.43)
N

where N is the normalization factor. A state yielding this distribution but containing
only even ms has been proposed to generate probabilistically with a success proba-
bility of half, where a single coherent light and a four-wave mixer are employed [26].

With this input, the final state of the system becomes

[\Ilf) = Z xfxglij) ® 2 f(m)[cos m6ijlm, 0) — isin m6z-j]0, m)]. (5.44)

1, j=0,1 m
By choosing N6 = 7r/ 2, similar to the single NOON state, upon detecting photons
from the lower port, the qubit will collapse to imbalanced subspace (5.38). Otherwise,
if no photon is detected from the lower port, the qubits will collapse to the state (5.36)

but with the false null rate given by

11’ = ~2}\7 :e_(m_N)2/2N(1+ cos m6) (5.45)

_ %(1_ 6—62N/2)
Since N6 = 1r / 2, we have 11' z 7r2/16N, which is negligible compare to the fidelity
reduction (z 350/N) due to the probability of spontaneous emission.

To conclude, in this section, we have shown that a NOON-state interferometer
can achieve similar performance with the TF state, yet without the requirement

for precisely measuring the output light field. While this scheme is not tolerant of a

222

 

 

 

single photon loss, this problem might be overcome by using a class of less extreme cat
states. Also, besides using a single NOON input state with definite photon number,
we showed the present scheme can also use a. class of states with indefinite photons

that correspond to the superposition of NOON states.

5.4 Examples of applications

The present interferometrical method of generating entanglement can serve as
a basic protocol in the quantum information processing, based on which quantum
computation and communication can be realized with the aid of local qubit operations.
As an example, here we first show how it can be used to teleport an arbitrary quantum
state from one qubit to another. We assume the :l:-qubit is the source qubit carrying
an unknown teleporting quantum state, and y—qubit is target qubit which the state

is transported to. The x-qubit is initially in the state

1102-) = X81011: + x‘lex. (5.46)

while the y—qubit is initially prepared as

I4.» = 91300.. +105). (547)

We first collapse the two qubits into entangled qubit-pair interferometrically using
our method. Once the qubit-pair is generated, completing the teleportation requires
that the qubits be disentangled. This can be accomplished in the following manner.
Conditional upon a null result, a 1r-pulse is applied to the source qubit, flipping
10):: H [1):r- When using a MZ interferometer with the coherent or TF state, in the

case of an odd measured 12, an additional relative 1r phase must be applied to the

223

 

 

state ”)2: (or [1)y). After these steps, the qubits’ state becomes
xfiIOO) + x‘flll). (5.48)

A 1r / 2-pulse is then applied to the source (or the target) qubit, transforming the state
into

[X8l00> — 1x300 — ix‘f |01> + x‘fl11>l/\/§- (5.49)

This is followed by a state measurement of :l:-qubit. If it is measured |0)x, the y-qubit

will collapse to

x8|0>y - ix’f |1>y, (5.50)

after which a 1r / 2 phase is imprinted onto |1)y. Otherwise, it is measured in |1)$, and
3 1r/ 2 phase is imprinted onto [0)y. After these conditional operations, the y-qubit

will end up in the desired state

X3|0>y + XTIDy. (5-51)

which accomplishes the teleportation.

Besides the state teleportation between two qubits, our scheme can be easily
generalized to generate many-qubit entanglement [269] as well as realize entanglement
swapping [270, 151]. For example, the three-particle Greenberger—Horne—Zeilinger
(GHZ) state,

(looms. + [111>xyz)/\/§a (5.52)

can be created by first preparing each qubit in the state
1
Wu) = $001+ |1>i). (5-53)

with i = 22,37, 2. Then the two—qubit protocol is used to collapse x and 37 into the

224

 

state

(100)333/ + Inns/(5 3 (I0). + |1>z)/>/\/§- (5.54)

If the same two-qubit procedure is applied to B and C, the GHZ state is obtained.
This simple scheme can be extended in a straightforward manner to producing an
N -particle Shriidinger cat state.

To realize entanglement swapping, we take an initially entangled qubit-pair,

 

 

1
2 2 (€00|00>xy + C11|11)acy), (5-55)
\/|000| + IC11|
and an uncorrelated third qubit
14 >— i<lo> +11>> (556)
Z — fl 2 Z .

and apply our protocol to qubits y and z to create a GHZ-like state. Then, by
disentangling y in the same manner as described for the source qubit in teleportation,

we arrive at the desired swapped state

1
\/|000|2 + |011|2

 

 

(6001001102 + 011110122). (5-57)

5.5 Conclusion

In conclusion, we have used the formalism of the optical interferometer to treat
the problem of creating entanglement among two single-atom qubits via a common
photonic channel. We have compared the results from a MZ-interferometer with a co—
herent input state and high-finesse cavity enhancement, with TF input and those from

a non-MZ interferometer based on the NOON state and nonlinear beamsplitter. Our

225

 

 

results suggest that high-fidelity entanglement can in principle be generated via any
of the interferometrical approaches. Experimentally feasible schemes under current
techniques are found by combining Heisenberg—limited interferometer with photon
resonators. In particular, we find that a two-photon input state has a fundamental
upper-limit to fidelity of .99, and provides the advantage that failure due to pho-
ton losses could be readily detected. Our interferometrical approaches of generating
entanglement is operated with close-to-unity success probabilities and are scalable,
and thus can serve as a universal protocol in quantum information processing, based
on which quantum computation and communication can be realized with the aid of

single-qubit operations.

5.6 Further discussions

5.6.1 MZ-interferometry with coherent input states

In this section, we examine the MZ-interferometric generation of entanglement
employing coherent input states. We consider the upper channel (with operator 65)
initially in a coherent state, while the lower channel a] is in the vacuum state. A
detector is used to count the photons coming from the upper output channel, while

output in the lower channel is unmeasured. A null result, meaning zero photons

detected, results in the qubits collapsing onto the balanced subspace,
x8x’fl01) + xfx3|10> + Is). (558)

where [5) is the intrinsic state error due to the possibility of a false null result. This
error, which adds imbalanced states to the desired balanced subspace, sets the upper

limit of the obtainable teleportation fidelity. If n aé 0 photons are detected, the qubits

226

 

will collapse onto imbalanced subspace

XSXE’IOO) + X‘fX31’(-1)"|11>. (559)

without intrinsic error. We note that the possibility of a dark count will introduce
an analogous error, but this error rate is governed by technical aspects of the photo—
detector, and is presumably not an intrinsic quantum error.

To derive these results for coherent input state, the initial state of the complete

system is given by equation (5.7), with

_ .1 .-
<1>,-(0[,,a[) =e “00+“ “0 (5.60)

Following equation (5.12), the state of the system at the interferometer output is

obtained as
[‘1' )= Z x‘fxylij) ® [d'sin6- ) ® [@0036 -) (5.61)
f 1 J 2] 0 Z] 1
i,j=0,l
where 6: = —iaei6 and the states [(1)0’1 indicate optical coherent states for the upper

and lower interferometer outputs, respectively. Expanding the upper channel onto

photon number-eigenstates and making the small-angle approximation gives

[‘I’fl = 2 [n)0 Q21501 ‘3 [(157023], (5-62)

n=0

where [11)0 indicates a state with n photons in the upper output, and

I40... = x3x1101>+xfx3110>+le> (5.63)

|¢n7é0>xy = fn 041ng + (—1)"xfx‘1’l11>l. (5.64)

227

 

 

where
14> = e—la'292/21x3x3100 + X16111» (5.65)
and
fn = 373640292”. (5.66)

The photon number in the upper channel is then measured with single-photon
resolution, while the output from the lower channel is left unmeasured. From equation
(5.62), the probability of detecting n photons P(n) is given by

N92)”

2
P(n) = 40,0 + (1 — A)e—N9 T (5.67)

where A = [xgx31/[2 + [xfxglz is the weight of balanced-space states in the initial

qubits’ state. The probability of detecting zero photons is thus

P(O) = A(1 — e) + e, (5.68)

2
where e = e—N‘9 indicates the probability of a false null result. On detecting the

null result, the qubits’ state will collapse onto

 

 

I‘I’Bl = [x‘6x31’|01>+ x‘fxg W) + #001766 I00) + x’fxlflllfll -

(5.69)

1
\/A(1—e)+e

The fidelity upon this null result fnul» which measures the weight of balanced states

in [\IIB), is thus

f _ A
"“1 _ A+(1—A)e’

 

(5.70)

which is non-unity due to the non-zero probability of a false null result. The condition

for faithful teleportation is then 5 << 1, or N 62 >> 1, characteristic of a standard-

228

 

 

 

quantum-limit interferometer.
The remaining time, a photon—number n 79 0 is detected, with the qubit-state

collapsing onto the imbalanced space with unit fidelity,

1
l—A

 

|‘1’U> = (x3x3l00>+(-1)nxfx’1’|11>)- (5.71)

The (—1)n term comes from the phase difference between number-states for the co-
herent states la) and | — oz), i.e. while measuring photon number can not distinguish
the states [00) and [11), it can introduce relative phase between them. If the photon
number is definitely non-zero, yet not measured exactly, then tracing over the photon
number creates a statistical mixture of [00) and [11). In this case, the protocol would
create an entangled state with non-unity success probability A, but success would
be heralded by the verification of zero photons in the upper output. Most likely,
the initial state x171, == 1/\/2 would be prepared so that A = 50%. For entangle-
ment on-demand, however, it is necessary to determine the photon number exactly.
This difficulty is somewhat mitigated by the fact that the average photon number is
1‘10 = — lne, i.e. only 5 photons must be counted for e = .01 and 7 for e = 0.001.
Leaving the lower output unmeasured means that computing the output state
requires tracing over the lower mode. In the proceeding derivation we have taken this

trace to be unity. In reality, it is less than unity due to the non-orthogonality of the

balanced and imbalanced lower output states, governed by the overlap

[1(51cos6~[57cos6~12 z 1—(1—(5-- 64N/8
12 17 17

= 1 —— 0(1/N). (5.72)

Here, N = [crl2 is the mean input photon number and the last equality is because our
scheme requires N 62 >> 1. The resulting error is then ~ 1 / N , which can be neglected

for large N. This result validates the small-angle approximation made for the final

229

 

 

state as in (5.62), where the lower-channel light field is assumed 6—independent and
factorized from the remaining system.
The overall fidelity due to state error in this interferometrical entanglement gen-

eration is obtained by averaging over the null— and not-null- results, giving

favg = 13(0) x fun, + Z P(n 94 0) x 1
= 1_ (1 — A)s (573)

Since A 2 0, it is always favg Z 1 - e, regardless of the quantum states of the two
qubits.

Aside from the technical challenge of single-photon counting, the fundamental
quantum-mechanical barrier to successful teleportation lies in finding a balance be-
tween phase-shift detection and spontaneous-emission avoidance, as a single spon-
taneously scattered photon can destroy the coherence of a qubit. The spontaneous
emission probability for a single qubit is 6N l" / A, which becomes negligible when
6N I‘ / A < 1. This condition must be satisfied without violating the shot-noise-
sensitivity condition N 62 >> 1. From equation (5.4) it follows that compatibility
requires 16(W/A)2 << 1, which clearly violates the standard optical diffraction limit.
That such a scheme can therefore not work is in agreement with common understand-
ing [262].

To overcome the effects of spontaneous emission, we can place the two qubits
in seperate high-finesse optical cavities, with mechanical Q-switching employed to
restrict the photon to M passes through each qubit. This will increase the phase-
shift 6 and the spontaneous emission probability PSP by a factor of M. This relaxes
the compatibility condition to 8(W/)\)2 < M, which can be satisfied without sub—
wavelength focussing.

The failure probabilities due to interferometry sensitivity and spontaneous emis-

230

 

—NM62

sion are then s = e and

P5P = 2NM6P/A, (5.74)

respectively. Setting Psp = e = .01, corresponding to a fidelity of .99, and taking
W/A = 3 gives

M = —144loge/e (5.75)

= 6.6 x 105,

which is large but not necessarily outside the range of current experimental techniques.
For these parameters, the mean number of photons in the upper output is 120 = 4, and
the input photon number is restricted only by the condition N (F/A)2 = 144 e / M =
4.4 x 10—6, together with the off-resonant condition A >> I‘.

A main difficulty in long-distance quantum communication is photon loss during
qubit-to-qubit transmission, where the loss probability increase exponentially with
the transport distance. In schemes based on cavity—QED [182, 183, 184, 185], atomic
qubits’ states are encoded in the internal (polarization) states of photons, and thus a
lost photon will immediately reveal the atomic states and destroy the qubits via de-
coherence. In contrast, during an interferometrical communication, the qubits’ state
information is encoded in a form of relative phase-shifts of photons propagating in
the upper and lower arms. Such a shift is not a measurable quantity until the two
channels are recombined at a second beamsplitter. Thus the lost photon cannot re-
veal the state of the qubit, and one might suspect that the qubit coherence would be
preserved. On the other hand, due to the photon-atom interaction, a lost photons
will introduce a small relative phase-shift to the qubits. The magnitude of the rela-
tive phase is 6, but the sign depends on which interferometer ‘arm’ lost the photon.

Tracing over which arm thus results in effective decoherence and thus a reduction in

231

 

 

 

the fidelity of entanglement.
To see this, we first consider one photon lost during propagating between the

first and the second qubits. This will alter the final state into

2 . . . .. .
W7) = V j—V‘UBSUyaquUBSI‘I’i), (5-76)

with q = 0,1 corresponding to the loss in upper and lower arms, respectively. The

identity (5.25) enables us to write,

- .1 . .
11173.) = ,/%e29°chy<1(aq—761_q)U|\17,-) (5.77)
. .7 .
= (—7)qe‘9%qcyq|rf), (5.78)

where in the last step we have used the fact that for the present input state (5.60),
dql‘lli) = aciq,0[\II,-). It is now clear that the net effect of one lost photon is equivalent
to introducing a relative phase :l:6 to the qubit, where 6 << 1. In the case of random
photon losses, such phase disturbances will lead to the unknown drift of the qubit’s
state and thus a reduction in the overall fidelity of the entanglement generation. To
estimate this fidelity reduction, we introduce the lost photon number distribution
f (k) Because each photon is lost independently, f (k) will exhibit a Poisson dis-
tribution, where for a mean loss number It, the variance is fl. For simplicity, we
approximate f (k) with a gaussian,

e—(k—E) /2E

The system’s density ploss after the loss is then a mixture of

4033 = Z(—0’“’f(k>f<k’>(1608131)”1700171< ‘l1k’(PJ)k, (5.80)
k,k’

232

 

 

where 1% is the y-qubit projector defined in (5.27). Defining the reduced fidelity due

to the photon loss,

floss = IMP/21033}. (5.81)

it is found

71: = Nln(2floss -1)
Inc '

 

(5.82)

Taking floss = 1 — e = .99 gives 71 = .004N, meaning about one photon can be lost
in every 250 photons.

5.6.2 An implementation of nonlinear beamsplitter

In this appendix, we supply a realization of the nonlinear beamsplitter for NOON
interferometry in section 5.3.2. This approach, first proposed in ref. [26], is imple-

mented with a Kerr-medium—type Hamiltonian,

_9.

4 (£1.16 + &0[)2, (5.83)

H

3

achievable in X nonlinear mediums. For convenience, we introduce the following spin

(Schwinger) operators,

Jx = $01113 + 61131)
_ _1_

27
J2 = $0716 —13T13), (5.84)

.79 (6’18 — 6161)

with which the Hamiltonian (5.83) is simply H = QJg. The goal is to investigate the

time evolution of the quantum state [‘Il)(t) under this Hamiltonian, given by

[110(7) = e-iQJ§t|\p)(t = 0). (5.85)

233

For an analytic solution, it is observed that

-7r _-7r
61214.7... “2'19 = .7... (5.86)
Hence,
_-7r . 2 ‘77
[\Il)(t)=e 27"ye“zmzte’2JKI|\1:)(1=0). (5.87)
Furthermore, it is found that
—z‘7IJ -7 NJ 1 .7 *7
e 2 gee? y=—a +b
fi( )
—17‘J ~ 7% l . .
e 2 97,167 y=_ bl—al
\/§( )
2"”.7 .7—1’7IJ 1 -7 «7
eYyae 7 y=—a —b
fi( )

17K] 4 —i"J 1 *7 .7
e7 ybe 2 y=—b +a

£1 1

e—i’ll'Jxal'e—iflJy 21:81

e—tflJzBTB—iflJy = 7:61.1-

(5.88)
Consider an initial state of [\II)(0) = [N / 2), we have
_-7r . 2 -7r
117m) = e zzjye-zQJzteZEJyW/n
.377 - 2
= e ngyZe—mm tomlm) (5.89)
m

“71'
where em = (mlehZJWN / 2). It is noticed in the above equation that if N is odd, m
must be half integers, with m = k + 1 / 2, with k an integer. Choosing Qt = 1r, we
have

e—iQm2t = e—i1r(k+1/2)2 = e—irr/4. (5.90)

234

 

 

 

Hence, the net action is to imprint a global —7r/ 4 phase shift, with the final state
given by
(117,.) = e—ir/4|N/2) (5.91)

Similar behavior is found for the case of [\II)(0) = | — N / 2). If N is even, then m = k,

with 11: an integer, so that

. 2 . 2 .
e—zflm t = e—tfl’k = (_1)m = e—irrJz (5.92)

As a result,

-1r
1%) = e“?"yZ(—1)mom1m>
m
'7!" . ‘TI’
= e"‘YJ"Z1m><mIe—“1J247Jy1N/2>
m

= e—ZYJye_sze27Jy[N/2). (5.93)
“7T . '71' .
Using the fact e_17Jye—’7TJZ elTIy = e—m‘lx, it is then obtained that
[1: f) = e—iWJxlN/2) = 7N1 — .01/2). (594)

A similar result is found for [\II)(0) = | — N / 2). Thus in summary, the action of this

beamsplitter is simply

air/4| :l: N/2), N odd;
| :t N/2) => (5.95)
iNI IF N/2), N even.

These results are trivial.

235

 

 

However, if we choose Qt = 1r / 2, for N being even, we have

 

2 e—i7r/4 + (_1)mei77/4

(“imi— — (5 96)
_ «2 .

This immediately leads to the result of

l 4 N/2) => T151947” 4) :t N/2) + ei’r/ 42'!" | :F N/2)

_l_e—i7r/4 ’lN 1
— fl lliN/Z)+ "LIT—M2). (5.97)

thus implementing the desired action (5.32).

5.6.3 NOON-interferometry with less-extreme cat states

As an example, we consider the following less-extreme cat state,

~ 1

|N) = W [|N — 117,111) + |N,N — 1(1)], (5.98)

with 0 < N < N / 2. For this state, a photon loss during transportation between the

first and second qubits alternates the system’s input state to

2 . ..
1970.9 = W471]. [1N> 6 14..) 6 12,/24>]
2 . ~ . .
= [/NU..[4j1N>®PJ-14.>6Iwy>] (5.99)

with j = 0,1 corresponding to the loss occurring in the upper and lower channel,
respectively. The :l:-qubit projector B, is defined in a similar manner as in (5.27),
with the qubit 7; replaced by 2:. From the equation, the net effect of a single photon-
loss on the atomic qubits is to introduce a phase shift to the :l:-qubit. Due to the
random nature of photon loss, such a phase disturbance will lead to the unknown drift

of the qubit’s state and thus a reduction in the overall fidelity of the entanglement

236

 

generation, as is the case for the TF-interferometer approach. The effect on the

probing light beam is that the quantum state after a single loss is alternated to

pphoton = [\PZOX’I’EOI 1' [WEIX‘IJHI (5.100)

with
[11770) = |N—N—1,N)+|N— 1,N—N), (5.101)
[11771) = |N—N,N—1)+|N,N—N—1). (5.102)

Such a light beam is still cat-like in sense that it is the uncertainty in number dif-
ference between the two modes is of the order of total particle number. Compared
to the extreme cat state, it yields a slightly reduced phase-measurement sensitiv-
ity. Therefore, although at a reduced fidelity, the entanglement before the qubit x
and y can be established in presence of photon loss, due to the same reason for the
MZ-interferometer approach.

Finally, the final state in the presence of multiple-photon loss is given by a

mixture of

p’ = Z f(k)f(k’)l\I"f(k. k'>><9}(k, k’>1 (5.103)
k,k’

where f (k) is the loss statistical given by (5.79). The resulting state [\Il’f(k,k')),
corresponding to k photons lost in upper and k' in lower, is given by

1

[337091 k,» = N

. . - I .. . . I
UNBSUyUI [656]“ IN) ‘8’ P(lfpik 11611:) ‘8’ [1631)] 1 (5-104)

with 7% the normalization factor. To maintain the cat-like nature, the number of lost
photons must be much smaller than N. Finally, we note that in presence of photon

loss, the particular kind of nonlinear beamsplitter presented in section 5.6.2 is no long

237

 

 

applicable, since it relies on even total number of photons in the beam.

 

 

238

 

 

Chapter 6

Interaction— and measurement-free
Quantum Zeno gates on—demand
for single-atom and single-photon

qubits

In this chapter, we investigate a quantum-zeno approach to implement quantum
information processing using single-atom and single-photon qubits. By extending
the concept of ‘interaction—free imaging’ to the single atom level, we show that on-
demand interaction- and measurement-free quantum logic gates can be realized for
both single-atom and single-photon qubits. The interaction-free feature suppresses
the possibility of qubit decoherence via atomic spontaneous decay, while the elimina-
tion of measurements can significantly reduce errors arising from detector inefficiency.
We present a general theory of universal quantum Zeno gates, and discuss physical
implementations for quantum information processing with single atoms and photons.
In addition, we propose a loss-tolerant protocol for long-distance quantum communi-

cation using quantum Zeno gates incorporated into a Mach-Zehnder interferometer.

239

 

 

The efliciency of our Zeno gates is limited primarily by the imprecise control of atom-
photon scattering and the finite number of feedback cycles N due to the limited finesse
of the optical ring-cavity. We find the success probability scales as l — C(11)), and for
realistic parameters could be as high as 98.4%. Successful generation of atom-atom
entanglement can be heralded by detection of the ancillary photon, upon which the
fidelity scales as 1 — 0(N12)’ with an achievable fidelity of 99.994%, which comes at
the cost of reducing the success probability by the detector efficiency. The major re-
sults of this part of work have been published in “Interaction- and measurement-free
quantum Zeno gates for universal computation with single-atom and single—photon
qubits ”, by Y. P. Huang and M. G. Moore in Phys. Rev. A. 77, 062332 (2008).

In collaborate with C. Piermarocchi’s ground also in the Physics and Astronomy
department of Michigan State University, we further propose a scheme for a two-qubit
conditional quantum Zeno phase gate for semiconductor quantum dots. The proposed
system consists of two charged dots and one ancillary neutral dot driven by a laser
pulse tuned to the exciton resonance. The primary decoherence mechanism is phonon-
assisted exciton relaxation, which can be viewed as continuous monitoring by the
environment. Because of the Zeno effect, a strong possibility of emission is sufl‘icient to
strongly modify the coherent dynamics, with negligible probability of actual emission.
We solve analytically the master equation and simulate the dynamics of the system
using a realistic set of parameters. In contrast to standard schemes, larger phonon
relaxation rates increase the fidelity of the operations. This work has been published
in “Two-Qubit Conditional Phase Gate in Laser-Excited Semiconductor Quantum
Dots Using the Quantum Zeno Effect”, by K. J. Xu, Y. P. Huang, M. G. Moore, and
C. Piermarocchi, in Phys. Rev. Lett. 103, 037401 (2009).

The organization of this chapter is as follows. In section 6.1, we give an intro-
duction on the background of quantum zeno effect and its applications in various

fields, as well as presenting an overview of this chapter’s main results. In section

240

 

 

6.2, we describe the general theory of quantum Zeno gates. In section 6.3, we pro-
pose the physical implementations of the Zeno gate for single atom and photons.
Specifically, we present two elementary Zeno gates built on the dissipative interaction
between photons and atoms in section 6.3.1. Then, in section 6.3.2, we build Zeno
CNOT gates for single atom and photon qubits. We then analyze the realistic suc-
cess probability and the fidelity upon heralded success of the present quantum Zeno
gates in section 6.4. Then, we briefly describe the two-qubit Zeno phase gate for
quantum dots in section 6.5. This is followed by conclusions in section 6.6. As ex-
tensions of our work, in the appendix, we additionally construct atom-to—photon and
photon-to-atom state transfer gates, as well as provide two long-distance quantum

communication protocols, one of which is tolerant of photon loss.

6.1 Introduction and Overview

In implementing quantum information processing using hybrid single-particle
qubits, i.e., single atoms and photons, perhaps the biggest challenge is to phase-
coherently mix them without incurring decoherence. If achieved, this could lead
to two—qubit operations between remote atomic qubits via a photonic channel, thus
eliminating the restriction to nearest-neighbor interactions when processing an array
of atomic qubits. As presented in chapter 5, a feasible scheme for such entanglement-
generation would involve passing a far-detuned light pulse over both atomic qubits,
whose polarization interacts only with one of the qubit internal states. Detection of
a certain phase-shift in the light pulse relative to an idler pulse could indicate that
one of the qubits was in the interacting state, but not which one, thus collapsing the
two-qubit state into an entangled state [150, 40]. While a light pulse containing many
photons can have a strong effect on a single atom, the back-effect of the atom on the

light-field is very small. The primary difficulty in creating entanglement between

241

 

 

two atomic qubits via a common photonic channel thus lies in the need to employ a
sufficiently large number of photons to detect the small phase distortion generated
by a single atom, while avoiding the loss of information due to atomic spontaneous
decay and / or imperfect optics. This problem is typically addressed by using high-
finesse Optical cavities to enhance the coupling while reducing the spontaneous decay
[271, 272], and / or replacing single atom qubits with collective ensemble qubits[150,
259, 260]. With atomic ensemble qubits, however, the coherence time will be limited
by information-destroying short-range interactions (collisions). In order to avoid this
limitation, one can use single isolated trapped atoms, which then requires a high-
Finesse optical resonator in the strong-coupling regime, and / or the use of highly
non-classical light pulses [40].

In cavity QED approaches, one usually seeks to suppress spontaneous emission by
significantly decreasing the density of states of the electromagnetic vacuum, so that a
coherent atom-photon interaction becomes possible over some timescale. Despite the
widely-held belief that decoherence must always be avoided in quantum information
processing, it has been known for some time that decoherence can instead be har-
nessed to implement high-efficiency coherent quantum logic gates for single photons
and atoms [273]. These gates rely on the fact that strong coupling to the environment
is equivalent to continuous measurement, and can therefore inhibit coherent quantum
dynamics, in analogy with the quantum Zeno effect [274, 275, 276, 277, 278, 279].
To best understand how such an effect can lead to coherent entanglement genera-
tion, consider a set of experiments where the Zeno effect was used to allow a sin-
gle photon to image an absorbing Object without being absorbed, knows as high-
efficiency interaction-free measurement (IFM), or alternatively as ‘quantum interro-
gation’, [280, 281, 282, 283, 284, 285, 286]. Replacing the classical absorber with an
atomic qubit, prepared in a superposition of absorbing and transparent states, can co-

herently change the quantum state of the probe photon conditioned on the state of the

242

 

 

atomic qubit. This leads to atom-photon entanglement, generated via a mechanism
in which the atom and photon arguably never interact directly, so that spontaneous
emission (decoherence) is avoided even for a resonant photon. In such a system it is
simply the possibility of a strong dissipative interaction which drives the Zeno effect
and creates entanglement, without dissipation actually occurring. The Zeno effect
forces the system to remain in a decoherence free subspace (DFS) [162, 163, 164], and
thus greatly suppresses the possibility of decoherence via spontaneous decay.

Several schemes for quantum entanglement manipulation and / or gate Operation
via IFM have been recently proposed [194, 195, 196, 197, 198]. Utilizing the dis-
sipative photon-atom interaction, Gilchrist et al propose implementing conditional,
post-selection protocols to generate Bell-, W- and GHZ—type states for single atoms
with a maximum success probability of 1 / 4 [194]. They also provided an approach to
generate n—photon superposition state with n IFM devices aided by an ancillary atom,
where the final atomic state is measured and a logical operation is performed condi-
tioned on the measurement outcome. In a closely related work, Azuma pIOposed to
generate Bell states Of an electron-positron pair, where pair annihilation supplies the
equivalent dissipative interaction to drive the Zeno effect [195]. Also demonstrated
for such a system are a Bell-measurement circuit and a Controlled-Not (CNOT)
gate, with a maximum success probability Of 3/4 and 9/16. By chaining multiple
IFM devices, Azuma was able to further push the success probability Of the Bell-
measurement circuit and consequently the CNOT gate to be near unity [196]. This
CN OT gate requires four ancillary entangled qubits, two Bell-measurements, together
with four classical measurements and multiple measurement-conditioned operations.
This is arguably too complex a construction relative to the simplicity of the task.
In a non-IFM-based yet related approach, Franson et al proposed using the quan-
tum Zeno effect to inhibit more than one photon occupying the same optical fiber

mode, and thus implement the VSWAP gate between two photonic modes, leading

243

 

to subsequent improvements in quantum computing with single photons [197]. Also
very interestingly, using ‘chained’ quantum Zeno effect, Hosten et al demonstrated
the counterfactual quantum computation, where the potential outcome of a quan-
tum computation can be inferred without actually running the computer [198]. In
a non-IFM-based yet related approach, Franson et al proposed using the quantum
Zeno effect to inhibit more than one photon occupying the same optical fiber mode,
and thus implement the m gate between two photonic modes, leading to sub-
sequent improvements in quantum computing with single photons [197]. Also very
interestingly, using ‘chained’ quantum Zeno effect, Hosten et al demonstrated the
counterfactual quantum computation, where the outcome of a quantum computation
can be inferred without actually running the computer [198].

In this chapter, we consider three important changes to the standard IFM gate
which allow us to design an extremely simple and elegant set of quantum logic gates
towards quantum information processing. First, in contrast to simply flipping the
internal state of the probe qubit, we allow arbitrary state rotation of the probe qubit.
Second, in addition to the usual IFM gate with photon as the probe and atom as
the Object, we construct a complementary gate where the role of atoms and photons
are exchanged. Third, we consider the placing multiple qubits in a single IFM cir-
cuit. With these changes, we are able to construct interaction- and measurement-free
quantum logic gates in a straight forward manner.

We begin by presenting a generalized theory of the Operating principle, and then
consider physical implementations of such Zeno gates. Specifically, we design atom-
photon, atom-atom and photon-photon CNOT gates, as well as atom-to-photon and
photon-tO-atom state transfer circuits. We also consider a long-distance quantum
communication protocol which is tolerant to photon transmission losses. All of our
quantum Zeno gates are high-efficiency in the sense that they deviate from unit

success probability by a factor ~ 717 and from unit fidelity upon heralded success by

244

 

 

only ~ N17 Here N is the (effective) measurement number, limited only by technical
considerations. One interesting feature of our design is that the IFM probe qubit acts
only as an ancillary qubit, which is disentangled from the logical qubits at the end of
the gate Operation, so that no measurement of this qubit is required. The elimination
Of this measurement greatly boosts the success probability by removing errors due to
detector inefficiency, as opposed to several previous schemes [182, 136, 168, 287, 288,
289, 170, 185, 290, 174]. On the other hand, detection of the ancillary qubit in the
output channel would ‘herald’ successful gate operation, thus improving the fidelity
of state creation by an additional factor l/N. In this way one can trade-off success
probability for fidelity, where success probability is decreased only by the detector
inefficiency. As is typical in cavity QED applications, the trade—Off for high-fidelity is

gate speed, as increasing N necessarily slows down the gate Operation.

6.2 Quantum Zeno Gate: General theory

The quantum Zeno effect occurs when a rapid sequence of strong measurements
is performed on a slowly evolving quantum system, with the result that the system
is ‘frozen’ in its initial state, i.e. the quantum watched-pot boils more slowly than
the unwatched pot. In the standard quantum Zeno effect, a two-level system (with
states denoted [0) and [1)) is rotated by a sequence of M rotations, each of angle
6 /M . In the absence of measurements, this results in a net rotation of 6, so that a
system initially prepared in state [\II,;) = [0) is transformed to the superposition state
[\Ilf) = cos 6|0) + sin 6|1). The Zeno effect occurs when the system is measured in
the {[0), [1)} basis after each 6/M rotation. It is readily found that the probability
to find the particle in state [0) after M cycles is [274, 275, 276, 277]

P0 = [1—sin2(6/M)]M

245

 

 

 

 

z 1—— . (6.1)

This probability approaches unity in the limit M —> 00, so that the system remains
‘frozen’ in the initial state.

An analogous effect occurs in the case of a two-level system evolving under con-
tinuous coherent evolution and dissipation, as illustrated in Figure 6.1. The coherent

evolution is governed by the Hamiltonian

. hf)

where 0y is a Pauli spin matrix in the y direction. In the absence of dissipation,
a system prepared initially in the state [\I/(0)) = [0) evolves after time t into the
superposition state

|\I'(t)) = cos(Qt/2)[0) + sin(Qt/2)[1). (6.3)

In a quantum Zeno system, however, spontaneous emission is introduced, coupling the
system to a reservoir. Standard quantum treatment of such systems is via a master
equation for the system density matrix. In the current model, the scattered state after
a spontaneous decay corresponds to the photon escaping from the system. Due to
irreversibility, this state is dynamically decoupled from the rest Of system during the
subsequent evolution Of the system. This allows us to map the master equation onto
a pure-state representation, in which the system state is represented by a pure state
in the unscattered subspace. The master equation dynamics can be then be exactly
reproduced if this pure state evolves under an effective non-Hermitian Hamiltonian.
Because of the non-Hermitian property, the state in this representation will no longer
be normalized to unity as time evolves. This loss of normalization corresponds to
the probability that the system has decayed. After a successful Zeno-gate operation,

however, the system decay was prohibited by the Zeno effect. The quantum state of

246

system is then the evolved pure state, which must be re—normalized to unity.

For system depicted in Fig. 6.1, the state [1) decays at rate 1" to a third state lg).
The effective non-Hermitian Hamiltonian in the pure-state representation of {[0), [1)}
is then,

. A .l‘lP
Heff=HQ—z-§-[1)(1[. (6.4)

Derivation of this effective Hamiltonian, as well as further explanations of pure-state
representation, is given in appendix 6.7.1. The reduction in normalization, P(t) =
(\Il(t)|\I!(t)), corresponds to the probability that the system has decayed to state lg).

This evolution can be solved analytically, and in the regime I‘ >> Q , we find
that the probability for the system to be found in the initial state [0) is given by

2
P0 (t) = e-Q Up. In the relevant case Qt ~ 1 and Ft >> 1, this becomes

2
P0109 1 — Lift/f4) . (6.5)

which approaches unity as I‘t is increased with Qt held fixed, in which limit the system
is again ‘frozen’ in the initial state. The discrete and continuous systems can be
mapped onto one another if we equate 6 4—1 Qt/2 and M H Meff = I‘t/4 = 6F/2Q.
The equivalence between the two systems can be understood by interpreting the
spontaneous emission in the continuous case as a source of effective ‘measurements’
by the reservoir. The measurements are implemented because the presence or absence
of spontaneous emitted photons will immediately reveal the state information and
collapse the system into either [1) or [0). For a decay rate 1", a single effective
measurement duration is 4I‘_1. The effective number Of measurements Meff during
a interval t is therefore t / 4F“1 = I‘t / 4. Equations (5) and (1) are in this sense exactly
equivalent. We note a large Meff corresponds to a large ratio of F/ Q, and hence a
stronger dissipation is favorable for the continuous system. This is exactly opposed

to the conventional approach, where dissipation must be negligible during a quantum

247

 

 

 

 

 

 

 

Figure 6.1: Level scheme for a two—state system with coherent coupling at Rabi
frequency Q and upper-state decay at rate F.

gate operation.

Using these results for the quantum Zeno effects, we now describe the general
theory of quantum Zeno gates. Our model applies to both the discrete and continuous
Zeno effects, as they can be exactly mapped onto each other. We will therefore treat
only the continuous Zeno effect, with the understanding that analogous results for
the discrete Zeno effect can be Obtained by substituting M for Meff- We show the
generalized two—qubit logic gates can be realized via a single IFM in a direct or indirect
manner. For the direct gate, one qubit acts as a ‘probe’, probing the quantum state of
the ‘object’ qubit repeatedly or continuously, while the Object qubit coherently driven
on a timescale slow compared to the measurement time. The coherent evolution of
the probe qubit will be allowed or prohibited depending on the state of the Object
qubit and thus create entanglement to implement the gate Operation. This gate,
however, requires the two qubits start inside the DFS, as states outside the DFS will
immediately decay, resulting in loss of quantum information. Such direct gates are
therefore incapable of implementing Unitary operations, which must be defined for
all possible input states. We have found that with the addition Of an ancillary qubit,
Unitary quantum phase gates can be realized, even without the need to measure the
final state of ancillary qubit. In these gates, the ancillary qubit acts as the probe

measuring the joint quantum state of the two logical qubits, during which the three

248

 

Figure 6.2: Scheme for the direct two-qubit quantum Zeno gate via a single IFM.
Each state is of the form |q1q2), where q1 and q2 are the probe and Object qubits.
The coherent Rabi oscillation is applied to the probe qubit as indicated by the blue
arrows, while the green arrow indicates spontaneous decay.

qubits remain in a DFS.

For the direct logic gate, only two qubits are involved. As shown in figure 6.2,
only the [00) state decays, so that the [01), [10) and [11) states form a DFS, where
the two-qubit states are in the form |q1q2), with q1 and q2 the probe and object
qubits, respectively. Achieving this state selective decoherence is the primary design
challenge in physical implementation of such gates. We will see, however, that the
DFS states correspond to states where there is no interaction between each other, i.e.
their wavefunction overlap is zero. ,

As the two logical qubits are required to be initially in the DFS, their initial
state is

I‘I’z') = 001|01> + 010|10> + 011l11), (6-6)

The coefficients 013': with i, j = 0, 1, are arbitrary and normalized to one. The goal is

to transform this state to the final state

[\Ilf) = c01[01) + (C10 cos6 — cu sin 6)[10)

+ (C10 sin6 + 011 cos 6)[11), (6.7)

249

 

 

with 6 arbitrarily adjustable. Applying a slow O'y pulse to the probe qubit rotates
its state according to Hamiltonian (6.2). The attempt to rotate the probe qubit will
therefore take the system out Of the DFS if the object qubit is in the states [0), in
which case the quantum Zeno effect will ‘freeze’ the system in its initial state and
prevent the rotation of the probe qubit. Only if the object qubit is in the [1) state
will the ancillary qubit undergo rotation and change its state. The final state of the
system will then be (6.7) with 6 = Qt / 2, and thus the desired target state is Obtained.
The success probability in this gate operation is given by the standard Zeno formula
2 92

Psuccess = 1 — [601' M ff, (6-8)
e

 

where the effective number of measurements is Meff = 61" / 2Q.
We now describe a quantum-Zeno phase-gate based on a three-qubit system,
where one qubit serves as an ancillary, and two-qubits are the logical qubits. The

operation of a phase gate on a two-qubit state is to transform an initial state of
I‘I’z') = CoolOO) + €01|01>+ 010|10> + C11|11> ((39)
into the final state
I‘I’f) = 000|00> +001|01l +010|10> - C11|11>- (6-10)

Our scheme consists of a pair of logical qubits in an arbitrary initial state, together
with a third ancillary qubit prepared in the [0) state. The initial state of the system
is therefore

[\Ilz'n) = CODIOOO) + €01[010)+ C10[100) + C11[110), (6.11)

where the three-qubit states are in the form [q1q20) , where ql and (12 are the logical

qubit quantum numbers and 0 describes the ancillary qubit. A single-qubit 217-pulse is

250

 

 

 

applied to the ancillary qubit, rotating its state from [0) through [1) and back to —[0) ,
which imprints a 7r-phase shift on the state Of the system in the absence of dissipation.
The complete state space of the three-qubit system consists of eight states. As shown
in Figure 6.3, the states [000), [100), [010), [110) and [111) form a DFS, while the
states [001), [101) and [011) decay at rates 2F, I‘ and I‘, respectively. The decay rate
for state [001) is doubled because there exist two possible decay channels, i.e., of q1, a
and q2, a. The attempt to rotate the ancillary qubit will therefore take the system
out of the DFS if the logical qubits are in the states [00), [01), or [10), in which
case the Zeno effect will prohibit the rotation. The ancillary qubit will thus undergo
rotation only if the logical qubits are in the [11) state, after which the 1r-phase shift

is imprinted. The final state of the system will thus be

Wow) = (COOIOO) + €01|01l+ 01000) - 611|11>)® |0)a (6-12)

with probability

 

2 2
c 7r
Psuccess = 1“ (_I 020' +l001l2 + [€10l2)M - (613)
6ff

Here Meff = 7rl"/ 2Q. The extra factor of % for the [00) state is due to that the
relevant decay rate is then 2I‘, resulting in doubled number of effective measurements.
Upon successful operation, the phase-gate has been applied and the state of the
ancillary qubit is not entangled with that of the logical qubits, so that no measurement
of the ancillary qubit and / or conditional operations are required.

Lastly, we note for both the direct and indirect quantum Zeno gates, successful
operation indicates that the system remained in the DFS throughout, so that en-
tanglement has been achieved without any interaction between qubits. Merely the
possibility of dissipative interaction and the constant monitoring of the system by its

environment are sufficient to project the system into an entangled state. As fast gate

251

[000) [000%0120) [0102100) [101))5410) 1111)

Figure 6.3: Scheme for a two-qubit phase gate with an ancillary qubit. Each state
is of the form [q1q2a), where ql and q2 are the logical qubits and a is the ancillary
qubit. Note that the coherent Rabi coupling, indicated by the blue arrows, is applied
to the ancillary qubit.

Operation at high success probability is desired for scalable quantum computation,
one should seek out a system with as large a decay rate I‘ as possible, which is exactly
the opposite regime one normally attempts to reach when quantum logic gates are

based on ordinary coherent qubit-qubit interactions.

6.3 Implementations for single atomic and pho-
tonic qubits

Now we consider physical implementations of quantum-Zeno logic gates based on
a single photon propagating in a high-finesse ring-cavity which interacts with a single
atom. In the first part of this section, we propose two elementary interaction-free
quantum gates. Then in the second part, we construct atom-atom, photon-photon
and atom-photon CNOT gates from the elementary gates. Further quantum logical
circuits, including atom-tO-photon and photon-to-atom state transfer gates, as well
long-distance quantum communication schemes, can be found in appendix 6.7.3 and

6.7.4, respectively.

252

 

 

|V0> |H1>

Figure 6.4: Level scheme for the elementary IFPG and IFRG gates. The states [V0)

and [H 1) are resonantly coupled to the excited state [e) with coupling strength 9,
while the state [e) spontaneously decays at rate 7.

(a) \ Mirror E PBS

A

 

   
  

  
    

   

 

 

 

 

 

 

 

r iOut after N cycles \
. ‘ a
rotator
W)
k j
(b) Atom \
Photon

 

J

Figure 6.5: Figure (a): a schematic illustration of the physical implementation of
IFPG; (b): its graphic representation.

253

(a) \ Mirror [3 PBS

Out after N cycles Raman pulse D

|H>
IL‘ \ Atom

IV)

)

 

sauna-lo

 
 
   
 
  

       
   
 

 

 

 

 

 

\
(b) Photon

Atom

 

 

Figure 6.6: Figure (a): a schematic illustration of the physical implementation of
IFRG; (b): its graphic representation.

 

254

6.3.1 Elementary atom-photon gates

The quantum logic circuits we propose are built from two elementary interaction-
free quantum gates, which we term the Interaction Free Polarization Gate (IFPG)
and the Interaction Free Raman Gate (IFRG). The IFPG is a generalization and
refinement of previously proposed interaction-free gates [195, 194], and induces a
rotation in the polarization of a photon conditioned on the state of a control atom.
The IFRG reverses the role Of atom and photon and induces a rotation of the atomic
hyperfine state conditioned on the polarization of a control photon. The atom-photon
interaction in these gates is based on a standard A-level scheme. The two atomic
ground hyperfine states are labeled [0) and [1), and are coupled to an excited state [6)
via absorption / emission of cavity photons with two orthogonal polarizations, labeled
IV) and [H ), respectively. We note that this choice of labels is purely symbolic,
and in practice the two states will most likely be circular polarization states. The
use of H and V is, however, in keeping with the early literature on interaction-free
quantum interrogation [281, 285]. For the case of a single photon in the ring-cavity,
the joint photon-atom states [V0) and [H 1) couple strongly to [e) which then decays
via spontaneous emission to a set of states having zero photons in the cavity, as shown
in Figure 6.4. Due to selection rules, the joint-states [H O) and [V1) are not coupled to
[e). Quantum back-reaction due to the non-emission Of a photon therefore collapses
the joint state onto the {[VI) [H 0)} subspace. For simplicity, we temporarily assume
that the states [H 0) and [V1) will immediately decay with unit probability.

The Interaction Free Polarization Gate is shown in figure 6.5 (a), with figure 6.5
(b) depicting a graphical representation of the gate. The circuit is modeled after the
standard high-efficiency quantum interrogation circuit [285], which we have general-
ized to allow to an arbitrary polarization rotation angle 6, rather than the standard
17 / 2 rotation. The gate consists of a high-finesse optical ring resonator, into which

a single photon is injected and then released after N cycles. Each cycle consists of

255

passage through a 6/N polarization rotator followed by a pair of polarized beam-
splitters (PBS), which spatially separate the H and V polarizations. The control

atom is placed in the V arm. The input state must be of the form

Wm) = CH0|H0> + CH1|H1> + CV1|V1>. (6.14)

as the [V0) state leads to immediate loss of the photon via spontaneous emission.
As only the V polarization interrogates the atom, it follows that the atomic state [1)
is transparent to the photon. In this case, the effect of the atom is negligible, and
the photon polarization rotates by the angle 6. If the atom is in state [0), however,
the quantum Zeno effect will freeze the photon in the [H ) state. Assuming successful

operation, this leads to the output state

[‘I’aut) = CHO[H0)+(cchos6—CV1sin6)[H1)

+ (CH1 sin 6 + CV1 cos 6)[V1), (6.15)

which shows that the photon polarization is rotated conditioned on the atom being
in state [1).

The complementary Interaction Free Raman gate is depicted in figure 6.6 (a)
and (b). It differs from the IFPG in that the atom is placed in the H arm, and
the polarization rotator is replaced by a Raman pulse which couples the [0) and [1)
atomic states via a rig rotation. The 63, operation is timed so that the atomic state
rotates by angle 6 during the time it takes for the photon to travel N ring-cavity

cycles. An arbitrary input state is of the form

Win) = CH0|H0> + CV0|V0) + CV1|V1>. (6-16)

where the state [H 1) is forbidden. If the photon is in [H ), the quantum Zeno effect

256

 

 

freezes the atom in the [0) state; otherwise, the atomic spin rotates by 6. Upon

successful operation, this results in the output state

Wow) = CH0|H0>+(CVOCOSQ—CWSiDQNVO)

+ (CV0 sin 6 + CV1 cos 6)[V1), (6.17)

which shows that the atomic state is rotated conditioned on the photon being in state

IV)-

6.3.2 Quantum Zeno CNOT gates

From these two elementary gates, we can construct the primary quantum logic
gates necessary for universal quantum computation. The first gate we consider is
the atom-atom Controlled-N OT gate, depicted schematically in figure 6.7 (a). In a
CNOT gate, the state of the target qubit is flipped if the control qubit is in the logical
[1) state, and is left unchanged otherwise. Our atom-atom CNOT uses a single IFPG
with 6 = 7r, but with two atoms placed in the IV) arm. One atom serves as the control
qubit and the other serves as the target. A single ancillary [H ) photon is injected
into the device to induce atom-atom entanglement via non-interaction with both
atoms. The control atom differs from the target in that Hadamard transformations
are applied to it before and after the interaction with the photon. For the IFPG with

two atoms and one photon, the initial state is

IW1n>=|H)®(000|00>+ 001|01> + C10|10)+ 011|11>)- (6-18)

If either atom is in the state [0), it will be sufficient to induce the quantum Zeno
effect and freeze the photon in the [H ) state. Thus only the [11) state is transparent

to the photon, in which case the photon undergoes a 1r-rotation from [H ) —-> —[H).

257

 

 

The output state of the two-atom 7r-IFPG is then

I‘l’out) = W) ® (COOIOO) + €01|01l+ C10|10) — 011111)), (6-19)

thus realizing a two-atom phase-gate. It is known that a phase-gate can be trans-
formed into a CN OT gate via single-qubit operations, such as the Hadamard trans-
form, defined as [0) —) ([0) + [1))/J2 and [1) —> ([0) — [1))/V2. Applying this
transform to the state of the target atoms before and after the action of the rr-IFPG

gives the output state
I‘I’out) = [Hi ® (Cool00> + C01|01> + C11|10> + 01001)). (620)

corresponding to a CN OT operation on the two atoms. Due to our use of a 17-
polarization rotation, as opposed to the 7r / 2 rotation of the standard IFM circuit, we
find that at the output the photon is not entangled with the atoms. Because of this
disentanglement, no detection of the photon and / or its polarization state is necessary.
We note that during the N -cycles of the IFPG, the atoms and photon move through
a highly entangled three-body state.

A photon-photon CNOT gate can be constructed similarly by injecting two pho-
tons into a 1r-IFRG containing a single ancillary atom. In this case, the control and
target photons can be input as time-separated wave packets, or counter-propagate in
the ring-cavity. This gate is depicted schematically in figure 6.7 (b). The analysis
is the same as the atom-atom CNOT, with H H 0 and V H 1. After successful
operation, the state of the atom is not entangled with the two-photon state, so that
no atomic-state measurement is required for successful photon-photon gate operation.

A hybrid atom-photon CN OT, in which a stationary atomic qubit and a flying
photonic qubit play the roles of control and target, can be constructed without the

use of a third ancillary particle. In figure 6.8, we depict an atom-photon phase gate.

258

 

 

0 Atom IV\ photon [E] Hadamard Gate
' ‘ “mm—mam ONO?“ 7'

 

 

 

 

 

 

 

 

 

 

 

 

 

Co A
T .
’V‘ 75 H
[11> IFPG I >
V (a) J
, Photon-PhotOn CNOT”
C'V‘ - [
T 7 [
. 71' I
~1 [0) ‘IFRG [0) l
7 (b)

Figure 6.7: Graphical logic circuits for atom-atom (a) and photon-photon (b) CN OT
gate. In both figures, we use ‘0’ for the control and ‘T’ for the target qubit.

0 Atom IV\ photon [E Hadamard Gate

 

Atom-Photon Phase Gate

 

 

 

 

 

 

, e i
, J [
, —7r/2 7t 7r/2 [

IFPG ‘
; ’V‘ V H [

Figure 6.8: Graphical logic circuits for the atom-photon phase gate.

259

The input state for this gate is

Wm) = CHolHO) + CH1|Hl> + CV0|V0> + CV1|V1>- (6-21)

The first PBS sends the IV) polarization state along an upper arm, where a —77/ 2

rotator converts it to [H ) polarization. The state of the system is then
I‘ll) = CH0|H0> + CH1|H1> + cvolH’O) + CV1IH'1), (622)

where the prime indicates the upper path. The n-IFPG then transforms the system

to the state

[‘10 = CH0[H0> + CH1|H1) + CVOIH’Ol — CV1IH’OI- (6-23)

The subsequent —7r / 2 rotation on the upper arm and the PBS restores the [H ’ ) state

to [V), resulting in the desired phase-gate output state
|‘1’outl = CHolHO) + CH1|H1) + CV0|V0> - CV1|V1>- (6-24)

This state can then be converted into a CN OT gate by the addition of single-qubit
Hadamard transformations onto the target qubit before and after the operation of
the phase-gate. Thus either the atom or the photon can play the role Of control qubit
simply by applying the Hadamard sequence to the other particle.

Lastly, we note that Azuma [195, 196] recently proposed an alternative CNOT
gate using essentially three IFM gates to prepare the required ancillary qubits, at least
two more IFM gates for the required twice Bell measurements, as well as a considerable
number of single-qubit measurements and measurement-conditioned operations. Thus
we believe that our gates represent a significant advance, as they are accomplished
within only a single IFM device and two logical Hadamard / Rotation Operations, and

most importantly, without any measurements or conditional operations.

260

6.4 Success Probability and Fidelity upon heralded
success

In describing each Of these quantum Zeno circuits we have assumed successful
operation, and shown that the desired output could be achieved. In actuality, the
descriptions we have provided are only exact in the limit N —> 00, where N is the
number of times the photon cycles through the ring-cavity. Three key factors de-
termine the success probability. The first is finite N, restricted primarily by the
reflectivity of the cavity mirrors and imperfections in whatever optical elements are
included in the circuit. The second effect is imperfect absorption of the [V0) state,
which we characterize by the parameter 60, so that the absorption probability for this
state is 1— 60. Due to this imperfectness, the effective number of measurements for N
feedback cycles is reduced to (1— 60)N. The third factor is imperfect transparency of
the [V1) state, characterized by the absorption probability 61. For an ideal system we
would have 60 = 61 = 0, indicating that the selection rules of Figure 6.4 are precisely
obeyed. In practice we expect finite selection-rule errors due to the combined effects
of tight-focussing of the photon, which warps the photon polarization vector; and
tight trapping of the atom, which may warp the atomic hyperfine spin vector. We
use the term ‘warp’ to indicate an undesired spatial dependence. A rigorous calcula-
tion of these errors will be investigated in future work. At present we only calculate
the effects of non-zero 60 and £1 and determine the acceptable upper-limits on these
two parameters.

As having been shown in section II, a quantum Zeno system, while being an
Open system, can nonetheless be described in the pure-state representation, governed
by a non-Hermitian Hamiltonian. For the single-atom IFPG, the quantum state of
system can be represented by a pure state in the {|H0), [H 1), [V0), [V1)} subspace.

The final state can be derived from the initial-state via a non-Unitary propagator

261

 

 

prom) = V(6, N, 60,61)[\P,;n), and the decay probability determined by Pdecay =
1 — (‘poutl‘l’outl- The exact form of the propagator V(6, N, 60, 61) is derived in the
appendix 6.7.2. The overall success probability, which is the probability for the system
to be found in the target state at the output, is given by Psuccess = [(Wtargetlwwt) [2.
Note since [‘I’out) is now not normalized to unity, Psuccess has implicitly taken into
account the decay possibility.

If we consider the 1r-IFPG with an input state of the form (6.14) we find that

the decay probability is given by

 

12...... = 16401240 + (lCHII2 +1cvr12)Ar (6.25)
where
2 1 2
7r + £6 77
A = — — — .
and
4
A1 = 1 — e—le/2 — (AZ—Bl) . (6.27)

The quantum gate can be considered ‘eflicient’ if the failure probability scales as
1 - - 1
Pdecay ~ N’ which therefore requires 1 — \/e_0 ~ I and 61 ~ N?" The overall success

probability is given to leading order by

1
Psuccess = 1 _ Pdecay + Olml -

= 1-0[%}, - (6.28)

characteristic of high-efficiency quantum interrogation. We note that the IFRG op-

erates on the same principles, and thus exhibits the same l/N scaling behavior.
The condition l—fl ~ 1 indicates that 60 need not be negligible, but rather only

not too close to one. This requirement is not very stringent, as previously described by

Azuma [195]. We note, however, that previous athors have not considered imperfect

262

 

 

 

 

transparency of the [V1), described by 61. We see that this is actually the critical
parameter upon which successful operation is sensitively dependent. For an N of
103, equation (6.27) implies that the absorption probability be ~ 10"6 in order to
maintain 1 / N scaling. Clearly this will require very precise control over the atom-
photon interaction. At present we do not know if it is possible to maintain both of
these error parameters in their Operational ranges simultaneously, although the weak
constraint on 60 leads us to be optimistic. In future research we will address the
interaction between tightly-trapped atoms and tightly—focussed photons in detail.

If we assume that absorption in the transparent state is negligible, the absorption
of the absorbing state can be estimated using the standard cross-section Of a two-
level atom scattering resonant light, a = A2/ 2. The absorption probability is thus
determined by the ratio of o to the focusing area A Of the photon. Assuming a tightly

focused light beam with transverse radius W, the absorption probability is thus

1 ,\ 2

Taking the diffraction limit W = A gives
A0 a: —. (6.30)

This means for N = 103, the imperfection in absorbtion degrades the success proba-
bility from ~ 99% to ~ 75%, while for N = 104 from 99.9% to 97.5%. On the other
hand, to compensate for this small absorption probability would require an increase
in the cycle parameter N by a factor of 25.

We can relate the maximum achievable cycling parameter N to the cavity finesse
by taking into account the round-trip net transmission probability T. The probability
to survive N cavity round-trips is RN, where R = 1 - T. Thus the total loss

probability due to imperfect Optics is Ploss = 1- RN = 1-(1— T)N z NT. In order

263

 

to maintain the 250 / N success rate imposed by realistic absorption probability, this
requires T ~ 250/N2, in analogy with the imperfect atomic transparency parameter
61. Presently, high-finesse super-mirrors [271, 272] can have T < 10‘6, which would
correspond to N ~ 1.5 x 104. An N of this size permit an overall success probability
for the quantum Zeno gates of 98.4%, which would be competitive against potential
competing methods. At present, achieving a net round-trip transmission probability
of 10‘6 for a single-photon pulse, while perhaps conceivable, would clearly require
heroic experimental efforts.

Lastly, we note that for conventional schemes based on cavity QED, the system
is Operated in the strong coupling regime, where vacuum-Rabi oscillations dominate
the dissipative atomic decay and cavity leakage [156, 271, 272]. This regime is char-
acterized by a large single-atom cooperativity parameter, e.g., C = gg/I‘cnc >> 1,
where go, PC and Kc represent the atom-cavity coupling strength, the atomic sponta-
neous emission rate inside the cavity, and the cavity decay rate, respectively. Unlike
the cavity-QED gates, the present Zeno gates are implemented in ring cavities. The
coupling strength and spontaneous emission rate in these systems are given by their
free-space values g and l". The lifetime Of photons inside the ring cavity is 7' 2 deT,
where c is the speed of light and LC is the coherent length of the photon. Physically,
this is the total amount of time the photon interacts with the atom. Correspondingly,
a ‘cavity decay’ rate is It = 1 / 7' = cT/Lc. The analogy of the cavity cooperativity C

mapped onto the current ring-cavity system is thus

CTZTlg = PK.
1 ,\ 21
- 2%(W') :7- (631)

For T < 1 and W m A, we have Cring >> 1. The present Zeno-gate system in this

sense also Operates in the strong coupling regime. Seemingly, the Zeno and cavity-

264

 

QED gates would therefore require similar experimental conditions. This conclusion
is, however, not necessarily true since the Zeno gates are operated in the regime
of 92/1" << gg/I‘c and It << 166. The large effective COOperative parameter Cring
is merely due to the ratio of 92/1“ to K remaining large. Thus, while cavity-QED
schemes demand 90 >> PC and gc >> 110 being satisfied simultaneously, the Zeno-gate
system only relies on g ~ P and T < 1. Taking the coherence length of the laser
pulse as c/I‘, the condition of 9 ~ I‘ would then require the single-photon pulse to be

tightly focused down to W ~ A, such that
= —— ~ 0.1. (6.32)

It is worth noting that while in the strong coupling regime, the present scheme is
still configured on free—space atom-photon interaction, in the sense that the photon
is coupled to the continuum atomic states.

In above discussions, we have considered the reduction in success probability for
IFPG and IFRG gates due to imprecise control of atom-photon scattering and / or
optics imperfections. In practice, the resulting photon loss can be detected by the
absence of the photon at the gate output. By adding a detector to the photon output
channel, we find that the detection of a photon necessarily indicates that the device
has operated successfully, as all failure mechanisms result in absorption or scattering
of the photon out of the resonator mode. Conditioned on this heralded success, the
fidelity Of the gate operation, measuring the overlap of the target state and the final

output state, will be given by

PS’UCCBSS (6 33)

Fherald = 1 _ Pd °
ecay

265

 

To leading order in if, F herald is found
2 2 2 (A0 - A02
6.4.7.. = 1— 1ch (1cH11 +|CV1| ) ——,,-—

= 1— mfij, (6.34)

compared to 1 - 0%,] for success probability. The small deviation from unity is due
to a slight imbalance which occurs in the final normalization Of the output state. The
fidelity upon heralded success of the present quantum Zeno gates can thus be made
extremely close to unity, e.g. for the ‘realistic’ N = 1.5 x 104 the success probability
is 98.4%, yet the corresponding fidelity is 99.994%.

6.5 Zeno phase gate for two quantum-dots

In this section, we extend the idea of the quantum Zeno gate to the system of
quantum dots. We consider a system with three vertically grown QDs. The schematic
view is shown in Fig.6.9, where QD1 is the central neutral dot, and QD2 and QD3 are
the two lateral charged dots. Photon absorption creates exciton states in QD1, and
trion states in QD2 and QD3. We assume that the ground trion energy in the two
lateral dots is the same and is lower than the ground exciton energy in the central dot.
QD1 is driven by the external laser field which can induce Rabi Oscillations between
zero and one exciton. We assume that the driving laser field is 0'- polarized with
energy resonant with the ground exciton transition on QD1. Due to the difference
between the exciton and trion energy, QD2 and QD3 are off-resonance. The standard
selection rules of semiconductor quantum dots apply, i.e. the 0. laser creates an
exciton with electron spin up(+%) and a heavy hole spin down(—%) in QD1 only.
For the selection rules, notice that if the electron in QD2 is spin down, the exciton

in QD1 can transfer to QD2 to form a trion. If the electron in QD2 is spin up, the

266

 

 

 

 

(a)

n
+4—1-

 

(b)

_+_ LII—+—

 

 

 

 

 

002 001 003 002 001 003
(c) r -H— (d)
V Y
Q
0
_7__ _.[._ ,_[__ _.[_
002 001 003 002 001 003

 

 

Figure 6.9: Scheme Of the dynamics Of the system under different initial states. The
blue (gray) arrow represents the electron (hole) spins. The energy levels for QD1
are the empty dot (lower) and the first exciton level (upper). The energI levels in
QD2 and QD3 are charged dot ground states (lower) and trion levels (upper). (a)-
(c) If either electron in QD2 and QD3 is spin down, the exciton in QD1 can decay
into the neighboring dots and be detected (detectors QD2 and/ or QD3 switched on).
Therefore the QZE prevents the Rabi Oscillation in QD1, no phase shift is present
after a 1r pulse. (d) Only if both detectors are switched 017, a 1r Rabi Oscillation can

be performed in QD1.

267

exciton in QD1 cannot transfer to QD2 due to the Pauli exclusion principle. In any
case, relaxation via phonon emission conserves the spin of the involved quasi-particles
The electrons in QD2 and QD3 (detectors) monitor the evolution of the exciton in
the QD1. Our electron-detectors in QD2 and QD3 have a quantum switch. In fact,
they can monitor the a- exciton by letting the system emit a phonon only if their
spin is down. If the electron spin in the detectors QD2 or QD3 is up, the detector is
switched off, since it will not be able to provide the information about the presence
of the polarized exciton. If we consider a. weak resonant 271' pulse with Rabi energy
(2 << I‘, where I" is the phonon emission rate, then it is sufficient to have one detector
on to induce the QZE and freeze the system. Only when both detectors are off, the
27r pulse will give an overall 7r phase to the wave-function of the system. In this
scheme, a 27r exciton Rabi flopping provides then a conditional phase gate on the two
spins. Note that the stronger the phonon dissipation, the better is for the gate. This
is a different approach with respect to current schemes which tends always to fight

dissipation.

6.6 Conclusion

In conclusion, we have presented a general theory of interaction- and measurement-
free quantum Zeno gates and discussed their physical implementation via high-finesse
optical resonators for systems of single-atom and single-photon qubits. We have con-
structed a useful set of logic circuits towards quantum information processing with
only one or at most two IFM devices. Because of the interaction- and measurement-
free features, as well as the lack of need for pre-existing entangled photon-pairs,
our gates are able to overcome some of the major obstacles in processing informa-
tion with single photons and atoms. This work is a significant extension and im-

provement of several previously studied IFM gates [194, 195, 196]. We have for

268

 

 

the first time 1) introduced the interaction-free raman gate, with which the photon-
photon CN OT gate is accomplished using only a single IFM device; 2) constructed
the atom-atom and hybrid atom-photon CNOT gate using circuits which are much
simpler and easier to implement; 3) constructed direct, reversible atom-photon state
transfer circuits; 4) solved the problem of photon loss during long-distance quan-
tum communication via lossy channels with a photon-loss-tolerant protocol. Most
importantly, compared to previously studied IFM gates, our gates do not require
any measurements and are operated asymptotically on-demand. This elimination
of measurement clearly distinguishes our Zeno logic gates from previous approaches
[182, 136, 168, 287, 288, 289, 194, 195, 196, 170, 185, 290, 174]. We have also shown
that the success probability and fidelity upon heralded success for our Zeno gates
scale at 1 — 0%,] and 1 — Obs—2] with N the (effective) number of measurements.
Experimentally, N is limited by three major factors, the rasonator finesse, the imper-
fect absorption of the absorbing state, and the imperfect transparency of transparent
state. While the limits to controlling the imperfect absorbtion and transparency
factors remain to be investigated, we have shown that with available mirror finesse
factors and for realistic atom-photon scattering, a success probability of 98.4% or a
fidelity of 99.994% upon heralded success may be attainable in the near future. To
this end, we note that while the present Zeno gates are discussed in the context of
single atomic and photonic qubits, they can also be implemented in other physical
systems in a straightforward manner. A quantum dot system, for example, is an
ideal alternative, due to its large dipole moment as well as the presence of ultra-fast
phonon-assisted dissipation mechanisms. Indeed, we have extended the idea of quan-
tum Zeno gates to the system of quantum dots, where in Ref. [291], we demonstrate

an implementation of the quantum Zeno phase gate employing phonon dissipations.

269

 

 

 

 

 

 

 

 

 

Figure 6.10: Decay parameters A0 and A1 versus the atom-photon scattering param-
eters 60 and 61. Figure (a) shows A0 versus 60 for N = 102 (solid line and circles),
N = 103 (dashed line and squares), and N = 104 (dahsed-pointed line and triangles).
All lines are the analytic approximations given in equation (6.26), while the points
correspond to exact numerical calculations. In figure (b) we similarly plot A1 versus
N 61, where because the analytical approximation of A1 (6.27) depends only on the
product N 61, three lines for different N s are overlapped. Both figures show excellent
agreements between analytic approximations and numerical calculations.

270

 

6.7 Derivations and further discussions

6.7.1 Pure-State Representation

In this appendix, we will show that the dynamical evolution of quantum-Zeno
systems, while involving dissipations into environment, can nonetheless be described
in a pure-state representation with non-Hermitian Hamiltonians. Our analyses are
based on the standard quantum treatment of an open system [59] , where the dynamics
is governed by a master equation for density matrix. We will take the system with
the level scheme drawn in Fig. 6.1 for an example. Similar results can be obtained
for other systems in a straightforward manner.

With | g) being the atomic state onto which the state |1) spontaneously decays,
the master equation for this system is read as

. z' . l"

p = —;,[Hn,pl - 5 [|1>(1|p + p|1)<1| - 2|g)<1|p|1>(9|]- (6535)
Here, p is the system’s density matrix, I‘ is the spontaneous emission rate. The
Hamiltonian HQ, given by Eq (6.2), is the Rabi coupling between the states IO) and
|1), which, however, does not act on lg). To solve this equation, we expand the density

p onto the {|g)} and {|0), |1)} subspaces, and obtain

p = p99 + P33 + pgs + Psg, (6.36)
with
P99 = |9><9|p|9><9|;
pss = Z |j><jlp|k><kl;
j,k=0,1
P93 = plg= Z Ig><glp|j)<jl;
j=0,1

271

 

 

 

(6.37)

The master equation (6.35) can now be separated into three equations

. 2' . r

pss = -;,[Ho,pss]-§[l1><1|pss+pssll)<lll, (638)
P99 = I‘|9><1|pss|1>(9|, (6-39)
. z‘ . r

Pgs = $109an ‘ Epgsllxll- (6-40)

Clearly, equation (6.38), which governs the motion of p33, is uncoupled from the
rest equations, meaning the states IO), l1) have formed a closed subspace. In this

subspace, for any pure initial state l\I!(O)), equation (6.38) has a. pure-state solution,

p33(t) = l‘I!(t))(\Il(t)l, where
W» = e‘iflefft/”Iw(0>>. (6.41)

with Heff defined in Eq (6.4). Since our Zeno logic gates are operated in the {l0}, l1)}
subspace, the state lg) becomes practically irrelevant, allowing us to describe the
system in a pure state representation of IO) and l1) states. Heff is then the effective
Hamiltonian governing the pure-state dynamics in this subspace. The system decay
is now governed by the non-Hermitian component of Heffa due to which the state

l\Il(t)) is no longer normalized to unity. Conservation of probability requires

Trfpgg} = 1 ‘ TTlPss}, (6-42)

which shows that the reduction in in the normalization of l\11(t)) corresponds to the
probability for the atom to have scattered into the state lg).
Finally, we note that if starting with an initial state in the {ID}, |1)} subspace,

the state lg) will always be uncoupled from the rest of system. This is because the

272

 

 

 

master equation (6.35) does not create any coherence between the two subspaces.
This can be easily seen from Eq (6.40), where with zero initial values, the cross terms

p93, psg will be identically zero at any later times.

6.7.2 Derivation of IFPG propagator

In this appendix we will derive the form of the propagator 17(6, N, 60, 61) which
describes the action of the elementary IFPG pictured in Fig. 6.4. In this system,
only the IV) photon interacts with the atom, and the dissipative states exposed to
spontaneous emission are lVO) and lVl). The standard master equation describing

the dissipative evolution of the system during a single measurement cycle is then [59]

. ’Y' . . . .
p = Z §‘[-|V2><Vzlp—pr2><Vzl
i=0,1

+ 2li)’(Vi|pri)(il’] , (6.43)

where the state lz')’ describes the scattered atomic qubit state with zero photons in
the cavity. The rate constants 70 and 71 will need to be determined from detailed
atom-photon scattering calculations. Following the previous appendix, this equation
has a pure state solution in the {|HO), |H1), lVO), lV1)} subspace, from which we find

that the initial state
I‘I’(0)> = CHOIHO) + CH1|H1>+ CV0|V0> + CV1|V1) (6-44)
evolves into the final state

|\P(t)) = cH0|H0) + cH1|H1) + cVOe’70t/2|VO) + e—‘Ylt/zIVl) (6.45)

273

 

 

 

The measurement stage of the IFPG corresponds to evolution under (6.43) for time
tm such that e—VOtm E 60 and e—7ltm E 1 — 61.
With 3(60, 61) as the propagator for the measurement stage and RM, N) as the

propagator for the rotation stage, the complete IFPG propagator is given by

A

- - N
we, N, 60,61) = [3(40, €1)R(9, ml . (6.46)

In the basis {|H1), lVl), lHO), lVO)} the matrix forms of the propagators are

 

 

{ cos 7% sin 7% O 0 \
- 0 0
. — sm cos 0 0
12(9, N) = N N (6.47)
O 0 cos 7% sin 7%-
\ O O — sin 7% cos 7% j
and
( 1 o o 0 )
. O l/l — 6 O O
B(€0,El) = 1 . (6.48)

0 010
(0 0 O\/e_0}

We have found that to excellent approximation, the full propagator can be expressed

 

 

 

 

as
/-€1 £2 0 0 l
- — — O O
V(6.N,eo.e1)= E2 51 (6.49)
0 0 771 772
K 0 0 -723 -774}
where

61 = e‘N 61/ 4. (6.50)

£2 = [Ml/1012, (6.51)

274

 

 

”Fl—l —2"1+‘/:—0l1/2 , (6.52)

 

N 1 — \/—0

W 1/2
42— — K,- [L + Q] . (6.53)
n3= rfl/N (6.54)
774 = 10\/€5/N2. (6.55)

The validity of these approximations is illustrated in figure 6.10, where the decay
parameters A0 and A1, given by Eqs (6.26) and (6.27), are plotted against the atom-
photon scattering parameters 60 and 61. The excellent agreement confirms our con-
clusions regarding the sensitivity of the success probability and fidelity with respect

to the scattering parameters 60 and 61.

6.7 .3 Extension I: State transfer circuits

In this extension, we construct state-transfer circuits, which map the quantum
state of an atom onto a photon and vice-versa. Unlike the CNOT gatas, state-transfer
circuits require two elementary interaction-free quantum-Zeno gates. The first serves
to entangle the atom and photon, while the second is required for disentanglement.
The atom-to—photon state-transfer circuit is depicted in Fig. 6.11 (a). It consists of
a 1r / 2 IFPG followed by a —7r / 2 IFRG. The photon is prepared in the lH) state, and
the atom is initially in the unknown quantum state colO) + clll). The goal of the
circuit is then to transfer this unknown quantum state onto the state of the photon.

With the input state taken as

I‘I’z'n) = |H>®(Col0>+61|1)) =60|H0>+61|H1), (6-56)

275

 

 

0 Atom ’V‘ photon E Hadamard Gate
3 Atom to Photon state transfer _

 

wig: Ia)

 

 

 

 

 

 

 

 

 

Figure 6.11: Graphical state-transfer circuits for atom—to—photon (a) and photon-
to—atom(b), where |<I>) and I‘ll) denote the arbitrary atomic and photonic states
respectively.

276

the action of the 7r / 2 IFPG is to rotate the polarization of the photon by 7r / 2 condi-

tioned on the atom being in state |1), resulting in the state
|\II) = colHO) + cllVl). (6.57)

The —1r/2 IF RG rotates the atomic state by —7r / 2 conditioned on the photon being

in state lV), resulting in the output state
I‘I’out> = 00|H0> + 61|V0)=(60|H>+ 61|V>) (8 M (6-58)

Thus the quantum state of the atom is written onto the photon, and the atom and
photon are successfully disentangled. The reverse photon-to-atom state transfer cir-
cuit is depicted in figure 6.11 (b). It operates identically to the atom-to—photon
circuit, except the roles of atomic and photonic qubits are reversed. We emphasize
here that the present state-transfer gates operate without the need to measure the
final state of the particle initially possessing the unknown state. This is clearly dif-
ferent and significantly improved from several previously studied schemes, where a
final (Bell) measurement is required [149, 175, 292, 136, 293, 186, 187, 294, 295]. We
note that a measurement-free state-mapping scheme between a coherent light field in

a high-finesse optical cavity and a single atom has been recently demonstrated [272].

6.7.4 Extension II: Long-distance quantum communication

This far, we have constructed the atom-atom phase (CNOT) gate using a single
IFM device, where the two atomic qubits are required to be in a common optical
resonator. This requirement restricts such a circuit to two qubits separated within one
resonator length. For the purpose of quantum communications and / or gate operations
between two distant atomic qubits, one alternative approach would be to indirectly

entangle them via mutual non-interaction with a common ancillary photon.

277

As a first example, we deviate from the measurement-free feature and show that
two uncorrelated qubits can be collapsed to the Bell (Pi or \IFt subspaces by using
two atom-photon Zeno CNOT gates and measuring the final photon state. We note
that for quantum communication via a lossy photonic channel, verifying the presence
of the transported photon is somewhat helpful as it can detect the photon loss and
thus improve the operation fidelity upon heralded success. In our scheme, the photon
is initially in lH) state and the two atoms are in arbitrary states. The initial state of

system is then written as
Win) = lH) ® (COOIOO) + c01|01) + 010|10> + 011|11>)- (6-59)

The photon non-interacts with the first atom via the atom-photon Zeno CNOT gate,
where its polarization reverses (lH) H lV)) only if the atomic qubit is in state |1).

The system afterwards will be in
N”) = lH) a (Cool00) + €01|01>)+ IV)®(010|10>+ €11|11))- (6.60)

After transmission, the photon then non-interacts with the second atom via a second

identical Zeno CN OT gate, alternating the system into

l‘yout) = lH) ‘8 (COOl00> + Cllllll)

+ lV) ® (C10l10) + 001l01)). (6.61)

Accomplishing the quantum communication then requires measuring the photon on
H / V basis. If it is measured in |H), the two atoms will be collapsed onto coolOO) +
c11l11) state, or the (Pi subspace. Otherwise, they will be collapsed to c01l01) +
c10|10) state, or the ‘11:]: subspace. In either way, entanglement between the two is

established, based on which long-distance logic gates can be realized with the addition

278

 

 

of single-qubit operations.

A main and common difficulty in quantum communication between distant qubits
is photon loss during transmission, where the loss probability increases exponentially
with the transport distance. In schemes based on cavity-QED [182, 183, 184, 185],
the quantum states of atomic qubits are encoded in the internal (polarization) states
of photons. A lost photon will therefore immediately collapse the atomic state(s) and
destroy the qubit(s) via decoherence. This problem also exists in the above scheme,
where the intermediate photon state is lH) or IV) depending on the first qubit being
in lO) or l1), as shown in (6.60). One possible way to overcome this difficulty is
to combine the atom-photon phase gate with interferometric phase measurements,
as a photon lost inside an interferometer does not reveal the quantum state of the
measured qubits [40]. In the following we neglect detector inefficiency and dark-count
rates at present, to focus on the problem of transmission loss.

As an example, here we describe a photon-loss-tolerant protocol using 7r-IFPGs
and a Mach-Zehnder (MZ) interferometer. The schematic setup is depicted in Fig.
6.12, where we introduce an ancillary qubit (a—qubit) to restore the quantum states
of the two logical qubits in case of photon loss. In the scheme, a photon in lH) state
is passed through a 50/ 50 beamsplitter and then guided through the 7r-IFPG gates
where it non-interacts successively with the ancillary and :1: atom in the upper arm,

or the y atom in the lower arm. The initial state of system is

I‘I’z'n) = lH) ® |‘1’)a ® (COOIOO) + €01|01)+ €10|10)+ 611|11>), (6-52)

with the ancillary atom prepared in the state

_1

Ma \/§

(|0>a + l1>a)' (6-63)
After the photon passes through the first beamsplitter and non-interacts with ancillary

279

 

and the first logical qubit (x-qubit) via 7r-IFPG gates, the system will be in the state

N”) = §uH>u 6 (10>. — I1).) 6 (6.64)
<coo|00> — 001|01> + C10l10> — 011|11>) +411» 6

(|0)a + |1>a) ® (€00|00> + €01|01l+ 010|10> + 011|11>)l,

where |H),“) indicates the photon in the upper and lower interferometer arm. In
absence of photon loss, the final state, obtained after the photon non-interacts with
the second qubit (y—qubit) via another identical 7r-IFPG and then passes the second

beamsplitter, is

l‘I’o’utl = (lHlulola —ilH>ll1>a) 8’ (001ml) - 010|10>)

-(|H)u|1)a - z'lle|0>a) ® (COOIOO) — 011|11>)- (6-65)

Measuring both the final states of the photon and the ancillary atom will thus project
the two atoms onto either c01lOl) — c10|10) or coolOO) — clllll) states, which estab-
lishes the desired entanglement. Otherwise, if neither of the photon detectors is
triggered, it is most likely that the photon is lost during transmission from either
of the interferometer arms, but without knowing from which. Assuming equal loss

possibilities for the two arms, the system will now be in a statistical mixture of
p‘oss = (Hluw’xw’lmu +<le‘1"><‘1"|H>z- (6.66)

We then need to measure the ancillary atom on Izh)a = (lO)a :l: |1)a)/\/2 basis, and
restore the quantum states of the qubits according to the measurement result. If it

is measured in I+), the state of two atomic qubits is automatically restored, as seen

280

 

 

 

a-qubit x-qubit = 50/50 33

M Detector
E

    

 

Figure 6.12: Schematic setup for long-distance entanglement generation with built-
in error correction. A photon in lH) polarization state is passed through a MZ
interferometer with three 7r-IFPGs, and is then measured by two detectors placed at
the two interferometer outputs. The MZ interferometer consists of two 50/50 linear
beamsplitters (BS) with a 7r / 2 phase imprinted on the reflected photons. In figure,
we label the ancillary and the two logical qubits as a-, x- and y-qubit, respectively.

in (6.64). Otherwise, if it is measured in l—), the atoms will be in state

COOIOO) - 001|01> + 610|10> - €11|11) (6-67)

and a relative 7r-phase applied on the first qubit immediately recovers the original
quantum state of the two qubits. Upon recovery, the whole procedure can be repeated,

until success.

281

 

Chapter 7

Summary and Outlook

In summary, in this dissertation we have studied the rich physics associated with
linear and nonlinear mixing of light and matter waves, while emphasizing applications
in quantum interferometry and information processing. A flow chart showing the
structure of the dissertation is presented in Fig. 7.1. In the following, I will give a
briefly summary on each project while listing the key innovations. I also discuss the

future work that can lead to improvements of the present results.

0 In chapter 2, we proposed a dynamical scheme to create and detect macroscopic
Schr6dinger-cat states in a double-well Bose-Einstein condensate (BEC) system.
We for the first time proposed to use the technique of Feshbach resonance to
suddenly change the inter-atom interaction from repulsive to attractive, thus
dynamically generating the Schr6dinger—cat state. In particular, we originally
studied the decoherence properties of such cat states induced by the trapping
lasers via the master-equation approach. Nonetheless, in the present work we
have used the zero—temperature approximation, assuming all atoms are in the
lowest-energy states. It remains as a future task to study the effect of finite
temperature on the system dynamics, and particularly, to find out the suscep—

tibility of our dynamical approach against possible collisional decoherence by

282

 

 

* mum-mm

 

 

 

Mbthg um and

 

Matter waves

 

 

single-atom
& laser

 

 

 

 

 

donate-well
BEC'

 

Creation, detection, and decoherence of macroscopic
quantum superposition states
in double-well Bose-Einstein condensates
Published: Phys. Rev. A 73, 023% (2006),

 

 

 

 

 

Optimized Dotble-Well Qumim Interferometry with '—
Gaussian Squeezed States

Published: Phys. Rev. Lett. 100, 250406 (2003)

 

 

 

two-component
BEC

 

 

 

 

 

Resonant Matter Wave Amplification in
mean-field theory
To be submitted for publication

 

 

 

Optical
interferometry

 

 

 

 

Orr-demand generation of entanglement of atomic
qubits via optical interferometry
Published: Phys. Rev. A. TI, 032349 (2008)

 

 

 

Quantum Zeno
effect

 

Interaction- and measuranent-free quantum Zeno
gates for universal computation
with single-atom and single-photon qubits
Published: Phys. Rev. A. 7?, 062332 (2008)

 

 

 

 

BEC‘: Bose-Einstein condensate

 

Two-thit Conditional Phase Gate in Laser-Embed
Semiconductor Quantum Dots Using the Quantim
Zeno Effect
Published: Phys. Rev. Lett. 103, 037401 (2009)

(collaborate with C. Plermarocchl’s group)

 

Figure 7.1: The flow chart of this dissertation.

283

 

thermal atoms. Further, we have assumed the localized left-well and right-well
condensate modes to be constant throughout the dynamics. While this is a
valid assumption as proven in recent experiments, it is more physical to employ

adaptive time-variant mode functions.

In chapter 3, we proposed to implement a Mach-Zehnder interferometer in a
double-well BEC system for ultra-efficient phase measurements. The input state
of the interferometer is the ground state of the system in presence of repulsive
interaction, and is thus easy to produce. We for the first time systematically
studied the optimal number-squeezing level of the input state for the best inter-
ferometer performance when measuring non-zero phases. The squeezing level
is easily tuned by using the Feshbach resonance technique. Combined with
adaptive measurement schemes, our apparatus can measure an arbitrarily un-
known phase at a Heisenberg-scaling resolution within only a few measurements.
This corresponds to an enormous improvement in the phase-measurement ef-
ficiency over the previous approaches, employing maximally number-squeezed
or entangled input states. In this work, however, we have again used the zero-
temperature and the time-invariant mode approximations, as in chapter 2. A
future improvement can be to go beyond these approximations. Also, here
we did not consider the possible collisional dephasing effect during the phase
measurement process. It will be helpful to characterize this dephasing effect
to determine the maximally allowed residual atom-atom interaction during the
phase measurement, and possibly, to use this effect to further reduce the phase

uncertainty.

In chapter 4, we studied the matter-wave amplification (MWA) process in a
two-component BEC system. Previous studies of this system were focused on

the off-resonance dynamic regime, whereas here we extended the studies to the

284

 

 

 

resonant regime. We found that the atom loss fraction during the MWA pro-
cess is a universal quantity independent of dynamical regimes. For a given laser
intensity, a significantly faster amplification process can be achieved employing
a resonant rather than off-resonance driving laser. Due to phase distortion by
the ac-Stark shift, it turns out an off-resonance three-level system is unsuitable
for state mapping between light and matter waves, while a resonant system is
not subjected to this problem. We also identified the laser depletion and spa-
tial effect by comparing the simulation results via quantum coupled-mode and
semiclassical multi-mode theories. In the present multi-mode model, however,
we only considered the light to propagate along the longitude axis of the con-
densates. While this is a valid approximation in the geometry limit of a large
Fresnel-number f, it is important to include the transversely-propagation light
in the model when .7: S, 1, especially when such a geometry helps to reduce the
atom loss fraction. Finally, in this work, the system dynamics is discussed in
the context of mean field theory. To study the associated quantum fluctuation
during the MWA process, a full quantum model is needed. This kind of work
is important when a MWA system is used for coherent state-mapping between

light and matter waves.

In chapter 5, we proposed to create entanglement between single-atom qubits
by putting them in a common optical interferometer. Previous approaches
using single-atom qubits rely on using high-finesse cavities to enhance atom-
photon interaction while suppressing spontaneous emission rates. We showed,
however, that the requirement for cavities can be circumvented by employ-
ing ultra-sensitive interferometers operating at Heisenberg-scaling sensitivities.
Specifically, we proposed to use a Mach-Zehnder interferometer with the maxi-
mally squeezed twin-Fock input state and the NOON-state interferometer with

nonlinear beamsplitters. These new approaches can lead to scalable elements for

285

 

 

quantum information processing, with a close-to-unity success probability and
fidelity. Yet, it remains as an experimental challenge to create highly squeezed
optical beams in the twin-Fock state, although there have been some steady
progresses. Towards this goal, we are now studying an optical four-wave mixing
approach in thermal gases to create such beams of high brightness. Further-
more, in the N OON-state approach, the nonlinear beamsplitter is demonstrated
in an atomic medium subjected to the Kerr-nonlinearity. It remains as the fu-
ture task to carefully examine the performance of such a beamsplitter for various

input states.

In chapter 6, we used the quantum Zeno effect to construct a series of interaction-
and measurement- free quantum logic gates. Applying to the system of single-
atom and single-photon qubits, we have for the first time 1) introduced the
interaction-free Raman gate, with which the photon-photon CNOT gate is ac-
complished using only a single interaction-free device; 2) constructed the atom-
atom and hybrid atom-photon CN OT gate using circuits which are much simpler
and easier to implement; 3) constructed direct, reversible atom-photon state
transfer circuits. Besides, we have solved the problem of photon loss during
long-distance quantum communication via lossy channels with a photon-loss-
tolerant protocol. In collaboration with the C. Piermarocchi group, we further
extended the two-qubit phase gate to the quantum-dot systems, where we for
the first time proposed a feasible interaction-free quantum logic gate between
two quantum-dot qubits. compared to previously studied quantum-Zeno gates,
our gates do not require any measurements and are operated asymptotically
on-demand. This elimination of measurement clearly distinguishes our Zeno
logic gates from previous approaches. An important remaining task in the
atom-photon project is to carefully characterize the property of single-atom

and single-photon scattering in free space, as our scheme is sensitive to the

286

 

atom-photon scattering probabilities. In addition, to further improve the gate
fidelity, a possible way is to replace the single-atom qubits with atomic—ensemble
qubits, thus significantly increasing the photon absorption probability. This idea
is, however, limited by the inherent collisional dephasing effect associated with
the ensemble qubits. A meaningful work in future is to overcome this limita-
tion, thereby greatly increasing the gate successful probability and fidelity. In
the quantum-dot project, on the other hand, the qubit-system is embedded in
a high-Q cavity to increase the lifetime of ancillary excitons. To obtain a better
gate fidelity, one may seek to further extend the exciton’s lifetime, for example,

by optimizing the shape of the trapping cavity.

287

 

Bibliography

[1] C. Davisson and L. H. Germer. The scattering of electrons by a single crystal
of nickel. Nature, 1192558, 1927.

[2] L. de Broglie. Researches on the quantum theory. PhD thesis, Paris, 1924.
[3] L. de Broglie. Ann. Phys, 3:22, 1925.

[4] S. N. Bose. Plancks gesetz und lichtquantenhypothese. Zeitschrift fr Physik,
262178, 1924.

[5] A. Einstein. Quantentheorie des einatomigen idealen gases. Sitzungsberichte
der Preussischen Akademie der Wissenschaften, 1:3, 1925.

[6] L. P. Pitaevskii and S. Stringari. Bose-Einstein condensation. Clarendon Press,
OXford, 2003.

[7] K. Huang. Statistical Mechanics. Wiley, New York, 1987.
[8] F. London. Supe'rfluids. Wiley, New York, 1950.

[9] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,
and W. Ketterle. Observation of interference between two bose condensates.
Science, 275(53OO)2637—641, 1997.

[10] T. A. Fulton, P. L. Gammel, D. J. Bishop, L. N. Dunkleberger, and G. J. Dolan.
Observation of combined josephson and charging effects in small tunnel junction
circuits. Phys. Rev. Lett., 63(12):1307—1310, Sep 1989.

[11] P. W. Anderson. Rev. Mod. Phys., 382298, 1966.
[12] H. Feshbach. H. Feshbach. Wiley, New York, Theoretical Nuclear Physics.

[13] S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo, R. Walser, and M. J.
Holland. Phys. Rev. A, 65:053617, 2002.

[14] M. Greiner, C. A. Regal, and D. S. Jin. Phys. Rev. Lett., 942070403, 2005.

[15] Jelena Stajic, J. N. Milstein, Qijin Chen, M. L. Chiofalo, M. J. Holland, and
K. Levin. Phys. Rev. A, 692063610, 2004.

[16] O. Mandel M. Greiner, T. Esslinger, T. W. Hasch, and I. Bloch. Nature (Lon-
don), 415z39, 2002.

288

 

[17] T. Kinoshita, T. Wegner, and D. S. Weiss. Science, 305:1125, 2004.

[18] C. D. Fertig, K. M. O’Hara, J. H. Huckans, S. L. Rolston, W. D. Phillips, and
J. V. Porto. Phys. Rev. Lett., 94:120403, 2005.

[19] B. L. Tolra, K. M. O’Hara, J. H. Huckens, W. D. Phillips, S. L. Rolston, and
J. V. Porto. Phys. Rev. Lett., 922190401, 2004.

[20] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fc'iling, G. V. Shlyapnikov
I. Cirac, T. W. Hfisch, and I. Bloch. Nature, 429:277, 2004.

[21] T. L. Ho. Phys. Rev. Lett., 87:060403, 2002.

[22] J. R. Abo—Shaeer, C. Raman, and W. Ketterle. Phys. Rev. Lett., 882070409,
2002.

[23] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen. Phys. Rev. A,
542R4649, 1996.

[24] J. J. Bollinger, Wayne M. Itano, D. J. Wineland, and D. J. Heinzen. Phys. Rev.
A, 542R4649, 1996.

[25] S. F. Huelga et al. Phys. Rev. Lett., 7923865, 1997.
[26] C. C. Gerry. Phys. Rev. A, 61:043811, 2000.

[27] W. J. Munro, K. Nemoto, G. J. Milburn, and S. L. Braunstein. Phys. Rev. A,
662023819, 2002.

[28] B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein. Atomic
bose and anderson glasses in optical lattices. Phys. Rev. Lett., 91(8)1080403,
Aug 2003.

[29] A. Sanpera, A. Kantian, L. Sanchez-Palencia, J. Zakrzewski, and M. Lewen-
stein. Atomic fermi-bose mixtures in inhomogeneous and random lattices: From

fermi glass to quantum spin glass and quantum percolation. Phys. Rev. Lett.,
93(4)204O401, Jul 2004.

[30] P. Bouyer and M. A. Kasevich. Phys. Rev. A, 56:RIO83, 1997.

[31] Y. Shin, M. Saba, W. Ketterle T. A. Pasquini, D. E. Pritchard, and A. E.
Leanhardt. Phys. Rev. Lett., 922050405, 2004.

[32] F. Dalfovo, S. Giogini, L. P. Pitaevskii, and S. Stringari. Rev. Mod. Phys, 71:
463, 1999.

[33] M. A. Kasevich. Science, 298:1363, 2002.
[34] M. Antezza, L. P. Pitaevskii, and S. Stringari. Phys. Rev. A, 702053619, 2004.

289

 

[35] G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio.
Phys. Rev. Lett., 922230402, 2004.

[36] D. M. Harber J. M. Obrecht, J. M. McGuirk, and E. A. Cornell. Phys. Rev. A,
72:033610, 2005.

[37] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, 1. Bar-
Joseph, and J. Schmiedmayer. Nature Physics, 1257, 2005.

[38] C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G.
Raizen. Direct observation of sub-poissonian number statistics in a degenerate
bose gas. Phys. Rev. Lett., 95(26):260403, Dec 2005.

[39] M. Shellekens, R. Hoppeler, A. Perrin, J .V. Gomes, D. Boiron, A. Aspect, and
Cl. Westbrook. Science, 3102648, 2005.

[40] Y. P. Huang and M. G. Moore. Optimized double-well quantum interferometry
with gaussian squeezed states. Phys. Rev. Lett., 100(25)22504O6, 2008.

[41] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell. Science, 269:198, 1995.

[42] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Bose-einstein condensation in a gas of sodium
atoms. Phys. Rev. Lett., 75(22):3969—3973, Nov 1995.

[43] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of bose-
einstein condensation in an atomic gas with attractive interactions. Phys. Rev.
Lett., 75(9):1687—1690, Aug 1995.

[44] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-einstein condensation of
lithium: Observation of limited condensate number. Phys. Rev. Lett., 78(6):
985—989, Feb 1997.

[45] Dale G. Fried, Thomas C. Killian, Lorenz Willmann, David Landhuis,
Stephen C. Moss, Daniel Kleppner, and Thomas J. Greytak. Bose-einstein
condensation of atomic hydrogen. Phys. Rev. Lett., 81(18):3811—3814, Nov
1998.

[46] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman.
Stable 85rb bose—einstein condensates with widely tunable interactions. Phys.
Rev. Lett., 85(9)21795—1798, Aug 2000.

[47] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio.
Bose-einstein condensation of potassium atoms by sympathetic cooling. Science,
294(5545):1320—1322, 2001.

[48] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I.
Westbrook, and A. Aspect. A bose-einstein condensate of metastable atoms.
Science, 292(5516):461—464, 2001.

290

 

[49] F. Pereira Dos Santos, J. Léonard, Junmin Wang, C. J. Barrelet, F. Perales,
E. Rasel, C. S. Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji. Bose-einstein
condensation of metastable helium. Phys. Rev. Lett., 86(16):3459—3462, Apr
2001.

[50] Tino Weber, Jens Herbig, Michael Mark, Hanns-Christoph Nagerl, and Rudolf
Grimm. Bose-einstein condensation of cesium. Science, 299(5604):232—235,
2003.

[51] Yosuke Takasu, Kenichi Maki, Kaduki Komori, Tetsushi Takano, Kazuhito
Honda, Mitsutaka Kumakura, Tsutomu Yabuzaki, and Yoshiro Takahashi.

Spin-singlet hose—einstein condensation of two-electron atoms. Phys. Rev. Lett.,
91(4)2O4O4O4, Jul 2003.

[52] Axel Griesmaier, Jorg Werner, Sven Hensler, Jiirgen Stuhler, and Tilman Pfau.
Bose-einstein condensation of chromium. Phys. Rev. Lett., 94(16):160401, Apr
2005.

[53] D. E. Pritchard, Helmerson K., and A. G. Martin. Atomic Phyiscs II, page 179.
World Scientific, Singapore, 1989.

[54] K. Helmerson, A. Martin, and D. E. Pritchard. J. Opt. Soc. Am. B, 9:1988,
1992.

[55] C. N. Cohen-Tannoudji and W. D. Phillips. New mechanisms of laser cooling.
Phys. Today, 43(10):33—40, 1990.

[56] Harald F. Hess. Evaporative cooling of magnetically trapped and compressed
spin-polarized hydrogen. Phys. Rev. B, 34(5):3476—3479, Sep 1986.

[57] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman. Pro-
duction of two overlapping bose—einstein condensates by sympathetic cooling.
Phys. Rev. Lett., 78(4):586—589, Jan 1997.

[58] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and
W. Ketterle. Output coupler for bose-einstein condensed atoms. Phys. Rev.
Lett., 78(4):582—585, Jan 1997.

[59] Pierre Meystre. Atom Optics. Springer-Verlag, New York, 2001.

[60] Jozsef Fortagh and Claus Zimmermann. Magnetic microtraps for ultracold
atoms. Reviews of Modern Physics, 79(1):235, 2007.

[61] J. D. Weinstein and K. G. Libbrecht. Microscopic magnetic traps for neutral
atoms. Phys. Rev. A, 52(5):4004—4009, Nov 1995.

[62] H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann.
Bose-einstein condensation in a surface microtrap. Phys. Rev. Lett., 87(23):
230401, Nov 2001.

291

 

 

[63] W. Hansel, J. Reichel, P. Hommelhoff, and T. W. Hansch. Trapped-atom in-
terferometer in a magnetic microtrap. Phys. Rev. A, 64(6):O63607, Nov 2001.

[64] A. Giinther, S. Kraft, M. Kemmler, D. Koelle, R. Kleiner, C. Zimmermann, and
J. Fortagh. Diffraction of a hose-einstein condensate from a magnetic lattice on
a microchip. Phys. Rev. Lett., 95(17):170405, Oct 2005.

[65] Y. Shin, C. Sanner, G.—B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, D. E.
Pritchard, M. Vengalattore, and M. Prentiss. Interference of bose-einstein con-
densates split with an atom chip. Phys. Rev. A, 72(2)2021604, Aug 2005.

[66] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls. Quantum dynamics
of an atomic hose-einstein condensate in a double-well potential. Phys. Rev. A,
55(6):4318—4324, Jun 1997.

[67] A. Smerzi, S. F antoni, S. Giovanazzi, and S. R. Shenoy. Quantum coherent
atomic tunneling between two trapped bose—einstein condensates. Phys. Rev.
Lett., 79(25):4950—4953, Dec 1997.

[68] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy. Coherent oscillations
between two weakly coupled bose—einstein condensates: Josephson effects, 7r

oscillations, and macroscopic quantum self-trapping. Phys. Rev. A, 59(1):620—
633, Jan 1999.

[69] A. J. Leggett. Rev. Mod. Phys, 73:307, 2001.

[70] A. P. Hines, R. H. McKenzie, and G. J. Milburn. Phys and Rev. A, 67:013609,
2003.

[71] Michael Albiez, Rudolf Gati, Jonas Féilling, Stefan Hunsmann, Matteo Cris-
tiani, and Markus K. Oberthaler. Direct observation of tunneling and nonlin-

ear self-trapping in a single bosonic josephson junction. Phys. Rev. Lett., 95(1):
010402, Jun 2005.

[72] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller. Phys. Rev. A, 67:013607, 2003.

[73] A. Sorenson, L. M. Duan, J. I. Cirac, and P. Zoller. Nature (London), 409:63,
2001.

[74] S. Raghavan, A. Smerzi, and V. M. Kenkre. Phys. Rev. A, 60:R1787, 1999.
[75] K. W. Mahmud, H. Perry, and W. P. Reinhardt. J. Phys. B, 36:L265, 2003.

[76] Matthew P. A. Fisher, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher.
Boson localization and the superfluid—insulator transition. Phys. Rev. B, 40(1):
546—570, Jul 1989.

[77] E. Schrédinger. Naturwissenschaften, 232807, 1935.
[78] T. J. Osborne and M. A. Nielsen. Phys. Rev. A, 662032110, 2002.

292

[79] M. C. Arnesen, S. Bose, and V. Vedral. Phys. Rev. Lett., 872017901, 2001.

[80] R. Gati, M. Albiez, J. F lling, B. Hemmerling, and MK. Oberthaler. Appl.
Phys. B, 822207, 2006.

[81] C. Lee. Phys. Rev. Lett., 972150402, 2006.

[82] L. Pezzé, A. Smerzi, G. P. Berman, A. R. Bishop, and L. A. Collins. Phys. Rev.
A, 742033610, 2006.

[83] G. B. Jo et al. Phys. Rev. Lett., 982030407, 2007.

[84] H. Fizeau. Ann. Phys. Chem, 3:457, 1853.

[85] L. Rayleigh. Philos. Mag., 112196, 1881.

[86] A. A. Michelson. Am. J. Sci, 22:20, 1881.

[87] L. Marton. Electron interferometer. Phys. Rev., 85(6):1057—1058, Mar 1952.
[88] H. Maier—Leibnitz and T. Springer. Z. Phys, 167:368, 1962.

[89] G. Badurek, H. Rauch, and A. Zeilinger. Physica B, 1512R9, 1988.

[90] Douglas S. Robertson. Geophysical applications of very-long-baseline interfer-
ometry. Rev. Mod. Phys, 63(4)2899—918, Oct 1991.

[91] Ojars J. Sovers, John L. Fanselow, and Christopher S. Jacobs. Astrometry and
geodesy with radio interferometry: experiments, models, results. Rev. Mod.
Phys, 70(4):1393-—1454, Oct 1998.

[92] R. Colella, A. W. Overhauser, and S. A. Werner. Observation of gravitationally
induced quantum interference. Phys. Rev. Lett., 34(23):1472—1474, Jun 1975.

[93] Mark Kasevich and Steven Chu. Atomic interferometry using stimulated raman
transitions. Phys. Rev. Lett., 67(2):181—184, Jul 1991.

[94] Markus K. Oberthaler, Stefan Bernet, Ernst M. Rasel, Jéirg Schmiedmayer, and
Anton Zeilinger. Inertial sensing with classical atomic beams. Phys. Rev. A, 54
(4):3165—3176, Oct 1996.

[95] Karin Sangster, E. A. Hinds, Stephen M. Barnett, and Erling Riis. Measurement
of the aharonov-casher phase in an atomic system. Phys. Rev. Lett., 71(22):
3641—3644, Nov 1993.

[96] A. G6rlitz, B. Schuh, and A. Weis. Measurement of the aharonov—casher phase
of aligned rb atoms. Phys. Rev. A, 51(6):R4305—R4308, Jun 1995.

[97] S. A. Werner, J. L. Staudenmann, R. Colella, and A. W. Overhauser. Neutron
Interferometry, page 209. Clarendon Press, OXford, 1979.

293

[98] A. Tonomura. Electron Holography. Springer-Verlag, New York, 1993.

[99] C. K. Law and N. P. Bigelow. Amplifying an atomic wave signal using a bose-
einstein condensate. Phys. Rev. A, 58(6)24791—4795, Dec 1998.

[100] S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard,
and W. Ketterle. Science, 2852571, 1999.

[101] Mikio Kozuma, Yoichi Suzuki, Yoshio Torii, Toshiaki Sugiura, Takahiro Kuga,
E. W. Hagley, and L. Deng. Phase-coherent amplification of matter waves.
Science, 286(5448):2309—2312, 1999.

[102] S. Inouye, T. Pfau, S. Gupta, A. P. Chikkatur, A. Géirlitz, D. E. Pritchard, and
W. Ketterle. Nature, 4022641, 1999.

[103] M. G. Moore and P. Meystre. Phys. Rev. Lett., 8325202, 1999.

[104] Konstantin V. Krutitsky, Frank Burgbacher, and Jiirgen Audretsch. Local-field
approach to the interaction of an ultracold dense bose gas with a light field.
Phys. Rev. A, 59(2):1517, Feb 1999.

[105] Ozgiir E. Miistecaplioglu and L. You. Superradiant light scattering from
trapped bose-einstein condensates. Phys. Rev. A, 62(6):O63615, Nov 2000.

[106] S. Inouye, R. F. L6w, S. Gupta, T. Pfau, A. Géirlitz, T. L. Gustavson, D. E.
Pritchard, and W. Ketterle. Amplification of light and atoms in a bose-einstein
condensate. Phys. Rev. Lett., 85(20)24225, Nov 2000.

[107] E. D. Trifonov. JETP, 93(4):969, Oct 2001.

[108] Dominik Schneble, Yoshio Torii, Micah Boyd, Erik W. Streed, David E.
Pritchard, and Wolfgang Ketterle. The onset of matter-wave amplification in a
superradiant bose—einstein condensate. Science, 300(5618):475, 2003.

[109] N. Piovella, V. Beretta, G. R. M. Robb, and R. Bonifacio. Photon echo in the
super-radiant light scattering from a bose-einstein condensate. Phys. Rev. A,
68(2)2021801, Aug 2003.

[110] D. Schneble, K. C. Gretchen, W. S. Erik, B. Micah, D. E. Pritchard, and
W. Ketterle. Phys. Rev. A, 692041601(R), 2004.

[111] Y. Yoshikawa, T. Sugiura, Y. Torii, and T. Kuga. Phys. Rev. A, 69:041603(R),
2004.

[112] Dominik Schneble, Gretchen K. Campbell, Erik W. Streed, Micah Boyd,
David E. Pritchard, and Wolfgang Ketterle. Raman amplification of matter
waves. Phys. Rev. A (Atomic, Molecular, and Optical Physics), 69(4)2041601,
2004.

294

[113] Yu. A Avetisyan and E. D. Trifonov. Multimode model of superradiant scat-
tering from bec of dilute gas. Laser Phys. Lett., 7:373, 2004.

[114] Mary M. Cola and Nicola Piovella. Theory of collective raman scattering from
a bose—einstein condensate. Phys. Rev. A, 70(4)2O45601, Oct 2004.

[115] E. D. Trifonov. Laser Phys. Lett., 2(3)2153, 2005.

[116] R. Bonifacio, N. Piovella, G. R. M. Robb, and M. M. Cola. Opt. Commun, 252:
381, 2005.

[117] A. Hilliard, F. Kaminski, R. 1e Targat, C. Olausson, E. S. Polzik, and J. H.
Miiller. Rayleigh superradiance and dynamic bragg gratings in an end-pumped

bose-einstein condensate. Phys. Rev. A (Atomic, Molecular, and Optical
Physics), 78(5):051403, 2008.

[118] S. L. Rolston and W. D. Phillips. Nature (London), 4162219, 2002.
[119] L. Pezzé and A. Smerzi. Phys. Rev. A, 73:011801(R), 2006.

[120] J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac,
and E. S. Polzik. Quantum teleportation between light and matter. Nature,
443:557—560, Dec 2006.

[121] Michael Fleischhauer, Atac Imamoglu, and Jonathan P. Marangos. Electromag—
netically induced transparency: Optics in coherent media. Reviews of Modern
Physics, 77(2):633, 2005.

[122] L.-M. Duan and C. Monroe. Robust probablistic quantum information process-
ing with atoms photons and atomic ensembles. Advances in atomic, molecular
and optical physics, 55:419, Dec 2008.

[123] W. Zhang and D. F. Walls. Quantum field theory of interaction of ultracold
atoms with a light wave: Bragg scattering in nonlinear atom optics. Phys. Rev.
A, 4923799, 1994.

[124] Tun Wang and S. F. Yelin. Theory for raman superradiance in atomic gases.
Phys. Rev. A, 72(4)2043804, Oct 2005.

[125] H. Uys and P. Meystre. Superradiant raman scattering in an ultracold bose gas
at finite temperature. Phys. Rev. A, 77(6)2O63614, 2008.

[126] E. D Trifonov and N. I Shamro. Physical and Quantum Optics, 962258, 2004.

[127] O. Zobay and Georgios M. Nikolopoulos. Dynamics of matter-wave and optical
fields in superradiant scattering from bose-einstein condensates. Phys. Rev. A,
72(4)2O41604, 2005.

[128] N. I Shamrov. Journal of Applied Spectroscopy, 73:68, 2006.

295

[129] O. Zobay and Georgios M. Nikolopoulos. Spatial effects in superradth rayleigh
scattering from bose—einstein condensates. Phys. Rev. A, 73(1):013620, 2006.

[130] O. Zobay and Georgios M. Nikolopoulos. Sequential superradiant scattering
from atomic bose—einstein condensates. Laser Physics, 172180, 2007.

[131] H. Uys and P. Meystre. Theory of coherent raman superradiance imaging of
condensed bose gases. Phys. Rev. A, 75(3)2033805, 2007.

[132] Yu. A Avetisyan and E. D. Trifonov. Maxwell-schrodinger equations for a dilute
gas hose-einstein condensate coupled to an electromeganetic field. Journal of
Experimental and Theoretical Physics, 1062426, 2008.

[133] Yu. A Avetisyan and E. D. Tfifonov. Superradiant reflection and scattering of
light from a hose-einstein condensate of a dilute gas. Optics and Spectroscopy,
105:557, 2008.

[134] Han Pu, Weiping Zhang, and Pierre Meystre. Phys. Rev. Lett., 912150407, 2003.

[135] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-
Kum. Coherenceenhanced imaging of a degenerate bose-einstein gas. Phys.
Rev. Lett., 98(11)2110401, 2007.

[136] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Infor-
mation. 2000.

[137] J. Audretsch, R. Blatt, M. Esfeld, C. Held, E. Joos, R. Léiw, T. Pfau, G. Rempe,
H. Weinfurter, and R. F. Werner. Entangled World. WILEY-VCH Verlag GmbH
& Co. KGaA, 2006.

[138] R. P. Feynman. Sitmulating physics with computers. Int. J. Theor. Phys, 21:
467, 1982.

[139] C. H. Bennett and G. Brassard. Proc. IEEE. Int. Conf. Computer Systems and
Signal Processing. IEEE, New York, 1984.

[140] David Deutsch. The church-turing principle and the universal quantum com-
puter. Proceedings of the Royal Society of London A, 400297, 1985.

[141] David Deutseh and Richard Jozsa. Rapid solutions of problems by quantum
computation. Proceedings of the Royal Society of London A, 4392553, 1992.

[142] Peter W. Shor. Algorithms for quantum computation: Discrete logarithms and
factoring. IEEE Symposium on Foundations of Computer Science, page 124,
1994.

[143] Lieven M. K. Vandersypen, Matthias Steffen, Gregory Breyta, Costantino S.
Yannoni, Mark H. Sherwood, and Isaac L. Chuang. Experimental realization of
shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature,

414(6866):883—~887, December 2001.

296

[144] L. K. Grover. A fast quantum mechanical algorithm for database search. Pro-
ceedings, 28th Annual ACM Symposium on the Theory of Computing, page 212,
1996.

[145] G. Vernam. J. Am. Inst. Elect. Eng., 452109, 1926.
[146] J. S. Bell. On the einstein podolsky rosen paradox. Physics, 1:195, 1964.
[147] Mark Fox. Quantum Optics: an introduction. Oxford University Press, 2006.

[148] Richard P. Feynman, Jr. Frank L. Vernon, and Robert W. Hellwarth. Geo-
metrical representation of the schrlo-umlautldinger equation for solving maser
problems. Journal of Applied Physics, 28(1):49—52, 1957.

[149] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Woot-
ters. Phys. Rev. Lett., 70:1895, 1993.

[150] L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik. Phys. Rev. Lett., 8525643,
2000.

[151] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller. long-distance quantum
communication with atomic ensembles and linear optics. Nature, 4142413, 2001.

[152] Gustavo A. N arvaez, Gabriel Bester, Alberto Franceschetti, and Alex Zunger.
Excitonic exchange effects on the radiative decay time of monoexcitons and
biexcitons in quantum dots. Physical Review B ( Condensed Matter and Mate-

rials Physics), 74(20)22O5422, 2006.

[153] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatiire, S. Gulde, S. Falt,
E. L. Hu, and A. Imamoglu. Quantum nature of a strongly coupled single
quantum dot-cavity system. Nature, 4452896, 2007.

[154] E. M. Purcell. Phys. Rev., 692681, 1946.

[155] C J. Hood, H. J. Kimble, and J.Ye. Phys. Rev. A, 642033804(R), 2001.
[156] J. M. Raimond, M. Brune, and S. Haroche. Rev. Mod. Phys, 732565, 2001.
[157] B. Yurke, S. L. McCall, and J. R. Klauder. Phys. Rev. A., 3324033, 1986.
[158] M. J. Holland and K. Burnett. Phys. Rev. Lett., 7121355, 1993.

[159] A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and
G. Camy. Phys. Rev. Lett., 5922555, 1987.

[160] B. Lantz, P. Fritschel, H. Rong, E. Daw, and G. Gonzalez. J. Opt. Soc. Am.
A, 19:91, 2002.

[161] C. B. Chin, E. C. G. Sudarshan, and B. Misra. Time evolution of unstable
quantum states and a resolution of zeno’s paradox. Phys. Rev. D, 16(2)252O—
529, Jul 1977.

297

 

[162] L.-M. Duan and G.—C. Guo. Phys. Rev. A, 5823491, 1998.
[163] P. Zanardi and F. Rossi. Phys. Rev. B, 5928170, 1999.

[164] A. Beige, D. Braun, B. Tregenna, and P. L. Knight. Phys. Rev. Lett., 8521762,
2000.

[165] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble. Phys.
Rev. Lett., 7524710, 1995.

[166] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland.
Phys. Rev. Lett., 7524714, 1995.

[167] A. Rauschenbeutel, G. Nogues, P. Bertet S. Osnaghi, M. Brune, J. M. Raimond,
and S. Haroche. Phys. Rev. Lett., 8325166, 1999.

[168] E. Knill, R. Laflamme, and G. J. Milburn. Nature, 409246, 2001.

[169] K. J. Resch, J. S. Lundeen, and A. M. Steinberg. Phys. Rev. Lett., 892037904,
2002.

[170] T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson. Phys. Rev. A, 682
032316, 2003.

[171] J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning. Nature,
4262264, 2003.

[172] D. Leibfried, B. DeMarco, V. Meyer, M. Barrett D. Lucas, J. Britton, W. M.
Itano, B. Jelenkovié, C. Langer, T. Rosenband, and D. J. Wineland. Nature,
4222412, 2003.

[173] T. Yamamoto, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai.
Nature, 4252941, 2003.

[174] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J.
Milburn. Rev. Mod. Phys, 792135, 2007.

[175] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and
A. Zeilinger. Experimental quantum teleportation. Nature, 390:575—579, Dec
1997.

[176] C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang,
J. Yang, J. Yin, Q. Zhang, N. Li, B. L.Tian, and J. W. Pan. Phys. Rev. Lett.,
942150501, 2005.

[177] T.-Y. Chen, J. Zhang, J.-C. Boileau, X.—M. Jin, B. Yang, Q. Zhang, T. Yang,
R. Laflamme, and J. W. Pan. Phys. Rev. Lett., 962150504, 2006.

[178] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Phys. Rev. Lett., 8424737,
2000.

298

 

[179] S.L. Braunstein and H.J. Kimble. Phys. Rev. A, 612042302, 2000.
[180] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg. Nature, 4292161, 2004.

[181] I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, and N. Gisin. Long-
distance teleportation of qubits at telecommunication wavelengths. Nature, 421
(6922)25O9—513, January 2003.

[182] S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral. Phys. Rev. Lett., 8325158,
1999.

[183] S.—B. Zheng and G.-C. Guo. Phys. Rev. A, 732032329, 2006.
[184] L.-M. Duan and H. J. Kimble. Phys. Rev. Lett., 902253601, 2003.

[185] T. Di, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy. Phys. Rev. A, 71:
062308, 2005.

[186] M. Riebe, H. Haffner, C. F. Roos, W. Hansel, J. Benhelm, G. P. T. Lancaster,
T. W. K6rber, C. Becher, F. Schmidt-Kaler, D. F. V. James, and R. Blatt.
Nature, 4292734, 2004.

[187] M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost,
E. Knill, C. Langer, D. Liebfried, R. Ozeri, and D. J. Wineland. Nature, 429:
737, 2004.

[188] N. A. Gershenfeld and I. L. Chuang. Science, 275:350, 1997.

[189] P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White. Grover’s
search algorithm: an optical approach. J. Mod. Opt, 472257, 2000.

[190] D. Cory, A. Fahmy, and T. Havel. Proc. Nat. Acad. Sci, 9421634, 1997.

[191] Isaac L. Chuang, Neil Gershenfeld, and Mark Kubinec. Experimental imple-
mentation of fast quantum searching. Phys. Rev. Lett., 80(15):3408—3411, Apr
1998. .

[192] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I.
Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf.
Demonstration of two-qubit algorithms with a superconducting quantum pro—
cessor. Nature, advanced online publication:—, June 2009.

[193] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G.Walker,
and M. Saffman. Observation of rydberg blockade between two atoms. Nat.
Phys, 5:110, 2009.

[194] A. Gilchrist, A. G. White, and W. J. Munro. Phys. Rev. A, 662012106, 2002.
[195] H. Azuma. Phys. Rev. A, 682022320, 2003.

299

[196] H. Azuma. Phys. Rev. A, 702012318, 2004.
[197] J. D. Franson, B. C. Jacobs, and T. B. Pittman. Phys. Rev. A, 702062302, 2004.

[198] O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat. Nature,
4392949, 2006.

[199] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu. Experimental
realization of teleporting an unknown pure quantum state via dual classical and
einstein-podolsky-rosen channels. Phys. Rev. Lett., 80(6):1121—1125, Feb 1998.

[200] Jian-Wei Pan, Matthew Daniel], Sara Gasparoni, Gregor Weihs, and Anton
Zeilinger. Experimental demonstration of four-photon entanglement and high-
fidelity teleportation. Phys. Rev. Lett., 86(20)24435—4438, May 2001.

[201] S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and
C. Monroe. Quantum teleportation between distant matter qubits. Science,
323(5913):486—489, 2009.

[202] W. H. Zurek. Phys. Today, 44236, 1991.

[203] J. R. Ffiedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens. Nature
(London), 60:43, 2000.

[204] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der. Zouw, and
A. Zeilinger. Nature (London), 4012680, 1999.

[205] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. Science, 27221131,
1996.

[206] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt,
M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe.
Experimental entanglement of four particles. Nature, 404(6775)2256—259, March
2000.

[207] J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo. Phys. Rev. Lett., 76:
3830, 1996.

[208] R. Rouse, S. Han, and J. E. Lukens. Phys. Rev. Lett., 75:1614, 1995.

[209] P. Silvestrini, V. G. Palmieri, B. Ruggiero, and M. Russo. Phys. Rev. Lett., 79:
3046, 1997.

[210] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai. Nature, 3982786, 1999.

[211] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell. Science, 269:198, 1995.

[212] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Phys. Rev. Lett.,
7521687, 1995.

300

 

[213] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Phys. Rev. Lett., 7523969, 1995.

[214] F. Dalfovo, S. Giogini, L. P. Pitaevskii, and S. Stringari. Rev. Mod. Phys, 712
463, 1999.

[215] J. Higbie and D. M. Stamper-Kurn. Phys. Rev. A, 692053605, 2004.

[216] M. Greiner, O. Mandel, T. Esslinger, T. W. H'ansch, and I. Bloch. Nature
(London), 415239, 2002.

[217] J. I. Cirac, M. Lewenstein, K. Molmer, and P. Zoller. Phys. Rev. A, 5721208,
1998.

[218] T. L. Ho and C. V. Ciobanu. Journal of Low Temperature Physics, 1352257,
2004.

[219] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy. Phys. Rev. Lett., 79:
4950, 1997.

[220] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy. Phys. Rev. A, 592620,
1999.

[221] D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek. Phys. Rev. A, 622013607,
2000.

[222] P. J. Y. Louis, P. M. R. Brydon, and C. M. Savage. Phys. Rev. A, 642053613,
2001.

[223] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls. Phys. Rev. A, 552
4318, 1997.

[224] J. K. Chin, J. M. Vogels, and W. Ketterle. Phys. Rev. Lett, 902160405, 2003.

[225] Yu. Kagan, A. E. Muryshev, and G. V. Shlyapnikov. Phys. Rev. Lett., 812933,
1997.

[226] R. H. Dicke. Phys. Rev., 93299, 1954.

[227] A. Imamoglu, M. Lewenstein, and L. You. Inhibition of coherence in trapped
boseeinstein condensates. Phys. Rev. Lett., 78(13)22511—2514, Mar 1997.

[228] W. Zhang and D. F. Walls. Phys. Rev. A, 4923799, 1994.

[229] V. Giovannetti, S. Lloyd, and L. Maccone. Science, 30621330, 2004.

[230] P. Carruthers and M. M. Nieto. Rev. Mod. Phys, 402411, 1968.

[231] M. Born and E. Wolf. Principles of Optics. Pergamon and Oxford, 1975.
[232] C. M. Caves. Phys. Rev. D, 2321693, 1981.

301

[233] R. S. Bondurant and J. H. Shapiro. Phys. Rev. D, 3022548, 1984.

[234] Y. P. Huang and M. G. Moore. On-demand generation of entanglement of
atomic qubits via optical interferometry. Phys. Rev. A, 77(3):032349, 2008.

[235] R. A. Campos, Christopher C. Gerry, and A. Benmoussa. Phys. Rev. A, 68:
023810, 2003.

[236] Ole Steuemagel and Stefan Scheel. J. Opt. B: Quantum Semiclass. Opt, 6:866,
2004.

[237] K. Eckert et al. Phys. Rev. A, 732013814, 2006.

[238] D. Denot, T. Bschorr, and M. Freyberger. Phys. Rev. A, 732013824, 2006.
[239] L. Pezzé and A. Smerzi. Phys. Rev. Lett., 1002073601, 2008.

[240] Christopher C. Gerry and Adil Benmoussa. Phys. Rev. A., 652033822, 2002.

[241] P. Walther, J .-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparon, and A. Zeilinger.
Nature (London), 4292158, 2004.

[242] T. Kim, J. Dunningham, and K. Burnett. Phys. Rev. A, 722055801, 2005.

[243] K. Eckert, P. Hyllus, D. BruB, U. V. Poulsen, M. Lewenstein, C. Jentsch,
T. Miiller, E. M. Rasel, and W. Ertmer. Phys. Rev. A, 732013814, 2006.

[244] z. Hradil and J. Rehaéek. Phys. Lett. A, 3342267, 2005.

[245] Alistair S. Lane, Samuel L. Braunstein, and Carlton M. Caves. Phys. Rev. A,
47:1667, 1993.

[246] Thomas B. Bahder and Paul A. Lopata. Phys. Rev. A, 742051801(R), 2006.

[247] Carlton M. Caves, Christopher A. Fuchs, and Riidiger Schack. Phys. Rev. A,
652022305, 2002.

[248] T. Kim and et al. Phys. Rev. A, 5724004, 1998.
[249] Y. P. Huang and M. G. Moore. Phys. Rev. A, 73:023606, 2006.
[250] L. Pezzé and A. Smerzi. quant-ph/0508158, 2005.

[251] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich.
Phys. Rev. A, 652033608, 2002.

[252] TheoryC. W. Helstrom. Quantum Detection and Estimation. Academic Press,
New York, 1976.

[253] B. C. Sanders and G. J. Milburn. Phys. Rev. Lett., 7522944, 1995.

302

[254] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe,
and D. J. Wineland. Phys. Rev. Lett., 8625870, 2000.

[255] L. Pezzé et al. Phys. Rev. A, 722043612, 2005.
[256] G. A. Durkin and J. P. Dowling. Phys. Rev. Lett., 992070801, 2007.

[257] M. O Scully and M. S. Zubairy. Quantum Optics. Cambridge University Press,
NewYork, USA, 1997.

[258] L. You, Maciej Lewenstein, and J. Cooper. Phys. Rev. A, 51(11):4712, 1995.
[259] B. Julsgaard, A. Kozhekin, and E. S. Polzik. Nature, 4132400, 2001.
[260] D. N. Matsukevich and A. Kuzmich. Science, 3062663, 2004.

[261] P. van Loock, T. D. Ladd, K. Sanaka, Kae Nemoto F. Yamaguchi, W. J. Munro,
and Y. Yamamoto. Phys. Rev. Lett., 962240501, 2006.

[262] S. J. van Enk and H. J. Kimble. Phys. Rev. A, 632023809, 2001.
[263] A. Kuhn, M. Hennrich, and G. Rempe. Phys. Rev. Lett., 892067901, 2002.

[264] W. T. M. Irvine, K. Hennessy, and D. Bouwmeester. Phys. Rev. Lett., 96:
057405, 2006.

[265] X.-B. Zou, J. Kim, and H.-W. Lee. Phys. Rev. A, 632065801, 2001.
[266] C. C. Gerry and R. A. Campos. Phys. Rev. A, 642063814, 2001.

[267] H. Lee, P. Kok, N. J. Cerf, and J. P. Dowling. Phys. Rev. A, 652030101(R),
2002.

[268] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller. Phys. Rev. A, 672013607, 2003.

[269] D. M. Greenberger, M. Horne, and A. Zeilinger. Bell’s Theorem and Quantum
Theory and Conceptions of the Universe. Kluwer and Dordrecht, 1989.

[270] J .-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger. Phys. Rev. Lett.,
8023891, 1998.

[271] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble. Phys. Rev.
Lett., 972083602, 2006.

[272] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble. Phys. Rev.
Lett., 982193601, 2007.

[273] S. P6tting, E. S. Lee, W. Schmitt, I. Rumyantsev, B. Mohring, and P. Meystre.
Phys. Rev. A, 622060101(R), 2000.

[274] B. Misra and E. C. G. Sudarshan. J. Math. Phys, 182756, 1977.

303

 

[275] A. Peres. Am. J. Phys, 482931, 1980.
[276] E. Joos. Phys. Rev. D, 2921626, 1984.

[277] W. M. Itano, D. J. Heingzen, J. J. Bollinger, and D. J. Wineland. Phys. Rev.
A, 4122295, 1990.

[278] S. H. Simon and P. M. Platzman. Phys. Rev. A, 612052103, 2000.
[279] J. Ruseckas and B. Kaulakys. Phys. Rev. A, 732052101, 2006.
[280] A. C. Elitzur and L. Vaidman. Found. Phys, 23 (7)2987, 1993.

[281] P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich. Phys.
Rev. Lett., 7424763, 1995.

[282] P. Kwiat, H. Weinfurter, and A. Zeilinger. Sci. Am., 2755272, 1996.

[283] T. Tsegaye, E. Goobar, A. Karlsson, G. BjOrk, M. Y. Loh, and K. H. Lim.
Phys. Rev. A, 5723987, 1998.

[284] A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat. Phys. Rev. A, 582605,
1998.

[285] P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter,
and A. Zeilinger. Phys. Rev. Lett., 8324725, 1999.

[286] T. Rudolph. Phys. Rev. Lett., 8522925, 2000.

[287] T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn. Phys. Rev. A, 65:
012314, 2002.

[288] T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White. Phys. Rev. A, 652
062324, 2002.

[289] H. F. Hofmann and S. Takeuchi. Phys. Rev. A, 662024308, 2002.

[290] Z. Zhao, A.-N. Zhang, Y.-A. Chen, H. Zhang, J.-F. Du, T. Yang, and J.-W.
Pan. Phys. Rev. Lett., 942030501, 2005.

[291] K. J. Xu, Y. P. Huang, M. G. Moore, and C. Piermarocchi. Two—qubit condi-
tional phase gate in laser-excited semiconductor quantum dots using the quan-
tum zeno effect. Physical Review Letters, 103(3):O37401, 2009.

[292] M. A. Nielsen, E. Knill, and R. Laflamme. Nature, 396252, 1998.
[293] A. Kuzmich and E. S. Polzik. Phys. Rev. Lett., 8525639, 2000.

[294] A. Dantan, N. Treps, A. Bramati, and M. Pinard. Phys. Rev. Lett., 942050502,
2005.

[295] N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa. Phys. Rev. Lett., 942220502,
2005.

304

 

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