ULTRAFAST TRANSIENT STATES IN NONEQUILIBRIUM QUANTUM SYSTEMS
By
Bin Hwang
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics â Doctor of Philosophy
Computational Mathematics, Science and Engineering â Dual Major
2017
ABSTRACT
ULTRAFAST TRANSIENT STATES IN NONEQUILIBRIUM QUANTUM SYSTEMS
By
Bin Hwang
Photo-induced phase transitions (PIPT) in quantum systems are the epitome of challenging
non-equilibrium many-body phenomena, that also have a wide range of potential applications. Recently interest in light-matter coupled states with an energy gap have yielded
evidence for Floquet topological states. In this study we demonstrate nonequilibrium Floquet band formation under ultrafast optical excitation using a one-dimensional topological
insulator. As an example, the effects are illustrated using a new Zig-Zag Su-SchriefferHeeger model of polyacetylene, which is a paradigmatic Hamiltonian exhibiting nontrivial
edge states. Our results indicate short optical pulses feasible in experiments can induce
novel topological states, local spectral selection and novel pseudospin textures in polyacetylene. Pump-probe photoemission spectroscopy is able to study these states by measuring
Floquet band formation and sizeable energy gaps on femtosecond time scales. We find that
optically activated nontrivial variations of sublattice mixing could lead to novel topological
phenomenon.
The rich variety of states induced by lasers have a wide range of potential applications so
that control of these states has become a key objective. We present a computational approach
to finding optimal ultrafast laser pulse shapes to induce target states and population inversion in pump-probe PIPT experiments. The Krotov approach for Quantum optimal control
theory (QOCT) is combined with a Keldysh Greenâs function calculation to describe experimental outcomes such as photoemission, transient single particle density of states and optical
responses. Results for a simple model charge density wave (CDW) system are presented, including generation of almost complete population inversion and negative temperature states.
Copyright by
BIN HWANG
2017
This thesis is dedicated to my family.
iv
ACKNOWLEDGEMENTS
It has been a long and exciting journey of pursuing my Ph.D. I would like to thank many
people who have guided me, helped me and supported me.
First of all, I must thank my advisor and mentor, Phillip Duxbury. I sincerely appreciate
him for his help in all aspects of my research career at Michigan State University. He is
not only one of the smartest teachers I know but also the most thoughtful friend as well.
Because of him, I have investigated deeply the physical world and also learned wisdom in
research and life. His wide knowledge and passion in physics kept me wanting to dig out
hidden truths behind nature, and his patience and support were great help when I faced
challenges in research and life.
Second, I appreciate greatly the help from the rest of the PA professors and members
of our research group. I would like to thank Jennie Portman, Xukung Xiang, Faran Zhou,
Professor Chong-Yu Ruan, and Professor Carlo Piermarocchi for very useful discussions in
all kinds of theoretical and computational physics research topics. I also would like to thank
Professor Jianliang Qian and Professor Sean Couch for their support during my defense process.
This work was supported by the College of Natural Science at Michigan State University
(MSU). Computational resources were provided by the High Performance Computer Center
at MSU.
Finally, I would like to thank my lovely wife Chi-Chun Hsieh and my family members
for their continued support and encouragement through all of this.
v
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . .
1.1 Time-resolved ARPES . . . . . . . . . . . . . . . . . .
1.1.1 Nonequilibrium Regime . . . . . . . . . . . . .
1.2 CDW Materials . . . . . . . . . . . . . . . . . . . . . .
1.2.1 CDW in 1T â T aS2 . . . . . . . . . . . . . . .
1.2.2 Optimal Laser Pulse Shaping . . . . . . . . . .
1.3 Floquet-Bloch States . . . . . . . . . . . . . . . . . . .
1.3.1 Laser Driven Topological Phases . . . . . . . . .
1.4 Chapters . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER 2 MODELS AND METHODS . . . . . . . . . .
2.1 Introduction to Nonequlibrium Greenâs Functions . . .
2.2 Introduction to Topology in Band Structure . . . . . .
2.2.1 Topology . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Berry Phase and Chern Number . . . . . . . . .
2.2.3 Bulk Edge Correspondence . . . . . . . . . . . .
2.3 Tight-binding Models at Equilibrium . . . . . . . . . .
2.3.1 SSH, Rice Mele, Platero Models . . . . . . . . .
2.3.2 Freerickâs Model . . . . . . . . . . . . . . . . . .
2.3.3 Greenâs Function Method . . . . . . . . . . . .
2.3.4 Graphene and Dirac Cone . . . . . . . . . . . .
2.4 Tight Binding Models in a Time Varying Field . . . . .
2.4.1 Peierlâs Substitution . . . . . . . . . . . . . . .
2.5 Introduction to Floquet Theory . . . . . . . . . . . . .
2.5.1 Periodic Hamiltonian . . . . . . . . . . . . . . .
2.5.2 Floquet Formalism . . . . . . . . . . . . . . . .
2.6 Calculation of Time Resolved Photoemission . . . . . .
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CHAPTER 3 THE ZIG-ZAG SSH MODEL: FROM FLOQUET TO TRPES
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Model and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Time-dependent Zig-Zag SSH Model . . . . . . . . . . . . . .
3.2.2 Floquet Hamiltonian for Zig-Zag SSH Model . . . . . . . . . .
3.2.3 Calculations from Non-equilibrium Greenâs Functions . . . . .
3.3 Analysis of the SSH Floquet Hamiltonian . . . . . . . . . . . . . . . .
3.3.1 Limit of High Frequency . . . . . . . . . . . . . . . . . . . . .
3.3.2 Simple Model for the Case of One Overlap . . . . . . . . . . .
3.3.3 Computational Results and Comparison to Analytic Results .
3.3.4 Effects of Varying Amplitude and Frequency . . . . . . . . . .
3.3.5 Topological Phase Diagram . . . . . . . . . . . . . . . . . . .
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vi
3.4
3.5
Pump-probe Results . . . . . . . . . . . . . . . . . . . . .
3.4.1 Pseudospin Content . . . . . . . . . . . . . . . . . .
3.4.2 Evolution of States and Their Occupancy . . . . . .
3.4.3 Dynamics of Phase Evolution . . . . . . . . . . . .
3.4.4 Effects of Varying Pump and Probe Pulsewidth and
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 4 KROTOV OPTIMAL CONTROL THEORY
4.1 Introduction to Optimal Control . . . . . . . . . . .
4.2 Preliminary Preparation of the Krotov Method . . .
4.3 The Tricks of Krotovâs Method . . . . . . . . . . .
4.3.1 Decomposition of Goal Function . . . . . . .
4.3.2 Iterative Algorithm . . . . . . . . . . . . . .
4.3.3 Monotonic Convergence of Krotov Method .
4.4 Construction of Ď . . . . . . . . . . . . . . . . . . .
4.4.1 First Order in x . . . . . . . . . . . . . . . .
4.4.2 Second Order in x . . . . . . . . . . . . . .
4.4.3 Algorithm . . . . . . . . . . . . . . . . . . .
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Example 1: First Order in x . . . . . . . . .
4.5.2 Example 2: Second Order in x . . . . . . . .
4.5.3 Example 3: A Two Level Quantum System .
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CHAPTER 5 QUANTUM OPTIMAL CONTROL OF A TRANSIENT CDW STATE 98
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Charge Density Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.1 Equations for the Nonequilibrium Solution . . . . . . . . . . . . . . . 102
5.3 Quantum Optimal Control Method . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Proof of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Application to a CDW Model . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
CHAPTER 6 SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . 116
CHAPTER 7 EXPERIMENTS AND APPLICATIONS . . . . . . . . . . . . . . . 118
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
vii
LIST OF FIGURES
Figure 1.1: The experimental design of time-resolved ARPES, and corresponding
ultrafast momentum-dependent dynamics of the melting of the charge
density wave state in T bT e3 monitored by trARPES. The figure is taken
from [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Figure 1.2: Schematic picture for the comparison between uniform electron density
and the Peierlâs transition of a charge density wave in a 1D system. . . .
6
Figure 1.3: (a) The resistivity versus temperature in different charge-density-wave
(CDW) phases of 1T â T aS2 . (b) Schematic of the T a atom layer in the
CCDW phase. 13-atom David-star clusters with the three T a sites a,
b, and c. The black arrows indicate the displacement of the T a atoms
from their original positions. Figure is take from [2]. . . . . . . . . . . .
7
Figure 1.4: Pump-induced gap magnitude oscillations in the CDW material 1T â
T aS2 near 30K. Using a 1.5eV pump pulse and 6eV probe pulse to induce instantaneous collapse of the Mott gap and to observe its recovery.
The oscillation in the band is interpreted as the CDW amplitude mode
related to electron-phonon coupling. Figure is taken from [3]. . . . . . . .
8
Figure 1.5: A closed-loop for an optimal control process in quantum systems. First,
input initial conditions to the quantum system, such as an initial random
field. Second, a current laser control field design is created with a pulse
shaper and then applied to the sample. Third, The system evolves
and the results are fed to a learning algorithm to suggest an improved
laser field for repeated excursions around the loop until the objective is
achieved. Figure is taken from [4]. . . . . . . . . . . . . . . . . . . . . .
10
Figure 1.6: An idealized band structure for a topological insulator. The bulk band
gap is traversed by topologically-protected surface states [5]. . . . . . .
11
Figure 1.7: Researchers recently showed experimental results of the Floquet-Bloch
states [6] through pump-probe trARPES measurements on the topological insulator Bi2 Se3 . Figure is taken from [6]. . . . . . . . . . . . . . . .
12
Figure 2.1: Kadanoff-Baym-Keldysh contour in the complex time plane. . . . . . . .
18
Figure 2.2: A coffee mug can smoothly deform to a donut since they both have one
hole or genus g = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Ë for a
Figure 2.3: The Berry phase is equal to half the sold angle swept out by d(k)
two band theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Figure 2.4: Edge states as skipping cyclotron orbits. . . . . . . . . . . . . . . . . . .
24
viii
Figure 2.5: The chiral edge states connect the valence band near K and K 0 with
linear or nonlinear dispersion for linear and nonlinear Hamiltonians respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Figure 2.6: Two distinct topological states of the SSH model. (a) is the case Î´Ď > 0
and (b) is the case Î´Ď < 0. . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Figure 2.7: Two distinct topological paths of the SSH model. (a) is the case Î´Ď > 0
and (b) is the case Î´Ď < 0. . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Figure 2.8: Dispersion relation of the SSH model based for R = Ď + Î´Ď and r = Ď â Î´Ď . 31
Figure 2.9: Topological path of the SSH model depends on R, r and ka. . . . . . . .
31
Figure 2.10: A dimers chain with two sublattices A and B, and Ď and Ď 0 are hopping
terms between two sublattices A and B. a0 is the cell size and b0 is the
distance between A-B atoms. . . . . . . . . . . . . . . . . . . . . . . . .
33
Figure 2.11: CDW model with two different sublattices A and B. . . . . . . . . . . . .
35
Figure 2.12: Dispersion relation ÎľkÂą of the CDW model when Âľ = 1/2 and U = 1. . .
38
Figure 2.13: Graphene is an atomic-scale hexagonal lattice made of carbon atoms.
.
40
Figure 2.14: Dirac coneâthe electrons in graphene behave like massless Dirac particles
that appear in the electronic band structure as gapless excitations with
a linear dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Figure 3.1: The Zig-Zag SSH model: (a) the trivial topological state Î´Ď > 0 and
(b) the nontrivial topological state Î´Ď < 0. Note that the polarization
is linear so the electric field is along one bond but perpendicular to the
other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Figure 3.2: Dispersion relation for the laser driven SSH model with laser amplitude
A0 = 1 and frequency âŚ = 5. Dashed lines: the exact solution for the
4Ă4 Floquet Hamiltonian. Solid lines: Numerical results for the 42Ă42
Floquet Hamiltonian with m = 10. Left is for the trivial condition
Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Figure 3.3: Scheme of band interactions for g0 , g1 and g2 . n is the Floquet band
index. g0 is the intraband process; while g1 and g2 are interband process.
65
ix
Figure 3.4: Dispersion relation for the laser driven SSH model with laser amplitude
A0 = 1 and frequency âŚ = 3. Dashed lines: the exact solution for the
4Ă4 Floquet Hamiltonian. Solid lines: Numerical results for the 42Ă42
Floquet Hamiltonian with m = 10. Left is for the trivial condition
Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Figure 3.5: Dispersion relation for the laser driven SSH model with laser amplitude
A0 = 1 and frequency âŚ = 1.5. Dashed lines: the exact solution for the
4Ă4 Floquet Hamiltonian. Solid lines: Numerical results for the 42Ă42
Floquet Hamiltonian with m = 10. Left is for the trivial condition
Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Figure 3.6: Dispersion relations of quasi-static Floquet Hamiltonian with drive frequency âŚ = 2.8. Left: trivial condition Î´Ď = 0.5 and Right: the nontrivial condition Î´Ď = â0.5. (a)(b) is for amplitude A0 = 2, (c)(d) is
for amplitude A0 = 4 and (e)(f) is for amplitude A0 = 6. . . . . . . . .
67
Figure 3.7: Quasienergy spectrum properties calculated from the Floquet matrix
with m = 10 for the original bands at k = Ď vs laser amplitude A0 for
Î´Ď = Âą0.5. Left: are for the trivial condition Î´Ď = 0.5 and Right: are
for the nontrivial condition Î´Ď = â0.5 for different driving frequencies âŚ.
68
Figure 3.8: Bandwidth calculated from quasi-static Floquet band structures vs laser
amplitude A0 for Î´Ď = Âą0.5. Left: are for the trivial condition Î´Ď = 0.5
and Right: are for the nontrivial condition Î´Ď = â0.5 for different
driving frequency âŚ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Figure 3.9: Topological phase diagram for different frequencies and amplitudes of
the drive pulses from effective Floquet Hamiltonian. (a)(b) shows the
cases for Î´Ď = 0.25 on the left and Î´Ď = â0.25 on the right. (c)(d)
shows the cases for Î´Ď = 0.5 on the left and Î´Ď = â0.5 on the right.
(e)(f) shows the cases for Î´Ď = 0.75 on the left and Î´Ď = â0.75 on the
right. The four red points indicate the frequencies and amplitudes we
discuss in Fig. (3.16) of the next subsection. . . . . . . . . . . . . . . .
70
Figure 3.10: Time evolution of time-resolved PES of the electronic SSH model is
presented for time t = â200, â100, 0, 100, 200 with Î´Ď = 0 under the
pulse frequency âŚ = 1, width Ď = 50 and amplitude A0 = 1. Dashed
lines show the corresponding quasi-static Floquet band structures at the
same frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
x
Figure 3.11: Time evolution of time-resolved PES of the electronic SSH model is
presented for time t = â200, â100, 0, 100, 200 with Î´Ď = 0.5 under the
pulse frequency âŚ = 3, width Ď = 50 and amplitude A0 = 2. Dashed
lines show the corresponding quasi-static Floquet band structures. . . .
75
Figure 3.12: Time evolution of time-resolved PES of the electronic SSH model is
presented for time t = â200, â100, 0, 100, 200 with Î´Ď = â0.5 under
the pulse frequency âŚ = 3 and amplitude A0 = 2. Dashed lines show
the corresponding quasi-static Floquet band structures. . . . . . . . . .
75
Figure 3.13: Berry phase mapping using the radius r0 = 1 + (1 + k)/2 with Î´Ď = 0.5
on the left and Î´Ď = â0.5 on the right under the pulse frequency âŚ = 3,
amplitude A0 = 2 and pulse width Ď = 50. (a)(b) is at time t = â200
with integral frequency window Ďl = 0.0 and Ďu = 3.0, (c)(d) is at time
t = 0 with integral frequency window Ďl = â1.5 and Ďu = 0.0 and (e)(f)
is at time t = 0 with integral frequency window Ďl = 1.5 and Ďu = 3.0. .
77
Figure 3.14: Time evolution of pseudospin phase Ď deriving from pseudospin context
P eiĎ = Px + iPy are presented for time t = â200, â100, â50, â25, 0
with Î´Ď = 0.5 on the top (a)-(e) and Î´Ď = â0.5 on the bottom (f)-(j)
under the pulse frequency âŚ = 3, amplitude A0 = 2 and pulse width
Ď = 50. Dashed lines show the corresponding quasi-static Floquet band
structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
Figure 3.15: Time-resolved PES of the electronic SSH model with Î´Ď = 0 at time
t = 0 with a pulse of frequency âŚ = 3, amplitude E = 1 and width (a)
Ď = 1, (b) Ď = 2, (c) Ď = 5, (d) Ď = 10, (e) Ď = 20, (f) Ď = 40. Dashed
lines show the corresponding quasi-static Floquet band structures. . . . .
79
Figure 3.16: Time-resolved PES with Î´Ď = 0.5 on the left and Î´Ď = â0.5 on the
right at time t = 0. (a)(b) with pulse of frequency âŚ = 1.6, amplitude
A0 = 0.5 and width is Ď = 50, (c)(d) with pulse of frequency âŚ = 1.6,
amplitude A0 = 2.0 and width is Ď = 50, (e)(f) with pulse of frequency
âŚ = 3, amplitude A0 = 0.5 and width Ď = 50, (g)(h) with pulse of
frequency âŚ = 3, amplitude A0 = 2 and width Ď = 50. Dashed lines
show the corresponding quasi-static Floquet band structures. . . . . . .
81
Figure 4.1: Left: fidelity versus iteration times. Right: optimal control sequence
with respect to time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
Figure 4.2: Left: Cost function versus iteration times. Right: Optimal evolution of
x with respect to time t. . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Figure 4.3: Left: Cost function versus iteration times. Right: Optimal control pulse
of Îľ respect to time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
Figure 4.4: Optimal probability evolution of Ď(t) respect to time t. . . . . . . . . . .
97
xi
Figure 5.1: Time-resolved PES of the electronic CDW system before, during and
after application of the optimal pulse (Ac (t), As (t), A(t)) from time t = 0
to t = 20 h/tâ with the initial state a filled lower band at temperature
T = 0. The target is excitation of a single state with the goal energy
1tâ . In this calculation Îť(t) = 2Ďh/â is a constant, with â = 0.02
and h = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Figure 5.2: Time-resolved PES, with the optimal pulse interacting with the system
from time t = 0 to t = 20 h/tâ . The initial state is a filled valence band
at temperature T = 0, and the goal is to excite all of the states in the
conduction band. The electron density at a given energy is plotted in
false color and is high near the band edges as this model has a square
root singularity in the density of states at both the valence band and
conduction band edges. In this calculation Îť(t) = 2Ďh/â is a constant,
with â = 0.02 and h = 10. . . . . . . . . . . . . . . . . . . . . . . . . . 114
Figure 5.3: Properties of the system with the optimal pulse acting on the system
from time t = 0 to t = 20 h/tâ (a) The average occupancy of the
conduction and valence bands nÂą (t), (b) The real space order parameter
âŚ(t), which measures the difference in occupancy of the two sublattices,
(c) The energy hH(t)i of the CDW system. . . . . . . . . . . . . . . . . 115
Figure 7.1: Angle-resolved photoemission spectra (APRES) of Bi2 Se3 . (A) A sketch
of the experimental geometry for the p-polarized case. kx is defined to be
the in-plane electron momentum parallel to the pump scattering plane.
(B to F) ARPES data for several pump-probe time delays t (values indicated in the figure) under strong linearly polarized mid-infrared (MIR)
excitation of wavelength l = 10 mm. Figure and caption are taken from [7].118
Figure 7.2: (a) Waveguide array fabricated out of PMMA on top of a Cr- and Aucoated glass substrate. Alternating center-to-center separations, 600
and 1000 nm, implement the bulk SSH model. (b) Plasmonic waveguide array incorporating a topological defect where the long separation
is repeated twice. Three different excitation sites, I, II, and III, are
highlighted. Figure and caption are taken from [8]. . . . . . . . . . . . . 120
Figure 7.3: Absorption images taken after 760Âľs(â 0.78h/t) of evolution following
the initialization and quench, corresponding to phases of Ď (top) and 0
(bottom), respectively, for â/t = 0.38(1). Figure and caption are taken
from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 7.4: (a) Sketch of the passive waveguide array acting like a PT-symmetric
structure with a topological interface. (b) End facet of the experimentally realized structure in fused silica glass. Image courtesy of Mark
Kremer, FSU Jena. Figure and caption are taken from [10]. . . . . . . . 121
xii
CHAPTER 1
INTRODUCTION
In physics, using spectroscopy to investigate interactions between light and matter has been
a long standing and broad area of study. Since the development of the laser as a source for
coherent light, many new methods have been utilized in a variety of fields, such as laser pulse
shaping and nonlinear optics. With advanced techniques, ultrafast pulses of durations from
picoseconds to a few femtoseconds are available; and more recently pulses in the attosecond
range have been generated. Source with frequencies from the terahertz to the X-ray are
now available. More intensive optical laser pulses have been employed recently to generate
tabletop X-ray and extreme-ultraviolet sources enabling the shortest pulses with a duration
in the range of attoseconds [11, 12, 13, 14, 15].
These tremendous improvements have enabled spectroscopy of matter with time resolution at the femtosecond scale. Some of the most interesting properties only happen at
such time-scales, particularly the dynamics of quantum states. In the last decade, many
researchers have used ultrafast spectroscopy to study correlated and nanoscale quantum materials; including superconductors, charge density wave materials and topological insulators,
and novel effects and new phases that only occur far from equilibrium are being discovered.
Angle-resolved photoemission spectroscopy (ARPES), is of special interest, as it is able
to resolve the occupied electronic structure in momentum space; yielding unique insight into
many-body correlations and the dispersion of quasiparticles in bulk and low-dimensional superconductors, strongly correlated materials and topological insulators. In the time domain,
applying an ultrafast laser pulse to a material and studying the related time-resolved ARPES
spectra (trARPES), one can study the dynamics of the electronic structure and quasiparticle
occupations; and discover the hidden structure of quantum states in physics.
1
Figure 1.1: The experimental design of time-resolved ARPES, and corresponding ultrafast
momentum-dependent dynamics of the melting of the charge density wave state in T bT e3
monitored by trARPES. The figure is taken from [1]
1.1
Time-resolved ARPES
Since the early 1980s, time-resolved experimental techniques have been under development
[16, 17, 18], and related technology keeps progressing in resolution of momentum, energy
and in the time domain. Some of the important developments include generating probe
photons that can access large values of electronic crystal momentum by using high-harmonic
sources [19, 20, 21, 22], use of 2D hemispherical analyzers to replace time-of-flight electron
spectrometers [23, 24], and aiding in selective nonequilibrium photoexcitation through the
incorporation of tunable pump frequencies [6].
Time and angular-resolved photoelectron spectroscopy (trARPES) combines ARPES
with the ultrahigh time resolution of femtosecond lasers. Related advanced researches in
capturing the fastest processes relevant to behaviors at surfaces in materials have been explored. For the experiments, two important processes are included in time-resolved ARPES,
2
namely photoemission and ultrafast pump-probe spectroscopy. First, in a photoemission
process, a crystalline material is pumped by ultraviolet (UV) photons in vacuum. When the
photon energy is larger than the work function of the system, electrons will be excited and
ejected into free space where their exit angles and energies can be observed. Because of the
energy and momentum conservation laws, one can ensure that the energy and momentum
of the photoelectrons in vacuum are directly related to the crystal momentum and energy
of the same electrons when they were still inside the material. Therefore the result can be
used to study electronic structure, and has been successfully applied to characterize many
different materials [25, 26, 27, 28, 29].
On the other hand, even though ARPES has many advantages and gives a great contribution in the field, there are a few drawbacks. First, some microscopic dynamic processes
cannot be decoupled from the basic analysis. Second, it can only be used to measure electronic bands that are occupied initially. Third, specific physical phases only occur after
driving the system far beyond the equilibrium state. One of the useful techniques scientists
already implemented is to incorporate pump-probe methods. By shining an ultraviolet pump
laser pulse on a material, the material will be irradiated into excited states at a nonequilibrium condition and, after that, the second laser pulse, as a probe laser pulse, provides a
photon energy within the range (6 to 60 eV) to detect photoemission of electrons from the
nonequilibrium state. By studying variations in the photoemission spectrum as the time
delay between pump and probe pulses changes, one is able to investigate nonequilibrium
dynamics in quantum systems (see Fig. (1.1)).
1.1.1
Nonequilibrium Regime
Using near-infrared or terahertz pump pulses to generate nonequilibrium states is one of the
most important topics for time-resolved spectroscopy nowadays. In an early finding a pumpprobe scheme was used to acquire the time resolved ARPES data on the high temperature
3
superconductor Bi2212 [30] yielding its quasiparticle dynamics. The results were related to
a quasi-thermal system of metallic quasiparticles [31], which was used to study the normalstate electron-phonon coupling. From Fig. (1.1), the experimental design of time-resolved
ARPES is shown, and the ultrafast momentum-dependent dynamics of the melting of the
charge density wave state in T bT e3 monitored by trARPES is illustrated [1].
Moreover, time-resolved ARPES can help scientists to study the phase transition induced by electron-phonon interactions that have no analogue in equilibrium ARPES. Gaps
in the electronic excitation spectrum usually open up simultaneously with the formation of
superconducting or charge density wave (CDW) order. With time-resolved ARPES in the
ultrafast regime, many novel experiments have revealed new nonequilibrium behaviors. For
instance, researchers have discovered a long-range CDW order persists in the nonequilibrium state but the local electronic spectrum becomes gapless for a transient period of time
[32, 33, 34, 3, 1, 35, 36, 37, 38, 39]. Similar results for other materials indicate a novel quasiuniversal behavior in the nonequilibrium state. Understanding of the momentum-dependent
structure of relaxation rates provides a unique probe of coupling effects in a variety systems;
including charge density waves, superconductors, magnetic states and Floquet-Bloch states.
Recently, extreme-UV (10-50 eV) pulses have been induced by high-harmonic generation
in gases driven by intense laser pulses. Time-resolved ARPES experiments making use of such
techniques have been applied to discover nonequilibrium behaviors in multiple materials [40,
41, 42, 43, 44]. For example, the chalcogenide based charge density wave (CDW) materials
such as 1T â T aS2 , 1T â T iSe2 and RT e3 [45, 37, 46]. Scientists now can also access the
timescales of charge-order collapse, and control the CDW gap through changing frequency
and amplitude of the laser pulse. The delay time is of order 600 fs for collapse of the
superconducting gap in Bi2212 while the experimental results on 1T â T iSe2 show that the
quenching procedure can rapidly appear within 20 fs [47, 45]. Time scales occurring for
various other charge density wave systems help to understand the difference between Peierls
and Mott effects driving CDW formation [46].
4
An intriguing result in experimental observations is that of oscillating spectral signatures
that indicate the coupling of electrons to collective modes. This pump-induced phenomena
lasts longer than what was originally expected and is a very interesting discovery; perhaps
due to the fact that the electron-phonon coupling and charge density wave and lattice response is far from the near equilibrium expectation. Coherently driven oscillations appear in
CDW materials as well. Early studies of these oscillations in KMO by Demsar et al. applying
optical techniques [48] were replicated in a wide range of materials, such as 1T â T aS2 to
T bT e3 [37, 34, 1].
1.2
CDW Materials
A CDW is an ordered quantum fluid of electrons normally in a linear chain compound or
layered crystal. In 1955, the CDW state was first mentioned by Peierls, and more recently
time-resolved ARPES (trARPES) has been widely applied to study several CDW materials
[49]. Peierls discovered a one-dimensional metal coupled to lattice vibrations is not stable
at low temperature. This instability was later called the Peierlsâ CDW state. The argument
first considers a one-dimensional metal with one conduction electron per lattice site and with
temperature ideally at T = 0 and lattice constant a. The ground state is a non-interacting
metal without the electron-electron and electron-phonon interaction. A periodic lattice distortion occurs when electron-phonon coupling is present, and will influence the Fermi wave
vector and the overall energy of the material. The reason is that lattice energy is increased
less than electronic energy is reduced. And, the size of the gap opening at the Fermi level is
related to the amplitude of the periodic lattice distortion. In Fig. (1.2), we show the uniform
electron density and lattice distortion of a one-dimensional charge density wave. Note that, in
a half filling situation, a dimerization transition occurs and the distortion will double the periodicity of the model as illustrated in Fig. (1.2). More discussion can be found in Ref. ([50]).
5
Figure 1.2: Schematic picture for the comparison between uniform electron density and the
Peierlâs transition of a charge density wave in a 1D system.
1.2.1
CDW in 1T â T aS2
One of the original 2D CDW materials discovered is the material 1T â T aS2 [3, 51, 32],
which has a sandwich structure consisting of S-Ta-S sheets and the sheets are bound together by Van der Waals forces. Because of the weak coupling between layers, the system
has a quasi two-dimensional property, and has a variety of stacking structures. Fig. (1.3)(a)
shows the measurements of conductance of 1T â T aS2 with temperature. At several critical temperatures, one can easily find multiple phase transitions between phases such as the
commensurate CDW phase; the incommensurate CDW phase and the metallic phase. A
commensurate CDW indicates a CDW state that is periodic as an integer times the periodicity of the underlying lattice system. Note that the material has a quasi commensurate
CDW phase at temperatures 180K-350K, and a commensurate CDW phase (CCDW) at temperature below 180K. Also an incommensurate CDW phase in the range 350K-550K, and a
metallic phase at temperature larger than 550K. In Fig. (1.3), we also show the structure of
6
the material 1T â T aS2 that is commonly studied with pump-probe experiments at room
temperature. The schematic shows the T a atom layer in the CCDW phase; consisting of
13-atom David-star clusters with the unique three T a sites a, b, and c. The black arrows
indicate the displacement of the T a atoms from their original undistorted positions.
Coherently driven oscillations appear in this CDW material as well. Perfetti et al. demonstrated the total photoemission signal in these phases by time resolved photo-emission measurement [52, 48]. They applied a 1.5 eV laser pulse with FWHM of 125 fs to pump the
material and a 6 eV near UV pulse to probe the effect. They discovered solid evidence that
the commensurate CDW 1T â T aS2 is a Mott insulator and the gap on the Fermi surface
has a completely electronic origin by comparing two spectra in the metallic phase and Mott
phase. In the Mott insulator phase, they found the quasi-instantaneous collapse of the Mott
gap and its recovery at 20 ps time scale. The result shown in Fig. (1.4) indicates coherent
oscillations in the photoemission signal with a fixed period. The corresponding frequency is
the phonon mode that forms the CDW when the outer lattice sites couple with the inner
lattice sites. The mode is normally called a CDW amplitude mode or a breathing mode.
Figure 1.3: (a) The resistivity versus temperature in different charge-density-wave (CDW)
phases of 1T â T aS2 . (b) Schematic of the T a atom layer in the CCDW phase. 13-atom
David-star clusters with the three T a sites a, b, and c. The black arrows indicate the
displacement of the T a atoms from their original positions. Figure is take from [2].
7
Figure 1.4: Pump-induced gap magnitude oscillations in the CDW material 1T â T aS2 near
30K. Using a 1.5eV pump pulse and 6eV probe pulse to induce instantaneous collapse of the
Mott gap and to observe its recovery. The oscillation in the band is interpreted as the CDW
amplitude mode related to electron-phonon coupling. Figure is taken from [3].
1.2.2
Optimal Laser Pulse Shaping
As discovered in the previous sections, high fidelity characterization of transient excited
many-body electron distributions in the ultrafast time domain is now available through a
variety of pump-probe experiments. Amongst the rich variety of non-equilibrium responses
observed, photo-induced phase transitions (PIPT) are particularly interesting for fundamental and applied reasons [39, 38, 53, 54, 55, 56, 57, 58, 59, 35, 60, 32, 33, 51, 3, 35, 36, 61, 60].
Important advances in optimal control theory were proposed in the twentieth century, and
our objective is to extend this approach to treat trARPES as a target. Fig. (1.5) shows the
scheme of pulse design, which is through a learning loop that comprises a computer, a pulse
shaper, the system and feedback spectroscopy [4]. The computer offers a particular pulse
shape, we shine it on the system and afterwards send probe light that analyses the effect of
the shaped pulse. An evolutionary algorithm iteratively refines the shape until an optimal
shape produces the desired effect.
Therefore, by adjusting the laser pulse properties it is possible to tune the non-equilibrium
PIPT response from the adiabatic to the non-adiabatic limits. Optimal laser pulse shaping
8
methods have not been applied to PIPT yet and have the potential to control non-equilibrium
response in order to isolate selected physical phenomena, and in order to tune response for
selected applications such as high speed electronics or optics. In Chapter 5 we introduce a
method, based on Krotov optimal control theory [62], to direct photo-induced phase transitions (PIPT) by combining non-equilibrium models with quantum optimal control theory
(QOCT).
QCOT is a powerful tool based on calculating the optimal pulse shape by minimizing a
physical cost function or maximizing a desired physical objective, and it has been developed
within a variety of variational frameworks to obtain control sequences [63, 64, 65, 66, 67, 68,
69, 70, 71, 62, 72, 73]. Quantum optimal control methods based on the classical gradient
optimization methods provide an alternative to iterative methods based on the Krotov approach. Krotov methods have been applied to the fields of Quantum computing and control
of charge transfer processes, [72, 74, 75, 62, 76, 73].
The Krotov approach has several appealing advantages over the gradient methods in the
following ways: First, a monotonic increase toward the objective with iteration number.
Second, no requirement for a line search and faster convergence to a given target. Third,
at each iteration it guarantees macrosteps of the time interval and it can reach the global
maximum.
To illustrate the approach, we consider PIPT in a simple model [56] for a transient
metal-insulator state in a charge density wave system. In experiments, a long-range chargedensity-wave (CDW) is formed in a variety of layered chalcogenide materials, in oxides, in
two dimensional materials, and in many other systems. The investigated layered CDW materials have disclosed a new nonequilibrium pattern where the long-range CDW is preserved
while the local electronic excitation spectrum becomes gapless (by having subgap states)
for a transient period of time, as has been elucidated using the simple model considered
here [56]. In the experimental systems the mechanism for gap-closing and population inversion remains an open question, though in some cases there has been significant theoretical
9
progress.
Figure 1.5: A closed-loop for an optimal control process in quantum systems. First, input
initial conditions to the quantum system, such as an initial random field. Second, a current
laser control field design is created with a pulse shaper and then applied to the sample.
Third, The system evolves and the results are fed to a learning algorithm to suggest an
improved laser field for repeated excursions around the loop until the objective is achieved.
Figure is taken from [4].
1.3
Floquet-Bloch States
The light-matter interaction between the coherent electric field of an intense laser pulse and
the electronic wave functions of a solid can induce intriguing phenomena such as modifications of the electronic structure and topology. In particular the finding of Floquet-Bloch
states in topological insulators has become an important area recently. A topological insulator is a material that has non-trivial topological order. It behaves as an insulator in the bulk
but its surface contains conducting states, indicating that electrons can only move along
the surface of the material. Fig. (1.6) shows an idealized band structure for a topological
insulator. The bulk band gap is traversed by topologically-protected surface states. On
the other hand, Floquet-Bloch states can be considered as a time domain version of Bloch
waves, where Bloch waves are defined as a periodic wave with fixed crystal momentum and
10
the corresponding Hamiltonian has a discrete translation invariance in space. Similarly,
Floquet-Bloch states in a material are periodic in both energy and momentum, and the
Hamiltonian has discrete invariance in both space and time.
Figure 1.6: An idealized band structure for a topological insulator. The bulk band gap is
traversed by topologically-protected surface states [5].
Researchers recently provided interesting experimental characterization of the effect [6]
through pump-probe trARPES studies of the topological insulator Bi2 Se3 as show in Fig. (1.7).
They found a ladder of replica bands appear in the time resolved photoemission spectrum
with equally spaced energies after a stable frequency of the pump laser pulse was input into
the system, and measured by a probe pulse within the duration of the pump pulse. Also,
the surface state bands are dispersive and lead to different overlaps for some bands. With a
strong enough polarized laser pulse, band gaps will open at the crossing points with induced
gap proportional to the amplitude of the applied electric field; and new topological properties may be induced [77, 78]. For pulse energies increasing to very high values, the gap will
shrink again and connect to the possibility of band gap opening depending on pump fluence.
11
The Floquet-Bloch phenomena is related to photon-dressed electron states in vacuum and
has the potential to disclose new physics in a variety of fields. It is also possible to control
these properties to design the next generation of quantum devices for new applications.
Figure 1.7: Researchers recently showed experimental results of the Floquet-Bloch states [6]
through pump-probe trARPES measurements on the topological insulator Bi2 Se3 . Figure
is taken from [6].
Moreover, nowadays ultrafast ARPES not only can study novel nonequilibrium states
but also can be used to map out the unoccupied band structure in solids by applying two
photon photoemission techniques. With these important techniques, one can study dispersion of correlated quasiparticles in energy-momentum space for quantum materials and the
conduction bands of nanoscale quantum systems. In topological insulators, the two photon
photoemission ARPES yields a mapping of the unoccupied state band structure on the surface, as discussed in Ref. [79].
12
1.3.1
Laser Driven Topological Phases
Optically induced phase transitions of topological states of matter by ultrafast lasers is becoming an important topic in condensed matter physics. By combining both of these lively
areas, recent work showed the coupling of short laser pulses to Dirac fermions in the topological insulator Bi2 Se3 as shown in Fig. (1.7) [80]; and in other recent studies [81, 82, 83].
A simple model of conducting polymers, the Su, Schrieffer, and Heeger (SSH) model,
describes the dimerization occurring in polyacetylene, and is a classic example of a onedimensional topological insulator [5]. While the topologically trivial or nontrivial character
of the dimerized chain is controlled by the relative strength of the nearest neighbor couplings,
it was only recently proposed that high-frequency laser light can turn trivial equilibrium
bands into topological nonequilibrium Floquet bands [6]. In Chapter 3, we introduce a new
Zig-Zag SSH model where a laser pulse can induce novel topological behavior.
1.4
Chapters
An outline of the thesis is as follows:
In chapter 2, the goal is to introduce key methods and ideas used, particularly nonequilibrium Greenâs functions for tight-binding models. The two main models used later in
the thesis; for charge density waves and topological insulators, will be introduced. Many
techniques for studying physical properties from nonequilibrium Greenâs function will be
mentioned, such as density of states, number density and order parameter.
In chapter 3, we simulate the real-time processing of single-particle energy gaps in polyacetylene coupled to short laser pulses through a Zig-Zag SSH model, applying trivial and
nontrivial topological parameters of the chain for time-resolved photoemission spectroscopy
(TRPES) analysis. We find that the TRPES band structure presents well-defined Floquet
13
bands and predict a novel band formation of the non-trivial topological chain. This chapter
is new work, and will lead to one first author paper and one or two co-auther papers.
In chapter 4, we first explain the complete foundation and schemes to solve the control
equation of motions and exhibit how to include additional constraints into the goal/cost
functions. Second, we introduce the tricks of Krotov optimal control method and the corresponding backpropagator. Third, we present the monotonic convergence property and
related proof. Fourth, we discuss the first order and second order cases for the backpropagator format, and the algorithm for implementation on a computing resource. Last, we will
show three examples to demonstrate the power of the optimal control method.
In chapter 5, we consider the hot-electron model following the simulation study of Ref. [56,
55]. They found ultrafast laser pulses can quickly heat electrons forming a hot quasithermal
gas that equilibrates with phonons on much longer time scales compared to the electron
relaxation time. We introduce a new QOCT method to find the possible shaped-laser pulses
to study the short-time transient phenomenon and to discover new photo-induced phase
transitions without melting the system. This method is applicable to general tight binding
models, and we illustrate it by controlling negative temperature states in the model of
Ref. [56].
In chapter 6, we summarize our progress in the field and give an outlook on promising
future directions.
14
CHAPTER 2
MODELS AND METHODS
As outlined in chapter 1, the use of lasers to direct photo-induced phase transitions in solids
has become a very important research subject in the field of condensed matter physics.
Techniques for the ultrafast studies of ferroelectricity, electrical conductivity, magnetism
and superconductivity have been widely utilized for different materials. Ultrafast study of
strongly correlated electron systems is one of the most interesting topic nowadays; especially
for the dynamics of an order parameterâs amplitude and phase, which determines the collective behaviour of novel states emerging in complex materials. Time-resolved photoemisssion
spectroscopy can decouple entangled degrees of freedom by visualizing their different responses in the time domain. Topological insulator (TI), a new quantum state of matter, is
another new field that can be studied with pump-probe spectroscopy. Their surfaces and
interfaces can act as a topological boundary to generate massless Dirac fermions with spinhelical textures and topological order. Investigation of the dynamics of the topology of these
materials is crucial and further development of theories and computational approaches is a
high priority.
In last decade, nonequilibrium physics has become an important area which is widely
studied; due to the observation of novel behaviours and phenomena that are not well explained by equilibrium models. Nonequlibrium problems often indicate time reverse symmetry breaking by a rapid disturbance in time, such as an interaction quench or a pulsed field.
Tight binding models of electronic structure are expected to capture many of the essential
features of the effects even in the nonequilibrium and ultrafast domains. Moreover, high fidelity characterization of transient excited many-body electron distributions in the ultrafast
time domain is now available through a variety of pump-probe experiments. Using nonequilibrium Greenâs functions, it is possible to fully calculate and understand the nonequilibrium
behaviours of interest.
15
In this chapter, the goal is to introduce key methods used, and in particular nonequilibrium Greenâs function for tight-binding models. The two main models studied in the thesis
for charge density waves and topological insulators are also included. Many techniques for
studying physical properties using nonequilibrium Greenâs function will be outlined, such as
density of states, number density and order parameters. The purpose for this chapter is to
cover the key tools and theories we use in the later chapters.
2.1
Introduction to Nonequlibrium Greenâs Functions
In general, interacting quantum and classical many-body systems in nonequilibrium states
can be completely represented by a set of N -particle wave functions entering the timedependent SchrĂśdinger equation (TDSE) as
i~
â
|Ď(t)i = H(t)|Ď(t)i
ât
(2.1)
where H(t) is a time-dependent Hamiltonian and |Ď(t)i is the corresponded time-dependent
wave function.
Next, it is important to know how expectation values can be calculated at finite temperatures, and how they change in time when the many-body system is affected by a timedependent disturbance leading to deviations of the system properties from equilibrium. To
this end, assume the system to be initially, at time t < t0 , in thermodynamic equilibrium
corresponding to the time-independent Hamiltonian H0 . The expectation value of an observable O at a given time point t > t0 is then accessible through the trace with the grand
canonical density operator as
hO(t)i = Tr{ĎOH (t)},
16
(2.2)
with
eâÎ˛H0
Ď =
Tr{eâÎ˛H0 }
.
(2.3)
Note that the operator OH (t) is in the Heisenberg picture, OH (t) = U (t0 , t)OU (t, t0 ), and
the time-evolution operator U is define by
U (t, t0 ) =
Â´
â ~i tt dtĚH(tĚ)
0
.
Te
(2.4)
From the above, one can also define the imaginary time operator for temperature as
U (t0 â iÎ˛, t0 ) = eâÎ˛H0 ,
(2.5)
and, by using the Trotter formula, the time evolution in numerical calculations becomes
U (t, t0 ) = U (t, t â ât)U (t, t â 2ât)...U (t0 + ât, t0 ),
(2.6)
where ât is a small time step. For each moment time t, we have
iât
U (t, t â ât) = exp â
H(t â ât/2) .
~
(2.7)
This operator propagates the system from the initial time t0 parallel to the imaginary
axis to t0 â iÎ˛, Further, this allows us to refine expression Eq. (2.3) for the time-dependent
expectation values. Inserting the definition Eq. (2.5) leads to the formula
hO(t)i =
Tr{U (t0 â iÎ˛, t0 )U (t0 , t)OU (t, t0 )}
.
Tr{U (t0 â iÎ˛, t0 )}
(2.8)
While the denominator is just the grand canonical partition function, the numerator of
Eq. (2.8) gives rise to the following interpretation: Reading the time-arguments from right
to left, one may say that the system first evolves along the real chronological time axis from
17
t0 to time t at which the operator O acts. Then the system anti-chronologically evolves back
along this axis from time t to t0 and finally gets propagated parallel to the imaginary axis
from t0 to t0 â iÎ˛. Graphically, this leads to a time contour in the imaginary time plane
which originally has been introduced by Keldysh [84] and is referred to as the KadanoffBaym-Keldysh contour as Fig (2.1) shows. The imaginary branch of this contour accounts
for the ensemble averaging at the given temperature Î˛.
Figure 2.1: Kadanoff-Baym-Keldysh contour in the complex time plane.
A more general expression for all time-dependent expectation values is consequently
Â´
Tr{TC exp(âi C dtĚH(tĚ))O(t) }
hO(t)i =
,
(2.9)
Tr{U (t0 â iÎ˛, t0 )}
where the exponential function is to be understood similarly to Eq. (2.4) as a Dyson series,
and TC is now the contour time-ordering operator satisfying
TC {a(t)b(t0 )} = Î¸(t â t0 )a(t)b(t0 ) â Î¸(t0 â t)b(t0 )a(t)
(2.10)
with Î¸(Ď ) the step function that equals to one when Ď > 0 and is equal to zero when Ď < 0.
Note that, operator O(t) is of the form
O(t) = U (t0 , t)OU (t, t0 ).
(2.11)
With the definitions and considerations above one is now ready to move on to the definition of the nonequilibrium Greenâs functions that we use. In the Heisenberg representation,
18
the one-particle time-ordered Greenâs function is defined by
G(x, x0 , t, t0 )
= âihTC
= âi
h
i
c(x, t)câ (x0 , t0 )
i
h
i
Tr{U (t0 â iÎ˛, t0 ) c(x, t)câ (x0 , t0 ) }
Tr{U (t0 â iÎ˛, t0 )}
(2.12)
where c(x, t) and câ (x0 , t0 ) are fermion creation and annihilation operators.
Through choosing t and t0 at different times on the Kadanoff-Baym-Keldysh contour, one
can obtain different kinds of Greenâs functions. In fact, ten different kinds of single particle
Greenâs function are mentioned in the literature [85, 56, 55] but, in the thesis, we will only
use two main independent Greenâs functions from which we are able to derive most other
Greenâs functions. One is the retarded Greenâs function defined as
GR (x, x0 , t, t0 ) = âiÎ¸(t â t0 )h{c(x, t), câ (x0 , t0 )}+ i,
(2.13)
and the other is the lesser Greenâs function defined as
G< (x, x0 , t, t0 ) = âihcâ (x0 , t0 )c(x, t)i.
(2.14)
Note that here {, }+ represents the anti-commutation relation.
2.2
Introduction to Topology in Band Structure
In condensed matter physics, a central topic is to identify phases of matter. Symmetry
breaking is one of the main reasons for certain phase transitions, for example, in magnets and
superconductors. These kind of phenomenon can be understood using ideas related to order
parameter symmetry breaking. Topological insulators have a different symmetry breaking
where the idea of topology or topological order is essential. In condensed matter physics,
many of these ideas developed through the quantum Hall effect which has fundamental
topological properties such as the quantized Hall conductivity, and the number of conducting
19
edge modes. These properties are insensitive to adiabatic changes of states and only change
when the system passes through a quantum phase transition point.
Even though the topological order of the quantum Hall effect has been studied for a long
time, the properties of topological insulators have been discovered recently. A topological
insulator has a bulk energy gap separating the highest occupied electronic band from the
lowest empty band. The system may have the same energy bands but have totally distinct
topological phases. One of the main objectives of topological band theory is to classify these
different electronic phases. New phenomena occur when there is a spatial interface between
two topologically different phases. Somewhere along the way between the two phases, the
energy gap has to go to zero. To characterize these gapless interfaces is a key goal. The
study of this goal brings us to the relation between boundary topological invariants and the
bulk topological invariants, which we will refer to as the bulk-boundary correspondence.
In this section, we give a short introduction to the relevant topics in topology, Berry
phase and Chern number.
2.2.1
Topology
In mathematics, topology is used to describe geometrical properties of objects that can
change shape smoothly. A famous example is that a coffee mug can smoothly deform to a
donut (as Fig. (2.2) shows) since they both have one hole or genus g = 1. For the condition
that one can only change the shape smoothly and cannot add a hole, a donut (g = 1) cannot
be the same topology as a soccer ball (g = 0) so they are topologically distinct. Manifolds
that can be deformed into one another are topologically equivalent.
For manifolds, a mathematical theory that defines an integer topological invariant is the
Gauss-Bonnet Theorem:
Ë
1
CdA,
g=
2Ď S
20
(2.15)
Figure 2.2: A coffee mug can smoothly deform to a donut since they both have one hole or
genus g = 1.
with Gaussian curvature C and S is a surface. Similar concepts are applied to band theory
wehre related quantities are the Berry phase and the Chern number, as discussed in the
following subsection.
2.2.2
Berry Phase and Chern Number
Berry phase is an important property in topological band theory [86, 87, 5] since it can be
used to understand the intrinsic phase of a quantum wavefunction. We know the Bloch
states are invariant under the shift
|Ď(k)i â eiĎ(k) |Ď(k)i.
(2.16)
This shift is an electromagnetic gauge transformation. We introduce the idea of the Berry
connection
~ = âihĎ(k)|âk |Ď(k)i.
A
(2.17)
Here A can be considered as analogous the electromagnetic vector gauge transformation.
Even though A is not gauge invariant, the corresponding flux in a closed loop is. One can
define Berry phase for a given closed loop C in momentum space as
Ë
Ë
~
ÎąC =
A Âˇ dk =
F d2 k,
C
S
21
(2.18)
where
~
F =âĂA
(2.19)
defines the Berry curvature. Note the surface and contour here can be in any dimension.
To be more specific, let us consider the simplest two level Hamiltonian that can be
represented in terms of Pauli matrices ~Ď as
ďŁŽ
~ Âˇ ~Ď = ďŁŻ
H(k) = d(k)
ďŁ°
ďŁš
dx â idy ďŁş
ďŁť.
dx + idy
âdz
dz
(2.20)
This Hamiltonian would give us eigenvalues Âą|d|. One could have an additional additive term
~ d|
~ so
but that would not affect the eigenvectors that depend only on the unit vector dË = d/|
we can skip it. dË here can be considered as a vector point on a sphere S 2 . If one considers
a loop C, let dË go over a 2Ď rotation in a plane, then the Berry phase is found to be Ď [88].
Ë
We therefore can say ÎąC is half the solid angle swept out by d(k).
Now we can define the
Ë as
Berry curvature as half the solid angle element for the mapping d(k)
1
Ë
F = ij dË Âˇ (âi dË Ă âj d).
2
(2.21)
From above, we have a relation that the Berry curvature integrated over a closed 2D space
Ë circles around the equator as k
is a multiple of 2Ď that is equal to the number of times d(k)
goes over a path in the Brillouin zone as Fig. (2.3) shows.
We can then define a topological invariant named the Chern number [86, 87, 5]
Ë
1
N=
F d2 k.
2Ď S
(2.22)
The result of quantization of the Chern number is not restricted to two band systems
but also generally applies to most cases. Note that the surface integral should be equal with
each other for the inside loop and outside loop up to a multiple of 2Ď, and quantization of
22
Ë for a two band
Figure 2.3: The Berry phase is equal to half the sold angle swept out by d(k)
theory.
the Dirac monopole are clearly related to the quantization, such that we can consider F as
a curvature number analogous to the Gaussian curvature K mentioned earlier.
2.2.3
Bulk Edge Correspondence
A topological insulator has gapped band structures in the bulk and gapless conducting states
at interfaces where the topological invariant changes. The states are classified into different
topological orders based on their fundamental properties. To understand the edge states, it
is easy to consider the interface between the integer quantum Hall state and vacuum [14].
In a semiclassical framework, these edge states can be understood in term of the skipping
orbits where electrons have cyclotron orbits that bounce off the edge (Fig. (2.4)).
Note that, the electronic states responsible for this behavior can propagate in two different
directions along the edge so they are chiral. There are no states available for backscattering
which underlies the perfectly quantized electronic transport in the quantum Hall effect, the
23
Figure 2.4: Edge states as skipping cyclotron orbits.
states are insensitive to disorder effects. The topology of the bulk quantum Hall state then
leads to the chiral surface states and quantized conductance.
A simple explanation of the chiral edge states can be shown by applying the two band
Dirac model, where at a Dirac point,
HD = mĎz ,
(2.23)
with Ďz is a Pauli matrix.
Now consider a mass m is a function of y as m(y) at one of the Dirac points of an
interface and will change sign with y. For y > 0, m(y) > 0 gives the insulator, and, for
y < 0, m(y) < 0 gives the quantum Hall state. Assume m0 > 0 is fixed. Since the states
satisfy translation symmetry in the x direction, one can write Ďqx = eiqx x Ď(y). The zero
energy mode Ď0 (y) is of the form
Â´
y
0
Ď0 (y) = eâ 0 m(y )/vF | âi
24
(2.24)
where | âi is the upper eigenvector of Ďz . The corresponding eigenenergy is
E(qx ) = ~vF qx .
(2.25)
The band of states has a positive group velocity dE/dqx = ~vF defining a right moving
chiral edge mode that intersects with the Fermi energy EF .
Figure 2.5: The chiral edge states connect the valence band near K and K 0 with linear or
nonlinear dispersion for linear and nonlinear Hamiltonians respectively.
In Fig. (2.5), the energy bands versus momentum kx along the edge is shown. The blue
areas indicate the bulk conduction and valence bands, which have continuum states and have
the energy gap near K and K 0 . Interestingly, a line connecting the valence band at K 0 to
the conduction band at K with a positive group velocity describes edge states.
The edge states may be different if the Hamiltonian near the surface is described by
a non-linear function. For example, the red line in Fig. (2.5) shows the edge states can
pass the Fermi level three times if the group velocity is non-linear. However the difference
NR â NL between the number of right and left chiral modes is the same, and is an integer
topological invariant characterizing the interface. The edge related value of NR â NL is
therefore decided by the topological structure of the bulk states. This is summarized by the
25
bulk edge correspondence:
ân = NR â NL .
(2.26)
Note that ân is the difference in the Chern number across the interface.
2.3
Tight-binding Models at Equilibrium
In solid-state physics, tight-binding models (or TB models) are used to compute electronic
band structure, using an approximate approach based upon superposition of localized wave
functions at each atomic site. The model is closely connected to the LCAO (Linear Combination of Atomic Orbitals) method used in chemistry. Since the electrons in the model are
quite tightly bound to the atoms and have limited overlap with surrounding atoms of the
solid, the model is particularly well-suited for calculations of solids for which the electronic
states remain relatively close to the unperturbed atomic orbitals. The energy of the electrons
will also be quite close to the ionization energy of the electron in the free atom or ion because
the interaction with potentials and states on neighboring atom is limited. Though complete
physical principles are involved, tight-binding models yield a relatively simple mathematical
formulation.
2.3.1
SSH, Rice Mele, Platero Models
In this subsection, we introduce an important one dimensional model that illustrates some
of the fundamental topological properties in solid state physics. The simple but elegant
Su Schrieffer Heegar (SSH) model was used as a basic model of the conducting polymer
polyacetylene [5]. At half filling it has a Peierls instability to a dimerized state. The most
intriguing part is that in a finite chain there are two different dimerized states, as shown in
Fig. (2.6). These two distinct topological states lead to different topological properties and
responses in the nonequilibrium situation. The SSH model here presents the simplest two
26
band model for introducing these special topological phenomena.
Figure 2.6: Two distinct topological states of the SSH model. (a) is the case Î´Ď > 0 and (b)
is the case Î´Ď < 0.
SSH used a 1D tight-binding Hamiltonian to model polyacetylene as
H=
X
â
â
(Ď + Î´Ď )cAi cBi + (Ď â Î´Ď )cAi+1 cBi + h.c.
(2.27)
i
with labels A and B referencing to the two different atoms/sites in a unit cell. The term Î´Ď
represents the dimerization and induces a gap opening in the energy band. The operators
satisfy the anti-commutation relations
â
{ci , cj }+ = Î´ij ,
{ci , cj }+ = 0,
â
â
{ci , cj }+ = 0,
(2.28)
Note that, Î´Ď can be positive or negative and we would have two distinct patterns or topologies determined by the sign of Î´Ď . Even though the real SSH model should consider spin, we
treat spinless electrons here for simplicity. To analyze the Hamiltonian in momentum space
through Fourier transform we impose periodic boundary conditions for a chain with an even
number of sites. In that case we get a two-by-two matrix for each momentum k, and we
27
define N as the number of sites:
H=
X
â
Hab (k)cak cbk ,
(2.29)
k
with
~ Âˇ ~Ď
H(k) = d(k)
(2.30)
and
dx (k) = (Ď + Î´Ď ) + (Ď â Î´Ď ) cos(ka),
dy (k) = (Ď â Î´Ď ) sin(ka),
dz (k) = 0.
(2.31)
Note that ~Ď = {Ďx , Ďy , Ďz } and
ďŁŽ
ďŁŻ0
Ďx = ďŁ°
1
ďŁŽ
ďŁŻ0
Ďy = ďŁ°
i
ďŁŽ
ďŁŻ1
Ďz = ďŁ°
0
ďŁš
1ďŁş
ďŁť
0
ďŁš
âiďŁş
ďŁť
0
ďŁš
0ďŁş
ďŁť
â1
(2.32)
(2.33)
(2.34)
P
~
so the inner product with d(k)
is H(k) = i={x,y,z} di (k)Ďi . From Eq. (2.31), one writes
~
the two band H(k) in terms of d(k)
enabling use of topological ideas introduced in section
2.2.1. An important point is that if dz = 0; the two bands have a chiral symmetry defined
by the operator Î = Ďz , which anticommutes with the Hamiltonian: {H(k), Î } = 0. Then
each eigenstate with energy E has a corresponding state |ĎE i = Î |ĎâE i with energy âE, so
that this chiral symmetry actually indicates a particle-hole symmetric spectrum. While this
symmetry is not preserved in real polyacetylene because there are second order effects such
as second neighbor hopping; it is still very useful to investigate the effects and topologies for
understanding some outstanding physical phenomena.
28
Consider the different signs of Î´Ď , which can lead to different topological phases or geo~
metric results. For Î´Ď > 0, then dx > 0 for all k so d(k)
has zero phase if d~0 = ~0 is the center
(degeneracy point). One can see this path from Fig. (2.7). On the other hand, for Î´Ď < 0,
~
dx (k â Ď/a) < 0, so that d(k)
circles around the center and has phase 2Ď. The phase results
can be understood as polarization in the strong coupling region, |Î´Ď | â Ď . In that limit, one
can think that the electrons stay in localized states so, when we shift Î´Ď from positive to
negative, the states will move electrons over by half a unit cell, inducing polarization and
topological phase.
Figure 2.7: Two distinct topological paths of the SSH model. (a) is the case Î´Ď > 0 and (b)
is the case Î´Ď < 0.
29
The polarization gains phase of integer n times phase 2Ď when the chiral symmetry is
kept. One can break the symmetry by considering a dz (k) > 0 so the polarization now can
vary continuously and topology is lost in this 1d model. Otherwise, keeping chiral symmetry
leads to topologically distinct states that are characteristic by their quantized polarization.
Note that, in order to get from the positive Î´Ď > 0 state to the negative Î´Ď < 0 state we
need to pass a point with d~ = ~0 to obey the chiral symmetry, which is a quantum phase
transition point.
For the SSH model the eigenenergies are derived from the Hamiltonian
ďŁš
ďŁŽ
0
R + reâika ďŁş
~ Âˇ ~Ď = ďŁŻ
H(k) = d(k)
ďŁť,
ďŁ°
ika
R + re
0
where R = Ď + Î´Ď and r = Ď â Î´Ď ; so that,
q
Ek = Âą R2 + r2 + 2Rr cos(ka).
(2.35)
(2.36)
Note that, Ek is invariant under r â R, but the topology changes. Here we use R and r to
show a relation between the topological phase and the dispersion relation in Eq. (2.36). We
also know that the length of R and r are decided by the dimerization term Î´Ď . The geometric
paths for the dispersion relation and the topological phase can be found in Fig. (2.8) and
Fig. (2.9). Note that when R = 0 or r = 0, we will have a flat band for dispersion relation,
and when |Î´Ď | is the same, the dispersion relation will be the same no matter for R > r or
R < r.
Another simple but significant model is the Rice-Mele model [89], which is the SSH model
with an extra staggered onsite potential. The Hamiltonian of the Rice-Mele model on a 1D
chain is written as
H=
N
X
â
â
â
â
u(cAi cAi â cBi cBi ) + vcAi cBi + wcAi+1 cBi + h.c.
i
30
(2.37)
Energy / Ď
2
1
4Î´Ď = 2|R â r|
0
4Ď = 2(R + r)
â1
â2
-1.0
-0.5
0.0
k(a/Ď)
0.5
1.0
Figure 2.8: Dispersion relation of the SSH model based for R = Ď + Î´Ď and r = Ď â Î´Ď .
Figure 2.9: Topological path of the SSH model depends on R, r and ka.
with labels A and B referencing to the two different atoms/sites in a unit cell, and the onsite
potential u, intercell hopping amplitude w, and the intracell hopping amplitude v. Note
that, for the SSH model, u = 0, v = Ď + Î´Ď and w = Ď â Î´Ď . For the Rice-Mele model, the
matrix of the Hamiltonian on a 1D chain with N = 4 sites is of the form
31
ďŁŽ
ďŁš
u v 0 0 0 0 0 0
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻv âu w 0 0 0 0 0 ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ0 w u v 0 0 0 0 ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ 0 0 v âu w 0 0 0 ďŁş
ďŁş.
H = ďŁŻ
ďŁŻ
ďŁş
ďŁŻ0 0 0 w u v 0 0 ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ 0 0 0 0 v âu v 0 ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ0 0 0 0 0 w u v ďŁş
ďŁ°
ďŁť
0 0 0 0 0 0 v âu
(2.38)
To analyze the Hamiltonian in momentum space through Fourier transform we impose
periodic boundary conditions for a chain with an even number of sites. In that case we get
a two-by-two matrix for each momentum k, and we define N as the number of sites:
H=
X
â
Hab (k)cak cbk ,
(2.39)
k
with
~ Âˇ ~Ď
H(k) = d(k)
(2.40)
and
dx (k) = v + w cos(ka),
dy (k) = w sin(ka),
dz (k) = u.
Note that ~Ď = {Ďx , Ďy , Ďz } are Pauli matrices. The eigenvalues are,
q
Ek = u Âą v 2 + w2 + 2vw cos(ka).
(2.41)
(2.42)
The bulk energy eigenstates of a band insulator are delocalized over the whole system.
We use as an example the bulk Hamiltonian of the Rice-Mele model. The energy eigenstates
are the plane wave Bloch states
|Î¨(k)i = |ki â |u(k)i,
32
(2.43)
with
N
1 X imk
â
|ki =
e
|mi,
N m=1
(2.44)
for k = {Î´k , 2Î´k , ..., N Î´k } with Î´k = 2Ď/N . The |u(k)i are eigenstates of the bulk momentumspace Hamiltonian from Eq. (2.40). Note that, the Bloch states |Î¨(k)i are spread over the
whole chain. They span the occupied subspace, defined by the projector
P =
X
|Î¨(k)ihÎ¨(k)|.
(2.45)
kâBZ
The phase of each Bloch eigenstate |Î¨(k)i can be set at will. A change of these phases, a
gauge transformation, |u(k)i â eiÎą(k) |u(k)i gives an equally good set of Bloch states, with
an arbitrary set of phases Îą(k) â R for k = Î´k , 2Î´k , .... Because of this freedom, we are able
to ensure that in the thermodynamic limit of N â â, the components of |Î¨(k)i are smooth
and continuous functions of k. While this gauge may not be easy to derive; it is topologically
trivial when u 6= 0 since a circle loop does not enclose degeneracy point.
Another interesting extension of the SSH model is a dimers chain coupled to an ac
electric field discussed by Platero [81] and illustrated in Fig. (2.10). Ď and Ď 0 are hopping
terms between two sublattices A and B, and with a periodic boundary conditions. a0 is the
cell size and b0 is the distance between A-B atoms.
Figure 2.10: A dimers chain with two sublattices A and B, and Ď and Ď 0 are hopping terms
between two sublattices A and B. a0 is the cell size and b0 is the distance between A-B
atoms.
33
The corresponding bulk Hamiltonian in momentum space is therefore of the form
ďŁŽ
ďŁš
ĎF (k)ďŁş
ďŁŻ 0
ĎĚn,m = Ď ďŁ°
(2.46)
ďŁť,
ĎĚF (k)
0
with
ĎF (k) = Îťeâikb0 + eik(a0 âb0 ) ,
(2.47)
ĎĚF (k) = ĎĚâF (k)
(2.48)
and
where Îť = Ď 0 /Ď .
2.3.2
Freerickâs Model
A simple model of an electronic CDW in a 2-D system is shown in Fig. (2.11) [56, 55]. Note
that, this is a special case of the Rice-Mele model described in the previous subsection on a
square lattice.
In the model, the charge density wave system is constituted by two different sublattices
A and B of equal size. The nearest neighbors for lattice sites in the A sublattice is from
the B sublattice and the nearest neighbors for lattice sites in the B sublattice is from the
A sublattice. To describe different sublattices, one can use the fixed site potential energy,
which is described by an onsite potential U on the sublattice A and onsite potential zero on
the sublattice B. Hopping term Ď is allowed between nearest sublattice A and sublattice B
and is the same for all directions. Note that here we use spinless electrons for simplicity.
To start with, let consider the Hamiltonian without an external field so the model now is
time-independent and in a second quantization form is
H=â
X
â
Ďij ci cj + (U â Âľ)
ij
X â
X â
ci ci â Âľ
ci ci ,
iâA
34
iâB
(2.49)
Figure 2.11: CDW model with two different sublattices A and B.
â
Note that here ci and ci are the creation, annihilation operators at site i for spinless electrons,
and Âľ is the chemical potential. To study the electrons in a periodic lattice, we work in
Fourier space and analyze it within the first Brillouin zone. We define the lattice vector
P
R = i ni ai eĚi where ni is an integer, ai is the lattice constant, and eĚi is the basis vector
direction in the lattice. We know any periodic function on the lattice R has nonzero Fourier
elements only for certain vectors {K}. In momentum space, we know these vectors {K}
form a reciprocal lattice. The sites of a reciprocal lattice represent wave vectors that satisfy
the condition eiKÂˇr = eiKÂˇ(R+r) for any real space vector r. Similar to the way one chooses
the primitive unit cell in real space. The first Brillouin zone is decided by the WignerSeitz primitive unit cell centered at the origin in reciprocal space. Note that we apply the
reduced zone scheme by reducing all wave vectors into first Brillouin zone so energy levels
originating from k + nK are now regarded as belonging to the nth band. We therefore have
the transformation from real space to momentum space in the reduce Brillouin zone scheme
35
satisfying
â
ci =
X
â
eâikÂˇRi ck ,
(2.50)
k
with k going over the first Brillouin zone, and Ri is a lattice vector for site i. Since the
translational symmetry is partially broken in the model so the momentum points k and
k + Q are coupled, with Q = (Ď, Ď, Ď, ...). Note that this holds true for hypercubic lattices
in general dimension, In the reduced Brillouin zone, the transformation equation becomes
â
ci =
X
â
â
(eâikÂˇRi ck + eâi(k+Q)ÂˇRi ck+Q ).
(2.51)
k
Because eâiQÂˇR is one for lattice sites on the A sublattices and minus one on the B sublattices,
we can find;
â
â
â
eikÂˇRi (ck + ck+Q ),
(2.52)
X ikÂˇR â
j (c â câ
=
e
k+Q ).
k
(2.53)
ciâA =
X
k
â
ciâB
k
By taking the Hermitian conjugate, the annihilation operator identities are derived. One
can show the electronic band structure at U = 0 is then;
Îľk = â
X
hiji
d
d
X
X
tâ
l
â cos(k l a),
Ďij exp âik Âˇ (Ri â Rj ) = â2Ďij
cos(k a) = â
d
l=1
l=1
(2.54)
then restricting k to the reduced Brillouin zone, is equivalent to having Îľk â¤ 0, and l is
the index for spatial component, d is the number of spatial dimensions, a is the lattice
constant, tâ is the re-defined hopping energy between nearest neighbors in each dimension
â
and Ďij â tâ /2 d. The rescaled hopping term tâ is the rescaled hopping energy between
nearest neighbors in each dimension to preventing divergence in large dimensions. To be
â
more specific, tij = tâ /2 d in the last equation allows us to approach the d â â limit.
Note that the reduced Brillouin zone is half the size of the original one (without the
sublattice modulation) since, due to the onsite chemical potential, the periodic size in real
36
space is doubled. Now one can write down the Hamiltonian in a two by two [ck , ck+Q ] basis,
H=
X
k
ďŁŽ
ďŁšďŁŽ
ďŁš
U/2
ďŁŻk + U/2 â Âľ
ďŁş ďŁŻ ck ďŁş
â
â
ďŁťďŁ°
ďŁť.
ck ck+Q ďŁ°
U/2
âk + U/2 â Âľ
ck+Q
(2.55)
If the system is in equilibrium, one is able to diagonalize the Hamiltonian matrix with the
eigenfunction basis,
ck+ = Îąk ck + Î˛k ck+Q ,
(2.56)
ckâ = Î˛k ck â Îąk ck+Q .
(2.57)
For the upper band and lower band eigenstates at the momentum point k in the reduced
Brillouin zone, we know that ck+ and ckâ are annihilation operators with Îąk and Î˛k given
by
q
Îąk = (U/2)/ 2(2k + U 2 /4 â 2k + U 2 /4)
q
q
Î˛k = ( 2k + U 2 /4 â k )/ 2(2k + U 2 /4 â 2k + U 2 /4)
(2.58)
(2.59)
Note that the Hamiltonian matrix is diagonalized so we are able to rewrite the Hamiltonian
in the two by two [ck+ , ckâ ] basis as
ďŁšďŁŽ
ďŁš
ďŁŽ
ďŁŻk+ 0 ďŁş ďŁŻck+ ďŁş
â
Hk = câ
ďŁťďŁ°
ďŁť.
ďŁ°
c
k+
kâ
0 kâ
ckâ
(2.60)
Where Îľk+ and Îľkâ are the eigenstates;
ÎľkÂą = U/2 â Âľ Âą
q
Îľ2k + U 2 /4.
(2.61)
Fig. (2.12) shows the corresponding dispersion relation ÎľkÂą for the CDW model. Note that,
in the Pauli matrix representations d~ Âˇ ~Ď , it is clear that the Chern number is always zero
and hence without light the model is not topological.
The corresponding dispersion relation for the CDW model is given in Fig. (2.11).
37
15
10
Îľk Âą
5
0
â5
â10
â15
â10
â5
0
5
Îľk
10
Figure 2.12: Dispersion relation ÎľkÂą of the CDW model when Âľ = 1/2 and U = 1.
2.3.3
Greenâs Function Method
We calculate the two-time retarded Greenâs function and two-time lesser Greenâs function
defined in subsection 2.1. Because the time-translation symmetry is not broken in the equilibrium system, one only need to consider the relative time t â t0 in the Greenâs functions.
The local density of states is found from the local retarded Greenâs function and defined
through
Ë
0
1
1
R
Ai (Ď) = â Im[Gi (Ď)] = â Im( eiĎ(tât ) )GR
(t, t0 )d(t â t0 )).
i
Ď
Ď
(2.62)
The two-time retarded Greenâs function is represented using the creation and annihilation
operators for sublattices A and B as a sum over all k points;
0
0
GR
i (t, t ) = âiÎ¸(t â t )
X
â
â
h{ck (t) Âą ck+Q (t), ck (t0 ) Âą ck+Q (t0 )}+ i,
(2.63)
k
when in equilibrium, OĚ(t) = eiHt OeâiHt for any operator. Note that the + sign represents
a local retarded Greenâs function on the A sublattice and the â sign represents that on the
B sublattice. One can re-write the Greenâs function in the diagonalized basis so the local
38
retarded Greenâs function on the A sublattice becomes
0
0
GR
A (t, t ) = âiÎ¸(t â t )
X
h{(Îąk + Î˛k )ck+ (t) + (Î˛k â Îąk )ckâ (t),
k
(Îąk + Î˛k )ck+ (t0 ) + (Î˛k â Îąk )ckâ (t0 )}i
(2.64)
By solving equation of motion for the time evolution of the ck+ (t) and ckâ (t) operators, one
finds;
â
â
â
â
ck+ (t) = exp[âik+ t]ck+
(2.65)
ckâ (t) = exp[âikâ t]ckâ
(2.66)
The local density of states (LDOS) is then of the following form
# q
"s
Ď Âą U/2
Ď( Ď 2 â U 2 /4).
AA,B (Ď) = Re
Ď â U/2
(2.67)
Through summing over the local DOS for each sublattice with weight 1/2, the total DOS
is calculated. Note that the two sublattice model at equilibrium gives a two band structure
with a band gap equal to the onsite interaction U . We also know the LDOS on the A
sublattice has a divergence at Ď = U/2 and the LDOS on the B sublattice has a divergence
at Ď = âU/2. Just like the retarded Greenâs function, the imaginary part of the Fourier
transform of the lesser Greenâs function gives
1
1
(Ď) = Ai (Ď)f (Ď) =
ImG<
Im
i
2Ď
2Ď
Ë
0
0
0
eiĎ(tât ) G<
i (t, t )d(t â t )
.
(2.68)
In our calculation, we set chemical potential Âľ = 0 which is the half-filling case. That is
the reason why the imaginary part of the lesser Greenâs function indicates that the electrons
fill the lower band. By definition, one can find that imaginary part of local lesser Greenâs
function corresponds to the number of local electrons at time t0 so the local electron number
density on each sublattice can be written as
nA,B (t) = Im(G<
A,B )(t, t)).
39
(2.69)
One can also define the order parameter at equilibrium, which is the difference between the
electron number density on the A and B sublattices as
âŚ(t) =
nB (t) â nA (t)
.
nB (t) + nA (t)
(2.70)
In equilibrium there are always more electrons on the B sublattice than on the A sublattice
as there is a repulsive potential on the A sublattice. In the half filling case, the upper band
is completely empty and the lower band is filled at T = 0. So we can conclude that the
larger the interaction U , the stronger the electronic CDW order is.
2.3.4
Graphene and Dirac Cone
The term graphene first appeared in 1987 to describe single sheets of graphite as a constituent
of graphite intercalation compounds. Graphene also is the most typical 2D material with
Dirac cones [12]. It has two C atoms per unit cell arranged in a hexagonal lattice (Fig. (2.13)).
Figure 2.13: Graphene is an atomic-scale hexagonal lattice made of carbon atoms.
40
Because the C atoms bind together with both Ď and Ď bonds, graphene is rather stable
and can be studied through a relatively simple model. A tight-binding model describes the
electronic structure of graphene quite well. The Hamiltonian is simplified into a 2 Ă 2 matrix
for each wave vector k using a nearest-neighbor hopping approximation as
ďŁŽ
ďŁš
ďŁŽ
ďŁš
X ďŁŻ
0
ti ei(kÂˇdi ) ďŁş
ďŁŻÎľ0 0 ďŁş
H(k) = ďŁ°
ďŁť+
ďŁ°
ďŁť,
âi(kÂˇd
)
i
0 Îľ0
0
i=1,2,3 ti e
(2.71)
where di with (i = 1, 2, 3) are vectors that connect a C atom to its three nearest neighbors,
Îľ0 is the on-site energy, and ti are the corresponding hopping energies. The energy bands
are thus found to be
EÂą (k) = Îľ0 Âą |
X
ti ei(kÂˇdi ) |.
(2.72)
i=1,2,3
For convenience, one can set the Fermi level to be Îľ0 = 0. For the equilibrium structure,
di = r0 , ti = t0 , and the valence and conduction bands contact at K and K 0 points of the
hexagonal Brillouin zone. One can expand the energy bands around K or K 0 as
EÂą (q) = Âą~vF |q|
(2.73)
where K = k â q, and vF = 3t0 r0 /2~ is the Fermi velocity. Near K and K 0 points,
Eq. (2.73) shows that graphene has a cone-like band structure with linear dispersion, similar
to a relativistic particle. The density of states (DOS) per unit cell with a degeneracy of 4
(2 Ă 2 spin and valley) near Fermi level is
â
Ď(E) = 4|E|/ 3Ďt20
(2.74)
Therefore graphene is a gapless semiconductor with zero DOS per site at Fermi level. The
Hamiltonian near the K and K 0 points can be transformed into
H(q) = vF p Âˇ Ď
(2.75)
where Ď are the Pauli matrices, and p = âi~â is the momentum operator. Note that,
Eq. (2.75) is identical to, by replacing c (speed of light) with vF , the massless Dirac equation or Dirac-Weyl equation with spin S = 1/2. For this reason, the K and K 0 points are
41
also called Dirac points, and the linear band structure is called a Dirac cone as Fig. (2.14)
shows. These graphene possesses massless Dirac fermions with pseudospins of Âą1/2. Here
pseudospins indicate which sublattice has a great occupancy.
Figure 2.14: Dirac coneâthe electrons in graphene behave like massless Dirac particles that
appear in the electronic band structure as gapless excitations with a linear dispersion.
Due to the Dirac-cone structure, many new electronic properties and physical phenomena
have been found in graphene. For instance, when a uniform perpendicular magnetic field B
is applied, a particular Landau level form in graphene is
q
2 |n|B
E(n) = sgn(n) 2e~vF
(2.76)
where n â Z (the Landau level index) and E(n) â n in normal semiconductors. The
Landau levels in 2D systems can be verified by quantum Hall effect (QHE) if the magnetic
field is strong enough. Different from the conventional 2D systems, graphene shows a novel
42
half-integer QHE with
Ďxy = (4e2 /h)(N + 1/2)
(2.77)
which can be described by the pseuospins and the degeneracy of Dirac fermions.
2.4
Tight Binding Models in a Time Varying Field
In the past decade, since precise nonequilibrium study is now possible in experiments, extensions of theory to treat pump-probe experiments has become very important. Normally,
for the case of pump-probe experiments, the Hamiltonian will be time-dependent due to the
effect of a time-varying electric field. We thus need to include electromagnetic fields into the
lattice models described earlier, and the way that we do this is described in the next section.
2.4.1
Peierlâs Substitution
To include the time-dependent electric field, we use the Peierlsâ substitution [90, 56, 55],
which is a widely employed approximation for describing tight-binding models in the presence
of a slowly changing magnetic vector potential. It is a simplified semi-classical treatment of
the electromagnetic field that is nonperturbative. The hopping matrix gains a phase factor
with the Peierlsâ substitution as
#
Ë
ie Rj
Ďij â Ďij exp â
A(r, t)dr .
~c Ri
"
(2.78)
The Peierls phase originates from the propagator of an electron in a magnetic field due to
the dynamical term qv Âˇ A appearing in the Lagrangian. In the path integral formalism,
which generalizes the action principle of classical mechanics from the Hamiltonian
H0 =
2
1
p â eA + Ë0 ,
2m
43
(2.79)
and one can derive the Peierlsâ phase from the path integral. A scalar potential term
P
â
âe i Ď(ri , t)ci ci is also added into the Hamiltonian. Ď(ri , t) is the scalar potential for
the external field and A(r, t) is the vector potential. In the following, we only work with the
gauge that has zero scalar potential and we also assume a spatially uniform time-dependent
vector potential so one can neglect the magnetic field effects. The is called the Hamiltonian
gauge. In Maxwellâs equations, one can find the corresponding electric field E(t) from the
derivative of the vector potential A(r, t) as
E(t) = â
1 âA(r, t)
.
c ât
(2.80)
Note that for a spatially uniform field, we have A(t) = A(t)(1, 1, ..., 1) for a vector potential
directed along the (1, 1, ..., 1) axis of a hypercubic lattice. In momentum space, the timedependent band structure for the U = 0 case is then
i
h
X
e
AB
Îľk (t) = â
Ďij exp âi(k â A(t)) Âˇ (RiA â RjB ) .
~c
(2.81)
hiji
Therefore, the influence of the Peierlsâ substitution is to induce a time-dependent shift to
the momentum in the non-interacting electron band structure. At U = 0 we have,
d
X
tâ
eA(t)
l
â cos a k â
.
Îľk (t) = â
~c
d
(2.82)
l=1
The time-dependent Hamiltonian for the CDW case becomes (U 6= 0),
X
ie
â
H(t) = â
Ďij exp
A(t) Âˇ (Ri â Rj ) ci (t)cj (t)
~c
hiji
X â
X â
+(U â Âľ)
ci (t)ci (t) â Âľ
ci (t)ci (t).
(2.83)
iâB
iâA
The Fourier transformation to momentum space can be applied to the above,
â
ci (t) =
X
â
â
[eâikÂˇRi ck (t) + eâi(k+Q)ÂˇRi ck+Q (t)],
(2.84)
k
and the corresponding hermitian conjugate. In momentum space and in the SchrĂśdinger
representation, the time-dependent Hamiltonian can be represented as
ďŁŽ
ďŁšďŁŽ
ďŁš
X
U/2
ďŁŻk (t) + U/2 â Âľ
ďŁş ďŁŻ ck ďŁş
â
â
Hs (t) =
ďŁťďŁ°
ďŁť
ck ck+Q ďŁ°
u/2
â
(t)
+
U/2
â
Âľ
c
k
k
k+Q
44
(2.85)
The band structure Îľk (t) at U = 0 in time-dependent form can be written with the difference
formula of the cosine,
Îľk (t) = cos(eaA(t)/~c)Îľ(k) + sin(eaA(t)/~c)ÎľĚ(k)
(2.86)
which depends on the band structure at U = 0
d
X
tâ
â cos(ak l )
Îľ(k) = â
d
l
(2.87)
d
X
tâ
â sin(ak l ).
ÎľĚ(k) = â
d
l
(2.88)
and
One can consider the equation of motion for the operators ck (t) and ck+Q (t) in the Heisenberg
picture, as
âck (t)
= [HH (t), ck (t)],
ât
(2.89)
âck+Q (t)
= [HH (t), ck+Q (t)],
ât
(2.90)
i
and
i
Note that HH (t) indicates the Heisenberg representation for the Hamiltonian. If one brings
in the time-dependent Hamiltonian and evaluates the commutators, we find
âck (t)
= (Îľk (t) + U/2 â Âľ)ck (t) + U/2ck+Q (t),
ât
(2.91)
âck+Q (t)
= U/2ck (t) + (âÎľk (t) + U/2 â Âľ)ck+Q (t),
ât
(2.92)
âi
and
âi
Then the time evolution for the creation and annihilation operators satisfy
ďŁŽ
ďŁš
ďŁŽ
ďŁš
ďŁŻ ck (t) ďŁş
ďŁŻ ck (t0 ) ďŁş
ďŁ°
ďŁť = Uk (t, t0 ) ďŁ°
ďŁť
ck+Q (t)
ck+Q (t0 )
45
(2.93)
Note that for each momentum the time-evolution operator Uk (t, t0 ) is a time ordered product
ďŁą
ďŁŽ
ďŁšďŁź
ďŁ´
ďŁ´
Ë
ďŁ˝
ďŁ˛
t
U/2
ďŁŻk (t) + U/2 â Âľ
ďŁş
0
0
dt ďŁ°
(2.94)
Uk (t, t ) = T exp i
ďŁť .
ďŁ´
ďŁ´
ďŁž
ďŁł t0
U/2
â (t) + U/2 â Âľ
k
Numerically, one needs to slice the continuous time-dependent terms into small pieces for
very short time steps ât to calculate the time evolution for Uk (t, t0 ) with the time-dependent
terms inside the exponential by changing
tĚ ââ t â â/2.
(2.95)
The corresponding result is more concise by employing an identity of the exponential of the
Pauli matrices {Ďx , Ďy , Ďz }. Since any two by two matrix has four degrees of freedom, we
know any two by two matrix can be represented as a linear combination of the Pauli matrices
Ďx , Ďy , Ďz and the unit matrix 1. So we expand
ďŁŽ
ďŁš
ďŁŻ a0 + az ax â iay ďŁş
A = a0 1 + ~a Âˇ ~Ď = ďŁ°
ďŁť.
ax + iay a0 â az
(2.96)
If we consider the case that the matrix A has a0 = 0, and define A = Îť~a Âˇ ~Ď , we can prove
the generalized Euler identity relation
eiÎť~aÂˇ~Ď = cos(Îť)1 + i(~a Âˇ ~Ď ) sin(Îť).
(2.97)
Here ~a is chosen to be a unit vector and Îť is the magnitude. A matrix exponential has its
Taylor expansion in general as
eA
â
X
An
=
.
n!
(2.98)
n=0
By separating the odd and even orders, we have
ďŁŤ
ďŁś
!
n
n
X
X
Îť
Îť
eiÎť~aÂˇ~Ď =
(i)n (a Âˇ Ď)n + i ďŁ
(i)nâ1 (a Âˇ Ď)nâ1 ďŁ¸ (~a Âˇ ~Ď ).
n!
n!
nâeven
(2.99)
nâodd
One also has the useful identity
(~a Âˇ ~Ď )(~b Âˇ ~Ď ) = ~a Âˇ ~b1 + i~Ď Âˇ (~a Ă ~b),
46
(2.100)
which includes the special case
(~a Âˇ ~Ď )2 = 1
(2.101)
with a unit vector ~a = ~b. We see that the expansion of the cosine is equal to the even terms
in the expansion, and the expansion of sine is equal to the odd terms in the expansion, hence
eiÎť~aÂˇ~Ď = cos(Îť)1 + i sin(Îť)(~a Âˇ ~Ď ).
(2.102)
In this thesis, we only work at half filling where Âľ = U/2 in momentum space k so the above
equations become
Uk (t, t â ât) = eiÎťk a~k Âˇ~Ď = cos(Îťk )1 + i sin(Îťk )(~ak Âˇ ~Ď )
(2.103)
with
ďŁŽ
ďŁš
U/2
ât ďŁŻk (t â ât/2)
ďŁş
a~k Âˇ ~Ď =
ďŁ°
ďŁť,
Îťk
U/2
âk (t â ât/2)
(2.104)
and
~ak = (0, U ât/2Îť, âtk (t â ât/2)/Îť)
q
Îťk = ât 2k (t â â/2) + U 2 /4.
(2.105)
(2.106)
In the real calculations, instead of the time at ââ, one must start from a minimum time t0
so one can calculate the time-evolution operator in the form
Uk (t, t0 ) = Uk (t, t â ât)Uk (t â ât, t â 2ât)...Uk (t + ât, t0 )
(2.107)
For each k, one can derive the two-time evolution operator from the relation
Uk (t, t0 ) = Uk (t, t0 )Uk (t0 , t0 ).
(2.108)
Once the time evolution at each time step is derived, we are able to calculate the nonequilibrium Greenâs functions and obtain the physical properties of the model.
47
2.5
Introduction to Floquet Theory
Floquet states can be considered as a time domain version of Bloch waves, where Bloch waves
are defined crystal momentum and the corresponding Hamiltonian has discretized translation invariance in space. Similarly, Floquet states in a material are periodic in energy, and
the Hamiltonian has discretized invariance in time [91].
2.5.1
Periodic Hamiltonian
Assume that time-dependent Hamiltonian HĚ(t) can be written as
HĚ(t) = f (t) + H(t),
(2.109)
where the scalar function f (t) depends only on time t so that it commutes with H(t). We
use |Ď, ti as the time dependent state vectors. One thus can reduce the time dependence
from the problem
i~
â
|Î¨, ti = HĚ(t)|Î¨, ti
ât
(2.110)
i~
â
|Ď, ti = H(t)|Ď, ti.
ât
(2.111)
to
Where the remaining non-trivial Hamiltonian H is a periodic function of time, i.e.
H(t + T0 ) = H(t)
(2.112)
where T0 is the period time. The related angular frequency âŚ is then
âŚ = 2Ď/T0 .
(2.113)
This periodicity allows us to rewrite H in the Fourier series as
H(t) =
inf
X
H n einâŚt ,
n=â inf
48
(2.114)
where the Fourier components H n are of the form
Hn
2.5.2
Ë T
0
1
=
dteâinâŚt H(t).
T0 0
(2.115)
Floquet Formalism
Floquet theory claims that the solutions of the time dependent SchrĂśdinger equation
i~
â
|Ď, ti = H(t)|Ď, ti
ât
(2.116)
can be written as
i
â t
|Ď, ti = e ~
XX
n
FnÎą einâŚt |Îąi.
(2.117)
Îą
Note that the set {|Îąi} is a complete orthonormal basis for the Hilbert space of the periodic
Hamiltonian H. To verify the claim, we substitute the expansion into the SchrĂśdinger
equation and derive
"
#
X
â
â
â ~i t
i~ |Ď, ti = i~
e
FnÎą einâŚt |Îąi
ât
ât
nÎą
X
X
i
â t
â i t
= e ~
FnÎą einâŚt |Îąi + i~e ~
FnÎą inâŚeinâŚt |Îąi
=
nÎą
X
i
â t
e ~
FnÎą einâŚt ( â n~âŚ)|Îąi
nÎą
nÎą
"
=
X
H m eimâŚt
â ~i t
e
m
#
X
FnÎą
einâŚt
nÎą
i
â t
= e ~
X
FnÎą ei(n+m)âŚt H m |Îąi.
(2.118)
nmÎą
One can make a summation index substitution as
i
â t
e ~
X
i
â t
FnÎą ei(n+m)âŚt H m |Îąi ââ e ~
nmÎą
X
nmÎą
49
FmÎą einâŚt H (nâm) |Îąi
(2.119)
which leads to,
i
â t
e ~
X
i
â t
FnÎą einâŚt ( â n~âŚ)|Îąi = e ~
nÎą
X
FmÎą einâŚt H (nâm) |Îąi.
(2.120)
nmÎą
Note that the exponentials {einâŚt } are an orthogonal set on the time interval [0, T0 ], and
that the basis {|Îąi} was considered to be an orthonormal one so one can have the equality
X
FnÎ˛ ( â n~âŚ)|Î˛i =
Î˛
X
FmÎ˛ H (nâm) |Î˛i.
(2.121)
mÎ˛
By taking the scalar product from the left with hÎą| one will have
( â n~âŚ)FnÎą =
X
hÎą|H (nâm) |Î˛iFmÎ˛ .
(2.122)
mÎ˛
n as
We can define the matrix elements HÎąÎ˛
n = hÎą|H n |Î˛i,
HÎąÎ˛
(2.123)
and the matrix elements ÎnÎą,mÎ˛ become
nâm + n~âŚÎ´
ÎnÎą,mÎ˛ = HÎąÎ˛
nm Î´ÎąÎ˛ .
(2.124)
One can then rewrite this equation as the eigenvalue problem
X
ÎFmÎ˛ = FnÎą ,
(2.125)
mÎ˛
and the matrix form is
ÎF = F.
(2.126)
By using the property
ân = hÎ˛|H ân |Îąi
HÎ˛Îą
Ë T
0 inâŚt
1
e
hÎ˛|H(t)|Îąidt
=
T0 0
Ë T
0 inâŚt
1
â
=
e
hÎą|H â (t)|Î˛i dt
T
#â
" 0 0Ë
T0
1
=
eâinâŚt hÎą|H(t)|Î˛idt
T0 0
n â
= HÎąÎ˛
(2.127)
50
it is easy to derive that
ÎnÎą,mÎ˛ = ÎâmÎ˛,nÎą
and therefore the matrix Î is Hermitian.
(2.128)
Because of the Hermiticity of Î, the eigen-
value problem ÎF = F has a solution. One also knows the eigenvalues are real and
the eigenvectors F form a complete orthonormal basis. Therefore the assumption |Ď, ti =
â i t P
inâŚt |Îąi solves the periodic time dependent problem with Floquet theory. In
e ~
nÎą FnÎą e
Chapter 3, Floquet theory is applied to the SSH model.
An example we show here is the effect of adding a time-dependent field into the Platero
Model mentioned in section 2.3.1. The time-dependent electric field is taken to be E(t) =
âât A(t) with A(t) = A0 sin(Ďt). At steady state the AC field can be interpreted as adding
an extra dimension to the model. The corresponding bulk Hamiltonian in momentum space
is therefore of the form
ďŁŽ
ďŁš
ĎF (k)ďŁş
ďŁŻ 0
ĎĚn,m = Ď ďŁ°
ďŁť,
ĎĚF (k)
0
(2.129)
ĎF (k) = Îťeâikb0 Jnâm (A0 b0 ) + eik(a0 âb0 ) Jmân (A0 (a0 â b0 )),
(2.130)
ĎĚF (k) = Îťeikb0 Jmân (A0 b0 ) + eâik(a0 âb0 ) Jnâm (A0 (a0 â b0 )),
(2.131)
with
and
where Îť = Ď 0 /Ď and (n, m) are integers. Note that, the spectrum depends on the intra-dimer
distance b0 , and the hopping terms are normalized by the field amplitude.
In the high frequency limit Ď > Ď, Ď 0 with the chosen band n = m = 0, the Hamiltonian
is block diagonal and can be described by a time independent 2 Ă 2 matrix
Hk0 = Ď~g (k) Âˇ ~Ď
51
(2.132)
with ~g (k) = (Re[ĎĚF ], Im[ĎĚF ], 0) for n = m = 0, and ~Ď = (Ďx , Ďy , Ďz ) are the Pauli matrices.
Since Hk0 here is similar to the SSH model and has chiral symmetry, one can calculate
the corresponding Chern number (winding number) directly
Ë
huÎą,k |iâk |uÎą,k idk
Î˝1 =
=
Ď
(1 + sign(J02 (y) â Îť2 J02 (x))),
2
(2.133)
where y = A0 (a0 â b0 ) and x = A0 b0 , and |uÎą,ki are the eigenfuctions of Hk0 . This result
indicates, in contrast with the undriven model, that one can induce non-trivial topological
phases even for Îť > 1 (the trivial phase). This is a great example to demonstrate topological
phases created by the a laser drive.
2.6
Calculation of Time Resolved Photoemission
Photoemission spectroscopy (PES) refers to energy measurement of electrons emitted from
materials, such as solids, by the photoelectric effect, in order to find the binding energies of
electrons in a substance. It is an important time-resolved method, now in ultrafast region
to study the effects of a pump laser in materials properties at the femtosecond time-scale.
It is straightforward to understand the pump-prob process in time resolved pump-probe
photoemission spectroscopy (TRPES). First the laser pulse pumps the surface of a material
to induce an initial nonequilibrium response of the system and, second, the probe pulse
would be used to study the excited system to generate the emitted photoelectrons. Here we
follow the derivation of the time-resolved photoemission signal from Freericks et al [92, 56].
Time resolved angle resolved photoemission spectroscopy (trARPES) experiments measure
the photoelectrons with a momentum ke collected at a solid angle dâŚkĚ within an energy
e
interval dE. The system evolves from an initial condition {|Ďn i} in equilibrium, then the
pump pulse will excite the system to an ensemble {|Î¨n i}. After that, the probe pulse is
turned on around time t = t0 . One can say {|Î¨n i} = {U (t, ââ)|Ďn i} with U (t, ââ) is the
time evolution of the system including the pump. The Hamiltonian has to be modified when
52
the probe pulse is coming as the following
H = Hsolid (t) + Hf ree + Hc (t).
(2.134)
Note that the first term Hsolid (t) only contains the creation câ (t) and annihilation c(t)
operators of electrons in the model without the effect of the probe pulse. Hf ree is for the
free electron Hamiltonian, and can be written as
Hf ree =
X
â
[E(ke ) + W ]ak ake .
e
(2.135)
k
â
Note that W is the work function, E(ke ) is the free electron kinetic energy and ak is the
e
creation operator and ake is the annihilation operator for a free electron with momentum
ke . The third term Hc shows the coupling between the electrons in the solid and the free
electrons out of the solid via an input photon with a wave number q. This term is of the
form
Hc = s(t â tp )
X
â
Mq (k, kq , q, t)ei~Ďq t ak (t)ck (t)Aq .
e
(2.136)
k
We use a matrix element Mq (k, ke , q, t) to describe the absorption of a photon with energy ~Ďq and ejection of an electron with momentum k on the surface of the material and
momentum ke outside, and Aq is the annihilation operator of the photon. Since we have
the time-varying pump laser pulse, the matrix element Mq (k, ke , q, t) is also time-dependent.
The interaction between the free electron that leaves the solid and the electrons in the solid is
assumed to be weak and can be ignored. The time evolution of the system is derived in terms
p
of the time-evolution operator UĚ (t, t0 ) and {|Î¨n i} = {UĚ (t, t0 )Î¨n (t0 )}. In the interaction
picture, we can write the time-evolution operator with a probe pulse as the form
UĚ (t, t0 ) =
Â´
âi/~ tt dt0 U (t,t0 )Ë(H)c U (t0 ,t0 )
0
U (t, t0 )Tt e
.
(2.137)
Note that Hf ree commutes with the propagator U (t, t0 ) so we do not need to include it.
Therefore, one can calculate the probability P (t) that an electron transfers from the ensemble
53
p
{|Î¨n i} to a free electron state for each time step t when the probe pulse is turned on. From
Fermiâs golden rule, the probability is
P (ke , t) =
X
p
|hke |HĚc |Î¨n (t)i|2 .
(2.138)
n
|ke i represents the free electron state with momentum ke . The system absorbs a photon of
wave vector q and ejects an electron with a wave vector k = (kxy , kz ) inside the system and
ke = (kexy , kez ) outside. In the process, since the momentum is conserved parallel to the
surface, one can define kexy = kxy . With this conservation rule, one can derive the matrix
element of the perturbation from HĚc between the final state with momentum ke and the
initial state |Î¨n (t0 )i
1
|hke |HĚc |Î¨n (t)i| = |
h
Ë
Ë t
dkz Mq (k, ke , q, t) Ă
0
dt0 s(t0 )eâiĎt Ă
t0
â
â
0
hke |a |ÎŚm ihÎŚm |UĚ (t , t0 )ck UĚ (t0 , t0 )|Î¨n (t0 )i|.
(2.139)
Here we made the assumption that once the electron is pumped to the excited state; inside
the material, we have hke |aâ |Î¨m i = 1 |Î¨m i with momentum k and a free electron with
momentum ke will be generated through the matrix Mq . Here we define Ď to be the energy
of the excitation left in the system and it satisfies ~Ď = ~Ďq â (~ke )2 /(2me ) â W . We can
expand the time-evolution matrix UĚ as
Ë
i t 0
UĚ (t, t0 ) = U (t, t0 ) â
dt U (t, t0 )HĚc U (t0 , t0 )
~ t0
(2.140)
Since in the experiments, the amplitude of the probe pulse is small and the pump pulse
is stronger than the probe pulse, one can apply the zeroth order perturbation in the time
evolution operator UĚ (t, t0 ) = U (t, t0 ). Then one gets the probability
Ë
Ë
1
0
P (t, ke ) â 2 dkz dkz I(t, w, eĚk ; kz , kz0 )
~
(2.141)
with
I(t, w, eĚk ; kz , kz0 )
Ë t
= âi
t0
dt00
Ë t
t0
dt0 Mq (kz , kez , kxy , t0 )Mq (kz0 , kez , kxy , t00 ) Ă
00 0
s(t00 )s(t0 )eiĎ(t ât ) G<
(t0 , t00 ).
k,k 0
54
(2.142)
Note that G< 0 (t0 , t00 ) represents the two time lesser Greenâs function in momentum space
k,k
with time-dependent creation and annihilation operators at k 0 and k. To be able to compare
the theoretical results with experimental data, one needs further assumptions. One of the
important assumptions is that the matrix element Mq will not change rapidly so we can
take it as time-independent and it conserves the momentum parallel to the surface of the
system. For these reasons, we take Mq to be constant in the calculations. With this assumption, we can directly calculate the time-resolved photoemission at site i and sum over
all k. The above function of G< 0 (t0 , t00 ) works both for the momentum-diagonal Greenâs
0 00
function G<
k (t , t ) =
k,k
0 00
G<
k,k (t , t ) and
for the local lesser Greenâs function since the local
Greenâs function is calculated by summing over different momentum points as
Ë t
Pii (Ď, t) = âi
0
dt0
Ë t
0
0 00
0 00
dt00 s(t0 )s(t00 )eiĎ(t ât ) G<
ii (t , t ).
(2.143)
Note that, in the following chapters, we take s(t) as a Gaussian shape
1
s(t) = â exp[â(t â tc )/Ď 2 ]
Ď Ď
Where tc is the central time of the probe pulse.
55
(2.144)
CHAPTER 3
THE ZIG-ZAG SSH MODEL: FROM FLOQUET TO TRPES
3.1
Introduction
In this chapter a detailed study of light driven PIPT in the Su, Schrieffer, and Heeger (SSH)
model is presented [93]. This model was introduced in subsection 2.3.1. It was originally
developed to describe transport in conducting polymers and more recently it has become a
fundamental model for topological insulators and Majorana bound states [5, 81]. While the
topologically trivial or nontrivial character of the SSH dimerized chain is controlled by the
relative strength of the nearest neighbor couplings, it was only recently proposed that highfrequency laser light could turn trivial (non-topological) equilibrium bands into topological
nonequilibrium Floquet bands [7].
We investigate these problems by simulating the real-time evolution of single-particle
spectra in polyacetylene coupled to short laser pulses. We introduce a modified SSH model
which allows us to tune topological states with polarized light. We compare the responses
of initially trivial and nontrivial topological states, and we also calculate the steady state
Floquet bands that are also used for comparisons.
3.2
Model and Methods
To understand the nonequilibrium Floquet bands of the SSH model mentioned in Chapter
2, we consider a one dimensional tight-binding model that can be divided into two sites A
and B with time-dependent hoping term Ďij only between the two closest sites as described
in section 2.3.1. The time independent Hamiltonian is
H=
X
â
â
(Ď + Î´Ď )cAi cBi + (Ď â Î´Ď )cAi+1 cBi + h.c.
i
In the following, a time-dependent effect will be included into the model.
56
(3.1)
Figure 3.1: The Zig-Zag SSH model: (a) the trivial topological state Î´Ď > 0 and (b) the
nontrivial topological state Î´Ď < 0. Note that the polarization is linear so the electric field
is along one bond but perpendicular to the other.
3.2.1
Time-dependent Zig-Zag SSH Model
We consider equal numbers of A and B sites with one electron per site, so that the electrons
are at half filling. In addition we include a polarized time-dependent pump pulse incorporated
into our model through the Peierls substitution, resulting in a time-dependent modulation
of the hopping term. Note that, since a space uniform laser field is not able to change
the original topological states [81], we consider a laser field parallel to (Ď â Î´Ď ) bond and
perpendicular to (Ď + Î´Ď ) bond or vise versa and we show that this leads to topological phase
transitions. This can be calculated using a Zig-Zag chain with light polarized along one of
the band directions as Fig. (3.1) shows. We call this model the Zig-Zag SSH model.
The Hamiltonian, using standard notation for creation and annihilation operators, is
H(t) =
X
â
â
(Ď + Î´Ď )cAi cBi + (Ď â Î´Ď )(t)cAi+1 cBi + h.c.
(3.2)
i
where the pump laser pulse is treated using the Peierls substitution as (see Section 2.4.1)
#
" Ë
ie Rj
(Ď â Î´Ď )(t) = (Ď â Î´Ď ) exp
A(r, t) Âˇ dr .
(3.3)
~c Ri
57
Here A(r, t) is the time dependent vector potential in the Hamiltonian gauge, Ri is the
position vector of the ith lattice site, and r is a real space vector. The Î´Ď is the dimerization
term and Ď is taken as the unit of energy in this chapter (Ď = 1).
In the presence of a pump field, through the vector potential A(r, t), the Floquet bands
form, and then disappear when the pump pulse amplitude goes to zero. In equilibrium, the
energy bands have a gap of width equal to 4Î´Ď that is symmetric about zero energy as shown
in Fig. (2.6). For the time dependent case, the eigenfunctions change as a function of time
and the quasi energy bands are modified by the presence of the external field. As described in
~ t) Âˇ ~Ď
Subsection 2.3.1, the Hamiltonian in momentum space can be written as H k (t) = d(k,
with H k (t) is 2 Ă 2 matrix written in the basis of A and B sites, ~Ď is the vector of Pauli
matrices (Ďx , Ďy , Ďz ), and d~k (t) is a vector as
dx (k, t) = (Ď + Î´Ď ) + (Ď â Î´Ď ) cos(ka â A(t)),
dy (k, t) = (Ď â Î´Ď ) sin(ka â A(t)),
dz (k, t) = 0,
(3.4)
Where a is unit cell length and, for simplicity, e = ~ = c = 1. Eigenenergies + and â
exist as a pair, and this symmetry ensures the existence of two distinct topological phases
(Î´Ď > 0 and Î´Ď < 0). We are able to calculate the time evolution of different momentum
k states, from which we derive the nonequilibrium Greenâs functions to obtain the physical
properties of the system at time t (see Chapter 2). In the following calculation, we use
ck (t) = (cAk (t), cBk (t)) and the equation of motion can be generally written as
i~ât ck (t) = H k (t)ck (t).
(3.5)
Since we have evolution equations, we can write down the time-evolution operator
Uk
(t, t0 )
Ë
i t
dtĚH k (tĚ) ,
= exp â
~ t0
58
(3.6)
and, by using the Trotter formula, the time evolution in numerical calculations becomes
Uk (t, t0 ) = Uk (t, t â ât)Uk (t, t â 2ât)...Uk (t0 + ât, t0 ),
(3.7)
where ât is a small time step. For each moment time t, we have
iât
H k (t â ât/2) .
Uk (t, t â ât) = exp â
~
3.2.2
(3.8)
Floquet Hamiltonian for Zig-Zag SSH Model
An effective Floquet Hamiltonian (see Section 2.5) can be used to derive Floquet spectra
where for the Zig-Zag SSH model we find,
HF = â
X
mâŚ|m, Îąihm, Îą|
mÎą
+
X
[gmân (k)|m, Aihn, B| + h.c.],
(3.9)
mn
where gmân (k) are the Fourier series expansion coefficients of the time-dependent hopping
term g(k, t) = dx (k, t) â idy (k, t) from Eq. (3.4) that can be written as (see Section 2.5)
Ë
1 T
gmân (k) =
dtei(mân)âŚt g(k, t)
T 0
Ď
= (Ď + Î´Ď )Î´mn + (Ď â Î´Ď )ei[kaâ 2 (mân)] Jmân (A0 )
(3.10)
with a periodic vector potential A(t) = A0 cos(âŚt) and T = 2Ď/âŚ. Note that Jmân (A0 )
is the Bessel function of the first kind and we achieve convergence of the corresponding
spectrum numerically when |m| âĽ 10 for all the cases discussed here. As an example, the
59
6 Ă 6 Floquet Hamiltonian matrix is
ďŁŽ
ďŁš
g0
0
g1
0
g2 ďŁş
ďŁŻâŚ â Îť
ďŁş
ďŁŻ
ďŁŻ gâ
0 ďŁş
0
g2â
âŚ â Îť g1â
ďŁş
ďŁŻ 0
ďŁŻ
ďŁş
ďŁş
ďŁŻ
ďŁŻ 0
g1
0âÎť
g0
0
g1 ďŁş
ďŁş,
ďŁŻ
M âÎť=ďŁŻ
ďŁş
â
â
â
ďŁş
ďŁŻ g
0
g
0
â
Îť
g
0
1
0
1
ďŁş
ďŁŻ
ďŁş
ďŁŻ
ďŁş
ďŁŻ 0
g
0
g
ââŚ
â
Îť
g
2
1
0 ďŁş
ďŁŻ
ďŁť
ďŁ°
g2â
0
g1â
0
g0â
ââŚ â Îť
(3.11)
where g0 = R + reika J0 (A0 ), g1 = âireika J1 (A0 ) and g2 = âreika J2 (A0 ). Note that
R = Ď + Î´Ď , r = Ď â Î´Ď and J0 , J1 are Bessel functions. The Bessel functions have the
integral form,
Ë Ď
1
Jn (x) =
ei(nĎ âx sin(Ď )) dĎ.
2Ď âĎ
3.2.3
(3.12)
Calculations from Non-equilibrium Greenâs Functions
By following the theory presented in Chapter 2 and Ref. [56, 55], the lesser Greenâs function
is defined as
â
0
0
G<
ij (t, t ) = ihcj (t )ci (t)i,
(3.13)
where we take a quantum statistical average of the time dependent creation and annihilation
â
operators. In the Heisenberg representation, ci (t) and ci (t) are creation and annihilation
operators for a spinless fermion at site i. The angle brackets represent a trace over all
quantum states in real or momentum space weighted by the equilibrium density matrix
~ t)|/[Îł(Ď 2 âÎ´Ď 2 )]
initialized in the far past; and the density of states is 2(âE(k)/âk)â1 = 2|d(k,
where Îł is the normalization term to make sure that the integral of density of states is one.
We can employ the Greenâs function method to calculate the time-resolved photoemission
spectroscopy signal (see Section 2.6), as a probe pulse weighted time Fourier transform of
60
the lesser Greenâs function centered at time tp ,
Ë
PI (k, Ď, tp ) = Im
Ë
dt
0
0
dt0 s(t)s(t0 )eâiĎ(tât ) G<
I (k, t + tp , t + tp ).
(3.14)
with lesser Green function
1 <
<
G<
I (k, t1 , t2 ) âĄ 2 [GAA (k, t1 , t2 ) + GBB (k, t1 , t2 )].
(3.15)
The response should be calculated for each sublattice and is averaged over both sites A and
B to compare with the experimental response. No extrapolation to large times is needed for
this calculation since the probe pulse provides a natural cutoff. The probe pulse is assumed
to be,
s(t) =
2 2
1
â eât /Îą
Îą Ď
(3.16)
with width Îą. The narrower the probe width, the better the time resolution and the worse
the energy resolution.
3.3
3.3.1
Analysis of the SSH Floquet Hamiltonian
Limit of High Frequency
In high frequency regime E0 /âŚ â 0 with E0 = A0 âŚ, we only need to consider the g0 (m = n)
term and all other terms can be set to zero. For any Floquet sideband centered at mâŚ if
we assume all gi6=0 (k) have a smaller effect than g0 (k), the energy band is approximately
Âą (m, k) ' mâŚ Âą g02 (k) where
g02 (k) = R2 + r2 J02 (A0 ) + 2RrJ0 (A0 ) cos(ka)
(3.17)
with R = (Ď + Î´Ď ) and r = (Ď â Î´Ď ). We find the gap â0 (k = Ď) = 2g02 (k = Ď) is
dependent on the electric field amplitude of the drive laser. An interesting feature is that A0
can be chosen so that flat sidebands occur, when J0 (A0 ) = 0, and reverse sidebands when
61
J0 (A0 ) < 0. We also note that gap opening depends on the amplitude of the light and also
on whether the system is in the trivial or non-trivial phase; i.e. Î´Ď > 0 or Î´Ď < 0. When
Î´Ď > 0 we have R > rJ0 (A0 ) so the behavior of the gap is dominated by R. On the other
hand, when rJ0 (A0 ) > R the effect of the light through rJ0 (A0 ) would strongly influence
the behavior of the gap opening. We show below that the different topological phases can
lead to completely different spectral responses when higher order terms gi6=0 (k) are included.
Since we know HF from above, we can calculate the Berry phase or winding number from
the corresponding eigenvectors uÎą,k with Îą the band index,
Ë
Î˝ =
huÎą,k |iâk |uÎą,k i.
(3.18)
Ignoring terms gi6=0 , it is possible to calculate the exact winding number as [81]
Î˝0 =
3.3.2
Ď
{1 + sgn[J02 (A0 ) â (R/r)2 ]}
2
(3.19)
Simple Model for the Case of One Overlap
To understand physics of the light driven SSH model for one overlap, deriving an exact
solution is a long standing challenge. Some high frequency limit results have been calculated
[81] recently but not an exact solution. While to solve a large Hamiltonian from Floquet
theory is not a reasonable approach, we found a small size Floquet Hamiltonian which only
considers two interbands is quite enough to describe the dispersion relation of light driven
SSH model for one overlap between bands. We would start with a 4 Ă 4 Hamiltonian for two
interband interactions as
ďŁŽ
ďŁš
g0
0 g1 ďŁş
ďŁŻâŚ â Îť
ďŁŻ
ďŁş
â
ďŁŻ gâ
ďŁş
âŚ
â
Îť
g
0
ďŁş
ďŁŻ 0
1
H = ďŁŻ
ďŁş,
ďŁŻ 0
g1
âÎť g0 ďŁş
ďŁş
ďŁŻ
ďŁ°
ďŁť
â
â
g1
0
g0 âÎť
62
(3.20)
For simplicity, one can first make the diagonals symmetric,
ďŁš
ďŁŽ
âŚ âÎť
g
0
g
0
1
ďŁş
ďŁŻ2
ďŁş
ďŁŻ
âŚ
â
â
ďŁŻ g
0 ďŁş
g1
ďŁş
ďŁŻ 0
2 âÎť
H = ďŁŻ
ďŁş.
ďŁş
ďŁŻ 0
âŚ âÎť
g
â
g
ďŁş
ďŁŻ
1
0
2
ďŁť
ďŁ°
ââŚ
0
g0â
g1â
â
Îť
2
(3.21)
The corresponding eigenvalues can then be derived directly as
Îť = Âą(Î˛ Âą Îą1/2 )1/2
(3.22)
âŚ
Î˛ = |g0 |2 + |g1 |2 + | |2
2
(3.23)
Îą = (âŚ|g0 |)2 + (g0 g1â + g0â g1 )2 .
(3.24)
with
and
Note that,
|g0 |2 = R2 + (rJ0 (E))2 + 2RrJ0 (E) cos(ka),
(3.25)
|g1 |2 = (rJ1 (E))2 ,
(3.26)
(g0 g1â + g0â g1 )2 = â4(rRJ1 (E))2 (cos(ka))2 .
(3.27)
|g0 |2 is a term that varies with cos(ka) and |g1 |2 is a constant with respect to ka. Interestingly, (g0 g1â + g0â g1 )2 changes with (cos(ka))2 and induces a second order modulation of the
band (a local minimum/maximum that will curve the band).
3.3.3
Computational Results and Comparison to Analytic Results
One can directly compute eigenenergies and eigenvectors by solving the Floquet Hamiltonian.
The Floquet Hamiltonian used in the calculations has size 2(2m + 1). m indicates a Floquet
frequency nâŚ in the range of [âmâŚ, mâŚ] and m, n â Z. We choose m = 10 to ensure that
63
7
6
6
5
5
4
4
Energy / Ď
Energy / Ď
7
3
2
1
3
2
1
0
0
â1
â1
â2
-1.0
-0.5
0.0
k(a/Ď)
0.5
â2
-1.0
1.0
-0.5
0.0
k(a/Ď)
0.5
1.0
Figure 3.2: Dispersion relation for the laser driven SSH model with laser amplitude A0 = 1
and frequency âŚ = 5. Dashed lines: the exact solution for the 4 Ă 4 Floquet Hamiltonian.
Solid lines: Numerical results for the 42 Ă 42 Floquet Hamiltonian with m = 10. Left is for
the trivial condition Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r).
the bands near m = 0 converge with respect to variations in m. In the following, comparison
between computational results for m = 10 and exact result from the 4 Ă 4 Hamiltonian will
be presented.
In Fig. (3.2), the dispersion relation for a no overlap case is presented; for laser amplitude
A0 = 1 and frequency âŚ = 5. The exact solution of the 4 Ă 4 Floquet Hamiltonian (dashed
lines) and results for the 42Ă42 Floquet Hamiltonian with m = 10 (solid lines) are compared.
One finds that the exact 4 Ă 4 solution is quite accurate in the high frequency regime. Only
for the nontrivial case the outer bands of the 4 Ă 4 solution are inaccurate indicating that
interband affects are important in this case.
Fig. (3.3) shows the scheme of band interactions for g0 , g1 and g2 , where n is the Floquet
band index. g0 is the intraband process; while g1 and g2 are interband process. g1 is related
to the one photon interaction and g2 is related to the two photons interaction.
In Fig. (3.4), a one overlap case is presented with laser amplitude A0 = 1 and frequency
âŚ = 3. The exact solution for the 4 Ă 4 Floquet Hamiltonian (dashed lines) and results
for the 42 Ă 42 Floquet Hamiltonian with m = 10 (solid lines) are compared for the trivial
condition Î´Ď = 0.5 (R > r) and the nontrivial condition Î´Ď = â0.5 (R < r). The inner
two bands for the exact 4 Ă 4 solution are very close to the computational results for the
64
Figure 3.3: Scheme of band interactions for g0 , g1 and g2 . n is the Floquet band index. g0
is the intraband process; while g1 and g2 are interband process.
one overlap situation. For the trivial case, however in the left figure in Fig. (3.4) the gap
at crossing points is not captured by the 4 Ă 4 model. By making gnâĽ2 = 0 in the m = 10
calculation we found that the gap closing effect is caused by higher order mixing through
the terms g0 and g1 terms which cannot be derived from the 4 Ă 4 Hamiltonian as Fig. (3.3)
shows. For the nontrivial case the outer bands of the 4 Ă 4 solution have some shifting
indicating effects due to bands above and below.
In Fig. (3.5), a two overlap case is presented with laser amplitude A0 = 1 and frequency
âŚ = 1.5. Neither the trivial nor nontrivial cases are fully described by the 4 Ă 4 Hamiltonian
in this case though the nontrivial case deviates the most, the need for higher order terms to
describe the gap of the nontrivial case is again evident.
65
5
4
4
3
3
Energy / Ď
Energy / Ď
5
2
1
0
â1
â2
-1.0
2
1
0
â1
-0.5
0.0
k(a/Ď)
0.5
â2
-1.0
1.0
-0.5
0.0
k(a/Ď)
0.5
1.0
4
4
3
3
2
2
Energy / Ď
Energy / Ď
Figure 3.4: Dispersion relation for the laser driven SSH model with laser amplitude A0 = 1
and frequency âŚ = 3. Dashed lines: the exact solution for the 4 Ă 4 Floquet Hamiltonian.
Solid lines: Numerical results for the 42 Ă 42 Floquet Hamiltonian with m = 10. Left is for
the trivial condition Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r).
1
0
â1
â2
-1.0
1
0
â1
-0.5
0.0
k(a/Ď)
0.5
â2
-1.0
1.0
-0.5
0.0
k(a/Ď)
0.5
1.0
Figure 3.5: Dispersion relation for the laser driven SSH model with laser amplitude A0 = 1
and frequency âŚ = 1.5. Dashed lines: the exact solution for the 4 Ă 4 Floquet Hamiltonian.
Solid lines: Numerical results for the 42 Ă 42 Floquet Hamiltonian with m = 10. Left is for
the trivial condition Î´Ď = 0.5 (R > r) and Right is for the nontrivial condition Î´Ď = â0.5
(R < r).
3.3.4
Effects of Varying Amplitude and Frequency
In Fig. (3.6), we show the dispersion relations of the quasi-static Floquet Hamiltonian with
m = 10 and drive frequency âŚ = 2.8. Left: trivial condition Î´Ď = 0.5 and Right: the
nontrivial condition Î´Ď = â0.5. (a)(b) is for amplitude A0 = 2, (c)(d) is for amplitude
A0 = 4 and (e)(f) is for amplitude A0 = 6. Clearly a variety of different dispersion relations
can be achieved through use of polarized light and a well designed lattice, Gap opening and
closing can also be changed by different frequencies, amplitudes, and polarization of light,
66
as well as lattice structure.
Energy / Ď
Energy / Ď
3
2
1
0
â1
â2
â3
Energy / Ď
Î´Ď = 0. 5
3
2
1
0
â1
â2
â3
3
2 (e)
1
0
â1
â2
â3
-1.0 -0.5
Î´Ď =
(a)
(b)
(c)
(d)
â 0. 5
(f)
0.0
k(a/Ď)
0.5
1.0 -1.0 -0.5
0.0
k(a/Ď)
0.5
1.0
Figure 3.6: Dispersion relations of quasi-static Floquet Hamiltonian with drive frequency
âŚ = 2.8. Left: trivial condition Î´Ď = 0.5 and Right: the nontrivial condition Î´Ď = â0.5.
(a)(b) is for amplitude A0 = 2, (c)(d) is for amplitude A0 = 4 and (e)(f) is for amplitude
A0 = 6.
Fig. (3.7) shows the quasienergy properties calculated from the quasi-static Floquet matrix of the original bands at k = Ď vs laser amplitude A0 for Î´Ď = Âą0.5. The left figure is for
the trivial condition Î´Ď = 0.5 and the right figure is the nontrivial condition Î´Ď = â0.5 for
different drive frequency âŚ. At high frequency, they oscillate as the Bessel function of the
first kind J0 (A0 ) but the bands have different trends in the trivial and nontrivial cases. For
the trivial case, the energy bands remain open and fluctuate with the laser amplitude while
the gap increases with increasing frequency. For large frequency the bands become similar to
67
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
âŚ = 2. 0
âŚ = 2. 4
âŚ = 2. 8
âŚ = 3. 2
âŚ = 3. 6
âŚ = 4. 0
â0.5
â1.0
â1.5
â2.0
Energy / Ď
Energy / Ď
2.0
0
2
4
6
A0
8
10
12
0.0
âŚ = 2. 0
âŚ = 2. 4
âŚ = 2. 8
âŚ = 3. 2
âŚ = 3. 6
âŚ = 4. 0
â0.5
â1.0
â1.5
â2.0
14
0
2
4
6
A0
8
10
12
14
Figure 3.7: Quasienergy spectrum properties calculated from the Floquet matrix with m =
10 for the original bands at k = Ď vs laser amplitude A0 for Î´Ď = Âą0.5. Left: are for the
trivial condition Î´Ď = 0.5 and Right: are for the nontrivial condition Î´Ď = â0.5 for different
driving frequencies âŚ.
each other. For the nontrivial case, the gap closes at laser amplitude A0 â 1.8, and is always
smaller than the trivial case when frequency âŚ > 2.8. When amplitude increases all bands
become similar (A0 > 6.2). Note that the chosen frequencies are in the range 2.0 â¤ âŚ < 4.0
where only one band overlap occurs.
1.0
0.6
0.4
0.2
0.0
âŚ = 1. 0
âŚ = 2. 0
âŚ = 3. 0
âŚ = 4. 0
âŚ = 5. 0
0.8
Energy / Ď
0.8
Energy / Ď
1.0
âŚ = 1. 0
âŚ = 2. 0
âŚ = 3. 0
âŚ = 4. 0
âŚ = 5. 0
0.6
0.4
0.2
0
2
4
A0
6
8
10
0.0
0
2
4
A0
6
8
10
Figure 3.8: Bandwidth calculated from quasi-static Floquet band structures vs laser amplitude A0 for Î´Ď = Âą0.5. Left: are for the trivial condition Î´Ď = 0.5 and Right: are for the
nontrivial condition Î´Ď = â0.5 for different driving frequency âŚ.
In Fig. (3.8), we show the bandwidth calculated from quasi-static Floquet band structures
vs laser amplitude A0 for Î´Ď = Âą0.5. The left are for the trivial condition Î´Ď = 0.5 and
the right are for the nontrivial condition Î´Ď = â0.5 for different driving frequency âŚ. Note
that, the N overlap regime can be decided by ( N4Ď+1 , 4Ď
N ), e.g. one overlap regime is in
the frequency range (2Ď, 4Ď ). In the high frequency domain âŚ > 4 (no overlap) the laser
68
amplitude dominates; so the bandwidths oscillate with the Bessel function of the first kind
|J0 (A0 )|. However the nontrivial bandwidths are larger than the trivial bandwidths since
the variation is decided by [Ď â (âÎ´Ď )]J0 (A0 ) > [Ď â (Î´Ď )]J0 (A0 ) with Î´Ď = 0.5. In the
low frequency domain, the influencies from other second order coupling through Ji (A0 ) with
i = 1, 2, 3, ... also plays a role in bandwidth variations so the behaviors change. Especially
for the case âŚ = 3 we have monotonic decreases in bandwidth as amplitude increases. It is
then clear that both the amplitude and frequency of the drive laser can be used to control
the electronic structure.
3.3.5
Topological Phase Diagram
We found the topological phase diagram by evaluating Eq. (3.18) using the Floquet matrix
eigenvectors uÎą,k . Here we discuss the topological phases for the trivial case Î´Ď > 0 and the
nontrivial case Î´Ď < 0. In Fig. (3.9), the topological phase diagrams for different frequencies
and amplitudes are presented. Note that different topological regions can have different
winding numbers Î˝ (0, Ď, 2Ď, ...) but the same Berry phase (0 or Ď). It is clear that the
topological phase diagrams are totally different between the trivial and the nontrivial cases
(See also [81]).
The original phase for the case Î´Ď = 0.25 is trivial and changes to a nontrivial phase at
low frequency when âŚ < 2.0. On the other hand, The original phase for the case Î´Ď = â0.25
is nontrivial and changes to a trivial phase in the high frequency domain when âŚ > 2.0. This
is also true for large values of Î´Ď as shown in Fig. (3.9) (c)-(f). (c)(d) shows the cases for
Î´Ď = 0.5 on the left and Î´Ď = â0.5 on the right. The high frequency results from Eq. (3.19)
describes this regime well. The four red points indicate the frequncies and amplitudes we
choose to discuss in Fig. (3.16) of the next subsection.
69
âŚ
âŚ
âŚ
(a)
Î´Ď = 0. 25
(b)
Î´Ď = â 0. 25
(c)
Î´Ď = 0. 50
(d)
Î´Ď = â 0. 50
(e)
Î´Ď = 0. 75
(f)
Î´Ď = â 0. 75
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
A0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
A0
Figure 3.9: Topological phase diagram for different frequencies and amplitudes of the drive
pulses from effective Floquet Hamiltonian. (a)(b) shows the cases for Î´Ď = 0.25 on the left
and Î´Ď = â0.25 on the right. (c)(d) shows the cases for Î´Ď = 0.5 on the left and Î´Ď = â0.5
on the right. (e)(f) shows the cases for Î´Ď = 0.75 on the left and Î´Ď = â0.75 on the right.
The four red points indicate the frequencies and amplitudes we discuss in Fig. (3.16) of the
next subsection.
3.4
Pump-probe Results
In this section we describe fully time dependent pump-probe calculations using nonequilibrium Greenâs functions (NEGF). The vector potential of the pump pulse is of the form
A(t) = A0 cos(âŚt) exp(ât2 /2Ď 2 ) where A0 is amplitude, âŚ is frequency and Ď is pulse width.
70
When Ď â â, this should lead to behavior like that observed in Floquet Theory. However,
there is a key difference. The NEGF calculation gives the occupancy of the nonequilibrium
states; which is not the same as the states themselves.
3.4.1
Pseudospin Content
We would like to use NEGE to make contact with the pseudospin representation of 2 Ă 2
Hamiltonians that reflect their orbital content and fully determine the Berry phase in and
out of equilibrium and in particular for low driving frequencies. When a description in terms
of a simple effective Floquet Hamiltonian is not available, one has to find an analogue of
a pseudospin analysis in terms of nonequilibrium Green functions. The pseudospin content
of Green functions from our numerical simulations is extracted by expanding the Green
function matrices in orbital representation using Pauli matrices [83],
1 <
[GAB (k, t1 , t2 ) + G<
BA (k, t1 , t2 )],
2
i <
<
G<
y (k, t1 , t2 ) âĄ â [GAB (k, t1 , t2 ) â GBA (k, t1 , t2 )],
2
1 <
<
[GAA (k, t1 , t2 ) â G<
Gz (k, t1 , t2 ) âĄ
BB (k, t1 , t2 )].
2
G<
x (k, t1 , t2 ) âĄ
(3.28)
The respective pseudospin content Px,y,z (k, Ď, tp ) is obtained by computing the analogue
<
of the PES response for G<
x,y,z (k, t1 , t2 ) instead of GI (k, t1 , t2 ) using Eq. (3.14). In or-
der to obtain the Berry phase with pseudospin vector S(k, tp ) = (Sx , Sy , Sz ) shown in
Fig.(3.13), we integrate Gx,y,z in a frequency window to cover a single band as Sx,y,z (k, tp ) =
Â´ Ďu
Ď dĎPx,y,z (k, Ď, tp ). We note that the pseudospin behavior is dependent on the choice of
l
Floquet sideband for which it is analyzed. Floquet bands may have different pseudospin
content, and the pseudospins in the upper and lower bands within a manifold always point
in opposite directions. The proof of pseudospin content can be found in the following.
71
First, consider a time-independent steady-state 2 Ă 2 Hamiltonian as
ďŁŽ
ďŁš
0
dx â idy ďŁş
~ = d~ Âˇ ~Ď = ďŁŻ
H(d)
ďŁ°
ďŁť
dx + idy
0
(3.29)
~ = d = (d2 + d2 )1/2 and corresponding eigenenergies,
with |d|
x
y
~
H(d)|Âąi
= Âąd|Âąi.
(3.30)
and time-dependent eigenvectors,
ďŁŽ
ďŁš
eâidt
ďŁŻ 1 ďŁş
|Âą, ti = eâidt |Âąi = â ďŁ°
ďŁť,
2 ÂąeiĎ
(3.31)
with deÂąiĎ = dx Âą idy .
Now we show how to derive di from the lesser Greenâs functions. First put Ďj with
j = {x, y} and eâidt on both sides of Eq. (3.30), we will have
~
Ďj H(d)|Âą,
ti = Ďj (Âąd)|Âą, ti.
(3.32)
Using Eq. (3.29) this equation can be written as
(dj 1 + idl Ďz )|Âą, ti = Ďj (Âąd)|Âą, ti,
(3.33)
where l = {x, y} is not equal to j. We take the inner product of the above equation with
another eigenvectors at a different time t0 to find,
hÂą, t0 |(dj + idl Ďz )|Âą, ti = ÂądhÂą, t0 |Ďj |Âą, ti.
(3.34)
hÂą, t0 |Ďz |Âą, ti = 0,
(3.35)
Since we know
the above equation can be rewritten as
dj hÂą, t0 |Âą, ti = ÂądhÂą, t0 |Ďj |Âą, ti.
72
(3.36)
In this formula the eigenvectors | âi and | âi of Ďz can be considered as electron occupancy components in sublattice A and B as shown, in the main text Eq. (3.5) so we can
represent the right-hand side in Eq. (3.36) as the pseudospin content of the lesser Greenâs
0
functions G<
k,j (t, t ; Ď(0)) and the left-hand side is an identity for the lesser Greenâs function
0
G<
k,I (t, t ; Ď(0)) with a given initial state Ď(0) for momentum k. Without loss of generality
Eq. (3.36) can then be written as
0
G<
k,j (t, t ; Âą) = Âą
dj (k) <
Gk,I (t, t0 ; Âą)
d(k)
(3.37)
where j = x, y, z. Note that any state can be a linear combination of eigenvectors so we can
apply Eq. (3.37) to any given state. One can also show that for the direction of pseudospin
content of the lesser Greenâs function is opposite for the two different eigenvectors |Âąi.
To have an exact calculation of pseudospin content from PES response we can put
Eq. (3.37) into Eq. (3.28) and assume that the laser probe pulse is of rectangular form
in the range of [â Ď2 , Ď2 ] so we have
Ë Ď
Pj (k, Ď, tp ) ' ÂąIm
Ă
2
âĎ
2
Ë Ď
dt
2
âĎ
2
0
dt0 eâiĎ(tât )
dj (k) <
G (t + tp , t0 + tp ).
d(k) k,I
(3.38)
For the initial state |+i we then derive the pesudospin content from PES response as
Pj (k, Ď, tp ) '
dj (k) sin2 [(d(k) â Ď)Ď/2]
.
d(k)
(d(k) â Ď)2
(3.39)
Since the integral term is positive, the only term that would affect the sign of the pseudospin
content from PES is the sign of dj ; and the initial state |Âąi. The intensity of pseudospin
content is then proportional to the ratio dj (k)/d(k) and the density of states. Hence, when
the Hamiltonian is time-dependent, we can still calculate psedudospin content from PES
and the results can be directly connected with an effective time-independent Hamiltonian.
Below it is shown that the average outcome shows consistency with steady state calculations
from the effective Floquet Hamiltonian.
73
3.4.2
Evolution of States and Their Occupancy
In this subsection we present trPES results for the dynamic generation of Floquet states.
The initial state in these calculation is a filled lower band and an empty upper band.
Energy / Ď
3
t=
â 200. 0
(a)
t=
â 100. 0
(b)
t = 0. 0
t = 100. 0
(c)
(d)
t = 200. 0
(e)
2
1.000
1
0.100
0.010
0
0.001
-1
-2
0.000
-3
-1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
Figure 3.10: Time evolution of time-resolved PES of the electronic SSH model is presented
for time t = â200, â100, 0, 100, 200 with Î´Ď = 0 under the pulse frequency âŚ = 1, width
Ď = 50 and amplitude A0 = 1. Dashed lines show the corresponding quasi-static Floquet
band structures at the same frequency.
In Fig. (3.10), we study time evolution of time-resolved PES of the electronic SSH model
for times t = â200, â100, 0, 100, 200 with Î´Ď = 0 for a pulse frequency âŚ = 3, width Ď = 50
and amplitude A0 = 1. Dashed lines show the corresponding quasi-static Floquet band
structures. At time t = â200 the pulse starts with a small amplitude so the Floquet states
are not generated with significant occupancy. At time t = â100 the occupation of the nearest
Floquet bands begins to be observed. At time t = 0 occupation of many Floquet bands can
be seen and, interestingly, their is strong intensity of occupation at the intersection points
of different bands. At times t = 100 and t = 200, we discover that the occupation at the
intersection points between the Floquet bands and the original upper band remains while
the Floquet states disappear. The anomalous occupation of the upper band is momentum
selected and can be considered as a resonant excitation enhanced by Floquet effects.
In Fig. (3.11), we study time evolution of time-resolved PES of the electronic SSH model
for time t = â200, â100, 0, 100, 200 with Î´Ď = 0.5 for pulse frequency âŚ = 3, width Ď = 50
74
Energy / Ď
3
t=
â 200. 0
(a)
t=
â 100. 0
(b)
t = 0. 0
t = 100. 0
(c)
(d)
t = 200. 0
(e)
2
1.000
1
0.100
0.010
0
0.001
-1
-2
0.000
-3
-1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
Figure 3.11: Time evolution of time-resolved PES of the electronic SSH model is presented
for time t = â200, â100, 0, 100, 200 with Î´Ď = 0.5 under the pulse frequency âŚ = 3, width
Ď = 50 and amplitude A0 = 2. Dashed lines show the corresponding quasi-static Floquet
band structures.
and amplitude A0 = 2. Dashed lines show the corresponding quasi-static Floquet band
structures. As for the case Î´Ď = 0, the generation of states is consistent with Floqeut theory
with the additional feature of resonant excitation at band crossing points.
Energy / Ď
3
t=
â 200. 0
(a)
t=
â 100. 0
(b)
t = 0. 0
t = 100. 0
(c)
(d)
t = 200. 0
(e)
2
1.000
1
0.100
0.010
0
0.001
-1
-2
0.000
-3
-1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0 -1.0-0.5 0.0 0.5 1.0
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
Figure 3.12: Time evolution of time-resolved PES of the electronic SSH model is presented
for time t = â200, â100, 0, 100, 200 with Î´Ď = â0.5 under the pulse frequency âŚ = 3
and amplitude A0 = 2. Dashed lines show the corresponding quasi-static Floquet band
structures.
In Fig. (3.12), we study time evolution of time-resolved PES of the electronic SSH model
75
for Î´Ď = â0.5 with the pulse frequency âŚ = 3, width Ď = 50 and amplitude A0 = 2. Dashed
lines show the corresponding quasi-static Floquet band structures. The time variation for this
nontrivial phase is totally different to the two previous cases. One can also see second order
mixing between bands near k = 1 and k = â1 when t = 0 since the electron occupancy near
k = 1 and k = â1 moves up with the second order mixing. Interestingly, at time t = 200,
the occupancy for the second order mixing stays in the upper band.
There are two reasons for these differences: (i) The Floquet bands change with amplitude
A0 and the occupation should be with the pulse amplitude at that time. (ii) Excitation only
occurs between states of the same phase. As in the cases Î´Ď = 0; and Î´Ď = 0.5; resonant
excitation at intersection points occurs.
3.4.3
Dynamics of Phase Evolution
Fig. (3.13) shows a Berry phase mapping using the renormalized radius r0 = 1 + (1 + k)/2
for Î´Ď = 0.5 on the left and Î´Ď = â0.5 on the right with pulse frequency âŚ = 3, amplitude
A0 = 2 and pulse width Ď = 50. Note that the renormalized radius r0 is for illustration
purposes in order to separate the paths that go around the origin more than once. (a)(b)
shows the trivial and nontrivial Berry phases before the laser pulse comes in at time t = â200
with integral frequency window Ďl = 0.0 and Ďu = 3.0. (c)(d) is at time t = 0 with integral
frequency window Ďl = â1.5 and Ďu = 0.0. For the trivial case, since the bands still mix,
the Berry connection moves up and down twice about the original line Sy = 0; leading to a
Berry phase of zero. (e) shows the trivial band with integral frequency window Ďl = 1.5 and
Ďu = 3.0 at time t = 0; in this case there is no phase flip. Note that it has reverse direction
from (c) since the band has the opposite phase Ď(k 0 ) + Ď. Some bands can have the same
Berry phase but different winding numbers Î˝, as we seen by comparing (b) and (f). In the
latter case the Berry connection goes around the origin twice so the winding number Î˝ = 4Ď.
To further illustrate the novel topological behaviours due to a laser pulse, in Fig. (3.14)
76
2
Î´Ď =0.5
(a)
(b)
Î´Ď = â0.5
Sy
1
0
â1
â2
2
(c)
(d)
(e)
(f)
Sy
1
0
â1
â2
2
Sy
1
0
â1
â2
â2
â1
0
Sx
1
2
â2
â1
0
Sx
1
2
Figure 3.13: Berry phase mapping using the radius r0 = 1 + (1 + k)/2 with Î´Ď = 0.5 on the
left and Î´Ď = â0.5 on the right under the pulse frequency âŚ = 3, amplitude A0 = 2 and
pulse width Ď = 50. (a)(b) is at time t = â200 with integral frequency window Ďl = 0.0 and
Ďu = 3.0, (c)(d) is at time t = 0 with integral frequency window Ďl = â1.5 and Ďu = 0.0
and (e)(f) is at time t = 0 with integral frequency window Ďl = 1.5 and Ďu = 3.0.
we show the time evolution of pseudospin phase Ď deriving from the pseudospin context
P eiĎ = Px + iPy that are presented for times t = â200, â100, 0, 100, 200 with Î´Ď = 0.5 on
the top (a)-(e) and Î´Ď = â0.5 on the bottom (f)-(j) with pulse frequency âŚ = 3, amplitude
A0 = 1 and pulse width Ď = 50. We can see in figures (a)(f) the upper band points in reverse
direction of P, which means the phase of the upper band is Ď(k 0 ) + Ď if lower band is at
phase Ď(k 0 ) with momentum k 0 . In (b), however, the occupied Floquet states cross with the
original band leading to a change of phase. In contrast, Floquet bands in (g) have the same
77
t=
â 100. 0
t=
â 50. 0
t=
â 25. 0
t = 0. 0
Energy / Ď
â 200. 0
Energy / Ď
t=
3 (a)
2
1
0
-1
-2
-3
3 (f)
(g)
(h)
(i)
(j)
Î˝=Ď
2
Î˝ = 2Ď
1
0
-1
Î˝=0
-2
-3
-1.0-0.50.0 0.5 1.0 -1.0-0.50.0 0.5 1.0 -1.0-0.50.0 0.5 1.0 -1.0-0.50.0 0.5 1.0 -1.0-0.50.0 0.5 1.0
Î˝=0
(b)
(c)
(d)
(e)
Î˝=0
3.0
Î˝=0
1.5
k(a/Ď)
k(a/Ď)
k(a/Ď)
k(a/Ď)
0.0
â1.5
â3.0
k(a/Ď)
Figure 3.14: Time evolution of pseudospin phase Ď deriving from pseudospin context P eiĎ =
Px + iPy are presented for time t = â200, â100, â50, â25, 0 with Î´Ď = 0.5 on the top (a)-(e)
and Î´Ď = â0.5 on the bottom (f)-(j) under the pulse frequency âŚ = 3, amplitude A0 = 2
and pulse width Ď = 50. Dashed lines show the corresponding quasi-static Floquet band
structures.
phase Ď(k 0 )+Ď as the phase of the upper band. We find the phase of these Floquet sidebands
are decided by the topological phase of the original band where Î´Ď > 0 (trivial case) and
Î´Ď < 0 (non-trivial case). From (b) we can find the phases are different at the crosspoints
so there is no gap opening at time t = 0, while we will have degeneracy at intersection and
would induce a state mixing or pauli exclusion between two chiral fermions so to open a gap
at time t = 0 as shows in (h). As time t increases the gap in the nontrivial case Î´Ď < 0
becomes larger and causes the bands to move apart. When the bands touch other bands
they mix again with a reverse phase band and form a new band with different Berry phase
as one can find in (i)(j). On the other hand, the trivial case will not open a gap so we can
only see the flattening effects from the pulse amplitude.
3.4.4
Effects of Varying Pump and Probe Pulsewidth and Amplitude
Fig. (3.15) shows several time-resolved PES results with Î´Ď = 0 for a pulse with frequency
âŚ = 3, amplitude A0 = 1, and for a range of different pump pulse widths; for full occupancy
of the lower band as the initial state. When the pulse width is small (a) Ď = 1 and (b)
78
(b)
Ď =2.0
1.000
Ď =5.0
Ď =10.0
Energy / Ď
Energy / Ď
Ď =1.0
3 (c)
2
1
0
-1
-2
-3
Energy / Ď
3 (a)
2
1
0
-1
-2
-3
3 (e)
(f)
2
1
0
-1
-2
-3
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
(d)
0.100
0.010
0.001
Ď =20.0
Ď =40.0
k(a/Ď)
0.000
k(a/Ď)
Figure 3.15: Time-resolved PES of the electronic SSH model with Î´Ď = 0 at time t = 0 with
a pulse of frequency âŚ = 3, amplitude E = 1 and width (a) Ď = 1, (b) Ď = 2, (c) Ď = 5, (d)
Ď = 10, (e) Ď = 20, (f) Ď = 40. Dashed lines show the corresponding quasi-static Floquet
band structures.
79
Ď = 2, we find that most of the electrons are only excited to the upper band and spread out
to the nearby energy states. As the pulse width becomes larger (c) Ď = 5 and (d) Ď = 10,
we can see a much clear generation of the Floquet bands with a weaker dispersion to the
nearby energy states. Note that for the case (c) the pulse width is about 10 times the period
of the probe. When the pulse width is larger (e) Ď = 20 and (f) Ď = 40, we find a clear
occupancy at the quasi-static Floquet band structures. Floquet physics is then accessible
for pulse widths of order 10 times the period of the probe.
Fig. (3.16) shows several TRPES results for full occupancy of the lower band with Î´Ď =
Âą0.5. (a)(b) gives the energy spectrum with laser pulse of frequency âŚ = 1.6, amplitude
A0 = 0.5 and width is Ď = 50. One can find similar responses both for Î´Ď = Âą0.5 but with
weaker excitation for Î´Ď = 0.5 of low frequency and small amplitude. (c)(d) shows Floquet
band formulation for low frequency but high amplitude region with laser frequency âŚ = 1.6,
amplitude A0 = 2.0 and width Ď = 50. We can see that the occupancy of the Floquet bands
increases with the intensity of the laser pulse, and the effect of a high amplitude pulse on
the Floquet spectra is clearly observed. (e)(f) shows a high frequency region, where the
different topological behaviours for Î´Ď = 0.5 and Î´Ď = â0.5 are evident. A gap opening only
occurs for the case Î´Ď = â0.5 at the intersection of the Floquet band with origin band. The
occupancy of the electrons in nontrivial case is also split at the crossing points. Note that
these different responses to the laser can also be found in (c)(d) but it is not as obvious.
3.5
Conclusion
We find that short optical pulses can lead to local spectral and novel pseudospin textures
in a one-dimensional topological insulator given by the Zig-Zag SSH model. Pump-probe
photoemission spectroscopy can probe these states by measuring sizeable energy gaps and
Floquet band formation on femtosecond time scales. Analysing band structures and pseudospin textures, we identify new states with optically induced nontrivial changes of sublattice
mixing that leads to novel topological phenomenon. This study reveals the possibility to dis-
80
cover new topological phases driven by optical pulses by turning the lattice structure and
polarization of the light.
Energy / Ď
(d)
Energy / Ď
3 (c)
2
1
0
-1
-2
-3
3 (e)
2
1
0
-1
-2
-3
(f)
Energy / Ď
Î´Ď = â0.5
Energy / Ď
Î´Ď =0.5
3 (a)
2
1
0
-1
-2
-3
3 (g)
(h)
2
1
0
-1
-2
-3
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
(b)
1.000
0.100
0.010
0.001
k(a/Ď)
0.000
k(a/Ď)
Figure 3.16: Time-resolved PES with Î´Ď = 0.5 on the left and Î´Ď = â0.5 on the right at
time t = 0. (a)(b) with pulse of frequency âŚ = 1.6, amplitude A0 = 0.5 and width is Ď = 50,
(c)(d) with pulse of frequency âŚ = 1.6, amplitude A0 = 2.0 and width is Ď = 50, (e)(f)
with pulse of frequency âŚ = 3, amplitude A0 = 0.5 and width Ď = 50, (g)(h) with pulse of
frequency âŚ = 3, amplitude A0 = 2 and width Ď = 50. Dashed lines show the corresponding
quasi-static Floquet band structures.
81
CHAPTER 4
KROTOV OPTIMAL CONTROL THEORY
4.1
Introduction to Optimal Control
In many physical areas, quantum control via specially tailored laser pulses is a long-standing
goal. In the most recent decade, this goal has been achieved, as sophisticated pulse shaping experiments can now coherently control some quantum states in quantum computing, quantum dots and phase transitions in materials. For example, laser pulses may be
applied to create and break a particular bond in a molecule, to control charge transfer
within molecules, or to optimize high harmonic generation and phase transitions in topology
[39, 38, 53, 54, 55, 56, 57, 58, 59, 35, 60, 32, 33, 51, 3, 35, 36, 61, 60].
Optimal control theory (OCT) can help to theoretically design the laser pulse to transfer
an initial state to a given final state. Especially Krotovâs optimal control theory or other
similar optimal control theories are widely used in recent years. This chapter provides an
introduction to Krotovâs optimal control theory. It indicates how the control parameters and
equation of motion define such an optimal control pulse [62].
Krotovâs method has several significant advantages over the traditional gradient methods.
First of all, monotonic increase in fidelity or decrease of cost/goal function with iteration
number. Secondly, no requirement for an exhaustive line search, and thirdly update in
macrosteps in time and at each iteration. Multiple versions of Krotovâs optimization method
have been used recently in many areas to deal with the Markovian and the non-Markovian
optimal control problem of a quantum Brownian motion model with an exact stochastic
equation of motion (master equation) in quantum computing. It has also been applied
to condensed matter physics in recent years enabled by progress in spectroscopy and laser
control [63, 64, 65, 66, 67, 68, 69, 70, 71, 62, 72, 73].
In this Chapter, We first explain the foundation and schemes to solve the control equation of motion and exhibit how to include additional constraints into the goal/cost functions.
82
Second, We introduce the tricks of the Krotov optimal control method and the corresponding
backpropagator. Third, We present the monotonic convergence property and related proof.
Fourth, We discuss the first order and second order cases for the backpropagator format,
and algorithms for implementation. Last, We will show a few examples to demonstrate the
power of the optimal control method. This is a review Chapter providing background for
the developments described in Chapter 5.
4.2
Preliminary Preparation of the Krotov Method
In this section the fundamentals of Krotov optimal control theory are sketched. For Krotov
optimal control theory, one needs to know the equation of motion of a given system, and
decide the cost/goal function to indicate the goal to be achieved. The formula of a goal
function depends on a target function of the system and a set of control parameters c.
Normally one can use square error as the target: (f (T, x, c) â y)2 or correlation metrics such
as: y T f (T, x, c) with y is a goal function, x are other parameters and T is terminal time. To
be more precise, one can consider an equation of motion of the form
âx
= f [t, x(t), c(t)],
ât
(4.1)
and assume we want to minimize the goal/cost function
Ë T
I[v] =
f 0 (t, x(t), c(t))dt + F [x(T )] ââ min.
(4.2)
0
Here x(t) is the time-evolution function or the trajectory of the system, c(t) is a timedependent control parameter. Note that v = (x(t), c(t)) â D with D is the set of permissible
process v = (x(t), c(t)) satisfying Eq. (4.2). The functionals f 0 (t, x(t), c(t)) and F [x(T )] are
defined for all t, x(t), c(t). Both need to be twice differentiable with respect to c and x. The
initial value x(0) = x0 is a given value and x(T ) is the value of x(t) at final time T. Note that
c(t) can be chosen within a closed domain U . The general functional F [x(T )] depends only
83
on the final time step of x(t) and f 0 (t, x(t), c(t)) depends on the intermediate time steps of
x(t) and c(t) so one can say the goal function I is a general functional based on the terminal
and intermediate time steps of x(t).
For a quantum system with a multi-dimensional vector space or multi-argument processes
and control parameters, we will have more than one equation of motion,
âxi
= f i [t, ~x(t), ~c(t)],
ât
i = 1, 2, ..., n.
(4.3)
And the minimization problem becomes
Ë T
I[v] =
f 0 [t, ~x(t), ~c(t)] + F [~x(T )]
(4.4)
0
Note that, ~c(t) and ~x(t) now are vectors, e.g. ~x = (x1 , x2 , ..., xn ).
4.3
4.3.1
The Tricks of Krotovâs Method
Decomposition of Goal Function
To implement the Krotov method, a real and twice differentiable function Ď[t, x(t)] is considered. The idea is to use the function Ď[t, x(t)] to help update the control parameters. Note
that this function would derive from a reverse time evolution start with the final time t = T
to the initial time t = 0. The control parameters are updated by comparing the difference
between the original forward evolution function and backpropagator Ď[t, x(t)]. We define the
constructions:
R[t, x, c] =
âĎ
âĎ
f [t, x, c] â f 0 [t, x, c] +
,
âx
ât
G[T, x] = F [T, x] + Ď[T, x],
(4.5)
(4.6)
Ë T
L[v, Ď] = G[T, x(T )] â
R[t, x(t), c(t)]dt â Ď[0, x(0)].
0
84
(4.7)
To be more specific, L[v, Ď] = I[v] for any function Ď[t, x(t)]. The following is the proof:
Ë T
R[t, x(t), c(t)] â Ď[0, x(0)]
L[v, Ď] = G[T, x(T )] â
0
Ë T
âĎ
âĎ
= G[T, x(T )] â
[ f [t, x(t), c(t)] â f 0 [t, x(t), c(t)] +
]dt
âx
ât
0
âĎ[0, x(0)]
Ë T
âĎ dx âĎ
= G[T, x(T )] â
[
+
â f 0 [t, x(t), c(t)]]dt
âx dt
ât
0
âĎ[0, x(0)]
Ë T
dĎ
dt â Ď[0, x(0)]
= F [T, x(T )] + Ď[T, x(T )] â
0 dt
Ë T
+
f 0 [t, x(t), c(t)]dt
0
Ë T
f 0 [t, x(t), c(t)]dt
= F [T, x(T )] +
0
= I[v].
(4.8)
Therefore minimizing I[v] can be achieved by minimizing L[v, Ď], and this also minimizes
G[x(T )] and maximizes R[t, x(t), c(t)].
For a multi-dimensional quantum system or for multi-argument processes, the equations
for R and G will be written as
R[t, ~x(t), ~c(t)] =
âĎ
âĎ ~
f [t, ~x(t), ~c(t)] â f 0 [t, ~x(t), ~c(t)] +
â~x
ât
(4.9)
and
G[T, ~x(T )] = F [T, ~x(T )] + Ď[T, ~x(T )].
(4.10)
For later use, it is convenient to define the function ÎŚ = âĎ
âx , and the functional
R[t, x(t), c(t)] = H[t, x(t), c(t), ÎŚ(t)] +
âĎ
,
ât
(4.11)
where
H[t, x(t), c(t), ÎŚ(t)] = ÎŚ(t)f [t, x(t), c(t)] â f 0 [t, x(t), c(t)].
85
(4.12)
Note that the parameters in H denoted by ÎŚ(t) emphasize that x and ÎŚ should be treated
as independent variables with respect to H.
4.3.2
Iterative Algorithm
The main purpose of the Krotov method is to find optimal control sequence ck+1 (t) to
improve the goal/cost function. In other words, the Krotov method hopes that I[v] is
monotonically decreasing with respect to ck (t) as k increases. That is
I[vk ] âĽ I[vk+1 ]
(4.13)
at every iteration. Since Ď[t, x(t)] is not restricted, we can freely choose the form of Ď[t, x(t)].
However, if we can construct the function Ď[t, x(t)] to maximize L[vk , Ď] at each k then we can
randomly choose the next control sequence ck+1 (t) and it will increase the value of L[v, Ď].
We therefore derive a smaller value of the goal function by the chosen Ď. Specifically, we
suppose that we already found the function Ď for a problem, then the complete processes
will be as follows:
(i.) Take an initial control sequence c0 (t), and calculate the trajectory x0 (t) from Eq. (4.1).
(ii.) Choose the functional Ď[t, x(t)] to make L[v0 , Ď] a maximum with the control c0 (t) and
trajectory x0 (t). This requirement is equivalent to the following two conditions:
R[t, x0 (t), c0 (t)] = min R[t, x, c0 (t)],
(4.14)
G[T, x0 (T )] = max G[T, x].
(4.15)
x
x
The conditions imply that the functionals R and G are calculated using the new Ď[t, x].
As a result the current control sequences c0 (t) will be the worst of all possible c(t) in
minimizing the goal functional L[v, ÎŚ] = I[v]. Any new c(t) changing from c0 (t) with
a new trajectory x will now improve the minimization of the goal function I[v].
86
(iii.) Finding a new control sequences c1 that maximizes the functional R. The corresponding conditions are
c1 [t, x] = max R[t, x, c]
c
= max H[t, x, c, ÎŚ],
c
(4.16)
where H is mentioned in Eq. (4.12). Note that the control sequence c1 [t, x(t)] depends
on the trajectory function x(t).
(iv.) With the new control sequence c1 [t, x] the new trajectory x1 (t) can be derived using
the equation of motion Eq. (4.1).
(v.) It is now guaranteed that the goal function in Eq. (4.2) has been minimized monotonically, which can be written as I[v1 ] â¤ I[v0 ]. The new control sequences and the
trajectory become a starting point of the next iteration and (i.)-(iv.) can be repeated
to further decrease the goal function.
4.3.3
Monotonic Convergence of Krotov Method
First we outline the proof that the new function I[v1 ] is indeed smaller than the previous
I[v0 ]. It is straightforward to show that
âI = I[t, x0 (t), c0 (t)] â I[t, x1 (t), c1 (t)]
= L[t, x0 (t), c0 (t), ÎŚ] â L[t, x1 (t), c1 (t), ÎŚ]
Ë T
=
R[t, x1 (t), c1 (t)] â R[t, x0 (t), c0 (t)]dt + G[T, x0 (T )] â G[T, x1 (T )]
0
= â 1 + â2 + â 3
(4.17)
where
â1 = G[T, x0 (T )] â G[T, x1 (T )]
Ë T
â2 =
R[t, x1 (t), c1 (t)] â R[t, x1 (t), c0 (t)]dt
0
Ë T
â3 =
R[t, x1 (t), c0 (t)] â R[t, x0 (t), c0 (t)]dt.
0
87
(4.18)
(4.19)
(4.20)
Using the conditions in Eq. (4.14) and Eq. (4.15) one can prove that â1 âĽ 0 and â3 âĽ 0,
and Eq. (4.16) also guarantees â2 âĽ 0. Therefore the new goal/cost functional I will be
smaller than the previous one and the monotonically convergence is proved.
4.4
Construction of Ď
To carry out the above iteration method, the most important and hardest task is finding a Ď
that satisfies the conditions in Eq. (4.14) and Eq. (4.15) which require the absolute minimum
of the functional R and maximum of the functional G with the old control sequences c0 and
the old trajectory x0 . In this section, we will show how to construct Ď to first order in x
and to second order in x to treat linear and non-linear problems respectively.
4.4.1
First Order in x
Consider that the equations of motion of the system are linear and can be written as
âx
= f [t, x, c] = a[t, c]x + b
ât
(4.21)
and the functions f 0 [t, x, c] and F [T, x] are concave with respect to x, which means
â 2 f 0 [t, x, c]
â¤ 0,
âxâx
â 2 F [T, x]
â¤ 0.
âxâx
(4.22)
In this case, we just need to consider Ď to first order in x since the second derivative is
guaranteed. To be more specific, The first order that implies the functional Ď needs to
satisfy Eq. (4.14) and Eq. (4.15). The function Ď needs to ensure that the first derivative
of the functions R and G with respect to x are equal to zero. Therefore, we can choose the
88
function Ď[t, x] = ÎŚ(t)x which satisfies the following conditions:
âR[t, x0 , c0 ]
âH[t, x0 , c0 , ÎŚ] âÎŚ[t, x0 ]
=
+
âx
âx
ât
= 0,
(4.23)
âF [T, x0 ] âĎ[T, x0 ]
âG[T, x0 ]
=
+
âx(T )
âx
âx
âF [T, x0 ]
=
+ ÎŚ[T, x0 ]
âx
= 0.
(4.24)
Therefore, Eq. (4.23) is the equation of motion for the function ÎŚ :
âH[t, x0 , c0 , ÎŚ]
âÎŚ[t, x0 ]
=â
ât
âx
(4.25)
with boundary conditions Eq. (4.24)
âF [T, x0 ]
âx
(4.26)
âx
âH[t, x0 , c0 , ÎŚ]
=
ât
âÎŚ
(4.27)
ÎŚ[T, x0 ] = â
From Eq. (4.1) and Eq. (4.12)
To satisfy the above requirements, the possible choice of Ď is Ď = ÎŚ(t)x. In the multiargument process, the similar choice of the functional Ď would be Ď[t, ~x(t)] = ÎŚi (t)xi (t).
Using the formula of Eq. (4.12), the conditions can be rewritten into the form
âÎŚ
âf 0 [t, x0 , c0 ]
= âJ T (t)ÎŚ(t) +
,
ât
âx
(4.28)
where
J =
âf [t, x0 , c0 ]
.
âx
Note that, in the multi-argument process, J T (t) is a transpose matrix.
89
(4.29)
4.4.2
Second Order in x
If the equations of motion of the system are not linear, one needs to consider a different form
of Ď. Since functional Ď needs to satisfy Eq. (4.14) and Eq. (4.15), the simplest choice of
functional Ď is of the form
1
Ď[t, x] = ÎŚ(t)x + âxÎŁ(t)âx,
2
(4.30)
where the â(x) âĄ x â x0 and both the function Ď[t, x] and the matrix ÎŁ(t) should be
found. Here ÎŁ(t) is the Hessian or matrix of the second derivatives of the function Ď(t, x).
The first necessary conditions for inequalities of Eq. (4.14) and Eq. (4.15) are equivalent to
Eq. (4.25) and Eq. (4.26), and the second necessary conditions for inequalities of Eq. (4.14)
and Eq. (4.15) yield the following differential inequalities:
d2 R âĽ 0,
d2 G â¤ 0,
â 2 R[t, x0 , c0 ]
âx,
âxâx
â 2 G[T, x0 ]
d2 G = âx
âx.
âxâx
d2 R = âx
(4.31)
(4.32)
Because the functional Ď can be chosen arbitrarily, in the multi-argument process, one can
require that the matrix ÎŁ(t) is a diagonal matrix and satisfies the conditions
â 2 R[t, x0 (t), c0 (t)]
= 0, i 6= j, i, j = 1, 2, ..., n,
âxi âxj
â 2 R[t, x0 (t), c0 (t)]
= Ďii (t), Ďii (t) âĽ 0, i = 1, 2, ..., n,
âxi âxi
(4.33)
and
â 2 G[T, x0 (T )]
= 0, i 6= j, i, j = 1, 2, ..., n,
âxi âxj
â 2 G[T, x0 (T )]
= Ďii (T ), Ďii (T ) â¤ 0, i = 1, 2, ..., n.
âxi âxi
(4.34)
One therefore determines the equation of motion of ÎŁ(t) with chosen boundary conditions
Ďii (t) and Ďii (T ) using one of the above linear differential equations.
90
4.4.3
Algorithm
In the previous section, Krotovâs optimal control method was introduced. Here I will summarize the algorithm:
(1) Choose an initial form of the control function c0 (t).
(2) Use Eq. (4.1) and initial condition x(0) = x0 to find the trajectory x0 (t).
(3) Find the functional ÎŚ(t) using Eq. (4.23) and Eq. (4.24) or Eq. (4.25) and Eq. (4.26).
(4) In the non-linear case, use Eq. (4.33) and Eq. (4.34) to find the matrix ÎŁ(t).
(5) With the functional Ď, the control c1 (t) is found according to Eq. (4.16).
(6) Derive the new trajectory x1 (t) using the control sequence c1 (t) by Eq. (4.1).
(7) Repeat processes (2) to (6) until the desired optimal value is achieved.
4.5
4.5.1
Examples
Example 1: First Order in x
Consider a linear problem with Ď chosen in the form given in subsection (4.4.1) for the
following optimal control problem:
xĚ(t) = i(1 + c(t))x(t),
x(0) = 1;
c(t) is real and one wants to minimize the cost function
Ë T
1
(c(t0 ) â c0 )2 dt0 ââ min.
I = Re[(1 + x(T ))] + b
2 0
(4.35)
(4.36)
where b > 0.
We choose the parameters b = 5, T = 2 and substitute the linear form of Ď = ÎŚ[t, x(t)]x
to derive R and G :
R = Re[ÎŚ(t)[i(1 + c(t))x(t)] +
â
1
(ÎŚ(t)x(t))] â b(c â c0 )2 (t),
ât
2
G = Re[(1 â x(T )(â1)) + ÎŚ(T )x(T )].
91
(4.37)
(4.38)
Using
â
âR
= Re[ÎŚ(t)[i(1 + c(t))] + ÎŚ(t)] = 0,
âx
ât
âG
= Re[1 + ÎŚ(T )] = 0,
âx
(4.39)
(4.40)
one can derive the equation of motion of ÎŚ in reverse time t â ât:
ÎŚ(T ) = â1.
ÎŚĚ(t) = i(1 + c(t))ÎŚ(t),
(4.41)
Performing the algorithm described in subsection (4.4.3), we require
â 2R
â¤ 0.
(âc)2
âR
= 0,
âc
(4.42)
We therefore obtain the result
c1 (t) = c0 (t) + Re[ÎŚ(t)ix(t)]/b.
(4.43)
Results are shown in Fig. 4.1, where we have used the Runge Kutta method with the segment
of integration partitioned into 200 intervals, and the fidelity is defined as F = Re[x(T )(â1)]
and error = 1 â F as shown in cost function.
4.5.2
Example 2: Second Order in x
Let us consider the approach from subsection (4.4.2) for the following optimal control problem. The functions x(t) and c(t) are constructed by
xĚ = c,
|c| â¤ 1,
x(0) = 0;
(4.44)
and one wants to minimize the cost function
Ë T
I =
1
(c2 â x2 )dt + bx2 (T ) ââ min,
2
0
where b > 0.
92
(4.45)
1.00
0.66
0.95
0.64
0.62
0.60
0.85
c(t)
Fidelity
0.90
0.80
0.56
0.75
0.54
0.70
0.65
0
0.58
0.52
5
10
15
Iteration
20
0.50
0.0
0.5
1.0
t
1.5
2.0
Figure 4.1: Left: fidelity versus iteration times. Right: optimal control sequence with
respect to time t.
Now we choose the parameters b = 20, T = 4 substitute Eq. (4.30) into Eq. (4.5) and
Eq. (4.6) to derive R and G of the form
1
R = ÎŚĚ(t)x(t) + ÎŁĚ(t)(âx)2 + 2ÎŚ(t)c(t)
2
+2ÎŁâx(t)c(t) â c2 (t) + x2 (t),
1
1
G = ÎŚ(T )x(T ) + ÎŁ(T )(âx(T ))2 + bx2 (T ).
2
2
(4.46)
(4.47)
Since
â 2R
= ÎŁĚ(t) + 1 âĽ 0,
(âx)2
â 2G
= ÎŁ(T ) + b â¤ 0,
(âx)2
(4.48)
we first choose that
ÎŁĚ(t) = 0,
ÎŁ(T ) = âb â 4.
93
(4.49)
We therefore bring it into
âR
= ÎŚĚ(t) + ÎŁĚ(t)âx + 2ÎŁc(t) + 2x(t) = 0,
âx
âG
= ÎŚ(T ) + ÎŁ(T )âx(T ) + bx(T ) = 0.
âx
(4.50)
(4.51)
Performing the algorithm described in subsection (4.4.3), we require
âR
= 0,
âc
â 2R
â¤ 0.
(âc)2
(4.52)
We obtain the result shown in Fig. 4.2, where we have used the Runge Kutta method with
the integration partitioned into 200 intervals. For comparison, the known solution of the
problem is shown below
ďŁą
ďŁ´
ďŁ´
Âąt,
t â¤ Ď1 ,
ďŁ´
ďŁ´
ďŁ˛
x(t) =
Âąkcos(t â T /2), Ď1 â¤ t â¤ Ď2 ,
ďŁ´
ďŁ´
ďŁ´
ďŁ´
ďŁł ÂąT â t,
Ď2 â¤ t,
(4.53)
where T is the final time and Ď1 , Ď2 and k are chosen according to smoothness conditions:
xĚ = Âą1 for t = Ď1 , xĚ = â1 for t = Ď2 , Âąt = Âąkcos(t â T /2) at t = Ď1 and Âąkcos(t â T /2) =
ÂąT â t at t = Ď2 . Note that the result of the Krotov optimal method in Fig. 4.2 is equal
to the known solution. We will extend the Krotov method to investigate a quantum gate in
the next example.
4.5.3
Example 3: A Two Level Quantum System
The time dependent SchrĂśdinger equation for the evolution equation (forward-propagator)
|Ď(t)i of a quantum system including a time-dependent control term ÂľÎľ(t) where Îľ(t) is the
control function
i~
â
|Ď(t)i = (H + ÂľÎľ(t))|Ď(t)i.
ât
Suppose that one want to minimize the cost function
Ë T
I = 1 â Re[hĎG |Ď(T )i] + Îť
(Îľ(t0 ) â Îľ0 )2 dt0 ââ min.
0
94
(4.54)
(4.55)
160
2.0
140
120
1.5
80
x(t)
Fidelity
100
60
1.0
40
0.5
20
0
â20
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1 2 3 4 5 6 7 8 9 10
Iteration
t
Figure 4.2: Left: Cost function versus iteration times. Right: Optimal evolution of x with
respect to time t.
where Îť > 0 and Îľ0 is an initial energy that can be considered as the restriction for the
optimal control sequences; furthermore, Îľ0 can also be time-dependent. |ĎG i is a target goal
for the forward-propagator |Ď(T )i.
Now we choose the parameters Îť = 1, T = 1 and substitute Ď = hÎ˛(t)|Ď(t)i into Eq. (4.5)
and Eq. (4.6) to derive R and G :
âhÎ˛(t)
R = Re hÎ˛(t)|(H + ÂľÎľ(t))|Ď(t)i +
|Ď(t)i â Îť(Îľ(t) â Îľ0 )2 ,
ât
i
h
G = Re hĎG |Ď(T )i â hÎ˛(t)|Ď(T )i T0 .
(4.56)
(4.57)
Using Eq. (4.25) and Eq. (4.26), one can derive the equation of motion of Î˛(t) (similar
process to Example 1):
i~
â
hÎ˛(t)| = hÎ˛(t)|(H + ÂľÎľ(t)),
ât
hÎ˛(T )| = hĎG |.
(4.58)
To optimize the control sequence, we require
âR
= 0,
âÎľ
â 2R
â¤ 0.
(âÎľ)2
95
(4.59)
Therefore the optimal control sequence is of the form
Îľ(t) = Îľ0 +
1
Re[hÎ˛(t)|Âľ|Ď(t)i].
2Îť
(4.60)
Using the algorithm in subsection (4.4.3) with the above conditions where hÎ˛(t)| depends
on the old Îľ(t), and |Ď(t)i is built from the new Îľ(t). Note that for a better performance, one
can substitute Îľ0 with the old Îľ(t) to derive new Îľ(t). The system under consideration is a
quantum qubit or a two-level quantum dot. Under appropriate conditions, the Hamiltonian
of the qubit reads
H(t) = âÎľ(t)Ďz /2 â âŚĎx /2
where H = âŚĎx /2 and Âľ = Ďz /2 and âŚ is a bias voltage. If we consider a target
ďŁŤ ďŁś
ďŁŹ 1 ďŁˇ
|ĎG i = | âi = ďŁ ďŁ¸ ,
0
(4.61)
(4.62)
we obtain the result shown in Fig. 4.3 and Fig. 4.4, where we use the Euler method with
the segment of integration partitioned into 320 intervals (dt = 0.01), âŚ = 1, and fidelity is
defined as Re[hĎG |Ď(T )i] with initial state
ďŁŤ
ďŁś
ďŁŹ 0 ďŁˇ
|Ď(t = 0)i = | âi = ďŁ ďŁ¸ .
1
(4.63)
We can achieve fidelity 10â4 or lower with learning rate Îť = 10. Note that this is a quantum
optimal control problem and is the fundamental in quantum computation. Chapter 5 will
use similar optimal techniques to a charge density wave (CDW) tight-binding model.
96
1.0
1.0
0.5
0.6
Îľ(t)
Fidelity
0.8
0.0
0.4
â0.5
0.2
0.0
â1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 20 40 60 80 100 120 140
Iteration
t
Figure 4.3: Left: Cost function versus iteration times. Right: Optimal control pulse of Îľ
respect to time t.
1.0
Probability
0.8
0.6
up
down
0.4
0.2
0.0
0.0
0.5
1.0
1.5
Time
2.0
2.5
Figure 4.4: Optimal probability evolution of Ď(t) respect to time t.
97
3.0
CHAPTER 5
QUANTUM OPTIMAL CONTROL OF A TRANSIENT CDW STATE
5.1
Introduction
As outlined in Chapter 1, 2, 3, High fidelity characterization of transient excited many-body
electron distributions in the ultrafast time domain is now available through a variety of
pump-probe experiments. Amongst the rich variety of non-equilibrium responses observed,
photo-induced phase transitions (PIPT) are particularly interesting for fundamental and
applied reasons [39, 38, 53, 54, 55, 56, 57, 58, 59, 35, 60, 32, 33, 51, 3, 35, 36, 61, 60]. For
example, by adjusting the laser pulse properties it is possible to tune the non-equilibrium
PIPT response from the adiabatic to the non-adiabatic limits. Optimal laser pulse shaping
methods have not been applied to PIPT yet and have the potential to control non-equilibrium
response in order to isolate selected physical phenomena, and in order to tune response for
selected applications such as high speed electronics or optics. We introduce a method,
based on Krotov optimal control theory, to direct photo-induced phase transitions (PIPT)
by combining non-equilibrium models with Quantum optimal control theory (QOCT).
To illustrate the approach, we consider PIPT in a simple model [56] for a transient
metal-insulator state in a charge density wave system. In experiments, a long-range chargedensity-wave (CDW) is formed in a variety of layered chalcogenide materials, in oxides, in two
dimensional materials, and in many other systems. The investigated layered CDW materials
have disclosed a new nonequilibrium pattern where the long-range CDW is preserved while
the local electronic excitation spectrum becomes gapless for a transient period of time, as
has been eluciated using the simple model considered here [56]. In the experimental systems
the mechanism for gap-closing and population inversion remains an open question, though
in some cases there has been significant theoretical progress.
QCOT is a powerful tool based on calculating the optimal pulse shape by minimizing a
physical cost function or maximizing a desired physical objective, and it has been developed
98
within a variety of variational frameworks to obtain control sequences [63, 64, 65, 66, 67, 68,
69, 70, 71, 62, 72, 73]. Quantum optimal control methods based on the classical gradient
optimization methods provide an alternative to iterative methods based on the Krotov approach. Krotov methods have been applied to the fields of Quantum computing and control
of charge transfer processes [72, 74, 75, 62, 76, 73]. The Krotov approach has several appealing advantages over the gradient methods in following ways: First, a monotonic increase
toward the objective with iteration number. Second, no requirement for a line search and
faster convergence to a given target. Third, at each iteration it guarantees macrosteps of the
time interval and it can reach the global maximum. As shown here, the Krotov approach
can easily be combined with CDW models for high efficiency calculations.
To illustrate the methods, we consider the hot-electron model following the simulation
study of Ref. [56, 55]. They found ultrafast laser pulses can quickly heat electrons forming a
hot quasithermal gas that equilibrates with phonons on much longer time scales compared to
the electron relaxation time. We introduce QOCT in this scenario to optimize shaped-laser
pulses to study the short-time transient phenomenon and to discover new photo-induced
phase transitions, and to control the outcome of pump-probe PIPT experiments to achieve
strong population inversion.
5.2
Charge Density Wave Model
To model the CDW system, we follow the work by Freericks et al. [56, 55]. We consider a
lattice that can be divided into two sublattices A and B with a hoping term Ďij only between
the two sublattices. For on site energy, sublattice A is chosen equal to U and sublattice B
is choose equal to zero. We consider equal numbers of A and B sites with one electron per
site, so that the electrons are at half filling and form an insulator. As outlined in Chapter
2, we include a spatially uniform time-dependent pump pulse incorporated into our model
through the Peierls substitution, resulting in a time-dependent modulation of the hopping
99
term. The Hamiltonian, using standard notation for creation and annihilation operators, is
H(t) = â
X
â
Ďij (t)ci cj + (U â Âľ)
ij
X â
X â
ci ci â Âľ
ci ci ,
iâA
(5.1)
iâB
where the chemical potential is Âľ = U/2 for half filling and the time-dependent hopping in
the presence of the laser pulse using Peierls substitution is
#
" Ë
ie Rj
A(r, t) Âˇ d~r .
Ďij (t) = Ďij exp
~c Ri
(5.2)
Here A(r, t) is the time dependent vector potential in the Hamiltonian gauge, Ri is the
position vector of the ith lattice site, and r is a real space vector.
~ t), the electronic band
In the presence of a pump field, through the vector potential A(r,
gap can be tuned [56], even though Ďij (t) only affects the phase but not the amplitude of
the hopping term of the Hamiltonian. In equilibrium, the density of states presents a gap
of width equal to U that is symmetric about zero energy. For the time dependent case,
the eigenfunctions change as a function of time and the nonequilibrium density of states is
modified by the presence of the external field. For this reason the non equilibrium energy
states are always influenced by the interaction of the system with the pulse field, and hence
it is possible to control the local charge density.
Translation symmetry is broken when U is nonzero, as the conversion from real space to
momentum space is more complicated than in a system with one atom per unit cell. The
momentum points k and k + Q are coupled, where Q = (Ď, Ď, Ď, ...) due to the presence of
the CDW order. The transformation from reciprocal space to real space becomes
â
ci =
X
â
â
(eâikÂˇRi ck + eâi(k+Q)ÂˇRi ck+Q ),
(5.3)
k
The electronic band in momentum space for the U = 0 case is time dependent and can
be written as
Îľk (t) = â
X
Ďij exp[âi(~k â
ij
100
e
A(t)) Âˇ (RiA â RjB )]
~c
(5.4)
where uniform vector potential is A(t) = A(t)(1, 1, ..., 1) and is directed along the diagonal
of the hypercubic lattice in d-dimensions, and we can rewrite as
Îľk (t) = Îľk Ac (t) + ÎľĚk As (t),
(5.5)
Ac (t) = cos(eaA(t)/~c),
(5.6)
As (t) = sin(eaA(t)/~c),
(5.7)
d
X
Ďâ
â cos(akl ),
d
l=1
(5.8)
d
X
Ďâ
â sin(akl ).
ÎľĚk = â
d
l=1
(5.9)
with
and
Îľk = â
In this work we take the limit that dimension d goes to infinity as in the model of Ref. [56].
In our work, the two functions Ac (t) and As (t) are used as control parameters for optimal
control.
To simplify the notation, we set ~ = e = c = 1 and choose U = tâ , where tâ is the
renormalized hopping for a hypercubic lattice in infinite dimensions. We also set the initial
temperature of the system equal to zero before the field is turned on. We can now derive
equations of motion for the creation operators in momentum space as
dck (t)
= [Îľk (t) + U2 â Âľ]ck (t) + U2 ck+Q (t)
dt
dck+Q (t)
i~
= U2 ck (t) + [âÎľk (t) + U2 â Âľ]ck+Q (t).
dt
i~
(5.10)
(5.11)
We are able to calculate the time evolution of different momentum k states, from which
we derive the nonequilibrium Greenâs functions to obtain the physical properties of the system at time t. In the following calculations, we use ck (t) = (ck (t), ck+Q (t)) and define a
P
superposition operator a(t) = k ck (t) in extended Hilbert space. Then the time evolution following Eq. (5.10) and (5.11) can be generally written as i~ât ck (t) = H k (t)ck (t) or
101
P
i~ât a(t) = H(t)~a(t) with H(t) = k H k (t). and
ďŁŤ
ďŁś
U
U
ďŁŹÎľk (t) + 2 â Âľ
ďŁˇ
2
H k (t) = ďŁ
ďŁ¸.
U
U âÂľ
âÎľ
(t)
+
k
2
2
(5.12)
Since we have evolution equations, we can write down the time-evolution operator as
Uk
(t, t0 )
Ë
i tÂŻ
dtH k (tĚ) ,
= exp â
~ t0
(5.13)
and, by using the Trotter formula, the time evolution in numerical calculation becomes
Uk (t, t0 ) = Uk (t, t â ât)Uk (t, t â 2ât)...Uk (t0 + ât, t0 ),
(5.14)
where ât is a small time step. For each moment time t, we have
iât
Uk (t, t â ât) = exp â
H k (t â ât/2) .
~
5.2.1
(5.15)
Equations for the Nonequilibrium Solution
In the paper, by following the theory of [56, 55], the retard and lesser Greenâs function are
defined as
â
0
0
0
GR
ij (t, t ) = âiÎ¸(t â t )h{ci (t), cj (t )}+ i,
â
0
0
G<
ij (t, t ) = ihcj (t )ci (t)i,
(5.16)
(5.17)
where we take a quantum statistical average of time dependent creation and annihilation
operators. Î¸(t â t0 ) is the unit step function. In the Heisenberg representation, ci (t) and
â
ci (t) are creation and annihilation operators for a spinless fermion at lattice site i. The angle
brakets represent a trace over all quantum states in real or momentum space weighted by
the equilibrium density matrix as is initialized in the far past. We can employ the Greenâs
function to calculate the election concentration on the A and B sublattices, the DOS, the
current, the total energy and the time-resolved PES signal.
102
We calculate the time-resolved PES response function as a probe pulse weighted time
Fourier transform of the lesser Greenâs function centered at time tp (see Section 2.6),
Ë
Ë
0
0
P (Ď, tp ) = âi dt dt0 s(t)s(t0 )eâiĎ(tât ) G<
ii (t + tp , t + tp ).
(5.18)
The response should be calculated for each sublattice and is then averaged over both sublattices to compare with the experimental response. No extrapolation to large times is needed
for this calculation since the probe pulse provides a natural cutoff. The probe pulse is
assumed to be a gaussian function
2 2
1
s(t) = â eât /Ď
Ď Ď
(5.19)
with width Ď. The narrower the probe width, the better the time resolution and the worse
energy resolution.
We also calculate the total energy, the current and the occupancy in the upper/lower
bands of the instantaneous band structure varies with time. Only equal time expectation
values are needed considered for these quantities, and They are found in a straightforward
manner from the equal time lesser Greenâs function. We write the calculations here in
terms of the equal time expectation values. Time-dependent number density for different
â
momentum k and k + Q are defined as nk1 ,k2 (t) = hck (t)ck2 (t)i with (k1 , k2 ) is (k, k + Q).
1
The current in the case satisfies
X
j(t) =
âÎľ(k; t) nk,k (t) â nk+Q,k+Q (t) ,
(5.20)
nk+Q,k (t) â nk,k+Q (t)
.
nk,k (t) + nk+Q,k+Q (t)
(5.21)
k:Îľ(k)<0
The order parameter is of the form
X
âŚ(t) =
k:Îľ(k)<0
and the total energy becomes
E(t) =
X
Îľ(k; t) nk,k (t) â nk+Q,k+Q (t)
k:Îľ(k)<0
U
+
n
(t) + nk,k+Q (t) .
2 k+Q,k
103
(5.22)
The occupancy of the upper and lower instantaneous bands become
n+ (t) =
X
h
Îą2 (k; t)nk,k (t) + Î˛ 2 (k; t)nk+Q,k+Q (t))
k:Îľ(k)<0
+ Îą(k; t)Î˛(k; t) nk+Q,k (t) + nk,k+Q (t)
(5.23)
and
nâ (t) =
h
X
Î˛ 2 (k; t)nk,k (t) + Îą2 (k; t)nk+Q,k+Q (t)
k:Îľ(k)<0
â Îą(k; t)Î˛(k; t) nk+Q,k (t) + nk,k+Q (t)
(5.24)
with the two time-dependent coefficients are given by
U/2
Îą(k; t) = q
p
2[Îľ2 (k; t) + U 2 /4 â Îľ(k; t) Îľ2 (k; t) + U 2 /4]
(5.25)
and
p
Îľ2 (k; t) + U 2 /4
Î˛(k; t) = q
.
p
2
2
2
2
2[Îľ (k; t) + U /4 â Îľ(k; t) Îľ (k; t) + U /4]
âÎľ(k; t) +
(5.26)
5.3
Quantum Optimal Control Method
In this section, we develop an optimal control method that maximizes the overlap between
the state of the system and a target (goal) state at time tf as implemented by QOCT. If
we denote an arbitrary target operator (state) as g, then the goal is to control the system
in such a way that the final operator is as close as possible to g. Fidelity is one way to
quantify performance of the optimal pulse. Fidelity is defined as the real part of the inner
product between the desired target operator g and the actual operator a(tf ) at time tf (see
e.g. [69, 72, 73, 71, 62]),
F = Re[hg â a(tf )i]N ,
104
(5.27)
with N denotes normalization, so that the maximum fidelity is F = 1.
In many optimal theories and machine learning, we restrict control parameters to a finite
range is necessary to prevent divergence or discontinuity, and, in physical problems, the
power of the pump pulse should be considered in a finite range as well. We are therefore
looking for a solution of the optimal control method that can achieve high fidelity while
keeping the total power transferred to the system from the control parameters finite. To
achieve the goal, we require that the power close to the power of the pulse that we use as
previous iteration. This is expressed by [72]
K=
Ë t
f
0
dt0 Îť(t0 )[
X
0
l 0 2
(Al+1
j (t ) â Aj (t )) ],
(5.28)
j=c,s
with l, l + 1 represent the l, l + 1 iteration times of the quantum optimal control algorithm.
We set A0c (t) â 1 and A0s (t) â 0 for the 0th iteration because of the initial condition of
A(t) = 0. Îť(t0 ) is a positive function chosen empirically that can be adjusted during the
optimal control procedure. It influences the rate of change of the control parameters at each
iteration step. We choose Îť(t0 ) as a constant proportional to the number of energy points
used in the simulation.
Since we know (Ac (t))2 + (As (t))2 = 1, we add an additional constraint through a
Lagrangian multiplier R to ensure that this equation is satisfied for all time steps t. Defining
Î˛(t) as another adjustable and positive function, we have
R=
Ë t
f
dt0 Î˛(t0 )((Ac (t0 ))2 + (As (t0 ))2 â 1).
(5.29)
0
When everything is put together, an objective function is introduced of the form
J = F â K â R,
(5.30)
where F , K and R are defined in Eqs. (5.27)-(5.29). Our goal is to maximize the objective
function given the equation of motion for the operator a(t).
In practice the algorithm consists of the following steps: First, choose an initial control
sequence Ac (t) and As (t), second, given the initial condition the operator a at very beginning,
105
apply the equations of motion to find the forward propagated operator al (t). Third, calculate
an auxiliary backward evolved operator bl (t) analogous to a(t) with the condition bâ (tf ) =
g â . Fourth, propagate al+1 (t) forward in time and update the control sequence iteratively
with the relation
l
Al+1
c (t) = (Ac (t) + Î´Ac (t))/Îą(t),
(5.31)
âH(t) l+1
1
l
â
Re hb (t)
a (t)i ,
Î´Ac (t) =
2Îť(t)
âAc (t)
(5.32)
and
2
where Îą(t) = 1+Î˛(t)/Îť(t) is an positive and adjustable term, which ensures that (Al+1
c (t)) +
2
(Al+1
s (t)) = 1 is satisfied for every time t.
Specifically Îą(t) is an arbitrary function greater or equal to one and it can be arbitrarily assigned at every iteration and every time t since the contribution of the Lagrangian
multiplier in the objective function J is 0. The equation for Al+1
s (t) has the same form,
in Eq. (5.32). Steps (3) and (4)
with Als , Al+1
and can be obtained by replacing Alc , Al+1
s
c
are repeated until a desired fidelity F is reached or until a given number of iterations has
been accomplished. Note that here we derive Î´Ac,s (t) base on the initial conditions, while
in other cases, one can derive state-independent results by using a trace instead of an inner
product. We terminate our program when 1-F is smaller than 10â4 or when the fidelity
|F l+1 â F l |/|1 â F l | is smaller than 10â3 .
5.3.1
Proof of Optimality
Using the cost function mentioned in the text
Ë t
X
f
â
J =Re[h~g ~a(tf )i] â
dtÎť(t)[
(Aj (t0 ) â A0j (t0 ))2 ]
0
â
Ë t
f
0
j=c,s
dtÎ˛(t)(A2c (t) + A2s (t) â 1),
(5.33)
we will now prove that the iteration algorithm for the cost function J exhibits monotonic
convergence. The simple proof for two-level quantum system can be found in Ref. [69]. First,
106
following Krotovâs method (see Chapter 4), we group the terms in J in the following way
JÂŻ = G(tf ) +
Ë t
f
dtR(t).
(5.34)
0
Here G(tf ) depends only on the terminal time tf and is defined as
G(tf ) âĄ Re
tf
h~g â ~a(tf )i â h~bâ (t)~a(t)i
0
,
(5.35)
where ~b(t) is an arbitrary continuously differentiable propagator which can be considered as
a Lagrange multiplier function constraining the system to obey the equation of motion. The
second term R(t) is related to the time integral part and is of the form
"
#
~bâ (t)
â
R(t) âĄ Re h~bâ (t)H(t)~a(t) +
~a(t)i
ât
X
(Aj (t0 ) â A0j (t0 ))2 ]
âÎť(t)[
j=c,s
âÎ˛(t)(A2c (t) + A2s (t) â 1).
(5.36)
In our case we have two control parameters in the Hamiltonian H(t) = H 0 + H c Ac (t) +
H s As (t). To maximize JÂŻ one can simply maximize G and R independently. Note that if R
is maximized at each time t the integral of R will be maximized as well. Second, to prove
that JÂŻ converges at every iteration, it is straightforward to show that
JÂŻ(l+1) â JÂŻ(l) = â1 + â2c + â2s + â3 âĽ 0
(5.37)
where
â1 âĄ G(~a(l+1) (tf )) â G(~a(l) (tf ))
= Re[h(~g â â ~bâ (tf ))â~a(tf )i],
107
(5.38)
â2c âĄ
Ë t
f
0
(l+1)
dt[R(t, ~a(l+1) (t), Ac
(l)
(t), As (t))
(l)
(l)
â R(t, ~a(l+1) (t), Ac (t), As (t))]
Ë t
f
â
(l+1)
= Re
dth~b (t)(HĚ c âAc (t))~a
(t)i
0
Ë t
f
(l+1)
dt2Îť(t)(Ac
(t) â A0c (t))âAc (t)
â
0
Ë t
f
(l+1)
dt2Î˛(t)Ac
(t)âAc (t)
â
0
+Îť(t)âA2c (t) + Î˛(t)âA2c (t),
(5.39)
and â2s also follows the same form by substituting Ac (t) with As (t).
â3 âĄ
Ë t
f
0
(l)
(l)
dt[R(t, ~a(l+1) (t), Ac (t), As (t))
(l)
(l)
â R(t, ~a(l) (t), Ac (t), As (t))]
Ë t
f
= Re
dth~bâ (t)(H 0 + H c Aic (t) + H s Ais (t))â~a(t)i
0
#
Ë t
f
â~bâ (t)
â
dth
)â~a(t)i .
ât
0
(l+1)
Here âAc = Ac
(5.40)
(l)
(t) â Ac (t) and â~a(t) = ~a(l+1) (t) â ~a(l) (t) , and in deriving these
expressions we also have chosen ~b(t) âĄ ~b(l) (t) in the expression for JÂŻ(l+1) as well as in that
for JÂŻ(l) . We can write the equation of motion and initial state of ~b(t) by making the choice
~bâ (tf ) = ~g â ,
â~bâ (t)
= ~bâ (t)(H 0 + H c Ac (t) + H s As (t)),
ât
(5.41)
(5.42)
Therefore we obtain the result
â1 = 0, â3 = 0.
(l+1)
Finally, the control parameter Ac
(5.43)
l
(t) can be chosen as Al+1
c (t) = (Ac (t)+Î´Ac (t))/Îą(t)
and
l
1
âH(t) l+1
~
â
Î´Ac (t) =
Re hb (t)
~a (t)i ,
2Îť(t)
âAc (t)
108
(5.44)
which implies that
â2 = (Îť(t) + Î˛(t))[(âAc )2 + (âAs )2 ] âĽ 0.
(5.45)
Here Îť(t) is a positive function that can be decided empirically and Îą(t) = 1 + Î˛(t)/Îť(t) is
2
l+1
2
a normalization term, which ensures that (Al+1
c (t)) + (As (t)) = 1 is satisfied at every
time t.
This proves that the iteration algorithm for cost function J exhibits monotonic convergence, given the proper choice of the equation of motion and initial condition for ~bâ (t), and
using Eq. (5.44) for the optimal control parameter at the next time step. Note that A0c (t) in
(l)
the l + 1 iteration can be substituted with the optimal control parameter Ac (t) acquired in
l
the l iteration. Similarly, the result of the As (t) is of the form Al+1
s (t) = (As (t)+Î´As (t))/Îą(t)
and
l
1
âH(t) l+1
~
â
Î´As (t) =
Re hb (t)|
|~a (t)i ,
2Îť(t)
âAs (t)
5.4
(5.46)
Application to a CDW Model
We illustrate the Krotov method above using a solvable spinless fermion model on hypercubic
lattices described by a hopping term Ďij and a periodic potential Ui , which is equal to U on
the A sublattice and equal to zero on the B sublattice (See chapter 2).
H(t) = â
X
â
Ďij (t)ci cj + (U â Âľ)
ij
X â
X â
ci ci ,
ci ci â Âľ
iâA
(5.47)
iâB
where the chemical potential is Âľ = U/2 for half filling, and the time dependence of the
hopping parameter is as given in Eq. (5.2). This model provides a useful starting point in
understanding of the electronic response of CDW systems to an ultrafast laser pulse [56, 55].
For U = 0, the time dependence of the energy levels in the problem is given by
~
Îľ(~k â eA(t)/~c)
= Îľk Ac (t) + ÎľĚk As (t)
109
(5.48)
where Ac , As are given in Eq. (5.6). In order to compare with previous work [56], we
P
tâ cos(r k ),
â
take the limit of a hypercubic lattice in infinite dimensions Îľk = â d=â
0 l
l=1
d
Pd=â tâ
ÎľĚk = â l=1 â sin(r0 kl ).
d
When U is finite the Hamiltonian breaks up into 2 Ă 2 blocks for each value of k, as the
unit cell is now of size 2. When the laser is off, the energy eigenvalues of the system are
q
given by ÎľkÂą = U/2 â Âľ Âą Îľ2k + U 2 /4, where Îľk are the eigenvalues for U = 0, and the
destruction operators after diagonalization may be written in terms of the U = 0 destruction
operators ck and ck+Q where Q = (Ď, Ď, Ď, ...). The energy spectrum breaks up into two
bands with band gap U , and the destruction operators are ck+ = Îąk ck + Î˛k ck+Q for states
in the conduction band and ckâ = Î˛k ck â Îąk ck+Q for states in the valence band. To impose
the condition that an arbitrary target state with quantum number q+ is occupied at the
â
â
final time tf we need hcq+ cq+ i = 1 and hcqâ cqâ i = 0. This is achieved by defining the goal
P â
operator ~g = q cq+ cqâ , where the sum is over the target excited states. In the remainder
of our discussion, the units are taken as r0 = tâ = U = 1, the speed of light and Planckâs
constant are in natural units, the chemical potential Âľ = U/2, the temperature of the initial
state is T = 0, while the time and energy steps are ât = 0.01 and âÎľ â¤ 0.02, as used in
previous work [56] on this model.
The equations of motion for the operators in momentum space are given by,
dck (t)
= [Îľk (t) + U2 â Âľ]ck (t) + U2 ck+Q (t)
dt
dck+Q (t)
i~
= U2 ck (t) + [âÎľk (t) + U2 â Âľ]ck+Q (t)
dt
i~
(5.49)
(5.50)
~
where k is a wave vector and and Îľk (t) âĄ Îľ(~k â eA(t)/~c).
It is useful to introduce the more
compact notation ~ck (t) = (ck (t), ck+Q (t)) and a superposition operator ~a(t) = âk~ck (t). The
time evolution in Eq. (5.49) and (5.50) can then be written as i~ât~ck (t) = H k (t)~ck (t) or
L
i~ât~a(t) = H(t)~a(t) with H(t) = k H k (t), where H k (t) is deduced by comparing with
Eq. (5.49) and Eq. (5.50) [55, 56]. The calculations are directed at optimizing the fidelity
from Eq. (5.27), and we terminate the calculations when 1 â F is smaller than 10â3 or when
the difference ratio |F l+1 â F l |/|1 â F l | is smaller than 10â3 . The initial condition before
110
arrival of the laser pulse is taken to be the half filled state so that the Fermi energy is at
1.0
(a)
0.5
0.0
0.5
1.0
1.0
(b)
0.5
0.0
0.5
1.0
0.4
0.2 (c)
0.0
0.2
0.4
0.6
0.8
40
â â
Energy
(unit/ tof
t )
Energy
A(t)
A s (t)
A c (t)
zero energy and in the middle of the band gap, which is U in this model.
20
0
20
40
60
80
3
0.368
2
0.135
1
0.050
0
0.018
-1
0.007
0.003
-2
-3
-40
-20
0
20
40
Tim
e (h/ tofâ)h/ t â )
Tim
e (unit
60
80
Figure 5.1: Time-resolved PES of the electronic CDW system before, during and after
application of the optimal pulse (Ac (t), As (t), A(t)) from time t = 0 to t = 20 h/tâ with
the initial state a filled lower band at temperature T = 0. The target is excitation of a
single state with the goal energy 1tâ . In this calculation Îť(t) = 2Ďh/â is a constant, with
â = 0.02 and h = 100.
We calculate several physical properties in the final state that is produced by the optimal
pulse, and we first focus on the time resolved photoemission spectra (tr-PES), P (Ď, t), where
i
Â´t
P hÂ´ t
iĎ(t
ât
)
<
2
1
P (Ď, t) = Im Îą ââ dt2 ââ dt1 s(t1 )s(t2 )e
GÎą,Îą (t2 , t1 ) , where s(t) is the probe
111
â
laser pulse shape and Îą = A, B. G<
Îą,Î˛ (t1 , t2 ) = ihcÎą (t2 )cÎ˛ (t1 )i is the lesser Greenâs function;
Îą and Î˛ can be either of the two sublattices A, B of a hypercubic lattice; and t1 , t2 are
two times on the Keldysh contour (see [55] for details). The ultrafast time resolved PES
presented in Fig. (5.1) is found when we choose an excitation band centered on Energy tâ .
In this calculation U = tâ which is the band gap. This solvable system consists of a set
of two-level systems (TLS) labeled by k with energy splitting (for q = k) Î´q = q+ â qâ
~
~ q Âˇ E/~
is the
with associated generalized Rabi frequency âŚq = (Ď2q + â2q )1/2 where Ďq = Âľ
~ the electric field and the detuning is
Rabi frequency, Âľ
~ q the transition dipole moment, E
âq = Ď â Î´q /~ with Ď the laser frequency. The optimal pulse found in this case (see
Fig. (5.1) is in the weak field limit so that the dominant frequency is 2 which corresponds
to the excitation energy Î´q for the target TLS. However some TLS near the target are also
excited and the optimal pulse attempts to minimize excitation of TLS other than the target.
The resulting optimal pulse is clearly not single mode with a Gaussian envelope, reflecting
the fact that the target state is embedded in a continuum.
Fig. (5.2) presents the tr-PES and optimal pulse found when the target is a fully excited
state, i.e. excitation of all electrons from the valence band to the conduction band. This is
an extreme excitation condition that can be interpreted as a state with a very large negative
temperature. The QOCT method yields a laser pulse where the primary mode no longer
corresponds to a single photon excitation of the TLS at the midband condition. Instead
the optimal pulse is in the strong field limit where higher order processes are dominant.
Moreover the optimal pulse is complex and asymmetric. An intriguing feature of this system
is that the optimal control procedure acts on the parameters Ac (t), As (t) which are harmonic
functions, and the form of the vector potential A(t) is deduced from them. This leads to a
multitude of solutions for A(t) â A(t) + 2nĎ (n is an integer) corresponding to the same
functions As (t) and Ac (t). The result for A(t) presented in Fig. (5.2) is chosen to be the
most continuous amongst this family of solutions.
In Fig.5.3 we present several quantities to measure the time evolution of the real space and
112
momentum space occupancies of the CDW model. The first panel shows that the occupancy
of the valence band (nâ (t)) and conduction band (n+ (t)) are almost completely inverted
by the optimal pulse. This is also evident in the real space occupancies, as in the initial
state the electron occupancy on sublattice A (nA (t)) is much lower than that on sublattice
B (nB (t)) due to the higher potential (U > 0) on the A sublattice. The order parameter
âŚ(t) = (nB (t) â nA (t))/(nB (t) + nA (t)) is presented in the second panel showing that the
optimal laser pulse fully âmelts" the electronic CDW state producing an almost complete
inversion of the sublattice occupancy. The third panel gives the total energy of the electronic
system as a function of time (hH(t)i), illustrating the fact that the energy in the electronic
system is not monotonically increasing as the ensemble of Rabi oscillators in the model both
absorb and emit photons into the radiation field.
5.5
Conclusion
To conclude, we developed a new QOCT for tight binding models. Effective laser pulses
have been found for the Photo-induced phase transitions (PIPT) in non-equilibruim timedependent charge density wave (CDW) systems based on a promising Quantum optimalcontrol theory (QOCT). The optical and structural properties as well as the temporal evolution of such states provide insight into the mutual dependence of electronic and atomic
structure. Through simulations based on non-equilibrium Keldysh Greenâs functions we find
that the optimal laser pulse can achieve population inversion PIPT. This novel study can
apply to many other complex materials and may-body dynamic problems.
113
A c (t)
A s (t)
A(t)
â
Energy
/ t of
Energy
(unit
tâ)
1.0
(a)
0.5
0.0
0.5
1.0
1.0
(b)
0.5
0.0
0.5
1.0
10
(c)
5
0
5
10
40
20
0
20
40
60
80
3
0.368
2
0.135
1
0.050
0
0.018
-1
0.007
0.003
-2
-3
-40
-20
0
20
40
e (h/ of
t â)h/ t â )
TimTim
e (unit
60
80
Figure 5.2: Time-resolved PES, with the optimal pulse interacting with the system from
time t = 0 to t = 20 h/tâ . The initial state is a filled valence band at temperature T = 0,
and the goal is to excite all of the states in the conduction band. The electron density at
a given energy is plotted in false color and is high near the band edges as this model has
a square root singularity in the density of states at both the valence band and conduction
band edges. In this calculation Îť(t) = 2Ďh/â is a constant, with â = 0.02 and h = 10.
114
n Âą (t)
0.50
n+
n-
(a)
0.25
(t)
0.00
0.5 (b)
0.0
0.5
H (t)
0.4
(c)
0.0
0.4
40
20
0
20
40
60
80
â
TimTim
e (unit
e (h/oft âh/) t )
Figure 5.3: Properties of the system with the optimal pulse acting on the system from time
t = 0 to t = 20 h/tâ (a) The average occupancy of the conduction and valence bands nÂą (t),
(b) The real space order parameter âŚ(t), which measures the difference in occupancy of the
two sublattices, (c) The energy hH(t)i of the CDW system.
115
CHAPTER 6
SUMMARY AND OUTLOOK
This thesis presents two main contributions to the field of photo-induced topological states
of matter. (i) A new variant of the SSH model is utilized to show how the interplay of
polarized light and lattice structure can lead to tunable topological properties. (ii) A new
quantum optimal control method was developed and demonstrated for a simple model of
charge density wave response at femtosecond time scales.
Moreover, we find that short optical pulses can lead to local spectral and novel pseudospin
textures in one-dimensional topological insulators such as the SSH model. Pump-probe photoemission spectroscopy can probe these states and can demonstrate tunable energy gaps
and Floquet band formation on femtosecond time scales. Analyzing band structures and
pseudospin textures, we identify new states with optically induced nontrivial changes of sublattice mixing that leads to novel topological phenomenon. This study reveals the possibility
to discover new topological phases driven by optical pulses in many insulators.
As a first step toward optimal control of topological states, effective laser pulses have been
found for the photo-induced phase transitions (PIPT) in non-equilibruim time-dependent
charge density wave (CDW) systems based on a new and promising Quantum Optimal
Control Theory (QOCT). Through simulations based on non-equilibrium Keldysh Greenâs
function we find the optimal laser pulse can achieve a high degree of population inversion.
This new QOCT approach is applicable to many other complex materials and many-body
dynamic problems.
Future work has a variety of possible directions: One can include disorder terms into
the tight-binding models [94, 95]. The effect of disorder may localize electrons and induce
116
Anderson localisation. Topological order may be influenced by the disorder effects if it is
large enough to break the topologically protected states. Moreover, the interaction between
light and disorder is also an interesting problem. Using disorder to describe noise in the
system can also lead to real applications in quantum devices.
Adding phonon effects to the tight binding models is already a very vivid field in physics.
The effect of electron-phonon interactions on optical absorption spectra requires a special
treatment. By incorporating key phonon modes into the charge density wave response
or in topological materials, many novel responses and new physics have been discovered
[96, 97, 98, 99].
Tuning these response with optimized light pulses is a promising future direction. Interaction effects between electrons is important for many charge density wave systems and for
some topological orders so this is an important and challenging topic to study [100, 101, 102].
By introducing the Hubbard model, which is an approximate model used in solid state physics
to describe the transition between conducting and insulating systems, one can discuss electron behaviors with Hubbard interaction and discover new phenomena in charge density
wave and topological phases; and the effect of light on these phenomena.
Finally quantum optimal control is important for operating quantum dots in quantum
computing and advanced quantum devices. One of the possible applications is to control
Majorana edge states in the time domain, and in general to understand and control FloquetBloch topological states of matter [103, 104].
117
CHAPTER 7
EXPERIMENTS AND APPLICATIONS
Recently many novel experiments have been designed for studying the field of photo-induced
phase transitions and topological states of matter. In this Chapter a brief overview of four
very different experimental systems are described. In Ref. [7], they used trARPES to study
photo-induced band gaps. When the photon energy of the laser is less than 300 meV, the coherent interaction between light and the TI surface states is the main effect. Therefore they
used polarized photons at midinfrared (MIR) wavelengths to investigate the photon-dressed
surface states in Bi2 Se3 . The laser pulses are focused to a 300 mm diameter spot on the
single-crystal Bi2 Se3 sample at an angle of 45âŚ , and the pulses are tunable in wavelength
from 4 mm to 17 mm with 1 mJ peak energy; with the amplitude of the electric field to be
E0 = 2.5 (T1) Ă 107 V/m and the pulsewidth estimated to be 250 fs (FWHM) on the surface
of the sample: Fig. 7.1 shows energy-momentum spectra of Bi2 Se3 obtained at several time
delays after arrival of the intense linearly polarized MIR excitation. Note that the probe
pulse is a linearly polarized ultraviolet (UV) pulse.
Figure 7.1: Angle-resolved photoemission spectra (APRES) of Bi2 Se3 . (A) A sketch of the
experimental geometry for the p-polarized case. kx is defined to be the in-plane electron
momentum parallel to the pump scattering plane. (B to F) ARPES data for several pumpprobe time delays t (values indicated in the figure) under strong linearly polarized midinfrared (MIR) excitation of wavelength l = 10 mm. Figure and caption are taken from
[7].
118
In Ref. [8], arrays of evanescently coupled dielectric-loaded surface-plasmon polariton
(SPP) waveguides was used to study the SSH model. The arrays are built by negativetone gray-scale electron beam lithography on top of a chromium (10 nm) and gold (60 nm)
coated glass substrate. The waveguides consist of polymethylmethacrylate (PMMA) ridges
with a height of 140 nm and a width of 250 nm. They used weak and strong bonds (1000
and 600 nm) as in the SSH model by alternating different separations between neighboring
waveguides, as depicted in Fig. 7.2, giving in a = 1600 nm for the SSH unit cell separation
distance. These geometrical parameters ensure, for the vacuum wavelength Îť = 980 nm,
single-mode operation of the waveguides and sufficient coupling among them.
In Ref. [9], they studied on the imaging and probing of topological bound states in
the SSH model through adiabatic preparation, quench dynamics and phase-sensitive injection by using an atom-optics realization of lattice tight-binding models. With an optical
lattice potential formed by lasers of wave number k = 2Ď/Îť and wavelength Îť = 1064
nm, momentum-space dynamics of Rb87 condensate atoms was started through controlled,
time-dependent driving. The lasers couple 21 discrete atomic momentum states coherently,
inducing a momentum-space lattice of states such that atomic population may change. The
momentum states are defined by momenta pn = 2n~k and site indices n(relative to the
lowest momentum value). Through 20 distinct two-photon Bragg diffraction processes, the
coupling between these states is fully controlled, letting them to simulate tight-binding models with local, arbitrary and time-dependent control of all tunnelling terms and site energies.
Fig. 7.3 is the absorption images taken after 760Âľs(â 0.78h/t) of evolution following the
initialization and quench, corresponding to phases of Ď (top) and 0 (bottom), respectively,
for â/t = 0.38(1).
In Ref. [10], photonic waveguide arrays were used (see Fig. 7.4), which are an exceptional system for implementing PT-symmetric non-hermitian Hamiltonians. By wiggling
119
Figure 7.2: (a) Waveguide array fabricated out of PMMA on top of a Cr- and Au-coated
glass substrate. Alternating center-to-center separations, 600 and 1000 nm, implement the
bulk SSH model. (b) Plasmonic waveguide array incorporating a topological defect where
the long separation is repeated twice. Three different excitation sites, I, II, and III, are
highlighted. Figure and caption are taken from [8].
the waveguides, which causes radiative loss (coupling to the continuum states), the loss in
photonic lattices can be precisely designed. The frequency and amplitude of the wiggling
can be tuned to generate a particular loss.
The four examples above are a small subset of experimental systems where photo-induced
phase transitions and photo-induced topological properties are being explored; as outlined
in Chapter 1.
120
Figure 7.3: Absorption images taken after 760Âľs(â 0.78h/t) of evolution following the
initialization and quench, corresponding to phases of Ď (top) and 0 (bottom), respectively,
for â/t = 0.38(1). Figure and caption are taken from [9].
Figure 7.4: (a) Sketch of the passive waveguide array acting like a PT-symmetric structure
with a topological interface. (b) End facet of the experimentally realized structure in fused
silica glass. Image courtesy of Mark Kremer, FSU Jena. Figure and caption are taken from
[10].
121
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