Out-of-equilibrium dynamics of open
quantum many-body systems
vorgelegt von
Leon Janek Droenner, M. Sc.
geb. in Achim
von der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Michael Kneissl, TU Berlin
1. Gutachter: Prof. Dr. Andreas Knorr, TU Berlin
2. Gutachter: Prof. Dr. Peter Rabl, TU Wien
Tag der wissenschaftlichen Aussprache: 17.12.2018
Berlin, 2019
Abstract
Many-body localization, which prevents a many-body quantum systems from reaching
thermal equilibrium has opened the possibility to study out-of-equilibrium dynamics of
isolated quantum systems, leading to new phases of matter such as the discrete time
crystal. However, little is known about these effects when the quantum many-body system
is coupled to an external environment. An intuitive assumption is that out-of-equilibrium
dynamics and quantum coherences would not survive due to thermalization with external
degrees of freedom. The present work focuses on open quantum many-body systems and
aims at the question how an external reservoir can preserve such out-of-equilibrium effects.
In the first part, common generic reservoirs are assumed to focus on many-body effects of
the systems of interest.
As a first many-body system, an optically driven many-emitter phonon laser is investigated.
The full quantum mechanical treatment reveals additional resonances caused by collective
effects of the many-emitter setup. Optically addressing these resonances results in an
enhancement of phonon intensities within the acoustic cavity.
The Heisenberg spin-chain has achieved the role of a standard model to study many-body
localization. As a second many-body system, the model is generalized to an open quantum
system via boundary reservoirs which are driving the system out-of equilibrium and a
spin-current is induced. When driven far from equilibrium, the transport is very much
dependent on the external reservoirs. In contrast, it is demonstrated that a setup with
long-range coupling between the spins shows transport behavior independent of the external
environment.
In the second part of this work, the coupling to the reservoir is structured by assuming a
boundary condition. Experimentally this is achieved by placing a distant mirror close to the
system. A common Born-Markov approximation leading to factorizing system and reservoir
states is insufficient to describe these feedback dynamics. In this work, an approach is used
to treat the reservoir states as part of the many-body system which is based on matrix
product states using the quantum stochastic Schrödinger equation.
Focusing on the reservoir statistics of a single two-level system, it is demonstrated that by
combining time-dependent optical excitation with time-delayed feedback via structuring
the reservoir, individual photon probabilities are controlled by changing external degrees
of freedom.
Finally, this method is generalized to a many-body system including a coupling to an
external reservoir beyond Born-Markov approximation. It is shown that feedback dynamics
stabilize a discrete time crystal against dissipation to the environment. Out-of-equilibrium
dynamics and many-body localization of the discrete time crystal survive for long times
even in the presence of an external environment and become independent of the coupling
to the reservoir. The developed method benefits from small entanglement entropy of the
many-body localized system as well as from a suppression of entanglement between system
and reservoir due to the feedback dynamics. This allows to consider up to forty spins with
individual structured reservoirs in a numerically exact manner.
III
Kurzfassung
Das Verständnis von Lokalisierung in Vielteilchensystemen unter Berücksichtigung von
Wechselwirkungen hat neue Möglichkeiten eröffnet, insbesondere die Erforschung von
Nichtgleichgewichtsdynamiken in isolierten Quantensystemen. Dies hat zur Entdeckung
neuer Nichtgleichgewichtsphasen, wie zum Beispiel den diskreten Zeitkristall, geführt.
Wenn das Vielteilchensystem mit einem externen Reservoir wechselwirkt, ist wenig bekannt
darüber, ob etwaige Nichtgleichgewichtszustände erhalten bleiben. Eine intuitive Annahme
wäre, dass Nichtgleichgewichtszustände und Quantenkohärenzen durch Thermalisierung
mit externen Freiheitsgraden zerstört werden. Diese Arbeit berücksichtigt offene quanten-
mechanische Vielteilchensysteme und zielt darauf ab, Nichtgleichgewichtszustände unter
Berücksichtigung von externen Reservoirs zu untersuchen.
Der erste Teil dieser Arbeit berücksichtigt einfache allgemeine Reservoirs, um genauer auf
Vielteilcheneffekte im jeweiligen System einzugehen.
Ein Phononlaser, bestehend aus mehreren Emittern als aktives Medium, wird als erstes
Vielteilchensystem untersucht. Eine voll quantenmechanische Beschreibung weist kollektive
Effekte auf, welche zusätzliche optisch adressierbare Resonanzen hervorrufen. Durch optis-
che Anregung dieser kollektiven Resonanzen, werden die erreichbaren Phonon-Intensitäten
innerhalb des akustischen Resonators deutlich verbessert.
Die Heisenberg Spin-Kette ist ein bekanntes Modell, welches zur Beschreibung von
Vielteilchenlokalisierung herangezogen wird. Als zweites untersuchtes Vielteilchensys-
tem wird dieses Modell zu einem offenen Quantensystem verallgemeinert. Durch zwei
Nichtgleichgewichts-Reservoirs an den Rändern der Kette wird ein Strom durch Spin-flips
induziert. Im starken Nichtgleichgewicht ist der Transport abhängig von den Eigenschaften
der externen Reservoirs. Dies ist nicht der Fall bei einer langreichweitigen Wechselwirkung
zwischen den Spins, welche den Transport unabhängig von den Reservoir Parametern
werden lässt.
Im zweiten Teil der Arbeit wird eine Randbedingung in der Kopplung ans Reservoir
berücksichtigt. Dies kann experimentell zum Beispiel durch einen Spiegel realisiert werden.
Das hat zur Folge, dass eine allgemeine Beschreibung des Reservoirs durch eine Born-
Markov-Näherung nicht mehr durchgeführt werden kann, da für Rückkopplungseffekte ein
Gedächtnis des Reservoirs angenommen werden muss. In dieser Arbeit wird das Reservoir
innerhalb des Vielteilchenproblems beschrieben, basierend auf Matrix-Produkt-Zuständen
und der Quantenstochastischen Schrödinger Gleichung. Mit Blick auf die Reservoirstatistik
eines einzelnen Zweiniveausystems wird gezeigt, dass durch die Kombination von zeitab-
hängiger Anregung mit zeitverzögerter Rückkopplung einzelne Photonwahrscheinlichkeiten
durch externe Kontrollparameter manipuliert werden.
Im letzten Teil wird diese Methode auf ein Vielteilchensystem verallgemeinert. Es wird
gezeigt, dass Rückkopplungsdynamik einen Zeitkristall gegenüber Dissipation ins Reservoir
stabilisiert. Nichtgleichgewichtszustände und Vielteilchenlokalisierung bleiben für lange
Zeit erhalten, auch mit Ankopplung an externe Reservoirs. Die diskreten Oszillationen
V
VI
des Zeitkristalls werden unabhängig von der Kopplung ans Reservoir. Die entwickelte
Methode profitiert sowohl von kleiner Verschränkungsentropie des lokalisierten Vielteilchen-
systems, als auch von einer Unterdrückung der Verschränkung mit dem Reservoir durch
zeitverzögerte Rückkopplung. Diese Kombination erlaubt eine Untersuchung von bis zu
vierzig Spins mit individuellen strukturierten Reservoirs innerhalb einer numerisch exakten
Beschreibung.
Contents
Abstract III
Kurzfassung V
1. Introduction 1
1.1. Motivation .................................... 1
1.2. Structureofthethesis .............................. 3
I. Theoretical background 5
2. Basics 7
2.1. Quantum mechanics in the Schrödinger picture . . . . . . . . . . . . . . . . 7
2.2. Spinoperator................................... 8
2.3. Quantization of the Maxwell field . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1. Hamiltonian................................ 9
2.3.2. Quantization ............................... 10
2.4. Two-level system in a classical electromagnetic field coupled to a continuum 13
2.4.1. Quantization ............................... 15
3. Quantum stochastic Schrödinger equation (QSSE) 17
3.1. Quantum-noise operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2. Itôcalculus .................................... 21
3.3. Lindblad form of the reduced density matrix . . . . . . . . . . . . . . . . . 23
4. Ergodicity versus many-body localization in closed quantum systems 27
4.1. Quantum thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2. Many-body localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
II. Factorized system-reservoir dynamics 33
5. Many-emitter phonon lasing 35
5.1. Model ....................................... 35
5.2. Collective phonon processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.1. Effective Hamiltonian approach . . . . . . . . . . . . . . . . . . . . . 41
5.2.2. Collective resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.3. Two-phonon resonances . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3. Non-identical emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4. Quantumyield .................................. 49
5.5. Conclusion .................................... 52
VII
Contents VIII
6. Boundary-driven Heisenberg spin-chain 55
6.1. Model ....................................... 56
6.2. Characterizing spin-transport . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.1. Weakdriving............................... 60
6.2.2. Maximaldriving ............................. 61
6.2.3. Reservoir dependency . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3. Absence of negative differential conductivity . . . . . . . . . . . . . . . . . . 64
6.4. Effectofdisorder................................. 66
6.5. Conclusion .................................... 70
III. Entangled system-reservoir dynamics 71
7. Introduction to matrix product states 73
7.1. Singular-value decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . . 74
7.2. Diagrammatic tensor representation . . . . . . . . . . . . . . . . . . . . . . 75
7.3. Canonical form of a matrix product state . . . . . . . . . . . . . . . . . . . 77
7.4. Matrix product operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.5. Expectationvalues................................ 81
8. Feedback controlled two-photon purification 85
8.1. Theoreticalmodel ................................ 87
8.1.1. Higher-order time-evolution operator . . . . . . . . . . . . . . . . . . 89
8.1.2. Feedback algorithm in the QSSE picture . . . . . . . . . . . . . . . . 91
8.1.3. Computing photon probabilities from the matrix product state . . . 94
8.2. Controlling photon statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2.1. Effect of the time-dependent pulse . . . . . . . . . . . . . . . . . . . 95
8.2.2. Effect of time-delayed feedback . . . . . . . . . . . . . . . . . . . . . 97
8.2.3. Controlling individual photon probabilities . . . . . . . . . . . . . . 99
8.3. Conclusion ....................................100
9. Feedback-stabilized time crystal 103
9.1. Closed system dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.1.1. Model...................................104
9.1.2. Computing many-body systems via matrix product states . . . . . . 105
9.1.3. Achieving a discrete time crystal . . . . . . . . . . . . . . . . . . . . 111
9.2. Opensystemdynamics..............................116
9.2.1. Reservoirmodel .............................116
9.2.2. Computing open quantum many-body systems in the QSSE picture 118
9.2.3. Stabilizing a dissipative discrete time crystal . . . . . . . . . . . . . 126
9.3. Conclusion ....................................131
10.Conclusion and outlook 133
Acknowledgments 135
Contents IX
IV. Appendices 137
A. Details on the Theoretical Background 139
A.1. Consitency with the Maxwell equations . . . . . . . . . . . . . . . . . . . . 139
A.2. Commutation relation of the mode operators . . . . . . . . . . . . . . . . . 141
A.3.Lorentzforce ...................................142
B. Details on Feedback-stabilized time crystal 143
B.1. Efficient application of a matrix product operator . . . . . . . . . . . . . . . 143
Bibliography 145
1.
Introduction
1.1. Motivation
Recent progress in experimental realizations of quantum simulators has enabled the possi-
bility to study many-body quantum systems in an almost isolated and well controllable
setup. In optical lattice systems, an ultra cold gas of atoms is loaded into the lattice
potentials and the interactions between the atoms is manipulated by external controllable
lasers [
1
,
2
]. Another approach is to use trapped ions, where interactions between the sites
are engineered via state-dependent optical dipole forces [
3
,
4
]. The advantages of these
experiments is that they open up the possibility to study out-of-equilibrium dynamics of
many-body quantum systems [
5
,
6
]. In solid-state setups, the loss of quantum coherences
destroys most out-of-equilibrium effects on a comparable short time-scale [
7
,
8
,
9
]. One of
the main reasons is pure dephasing resulting from phonon interactions. Due to the well
controlled setup in optical lattices and trapped ions, decoherence is strongly suppressed.
This leads to possible devices for quantum information protocols for quantum computation
[10, 11].
However, even if a many-body quantum system is well isolated from the external environ-
ment it acts as its own heat bath for small subsystems as, e.g., a single spin. This leads to
a thermalization of the subsystems with the rest of the many-body system [
12
,
13
]. This
implies that quantum information of local subsystems becomes highly non-local which is
described by the spreading of entanglement entropy [
14
]. This process is called quantum
thermalization, where expectation values of local observables become thermal in the long
time limit, meaning that local observables are described by a thermodynamic ensemble.
This process is described by the Eigenstate thermalization hypothesis which states that
if thermalization occurs for some local Eigenstates, all local Eigenstates would become
thermal [
15
]. The Eigenstate thermalization hypothesis is often referred to as quantum
ergodicity hypothesis in analogy to the classical ergodicity hypothesis which states that a
thermalizing system explores the whole phase space [16].
For the purpose of quantum information, thermalization of local observables is an unwanted
process as it is equivalent to the loss of information about the initial state, which is
fundamental for quantum computation protocols [
17
]. The question arises, how the process
of thermalization can be prevented to preserve information about the initial state. A
prominent example for the breaking of ergodicity is Anderson localization [
18
]. Originally
investigated for a solid-state setup, Anderson found that disordered lattice potentials lead
to the localization of particles at certain lattice positions. Disorder occurs naturally in
solid-state systems due to impurities which deform the lattice potential. The scattering of
particles on the deformed potentials leads to interference effects which localize the particle.
This means when being localized, the initial information about the position of the particle
is preserved which is in contrast to a thermalization process.
1
Chapter 1. Introduction 2
However, it was long believed that any interaction between particles would destroy this lo-
calization behavior [
19
]. The breakthrough discovery of localization for interacting systems,
called many-body localization, opened a new field of present research [
20
,
21
]. In contrast
to Anderson localization, the entanglement entropy in many-body localized systems grows
logarithmically as a function of time [
22
,
23
]. This area-law scaling of entanglement [
24
]
stands in strong contrast to the volume-law scaling of thermalizing systems [
14
,
16
]. Thus,
a many-body localized system provides a platform where initial conditions are preserved
for long times even when information is exchanged between the local subsystems, making
it a promising device for quantum memory storage.
From a theoretical point of view, the computation of many-body systems is very challenging
as the number of states of the Hilbert space grows exponentially with the system size.
The development of matrix product states [
25
] based on the powerful density matrix
renormalization group methods [
26
] for the investigation of ground states provides a tool to
explore larger system sizes in finding hints for the behavior in the thermodynamical limit.
However, when simulating time evolution with matrix product states, it is limited to very
small times due to the build up of entanglement entropy between local subsystems [
27
,
28
].
The matrix product state method is based on the truncation of the Hilbert space, the idea
is to truncate all states of the many-body system which are not relevant for the system
dynamics. Thus, all states which are not entangled with relevant states for the dynamics
are truncated [
29
]. The discovery of many-body localization opened up the possibility to
explore much larger system sizes and/or times with matrix product state methods due to
the strong suppression of entanglement growth [22, 23, 30].
Many-body localized systems allow to study out-of-equilibrium dynamics on long time-
scales both theoretically [
14
,
30
] and experimentally [
31
,
32
,
33
,
34
] as ergodicity is broken
due to the violation of the Eigenstate thermalization hypothesis. This is even true for
systems which are periodically driven out-of-equilibrium with external lasers [
35
,
36
,
37
].
Without many-body localization the system would absorb the energy of the drive, resulting
in thermal equilibrium [
38
,
39
]. This was followed by the discovery of new phases of matter
such as the discrete time crystal, where a many-body system breaks the time reversal
symmetry of the governing Hamiltonian and returns back to its initial state after discrete
periods [
40
,
41
]. Time reversal symmetry breaking has recently been demonstrated experi-
mentally on different platforms [42, 43, 44, 45].
Despite its success in many different areas, it is still under debate which aspects of many-
body localization survive in case of open quantum systems [
46
,
47
,
48
,
49
,
34
]. Even when
the system itself is many-body localized, interactions with external degrees of freedom
naturally lead to decoherence and loss of initial information [
50
]. This is especially true for
the common description of open quantum systems, where a Born-Markov approximation
allows to formulate a trace-preserving Lindblad master equation [51].
The question of the suppression of decoherence is not only relevant for many-body localiza-
tion, but also for any experimental setup and goes as far as fundamental quantum physics
itself, because any measurement process is an interaction with external degrees of freedom
[
52
,
53
]. This is also true for the above mentioned experiments which are very well isolated
from external environments.
The aim of this thesis is to formulate a theory based on matrix product state methods
for open quantum many-body systems which can describe reservoirs beyond Born-Markov
approximation [
54
]. The idea is to tailor the external environment such that decoherence
is suppressed. This can be seen as a suppression of entanglement between states of the
Chapter 1. Introduction 3
system and states of the reservoir in analogy to many-body localization which suppresses
entanglement growth within the many-body system itself.
1.2. Structure of the thesis
This thesis is divided into three parts. Part I describes the theoretical and phenomenological
framework which is used in the following two parts. Part II focuses on many-body effects
of two different systems with factorizing system-reservoir dynamics, which is describe by
the common Lindblad formalism. In Part III, reservoirs beyond the Lindblad formalism
are investigated without performing a Born-Markov approximation, describing entangled
system-reservoir dynamics.
In chapter 2, the basic theoretical framework is introduced. In particular, the Hamiltonian
for a two-level system interacting with an external mode continuum is derived, based on
the Maxwell equations and the Lorentz force.
In chapter 3, the quantum stochastic Schrödinger equation is introduced. This formalism
is especially relevant for Part III to describe entangled system-reservoir interactions.
Furthermore, the Lindblad formalism is derived from the quantum stochastic Schrödinger
equation which is used for Part II of this thesis.
In chapter 4, a phenomenological introduction to quantum ergodicity and many-body
localization is given, which will be especially relevant for chapter 6 and chapter 9.
As a first many-body system, the many-emitter phonon laser is investigated in chapter 5,
based on factorizing system-reservoir interactions via the Lindblad formalism. The focus
lies on collective effects of the many-emitter setup.
The transport within the ergodic phase of the Heisenberg spin-chain is investigated in
chapter 6. This is the most standard model to describe many-body-localization. The system
is coupled to two boundary reservoirs, described by the Lindblad formalism, which induce
an out-of-equilibrium situation. The focus lies on effects induced by the external reservoirs.
Matrix product states are introduced in chapter 7 as all computations in Part III are based
on the matrix product state formalism.
As a first example, a single two-level system in front of a mirror is investigated in chapter 8.
The self-feedback renders a description based on Lindblad dynamics impossible. This is
why the surrounding reservoir is considered as a many-body problem, to describe it via
matrix product states based on the quantum stochastic Schrödinger formalism. It will be
shown that the reservoir statistics are controlled by external achievable parameters.
This formalism is generalized to a many-body system in chapter 9. An efficient algorithm is
developed to stabilize a discrete time crystal against dissipation. The formalism based on
matrix product states and the stochastic Schrödinger equation allows to consider efficiently
a large system size together with a structured reservoir. Final remarks and an outlook
conclude this thesis in chapter 10.
PART I
Theoretical background
5
2.
Basics
2.1. Quantum mechanics in the Schrödinger picture
The fundamental difference of quantum mechanics in contrast to classical mechanics lies
in the description of systems via a probability distribution and the quantization of states.
This implies that a theorist can well describe the probability of states of a given system
but never predict with certainty the output of a single measurement. The probability
distribution ρ=|ψihψ|of a given system is described by its wavefunction |ψi, where
|ψi=X
i
ci|ii(2.1.1)
consists of a superposition of all possible states. The coefficients
ci
describe the probability
of the basis state |ii.
Probably the most famous example of the superposition principle with a correctly deter-
mined probability distribution but uncertain result in a simple measurement process is
Schrödinger’s cat. The cat is put into a closed box with a mechanism which breaks a vial
of poison with a probability of 50%. The system of the cat consists of two states where the
cat is found dead in the state
|
0
i
with probability
|c0|2
= 1
/
2or found alive in the state
|
1
i
with probability
|c1|2
= 1
/
2. The physicist as external observer does not know if the
cat is alive or dead before opening the box, but he knows well that he will be sadly finding
the cat dead with probability
|c0|2
= 1
/
2. Before opening the box, to him, the cat is both
dead and alive at the same time. Opening the box is a measurement process where the
Hermitian operator ˆ
Omeasures the state of the cat
hˆ
Oi=hψ|ˆ
O|ψi.(2.1.2)
In the Copenhagen interpretation of quantum mechanics, this measurement process results
in a collapse of the wavefunction to either the state
|
0
i
or
|
1
i
and the cat remains either
dead or alive after opening the box.
How does the physicist know the probability distribution before opening the box? The
answer to that lies in the Schrödinger equation. By solving the Eigenvalue problem of the
time-dependent Schrödinger equation
i~∂t|ψ(t)i=H|ψ(t)i,(2.1.3)
the coefficients of the wavefunction, i.e. probabilities of states, are determined. The solution
is given by the expansion of the wavefunction in Eigenstates |ϕni
|ψ(t)i=X
n
cn(t)ei
~Ent|ϕni(2.1.4)
7
Chapter 2. Basics 8
with quantized Eigenenergies Ento the corresponding Hamiltonian H.
With a given Hamiltonian which describes the energies of the system of interest, the
dynamics of the system are derived by solving the time-dependent Schrödinger equation.
2.2. Spin operator
Most of the magnetic moments measured in experiment result from the angular moment
of electrons. Besides orbital momentum, there is also an intrinsic angular momentum of
electrons which is called the spin [55]. The magnetic moment based on spins reads
µ=gβS(2.2.1)
with the Bohr magneton β, the spin Sin units of ~and the gyromagnetic ratio g[55].
The operator
S
describes the spin of the electron. Considering a single spin-1
/
2it consists
of the states spin up | ↑i and spin down | ↓i. An arbitrary state of a single spin reads
|ψi=α| ↑i+β| ↓i .(2.2.2)
The spin operator consists of
S
=
{Sx, Sy, Sz}
, where the
z
-component gives the Eigenvalues
Sz| ↑i =~
2| ↑i ,(2.2.3)
Sz| ↓i =−~
2| ↓i .(2.2.4)
The action of the
x
-component and
y
-component on an Eigenstate results in a superposition
state. However, by defining the raise S+and lower operator S−[56]
S+=Sx+iSy,(2.2.5)
S−=Sx−iSy,(2.2.6)
its application on the Eigenstate results in
S+| ↑i = 0 S+| ↓i =| ↑i (2.2.7)
and the opposite for the lower operator S−
S−| ↓i = 0, S−| ↑i =| ↓i .(2.2.8)
The spin components have a cyclic commutation relation
hSx
i, Sy
ji=i~Sz
iδij (2.2.9)
and, based on this, one yields for the raise and lower operators [55]
hSz, S+i=~S+,(2.2.10)
Sz, S−=−~S−,(2.2.11)
hS−, S+i=−2~Sz.(2.2.12)
Chapter 2. Basics 9
Within this thesis, the spin operators will be described via
S=~
2σ,(2.2.13)
where σare the typical Pauli-matrices
σx= 0 1
1 0!, σy= 0−i
i0!, σz= 1 0
0−1!.(2.2.14)
Note that a single spin is formally the same as a single two-level system which is why the
Pauli-matrices will also be the basis to describe two-level systems later on.
2.3. Quantization of the Maxwell field
One of the main aspects within this thesis will be the consideration of open quantum systems.
The quantum system of interest is coupled to an external reservoir, where interaction with
the external degrees of freedom leads to dissipation.
The concept of an external mode continuum which is in a vacuum state forms the basis
to describe the most simple open quantum system. The external vacuum is described
on the basis of the microscopic Maxwell equations without source fields. For the open
quantum system dynamics investigated in this thesis, a quantized (many-body) system and
a quantized reservoir is considered. Thus, the quantization of the vacuum Maxwell field
forms the theoretical background for the mode continuum of the surrounding reservoir. In
this section, the Hamiltonian for the free evolution of the surrounding reservoir is derived.
The starting point forms the Lagrange function LMof the electromagnetic field [57]
LM=Zd3rL,L=−1
4µ0
FαβFαβ =0
2E2−c2B2,(2.3.1)
with Lagrange density
L
, electromagnetic tensor
Fαβ
(cp. Equation A.1.2), electric field
E
and the magnetic field
B
. In the Appendix in section A.1 it is shown that
LM
leads to the
vacuum Maxwell equations. Thus, it is justified to take this Lagrangian as the fundamental
equation for the quantization of the vacuum Maxwell field.
2.3.1. Hamiltonian
In order to derive the Hamiltonian, the Lagrangian in Equation 2.3.1 is transformed
with a Legendre transformation. For the Legendre transformation of the Maxwell field,
the canonical coordinates are defined. The first canonical coordinate according to the
Lagrangian is
rα=∂0Aα=1
c∂tAα.(2.3.2)
Without any charges, as assumed for the Maxwell field, this yields the electrical field as
E=−∂tA. The canonical momentum is defined as
Πα=∂L
∂(∂0Aα)=−1
µ0cEα,(2.3.3)
Chapter 2. Basics 10
Thus, both coordinates include the
E
-field but with different prefactors. Therefore, the
Hamiltonian for the Maxwell field reads [57]
HM=Zd3rH=Zd3r(r·Π−L)
=Zd3r0E2−L
=Zd3r0
2E2+1
2µ0
B2.(2.3.4)
This classical Hamiltonian describes the energy of the vacuum Maxwell field and is used in
the following for the quantization.
2.3.2. Quantization
The task now is to perform the quantization of the Maxwell field. For simplicity, the
representation of the fields in matter is used
D=0rE,(2.3.5)
H=µ0µrB,(2.3.6)
but a vacuum is assumed (
µr
=
r
= 1). With this, the second Maxwell equation in
Equation A.1.12 reads
˙
D=∇×H.(2.3.7)
In a first step, it is assumed that the fields are quantized and therefore operators
D→ˆ
D
,
H→ˆ
H
. For quantum mechanical operators, the equation of motion of an arbitrary (not
explicitly time-dependent) operator reads
−i~d
dthˆ
Oi=hhˆ
H, ˆ
Oii.(2.3.8)
This equation for the field operators should be in correspondence to the classical Maxwell
equations. Thus, the idea is that the commutation relation of
ˆ
D
with the Hamiltonian
shall yield the second Maxwell equation
−i~˙
ˆ
D=hˆ
HM,ˆ
Di=∇× ˆ
H.(2.3.9)
By defining this commutation relation, the field operators are derived to generate corre-
spondence with the second Maxwell equation
hˆ
HM,ˆ
Dii=Zd3r01
20hˆ
D2(r0),ˆ
Di(r)i+1
2µ0hˆ
B2(r0),ˆ
Di(r)i.(2.3.10)
To arrive at the second Maxwell equation, the first summand shall vanish. Therefore it is
requested that the commutator
hˆ
D2(r0),ˆ
Di(r)i
= 0. This is the case if all components are
linearly independent, thus the first condition on the commutator reads [58]
hˆ
Di(r, t),ˆ
Dj(r0, t)i= 0 .(2.3.11)
Chapter 2. Basics 11
The second summand should yield the Maxwell equation, therefore the outcome shall be
the rotation with the Nabla operator. The square of the magnetic field is rewritten as
1/µ0ˆ
B2
i=ˆ
Hiˆ
Biwhich results in
hˆ
HM,ˆ
Dii=Zd3r01
2X
jˆ
Hj(r0)hˆ
Bj(r0),ˆ
Di(r)i+hˆ
Bj(r0),ˆ
Di(r)iˆ
Hj(r0).(2.3.12)
The second Maxwell eq. is then fulfilled if the commutator obeys the condition [58]
hˆ
Bj(r0, t),ˆ
Di(r, t)i=i~ijk
∂
∂kδ(r−r0).(2.3.13)
Thus, a condition for the commutator of the field operators is derived on the basis of
the second Maxwell equation. However, in this thesis the formalism will be the second
quantization where the light field is described by mode operators. In order to arrive at the
Hamiltonian in second quantization, a mode expansion Ansatz of the fields is assumed [
52
]
E(r, t) = −˙
A=X
λk"s~
20ωkVελkˆ
bλk(iωk)e−iωkt+ikr +h.c.#,(2.3.14)
B(r, t) = ∇×A=X
λk"s~
20ωkVελk×(ik)ˆ
bλke−iωkt+ikr +h.c.#,(2.3.15)
with frequency
ωk
in mode
k
, quantization volume
V
, and polarization
ελk
. Note that
the quantization is done in assuming the coefficients for the amplitude to be operators
bλk→ˆ
bλk
. The task now is to derive the commutation relation in Equation 2.3.13 for the
mode operators ˆ
bλk. For simplicity, this will be done for the xand ycoordinates
hˆ
Dx(r),ˆ
By(r0)i=ˆ
Dx(r),∇0׈
A(r0)y
=∂z0hˆ
Dx(r),ˆ
Ax(r0)i−∂x0hˆ
Dx(r),ˆ
Az(r0)i.(2.3.16)
In order to know if the mode operators obey a bosonic or fermionic commutation relation,
the summands are derived by inserting the mode expansion
hˆ
Di(r),ˆ
Aj(r0)i=X
λλ0kk0
0iωkαk(ελk)iαk0(ελ0k0)jhˆ
bλkeikr −h.c.,ˆ
bλ0k0eik0r0+h.c.i ,
(2.3.17)
with
αk
=
q~
20ωkV
. For simplicity, the time is set
t
= 0, as the commutation relation
shall hold for all times. The commutation relation in Equation 2.3.13 is independent of
the amplitude, thus the commutator in Equation 2.3.17 shall be independent of the mode
operators
ˆ
bλk
. Therefore, it is postulated that the commutation relation of the mode
operator reads
ˆ
bλkˆ
b†
λ0k0=δλλ0δ(k−k0)±ˆ
b†
λ0k0ˆ
bλk.(2.3.18)
Chapter 2. Basics 12
Inserting this into Equation 2.3.17, it becomes clear that the mode operators only vanish
for a bosonic commutation relation
hbλk, b†
λ0k0i=δλλ0δ(k−k0),(2.3.19)
[bλk, bλ0k0] = hb†
λk, b†
λ0k0i= 0 .(2.3.20)
This is shown in detail in the Appendix in section A.2.
Now it is the task to show that the mode expansion Ansatz combined with the bosonic
commutation relation for the mode operators yields the commutation relation for the field
operators in Equation 2.3.13. For this, the bosonic commutation relation is applied to
Equation 2.3.17
hˆ
Di(r),ˆ
Aj(r0)i=X
λλ0kk0
0iωkαk(ελk)iαk0(ελ0k0)j×
×ei(kr−k0r0)δλλ0δ(k−k0) + e−i(kr−k0r0)δλλ0δ(k−k0)
=X
λk
0iωkα2
k(ελk)i(ελk)jeik(r−r0)+e−ik(r−r0).(2.3.21)
For light, the polarization
λ
is transversal to its propagation. Thus,
λ
= 1
,
2, while the
third unit vector points in the direction of the propagation
ε3k
=
k/|k|
. A summation over
all three unit vectors yields
3
X
λ=1
(ελk)i(ελk)j=δij .(2.3.22)
Thus, a summation over the two polarization yields [59]
2
X
λ=1
(ελk)i(ελk)j=δij −kikj
k2.(2.3.23)
Inserting this into Equation 2.3.21 reads
hˆ
Di(r),ˆ
Aj(r0)i=X
k
0iωkα2
kδij −kikj
k2eik(r−r0)+e−ik(r−r0).(2.3.24)
The discrete sum is assumed to be a continuous integral. Furthermore,
k
is integrated from
−∞
to
∞
and the integrand
f
(
k
)is a symmetric function around the origin
f
(
k
) =
f
(
−k
).
Note that αkis inserted as well. With this, the commutator reads
hˆ
Di(r),ˆ
Aj(r0)i=i~
(2π)3Zd3kδij −kikj
k2eik(r−r0)=δT
ij(r−r0),(2.3.25)
which is the transversal delta function [
58
]. By inserting the transversal delta function
into the commutation relation of the field operators, it is shown that the mode expansion
Chapter 2. Basics 13
Ansatz with a bosonic commutation relation for the mode operators is in correspondence
to the second Maxwell equation
hˆ
Dx(r),ˆ
By(r0)i=∂z0δT
xx(r−r0)−∂x0δT
xz(r−r0).
=i~
(2π)3Zd3k1−kxkx
k2(−ikz)eik(r−r0)
−i~
(2π)3Zd3kkxkz
k2(−ikx)eik(r−r0)
=i~
(2π)3Zd3k(−ikz)eik(r−r0)=−i~
(2π)3∂zZd3keik(r−r0)
=−i~∂zδ(r−r0).(2.3.26)
This proves that the mode expansion Ansatz combined with a bosonic commutation relation
for the mode operators
ˆ
bλk
is in correspondence to the second Maxwell equation as this
is in agreement with the conditions for the commutation relation of the field operators.
Furthermore, the Hamilton function
HM
in Equation 2.3.4 yields the Maxwell equations
as shown in the Appendix in section A.1. The mode expansion in Equation 2.3.14 and
Equation 2.3.15 with the mode operators
ˆ
bλk
in second quantization are then inserted into
the Hamiltonion of the vacuum Maxwell field in Equation 2.3.4. This leads directly to the
Hamiltonian of the vacuum Maxwell field in second quantization [60]
HM=X
λk
~ωkˆ
b†
λkˆ
bλk+1
2.(2.3.27)
This Hamiltonian forms the basis to describe an open quantum system with a surrounding
mode continuum. However, this Hamiltonian only describes the free evolution of the
mode continuum. The coupling of the quantum system to its environment remains to be
determined.
2.4. Two-level system in a classical electromagnetic field
coupled to a continuum
To derive the coupling of an electronic system to a photonic environment, a single two-level
system (TLS) is considered. This exemplary system is especially relevant as it forms the
basis for the many-emitter phonon-laser in chapter 5 in Part II. The same exemplary
system is considered in chapter 8 to demonstrate the effects of entangled system-reservoir
interactions in Part III.
The basis to describe a single electron in an electromagnetic field forms the Lagrangian
LF(r,˙
r) = 1
2m˙
r2+q˙
r·A−qφ . (2.4.1)
In the Appendix in section A.3 is is shown that this Lagrangian is justified as it leads
directly to the fundamental Lorentz force.
Above, it has been shown that the Lagrangian
LM
in Equation A.1.4 leads to the vacuum
Maxwell field in second quantization. This justifies to take
LM
to describe the surround-
ing electromagnetic field. In this section, these two Lagrangians are used to derive the
Hamiltonian of an electron dipole, oscillating between two energies due to an external
Chapter 2. Basics 14
classical driving field. This Hamiltonian is derived from the Lagrangian
LF
. Furthermore,
the electron is coupled to a bosonic mode continuum, which acts as a photonic reservoir
and is described by LM. The total Lagrangian with electron charge ethen reads,
L=1
2m˙
r+e(A·˙
r−φ) + Zd3r0
2E2−c2B2.(2.4.2)
In a first step, the dipole approximation is applied [
59
]. The position of the electron only
varies on small distances compared to the wavelength of the external field. The reason
is that the size of atoms or quantum dots is in the orders of Ångström or nanometers
in contrast to the driving field, which is considered to be in the orders of hundreds of
nanometers. Therefore, the interaction can be expanded around the position of the electron
R, therefore r=R+rsand the Lagrangian
L=m
2˙
r2
s+e[A(R+rs, t)·˙
rs−φ(R+rs)] + Zd3r0
2E2−c2B2.(2.4.3)
The relation between the amplitudes of the electric and magnetic field reads
B0
=
E0/c
.
The amplitude of the electric field is orders of magnitudes larger than the amplitude of the
magnetic field [
61
]. The main contribution to the Lorentz force in Equation A.3.6 will thus
result from the electric field. For this reason, the magnetic contribution will be neglected
A
(
R
+
rs, t
)
≈A
(
R
)and the expansion in the scalar field is truncated after the first order
φ(r) = φ(R) + rs·∇rsφ(R).(2.4.4)
With this, the Lagrangian reads
L=m
2˙
r2
s+e[A(R)·˙
rs−φ(R)−rs·∇rsφ(R)] + Zd3r0
2E2−c2B2.(2.4.5)
Before deriving the Hamilton function, the gauge freedom of the Lagrangian is used to
obtain an equation with explicit dependence on the electrical field strength. With the
gauge-function
G=−qrsA(R)(2.4.6)
˙
G=−e˙
rsA(R) + −ers˙
A(R),(2.4.7)
and
L0
=
L
+
˙
G
, the Lagrangian is expressed in terms of the electric field, because
−˙
A(R)−∇rsφ(R) = E(R). Thus, the gauged Lagrangian reads
L0=m
2˙
r2
s−eφ(R) + d·E(R) + Zd3r0
2E2−c2B2,(2.4.8)
with dipole-operator d=ers.
This has the advantage that for
LF
, the separation of kinetic and potential energy is
already done, so the Hamilton function for this part can be written down directly. With
the known Maxwell field Hamilton function
HM
in Equation 2.3.4, combined with
LF
, the
total Hamiltonian reads
H=p2
2m+eφ(R)−d·E(R) + Zd3r0
2E2+1
2µ0
B2.(2.4.9)
Chapter 2. Basics 15
2.4.1. Quantization
The quantization of the Maxwell field was already shown in Equation 2.3.27. The electronic
part of the Hamilton function still remains to be quantized. The Hamiltonian reads
H0=Zd3rψ∗(r, t) p2
2m+eφ(R)−d·E(R)!ψ(r, t) + HM,(2.4.10)
where ψ(r, t)represents the Schrödinger field of the electronic degrees of freedom.
For the second quantization, the Schrödinger field is quantized
ψ(∗)
(
r, t
)
→ˆ
ψ(∗)
(
r, t
)[
62
]
with the mode expansion Ansatz
ˆ
ψ(∗)(r, t) = X
λ
ˆa(†)
λ(t)u(∗)
λ(r),(2.4.11)
where
ˆa(†)
λ
is a fermionic annihilation (creation) operator for an electron with mode
λ
and
u(∗)
λ
(
r
)are orthonormal functions which solve the Eigenvalue problem of the Schrödinger
equation. The energy Eigenvalues are obtained by solving
ελ=X
λ0Zd3r u∗
λ(r) p2
2m+eφ(R)!uλ0(r).(2.4.12)
With this, the Hamiltonian in second quantization of a bound electron reads
He=X
λ
ελˆa†
λˆaλ.(2.4.13)
If the number of states is restricted to a two-level system (
λ
= 0
,
1,
|
0
i
is the ground- and
|
1
i
the excited state) and the origin is set to the ground state, the Hamiltonian can be
expressed with the Pauli-spin matrices in Equation 2.2.14
He=~ω01ˆσ+ˆσ−,(2.4.14)
where
~ω01
is the energy gap between the two levels. The dipole moment of the driving
term reads
Zd3rˆ
ψ∗(r, t) (d·E(R)) ˆ
ψ(r, t) = X
λλ0
ˆa†
λˆaλ0Zd3r u∗
λ(r) (er·E(R)) uλ0(r),(2.4.15)
where the integral over the unit cell reduces to the c-number of the dipole moment
dλλ0
.
With this the driving term of the Hamiltonian reads
HI=d01ˆσ++d10ˆσ−·E(R).(2.4.16)
Within this section, the interaction of the electron with a classical field as well as with
a mode continuum is considered. The reason to differentiate between a classical and a
quantized field is that the two-level system is pumped with a laser light of a single frequency
ωL
and high intensity. On the other hand, the mode-continuum is quantized as the photon
number is very small. It will be shown in chapter 3 that the coupling to a quantized mode
continuum results in the spontaneous decay of the electronic excitation. Thus, there are
the quantized field and the external classical field Eext(R).
Chapter 2. Basics 16
To begin with the interaction with the quantized light field, the mode-expansion of the
electric field in Equation 2.3.14 is inserted
He−p=−iX
λkd01ˆσ++d10ˆσ−"s~ωk
20Vε
ε
ελkˆ
bλke−iωkt+ik·r0−h.c.#.(2.4.17)
By defining the coupling gλ(k) = iqωk
2~0Vd10 ·ε
ε
ελke−iωkt+ik·r0, the Hamiltonian reads
He−p=X
λk
~gλ(k)ˆσ+ˆ
bλk+g∗
λ(k)ˆσ+ˆ
b†
λk+gλ(k)ˆσ−ˆ
bλk+g∗
λ(k)ˆσ−ˆ
b†
λk.(2.4.18)
At this point, the rotating wave approximation (RWA) is applied [
63
]. By transforming
the Hamiltonian He+He−pinto a rotating frame with respect to He, fast-rotating terms
like
ei(ωk+ω01)t
are neglected. These terms appear in the summands
ˆσ+b†
and
ˆσ−b
while
the summands (
ˆσ+b
)
(†)
rotate with
ei(ωk−ω01)t
. If the energy gap is much larger than the
respective coupling
ω01 gλ
(
k
)the fast-rotating terms do not appear in the dynamics as
they average out for the equations of motion on the relevant time-scale. If the RWA is valid
for the respective system of the proposed experiment, the electron-photon Hamiltonian
reads
He−p=X
λk
~gλ(k)ˆσ+ˆ
bλk+g∗
λ(k)ˆσ−ˆ
b†
λk.(2.4.19)
For the classical driving field, the Rabi-frequency is defined as Ω(
t
) =
d10 ·Eext
(
r0
)
/~
.
Inserting this, as well as the quantized vacuum Maxwell field, the final Hamiltonian for a
classical driven two-level system coupled to a continuum is obtained
H=~ω01ˆσ+ˆσ−+X
λk
~ωkˆ
b†
λkˆ
bλk+ Ω(t)ˆσ++ ˆσ−+X
λk
~gλ(k)ˆσ+ˆ
bλk+g∗
λ(k)ˆσ−ˆ
b†
λk.
(2.4.20)
In this thesis, this Hamiltonian forms the basis to describe open system dynamics. In chap-
ter 3, the quantum stochastic Schrödinger equation is introduced to solve the Schrödinger
equation for the Hamiltonian in Equation 2.4.20. By tracing out the reservoir degrees of
freedom in a reduced density matrix picture, the Lindblad form is obtained. However, this
Hamiltonian also serves as a first example to describe entangled system-reservoir interaction
on the basis of the quantum stochastic Schrödinger equation in chapter 8 in Part III of
this thesis.
3.
Quantum stochastic Schrödinger equation
(QSSE)
To calculate dynamics of a quantum system, the time-dependent Schrödinger equation
forms the basis to solve the dynamics of the wavefunction
|ψ
(
t
)
i
for a given system which is
described by the Hamiltonian
H
. As an example the derived Hamiltonian in Equation 2.4.20
for a driven two-level system (TLS) is considered. If the coupling to the continuum described
by the quantized Maxwell field is neglected, the solution of the Schrödinger equation leads
straightforward to the Rabi model [
59
]. In this model, the wavefunction of the electronic
system is a two-state system
|ψ(t)i=c0(t)|0i+c1(t)|1i,(3.0.1)
where the probability to measure the electron in the state
|ii
is given by
|ci
(
t
)
|2
. The
coefficients at time
t
are determined by the solution of the Schrödinger equation in
Equation 2.1.3. However, if the quantized Maxwell field is included in the calculation, the
solution is not as straightforward. The surrounding photonic reservoir for the electronic
TLS is then described by a mode continuum and the Ansatz for the wavefunction reads
|ψ(t)i=
1
X
is=0
∞
X
nω1...,nωi,...,nωN=0
cis,nω1...,nωi,...,nωN(t)|is, nω1...,nωi, . . . , nωNi.(3.0.2)
The electronic degrees of freedom are described by the TLS state
|isi
. The photonic
reservoir is described by Fock states, where
nωi
is the number of photons with frequency
ωi
.
Thus, the electronic system couples to an infinite number of modes, resulting in an infinite
number of coupled differential equations for the time-dependent coefficients by inserting
|ψ
(
t
)
i
in the Schrödinger equation. To generate a solvable set of differential equations, the
number of considered photons
Ni
in the state states
nωi
is usually cut off as well as the
number of considered modes
K
. The relevant modes are in many cases close to the TLS
frequency. However, the modes have to be discretized, resulting still in an exponential
number ((2
Ni
)
K
) of differential equations. To deal with this problem in this thesis, the
photon creation (annihilation) operators for the mode
ω
are treated as noise operators.
This is done in transforming the basis as well as the Hamiltonian to a time-discrete picture
where the creation (annihilation) of a photon takes place in a discrete time interval ∆
t
.
The information about the mode is hidden due to the transformation. This formulation
is called the quantum stochastic Schrödinger equation (QSSE) [
64
], which will be the
basis for calculating dynamics within this thesis. By using the Itô calculus and assuming a
Wiener process of the noise operators [
52
], the derived wavefunction allows to trace out
the reservoir degrees of freedom when going to the reduced density matrix formalism. This
17
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 18
leads to the common Lindblad form [
65
], where the system of interest is coupled via rate
equations to the surrounding reservoir. This Lindblad formalism forms the theoretical
background for the treatment of open quantum systems with factorizing system-reservoir
dynamics in Part II of this thesis. However, if the system is entangled with the reservoir,
i.e. for structured reservoirs where for example the output of the system is fed back after a
time delay, the QSSE with the time-discrete basis forms the theoretical background for
calculating these entangled system-reservoir dynamics in Part III.
3.1. Quantum-noise operators
Starting with the Hamiltonian in Equation 2.4.20 in RWA, the polarization degrees of
freedom are neglected (
λ
= 1). Furthermore, the discrete sum is converted into a continuous
integral
H=~ω01σ+σ−+Zdω ~ωb†(ω)b(ω) + Ω(t)σ++σ−
+Zdω ~g(ω)ˆσ+ˆ
b(ω) + g∗(ω)ˆσ−ˆ
b†(ω).(3.1.1)
In a first step, this Hamiltonian is transformed into a rotating frame via a unitary transfor-
mation. Due to the different orders of magnitude of the free evolution
ω01
and
ω
compared
to the coupling frequencies
g
(
ω
)and Ω, two different time-scales are involved. The relevant
time-scale shall be the one of the coupling frequencies. Thus, the whole set of equations
of motion, based on the Schrödinger equation, is transformed with respect to the free
evolution. This is possible as a unitary transformation with an arbitrary unitary operator
U(U†U=1) leaves the Schrödinger equation invariant
HU†U|ψ(t)i=i~∂tU†U|ψ(t)i
HU†U|ψ(t)i=i~∂tU†U|ψ(t)i+i~U†(∂tU|ψ(t)i).(3.1.2)
By defining the transformed wavefunction
|˜
ψ
(
t
)
i
=
U|ψ
(
t
)
i
, and multiplying the equation
from left with U, the transformed Hamiltonian ˜
His obtained
UHU†−i~U∂tU†
|{z }
˜
H
|˜
ψ(t)i=i~∂t|˜
ψ(t)i.(3.1.3)
To transform the Hamiltonian in Equation 3.1.1 into a rotating frame with respect to the
free evolution, the following unitary operator is defined
U=ei
~t(~ω01σ+σ−+Rdω ~ωb†(ω)b(ω)).(3.1.4)
The derivative of
U†
cancels out with the free evolution of the Hamiltonian in Equation 3.1.1
as the free part commutes with
U
. Combined with the first summand in Equation 3.1.3,
the transformed Hamiltonian reads
˜
H=E0cos(ωLt)Uσ++σ−U†+Zdω ~Ug(ω)ˆσ+ˆ
b(ω) + g∗(ω)ˆσ−ˆ
b†(ω)U†
=E0σ++σ−+Zdω ~g(ω)ˆσ+ˆ
b(ω)e−i(ω−ω01)t+g∗(ω)ˆσ−ˆ
b†(ω)ei(ω−ω01)t.(3.1.5)
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 19
The external driving field was expressed as Ω(
t
) =
E0cos
(
ωLt
). In deriving this Hamiltonian,
the Baker-Campbell-Hausdorff equation was used and, for simplicity, a resonant excitation
of the external driving field with the TLS
ωL
=
ω01
was assumed. If the amplitude of the
external driving field
E0
is time independent (continuous wave), the only time dependency
of
˜
H
remains in the interaction with the quantized reservoir. As an approximation, the
coupling constant is assumed to be frequency independent
g
(
ω
)
→g0
. This is justified
as long as the reservoir is not structured in an unusual way (e.g. some modes are highly
occupied) because most of the coupling to the reservoir is close to the TLS frequency
and thus, only small differences in
ω
are assumed. With this it is possible to define
time-dependent operators
ˆ
b(t) = 1
√2πZdω ˆ
b(ω)e−i(ω−ω01)t,(3.1.6)
and the Hamiltonian simplifies to
˜
H=E0σ++σ−+~√2πg0ˆσ+ˆ
b(t) + √2πg∗
0ˆσ−ˆ
b†(t).(3.1.7)
This can be interpreted as follows: The driven TLS interacts with a mode continuum via
g0
.
Due to the time representation of
˜
H
, the information with which mode the TLS interacts
is hidden within the time-dependent operators
ˆ
b
(
t
). Thus,
˜
H
only provides information at
which time a photon was emitted (absorbed) and not in which mode. The commutation
relation of the time-dependent operators
ˆ
b
(
t
)is determined from the bosonic commutation
relation of the mode operators in Equation 2.3.19
hˆ
b(t),ˆ
b†(t0)i=δ(t−t0).(3.1.8)
In order to arrive at this commutation relation, the integral was taken from minus infinity
to infinity. This is called narrow bandwidth approximation and is in agreement with the
approximation of a mode independent coupling constant and that only modes close to the
TLS transition frequency are relevant. With these approximations, the operator
ˆ
b
(
t
)is
referred to as Gaussian white noise [
52
]. It will be shown that by assuming the operator
ˆ
b(t)as a stochastic element, the Lindblad equation is generated.
The idea is to calculate the time evolution of the wavefunction
|˜
ψ
(
t
)
i
by stroboscopic
application of the time-evolution operator. This time-evolution operator will be discretized
to infinitesimal small time steps.
A solution of the wavefunction
|˜
ψ
(
t
)
i
is given by a formal integration of the Schrödinger
equation
|˜
ψ(t)i=U(t, t0)|˜
ψ(t0)i,(3.1.9)
with time-evolution operator
U(t, t0) = ˆ
Texp −i
~Zt
t0
dt0˜
H(t0),(3.1.10)
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 20
where the operator ˆ
Tdenotes the time ordering operator for time from t0to t.
The discretized time-evolution operator with time increment ∆t=tk+1 −tkthen reads
U(∆t) = ˆ
Texp −i
~Ztk+1
tk
dt0˜
H(t0),(3.1.11)
and the system evolves in time with time-step ∆tas
|˜
ψ(tk+1)i=U(∆t)|˜
ψ(tk)i.(3.1.12)
In principle, this suffices to compute time evolution of the system, described by
H
, via
stroboscopic application of the time-evolution operator
U
(∆
t
)on the state
|˜
ψ
(
t
)
i
. However,
|˜
ψ
(
t
)
i
is still in the basis of the mode continuum with infinite frequency modes
ω
in
Equation 3.0.2, as without adapting the basis, the time-dependent operators
ˆ
b
(
t
)are only
an abbreviation for the integral over the mode operators. From this point on, there are two
options, on the one hand, the basis can be adapted by going into a time-discrete basis. The
degrees of freedom of the mode continuum are then reduced by the time representation.
There would be still a possible very large number of time steps, but this basis is in favor of
the computation of time evolution. On the other hand, it is possible to reduce the bath
degrees of freedom by tracing out the reservoir. This has to be done in the formalism of
the reduced density matrix, but it is important that the tracing out of the reservoir is
performed within the basis of the mode continuum. This will be discussed in more detail
later on.
Either way will be used in this thesis. First, some general remarks about the QSSE are
important. As the time-discrete basis allows an easier interpretation it will be introduced
together with the quantum-noise operators before introducing the reduced density matrix
formalism. These quantum-noise operators combined with the time-discrete basis forms
the theoretical background for computing entangled system-reservoir dynamics in Part III
of this thesis while the reduced density matrix formalism is used in Part II of this thesis.
The quantum-noise operators are defined over the creation (annihilation) of photons in the
time interval ∆t
∆B(†)(∆t) = Ztk+1
tk
dt0ˆ
b(†)(t0).(3.1.13)
With this, the discrete time-evolution operator reads
U(∆t) = exp −i
~hE0∆tσ++σ−+~√2πg0ˆσ+∆B(∆t) + √2πg∗
0ˆσ−∆B†(∆t)i.
(3.1.14)
Note that due to the time increment, the time ordering
ˆ
T
is redundant, because due to the
definition of the quantum-noise operators there are no time-dependent operators in the
time interval ∆
t
as the quantum-noise operators are only defined over the time interval ∆
t
.
Without adapting the basis, also the quantum-noise operators are only an abbreviation
for the mode operators
ˆ
b
(
ω
). Thus, the basis of the wavefunction needs to be adapted. To
define the basis, the commutation relation of the noise operators is determined
h∆B(tk),∆B†(tj)i=Ztk+1
tk
dt Ztj+1
tj
dt0δ(t−t0)=∆tδk,j .(3.1.15)
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 21
Thus, the quantum-noise operators form an orthogonal set of basis elements at different
discrete times
tk
. However, they are not normalized as the commutator is proportional to
∆
t
. This leads to the assumption, that the noise operators are proportional to
√∆t
. By
creating a time-discrete number basis, the proportionality to
√∆t
is stored in the basis to
obtain normalization [54]
|iki=∆B†(tk)ik
qik!(∆t)ik|vaci.(3.1.16)
The repeated action of the time-discrete quantum-noise operator ∆
B†
(
tk
)within ∆
t
=
tk+1 −tk
creates photons in the number basis in the state
|iki
. This state is called time-bin
at time interval
tk+1 −tk
in the following. Thus, the time-bin
|iki
represents the number of
created photons in the reservoir at time interval ∆
t
. In order to maintain normalization of
the basis, the action of (∆
B†
(
tk
))
ik
on the state is divided by
q(∆t)ik
. By expanding
|ψi
in the time-discrete basis and let the time-evolution operator act subsequently on
|ψi
, time
evolution in the QSSE picture can be calculated. In the following it will be shown that
the time-evolution operator can be brought to a similar form as the Langevin equation
which justifies a stochastic calculus. This is important for the derivation of the Lindblad
formalism in the reduced density matrix picture.
3.2. Itô calculus
One of the standard stochastic processes is the Brownian motion of a particle. This simple
model is described by the Langevin equation with [66]
dvt=−βvtdt +σdBt.(3.2.1)
This can be interpreted as the motion of a Brownian particle which interacts with its
environment. The constant
β
describes a drift, where the velocity
vt
of the particle is
damped. the parameter
σ
is called the volatility to the stochastic process
dBt
. The idea
is that the change of momentum of the particle obeys a stochastic process. It is assumed
that the number of collisions obeys a normal distribution in the interval ∆
t
, such that at
the end of each time interval, only the direction of the particle changes, corresponding to a
random walk. This is one of the standard and most simple stochastic differential equations.
In this section, it will be shown, that the time evolution of the state
|ψ
(
t
)
i
can be brought
to a similar form which justifies the use of the Itô calculus.
In order to calculate the time evolution of the system, the action of the time-evolution
operator on a given initial state |ψ(0)iis calculated
|ψ(∆t)i=U(∆t)|ψ(0)i.(3.2.2)
The time-step ∆
t
is a time interval which is chosen to be infinitesimal small. With this,
the exponential of the time-evolution operator is expanded in orders of ∆t
|ψ(∆t)i ≈ 1−i
~(Usys +Ures)−1
2~2(Usys +Ures) (Usys +Ures)|ψ(0)i,(3.2.3)
where Usys is the first part of Uin Equation 3.1.14 and Ures the last part which contains
the quantum-noise operators ∆
B
. The following consideration simplifies this calculation: As
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 22
∆
t
is arbitrarily small, all terms proportional to (∆
t
)
2
can be neglected and the expansion
is truncated after the first order of ∆
t
. However, it is assumed that the quantum-noise
operators are proportional to
p(∆t)
due to the commutation relation. With this, the
second order of the exponent has to be taken into account as the second-order term with
the quantum-noise operators is proportional to ∆
t
. In contrast, all terms of the second
order including the system can be neglected as
Usys ∼
∆
t
. Together with the quantum-noise
operators this would correspond to a proportionality of (∆
t
)
3/2
of a term such as
UsysUres
and
UsysUsys ∼
(∆
t
)
2
. Thus, the only relevant part of the last summand is the term
UresUres which reads
UresUres =~22π|g0|2σ+σ−∆B∆B†+|g0|2σ−σ+∆B†∆B
+g0g0σ+σ+∆B∆B+g∗
0g∗
0σ−σ−∆B†∆B†
=~22π|g0|2σ+σ−(∆t+ ∆B†∆B) + |g0|2σ−σ+∆B†∆B
+g0g0σ+σ+∆B∆B+g∗
0g∗
0σ−σ−∆B†∆B†(3.2.4)
To simplify Equation 3.2.3 further, it is possible to set conditions on
|ψ
(0)
i
. Initially, it is
assumed that the surrounding reservoir is in a vacuum state. This means, the occupation
in all time-bins
|iki
is initially set to zero. The state of the system is left at an arbitrary
state. The vacuum has occupation zero, thus initially, the time-bin occupation at all times
tkreads
D∆B†(tk)∆B(tk)E= 0 ,(3.2.5)
before the system has interacted with time-bin
|iki
. The same holds for the action of
the time-bin occupation operator on the state ∆
B†
(
tk
)∆
B
(
tk
)
|iki
= 0 as well as for the
action of the annihilation operator ∆
B
(
tk
)
|iki
= 0. Furthermore, the action of
σ−σ−
on the arbitrary system state yields zero as in this case it is a single TLS. With these
considerations, Equation 3.2.3 reduces to
|ψ(∆t)i=|ψ(0)i+ ∆t−i
~E0σ++σ−−π|go|2σ+σ−
|{z }
Ueff
|ψ(0)i−i√2πg∗
0σ−∆B†|ψ(0)i.
(3.2.6)
The first part contains only system operators and is defined as
Ueff
. Note that the only
relevant part of the second-order expansion is the second term in
Ueff
. It will be shown that
this term is crucial to generate the Lindblad equation, which clarifies the proportionality
∆
B∼√∆t
. This equation is now generalized to arbitrary times
t
. This is justified as the
time-bin basis was introduced and initially all time-bins where set to the vacuum state.
Thus, the time evolution for the next time step always acts on a vacuum state and only the
past time-bins
|ili
with
l < k
may differ from a vacuum. Defining the differential
d|ψ
(
t
)
i
results in
d|ψ(t)i:= |ψ(t+ ∆t)i−|ψ(t)i=Ueff|ψ(t)i∆t−i√2πg∗
0σ−∆B†|ψ(t)i.(3.2.7)
This equation is in strong analogy to the Langevin equation in Equation 3.2.1. The first
part describes the evolution of the system and does not contain a stochastic element.
However, the second summand contains the quantum-noise creation operator ∆
B†
. This
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 23
also justifies the interpretation of ∆
B†
as Gaussian white noise as it formally looks like
a Langevin equation [
52
]. Furthermore, ∆
B†
can be interpreted as a Wiener process [
64
]
combined with the assumption that all future time-bins are initialized in the vacuum state.
The time increment is independent of the past and the mean value vanishes while the
variance is defined
h∆B†(tk)i= 0,h∆B(tj)∆B†(tk)i=δj,k .(3.2.8)
This formal analogy to the Langevin equation suffices at this point to use the product rule
of the Itô calculus [
66
]. In the following section it will be applied for the density operator
ˆρ
=
|ψihψ|
, thus it will be derived with the differential of the density matrix. Defining the
differential of ˆρ(t)it reads
d(|ψ(t)ihψ(t)|) = |ψ(t+dt)ihψ(t+dt)|−|ψ(t)ihψ(t)|.(3.2.9)
By adding zero twice into the first summand, the Itô product rule is derived
d(|ψ(t)ihψ(t)|) = (|ψ(t+dt)i−|ψ(t)i+|ψ(t)i) (hψ(t+dt)|−hψ(t)|+hψ(t)|)−|ψ(t)ihψ(t)|
= (d|ψ(t)i+|ψ(t)i) (dhψ(t)|+hψ(t)|)−|ψ(t)ihψ(t)|
=d(|ψ(t)i)hψ(t)|+|ψ(t)id(hψ(t)|) + d(|ψ(t)i)d(hψ(t)|),(3.2.10)
where the last term is in addition to the standard product rule and results from the
stochastic process.
3.3. Lindblad form of the reduced density matrix
In the last section, the quantum stochastic Schrödinger equation was derived. Thus, the
time evolution of the wavefunction is described by Equation 3.2.7. However, even if the
information about the modes is hidden in the time-discrete picture, one deals with a large
Hilbert space, as the photon number states of the discrete time-bins have to be taken into
account. In most cases, the reservoir is not of interest and the important dynamics take
place at the system of interest. This will be the case in Part II of this thesis. In order to
reduce the complexity of the problem in these cases, the reservoir is traced out by going
into the reduced density matrix picture. The dynamics of this reduced density matrix
allows to draw conclusion about the system of interest by describing the interaction with
the surrounding reservoir via a rate. In this section, it will be shown that by calculating
the differential of the reduced density matrix with the product rule of the Itô calculus
in Equation 3.2.10, the von-Neumann equation with the common Lindblad dissipator is
obtained. The reduced density-matrix of the system is defined as
ˆρs(t) = trR(ˆρ(t)) =
∞
X
nω1...,nωi,...,nωN=0hnω1...,nωi, . . . , nωN|ψ(t)i×
×hψ(t)|nω1...,nωi, . . . , nωNi.(3.3.1)
In order to derive the Lindblad form, the most easiest and typical way is to perform the
trace in the basis of the frequency modes
ω
. It is assumed that system and reservoir states
factorize initially which is the Born approximation.
To perform the trace, the starting point is before the definition of the quantum-noise
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 24
operators in Equation 3.1.13. However, the same notation is used here to simplify the
notation of the integrals including the integral over modes
ω
and the time increment ∆
t
.
This becomes clear in the following.
In the same manner as before, the time-evolution operator is expanded to the first order
in ∆
t
. By setting the initial reservoir state to a vacuum, only the action of the creation
operators of the time evolution are unequal to zero (cp. Equation 3.2.6). With this, the
differential of the state reads (but still in the basis of the modes ω)
d|ψ(t)i=|ψ(t+ ∆t)i−|ψ(t)i=dtUeff|ψ(t)i+σ−∆B†
0(dt)|ψ(t)i
+Usysσ−∆B†
1(dt)|ψ(t)i+ ∆B†
2(dt)σ−Usys|ψ(t)i,
(3.3.2)
with the definitions
∆B†
0(dt) = −i√2πg∗
0Z∆t
0
dtb†(t),(3.3.3)
∆B†
1(dt) = −√2πg∗
0
~Z∆t
0
dt Zt
0
dt0b†(t0),(3.3.4)
∆B†
2(dt) = −√2πg∗
0
~Z∆t
0
dtb†(t)Zt
0
dt0.(3.3.5)
It will be shown that the Lindblad equation is obtained when tracing out the reservoir
degrees of freedom with the consideration of only terms proportional to dt.
As before, the reservoir is initially assumed to be in a vacuum state
|vaci
with all
nωi
= 0.
Thus, the reduced density matrix reads
ˆρs(t) = X
{nω}h{nω}|ψ(t)ihψ(t)|{nω}i =hvac |vaci⊗|ψ(t)ishψ(t)|s⊗hvac |vaci,(3.3.6)
with
{nω}
=
nω1...,nωi, . . . , nωN
. The reduced density matrix in differential form to
compute time evolution is obtained by using the Itô-product rule derived in Equation 3.2.10
for the differential of |ψ(t)iin Equation 3.3.2
dˆρs(t) = trR(dˆρ(t)) = trR{[d(|ψ(t)i)hψ(t)|+h.c.] + d(|ψ(t)i)d(hψ(t)|)}.(3.3.7)
The first summand in Equation 3.3.7 reads
trR{d(|ψ(t)i)hψ(t)|} =X
{nω}h{nω}|d(|ψ(t)i)hψ(t)|s⊗hvac |{nω}i
=hvac |d(|ψ(t)i)hψ(t)|s
=hvac |dtUeff +σ−∆B†
0(dt)|vaci⊗|ψ(t)ishψ(t)|s
+hvac |Usysσ−∆B†
1(dt)+∆B†
2(dt)σ−Usys|vaci⊗|ψ(t)ishψ(t)|s
=dtUeff ˆρs,(3.3.8)
where each summand including a reservoir creation operator
b
(
ω
)
†
yields zero as a a photon
is created in the ket state whereas the bra state remains the same.
To calculate the last summand of Equation 3.3.7, some considerations simplify the calcu-
lation. Each summand of higher order than
dt
equals zero as
dt
is arbitrarily small and
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 25
these terms can be neglected. This is the case for each summand already including
dt
due
to the multiplication with the first summand of Equation 3.3.2.
With these considerations, the last summand of Equation 3.3.7 reads
trR{d(|ψ(t)i)d(hψ(t)|)}=trRnσ−∆B†
0(dt)ˆρσ+∆B0(dt) + σ−∆B†
0(dt)ˆρUsysσ+∆B1(dt)
+Usysσ−∆B†
1(dt)ˆρσ+∆B0(dt)
+σ−∆B†
0(dt)ˆρ∆B2(dt)σ+Usys
+∆B†
2(dt)σ−Usys ˆρσ+∆B0(dt)
+Usysσ−∆B†
1(dt)ˆρUsysσ+∆B1(dt)
+Usysσ−∆B†
1(dt)ˆρ∆B2(dt)σ+Usys
+∆B†
2(dt)σ−Usys ˆρUsysσ+∆B1(dt)
+∆B†
2(dt)σ−Usys ˆρ∆B2(dt)σ+Usyso.(3.3.9)
To analyze the trace of these terms, it is convenient to have a look each combination of
reservoir operators to obtain its proportionality to
dt
. Under cyclic permutation of the
trace, the action of the reservoir operators on the vacuum state is investigated. By inserting
the definitions of the terms ∆
Bi
(
dt
), it is shown that only one term is in the order of
dt
which is
∆B0(dt)∆B†
0(dt)|vaci=|g0|2Z∆t
0
dt Zt
0
dt0Z∞
0
dωˆ
b(ω)e−i(ω−ω01)t×
×Z∞
0
dω0ˆ
b(ω0)ei(ω0−ω01)t0|vaci.(3.3.10)
The action of the reservoir operators on the vacuum yields
δ
(
ω−ω0
). Together with the
integral over ω0this yields ω=ω0
∆B0(dt)∆B†
0(dt)|vaci=|g0|2Z∆t
0
dt Zt
0
dt0Zdωe−iω(t−t0)e−iω01(t−t0)|vaci
= 2π|g0|2dt|vaci.(3.3.11)
The combination of ∆
B0
(
dt
)∆
B†
0
(
dt
)within the trace results in a term proportional to
dt
.
This term is crucial for the Lindblad dynamics. However, all other combinations are of
higher order in dt which is exemplarily shown for
∆B0(dt)∆B†
1|vaci=−i2π|g0|2
~Z∆t
0
dtb(t)Zt
0
dt0Zt0
0
dt00b†(t00)|vaci
=−i2π|g0|2
~Z∆t
0
dt Zt
0
dt0|vaci=−i2π|g0|2
~O(dt2)|vaci.(3.3.12)
The same holds for terms such as ∆
B0
(
dt
)∆
B†
2
(
dt
)and ∆
B1
(
dt
)∆
B†
2
(
dt
). With this, the
last summand in Equation 3.3.7 in first order of dt reads
trR{d(|ψ(t)i)d(hψ(t)|)}= 2π|g0|2dtσ−ˆρsσ+.(3.3.13)
Chapter 3. Quantum stochastic Schrödinger equation (QSSE) 26
Inserting Equation 3.3.8 and Equation 3.3.13 into Equation 3.3.7 results in
dˆρs(t) =Ueff ˆρs(t)dt + ˆρs(t)U†
eff dt + 2π|g0|2σ−ˆρs(t)σ+dt
=−i
~E0σ++σ−dt −π|go|2σ+σ−ˆρs(t) + ˆρs(t)i
~E0σ++σ−−π|go|2σ+σ−dt
+ 2π|g0|2σ−ˆρs(t)σ+dt
=−i
~[Hsys,ˆρs(t)] dt −π|g0|2nσ+σ−,ˆρs(t)odt + 2π|g0|2σ−ˆρs(t)σ+dt , (3.3.14)
with
Hsys
being the Hamiltonian of the semi-classical drive, i.e. the first term from
Equation 3.1.7. By dividing the equation by dt , the von-Neumann equation is obtained
dˆρs(t)
dt =−i
~[Hsys,ˆρs(t)] + ΓRD[σ−]ˆρs(t),(3.3.15)
with ΓR= 2π|g0|2and Lindblad dissipator
D[σ−]ˆρs(t) = σ−ˆρs(t)σ+−1
2nσ+σ−,ˆρs(t)o.(3.3.16)
Thus, by tracing out the reservoir in the reduced density matrix picture, the Lindblad
form is obtained on the basis of the QSSE. This Lindblad form simplifies the calculation
of open quantum system. The system-reservoir interaction is described by the rate Γ
R
,
where excitations of the system dissipate into the environment. This Lindblad dissipator
forms the basis for calculating open quantum systems with factorized system-reservoir
dynamics in Part II of this thesis. However, in Part III of this thesis, the entanglement
between system and reservoir becomes relevant, which is why the reservoir states are then
considered as part of the open quantum many-body system.
4.
Ergodicity versus many-body localization
in closed quantum systems
When a many-body system reaches thermal equilibrium, the microscopic dynamics becomes
irrelevant to describe the whole system and the description with macroscopic observables
such as temperature or free energy suffices to govern the macroscopic dynamics. However,
when studying systems out-of-equilibrium, the law of thermodynamics are not applicable
and the description on a microscopic scale is necessary.
A system reaching thermal equilibrium is characterized by its ergodic behaviour, where the
whole phase space is explored and thus, initial information is hidden in thermodynamic
observables [
16
]. This leads to the question if an out-of-equilibrium many-body system
remains out-of equilibrium for long times or if it necessarily reaches a thermal equilibrium.
In the past, it has been shown that a broad class of systems shows ergodicity breaking
when being subject to random disorder. This is known as many-body localization (MBL) a
generalization from Anderson localization of non-interacting systems to interacting systems.
These two competing concepts are sketched in Figure 4.1. As a model system it is convenient
to consider electrons on a lattice potential. The electrons interact with each other due
to nearest-neighbor Coulomb interaction and are allowed to tunnel through the potential
barrier and thus, hop from site to site. Initially, one electron is located at each even site
whereas the odd sites are unoccupied. Thus, initially the electrons obey a non-uniformly
distributed density pattern which is an out-of-equilibrium situation. If the system is ergodic
(top, right), at long times, the system has equilibrated and the probabilities of finding an
electron at a specific site are uniformly distributed which is in agreement with a micro-
canonical ensemble. In contrast, if there is strong random on-site disorder, it prevents
the electron hopping and the electrons are localized at their initial position on the even
sites. Thus, the out-of-equilibrium situation survives for long times. This localization also
takes place in interacting systems which is surprising as the electrons constantly exchange
information with each other and the entanglement entropy is growing with time. However,
this entanglement growth is strongly suppressed and obeys an area-law scaling. This
characteristic is referred to as MBL phase and is a new and unexpected type of ergodicity
breaking [16].
The main aspect of this thesis is not to investigate MBL in detail. However, as many-body
systems out-of-equilibrium form the main part of this thesis, an introduction to the topic of
ergodicity and MBL is given within this chapter. In chapter 6, the transport of a many-body
localizable system is investigated within the ergodic phase, already showing anomalous
characteristics.
In chapter 9, the MBL phase is used to create discrete time translational symmetry breaking
for a many-body system to create a robust discrete time crystal, being out-of-equilibrium
for long times.
27
Chapter 4. Ergodicity versus many-body localization in closed quantum systems 28
Thus, this chapter serves as an introduction to these two phases of many-body quantum
systems in general.
even even evenodd odd odd
Ergodic
Localized
Figure 4.1.:
Interacting particles are initialized in a non-uniform density pattern (only even
sites are occupied, left). For an ergodic system, after unitary time evolution
of the many-body quantum system, all subsystems show the same density
pattern (top, right), due to thermalization with the large many-body system.
In contrast, the initial information of all subsystems remains accessible in
a localized system (bottom, right). This ergodicity breaking is achieved by
random on-site disorder, indicated by the different on-site potentials.
4.1. Quantum thermalization
The thermalization process of classical systems in statistical mechanics is based on the
ergodicity hypothesis. This predicts that over a long period of time all microscopic states
of the classical system are accessed with the same probability [
16
]. The whole system can
thus be described by thermodynamical quantities.
However, in a quantum context this concept of ergodicity cannot be adapted directly
due to the linear time evolution. In classical systems, ergodic behavior is often caused
by nonlinearities leading to chaotic behavior and initially stored information is lost. In
contrast, nonlinear effects are not present in quantum system on a global scale. This implies
that the global many-body quantum system never explores the whole phase space. This
intuitive behavior of evolving a quantum system in time is demonstrated on a general
example: Expanding the initial states in Eigenstates of the many-body system
|αi
it can be
written as
|ψ
(0)
i
=
Pαcα|αi
. Time evolving an initial state which is one of the many-body
Eigenstates results in an additional phase factor which is the energy of that Eigenstate
Eα
|ψ(t)i=e−iHt|ψ(0)i=X
α
cαe−iEαt|αi.(4.1.1)
This shows that the probability of finding the system in a given Eigenstate
|αi
is only
described by the probability
|cα|2
and does not change over time. Thus, the probability of
finding a many-body quantum system in a certain Eigenstate only depends on the initial
Chapter 4. Ergodicity versus many-body localization in closed quantum systems 29
conditions which is clearly non-ergodic in a classical sense.
It becomes clear that globally, a quantum system never satisfies the classical ergodicity
hypothesis. However, a broad class of many-body quantum system shows ergodic behavior,
even under unitary time evolution, when considering small subsystems as for example a
single spin in a many-body system. When describing the rest of the many-body system as
a reservoir for the single spin, interactions with the many degrees of freedom results in a
thermalization of the single spin with the rest of the many-body system. The reason is
that the rest of the many-body quantum system acts as a heat bath for the subsystem.
Thus, even when global observables cannot be described by thermodynamical quantities,
local observables such as the single spin magnetization indeed show thermalizing behavior
[67, 15].
An intuitive explanation of quantum ergodicity is given by the infinite time average
expectation value of Eigenstates. The infinite time average of an operator
ˆ
O
is defined as
[68]
hˆ
Oi∞= lim
τ→∞
1
τZτ
0
dthψ(t)|ˆ
O|ψ(t)i=X
α|cα|2hα|ˆ
O|αi,(4.1.2)
where it is assumed that off-diagonal terms of the operator oscillate at different frequencies
than the diagonal terms and average out in the infinite integral [
16
]. Thus, the expectation
value of an operator acting on global Eigenstates
|αi
is given only by the choice of the initial
state via
hˆ
Oi∞
=
Pα|cα|2hα|ˆ
O|αi
. However, the expectation value of an operator
hβ|ˆ
O|βi
with local Eigenstates
|βi
might become thermal [
67
,
15
]. Note that the local Eigenstate
|βi
is referred to the Eigenstates of the subsystem. If a system quantum thermalizes, the
assumption is that the infinite time average of an expectation value with local Eigenstates
|βi
agrees with the micro-canonic ensemble
hˆ
Oi∞
=
hˆ
Oimc
. This means a local expectation
value is uniformly distributed for all local subsystems of the many-body system which
is sketched in Figure 4.1 (top, right). In other words, a global expectation value of a
many-body quantum system is given by the choice of the initial state. In contrast, a local
expectation value becomes thermal even if initialized in an out-of-equilibrium state.
This is described by the Eigenstate thermalization hypothesis (ETH) which states that
in a thermalizing quantum system, local Eigenstates have thermal expectation values for
t→ ∞
. In particular, this means, even when the global system is prepared in a many-
body Eigenstate, expectation values of local subsystems agree with a thermodynamical
ensemble in the infinite time average. In quantum systems, the ETH is often referred to
as quantum ergodicity. Note that it is not clear if the ETH is a necessary condition for
quantum thermalization. However, it has been shown that for all thermalizing quantum
systems, almost all Eigenstates obey the ETH [
12
,
69
]. The ETH has been experimentally
demonstrated in the Ref. [
13
], where they showed that local observables indeed obey a
thermal distribution while a global observable remains pure for all times due to the unitary
time evolution.
Further information about quantum fluctuations in the infinite time average are necessary.
Above it has been assumed that off-diagonal matrix elements of local operators average
out in the infinite time average. However, strong fluctuations might be present disobeying
a thermal behavior. Without going into detail, Ref. [
68
] introduced an Ansatz where
off-diagonal matrix elements of local operators are damped by the entanglement entropy of
the local subsystem which is defined as
SA=−tr ˆρAlog2ˆρA,(4.1.3)
Chapter 4. Ergodicity versus many-body localization in closed quantum systems 30
where
ρA
is the reduced density matrix of the local subsystem. If the system obeys the ETH,
the local entanglement entropy equals the thermal entropy
SA
=
Sth
A
. Because
Sth
A
is an
extensive quantity, it obeys a volume-law scaling
Sth
A∼vol
(
A
). This implies that quantum
fluctuations in the infinite time average are bound by a volume-law scaling thermal entropy.
This stands in strong contrast to an area-law scaling of the entropy for many-body localized
systems which will be explained in the following section.
To conclude, the ETH states that if some local Eigenstates have thermal expectation values
in the infinite time average, all local observables tend to a thermodynamical ensemble. The
spreading of entanglement obeys a volume-law scaling. This is closely connected to the
loss of the local initial information, which is hidden in local observables due to the strong
entanglement spreading.
4.2. Many-body localization
A phenomenological explanation of quantum thermalization of a closed quantum many-
body system has been given above. One broad class of systems which violate the ETH
and therefore break ergodicity are Anderson localized systems [
18
]. These systems describe
non-interacting particles in a random potential. Due to the random potential, hopping of
the particles is strongly suppressed and the particles are localized at their initial position.
Before explaining the concept of Anderson localization, it is convenient to introduce a
standard model to describe the terms resulting in localization. The Heisenberg spin-chain
with random on-site disorder reads
H=
N−1
X
i=1
J⊥σx
iσx
i+1 +σy
iσy
i+1+
N−1
X
i=1
Jzσz
iσz
i+1 +
N
X
i=1
hiσz
i.(4.2.1)
This is a model system of a one-dimensional spin-chain with
N
spins, where at each
site
i
a single spin is considered. The first two terms with coupling
J⊥
describe spin-flips
between neighboring sites. These terms do not describe interactions. This can be seen, when
transforming the Heisenberg spin-chain to a model of spinless fermions via a Jordan-Wigner
transformation [
70
]. The first two terms then describe a hopping of a single fermion from
site to site. In contrast, the second term in Equation 4.2.1 describes interactions between
the spins via
Jz
. This term is neglected when describing Anderson localization
Jz
= 0, but
for MBL this is the crucial term to differentiate it from single particle localization. The last
term describes random on-site disorder, which can be interpreted as a random deformation
of the potentials in the particle picture (cp. Figure 4.1, bottom right). The Heisenberg
spin-chain will be introduced in more detail in chapter 6.
Turning now to the explanation of Anderson localization, where the interactions are
neglected
Jz
= 0. Assuming the initial state consists of a single spin-up at site
i
and all
other spins pointing down. Considering the dynamics of this spin, if there is an Eigenstate
which is localized with weight on site
i
, spin-flips are strongly suppressed and the spin-up
survives for long times. Thus, at infinite times there is a non-zero probability of finding
a spin-up at site
i
[
14
]. This stands in strong contrast to the ETH as initial information
is still preserved. Without the disorder in Equation 4.2.1, within each time interval the
spin might flip to neighboring sites. Thus, for
t→ ∞
all sites have equal probability for a
spin-up which is described by the micro-canonic ensemble.
If the disorder
hi
is strong enough, all Eigenstates are localized which means the spin-up
survives for all times and no information is exchanged with other Eigenstates. This means
Chapter 4. Ergodicity versus many-body localization in closed quantum systems 31
that the entanglement entropy remains constant. However, this changes drastically if
interactions are considered with
Jz6
= 0. Even if spin-flips are suppressed by localized
Eigenstates, interactions might suffice to close the energy gap between localized Eigenstates.
The discovery of localization in the presence of interactions
Jz
was therefore a surprising
feature and opened a new field of research called many-body localization (MBL), including
experimental realizations [
20
,
21
,
71
,
72
,
14
,
30
,
31
,
33
,
32
,
73
]. Although it has similarities
to Anderson (single particle,
Jz
= 0) localization, it has strikingly different properties.
Both types of localization are detected by a vanishing conductivity, which means there is
no transport if the system is localized, describing a perfect insulator. Furthermore, the
Eigenstate spectrum of both types is similar (at least deep in the MBL regime). However,
the entanglement growth of MBL systems spreads logarithmically slow [
23
,
74
,
24
] which
stands in strong contrast to the constant entanglement entropy of Anderson localization.
The concept of MBL systems, including the logarithmic growth of entanglement, is explained
by the so called local integrals of motions (LIOM) [
75
,
72
,
76
]. Assuming the system is
deep within the MBL regime, where all Eigenstates are localized such that the Hamiltonian
is diagonalized with quasi-local unitary transformations
τz
i
=
Uσz
iU†
. This results in the
LIOM [16]
τz
i=Zσz
i+X
n
V(n)
iˆ
O(n)
i,(4.2.2)
where
Z
is the overlap with the physical operator
σz
i
and the operator
ˆ
O(n)
i
contains
contributions from neighboring operators with distance
n
. However, the overlap with the
distant operators
V(n)
i
is exponentially small due to the localization. This is why the
τz
i
are
called quasi-local bits (l-bit) as they have a large overlap with
σz
i
which is the key feature
of localized systems. Note that this is only true for localized systems and for a system
obeying the ETH, the l-bits would be highly non-local with vanishing overlap Zσz
i.
The exact construction of such LIOM, especially the choice of the unitary transformation
U
, is challenging [
77
,
78
,
79
]. However, the LIOM serve as a phenomenological model to
explain MBL phenomena. Assuming the Hamiltonian can be written in terms of LIOM,
the Hamiltonian the reads [75]
HLIOM =X
i
τz
i+X
i,j
Ji,jτz
iτz
j+X
i,j,k
Ji,j,kτz
iτz
jτz
k+. . . (4.2.3)
In principle, every system can be diagonalized in such manner. Considering for example
a many-body system obeying the ETH, each l-bit would have weight on all sites and
the higher orders with couplings
Ji,j
etc. would be very relevant. In contrast, for a MBL
system, a description with LIOM is advantageous as each l-bit is quasi local. For example
in a non-interacting localized system (
Jz
= 0), the l-bits would be the occupation in the
corresponding local Eigenstate. Turning on small interactions, the l-bits become quasi
local with small weight on other sites. This implies that the higher order couplings fall of
exponentially with the distance due to the localization [16]
Ji,j =Jze−|i−j|
ξ, Ji,j,k =Jze−max(|i−j|,|i−k|,|j−k|)|
ξ,(4.2.4)
where the length scale ξis connected to the localization length.
The effective Hamiltonian in Equation 4.2.3 can be used to understand the Entanglement
spreading. In a non-interacting system, the LIOM are truly local. Thus, they do not
Chapter 4. Ergodicity versus many-body localization in closed quantum systems 32
exchange information with each other and the entanglement entropy remains constant [
14
].
Considering interactions, the LIOM become quasi local with exponentially small weight
on distant sites. Considering two distant spins which are not nearest-neighbors, the two
spins do not interact directly with each other but indirectly via all spins in between. The
effective coupling between the two spins consists of all couplings of the spins in between.
However, all couplings are exponentially small, thus, logarithmic growing entanglement
is induced at time
t
between all spins within a distance
L
via
L∼ξln
(
Jzt
)[
14
]. The
logarithmic spreading of entanglement can be seen as an area-law scaling, where only the
boundaries of each spin contribute to the entanglement as the coupling to further spins
falls off exponentially with the distance. This stands in strong contrast to the volume-law
scaling of thermalizing systems but also in contrast to constant entanglement of Anderson
localized systems.
Thus, MBL describes a new robust dynamical phase of matter as it remains stable in
the presence of interactions together with logarithmic growth of entanglement. Similar
to Anderson localization, MBL describes a perfectly insulating system. Therefore, the
transport of spin-current is an important quantity to detect localization. The transport
behavior in the ergodic phase for an open quantum system will be investigated in chapter 6
of this thesis.
To truly distinguish an MBL system from Anderson localization, the logarithmic entangle-
ment spreading provides a useful tool. From a computational point of view, the slow growth
of entanglement is advantageous when simulating with matrix product state methods. As
matrix product states provide a formalism which allows to truncate large parts of the
Hilbert space, depending on the entanglement entropy, large system sizes in the MBL phase
become numerically accessible. Matrix product states will be introduced in chapter 7 of
this thesis.
In chapter 9, MBL connected with MPS methods is used to simulate a large many-body
system. In contrast to isolated MBL systems introduced in this chapter, an open quantum
system will be studied. It is still under debate what survives of MBL in case of an open
quantum system [
34
]. In chapter 9 it will be shown that the time crystal within a MBL phase
is stabilized against external dissipation due to feedback dynamics which are introduced in
chapter 8.
PART I I
Factorized system-reservoir dynamics
33
5.
Many-emitter phonon lasing
In this part of the thesis, open many-body systems are investigated. The focus does not
lie on the reservoir degrees of freedom but on the many-body system itself. It is assumed
that the investigated many-body system and external reservoir states factorize and the
Born-Markov approximation is valid. Thus, the Lindblad form in Equation 3.3.16 suffices
to describe the coupling to the surrounding reservoir.
As a first many-body system, quantum dots placed inside a cavity are considered with
focus on many-body effects. In analogy to the standard optical laser [
80
,
81
], the concept
of coherent amplification by stimulated emission is adapted to sound waves. The idea is to
create coherent vibrations, which could lead to new types of non-demolishing measurement
devices [
82
]. Quantized vibrations of the lattice ions in a solid form the quasi-particle
called phonon. The generation of coherent phonon statistics is furthermore interesting
for fundamental physics itself. The idea of the so called phonon laser has led to a variety
of experimental and theoretical proposals to generate coherent phonons in e.g. trapped
ions [
83
,
84
], compound microcavities [
85
], NV-centers [
86
], electromagnetic resonators
[
87
] and semiconductor devices [
88
,
89
,
90
,
91
,
92
,
93
]. In order to achieve lasing, the
necessary ingredients are the active medium, an external pump mechanism, inversion and
a cavity to confine a single mode. Thus, the design of the phonon or acoustic cavities
[
94
,
95
,
96
,
97
,
98
,
99
] forms the basis for phonon lasing. Via a combination of different
lattice constants of the surrounding solid, a superlattice allows to confine a single phonon
mode inside the cavity [
94
,
95
,
96
]. Quality factors up to
Q
= 10
5
have been achieved
in the past [
97
,
98
,
99
]. The investigated model setup is based on a single quantum dot
as an active medium [
100
] embedded within an acoustic cavity. Via external coherent
optical excitation of the quantum dot [
101
], the induced Raman process [
91
,
92
] leads to a
coherent phonon population inside the acoustic cavity. The focus within this chapter lies
on a generalization of the single-emitter quantum dot to a many-emitter system [
102
,
103
].
The quantum dots are assumed to be identical and not coupled directly with each other,
but via the cavity phonon field. For optical cavities this is known as the Tavis-Cummings
model [
104
,
105
]. Due to the many-emitter setup, collective effects are present. One example
is superradiance, discovered by Dicke [
106
,
107
,
108
,
109
]. Similar collective phenomena
have been found recently for phonons as well [
110
,
111
]. These collective processes due
to a many-body setup, combined with the generation of coherent phonons by stimulated
emission will be the focus of this chapter.
5.1. Model
As a model system, the semi-classical coherently driven TLS in Equation 3.1.1 forms the
basis for a single emitter. For an application, the idea is to use a semiconductor quantum
dot, which is modelled by a TLS [
91
]. Here, the single-emitter setup is generalized to many
35
Chapter 5. Many-emitter phonon lasing 36
emitters. To start with a simple model, the TLSs are assumed to be identical with frequency
difference
ω01
between the ground
|
0
ii
and excited state
|
1
ii
of emitter
i
. This will be
generalized to non-identical emitters in section 5.3. The coupling to the external photon
reservoir will be described by the Lindblad master equation due to factorized system-
reservoir dynamics, derived in Equation 3.3.16, resulting in a rate for the spontaneous
emission Γ
R
of the TLS. In addition, a surrounding phonon reservoir is assumed leading to
a cavity loss κwhich is also described via a Lindblad dissipator.
As mentioned before, the quantum dots are embedded within a phonon cavity. This results
in a single frequency
ωph
for the free evolution of the phonons. In this thesis, phonons are
described by the operators
ˆc(†)
to clarify the presence of a single-mode phonon cavity. Note
that phonons and photons obey the same bosonic commutation relations. However, the
coupling to the electronic degrees of freedom is fundamentally different, giving rise to the
effects investigated within this chapter.
The Hamiltonian for the free evolution of the TLS together with the free evolution of the
phonon cavity reads
H0=~ω01
N
X
i=1
σ+
iσ−
i+~ωphˆc†ˆc . (5.1.1)
The pump mechanism is described as a semi-classical laser driving each TLS. The according
Hamiltonian was derived in section 2.4. The external driving field is expressed as Ω(
t
) =
2
~E0cos
(
ωLt
), where
ωL
is the frequency and
E0
the Rabi-frequency of the driving laser. in
the following, the Rabi-frequency
E0
is denoted by Ω. The rotating wave approximation
(RWA) was already performed, assuming the pump strength Ωto be small in comparison
to the transition frequency
ω01
. Furthermore, the driving frequency
ωL
is assumed to be
close to the TLS frequency
ω01
. Note that the driving laser is always detuned by several
phonon frequencies. The RWA is still justified as the phonon cavity frequency is orders of
magnitudes smaller than the TLS frequency. Introducing the electron-phonon coupling,
the interaction Hamiltonian reads
HI=
N
X
i=1 h~Ωσ−
ieiωLt+σ+
ie−iωLt+~gph ˆc+ ˆc†σ+
iσ−
ii.(5.1.2)
The second part describes the interaction between the TLSs and the cavity phonons via
coupling
gph
. This diagonal coupling is responsible for the collective effects investigated
within this chapter.
The frequency of the energy-gap
ω01
deviates orders of magnitudes from the other involved
frequencies. In order to deal numerically with these two time-scales, the Hamiltonian is
transformed into a rotating frame with respect to ω01 as shown in Equation 3.1.3
H=~∆
2
N
X
i=1
σz
i+~ωphˆc†ˆc+
N
X
i=1 h~gph ˆc+ ˆc†σ+
iσ−
i+~Ωσx
ii.(5.1.3)
The difference to Equation 3.1.5 is that the external driving laser is detuned from the TLS
transition frequency with ∆ =
ωL−ω01
. This detuning is the key parameter to trigger
coherent phonon excitation in this setup, which is shown in Figure 5.1. The state of the
system is expressed as NN
i|nTLSii⊗|ni, where nT LS ∈ {0,1}is the state of the ith TLS
and
n∈
[0
,∞
)is the occupation of the phonon cavity mode, expressed in the phonon
number basis. The excitation laser is blue detuned at the phonon cavity frequency ∆
≈ωph
.
Chapter 5. Many-emitter phonon lasing 37
...
(a) Energetic excitation scheme (b) Phonon lasing cycle
Figure 5.1.:
Illustration of the many-emitter setup leading to coherent phonon generation.
(a) Solid lines denote the electronic energy levels with transition frequency
ω01
.
Dashed lines correspond to the virtual phonon energies
ωph
. The distributed
Bragg-reflector structure on the level of the phonon energies symbolizes the
acoustic cavity for this frequency. The cavity is subject to a phonon loss
κ
.
The external laser
ωL
is detuned at the anti-Stokes resonance ∆
≈ωph
. (b)
Starting in the ground state with
n
phonons, the TLS is excited by Ω. Due to
the electron-phonon coupling
gph
a phonon is emitted into the cavity. The loop
is closed due to a radiative decay Γ
R
where the TLS ends up in the ground
state again but with an additional phonon in the cavity n+1.
Thus, in exciting one TLS from
|
0
ii
to
|
1
ii
there is additional energy left resulting in the
creation of a phonon via
gph
. Via couplings Ωand
gph
, the state is brought from
|
0
i,ni
to
|
1
i,n
+1
i
. If the cavity is already populated, the phonon is created due to a stimulated
emission process [91]. This process is called the induced Raman process.
These processes are both reversible, indicated by the arrows in Figure 5.1(b). However,
if an external mode continuum is assumed, as done in Equation 3.1.1, the TLS interacts
with the photonic environment as well. For weak coupling strengths and general reservoirs,
usually the Born-Markov approximation is performed to arrive at the Lindblad equation.
This was shown in section 3.3 with a stochastic calculus. This stochastic calculus implies
that the process in Equation 3.3.16 is an irreversible process. However, this irreversible
spontaneous emission of photons to the surrounding reservoir is advantageous when it
comes to phonon lasing. This process prevents the system from just going the way back
from
|
1
i,n
+1
i
to
|
0
i,ni
by reabsorbing a cavity phonon. Thus, a radiative decay from
|
1
i,n
+1
i
to
|
0
i,n
+1
i
is a wanted process as it ensures that the lasing cycle starts from
the beginning with an additional phonon inside the cavity in |0i,n+1i.
It was shown in section 3.3 that the Lindblad master equation is justified for factorized
system-reservoir dynamics. Thus, the surrounding photon reservoir is described by Equa-
tion 3.3.16. Additionally, a surrounding phonon reservoir is assumed. This phonon reservoir
is described by the Lindblad master equation as well and damps the cavity phonons via
rate κ. Thus, in total the master equation for the reduced density matrix ˆρsreads
˙
ˆρs=−i
[H,ˆρs]+2κD[ˆc]ˆρs+ 2ΓR
N
X
i=1 D[σ−
i]ˆρs,(5.1.4)
Chapter 5. Many-emitter phonon lasing 38
with the dissipator Dintroduced in Equation 3.3.16.
The number of differential equations
NDGL
of the reduced density matrix therefore scales
as
NDGL =2NT LS Nph2,(5.1.5)
where
NTLS
is the number of emitters and
Nph
the number state of considered phonons.
Due to the open system dynamics, the phonon number of the cavity reaches a steady
state for
t→ ∞
. The maximal number of considered phonons
Nph
needs to be chosen
according to the parameters such that all relevant phonon states are included. This cutoff
is chosen such that the dynamics do not change for increasing
Nph
which is in practice
around
Nph ∼
50. Due to the number state for the cavity phonons and the resulting number
of differential equations, the number of included emitters is restricted to a maximum of
NTLS
= 3 in this chapter. However, this already results in
NDGL
= 16
×
10
4
which takes a
long computation time to reach the steady state.
Initially, all TLS are set to their ground state
|
0
ii
. The phonons are assumed to obey a
thermal distribution at temperature
T
= 4
K
. The set of equations is solved numerically
up until all entries of the density matrix converge to their steady state value
ˆρs
(
∞
). The
relevant expectation value is then computed for the respective parameter set. The most
important one is the expectation value of the phonon number
hnphi
of the acoustic cavity
hnphi= lim
t→∞ tr ˆρs(t)ˆc†ˆc.(5.1.6)
This quantity measures the occupation of the phonon mode of the acoustic cavity. However,
to determine whether the phonons obey a coherent statistics, the autocorrelation function
g2
(
τ
)[
112
,
113
] is another important quantity of this chapter. Its steady state value is
defined as
g2(τ) = lim
t→∞ Dˆc†(t)ˆc†(t+τ)ˆc(t+τ)ˆc(t)E
hˆc†(t)ˆc(t)i2.(5.1.7)
In this case,
τ
= 0 is sufficient as
g2
(0) = 1 gives a hint on coherent statistics, whereas
a value
g2
(0)
>
1is identified with thermal statistics. The autocorrelation function is
computed as
g2(0) = lim
t→∞
tr ˆρs(t)ˆc†ˆc†ˆcˆc
hnphi2.(5.1.8)
5.2. Collective phonon processes
The induced Raman process takes place, when the TLS is excited with a blue detuned laser
close to the phonon frequency ∆
≈ωph
. In the Refs. [
92
,
93
] it was shown that the exact
frequency is subject to a shift with respect to the involved couplings. Maximal output is
obtained for
∆ = ωph −g2
ph
ωph
.(5.2.1)
Chapter 5. Many-emitter phonon lasing 39
Thus, in addition to the phonon cavity frequency
ωph
, the energy to create coherent phonons
is shifted to lower energies according to the electron-phonon coupling
gph
. The goal of this
chapter is to show that this is only true for a single emitter and the many-emitter setup
is subject to collective phenomena which include additional energy shifts. Note that in
principle, the laser pump Ωalso results in a shift of the driving frequency [
92
,
93
]. To
clarify the collective effects, Ωis chosen a magnitude smaller than the electron-phonon
coupling
gph
which is why the energy shift Ω
2/
∆is neglected for the interpretation. In
section 5.4 also higher pumping strengths are investigated.
In this section, the resonances for which collective phonon generation takes place will be
investigated by varying ∆. In Figure 5.2 (top), the phonon number
hnphi
is shown for
varying the optical detuning, normalized by the cavity frequency ∆
/ωph
. Thus, ∆
/ωph
= 1
would indicate the anti-Stokes resonance without additional frequency shifts. For one
emitter (yellow, dashed), the resonance is red-shifted with respect to Equation 5.2.1 and
reproduces the findings of Refs. [
92
,
93
]. This resonance is called the single-emitter resonance
in the following. However, for two emitters (red, dotted), the maximal phonon number is
not obtained at the same resonance. The single-emitter resonance of the two-emitter case
also results in lasing (cp. Figure 5.2 bottom) with a slightly higher phonon number, but
does not result in double the phonon number as expected for adding a second emitter. This
output is obtained at a second resonance which is even more red-shifted. This resonance is
labeled the two-emitter collective resonance in the following. For three emitters (blue, solid),
the single-emitter resonance as well as the two-emitter collective resonance is included,
resulting in comparable phonon numbers. Furthermore, the three-emitter setup results in a
third resonance which shows thrice the number of cavity phonons than the single-emitter
resonance. These additional resonances are a characteristic and unexpected feature of
the many-emitter phonon laser [
114
,
115
] and are the subject of investigation of this
chapter. Note, that the linewidth of the collective resonances narrow in comparison to the
single-emitter resonance.
In Figure 5.2 (bottom), the steady state value of the
g2
(0)-function is shown for varying
∆. At each resonance (the single and collective resonances), the statistics show
g2
(0) = 1
which is associated to coherent phonon statistics. Thus, when the phonon emission is
triggered by the induced-Raman process, the statistics of the cavity phonons is coherent
which proves a stimulated emission of phonons. Surprisingly, this is also the case for the
collective resonances including a higher number of coherent cavity phonons.
When the detuning is increased to higher driving frequencies, i.e. approximately the doubled
cavity phonon frequency ∆
≈
2
ωph
, additional resonances appear with coherent phonon
statistics
g2
(0) = 1. Those resonances are two-phonon resonances which means that in a
single laser loop two phonons are emitted due to the energy mismatch (cp. Figure 5.1).
Thus, starting in
|
0
i, ni
results in
|
0
i, n
+ 2
i
in a single loop for a single emitter. In contrast
to the one phonon resonances at ∆
≈ωph
, the linewidth of the two-phonon resonances
is narrowing due to the two-phonon process. However, the output of the two-phonon
single-emitter resonance is twice as high as for the one-phonon single-emitter resonance.
The collective two-emitter (two-phonon) resonance is also apparent for two and three
emitters and results in a doubled phonon number compared to the collective two-emitter
resonance at ∆
≈ωph
. Note, that the three-emitter collective resonance does not appear for
the investigated parameter set. This may have two reasons, on the one hand it is possible
that the linewidth is too narrow to detect it by varying ∆numerically. Or, on the other
hand, the pumping strength Ωmight be too small to trigger this process. This will become
Chapter 5. Many-emitter phonon lasing 40
0
5
10
15
20
25
0.85 0.9 0.95 1 1.05 1.85 1.9 1.95 2 2.05
hnphi
1 Emitter
2 Emitter
3 Emitter
1
10
100
1000
0.85 0.9 0.95 1 1.05 1.85 1.9 1.95 2 2.05
g2(0)
∆/ωph
1 Emitter
2 Emitter
3 Emitter
Figure 5.2.:
Detuning of the TLS transition frequency ∆versus phonon number
hnphi
(top)
and
g2
(0) (bottom). One emitter (yellow, dashed) shows a single resonance at
∆
≈ωph
(top, left). However, increasing the number of emitters, for two emitters
(red, dashed) there are two resonances and for three emitters (blue, solid) there
are three resonances. A similar pattern appears close to ∆
≈
2
ωph
. The second-
order correlation function (bottom) shows coherent phonon statistics
g2
(0) = 1
at the respective resonances. Parameters:
ω01
= 2
.
28 1
/fs
,
ωph
= 0
.
011 1
/fs
,
Ω = 4
.
56
·
10
−4
1
/fs
,
gph
= 2
·
10
−3
1
/fs
,Γ
R
= 1
·
10
−5
1
/fs
,
κ
= 5
·
10
−7
1
/fs
.
Chapter 5. Many-emitter phonon lasing 41
clear in the following, where the reason for the appearance of these collective resonances is
investigated.
5.2.1. Effective Hamiltonian approach
To unravel the origin of the collective resonances for the many-emitter phonon laser, the
idea is to transform the Hamiltonian in Equation 5.1.3 to compare it to a well-known
many-emitter Hamiltonian which is the Tavis-Cummings model [
104
,
105
]. It will become
clear that when bringing the Hamiltonian in Equation 5.1.3 into a form, comparable to the
Tavis-Cummings Hamiltonian, an additional term is included which is responsible for the
collective resonances, resulting from the diagonal electron-phonon interaction. Similar to
the Refs. [
92
,
116
], the idea is to eliminate the first-order electronic processes and restrict it
to the second-order process of cavity phonon generation. Thus, with one effective coupling
constant, the electron is brought from the ground to the excited state and a phonon is
created. Furthermore, the derived effective Hamiltonian will be restricted to the excitation
condition ∆
≈ωph
. To transform the Hamiltonian, a unitary operator
S
is defined such
that
Heff =eiSHe−iS .(5.2.2)
A typical second-order perturbation treatment [
92
,
63
,
59
] is performed in expanding the
exponential and cut the expansion such that
Heff =H0+HI+ [iS, H0]+[iS, HI] + 1
2[iS [iS, H0]] .(5.2.3)
With this, an Ansatz for the unitary operator is made such that it contains all relevant
operators
S=
N
X
i=1 αiσ−
i+α∗
iσ+
i+γiσ+
iσ−
iˆc+γ∗
iσ+
iσ−
iˆc†.(5.2.4)
The first commutator of the expansion then reads
[iS, H0] = X
i
i~∆αiσ−
i−α∗
iσ+
i+X
i
i~ωphσ+
iσ−
iγiˆc−γ∗
iˆc†.(5.2.5)
The first-order process is then eliminated by choosing appropriate coefficients. With
αi=iΩ
∆, γi=igph
ωph
,(5.2.6)
the first commutator cancels out with the interaction Hamiltonian such that
[iS, H0] = −HI,(5.2.7)
[iS [iS, H0]] = −[iS, HI].(5.2.8)
With this, the effective Hamiltonian is derived via
Heff =H0+1
2[iS, HI].(5.2.9)
Chapter 5. Many-emitter phonon lasing 42
The commutator with the interaction Hamiltonian reads
[iS, HI] =
N
X
i"−~Ωgph
∆iσ−
i+σ+
iˆc†+ ˆc+2~Ω2
∆i
σz
i#+~Ωgph
ωph
N
X
iσ+
iˆc†−σ−
iˆc†+h.c.
+
~g2
ph
ωph
N
X
i,j n−hσ+
iσ−
iˆc, σ+
jσ−
jˆc†+ ˆci+hσ+
iσ−
iˆc†, σ+
jσ−
jˆc†+ ˆcio .
(5.2.10)
The last term describes the interaction between two emitters
i
and
j
. In general, the
spin-matrices for different emitters commute with each other. However, the cavity operators
do not commute. This can be interpreted as an interaction between emitters, mediated
by the cavity phonons. Thus, there is a part of the last term in Equation 5.2.10 which
does not reduce to a sum of a single emitter which is the crucial term for the collective
resonances. Rearranging Equation 5.2.10 results in
[iS, HI] =~
N
X
i 2Ω2
∆σz
i−2g2
ph
ωph
σ+
iσ−
i!−
N
X
i,j
i6=j
2g2
ph
ωph
σ+
iσ−
i⊗σ+
jσ−
j
+~
N
X
i
Ωgph 1
∆+1
ωph !σ+
iˆc†+σ−
iˆc
+~
N
X
i
Ωgph 1
∆−1
ωph !σ−
iˆc†+σ+
iˆc.(5.2.11)
The second term in the first line is the interaction between emitters. The last line can be
neglected for driving close to the anti-Stokes resonance ∆
≈ωph
. Inserting Equation 5.2.11
into Equation 5.2.9 and shifting the origin of the
σz
i
terms results in the effective Hamiltonian
Heff =~ωphˆc†ˆc+
N
X
i
~ −∆−2Ω
∆−g2
ph
ωph !σ+
iσ−
i−
N
X
i,j
i6=j
g2
ph
ωph
σ+
iσ−
i⊗σ+
jσ−
j
+~
N
X
i
Ωgph
2 1
∆+1
ωph !σ+
iˆc†+σ−
iˆc.(5.2.12)
By defining the effective transition frequency
ωeff =−∆−2|Ω|2
∆−g2
ph
ωph
(5.2.13)
and effective coupling strengths
geff =Ωgph
2 1
∆+1
ωph !,(5.2.14)
Chapter 5. Many-emitter phonon lasing 43
(a) Jaynes-Cummings model (b) Effective phonon laser model
Figure 5.3.:
Comparison of the Jaynes-Cummings model with the effective phonon laser for
a single emitter
i
. (a) The interaction of a TLS with cavity photons in RWA
(counter rotating terms are neglected) results in the annihilation of cavity
photons when the TLS is excited via
gJCM
and vice versa. (b) In contrast, for
the effective Hamiltonian, a phonon is created when the TLS is excited via
geff and vice versa.
the effective Hamiltonian becomes formally close to the Tavis-Cummings Hamiltonian.
The effective Hamiltonian is split into three parts to highlight the additional interaction
between emitters
Heff =Heff
0+Heff
I+Heff
E−E,(5.2.15)
with Heff
E−Eas the many-particle interaction term
Heff
0=
N
X
i=1
~ωeff σ+
iσ−
i+~ωphˆc†ˆc , (5.2.16)
Heff
I=
N
X
i=1
~geff σ−
iˆc+σ+
iˆc†,(5.2.17)
Heff
E−E=−
N
X
i6=j
g2
ph
ωph σ+
iσ−
i⊗σ+
jσ−
j.(5.2.18)
Initially, it was assumed that all emitters are identical. Thus, Equation 5.2.16 and Equa-
tion 5.2.17 do not differ from the case of a single emitter [
92
] except for the sum over
the number of emitters. However, Equation 5.2.18 is an additional term resulting from
the interaction between emitters, mediated by the cavity phonons. This term is crucial
for the resonances of the many-emitter phonon laser. Before analyzing this term in detail,
the analogy to the Tavis-Cummings Hamiltonian is discussed: Although, formally it looks
similar, there is a fundamental difference to the Tavis-Cummings model and photons in
general. The creation and annihilation of cavity excitations is reversed for phonons as
it can be seen in Figure 5.3(b). By exciting the TLS, a phonon is created via
geff
and
vice versa. As the first-order process was eliminated, now a single process combines the
excitation of the TLS via Ωas well as the creation of a phonon via
gph
. In contrast, for the
interaction of a TLS with cavity photons in RWA, an excitation within the TLS is created
by annihilating a cavity photon via coupling
gJCM
as it can be seen in Figure 5.3(a). This
illustrates the difference between a laser and the phonon laser as the electron-phonon
interaction is diagonal. When comparing the effective Hamiltonian to the Tavis-Cummings
Chapter 5. Many-emitter phonon lasing 44
0
2
4
6
8
10
12
0.7 0.75 0.8 0.85 0.9 0.95
hnphi
∆/ωph
Hfull
Heff
Heff without Heff
E−E
Figure 5.4.:
Comparison of the full (red, solid) and the effective Hamiltonian (orange,
dotted) for two emitters. Both models are in good agreement. The maximum
is slightly red-shifted for the effective Hamiltonian with a smaller phonon
number and a broader linewidth. When
Heff
E−E
is not considered, the effective
Hamiltonian (yellow, dashed) shows only the single-emitter resonance with
twice the phonon number. The coupling strength
g
is chosen twice as large as
before to visualize the splitting of the resonances.
Hamiltonian beyond RWA [
117
], the effective Hamiltonian is comparable to the counter
rotating terms which are not energy conserving. For the lasing cycle, this implies that the
ground state
|
0
ii
can be seen as the state which has to be populated to create inversion
for achieving lasing. This again clarifies the importance of a radiative decay Γ
R
for the
phonon lasing cycle as it is the parameter to create population inversion [91, 92].
For the many-emitter phonon laser, the most important difference is the additional term
Heff
E−E
, resulting from the interaction between emitters via the cavity phonon field. In
Figure 5.4, the effective Hamiltonian for two emitters is evaluated. Comparing
Hfull
and
Heff
, both are in a good agreement. Both, the single-emitter as well as the collective
resonance are well described by the effective Hamiltonian, although the resonances are
slightly red-shifted and the maximum and linewidth is different. This might be due to the
neglected term in Equation 5.2.11 in setting ∆ =
ωph
. As the resonances are red-shifted
from the anti-Stokes resonance, this term might have small relevance. By neglecting
Heff
E−E
,
it becomes clear in Figure 5.4 that this term is responsible for the splitting of the single-
emitter and collective resonances. Without
Heff
E−E
, the two-emitter setup only shows the
single-emitter resonance with approximately twice the number of phonons compared to
the single-emitter setup.
The effective transition frequency
ωeff
in Equation 5.2.13 consists of two terms which
add up to the detuning. On the one hand 2
|
Ω
|2/
∆which is negligible due to the chosen
parameters and on the other hand
g2
ph/ωph
which is responsible for the red-shift of the
single-emitter resonance. In addition, the many-body interaction
Heff
E−E
has the same
prefactor
g2
ph/ωph
and also contributes to the effective transition frequency if both emitters
are found in the excited state. This explains why the collective resonance for two emitters is
Chapter 5. Many-emitter phonon lasing 45
0
2
4
6
8
10
12
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
hnphi
∆/ωph
1 Emitter
2 Emitter
0
2
4
6
8
10
12
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
hnphi
∆/ωph
1 Emitter
2 Emitter
0
2
4
6
8
10
12
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
hnphi
∆/ωph
1 Emitter
2 Emitter
(a) Ω=0.03 meV (b) Ω=0.1meV (c) Ω=0.25 meV
Figure 5.5.:
Phonon number versus optical detuning for three different pump strengths Ω.
For low pumping strengths (a), only the single-emitter resonance is apparent.
Increasing the pumping strength (b), for two emitters it is probable for both
emitters being excited at the same time such that eq.
(5.2.18)
becomes relevant.
For higher pumping strengths (c), the collective resonance shows twice as much
phonons at the collective resonance than at the single-emitter resonance.
Furthermore, the resonances are broadening due to the stronger pumping, such
that both resonances overlap.
red-shifted with 2
g2
ph/ωph
. Knowing that
Heff
E−E
is responsible for the collective resonances,
it will be investigated in more detail in the following.
5.2.2. Collective resonances
The lasing cycle at the respective resonance is analyzed in analogy to Figure 5.1(b). For
simplicity, the interpretation is limited to two emitters. For the single-emitter resonance, one
of the emitters is excited at
|
1
i, ni
. By interacting with a cavity phonon, a second phonon
is generated via a stimulated emission process resulting in
|
1
i, n
+ 1
i
. Due to the involved
couplings, this process is most effective for a detuning calculated via Equation 5.2.13.
The same idea holds for the two-emitter collective resonance where two emitters are
involved. Both emitters are excited in
|
1
i,
1
j, ni
and create collectively two cavity phonons
via stimulated emission resulting in
|
1
i,
1
j, n
+ 2
i
. In other words, both emitters collectively
generate a phonon, resulting in two more phonons in the cavity. When both emitters are
excited at the same time, an additional red-shift has to be considered for the excitation
frequency according to Equation 5.2.18 resulting in
∆collective ≈ωph −
2g2
ph
ωph
.(5.2.19)
Due to this process, the phonon number is doubled and results in a linear scaling of the
phonon number with the number of emitters. In Figure 5.2 a narrowing of the linewidth is
observed which indicates a longer lifetime of the collective excitation of the emitters.
In order to verify the collective red-shift in Equation 5.2.18, its dependence on the pumping
strength is investigated. As it is required that both participating emitters are in the excited
state, the pumping strength would have a huge impact on the collective resonance as
Ωbrings the TLS to the excited state. Indeed, the collective resonance disappears for
small pumping strengths where the single-emitter resonance is lasing which is shown in
Figure 5.5(a). For higher pumping strengths, Figure 5.5(b) shows that also the collective
resonance starts lasing as it becomes probable for both emitters being in the excited state.
After the resonances reach the maximal phonon number which is linear to the number
Chapter 5. Many-emitter phonon lasing 46
of participating emitters, a further increase of the pumping strength results in linewidth
broadening Figure 5.5(c).
Thus, the recipe to address collective phonon generation is to adjust the excitation frequency
according to the number of participating emitters as well as ensuring a high pumping
strength Ωsuch that it is probable for all participating emitters to be in the excited state at
the same time. However, a detuning at the single-emitter resonance only results in phonon
generation of an individual emitter including a limit for the maximal phonon number. The
advantage is that without knowing the exact shift-dependencies, the single-emitter limit is
included in the many-emitter setup as well. Furthermore, varying the detuning allows to
choose between different phonon intensities in a many-emitter setup, as long as
gph
is large
enough that the resonances split up.
This brings up an important point: The position of the resonances is very sensible to the
electron-phonon coupling
gph
. A comparable small
gph
would smear out all resonances
as the position scales as
Ng2
ph/ωph
and every resonance would be addressed close to the
anti-Stokes resonance. In practice, the exact value of the electron-phonon coupling
gph
is
often unclear. As the position of the resonances is highly sensible on
gph
, these findings
could lead to an experimental setup to deduce the electron-phonon coupling for the involved
materials as well.
5.2.3. Two-phonon resonances
Having identified the additional resonances in a many-emitter setup close to ∆
≈ωph
(cp.
Figure 5.2) with collective phonon generation due to Equation 5.2.18, the resonances close
to ∆
≈2ωph
remain to be interpreted. In contrast to ∆
≈ωph
, where additional resonances
appear due to collective phonon generation of the respective number of emitters, only one
additional collective resonance appears close to the two-phonon resonance. Furthermore,
the phonon number for driving at the two-phonon resonance is increased approximately by
a factor of two.
The first resonance close to ∆
≈2ωph
is called the single-emitter two-phonon resonance.
The reason is that an individual emitter is excited by the driving laser with a detuning
almost twice the cavity frequency. Thus, an individual emitter is brought from
|
0
i, ni
to
|
1
i, n
+ 2
i
in a single lasing loop (cp. Figure 5.1(b)). This means, in a single loop, two
phonons are created in the cavity by a single emitter. This is in contrast to the two-emitter
collective resonance close to ∆
≈ωph
where two phonons are created by two emitters
collectively
|
1
i,
1
j, n
+ 2
i
. On the one hand, two phonons are created by an individual
emitter in a single loop and on the other hand, two phonons are created by two emitters
collectively in a single loop. However, comparing both resonances, the phonon number as
well as the linewidth is similar for both excitation schemes.
The collective resonance is also present for driving at the two-phonon resonance. This
process results in a collective generation of four phonons by two emitters in a single loop.
Two emitters are excited
|
1
i,
1
j, ni
and create four phonons by interacting with the cavity
field
|
1
i,
1
j, n
+ 4
i
. In Figure 5.2 it becomes clear that a detuning at this resonance results
in four times the phonons compared to the single-emitter resonance and twice the number
of phonons compared to the two-emitter collective / single-emitter two-phonon resonance.
Furthermore, its linewidth is very narrow compared to the other resonances. This leads to
the three-emitter collective two-phonon resonance which is not apparent in Figure 5.2. This
resonance is identified with a process of three emitters
|
1
i,
1
j,
1
k, ni
collectively generating
six phonons in a single loop
|
1
i,
1
j,
1
k, n
+ 6
i
. Thus, each emitter creates two phonons
simultaneously. Its linewidth is very narrow and it becomes only visible when increasing the
Chapter 5. Many-emitter phonon lasing 47
0
2
4
6
8
10
12
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
collective resonance
single emitter resonance
single emitter resonance
Figure 5.6.:
Detuning versus phonon number for two emitters
i
and
j
which differ in the
transition frequency. The second emitter has either 5% (red,solid) or 10%
(yellow, dashed) difference in
ωj
01
, while the first one is kept at the same
transition frequency as before. Both emitters have different single-emitter
resonances. In both cases, the collective resonance is still visible. The linewidth
of the collective resonance narrows and the phonon number decreases for
increasing difference in the transition frequencies. The position of the collective
resonance is a mixture of both transition frequencies.
pump strength. For the investigated pump strength in Figure 5.2, no according resonance
was identified. For higher pump strength it became apparent as well.
5.3. Non-identical emitters
One requirement for the Hamiltonian in Equation 5.1.3 was that all emitters are identical
with respect to their transition frequencies and the electron-phonon coupling. However,
realistic emitters such as quantum dots differ in size. This results in different transition
frequencies
ωi
01
and electron-phonon couplings
gi
ph
for each emitter
i
[
118
,
119
,
120
,
9
]. The
purpose of this section is to analyze, if the collective effects found in section 5.2 also hold
for non-identical emitters. In particular, it will be shown which differences in transition
frequencies or electron-phonon couplings can be tolerated to still observe collective phonon
generation.
Starting with a difference in the transition frequencies, the positions of the resonances are
sensible for non-identical emitters as the detuning itself has to be adjusted. It is assumed,
that one emitter
i
has the same transition frequency
ωi
01
as before and a second emitter is
assumed to have a different transition frequency ωj
01. The resulting resonances are shown
in Figure 5.6 for
ωj
01
=0
.
95
ωi
01
(red, solid) and
ωj
01
=0
.
90
ωi
01
(yellow, dashed). The
single-emitter resonance of emitter
i
does not change as it still has the same transition
frequency. As expected, the single-emitter resonance of emitter
j
is red-shifted to lower
frequencies. However, the collective resonance is still apparent and for both cases found
between the two single-emitter resonances. For a deviation of 10%, the collective resonance
is narrowing remarkably such that it can be concluded that a higher deviation in the
Chapter 5. Many-emitter phonon lasing 48
transition frequencies might eliminate collective effects. As it can be seen well in Figure 5.6,
the collective resonance is located at a driving frequency which is a mixture of both
transition frequencies as both emitters participate collectively. The position of the collective
resonance is estimated by
∆collective =ωph −ωcv
1−ωcv
2
2−2g2
ph
ωph
(5.3.1)
and agrees well with both investigated cases. For identical emitters, the second summand
vanishes and yields the position of the collective resonance found in section 5.2.
0
2
4
6
8
10
12
0.4 0.5 0.6 0.7 0.8 0.9 1
single emitter resonance
collective resonance
single emitter resonance
Figure 5.7.:
Detuning versus phonon number for two emitters
i
and
j
which differ in the
electron-phonon coupling
gph
. The first emitter is kept at the same electron-
phonon coupling than before
gi
ph
. As for the transition frequencies, a difference
in
gph
results in a different single-emitter resonance, respectively. For small
differences
gj
ph
=1
.
6
gi
ph
(red, solid) and
gj
ph
=2
gi
ph
(yellow, dashed), the
single-emitter resonance of emitter
j
overlaps with the collective resonance.
For high difference
gj
ph
=4
gi
ph
(orange, dotted), all resonances split up and the
single-emitter resonance is at lower ∆than the collective resonance. The output
of the collective resonance is smaller than for the single-emitter resonance for
high difference in gph.
In contrast to the transition frequencies, where a difference of 10% already results in a
narrowing of the collective resonance, collective effects are more robust against a deviation
in electron-phonon coupling
gph
. Again one emitter
i
is kept at the same electron-phonon
coupling as before whereas the second emitter
j
is assumed to be different. In Figure 5.7
three different cases are investigated:
gj
ph
=1
.
6
gi
ph
(red, solid),
gj
ph
=2
gi
ph
(yellow, dashed)
and
gj
ph
=4
gi
ph
(orange, dotted). Due to the different electron-phonon coupling, the single-
Chapter 5. Many-emitter phonon lasing 49
emitter resonances split up according to Equation 5.2.13 (neglecting the comparably small
shift due to the pump Ω)
∆single
j≈ωph −(gj
ph)2
ωph
.(5.3.2)
The single-emitter resonance of emitter
i
remains at the same position as before as collective
effects do not play a role for the single-emitter resonances. In contrast to a difference
in the transition frequency, the red-shift of the single-emitter resonance is comparably
small such that the single-emitter resonances overlap with the collective resonances for
gj
ph
= 1
.
6
gi
ph
(red, solid) and
gj
ph
= 2
gi
ph
(yellow, dashed). For
gj
ph
= 4
gi
ph
(orange, dotted),
all three resonances, the two single-emitter resonances and the collective resonance, are
well separated. The collective resonance is located at a higher driving frequency than
the single-emitter resonance of emitter
j
. Furthermore, the output of the single-emitter
resonance is higher than the collective resonance for
gj
ph
= 4
gi
ph
which has narrowed
remarkably. The reason might be that
gj
ph
is now much higher and collective effects start
to vanish for a comparable high difference in the electron-phonon coupling.
The position of the collective resonances is determined for all three cases by a mixture of
both electron-phonon couplings
∆collective ≈ωph −gj
ph +gi
ph2
2ωph
.(5.3.3)
This agrees with the collective resonance found in section 5.2 for identical emitters.
In conclusion, non-identical emitters do not destroy the effect of collective phonon generation.
For a difference in the transition frequencies of 10%, a collective resonance was still observed,
although it is narrowing for increasing differences. In contrast, collective effects for emitters
which differ in the electron-phonon coupling are robust up to a deviation of 400%.
5.4. Quantum yield
The different resonances close to ∆
≈ωph
in Figure 5.2 are identified with collective
phonon emission. Driving at the respective collective resonance and bringing a number of
emitters
i
to the excited state
|
1
1, ...,
1
i, ni
, they collectively generate coherent phonons
|
1
1, ...,
1
i, n
+
ii
, where the cavity phonon number in the steady state scales as
hnphi ∼ i
.
The quantum yield, i.e. the increase of the phonon number with an increasing number of
emitters, is the subject of investigation within this section. However, collective effects only
occur if the pumping strength Ωis above a certain threshold. Furthermore, increasing Ω
also results in a shift of the resonances. Thus in particular, the dependence of the quantum
yield on Ωis the focus of this section.
In analogy to section 5.2, the number of emitters is restricted to two to simplify the
interpretation. To quantify the quantum yield, a quantity called phonance witness is
defined in analogy to the radiance witness of Ref. [121]
R=hˆc†ˆci2−2hˆc†ˆci1
2hˆc†ˆci1
.(5.4.1)
Chapter 5. Many-emitter phonon lasing 50
−1
0
1
2
3
4
5
0.01 0.1 1 10
Figure 5.8.:
Phonance witness
R
of the two-emitter phonon laser versus the external
pumping strength Ω. Comparing driving at the single-emitter resonance (red,
solid) with driving at the collective resonance (yellow, dashed), results in
different collective behavior if Ωis increased. The one emitter case
hˆc†ˆci1
is
always driven at the single-emitter resonance. The collective resonance shows
a higher quantum yield
R≥
0after Ω = 0
.
1meV. Both cases show collective
enhancements with
R >
0, including two peaks with even
R >
1, for Ω
≥
2
meV .
The phonance witness compares the expectation value of phonon numbers for the two-
emitter case
hˆc†ˆci2
with the one emitter case
hˆc†ˆci1
. The idea is that
hˆc†ˆci2
represents the
correlated number of phonons for the two-emitter case, whereas 2
hˆc†ˆci1
represents the
expected uncorrelated linear scaling with the number of emitters. Thus,
R
= 0 signifies
no collective enhancements but a linear increase of the phonon number with the number
of emitters. If
R <
0it is interpreted as a subradiant-like behavior, where the number
of emitters suppresses the output [
122
] and
R
=
−
0
.
5reflects the single-emitter scenario
even if two emitters participate in phonon emission. In this section, the focus lies on many-
emitter enhancements in increasing the pumping strength. Thus,
R >
0is advantageous
and represents a superradiant-like behavior where the number of emitters result in an
increase of the phonon number higher than the expected linear scaling.
In the previous section it has been shown that the two-emitter setup shows two resonances
close to ∆
≈ωph
, the single-emitter resonance and the collective resonance, whereas the
one-emitter setup only shows the single-emitter resonance. The question is, how does the
phonance witness change for a detuning at the collective resonance compared to a detuning
at the single-emitter resonance. In Figure 5.8, the phonance witness for driving at the
collective resonance (red, solid) is compared to the case where the optical laser is detuned
at the single-emitter resonance (yellow, dashed) for increasing pump powers Ω. For low
pump powers, driving at the single-emitter resonance shows
R >
0, then drops below zero
and for Ω
>
1meV again shows superradiant-like behavior including two peaks with high
quantum yield. In contrast, driving at the collective resonance shows
R
=
−
1up until
Ω = 0
.
1meV and then immediately shows
R
= 0. For higher pump powers, it shows the
same behavior than for driving at the single-emitter resonance, but enters the respective
regime at lower pump powers.
Chapter 5. Many-emitter phonon lasing 51
0
2
4
6
8
10
12
14
0.01 0.1 1 10
hnphi
Ωin meV
Driving one emitter at single emitter resonance
Driving two emitter at single emitter resonance
Driving two emitter at collective resonance
Figure 5.9.:
Phonon number versus pump power for one emitter (orange, dotted), two
emitters and detuning at the single-emitter resonance (red, solid) and detuning
at the collective resonance (yellow, dashed). Detuning at the single-emitter
resonance for one and two emitters is comparable but the two-emitter case
shows a larger lasing window and an additional peak for high pump powers.
Driving two emitters at the collective resonance only starts lasing at high
pump powers but shows twice the phonon number.
To understand the difference in
R
for increasing the pump power it is useful to take a
look at the phonon number of the different cases which is shown in Figure 5.9. For one
emitter with a detuning at the single-emitter resonance (orange, dotted), the phonon
number increases with the pump power until a maximum is reached. For high pump powers
Ω
>
1, the phonon number decreases again until it stops lasing due to the self-quenching
behavior [
91
,
92
,
93
]. In principle, the self-quenching results from the additional energy
shift with Ω
2/
∆in Equation 5.2.13. Due to this shift, the resonance is brought out of
the chosen detuning. This happens only for high pump powers, where the broadening
of the resonance does not counteract the shifting of the resonance. However, for Ω
≈
9
meV, the broadening combined with the shifting of the resonance results in addressing the
two-phonon resonance as already shown in Ref. [
93
]. Thus, after entering the self-quenching
regime, the one-emitter setup starts lasing again for certain pump powers as long as the
single-emitter two-phonon resonance is addressed.
Adding a second emitter, but keeping the detuning at the single-emitter resonance, up
until Ω=0
.
1meV, the phonon number is comparable to the one-emitter setup. One
difference is that the lasing window is increased and the two-emitter setup enters the lasing
regime at lower pump powers. This explains the phonance witness
R≥
0for small pump
powers in driving at the single-emitter resonance in Figure 5.8 (red, solid). In reaching the
maximal phonon number, the two-emitter setup shows the same phonon number than for
one emitter and the phonance witness drops below zero
R <
0. However, for Ω
>
0
.
1, the
higher pump power results in linewidth broadening and the peaks of the single-emitter
and collective resonances overlap for the investigated electron-phonon coupling. Thus,
due to the linewidth broadening, also collective processes are addressed and the phonon
Chapter 5. Many-emitter phonon lasing 52
number increases (cp. Figure 5.9 (red,solid) for Ω
∈
[0
.
2
,
1] meV). This is the reason why
R
approaches zero again for driving at the single-emitter resonance as for the one-emitter
case no collective resonance is apparent.
For Ω
>
1meV, the one-emitter setup is already within the self-quenching regime whereas
the two-emitter case at the single-emitter resonance still overlaps with the collective
resonance and shows a wider lasing window. This is why the phonance witness shows
superradiant-like behavior with
R >
0for these pumping strength. Around Ω
≈
3meV, the
two-emitter setup enters a regime, where the collective two-phonon resonance is addressed.
Thus, the two-emitter setup shows a high phonon number at these pumping strengths.
This resonance does not exist for the one-emitter setup which is why the phonance witness
shows hyperradiance-like behavior with R > 1.
For Ω
≈
9meV, also the single-emitter two-phonon resonance is addressed. The one-emitter
setup addresses this resonance for slightly higher pump powers which is why the phonance
witness again shows hyperradiant-like behavior.
A detuning at the collective resonance ∆ =
ωph −
2
g2
ph/ωph
shows a slightly different
picture as for intermediate pumping strengths (Ω
∈
[0
.
1
,
1]), the expected quantum yield
is obtained with
R
= 0. However, for low pump powers, the collective resonance is not
lasing as it is not probable that both emitters are excited at the same time as shown in
Figure 5.5. For Ω
>
0
.
5meV, also the collective resonance enters the lasing regime and
reaches the maximal phonon number which is twice as high as for the one-emitter setup.
This results in the expected
R
= 0. As already explained for the single-emitter resonance,
the lasing window is wider, which is why for Ω
>
1the phonance witness increases and
shows superradiant-like behavior, because the one-emitter setup is already within the
self-quenching regime. As for driving at the single-emitter resonance, high pump powers
address the collective and single-emitter two-phonon resonance which results in a high
quantum yield
R >
1. Note that due to the linewidth broadening for high pump powers,
there is not much difference between driving at the single-emitter or the collective resonance
for the investigated electron-phonon coupling.
5.5. Conclusion
As a first many-emitter setup, a set of two-level systems in an acoustic cavity was investi-
gated. The system was computed with a reduced density matrix method and factorizing
system-reservoir dynamics. However, the cavity phonons were considered as part of the
system. Due to the combination of many emitters as well as the exact treatment of cavity
phonon statistics, the number of emitters was limited to three.
In blue-detuning the external optical laser to almost the phonon cavity frequency, coherent
phonon statistics were created in the acoustic cavity. It was shown that the many-emitter
setup results in a variety of resonances to address coherent phonon generation by detuning
the optical laser.
By mapping the full Hamiltonian to a Tavis-Cummings like Hamiltonian via a second-order
perturbative treatment, it was demonstrated that the additional resonances result from
collective effects of the many-emitter setup. If many emitters participate collectively in a
stimulated phonon generation process, the corresponding resonance frequency is lowered
by ∆ =
ωph −Ng2
ph/ωph
. Thus, an
N
-emitter setup results in
N
corresponding resonances
which positions scale with the electron-phonon coupling. A detuning at the respective
collective resonance results in a linear scaling of the phonon number with the number of
participating emitters. However, the linewidth is narrower which makes it more difficult to
Chapter 5. Many-emitter phonon lasing 53
address them for increasing number of emitters.
It was demonstrated that a detuning at the two-phonon resonance also leads to coherent
phonon statistics. The collective resonances also appear in a many-emitter setup close to
the two-phonon frequency. At the two-phonon resonances, the phonon number is doubled
compared to the single-emitter phonon resonances but the linewidth is narrowing as well.
Furthermore, it has been shown that the linewidth depends on the pump power of the
external optical laser and lasing at the collective resonances disappears for decreasing pump
powers, where the single-emitter resonance is still lasing.
For non-identical emitters which differ in transition frequency or electron-phonon coupling,
collective effects have been demonstrated to exist as well. The position of the collective
resonance depends on a mixture of the participating emitters. It has been shown that
collective effects are robust for different transition frequencies up to 10% whereas for the
electron-phonon coupling a difference up to 400% can be tolerated.
Investigating the quantum yield, it has been shown that the two-emitter setup has a wider
lasing window and shows collective enhancements when the single-emitter setup is already
within a self-quenching regime. For high pump powers, the resonances are broadening as
well as shifting with the pump power such that the two-phonon resonance is addressed and
a hyperradiant-like quantum yield is obtained.
6.
Boundary-driven Heisenberg spin-chain
One of the standard models to investigate generic quantum many-body interactions is the
Ising spin-chain. A generalization of this model to all three spin dimensions is the Heisenberg
model. The model was developed to explain ferromagnetic and anti-ferromagnetic phases
of matter. However, this model also has actual relevance as the underlying quantum phase
transitions are more complex in their behavior [
123
]. Therefore, as a second many-body
system, the Heisenberg model is investigated in this thesis.
In recent experiments it became accessible to tune long-range interactions of such generic
quantum spin-models [
124
,
125
,
126
,
127
,
128
,
129
]. Long-range coupling as e.g. Coulomb
interaction is far more general than the assumed nearest-neighbor coupling for this model.
Thus, a generalization from nearest-neighbor coupling to long-range interactions and the
connection to quantum phase transition is of fundamental importance and has drawn a lot
of interest in the recent years [130, 131, 132].
One highly debated phase transition is the many-body localization (MBL) transition
[
20
,
21
,
71
,
72
,
14
,
30
,
31
,
33
,
32
,
73
], which is the generalization of Anderson localization
[
18
] for interacting systems. The disordered Heisenberg-chain has become one of the
standard models to investigate MBL. The existence of MBL in long-range interacting
systems is topic of recent publications [32, 133, 134, 135].
The focus of this thesis lies on open quantum many-body systems. While the existence of
MBL in closed many-body quantum systems has been demonstrated on many platforms, it
is still under debate what survives of MBL in open quantum many-body systems. In closed
quantum many-body systems, the MBL phase prevents a subsystem from thermalizing
with the rest of the closed system. However, in case of an open quantum system, the
interaction with surrounding reservoirs leads to thermalization of the quantum system with
its environment. Thus, it is a challenge to prevent thermalization if surrounding thermal
reservoirs are present as well [
46
,
47
]. Especially dephasing is induced in the measurement
process of optical lattice systems [
48
,
50
,
49
]. However, a well-studied approach to investigate
an open quantum system in case of the Heisenberg chain is to couple the system to two
magnetic reservoirs at the boundaries [
136
,
137
,
138
,
139
,
140
,
141
,
142
,
143
]. The idea
is to apply a voltage where on one side charge is injected and on the other side it is
extracted. The generic theoretical model is described by the Lindblad master equation,
where factorized system and reservoir states are assumed. Both reservoirs are kept at
different potentials, e.g. spin-up at the left side and spin-down at the right side. Due
to the Born-Markov approximation of the external reservoirs, the system is then kept
out-of-equilibrium for all times. For nearest-neighbor coupling between the spins it has
been shown that such systems exhibit negative differential conductivity (NDC) [
144
,
145
]
and anomalous transport [146, 147].
It is an open question what happens in case of strong disorder, including a possible MBL
phase when boundary reservoirs are present. However, for weak disorder it has been
55
Chapter 6. Boundary-driven Heisenberg spin-chain 56
demonstrated that the transport of spin-current contains a transition from diffusive to
subdiffusive transport [
148
]. This subdiffusive behavior was also found for the isolated
Heisenberg chain (which is a closed quantum system) with different results: The subdiffusive
regime was found close to the MBL transition [
149
,
150
] or already for disorder approaching
the clean limit [151, 152, 153].
In this chapter of the thesis, the impact of external reservoirs on the transport is investigated
in detail with focus on the subdiffusive behavior. There are two processes leading to
subdiffusive transport. On the one hand, weak disorder results in rare regions where
disorder is locally stronger and therefore acts as a bottleneck for the transport. These
are called Griffiths effects and it is under debate if this is the reason for the subdiffusive
transport [
149
,
154
]. In this chapter it will be demonstrated that on the other hand, a high
potential difference of the external reservoirs also results in subdiffusive behavior. Due
to the far-from-equilibrium driving, the system builds up ferromagnetic domains at the
boundaries which is called a spin-blockade [
145
,
155
]. This leads to the NDC found by
Ref. [
144
] and results in subdiffusive transport. However, it will be demonstrated that this
spin-blockade and the resulting NDC is not existent for long-range coupling. The transport
of the long-range coupling scenario is independent of the external reservoir parameters
[
156
]. As a consequence, also the transition to subdiffusive behavior in case of disorder is
assumed to be independent of the open system character which would make the long-range
coupled Heisenberg chain an ideal candidate to investigate the transition to MBL for open
quantum systems.
6.1. Model
Figure 6.1.:
Illustration of the Heisenberg spin-chain with two boundary reservoirs: Spin-
flips are illustrated as excitation hopping via (
σ−
iσ+
i+1
)
(†)
. The Ising-like inter-
action
σz
iσz
i+l
will be either a nearest-neighbor (
l
=1) or a long-range coupling.
The system-reservoir interaction is described by four Lindblad dissipators with
excitation in- and outscattering rates Γ. The two reservoirs are kept at a
different potential, described by µ.
As a model system, the Heisenberg quantum spin-chain is investigated. The Heisenberg
spin-chain considers interactions in all three spin dimensions. Except for the long-range
coupling, only the isotropic case is considered, where all spin-coupling constants
J
are
equal. The spin operator
S
=
2σ
is described by the Pauli spin-matrices
{σx,σy,σz}
,
introduced in Equation 2.2.14.
In correspondence to the previous chapters, a single spin is imagined as a TLS. The reason
is that the Heisenberg model can be transformed to a spinless fermion model [
145
] via a
Jordan-Wigner transformation [
70
]. A single site is then described by a TLS with a single
Chapter 6. Boundary-driven Heisenberg spin-chain 57
fermion in analogy to chapter 5. The interactions in
x
and
y
dimension can be mapped to
raising
σ+
and lowering operators
σ−
. Thus, the interactions in
x
and
y
dimension describe
spin-flips where the excitation at one side is annihilated and a neighboring site is excited.
However, in the particle picture this is an excitation hopping and therefore not considered
as an interacting term. Note that this term is also present in systems which are described
by Anderson localization. When talking about interactions, the
σz
term is crucial as it
describes the interactions in the spinless fermion model between sites. The Hamiltonian
reads (~= 1)
H=
N−1
X
i=1
J
4σx
iσx
i+1 +σy
iσy
i+1
+1
4
J
A
N−1
X
i=1
N
X
l>i
1
|l−i|ασz
iσz
l+
N
X
i=1
hi
2σz
i(6.1.1)
A=1
(N)
N−1
X
i=1
N
X
l>i
1
|l−i|α.(6.1.2)
As mentioned before the first two terms describe spin-flips, as it can be seen when mapping
this term to spin-raise and lower operators. The first term in the second line is an Ising-like
interaction between the spins. The model system is schematically shown in Figure 6.1.
In contrast to the standard Heisenberg model, the Ising interaction is assumed to be
long-ranged and decays with the distance. The parameter
α
differs between the coupling
scenarios. When
α→ ∞
, the standard isotropic Heisenberg model is obtained. In this
chapter the case
α
= 1000 is considered as the nearest-neighbor scenario. The case
α
= 0
.
5
is the investigated long-range scenario which is a square root decaying coupling with the
distance. Note that the Ising part is normalized by
A
in Equation 6.1.2 to compare both
coupling scenarios. The nearest-neighbor case yields A= 1 and α= 0 yields A=N/2.
The last term in Equation 6.1.1 describes on-site random disorder of the sites. This is
the crucial term which is responsible for the subdiffusive behavior resulting from Griffiths
effects [
149
,
154
,
148
] and leads to MBL for the closed system [
14
]. The disorder is chosen
randomly
hi∈
[
−h, h
]at each site and averaged out by simulating thousands of realizations
with random on-site disorder. The disorder term is only investigated in the last section of
this chapter where the findings of this chapter are compared to Ref. [148].
As the focus of this thesis lies on open quantum many-body systems, the Heisenberg model
is investigated for a coupling to external reservoirs. In this part of the thesis factorized
system-reservoir interactions are investigated, where the coupling to the external reservoir is
described via a Lindblad master equation which was derived for a single TLS in section 3.3.
As a generic model, the Heisenberg spin-chain is assumed to couple only at the boundaries
to external magnetic reservoirs. Thus, only the boundary spins
{
1
, N}
are subject to system-
reservoir interaction. In the spin-picture the left reservoir contains spin-up and spin-down
magnetization as well as the right reservoir. However, different magnetizations are assumed,
i.e. the left reservoir contains more spin-up magnetization while the right reservoir contains
more spin-down magnetization. If the spins are imagined as TLSs this can be viewed as two
charge reservoirs, where the left reservoir contains a higher number of excitations. Thus, on
the left side inscattering dominates, whereas at the right side outscattering dominates. Due
Chapter 6. Boundary-driven Heisenberg spin-chain 58
to this potential difference, the system is always kept out-of-equilibrium as the external
reservoirs never change. This is modeled as
∂tρ(t) = −i[H, ρ(t)] + X
j∈{L,R}
k∈{+,−}
DhLk
jiρ(t),(6.1.3)
where
D
is the Lindbald dissipator defined in Equation 3.3.16. At the left side, the operators
for the Lindblad dissipator are [148]
L+
L=qΓ(1 + µ)σ+
1, L−
L=qΓ(1 −µ)σ−
1(6.1.4)
and the right side is the complex conjugate
L+
R=qΓ(1 −µ)σ+
N, L−
R=qΓ(1 + µ)σ−
N.(6.1.5)
The parameter
µ
is introduced phenomenologically to generate the bias between in and
outscattering at the left and right side respectively as it can be seen in Figure 6.1. Note
that
µ∈
[0
,
1]. If
µ
= 0, in- and outscattering are the same, whereas for
µ
= 1 at the
left side only inscattering is present and on the right side only outscattering. In other
words, the left reservoir contains only spin-up magnetization and the right reservoir only
spin-down magnetization.
The system is initialized with spin-down magnetization
ρ
(0) =
| ↓↓↓ ...ih↓↓↓ ...|
. As
mentioned before, due to the potential difference of the external reservoirs, the system
is always kept out-of-equilibrium. However, for
limt→∞ ρ
(
t
), the system reaches a steady
state. This steady state is called a non-equilibrium steady state as the bias due to the
different reservoirs is still present. Thus, in the non-equilibrium steady state, the system
has a specific magnetization but also a spin-current from left to right. This non-vanishing
spin-current in the non-equilibrium steady state is the observable of interest in this chapter.
The spin-current is derived via the continuity equation, where
˙
σz
k
is the time derivative of
the on-site magnetization
−˙
σz
k=jk−jk−1
k−(k−1) =jk−1−jk.(6.1.6)
The time derivative is calculated with Heisenberg equation of motion
˙
σz
k
=
i[σz
k, H]
. The
spin-current per site kthen reads
jk=J
4σx
kσy
k+1 −σy
kσx
k+1.(6.1.7)
In the non-equilibrium steady state
limt→∞ ρ
(
t
), the spin-current does not change. Fur-
thermore, it becomes independent of the site index kas it is then equal on all sites
hji= lim
t→∞ tr (ρ(t)jk).(6.1.8)
In the following, the investigated observables of interest will be the magnetization of the
respective site
hσz
ki= lim
t→∞ tr (ρ(t)σz
k),(6.1.9)
Chapter 6. Boundary-driven Heisenberg spin-chain 59
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
Nhji
µ
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(a) α= 1000
0
1
2
3
0 0.2 0.4 0.6 0.8 1
Nhji
µ
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(b) α= 0.5
Figure 6.2.:
Absolute current
Nhji
versus the driving strength
µ
for the nearest-neighbor
scenario (red) (a)
α
= 1000 and long-range coupling (blue) (b)
α
= 0
.
5for
different chain lengths. Both are in a linear response regime for weak driving.
Increasing the driving, nearest-neighbor coupling (a) shows a maximum. Thus,
nearest-neighbor coupling results in NDC for far-from-equilibrium driving.
There is a crossing of the current for different system sizes signifying diffusive
transport at
µdiff ≈
0
.
6. For all driving strengths
µ
, long-range coupling (b)
shows linear response.
the single site current
hji
in Equation 6.1.8 and the absolute current which is a summation
of all single-site currents Nhji.
6.2. Characterizing spin-transport
In this section, the transport for different external driving strengths is investigated. In
particular, the interest lies on the far-from-equilibrium situation, where the strong driving
µ
= 1 has an influence on the transport for nearest-neighbor coupling (
α
= 1000). This case
is compared to the long-range scenario with
α
= 0
.
5. It will be shown that the transport for
long-range coupling for all investigated reservoir parameters acts within a linear response
regime. For all following plots, the spin coupling constant is set equal to the coupling to
the external reservoir
J
= Γ = 1. In Figure 6.2, the absolute current
Nhji
is investigated
for increasing driving strength
µ
. The nearest-neighbor case (a) (red) is compared to
the long-range coupling scenario (b) (blue). As already shown by Refs. [
145
,
144
], the
nearest-neighbor scenario exhibits NDC for increasing the driving strength. This is a
surprising feature of the far-from-equilibrium situation. Why the NDC is surprising can be
understand by a simple model: When increasing the external bias, one would expect that
the current of the system in between responses linearly. Considering as an example Ohms
law, when one increases the voltage, the current increases linearly when the resistance
is held constant. When this example is compared to the investigated setup, the driving
would correspond to the voltage and the inner spin dynamics described by
J
correspond to
the resistance. Thus, when increasing the driving, the inner dynamics change at a certain
point and therefore the resistance is increased which is why NDC is present. In the Refs.
[
145
,
144
] this was explained by the spin-blockade effect, which will be explained in this
chapter later on.
Chapter 6. Boundary-driven Heisenberg spin-chain 60
However, in Figure 6.2 (b) it is shown that long-range coupling acts within a linear response
regime for all driving strengths. Thus, long-range coupling prevents the system from
changing its resistance due to the absence of a spin-blockade.
The focus of this chapter lies on the transport behavior, i.e. the current for increasing
system size N. It is assumed that the current scales as
hji ∼ 1
Nγ,(6.2.1)
where the power-law exponent
γ
defines the transport scenario. The case
γ
= 1 is referred
to as diffusive transport which corresponds to Fick’s law [157]
j=D∇σz,(6.2.2)
where
D
is a diffusion constant independent of the position and chain length. However, for
most cases of this chapter
γ
differs from one which is referred to as anomalous transport.
Any
γ <
1is called superdiffusive transport, whereas any
γ >
1is called subdiffusive
transport. Note that
γ
= 0 is called ballistic transport where the relative current becomes
independent of the system size.
This brings up the point for differentiating between the relative current
hji
and the absolute
current
Nhji
. The reason is that on the one hand, the transport is determined by the
scaling of the relative current with the system size. On the other hand, the crossing from
superdiffusive (
γ <
1) to subdiffusive (
γ >
1) is the most relevant change of the transport
investigated here. When multiplying Equation 6.2.1 by the system size
N
, it becomes
apparent that diffusive transport is obtained where different system sizes show the same
absolute current
Nhji
. In contrast, ballistic transport is present when different system
sizes show the same relative current
hji
. Thus, an intersection of curves which show the
relative current for different system sizes signifies ballistic transport. An intersection of
curves which show the same absolute current for different system sizes signifies diffusive
transport.
This is the case in Figure 6.2 (a) where the NDC sets in. At
µdiff ≈
0
.
6, nearest-neighbor
coupling shows diffusive transport which can be seen by the intersection of the curves of
the absolute current. For smaller driving strengths, the absolute current increases with the
system size, which means the transport is superdiffusive with
γ <
1. For higher driving
strengths, the absolute current decreases with the system size, meaning the transport is
subdiffusive with
γ >
1. In contrast, for long-range coupling and all driving strengths
µ
(Figure 6.2 (b)), the absolute current increases with the system size, meaning
γ <
1.
The scaling of the relative current with the system size according to Equation 6.2.1 is
investigated in the following for the different driving scenarios to determine the value of
γ
.
6.2.1. Weak driving
Starting with the weak driving regime with
µ
= 0
.
02, the power-law exponent
γ
is obtained
by fitting Equation 6.2.1 to the values obtained for the relative current for system sizes
up to
N
= 11. This is shown in Figure 6.3 (a) (red) for the nearest-neighbor case and in
Figure 6.3 (b) (blue) for the long-range scenario. For nearest-neighbor coupling, the fit
yields
hjiµ=0.02 ∼
1
/N0.48
. This value agrees well with the findings of the Refs. [
138
,
148
]
where they obtained
γ
= 0
.
5for system sizes up to
N
= 250. Thus, within the linear
response regime, the nearest-neighbor scenario shows anomalous superdiffusive transport.
For the long-range scenario, the fit yields hjiµ=0.02 ∼1/N0.01. The power-law exponent γ
Chapter 6. Boundary-driven Heisenberg spin-chain 61
0.001
0.01
0.1
1
6 8 10 12
hji
N
Weak driving µ= 0.02
hjiµ=0.02 = 0.01 ∗N−0.48
Strong driving µ= 0.02
hjiµ=1 = 2.08 ∗N−1.87
hjiµ=1 = 0.38 ∗e(N∗−0.27)
(a) α= 1000
0.001
0.01
0.1
1
6 8 10 12
hji
N
Weak driving µ= 0.02
hjiµ=0.02 = 0.01 ∗N−0.01
Strong driving µ= 1
hjiµ=1 = 0.24 ∗N−0.01
(b) α= 0.5
Figure 6.3.:
Comparison of the transport for nearest-neighbor (red) (a) and long-range
coupling (blue) (b): The scaling of the relative current
hji
with the system
size
N
is shown for weak (
µ
= 0
.
02) and strong (
µ
= 1) external driving.
(a) Nearest-neighbor coupling: Weak driving yields the exponoent
γ
= 0
.
48
which is close to the known value
γ
= 0
.
5[
138
] with superdiffusive transport.
Strongest driving results in a change of the transport to either subdiffusive
transport (γ= 1.87) or an exponential decay.
(b) Long-range coupling: The transport for long-range coupling is independent
of the external driving in contrast to the nearest-neighbor scenario. For weak
and strong driving, the transport is nearly ballistic with γ= 0.01.
is close to zero which is nearly ballistic transport. Already for the weak driving regime
there is a difference in the transport between nearest-neighbor and long-range coupling.
6.2.2. Maximal driving
For far-from-equilibrium driving, the nearest-neighbor scenario shows NDC. After crossing
µdiff ≈
0
.
6, the transport changes from superdiffusive to subdiffusive transport with
γ >
1. The fit in Figure 6.3 (a) for maximal driving yields
hjiµ=1 ∼
1
/N1.87
which signifies
subdiffusive transport. In addition, an exponential fit is also shown which would indicate
an insulating system. In the Ref. [
145
] an exponential fit was more adequate. The data
in Fig. Figure 6.3 (a) suggests subdiffusive transport. However, finite size effect might be
present for the data at hand.
The data fit for long-range coupling yields
hjiµ=1 ∼
1
/N0.01
which is the same than for
the weak driving regime with nearly ballistic transport. This means that the transport
for long-range coupling is independent of the external bias of the reservoirs
µ
which is a
remarkable result. To further investigate the dependence of the transport on the external
reservoirs, the current for different coupling strengths Γis investigated.
Chapter 6. Boundary-driven Heisenberg spin-chain 62
6.2.3. Reservoir dependency
1×10−3
2×10−3
3×10−3
12345
hji
Γ/J
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(a) α= 1000,µ= 0.02
1×10−3
2×10−3
3×10−3
4×10−3
5×10−3
12345
hji
Γ/J
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(b) α= 0.5,µ= 0.02
0
4×10−2
8×10−2
12 ×10−2
12345
hji
Γ/J
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(c) α= 1000,µ= 1
0
4×10−2
8×10−2
12 ×10−2
16 ×10−2
20 ×10−2
24 ×10−2
12345
hji
Γ/J
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(d) α= 0.5,µ= 1
Figure 6.4.:
Relative current
hji
versus reservoir coupling strength Γnormalized by the spin-
coupling
J
. Nearest-neighbor coupling (red) is compared to long-range coupling
(blue) for weak driving (a, b) and strong driving (c, d). Nearest-neighbor
coupling shows a different scaling with the system size with a maximum at
Γ =
J
(a). It changes drastically for strong driving, where the maximum tends
to smaller Γfor increasing system size (c). In contrast, for long-range coupling,
the relative current is the same for all coupling strength Γ, with the maximum
at Γ = Jfor weak (b) and strong driving (d).
It was shown that the transport for long-range coupling is robust against far-from-
equilibrium effects induced by the driving strength
µ
. It will be shown that this is also the
case for all investigated reservoir coupling strength Γ, whereas the transport for nearest-
neighbor coupling changes drastically with both
µ
and Γ. In Figure 6.4, nearest-neighbor
coupling (red) (a, c) is compared to long-range coupling (b, d). It is differentiated between
weak driving µ= 0.02 (a, b) and the far-from-equilibrium driving µ= 1 (c, d).
If the potential difference is small (i.e. weak driving
µ
= 0
.
02), both inscattering via
σ+
and outscattering
σ−
is present at both side with a small bias to inscattering at the left
side and outscattering at the right side. For the interpretation of the physical processes for
brevity only the left side is considered. However, the same holds for the right side with
dominant outscattering σ−.
Chapter 6. Boundary-driven Heisenberg spin-chain 63
For both, nearest-neighbor and long-range coupling, the maximum is obtained at Γ =
J
in
the weak driving regime. At the left side, spin-up (Γ(1 +
µ
)) dominates over spin-down
(Γ(1
−µ
)) in the reservoir. Thus, for Γ =
J
, the spin coupling and the resulting spin-flips
dominate over the outscattering
J >
Γ(1
−µ
). In contrast, the inscattering (Γ(1 +
µ
))
dominates over the spin-flips
J <
Γ(1 +
µ
)which is why the left side of the chain favors
spin-up magnetization. This is advantageous for the current as a constant bias is present
also in the non-equilibrium steady state.
When Γ
< J
, lesser inscattering takes places and reduces the current. The inscattering is a
bottleneck for the transport. This is why with decreasing Γin Figure 6.4 (a), the current
becomes more independent of the system size. In contrast, increasing Γalso results in
a higher outscattering
J <
Γ(1
−µ
). For an inscattered excitation, it is more probable
to scatter out before it gets transported to further sites. For this reason, the curves of
different system sizes move together for increasing Γas this process is independent of the
system size. This has consequences for the transport: At the maximum, the power-law
exponent shows
γ
= 0
.
5which is superdiffusive transport. Either decreasing or increasing
Γresults in a bottleneck at the boundary spins which is why the transport becomes more
independent of the system size and thus, approaches the ballistic limit
γ→
0. This can be
seen in Figure 6.4 (a) that the curves for different system sizes move together away from
the maximum.
However, long-range coupling tells a different story. At the maximum Γ =
J
, different
system sizes already intersect, signifying ballistic transport. Increasing or decreasing Γ
changes the value of the current, but all system sizes show the same relative current. Thus,
the transport is independent of the reservoir coupling Γin the weak driving regime.
When increasing the potential difference to maximal driving
µ
= 1, it means that the left
reservoir only contains spin-up and the coupling to the reservoir becomes 2Γ
σ+
. There is no
spin-down magnetization in the left reservoir, or in other words, outscattering is forbidden
at the left side. The consequence is that each inscattered excitation has to be transported to
the right side of the chain. The decreasing current for higher scattering rates has a different
origin than in the weak driving regime. Due to the spin-up reservoir, the boundary spins of
the system become polarized and reduce further interaction with the reservoir as they are
already polarized with respect to the reservoir. For nearest-neighbor coupling, the spin-up
polarization accumulates up to the central site which is the spin-blockade effect. Thus, the
bottleneck is the central site, where the two different spin-polarizations interact with each
other. This bottleneck and the accumulation of polarizations is reduced if the coupling
strength is chosen smaller, as it prevents the system from building up a spin-blockade. This
is why for larger chain length, the maximum moves towards smaller Γin Figure 6.4 (c) as
the spin-blockade becomes larger for larger chain lengths in the non-equilibrium steady
state. At a specific coupling, e.g. Γ =
J
, this has a crucial consequence: The transport
changes from superdiffusive to subdiffusive as for larger chain lengths the spin-blockade
effect is intensified.
In case of long-range coupling, one would expect that the situation changes for the far-
from-equilibrium driving. Due to the missing spin-down magnetization in the left-reservoir,
the inscattering is increased and one would expect a higher current for increasing the
scattering rate Γ. In contrast, the dependence on Γdoes not change for maximal driving as
it can be seen in Figure 6.4 (d). The curve does not change compared to the weak driving
regime, but for the value of the relative current and the maximum remains at Γ =
J
. This
is a surprising effect as one would expect a higher current for higher coupling Γas NDC
was not observed. However, even if there is no NDC, the boundary spin polarizes with
Chapter 6. Boundary-driven Heisenberg spin-chain 64
respect to the reservoir magnetization and reduces further inscattering when increasing Γ.
Still, it is surprising that the transport behavior does not change for any investigated Γor
µ
and remains always nearly ballistic. The reason for this effect is investigated in detail in
the following section.
6.3. Absence of negative differential conductivity
0.49
0.5
0.51
0 0.2 0.4 0.6 0.8 1
0.49
0.5
0.51
0 0.2 0.4 0.6 0.8 1
hσz
ii
(i−1)/(N−1)
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
hσz
ii
(i−1)/(N−1)
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(a) Weak driving µ= 0.02
0
0.5
1
0 0.2 0.4 0.6 0.8 1
0
0.5
1
0 0.2 0.4 0.6 0.8 1
hσz
ii
(i−1)/(N−1)
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
hσz
ii
(i−1)/(N−1)
N= 5
N= 6
N= 7
N= 8
N= 9
N= 10
N= 11
(b) Maximal driving µ= 1
Figure 6.5.:
Spin-polarization (occupation probability)
hσz
ii
of the respective site
i
normal-
ized by the chain length to compare different system sizes (zero is the left spin
and one the right spin). For weak driving (a) both profiles (nearest-neighbor
coupling red, long-range coupling blue) look qualitatively similar. All spins are
close to
hσz
ii ≈
0
.
5but there is a small linear decrease from the left side to the
right side due to the driving. The first and the last spin show a polarization
with respect to the reservoir. The gradient is different for nearest-neighbor and
long-range coupling defining superdiffusive and nearly ballistic transport. For
maximal driving (b), both couplings show a qualitative difference. Long-range
coupling shows qualitatively the same, only the boundary spins are aligned
more strongly. Nearest-neighbor coupling has changed significantly. Further
sites are also aligned with respect to the reservoir polarization and a wide
spin-blockade is present, resulting in subdiffusive transport.
In this section it is investigated in detail why the long-range coupling scenario does not
show NDC and the transport is independent of the external reservoir parameters. The
magnetization profile for nearest-neighbor-coupling (red) and long-range coupling (blue) is
compared in Figure 6.5. It is differentiated between the weak driving
µ
= 0
.
02 regime (a)
and the far-from-equilibrium situation with maximal driving µ= 1 (b).
For weak driving, all spins do not have a specific alignment
hσz
ii ≈
0
.
5. Only the boundary
spins are aligned weakly with respect to the two reservoir polarizations (spin-up left and
spin-down right).
Due to the potential difference of the external reservoirs, a small gradient from left to
right is induced. Long-range coupling and nearest-neighbor coupling already differ in the
slope of the gradient. Long-range coupling is close to horizontal alignment which signifies
nearly ballistic transport. Nearest-neighbor coupling has a steeper slope which leads to
superdiffusive transport which was demonstrated in section 6.2. With the magnetization
Chapter 6. Boundary-driven Heisenberg spin-chain 65
profile at hand, the transport behavior can be explained properly: The phenomenological
transport law in Equation 6.2.2 shows that a linear gradient of the magnetization leads
to diffusive transport. Both, long-range coupling and nearest-neighbor coupling, do not
show the linear gradient which is why the transport is anomalous. Note that at
µdiff
,
the magnetization profile shows exactly this linear decrease of magnetizations for nearest-
neighbor coupling.
For maximal driving in Figure 6.5 (b), the magnetization profile for nearest-neighbor
coupling changes qualitatively. All spins up to the central site are polarized with respect to
the two reservoirs at the boundary. This is the spin-blockade effect found by Ref. [
145
,
144
].
Thus, the gradient is site-dependent and beyond the linear decrease of Equation 6.2.2.
The transport changes from superdiffusive transport for weak driving to subdiffusive
transport for strong driving. The spins align due to the magnetization of the reservoir. This
polarization accumulates up to the central site and serves as a bottleneck for the transport.
In Figure 6.5 (b) it is clearly visible that only the boundary spins are aligned and further
spins are not affected by the reservoirs in case of long-range coupling. Thus, the transport
remains nearly ballistic and NDC is absent. The reason is that long-range coupling enables
interactions beyond the central site. At both sides of the chain, there are ferromagnetic
domains with opposite alignment. Thus, due to the long-range coupling, both domains
interact with each other and the strong alignment is reduced. This is illustrated in Figure 6.6.
Due to the interaction between opposite polarized ferromagnetic domains, the strong
Figure 6.6.:
Sketch to illustrate the spin-blockade effect for nearest-neighbor (top) as well
as the missing NDC for long-range coupling (bottom) in case of maximal
driving. Spin-polarizations accumulate up to the central site resulting in NDC
for nearest-neighbor coupling. Long-range coupling results in an interaction
between both magnetic domains and thus reducing the respective polarizations.
alignment is reduced and enables further interaction with the reservoirs. This is why the
transport remains nearly ballistic for strongest driving. However, the boundary spins are
still affected and counteract the interaction with the reservoir. This is the reason why
increasing Γalso results in a current decrease, but not in a change of the transport (cp.
Figure 6.4 (d)).
Chapter 6. Boundary-driven Heisenberg spin-chain 66
To conclude, nearest-neighbor coupling is subject to the spin-blockade effect, where spin-
polarizations accumulate up to the central site and serve as a bottleneck for the transport,
resulting in NDC. In contrast, long-range coupling enables interactions between the opposite
polarized ferromagnetic domains and reduces the strong alignment. As a consequence, the
long-range coupling scenario is robust against far-from-equilibrium effects and does not
show NDC. The transport is independent of the driving and the reservoir coupling strength.
6.4. Effect of disorder
In this section, the effects found in the previous sections, namely the subdiffusive transport
for nearest-neighbor coupling and the absence of NDC for long-range coupling, are combined
with random disorder. This is especially relevant for ongoing research as random disorder
leads to a transition to MBL in case of a closed system [
14
]. However, the transport for
weak disorder before the MBL transition is still under discussion [
149
,
150
,
151
,
152
,
153
].
In case of an open quantum system it was shown by Ref. [
148
] that the boundary-driven
nearest-neighbor coupled XXZ-chain contains a transition from diffusive to subdiffusive
transport with increasing disorder. This was found for the linear response regime with
weak driving. This section serves as a demonstration that this transition to subdiffusive
transport not only might results from Griffiths effects but also from a spin-blockade effect
in case of far-from-equilibrium driving.
Griffiths effects occur due to the randomness of disorder. This results in rare regions, where
locally, between two sites, the disorder is stronger than in the rest of the system. As strong
disorder suppresses spin-flips and thus the current, these regions with locally strong disorder
serve as a bottleneck for the transport. Due to the averaging of random disorder realizations,
for increasing disorder strengths these regions occur more often. It was assumed that these
are the reason for the transition to subdiffusive transport already for disorder strengths
far away from the MBL transition [
149
,
154
]. In this section, both subdiffusive transport
resulting from Griffiths effects due to disorder as well as subdiffusive transport resulting
from a spin-blockade due to far-from-equilibrium driving are investigated.
In addition to the previous sections, disorder is applied which is the last term in Equa-
tion 6.1.1. For each data point in the following plots, the disorder is chosen randomly at
each site with
hi∈
[
−h, h
]. Thus, each realization has a specific Hamiltonian. For each
Hamiltonian the current in the non-equilibrium steady state is calculated by evaluating the
density-matrix dynamics. This procedure is repeated with new random disorder realizations
and the averaged non-equilibrium steady state current is calculated, including all performed
realizations. The simulation is completed when the averaged current does not change as
well as each on-site disorder shows
¯
hi≈
0. In practice, this is the case for around 6000
disorder averages.
It is important to mention that the density-matrix simulations are not advantageous when
it comes to disorder. The fact that the number of differential equations scales as 2
2N
is already limiting the system size to a maximum of
N
= 12 for the previous sections.
However, increasing disorder needs a significant longer integration time as the current
is suppressed and thus, the time to reach the non-equilibrium steady state is increased.
As a further complication, the different on-site potentials due to
hiσz
i
result in different
dynamics between all sites. As a consequence a careful adjustment of the numerical step
size is important to cover all random disorder realizations for
hi∈
[
−h, h
]. The step size
to yield exact dynamics can be orders of magnitudes smaller than for the case without
disorder of the previous sections. Note that as a numerical control parameter, the trace of
Chapter 6. Boundary-driven Heisenberg spin-chain 67
4.0×10−4
6.0×10−4
8.0×10−4
1.0×10−3
0.5 0.6 0.7 0.8 0.9 1
Nhji
h/J
N=5
N=6
N=7
Figure 6.7.:
Absolute current versus random disorder. At disorder strength
hdiff ≈
0
.
6the
curves for different system sizes intersect, signifying diffusive transport. For
h < hdiff
the system shows anomalous transport with superdiffusive transport,
whereby for
h > hdiff
it changes from superdiffusive to subdiffusve transport.
This plot serves as a comparison to Ref. [
148
], where the disorder is weaker at
the boundaries h0, hN−1∈[−h/2, h/2].
the density matrix was carefully observed to yield one.
A more advantageous form to simulate disorder realizations is the matrix product state
method which will be introduced in Part III of this thesis in chapter 7. The advantage
is that due to random disorder, the entanglement between subsystems of the many-body
system is suppressed (the consequence is that the transport changes to subdiffusive trans-
port). This is advantageous for matrix product state simulations as small entanglement
allows to cut off a large part of the Hilbert space. However, for the system at hand with
factorized system-reservoir dynamics, a matrix product simulation in the Liouville space
has to be performed to trace out the reservoir degrees of freedoms as done in Ref. [
148
].
This is beyond the scope of this thesis as the simulations within the matrix product state
formulation are performed based on the QSSE, introduced in chapter 3.
Altogether, this is the reason why in this approach, the system size is limited to a maximum
of
N≤
8when considering random disorder. Note that for
N
= 8 only 1000 disorder
realizations are performed due to the long integration time. The considered sytem sizes are
much smaller than the thermodynamical limit and finite size effects are present. For this
reason, the results obtained with the density matrix are compared to the results obtained
for matrix product state simulations in the Liouville space of Ref. [
148
]. The focus of this
section lies on a comparison of nearest-neighbor with long-range coupling, whereas the
exact crossing to subdiffusive transport might differ from the thermodynamical limit.
In Figure 6.7, the absolute current
Nhji
for increasing random disorder is shown for
different system sizes. In analogy to section 6.2 in Figure 6.2, the transition to subdiffusive
transport is obtained where the curves for different system sizes intersect at
hdiff
. At
this point, the absolute current does not change for increasing system size which signifies
diffusive transport (cp. Equation 6.2.1). For
h<hdiff
, the power-law exponent is smaller
than one, whereas right of the intersection the power-law exponent shows
γ >
1signifying
subdiffusive transport. Note that in order to compare the transition with Ref. [
148
], the
disorder at the boundary spins is chosen as h0, hN−1∈[−h/2, h/2] for this plot.
Comparing the obtained point of the transition
hdiff ≈
0
.
6with
hdiff
= 0
.
55 of Ref. [
148
],
the findings of this chapter are relative close to the reference, where they simulated up to
Chapter 6. Boundary-driven Heisenberg spin-chain 68
400 spins. Furthermore, in Ref. [
148
], they showed that finite size effects are present up to a
certain system size
N∗
. The here investigated system sizes are much smaller than
N∗
which
is why it is surprising that such small system sizes show nearly the same
hdiff
. The reason
might be the different method to determine
hdiff
. In the Ref. [
148
] it was obtained with a
fit of the relative current in analogy to Figure 6.3, whereas the here proposed method is to
determine
hdiff
with an intersection of curves for different system sizes for the absolute
current.
This gives rise to the assumption that the here used method to obtain the transition
hdiff
via an intersection of curves is a good method to predict
hdiff
for systems which are subject
to finite size effects.
It is interesting to mention that in case of a closed system there exists a second transition
at
hMBL ≈
3
.
7[
71
,
158
] where it is assumed that the scaling of the current with the
system size changes from
j∼
1
/Nγ
(
γ >
1) to
j∼exp
(
−κN
). This is referred to as the
MBL transition. The scope of this chapter is not to address the question if
hMBL
exists in
boundary-driven systems, but to clarify the influence of the external reservoirs, especially
for the far-from-equilibrium situation.
In section 6.2, it was shown that nearest-neighbor coupling is subject to NDC resulting from
a spin-blockade effect. This leads to subdiffusive transport already for the case without
disorder. However, the transport for long-range coupling is independent of the driving
strength and the coupling to the external reservoirs (cp. section 6.3). This different behavior
is now investigated for the effect of disorder.
In Figure 6.8, the nearest-neighbor scenario (a, c) is compared to the long-range coupled
chain (b, d) for weak driving
µ
= 0
.
02 (a, b) and maximal driving
µ
= 1 (c, d). As men-
tioned before, the nearest-neighbor scenario shows a transition to subdiffusive transport
at
hdiff ≈
0
.
6(Figure 6.8 (a)) in the linear response regime, where no NDC is present
due to Griffiths effects. In contrast, long-range coupling shows a transition to subdiffusive
transport at stronger disorder hdiff ≈0.8(Figure 6.8 (b)). The reason might be that the
transport for the case without disorder is already different. Nearest-neighbor coupling
shows superdiffusive transport with
γ
= 0
.
5, whereas long-range coupling shows nearly
ballistic transport with
γ
= 0
.
01 (Figure 6.3). Thus, in case of long-range coupling, a higher
disorder is needed to suppress the transport such that it becomes diffusive.
When the external driving is increased to maximal driving
µ
= 1, the situation changes
drastically for the nearest-neighbor coupling. The transition to subdiffusive transport
already takes place due to the spin-blockade effect, which is why the transport is al-
ready subdiffusive for zero disorder and no
hdiff
exists (Figure 6.8 (c)). In contrast, for
long-range coupling, the transition to subdiffusive transport remains at nearly the same
disorder strength (Figure 6.8 (d)). The reason is that long-range coupling is robust against
far-from-equilibrium effects such as NDC.
These findings prove that for nearest-neighbor coupling, the transition to subdiffusive
transport is highly dependent on the external reservoir parameters. Far-from-equilibrium
driving results in a spin-blockade with NDC, which is in addition to Griffiths effects in case
of disorder. It might become difficult to unravel the effect of disorder from the spin-blockade
effect in the presence of boundary reservoirs.
Chapter 6. Boundary-driven Heisenberg spin-chain 69
0.01
0.02
0.03
0.5 0.6 0.7 0.8 0.9 1
Nhji
h/J
N=5
N=6
N=7
N=8
(a) α= 1000,µ= 0.02
0.01
0.02
0.03
0.5 0.6 0.7 0.8 0.9 1
Nhji
h/J
N=5
N=6
N=7
N=8
(b) α= 0.5,µ= 0.02
0.1
0.5
1
0 0.2 0.4 0.6 0.8 1
Nhji
h/J
N=5
N=6
N=7
(c) α= 1000,µ= 1
0.1
0.5
1
0.5 0.6 0.7 0.8 0.9 1
Nhji
h/J
N=5
N=6
N=7
N=8
(d) α= 0.5,µ= 1
Figure 6.8.:
Comparison between nearest-neighbor (a, c) and long-range coupling (b, d) for
weak (a, b) and strong driving (c, d) to determine the transition to subdiffusve
transport
hdiff
. Both, nearest-neighbor (a) and long-range coupling (b) show
a transition to subdiffusive transport at a certain disorder strength for weak
driving. For long-range coupling, the transition
hdiff
is at a higher disorder
strength. However, for maximal driving
µ
= 1, the nearest-neighbor scenario
shows subdiffusive transport up until
h
= 0 (c). In contrast, for long-range
coupling,
hdiff
does not change significantly (d). Due to the numerical effort,
N= 8 was excluded in (c) as the effect becomes already clear.
However, the transport for long-range coupling is unaffected by the external reservoirs,
even in the far-from-equilibrium situation. As a consequence, the transition to subdiffusive
transport results purely from disorder effects and is independent of the open quantum
system setup. In the case of far-from-equilibrium reservoirs, a possible transition to MBL
would be unaffected by the external driving which makes the long-range coupled Heisenberg
spin-chain an ideal candidate to study such effects in case of open quantum systems.
Chapter 6. Boundary-driven Heisenberg spin-chain 70
6.5. Conclusion
As a second many-body system, the isotropic Heisenberg quantum spin-chain was inves-
tigated in this thesis. This has become a standard model when investigating effects of
disorder on interacting systems leading to many-body localization.
To generalize the standard model to an open quantum system, the spin-chain is coupled
to two external magnetic reservoirs at the boundaries. The coupling to the reservoir is
described by the Lindblad formalism, where factorized system-reservoir states are assumed.
Due to strongly-correlated spins with long-range interactions, rendering typical numerical
methods ineffective, the full density matrix equations of the many-body system were
evaluated numerically.
In assuming a potential difference of the external reservoirs, i.e. the left reservoir contains
a higher spin-up and the right reservoir a higher spin-down magnetization, a spin-current
through the chain is induced. This spin-current was investigated in the non-equilibrium
steady state for different system sizes to define the transport properties of the system.
For small potential differences, both, nearest-neighbor and long-range coupling, act within
a linear response regime where the current increases linearly with the driving induced by
the potential difference. Nearest-neighbor coupling shows superdiffusive transport, whereas
long-range coupling shows nearly ballistic transport. However, for far-from-equilibrium
driving, nearest-neighbor coupling exhibits negative differential conductivity resulting in a
change to subdiffusive transport. In contrast, long-range coupling still acts within a linear
response regime with nearly ballistic transport.
By investigating the driving strength as well as the coupling to the external reservoirs,
it has been shown that the transport of the long-range coupled Heisenberg spin-chain is
independent of the external reservoir parameters, whereas for nearest-neighbor coupling
the transport is highly dependent on both.
These differences are explained by a spin-blockade effect which is induced by the far-from-
equilibrium driving. Nearest-neighbor coupling builds up ferromagnetic domains which
serve as a bottleneck for the current. For long-range coupling, these ferromagnetic domains
interact with each other and vanish.
When disorder is added, both systems undergo a transition to subdiffusive transport for
increasing disorder strength in the regime of a small potential difference between the
reservoirs. However, the transition to subdiffusive transport is at a higher disorder strength
for long-range coupling.
In the far-from-equilibrium situation, nearest-neighbor coupling shows subdiffusive trans-
port already for zero disorder. Surprisingly, for long-range coupling, the transition to
subdiffusive transport takes place at the same disorder strength than for a small potential
difference.
The transport of the nearest-neighbor coupled Heisenberg chain is highly dependent on
the coupling to the external reservoirs in case of an open quantum system. This makes
it difficult to distinguish the effect of disorder from the reservoir induced spin-blockade.
Long-range coupling shows robust transport for changing the external reservoirs. Thus,
long-range coupling provides a clear understanding of disorder effects when going to a
regime where MBL might be present in case of open quantum systems.
PART I II
Entangled system-reservoir dynamics
71
7.
Introduction to matrix product states
Considering a typical one-dimensional spin-chain with
N
spins. Each site
ni
with
i∈
{1, . . . , N}
has a local state dimension of
d
. In practice, the local state dimension will be
d= 2 in most cases. In general, the wave-function of an arbitrary state reads
|ψi=
d
X
n1,...,nN
cn1,...,nN|n1, . . . , nNi,(7.0.1)
where
|ψi
can be either in a product or in a superposition state. The most simple product
state is where one coefficient is nonzero and all others are zero. For example, the Néel
state
|
1
,
0
,
1
,
0
, . . . i
would read
|ψi
=
c1,0,1,0,...|
1
,
0
,
1
,
0
, . . . i
, with
c1,0,1,0,...
= 1 if
|ψi
is
normalized. However, for a many-body system the total dimension scales exponentially
with the system size
N
. This means, the total number of coefficients
cn1,...,nN
scales as
dN
. This makes it difficult to treat arbitrary superposition states as they consist of many
coefficients of the many-body system.
Calculating dynamics via a solution of the time dependent Schrödinger equation is numeri-
cally expensive as this corresponds to
dN
coupled differential equations. With growing
N
,
the total number of differential equations quickly reaches the limit of todays computational
possibilities.
However, for many cases, not all states are occupied and most coefficients are zero as
it is easy to see for the Néel state. Still, the time evolution with a typical many-body
Hamiltonian induces correlations between subsystems and simple product states end up in
a complicated superposition state after few time steps. The question is, how is it possible
to judge which coefficients are relevant for the dynamics and which subsystems of the
given many-body problem are truly coupled to each other. The subsystems, i.e. sets of
coefficients which do not become relevant for the system dynamics, decouple and can be
neglected when solving the differential equations. Neglecting these coefficients, directly
reduces the exponentially many coefficients to an algebraic scaling with
N
, provided that
the entanglement between subsystems is small.
The answer to that lies in the matrix product state (MPS) formalism and the most im-
portant operation called singular-value decomposition (SVD). It is very closely related to
the Schmidt decomposition which allows to calculate bipartite entanglement between two
quantum subsystems. This chapter is written in close analogy to the review on matrix
product states by Ulrich Schollwöck [29].
73
Chapter 7. Introduction to matrix product states 74
7.1. Singular-value decomposition (SVD)
The singular-value decomposition can be viewed as a diagonalization of arbitrary rectangular
matrices. Considering a matrix
M
of dimension
NA×NB
, the SVD is a decomposition of
M=USV †.(7.1.1)
Thus, the matrix
M
is decomposed into three matrices where the centering matrix
S
is
diagonal and of dimension
min
(
NA, NB
)
×min
(
NA, NB
). The entries on the diagonal are
the singular values with non-negative entries Sαα =sα. In the following, the SVD will be
done such that the singular values are of descending order
s1≥s2≥ ··· ≥ sr
. The Schmidt
rank rof Sdenotes the number of non-zero singular values.
Before making the connection of singular values and entanglement of subsystems, it is
important to note some properties of the matrices Uand V.
The matrix
U
is of dimension
NA×min
(
NA, NB
)with orthonormal columns. This implies
that
UU†=1.(7.1.2)
This property is important when it comes to the gauge of the MPS and is called left-
normalized in the following. The matrix
V†
is right-normalized as it is of dimension
min(NA, NB)×NBand has orthonormal rows with
V†V=1.(7.1.3)
From the context of quantum systems, the SVD is also called a Schmidt decomposition,
where the Schmidt values display the entanglement between two subsystems. Getting back
to the wavefunction in Equation 7.0.1, it can be divided into two subsystems Aand B
|ψi=X
ij
Ci,j|iiA|jiB,(7.1.4)
where
i
and
j
contain several site indices. This grouping of indices into a single one (
i
and
j
) is called a compound index. For simplicity, the chain is divided at the half, so
A
contains
the left half of the chain i∈nn1, . . . , nN/2oand Bthe right half j∈nnN/2+1, . . . , nNo.
Decomposing the coefficient matrix Ci,j it reads
|ψi=
min(NA,NB)
X
α=1 X
i
Uiα|iiA!sα
X
j
V†
jα|jiB
.(7.1.5)
As the matrices Uand V†are orthonormal (left and right), a new basis is introduced
|ψi=
min(NA,NB)
X
α=1
sα|αiA|αiB.(7.1.6)
In Figure 7.1, it is exemplarily shown, how a decomposition of the coefficient matrix is
performed. The
sα
are connected to the bipartite entanglement: From this new basis, the
Chapter 7. Introduction to matrix product states 75
=
Figure 7.1.:
Singular-value decomposition of the coefficient matrix
Ci,j
. The singular values
are the diagonal entries of the diagonal central matrix with descending order.
The smallest singular values will be discarded. The left matrix
Ui,α
is left-
normalized and the right matrix V†
α,j is right-normalized.
reduced density operators of each half respectively is given by
ˆρA=
min(NA,NB)
X
α=1
s2
α|αiA Ahα|ˆρB=
min(NA,NB)
X
α=1
s2
α|αiB Bhα|,(7.1.7)
from which the von-Neumann entropy is calculated via
SAB(ψ) = −tr ˆρAlog2ˆρA=−
min(NA,NB)
X
α=1
s2
αlog2s2
α.(7.1.8)
Thus, the SVD allows to decompose a large coefficient matrix into a set of three matrices
where one matrix displays the entanglement between the two subsystems
A
and
B
via the
singular values
sα
. The advantage in doing so is that when the entanglement
SAB
is small,
many singular values are zero as the Schmidt rank
r≤min
(
NA, NB
). Thus, it reduces
the sum in Equations 7.1.5-7.1.7 to
r
summands. Furthermore, in numerical practice, one
defines an approximate matrix
˜
S
with Schmidt rank
˜r
. This approximation is performed
such that the norm of the approximate coefficient matrix
˜
Ci,j
will only deviate marginally
from Ci,j
||Ci,j|| ≃ ||˜
Ci,j|| .(7.1.9)
Now the great advantage of performing a SVD becomes clear: One can truncate the matrix
S
with rank
min
(
NA, NB
)to an approximate matrix
˜
S
with rank
˜r
. This allows to reduce
the number of columns of
U
and the number of rows of
V†
. Thus, the SVD grants a tool to
measure the entanglement between subsystems and reduce the matrix dimension according
to that. If for example |ψiis in a product state, the rank of Swill be r= 1, thus for this
trivial example, the dimension of
Ci,j
is reduced drastically by performing a SVD. The
goal of the MPS formalism is to bring any superposition state as close as possible to a
product state.
7.2. Diagrammatic tensor representation
When it comes to more complex matrices and one performs more than a single SVD,
the display of the corresponding tensors in a single equation becomes confusing as with
each SVD, the number of indices is increased by one. To simplify the display of more
complex MPS, there is the common diagrammatic form. Considering the coefficient matrix
Ci,j
in Equation 7.1.4, it consists of two indices,
i
and
j
. These indices are referred to as
physical indices, as they have a physical relevant label, in most cases the site index
nk
.
In this particular example
i
is a compound index of all site indices of the left half and
j
Chapter 7. Introduction to matrix product states 76
Figure 7.2.:
Diagrammatic representation of the coefficient matrix. It has two physical
indices iand jwhich are displayed as outgoing vertical lines.
of all site indices of the right half. In Figure 7.2, the diagrammatic representation of the
coefficient matrix is shown. The two vertical lines represent the two physical indices which
is a compound index in this example. Note that vertical lines throughout this thesis are
always denoted as physical indices with local Hilbert-space dimension
d
(
d
= 2 for spins).
The number of indices of a tensor defines its rank. The coefficient matrix in Equation 7.1.4
and Figure 7.2 has two indices and thus two outgoing lines. This makes it a tensor of rank
two which is a matrix. A tensor with one single index would be a vector. The diagrammatic
representation becomes really advantageous when dealing with high rank tensors
r≥
3.
It will be shown that a common MPS mostly consists of rank three tensors. However,
tensors of higher ranks are considered in this thesis as well, especially when it comes to
the connection of feedback dynamics with a many-body system (cp. chapter 9).
Another advantage of the diagrammatic representation is the display of a SVD or other
operations performed on tensors. The SVD of Equation 7.1.5 is shown in Figure 7.3 in
diagrammatic form. The coefficient matrix with two outgoing physical indices is decomposed
into three matrices: A left-normalized Matrix
U
with physical index
i
which is represented
with round edges at the left side, a diagonal matrix
S
and a right-normalized matrix (round
edges at the right side)
V†
with physical index
j
. In contrast to the two vertical physical
indices, there are now two horizontal indices
α
which connect the matrices
U
,
S
and
V†
.
These indices are referred to as bond or link indices as they link the decomposed matrices.
If one would contract the tensors over αby performing the sum
r
X
α=1
UiαSααV†
αj =Ci,j ,(7.2.1)
one obtains the initial coefficient matrix. Thus, the links can be seen as an Einstein
summation where in a product of tensors, a sum over the shared indices is performed. The
SVD results in additional link indices which is a rewriting of the initial matrix into a set of
tensors which are connected with this link index. In this thesis horizontal lines are always
referred to as link indices.
It is important to mention that the MPS representation is not unique. The dimension
r
of
the link index
α
is not fixed and especially by performing a time evolution, its dimension
grows with each time step. Thus, depending on the problem, different representations and
series of SVDs could be advantageous. This gauge of the MPS will be discussed in the
following.
Chapter 7. Introduction to matrix product states 77
Figure 7.3.:
Diagrammatic representation of the SVD,
Ci,j
with two physical indices is
decomposed into three matrices where the left-normalized matrix
U
(round
edges at the left side, green) gets the vertical physical index
i
and the right-
normalized matrix
V†
(round edges at the right side, blue) gets the vertical
physical index
j
. Horizontal lines represent link indices between the diagonal
singular-value matrix
S
and
U
and
V†
. Connected lines are shared indices of
two tensors. The two tensors will be contracted over these indices.
7.3. Canonical form of a matrix product state
In the previous section a decomposition of two subsystems was considered. The matrix
dimension was reduced by neglecting all singular values
sα
with
α > ˜r
. Thus, due to the
descending order, small singular values will then be neglected. The real advantage of the
MPS representation lies in performing series of SVDs and at each decomposition cut off
the negligible part of the tensors. This is done by controlling the error with the norm as
shown in Equation 7.1.9. In practice, this will be done for each physical site index.
Instead of writing down the equations, the advantageous diagrammatic representation will
be used to display operations with tensors in the following. Note that the diagrammatic
representation contains all the relevant information, but for the explicit entries of the
tensors. However, the explicit entries are usually not of interest. A different gauge of the
MPS results in different entries. This does not cause a problem, because a different gauge
leaves the important expectation values invariant. Although, the gauge does become relevant
in calculating the expectation values as this makes a huge difference in the computation
time. The different relevant gauges of the MPS will be discussed within this section.
In Figure 7.4, an arbitrary coefficient tensor with
N
physical site indices
ni
is shown.
Note that physical indices are represented as vertical outgoing lines. In a first step, two
subsystems are defined, one contains the first physical index
n1
and the second subsystem
contains all other indices
n2, . . . , nN
. Now both subsystems are decomposed as shown in
Equation 7.1.5, resulting in a tensor containing only the physical index
n1
and the link to
the singular-value matrix. A second tensor contains all other physical indices as well as the
link index resulting from the SVD.
In the next step, the singular-value matrix and the right-normalized tensor containing
indices
n2, . . . , nN
are contracted over the link
l2
. Note that this step is not necessary
but just a specific gauge of the MPS used in this thesis. The advantage is that the first
tensor containing the physical index
n1
is left-normalized. The right tensor containing all
other indices is again orthonormal and the next step can be performed. This is shown in
Figure 7.5.
Chapter 7. Introduction to matrix product states 78
Figure 7.4.:
Diagrammatic representation of the coefficient tensor of
|ψi
. It has
N
physical
site indices
ni
. In a first step, the decomposition takes place between the first
site index n1and the rest of the system n2, . . . , nN.
Figure 7.5.:
Diagrammatic representation of the construction of a matrix product state.
The MPS is constructed from the left, resulting in a left-canonical MPS. After
each SVD, the singular-value matrix is contracted with the rest of the right
system to create an orthogonal tensor (black box). Then the next SVD of the
following physical index takes place.
Chapter 7. Introduction to matrix product states 79
Figure 7.6.:
Right-canonical MPS. The orthogonality center (OC) is at the left side (black
box, round edges) and all other tensors are right-normalized (blue boxes, right
round edges).
In the next step the first site tensor containing
n1
is put aside and two subsystems are
defined, one containing
n2
as well as the link index
l1
and the other subsystem contains
the indices
n3, . . . , nN
. The SVD again decomposes the tensor and after contracting the
right side with the singular-value matrix, there are two linked left-normalized tensors, one
containing the index
n1
and the other the index
n2
. Both share the link index
l1
resulting
from the first SVD. Furthermore, the tensor containing the physical index
n2
also share a
link index l2with the rest of the system resulting from the second SVD.
This procedure is repeated until the orthonormal matrix called orthogonality center (OC,
black box) is at the right end of the system and contains only nNas physical index.
At a first glance, the MPS formalism seems to blow up the dimension of the given problem.
On a closer look, it becomes clear that indeed the formalism allows to truncate the Hilbert
space: The link indices reflect the entanglement between the subsystems which are now the
different sites of the chain. Within each step, the entanglement between the subsystems is
reflected in the singular values. If there is no entanglement between states of the subsystems,
the singular value is zero and the matrices can be truncated accordingly. In particular,
after the truncation in the SVDs, all
li
have dimension one if
|ψi
was initially in a product
state. It becomes clear that series of SVDs reduce the dimension of
|ψi
when written in a
MPS as with each SVD, the dimension of the Hilbert space is reduced, depending on the
entanglement of the system.
The MPS form in Figure 7.5 is called canonical form and in particular left-canonical form
as all tensors are left-normalized but for the last site
nN
which is the OC. This specific
gauge of the MPS has advantages when it comes to calculating observables. This will be
shown in detail later on. An obvious example is the calculation of the norm
hψ|ψi
. The
contraction of all left-normalized matrices with its complex conjugated is obsolete as they
are
1
by construction. Thus, the only operation which has to be performed is a contraction
of the OC (more details in the next section).
This not only holds for the left-canonical form. One can do the construction of the MPS,
starting from the right side with
nN
. This will end up in a right-canonical form as shown
in Figure 7.6. Here, the OC is at the left side of the MPS at
n1
and all other tensors are
right-normalized.
The last and most important form of used MPS in this thesis is the mixed-canonical form,
shown in Figure 7.7. There, the OC is at an arbitrary position. All tensors left of the OC
are left-normalized and all right of it are right-normalized. It is straightforward to arrive
at the mixed canonical form from e.g. right-canonical form: The tensors
n1
and
n2
are
contracted and decomposed, but now the singular-value matrix is contracted with the
tensor of
n2
. Thus, the OC is now at the tensor of
n2
and the
n1
-tensor is left-normalized.
By repeating this procedure, the OC is brought to the desired position. It becomes clear
that the position of the OC is crucial for MPS algorithms.
Chapter 7. Introduction to matrix product states 80
Figure 7.7.:
Mixed-canonical MPS. The OC (black box, round edges) is at an arbitrary
position. All matrices left from the OC are left-normalized (green box, left
round edges) and all right from the OC are right-normalized (blue boxes, right
round edges).
Figure 7.8.:
General form of an MPO which acts on all sites. Each site has one incoming
and one outgoing physical index acting as a projector for the MPS.
In practice, the construction of the MPS does not start as shown in Figure 7.4. This has
the disadvantage that the initial coefficient matrix is already an exponentially large tensor,
which for a common many-body problem is already at computational limits. The idea
is to start in a specific product initial state as for example the Néel-state. This is easy
to write down in tensor notation as each tensor only has one occupied physical index
without being entangled with the other sites. The gauge is then achieved by contracting
the respective tensors with each other and then decomposing them before getting on with
the next index. This results in a dummy link index of dimension one between the two
tensors. This procedure is the same as for shifting the OC.
7.4. Matrix product operators
In the matrix product formalism an operator
ˆ
O
can be seen as a projector which projects
one physical index nto another one n0
ˆ
O=X
n0n
Wn0
1n1. . . Wn0
NnN|n0ihn|(7.4.1)
with coefficients
Wnin0
i
. The difference to the coefficients of a matrix product state is that a
matrix product operator (MPO) has two physical indices per site, one in- and one outgoing.
In Figure 7.8, an MPO acting on all sites is demonstrated in diagrammatic form. In practice,
any operator acting on several sites can be brought in the form of Equation 7.4.1 by a
series of SVDs, but with the two physical site indices per tensor. Note that this can become
complicated for long-range coupling. However, for nearest-neighbor coupling, the MPO
formalism has the advantage that it is not necessary to contract the whole MPS when
the operator is applied. Instead, one can apply the MPO site wise, starting e.g. from the
left side. This will be explained in detail when explaining the MPS algorithms for the
respective investigated systems.
Chapter 7. Introduction to matrix product states 81
Figure 7.9.: Diagrammatic form of a (a) single-site and (b) two-site MPO.
Figure 7.10.:
Calculation of the norm
hψ|ψi
. One profits from a MPS in mixed canonical
form as only the respective tensors at the position of the OC have to be
contracted.
In this thesis, an operator acting on all sites will be the time-evolution operator
ˆ
U
includ-
ing the respective Hamiltonian. Other important operators are single-site operators (e.g.
magnetization) or two-site operators for the computation of e.g. current or correlation
functions as it is exemplarily shown in Figure 7.9.
Note that each tensor of the MPO contains two physical indices
ni
and
n0
i
. The indices
act on the respective MPS,
ni
acts on
|ψi
, whereas
n0
i
acts on the complex conjugate
hψ|
which is denoted for clarity by the prime sign. This is just a new label for the respective
index to avoid confusion and prevent contractions over the wrong physical indices.
7.5. Expectation values
The most simple and crucial expectation value is the norm
||ψ|| =hψ|ψi.(7.5.1)
Assuming
|ψi
to be already a MPS in mixed-canonical form as in Figure 7.7 with vertical
physical indices at the bottom, the complex conjugated
hψ|
is the same MPS with complex
conjugated values. In diagrammatic form it is displayed with vertical physical indices at
the top. To compute the norm, the two MPS are multiplied with each other. Thus, the
MPS is contracted via a summation over shared indices which are the vertical physical
indices as shown in Figure 7.10 (left). For simplicity, the tensor at the respective site
ni
is
named
Ani
and its complex conjugated is named
Ani†
. Note that the tensors have also link
indices in addition to the physical index appearing in the name.
When calculating expectation values, one benefits from a gauged MPS. Assuming the OC is
at the tensor
An2
of the MPS. As the MPS is in mixed canonical form, the left tensor
An1
Chapter 7. Introduction to matrix product states 82
Figure 7.11.:
Calculation of the single-site magnetization for a MPS in mixed-canonical
form.
with physical index
n1
is left-normalized. From the property of left-normalized matrices in
Equation 7.1.2, it becomes clear that the multiplication with its complex conjugated yields
the identity. The same holds for all right-normalized tensors to the right of the OC as it
can be seen in Equation 7.1.3. Thus, the only part of the expectation value in Figure 7.10
which has to be contracted is the OC with the two link indices at the left and right side, as
the two tensors
An2
and
An2†
share the same link indices. This is the main reason why the
mixed canonical form is a great advantage: One does not have to contract the whole MPS
when calculating an expectation value, but just the part with the OC as it is known per
construction that all left- and right-normalized matrices yield the identity when contracted
in the right order. Note that a contraction over all indices yields a scalar which will be the
norm in Figure 7.10.
In numerical practice, this operation is performed
N
-times for the norm. Starting with
the OC at the first position
An1
, multiply the MPS with its complex conjugate, contract
the two tensors
An1
and
An1†
and thus compute the norm. Note that for this computation
the physical index of the complex conjugate
n0
1
is unprimed to clarify the contraction over
the same index. In the next step bring the OC to position two (
An2
), square the MPS,
contract the tensors and add the result to the norm. Repeat this procedure up to the last
site results in the norm
||ψ|| =1
N
N
X
i=1
An†
iAni.(7.5.2)
In practice, Equation 7.5.2 will deviate from one even when
|ψi
is normalized, as each
truncation after a SVD involves an error. If the neglected singular values are too high, this
will be reflected in the norm
||ψ||
. Thus, the norm is the most important expectation value
when performing MPS algorithms, as it is the control variable to check the numerical error.
The expectation value of a single-site operator is straightforward. As an example, the
computation of the single-site magnetization is shown in Figure 7.11. The magnetization is
calculated via the Pauli Spin-matrix σz
i
Mi=hni|σz
i|nii(7.5.3)
The Pauli-matrix has two physical indices, one for the rows and one for the lines, both
with the same physical index
ni
. The spin-matrix was defined in Equation 2.2.14. As for
Chapter 7. Introduction to matrix product states 83
Figure 7.12.:
Current
ˆ
ji,i+1
as exemplary two-site operator. The tensors on which the
operator acts (i.e.
Ani+1
even if its right-normalized) have to be included in
the operation.
Figure 7.13.:
Performing a swap operation to switch the two physical indices
nk−1
and
nk
while maintaining the link indices
lk−i
and
lk+1
. Note that the dimension
of the link indices might change by performing this operation. After this
operation nkmay be swapped with the next left tensor Ank−2.
the system norm, the multiplication of a MPS in mixed-canonical form with its complex
conjugate yields the identity for left- and right-normalized tensors. However, if an operator
is involved this is not the case. Thus, if an operator is applied to the MPS, the OC has to
be involved in the operation. The most effective way is to bring the OC to the position
where the single-site operator is acting on by performing a series of SVDs. Then, all other
products of normalized tensors yield the identity and only the local tensors are involved in
the operation (Figure 7.11).
Another important operation is a two-site operator, as for example the local current
ˆ
ji,i+1
in Equation 6.1.7. As for the single-site operators, it is crucial for an efficient computation
that the OC is at one of the tensors the operator acts on. Otherwise, all tensor between
the OC and the operator need to be contracted as well. Thus, a more efficient way is
to bring the OC to the tensors involved in the operation. It is irrelevant if the OC is
at
Ani
or at
Ani+1
. Note that the two-site operator
ˆ
ji,i+1
has four physical indices two
for site
ni
and two for site
ni+1
as shown in Figure 7.12. As the local current operator
acts on nearest-neighbors, only the respective neighboring tensors have to be included in
the operation. However, if a two-site operator acts at distant tensors
ˆ
Oi,k
(e.g. for the
computation of a correlation function), all tensors between
Ani
and
Ank
have to be included
and contracted for the operation.
Another possibility for a two-site operator acting on distant tensors, is to swap the tensor
Ank
with its left neighbors until it is next to
Ani
. In doing so, the OC is kept at
Ank
. This
is shown in Figure 7.13. The tensors
Ank−1
and
Ank
are contracted over the link index
lk
resulting in a tensor of rank four with the two physical indices and the two link indices
to the left and the right. Then a SVD is performed in maintaining the link indices at the
Chapter 7. Introduction to matrix product states 84
left and right position but switch the physical indices such that the index
nk
now is left
of the physical index
nk−1
. Note that the link index
lk
might change its dimension after
the operation. This is repeated until
Ank
is the tensor next to
Ani
. Then, the operator
ˆ
Oi,k
can be evaluated similar to
ˆ
ji,i+1
in Figure 7.12. This might be more effective for a
two-site operator acting on distant sites (and depending on the size of the MPS the only
possible) because otherwise all tensors in between have to be contracted as well for this
operation. This swap operation will especially become relevant when performing a feedback
algorithm.
8.
Feedback controlled two-photon
purification
In this part of the thesis, entangled system-reservoir interaction is investigated. In this
chapter, the system of interest consists of only a single TLS. However, in contrast to
Part II, tracing out the reservoir degrees of freedom is not possible as system and reservoir
are entangled due to a quantum feedback mechanism. The idea is to treat the reservoir
as a many-body system with the matrix product state (MPS) formalism, introduced in
chapter 7. It is assumed that the reservoir is structured and includes a distant mirror.
Photons emitted from the TLS will interact again with the system after a time-delay, which
is why any entanglement becomes relevant again in the future. The key-mechanism is, by
varying the mirror distance, it is possible to control the photon statistics of the reservoir
which is treated in a numerically exact manner within the MPS formalism.
Single photons are used as qubits for quantum computation protocols [
159
] or quantum
key distribution [
160
]. An almost ideal single-photon source is achieved, by using a single
TLS [
59
,
161
], which is experimentally realized in, e.g., atoms [
162
], single quantum dots
[
163
,
164
,
165
,
166
,
167
,
168
,
169
], molecules [
170
] and defects in solids [
171
]. The TLS is
initialized in its ground state and inverted by excitation with a Gaussian pulse, resulting in
half a Rabi-oscillation (
π
-pulse). After decaying radiatively, the TLS emits a single photon
into the environment under spontaneous emission. However, by driving the TLS with a full
Rabi-oscillation (2
π
-pulse), it was shown recently that the two-photon probability becomes
higher than that of a single photon. This forms the idea that a single TLS acts as a source
for a multi-photon state. To achieve a reliable two-photon source [
172
,
173
,
174
,
175
], a
high degree of control is essential. This is especially relevant if the same source should act
as source for both, single and multi-photon generation [176, 177, 178, 179].
The idea of this chapter is to use a mirror, where quantum interferences manipulate the
photon statistics due to an all-optical feedback mechanism.
In classical chaotic systems, time-delayed Pyragas control is used to stabilize otherwise
unstable steady-states [
180
,
181
,
182
,
183
]. In the quantum-regime it was successfully
demonstrated that in measurement-based feedback setups non-classical light states are
stabilized [
184
,
185
], single trapped ions are cooled [
186
] or quantum correlations are
controlled [
187
]. A different approach is to use structured baths [
188
], where reservoir
engineering leads to the desired dynamics [
189
,
190
,
191
,
192
,
193
]. The advantage is that
the quantum system is not perturbed by a measurement process.
However, the here investigated approach is to adapt the classically successful concept of a
finite delay time
τ
of Pyragas control to quantum protocols [
194
,
195
]: The signal of the
system
s
(
t
)acts as a control field, where the time-delayed signal
s
(
t−τ
)allows all-optical
control while no measurement is needed.
Adapting the concept of time-delayed feedback in the quantum regime, it was shown that the
85
Chapter 8. Feedback controlled two-photon purification 86
entanglement between photons emitted from a biexciton [
196
] as well as between nodes in a
quantum network is controlled [
197
]. In parametric oscillators, quantum feedback enhances
squeezing [
198
,
199
]. Furthermore, unstable branches of bistabilities are stabilized using
feedback in optomechanical systems [
200
,
201
]. It was also experimentally demonstrated
that feedback can manipulate the emission statistics of photonic devices [202, 203].
To achieve all-optical feedback, the light is reflected by, e.g., an external mirror or an
integrated semi-infinite waveguide. The photons emitted by the system are fed back after a
time-delay which is why a non-Markovian treatment is necessary. The emitted photons
interact subsequently with the system after a time-delay due to the structured surrounding
reservoir. Thus, a memory kernel of the reservoir is the basis to describe all-optical feedback
in the theoretical approach. In this chapter, the description of quantum feedback is based
on the quantum stochastic Schrödinger equation [
204
,
54
,
205
], introduced in chapter 3.
The idea is to use an MPS representation as introduced in chapter 7 to treat only the
most relevant part of the Hilbert space to deal with the large memory kernel. However,
in the here investigated setup, there is a further complexity. Due to the time-dependent
Gaussian shaped excitation, two time-scales are involved, i.e. the short pulse and the long
radiative decay. The solution is to extend the MPS method based on the QSSE by going
beyond the Euler-like expansion of the time evolution. The speedup in the computation also
allows a non-Markovian treatment for more complex systems such as strongly-correlated
[
14
,
148
,
48
,
156
] or many-emitter setups [
102
,
103
,
93
,
114
] which have been subject of
investigation in chapter 5 and chapter 6.
However, in this chapter the focus lies on a comparable simple system and a complex
structured reservoir. The findings of this chapter demonstrate a quantum interference
between the system and the photon-field. In changing the distance between emitter and
mirror, these interferences allow the control of the photon statistics [
206
]. Destructive
interference results in a suppression of the one-photon probability at the simultaneous
increase of the two-photon probability. For constructive interference, two-photon generation
is suppressed and a single-photon emission process becomes more probable resulting in a
single-photon source for a 2π-pulse.
The proposal is that a combination of the two control concepts, i.e. time-dependent
excitation and time-delayed feedback allows to achieve statistics on demand, ranging from
enhancing the total photon output to an increase of the two-photon probability as well as
two-photon purification by suppressing single-photon events. The proposed setup allows
to all-optically address individual photon probabilities in varying the external control
parameters.
Chapter 8. Feedback controlled two-photon purification 87
Figure 8.1.:
A TLS with frequency
ω01
is placed inside a semi-infinite waveguide. The TLS
is pumped via a time-dependent external laser Ω(
t
). Due to spontaneous decay,
a photon is emitted either to the left side with spontaneous emission rate Γ
L
or
to the right side with rate Γ
R
. The photon emitted to the right side is reflected
at the closed end and interacts again with the TLS after a time-delay τ.
8.1. Theoretical model
In chapter 3, the quantum stochastic Schrödinger equation was introduced to generate the
Lindblad equation for a continuously driven TLS coupled to a continuum (Equation 3.1.1). In
this chapter, an additional complexity is considered which is a structured external reservoir.
As a model system, in contrast to Part II, a simple two-level system is considered. However,
the reservoir degrees of freedom cannot be traced out as the TLS and the surrounding
reservoir become entangled. Thus, the whole system is considered, i.e. electronic TLS and
bosonic reservoir. The idea is to treat the reservoir combined with the TLS as a many-body
system and describe it with the matrix product state formalism, introduced in chapter 7.
In order to model the structured reservoir, a semi-infinite waveguide [
207
,
208
] is assumed
as surrounding reservoir. The TLS is placed at a specific position within the waveguide. As
the waveguide is closed at one end, a boundary condition for the photon field is present.
Thus, the system-reservoir interaction is modeled differently in contrast to Equation 3.1.1
and reads
Hfb =Zdω~ωˆ
b†(ω)ˆ
b(ω)
+Zdω~hGfb(ω)ˆ
b†(ω)σ−+h.c.i.(8.1.1)
In analogy to Equation 3.1.1, the first term describes the free evolution of the reservoir
resulting from the quantized vacuum Maxwell field from section 2.3. The crucial difference
lies within the TLS-reservoir interaction, where the boundary condition of a closed waveguide
is modeled as a frequency-dependent coupling with [209, 210, 211]
Gfb(ω) = g0sin(ωL/c0),(8.1.2)
where
c0
is the speed of light in vacuum and
L
the distance between the TLS and the
closed end of the waveguide. The boundary condition can be interpreted as a reflecting
mirror where the light field within the waveguide has a node at the closed end of the
waveguide. The reflecting mirror results in a feedback loop, where photons emitted from
the TLS interact again with the system after time-delay
τ
= 2
L/c0
. Note that besides this
boundary condition, the coupling strengths is assumed to be a constant decay parameter
g0
=
pΓ/2π
of the TLS excitation, which is associated to a Markov approximation in
Chapter 8. Feedback controlled two-photon purification 88
analogy to section 3.1. As a further complexity, a pulsed driven TLS is considered, where the
amplitude of the external driving laser becomes time dependent
E0→
Ω(
t
). The external
coherent pulse is assumed to be resonant with the TLS frequency
ωL
=
ω01
. The whole
model system is sketched in Figure 8.1. With these considerations, the total Hamiltonian
reads
H(t) = ~ω01σ+σ−+~Ω(t)σ+e−iωLt+σ−eiωLt+Hfb .(8.1.3)
Note that the Hamiltonian of the pumped TLS is described in energy conserving rotating
wave approximation as explained in section 2.4. The time-dependent external laser is
modeled as a Gaussian pulse with frequency ωLand time-dependent amplitude
Ω(t) = A
√ν2πe−t2/ν2.(8.1.4)
One of the crucial parameter within this chapter will be the pulse area
A
as it defines the
number of Rabi-oscillations, the TLS undergoes during the pulse, i.e. for
A
=
π
the TLS
excitation is inverted due to a half Rabi-oscillation. For
A
= 2
π
, the TLS undergoes a full
Rabi-Oscillation, ending up in the electronic initial state after the pulse. The parameter
ν
describes the linewidth of the Gaussian pulse. Due to the finite pulse width combined with
a finite radiative decay to the surrounding environment, the TLS does not end up in its
initial state after a full Rabi-oscillation [
212
] which will be described more detailed in the
following section.
In contrast to chapter 3, where the Lindblad equation was derived with a reduced density
matrix formalism in section 3.3, the frequency-dependent coupling
Gfb
(
ω
)does not allow a
straightforward evaluation of the trace over the reservoir as system and reservoir are highly
entangled due to the boundary condition. However, the problem can be solved numerically
in the Schrödinger picture by writing the combined state of the system and reservoir as
an MPS. Equation 8.1.3 is again transformed into a rotating frame via Equation 3.1.3
with respect to the free evolution
ω01
of the system and
ω
of the reservoir as done in
Equation 3.1.5
HTLS,rf(t) = ~Ω(t)σ++σ−,(8.1.5)
Hfb,rf(t) = Zdω ~Gfb(ω)σ+ˆ
b(ω)e−i(ω−ω01)t+G∗
fb(ω)σ−ˆ
b†(ω)ei(ω−ω01)t.(8.1.6)
It is differentiated between photons emitted to the left Γ
L
and photons emitted to the
right side Γ
R
. Thus,
g0
is split into the left propagating wave with
pΓL/2π
and
pΓR/2π
for the right propagating wave. The structured coupling in exponential form then reads
Gfb(ω) = i
sΓR
2πe−iωτ/2−sΓL
2πeiωτ/2
.(8.1.7)
Inserting this into the Hamiltonian in Equation 8.1.5 and Equation 8.1.6 and performing
again a rotating frame transformation with Equation 3.1.3 to shift the
τ
dependence into
one of the summands. With the unitary operator which fixes the spatial coordinates
U=e−iRdωωτ/2ˆ
b†(ω)ˆ
b(ω),(8.1.8)
Chapter 8. Feedback controlled two-photon purification 89
the transformed Hamiltonian reads
HTLS,rf(t) = ~Ω(t)σ++σ−,(8.1.9)
Hfb,rf(t) = −i~
sΓ
2ˆ
b(t−τ)e−iφ +sΓ
2ˆ
b(t)
σ++h.c.. (8.1.10)
It is assumed that the radiative decay to the left and to the right side of the waveguide
are equal Γ
R
= Γ
L
= Γ. Furthermore, the time-dependent reservoir operators in Equa-
tion 3.1.6 have been inserted. A new parameter
φ
=
π−ω01τ
is introduced, which is
called the feedback phase. The fast rotating phase
φ
determines whether the feedback
field is constructive or destructive with the TLS excitation, where
ω01τ
= 2
nπ
is called
constructive and
ω01τ
= (2
n−
1)
π
is called destructive interference for integer numbers
n
.
In the following, the nomenclature constructive/destructive will become clear.
This Hamiltonian shows system-reservoir interaction at two different times. If a basis set
such as in Equation 3.1.16 is introduced, the resulting time-evolution operator interacts
with two different time-bins at each time step ∆
t
. The time-bin
|iki
is called the current
time-bin which corresponds to all operators
ˆ
b(†)
(
t
). Without the boundary condition this is
the term resulting in the Lindblad dissipation in Equation 3.3.15. However, all operators
ˆ
b(†)
(
t−τ
)interact with past time-bins
|ik−li
, where
k−l
=
τ/
∆
t
and
k > l
. This means,
any photon emitted to the right side of the waveguide interacts again with the TLS after
delay τ. This is why this Hamiltonian is called the feedback Hamiltonian Hfb.
This Hamiltonian already clarifies why the tracing out of reservoir degrees of freedom is not
possible in this case. The resulting time-evolution operator acts on two distant time-bins.
Thus, an entangled state between system and the two time-bins
|is, ik, ik−li
is created
at each time step ∆
t
. A Born-Markov approximation, resulting in factorized reservoir
states, as performed in section 3.3 would destroy any interference induced by the boundary
condition. This is the reason, why the QSSE is used and |ψiis expressed as an MPS.
8.1.1. Higher-order time-evolution operator
As in chapter 3, the time-evolution operator governs the dynamics of
|ψi
. The time-ordering
operator
ˆ
T
is redundant for the feedback Hamiltonian due to the definition of the time-
increment quantum-noise operators ∆
B(†)
(
tk
)and ∆
B(†)
(
tk−l
). However it is important
that both commute with each other in the interval ∆
t
. This is fulfilled as long as ∆
t < τ
.
By adapting the numerical time step accordingly, this can be achieved in the simulation.
By inserting the noise operators, defined in Equation 3.1.13, the time-evolution operator
reads
U(tk+1, tk) = exp h−i∆tΩ(∆t)σ++σ−
−
sΓ
2∆B(tk−l)e−iφ +sΓ
2∆B(tk)
σ++h.c.i,(8.1.11)
where
tk
=
k
∆
t
and
tk−l
=
tk−τ
. It was assumed that the amplitude of the driving laser
Ω(∆
t
)at each time step only changes marginally Ω(
t
)
→
Ω(∆
t
), such that it is excluded
from the integral over ∆
t
. In contrast to chapter 3 and the reference introducing the
method to compute the feedback Hamiltonian with the QSSE [
54
], in this chapter, the
Chapter 8. Feedback controlled two-photon purification 90
time-evolution operator is expanded in second order of ∆
t
. This has several computational
reasons, one of them is that two different time-scales are involved due to the time-dependent
pump Ω(
t
)which will be chosen comparable fast to the radiative decay Γof the TLS. Thus,
the system has to be integrated for a long time while still governing the exact dynamics
during the pulse. A higher order in the time-evolution operator allows to chose a larger ∆
t
and thus, reduce the total number of time steps. The other reasons to expand
U
will become
clear when the feedback algorithm and the computed observables will be introduced.
To give an explicit construction of the time-evolution operator for numerical implementation,
U
is expressed in matrices. The operators in the exponent acting on the basis states are
expressed in matrices MTLS and Mfb, respectively. Considering the basis states, the only
time dependence is the amplitude of the driving laser Ω(
tn
)at the respective time step
tn
.
An explicit expansion of the time-evolution operator then reads
U(tn) = exp (Ω(tn)MTLS +Mfb)
=
∞
X
p=0
1
p!(Ω(tn)MTLS +Mfb)p,(8.1.12)
where
p
is the order of the expansion and not the order in ∆
t
. Note that the amplitude of
the driving laser is separated from the Hamiltonian to deal with time-independent matrices.
The reason is that with this expression, the large matrices only have to be initialized once
in the beginning of the simulation.
The exponent of the time evolution is written as a matrix by multiplying the respective
basis elements from left and right. The basis is labeled as
|iS, ik, iτi
, where
|iSi
is the
state of the TLS (either
|
0
i
or
|
1
i
),
|iki
is the time-bin occupation with
ik
photons at time
tk
=
k
∆
t
and
|iτi
is the feedback-bin occupation with
τ
= (
k−l
)∆
t
. Multiplying the basis
states from left and right results in
MTLS =−i
~hjS, jn, jτ|Ztk+1
tk
HTLS,rf
Ω(∆t)dt|iS, in, iτi
=−i∆t(δjS,1δiS,0+δjS,0δiS,1)δjn,inδjτ,iτ.(8.1.13)
Mfb =−i
~hjS, jn, jτ|Ztk+1
tkHfb,rfdt|iS, in, iτi
=−
sΓ
2√iτδjτ+1,iτe−iφ +sΓ
2√inδjn+1,in
δjS,1δiS,0√∆t
+
sΓ
2pjτδjτ,iτ+1e−iφ +sΓ
2pjnδjn,in+1
δjS,0δiS,1√∆t . (8.1.14)
As mentioned in chapter 3, the action of the time-bin operator ∆
B†
(
tk
)on the state
|iki
creates a
√∆t
to maintain the commutation relation in Equation 3.1.15. Thus, the matrix
Mfb ∝√∆t
, while
MTLS,env
(
tn
)
∝
∆
t
. This is the reason, why a higher expansion in ∆
t
of the time-evolution operator is not straightforward in the order of the expansion as both
terms obey a different proportionality to ∆
t
. For the first order in ∆
t
, terms up to second
Chapter 8. Feedback controlled two-photon purification 91
order of
Mfb
contribute as shown in chapter 3. For an expansion in the orders of (∆
t
)
2
,
terms up to the fourth order in Mfb become relevant
U(tn)≈U0+ Ω(tn)U1+1
2Ω(tn)2U2
=1+Mfb +1
2M2
fb +1
6M3
fb +1
24M4
fb
+ Ω(tn)MTLS +1
2(MTLSMfb +MfbMTLS)
+1
6MTLSM2
fb +MfbMTLSMfb +M2
fbMTLS
+ Ω(tn)21
2M2
TLS .(8.1.15)
The expansion is differentiated in orders of the pump amplitude Ω(
t
). The reason is that the
pump matrix changes with each time step and thus has to be adapted and multiplied with
the corresponding matrices at each time step. By computing the matrix multiplications
once in the beginning of the algorithm, at each time step only the multiplication with
Ω(
t
)
U1
(Ω(
t
)
2U2
) has to be performed. As especially the matrix
Mfb
might become very
large, this optimizes the computation time drastically.
8.1.2. Feedback algorithm in the QSSE picture
Due to the feedback system-reservoir interaction, system and reservoir do not factorize.
However, the state
|ψi
is brought formally close to a product state when written as an
MPS. As explained in chapter 7, the coefficient matrix of
|ψi
is decomposed into a train of
tensors. Thus initially,
|ψi
is already written in the form of an MPS, even if it is initialized
in a product state. The used basis is the time-bin basis of Equation 3.1.16, therefore
|ψi
consist of infinite many future time-bins. However, it is assumed that initially all future
time-bins are in the photon number state zero and are not entangled with the system
or with other time-bins. Thus in practice, only one future time-bin becomes relevant at
each time step. Furthermore, initially all past time-bins are in a vacuum state as well. As
long as
t<τ
, the interaction with past time-bins is not present. In Figure 8.2, the time
evolution of the MPS is sketched for
t < τ
. Note, that the MPS is always assumed to be
in mixed-canonical form. If this is not the case it can be brought to the mixed-canonical
gauge by a series of SVDs as explained in chapter 7.
The most relevant tensor is the system-bin with physical index
is
which describes the state
of the TLS. Initially, all time-bins are set to zero and the system-bin is initialized at an
arbitrary state. The MPS consists of the system- and the time-bins, where future time-bins
are placed left of the system-bin and past time-bins right of the system-bin. Before
t
=
τ
,
the time-evolution operator is a tensor of rank four, which acts on the system-bin and
the future time-bin. Thus, two indices are shared with the MPS when the time-evolution
tensor is contracted with it. The primed indices are then the new state of the MPS after
the first time step ∆
t
. The resulting tensor contains the two primed indices and the link
index to the past time-bins. The primed indices are unprimed to be consistent with the
rest of the MPS and the tensor is decomposed via a SVD and the MPS is restructured
such that the former future and now past time-bin
i0
is written right of the system-bin.
Note that the OC is always kept at the system-bin. Furthermore, the link index between
the system-bin and past time-bin
i0
now might be occupied, depending on the parameters
Chapter 8. Feedback controlled two-photon purification 92
Figure 8.2.:
Time evolution of the first two time steps for
t<τ
. The time-evolution operator
acts on the MPS
U
(
t1, t0
)
|ψ
(0)
i
. By contracting over the two physical indices
is
(TLS index) and
i0
(first time-bin) with the time-evolution tensor, as well
as the link index between the system and first time-bin, the time evolution
is computed. The resulting tensor is decomposed and the outgoing physical
indices
i0
s
and
i0
0
are unprimed to restore the form of the MPS. The tensor
containing the system index
is
is now left of the time-bin
i0
. The system
has already interacted with time-bin
i0
which is now a past time-bin and
therefore right of the system-bin. It becomes clear that all time-bins right of
the system-bin are past time-bins and all indices left of the system-bin are
future time-bins. In the next time step the time-evolution operator acts on the
indices is(system-bin) and i1(next time-bin).
in
U
and the cutoff of the SVD, representing entanglement between system and reservoir.
Between each time step observables are derived as, e.g., the norm or the occupation of
the system. This is straightforward as the OC is always at the system-bin for
t < τ
.
Afterwards, the MPS is ready to compute the next time step. Thus, the time-evolution
operator now changes its index. The system index
is
remains the same but the reservoir
index
i0
has changed to
i1
as
U
(
t2, t1
)acts on the next time-bin. This is in practice realized
by multiplicating the time-evolution operator with an identity tensor with two physical
indices
i0
and
i1
. Without changing the entries of the tensors the index
i0
is replaced with
i1by contracting over i0. The same is performed for the primed indices i0
0and i0
1.
Thus within each time step, the system-bin moves one position to the left in the MPS and
thus also forward in time. This procedure is repeated until t=τis reached.
When
t>τ
, the time-delay becomes relevant as past time-bins interact again with the
system. Thus, the time-evolution operator now acts on the system-bin
is
, on the future
time-bin
ik
and on the feedback time-bin
ik−l
=
τ/
∆
t
. This poses a problem as between
system-bin and feedback time-bin might be a high number of time-bins depending on
∆
t
and
τ
. In order to apply the time-evolution tensor on these three tensors, one would
have to contract all the time-bins between
is
and
ik−l
as well. This can result in a very
large tensor and is not practical. However, a faster possibility to deal with this long-range
Chapter 8. Feedback controlled two-photon purification 93
Figure 8.3.:
Time evolution after time-delay
τ
for applying
U
(
tk, tk−1
)
|ψ
(
tk−1
)
i
. The tensor
with time-bin
τ/
∆
t
=
ik−l
is brought next to the system-bin by SWAP-
operations. Then the time-evolution tensor being a tensor of rank six acts on
the future time-bin, the system-bin and the past time-bin corresponding to the
delay time
τ
. The resulting tensor is decomposed by SVD and the feedback-bin
ik−lis brought back to the original position.
coupling is by performing series of swap operations until the feedback time-bin is right to
the system-bin [
54
]. This procedure, including the application of
U
, is shown in Figure 8.3.
The swapping of two physical indices was introduced in Figure 7.13 in section 7.5: The OC
is brought to the feedback time-bin by contractions and SVDs. Between feedback time-bin
ik−l
and system-bin are
τ/
∆
t
past time-bins right of the system-bin. Thus, in a first step
the feedback time-bin
ik−l
is contracted with its left neighbor
ik−l+1
. For the SVD, the
physical index is swapped while maintaining the link indices to the left and right. The new
link index between
ik−l
and
ik−l+1
might change in performing this operation. Then the
procedure is repeated until the feedback time-bin is next to the system-bin, by keeping the
OC at the feedback time-bin. In a last step, the OC is brought from the feedback time-bin
to the system-bin. Observables concerning the system-bin might be computed at this point
before applying the time-evolution operator.
When the feedback time-bin is next to the system-bin, the application of the time-evolution
operator is the same than for
t < τ
. The only difference is that
U
acts on three tensors.
Thus, the resulting tensor is decomposed twice to restore the three bins. Furthermore, the
tensor is decomposed such that the two time-bins are positioned right of the system-bin,
because both now represent past time-bins. The OC is stored in the feedback time-bin
ik−l
.
Afterwards, the feedback time-bin is swapped back to its original position. After the last
swap operation, the OC is stored in the feedback time-bin for the next time step
ik−l+1
and the procedure is repeated until the end of the time integration is reached.
Chapter 8. Feedback controlled two-photon purification 94
Another advantage of the higher-order time-evolution operator becomes apparent. Each
time step an error is induced by a non-unitary
U
. This results from the expansion of the
exponential in orders of ∆
t
. When the order in ∆
t
is higher, the same accuracy is obtained
with a higher chosen ∆
t
. For constant time-delay
τ
, a lower number of time-bins are needed
in the delay-interval
τ/
∆
t
as ∆
t
is higher. This directly leads to fewer swap operations
during each time step and thus reduces drastically the computation time.
8.1.3. Computing photon probabilities from the matrix product state
In this chapter, the focus does not lie on the system observables, but on the statistics
of the reservoir. The quantity of interest are the photon statistics described by photon
number probabilities
p
(
n
)for
t→ ∞
after the time-dependent pulse has excited the system
and the TLS is equilibrated. The MPS does not lead directly to the photon probabilities
of interest. However, the photon probabilities can be obtained by calculating correlation
functions [
213
] which are multi-site operators acting on the MPS. The operator for the
total intensity emitted into the environment is defined as
ˆ
I
=
P∞
j=0
∆
B†
(
tj
)∆
B
(
tj
). This
results from the total intensity inside each corresponding time-bin
tj
. For the numerical
evaluation, it is assumed that after a large enough time
tend
=
N
∆
t
, all excitation from
the TLS is emitted into the environment, so that afterwards no photons will be observed.
The
m
th order intensity correlation function is defined as
h
:
ˆ
Im
:
i
, where :indicates the
normal ordering of the operators. Thus, in the time-bin basis, the correlation functions up
to the third order are obtained via
C1=
N
X
k=0h∆B†(tk)∆B(tk)i(8.1.16)
C2=
N
X
k=0
N
X
l=0h∆B†(tk)∆B†(tl)∆B(tl)∆B(tk)i(8.1.17)
C3=
N
X
k=0
N
X
l=0
N
X
m=0h∆B†(tk)∆B†(tl)∆B†(tm)∆B(tm)∆B(tl)∆B(tk)i.(8.1.18)
The total intensity inside the reservoir is of interest, thus the photon occupations of all
time-bins from the start of the integration until the end
ti∈
[0
, tend
]have to be computed.
For the photon intensity in Equation 8.1.16 this is straightforward, because the intensity
operator
h
∆
B†
(
tk
)∆
B
(
tk
)
i
is local. Thus, the OC is shifted to the respective time-bin
and the local intensity operator is computed in analogy to Figure 7.11. Then, the local
intensities of the time-bins are summed up and the total photon intensity of the reservoir is
obtained. However, Equation 8.1.17 and Equation 8.1.18 are two and three site operations
on the MPS and the computation can be very costly. The computation of a single integrand
of
C3
is shown in Figure 8.4. Note that for each operation, the OC has to be at one
of the involved contracted tensors. Furthermore, all tensors between the three involved
time-bins have to be contracted as well. At worst this corresponds to a contraction of
tend/
∆
t
time-bins. In order to avoid to perform this costly operation multiple times, the
symmetry of the correlation function is used. Furthermore, it becomes clear that the higher
order in the time-evolution operator is very advantageous when dealing with higher-order
correlation functions, as the number of total time-bins is reduced by choosing a higher ∆
t
.
After computing all relevant integrands once, the third-order correlation function is then
obtained by summing up all integrands in the right manner to be in correspondence with
Equation 8.1.18.
Chapter 8. Feedback controlled two-photon purification 95
Figure 8.4.:
Computation of a single integrand of the third-order correlation function
C3
.
The OC has to be included in the operation.
The relation between the
m
th order intensity correlation function and the probability of
n
photons p(n)reads
Cm=h:ˆ
Im:i=
∞
X
n=0
n!
(n−m)!p(n).(8.1.19)
The physical mechanism to generate multi-photon states in the present setup is a sponta-
neous decay of electronic excitation during the time-dependent pulse. Thus, during the
pulse, multiple photons are created via spontaneous emission rate Γ. If the pulse width
ν
is chosen small in comparison to Γ, high photon probabilities can be neglected. In the
investigated pump regimes,
p
(2) dominates and
p
(3) is very small compared to
p
(1) and
p
(2). This justifies, to assume any correlations higher than third order to be negligible.
Thus, the expansion in Equation 8.1.19 can be cut off after the third order. By rearranging
the set of equations, the photon probabilities are calculated from the correlation functions
up to the third order via
p(1) = C1−C2+C3
2,(8.1.20)
p(2) = C2−C3
2,(8.1.21)
p(3) = C3/6.(8.1.22)
8.2. Controlling photon statistics
8.2.1. Effect of the time-dependent pulse
When a single TLS is excited with a resonant pulse, the TLS undergoes Rabi-oscillations.
The pulse area
A
determines the number of oscillations. If
A
=
π
, the TLS excitation is
inverted as shown in Figure 8.5. If the TLS is initially in its ground state
|
0
i
, after the
pulse, the TLS ends up in the exited state
|
1
i
. When the TLS is coupled to an external
reservoir it is subject to spontaneous radiative decay. Thus, a photon is emitted on a long
time-scale 1
/
Γ. The inverted TLS now being in the state
|
1
i
, decays radiatively under
emission of a single photon. An excitation with a
π
-pulse results in an almost perfect
single-photon source, as for
t→ ∞
one would detect a single photon coming from the TLS
(inset Figure 8.5).
However, if the pulse induces a full Rabi-oscillation of the TLS excitation for
A
= 2
π
, the
Chapter 8. Feedback controlled two-photon purification 96
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
0
0.02
0.04
0.06
0.08
0.1
0
0.2
0.4
0.6
0.8
1
0 1 2 3
hσ+σ−i
Ω(t)
tΓ
Excited state density
Trajectory
π-pulse
p(n)
n
Figure 8.5.:
Time evolution of the TLS excited with a pulse of
A
=
π
. The TLS excitation
(blue, solid) is inverted after the pulse (red, dotted). On a long time-scale, a
photon is emitted resulting in a single-photon source (see inset where
p
(1) is
dominant). An exemplary trajectory of the TLS excitation is sketched (yellow,
dashed).
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
0
0.02
0.04
0.06
0.08
0.1
0
0.2
0.4
0.6
0.8
1
0 1 2 3
hσ+σ−i
Ω(t)
tΓ
Excited state density
Trajectory
π-pulse
p(n)
n
Figure 8.6.:
Time evolution of the TLS excited with a pulse of
A
= 2
π
. The TLS undergoes
a full Rabi-oscillation (blue, solid) after the pulse (red, dotted). Zero photons
are most probable (inset for photon probabilities
p
(
n
)). However, due to a
finite pulse width, the TLS might decay during the pulse and be re-excited
again and a second photon is emitted on a long time-scale. An exemplary
trajectory of the TLS excitation is sketched (yellow, dashed). This procedure
results in p(2) > p(1).
Chapter 8. Feedback controlled two-photon purification 97
two-photon probability is higher than the single-photon probability [
212
,
214
] as shown in
Figure 8.6. If the pulse width would be infinitely small, the TLS undergoes a perfect full
Rabi-oscillation without a photon emitted to the environment and ending up again in its
initial state. Due to a finite pulse width and a finite radiative decay, there is a probability
of a photon emission process during the pulse. The TLS is excited by the pulse, bringing it
to the state |1i. Due to the finite decay, there is a probability that one photon is emitted
into the environment and the TLS is found in the state
|
0
i
. However, the remaining pulse
might re-excite the TLS, bringing it again to the excited state
|
1
i
. A second photon is then
emitted on a long time-scale as for driving with
A
=
π
. This results then in a probability of
p
(2)
> p
(1). Thus, for a full Rabi-oscillation zero photons
p
(0) are most probable. However,
the probability in finding the TLS in the ground state after the pulse is not one as can be
seen on the offset of the TLS excitation density (blue, solid) in Figure 8.6. This is due to
the emission of a photon and a re-excitation during the pulse and the reason for
p
(2)
> p
(1).
8.2.2. Effect of time-delayed feedback
So far, the results shown are without a structured reservoir. The key idea of this chapter
is to trigger the two-photon process with a time-delay during the pulse. In the inset of
Figure 8.7, a sketch of the TLS excitation density is shown for a structured reservoir
with a mirror as boundary condition. In yellow, the expected TLS decay is shown for
a reservoir without boundary condition and therefore without feedback field. When the
feedback phase is constructive (
ω01τ
= 2
nπ
), the back-action of photons results in revivals
of the TLS excitation after
τ
as shown in blue. In contrast, when the phase is destructive
(
ω01τ
= (2
n−
1)
π
), the radiative decay is effectively increased after delay
τ
as the
interference with the feedback field triggers the photon emission process of the TLS. In
Figure 8.7, the two-photon emission for different phases
φ
and delay-times
τ
is shown for a
pulse area of A= 2π. The ratio
r=p(2)
p(1)
p(1)nofeedback
p(2)nofeedback
(8.2.1)
quantifies the two-photon purification obtained from the structured reservoir. If
r
= 1,
the photon probabilities
p
(
n
)of the feedback case are the same than without feedback
p
(
n
)
nofeedback
. If
r >
1, the ratio
p
(2)
/p
(1), signifying two-photon purification, is higher
than without feedback. If
r <
1, single photons dominate compared to the case without
feedback.
It becomes apparent that the feedback phase
φ
has a huge impact on the two-photon
emission. In particular, it switches between single and two-photon emission. For constructive
feedback, where the TLS excitation shows revivals after
τ
, single-photon emission dominates
over two-photon emission with
r→
0. However, destructive feedback results in a higher
two-photon probability compared to the case without feedback. The ratio
r
is increased
up to a factor of two. This factor is dependent on the time-delay
τ
. The reason behind
this increase in the two-photon probability lies within the effectively increased radiative
decay during the pulse. As the dominant two-photon emission results from a spontaneous
decay during the pulse, an effective increase of Γdue to feedback interference increases the
two-photon emission process. Note that the ratio shows
r
= 1 if
φ
=
π/
2. This means the
feedback dynamics are then almost equal to the case without feedback.
Chapter 8. Feedback controlled two-photon purification 98
0
1
0 1
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
τΓ
−1
−0.5
0
0.5
1
φ/π
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
r
hσ+σ−i
t/τ
0
1
0 1
Figure 8.7.:
Dependence of two-photon emission on the feedback phase
φ
and the time-delay
τ
for
A
= 2
π
. The color denotes the ratio
r
=
p
(2)
/p
(1) normalized by the
ratio without feedback. Thus, yellow color reproduces two-photon emission
without feedback. Red color represents two-photon enhancement which is the
case for destructive feedback (
φ
= 0). Blue denotes single-photon emission.
The inset sketches the effect of feedback on the TLS decay, the color matches
the respective emission process.
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
0
0.05
0.1
0 1 2 3
n
p(n)
0.8
0.9
0 1 2 3
n
p(n)/p(n)nofeedback
τΓ
p(1)
p(2)
Figure 8.8.:
Photon probabilities versus varying time-delay
τ
, scaled with Γfor a pulse
area
A
= 2
π
. The probabilities are normalized by the case without feedback,
the black line at one indicates the probabilities at
A
= 2
π
for no feedback.
Note that
p
(1) (blue) and
p
(2) (red) respond non-trivial on varying
τ
. Inset:
Photon probabilities for
τ
Γ=0
.
06 for no feedback (yellow) and destructive
feedback (red). Destructive feedback increases
p
(2) by almost 50% while
p
(1)
is unaffected.
Chapter 8. Feedback controlled two-photon purification 99
8.2.3. Controlling individual photon probabilities
This effect of triggered two-photon emission due to feedback interference enables the control
of the photon emission statistics. The external control parameters for this scenario are
the pulse area
A
and the distance
L
between mirror and TLS which affects both
τ
and
φ
.
The phase
φ
represents fast oscillations with the distance and is very sensitive on varying
L
. As shown in Figure 8.7, destructive interference takes place in
φ∈
[
−π/
2
, π/
2] which
corresponds to a distance of ∆
L≈
0
.
3
µ
m for a typical quantum dot with band gap of 1eV.
If the system is a superconducting circuit which has usually a smaller transition frequency
of, e.g.,
ω01/
2
π
= 6 GHz [
215
], two-photon enhancement is robust for ∆
L≈
1
.
3cm. The
time-delay
τ
has to be adjusted dependent on the radiative decay of the respective system
and the chosen pulse width
ν
. The pulse width
ν
is also a relevant control parameter. By
increasing
ν
, higher photon probabilities become relevant. This case is beyond the scope of
this chapter as higher-order correlation functions would become relevant. In this chapter
the pulse width is fixed at ν=1
10Γ
1
√2ln(2) .
In Figure 8.7 it was shown that the phase
φ
determines whether two-photon creation is
enhanced or single-photon emission is dominant compared to the case without feedback.
The question is, how much the individual probabilities can be influenced by varying the
control parameters for destructive feedback in order to achieve a two-photon purification.
Thus,
φ
will be chosen to yield destructive interference from now on to trigger the two-
photon emission process.
In Figure 8.8, the photon probabilities are shown for varying
τ
. The one-photon (blue, solid)
and two-photon (red, dashed) probabilities are normalized by the probabilities without
feedback, respectively. The horizontal black line corresponds to the case without feedback
for both probabilities. On a first glance, it becomes apparent that
p
(2) is enhanced for
all investigated time-delays
τ
whereas
p
(1) is either enhanced or suppressed. For short
time-delay (
τ
Γ
<
0
.
06), both probabilities are increased compared to the case without
feedback. As the delay is very short, the radiative decay is already effectively increased at
the beginning of the pulse. Thus, this can be interpreted as a higher global radiative decay
of the TLS, induced by short delay-times τ.
The full power of the feedback control setup becomes apparent for time delays in the orders
of the pulse width
ν
: The radiative decay is effectively increased at a certain time during
the pulse. This controlled increase of the decay allows to only increase
p
(2) while
p
(1) is
still the same than without feedback. The two-photon probability is enhanced by 50% at
τΓ=0.06 whereas p(1) remains the same.
The interplay between finite pulse width
ν
and time-delay
τ
allows to manipulate an
individual photon probability and results in a two-photon purification. For higher time-
delays, the two-photon probability decreases but is still higher than without feedback. In
contrast,
p
(1) is suppressed compared to no feedback and reaches a minimum for
τ
Γ
≈
0
.
1.
This explains the highest purification of the ratio
p
(2)
/p
(1) in Figure 8.7, visible as the
red center at
τ
Γ
≈
0
.
1. For increasing time-delay, both
p
(1) and
p
(2) approach the case
without feedback. The reason is that the backaction takes place after the pulse and the
two-photon emission process is not triggered by the feedback dynamics. The only difference
is that the remaining excitation of the TLS decays faster.
When the amplitude of the driving laser is increased such that the pulse area corresponds
to
A
= 4
π
, the TLS undergoes an additional full Rabi-oscillation. As the pulse width
remains fixed, the same window of time-delays allows for a higher control, which is shown
in Figure 8.9. E.g., the two-photon probability is increased by almost 50% while
p
(1)
Chapter 8. Feedback controlled two-photon purification 100
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
0
0.05
0.1
0 1 2 3
n
p(n)
0.8
0.9
0 1 2 3
n
p(n)/p(n)nofeedback
τΓ
p(1)
p(2)
Figure 8.9.:
Photon probabilities normalized by the case without feedback for a pulse
area
A
= 4
π
. The TLS undergoes an additional Rabi-oscillation. This is
reflected in the photon probabilities, where
p
(1) (blue) and
p
(2) (red) change
their dependence on
τ
compared to
A
= 2
π
. Inset: Photon probabilities for
Γ
τ
= 0
.
05 for no feedback (yellow) and destructive feedback (red). Destructive
feedback increases p(2) by almost 50% and simultaneously decreases p(1).
is minimal at
τ
Γ=0
.
05. This is an even higher achieved purification as for
A
= 2
π
.
Furthermore, for
τ
Γ
∈
[0
.
07
,
0
.
12], both probabilities are enhanced compared to the case
without feedback. Again, the two-photon probability
p
(2) is increased for all investigated
time-delays while the one-photon probability responds non-trivially to a change of the
time-delay.
8.3. Conclusion
In this chapter, it has been demonstrated how a structured reservoir leading to entan-
gled system-reservoir dynamics can be simulated on the basis of the quantum stochastic
Schrödinger equation. By writing the state of the system and the state of the reservoir as
a matrix product state and by performing stroboscopic simulation of the time-evolution
operator, the state of the system and reservoir is evolved in time. By considering the state
of the reservoir as a many-body problem, it has been demonstrated how photon statistics
can be evaluated, leading to the photon probabilities of the reservoir within a matrix
product state representation close to an experimentally photon counting setup.
By assuming a distant mirror, it has been shown how feedback dynamics including a
memory kernel are evaluated in the quantum stochastic Schrödinger picture using matrix
product states. The findings of this chapter demonstrate a quantum interference effect
between the photon-field and the two-level system induced by the mirror.
Having two external control parameters, i.e. the time-dependent excitation as well as the
mirror distance with time-delay
τ
, a setup to control the photon statistics of the reservoir
is proposed. By adjusting the amplitude of the driving laser as well as the mirror distance
Chapter 8. Feedback controlled two-photon purification 101
it becomes possible to manipulate and tailor feasible parts of the photon probabilities.
Constructive feedback results in a single-photon source for a pulse-area of 2
π
, where without
feedback, the two-photon probability would be higher than the one-photon probability.
However, for destructive feedback, the two-photon emission process is triggered. If the
time-delay is very short in comparison to the driving pulse, both single and two-photon
probabilities are increased. This is due to an effectively increased radiative decay which is
global for short delay times.
However, by adjusting the pulse-duration and the time-delay it becomes possible to ad-
dress individual photon probabilities. For a time-delay of the order of the pulse width,
the one-photon probability responds differently than the two-photon probability. As an
example, the two-photon emission is increased by 50%, whereas the one-photon probability
remains the same than without feedback. In general, for destructive feedback, two-photon
emission is enhanced or remains the same than without feedback whereas the one-photon
probability is either enhanced or suppressed, depending on the time-delay.
In choosing a higher amplitude, i.e. a driving laser which induces two full Rabi-oscillations
of the two-level system, the same window of time-delays allows for more control. This
results in an even purer two-photon source as the two-photon probability is enhanced up
to 50% with a simultaneous decrease of the one-photon probability.
9.
Feedback-stabilized time crystal
The concept of spontaneous symmetry breaking is uniformly used to explain phase transi-
tions [
123
,
216
]. The standard example in condensed matter physics is the Ising transition.
The Hamiltonian displays a
Z2
-symmetry, which is spontaneously broken at zero temper-
ature as the ground state consists of ordered spins. Another example is the continuous
translational symmetry of liquids. If the symmetry is broken, the system forms a spatial
crystal which is only translational invariant in discrete positions in space.
These fundamental concepts of symmetry breaking gave rise to the question, whether
continuous time translational symmetry might be broken as well, forming a so called time
crystal [
217
,
218
]. Adapting the analogy of spatial crystals, a time crystal would be a
many-body system with periodic behavior in time. This implies that the system returns to
its initial state after discrete periods.
These considerations were followed by no-go theorems [
219
,
220
] which stated that a time
crystal can neither exist in the ground state nor in thermodynamic equilibrium due to the
periodic oscillations.
However, this leaves the possible existence of a time crystal in systems being out-of-
equilibrium, far away from their ground state [
40
,
41
,
221
,
222
]. These so called Floquet
systems are periodically driven, where the Hamiltonian is periodic in time
H
(
t
) =
H
(
t
+
T
)
with period
T
. The external driving force induces an out-of-equilibrium situation. However,
for long times, the system absorbs the energy of the driving and is heating to infinite
temperatures [
38
,
39
]. This process would destroy any time crystal phenomena in reaching
a thermal equilibrium, prohibiting time-dependent observables.
Ergodicity breaking due to MBL (introduction in chapter 4), prevents a system from
reaching thermal equilibrium. It has been demonstrated that MBL also exists in Floquet
systems [35, 36, 37].
Based on a proposal in Ref. [
223
], this paved the way for the demonstration of discrete
time crystals (DTC) in trapped ions [
42
], diamond impurities [
43
], in a solid crystal [
44
] or
in molecules [
45
]. The DTC shows oscillations in the observable with an integer number of
the Floquet period. This breaks the time translation symmetry of the Floquet Hamiltonian,
because the system follows a different period.
Altogether, due to the new phase of matter in an out-of-equilibrium system, the time crystal
forms an application of the findings of this thesis. Similar to chapter 5 and chapter 8, the
driving induces out-of equilbrium dynamics. Many-body localization, discussed in chapter 4
and chapter 6, is an important requirement to prevent thermalization in the time crystal
phase due to interactions.
However, the fate of a time crystal as an open quantum system is still under debate [
43
].
It was argued that for Lindblad dissipators, the time crystal necessarily thermalizes with
the environment [224].
In this chapter it will be demonstrated that in contrast, feedback dynamics similar to
103
Chapter 9. Feedback-stabilized time crystal 104
chapter 8, stabilize a discrete time crystal against dissipation for long times.
In this thesis, the time crystal is the first many-body spin system investigated with MPS
methods, introduced in chapter 7. As many-body spin systems form the standard applica-
tion for the MPS method, the important aspects when simulating time evolution will be
discussed in a first section with focus on the time crystal behavior.
In the second section, the system will be generalized to an open quantum system via the
QSSE method introduced in chapter 3. It will be shown that the time crystal thermalizes
due to the open system dynamics. A time-delayed feedback mechanism, in analogy to
chapter 8, prevents the thermalization with the external reservoir. Due to the feedback
dynamics, the entanglement between system and reservoir states is important and the
dynamics are evaluated in the full QSSE picture without tracing out the reservoir degrees
of freedom.
9.1. Closed system dynamics
9.1.1. Model
The time crystal is modeled as a driven one-dimensional Ising spin-1/2 chain. The idea is
to apply different Hamiltonians at different times. The overall period of the system reads
T
=
T1
+
T2
and the Hamiltonian is periodic in time
H
(
t
) =
H
(
t
+
T
). During period
T1
,
the Hamiltonian
HT= (Ω −)
N
X
i=1
σx
i,0< t < T1(9.1.1)
is applied. Note that for spin-models this term is called a transverse field. When mapping
σx
i
to raise
σ+
i
and lower operators
σ−
i
, the transverse field of a single spin is comparable
to the coherently driven two-level system of chapter 8. When choosing a pulse-area
A
=
π
,
the TLS excitation is inverted after applying the pulse. This is formally the same for a
single-site of the spin-flips in Equation 9.1.1 with Ω = π/2.
Thus, when Ωis chosen as
π/
2and the perturbation set to zero
= 0, each spin of the chain
is flipped. If a spin is initialized with spin-up, after
T1
it will be flipped to spin-down and
vice versa. The parameter
describes a perturbation of the driving resulting in spins which
are not ending up exactly in the opposite polarization after
T1
. So far, the Hamiltonian
does not describe a many-body system as the spins do not interact with each other. The
interaction is described by the disordered Ising Hamiltonian
HI=
N−1
X
i=1
Jσz
iσz
i+1 +
N
X
i=1
hiσz
iT1< t < T2.(9.1.2)
This Hamiltonian is formally the same as the second line in Equation 6.1.1 with nearest-
neighbor coupling (
α
= 1000). The nearest-neighbor interaction between different sites
results in entanglement between subsystems of the many-body system under time evolution.
The interaction is the crucial term for the time crystal to act against perturbation in the
driving. Furthermore, the on-site random disorder is chosen such that the spins are in a
many-body localized phase to prevent thermalization between subsystems of the many-body
system. This will be analyzed in detail in the following sections.
In Figure 9.1, the idea of the time crystal is shown. The spins are initialized in a random
initial state where all spins are polarized to either spin-up or spin-down. After applying
Chapter 9. Feedback-stabilized time crystal 105
Figure 9.1.:
Illustration of the 2
T
discrete time crystal. The spins are initialized in a random
initial state where each spin is polarized with either spin-up or spin-down.
After applying
HT
each spin is flipped to nearly the opposite polarization with
a small perturbation indicated by the tilted spins. When applying
HI
, the
interaction via
J
counteracts the perturbation in the driving and the spins are
polarized again to either spin-up or spin-down but with opposite polarization
compared to the initial state. The random on-site disorder
hi
prevents the
system from thermalizing with itself due to the interaction. After applying
again HTand HI, the system is found again in its initial configuration.
HT
(top left to top right), all local spins flip to the opposite polarization with a small
perturbation
, indicated by the tilted spins. Applying the Ising Hamiltonian
HI
(top right to
bottom right), interactions are induced by
J
(cp. first term in Equation 9.1.2). Furthermore,
the local spins are randomly disordered via
hi
. The interaction counteracts the perturbation
of the driving as the Ising Hamiltonian favors energetically spin-up or spin-down polarization
and not a superposition state. After applying again the driving Hamiltonian
HT
(bottom
right to bottom left), the spins are close to their initial configuration. However, the small
perturbation still results in not exactly the initial state. When
HI
is applied again, the
initial configuration is restored due to the Ising interaction. Thus, after two periods 2
T
, the
many-body system is found again in its initial configuration even if there is a perturbation
in the driving. This is the key idea of the discrete time crystal.
Before analyzing this in detail and explaining the physics behind this mechanism some
remarks about the computation of many-body spin systems in the matrix product state
description are necessary.
9.1.2. Computing many-body systems via matrix product states
The advantage in using the MPS formalism is that depending on the entanglement between
subsystems of the many-body problem, the scaling of the Hilbert space 2
N
is reduced to
an algebraic scaling. Hereby, the cutoff of small singular values is controlled within each
step of the computation such that the considered Hilbert space is adapted, depending on
the complexity of the given dynamics. Thus initially, all spins are written in an MPS as
introduced in chapter 7.
Chapter 9. Feedback-stabilized time crystal 106
To perform the time evolution of a closed system, the time-evolution operator acts on the
state
|ψ(tk+1)i=U(tk+1, tk)|ψ(tk)i.(9.1.3)
With the Hamiltonian in Equation 9.1.1 and Equation 9.1.2,
U
(
tk+1, tk
)is an operator
acting on all states of the MPS. This is in contrast to chapter 8, where the time-evolution
operator only acts on a single TLS and at most on two other physical states which are the
time-bins of the reservoir. There,
U
(
tk+1, tk
)was a tensor of rank six in case of feedback
dynamics (three ingoing and three outgoing physical indices).
In case of the many-body spin system, a straightforward tensor formulation of
U
(
tk+1, tk
)
would result in a tensor of rank 2
N
. It is clearly visible that nothing is gained in formulating
the Hamiltonian as a single tensor because the complexity still scales as 2
N
as each index
of the tensor of rank 2Nhas dimension of two.
The answer to that problem also lies in singular value decompositions (SVDs) of the
time-evolution operator. Similar to the decomposition of the coefficient matrix of the
corresponding many-body state
|ψi
to an MPS, the operator
U
(
tk+1, tk
)can be decomposed
to a matrix product operator (MPO), where each tensor only contains two physical indices
which was illustrated in Figure 7.8. Similar to an MPS formulation of
|ψi
, the Hilbert
space of
U
(
tk+1, tk
)is truncated by neglecting small singular values. Note that this induces
an error within each time step.
The advantage of formulating
U
(
tk+1, tk
)as an MPO is that it does not destroy the form
of the MPS when acting on the state
|ψi
. This is especially relevant when dealing with
a many-body problem, as with increasing
N
the Hilbert space grows exponentially and
it becomes impossible to write down the whole coefficient matrix of the state
|ψi
. Thus,
in practice, one never deals with the whole coefficient matrix but initializes it already
from the beginning as an MPS. The same holds for
U
(
tk+1, tk
)as it becomes exponentially
difficult to write down the whole tensor.
For the driving Hamiltonian
HT
this is straightforward as it only consists of local operators
σx
i
UT= exp −i∆t(Ω −)
N
X
i=1
σx
i!=
N
Y
i=1
exp (−i∆t(Ω −)σx
i).(9.1.4)
This means, to evolve a spin system in time with the driving Hamiltonian, one has to
perform
N
single-site operations as shown in Figure 7.9(a). The MPO contains
N
single-site
operators without links in between. This clarifies that
HT
does not introduce entanglement
between the spins as the link dimension of the MPS is not increased when UTis applied.
However,
HI
consists of two-site operators. Furthermore, all two-site operators are connected
with each other,
σz
1σz
2
is connected to
σz
2σz
3
and so on. This makes it impossible to write
down
UI
exactly in terms of single-site operators. As a solution, one could initialize the
whole tensor and then decompose it as shown for the MPS in Figure 7.5 but with two
physical indices per site. However, this involves the problem of an exponentially large initial
tensor UIfor large system sizes.
A solution to this problem is the Suzuki-Trotter decomposition. The idea is to write the
Chapter 9. Feedback-stabilized time crystal 107
sum in the exponential as a product of exponentials similar to Equation 9.1.4. To simplify
the notation, the two-site operator is written as
Zi
=
σz
iσz
i+1
. The exponential is written as
UI= exp −iJ∆t
N−1
X
i=1 Zi−i∆t
N
X
i=1
hiσz
i!
=
N−2
Y
i=1
exp (−iJ∆tZi−ihi∆tσz
i) exp −iJ∆tZN−1−ihN−1∆tσz
N−1−ihN∆tσz
N.
(9.1.5)
Note that the disorder is a single-site operation and thus for the last two-site operator
ZN−1, the disorder term hNσz
Nis included as well.
In general, for Hamiltonians such as the Heisenberg spin-chain in Equation 6.1.1, the
Suzuki-Trotter decomposition introduces an error, because it is required that the exponents
commute with each other to write it as a product of exponentials. However, for the problem
at hand, the approximation is exact as the different exponents commute with each other.
The error of the Suzuki-Trotter decomposition is given by
exp ˆ
A∆t+ˆ
B∆t−exp ˆ
A∆texp ˆ
B∆t=1
2ˆ
Bˆ
A−ˆ
Aˆ
B+. . . . (9.1.6)
By inserting
ˆ
A
=
−iJZi−ihiσz
i
and
ˆ
B
=
−iJZi+1 −ihi+1σz
i+1
, it can be shown that
Equation 9.1.6 yields zero as σz
icommutes for different sites i.
With this, the many-body Hamiltonian is expressed as a product of two-site operators.
However, in the following, the goal is to construct an MPO which only acts on single sites.
The reason is that this simplifies the application of
UI
on
|ψi
as it does not destroy the
MPS form and fewer SVDs are necessary for each time step.
With the product formulation in Equation 9.1.5 it is possible to split the time-evolution
operator into even and odd parts of Zi.
UI=
(N−1)/2
Y
i=1
UI(Z2i)
(N−1)/2
Y
i=1
UI(Z2i−1),(9.1.7)
where
UI(Zi)
are the exponentials which include the respective operators including the
part with the disorder. Note that the last term which includes the last sites
ZN−1
includes
two disorder terms as shown in Equation 9.1.5.
As the even and odd exponentials are written in a product form they can be applied after
each other on the MPS. This is exemplarily shown in Figure 9.2 for
N
= 6. First, all odd
parts of
UI
are applied on the MPS and it is contracted over all site indices
ni
as well as
the link indices which are part of the two-site operation. If one would directly apply the
even parts afterwards as shown in Figure 9.2 and contract over all primed site indices
n0
i
as well as the involved link indices, the MPS is destroyed and ends up in a tensor of rank
N
. This is not practical and becomes numerical impossible for growing system size
N
. One
possibility is to decompose the tensors after applying the odd parts of
UI
(
Zi
). However,
the approach chosen here is to further decompose the time-evolution operator to write it
as an MPO.
The decomposition is demonstrated in Figure 9.3. Starting with the odd part which is
the second product in Equation 9.1.7, each two-site operator
UI
(
Z2i−1
)is decomposed via
a SVD into two tensors acting on a single site
ni
. The resulting tensors are denoted as
Chapter 9. Feedback-stabilized time crystal 108
Figure 9.2.:
Application of the time-evolution operator on the MPS. When written in a
product form, it allows to apply even and odd parts of
UI
(
Zi
)after each other.
Each
UI
(
Zi
)contains a two-site operator acting on site
ni
and
ni+1
. Note that
for clarity for the even part, all indices are primed to avoid contraction over
the wrong indices. After the application of
UI
, all indices are unprimed to
restore the MPS.
Figure 9.3.:
Construction of the MPO for
UI
. The time-evolution operator is split into
even
UE
and odd
UO
parts of the two-site operators
Zi
. Each two-site operator
is decomposed via SVDs into local tensors. Due to the decomposition, the
decomposed tensors share a link index. This is between different tensors for
the even part (first line) and the odd part (second line). The MPO is then
constructed by contracting over all physical indices
n0
i
. Afterwards, the prime
level of
n00
i
is decreased by one to obtain the final MPO for
UI
. Note that on
the boundaries of
UE
, an identity tensor is applied for a simpler multiplication
within the numerics.
Chapter 9. Feedback-stabilized time crystal 109
UO
(
ni
), because each tensor now acts on a single site. For numerical practice, the diagonal
matrix containing the singular values
Sα
is multiplicated into both tensors with the entries
√Sα
which is why there is no OC in the MPO. Due to the SVD, the two tensors
UO
(
n2i−1
)
and
UO
(
n2i
)corresponding to the two-site operator
UI
(
Z2i−1
)now share a link index. This
can be seen in Figure 9.3 (first line), where every second pair of tensors share a link index.
The same procedure is repeated for the even part of
UI
. It is important to mention that
the link indices are between different pairs of tensors than for the odd part. Note that in
the diagrammtic representation, a contraction over the same indices is performed. This
is why the priming of the indices is relevant in this context. For this reason, the prime
level of all indices of
UE
(
ni
)is increased by one. Thus, with the notation here, the whole
time-evolution operator is reproduced when contracting over all shared indices.
Another important point for the numerical implementation are the identities at the
boundaries for the even part (cp. second line in Figure 9.3). When the total number of
spins
N
is even, the even part does not have any tensors for the boundary sites. When the
total number of spins
N
is odd, the first site of
UE
has no tensor and needs to be filled
with an identity as well as the last site of UO.
The MPO for
UI
is then constructed by contracting over all primed site indices
n0
i
as shown
in the last line of Figure 9.3. The identities at the boundaries leave the corresponding
tensors invariant but the multiplication can be formalized by writing
UO
and
UE
already as
MPOs. With this, an MPO is constructed with links between all tensors and two physical
indices at each site, one ingoing
ni
and one outgoing
n00
i
. The prime level of all outgoing
indices n00
iis then decreased by one for the final MPO.
Now both MPOs have been constructed. On the one hand
UT
(∆
t
), which consists of
N
local operations, and on the other hand
UI
(∆
t
), which is a standard MPO with local
tensors for each site and a link index between all neighboring sites. With this, the time
evolution can be computed by repeatedly applying the MPOs on the MPS. During
T1
, the
time evolution MPO of the driving Hamiltonian
UT
(∆
t
)and during
T2
the time-evolution
MPO of the Ising Hamiltonian UI(∆t)are applied on the MPS for each time step.
For
|ψ
(0)
i
, all spins are initialized in a polarized state with either spin-up or spin-down with
no entanglement between the sites. Thus, until the end of
T1
there exists no entanglement
between the spins and one deals with Nlocal operations at each time step.
The simulation of time evolution is shown in Figure 9.4. In a first step,
UT
(∆
t
)is applied
on the MPS of
|ψ
(0)
i
. Note that there are no links between the local operators of the
MPO. The MPS of the next time step
|ψ
(∆
t
)
i
is thus obtained by contracting over all
indices
ni
. Afterwards all indices are unprimed for the next application of
UT
(∆
t
). This
is repeated
T1/
∆
t
times until the driving period is finished and the state has evolved to
|ψ
(
T1
)
i
. Then the Ising Hamiltonian in
UI
(∆
t
)is applied on the system. In general, this is
the same as for
UT
(∆
t
), by repeatedly applying
UI
(∆
t
), the system is evolved to
|ψ
(
T
)
i
and a full period has been computed. However, due to the link indices between the tensors
of
UI
(∆
t
), its application on the MPS is not as straightforward as for
UT
(∆
t
)to maintain
the MPS form of the state.
An efficient application of
UI
(∆
t
)on
|ψ
(
T1
)
i
is explained in the appendix in section B.1.
The algorithm is shown in diagrammatic form in Figure B.1.
By repeating this procedure, an arbitrary number of periods can be simulated with the
MPS formalism. However, with each application of
UI
, the link dimension of the MPS grows.
By truncating the number of singular values accordingly, a large number of periods can be
simulated, especially for the MBL phase where the entanglement only grows logarithmically.
Of course, the total number of periods is still limited as within each time step an error is
Chapter 9. Feedback-stabilized time crystal 110
induced by the truncation of singular values and the finite order of the expansion of the
time-evolution operator. However, with the method introduced in this section, it is possible
to simulate a satisfying number of periods to discuss the DTC in a closed quantum system.
Figure 9.4.:
Time evolving the MPS for one period
T
=
T1
+
T2
. First, the initial state
|ψ
(0)
i
is evolved to
|ψ
(∆
t
)
i
in applying
UT
(∆
t
).
UT
(∆
t
)consists of
N
single-
site operators, which is why no entanglement between the sites is induced
and the links between the tensors do not change. However, the respective
site tensors change according to
UT
(∆
t
). The time-evolved state
|ψ
(∆
t
)
i
is
simply obtained by contracting over all
ni
. The prime level of the indices of the
resulting MPS is decreased by one afterwards. The MPS is ready for the next
time step. By repeatedly applying
UT
(∆
t
)on the MPS, the time is evolved to
the end of the first period
T1
. Afterwards, during
T2
,
UI
(∆
t
)acts repeatedly
on
|ψ
(
T1
)
i
to arrive at the end of the period
T
=
T1
+
T2
. Due to the link
indices of
UI
(∆
t
), its application is not as straightforward as for
UT
(∆
t
)and
is explained detailed in Figure B.1. Note that within each time step ∆
t
, the
link dimension of
UI
(∆
t
)is added to the existing link dimension between each
elements of MPS.
Chapter 9. Feedback-stabilized time crystal 111
9.1.3. Achieving a discrete time crystal
Having demonstrated the numerical tools to simulate the time evolution of the driven
disordered Ising spin-chain, the behavior of the system for different parameter setups is
investigated with focus on time crystal behavior.
The system size considered is
N
= 10. To simplify the display of all spins, the initial state
is not set to a random polarization of each spin, but the Néel state where spin-up and
spin-down polarization are alternating from site to site
|ψ
(0)
i
=
| ↑↓↑ . . . i
. The observable
of interest is the magnetization of each spin, but to consider all spins in a single observable,
the staggered magnetization is convenient
hMi=1
N
N
X
i=1
(−1)ihσz
ii
2.(9.1.8)
The alternating minus sign is chosen in correspondence to the initial Néel state. Thus,
initially hMi=−1/2.
The single-site magnetization is computed according to Figure 7.11. Note that for the
computation of each spin magnetization, the orthogonality center (OC) is shifted to the
respective position before computing the observable. Afterwards, all single-site magnetiza-
tions hσz
iiare added up according to Equation 9.1.8.
In this section, the parameters leading to the time crystal phase of the Hamiltonians
in Equation 9.1.1 and Equation 9.1.2 are investigated in detail. It is convenient to start
with the most simple scenario with Ω =
π/
2and all other parameters set to zero, i.e the
perturbation
= 0, the interaction
J
= 0 and the disorder
hi
= 0. Initially, all spins show
hMi
=
−
1
/
2. During the driving period
T1
with Ω =
π/
2, all spins flip to the opposite
polarization and the system shows
hMi
= 1
/
2. During the period
T2
, where the Ising
Hamiltonian is applied, the MPO consists of identities and the system is left in its state
with
hMi
= 1
/
2, because
J
= 0 and
hi
= 0. Thus, after a full period, the system is in the
state of opposite polarization. After another period at
t
= 2
T
, the system is again found
in the initial state with
hMi
=
−
1
/
2as the driving Hamiltonian induces a spin-flip of
each spin and the Ising Hamiltonian has no effect as all parameters are set to zero. Thus,
after 2
T
, the system always ends up in the initial state. This effect of the driving with
all other parameters set to zero is shown in Figure 9.5. In the inset, the time evolution
of the staggered magnetization
hMi
is shown for
T
= 10 periods. It becomes clear that
after 2
T
the spins are found in the same state. The main plot shows the Fourier spectrum,
computed over
T
= 500 periods. The system locks to the subharmonic frequency of
ω
= 1
/
2
according to the 2Tperiodicity of the driving.
The question is, if this is already a time crystal as a many-body spin system returns after a
discrete period back to its initial state. The answer is no, for two reasons. On the one hand,
it is not really a many-body system as it consists of isolated spins which do not interact
with each other as
J
= 0. Each individual spin is flipped and the observable is computed
adding up spin magnetizations, but due to the lack of interactions, the individual spins do
not produce entanglement entropy.
On the other hand, it is not a time crystal because of the sensibility to perturbations
. In
Figure 9.6, the response of the systems to a perturbation in the driving with
= 0
.
015 is
shown. In the inset it becomes clear that there is another enveloping oscillation of
hMi
due to 6= 0. The Fourier spectrum is again computed over T= 500 periods. The system
is no longer locked at the subharmonic frequency
ω
= 1
/
2, because the perturbation in
the driving results in a peak splitting. Furthermore, the height of the peak decreases. The
Chapter 9. Feedback-stabilized time crystal 112
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 2 4 6 8 10
Fourier spectra
ω
hMi
periods T
-0.5
0
0.5
0 2 4 6 8 10
Figure 9.5.:
Fourier spectrum of the Floquet driven spin-chain. During
T1
,
UT
is applied
with Ω =
π/
2and
= 0. During
T2
, where usually
UI
is applied nothing
happens as
J
= 0,
h
= 0 and
UI
consists of identities. The spin system shows
perfect oscillations, where after each 2
T
, the system is found again in its initial
state (inset). Thus, the Fourier spectrum shows
ω
= 0
.
5. Note that this is not
a many-body system as the spins are not interacting.
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 20 40 60 80 100
Fourier spectra
ω
hMi
periods T
-0.5
0
0.5
0 20 40 60 80 100
Figure 9.6.:
When perturbations are turned on
= 0
.
015, the spins do not end up in their
initial state after
T
= 2. This results in a peak-splitting of the Fourier spectra.
The enveloping oscillation induced by the perturbation can be seen in the
inset.
Chapter 9. Feedback-stabilized time crystal 113
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 20 40 60 80 100
Fourier spectra
ω
hMiav
periods T
-0.5
0
0.5
0 20 40 60 80 100
Figure 9.7.:
Investigating the effect of disorder without interactions
J
= 0, it becomes clear
that the system is also perturbed by the random disorder. The perturbation
changes (inset) and the peak height decreases with still a splitting of the peak.
system simply follows the external dictated frequency of the driving (Ω −).
Focusing now on the effect of the Ising Hamiltonian with the driving Hamiltonian remain-
ing at Ω =
π/
2and
= 0
.
015, the starting point is to investigate the effect of disorder
hi
without interactions
J
= 0. Each single-site disorder is chosen randomly with
hi∈
[0
,
7],
where it is known that the spin system in case of interactions is in a many-body localized
phase [
223
]. As in section 6.4, for each random disorder realization, the time evolution and
the according observable
hMi
of the system is computed until
T
= 500 periods. Then, a
new system is initialized with random disorder and the time evolution is computed. After
finishing the computation, the observable
hMi
is averaged over the performed disorder
realizations Nav for every time step
hMiav(t) = 1
Nav
Nav
X
i=1hMii(t).(9.1.9)
If not denoted otherwise, the number of disorder realizations will be chosen as
Nav
= 10
3
.
In Figure 9.7, the effect of disorder on the non-interacting driven system is shown. Due
to the random disorder on each site, the spins precess with different Larmor rates and
dephase with respect to each other [
42
]. This can be observed in the inset of Figure 9.7,
that
hMiav
is no longer only following the driving but has another enveloping oscillation in
comparison to Figure 9.6. Thus, the splitting of the peaks is slightly different. Furthermore,
the averaged staggered magnetization approaches zero for long times
hMiav →
0due to
the random disorder and a different response of each spin to the driving. At a first glance
it seems like random on-site disorder does not favor the periodic Floquet driving and
counteracts any time crystal behavior. However, it will be shown in the following that the
opposite is the case: Random disorder leading to many-body localization is an important
ingredient to achieve a stable DTC for long times [223].
Before coming to that, disorder is turned off with
hi
= 0 and the interactions of the
Ising Hamiltonian are turned on with
J
= 0
.
02 to deal with a many-body system. These
Chapter 9. Feedback-stabilized time crystal 114
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 20 40 60 80 100
Fourier spectra
ω
hMi
periods T
-0.5
0
0.5
0 20 40 60 80 100
Figure 9.8.:
Considering a real many-body system in turning on the interactions
J
= 0
.
02,
the Ising interaction counteracts the perturbation and the frequency is locked
at
ω
= 0
.
5. However, interaction results in a build up of entanglement between
the spins which is why they deviate from a polarized state. The peak height
decreases and
hMi →
0(inset). Note that due to the small system size (
N
= 10)
this thermalization process is not as strong as for larger systems.
interactions are crucial to form the DTC as on the one hand it is now a many-body system
and on the other hand, the interactions counteract the small perturbation in the driving.
In Figure 9.8, the observation of a time crystal is shown. Even in the presence of per-
turbations in the driving
6
= 0, the Fourier spectrum is locked at the subharmonic
frequency
ω
= 1
/
2, meaning that after 2
T
, the system is found again in its initial state
with
hMi
=
−
1
/
2. Thus, collective synchronization of all spins take place due to the Ising
interaction
J
, which counteracts the perturbation in the driving. The system favors energet-
ically a polarization of spin-up or spin-down after applying the Ising Hamiltonian. However,
it becomes apparent that the peak height of the Fourier spectrum is smaller than the one
without perturbation and interactions in Figure 9.5. This is caused by the interaction
J
.
After the Floquet period
T1
, the spins are not in a polarized state due to the perturbation
6
= 0. Therefore, the Ising Hamiltonian induces entanglement between all spins due to the
interaction. This buildup of entanglement results in
hMi →
0(inset Figure 9.8) as all spins
approach a maximally entangled state for
t→ ∞
. This is a thermalization process within
the many-body system. The Ising Hamiltonian introduces entanglement of a single spin
with the rest of the many-body system. In the simulation, this can be directly observed
in the growing number of relevant singular values and the resulting dimension of the link
indices between the sites. Thus, when measuring a single-site observable such as
hσz
ii
,
the total information is hidden within all other involved spins. This results in a state
of maximal entanglement entropy for a large number of periods
T→ ∞
. Approaching
the thermodynamical limit with
N→ ∞
, it is a thermalization process of all subsystems
with the rest of the many-body system and the staggered magnetization approaches zero
hMi →
0. This would destroy the time crystal already at comparable small times which
can be seen in the comparably small peak height of the Fourier spectrum and the decay of
hMi
in Figure 9.8. Note that for the considered system size
N
= 10, this thermalization
Chapter 9. Feedback-stabilized time crystal 115
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 20 40 60 80 100
Fourier spectra
ω
hMiav
periods T
-0.5
0
0.5
0 20 40 60 80 100
Figure 9.9.:
Turning on random disorder such that the system is in a many-body localized
phase and considering interactions
J
= 0
.
02, a stable 2
T
discrete time crystal
is created even with perturbation in the driving. The frequency is locked at
ω
= 0
.
5and the peak height is almost as high as for the non-interacting case
without perturbation (Figure 9.5). The many-body localization counteracts
the thermalization due to spin interactions and the DTC is stable for long
times (up to T= 500 investigated).
process is not as grave as for larger systems, as the total entanglement is limited due to
the small system size compared to
N→ ∞
. Furthermore, the magnetization is subject to
revivals for longer integration times which is also due to finite size effects.
To prevent this thermalization, the random on-site disorder comes into play. Now turning on
interactions as well as random on-site disorder, a stable DTC is created for long times which
can be seen in Figure 9.9. The Fourier spectrum shows a clear peak at the subharmonic
frequency
ω
= 1
/
2, corresponding to the return to the initial state of the many-body system
after 2
T
periods. The peak height is almost as high as for the case without interactions
and perturbations in Figure 9.5. The reason for a slightly smaller peak height is a small
decay of
hMi
which can be seen in the inset of Figure 9.9. However, the decay is small in
comparison to the case without disorder in Figure 9.8 and stabilizes at high values of
|hMi|
after
T≈
40 periods. Note that this oscillations are stable up until the end of the time
integration which is
T
= 500. Thus, the Ising interaction counteracts the perturbation in
the driving
via
J
. A reason might be that the Ising Hamiltonian favors energetically full
polarized spins. However, when the spins are not exactly in a full polarized state, the Ising
interaction introduces entanglement between the subsystems. MBL achieved by turning on
random on-site disorder, prevents this entanglement growth and the system shows time
crystal behavior for long times.
To conclude, having turned on all parameters and chosen such that the system exhibits a
DTC [
223
] it is observed that the interacting many-body system shows a discrete periodicity
where it returns always after 2
T
back to its initial state. The time translational symmetry
of the Hamiltonian
H
(
t
) =
H
(
t
+
T
)is broken as the expectation value shows periodicity
for twice the period
hMi
(
t
) =
hMi
(
t
+ 2
T
). This is stable against perturbations in the
driving
6
= 0, where without interactions the system would follow the period dictated by
Chapter 9. Feedback-stabilized time crystal 116
the driving (Figure 9.6). Due to the interaction, all spins synchronize collectively and favor
the polarized state. Many-body localization due to random on-site disorder prevents the
system from going into thermal equilibrium for long times.
9.2. Open system dynamics
Having shown how a stable time crystal is achieved in case of a closed quantum system, the
question of interest is what happens in case of an open quantum system. It has been shown
that without MBL, the closed quantum system is subject to thermalization with the closed
many-body quantum system itself due to the Ising interaction. This results in a decay
of the staggered magnetization
hMi
for long times due to the buildup of entanglement
between the spins.
It has been shown by Ref. [
224
] that a time crystal as open quantum system with any
reasonable Lindblad dissipator is subject to thermalization with its environment and melts
for long times with
hMi →
0. Systems subject to Lindblad dissipators were investigated in
Part II of this thesis for the many-emitter phonon laser in chapter 5 and the Heisenberg spin-
chain in chapter 6. The focus of this chapter lies in the computation of an open quantum
system in the QSSE picture introduced in chapter 3 without tracing out the reservoir
degrees of freedom leading to Lindblad dissipators. Thus, the subject of investigation is
the entanglement between many-body system and the surrounding reservoirs. It has been
shown in the previous section and argued in chapter 4 that many-body localization prevents
the many-body system from building up entanglement within the many-body system itself
and thus, prevents it from reaching thermal equilibrium.
An interaction of the many-body system with a surrounding reservoir, which is modeled as
a mode continuum, leads to a build up of entanglement of the many-body system with
the reservoir states. The idea of this chapter is to prevent the buildup of entanglement
by structuring the external reservoir. In including a boundary condition and a sinusoidal
coupling to the environment, similar to chapter 8, it will be shown that feedback dynamics
stabilize a dissipative time crystal which interacts with the surrounding environment.
Feedback dynamics can counteract a building up of entanglement of the many-body system
with the surrounding reservoir in addition to many-body localization which prevents the
build up of entanglement within the many-body system itself.
9.2.1. Reservoir model
To model an external reservoir, the idea is to assume the spins as single TLS which are
subject to a radiative decay Γ, similar to the investigated TLS in chapter 8. Thus, each spin
is coupled to an external mode continuum in analogy to Equation 8.1.6. The dissipative
Hamiltonian is therefore modeled as
HD=
N
X
j=1 Zdω~ωb†
j(ω)bj(ω) +
N
X
j=1 Zdω~hGj(ω)b†
j(ω)σ−
j+h.c.i.(9.2.1)
Initializing the reservoir in a vacuum state, the excitation of each single TLS is subject
to radiative decay, where an excitation of an individual TLS is annihilated and inside
the respective reservoir an excitation is created. In the spin-picture, the reservoir would
correspond to a spin-down reservoir where any spin-up polarization of the system is subject
to interaction with the spin-down polarization of the external reservoir. Note that by
introducing the time-bin basis in Equation 3.1.16, the number basis in the computation
Chapter 9. Feedback-stabilized time crystal 117
Figure 9.10.:
Model of the spin-chain coupled to external structured reservoirs. Each spin
is subject to a decay into its individual environment which would be a
spontaneous emission process in case of a TLS coupled to a photonic reservoir.
The spins are coupled to each other with Ising interaction
J
. Each individual
reservoir contains a boundary condition which is sketched as a semi-infinite
waveguide, where excitations are reflected and interact again with the system
after time-delay
τ
. It is differentiated between a decay to the infinite side of
the waveguide ΓLand the side with the closed end ΓR.
is restricted to two states. Thus, it is not differentiated between bosons or fermions in
modeling the reservoirs. However, for simplicity and in analogy to chapter 8 it is convenient
to imagine the spins as TLS with a photonic reservoir.
The coupling is modeled in analogy to chapter 8 as
Gj(ω) = i
sΓR
2πe−iωτj/2−sΓL
2πeiωτj/2
,(9.2.2)
where a boundary condition leads to the sinusoidal dependence on the distance. This results
in an exponential form of the dependence on the time-delay
τj
= 2
Lj/co
, where
c0
would
be the speed of light in vacuum in case of a photonic reservoir. Note that this changes if
the reservoir consists of magnetizations. In case of a single TLS with a photonic reservoir,
the boundary condition is realized as a reflecting mirror. This chapter serves as a proof of
principle that such a sinusoidal coupling constant leads to a stabilization of a dissipative
time crystal for long times.
The model setup is sketched in Figure 9.10. Each individual spin is put into a semi-infinite
waveguide, to model the individual reservoir including a boundary condition which is the
closed end of the waveguide. The Ising interaction is denoted as
J
between the different
TLS. The interaction to the environment takes place during the whole period
T
=
T1
+
T2
,
which includes the driving Hamiltonian during T1and the Ising Hamiltonian during T2.
Chapter 9. Feedback-stabilized time crystal 118
9.2.2. Computing open quantum many-body systems in the QSSE picture
By going into the QSSE picture in introducing quantum noise operators ∆
B(†)
nj
(∆
t
)with
time increment ∆
t
=
tk+1 −tk
, the time-evolution operator for the dissipation reads similar
to Equation 8.1.11
UD(tk+1, tk) =
N
Y
j=1
exp h−pΓR∆Bnj(tk−l)e−iφnj+pΓL∆Bnj(tk)σ+
j+h.c.i.(9.2.3)
In introducing the quantum noise operators, the time-bin basis is constructed similar to
Equation 3.1.16
|inj
ki=∆B†
nj(tk)inj
k
qinj
k!(∆t)inj
k|vaci.(9.2.4)
The difference to chapter 8 is that more than a single reservoir is considered, i.e. one
reservoir for each spin of the chain. Thus,
N
reservoirs are considered, where the upper
index njlabels the corresponding reservoir for the j-th spin.
Due to the additional reservoirs for each spin and the exact consideration of the reservoir
degrees of freedom within the MPS formalism, the total MPS is now two-dimensional. Each
spin of the chain and with this, each tensor of the system MPS is coupled to an additional
MPS which represents the reservoir. This is shown in Figure 9.11, where the tensors of
the reservoir MPSs are sloped and dashed in the diagrammatic representation. Due to
the two-dimensional form of the MPS it is even more important to keep track of the OC
to calculate expectation values. Before explaining the algorithm and the computation of
observables, some remarks about the construction of the MPOs when including dissipation
in the QSSE picture are in order.
By applying ∆
B†
nj
(
tk
)on the state, an excitation is created in the time-bin
inj
k
corresponding
to the reservoir of the spin
nj
at time
tk
. Note that the first summand in the exponent
of Equation 9.2.3 corresponds to an interaction with the feedback time-bin
inj
k−l
with Γ
R
as the radiative decay to the right side of the waveguide which includes the reflecting
element. Thus, when the reservoir is assumed to be in general form, i.e. only dissipation
without feedback dynamics, Γ
R
is set to zero such that the system only interacts with
future time-bins inj
kand not with past time bins such as inj
k−l.
The starting point is the construction of the MPO for general (unstructured) reservoirs
without a feedback algorithm. During the driving period this is straightforward as the
driving Hamiltonian only consists of local operators
UT−D =
N
Y
j=1
exp −i∆t(Ω −)σx
j−pΓR∆Bnj(tk−l)e−iφnj+pΓL∆Bnj(tk)σ+
j+h.c.,
(9.2.5)
where Γ
R
is set to zero for a general (unstructured) reservoir coupling. The time-evolution
operator is then expanded until the second order in ∆
t
, similar to Equation 8.1.15, due to
the proportionality of ∆
B(†)
nj
(∆
t
)
∼√∆t
. The MPO then consists of
N
local operations
on the system MPS. Note that local is only referred to the spin system as each operator
does interact with two tensors: One is the respective system-bin of the spin system
nj
and
the other one is the respective time-bin of the reservoir
inj
k
. Thus, each local operator is a
Chapter 9. Feedback-stabilized time crystal 119
Figure 9.11.:
General form of the MPS in mixed canonical gauge in case of an individual
reservoir for each spin. The notation of the many-body spin system is chosen
as explained within the previous chapters, with vertical lines corresponding
to physical indices and horizontal lines corresponding to link indices. The
black box denotes the OC (at the tensor corresponding to the physical index
n1
), blue boxes show right orthogonality and green boxes left orthogonality.
As each spin is coupled to a reservoir, the MPS is no longer one-dimensional,
thus each spin is linked to an additional MPS which represents the respective
reservoir. The reservoir MPSs are denoted as dashed sloped boxes. The
physical indices are labeled with the time-bin bases
inj
k
, whereas the link
indices are not labeled in this picture. Orthogonality is denoted as for the
many-body system. Note that the feedback time-bins have the physical index
inj
k−l.
Chapter 9. Feedback-stabilized time crystal 120
Figure 9.12.:
MPO for the feedback driving time-evolution operator
UT−D
(top) and for the
feedback Ising time-evolution operator
UI−D
(bottom). Each local operator
(acting only on one site of the spin-chain) consists of six physical indices, three
ingoing and three outgoing. One physical index describes interaction with
the system-bin
nj
, one describes interaction with the future time-bin
inj
k
and
one describes interaction with the feedback time-bin
inj
k−l
. Note that
UI−D
has link indices between each local operator due to the Ising interaction.
tensor of rank four with two ingoing and two outgoing physical indices.
The Ising time evolution during
T2
is slightly more complicated, as it consists of two-site
operations. However, due to the assumption of individual reservoirs for each spin for the
dissipation in Equation 9.2.3, the dissipation consists of local operations on the spin-chain.
Thus, in performing the Suzuki-Trotter expansion as depicted in Figure 9.3, the MPO is
constructed from
UI−D = exp
−iJ∆t
N−1
X
j=1
σz
jσz
j+1 −i∆t
N
X
j=1
hjσz
j
−
N
X
j=1 pΓR∆Bnj(tk−l)e−iφnj+pΓL∆Bnj(tk)σ+
j+h.c.
,(9.2.6)
where the dissipation is included for each local operator acting on the spin-chain. The time-
evolution operator is then expanded until the second order in ∆
t
, similar to Equation 8.1.15,
to have the same error in ∆
t
as for applying the driving time-evolution operator. As for the
driving time-evolution operator
UT−D
, each operator of the MPO of
UI−D
now consists
of four physical indices, two ingoing and two outgoing. One is acting on the respective
system-bin of the spin-chain and one on the future reservoir time-bin
inj
k
. In contrast to
the dissipative driving time-evolution operator
UT−D
, the dissipative Ising time-evolution
operator
UI−D
has a link index between each operator due to the two-site operation of the
Ising Hamiltonian as discussed in subsection 9.1.2.
If a structured reservoir is considered with Γ
R>
0, the construction of the MPOs is
basically the same as for a general reservoir. The difference is that each tensor has a higher
rank due to the additional interaction with past time-bins. Each local tensor consists of
six physical indices similar to Figure 8.3. The MPOs used for the feedback algorithm for
Chapter 9. Feedback-stabilized time crystal 121
Figure 9.13.:
MPS for the application of a dissipative time-evolution operator
UD
without
feedback dynamics. Each local tensor consists of two physical indices, one
for the respective site index
nj
(vertical) and one for the future time-bin
inj
k
(sloped). Furthermore, each tensor has four link indices (three if it is a
boundary site). Two for the neighboring sites (horizontal) and two to the
respective reservoir MPS (sloped). The upper link index to the reservoir
connects to future time-bins which is why this link index has dimension one.
The sloped index pointing down right connects to past time-bins.
UT−D and UI−D are shown in Figure 9.12.
Having now constructed the MPOs for the two time-evolution operators, including dissipa-
tive dynamics, it is possible to compute the time evolution of the system by subsequent
application of the MPOs. The starting point is a general reservoir with Γ
R
= 0, where each
operator of the MPOs acts on one site index
nj
and on the respective future time-bin
inj
k
.
To apply the time-evolution operator, the future time-bins of the reservoirs are multiplied
into the respective tensor of the spin MPS. Thus, each local tensor with site index
nj
is
multiplied with the future time bin
inj
k
by contracting over the shared link index. The
resulting MPS is shown in Figure 9.13. Each local tensor has the respective site index
nj
and the future time-bin index
inj
k
. Furthermore, each tensor shares two link indices
with the respective reservoir, one to future time-bins and one to past time-bins. Note that
orthogonality is not guaranteed in this case as the future time-bins have been multiplied
into the system MPS. Besides that, the MPS is in a standard form as in Figure 9.4. Thus,
the application of
UT−D
during
T1
and
UI−D
is straightforward. However, there is still
an important difference, compared to closed system dynamics, after each application of a
time-evolution operator. The tensors still include the reservoir time-bin which is now the
past time-bin. Thus, each tensor is decomposed and the tensor including the past time-bin
index is stored in the respective reservoir MPS. The OC is kept at the system-bin during
this decomposition such that the MPS is re-gauged.
After applying the respective time-evolution operator and decomposing the tensors of the
system MPS, the total two-dimensional MPS should be again in the form of Figure 9.11.
This is the point to compute observables concerning the system such as the magnetization
hσz
ji
in Figure 7.11 as the MPS is in a mixed canonical form and the OC is at the system
MPS.
There is still one step missing before computing the next time step. The MPO for the
time-evolution operator still acts on the time-bin
inj
k
, which is now a past time-bin. For the
next time step, they are supposed to act on the future time-bin
inj
k+1
. As the MPO itself
does not change but for the physical indices, this is achieved by multiplicating each tensor
of the MPO with two identity tensors, each containing two indices one with
inj
k
and
inj
k+1
Chapter 9. Feedback-stabilized time crystal 122
and the other identity tensor contains
i0nj
k
and
i0nj
k+1
. The past time-bin indices
inj
k
and
i0nj
k
are then replaced by the future time-bin indices
inj
k+1
and
i0nj
k+1
, respectively. With this, the
MPO then act on the future time-bin inj
k+1 after contracting over inj
kand i0nj
k.
With these considerations, it is possible to compute dissipative time evolution for the
DTC in the QSSE picture. However, for feedback dynamics, the algorithm becomes more
complicated as swap operations have to be performed for each reservoir.
Before
tk
=
τ/
∆
t
, the time evolution is computed in the same manner as for the general
reservoir, described above. As all reservoirs are initially assumed to be in a vacuum state,
all past time-bins are not occupied and there is no interaction with feedback time-bins.
However, after
tk
=
τ/
∆
t
is reached, the past time-bins interact subsequently again with
the system due to the reflecting element. Thus, the algorithm has to be adapted for the
structured reservoir to deal with the memory kernel. The algorithm is comparable to sub-
section 8.1.2, with the difference that the considered system is an interacting many-body
system. Therefore, as mentioned above, the MPS becomes two-dimensional and for each
reservoir, swap operations have to be performed, as shown in Figure 7.13, to deal with the
long-range interaction of the system with feedback time-bins.
Assuming at
tk
=
τ/
∆
t
, the two-dimensional MPS is in mixed canonical gauge in the form
of Figure 9.11. If this is not the case it can be brought to this form by performing series
of contractions and SVDs of neighboring tensors. Thus initially, the OC is at the tensor
with physical index
n1
. The OC is brought to the feedback time-bin
in1
k−l
by contracting
neighboring tensors of the reservoir MPS of the first site and shifting the OC in direction
of the feedback time-bin by performing SVDs. When the OC is at the feedback time-bin
in1
k−l
,
τ/
∆
t
swap operations (because the memory kernel consists of
τ/
∆
t
time-bins) are
performed to bring the feedback time-bin next to the system-bin
n1
. In doing so, the OC
is kept at the feedback time-bin
in1
k−l
. When the feedback time-bin is next to the system,
the OC is brought first to the system-bin
n1
, then to the system-bin
n2
and then to the
feedback time-bin of the second site
in2
k−l
. Then this procedure is repeated until arriving at
the last site
nN
. The MPS is then in the form of Figure 9.14 (top), where each system-bin
njhas the future time-bin inj
kand the feedback time-bin inj
k−las neighbors.
The task then is to bring the MPS in a form to apply the respective time-evolution MPO,
either
UT−D
or
UI−D
in Figure 9.12. Thus, each tensor of the system MPS with index
nj
is multiplied with the tensor of the future time-bin
inj
k
and the feedback time-bin
inj
k−l
in
contracting over the shared link index. The system MPS is then in the form of Figure 9.14
(bottom), where each tensor has three physical indices (
nj
,
inj
k
and
inj
k−l
) as well as two
link indices to neighboring sites of the many-body system (only one if its a boundary
site) and two link indices to the respective reservoir, one to future time-bins (top-left
for each tensor in Figure 9.14) and one to past time-bins (bottom-right for each tensor
in Figure 9.14). With this, the respective MPO is applied on the MPS, shown for
UI−D
in Figure 9.14 (bottom). The application of the MPO is similar to Figure B.1, but with
three physical indices per tensor. After applying the MPO, the prime level of all physical
indices is decreased by one. Then, each tensor is decomposed according to its physical
indices. Thus, each tensor is decomposed into three tensors, where one contains the system
index
nj
as well as the links to neighboring sites
nj−1
and
nj+1
and the link to the now
future time-bin
inj
k+1
. The second tensor consists of the physical index
inj
k
and two new link
indices resulting from the decomposition. The third tensor contains the feedback time-bin
index
inj
k−l
, and the link to the time-bin
inj
k−1
which is the neighboring past time-bin in
the reservoir MPS. After the decomposition, the MPS should look like in Figure 9.15,
where each system-bin is linked to the now past time bin
inj
k
, the time-bins
inj
k
are linked
Chapter 9. Feedback-stabilized time crystal 123
Figure 9.14.:
Feedback algorithm for the many-body system coupled to
N
reservoirs. First,
perform swap operations to bring each feedback time-bin
inj
k−l
next to the
respective system-bin
nj
(the starting point is the MPS in Figure 9.11). Note
that the OC is kept at the respective feedback time-bin and then multiplied
back into the system. Then for computing the swap operations of the next
reservoir it is put into the following site
ni+1
and brought to the feedback
time-bin
inj+1
k−l
. Afterwards, each feedback time-bin
inj
k−l
as well as each future
time-bin
inj
k
is multiplied into the system MPS by contracting over the shared
link indices. The respective MPO is applied, here shown for the Ising time-
evolution operator
UI−D
. Then by contracting over all
nj
,
inj
k
and
inj
k−l
, the
MPS for the next time step is computed in an efficient manner as depicted
in Figure B.1. The prime level of all indices is decreased by one after the
application of the MPO. Afterwards each tensor is decomposed to split the
now past time-bin
inj
k
and the past feedback time-bin
inj
k−l
from the system
MPS. Both are then stored in the respective MPS of the reservoirs.
Chapter 9. Feedback-stabilized time crystal 124
Figure 9.15.:
MPS after the application of either
UT−D
or
UI−D
. The former future time-
bins
inj
k
are now past time-bins after the application of the MPO and are thus
stored in the reservoir MPS below the system-bin. The feedback time-bin
inj
k−l
is stored below the time-bin
inj
k
and is linked to the past time-bin
inj
k−1
.
Now the MPS is re-gauged after the application of the MPO and ready for
performing swap operations to bring the feedback time-bins
inj
k−l
back to their
original positions.
to the feedback time-bins
inj
k−l
and the feedback time-bins are linked to
inj
k−1
. Note that
for simplicity, the OC is at the system-bin
n1
but might also be at the system-bin
nN
dependent on from which side the MPO has been applied.
Before calculating observables, it is convenient in the algorithm to perform first the swap
operations. Thus, the OC is brought to the feedback time-bin
in1
k−l
and afterwards, the
feedback time-bin is brought to its original position by performing swap operations. In the
last step, the OC is stored in the future feedback time-bin
in1
k−l+1
. This future time-bin is
then brought next to the system-bin
n1
by performing swap operations. Note that this
speeds up the algorithm as one saves two times shifting the OC through the reservoir MPSs.
The same procedure is repeated for each reservoir. After the swap operations, the MPS
looks like in Figure 9.16. Each system-bin
nj
has the future time-bin
inj
k+1
and the future
feedback time-bin
inj
k−l+1
as neighboring tensors for the computation of the next time step.
Furthermore, the MPS is in mixed canonical gauge which is why system observables such
as the magnetization
hσz
ji
, shown in Figure 7.11, are computed for the time
tk
at this point.
As done for a general reservoir, the MPOs have to be adapted for the computation of the
next time step. This is done by replacing the index
inj
k
with the index
inj
k+1
and replacing
inj
k−l
with
inj
k−l+1
and the same for the primed outgoing indices. This is achieved in multiplying
the MPOs with identities containing the index of the past time step as well as the index
for the future time step as explained above. Then, the MPS and the MPO are ready for
the next time step and the procedure is repeated starting with Figure 9.14.
Chapter 9. Feedback-stabilized time crystal 125
Figure 9.16.:
After the application of the MPO in Figure 9.14 and the decomposition of
the tensors, the feedback time-bin
inj
k−l
is brought back to its original position
by performing swap operations. In doing so, the OC is kept at the feedback
time-bin
inj
k−l
. When arriving at its original position, the OC is then multiplied
into the future feedback time-bin
inj
k−l+1
. Then, the future feedback time-bin is
brought next to the system bin
nj
by performing swap operations. Afterwards,
the OC is brought to the system-bin and then brought to the past feedback
time-bin of the next reservoir
inj+1
k−l
and the same procedure is repeated until
arriving at the end of the chain
nN
. Then, observables concerning the system
are computed as the MPS is in a mixed canonical gauge. The MPS is then
ready for the application of the MPO for the next time step.
Chapter 9. Feedback-stabilized time crystal 126
9.2.3. Stabilizing a dissipative discrete time crystal
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 25 50 75
Fourier spectra
ω
Γ=0.001
Γ=0.005
Γ=0.010
Γ=0.015
hMi
periods T
-0.5
0
0.5
0 25 50 75
Figure 9.17.:
Computation of a dissipative time crystal within the QSSE picture for a
general (unstructured) coupling to the reservoir. The staggered magnetization
hMi
is subject to a decay with
hMi →
0. This decay increases for increasing
coupling to the reservoir Γ(inset). As a consequence, the peak height of the
Fourier spectrum decreases with increasing Γ. The decay of
hMi
results from
building up entanglement between the system and the reservoir states which
is why the computation becomes very challenging already for small numbers
of total time steps (
T
= 75 periods computed). Parameters are chosen as
before
N
= 10,∆
t
= 0
.
005, svdcutoff= 10
e−
9for the spin system and
svdcutoff= 10
e−
6between system and reservoir states, Ω =
π/
2,
= 0
.
015,
J
= 0
.
02,
hi∈
[0
,
7] and the number of averages between
Nav
= 10
2
and
Nav = 103.
Having developed an algorithm to compute a dissipative time crystal in the QSSE picture,
the interaction and buildup of entanglement between many-body spin system and the
reservoir states are now investigated. First, a general reservoir is considered, where the
coupling to the reservoir is constant with Γ
R
= 0 and Γ
L
= Γ. Thus, excitations are only
spontaneously generated to the infinite side of each waveguide and do not interact again
with the spin system. In Figure 9.17, a dissipative discrete time crystal is shown with a
generic general (unstructured) coupling to
N
external reservoirs. In the inset of Figure 9.17,
the staggered magnetization is shown. It becomes clear that
hMi
(Equation 9.1.8) decays
faster for increasing reservoir coupling rates Γwith
hMi →
0. This results in smaller peak
heights with increasing Γin the Fourier spectrum. These findings are in agreement with
Ref. [
224
]. For a comparable small number of periods (
T
= 75), the DTC thermalizes with
its environment due to build up of entanglement with reservoir states via Γ.
In contrast to computations in the Liouville space or with the density-matrix formalism
as done in Part II of this thesis, the reservoir degrees of freedom are not traced out. This
means that all entanglement between system and reservoir is kept during the computation
within the MPS. As a consequence, the dimension of each link index grows with increasing
computation time because the entanglement grows and more and more singular values
Chapter 9. Feedback-stabilized time crystal 127
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 150 300
Fourier spectra
ω
Γ=0.001
Γ=0.005
Γ=0.010
Γ=0.015
hMi
periods T
-0.5
0
0.5
0 150 300
Figure 9.18.:
Computation of a dissipative time crystal within the QSSE picture with
constructive feedback dynamics (
φni
=
π
). The subsequent interaction of the
system with self-generated reservoir excitations counteracts the building up
of entanglement between system and reservoir. Thus, the DTC dynamics are
stabilized for a long time (
T
= 300, inset). The subharmonic frequency is
locked at
ω
= 1
/
2with almost the same peakheight as without dissipation
(Figure 9.9). This is also the case if the coupling to the reservoir Γis increased.
Note that Γ = Γ
R
= Γ
L
and
τ
= 0
.
02. The other parameters are chosen as
before.
become relevant. As this setup has a high amount of link indices, i.e. one between each site
index as well as link indices to each reservoir, the simulation quickly reaches a numerical
limit. For this reason, the time evolution with dissipation to a general reservoir has only
been computed up until
T
= 75 periods. Furthermore, ∆
t
has to be chosen small enough to
reduce the error due to the expansion to second order in ∆
t
of the time-evolution operators.
It is assumed that ∆
t
is not small enough for Γ = 0
.
001, as the radiative decay is not
covered by the chosen ∆
t
. However, in choosing a smaller ∆
t
, the truncation in the singular
values has to be chosen appropriate. With smaller ∆
t
also smaller singular values become
relevant. When the truncation in the singular values is not chosen appropriate, the time
crystal behavior disappears because the important Ising interaction is not covered anymore
by the considered singular values. Thus, if these important singular values are neglected in
the computation, the system is following the external perturbation in the driving. When
this happens within a parameter setup where time crystal behavior is expected according
to Ref. [223], it is a strong indicator for a too high truncation in the singular values.
However, if feedback dynamics are considered with Γ = Γ
R
= Γ
L
it is possible to compute
far more periods. In Figure 9.18 the dissipation including feedback dynamics is shown. It
becomes clear that the feedback dynamics counteract the general dissipation in Figure 9.17.
This can be seen on two things. On the one hand, the peak height is almost the same as
without any dissipation (cp. Figure 9.9), because the staggered magnetization stabilizes
at the same value. On the other hand, if the coupling to the surrounding reservoirs Γis
increased, the peak height as well as the oscillations of
hMi
remain at nearly the same
value. This strongly indicates that the dissipation via Γis suppressed by the feedback
Chapter 9. Feedback-stabilized time crystal 128
dynamics. Another indicator is that the dimension of the link indices does not grow as
much as for the general dissipation in Figure 9.17 which is why it is possible to compute
up to T= 300 periods. This is a remarkable observation, because one would assume that
the feedback algorithm would take more time. On the one hand, with each time step a lot
of swap operations have to be performed, the exact number depends on the time delay
τ/
∆
t
and the system size. On the other hand, the dimension of each tensor of the MPS is
of one higher rank and each tensor of the MPO is of higher rank two. However, the smaller
dimension of the link indices, due to the suppression of entanglement with the reservoirs,
counteracts this further complexity and the computation is more feasible in the QSSE
picture with feedback dynamics as for general dissipation.
Having now demonstrated that feedback dynamics stabilize the dissipative DTC, a different
model of the DTC is considered. The driving Hamiltonian with the transverse field
HT
during
T1
is the same as before, but the interacting Hamiltonian
HI
during
T2
is replaced
by
HI=
N−1
X
i=1
Jz
iσz
iσz
i+1 +
N−1
X
i=1
Jx
iσx
iσx
i+1 +
N
X
i=1
hx
iσx
iT1< t < T2.(9.2.7)
The couplings
Jz
i
,
Jx
i
and the transverse field
hx
i
are randomly distributed between
[−χ, χ]
,
where
χ
is the respective parameter. This model is known to exhibit localization protected
spatial order which is also called a MBL spin-glass [
225
]. Thus, a combination of spatial
and temporal order is formed in this model when
HT
and
HI
are applied periodically [
226
].
Without addressing the question whether feedback dynamics also stabilize spatial order
against dissipation, this might be an interesting future investigation. A possible detection
of spatial order with the here used method is a modification of the Edward-Anderson order
parameter, based on reduced density matrices [227].
However, the scope of this chapter is to demonstrate stabilization of temporal order against
dissipation, namely the persistent oscillations of the DTC returning to its initial state after
two periods. Another advantage of the model in Equation 9.2.7 is the MBL phase without
using a random field in the
z
-dimension. Temporal order is only robust against dissipation
for constructive interference
φni
=
π
of each spin
ni
. The feedback phase
φ
=
π−ω01τ
was introduced for a single TLS in chapter 8. The random disorder
hiσz
i
would modify the
energy gap of each individual TLS of the many-body system according to the disorder
realization. In the simulation it is possible to force each individual
φi
=
π
as done above.
However, in an experiment, each individual mirror distance
Li
would have to be adjusted
dependent on the disorder realization. To overcome this further difficulty, the model in
Equation 9.2.7 shows DTC behavior combined with MBL, without modifying the energy
gap of each individual TLS and is therefore much more feasible in this context. Furthermore,
it becomes possible to study the effect of a shared reservoir as all TLS have the same
constructive feedback phase for this model.
The simulation of a DTC with a shared reservoir of all spins becomes very involved. The
reason is that all spins share a link index to the same reservoir. Thus, a decomposition to
an MPO, as explained in the previous section, is not possible. The reason is that an MPO
formulation would implicate that the spins act after each other on the reservoir, starting
with the first spin, then the second and so on.
Due to this, the simulation with a shared reservoir is comparable to the algorithm in
chapter 8. All physical indices of the many-body system are expressed in a single compound
index for the system
is
. This includes the exponential scaling of the Hilbert space with
Chapter 9. Feedback-stabilized time crystal 129
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 50 100
Fourier spectra
ω
Γ=0.005
Γ=0.010
Γ=0.015
hMi
periods T
-0.5
0
0.5
0 50 100
Figure 9.19.:
A dissipative DTC with general (unstructured) reservoir coupling with a
single reservoir for all
N
= 8 spins. The staggered magnetization decays
similar to the case of individual reservoirs. Note that for computational
reasons, the perturbation was set to zero
= 0. Parameters: ∆
t
= 0
.
005,
svdcutoff= 10
e−
9for the spin system and svdcutoff= 10
e−
6between system
and reservoir states, Ω =
π/
2,
= 0,
Jz
= 0
.
1,
Jx
= 0
.
01,
hx
= 0
.
01 and the
number of averages around Nav = 10.
growing system size, which is why with this approach the maximal system size is N= 8.
However, the overall entanglement with the reservoir is smaller than for individual reservoirs
for this system size, as not so many link indices to the reservoirs are involved. For this
reason, the time evolution without feedback dynamics can be evaluated for a larger number
of periods. The dissipation of the DTC with a shared reservoir for all spins is shown in
Figure 9.19. Note that the perturbation
is set to zero in this and all following plots. The
reason is that this speeds up the computation time as lesser entanglement is generated
within the spin system. However, the robustness of the DTC against perturbations
>
0
was tested for all this cases as well. It becomes clear that there is no visible difference
between the dissipation to a shared reservoir compared to the individual reservoirs in
Figure 9.17. For a general dissipation this is also expected as without a structured reservoir,
the decay Γis the same for all spins. As any dissipation is lost to the infinite side of the
waveguide, it is not to be expected that there is a difference between individual or shared
reservoirs.
In Figure 9.20 it is shown that also constructive feedback dynamics are the same as for
individual reservoirs in Figure 9.18. The constructive feedback interference counteracts the
buildup of entanglement and prevents the system from thermalizing with the environment,
also for a shared reservoir of all spins. This is a strong finding and allows to take the case
of individual reservoirs as a general example to explore much larger system sizes. This is
because individual reservoirs can be computed very efficiently with the algorithm developed
within previous section.
To end this chapter, an example for a large system of
N
= 40 which is beyond computation
with conventional open quantum system methods is shown in Figure 9.21. This proves
that temporal order for the MBL spin-glass model in Equation 9.2.7 is also stabilized
Chapter 9. Feedback-stabilized time crystal 130
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 50 100
Fourier spectra
ω
Γ=0.005
Γ=0.010
Γ=0.015
hMi
periods T
-0.5
0
0.5
0 50 100
Figure 9.20.:
Feedback dynamics for a shared reservoir. As for the individual reservoirs,
constructive feedback stabilizes temporal order acting against dissipation.
The staggered magnetization becomes independent of the coupling to the
reservoir Γ. All parameters are chosen as before.
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
-0.5
0
0.5
0 50 100
Fourier spectra
ω
Γ=0.005
Γ=0.010
Γ=0.015
hMi
periods T
-0.5
0
0.5
0 50 100
Figure 9.21.:
Simulation of a large system of
N
= 40 with individual reservoirs. This
proves that the developed algorithm has an efficient scaling with the system
size. Furthermore, temporal order is stabilized against dissipation using
constructive feedback for the MBL spin-glass system.
Chapter 9. Feedback-stabilized time crystal 131
via feedback dynamics. Furthermore, this serves as a demonstration that the algorithm
has a very efficient scaling with the system size. The main reason is that MPS methods
are very efficient for systems with a small amount of entanglement. On the one hand the
entanglement growth is strongly suppressed by MBL, allowing to explore large systems. On
the other hand, constructive feedback suppresses entanglement with the reservoir, allowing
to consider the reservoir degrees of freedom in a numerically exact method. Furthermore,
this allows to draw the conclusion that not only the DTC is stabilized for long times
against dissipation, but also MBL survives for an open quantum system with a structured
reservoir.
9.3. Conclusion
In this chapter, an open many-body spin system has been investigated using matrix product
state methods. In the first section, a closed quantum system was considered to explain
standard matrix product state methods. The system of interest is a discrete time crystal,
where the time translational symmetry of the Floquet Hamiltonian
H
(
t
) =
H
(
t
+
T
)is
broken, i.e. the staggered magnetization as observable shows oscillations at twice the period
hMi
(
t
) =
hMi
(
t
+ 2
T
), meaning the many-body system returns to its initial state after
two periods of the Floquet Hamiltonian. This behavior is stable even against perturbations
in the driving. However, the external driving results in heating of the spin system, bringing
it to thermal equilibrium. Many-body localization prevents the system from absorbing the
energy of the driving, and a stable discrete time crystal is achieved for long times. This
makes the system an ideal candidate to use matrix product state methods due to small
entanglement between subsystems.
In the second section, the many-body system was generalized to an open quantum system,
adapting the method of chapter 8, where the environment was included within the many-
body formulation, based on the quantum stochastic Schrödinger equation. This results in
a two-dimensional matrix product state where each spin is interacting with an individual
reservoir. The formulated method can be adapted to generic one-dimensional many-body
quantum systems and is efficient as long as the entanglement within the many-body system
is very small (i.e. any many-body localized system).
It has been demonstrated that the decay due to Lindblad dissipators is reproduced, resulting
in melting of the discrete time crystal due to thermalization with the environment.
When the reservoir is structured with a reflecting element, e.g. a distant mirror, it has
been demonstrated that for constructive interference, the DTC is stabilized and becomes
independent of the coupling to the reservoir. The interference induced by the mirror
counteracts thermalization with the environment and a stable open discrete time crystal is
achieved.
Furthermore, a second model was investigated, exhibiting time translational symmetry
breaking, which is a many-body localized spin-glass. In this model, a shared reservoir
has been considered, showing also stabilization due to feedback dynamics. This allows to
assume that the method based on individual reservoirs for each spin suffices to explain the
behavior for a shared reservoir as well.
The advantage of the proposed method to simulate open many-body spin systems including
feedback dynamics is that entanglement between the many-body system and the reservoir
is strongly suppressed. As the many-body system itself is within a many-body localized
phase, the proposed method using matrix product states has a very efficient scaling with
the system size. An open system up to a number of forty spins including feedback dynamics
Chapter 9. Feedback-stabilized time crystal 132
has been simulated, showing oscillations independent of the coupling to the reservoir.
Thus, a large open quantum many-body system has been stabilized against dissipation
and temporal order has been observed for a long time. This also includes that many-body
localization is stable against dissipation for constructive feedback interference.
10.
Conclusion and outlook
In this thesis, different out-of-equilibrium open quantum many-body systems have been
investigated. In the first part, the theoretical and physical framework for the investigated
methods and effects has been provided.
The second part considered couplings to general (unstructured) reservoirs described by
Lindblad dissipators based on the Born-Markov approximation via the reduced density
formalism.
As a first many-body system, the many-emitter phonon laser has been investigated. Due
to the coupling between different emitters over the cavity phonon mode, collective ef-
fects are present. A mapping of the Hamiltonian to a Tavis-Cummings type model has
revealed additional resonances caused by collective phonon emission of many emitters.
It has been demonstrated that by optically addressing these collective resonances, the
phonon intensities are enhanced in the many-emitter setup. A possible future direction
is to investigate interactions with reservoirs beyond Born-Markov approximation as done
for different systems in the third part of this thesis. This might allow to manipulate the
phonon statistics in a controllable setup.
As second many-body system, the Heisenberg spin-chain was considered. In driving the
system out-of-equilibrium via two boundary reservoirs, a spin current is induced. It has
been shown that the transport for the conventional nearest-neighbor coupling between the
spins is very sensible on the reservoir parameters due to far-from-equilibrium effects. In
contrast, the transport for long-range coupling is independent of the external reservoirs.
When considering small disorder strengths, where the system is still within an ergodic
phase, as well nearest-neighbor as long-range coupling show a transition to subdiffusive
transport. However, the transition to subdiffusive transport for long-range coupling is
independent of the external driving and thus results purely from disorder effects. This
could help to understand the reason for the existence of subdiffusive transport due to
disorder which is still under debate. Furthermore, it remains an open question under which
conditions many-body localization exists in open boundary driven systems.
In the third part of this thesis, reservoirs with memory effects have been considered, where
a common Born-Markov approximation is not possible. Dynamics have been computed via
the quantum stochastic Schrödinger equation in formulating the system and reservoir as a
matrix product state.
A single two-level system coupled to a structured reservoir has been considered. In exciting
the emitter with a time-dependent pulse and inducing optical self-feedback via the distance
to a mirror, it has been shown that individual photon probabilities are controlled. For exam-
ple, the two-photon is higher than the one-photon probability and shows an enhancement
of up to 50% compared to the case without feedback. The two photons generated from a
single emitter could be entangled in time which remains to be shown. A computation of an
observable based on Franson interferometry for matrix product state methods could be
133
Chapter 10. Conclusion and outlook 134
interesting to prove time entanglement for setups with feedback dynamics, as two-photon
probabilities are explicitly controlled via time-delayed feedback.
In the last chapter, a theory has been developed to consider reservoirs with memory effects
combined with a many-body system. The common description via matrix product state
methods has been generalized to a two-dimensional matrix product state with an individual
reservoir for each site of the many-body system.
The derived methods scales very efficiently with the system size due to small entangle-
ment within a many-body localized system. Furthermore, entanglement with the external
reservoirs is suppressed using feedback dynamics. As an example, it has been shown that a
discrete time crystal is stabilized against dissipation in structuring the external reservoirs.
Thus, temporal order of the time crystal is stable for a long time even in the presence of an
external environment. It is an interesting question, if spatial order, which is also present in
these systems, is also stabilized using feedback dynamics. This might be closely connected
to the performed rotating wave approximation for the coupling to the reservoir.
For the discrete time crystal up to forty spins have been investigated. However, the proposed
method can be optimized in the future to even explore larger systems. Furthermore, the
method is not limited to the proposed system. An investigation of other systems showing
many-body localization is a promising future direction. Especially to show in detail a
robustness of many-body localization for open quantum many-body systems to explore
further out-of-equilibrium effects.
Acknowledgments
I would like to thank Prof. Dr. Andreas Knorr for giving me the opportunity to work on
these interesting projects and the constant support he gave me in developing my research
subject. Furthermore, I would like to thank him for sparking my interest in quantum
mechanics already in the beginning of my studies with his inspiring lectures in the morning
hours.
My thanks go to Prof. Dr. Peter Rabl for being the second reviewer of my thesis. Further-
more, I would like to thank Prof. Dr. Michael Kneissl for chairing the board of referees.
Special thanks go to Dr. Alexander Carmele. His constant support and trust in my work
not only influenced the major part of this thesis, but also personally I benefited a lot from
his experience. The many discussions and conversations deepened my interest in physics
and created an enjoyable atmosphere which I will definitely miss in the future.
I want to thank Dr. Markus Heyl for opening my interest in many-body physics already
during my master thesis. The fruitful discussions helped me developing the ideas of this
thesis. Furthermore, I want to thank him and the Max-Planck-Institut für Physik komplexer
Systeme for giving me the opportunity to run simulations on their cluster.
Furthermore, I would like to thank Dr. Julia Kabuss for supervising me in the beginning of
my research and the development of the ideas concerning the many-emitter phonon laser.
For being the perfect collaborator, I would like to thank Dr. Nicolas Naumann. Working
together on the phonon laser and developing the first simulation for feedback dynamics
has improved a major part of this thesis.
I want to thank all the former and current members of the AG Knorr for the many funny
moments during lunch and coffee breaks, conferences and symposias. Especially my office
colleagues over the years, Shahabedin Chatraee Azizabadi, Alexander Carmele, Michael
Gegg, Manuel Kraft, Nicolas Naumann and Judith Specht have made a perfect atmosphere
with helpful discussions and humorous chats.
Special thanks go to Alexander Carmele, Regina Finsterhölzl, Manuel Kraft, Julian Schleib-
ner, Malte Selig and Judith Specht for proofreading this thesis.
For the technical support and the management of the computer cluster, which I was
constantly using, I thank Dr. Marten Richter and Peter Orlowski.
135
Chapter 10. Conclusion and outlook 136
This thesis was made possible by financial support of the SFB 910 as part of the DFG.
The School of Nanophotonics of the SFB 787 enabled many national and international
conference participations.
For their emotional support and constantly listening to my progress in research, I would
like to thank my roommates Juliette Bersou and Marie-Annick Schmidt. Further thanks
go to all my friends. Writing this thesis would not have been possible without the many
beautiful shared moments outside the world of physics.
Especially, I want to thank my parents, my brother as well as my whole family for their
constant support, for listening and that I always could rely on them over all the years.
PART IV
Appendices
137
A.
Details on the Theoretical Background
A.1. Consitency with the Maxwell equations
The fundamental equations of the classical description of the electromagnetic field are
the Maxwell equations. In this thesis, the equations are formulated, based on a covariant
formulation of the Lagrange function. In correspondence to the Minkowski tensor, the
metric tensor is defined as
ηαβ =
1 0 0 0
0−1 0 0
0 0 −1 0
0 0 0 −1
.(A.1.1)
The components of the electric field
E
and the magnetic field
B
are written into the
contravariant electromagnetic tensor [61]
Fαβ =
0Ex/c Ey/c Ez/c
−Ex/c 0−BzBy
−Ey/c Bz0−Bx
−Ez/c −ByBx0
,(A.1.2)
with
c
=
√0µ0
as the speed of light in vacuum with the corresponding electric constant
0
and magnetic permeability
µ0
. The covariant form
Fαβ
is obtained by raising its indices
with the metric tensor (using Einstein notation)
Fγδ =ηγαFαβηβδ .(A.1.3)
The Lagrange function
LM
of the electromagnetic field with the electromagnetic tensor
then reads [57]
LM=Zd3rL,L=−1
4µ0
FαβFαβ =0
2E2−c2B2,(A.1.4)
with Lagrange density L. By defining the four-gradient and the four-potential
∂α=1
c
∂
∂t,−∇, Aα= (φ/c, A),(A.1.5)
with scalar potential
φ
and vector potential
A
, the differential of the electromagnetic
potentials yields the electromagnetic tensor [57]
Fαβ =∂αAβ−∂βAα.(A.1.6)
139
Appendix A. Details on the Theoretical Background 140
The Maxwell equations are obtained by deriving the conditions for the Euler-Lagrange
equation of the Lagrangian density L(Aα, ∂βAα)
∂β
∂L
∂(∂βAα)−∂L
∂Aα
= 0 .(A.1.7)
Calculating the first summand
∂L
∂(∂βAα)=−1
4µ0
∂FµνηµληνσFλσ
∂(∂βAα)
=−1
4µ0
∂h(∂µAν−∂νAµ)ηµληνσ(∂λAσ−∂σAλ)i
∂(∂βAα)
=−1
4µ0
ηµληνσ h(δβ
µδα
ν−δβ
νδα
µ)Fλσ +Fµν(δβ
λδα
σ−δβ
σδα
λi
=−1
4µ0h(δβλδασ −δβσδαλ)Fλσ +Fµν(δβµδαν −δβνδαµi
=−1
4µ0hFβα −Fαβ +Fβα −Fαβi
=−1
µ0
Fβα ,(A.1.8)
where in the first step the indices of the left electromagnetic tensor were lowered, then in a
second step the derivative yields delta tensors. Finally the delta and the metric tensor act
on the electromagnetic potentials.
The second part of Equation A.1.7 yields
∂L
∂Aα
= 0 ,(A.1.9)
as the Lagrangian in Equation A.1.4 without the four-current has no dependency on the
four-potential itself. With the condition of the Euler-Lagrange equation in Equation A.1.7
∂β
1
µ0
Fβα !
= 0 ,(A.1.10)
the first two Maxwell equations are obtained. For
β
= 0, the first line of the covariant
electromagnetic tensor is derivated yielding
∇·E= 0 .(A.1.11)
For
β∈ {
1
,
2
,
3
}
, it yields the respective part of the rotation of
Bβ
and the time derivative
of Eβresulting in
∇×B−1
c2
∂E
∂t = 0 .(A.1.12)
Appendix A. Details on the Theoretical Background 141
Note that in the Lagrangian in Equation A.1.4 the four-current was set to zero, thus the
here derived Maxwell equations do not include charges or currents.
The other two Maxwell equations are derived by the Jacobian identity
∂αFβγ +∂βFγα +∂γFαβ = 0 .(A.1.13)
The Jacobian identity has a compact notation for the dual electromagnetic tensor
˜
Fαβ
which is a transformation
1
cE→B,B→ −1
cE(A.1.14)
of the electromagnetic tensor. With this the Jacobian identity reads
∂α˜
Fαβ .(A.1.15)
For α= 0 the third Maxwell equation is obtained
∇·B= 0 (A.1.16)
and for α∈ {1,2,3}, the rotation of Eand time derivative of Bis obtained
∇×E+∂B
∂t = 0 .(A.1.17)
With this it is shown that the Lagrangian
LM
in Equation A.1.4 leads to the basic Maxwell
equations for the classical electromagnetic field and will therefore be used in the following
to quantize the Maxwell field.
A.2. Commutation relation of the mode operators
To determine whether the commutation relation of the mode operators is bosonic or
fermionic, a general commutator is calculated with either a plus or a minus sign. Calculating
the commutator in Equation 2.3.17 it reads
h(bλkeikr −h.c.),(bλ0k0eik0r0+h.c.)i=hbλkeikr, bλ0k0eik0r0i−hb†
λke−ikr, b†
λ0k0e−ik0r0i
+hbλkeikr, b†
λ0k0e−ik0r0i−hb†
λke−ikr, bλ0k0eik0r0i
=hbλkeikr, bλ0k0eik0r0i−hb†
λke−ikr, b†
λ0k0e−ik0r0i
+ei(kr−k0r0)δλλ0δ(k−k0)±b†
λ0k0bλk−b†
λ0k0bλk
+e−i(kr−k0r0)δλλ0δ(k−k0)±b†
λkbλ0k0−b†
λkbλ0k0
(A.2.1)
With this it becomes clear that the mode operators vanish from the equation, if the plus
sign is correct. With this it is shown that the mode operators obey a bosonic commutation
relation
Appendix A. Details on the Theoretical Background 142
hbλk, b†
λ0k0i=δλλ0δ(k−k0),(A.2.2)
[bλk, bλ0k0] = hb†
λk, b†
λ0k0i= 0 .(A.2.3)
A.3. Lorentz force
In this section, the Lorentz force will be derived from the Lagrangian of a charged particle
with mass mand charge qin a field with vector potential Aand scalar potential φ
LF(r,˙
r) = 1
2m˙
r2+q˙
r·A−qφ . (A.3.1)
This Lagrangian will be used later on to derive the interaction between an electron and
a light field (classical as well as quantized). This section serves to prove the relevance of
this Lagrangian in deriving the fundamental Lorentz force from it. The Euler-Lagrange
equation reads
∂L
∂ri−d
dt
∂L
∂˙ri
= 0 .(A.3.2)
The first part results in
∂LF
∂ri
=qX
l
∂Ai
∂rl
drl
dt −q∂φ
∂ri
,(A.3.3)
and the second part yields
d
dt
∂LF
∂˙ri
=d
dt (m˙ri+qAl(r(t), t))
=m¨ri+q∂Ai
∂t +qX
l
∂Al
∂ri
drl
dt .(A.3.4)
Inserting Equation A.3.3 and Equation A.3.4 into Equation A.3.2 already results in the
Lorentz force
m¨ri=q−∂φ
∂ri−∂Ai
∂t +qX
l∂Ai
∂rl−∂Al
∂ridrl
dt .(A.3.5)
The first part results in the respective part of the electric field
E
=
−∇φ−∂A
∂t
and the
second part in the respective rotation of
B×˙
r
due to
B
=
∇×A
. Thus, including all
three dimensions it results in the Lorentz force
FL=qE+q(˙
r×B).(A.3.6)
B.
Details on Feedback-stabilized time
crystal
B.1. Efficient application of a matrix product operator
Figure B.1.:
Efficient application of
UI
on the MPS. First (top left), it is contracted over the
physical index
n1
. In a second step (top central) the link index
l1
is contracted.
In a third and fourth step it is contracted over the link index
b1
of the MPO
and then over the physical index
n2
. The resulting tensor is of rank four with
two physical and two link indices (top right). The tensor is decomposed such
that the OC is at the tensor containing
n0
2
and the tensor of the first site
n0
1
is
of rank two and left-normalized (bottom right). Note that the link index
l0
1
is
now of higher dimension according to
l1
and
b1
. The tensor at the second site
now shares a link index
l2
with the tensor of the third site
n3
and a link index
b2
with
UI
(
n3
). The procedure is repeated for the following sites starting with
contracting over l2(bottom left).
An efficient application of
UI
(∆
t
)on
|ψ
(
T1
)
i
is shown in Figure B.1. The starting point is
the second diagram in Figure 9.4. At first, a contraction over the index
n1
is performed.
Then the resulting tensor of rank three is contracted with the tensor of the MPS of the
second site which is done in contracting over
l1
. The resulting tensor of rank four is then
143
Appendix B. Details on Feedback-stabilized time crystal 144
contracted with the MPO tensor of the second site by contracting first over
b1
and then
over
n2
. The resulting tensor now includes both outgoing physical indices
n0
1
and
n0
2
. Now
it is important to decompose this tensor to avoid dealing with high rank tensors in the
following. Thus, the MPS form is constructed from the left side in separating the tensor
containing
n0
1
. The resulting link index
l0
1
now includes the link dimension of
l1
of the MPS
and of
b1
of the MPO. Without truncation in the SVD, the link dimension
b1
would add
up for each time step. Note that the OC is multiplicated into the tensor containing
n0
2
to
proceed with the algorithm and maintain the gauge of the MPS. The tensor containing
n0
2
now shares the link index
l2
with the tensor of the MPS containing
n3
and the link
index
b2
with the tensor of the MPO containing
n3
. In now contracting first over
l2
, then
b2
, then
n3
and decomposing the resulting tensor, the procedure is repeated until the end
of the MPS is reached. The resulting MPS is then in left canonical gauge and has evolved
to
|ψ
(
T1
+ ∆
t
)
i
. In decreasing the prime level of all indices by one and in either shifting
the OC back to the left side by series of SVD or performing the application of
UI
(∆
t
)from
the right, the MPS is ready for the next time step.
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