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NONLINEAR RESONANT PHENOMENA
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Christian Hicke

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NONLINEAR RESONANT PHENOMENA
IN MULTILEVEL QUANTUM SYSTEMS

By

Christian H icke

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department Of Physics and Astronomy

2008

ABSTRACT

NONLINEAR RESONANT PHENOMENA
IN MULTILEVEL QUANTUM SYSTEMS
By
Christian Hicke

We study nonlinear resonant phenomena in two-level and multilevel quantum
systems. Our results are of importance for applications in the areas of quantum
control, quantum computation, and quantum measurement.

We present a method to perform fault-tolerant Single—qubit gate operations using
Landau-Zener tunneling. In a single Landau-Zener pulse, the qubit transition fre-
quency is varied in time so that it passes through the frequency of a radiation ﬁeld.
We Show that a Simple three-pulse sequence allows eliminating errors in the gate up
to the third order in errors in the qubit energies or the radiation frequency.

We study the nonlinear transverse response of a spin S > 1/2 with easy-axis
anisotropy. The coherent transverse response displays sharp dips or peaks when the
modulation frequency is adiabatically swept through multiphoton resonance. The
effect is a consequence of a certain conformal property of the spin dynamics in a
magnetic ﬁeld for the anisotropy energy oc SE. The occurrence of the dips or peaks
is determined by the spin state. Their shape strongly depends on the modulation

amplitude. Higher—order anisotropy breaks the symmetry, leading to sharp steps in

the transverse response as function of frequency. The results bear on the dynamics
of molecular magnets in a static magnetic ﬁeld.

We Show that. a modulated large-spin system has special symmetry. III the pres-
ence of dissipation it leads to characteristic nonlinear effects. They include abrupt
switching between transverse magnetization branches with varying modulating ﬁeld
without hysteresis and a speciﬁc pattern of switching in the presence of nmltistability
and hysteresis. Along with steady forced vibratirms the transverse spin components
can display transient vibrations at a combination of the Larmor frequency and a
slower frequency determined by the anisotropy energy. The analysis is based on a mi-
croscopic theory that takes into account relaxation mechanisms important for Single-
molecule magnets and other large-spin systems. we ﬁnd how the Landau-Lifshit-z
model should be modiﬁed in order to describe the classical Spin dynamics. The oc-
currence of transient oscillations depends on the interrelation between the relaxation
parameters.

We extend the analysis to the quantum regime by developing a formalism which
allows to transform the system’s quantum kinetic operator equation into a partial
differential equation of motion of the system’s probability density distribution in the
spin coherent state representation. Using the spin density distribution, we analyze
the quantum corrections of classical limit cycles. We Show that the stationary dis—
tribution of the system coincides with the positions of stable stationary states in
the semiclassical limit. We discuss the mechanism of quantum activation and Show
that it leads to switching in the system, where the transition between globally stable
states happens in a comparatively small range of the parameter space. we analyze
the quantum behavior of switching in the absence of hysteresis and Hamiltonian-like

dynamics.

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my advisor and teacher Mark Dykman
for his outstanding support during the last ﬁve years at Michigan State University.
Apart from having been a constant source of insight into scientiﬁc techniques at the
highest level, Mark was always very supportive in personal matters. for which I am
very grateful.

I would also like to thank my Dissertation Committee members, Norman Birge,
Bhanu Mahanti, Carlo Piermarocchi, and C.-P. Yuan, who have been enthusiastic
about my work and helpful with their comments.

The staff at the Physics and Astronomy Department and at the Office for Inter-
national Students and Scholars helped to make the stay at Michigan State University
a very pleasant experience.

Many thanks go to Dmitry Ryvkin and Lea Santos, with whom I was lucky to
Share the oﬂice as well as to my long-time roommate, Johannes Grote, without whom
the time in Michigan would have been only half the fun.

Special thanks go to my wife, Mina, my son, h-‘Iarvin, and my parents, Gabriele
and Hans-Georg, my brothers, Martin and Konstantin, my parents—in-law, Young Ja
Kim and Suk Doo Yoon, and all my friends and family members in East Lansing, back
in Germany, and in many other corners of the world for all the beautiful moments in

between.

iv

TABLE OF CONTENTS

LIST OF FIGURES vii
1 Introduction 1
2 Fault-Tolerant Landau-Zener Quantum Gates 9
2.1 Introduction .................................. 9
2.2 Landau-Zener transformation in the modiﬁed adiabatic basis ....... 12
2.3 Rotation matrix representation ....................... 16
2.4 Composite Landau-Zener pulses ....................... 18
2.4.1 Error compensation with 7r-pulses ..................... 19
2.4.2 Maximal error of the three-pulse sequence ................ 22
2.5 Conclusions .................................. 25
3 Multiphoton Antiresonance in Large-Spin Systems 28
3.1 Introduction .................................. 28
3.2 Low-ﬁeld susceptibility crossing ....................... 32
3.2.1 The quasienergy spectrum ......................... 32
3.2.2 Susceptibility and quasienergy crossing .................. 35
3.3 Antiresonance of the transverse multiphoton response ........... 36
3.3.1 Two-photon resonance ........................... 40
3.4 Susceptibility crossing for a. semiclassical spin ............... 41
3.4.1 Conformal property of classical trajectories ............... 43
3.4.2 The WKB picture in the neglect of tunneling .............. 44
3.5 Degeneracy lifting by higher order terms in S 3 ............... 46
3.6 Conclusions .................................. 47
4 Hysteresis, Transient Oscillations, and Nonhysteretic Switching in
Resonantly Modulated Large-Spin Systems 52
4.1 Introduction .................................. 52
4.2 The model ................................... 56
4.2.1 Rotating wave approximation ....................... 58
4.2.2 Quantum kinetic equation ......................... 59
4.3 Classical motion of the modulated spin ................... 61
4.3.1 Stationary states in the rotating frame for weak damping ........ 63
4.3.2 Saddle-node bifurcations .......................... 65
4.3.3 Periodic states and Hopf bifurcations ................... 68
4.4 Hamiltonian—like motion at exact resonance ................ 69
4.5 Spin dynamics in the absence of limit cycles ................ 71

4.5.1 Hysteresis of spin response in the absence of limit cycles ........ 72

4.5.2 Interbranch switching without hysteresis ................. 74
4.6 Spin dynamics in the presence of limit cycles ................ 75
4.6.1 Phase portrait far from the astroid .................... 77
4.6.2 Other bifurcations of limit cycles ..................... 79
4.6.3 Hysteresis of spin response in the presence of limit. cycles ........ 82
4.7 Conclusions .................................. 85
5 Quantum-Classical Transition and Quantum Activation in Modu-
lated Large-Spin Systems 88
5.1 Introduction .................................. 88
5.2 Dynamics of modulated large-spin systems ................. 90
5.2.1 Some properties of spin coherent states .................. 91
5.2.2 The master equation in the spin coherent state representation and its
semiclassical limit ......................... 92
5.2.3 Dynamics of the system in the presence of limit cycles ......... 96
5.3 The stationary limit of modulated large-spin systems ........... 101
5.3.1 The stationary distribution in the quasienergy representation ..... 102
5.3.2 Switching and hysteresis .......................... 106
5.3.3 Uniform distribution of quasienergy levels and non-hysteretic switching 108
5.4 Conclusions .................................. 1 10
6 Conclusions 114
APPENDICES 119
A Symmetry Of classical spin dynamics: a feature Of the conformal
mapping 120
B Energy change near a Hopf bifurcation 123
C Spin Coherent State Representation of the Master Equation 126
BIBLIOGRAPHY 129

vi

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3.1

3.2

3.3

3.4

3.5

3.6

3.7

4.1

4.2

4.3

4.4

LIST OF FIGURES

System of spatially separated qubits with individually tunable energies

Landau-Zener transitions ...........................
Trajectory of system on modiﬁed Bloch sphere ...............
Rotation angles as function of the control parameter ...........
Landau-Zener composite pulse sequence ..................
Gate error of Landau-Zener pulse sequences ................

Gate error of composite on-resonance pulse sequences ...........

Three-photon resonance in a spin-2 system .................
Quasienergies of a spin-2 system as a function of the detuning ......
Level anticrossing and antiresonance of the susceptibilities ........
Scaling of the multiphoton susceptibility splitting .............
Quasienergy surface in the semiclassical limit ...............
Susceptibility of a spin with quartic anisotropy ..............

Independence of hysteresis steps on transverse ﬁeld ............

Phase portraits of the spin ..........................
Saddle—node bifurcation lines in the limit of zero damping ........
Hysteresis of spin response in the absence of periodic states ........

Non-hysteretic switching ...........................

vii

10

14

17

27

30

34

38

39

42

51

64

67

74

75

4.5 Saddle—node and Hopf bifurcation lines ................... 76

4.6 Full bifurcation diagram in the zero damping limit ............ 80
4.7 Hopf, saddle-node, and saddle—loop bifurcation point ........... 81
4.8 Hysteresis of the spin dynamics in the presence of limit cycles ...... 83
5.1 Spin density distribution of a limit cycle in a spin 10 system ....... 97
5.2 Spin density distribution of a limit cycle in a spin 30 system ....... 98
5.3 Cross section through spin density distribution (spin 10) ......... 99
5.4 Cross section through spin density distribution (spin 30) ......... 100
5.5 Stationary distribution over quasienergy levels I .............. 102
5.6 Hysteresis of the magnetization for different values of S (ﬁxed points) . . 104
5.7 Hysteresis of the magnetization for different values of S (limit cycles) . . 105
5.8 Switchng in the presence of hysteresis ................... 107
5.9 Hysteresis in the case of small damping ................... 108
5.10 Stationary distribution over quasienergy levels II ............. 109
5.11 Switching in the absence of hysteresis .................... 110
5.12 Spin density distribution in the range of non-hysteretic switching . . . . 111
A1 Contour of integration ............................ 122

viii

Chapter 1

Introduction

The investigation of modulated multilevel quantum systems with almost equidistant
energy levels has attracted much theoretical and experimental interest in recent years.
In such systems damping is often weak and even a comparatively small resonant ﬁeld
can lead to interesting nonlinear effects. This happens because the ﬁeld is in resonance
with many transitions at a time. Due to the nonlinear character of these multilevel
systems, different states of forced vibrations can coexist. Quantum and classical
ﬂuctuations cause transitions between the stable states. This makes the analysis of
modulated systems far more complicated than Of systems at thermal equilibrium.
Their dynamics remains poorly understood. It. is therefore important to improve the
theoretical understanding of these systems in order to ﬁnd new effects, explain the
existing experiments, give guidance to new experiments, and to ﬁnd new applications.

The multilevel character of the energy spectrum leads to often unexpected and
unusual coherent and incoherent effects. Coherent effects are of immediate interest
for quantum control and quantum computation. Understanding them signiﬁcantly
extends the current theory of coherent quantum processes, which is largely focused

on the dynamics of two-level systems. At the same time, understanding incoherent

effects, and in particular phenomena related to switching between classically stable
vibrational states, would substantially contribute to the broad ﬁeld of quantum
physics far from thermal equilibrium. It has become clear recently that the mech-
anism of interstate switching opens a new and very promising approach to sensing

and in particulat to quantum measurements.

In this study we concentrate our investigation on single-spin systems which
are modulated by an external radiation ﬁeld. we start with the analysis of a spin
S = 1 / 2 system and investigate how its operation as a quantum gate can be made
robust under the influence of an environment that exerts low frequency noise on
the qubit. We then move to large spin systems, where S > 1 / 2, and investigate
multiphoton resonant phenomena and nonlinear dissipative effects which play an

important role in the research ﬁeld of molecular magnets and other large spin systems.

In many proposed implementations of a quantum computer single-qubit opera-
tions are performed by applying pulses of radiation. These pulses cause resonant
transitions between qubit states, that is between the two states of a two—level
system. The operation is determined by the pulse amplitude and duration. In
many proposals, particularly in the proposed scalable condensed—matter based
systems [1], control pulses will be applied globally, to many qubits at a time. A
target qubit can be addressed individually by tuning it in resonance with the
radiation. The corresponding gate operations invariably involve errors which come
from the underlying errors in the radiation frequency, amplitude, and length of the
radiation pulse as well as in the qubit transition frequency tuning.

Improving the accuracy of quantum gates and reducing their sensitivity to errors

from different sources is critical for a successful operation of a quantum computer.
Much progress has been made to reduce the impact of the errors on the gate operation
ﬁdelity by using active control and the design of composite pulse sequences [2, 3, 4.
5, 6].

Of special interest are frequency offset errors in the qubit energy and radiation
frequency, 5. Such errors come from various sources. AII example is provided by sys-
tems where the qubit-qubit interaction is not turned off, and therefore the transition
energy of a qubit depends 011 the state of other qubits. Typically, the error affects
the ﬁdelity of the gate operation linearly. The goal is to ﬁnd ways to increase the
ﬁdelity of the gate. For conventional single-qubit gate implementations, where the
qubit transition frequency, ideally, stays in exact resonance with the radiation pulse
for a speciﬁed period of time, the best known error compensating pulses still produce
errors that scale as 52 [7].

Here, we will investigate gate operations that are based on a non—standard way
of operating qubits, where the qubit transition frequency is swept through resonance
with a radiation ﬁeld. This process is well known as Landau-Zener crossing [8, 9]
and plays an important role in different areas of quantum physics. Far away from
resonance the interaction between qubit and radiation ﬁeld is weak. At the same
time, for a sufficiently broad range over which the transition frequency is swept,
even in the presence of an unknown frequency offset, the system will always go
through resonance. Therefore, one may expect that Landau-Zener crossings have an
advantage over conventional on-resonance gates. In turn, this may lead to better
composite pulses and a far more efﬁcient error correction. In this study we will show

that this is indeed the case.

Large-spin systems have been attracting much attention recently. Examples
are S = 3/ 2 and S = 5 / 2 Mn impurities in semiconductors as well as nuclear spins
I = 3/ 2 where radiation—induced quantum coherence between the spin levels was
observed [10]. The interest in nuclear spins has renewed in view of their possible use
in quantum computing [11].

An important class of large-spin systems is single-molecule magnets (SMMS).
SMMS display an extremely rich behavior and have been attracting much attention in
recent years. A variety of SMMS has already been discovered and investigated theo-
retically and experimentally [12, 13, 14, 15, 16, 17, 18, 19, 20, 21] (see Refs. [22, 23, 24]
for a review) and new systems are being found [25, 26]. Mn- and Fe—based molecular
magnets exhibit electron spins of S = 10 and higher.

In a broader sense, the interest in quantum spin dynamics comes also from ﬁeld
theory. The dynamics is closely related to the Lipkin—h’leshkov-Glick model [27, 28,
29, 30], which is used in various areas, from condensed matter physics to nuclear
theory.

Large-spin systems have a ﬁnite but comparatively large number of quantum
states. Therefore, a single system can be used to study a broad range of phenomena,
from purely quantum to semiclassical where the spin behaves almost like a classical
top. An important feature of large-Spin systems is that, in a strong static magnetic
ﬁeld, their energy levels become almost equidistant, with level spacings close to ﬁrm,
where tag is the Larmor frequency. As a result, radiation at frequency z too is reso—
nant Simultaneously for many interlevel transitions. This leads to new quantum and
classical nonlinear resonant effects.

The effects of a strong resonant ﬁeld on systems with nearly equidistant energy

levels, in the absence of dissipation, have been studied for weakly nonlinear oscillators

[31, 32, 33, 34]. However, spins are qualitatively different from oscillators. The
number of spin states is ﬁnite, 25 + 1, and the algebra of spin operators differs from
that of the oscillator operators.

An outstanding question is whether the coherent semiclassical spin dynamics
has conformal properties. Such properties may lead to interesting observable
consequences. As we show, the model with uniaxial anisotropy, which describes
many systems of current interest, indeed has such properties. This provides an
explanation to a number of experiments on molecular magnets, where an unexpected

hysteretic behavior was observed [16] (see Fig. 3.7 below).

The classical dynamics of a large—spin system in a resonant ﬁeld would be ex-
pected to have similarities with the dynamics of a modulated magnetic nanoparticle
near ferromagnetic resonance. It was understood back in the 1950’s [35, 36] that
the response near ferromagnetic resonance becomes strongly nonlinear already for
a comparatively weak radiation strength due to the magnetization dependence
of the effective magnetic ﬁeld. The resonant response may become multivalued
as a function of the modulating ﬁeld amplitude [37, 38]. A detailed analysis of
nonlinear magnetization dynamics in uniaxial nanoparticles modulated by a strong
circularly polarized periodic ﬁeld was done recently [39]. These studies as well as
many other studies of magnetization dynamics in ferromagnets were based on the
phenomenological Landau—Lifshitz-Gilbert equation.

In contrast to magnetic nanoparticles, for large—Spin systems quantum effects are
substantial. A distinction which remains important in the classical limit. concerns
relaxation mechanisms. Spin relaxation occurs via transitions between discrete energy

levels with emission, absorption, or inelastic scattering of excitations of a thermal

reservoir to which the spin is coupled. Relevant relaxation mechanisms depend on
the speciﬁc system but as we Show, even in the classical limit relaxation is generally
not described by the Landau-Lifshitz damping. As a result the classical spin dynamics
strongly differs from the dynamics of a magnetic nanoparticle.

The microscopic analysis of relaxation is Simpliﬁed by the near equidistance of
the energy levels in a strong static magnetic ﬁeld. As a consequence, excitations
of the thermal bath emitted, for example, in transitions within different pairs of
neighboring levels have almost the same energies. Therefore, relaxation is described
by a small number of constants independent of the form of the weighted with
the interaction density of states of the bath, and the analysis applies for an ar-

bitrary ratio between the level nonequidistance and their relaxational broadening [40].

Large—spin systems are of great interest for the study of quantum to classical
transitions. The inverse size of the spin, 1/S, can be seen as an effective Planck
number which controls the “quantumness” of the system. Of particular interest
in this context is hysteresis in a modulated large spin system, which occurs in the
classical limit.

Switching between coexisting stable states underlies many phenomena in physics,
from diffusion in solids to protein folding. For classical systems in thermal equilib-
rium switching is often described by the activation law, with the switching probability
being W oc exp (—AU/kT), where AU is the activation energy. As temperature is
decreased, quantum fluctuations become more and more important, and below a cer-
tain crossover temperature switching occurs via tunneling [41, 42, 43]. The behavior
of systems away from thermal equilibrium is far more complicated. Still, for classical

systems switching is often described by an activation type law, with the tempera-

ture replaced by the characteristic intensity of the noise that leads to ﬂuctuations
[44, 45, 46, 47, 48, 49, 50, 51, 52]. Quantum nonequilibrium systems can also switch
via tunneling between classically accessible regions of their phase space [32, 53, 33, 54].

De tay of a metastable state is usually considered as resulting from tunneling or
thermal activation. Besides classical activation and quantum tunneling, nonequi-
librium systems have another somewhat counterintuitive mechanism of transitions
between stable states. This mechanism is called quantum activation and has been
explained recently for the cases of a parametrically driven oscillator [55] and a non-
linear oscillator [56]. It describes escape from a metastable state due to quantum
fluctuations that accompany relaxation of the system [57]. These fluctuations lead
to diffusion away from the metastable state and, ultimately, to transitions over the
classical ”barrier”, that is, the boundary of the basin of attraction of the metastable
state in phase space. Quantum activation is often more probable than tunneling even
at zero temperature.

The counterintuitive nature of the effect of quantum activation requires studying
whether this effect occurs in systems other than an oscillator. Large—spin systems
provide an ideal example. They are particularly interesting and challenging because
they may display features that have no counterpart for oscillators. One of such
features is the onset of periodic states in the rotating frame, which has been previously
found in the phenomenological analysis of magnetic nanoparticles [39].

It is a challenge to ﬁnd limit cycles for large spins in the classical limit, using
a microscopic model of coupling to a bath, and to determine the conditions where
they emerge. The natural next step is to investigate the smearing of the cycles by
quantum fluctuations and the study of quantum activation in systems with limit

cycles.

The thesis is organized as follows. I11 cha )ter 2 we develo a theorr that al—
C
lows us to analyze single-(11bit o erations where the (ubit transition fre uenc '
. o l
w0(t) is swept through the frequency of the resonant ﬁeld tap and by that inducing
Landau-Zener tunneling in the qubit. we propose a composite Landau-Zener pulse
sequence and demonstrate that it compensates energy Offset errors to a much higher
decree than conventional uantum ates. In cha)ter 3 we stud r the uasienergv
0 Cu
spectrum and the response of a modulated large spin with quadratic in S 2 anisotropy
energy. We Show that, at multiphoton resonance, the susceptibilities in the resonating
states display an interesting coherent antiresonant behavior. In chapter 4 we study
the dynamics of such a spin in the semiclassical limit in the presence of relaxation
and present a stability analysis which predicts a rich hysteretic behavior. In chapter
5 we extend the analysis from the semiclassical limit to the the quantum regime by
introducing a formalism that makes use of spin coherent states and the stationary
quas1energy dlstrIbutlon. e explain how class1ca.l lImIt cycles are mamfested In the
quantum regime and show that in modulated large spin system switching occurs via

quantum activation.

Chapter 2

Fault-Tolerant Landau-Zener

Quantum Gates

2. 1 Introduction

In many proposed implementations of a quantum computer (QC) single-qubit op-
erations are performed by applying pulses of radiation. The pulses cause resonant
transitions between qubit states, that is between the states of the system that com-
prises a qubit. The operation is determined by the pulse amplitude and duration. In
many proposals, particularly in the proposed scalable condensed-matter based sys-
tems [1], control pulses will be applied globally, to many qubits at a time. A target
qubit is chosen by tuning it in resonance with the radiation. The corresponding gate
operations invariably involve errors which come from the underlying errors in the
frequency, amplitude, and length of the radiation pulse as well as in the qubit tuning.

Improving the accuracy of quantum gates and reducing their sensitivity to errors
from different sources is critical for a successful operation of a QC. Much progress has

been made recently in this direction by using radiation pulses of special shape and

composite radiation pulses [2]. In the analysis or resonant pulse shape it is usually
assumed that the qubit transition frequency is held constant during the pulse.

An alternative approach to single-qubit operations is based on Landau-Zener tun-
neling (LZT) [8, 9]. In this approach the qubit transition frequency w0(t) is swept
through the frequency of the resonant ﬁeld w F [58]. The change of the qubit state
depends on the ﬁeld strength and the speed at which w0(t) is changed when it goes
through resonance [59]. The LZT can be used also for a two-qubit operation in which

qubit frequencies are swept past each other leading to excitation swap [58, 60, 61].

TTTT”

\J“

J-L-__ __________
I [are [are “830) was)
I.

 

 

Figure 2.1: System of spatially separated qubits with individually tunable energies.
In between operations transition frequencies are detuned to avoid excitation hopping.

In this chapter we study the robustness of the LZT-based gate operations. We
develop a simple pulse sequence that is extremely stable against errors in the qubit
transition frequency or equivalently, the radiation frequency. Such errors come from
various sources. An example is provided by systems where the qubit-qubit interaction
is not turned off, and therefore the transition energy of a qubit depends on the state
of other qubits. Much effort has been put into developing means for correcting them
using active control [4, 5, 6].

An advantageous feature of LZT is that the change of the qubit state populations

10

depends on the radiation amplitude and the speed of the transition frequency change
(1)0, but not on the exact instant of time when the frequency coincides with the
radiation frequency, w0(t) = cup. However, the change of the phase difference between
the states depends on this time. Therefore an error in mg or as F leads to an error in
the phase difference, i.e., a phase error. This error has two parts: one comes from the
phase accumulation before crossing the resonant frequency, and the other after the
crossing. Clearly, they have opposite signs.

A natural way of reducing a phase error is to make the system accumulate the
appropriate opposite in sign phases before and after the “working” pulse. To do this,
we ﬁrst apply a strong radiation pulse that swaps the states, which can be done with
exponentially high efﬁciency using LZT. Then we apply the “working” pulse, and
then another swapping pulse. The swapping pulses effectively change the sign of the
accumulated phase. As we Show, by adjusting their parameters we can compensate
phase errors with a high precision.

In Sec. II below we give the scattering matrix for LZT in a modiﬁed adiabatic
basis which turns out to be advantageous compared to the computational basis. The
scattering matrix describes the quantum gate. In Sec. III it is presented in more
conventional for quantum computation terms of the qubit rotation matrix. In Sec. IV,
which is the central part of the chapter, we propose a simple composite Landau—Zener
(LZ) pulse and demonstrate that it efﬁcierrtly compensates energy offset errors even

where these errors are not small. Sec. V contains concluding remarks.

11

2.2 Landau-Zener transformation in the modiﬁed

adiabatic basis

A Simple implementation of the L2 gate is as follows. The amplitude of the radiation
pulse is held ﬁxed, while the difference between the qubit transition frequency and

the radiation frequency
A = A(t) 2 cup — w0(t) (2.1)

is swept through zero. If w0(t) is varied slowly compared to tap, i.e., [wol << (4112;, the
qubit dynamics can be described in the rotating wave approximation, with Hamilto-
nian
A/2 7
H = H (t) = . (2.2)
’7 -A/2

Here, 7 is the matrix element of the radiation-induced interstate transition. The

Hamiltonian H is written in the so-called computational basis, with wave functions

1
[0) = and [1) =
0 1
We assume that well before and after the frequency crossing the values of [Al

largely exceed 7 and A slowly varies in time, [A/A2l << 1. Then the wave func—
tions of the system are well described by the adiabatic approximation, i.e., by the

instantaneous eigenfunctions of the Hamiltonian (2.2),

cos(6/2) —Sin(6/2)
we = , It’s) = , (2-3)
sin(6/2) cos(9/2)
1/2
_ .—1 IAI _ A: .2
6—(sgnA)cos 2E’ E—(4 +x) ,

where A E A(t) and (—1)"EsgnA is the adiabatic energy of the states W’n) =

|¢0,1>- The adiabatic approximation for E and 6 is accurate to 72A/A3 and 'yA/A3,

12

respectively.

 

In contrast. to the standard adiabatic approximation, we chose the states 7,001) and
their energies in such a way that [1,1'0) and Idl'l) go over into [0) and |1), respectively,
for |A|/y —> 00. As a result 6 is discontinuous as a function of A for A = 0, but the
adiabatic approximation does not apply for such A anyway.

For the future analysis it is convenient to introduce the Pauli matrices X, Y, Z in

the basis (2.3), with
Zl¢”72> = (1 “ 270M570, Xl’lv’ffrd : [Uh—n) (n = 0a 1):

and Y = iX Z. In these notations, the operator of the adiabatic time evolution

t
U(tf, ti) = Texp[—if,if dtH(t)] has the form

, (2-4)

'i

t
L7(tf,t,) = exp [—i (sgnA) Z / f E(t)dt
t

 

with sgnA E sgnA(t,-) E sgnA(tf) [the Sign of A(t) is not changed in the range
where Eq. (2.4) applies].

The LZ transition can be thought of as occurring between the states (2.3). Fol-
lowing the standard scherne [8, 9] we take two values A12 of A(t) such that they have
opposite signs, A1A2 < 0. We choose [A12] sufﬁciently large, so that the adiabatic
approximation (2.3) applies for A(t,:) = A,,i = 1,2. At the same time, [A12] are
sufﬁciently small, so that A(t) can be assumed to be a linear function of time between

A1 and A2,

A(t) % —-r/(t —— ta), 7} = —A(t(;), (2.5)

where the crossing time tc is given by the condition AUG) 2 0. The adiabaticity for

t = t1,2 requires that [A12] >> ’7, 711/2. We will consider the LZ transition ﬁrst for

the case A1 > 0 and A2 < 0, when 77 > 0.

13

The modiﬁed adiabatic basis (2.3) is advantageous, because in this basis the tran-
sition matrix S has a particularly Simple form. For A(t) of the form (2.5) the error
in S is determined by the accuracy of the adiabatic approximation itself and is of
order y/n2lt13 — tc|3, in contrast to the computational basis, where the error is
~ 0(7/77It1,2 — tel). This latter error is comparatively large for the values of 'y/ |A1,2|
of interest for quantum computing. It leads to the well-known oscillations of the tran-

sition amplitude with increasing [AI [59], whereas in the basis (2.3) such oscillations

do not arise, see Fig.2.2.

 

 

 

 

 

1.0 : ‘ , yyyygft
a? i 3: ‘i‘!c)(
E [ (f Y, t,(¢£
(U ‘ s'
‘8 [32’
g. 0.5"“ f "Tfftfl
9 l
5
E ][ ﬁ’vftltrr"

0.0 "‘11:

-1O 0 10

1/2
(Ho) 11

Figure 2.2: Landau-Zener transitions MO) ——> W1) and [0) ——> [1) in the modiﬁed
adiabatic basis (2.3) and in the computational basis for linear A(t) (2.5). Solid and
dotted lines show the squared amplitude of the initially empty states WI) and |1),
respectively. The lines refer to g = 1, 0.47, 0.33, and 0.21, in the order of decreasing
transition probability for (t — tc)n1/2 = 10. As long as A(t) is large and negative,
the system stays in the initially occupied adiabatic state Mg), and therefore the solid
curves for different 9 cannot be resolved for (t -— tc)171/2 g —-—1. For large (t -- tc)n1/2

2
the solid lines quickly approach the Landau-Zener probabilities 1 — 6'2” .

The energy detuning |A1,2| cannot be made too large, because this would make the

14

gate operation long. If we characterize the overall error of the adiabatic approximation
as the sum 23,-=12 7/772lt,-_ — tel3 and impose the condition that the overall duration of
the operation t2 — 151 be minimal, we see that the error is minimized when the pulses
A(t) are symmetrical, t2 — tc = tc — t1, i.e., [A1] = [A2].

The matrix S (t2, t1) 2 S in the basis (2.3) can be obtained using the parabolic
cylinder functions that solve the Schriidinger equation with the Hamiltonian (2.2),

(2-5).

511 512
S(t2a t1) = 7 (26)

521 S22

511: 0X1) [—7T92 + “992 — 991)],

(27r)1/2 7r 2 ,7r ,
S = — 34 [—— — .- 'y ],
12 gF (1.92) WP 29 14 + 1091 + 992)
(2701/2 7r 2 .vr .
S =————-. [—- +1—-2'"+" ],
21 gr (_igz) p 29 (4 [(WI 9’2)

522 = exp [4.92 - W2 - 991)] ,

where F(.T) is the gamma. function.

The dimensionless coupling parameter g = y/l'r][1/2 in Eq. (2.6) is the major
parameter of the theory, it determines the amplitude of the [11571) —+ [iv/IF") transition.

The phases 9912 are

 

 

A2 IA] 94l77|
’1- : -—Z + 211] ( I )+ . 1:1, 2. 2.7
M 4W 9 [rill/2 2A.?) “ ( )

Here we have disregarded the higher order terms in |A132|_1. The constants in 991,2 are
chosen so as to match the corresponding constants in the parabolic cylinder functions
[62].

The matrix S for a transition from the initial state with A1 < 0 to the ﬁnal state

with A2 > 0 is given by the transposed matrix (2.6) in which the phases cpl and 992

15

are interchanged. In this case r} < 0 in Eq. (2.5); the expressions for 4,912 and 9 do

not change.

2.3 Rotation matrix representation

The LZ transition can be conveniently described using the standard language of gate
operations in quantum computing. To do this we express the transition matrix in
terms of the operators RX(6) = exp(—i6X/2) and R209) 2 exp(—i9Z/2) of rotation
about :1: and z axes in the basis (2.3). The rotation matrices can be written using
the “adiabatic” phases ¢(t,:) that accumulate between the time t,- and the time tc at

which the levels would cross in the absence of coupling. From Eq. (2.7)

992' =¢(ti)+900 0:192), t1<tc<t2,

’2'
/ Edt
1c

where we have disregarded corrections o< |A1~2|_4, in agreement with the approxima—

1 .
, 990 = 59201192 - 1), (2-8)

 

 

tions made in obtaining Eq. (2.6).
For the case A1 > 0 > A2 the dependence of the transition matrix S (2.6) on the

phases q’9(t1,2) has the form
S(1‘2. t1) = 32 l—2¢5(t2)l SIR-z l2¢>(t1)l- (2-9)
A direct calculation shows that the matrix S’ is
S’ = R,(<I>)R,.(a)Rz(—<I>). (2.10)

The rotation angles 4), a are given by the expressions

3
(I) = —2¢0 + arg 17(ig2) + TW’ (2.11)

a = 2cos‘1 [exp(-—7rg2)] .

16

   

l1>

 

Figure 2.3: Trajectory of system in 1/10) , [tr/)1) and [0) , [1) basis, respectively.
Initial state: [iv/1,) = \/0-01 [1110) + v0.99 [1111), g = 0.5. Note that instead
of using basis (2. 3) the slightly different basis [1450) = 1/\/1 + 72/A2 (1, y/A)T

— —1/\/1 + 72/A2( —,y/A 1)T was used to draw the ﬁgure. For large values of
[7/1]>A[ this basis 1S asymptotically the same as the original basis (2.3). The beneﬁt of
using this basis is solely for illustration purposes; in contrast to using basis (2.3) the
trajectories of the system have no discontinuity at A = 0 (cf. Fig. 2.2).

A minor modiﬁcation of these equations allows using them also for the case A1 <
0 < A2 when the frequency difference is increased in time in order to bring the states
in resonance. It was explained below Eq. (2.7) how to relate the matrix S in this case

to the matrix S for A1 > 0 > A2. Following this prescription we obtain
5(t21t1) : Rz l2¢ft2) _ (1)] Rz(a)R2 l—2¢(t1) + (1)] - (2-12)

In the rotation matrix representation, the only difference from the S matrix from
the case of decreasing A(t) is that (I) and ¢(t1,2) change signs. Eqs. (2.9)-(2.12)
express the LZ transition matrix in the form of rotation operators in the basis of the
modiﬁed adiabatic states [1,00) and [1.21) (2.3). For strong coupling, exp(—7rg2) << 1,
the rotation angle a approaches 7r, which corresponds to a population swap between

the adiabatic states. It is well known from the LZ theory [8, 9] that the swap operation

17

 

 

7272

I

 

 

 

 

Figure 2.4: The rotation angles a (solid line) and Q (dashed line) in the rotation-
matrix representation of the Landau-Zener gate operation as functions of the control
parameter 9. The 7r/2 gate, a = 7r/2, requires 9 = [In 2/(27r)]‘1/2 % 0.33.

is exponentially efﬁcient, 7r —— a z 2exp(——7r92) for large 9. In the opposite limit of

weak coupling, 9 << 1, the change of the state populations is small, a z (87r)1/2g. In
addition to the change of state populations there is also a phase shift that accumulates
during an operation. The dependence of the angles a and Q on the coupling parameter

9 is shown in Fig.2.4.

2.4 Composite Landau-Zener pulses

For many models of quantum computers an important source of errors are errors in
qubit transition frequencies (so. They may be induced by a low-frequency external
noise that modulates the interlevel distance. They may also emerge from errors in
the control of the qubit-qubit interaction: if the interaction is not fully turned off

between operations, the interlevel distance is a function of the state of other qubits.

18

In addition there are systems where the interaction is not turned off at all, like in
liquid state NMR—based QC‘S. In all these systems it is important to be able to
perform single—qubit gate operations that would be insensitive to the state of other
qubits.

The rotation—operator representation suggests a way to develop fault tolerant. corn—
posite LZ pulses with respect to errors in the qubit transition frequency tag and in
the radiation frequency tap. We will assume that there is a constant error e in the
frequency difference A(t) = (.0 F — w()(t), but that no other errors occur during the
gate operation. From Eq. (2.5), the renormalization A(t) —> A(t) + 5 translates into
the change of the adiabatic energy E and the crossing time tc, with tc —> tc + 5/77.

As a result the phases O(t1,2) as given by Eq. (2.8) are incremented by

_ E(t,-)A(t,~) [Null 2

6e t: ————5+—_-——€, =1,2, 2.13
(J was): was) ( )

to second order in 6.

2.4.1 Error compensation with 7r-pulses

A simple and robust method of compensating errors in ab is based on a composite
pulse that consists of the desired pulse sandwiched between two auxiliary pulses.
Using 7r-pulses in which A(t) is linear in t, as shown in Fig. 2.5, it is possible to
eliminate errors of ﬁrst and second order in e. The goal is to compensate the factors
Rz[:l:26gb(t1,2)] in the S—matrix (2.9). We note that all other factors in S are not
changed by the energy change 5, which is one of the major advantageous features of
the LZ gate operation. A 7r-pulse is obtained if exp(7r92) >> 1, which is met already

for not too large 9: for example, exp(—7rgQ) < 10”5 for g > 1.92.

19

Disregarding corrections ~ exp(—7rg2) we can write the S-matrix for the 7r-pulse

S7,(t’,t) a: —2iXR,[2a,,(t)+2a,,(t’) -—2<I>]

E —iR,[—2a,(t) — 2a,,(t’) + 2Q[X, (2.14)

where t,t’ are the initial and ﬁnal times, and the subscript 7r indicates that. the
corresponding quantities refer to a 7r-pulse. We assume that A(t) > 0 > A(t’).

The overall gate operation is now performed by a composite pulse
520,251) = SwUfz. @502, t1)5w(f1~.ti)- (2-15)

In writing this expression we assumed that the system is switched instantaneously
between the states that correspond to the end (beginning) of the correcting pulse
and the beginning (end) of the working pulse S (t2, t1). The overall composite pulse
is Shown in Fig. 2.5. The ﬁrst and the second 7r-pulses correct the errors (Sc/9 (2.13)
in the phases c')(t1) and (Mtg), respectively. 'We show how it works for 93052). From

Eqs. (2.9), (2.14), the error in $02) will be compensated if

was) + was) — 6M) = 0.

To second order in 5, the errors (595 here are given by Eq. (2.13) with appropriate t,.

The total error will be equal to zero provided

EnU'g) _ E(?‘-2) _ En(t2) = 0.

777T Tl 7hr

[A7,(t’2)| _ [AWN A7r(t2)
7I7rE7r(tgz) 71E(t2) 777rE7r(f2)

 

 

= 0. (2.16)

 

Equations (2.16) are simpliﬁed if we keep only the lowest order terms with respect

to 72/A2, in which case E(t,j) z [A(t.,-_)[ / 2 both for the working and the correcting

20

 

 

 

Figure 2.5: An idealized composite pulse. The ﬁrst and third pulses are 7r-pulses,
the pulse in the middle performs the required gate operation. The overall pulse
compensates errors in the qubit energy to 3rd order.

pulse. This gives
77a = 277. [Arr(t,2)[ — 2IAN/2)] — An(t2) = 0. (2-17)

An immediate consequence of Eq. (2.17) is that the coupling constant 77f for the it-
pulse should exceed the value of ”y for the working pulse, because g7r 2 g and 717T > 7).
Another consequence is that the 7r—pulse amplitude should exceed that of the working
pulse. If we choose A7r so that the error of the adiabatic approximation in the 7r-pulse

3 S ”)"77/[A(t2)[3, we obtain from

 

does not exceed that of the working pulse, efﬁng / [A7r
Eq. (2.17) the condition Aﬁ(t2) Z [A(t2)[21/2(g7r/g)1/3.

We note that the correcting pulse is asymmetric, with [Aw(t52)[ > A7,(t2), 2[A(t2)[,
as shown in Fig. 2.5. Another important comment. is that the proposed simple single
pulse does not allow us to correct errors of higher order in 5. It is straightforward
to see that the equation for Aw(t2), An(t’2) that follows from the condition that the

error ~ 53 vanishes is incompatible with Eqs. (2.17). However, the terms oc (€/’)’)3

21

II 11111

 

 

contain a small factor g2y/A(t1,2)3 << 1. The higher—order terms in e/y contain
higher powers of the parameter 'y/A(t1,2). This is why compensating errors only up
to the second order in 5 turns out efﬁcient.

The analysis of the ﬁrst correcting 7r-pulse, Sr,(t1,t'1), is similar to that given
above. The amplitude of this pulse also exceeds the amplitude of the working pulse.
The duration of the correcting pulses is close to the duration of the working pulse,
for 9 ~ 1 and 9” 2.2.

The pulse sequence (2.15) is written assuming that the radiation is switched off
between the pulses and that the switching between the working and correcting pulses
is instantaneous. A generalization to a more realistic case of a nonzero switching
time is straightforward. The time evolution between the pulses can be described by
extra terms in the phases (bail), $74112), leading to the appropriate modiﬁcation of
the equations for error compensation (2.16). The analysis can be also extended to
the case where A(t) is a nonlinear function of time and the coupling 9 depends on
time. This extension requires numerical analysis; we have found for several types of

A(t), g(t) that good error correction can still be achieved with a three-pulse sequence.

2.4.2 Maximal error of the three-pulse sequence

In order to demonstrate the error correction we will consider single working pulses
S (t2, t1) with the overall phases Q — 2¢(t1,2) E 0 (mod 2a) in the absence of errors,
which we will denote by S (0)02, t1). Such pulses describe transformations between
the states (2.3) with no extra phase, that is pure X rotations. we will also choose
the correcting 7r—pulses with the overall phase 2on0) + 296(t’) — 2Q E 0 (mod 27r) in
the absence of errors, with t, 13’ being 15’ , t1 and t2, t'2 for the ﬁrst and second pulse,

respectively. Then in the absence of errors the overall gate is either not affected by

22

the correcting pulse or its Sign is changed.

The restriction on the phases provides extra constraints on the parameters of
the correcting pulses. First of all, it ”discretizes” the total duration of the pulses.
For the correcting pulses we still have a choice of Ayr(t2) and Afr(t1). They will be
chosen maximally close to [A(tg)|21/2(g7r/g)1/3 and A(t1)21/2(g7r/g)1/3, respectively,
in order to minimize the error of the adiabatic approximation (2.3) and to minimize
the overall pulse duration.

We will characterize the gate error 8 by the spectral norm of the difference of the

operator S in the presence of errors and the “ideal” gate operator S (0),
8 = [[S — 5(0)“? (2.18)

Here, [[A[[2 is the square root of the maximal eigenvalue of the operator AIA. For
uncorrected pulses S = S (t2,t1), whereas for corrected pulses S = Sc(t’2,t’l). For
simple symmetric composite pulses described below, the overall Sign of the composite
pulse is opposite to that of the original pulse in the absence of errors. In this case we
set S = —Sc(t'2, ii) in Eq. (2.18).

For uncorrected pulses we have
8 —— 21/2l1— 72,, n1. — 72 n cos 041/2 (219)
” 1 22 ’y1 ‘y2 1 '

where n,- = (cos[6q§(t.i)], sin[6¢)(t.,-)]) is an auxiliary 2D unit vector (i = 1,2). Eq. (2.18)
applies also in the case of corrected pulses, but now we have to replace in the deﬁnition

Of the HI vector
(Saber) —> 6M) — was) — 5¢7rftil (2.20)

A similar replacement must be done in the deﬁnition of the vector 112.
For small phase errors [6¢(t1,2)[ the function 5 for uncorrected pulses is linear in

the error. In particular, to ﬁrst order in e for a symmetric pulse we have [6e(t1)[ z

23

[6¢(t2)[ z 5E(t1)/r}, and 8 % 2[e[E(t1)n_1sin(a/2). In contrast, by applying the
same arguments to a corrected pulse, we see that the gate error is oc 53. As noted
above, the terms cc 53 and higher-order terms in 5 contain a small factor. They
become very small already for not too small 5.

To illustrate how the composite pulse works we compare in Fig. 2.6 the error of
an uncorrected LZ gate with the gate error of the composite pulse. The data refer
to different values of g of the working pulse; the corresponding values of a are given
in Fig. 2.4. we used A(tl) = —A(t2) % 10771/2 [the precise value of A(t1,2) was
adjusted to make S (t2, t1) an X -rotation, S (t2, t1) = RI(O)]. The compensating 7r-
pulses where modeled by pulses with g7r = 3. Based on the arguments provided at
the end of Sec. IV A, we took Aﬂ(t2) % [A(t2)[21/2(g7r/g)1/3, whereas An(t'2) was
found from Eq. (2.16); we used Aﬁ(t'l) = —A7r(t’2) and Aw(t1) = —A7r(t2).

It is seen from Fig. 2.6 that the proposed composite pulses are extremely efﬁcient
for compensating gate errors. Even for the energy error 5 = y, where the error of an
uncorrected pulse is close to 1, for the composite pulse 8 3 10‘3. For g g 1 the error
Of the single pulse scales as 5, whereas the error of the composite pulse scales as 83, in
agreement with the theory. For large 9, when the gate is almost an X -gate (7r-pulse),
in the case of symmetric pulses that we discuss, the coefficients at the terms cc 5
and cc 53 become small; they become equal to zero for a = 7r. Therefore for large 9

2 and e4,

and for not too small 5 the errors of single and composite pulses scale as 5
respectively. On the other hand, for 5/7 close to 1, errors of the composite pulses
with larger 9 are larger than for smaller 9. This is because the calculations in Fig. 2.6

refer to the same A / 711/ 2, in which case the errors oc 53, e4 are proportional to 92.

24

 

0.01

1 E4

1 EB

Gate error (8)

 

1 E-8

 

 

 

S/Y

Figure 2.6: Gate errors 8 for Landau-Zener pulses as a function of the frequency error
e. The upper and lower curves refer to the single LZ pulse and the composite pulse,
respectively. The dash-dotted, dotted, solid, and dashed lines Show 8 for g = 2, 1.2, 1,
and 0.3.

2.5 Conclusions

In this chapter we have developed a theory of quantum gates based on L2 pulses.
In these pulses the control dc ﬁeld is varied in such a way that the qubit frequency
passes through the frequency of the external radiation ﬁeld. In the adiabatic basis an
LZ gate can be expressed in a simple explicit form in terms of rotation matrices. Our
central result is that already a sequence of three LZ pulses can be made fault—tolerant.
The error of the corresponding composite pulse 8 scales with the error e in the qubit
energy or radiation frequency at least as 53. In addition, the coefﬁcient at 53 has an
extra parametrically small factor. The duration of the 3-pulse sequence is about 4
times the duration of the single pulse, for the parameters that we used.

Fault tolerance of LZ gates is partly due to the change of state populations be-

ing independent of precise frequency tuning. In particular, LZ tunneling makes it

25

possible to implement simple 7r-pulses with an exponentially small error in the state
population.

The approach developed here can be easily generalized to more realistic smooth
pulses, as mentioned above. It can be applied also to two-qubit gate operations
in which the frequencies of interacting qubits are swept past each other, leading to
excitation transfer [58]. Such operations are conIplementary to two—qubit phase gates
and require a different qubit-qubit interaction.

LZ pulses provide an alternative to control pulses where qubits stay in resonance
with radiation for a specified time [2]. In this more conventional approach it is
often presumed that qubits are addressed individually by tuning their frequencies.
In contrast to this technique, LZ pulses do not require stabilizing the frequency at a
ﬁxed value during the operation. As a consequence, calibration of LZ pulses is also
different, which may be advantageous for some applications, in particular in charge-
based systems [60, 63]. The explicit expressions discussed above require that the
qubit transition frequency vary linearly with time, but the linearity is needed only
for a short time when the qubit and radiation frequencies are close to each other, as
seen from Fig.2.2, which should not be too difficult to achieve.

For pulses based on resonant tuning for a ﬁxed time, much effort has been put.
into developing fault-tolerant pulse sequences, see Ref. [3] and papers cited therein.
In particular, for energy offset errors it has been shown that a three-pulse sequence
can reduce the error to 5 ~ 52 [7] (the ﬁdelity F evaluated in Ref. [7] is related to 5
discussed in Ref. [3] and in this chapter by the expression 1 — F cc 82 for small 5;

2 corresponds to the estimate [7] 1 — F ~ 54). As illustrated

therefore an error 5' ~ 5
in Fig.2.7, this error is parametrically larger, for small 5, than the error of the three-

pulse sequence proposed here, 5 oc 53. We note that, with two correcting pulses of a

26

more complicated form, it is possible to eliminate errors of higher order in 5. It follows

 

 

 

 

Gate error (8)

1E’6 “‘ .1

I
l

1 EB

 

 

 

0.1 A 1
s/y

Figure 2.7: Gate errors 5 for CORPSE pulses [7] as a function of the off-resonance
error 5. The upper and lower curves refer to uncorrected pulses and composite
pulses, respectively. The dash-dotted, dotted, solid, and dashed lines Show 8 for
rotation angles 6 = 7r,7r/2,7r/4, and 7r/8. Here, the ideal gate operation, related
to the ideal Hamiltonian H = 7X, is U = 1233(6), where 6 = 27T and T is the
duration of the pulse. Off-resonance errors lead, as in the case of Landau-Zener
gates, to H —> H’ = H + EZ/2. As a consequence, U —-> U’ = R,~,(0’), where
ii 2: 1/\/1+ 82/472 (1, Owe/27)T and 0’ = 6\/1+ 232/472. Note that in the displayed
range of 5/7, the error of the composite pulse is practically independent of 9 (which
is not the case for smaller values of 5/7). The CORPSE composite pulse sequence
has the form U = Rx(63)R_x(62)RI(91), where 61 = 27r + 6/2 — sin-1(sin(l9/2)/2),
62 = 261 - 27r — l9, and 63 = 61 — 27r.

from the results of this chapter that fault-tolerant LZ gates can be implemented using

the standard repertoire of control teclmiques and may provide a viable alternative to

the conventional single qubit gates.

27

Chapter 3

Multiphoton Antiresonance in

Large-Spin Systems

3. 1 Introduction

Large-spin systems have been attracting much attention recently. Examples are S =
3/ 2 and S = 5/ 2 Mn impurities in semiconductors and l\»’In— and Fe-based molecular
magnets with electron spin S = 10 and higher. Nuclear spins I = 3/ 2 have been
also studied, and radiation-induced quantum coherence between the spin levels was
observed [10]. An important feature of large-spin systems is that their energy levels
may be almost equidistant. A familiar example is spins in a strong magnetic ﬁeld
in the case of a relatively small magnetic anisotropy, where the interlevel distance is
determined primarily by the Larmor frequency. Another example is low-lying levels
of large—S molecular magnets for small tunneling. As a consequence of the structure
Of the energy spectrum, external modulation can be close to resonance with many
transitions at a time. This should lead to coherent nonlinear resonant effects that

have no analog in two-level systems.

28

The effects of a strong resonant ﬁeld on systems with nearly equidistant energy
levels have been studied for weakly nonlinear oscillators. These studies concern both
coherent effects, which occur without dissipation [31, 32, 33], and incoherent effects, in
particular those related to the oscillator bistability and transitions between coexisting
stable states of forced vibrations. In the absence of dissipation, a nonlinear oscillator
may display multiphoton antiresonance in which. the susceptibility displays a dip or
a peak as a function of modulation frequency [34].

In this chapter we study resonantly modulated spin systems with S > 1 / 2. Of pri-
mary interest are systems with uniaxial magnetic anisotropy, with the leading term in
the anisotropy energy of the form of —DS§ / 2. We Show that the coherent transverse
response of such Spin systems displays peaks or dips when the modulation frequency
adiabatically passes through multiphoton resonances. The effect is nonperturbative
in the ﬁeld amplitude. It is related to the special conformal property of the spin
dynamics in the semiclassical limit. It should be noted that the occurrence of an—
tiresonance for a spin does not follow from the results for the oscillator. A Spin can
be mapped onto a system of two oscillators rather than one; the transition matrix
elements for a spin and an oscillator are different as are also the energy spectra.

We Show that the coherent transverse response of a spin is sensitive to terms of
higher order in SZ in the anisotropy energy. In addition, there is a close relation
between the problem of resonant high—frequency response of a spin and the problem
of static spin polarization transverse to the easy axis. Spin dynamics in a static
magnetic ﬁeld has been extensively studied both theoretically and experimentally [12,
13, 14, 15, 16, 17, 18, 19, 20, 21]. One of the puzzling observations on magnetization
switching in molecular magnets, which remained unexplained except for the low—order

perturbation theory, is that the longitudinal magnetic ﬁeld at which the switching

29

occurs is independent of the transverse magnetic ﬁeld [16]. The analysis presented
below provides an explanation which is nonperturbative in the transverse ﬁeld and
also predicts the occurrence of peaks or dips in the static polarization transverse to

the easy axis as the longitudinal magnetic ﬁeld is swept through resonance.

(a) (b)

 

 

 

 

 

A
A
2 — Ith 8(0)(-1) 8(0)(0)
1 — A (0) - (0)
Tirol: w ................. w
0 AL—
A
fire
I hm .................................... —
F
E-z ‘_ —

 

Figure 3.1: Three-photon resonance in a S = 2 system in the limit of a weak ac ﬁeld.
(a) Spin energy levels Em and n-photon energies nﬁwp. (b) Quasienergies in the limit

of zero modulation amplitude, 5(0) (m) = Em — mﬁwp; the pairwise degenerate levels
correspond to one- and three-photon resonance, respectively.

The onset of strong nonlinearity of the transverse response due to near equidis-
tance of the energy levels can be inferred from Fig. 3.1(a). It presents a sketch of the
Zeeman levels of a spin Em (—S S m S S) in a strong magnetic ﬁeld along the easy

magnetization axis 2. The spin Hamiltonian is
H0 = tags, — $033 (h :1), (3.1)

where rug is the Larmor frequency. For comparatively weak anisotropy, DS << tag,
the interlevel distances Em+1 — Em are close to each other and change linearly with
m.

A transverse periodic ﬁeld leads to transitions between neighboring levels. An

interesting situation occurs if the ﬁeld frequency (up is close to am and there is

30

multiphoton resonance in the mth state: N w F coincides with the energy difference
Em+ N — Em, N > 1. The amplitude of the resonant N -photon transition in this
case is comparatively large, because the transition goes via N sequential one-photon
virtual transitions which are all almost resonant. Therefore one should expect a com-
paratively strong multiphoton Rabi splitting already for a moderately strong ﬁeld.

A far less obvious effect occurs in the coherent transverse response of the system,
that is in the magnetization at the modulation frequency or, equivalently, the suscep-
tibility. As we Show, the expectation value of the susceptibility displays sharp spikes
at multiphoton resonance. The shape of the spikes very strongly depends on the ﬁeld
amplitude.

The chapter is organized as follows. In Sec. II we study the quasienergy spectrum
and the transverse response of a spin with quadratic in Sz anisotropy energy. We
Show that, at multiphoton resonance, not only multiple quasienergy levels are crossing
pairwise, but the susceptibilities in the resonating states are also crossing. In Sec. III
we Show that multiphoton transitions, along with level repulsion, lead to the onset of
Spikes in the susceptibility and ﬁnd the shape and amplitude of the spikes as functions
of frequency and amplitude of the resonant ﬁeld. In Sec. IV we present a VVKB
analysis of spin dynamics, which explains the Simultaneous crossing of quasienergy
levels and the susceptibilities beyond perturbation theory in the ﬁeld amplitude. In
Sec. V the role of terms of higher order in S 2 in the anisotropy energy is considered.

Section VI contains concluding remarks.

31

3.2 Low-ﬁeld susceptibility crossing

3.2.1 The quasienergy spectrum

We ﬁrst consider a spin with Hamiltonian H0 (31), which is additionally modulated
by an almost resonant ac ﬁeld. The modulation can be described by adding to H0
the term —SIA cos wpt, where A characterizes the amplitude of the ac ﬁeld. As

mentioned above, we assume that the ﬁeld frequency cup is close to am and that

 

wp,w0 >> D,A, wF — w0|.
It is convenient to describe the modulated system in the quasienergy, or Floquet

1,’,.15(t + TF)) =

 

representation. The Floquet eigenstates [1115(t)) have the property
exp(—iETF)[1,/=5(t)), where 7F = 27r/u1p is the modulation period and 5 is quasienergy.
For resonant modulation, quasienergy states can be found by changing to the rotating
frame using the canonical transformation U (i) = exp(—iprzt). III the rotating wave

approximation the transformed Hamiltonian is

H = 41.15,, — 51333 —- 1A3... (3.2)

(511) 21.111: —w0.

Here we disregarded fast-oscillating terms oc A exp(j:2iwpt).

The Hamiltonian H has a familiar form of the Hamiltonian of a spin in a scaled
static magnetic field with components (in; and A / 2 along the 2 and 5c axes, respectively.
Much theoretical work has been done on spin dynamics described by this Hamiltonian
in the context of molecular magnets.

The eigenvalues of H give quasienergies of the modulated spin. In the weak
modulating ﬁeld limit, A —> 0, the quasienergies are shown in Fig.3.1(b). In this limit

spin states are the Zeeman states, i.e., the eigenstates [171.)(0) of 5;, with —S S m S

32

S. The interesting feature of the spectrum, which is characteristic of the magnetic
anisotropy of the form DSE, is that several states become simultaneously degenerate
pairwise for A = 0 [16, 17]. From Eq. (3.2), the quasienergies 5(0)(m) and 5(0)(m+N)

are degenerate if the modulation frequency is

(511,1 = (5w,,,,N, (Ito'.,-n_;N = —D (m + $N) . (3.3)
The condition (3.3) is siirmltaneously met for all pairs Of states with given 2m + N.
It coincides with the condition of N-plroton resonance Em+N — Em = N a) F- In what.
follows N can be positive and negative. There are 43 — 1 frequency values that satisfy
the condition (3.3) for a given S.

The ﬁeld oc A leads to transitions between the states [111.)(0) and to quasienergy

splitting. The level splitting for the Hamiltonian (3.2) was calculated earlier [17]. For

multiphoton resonance, it is equal to twice the multiphoton Rabi frequency QR(m; N),

123011.; N) = [A/2D[[NI [D[

(s + m + N)!(S — m)! is“ 1
(S + m)!(S — m — N)! 2([N[ —1)!2

 

 

(3.4)

 

The N -photon Rabi frequency (3.4) is o< A]N , as expected. we note that the ampli-
tude A is scaled by the anisotropy parameter D, which characterizes the nonequidis-
tance of the energy levels and is much smaller than the Larmor frequency. Therefore
93 becomes comparatively large already for moderately weak ﬁelds A ~ D.

We denote the true quasienergy states as [l/), with integer or half—integer 1/ such
that —S g l/ S S. The quasienergies 5,, do not cross. One can enumerate the states
IV) by thinking of them as the adiabatic states for slowly increasing (5w, starting
from large negative (is). For —(5w/DS >> 1, [A[ / D the states IV) are very close to the

Zeeman states [V)(0) , with V being the eigenvalue of S ,3. This then speciﬁes the values

Of I/ for all do}.

33

If the ﬁeld is weak, the states IV) are close to the corresponding Zeeman states,
IV) 2 Im)(0), for all 6w except for narrow vicinities of the resonant values 6112,",N

given by Eq. (3.3). The relation between the numbers V and m for IV) % [171)(0) is
I
V = m + ZN 6(6111 — 6w,n,,1\;)sgnN, (3.5)

where N runs from —S — m to S —m; the term N = 0 is eliminated, which is indicated
by the prime over the sum; 6(17) is the step function. III obtaining Eq. (3.5) we took
into account that, for weak ﬁelds, only neighboring quasienergy levels 5,, and 5,,i1

come close to each other. Eq. 3.5) deﬁnes the state enrn'nerating function m(V).

 

 

 

 

 

-2 -1 0 1 2
503/ D

Figure 3.2: Quasienergy levels 5,, for a spin—2 system as functions of detuning 615/ D for
the scaled ﬁeld amplitude A / D = 0.3. The levels anticross pairwise at multiphoton
resonances given by Eq. (3.3). The unperturbed quasienergies (the limit A = 0)
correspond to straight lines 5(0)(m) 2 —6w m — Dn12/2.

The enumeration scheme and the avoided crossing of the quasienergy levels are
illustrated in Fig. 3.2. For the chosen S = 2 the anticrossing occurs for 7 frequency

values, as follows from Eq. (3.3). The magnitude of the splitting strongly depends

34

on N: the largest splitting occurs for one-photon transitions. It is also obvious from

Fig. 3.2 that several levels experience anticrossing for the same modulation frequency.

3.2.2 Susceptibility and quasienergy crossing

Of central interest to us it the nonlinear susceptibility of the spin. We deﬁne the

 

dimensionless susceptibility XV in the quasienergy state V) as the ratio of the expec-
tation value of the appropriately scaled transverse magnetization at the modulation

frequency to the modulation amplitude,

XVfWF) = (VlS—IV>/A- (3-6)

In the weak ﬁeld limit, A ——> 0.

m(26w + Dm) + DS(S + 1)
4(6w + Dm)2 — 02

 

XV('W'F) : (317)

where m and V are related by Eq. (3.5); in fact. Eq. (3.7) gives the susceptibility in
the perturbed to ﬁrst order in A Zeeman state [m)(0).

A remarkable feature of Eq. (3.7) is the susceptibility crossing at multiphoton
resonance. The susceptibilities in Zeeman states Im)(0) and [m + N )(0) are equal
where the unperturbed quasienergies of these states are equal, 5(0) (111) = 5(0) (m+N),
i.e., where the frequency detuning is (in; = 6w,,,,N. In terms of the adiabatic states
IV), for such (in) we have from Eqs. (3.5), (3.7) x,,(wF) = XV/(LUF) for V' = V + sgnN.

A direct calculation shows that simultaneous crossing of the susceptibilities and
quasienergies occurs also in the fourth order of the perturbation theory provided
N Z 3. Numerical diagonalization of the Hamiltonian (3.2) indicates that it persists
in higher orders, too, until level repulsion due to multiphoton Rabi oscillations comes

into play.

35

The susceptibility XV is immediately related to the ﬁeld dependence of the
quasienergy 5,,. Since (VIS+[V) = (V[S_[V), from the explicit form of the Hamil-
tonian (3.2) we have

Simultaneous crossing of the susceptibilities and quasienergies means that, for an N-
photon resonance, the Stark shift of resonating states is the same up to order N — 1
in A; only in the Nth order the levels 5,, and 5V+sgnN become split [by 2f2R(1‘n;N)].
Respectively, the susceptibilities XV and XV+sgnN coincide up to terms o< AIM—3.
The physical mechanism of this special behavior is related to the conformal property
of the spin dynamics, as explained in Sec. 3.4.

Equation (3.7) does not apply in the case of one-photon resonance, N = 1: it
gives [XVI —> 00 for (in; —> 61.11,”;1. This is similar to the case of one-photon resonance
in a two-level system, where the behavior of the susceptibility is well understood
beyond perturbation theory. Interestingly, the l(')wr'3st-order perturbation theory does

not apply also at exact two—photon. resonance, (in) = drama, as discussed below, even

though Eq. (3.7) does not diverge.

3.3 Antiresonance of the transverse multiphoton

response

The ﬁeld-induced anticrossing of quasienergy levels at multiphoton resonance is ac-
companied by lifting the degeneracy of the susceptibilities. It leads to the onset
Of a resonant peak and an antiresonant dip in the susceptibilities as functions of fre-
quency (5a). The behavior of the quasienergy levels and the susceptibilities is seen from

Fig. 3.3. For small ﬁeld amplitude A the multiphoton Rabi frequency QR cx AINl is

36

small, the quasienergies of interest 5,, and 5,,+1 (with m(V + 1) — m(V) = N) come
very close to each other at resonant 6w, as do also the susceptibilities XV and XV-I-I'

With increasing A the level splitting rapidly increases in a standard way. The
behavior of the susceptibilities is more complicated. They cross, but sufﬁciently close
to resonance they repel each other, forming narrow dips (antiresonance) or peaks
(resonance). The widths and amplitudes of the dips / peaks display a sharp dependence
on the amplitude and frequency of the ﬁeld.

For weak ﬁeld it is straightforward to ﬁnd the splitting of the susceptibilities

AXV;N (WF) 2 [XI/(WF) _ XV+sgnN(wF)[

close to N -photon resonance between states Im)(0) and [m + N )(0). III this region the

frequency detuning from the resonance
Aw(m; N) = N(6w — 5Wm;N)/2 (3.9)

is small, [Aw(m; N )I g QR(m; N). To the lowest order in A but for an arbitrary
ratio 9 R(m; N ) / [Aw(m; N )I the quasienergy states IV) and [V + sgnN) are linear
combinations of the states [171.)(0) and [m + N )(0). Then from Eq. (3.2) it follows that

the splitting of the quasienergies A5,,,N = [5,, — 5u+sgan is

A5,,,N = Aw2(m; N) + 40%(171; N)

[1/2. (3.10)

From this expression and Eqs. (3.4), (3.8) it follows that the susceptibility splitting
is
amnion; N)
A2 [Aw2(m; N) + 49%(171; N)]

The splitting AXV;N as a function of frequency 600 is maximal at N -photon reso-

(3.11)

 

AXV;N : 1/2'

nance, 6w 2 610m; N1 The half—width of the peak of AXV;N at half height is determined

37

 

1.5

 

(a) A I A (b)

 

 

 

 

 

 

 

 

0 4
4.00
/ x i X0 ,
\ 1.o.\<——-—'-
-1.05‘ l/ ' V» 96.1
// i
4.10 - . - 0.5
(C) A ' ‘ (d) is
8O
-1.00- X_1
I \ .
// ‘5‘ 1'0““ —-.t
4.05‘ \ »
/ '1 \‘ X0
//
4.10 I . r - 0.5 g . -

 

 

 

 

 

 

(e) U) ‘5

4.00-

4.05

   

 

 

 

 

 

 

0.45 V 0.50 ' 0.55 0.45 0.250 0.55

5co/ D owl D

Figure 3.3: Level anticrossing and antiresonance of the susceptibilities. The ﬁgure
refers to a. 3-photon resonance, N = 3, in an S = 2 system. The involved quasienergy
states are V = —1 and V = 0. The resonating Zeeman states for A = O are m = ——2
and m = 1 (the ground and 3rd excited state). Left and right panels show the
quasienergies 5V and susceptibilities Xv for the same reduced ﬁeld A/ D. Panels (a)
and (b), (c) and (d), and (e) and (f) refer to A/D = 0,02, and 0.3, respectively.

38

 

 

 

 

 

 

0.01 .......(.),1 .. ....1
ND

Figure 3.4: The multiphoton susceptibility splitting for S = 2. The curves refer
to exact N -photon resonances, with N = 1, . . . ,4, for transitions from the ground
Zeeman state m = —2 to the excited states 772. = —1, . . . ,2, respectively.

by the Rabi splitting and is equal to J39 R /N . The peak is strongly non-Lorentzian,
it is sharper than the Lorentzian curve with the same half-width. This sharpness is

indeed seen in Fig. 3.3. Our numerical results show that Eq. (3.11) well describes the

 

splitting in the whole frequency range Aw! g 93.

For small A, the susceptibility splitting is stronger than the level repulsion. It
follows from Eqs. (3.10), (3.11) that at exact N ~photon resonance A5 o< AlNl whereas
AX oc AIM—2. This scaling is seen in Fig. 3.4. For A/D >> 1, on the other hand, the
eigenstates |1/) become close to the eigenstates of a spin with Hamiltonian —AS I / 2.
As a result, the susceptibility splitting decreases with increasing A, lAXz/Wl o< A—l;

the proportionality coefﬁcient here is independent of N. Therefore, for N Z 3 AXV

displays a maximum as a function of A, as seen from Fig. 3.4.

39

3.3.1 Two-photon resonance

As mentioned above, the lowest. order perturbation theory (3.7) does not describe
resonant susceptibility for two-photon resonance. Indeed, it follows from Eq. (3.11)

that at exact resonance, 6w 2 6w(m; 2), the susceptibility splitting for weak ﬁelds is

Aka/2 = DA [(5 - m -1)(5 — m)

x (S+7n+1)(S+m+2)]1/2. (3.12)

This splitting is independent of A. The expression for the susceptibility (3.7) is also
independent of A, yet it does not lead to susceptibility splitting and therefore is
incorrect at two-photon resonance.

The inapplicability of the simple perturbation theory (3.7) is a consequence
of quantum interference of transitions, the effect known in the linear response of
multilevel systems [40]. To the leading order in A, the susceptibility is deter-
mined by the squared amplitudes of virtual transitions to neighboring states. For
a two—photon resonance, (in) = 6w7m2, the distances between the levels involved
in the transitions I'm.)(0l —> [m + 1)(0) and Im + 2)(0) —> |m + 1)(0) are equal,
8(0)(m + 1) —- 5(0)(m) = 5(0)(m + 1) — 5(0)(m + 2). Therefore the transitions res-
onate and interfere with each other.

To calculate the susceptibility it is necessary to start with a superposition of
states Im)(0) and |m+2)(0), add the appropriately weighted amplitudes of transitions
lm)(0) —> Im + 1)(0) and Im + 2)(0) -—> Im + 1)(0), and then square the result. This
gives the correct answer. The independence of the susceptibility splitting from A for

two—photon resonance in the range of small A as given by Eq. (3.12) is seen in Fig. 3.4.

40

3.4 Susceptibility crossing for a semiclassical spin

The analysis of the simultaneous level and susceptibility crossing is particularly inter—
esting and revealing for large spins and for multiphoton transitions with large N. For
S >> 1 the spin dynan‘iics can be described in the VVKB approximation. We will start
with the classical limit. In this limit it is convenient to use a unit vector 8 = S/ S,
with s E (31-, 3,), sz) E (si116cos qﬁ, si116sin c5, cos 9), where 6 and (f) are the polar and
azimuthal angles of the vector 5. To the lowest order in S ‘1 equations of motion for

the spin components can be written as

5117: Sy(3z + [1)), 3y 2 —'Sx(83 + H) + fSZ, (3.13)

3'2 2 —fs,,, f = A/2SD, [,L = (Sta/SD.

Here, overdot implies differentiation with respect to dimensionless time 7' = S Dt, that
is, s E (ls/(1T = (SD)—1ds/dt. Equations (3.13) preserve the length of the vector 8

and also the reduced Hamiltonian 9 = H / 52D,

, 1
9 E Hwy 99) : _§(3z + (1)2 " fsx (3-14)

For convenience, we added. to g the term —,u2 / 2.

The effective energy 9(0, gb) is shown in Fig. 3.5. Also shown in this ﬁgure are
the positions of the stationary states s = O and examples of the phase trajectories
described by Eqs. (3.13).

An insight into the spin dynamics can be gained by noticing that g has the form
of the scaled free energy of an easy axis ferromagnet [64], with 5 playing the role of
the magnetization M/ZW, and with it. and f being the reduced components of the

magnetic ﬁeld along the easy axis 7: and the transverse axis :L', respectively. In the

41

 

 

 

Figure 3.5: The effective energy g((). a) as a function of the polar and azimuthal
angles of the classical spin (9 and (b. The lines 9(0, 94)) = const describe classical spin
trajectories. The points A1 and A2 are the minima of g, B is the maximum, and S
is the saddle point. In the region gs > g > 9A1 there are two coexisting types of
trajectories. They lie on the opposite sides of the surface 9(6, (15) with respect to 95-
The plot refers to ,u = 0.125. f = 0.3.

region
If!” +1142“ <1 (3.15)

the function g has two minima, A1 and A2, a maximum B, and a saddle point 8. We

will assume that the minimum A2 is deeper than A1, that is
93 > 95 > 9A1 > 9A2- (3-16)
As seen from Eqs. (3.13) and (3.14) and Fig. 3.5, for f > 0 the minima and the saddle

42

point are located at (D = O and the maximum is at c9 2 7r; the case f < 0 corresponds
to a replacement ob —-> gb + 7r. On the boundary of the hysteresis region (3.15) the
shallower minimum A1 merges with the saddle point 8.

In the case of an easy-axis ferromagnet with free energy 9, the minima of g corre-
spond to coexisting states of magnetization within the hysteresis region (3.15). For
multiphoton absorption 9 is the scaled quasienergy, not free energy, and stability is
determined dynamically by balance between relaxation and high—frequency excita-
tion. One can show that, for relevant energy relaxation mechanisms, the system still
has two coexisting stable stationary states albeit only in a part of the region (3.15).
The states correspond to the shallower minimum and the maximum of g; for small
damping the actual stable states are slightly shifted away from the extrema of g 011

the (6, gb)-plane. We will not discuss relaxation effects in this chapter.

3.4.1 Conformal property of classical trajectories

Dynamical trajectories of a classical spin on the plane (6, 96) are the lines
g(6,¢) = const. They are either closed orbits around one of the minima A1,A2
or the maximum B of g. or open orbits along the (b axis, see Fig. 3.5. On the Bloch
sphere $2 = 1, closed orbits correspond to precession of the unit vector 5 around the
points SAPS/12, or 83, in which 3 does not make a complete turn around the polar
axis. Open orbits correspond to spinning of 5 around the polar axis accompanied by
oscillations of the polar angle 6. Even though the spin has 3 components, the spin
dynamics is the dynamics with one degree of freedom, the orbits on the Bloch sphere
do not cross.

An important feature of the dynamics of a classical spin in the hysteresis region

is that, for each 9 in the interval (9A1, gs), the spin has two coexisting orbits, see

43

Fig. 3.5. One of them corresponds to spin precession around SA 1' It can be a closed
loop or an open trajectory around the point .41 on the (6, c5)-plane. The other is an
open trajectory on the opposite side of the g-surface with respect to the saddle point.
We will classify them as orbits of type I and II, respectively.

We show in Appendix that classical equations of motion can be solved in an explicit
form, and the time dependence 8(7') is described by the Jacobi elliptic functions. The
solution has special symmetry. It is related to the conformal property of the mapping
of 3;, onto 7'. The major results of the analysis are the following features of the
trajectories 5(7') of types I and II: for equal g, (i) their dimensionless oscillation
frequencies w(g) are equal to each other, and (ii) the period averaged values of the

component 337(7) are equal, too,

w1(9) = 0011(9): (317(7))1 = (8.217))!!- (317)

Here, the subscripts I and II indicate the trajectory type. The angular brackets (. . .)
imply period averaging on a trajectory with a given 9.

The quantity (53(7)) gives the classical transverse response of the spin to the
ﬁeld or A. Equation (3.17) shows that this response is equal for the trajectories with
equal values of the effective Hamiltonian function g. This result holds for any ﬁeld
amplitude A, it is by no means limited to small A / D where the perturbation theory

in A applies.

3.4.2 The WKB picture in the neglect of tunneling

In the WKB approximation, the values of quasienergy 5V in the neglect of tunneling
can be found by quantizing classical orbits 9(9, gb) = const, see Ref. [65] and papers

cited therein. Such quantization should be done both for orbits of type I and type

44

II, and we classify the resulting states as the states of type I and II, respectively.
The distance between the states of the same type in energy units is hw(g)S D [66].
Transitions between states of types I and II with the same 9 are due to tunneling.

If we disregard tunneling, the quasienergy levels of states I and II will cross, for
certain values of ,u. Remarkably, if two levels cross for a given ,u, then all levels in the
range 9A1 < 9 < 95 cross pairwise. This is due to the fact that the frequencies w(g)
and thus the interlevel distances for the two sets of states are the same, see Eq. (3.17).
Such simultaneous degeneracy of multiple pairs of levels agrees with the result of the
low-order quantum perturbation theory in A and with numerical calculations.

In the W KB approximation, the expectation value of an operator in a quantum
state is equal to the period-averaged value of the corresponding classical quantity
along the appropriate classical orbit [66]. Therefore if semiclassical states of type I
and II have the same 9, the expectation values of the operator SJ. in these states
are the same according to Eq. (3.17). Thus, the WKB theory predicts that, in the
neglect of tunneling, there occurs simultaneous crossing of quasienergy levels and
susceptibilities for all pairs of states with quasienergies between 9A1 and 95- This
is in agreement with the result of the perturbation theory in A and with numerical
calculations. However, we emphasize that the WKB theory is not limited to small A,
and the WKB analysis reveals the symmetry leading to the sin'iultaneous crossing of
quasienergy levels and the susceptibilities.

Tunneling between semiclassical states with equal 9 leads to level repulsion and
susceptibility antiresonance. The level splitting 252 R can be calculated by appropri-
ately generalizing the standard W KB technique, for example as it was done in the
analysis of tunneling between quasienergy states of a modulated oscillator [67]. Then

the resonant susceptibility splitting can be found from Eq. (3.8). The corresponding

45

calculation is beyond the scope of this chapter.

3.5 Degeneracy lifting by higher order terms in 5;,

The simultaneous crossing of quasienergy levels and susceptibilities in the neglect of
tunneling is a feature of the spin dynamics described by Hamiltonian (3.2). Higher-
order terms in S 2 lift both this degeneracy and the property that many quasienergy
levels are pairwise degenerate for the same values of the frequency detuning 6a). The
effect is seen already if we incorporate the term Sf,1 in the anisotropy energy, i.e. for

a spin with Hamiltonian
.. 1 4
H = H — zG’Sz. (3.18)

The Hamiltonian H is written in the rotating wave approximation, H is given by
Eq. (3.2), and G is the parameter of quartic anisotropy. The terms SE, SE in the
spin anisotropy energy do not show up in H even if they are present in the spin
Hamiltonian H0 but the corresponding anisotropy parameters are small compared to
cm. In the rotating frame these terms renormalize the coefﬁcient at S? and lead to
fast oscillating terms oc 5:2,: exp(:l:2iwpt) that we disregard.

Multiple pairwise degeneracy occurs where the condition on Zeeman quasienergies
5(O)(m) = 5(0)(m’) is simultaneously met for several pairs (m, m’). For C 75 0 this
happens only for (5w 2 0, that is when the modulation frequency cap is equal to
the Larmor frequency cm. In this case the resonating Zeeman states are |m)(0) and
I — m)(0) with the same m. The susceptibilities of these states are equal by symmetry
with respect to reﬂection in the plane (a: y).

N -photon resonance for nonzero G and (up 7é (.00 occurs generally only for one

pair of states |m)(0) and Im + N)(0). This is seen from panel (a) in Fig. 3.6. With

46

increasing [G] the difference in the resonant values of frequency increases, as seen
from panel (c) in the same ﬁgure.

The susceptibilities in resonating states are different in the weak-ﬁeld limit. When
the frequency w F adiabatically goes through resonance, there occurs an interchange
of states, for weak ﬁeld A: if the state IV) was close to [771)(0) on one side of resonance,
it becomes close to [m + N)(O) on the other side. Respectively, the susceptibility XV

sharply switches from its value in the state [771)(0) to its value in the state [m + N )(0).

Susceptibility switching is seen in panels (b) and (d) in Fig. 3.6. For a weak ﬁeld
the frequency range where the switching occurs is narrow and the switching is sharp
(vertical, in the limit A —+ 0). As the modulation amplitude A increases the range
of frequency detuning (in) over which the switching occurs broadens. In addition, for
small G the susceptibility displays spikes. They have the same nature as for G = 0.
However, they are much less pronounced, as seen from the comparison of panel (d)

in Fig. 3.6 and panel (f) in Fig. 3.3 which refer to the same value of A/ D.

3.6 Conclusions

In this chapter we have considered a large spin with an easy axis anisotropy. The
spin is in a strong magnetic ﬁeld along the easy axis and is additionally modulated
by a transverse ﬁeld with frequency an F close to the Larmor frequency am. We have
studied the coherent resonant transverse response of the spin. It is determined by the
expectation value of the spin component transverse to the easy axis. We are interested
in multiphoton resonance where Nw F coincides or is very close to the difference of

(D) Em)

the Zeeman energies Em + N — m in the absence of modulation.

47

1.0

0.5 ,

 

 

 

 

 

 

 

 

 

 

 

0.95 ' 1.00 1.05

 

 

 

 

 

 

 

 

 

0.0 . i .
0.95 1.00 1.05

600/ D

 

 

 

Figure 3.6: Quasienergy and susceptibility switching for a spin S = 2 with quar-
tic in 3;, anisotropy. Panels (a), (b), and ((1) refer to the dimensionless quartic
anisotropy parameter G / D = 0.4 in Eq. (3.18). Panels (b) and ((1) refer to the 3—
photon resonance | — 2)(0) —> mm with the scaled modulation amplitude A/ D —+ 0
and A/ D = 0.3, respectively; the dotted line shows the position of the resonance
6w/ D = 1. Panel (c) shows the dependence of the resonant frequency detuning
602m ; N on the higher-order anisotropy parameter G in the limit A —-> 0.

48

The major results refer to the case where the anisotropy energy is of the form
—DS§ / 2. In this case not only the quasienergies of the resonating Zeeman states
lm)(0) and [m+N)(0l cross at multiphoton resonance. but the susceptibilities in these
states also cross. in the weak-11iodulation limit. Such crossing occurs simultaneously
for several pairs of Zeeman states. As the modulation amplitude A increases, the
levels are Stark—shifted and the susceptibilities are also changed. However, as long
as the Rabi splitting due to resonant multiphoton transitions (tunneling) can be
disregarded, for resonant frequency the quasienergy levels ren'iain pairwise degenerate
and the susceptibilities remain crossing. We show that this effect is nonperturbative
in A, it is due to the special conformal property of the classical spin dynamics.

Resonant multiphoton transitions lift the degeneracy of quasienergy levels, lead-
ing to a standard level anticrossing. In contrast, the susceptibilities as functions of
frequency cross each other. However, near resonance they display spikes. The spikes
of the involved susceptibilities point in the opposite direction, leading to decrease
(antiresonance) or increase (resonance) of the transverse response. They have a pro-
foundly non-Lorentzian shape (3.11), with width and height that strongly depend
on A. The spikes can be observed by adiabatically sweeping the modulation fre-
quency through a multiphoton resonance. If the spin is initially in the ground state,
a sequence of such sweeps allows one to study the susceptibility in any excited state
provided the relaxation time is long enough.

The behavior of the susceptibilities changes if terms of higher order in 52 in
the anisotropy energy are substantial. In this case crossing of quasienergy levels is
not accompanied by crossing of the susceptibilities in the limit A —> 0. Resonant
multiphoton transitions lead to step-like switching between the branches of the sus-

ceptibilities of the resonating Zeeman states. Still. the susceptibilities display spikes

49

as functions of frequency for a. sufficiently strong modulating ﬁeld.

The results of the chapter can be applied also to molecular magnets in a static
magnetic ﬁeld. The spin Hamiltonian in the rotating wave approximation (3.2)
is similar to the Hamiltonian of a spin in a comparatively weak static ﬁeld, with
the Larmor frequency 60) of the same order as the anisotropy parameter D. The
susceptibility then characterizes the response to the ﬁeld component transverse
to the easy axis. Quasienergies 5(0l(m) are now spin energies in the absence
of the transverse ﬁeld, and instead of multiphoton resonance we have resonant
tunneling. Our results show that a transverse ﬁeld does not change the value of
the longitudinal ﬁeld for which the energy levels cross, in the neglect of tunneling.
This explains the experiment [16] where such behavior was observed. Fig. 3.7 shows
this independence for the case of switching steps in magnetization hysteresis loops

of Mnlg, where switching occurs because of tunneling through the anisotropy barrier.

In conclusion, we have studied multiphoton resonance in large—spin systems.
We have shown that the coherent nonlinear transverse response of the spin displays
spikes when the modulation frequency goes through resonance. The spikes have
non-Lorentzian shape which strongly depends on the modulation amplitude. The
results bear on the dynamics of molecular magnets in a static magnetic ﬁeld and

provide an explanation of the experiment.

50

 

 

F‘Sll

 

 

0.06

 

 

 

 

   
  
 

 

 

 

000‘
'3 0.02
E -
0)
v
5" 0
0.02 ._-.
o0.04_:1::%‘r¢: :4::—;—+-]--4+75§11:5.
(b) 1
i
’5‘ 5 -” 7
* I . 1
O : '- 1
3 4 f _, i. j
E r '1. L
a) l . i
'9 3 i ‘ 1
o .
v- _ 3' - 1 j
"j: 2 ,1. ,1 1
I l . ‘ T” ,‘i'i .4
t: ' .: 11.1» . .
\z 1 3'; L-li‘ii. )1 [ _, ' 1 1
‘O f.“ A ‘1 " J%'Iiffy;fl'1ccttt".5r,;2‘ ' i. ' ' 1 ' \\
o ' o u N“‘.’.‘"“'"""~"""¢""“"'t"~I'vu'u'u'uuu'ﬂn’u1Igu-u(um:11¢mufﬁn-u...
_1 i . A L l . . . 1 1 i 1 I ”a 1 A mam 1 . . . -
0 5 1O 15 20 25
1-1,I (kOe)

Figure 3.7: (Reproduced from Ref. [16]. Independence of hysteresis steps on trans—
verse ﬁeld. (a) Magnetic moment of Mn12 as a function of the external longitudinal
ﬁeld at 2.0 K for several values of the transverse ﬁeld. Note that this is one quarter of
the hysteresis loops. The steps always occur at the same values of longitudinal ﬁeld.
(b) The derivative of the curves in (a).

51

Chapter 4

Hysteresis, Transient Oscillations,
and Nonhysteretic Switching in
Resonantly Modulated Large-Spin

Systems

4. 1 Introduction

Large-spin systems have a ﬁnite but con'iparatively large number of quantum states.
Therefore a single system can be used to study a broad range of phenomena, from
purely quantum to semiclassical where the spin behaves almost like a classical top.
One of the interesting features of large-spin systems is that, in a strong static mag-
netic ﬁeld, their energy levels become almost equidistant, with level spacing close to
5020, where 010 is the Larmor frequency. As a result, radiation at frequency 2 1.00 is

resonant simultaneously for many interlevel transitions. This leads to new quantum

52

and classical nonlinear resonant effects.

An important class of large-spin systems is single—molecule magnets (SMMs).
SMMS display an extremely rich behavior and have been attracting much attention
in recent years. A variety of SMMs has already been discovered and investigated, see
Refs. [22, 23, 24] for a review, and new systems are being found [25, 26]. Another
example of large-spin systems is provided by large nuclear spins, the interest in which
has renewed in view of their possible use in quantum computing [11].

In this paper we study the dynamics of large-spin systems, S >> 1, in the classical
limit. we assume that the system is in a strong static magnetic ﬁeld along the easy
magnetization axis and in an almost resonant transverse ﬁeld. For a small relaxation
rate, even a weak transverse resonant ﬁeld can lead to hysteresis of the response. As
we show, the hysteresis is quite unusual.

It is convenient to describe the dynamics of a resonantly modulated spin in the
rotating wave approximation (RVVA). The corresponding analysis in the absence of
relaxation has revealed a special quantum feature, an antiresonance of the response
which accompanies anticrossing of quasienergy levels [68]. Quantum spin dynamics
in the rotating frame bears also on the dynamics of the Li}’)kin-l\v’Ieshkov-Glick model
[27,28,29,30]

One may expect that the features of the coherent quantum dynamics should have
counterparts in the classical spin dynamics in the presence of dissipation. As we
show, this is indeed the case. The system displays an unusual behavior in a certain
range of modulation parameters. This behavior is due to a special symmetry. It leads
to speciﬁc features of hysteresis and to discontinuous (in the neglect of ﬂuctuations)
switching between different response branches even in the absence of hysteresis.

Classical dynamics of a large-spin system in a resonant ﬁeld would be expected to

have similarities with the dynamics of a modulated magnetic nanoparticle near ferro-
magnetic resonance. It was understood back in the 1950’s [35, 36] that the response
near ferromagnetic resonance becomes strongly nonlinear already for comparatively
weak radiation strength due to the magnetization dependence of the effective magnetic
ﬁeld. Resonant response may become multivalued as a function of the modulating
ﬁeld amplitude [37, 38]. A detailed analysis of nonlinear magnetization dynamics in
uniaxial nanoparticles modulated by a strong circularly polarized periodic ﬁeld was
done recently [39]. These studies as well as many other studies of magnetization
dynamics in ferromagnets were based on the Landau-Lifshitz-Gilbert equation.

111 contrast to magnetic nanoparticles, for large-spin systems quantum effects are
substantial. A distinction which remains important in the classical limit concerns
relaxation mechanisms. Spin relaxation occurs via transitions between discrete energy
levels with emission, absorption, or inelastic scattering of excitations of a thermal
reservoir to which the spin is coupled. Relevant relaxation mechanisms depend on the
speciﬁc system but as we show, even in the classical limit relaxation is not described,
generally, by the Landau—Lifshitz damping. As a result the classical spin dynamics
strongly differs from the dynamics of a magnetic nanoparticle.

The microscopic analysis of relaxation is simpliﬁed in the presence of a strong
static magnetic ﬁeld. Here, all spin energy levels are almost equidistant. Therefore
excitations of the thermal bath emitted, for example, in transitions within differ-
ent pairs of neighboring levels have almost the same energies. As a consequence,
relaxation is described by a small number of constants independent of the form of
the weighted with the interaction density of states of the bath, and the analysis ap-
plies for an arbitrary ratio between the level nonequidistance and their relaxational

broadening [40].

54

we consider three relaxation mechanisms. Two of them correspond to transitions
between neighboring and next neighboring spin levels, with the coupling to bosonic
excitations quadratic in the spin operators. Such coupling is important, in particular,
for Sl\'-"I;\/Is where energy relaxation is due to phonon scattering. The theory of relax-
ation of SMMs was developed earlier [17, 69] and has been tested experimentally, see
Refs. [70, 71] and papers cited therein. We also consider coupling to a bosonic bath
linear in spin operators. It leads to relaxation that in the classical limit has the form
of the Landau—Lifshitz damping provided the modulation ﬁeld is weak compared to
the static ﬁeld, as assumed in the RVVA.

The typical duration of scattering events that lead to spin relaxation is often
~ 020— 1. In the RVVA they appear instantaneous. The operator that describes re-
laxation has a simple functional form, with no retardation in the “slow” time. This
is advantageous for studying the classical limit and allows us to obtain analytical
results.

In the classical limit, a spin is characterized by two dynamical variables, for ex-
ample, azimuthal and polar angles. In the RW'A, they satisfy autonomous equations
of motion, the coefficients in these equations do not depend on time. A two—variable
nonlinear dissipative system can have both stationary and periodic states [72]. As
we show, such states indeed emerge for a resonantly modulated spin. They were pre—
dicted also for a magnetic nanoparticle with Landau-Lifshitz damping in the case of
a generally nonresonant modulation [39].

For a spin, the occurrence of periodic states in the rotating frame critically depends
on the interrelation between the relaxation parameters. In particular, these states do
not emerge for a resonantly modulated spin with microscopic relaxation that reduces

to the Landau-Lifshitz damping in the RWA. Moreover, quantum fluctuations lead to

phase diffusion which results in decay of periodic states in the rotating frame, making
the corresponding vibrations transient.

This chapter is organized as follows. In Sec. 4.2 we introduce a model of the
spin and its interaction with a thermal bath and derive the quantum kinetic equation
with account taken of different relaxation mechanisms. In Sec. 4.3 we obtain classical
equations of motion and discuss the symmetry of the system. We ﬁnd analytically. for
weak damping, the positions of the bifurcation curves where the number of stationary
states in the rotating frame changes (saddle-node bifurcations) and where periodic
states are split off from stationary states (Hopf bifurcations). Sec. 4.4 describes
the speciﬁc and, perhaps, most unusual feature of the system, the occurrence of
Hamiltonian-like dynamics in the presence of dissipation. In Sec. 4.5 spin dynamics
and hysteresis are described for the relation between relaxation parameters where the
system does not have periodic states in the rotating frame. In Sec. 4.6 we consider
the opposite case. The onset of periodic states and their stability are analyzed and
the features of the hysteresis related to the occurrence of periodic states are studied.
Details of the calculations are outlined in Appendix. Sec. 4.7 contains concluding

remarks.

4.2 The model

We consider a large spin, 5' >> 1, in a strong stationary magnetic ﬁeld along the easy
axis 2. The spin is modulated by a transverse magnetic ﬁeld with frequency w F close

to the Larmor frequency tag. The Hamiltonian of the spin has the form

H0 2 01052 — $053 — SIAcoswpt (’1 =1) (4.1)

56

This Hamiltonian well describes many single—molecule magnets, including Mnlg crys—
tals; D characterizes the magnetic anisotropy and A is the modulation amplitude. It
also describes a nuclear spin, with D characterizing the quadrupolar coupling energy
to an electric ﬁeld gradient in the crystal with an appropriate symmetry.

We assume that. the Zeeman energy levels in the absence of modulation are almost
equidistant. We also assume that the resonant modulation is not too strong. These

conditions are met provided
[000 -wF],DS,A << 0110. (4.2)

For many SMMS the inequality DS << cm is fairly demanding and requires strong
static magnetic ﬁelds; for example D x 0.6 K for Feg (where S = 10) [24]. On the
other hand, for more isotropic SMMs the anisotropy is much smaller; for example,
D z 0.04 K for Fe” where S = 35/2, see Ref. [26] (our deﬁnition of D differs by a
factor of 2 from the deﬁnition used in the literature on SMMs). The anisotropy is
usually much weaker for large-S nuclei and the condition (4.2) is not restrictive.
The quantum dynamics of an isolated spin with Hamiltonian Eq. (4.1) was con-
sidered earlier [68]. Here we are interested in the spin dynamics in the presence of
dissipation. Different dissipation mechanisms are important for different systems. For
single—molecule magnets, energy dissipation is due primarily to transitions between
spin energy levels accompanied by emission or absorption of phonons. The transitions
between both nearest and next nearest spin levels are important [17, 69, 73]. The

corresponding interactions are
1
H)” = 2,, v] ) ((3.5z + as.) b), + 11.0.] (4.3)

Hi2) = 2k V152) (S30)c + 11.0.), Si 2 S17 :t 2S ,

Where It enumerates phonon modes, bk is the annihilation operator for the k-th mode,

57

1 2 . . . .
and Vk( ) and V1: ) are the coupling parameters respons1ble for trans1tions between

nearest and next nearest Zeeman levels. The phonon Hamiltonian is

Hp}, = :15 kazrbk- (4.4)

A similar interaction Hamiltonian describes the coupling of a nuclear spin to phonons,
cf. Ref. [74] and the early work [75, 76].
Along with the interaction (4.3) we will consider the interaction that is linear in

Such interaction is allowed by time—reversal symmetry in the presence of a strong
static magnetic ﬁeld, with the coupling constants V153) proportional to the off power
of the ﬁeld. It can be thought of as arising from phonon-induced modulation of
the spin g—factor. The interaction Eq. (4.5) is also important for impurity spins in

magnetic crystals, in which case bk is the annihilation operator of a magnon [77, 78].

4.2.1 Rotating wave approximation

The dynamics of a periodically modulated spin can be conveniently described in the
rotating wave approximation. To do this we make a canonical transformation U (t) =

exp(—iprZt). The transformed Hamiltonian H0 then becomes H0 2 U IHOU —iU 1LU ,

1% = —6wSz — $033 — $.43,”
6w = 0111? -— (do. (4.6)
Here we disregarded fast-oscillating terms o< Aexp(j:2iwpt).

We note that the Hamiltonian (4.6) has the form of the free energy of a magnetic

moment in an easy axis ferromagnet, with S playing the role of the magnetization

58

and 61.0 and A giving the components of the effective magnetic ﬁeld (in energy units)
along the z and :1: axes, respectively.
It is convenient to change to dimensionless variables and rewrite the Hamiltonian

as H0 2 S2D(Q +112/2), with

A I
g : _§(SZ +1“)2 — f'SlH

s = 8/8, )1 = 5.1/SD, f z A/QSD. (4.7)

The Hamiltonian Q describes the dynamics of an isolated spin in “slow” dimensionless
time 7' = S Dt. It gives dimensionless quasienergies of a. periodically modulated spin
in the RWA. From Eq. (4.7), the spin dynamics is determined by the two dimension-
less parameters, ” and f, which depend on the interrelation between the frequency
detuning of the modulating ﬁeld 610, the anisotropy parameter DS, and the modula-
tion amplitude A. The spin variables 5 are advantageous for describing large spins,
since the commutators of their components are oc S ‘1, which simpliﬁes a transition

to the classical limit for S >> 1.

4.2.2 Quantum kinetic equation

We will assume that the interaction with phonons (magnons) is weak. Then under
standard conditions the equation of motion for the spin density matrix p is Markovian
in slow time 7', i.e., on a time scale that largely exceeds 00;} and the typical correlation
time of phonons (magnons). We will switch to the interaction representation with
respect to the Hamiltonian wFSZ + lei' Then to leading order in the spin to bath

coupling the quantum kinetic equation can be written as

A

5‘10 = III), 9] — Imp - imp — Fmp. (48)
where A E BA/aT.

59

(j)

The operators I‘m describe relaxation due to the interactions Hi. , with j =

1, 2, 3. They can be written schematically as

rp = r((n + 1) (L+Lp — 2LpL+ + pL+L)

+71 (LL+p — 2L+pL + pLL+)] (4.9)

Here we have taken into account that all transitions between spin states with emission
or absorption of phonons (magnons) involve almost the same energy transfer AE,
with AB 3 tap for terms o< I‘(1),F(3) and AE % 2001: for the term o< I‘m. In
this sense, equation for spin relaxation (4.9) resembles the quantum kinetic equation
for a weakly nonlinear oscillator coupled to a bosonic bath [40]. Respectively, ﬂ
is the Planck number of the emitted/ absorbed phonons, ﬁ. = [exp(AE/kT — 1)]_1.
Because of the same transferred energy, different transitions are characterized by the
(ll—(3)

same rate constants, which for the interactions Hi have the following form, in

dimensionless time:

[41): 7rD_1S2 Z}; [Vk(1)[26(wp — wk),

11(2): 7rD—isz :1. |Vl§2ll25 (20117 — wk),

2
PB) :WD—lzklvlf3)] 6(wF—wk). (4.10)
The operators L for the interactions H 2(1l-(3) are
L“) = 3-3,. + 3.3-, L9) = 53, L(3) = 5-, (4.11)

where Si 2 Si/S.
It is important to note that, along with dissipation, coupling to phonons
(magnons) leads to a polaronic effect of renormalization of the spin energy. A stan-

( )

. . . 3 . 3
dard ana1y81s shows that renormalization due to H 2‘( ), to second order in H 2. comes

6O

to a change of the anisotropy parameter D and the Larmor frequency. A similar
l " ‘f" "SbJr H’I‘ 1‘
crange comes iom nonresonant terms o< + k + .c.. n contrast, renormalization
(1)42) . .- . . 2 . .. -. - -
from H,- , along mth terms oc SZ, S3, leads to terms of higher order in S z in the
spin Hamiltonian, in particular to terms or S3. The condition that they are small
compared to the anisotropy energy DS,? imposes a constraint on the strength of the

coupling HIGH”.

Respectively, we will assume that the dimensionless decay rates
FUMQ) are small, I‘UMQ) << 1. It is not necessary to impose a similar condition on

the dimensionless rate PB). Still we will be interested primarily in the spin dynamics

in the underdainped regime, where I‘m—(3) are small.

4.3 Classical motion of the modulated spin

The analysis of spin. dynamics is signiﬁcantly simpliﬁed in the classical, or mean-
ﬁeld limit. Classical equations of motion for the spin components can be ob-
tained by multiplying Eq. (4.8) by 3,: (2'. = 3:, y, z), taking the trace, and decoupling
Tr (31115120) —> 8,13,:2. The decoupling should be done after the appropriate com-
mutators are evaluated; for example, Tr ([33,53] p) ——> ~27 f 3y. From Eqs. (4.7), (4.8),

(4.11) we obtain

S = -s x 059 + (SM. (SM = Fd(Sz)S X (S X 2).

Fd(sz) = 2 (4r(1)s§ + 2r(2)(1—s§)+ rm) , (4.12)

where 2 is a unit vector along the z-axis, which is the direction of the strong dc
magnetic ﬁeld.

We have assumed in Eq. (4.12) that S >> 71. Note that, in dimensional units,
S = [LI/h, where L is the angular momentum, whereas in the classical temperature

limit 7'1 2 kT/ﬁwp or kT/ 2m F depending on the scattering mechanism. Therefore

61

the condition S >> 7'1. imposes a limitation on temperature.

Equation (4.12) is reminiscent of the Landau-Lifshitz equation for magnetization
of a ferromagnet. However, in contrast to the Landau-Lifshitz equation a. retardation-
free equation of motion for a classical spin could be obtained only in the rotating
frame, that is, in slow time 7'. The term with 039 describes precession of a spin with
energy (quasienergy. in the present case) 9. The term (5) d describes the effective
friction force. It is determined by the instantaneous spin orientation, but its form is
different from that of the friction force in the Landau-Lifshitz equation.

We emphasize that Eq. (4.12) is not plienor‘nenological, it is derived for the nii-
croscopic model of coupling to the bath (4.3), (4.5). We now consider what would
happen if we start from the Landau-Lifsliitz equation and switch to the rotating
frame using the RMI’A in the assumption that the resonant driving is comparatively
weak, A << 000 [cf. Eq. (4.2)]. In this case one should keep in the expression for the
friction force only the leading term in the effective magnetic ﬁeld, i.e., assume that
H I] 2. The result would be Eq. (4.12) with a dissipative term of the same form as
the term oc F(3) but without dissipative terms that have the structure of the terms
oc F (1), F(2). However, these latter terms play a major role for Sl\='I.\*Is [17, 69, 70, 71]
and for phonon scattering by nuclear spins.

As mentioned in the Introduction, the dynamics of a single-domain magnetic
nanoparticle in a circularly polarized ﬁeld was studied using the Landau-Lifsliitz-
Gilbert equation in a series of papers [39, 79, 80, 81]. It is clear from the above
comment that the results of this analysis do not generally describe resonant behavior
of SMMS. l\/Ioreover, periodic states in the rotating frame predicted in Ref. [39] do

not arise in resonantly excited spin systems with relaxation o< 143), as shown below.

62

4.3.1 Stationary states in the rotating frame for weak damp-
ing
A classical spin is characterized by its azimuthal and polar angles, (25 and 0, with

sz 2 cos 6, 317 = si116cos 05, 8y = sindsinrf). In canonically conjugate variables 06,53

equations of motion (4.12) take the form

.4. = —,.g—rd(sz><1—s3>,

”2 cos (0, cf.

where g as a function of .32, 90 has the form 9 = —(sz + n)2/2 — f(1 — 3%)
Eq. (4.7). we note that the dissipation term is present only in the equation for .92.

In the absence of relaxation, precession of a spin with given 9 corresponds to
moving along orbits on the (06, sz)-plane. The orbits can be either closed or open; in
the latter case c0 varies by 27r over a period, cf. Fig. 4.1. There are also stationary
states where the spin orientation does not vary in time. Generally, relaxation breaks
this structure. If it is weak it makes some of the stationary states asymptotically
stable or unstable and can also transform some of the orbits into stable or unstable
limit cycles, which correspond to periodic oscillations of sz and 0’) in the rotating
frame. The frequency of these oscillations is determined by the system nonlinearity
and is not immediately related to a combination of the modulation frequency and the
Larmor frequency, for example.

Stationary states of Eq. (4.13), which is written in the rotating frame, correspond
to the states of forced vibrations of the spin components 33;, 8,, at frequency L.) F in the
laboratory frame. Periodic states in the rotating frame correspond, in the laboratory

frame, to periodic vibrations of 8;; and to vibrations of 31-, 8,, at combination frequen-

cies equal to w F with added and subtracted multiples of the oscillation frequency in

63

 

 

 

 

 

 

Figure 4.1: Phase portraits of the spin on (0, (23)—plane (32 = cos 0). The data refer
to [‘(1) = Fl?) = 0, 143) = 0.1, and f = 0.3. In panels (a)—(d) ii = —0.6, -0.2,0,0.2,
respectively.

the rotating frame (which is small compared to (up). In what follows we keep this
correspondence in mind, but the discussion refers entirely to the rotating frame.
The analysis of stability of stationary states is based on linearizing the equations
of motion near these states and looking at the corresponding eigenvalues A1, A2 [72].
In the absence of damping the stationary states are either hyperbolic points (saddles)
with real A13 or elliptic points (centers) with imaginary A12. From Eq. (4.13), a

ﬁxed point is hyperbolic if /\1/\2 = ’D < 0, where
2
D = egg 63,9 — (6,6,,g) (4.14)
(the derivatives are calculated at the stationary state). On the other hand, if D > 0

64

the stationary state corresponds to an elliptic point, orbits g = const are circling
around it.

For weak damping, hyperbolic points remain hyperbolic. On the other hand, a
center becomes asymptotically stable (an attractor) or unstable (a repeller) for T < 0

or T > 0, respectively. Here T = —E)[I‘ d(sz)(1 — SEN/052, or in explicit form
’1' = —4sz [4r<1>(1— 2.5-3) — 4r(2)(1— 33,) — rm] , (4.15)

where 82 is taken for the appropriate center; A1 + A2 = T. The sign of T determines
stability of a stationary state also where dissipation is not small.

The quasienergy g has symmetry properties that the change f —> — f can be
accounted for by replacing 05 —> 05+7r, 3,, —> 32. This replacement preserves the form of
equations of motion (4.13) also in the presence of damping. Therefore in what follows
we will concentrate on the range f 2 0. On the other hand, the change [1. —> —,u
would not change 9 if we simultaneously replace 0‘) —-> 05, sz —> —32. In equations of
motion one should additionally change T —> —T. Therefore, if for u 2 14(0) < 0 the

(0)

system has an attractor located at a given (05(0),.92 ), then for p. = —;i.0 it has a
(0)

repeller located at ¢(0), —sz . This behavior is illustrated in Fig. 4.1, where panels

(b) and (d) refer to opposite values of 11..

4.3.2 Saddle—node bifurcations

The function g(s) has a form of the free energy of a magnetic moment of an easy axis
ferromagnet, as mentioned earlier, with p. and f corresponding to the components
of the magnetic ﬁeld along and transverse to the easy axis, respectively. It is well
known that 9 may have either two or four extreme points where 89/833 = 89/805 = 0.

The region where there are four extrema lies inside the Stoner-Wohlfarth astroid

65

‘ uh --—— ~1‘**‘—“ b‘-f,ﬁ*—,‘:t\.u‘;:._“; -1. .

[82] I f [2/ 3 + [142/3 = 1 on the plane of the dimensionless parameters 11, and f, see
Fig. 4.2(a). The extrema. of g outside the astroid are a minimum and a maximum,
whereas inside the astroid g additionally has a saddle and another minimum or max-
imum. All of them lie at (0 = 0 or c") = 7r.

In the presence of weak damping, the minima and maxima of g become stable
or unstable stationary states. \Ve note that there are no reasons to expect that the
stable states lie at the minima of 9, because 9 is not an energy but a quasienergy of
the spin. The number of stable / unstable stationary states changes on the saddle—node
bifurcation curve on the (f. ir)-plane. The condition that two stationary states merge

[72] has the form
D + 70,0,.,g = 0. (4.16)

For weak damping a part of the curve given by this equation is close to the astroid.
On the astroid 3,: = —sgn(/1)|ir|1/3. Then from Eq. (4.15) for the merging saddle and

node

T = —4Sgll(ﬂ.)‘/1—[f|2/3

x (4r(1)(1— 2[f[2/3) + 4rl2)|f|2/3 + 143)). (4.17)

If damping o< 141) is weak, the node is stable for ,u > 0 and unstable for it < 0. On
the other hand, if F (1) is large, the stability depends on the value of f.

The most signiﬁcant difference between the saddle-node bifurcation curve and
the Stoner-W’ohlfarth astroid is that the bifurcation curve consists of two curvilinear
triangles, that is, the astroid is “split”, see Fig. 4.2(b) and Fig. 4.5 below. This is also
the case for a modulated magnetic nanoparticle [39]. The triangles are obtained from

Eqs. (4.13) and (4.16). After some algebra we ﬁnd that. the “bases” of the bifurcation

66

 

 

 

 

 

-1

Figure 4.2: Saddle—node bifurcation lines. Panel (a): zero-damping limit, the lines
have the form of the Stoner-Wolfarth astroid in the variables of reduced amplitude
f and frequency detuning n of the resonant ﬁeld. Panel (b): nonzero damping,

Fm = 0.1, I‘(1) = {42) = 0. In the dashed region the spin has two coexisting stable
equilibria in the rotating frame.

triangles are given by expression
13 z H.000 — We, (4.18)

to leading order in Pd. This expression applies not too close to the vertices of the
triangles. we note, however, that Eq. (4.18) gives the exact bifurcational value of f B
for ,u = 0 and arbitrary F(1(0).

The shape of the gap between the upper and lower curvilinear bifurcation triangles
depends on the damping mechanism. In particular, the damping o< [‘(1) does not
contribute to the gap for small [11.] (cf. Fig. 4.5), whereas the damping o< ﬂ?) does
not contribute to the gap at small 1 — | [1.]. The damping-induced change of the sides
of the triangles compared to the astroid is quadratic in Pd, far from the small- f range.

The positions of the small- f vertices of the bifurcation triangles fC, .“C for small

damping can be found from Eqs. (4.13) and the condition that Eq. (4.16) has a

67

degenerate root, which gives

12
11.Czi[1—x/3(—F3+Tl“d)/],

f0 z 4054/27)“ 1‘3” (D. + <1/2m1/2 <—rd + 7W4.

where F d and. T are calculated for 33 = 1.

4.3.3 Periodic states and Hopf bifurcations

An important property of the modulated classical spin is the possibility to have
periodic states in the rotating frame. Such states result from Hopf bifurcations in
which a stationary state transforms into a limit cycle [72]. A Hopf bifurcation occurs
if

T = 0, D > 0

in the stationary state. Besides the special case 3; = 0 discussed in Sec. 4.4, the

corresponding stationary state is at sz = 3211: where

1 nﬂtaﬂm—HSIﬂ
'53” _ 02 2141) .. ﬂ?) ’

 

aw)
aziL ﬂ03r®+iﬂw

(the inequality on the damping paran‘ieters follows from the condition (53) H S 1)
The ﬁeld f” on the Hopf bifurcation lines as a. function of the reduced detuning
,u is given by a particularly simple expression for weak damping. In this case, from
second equation (4.13) the phase 0‘) H for the bifurcating stationary state is close to
either 0 or 7r with the additional constraint figzgagg > 0. Then from ﬁrst equation

(4.13) and Eq. (4.19) we ﬁnd that Hopf bifurcation curves are straight lines, in the

68

limit of vanishingly small damping.

_ ,2 1/2 —1 ’
fII—i 1—8211 Hus,” . (4.20)

3/2
|f11| Z [1 "5311] 01‘ |1le Z lszill- (4-21)

The structure of these lines is seen in Fig. 4.5 below. They end on the sadr‘lle-node
bifurcation curves and are tangent to these curves at the end points. A detailed

analysis is presented in Sec. 4.6.

4.4 Hamiltonian-like motion at exact resonance

The spin dynamics (4.12) displays an unusual and unexpected behavior where the
n'iodulation frequency WF coincides with the Larmor frequency 0120, in which case
it = 0. This is a consequence of the symmetry of the quasienergy and the dissipation
operator. In a certain range of dynamical variables (b, .93, the spin behaves as if there
were no dissipation, even though dissipation is present. This behavior is seen in the
pattern of phase trajectories of the spin. An example of the pattern is shown in
Fig. 4.1(c) for the case 1“le = F(3) = 0, but the behavior is not limited to this case.
As seen from Fig. 4.1(c), phase trajectories form closed loops, typical for Hamiltonian
systems.

For | f | lying inside the bifurcation triangles, the Hamiltonian-like dynamics occurs
only in a part of the phase plane. This region of f corresponds to I‘d(0) < | fl < [1 +
I‘(21(0)]1/2 [the upper bound on [f| for 11. = 0 can be easily obtained from Eqs. (4.13),
(4.16)]. Here, the spin has four stationary states. For small [n] two of them have small
[sz|, .32 z —,u/(1 — fcos (p) where sinab z —I‘d(0)/f. One of these states is a saddle
point [05 z —— zircsiri[Fd(0)/f]] and the other is a focus [05 z 7r + arcsiii[Fd(0)/f]].

For 11. = 0 there occurs a global bifurcation, a homoclinic saddle-saddle bifurcation

69

(saddle loop [72]) where the separatrix coming out from the saddle goes back into
it, forming a homoclinic orbit. Simultaneously. the focus inside the loop becomes a
center, T = 0 for .32 = 0. All trajectories inside the homoclinic orbit are closed loops.
In contrast to the case of the vicinity of the double-zero eigenvalue bifurcation [72],
the pattern persists throughout a broad region of f.

We show how a Hamiltonian-like region in phase space emerges ﬁrst for weak
damping. For [11, = 0 the quasienergy 9 corresponds to the Hamiltonian of a spin with
anisotropy energy o< S3, which is in a transverse ﬁeld oc f. Such spin in quantum
mechanics has special symmetry, it can be mapped onto a particle in a symmetric
potential [28, 29]. A part of the classical g = const orbits are closed loops on the
(05, sz)-plane. They surround the center (3;, = 0, 05 = 7r). The orbits are symmetric

with respect to the replacement
32 —’ “32: C9 —* Cf): (4'22)

which leads to 05 —+ —(D, .4, —> 53.
\Neak damping would normally cause drift of quasienergy. The drift velocity

averaged over the period Tp(g) of motion along the orbit is

TI)
<g> = —T,;1 /0 (1705.9 rd<sz><1 — .3) (4.23)

From the symmetry (4.22) and the relation I‘d(sz) = Fd(—sz), we have (9) = 0 on a
closed orbit for ,u. = 0. Therefore a closed orbit remains closed to ﬁrst order in I‘d. Of
course, for open orbits, where 00 is incremented by 27r over a period, (9) 75 0. These
orbits become spirals in the presence of damping.

Spirals and closed orbits should be separated by a separatrix, which must be

a closed orbit itself. Since the separatrix must start and end at the saddle point,

70

we understand that at ,u = 0 for small Pd there occurs a saddle-saddle homoclinic
bifurcation.

The topology discussed above persists as I‘ d increases. The symmetry (4.22) is
not broken by Fd. Indeed, from equations of motion (4.13), any orbit that crosses
32 = 0 twice per period for [1. = 0 has the property (4.22) and therefore is closed.
The closed orbits surround the center 5;, = 0, (25 = 7r — aI‘CSIII(Fd(O) / f ) and ﬁll out the
whole interior of the separatrix loop.

The Hamiltonian-like behavior is displayed also for n = 0 and f lying outside the
bifurcation triangles. Here, the system has two stationary states, both with 32 = 0
but with different g0. From Eq. (4.15), for both of them T changes sign as ii goes
through zero. Because there is no saddle point, for small In] there is no separatrix,
trajectories spiral toward or away from stationary states and possibly limit cycles. It
follows from the arguments above that for ,u = 0 all trajectories become closed orbits.
This is conﬁrmed by numerical calculations for different relaxation mechanisms.

It is convenient to analyze the overall dynamics of the spin system for ,u 71$ 0
separately for the cases where the system does or does not have stable periodic states
in the rotating frame. In turn, this is determined by the interrelation between the

damping parameters, cf. Eq. (4.19). Such analysis is carried out in Secs. 4.5 and 4.6.

4.5 Spin dynamics in the absence of limit cycles

We start with the case where the system does not have limit cycles. It corresponds
to the situation where the damping parameter Fm is comparatively small and the
interrelation between the damping parameters (4.19) does not hold. To simplify the

analysis we set F (1) = Ff?) = 0, i.e., we assume that the coupling to the bath is

71

()

linear in the spin operators and is described by the ii‘iteraction Hamiltonian Hi .

The qualitative results of this Section apply also for nonzero F(1),F(2)

as long as
F(3) + 4F(2) > 4F(1). The bifurcation diagram for this case is shown in Fig. 4.2.
From the form of the function T, Eq. (4.15), it follows that the damping oc I43)
transforms centers of conservative motion with 32 > 0 into unstable foci (repellers),
whereas centers with .52 < 0 are transformed into stable foci (attractors). Therefore
for n < 0 the spin has one stable state. It also has one stable state in the unshaded
region of the half-plane ,u. > 0 (outside the bifurcation triangles in Fig. 4.2). Inside
the shaded regions within the triangles the spin has two coexisting stable states.
Examples of the phase portrait are shown in Fig. 4.1. As expected, for weak
damping the system has a stable and an unstable focus outside the bifurcation tri-
angles, Fig. 4.1(a). In the shaded region inside the triangle it has two stable foci,

an unstable focus, and a saddle point, Fig. 4.1(d). In the unshaded region inside the

triangle there is one stable and two unstable foci, Fig. 4.1(b) (the values of n in panels

(b) and (d) differ just by sign).

4.5.1 Hysteresis of spin response in the absence of limit cy-

cles

The presence of two coexisting stable states leads to hysteresis of the spin response
to the external ﬁeld. Such hysteresis with varying dimensionless parameter it, which
is proportional to the detuning of the ﬁeld frequency, is shown in Fig. 4.3. For
large negative [i the system has one stable state with negative 32, cf. Fig. 4.1(a).
As it increases the system stays on the corresponding branch (the lowest solid line

in Fig. 4.3) until the stable state merges with the saddle point (the saddle-node

72

bifurcation). This happens for ,u > 0 as 11. reaches the bifurcation triangle. As [.0
further increases the system switches to the branch with larger .92 and then moves
along this branch. If it. decreases starting with large values where the system has only
one stable state, the switching to the second branch occurs for ii. = 0.

The hysteresis pattern in Fig. 4.3 differs from the standard S—shape characteristic.
This is the case for any f lying between the minimum and maximum of the bifurcation
triangle for )1. = 0, i.e., for 2T(3) < [f] < (1 + 4F(3)2)1/2. It is a consequence of the
fact that the bifurcation at ,u = 0 is not a saddle-node bifurcation, whereas a most
frequently considered S-shape hysteresis curve arises if both bifurcations are of the
saddle—node type. In our case, for p. = 0 the branch which is stable in the range of
large positive 11 (the upper stable branch in Fig. 4.3) becomes unstable as a result
of the motion becoming Hamiltonian-like. The value of sz on this branch for [u = 0
is 33 = 0, it coincides with the value of 32 at the saddle (but the values of 33, are
different). Therefore when .92 is plotted as a function of ii the branch, which is stable
for large positive ii crosses with the branch that corresponds to the saddle point.
For negative 11 the branch, which is stable for large positive It, becomes unstable, cf.
Fig. 4.1. As 1i decreases and reaches the bifurcation triangle for ,u < 0, the saddle
merges with an unstable equilibrium as seen in Fig. 4.3.

The spin components display hysteresis also if the shaded area of the bifurcation
triangle in Fig. 4.2(b) is crossed in a different way, for example, by varying f. If
the crossing occurs so that the line it = 0 is not crossed, the hysteresis curves have
a standard S shape. We note that hysteresis of 34,3, 3,, corresponds to hysteresis of

amplitude and phase of forced vibrations of the spin.

73

 

 

 

 

 

u

 

Figure 4.3: Hysteresis of spin response in the absence of periodic states in the rotating
frame. The data refer to I‘m = I‘m = 0, I‘m = 0.1, and f = 0.3. The solid and
dashed lines show, respectively, stable and unstable stationary states, the dotted line
shows the saddle point.

4.5.2 Interbranch switching Without hysteresis

The occurrence of Hamiltonian dynamics for it = 0 leads to an interesting and unusual
behavior of the system even outside the bifurcation triangles, i.e. in the region where
the system has only one stable state. In the small damping limit and for I f | > 1 and
[pl << 1 the stationary states are at ct = 0 and (b = 7r, with 32 = u/(f cosq’) — 1). The
stable state is the one with 8,, < 0, whereas the one with .92 > 0 is unstable. As 11.
goes through zero the states with q’) = 0 and 0‘) = 7r interchange stability. This means

that 31; a: cos 05 jumps between —1 and 1. Such switching is seen in Fig. 4.4.

74

 

*~~.. l
-1 ,---.,“—-I-l/, H

-1 0 1

 

 

 

 

Figure 4.4: Frequency dependence of the transverse spin component for ﬁeld ampli-
tudes f where the system has one stable state. The solid and dashed lines show
the stable and unstable values of .92: in the rotating frame. The data refer to
I‘m = [‘(2) = 0, I‘m = 0.1, and f = 1.1. As the scaled frequency detuning )1
goes through ,u = 0 the value of am changes to almost opposite in sign.

4.6 Spin dynamics in the presence of limit cycles

The classical dynamics of the spin changes signiﬁcantly if the spin has stable peri-
odic states in the rotating frame. This occurs where condition (4.19) on the damp-
ing parameters is met. The features of the dynamics can be understood by setting
ff?) = PB) = 0, [‘(1) > 0, i.e., by assuming that damping is due primarily to
coupling to a bath Hill), which is quadratic in spin components, with elementary
scattering processes corresponding to transitions between neighboring Zeeman levels.
This model is of substantial interest for single—molecule magnets [17, 71].

The saddle-node bifurcation curves for weak damping :x F“) are shown in Fig. 4.5.
Inside the curvilinear triangles the spin has four stationary states, whereas outside

the triangles it has two stationary states. In contrast to the case of damping o< F(3)

75

 

Figure 4.5: (a) Saddle-node bifurcation lines for 1‘“) = 0.05,I‘(2) = PB) 2 0. (b)
Saddle-node (solid lines) and Hopf bifurcation (dotted lines) in the limit of small
damping oc I‘ (1). Not too close to the astroid (see Sec. 4.6.2) for weak damping the
system has the following states: (i) a stable and an unstable focus; (ii) two unstable
foci and a stable limit cycle; (iii) a stable and an unstable focus and a stable and an
unstable limit cycle; (iv) two stable foci and an unstable limit cycle.

shown in Fig. 4.2, in the present case the bases of the triangles touch at ,u = 0.
From Eq. (4.17), one of the states emerging on the sides of the triangles is stable for
a > 0, I f | < 2_3/2 and is unstable otherwise; note that the stability changes in the
middle of the bifurcation curves.

The occurrence of periodic oscillations of the spin is associated with Hopf bifur-
cations. In the present case, from Eq. (4.19) the Hopf bifurcational values of 32 are
32 H = i1 / \/2 Therefore Eq. (4.20) for the Hopf bifurcation lines for weak damping

takes a simple form

fl! = 2-1/2 4: H. fH if (02—3/2); (424)

fH = -2_1/2 i1“. f1] 5? (—2—3/2.0)-

These lines are shown in Fig. 4.5(b). For [ f | ~ 1 and far from the end points of
the bifurcation lines, the typical frequency of the emerging oscillations is ~ 1 in

dimensionless units, or ~ DS/h in dimensional units.

76

4.6.1 Phase portrait far from the astroid

Evolution of the spin phase portrait with varying parameters far away from the as-
troid, [it] >> 1, can be understood by looking at what happens as the Hopf bifurcation
curves are crossed, for example by varying f. The result is determined by two char-
acteristics. One is stability of the stationary state for f close to the bifurcational
value f H- The stability depends on the sign of T for small f -— f H (note that T
changes sign for f = f II)- The other characteristic is the sign of the quasienergy drift
velocity (9) for f = f H and for 9 close to its bifurcational value 9 H at the stationary
state. It is given by Eq. (4.23) [note that, generally, (9) oc (g — g”)2 for f = f Hl- A
combination of these characteristics tells on which side of the bifurcation point there
emerges a limit cycle and whether this cycle is stable or unstable.

We write the value of sz at the Hopf bifurcation point as sz H = o/x/2, where
a = :l:1, cf. Eq. (4.19). The bifurcational value of the ﬁeld (4.24) is fy =
i (2‘1/2 + an) cos 0511, where (7)11 is the phase of the bifurcating stationary state.
Linearizing Eq. (4.15) in 32 —.9 Z H and using the explicit form of the determinant D one
can show that, for small f — f H: in a stationary state 5in [T/ (f — f 11)] = —sgn[a f Hl-

Then

854117 = —(C¥Sgnf11) Sgnff — fH)- (425)

The analysis of the quasienergy drift velocity near a Hopf bifurcation point is done

in Appendix B. It follows from Eqs. (BI), (82) that

(Q) = COF(1)(9 - 911)2(1’3|f11|— 15).

 

le — x/i) , (4.26)

[ca/(g — gm] = 43451 (,8
where C > 0 is a constant and ,8 = sgn(fH cos 0511) E sgn(2—1/2 + 0,11.) = i1 [[1 is
related to fH by Eq. (4.24)].

77

The sign of (g) / (g — 9H) shows whether g approaches 91! as a result of damping or
moves away from g H- If sgn [(g) (g - 911)] < 0, the vicinity of the stationary state and
the nascent limit cycle attracts phase trajectories. Therefore at a Hopf bifurcation a
stable focus becomes unstable and a stable limit cycle emerges. On the other hand,
if sgn [(g)(g — 911)] > 0, at a Hopf bifurcation an unstable focus transforms into a
stable one and an unstable limit. cycle emerges.

Equation (4.25) allows one to say on which side of f H- i.e., for what sign of f — f H
the stationary state is stable, since for a stable state T < 0. Therefore together
Eqs. (4.25) and (4.26) fully determine what happens as f crosses the bifurcational
value.

We are now in a position to describe which states exist far from the astroid in
different sectors (i)-(iv) in Fig. 4.5(b). For small I f I and large In], regions (i) in
Fig. 4.5(b), the system is close to a spin in thermal equilibrium, it has one stable and
one unstable stationary state. We now start changing f staying on the side of large
positive 11.. W" hen f crosses one of the bifurcation curves f H = :t (2‘1/2 — I1), the
system goes to one of the regions (ii) in Fig. 4.5(b). On the both bifurcation curves
01 = [3 = —1. Therefore, from Eqs. (4.25), (4.26), when one of these curves is crossed
as I f I increases, there emerges a stable limit cycle, and the stable focus becomes
unstable. As I f I further increases it crosses the bifurcation curves :i:(2—1/2 + [.l.) and
the system goes to one of the regions (iii) in Fig. 4.5(b) (we assume that the crossing
occurs in the region I f HI > 21(2). On these bifurcation curves (1 = S = 1. Therefore,
from Eqs. (4.25), (4.26), when they are crossed with increasing If] there emerges an
unstable limit cycle and the unstable focus becomes stable.

we now start from the range of large negative 11. and small I f I. As we increase If]

and cross the bifurcation curves f H = :t(,u+2_1/2) the system goes from region (i) to

78

one of the regions (iv) in Fig. 4.5(b). From Eqs. (4.25), (4.26), in this case an unstable
focus goes over into a stable focus and an unstable limit cycle emerges. Further
crossing into one of the regions (iii) with increasing I f I leads to a transformation of
a stable focus into an unstable focus and an onset of a stable limit cycle. These
arguments were used to establish the nomenclature of states in regions (i)-(iv) in

Fig. 4.5(b). They agree with the results of direct numerical calculations.

4.6.2 Other bifurcations of limit cycles
Merging of stable and unstable limit cycles

The number of periodic states in the rotating frame may change not only through Hopf
bifurcations, but. also through other bifurcations, where the radius of the bifurcating
limit cycle does not go to zero. The simplest is a bifurcation where a stable limit
cycle merges with an unstable limit cycle (saddle-node bifurcation of limit cycles).
The onset of such bifurcations is clear already from Eq. (4.26). Indeed, at a Hopf
bifurcation point the equation for the period-averaged quasienergy has a form (g) =
0(9 —gH)2 +. .. with c oc [3|le — \/2. For Ile = \/2 on the bifurcation curves (4.24)
with 3 = 1 [the top and bottom dotted lines in Fig. 4.5(b)] the coefficient 0 = 0. This
is a generalized Hopf bifurcation [72], see Fig. 4.6.

At the generalized Hopf bifurcation, in phase space (3,53) a stationary state
merges simultaneously with a stable and an unstable limit cycle. In parameter space
(,u, f), the Hopf bifurcation curve coalesces with the curve where stable and unstable
limit cycles are merging, and the latter curve ends. The bifurcation curves are tangent,
the distance between them scales as a square of the distance to the end point 3 I f HI =
\/2 if the latter distance is small. This is seen in Fig. 4.6. In the comparatively narrow

region between the Hopf bifurcation curve and the corresponding limit—cycle merging

79

curve the system has three limit cycles. One of these cycles disappears on the Hopf
bifurcation curve, so that in regions (iii) in Fig. 4.5(b) there are two limit cycles and
deeper in regions (ii) and (iv) there is one limit cycle. On its opposite end, the curve

of merging limit cycles coalesces with the saddle-loop bifurcation curve.

 

 

 

 

 

 

 

 

 

Figure 4.6: Bifurcation diagram in the limit I‘m —> 0. The diagram is symmetric
with respect to p. = 0 and f = 0 axes, and therefore only the quadrant f 2 0,11 5 0
is shown. Saddle-node, Hopf, and saddle-loop bifurcation curves are shown by the
solid, dotted, and long-dash lines, respectively, whereas the short-dash line shows the
curve on which stable and unstable limit cycles merge.

Saddle loops

Spin dynamics for damping o< I‘m is characterized also by global bifurcations of the
type of saddle loops. This is clear already from the analysis of the end points of the
Hopf bifurcation curves. These points lie on the curves of saddle—node bifurcations.

The corresponding equilibrium point has double—zero eigenvalue, and the behavior of

80

the system near this point is well-known [72]. The Hopf bifurcation curve is tangent
to the saddle-node bifurcation curve at the end point. In addition, there is a saddle-
loop bifurcation curve coming out of the same end point and also tangent to the
saddle-node bifurcation curve at. this point. At a. saddle-loop bifurcation the system
has a homoclinic trajectory that starts and ends at the saddle point.

The structure of vicinities of the end points of the Hopf bifurcation curves is shown
in Figs. 4.6 and 4.7 for the curves ending on the sides and the bases of the saddle-
node bifurcation triangles, respectively. Note that the Hopf bifurcation curves that
crossed at f = 0 in the limit ffll —> 0 are separated for finite F“). They end on the

saddle-node bifurcation curves.

 

 

 

 

 

-0.'73 -0.'71

 

 

 

-1 ' -0.5 ',, 0

Figure 4.7: Bifurcation diagram near the end point of the Hopf bifurcation line which
in the limit I‘fll —> 0 has the form fH = —u—2-1/2. For nonzero F“) this bifurcation
line ends on the saddle-node bifurcation line (4.18). The plot refers to I‘m = 0.0125.
The inset shows a close vicinity of the end point. Hopf, saddle—node, and saddle-loop
bifurcation curves are shown by dotted, solid, and long-dashed lines, respectively.
Other Hopf bifurcation curves that go to fH = 0 for 1‘0) ——> 0 display a similar
behavior near their end points.

81

We have found numerically a fairly complicated pattern of saddle—loop bifurcation

curves. Full analysis of this pattern is beyond the scope of this paper chapter.

4.6.3 Hysteresis of spin response in the presence of limit cy-

cles

Coexistence of stable stationary states and stable limit cycles in the rotating frame
leads to hysteresis of the response of a spin when the modulating ﬁeld parameters are
slowly varied. Examples of such hysteresis with varying scaled frequency detuning )u
and the characteristic phase portraits are shown in Fig 4.8.

The hysteretic behavior is unusual. This is a consequence of the feature of the
spin dynamics for it = 0 where either all phase trajectories are closed loops (for f
outside the curvilinear saddle-node bifurcation triangles in Fig. 4.5) or all trajectories
in a part of the phase plane are closed loops (for f inside the triangles in Fig. 4.5). As
a result two or more states (stationary or periodic) simultaneously loose or acquire
stability as it. goes through 0. This leads to an ambiguity of switching, a “Buridan’s
ass” type situation. W here a stable branch looses stability for u = 0, the system has
more than one stable state to switch to. Also, in contrast to the situation of Sec. 4.5
where the system had no limit cycles, hysteresis emerges whether the varying ﬁeld
parameter does or does not cross the saddle—node bifurcation lines.

Figures 4.8(a) and (b) show the behavior of the system with varying It for f inside
and outside the saddle—node bifurcation triangles, respectively. It should be noted that
we chose f in Fig. 4.8(a) so that the saddle-loop bifurcation line is not encountered,
which provides an insight into the most basic features of the hysteresis. In addition,
in Fig. 4.8(b) we do not show an extremely narrow region near Hopf bifurcation lines

)1 x :l:( f — 2’1/2) where the system has small-radii stable and unstable cycles which

82

 

 

 

 

 

 

 

 

Figure 4.8: Panels (a) and (b): hysteresis of the spin dynamics with varying scaled
detuning of the modulating ﬁeld frequency 14. In (a) f = 0.4, so that 11 goes through
the curvilinear bifurcation triangle in Fig. 4.5. In (b) f = 1.2, it lies above the
triangles. Bold solid lines, long dashed lines, and dotted line show stable and unstable
equilibria and the saddle stationary state, respectively. Pairs of thin solid lines and
short dashed lines show the boundaries (with respect to sz) of stable and unstable
limit cycles. Panels (c) and ((1): phase portraits for M = 0.2. In (c) and (d) f = 0.4
and 1.2, respectively. The arrows show the direction of motion along the trajectories.
The data refer to I‘m = 0.05.

merge on the short-dash bifurcation line in Fig. 4.6.

In Fig. 4.8(a), for large negative 11 the system has one stable state (with negative
32). As it increases this state disappears via a saddle—node bifurcation and the system
switches to a stable limit cycle. For chosen f = 0.4 this happens for 11 z 0.33. With
further increase of ii the limit cycle shrinks and ultimately disappears via a Hopf
bifurcation, and then the stationary state inside the cycle becomes stable.

On the other hand, if we start in Fig. 4.8(a) from large positive [,4 and decrease

83

,u, the stable stationary state via a supercritical Hopf bifurcation becomes a stable
limit cycle. The cycle looses stability at ,u. = 0, and as H becomes negative the system
can switch either to the stable stationary state inside the cycle (with 52 —-> +0 for
p. —> —0) or to a stable stationary state outside the cycle with negative 32. The stable
state with s; —> +0 for p, —> —0, ultimately looses stability with decreasing 11 via a
Hopf bifurcation (at it s — f — 2‘1/2, for small damping, cf. Fig. 4.5). If the system
is in this state, it switches to the stable equilibrium with negative .33.

A typical phase portrait for f = 0.4, 0 < ,u < 0.33 is shown in Fig. 4.8(c). It gives
an insight into the behavior described above. The system has a stable limit cycle with
an unstable focus inside and with stable and unstable equilibria and a saddle point
outside the limit cycle. For 11 = 0 the system has a homoclinic saddle connection,
and all trajectories inside the homoclinic trajectory are closed loops, cf. Fig. 4.1(c)

In Fig. 4.8(b), for large negative ,u the system also has one stable state (with
negative .92). As [1 increases this state looses stability via a Hopf bifurcation (at
it x — f + 2-1/2, for small damping). The emerging state of stable oscillations looses
stability for n = 0. For larger )1. the system switches either to the stationary state
inside the limit cycle (with .92 —> +0 for H ——> +0) or to another stable periodic state.
The coexistence of stable and unstable limit cycles with stationary states inside of
them is seen in Fig. 4.8(d).

As )2 becomes positive and further increases, the stable stationary state inside the
unstable cycle looses stability by merging with this cycle, and the system switches
to the periodic state corresponding to the stable limit cycle in Fig. 4.8(d). For still
larger )2 (it m f + 2—1/2, for weak damping) this state experiences a Hopf bifurcation
and becomes a stable stationary state. The behavior with a decreasing from large

positive values can be understood from Fig. 4.8 in a similar way.

84

4. 7 Conclusions

We have developed a microscopic theory of a resonantly modulated large spin in a
strong static magnetic ﬁeld and studied spin dynamics in the classical limit. We have
taken into account relaxation processes important for large-spin systems of current
interest. They correspond to transitions between neighboring and next-neighboring
Zeeman levels with emission or absorption of excitations of a bosonic thermal bath.
Classical spin dynamics depends signiﬁcantly on the interrelation between the rates
of different relaxation processes. Generally it is not described by the Landau-Lifshitz
equation for n‘iagnetization in a ferromagnet, although one of the coupling mechanisms
that we discuss leads to the Landau-Lifshitz damping in the rotating frame.

We found that the spin dynamics has special symmetry at exact resonance where
the modulation frequency is equal to the Larmor frequency, w F = 020. This symmetry
leads to a Hamiltonian-like behavior even in the presence of dissipation. In the
rotating frame, phase trajectories of the spin form closed loops in a part of or on
the whole phase plane. Therefore when 00 F goes through (.00 several states change
stability at a time.

The simultaneous stability change leads to unusual observable features. Where
the system has only one stable state for a given parameter value, as L.) F goes through
(.00 there occurs switching between different states that leads to an abrupt change
of the magnetization. The behavior is even more complicated where several stable
states coexist for an F close but not equal to am. Here, where w F — r00 changes sign,
the state into which the system will switch is essentially determined by fluctuations
or by history (if u) F is changed comparatively fast).

We found the conditions where the spin has more than one stable stationary state

in the rotating frame. Such stable states correspond to oscillations of the transverse

85

magnetization at the driving frequency in the laboratory frame. Multistability leads
to n'iagnetization hysteresis with varying parameters of the modulating ﬁeld. If the
fastest relaxation process is transitions between neighboring states due to coupling
quadratic in spin operators, the resonantly modulated spin can have periodic nonsinu-
soidal states in the rotating frame with frequency oc DS/h, where D is the anisotropy
energy. In the laboratory frame. they correspond to oscillations of the transverse
magnetization at. combinations of this frequency (and its overtones) and the Larmor
frequency.

Quantum fluctuations of the spin lead to phase diffusion of the classical periodic
states in the rotating frame. As a result, classical oscillations decay. The intensity of
quantum fluctuations and the related decay rate depend on the value of S ‘1. We have
found [83] that the oscillations decay comparatively fast even for S = 10. Therefore
they are transient. Still the classically stable vibrations lead to pronounced features
of the full quantum spin dynamics.

The present analysis can be innnediately extended to a more general form of the
spin anisotropy energy, in particular to the case where along with DSE this energy has
a term E (SE — S3), which is important for some types of single-molecule magnets [24].
In the RVVA, the corresponding term renormalizes D and (no. The analysis applies also
to decay due to two-pl'ionon or two-magnon coupling, which often plays an important
role in spin dynamics and leads to energy relaxation via inelastic scattering of bath
excitations by the spin. Another important generalization is that the results are not
limited to linearly polarized radiation. It is easy to show that they apply for an
arbitrary polarization as long as the radiation is close to resonance.

In conclusion, starting from a microscopic model, we have shown that the classi-

cal dynamics of a resonantly modulated large spin in a strong magnetic ﬁeld displays

86

several characteristic features. They include abrupt switching between magnetization
branches with varying parameters of the modulating ﬁeld even where there is no hys-
teresis, as well as the occurrence of hysteresis and an unusual pattern of hysteretic
inter—branch switching. These features are related to the Hamiltonian—like behavior
of the dissipative spin for modulation frequency equal to the Larmor frequency in the
neglect of spin anisotropy. Along with forced vibrations at the modulation frequency,
the transverse spin components can display transient vibrations at a combination of
the modulation frequency and a slower frequency o< DS/h and its overtones. They
emerge if the fastest relaxation mechanism corresponds to transitions between neigh-
boring Zeeman levels with the energy of coupling to a thermal bath quadratic in the

spin operators.

87

Chapter 5

Quantum-Classical Transition and
Quantum Activation in Modulated

Large-Spin Systems

5.1 Introduction

Large—spin systems are of great ii‘iterest for the study of quantum to classical transi-
tions. The inverse size of the spin, 1 / S , can be seen as an effective Planck number
which controls the “quantumness” of the system. Of particular interest in that con-
text is the effect of hysteresis in a modulated large spin system. As detailed in
chapter 4, a modulated large spin system can have several coexisting stable states in
the semiclassical limit. In the quantum regime for a given set of parameters, even
at zero temperature, all of these states but one become metastable. This leads to
switching between the states and hence hysteresis.

For classical systems in thermal equilibrium switching is often described by the

88

activation law, with the switching probability being IV oc exp (—AU/kT), where AU
is the activation energy. As temperature decreases, quantum fluctuations become
more and more important, and below a certain crossover temperature switching occurs
via tunneling [41, 42, 43]. The behavior of systems away from thermal equilibrium
is far more complicated. Still, for classical systems switching is often described by
an activation type law, with the temperature replaced by the characteristic intensity
of the noise that leads to fluctuations [44, 45, 46, 47, 48, 49, 50, 51, 52]. Quantum
nonequilibrium systen‘is can also switch via tunneling between classically accessible
regions of their phase space [32, 53, 33, 54].

Besides classical activation and quantum tunneling, nonequilibrium systems have
another somewhat counterintuitive mechanism of transitions between stable states.
This mechanism is called quantum activation and has been explained recently for
the cases of a parametrically driven oscillator [55] and a nonlinear oscillator [56]. It
describes escape from a metastable state due to quantum ﬂuctuations that accompany
relaxation of the system [57]. These fluctuations lead to diffusion away from the
metastable state and, ultimately, to transitions over the classical ”barrier”, that is,
the boundary of the basin of attraction of the metastable state.

Quantum activation in periodically modulated systems can be understood by not-
ing that metastable states are formed as a result of the balance between external
driving and dissipation due to coupling to a thermal bath. Dissipation corresponds
to transitions to lower energy states with emission of excitations of the bath. Because
energy of modulated systems is not conserved even without dissipation, it is more
convenient to describe them by the F loquet (quasienergy) states rather than the en-
ergy eigenstates. Emission of bath excitations may result in transitions to both higher

and lower quasienergies, albeit with different probabilities [57, 55, 56]. The higher-

89

probability transitions lead to relaxation towards a metastable state, whereas the
lower—probability transitions lead to effective diffusion away from it, a ﬁnite—width
distribution over quasienergy, and metastable decay even in the zero temperature
limit.

Because of the similarities of a ii’i(_)(‘lulated spin system with a non-linear oscilla-
tor it. can be expected that the dynamics of the spin is also governed by quantum
activation.

In this chapter we extend the analysis of the previous chapter 4 of a modulated
large spin system from the semiclassical limit to the the quantum regime. In section
5.2 we introduce spin coherent states and reformulate the quantum kinetic equation
in the spin coherent state representation. This helps us to obtain a better under—
standing of the dynamics of the system in the presence of limit cycles. In section 5.3
we investigate the stationary quasienergy distribution of the system which enables
us to get insight. into various phenomena such as switching between coexisting stable
states through quantum activation as well as abrupt switching between magnetiza-
tion branches even in the absence of hysteresis, and Hamiltonian—like motion in the

quantum regime.

5.2 Dynamics of modulated large-spin systems

To investigate the dynamics of large-spin systems and their transition from the quan-
tum to the semiclassical limit it is beneﬁcial to express the density matrix of the

system in the overcomplete basis of spin coherent states.

90

5.2.1 Some properties of spin coherent states

A spin coherent state is uniquely deﬁned by relating it to a point on the surface of a

sphere with integer or half integer radius S [84],
14> = (1+€€*)’Sexp(€S—)IS>
s 1/2
:1: —S 25 s—m
.2 1 ".
< +54 1 ”2:5 ( _ m) a 1m) 0 1)

where S 3 Im) = 711. Im) and E = eff tan 6/ 2. The azimuthal and polar angles, 05 and 6,

parameterize the spin coherent state and determine its average direction,

(£IS$I€) = Ssinqbsin 6, (5,2)
(EISylé) = Scosqﬁsin 6, (5.3)
(HS/Zia = SCOSQ- (5.4)

Another important property of spin coherent states is that they satisfy the minimum
Heisenberg uncertainty relation in the sense that spin components orthogonal to the

mean spin vector have equal and minimal dispersion [85, 86, 87],

I
ASx/ASy/ = -2- I<Sz’>l’ (5.5)

 

where AS = \/(Sz) — (S)2, 2’ is the direction of the mean spin vector, and 513’, y’ are
the directions of the respective orthogonal spin components. Because of this property
spin coherent states are states that come closest to the classical points in phase space
of the system in the large spin limit. The use of spin coherent states is therefore
appropriate to analyze the transition from quantum to classical behavior.

Spin coherent states form an overcomplete set in Hilbert space; they are generally
non-orthogonal to each other [84]. The overlap probability between two states is given

by
1+ n1n2)25

[(6352)]? = ( 2

91

 

where n1 and n2 are unit vectors in the directions speciﬁed by (091, 61) and (902, 62),
respectively. Hence, more distant states have smaller overlaps. The completeness

relation includes a weight factor and has the form

2S+i (12g _
,, / ——,1+,,,), 15> <41 — 1. (57>

 

If integrating over 6 and (f) the completeness relation transforms into

2S+1
47r

 

/sin(6)dc’9d6 I0), 6) (05,6I = 1. (5.8)

5.2.2 The master equation in the spin coherent state repre-

sentation and its semiclassical limit

Starting from the quantum kinetic equation (4.8), we will now demonstrate an alter-
native way of deriving the classical equations of motion as they are presented in Eq.
(4.13). For that purpose we switch to a representation of the system’s density matrix
in terms of spin coherent states. For us it is sufficient to consider the expectation
value of the density matrix p for a given spin coherent state [6). This expectation

value is the probability to ﬁnd the system at time t in the state [6),

MM) = (El/40K). (5-9)

In the semiclassical large spin limit, S —+ 00, this quantity becomes the delta function

6(qb — ¢0(t))6 (6 — 6(t)). To proceed we introduce the unnormalized spin coherent state

 

é)=(1+5s*>514> (5.10)

92

and make the following observatirms:

5.5) = (S— 5— 5,)I6‘)
s- 5) = $5) (5.11)
5+ E) = (285— 4 %)IE)

which can be veriﬁed directly by plugging in the sum from Eq. (5.1). Note that

 

for a sequence of spin operators the sequence of their respective differential operator
‘b"""l‘ ss~_3S 0"
expressions must. ,e 111 reverse orr er. e.g. z _ 5 — U—C -— 5 '35 5 .
S
As a consequence and if we treat 5 and 5* as two independent variables the

following expressions hold true:

 

 

(SI/155) _ (S '5 To)“
( 25— 5) = 0% (5.12)
( 28+ 5) = (285— a —,,,)2

where 15 = 13(5,5*,t) = <5 pI5>. Note that in expressions where a spin operator

is positioned to the left of the density matrix its hermitian conjugate acts on the
bra component of the scalar product, e.g. <5IS+pI5> = 5%)} or as another exam-

ple (5 z i) = 034—. (-5 Eag)z3 Since (6 €)= Em=—SC77"(.s?§n)1/2€S_ma

1 2 , . .
where cm— — Z,,__ S (83,) / (5 *)S_”, this formalism allows us to express all relevant

        

 

 

gpg

quantities by differential operators which act on polynomials of order 2S in 5 or 5*,
respectively.

In order to relate expressions (5.12) to normalized spin coherent. states we note
that (5|SlpSgl5) = (1+55*)_2S<5ISlpS2I5> and 13 2 (1+ 55*)25 p. This leads to
the following replacement rules for Eqs. (5.12):

343+ 255*
ag a: 1+5?“

 

It) —> 14>. 2 —> p (5.13>

93

With these results we are now prepared to transform the master equation (4.8) to
a differential equation. We will here rewrite the master equation in a slightly different

form:
p'=1'l/).H0/SD] —SF(1)p—SF(2)p—SF(3)p, (5.14)

where H0 / SD = —11SZ — 21353 — f Sr and p denotes the partial derivative of the
system s density matrix with respect to the slow dimensionless time T— - S Dt p= _ 3?.
Starting with the coherent part of the equation we obtain the following second

order partial differential equation:

13110 = ifEllpaHo/SDIIQ (5-15)
_ ,- 1 1-66’“ i 2_ g2_.§2_32£
_ '((“_2.5‘+i+gg=1)f+2(g 1)) 0g 5588+”

Terms of order 1 / S are quantum corrections to the semiclassical equations of motion.

 

 

In the semiclassical limit, S —> 00, we can neglect these terms which leaves us with

the ﬁrst order differential equation

1— * 0
p110=i((11+1+::*)5+§(52 —1)) é—:+C .c., S>> I. (5.16)

Using the method of characteristics or simply by comparing coefficients with p =

 

60? = {—113 — (213151— 06*4 J—S— we ﬁnally obtain an ordinary differential equation for 5:

 

(17' ()5dT
. d5 . _ *
{HOE%=—z((p+1+::*)5+f(52—1)>, S>>1. (5.17)

For brevity, we analyze the dissipation terms (4.9) only for the case of zero tempera-

ture.

sr<1>p = 3131““)(Lle—2Lle+leL),L=S_SZ+SzS_
sr<2>p = 31-3r<2>(s_2,s2p— 2S2pS++ ps2s2) (5.18)
sr<3>p = §r<3 >(s+s_ p— 25 pS++pS+S_ )

94

In order to determine the contributions of the dissipation terms to the ordinary dif-

ferential equation in 5 we follow the same steps as above, i.e. calculate the coefficient

of g? in 2,025,) = — <5ISF(1’2’3)pI5> and drop terms of order l/S or smaller.

In the large spin limit we obtain the following ﬁrst order partial differential equa-

tions (the full equations which contain higher order terms in 1 / S are presented in

Appendix C; there it. is shown that in fact pm) and [21.02) are partial differential

equations of second and fourth order, respectively):

2 (1)=—8F(1) < 2(1—££*>(1—5£*<6—££*>>
I‘

 

(1 +5503

 

1—£4* 2 62 .82
+(1+e*) ($525)): 221’

 

. = _ (2) 46841 " 155*),
Pm) 16F ( (1+€€*)3 P

 

5* 01) .01?
+(1+551)2( EH 83)) S>>1’

21— * a a.
2,2.) = —2r<3> (—(—E€JP+ a—p + 15* 7’ ), S >> 1.

 

1+55* ’05 05*

Hence,

 

. 1_ * 2
£2“) = 8F(1)€(1+::*) ’

€26“
(1+€£*>2’
1515(3) 2 2I‘(3)5.

51,2, 2 16142) S>>1

The resulting complete classical equation of motion is

5 Z5110 + 52(1) + 52(2) +5115»-

(5.19)

(5.21)

(5.22)

(5.23)

In order to compare this result with Eq. (4.13) we have to replace 5 (t) by ¢>(t) and

6(t). Therefore, with
eff)

. _ . "Cb .
5 —1.el tan (6/2)0’) + ——cos2 6/20

95

(5.24)

and equating real and imaginary part of Eqs. (5.23) and (5.24) we ﬁnally obtain

5'9 = —,u. — cos6 —+— f cos qbcot 6, (5,25)

6 = fsin (25 + 2sin6 (4F(1) cos2 6 + 2F(2) sin2 6 + PC”).

These two equations are identical with Eqs. (4.13) as one can see by substituting 82
by cos6 in Eq. (4.13).

Note that in order to check the validity of Eqs. (5.16) and (519-521) one can for
example inspect whether Tr(p) is conserved. Using the completeness relation (5.7)

we obtain that.

 

Tr(p)

=2s+1f d5d5* (5.25)
(

——p.
71 1 + 56*)2
Plugging in Eqs. (5.16) and (5.19-5.21), integrating by parts, and making use of the

periodic boundary conditions one can easily show that indeed %Tr(p) = 0.

5.2.3 Dynamics of the system in the presence of limit cycles

Of particular interest is the quantum-classical transition of limit cycles in the system.
Limit cycles can occur in the semiclassical limit if the relaxation parameters in Eqs.
(5.14) and (5.18) satisfy the condition I‘m 2 I‘m + if“), as has been derived in
4.3.3. In order to describe limit cycles in the quantum regime it is useful to switch
to a new set of real local coordinates, 5H and 51; In the vicinity of the limit cycle, 5“
describes the length along a path on the classical limit cycle, whereas £1. measures
the distance locally perpendicular to that path. Therefore, on the limit cycle it is
ii = O-

A similar analysis has been done for the case of limit cycles in arbitrary dimen-
sional classical Markovian systems described by the Fokker-Planck equation [88].

There the dynamics results in a stationary probability distribution in the shape of a

96

nearly circular crater where the ridge of the crater corresponds to the deterministic

trajectory of the limit cycle. Here, we are interested in analyzing the stationary

7Z'/2

 

 

Figure 5.1: Spin density distribution, [35,, as a function of the azimuthal angle <2
and polar angle 6. S = 10, I‘m = 0.01,I‘(2) : TB) = 0, ,u = 0.6, and f = 0.4.
The dotted line indicates the position of the corresponding classical stable limit cycle
which coincides with the maximum of Pat-

solution of the system’s density matrix. In order to relate the properties of the density
matrix, p, to the classical language of limit cycles we investigate its corresponding
spin coherent state probability density distribtution p = (05, 6I p [05, 6).

We now assume that p is expressed in form of two real coordinates, (3:1, (1:2) which
are locally orthogonal to each other, e.g. (<2, 6) or (Re(5), Im(5)). The transformation
to the new coordinates 51 E 5“ and 52 E 5i is then a unitary transformation given
by

(905- . . —
B12 : f; (m = 1.2). where B 1 = 3T (5'27)

97

 

 

 

 

 

 

 

 

Figure 5.2: Same as in Fig. 5.1 except that here S = 30.

In the large spin limit, S >> 1, 11:1 and :32 are state variables and we know how to obtain
their deterministic equations of motion (cf., for example, derivation of Eq. (5.25))

E

d, =Az'(a=1.:v2). i=1.2. (5.28)

The dynamic equations of the new variables are therefore

(1152' 552' 61132 —1 .

For motion directly on the limit cycle the ﬁrst coordinate, 5”, changes with velocity

V along the cycle and the second coordinate is zero

al€11
__ _ 2 2 _
d, —V— I/Al+A , £1 —0. (5.30)

From Eqs. (5.27) - (5.30) we obtain the matrix elements

A A A A
1 — 2 311: 72. B22 = 7% (5-31)

0.2 1 L

 

0.15 _

 

 

O . .
-7T 0
¢ 72'

 

 

Figure 5.3: Cross section through spin density distribution, pst(¢,6 = 2). S = 10,
1‘”) = 0.01, I‘m = I‘(3) = 0, 11 = 0.6, and f = 0.4. The two peaks have the shape of

a normal distribution.

In a close vicinity to the limit cycle, [5]] is small and we can approximate d5 T / (IT by
a term that is proportional to 5]. To calculate this term we carry out the following

Taylor expansion:

d_€_i_ 63A
=2 B,.2 =ZB, ,.2,,—, a: :13 ,2Bz.2%—,l 21.: (5.32)
2': 1 2 2:12 j,l= 1,2
where the derivatives are evaluated on the limit cycle. Therefore,
dfi
— = —L 5. 3
where
EB B 8A2 (5 34)
_ — j,2 1,2—01‘1- '
31—1.?

The system’s master equation (C.2) which has the schematic form of

p = Aop — Z Ar—I—l. + 0(1/5) (5.35)

 

0.1— _

 

 

 

0 . , .
-7Z' 0
¢ 72'

Figure 5.4: Same as in Fig. 5.3 except that here S = 30. The peaks are smaller in
width and height by a factor of \/3 than those in Fig. 5.3.

can now be expressed in {H and éi coordinates for small values of |g,| and brought

into the form

. 0' 0
p = Aop- V—p + 141-3
051

22$” + 0(1/5) (5.35)

The solution of
1') = 0 (5.37)

is the spin density distribution of the system in the stationary limit, Pst- If terms of
order 1 /S and higher are disregarded Eq. (5.37) leads, in the vicinity of the limit cycle,
to a delta function that is zero everywhere but on the limit cycle itself. Quantum
corrections, i.e. terms proportional to powers of 1 / S in (5.37), include higher order
derivatives of p which lead to a smearing of the delta function. Applying the general

theory that has been developed in [88], one sees that the delta function becomes a

distribution that is maximal on the limit cycle. Its cross section perpendicular to

100

n“- 1!! '1‘ f

 

the cycle has a Gaussian peak which has a width that scales as 1/\/S. These results
are illustrated in Fig. 5.1-5.4 where pg, is plotted as a function of 6 and (b for two
different sizes of the spin system, S = 10 and S = 30. The graphs show clearly that
the maximum of the distribution coincides with the location of the classical limit
cycle. Fig. 5.3 and 5.4 display the cross sections pg, ((2, 6 = 2), for the cases S = 10
and S = 30 respectively. Indeed, these are normal distributions and the width of the
Gaussian peak of the spin 10 system is W = M3 times larger than the width of

the normal distribution of the spin 30 system.

5.3 The stationary limit of modulated large-spin

systems

we will now show that an almost resonantly driven large spin system can, even at zero
temperature and in the absence of tunneling, switch from a metastable to a globally
stable state. we will see that this feature is due to the mechanism of quantum
activation.

For not too large values of the relaxation parameter, the system can be conve-
niently described in its quasienergy representation. There, relaxation is described
by transitions between nearest and next nearest quasienergy levels. As explained
in chapter 3, in the semiclassical limit, quasienergy levels correspond to contours
on the quasienergy surface which are orbits of motion in the limit of zero damping.
Therefore, in addition to the spin coherent state representation also the quasienergy
representation of the system is very suitable to investigate the quantum to classical

transition of a modulated large spin.

101

5.3.1 The stationary distribution in the quasienergy repre-

sentation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 VT 00"...
logpn ------ 00000 0' .....
-20. .°
-40. ° , .
(C) (d)
0 . A , . ‘ .
logpn 0. ...° 0 o...
Woooooooooo....... .' .0. 2
-10. , . 4
-o.5 0.0 g” -o.5 0.0 9"
Figure 5.5: Stationary distribution over quasienergy levels, p7, = (nlpstln), as a

function of quasienergy, (nlmn), for f = 0.4 in the weak damping limit for S = 20 and
T = 0. In panels (a),(c) and (b),(d) it is a = 0.07, 0.5, respectively. In panels (a) and
(b) Fm = I‘m = 0, I‘m —> 0 and in panel (c) and (d) it is Fm -—> 0, I‘m 2 N3) = 0.
In panel (a) the position of the globally stable state corresponds to the position of
the local minimum in g. The maximum of g coincides with the metastable state of
the system. In panel (b) the system has only one stable state which corresponds to
the maximum of g. Panel (c) shows the emerging metastable limit cycle. In panel
((1) the maximum of the distribution corresponds to the position of the classical limit
cycle. In panel (a) and (c) the position of the classical saddle point coincides with
the position where the two quasienergy branches meet each other.

In chapter 3 we calculated the distance between quasienergy levels, 9mm“ E
9(9) of the spin. To analyze the stationary distribution of the system we start with

the case where relaxation is slow so that the broadening of quasienergy levels is much

102

smaller than the distance between them, F << 9.. Then off-diagonal matrix elements
of p in the basis of quasienergy states |m) are small. We note that, at the same time,
off-diagonal matrix elements of p in the basis of Zeeman spin states in the laboratory
frame do not have to be small.

To the lowest order in F / f2 relaxation of the diagonal matrix elements

pm E (m|p|m) is described by the balance equation

0p , ,
8:1 = _2P 2 (”nun/pm _ Wm’mpm’) (5‘38)
ml

 

with the dimensionless transition probabilities

IV I 2 (7": + 1) KHz/(S- lm)|2 + 72. |(m|S_ lm')|2. (5.39)

"I 77?.

7

From the explicit expression of WWW and in accordance with the semiclassical results
in chapter 4, it follows that, even for T = 0, the spin can make transitions to states
with both higher and lower quasienergy g, with probabilities VII/mm“ where m' > m
and m' < m, respect.i\-'ely.

Depending on the values of the driving field strength, f, and the detuning, a, the
probability of a transition to a lower level of g is either larger or smaller than the

probability of a transition to a higher level, that is, W > Wn

2/7 ’7
"In I or II n’n < ”7112’

n,
for n’ > 77., respectively. This agrees with the classical limit in the underdamped
regime in which the stable states of a modulated large spin can be both in minima
and maxima of the quasienergy surface g((j),6) or, in the case of limit cycles, even
somewhere in between. However, along with the drift down or up the quasienergy 9,
even for T = 0, there is also diffusion over quasienergy away from the stable states on
9. due to nonzero transition probabilities ”2,1,”; with n’ > n. This diffusion leads to

a quasienergy distribution which has maxima at stable points or stable limit cycles

on g and falls of rapidly if moved away from them. In the range of multistability,

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.6: Hysteresis of the magnetization for different values of S and T = 0. The
thick solid line shows the normalized expectation value, (S2) / S , as a function of
the detuning parameter, a, for the system in the stationary limit for f = 0.4 and
F (1) = I‘m 2 0, F (3) = 0.1. Thin solid lines show the position of classically stable
states (cf. Fig. 4.3). The dashed thick line shows the uncertainty in the magnetization,

\/(S§) — (Sz)2/S—1. In panels (a)-(d) it is S = 2, 5, 10, 30, respectively. In the range
where the system switches between the two stable states the uncertainty in S 2 has a
peak. The width of the transition region depends sensitively on S.

 

generally, only one of the stable states is predominantly occupied. This state plays
the role of the globally stable state, all other maxima in the quasienergy distribution
correspond to metastable states.

In order to analyze the stationary limit in the case of finite damping the full master

equation 5.14 needs to be solved:
[2 = 0. (5.40)

Fig. 5.5 shows examples of the quasienergy distribution for the cases of relaxation

104

 

(a) (b)

 

 

 

 

 

 

 

Figure 5.7: Hysteresis of the magnetization for different values of S in the presence
of limit cycles. T = O. The thick solid line shows the normalized expectation value,
(S 3) / S, as a function of the detuning parameter, ,u, for the system in the stationary
limit for f = 0.4 and P“) = 0.05, Fm = F(3) = 0. Thin solid lines Show the position
of classically stable states (cf. Fig. 4.8). The dashed thick line shows the uncertainty

in the magnetization, (RS?) — (Sz)2/S — 1. In panels (a) and (b) it is S = 10 and
S = 30, respectively. There where the system changes from a stable ﬁxed point to a
stable limit cycle a step in the uncertainty in S 3 can be observed.

 

of type F“) and 143), respectively. From there it can be seen that the distribution
coincides with the positions of stable stationary states in the semiclassical limit.

It can be seen that while the distribution falls off exponentially in the vicinity of
stable fixed points it only falls off subexponentially in the vicinity of limit cycles.

If the parameter pair (a, f) is located inside the astroid curve the quasienergy
surface, 9, has a separate well which is connected to the rest of the systen‘i via a
saddle point (cf. Fig. 3.5 and chapter 4.32.) In the zero damping limit, contours on
9 inside and outside the well correspond to different states. In the quantum regime, in
the stationary limit, levels inside the well around the stable state at the minimum of
the well are much stronger populated than levels with the same quasienergy outside

the well.

105

 

5.3.2 Switching and hysteresis

If the system is in the basin of attraction of a metastable state it will ﬁrst decay
towards this state on the time scale of the relaxation time, 1 / F. Quantum diffusion
over quasienergy described by Eq. (5.38) eventually leads to switching to the globally
stable state of the spin. The switching rate WEN, is determined by the probability to
reach the top of the barrier which is located at the boundary of the basin of attraction,

i.e. the saddle point of g(qz'), 6).
IVS“, = W x exp (—SRA) (5.41)

The parameter CSW is of the order of the relaxation rate F. R A plays the role of the
activation energy of escape in the process of quantum activation. It originates from
quantum fluctuations that accompany relaxation of the spin. Note that R A is not
the difference in quasienergies but the difference of the logarithms of populations of
the metastable state and the state close to the classical saddle point.

For large enough values of the relaxation rate, F, the switching between stable
states occurs via quantum activation and not via tunneling. Because of this fact the
dynamics for a modulated large spin is fundamentally different of the dynamics of a
large spin in a static magnetic ﬁeld where tunneling and thermal activation play the
major role.

There are narrow parameter ranges where there occurs switching from one globally
stable state to another. These parameter ranges are analogs of first order phase
transitions. In Fig. 5.6 and 5.7 it can be seen that these transitions are sharp even
for comparatively small values of S. If the detuning parameter, p, is swept non-
adiabatically through this region one can observe hysteresis in the magnetization of

the spin. Fig. 5.8 shows the spin density distribution of the system in a case where

106

C)

WW
23‘ WW

-7Z' 0¢7I-7T

 

 

 

 

 

Figure 5.8: Switching in the presence of hysteresis. Panels (a)-(c) and (d)—(f) show
the spin density distribution for S : 10 and S = 30, respectively. For all panels
T = 0, f = 0.4 and Fm = 0.01,F(2) = f‘(3) = 0. In panels (a) and (d), (b) and (e),
(c) and (f) the detuning has the value of ,u = 0.1, 0.23, and 1.0, respectively. In panel
(a) and (b) the system is inside the Stoner-Wohlfarth astroid where there are two
classical stable solutions. The dominant stable branch in panel (a) is a ﬁxed point
centered roughly at (b = 0. In panel (b) the system is in its switching region; the
second metastable stable branch in form of a limit cycle becomes populated and the
population of the ﬁxed point is reduced. In panel (c) the system is outside the astroid;
the limit cycle remains as the only stable solution. It has a relatively small radius
because the system is close to the Hopf bifurcation where the limit cycle transforms
into a ﬁxed point.

both stable states are nearly equally populated.

For small spin systems quantum tunneling effects become prominent. In Fig. 5.9
the hysteresis of a spin 10 system is compared for two different values of the relaxation
parameter, F(3). Close to multiphoton resonance, if the relaxation parameter becomes
of the order of the multiphoton Rabi splitting (3.4), (23, tunneling effects play a.
dominant role. They are expressed as sharp spikes in in the hysteresis curve. The

width of the spikes depends sensitively on the order of the multiphoton resonance.

107

(b)

 

 

 

 

 

 

 

 

Figure 5.9: Hysteresis of the magnetization for different values of S in the case of
small damping. The thick solid line shows the normalized expectation value, (S 2) / S ,
as a function of the detuning parameter, a, for the system in the stationary limit
for f = 0.4 and P“) = PM = 0,r<3) = 0.01. T = O.Thin solid lines show the
position of classically stable states (cf. Fig. 4.3). The dashed thick line shows the

 

uncertainty in the magnetization, \/(S§) — (Sz)2 / S — 1. In panels (a) and (b) it is
S = 5 and S = 10, respectively. Close to multiphoton resonances tunneling effects
become prominent. There, the system forms superpositions of states inside and out-
side the well of the quasienergy surface 9. As a consequence, the magnetization and
its standard deviation display peaks.

5.3.3 Uniform distribution of quasienergy levels and non-

hysteretic switching

Another interesting feature of the quasienergy distribution can be seen at the symme-
try line a = 0. There the distribution is uniform in a range of quasienergy states which
correspond to the part of phase space where in the semiclassical limit a Hamiltonian-
like behavior can be observed (cf. chapter 4.4). Fig. 5.10b and 5.10d show such
distributions in the region of monostability and bistability, respectively.

As shown in chapter 4.5.2, for p. = 0 and for values of the driving strength f out—
side the astroid region, there occurs non-hysteretic switching between two different
solution branches. Quantum corrections, similar to the case of hysteretic switching,

lead to a narrow parameter range where the two different branches of classical so—

108

mp: “ts-
It

 

(a A . <b2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 4
Iogpn
-2 00.00.0000... -24 L
.4 . . . .4 . . .
-1.0 -0.5 0.0 g -1.0 -0.5 0.0 g
n d n
(C) ( )
O 1 . 1 0 .
Iogpn °
_2 ‘ o...00000°. 1 -20 : _:oooooo _
-4 . . . «40 ' .
-1.o -o.5 0.0 gn as 0.0 92

Figure 5.10: Stationary distribution over quasienergy levels, pn = (n|p3,|n), as a
function of quasienergy, (nlgln), in the weak damping limit, Fm 2 FM = 0, F(3) —> 0,
for S = 20 and T = 0. In panels (a)-(d) it is ,u = —0.01,0,0.01,0.001, respectively.
In panels (a)-(c) it is f = 1.1 whereas in panel ((1) it is f = 0.4. Panels (a)-(c)
show non-hysterestic switching from one stable state to another. Panels (b) and
(d) show a completely or partially ﬂat distribution which corresponds to the part of
the quasienergy surface g where the system displays Hamiltonian-like motion in the
semiclassical limit, i.e. either over the whole phase plane as in panel (b) or from the
maximum in g down to the saddle as in panel ((1). In panel ((1) u = 0.001 instead of
a = 0 was chosen in order to avoid multiphoton resonance.

lutions have almost equal population. However, as shown in Fig. 5.10 for any given
value of the detuning, there is only one maximum in the quasienergy distribution,
hence only one stable state. This is made possible by the symmetry at p = 0, where
the distribution is completely uniform. Fig. 5.11 shows such a non-hysteretic switch-
ing for a component of the spin that is perpendicular to the magnetization axis. In

Fig. 5.12 the switching is illustrated by plots of the spin density distribution.

109

 

 

 

 

 

 

1 1 l 1
\\
sx
1 N .........
0- as _
'1 I “-f ....... I ' l y
-1 0 1

Figure 5.11: Switching in the absence of hysteresis. The thick solid line shows the
normalized expectation value of the transverse magnetization in the rotating frame,
(S1) / S as a function of the detuning parameter, a, for the system in the stationary
limit for S = 10, f = 1.1 and Fm = 1‘9) = 0,F(3) = 0.1. T = 0. Thin solid and
dashed lines show the position of classical stable and unstable states, respectively.
The spin switches between the two states within a ﬁnite transition region.

5.4 Conclusions

In the representation of spin coherent states the state of a modulated large—spin
system in the rotating wave approximation can be expressed in terms of its spin
density distribution which is the expectation value of the system to be in a given spin
coherent state which by itself is represented by a point on the surface of a sphere
with integer or half integer radius S. Since in the large spin limit the spin density
distribution becomes the classical statistical probability distribution of the system
it is well suited for analyzing the transition of the spin from the quantum to the
semiclassical regime. The inverse size of the spin, 1 / S , plays the role of an effective
Planck number which controls the “quantumness” of the system.

As an example we analyzed the role of quantum corrections for the case where

110

 

(a) (b)

 

 

 

 

 

   

0 v
-7Z' 0¢7I-ﬂ' 0¢7I-7Z' 0¢7I

Figure 5.12: Switching in the absence of hysteresis. Panels (a)-(c) and (d)-(f) show
the stationary spin density distribution for F“) = ﬂ?) = 0, F(3) = 0.01 and PU) =
0.01, [‘(2) : F(3) = 0, respectively. T = 0. In all panels S = 30 and f = 1.1, hence
the system lies outside astroid where no coexisting stable solutions exist. In panels
(a) and (d), (b) and (e), (c) and (f) the detuning has the value ofp = ——0.1, 0, and 0.1,
respectively. Classically, the stable solutions in panels (a) and (d) abruptly switch to
another branch at p = 0. In the quantum regime the sharp switching line is smeared
out as can be seen from panels (b) and (e). Note that in panels (b) and (e) the system
is in its Hamiltonian-like state.

the system’s classical motion is along a limit cycle. We showed that in the quantum
regime the spin density distribution displays a crater shaped ring with its maximum
coinciding with the position of the classical limit cycle. This stationary behavior
is a consequence of the diffusion of the system due to quantum ﬂuctuations. The
uncertainty property of the quantum system is manifested in a cross section of the
spin density distribution that has a Gaussian peak with a width that scales as 1 / \/S .

To analyze the system’s stationary limit it is convenient to determine its
quasienergy level distribution. In the zero damping limit, this can easily be done

by solving the balance equation. We showed that the distribution coincides with

111

the positions of stable stationary states in the semiclassical limit. The analysis has
shown that in the range of multistability, generally. only one of the stable states is
predominantly occupied. However, there are narrow parameter ranges where there
occurs switching from one globally stable state to another. These parameter ranges
are analogs of ﬁrst order phase transitions. We show that these transitions are sharp
even for comparatively small values of S. We established that the method of interstate
switching is quantum activation which corresponds to diffusion over quasienergy levels
and found that the switching probability depends on S expor'ientially as exp {—S RA},
where R A is the quantum activation energy.

We demonstrated that switching in the absence of hysteresis also occurs within
a sharp transition range where the two different branches of classical solutions are
almost equally strongly occupied. However, here, in contrast to hysteretic switching
the quasienergy distribution exhibits only one maximum for any given value of the
detuning parameter.

We showed that the semiclassical feature of Hamiltonian-like behavior of a part
of phase space corresponds to a flat distribution of the respective quasienergy levels.

It is interesting to note that the effect of quantum activation only requires the
generic property of a system to have an almost equidistant energy spectrum where
transitions to neighboring levels are caused by emission and absorption of quanta
to and from the environment and pumping to higher levels is caused by an almost
resonant radiation source. Because of that it is that large modulated spins behave sim-
ilarly to oscillators where quantum activation has previously been observed [55, 56].
However, important differences between modulated large-spin systems and oscillators
remain. Both systems follow different Lie algebras and have different syn'nnetry prop-

erties. In contrast to the oscillator the spin has finite dimensionality and undergoes

112

 

different relaxation n’iechanisms. In summary, this leads to differentiating features
of the spin such as limit cycles, Hamiltonian-like dynamics, multistability, and non-
hysteretic switching. Interestingly, the classical character of these features is visible

already for comparatively small spin systems, S 2. 3.

113

 

 

 

Chapter 6

Conclusions

We have studied nonlinear resonant phenomena in two-level and multilevel quantum
systems. we showed that modulated multilevel systems display new coherent
and incoherent effects hat can be observed with currently available experimental
techniques. The results bear on quantum control, quantum computation, and

quantum measurement.

'We have developed a theory of quantum gates based on Landau-Zener (LZ)
pulses. In these pulses the control dc ﬁeld is varied in such a way that the qubit
frequency passes through the frequency of an external radiation ﬁeld. We showed
that an LZ gate can be expressed in a simple explicit form in terms of rotation
matrices. LZ pulses allow one to implement arbitrary single qubit gate operations.
One of our central results is that already a sequence of three LZ pulses can be made
fault-tolerant. The duration of this error compensating pulse sequence is about 4
times the duration of the single pulse and the error of the corresponding gate 8

3

scales with the error 5 in the qubit energy or radiation frequency at least as E .

This is parametrically better than in the case of standard error compensating pulse

114

 

 

2. For typical values of frequency offsets

sequences where the gate error scales as E
LZ gates have errors that are at least 104 times smaller than those of conventional

fault-tolerant on-resonance gates.

We have developed a microscopic theory of a resonantly modulated large spin
in a strong static magnetic ﬁeld. We studied coherent quantum effects, which
occur on times smaller than the relaxation time, as well as the stationary and
quasistationary behavior in the presence of of dissipation. We analyzed the dynamics
of a large-spin system in the semiclassical limit and established a picture of the
quantum to classical transition in such type of systems.

In our analysis we concentrated on the important case of a large spin with an
easy axis anisotropy, where the anisotropy energy is of the form —DS§/2. This
is characteristic of large nuclear spins and molecular magnets. we considered the
standard geometry where a static magnetic ﬁeld is applied along the easy axis
whereas an ac ﬁeld points in the transverse direction. The ac ﬁeld frequency w F is

close to the Larmor frequency (.00.

For the case where the system can be regarded as being isolated from its en-
vironment we studied the coherent resonant response of the spin.

At multiphoton resonance, in the weak-modulation limit, the quasienergies of the
resonating Zeeman states cross. We found that this is also true for the susceptibili-
ties in these states. Such crossing occurs simultaneously for several pairs of Zeeman
states. Resonant multiphoton transitions lift the degeneracy of quasienergy levels,
leading to level anticrossing. In contrast, near resonance, the susceptibilities as func-

tions of frequency display spikes. The spikes of the susceptibilities point in opposite

115

 

 

 

directions, leading to a decrease (antiresonance) or an increase (resonance) of the
response. They have a profoundly non-Lorentzian shape with width and height that
strongly depend on the driving ﬁeld amplitude A. The spikes can be observed by
adiabatically sweeping the modulation frequency through a multiphoton resonance.
We showed that this effect is nonperturbative in A, it is due to a special conformal
property of the classical spin dynamics. Our results bear also on the dynamics of
molecular magnets in a static magnetic ﬁeld and provide an explanation of a number

of experiments that have been done on such systems but remained unexplained.

For the case where the system is coupled to an environment we have taken
into account relaxation processes important for large-spin systems of current interest.
They correspond to transitions between neighboring and next-neighboring Zeeman
levels with emission or absorption of excitations of a bosonic thermal bath. Starting
from the microscopic model we derived the quantum kinetic equation. This equation
is simpliﬁed in the semiclassical limit, leading to classical equations of motion of the
modulated spin. We showed that the classical spin dynamics depends strongly on
the interrelation between the rates of different relaxation processes. Although one
of the coupling mechanisms that we discuss leads formally to the Landau-Lifshitz
equation for magnetization dynan‘iics in the rotating frame, generally the dynamics
in the rotating frame is not described by the Landau-Lifshitz damping.

we found that the spin dynamics has special symmetry at exact resonance where
the modulation frequency is equal to the Larmor frequency, w F = (.120. This symmetry
leads to a Hamiltonian-like behavior even in the presence of dissipation. In the
rotating frame, phase trajectories of the spin form closed loops in a part of or on the

whole phase plane. Therefore when when as F goes through (.00 several states change

116

 

stability at a time. The simultaneous stability change leads to unusual observable
features. Where the system has only one stable state for a given parameter value, as
as w F goes through (.00 the state changes discontinuously, leading to an abrupt change
of the transverse magnetization.

We found the conditions where the spin has more than one stable stationary state
in the rotating frame. Such stable states correspond to oscillations of the transverse
magnetization at the driving frequency in the laboratory frame. Multistability leads
to magnetization hysteresis with varying parameters of the modulating ﬁeld. If the
fastest relaxation process is transitions between neighboring states due to coupling
quadratic in spin operators, the resonantly modulated spin can have periodic
nonsinusoidal states in the rotating frame with frequency oc DS/h, where D is the
anisotropy energy. They are described by limit cycles on the spin phase plane in the
rotating frame. In the laboratory frame, these states correspond to oscillations of the
transverse magnetization at combinations of the limit-cycle frequency, its overtones,

and the Larmor frequency.

In order to extend the analysis from the semiclassical limit to the full quan-
tum regime and describe the effects of quantum ﬂuctuations we developed a
formalism which allows us to transform the system’s quantum kinetic operator
equation into a partial differential equation of motion of the system’s probability
density distribution in the spin coherent state representation. The spin density
distribution is the expectation value of the system to be in a given spin coherent state
which by itself is represented by a point on the surface of a sphere with radius S.
In the large spin limit this probability density goes over into the classical statistical

probability distribution of the system.

117

Spin coherent states are particularly convenient for the analysis of the quantum
analog of a classical limit cycle. There, the spin density distribution displays a crater
shaped ring with its maximum coinciding with the position of the limit cycle. The
cross section of the spin density distribution has a Gaussian peak and the width of
the peak scales as 1/\/S.

We showed that the full stationary distribution of the system has peaks at the
positions of stable stationary states in the semiclassical limit. The analysis demon-
strates that in the range of multistability, generally, only one of the stable states is
predominantly occupied. However, there are narrow parameter ranges where there
occurs switching from one stable state to another. These parameter ranges are analogs
of ﬁrst order phase transitions. We showed that these transitions are sharp even for
comparatively small values of S. We established that the mechanism of interstate
switching is quantum activation which corresponds to diffusion over quasienergy lev-
els. The switching probability depends on S exponentially as exp {—S RA}, where
R A is the quantum activation energy.

We found that switching in the absence of hysteresis also occurs within a narrow
transition range. However, here, in contrast to switching in the region where there
are two or more classically stable states the quasienergy distribution exhibits only
one maximum for any given value of the detuning parameter. We showed that the
semiclassical feature of Hamiltonian-like behavior of a part of phase space corresponds

to a flat distribution over the respective quasienergy levels.

118

 

 

APPENDICES

119

Appendix A

Symmetry of classical spin
dynamics: a feature of the

conformal mapping

Classical equations of motion for the spin components (3.13) can be solved in the
explicit form, taking into account that s2 = 1 and that 9(6, o) = const on a classical

trajectory. For time evolution of the z-component of the spin we obtain

 

22(21 — r3) — r301 - 7'2)Sn2(u% mJ) (A.1)

S T Z
Z( ) 7'1 _ 7'3 —- (7'1 — 7‘2)Sn2(ui m.])

where 1‘1 > T2 > 73 > 7‘4 are the roots of the equation
2 2 2 2
[(7‘ +11.) + 29] + 4f (7" —— 1) = 0 (A2)

and sn(u; m J) is the Jacobi elliptic function. The argument u and the parameter m J

are

~

21 2 an, (I): 1/2,

[(7‘1 - 7‘3)(7‘2 - 74)]

i-D-IH

"U = (7‘1 — 7‘2)(‘7‘3 - T4)/(7‘1 — 7‘3)("‘2 - 7‘4): (A3)

120

 

 

Equation (A.1) describes an orbit which, for a given 9. oscillates between 82 = 7‘1 and
.32 2 7'2; the corresponding oscillations of 81.3,, can be easily found from Eqs. (3.13),
(3.14).

Oscillations of 3,: between 73 and T4 for the same 9 are also described by Eq. (A.1)
provided one replaces u —> u+K(mJ) +iK’ ('"LJ), where K(mJ) is the elliptic integral
and K'(m,]) = K(1 — mJ). Clearly. both types of oscillations have the same period
over 7' equal to 2K(mJ)/JJ. They correspond to the trajectories of types I and II
in Fig. 3.5 that lie on different sides of g(6,d>)-surface. Respectively, the vibration
frequencies for the corresponding trajectories (221(9) and (4211(9) are the same. This
proves the ﬁrst relation in Eq. (3.17).

The Jacobi elliptic functions are double periodic, and therefore 32 is also double

periodic,

32(7) 2 .92

T + (SJ-1(2nK + 22mK’)] (A.4)

with integer n. m. Ultimately, this is related to the fact that equations of motion
(3.13) after simple transformations can be put into a form of a Schwartz-Christoffel
integral that performs conformal mapping of the half—plane 1111 3,, > 0 onto a rectangle
on the u-plane. W" will show now that the mapping has a special property that leads
to equal period-averaged values of 31(7) 011 trajectories of different types but with
the same 9. Because 33(7) is double periodic. cf. Eq. (A4), so is also the function
517(7) 2 —(2f)_1 [29 + (83(7') + m2]. Keeping in mind that the transformation 21 ——>
u + K (my) + z'K’(mJ) moves us from a trajectory with a given 9 of type I to a
trajectory of type II, we can write the difference of the period-averaged values of

333(7) on the two trajectories as

711d

we»; — (2.-re)»; = “,ﬂ 3% Wu (AB)

121

 

iK' ..............

 

 

 

r
K 2K 3K

Figure A1: The contour of integration in the u o< T plane. The horizontal parts
correspond to two trajectories S(7') with the same 9. The values of S(T) on the tilted
parts of the parallelogram are the same. The plot refers to u = 0.125, g = —0.366.

where the contour C is a parallelogram on the u—plane with vortices at 0, 2K, 3K +
iK', K + Hi". It is shown in Fig. A.1.

An important property of the mapping (Al) is that 32(7) has one simple pole
inside the contour C, as marked in Fig. A.1. Respectively, 3,,(7) has a second-order
pole. The explicit expression (A.1) allows one to ﬁnd the corresponding residue. A
somewhat cumbersome calculation shows that it is equal to zero. This shows that
the period-averaged values of s,r on the trajectories with the same 9 coincide, thus

proving the second relation in Eq. (3.17).

122

 

Appendix B

Energy change near a Hopf

bifurcation

In this Appendix we outline the calculation of the relaxation of quasienergy 9 near
a Hopf bifurcation point. For concreteness we assume that Fm = F(3) = 0 and
the only nonzero damping parameter is F (I). For small damping a stationary state
that experiences a bifurcation has phase Q5 H close to either 0 or 7r, whereas .32 H z
i2‘1/2. The dynamics is characterized by two parameters, a = sgnsz H and ﬂ =
sgn[fH cos $11]. The bifurcational value of the ﬁeld for F“) —> 0 is fH = (2-1/2 +
an) cos (2H [cf. Eq. (4.24)].

At the bifurcating stationary state the quasienergy is g H = g (@511, 32H)? it is easy
to see that this is a local minimum of g(c,b, 3,) for ,3 > 0 or a maximum for )3 < 0.
Respectively, on phase plane (g5, .93) the constant—g trajectories close to the bifurcating
stationary state rotate about this state clockwise for )3 > 0 and counterclockwise for
)3 < 0. The angular frequency of this rotation is a: 27r/Tp(gH) = 731/2, where ’D is
given by Eq. (4.14).

We now consider dissipation-induced drift over quasienergy (g). It is given by

123

 

Eq. (4.23). Noting that (95,9 2 q) and using the Stokes theorem we can rewrite this

equation as

(g) 2 5,,-1(2) / (1222.7, (B.1)

where the integral is taken over the interior of the constant-g orbit on the ((10.32)
plane and T E T (.92) is given by Eq. (4.15). At a Hopf bifurcation point T = 0.
Therefore T(sz) in Eq. (B.1) must be expanded in ()32 = 3,3 — SzH-

It is convenient to calculate integral (B.1) by changing to integration over action—
angle variables (1,111). which are canonically conjugate to (3,3, (0), with 9 being
the effective Hamiltonian. The angle 112 gives the phase of oscillations with given
quasienergy g. The action variable I = (2%)"1 f szdgb is related to g by the standard
expression (II/(lg = 'rp(g)/27r % D_1/2; we note that I becomes negative away from
the stationary state for ,8 < 0.

In evaluating expression (8.1) it is further convenient to start with integration
over (,0. It goes from 0 to 27r and corresponds to period averaging for a given I OC
()9 = g —- gH (integration over I corresponds to integration over (59).

If vibrations about ((0H, 32H) were harmonic, the lowest-order term in (53,, that
would not average to zero on integration over 20 would be ((12T/dsg)((533)2 /2 or |ng
(the derivative of T is calculated at the bifurcating stationary state). However, it is
easy to see that the integral over 12’) ()f the linear in (5.93 term in T is also ~ ()g. It can
be calculated from equation of motion (,0 = (9,9,9 by expanding the right-hand side to

second order in (53,, (5(0 and noting that (,0 = 0, where the overline means averaging

over 122'. This gives, after some algebra,

_ —2
T: 64F(1)O((dg) (23/2,3|f[{| — I)
x (Bum — 22/2) . (13.2)

124

This expression combined with Eq. (B.1) shows how the energy relaxation rate de-

pends on the ﬁeld f”. It is used in Section 4.6 to establish the full bifurcation

diagram.

125

 

Appendix C

Spin Coherent State
Representation of the Master

Equation

Eqs. (5.16) and (519-521) describe the system’s master equation in the coherent
state representation in the large spin limit, S >> 1. Here, we present the full equation,
including all quantum corrections, i.e. terms of order 1 / S and higher.

The original operator master equation (5.14) has the form

Switching to the coherent state representation leads to the partial differential equation

1') = (503(5) = PHD +25P(1) +PF(2) +I3F(3). (C?)
where p = p(£,§*, t), 1') E 31%, and

 

. _. 1 1—££* f 2 a, (28212
pffo _z((#_%+1+€€*)€+§( _1))aE—2Ea—€2+C.C. ((3.3)

126

82+ j

_1“(1) J. ______
Z ”05‘0(£*)2p

0324354

4(25 — 1)(1— 66*)(€€*(14 - (56*) — 1+ 25(1 + 66*(652‘ - 6)))

 

52(1 + 33*)3
2((4S3ec‘ — 1)2 — 9(1+((*)2 — 382cc — 3)(3€§* — 1))

 

83(1 +533“)2

26(29 — 65*(50 +195?»
52(1+€£*)2

 

_2(£(*)2(4S2(£(* — 3)2 + 9(1+(3*)2 + 4S(££*<2 + 305*) — 9))

 

52(1+€€*)2

_(2(57<1+ (8)2 + 452(5) + €€*(€€* — 10)) + were +1938) — 19))

 

S~"‘(1+€£*)2
_4(3<(*)2(3 + 366* + 28(38‘ — 3))

 

53(1 + (5*)
SW)"
S3
1653(5 — 2 — (2 + S)£§*)
83(1 + 66*)

 

 

127

 

0i+j

2 . r

094234

3(28— 1)(1—££*)<1—(€*(2 — 4S —((-‘*))
52(1 +5603
4&3 + 4S(S — 2) + (6 + 25(4 + 5(25 — 9)))£€* + 3(1+ S + 282)(££*)2)
53(1 +6502
8(£(*)3(2s(2 + (5*) — 432 — (1+ (3*)2)
83(1 +58)2
2«52(12(1+ 35*)? — 3S2(££*(4 + (5*) — 1) + sew +1338) — 13))
53(1 +6602

 

 

 

 

 

454((*)3(2S — 1 — (8‘)
S3(

 

 

 

1+ 64*)
_2<((*)4
53
253(5 — 25 + (5 + QS)££*)
53(1 + 43*)
0
£4
5’3
K},

 

(£52)2 527) 62 82p (8“)2 3211
S 0609 28052 25 6(8)?

Note, that the terms 15,10 and 12,23) are partial differential expressions of second order

whereas 101.02) are of fourth order.

128

 

BIBLIOGRAPHY

129

 

 

Bibliography

[1] B. E. Kane, Nature 393(6681), 133 (1998). l
[2] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. 76(4), 1037 (2004).

[3] K. R. Brown, A. W. Harrow, and I. L. Chuang, Phys. Rev. A 70(5), 052318
(2004).

 

[4] L. Viola. J. Mod. Opt. 51(16-18). 2357 (2004). J [

[5] P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. A. Lidar, Phys.
Rev. A 71(2), 022302 (2005).

[6] P. Sengupta and L. P. Pryadko, Phys. Rev. Lett. 95(3), 037202 (2005).

[7] H. K. Cummins, G. Llewellyn, and J. A. Jones, Phys. Rev. A 67(4), 042308
(2003).

[8] L. Landau, Phys. Z. Sowjetunion 2, 46 (1932).
[9] C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).

[10] G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, Nature
434(7036), 1001 (2005).

[11] M. N. Leuenberger and D. Loss, Phys. Rev. B 68(16), 165317 (2003).
[12] E. M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60(8), 661 (1988).
[13] D. A. Garanin, J. Phys. A 24(2), L61 (1991).

[14] A. Garg, Europhys. Lett. 22(3), 205 (1993).

[15] J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, Phys. Rev. Lett. 76(20),
3830 (1996).

[16] J. R. Friedman, M. P. Sarachik, J. M. Hernandez, X. X. Zhang, J. Tejada,
E. Molins, and R. Ziolo, J. Appl. Phys. 81(8), 3978 (1997).

[17] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 56(17), 11102 (1997).

130

[18 W. Wernsdorfer and R. Sessoli, Science 284(5411), 133 (1999).

 

l

[19] A. Garg, Phys. Rev. Lett. 83(21), 4385 (1999).

[20] J. Villain and A. Fort, Eur. Phys. J. B 17(1), 69 (2000).
l

[21 W. Wernsdorfer, M. lVlurugesu, and G. Christou, Phys. Rev. Lett. 96(5), 057208
(2006)

[22] W. W'ernsdorfer, Adv. Chem. Phys. 118, 99 (2001).

[23] J. R. Friedman, Exploring the Quantum/Classical Frontier: Recent Advances in
Macroscopic and Mesoscopic Quantum Phenomena (Nova Science, Huntington, I
NY, 2003), chap. Resonant Magnetization Tunneling in Molecular Magnets. :3...

[24] D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nanomagnets (Oxford Uni-
versity Press, 2006).

 

[25] A. Sieber, G. Chaboussant, R. Bircher, C. Boskovic, H. U. Giidel, G. Christou, ﬂ
and H. Mutka, Phys. Rev. B 70(17), 172413 (2004).

[26] M. Evangelisti, A. Candini, A. Ghirri, M. Affronte, G. W. Powell, I. A. Cass,
P. A. Wood, S. Parsons, E. K. Brechin, D. Collison, and S. L. Heath, Phys. Rev.
Lett. 97(16), 167202 (2006).

[27] H. J. Lipkin, N. Meshkov, and A. J. Click, Nucl. Phys. 62(2), 188 (1965).
[28] V. V. Ulyanov and O. B. Zaslavskii, Phys. Rep. 216(4), 179 (1992).

[29] D. A. Garanin, X. M. Hidalgo, and E. M. Chudnovsky, Phys. Rev. B 57 (21),
13639(1998)

[30] P. Ribeiro, J. Vidal, and R. Mosseri, Phys. Rev. Lett. 99(5), 050402 (2007).
[31] D. M. Larsen and N. Bloembergen, Opt. Commun. 17(3), 254 (1976).

[32] V. N. Sazonov and V. I. Finkelstein, Doklady Akad. Nauk Sssr 231(1), 78 (1976).
[33] A. P. Dmitriev and M. I. Dyakonov, Zh. Eksp. Teor. Fiz. 90(4), 1430 (1986).
[34] M. I. Dykman and M. V. Fistul, Phys. Rev. B 71(14), 140508 (2005).

[35] P. W. Anderson and H. Suhl, Phys. Rev. 100(6), 1788 (1955).

[36] H. Suhl, J. Phys. Chem. Solids 1(4), 209 (1957).

[37] G. V. Skrotskii and Y. I. Alimov, Sov. Phys. JETP 9(4), 899 (1959).

[38] D. J. Seagle, S. H. Charap, and J. O. Artman, J. Appl. Phys. 55(6), 2578 (1984).
[39] G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86(4), 724 (2001).

131

[40] M. I. Dykman and M. A. Krivoglaz, Soviet Physics Reviews (Harwood Academic,
New York, 1984), vol. 5, pp. 265—441.

[41] I. Affleck, Phys. Rev. Lett. 46(6), 388 (1981).

[42] H. Grabert and U. Weiss, Phys. Rev. Lett. 53(19), 1787 (1984).

[43] A. I. Larkin and Y. N. Ovchinnikov, J. Stat. Phys. 41(3—4), 425 (1985).
[44] R. Landauer, J. Appl. Phys. 33(7), 2209 (1962).

[45] A. D. Ventcel" and M. I. Freidlin, Uspehi Mat. Nauk 25(1), 3 (1970), ISSN
0042-1316.

[46] M. I. Dykman and M. A. Krivoglaz, Zh. Eksp. Teor. Fiz. 77(1), 60 (1979).
[47] R. Graham and T. Tel, J. Stat. Phys. 35(5-6), 729 (1984).

[48] A. J. Bray and A. J. McKane, Phys. Rev. Lett. 62(5), 493 (1989).

[49] M. I. Dykman, Phys. Rev. A 42(4), 2020 (1990).

[50] R. S. Maier and D. L. Stein, Phys. Rev. E 48(2), 931 (1993).

[51] R. L. Kautz, Rep. Prog. Phys. 59(8), 935 (1996).

[52] O. A. Tretiakov, T. Gramespacher, and K. A. l\-"Iatveev, Phys. Rev. B 67(7),
073303 (2003).

[53] M. J. Davis and E. J. Heller, J. Chem. Phys. 75(1), 246 (1981).

[54] E. J. Heller, J. Phys. Chem. A 103(49), 10433 (1999).

[55] M. Marthaler and M. I. Dykman, Phys. Rev. A 73(4), 042108 (2006).

[56] M. I. Dykman, Phys. Rev. E 75(1), 011101 (2007).

[57] M. I. Dykman and V. N. Smelyansky, Zh. Eksp. Teor. Fiz. 94(9), 61 (1988).
[58] M. Dykman and P. Platzman, Quantum Inf. Comput. 1, 102 (2001).

[59] V. Benza and G. Strini, Fortschr. Phys. 51(1), 14 (2003).

[60] P. Platzman and M. I. Dykman, Science 284(5422), 1967 (1999).

[61] K. Saito and Y. Kayanuma, Phys. Rev. B 70(20), 201304 (2004).

[62] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th.
edition (San Diego, CA: Academic Press, 2000).

[63] V. N. Smelyanskiy, A. G. Petukhov, and V. V. Osipov, Phys. Rev. B 72(8),
081304 (2005).

132

 

 

[64] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier
Butterworth-Heinemann, Oxford, 2004), second ed.

[65] A. Garg and M. Stone, Phys. Rev. Lett. 92(1), 010401 (2004).

[66] L. D. Landau and E. M. Lifshitz, Quantum mechanics. Non-relativistic theory
(Butterworth-Heinemann, Oxford, 1981), third ed.

[67] A. P. Dmitriev and M. I. Dyakonov, JETP Lett. 44(2), 84 (1986).
[68] C. Hicke and M. I. Dykman, Phys. Rev. B 76(5), 054436 (2007).
[69] M. N. Leuenberger and D. Loss, Phys. Rev. B 61(2), 1286 (2000).

[70] S. Bahr, K. Petukhov, V. Mosser, and W. “fernsdorfer, Phys. Rev. Lett. 99(14),
147205 (2007).

[71] M. Bal, J. R. Friedman, W. Chen, M. T. Tuominen, C. C. Beedle, E. M. Rum-
berger, and D. N. Hendrickson, arXiv.org:0706.3471 (2007).

[72] J. Guckenheimer and P. Holmes, Nonlinear Oscillators, Dynamical Systems and
Bifurcations of Vector Fields (Springer-Verlag, New York, 1997).

[73] J. Villain, F. Hartn’ian-Boutron, R. Sessoli, and A. Rettori, Europhys. Lett.
27(2), 159 (1994).

[74] R. L. Melcher, in Proceedings of the 1970 Ultrasonics Symposium (IEEE, San
Francisco, 1971), pp. 35—47.

[75] R. G. Shulman, B. J. \Nyluda, and P. W. Anderson, Phys. Rev. 107(4), 953
(1957)

[76] E. F. Taylor and N. Bloembergen, Phys. Rev. 113(2), 431 (1959).
[77] Y. Izyumov, Proc. Phys. Soc. 87, 505 (1966).
[78] M. A. Krivoglaz and L. B. Kvashnina, Phys. Stat. Sol. 29(1), 53 (1968).

[79] A. Magni, G. Bertotti, C. Serpico, and I. D. 1\’Iayergoyz, J. Appl. Phys. 89(11),
7451(2001)

[80] G. Bertotti, I. l\Iayergoyz, and C. Serpico, J. Appl. Phys. 95(11 II), 6598 (2004).

[81] C. Serpico, M. d’Aquino, G. Bertotti, and I. D. l\/Iayergoyz, J. Appl. Phys. 95(11),
7052 (2004).

[82] E. C. Stoner and E. P. \Nohlfarth, Phil. Trans.Roy. Soc. A 240(826), 599 (1948).
[83] C. Hicke and M. I. Dykman, in preparation.
[84] J. M. Radcliff, J. Phys. A 4(3), 313 (1971).

133

[85] F. T. Arecchi, H. Thomas, R. Gilmore, and E. Courtens, Phys. Rev. A 6(6),
2211(1972)

[86] C. Aragone, E. Chalbaud, and S. Salamo, J. l\‘-’Iath. Phys. 17(11), 1963 (1976).
[87] G. Vetri, J. Phys. A 8(6), L55 (1975).

[88] M. I. Dykman, G. P. Golubev, D. G. Luchinsky, P. V. E. l\*IcClintock, N. D.
Stein, and N. G. Stocks, Phys. Rev. E 49(3), 1935 (1994).

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