Open quantum system theory from an
information perspective
vorgelegt von
Master of Science
Oliver Kästle
an der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
— Dr. rer. nat. —
genehmigte Dissertation
Promotionsausschuss:
Vorsitzende: Prof. Dr. rer. nat. Ulrike Woggon, TU Berlin
Gutachter: Prof. Dr. rer. nat. Andreas Knorr, TU Berlin
Gutachter: Prof. Dr. rer. nat. Peter Rabl, TU Wien
Tag der wissenschaftlichen Aussprache: 28. Juni 2021
Berlin 2021
für Alex, den großen Komprimator
The first principle is that you must not fool yourself –
and you are the easiest person to fool.
—R. Feynman
Abstract
Theoretical investigations of open quantum systems have uncovered a wide variety of
unique dynamical phenomena. For instance, current research in quantum information tech-
nology aims to control and utilize time-delayed system-reservoir interactions and phonon-
induced dynamical dissipative processes for the design of novel chip-integrated quantum
optical devices. This thesis develops and advances multiple theoretical approaches for
the accurate and efficient description of open quantum systems, where a high degree of
information compression is essential to extract crucial information from the open system
degrees of freedom which scale exponentially with the system size. The presented methods
range from correlation expansions to second-order perturbative master equations, real-time
path integral formulations, numerical realizations of high-dimensional tensor networks and,
based on the most recent theoretical developments, machine learning implementations in
the form of artificial neural networks.
We explore all of these techniques with respect to their compression efficiency, performance
and representational limits by applying them to a variety of physical setups and scenarios.
Firstly, emerging non-Markovian memory effects in open quantum systems are investi-
gated, e.g., resulting from interactions with a structured phonon environment or time-
discrete coherent quantum feedback, leading to information exchange and a time-delayed
dynamical interplay between system and reservoir. Examined systems range from solid
state p-wave topological superconductors coupled to a structured phonon environment,
where a memory-induced stabilization and recovery of topological properties is observed,
to semiconductor quantum dot nanostructures exhibiting complete population inversion
and unidirectional quantum transport as a result of dissipation-induced non-Markovian
system-reservoir interactions, and quantum emitters simultaneously subjected to coherent
quantum feedback and a decoherence-inducing structured phonon reservoir, leading to the
formation of dynamical dissipative structures and population trapping.
Secondly, novel machine learning techniques based on artificial neural networks are inves-
tigated, enabling simulations of symmetric open quantum spin-1/2systems with Marko-
vian dynamics. Combining their potential for parallelization and efficient Hilbert space
truncation, they facilitate unmatched degrees of information compression and numerical
performance for large systems. In this thesis, we expand the representational limits of
the restricted Boltzmann machine neural network architecture by developing novel hybrid
sampling strategies for a highly customizable compression of configuration space, rendering
accurate and efficient simulations of asymmetric systems feasible. Finally, the applicability
of the neural network is extended beyond pure spin-1/2systems by introducing a neural
bit encoding scheme for Fock number states, facilitating high performing calculations of
large hybrid open quantum systems with bosonic degrees of freedom.
v
Kurzfassung
Die theoretische Beschreibung offener Quantensysteme erschließt ein weites Feld einzigar-
tiger physikalischer Phänomene und ermöglicht die Erforschung neuartiger Anwendungen
von zeitretardierten System-Bad-Wechselwirkungen und phononinduzierten dissipativen
Prozessen im Bereich der Quanteninformationstechnologie. In dieser Arbeit werden ver-
schiedene theoretische Methoden zur Beschreibung offener Quantensysteme analysiert,
angewandt und weiterentwickelt, für die ein hoher Grad an Informationskompression
entscheidend ist, um effizient relevante Information aus den Systemfreiheitsgraden zu ex-
trahieren, die mit wachsender Systemgröße exponentiell skalieren. Die dargelegten Ansätze
umfassen Korrelationsentwicklungen, Mastergleichungen in zweiter Ordnung Störungsthe-
orie, Pfadintegralformalismen, numerische Implementierungen hochdimensionaler Tensor-
netzwerke sowie Anwendungen künstlicher Intelligenz auf Basis neuronaler Netzwerke.
Die Techniken werden bezüglich ihrer Kompressions- und Leistungsfähigkeit sowie repräsen-
tativer Limitationen beleuchtet. Zunächst werden dazu nicht-Markoffsche Gedächtnisef-
fekte in offenen Quantensystemen untersucht, die durch zeitretardierte Interaktionen zwis-
chen dem betrachteten System und seiner Umgebung entstehen können, etwa in Form von
Wechselwirkungen mit einem strukturierten Phononreservoir oder kohärentem Quanten-
feedback. Die untersuchten Systeme umfassen hierbei einen topologischen Supraleiter,
der mit einem strukturierten Phononreservoir interagiert. Es zeigt sich, dass auftre-
tende dissipative Gedächtniseffekte zu einer Stabilisierung und Verstärkung topologischer
Eigenschaften führen können. Weiterhin werden Quantenpunktsysteme untersucht, in
denen phonon-induzierte nicht-Markoffsche Wechselwirkungen zu vollständiger Popula-
tionsinversion und unidirektionalem Quantentransport führen können. Zuletzt wird ein
Quantenemitter gleichzeitig einem strukturierten Phononreservoir und kohärentem Quan-
tenfeedback ausgesetzt, woraus die Formation von dynamisch-dissipativen Strukturen mit
stabilisierter angeregter Systembesetzung resultiert.
Weiterhin kommen in dieser Arbeit neuartige neuronale Netzwerke zur Beschreibung
offener und symmetrischer Spin-1/2Quantensysteme mit Markoffscher Dynamik zum
Einsatz. Ihre unbegrenzte Parallelisierbarkeit und effiziente Hilbertraumtrunkierung er-
möglichen hohe Grade an Informationskompression und hochperformante Simulationen
großer Systeme. Konkret werden neuartige Hybridmethoden zur approximativen Hilber-
traumabtastung entwickelt, womit präzise Simulationen asymmetrischer Spin-1/2Systeme
auf Basis der restricted Boltzmann machine Architektur erstmals möglich werden. Zu-
dem wird ein neuronales Bitkodierungsformat für Fockzustände entwickelt, wodurch eine
hocheffiziente Beschreibung offener Hybridsysteme mit bosonischen Freiheitsgraden auf
Basis neuronaler Netzwerke ermöglicht wird.
vii
Publications
■O. Kaestle, Y. Sun, Y. Hu, and A. Carmele, “Memory-critical dynamical buildup
of phonon-dressed Majorana fermions,” Phys. Rev. B 102, 245115 (2020).
■O. Kaestle, E. V. Denning, J. Mørk, A. Knorr, and A. Carmele, “Unidirectional
quantum transport in optically driven V-type quantum dot chains,” Phys. Rev. B
103, 115418 (2021).
■O. Kaestle, R. Finsterhoelzl, A. Knorr, and A. Carmele, “Continuous and time-
discrete non-Markovian system-reservoir interactions: Dissipative coherent quantum
feedback in Liouville space,” Phys. Rev. Research 3, 023168 (2021).
■O. Kaestle and A. Carmele, “Sampling asymmetric open quantum systems for
artificial neural networks,” Phys. Rev. B 103, 195420 (2021).
■O. Kaestle and A. Carmele, “Efficient bit encoding of neural networks for Fock
states,” Phys. Rev. A 103, 062409 (2021).
ix
Contents
Abstract v
Kurzfassung vii
Publications ix
Acronyms xv
1 Introduction 1
1.1 Motivation .................................... 1
1.2 Structure of the thesis .............................. 2
I Theoretical background 5
2 Open quantum systems 7
2.1 Density matrix theory .............................. 8
2.1.1 Von Neumann equation ......................... 9
2.2 Interaction picture ................................ 10
3 Master equation theory 13
3.1 Redfield master equation ............................. 13
3.1.1 Bath assumption and Born approximation ............... 14
3.1.2 First Markov approximation ....................... 14
3.2 Lindblad master equation ............................ 15
3.2.1 Second Markov approximation ..................... 17
3.3 Polaron master equation ............................. 19
3.3.1 Polaron transformation ......................... 20
3.4 Franck-Condon renormalization ......................... 21
3.5 Numerical evaluation ............................... 21
4 Real-time path integrals 23
4.1 From classical action to quantum mechanics .................. 23
4.1.1 Principle of least action ......................... 23
4.1.2 Probability amplitude .......................... 24
4.1.3 Path integral formalism ......................... 25
4.2 Path integral formulation of the spin-boson model .............. 26
4.2.1 Spin-boson model ............................ 26
4.2.2 Time-discrete expansion of the density matrix ............. 27
xi
xii Contents
4.2.3 Evaluation of discrete time steps .................... 29
4.2.4 Extremal action ............................. 30
4.2.5 Initial state ................................ 30
4.2.6 Tracing out the reservoir ......................... 31
4.2.7 Time-discrete influence functional ................... 32
4.2.8 Finite memory approximation ...................... 33
4.3 Efficient tensor network implementation .................... 34
4.3.1 Matrix product states and matrix product operators ......... 34
4.3.2 Tensor network realization of path integrals .............. 35
5 Artificial neural networks 39
5.1 Basic principles and definitions ......................... 40
5.1.1 Supervised and unsupervised learning ................. 40
5.1.2 Models .................................. 41
5.1.3 Cost functions .............................. 41
5.1.4 Stochastic gradient descent ....................... 42
5.2 Restricted Boltzmann machines ......................... 42
5.3 Neural density operator ............................. 44
5.4 Implementation of symmetries .......................... 46
5.5 Metropolis sampling ............................... 48
5.6 Stochastic reconfiguration for Liouvillians ................... 49
5.6.1 Stationary states ............................. 50
5.6.2 Time evolution dynamics ........................ 52
5.7 Numerical execution ............................... 54
5.7.1 Observables ................................ 55
5.7.2 Sampling ................................. 55
5.7.3 Logarithmic derivatives ......................... 56
5.7.4 Local Liouvillian ............................. 57
5.7.5 Pre-conditioning ............................. 58
II Non-Markovian phenomena in open quantum systems 61
6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions 63
6.1 Introduction .................................... 63
6.2 Polaronic Kitaev chain .............................. 64
6.2.1 p-wave superconducting chain coupled to phonons .......... 65
6.2.2 Dressed-state representation ....................... 67
6.2.3 Polaron master equation ......................... 68
6.3 Memory-induced loss and rephasing of topological properties ........ 70
6.4 Critical memory depth .............................. 72
6.5 Discussion ..................................... 73
6.5.1 Impact of interparticle interactions ................... 74
6.5.2 Polaron energy shift ........................... 75
6.5.3 Topological correlation .......................... 76
6.5.4 Nonideal Kitaev chain .......................... 77
Contents xiii
6.6 Conclusion .................................... 78
7 Unidirectional quantum transport in optically driven V-type quantum
dot chains 81
7.1 Introduction .................................... 81
7.2 V-type emitter model .............................. 83
7.3 Complete population inversion by non-Markovian reservoir interaction . . . 87
7.4 Unidirectional quantum transport in a quantum dot chain .......... 89
7.5 Discussion ..................................... 92
7.5.1 Comparison between Heisenberg and polaron description . . . . . . 92
7.5.2 Dependence on intraband phonon coupling and temperature ..... 94
7.5.3 Linear absorption spectrum ....................... 94
7.5.4 Bare eigenstate dynamics ........................ 95
7.5.5 Alternative interdot coupling mechanisms: Dexter coupling . . . . . 97
7.5.6 Alternative interdot coupling mechanisms: Förster coupling ..... 98
7.6 Conclusion .................................... 99
8 Dissipative coherent quantum feedback in Liouville space 101
8.1 Introduction ....................................101
8.2 Path integral formulation for continuous reservoirs ..............103
8.3 Time-discrete memory in Liouville space ....................105
8.4 Quasi-2D tensor network .............................109
8.5 Memory-induced dynamical population trapping ...............111
8.6 Convergence analysis ...............................113
8.7 Conclusion ....................................116
III Information compression in open quantum systems via artificial
neural networks 117
9 Sampling asymmetric open quantum systems for artificial neural net-
works 119
9.1 Introduction ....................................119
9.2 Incoherently driven isotropic Heisenberg chain .................120
9.3 Neural network implementation .........................121
9.4 Symmetric systems: Improving sampling efficiency ..............123
9.5 Asymmetric systems: Systematic sampling errors ...............125
9.6 Hybrid sampling for asymmetric systems ....................127
9.7 Conclusion ....................................130
10 Efficient bit encoding of neural networks for Fock states 131
10.1 Introduction ....................................131
10.2 Neural encoding of Fock states .........................132
10.3 Training procedure ................................134
10.4 Neural network efficiency gain ..........................136
10.5 Accuracy .....................................138
xiv Contents
10.6 Scalability .....................................140
10.7 Conclusion ....................................141
11 Conclusion and outlook 143
Appendices 145
A Calculations: Path integral formalism 147
A.1 Trace over coherent states ............................147
A.2 Evaluation of Gaussian integrals ........................148
A.2.1 First integration .............................149
A.2.2 Second integration ............................149
A.2.3 Third integration .............................150
B Calculations: Polaronic Kitaev chain 153
B.1 Polaron transformation ..............................153
B.2 Franck-Condon renormalization .........................155
B.3 Polaron master equation .............................156
B.3.1 Reservoir correlations ..........................158
B.4 Majorana edge correlation ............................160
B.5 Equations of motion ...............................161
C Calculations: Chain of V-type emitters 165
C.1 Parameters ....................................165
C.2 Heisenberg picture: Equations of motion ....................165
C.3 Polaron master equation: Derivation ......................166
C.3.1 Polaron transformation .........................166
C.3.2 Franck-Condon renormalization .....................168
C.3.3 Polaron master equation .........................169
C.4 Polaron master equation: Equations of motion ................180
C.4.1 Single V-type emitter ..........................180
C.4.2 Chain of V-type emitters ........................181
Bibliography 183
Selbständigkeitserklärung 207
Danksagung 210
Acronyms
1D One-dimensional
2D Two-dimensional
3D Three-dimensional
MPS Matrix product state
MPO Matrix product operator
xv
1 Introduction
1.1 Motivation
In contrast to isolated systems thoroughly decoupled from their environment, open quan-
tum systems denote setups featuring crucial interactions with their surroundings, resulting
in unique dynamical phenomena [1–3]. Current research aims to control and utilize such
dynamical system-reservoir interactions for the design and development of novel quantum
optical devices and quantum information technology [4–10]. In particular, phonon-assisted
dynamical dissipative processes arising in semiconductor nanostructures are promising
candidates for potential chip-integrated applications [11–24]. Since the environmental
degress of freedom are often times unknown and comprise a potentially infinite number of
elements, the precise description of open quantum systems constitutes an immense the-
oretical challenge [25–27]: For increasing numbers of system elements, the dimension of
the corresponding Hilbert space scales exponentially in size. In consequence, the crucial
information contained in the system of interest and its surrounding environment must be
extracted and efficiently compressed to allow for an accessible theoretical description.
Multiple theoretical methods and numerical approaches have been fathomed to derive the
most efficient representation of a wide variety of open quantum systems without losing
crucial information on the dynamical interplay between system and environment. They
include correlation expansions of the Heisenberg equations of motion [28,29], a variety of
second-order perturbative master equations based on numerous theoretical assumptions
and approximations [1–3], real-time path integral formulations [30–38], numerical realiza-
tions of high-dimensional tensor networks [39–45] and, in recent developments, machine
learning implementations based on artificial neural networks [46–59]. All of these tech-
niques facilitate varying degrees of information compression based on specific assumptions
about the investigated open quantum system. Depending on the underlying approach,
they each have individual advantages and shortcomings with respect to the achievable
compression efficiency, numerical performance and scalability, and representational lim-
its. In this thesis, we review and compare all of the aforementioned approaches from an
information perspective and apply them to a variety of physical setups and scenarios.
While a quantum system is never completely isolated from its environment, often times
the interplay with an external reservoir merely results in constant dissipative processes
such as decoherence or thermal loss of system excitation. The time evolution dynam-
ics of such systems is considered Markovian, i.e., a system memory is not required for
its accurate depiction, drastically decreasing the complexity of the system-reservoir in-
teraction. This is not the case if the considered system and its environment evolve on
similar time scales. There, the theoretical description must incorporate a system memory
1
21 Introduction
to accurately capture the unfolding non-Markovian dynamical interplay between system
and reservoir. We explore emerging dissipation-based non-Markovian memory effects in-
duced, e.g., by system-reservoir interactions with a structured phonon environment or
time-discrete coherent photonic quantum feedback, resulting in time-delayed dynami-
cal information exchange between system and environment [60–68]. Our investigations
range from p-wave topological superconductors coupled to a structured phonon environ-
ment, where a memory-induced stabilization and recovery of topological properties is
observed [69], to semiconductor quantum dot nanostructures exhibiting complete popu-
lation inversion and unidirectional quantum transport as a result of dissipation-induced
non-Markovian excitation transfer [70], and quantum optical systems exposed to both co-
herent quantum feedback and phonon-induced decoherence, leading to the formation of
dynamical dissipative structures and population trapping [71].
In terms of achievable information compression, machine learning applications in the form
of artificial neural networks have recently emerged as a novel technique for the approximate
description of open quantum systems, offering high performing simulations of Markovian
scenarios with a potential impact on all physical research fields [46–59]. Combining their
boundless capacity for parallelization with precise and customizable Hilbert space com-
pression via Metropolis sampling algorithms, very large system sizes become accessible.
In this thesis, we specifically investigate and advance the restricted Boltzmann machine
architecture, representing a natural and highly efficient neural network realization of the
density matrix of symmetric open spin-1/2systems. We extend the representational limits
of the established neural network by developing novel adaptive sampling strategies for a
highly tailored compression of configuration space, enabling accurate simulations of asym-
metric systems [72]. Furthermore, we push the representational power of the restricted
Boltzmann machine architecture beyond spin-1/2systems by introducing a neural bit
encoding scheme for Fock number states, facilitating supremely efficient calculations of
hybrid systems with bosonic degrees of freedom [73].
1.2 Structure of the thesis
The thesis is divided into three main parts. The first part comprises the foundation of
all employed theoretical approaches and numerical techniques: In Chapter 2, we intro-
duce the basic concepts of open quantum systems and density matrix theory. Chapter 3
provides details and derivations for the most important Markovian and non-Markovian
master equations. In Chapter 4, we discuss the concepts of real-time path integrals as an
alternative approach to quantum mechanics and derive the path integral description for
a continuously driven quantum emitter subjected to a structured continuous reservoir of
harmonic oscillators, known as the spin-boson model. Lastly, we introduce the core ideas
of artificial neural networks and derive theoretical descriptions and numerical techniques
to enable neural network calculations of the stationary states and time evolution dynamics
of open quantum systems in Chapter 5.
In the second part of this thesis, we investigate the emergence of dissipative non-Markovian
phenomena in open quantum systems. In Chapter 6, we employ the polaron master equa-
1.2 Structure of the thesis 3
tion for the description of a topological p-wave superconductor surrounded by a struc-
tured phonon environment and demonstrate a memory-induced recovery and long-term
stabilization of topological properties. In Chapter 7, both a Heisenberg correlation ex-
pansion and the polaron master equation are used to investigate optically driven V-type
quantum dots interacting with a structured phonon reservoir. We predict the occur-
rence of a non-reciprocal phonon-assisted energy transfer in a single emitter, resulting in
complete population inversion. Moreover, when combining multiple quantum dots to a
chain-like spatial distribution, this mechanism is shown to facilitate non-Markovian undi-
rectional quantum transport. Lastly, in Chapter 8we develop a numerically exact tensor
network-based approach to describe two non-Markovian processes with continuous and
time-discrete retardation effects simultaneously, supporting both interactions with and
without energy exchange between system and reservoir. Here, the implementation of the
continuous reservoir is based on a real-time path integral approach. As an example ap-
plication of the presented method, we consider a two-level quantum emitter featuring a
time-discrete quantum memory via coherent quantum feedback and simultaneously cou-
pled to a structured reservoir of independent oscillators. The resulting non-Markovian
interplay between the two structured reservoirs leads to a dynamical protection of coher-
ence against destructive interference processes, facilitating dynamical population trapping
in the system.
The third part of this thesis is focused on the development and investigation of artifi-
cial neural networks to achieve highly efficient information compression in open quantum
systems. In Chapter 9, we present an adjustment to the commonly employed Metropolis
algorithm for the approximate sampling of configuration space and derive a novel hybrid
sampling approach, facilitating accurate neural network simulations of asymmetric open
spin-1/2quantum systems. As an example application, we calculate the stationary state of
a boundary-driven isotropic Heisenberg chain without symmetries of translational invari-
ance. In Chapter 10, the representational capabilities of the restricted Boltzmann machine
architecture are further advanced via a neural bit encoding scheme for Fock number states,
allowing for the simulation of open hybrid systems comprising bosonic degrees of freedom.
We confirm the accuracy, scalability and supreme compression efficiency of the presented
method, outperforming even maximally optimized density matrix approaches.
Finally, we briefly summarize our findings in Chapter 11 and give an outlook on possible
future investigations.
Part I
Theoretical background
5
2 Open quantum systems
In the simplest theoretical conception of a quantum system, it is assumed that the system
of interest is completely isolated from its environment. Without external perturbations
such as noise or decoherence processes, the system state remains pure for all times and
the corresponding Hilbert space dimension depends only on the system degrees of free-
dom. However, in most conceivable quantum optical scenarios the system of interest
interacts with its environment, e.g., via thermal dissipation processes or by performing
measurements on the system, influencing the dynamical system state and resulting in in-
formation loss. In open quantum system theory, environmental influences are explicitly
taken into account for the description of the system. Since we are only interested in a
clearly defined subsystem and the environmental degrees of freedom usually remain un-
determined, we differentiate between the relevant and irrelevant components of the open
system. The irrelevant part is then compressed or partly disregarded, resulting in a mixed
state representation and allowing for an efficient and accurate description of the open
system potentially comprising an infinite number of degrees of freedom.
From a theoretical perspective, we declare the relevant components of the open system
as the system and all remaining parts are collectively described as the environment. The
Hamiltonian describing such an open system consists of freely evolving contributions H0
and a coupling term HIaccounting for the interactions between system and environ-
ment [1],
H=H0+HI.(2.1)
A schematic of a generic open quantum system is shown in Fig. 2.1. The main challenge
for the description of open quantum systems is the fact that the corresponding Hilbert
space dimension scales exponentially with the number of involved particles. While this
issue of course also limits the theoretical accessibility of large isolated systems, in case
of open systems the number of contributing environmental degrees of freedom is often
undetermined. Hence, for system-reservoir interactions with a continuum of environmental
modes, e.g., arising in a harmonic structured reservoir of independent oscillators, an exact
analytical description is unavailable in most cases, even for small systems consisting of a
single particle [74].
In order to address the exponential scaling of the Hilbert space dimension and to render
the description of open quantum systems feasible, multiple analytical truncation-based
approaches and novel numerical compression schemes have been developed. In the course
of this thesis, we will explore and apply the most important state-of-the-art methods for
the description of open quantum systems: Analytical techniques based on perturbation
theory range from correlation expansions [28,29] to second-order perturbative master
equations [1–3], which exploit the fact that the environment is usually large compared
7
82 Open quantum systems
Figure 2.1: Sketch of an open quantum system, consisting of a system interacting with an envi-
ronment.
to the system of interest. Hence, when assuming a weak system-reservoir coupling, the
environment is mostly unperturbed by the system and remains in its initial state through-
out the open system time evolution. In consequence, it is often sufficient to trace out
the reservoir degrees of freedom for an effective reduced system description. The numer-
ically exact real-time path integral formalism represents an alternative approach [30–38]
and has been efficiently implemented using high-dimensional tensor networks, allowing
for on-the-fly compression [39,75–77]. In most recent developments, machine learning
applications based on artificial neural networks have been employed for the description
of symmetric open spin-1/2quantum systems [46,47,51,56–59,78,79], achieving vast
performance gains via an efficient compression of configuration space and the required
degrees of freedom.
Depending on the investigated open system and the architecture employed for its descrip-
tion, time-delayed information backflow between system and reservoir may occur, resulting
in an effective system memory and unfolding non-Markovian dynamics [1–3,60–66,80].
Utilizing and controlling such non-Markovian memory effects has emerged as a major field
of research, with a broad range of potential applications in the fields of quantum optical
devices and quantum information technology [4–10], and represents the field of research
of the second part of this work.
2.1 Density matrix theory
A pure quantum state is described by a state vector |ψ(t)⟩, with its time evolution provided
by the Schrödinger equation. In case parts of the considered system are deemed irrelevant
and disregarded from the description, the system is no longer in a pure state. Rather, the
system takes on a mixed state which can be represented in terms of the density matrix ρ,
defined by [1,81,82]
ρ(t) = X
i,j
ρij(0) |ψi(t)⟩⟨ψj(t)|.(2.2)
Here, diagonal elements ρii(0) |ψi(t)⟩⟨ψi(t)|denote system state densities and off-diagonal
elements describe coherences. We distinguish between three perspectives: Firstly, the
density matrix allows for a many-body ensemble description of mixed states. Here, it
is unknown in which possible realization the current system manifests itself, rendering a
2.1 Density matrix theory 9
superposition of diagonal state spaces necessary. For instance, in a thermal ensemble of a
continuum of modes k,
ρB(T) = 1
Zexp −1
kBTZd3kℏωkr†
krk,(2.3)
with bosonic creation and annihilation operators r†
k,rkof modes at frequencies ωk=c|k|
and obeying the commutation relation [rk, r†
k′] = δ(k−k′), with cthe mode propagation
velocity, Zthe partition function and kBthe Boltzmann constant, many realizations must
be considered to define a temperature T. Secondly, when considering a system without
an ensemble description, the density matrix still constitutes a more general representation
than the corresponding pure state, as it allows for the description of purely thermalized
states without off-diagonal elements [1], and is therefore the least restrictive available
ansatz for the description of arbitrary quantum states. Lastly, from a statistical reasoning
perspective the density matrix is the most general foundation of the probability amplitude
for a certain measurement outcome: Using techniques such as quantum state tomography,
it is possible to reconstruct the shape of the density matrix from experimental measure-
ments on a quantum system [83–85], allowing to deduce an underlying pure state via
quantum state estimates. On the other hand, it is not possible to reconstruct a pure
system state directly from experimental measurements, since the pure state representation
compresses the outcome of many measurements of system observables ⟨ψ(0)|ˆ
A(t)|ψ(0)⟩
and therefore can only be obtained after the fact.
The density matrix is self-adjoint and trace-preserving,
ρ†(t) = ρ(t),(2.4a)
tr{ρ(t)}= 1,(2.4b)
tr{ρ2(t)} ≤ 1,(2.4c)
and the equal sign in Eq. (2.4c) holds true if the respective density matrix describes a
pure state. Since the eigenvalues of a matrix are invariant under unitary transformations,
the choice of basis is arbitrary as long as it is complete [1]. Therefore, a density matrix
with more than one eigenvalue varying from zero cannot be described as a state vector and
thus represents a mixed state. To calculate the expectation value of arbitrary Hermitian
quantum operators ˆ
A, the trace over the operator product of ˆ
Aand ρ(t)is taken,
⟨ˆ
A(t)⟩= tr nˆ
Aρ(t)o.(2.5)
The time evolution of the density matrix is governed by the von Neumann equation, which
is derived next.
2.1.1 Von Neumann equation
The time evolution of a pure state is provided by the Schrödinger equation
iℏd
dt|ψ(t)⟩=H(t)|ψ(t)⟩,(2.6)
10 2 Open quantum systems
with H(t)denoting an arbitrary system Hamiltonian. To calculate the time evolution
of the density matrix, we replace the state vector in the Schrödinger equation by ρ(t),
yielding
iℏd
dtρ(t) = iℏd
dt
X
i,j
ρij(0) |ψi(t)⟩⟨ψj(t)|
=iℏX
i,j
ρij(0) h|˙
ψi(t)⟩⟨ψj(t)|+|ψi(t)⟩⟨ ˙
ψj(t)|i
=X
i,j
ρij(0) [H(t)|ψi(t)⟩⟨ψj(t)|−|ψi(t)⟩⟨ψj(t)|H(t)]
=H(t)ρ(t)−ρ(t)H(t)
= [H(t), ρ(t)] .(2.7)
Rewriting Eq. (2.7) results in the von Neumann equation
d
dtρ(t) = −i
ℏ[H(t), ρ(t)] ,(2.8)
providing the time evolution dynamics of the density matrix. Alternatively, the density
matrix can be propagated in time by application of the time evolution operator U(t, 0) =
exp[−i/ℏRt
0dτH(τ)] via [1,82]
ρ(t) = U(t, 0)ρ(0)U†(t, 0).(2.9)
2.2 Interaction picture
For the calculation of open system time evolution dynamics, it is often advantageous to
transform into the interaction picture. This is performed via the unitary transformation
matrix U0(t, 0) = exp(−iH0t/ℏ),
˜ρ(t) = U†
0(t, 0)ρ(t)U0(t, 0),(2.10)
with H0denoting an arbitrary element of H. To obtain the interaction picture represen-
tation, it is chosen as the freely evolving contribution H0=H−HIof the open system,
excluding system-reservoir interactions HI. To derive the equation of motion governing
the time evolution of ˜ρ(t), we take the time derivative of the density operator,
d
dtρ(t) = d
dthU0(t, 0)U†
0(t, 0)ρ(t)U0(t, 0)U†
0(t, 0)i
=˙
U0(t, 0)˜ρ(t)U†
0(t, 0) + U0(t, 0) ˙
˜ρ(t)U†
0(t, 0) + U0(t, 0)˜ρ˙
U†
0(t, 0)
=U0(t, 0) ˙
˜ρ(t)U†
0(t, 0) −i
ℏhH0U0(t, 0)˜ρ(t)U†
0(t, 0) −U0(t, 0)˜ρ(t)U†
0(t, 0)H0i.(2.11)
Applying U†
0(t, 0) from the left and U0(t, 0) from the right of Eq. (2.11) results in
U†
0(t, 0) ˙ρ(t)U0(t, 0) = ˙
˜ρ(t)−i
ℏh˜
H0(t),˜ρ(t)i,(2.12)
2.2 Interaction picture 11
with ˜
H0(t) = U†
0(t, 0)H0U0(t, 0).
Next, the von Neumann Eq. (2.8) is employed. Again, we apply U†
0(t, 0) from the left and
U0(t, 0) from the right, yielding
U†
0(t, 0) ˙ρ(t)U0(t, 0) = −i
ℏh˜
H(t),˜ρ(t)i,(2.13)
with ˜
H(t) = U†
0(t, 0)HU0(t, 0). Finally, equating Eqs. (2.12) and (2.13) results in the
equation of motion for the interaction picture density matrix,
d
dt˜ρ(t) = −i
ℏnh˜
H(t),˜ρ(t)i−h˜
H0(t),˜ρ(t)io
=−i
ℏh˜
HI(t),˜ρ(t)i.(2.14)
To conclude, in the interaction picture representation the time evolution dynamics of
the density operator is solely provided by the interaction Hamiltonian ˜
HI(t)of the open
quantum system. It is noted, however, that the transformation must be reversed for the
calculation of expectation values, since in general ⟨ˆ
A(t)⟩= tr{ˆ
Aρ(t)} = tr{ˆ
A˜ρ(t)}.
3 Master equation theory
Master equations denote differential equations to describe the time evolution dynamics of
open quantum systems in terms of the density matrix. They are derived in second-order
perturbation theory and under the application of numerous assumptions with regard to
the environmental state, the system-reservoir coupling and the history of the system.
Depending on the extent of applied truncations, different forms of master equations can
be derived, each with their own advantages and limitations.
In this Chapter, we outline the concepts of master equation theory and derive the most
common Markovian and non-Markovian master equations. Using second-order perturba-
tion theory, the non-Markovian Redfield master equation is derived in Sec. 3.1. Advantages
and shortcomings of the Redfield form are discussed, before additional approximations are
applied to derive the Markovian Lindblad form in Sec. 3.2. While the mathematical struc-
ture of the Lindblad master equation is probability conserving under all conditions, it fails
to capture memory effects with potential impact on the system evolution. In Sec. 3.3, the
polaron transformation is introduced and applied to reshape the Redfield equation into
the non-Markovian polaron master equation. In Sec. 3.4, we discuss the Franck-Condon
renormalization, a routinely applied renormalization of the open system Hamiltonian to
simplify the structure of the master equation. Lastly, Sec. 3.5 provides details on the
numerical implementation of the presented master equations.
3.1 Redfield master equation
The exponential scaling of the Hilbert space dimension for increasing system degrees of
freedom represents the key challenge for the accurate description of open quantum systems
(see Ch. 2). Master equation theory tackles this problem from an analytical perspective
to obtain a compressed and efficient representation of the considered system. Any master
equation must maintain the mathematical properties of the density matrix [Eq. (2.4)]
throughout the entire time evolution. Moreover, conservation laws apply when excitation
is transferred between the system and reservoir. In the density matrix formalism (see
Sec. 2.1) this translates to trace conservation of the full open system density operator for
all times, tr{ρ(t)}= 1 ∀t.
We start in the interaction picture (see Sec. 2.2) and formally solve the von Neumann
Eq. (2.14) by formal integration,
˜ρ(t) = ˜ρ(0) −i
ℏZt
0
dt′h˜
HI(t′),˜ρ(t′)i.(3.1)
13
14 3 Master equation theory
For a second-order perturbative treatment, Eq. (3.1) is reinserted once more into the von
Neumann equation, yielding
d
dt˜ρ(t) = −i
ℏh˜
HI(t),˜ρ(0)i−1
ℏ2Zt
0
dt′h˜
HI(t),h˜
HI(t′),˜ρ(t′)ii.(3.2)
The first term in Eq. (3.2) is renormalized to zero by application of a Franck-Condon
renormalization of the full open system Hamiltonian (see Sec. 3.4). In the second term,
the integration variable is substituted by τ=t−t′, resulting in
d
dt˜ρ(t) = −1
ℏ2Zt
0
dτh˜
HI(t),h˜
HI(t−τ),˜ρ(t−τ)ii.(3.3)
The structure of Eq. (3.3) uncovers the influence of the past τ= 0, . . . , t on the present
state and the evolution of the open system. Due to the time retardation of ˜
HIand ˜ρin the
integrand, the evaluation of Eq. (3.3) in its current form becomes increasingly expensive
for each incremental time step, drastically limiting the accessible time scales. Hence, as
a next step several approximations are applied to Eq. (3.3) to obtain a more accessible
representation.
3.1.1 Bath assumption and Born approximation
The first approximation to simplify Eq. (3.3) is based on the assumption that the environ-
ment is large compared to the system and initially in a thermal equilibrium state, often
referred to as the bath assumption. Under these conditions the Born approximation is
applied, assuming a weak system-reservoir interaction. In consequence of a weak coupling
between system and environment, the latter is not perturbed by excitation transfer from
the system or re-equilibrates on a much faster time scale than the time for the system
to evolve from ˜ρ(t)→˜ρ(t+ ∆t). Hence, it remains in its initial equilibrium state for all
times, ˜ρB(t)≈˜ρB(0), and the density matrix factorizes [1]:
˜ρ(t−τ)≈˜ρS(t−τ)⊗˜ρB(0).(3.4)
As a result, Eq. (3.3) takes the form
d
dt˜ρ(t) = −1
ℏ2Zt
0
dτh˜
HI(t),h˜
HI(t−τ),˜ρS(t−τ)⊗˜ρB(0)ii.(3.5)
3.1.2 First Markov approximation
As a next step, the first Markov approximation is applied. It states that the time evolution
of the system is mainly affected by its present state rather than by its past. Thus, the
history of the system state is disregarded in the integrand,
˜ρS(t−τ)≈˜ρS(t),(3.6)
3.2 Lindblad master equation 15
resulting in
d
dt˜ρ(t) = −1
ℏ2Zt
0
dτh˜
HI(t),h˜
HI(t−τ),˜ρS(t)⊗˜ρB(0)ii,(3.7)
which can of course be wrong to an arbitrary degree, e.g., when considering feedback
effects. Eq. (3.7) is transformed back into the Schrödinger picture by application of the
transformation matrix U0(t, 0) from the left and U†
0(t, 0) from the right (see Sec. 2.2).
Moreover, since we are only interested in the dynamics of the system S, the reservoir
contributions Bare traced out from the equation. Therefore, Eq. (3.7) takes the form
d
dttrB{ρ(t)}=−i
ℏtrB{[H0, ρS(t)⊗ρB(0)]}
−1
ℏ2Zt
0
dτtrB{[HI,[HI(−τ), ρS(t)⊗ρB(0)]]},(3.8)
and we arrive at the Redfield master equation [1–3]
d
dtρS(t) = −i
ℏ[HS, ρS(t)] −1
ℏ2Zt
0
dτtrB{[HI,[HI(−τ), ρS(t)⊗ρB(0)]]},(3.9)
where HSdenotes the free evolution contribution of the system with traced out reservoir
degrees of freedom.
The second term in Eq. (3.9) is referred to as the systems’ memory kernel, representing
system-reservoir interactions of past times HI(−τ)and their effect on the current system
state. Hence, the Redfield equation is of a non-Markovian nature, allowing for the descrip-
tion of environment-induced memory effects. Its validity and applicability is limited by
the assumptions made in this Section. However, it is noted that mathematical issues may
arise from the structure of the Redfield equation. While the trace of ρS(t)is preserved
under all conditions, the density matrix is no longer necessarily positive-semidefinite. In
consequence, unphysical behavior such as negative densities may occur in the course of
the time evolution, specifically depending on the parameter regime of the system-reservoir
interaction strength [86], restricting the accessible parameter range to the weak coupling
limit. To overcome this problem and to ensure physical results under all conditions,
additional approximations can be performed. The master equation resulting from this
treatment is the so-called Lindblad form, which is derived next. However, enforcing a
positive-semidefinite density matrix comes at a cost: The Lindblad master equation is of
aMarkovian nature, i.e., in contrast to the Redfield equation memory effects arising from
time-delayed system-reservoir backaction are no longer encompassed.
3.2 Lindblad master equation
Provided that the system observables evolve on much larger time scales than the bath cor-
relations, memory effects arising from system-reservoir interactions become insignificantly
small and can be neglected for a more efficient formalism. Hence, for a wide variety of
scenarios a Markovian description suffices for an accurate representation of the considered
16 3 Master equation theory
system. In this Section, the most successful and commonly applied Markovian master
equation in quantum optics is derived: The Lindblad form [1].
We start from the Redfield Eq. (3.9) and assume generic fermionic system operators of
the form J(†)(t), where we differentiate between two cases,
(i)J(t) = J†(t),(3.10a)
(ii)J(t)=J†(t).(3.10b)
Provided that the operator Jrepresents a single operator rather than a sum of operators
and under application of the rotating wave approximation, a generic system-reservoir
interaction Hamiltonian featuring a structured 3D reservoir is given by
HI(t) = ℏZd3khgkrkJ†(t)e−iωkt+ H.c.i,(3.11)
with bosonic creation and annihilation operators r†
k,rkof reservoir modes at frequencies
ωk=cs|k|, with csdenoting the propagation velocity in a given material, and mode-
dependent system-reservoir coupling elements gk. Defining collective bosonic operators
R(t) = Zd3k gke−iωktrk(0),(3.12)
Eq. (3.11) takes the form
HI(t) = ℏhR†(t)J(t) + R(t)J†(t)i.(3.13)
Eq. (3.13) is inserted into the integrand of the Redfield Eq. (3.9). The evaluation of the
double commutator yields
d
dtρS(t) = −i
ℏ[HS, ρS(t)]
−1
ℏ2Zt
0
dτn⟨R†R(−τ)⟩hJJ†(−τ)ρS(t)−J†(−τ)ρS(t)Ji
+⟨R(−τ)R†⟩h−JρS(t)J†(−τ) + ρS(t)J†(−τ)Ji
+⟨RR†(−τ)⟩hJ†J(−τ)ρS(t)−J(−τ)ρS(t)J†i
+⟨R†(−τ)R⟩h−J†ρS(t)J(−τ) + ρS(t)J(−τ)J†io,(3.14)
with ⟨. . .⟩= trB(. . . ρB). Under the bath assumption (see Sec. 3.1) the reservoir is modeled
as a thermal ensemble,
ρB=1
Zexp −1
kBTZd3kℏωkr†
krk,(3.15)
with Zthe partition function, kBthe Boltzmann constant and Tthe reservoir temperature.
With this, the reservoir correlations occurring in Eq. (3.14) can be calculated explicitly.
3.2 Lindblad master equation 17
For the simplest conceivable case of an unstructured 1D reservoir in a vacuum state, i.e.,
gk≈g0and T= 0 K, no further approximations must be applied for the derivation of the
Lindblad form. In vacuum, it is ⟨r†
krk⟩= 0 and calculating, e.g., ⟨RR†(−τ)⟩yields
⟨RR†(−τ)⟩=Z∞
0
dkZ∞
0
dk′g0g∗
0e−iωk′τ⟨rkr†
k′⟩
=Z∞
0
dk|g0|2e−icskτ
=π|g0|2
cs
δ(τ)−iP 1
csτ,(3.16)
where P(. . .)denotes the Cauchy principal value [1] which leads to a phase shift and renor-
malization of the systems’ transition frequencies. However, since the transition frequencies
can be chosen freely and are not crucial for our investigations, the contribution from the
Cauchy principle value is omitted in the following. Under these presumptions, we get
⟨R(−τ)R†⟩=⟨RR†(−τ)⟩and ⟨R†R(−τ)⟩=⟨R†(−τ)R⟩= 0. Defining a constant damp-
ing rate π|g0|2/(ℏ2cs) =: γand inserting the evaluated bath correlations into Eq. (3.14)
immediately results in the Lindblad master equation,
d
dtρS(t) = −i
ℏ[HS, ρS(t)] + γh2JρS(t)J†−J†JρS(t)−ρS(t)J†Ji.(3.17)
To conclude, for an unstructured 1D reservoir with energy-conserving interactions in a
vacuum state (T= 0 K), only the Born approximation is required for the derivation of
the Lindblad form. The delta function arising from the bath correlations renders the first
Markov approximation unnecessary.
3.2.1 Second Markov approximation
Next, we consider the non-vacuum case T > 0K and a structured reservoir. Under these
conditions, the bath correlations are evaluated as
⟨R(−τ)R†⟩=Z∞
0
d3k|gk|2eiωkτ(nk+ 1) ,(3.18a)
⟨RR†(−τ)⟩=Z∞
0
d3k|gk|2e−iωkτ(nk+ 1) ,(3.18b)
⟨R†(−τ)R⟩=Z∞
0
d3k|gk|2e−iωkτnk,(3.18c)
⟨R†R(−τ)⟩=Z∞
0
d3k|gk|2eiωkτnk,(3.18d)
with the mean boson number modeled by a Bose distribution,
nk=⟨r†
krk⟩ ≈ exp ℏωk
kBT−1−1
.(3.19)
18 3 Master equation theory
Inserting these expressions into Eq. (3.14) results in terms containing both integrals over
τand k,
d
dtρS(t) = −i
ℏ[HS, ρS(t)]
−Zt
0
dτZ∞
0
d3k|gk|2neiωkτnkhJJ†(−τ)ρS(t)−J†(−τ)ρS(t)Ji
+eiωkτ(nk+ 1) h−JρS(t)J†(−τ) + ρS(t)J†(−τ)Ji
+e−iωkτ(nk+ 1) hJ†J(−τ)ρS(t)−J(−τ)ρS(t)J†i
+e−iωkτnkh−J†ρS(t)J(−τ) + ρS(t)J(−τ)J†io.(3.20)
Due to non-vanishing reservoir occupation numbers, in this case the second Markov ap-
proximation must be applied to obtain a simplified description. It states that the system-
reservoir interaction is independent of its past, HI(−τ)≈HI. In consequence, the system
operators lose their time dependence, J(†)(−τ)≈J(†), and the time integration to the
present time tcan be expanded to all future times,
Zt
0
dτ(. . .)≈Z∞
0
dτ(. . .).(3.21)
The second Markov approximation is valid if the open system effectively has no mem-
ory, meaning that its time evolution dynamics is invariant against its history. The as-
sumption made in Eq. (3.21) is only justified if the integrand decays sufficiently fast
for times τ > t. Therefore, the second Markov approximation is acceptable if the time
scales on which the system observables vary are large compared to the decay time of
the reservoir correlations [1]. The application of the full set of approximations made so
far, i.e., the bath assumption, Born approximation and first and second Markov approx-
imation, is commonly referred to as the Born-Markov approximation. Using the identity
R∞
0dτ e−iωkτ=π/csδ(k)−iP (1/ωk), the time integral can be explicitly evaluated and
Eq. (3.20) takes the form
d
dtρS(t) = −i
ℏ[HS, ρS(t)]
−π
csZ∞
0
d3k|gk|2nδ(k)nkhJJ†(−τ)ρS(t)−J†(−τ)ρS(t)Ji
+δ(k) (nk+ 1) h−JρS(t)J†(−τ) + ρS(t)J†(−τ)Ji
+δ(k) (nk+ 1) hJ†J(−τ)ρS(t)−J(−τ)ρS(t)J†i
+δ(k)nkh−J†ρS(t)J(−τ) + ρS(t)J(−τ)J†io.(3.22)
Again, we first consider the case of a 1D reservoir, k=k. Evaluation of the delta function
3.3 Polaron master equation 19
results in a generalized Lindblad form
d
dtρS(t) = −i
ℏ[HS, ρS(t)]
+γ(n0+ 1) h2JρS(t)J†−J†JρS(t)−ρS(t)J†Ji
+γn0h2J†ρS(t)J−JJ†ρS(t)−ρS(t)JJ†i,(3.23)
with γ=π|g0|2/cs. Setting n0= 0 recreates the Lindblad equation for the vacuum case
[Eq. (3.17)]. Secondly, Eq. (3.22) is evaluated for a structured 3D reservoir. Spherical
coordinates are employed,
Z∞
−∞
d3k= 4πZ∞
0
dk k2,(3.24)
and to obtain a nontrivial result, the factor k2arising from the functional determinant must
be compensated by the coupling element. Hence, we must assume a super-Ohmic system-
reservoir coupling element, gk∼k−1, and define ˜gk:= gk/k. With this presumption,
Eq. (3.23) can be reproduced for the 3D reservoir case at γ= 4π2|˜g0|2/(ℏ2cs). To conclude,
when considering a structured reservoir at T > 0K, the second Markov approximation is
required for the derivation of a Markovian Lindblad form.
3.3 Polaron master equation
We conclude the discussion of master equations with the non-Markovian polaron master
equation. While the Lindblad and Redfield equations can for instance also be applied to
photonic reservoirs to model processes such as spontaneous emission, the polaron mas-
ter equation is exclusively employed for the description of systems coupled to harmonic
phonon reservoirs featuring a diagonal system-reservoir coupling without energy exchange
between them [1,14,80,87–92]. In these scenarios, the diagonal system-reservoir coupling
Hamiltonian describes decoherence processes caused by interactions with an environment
of independent harmonic oscillators. For the case of diagonal 3D fermion-phonon interac-
tion it is of the general form [93–96]
HI=Zd3r g(r)⟨ψ(r)|φ(r)|ψ(r)⟩,(3.25)
where |ψ(r)⟩denotes the fermionic field vector, φ(r)is the phonon field and g(r)refers
to the microscopic fermion-phonon coupling coefficient, respectively. In typical quan-
tum optical setups, the fermionic particles are assumed distinguishable and located at
fixed positions in space, such that the field intensity can be modeled as ⟨ψ(r)|ψ(r)⟩=
PN
l=1 c†
lclδ(r−rl)with c(†)
ldenoting fermionic creation and annihilation operators acting
on the l-th particle at position rl. A generic 3D phonon field is given by [95]
φ(r) = Zd3kscsk
2r†
ke−ik·r+ H.c.,(3.26)
20 3 Master equation theory
with r(†)
kagain denoting bosonic annihilation (creation) operators of phonon modes at
frequencies ωk=cs|k|, with csthe speed of sound for a given material [97,98]. Inserting
these identities into Eq. (3.25) and defining
gk:= scsk
2Zd3r g(r)e−ik·rδ(r−rl),(3.27)
the fermion-phonon interaction Hamiltonian for a set of Ndistinguishable and localized
fermionic particles takes the form
HI=
N
X
l=1 Zd3k c†
lclgkr†
k+ H.c..(3.28)
It is noted that when considering closely localized particles, the spatial dependence of
the fermion-phonon coupling element gkis commonly disregarded, i.e., g(rl) = g(r0)∀l,
together with phonon-induced momentum transfer. The main objective of the polaron
master equation approach is to include as much information about the electron-phonon
interaction as possible while still remaining in a reduced density matrix description in
second-order perturbation theory [1,14,80,87–92,99,100]. For the derivation of the
polaron master equation, the open system Hamiltonian of the investigated system is trans-
formed into the polaron picture.
3.3.1 Polaron transformation
A polaronic state describes a fermion in interaction with a harmonic phonon continuum,
resulting in a self-reorganization of the collective ground state energy. Using collective
bosonic operators R(†)=Rdk(gk/ωk)r(†)
k, we apply the unitary polaron transformation
to the full open system Hamiltonian H=H0+HI,
Hp=UpHU−1
p,(3.29)
with
Up= exp "N
X
l=1
c†
lcl(R†−R)#,(3.30)
representing an exact and reversible transformation explicitly carried out via the Baker-
Campbell-Hausdorff formula
eXY e−X=
∞
X
n=0
1
n![X, Y ]n,(3.31a)
[X, Y ]n=hX, [X, Y ]n−1i,(3.31b)
[X, Y ]0=Y. (3.31c)
The resulting polaron Hamiltonian Hp=Hp,0+Hp,I is inserted into the Redfield Eq. (3.9),
yielding the polaron master equation.
3.4 Franck-Condon renormalization 21
3.4 Franck-Condon renormalization
During the derivation of the Redfield Eq. (3.9), we have assumed throughout that
trB{[HI, ρ(t)]}= 0 ∀t, (3.32)
such that the first term of the master equation only contains the freely evolving contribu-
tion of the open system Hamiltonian, restraining contributions from the system-reservoir
interaction Hamiltonian solely to the memory kernel and significantly simplifying the
structure of the equation. However, in general Eq. (3.32) does not hold true for arbi-
trary Hamiltonians. To ensure its validity, a Franck-Condon renormalization is commonly
performed on the considered open system Hamiltonian, where the equilibrium position
is shifted according to the interaction strength: First, the interaction Hamiltonian HI
is modified by subtracting an artificial term HF C, such that Eq. (3.32) is fulfilled. To
compensate the artificial term, it is in turn added to the free Hamiltonian H0. As a result,
the full open system Hamiltonian H=H0+HI= (H0+HF C) + (HI−HF C)remains
unchanged.
3.5 Numerical evaluation
The non-Markovian master equations derived in this Chapter represent regular integro-
differential equations and can be numerically evaluated using two nested fourth order
Runge Kutta algorithms, where the full system-reservoir memory kernel is evaluated iter-
atively during each time step. In case of the Markovian Lindblad master equation, only a
single fourth order Runge Kutta algorithm is required.
Non-Markovian master equations feature a memory kernel, i.e., an integral over all past
times with time-dependent system and bath operators [see Eq. (3.9)]. It is in general
possible to derive analytical expressions for the time-dependent reservoir correlations.
Analytical solutions of time-dependent system correlations X(−τ), consisting of combina-
tions of system operators J(†)(−τ), are available in some elementary models such as the
spin-boson model [74,80]. Aside from these special cases, the system correlations must be
evaluated numerically. For the inclusion of the full memory kernel featuring both time-
dependent system and reservoir operators, the time evolution of the system correlations
X(−τ)is determined by a closed set of differential equations depending only on the free
evolution contribution H0. Using the time evolution operator U0(t, 0) = exp(−iH0t/ℏ),
22 3 Master equation theory
we get for a specific matrix element
⟨m|X(−τ)|n⟩=⟨m|U†
0(−τ, 0)XU0(−τ, 0) |n⟩
=⟨m|U0(τ, 0)XU†
0(τ, 0) |n⟩
=X
{s}⟨m|U0(τ, 0)X|s⟩⟨s|U†
0(τ, 0) |n⟩
=X
{s}⟨s|U†
0(τ, 0) |n⟩⟨m|
|{z }
=:ρc(0)
U0(τ, 0)X|s⟩
=X
{s}⟨s|U†
0(τ, 0)ρc(0)U0(τ, 0)
|{z }
=ρc(−τ)
X|s⟩
= tr{ρc(−τ)X}.(3.33)
Here we introduced the conditional density matrix ρc(−τ)whose time evolution dynam-
ics is prescribed by the free energy contribution, ˙ρc(t) = −i/ℏ[Hp,0, ρc(t)], and must be
determined for all possible initial conditions ρc(0) = |n⟩⟨m|, i.e., ⟨n|ρc(0) |m⟩= 1 and all
other entries set to zero. It is noted that in difference to the system density matrix, ρc
does not necessarily obey conservation laws, tr{ρc(t)} = 1 ∀t.
For the numerical solution of non-Markovian master equations, we first solve and store
ρc(−τ)for all possible initial conditions using a fourth order Runge Kutta implementa-
tion. In a second step, the conditional density matrix is employed to calculate the system
correlation elements ⟨m|X(−τ)|n⟩to evaluate the integrand during each time step. Af-
terwards, the integral contributions are iteratively summed up over all time steps and
supplied to a second fourth order Runge Kutta algorithm to solve the time evolution
dynamics of the system density matrix ρS(t).
4 Real-time path integrals
The real-time path integral formalism, first developed by Richard P. Feynman [30], con-
stitutes an alternative approach to the description of quantum mechanical phenomena.
Here, the key concept is to apply the principle of least action to the quantum realm, fol-
lowing from the idea that classical statistical mechanics may arise as a natural limit of
quantum mechanics. As a core advantage over established techniques, the path integral
formalism allows for a numerically exact description of open quantum systems by taking
into account the usually finite time-resolved system-reservoir correlations of the respective
open system [28,31–38]. In recent developments of novel tensor network-based descrip-
tions for time-discrete path integrals, highly efficient and numerically exact calculations
of such systems have become feasible [75–77].
In this Chapter, we introduce the core concepts of the path integral formalism. Sec. 4.1 mo-
tivates the transition from the principle of least action in classical mechanics to determin-
ing the time-evolving state of a quantum mechanical system. As an example application,
Sec. 4.2 provides a detailed derivation of the time-discrete path integral representation for
the spin-boson model [74], which has been established as a proving ground for open system
methods. We conclude this Chapter by introducing the numerically exact tensor network
representation for discrete path integrals in Sec. 4.3.
4.1 From classical action to quantum mechanics
As a motivation for the path integral approach, we start from classical mechanics where
the trajectory of a particle in principle can be determined to arbitrary precision. This is
no longer the case for the description of quantum mechanical particles: Here, a complex
probability amplitude serves as the most accurate available description of the system state.
Based on the concept of superposition, these amplitudes can be calculated by taking the
integral over all possible trajectories or paths during finite time steps. In the following,
we derive a general path integral description by applying the principle of least action to
the quantum realm.
4.1.1 Principle of least action
In the Lagrangian approach to classical mechanics, the trajectory of a particle is deter-
mined by the principle of least action. The action of a particle traversing space on an
23
24 4 Real-time path integrals
Figure 4.1: (a) Sketch of the double-slit experiment. A quantum mechanical particle emitted by
a source (red shape) traverses through a double-slit plane, going through either the
first or the second hole and is detected on a screen located at a distance behind it. (b)
Extension of the double-slit setup with multiple screens in between source and detector
plane, each of them featuring many holes. A particle emitted from the source may take
one of many paths (blue lines) to arrive at the detector, each of them associated with
its own probability amplitude.
arbitrary trajectory from an initial time tato time tbis calculated via [30]
S=Ztb
ta
dt L(˙
r,r, t),(4.1)
with L(˙
r,r, t)denoting the Lagrangian of the considered system and r(t)=[rx(t), ry(t), rz(t)]T
the particles’ position at a given time t. Now, the physical trajectory or extremum path
¯
r(t)is calculated by application of the variational principle: Assuming fixed start and end
positions, δr(ta) = δr(tb)=0, the principle of least action states that the extremum path
¯
r(t)minimizes the action, i.e.,
δS =S[¯
r(t)−δr(t)] −S[¯
r(t)] = 0,(4.2)
for a small variation δr(t). Evaluation of Eq. (4.2) yields the classical Euler-Lagrange
equation of motion (i={x, y, z}),
d
dt∂L
∂˙ri−∂L
∂ri
= 0,(4.3)
determining the time-resolved physical trajectory ¯
r(t)of a classical particle for a given
initial condition.
4.1.2 Probability amplitude
In difference to classical mechanics, the trajectory of a quantum mechanical particle cannot
be determined to arbitrary precision, as described by the uncertainty principle. Rather,
following the Kopenhagen interpretation of quantum mechanics, since the particles’ time-
resolved trajectory is affected by the measurement process itself, only its averaged detec-
tion at a fixed time and position in space is deterministic [101]. Thus, many measurements
4.1 From classical action to quantum mechanics 25
of the same event must be performed to calculate a probability amplitude ψ(r, t), denoting
the probability of the particle to be detected at position rat time t.
The famous double-slit experiment constitutes a prime example of this fact [see Fig. 4.1(a)]:
A source of quantum mechanical particles, e.g., electrons or photons, emits in all directions
towards a double-slit screen. Positioned at a distance behind it, a detector plane measures
incoming particles. If the detector has a high sensitivity and the intensity of the source is
low, individual particles will be measured in the form of time-resolved pulses. Measuring
the mean number of pulses at a given position in the detector plane determines the relative
probability P(r)for a particle to follow the path from its source to the detector location.
Since the electrons or photons can be measured individually and therefore behave like
particles, one might assume that they traverse either the first or the second hole of the
double-slit plane on their way to the detector. Following this argument, the probability
of arrival at position rwould be given by the sum P(r) = P1(r) + P2(r)of probabilities
to go through either the first or the second hole while the respective other hole is closed.
However, experimental realizations of this setup have proven otherwise [30,101]: The
measured probability P(r)for the arrival of a particle in the detector plane is given by an
interference pattern, as it would be expected for a wave passing through the double slit.
The simplest representation of a wave amplitude is constituted by complex numbers. Thus,
to correctly describe the probability distribution P(r)we assume a complex probability
amplitude ψ(r) = ψ1(r) + ψ2(r)consisting of the amplitudes for passing through either
hole 1 or 2, and the correct probability distribution is given by the wave intensity
P(r) = |ψ(r)|2.(4.4)
In other words, quantum mechanical phenomena are described by a statistical theory
with complex numbers, allowing to describe interference effects between measurement
events. The probability amplitude or wave function ψ(r, t)is a representation of many
deterministic measurements, assessing all possible outcomes and giving the most accurate
description available for the time-resolved trajectory of a quantum mechanical ensemble.
Following the principle of superposition, the complete probability amplitude is determined
by summation over the amplitudes of all possible paths.
4.1.3 Path integral formalism
With the fundamental properties of the probability amplitude established, we extend the
double-slit setup by considering multiple screens in between the source and the detector,
each of them featuring numerous holes [see Fig. 4.1(b)]. In this scenario, a particle emitted
from the source (red shape) can take one of many paths to arrive at the detector plane (blue
lines), each of them associated with its own amplitude. As a result of the superposition
principle, the total probability amplitude is again given by the sum of the amplitudes of
all possible paths. Going one step further, we imagine infinitely many screens in between
source and detector plane, each perforated by an increasing amount of holes until they
consist of nothing but holes. In consequence, a single path from the source to the detector
becomes a continuous function of space and time. The resulting probability amplitude
26 4 Real-time path integrals
as a sum of amplitudes of all possible paths is then determined by integration over all
possible paths [30]. This concept is the foundation of the path integral formalism. In the
following, we explore its application in detail.
While all possible paths contribute to the full probability amplitude by the same magni-
tude, they differ in phases: The phase of a specific path is determined by the action Sin
units of the quantum of action ℏ[30]. In the classical limit, Sis much larger than ℏ, such
that the phase contribution S/ℏis a large angle which oscillates rapidly when perturbing
the path by a small amount δron a classical scale. As a result, a small change in Sis
accompanied by a drastic change of the phase, whose total contribution will average to
zero. On a quantum scale, however, the phase only slightly changes by perturbation of
S∼ℏand crucially determines the contribution of a respective path to the full proba-
bility amplitude. The path integral to calculate the probability amplitude of a particle
traversing from point rato point rbis given by [30]
ψ(rb, tb) = Zrb
ra
e(i/ℏ)S[rb,ra]ψ(ra, ta)Dr(t),(4.5)
with RDr(t) = RR. . . Rdr1(t)dr2(t). . . drN−1(t)denoting the integration over all possi-
ble paths traversing points ri(t)between the fixed start and end points raand rb[see
Fig. 4.1(b)]. Analogous to classical mechanics, the action is given by
S[rb,ra] = Ztb
ta
dt L(˙
r,r, t).(4.6)
While the form of Eq. (4.5) in principle allows to calculate the wave function, depending
on the considered system it is not always the most viable representation: If all variables of
the action Sappear only up to second order, the path integral can be described by consid-
ering the deviation from the classical trajectory prescribed by the extremum action [30].
Importantly, this condition applies to a harmonic bath of independent oscillators. Here,
rather than calculating the path integral for a finite number of points ri(t)explicitly, we
minimize the action once more, introducing Scl[rb,ra] := S[¯r(t)]. As a result, the integrals
in Eq. (4.5) take the form of Gaussian integrals and can be carried out explicitly, resulting
in a much simpler expression
ψ(rb, tb) = e(i/ℏ)Scl[rb,ra]F(tb, ta)ψ(ra, ta),(4.7)
with a new function F(tb, ta)arising from the integrations, whose specific form depends
on the investigated system. Notably, in this representation the spatial dependence rhas
been carried out completely.
4.2 Path integral formulation of the spin-boson model
4.2.1 Spin-boson model
Despite its seemingly simple structure, the spin-boson model has been shown to exhibit a
rich variety of physical phenomena [74,75] and has been established as a common testing
4.2 Path integral formulation of the spin-boson model 27
Figure 4.2: Sketch of the spin-boson model. A two-level emitter with a single electron (red shape)
and transition frequency ω0between ground and excited state is continuously driven
by an incoming light field at Rabi frequency Ω0. The excited state |2⟩is subject
to decoherence processes by diagonal coupling to a structured harmonic reservoir of
independent oscillators, with a mode (q)-dependent coupling element gq.
ground for open system methods. The spin-boson model is shown schematically in Fig. 4.2:
It consists of a two-level quantum emitter with a single electron, featuring a transition
frequency ω0between ground and excited state and continuously driven by an external
laser field. The excited state features diagonal coupling to a structured harmonic reservoir
of independent oscillators such as phonons, resulting in decoherence processes without
affecting the level occupation. Under the rotating wave approximation, the open system
Hamiltonian of the spin-boson model reads
H/ℏ=ω0σ22 +Ω0hσ21ei(ω0−ωL)t+ H.c.i+Zd3qhωqb†
qbq+gqσ22 b†
qeiωqt+ H.c.i,(4.8)
with system operators σij =|i⟩⟨j|, bosonic annihilation (creation) operators b(†)
qof phonon
modes at frequency ωq=cs|q|,csthe speed of sound, and a structured system-reservoir
coupling element gq.Ω0denotes the Rabi frequency of the external driving field at a
laser frequency ωL. We note that in difference to the independent boson model where
no external driving field is applied, there exists no analytical solution to the spin-boson
model for arbitrary coupling elements qq. In the following, we derive a numerically exact
solution for the reduced system time evolution dynamics of the spin-boson model using
the real-time path integral approach.
4.2.2 Time-discrete expansion of the density matrix
For a full description of the open quantum system, the density matrix formalism is applied.
Hence, the time evolution dynamics is governed by the von Neumann equation (see Ch. 2).
Assuming an initially separable state between system and reservoir, the formal solution
to the von Neumann equation is given by
ρS(t) = trBhU(t, 0)ρ(0)U†(t, 0)i,(4.9)
with the time evolution operator
U(t, 0) = Texp −i
ℏZt
0
dt′H(t′),(4.10)
28 4 Real-time path integrals
and Tthe time ordering operator [1]. However, for the numerical evaluation of the density
matrix a discretized representation is required. Using the Trotter decomposition [39], one
obtains an approximate representation of the time-ordered time evolution operator,
U(t, 0) ≈
N
Y
n=1
exp −i
ℏ∆tH(n∆t),(4.11)
which becomes exact for N→ ∞ and ∆t→0, where N∆t=tand ∆tthe time discretiza-
tion [28,31,33]. In the following, we abbreviate n∆t=: tn.
By insertion of multiple identity operators
1
n, the product over time steps is converted
into a summation over a multidimensional configuration space, yielding
U(t, 0) = U(tN, tN−1)
1
N−1U(tN−1, tN−2)
1
N−2. . .
1
1U(∆t, 0).(4.12)
As a result, the density matrix takes the form
ρ(t) = e−iH(tN)∆t/ℏ
1
N−1. . .
1
1e−iH(0)∆t/ℏρ(0)eiH(0)∆t/ℏ
1
1. . .
1
N−1eiH(tN)∆t/ℏ,(4.13)
with the identity operators constructed as a direct product of the spin system states |in⟩
and unnormalized coherent phonon states |αqn⟩at times tn,
1
n=
2
X
in=1 |in⟩⟨in|⊗Y
qZd2αqn
π|αqn⟩⟨αqn|,(4.14)
where
|αq⟩=e−|αq|2/2
∞
X
nq=0
αnq
q
√nq|nq⟩,(4.15)
and |nq⟩denoting thermal states with occupation numbers nqof modes q. With this, we
have derived a time-discrete representation of the density matrix with time ordering. In
the course of the following path integral derivation, the integrals arising from the identity
operators are carried out explicitly.
As a first step, the reservoir degrees of freedom are formally traced out, i.e., ρS(t) =
trB{ρ(t)}, yielding (see Appendix A)
trB{. . .}=Y
q
∞
X
nq=0 ⟨nq|. . . |nq⟩ ≡ Y
qZd2αq
π⟨αq|. . . |αq⟩.(4.16)
Hence, the reduced system density matrix elements at time tN=N∆tare given by
⟨iN|ρS(tN)|i′
N⟩=Y
qZd2αqN
πZd2αq0
πZd2βq0
π
×⟨αqN, iN|U(tN, tN−1)
1
N−1. . .
1
1U(∆t, 0) |αq0, i0⟩
×⟨αq0, i0|ρ(0) |βq0, i′
0⟩
×⟨βq0, i′
0|U†(0,∆t)
1
1. . .
1
N−1U†(tN−1, tN)|αqN, i′
N⟩.(4.17)
4.2 Path integral formulation of the spin-boson model 29
4.2.3 Evaluation of discrete time steps
In order to evaluate Eq. (4.17) with respect to the spin-boson model Hamiltonian [Eq. (4.8)],
we first calculate the time evolution operator [Eq. (4.11)] for a single time step, namely
⟨αqn, in|U(tn, tn−1)|αqn−1, in−1⟩=Y
q⟨αqn, in|e−i∆tH(tn)/ℏ|αqn−1, in−1⟩
=Y
q⟨αqn|αqn−1⟩e−i∆t[ωqα∗
qnαqn−1+ingq(α∗
qn+αqn−1)] ⟨in|e−i∆tHΩ(tn)|in−1⟩.(4.18)
Here the Trotter splitting has been applied once more [28,33], and HΩ= Ω0[σ21ei(ω0−ωL)t+
H.c.]denotes the external driving term of the open system Hamiltonian. Using the defi-
nition from Eq. (4.15), the first resulting product is evaluated as
⟨αqn|αqn−1⟩= exp −|αqn|2
2−|αqn−1|2
2+α∗
qnαqn−1!.(4.19)
For the examination of the driving term ⟨in|e−i∆tHΩ(tn)|in−1⟩=: Minin−1, the exponential
function is expanded as a series, yielding
Minin−1=⟨in| cos(Ω0∆t)−iei(ω0−ωL)∆tsin(Ω0∆t)/ℏ
−ie−i(ω0−ωL)∆tsin(Ω0∆t)/ℏcos(Ω0∆t)!|in−1⟩,(4.20)
and resulting in Minin−1=
1
δin,in−1for the case Ω0= 0, corresponding to the independent
boson model. Minin−1is referred to as the field transformation matrix. In total, Eq. (4.18)
takes the form
⟨αqn, in|U(tn, tn−1)|αqn−1, in−1⟩=Minin−1Y
q
exp α∗
qnαqn−1−|αqn|2
2−|αqn−1|2
2
−i∆thωqα∗
qnαqn−1+ingqα∗
qn+αqn−1i.(4.21)
As a next step, we evaluate Ntime steps while taking the limit ∆t→0, resulting in
⟨αqN, iN|U(tN, tN−1)
1
N−1. . .
1
1U(∆t, 0) |αq0, i0⟩
=Y
qZd2αqN−1
π. . . Zd2αq1
π
2
X
i1,...,iN−1=1
MiNiN−1. . . Mi1i0
×exp (N
X
n=1 "α∗
qnαqn−1−|αqn|2
2−|αqn−1|2
2−i∆tωqα∗
qnαqn−1−i∆tgqα∗
qn+αqn−1in#)
∆t→0
=Y
qZd2αqN−1
π. . . Zd2αq1
π
2
X
i1,...,iN−1=1
MiNiN−1. . . Mi1i0exp (−|αq(t)|2
2−|αq(0)|2
2
+α∗
q(t)αq(t)−Zt
0
dτ"α∗
q(τ) ˙αq(τ) + iωqα∗
q(τ)αq(τ) + igqαq(τ) + α∗
q(τ)j(τ)#)
=
2
X
i1,...,iN−1=1
MiNiN−1. . . Mi1i0Y
q
exp (−|αq(t)|2
2−|αq(0)|2
2)ZDα T heS(αq,α∗
q)i,
(4.22)
30 4 Real-time path integrals
with
j(′)(τ) :=
N
X
n=1
i(′)
nΘ[τ−(n−1)∆t] Θ[n∆t−τ],(4.23)
and Θ(t)the Heaviside function. In this limit, the time evolution operator matrix elements
are transformed to a path integral over phonon variables, compactly written in the last
line in Eq. (4.22) with the action S(αq, α∗
q). Note that an exponential prefactor is excluded
from S, since it belongs to the identity of the unnormalized coherent phonon states.
4.2.4 Extremal action
As outlined in Sec. 4.1, the path integral for the time evolution operator can be evaluated
by calculating deviations from the classical trajectory via the extremal action Scl. There-
fore, as a next step we minimize the action S(αq, α∗
q), leading to the Euler-Langrange
Eq. (4.3). Assuming fixed start and end positions, the Lagrangian takes the form
L(αq, α∗
q) = −nα∗
q(τ) ˙αq(τ) + iωqα∗
q(τ)αq(τ) + igqhαq(τ) + α∗
q(τ)ij(τ)o,(4.24)
and the resulting differential equations for αqand α∗
qread
−ihωqα∗
q(τ) + j(τ)gqi+ ˙α∗
q(τ)=0,(4.25a)
−i[ωqαq(τ) + j(τ)gq]−˙αq(τ)=0,(4.25b)
with boundary conditions α∗
q(t) = α∗
qN,αq(0) = αq0resulting from the transition from the
discrete to the continuous form in Eq. (4.22). Formally solving these differential equations
and inserting the solutions for αq(τ)and α∗
q(τ)into S(αq, α∗
q)yields the extremal action,
Scl(αq, α∗
q) = αq0α∗
qNe−iωqt−igqZt
0
dτ′hαq0e−iωqτ′+α∗
qNe−iωq(t−τ′)ij(τ′)
−g2
qZt
0
dτZτ
0
dτ′e−iωq(τ−τ′)j(τ)j(τ′),(4.26)
and determines the path integral for the time evolution operator,
ZDα T heS(αq,α∗
q)i≡eScl(αq,α∗
q).(4.27)
4.2.5 Initial state
Apart from the time evolution operator, the density matrix elements [Eq. (4.17)] depend on
the initial open system state ⟨αq0, i0|ρS(0) ⊗ρB(0) |βq0, i′
0⟩. Assuming a thermal reservoir
state and using Wick’s theorem, we can write ρBas a super-Poissonian distribution [28,
33],
ρB=1
Ze−βPqℏωqb†
qbq=Y
q1−e−βℏωqe−βℏωqb†
qbq=Y
q
¯nnq
q
(1 + ¯nq)nq+1 ,(4.28)
4.2 Path integral formulation of the spin-boson model 31
with ¯nq=⟨b†
qbq⟩and Zthe partition function. The initial state then takes the form
⟨αq0, i0|ρS(0) ⊗ρB(0) |βq0, i′
0⟩
=⟨i0|ρS(0) |i′
0⟩⟨αq0|
∞
X
nq′=0 Y
q′
¯nnq′
q′
(1 + ¯nq′)nq′+1 |nq′⟩⟨nq′|βq0⟩
=⟨i0|ρS(0) |i′
0⟩
∞
X
nq′=0 Y
q′
¯nnq′
q′
(1 + ¯nq′)nq′
1
(1 + ¯nq′)⟨αq0|nq′⟩δq0q′
βnq0
q0
pnq0!e−|βq0|2/2
=⟨i0|ρS(0) |i′
0⟩
∞
X
nq′=0 Y
q′
¯nnq0
q0
(1 + ¯nq0)nq0
1
(1 + ¯nq0)
α∗
q0
nq0βnq0
q0
nq0!e−|αq0|2/2−|βq0|2/2
=⟨i0|ρS(0) |i′
0⟩(1 −ξq0) exp ξq0α∗
q0βq0−|αq0|2
2−|βq0|2
2!,(4.29)
with ξq:= ¯nq/(1 + ¯nq). Having determined the initial state, we can explicitly trace out
the reservoir by carrying out the remaining three integrals in Eq. (4.17).
4.2.6 Tracing out the reservoir
Inserting the initial state representation of Eq. (4.29), the time evolution operator in the
continuous limit [Eq. (4.22)] and the extremal action [Eq. (4.26)] into Eq. (4.17), the
density matrix elements take the form
⟨iN|ρS(t)|i′
N⟩=Y
qZd2αqN
πZd2αq0
πZd2βq0
π
2
X
i1,...,iN−1=1
2
X
i′
1,...,i′
N−1=1
×MiNiN−1. . . Mi1i0M∗
i′
0i′
1. . . M∗
i′
N−1i′
N(1 −ξq0)⟨i0|ρS(0) |i′
0⟩
×exp (−|αqN|2−|αq0|2−|βq0|2+ξq0α∗
q0βq0+αq0α∗
qNe−iωqt+β∗
q0αqNeiωqt
−igqZt
0
dτ′hαq0e−iωqτ′+α∗
qNe−iωq(t−τ′)ij(τ′)−g2
qZt
0
dτZτ
0
dτ′e−iωq(τ−τ′)j(τ)j(τ′)
+igqZt
0
dτ′hβ∗
q0eiωqτ′+αqNeiωq(t−τ′)ij′(τ′)−g2
qZt
0
dτZτ
0
dτ′eiωq(τ−τ′)j′(τ)j′(τ′)).
(4.30)
The remaining three integrals have the form of Gaussian integrals and are solved explicitly
in Appendix A. The resulting final expression for the density matrix elements in continuous
representation reads
⟨iN|ρS(t)|i′
N⟩=
2
X
i1,...,iN−1=1
2
X
i′
1,...,i′
N−1=1
MiNiN−1. . . Mi1i0M∗
i′
0i′
1. . . M∗
i′
N−1i′
N
×exp [Sinf (t)] ⟨i0|ρS(0) |i′
0⟩,(4.31)
32 4 Real-time path integrals
where we introduced the influence functional
Sinf (t) = −Zt
0
dτZτ
0
dτ′j(τ)−j′(τ)η(τ−τ′)j(τ′)−η∗(τ−τ′)j′(τ′),(4.32)
and the reservoir autocorrelation function [28,31–33,74]
η(τ−τ′) := X
q
g2
q{(2¯nq0+ 1) cos ωq(τ−τ′)−isin ωq(τ−τ′)}.(4.33)
4.2.7 Time-discrete influence functional
For the numerical evaluation of Eq. (4.31), we have to go back to a time-discrete repre-
sentation by reinserting the definition j(′)(τ) := PN
n=1 i(′)
nΘ[τ−(n−1)∆t]Θ[n∆t−τ]. The
influence functional [Eq. (4.32)] then takes the form
Sinf (tN) = −
N
X
n=1
n
X
m=1 in−i′
nηn−mim−η∗
n−mi′
m
=:
N
X
n=1
n
X
m=1
Sinim
i′
ni′
m,(4.34)
with
ηn−m:= Zn∆t
(n−1)∆t
dτZm∆t
(m−1)∆t
dτ′η(τ−τ′).(4.35)
For the evaluation of the time integrals in Eq. (4.35), we have to distinguish between the
cases n > m and n=m. For the case n > m, the elements of the influence functional
read [28,31,33]
Sinim
i′
ni′
m=−X
q
2g2
q
ω2
q
[1 −cos(ωq∆t)] in−i′
n
×{(2¯nq0+ 1) cos [ωq(n−m)∆t]−isin [ωq(n−m)∆t]}im
−{(2¯nq0+ 1) cos [ωq(n−m)∆t] + isin [ωq(n−m)∆t]}i′
m.(4.36)
At n=m, they are given by
Sinim
i′
ni′
m=−X
q
g2
q
ω2
qin−i′
n
×{(2¯nq0+ 1) [1 −cos(ωq∆t)] −iωq∆t+isin (ωq∆t)}im
−{(2¯nq0+ 1) [1 −cos(ωq∆t)] + iωq∆t−isin (ωq∆t)}i′
m.(4.37)
4.2 Path integral formulation of the spin-boson model 33
Finally, this yields the time-discrete path integral representation of the density matrix
describing the reduced system time evolution dynamics of the spin-boson model,
⟨iN|ρS(tN)|i′
N⟩=
N
Y
n=1
2
X
in−1=1
2
X
i′
n−1=1
Minin−1M∗
i′
n−1i′
n
×
n
Y
m=1
exp Sinim
i′
ni′
m⟨i0|ρS(0) |i′
0⟩,(4.38)
with the field transformation matrix
Minin−1=⟨in| cos(Ω0∆t)−iei(ω0−ωL)∆tsin(Ω0∆t)/ℏ
−ie−i(ω0−ωL)∆tsin(Ω0∆t)/ℏcos(Ω0∆t)!|in−1⟩.(4.39)
Again, it is noted that for Ω0= 0 and therefore Minin−1=
1
δin,in−1, Eq. (4.38) directly
yields the time-discrete path integral representation of the independent boson model.
4.2.8 Finite memory approximation
The evaluation of Eq. (4.38) becomes increasingly expensive for increasing numbers of
time steps, since the history of all preceding paths at times 0, . . . , tn−1must be taken
into account for the calculation of time step tn. In order to make the time-discrete path
integral representation numerically accessible, the augmented density tensor scheme is
introduced [37,38]. Here, the essential approximation is based on the fact that often
times the reservoir has a sharp finite memory, corresponding to finite system-reservoir
correlations in time.
Exploiting the finite memory length of environment-induced correlations for an efficient
approximation of the full path integral, only the last nctime steps are taken into account
for the calculation of the current path, while disregarding all prior paths with zero or close
to zero contributions. This treatment is known as the finite memory approximation and
results in the augmented density tensor representation, reading at time tN=N∆t
⟨iN|ρS(tN)|i′
N⟩=
N
Y
n=1
2
X
in−1=1
2
X
i′
n−1=1
Minin−1M∗
i′
n−1i′
n
×
n
Y
m=n−nc
exp Sinim
i′
ni′
m⟨i0|ρS(0) |i′
0⟩.(4.40)
In an improved version of the truncation scheme, rather than cutting off the memory after
ncsteps, in the case of n−m≡ncall former paths up to tnc=nc∆tare additionally
incorporated in the time integration in Eq. (4.35) [102],
ηnc:= ηn−m+
n−nc−1
X
k=1
ηn−k.(4.41)
34 4 Real-time path integrals
4.3 Efficient tensor network implementation
4.3.1 Matrix product states and matrix product operators
Matrix product states (MPS) are representations of arbitrary quantum states in the form
of one-dimensional arrays of contracted tensors (see Fig. 4.3). The tensors may, e.g.,
represent the physical sites of a quantum many-body state, or the full system state during
one step of its time evolution dynamics. Information stored in the tensors can be accessed
and manipulated via their respective site indices, illustrated by unconnected vertical links
in Fig. 4.3. The network is conjoined via contracted link indices (horizontal lines) allowing
for information exchange between the individual tensors. This representation features a
high degree of compression of the corresponding Hilbert space dimension, combining a
reduction from exponential to polynomial scaling of the respective degrees of freedom
with the ability to perform operations directly on the compressed states [39–45]. For the
construction of an MPS the singular value decomposition is employed, where an arbitrary
matrix Mof dimension NA×NBis decomposed into a product of matrices according
to [39]
M=USV †.(4.42)
Here, Udenotes a matrix of dimension NA×min(NA, NB)featuring orthonormal columns,
V†has orthonormal rows and is of dimension min(NA, NB)×NB, and Srepresents a
diagonal matrix of dimension min(NA, NB)×min(NA, NB)with non-negative entries Sii =
si≥0, which are referred to as the singular values of matrix M. When Mrepresents
a quantum state, the latter can be interpreted as entanglement weights [39,40]. In this
case, the singular value decomposition corresponds to a Schmidt decomposition, i.e., the
decomposition of a general quantum state living in a bipartite product space into a product
of orthonormal basis vectors of the respective product spaces [103]. The number of nonzero
singular values determines the Schmidt rank of matrix Mand is at the core of the employed
compression scheme [39,40,42]:
An arbitrary matrix Mwith Schmidt rank dcan be efficiently and accurately approximated
by a matrix M′with rank d′< d via M′=US′V†and
S′=diag(s′
1, s′
2, . . . , s′
d′,0,...,0),(4.43)
where the d−d′smallest occurring singular values in Shave been set to zero, effectively
truncating the column and row dimension of Uand V†, respectively. As a result, the
complexity of matrix Mis decreased without loss of crucial entanglement information.
In order to perform operations on the decomposed quantum state, quantum operators
are restated in the same fashion as matrix product operators (MPOs). Here, the resulting
tensors feature two physical site indices accounting for the input and corresponding output
state as a result of the operator action. To apply an MPO to an MPS, each tensor of
the MPO is contracted with a corresponding tensor of the MPS with matching input
site indices. In consequence of the contraction, a new MPS with updated site indices is
constituted.
4.3 Efficient tensor network implementation 35
Figure 4.3: Diagrammatic representation of a matrix product state, constituting a decomposed
and compressed description of an arbitrary quantum state. Vertical lines emerging
from each tensor (red shapes) represent the physical site indices of the compressed
quantum state. Horizontal lines are link indices conjoining the network and allowing
for information transfer between the tensors.
4.3.2 Tensor network realization of path integrals
Under the improved finite memory approximation, the time evolution of the augmented
density tensor can be reformulated as a tensor network [75,76]. Using standard ten-
sor compression techniques [39], this architecture enables efficient and numerically exact
calculations of the reduced open system dynamics. For the explicit tensor network imple-
mentation, the augmented density tensor [Eq. (4.40)] is first mapped to a vector ρjnin
Liouville space,
ρjN(tN) =
N
Y
n=1
n
Y
m=n−nc
I(jn, jm)ρj0(0),(4.44)
with
I(jn, jm) :=
4
X
jn−1=1
˜
Mjnjn−1exp ˜
Sjnjm.(4.45)
Here, left and right system indices ik,i′
khave been combined to a single index jkfor
each time step, resulting in Liouville space representations ˜
Mjnjn−1and ˜
Sjnjmof the
field transformation matrix and the influence functional, respectively. Afterwards, the
augmented density tensor is rewritten as an MPS, storing both present and past system
states in individual tensors,
ρjn(tn) = X
j0,...,jn−1
Aj0,j1,...,jn−1,jn,(4.46)
here shown for the case n≤nc. In this representation, stored information is compressed
efficiently using standard tensor network techniques, i.e., by consecutive applications of
the singular value decomposition [39–42]. In consequence, memory requirements for the
augmented density tensor are reduced from exponential to polynomial scaling with respect
to the number of preceding paths nctaken into account for the calculation of the current
system state [75], enabling simulations of memory depths inaccessible in traditional path
integral implementations.
The time evolution is carried out by a network of MPOs, shown schematically in Fig. 4.4(a)
(green shapes). The augmented density tensor is stored as an MPS, containing the present
system state and up to nc−1past states stored in individual tensors, with the oldest state
located at the left end of the MPS. During the first time step, the initial system state ρj0(0)
(red shape) is contracted with the first MPO in the network [dashed frame in Fig. 4.4(a)].
36 4 Real-time path integrals
Figure 4.4: (a) Efficient tensor network implementation of real-time path integrals. (b) System
MPS containing the current (red) and past system state (grey) after the first network
contraction. (c) MPO structure for the calculation of the n-th time step.
As a result, the system state is updated and the preceding path is stored to its left,
increasing the length of the MPS by one [see Fig. 4.4(b)]. Once step n=ncis reached, the
oldest path in the MPS is summed over by application of a delta tensor [semicircular shape
in Fig. 4.4(a)], corresponding to the improved finite memory approximation [75,102]. At
this stage, the MPS length remains fixed for the rest of the time evolution. Moreover,
for time-independent problems such as the spin-boson model the structure of the MPO
remains unchanged for all time steps n≥ncapart from the index nomenclature. In
consequence, the MPO constructed for the calculation of time step n=nccan be reused for
the execution of all following steps, resulting in an additional performance gain. Fig. 4.4(c)
shows the structure of the MPO during the n-th time step with n≥nc, given by [75]
Bj′
n−(nc−1),...,j′
n−1,j′
n
j′
n−(nc−1),...,j′
n−2,j′
n−1=hbn−(nc−1)iαn−(nc−1),j′
n−(nc−1)
jn−(nc−1)
n−1
Y
m=n−(nc−2)
[bm]αm,j′
m
αm−1,jm
[bn]j′
n
αn−1,
(4.47)
with αkand j′
kdenoting MPO link indices and updated system states, respectively, and
new tensors
[bm]α′,j′
α,j =δα′
αδj′
jI(α′, j′),(4.48a)
hbn−(nc−1)iα′,j′
j=X
αhbn−(nc−1)iα′,j′
α,j =δj′
jI(α′, j′),(4.48b)
[bn]j′
α=δj
α′[bn]α′,j′
α,j =δj′
αI(j′, j′).(4.48c)
Lastly, the network contraction for the calculation of the n-th time step reads
Aj′
n−(nc−1),...,j′
n−1,j′
n=Bj′
n−(nc−1),...,j′
n−1,j′
n
jn−(nc−1),...,jn−2,jn−1Ajn−(nc−1),...,jn−2,jn−1.(4.49)
4.3 Efficient tensor network implementation 37
In summary, we have introduced the concept of real-time path integrals and used it to
derive a highly efficient and numerically exact representation of the spin-boson model.
Using state-of-the-art tensor network architectures, numerically exact calculations of the
reduced system time evolution dynamics with deep system memory become feasible, en-
abling high-performing simulations of open quantum systems interacting with harmonic
structured reservoirs and diagonal system-reservoir couplings.
5 Artificial neural networks
During the past decade, data-based computation, broadly referred to as machine learning,
has become one of the most important and rapidly growing fields in technological research.
With implications for all major industries, similar to the rise of personal computers during
the past century, it is considered the paramount disruptive technology of our time. The
fundamental idea of machine learning is to recognize recurrent patterns in large amounts of
data via iterative training. Based on the structure of the input data, the algorithm makes
predictions on problems that are as of yet unresolved. Already, applications range from
image and facial recognition to resource management optimization, autonomous driving,
estimations on future stock market development and countless others [46,78]. In one of
the most successful realizations of machine learning, a network of interconnected nodes
is optimized to extrapolate new data from an existing training data set. These so-called
artificial neural networks are reminiscent of the way biological neurons process information
by interacting with each other over synapse formation.
Aside from the aforementioned industrial applications, interest in machine learning tech-
niques has recently grown in physical research fields. In the context of open quantum
systems and their notoriously inaccessible Hilbert space dimensions, the concept of ef-
ficiently sampling core information from large amounts of data appears as a promising
new approach for an approximate description. In recent breakthroughs, artificial neu-
ral networks have been successfully employed for the description of quantum states [47–
55] and open spin-1/2quantum systems with Markovian dynamics [56–59]. Exploiting
an almost limitless potential for parallelization and accurate information compression of
Hilbert space by Metropolis sampling of possible system configurations in a Markov chain
Monte Carlo approach, simulations of very large systems have been realized [46,104–106].
Specifically, the restricted Boltzmann machine neural network architecture has emerged
as a natural and highly efficient representation of the density matrix for small molecular
and spin-1/2quantum systems [47,51,107–112], since it features a one-to-one mapping
of spins to artificial neurons and facilitates direct access to the stationary state via itera-
tive application of a variational principle [113,114]. High numerical performance and fast
convergence times have been achieved for the simulation of symmetric and periodic spin
systems, outperforming established numerical techniques such as quantum Monte Carlo
or tensor network approaches for certain scenarios and including calculations for both
stationary states [56,57] and real-time evolution dynamics [58,59].
In this Chapter, we introduce the core concepts of machine learning and artificial neural
networks for open quantum systems. In Sec. 5.1, some of the basic ideas and principles
of machine learning are introduced. Afterwards, in Sec. 5.2 the restricted Boltzmann ma-
chine architecture is established as a specific class of artificial neural networks. These
39
40 5 Artificial neural networks
networks allow for an straightforward representation of the density matrix for spin-1/2
quantum systems, which is demonstrated in detail in Sec. 5.3. Due to the structure of this
representation, symmetries inherent to the considered system can be exploited to drasti-
cally decrease the required number of connections in the network. As a result, even large
symmetric systems can be described efficiently, constituting a key advantage of the neural
network approach. The implementation of symmetries is discussed in Sec. 5.4. Having
derived an efficient neural network representation of the density matrix, we introduce the
core algorithms required for the training of the probabilistic network. For the sampling of
input data, the Metropolis algorithm is presented in Sec. 5.5. In Sec. 5.6, the stochastic
reconfiguration approach is established to approximate system observables and occurrence
probabilities as statistical expectation values over the sampled training data. Afterwards,
we derive optimization schemes for calcualting both the stationary states and the full time
evolution dynamics of open quantum systems. Lastly, we provide details on the numerical
implementation and the training procedure in Sec. 5.7.
5.1 Basic principles and definitions
5.1.1 Supervised and unsupervised learning
In any machine learning application, an algorithm is fed with input data to produce an
answer to a given problem. We note that in the following, the terms training and learning
are used synonymously and refer to the same process. In the case of supervised learning,
the algorithm is fed with a training data set D={(x1, y1),...,(xNs, yNs)}consisting of
Nssamples of inputs xn∈ X and corresponding target outputs yn∈ Y, where Xand Y
denote the input and output domains, respectively. To extrapolate from the existing data,
the algorithm searches for underlying patterns in the training data set. The goal is to find
a rule connecting inputs and outputs, for instance in the form of a deterministic model
function f(x, ϑ) = y. Here, a set of latent parameters ϑis varied and optimized until
the differences between the training data and values obtained from the model function are
minimized [46]. Once training is finished, the algorithm is fed with new and unclassified
input ˜xand produces corresponding output ˜yusing the optimized model function. It
is apparent that increasing the sample size Nsof the training data likely improves the
accuracy of the answers obtained from the machine. Perhaps the most prominent example
of supervised machine learning is modern image recognition: The algorithm is fed with
a large number of images as input samples xntogether with labels naming the depicted
objects, acting as the corresponding target outputs yn. The goal of the learning procedure
is to find a relationship between the pixel matrices of the images and their respective
contents. Once the learning process is finished, the algorithm is fed with new images of
varying objects. If training was successful, the algorithm will be able to correctly identify
images of training objects [46].
Unsupervised learning, on the other hand, describes training procedures where only input
data is provided, with no explicit target outputs available. Again, the goal is to find
underlying structures and patterns in the input data. A prime example of unsupervised
5.1 Basic principles and definitions 41
learning and the main focus in this work is the search for a model probability distribution
reproducing the statistics of a given input data set, often referred to as generative model-
ing [78]. As in the supervised learning case, the model probability distribution depends on
a set of latent parameters ϑ. To obtain a probability function which fits the distribution
of the input data, these parameters are adjusted until the likelihood of reproducing the
training data is maximized. In particular, here we estimate the density matrix ρof a
considered system by an iteratively optimized model distribution ρϑ.
5.1.2 Models
In the context of machine learning, a model defines the relationship between input and
output data and is obtained by training [46]. We differentiate between deterministic and
probabilistic models. As mentioned in the previous Section, a deterministic model denotes
a function
f(x, ϑ) = y, (5.1)
with inputs x∈ X, outputs y∈ Y and a set of training parameters ϑ. To obtain a deter-
ministic model function, the variational parameters are optimized by an existing training
data set of inputs and corresponding outputs. Alternatively, unsupervised machine learn-
ing can be employed to produce a probabilistic model. Here, the input training data is
produced, e.g., by drawing samples from the configuration space of an existing probability
distribution P(x, y), which is referred to as the target distribution. These sample configu-
rations are used to approximate the target function by a probabilistic model distribution
p(x, y, ϑ),(5.2)
again depending on a set of latent parameters ϑto be optimized. In the context of open
quantum systems, the full density matrix ρtakes the role of the target distribution to be
approximated, since it is often times inaccessible due to exponential scaling of the Hilbert
space dimension. Using the unsupervised learning approach, we solve this problem by
introducing a model distribution ρϑwith vastly reduced degrees of freedom ϑ. By draw-
ing sample configurations from the systems’ Hilbert space as input data, the variational
parameters ϑare optimized to obtain a compressed representation of the full density
matrix.
5.1.3 Cost functions
Having established the core ideas of machine learning, we still require a strategy to obtain
a model function and quantify its ability to generalize from the input data. The model
function is determined by the learning parameters ϑ, which are optimized by the training
data set. To this end, we define a cost function
C(ϑ) = L(x, y, ϑ) + R(ϑ).(5.3)
Most importantly, it consists of a loss function L(x, y, ϑ)measuring the differences between
predictions from the model function and samples from the training data. Secondly, a
42 5 Artificial neural networks
regularizer function R(ϑ)can be included to prevent the model from overfitting, e.g., by
putting restraints on the length of the parameter vector ϑ[46]. Once a cost function which
fits the objective has been constructed, the training procedure is carried out as follows:
Samples from the training data set are fed into the cost function, with the goal of finding
the optimal set of parameters ϑwhich minimize C(ϑ). To find the model parameters
resulting in the most efficient cost function, they are iteratively updated and reinserted
into the cost function until a minimum is reached.
5.1.4 Stochastic gradient descent
The optimization of the cost function lies at the core of any machine learning application.
Stochastic gradient descent describes the standard optimization algorithm for probabilis-
tic models. It represents an iterative and stepwise search for the cost function minimum,
based on samples drawn from the training data. Especially in the field of artificial neural
networks, gradient-based minimization methods such as stochastic gradient descent are
standard practice [46]. The basic idea of gradient descent is to update the learning coef-
ficients ϑsuccessively towards the direction of steepest descent of the cost function. The
update rule for iteration t→t+ 1 is given by
ϑ(t+1) =ϑ(t)−ν∇ϑC(ϑ(t)),(5.4)
with an external parameter νreferred to as the learning rate in the context of machine
learning. Stochastic gradient descent refers to a special case of gradient descent where only
a subset of the training data set is used for the calculation of each update. The resulting
stochastic nature of the gradient direction helps to find the global minimum faster, while
reducing the risk of getting stuck in local minima [46].
5.2 Restricted Boltzmann machines
The model function is determined by the set of training parameters ϑ. In linear regression
models, the desired model function is designed to be linearly dependent on the parameters
ϑ, but may have nonlinear dependencies on the input data. Nonlinear regression models,
on the other hand, are nonlinear in both the input data and the parameters [46]. In
this Section, we introduce the concept of artificial neural networks, which are the most
successful realization of nonlinear regression models. Artificial neural networks are inspired
by biological neurons interacting with each other via synapses. They consist of a set of
interconnected artificial neurons which model the state space of spins up or down, i.e., each
taking one of two possible configurations +1 or −1[46]. The set of training parameters ϑ
determines the shape of the neural network and is split up into weights and biases. Weights
denote the connection strengths between interconnected pairs of neurons, whereas each
neuron is assigned an individual bias or local field strength. The neural network is trained
by optimization of the parameters ϑ, until an optimal representation of the underlying
target function is achieved.
5.2 Restricted Boltzmann machines 43
Figure 5.1: Graphical representation of a restricted Boltzmann machine featuring Nvisible and J
hidden units with biases a,b, and connected by weights W.
Here we focus on a specific class of artificial neural networks, which allows for a direct
mapping of the density matrix for spin-1/2systems: The restricted Boltzmann machine.
This architecture defines a probabilistic model distribution over the states of a set of
binary neurons or units s= (s1, . . . , sG), where each of the neurons carries a probability
of being in either state +1 or −1.sis further divided into a set of visible neurons v=
(v1, . . . , vN)and a set of hidden neurons h= (h1, . . . , hJ). While the visible neurons carry
the information of the model distribution and act as input and output gates, the hidden
units are auxiliary degrees of freedom to improve the networks’ ability to approximate the
target function. Boltzmann machines belong to the class of recurrent neural networks, i.e.,
representations where all neurons are connected to each other by weights. In a restricted
Boltzmann machine, visible neurons are only connected to hidden neurons and vice versa.
A graphical representation is shown in Fig. 5.1. Since it has been found that this network
architecture can be trained much more efficiently than regular Boltzmann machines [46],
current research on machine learning for open quantum systems focuses on restricted
Boltzmann machines [47,51,56–59,79].
The model distribution of a restricted Boltzmann machine is given by a Boltzmann prob-
ability distribution [46],
p(v,ϑ) = 1
ZϑX
h
e−E(v,h,ϑ),(5.5)
with a partition function
Zϑ=X
v,h
e−E(v,h,ϑ),(5.6)
and an Ising-type energy contribution Edepending on the optimization parameters,
E(v,h,ϑ) = −
N
X
i=1
J
X
j=1
Wjihjvi−
N
X
i=1
aivi−
J
X
j=1
bjhj.(5.7)
The parameter vector ϑconsists of biases aiand bjof visible and hidden neurons, respec-
tively, and of weights Wij connecting the visible and hidden units to each other. Since
only the visible units are of interest, all hidden neurons are traced out from the model
distribution in Eq. (5.5). Lastly, the cost function of a restricted Boltzmann machine is
defined by a maximum log-likelihood estimation,
C(ϑ) =
Ns
X
n=1
log p(vn,ϑ),(5.8)
44 5 Artificial neural networks
with vndenoting the n-th sample of a visible unit configuration drawn from the training
data.
5.3 Neural density operator
As a next step, a neural network representation of arbitrary spin-1/2quantum states is
derived. We start with the simpler case of pure wave functions. As motivated in the last
Section, a Boltzmann probability distribution can be constituted by a neural network in the
form of a restricted Boltzmann machine. The restricted Boltzmann machine representation
of a pure wave function featuring Nvisible spins (or neurons) σand Jauxiliary hidden
spins his provided by [47]
ψϑ(σ) = X
h
exp
N
X
i=1
J
X
j=1
Wjihjσi+
N
X
i=1
aiσi+
J
X
j=1
bjhj
.(5.9)
Note that this representation cannot constitute the wave function exactly. Rather, it is de-
signed as an approximative architecture with the achievable accuracy limited by the num-
ber of neurons. For an analytically exact neural network depiction of pure wave functions,
so-called deep networks featuring multiple layers of hidden neurons are required [115].
Since the network does not feature any intralayer connections, the hidden layer can be
traced out explicitly:
ψϑ(σ) = X
h1=±1
. . . X
hJ=±1
exp
N
X
i=1
J
X
j=1
Wjihjσi+
N
X
i=1
aiσi+
J
X
j=1
bjhj
= exp N
X
i=1
aiσi!X
h1=±1
exp N
X
i=1
W1ih1σi+b1h1!. . . X
hJ=±1
exp N
X
i=1
WJihJσi+bJhJ!
= exp N
X
i=1
aiσi!J
Y
j=1 "exp N
X
i=1
Wjiσi+bj!+ exp −
N
X
i=1
Wjiσi−bj!#
= 2 exp N
X
i=1
aiσi!J
Y
j=1
cosh N
X
i=1
Wjiσi+bj!.(5.10)
However, an arbitrary mixed state cannot be represented analogously due to its nondiag-
onal elements. To resolve this problem, an additional set of hidden units, often referred to
as an ancillary layer, is incorporated in the network to create a diagonalized representation
of the mixed state [51]. In other words, to depict an arbitrary mixed state in terms of a
restricted Boltzmann machine, it must first be purified by introducing ancillary degrees of
freedom. As a result, the density matrix representation of a mixed state is estimated by
a mapping ρϑ, called neural density operator. Given two input states σand η, it returns
the corresponding matrix element ρϑ(σ,η) = ⟨σ|ρϑ|η⟩. Moreover, the neural density
operator must fulfil the same properties as the regular density matrix (see Ch. 2),
tr{ρϑ}= 1,(5.11a)
5.3 Neural density operator 45
ρ†
ϑ=ρϑ,(5.11b)
⟨σ|ρϑ|σ⟩ ≥ 0∀σ.(5.11c)
In general, the neural density operator can be written in terms of the pure state restricted
Boltzmann machine representation,
ρϑ(σ,η) = X
hµ
ψϑ(σ,hµ)ψ∗
ϑ(η,hµ),(5.12)
where we introduced the ancillary hidden layer hµto index the different wave func-
tions forming the mixed state [51]. In total, this results in three hidden layers: hσ=
(hσ
1, . . . , hσ
M)and hη= (hη
1, . . . , hη
M)with Munits each for the wave functions ψϑ(σ,hµ)
and ψ∗
ϑ(η,hµ), respectively, and hµ= (hµ
1, . . . , hµ
K)featuring Kunits for the mixing of
the states.
Inserting the restricted Boltzmann machine representation for pure states [Eq. (5.10)], the
neural density operator takes the form
ρϑ(σ,η) = X
hσX
hηX
hµ
exp "K
X
k=1
(ck+c∗
k)hµ
k#
×exp N
X
i=1
aiσi+
M
X
m=1
bmhσ
m+
N
X
i=1
M
X
m=1
Wmihσ
mσi+
N
X
i=1
K
X
k=1
Ukihµ
kσi!
×exp N
X
i=1
a∗
iηi+
M
X
m=1
b∗
mhη
m+
N
X
i=1
M
X
m=1
W∗
mihη
mηi+
N
X
i=1
K
X
k=1
U∗
kihµ
kηi!.(5.13)
The training parameters ϑ= (a,b,c,W,U)are complex-valued and contain biases afor
visible units σand η,bfor auxiliary hidden units hσand hη, and cfor the ancillary hidden
layer hµ, respectively. The complex weights Wconnect the visible layers σand ηto their
auxiliary hidden counterparts hσand hη. In addition, weights Udenote the coupling
elements between the visible layers and the ancillary mixing layer hµ. Fig. 5.2 shows
a graphical representation of the restricted Boltzmann machine as stated in Eq. (5.13),
constituting the neural density operator. Once again, all hidden layers can be traced out
explicitly, yielding [51,58,59]
ρϑ(σ,η) = 8 exp N
X
i=1
aiσi!exp N
X
i=1
a∗
iηi!
×
M
Y
m=1
cosh bm+
N
X
i=1
Wmiσi!cosh b∗
m+
N
X
i=1
W∗
miηi!
×
K
Y
k=1
cosh ck+c∗
k+
N
X
i=1
Ukiσi+
N
X
i=1
U∗
kiηi!.(5.14)
This parametrization fulfills all properties of the density matrix [Eq. (5.11)]. Note that
a normalizing partition function Zas introduced in Eq. (5.5) is so far missing from the
representation: In its current form, the neural density operator remains unnormalized
46 5 Artificial neural networks
Figure 5.2: Restricted Boltzmann machine representation of the neural density operator ρϑ.
since the partition function is often times inaccessible in large quantum systems. We will
resolve this problem by approximating Zusing the system sample configurations drawn for
training, which is explained in detail in the upcoming Sections. Moreover, the networks’
ability to accurately represent an arbitrary density matrix is given by the number of hidden
units Mand K. Specifically, we define the hidden node densities
α=M
N, β =K
N,(5.15)
which determine the representational power of the network [46,78]. In the numerical
implementation, the real and imaginary part of all training parameters are stored and
calculated separately. Due to the structure of Eq. (5.14), only the real part of the ancillary
bias cis required. The total number npof variational parameters stored in ϑis thus given
by
np= 2N+ 2M+K+ 2MN + 2KN, (5.16)
representing the number of degrees of freedom of the neural density operator. In compar-
ison to the N(N+ 1)/2independent density matrix elements required for the calculation
of a system of Nspin-1/2particles, this mapping yields a high grade of compression of
the relevant system information and lies at the core of the imposed truncation scheme.
5.4 Implementation of symmetries
Often times, physical systems and their corresponding Hamiltonians feature intrinsic sym-
metries, for instance in the form of site-translation invariance in a spin chain system. These
symmetries are also inherent to the states of the respective system and maintained dur-
ing its time evolution dynamics. Using the artificial neural network implementation, this
property can be exploited to further decrease the computational cost for the calculation of
symmetric open spin-1/2quantum systems. Specifically, the number of variational param-
eters can be reduced by taking symmetries inherent to the Hamiltonian into account.
5.4 Implementation of symmetries 47
To this end, we consider a set of linear transformations Ts,s={1, . . . , S}, defining a
symmetry group [47]. The visible units of the restricted Boltzmann machine are then
transformed via
Tsσi= ˜σi(s),(5.17a)
Tsηi= ˜ηi(s).(5.17b)
As a result, the neural density operator can be rewritten to be invariant against transfor-
mations imposed by Ts:
ρα,β
ϑ(σ,η) = X
hσ
sX
hη
sX
hµ
s′
exp
β
X
f′=1 c(f′)+c(f′)∗S′
X
s′=1
hµ
f′,s′
×exp "α
X
f=1
a(f)
S
X
s=1
N
X
i=1
˜σi(s) +
α
X
f=1
b(f)
S
X
s=1
hσ
f,s
+
α
X
f=1
S
X
s=1
hσ
f,s
N
X
i=1
W(f)
i˜σi(s) +
β
X
f′=1
S′
X
s′=1
hµ
f′,s′
N
X
i=1
U(f′)
i˜σi(s′)#
×exp "α
X
f=1
a(f)∗
S
X
s=1
N
X
i=1
˜ηi(s) +
α
X
f=1
b(f)∗
S
X
s=1
hη
f,s
+
α
X
f=1
S
X
s=1
hη
f,s
N
X
i=1
W(f)∗
i˜ηi(s) +
β
X
f′=1
S′
X
s′=1
hµ
f′,s′
N
X
i=1
U(f′)∗
i˜ηi(s′)#.(5.18)
The dimensions of the variational parameters ϑ(f)have been modified in the process.
Biases a(f),b(f)and c(f′)are vectors in feature space with f= 1, . . . , α,f′= 1, . . . , β, and
αand βdenote the auxiliary and ancillary hidden unit densities, respectively [Eq. (5.15)].
The restated weights W(f)
iand U(f′)
ihave α×Nand β×Nelements, respectively. This
representation corresponds to a neural density operator featuring M=S×αauxiliary
units and K=S′×βancillary units. As before, all hidden spins can be traced out
explicitly, yielding
ρα,β
ϑ(σ,η) = 8 exp
α
X
f=1
S
X
s=1
N
X
i=1
a(f)˜σi(s)
exp
α
X
f=1
S
X
s=1
N
X
i=1
a(f)∗˜ηi(s)
×
α
Y
f=1
S
Y
s=1
cosh "b(f)+
N
X
i=1
W(f)
i˜σi(s)#cosh "b(f)∗+
N
X
i=1
W(f)∗
i˜ηi(s)#
×
β
Y
f′=1
S′
Y
s′=1
cosh "c(f′)+c(f′)∗+
N
X
i=1
U(f′)
i˜σi(s′) +
N
X
i=1
U(f′)∗
i˜ηi(s′)#.(5.19)
For the specific case of site-translation invariance, the symmetry group defined by Tshas
an orbit of S=Nelements [47]. This means that using the symmetry propoerties of the
system, a given spin configuration is translated into Ncopies. The translation-invariant
parameters ϑ(f)simultaneously act on all copies of the configuration for a given feature
f. As a result, far fewer parameters are required for the same total number of units as
before.
48 5 Artificial neural networks
5.5 Metropolis sampling
In this Section, we take first steps towards the numerical realization of the neural network.
The training of the network is carried out by optimization of the variational coefficients ϑ,
until an optimal mapping of the full density matrix is obtained. In the limit case of small
systems where the full Hilbert space is accessible, all possible configurations of the density
matrix can be taken into account for the optimization of the training parameters. Yet,
it is noted that the resulting neural density operator is still an approximative represen-
tation due to the underlying restricted Boltzmann machine architecture [115]. However,
in general the Hilbert space of open quantum systems quickly becomes inaccessibly large,
making the full density matrix inaccessible as well. To find an accessible approximation,
Nssample configurations of the system, i.e., visible spin configurations (σ,η), are drawn
from the configuration space of the target distribution. The samples are drawn using the
Metropolis algorithm [104], a Markov chain Monte Carlo method to obtain a sequence
of samples from the phase space of a probability distribution, corresponding to a ran-
dom walk in configuration space [46,105,106]. The Metropolis algorithm is applicable
to symmetric probability distributions while its generalization, the Metropolis-Hastings
algorithm, can be used to sample arbitrary distributions [116].
In the following, we briefly outline the steps of the Metropolis algorithm for symmetric dis-
tributions. Its goal is to approximate an inaccessible target distribution P(x)by a model
distribution p(x), which is proportional to the target distribution. This is performed by
drawing samples xfrom the configuration space of the target distribution. For the initial-
ization of the Metropolis algorithm, a random sample xtis drawn from the configuration
space of the target distribution P(x), acting as the current sample. As a first step, a new
or proposed sample x′is drawn on the basis of xt, based on a proposal distribution g(x′|xt)
denoting the probability of drawing sample x′given sample xt. Next, the proposed sam-
ples’ target probability P(x′)is compared to the target probability of the current sample
P(xt)via the acceptance probability function
A(x′, xt) = min 1,P(x′)
P(xt)
g(xt|x′)
g(x′|xt)= min 1,p(x′)
p(xt)
g(xt|x′)
g(x′|xt),(5.20)
measuring the occurrence probability of the proposed sample x′in comparison to xt.
Taking the ratio of target probabilities in Eq. (5.20) bypasses the issue of their factual
inaccessibility, since the model distribution p(x)is proportional to P(x). Hence, p(x)can
be used in its place for the calculation of the acceptance ratio. Lastly, the algorithm has
to choose whether it accepts or rejects the proposed sample x′as the next sample xt+1 in
the Markov chain. To this end, a uniform random number r∈[0,1] is generated and its
value is compared to the acceptance probability [105]:
if r≤A(x′, xt) : xt+1 =x′,
if r > A(x′, xt) : xt+1 =xt.(5.21)
In summary, the Metropolis algorithm performs a random walk in configuration space,
sometimes accepting and sometimes rejecting proposed steps. Due to the selection rule
5.6 Stochastic reconfiguration for Liouvillians 49
of Eq. (5.21), drawn samples with a higher occurrence probability according to the target
distribution P(x)are more likely to be accepted, whereas samples with a low probability
are more often rejected. In consequence, the algorithm chooses a selection of samples which
are used to efficiently approximate the target distribution by the model distribution.
5.6 Stochastic reconfiguration for Liouvillians
At the core of all machine learning algorithms lies the optimization of a predefined cost
function towards a global minimum. In this Section, we introduce the stochastic recon-
figuration algorithm [117–119], which allows for the application of this approach to the
Liouvillian of open spin-1/2quantum systems by approximating system observables and
occurrence probabilities as statistical expectation values over the sample configurations
drawn for the training of the network. While it is in principle possible to simulate real-
time dynamics of open systems using artifical neural networks [58,59], the approach excels
in finding stationary states, such as nonequilibrium steady states or ground state energies
of the system. Here we derive strategies to calculate both stationary states and Marko-
vian time evolution dynamics of open spin-1/2quantum systems, imposed by a Lindblad
master equation as introduced in Ch. 3. We start from the neural density operator rep-
resentation [Eq. (5.14)]. Given this parametrization of the density matrix, our goal is to
find an approximate solution of the Lindblad master equation describing the Markovian
time evolution dynamics of a considered system. To enable the application to established
machine learning strategies, the master equation is first stated as a variational optimiza-
tion problem. In the following it is solved using the stochastic reconfiguration method and
time-dependent variational Monte Carlo techniques, which are extended to the dissipative
case [56–59].
We start by rewriting the density matrix as a vector, such that the Lindblad master
equation of the general form (see Ch. 3)
∂tρ=−i
ℏ[H, ρ] + γ
2
N
X
i=1 2JiρJ†
i−J†
iJiρ−ρJ†
iJi(5.22)
can be stated by means of a non-Hermitian Liouvillian superoperator,
∂tρ=Lρ.(5.23)
As mentioned in Sec. 5.3, all training parameters are split up into their real and imaginary
parts and written in a single vector,
ϑ= (Re{a},Im{a},Re{b},Im{b},Re{c},Re{W},Im{W},Re{U},Im{U}),(5.24)
containing npreal-valued elements [Eq. (5.16)].
50 5 Artificial neural networks
5.6.1 Stationary states
When the training objective is to find the stationary state of an open quantum system,
the variational parameters must be trained to fulfill the condition
∂tρ=Lρ= 0.(5.25)
To this end, we define a corresponding cost function [57,59],
C(ϑ) = ∥Lρϑ∥2
2,(5.26)
and rewrite it by inserting multiple identities, yielding
C(ϑ) = X
σ1,η1⟨η1|{Lρϑ}†|σ1⟩⟨σ1|{Lρϑ}|η1⟩
=X
σ1,η1X
σ2,η2X
σ3,η3
ρ†
ϑ(σ3,η3)L†(σ1,η1,σ3,η3)L(σ1,η1,σ2,η2)ρϑ(σ2,η2),(5.27)
with
⟨σ1|{Lρϑ}|η1⟩
=⟨σ1|−i
ℏ[H, ρϑ]+2JρϑJ†−J†Jρϑ−ρϑJ†J|η1⟩
=X
σ2,η2⟨σ1|−i
ℏ[H, |σ2⟩⟨σ2|ρϑ|η2⟩⟨η2|]+2J|σ2⟩⟨σ2|ρϑ|η2⟩⟨η2|J†
−J†J|σ2⟩⟨σ2|ρϑ|η2⟩⟨η2|−|σ2⟩⟨σ2|ρϑ|η2⟩⟨η2|J†J|η1⟩
=X
σ2,η2⟨σ1|−i
ℏ[H, |σ2⟩⟨η2|]+2J|σ2⟩⟨η2|J†
−J†J|σ2⟩⟨η2|−|σ2⟩⟨η2|J†J|η1⟩⟨σ2|ρϑ|η2⟩
=X
σ2,η2L(σ1,η1,σ2,η2)ρϑ(σ2,η2).(5.28)
Again we are faced with the issue of inaccessiblity, since C(ϑ)cannot be computed without
the knowledge of ρϑ. However, it can be approximated as a statistical expectation value
over the probability distribution
pϑ(σ,η) = |ρϑ(σ,η)|2.(5.29)
This approach is analogous to the concept of variational Monte Carlo techniques and
constitutes the stochastic reconfiguration method [119]. We introduce the estimator of
the Liouvillian superoperator,
˜
L(σ1,η1) := X
σ2,η2
L(σ1,η1,σ2,η2)ρϑ(σ2,η2)
ρϑ(σ1,η1),(5.30)
5.6 Stochastic reconfiguration for Liouvillians 51
and rewrite the cost function accordingly,
C(ϑ) = X
σ,η
pϑ(σ,η)˜
L†(σ,η)˜
L(σ,η).(5.31)
The statistical expectation values are obtained by Monte Carlo sampling of the Hilbert
space, using the Metropolis algorithm where Nssamples of visible unit configurations
(σn,ηn)are drawn (see Sec. 5.5). To construct statistical expectation values, the occur-
rence probability pϑ(σn,ηn)of each drawn sample is calculated [see Eq. (5.29)]. However,
at this stage the neural density operator is still unnormalized due to the inaccessibility of
the full partition function (see Sec. 5.3). To represent a probability, pϑ(σ,η)must still be
normalized. To this end, an approximate partition function is constructed from all drawn
samples,
Zpϑ=
Ns
X
n=1
pϑ(σn,ηn),(5.32)
and the occurrence probability of sample (σn,ηn)is given by
˜pϑ(σn,ηn) = 1
Zpϑ
pϑ(σn,ηn).(5.33)
As a result, the cost function is approximated as
C(ϑ)≈
Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)˜
L(σn,ηn).(5.34)
Inserting the identity for the normalized probability distribution ˜pϑ(σn,ηn), the cost func-
tion takes the form
C(ϑ) =
Ns
X
n=1
|ρϑ(σn,ηn)|2
PNs
m=1 |ρϑ(σm,ηm)|2
×
Ns
X
k=1
ρ†
ϑ(σk,ηk)
ρ†
ϑ(σn,ηn)L†(σn,ηn,σk,ηk)
Ns
X
p=1 L(σn,ηn,σp,ηp)ρϑ(σp,ηp)
ρϑ(σn,ηn)
=1
PNs
m=1 |ρϑ(σm,ηm)|2
×
Ns
X
n,k,p=1
ρ†
ϑ(σk,ηk)L†(σn,ηn,σk,ηk)L(σn,ηn,σp,ηp)ρϑ(σp,ηp).(5.35)
In order to apply the standard stochastic gradient descent procedure for the optimization
of the artificial neural network, we require the gradient of the cost function ∇ϑC(ϑ)with
respect to all parameters ϑ(see Sec. 5.1). The cost function gradient is formally calculated
52 5 Artificial neural networks
using the product rule, yielding [57]
∇ϑlC(ϑ)
= 2Re(Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)
Ns
X
m=1 L(σn,ηn,σm,ηm)ρϑ(σm,ηm)
ρϑ(σn,ηn)Oϑl(σm,ηm)
−"Ns
X
n=1
˜pϑ(σn,ηn)Oϑl(σn,ηn)#"Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)˜
L(σn,ηn)#),(5.36)
where we introduced logarithmic derivatives in the form of diagonal matrices Oϑlwith
elements
[Oϑl]ση,ση =Oϑl(σ,η) = ∂ln ρϑ(σ,η)
∂ϑl
,(5.37)
corresponding to the gradients of the neural density operator with respect to all variational
parameters ϑland for a specific sample configuration (σ,η). The network is trained via
the standard stochastic gradient descent formula, updating the variational parameters
during each iteration t→t+ 1 via
ϑ(t+1)
l=ϑ(t)
l−ν∇ϑlC(ϑ(t)),(5.38)
at learning rate ν.
5.6.2 Time evolution dynamics
As a second application, we demonstrate the ability of artificial neural networks to model
Markovian dynamics of open spin-1/2systems. For the full time evolution dynamics
prescribed by Eq. (5.23), we first calculate the formal time derivative of the neural density
operator,
∂tρϑ=
np
X
l=1
∂ρϑ
∂ϑl
˙
ϑl=
np
X
l=1
˙
ϑlOϑlρϑ.(5.39)
Note that both the density matrix ρand the neural density operator ρϑare unknown,
representing the target and model distribution, respectively. Again, the only information
available are the neural density operator elements ρϑ(σn,ηn)for individual sample con-
figurations (σn,ηn)and the current parameter set ϑ. To find an approximation of the
Lindblad dynamics, we require a closed set of equations of motion for the time-dependent
variational parameters ϑl(t). They are optimized by minimizing the difference between
the approximate variational evolution of the neural density operator [Eq. (5.39)] and the
equation of motion for the density matrix as prescribed by Eq. (5.23). Taking the square
norm difference between these two expressions defines the cost function [58,59],
C(ϑ) =
X
l
˙
ϑlOϑlρϑ−Lρ
2
2
= X
l
˙
ϑlρ†
ϑO†
ϑl−ρ†L†! X
l
˙
ϑlOϑlρϑ−Lρ!.(5.40)
5.6 Stochastic reconfiguration for Liouvillians 53
Here we minimize Eq. (5.40) with respect to the time derivative of a specific coefficient ˙
ϑl′
by solving ∂C(ϑ)/∂ ˙
ϑl′= 0, yielding
X
l
˙
ϑlρ†
ϑO†
ϑl−ρ†L†!Oϑl′ρϑ+ρ†
ϑO†
ϑl′ X
l
˙
ϑlOϑlρϑ−Lρ!= 0.(5.41)
Eq. (5.41) is rearranged as a system of linear equations [117–119]
X
l
Sl′l˙
ϑl=fl′,(5.42)
where we introduced the covariance matrix Swith elements
Sl′l=ρ†
ϑO†
ϑl′Oϑlρϑ+ρ†
ϑO†
ϑlOϑl′ρϑ,(5.43)
and the vector of forces fwith entries
fl=ρ†
ϑO†
ϑl
Lρ+ρ†L†Oϑlρϑ.(5.44)
In correspondence to the stationary case, we evaluate these quantities using the stochastic
reconfiguration approach, yielding statistical expectation values as approximate expres-
sions for the covariance matrix and vector of forces [57,119],
Sl′l≈2Re(Ns
X
n=1
˜pϑ(σn,ηn)O†
ϑl′(σn,ηn)Oϑl(σn,ηn)
−"Ns
X
n=1
˜pϑ(σn,ηn)O†
ϑl(σn,ηn)#"Ns
X
n=1
˜pϑ(σn,ηn)Oϑl(σn,ηn)#),(5.45)
and
fl≈2Re(Ns
X
n=1
˜pϑ(σn,ηn)O†
ϑl(σn,ηn)˜
L(σn,ηn)
−"Ns
X
n=1
˜pϑ(σn,ηn)O†
ϑl(σn,ηn)#"Ns
X
n=1
˜pϑ(σn,ηn)˜
L(σn,ηn)#).(5.46)
Inverting the covariance matrix in Eq. (5.42) finally results in the equation of motion for
the variational parameters,
∂tϑl′=X
l
S−1
l′lfl.(5.47)
Finally, when assuming discrete time steps, ∂tϑl′≈∆ϑl′/∆t, the learning parameters are
updated during each time step via
ϑ(t+1)
l′=ϑ(t)
l′+ ∆tX
l
S−1
l′lfl,(5.48)
with a step size or learning rate ∆t=νand in resemblance to the update procedure
prescribed by the stochastic gradient descent method [see Eq. (5.4)]. Fig. 5.3 schematically
54 5 Artificial neural networks
Figure 5.3: Flow diagram schematically depicting the network training algorithm for the calcula-
tion of time evolution dynamics.
shows a flow diagram of all steps taken during the time evolution of the artificial neural
network. Lastly, we note that it is in principle possible to find the stationary state of
a system using Eq. (5.47) and setting Sl′l≡1, such that ∂tϑl′=fl′. However, in this
description information is lost during the initial steps of the derivation, namely the adjunct
Liouvillian operator, resulting in decreased stability of the optimization procedure for the
steady state [57,59].
5.7 Numerical execution
We conclude this Chapter by providing details on the numerical implementation. The
interplay of all introduced algorithms and their specific execution is explained. Moreover,
we provide some additional techniques to increase numerical stability and performance.
5.7 Numerical execution 55
5.7.1 Observables
As a last step for the calculation of open quantum systems, we require a strategy for
the evaluation of observables. In the density matrix formalism, the expectation value of
arbitrary observables Xis provided by
⟨X⟩= tr{Xρ}.(5.49)
Again following the stochastic reconfiguration approach, ⟨X⟩is interpreted as a statistical
expectation value over a probability distribution (see Sec. 5.6). For diagonal observables
we can additionally define the probability distribution
qϑ(σ) = ρϑ(σ,σ),(5.50)
which again must be normalized by approximating the partition function over all drawn
samples. The occurrence probability of sample (σn)is thus given by
˜qϑ(σn) = 1
Zqϑ
qϑ(σn),(5.51)
with
Zqϑ=
Ns
X
n=1
qϑ(σn).(5.52)
Now the expectation value obtained from the density matrix can be approximated as a
statistical expectation value over all sample configurations, ⟨X⟩≈⟨X⟩q, with
⟨X⟩q=
Ns
X
n=1
˜qϑ(σn)X
ξ
X(σn,ξ)ρϑ(ξ,σn)
ρϑ(σn,σn).(5.53)
5.7.2 Sampling
As a first step of the implementation, the Metropolis algorithm is used to draw sample con-
figurations (σn,ηn)from configuration space of the target distribution. Specifically, neural
density operator elements ρϑ(σn,ηn)are generated to estimate the target distribution ρ
by the model distribution ρϑvia occurrence probabilites pϑ(σ,η). For the calculation of
diagonal system observables, diagonal sample configurations (σn,σn)are simultaneously
drawn to generate occurrence probabilities qϑ(σ). During each step of the sampling pro-
cedure, the new proposed sample is compared to the last sample configuration. Depending
on their probability ratio, the proposed sample is either accepted or rejected following a
Markov chain Monte Carlo approach (see Sec. 5.5). The acceptance function can, e.g.,
take the form of an exponential function [57],
Aexp[(σn+1,ηn+1); (σn,ηn)] = exp −pϑ(σn,ηn)
pϑ(σn+1,ηn+1),(5.54)
56 5 Artificial neural networks
or is modeled by a linear ratio [58],
Alin[(σn+1,ηn+1); (σn,ηn)] = min 1,pϑ(σn+1,ηn+1)
pϑ(σn,ηn).(5.55)
For the easiest realization, the proposed samples pϑ(σn+1,ηn+1)are chosen randomly
from full configuration space. For an improved mapping, it is also possible to employ
a set of selection rules taking the role of to the proposal distribution function g(x′|xt)
in Eq. (5.20) [58]. Here we provide an example of such a selection rule set for a spin-
1/2quantum system of Nsites. When sampling from Hilbert space to approximate the
distribution pϑ(σ,η), four types of moves are allowed to form a new proposed sample
based on the current sample (σn,ηn)=(σ1,n, . . . , σi,n, . . . , σN,n;η1,n, . . . , ηi,n, . . . , ηN,n)
with σi,n =±1,ηi,n =±1:
1. a single index of a random site iis flipped, either σi,n or ηi,n.
2. both indices σi,n and ηi,n of a random site iare flipped.
3. two neighboring indices of a random site iare flipped, either σi,n and σi+1,n or ηi,n
and ηi+1,n.
4. a random new configuration (σ˜n,η˜n)is drawn from a uniform distribution.
While the last move only occurs at a probability of 1%, the remaining moves are set equally
probable. For the diagonal sampling procedure to generate the distribution qϑ(σ), three
possible moves are allowed given a current sample (σn):
1. the index σi,n of a random site iis flipped.
2. two neighboring indices σi,n and σi+1,n of a random site iare flipped.
3. a random new configuration (σ˜n)is drawn from a uniform distribution.
Again, while the first two moves have the same probability of occurring, a completely new
configuration is only drawn at a probability of 1%.
5.7.3 Logarithmic derivatives
Once a drawn sample is either accepted or rejected as new current sample (σn,ηn), we have
to calculate all corresponding neural density operator gradients Oϑl(σn,ηn)[Eq. (5.37)].
The logarithmic derivatives of the neural density operator with respect to all variational
5.7 Numerical execution 57
parameters ϑ= (a,b,c,W,U)are given by
∂ln[ρϑ(σn,ηn)]
∂Re{ai}=σi,n +ηi,n,(5.56a)
∂ln[ρϑ(σn,ηn)]
∂Im{ai}=i(σi,n −ηi,n),(5.56b)
∂ln[ρϑ(σn,ηn)]
∂Re{bm}=ξ1(m, σn) + ξ∗
1(m, ηn),(5.56c)
∂ln[ρϑ(σn,ηn)]
∂Im{bm}=i[ξ1(m, σn)−ξ∗
1(m, ηn)] ,(5.56d)
∂ln[ρϑ(σn,ηn)]
∂Re{Wmi}=σi,nξ1(m, σn) + ηi,nξ∗
1(m, ηn),(5.56e)
∂ln[ρϑ(σn,ηn)]
∂Im{Wmi}=i[σi,nξ1(m, σn)−ηi,nξ∗
1(m, ηn)] ,(5.56f)
∂ln[ρϑ(σn,ηn)]
∂Re{ck}= 2ξ2(k, σn,ηn),(5.56g)
∂ln[ρϑ(σn,ηn)]
∂Re{Uki}= (σi,n +ηi,n)ξ2(k, σn,ηn),(5.56h)
∂ln[ρϑ(σn,ηn)]
∂Im{Uki}=i(σi,n −ηi,n)ξ2(k, σn,ηn),(5.56i)
with
ξ1(m, πn) = tanh "bm+
N
X
i=1
Wmiπi,n#,(5.57a)
ξ2(k, σn,ηn) = tanh "ck+c∗
k+
N
X
i=1
(Ukiσi,n +U∗
kiηi,n)#.(5.57b)
This procedure is repeated for Nsindividual samples (σn,ηn). Once all gradients are
calculated and stored, the covariance matrix [Eq. (5.45)] can be calculated for the time
evolution dynamics case. It is noted that the variational coefficients must be initialized at
small nonzero values, e.g., ϑl∈[−0.01,0.01]\{0}. When initialized at zero, the evaluation
of Eq. (5.56) becomes trivial and the respective parameters remain zero throughout the
entire training procedure. In addition, choosing large values for the initial variational
coefficients may result in numerical divergence.
5.7.4 Local Liouvillian
Aside from the logarithmic derivatives, the local estimator of the Liouvillian ˜
L(σn,ηn)
[Eq. (5.30)] must be calculated individually for each sample. For a given Hamiltonian H,
the first term of the local Liouvillian reads [58]
−i
ℏ⟨σn|[H, ρ]|ηn⟩
ρϑ(σn,ηn)=−i
ℏX
ξH(σn,ξ)ρϑ(ξ,ηn)
ρϑ(σn,ηn)−ρϑ(σn,ξ)
ρϑ(σn,ηn)H(ξ,ηn),(5.58)
58 5 Artificial neural networks
with
H(σ,η) = ⟨σ|H|η⟩.(5.59)
The second term of ˜
L(σn,ηn)arises from the dissipator
D[qγ/2J]ρ=γ
2
N
X
i=1 2JiρJ†
i−J†
iJiρ−ρJ†
iJi,(5.60)
resulting in
⟨σn|D[pγ/2J]ρ|ηn⟩
ρϑ(σn,ηn)=γ
2
N
X
i=1 "2X
σ′,η′⟨σn|Ji|σ′⟩⟨η′|J†
i|ηn⟩ρϑ(σ′,η′)
ρϑ(σn,ηn)
−X
ξ⟨σn|J†
iJi|ξ⟩ρϑ(ξ,ηn)
ρϑ(σn,ηn)−X
ξ
ρϑ(σn,ξ)
ρϑ(σn,ηn)⟨ξ|J†
iJi|ηn⟩#.
(5.61)
5.7.5 Pre-conditioning
The current training iteration is completed by updating the variational parameters ϑ. In
the time evolution dynamics case, their time derivatives are evaluated as prescribed by
Eq. (5.47), requiring the inversion of the positive semi-definite covariance matrix S−1.
This poses a numerical challenge since Smay be ill conditioned, e.g., featuring very small
eigenvalues, with the inversion resulting in dramatically increased statistical noise [118].
Hence to ensure invertibility and improve the accuracy of the calculation, the covariance
matrix is modified first.
To avoid the issue of large statistical noise, the diagonal elements of Sare regularized
according to [47,118,119]
˜
Sl′l=Sl′l+λ(t)δll′Sl′l,(5.62)
with a regularizer function
λ(t) = max λ0bt, λmin,(5.63)
which decays at increasing time steps tand features parameters b,λ0and λmin. Moreover,
the covariance matrix may feature eigenvalues that differ in several orders of magnitude.
To stabilize the inversion, it is helpful to pre-condition Sby means of a rescaling [119],
Spc
l′l=1
√Sl′l′Sll
Sl′l.(5.64)
The vector of forces and the time derivatives of the learning parameters are rescaled
accordingly,
fpc
l=1
√Sll
fl,(5.65a)
˙
ϑpc
l′=pSl′l′˙
ϑl′.(5.65b)
5.7 Numerical execution 59
S,fand ˙
ϑare pre-conditioned first. Afterwards, the covariance matrix is regularized.
After solving the set of linear equations of motion, the resulting variational parameter
derivatives are rescaled back again via
˙
ϑl′=1
√Sl′l′
˙
ϑpc
l′.(5.66)
Finally, the parameters are updated for the next step of the time evolution using Eq. (5.48),
concluding the current training iteration of the artificial neural network.
Part II
Non-Markovian phenomena in
open quantum systems
61
6 Memory-critical dynamical buildup of
phonon-dressed Majorana fermions
In this Chapter, we explore the dynamical impact of non-Markovian dissipative interac-
tions on topological state of matter, giving rise to a quantum memory. Specifically, the
open system dynamics of a one-dimensional polaronic topological superconductor with
phonon-dressed p-wave pairing is investigated, where the dynamics is induced by a fast
temperature increase. For rising memory depths, the topological properties are shown to
transit from monotonic relaxation to a plateau of substantial value into a collapse-and-
buildup behavior, even when the system is close to the topological phase boundary. Above
a critical memory depth, the system can approach a dressed state in dynamical equilib-
rium with phonons with close to perfect buildup of topological correlation. The Chapter
is closely based on the publication “Memory-critical dynamical buildup of phonon-dressed
Majorana fermions” by O. Kästle et al. [69]
6.1 Introduction
Exploring topological properties of nonequilibrium open quantum systems represents a
major objective for the realization, probing and utilization of topological states of mat-
ter [120–134]. Recent investigations of topological systems interacting with an environment
have focused on Markovian reservoir couplings described by a Lindblad master equation
for the reduced system density matrix, resulting in open-dissipative time evolution dy-
namics [135–148]. However, experimental solid-state realizations of topological systems
are often based on semiconductor nanostructures, which inevitably interact with a struc-
tured phonon environment, resulting from the deformation of host materials and lattice
vibrations. The most prominent examples of topological solid-state setups are based on
topological superconductors [149–159]. Due to the arising dynamical interplay with the
structured phonon reservoir, the Markovian approach based on the Lindblad formalism
often fails to correctly describe these scenarios: Dissipation-induced decoherence to the
structured environment enables quantum memory effects via time-delayed information
backflow to the system [2,3], resulting in a time scaling of the system dynamics which
is absent in a Markovian context. In consequence, a unique dynamical interplay between
topological properties and non-Markovian dissipation may arise.
In this Chapter, we demonstrate that quantum memory effects resulting from a non-
Markovian parity-preserving interaction of a topological p-wave supercondutor surrounded
by a phonon environment can result in a recovery and long-term stabilization of topolog-
ical properties not exhibited in a corresponding Markovian setup [69]. We employ the
63
64 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
polaron master equation (see Ch. 3) for the description of open-dissipative dynamics of a
polaronic topological superconductor with phonon-renormalized Hamiltonian parameters.
Specifically, we investigate a Kitaev superconducting wire - the paradigmatic example of a
setup supporting Majorana fermions - weakly coupled to a 3D bulk phonon reservoir [160].
A perfect realization of the Kitaev wire features a gapped energy spectrum, causing the
emergence of Majorana zero modes. They are robust against external perturbations and
exhibit permanent and topologically protected edge-edge correlations. In the presence of
a phonon reservoir with energies surpassing the spectral gap, dephasing processes arise.
In this scenario, the otherwise topologically protected Majorana edge correlations typi-
cally decay if the phonon reservoir is treated in a Markovian description, even if parity
is preserved [144]. Here, we show that this is not the case if the microscopic proper-
ties of the electron-phonon interactions are taken fully into account, as they give rise to
a non-Markovian and partially time-reversible dynamics [69]. In contrast to Markovian
decoherence destroying topological properties in the long term limit, we show that a fi-
nite quantum memory enables substantial preservation of topological properties far from
equilibrium, even when the polaron Hamiltonian approaches the topological phase bound-
ary. Depending on the memory depth, i.e., the characteristic time scale of the quantum
memory, the dynamical Majorana edge correlation can perform a collapse-and-buildup
relaxation. Strikingly, above a critical value of the memory depth, the edge correlation
can be recovered almost completely, corresponding to a polaronic state of the topological
superconductor in a phonon-mediated dynamical equilibrium [69].
The polaronic Kitaev chain and its theoretical background are introduced in Sec. 6.2. First,
the bare Kitaev chain and the considered fermion-phonon interaction are discussed, before
the open system Hamiltonian is transformed into the polaronic frame. Afterwards, we
introduce the non-Markovian polaron master equation for the dissipative Kitaev wire and
discuss the impact of time-resolved phonon correlations on the open system dynamics. In
Sec. 6.3, the unfolding non-Markovian Majorana edge correlation dynamics are presented
and compared to calculations using a Markovian Lindblad-type master equation with a
time-independent dephasing rate. We discuss the emergence of a critical memory depth
in the non-Markovian framework in Sec. 6.4, enabling a recovery and stabilization of
topological properties via time-delayed system-reservoir correlations. Sec. 6.5 includes
further discussions on the impact of additional p-wave interparticle interaction on the
Majorana edge correlation, the inclusion of the polaron energy shift in the open system
Hamiltonian, the nature of the observed correlation and additional calculations for an
initially nonideal Kitaev chain. Lastly, we summarize our findings and discuss resulting
implications and perspectives for the field in Sec. 6.6.
6.2 Polaronic Kitaev chain
Symmetry-protected topological states exist in a wide variety of quantum systems, such as
spin chains and fermionic systems [161–168]. These non-Abelian anyonic particles are ro-
bust against microscopic imperfections, making them promising candidates for topological
quantum computing [169–173]. The most prominent example of such topological states are
6.2 Polaronic Kitaev chain 65
Majorana zero modes [174–178] emerging at the edges of the 1D Kitaev superconducting
wire [160]. Several setups for the physical realization of Majorana edge states based on the
Kitaev model have been proposed both in solid-state structures [179–182] and setups of ul-
tracold atoms and molecules [136,183–185]. Recent experiments based on these proposals
strongly imply the emergence of Majorana states [151–154,156,186–188], with ongoing
discussions regarding the role of dissipative processes [93,94,137,140,141,143,144,189–
196]. Current studies of dissipative Kitaev superconductors have demonstrated, e.g., that
the absorption of thermal phonons at finite temperatures allows for the excitation of the
electronic ground state to a delocalized state whose energy surpasses the topological en-
ergy gap, resulting in an energy level broadening [93]. Moreover, it has been shown that
a finite tunneling rate of superconducting quasiparticles from the p-wave superconductor
to the wire results in the decay of Majorana bound states [137]. Coupling the chain to an
ungapped fermionic bath has the same effect, since these sorts of environmental interac-
tions explicitly break the parity of the fermionic ground state [197]. Using classical noise
as a model of dissipation, it has been demonstrated that due to motional narrowing in
fast noise, dephasing processes are suppressed in the limit of fast fluctuations, enabling
quantum operations on the Majorana modes [143]. Furthermore, in a Markovian phonon
environment, the edge-edge correlation decays exponentially even if parity is preserved,
while an incorporation of disorder can be used to increase the Majorana lifetime [144].
With respect to non-Markovian system-reservoir interactions, it has been shown that an
impurity or a qubit can be employed as a non-Markovian quantum probe of a topologi-
cal reservoir [198–200] and that topological phenomena induced by Markovian dissipation
processes can sustain in non-Markovian regimes [201].
In contrast, in the following we demonstrate that tailoring the coupling to a super-
Ohmic reservoir may suppress the decay of and even fully restore topological properties
of a phonon-dressed Kitaev wire, opening up new prospects for the control of memory-
dependent topological phenomena. Specifically, we investigate the fate of Majorana edge
modes in the presence of a parity-preserving, structured phonon reservoir. For its de-
scription we choose the polaron master equation [1,14,80,87–92] to include as much
information about the electron-phonon interaction as possible, while still remaining in a
reduced density matrix description in second order [14,80,89].
6.2.1 p-wave superconducting chain coupled to phonons
We consider the paradigmatic Kitaev p-wave superconductor [160], with a super-Ohmic
coupling to a structured 3D phonon reservoir (see Fig. 6.1). The bare Kitaev Hamiltonian
reads
Hk/ℏ=
N−1
X
l=1 h−Jc†
lcl+1 + ∆clcl+1+ H.c.i−µ
N
X
l=1
c†
lcl,(6.1)
describing spinless fermions cl,c†
lon a chain of Nsites l, with a nearest-neighbor tunneling
amplitude J∈R, superconductive pairing amplitude ∆∈R, and chemical potential µ. For
the case |µ|<2Jand ∆= 0, the system exhibits a topological phase featuring unpaired
66 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
Figure 6.1: Sketch of a polaronic Kitaev chain featuring phonon-dressed spinless fermions (blue
circles) with nearest-neighbor tunneling Jand coupled to a structured reservoir of bulk
phonons (green circles). They are arranged in close proximity to a superconducting
wire (red shape), exhibiting a renormalized p-wave pairing ⟨B⟩∆at temperature T[see
Eq. (6.4)]. In the topological ground state of the ideal chain, two unpaired Majorana
edge modes γL,γRemerge at the boundary sites (grey shapes). The system-reservoir
coupling gkfeatures a mode-dependence with respect to the spectral width σ. In an
initially nonideal scenario, a chemical potential µacts as a perturbation on the chain.
Majorana edge modes γL/R =PjfL/R,jbjlocated at the boundary sites of the chain, with
Majorana operators b2j−1=cj+c†
j,b2j=−i(cj−c†
j)and fL/R,j denoting Majorana wave
functions exponentially localized near the left (L) and right (R) edges. As an observable,
we consider the nonlocal correlation θ=−i⟨γLγR⟩=±1exhibited by the Majorana modes
which corresponds to the fermionic parity of the ground states.
The entire chain is coupled to a structured 3D phonon reservoir with parity-preserving
interaction, described by [96–98,202–207]
Hb/ℏ=Zd3k"ωkr†
krk+
N
X
l=1
gkc†
lcl(r†
k+rk)#.(6.2)
Here, operators r†
k,rkdenote bosonic creation and annihilation operators of phonons with
momentum kand frequencies ωk=cs|k|, where csis the sound velocity of the considered
environment. This form of coupling is commonly applied to describe the formation of
polarons (see Sec. 3.3) [14,87–89,92,100]. Specifically, in solid-state setups it can be
used to represent the coupling of semiconductor nanostructures to longitudinal acoustic
phonons which typically feature a long wavelength. Without loss of generality, here we
choose a generic super-Ohmic fermion-phonon coupling gkwith the frequency-dependence
modeled by a Gaussian function,
gk=fphs|k|
σ2exp −k2
σ2!,(6.3)
where σand fph denote a spectral width and a dimensionless amplitude, respectively.
In the considered scenario, the topological superconductor and reservoir are initially in
6.2 Polaronic Kitaev chain 67
equilibrium at low temperatures, before a fast and sudden temperature increase induces
the open system dynamics.
The total Hamiltonian of the dissipative Kitaev chain is denoted by H0=Hk+Hband can
arise in condensed matter systems ranging from from atom-molecule and solid-state setups,
where the reservoir is realized via a superconducting wire and the coupling amplitude ∆
is induced by the proximity effect, to setups with ultracold fermionic atoms in an optical
lattice, where the interaction arises via coupling to an s-wave superfluid with Raman
lasers [190].
6.2.2 Dressed-state representation
Due to the exponential scaling of the Hilbert space dimension, the inclusion of a struc-
tured phonon reservoir Hbrenders calculations of open system dynamics very expensive,
e.g., in a Heisenberg picture approach, and thus requires an approximative treatment.
Here we choose a polaron representation of the coupled system (see Fig. 6.1), which can
be effectively described by a second-order perturbative polaron master equation for the
polaronic dressed-state system-reservoir Hamiltonian [1,14,15,80,87–92,208,209]. The
phonon-renormalized Hamiltonian in dressed-state representation retains the coherent pro-
cess constituted by higher-order contributions from the fermion-phonon interaction. By
combining the polaronic picture with second-order perturbation theory, dynamical non-
Markovian features arising in the long term limit are accounted for. In contrast, merely
applying the typical second-order Born approximation of the bare reservoir Hamiltonian
Hbis not sufficient [96,202–204].
As a first step, the dissipative Kitaev Hamiltonian H0=Hk+Hbis transformed into
the dressed-state representation. To this end, we define collective bosonic operators
R†=Rd3k(gk/ωk)r†
kand apply the unitary polaron transformation Hp=UpH0U−1
pvia
transformation matrices Up= exp[PN
l=1 c†
lcl(R†−R)]. Detailed calculations are provided
in Appendix B, resulting in a phonon-dressing of fermionic operators c†
l→e−(R−R†)c†
l.
The polaron-transformed total Hamiltonian is derived as
Hp/ℏ=
N−1
X
l=1 h−Jc†
lcl+1 + ∆e−2(R−R†)c†
l+1c†
l+ H.c.i−µ
N
X
l=1
c†
lcl+Zd3k ωkr†
krk.(6.4)
In the dressed-state representation, the considered fermion-phonon interaction constitutes
a polaronic Kitaev chain featuring phonon-dressed p-wave pairing with phonon-induced
quantum fluctuations, decreasing the tunneling capability. This representation intuitively
shows that arising Majorana modes are damped as a consequence of the environmen-
tal coupling, since p-wave pairing is essential for the formation of the topological edge
states. When considering a scenario with individual reservoirs for each chain site, the
nearest-neighbor tunneling is also subject to the phonon-dressing, weakening its coupling
amplitude and further decreasing the potential of Majorana emergence in the system. For
the moment, we have neglected an energy renormalization term arising in the derivation
68 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
of Eq. (6.4), given by
Hshift/ℏ=Zd3kg2
k
ωk N
X
l=1
c†
lcl!2
,(6.5)
known as the polaron shift. It is regarded as standard practice to disregard this contribu-
tion if the initial state of the respective system is an equilibrium [1,32]. Since we consider
the Kitaev chain to be ideal and initially in ground state configuration, neglecting the
term is justified. However, as a comparison and to ensure validity of the results in the
presence of this energy renormalization, in Sec. 6.5 we reproduce all of our key findings
in the presence of the polaron energy shift in a numerically exact fashion, demonstrating
that its inclusion does not affect the qualitative physics.
Before continuing with the derivation of the polaron master equation, we rewrite Eq. (6.4)
as Hp=Hp,s +Hp,I +Hp,b with Hp,b/ℏ=Rd3k ωkr†
krkfor the free reservoir contribution.
In the limiting case without coupling, gk→0, the bare Kitaev Hamiltonian dynamics can
be recovered by introducing a Franck-Condon renormalization to the dressed-state interac-
tion Hamiltonian Hp,I, such that trB{[Hp,I, ρ(t)]}= 0, with ρ(t)denoting the total density
operator (see Sec. 3.4). Tracing out the reservoir, the renormalized system Hamiltonian
is given by
Hp,s/ℏ=
N−1
X
l=1 h−Jc†
lcl+1 +⟨B⟩∆c†
l+1c†
l+ H.c.i−µ
N
X
l=1
c†
lcl,(6.6)
with a pairing renormalization factor ⟨B⟩= trB{exp[−2(R−R†)]}determined explicitly
below. Lastly, the renormalized system-reservoir interaction in the polaron picture reads
Hp,I/ℏ= ∆
N−1
X
l=1 he−2(R−R†)−⟨B⟩c†
l+1c†
l+H.c.i.(6.7)
The Franck-Condon renormalization is carried out in detail in Appendix B.
6.2.3 Polaron master equation
In the present study, we focus on the limit ⟨B⟩ ≪ 1, which allows to treat Hp,I perturba-
tively in second-order Born theory, as dynamical decoupling effects cannot occur [1,89].
In Appendix B, we present a detailed derivation of the polaron master equation for the
reduced system density matrix ρS(t)of the polaronic Kitaev chain, yielding
˙ρS(t) = −i[Hp,s/ℏ, ρS(t)] −⟨B⟩2
ℏ2Zt
0
dτn(cosh [ϕ(τ)] −1) [Xa, Xa(−τ)ρS(t)]
−sinh [ϕ(τ)] [Xb, Xb(−τ)ρS(t)] + H.c.o.(6.8)
Here,
Xa=−J
N−1
X
l=1 c†
lc†
l+1 +cl+1cl,(6.9a)
Xb=J
N−1
X
l=1 c†
lc†
l+1 −cl+1cl,(6.9b)
6.2 Polaronic Kitaev chain 69
-5
0
5
10
0 25 50 75Jτ
0.025
0.05
0 1 2
k [nm-1]
Abs(Φ(τ))
Re(Φ(τ))
Im(Φ(τ))
Figure 6.2: Phonon correlation function ϕ(τ)at spectral widths σ= 0.2(blue lines) and σ= 0.6
(orange lines) of the Gaussian coupling element gkshown in the inset.
denote collective system operators, whose dynamics obeys a time-reversed unitary evolu-
tion governed by the renormalized system Hamiltonian Hp,s,Xa,b(−τ)≡e−iHp,s τXa,beiHp,sτ.
Moreover, ϕ(τ)represents the phonon correlation function
ϕ(τ) = Zd3k
2gk
ωk
2coth ℏωk
2kBTcos (ωkτ)−isin(ωkτ),(6.10)
with kBdenoting the Boltzmann constant. The renormalization factor ⟨B⟩is determined
by the initial phonon correlation, ⟨B⟩= exp[−ϕ(0)/2], and features temperature depen-
dence.
Eq. (6.8) provides the time evolution dynamics for the polaronic Kitaev chain. It features
a phonon-renormalized Hamiltonian Hp,s and a memory kernel that involves both reservoir
and system correlators ϕ(τ)and Xa,b(−τ). Crucially, the correlation of a super-Ohmic
phonon reservoir only has a finite lifetime τM(see Fig. 6.2), such that ϕ(τ)≈0for
τ > τM. In consequence of the finite memory depth, once the time evolution surpasses
t>τM, only system correlators Xa,b(−τ)within the past times τ≤τMcontribute to the
memory kernel in Eq. (6.8) and the integral becomes effectively time-independent for times
t>τM. Hence, the contribution of the memory effect depends on the time scale on which
the system correlators Xa,b(−τ)evolve, yielding a substantial impact when it is comparable
to τM. It is typically determined by the inverse E−1
∆of the bulk energy gap of Hp,s. For
a given temperature and sound velocity cs, the phonon correlation function ϕ(τ)and τM
critically depend on the spectral bandwidth σof the fermion-phonon coupling element gk,
as exemplified in Fig. 6.2. The bandwidth and mode localization of the coupling element
(inset in Fig. 6.2) critically determine the depth of the memory kernel: Choosing a large
spectral width σresults in short lifetimes τM(orange lines in Fig. 6.2) and the integral
in Eq. (6.8) quickly converges to a constant value. At the same time, the initial phonon
correlation ϕ(0) is decreased, yielding a larger renormalization parameter ⟨B⟩. Vice versa,
a smaller Gaussian width σ(blue lines) leads to increasing correlation lifetimes and a
decreased renormalized coupling.
70 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
As a measure of topological properties we choose the Majorana mode correlation [143]
θ(t) = −itr{ρS(t)γLγR}=−
2N
X
i,l=1
fL,ifR,lΓil(t),(6.11)
where Γil(t) = itr{ρ(t)[bi, bl]}/2and fL/R,i denote the Majorana wave functions of the
initial Kitaev Hamiltonian in the Majorana basis. Specifically, we consider an ideal Kitaev
chain initially prepared in its even-parity ground state with ∆ = Jand µ= 0, where
fL,i =δi,1and fR,l =δl,2N, such that the Majorana correlation θ(t)reduces to θ(t) =
−Γ1,2N(t)and θ(0) = 1. To induce the time evolution dynamics of the polaronic Kitaev
chain for times t > 0, we assume a fast temperature increase to T= 4 K, resulting in the
parameter dressing ∆→∆⟨B⟩of the Hamiltonian. Here, the coupling of the chain to
the environment acts as a time-reversible perturbation to the closed system. In response,
system and reservoir synchronize and a new polaronic equilibrium state emerges. We solve
the integro-differential equation numerically via two nested fourth order Runge-Kutta
algorithms: The first time integration is required to evaluate the integral over all past
times τ. In a second time integration, the reduced system dynamics are calculated using
the previously determined integral up to the current time step. Due to the numerically
very expensive size of the density matrix and its full memory kernel, all calculations are
performed for N= 4 sites. For the electron-phonon coupling element, we consider phonon
modes in the range of k∈[0.0,4.0] nm−1and fph = 0.1.
6.3 Memory-induced loss and rephasing of topological
properties
As a first step, we identify the non-Markovian character of the unfolding time evolution
dynamics by comparing results obtained from the polaron master Eq. (6.8) with a Marko-
vian Lindblad master equation for phenomenological decoherence at a constant dephasing
rate γ, given by (see Sec. 3.2)
˙ρS=−i[Hk/ℏ, ρS] + γ
2
N
X
l=1 2c†
lclρSc†
lcl−{c†
lcl, ρS}.(6.12)
In this description, a structured reservoir with super-Ohmic coup ling gk∼k−1must be
presumed when considering 3D bulk phonons, resulting in a constant coupling gk=g0
incorporated in the damping constant γ. The green line in Fig. 6.3 depicts the non-
Markovian polaronic time evolution dynamics of the Majorana correlation θ(t)at a spec-
tral coupling width σ= 0.6, corresponding to a superconducting gap renormalization
parameter ⟨B⟩= 0.07. Taking full account of the system-reservoir memory allows a sub-
stantial Majorana edge correlation to be retained at long times. After an initial decay, the
Majorana edge correlation stabilizes well above zero. The long lived and substantial occur-
rence of Majorana modes seen in the non-Markovian dynamics is quite remarkable. This
is particularly true since Hp,s is near the topological phase boundary due to the strongly
6.3 Memory-induced loss and rephasing of topological properties 71
0
0.5
1
0 0.25 0.5
θ(t)
100 Jt
Figure 6.3: Time evolution dynamics of the polaronic topological Kitaev chain, showing a compari-
son of the Majorana correlation θ(t)calculated using a time-independent Lindblad-type
master equation for dephasing (blue line) and non-Markovian dynamics obtained from
the polaron master Eq. (6.8) with full account of memory (green line). The dashed
grey line shows the asymptotic Majorana correlation in a coherent quench scenario to
the phonon-dressed state ∆→∆⟨B⟩.
suppressed renormalized pairing ∆⟨B⟩ ≪ ∆. In comparison, in the Markovian descrip-
tion of decoherence the Majorana correlation inevitably collapses to zero in an exponential
decay (blue line in Fig. 6.3) [144].
When the dissipative contribution in Eq. (6.8) is disregarded, the dynamics formally re-
duces to that of a coherent quench of the pairing, leading to a renormalized superconduct-
ing gap from ∆to ∆⟨B⟩. In this case, the Majorana correlation approaches an asymptotic
value determined by the overlap of the Majorana wave functions for the pre- and post-
quench topological Hamiltonians [143], which is small if the post-quench Hamiltonian is
near the phase boundary (dashed grey line). The resulting stationary correlation differs
significantly from the non-Markovian behavior in Fig. 6.3 and underlines the crucial role of
the memory effect. Due to the dependence of the memory on both the phonon correlation
ϕ(τ)and the reversed dynamics of system correlations Xa,b(−τ), it simultaneously intro-
duces decoherence and backflow of coherence to the system. The interplay of these two
competing processes defines the non-Markovian character of the dynamics: The polaronic
Kitaev chain is initially prepared in its ground state and then perturbed by a temperature
increase to T, yielding a renormalization of the chain towards the topological phase bound-
ary via the initial phonon correlation ϕ(0) = Rd3k|2gk(σ)/ωk|2coth [ℏωk/(2kBT)]. This
quench generates a significant amount of bulk excitations and populates the Majorana edge
mode, effectively changing the parity of Majorana states. In addition, phonon-induced de-
phasing processes set in, resulting in strong decoherence in the dressed Kitaev wire. In
contrast, the time-reversed dynamics of system correlations Xa,b(−τ)reinstate coherence
of the superconducting p-wave pairing via the system memory, which is the key ingredient
for the emergence of a topological phase. During the initial memory times τ < τM, this
rephasing process is in the buildup stage and only has a marginal impact on the evolution
of the system. In consequence, an irreversible loss of parity information occurs during
72 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
-0.5
0
0.5
1
0 0.5 1
θ(t)
100 Jt
σ = 0.4
σ = 0.2 0.2
1
0.2 0.8
θ(t∞)
σ
Figure 6.4: Non-Markovian time evolution dynamics of the Majorana correlation θ(t)for several
bandwidths σof the fermion-phonon coupling gk. The inset shows the corresponding
steady-state value θ(t∞)as a function of σ.
the short-time dynamics. However, once the phonon correlation has decayed to zero at
times τ > τM, the full potential of the memory kernel unfolds, leading to strong rephasing
and a growth of topological properties due to Xa,b(−τ). In return, considerable Majorana
correlation is sustained in the long term in Fig. 6.3. It is noted that in spite of the small
size of the considered system, the asymptotic nonlocal Majorana edge correlation is indeed
of topological origin rather than resembling a phonon-mediated long-range correlation. As
verification of the topological nature of the observed correlation, in Sec. 6.5 we calculate
the Majorana correlations between all chain sites, showing increasing decay with relative
spacing, whereas the edge-edge correlation is significant.
6.4 Critical memory depth
The distinct non-Markovian relaxation behavior of the Majorana edge correlation depends
crucially on the memory depth, which can be tuned through the spectral fermion-phonon
coupling bandwidth σ. Fig. 6.4 shows the non-Markovian time evolution dynamics of
θ(t)for various spectral widths σ. In comparison to the case σ= 0.6(green line in
Fig. 6.3), an initial decrease of σleads to a steeper monotonic decay of θ(t), resulting
in a decreased asymptotic stationary state (dark blue line in Fig. 6.4). However, when
further decreasing the coupling bandwidth, the previously monotonic relaxation becomes
non-monotonic. While the initial short-time decoherence caused by phonon correlations
is accelerated, the Majorana edge correlation recovers substantially in the long term limit
(light blue line). This buildup behavior becomes even more distinct for further decreasing
values of σ, leading to stationary state correlations larger than the σ= 0.6case (green
and orange lines in Fig. 6.4). Strikingly, once the coupling width σsurpasses a critical
6.5 Discussion 73
value, the asymptotic edge correlation approaches unity, θ(t∞)→1(red line). The inset
of Fig. 6.4 shows the non-monotonic progression of θ(t∞)as a function of σ. Starting at a
large value of the coupling width, the asymptotic edge correlation first decreases towards
a minimum, before striving towards a perfect topological correlation in the limit of low
σ.
This intriguing phenomenon can be explained by the dependence of ϕ(τ)on the spectral
coupling bandwidth: As described above, reducing σincreases the lifetime of the phonon
correlation ϕ(τ)and therefore the memory depth. However, this also comes at the cost
of a decreased renormalization parameter ⟨B⟩= exp[−ϕ(0)/2] (see Fig. 6.2). While the
former enhances the rephasing of superconductive pairing via an increasing time scale of
the time-reversed evolution of Xa,b(−τ), the latter further suppresses the bulk gap E∆of
Hp,s and decreases the impact of the memory. When σis initially decreased from 0.6, the
suppressive effect dominates and aggravates the decay. However, by further reducing σthe
influence of the memory-enabled rephasing effect increases, allowing phonons and fermions
to synchronize and thereby inducing a backflow of parity information. As a result, a new
dressed polaronic state emerges, where the buildup of Majorana correlation starts to dom-
inate over decoherence, resulting in a recovery of correlation. In general, this recovery
sets in at times t>τMgiven that E∆τM≳1is satisfied. Importantly, the existence of a
critical σcorresponds to a critical memory depth, which allows for the formation of a new
polaronic steady state in dynamical equilibrium with the phonon reservoir at T= 4 K.
Remarkably, above this critical bandwidth σthe Majorana edge mode correlation exhibits
θ≈1. In the considered scenario, the critical value of σis between 0.21 and 0.20, cor-
responding to a superconducting pairing renormalization ⟨B⟩= 0.01, but it is dependent
on the specific model and the remaining system parameters. These findings open up new
perspectives for the design and control of topological superconductors, since the fermion-
phonon coupling bandwidth can be controlled in solid-state setups by nanotechnological
design, e.g., using alloys, impurities and confinement potentials [210–213].
6.5 Discussion
We have demonstrated the emergence of memory-critical Majorana edge correlation dy-
namics in a topological Kitaev superconductor, resulting from non-Markovian interaction
with phonons and leading to a revival of topological properties. We consolidate these
findings by considering the impact of an additional p-wave interparticle interaction to the
chain. To demonstrate the negligible impact of the polaron energy renormalization on the
system dynamics [see Eq. (6.5)], we provide additional calculations of the edge dynamics
including the polaron shift. Moreover, the arising correlations between all pairs of sites
in the chain are calculated, implicating their topological origin rather than merely repre-
senting phonon-mediated long-range correlations. Lastly, we calculate the Majorana edge
correlation dynamics for an initially nonideal Kitaev chain setup with a nonzero chemical
potential, recreating the memory-critical behavior.
74 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
0
0.5
1
0 0.25 0.5
θ(t)
100 Jt
U = 0.002
U = 0.000
U = -0.002
U = -0.008
Figure 6.5: Non-Markovian time evolution dynamics of the Majorana edge correlation for the renor-
malized system Hamiltonian Hp,s, calculated in the presence of a weak attractive p-
wave interparticle interaction U. Calculations are performed at parameters N= 4,
J= ∆ = 0.01,µ= 0 and σ= 0.6and for various amplitudes U.
6.5.1 Impact of interparticle interactions
As demonstrated in the polaron picture, the environmental coupling weakens fermionic
p-wave pairing to the superconducting wire [Eq. (6.4)], thus corrupting the stability of
the Majorana edge correlation. We have shown that non-Markovian phonon interactions
can counteract this effect, providing a substantial stabilization mechanism for topological
properties. To further investigate the role of attractive coupling for the longevity of Majo-
rana correlations, here we consider an additional perturbation caused by weak attractive
p-wave interparticle interactions. Therefore, Eq. (6.4) is expanded by a fermionic p-wave
interaction term, given by
Hint/ℏ=U
N−1
X
l=1 c†
lcl−1
2c†
l+1cl+1 −1
2,(6.13)
with a weak interaction strength |U| ≪ 1. We note that the interaction must be treated
numerically exact, since resorting to approximate descriptions such as a mean field ap-
proach would inevitably lead to a decay of entanglement over time. The exact treatment
is feasible for the chosen system size N= 4.
Fig. 6.5 shows the resulting non-Markovian time evolution dynamics of the Majorana edge
correlation θ(t)at a fermion-phonon coupling width σ= 0.6and for varying interaction
amplitudes U. The light blue line shows the case without additional p-wave interaction
U= 0. In comparison, introducing a repulsive interparticle interaction U > 0results in
a significant decline of the stationary topological correlation (dark blue line), since repul-
sive interactions energetically suppress fermionic pairing. On the contrary, adding a weak
attractive interaction U < 0can further enhance the asymptotic Majorana correlation
for a sub-critical memory depth (orange and red lines), which is given for the considered
bandwidth σ= 0.6. Depending on the amplitude |U|, attractive pairing not only shortens
6.5 Discussion 75
-0.5
0
0.5
1
0 0.5 1
θ(t)
100 Jt
σ = 0.4
σ = 0.2 0.2
1
0.2 0.8
θ(t∞)
σ
Figure 6.6: Non-Markovian dynamics of the polaronic Majorana edge correlation θ(t)for various
phonon coupling bandwidths σ, now including the polaron energy renormalization term
Hshift. The resulting steady state values θ(t∞)are shown as a function of σin the inset,
where a comparison of the cases including the polaron shift (solid line) and disregarding
it (dashed line) is provided. As before, the results are obtained using the parameters
N= 4,J= ∆ = 0.01,µ= 0,fph = 0.1and phonon modes within k∈[0.0,4.0] nm−1.
the required time for a revivial of topological correlation (see Fig. 6.4), but can even result
in a complete recovery in the sub-critical memory regime (red line). This behavior can be
intuitively understood, since the attractive interparticle interaction is energetically favor-
able for the coherent formation of superconductive pairing. As demonstrated before, the
latter provides a counteracting mechanism against phonon-induced dephasing processes
and thereby stabilizes the systems’ topological phase.
6.5.2 Polaron energy shift
During the polaron transformation of the dissipative Kitaev Hamiltonian, an additional
energy renormalization term arises [see Eq. (6.5)] which has been neglected so far. It is
justified to disregard this so-called polaron shift under certain conditions, which are ful-
filled in the investigated scenario: With respect to non-Markovian dynamical phenomena,
neglecting the polaron shift is standard practice if the initial condition is an equilibrium,
as established by Leggett [1,32]. As an initial condition, we have assumed the even parity
ground state of the Kitaev chain which constitutes an equilibrium state. Therefore, we
have disregarded Eq. (6.5) in the full polaron Hamiltonian according to the convention.
However, to this day the question whether it is necessary to take generic account of the
polaron shift remains an open question in non-Markovian physics. In order to demonstrate
that the presented results are invariant against this energy renormalization regardless of
76 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
0
0.5
1
2 3 4
correlation
site
σ=0.6
σ=0.2
Figure 6.7: Asymptotic site-site correlations −i⟨b1b2j⟩(t∞) = limt→+∞(−itr{ρS(t)b1b2j})between
the Majorana operators b1and b2jfor lattice sites j= 2,3,4for the considered Kitaev
wire at parameters N= 4,J= ∆ = 0.01,µ= 0 and fph = 0.1. Calculations are shown
for two fermion-phonon coupling bandwidths σ={0.6,0.2}.
the initial condition and the aforementioned convention, we here provide additional cal-
culations of the non-Markovian Majorana edge correlation in the presence of the polaron
energy shift. The considered system size of N= 4 allows for a numerically exact treatment
of Eq. (6.5). Fig. 6.6 shows the corresponding non-Markovian dynamics for θ(t), now in-
cluding Eq. (6.5) in the full polaron Hamiltonian. The inset again provides the asymptotic
Majorana edge correlation as a function of the spectral reservoir coupling bandwidth σ,
showing a comparison between the cases of including the polaron shift (solid line) and dis-
regarding it (dashed line). When comparing Fig. 6.6 to Fig. 6.4, it becomes apparent that
the inclusion of the polaron shift does not alter the qualitative dynamics and the memory-
critical recovery mechanism for the topological correlation. The polaron renormalization
term is proportional to the coupling bandwidth σ, albeit this only impacts the number
of nonzero phonon modes taken into account for the energy shift. This dependence is of
a much lower scale than the influence of σon the lifetime τMof the phonon correlation
function ϕ(τ), which has been shown to critically determine the stationary topological
properties of the superconducting Kitaev wire. In consequence, the main difference when
taking the polaron shift into account is the formation of a plateau of the stationary Majo-
rana edge correlation at intermediate σand slightly decreased values in the limit of large
bandwidths (solid line in inset of Fig. 6.6).
6.5.3 Topological correlation
In the present investigation, the numerically accessible size of the Kitaev chain is severely
limited by the expensive calculation of the density matrix and the finite memory kernel
of the integro-differential polaron master equation. Therefore, the considered chain size
is chosen at N= 4 sites. In order to demonstrate that the reported stationary Majorana
edge correlation is genuinely of topological origin, we calculate the stationary state site-
site correlations limt→+∞(−itr[ρS(t)b1b2j]) between the Majorana operators b1and b2j
6.5 Discussion 77
0
0.5
1
0 1 2
θ(t)
100 Jt
σ = 0.6
σ = 0.4
σ = 0.3
0
0.5
1
1 2 3 4site
fL
fR
Figure 6.8: Majorana edge correlation dynamics −i⟨γLγR⟩(t)for a nonideal Kitaev chain at pa-
rameters J= ∆ = 0.01,µ= 0.1J,N= 4 and various spectral coupling widths σ. The
inset shows the exponential model functions employed for the Majorana wave functions
fLand fR.
for all chain sites j= 2,3,4. Fig. 6.7 shows the resulting asymptotic correlations for
two reservoir coupling bandwidths σ={0.6,0.2}. In the figure, the site-site correlation
decays with increasing separation lengths d=j−1and vanishes almost entirely at d= 2,
underlining that phonon-mediated long-range correlations do not play a significant role
for the formation of the correlation. However, the site-site correlation increases to a
substantial value at the right boundary d= 3, representing a topological signature of the
nonlocal Majorana edge modes: If the observed correlation beyond nearest-neighbor sites
at separation lengths d > 1were to originate from phonon-mediated effects, one would
expect a monotonous decay with increasing distances d.
6.5.4 Nonideal Kitaev chain
The presented results are based on an initially ideal Kitaev wire, i.e., featuring strictly
zero overlap between the Majorana wave functions fL/R,j located at the left and right
boundaries of the chain, even in the case of a system of N= 4 sites [143]. In the presence
of a small perturbation caused, e.g., by a nonzero chemical potential |µ|>0, the initial
Kitaev chain is no longer ideal, corresponding to an extension of the Majorana wave
functions into the bulk [144,150,214]. In this scenario, the Majorana edge correlation
is given by −i⟨γLγR⟩, with γL/R =PjfL/R,jbjand Majorana operators b2j−1=cj+c†
j,
b2j=−i(cj−c†
j). Fig. 6.8 shows the resulting Majorana edge correlation dynamics for an
initially nonideal polaronic Kitaev Hamiltonian with parameters N= 4,J=∆=0.01,
µ= 0.1Jand various spectral coupling widths σ. Since only qualitative differences with
respect to the ideal chain dynamics are of interest, the Majorana wave functions fL/R with
78 6 Memory-critical dynamical buildup of phonon-dressed Majorana fermions
finite localization lengths are modeled via normed exponential decay functions, depicted
in the inset in Fig. 6.8 and featuring a small nonzero overlap between them.
The nonideal setup poses an additional perturbation to the polaronic state. As a result, for
a given parameter configuration the stationary edge correlation is decreased with respect to
the ideal case (blue and green lines). However, the memory-enabled recovery of topological
properties can still be observed when σis tuned below a threshold value (orange line).
Hence, for sufficiently small deviations from the ideal setup the dynamical recovery of
Majorana correlation is preserved under nonideal initial conditions, demonstrating the
robustness of the effect and the generality of the presented results.
6.6 Conclusion
In summary, we have demonstrated the emergence of memory-critical Majorana edge cor-
relation dynamics in a topological superconductor, enabled by non-Markovian fermion-
phonon interactions and resulting in a revival of topological properties. This mechanism
uniquely arises from the phonon-renormalized topological Hamiltonian, where the phonon
environment leads to a suppression of superconductive tunneling, and from its interplay
with a quantum memory effect. As a result, dephasing and information backflow processes
are simultaneously induced. While the presented analysis is based on the ideal Kitaev
chain, we expect the essential physics to emerge in a wide class of topological materials
interacting with a super-Ohmic reservoir. In this context, we have reproduced our key
findings in an initially nonideal Kitaev chain setup with Majorana wave function overlap.
Moreover, we have shown that the reservoir-induced suppression of superconductive pair-
ing can be further counteracted via attractive p-wave interparticle interactions, providing
an additional mechanism for the recovery and stabilization of topological properties.
The experimental realization of Majorana fermions in condensed matter setups, e.g., based
on hybrid superconductor and semiconductor nanowires, is currently subject to significant
research efforts. The fermion-phonon reservoir coupling considered in the present inves-
tigations can for instance occur in InAs nanowires [149–154], where the acoustic phonon
wavelengths are in the 100 nm length range of typical nanowire realizations and feature
super-Ohmic coupling. Aside from solid-state setups, the fundamental non-Markovian
mechanism of a reservoir-induced recovery of topological properties is expected to also
be observable in realizations of ultracold quantum gases coupled to a superfluid reservoir,
where excitations result in the emergence of phonon interactions [183,190,215]. In a
broader context, one can also consider experimental scenarios where interactions with a
structured bosonic reservoir are intentionally induced, as long as the coupling leads to
non-Markovian system-reservoir correlations. Recent related studies have shown that an
impurity or a qubit can be employed as a non-Markovian quantum probe for a topologi-
cal reservoir [198–200], and that topological system properties stemming from Markovian
dissipation can be sustained in non-Markovian regimes [201]. In difference to these in-
vestigations on topological systems with non-Markovian system-reservoir dynamics, we
have demonstrated that tailoring the coupling to a super-Ohmic reservoir can improve the
longevity and grade of topological properties, and even fully restore them. Our findings
7 Unidirectional quantum transport in
optically driven V-type quantum dot
chains
In this Chapter, we present a mechanism to achieve complete population inversion in
an optically driven InAs/GaAs semiconductor quantum dot featuring a V-type energy
structure. The unfolding nonequilibrium stationary state is induced by the interaction
of a non-Markovian decoherence mechanism introduced by acoustic phonons with the
systems’ V-type interband transitions. In a second step, we apply the population inversion
mechanism to create unidirectional excitation transport in a chain of coupled quantum dots
without application of an external bias and independent of the unitary interdot coupling
mechanism. The Chapter is closely based on the publication “Unidirectional quantum
transport in optically driven V-type quantum dot chains” by O. Kästle et al. [70]
7.1 Introduction
The control and utilization of non-Markovian phenomena has become a main focus of
attention for the development of quantum optical devices and quantum information tech-
nology [4–10]. Specifically, investigations of the interplay of phonon-assisted coherence and
dissipative processes in semiconductor nanostructures such as quantum dots have emerged
as a growing field of research [11–24]. For instance, non-Markovian system-reservoir cou-
plings have been tailored to create population inversion in such emitters to a certain
degree [29,34,35,217,218], making them potential candidates for chip-integrated single-
exciton light sources. Approaches to achieve population inversion encompass adiabatic
rapid passage in quantum emitters [219] and hybrid setups of quantum dots interacting
with metal nanoparticles [220] and cavities [221–223]. Most recently, population inversion
has been achieved in a single InAs/GaAs quantum dot using pulsed excitations tuned
within the exciton phonon sideband, enabled by a phonon-mediated thermalization of the
optically dressed states [224,225]. Already, these examples demonstrate the potential of
tailored electron-phonon interactions for controlled optical excitations.
In this Chapter, we describe non-Markovian undirectional quantum transport enabled
by electron-phonon interactions in quantum dots featuring V-type transitions and con-
tinuously driven by a single laser field [226–228]. We demonstrate the occurrence of a
non-reciprocal phonon-assisted energy transfer in single quantum dots and quantum dot
chains as a result of resonant excitation of an electronic state and its interaction with a
lower-energy state [70]. Specifically, the non-Markovian description of the environment
81
82 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
Figure 7.1: (a) Schematic of the considered V-type emitter model with two excited states featuring
an energy detuning δϵ =ℏ(ω3−ω2). The two transitions are driven by a continuous
laser field at a Rabi frequency Ωand in resonance with the transition between states
|1⟩and |2⟩. The excited states are interacting with a structured 3D acoustic phonon
reservoir via coupling elements gk. Initially, a single electron is located in state |1⟩. (b)
Schematic quantum dot energy level structure of InAs embedded in GaAs, enabling
V-type transitions between electron and hole states.
enables information backflow, such that excitation transfer within the system is supported
by the structured reservoir [1,3,80]. The observed effects are non-Markovian in nature,
since a Lindblad form for pure dephasing derived from the considered full Hamiltonian
fails to capture the presented mechanism, even in the case of a time-dependent pure de-
phasing rate. Moreover, the reported inversion mechanism also cannot be reproduced by
a Redfield master equation subjected to the first and second Markovian approximation.
The implications of this effect on the considered V-type systems are demonstrated in two
subsequent steps:
(i) For a single quantum dot, it results in the preparation of a highly nonequilibrium
stationary state. For certain parameter choices, complete population inversion is achieved
in the detuned excited state and can be maintained for a wide range of coupling and driving
amplitudes. In difference to previously reported mechanisms [12,29,34,217–219,222],
the reported inversion is induced in a V-type emitter via continuous optical excitation
and without a cavity mode.
(ii) The observed population inversion is applied to create excitation transport in a chain-
like spatial distribution of several quantum dots. Exploiting the population trapping in
the red-detuned excited state of each emitter, we predict the emergence of unidirectional
excitation transport from one end of the quantum dot chain to the other. Unidirectional
energy transport via electronic excitation transfer has been widely investigated in semi-
conductor nanostructures [229–236], biological systems [237] and molecules [238,239].
Moreover, it has been shown that non-Markovian system-reservoir interactions can af-
fect non-reciprocal quantum transport [240] and increase its efficiency [241] on the basis
of well-established transfer protocols, e.g., enabled by an external bias [242]. In differ-
ence to these investigations, here we provide an example in which unidirectional quantum
transport results without the application of an external potential, but from the inter-
play between incoherence and coherence in the system alone. Due to the robustness of
the underlying mechanism, the transport endures the perturbative effects such as radia-
7.2 V-type emitter model 83
tive decay phenomena or intraband phonon couplings, and emerges independently of the
specific interdot coupling mechanism.
Two independent theoretical approaches are employed for the description of the system.
In a first step, the system dynamics is calculated using a polaron master equation in
second-order perturbation theory with respect to the polaron Hamiltonian [1,14,80,87–
92,99,100]. Afterwards, we reproduce the dynamics in a Heisenberg picture correlation
expansion to gain insight into the key interaction processes enabling the population in-
version mechanism and the specific channels of information backflow between system and
reservoir.
In Sec. 7.2, we introduce the investigated V-type system in detail and derive both the
polaron master equation and the Heisenberg equations of motion for its description. Af-
terwards, we analyze the emergence of complete population inversion in the Heisenberg
picture description in Sec. 7.3. In Sec. 7.4, multiple V-type emitters are combined to a
quantum dot chain exhibiting unidirectional quantum transport. We discuss our findings
in Sec. 7.5, where we compare the solutions obtained from the Heisenberg picture and
polaron master equation approach, incorporate extensions to our model and discuss the
linear absorption spectrum of the V-type emitter and its eigenstate dynamics. In addition,
we provide calculations for alternative interdot coupling mechanisms for the quantum dot
chain, demonstrating the emergence of unidirectional transport regardless of the specific
coupling type. Lastly, we conclude our investigation in Sec. 7.6.
7.2 V-type emitter model
The investigated V-type emitter model system shown in Fig. 7.1(a) consists of a single
ground state |1⟩and two excited states |2⟩and |3⟩, with an energy detuning δϵ =ℏ(ω3−ω2)
between them. The two allowed transitions are driven by a single continuous laser field at a
Rabi frequency Ω, which is tuned in resonance to the transition between states |1⟩and |2⟩.
Such a scenario is constituted, e.g., by conical InAs quantum dots surrounded by a bulk of
GaAs [243]. The energy level structure for this case is shown schematically in Fig. 7.1(b).
Here, the electronic structure comprises six confined electron and hole states denoted by
|e0⟩,...,|e5⟩and |h0⟩,...,|h5⟩within the envelope wave function and effective mass ap-
proximation [226,227,244,245]. Due to the rotational symmetry of the conical quantum
dot structure, the first and second as well as the third and fourth states in both bands
are degenerate [226,227]. As a result, the dipole selection rules permit two possible inter-
band transitions from the heavy hole state |h0⟩to the electronic states |e0⟩and |e5⟩[see
Fig. 7.1(b)], manifesting the V-type transition pattern in the here considered three-level
system [226]. We note that interband transitions to the states |e1⟩-|e4⟩are not explicitly
dipole-forbidden, however, they are impeded by many-body Coulomb effects giving rise
to an energy renormalization with strong impact on the oscillator strengths [226]. In the
following, the transitions between |h0⟩and |e0⟩and between |h0⟩and |e5⟩are assumed to
be dominant, which is for instance achieved via tuning of the excitation strength to control
Coulomb renormalization effects or doping of the semiconductor material. We choose the
laser frequency as the rotating frame and apply the rotating wave approximation. As a
84 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
result, the Hamiltonians describing the quantum dot Hel and the electron-light coupling
Hlread
Hel =ℏ(∆2σ22 + ∆3σ33),(7.1)
Hl=ℏΩ(σ12 +σ13 + H.c.),(7.2)
with operators σij =|i⟩⟨j|.∆i=ωi−ωLis the frequency detuning between the i-th excited
state and the incoming laser field, cooresponding to an energy detuning δϵ =ℏ(∆3−∆2)
between the excited states, and Ωdenotes the real-valued and slowly varying envelope of
the Rabi frequency of transitions between the ground and excited states.
In addition, decoherence effects caused by the surrounding GaAs material are accounted
for by coupling a structured phonon reservoir to the electron states |2⟩and |3⟩of the
quantum emitter [see Fig. 7.1(b)]. In our investigation, we consider a generic model of
diagonal electron-phonon interactions Hel,ph for 3D bulk phonons of the form [95,96]
Hel,ph =ℏZd3khg(2)
kσ22 +g(3)
kσ33ir†
k+rk,(7.3)
with bosonic creation and annihilation operators r†
k,rk. The mode-dependent fermion-
phonon coupling elements at states |2⟩and |3⟩are given by the overlap integrals with state
|1⟩[80,97,98,246],
g(2)
k:= sℏk
2ρcs"D2exp −ℏk2
4m2ω2!−D1exp −ℏk2
4m1ω1!#,(7.4a)
g(3)
k:= sℏk
2ρcs"D3exp −ℏk2
4m3ω3!−D1exp −ℏk2
4m1ω1!#,(7.4b)
where cs,Di,mi,ℏωiand ρdenote the sound velocity, deformation potentials, effec-
tive masses, confinement energies and the mass density of GaAs, respectively. Moreover,
we assume identical overlaps between level |1⟩and both excited states, yielding a single
fermion-phonon coupling element gk:= g(2)
k=g(3)
k. The spectral density is therefore given
by J(ω) = Rd3k|gk|2δ(ω−ωk), with ωk=cs|k|denoting the acoustic phonon frequen-
cies [80]. We emphasize that the following investigations describe a generic effect which
is robust against the employed coupling element, as long as Stokes processes can emerge
via phonon-assisted resonances. In addition to the here applied coupling, Gaussian and
Lorentzian shaped couplings gkhave been confirmed to result in qualitative and quanti-
tative comparable results. For now, phonon-induced intraband couplings ∼g23
kσ23 and
radiative decay are neglected, since all key results can be shown to be robust against their
presence (see Sec. 7.5). Together with the free phonon evolution
Hph =ℏZd3k ωkr†
krk,(7.5)
we arrive at the full open system Hamiltonian H=Hel +Hl+Hph+Hel,ph. In Appendix C,
we provide a table containing all parameters employed for calculations.
7.2 V-type emitter model 85
For the numerical evaluation, we employ the standard second-order perturbative polaron
master equation (see Ch. 3) [1,14,80,87–92,99,100],
˙ρS(t) = −i
ℏ[Hp,0, ρS(t)] −1
ℏ2Zt
0
dτ trB{[Hp,I,[Hp,I (−τ), ρS(t)⊗ρB]]},(7.6)
where Hp,0and Hp,I refer to the polaron-transformed free evolution and system-reservoir
interaction Hamiltonians, respectively,
Hp,0=ℏX
i=2,3h¯
∆iσii +¯
Ω(σ1i+σi1)i,(7.7a)
Hp,I =ℏX
i=2,3n(σ1i+σi1)hΩ cosh (R†−R)−¯
Ωi+ Ω(σi1−σ1i) sinh (R†−R)o.(7.7b)
Here we have defined collective bosonic operators R(†)=Rd3k(gk/ωk)r(†)
k,¯
∆idenotes a
polaron-shifted detuning and a Franck-Condon renormalization is applied by introducing
¯
Ω = Ω exp −Zd3kgk
2ωk
coth ℏωk
2kBT,(7.8)
such that trB{[Hp,I , ρ(t)]}= 0. Moreover, Hp,0governs the time-reversed unitary evolution
of the interaction Hp,I(−τ) = U†(−τ, 0)Hp,IU(−τ, 0) via U(t, 0) = exp(−i/ℏHp,0t)(see
Ch. 3). As a result, the equations of motion for the density matrix elements ρmn are given
by
˙ρmn =X
i=2,3hi¯
∆i(ρmiδni −ρinδmi) + i¯
Ω(ρm1δni +ρmiδn1−ρinδm1−ρ1nδmi)i
−¯
Ω2
ℏ2X
i,j=2,3Zt
0
dτ ⟨m|χij(τ)|n⟩,(7.9)
with
⟨m|χij(τ)|n⟩=
3
X
q=1 nρqnhG+(τ)Xiq
+,j(−τ)−iG−(τ)Xiq
−,j(−τ)δm1
+G+(τ)X1q
+,j(−τ) + iG−(τ)X1q
−,j(−τ)δmii
+ρmqhG∗
+(τ)Xq1
+,j(−τ)−iG∗
−(τ)Xq1
−,j(−τ)δni
+G∗
+(τ)Xqi
+,j(−τ) + iG∗
−(τ)Xqi
−,j(−τ)δn1i
+ρq1h−G+(τ)Xmq
+,j(−τ) + iG−(τ)Xmq
−,j(−τ)iδni
+ρqih−G+(τ)Xmq
+,j(−τ)−iG−(τ)Xmq
−,j(−τ)iδn1
+ρ1qh−G∗
+(τ)Xqn
+,j(−τ)−iG∗
−(τ)Xqn
−,j(−τ)iδmi
+ρiqh−G∗
+(τ)Xqn
+,j(−τ) + iG∗
−(τ)Xqn
−,j(−τ)iδm1o.(7.10)
86 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
Here, we have abbreviated ⟨m|X±,i(−τ)|n⟩=Xmn
±,i (−τ)and defined system correlations
X+,i(τ) := [σ1i(τ) + σi1(τ)],X−,i(τ) := i[σi1(τ)−σ1i(τ)] and the phonon correlation
function
ϕ(τ) = Zd3kg2
k
ω2
kcoth ℏωk
2kBTcos(ωkτ)−isin(ωkτ),(7.11)
with G+(τ) := cosh [ϕ(τ)]−1,G−(τ) := sinh [ϕ(τ)] and ¯
Ω = Ω exp [−ϕ(0)/2]. The polaron
master equation approach offers the advantage of high numerical performance and stability,
allowing for long simulation times and accurate results in the weak and moderate electron-
phonon coupling regime [29,207,247,248]. A detailed derivation of the polaron master
equation for the V-type emitter model is presented in Appendix C.
In addition, we calculate the system dynamics using a Heisenberg picture correlation
expansion up to second order in the phonon contributions, which is better suited to unravel
the underlying physical interactions [19,97,205,207,249–251]. The equivalence of the
two descriptions in the here considered parameter regime is demonstrated in Sec. 7.5.
The Heisenberg equations of motion for the electronic contributions are prescribed by
−iℏ˙σmn(t)=[H, σmn(t)], resulting in
d
dt⟨σmn⟩=ih∆2(⟨σ2n⟩δm2−⟨σm2⟩δn2)+∆3(⟨σ3n⟩δm3−⟨σm3⟩δn3)
+ Ω⟨σ1n⟩δm2+⟨σ2n⟩δm1−⟨σm2⟩δn1−⟨σm1⟩δn2
+⟨σ1n⟩δm3+⟨σ3n⟩δm1−⟨σm3⟩δn1−⟨σm1⟩δn3
+Zd3k gk⟨σ2nr†
k⟩δm2+⟨σ3nr†
k⟩δm3−⟨σm2r†
k⟩δn2−⟨σm3r†
k⟩δn3
+⟨σ2nrk⟩δm2+⟨σ3nrk⟩δm3−⟨σm2rk⟩δn2−⟨σm3rk⟩δn3i,(7.12)
where we have disregarded the time argument of the operators for brevity and assigned
indices m, n ={1,2,3}to the emitter levels. The remaining higher-order equations of
motion for the phonon-assisted transitions (m=n) and occupations (m=n)⟨σmnr(†)
k⟩
are provided in Appendix C. To close the system of equations, we employ a second-order
Born factorization in the limit where the correlation expansion manages to reproduce the
full independent boson model. Here, the phonon number expectation value is assumed as
a thermal Bose distribution [1], i.e.,
⟨r†
krk⟩ ≈ exp ℏωk
kBT−1−1
,(7.13)
with kBdenoting the Boltzmann constant. For an explicit illustration of the involved
oscillator strengths, we additionally discuss the resulting linear absorption spectrum of
the considered V-type emitter model in Sec. 7.5.
7.3 Complete population inversion by non-Markovian reservoir interaction 87
Figure 7.2: V-type emitter population dynamics obtained from the polaron master equation, ex-
hibiting complete population inversion in the red-shifted excited state |3⟩. The corre-
sponding coherence dynamics is shown in the inset.
7.3 Complete population inversion by non-Markovian
reservoir interaction
We start with the single emitter case, where an electron is initially located in the ground
state |1⟩and the laser frequency is chosen in resonance with the transition between the
ground state |1⟩and the excited state |2⟩, such that ∆2= 0 and therefore δϵ =ℏ∆3= 0,
corresponding to a negative energy detuning (ω3< ωL) within range of the acoustic phonon
spectral density maximum. Fig. 7.2 shows the polaron master equation dynamics for the
level populations σ11,σ22 and σ33 after switching on the driving field at a Rabi frequency
Ω. In the stationary state, the system exhibits complete population inversion: The laser-
detuned excited state |3⟩becomes almost fully occupied, ⟨σ33⟩ → 1, while the populations
of the resonantly pumped levels |1⟩and |2⟩decline to zero. Due to the more intuitive
nature of the Heisenberg picture description, the following analysis of this behavior is
based on the Heisenberg equations of motion [Eq. (7.12)].
Fig. 7.3 shows a diagrammatic representation of the interactions between coherences, oc-
cupations and their phonon-assisted amplitudes in correspondence to their occurrence in
the Heisenberg equations of motion. As illustrated in the diagram, a direct optical tran-
sition from the resonantly driven excited state |2⟩to the detuned excited state |3⟩is not
allowed. Therefore, the population transfer to level |3⟩must be carried out by the struc-
tured reservoir. Red shapes and arrows in Fig. 7.3 highlight the crucial interactions of
populations and coherences facilitating the excitation transfer to the laser-detuned level
|3⟩. As a result of the continuous resonant driving of transitions between levels |1⟩and |2⟩,
energy is dissipated to the phonon environment via the coherence σ12: The non-Markovian
nature of the reservoir enables backflow of excitation to the system via phonon-assisted
transitions σijr(†)
k, which form a gateway from the optically driven amplitude σ12 to the
88 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
Figure 7.3: Diagrammatic representation of the Heisenberg equations of motion up to second order
in phonon contributions for the V-type emitter. Red shapes and arrows highlight the
key interactions enabling the observed population inversion. Green shapes represent
the coherence σ23 and its phonon-assisted amplitudes.
detuned state σ33 (see Fig. 7.3). More explicitly, the energy is transported from σ12r(†)
k
via σ11r(†)
k,σ13r(†)
kand σ33r(†)
kto the detuned excited state σ33. Once the excitation is
transferred to σ33, laser-driven transitions back to the ground state can in principle take
place via the coherence σ13. However, dissipative losses to the phonon mode continuum oc-
cur on a much faster time scale: Enabled by non-Markovian system-reservoir interactions
via phonon-assisted amplitudes σ13r(†)
kand σ33r(†)
k, the excitation becomes dynamically
trapped in the laser-detuned state σ33 (dashed frame in Fig. 7.3). Since energy localized in
the two resonantly driven states |1⟩and |2⟩is permanently dissipated to level |3⟩through
the reservoir, their population uniformly decays towards zero.
The relevant time scale enabling the population inversion is highly dependent on the re-
lation between the driving strength Ωand the fermion-phonon coupling amplitude gk. In
Fig. 7.2, we employ a set of parameters with an energy detuning δϵ =−1meV (see Ap-
pendix C), resulting in the observed complete population inversion. However, the unfolding
grade of inversion is highly susceptible to the choice of parameters and can be tuned, e.g.,
via the laser field amplitude and reservoir temperature. Moreover, the fermion-phonon
coupling gkcan be tailored by nanotechnological engineering, e.g., using alloys, impurities
and confinement potentials [210–213]. The coherence σ23 and its phonon-assisted ampli-
tudes (green shapes in Fig. 7.3) can be suspected to additionally facilitate the transfer,
since they constitute a direct connection between the two excited states via the transi-
tion amplitudes σ12 and σ13. However, manually turning off all contributions from σ23
and σ23r(†)
k′does not prohibit the complete inversion exhibited in Fig. 7.2. The coher-
ence dynamics σ12,σ13 and σ23 of the V-type emitter is depicted in the inset: While
σ12 and σ23 decay to zero in the course of time, σ13 remains finite in the steady state
7.4 Unidirectional quantum transport in a quantum dot chain 89
fff
Figure 7.4: Multiple V-type emitters arranged in a chain and coupled by Dexter-type electron
transfer at a coupling amplitude f, enabling a single interdot carrier transition via
electronic wave function overlap.
since excitation in the laser-detuned level is constantly dissipated to the reservoir, dynam-
ically trapping the energy and allowing for the formation of the observed nonequilibrium
stationary state, which is not observed in a Lindblad-based evaluation of the here consid-
ered Hamiltonians. Furthermore, the grade of stationary population inversion is highly
temperature-dependent: Increasing the temperature improves the potential for phonon-
assisted transitions of energy back to the ground state and therefore perturbs the dy-
namical trapping of excitation, resulting in decreasing inversion. In the high-temperature
regime, the level populations start to equilibrate until the inversion completely vanishes.
We further elaborate on the temperature dependence in Sec. 7.5, where we also discuss
the emergence of the effect from a different perspective via the bare eigenstate dynamics
of the V-type model system.
7.4 Unidirectional quantum transport in a quantum dot
chain
As an example application of the reported population inversion mechanism for quantum
dot-based coherent transport devices, we arrange multiple V-type emitters in a closely
spaced chain-like distribution schematically shown in Fig. 7.4. Here, we consider a generic
Dexter-type bidirectional electron transfer as the enabling resonance energy transfer mech-
anism between the quantum dots [252]. This process describes a Coulomb-induced direct
and spin-preserving electron exchange between semiconductor nanostructures and thus
requires electronic wave function overlap between the respective donor and acceptor lev-
els [252,253]. Dexter transfer has been shown to take place, e.g., in self-assembled InAs
quantum dot chains [254–260].
For a straightforward realization, we first consider the case where only two levels of neigh-
boring quantum dots in the chain have an electronic wave function overlap, namely between
the detuned excited state of the first and the resonantly driven excited state of the second
emitter (see Fig. 7.4). In an experimental scenario, such a coupling can be achieved, e.g.,
by engineering the wave function overlap via the potential barriers between neighboring
quantum emitters [261], or by tailoring of the relevant confinement potentials via electrical
gating [262]. Furthermore, it is possible to utilize multiple type-2 quantum dots where the
electrons and holes are not co-localized in the exciton states [263]. This simple coupling
90 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
0
0.5
1
0 200 400
ρ11
ρ22
ρ33
(a) Emitter 1
0 200 400
ρ11
ρ22
ρ33
(b) Emitter 2
0
0.5
1
0 200 400
t [ns]
ρ11
ρ22
ρ33
(c) Emitter 3
0 200 400
t [ns]
ρ11
ρ22
ρ33
(d) Emitter 4
Figure 7.5: Population dynamics for a chain of N= 4 V-type emitters coupled via Dexter-type
carrier transfer enabling a single interdot transition. All parameters are unchanged
with respect to the single emitter calculations shown in Fig. 7.2.
scheme results in the most distinct unidirectional transport effect, however, in Sec. 7.5
we show that the mechanism still arises if all excited states of adjacent quantum dots are
Dexter-coupled, albeit at a considerably lower degree. Moreover, we discuss Förster-type
excitation transfer as an additional interdot coupling mechanism, yielding comparable re-
sults. Here we assume the single carrier case with a single electron initially localized in
the ground state of the first emitter in the chain (see Fig. 7.4). Physical realizations of
such a scenario can be achieved, e.g., by a doping of the first quantum dot and utilizing
empty intersubband transitions in the conduction bands to facilitate V-type transitions.
The resulting Dexter interdot coupling Hamiltonian is given by
HD=ℏf
N−2
X
l=0 hσ(3+3l)(5+3l)+ H.c.i,(7.14)
with fdenoting the coupling amplitude between the emitters. Moreover, each emitter is
assigned a separate phonon reservoir with independent coupling elements gkl. For a chain
of Nquantum emitters, the resulting polaron master equation reads
˙ρS(t) = −i
ℏ[Hp,0, ρS(t)]
−¯
Ω2
ℏ2
N−1
X
l,l′=0 X
i,j=2,3Zt
0
dτ X
s=±Gs(τ)X[l]
s,j, X[l′]
s,i
′ρS+ H.c.,(7.15)
7.4 Unidirectional quantum transport in a quantum dot chain 91
with
X[l]
+,i(τ) = hσ(1+3l)(i+3l)(τ) + σ(i+3l)(1+3l)(τ)i,(7.16a)
X[l]
−,i(τ) = ihσ(i+3l)(1+3l)(τ)−σ(1+3l)(i+3l)(τ)i,(7.16b)
and Hp,0now comprising the Dexter interdot coupling Hamiltonian [Eq. (7.14)]. A detailed
derivation of the polaron master equation for a chain of NV-type emitters is provided in
Appendix C.
The unfolding population dynamics for a chain of N= 4 emitters is shown in Fig. 7.5.
Analogous to the single emitter case, a non-reciprocal, phonon-assisted excitation transfer
to the detuned excited state takes place in each quantum dot, resulting in excitation
transfer to the detuned state of the last emitter in the chain and corresponding to perfect
unidirectional electron transport. This mechanism is facilitated by the population inversion
effect rather than the here employed interdot coupling scheme: Initially, the population
in level |3⟩of the first emitter rises at the same rate as in the single emitter case (see
Fig. 7.2). Once a threshold occupation is attained, the Dexter coupling to the neighboring
system begins to dominate, resulting in carrier transfer and an asymptotic decline of |3⟩
towards zero, while the resonantly driven levels |4⟩and |5⟩of the second emitter start to
become uniformly occupied. Once the excitation transfer sets in, the population inversion
commences in the second emitter and in turn leads to a fast population decline in levels
|4⟩and |5⟩and rising occupation in the laser-detuned level |6⟩. This pattern repeats
within each chain link until all excitation is transferred to the detuned excited state of the
last emitter. Due to the temperature-dependent grade of population trapping, increasing
temperatures result in a decreased stationary state inversion in the last emitter and overall
slowed down transport, however, the unidirectional transport mechanism endures as long
as population inversion is achieved.
The transport mechanism also withstands perturbations such as radiative decay processes,
demonstrating the robustness of the underlying inversion mechanism. To account for
incoherent radiative dissipation from the excited states to the ground state, we introduce
a Lindblad dissipator of the form
Lradρ=γr
N−1
X
l=1 X
i=2,3h2σ(1+3l)(i+3l)ρσ(i+3l)(1+3l)−{σ(i+3l)(i+3l), ρ}i,(7.17)
where γrdenotes the radiative decay rate. In correspondence to Fig. 7.5, Fig. 7.6 shows
the quantum dot chain population dynamics for N= 4 emitters including radiative decay
at rate γr= 0.1 ns−1. As a result of incoherent dissipation, the grade of inversion in the
last emitter is decreased, however, the transfer mechanism itself is not suppressed by the
radiative decay. Driven by the continuous laser field, the excitation is still transmitted to
the last site of the quantum dot array.
In realistic scenarios, non-diagonal fermion-phonon interactions between the excited states
of the emitters may impose an additional perturbation on the system. In Sec. 7.5, we
demonstrate that the inclusion of intraband coupling processes slows down the popu-
lation inversion process, but does not change the qualitative outcome. Moreover, we
92 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
0
0.5
1
0 200 400
ρ11
ρ22
ρ33
(a) Emitter 1
0 200 400
ρ11
ρ22
ρ33
(b) Emitter 2
0
0.5
1
0 200 400
t [ns]
ρ11
ρ22
ρ33
(c) Emitter 3
0 200 400
t [ns]
ρ11
ρ22
ρ33
(d) Emitter 4
Figure 7.6: Population dynamics for a chain of N= 4 V-type emitters including radiative decay.
All system parameters are set equal to the calculations presented in Fig. 7.5.
demonstrate the emergence of unidirectional quantum transport regardless of the em-
ployed interdot coupling by additionally considering (i) Dexter coupling with electronic
wave function overlap between all excited states of adjacent emitters and (ii) Förster-type
coupling between neighboring quantum dots [264], both exhibiting comparable unidirec-
tional transport dynamics. In the latter scenario, excitation energy transfer is induced via
dipole-dipole interactions and has been shown to arise in quantum dot arrays [230,265–
267]. Based on the predicted population inversion towards the red-detuned excited states
in V-type quantum emitters, we have established a mechanism for unidirectional spatio-
temporal transport of both excitation and carriers in quantum dot chains, constituting an
example in which unidirectional, non-reciprocal quantum transport is achieved without
application of an external bias and enabled solely by the interplay between coherence and
incoherence in the system itself.
7.5 Discussion
7.5.1 Comparison between Heisenberg and polaron description
Aside from the second-order perturbative polaron master equation, we have employed
a correlation expansion up to second order in phonon contributions in the Heisenberg
7.5 Discussion 93
0
0.5
1
0 0.5 1
t [ns]
σ11
σ22
σ33
(a) Heisenberg
0 0.5 1
t [ns]
ρ11
ρ22
ρ33
(b) Polaron
Figure 7.7: V-type emitter population dynamics calculated (a) using the Heisenberg picture appo-
rach and (b) using the polaron master equation.
0
0.5
1
0 0.5 1
t [ns]
σ11
σ22
σ33
(a)
0 100 200
t [ns]
4 K
10 K
25 K
(b)
Figure 7.8: (a) V-type emitter population dynamics calculated in the Heisenberg picture including
intraband couplings at amplitude gintra
k= 0.1gkand otherwise unchanged parameters
with respect to Fig. 7.7. (b) Population dynamics of the laser-detuned excited state
for increasing temperatures obtained from the polaron master equation.
picture to backtrace the key interactions responsible for the observed population inver-
sion. The polaron master equation has been used for numerical calculations, since it
allows for long simulation times and offers high performance and stability. The Heisen-
berg approach, on the other hand, requires a very fine time discretization and breaks down
after a few million time steps, therefore limiting the accessible time evolution dynamics.
Here we demonstrate the equivalence of the two approaches in a parameter regime where
both implementations are stable and the systems’ stationary state is obtained within ac-
cessible time frames. Fig. 7.7 shows the unfolding population dynamics at T= 4 K,
δϵ =−1.0meV, ℏΩ=0.3291 meV and a rescaled coupling element ˜gk= 2.5gk, obtained
from (a) the Heisenberg correlation expansion and (b) the polaron master equation. Dur-
ing the polaron transformation of the system Hamiltonian, a polaron energy shift arises
in the energy detuning between the excited states, i.e., ∆→¯
∆[see Eq. (7.7)]. This en-
ergy shift does not affect the qualitative physical behavior with respect to the Heisenberg
description, however, it leads to a small phase shift with minor impact on the population
dynamics which becomes visible when comparing Figs. 7.7(a) and (b). Otherwise, the
results obtained from the two descriptions are in excellent agreement with each other.
94 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
7.5.2 Dependence on intraband phonon coupling and temperature
As of yet, we have neglected possibly arising intraband phonon couplings between the two
excited states of the V-type emitter [see Eq. (7.3)]. Since couplings of this form may occur
in experimental scenarios featuring excited states with a low energy detuning, we have to
validate their absence in our model. To this end, we calculate the Heisenberg picture
population dynamics for a single emitter using an extended interaction Hamiltonian of
the form
Hel,ph =ℏZd3khgk(σ22 +σ33) + gintra
k(σ23 +σ32)ir†
k+rk,(7.18)
which explicitly includes real-valued intraband couplings gintra
kbetween the two excited
states. Fig. 7.8(a) shows the resulting occupation dynamics including intraband phonon
couplings and otherwise unchanged parameters with respect to Fig. 7.7. Moreover, we
have set the intraband coupling strength to a tenth of the interband coupling amplitude,
i.e., gintra
k= 0.1gk. As can be seen in Fig. 7.8, the population inversion slightly slows
down while the overall qualitative behavior is maintained. Hence, disregarding the impact
of possible intraband couplings is justified in the present investigation.
Fig. 7.8(b) illustrates the temperature-dependence of the stationary state population in-
version, showing the V-type emitter population dynamics of the detuned excited state
for increasing temperatures. Due to an increased potential for phonon-assisted transitions
back to the ground state, the grade of inversion decreases for increasing temperatures. In
the high-temperature regime, the system starts to equilibrate until the population inver-
sion has vanished.
7.5.3 Linear absorption spectrum
As an illustration of the involved oscillator strengths, in Fig. 7.9 we show the linear ab-
sorption spectrum α(ω)∼Im[σ12(ω)] + Im[σ13(ω)] [80,246,268] (green line) calculated
in the limit Ω→0at an incoherent decay rate γ= 500 ps−1,T= 4 K, ℏω12 = 1 eV and
ℏω13 = 0.999 eV, corresponding to an energy detuning δϵ =−1meV. Here, σmn(ω)repre-
sents the Fourier transform of the time-dependent coherence σmn(t)(m=n), prescribed
by σmn(ω) = R∞
−∞ dt σmn(t) exp(iωt). The blue line shows a comparison calculation with-
out reservoir coupling, gk= 0, to highlight the impact of phonon interactions. The inset
in Fig. 7.9 shows the absorption spectra on a logarithmic scale.
In the weak coupling and low temperature limit where the second-order Heisenberg corre-
lation expansion is valid, the shape of the resulting absorption spectrum is determined by
the employed oscillator strengths and the energy detuning δϵ between the excited states of
the V-type emitter. Fig. 7.9 features two peaks at ℏω12 = 1 eV and ℏω13 = 0.999 eV, corre-
sponding to the two allowed transitions between the ground state and excited states (green
line). Compared to the case without phonon interactions (blue line), phonon-induced fluc-
tuations in the form of absorption and emission processes give rise to phonon resonances
with the system coherences and a modulation of the eigenenergies. As a result, the side-
bands of the peaks feature an asymmetric broadening visible on a logarithmic scale (green
7.5 Discussion 95
0
1
2
3
0.999 1
Absorption [a.u.]
Energy [eV]
no phonons
with phonons
IBM
0
0.01
0.995 1 1.005
Figure 7.9: Linear absorption spectrum of the considered V-type emitter, calculated in the Heisen-
berg picture up to second order (blue line). For comparison, the spectrum resulting
without the presence of a phonon reservoir is additionally shown (green line). The cor-
responding absorption spectrum of the analytically solvable independent boson model
for a two-level system is depicted in orange. The inset shows the same spectra on a
logarithmic scale, where a phonon-induced asymmetric broadening of the absorption
peaks becomes visible.
line in inset). As an additional comparison, the dashed orange line in Fig. 7.9 shows the
linear absorption spectrum of the exactly solvable independent boson model [96,205,269],
describing a two-level system coupled to a structured phonon reservoir in the linear regime.
The spectrum is obtained with the same set of parameters used for the V-type emitter
case. Here, only a single optical transition is present, resulting in a single absorption peak.
We note that the shape of the phonon sidebands visible on a logarithmic scale is dependent
on the specific confinement potentials and the electronic wave function overlap (see inset).
Overall, the spectrum is in very good agreement with the V-type model case, verifying the
validity of the second-order Heisenberg correlation expansion in the employed parameter
regime, with only minor variations on the logarithmic scale caused by superpositions with
the second absorption peak.
7.5.4 Bare eigenstate dynamics
In addition to the interaction processes visible in the Heisenberg correlation expansion
approach (see Sec. 7.3), the emergence of population inversion in the considered V-type
emitter can be examined from a different perspective, i.e., via the dynamics of the bare
density matrix eigenstates which are shown to be highly dependent on the energy detuning
δϵ. In the absence of a phonon reservoir, the system Hamiltonian in matrix form is given
by
H= Ω
0 1 1
1 0 0
1 0 δϵ
Ω
.(7.19)
96 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
0
0.5
1
0 100 200
t [ns]
ρ--
ρ++
ρDD
Figure 7.10: Population of density matrix eigenstates |−⟩,|+⟩and |D⟩at δϵ < 0, corresponding
to the single V-type emitter setup calculated in Fig. 7.2.
f
Figure 7.11: Dexter-coupled chain of V-type emitters with carrier transfer between all excited
states of adjacent emitters at amplitude f.
When considering δϵ = 0.0meV, the resulting eigenvalues can be calculated analytically,
yielding
λ−=−√2Ω, λ+=√2Ω, λD= 0.(7.20)
Therefore, at zero energy detuning and without phonon interactions, the bare eigenstates
of the system feature a vanishing eigenvalue λD= 0, corresponding to a dark state. In
terms of the original basis states, the respective eigenstates read
|−⟩ = (−√2|1⟩+|2⟩+|3⟩)/2,(7.21a)
|+⟩= (√2|1⟩+|2⟩+|3⟩)/2,(7.21b)
|D⟩= (|2⟩−|3⟩)/√2.(7.21c)
In correspondence to its vanishing eigenvalue, the asymmetric state |D⟩is completely
unoccupied for the case δϵ = 0. The occupation dynamics of the system density matrix
eigenstates ρ−−,ρ++ and ρDD for the here considered negative detuning δϵ < 0are shown
in Fig. 7.10, corresponding to the V-type emitter dynamics shown in Fig. 7.2. In this case,
the asymmetric eigenstate |D⟩begins to weakly interact with the phonon reservoir and
the laser field and becomes fully occupied via a non-reciprocal Stokes process (orange line
in Fig. 7.10). Such effects have been shown to also emerge, e.g., in laser-pulsed exciton
and biexciton systems [29,34,35,217,218].
7.5 Discussion 97
0
0.5
1
0 50 100
ρ11
ρ22
ρ33
(a) Emitter 1
0 50 100
ρ11
ρ22
ρ33
(b) Emitter 2
0
0.5
1
0 50 100
t [ns]
ρ11
ρ22
ρ33
(c) Emitter 3
0 50 100
t [ns]
ρ11
ρ22
ρ33
(d) Emitter 4
Figure 7.12: V -type emitter population dynamics for a Dexter-coupled chain of length N= 4 with
carrier transfer between all excited states of neighboring emitters.
7.5.5 Alternative interdot coupling mechanisms: Dexter coupling
As an alternative interdot coupling mechanism enabling resonance energy transfer in the
chain, we additionally consider Dexter-type electron transfer where all excited states of
neighboring emitters feature wave function overlap (see Fig. 7.11) [252,253]. Again,
the single excitation case with a single electron initially localized in the ground state of
the first emitter in the chain is considered. The resulting Dexter-type interdot coupling
Hamiltonian is given by
HD=ℏf
N−2
X
l=0 hσ(2+3l)(5+3l)+σ(2+3l)(6+3l)+σ(3+3l)(5+3l)+σ(3+3l)(6+3l)+ H.c.i,(7.22)
where fagain denotes the Dexter coupling amplitude. Increasing the wave function overlap
has a considerable effect on the chain dynamics: Fig. 7.12 shows the resulting population
dynamics for N= 4 emitters, corresponding to the parameters employed in Fig. 7.5. Since
the excited states of all emitters support interdot coupling, energy is also transported from
the end of the chain back to the first site, resulting in a periodic stationary state current in
the quantum dot chain. Crucially, the population inversion still imposes unidirectionality,
such that the total current is still directed from the first to the last site in the emitter
array. The laser-detuned levels feature the majority of population, with the largest amount
located in the last site of the chain. To conclude, unidirectional electron transfer can still
be observed in the form of a directed stationary state current.
98 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
0
0.5
1
0 50 100
ρ11
ρ22
ρ33
(a) Emitter 1
0 50 100
ρ11
ρ22
ρ33
(b) Emitter 2
0
0.5
1
0 50 100
t [ns]
ρ11
ρ22
ρ33
(c) Emitter 3
0 50 100
t [ns]
ρ11
ρ22
ρ33
(d) Emitter 4
Figure 7.13: V -type emitter population dynamics at zero energy detuning δϵ = 0 and otherwise
corresponding to the setup shown in Fig. 7.12.
To demonstrate the key role of the population trapping for the unidirectional transport
mechanism, we additionally calculate the Dexter coupling setup shown in Fig. 7.12 once
more at zero energy detuning δϵ = 0.0meV, where no population inversion is observed. The
resulting chain dynamics is depicted in Fig. 7.13. In this case, carrier transport still takes
place, however, without the emergence of a stationary state current: All quantum dots
exhibit the same stationary state occupations, with the majority of excitation localized
in the ground states (red lines). Hence, the here employed interdot coupling is not the
crucial ingredient for the emergence of unidirectional transport, but rather the population
trapping mechanism enabled by non-Markovian reservoir interactions.
7.5.6 Alternative interdot coupling mechanisms: Förster coupling
As a second alternative realization of interdot coupling, we employ a bidirectional dipole-
dipole induced Förster excitation transfer mechanism [264]. Here, dipole-forbidden ex-
citation transfer between closely located quantum dots can be enabled, e.g., via optical
near-field interactions caused by exciton-polariton coupling [232,270–274]. Here we con-
sider such dipole-forbidden Förster-type couplings to enable excitation transport in the
quantum dot chain via a wetting layer modeled by an additional level |0⟩in each emitter
(see Fig. 7.14). The system is initialized with a single electron located in the ground state
of the first emitter. In the remaining sites, a single excitation is initially localized in state
7.6 Conclusion 99
f
Figure 7.14: Realization of Förster-type dipole-dipole excitation transfer in a quantum dot chain
at amplitude fvia an additional electron reservoir state |0⟩.
|0⟩. The resulting Förster-type interdot coupling Hamiltonian is given by
HF=ℏf
N−1
X
l=1 σl
03σl+1
20 + H.c.,(7.23)
where the index ldenotes the emitters and frefers to the Förster coupling amplitude.
Fig. 7.15 shows the population dynamics for a Förster-coupled V-type emitter chain of
length N= 2, consistent with the chain dynamics observed for the case of a single Dexter-
type transition between neighboring quantum emitters (see Fig. 7.5): Once more, the
excitation is transferred through the reservoir to the detuned excited state on the first
site and transported to the second emitter via dipole-dipole induced Förster interactions,
where it becomes dynamically trapped in the laser-detuned level. Initially, the occupation
of the first detuned level |3⟩1rises in accordance with the single emitter setup (see Fig. 7.2),
before dipole-dipole interactions with the neighboring site facilitate Förster resonance en-
ergy transfer and lead to an asymptotic decline of |3⟩1towards zero. In consequence, the
resonantly pumped levels |1⟩2and |2⟩2on the second site become uniformly occupied. Af-
terwards, the population inversion mechanism engages in the second quantum dot, yielding
a quick rise of population in the laser-detuned level |3⟩2accompanied by a decline in the
resonantly driven levels |1⟩2and |2⟩2. In the stationary state, all excitation is transported
to the detuned excited level of the second emitter. In conclusion, we have demonstrated
that the observed unidirectional quantum transport is robust against the specific interdot
coupling mechanism in the quantum dot array and uniquely dependent on the underlying
population inversion effect.
7.6 Conclusion
We have predicted the preparation of a nonequilibrium stationary state in a continu-
ously driven V-type emitter featuring complete population inversion and enabled by non-
Markovian system-reservoir interactions with the detuned excited states. Strikingly, the
observed energy transfer mechanism has been shown to be robust against perturbations
and external conditions such as incoherent radiative decay processes or phonon-induced
intraband couplings, which may arise in V-type systems featuring a small energy detuning
100 7 Unidirectional quantum transport in optically driven V-type quantum dot chains
0
0.5
1
0 200 400
t [ns]
ρ11
ρ22
ρ33
(a) Emitter 1
0 200 400
t [ns]
ρ11
ρ22
ρ33
(b) Emitter 2
Figure 7.15: V -type emitter dynamics for a chain of length N= 2 coupled by Förster-type res-
onance energy transfer using the same set of parameters as for the single V-type
emitter dynamics shown in Fig. 7.2.
between the excited states, allowing for complete inversion in a wide regime of tunable
coupling and driving parameters.
As an example application of the reported inversion mechanism, we have predicted the
emergence of unidirectional quantum transport of both carriers and excitation in an array
of InAs/GaAs quantum dots, depending on the employed bidirectional interdot coupling
mechanism. In summary, the presented non-Markovian population inversion can be ac-
complished in a wide array of realistic experimental conditions, giving rise to a range of
potential applications utilizing carrier or excitation transport in quantum optical devices
analogous to propositions of photonic unidirectional quantum transport in comparable sce-
narios [275–281]. The reported effect is also relevant for biological systems in the context
of excitation transport processes for light harvesting mechanisms [237].
8 Dissipative coherent quantum feedback
in Liouville space
In this Chapter, we consider a quantum optical system simultaneously coupled to both a
continuous and a time-discrete structured reservoir. For the description of the continuous
environment, a tensor network implementation of real-time path integrals is employed,
improving the memory scaling from exponential to polynomial efficiency. Combined with
a tensor network implementation of a time-discrete quantum memory, we construct a
quasi-2D tensor network accounting for both diagonal and off-diagonal system-reservoir
interactions with a twofold memory for continuous and discrete retardation effects. As
an example application, we investigate the non-Markovian dynamical interplay between
a structured acoustic phonon environment and time-discrete coherent photon feedback
acting on an initially excited two-level system. As a result of emerging inter-reservoir
correlations, long-living population trapping in the emitter is observed. The Chapter is
closely based on the publication “Continuous and time-discrete non-Markovian system-
reservoir interactions: Dissipative coherent quantum feedback in Liouville space” by O.
Kästle et al. [71]
8.1 Introduction
Given the exponential scaling of the Hilbert space dimension for increasing system sizes,
the accurate description of open quantum systems poses an immense challenge [25–27,44].
Various theoretical approaches have been proposed to reduce their complexity, ranging
from correlation expansions [28,29] and second-order perturbative master equations [2,3]
to numerically exact path integral formulations for pure decoherence effects [30–38]. In
general, all of these approaches have focused on system-reservoir interactions with a single
structured environment, for instance in the form of a continuous reservoir or a time-
discrete quantum memory, leading to time-delayed information backflow to the system
and emerging non-Markovian phenomena [60–68].
In this Chapter, we develop a numerically exact tensor network implementation to describe
two non-Markovian processes in the form of continuous and time-discrete retardation ef-
fects simultaneously, where both off-diagonal and diagonal system-reservoir couplings are
accounted for, i.e., interactions with or without energy exchange between system and en-
vironment. Established tensor network techniques for the description of a time-discrete
quantum memory focus on systems where decoherence effects are not crucially important,
and thus rely on a wave function ansatz to solve the quantum stochastic Schrödinger equa-
tion [282–284]. Here, we first extend this architecture to a time bin-based density matrix
101
102 8 Dissipative coherent quantum feedback in Liouville space
Figure 8.1: Sketch of an open quantum system coupled to both a time-discrete and a continuous
structured reservoir.
formalism in Liouville space, allowing for the inclusion of Markovian and non-Markovian
decoherence effects. Secondly, we combine this generalized MPS algorithm with a tensor
network implementation of real-time path integrals (see Ch. 4) to incorporate the inter-
play with a continuous harmonic reservoir of independent oscillators [75–77,102,285],
resulting in a novel quasi-2D tensor network architecture [286–289]. In this formalism,
simulations of quantum systems coupled to two structured reservoirs become feasible (see
Fig. 8.1), equally accounting for diagonal and off-diagonal interactions and maintaining
crucial entanglement information in between system and reservoirs. Potential applications
encompass, e.g., waveguide quantum electrodynamical setups with dephasing [67,290–292]
realized via additional decay channels, or multiple spatially separated solid-state quantum
emitters coupled to an environment and initially prepared in a dark state [293]. To demon-
strate the potential of the presented method, we specifically consider a two-level quantum
emitter interacting with a structured reservoir of independent oscillators and additionally
featuring a time-discrete quantum memory in the form coherent quantum feedback, ex-
tending the paradigm of the spin-boson model to feedback-related phenomena [74]. The
unfolding non-Markovian interplay between decoherence and relaxation effects is shown
to result in a dynamical protection of coherence against destructive interference processes,
facilitating dynamical population trapping in the system. This observation expands upon
the localized phase stabilization effect in the spin-boson model [74,75] from an incoherent
feedback-induced perspective, where the coherent external driving is replaced by another
structured reservoir.
In Sec. 8.2, we briefly recall the numerically exact tensor network implementation of real-
time path integrals to account for continuous reservoirs (see Ch. 4) and provide benchmark
calculations for the independent boson model and the spin-boson model. Afterwards, the
MPS realization of a time-discrete quantum memory in Liouville space is introduced in
Sec. 8.3. In Sec. 8.4, we combine the two MPS algorithms to construct a quasi-2D ten-
sor network architecture, enabling numerically exact calculations of a quantum system
simultaneously exposed to two non-Markovian system-reservoir interactions. As an exam-
ple application, in Sec. 8.5 we demonstrate the dynamical protection of feedback-induced
coherence, resulting in population trapping in a two-level system. In Sec. 8.6, a conver-
gence analysis of the reported results with respect to all relevant physical and numerical
parameters is performed. Lastly, we summarize our findings in Sec. 8.7.
8.2 Path integral formulation for continuous reservoirs 103
8.2 Path integral formulation for continuous reservoirs
As a first step, we calculate the time evolution dynamics of both the independent boson
model and the spin-boson model using a numerically exact implementation of real-time
path integrals [30–38] in a tensor network approach (see Ch. 4). In this framework, first in-
troduced by Strathearn et al. [75,76,102,285], the time-discrete path integral formulation
of the reduced system density matrix is efficiently encoded in high-dimensional tensors.
The first objective is to derive a path integral solution of the von Neumann equation given
a Hamiltonian H(t)describing a time-dependent system-reservoir interaction [1,268],
˙ρ(t) = L(t)ρ(t) = −i
ℏ[H(t), ρ(t)],(8.1)
where ρ(t)denotes the density matrix and L(t)the considered Liouvillian superoperator.
Since the history of all preceding paths at times 0, . . . , tn−1has to be taken into account for
the evaluation of the current time step tn, the numerical calculation of the path integral
solution becomes very demanding after only a few time steps. To enhance numerical
accessibility, in the case of finite system-reservoir correlations the augmented density tensor
scheme can be applied [37,38]. Here, a finite reservoir memory depth allows to only
consider the last nctime steps for the calculation of the current path. This approach
is referred to as the finite memory approximation and leads to the augmented density
tensor representation, constituting a discrete path integral formulation as a solution to
the system part of Eq. (8.1) with traced out reservoir contributions. At time tN=N∆t
it is given by
ρiNi′
N(tN) =
N
Y
n=1 X
in−1X
i′
n−1
Minin−1M∗
i′
n−1i′
n
n
Y
m=n−nc
exp Sinim
i′
ni′
mρi0i′
0(0),(8.2)
with indices in,i′
nstoring the left and right configuration of the system at time tn=n∆t
and Minin−1denoting the field transformation matrix which, e.g., accounts for an external
driving field Ω0present in the spin-boson model case. In the absence of external driving,
Ω0= 0, it is simply given by Minin−1=1δin,in−1. The influence functional reads
Sinim
i′
ni′
m=−in−i′
nηn−mim−η∗
n−mi′
m,(8.3)
with
ηn−m=Zn∆t
(n−1)∆t
dτZm∆t
(m−1)∆t
dτ′ϕ(τ−τ′),(8.4)
and ϕ(τ−τ′)denoting the reservoir autocorrelation function [31,32,74]. Under the
improved finite memory approximation, for n−m≡ncall preceding paths up to tnc=nc∆t
are additionally included in the integration, yielding ηnc:= ηn−m+Pn−nc−1
k=1 ηn−k(see
Ch. 4) [102]. With this truncation, the augmented density tensor and its time evolution
dynamics can be stated efficiently in the form of a tensor network [75]: As a first step,
Eq. (8.2) is mapped to a vector ρjnin Liouville space,
ρjN(tN) =
N
Y
n=1
n
Y
m=n−nc
I(jn, jm)ρj0(0),(8.5)
104 8 Dissipative coherent quantum feedback in Liouville space
0
0.5
1
0 1 2
t [ps]
T = 4 K
T = 30 K
T = 77 K
Abs[ρ01]MPS
Abs[ρ01]ana
(a)
0
0.5
1
0 100 200 300 400 500
t [ps]
ρ00
Im[ρ01]
Re[ρ01]
(b)
Figure 8.2: (a) Path integral solution of the independent boson model calculated at varying temper-
atures (solid lines) and compared to corresponding analytical solutions (dashed lines).
(b) Path integral solution of the spin-boson model time evolution dynamics at a reser-
voir memory depth nc= 100, driving field strength Ω0= 0.5 ps−1and temperature
T= 77 K.
with I(jn, jm) := Pjn−1˜
Mjnjn−1exp ( ˜
Sjnjm). The left and right system indices ik,i′
kare
rewritten as a single index jkfor each time step, yielding Liouville space representations
˜
Mjnjn−1and ˜
Sjnjmof the field transformation matrix and the influence functional. In
addition, the augmented density tensor is decomposed into MPS format, with the present
and up to nc−1past states stored in individual tensors and the oldest state located at the
left MPS boundary site. As a result of tensor compression via consecutive applications
of the singular value decomposition at a finite Schmidt value cutoff precision [39], the
required memory is reduced from exponential to polynomial scaling with respect to the
number of incorporated paths nc[75].
The system time evolution dynamics is imposed by application of a network of MPOs
(see Ch. 4). During each time step n, the current system state ρjn(tn)is contracted with
an MPO in the network, resulting in an updated system state MPS with the preceding
path stored to the left of the system tensor, increasing the MPS length by one. After
reaching step n=nc, the oldest path in the MPS is summed over by application of a delta
tensor in correspondence to the improved finite memory approximation [75,102]. From
this point on, the length of the MPS remains constant for the rest of the time evolution. In
addition, for time-independent problems, aside from the index nomenclature the structure
of the MPO remains unchanged for all time steps n≥nc, further improving performance.
For the description of a continuous reservoir of noninteracting harmonic oscillators, we
consider the Hamiltonian [96]
HC/ℏ=Zd3qhωqb†
qbq+gqσ11 b†
qeiωqt+ H.c.i,(8.6)
imposing a diagonal system-reservoir coupling without energy transfer. Here, σij =|i⟩⟨j|
denotes system operators, b(†)
qare bosonic annihilation (creation) operators of reservoir
modes at frequency ωq=cs|q|and csthe sound velocity, and gqdenotes the mode q-
dependent system-reservoir coupling amplitude. The corresponding correlation function
8.3 Time-discrete memory in Liouville space 105
is given by
ϕ(τ−τ′) = Zd3q g2
qcoth ℏωq
2kBTcos[ωq(τ−τ′)] −isin[ωq(τ−τ′)],(8.7)
with Tthe reservoir temperature and kBthe Boltzmann constant. For our investigation,
we employ a generic coupling element describing, e.g., acoustic bulk phonons interacting
with a quantum emitter. Specifically, we choose the acoustic bulk phonon coupling ele-
ment of GaAs, given by gii
q=pℏq/(2ρcs)Diexp[−ℏq2/(4miωi)], resulting in a transition
coupling element gq:= g22
q−g11
q[80,97,98,251], with deformation potentials Di, effective
masses mi, confinement energies ℏωiand the mass density of GaAs ρ.
To verify the accuracy of the employed MPS path integral implementation, we first cal-
culate the independent boson model where an analytical solution is available. It consists
of a single two-level emitter embedded in a structured harmonic reservoir and subjected
to pure dephasing, as described by Eq. (8.6). Fig. 8.2(a) shows the unfolding time evo-
lution dynamics of the polarization ρ01(t)at varying temperatures T, initially prepared
at ρ01(0) = 0.5i(solid lines). The corresponding analytical solution (dashed grey lines) is
prescribed by [80]
ρ01(t)=exp(−Zd3qg2
q
ω2
qiωqt−isin(ωqt) + coth ℏωq
2kBT[1 −cos(ωqt)])ρ01(0),(8.8)
and exhibits excellent agreement with the numerical results for all featured tempera-
ture regimes. Moreover, due to the high numerical performance of the employed tensor
network implementation, reservoir memory depths of nc= 100 and beyond become ac-
cessible. As a second benchmark and to underline the performance capabilities of the
MPS approach, we additionally calculate the time evolution dynamics of the spin-boson
model by adding an additional continuous driving term at amplitude Ω0to Eq. (8.6),
i.e., HSBM =HC+ Ω0(σ01 +σ10). The resulting time evolution dynamics of the emitter
ground state population and the system coherence are shown in Fig. 8.2(b), calculated
at parameters nc= 100,Ω0= 0.5 ps−1and T= 77 K for the memory, driving field and
temperature, respectively.
8.3 Time-discrete memory in Liouville space
In addition to the continuous structured reservoir, we consider a discrete non-Markovian
time-bin based quantum memory. Established realizations of such a time-discrete reservoir
rely on an MPS-encoded wave function ansatz to solve to quantum stochastic Schrödinger
equation [282–284]. However, we require a compatible formulation to the previously intro-
duced path integral formalism defined in terms of the density matrix in Liouville space.
Therefore, here we present an MPS implementation of a time-discrete quantum memory
in Liouville space. The dynamics of the system density matrix is once more prescribed
by a Liouvillian superoperator [see Eq. (8.1)], where a Hamiltonian HDaccounts for the
time-delayed system-reservoir coupling, such that interactions taking place at a given time
tcouple back into the system and influence its state at a later time t+τ, where τdenotes
106 8 Dissipative coherent quantum feedback in Liouville space
the retardation time. Such a time-discrete coupling can arise, e.g., in a two-level system
with states |0⟩,|1⟩and an energy difference ℏω0between them, located in front of a mirror
with a round trip time τ. The corresponding Hamiltonian is given by
HD/ℏ=ω0σ11 +s2Γ
πZdksin ωkτ
2hσ10ckei(ω0−ωk)t+ H.c.i,(8.9)
featuring off-diagonal coupling and energy exchange between system and environment,
with system operators σij =|i⟩⟨j|, bosonic annihilation (creation) operators c(†)
kof photon
modes at frequency ωk=ck with cthe speed of light, and a constant electron-photon
coupling amplitude Γ.
The time evolution dynamics prescribed by the Liouvillian L(t)is translated into a time
bin-based MPS realization [282–284,288], maintaining crucial system-reservoir correla-
tions and scaling with the feedback time τ. When considering additional phenomenological
dissipative channels, the Liouvillian is extended by the standard Lindblad operator [1,268].
We start from the formal solution of the system part of Eq. (8.1) for the density matrix,
ρ(t) = Texp Zt
0
dt′L(t′)ρ(0),(8.10)
with Tdenoting the time-ordering operator. As a first step to restate Eq. (8.10) in terms
of a tensor network algorithm it is rewritten in a time-discrete basis, yielding at time
tN=N∆t
ρ(tN) = L(N, N −1)L(N−1, N −2) . . . L(1,0),(8.11)
at a time discretization ∆tand using time-bin normalized operators
L(n, n −1) = exp "√∆tZn∆t
(n−1)∆t
dt′L(t′)#.(8.12)
The Liouvillian time step operator L(n, n −1) prescribes the time evolution dynamics
during the n-th time step tn−1→tn. As a next step, it is approximated by a tenth order
series expansion,
L(n, n −1) ≈
10
X
m=0
√∆tm
m!"Zn∆t
(n−1)∆t
dt′L(t′)#m
.(8.13)
In this format, Eq. (8.13) can be incorporated in a tensor network-based implementation
of the time-discrete quantum memory, schematically shown in Fig. 8.3: Fig. 8.3(a) shows
the time-bin based MPS at the start of the time evolution (t= 0), with the square red
tensor storing the initial system density matrix. To account for a time-discrete memory,
here nd= 4 circular tensors located to the left of the system state contain the state of
the reservoir in Liouville space at preceding times, with the oldest state located at the
left boundary of the MPS (blue). In the here considered scenario, these memory bins
are initially empty. The circular tensors located to the right of the system bin store the
reservoir states for all future time steps, which have not been carried out yet and contain
8.3 Time-discrete memory in Liouville space 107
Figure 8.3: Tensor network realization of a time-discrete memory in Liouville space. (a)-(d) depict
the operations performed during the initial time step, where the system state (red
square) interacts with both present and past reservoir bins (orange and blue circles)
by application of the Liouvillian L(t). (e) shows the second time step with analogous
operations.
the full reservoir entanglement information, e.g., at a finite temperature. Since we assume
an initial vacuum state, a new empty reservoir bin (orange) is added to the right of the
system bin and represents the current state of the reservoir [see Fig. 8.3(a)].
The implementation of the time-discrete memory is explained next. Here, we introduce
the memory retardation time τ=nd∆tgiven by the number of initial memory bins nd.
For the first step of the time evolution [see Fig. 8.3(a)], the first memory bin (blue) is
pushed to the left of the system bin (red) by consecutive applications of the singular value
decomposition, maintaining crucial entanglement information (see Ch. 4) [39]. Afterwards,
the time-discrete Liouvillian operator L(1,0) prescribing the evolution during the first time
step is applied to the system bin (red), the current memory bin (blue) and the current
reservoir bin (orange), as depicted in Fig. 8.3(b). After contraction with the Liouvillian
operator, the three bins are updated and the processed memory bin [dark grey bin in
Fig. 8.3(c)] is swapped back to its original position on the left end of the MPS, where it
remains for the rest of the time evolution [see Fig. 8.3(d)]. Crucially, the updated reservoir
bin (green) is pushed to the left of the system bin and takes the role of a memory bin for a
subsequent time step of the evolution [see Figs. 8.3(c),(d)]. In Fig. 8.3(d), the execution of
the first time step is complete. All following time steps are carried out in the same fashion,
as shown schematically for the second time step in Fig. 8.3(e). After the calculation of nd
time steps, all memory bins initialized at the left of the system bin have been processed
and pushed to the end of the MPS. At step nd+ 1, the reservoir bin employed during the
first time step and pushed to the left of the MPS [green bin in Fig. 8.3(d)] becomes the
current memory bin, storing information of a previous system state. By application of the
corresponding Liouvillian time step operator, reservoir-induced memory effects unfold in
the system according to the time-ordered problem.
108 8 Dissipative coherent quantum feedback in Liouville space
0
0.5
1
0 3 6 9
t [ps]
MPS
analytical
(a)
0
0.5
1
0 10 20 30
t [ps]
γ = 0.0
γ = 0.5
(b)
Figure 8.4: (a) Population dynamics of a two-level system subjected to time-discrete coherent
quantum feedback [Eq. (8.9)] at parameters τ= 3 ps and Γ = 31.6 ps−1. Numerical
calculations carried out via the MPS algorithm (solid blue line) are in excellent agree-
ment with the analytical result for the single excitation case (dashed grey line). (b)
MPS population dynamics under time-discrete feedback calculated with an additional,
analytically not accessible phenomenological dephasing rate γ= 0.5 ps−1(green line)
and compared to the case without dephasing (blue line).
To verify the accuracy of the presented time-discrete memory algorithm in Liouville space,
we first calculate the time evolution dynamics resulting from Eq. (8.9). Fig. 8.4(a) depicts
the resulting emitter population dynamics at a retardation time τ= 3.0ps and Γ =
31.6 ps−1obtained from the numerical MPS implementation (solid blue line) and compared
to the analytical solution up to t= 3τwhich is only available in the single excitation regime
(dashed grey line). The latter is prescribed by ⟨σ11(t)⟩=|⟨σ01(t)⟩|2with [294–296]
⟨σ01(t)⟩=
∞
X
n=0
e−Γt
n!hΓe(Γ−iω0)τ(t−nτ)inΘ(t−nτ),(8.14)
exhibiting excellent agreement with the numerical result. In the here considered regime
Γτ≫1, the delay in the amplitude dominates the time evolution dynamics and leads to
emitter re-excitations at integer multiples of the feedback time τ. However, the phase of
the amplitude φ=ω0τbecomes irrelevant after the initial τ-intervals due to increased
decay of the mixing terms in the absolute square of Eq. (8.14). In previous realizations of a
time-discrete quantum memory, investigations of the impact of phase-destroying processes
have been limited to a special case, i.e., the emergence of an Ornstein-Uhlenbeck process
during the initial feedback intervals [296]. Moreover, only an analytical approach had been
available, limiting calculations to a small number of τ-intervals and prohibiting access to
the stationary state. The here presented numerical approach overcomes these limitations:
The proposed Liouville space architecture allows to incorporate Markovian decoherence
in the form of Lindblad dissipators without increased numerical costs.
Fig. 8.4(b) shows the time evolution dynamics of an initially excited two-level emitter
subjected to coherent quantum feedback and additional phenomenological dephasing at
rate γ, realized by embedding a Lindblad dissipator into the Liouvillian [Eq. (8.1)] and
setting H=HD, [1,268]
Dqγ/2˜σ11ρ(t) = γ
2[2˜σ11ρ(t)˜σ11 −{ρ(t),˜σ11}],(8.15)
8.4 Quasi-2D tensor network 109
with the system operator rewritten in full configuration space, ˜σ11 =1Dσ111D, via the
time-discrete reservoir basis 1D=RdkP∞
n=0 |{nk}⟩⟨{nk}|. For comparison, we show the
cases γ= 0 (blue line) and γ= 0.5 ps−1(green line). Crucially, the impact of pure
dephasing begins to govern the dynamics only after the time-discrete memory signal re-
excites the system and the stabilization of the incoming and outgoing phase becomes
a factor. When considering an additional pure dephasing γ= 0, the initial decay of
population remains unchanged, but the re-excitation becomes less distinct after the first
feedback interval. After a few τ-intervals only incoherent re-excitation occurs, resulting in
a faster decay to zero without population trapping and regardless of the feedback phase
φ.
These findings highlight the robustness of quantum feedback processes when accounting
for additional Markovian dissipative channels. Due to the inevitable loss of quantum
feedback-induced coherence, the presence of additional Markovian decoherence results
in a degrading of the feedback signal. However, this is not necessarily the case when
considering non-Markovian dissipation channels, which are incorporated next.
8.4 Quasi-2D tensor network
For a joint description of non-Markovian decoherence and time-discrete system-reservoir
interactions, we combine the MPS implementation of the time-discrete quantum memory
in Liouville space with the tensor network realization of real-time path integrals for con-
tinuous harmonic reservoirs, resulting in a quasi-2D tensor network architecture. The two
networks are connected by link indices containing the arising entanglement information
to enable a numerically exact description of correlation buildup in between the reservoirs.
In scenarios involving multiple non-Markovian reservoirs connected to each other, such
inter-reservoir correlations may have a crucial influence on the time evolution dynamics.
Therefore, the arising inter-reservoir entanglement must be taken into account without
strict truncation, for instance in the form of a high Schmidt value cutoff precision dcut.
The overall grade of entanglement in the system rises intensively with respect to the single
reservoir setup. However, setups featuring two non-Markovian environments without cru-
cial interactions between them, e.g., due to dynamical decoupling, allow for a much more
restrictive truncation without loss of accuracy. In the here considered scenario, we have
chosen a high Schmidt value cutoff precision dcut = 10−12, resulting in the conservation of
relevant entanglement information during the time evolution.
Accurately depicting the dynamical interplay of the two reservoirs with each other and the
system is an immense numerical challenge and poses strict limitations on the accessible
memory depths for the here considered setup: While both of the presented MPS algorithms
individually enable simulations of a single reservoir with deep memories, their combination
results in restrictions caused by arising inter-reservoir entanglement. In consequence, the
total number of memory bins in the quasi-2D network is constricted to nc+nd<20 for
the here considered model, as is the case in established path integral implementations for
a single reservoir [32–36,74]. It is noted once more that these limitations are a natural
result of the high correlation degree in between system and reservoirs. The here presented
110 8 Dissipative coherent quantum feedback in Liouville space
Figure 8.5: Implementation of the quasi-2D tensor network. (a) MPS-based path integral imple-
mentation for the simulation of continuous structured reservoirs. (b) MPS containing
the current (red) and preceding system states (grey) after completion of the first net-
work contraction [dashed square in (a)]. For the realization of the quasi-2D tensor
network, an additional link index connects the current state tensor to the discrete
memory MPS (blue diagonal line). (c) Time-discrete memory MPS during the ini-
tial time step, with the shared current system state tensor (red) acting as a junction
between the reservoirs and the two networks (dashed circles).
quasi-2D tensor network is a first step towards investigations of the mostly unexplored field
of multiple non-Markovian reservoir interactions, explicitly enabling information backflow
in between the twofold reservoir memories.
Fig. 8.5 illustrates the construction of the quasi-2D tensor network via combination of
the two previously introduced MPS algorithms for the continuous [Figs. 8.5(a),(b)] and
time-discrete structured reservoirs [Fig. 8.5(c)]. They are connected to each other via a
shared current system state tensor (red shape), acting as a junction link between the two
reservoirs [dashed circles and blue lines in Figs. 8.5(b),(c)] and allowing for the storage
of arising inter-reservoir correlations. During each step of the time evolution, the system
bin is first subjected to the action of the continuous reservoir by a single contraction in
the network shown in Fig. 8.5(a) (dashed frame). Afterwards, the updated current system
bin [red shape in Fig. 8.5(b)] is inserted into the second MPS algorithm responsible for
the action of the time-discrete reservoir [Fig. 8.5(c)]. The resulting quasi-2D tensor net-
work effectively stores the history of both system-reservoir interactions, maintaining the
relevant entanglement information in the open system and allowing for the calculation of
two dynamically interacting time-delayed processes. As a first application and to demon-
strate the capabilities of the presented quasi-2D network, we investigate the interplay of
a continuous reservoir of independent oscillators with a time-discrete quantum memory
realized by coherent quantum feedback. For specific memory depths and initial states,
this scenario is shown to result in a dynamical protection of coherent quantum feedback
properties in the open system and population trapping in the presence of non-Markovian
decoherence processes.
8.5 Memory-induced dynamical population trapping 111
0
0.5
1
0 25 50
t [ps]
ϕ = 1.00, γ = 0.000
ϕ = 1.00, γ = 0.001
ϕ = 1.17, γ = 0.000
ϕ = 1.17, γ = 0.001
Figure 8.6: Excited state occupation dynamics of the two-level system subjected to time-discrete
photon feedback at varying feedback phases φ=ω0τ/(2π)and constant phenomeno-
logical dephasing rates γ.
8.5 Memory-induced dynamical population trapping
Coherent quantum feedback mechanisms have been shown to feature a wide variety of non-
Markovian phenomena [295,297–302], e.g., leading to Ornstein-Uhlenbeck-type events in
the presence of white noise [296], coherent population trapping [206,283,293,303–305], or
the formation of large entangled photon states [306]. However, until now these effects have
not been investigated in the presence of additional non-Markovian dissipative channels
or decoherence mechanisms. To explore the influence of dephasing on feedback-induced
decoherence effects, we consider a two-level system placed in front of a mirror with a round
trip time τ, representing the time-discrete reservoir schematically shown in Fig. 8.1 and
explicitly given by Eq. (8.9). The system is imprinted with a time-delayed coherence by
the photon-induced feedback, taking the form of a feedback phase φ=ω0τ/(2π). This
phase has a crucial impact on the system dynamics and is given by the feedback time τ
and the transition frequency ω0of the electronic coherence operator σ12. We specifically
consider a scenario which exhibits a pronounced quantum optical effect, i.e., coherent
population trapping in an initially excited two-level emitter resulting from a continuum
bound state at feedback phases φ∈Z[206,283,293,296,303–305]. The relevant system
parameters are chosen at Γ=0.9 ps−1,τ= 1.2ps and nd= 4, yielding a time discretization
∆t= 0.3ps, for instance common for semiconductor quantum dot devices [206,302,305],
and Γτ≈1.1, corresponding to the strong non-Markovian regime [282,290,296]. The
coarse time discretization is justified by the tenth order series expansion of the time-
discrete Liouvillian operator [see Eq. (8.13)], such that the employed nd= 4 time bins are
sufficient to achieve convergent results (see Sec. 8.6).
To demonstrate the potential of the presented quasi-2D tensor network approach, we first
calculate the system dynamics prescribed by Eq. (8.9) for the case of Markovian dephasing
introduced by the Lindblad dissipator in Eq. (8.15) at a phenomenological dephasing
rate γ. Afterwards, it is compared to the case of non-Markovian decoherence imposed
112 8 Dissipative coherent quantum feedback in Liouville space
0
0.5
1
0 25 50
t [ps]
ϕ = 1.17, T = 77 K
ϕ = 1.17, T = 30 K
ϕ = 1.17, T = 4 K
0
100
200
300
0 50t [ps]
Figure 8.7: Excited state emitter occupation dynamics resulting from time-discrete photon and
continuous phonon system-reservoir interactions at φ /∈Zand varying temperatures
T. The inset shows the dimension of the link index connecting the system bin to the
time-discrete memory MPS in the presence of a structured phonon reservoir (blue),
without phonon coupling (green) and for phenomenological dephasing (orange).
by the continuous reservoir schematically shown in Fig. 8.1, resulting in an additional
diagonal system-reservoir coupling. Fig. 8.6 shows the unfolding excited state emitter
occupation dynamics for the case of Markovian dephasing at varying γand feedback
phases φ. Without dissipation, γ= 0, the feedback phase can be periodically tuned to
an ideal value φ∈Z, such that constructive interference leads to a decoupling of the
system from the environment, resulting in coherent population trapping (solid green line
in Fig. 8.6). Introducing a nonzero dephasing γ= 0.001 ps−1to the system does not
affect the population dynamics until the time-discrete feedback sets in, since the radiative
decay is frequency-independent until the first feedback interval at t=τ(dashed green
line). At this point, phenomenological dephasing destroys the coherent phase interference,
resulting in an asymptotic decline of population to zero without population trapping.
When choosing a nonideal feedback phase, here φ= 1.17, and zero dephasing, destructive
phase interference also results in an asymptotic decline towards zero (solid orange line).
Setting γ > 0further accelerates the population decay (dashed orange line) due to the
phenomenological decoherence perturbing the phase relation φ. To conclude, without the
presence of a structured continuous reservoir and at feedback phases φ /∈Z, asymptotic
population decay via thermalization inevitably occurs, and a Lindblad-based incorporation
of Markovian decoherence cannot preserve quantum correlations between the reservoir and
system states under any conditions.
As a next step, we consider the case of a non-Markovian system-reservoir interaction induc-
ing decoherence to the system. Employing the quasi-2D tensor network architecture, we
calculate two-level emitter population dynamics prescribed by HD+HC[Eqs. (8.6), (8.9)]
at a continuous reservoir memory depth nc= 4 and leaving all remaining parameters
unchanged with respect to the results shown in Fig. 8.6. Fig. 8.7 shows the unfolding
time evolution dynamics at varying temperatures T. For the here employed parameters,
population trapping can be achieved at T= 4 K and a nonideal feedback phase φ= 1.17,
8.6 Convergence analysis 113
i.e., φ /∈Z(solid blue line). Here, time-delayed backflow of excitation from the continuous
structured environment to the emitter leads to a decoupling from destructive phase inter-
ference with the time-discrete photon reservoir, thereby facilitating information exchange
and correlation buildup in between the reservoirs. As a result, feedback-induced coherence
in the system is dynamically protected for long times. After a typical excitation back-
flow time, dynamical population trapping at a finite temperature can always be found by
tuning of the feedback phase φ, if the relevant diffusion processes occur on a comparable
time scale as the coherence-inducing feedback dynamics. For increasing temperatures, the
dynamics becomes mainly governed by the thermal properties of the continuous phonon
reservoir: Dashed and dotted blue lines in Fig. 8.7 depict resulting thermalization dy-
namics with population decay at T= 30 K and T= 77 K, respectively. The observed
temperature dependence evidences the importance of the correlation lengths within the
full system-reservoir dynamics. Moreover, these otherwise inaccessible microscopic en-
vironmental properties can be effectively probed via the observed population trapping
effect.
In close analogy to the observed formation of a self-stabilizing dissipative structure, a
localized phase stabilization can be observed in the spin-boson model with coherent driving
Ω0= 0 at Ohmic spectral densities and without feedback effects [74,75], where the open
system transitions into a localized phase with a nonzero stationary emitter occupation
above a critical system-reservoir coupling amplitude. In the here considered scenario,
time-discrete photonic feedback replaces the coherent external driving field, resulting in a
localized phase stabilization via dissipative non-Markovian interactions. As an estimation
of the correlation buildup in between the two reservoirs, the inset in Fig. 8.7 shows the
dynamical evolution of the link index dimension connecting the current system bin to
the time-discrete memory MPS (see Fig. 8.5). In the case of a structured continuous
reservoir (blue line in inset), after a slow buildup phase during times t < τ the index
dimension swiftly increases to a maximum determined by the employed memory depths.
The unfolding high grade of inter-reservoir entanglement at comparably low finite memory
sizes nd= 4,nc= 4 highlights the essential role of the interplay between the two non-
Markovian reservoirs for the observed protection of coherence. Switching off the continuous
reservoir coupling, gq= 0, leads to a vastly decreased maximum link dimension (green
line in inset), since no entanglement buildup occurs between the reservoirs. In the case
of a phenomenological dephasing (orange line in inset), the link index dimension between
system and reservoir bin in the resulting 1D network is even further reduced due to its
highly decreased complexity.
8.6 Convergence analysis
In this Section, we provide a thorough convergence analysis of the presented results with
respect to the relevant system parameters. Due to the memory limitations arising from
the intensive scaling of entanglement when describing two non-Markovian reservoirs si-
multaneously, we first verify numerical convergence with respect to the memory depth of
the continuous harmonic reservoir nc, corresponding to the validity of the finite memory
114 8 Dissipative coherent quantum feedback in Liouville space
0
0.5
1
0 25 50
t [ps]
nc = 2
nc = 3
nc = 4
nc = 5
(a) 0
0.5
1
0 25 50
t [ps]
dcut = 10-08
dcut = 10-10
dcut = 10-12
dcut = 10-14
(b)
0
400
800
1200
0 50t [ps]
0.2
0.25
25 50t [ps] 0.2
0.21
25 50t [ps]
Figure 8.8: Convergence analysis of the calculations shown in Fig. 8.7 regarding (a) the continuous
reservoir memory depth ncand (b) the Schmidt value cutoff precision dcut employed
during the singular value decomposition. In (a), the left and right inset show a zoom-
in on the dynamics and the resulting dimension of the link index connecting the two
tensor networks, respectively. The inset in (b) depicts a zoom-in on the long term
dynamics.
approximation for the considered setup. Fig. 8.8(a) shows the population dynamics cor-
responding to Fig. 8.7 at φ= 1.17,T= 4 K and increasing memory depths ncof the
continuous reservoir. The left and right inset show a zoom-in on the population dynamics
and the resulting dimension of the link index connecting the two tensor networks, re-
spectively. The latter again highlights the occurring exponential growth of inter-reservoir
entanglement for increasing memory depths. At nc= 2 [dark blue line in Fig. 8.8(a)], the
results clearly do not converge, however, slightly increasing the memory depth to nc= 3
and nc= 4 (light blue and green lines) already leads to results in very good agreement
with each other. Further increasing the memory depth to nc= 5 (dashed orange line)
leads to no visible deviation from the case nc= 4 even at a close range (see left inset),
verifying the numerical convergence at nc= 4. In Fig. 8.8(b), we calculate the multiple
reservoir case for varying Schmidt value cutoffs dcut applied during the tensor truncation
procedure of the singular value decomposition [39]. Choosing cutoffs between dcut = 10−8
(dark blue line) and dcut = 10−14 (dashed orange line) leads to identical results even at a
close range (see inset), demonstrating the numerical convergence with respect to the cutoff
precision dcut = 10−12 employed for all calculations (green line). Therefore, no relevant
entanglement information has been disregarded during the time propagation.
As a next step, we confirm numerical convergence of all presented results with respect to
the employed time evolution step size ∆tto ensure the validity of the Trotter decomposi-
tion. In Fig. 8.9(a), we calculate the two-level emitter occupation dynamics when exposed
to the time-discrete reservoir at φ= 1.17 (see Fig. 8.6) and decreasing step sizes ∆t. Aside
from small deviations during the initial steps of the time evolution, the results are in ex-
cellent agreement for all employed values of ∆t, verifying the numerical convergence of the
time-discrete MPS feedback algorithm at the chosen time discretization ∆t= 0.3ps (light
blue line). Fig. 8.9(b) shows the polarization dynamics of a two-level emitter coupled to
a continuous reservoir, corresponding to the results shown in Fig. 8.2(b) and calculated
at T= 4 K and varying step sizes ∆t. Once more, the unfolding time evolution dynamics
8.6 Convergence analysis 115
0
0.5
1
0 25 50
ρ11
∆t = 0.600 ps
∆t = 0.300 ps
∆t = 0.150 ps
∆t = 0.075 ps
(a)
0
0.5
1
0 1 2
Abs[ρ01]
∆t = 0.600 ps
∆t = 0.300 ps
∆t = 0.150 ps
∆t = 0.075 ps
(b)
0
0.5
1
0 5 10
ρ11
t [ps]
∆t = 0.300 ps
∆t = 0.240 ps
(c)
0
0.5
1
0 25 50
ρ11
t [ps]
Order 8
Order 9
Order 10
(d)
0.2
0.21
25 50t [ps]
Figure 8.9: Convergence analysis of all calculations with respect to the time evolution step size ∆t
applied in (a) the time-discrete reservoir algorithm (see Fig. 8.6), (b) the coherence
time evolution dynamics for the continuous reservoir (see Fig. 8.2) and (c) the combined
scenario (see Fig. 8.7). (d) shows a convergence analysis of the multiple reservoir case
regarding the order of the series expansion applied to the time-discrete Liouvillian
operator, with the inset showing a zoom-in on the long term dynamics.
at time discretizations ∆t={0.3 ps,0.15 ps,0.075 ps}are in good agreement. Since the
two individual algorithms for the time-discrete and continuous structured reservoir cou-
pling both converge with respect to the time evolution step size, the combined setup in the
quasi-2D tensor network can be expected to converge as well, since the involved time scales
remain unchanged. In Fig. 8.9(c), we calculate the emitter occupation dynamics for the
twofold reservoir case and varying time discretizations ∆t={0.3 ps,0.24 ps}, correspond-
ing to nd={4,5}time-discrete memory bins. Restricted by the accessible memory depths
of both reservoirs and therefore the size of the time discretization at a given feedback time
τ, the convergence analysis with respect to the step size is limited to t= 10 ps, where good
qualitative agreement can be observed. Finally, Fig. 8.9(d) shows a convergence analysis
with respect to the series expansion order of the time-discrete Liouvillian operator [see
Eq. (8.13)]. A small deviation between the series expansion orders eight and nine can
be observed on a small scale (blue and green lines, see inset). However, comparing the
dynamics corresponding to ninth and tenth order expansions of the Liouvillian exhibits
excellent agreement, thus confirming the numerical convergence of the employed tenth
order series expansion.
116 8 Dissipative coherent quantum feedback in Liouville space
8.7 Conclusion
We have derived an MPS-based tensor network algorithm for the simulation of a time-
discrete quantum memory based in Liouville space. Combining this method with a tensor
network implementation of real-time path integrals for continuous structured reservoirs,
we have established a novel quasi-2D tensor network architecture to enable simulations
of quantum systems coupled to two non-Markovian environments with both diagonal and
off-diagonal system-reservoir interactions and maintaining relevant system-reservoir and
inter-reservoir entanglement. Arising correlations in between the structured reservoirs
result in an intensive scaling of correlations with respect to the single reservoir setups. As
a result, accessible memory depths are restricted to nc+nd<20 in the here considered
scenario. However, we stress that numerical convergence can be achieved in a wide range
of conceivable setups, e.g., by tuning of the respective system and reservoir time scales. To
further improve numerical efficiency with respect to memory scaling, it may be practical
to trace out the time-discrete memory bins after their interaction with the system state
via the Liouvillian operator. This treatment would allow for increased feedback times τ
accompanied by smaller accessible time discretizations ∆twithout altering the qualitative
physics. The here established quasi-2D tensor network algorithm provides a numerically
exact tool to gain access to the as of yet mostly unexplored field of multiple interacting
non-Markovian environments.
As a first application, we have shown that the dynamical interaction and correlation
buildup in between a structured phonon reservoir and time-discrete coherent photon feed-
back can result in a dynamical protection against destructive interference effects via time-
delayed backflow of coherence, enabling the formation of dynamical population trapping.
The formation of inter-reservoir correlations can be tuned via the relevant time scales of
the non-Markovian system-reservoir interactions. The presented methods and results have
potential applications in the fields of nonequilibrium physics, quantum thermodynamics
and dynamical quantum phase transitions with ergodic, entropic or negentropic infor-
mation exchange, where considering dynamical dissipative structures may unravel new
phenomena. In future investigations, the presented quasi-2D network architecture may
be further advanced to a true 2D implementation using projected entangled pair states
with combined reservoir memory bins to enable simulations of multi-level systems and
improving numerical performance.
Part III
Information compression in open
quantum systems via artificial
neural networks
117
9 Sampling asymmetric open quantum
systems for artificial neural networks
In this Chapter, we extend the capabilities of the restricted Boltzmann machine neural
network architecture using novel Hilbert space sampling techniques. Established artificial
neural network methods based on the restricted Boltzmann machine and Metropolis sam-
pling of configuration space have been shown to enable precise estimations of symmetric
open spin-1/2quantum systems. However, for systems without symmetries of translational
invariance these techniques lead to systematic errors and poor scalability, independent of
network parameters such as the sample size. To resolve this representational limit, we first
propose a adjustment to the standard Metropolis algorithm, leading to enhanced conver-
gence of the neural network. Secondly, we introduce a novel hybrid sampling strategy
where asymmetric system properties are taken explicitly into account, resulting in high
scalability and fast convergence times for asymmetric open spin-1/2systems. The Chap-
ter is closely based on the publication “Sampling asymmetric open quantum systems for
artificial neural networks” by O. Kästle and A. Carmele [72].
9.1 Introduction
In recent breakthroughs, artificial neural networks have been successfully employed for
the description of Markovian symmetric open spin-1/2quantum systems [47–50,52–54].
These network architectures offer a nearly limitless potential for numerical parallelization
and can be iteratively optimized by application of a variational principle, giving direct
access to the stationary state (see Ch. 5) [113,114]. Combined with efficient compression
of Hilbert space by application of the Metropolis algorithm where possible system configu-
rations are sampled in a Markov chain Monte Carlo method, high-performing simulations
of large symmetric systems have been fathomed [46,104–106]. In particular, the restricted
Boltzmann machine neural network architecture has emerged as a natural and highly ef-
ficient mapping of the density matrix of spin-1/2quantum systems, since it allows for a
one-to-one mapping of individual spins to artificial neurons while drastically reducing the
required degrees of freedom [47,51]. On its basis, simulations of large symmetric and pe-
riodic open spin-1/2systems have been realized, including calculations for both stationary
states [56,57] and real-time evolution dynamics (see Ch. 5) [58,59].
In this Chapter, we demonstrate that established configuration space sampling procedures
are uneligible for the training of restricted Boltzmann machines constituting asymmetric
open spin-1/2quantum systems, i.e., setups without symmetries of translational invari-
ance. As an example application, we investigate the stationary state of a boundary-driven
119
120 9 Sampling asymmetric open quantum systems for artificial neural networks
isotropic Heisenberg spin chain with open boundary conditions, featuring asymmetric
properties [288,289,307–321]. As a first step, we provide converging neural network
results for a symmetric Heisenberg chain and propose an adjustment to the standard
Metropolis algorithm for the sampling of system configurations, further improving conver-
gence and computational efficiency. Afterwards, we demonstrate that common Metropolis
techniques for Hilbert space sampling result in a systematic overestimation of asymmetric
system properties, independent of neural network parameters such as sample size or learn-
ing rate. To overcome this representational limitation, we introduce a hybrid sampling
strategy to explicitly account for asymmetric system properties during Hilbert space com-
pression. To this end, we combine the accuracy of exact Hilbert space mapping for sites
with asymmetric properties with the performance advantage of regular sampling for the
remaining subsystem. The presented sampling methods establish a powerful new ansatz
for the optimization and training of artificial neural networks while maintaining a high
scalability potential and fast convergence times at decreased noise.
The incoherently driven isotropic Heisenberg chain is introduced in Sec. 9.2. In Sec. 9.3,
we discuss the implementation and training procedure of the employed neural network.
Afterwards, we present an adjustment to the regular Metropolis algorithm in Sec. 9.4 and
demonstrate resulting improvements in convergence using the example of a symmetric
Heisenberg chain setup. In Sec. 9.5, we demonstrate that regular sampling strategies are
unsuitable for the description of systems with asymmetric properties. As a solution, a
hybrid sampling method is presented in Sec. 9.6, enabling accurate neural network simu-
lations of large asymmetric systems. Lastly, we summarize our findings in Sec. 9.7.
9.2 Incoherently driven isotropic Heisenberg chain
To investigate the representational power of the restricted Boltzmann machine for sys-
tems without symmetries of translational invariance, we consider the incoherently driven
isotropic Heisenberg chain shown schematically in Fig. 9.1, featuring Nspin-1/2systems
with next-neighbor coupling at amplitude Jand open boundary conditions. All sites iof
the chain are subjected to incoherent driving at rates γi
in and dissipation at rates γi
out.
The corresponding Hamiltonian reads [288,289,307–321]
H/ℏ=J
4
N−1
X
i=1 σx
iσx
i+1 +σy
iσy
i+1 +σz
iσz
i+1,(9.1)
where σidenote Pauli spin matrices acting on individual sites i. To account for incoherent
excitation and decay, we introduce the Lindblad dissipators [1,268]
Dqγi
out/2σ−
iρ(t) = γi
out
2h2σ−
iρ(t)σ+
i−{σ+
iσ−
i, ρ(t)}i,(9.2a)
Dqγi
in/2σ+
iρ(t) = γi
in
2h2σ+
iρ(t)σ−
i−{σ−
iσ+
i, ρ(t)}i.(9.2b)
To incorporate asymmetric properties, a boundary-driven scenario with increased driving
on the first chain site, γ1
in > γ1
out (red shape in Fig. 9.1), increased decay on the right
9.3 Neural network implementation 121
Figure 9.1: Schematic of a boundary-driven isotropic Heisenberg chain with increased driving on
the left boundary, γ1
in > γ1
out (red shape), increased decay on the right boundary,
γN
out > γN
in (dark blue shape), and equal driving and dissipation γi
in =γi
out in the bulk
(light blue shape).
boundary, γN
out > γN
in (dark blue shape), and equal driving and dissipation γi
in =γi
out in
the bulk (light blue shape) is assumed. In total, the resulting time evolution dynamics in
Liouville space is prescribed by
˙ρ(t) = Lρ(t) = −i[H/ℏ, ρ(t)] +
N
X
i=1 Dqγi
out/2σ−
i+Dqγi
in/2σ+
iρ(t),(9.3)
and can be solved, e.g., in a common fourth order Runge Kutta approach where system
sizes of typically N≲14 can be accessed without requiring high-performing computational
resources. In the following investigations, we employ such brute force density matrix cal-
culations as a benchmark for the results obtained from the restricted Boltzmann machine
neural network.
9.3 Neural network implementation
For a neural network realization of open spin-1/2quantum systems, the density matrix is
estimated by a probabilistic architecture, replacing 22Ndensity matrix elements ⟨σ|ρ|η⟩=
⟨σ1, . . . , σN|ρ|η1, . . . , ηN⟩of a setup consisting of Nspin-1/2systems by a set of variational
training parameters as degrees of freedom. The restricted Boltzmann machine architecture
has recently emerged as a natural mapping of the density matrix for symmetric open spin-
1/2quantum systems, since it enables a straightforward mapping of spins to artificial
neurons [47,51,56–59,78,79,115,322]. It generates a model distribution called neural
density operator shown schematically in Fig. 9.2, featuring a visible layer of 2Nsites σ
and ηfor the representation of the left and right side of the density matrix, two auxiliary
hidden layers each consisting of Msites hσand hη, respectively, and an ancillary mixing
layer of Ksites hµ. Tracing out the hidden and ancillary degrees of freedom, the neural
density operator elements are given by (see Ch. 5)
ρϑ(σ,η) =
M
Y
m=1
K
Y
k=1
8 exp N
X
i=1
aiσi!exp N
X
i=1
a∗
iηi!cosh bm+
N
X
i=1
Wmiσi!
×cosh b∗
m+
N
X
i=1
W∗
miηi!cosh ck+c∗
k+
N
X
i=1
Ukiσi+
N
X
i=1
U∗
kiηi!,(9.4)
122 9 Sampling asymmetric open quantum systems for artificial neural networks
Figure 9.2: Schematic of a restricted Boltzmann machine constituting the neural density operator
with variational parameters ϑ= (a,b,c,W,U)as degrees of freedom, consisting of a
visible layer (blue), two auxiliary hidden layers (green) and an ancillary mixing layer
(red) connecting the left and right side.
depending on a set of complex training parameters ϑ= (a,b,c,W,U)divided into real
and imaginary parts, yielding a total of np= 2N+ 2M+K+ 2MN + 2KN elements
representing the neural network degrees of freedom. The training parameters consist of
biases afor visible neurons, bfor auxiliary hidden sites and cfor the ancillary mixing
layer, respectively. The complex weights Wconnect visible neurons σand ηto their
auxiliary hidden counterparts hσand hη, whereas weights Ucouple visible units to the
ancillary mixing layer hµ(see Fig. 9.2).
The training objective is to estimate the unknown density matrix ρby the neural density
operator ρϑvia iterative variation of the parameter set ϑ. During each training iteration,
input data is generated by drawing Nssamples of possible system configurations from
Hilbert space. Using the stochastic reconfiguration approach [117–119], occurrence proba-
bilities and system observables are approximated as statistical expectation values over all
drawn sample configurations. The normalized occurrence probability of the n-th drawn
sample (σn,ηn)is obtained via the estimated neural density operator partition function
by summing over all Nssamples drawn during one iteration, resulting in the probability
˜pϑ(σn,ηn) = |ρϑ(σn,ηn)|2
PNs
n=1 |ρϑ(σn,ηn)|2.(9.5)
In the same fashion, expectation values of diagonal observables are approximated as sta-
tistical averages [57–59,117–119]
⟨X(σ,σ)⟩ ≈ ⟨⟨X(σ,σ)⟩⟩q=
Ns
X
n=1
˜qϑ(σn)X
ξ
X(σn,ξ)ρϑ(ξ,σn)
ρϑ(σn,σn),(9.6)
where we have introduced a probability distribution for diagonal samples
˜qϑ(σn) = ρϑ(σn,σn)
PNs
n=1 ρϑ(σn,σn).(9.7)
9.4 Symmetric systems: Improving sampling efficiency 123
Since only diagonal observables are figures of merit in the here considered setup, ˜qϑ(σ)
is employed to calculate diagonal expectation values for further improved numerical effi-
ciency, while ˜pϑ(σ,η)is used for the training of the network.
The training goal is to estimate the stationary state of the considered model system, pre-
scribed by the condition ˙ρ(t) = Lρ(t)=0. For the variational optimization of the network,
we define a corresponding cost function C(ϑ) = ∥Lρϑ∥2
2[57,59]. During each training
iteration t→t+1, the variational parameters ϑconstituting the degrees of freedom of the
neural density operator are optimized and updated via the standard stochastic gradient
descent procedure, obeying the rule
ϑ(t+1)
l=ϑ(t)
l−ν∇ϑlC(ϑ(t)),(9.8)
with νthe learning rate [46]. During the initialization of the network, the variational
parameters are first set to small nonzero random values, ϑ(0)
l∈[−0.01,0.01]\{0}. The
corresponding cost function gradient is given by (see Ch. 5) [57]
∇ϑlC(ϑ)
= 2Re(Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)
Ns
X
m=1 L(σn,ηn,σm,ηm)ρϑ(σm,ηm)
ρϑ(σn,ηn)Oϑl(σm,ηm)
−"Ns
X
n=1
˜pϑ(σn,ηn)Oϑl(σn,ηn)#"Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)˜
L(σn,ηn)#),(9.9)
where we have defined logarithmic derivatives as diagonal matrices with elements
[Oϑl]σnηn,σnηn=Oϑl(σn,ηn) = ∂[ln ρϑ(σn,ηn)]
∂ϑl
,(9.10)
corresponding to the gradients of the neural density operator with respect to all lelements
of the variational parameter vector ϑand for a given sample configuration (σn,ηn). Fur-
thermore, we have introduced the estimator of the Liouvillian, reading
˜
L(σn,ηn) := X
σm,ηm
L(σn,ηn,σm,ηm)ρϑ(σm,ηm)
ρϑ(σn,ηn).(9.11)
9.4 Symmetric systems: Improving sampling efficiency
To underline the representational capabilities of the neural network approach for sys-
tems with symmetries of translational invariance, we start with the symmetric isotropic
Heisenberg chain with open boundary conditions, featuring identical incoherent driving
and dissipation rates γi
in = 1.05γi
out on all sites i. In recent studies, similar scenarios
with periodic boundary conditions, coherent driving and symmetric dissipation have been
successfully realized using the restricted Boltzmann machine architecture, rendering sim-
ulations of large systems with N≥15 feasible [57–59]. For these system sizes, an exact
mapping of the density matrix becomes increasingly expensive due to the exponential
124 9 Sampling asymmetric open quantum systems for artificial neural networks
0.48
0.5
0.52
0 200Iteration
0
0.02
0.04
0.06
0 200
Figure 9.3: Neural network estimation for the stationary state of the symmetric Heisenberg chain
of length N= 10, calculated via regular Metropolis sampling (grey lines) and the
adjusted accept-only sampling method (colored lines). Orange and blue lines show the
stationary ground and excited state populations, which are identical on all sites in
the symmetric case. For improved visibility, ground and excited state populations are
averaged over all sites. Benchmark results are indicated by dashed black lines. The
inset shows the stationary state magnetization mz.
growth of the Hilbert space dimension. In established approaches, the Metropolis algo-
rithm is employed for an efficient compression of configuration space (see Ch. 5) [104].
It constitutes a Markov chain Monte Carlo method where Nssystem configurations are
drawn as samples, corresponding to a random walk in Hilbert space [46,105,106]. A new
sample configuration is drawn based on a set of selection rules imposed on the current
sample and accepted or rejected at a certain acceptance probability. Here, we set the
selection rule to flip the spin of each site at a probability of 50%. Moreover, we employ a
linear acceptance probability (see Ch. 5)
A(n+ 1, n) = min 1,˜pϑ(σn+1,ηn+1)
˜pϑ(σn,ηn),(9.12)
with (σn,ηn)and (σn+1,ηn+1)referring to the current and proposed sample, respectively.
In case the proposed sample is rejected by the acceptance function, the standard Metropolis
algorithm proceeds with the current sample. This strategy leads to convergent results for
the case of symmetric systems and which be further improved for increasing sample sizes
Ns, as demonstrated in the following.
As a first numerical test, the default Metropolis sampling method is employed. The re-
sulting neural network estimations are compared to benchmarks obtained from a common
Runge Kutta master equation implementation. Calculations are performed for system
parameters N= 10,γin = 0.21,γout = 0.20 and J= 2γin. For the neural network,
we choose to draw Ns= 20000 samples per iteration and employ hidden layer densities
M/N =K/N = 1 at a learning rate ν= 0.1. Using these parameters, the restricted
Boltzmann machine compresses the system information onto np= 450 variational param-
eters compared to the ≈500000 density matrix elements which must be calculated in a
brute force approach. Grey lines in Fig. 9.3 depict neural network results for the mean
9.5 Asymmetric systems: Systematic sampling errors 125
stationary ground and excited state populations using the regular Metropolis sampling
algorithm. Since all sites of the chain take on the same steady state occupation in the
symmetric scenario, the depicted results are averaged over all N= 10 sites for improved
visibility. The inset shows the stationary state magnetization mz= (1/N)PN
i=1 ⟨σz
i⟩of the
symmetric Heisenberg chain. Benchmark populations are indicated by dashed black lines
and exhibit good agreement with the results obtained from regular Metropolis sampling.
It is noted that the sample size Nscan be further increased to improve convergence and
reduce noise.
As a first improvement to the standard Metropolis approach, we propose an adjusted sam-
pling method breaking the detailed balance condition in the Markov chain and resulting
in a net stochastic flow and suppressed random walk behavior in Hilbert space [323]. In
consequence, we achieve a considerable improvement of convergence and noise. Rather
than proceeding with the current sample configuration in case of a rejection from the
acceptance function [Eq. (9.12)], we propose an accept-only strategy: The procedure of
drawing new sample configurations (σn+1,ηn+1)is repeated until one of them is accepted
as the new current sample by the acceptance function. In consequence, no sample is used
more than once in a row unless it is chosen by the selection rule at a probability of 1/2N,
strongly reducing autocorrelations in the Markov chain. Colored lines in Fig. 9.3 depict
corresponding neural network results for the symmetric Heisenberg chain obtained via the
accept-only sampling strategy. Again, the stationary ground and excited states of all sites
have been averaged over for improved visibility (orange and blue lines). The red line in
the inset shows the corresponding magnetization mz. Compared to the regular Metropolis
sampling method (grey lines), the adjusted approach results in smoother lineshapes with
fewer kinks and a considerably decreased standard deviation at the same sample size Ns,
even more so when considering increasing chain lengths N. After 100 iterations, the maxi-
mum relative deviation of the results obtained from regular sampling from the benchmark
is 3.35% compared to 0.99% for the accept-only strategy, corresponding to an improvement
ratio of 3.35/0.99 ≈3.38.
9.5 Asymmetric systems: Systematic sampling errors
Next, we investigate the asymmetric Heisenberg chain at incoherent driving and decay
parameters γi
in =γi
out = 0.20 on all bulk sites i,γ1
in = 0.21,γ1
out = 0.20 on the left
boundary (red shape in Fig. 9.1) and γN
in = 0.20,γN
out = 0.21 on the right boundary (dark
blue shape in Fig. 9.1), constituting a boundary-driven isotropic chain without symmetries
of translational invariance. All other parameters remain unchanged with respect to the
previously discussed symmetric setup. To investigate the representational power of the
restricted Boltzmann machine for asymmetric systems, we start at a chain length N= 6
where an exact mapping of Hilbert space is straightforward: Instead of applying a selective
sampling method, the neural network takes all possible system configurations into account
to minimize the cost function. The resulting stationary state populations of the boundary
and bulk sites of the asymmetric Heisenberg chain are shown in Fig. 9.4 (colored lines),
perfectly matching their respective benchmark values (dashed grey lines). This proves that
126 9 Sampling asymmetric open quantum systems for artificial neural networks
0.48
0.5
0.52
0 400Iteration
Figure 9.4: Stationary ground state populations for an asymmetric boundary-driven Heisenberg
chain of length N= 6, calculated via an exact Hilbert space mapping. Dashed grey
lines depict benchmark occupations. Orange and dark blue lines show the ground state
populations of the left and right boundary sites.
the restricted Boltzmann machine architecture is not inherently limited to the description
of symmetric setups and manages to accurately represent asymmetric systems in case of
an exact Hilbert space mapping.
However, the numerical performance is drastically decreased when regular sampling is ap-
plied, as shown in Fig. 9.5: For improved visibility, the unfolding stationary ground state
populations (grey lines) are averaged over 50 iterations (solid colored lines) and we have
taken the mean values of all bulk site populations as well (solid red line). Dashed grey
lines again depict the corresponding benchmark populations. While the populations of
all sites in the symmetric bulk match their respective Runge Kutta benchmark results,
the stationary state populations of the asymmetric boundary sites exhibit an overshooting
behavior (solid orange and blue lines in Fig. 9.5), occurring independently of neural net-
work parameters such as system and sample sizes Nand Ns, learning rate νand hidden
layer densities M/N and K/N: Dashed colored lines in Fig. 9.5 show comparison results
obtained at Ns= 40000,ν= 0.05,M/N =K/N = 2 and averaged over 50 iterations, ex-
hibiting the same overshooting behavior. Furthermore, this erroneous convergence emerges
for both the standard Metropolis and the previously introduced accept-only sampling al-
gorithm and occurs independently of the employed acceptance function and selection rules
for sampling: The same behavior is still observed when choosing, e.g., an exponential ac-
ceptance function of the form A(n+ 1, n) = exp[−˜pϑ(σn,ηn)/˜pϑ(σn+1,ηn+1)]. Given the
success of training for an exact Hilbert space mapping (see Fig. 9.4), this only leaves the
sampling procedure as the cause of error: Due to the asymmetric properties of the system,
Hilbert space elements with a low occurrence probability can still impose a large effect
on the steady state under certain conditions. In this case, the application of standard
sampling methods established for symmetric systems leads to a systematic error, since
Hilbert space configurations with a higher probability of occurrence are favored per se,
resulting in a selection bias [see Eq. (9.12)]. In conclusion, we have demonstrated that reg-
ular Metropolis sampling methods are unsuitable for the optimization of artificial neural
9.6 Hybrid sampling for asymmetric systems 127
0.48
0.5
0.52
0 1200Iteration
Figure 9.5: Results for an asymmetric chain of length N= 6, using the accept-only sampling
strategy. Grey lines indicate raw results, with colored lines showing their averages over
50 iterations. Orange and blue lines depict the boundary site ground state populations.
Occupations of all bulk sites are also averaged over for improved visibility (red line).
Dashed colored lines depict averaged comparison results obtained at half the learning
rate and doubled hidden layer densities and samples per iteration.
networks describing systems with asymmetric properties. To resolve this representational
limitation, in the following we present a novel hybrid sampling approach to explicitly incor-
porate system asymmetries, allowing for an efficient and accurate compression of Hilbert
space for asymmetric open spin-1/2quantum systems.
9.6 Hybrid sampling for asymmetric systems
To resolve the systematic error occurring during the sampling of asymmetric systems, we
introduce a hybrid sampling strategy, combining the precision of an exact configuration
space mapping (see Fig. 9.4) with selective sampling to facilitate computational access
to large systems. In the here considered boundary-driven isotropic Heisenberg chain, the
asymmetric system properties are located at the boundary sites. The core concept of
the proposed hybrid sampling method is to employ a regular Metropolis sampling ap-
proach to the symmetric bulk subsystem, while manually selecting the configurations of
the asymmetric boundary sites during the sampling procedure: In analogy to the standard
sampling algorithm, a new proposed sample (σn+1,ηn+1)is drawn first. To prevent the
acceptance function from making arbitrary choices, the occurrence probability of the bulk
site configuration, i.e., the symmetric subsystem of the chain, is calculated independently
of the asymmetric boundary sites. To this end, we determine the mean probability of a
given bulk configuration by averaging over all 16 possible boundary site configurations.
Afterwards, the edge sites of the new proposed sample are manually set to one of the
16 possible settings. Once 16 different accepted bulk configurations have been applied,
one cycle of all possible boundary configurations has been finished. It is noted that in the
case of sampling diagonal configurations (σn,σn), only four possible boundary settings are
varied over. This procedure is repeated for all Nssamples drawn during each iteration.
128 9 Sampling asymmetric open quantum systems for artificial neural networks
0.48
0.5
0.52
0 500Iteration
Figure 9.6: Stationary ground state populations of the boundary-driven asymmetric Heisenberg
chain, calculated via the hybrid sampling approach for a chain of length N= 10.
Orange and blue lines show the left and right boundary site populations. For improved
visibility, ground state populations of all bulk sites are averaged over (red line, see
Fig. 9.7). Dashed grey lines show corresponding benchmark values. Grey lines show
comparison results obtained via regular Metropolis sampling.
This hybrid sampling approach combines the accuracy of exact Hilbert space mapping
for asymmetric system parts with the high performance of selective Metropolis sampling
for the remaining symmetric subsystem, counteracting the systematic overestimation of
asymmetric properties during Hilbert space compression. It is noted that the mapping of
the edge sites is still approximate, since we do not go through all 16 possible boundary
setting for a single bulk configuration. Doing so would result in severely decreased bulk
convergence, unless the sample size is drastically increased. The quality of convergence
resulting from the hybrid sampling approach is limited only by the choice of neural net-
work parameters. In Fig. 9.6, we show the stationary ground state populations of an
asymmetric Heisenberg chain of length N= 10, calculated using Ns= 40000 samples
per iteration and at learning rate ν= 0.05. The remaining system and network param-
eters are unchanged with respect to Fig. 9.5. Strikingly, in difference to the calculations
obtained via regular Metropolis sampling (solid grey lines), the stationary boundary site
occupations (orange and blue lines) are in excellent agreement with their corresponding
benchmark results (dashed grey lines). The applied hybrid sampling strategy prohibits the
previously observed overshooting behavior by explicitly accounting for asymmetric system
properties. At the same time fluctuations and noise are drastically reduced, yielding a
maximum relative deviation of 0.45% between the boundary site occupations and their
respective benchmark values after 300 iterations.
Again, all bulk site occupations in Fig. 9.6 have been averaged over for improved visibil-
ity (red line), since they are located very closely to each other. As a result, the varying
neural network estimations of the bulk site populations overlap significantly, making a
visual distinction difficult. However, they do converge to their respective benchmark oc-
cupations within the margin of error: As an example to demonstrate the convergence of
the symmetric bulk subsystem, Fig. 9.7 shows all ground state occupations depicted in
9.6 Hybrid sampling for asymmetric systems 129
0.49
0.5
0.51
0 500Iteration
site 1
0.49
0.5
0.51
0 500Iteration
site 2
0.49
0.5
0.51
0 500Iteration
site 3
0.49
0.5
0.51
0 500Iteration
site 4
0.49
0.5
0.51
0 500Iteration
site 5
0.49
0.5
0.51
0 500Iteration
site 6
0.49
0.5
0.51
0 500Iteration
site 7
0.49
0.5
0.51
0 500Iteration
site 8
0.49
0.5
0.51
0 500Iteration
site 9 0.49
0.5
0.51
0 500Iteration
site 10
Figure 9.7: Unaveraged ground state occupations of all N= 10 sites in the asymmetric chain
scenario and using hybrid sampling corresponding to Fig. 9.6, explicitly including the
unaveraged stationary ground state occupations of all eight bulk sites.
Fig. 9.6 individually for each site (colored lines), including the unaveraged ground state
populations of all eight bulk sites, exhibiting excellent agreement with their corresponding
benchmarks (dashed grey lines). Again, the observed oscillations over iterations can be
further reduced by increasing the sample size Ns. Lastly, we demonstrate the high perfor-
mance and scalability potential of the presented hybrid sampling approach. Fig. 9.8 shows
the stationary ground state populations of an asymmetric Heisenberg chain with N= 16
sites at neural network parameters Ns= 40000,ν= 0.05 (grey lines) and ν= 0.025
(colored lines). Bulk site occupations are again averaged over for improved visibility (red
lines). As a guide for the eye, dashed grey lines indicate Runge Kutta benchmark results
for the case N= 10. As expected, due to decreasing asymmetric system properties at
unmodified incoherent driving and decay rates, for increasing chain lengths Nthe bound-
ary site occupations begin to strive towards the bulk occupation values. The boundary
site populations (orange and blue lines) are once more in excellent agreement with their
respective benchmark results after approximately 100 training iterations, underlining the
capabilities of the presented hybrid sampling approach.
130 9 Sampling asymmetric open quantum systems for artificial neural networks
0.48
0.5
0.52
0 140Iteration
Figure 9.8: Stationary ground state populations of the boundary-driven asymmetric Heisenberg
chain, calculated via the hybrid sampling approach for a chain of length N= 16 and
network parameters Ns= 40000 and ν= 0.025. Orange and blue lines show the left
and right boundary site populations, with the ground state populations of all bulk sites
averaged over for improved visibility (red line). Dashed grey lines depict benchmark
occupations. Grey lines show comparison results at a doubled learning rate ν= 0.05.
9.7 Conclusion
We have investigated the representational limits of established neural network sampling
strategies. Based on the example of the boundary-driven isotropic Heisenberg chain with-
out symmetries of translational invariance, we have shown that regular Metropolis-based
compression approaches result in a systematic overestimation of asymmetric system prop-
erties during the selective Hilbert space sampling. To improve overall convergence and
efficiency in the limit of low sample sizes, we have first introduced an accept-only strategy
as an adjustment to the regular Metropolis algorithm, explicitly breaking the detailed
balance condition in the Markov chain and suppressing the random walk behavior in
configuration space during sampling. Afterwards, we have presented a hybrid sampling
algorithm to combine the precision of exact Hilbert space mapping for asymmetric proper-
ties with high performing information compression via selective sampling of the remaining
symmetric subsystem. As a result, we have achieved fast converging results with reduced
noise even for large asymmetric open spin-1/2systems. These adaptive sampling strate-
gies represent a novel access point for the optimization and individual tailoring of artificial
neural networks for a wide variety of open quantum systems, further improving upon their
accuracy and performance. In conclusion, we have expanded the representational power of
the restricted Boltzmann machine architecture beyond purely symmetric open quantum
systems, further advancing its versatility and applicability in quantum simulations.
10 Efficient bit encoding of neural
networks for Fock states
In this Chapter, we derive a bit encoding approach for a highly scalable and accurate
representation of bosonic Fock number states in the restricted Boltzmann machine neu-
ral network architecture, for the first time expanding its representational power beyond
the description of pure spin-1/2open quantum systems. In regular master equation im-
plementations of bosonic open systems, the number of degrees of freedom of the density
matrix scales with the predetermined maximum boson number. In contrast, the complex-
ity of the here presented neural network implementation rises only with the number of
bit-encoded neurons, resulting in a vast degree of information compression. In the high
boson number regime, the resulting compression efficiency is demonstrated to outperform
even maximally optimized density matrix implementations based on a projector method to
access the sparsest available representation of configuration space. The Chapter is closely
based on the publication “Efficient bit encoding of neural networks for Fock states” by O.
Kästle and A. Carmele [73].
10.1 Introduction
As discussed in previous Chapters, the restricted Boltzmann machine has emerged as a
default neural network architecture for the description of quantum states [47–50,52–55]
and open spin-1/2quantum systems with Markovian dynamics [56–59], since it facilitates
a direct and highly efficient mapping of spin-1/2systems to artificial neurons. In the
last Chapter, we have extended the established representational limits of the restricted
Boltzmann machine beyond symmetric and periodic spin systems to asymmetric setups
by introducing adaptive strategies for the sampling of configuration space [72].
In this Chapter, the applicability of the restricted Boltzmann machine is further advanced
to enable simulations of hybrid quantum systems featuring bosonic degrees of freedom.
This is achieved by application of a bit encoding scheme to the visible neuron layer, allow-
ing for a direct mapping of Fock number states to the neural network degrees of freedom in
analogy to the established one-to-one assignment of spin-1/2systems to artificial neurons
and without modification of the underlying restricted Boltzmann machine architecture it-
self. In the high boson number regime, the resulting bit-encoded neural network is shown
to accomplish vast degrees of information compression, surpassing even a maximally op-
timized density matrix implementation based on a fourth order Runge Kutta algorithm
where a projector method is employed to attain the sparsest representation of Hilbert space
available. We demonstrate both the accuracy and scalability potential of the proposed
131
132 10 Efficient bit encoding of neural networks for Fock states
neural encoding of Fock states. To this end, we first present neural network calculations
of the stationary boson number statistics for a generic one-atom laser model [24,324–
327] in comparison to benchmark results obtained in a common density matrix approach.
Afterwards, the method is benchmarked in the large boson number regime where the bit-
encoded neural network achieves supremely efficient information compression, underlining
the scalability potential of the approach. Applications of the presented technique include,
e.g., neural network implementations of boson sampling algorithms [328,329] or of recent
efforts to measure quantum coherences via Fock state superposition [330].
The proposed bit encoding scheme for Fock states is introduced in Sec. 10.2, allowing
for a direct mapping of boson occupation numbers to the visible neuron layer of the
restricted Boltzmann machine. In Sec. 10.3, we briefly recapitulate the realization and
training of the bit-encoded neural network. Afterwards, we discuss the attainable degrees
of information compression compared to both a regular and a maximally optimized master
equation implementation with respect to the required dimension of the Fock state basis
in Sec. 10.4. As an example application, we specifically investigate the scaling of required
degrees of freedom in a one-atom laser model featuring a sparse Hilbert space in stationary
state. Using a Heisenberg projection method, the latter can be truncated for maximum
efficiency. In Sec. 10.5, the accuracy of the bit-encoded neural network is demonstrated
before confirming its scalability potential in Sec. 10.6. Lastly, we conclude our investigation
and summarize our findings in Sec. 10.7.
10.2 Neural encoding of Fock states
Artificial neural networks consist of binary neurons, i.e., each neuron is configured in
one of two possible settings +1 or −1. In case of the restricted Boltzmann machine
architecture, this allows for a straightforward mapping of spin-1/2systems, enabling a
natural representation of the corresponding system density matrix [47,51,56–59,72,78,
79,107,115,322]. When considering a system of Nspins σn, ηn={−1,1}, the 22N
elements of the density matrix ⟨σ1, . . . , σN|ρ|η1, . . . , ηN⟩are stated in terms of a model
distribution called neural density operator, which is trained to estimate the stationary
state by iterative variation of a set of neural network parameters (see Ch. 5). In order to
unleash the full potential of the restricted Boltzmann machine architecture as a universal
tool for the description of Markovian open quantum systems, in the following we develop a
supremely efficient and highly scalable mapping of Fock number states via a bit encoding
scheme [44,45]:
Specifically, we derive a neural network representation of a quantum system comprising
Nspin-1/2systems and a single bosonic mode, corresponding to density matrix elements
⟨σ1, . . . , σN;nσ
β|ρ|η1, . . . , ηN;nη
β⟩where σn, ηn={−1,1}are the left and right spin con-
figurations and nσ
β, nη
β∈N0denote the left and right Fock state occupation. The boson
number populations nσ
β,nη
βare each encoded in Nβbits (βσ
1, . . . , βσ
Nβ)and (βη
1, . . . , βη
Nβ)
10.2 Neural encoding of Fock states 133
Figure 10.1: Restricted Boltzmann machine constituting the neural density operator, with a visi-
ble neuron layer comprising Nspin-1/2systems (orange) and the bit-encoded boson
number occupation via Nβneurons (blue). In addition, it features two hidden layers
(green) and an ancillary mixing layer (red). A set of variational training parameters
ϑ= (a,b,c,W,U)represents the neural network degrees of freedom.
via the decomposition rule
nβ=
Nβ
X
i=1
2i−1δβi,1,(10.1)
facilitating the representation of nβ={0,1,...,2Nβ−1}indistinguishable bosons on each
side of the density matrix. This bit representation allows for a direct mapping of the Fock
state basis onto the visible neuron layer of the restricted Boltzmann machine, analogous
to the assignment of spin-1/2systems and without further modifications to the underlying
neural network structure. Here, the maximum representable boson occupation number is
determined by the number of bits. When employing, e.g., a total of Nβ= 4 artificial
neurons as bits, 24possible Fock state configurations can be depicted by the resulting
network, with the bit-encoded Fock state number given by
nβ= 20δβ1,1+ 21δβ2,1+ 22δβ3,1+ 23δβ4,1.(10.2)
Fig. 10.1 depicts the unfolding bit-encoded restricted Boltzmann machine, consisting of a
visible layer of 2(N+Nβ)sites σ= (σ1, . . . , σN, βσ
1, . . . , βσ
Nβ)and η= (η1, . . . , ηN, βη
1,...,
βη
Nβ)storing the configuration of the left and right side of the density matrix and split up
into Nneurons each representing a spin-1/2system (orange) and Nβneurons employed as
bits to encode the boson number occupation (blue). Moreover, the network includes two
auxiliary hidden layers, each featuring Mneurons hσand hη, respectively (green shapes),
and an ancillary mixing layer of Ksites hµlinking the left and right side of the density
matrix (red shapes). The hidden and ancillary neural network layers are once more traced
134 10 Efficient bit encoding of neural networks for Fock states
out (see Ch. 5), resulting in neural density operator elements [57–59,72,78]
ρϑ(σ,η) =
M
Y
m=1
K
Y
k=1
8 exp "N
X
i=1
(aiσi+a∗
iηi) +
N+Nβ
X
i=N+1 aiβσ
i−N+a∗
iβη
i−N#
×cosh bm+
N
X
i=1
Wmiσi+
N+Nβ
X
i=N+1
Wmiβσ
i−N!cosh b∗
m+
N
X
i=1
W∗
miηi+
N+Nβ
X
i=N+1
W∗
miβη
i−N!
×cosh "ck+c∗
k+
N
X
i=1
(Ukiσi+U∗
kiηi) +
N+Nβ
X
i=N+1 Ukiβσ
i−N+U∗
kiβη
i−N#,(10.3)
with ϑ= (a,b,c,W,U)denoting the set of complex training parameters which constitute
the neural network degrees of freedom. They are split up into real and imaginary parts,
resulting in a total of 2(N+Nβ)+2M+K+ 2M(N+Nβ)+2K(N+Nβ)elements, and
consist of biases afor visible neurons, bfor hidden sites and cfor the mixing layer, and of
complex weights Wand Ulinking the visible sites (σ,η)to the hidden neurons hσ,hη
and the ancillary mixing layer hµ, respectively [see Fig. 10.1].
10.3 Training procedure
In common master equation implementations of bosonic and hybrid quantum systems, the
exponential growth of the Hilbert space dimension for rising system sizes renders an exact
mapping of the density matrix increasingly expensive in the limit of large boson numbers.
As discussed in previous Chapters, the neural network ansatz for open quantum systems
solves this problem by estimating the density matrix ρby the neural density operator ρϑ
[Eq. (10.3)] via iterative training of variational parameters ϑ. For the following imple-
mentation, we employ the standard Metropolis algorithm for the sampling of Nselements
of configuration space, where a new sample (σ,η)=(σ1, . . . , σN, βσ
1, . . . , βσ
Nβ;η1, . . . , ηN,
βη
1, . . . , βη
Nβ)is proposed based on the current system configuration and either accepted or
rejected at a certain acceptance probability (see Ch. 5) [104].
When considering a setup comprising a number of spin-1/2quantum systems interacting
with a bosonic mode, the amount of nonzero density matrix elements is severely restricted
by the properties of the spin-boson interaction. As a result, in these scenarios the unfolding
steady state density matrix is often times highly sparse. Since we are only interested
in the stationary system state, we make use of this condition to improve the efficiency
and accuracy of the Hilbert space sampling by only drawing system configurations from
the subspace of nonzero stationary state density matrix elements. Hence, during the
proposition of a new sample configuration we employ a random selection rule where the
left and right setting of each spin σ1, . . . , σN, η1, . . . , ηNis flipped at a probability of
50% each. Based on the new spin configuration, a new random and bit-encoded Fock
number configuration βσ
1, . . . , βσ
Nβ, βη
1, . . . , βη
Nβis drawn, only allowing for combinations
corresponding to nonzero elements of the stationary density matrix. To determine whether
10.3 Training procedure 135
to accept or reject a newly proposed sample configuration, we employ the linear acceptance
function
A(n+ 1, n) = min 1,˜pϑ(σn+1,ηn+1)
˜pϑ(σn,ηn),(10.4)
where (σn,ηn)denotes the current and (σn+1,ηn+1)the proposed Hilbert space sample.
Again, we employ the stochastic reconfiguration approach [117–119] to estimate the oc-
currence probability of a given sample configuration (σn,ηn)with n={1, . . . , Ns}and
system observables as statistical expectation values over the Nssample configurations
drawn during each training iteration. As a result, the normalized occurrence probability
is given by (see Ch. 5)
˜pϑ(σn,ηn) = |ρϑ(σn,ηn)|2
PNs
n=1 |ρϑ(σn,ηn)|2,(10.5)
and diagonal observables are calculated as statistical averages ⟨X(σ,σ)⟩ ≈ ⟨⟨X(σ,σ)⟩⟩q[57–
59,117–119] with
⟨⟨X(σ,σ)⟩⟩q:=
Ns
X
n=1
˜qϑ(σn)X
ξ
X(σn,ξ)ρϑ(ξ,σn)
ρϑ(σn,σn).(10.6)
Since only diagonal observables are figures of merit in the here considered scenario, the
normalized occurrence probability of diagonal system configurations
˜qϑ(σn) = ρϑ(σn,σn)
PNs
n=1 ρϑ(σn,σn)(10.7)
is used to improve numerical efficiency: During each training iteration, Nsdiagonal system
configurations (σn,σn)are sampled for the calculation of diagonal observables and Ns
general samples (σn,ηn)are drawn to determine ˜pϑ(σ,η)for the optimization of the
neural network.
The training objective is to estimate the stationary state of the considered system, given
by the condition ˙ρ=Lρ= 0, with Ldenoting the Liouvillian superoperator [1,268]. For
the optimization of the variational parameter set ϑtowards this condition, we define a cost
function C(ϑ) = ∥Lρϑ∥2
2(see Ch. 5) [57,59]. The variational parameters are initialized at
small nonzero random values, ϑ(0)
l∈[−0.01,0.01]\{0}and updated during each iteration
t→t+ 1 using the standard stochastic gradient descent approach,
ϑ(t+1)
l=ϑ(t)
l−ν∇ϑlC[ϑ(t)],(10.8)
at learning rate ν[46]. The gradient of the corresponding cost function is again determined
as (see Ch. 5) [57,72]
∇ϑlC(ϑ)
= 2Re(Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)
Ns
X
m=1 L(σn,ηn,σm,ηm)ρϑ(σm,ηm)
ρϑ(σn,ηn)Oϑl(σm,ηm)
−"Ns
X
n=1
˜pϑ(σn,ηn)Oϑl(σn,ηn)#"Ns
X
n=1
˜pϑ(σn,ηn)˜
L†(σn,ηn)˜
L(σn,ηn)#),(10.9)
136 10 Efficient bit encoding of neural networks for Fock states
with the estimator of the Liouvillian
˜
L(σn,ηn):= X
σm,ηmL(σn,ηn,σm,ηm)ρϑ(σm,ηm)
ρϑ(σn,ηn),(10.10)
and logarithmic derivatives written as diagonal matrices with elements
[Oϑl]σnηn,σnηn=Oϑl(σn,ηn) = ∂[ln ρϑ(σn,ηn)]
∂ϑl
,(10.11)
representing the neural density operator gradients with respect to all lelements of ϑand
for a specific sample configuration (σn,ηn).
10.4 Neural network efficiency gain
The core advantage behind the application of the restricted Boltzmann machine approach
to open quantum systems is its high degree of information compression: The 2(N+
Nβ) + 2M+K+ 2M(N+Nβ) + 2K(N+Nβ)variational training parameters stored
in ϑrepresent the neural network degrees of freedom and only depend on the number
of visible neurons N+Nβconstituting the spin-1/2systems and the employed bits to
encode the Fock state basis. In a common master equation implementation of a system
featuring ddegrees of freedom, d(d+ 1)/2density matrix elements must be determined to
solve the system of differential equations ˙
ρ(t) = Lρ(t)until the steady state is attained.
To compare the neural network information compression to these common density matrix
approaches with respect to the efficiency scaling of the Fock state basis dimension, we
investigate the paradigmatic open Jaynes-Cummings model, consisting of a single spin-1/2
system interacting with a bosonic cavity mode, constituting a one-atom laser system [24].
Under the rotating wave and dipole approximation, the respective system Hamiltonian
reads [324–327]
H/ℏ=ω0σ+σ−+ωcc†c+g0σ+c+σ−c†,(10.12)
where c†,care bosonic creation and annihilation operators and σ±denote Pauli spin op-
erators. ω0and ωcrefer to the spin and cavity mode frequencies and g0corresponds to
the coupling strength between system and cavity mode. Moreover, the system is incoher-
ently excited at a driving rate Γand subjected to incoherent decay of the bosonic mode
occupation at rate κin a Markovian Lindblad description. As a result, the time evolution
dynamics for the density operator is given by
˙
ρ(t) = Lρ(t) = −i[H/ℏ,ρ(t)] + Dqκ/2cρ(t) + DqΓ/2σ+ρ(t),(10.13)
with Lindblad dissipators [1,268]
Dqκ/2cρ(t) = κ
2h2cρ(t)c†−{c†c, ρ(t)}i,(10.14a)
DqΓ/2σ+ρ(t) = Γ
2h2σ+ρ(t)σ−−{σ−σ+,ρ(t)}i,(10.14b)
10.4 Neural network efficiency gain 137
0
4
8
12
0 250〈nβ(ts)〉
RK (regular)
RK (optimized)
RBM
400
1200
100 250
Figure 10.2: Required degrees of freedom to achieve numerical convergence with respect to the
average stationary state boson occupation number, depicted on a logarithmic scale.
The restricted Boltzmann machine implementation (solid light blue line) is compared
to both a regular unoptimized master equation approach (solid dark blue line) and a
maximally optimized approach with a truncated configuration space only comprising
nonzero stationary state density matrix elements (dashed dark blue line). The inset
shows a zoom-in on a linear scale, where the neural network implementation becomes
supremely efficient.
resulting in incoherent driving and decay of system and cavity mode, respectively. In the
following calculations, we employ the parameters g0= 0.2 ps−1,Γ = 0.4 ps−1,ω0=ωc
and various bosonic dissipation rates κ. Moreover, we only consider the stationary state
attained at time ts, where the condition ˙
ρ(ts) = Lρ(ts)=0is fulfilled within numerical
precision.
This model is chosen due to the high sparsity of the resulting stationary state density ma-
trix, allowing for the application of a Heisenberg projector method to achieve a profound
grade of optimization for the regular density matrix implementation: Using a Heisenberg
operator, we project the full configuration space onto the relevant subspace consisting of
only 2(d−1) nonzero steady state density matrix elements [1,331]. In the here con-
sidered system, this corresponds to 2(2nmax
β−1) elements with nmax
βdenoting the cho-
sen upper numerical limit for the boson number occupation. While this corresponds to
2nmax
β(2nmax
β+ 1)/2degrees of freedom in the unoptimized master equation approach,
the complexity of the neural network implementation increases with the number of bits
Nβrepresenting the Fock state basis. Here, the maximum boson number occupation is
determined by nmax
β= 2Nβ−1. We note that regardless of the specific implementation
and parameter choices, convergence with respect to the numerical boson limit given by
nmax
βor Nβmust always be ensured.
Fig. 10.2 shows the required degrees of freedom to achieve numerical convergence with
respect to the average steady state boson number occupation ⟨nβ(ts)⟩, which is varied
via tuning of the bosonic dissipation rate κ. The resulting numerical complexity is de-
138 10 Efficient bit encoding of neural networks for Fock states
picted on a logarithmic scale. In case of the neural network implementation, convergence
is achieved for a sufficiently large number of bits Nβ. At this point, statistical noise
can be reduced by increasing the number of sample configurations Ns, further improving
convergence. For the master equation approach, we assume numerical convergence once
an incremental expansion of the Fock state basis yields a relative deviation of less than
0.1% in ⟨nβ(ts)⟩. Moreover, common dynamical Runge Kutta implementations typically
demand increasingly small time steps for increasing system sizes to obtain numerically
convergent results. In addition, the number of density matrix elements for the solution
of Eq. (10.13) scales polynomially with the system size, leading to a polynomial increase
of complexity for increasing stationary Fock state occupations (solid dark blue line in
Fig. 10.2). For a maximally optimized density matrix implementation, we exploit the
sparsity of the stationary state configuration space, reducing its dimension via application
of a Heisenberg projector method. As a result, the increase in complexity is reduced to
scale linearly with the bosonic degrees of freedom (dashed dark blue line in Fig. 10.2). In
the neural network approach, the required degrees of freedom scale only with the number
of visible neurons Nβ+ 1. In our experience, it has proven more successful to increase the
number of bosonic bits Nβrather than the hidden layer densities to achieve improvements
in convergence. In consequence, the solid light blue line in Fig. 10.2 depicts the neural
network degrees of freedom to attain numerical convergence at fixed hidden layer densities
M/(Nβ+ 1) = K/(Nβ+ 1) = 1, exhibiting a slow linear increase for rising stationary
boson occupation numbers.
As illustrated in Fig. 10.2, the bit-encoded restricted Boltzmann machine architecture fa-
cilitates a much more efficient information compression for open hybrid systems compared
to the common unoptimized master equation approach. The inset shows a zoom-in on
the scaling of degrees of freedom depicted on a linear scale. In the neural network im-
plementation, the bit encoding of Fock states leads to a stepwise increase in complexity
(solid light blue line in inset). Comparable efficiency is only achieved via the maximally
optimized density matrix approach, whose efficiency even surpasses the restricted Boltz-
mann machine implementation in the limit of low boson number occupations. Crucially,
the degree of information compression in the neural network approach becomes even more
effective above Fock state occupations of ⟨nβ(ts)⟩ ≈ 160 (see inset). Considering the al-
ready outstanding degree of Hilbert space compression in the maximally optimized master
equation implementation, the here achieved neural network efficiency is remarkable. In
the following Sections, we verify the accuracy of the bit-encoded restricted Boltzmann
machine and its scalability potential with respect to large boson number regime.
10.5 Accuracy
As an example application to confirm the accuracy of the presented neural encoding of
Fock states, we consider the stationary boson number statistics Pn(ts)for the model system
prescribed by Eq. (10.13), where
Pn(t) = 1
n!⟨c†ncn(t)⟩− 1
n!
nmax
X
m=1
(n+m)!
m!Pn+m(t)(10.15)
10.5 Accuracy 139
5
10
15
0 4000 8000
Iteration
RBM
RK
(a) 〈nβ(ts)〉
0
0.1
0.2
0 1 2 3 4 5 6 7 8 9 10
nβ
RK
RBM
(b) Pnβ(ts)
Figure 10.3: Demonstration of the accuracy of the bit-encoded restricted Boltzmann machine im-
plementation of bosonic number states. (a) Mean stationary boson occupation number
⟨nβ⟩(ts)calculated using the neural network (solid blue line), compared to a calcu-
lation using fewer samples per iteration (solid grey line) and to the common density
matrix benchmark (dashed line). (b) Stationary boson number statistics obtained
from the restricted Boltzmann machine approach (light blue bars) and compared to
benchmark calculations (dark blue bars).
denotes the probability of detecting nbosons in the system at time t, calculated up to the
highest included correlation degree nmax [21,332]. We employ a low cavity dissipation
rate κ= 0.04 ps−1and choose Nβ= 5 bits and hidden layer densities M/(Nβ+ 1) =
K/(Nβ+ 1) = 1 to achieve numerical convergence in accordance to Fig. 10.2. Moreover,
the presented results are obtained using Ns= 5000 sample configurations per iteration at
a learning rate ν= 0.01. In addition, as a benchmark we employ a regular master equation
approach to calculate the stationary state using the same set of parameters, nmax
β= 14
and a time discretization ∆t= 0.02 ps.
Fig. 10.3(a) depicts the estimated mean steady state of the boson number occupation
⟨nβ⟩(ts)over the number of neural network training iterations (solid blue line) and com-
pared to the density matrix implementation benchmark ⟨nβ(ts)⟩ ≈ 4.56 (dashed blue line),
showing excellent agreement after approximately 4000 iterations. The light statistical fluc-
tuations exhibited in the restricted Boltzmann machine solution can be further suppressed
by increasing the number of training samples Ns: The solid grey line in Fig. 10.3(a) shows
a comparison calculation using five times fewer samples, resulting in increased oscillations.
Fig. 10.3(b) shows the stationary Fock state number statistics Pnβ(ts)[Eq. (10.15)] obtained
140 10 Efficient bit encoding of neural networks for Fock states
200
400
600
800
1000
0 800Iteration
nβ
σ00
σ11
0
0.5
1
0 400 800
Figure 10.4: Demonstration of the scalability potential of the bit-encoded restricted Boltzmann
machine in the limit of large boson number occupations. The solid blue line shows
the average Fock state occupation number ⟨nβ(ts)⟩over network training iterations.
The inset depicts the spin up and spin down populations of the single spin system
over iterations (green and orange lines). Dashed blue lines correspond to benchmark
results.
in the neural network approach (light blue bars) and compared to a benchmark density
matrix implementation (dark blue bars). The two unfolding statistics are in overall very
close qualitative agreement, with both their highest boson number probability located at
nβ= 4. However, we note that the statistics resulting from the neural network implemen-
tation is prone to error accumulation in the boson number regime nβ>10: In consequence
of the stochastic reconfiguration method applied to the raw restricted Boltzmann machine
output to calculate occurrence probabilities, the latter are afflicted by statistical deviations
by design. During the evaluation of Eq. (10.15), the occurring statistical error multiplies
for each increasing correlation order of nβ, leading to error propagation displayed in the
form of oscillating statistics in the large occupation regime. This limits the neural network
accuracy for the estimation of Fock state occupation statistics to the low boson number
regime within reasonable computation times.
10.6 Scalability
To access the large Fock state occupation regime, we calculate the stationary state of
the considered model system [Eq. (10.13)] once more using a small cavity dissipation
rate κ= 0.001 ps−1, yielding an average steady state occupation ⟨nβ(ts)⟩ ≈ 199. In
this regime, the information compression efficiency of the neural network implementation
has been demonstrated to be superior even to a maximally optimized master equation
implementation, as shown in Fig. 10.2. For the training of the network, Nβ= 13 bits,
hidden layer densities M/(Nβ+1) = K/(Nβ+1) = 1 and Ns= 5000 samples per iteration
are employed at a learning rate ν= 0.003. Even though the mean Fock state occupation
number ⟨nβ(ts)⟩is valued well below the maximum boson number nmax
β= 2Nβ−1, applying
fewer bits Nβin the network leads to non-converging results, highlighting the necessity for
10.7 Conclusion 141
sufficient degrees of freedom to allow for accurate outcomes [112]. However, the supremely
efficient scaling of the neural network degrees of freedom for increasing system sizes still
enables high-performing calculations in this regime.
In Fig. 10.4, we show results for the average boson number occupation ⟨nβ(ts)⟩over
training iterations (solid blue line), obtained using the restricted Boltzmann machine ar-
chitecture. Strikingly, after approximately 400 iterations it already approaches the density
matrix benchmark result at ⟨nβ(ts)⟩ ≈ 199 (dashed blue line). The inset depicts the aver-
age stationary state spin up and spin down populations of the single spin system (green
and orange lines), exhibiting very good agreement with the corresponding benchmark re-
sults (dashed blue lines). We conclude that while the required bosonic degrees of freedom
by far exceed the average stationary boson occupation taken in the considered parameter
setup, the amount of training iterations to achieve numerical convergence is drastically
decreased for increasing visible neurons. This behavior is explained by the reduced asym-
metry of the employed spin-boson interaction [Eq. (10.12)] in the regime of large Fock state
occupations n≫1where √n≈√n+ 1, since the restricted Boltzmann machine has been
shown to excel with regard to performance and convergence for the simulation of sym-
metric systems [72]. Moreover, the bit-encoded neural network has been demonstrated
to achieve better degrees of information compression than even a maximally optimized
density matrix implementation where a Heisenberg projector method has been used for
optimal truncation of the corresponding Hilbert space, underlining the efficiency and per-
formance of the bit-encoded restricted Boltzmann machine implementation of Fock states
in the high boson number regime.
10.7 Conclusion
We have developed a neural encoding of Fock number states for the restricted Boltzmann
machine architecture, expanding its capabilities from high-performing approximate map-
pings for open spin-1/2systems to hybrid systems featuring bosonic degrees of freedom.
Using the presented method, we have further advanced the paradigm of a universally appli-
cable neural network architecture for the simulation of open quantum systems. Strikingly,
in the limit of large boson number occupations the restricted Boltzmann machine imple-
mentation requires severely fewer degrees of freedom than regular master equation based
approaches and even outperforms the information compression efficiency of a maximally
optimized realization, where the corresponding configuration space has been truncated to
the sparsest possible subspace via a Heisenberg projector method. We have confirmed the
accuracy of the bit-encoded neural network approach for Fock states by estimating the
steady state boson number statistics of a model system, yielding good qualitative agree-
ment with comparison benchmark results. Moreover, to underline the scalability potential
and performance of the presented method, we have calculated the average stationary Fock
state occupation in the high boson number regime, where the information compression
of the neural network has been shown to become supremely efficient. Once numerical
convergence is attained by variation of the number of network degrees of freedom, it can
11 Conclusion and outlook
In summary, in this thesis we have derived and further advanced multiple state-of-the-art
theoretical open system methods and applied them to a wide variety of physical scenarios.
After introducing all of the employed techniques in the first part of this thesis, the second
part has focused on the emergence of dissipation-induced non-Markovian phenomena in
open quantum systems:
In Chapter 6, we have predicted arising memory-critical Majorana edge correlation dy-
namics in a topological superconductor, facilitated by non-Markovian fermion-phonon
interactions and leading to a recovery of topological properties. Aside from the considered
solid-state scenarios, the observed non-Markovian information backflow is expected to also
occur in realizations of ultracold quantum gases coupled to a superfluid reservoir where
excitations result in the emergence of phonon interactions, or in experimental setups with
intentionally induced system-reservoir interactions with a structured bosonic reservoir, as
long as non-Markovian correlations can arise. We have shown that a tailored reservoir
coupling can improve the stability of topological properties and even lead to their full
recovery, giving rise to new prospects for the utilization and control of memory-dependent
topological phenomena.
In Chapter 7, we have investigated the emergence of a non-equilibrium steady state in an
optically driven V-type quantum emitter, exhibiting complete population inversion due to
phonon-induced non-Markovian system-reservoir interactions. The observed mechanism
can be exploited to create unidirectional quantum transport of carriers or excitation in
an array of quantum dots, depending on the considered bidirectional interdot coupling.
The reported population inversion has been shown to be robust against a wide range of
perturbations and may be applied to enable carrier or excitation transport in quantum
optical devices or in the context of light harvesting processes in biological systems.
In Chapter 8, we have developed an MPS-based tensor network approach for a time-
discrete quantum memory in Liouville space. In combination with a tensor network im-
plementation of real-time path integrals for continuous structured reserovirs, we have
created a quasi-2D tensor network architecture to facilitate numerically exact calculations
of quantum systems simultaneously coupled to two non-Markovian environments, support-
ing both diagonal and off-diagonal system-reservoir interactions and maintaining crucial
system-reservoir and inter-reservoir entanglement information. As an example application,
we have demonstrated that dynamical correlation buildup and non-Markovian interactions
in between a structured phonon reservoir and time-discrete coherent photon feedback can
counteract destructive interference, resulting in dynamical population trapping. The here
established quasi-2D tensor network provides a numerically exact tool to gain access to the
143
144 11 Conclusion and outlook
mostly unexplored field of multiple interacting non-Markovian environments. Its perfor-
mance may be further enhanced by tracing out the time-discrete memory bins after their
interaction with the system, giving access to improved time discretizations. Furthermore,
in future works the quasi-2D network may be advanced to a true 2D architecture using
projected entangled pair states with combined reservoir memory bins to enable simulations
with further improved numerical performance.
In the third part of this thesis, novel artificial neural network techniques for efficient
information compression in open quantum systems have been developed and investigated.
In Chapter 9, we have expanded the representational limits of established neural network
implementations by deriving adaptive sampling strategies, enabling accurate and highly
efficient calculations of large open quantum spin-1/2systems with asymmetric properties.
These hybrid sampling techniques open up new perspectives for the optimization and
customization of the restricted Boltzmann machine architecture in a wide variety of open
quantum systems, while further improving accuracy and performance.
Lastly, in Chapter 10 we have derived a neural bit encoding scheme for Fock number states
in the restricted Boltzmann machine architecture to facilitate high performing calculations
of hybrid systems featuring bosonic degrees of freedom. Using the presented approach, we
have further advanced the paradigm of a universally applicable neural network architec-
ture for open quantum systems beyond pure spin-1/2setups. We have demonstrated the
accuracy and scalability of the approach by estimating both the stationary state boson
number statistics of a model system and the average stationary Fock state occupation in
the large boson number regime. Crucially, in the high occupation limit the bit-encoded
neural network requires severely fewer degrees of freedom than common density matrix
based approaches and even surpasses the information compression efficiency of a maxi-
mally optimized master equation implementation, where the corresponding configuration
space has been truncated to the sparsest possible subspace via a Heisenberg projector
method.
Appendices
145
A Calculations: Path integral formalism
A.1 Trace over coherent states
The trace over the reservoir is defined in terms of thermal phonon states |nq⟩,
trB{. . .}=Y
q
∞
X
nq=0 ⟨nq|[. . .]|nq⟩,(A.1)
which can be written in terms of unnormalized coherent states,
|nq⟩=Zd2αq
π|αq⟩⟨αq|nq⟩=Zd2αq
π|αq⟩
∞
X
mq=0
e−|αq|2/2α∗
qmq
√mq!⟨mq|nq⟩.(A.2)
Using this representation, Eq. (A.1) takes the form
trB{. . .}=Y
q
∞
X
nq=0 ⟨nq|
1
[. . .]
1
|nq⟩
=Y
q
∞
X
nq=0 Zd2αq
πZd2βq
π⟨nq|βq⟩⟨βq|[. . .]|αq⟩⟨αq|nq⟩
=Y
qZd2αq
πZd2βq
π⟨βq|[. . .]|αq⟩e−|αq|2/2−|βq|2/2+α∗
qβq
|{z }
=⟨αq|βq⟩
=Y
qZd2αq
π⟨αq|[. . .]|αq⟩.(A.3)
147
148 A Calculations: Path integral formalism
A.2 Evaluation of Gaussian integrals
Starting from the density matrix element representation of Eq. (4.30),
⟨iN|ρS(t)|i′
N⟩=Y
qZd2αqN
πZd2αq0
πZd2βq0
π
2
X
i1,...,iN−1=1
2
X
i′
1,...,i′
N−1=1
×MiNiN−1. . . Mi1i0Mi′
0i′
1. . . Mi′
N−1i′
N(1 −ξq0)⟨i0|ρS(0) |i′
0⟩
×exp (−|αqN|2−|αq0|2−|βq0|2+ξq0α∗
q0βq0+αq0α∗
qNe−iωqt+β∗
q0αqNeiωqt
−igqZt
0
dτ′hαq0e−iωqτ′+α∗
qNe−iωq(t−τ′)ij(τ′)−g2
qZt
0
dτZτ
0
dτ′e−iωq(τ−τ′)j(τ)j(τ′)
+igqZt
0
dτ′hβ∗
q0eiωqτ′+αqNeiωq(t−τ′)ij′(τ′)−g2
qZt
0
dτZτ
0
dτ′eiωq(τ−τ′)j′(τ)j′(τ′)),
(A.4)
we have yet to trace out the reservoir explicitly by carrying out the remaining three
integrals. For an easier notation, we abbreviate the time-dependent integrals via
x:= −igqZt
0
dτ′e−iωqτ′j(τ′),(A.5a)
x′:= igqZt
0
dτ′eiωqτ′j′(τ′),(A.5b)
y:= −igqZt
0
dτ′e−iωq(t−τ′)j(τ′),(A.5c)
y′:= igqZt
0
dτ′eiωq(t−τ′)j′(τ′),(A.5d)
and extract all phonon contributions from Eq. (4.30), defining
A:= Zd2αqN
πZd2αq0
πZd2βq0
πexp hξq0α∗
q0βq0−|αqN|2−|αq0|2−|βq0|2
+αq0α∗
qNe−iωqt+β∗
q0αqNeiωqt+αq0x+α∗
qNy+β∗
q0x′+αqNy′i.(A.6)
In the following calculations, we make use of Gaussian integrals,
Z∞
−∞
dz e−µ(z+ν)2=rπ
µ.(A.7)
A.2 Evaluation of Gaussian integrals 149
A.2.1 First integration
For the first integration, we define αq0=: a+ib and extract all αq0-dependent terms from
Eq. (A.6), yielding
B:= ZdaZdbexp hξq0βq0(a−ib)−a2−b2+ (a+ib)x+α∗
qNe−iωqti
=Zdaexp h−a2+aξq0βq0+x+α∗
qNe−iωqti
Zdbexp h−b2+ib −ξq0βq0+x+α∗
qNe−iωqti.(A.8)
Extending the exponential arguments by a zero addition of the form
−(a2−a[. . .]) = −a2−2
2a[. . .] + 1
4[. . .]2−1
4[. . .]2=−a−1
2[. . .]2
+1
4[. . .]2,(A.9)
and making use of the Gaussian identity in Eq. (A.7), the integrals are evaluated as
B=πexp hξq0βq0x+α∗
qNe−iωqti,(A.10)
and Eq. (A.6) takes the form
A=Zd2αqN
πZd2βq0
πexp h−|αqN|2−|βq0|2
+β∗
q0αqNeiωqt+ξq0βq0α∗
qNe−iωqt+ξq0βq0x+α∗
qNy+β∗
q0x′+αqNy′i.(A.11)
A.2.2 Second integration
The second integral is carried out in the same fashion. We define βq0=: a+ib and
introduce a new expression containing all βq0-dependent terms of Eq. (A.11),
C:= ZdaZdbexp h−a2−b2+ (a−ib)x′+αqNeiωqt+ (a+ib)ξq0x+ξq0α∗
qNe−iωqti
=Zdaexp h−a2+ax′+αqNeiωqt+ξq0x+ξq0α∗
qNe−iωqti
Zdbexp h−b2+ib −x′−αqNeiωqt+ξq0x+ξq0α∗
qNe−iωqti.(A.12)
The arguments of the exponential functions are once more expanded by a zero addition
[Eq. (A.9)]. Afterwards, the Gaussian identity is used, yielding
C=πexp hξq0xx′+x′α∗
qNe−iωqt+xαqNeiωqt+|αqN|2i,(A.13)
and the updated expression for Eq. (A.11) reads
A=Zd2αqN
πexp h−|αqN|2+α∗
qNy+αqNy′
+ξq0xx′+x′α∗
qNe−iωqt+xαqNeiωqt+|αqN|2i.(A.14)
150 A Calculations: Path integral formalism
A.2.3 Third integration
For the last remaining integral, we define αqN=: a+ib and perform the same operations
as before, resulting in
A=1
1−ξq0
exp hξq0xx′+1
1−ξq0yy′+ξq0xyeiωqt+ξq0x′y′e−iωqt+ξ2
q0xx′i.(A.15)
As a next step, ξq0= ¯nq0/(1 + ¯nq0)and the definitions of Eq. (A.5) are reinserted into
Eq. (A.15). Together with all phonon-independent terms, the resulting expression for the
density matrix elements reads
⟨iN|ρS(t)|i′
N⟩=
2
X
i1,...,iN−1=1
2
X
i′
1,...,i′
N−1=1
MiNiN−1. . . Mi1i0Mi′
0i′
1. . . Mi′
N−1i′
N
×exp X
q(−g2
qZt
0
dτZτ
0
dτ′he−iωq(τ−τ′)j(τ)j(τ′) + eiωq(τ−τ′)j′(τ)j′(τ′)i
+g2
qZt
0
dτZt
0
dτ′eiωq(τ−τ′)j(τ)j′(τ′)
−¯nq0g2
qZt
0
dτZt
0
dτ′eiωq(τ−τ′)j(τ)−j′(τ)j(τ′)−j′(τ′))!⟨i0|ρS(0) |i′
0⟩.(A.16)
The argument of the exponential function is referred to as the influence functional Sinf
and can be further simplified for the case gq=g∗
qand two carrier states. Inserting
1 = [Θ(τ−τ′) + Θ(τ′−τ)], the last two terms can be rewritten,
Zt
0
dτZt
0
dτ′Θ(τ−τ′) + Θ(τ′−τ)[. . .] = Zt
0
dτZτ
0
dτ′[. . .] + Zτ′
0
dτZt
0
dτ′[. . .].
(A.17)
Substituting τwith τ′in the second term of Eq. (A.17), the influence functional can be
written as
Sinf (t) = X
q(−g2
qZt
0
dτZτ
0
dτ′he−iωq(τ−τ′)j(τ)j(τ′) + eiωq(τ−τ′)j′(τ)j′(τ′)i
+g2
qZt
0
dτZτ
0
dτ′heiωq(τ−τ′)j(τ)j′(τ′) + e−iωq(τ−τ′)j(τ′)j′(τ)i
−¯nq0g2
qZt
0
dτZτ
0
dτ′eiωq(τ−τ′)j(τ)−j′(τ)j(τ′)−j′(τ′)
−¯nq0g2
qZt
0
dτZτ
0
dτ′e−iωq(τ−τ′)j(τ′)−j′(τ′)j(τ)−j′(τ)).(A.18)
Defining the reservoir autocorrelation function
η(τ−τ′) := X
q
g2
q{(2¯nq0+ 1) cos ωq(τ−τ′)−isin ωq(τ−τ′)},(A.19)
A.2 Evaluation of Gaussian integrals 151
we arrive at the final expression for the influence functional,
Sinf (t) = −Zt
0
dτZτ
0
dτ′j(τ)−j′(τ)η(τ−τ′)j(τ′)−η∗(τ−τ′)j′(τ′),(A.20)
and
⟨iN|ρS(t)|i′
N⟩=
2
X
i1,...,iN−1=1
2
X
i′
1,...,i′
N−1=1
MiNiN−1. . . Mi1i0Mi′
0i′
1. . . Mi′
N−1i′
N
exp [Sinf (t)] ⟨i0|ρS(0) |i′
0⟩.(A.21)
B Calculations: Polaronic Kitaev chain
B.1 Polaron transformation
To derive the polaron-transformed Kitaev Hamiltonian, we start from Eqs. (6.1) and (6.2),
yielding the total dissipative Kitaev Hamiltonian H0=Hk+Hb,
H0/ℏ=
N−1
X
l=1 h−Jc†
lcl+1 + ∆clcl+1+ H.c.i−µ
N
X
l=1
c†
lcl
+Zd3k"ωkr†
krk+
N
X
l=1
gkc†
lcl(r†
k+rk)#.(B.1)
As a first step, we define collective bosonic operators R†=Rd3k(gk/ωk)r†
kand apply the
unitary polaron transformation Hp=UpH0U−1
pto the dissipative Kitaev Hamiltonian via
transformation matrices Up= exp[PN
l=1 c†
lcl(R†−R)]. To carry out the transformation,
the Baker-Campbell-Hausdorff formula is applied,
eXY e−X=
∞
X
n=0
1
n![X, Y ]n,[X, Y ]n= [X, [X, Y ]n−1],[X, Y ]0=Y. (B.2)
We start with the transformation of the first term of the dissipative Kitaev Hamiltonian,
1
ℏUpH0,(i)U−1
p=
∞
X
n=0
1
n!"N
X
l=1
c†
lcl(R†−R),
N−1
X
l′=1 h−Jc†
l′cl′+1 + ∆cl′cl′+1+ H.c.i#n
.
(B.3)
Evaluating the first four steps of the Baker-Campbell-Hausdorff formula yields
1
ℏupH0,(i)u−1
p=−
N−1
X
l=1
Jc†
lcl+1 +c†
l+1cl
−1
2∆
N−1
X
l=1 c†
lc†
l+1 +cl+1cl1
2!22(R†−R)2+1
4!24(R†−R)4+. . .
+1
2∆
N−1
X
l=1 c†
lc†
l+1 −cl+1cl1
1!(−2)(R†−R) + 1
3!(−2)3(R†−R)3+. . ..(B.4)
153
154 B Calculations: Polaronic Kitaev chain
At this point, a recurrent pattern of fermionic operators can be identified and expanded
into a representation of hyperbolic functions, resulting in the form
1
ℏupH0,(i)u−1
p=
N−1
X
l=1 (−Jc†
lcl+1 +c†
l+1cl−∆c†
lc†
l+1 +cl+1clcosh h2(R−R†)i
+ ∆ c†
lc†
l+1 −cl+1clsinh h2(R−R†)i).(B.5)
The transformation of the second contribution of Eq. (B.1) yields
1
ℏupH0,(ii)u−1
p=
∞
X
n=0
1
n!"N
X
l=1
c†
lcl(R†−R),−µ
N
X
l′=1
c†
l′cl′#n
=−µ
N
X
l=1
c†
lcl.(B.6)
For the evaluation of the third term, the first two steps of the Baker-Campbell-Hausdorff
formula are carried out explicitly. The transformation is prescribed by
1
ℏupH0,(iii)u−1
p=
∞
X
n=0
1
n!"N
X
l=1 Zd3k c†
lcl
gk
ωk
(r†
k−rk),Zd3k′ωk′r†
k′rk′#n
,(B.7)
leading to
1
ℏupH0,(iii)u−1
p=Zd3k′
ωk′r†
k′rk′−
N
X
l=1
c†
lclgk′(r†
k′+rk′) + 2
2!
N
X
l,l′=1
g2
k′
ωk′
c†
lclc†
l′cl′
.
(B.8)
From the result of the second step, i.e., the last term in Eq. (B.8), it follows that [. . .]n≥3=
0. The transformation of the fourth term is prescribed by
1
ℏupH0,(iv)u−1
p=
∞
X
n=0
1
n!"N
X
l=1 Zd3k c†
lcl
gk
ωk
(r†
k−rk),
N
X
l′=1 Zd3k′gk′c†
l′cl′r†
k′+rk′#n
.
(B.9)
Carrying out the first step of the Baker-Campbell-Hausdorff formula results in
1
ℏupH0,(iv)u−1
p=
N
X
l′=1 Zd3k′"gk′c†
l′cl′r†
k′+rk′−2
N
X
l=1
g2
k′
ωk′
c†
lclc†
l′cl′#.(B.10)
Again, from the last term in Eq. (B.10) it follows that [. . .]n≥2= 0.
With this, we have transformed all terms of Eq. (B.1). Adding up all partial results finally
yields Hp=Hp,s +Hp,b +Hp,I with
Hp,s/ℏ=−J
N−1
X
l=1 c†
lcl+1 + H.c.−µ
N
X
l=1
c†
lcl,(B.11)
B.2 Franck-Condon renormalization 155
Hp,I/ℏ= ∆
N−1
X
l=1 (−c†
lc†
l+1 +cl+1clcosh h2(R−R†)i
+c†
lc†
l+1 −cl+1clsinh h2(R−R†)i),(B.12)
and Hp,b/ℏ=Rd3k ωkr†
krk. Here we have neglected the arising polaron energy shift
term
Hshift/ℏ=Zd3kg2
k
ωk N
X
l=1
c†
lcl!2
.(B.13)
B.2 Franck-Condon renormalization
The polaron Hamiltonian is renormalized such that
trB{[Hp,I, ρ(t)]}= 0 ∀t. (B.14)
This is achieved via a Franck-Condon renormalization, where a zero term is added to the
Hamiltonian. For the polaronic Kitaev chain, Eq. (B.14) can be fulfilled by adding the
term
N−1
X
l=1
∆(c†
lc†
l+1 +cl+1cl)⟨B⟩(B.15)
to the interaction Hamiltonian Hp,I with a newly introduced Franck-Condon renormaliza-
tion factor
⟨B⟩= tr nexp h−2(R−R†)io= exp [−ϕ(0)/2] .(B.16)
In return, Eq. (B.15) is subtracted from the free contribution of the polaron Hamiltonian.
Hence, the total Hamiltonian Hpremains unaltered by this transformation. The Franck-
Condon renormalized polaron Hamiltonians read
Hp,s/ℏ=−
N−1
X
l=1 h∆c†
lc†
l+1 +cl+1cl⟨B⟩+Jc†
lcl+1 +c†
l+1cli−µ
N
X
l=1
c†
lcl,(B.17)
and
Hp,I/ℏ= ∆
N−1
X
l=1 h−c†
lc†
l+1 +cl+1clcosh 2(R−R†)−⟨B⟩
+c†
lc†
l+1 −cl+1clsinh 2(R−R†)i.(B.18)
156 B Calculations: Polaronic Kitaev chain
B.3 Polaron master equation
The polaron master equation is derived using standard second-order perturbation theory.
The von Neumann equation is formally solved by integration and reinserted into itself.
The Born approximation and the first Markov approximation are applied,
ρ(t−τ)≈ρS(t)⊗ρB(0) (B.19)
where indices Sand Bdenote the system and environmental part of the full system
density matrix ρ(t), respectively. Tracing out the phononic degrees of freedom results in
the non-Markovian Redfield master equation
d
dtρS(t) = −i[Hp,s, ρS(t)] −Zt
0
dτtrB{[Hp,I,[Hp,I(−τ), ρS(t)⊗ρB]]}.(B.20)
We define collective operators
Xa(t) = −∆
N−1
X
l=1 c†
lc†
l+1 +cl+1cl, Xb(t)=∆
N−1
X
l=1 c†
lc†
l+1 −cl+1cl,
Ba(t) = exp n2hR(t)−R†(t)io+ exp n−2hR(t)−R†(t)io,
Bb(t) = exp n2hR(t)−R†(t)io−exp n−2hR(t)−R†(t)io,(B.21)
and the Franck-Condon renormalized fermion-phonon interaction term of the polaron-
transformed Hamiltonian takes the form
Hp,I(t) = 1
2Xa(t) [Ba(t)−⟨Ba⟩] + 1
2Xb(t)Bb(t).(B.22)
Next, the integrand of Eq. (B.20) is evaluated. For a simpler notation we denote Xa,b(−τ) =
X′
a,b,Ba,b(−τ) = B′
a,b, resulting in
[Hp,I,[Hp,I(−τ), ρ]]
=Hp,I,1
2X′
aB′
a, ρ+1
2X′
bB′
b, ρ−⟨Ba⟩
2X′
a, ρ
=1
2Hp,I,X′
aB′
a, ρ+1
2Hp,I,X′
bB′
b, ρ−⟨Ba⟩
2Hp,I,X′
a, ρ
=1
4XaBa,X′
aB′
a, ρ+1
4XbBb,X′
aB′
a, ρ−⟨Ba⟩
4Xa,X′
aB′
a, ρ
+1
4XaBa,X′
bB′
b, ρ+1
4XbBb,X′
bB′
b, ρ−⟨Ba⟩
4Xa,X′
bB′
b, ρ
−⟨Ba⟩
4XaBa,X′
a, ρ−⟨Ba⟩
4XbBb,X′
a, ρ+⟨Ba⟩2
4Xa,X′
a, ρ.(B.23)
B.3 Polaron master equation 157
The most complicated terms of Eq. (B.23) are of the form [XB, [X′B′, ρ]]. Taking the
trace over the reservoir yields
trBXB, X′B′, ρ
= trBXBX′B′ρ−XBρX′B′−X′B′ρXB +ρX′B′XB
=XX′ρS⟨BB′⟩−XρSX′⟨B′B⟩−X′ρSX⟨BB′⟩+ρSX′X⟨B′B⟩
=⟨BB′⟩XX′ρS−X′ρSX+⟨B′B⟩ρSX′X−XρSX′
=⟨BB′⟩X, X′ρS−⟨B′B⟩X, ρSX′,(B.24)
with ⟨. . .⟩= trB{ρ(t). . .}. The other contributions read
trBX, X′B′, ρ=⟨B′⟩X, X′ρS−X, ρSX′,
trBXB, X′, ρ=⟨B⟩X, X′ρS−X, ρSX′,
trBX, X′, ρ=X, X′ρS−X, ρSX′.(B.25)
It can be shown that ⟨BaB′
b⟩=⟨BbB′
a⟩= 0.⟨B′
a⟩is calculated as
⟨B′
a⟩= trBρBne2[R(−τ)−R†(−τ)] +e−2[R(−τ)−R†(−τ)]o
= 2trBρB1 + 1
2! n2hR(−τ)−R†(−τ)io2+...
= 2trB ρB(1 + 1
2! 2Zd3kgk
ωkrkeiωkτ−r†
ke−iωkτ2
+...)!
= 2trBρB1 + 1
2! −Zd3k|2gk/ωk|2(1 + 2nk)+...
= 2trBρB1 + 1
1! −1
2Zd3k|2gk/ωk|2(1 + 2nk)+...
= 2 exp −1
2Zd3k|2gk/ωk|2(2nk+ 1)=⟨Ba⟩,(B.26)
and Eq. (B.23) takes the form
trBhHp,I,h˜
H′
p,I, ρS(t)⊗ρBii
=1
4⟨BaB′
a⟩Xa, X′
aρS(t)−1
4⟨B′
aBa⟩Xa, ρS(t)X′
a+1
4⟨BbB′
b⟩Xb, X′
bρS(t)
−1
4⟨B′
bBb⟩Xb, ρS(t)X′
b−1
4⟨Ba⟩2Xa, X′
aρS(t)−Xa, ρS(t)X′
a.(B.27)
158 B Calculations: Polaronic Kitaev chain
B.3.1 Reservoir correlations
The reservoir correlations ⟨Ba,bBa,b(−τ)⟩,⟨Ba,b(−τ)Ba,b⟩are calculated next. We re-
quire
h2(R−R†),2(R(−τ)−R†(−τ))i
=Zd3kZd3k′22gkgk′
ωkωk′hrk−r†
k, rk′eiωk′τ−r†
k′e−iωk′τi
= 2iZd3k|2gk/ωk|2sin (ωkτ) =: 2iϕs(τ),(B.28)
and make use of the simplified Baker-Campbell-Hausdorff formula eXeY=eX+Ye1/2[X,Y ],
which is valid if [X, [X, Y ]] = 0 and [Y, [Y, X]] = 0. We start with the correlation
⟨BaBa(−τ)⟩,
⟨BaBa(−τ)⟩= trBnρBhe2(R−R†)+e−2(R−R†)ihe2[R(−τ)−R†(−τ)] +e−2[R(−τ)−R†(−τ)]io
= trB(ρBhe2(R−R†)e2[R(−τ)−R†(−τ)] +e−2(R−R†)e−2[R(−τ)−R†(−τ)]
+e2(R−R†)e−2[R(−τ)−R†(−τ)] +e−2(R−R†)e2[R(−τ)−R†(−τ)]i)
= trB(ρBhe2[R†+R†(−τ)]−2[R+R(−τ)]e1/2[2(R−R†),2(R(−τ)−R†(−τ))]
+e−2[R†+R†(−τ)]+2[R+R(−τ)]e1/2[2(R−R†),2(R(−τ)−R†(−τ))]
+e2[R†−R†(−τ)]−2[R−R(−τ)]e1/2[2(R−R†),2(R(−τ)−R†(−τ))](−1)
+e−2[R†−R†(−τ)]+2[R−R(−τ)]e1/2[2(R−R†),2(R(−τ)−R†(−τ))](−1)i),(B.29)
with e1/2[2(R−R†),2(R(−τ)−R†(−τ))]=eiϕs(τ). Using the identity cosh (x) = 1
2(ex+e−x) =
1 + x2
2! +x4
4! +. . ., Eq. (B.29) is rewritten as
⟨BaBa(−τ)⟩= 2trBρB" 1 + 1
2! n2hR†+R†(−τ)i−2 [R+R(−τ)]o2
+1
4! n2hR†+R†(−τ)i−2 [R+R(−τ)]o4+. . . !eiϕs(τ)
+ 1 + 1
2! n2hR†−R†(−τ)i−2 [R−R(−τ)]o2
+1
4! n2hR†−R†(−τ)i−2 [R−R(−τ)]o4+. . . !e−iϕs(τ)#.(B.30)
B.3 Polaron master equation 159
We define
[R±R(−τ)] = Zd3kgk
ωk
(1 ±eiωkτ)rk=: Zd3k f±(k)rk,
hR†±R†(−τ)i=Zd3kgk
ωk
(1 ±e−iωkτ)r†
k=: Zd3k f∗
±(k)r†
k,(B.31)
and the first product in Eq. (B.30) takes the form
trBρBn2hR†+R†(−τ)i−2 [R+R(−τ)]o2
= trB ρB2Zd3khf∗
+(k)r†
k−f+(k)rki2!
= 22Zd3kZd3k′trBnρBh−f∗
+(k)f+(k′)r†
krk′−f+(k)f∗
+(k′)rkr†
k′io
=−22Zd3k|f+(k)|2(2nk+ 1).(B.32)
The next term is factorized using Wick’s theorem, yielding
trBρBn2hR†+R†(−τ)i−2[R+R(−τ)]o4
≈243Zd3kZd3k′|f+(k)|2|f+(k′)|2(2nk+ 1)(2nk′+ 1).(B.33)
Factorizing all terms in Eq. (B.30) in this fashion results in two exponential series, yield-
ing
⟨BaBa(−τ)⟩=2 exp −1
2Zd3k|2f+(k)|2(2nk+ 1) + iϕs(τ)
+2 exp −1
2Zd3k|2f−(k)|2(2nk+ 1) −iϕs(τ),(B.34)
with |f±(k)|2=|gk/ωk|2[2 ±2 cos (ωkτ)]. The phonon number nkis modeled as a Bose
distribution, resulting in (2nk+1) = coth [ℏωk/(2kBT)]. We define the phonon correlation
function
ϕ(τ) := Zd3k
2gk
ωk
2coth ℏωk
2kBTcos (ωkτ)−isin(ωkτ),(B.35)
and arrive at the final form of the reservoir correlation
⟨BaBa(−τ)⟩= 4 exp [−ϕ(0)] cosh [ϕ(τ)] .(B.36)
⟨BbBb(−τ)⟩is calculated in the same fashion. The calculation of ⟨Ba(−τ)Ba⟩has already
been done up to a sign in ϕs(τ). The resulting terms read
⟨Ba(−τ)Ba⟩= 4 exp [−ϕ(0)] cosh [ϕ∗(τ)] = ⟨BaBa(−τ)⟩∗,
⟨BbBb(−τ)⟩=−4 exp [−ϕ(0)] sinh [ϕ(τ)],
⟨Bb(−τ)Bb⟩=−4 exp [−ϕ(0)] sinh [ϕ∗(τ)] = ⟨BbBb(−τ)⟩∗.(B.37)
160 B Calculations: Polaronic Kitaev chain
Finally, inserting the correlation functions into Eq. (B.27) yields
trBhHp,I,hH′
p,I, ρS(t)⊗ρBii={exp [−ϕ(0)] (cosh [ϕ(τ)] −1) Xa, X′
aρS(t)
−exp [−ϕ(0)] sinh [ϕ(τ)] Xb, X′
bρS(t)+ H.c.},(B.38)
and the polaron master equation takes the form
d
dtρS(t) = −i[Hp,s, ρS(t)] −e−ϕ(0) Zt
0
dτ{(cosh [ϕ(τ)] −1) [Xa, Xa(−τ)ρS(t)]
−sinh [ϕ(τ)] [Xb, Xb(−τ)ρS(t)] + H.c.}.(B.39)
B.4 Majorana edge correlation
To investigate the fate of topological properties in the polaronic Kitaev chain, the Majo-
rana edge correlation ⟨θ⟩(t)is calculated. In the case of an ideal Kitaev chain it is given
by [143,144]
−i⟨γLγR⟩(t) = tr{ρ(t)(c1+c†
1)(c†
N−cN)}
=X
{n}⟨n1, . . . , nN|ρ(t)(c1+c†
1)(c†
N−cN)|n⟩
=X
{n}h−⟨n1, . . . , nN|ρ(t)|0,...,1⟩δn1,1δnN,0−⟨n|ρ(t)|1,...,0⟩δn1,0δnN,1
+⟨n1, . . . , nN|ρ(t)|1,...,1⟩δn1,0δnN,0+⟨n|ρ(t)|0,...,0⟩δn1,1δnN,1ieiπ PN−1
l=2 nl
=X
{n}h−⟨1, n2, . . . , nN−1,0|ρ(t)|0, n2, . . . , nN−1,1⟩
−⟨0, n2, . . . , nN−1,1|ρ(t)|1, n2, . . . , nN−1,0⟩
+⟨0, n2, . . . , nN−1,0|ρ(t)|1, n2, . . . , nN−1,1⟩
+⟨1, n2, . . . , nN−1,1|ρ(t)|0, n2, . . . , nN−1,0⟩ieiπ PN−1
l=2 nl,(B.40)
with nl={0,1}and l={1, . . . , N}. When assuming even parity conditions, i.e.,
PN
l=1 nl= 2ν,ν∈N0, this yields
⟨θ⟩(t) = X
{n}h⟨1, n2, . . . , nN−1,0|ρ(t)|0, n2, . . . , nN−1,1⟩
+⟨0, n2, . . . , nN−1,1|ρ(t)|1, n2, . . . , nN−1,0⟩
+⟨0, n2, . . . , nN−1,0|ρ(t)|1, n2, . . . , nN−1,1⟩
+⟨1, n2, . . . , nN−1,1|ρ(t)|0, n2, . . . , nN−1,0⟩i.(B.41)
B.5 Equations of motion 161
In the case of odd parity conditions, PN
l=1 nl= 2ν+ 1,ν∈N0, the Jordan-Wigner phase
results in negative signs,
⟨θ⟩(t) = −X
{n}h⟨1, n2, . . . , nN−1,0|ρ(t)|0, n2, . . . , nN−1,1⟩
+⟨0, n2, . . . , nN−1,1|ρ(t)|1, n2, . . . , nN−1,0⟩
+⟨0, n2, . . . , nN−1,0|ρ(t)|1, n2, . . . , nN−1,1⟩
+⟨1, n2, . . . , nN−1,1|ρ(t)|0, n2, . . . , nN−1,0⟩i.(B.42)
B.5 Equations of motion
The equations of motion are stated for the system density matrix elements
⟨m1, . . . , mN|ρS|n1, . . . , nN⟩(t) = ⟨m|ρS|n⟩(t).(B.43)
For a shorter notation we henceforth write ρS(t) = ρ(t). The equations of motion resulting
from the first term of the polaron master equation read
−i⟨m|[Hp,s, ρ(t)] |n⟩
= 2iµ
N−1
X
l=1
(ml−nl)⟨m|ρ(t)|n⟩
+iU
N−1
X
l=1 h(2nl−1) (2nl+1 −1) −(2ml−1) (2ml+1 −1) i⟨m|ρ(t)|n⟩
+iJe−ϕ(0)/2
N−1
X
l=1 h⟨...,ml+ 1, ml+1 + 1, . . .|ρ(t)|n⟩δml,0δml+1,0
+⟨...,ml−1, ml+1 −1, . . .|ρ(t)|n⟩δml,1δml+1,1
−⟨m|ρ(t)|...,nl+ 1, nl+1 + 1, . . .⟩δnl,0δnl+1,0
−⟨m|ρ(t)|...,nl−1, nl+1 −1, . . .⟩δnl,1δnl+1,1i
+iJ
N−1
X
l=1 h⟨...,ml+ 1, ml+1 −1, . . .|ρ(t)|n⟩δml,0δml+1,1
+⟨...,ml−1, ml+1 + 1, . . .|ρ(t)|n⟩δml,1δml+1,0
−⟨m|ρ(t)|...,nl+ 1, nl+1 −1, . . .⟩δnl,0δnl+1,1
−⟨m|ρ(t)|...,nl−1, nl+1 + 1, . . .⟩δnl,1δnl+1,0i.(B.44)
We define Ga(τ) := cosh [ϕ(τ)] −1,Gb(τ) := −sinh [ϕ(τ)], and
χ(τ) := Ga(τ) [Xa, Xa(−τ)ρS(t)] + Gb(τ) [Xb, Xb(−τ)ρS(t)] + H.c.. (B.45)
162 B Calculations: Polaronic Kitaev chain
The second term of the master equation then takes the form −exp [−ϕ(0)] Rt
0dτχ(τ),
with
⟨m|χ(τ)|n⟩
=X
i=a,b hGi(τ)⟨m|XiXi(−τ)ρ(t)|n⟩−Gi(τ)⟨m|Xi(−τ)ρ(t)Xi|n⟩
+G∗
i(τ)⟨m|ρ(t)Xi(−τ)Xi|n⟩−G∗
i(τ)⟨m|Xiρ(t)Xa(−τ)|n⟩i
=X
i=a,b X
{s}hGi(τ)⟨m|XiXi(−τ)|s⟩⟨s|ρ(t)|n⟩−Gi(τ)⟨m|Xi(−τ)|s⟩⟨s|ρ(t)Xi|n⟩
+G∗
i(τ)⟨m|ρ(t)|s⟩⟨s|Xi(−τ)Xi|n⟩−G∗
i(τ)⟨m|Xiρ(t)|s⟩⟨s|Xi(−τ)|n⟩i.(B.46)
Inserting the definitions of the system correlations Xa,b yields
⟨m|χ(τ)|n⟩= ∆ X
{s}X
ln⟨s|ρ(t)|n⟩
h−Ga(τ)⟨...,ml+ 1, ml+1 + 1, . . .|Xa(−τ)|s⟩δml,0δml+1,0
−Ga(τ)⟨...,ml−1, ml+1 −1, . . .|Xa(−τ)|s⟩δml,1δml+1,1
+Gb(τ)⟨...,ml−1, ml+1 −1, . . .|Xb(−τ)|s⟩δml,1δml+1,1
−Gb(τ)⟨...,ml+ 1, ml+1 + 1, . . .|Xb(−τ)|s⟩δml,0δml+1,0i
+⟨m|ρ(t)|s⟩
h−G∗
a(τ)⟨s|Xa(−τ)|...,nl+ 1, nl+1 + 1, . . .⟩δnl,0δnl+1,0
−G∗
a(τ)⟨s|Xa(−τ)|...,nl−1, nl+1 −1, . . .⟩δnl,1δnl+1,1
+G∗
b(τ)⟨s|Xb(−τ)|...,nl+ 1, nl+1 + 1, . . .⟩δnl,0δnl+1,0
−G∗
b(τ)⟨s|Xb(−τ)|...,nl−1, nl+1 −1, . . .⟩δnl,1δnl+1,1i
+⟨s|ρ(t)|...,nl+ 1, nl+1 + 1, . . .⟩δnl,0δnl+1,0
hGa(τ)⟨m|Xa(−τ)|s⟩−Gb(τ)⟨m|Xb(−τ)|s⟩i
+⟨s|ρ(t)|...,nl−1, nl+1 −1, . . .⟩δnl,1δnl+1,1
hGa(τ)⟨m|Xa(−τ)|s⟩+Gb(τ)⟨m|Xb(−τ)|s⟩i
+⟨...,ml+ 1, ml+1 + 1, . . .|ρ(t)|s⟩δml,0δml+1,0
hG∗
a(τ)⟨s|Xa(−τ)|n⟩+G∗
b(τ)⟨s|Xb(−τ)|n⟩i
+⟨...,ml−1, ml+1 −1, . . .|ρ(t)|s⟩δml,1δml+1,1
hG∗
a(τ)⟨s|Xa(−τ)|n⟩−G∗
b(τ)⟨s|Xb(−τ)|n⟩io.(B.47)
B.5 Equations of motion 163
As described in Sec. 3.5, the time-dependent system correlations Xa,b(−τ)are expanded
via the time-evolution operator U0(t, 0) = exp (−iHp,st/ℏ), resulting in
⟨m|Xa(−τ)|n⟩=−∆X
lX
{s}h⟨s1,...,0,0, . . . , sN|ρc(−τ)|s1,...,1,1, . . . , sN⟩
+⟨s1,...,1,1, . . . , sN|ρc(−τ)|s1,...,0,0, . . . , sN⟩i,(B.48)
⟨m|Xb(−τ)|n⟩=+∆X
lX
{s}h⟨s1,...,0,0, . . . , sN|ρc(−τ)|s1,...,1,1, . . . , sN⟩
−⟨s1,...,1,1, . . . , sN|ρc(−τ)|s1,...,0,0, . . . , sN⟩i,(B.49)
with ρcthe conditional density matrix.
C Calculations: Chain of V-type emitters
C.1 Parameters
Table C.1 shows the parameters employed for all numerical calculations in Ch. 7.
Parameter Single emitter Chain (single) Chain (all-to-all)
δϵ −1.0meV −1.0meV −1.0meV
ℏΩ 0.0658 meV 0.0658 meV 0.3291 meV
T4K4K4K
ℏf—0.0658 meV 0.0658 meV
Table C.1: Parameters employed for numerical calculations of the V-type emitter model, including
the single emitter case and the quantum dot chain with single Dexter- and Förster-type
interdot couplings, as well as Dexter-type coupling between all excited states of adjacent
emitters.
C.2 Heisenberg picture: Equations of motion
The full set of Heisenberg equations of motion up to second order in phonon contributions
for a single V-type emitter read
d
dt⟨σmn⟩=ih∆2(⟨σ2n⟩δm2−⟨σm2⟩δn2)+∆3(⟨σ3n⟩δm3−⟨σm3⟩δn3)
+ Ω (⟨σ1n⟩δm2+⟨σ2n⟩δm1−⟨σm2⟩δn1−⟨σm1⟩δn2)
+ Ω (⟨σ1n⟩δm3+⟨σ3n⟩δm1−⟨σm3⟩δn1−⟨σm1⟩δn3)
+Zd3k gk⟨σ2nr†
k⟩δm2+⟨σ3nr†
k⟩δm3−⟨σm2r†
k⟩δn2−⟨σm3r†
k⟩δn3
+⟨σ2nrk⟩δm2+⟨σ3nrk⟩δm3−⟨σm2rk⟩δn2−⟨σm3rk⟩δn3i,(C.1)
165
166 C Calculations: Chain of V-type emitters
d
dt⟨σmnr†
k⟩=ih∆2⟨σ2nr†
k⟩δm2−⟨σm2r†
k⟩δn2+ ∆3⟨σ3nr†
k⟩δm3−⟨σm3r†
k⟩δn3
+ Ω ⟨σ1nr†
k⟩δm2+⟨σ2nr†
k⟩δm1−⟨σm2r†
k⟩δn1−⟨σm1r†
k⟩δn2
+ Ω ⟨σ1nr†
k⟩δm3+⟨σ3nr†
k⟩δm1−⟨σm3r†
k⟩δn1−⟨σm1r†
k⟩δn3
+ωk⟨σmnr†
k⟩
+gk⟨r†
krk⟩+ 1(⟨σ2n⟩δm2+⟨σ3n⟩δm3)−gk⟨r†
krk⟩(⟨σm2⟩δn2+⟨σm3⟩δn3)i,(C.2)
d
dt⟨σmnrk⟩=ih∆2(⟨σ2nrk⟩δm2−⟨σm2rk⟩δn2)+∆3(⟨σ3nrk⟩δm3−⟨σm3rk⟩δn3)
+ Ω (⟨σ1nrk⟩δm2+⟨σ2nrk⟩δm1−⟨σm2rk⟩δn1−⟨σm1rk⟩δn2)
+ Ω (⟨σ1nrk⟩δm3+⟨σ3nrk⟩δm1−⟨σm3rk⟩δn1−⟨σm1rk⟩δn3)
−ωk⟨σmnrk⟩
+gk⟨r†
krk⟩(⟨σ2n⟩δm2+⟨σ3n⟩δm3)−gk⟨r†
krk⟩+ 1(⟨σm2⟩δn2+⟨σm3⟩δn3)i.(C.3)
C.3 Polaron master equation: Derivation
Since the single V-type emitter is a special case of the chain of Nemitters, we only derive
the polaron master equation for the elaborate case. In difference to the derivation of the
polaron master equation for the polaronic Kitaev chain (see Appendix B), here we assume
separate baths for each emitter.
C.3.1 Polaron transformation
As a first step, the open system Hamiltonian is transformed into a polaronic frame. We
apply the unitary polaron transformation Hp=UpHU−1
pwith
Up= exp "N−1
X
l=0 Zd3k1
ωklgklσl
22 +gklσl
33r†
k−rk#,(C.4)
and define collective bosonic operators
Rl=Zd3kl
gkl
ωkl
rkl,(C.5)
resulting in
Up= exp hN−1
X
l=0 X
i=2,3
σl
ii(R†
l−Rl)i.(C.6)
The transformation is performed via multiple applications of the Baker-Campbell-Hausdorff
formula
eXY e−X=
∞
X
n=0
1
n![X, Y ]n,(C.7)
C.3 Polaron master equation: Derivation 167
with [X, Y ]n= [X, [X, Y ]n−1],[X, Y ]0=Y. We start with the free electronic contribu-
tion
Hel,0=ℏ∆2σl
22 +ℏ∆3σl
33.(C.8)
The transformation reads
1
ℏUpHel,0U−1
p=
∞
X
n=0
1
n!"N−1
X
l=0 nσl
22(R†
l−Rl) + σl
33(R†
l−Rl)o,
N−1
X
l′=0 ∆2σl′
22 + ∆3σl′
33#n
.
(C.9)
Since Upσl
22U−1
p=σl
22,Upσl
33U−1
p=σl
33, the shape of the Hamiltonian does not change.
Next, we transform the electron-light interaction Hamiltonian Hlwith
1
ℏUpHlU−1
p=
∞
X
n=0
1
n!"N−1
X
l=0 nσl
22(R†
l−Rl) + σl
33(R†
l−Rl)o,
N−1
X
l′=0
Ωσl′
12 +σl′
21 +σl′
13 +σl′
31#n
.
(C.10)
We calculate the first three terms of the sum and identify an exponential factorization of
the entire expression, resulting in
1
ℏUpHlU−1
p=
N−1
X
l=0 X
i=2,3
Ωh(σl
1i+σl
i1) cosh(R†
l−Rl)+(σl
i1−σl
1i) sinh(R†
l−Rl)i.(C.11)
The homogeneous phonon contribution yields
1
ℏUpHph,0U−1
p=
N−1
X
l=0 Zd3kl
ωklr†
klrkl−X
i=2,3
gklσl
ii(r†
kl+rkl) + X
i=2,3
g2
kl
ωkl
σl
ii
,(C.12)
and the transformed electron-phonon interaction Hamiltonian is given by
1
ℏUpHel,phU−1
p=
N−1
X
l=0 X
i=2,3Zd3kl"gklσl
ii(r†
kl+rkl)−2g2
kl
ωkl
σl
ii#.(C.13)
Lastly, the transformation of the Dexter interdot coupling Hamiltonian yields
1
ℏUpHDU−1
p=
N−2
X
l=0 nhf(t)σl
35 +f∗(t)σl
53icosh hR†
l+1 −Rl+1 −R†
l−Rli
+hf∗(t)σl
53 −f(t)σl
35isinh hR†
l+1 −Rl+1 −R†
l−Rlio.(C.14)
We define Hp=Hp,0+Hp,I and get the full polaron-transformed open system Hamiltonian
with
Hp,0/ℏ=
N−1
X
l=0
X
i=2,3
¯
∆iσl
ii +Zd3klωklr†
klrkl
,(C.15)
168 C Calculations: Chain of V-type emitters
and
Hp,I/ℏ=
N−1
X
l=0 X
i=2,3
Ωh(σl
1i+σl
i1) cosh(R†
l−Rl)+(σl
i1−σl
1i) sinh(R†
l−Rl)i
+
N−2
X
l=0 nhf(t)σl
35 +f∗(t)σl
53icosh hR†
l+1 −Rl+1 −R†
l−Rli
+hf∗(t)σl
53 −f(t)σl
35isinh hR†
l+1 −Rl+1 −R†
l−Rlio,(C.16)
where we have introduced the polaron-shifted detuning
¯
∆l
i= ∆i−Zd3kl
g2
kl
ωkl
.(C.17)
C.3.2 Franck-Condon renormalization
To simplify the evaluation of the upcoming polaron master equation, we renormalize the
polaron interaction Hamiltonian such that
trB{[Hp,I, ρ(t)]}= 0.(C.18)
We define
X[l]
+,i := σl
1i+σl
i1, X[l]
−,i := iσl
i1−σl
1i,(C.19a)
X[l]
+,f := eiδϵt/ℏσl
35 +e−iδϵt/ℏσl
53, X[l]
−,f := ie−iδϵt/ℏσl
53 −eiδϵt/ℏσl
35,(C.19b)
B[l]
p:= exp hR†
l−Rli, B[l]
m:= exp h−R†
l−Rli,(C.19c)
and
¯
Ωl:= Ωtr nB[l]
po= Ω ⟨eR†
l−Rl⟩,
¯
f0l:= f0tr nB[l+1]
pB[l]
mo=f0⟨eR†
l+1−Rl+1−(R†
l−Rl)⟩.(C.20)
The condition in Eq. (C.18) is fulfilled by subtracting PN−1
l=0 Pi=2,3¯
ΩlX[l]
+,i+PN−2
l=0 ¯
f0lX[l]
+,f
from Hp,I and in turn adding it to Hp,0. The Franck-Condon renormalized Hamiltonians
read
Hp,0/ℏ=
N−1
X
l=0
X
i=2,3¯
∆iσl
ii +¯
ΩX[l]
+,i+¯
f0lX[l]
+,f h1−δl,(N−1)i+Zd3klωklr†
klrkl
,
(C.21)
and
Hp,I/ℏ=
N−1
X
l=0 X
i=2,3nX[l]
+,i hΩ cosh R†
l−Rl−¯
Ωli−iΩX[l]
−,i sinh R†
l−Rlo
+
N−2
X
l=0 X[l]
+,f nf0cosh hR†
l+1 −Rl+1 −R†
l−Rli−¯
f0lo
−if0X[l]
−,f sinh hR†
l+1 −Rl+1 −(R†
l−Rl)i.(C.22)
C.3 Polaron master equation: Derivation 169
C.3.3 Polaron master equation
For the derivation of the polaron master equation we start from the Redfield equation,
d
dtρS(t) = −i
ℏ[Hp,0, ρS(t)] −1
ℏ2Zt
0
dτtrB{[Hp,I,[Hp,I (−τ), ρS(t)⊗ρB]]},(C.23)
and first evaluate the integrand [Hp,I ,[Hp,I(−τ), ρ]]. We define new collective bath oper-
ators in the interaction picture,
B[l]
+(t) = exp hRl(t)−R†
l(t)i+ exp n−hRl(t)−R†
l(t)io,(C.24a)
B[l]
−(t) = exp hRl(t)−R†
l(t)i−exp n−hRl(t)−R†
l(t)io,(C.24b)
and
C[l]
+(t) = exp nhRl+1(t)−R†
l+1(t)i−hRl(t)−R†
l(t)io
+ exp n−hRl+1(t)−R†
l+1(t)i+hRl(t)−R†
l(t)io,(C.25a)
C[l]
−(t) = exp nhRl+1(t)−R†
l+1(t)i−hRl(t)−R†
l(t)io
−exp n−hRl+1(t)−R†
l+1(t)i+hRl(t)−R†
l(t)io,(C.25b)
and rewrite the interaction Hamiltonian accordingly, resulting in
Hp,I(t)/ℏ=
N−1
X
l=0 X
i=2,3Ω
2X[l]
+,i(t)B[l]
+(t) + iΩ
2X[l]
−,i(t)B[l]
−(t)−¯
ΩlX[l]
+,i(t)
+
N−2
X
l=0 f0
2X[l]
+,f (t)C[l]
+(t) + if0
2X[l]
−,f (t)C[l]
−(t)−¯
f0lX[l]
+,f (t).(C.26)
For a simpler notation, we denote Xi(−τ) = X′
i,B(−τ) = B′,C(−τ) = C′. The integrand
is evaluated as
[Hp,I,[Hp,I (−τ), ρ]]
=
N−1
X
l′=0 X
i=2,3Hp,I,Ω
2X[l′]
+,i
′B[l′]
+
′, ρ+iΩ
2X[l′]
−,i
′B[l′]
−
′, ρ−¯
Ωl′X[l′]
+,i
′, ρ
+
N−2
X
l′=0 Hp,I,f0
2X[l′]
+,f
′C[l′]
+
′, ρ+if0
2X[l′]
−,f
′C[l′]
−
′, ρ−¯
f0l′X[l′]
+,f
′, ρ
170 C Calculations: Chain of V-type emitters
=
N−1
X
l,l′=0 X
i,j=2,3 Ω2
4X[l]
+,jB[l]
+,X[l′]
+,i
′B[l′]
+
′, ρ+iΩ2
4X[l]
−,jB[l]
−,X[l′]
+,i
′B[l′]
+
′, ρ
−Ω
2¯
ΩlX[l]
+,j,X[l′]
+,i
′B[l′]
+
′, ρ+iΩ2
4X[l]
+,jB[l]
+,X[l′]
−,i
′B[l′]
−
′, ρ
−Ω2
4X[l]
−,jB[l]
−,X[l′]
−,i
′B[l′]
−
′, ρ−iΩ
2¯
ΩlX[l]
+,j,X[l′]
−,i
′B[l′]
−
′, ρ
−Ω
2¯
Ωl′X[l]
+,jB[l]
+,X[l′]
+,i
′, ρ−iΩ
2¯
Ωl′X[l]
−,jB[l]
−,X[l′]
+,i
′, ρ
+¯
Ωl¯
Ωl′X[l]
+,j,X[l′]
+,i
′, ρ!
+
N−1
X
l,l′=0 X
i=2,3
[1 −δl,(N−1)] Ωf0
4X[l]
+,f C[l]
+,X[l′]
+,i
′B[l′]
+
′, ρ+iΩf0
4X[l]
−,f C[l]
−,X[l′]
+,i
′B[l′]
+
′, ρ
−Ω
2¯
f0lX[l]
+,f ,X[l′]
+,i
′B[l′]
+
′, ρ+iΩf0
4X[l]
+,f C[l]
+,X[l′]
−,i
′B[l′]
−
′, ρ
−Ωf0
4X[l]
−,f C[l]
−,X[l′]
−,i
′B[l′]
−
′, ρ−iΩ
2¯
f0lX[l]
+,f ,X[l′]
−,i
′B[l′]
−
′, ρ
−f0
2¯
Ωl′X[l]
+,f C[l]
+,X[l′]
+,i
′, ρ−if0
2¯
Ωl′X[l]
−,f C[l]
−,X[l′]
+,i
′, ρ
+¯
f0l¯
Ωl′X[l]
+,f ,X[l′]
+,i
′, ρ!
+
N−1
X
l,l′=0 X
j=2,3
[1 −δl′,(N−1)] Ωf0
4X[l]
+,jB[l]
+,X[l′]
+,f
′C[l′]
+
′, ρ+iΩf0
4X[l]
−,jB[l]
−,X[l′]
+,f
′C[l′]
+
′, ρ
−f0
2¯
ΩlX[l]
+,j,X[l′]
+,f
′C[l′]
+
′, ρ+iΩf0
4X[l]
+,jB[l]
+,X[l′]
−,f
′C[l′]
−
′, ρ
−Ωf0
4X[l]
−,jB[l]
−,X[l′]
−,f
′C[l′]
−
′, ρ−if0
2¯
ΩlX[l]
+,j,X[l′]
−,f
′C[l′]
−
′, ρ
−Ω
2¯
f0l′X[l]
+,jB[l]
+,X[l′]
+,f
′, ρ−iΩ
2¯
f0l′X[l]
−,jB[l]
−,X[l′]
+,f
′, ρ
+¯
Ωl¯
f0l′X[l]
+,j,X[l′]
+,f
′, ρ!
+
N−2
X
l,l′=0 f2
0
4X[l]
+,f C[l]
+,X[l′]
+,f
′C[l′]
+
′, ρ+if2
0
4X[l]
−,f C[l]
−,X[l′]
+,f
′C[l′]
+
′, ρ
−f0
2¯
f0lX[l]
+,f ,X[l′]
+,f
′C[l′]
+
′, ρ+if2
0
4X[l]
+,f C[l]
+,X[l′]
−,f
′C[l′]
−
′, ρ
−f2
0
4X[l]
−,f C[l]
−,X[l′]
−,f
′C[l′]
−
′, ρ−if0
2¯
f0lX[l]
+,f ,X[l′]
−,f
′C[l′]
−
′, ρ
−f0
2¯
f0l′X[l]
+,f C[l]
+,X[l′]
+,f
′, ρ−if0
2¯
f0l′X[l]
−,f C[l]
−,X[l′]
+,f
′, ρ
+¯
f0l¯
f0l′X[l]
+,f ,X[l′]
+,f
′, ρ!.(C.27)
C.3 Polaron master equation: Derivation 171
The most complicated terms in Eq. (C.27) are of the form [XjB, [X′
iC′, ρ]] for i, j =
{2,3, f}. Taking the trace over the bath yields
trB{XjB, X′
iC′, ρ}
= trB{XjBX′
iC′ρ−XjBρX′
iC′−X′
iC′ρXjB+ρX′
iC′XjB}
=XjX′
iρS⟨BC′⟩−XjρSX′
i⟨C′B⟩−X′
iρSXj⟨BC′⟩+ρSX′
iXj⟨C′B⟩
=⟨BC′⟩XjX′
iρS−X′
iρSXj+⟨C′B⟩ρSX′
iXj−XjρSX′
i
=⟨BC′⟩Xj, X′
iρS−⟨C′B⟩Xj, ρSX′
i.(C.28)
Moreover, we have
trB{XjB, X′
iB′, ρ}=⟨BB′⟩Xj, X′
iρS−⟨B′B⟩Xj, ρSX′
i,
trB{Xj,X′
iB′
i, ρ}=⟨B′
i⟩Xj, X′
iρS−Xj, ρSX′
i,
trB{XjBj,X′
i, ρ}=⟨Bj⟩Xj, X′
iρS−Xj, ρSX′
i,
trB{Xj,X′
i, ρ}=Xj, X′
iρS−Xj, ρSX′
i.(C.29)
It can be shown that ⟨B[l]
+B[l]
−
′⟩=⟨B[l]
−B[l]
+
′⟩= 0 ∀l, l′. Since ⟨eR−R†⟩=⟨e−(R−R†)⟩we
have
⟨B[l]
+⟩= 2 ⟨eRl−R†
l⟩,⟨B[l]
−⟩= 0.(C.30)
In the same fashion, we obtain ⟨C[l]
+C[l]
−
′⟩=⟨C[l]
−C[l]
+
′⟩= 0 ∀l, l′and
⟨C[l]
+⟩= 2 ⟨e(Rl+1−R†
l+1)−(Rl−R†
l)⟩,⟨C[l]
−⟩= 0.(C.31)
Lastly, this also yields ⟨B[l]
+C[l]
−
′⟩=⟨B[l]
−C[l]
+
′⟩= 0 ∀l, l′and ⟨C[l]
+B[l]
−
′⟩=⟨C[l]
−B[l]
+
′⟩= 0 ∀l, l′.
Next, we calculate ⟨B[l]
+
′⟩,
⟨B[l]
+
′⟩= trBρBe˜
Rl−˜
Rl
†+e−(˜
Rl−˜
Rl
†)
= 2trBρB1 + 1
2! ˜
Rl−˜
Rl
†2+. . .
= 2trB
ρB
1 + 1
2! "Zd3kl
gkl
ωklrkleiωklτ−r†
kle−iωklτ#2
+. . .
= 2 exp −1
2Zd3kl|gkl/ωk|2(2nkl+ 1)=⟨B[l]
+⟩.(C.32)
⟨C[l]
+
′⟩is calculated analogously, resulting in
⟨C[l]
+
′⟩= trBnρBhe(˜
Rl+1−˜
R†
l+1)−(˜
Rl−˜
R†
l)+e−(˜
Rl+1−˜
R†
l+1)+( ˜
Rl−˜
R†
l)io
= 2trBnρBh1 + 1
2! ˜
Rl+1 −˜
R†
l+1 −˜
Rl+˜
R†
l2+. . . io
= 2trBnρB[1 + 0 + . . .]o=2=⟨C[l]
+⟩.(C.33)
172 C Calculations: Chain of V-type emitters
However, [˜
Rl+1 −˜
R†
l+1 −˜
Rl+˜
R†
l)]2= 0, and therefore all terms depending on phonon
operators vanish in the expectation value. With this and Eqs. (C.30) and (C.31) we have
also evaluated
⟨e(Rl−R†
l)⟩= exp "−1
2Zd3kl
gkl
ωkl
2(2nkl+ 1)#,(C.34)
and ⟨e(Rl+1−R†
l+1)−(Rl−R†
l)⟩= 1. Moreover, it follows ¯
f0l=f0.
trB{[Hp,I,[Hp,I (−τ), ρ]]}now takes the form
N−1
X
l,l′=0 X
i,j=2,3(Ω2
4⟨B[l]
+B[l′]
+
′⟩X[l]
+,j, X[l′]
+,i
′ρS−⟨B[l′]
+
′B[l]
+⟩X[l]
+,j, ρSX[l′]
+,i
′
−Ω
2¯
Ωl⟨B[l′]
+
′⟩X[l]
+,j, X[l′]
+,i
′ρS−X[l]
+,j, ρSX[l′]
+,i
′
−Ω2
4⟨B[l]
−B[l′]
−
′⟩X[l]
−,j, X[l′]
−,i
′ρS−⟨B[l′]
−
′B[l]
−⟩X[l]
−,j, ρSX[l′]
−,i
′
−Ω
2¯
Ωl′⟨B[l]
+⟩X[l]
+,j, X[l′]
+,i
′ρS−X[l]
+,j, ρSX[l′]
+,i
′
+¯
Ωl¯
Ωl′X[l]
+,j, X[l′]
+,i
′ρS−X[l]
+,j, ρSX[l′]
+,i
′)
+
N−1
X
l,l′=0 X
i=2,3
[1 −δl,(N−1)](Ωf0
4⟨C[l]
+B[l′]
+
′⟩X[l]
+,f , X[l′]
+,i
′ρS−⟨B[l′]
+
′C[l]
+⟩X[l]
+,f , ρSX[l′]
+,i
′
−Ω
2¯
f0l⟨B[l′]
+
′⟩X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′
−Ωf0
4⟨C[l]
−B[l′]
−
′⟩X[l]
−,f , X[l′]
−,i
′ρS−⟨B[l′]
−
′C[l]
−⟩X[l]
−,f , ρSX[l′]
−,i
′
−f0
2¯
Ωl′⟨C[l]
+⟩X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′
+¯
f0l¯
Ωl′X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′)
+
N−1
X
l,l′=0 X
j=2,3
[1 −δl′,(N−1)](Ωf0
4⟨B[l]
+C[l′]
+
′⟩X[l]
+,j, X[l′]
+,f
′ρS−⟨C[l′]
+
′B[l]
+⟩X[l]
+,j, ρSX[l′]
+,f
′
−f0
2¯
Ωl⟨C[l′]
+
′⟩X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′
−Ωf0
4⟨B[l]
−C[l′]
−
′⟩X[l]
−,j, X[l′]
−,f
′ρS−⟨C[l′]
−
′B[l]
−⟩X[l]
−,j, ρSX[l′]
−,f
′
−Ω
2¯
f0l′⟨B[l]
+⟩X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′
C.3 Polaron master equation: Derivation 173
+¯
Ωl¯
f0l′X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′)
+
N−2
X
l,l′=0 (f2
0
4⟨C[l]
+C[l′]
+
′⟩X[l]
+,f , X[l′]
+,f
′ρS−⟨C[l′]
+
′C[l]
+⟩X[l]
+,f , ρSX[l′]
+,f
′
−f0
2¯
f0l⟨C[l′]
+
′⟩X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′
−f2
0
4⟨C[l]
−C[l′]
−
′⟩X[l]
−,f , X[l′]
−,f
′ρS−⟨C[l′]
−
′C[l]
−⟩X[l]
−,f , ρSX[l′]
−,f
′
−f0
2¯
f0l′⟨C[l]
+⟩X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′
+¯
f0l¯
f0l′X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′).(C.35)
From ¯
Ωl= Ω ⟨e−(Rl−R†
l)⟩, it follows
⟨B[l]
+⟩=2¯
Ωl
Ω.(C.36)
Together with ¯
f0l=f0and ⟨C[l]
+⟩= 2 this results in
trB{[Hp,I,[Hp,I (−τ), ρ]]}
=
N−1
X
l,l′=0 X
i,j=2,3(Ω2
4⟨B[l]
+B[l′]
+
′⟩X[l]
+,j, X[l′]
+,i
′ρS−⟨B[l′]
+
′B[l]
+⟩X[l]
+,j, ρSX[l′]
+,i
′
−Ω2
4⟨B[l]
−B[l′]
−
′⟩X[l]
−,j, X[l′]
−,i
′ρS−⟨B[l′]
−
′B[l]
−⟩X[l]
−,j, ρSX[l′]
−,i
′
−¯
Ωl¯
Ωl′X[l]
+,j, X[l′]
+,i
′ρS−X[l]
+,j, ρSX[l′]
+,i
′)
+
N−1
X
l,l′=0 X
i=2,3
[1 −δl,(N−1)](Ωf0
4⟨C[l]
+B[l′]
+
′⟩X[l]
+,f , X[l′]
+,i
′ρS−⟨B[l′]
+
′C[l]
+⟩X[l]
+,f , ρSX[l′]
+,i
′
−Ωf0
4⟨C[l]
−B[l′]
−
′⟩X[l]
−,f , X[l′]
−,i
′ρS−⟨B[l′]
−
′C[l]
−⟩X[l]
−,f , ρSX[l′]
−,i
′
−f0¯
Ωl′X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′)
+
N−1
X
l,l′=0 X
j=2,3
[1 −δl′,(N−1)](Ωf0
4⟨B[l]
+C[l′]
+
′⟩X[l]
+,j, X[l′]
+,f
′ρS−⟨C[l′]
+
′B[l]
+⟩X[l]
+,j, ρSX[l′]
+,f
′
−Ωf0
4⟨B[l]
−C[l′]
−
′⟩X[l]
−,j, X[l′]
−,f
′ρS−⟨C[l′]
−
′B[l]
−⟩X[l]
−,j, ρSX[l′]
−,f
′
−¯
Ωlf0X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′)
174 C Calculations: Chain of V-type emitters
+
N−2
X
l,l′=0 (f2
0
4⟨C[l]
+C[l′]
+
′⟩X[l]
+,f , X[l′]
+,f
′ρS−⟨C[l′]
+
′C[l]
+⟩X[l]
+,f , ρSX[l′]
+,f
′
−f2
0
4⟨C[l]
−C[l′]
−
′⟩X[l]
−,f , X[l′]
−,f
′ρS−⟨C[l′]
−
′C[l]
−⟩X[l]
−,f , ρSX[l′]
−,f
′
−f2
0X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′).(C.37)
Bath correlations
In the upcoming calculations of the bath correlations, we require
hRl−R†
l,˜
Rl′(−τ)−˜
R†
l′(−τ)i= 2iZd3kl(g2
kl/ω2
kl) sin (ωklτ) =: 2iϕs(τ),(C.38)
and apply the simplified Baker-Campbell-Hausdorff formula eXeY=eX+Ye1/2[X,Y ], which
is valid if [X, [X, Y ]] = 0 and [Y, [Y, X]] = 0. The first reservoir correlation is evaluated
as
⟨B[l]
+B[l′]
+(−τ)⟩= trBnρBhe(Rl−R†
l)+e−(Rl−R†
l)ihe(˜
Rl′−˜
R†
l′)+e−(˜
Rl′−˜
R†
l′)io
= trBnρBhe(R†
l+˜
R†
l′)−(Rl+˜
Rl′))e1/2(Rl−R†
l),(˜
Rl′−˜
R†
l′)
+e−(R†
l+˜
R†
l′)+(Rl+˜
Rl′))e1/2(Rl−R†
l),(˜
Rl′−˜
R†
l′)+e(R†
l−˜
R†
l′)−(Rl−˜
Rl′))e1/2(Rl−R†
l),(˜
Rl′−˜
R†
l′)(−1)
+e−(R†
l−˜
R†
l′)+(Rl−˜
Rl′))e1/2(Rl−R†
l),(˜
Rl′−˜
R†
l′)(−1)io,(C.39)
with e1/2(Rl−R†
l),(˜
Rl′−˜
R†
l′)=eiϕs(τ), leading to
⟨B[l]
+B[l′]
+(−τ)⟩
= 2trBρBn1 + 1
2! hR†
l+˜
R†
l′−Rl+˜
Rl′i2+1
4! hR†
l+˜
R†
l′−Rl+˜
Rl′i4+. . . oeiϕs(τ)
+n1 + 1
2! hR†
l−˜
R†
l′−Rl−˜
Rl′i2+1
4! hR†
l−˜
R†
l′−Rl−˜
Rl′i4+. . . oe−iϕs(τ),
(C.40)
with
Rl±˜
Rl′(−τ) = Zd3kl
gkl
ωkl
rkl±Zd3k′
l
gkl′
ωkl′
eiωkl′τrkl′,
R†
l±˜
R†
l′(−τ) = Zd3kl
gkl
ωkl
r†
kl±Zd3k′
l
gkl′
ωkl′
e−iωkl′τr†
kl′.(C.41)
C.3 Polaron master equation: Derivation 175
Hence, we arrive at
trBn˜ρBh(R†
l+˜
R†
l′)−(Rl+˜
Rl′)i2o
=−Zd3kl
g2
kl
ω2
klh(2nkl+ 1) + nkleiωklτ+ (nkl+ 1)e−iωklτi
−Zd3k′
l
g2
kl′
ω2
kl′h(2nkl′+ 1) + nkl′e−iωkl′τ+ (nkl′+ 1)eiωkl′τi
=−Zd3kl
g2
kl
ω2
kl{(2nkl+ 1) [2 + 2 cos (ωklτ)]}.(C.42)
In the last step we have assumed identical baths. Since all that remains are complex
numbers and the sums over kland kl′contain the same elements, they can be summarized
in a single sum. We factorize the next term using Wick’s theorem and arrive at
trB˜ρBhR†
l+˜
R†
l′−Rl+˜
Rl′i4
≈3 −Zd3kl
g2
kl
ω2
kl{(2nkl+ 1)[1 + cos (ωklτ)] −sin (ωklτ)}
−Zd3k′
l
g2
kl′
ω2
kl′n(2nkl′+ 1)[1 + cos (ωkl′τ)] + sin (ωkl′τ)o!
−Zd3kl2
g2
kl2
ω2
kl2{(2nkl2+ 1)[1 + cos (ωkl2τ)] −sin (ωkl2τ)}
−Zd3k′
l2
g2
kl2′
ω2
kl2′n(2nkl2′+ 1)[1 + cos (ωkl2′τ)] + sin (ωkl2′τ)o!
= 3 Zd3klZd3k′
l
g2
kl
ω2
kl
g2
kl′
ω2
kl′n(2nkl+ 1)[2 + 2 cos (ωklτ)](2nkl′+ 1)[2 + 2 cos (ωkl′τ)]
+ (2nkl+ 1)[2 + 2 cos (ωklτ)] sin (ωkl′τ)−(2nkl′+ 1)[2 + 2 cos (ωkl′τ)] sin (ωklτ)o.
(C.43)
In total, we get
⟨B[l]
+B[l′]
+(−τ)⟩= 2 exp −1
2Zd3k|fl
+|2(2nk+ 1) + iϕs(τ)
+ 2 exp −1
2Zd3k|fl
−|2(2nk+ 1) −iϕs(τ),(C.44)
with
|fl
±|2=g2
kl
ω2
kl
[2 ±2 cos (ωkτ)] .(C.45)
176 C Calculations: Chain of V-type emitters
Moreover, we have assumed the reservoir in a thermal equilibrium state, allowing for its
description in terms of a Bose distribution, nkl= 1/(eℏωkl/kBT−1), resulting in
(2nkl+ 1) = coth ℏωkl
2kBT.(C.46)
Inserting Eqs. (C.45) and (C.46) into Eq. (C.44) and making use of the definition ϕs(τ)
[Eq. (C.38)] yields the final expression for the bath correlation
⟨B+,jB+,i(−τ)⟩= 4 exp "−Zd3kl
g2
kl
ω2
kl
coth ℏωkl
2kBT#
×cosh (Zd3kl
g2
kl
ω2
klcos (ωklτ) coth ℏωkl
2kBT−isin(ωklτ)).(C.47)
We define the reservoir correlation function
ϕ(τ) := Zd3kl
g2
kl
ω2
kcoth ℏωkl
2kBTcos (ωklτ)−isin(ωklτ),(C.48)
and finally arrive at
⟨B[l]
+B[l′]
+(−τ)⟩= 4 exp [−ϕ(0)] cosh [ϕ(τ)] .(C.49)
⟨B[l]
−B[l′]
−(−τ)⟩is calculated in the same fashion. The calculation of ⟨B[l′]
+(−τ)B[l]
+⟩has
already been done up to a sign in ϕs(τ). The resulting terms read
⟨B[l′]
+(−τ)B[l]
+⟩= 4 exp [−ϕ(0)] cosh [ϕ∗(τ)] = ⟨B[l]
+B[l′]
+(−τ)⟩∗,
⟨B[l]
−B[l′]
−(−τ)⟩=−4 exp [−ϕ(0)] sinh [ϕ(τ)],
⟨B[l′]
−(−τ)B[l]
−⟩=−4 exp [−ϕ(0)] sinh [ϕ∗(τ)] = ⟨B[l]
−B[l′]
−(−τ)⟩∗.(C.50)
We still have to calculate correlations including C[l]. In the upcoming calculations, we
require
hRl+1 −R†
l+1−Rl−R†
l,˜
Rl′+1 −˜
R†
l′+1−˜
Rl′−˜
R†
l′i= 0,(C.51)
and hRl+1 −R†
l+1−Rl−R†
l,˜
Rl′−˜
R†
l′i= 0.(C.52)
Again we make use of the simplified Baker-Campbell-Hausdorff formula eXeY=eX+Ye1/2[X,Y ],
which is valid if [X, [X, Y ]] = 0 and [Y, [Y, X]] = 0. The next reservoir correlation is eval-
uated as
⟨C[l]
+C[l′]
+(−τ)⟩= trBnρBhe(Rl+1+˜
Rl′+1)−(R†
l+1+˜
R†
l′+1)+(R†
l+˜
R†
l′)−(Rl+˜
Rl′)
+e(Rl+1−˜
Rl′+1)−(R†
l+1−˜
R†
l′+1)+(R†
l−˜
R†
l′)−(Rl−˜
Rl′)
+e−(Rl+1−˜
Rl′+1)+(R†
l+1−˜
R†
l′+1)−(R†
l−˜
R†
l′)+(Rl−˜
Rl′)
+e−(Rl+1+˜
Rl′+1)+(R†
l+1+˜
R†
l′+1)−(R†
l+˜
R†
l′)+(Rl+˜
Rl′)io.(C.53)
C.3 Polaron master equation: Derivation 177
Once again the series expansion of the cosh function is used, resulting in
trBn˜ρBhRl+1 +˜
Rl′+1−R†
l+1 +˜
R†
l′+1+R†
l+˜
R†
l′−Rl+˜
Rl′i2o
=−Zd3kl
g2
kl
ω2
klh(2nkl+ 1) + nkleiωklτ+ (nkl+ 1)e−iωklτi
−Zd3k′
l
g2
kl′
ω2
kl′h(2nkl′+ 1) + nkl′e−iωkl′τ+ (nkl′+ 1)eiωkl′τi
+Zd3kl
g2
kl
ω2
klh(2nkl+ 1) + nkleiωklτ+ (nkl+ 1)e−iωklτi
+Zd3k′
l
g2
kl′
ω2
kl′h(2nkl′+ 1) + nkl′e−iωkl′τ+ (nkl′+ 1)eiωkl′τi= 0.(C.54)
Analogous to before, we find
hRl+1 +˜
Rl′+1−R†
l+1 +˜
R†
l′+1+R†
l+˜
R†
l′−Rl+˜
Rl′i2= 0,(C.55)
resulting in
⟨C[l]
+C[l′]
+(−τ)⟩=⟨C[l′]
+(−τ)C[l]
+⟩= trB{ρB(2 + 2)}= 4.(C.56)
Applying the same treatment to ⟨C[l]
−C[l′]
−(−τ)⟩results in
⟨C[l]
−C[l′]
−(−τ)⟩=⟨C[l′]
−(−τ)C[l]
−⟩= 0.(C.57)
Due to the structure of the reservoir correlations, we once again find ⟨C+C−⟩=⟨C−C+⟩=
0.
We still have to calculate the mixed bath contributions ⟨CB⟩and ⟨BC⟩:
⟨C[l]
+B[l′]
+(−τ)⟩
= trBnρBhe(Rl+1−R†
l+1)+(R†
l+˜
R†
l′)−(Rl+˜
Rl′)+e(Rl+1−R†
l+1)+(R†
l−˜
R†
l′)−(Rl−˜
Rl′)
+e−(Rl+1−R†
l+1)−(R†
l−˜
R†
l′)+(Rl−˜
Rl′)+e−(Rl+1−R†
l+1)−(R†
l+˜
R†
l′)+(Rl+˜
Rl′)io.(C.58)
Using the series expansion of the cosh function yields
trBn˜ρBhRl+1 −R†
l+1+R†
l+˜
R†
l′−Rl+˜
Rl′i2o
=Zd3kl+ 1 g2
kl+1
ω2
kl+1 hnkl+1 eiωkl+1 τ+ (nkl+1 + 1)e−iωkl+1 τi
−Zd3kl
g2
kl
ω2
klhnkleiωklτ+ (nkl+ 1)e−iωklτi−Zd3k′
l
g2
kl′
ω2
kl′2nkl′+ 1
=−Zd3kl
g2
kl
ω2
kl
(2nkl+ 1).(C.59)
178 C Calculations: Chain of V-type emitters
In a completely analogous calculation, we obtain
trB˜ρBhRl+1 −R†
l+1+R†
l−˜
R†
l′−Rl−˜
Rl′i2=−Zd3kl
g2
kl
ω2
kl
(2nkl+1).(C.60)
Following the same steps as before, this results in
⟨C[l]
+B[l′]
+(−τ)⟩=⟨B[l′]
+(−τ)C[l]
+⟩= 4 exp −1
2ϕ(0).(C.61)
The other mixed reservoir contribution is evaluated as
⟨C[l′]
+(−τ)B[l]
+⟩
= trBnρBhe(˜
Rl′+1−˜
R†
l′+1)+( ˜
R†
l′+R†
l)−(˜
Rl′+Rl)+e(˜
Rl′+1−˜
R†
l′+1)+( ˜
R†
l′−R†
l)−(˜
Rl′−Rl)
+e−(˜
Rl′+1−˜
R†
l′+1)−(˜
R†
l′−R†
l)+( ˜
Rl′−Rl)+e−(˜
Rl′+1−˜
R†
l′+1)−(˜
R†
l′+R†
l)+( ˜
Rl′+Rl)io.(C.62)
Again, we employ the series expansion of the cosh function, resulting in
trB˜ρBh˜
Rl′+1 −˜
R†
l′+1+˜
R†
l′+R†
l−˜
Rl′+Rli2=−Zd3kl
g2
kl
ω2
kl
(2nkl+ 1),
(C.63)
and
trB˜ρBh˜
Rl′+1 −˜
R†
l′+1+˜
R†
l′−R†
l−˜
Rl′−Rli2=−Zd3kl
g2
kl
ω2
kl
(2nkl+ 1).
(C.64)
Following the same steps as before, this leads to
⟨C[l′]
+(−τ)B[l]
+⟩=⟨B[l]
+C[l′]
+(−τ)⟩= 4 exp −1
2ϕ(0).(C.65)
The remaining contributions are given by
⟨C[l]
−B[l′]
−(−τ)⟩=⟨B[l′]
−(−τ)C[l]
−⟩= 0,
⟨C[l′]
−(−τ)B[l]
−⟩=⟨B[l]
−C[l′]
−(−τ)⟩= 0,(C.66)
and all mixed contributions of the form ⟨B+C−⟩,⟨B−C+⟩are once again zero.
C.3 Polaron master equation: Derivation 179
Resulting master equation
Making use of ¯
Ω = Ω exp(−ϕ(0)/2),¯
f0=f0, and inserting the correlation functions into
Eq. (B.27) yields the final result
trB{[Hp,I,[Hp,I (−τ), ρ]]}
=
N−1
X
l,l′=0 X
i,j=2,3
Ω2e−ϕ(0)((cosh[ϕ(τ)] −1) X[l]
+,j, X[l′]
+,i
′ρS−(cosh[ϕ∗(τ)] −1) X[l]
+,j, ρSX[l′]
+,i
′
+ sinh[ϕ(τ)] X[l]
−,j, X[l′]
−,i
′ρS−sinh[ϕ∗(τ)] X[l]
−,j, ρSX[l′]
−,i
′)
+
N−1
X
l,l′=0 X
i=2,3h1−δl,(N−1)i(Ωf0e−ϕ(0)/2X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′
−Ωf0e−ϕ(0)/2X[l]
+,f , X[l′]
+,i
′ρS−X[l]
+,f , ρSX[l′]
+,i
′)
+
N−1
X
l,l′=0 X
j=2,3h1−δl′,(N−1)i(Ωf0e−ϕ(0)/2X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′
−Ωf0e−ϕ(0)/2X[l]
+,j, X[l′]
+,f
′ρS−X[l]
+,j, ρSX[l′]
+,f
′)
+
N−2
X
l,l′=0 (f2
0X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′
−f2
0X[l]
+,f , X[l′]
+,f
′ρS−X[l]
+,f , ρSX[l′]
+,f
′)
=
N−1
X
l,l′=0 X
i,j=2,3
Ω2e−ϕ(0)((cosh[ϕ(τ)] −1) X[l]
+,j, X[l′]
+,i
′ρS
−(cosh[ϕ∗(τ)] −1) X[l]
+,j, ρSX[l′]
+,i
′
+ sinh[ϕ(τ)] X[l]
−,j, X[l′]
−,i
′ρS−sinh[ϕ∗(τ)] X[l]
−,j, ρSX[l′]
−,i
′),(C.67)
and the master equation takes the form
d
dtρS(t) = −i
ℏ[Hp,0, ρS(t)] −¯
Ω2
ℏ2
N−1
X
l,l′=0 X
i,j=2,3
×Zt
0
dτ G+(τ)X[l]
+,j, X[l′]
+,i
′ρS+G−(τ)X[l]
−,j, X[l′]
−,i
′ρS+ H.c.,(C.68)
180 C Calculations: Chain of V-type emitters
with
G+(τ) = cosh [ϕ(τ)] −1,(C.69a)
G−(τ) = sinh [ϕ(τ)] ,(C.69b)
ϕ(τ) = Zd3kl
g2
kl
ω2
klcoth ℏωkl
2kBTcos(ωklτ)−isin(ωklτ),(C.69c)
and
X[l]
+,i(t) = hσl
1i(t) + σl
i1(t)i,(C.70a)
X[l]
−,i(t) = ihσl
i1(t)−σl
1i(t)i,(C.70b)
where σl
mn =σ(m+3l)(n+3l).
C.4 Polaron master equation: Equations of motion
C.4.1 Single V-type emitter
The polaron master equation for a single V-type emitter reads
d
dtρS(t) = −i
ℏ[Hp,0, ρS(t)]
−¯
Ω2
ℏ2X
i,j=2,3Zt
0
dτ X
s=±{Gs(τ) [Xs,j, Xs,i(−τ)ρS(t)] + H.c.},(C.71)
with
X+,i(τ)=[σ1i(τ) + σi1(τ)] ,(C.72a)
X−,i(τ) = i[σi1(τ)−σ1i(τ)] ,(C.72b)
¯
Ω = Ω exp [−ϕ(0)/2] .(C.72c)
We calculate the equations of motion for all density matrix elements ρmn := ⟨m|ρS|n⟩.
For a shorter notation, we denote ⟨m|Xl,i(−τ)|n⟩=: Xmn
l,i (−τ). The resulting equations
of motion read
d
dtρmn =X
i=2,3hi¯
∆l
i(ρmiδni −ρinδmi) + i¯
Ω(ρm1δni +ρmiδn1−ρinδm1−ρ1nδmi)i
−¯
Ω2
ℏ2X
i,j=2,3Zt
0
dτ ⟨m|χij(τ)|n⟩,(C.73)
C.4 Polaron master equation: Equations of motion 181
where
⟨m|χij(τ)|n⟩=
3
X
q=1 ρqnnhGij
+(τ)Xiq
+,j(−τ)−iGij
−(τ)Xiq
−,j(−τ)iδm1
+hGij
+(τ)X1q
+,j(−τ) + iGij
−(τ)X1q
−,j(−τ)iδmio
+ρmqnhGij∗
+(τ)Xq1
+,j(−τ)−iGij∗
−(τ)Xq1
−,j(−τ)iδni
+hGij∗
+(τ)Xqi
+,j(−τ) + iGij∗
−(τ)Xqi
−,j(−τ)iδn1o
+ρq1h−Gij
+(τ)Xmq
+,j(−τ) + iGij
−(τ)Xmq
−,j(−τ)iδni
+ρqih−Gij
+(τ)Xmq
+,j(−τ)−iGij
−(τ)Xmq
−,j(−τ)iδn1
+ρ1qh−Gij∗
+(τ)Xqn
+,j(−τ)−iGij∗
−(τ)Xqn
−,j(−τ)iδmi
+ρiqh−Gij∗
+(τ)Xqn
+,j(−τ) + iGij∗
−(τ)Xqn
−,j(−τ)iδm1.(C.74)
C.4.2 Chain of V-type emitters
The equations of motion resulting from Eq. (C.68) are given by
d
dtρmn =
N−1
X
l=0 X
i=2,3ni¯
∆l
ihρm(i+3l)δn,(i+3l)−ρ(i+3l)nδm,(i+3l)i
+i¯
Ωhρm(1+3l)δn,(i+3l)+ρm(i+3l)δn,(1+3l)−ρ(i+3l)nδm,(1+3l)−ρ(1+3l)nδm,(i+3l)i
−¯
Ω2
ℏ2Zt
0
dτ ⟨m|χij
l(τ)|n⟩o+i
N−2
X
l=0 nf(t)hρm(5+3l)δn,(3+3l)−ρ(3+3l)nδm,(5+3l)i
+f∗(t)hρm(3+3l)δn,(5+3l)−ρ(5+3l)nδm,(3+3l)io,(C.75)
where
⟨m|χij
l(τ)|n⟩=
N−1
X
l′=0
3
X
q=1 ρqn
×nhG+(τ)⟨i+ 3l|X[l′]
+,j(−τ)|q+ 3l⟩−iG−(τ)⟨i+ 3l|X[l′]
−,j(−τ)|q+ 3l⟩iδm,(1+3l)
+hG+(τ)⟨1+3l|X[l′]
+,j(−τ)|q+ 3l⟩+iG−(τ)⟨1+3l|X[l′]
−,j(−τ)|q+ 3l⟩iδm,(i+3l)o
+ρmqnhG∗
+(τ)⟨q+ 3l|X[l′]
+,j(−τ)|1+3l⟩−iG∗
−(τ)⟨q+ 3l|X[l′]
−,j(−τ)|1+3l⟩iδn,(i+3l)
+hG∗
+(τ)⟨q+ 3l|X[l′]
+,j(−τ)|i+ 3l⟩+iG∗
−(τ)⟨q+ 3l|X[l′]
−,j(−τ)|i+ 3l⟩iδn,(1+3l)o
+ρq1h−G+(τ)⟨m|X[l′]
+,j(−τ)|q+ 3l⟩+iG−(τ)⟨m|X[l′]
−,j(−τ)|q+ 3l⟩iδn,(i+3l)
+ρqih−G+(τ)⟨m|X[l′]
+,j(−τ)|q+ 3l⟩−iG−(τ)⟨m|X[l′]
−,j(−τ)|q+ 3l⟩iδn,(1+3l)
182 C Calculations: Chain of V-type emitters
+ρ1qh−G∗
+(τ)⟨q+ 3l|X[l′]
+,j(−τ)|n⟩−iG∗
−(τ)⟨q+ 3l|X[l′]
−,j(−τ)|n⟩iδm,(i+3l)
+ρiqh−G∗
+(τ)⟨q+ 3l|X[l′]
+,j(−τ)|n⟩+iG∗
−(τ)⟨q+ 3l|X[l′]
−,j(−τ)|n⟩iδm,(1+3l).(C.76)
For the solution of the resulting integro-differential equation, we employ the time-evolution
operator U0(t, 0) = exp(−iHp,0t/ℏ)to calculate the time-dependent system correlations,
yielding
⟨m|X[l′]
+,i(−τ)|n⟩=
3N
X
q=1 h⟨q|ρc(−τ)|i+ 3l′⟩δq,(1+3l′)+⟨q|ρc(−τ)|1+3l′⟩δq,(i+3l′)i
=⟨1+3l′|ρc(−τ)|i+ 3l′⟩+⟨i+ 3l′|ρc(−τ)|1+3l′⟩,(C.77)
and
⟨m|X[l′]
−,i(−τ)|n⟩=i
3N
X
q=1 h⟨q|ρc(−τ)|i+ 3l′⟩δq,(1+3l′)−⟨q|ρc(−τ)|1+3l′⟩δq,(i+3l′)i
=i⟨1+3l′|ρc(−τ)|i+ 3l′⟩−⟨i+ 3l′|ρc(−τ)|1+3l′⟩,(C.78)
with ρcthe conditional density matrix.
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Selbständigkeitserklärung
Hiermit erkläre ich, dass ich die vorliegende Arbeit selbständig und eigenhändig sowie ohne
unerlaubte fremde Hilfe und ausschließlich unter Verwendung der aufgeführten Quellen
und Hilfsmittel angefertigt habe.
Berlin, 5. Mai 2021
Oliver Kästle
209
Danksagung
Mein erster Dank gilt meinem Doktorvater Prof. Andreas Knorr, der diese Arbeit erst
ermöglicht hat und mich in den letzten Jahren stets in allen Bereichen nach Kräften
unterstützt und inspiriert hat. Mein herzlicher Dank geht zudem an Prof. Peter Rabl für
die Übernahme des Zweitgutachtens sowie an Prof. Ulrike Woggon für die Übernahme des
Vorsitzes des Promotionsausschusses.
Besonders möchte ich meinem Betreuer und Bürokollegen Dr. Alexander Carmele danken,
ohne den es die vorliegende Arbeit in dieser Form nicht gäbe. Sein Forscherdrang, seine
Erkenntniswut und sein Streben nach neuen theoretischen Fertigkeiten haben mich jeden
Tag aufs neue inspiriert und stetig wachsen lassen. Sein Vertrauen in mich und unsere
enorm produktive Zusammenarbeit haben die letzten zwei Jahre zu einer außergewöhn-
lichen und prägenden Zeit in meinem Leben gemacht, an die ich mich sehr gerne erinnern
werde. Gemeinsam haben wir die Fesseln der Physik hinter uns gelassen und sind in neue
Zeitalter aufgebrochen.
Mein großer Dank gebührt außerdem meinen Co-Autoren: Prof. Andreas Knorr, Dr.
Alexander Carmele und Dr. Regina Finsterhölzl danke ich für eine immer konstruktive
und schöne Zusammenarbeit. Außerdem danke ich Prof. Ying Hu und Yue Sun für viele
lehrreiche Diskussionen über topologische Systeme und eine tolle Kollaboration. Ebenso
möchte ich Prof. Jesper Mørk und Dr. Emil Denning für unsere gelungene Zusamme-
narbeit und zahlreiche impulsgebende Diskussionen im Bereich der Halbleiter danken.
Bedanken möchte ich mich außerdem bei Dr. Marten Richter, der mich in meiner Zeit
als Masterstudent betreute und in die Welt der Informationskompression einführte. Auch
dadurch ist diese Arbeit zu dem geworden, was sie ist.
Weiterhin möchte ich mich herzlich bei der gesamten AG Knorr dafür bedanken, dass sie
die letzten Jahre mit unzählbaren Konferenzen, Seminaren, Kaffee- und Mittagspausen
und einer immer netten Atmosphäre zu einer der schönsten Zeiten in meinem Leben
gemacht hat. Insbesondere möchte ich mich bei Florian Katsch, Sebastian Franke und
Dominik Christiansen bedanken, die mir immer zur Seite standen und zu engen Freunden
geworden sind. Mein Dank geht außerdem an Malte Selig und Irena Koschinski für schöne
und unterhaltsame Mittagsrunden.
Meinen Eltern möchte ich herzlich danken, dass sie mir dieses Studium ermöglicht haben
und immer zu mir und meinen Interessen standen. Auch meiner Schwester Nikola und
ihrem Verlobten Steve danke ich, dass sie in all der Zeit stets ein offenes Ohr für mich
hatten. Zuletzt und vor allen anderen bedanke ich mich bei Tina – dafür, dass sie in
diesen turbulenten Jahren immer für mich da war und an mich geglaubt hat. Und weil
das Leben mit ihr jeden Tag ein neues Wunder ist.
211
