Iden tical Emitters, Collectiv e Effects and Dissipation in
Quan tum Optics
No v el n umerical approac hes for quan tum master equations
v orgelegt v on
M.Sc. Mic hael Gegg
geb. in Münc hen
v on der F akultät I I – Mathematik und Naturwissensc haften
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des akademisc hen Grades
Doktor der Naturwissensc haften
– Dr.rer.nat. –
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr. Thomas Möller
Gutac h ter: Prof. Dr. Andreas Knorr
Gutac h ter: Prof. Dr. Jan Wiersig
Gutac h ter: Dr. Marten Ric h ter
T ag der wissensc haftlic hen A ussprac he: 6. Oktob er 2017
Berlin 2017

ii

iii
Abstract
In this thesis a formalism for indistinguishable m ulti-lev el quantum emitters in quan tum master
equations is dev elop ed. The complexit y of the approac h scales only p olynomially in the n um b er
of quan tum emitters. Complexit y here means the n um b er of coupled equations or rather the
dimension of the Liouville space. This approac h op ens new p ossibilities for calculating op en
quan tum systems. The metho d is implemen ted in the PsiQuaSP library , whic h allo ws to setup
and solv e arbitrary master equations, in particular master equations with the reduced p olyno-
mial scaling. The in tro duced to ols are utilized to study subradiance in the op en Dic k e mo del
and v arious cQED lasers.
One of the main curren t researc h goals in quan tum optics and quan tum information science
is the generation and stabilization of quan tum coherence b et w een distinct quantum systems.
These systems can for instance b e photons or quan tum emitters, qubits. F rom a theoretical
stance the difficult y with these systems is that their complexit y scales in general exp onen tially
with the system size, like the n um b er of quan tum emitters. The reduced scaling of the formalism
in tro duced in this thesis stems from the p ermutation symmetry of the indistinguishable emit-
ters. The metho d is deriv ed in the con text of Lindblad quan tum master equations. Lindblad
quan tum master equations for man y identical m ulti-lev el systems ha v e b een used to study funda-
men tal quan tum optical systems, such as lasing and laser lik e action, v arious phase transitions,
optical bistabilit y , co op erativ e resonance fluorescence, en tanglemen t, quantum ligh t generation,
quan tum to classical transitions, sup er- and subradiance etc. In most of these cases the single
emitter, the man y emitter and the w eak correlation limit can b e satisfactorily treated with ex-
isting tec hniques, suc h as direct in tegration, phase space metho ds or cluster expansion. How ev er
these tec hniques are not w ell suited for the few emitter case with strong correlations. The the-
ory dev elop ed in this thesis allo ws to study all these systems for mo derate n um b ers of quan tum
emitters and arbitrary correlation strengths, th us filling the gap left op en b y con v entional meth-
o ds. Ev en though the theory is dev elop ed around Lindblad quan tum master equations it is not
limited to this case – the requiremen t of indistinguishabilit y is used to construct symmetrized
Liouville space basis states and op erators whic h can also b e used to construct other master
equations.
In this thesis the general metho dology for the p olynomial scaling of the man y emitter master
equation is deriv ed b y t w o differen t approaches. Generalized p erm utation symmetric Liouville
space states and op erators are intr o duced that allow for the construction of arbitrary mas-
ter equations and observ ables. These Liouville space states and op erators are translated into a
graphical represen tation that greatly facilitates their usage. This represen tation is found to hav e
a close connection to Lie algebras as w ell as graph theory . The whole metho d is implemen ted in a
general and mo dular w a y in a C++ library called PsiQuaSP , whic h allo ws to simulate arbitrary
master equations based on a n um b er state represen tation. The design of the library hea vily relies
on the in tro duced graphical represen tation, whic h greatly facilitates the setup of a sim ulation.
The treatmen t of non-iden tical systems with PsiQuaSP is illustrated and further techniques for
complexit y reduction and computational sp eedup are discussed. These to ols are applied to the
t w o main cases of cQED lasers/spasers and super-, subradiance. In the latter case a new t yp e
of phase transition in the op en Dic k e mo del is predicted that leads to a deterministic generation
of subradian t steady state coherences, with applications in quan tum information science.
F urthermore the in teraction of quan tum dots with the dissipativ e mo des at metal in terfaces is
in v estigated in order to explain exp erimen tal findings.

iv

v
Zusammenfassung
In dieser Arb eit wird ein F ormalismus für un un tersc heidbare Multi-Niv eau Quan tenemitter in
Quan ten Mastergleic hungen en t wic k elt. Die K omplexität dieses Ansatzes skaliert n ur p oly-
nomiell und nic h t exp onen tiell in der Anzahl der Emitter. K omplexität b edeutet hier die Anzahl
der gek opp elten Gleic h ungen bzw. die Dimension des Liouville Raumes. Dieser Ansatz erlaubt
neuartige Betrac h tungen in offenen, quan tenoptisc hen Systemen. Die Metho de wurde in der Psi-
QuaSP Programm bibliothek implemen tiert. Diese erlaubt es generelle Quan ten-Mastergleic hungen
n umerisc h zu lösen, insb esondere Mastergleic h ungen mit der reduzierten p olynomiellen Skalierung.
Mit Hilfe der eingefürten Metho den w erden Subradianz im offenen Dic k e Mo dell und v er-
sc hiedene cQED Laser un tersuch t.
Eines der zen tralen, aktuellen F orsc h ungsziele in der Quantenoptik und der Quan teninforma-
tions T ec hnologie ist die Generierung und Stabilisierung v on Quan tenkorrelationen zwisc hen
un tersc hiedlichen Quan tensystemen. Beispiele für solc he Quan tensysteme sind Photonen o der
Quan tenemitter, Qubits. Aus theoretisc her Sic h t liegt die Sc hwierigk eit mit diesen Systemen in
der T atsac he, dass im Allgemeinen ihre K omplexität exp onen tiell mit der Systemgröße, wie etw a
der Anzahl der Quan tenemitter, skaliert. Die Reduzierung der K omplexität in dem hier einge-
fürten F ormalism us resultiert aus der V ertausc h ungssymmetrie der un un terscheidbaren Emit-
ter. Die Metho de wird im K on text v on Lindblad Quan ten-Mastergleic h ungen en t wic k elt. Lind-
blad Quan ten-Mastergleic hungen für iden tisc he Multi-Niv eau Systeme w erden v erw endet um
fundamen tale quan tenoptische Systeme zu un tersuc hen, wie et w a Lasing und laserartiges V er-
halten, v ersc hiedene Phasen üb ergänge, optisc he Bistabilität, K o op erativ e Resonanzfluoreszenz,
V ersc hränkung, Quan tenlic h t, Üb ergang v on der Quan ten- zur klassisc hen Ph ysik und Sup er-,
Subradianz. In den meisten dieser F älle k önnen die Grenzfälle v on einzelnen Emittern, vie-
len Emittern und sc h w ac hen K orrelationen mit b estehenden Metho den zufriedenstellend gelöst
w erden. Diese Metho den sind et w a direkte In tegration, Phasenraummetho den o der Cluster
En t wicklung. Diese Metho den sind jedo c h nic ht geeignet für den F all v on mittleren Emitter-
Anzahlen mit stark en K orrelationen. Der hier en t wic kelte F ormalism us erlaubt es all diese Sys-
teme mit mittleren Emitter-Anzahlen und b eliebigen K orrelationsstärk en zu b erec hnen. Somit
füllt der F ormalism us die Lüc k e w elc he v on herkömmlic hen Metho den zurüc kgelassen wird.
In dieser Arb eit wird der F ormalismus anhand v on zw ei v ersc hiedenen Betrac h tungsw eisen en-
t wic kelt. Generelle Liouville Raum Op eratoren und symmetrisierte Liouville Raum Basiszustände
w erden eingeführt mit denen b eliebige Mastergleic h ungen und Observ ablen k onstruiert w er-
den k önnen. Diese Zustände und Op eratoren w erden in eine graphisc he Darstellung üb ersetzt
w elc he die Ben utzung stark v ereinfac ht. Die graphisc he Darstellung hat V erkn üpfungen zu Lie
Algebren und der Graphen theorie. Der gesam te F ormalism us ist in der Bibliothek PsiQuaSP
implemen tiert, welc he Sim ulationsrec hn ungen für b eliebige Mastergleic h ungen auf Basis v on
Nummernzuständen erlaubt. Die Bibliothek basiert auf der graphisc hen Darstellung, w as das
Implemen tieren v on Sim ulationen stark v ereinfac ht. Des w eiteren w erden die Behandlung v on
nic h t-identisc hen Emittern in PsiQuaSP so wie zusätzlic he Methoden zur Reduzierung der Kom-
plexität v orgestellt. Der F ormalism us und die Bibliothek w erden auf die zw ei Gebiete cQED
Laser und Sup er-, Subradianz angew endet. Im letzteren F all wird ein neuartiger Phasen üb er-
gang im offenen Dic k e Mo dell v orhergesagt, der zur A usbildung v on subradian ten K ohärenzen
im stationären Zustand führt.
Des w eiteren wird die W ec hselwirkung v on einzelnen Quan tenpunkten mit den dissipativ en op-
tisc hen Mo den an Metallgrenzfläc hen un tersuc h t um exp erimen telle Befunde zu erklären.

vi

Con ten ts
1 In tro duction 1
I Theoretical Bac kground 5
2 Mo del systems 7
2 . 1 Q u a n t u m e m i t t e r s ................................... 8
2 . 1 . 1 T w o - l e v e l S y s t e m s ............................... 8
2.1.2 Three and more lev els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Electromagnetic mo des in quantum optics . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Electromagnetic mo des in v acuum . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Dissipativ e mo des – Quan tum optics in media . . . . . . . . . . . . . . . . 12
3 Op en system dynamics 15
3 . 1 D e n s i t y m a t r i x ..................................... 1 5
3.2 Closed quantum systems - v on Neumann equation . . . . . . . . . . . . . . . . . 16
3.3 Op en quan tum systems - Lindblad equation . . . . . . . . . . . . . . . . . . . . . 17
3.4 Steady states and the Liouvillian sp ectrum . . . . . . . . . . . . . . . . . . . . . 19
I I F ormalism 21
4 Man y emitters – Complex quan tum systems 23
4.1 Iden tical t w o-lev el emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 The pro duct state basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 . 1 . 2 T h e D i c k e b a s i s ................................. 2 5
vii

viii CONTENTS
4.2 Iden tical m ulti-lev el systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 . 2 . 1 P r o d u c t s t a t e b a s i s ............................... 2 9
4.2.2 The Lie algebras su ( d ) ............................. 3 0
4.2.3 Dic k e states for m ulti-lev el systems – su ( d ) m ultiplets . . . . . . . . . . . 31
4 . 3 C o n c l u s i o n ....................................... 3 2
5 P erm utation symmetry in quan tum master equations 33
5.1 Symmetries, size and a v erages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Dissipation vs. symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Time evolution of identical m ulti-lev el systems . . . . . . . . . . . . . . . . . . . 37
5.3.1 Time ev olution of a simple master equation . . . . . . . . . . . . . . . . . 37
5.3.2 Time ev olution – Simplification . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.3 Prev en ting n umerical instabilit y . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.4 Open Dic k e mo del equations of motion . . . . . . . . . . . . . . . . . . . . 43
5 . 4 M u l t i - l e v e l s y s t e m s................................... 4 5
5 . 4 . 1 T h r e e - l e v e l s y s t e m s ............................... 4 5
5 . 4 . 2 M u l t i - l e v e l s y s t e m s ............................... 4 7
5.4.3 T w o-, three- and four-level laser examples . . . . . . . . . . . . . . . . . . 49
5.5 Symmetrized eigenstates of p erm utation symmetric Liouville space op erators . . 54
5.5.1 P erm utation symmetric Liouville space op erators – Elemen tary sk etc hes . 55
5.5.2 Building ph ysically meaningful Liouville space op erators . . . . . . . . . . 56
5 . 5 . 3 L i e a l g e b r a c o n t e x t ............................... 5 8
5.6 Reco v ering the Dick e states – Ho w to diagonalize the densit y matrix . . . . . . . 59
5 . 7 C o n c l u s i o n ....................................... 6 0
A The PsiQuaSP Library 61
A.1 Using PsiQuaSP – Basic structure of the library . . . . . . . . . . . . . . . . . . 62
A . 2 E x a m p l e s ........................................ 6 3
A.2.1 Example 1: Op en T avis-Cummings relaxation . . . . . . . . . . . . . . . . 63
A.2.2 Example 2: Three-lev el systems . . . . . . . . . . . . . . . . . . . . . . . . 67
A.3 T emplate functions v ersus custom Liouvillians . . . . . . . . . . . . . . . . . . . . 70
A.4 Building arbitrary Liouvillians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A . 5 P e r f o r m a n c e ....................................... 7 5
A . 6 S o m e n o t e s o n s o l v e r s ................................. 7 6
A . 6 . 1 K r y l o v s u b s p a c e s................................ 7 6
A.6.2 Sp ectral transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A . 7 C o n c l u s i o n ....................................... 7 8
I I I Results 79
6 Ca vit y QED Lasers and Spasers 81
6.1 Cluster expansion – Rate equation theory . . . . . . . . . . . . . . . . . . . . . . 83
6 . 2 T w o - l e v e l l a s e r s..................................... 8 5
6.3 Three- and F our-lev el bad ca vit y lasers – Spasers . . . . . . . . . . . . . . . . . . 87
6 . 4 C o n c l u s i o n ....................................... 9 0

CONTENTS ix
7 The op en Dic k e mo del 91
7 . 1 D i c k e m o d e l p h y s i c s .................................. 9 3
7.2 Bistable effects in quan tum optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.3 Sup erradian t to subradian t phase transition . . . . . . . . . . . . . . . . . . . . . 98
7.3.1 Collectivit y measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3.2 Nature of the phase transition . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.3 Robustness test and en tanglemen t prop erties . . . . . . . . . . . . . . . . 104
7 . 4 D a r k s t a t e c a s c a d e s .................................. 1 0 5
7 . 5 C o n c l u s i o n ....................................... 1 0 7
8 Quan tum dots on a thin metal film 109
8 . 1 S p e c t r a l r e s p o n s e.................................... 1 1 1
8.2 Mo deling the surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8 . 3 C o n c l u s i o n ....................................... 1 1 6
9 F urther usages of PsiQuaSP 117
9 . 1 N o n i d e n t i c a l s y s t e m s .................................. 1 1 7
9.2 Graph theory based optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2.1 Graph theory and partitioning in a n utshell . . . . . . . . . . . . . . . . . 119
9.2.2 Reduction of degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . 121
9 . 3 C o n c l u s i o n ....................................... 1 2 4
IV Epilogue 125
10 Summary and Outlo ok 127
11 A c kno wledgemen ts 129
B Prop erties of the symmetrized Liouville space basis 131
B . 1 S c a l i n g .......................................... 1 3 1
B.2 Prop erties of the symmetrized Liouville space basis states . . . . . . . . . . . . . 132
B.3 P erm utation symmetric Liouville space op erators Γ ................. 1 3 5
B.4 Blo c k diagonal represen tation of the densit y matrix in su ( d ) m ultiplets . . . . . . 136
B . 4 . 1 T w o - l e v e l s y s t e m s ............................... 1 3 6
B . 4 . 2 M u l t i - l e v e l s y s t e m s ............................... 1 3 8
C Spin squeezing inequalities 143
C . 1 G e n e r a l r e m a r k s .................................... 1 4 4
C.2 Definitions and necessary inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 145
C . 3 P r o o f s .......................................... 1 4 7
C . 4 C o m m e n t s ........................................ 1 4 8
C.5 Differen t representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
D Equations of motion for the laser examples 151
D . 1 T h r e e - l e v e l s y s t e m s ................................... 1 5 1
D . 2 F o u r - l e v e l s y s t e m s ................................... 1 5 4
E P ossible Dissipators 157
Bibliograph y 159

x CONTENTS

1 In tro duction
The researc h quest of non-relativistic quan tum theory can b e regarded as ha ving roughly three
categories. The microscopic descriptions of single, elemen tary en tities are the first category:
The h ydrogen atom, the photon or p erhaps for quantum optics the Ja ynes-Cummings mo del [1].
These problems w ere historically the first to b e in v estigated – they are relativ ely simple, pro vide
the theoretical fo oting for a manifold of phenomena and laid the foundation for the success of
quan tum ph ysics. Second the researc h focus was widened to collections of (infinitely) man y suc h
elemen tary en tities, structured in an very symmetric w a y . One example is solid state ph ysics,
where p erfectly iden tical atoms are placed in p erfectly symmetric Brav ais lattices 1 . This allo ws
to reduce the problem of an in teracting man y particle system to the study of a single unit cell
– reducing it to a few particle system. F or quan tum optics the related fields and tec hniqu es are
Glaub er’s photonic and Arecc hi’s atomic coheren t states [2, 3], the related phase space form u-
lation of quan tum optics lik e the p ositiv e P represen tation and the whole notion of quan tum to
classical transition in quan tum optics [4, 5, 6]. These tec hniques describ e large quan tum optics
setups – man y atoms and large photon n umbers. In general these theories and concepts of this
second category aim to explain our macroscopic w orld from a microscopic quan tum viewpoint.
Ho w ever in order to be ab le to describ e large quan tum systems the requiremen t of symmetry
and large size is crucial: These requiremen ts are used to reduce the exponential complexit y of
in teracting quan tum systems to something manageable.
Third, after the micro and the macro limits hav e b een established w e are no w faced with the
c hallenge of the w orld in b et w een: Mesoscopic systems are no w at the fo cus of atten tion since
micro- and nanofabrication to ols ha v e made tremendous progress and ha ve opened a world of
size dep enden t ph ysics. Omitting the limits in ven ted for w ell ordered macro ob jects requires new
1 There is also disorder in solid state ph ysics, but this disorder is usually treated as a p erturbation, th us a small
deviation from the symmetric case.
1

2 In tro duction
metho dologies. At first sigh t when dealing with a mesoscopic in teracting quan tum system w e
are left with the exp onen tially scaling Hilb ert space. This limits full, straigh tforw ard mo deling
of in teracting N b o dy problems to v ery small n um b ers N .
Ho w ever progress has been made in v arious directions: DMR G and other matrix pro duct state
based metho ds [7, 8], correlation and cluster expansion, mean field theory [9], p erturbativ e
metho ds [10, 11], etc. and sometimes exactly solv able to y mo dels [10, 12]. These metho ds eac h
harv est sp ecialized symmetries, facilitating prop erties and/or use truncation in order to b ypass
solving a system of exp onen tial complexit y . Almost all these metho ds are numerical recipes,
hea vily relying on the p o w er of mo dern computers.
A mesoscopic quan tum optical system ma y consist of a set of quantum e mitters coupled to quan-
tized and/or classical radiation mo des. In these systems not only the problem of an exp onen tial
Hilb ert space needs to b e addressed but also system bath in teraction is generally imp ortan t
[13]. Thus an open system description is needed whic h further complicates the mathematical
problem. The metho ds commonly used in the field ma y solv e these systems in certain limits –
most prominen tly the w eak correlation and large system size limits [4, 6, 9]. Ho w ev er for quan-
tum optical systems with mo derate emitter n um b ers and strong correlations these approac hes
are not suitable. Since quan tum correlations are b eliev ed to b e essen tially imp ortan t in exactly
this few particle limit [14] a metho d for inv estigating strong correlations in few particle op en
quan tum systems is highly desirable.
In this thesis an exact formalism for indistinguishable quan tum emitters is dev elop ed that fills
the gap b et w een the large and the small system size limits and w orks up to arbitrary correlation
strengths. The formalism utilizes the p ermutation symmetry induced b y the indistinguishabilit y
of the emitters and is form ulated in a Lindblad quan tum master equation con text. This treat-
men t reduces the exp onen tial complexit y of the in teracting man y emitter setup to a p olynomial
complexit y , th us greatly enhancing the application range of n umerical treatmen ts in these sys-
tems.
In quan tum optics Lindblad quan tum master equations for setups in v olving man y quantum
emitters ha v e b een studied in a v ast n um b er of differen t con texts, suc h as lasing and laser like
action [15, 16, 17], v arious phase transitions [18, 19, 20], optical bistabilit y [21], co op erativ e
resonance fluorescence [19], en tanglemen t [22], quan tum ligh t generation [23, 24], quan tum to
classical transitions [5], sup er- and subradiance etc [16, 5, 25]. The presen ted formalism allo ws
to treat all these phenomena on an equal theoretical fo oting and is esp ecially adv an tageous o v er
existing tec hniques in the few particle and strong correlation limit.
The formalism is deriv ed b y directly in v estigating the equations of motion and is then reform u-
lated using symmetrized Liouville space basis states and elemen tary op erators acting on these
basis states. This allo ws for maximal flexibilit y in the construction of master equations to fully
harv est all the p ossibilities of this formalism. This treatmen t in principle also allo ws for op en
system form ulations other than the Born-Mark ov appro ximation in the Lindblad equation. One
of the main features in the presen t discussion is the translation of these Liouville space states
and op erators in to a graphical represen tation that greatly facilitates the usage. The resulting
sk etc hes are found to hav e close connections to the Lie algebras su ( N ) and graph theory . The
whole metho d is implemen ted in a general and mo dular w a y in a C++ library called PsiQuaSP ,
whic h allo ws to simulate arbitrary master equations based on a n um b er state represen tation,
esp ecially those in v olving indistinguishable quan tum emitters. PsiQuaSP is an acron ym for
Permutation symmetry for identic al Quantum Systems Package and the library has b een made
publicly a v ailable [26, 27].
The metho d and the library are applied to the t w o main cases of cQED lasers and sup er-
/subradiance. In the latter case a new t yp e of phase transition is predicted that leads to a de-
terministic generation of dark states through ca vit y assisted coherences. A dditional symmetries

3
in quan tum master equations and v arious tec hniques for complexit y reduction and computational
sp eedup that are compatible with the metho d and the library are also discussed. In addition
the effects of dissipativ e quan tum optical mo des in media, more precise at metal in terfaces, on
the emission prop erties of individual quan tum dots are in v estigated using a non-p erturbativ e
system bath in teraction theory . F urther p ossible applications, esp ecially the connection of the
indistinguishable emitter formalism and the PsiQuaSP library to dissipativ e quan tum modes in
media are discussed.
Structure of the thesis
This thesis is structured in four parts: In P art I the theoretical foundations for the follo wing
discussion are laid out: In Chapter 2 the theoretical description of the mo del systems considered
in this thesis are presen ted, namely m ulti-lev el systems for quan tum emitters and v arious treat-
men ts for quan tized electromagnetic fields. In Chapter 3 the standard op en system description
of quan tum optics, namely the Lindblad equation, is in tro duced and some prop erties and con-
cepts of this theory are explained.
In P art I I the p erm utation symmetric, op en few and man y emitter formalism is in tro duced and
the usage of the PsiQuaSP library is explained. Chapter 4 is an in tro duction to the closed
system form ulation of iden tical quan tum emitters in quan tum optics. Also basic concepts of
Lie algebras are in tro duced that are relev an t to the presen t discussion. The deriv ation of the
op en system theory is presen ted in Chapter 5. Here tw o differen t approac hes are presen ted that
together giv e a comprehensible view on the p ossibilities and the mathematical as w ell as ph ysi-
cal prop erties of the approac h. In this discussion the sk etch represen tation is in tro duced using
man y examples. The extensiv e discussion of the formalism, esp ecially the sk etc hes, allo ws to
completely omit an y equations of motion thereafter. Directly after this discussion in App endix
A the direct translation of this formalism in to n umerical sim ulations using the PsiQuaSP li-
brary is explained. F urthermore a discussion on p erformance and differen t solution strategies
for quan tum master equations is presen ted that cov ers mo dern, state-of-the-art Krylov subspace
metho ds.
The application of the formalism and the library is presen ted in P art I I I. The discussion starts
with v arious cQED laser and spaser examples in Chapter 6. Thereafter collectiv e effects in the
op en Dic k e mo del are in v estigated in Chapter 7. In this chapter a new t yp e of phase transition
is predicted that leads to a deterministic generation of subradian t states through ca vit y assisted
coherences. In Chapter 8 a collab oration with an exp erimen tal group is presen ted, in v estigating
the effect of thin metal films on the radiativ e lifetimes of individual quan tum dots. In this
c hapter a theory for dissipativ e quan tized electromagnetic mo des, including strong system bath
in teractions is used to explain the exp erimen tal findings. In Chapter 9 further usages of the
library are presen ted: The treatmen t of non-iden tical quan tum emitters and a graph theory
based algorithm for reduction of n umerical effort are presen ted.
The last P art IV summarizes the findings and pro vides additional detailed information through
the app endices.

4 In tro duction

I Theoretical Bac kground
5

2 Mo del systems
The field of quan tum optics is concerned with the generation and manifestation of quan tum ligh t
or quan tum ligh t matter coupling [28]. Ligh t is emitted and absorb ed b y matter, suc h as atoms,
molecules, quan tum dots, lattice defects in solids, etc. In full generalit y these emitters usually
liv e in an infinite Hilb ert space with a complex o v erall structure. Ho w ev er for the understanding
of the ligh t matter in teraction it is often sufficien t to use simple appro ximations of these systems:
In order to describ e the in teraction of suc h a system with a single mo de of the electromagnetic
field one ma y just consider the t w o states of the Hilb ert space that form a resonan t transition
with this mo de. Hence, dep ending on exp erimen tal conditions, a generic quan tum emitter can
b e appro ximated b y a t w o-lev el system. If needed the mo del can alw a ys b e extended to more
lev els, forming three-, four- and m ulti-level systems.
The mo des of the electromagnetic field considered in quan tum optics w ere historically mostly
v acuum fields: Classical coheren t electromagnetic w a ves, as appro ximately produced by a sin-
gle mo de laser, and canonically quantized cavit y mo des. T ogether with the m ulti-lev el system
description of matter this leads to a non-relativistic or lo w-energy v ersion of quan tum electro-
dynamics, whic h is called ca vit y quan tum electro dynamics (cQED). Spark ed b y the adv ances in
nanofabrication to ols this fo cus has widened and no w ada ys also electromagnetic fields and mo des
in media are considered. F or example strips of dielectrics – also called w a v eguides – supp ort tra v-
eling electromagnetic or photonic mo des, t wo-dimensional dielectric slabs with holes arranged
in a regular Bra v ais lattice form photonic crystals, where photon mo des ha v e a t wo-dimensional
band structure. Also metallic nanostructures exhibit classical and quan tized elementar y excita-
tion mo des: Surface plasmons and surface plasmon p olaritions are quasiparticle like excitations
at metal in terfaces and metal nanoparticles b ear lo calized surface plasmons. In all these systems
disp ersion and dissipation is presen t, whic h giv es rise to new in teresting ph ysics but also p oses
a more c hallenging mathematical problem.
In this c hapter the theoretical quan tum optical concepts and mo del systems used in this thesis
7

8 Mo del systems
are in tro duced: Multi-lev el systems for quan tum emitters an d classical electromagnetic w a v es,
photon mo des as w ell as disp ersive modes for surface plasmons.
2.1 Quan tum emitters
The simplest mo del for a quan tum emitter is the t w o-level system. T w o-lev el systems can b e
used to describ e a h uge v ariet y of different ph ysical systems, lik e quan tum dots [29], nitrogen
or silicon v acancy cen ters in diamond [30], dy e molecules and electron or n uclear spins in solid
state en vironmen ts [31], see Fig. 2.1. The in teraction of single t wo-lev el systems with the elec-
tromagnetic field is one of the most fundamen tal problems in quan tum optics. The dressed
states of the Ja ynes-Cummings mo del, Rabi oscillations and the Purcell effect can b e explained
on this lev el of theory [1, 32]. W ell kno wn ph ysical effects in three-lev el systems are coheren t
p opulation trapping, stimulated Raman adiabatic passage (STIRAP) and electromagnetically
induced transparency [15, 33]. An example for a four-level system effect is the quan tum dot
biexciton cascade [34, 24].
These quan tum emitters can either couple to the electromagnetic field, but also directly cou-
pled emitters without considering photonic degrees of freedom are studied. Examples range
from man y b o dy mo dels suc h as spin c hains [35] or the Lipkin-Meshko v-Glic k mo del [10] to
electronically addressed spins in quan tum information science [36]. Material platforms that are
of in terest for related exp erimen ts are for instance adatoms on top ological sup erconductors or
electronic and n uclear spins in silicon.
Ov erall there is a h uge v ariet y of actual ph ysical lab oratory systems and theoretical questions
cen tered around quan tum emitters or m ulti-lev el systems.
2.1.1 T w o-lev el Systems
The simplest non trivial quan tum mechanical system is the t w o-lev el system. Prob ably due to
its simplicit y it is one of the most p opular systems in quan tum optics and quan tum information
theory . It is describ ed b y t w o quantum states | 0 ⟩ and | 1 ⟩ , whic h are dra wn in Fig. 2.2 (a). The
dynamics of a single t w o-level system can be describ ed b y the four spin matrices
σ 11 = | 1 ⟩⟨ 1 | , σ 01 = | 0 ⟩⟨ 1 | ,
σ 10 = | 1 ⟩⟨ 0 | , σ 00 = | 0 ⟩⟨ 0 | . (2.1.1)
The matrices σ 11 and σ 00 describ e the p opulation in the upp er and lo w er level. Thus the
Hamiltonian of a single t w o-level system can be written as
H 0 = ℏ ω σ 11 . (2.1.2)
This is equiv alen t to setting the energy of the lo w er state | 0 ⟩ to zero 1 .
The matrices σ 10 , σ 01 describ e the transition or flip pro cesses b etw een the t w o states
σ 01 | 0 ⟩ = 0 , σ 01 | 1 ⟩ = | 0 ⟩ , σ 10 | 0 ⟩ = | 1 ⟩ , σ 10 | 1 ⟩ = 0 .
These matrices ob ey the comm utation relations
[ σ 10 , σ 01 ] = σ 11 − σ 00 , [ σ 10 , σ 11 ] = − σ 10 , [ σ 01 , σ 11 ] = σ 01 , (2.1.3)
whic h are the comm utation relations of the su (2) Lie algebra [42].
1 Some authors prefer to set the zero of energy b etw een these levels, arriving at H 0 = ℏ ω
2 ( σ 11 − σ 00 ) = ℏ ω σ z .
Since in quan tum mec hanics one usually only deals with relativ e energies this c hoice is arbitrary and in this work
the con v en tion Eq. (2.1.2) will b e used throughout.

2.1. Quan tum emitters 9
(a) (b)
(c) (d) (e)
(f ) (g)
Figure 2.1 – Material platforms for quan tum optics and quan tum information:
(a) Sk etc h of a colloidal core-shell quantum dot. (b) Photoluminescence of colloidal quantum
dot solutions. Dep ending on size and comp osition the energy sp ectrum of these nano-ob jects
c hanges. T ak en from [37]. (c) Nitrogen v acancy cen ter in diamond and (d) Silicon v acancy
cen ter in diamond. T ak en from Ref. [38]. Diamond v acancy cen ters are p opular in quan tum
optics due to their long coherence times and small incoheren t broadening. (e) Oregon Green
organic dy e molecule. This dy e was used b y Nogonio v et al. in their spaser setup [39]. T ak en
from Ref. [40]. (f ) T wo electronic spins close to eac h other in a solid state setup. T ak en
from Ref. [41]. (g) A single phosphorous atom in a solid state atom allo ws to couple the
electron and n uclear spin and to prepare them in en tangled states. T ak en from Ref. [36].
Dep ending on exp erimen tal conditions, all these systems can b e treated as either t w o-,
three-, or m ulti-lev el systems.

10 Mo del systems
2.1.2 Three and more lev els
Three and more lev els in tro duce more complexit y in to the theory , giv e rise to new ph ysical
phenomena and also pro vide a more realistic picture in most cases 2 .
Three lev el systems are represen ted by three lev els | 0 ⟩ , | 1 ⟩ and | 2 ⟩ . The dynamics of suc h a
three-lev el system is then describ ed b y the nine matrices
σ 22 = | 2 ⟩⟨ 2 | , σ 12 = | 1 ⟩⟨ 2 | , σ 02 = | 0 ⟩⟨ 2 | ,
σ 21 = | 2 ⟩⟨ 1 | , σ 11 = | 1 ⟩⟨ 1 | , σ 01 = | 0 ⟩⟨ 1 | ,
σ 20 = | 2 ⟩⟨ 0 | , σ 10 = | 1 ⟩⟨ 0 | , σ 00 = | 0 ⟩⟨ 0 | . (2.1.4)
These matrices ob ey the comm utation relations,
[ σ xy , σ y x ] = σ xx − σ y y ,
[ σ xy , σ y z ] = σ xz ,
[ σ xy , σ xx ] = − σ xy ,
whic h, in group theoretic language, corresp onds to a su (3) algebra [42]. Th us the matrices Eq.
(2.1.4) are equiv alen t to the Gell-Mann matrices of quan tum c hromodynamics. In quan tum
optics these three-lev el systems are usually distinguished as Λ , V and Ξ systems, dep ending on
the relativ e alignmen t of the energies of the three states, see Fig. 2.2
Generally , a d -lev el system is describ ed b y the state | 0 ⟩ , . . . , | d − 1 ⟩ and its dynamics can b e
describ ed using the d 2 matrices
σ xy = | x ⟩⟨ y | , (2.1.5)
whic h ob ey the comm utation relations
[ σ xy , σ y x ] = σ xx − σ y y , (2.1.6)
[ σ xy , σ z z ] = σ xz δ y z − σ z y δ xz , (2.1.7)
(2.1.8)
and zero otherwise, whic h is equiv alen t to the su ( d ) algebra. As an example consider the
quan tum dot biexciton cascade whic h is usually represen ted b y a four-lev el system, see Fig. 2.2
d).
2 In principle the term m ulti-lev el system just refers to a quan tum system with a finite dimensional Hilb ert
space. So essen tially , whenever a computer sim ulation is run for an arbitrary quan tum system based on discr ete
basis states, whatev er quan tum system is considered is appro ximated b y a m ulti-level system, since the computer
alw a ys uses a finite excerpt of Hilb ert space. Hence also an y sim ulation in v olving photon num b er states essen tially
appro ximates that photon mo de b y a m ulti-lev el system. F or iden tical t w o-lev el systems this has a connection to
the Holstein-Primak off appro ximation and the gian t quantum oscillator [43].

2.2. Electromagnetic mo des in quan tum optics 11
Figure 2.2 – Multi-lev el systems: a) Sk etc h of a t w o-lev el system. The upp er and lo w er
lev el o ccupations can b e expressed with the sigma matrices σ 11 and σ 00 , the transition
b et w een the lev els can b e represen ted b y the spin matrices σ 01 and σ 10 . b) Ξ system with,
whic h is used for e.g. three-lev el lasers. c) Λ system whic h is imp ortan t for effects lik e
electronically induced transparency and coheren t p opulation trapping. d) F our lev el system
with energy structure t ypical for a quan tum dot biexciton cascade.
2.2 Electromagnetic mo des in quan tum optics
In the early da ys of quan tum optics the considered mo des of the electromagnetic field usually
w ere classical coheren t w a v es, as appro ximately pro duced b y a single mo de laser, and ca vit y
mo des, b oth classically and quantum mec hanically [28, 15, 32]. All these mo des are v acuum
mo des and laid the foundation to the field of ca vit y quan tum electrodynamics (cQED). The
exp erimen tally a v ailable systems at that time w ere lasers and macroscopic mirror ca vities and
these mo des pro vide the prop er theoretical description for these setups.
Confining free space mo des is limited b y the diffraction limit, therefore in the quest for minia-
turization in quan tum optical and photonic systems dielectric and metal nanostructures ha v e
gained a lot of in terest [44, 45]. Metal in terfaces hav e the p ossibilit y of confining electromagnetic
fields w ell b elo w the diffraction limit and some of the elemen tary excitations at these in terfaces
ha v e quantum properties, suc h as surface plasmons, surface plasmon p olaritons and lo calized
surface plasmons [45, 46, 47].
The quan tum effects of the quan tized electromagnetic mo des are in fact quite elusiv e [48]. Man y
of the effects in quan tum optics can b e describ ed with classical ligh t, just the matter needs to
b e quan tized. A famous and ironical example for this is the photo effect, whic h w as one of
the first effects that w ere "explained" using the photons p ostulated b y Einstein [49]. The quest
of finding "true" quan tum b eha vior is ev en more c hallenging in the field of plasmonics where
dissipation and strong dephasing induced b y in trinsic system scattering ev en ts and confinemen t
effects causes rapid deca y of quan tum coherences [47].
In this section the theory for the t w o scenarios of free space mo des and mo des supp orted b y
metal nanostructures are sk etc hed, not trying to giv e an exhaustiv e discussion but highligh ting
the differences b et w een them.
2.2.1 Electromagnetic mo des in v acuum
In free space the mo des of the electromagnetic field satisfy the homogeneous Maxw ell equations.
It is con v enien t to w ork in the Coulom b gauge, where the v ector p oten tial A ( r , t ) is c hosen to
b e div ergence free [28, 32]
∇ A ( r , t )=0 . (2.2.1)
The remaining part of the v ector p oten tial is called the transv erse v ector p oten tial, whic h satisfies
the w a ve equation
∇ 2 A ( r , t ) − 1
c 2 ∂ 2
t A ( r , t )=0 . (2.2.2)

12 Mo del systems
The solutions of this w a ve equation are plane w a v es, normalized using p erio dic b oundary con-
ditions
A ( r , t ) = 1
L 3 / 2 ε 1 / 2
0 ∑
k A k ( t ) e i kr , (2.2.3)
where L is the p erio dicit y length of the normalization condition that fixes the normalization
and w e sum o v er all allo w ed F ourier comp onen ts k . By replacing th e canonical momen tum
and p osition v ariables in these w a ves b y the canonical p osition and momentum operators and
p ostulating comm utation relations these mo des can b e quan tized. Using the relation for the
in ternal energy of the free field
U = 1
2 ∫ L 3 ( ε 0 E 2 ( r , t ) + 1
µ 0
B 2 ( r , t ) ) (2.2.4)
together with the relations
E ( r , t ) = − ∂ t A ( r , t ) , B ( r , t ) = ∇ × A ( r , t ) , (2.2.5)
one can deriv e the Hamiltonian of the free field
H = ∑
k
ℏ ω k ( b †
k b k + 1
2 ) , (2.2.6)
where b k , b †
k are the usual b osonic raising and low ering op erators for the mo de k . Considering
a one dimensional ca vit y is equiv alen t to imp osing the b oundary conditions in one dimension
only and the problem reduces to
H = ∑
k
ℏ ω k ( b †
k b k + 1
2 ) , (2.2.7)
where the k index is no w a scalar instead of a v ector. Ob viously that remo v es the electromagnetic
degrees of freedom in the other t w o dimensions whic h ha v e to b e treated differen tly , for instance
using a Lindblad quan tum master equation, see Chapter 3.
2.2.2 Dissipativ e mo des – Quan tum optics in media
In the discussion ab o v e the approac h for mo de quan tization of the free, transv erse field in v ac-
uum w as presen ted. How ev er v acuum is often not a justified assumption in mo dern exp erimen tal
setups. Setups lik e DBR micropillars [50], wa v eguides [44], photonic crystals [51], metal nanopar-
ticles [52] and the v ast field of meta materials [53] op en material platforms with ric h (quan tum)
optical prop erties that do not liv e in mere v acuum. Therefore new theoretical approac hes are
needed.
In this w ork, aside from the v acuum mo des, mainly the field of plasmonics or rather metal
nanostructures is considered. There are roughly t w o differen t approac hes in literature on ho w
to treat metal nanostructures in a quan tum dynamical w ay . These tw o will b e shortly outlined
in the follo wing. As in the v acuum case, all these deriv ations start from the Maxw ell equations
and then in tro duce comm utation relations.
Phenomenological quan tization
The bulk plasmon can b e deriv ed as a collectiv e excitation of the F ermi sea from microscopic
in teraction Hamiltonians in solid state theory – it has a sound theoretical fo oting [54]. The
surface plasmon coun terpart lac ks this theoretical fo oting, up to date it could not b e deriv ed from

2.2. Electromagnetic mo des in quan tum optics 13
suc h a microscopic theory , it remains more or less a fact from exp erimen tal exp erience [46]. Since
there is no microscopic deriv ation there ha ve been v arious approac hes for a phenomenological
quan tization or quan tum description of the surface plasmon of v arying sophistication. In the
follo wing the most straigh tforward approac h will b e briefly outlined. The details of the deriv ation
v ary from author to author but the o verall assumptions are the same. Due to its simplicit y this
approac h has b een widely used [45, 55, 56, 46, 57, 17].
Assuming a spherical metal nanoparticle with radius r ∼ 10 0 − 10 1 nm, that is excited at optical
frequencies λ ∼ 10 2 nm: The field that the metal nanoparticle exp eriences can b e assumed to
b e spatially homogeneous, which is called time harmonic appro ximation. The field of the metal
nanoparticle is then appro ximated b y a simple dip ole field along the external field axis, which
amoun ts to truncating the m ultip ole expansion at the dip ole level and then neglecting the t w o
dip ole mo des that are p erp endicular to the external field. The field of the metal nanoparticle
E np is then giv en b y
E np ∝ ϵ h − ϵ ( ω )
2 ϵ h + ϵ ( ω )
  
= g ( ω )
G i ( r ) , (2.2.8)
where ϵ h is the dielectric constan t of the medium surrounding the metal nanoparticle and ϵ ( ω )
is the frequency dep enden t dielectric function of the metal, whic h can b e describ ed b y Drude
theory [54] but is usually tak en from exp erimen tal data [58], and G i ( r ) is the spatial dep endence
of a electric dip ole field. The resp onse co efficient g ( ω ) can b e appro ximated b y a Lorentzian
around the dip ole, surface plasmon resonance
ϵ ( ω = ω sp ) = − 2 ϵ h , (2.2.9)
whic h minimizes the real part of the denominator in g ( ω ) . This is called the F röhlic h condition
for dip olar plasmon resonance [45]. A Loren tzian in frequency domain is represen ted b y a
damp ed harmonic oscillator in time domain. F or the quan tization one neglects the damping,
computes the in ternal energy of the oscillation and in tro duces comm utation relations as in the
v accum case, arriving at
H 0 = ℏ ω sp b † b (2.2.10)
and
E np ∝ G i ( r )( b † + b ) . (2.2.11)
The finite lifetime can b e rein tro duced using op en system theory e.g. the Lindblad equation.
F or more information on this approac h please refer to Ref. [57]. In Chapter 6 this approac h will
b e used to in v estigate the influence of three- and four-lev el emitters on the threshold b eha vior
of the spaser – surface plasmon amplification b y stim ulated emission of radiation [59, 60].
Dissipativ e mo des
The approac h describ ed ab o v e is simple and giv es a first estimate on the system behavior, but
it is lac king in man y regards. F or instance it is not applicable for larger and nonspherical
structures and cannot describ e effects lik e the size dep endence of the plasmon resonance of
metal nanoparticles. Also the inclusion of higher m ultip ole mo des in this quan tum theory is
not straigh tforw ard. F urthermore the strong system bath in teraction is not w ell described by a
Lindblad equation, whic h is based on a second order p erturbation theory . Th us another more
sound and sophisticated theoretical approac h is needed. The metho dology presen ted in the
follo wing w as introduced by W elsc h et al. [61, 62].
In the v acuum setting the starting p oin t w ere the free Maxwell equations. Here the starting

14 Mo del systems
p oin t are the Maxw ell equations in media [63]. One then in tro duces the electric and magnetic
field op erators ˆ
E and ˆ
B that satisfy the Maxw ell equations
∇ · ˆ
B ( r , ω )=0
∇ [ ϵ 0 ϵ ( r , ω ) ˆ
E ( r , ω )] = ˆ ρ ( r , ω )
∇ × ˆ
E ( r , ω ) = iω ˆ
B ( r , ω )
∇ × ˆ
B ( r , ω ) = − i ω
c 2 ϵ ( r , ω ) ˆ
E ( r , ω ) + µ 0 ˆ
j ( r , ω ) . (2.2.12)
Here ϵ ( r , ω ) is the complex dielectric function of the system, explicitly allo wing general geome-
tries. The op erators ˆ ρ ( r , ω ) and ˆ
j ( r , ω ) are the noise c harge and noise curren t densities, whic h
are related to the noise p olarization op erator ˆ
P ( r , ω ) as
ˆ ρ ( r , ω ) = −∇ · ˆ
P ( r , ω )
ˆ
j ( r , ω ) = − iω ˆ
P ( r , ω ) . (2.2.13)
The noise p olarization op erator ˆ
P ( r , ω ) is defined as
ˆ
P ( r , ω ) = i √ ℏ ϵ 0
π ϵ I ( r , ω ) ˆ
f ( r , ω ) . (2.2.14)
Th us all noise op erators are prop ortional to the imaginary part of the dielectric function ϵ I ( r , ω ) ,
whic h indicates an absorptiv e, dissipativ e pro cess that cannot b e describ ed in the v acuum setting.
The ˆ
f ( r , ω ) are the actual underlying op erators of the quantization procedure in this approach,
their three spatial comp onen ts ob ey the standard b osonic comm utation relations
[ ˆ
f i ( r , ω ) , ˆ
f †
j ( r ′ , ω ′ )] = δ ij δ ( r − r ′ ) δ ( ω − ω ′ )
[ ˆ
f i ( r , ω ) , ˆ
f j ( r ′ , ω ′ )] = 0 . (2.2.15)
This approac h is more tec hnical but allows to treat a v ariet y of differen t systems on a solid
theoretical fo oting. The expression for the electric field reads
ˆ
E ( r , ω ) = ˆ
E (+) ( r , ω ) + ˆ
E ( − ) ( r , ω ) , (2.2.16)
with ˆ
E (+) ( r , ω ) = ( ˆ
E ( − ) ( r , ω ) ) † and
ˆ
E ( − ) ( r , ω ) = i √ ℏ
π ϵ 0
ω 2
c 2 ∫ d 3 r ′ √ ϵ ( r ′ , ω ) G ( r , r ′ , ω ) ˆ
f ( r , ω ) , (2.2.17)
where the ω 2 /c 2 is a spherical co ordinates prefactor. The strength of this formalism lies in the
fact that the electric field op erator dep ends on the classic al dy adic Green’s function G ( r , r ′ , ω )
satisfying the equation
[ ω 2
c 2 ϵ ( r , ω ) − ∇ × ∇× ] G ( r , r ′ , ω ) = − δ
δ
δ ( r − r ′ ) . (2.2.18)
These classical dy adic Green’s functions ha ve been deriv ed for a m ultitude of differen t geometries
and can also b e calculated n umerically using partial differen tial equation solv ers [64, 65, 66].
Deriving a full quan tum theory , including F o c k states for the elemen tary b osonic excitations in
the system, is quite in v olved, but there has been recent progress in this direction [67]. Ho w ev er
in the w eak excitation limit the b osonic op erators ˆ
f ( r , ω ) can b e eliminated and closed equations
of motion can b e obtained. This approac h will b e used in Chapter 8 to explain exp erimen tal
findings of lifetimes of quan tum dots in the vicinit y of differen t silv er surfaces.

3 Op en system dynamics
The dynamics of non-relativistic, closed quan tum systems can b e describ ed with a Sc hrö dinger
equation [11]
i ℏ ∂ t | ψ ( t ) ⟩ = H ( t ) | ψ ( t ) ⟩ . (3.0.1)
There exists a scalar pro duct for the eigenfunctions of this equation and differen t eigenfunctions
to differen t eigen v alues are orthonormal, since the Hamiltonian H ( t ) is hermitian. Th us, if a
coun table n umber of eigenfunctions is considered, the eigenfunctions form a Hilb ert space basis
and closed system quan tum mec hanics essen tially reduces to linear algebra. In this picture the
state of the system is alw a ys describ ed b y a Hilb ert space v ector | ψ ( t ) ⟩ , a pure state. Statistical
mixtures, suc h as thermal distributions cannot b e describ ed at this level.
In this c hapter the densit y matrix is in tro duced and its necessit y when dealing with statistical
mixtures is explained. This leads to the concepts of mixed states and op en quantum systems.
The v on-Neumann and the Lindblad quan tum master equations are introduced, which describe
the time ev olution of densit y matrices in closed and op en systems. F urthermore some general
prop erties of the Lindblad equation are explained.
3.1 Densit y matrix
In quan tum mec hanics pure states are describ ed b y Hilb ert space v ectors | ψ ⟩ and their time
ev olution is go verned b y the Sc hrö dinger equation. A mixed state cannot b e describ ed by a
state v ector, as b y definition it liv es in m ultiple states sim ultaneously . In this con text one uses
a state matrix – the densit y matrix ρ [15, 68, 13, 57]. It can b e defined as follo ws
ρ = ∑
i,j
ρ ij | ψ i ⟩⟨ ψ j | , (3.1.1)
15

16 Op en system dynamics
where the | ψ i ⟩ form a complete set of basis states. The densit y matrix is hermitian and has
trace class 1
ρ † = ρ, tr ( ρ )=1 . (3.1.2)
Since the densit y matrix is hermitian it can alw ays be diagonalized. The exp ectation v alue of
an y op erator ˆ
O can b e calculated as
⟨ ˆ
O ⟩ = tr ( ˆ
O ρ ) . (3.1.3)
Coming bac k to the distinction b et w een pure and mixed states: It is p ossible to define a densit y
matrix for a pure state, sa y the state | ϕ ⟩ . The probabilit y of finding a pure state | ϕ ⟩ in the state
| ϕ ⟩ is, of course, equal to 1 . Therefore from Eqs. (3.1.1) w e can write
ρ pur e = | ϕ ⟩⟨ ϕ | . (3.1.4)
This is a matrix of p oten tially infinite dimension (dep ending on the n um b er of basis states),
whic h has one diagonal en try equal to 1 and all other entries are 0 . Ho w ev er the c hoice of
basis is arbitrary and w e could transform this densit y matrix in to an arbitrary basis via unitary
transformations. Since the prop ert y of diagonalizabilit y and the eigenv alues p ersevere under
unitary transformations a pure state can b e defined b y lo oking at the eigen v alues of the densit y
matrix. If the densit y matrix has only one nonzero eigen v alue, whic h has to b e 1 due to Eq.
(3.1.2), it represen ts a pure state. A mixed state on the other hand is an y densit y matrix with
more than one nonzero eigen v alue. As an example for a mixed state w e could think of a quan tum
system of finite dimension N with uniform distribution along the diagonal, i.e.
ρ mixed =
N
∑
i =1
1
N | ψ i ⟩⟨ ψ i | , (3.1.5)
whic h is a matrix that has N nonzero eigen v alues.
Diagonalization/diagonalizabilit y is a nice theoretical concept, ho w ev er actual, n umerical diag-
onalization of densit y matrices migh t b e (v ery) exp ensiv e if the dimension of the densit y matrix
is large. T o circum v ent exact diagonalization it is a popular approach to just lo ok at the purit y
γ whic h is defined as
γ ≡ tr ( ρ 2 ) . (3.1.6)
The purit y has an upp er and lo w er bound
1
N ≤ γ ≤ 1 , (3.1.7)
where the upp er b ound holds for pure states and the lo w er b ound holds for equipartitioned
states, with N b eing the dimension of the Hilb ert space, c.f. Eqs. (3.1.4) and (3.1.5).
3.2 Closed quan tum systems - v on Neumann equation
As stated ab o v e, the dynamics of a pure state is describ ed b y the Sc hrödinger equation [13, 57]
i ℏ ∂ t | ψ ( t ) ⟩ = H ( t ) | ψ ( t ) ⟩ . (3.2.1)
F rom this it follo ws that the time ev olution of a state | ψ ( t ) ⟩ can b e written in terms of a unitary
time-ev olution op erator U ( t, t 0 )
| ψ ( t ) ⟩ = U ( t, t 0 ) | ψ ( t 0 ) ⟩ (3.2.2)

3.3. Op en quan tum systems - Lindblad equation 17
with
U ( t, t 0 ) = T ← exp [ − i
ℏ ∫ t
t 0
H ( t ′ ) dt ′ ] , (3.2.3)
s where T ← refers to time ordering. W e assume that at some initial time t 0 the densit y matrix
of the system can b e written as
ρ ( t 0 ) = ∑
α
ω α | ψ α ( t 0 ) ⟩⟨ ψ α ( t 0 ) | , (3.2.4)
with the | ψ α ⟩ forming a complete basis in the Hilb ert space. Multiplying this expression with
U ( t, t 0 ) from the left and U † ( t, t 0 ) from the righ t, the time ev olution of the densit y matrix is
giv en b y
ρ ( t ) = ∑
α
ω α U ( t, t 0 ) | ψ α ( t 0 ) ⟩⟨ ψ α ( t 0 ) | U † ( t, t 0 )
= U ( t, t 0 ) ρ ( t 0 ) U † ( t, t 0 ) . (3.2.5)
Since the eigen v alues of a matrix are in v arian t under unitary transformations, this result implies
that a closed system in a pure (mixed) state will remain in a pure (mixed) state for all times.
Ev en the purit y as a quantitativ e measure for mixedness is in v arian t under unitary time ev olu-
tion.
Differen tiating Eq. (3.2.5) with resp ect to t immediately results in the v on-Neumann equation
∂ t ρ ( t ) = i
ℏ [ ρ ( t ) , H ( t ) ] . (3.2.6)
3.3 Op en quan tum systems - Lindblad equation
In the previous section the theory for time ev olution of a closed quan tum system w as outlined.
Ho w ever a closed system approac h is neither realistic, as ev ery real system is em b edded in an
en vironmen t, nor is it able to describ e ph ysical pro cesses lik e dissipation. An atom sp on taneously
emitting a photon in to the photonic en vironment, driving the system with an external source,
coupling the system to a thermal reserv oir are all pro cesses that cannot b e describ ed using a
state v ector represen tation of the system alone. T aking in to accoun t the full Hilb ert space of
system and en vironmen t w ould allo w to treat these pro cesses, ho w ev er this would result in an
unacceptably h uge Hilb ert space and th us complexit y of the associated equations.
The standard approac h in order to catc h these ph ysical pro cesses while not treating the bath
explicitly is the Lindblad quan tum master equation formalism. Here the deriv ation is briefly
outlined, for a more thorough treatmen t please refer to the literature [4, 15, 13, 69].
The complete Hamiltonian of b oth system and envi ronmen t reads
H = H S + H E + H I . (3.3.1)
Here H S , H E and H I are the system, en vironmen t and in teraction Hamiltonians, where the
in teraction Hamiltonian can b e explicitly time dep enden t, i.e. H I = H I ( t ) , c.f. Fig. 3.1.
The densit y matrix of the com bined sys tem and en vironmen t shall b e denoted as χ and R 0
resp ectiv ely , the latter is assumed to b e constan t or rather the en vironment should b e large
enough to b e unaffected b y the coupling to the system. The density matrix ρ of the system is
reco v ered by tracing o v er the en vironmen t degrees of freedom
ρ = tr E ( χ ) . (3.3.2)

18 Op en system dynamics
Figure 3.1 – Sc heme of an op en quan tum system: System and bath are eac h de-
scrib ed b y a Hamillonian and densit y matrix, the in teraction is describ ed b y an in teraction
Hamiltonian and tracing out the bath degrees of freedom in the Born-Mark o v appro ximation
results in the Lindblad equation.
Ev en if the total state χ of system and en vironmen t is in a pure state, the action of the partial
trace generally results in mixed state of the system. The partial trace only then results in a
pure state of the system if the reduced densit y matrices of system and bath factorize, whic h
is equiv alen t to the absence of quan tum correlations. F or finite in teraction strengths b et w een
system and bath this will nev er o ccur. F ormally integrating the v on-Neumann equation (Eq.
(3.2.6)) in the in teraction picture, reinserting the result and tracing o v er the en vironmen t results
in
˙
˜ ρ = − 1
ℏ 2 ∫ t
0
dt ′ tr E [ H I ( t ) , [ H I ( t ′ ) , ˜ χ ( t ′ )] ] , (3.3.3)
where the ˜ . . . indicates in teraction picture represen tation. The integration constan ts can b e
sho wn to b e zero (see for example Ref. [4]).
The next step in the deriv ation is a Born appro ximation (second order p erturbation theory):
Con tributions higher than second order in H I in this equation are neglected, i.e. one writes
˜ χ ( t ) = ˜ ρ ( t ) R 0 + O ( H I ) and drops the O ( H I ) term. F urthermore the time ev olution of the
system due to the system-bath in teraction is assumed to b e Mark o vian, i.e. the replacemen t
˜ χ ( t ′ ) → ˜ χ ( t ) in the ab o v e equation is made. This results in the Born-Mark o v quan tum master
equation in the in teraction picture
˙
˜ ρ = − 1
ℏ 2 ∫ t
0
dt ′ tr E [ H I ( t ) , [ H I ( t ′ ) , ˜ ρ ( t ) R 0 ] ] . (3.3.4)
Ev aluating the in tegral in its most general form results in the Lindblad equation [70]
˙ ρ = L ρ = i
ℏ [ ρ, H ] +
N 2 − 1
∑
k =1
γ k
2 ( 2 A k ρA †
k − A †
k A k ρ − ρA †
k A k )
  
D k ( ρ )
. (3.3.5)
Here A †
k and A k are general system op erators, the damping constan ts γ k are p ositiv e, real v alued
and N 2 − 1 is the dimension of the Liouville space asso ciated to the system degrees of freedom
( N = dim ( H S ) ). The terms D k ( ρ ) are often called Lindblad dissipators and describ e the system
bath in teraction. The general (sup er-) op erator or Liouvillian L defines a linear op erator on
the Liouville space – the space of densit y matrices. Linear mappings are the topic of linear
algebra, th us also finding the solution of a Lindblad equation b oils do wn to linear algebra, see
next Section 3.4.

3.4. Steady states and the Liouvillian sp ectrum 19
Lo oking at the exp ectation v alue of a general system op erator J and using the cyclic prop ert y
of the trace i.e. tr ( J ρ ) = tr ( ρJ ) it is p ossible to write do wn the Heisen b erg picture Lindblad
equation [71]
˙
J = L † J = − i
ℏ [ J, H ] +
N 2 − 1
∑
k =1
γ k
2 ( 2 A †
k J A k − A †
k A k J − J A †
k A k ) . (3.3.6)
A direct consequence of this expression is that the trace of the densit y matrix is conserv ed
[72]: The trace is the exp ectation v alue of the Hilb ert space iden tit y I : tr ( ρ ) = tr ( I ρ ) . Since
˙
I = L † I = 0 it follo ws that ∂ t tr ( I ρ ) = 0 . Generally , any operator O that comm utes with the
Hamiltonian and the op erators in the Lindblad dissipators represen ts a conserv ed quan tity of
the system, since its exp ectation v alue has deriv ativ e zero
[ O , H ]=[ O , A k ] = [ O , A †
k ] = 0 → ∂ t tr ( O ρ )=0 . (3.3.7)
This observ ation will b ecome imp ortant in P art I I when iden tifying the actual degrees of freedom
of quan tum optical systems describ ed b y a Lindblad equation. Please note that the rev erse
argumen t do es not hold, it is p ossible that ∂ t tr ( O ρ )=0 ev en though the op erator do es not
comm ute with all the ingredien ts of the Liouvillian H , A k and A k .
3.4 Steady states and the Liouvillian sp ectrum
F or a closed system describ ed by the v on-Neumann equation, each eigenstate of the Hamiltonian
represen ts a steady state of the system: If H | ψ i ⟩ = E i | ψ i ⟩ then | ψ i ⟩ is a steady state of the
system, since [ | ψ i ⟩⟨ ψ i | , H ] = 0 . If the there are N eigenstates of the Hamiltonian, there are N
p ossible distinct steady states 1 .
A single t w o-level system subject to sp on taneous emission is describ ed b y the Lindblad equation
˙ ρ = L ρ = γ
2 (2 σ 01 ρσ 10 − σ 11 ρ − ρσ 11 ) . (3.4.1)
F rom ph ysical in tuition alone it is clear that the steady state of this system has to b e the ground
state. Th us ev en tough the dimension of the Liouville space is 3 the steady state is unique [72].
This raises the question when, wh y and ho w the steady state of a Lindblad master equation is
unique, whether there are p ossibilities of m ultiple steady states and ho w they o ccur.
Con trary to the Hamiltonian the Liouvillian L is not hermitian (compare L and L † in Eqs.
(3.3.5) and (3.3.6)). Th us L it is not necessarily diagonalizable, its eigen v alues do not need to
b e real and are generally not real for ph ysically relev an t systems [69]. Ho w ev er there are some
general kno wn prop erties ab out the eigen v alues of the Liouvillian L [72, 69, 71]: (i) the real
parts of the eigen v alues are zero or negativ e, (ii) there is alw a ys at least one zero eigenv alue and
the asso ciated densit y matrix is the steady state densit y matrix ρ ss
˙ ρ ss = L ρ ss = 0 (3.4.2)
and (iii) there alw a ys exists a real v alued eigen v alue λ 1 with largest, non-zero magnitude, whic h
represen ts the slo west steady state con v ergence time λ − 1
1 , see Fig. 3.2. The absolute v alue of
this eigen v alue, whic h is the difference b et w een λ 0 = 0 and λ 1 is called the Liouvillian gap
∆ . If this eigen v alue λ 1 go es to zero (the Liouvillian gap closes) a new steady state is formed.
If this happ ens due to a c hange in system parameters this phenomenon is called a dissipativ e
phase transition (DPT). This nomenclature is inspired b y the field of quan tum phase transitions
1 If there are degenerate eigenstates | ψ ⟩ and | ϕ ⟩ , there ma y b e steady state coherences | ϕ ⟩⟨ ψ | .

20 Op en system dynamics
Re [ λ ]
Im [ λ ]
∆
λ 0
λ 1
Figure 3.2 – Liouvillian sp ectrum: All the eigen v alues are lo cated in the negative, com-
plex half-plane. The red circle is the λ 0 = 0 eigen v alue, corresp onding to the steady state(s).
The Liouvillian gap ∆ is determined b y the smallest magnitude non-zero eigen v alue λ 1 . Red
dots corresp ond to real eigen v alues describing monotonous deca y , blue dots corresp ond to
complex eigen v alues describing oscillatory deca y and blue, dotted circles are p ossible steady
state coherences (Did not o ccur in this thesis, but included here for completeness sake).
whic h o ccur, when the first excited state and the ground state b ecome degenerate, i.e. when the
Hamiltonian gap closes [73].
Ha ving m ultiple steady states results in quan tum memory: Eac h steady state liv es in a subspace
of Liouville space and the subspaces of differen t steady states do not in teract with eac h other
according to the master equation. The existence of suc h non-in teracting subspaces is a necessary
condition for the existence of m ultiple steady states – the steady state is determined b y the
initial conditions [72]. If there are non-in teracting subspaces then in the final state of the
system information ab out the initial state is preserv ed and this information can b e asso ciated
to a conserv ed quan tity in the system.

I I F ormalism
21

4 Man y emitters – Complex quan tum systems
In quan tum optics and quan tum information theory man y w orks in v olv e not single quan tum
emitters but collections of quan tum emitters. There are man y exp erimen tal setups where man y
emitters are presen t: In lasers and related devices more emitters lead to more gain and th us to
higher p ossible output p o w ers and sometimes more emitters are needed just to o v ercome the
lasing threshold. Irradiating a sample with randomly distributed quan tum dots on a substrate
or in solution using a diffraction limited laser sp ot will result in measuring multiple dots sim ul-
taneously .
F rom a theoretical p ersp ectiv e these systems can b e split in tw o categories: The laser example
corresp onds to an additiv e logic, more emitters lead to more yield [15, 74], or more philosophical
the sum is just the sum and nothing more. Ho w ev er there are also situations where more emitters
lead to not only more yield but rather new ph ysics altogether. Suc h b ehavior is usually referred
to as collectiv e or sometimes emergen t b eha vior. Examples for suc h truly collectiv e b eha vior in
quan tum optical setups are sup er- and subradiance [25, 75, 76, 77, 78], (m ultipartite) en tangle-
men t [79, 80, 81, 82] or the scaling b eha vior at phase transitions [83, 18, 19, 14, 84, 85, 86, 20].
In this c hapter the op erators and Hilb ert space basis states for collections of identical, indis-
tinguishable m ulti-lev el systems are in tro duced. There are tw o differen t t yp es of basis states,
individual and collectiv e. F or t w o-level systems these t w o are the direct pro duct basis and the
Dic k e basis.
23

24 Man y emitters – Complex quan tum systems
1 2 1 2 1 2 1 2
Figure 4.1 – Sc hematic represen tation of the pro duct state basis for t w o t wo-lev el
systems. One writes do wn all p ossible excitation configurations and assigns a state to eac h
of these configurations.
4.1 Iden tical t w o-lev el emitters
A set of N iden tical t w o-lev el systems is describ ed b y the collectiv e op erators
J 11 =
N
∑
i =1
σ i
11 , J 01 =
N
∑
i =1
σ i
01 ,
J 10 =
N
∑
i =1
σ i
10 , J 00 =
N
∑
i =1
σ i
00 . (4.1.1)
These matrices ob ey the same comm utation relations as the individual σ xy matrices, whic h w ere
in tro duced in Chapter 2, th us also these op erators form a su (2) algebra [42]. Ho w ev er comparing
pro ducts of the single σ matrices, e.g.
σ 10 σ 01 = σ 11 (4.1.2)
with the pro ducts of their collectiv e coun terparts J
J 10 J 01 = ∑
i,j
σ i
10 σ j
01 = J 11 + ∑
i  = j
σ i
10 σ j
01
  
A
 = J 11 (4.1.3)
it b ecomes apparen t that these iden tities are violated for collections of t w o-lev el systems: The A
term distinguishes a collection of t w o-lev el systems from a single t w o-lev el system – it in tro duces
a coupling or crosstalk b et w een differen t tw o-lev el systems, whic h lies at the heart of collectiv e
effects suc h as sup erradiance.
In this thesis a n umerical metho d for solving master equations with man y m ulti-lev el systems is
in tro duced. In order to solv e a master equation n umerically , first one needs to define a basis in
whic h to expand the densit y matrix and the whole master equation. Thus replacing the operator
equation b y a c n um b er equation, since this is all that n umerics can handle.
The start of the discussion is the definition of the collectiv e ground state of the system 1
| 0 ⟩ N =
N
⨂
i =1 | 0 ⟩ i . (4.1.4)
Excited states can b e constructed from the collectiv e ground state b y applying the raising op-
erator. Since there are t w o differen t raising op erators, the individual σ i
10 and the collectiv e J 10 ,
there are t w o differen t basis represen tations. Constructing excited states from the individual
spin matrices results in the pro duct state basis and constructing basis states from the collectiv e
spin matrices results in the collectiv e Dic ke basis states. These t w o t yp es of states b ehav e quite
1 In the presence of strong coupling or quan tum phase transitions this state ma y not be the ground state
an ymore. How ever this is not important, an y state can b e used in order to construct a basis, if the raising and
lo w ering operators are known. In group theory this state would be called highest weight state – it is a standard
pro cedure to construct the Hilb ert space from a single kno wn highest w eigh t state. See Ref. [42].

4.1. Iden tical t w o-lev el emitters 25
differen tly when in teracting with a radiation mo de.
F or the closed system the prop er choice of basis is the Dic k e basis, it reduces the complexit y
dramatically and directly explains effects lik e sup er- and subradiance. Ho w ever including de-
phasing in an op en system description seemingly do es not allo w to use the Dic k e states, fa v oring
the pro duct state basis. In the following the t w o t yp es of basis states are in tro duced.
4.1.1 The pro duct state basis
Applying the individual raising op erator σ i
10 once to the collectiv e ground state results in a
singly excited state
σ i
10 | 0 ⟩ N = | 1 ⟩ i ⨂
j  = i | 0 ⟩ j ≡ | 1 , { i }⟩ N . (4.1.5)
The last expression means that a single t w o lev el-system is excited and the excited t w o-lev el
system is the t w o-level system i . and applying the raising op erator to n differen t t w o-lev el
systems results in higher excited states
⨂
i ∈ u n
σ i
10 | 0 ⟩ N = ⨂
i ∈ u n | 1 ⟩ i ⨂
j / ∈ u n | 0 ⟩ j ≡ | n, u n ⟩ N (4.1.6)
where u n = { i 1 ,...i n } is the set con taining all t wo lev el systems that are in the excited state
and n is the n um b er of t w o-lev el systems that are excited. If the t w o-lev el systems are iden tical
there is a binomial degeneracy D for | n, u n ⟩ N
D = ( N
n ) (4.1.7)
and the total size of the Hilb ert space scales exp onen tially
N
∑
n =0 ( N
n ) = 2 N , (4.1.8)
whic h is a direct consequence of the binomial theorem [87]. The complete pro duct space basis
for t w o tw o-lev el systems is dra wn in Fig. 4.1.
Lo oking at the sp on taneous emission rate from suc h a state
A ∝ ⟨ n, u n | J 01 J 10 | n, u n ⟩ = N for n > 0 (4.1.9)
it is clear that N emitters prepared in suc h a state deca y individually . The rate of photon
emission is constan t for all states, resulting in a simple exp onen tial deca y , see Fig. 4.2.
4.1.2 The Dic k e basis
Dic k e p oin ted out that the radiation of a t w o-lev el atom is affected if additional, resonan t t w o-
lev el atoms are placed in its vicinit y [25, 28]. In this case the t w o-lev el atoms in teract via
the common radiation mo de and states analogous to the spin m ultiplet states of molecular and
atomic ph ysics are created. In the con text of quan tum optics these states are called Dic k e states.
The Dic k e basis states are simultaneous eigenstates of the t w o op erators
J z = 1
2 ( J 11 − J 00 ) , (4.1.10)
J 2 = 1
4 ( J 01 J 10 + J 10 J 01 ) + J 2
z , (4.1.11)

26 Man y emitters – Complex quan tum systems
Figure 4.2 – Sup erradian t burst vs. individual deca y: Rate of photon emission Eqs.
(4.1.9) and (4.1.15) for N = 10 t w o-lev el systems. The system is prepared in the fully excited
state and relaxes sp on taneously to w ards the ground state. The sup erradian t deca y results
in a burst since the collectiv e dip ole elemen t is maximal for the half excited states ( m = 0 ).
Ov erall this leads to a fast depletion of excitation. The individual deca y reproduces the
naiv e exp ectation that N emitters radiate at N times the individual emitters deca y rate,
but still follo w a simple exp onen tial deca y . Curv es generated with the PsiQuaSP library
[26, 27], see App endix A.
with corresp onding quan tum n um b ers m and l ( l + 1) . J 2 is often called the pseudo spin op erator
since its eigen v alue in molecular and atomic ph ysics giv es the total spin of the m ulti-electron
system and l is th us called the total (pseudo) spin quan tum num b er. Dick e states are the N
particle generalization of the Bell states.
Dic k e states are usually lab eled as | l , m ⟩ . The quan tum n umber m is called in v ersion quan tum
n um b er: It equals − N / 2 in the ground state and N / 2 in the fully excited state. m and l are
b ounded b y
0 , 1
2 ≤ l ≤ N
2 ,
− l ≤ m ≤ l , (4.1.12)
where the zero holds for ev en and the 1 / 2 holds for o dd N . The actions of the collectiv e raising
and lo w ering op erators on the Dic k e states are
J 01 | l , m ⟩ = √ ( l + m )( l − m + 1) | l , m − 1 ⟩
J 10 | l , m ⟩ = √ ( l − m )( l + m + 1) | l , m + 1 ⟩ . (4.1.13)
In the closed system the in teraction of a set of identic al t w o-lev el systems to an electromagnetic
mo de is mediated b y the collectiv e raising and lo w ering op erators 2 . Th us in the closed system
the optical selection rules are
∆ m = ± 1 , ∆ l = 0 . (4.1.14)
Hence if only collectiv e optical transitions are considered the Hilb ert space splits in to non-
in teracting subspaces of dimension 2 l + 1 , with l b eing the eigen v alue of the J 2 operator. All
2 This also requires that short ranged, direct coupling b et w een the t w o-level systems, such as dipole-dip ole
coupling, can b e neglected. This induces some restrain ts on exp erimen tal realizations.

4.1. Iden tical t w o-lev el emitters 27
m
l
N = 2
1
0
− 1
1 0
N = 3
3
2
1
2
− 1
2
− 3
2
3
2
1
2
N = 4
2
1
0
− 1
− 2
2 1 0
Figure 4.3 – Sc hematic represen tation of the Dic k e state basis for 2 , 3 and 4 t w o-
lev el systems. The lo w est state of the l max = N / 2 subspace is the ground state Eq. (4.1.4)
and the lo w est state in eac h subspace is dark, if the t w o-lev el systems couple collectiv ely
to the electromagnetic surroundings. The differen t l subspaces do not couple to eac h other
through collectiv e in teractions, therefore in a closed system description the dynamics of the
system is generally restricted to the sup erradian t subspace, c haracterized b y l max = N / 2 .
collectiv e op erators comm ute with this op erator. Therefore, if the dynamics of the system is
expressible in collectiv e op erators only , the system will stay in the resp ectiv e l subspace. This is
equiv alen t to l b eing a conserved quan tit y as explained in Chapter 3. This also holds for op en
systems if only collectiv e deca y is considered.
Lo oking at the sp on taneous emission rate from a Dic k e state
A ∝ ⟨ l , m | J 01 J 10 | l , m ⟩ = ( l + m )( l − m + 1) ∝ { N 2 for l = N
2 , m = 0 ,
0 for m = − l . (4.1.15)
it b ecomes clear that the in teraction b et w een a set of N emitters prepared in a Dick e state
b eha v es highly collectiv e: The half excited states in the l = l max = N / 2 subspace deca y with
rates that scale with N 2 while the lo w est states in eac h l subspace ( m = − l ) are dark, ev en though
only the lo w est state in the l max subspace is the ground state, and the other m = − l states are
excited states. This observ ation coins the concepts sup erradiance and subradiance: In general
a sup erradian t emission pro cess of a collection of N emitters is c haracterized b y a sp on taneous
emission rate that scales sup erlinear in N and a subradian t emission pro cess is c haracterized b y
a sp on taneous emission rate that scales sublinear in N . Since the collectiv e dip ole momen t is
maximal for half excited states, a set of initially excited t w o-lev el systems deca ys in a burst if
the t w o-level systems in teract collectiv ely with the electromagnetic surrounding, see Fig. 4.2.
Since the ground state is part of the l max = N
2 subspace – also called sup erradian t subspace – it
is a p opular approac h to restrict the dynamics of the N tw o-lev el system setup to this subspace.
This amoun ts to a dramatic reduction in the degrees of freedom, from 2 N to N + 1 . Ho w ev er,
in the op en system, when dephasing and individual deca y is included these subspaces couple
to eac h other and this restriction no longer holds. In this case the straigh tforw ard approach
w ould b e to expand the densit y matrix in the direct pro duct states Eq. (4.1.6) resulting in an
exp onen tial complexit y . Luc kily this can b e circum v ented as will be seen in the next c hapter.
The m>l Dick e states can b e constructed from the l subspace ground states | l , − l ⟩ b y rep eated
application of the collectiv e raising op erator
| l , m ⟩ = ( ( l − m )!
(2 l )!( l + m )! ) J ( m + l )
10 | l , − l ⟩ . (4.1.16)

28 Man y emitters – Complex quan tum systems
Lo oking at the total n um b er of differen t Dic k e states
∑
l
2 l + 1 ∝ N 2 (4.1.17)
it is clear that some of these states ha v e to b e highly degenerate, since the total size of the
Hilb ert space is still exp onen tial, see Eq. (4.1.8). The degeneracy D l of a Dic k e state just
dep ends on l and N and is giv en b y
D l = (2 l + 1) N !
( 1
2 N + l + 1)!( 1
2 N − l )! . (4.1.18)
F rom this expression it follo ws that the sup erradian t subspace l = N / 2 alw a ys has degeneracy
1 . This immediately sho ws that the superradiant subspace is v ery small compared to the whole
Hilb ert space for mo derate or large N ( N + 1 vs 2 N ), whic h will b ecome imp ortant in Chapter
7.
The construction of arbitrary Dic k e states even for moderate N is a tedious task, since ev en
the explicit form of the lo w est Dick e states | l , − l ⟩ is complicated for l < l max = N / 2 . Ho w ev er
constructing the states of the sup erradian t subspace is quite simple, they are just the normalized,
totally symmetric sup erp osition states of matc hing excitation, i.e.
| N
2 , n − N
2 ⟩ = ( N
n ) − 1
2
S | n, u n ⟩ N . (4.1.19)
Here S is the symmetrization op erator [11]
S = ∑
P
ˆ
P , (4.1.20)
whic h is a sum o ver all permutations P generated b y the p erm utation op erator ˆ
P . As an
example consider the t w o t w o-lev el system pro duct state | 1 , { 1 }⟩ = | 1 ⟩ 1 | 0 ⟩ 2 : The action of the
symmetrization op erator pro duces the symmetric Bell state: Using Eq. (4.1.19)
( 2
1 ) − 1
2
S | 1 , { 1 }⟩ = 1
√ 2 ∑
P
ˆ
P | 1 ⟩ 1 | 0 ⟩ 2 = 1
√ 2 ( ˆ
P 12 + ˆ
P 21 ) | 1 ⟩ 1 | 0 ⟩ 2 = 1
√ 2 ( | 1 ⟩ 1 | 0 ⟩ 2 + | 0 ⟩ 1 | 1 ⟩ 2 ) .
The other Dic k e states for l < N / 2 in v olv e an ti-symmetric sup erp ositions generated b y the
an ti-symmetrization op erator A which is kno wn from fermionic F o c k states [11]. One of the
difficulties of constructing arbitrary Dic k e states stems from the fact that for t w o-lev el systems
there are no totally an ti-symmetric states for N > 2 . Mixed symmetric an ti-symmetric states
are complicated as p oin ted out b efore.
4.2 Iden tical m ulti-lev el systems
As for the t w o-lev el systems collections of iden tical d -lev el systems can b e describ ed with the
collectiv e op erators
J xy =
N
∑
i =1
σ i
xy , (4.2.1)
where the σ i
xy are the d 2 individual spin matrices or su ( d ) generators [42]. Again, the collectiv e
d -lev el op erators ob ey the same comm utation relations as their individual σ matrix coun terparts

4.2. Iden tical m ulti-level systems 29
1 2 3 4 1 2 3 4
Figure 4.4 – Three-lev el system pro duct states: Sc hematic represen tation of t w o
pro duct states for four three-lev el systems. The left state is giv en b y Eq. (4.2.6).
and again the difference b et w een individual and collectiv e op erators can b e seen at the level of
the op erator pro ducts
σ xy σ y x = σ xx (4.2.2)
with the pro ducts of their collectiv e coun terparts
J xy J y x = ∑
i,j
σ i
xy σ j
y x = J xx + ∑
i  = j
σ i
xy σ j
y x
  
A
 = J xx . (4.2.3)
The collectiv e ground state can b e written as (no ultra-strong coupling effects are considered)
| 0 ⟩ N =
N
⨂
i =1 | 0 ⟩ i (4.2.4)
and a complete basis can b e defined b y the rep eated application of the raising op erators, σ i
xy or
J xy . The individual raising op erators again result in a direct pro duct state basis. The collectiv e
raising op erators for general d -level systems lead to collectiv e states. In a Lie algebraic con text
these collectiv e states are called m ultiplet states, Arecchi et al. called these states Gelfand-
T setlein basis states in the con text of quan tum optics [88]. Ho w ev er these basis states ha v e only
b een explicitly deriv ed in the fully symmetrical case in the quan tum optics con text, to the b est
of m y kno wledge, and furthermore hav e hardly receiv ed an y atten tion at all. Ma yb e with the
adv en t of cold atom, NV cen ter and R ydb erg atom exp eriments and ev er more con trol in these
systems this will c hange in the future. An example where the symmetric three-lev el system
states ha v e b een used is Ha yn et al. [89]. There the closed system three-lev el coun terpart of
the ultra-strong coupling phase transition in the Dic k e mo del – called the sup erradian t phase
transition – w as studied in the fully symmetrical subspace of the collectiv e three-lev el system
basis states using su (3) Holstein-Primak off b osons 3 .
4.2.1 Pro duct state basis
The basis states of the individual d -level system are | 0 ⟩ i , . . . | d − 1 ⟩ i . As for the tw o-lev el sys-
tems the simplest c hoice of basis is the product state basis which can be constructed using the
individual flip op erators σ i
xy
| n 1 , u 1 , n 2 , u 2 ,...n ( d − 1) , u ( d − 1) ⟩ = ⨂
i ∈ u 1
σ i
10 ⨂
j ∈ u 2
σ j
20 . . . | 0 ⟩ N
= ⨂
i ∈ u 1 | 1 ⟩ i ⨂
j ∈ u 2 | 2 ⟩ j . . . ⨂
k / ∈ u 1 ∪ u 2 ...
| 0 ⟩ k , (4.2.5)
3 Bosonization tec hniques are p opular algebraic tec hniques since comm utation relations of bosons are generally
simpler than su ( n ) , fermionic comm utation relations. Therefore these tec hniques often allo w for analytic solutions.
Other b osonization tec hniques are the Jordan-Sc h winger bosonization and the hard sphere b osonization [90, 91].

30 Man y emitters – Complex quan tum systems
where the sets u x are disjoin t, e.g. u x ∩ u y = ∅ . This sligh tly length y lo oking expression
is actually not v ery complicated, one example for a pro duct state of a four three-lev el system
collection is
| 2 , { 1 , 4 } , 1 , { 3 }⟩ = | 1 ⟩ 1 | 0 ⟩ 2 | 2 ⟩ 3 | 1 ⟩ 4 , (4.2.6)
whic h is sho wn in Fig. 4.4. The degeneracy of one of these states is given b y a m ultinomial
co efficien t ( N
{ n d } ) = N !
n 0 ! n 1 ! . . . n ( d − 1) ! , (4.2.7)
where { n d } = { n 0 , n 1 , n 2 ,...n ( d − 1) } is the set con taining all n um b ers n x that ob ey the relation
N =
d − 1
∑
x =0
n x , (4.2.8)
whic h stems from the fact that all the sets u x are m utually disjoin t, or rather that eac h m ulti-
lev el system is represen ted b y exactly one Ket in Eq. (4.2.5). This can b e used to reco v er the
n um b er of m ulti-lev els in the ground state
n 0 = N − n 1 − n 2 . . . (4.2.9)
The total Hilb ert space dimension is reco v ered from
∑
n d + ··· + n 0 = N ( N
{ n k } ) = ( d + 1) N , (4.2.10)
whic h is a direct consequence of the m ultinomial theorem [87]. Here the sums runs o v er all
p ossible sets/n um b ers { n d } = { n 0 , n 1 , n 2 ,...n ( d − 1) } that ob ey Eq. (4.2.8).
4.2.2 The Lie algebras su ( d )
In the in tro duction it w as stated that the d -lev el system op erators form a su ( d ) Lie algebra 4 .
A t the lev el of individual spins this statement is somewhat trivial: The d -lev el system op erators
can b e represen ted b y d × d matrices and any basis in the (Liouville) space spanned b y d × d
matrices forms a su ( d ) algebra – there is nothing to b e learned from Lie algebras for individual
spins. The situation c hanges when collections of iden tical d -lev el systems are considered: In
the last section the collectiv e t wo-lev el system op erators w ere in tro duced, whic h still ob ey the
same comm utation relations as the single spin matrices. This means that also the collections of
t w o-level systems form a su (2) algebra. It b ecame clear that there are t w o w a ys to construct a
Hilb ert space basis for these collections – direct pro duct states and Dic k e states.
The Dic k e states w ere in tro duced around the realization that the op erator J 2 comm utes with all
the other collectiv e op erators. Therefore its eigenstates can b e used to construct sim ultaneous
eigenstates with one of the other op erators. Lo oking at this op erator
J 2 = 1
4 ( J 01 J 10 + J 10 J 01 ) + J 2
z = J 2
x + J 2
y + J 2
z (4.2.11)
it b ecomes clear that it is a second order p olynomial of all the collectiv e spin op erators. The
collectiv e op erators are also called the generators and the J 2 op erator is called the Casimir
4 The con v en tion in group theoretic, Lie algebraic con texts is to label the algebras with a capital N , i.e. su ( N ) ,
ho w ev er N usually refers to the total n umber of multi-lev el systems in this thesis. Therefore I write su ( d ) instead
of su ( N ) .

4.2. Iden tical m ulti-level systems 31
op erator C 1 of the Lie algebra su (2) .
Generally a su ( d ) Lie algebra represen ting a collection of N d -lev el systems has d − 1 Casimir
op erators C 1 , C 2 , ... C ( d − 1) that are p olynomials of order 2 , 3 ,...d in the generators/collectiv e
op erators [42]. All these Casimirs commute with eac h other and all other generators or collectiv e
op erators of the Lie algebra. Therefore these Casimirs can b e used to construct sim ultaneous
eigenstates with the Casimirs and d − 1 J k k collective densit y op erators 5 . Completely analogous
to the Dic k e basis the collective operators J kl do not coup le the subspaces spanned b y the
eigen v alues of the Casimir op erators since they comm ute with the Casimirs. These subspaces
are called the m ultiplets of su ( d ) . These are the same m ultiplets kno wn from e.g. sp ectroscop y
in atomic and molecular ph ysics. Therefore a system of indistinguishable d -lev el systems in a
closed system, go v erned by a v on-Neumann time ev olution, will alw a ys live in only one of these
m ultiplets as long as the d -lev el systems are indistinguishable. The dimension of these m ultiplets
scales only p olynomially in the n um b er of individual systems as compared to the exp onen tial
scaling of the direct pro duct basis, whic h mak es these considerations v ery pow erful.
4.2.3 Dic k e states for m ulti-lev el systems – su ( d ) m ultiplets
The Dic k e states are the m utiplets of su (2) : By applying the collectiv e raising and lo w ering
op erators only a subspace of the Hilb ert space can b e constructed, the su (2) m ultiplet. These
subspaces are lab eled b y the quan tum n um b er l , whic h results from the Casimir op erator J 2 .
States from differen t m ultiplets cannot b e transferred in to eac h other b y the actions of the col-
lectiv e op erators alone, th us states from differen t m ultiplets are noninteracting if only collectiv e
in teractions are considered. This is realized e.g. in the closed system dynamics of indistin-
guishable t w o-lev el systems. These multiplets do not only occur in quantum optics, but also
in particle ph ysics and atomic/molecular sp ectroscop y: Consider t w o electrons in a molecular
orbital with an tiparallel spin orien tation: Exciting the electron represen ts a singlet state, the
total spin of b oth electrons is zero. If, b y some pro cess, the spin in the excited state is flipp ed,
a spin triplet is formed This triplet can ha v e three differen t spin orien tations or three differen t
states, therefore the name triplet. The spin triplet cannot return to the spin singlet ground state
unless another spin flip o ccurs, whic h usually happ ens on slo w time scales, whic h is wh y these
states are long liv ed. This is completely analogous to the N = 2 Dic k e (or Bell) states, with
the difference that in the sp ectroscop y con text all states are excited states. In the Dick e mo del
con text the lo w est state in the triplet subspace represen ts the total ground state. Considering
more electrons with spin 1 / 2 eac h results in more differen t m ultiplets, as in the Dic k e case, see
Fig. 4.3.
The notion of spin m ultiplets is not restricted to the spin- 1 / 2 or t w o-level system case. Arecc hi
et al. called the generalization of the Dic k e states to m ulti-lev el systems in quan tum optics
the Gelfand-T setlein states [88]. They ha v e b y far not attracted the same p opularit y as their
t w o-level system coun terparts. As seen from the discussion ab o v e from a Lie algebraic, group
theoretic stance, it is quite straigh tforw ard that these states hav e to exist. Ho w ev er even for
t w o-level systems the explicit construction of the l < N / 2 subspaces is difficult and just the
form of the higher Casimir op erators C > 2 is not trivial. Therefore one usually constructs these
m ultiplets b y finding a single state in the m ultiplet and b y applying the raising and lo w ering
op erators on this state all other states of the m ultiplet can b e constructed. The collectiv e ground
state is part of the totally symmetric m ultiplet and the symmetric m ultiplet is just the normal-
ized sup erp osition of all direct pro duct states of equal excitations n k in lev el k . Therefore this
5 One can lea v e one of the d J kk op erators out in the definition of the basis states since the total n um b er of
m ulti-lev el systems is fixed. In the tw o-level system case there is one quantum n umber for the excitation.

32 Man y emitters – Complex quan tum systems
m ultiplet is easily constructed:
| n 1 , n 2 ,...n d − 1 ⟩ = ( N
{ n k } ) − 1
2
S | n 1 , u 1 , n 2 , u 2 . . . n ( d − 1) , u ( d − 1) ⟩ . (4.2.12)
The dimensionalit y of this subspace is giv en by the n um b er of p ossible n um b ers n 0 , n 1 , . . . that
satisfy
N =
d − 1
∑
k =0
n k , (4.2.13)
whic h is giv en by ( N + d − 1
N ) ∝ 1
( d − 1)! N d − 1 . (4.2.14)
This observ ation has t wo important consequences: First, if a master equation with only collec-
tiv e in teractions is considered, the dynamics can alw a ys b e reduced to the totally symmetric
m ultiplet. This greatly reduces n umerical effort. Second, in the next section w e will find that
ev en if non-collectiv e effects are considered, the dimension of the Liouville space do es not scale
exp onen tially but p olynomially , if the individual emitters are indistinguishable. This is then
found to b e equiv alen t to the densit y matrix being blo c k diagonal in the spin m ultiplet states.
This result is v ery useful for densit y matrix diagonalization, since in a blo c k diagonal matrix eac h
blo c k can b e diagonalized indep enden tly and the size of these blo c ks only scales p olynomially .
This dramatically reduces n umerical effort.
4.3 Conclusion
In this c hapter the Hilb ert space basis states for collections of iden tical m ulti-level systems
w ere in tro duced. It b ecame clear that dep ending on the c hoice of raising and lo w ering op erators
there are t w o differen t t yp es of basis states – individual and collectiv e basis states. The c hoice of
op erators is closely link ed to the considered in teraction, whether the in teraction Hamiltonian can
b e constructed from individual or collectiv e op erators. The collectiv e basis states w ere found to
b e the su ( d ) multiplets and their in teraction strength with e.g. a b osonic mo de strongly dep ends
on the quan tum n umbers.

5 P erm utation s ymmetry in quan tum master equations
While from an exp erimen tal p ersp ectiv e it is often more c hallenging to reduce the n um b er of
emitters, or rather ha v e a more con trolled sample/setup, it is usually more c hallenging from a
theoretical p ersp ectiv e to go to higher n umbers of emitters. In order for exp erimen t and theory
to meet in the mesoscale b oth sides ha v e to mak e an effort.
F rom the theoretical side the difficult y to treat more systems stems from the fact that just
straigh tforw ardly extending the microscopic, single emitter theory to a few or man y emitter
theory results in an o v erwhelming complexit y and information con ten t. There are plen t y of
examples for this explosion in complexit y: F or instance the memory requiremen t for storing a
n umerically computed w av efunction for a man y particle system ev en using a mo derate spacial
grid easily exceeds to storage capacit y of a normal computer, serv er or of all hard driv es in the
w orld [92]. This is due to the fact that the complexity for straigh tforw ardly solving these sys-
tems scales exp onen tially in system size. Scaling can mean b oth scaling in storage requirement
and computation time.
5.1 Symmetries, size and a v erages
Luc kily there are go o d and v ery general argumen ts as to wh y one should not try to approac h
these problems in suc h a straigh tforward w a y . They can b e understo o d e.g. from classical sta-
tistical mec hanics: The first argumen t is that ev en if explicitly solving a system of o v erwhelming
complexit y w as feasible it would not be a go o d idea to do so since w e w ould not understand
the answ er. If a simulation prin ts an output on the order of T erab ytes at eac h run h umans will
ha v e a very hard time to get an in tuitiv e understanding of what is going on. Or, more sp ecific,
calculating the tra jectory of a n um b er of gas particles on the order of the A v ogadro constan t will
33

34 P erm utation symmetry in quantum master equations
hardly tell us what w e w ant to kno w, what w e are able to p erceiv e and grasp, things lik e pressure
and temp erature. Another argumen t, again from classical statistical mec hanics, form ulated as
a question is: Ho w come the tra jectories of a unfathomable n um b er of w ater molecules can all
together b e describ ed b y an equation dep ending on only three spatial co ordinates – the Na vier-
Stok es equation [93]? The short answ er is: Collective behavior, symmetries and a veraging leads
to a reduction of degrees of freedom. Collective behavior suc h as temp erature, pressure and
v olume emerge from the b eha vior of the individual particles and w ould b e imp ossible to grasp
b y only lo oking at full microscopic treatmen ts.
In quan tum mec hanics the quan tities of in terest are differen t from classical statistical mec hanics
– there are quan tities lik e states, p opulations and quan tum correlations but also classical a v er-
ages and correlations. The underlying principles of the scaling in complexit y are similar alb eit
more sev ere in quan tum mechanics [94]: The Hilb ert space of an in teracting quan tum system
scales exp onen tially in system size or is alw a ys infinite in the case of e.g. photons. Ho w ever,
also in quan tum mec hanics, the solution of this o v erwhelmingly (or infinitely) complex system
is neither needed nor w ould it b e a go o d idea ev en if p ossible. F or instance dynamical pro cesses
of electrons and holes in semiconductors can b e v ery efficien tly describ ed b y a truncation in
the hierarc h y expansion of Heisen b erg equations of motion, called cluster expansion. In man y
instances this in principle infinite expansion series can b e truncated at the lo w est order, called
singlet, and still give remarkably accurate results [9]. In this con text the man y b o dy bac kground
of the solid state induces effectiv e dephasing at the single and few particle lev el th us leading
to the destruction of higher order correlations in the hierarc h y expansion and th us allo wing to
truncate at a lo w lev el [9]. A p opular approac h in metals is the use of the Boltzmann equation
for e.g. electrical transp ort, which is a tec hnique comparable to a lo w order cluster expansion
[95]. Th us in these systems the man y b o dy con tributions reduce the complexit y of the studied
effect.
There are uncon trolled, disordered, c haotic, non-symmetric systems that can b e understo o d
b y lo oking at a v erages (electron dynamics in solids [9, 95]) and there are highly symmetrical
systems, where the symmetry induces a reduction of degrees of freedom (fluids and the Na vier-
Stok es equation [93]). Both scenarios reduce the complexit y of the straigh tforw ard solution.
In this c hapter a metho d for treating iden tical m ulti-lev el systems in an op en system setting
including dephasing is deriv ed. The metho d is exact and non-appro ximate – it uses a symme-
try to reduce the n um b er of degrees of freedom. The exploited symmetry is the p erm utation
symmetry: In terc hanging an y t w o indistinguishable m ulti-lev el systems lea v es the equations of
motion in v arian t. This leads to the observ ation that a m ultinomial n um b er of densit y matrix
elemen ts are iden tical, whic h in turn reduces the n umerical complexit y from exp onen tial to
p olynomial. The strength of the metho d is that it is able to treat al l correlations – in fact it is
equiv alen t to a cluster/correlation expansion to maximal order in the multi-lev el system degrees
of freedom [4]. The p erm utation symmetry and the resulting reduction in degrees of freedom
for t w o-level system master equations has been indep enden tly observ ed b y a v ariet y of authors
using v arying lev els of sophistication [96, 97, 98, 99, 100, 17, 101, 102]. The strength of the
discussion presen ted in this c hapter compared to treatmen ts of other authors is the dev elop ed
sk etc h representation: After man y pages of deriv ations the sk etc hes allo w for a simple and in-
tuitiv e treatmen t of the formalism and asso ciated master equations. The sk etc hes are simple
to dra w, allo w to omit the explicit deriv ation of the equations of motion and at the same time
pro vide a deep er understanding of the in v olved processes. The sk etch represen tation pro vides
the foundation of the PsiQuaSP library . This library allo ws to directly translate the sk etc hes
in to co de, which greatly speeds up co de dev elopmen t time while still pro viding a clear and direct
view on the actual pro cesses in the master equation. PsiQuaSP is an acron ym for Permutation
symmetry for identic al Quantum Systems Package . The library is in tro duced in App endix A

5.2. Dissipation vs. symmetry 35
directly after this c hapter.
The c hapter is organized as follo ws: First in Section 5.2 the problem is motiv ated b y lo oking
at a simple master equation for t w o-lev el systems. In Section 5.3 the p erm utation symmetry
is iden tified in the explicit equations of motion, the reduction of degrees of freedom is deriv ed
for t w o-leve l systems and the sk etc h es are in tro duced. In Section 5.4 the findings are gener-
alized to m ulti-lev el systems. In Section 5.5 the findings are generalized and translated into a
mathematically more formal con text that allo ws for more flexibilit y . This is imp ortan t for the
library since it pro vides the foundation for implemen ting arbitrary master equations. Finally
in Section 5.6 the connection to the collectiv e Dic k e and general m ulti-lev el system m ultiplet
states is presen ted, whic h not only pro vides an in teresting viewp oin t on the whole metho dology
but also is v ery useful for densit y matrix diagonalization.
5.2 Dissipation vs. symmetry
In the last c hapter the Hilb ert space basis states for iden tical m ulti-level systems and the oper-
ators acting on these w ere in tro duced. F or a closed system description this w ould b e sufficien t.
In this c hapter the theory will b e extended to the op en system scenario. Therefore Liouville
space basis states and Liouville space op erators are introduced.
Generally in this thesis Lindblad quan tum master equations for indistinguishable emitters are
considered, whic h means that all the parameters describing the ph ysical prop erties of these
emitters in the master equation are iden tical. As an example N t w o-lev el systems coupled to
a single b osonic mo de are considered. This system is called Dic k e or T avis-Cummings model
[25, 12]. The whole setup is coupled to the electromagnetic en vironmen t, leading to sp ontaneous
emission and dissipation, see Fig. 5.1. The master equation for this setup reads
˙ ρ = i
ℏ [ ρ, H ] + D 1 → 0 ( ρ ) + D ph ( ρ ) (5.2.1)
with the Hamiltonian
H = ℏ ω 0 b † b + ℏ ω 1 J 11 + ℏ g ( J 10 b + J 01 b † ) (5.2.2)
and the Lindblad dissipators
D 1 → 0 ( ρ ) = γ
2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) , (5.2.3)
D ph ( ρ ) = κ
2 (2 bρb † − b † bρ − ρb † b ) . (5.2.4)
Here D 1 → 0 ( ρ ) describ es individual sp on taneous deca y of the t w o-lev el systems from the excited
state to the ground state, through e.g. spontaneous photon emission in to the surrounding
v acuum and D ph ( ρ ) describ es the ca vit y loss when the cavit y has non-p erfect mirrors. Using
the bare dissipators and the rotating w a v e appro ximation (R W A) in this master equation means
that it is only v alid as long as ultra-strong coupling effects can b e neglected, i.e. g ≪ ω 0 , ω 1
[103].
Lo oking at these expressions it b ecomes apparen t that all con tributions in this master equation
acting on the t w o-lev el systems can b e expressed in collectiv e t w o-level system op erators J k l ,
Eq. (4.1.1), except for the first term in Eq. (5.2.3):
D 1 → 0 ( ρ ) = γ
2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) = γ
2 (2 ∑
i
σ i
01 ρσ i
10
  
symmetry
breaking
− J 11 ρ − ρJ 11 ) .

36 P erm utation symmetry in quantum master equations
Figure 5.1 – Illustration of the op en Dic k e mo del: a) A v ariable n um b er of t w o-lev el
emitters in teracts with a b osonic ca vit y mo de and b oth mo de and t w o-lev el systems are
sub ject to loss and deca y . b) The Dic k e states and the actions of the differen t con tributions in
the master equation are sho wn. Only the individual sp on taneous emission couples differen t
Dic k e subspaces, b ecause of the J 2 symmetry breaking.
Since the collectiv e op erators comm ute with the Casimir op erator J 2 , all con tributions in the
master equation except the first term in this equation comm ute with the J 2 op erator Eq.
(4.1.11). As explained Chapters 3 and 4, without this term the exp ectation v alue of J 2 w ould
b e a conserv ed quan tit y (see Eq. (3.3.7)) and the dynamics of the whole setup could b e confined
to the sup erradian t subspace. The J 2 symmetry breaking term is the only term in the master
equation that couples the sup erradian t Dic k e states to other states and therefore this master
equation cannot b e solv ed b y expanding the master equation only in the symmetric Dic k e states.
A t first glance expanding this master equation in the direct pro duct states Eq. (4.1.6) results
in an exp onen tial complexit y , the dimension of the Hilb ert space Eq. (4.1.8) squared:
(2 N ) 2 = 4 N . (5.2.5)
This complexit y w ould limit numer ical sim ulations to v ery few emitters. F ortunately it is p os-
sible to exploit the fact that the t w o-level systems are indistinguishable in order to reduce the
exp onen tial complexit y to a p olynomial one. Dep ending on parameters and ho w man y bosonic
mo des are considered this allo ws to sim ulate up to h undreds of t w o-lev el systems on mo dern
computers. Ho w ev er the application of this metho d is only advisable in the range up to 50
emitters, since for larger emitter n um b ers the phase space form ulation of quan tum optics, lik e
the p ositiv e P represen tation, is the prop er c hoice of metho d [104, 14, 4].
The underlying symmetry that leads to this reduction in degrees of freedom is the p erm utation
symmetry of the indistinguishable emitters. The argumen ts are v ery general and can directly
b e applied to arbitrary m ulti-lev el systems. In the following this scaling will b e deriv ed using
t w o different approac hes: First by expanding the densit y matrix and the master equation in the
direct pro duct states and lo oking at the resulting equations of motion and second b y in tro ducing
symmetrized Liouville space basis states and general op erators acting on these states. These
op erators then again form a Lie algebra and the symmetrized eigenstates form the symmetric
m ultiplet of this algebra. Both approac hes are equiv alent, ho w ev er the first approac h is ph ys-
ically more in tuitiv e and simple and the second approac h is mathematically more general and
allo ws for more flexibilit y and a higher theoretical p ersp ectiv e. In v estigating the connection of
the symmetrized Liouville space basis states to the Dic k e states or su ( d ) m ultiplets rev eals that
a densit y matrix describing indistinguishable m ulti-level systems is block diagonal in this basis.
This then rev eals the close in trinsic connection b et w een the closed system Hilb ert space m ulti-
plets discussed in the last c hapter and the formalism in tro duced in this c hapter. F or t w o-lev el
systems the statemen t that the densit y matrix is blo c k diagonal in the Dic k e basis is equiv alen t
to the statemen t that the J 2 symmetry breaking con tribution in Eq. (5.2) in tro duces (diagonal)

5.3. Time ev olution of iden tical multi-lev el systems 37
coupling b et w een differen t Dick e subspaces, see Fig. 5.1 b).
Here the R W A is used for reasons of brevity , Eq. (5.2.1) is only v alid for mo derate coupling
strengths g . The R W A has no influence on the scaling b eha vior of the p erm utation symmetric
metho d. The Non-R W A terms simply lead to more terms and require finer time discretization in
n umerical in tegration algorithms. F or direct steady state computations the difference in run time
is negligible 1 . Also the ground state ma y c hange in systems where the R W A fails, ho w ev er the
existence of the quan tum phase transition in the Dic k e mo del, the sup erradian t phase transi-
tion [18, 85], is still a matter of activ e debate [105, 106]. F urthermore the system en vironmen t
coupling should then b e treated in the prop er eigen basis of the Hamilitonian [103]. Nonetheless
the Non-R W A terms do not affect the applicabilit y of the p erm utation symmetric metho d.
5.3 Time ev olution of iden tical m ulti-lev el systems
In the follo wing the master equation Eq. (5.2.1) will b e expanded in the direct pro duct states
Eq. (4.1.6) and b y carefully lo oking at the resulting equations of motion the reduction in the
degrees of freedom will b e deriv ed. The arguments dev elop ed in this discussion are then applied
to the general m ulti-lev el system case. The resulting quantities are discussed and a sk etc h
represen tation is dev elop ed that pro vides a more in tuitiv e view on the p ossible pro cesses and
their actions. F urthermore these sk etc hes allo w to iden tify additional symmetries that further
reduce the complexit y without an y approximation. The deriv ation and discussion presen ted in
this section w ere published in M. Gegg, M. Ric hter, New J. Phys. 043037 (2016) [101].
5.3.1 Time ev olution of a simple master equation
The expansion of the densit y matrix (and th us the Lindblad equation) in the tw o-lev el system
pro duct state basis in tro duced in the last chapter, Eq. (4.1.6), is giv en b y
⟨ m, u m | ρ | n, u n ⟩ . (5.3.1)
Here m and n are again the n um b er of excited t w o-lev el systems in the resp ectiv e state and u n ,
u m are the asso ciated sets of t w o-lev el system indices. The aim is to arriv e at a Liouville space
basis, whic h is a basis in the space of square matrices. The expansion of the densit y matrix in
this basis is p erformed using a Hilb ert-Sc hmidt inner pro duct:
⟨ m, u m | ρ | n, u n ⟩ = tr [ | n, u n ⟩⟨ m, u m | ρ ]
≡ ⟨ | n, u n ⟩⟨ m, u m | ⟩ . (5.3.2)
Here the matrices
| n, u n ⟩⟨ m, u m | (5.3.3)
form a complete basis in the Liouville space of the set of N t w o-lev el systems. The total n um b er
of these op erators is (2 N ) 2 = 4 N , whic h is the brute force complexit y of this problem.
In Eq. (5.3.3) ev ery t w o-lev el system is represen ted b y one ket | k ⟩ i and one bra ⟨ l | i , whic h results
in a spin matrix | k ⟩ i ⟨ l | i for t w o-lev el system i :
| n, u n ⟩⟨ m, u m | = . . . | i ⟩ k . . . ⟨ j | k · · · = . . . | i ⟩ k ⟨ j | k
  
= σ k
ij
. . . (5.3.4)
1 This is only true as long as b oth R W A and Non-R W A Liouvillians do not ha v e an explicit time dep endence.
The use of eigen v alue based steady state solv ers requires a constan t Liouvillian. If there is an explicit time
dep endence then direct steady state computation requires tec hniques suc h as pseudo time stepping (or direct
in tegration using R unge-Kutta), see Section A.6.

38 P erm utation symmetry in quantum master equations
There are four p ossible single t w o lev el system op erators: σ i
11 , σ i
10 , σ i
01 , and σ i
00 , see Eq. (2.1.1).
It is b eneficial to pro ceed b y coun ting the n um b er of t w o-lev el systems that are represented b y
the four differen t spin matrices: let n 11 b e the n um b er of t w o-level systems that are represen ted
b y σ i
11 in Eqs. (5.3.3), (5.3.4) and let u 11 b e the set of the resp ective labels. Analogous n 10 , n 01 ,
n 00 and u 10 , u 01 , u 00 are the n um b ers and sets for the other three cases. Since ev ery t w o-lev el
system is represen ted b y exactly one spin matrix these quantities ob ey the relations
N = n 11 + n 10 + n 01 + n 00 (5.3.5)
and
u N = u 11 ∪ u 10 ∪ u 01 ∪ u 00 , (5.3.6)
where u N = { 1 ,...N } is the set con taining all tw o-lev el system indices/lab els. Rewriting the
basis elemen t (5.3.3) using these quan tities yields
| n, u n ⟩⟨ m, u m | = | n 11 + n 10 , u 11 ∪ u 10 ⟩⟨ n 11 + n 01 , u 11 ∪ u 01 |
≡ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | . (5.3.7)
So far ev erything amoun ts to relab eling. Please note that n 00 and u 00 do not en ter this ex-
pression: As in the definition of the excited state (v ector) basis (4.1.6), the t w o-lev el systems
in the ground state can b e omitted, b ecause the total n um b er of t w o-level systems is fixed and
th us the information ab out the ground state t w o-lev el systems can b e reco v ered (from (5.3.5)
and (5.3.6)). An imp ortan t quan tit y is the n um b er of basis elemen ts for fixed n 11 , n 10 , n 01 but
v ariable sets u 11 , u 10 , u 01 . It is giv en b y the multinomial coefficient [87], using Eq. (5.3.5)
C ( n 11 , n 10 , n 01 ) = ( N
n 11 , n 10 , n 01 , n 00 )
= N !
n 11 ! n 10 ! n 01 !( N − n 11 − n 10 − n 01 )! . (5.3.8)
The Hilb ert space completeness relation in this notation is given b y
I H =
N
∑
n =0 S | n, u , 0 , ∅ ⟩⟨ n, u , 0 , ∅ | , (5.3.9)
where ∅ = {} is the empt y set and the symmetrization op erator S creates a sum o v er all possible
sets u .
In the follo wing the time ev olution of the densit y matrix elemen ts asso ciated to the basis in
(5.3.7) according to the master equation Eq. (5.2.1) will b e deriv ed and the effects of the
p erm utation symmetry will b e iden tified, whic h results in the p olynomial complexit y .
5.3.2 Time ev olution – Simplification
The equations of motions are obtained b y expanding the quan tum master equation in the excited
state basis b y using
∂ t ⟨ ˆ
O ⟩ = tr [ ˆ
O ∂ t ρ ] = i
ℏ tr [ ˆ
O [ ρ, H ] ] + tr [ ˆ
O ∑
i D [ A i ] ] , (5.3.10)
where ˆ
O will b e the basis elemen ts defined in (5.3.7). The deriv ation will b e illustrated b y
discussing the sp on taneous emission con tribution Eq. (5.2.3) in the quan tum master equation
Eq. (5.2.1), since it is the only term that breaks the symmetry of the Dic k e states and therefore

5.3. Time ev olution of iden tical multi-lev el systems 39
the only reason wh y the whole discussion of this c hapter has an y b enefit. The whole set of
equations arising from the op en Dic k e mo del example will b e discussed in Section 5.3.4. Here
the ca vit y mo de is omitted as it is not necessary for sp on taneous emission in to v acuum 2 .
Inserting Eqs. (5.2.3) and (5.3.7) in to Eq. (5.3.10) and rearranging the matrices in the trace
using tr [ AB C ] = tr [ C AB ] yields
∂ t ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ ⏐ ⏐ ⏐ ⏐ D 1 → 0 ( ρ )
=
γ ∑
j [ ⟨ σ j
10 | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | σ j
01 ⟩
− 1
2 ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | σ j
11 ⟩
− 1
2 ⟨ σ j
11 | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ ] . (5.3.11)
This equation of motion has three con tributions: The first term on the rhs of Eq. (5.3.11) is
the J 2 symmetry breaking con tribution, the single system op erators act on b oth sides (bra and
k et) of the basis elemen t simultaneously . The other t w o terms act on only one side eac h (bra or
k et). Lo oking at the op erator actions of the first, symmetry breaking term in Eq. (5.3.11) for
t w o-level system j b y rearranging the expression as in Eq. (5.3.4)
σ j
10 | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | σ j
01 = . . . σ j
10 σ j
k l σ j
01 · · · = . . . σ j
11 δ k 0 δ l 0 . . .
it b ecomes clear that eac h t w o-lev el system is represen ted b y a pro duct of three spin matrices.
This pro duct is nonzero only if σ j
k l = σ j
00 and yields σ j
11 , therefore summing o v er all t w o-lev el
systems results in exactly n 00 terms where in each term the single t w o-lev el system j has b een
raised to the σ j
11 . The remaining t w o terms are non-zero if the t w o-lev el system is excited on
the righ t or on the left side resp ectiv ely . Hence the J 2 symmetry breaking term acts on b oth
sides of the individual spin matrices and the J 2 symmetry preserving terms act only on one side
at a time 3 .
This can b e expressed in terms of the differen t sets of t w o-lev el systems represen ted b y the
differen t spin matrices u 11 , . . . (cf. (5.3.5), (5.3.6) and (5.3.7)):
∂ t ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ ⏐ ⏐ ⏐ ⏐ D 1 → 0 ( ρ )
=
γ [ ∑
j ∈ u 00 ⟨ | n 11 + 1 , u + j
11 , n 10 , u 10 ⟩⟨ n 11 + 1 , u + j
11 , n 01 , u 01 | ⟩
− 1
2 ∑
j ∈ u 11 ∪ u 10 ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩
− 1
2 ∑
j ∈ u 11 ∪ u 01 ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ ] , (5.3.12)
where u + j
11 = { i 1 , . . . , i n 11 , j } is the set u 11 including the additional element j . The rhs of (5.3.12)
consists of three sums o v er tw o-lev el system lab els j .
2 This amoun ts to taking the partial trace of the densit y matrix tr b [ ρ ] o v er the bosonic degrees of freedom.
3 This is the basic difference b et w een the closed system symmetry preserving con tributions and the con tributions
that break this symmetry . Closed system symmetry means that the Hamiltonian of identical d -level systems
comm utes with the su ( d ) Casimir op erators. Th us the eigen v alues corresp onding to these Casimirs are constan ts
of motion and the dynamics of the system is confined to the symmetric m ultiplet.

40 P erm utation symmetry in quantum master equations
These sums ha v e n 00 , n 11 + n 10 , and n 11 + n 01 summands regardless of the sp ecific sets of unique
t w o-level system labels u 11 , . . . . In fact the whole equation do es not dep end on the sp ecific sets
as long as the parameter of the Liouvillian γ is the same for all t w o-lev el systems. This holds
for all con tributions arising from the Lindblad equation. Up on further requiring that at some
initial time all densit y matrix en tries of the form ⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ with the
same n um b ers n 11 , n 10 , n 01 but differen t sets u 11 , . . . are equal, the information ab out the sets
is redundan t, since the equations of motion for all these states are iden tical. This requiremen t
is fulfilled if the system starts in the ground state or a thermal equilibrium state.
If the information ab out the sets is redundant it ma y b e omitted and the densit y matrix elemen ts
can b e replaced b y
⟨ | n 11 , u 11 , n 10 , u 10 ⟩⟨ n 11 , u 11 , n 01 , u 01 | ⟩ ≡ ρ [ n 11 , n 10 , n 01 ] . (5.3.13)
These quan tities con tain all information ab out the system. The action of all Liouvillians on
the densit y matrix elemen ts defined by (5.3.13) ( n xy fixed, u xy v ariable) is the same. The total
n um b er of these elemen ts (for n xy fixed) is given b y (5.3.8).
The n um b er of differen t ρ [ n 11 , n 10 , n 01 ] , i.e. the total n um b er degrees of freedom of the system
is 1
6 ( N + 1)( N + 2)( N + 3) ∼ 1
3! N 3 (5.3.14)
The p erm utation symmetry of quan tum master equations of indistinguishable t w o-lev el systems
has b een used b y other authors as w ell: T o our knowl edge the first t w o publish ed men tions of
this metho d w ere rep orted b y Sarkar and Satc hell in 1987 [96, 97]. They contributed to the
discussion of optical bistabilit y in the few emitter case, since the phase space form ulation of mas-
ter equations lik e Eq. (5.2.1) breaks do wn as emitter n um b ers b ecome as small as 50 [14, 97].
Therefore for the few emitter regime ( ≤ 50 ) the formalism in tro duced here is the prop er c hoice
of metho d and in the man y emitter regime ( > 50 ) the phase space metho ds, lik e the p ositiv e
P represen tation [104], are the prop er c hoice of metho d. The phase space metho ds result in
a F okk er-Planc k equation or associated quantum Langevin equation of fiv e dimensions for the
op en Dic k e mo del example, whic h can ev en b e solv ed analytically in some cases [4]. In reference
to the w orks of Sarkar and Satc hell, Carmic hael pro vides an explanation of this scaling in his
b o ok based on an op erator exp ectation v alue hierarc h y expansion [4]. Ho w ev er this metho d did
not receiv e m uch atten tion un til in 2012 Hartmann found this scaling b eha vior b y exploiting a
su (4) symmetry of the quan tum master equation using group theory [100, 107]. W e deriv ed this
scaling for t w o-lev el systems indep endently using the equation of motion approac h presen ted
ab o v e [17]. Also Chase, Geremia and Baragiola indep enden tly disco v ered this metho dology in
2008 b y directly expanding the densit y matrix in the full Dick e basis and realizing that the den-
sit y matrix is blo c k diagonal in this basis [98, 99]. Also in the con text of quan tum tomograph y
it w as p oin ted out that a densit y matrix describing indistinguishable, p erm utation symmetric
t w o-level systems leads to similar reductions in degrees of freedom [108, 109]. In recen t y ears this
metho d has receiv ed more atten tion and is no w used in v arious con texts [110, 111, 112, 102, 113].
The deriv ation in this section comes from simple and accessible argumen ts and the direct pro duct
state represen tation giv es a clear view on the underlying physics, esp ecially in the con text of co-
herences/collectivit y vs dephasing/individualization [20]. F urthermore the equations of motion
are easily deriv ed and the generalization to m ulti-level systems is straigh tforw ard. Esp ecially
the sk etc h represen tation in tro duced in the follo wing will pro ve useful compared to other treat-
men ts.
The lac k of p opularit y bac k then and the p opularit y of this metho d no w can probably b e ex-
plained b y the increase of computational p o w er in the mean time: Sarkar and Satc hell adiabati-
cally eliminated the ca vit y mo de in an equation similar to Eq. (5.2.1) and w ere able to compute

5.3. Time ev olution of iden tical multi-lev el systems 41
the Liouvillian gap for up to N = 12 t w o-lev el systems on a Cra y-1 sup ercomputer in 1987 [97].
This corresp onds to a partial diagonalization of a 455 × 455 non-hermitian matrix. Thirt y y ears
later, using our co de and mo dern Krylo v subspace metho ds w e are able to (partially) diagonalize
these same non-hermitian matrices up to order 10 6 × 10 6 on a standard p ersonal computer.
Inserting (5.3.13) into (5.3.12) yields
∂ t ρ [ n 11 , n 10 , n 01 ] ⏐ ⏐ ⏐ ⏐ D 1 → 0 ( ρ )
= γ
2 [ 2( N − n 11 − n 10 − n 01 ) ρ [ n 11 + 1 , n 10 , n 01 ]
− (2 n 11 + n 10 + n 01 ) ρ [ n 11 , n 10 , n 01 ] ] . (5.3.15)
The action of this Liouvillian on the densit y matrix elemen t ρ [ n 11 , n 10 , n 01 ] con tains in- and
out-scattering con tributions: The in-scattering con tribution (first term, J 2 symmetry breaking)
stems from a densit y matrix elemen t that has a higher n um b er of emitters in the excited state,
i.e. n 11 + 1 . The out-scattering con tribution (second term, J 2 symmetry preserving) is prop or-
tional to the same densit y matrix elemen t ρ [ n 11 , n 10 , n 01 ] . Suc h a pro cess, where a state with a
higher excitation n um b er scatters in to a state with a lo w er excitation num b er, describ es an ex-
p onen tial deca y of excitation. The part prop ortional to n 10 + n 01 describ es the dephasing. Both
con tributions together describ e sp on taneous radiativ e deca y . The decrease in excited t w o-level
systems is reflected in the diagonal elemen ts ( n 10 = n 01 = 0 ) and the loss of coherence in the
off-diagonals ( n 10  = 0  = n 01 ).
Using the definition Eq. (5.3.13) it is p ossible to deriv e closed equations of motion for all con-
ceiv able p erm utation symmetric t wo-lev el system quan tum master equations. In principle one
could stop here and implemen t a sim ulation based on the quantit y ρ [ n 11 , n 10 , n 01 ] . Ho w ev er at
this lev el there are issues concerning the n umerical stability , whic h will b e explained and solved
in the follo wing.
5.3.3 Prev en ting n umerical instabilit y
The densit y matrix con tains all accessible information ab out a quan tum system [5]. Ho w ev er
this information is in general only indirectly accessible to an exp erimen ter through measurable
observ ables. Thus in order to mak e meaningful predictions op erator exp ectation v alues ha v e
to b e calculated. F or instance lo oking at the excited state p opulation exp ectation v alue ⟨ J 11 ⟩ ,
using (5.3.9)
⟨ J 11 ⟩ = ⟨ ∑
i
σ i
11
N
∑
n =0 S | n, u , 0 , ∅ ⟩⟨ n, u , 0 , ∅ | ⟩
=
N
∑
n =0 ( N
n ) ⟨ ∑
i
σ i
11 | n, u , 0 , ∅ ⟩⟨ n, u , 0 , ∅ | ⟩
=
N
∑
n =0 ( N
n ) n ρ [ n, 0 , 0] . (5.3.16)
Ev en though the information ab out the sets is redundan t there is still a m ultinomial num b er of
basis elemen ts for eac h ρ [ i, j, k ] , whic h is a binomial in this case, since j = k = 0 .
F rom a n umerical p oin t of view this expression is problematic: When ev aluating the binomial
for e.g. 100 tw o-lev el systems one needs to calculate and store n um b ers of dramatically differen t

42 P erm utation symmetry in quantum master equations
magnitudes i.e. ( 100
50 ) ∼ 10 29 vs. ( 100
100 ) = 1 . This reduces numerical accuracy or mak es compu-
tation imp ossible altogether: The larger binomial co efficien ts gro w faster than exp onen tially in
N , th us quic kly leaving reasonable n um b er storage formats. F urthermore since the trace of the
densit y matrix is conserv ed, i.e.
tr [ ρ ] =
N
∑
n =0 ( N
n ) ρ [ n, 0 , 0] = 1 , (5.3.17)
it is clear that the magnitudes of those elemen ts ρ [ n, 0 , 0] that are asso ciated with large binomial
co efficien ts b ecome (less than) exp onen tially small. Multiplying larger than exp onen tially large
n um b ers with smaller than exp onentially small n um b ers on a finite precision mac hine results in
enormous n umerical errors.
F ortunately , this problem can b e con v enien tly circum v en ted b y accoun ting for the m ultinomial
n um b er of ρ [ . . . ] at the lev el of the equations of motion. Simply m ultiplying the equations of
motion (lik e (5.3.15)) b y the m ultinomial n um ber of elements ρ [ n 11 , n 10 , n 01 ] Eq. (5.3.8)) and
defining
P [ n 11 , n 10 , n 01 ] = ( N
n 11 , n 10 , n 01 , n 00 ) ρ [ n 11 , n 10 , n 01 ] (5.3.18)
transforms Eq. (5.3.15) in to
∂ t P [ n 11 , n 10 , n 01 ] ⏐ ⏐ D 1 → 0 ( ρ ) = γ
2 [ 2( n 11 + 1) P [ n 11 + 1 , n 10 , n 01 ]
− (2 n 11 + n 10 + n 01 ) P [ n 11 , n 10 , n 01 ] ] . (5.3.19)
Th us only the prefactor of the first rhs term c hanges (5.3.15), whic h is due to
( N
n 11 , . . . )( N
n 11 + 1 , . . . , n 00 − 1 ) − 1
P [ n 11 + 1 , . . . ] = n 11 + 1
n 00 P [ n 11 + 1 , . . . ] .
This definition fixes the n umerical issues since the excited state exp ectation v alue
⟨ J 11 ⟩ =
N
∑
n =0
n P [ n, 0 , 0] . (5.3.20)
and the trace
tr ( ρ ) =
N
∑
n =0 P [ n, 0 , 0] . (5.3.21)
are n umerically w ell b eha v ed. The ph ysical in terpretation of P [ n 11 , n 10 , n 01 ] is also more acces-
sible than ρ [ n 11 , n 10 , n 01 ] : P [ n, 0 , 0] giv es the full, incoheren t probabilit y of finding the system in
a state with n t w o-lev el systems excited and N − n unexcited and for n 10 , n 01  = 0 P [ n 11 , n 10 , n 01 ]
giv es the full p olarization/transition probabilit y b et w een adjacen t states.
Our aim with this metho d is to use it for n umerics. The metho d is quite general – any per-
m utation symmetric master equation of t w o- (or m ulti-) lev el systems can b e solv ed. It w ould
b e tedious and error prone to deriv e equations of motion for ev ery sp ecific master equation and
then implemen t these equations b y hand. Therefore w e dev elop ed a sk etc h represen tation of the
elemen ts P [ . . . ] in whic h w e can visualize the pro cesses of the master equation and designed
the implemen tation of this metho d in PsiQuaSP in a w a y that allo ws to directly translate these
sk etc hes into code. The sk etc h illustrating the sp on taneous deca y Eq. (5.3.19) is sho wn in Fig.

5.3. Time ev olution of iden tical multi-lev el systems 43
Figure 5.2 – Sk etc h represen ting the action of the sp on taneous emission Liou-
villian: Eac h bubble represen ts a degree of freedom, eac h arro w a pro cess. Arro ws and
corresp onding terms ha v e the same color. The green (blac k) arro w depicts the loss of exci-
tation, states with higher n um b ers n 11 deca y in to states with higher n um b ers n 00 . There are
t w o terms resp onsible for this pro cess – one in- and one out-scattering term – since the total
probabilit y has to b e conserv ed. The yello w and purple (gra y) arro ws depict the dephasing
or "coherence out-scattering". The offdiagonal elemen ts ( n 10 , n 01  = 0 ) are purely damp ed.
The corresp onding arro ws p oin t to the outside, indicating this loss.
5.2. The arro w p ointing from the n 11 to the n 00 circle depicts the pro cess in whic h states with
higher n um b ers n 11 of excited t w o-lev el systems (and th us lo w er n 00 ) deca y to the states with
higher n 00 (and th us lo wer n 11 ), whic h results in a reduction of the excited state p opulation. The
arro ws p oin ting from the n 10 and n 01 circles to the outside depict the destruction of quan tum
coherence. In the next section the full equations of motion for op en Dic k e mo del example Eq.
(5.2.1) are discussed.
5.3.4 Op en Dic k e mo del equations of motion
In order to discuss the full time ev olution of the op en Dic k e mo del example Eq. (5.2.1) a basis
for the photon mo de needs to b e in tro duced. This is treated b y the usual photon num b er states
| k ⟩ and the basis for the join t system is constructed via
| k ⟩| n, u n ⟩→P [ n 11 , n 10 , n 01 ; k , p ] , (5.3.22)
where k and p are the photon degrees of freedom, i.e. ⟨ . . . | k ⟩⟨ p | ⟩ = P [ . . . ; k , p ] .
The con tributions of the Lindblad dissipators to the equations of motion are the spontaneous
emission Eq. (5.3.19) and
∂ t P [ n 11 , n 10 , n 01 ; k , p ] ⏐ ⏐ D ph ( ρ ) = κ
2 [ 2 √ ( k + 1)( p + 1) P [ n 11 , n 10 , n 01 ; k + 1 , p + 1]
− ( k + p ) P [ n 11 , n 10 , n 01 ; k , p ] ] (5.3.23)

44 P erm utation symmetry in quantum master equations
for the ca vit y photon deca y . The con tribution of the Dic k e in teraction Hamiltonian in the R W A
Eq. (5.2.2) to the quan tum master equation is giv en b y
∂ t P [ n 11 , n 10 , n 01 ; k , p ] ⏐ ⏐ H I
= ig [ ( n 01 + 1) √ k + 1 P [ n 11 − 1 , n 10 , n 01 + 1; k + 1 , p ]
+ ( n 00 + 1) √ k + 1 P [ n 11 , n 10 − 1 , n 01 ; k + 1 , p ]
+ ( n 10 + 1) √ k P [ n 11 , n 10 + 1 , n 01 ; k − 1 , p ]
+ ( n 11 + 1) √ k P [ n 11 + 1 , n 10 , n 01 − 1; k − 1 , p ]
− ( n 10 + 1) √ p + 1 P [ n 11 − 1 , n 10 + 1 , n 01 ; k , p + 1]
− ( n 00 + 1) √ p + 1 P [ n 11 , n 10 , n 01 − 1; k , p + 1]
− ( n 01 + 1) √ p P [ n 11 , n 10 , n 01 + 1; k , p − 1]
− ( n 11 + 1) √ p P [ n 11 + 1 , n 10 − 1 , n 01 ; k , p − 1] ] , (5.3.24)
where n 00 = N − n 11 − n 10 − n 01 is used for visual clarit y . The terms are all asso ciated to
pro cesses where one excitation in the t w o-lev el systems is destro y ed (created) while a photon is
created (destro y ed). The eigh t terms corresp ond to t w o op erators in the in teraction Hamiltonian
Eq. (5.2.2), eac h acting from the left and the righ t and making a distinction b et w een t w o sets
eac h n 11 , n 10 or n 11 , n 01 . This results in 2 × 2 × 2 = 8 terms. Including the Non-R W A terms
therefore leads to 4 × 2 × 2 = 16 terms. F urther including a pure dephasing con tribution
D z ( ρ ) = γ ′
N
∑
i =1
( σ i
z ρσ i
z − ρ ) (5.3.25)
whic h can b e used to mo del e.g. line broadening due to phonons [114] results in
∂ t P [ n 11 , n 10 , n 01 ; k , p ] ⏐ ⏐ D z ( ρ ) = − γ ′ ( n 10 + n 01 ) P [ n 11 , n 10 , n 01 ; k , p ] . (5.3.26)
Again in Eq. (5.3.25) the con tribution where individual spin matrices act on b oth sides of the
densit y matrix sim ultaneously (the first rhs term) is breaking the J 2 symmetry of the closed
system. The sk etc h of the action of the Dick e in teraction Hamiltonian on the t w o-lev el systems is
giv en in Fig. 5.3 b): Ev ery arro w corresp onds to one term in (5.3.24). Occupation is exc hanged
b et w een the t wo lev els 0 and 1 via the build up of quan tum coherences, as opp osed to the action
of the Lindblad dissipators, whic h are inheren tly incoherent con tributions, cf. Fig. 5.3 a).

5.4. Multi-lev el systems 45
Figure 5.3 – Sk etc h of the actions of the differen t con tributions of the op en Dic k e
mo del: Arrows pointing from circle A to circle B corresp ond to a build up of densit y matrix
en tries with higher n umber in B and lo w er n um b er in A . Arro ws p oin ting from a circle to
the outside corresp ond to dephasing, i.e. a pure deca y of the resp ectiv e (offdiagonal) densit y
matrix en try . a) A ction of the Lindblad dissipators: blue (solid) arro ws describ e sp on ta-
neous emission, purple (dashed) arrows incoheren t pumping and y ello w (dot-dashed) arro ws
pure dephasing. b) T a vis-Cummings Hamiltonian (electron-photon coupling): densities are
exc hanged b et w een n 11 and n 00 via the build up of quan tum coherence n 10 and n 01 . c) The
com bination of all pro cesses describ es the gain medium dynamics of a laser setup [17].
5.4 Multi-lev el systems
The dimension of the full Liouville space asso ciated to N d -lev el systems is d 2 N , it scales only
p olynomially in d , but still this renders simulations impossible even for moderate d and N .
F ortunately the argumen ts dev eloped in the previous section are general and not limited to the
t w o-level system case: The action of the quan tum master equation on groups of elemen ts of
the densit y matrix w as found to b e iden tical and the n umber of degrees of freedom w as greatly
reduced. Th us also for m ulti-level systems the exponential complexit y can b e reduced to a
p olynomial one. This section starts b y outlining the formal deriv ation for the three- and general
m ulti-lev el systems. Thereafter the sk etc hes and the application of the metho d is discussed using
sev eral laser examples.
5.4.1 Three-lev el systems
The simplest extension of the theory is to consider three-lev el systems. Three level systems are
often divided in to the three categories V , Λ , Ξ according to the relativ e energies of the lev els
[15, 9]. Three lev el systems are used for realistic laser theories [115], noise induced coherences
[116, 117], electromagnetically induced transparency [118], coheren t population trapping [15],
STIRAP [33] or for more realistic quan tum dot mo dels including e.g. m ultiple single exciton or
the trion state [119, 120].
In the previous c hapters the spin matrices for three-lev el systems σ i
k l and the corresp onding
man y emitter pro duct basis states
| n 1 , u n 1 , n 2 , u n 2 ⟩ = ⨂
i ∈ u n 1
σ i
10 ⨂
j ∈ u n 2
σ j
20 | 0 ⟩ N , (5.4.1)
w ere in tro duced. Here the three-lev el systems in u n 1 and u n 2 are in state | 1 ⟩ and | 2 ⟩ resp ectiv ely .
The t w o sets u n 2 and u n 1 are disjoin t u n 2 ∩ u n 1 = ∅ . A conceiv able Hamiltonian is
H 0 = ℏ ∑
m
ω m b †
m b m + ℏ
2
∑
k =0
ω k J k k (5.4.2)
H I = ℏ ∑
k  = l ∑
m
g m
k l ( J k l + J lk )( a †
m + a m )
  
H kl
, (5.4.3)

46 P erm utation symmetry in quantum master equations
including and arbitrary n um b er of electromagnetic (ca vit y) modes. P ossible Lindblad dissipators
are again sp on taneous radiativ e and non-radiativ e decay betw een individual lev els, incoheren t
pumping and pure dephasing, where again the parameter needs to b e identical for all individual
three-lev el systems. F or more information please refer to App endix E.
The asso ciated Liouville space is spanned b y the basis elemen ts
| n 1 , u n 1 , n 2 , u n 2 ⟩⟨ m 1 , u m 1 , m 2 , u m 2 | . (5.4.4)
Here n 2 giv es the n umber of three-lev el systems that are in the Ket state | 2 ⟩ i in ab o v e expression,
the asso ciated lab els are con tained in set u n 2 . The asso ciated Bra state to three-lev el system i
can either b e ⟨ 2 | i , ⟨ 1 | i , or ⟨ 0 | i , hence the set and the n um bers will b e split in to three. Analogously
to the previous section distinguishing these cases results in
n 1 = n 10 + n 11 + n 12 , u n 1 = u n 10 ∪ u n 11 ∪ u n 12 . (5.4.5)
The other n um b ers and sets split in to
n 2 = n 20 + n 21 + n 22 , u n 2 = u n 20 ∪ u n 21 ∪ u n 22
m 1 = n 01 + n 11 + n 21 , u m 1 = u n 01 ∪ u n 11 ∪ u n 21
m 2 = n 02 + n 12 + n 22 , u m 2 = u n 02 ∪ u n 12 ∪ u n 22 .
The con tributions for the ground state p opulation ( σ i
00 ) n 00 and u n 00 can b e defined through
the equalities
u N = u 00 ∪ u 01 ∪ · · · ∪ u 22 ,
N = n 00 + n 01 + · · · + n 22 . (5.4.6)
The fact that the n um b er n 00 and set u 00 can b e expressed through the other 8 n um b ers and
sets allo ws the omission. As long as the n um b er of three- (or general m ulti-) lev el systems is
fixed this elimination is p ossible and the selection of σ 00 is merely conv en tion 4 .
Again if the parameters in the master equation are iden tical for all three-lev el systems, the
information ab out the sets is redundant. In complete analogy to the previous section closed
equations can b e obtained b y in tro ducing the quan tities
tr [ | n 1 , u n 1 , n 2 , u n 2 ⟩⟨ m 1 , u m 1 , m 2 , u m 2 | ρ ]
≡ tr [ | n 10 , u 10 , n 11 , u 11 . . . n 22 , u 22 ⟩⟨ n 01 ,...n 22 , u 22 | ρ ]
≡ ρ [ n 22 , n 21 ,...n 01 ] (5.4.7)
and
P [ n 22 , n 21 ,...n 01 ] = C ( n 22 , n 21 ,...n 01 ) ρ [ n 22 , n 21 ,...n 01 ] (5.4.8)
with the m ultinomial co efficien t
C ( n 22 , n 21 ,...n 01 ) = N !
n 22 ! n 21 ! . . . n 01 ! n 00 ! . (5.4.9)
The total n um b er of degrees of freedom of this theory is
( N + 8
N ) = 1
8! ( N + 8)( N + 7) . . . ( N + 1) ∝ 1
8! N 8 . (5.4.10)
4 The lab el 00 do es not hav e to refer to the ground state, b y relab eling the lev els in the m ulti-lev el system this
can refer to an y lev el.

5.4. Multi-lev el systems 47
Figure 5.4 – Sk etc h of the action of a three-lev el system in teraction Hamiltonian:
It is apparen t that there are, additionally to the t w o-lev el system case, con tributions ex-
c hanging quan tum coherence b et w een n 20 , n 21 and n 02 , n 12 , which are completely separated
from the other dynamics of the system. The represen tation is simplified with resp ect to Fig.
5.3 to a single arro w with t w o heads. This also corresp onds to the usage in PsiQuaSP . In the
R W A this corresp onds to t w o terms in the equations of motion (cf. (5.3.24) and Fig. 5.3 b)),
without R W A there are four. The dots indicate the transition to higher m ulti-lev el system
dynamics. The prop er rules for constructing these sketc hes are giv en in Section 5.4.3.
The in teraction of lev els 0 and 1 with a single ca vit y mo de is describ ed b y the Hamiltonian (cf.
(5.4.3))
H 10 = ℏ g 10 ( a † + a )( J 10 + J 01 ) . (5.4.11)
In Fig. 5.4 the sk etch for this pro cess is sho wn. The sketc h has 3 2 = 9 circles and from Eq. (5.4.6)
one degree of freedom can b e eliminated leading to a ∝ N 9 − 1 = N 8 scaling. Additional degrees
of freedom (bubbles) arise compared to the t w o-lev el system case and additional exc hange of
quan tum coherence (arro ws). Please note that the additional exc hange of coherence is completely
decoupled from the other dynamics. This will b ecome imp ortan t in Section 5.4.3, where the
application of the p erm utation symmetric expansion sc heme is discussed at the lev el of the
sk etc hes by looking at different laser setups.
5.4.2 Multi-lev el systems
There is a large application range for m ulti-lev el systems: F our-lev el systems describ e gain
in optical devices with reduced thresholds or are mo del systems to study the quan tum-dot
biexciton cascade [34, 24]. Generally more realistic descriptions of quan tum optical systems suc h
as (coupled) quan tum dots require m ulti-level systems [120, 121], whic h leads to increasingly
complex quan tum dynamics [122, 123].
Again starting form the pro duct space basis for the Hilb ert space of the collection of d -lev el
systems Eq. (4.2.5)
| n 1 , u n 1 ,...n ( d − 1) , u n ( d − 1) ⟩ = ⨂
i ∈ u n 1
| 1 ⟩ i ⨂
j ∈ u n 2
| 2 ⟩ j . . . ⨂
k / ∈ u n 1 ∪ u n 2 ...
| 0 ⟩ k , (5.4.12)
the direct pro duct Liouville space basis is giv en b y
| n 1 , u n 1 ,...n ( d − 1) , u n ( d − 1) ⟩⟨ m 1 , u m 1 ,...m ( d − 1) , u m ( d − 1) | . (5.4.13)
Analogous to t w o- and three-lev el systems the n um b ers n 1 , . . . , m 1 , . . . and the asso ciated sets
u n 1 , . . . , u m 1 , . . . are divided in to d 2 differen t num b ers n k l and sets u k l – on e p er spin matrix

48 P erm utation symmetry in quantum master equations
σ i
k l Eq. (2.1.5):
n 1 = n 10 + n 11 . . . n 1( d − 1) , u n 1 = u n 10 ∪ u n 11 . . . u n 1( d − 1)
.
.
.
n ( d − 1) = n ( d − 1)0 + n ( d − 1)1 . . . , u n ( d − 1) = u n ( d − 1)0 ∪ u n ( d − 1)1 . . .
m 1 = n 01 + n 11 . . . n ( d − 1)1 , u m 1 = u n 01 ∪ u n 11 . . . u n ( d − 1)1
.
.
.
m ( d − 1) = n 0( d − 1) + n 1( d − 1) . . . , u m ( d − 1) = u n 0( d − 1) ∪ u n 1( d − 1) . . . (5.4.14)
and
u N = u 00 ∪ u 01 ∪ . . . u ( d − 1)( d − 1) ,
N = n 00 + n 01 + . . . n ( d − 1)( d − 1) , (5.4.15)
whic h ensure conserv ation of the n um b er of m ulti-lev el systems and allo w to eliminate one degree
of freedom, n 00 , whic h reduces the scaling b y one p o w er i.e. N m → N m − 1 . Closed equations for
p erm utation symmetric Lindblad equations are obtained for
tr [ | n 1 , u n 1 , . . . ⟩⟨ m 1 , u m 1 , . . . | ρ ]
≡ tr [ | n 10 , u 10 , . . . ⟩⟨ n 01 , u 01 , . . . | ρ ]
≡ ρ [ n ( d − 1)( d − 1) , . . . ] . (5.4.16)
Equiv alen tly , n umerical stabilit y fav ors the P represen tation
P [ n ( d − 1)( d − 1) ,...n 01 ] = C ( n ( d − 1)( d − 1) ,...n 01 ) ρ [ n ( d − 1)( d − 1) ,...n 01 ] , (5.4.17)
where C ( n ( d − 1)( d − 1) ,...n 01 ) is again the m ultinomial co efficien t
C ( n ( d − 1)( d − 1) ,...n 01 ) = N !
n ( d − 1)( d − 1) ! . . . n 01 ! n 00 ! . (5.4.18)
The n um b er of degrees of freedom of the man y d -level system solution is fixed b y the n um b er of
indices coun ting the basis elemen ts, i.e. the n um b er of differen t sets { n k l } = { n ( d − 1)( d − 1) , . . . }
satisfying the relation
N = ∑
k ,l
n k l
  
m summands
, (5.4.19)
whic h is ( N + m
N ) ∝ 1
( m − 1)! N m − 1 . (5.4.20)
The sk etc h of such a basis represen tation/master equation w ould ha v e m circles, and the scaling
of the p erm utation symmetric metho d is alwa ys prop ortional to N to the p ow er of the n um b er
of circles in the sk etc h minus one, i.e. ∝ N m − 1 . Hence the full solution of the man y d -level
system scales as ( N + d 2 − 1
N ) (5.4.21)
since d 2 is the n um b er of spin matrices for the individual d -lev el system. This scaling relation
lo oks strikingly familiar when lo oking bac k at the discussion of the Gelfand-T setlein basis states

5.4. Multi-lev el systems 49
Figure 5.5 – Lev el sc hemes for the differen t laser setups: a) t w o-lev el, b) three-lev el
and c) four-lev el optical emitters (cf. (5.4.23) and (5.4.29)). The dotted arro ws indicate
optical pumping.
or symmetric su ( d ) m ultiplets in Chapter 4. In fact it will turn out that the Liouville space
basis discussed in this section also represen ts a symmetric m ultiplet, just in Liouville instead of
Hilb ert space, and the asso ciated Lie algebra is su ( m ) , with m as in Eq. (5.4.19). This will b e
explained in Section 5.5.
In the next section the application of the p erm utation symmetric formalism is illustrated b y
lo oking at differen t laser setups. The discussion will b e cen tered around the sk etc hes, whic h
illustrates that equations and form ula can b e omitted.
5.4.3 T w o-, three- and four-lev el laser examples
In this section the application of the p erm utation symmetric metho d is illustrated and esp e-
cially the use of the sk etc h representation using differen t setups for t w o-, three- and four-lev el
system based laser setups. This serv es as an in tro duction to the sk etc h usage. The aim of this
treatmen t is to completely omit deriving an y equations of motion: The p erm utation symmetric
metho d w as implemen ted in the PsiQuaSP library [26, 27] based on this sk etc h represen tation.
There the user do es not deriv e an y equations of motion but translates the master equation
in to the sk etch represen tation in tro duced ab o v e – a matter of a few minutes – and then di-
rectly translates these sk etc hes in to co de. The library w as made publicly a v ailable on GitHub:
h ttps://gith ub.com/mo dmido/psiquasp and a man ual/in tro duction w as published in M. Gegg,
M. Ric h ter arXiv:1707.01079 (2017). Please refer to App endix A directly after this c hapter for
details on PsiQuaSP .
The range of applicabilit y of the p erm utation symmetric metho d to laser theory is the cQED
laser limit: F ew emitters ( 1 − 100 ) and relativ ely lo w ca vity photon n um b ers, the range where
in principle quan tum correlations are exp ected to b e imp ortan t. The ca vit y degrees of freedom
and the full equations of motion are omitted in this section, as they do not b enefit the under-
standing of the sk etc hes. F urther, some sp ontaneous emission con tributions and all conceiv able
pure dephasings are omitted for reasons of brevit y . The scalings/complexities of the solutions
of the resp ectiv e quan tum master equation do not c hange if these con tributions w ere included.
The only difference in the results are more terms in the equations of motion, but it do es not
affect the n umerical scaling. The sp on taneous emission of the lasing transition in to non-lasing
mo des is included, as this rate is crucial for a definition of the β factor, a central parameter in
laser theory [17, 84, 124].
The full equations are giv en in App endix D. Please note that the results or rather the n umerical
scalings observ ed in this section are v alid for b oth R W A and Non-R W A treatmen ts, th us all
conclusion of this section are v alid in b oth cases.
Fiv e differen t setups are discussed: incoheren tly driv en t w o-lev el systems and incoheren tly as
w ell as coheren tly driven three- and four-lev el systems. F or a t w o-lev el system laser theory in-
coheren t pumping is mandatory since otherwise no p opulation in v ersion can be achiev ed.
The master equation for the t w o-level laser is giv en b y (omitting the ca vit y decay and H 0 for

50 P erm utation symmetry in quantum master equations
Figure 5.6 – T w o-lev el laser sk etc h: The only difference to the op en Dic k e model Fig.
5.3 is the purple up w ard arrow that results from the incoheren t driving. The scaling is ∼ N 3 ,
N to the p o w er of the n umber of circles min us one. The s k etc h can b e directly translated
in to co de using the PsiQuaSP library .
Figure 5.7 – Sk etc h represen tation of the coheren tly pump ed three-lev el laser:
Compare to Eq. (5.4.23) and Fig. 5.5 b). a) The Hamiltonian of the lasing transition H 10
Eq. (5.4.11), b) the system-bath con tributions of D 2 → 1 ( ρ ) (yello w) and D 1 → 0 ( ρ ) (blue), c)
the pumping Hamiltonian H P Eq. (5.4.24), and d) the full dynamics. The com bination
of H 10 and H 20 couples all p olarization degrees of freedom to densit y degrees of freedom,
hence the full ∼ N 8 scaling is needed to describ e these dynamics.
brevit y)
˙ ρ = i
ℏ [ ρ, H 10 ] + γ 10
2 D 1 → 0 ( ρ ) + P
2 D 0 → 1 ( ρ ) , (5.4.22)
where H 10 is the in teraction Hamiltonian b et w een the tw o-lev el systems and the mo de, the term
prop ortional to γ 10 is the sp on taneous emission and the term prop ortional to P is the incoheren t
pumping term, see Fig. 5.5 a). The scaling of this master equation is ∼ N 3 , see Fig. 5.6.
The quan tum master equation for an optically pump ed three-lev el laser sc heme is
˙ ρ = i
ℏ [ ρ, H 10 + H P ] + γ 21
2 D 2 → 1 ( ρ ) + γ 10
2 D 1 → 0 ( ρ ) , (5.4.23)
Here H 10 is the in teraction with the ca vit y mo de (cf. Eq. (5.4.11)), H P is the optical pumping
of the system (using a suitable rotating frame)
H P = ℏ E ( J 20 + J 02 ) . (5.4.24)
The term prop ortional to γ 21 is the incoheren t relaxation pro cess that brings o ccupation into
the upp er lasing lev el and the term prop ortional to γ 10 is again the sp ontaneous emission in to

5.4. Multi-lev el systems 51
Figure 5.8 – Sk etc h represen tation of the incoheren tly driv en three-lev el laser:
Eq. (5.4.25). a) The dynamics of D [ σ 20 ] ρ , b) the full dynamics of the system: Some
p olarizations are completely decoupled from the rest of the dynamics and can therefore b e
omitted. c) The full laser dynamics can b e describ ed b y a ∼ N 4 theory .
non-lasing mo des. The lev el sc heme of this setup is sho wn in Fig. 5.5 b). The asso ciated
sk etc hes are shown in Fig. 5.7. The dissipator con tributions – Fig. 5.7 b) – connect densit y
degrees of freedom ( ∼ n ii ) with densities, p olarization degrees of freedom ( ∼ n ij , i  = j ) are only
dephased. A t this lev el the p olarizations are decoupled from the densities and one could omit
the p olarizations en tirely , as will b ecome clear b elo w. Ho w ev er, the Hamiltonians H 10 and H P
connect all p olarization degrees of freedom to densities, see Figs. 5.7 a) and c), so that the
system has full complexit y .
This b eha vior c hanges when replacing the pumping Hamiltonian H P b y an incoheren t pumping
Lindblad dissipator, omitting in termediate quan tum coherences. The corresp onding master
equation is
˙ ρ = i
ℏ [ ρ, H 10 ] + γ 21
2 D 2 → 1 ( ρ ) + γ 10
2 D 1 → 0 ( ρ ) + P
2 D 0 → 2 ( ρ ) . (5.4.25)
The incoheren t setup corresp onds to the solid up w ard arro w in Fig. 5.5 b). This emphasizes
that densities are directly transferred b et w een the lev els by the Lindblad dissipator and not via
the build up of p olarizations as in the H P case.
In Fig. 5.8 a) the sk etc h for this Liouvillian is sho wn. Com bining all o ccurring pro cesses in
a single sk etc h sho ws that four p olarization degrees of freedom (i.e. n 20 , n 21 , n 02 , n 21 ) are
completely decoupled from the other system dynamics, see Fig. 5.8 b). This can b e used to
greatly reduce the n umerical complexit y and the scaling with the system size N : Coherences
can only b e driv en b y coupling to densities. Starting in a state that has no quan tum coherences
implies that the densit y matrix en tries for n 20 , n 21 , n 02 , n 21  = 0 remain zero throughout the
whole time ev olution. F urthermore, regardless of the initial state the densit y matrix elemen ts
for n 20 , n 21 , n 02 , n 21  = 0 will b e zero in the steady state since they are dephased. Therefore
this quan tum master equation for three-lev el systems is exactly describ ed b y the quan tit y
P [ n 22 , n 11 , n 10 , n 01 ] , (5.4.26)
where no w the new relation

52 P erm utation symmetry in quantum master equations
N = n 22 + n 11 + n 10 + n 01 + n 00 (5.4.27)
holds. F rom Eq. (5.4.20) it follo ws that this solution scales with
( N + 4
N ) ∼ 1
4! N 4 . (5.4.28)
This scaling is considerably lo w er than that of the coherently driv en three lev el systems Eq.
(5.4.23), ∼ N 8 and of course also the brute force solution ∼ 3 2 N . Please b ear in mind that this
do es not corresp ond to a Hilb ert/Liouville space truncation or appro ximation of an y kind – the
argumen t is solely that some off-diagonal densit y matrix elemen ts are not connected to diagonal
elemen ts and th us remain zero and th us do not need to b e computed. This still represen ts the
full non-appro ximate densit y matrix.
With the help of the sk etc hes the exact solution can b e directly implemen ted (in PsiQuaSP)
and the additional simplification can b e seen without deriving a single equation of motion. This
mak es these sk etc hes v ery useful. Bolaños and Barb eris-Blostein [125] came to a similar decou-
pling in complexit y: By calculating the comm utation relations of the 9 2 = 81 su (9) generators
for three-lev el systems they observ ed that the master equation could b e describ ed by a su (3)
subalgebra, if they only considered sp on taneous emission con tributions b et w een the lev els (see
Section 5.5.3 for more information). This is exactly the same situation as the sk etch in Fig. 5.7
b): This sk etc h itself could b e reduced to a ∼ N 3 theory . How ev er the sk etc hes are v ery simple
to dra w – at least compared to calculating comm utation relations of 81 abstract op erators – and
furthermore giv e an in tuitive picture. Therefore also for a more formal Lie algebra approac h
these sk etc hes are very useful, this connection will be explained in more detail b elo w.
The coheren tly driv en four-level laser theory w ould b e constructed with the master equation
˙ ρ = i
ℏ [ ρ, H P + H 21 ] + γ 32
2 D 3 → 2 ( ρ ) + γ 21
2 D 2 → 1 ( ρ ) + γ 10
2 D 1 → 0 ( ρ ) , (5.4.29)
where the pump H P is no w b et w een lev els 3 and 0 . F or the incoheren t driving the master
equation reads
˙ ρ = i
ℏ [ ρ, H 21 ] + γ 32
2 D 3 → 2 ( ρ ) + γ 21
2 D 2 → 1 ( ρ ) + γ 10
2 D 1 → 0 ( ρ ) + P
2 D 0 → 3 ( ρ ) . (5.4.30)
The lev el sc heme for b oth setups is sho wn in Fig. 5.5 c ). In Fig. 5.9 a) and b) the pro cesses
included in the Hamiltonians and Lindblad dissipators of the four-lev el laser with coheren t pump
are sk etc hed. Again some of the p olarization degrees of freedom decouple. The corresp onding
elemen ts P [ n kl , . . . ] are zero, cf. Figs. 5.9 c) and d) and th us do not need to b e computed.
Consequen tly the complexit y of this solution do es not scale with ∼ N 15 but with ∼ N 7 instead.
In terestingly this is ev en low er than in the three-lev el system case: The symmetry of the four
lev els allo ws for a more efficient decoupling of polarization degrees of freedom compared to the
three-lev el systems, where all p olarization degrees of freedom are connected to densities.
Again a further reduction is ac hiev ed via an incoheren t pump term Eq. (5.4.30). The solution
of this setup scales with ∼ N 5 , see Fig. 5.10.
The general rules for constructing the sk etc hes are: (i) an in teraction Hamiltonian connecting
the lev els x and y results in arro ws connecting all circles n xz , n y z and n z x , n z y . F or R W A
treatmen ts there are alw a ys t w o arrows connecting t w o circles in bac k and forth direction. F or
Non-R W A treatmen ts there are four arro ws eac h. (ii) a individual sp on taneous deca y from lev el
x to lev el y results in a single arro w from n xx to w ards n y y and dephasing arro ws for all circles n xz
and n z x . These rules b ecomes apparen t b y lo oking at the full equations of motion in App endix
D or b y the discussion in the follo wing section.

5.4. Multi-lev el systems 53
Figure 5.9 – Sk etc h for the coheren tly driv en four-lev el laser: Eq. (5.4.29) and
Fig. 5.5 b): a) Pumping Hamiltonian H 30 (purple) and in teraction with the lasing mo de
H 21 (green), b) depletion of lo w er lasing lev el D 1 → 0 ρ (blue), sp on taneous emission in to non-
lasing mo des D 2 → 1 ( ρ ) (green), and filling of the upp er lasing lev el D 3 → 2 ( ρ ) (y ello w). c) The
full dynamics: Again some p olarizations decouple from the rest and thus d) describes the
full, non-appro ximate dynamics. The solution then scales as N 7 as opp osed to the full N 15
scaling of the man y four-lev el system metho d.
Figure 5.10 – Sk etc h for the incoheren tly driv en four-lev el laser: Eq. (5.4.30):
a) The dynamics of the incoheren t pump term D 0 → 3 ρ . b) The full laser dynamics can b e
describ ed b y a ∼ N 5 theory .

54 P erm utation symmetry in quantum master equations
In this section the usefulness of the sk etc h represen tation of the p erm utation symmetric m ulti-
lev el system metho d w as illustrated. The sk etc hes for the individual con tributions in the master
equation are easy to dra w and sk etches for the full quan tum master equation pro vide an easy
and quic k to ol to iden tify simplifications and to calculate the scaling b eha vior for a sp ecific
setup.
Ev en if the diagrams for the full quan tum master equation dynamics ma y app ear complicated
at first sigh t, their construction is straigh tforw ard. F urthermore the PsiQuaSP library allo ws to
directly translate these sk etc hes in to n umerical co de, whic h completely remo v es the necessit y to
deriv e equations of motion. Please b ear in mind that, from a mathematical p oin t of view, all
presen ted solutions, represen tations are (numerically) exact.
The metho dology presen ted in the last sections pro vides a ph ysically in tuitiv e picture. Ho w ev er
it is not v ery general since for eac h Hamiltonian, or rather, eac h Liouville space op erator a sk etc h
represen tation needs to b e in tro duced and also implemen ted in the PsiQuaSP library . F or the
Hamiltonians and dissipators discussed ab o v e this is not v ery difficult, ho w ev er for op erators
lik e J n
xy these sk etc hes can b ecome quite cum b ersome. Therefore in order to utilize the full
generalit y of the p erm utation symmetric metho d for quan tum master equations a more flexible
approac h is needed. This approac h will b e presen ted in the follo wing.
5.5 Symmetrized eigenstates of p erm utation symmetric Liou-
ville space op erators
In the previous section the p erm utation symmetry in the master equations w as iden tified and
exploited b y directly lo oking at the equations of motion resulting from an expansion of the den-
sit y matrix in a direct pro duct state basis. General densit y matrix elemen ts ρ [ . . . ] and P [ . . . ]
w ere in tro duced and the action of the Liouvillians w ere found to either lea v e these elemen ts
in v arian t or transform them in to another one of these elemen ts. Therefore closed equations of
motion could b e ac hiev ed.
In this section the findings of the previous section will b e generalized: The expansion of the
master equation in the P [ . . . ] representation is equiv alen t to expanding the master equation
in symmetrized Liouville space states. These states are the eigenstates of sp ecial p erm utation
symmetric Liouville space op erators and other p erm utation symmetric op erators serv e as flip
op erators b et w een these states. These op erators form a su ( n ) Lie algebra, with n b eing the
n um b er of necessary individual spin matrices σ i
k l . The symmetrized eigenstates represen t the
asso ciated totally symmetric m ultiplet, similar to the (totally symmetric) sup erradian t Dic k e
or Gelfand-T setlein states but in Liouville space instead of Hilb ert space. These states and the
actions of the symmetrized Liouvlle space op erators can again b e represen ted b y simple sketc hes.
This more formal treatmen t allo ws to realize that the p erm utation symmetric Liouville space op-
erators serv e as elemen tary building blo c ks that can b e utilized to construct arbitrary Liouville
space op erators. This can b e used to construct arbitrary quan tum master equations and observ-
ables (that are p erm utation symmetric). This is implemented i n the PsiQuaSP library [26] to
pro vide maximal flexibilit y in constructing arbitrary master equations. This approac h is neces-
sary b ecause the theory of the last section allo ws to implemen t arro ws for sp e cific Hamiltonians
and dissipators but the p erm utation symmetric metho d is in principle applicable to any p erm u-
tation symmetric master equation. Since it is not p ossible to write ready made functions for all
conceiv able p erm utation symmetric Liouville space op erators there needs to b e another, more
flexible approac h. Exactly this is pro vided b y the approac h presented in this section. F urther-
more the more formal treatmen t of this section also allo ws to iden tify generalizations and more
general, theoretical prop erties and p ossibilities of the metho d. P arts of the discussion in this
section w ere published in the PsiQuaSP publication M. Gegg and M. Ric h ter, arXiv :1707.01079

5.5. Symmetrized eigenstates of p erm utation symmetric Liouville space op erators 55
[27].
The definition of the P [ . . . ] representation deriv ed in the last sections is
P [ { n k l } ] = ( N
{ n k l } ) ρ [ { n k l } ] = ( N
{ n k l } ) tr [ | . . . n k l , u k l . . . ⟩⟨ . . . | ρ ] , (5.5.1)
where { n k l } = { n ( d − 1)( d − 1) , . . . } is a short hand notation, it is the set of all n um b ers n kl . In this
expression the Hilb ert-Sc hmidt pro duct of a single, direct pro duct Liouville space state with the
densit y matrix is computed and m ultiplied with the total num b er of direct pro duct states that
yield the same Hilb ert-Sc hmidt pro duct. In the follo wing this expression will b e rewritten as a
sum o v er all these direct pro duct basis states.
Expressing these Liouville space basis state directly in terms of the spin matrices results in
| . . . n k l , u k l . . . ⟩⟨ . . . | = . . . σ i 1
k l ⊗ σ i 2
k l · · · ⊗ σ i n k l
k l
  
≡ σ ⊗ n kl
kl
. . .
= ⨂
k ,l
σ ⊗ n kl
k l . (5.5.2)
This expression do es not con tain the information ab out the sets, the ordering of the individual
spin matrices is not uniquely defined – there are man y p erm utations of spin matrix indices
that can b e written as suc h a pro duct of spin matrices, c haracterized b y the n um b ers { n k l } .
Ho w ever it is possible to construct a unique basis state by forming a symmetric sup erp osition
of all p ossible basis states for fixed { n k l } :
ˆ
P [ { n k l } ] ≡ S ⨂
k ,l
σ ⊗ n kl
k l , (5.5.3)
where S is again the symmetrization op erator
S = ∑
P
ˆ
P , (5.5.4)
with ˆ
P b eing the p erm utation op erator and the sum runs o v er all p ossible p ermutations P of
m ulti-lev el system indices i . The num b er of p ossible p erm utations is again the m ultinomial
co efficien t Eq. (5.4.18) ( N
{ n k l } ) = N !
n 00 ! n 01 ! . . . . (5.5.5)
Since the trace has the prop ert y tr [ A + B ] = tr [ A ] + tr [ B ] it follows that
P [ { n k l } ] = tr [ ˆ
P [ { n k l } ] ρ ] . (5.5.6)
Hence the expansion of the densit y matrix found in the previous section is equiv alent to defining
a basis of totally symmetric sup erp ositions of direct pro duct Liouville space basis states.
5.5.1 P erm utation symmetric Liouville space op erators – Elemen tary sk etc hes
As stated ab o v e there are t wo t yp es of p ermutation symmetric Liouville space operators Γ :
Op erators to whic h the symmetrized basis states Eq. (5.5.3) are eigenstates and general flip
op erators that transform one eigenstate in to the other. These tw o t yp es of Γ op erators can b e
defined via their action on the symmetrized basis states ˆ
P [ { n k l } ]
Γ nn
mm ˆ
P [ { n k l } ] ≡ ∑
i
σ i
mm ˆ
P [ { n k l } ] σ i
nn (5.5.7)

56 P erm utation symmetry in quantum master equations
Figure 5.11 – Mo dular sk etc hes for iden tical m ulti-lev el systems: a) The noncon-
necting arro w can represen t the phase oscillations arising from the self energy Hamiltonians
(curv ed arro w) and it can describ e dephasing (straigh t arro w). The self energy Hamiltonian
w as omitted in the discussion of the last section. b) The connecting arro w connects t w o
differen t circles. It can represen t density relaxation c) and elemen ts of flip op erators d).
and
Γ op
mn ˆ
P [ { n k l } ] ≡ ∑
i
σ i
mn ˆ
P [ { n k l } ] σ i
op , (5.5.8)
requiring that m  = n and o  = p . It can b e sho wn that
Γ nn
mm ˆ
P [ { n k l } ] = n mn ˆ
P [ { n k l } ] , (5.5.9)
hence the symmetrized Liouville space states are in fact eigenstates of the p erm utation symmetric
Liouville space op erators. The action of the other op erators is giv en b y
Γ op
mn ˆ
P [ { n k l } ] = ( n mp + 1) ˆ
P [ . . . n mp + 1 . . . n no − 1 . . . ] , (5.5.10)
hence they serv e as generalized flip op erators. The comm utation relations for these op erators
are giv en b y
[Γ op
mn , Γ r r
q q ] = +Γ op
mn δ nq δ or − Γ op
mn δ mq δ pr ,
[Γ op
mn , Γ st
q r ] = +Γ sp
mr δ nq δ to − Γ ot
q n δ r m δ ps (5.5.11)
and zero otherwise. The corresp onding pro ofs can b e found in App endix B.2.
The corresp onding sk etc hes are sho wn in Fig. 5.11: In all the sk etches so far there w ere t w o
differen t t yp es of arro ws – connecting and non-connecting arro ws. Non-connecting arro ws just
p oin t from one circle a w ay , represen ting a dephasing pro cess and the connecting arro ws connect
t w o circles. It turns out that a single non-connecting arrow corresponds to a Γ nn
mm op erator
and a single connecting arro w corresp onds to a Γ op
mn connecting arro w. Th us these complicated
lo oking op erators are just the formal, mathematical form ulation of the individual arro ws in the
sk etc hes 5 , whic h again is not v ery complicated and the (PsiQuaSP) user just needs to dra w
these sk etc hes. These sk etc hes already imply some in teresting iden tities/sum rules, wh ic h are
needed to construct ph ysically meaningful op erators (sk etc hes) from these elementary operators
(arro ws).
5.5.2 Building ph ysically meaningful Liouville space op erators
In the discussion ab o v e the elemen tary Liouville space op erators and the corresp onding sk etc hes
w ere in tro duced. Here the necessary relations to construct ph ysically meaningful operators are
5 In the case of the emitter-ca vit y coupling the arro ws also implicitly represen ted actions on the mode degrees
of freedom, whic h is not the case here.

5.5. Symmetrized eigenstates of p erm utation symmetric Liouville space op erators 57
Figure 5.12 – Building complex Liouvillians from elemen tary sk etc hes: a) and b)
corresp ond to the t w o op erators J L
11 and J R
11 . c) The sp on taneous emission Liouvillian in
the elemen tary picture. d) - g) The sk etc hes corresp onding to J L
10 , J L
01 , J R
10 , J R
01 for t w o-level
systems. h) - j) three-lev el system op erators: J L
10 , J R
10 and J L
10 for reduced complexit y .
in tro duced. F or this purp ose it is useful to use the R , L algebra notation for Liouville op erators
[126], whic h is defined b y
O ρ ≡ O L ρ, ρ O ≡ O R ρ. (5.5.12)
Using this notation there are four relev an t Liouville space op erators based on the collectiv e spin
op erators J ... : J L
k k , J R
k k and J L
k l , J R
k l . The relations b et w een these op erators and the elemen tary
Γ op erators are
J L
k k = ∑
n
Γ nn
k k = ∑
n ∑
i
σ i
k k . . . σ i
nn ,
J R
k k = ∑
n
Γ k k
nn = ∑
n ∑
i
σ i
nn . . . σ i
k k ,
J L
k l = ∑
n
Γ nn
k l = ∑
n ∑
i
σ i
k l . . . σ i
nn ,
J R
k l = ∑
n
Γ k l
nn = ∑
n ∑
i
σ i
nn . . . σ i
k l . (5.5.13)
The corresp onding pro of of these relations are sho wn in App endix A, while explaining ho w to
construct arbitrary op erators in PsiQuaSP . With this set of results it is clear ho w to construct
e.g. a v on-Neumann equation for a general self energy Hamiltonian H 0 = ℏ ω J k k
˙ ρ ∼ i/ ℏ [ ρ, H 0 ] = i/ ℏ ( H R
0 − H L
0 ) = iω ∑
n
(Γ k k
nn − Γ nn
k k ) ρ (5.5.14)

58 P erm utation symmetry in quantum master equations
or a general individual dissipator
D ρ = γ
2 ( ∑
i
2 σ i
k l ρσ i
lk − J l l ρ − ρJ ll ) = γ
2 ( ∑
i
2 σ i,L
k l σ i,R
lk − J L
ll − J R
ll ) ρ,
= γ
2 (2Γ lk
k l − ∑
n
Γ nn
ll − ∑
n
Γ ll
nn ) ρ. (5.5.15)
In Fig. 5.12 a) - c) the corresp onding sk etc hes are shown for t w o-lev el systems. Fig. 5.12 d) -
g) represen t the t wo-lev el system flip op erators J L,R
10 J L,R
01 and Fig. 5.12 h) - j) sho w differen t
flip op erators for three-lev el systems.
The construction of equations of motion from these expressions is done b y multiplying with
ˆ
P [ . . . ] from the left p erforming the trace op eration, see App endix A for details and the corre-
sp onding PsiQuaSP implemen tation.
5.5.3 Lie algebra con text
The comm utation relations Eq. (5.5.11) are the same comm utation relations as the ones found
for the m ulti-lev el systems (Eq. (2.1.8)), th us the Γ op erators form a Lie algebra and the
symmetrized basis states Eq. (5.5.3) are the totally symmetric m ultiplet (or irreducible repre-
sen tation) of this algebra. The order of the Lie algebra is su ( d 2 ) for iden tical d -lev el systems,
or lo w er if coherences decouple as in Section 5.4.3. The only difference b etw een the ˆ
P basis and
the Gelfand-T setlein basis states of Chapter 4 is that the ˆ
P are Liouville space basis states and
not Hilb ert space basis states. This raises the question whether an tisymmetric Liouville space
basis states are of in terest: The definition of the trace or rather the Hilb ert space iden tit y
I H = ∑
{ n kk }
ˆ
P [ { n k k } ] , (5.5.16)
is included in the symmetric basis states and not in the (partially) an ti-symmetric basis states.
Here { n k k } = { n 11 , n 22 , . . . } is the set of all densit y lik e quan tum n um b ers. Also the ground state
ˆ
P [0 , 0 , . . . ] is part of the symmetric and therefore not the (partially) an ti-symmetric m ultiplets.
Th us confining the system to an an ti-symmetric multiplet is probably unph ysical. Including all
m ultiplets symmetric and an ti-symmetric should reco v er the full exp onen tial complexit y of the
individual direct pro duct Liouville space basis.
A direct consequence of this form ulation is that it is p ossible to apply analytic Lie algebra
tec hniques lik e b osonization tec hniques, suc h as the Holstein-Primak off b osons or the Jordan-
Sc h winger b osons [91]. In the Holstein-Primak off b osonization the su ( n ) op erators are replaced
b y n − 1 b osonic mo de op erators, while the Jordan-Sc h winger b osonization requires n b osons.
These tec hniques often allo w for analytic solutions. Please note that this tec hnique is usually
used for Hilb ert space op erators, applying these transformations to the op erators in tro duced
here corresp onds to a Liouville space b osonization ansatz. F or a simple three-lev el system setup
the Jordan-Sc h winger b osons w ere successfully used in Ref. [125].
In general the minimal group for the p erm utation symmetric formalism is su (2) e.g.
ˆ
P [ n 11 ] = S σ ⊗ n 11
11 σ ⊗ n 00
00 (5.5.17)
whic h is a v alid basis for e.g. individually deca ying t w o-lev el systems describ ed b y the master
equation
˙ ρ = γ
2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) . (5.5.18)
This can b e seen from the sk etc hes, e.g. Fig. 5.11 c). There the p olarization degrees of freedom
decouple. This result is obtained without explicitly computing comm utation relations, th us also

5.6. Reco v ering the Dick e states – Ho w to diagonalize the densit y matrix 59
in the group theoretic con text the sk etch represen tation is useful.
This basis is the symmetric su (2) multiplet in Liouville space. Th us the individual sp on taneous
deca y of a set of indistinguishable t w o-lev el systems is go v erned b y the Liouville space analogue
of the sup erradian t Dic k e states (symmetric su (2) m ultiplets). Inclusion of coheren t, collectiv e
prop erties alw a ys results in adding t wo quan tum n um b ers n xy and n y x . Of course this is merely
a to y example but this illustrates the flexibilit y of the approac h.
5.6 Reco v ering the D ic k e states – Ho w to diagonalize the den-
sit y matrix
The results presen ted so far are useful if one seeks the solutions to a quan tum master equation:
The n um b er of differen t symmetrized basis states is giv en b y Eq. (5.4.20)
D = ( N + m
N ) ∝ 1
m ! N m , (5.6.1)
therefore w e can implemen t equations of motion whic h represen t a set of D coupled linear
homogeneous differen tial equations, whic h means that the Liouvillian L of the master equation
˙ ρ = L ρ (5.6.2)
can b e written as a D × D matrix, see App endix A and B. This can b e used for time-in tegration
or direct steady state computation, e.g. via iterativ e Krylo v-subspace metho ds. This corre-
sp onds to a v ectorized represen tation of the master equation, meaning the densit y matrix is
represen ted b y a v ector in this picture, whic h is also the underlying represen tation in PsiQuaSP
(see App endix B.2 for details on v ectorized represen tation).
If one wishes to compute for example the v on-Neumann en trop y [94]
S = − tr ( ρ l nρ ) (5.6.3)
it is necessary to diagonalize the densit y matrix since the logarithm of a matrix is only defined
in the diagonal form. Using the ˆ
P represen tation to reconstruct the densit y matrix in the direct
pro duct state basis Eqs. (5.3.3) and 5.4.13 w ould again result an exponentially scaling densit y
matrix d N × d N for d -lev el systems. Man y of the en tries in this matrix are iden tical, but still, the
diagonalization or ev en storage of suc h a matrix is not feasible ev en for mo derate N . F ortunately
there is a w a y to find a blo c k diagonal represen tation of the densit y matrix
ρ = ⎛
⎜
⎜
⎜
⎜
⎝
ρ 1 0 0 . . .
0 ρ 2 0 . . .
0 0 ρ 3 . . .
.
.
. .
.
. .
.
. . . .
⎞
⎟
⎟
⎟
⎟
⎠ , (5.6.4)
where eac h of the sub-blo c ks scales at most p olynomially in N for iden tical d lev el systems.
The basis states in whic h the densit y matrix is blo c k diagonal are the su ( d ) m ultiplet states,
i.e. for t w o-le v el systems the densit y matrix is block diagonal in the Dic k e basis. F or t w o-lev el
systems this w as sho wn by Xu and Holland [110] and Chase, Geremia and Baragiola [98, 99].
F or m ulti-lev el systems this result has not been published so far in quantum optics, to the best
of m y kno wledge 6 . The details of ho w to reconstruct the Dic ke states and the proof for the
6 F rom a group theoretic stance this relation seems v ery fundamen tal, th us it is lik ely that this relation do es
exist somewhere in group theory . The group theoretic formulation of the statemen t would be: The blo c k diagonal
represen tation a matrix of N iden tical d -lev el systems in the su ( d ) m ultiplets is equiv alen t to the totally symmetric
m ultiplet of su ( d 2 ) .

60 P erm utation symmetry in quantum master equations
equalit y for m ulti-lev el systems are outlined in App endix B.2. Please note that this result is
indep enden t of what kind of master equation is considered, the only requiremen t is that the
m ulti-lev el systems are indistinguishable.
F or t w o-lev el systems this means that only matrices of order N × N need to b e diagonalized,
for m ulti-lev el systems the blo c ks scale at most p olynomially , which is a dramatic reduction in
complexit y . The blo c k size in the Dic k e basis is defined b y the quan tum num b er l , eac h blo c k
corresp onds to the n um b er of p ossible m v alues 2 l + 1 , meaning that each block has (2 l + 1) 2
en tries. In fact the sum o v er all the blo c k en tries reco vers the scaling found from the permutation
symmetry
N/ 2
∑
l =0 , 1 / 2
(2 l + 1) 2 = 1
6 ( N + 1)( N + 2)( N + 3) . (5.6.5)
Hence the t w o statements (i) "the densit y matrix is blo c k diagonal in the Dic ke basis", (ii)
"the master equation b ears p erm utation symmetry" and (iii) "the m ulti-lev el systems are indis-
tinguishable" are equiv alen t. F rom an in tuitiv e p ersp ective this can be understo o d as follo ws:
The con tributions of the Hamiltonian part of the master equation do not couple differen t Dic k e
subspaces, only the J 2 symmetry breaking dissipator terms do so. Dissipators should not in-
duce coherences therefore the densit y matrix should b e blo c k diagonal, since if it w asn’t blo ck
diagonal the dephasing w ould induce nonzero off-diagonal elemen ts.
5.7 Conclusion
The discussion in this c hapter sho w ed that the inclusion of (individual) dissipation of the m ulti-
lev el systems breaks the symmetry of the su ( n ) Casimir op erators. This means that the system
of iden tical m utli-lev el systems cannot b e describ ed b y the symmetric su ( n ) m ultiplet states
alone, whic h at first glance seems to result in an exp onen tial complexit y of the underlying equa-
tions of motion/Liouville space. Ho w ev er it w as sho wn that the p erm utation symmetry imp osed
b y the requiremen t of indistinguishable m ulti-lev el systems reduces this complexit y again to a
p olynomial complexit y . This can b e sho wn b y directly lo oking at the equations of motion or
b y in tro ducing symmetrized Liouville space states and p erm utation symmetric Liouville space
op erators. The first approac h pro vides a simple and accessible treatmen t, while the second ap-
proac h is more general and mathematically sound. This treatmen t is then found to b e iden tical
to a blo c k diagonal represen tation of the densit y matrix in the su ( d ) m ultiplet states for d -lev el
systems. This is an incredibly useful relation for densit y matrix diagonalization.
The sk etc h represen tation in tro duced in this c hapter pro vides a simple and intuitiv e approac h
to the metho dology and mak es it p ossible to completely omit an y equations of motion. The
sk etc hes also provide a v ery p o w erful to ol to iden tify further symmetries in the setup, which
could in principle also b e deriv ed from the Lie algebraic considerations, but the sk etc h repre-
sen tation mak es these considerations muc h more simple. Th us b oth for in tuition and sound
mathematical treatmen ts the sk etch represen tation pro vides a ma jor b enefit. F urthermore the
sk etc h representation w as used to design the PsiQuaSP library that allo ws to translate these
sk etc hes directly in to co de, whic h greatly reduces co de dev elopmen t time.

A The PsiQuaSP Library
In the last c hapter the p erm utation symmetric metho d for quan tum master equations w as in-
tro duced and a sk etc h represen tation was dev elop ed. In this c hapter the library PsiQuaSP is
presen ted, whic h is an ob ject orien ted C++ implemen tation of this metho d. PsiQuaSP is an
acron ym for Permutation symmetry for identic al Quantum Systems Package . It is based on the
PETSc library for iterativ e sparse matrix metho ds and differen tial equations. The library w as
made publicly a v ailable on GitHub: h ttps://gith ub.com/mo dmido/psiquasp and an in tro duc-
tion/man ual w as published: M. Gegg and M. Ric h ter, arXiv :1707.01079 [27]. The follo wing
discussion is a close adaptation from this publication.
The main design feature of PsiQuaSP is that it allo ws to translate the sk etc h represen tation
of the p erm utation symmetric metho d directly in to co de. The user do es not need to deriv e
an y equation of motion. The bubbles and arro ws can eac h b e set b y single function calls and
the whole setup of the equations of motion is handled in ternally b y PsiQuaSP . Solving these
equations is then en tirely handled b y PETSc and related pac kages suc h as SLEPc. The in ternal
design of PsiQuaSP allo ws to write scalable, efficien t co de while b eing v ery user friendly . There
are sev eral examples in the example/ directory in tro ducing the basic and adv anced p ossibilities
that PsiQuaSP pro vides. Ov erall, library and examples, PsiQuaSP con tains o v er 20000 lines of
co de in ∼ 50 source co de files, it has b een implemen ted for > 10 different master equations and
successfully tested on Lin ux and macOS op erating systems. It uses recursiv e GnuMak e to facil-
itate the installation pro cedure. The commen ts in the source co de are written in Do xygen [127]
formatting, pro viding a comprehensible and easily accessible do cumentation in a h tml w ebsite
format, see Fig. A.1.
61

62 The PsiQuaSP Library
Figure A.1 – File dep endency graph of the system.hpp file, which is the heart of the
library . The upp er gra y b o x represen ts this file and the white b o xes are all source co de files
in PsiQuaSP that directly or indirectly access this file. The plot is generated b y Do xygen.
A.1 Using PsiQuaSP – Basic structure of the library
PsiQuaSP is designed in a w a y that provides maximal flexibilit y for setting up sim ulations. The
aim is to find a n umerical solution to a master equation
˙ ρ = L ρ. (A.1.1)
PsiQuaSP uses a v ectorized represen tation of this equation, meaning that the density matrix ρ
is represen ted b y a v ector and the Liouvillian L is represen ted b y a matrix. The equation can
either b e solv ed b y standard time in tegration or b y linear algebra metho ds, using the b enefits
of the v ectorized represen tation. F or instance the steady state is giv en b y the null space of
the Liouvillian L , whic h can b e calculated using e.g. mo dern Krylo v subspace based iterativ e
solv ers.
The densit y matrix and the Liouvillian are expanded in the ˆ
P basis states in tro duced in Chapter
5 through the Hilb ert-Sc hmidt inner pro duct
P [ . . . ] = tr [ ˆ
P [ . . . ] ρ ] , (A.1.2)
hence the quan tities P [ . . . ] are directly b eing computed. In order to allo w for maximal flexibilit y
PsiQuaSP only pro vides setup routines for constructing the densit y matrix and the Liouvillians
L and it allo ws to define observ ables, distributions, correlation functions etc. and encapsu-
lates all these ob jects in a user friendly wa y . PsiQuaSP itself do es not con tain an y solv ers, the
n umerical solution relies on PETSc, deriv ed pac kages suc h as SLEPc and/or external pac kages
that can b e used with PETSc suc h as MUMPS, Sup erLU, Metis/ParMetis, PTScotc h and others
[128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138]. All these pac kages are incredibly v aluable,
v ersatile and p o w erful to ols and PETSc pro vides a unified and clean in terface to all of these
pac kages. This is one of the reasons that mak es PETSc an excellen t foundation for PsiQuaSP .
Getting to kno w all these pac kages requires a lot of time and effort, but the a v erage user can
use PsiQuaSP without kno wing these additional pac kages. Ho w ev er these lines are in tended to
encourage the readers of this thesis to lo ok in to these pac kages and related n umerical literature
and find out what they can do in order to b o ost the p erformance of the n umerical co de. The
righ t c hoice of metho d can reduce computing time b y orders of magnitude, see Fig. A.8 (b).
The heart of PsiQuaSP is the System class, which is con tained in the system.hpp file, see Fig.
A.1. The user first sp ecifies whether t w o-, three- or d-lev el systems 1 and ho w man y b osonic
mo des are used. F or t w o-lev el systems there is a sp ecial TLS class that pro vides further encapsu-
lation and therefore simplification for standard t w o-level system Hamiltonians and dissipators.
Based on the information on m ulti-lev el sytems and bosonic mo des the System / TLS class pro-
vides initialization functions for the densit y matrix and Liouvillians th us ev erything required for
1 Curren tly only one t yp e of m ulti-lev el system at the same time is supp orted, this will c hange in the future.

A.2. Examples 63
Figure A.2 – Structure of the PsiQuaSP library: a) Sc hematic represen tation of the
general structure of a PsiQuaSP application co de: MySystem con tains all the relev an t infor-
mation ab out the system and is used to construct the master equation and the output. The
output is organized in three la y ers, the first lay er consists of ob jects that can compute the
desired prop erties of the system, like Observable , Distribution , the correlation functions
Gnfct and the custom t yp es MyObs and MyDist . The second la y er groups these ob jects
in to output files, eac h managed b y another ob ject. The third la yer consists of the MyOut
class, whic h groups all output files and pro vides a clean in terface to PETSc. Classes that
need to b e deriv ed from base classes ha v e blue b o xes, green b o xes indicate ready to use
classes. b) Base class diagram for the deriv ed classes in a). Only for MySystem there are t w o
p ossibilities: TLS for t w o-lev el system setups and System for all other purp oses like d -lev el
systems.
setting up the master equation. The output of the program is managed by the Output class,
whic h can manage a set of user defined output files, con taining observ ables, correlation func-
tions, distributions, etc, see Fig. A.2 a). Please note that ev en though PsiQuaSP is in tended and
designed for solving p erm utationally symmetric master equations, the library is not limited to
this application. It ma y also b e used for efficien t treatmen ts of noniden tical m ulti-lev el systems
as w ell as Hamiltonian diagonalizations, see Chapter 9.
Installation instructions for PsiQuaSP and PETSc are giv en in the README.md and INSTALL.md
files in the PsiQuaSP folder. PsiQuaSP uses Doxygen commen ting. Do xygen translates the
commen ts in the source co de in to a structured w ebsite representation, whic h is extremely useful
for getting to kno w the library . Read doc/README.md for further information.
A.2 Examples
The setup of PsiQuaSP sim ulations will no w b e illustrated b y discussing examples. All source
co des of the examples and man y more can b e found in the example/ directory in the PsiQuaSP
directory .
A.2.1 Example 1: Op en T a vis-Cummings relaxation
The first example will b e the master equation in tro duced at the b eginning of Chapter 5: Eqs.
(5.2.1), (5.2.2) and (5.2.3), whic h w as the starting p oin t of the whole deriv ation of the ˆ
P formal-
ism. The equation describ es the Dic k e Hamiltonian, including sp on taneous emission and ca vit y
loss
˙ ρ = i
ℏ [ ρ, H ] + D 1 → 0 ( ρ ) + D ph ( ρ ) , (A.2.1)

64 The PsiQuaSP Library
Figure A.3 – Op en T a vis-Cummings relaxation: a) The lev el sc heme for the individual
t w o-level system and b) the sk etc h for the P represen tation of this master equation.
with the Dic k e Hamiltonian in R W A
H = ℏ ω 0 b † b + ℏ ω 1 J 11 + ℏ g ( J 10 b + J 01 b † ) (A.2.2)
and the Lindblad dissipators
D 1 → 0 ( ρ ) = γ
2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) , (A.2.3)
D ph ( ρ ) = κ
2 ( bρb † − b † bρ − ρb † b ) . (A.2.4)
In Fig. A.3 the lev el sc heme and the corresp onding sk etc h are shown. It is the setup of a basic
T a vis-Cummings/Dic k e mo del including individual sp ontaneous deca y of the t w o-lev el systems
and a ca vit y loss term. The example co de computes the temp oral dynamics of this master equa-
tion using direct R unge-Kutta time in tegration. The source co de can b e found in example/ex1a .
example/ex1b solv es the same equation with an adaptiv e step width Runge-Kutta and at the
same time sho ws the application of more adv anced PETSc routines. Since there is no pump
term in this master equation the steady state will b e the ground state and w e need to prepare
the system initially in an excited state in order to observ e non trivial dynamics.
System/Master e quation setup: First a deriv ed class for the system under consideration is de-
clared:
class OTC: public TLS
{
public:
void Setup(Vec * dm, Mat * L);
};
This class just defines a setup function. This is the standard pro cedure for all user deriv ed
classes in PsiQuaSP , i.e. in most cases user deriv ed classes just define a setup function. Here
the base class TLS is used, which pro vides enhanced to ols for master equations only in v olving
t w o-level systems. Here the setup function will create a v ector Vec * dm and a matrix Mat *
L , whic h are the densit y matrix and the Liouvillian of the system. PsiQuaSP uses a v ectorized
v ersion of the master equation. The t w o t yp es Vec and Mat are defined by PETSc. Both can
b e either serial or parallel, Mat is sparse b y default (but dense t yp es are a v ailable if needed),
leading to efficien t memory usage and reduction in computation time.
In the OTC::Setup(...) function w e call the functions
TLSAdd(ntls,ntls,ntls,tlsenergy);
ModeAdd(m0+1,dm0,modeenergy);
PQSPSetup(dm,1,L);
to tell PsiQuaSP that nlts t w o-lev el systems and one b osonic mo de with maxim um F o c k state
m0 are considered. TLSAdd(...) adds the tw o-lev el system quan tum n um b ers n 11 , n 10 and
n 01 , c.f. Fig. A.3 b). The n 00 quan tum n umber is not needed, as explained in the previous
c hapter. The three argumen ts ntls,ntls,ntls sp ecify the maxim um n um b er for the three
indices n 11 , n 10 , n 01 . This allo ws for a truncation of the three individual quan tum n um b ers.

A.2. Examples 65
tlsenergy and modeenergy are the transition energies for exciting a t w o-lev el system and the
photon energy , which are needed for preparing the system in a thermal equilibrium state. These
energy parameters are indep enden t of the parameters used for the equation of motion since
rotating frame represen tations migh t b e used. After this the user needs to call PQSPSetup() ,
the setup function for all in ternal structures whic h creates the densit y matrix v ector dm and the
Liouvillian matrix L . No w the master equation is sp ecified. This is done b y calling
AddTLSH0(*L,NULL,NULL,1,domega_tls*PETSC_i);
AddTavisCummingsHamiltonianRWA(*L,NULL,NULL,1,0,gcouple*PETSC_i);
AddTLSSpontaneousEmission(*L,NULL,NULL,1,gamma/2.0);
AddLindbladMode(*L,NULL,NULL,1,0,kappa/2.0);
Here eac h line adds the con tributions of a different term of the master equation to the Liouvillian
matrix L . The sk etc h for AddTLSSpontaneousEmission(..) is represen ted b y the blue arro ws
in Fig. A.3 b) and AddTavisCummingsHamiltonianRWA(..) is represen ted b y the green arro ws
in Fig. A.3 b).
Mo de related Liouvillians lik e AddLindbladMode(...) are not represen ted in sk etc hes and the
AddTLSH0(..) is giv en b y the com bination of sk etc hes Fig. A.7 a) and b). In this example a
rotating frame represen tation is used and domega_tls is the detuning of the t w o-lev el systems
from the ca vit y mo de, on resonance domega_tls = 0 . 0 holds 2 . The next step is to sp ecify initial
conditions, here w e prepare the system in an excited state:
PetscInt qnumbers [5] = {n11,n10,n01,mket,mbra};
DMWritePureState(*dm,qnumbers);
The qnumbers arra y con tains the quantum n um b ers of the desired state. This setup function can
prepare the densit y matrix in an y of the p erm utation symmetric basis states equation (5.5.3).
Ho w ever only for n10 = n01 = 0 and mket = mbra the state corresp onds to a physically
meaningful p opulation. This function addresses the differen t quan tum n um b ers in the order
they ha v e b een set: As stated ab o v e the TLSAdd(...) function call adds the t w o-lev el system
quan tum n umbers in the order n 11 , n 10 , n 01 and the fun ction ModeAdd(...) alw a ys adds first
the k et and then the bra quan tum n um b er of | mket ⟩⟨ mbra | . A dding t w o mo des in this example
via t w o successive calls to ModeAdd(...) w ould require to address an individual state with an
arra y lik e
PetscInt qnumbers [7] = {n11,n10,n01,m0ket,m0bra,m1ket,m1bra};
PsiQuaSP in ternally lab els the mo des with num b ers starting from 0 in the order of creation.
T o create an ob ject of the system sp ecification class OTC , e.g. in the main routine the constructor
and the setup function are called
OTC otc;
otc.Setup(&dm,&L);
The otc ob ject has t w o purp oses: It creates all ingredien ts to the master equation and after
successful setup it con tains all necessary information ab out the system. Afterw ards the ob ject
is used to build observ ables and to sp ecify the output data.
Defining the output: The exp ectation v alue of a collectiv e op erator lik e J 11 (Eq. (4.1.1)), whic h
represen ts the mean o ccupation of the excited states of all t w o-lev el systems
⟨ J 11 ⟩ = tr [ J 11 ρ ] =
N
∑
n =0 ∑
m
n P [ n, 0 , 0; m, m ] (A.2.5)
can b e defined using the Observable class:
2 On resonance w e could in principle lea v e out the AddTLSH0(..) setup function, ho w ev er PETSc solv ers require
that the diagonals of ev ery sparse matrix are explicitly set, ev en if they are zero. Since AddTLSH0(..) adds all
these diagonal elemen ts it should alw a ys be included, or the user calls the function AddDiagZeros(...) that
could b e used instead of AddTLSH0(...) on resonance.

66 The PsiQuaSP Library
Observable *pdens11 = new Observable();
MLSDim n11 (1,1);
pdens11->SetupMlsOccupation(otc,n11);
Here n11 is an iden tifier referring to the n 11 degree of freedom and the function SetupMlsOccupation()
can b e used to define all ⟨ J k k ⟩ observ ables for t w o-lev el systems
⟨ J k k ⟩ =
N
∑
n kk =0 ∑
...
n k k P [ . . . n k k . . . ] , (A.2.6)
where the second sum runs o v er all density degrees of freedom, e.g. a partial trace. The
MLSDim and ModeDim classes pro vide a w ay to access differen t degrees of freedom within the
application co de. Output files that prin t observ ables, distributions etc. at ev ery n th time
step are also managed b y classes, n is equal to 30 b y default and can b e c hanged with the
-tev_steps_monitor newvalue command line option. F or files printing observ ables lik e J kk
the user creates a deriv ed class lik e
class ObservablesFile: public PropFile
{
public:
void SetupMyObsFile(OTC * otc, std::string name);
};
As in the OTC class only the definition of a setup function is required. name is the name for the
output file. This class is deriv ed from the PropFile class. Classes deriv ed from this class allo w
the user to prin t an arbitrary n um b er of user sp ecified prop erties that are related to op erator ex-
p ectation v alues. This includes standard (already implemen ted) observ ables of the Observable
class, correlation functions g ( n ) ( τ ) ( Gnfct class) and user defined custom observ ables ( PModular
class). Within this setup function the Observable ob ject needs to b e created as ab o v e and
added to the output file with the command
AddElem(pdens11,"<J_11>");
The second argumen t is the name of this quan tity will ha v e in the header of the output file. This
observ ables file including an arbitrary n um b er of other user sp ecified output files is bundled into
the MyOut class
class MyOut: public Output
{
public:
void SetupMyOut(OTC * system);
};
The setup function includes the follo wing function calls
ObservablesFile *obsfile = new ObservablesFile;
obsfile->SetupMyObsFile(system,"observables.dat");
AddOFile(obsfile);
The user can sp ecify an arbitrary n um b er of differen t output files for customized purp oses
b y either pro viding m ultiple setup functions in one class or deriving a new class with a single
setup function for eac h file. Aside from files managing observ ables the DistFile class is used for
n um b er state distributions of the mo des and the m ulti-lev el systems, as w ell as more complicated
(also custom made) distributions lik e the DickeDistribution . The usage of the DistFile class
is straigh tforw ard and can b e seen in the example co des.
As for the OTC class e.g. in the main file w e need to call
MyOut *out = new MyOut;
out->SetupMyOut(&otc);
These function calls create the whole output structure of the program bundled in to one object.

A.2. Examples 67
Generally PsiQuaSP pro vides functionalit y for setting up v ectors and matrices and to create the
output ob ject ( out ). These three t yp es of ob jects ( Vec , Mat and MyOut ) then pro vide the input
fed in to the PETSc (SLEPc, ...) solution routines. Please note that PETSc vectors are not
alw a ys density matrices and matrices are not alw a ys Liouvillians for the master equation. F or
example the trace op eration and therefore an y computation of an observ able can b e defined as
a PETSc v ector whic h is subsequently applied to the densit y matrix via a scalar pro duct using
the PETSc routine VecDot(...) : Defining a custom observ able ⟨ ˆ
O ⟩ usually is done b y setting
the matrix for the Liouvillian corresp onding to the action of ˆ
O ρ , m ultiplying it with the trace
v ector and storing the resulting v ector. The computation of the observ able is then giv en b y the
scalar pro duct of this v ector with the densit y matrix v ector ( VecDot(...) ). This is sho wn in
example/ex2a . This is also the prop er choice for m ulti-lev el system observ ables.
Solution using PETSc: The n umerical solution is handled b y PETSc, which in this example is
done b y simple time in tegration using a normal fourth order R unge-Kutta in example/ex1a and
an adaptiv e time step R unge-Kutta in example/ex1b . The basic setup of a time in tegration
using PETSc is as follo ws:
TS ts;
TSCreate(PETSC_COMM_WORLD,&ts);
TSSetType(ts,TSRK);
This creates the PETSc time stepp er con text TS and sets it to R unge-Kutta. PETSC_COMM_WORLD
is the PETSc MPI comm unicator. PsiQuaSP is fully parallelized b y default b y using the PETSc
routines, but it can of course alw a ys b e run on a single pro cessor. With the commands
TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,NULL);
TSSetRHSJacobian(ts,L,L,TSComputeRHSJacobianConstant,NULL);
one tells PETSc that the righ t hand side of the differen tial equation ( ˙ ρ = L ρ ) is giv en b y a
constan t matrix and that this matrix is the L matrix. The output of the PETSc time stepp ers
is handled b y a monitor function, whic h is a function with a defined in terface that PETSc calls
at ev ery in tegration step:
TSMonitorSet(ts,MyOut::GenMonitor,out,NULL);
The function MyOut::GenMonitor is the general purp ose monitor function of PsiQuaSP . It prin ts
a single line in to eac h sp ecified output file b y computing all user sp ecified observ ables and
distributions in eac h individual file. By default this function prin ts a single line at ev ery 30 th
time step. This can b e c hanged with the command line option -tev_steps_monitor newvalue .
The command
TSSolve(ts,dm);
solv es the time dynamics, dm con tains alw ays the curren t time step densit y matrix. In Fig. A.4
(a) the mean excitation in the t w o-lev el systems and the mo de during this relaxation is sho wn:
Initially the dynamics are fast due to Rabi oscillations b et w een brigh t Dick e states and the
mo de. Afterw ards the dynamics is go v erned b y the slo w, monotonous sp on taneous emission,
since only the dark Dic k e states remain excited. In Fig. A.4 (b) the p opulation in these Dic k e
states is sho wn.
A.2.2 Example 2: Three-lev el systems
In the t w o-lev el system example ab ov e the base class TLS w as used. F or three- and general
m ulti-lev el systems sp ecialized classes are not pro vided, instead there is the m ulti purp ose class
System (the base class of TLS ). In Figs. A.5 b) and d) t wo differen t three-lev el system sk etc hes
are sho wn, whic h corresp ond to the three-lev el system setups Fig. A.5 a) and c): A.5 b) connects
all degrees of freedom while in A.5 d) four degrees of freedom can b e eliminated, resulting in a
∼ N 4 scaling instead of an ∼ N 8 scaling for A.5 b). The decoupling of some basis states and

68 The PsiQuaSP Library
Figure A.4 – Time dynamics of the op en T a vis-Cummings relaxation: Using the
co de of example/ex1b : a) mean excitation in the t w o-lev el systems n = ⟨ J 11 ⟩ and mean
photon n um b er m = ⟨ b † b ⟩ for 2 t w o-lev el systems prepared in the state P [ 1 , 0 , 0; 0 , 0] . This
corresp onds to the en tanglemen t distillation setup[139]. The brigh t sup erradiant states
couple to the ca vit y mo de and cause Rabi oscillations, while the dark subradiant state does
not couple to the ca vit y and just decays via individual sp on taneous emission[28, 20], c.f. Eq.
(A.2.3). b) Dic ke state occupations ⟨| l , m ⟩⟨ l , m |⟩ : temp oral dynamics of the states of the
sup erradian t subspace (green) vs the single dark state in the subradian t subspace (blue).
the resulting reduction in degrees of freedom is the main reason wh y PsiQuaSP do es not pro vide
sp ecialized classes for m ulti-lev el systems.
F or the t w o-lev el system example w e called TLSAdd(a,b,c,energy) which in ternally calls
MLSAddDens(n11,a+1,energy);
MLSAddPol(n10,b+1);
MLSAddPol(n01,c+1);
where MLSAddDens(...) adds a densit y degree of freedom, corresp onding to a quan tum n um b er
n xx , and MLSAddPol(...) adds a p olarization degree of freedom, corresp onding to a quan tum
n um b er n xy , x  = y . Th us setting the degrees of freedom for three-level systems (Fig. A.5 b)) is
done with the function calls (without truncation)
MLSAddDens(n22,n+1,energy2);
MLSAddPol(n21,n+1);
MLSAddPol(n20,n+1);
MLSAddPol(n12,n+1);
MLSAddDens(n11,n+1,energy1);
MLSAddPol(n10,n+1);
MLSAddPol(n02,n+1);
MLSAddPol(n01,n+1);
or for Fig. A.5 d) with
MLSAddDens(n22,n+1,energy2);
MLSAddDens(n11,n+1,energy1);
MLSAddPol(n10,n+1);
MLSAddPol(n01,n+1);
Here n represen ts the n um b er of three-level systems considered. This n um b er can also b e lo w er
than the n um b er of treated three-lev el systems, whic h corresp onds to a truncation of the n um b er
of three-lev el system basis states. A truncation should alw a ys b e tested if it is applicable in
the giv en situation (parameter dep enden t), but it can reduce the n umerical cost considerably
(Example: strong dephasing in driv en systems can reduce the n um b er of needed offdiagonals
( nxy ) considerably). The nxy ob jects are again the MLSDim iden tifiers and are created with e.g.
MLSDim n21 (2,1);

A.2. Examples 69
Figure A.5 – Differen t three-lev el system setups: a), b) The lev el sc heme and the
sk etc h for the Lambda system setup. c), d) The three lev el laser of Section 5.4.3.
As in the t w o-lev el system example, after setting all m ulti-lev el system degrees of freedom the
user can add b osonic mo des with the command
ModeAdd(m0+1,dm0,modeenergy);
This order of ModeAdd(...) after the MLSAdd... function calls is mandatory , PsiQuaSP returns
an error message if these routines are not called in the righ t order. Setting e.g. the sp on taneous
emission b et w een lev els 1 − 0 for Fig. A.5 b) is done with
AddLindbladRelaxMLS(L,NULL,NULL,1,n11,n00,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n10,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n01,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n21,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n12,gamma/2.0);
and for Fig. A.5 d) it is
AddLindbladRelaxMLS(L,NULL,NULL,1,n11,n00,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n10,gamma/2.0);
AddLindbladDephMLS(L,NULL,NULL,1,n01,gamma/2.0);
The parameter gamma/2.0 is the same parameter as app earing in the master equation and each
function call corresp onds to exactly one of the arrows in the sk etc hes. The incoheren t pumping
is added b y calling
AddLindbladRelaxMLS(L,NULL,NULL,1,n00,n22,pump/2.0);
and the resp ectiv e calls to AddLindbladDephMLS() . The in teraction of the three-lev el systems
with the mo de for Fig. A.5 b) is added by calling
AddMLSModeInt(AA,NULL,NULL,1,n20,n21,mbra,-gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n10,n11,mbra,-gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n00,n01,mbra,-gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n02,n12,mket,gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n01,n11,mket,gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n00,n10,mket,gcouple*PETSC_i);
and for Fig. A.5 d) omitting the arro ws of the disconnected part of the sk etc h
AddMLSModeInt(AA,NULL,NULL,1,n10,n11,mbra,-gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n00,n01,mbra,-gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n01,n11,mket,gcouple*PETSC_i);
AddMLSModeInt(AA,NULL,NULL,1,n00,n10,mket,gcouple*PETSC_i);
mket and mbra are the iden tifiers for the mo de degrees of freedom and are created b y calling
ModeDim mket (0,photonnumber);
ModeDim mbra (1,photonnumber);
photonnumber is the index of the mo de. Mo des are n um b ered in ternally starting from zero
in the order they are created with an AddMode() call. Hamiltonian con tributions that c hange
the righ t index of the MLSDim and/or act on the bra side of the mo de expansion come with a
min us sign. This stems from the commutator in the v on-Neumann part of the quan tum master
equation, see section A.4 for more details. The generation of the output as w ell as the solution

70 The PsiQuaSP Library
example/ System, concepts, tec hniques
ex1a Op en T avis-Cummings model, simple observ ables, distributions, time-in tegration
ex1b ex1a with thermal bath, PETSc concepts, adaptiv e time in tegration,
Dic k e distribution
ex2a T w o-lev el laser, incoheren t pump, custom observ ables
ex2b Direct steady state/n ull space computation using SLEPc Krylo v-Sch ur algorithm
ex2c T w o-lev el laser with Non-R W A terms
ex3a Lam b da system setup, m ulti-lev el system usage
ex3b Three-lev el laser
ex4a Phonon laser/laser co oling setup, custom Liouvillians
ex5 Same as ex3a , using P arMETIS graph partitioning to exploit U (1) symmetry ,
leading to a reduction from N 8 to ∼ N 7
T able A.1 – Ov erview o v er the example co des and the concepts explained/in tro duced
in these examples. ex2b requires an additional SLEPc installation and for ex5 it is necessary
to build PETSc with the –download-parmetis flag.
stage is completely analogous to the t w o-lev el system example. F urther examples, illustrating
other master equations, custom observ ables, custom distributions, custom Hamiltonians and
Liouvillians as w ell as other solution tec hniques and adv anced, graph theory based reduction of
degrees of freedom are pro vided in the examples in the example/ folder as well as in section A.4.
An o v erview of the examples and the concepts explained in them is giv en in T able A.1.
A.3 T emplate functions v ersus custom Liouvillians
PsiQuaSP has roughly t w o types of usages. The first usage w as presen ted in the previous sec-
tion: There are ready-made functions for setting arrows of common Hamiltonians and Lindblad
dissipators. Generally a single function call to one of these functions represen ts a single arro w
in one of the sk etc hes. First the user dra ws the sk etch represen tation of the master equation
and then directly translates the sk etc h in to co de. In the case of t w o-lev el systems it is ev en
simpler – a single function call is sufficien t to set a Hamiltonian or dissipator con tribution. The
implemen ted con tributions are shown in T able A.2.
In the second usage form the user defines elemen tary Liouville space operators and constructs
arbitrary master equations, observ ables, distributions, etc. from these elemen tary op erators:
The p erm utation symmetric metho dology is in principle applicable to any p erm utation sym-
metric quan tum master equation and using the general framew ork of PsiQuaSP one can solv e
in principle any quan tum master equation in a n um b er state represen tation (there is no sup-
p ort for coheren t state basis etc.). Since w e cannot pro vide template setup functions for ev ery
conceiv able Liouvillian matrix there needs to b e another, more flexible approac h for this: In the
second t yp e of usage the user defines elemen tary Liouville op erators lik e
J xy ρ ˆ = J L
xy ρ, ρJ xy ˆ = J R
xy ρ. (A.3.1)
Here again the L, R algebra w as used [126]: F or an y Hilb ert space op erator one defines a Liouville
space op erator b y distinguishing whether it acts on the left or righ t side of the densit y matrix,
i.e. Aρ = A L ρ and ρA = A R ρ . As in the first t yp e of usage the setup of these elemen tary
Liouville op erators is done b y first dra wing a sk etc h for eac h needed op erator and then adding
all needed arro ws b y single function calls. Based on these elemen tary operators the user then
can define arbitrary in teraction Hamiltonians and dissipators as w ell as custom observ ables,

A.4. Building arbitrary Liouvillians 71
Liouvillian System function Examples
H = ℏ ω 0 b † b AddModeH0() ex3a
H = ℏ ω xx J xx AddMLSH0() ex1a , ex1b
H = ℏ g ( J xy + J y x )( b † + b ) AddMLSModeInt() ex2c
H = ℏ g ( J xy b † + J y x b ) AddMLSModeInt() ex1a , ex1b
H = ℏ E ( J xy e iω t + J y x e − iω t ) AddMLSCohDrive() ex3a
H = ℏ E ( be iω t + b † e − iω t ) AddModeCohDrive() none
D = γ
2 ∑ i (2 σ i
xy ρσ i
y x − σ i
y y ρ − ρσ i
y y )
D = δ ∑ i ( σ z ,i
xy ρσ z ,i
y x − ρ )
AddLindbladRelaxMLS()
AddLindbladDephMLS() ex1a , ex1b
D = κ
2 ( bρb † − b † bρ − ρb † b ) AddLindbladMode() ex1a
D = κ
2 ( ( ¯ m + 1)( bρb † − b † bρ − ρb † b )
+ ¯ m ( b † ρb − bb † ρ − ρbb † ) ) AddLindbladModeThermal() ex1b
T able A.2 – Ov erview o v er the general ready-made Liouvillian setup functions
of the System class. Please lo ok in to the TLS class do cumen tation to see the deriv ed,
sp ecialized t w o-lev el system functions. The Hamiltonian con tributions alw a ys refer to the
i/ ℏ [ ρ, H ] terms. Using σ z ,i
xy = 1 / 2( σ xx − σ y y ) .
distributions, basis transformations etc. F or instance using Eq. (A.3.1) the definition of a
collectiv e sp on taneous emission Liouvillian from lev el x to lev el y w ould be
D ( ρ ) = Γ
2 ( J y x ρJ xy − J xy J y x ρ − ρJ xy J y x ) = Γ
2 ( J L
y x · J R
xy − J L
xy · J L
y x − J R
y x · J R
xy ) ρ, (A.3.2)
here the com bination of the R/L op erators · is p erformed b y the standard matrix-matrix pro duct,
pro vided b y the PETSc function MatMatMult() . Hence the user first defines elemen tary matrices
and then uses the PETSc matrix m ultiplication and addition to ols to construct ev ery conceiv able
Liouville op erator. The details for this type of application are explained in the next s ection.
A.4 Building arbitrary Liouvillians
In this section the formalism that allo ws to setup all p ossible Liouvillians that are consisten t
with the p erm utation symmetric metho d is discussed. It closely resem bles the treatmen t in the
Sections 5.5.1 and 5.5.2. There are separate setup functions for m ulti-lev el system and mo de
degrees of freedom, e.g. for J L
xy or b R . These elementary matrices can b e used to construct
more complicated op erators suc h as J L
xy + J R
y x and J L
xy b L b y using the PETSc to ols for matrix
m ultiplication and addition. Defining suc h setup functions for the mo de degrees of freedom
is straigh tforw ard and is based on textb o ok ph ysics [140]. F or the symmetric basis states of
PsiQuaSP the treatmen t is a bit more difficult. The follo wing discussion is tec hnical, it in v olv es
the p erm utation symmetric Liouville space op erators of the previous section. The usage how ev er
is then v ery simple, it again results in dra wing simple sketc hes and directly implemen ting single
arro ws b y single function calls.
T e chnic al details: As defined in Eq. (A.1.2) PsiQuaSP uses an expansion of the densit y matrix
in Liouville space. Expansion co efficients can be obtained via the Hilb ert-Sc hmidt inner pro duct
P [ { n k l } ] = tr [ ˆ
P [ { n k l } ] ρ ] (A.4.1)
The actions of an y op erators A , B on the density matrix AρB are handled b y PsiQuaSP lik e
applying these op erators to ˆ
P [ { n k l } ] :
tr [ ˆ
P [ { n k l } ] AρB ] = tr [ B ˆ
P [ { n k l } ] Aρ ] . (A.4.2)

72 The PsiQuaSP Library
Hence one needs to find a general recip e to construct arbitrary op erators B ˆ
P [ { n k l } ] A expressed
in the p erm utation symmetric basis, for all AρB that liv e in the p erm utation symmetric sub-
space. In order to do this there are t w o steps: First the elemen tary pro cesses/Liouville op erators
need to b e iden tified and second the recip es ho w to construct ph ysically relev an t op erators, lik e
e.g. a collectiv e raising op erator for a four-lev el system acting from the left, from these ele-
men tary op erators need to b e form ulated. The p erm utation symmetry demands that only those
pro cesses can b e included that are based on spin matrices acting indistinguishably on the left
and/or righ t side of the densit y matrix. These elemen tary op erators should b e represen table b y
arro ws.
Defining elementary pr o c esses/arr ows: Lo oking at the sk etc hes tw o general t yp es of arro ws ap-
p ear: Connecting and nonconnecting arrows. A connecting arro w represen ts a coupling b et w een
t w o different symmetric basis states Eq. (5.5.3), corresp onding to an in- or out-scattering pro-
cess, and a nonconnecting arro w just acts on the state itself, lea ving it unc hanged. This is quite
analogous to the actions of the in teracting and non-in teracting parts of a Hamiltonian acting on
a Hilb ert space state, or rather that the symmetrized basis states ˆ
P Eq. (5.5.3) are eigenstates
of the op erators corresp onding to the nonconnecting arro ws. It turns out that these are the
only p ossible t w o t yp es. The general mathematical expressions are giv en b y the Γ matrices
in tro duced in the previous c hapter
∑
i
σ i
xx ˆ
P [ . . . ] σ i
y y = n xy ˆ
P [ . . . ] (A.4.3)
for a single nonconnecting arro w and
∑
i
σ i
xy ˆ
P [ . . . ] σ i
k l = ( n xl + 1) ˆ
P [ . . . n xl + 1 . . . n y k − 1 . . . ]Θ( n y k ) , (A.4.4)
for a connecting arro w, where Θ( n ) is equal to one for n > 0 and zero otherwise. Here w e
write do wn only the c hanged n um b ers n xl and n y k . Multiplying these equations with the den-
sit y matrix and taking the trace results again in equations of motion. Using these t w o t yp es
of arro ws it is p ossible to construct ev ery p erm utationally symmetric m ulti-level system Li-
ouville op erator. The PsiQuaSP functions for adding one of these arro ws to a giv en matrix
are AddMLSSingleArrowNonconnecting(...) and AddMLSSingleArrowConnecting(...) . The
sk etc h represen tation for these t w o types is shown in Fig. A.6 a) and d): Eq. (A.4.3) describing
nonconnecting arro ws can represen t t w o differen t types of pro cesses dep ending on the corre-
sp onding prefactor in the master equation. If the prefactor is imaginary the term corresp onds
to a Hamiltonian part H 0 , or if it is negativ e and real it corresp onds to dephasing, caused e.g.
b y a dissipator (Fig. A.6 b) and c)). The t w o arro ws are the lo op ed and the out w ard p oin ting
arro ws in Fig. A.6 a). The connecting arro w Eq. (A.4.4) usually also represen ts t w o differen t
pro cesses: One sided flip op erator actions arising from in teraction Hamiltonians (Fig. A.6 e) and
f )) and densit y relaxation caused b y individual sp on taneous emission, decay dissipators (Fig.
A.6 g)).
Constructing physic al op er ators: Lo oking at the collectiv e flip op erator acting from the righ t
tr [ ˆ
P [ . . . ] ρJ xy ] = tr [ J xy ˆ
P [ . . . ] ρ ]
= tr [ ∑
i
σ i
xy ˆ
P [ . . . ] ∑
k
σ i
k k ρ ]
= ∑
k
( n xk + 1) P [ . . . n xk + 1 . . . n y k − 1 . . . ]Θ( n y k ) (A.4.5)
amoun ts to summing o ver all possible individual connecting arrows, see Fig. A.6 e). Here in the
second line the Hilb ert space iden tit y for eac h individual d -level system w as inserted
I i = ∑
k
σ i
k k . (A.4.6)

A.4. Building arbitrary Liouvillians 73
Figure A.6 – Mo dular sk etc hes for m ulti-lev el systems: a) The nonconnecting arro w
can represen t the phase oscillations arising from the self energy Hamiltonians (curv ed arro w)
and it can describ e dephasing (straigh t arro w). b) and c) the sk etc hes corresp onding to
dephasing ˙ ρ ∝ J R
xx ρ and ˙ ρ ∝ J L
y y ρ . d) The connecting arro w can represen t flip op erators
and densit y relaxation. e) and f ) the arro ws corresp onding to the flip op erators ˙ ρ ∝ J R
xy ρ
and ˙ ρ ∝ J L
xy ρ , c.f. Eqs. (A.4.5) and (A.4.7). g) The density relaxing arro w caused b y an
individual sp on taneous emission lik e dissipator ˙ ρ ∝ ∑ i σ i
xy ρσ i
y x = Γ y x
xy ρ . h) The densit y
relaxation arro w in tro duced in Fig. 5.2 called by the function AddLindbladRelaxMLS()
consists of three arro ws in the elemen tary picture, t w o nonconnecting and one connecting
arro w.
The actions of the σ i
xy matrices in Eq. (A.4.5) c hange eac h individual spin matrix σ i
y k in to a
spin matrix σ i
xk . The k sum of the σ i
k k matrices results in a sum o v er all p ossible righ t k indices
in n y k and n xk . In the last step Eq. (A.4.4) w as inserted and the trace w as computed. F rom
this expression it can already b e seen that the resulting matrix will b e sparse, since there are at
most k nonzero en tries in eac h line of this matrix.
The same op erator acting from the left results in a sum ov er all p ossible left k indices
tr [ ˆ
P [ . . . ] J xy ρ ] = tr [ ∑
k ∑
i
σ i
k k ˆ
P [ . . . ] σ i
xy ρ ]
= ∑
k
( n k y + 1) P [ . . . n k y + 1 . . . n k x − 1 . . . ]Θ( n y k ) . (A.4.7)
These t w o op erators can b e implemen ted b y rep eatedly calling the single connecting arro w
function AddMLSSingleArrowConnecting(...) – once for ev ery p ossible k v alue, see Fig. A.6
e) and f ). The action of a collective projection or diagonal op erator J xx is giv en b y
tr [ ˆ
P [ . . . ] ρJ xx ] = tr [ ∑
k ∑
i
σ i
xx ˆ
P [ . . . ] σ i
k k ρ ]
= ∑
k
n xk P [ . . . ] (A.4.8)
and
tr [ ˆ
P [ . . . ] J xx ρ ] = tr [ ∑
k ∑
i
σ i
k k ˆ
P [ . . . ] σ i
xx ρ ]
= ∑
k
n k x P [ . . . ] , (A.4.9)
whic h can b e implemen ted b y rep eatedly calling AddMLSSingleArrowNonconnecting(...) –
again once for ev ery p ossible k v alue, see Fig. A.6 b) and c). This discussion also serv es as a

74 The PsiQuaSP Library
bρ AddModeLeftB(...) ρb AddModeRightB(...)
b † ρ AddModeLeftBd(...) ρb † AddModeRightBd(...)
b † bρ AddModeLeftBdB(...) ρb † b AddModeRightBdB(...)
bb † ρ AddModeLeftBBd(...) ρbb † AddModeRightBBd(...)
bρb † AddModeLeftBRightBd(...) b † ρb AddModeLeftBdRightB(...)
T able A.3 – List of all a v ailable functions for setting elemen tary mo de Liou-
villians: The redundan t functions allo w faster and easier co de dev elopmen t – actually all
Liouvillians could b e constructed from the first t w o.
Figure A.7 – Sk etc hes needed for the phononlaser example co de: F rom a) to f ):
Sk etc hes corresp onding to J L
11 , J R
11 , J R
10 , J R
01 and J L
10 , J L
01 . When the op erator acts on the left
(righ t) side of the densit y matrix, it acts on the righ t (left) index of the n xy , c.f. Eq. (A.4.2).
T w o v ersions of the J L
10 op erators for the full and reduced three lev el system dynamics, c.f
Fig. A.5 b) and d).
pro of for the sum rules for the Γ op erators in the last c hapter.
With this set of results it is clear ho w to construct a general self energy Hamiltonian ˙ ρ ∼
i/ ℏ [ ρ, H 0 ] or a general individual dissipator
D ρ = γ
2 ( ∑
i
σ i
xy ρσ i
y x − J y y ρ − ρJ y y ) , (A.4.10)
where the first term is set b y a single call to AddMLSSingleArrowConnecting(...) , see Eq.
(A.4.4), and the second and third term are set as in Eqs. A.4.8 and A.4.9. Please note that the
p ossibilit y of a decoupling of some coherence degrees of freedom as in Fig. A.5 d) is the main
reason wh y PsiQuaSP do es not pro vide generalized setup functions for op erator actions of J xy
and J xx , since it w ould result in unnecessary n umerical cost, if the decoupled basis elemen ts w ere
included. The other reason is that the elemen tary arro w representation also pro vides maximal
freedom for the application programmer, whereas an y encapsulation/facilitation w ould alw ays
b e asso ciated with a cut in generalit y .
The sk etc hes corresp onding to simple op erators lik e J xy and J xx are simple to draw. Sk etches
corresp onding to Liouville op erators lik e J xy ρJ y x or J n
xy ρ are more complicated and it is not
recommended to implemen t these sk etches b y hand. Rather w e recommend to define the ele-
men tary pro cesses lik e J xy and J xx , set the corresp onding matrices and then use the PETSc
to ols MatMatMult() and MatAXPY(...) to construct more complicated op erators. In order to

A.5. P erformance 75
do so the follo wing iden tities are useful
AρB ˆ = A L · B R ρ = B R · A L ρ,
AB ρ ˆ = A L · B L ρ, ρAB ˆ = B R · A R ρ, (A.4.11)
where the · op eration is given b y the MatMatMult() op eration. The elemen tary Liouville space
op erators for the b osonic mo des can b e set b y calling the functions sho wn in T able A.3.
Simple example: In example/ex4a the phonon laser/laser co oling master equation from Refs.
141, 142 is implemen ted
H = ℏ ∆ J 11 + ℏ ω ph b † b + ℏ g J 11 ( b + b † ) + ℏ E ( J 10 + J 01 ) . (A.4.12)
Here ∆ = ω 11 − ω L is the detuning of the t w o-lev el systems from the driving laser. F or p ositive
detuning near the Stok es resonance this corresp onds to laser co oling and for negativ e detuning
at the an ti-Stok es resonance this corresp onds to phonon lasing. The master equation includes
individual sp on taneous emission and finite phonon lifetime through the dissipators
D 1 ( ρ ) = γ
2 ∑
i
( σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) ,
D 2 ( ρ ) = κ
2 ( bρb † − b † bρ − ρb † b ) . (A.4.13)
In this example six t w o-lev el system op erators are needed to construct the master equation:
J L,R
11 , J L,R
10 and J L,R
01 . Eac h of these matrices are defined b y t w o calls to the single nonconnecting
arro w functions AddMLSSingleArrowNonconnecting(...) for J L,R
11 and the connecting arro w
functions AddMLSSingleArrowConnecting(...) for J L,R
10 and J L,R
01 . The sketc hes for these
matrices are sho wn in Fig. A.7. F rom these matrices and the resp ective phonon matrices w e
can construct the Liouvillian and also p ossible additional observ ables.
A.5 P erformance
The t w o main adv an tages of PsiQuaSP are the reduction of complexit y due to the symmetrized
basis states and the manifold of solv ers pro vided through PETSc and e.g. SLEPc.
Over al l c omplexity: In Fig. A.8 a) th e n um b er of basis elemen ts of the densit y matrix for the
full exp onen tial densit y matrix is compared to the p olynomial, symmetrized PsiQuaSP densit y
matrix for t w o- and three-level systems. This corresp onds to the o v erall complexit y since b oth
the storage requiremen t and the n um b er of coupled equations scale like the n um b er of basis
elemen ts.
Ste ady state c omputation: In Fig. A.8 b) the con v ergence time for steady state calculations for a
t w o-level laser as discussed in the last c hapter and implemen ted in the examples example/ex2a
and example/ex2b for differen t solv ers is shown: the fixed time step fourth order Runge-Kutta
is b y far the slo west solv er. The adaptiv e time step and the direct n ull space computation
using the SLEPc pac kage outp erform this standard routine. The sp eedup of the shift and in v ert
sp ectral transformation solv er compared to the RK4 metho d amoun ts almost to a factor of 5000 .
Please note that these n um b ers and the relativ e p erformance of these solv ers is parameter and
system size dep enden t, it is p ossible to find examples where the difference is ev en higher but it
is also p ossible to find examples where the difference is less pronounced. Esp ecially for iterative
solv ers lik e the SLEPc Krylov-Sc h ur eigen v alue solv er con v ergence time is highly dep enden t on
the sp ectrum of the matrix and on c hosen solv er sp ecific parameters. Please refer to the PETSc
and SLEPc do cumen tations for the sp ecifics of these metho ds.

76 The PsiQuaSP Library
Figure A.8 – PsiQuaSP p erformance: a) The scaling of storage space and o v erall
computation time for t w o-level systems and three-lev el systems using the full exp onen-
tial approac h vs. the p erm utation symmetric PsiQuaSP approac h. b) Run time compar-
ison b et w een differen t solution metho ds for steady state calculations for a t w o-lev el laser
setup: fixed time step fourth order R unge-Kutta (RK4), adaptiv e time step R unge-Kutta
(TSRK3BS), SLEPc Krylo v-Sc hur n ull space computation (EPS KS) and SLEPc Krylo v-
Sc h ur null space computation with exact shift and in v ert sp ectral transformation (sin v ert).
Please refer to the PETSc and SLEPc do cumen tation and the discussion in this section for
details to these solv ers.
A.6 Some notes on solv ers
In the last section the p erformance of differen t solv ers for steady state computation w as com-
pared. There are in principle t w o differen t questions that can b e answered b y n umerically solving
a quan tum master equation
˙ ρ = L ρ. (A.6.1)
The first question is "Ho w do es the system ev olv e in time?" and the second question is "What
is the steady state?". Dep ending on the question the solver should be chosen.
F or a constan t Liouvillian steady states should b e computed using steady state solvers and only
if the time dep endency is of in terest then one should resort to direct in tegration. Time ev olution
should b e calculated with an adaptiv e step width R unge-Kutta: The step width is adaptiv ely
c hosen b y comparing the solutions of t w o differen t R unge-Kutta algorithms of differen t order,
for instance a second and a third order R unge-Kutta. As seen in Fig. A.8 b) this alone pro vides
a dramatic sp eedup.
There is a large v ariet y of different steady state solv ers: PETSc pro vides generally t w o differen t
approac hes: (i) pseudo time stepping, whic h is a fak e time in tegration that directly computes
the steady state and the in termediate time steps are unph ysical. This in principle allo ws also for
explicitly time-dep enden t Liouvillians (not used in this thesis), and (ii) a h uge v ariet y of iterative
Krylo v subspace metho ds (through the KSP ob ject). Krylov subspace metho ds are somewhat
the standard for iterativ e sparse matrix diagonalization and ha v e b een used a lot in this thesis.
A.6.1 Krylo v subspaces
A Krylo v subspace K of a complex v ector space C n , is constructed from a matrix/op erator
A ∈ C n × n and a column v ector q ∈ C n [143]
K = K ( A, q ) = K m ( A, q ) = span { q , Aq , A 2 q , . . . A m − 1 q } . (A.6.2)

A.6. Some notes on solv ers 77
Re [ λ ]
Im [ λ ]
∆
λ 0
λ 1
(a)
Re [ λ ]
Im [ λ ]
∆
λ 0
λ 1
(b)
Figure A.9 – T w o differen t Liouvillian sp ectra: a) This sp ectrum mak es the gap ∆
computation exp ensiv e. b) Gap and steady state computation are (relativ ely) exp ensiv e.
The main idea is that the eigen v ectors of A are usually part of the Krylo v subspace and if
one can find the desired eigen v ector in a Krylov subspace with m ≪ n then the dimensional-
it y/complexit y of the problem is greatly reduced. Also the in v erse of a non-singular matrix is
part of the Krylo v subspace. The n ull space is ho w ev er not con tained in the Krylo v-subspace,
a singular matrix also do es not ha v e an in v erse (exept for the Mo ore-P enrose pseudoin v erse).
Therefore Krylo v subspace algorithms often ha v e a n ull space correction included. There is a
large v ariet y of different iterativ e algorithms that utilize exactly this prop erty , one rather well
kno wn example is GMRES – generalized minimal residual metho d. This solv er is ho w ev er not
suited for the Liouvillians considered in this w ork. Throughout this work the Krylo v-Sc h ur
algorithm of the SLEPc pac kage is used for steady state computations.
A.6.2 Sp ectral transformation
Normally for sparse matrices only an excerpt of the sp ectrum is computed, since a large ma-
trix has a lot of eigen v alues and usually only few of those are of actual interest. In the case
of quan tum master equations it is usually the n ull space and the smallest magnitude non-zero
eigen v alue/v ector that are of in terest. As stated in Section 3.4 the sp ectrum of the matrix in-
fluences the con v ergence time of the iterative solv ers: Generally , if there are other eigen v alues
in the vicinit y of the desired eigen v alue the con v ergence is slo w: In Fig. A.9 a) and b) tw o
differen t sp ectra are sho wn. In Fig. A.9 a) there are man y eigen v alues close to the λ 1 eigen v alue
making the computation of the Liouvillian gap ∆ exp ensive, because it requires λ 1 , computing
only the n ull space is c heap. In Fig. A.9 b) λ 1 and λ 0 are close to eac h other (dissipativ e phase
transition), whic h mak es the steady state computation more exp ensiv e, but computing the gap
is equally exp ensiv e.
In order to b ypass these difficulties there are the sp ectral transformation tec hniques: One trans-
forms the eigen v alue problem
Ax = λx (A.6.3)
in to a problem that has a "nicer" sp ectrum but is still equiv alen t to the original problem. The
easiest sp ectral transformation is the shift transformation
Ax = λx → ( A − σ I ) x = ( λ − σ ) x. (A.6.4)

78 The PsiQuaSP Library
This ho w ever does not change the fact that the eigen v alues are to o close to eac h other, therefore
one uses shift and in v ert
Ax = λx → ( A − σ I ) x = ( λ − σ ) x → ( A − σ I ) − 1 x = ( λ − σ ) − 1 x (A.6.5)
The 1 /x function is w ell suited to separate the eigen v alues ev en if the λ i are close to eac h other.
In the case of Liouvillians the shift is mandatory since the Liouvillian is singular and th us do es
not ha v e an inv erse without a shift. F urthermore the shift σ should b e chosen positive to ensure
that all eigen v alues are unequal to zero (b ecause the Liouvillian sp ectrum lives in the negativ e
half plane).
Th us shift and in v ert sp eeds up the conv ergence, see Fig. A.8 b). The dra wbac k of this metho d
is that computing the in v erse requires a LU factorization, whic h in requires a lot of storage and
the standard PETSc LU factorization only w orks in single pro cessor op eration. F or instance for
the steady states computed in Chapter 7 the shift and in v ert sp ectral transformation w as used
and for larger systems ( N = 9 ) with photon n um b er states up to 50 the storage requiremen t
exceeded 100 giga b ytes. This corresp onds to a matrix of the order 10 5 × 10 5 to 10 6 × 10 6 .
Pro cessing 100 giga b ytes on a single cpu is far from efficien t. Therefore in order to go to ev en
larger systems/matrices parallel LU factorization provided b y the external pac kages MUMPS
or Sup erLU should b e used.
A.7 Conclusion
A library w as in tro duced that enables the setup of master equations for iden tical m ulti-level
systems. The library pro vides ready-made setup functions for densit y matrices as w ell as Liou-
ville op erators. The design of these functions is cen tered around the sk etc h represen tation of
the Liouville op erators or master equation in tro duced in in Chapter 5. This has the adv an tage
that implemen ting an arbitrary master equation do es not require calculating an y equations of
motion but can b e done b y directly implemen ting the sk etc hes. There is a simplified usage for
t w o-level systems and ready-made Liouvillian setup routines and an adv anced usage where the
user can construct arbitrary p erm utation symmetric Liouvillians from simple sk etc hes.

I I I Results
79

6 Ca vit y QED Lasers and Spasers
In Chapter 5 the usage of the sk etc h represen tation w as illustrated using differen t laser exam-
ples. In this c hapter these examples are applied. The range of application of the p erm utation
symmetric metho d in the con text of lasers is the small system size, cQED laser – meaning one
to few emitters and mo derate photon num b ers. The cQED lasers are the small or mesoscopic
coun terparts to the classical lasers, whic h usually in v olv e macroscopic setups: T wo mirrors and
a macroscopic gain medium lik e an atomic gas disc harge lamp in b et w een, see Fig. 6.1 (a). The
classical, con v en tional lasers usually inv olv e thousands, millions or more emitters/electrons that
serv e as gain medium and usually ha ve v ery high photon output rates. Con trary examples for
cQED lasers are quan tum dots in micropillar structures: These micropillars contain only few
quan tum dots that are on resonance with the ca vit y mo de and the cavit y mo de is not formed
b et w een t wo standard macroscopic mirrors but rather b y the distributed Bragg reflector (DBR)
structure of the micropillar, see Fig. 6.1 (b).
Apart from the t ypical output p o w ers and the size of the setup there is one striking difference
in the b eha vior of cQED lasers compared to con v en tional, macroscopic lasers: In con v en tional
lasers the lasing threshold and the start of the stim ulated emission pro cess can b e seen in a sim-
ple input-output curv e – Increasing the pump rate leads to a sudden "explosion of stim ulated
emission" [84], which manifests itself in a stark sudden increase in the ca vit y output. This can b e
observ ed exp erimen tally and serv es as a demonstration of lasing action. Ho wev er in cQED lasers
this sudden increase is (almost) absen t, the ca vit y outpu t increases linearly with the pump rate,
there is no real indication where the lasing action starts in a simple input-output curv e. This
has spark ed a vivid discussion ab out threshold-less lasing, since at first sigh t it seems that the
threshold for lasing action go es to zero pump p o w er [84]. How ev er b y lo oking at the statistics
of the emitted ca vit y photons it b ecomes clear that this is not the case, there remains a finite
threshold at whic h the photons b ecome coheren t or rather P oissonian and the absence of the
threshold is attributed to a pronounced sp on taneous emission con tribution. The quan tit y that
81

82 Ca vit y QED Lasers and Spasers
(a) (b)
Figure 6.1 – Differen t t yp es of lasers: (a) A Helium-Neon Laser is a macroscopic setup.
Image tak en from [144]. (b) A DBR Micropillar with corresp onding mo de structure. Suc h
a micropillar has usually a spatial dimension of a few microns, it is not visible to the nak ed
ey e. Image tak en from [145]
Figure 6.2 – Lev el sc hemes for the a) t w o-, b) three and c) four-lev el laser setup considered
in this c hapter.
distinguishes b et w een these t wo cases is the β factor, whic h is defined as the ratio of sp on taneous
emission in to the lasing mo de γ l to the total sp on taneous emission rate γ l + γ nl
β = γ l
γ l + γ nl
. (6.0.1)
In macroscopic lasers usually γ nl ≫ γ l holds and in cQED lasers usually γ nl ∼ γ l and ev en
γ nl ≪ γ l , whic h results in β factors of β = ∼ 10 − 6 for con v en tional lasers and β = ∼ 0 . 1 − 1 for
cQED lasers [84]. In fact the β factor can b e used as a measure to define the system size in
lasers, with the β = 1 defining the atomistic laser and β → 0 defining the thermo dynamic limit,
the macroscopic laser [84].
In this c hapter three differen t laser setups are discussed, namely the incoheren tly driv en t w o-,
three- and four-lev el laser. The t w o-level laser theory will be used to explore the standard range
of cQED lasers. The three- and four-lev el laser theories will b e used to further in v estigate the
findings of M. Ric h ter, M. Gegg, T.S. Theuerholz, A. Knorr published in Phys. Rev. B 035306
(2015) [17]: The spaser – surfac e plasmon amplific ation by stimulate d emission of r adiation
[59, 60] can b e appro ximated b y a bad ca vit y (short photon lifetime) laser with relativ ely high
emitter-ligh t coupling strength [57]. In Ref. [17] it w as found that threshold pump rates for
spasing action based on a t w o-lev el system bad ca vit y laser mo del are unrealistically high, since
they need to o v ercome the tremendous plasmonic losses. Therefore it migh t be interesting to
in v estigate three- and four-level laser theories.
The sk etc hes and level sc hemes for the t w o-, three- and four-lev el laser setups considered in this
c hapter are sho wn in Figs. 6.2 and 6.3. They all include an incoheren t driv e, whic h reduces

6.1. Cluster expansion – Rate equation theory 83
Figure 6.3 – Sk etc hes for the a) t w o-, b) three and c) four-lev el laser setup considered in
this c hapter. The four-lev el laser do es not ha v e a pump-rate dep enden t dephasing. These
sk etc hes we re directly translated in to co de using the PsiQuaSP library [26, 27], see App endix
A.
n umerical cost and enhances comparabilit y .
P arts of the discussion in this c hapter were presen ted at the PQE 2016 and NOEKS 2016 con-
ferences.
6.1 Cluster expansion – Rate equation theory
Ev en though the p erm utation symmetric metho d is efficien t compared to the brute force solution
it still can b e quite in v olv ed. Therefore it is b eneficial to first deriv e a simplified theory giv es
estimates for parameter ranges. Here, esp ecially the lasing threshold is of in terest. F or this
purp ose it is customary to deriv e a rate equation theory , whic h is equiv alent to a lo w order
cluster expansion.
In the follo wing the deriv ation is shortly outlined: The starting p oin t are exp ectation v alues
of op erators lik e b † b and J xx , the mean ca vit y and excited state p opulations. Here x refers
to the upp er lev el of the lasing transition, see Figs. 6.2 and 6.3. Equations of motion for
the exp ectation v alues can b e derived b y lo oking at the time-deriv ativ e of these quantities and
inserting the Lindblad equation (Sc hrö dinger picture, see Section 3.3)
∂ t tr [ O ρ ] = tr [ O ˙ ρ ] = tr [ O L ρ ] (6.1.1)
and then calculating all necessary comm utation relations. F or the mean ca vit y mo de p opulation
this yields
∂ i ⟨ b † b ⟩ = 2 g I m ⟨ J xy b † ⟩ − κ ⟨ b † b ⟩ , (6.1.2)
where g is the ligh t matter in teraction strength, κ is the ca vit y deca y rate, J xy is the flip op erator
corresp onding to the lasing transition, where y is the upp er and x is the low er lev el of the lasing
transition, see Figs. 6.2 and 6.3. In order to pro ceed one then derives the equation of motion
for the new op erator ⟨ J xy b † ⟩
⟨ J xy b † ⟩ ∝ ig ⟨ ( J y y − J xx ) b † b ⟩ , (6.1.3)

84 Ca vit y QED Lasers and Spasers
whic h in turn couples to another op erator pro duct. This is called hierarc h y problem [9]: Through
the in teraction Hamiltonian the exp ectation v alues of op erators couple to exp ectation v alues of
higher op erators, where higher means a pro duct of (more) op erators. In principle this hierarc hy
w ould go to infinit y (since the Hilb ert space of the b osons is not b ounded), but in the rate
equation theory one assumes that quan tum correlations are small and th us the assisted quan tities
are factorized
⟨ ( J y y − J xx ) b † b ⟩ ∼ ⟨ J y y − J xx ⟩⟨ b † b ⟩ . (6.1.4)
This results in closed equations of motion. F or t w o-, three- and four-level systems this approac h
results in three, four and fiv e coupled equations resp ectiv ely . These can b e solved analytically
in the stationary limit
∂ t ⟨ O ⟩ = 0 (6.1.5)
and results in a second order p olynomial equation for the mean cavit y n um b er m = ⟨ b † b ⟩
( κm ) 2 + b ( P )( κm ) + c ( P )=0 , (6.1.6)
where κ − 1 is the photon lifetime and the parameters b ( P ) and c ( P ) are functions of all system
parameters, esp ecially of the pump rate P . Eq. (6.1.6) defines a parab ola and the zero es of the
parab ola determine the ca vit y photon n umber: One solution is alw ays negativ e, th us there is
alw a ys exactly one ph ysical solution. The system starts lasing if the parameter b ( P ) b ecomes
negativ e, see Fig. 6.4 (a) and (b). Please note that there are parameter ranges in whic h b ( P )
is strictly p ositiv e for all P , whic h means that some systems just nev er will cross the lasing
threshold. This is for example the case if the ca vit y lifetime is to o short, or if the emitter ca vit y
coupling or the n um b er of emitters is to small, or for three- and four-lev el system setups if the
incoheren t pro cesses that p opulate the upp er lasing lev el and dep opulate the lo w er lasing lev el
are to o slo w compared to the other parameters. Generally there needs to b e a stable p opulation
in v ersion and a sufficien t assisted p olarization I m ⟨ J xy b † ⟩ in order to spark and main tain the
stim ulated emission of the laser/spaser.
The functional dep endency of this b parameter strongly influences the b eha vior of the laser and
v aries for the three differen t setups: F or a t w o-lev el system it is a parab ola, whic h is alw ays
op ened to the top
b ( P ) = q P 2 + r P + s, q > 0 (6.1.7)
for a three lev el system it asymptotically b eha v es lik e a linear function of p ositiv e slop e
b ( P ) = q P 2 + r P + s
tP + u , q > 0 . (6.1.8)
The constan ts in these expressions q , r , s , t , u dep end on the parameters in the master equa-
tion, how ev er their explicit form is not of in terest for the presen t discussion. These t w o laser
theories will alw a ys only ha v e a lasing windo w with resp ect to the pump-rate P : The dephasing
in tro duced b y the incoheren t pump term ev en tually leads to a strong damping of the assisted
p olarization I m ⟨ J xy b † ⟩ , b ( P ) is alw a ys p ositiv e for large P and the lasing action dies. Con trary
the b parameter of the four-lev el laser theory b eha v es asymptotically lik e a constan t function
b ( P ) = q P + r
sP + t , s > 0 , (6.1.9)
where q in this case can b e negativ e and also should b e for the lasing action. Th us for a negativ e
q (of sufficien t magnitude) the system en ters the lasing regime and then sta ys in the lasing
regime for increasing P . The q parameter dep ends on the difference b etw een the sp on taneous

6.2. T w o-lev el lasers 85
Figure 6.4 – Iden tifying the lasing threshold: (a) The zero of the parab ola only go es
to high p ositiv e v alues if b ( P ) b ecomes negativ e. (b) b as a function of P for t w o-, three-
and four-lev el lasers. It is apparen t that for t w o- and three-lev el systems there is a windo w
of lasing whic h is due to the fact that for these t w o cases the pump P in tro duces strong
dephasing in the lasing transition, which in ultimately quenc hes photon generation. The
four-lev el laser theory do es not ha v e this dephasing effect and thus does not turn off for
large P . This can also b e seen in the sketc hes Fig. 6.3.
emission rate in to nonlasing mo des γ 2 → 1 and the dep opulation rate of the lo w er lasing transition
γ 1 → 0 1
q ∝ N g 2 ( γ 2 → 1 − γ 1 → 0 ) , (6.1.10)
with N b eing the n um b er of four-lev el systems. This indicates that the relativ e v alues of these
t w o deca y pro cesses are crucial for the system to start lasing. The three differen t functional
dep endencies of the b parameter are sho wn in Fig. 6.4 (b). F or tw o- and three-lev el lasers
b alw a ys approaches + ∞ for P → ∞ since the pumping introduces dephasing in the lasing
transition, whic h leads to quenc hing for high pump rates. Th us these t w o systems alw a ys only
ha v e a lasing windo w. The four-lev el laser do es not exp erience this dephasing and b is constan t
for P → ∞ and therefore the laser do es not turn off for high pump p o w ers.
6.2 T w o-lev el lasers
The t w o-level laser is somewhat a limiting case for the three-lev el laser: If the transition rate
from lev el 2 to lev el 1 (the y ellow arro w in Figs. 6.2 b) and 6.3 b)) is fast compared to all other
pro cesses, the three-lev el laser essen tially reduces to a t w o-lev el laser since p opulation in upp er
lev el should b e close to zero and correlations are not in v olved. Th us the t w o-lev el laser is a
con v enient theoretical concept but in real ph ysical systems more than t w o-lev els are needed in
order to realize lasing action, since the stable generation of a p opulation inv ersion is generally
not p ossible with only t w o lev els 2
In Fig. 6.5 (a) the normalized mean excitation ⟨ J 11 ⟩ / N is plotted: As the n um b er of emitters
is increased the system passes the lasing threshold and the shap e of the curv e c hanges to a
c haracteristic double S shap e, whic h indicates lasing [147]. The mean in traca vit y photon n um b er,
Fig. 6.5 (b), exp eriences a drastic increase in the range of the "second" S, b et w een the pump
1 Here, for simplicit y , it w as assumed that the rate of p opulation of the upp er lasing lev el γ 3 → 2 and the rate of
dep opulation of the lo w er lasing lev el γ 1 → 0 are iden tical.
2 There is also the notion of in v ersion less lasing, whic h is particularly in teresting for high energy radiation
lasers, i.e. ultra-violett and x-ray , since the in v ersion less lasing sc hemes are more energy efficien t and require
lo w er pump rates [146]. How ever this rather specific topic is not considered in this thesis.

86 Ca vit y QED Lasers and Spasers
Figure 6.5 – cQED t w o-lev el laser: (a) The normalized mean excitation of the t w o-lev el
systems n/ N = ⟨ J 11 ⟩ / N for v arying N shows the emergence of a c haracteristic double S
shap e, whic h indicates the lasing action. (b) The intraca vit y photon n um b er m = ⟨ b † b ⟩
gro ws in the lasing windo w for increasing emitter num b ers. The quenc hing of the photons
in the high pump limit results from the strong dephasing of the lasing transition due to the
incoheren t pump. (c) The second order correlation function g (2) (0) sho ws that in the lasing
windo w for N > 1 the system indeed reac hes a P ossonian distribution. (d) The imp ortance
of the offdiagonal elemen ts for N = 10 : T runcating the elemen ts P [ n, k , l ] with k + l < x
for v arious x . Belo w the lasing threshold the t w o-lev el system coherences are imp ortan t,
ho w ever abov e the lasing threshold the coherences can almost b e en tirely neglected.
rates P = 0 . 1 ps − 1 and P = 10 ps − 1 . This is merely an indication for lasing action since the
rate equation do es not pro vide an y information ab out the statistics of the photons. Ho w ev er
lo oking at the equal time second order correlation function [32]
g (2) (0) = ⟨ b † b † bb ⟩
⟨ b † b ⟩ 2 (6.2.1)
in Fig. 6.5 c) it b ecomes clear that the system indeed starts lasing. Coheren t radiation implies
constan t in tensity , whic h in a photon detection exp erimen t should yield a P oissonian n um b er
state distribution, see e.g. Ref. [148]. The correlation functions [28]
g ( n ) (0) = ( ⟨ b † ) n b n ⟩
⟨ b † b ⟩ n (6.2.2)
are the momen ts of this distribution. Ev en though a second order correlation function equal
to one is in principle not a sufficien t criterion for a P oisson ian n um b er state distribution, from
practical exp erience it b ecomes clear that in the regions where g (2) (0) = 1 usually also the other
correlation functions g ( n ) (0) approac h unit y , whic h is a sufficien t criterion.
In the discussion of the rate equation theory in b ecame clear that the assisted p olarization

6.3. Three- and F our-lev el bad ca vit y lasers – Spasers 87
I m ⟨ J xy b † ⟩ is a crucial quan tit y in laser theory . Ho w ever the strength of the p erm utation sym-
metric metho d lies in the fact that it is able to treat all in ter t w o-lev el system correlations
exactly: Lo oking at the imp ortance of quantum coherence betw een the individual t w o-lev el sys-
tems represen ted b y the offdiagonal elemen ts of the t w o-level system part of the densit y matrix
– the P [ n, k , l ] terms for k , l > 0 – w e see that ab o v e the lasing threshold these coherences
are not imp ortan t, see Fig. 6.5 d). Artificially truncating the offdiagonal elemen ts P [ n, k , l ]
with k + l < x for v arying x shows that below the lasing threshold these off diagonal elemen ts
are imp ortan t in order to ac hiev e conv erged results, it ev en can pro duce completely unph ysical
b eha vior lik e negativ e g (2) (0) . Ho w ev er ab o v e the transition no influence can b e detected. This
indicates that the lasing action is essen tially a classical effect, at least at the lev el of the gain
medium, whic h ma y explain the success of (semi-) classical laser theories suc h as the rate equa-
tion theory presen ted in the previous section.
In fact the full quan tum solution more or less just v erified the expectations of the rate equation
theory: If the in tra-cavit y photon n um b er b ecomes large the system ev en tually starts lasing.
Ho w ever quan titativ ely determining the threshold is not p ossible with the rate equation the-
ory , since it is unclear at which photon n um b er the second order correlation function actually
reac hes unit y . Sometimes intra-ca vit y photon n um b ers as large as 20 are needed f or the g (2) (0)
function to reac h unit y [17]. Therefore common estimates lik e "The system starts lasing if the
in tra-ca vity mean photon n um b er reac hes/exceeds unit y" are v ery rough estimates in the cQED
laser limit. The findings in this section repro duce the findings of Ref. [84] that in the cQED
limit the second order correlation function is indisp ensable for identifying the lasing threshold.
Murra y Holland et al. [110, 149] call the sub-threshold regime of the t w o-lev el laser the sup er-
radian t regime, since here the in ter t w o-lev el system correlations are imp ortan t, whic h indicates
collectiv e b eha vior (see Chapter 7 for details). They hav e in v estigated the ph ysics of the sub-
threshold laser using a quan tum tra jectory v ersion of the formalism of Hartmann [100, 107]. In
recen t y ears also the connection of lasing action and collectiv e effects suc h as sup er-, subradi-
ance in quan tum dot based micropillars has b een in v estigated explicitly including semiconductor
effects [16, 150].
6.3 Three- and F our-lev el bad ca vit y lasers – Spasers
In the pap er of M. Ric h ter, M. Gegg, T.S. Theuerholz, A. Knorr, published in Ph ys. Rev. B
035306 (2015) [17] the spaser – surface plasmon amplification b y stim ulated emission of radiation
[59, 60] w as in vestigated using the permutation symmetric method: A set of iden tical t w o-lev el
systems coupled to a b osonic surface plasmon mo de of a spherical metal nanoparticle [151]. The
whole setup is inspired b y the exp erimen t of Nogino v et al. from 2009 [39]. This pap er receiv ed
considerable atten tion as it promised a new laser lik e light source with spatial dimensions below
the diffraction limit, which is imp ossible using con v entional ligh t resonator designs. Ho wev er,
no w adays, their findings are at best contro v ersially discussed in the field [152, 153].
The treatmen t in Ref. [17] is equiv alen t to a bad ca vit y (short b oson lifetime) t w o-lev el laser
setup with (relativ ely) high t w o-lev el system mo de in teraction strength. The main reason for the
short plasmon lifetime are ohmic losses in the metal, whic h in turn results in heat generation.
It w as found that the system could in principle start spasing, ho wev er the pump rates and gain
required to o v ercome the plasmonic losses are so high that the system very lik ely w ould melt
b efore coming near the spasing threshold [154]. Th us the spaser setup proposed by Bergman,
Sto c kman and measured b y Nogino v et al. can b e regarded as an efficien t nanosized optical to
thermal energy con v erter.
Ho w ever the question occurred whether three- or four-level laser theories migh t b e able to
partly o v ercome the intrinsic shortcomings of the spaser design. In fact Nogino v et al. used dy e

88 Ca vit y QED Lasers and Spasers
Figure 6.6 – Three- and four-lev el spasers: Left column three-lev el spaser and righ t
column four-lev el spaser. (a) and (b) The mean, normalized p opulation in v ersion of the
lasing transition n/ N . Double and s ingle S shap e curv es for three- and four-lev els resp ec-
tiv ely . (c) and (d) the plasmon n um b er m = ⟨ b † b ⟩ . Quenc hing for three-lev el systems and
no quenc hing for four-lev el systems, due to the differen t pump dep enden t dephasing actions
in these t w o setups. (e) and (f ) the second order correlation function g (2) (0) . In the lasing
windo w of the three-lev el systems the system indeed reac hes a P ossonian distribution. F or
the four-lev el systems it is not en tirely clear from the second order correlation function alone
whether this should b e called spasing action.

6.3. Three- and F our-lev el bad ca vit y lasers – Spasers 89
molecules as gain medium in their setup – dy e lasers are the standard example for four-lev el
lasers. Th us for a correct quan tum description of their findings a four-lev el gain theory should
b e applied. F urthermore this serv es as a pro of of principle that the p ermutation symmetric
metho d and PsiQuaSP [26, 27] are in fact able to handle multi-lev el systems.
As stated in the discussion of the rate equation theory exp ectations the three-lev el laser and
the t w o-lev el laser are similar setups, th us there is not m uc h difference to b e exp ected. Ho w ev er
the four-lev el laser is in trinsically different since the pump does n ot in tro duce dephasing in the
lasing transition and therefore the quenc hing effect should b e absen t. It is crucial to realize that
in three- and four-lev el laser theories the additional relaxation pro cesses that are related to the
p opulation of the upp er lasing lev el and the dep opulation of the lo w er lasing lev el (Fig. 6.3 b)
and c)) are material parameters and should therefore b e k ept constan t in the sim ulation. This
ho w ever introduces material limitations to the gain and thus lasing action that are absent in the
t w o-level laser theory .
As exp ected from the rate equation considerations, the three-level laser b eha v es quite similar to
the t w o-level laser, Fig. 6.6: There is a maxim um in the plasmon output ∝ m and the system
starts spasing for sufficien tly high emitter n um b ers. A t high pump rates the dephasing of the
lasing transition quenc hes plasmon generation (compare to Figs. 6.4 (b) and 6.5 (b)). Con trary
the four-lev el spaser do es not exp erience this dephasing and th us do es not ha v e a maxim um in
the plasmon n um b er. Ho w ev er the plasmon generation saturates for high pump p o w ers whic h
is a result of the material limitations due to the constan t rates for γ 3 → 2 and γ 1 → 0 , i.e. the
p opulation rate of the upp er lasing lev el and the dep opulation rate of the lo w er lasing lev el.
In terestingly it is not clear at all whether the four-lev el setup crosses the threshold at all: F or
N = 6 it seems as if the setup starts spasing, the n um b er of plasmons is ab ov e unit y (whic h is a
rather common criterion for lasing action, as discussed ab o v e) and the g (2) (0) function equal to
one for P > 0 . 1 fs − 1 . Normally this w ould be asso ciated to spasing/lasing. Ho w ev er increasing
the n um b er of emitters, whic h is equiv alent to increasing the gain results in higher num b ers of
plasmons and g (2) (0) v alues. So ev en though the n um b er of plasmons increases the statistics
b ecomes less P oissonian when increasing the gain. F or higher n um bers of emitters it might ev en
gro w further and th us completely lea ving the coheren t regime. This should clearly b e attributed
to sub-threshold b eha vior, b ecause ab ov e the threshold an increase in gain should lead to b etter
lasing p erformance rather than w orse. This further supp orts the observ ation that iden tifying the
lasing threshold for cQED lasers is a complex task and should therefore b e done with caution
and prop er theoretical metho ds.
Ov erall the three- and four-lev el spaser schemes do not pro vide ma jor b enefits compared to the
t w o-level spaser: The threshold is still unrealistically high and the problem of high losses remains
the same. Generally it is theoretically exp ected that the metal nanoparticle melts when the
plasmon n um b er roughly reac hes one [154], whic h is also confirmed b y exp erimen talists w orking
in the field [152]. Ho w ev er the three-lev el spaser theory exp ects spasing at plasmon n um b ers of
m > 5 , th us in an unrealistic parameter regime. Generally there is an activ e discussion in the
plasmonics comm unit y of how to o v ercome the problem of high plasmonic losses. Jacob Kh urgin
prop osed to use the "high Q windo w" of the noble metals gold and silv er: As stated in Chapter 2
the dielectric constan t of the surrounding medium ϵ h has a large impact on the dip ole plasmon
resonance ω sp , due to the F röhlic h condition [155, 45]
Re ( ϵ ( ω = ω sp )) = − 2 ϵ h . (6.3.1)
here ε ( ω ) is the dielectric function of the metal. Due to this resonance condition the plasmon
frequency shifts to lo w er energies for higher host dielectric constan ts ε h . Ho w ev er the dielectric
constan ts needed to pro vide Q factors of ∼ 100 are of the order ϵ h ≥ 10 , whic h is a v alue that is
difficult to obtain. F urthermore Kh urgin p oin ted out that surface scattering pla ys an imp ortan t

90 Ca vit y QED Lasers and Spasers
role in nanosized metal particles, whic h also reduces plasmon lifetimes [47].
Generally the mo del assumptions made in the presen t spaser theory are hardly justified but
serv e as an extreme v ersion of a b est-case scenario – there is no reason to exp ect that a more
noisy/complicated/realistic mo del w ould giv e a b etter p erformance, bypass the in trinsic short-
comings and o v ercome the losses. This exp ectation is supp orted b y literature: As stated in
Chapter 2 the plasmonic theory emplo y ed is very rough, considering only one mo de is not jus-
tified since the plasmonic structure supp orts m ultip ole mo des that are esp ecially imp ortan t in
the closely pac k ed geometry of the Nogino v design. In fact the measured threshold b eha vior
could b e repro duced theoretically as a sub threshold mo de comp etition effect b et w een these
m ultip ole mo des, whic h ho wev er further degrades the prosp ects of spasing action [153]. F ur-
thermore the assumption of iden tical coupling is not justified for randomly distributed emitters
in a strongly spatially v arying near field mo de. Y uang Zhang, Klaus Mølmer and V olkhard May
in v estigated this spaser geometry using a three-lev el gain theory as in the discussion ab o v e [156]
and randomly orien ted emitters using a explicitly spatially dependent coupling parameter [157].
Ho w ever the authors found that the inclusion of the spatial dependence further degrades the
p erformance of the device.
6.4 Conclusion
In this c hapter the p erm utation symmetric metho d w as successfully applied to differen t laser
setups, namely t w o-, three- and four-lev el laser theories. It w as iden tified at the rate equation
theory lev el that t w o- and three-lev el lasers should b eha v e similarly , they b oth exp erience pump
dep enden t quenc hing of the lasing output, while this effect is absen t in a four-lev el laser. Gen-
erally the qualitativ e exp ectations of the rate equation theory w ere confirmed using the exact
p erm utation symmetric metho d and PsiQuaSP . This serv es as a pro ve of principle that the li-
brary is in fact able to handle m ulti-lev el system s etups.
In the discussion of the t w o-level cQED laser it w as found that the strength of the p erm utation
symmetric metho d in fact lies in describing the sub-threshold b eha vior of the system: The in-
clusion of the off-diagonal elemen ts in the t wo-lev el system part of the densit y matrix is only
imp ortan t b elo w the lasing threshold. This is the so-called sup erradiant regime of the laser
[110, 149]. Generally this setup is somewhat the simplest meaningful setup for the p erm utation
symmetric metho d. All results in this c hapter w ere obtained using direct time-in tegration, the
transien t dynamics con verge fast, th us there is no need for steady state solv ers. F urthermore it
turns out that the t w o-lev el system part of the densit y matrix of the t w o-lev el system laser only
scales lik e ∝ N 2 instead of the exp ected ∝ N 3 , th us further simplifying this setup. The details
of this reduction of degrees of freedom are discussed in Chapter 9.
The in v estigation of the m ulti-lev el spaser theories rev ealed that even the inclusion of more re-
alistic and in v olv ed gain medium theories cannot o v ercome the intrinsic design shortcomings of
the spaser setup: Plasmonic losses are to o high, heating destro ys the samples and do wnscaling
in tro duces further losses due to surface or b ound state scattering [47].

7 The op en Dic k e mo del
In this c hapter the p erm utation symmetric metho d will b e applied to the op en Dic k e mo del: A
set of iden tical t w o-lev el systems coupled to a b osonic mo de. This is in fact the same setup as
in the t w o-lev el laser discussed in the previous c hapter. Ho w ever in this c hapter the t w o-lev el
systems will b e coheren tly driv en b y an external, cw laser and the fo cus of the in v estigation
will b e the p opulation of the v arious Dic k e states, esp ecially the subradian t and dark states as
discussed in Chapter 4. F urthermore the distinction of individual and collectiv e b eha vior in this
system under the presence of dephasing is elucidated.
A simple rate equation description of the mo del predicts a bistable b eha vior, whic h is closely
related to the system kno wn as absorptiv e optical bistabilit y and co op erativ e resonance flu-
orescence. How ev er the master equation has alw a ys a unique steady state, which is an ap-
paren t con tradiction. It is kno wn from adv anced theoretical analysis that strong quan tum
coherences/collectiv e b eha vior lifts this bistabilit y and allows tunneling betw een the t w o states
predicted b y rate equation theory . Therefore it is to b e exp ected that coheren t effects pla y a
dominan t role in this setup.
The influence of the incoheren t driv e of the t w o-lev el laser w as found to result in strong dephas-
ing of the in ter t wo-lev el system coherences and th us leads to individual b eha vior. In Chapter
4 the collectiv e Dic k e basis w as in tro duced and it w as seen that this collectivit y has a strong
influence on the ligh t matter in teraction strength: The in teraction strength of the sup erradian t
states with a ca vit y mo de scales with ∝ N 2 while some of the subradian t states are dark, or
rather do not couple to the ca vit y at all. If the t w o-lev el systems in teract individually with the
ca vit y mo de the coupling scales strictly lik e ∝ N .
While the closed system is completely describ ed b y the sup erradian t subspace whic h has dimen-
sion N + 1 the individual system bath in teractions lik e sp on taneous emission, pure dephasing
and incoheren t driving w ere found to break the J 2 symmetry of the Dic k e Hamiltonian and th us
do es not allo w a treatmen t in the symmetric Dic k e states. This observ ation lead to the formu-
91

92 The op en Dic k e mo del
lation of the symmetrized Liouville space states and the observ ation that the densit y matrix is
blo c k diagonal in the Dic k e basis in Chapter 5. In this c hapter collectiv e effects and esp ecially
the p opulation of the subradian t and dark Dic k e states are in v estigated. Therefore, instead of
driving the t w o-level systems incoheren tly , the systems are no w driv en coheren tly via an external
classical laser, whic h fa vors collectiv e b eha vior. Driving is necessary since subradian t states are
excited states, see Fig. 7.1 (b).
In a frame rotating at the external laser frequency , using the rotating w a v e approximation the
system Hamiltonian reads
H = ℏ ∆ 0 b † b + ℏ ∆ 1 J 11 + ℏ g ( J 10 b + J 01 b † ) + ℏ E ( J 10 + J 01 ) , (7.0.1)
where ∆ 0 , ∆ 1 are the mo de and TLS detuning, g is the TLS-mo de coupling, E is the optical
driving. Both ca vity and TLS are sub ject to loss and dephasing, using Lindblad formalism [13].
The master equation reads
∂ t ρ = L ρ = i
ℏ [ ρ, H ] + D de + D pd + D ph . (7.0.2)
The Lindblad dissipators describ e deca y pro cesses lik e individual radiativ e and non-radiativ e
deca y
D de = γ / 2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) , (7.0.3)
pure dephasing
D pd = δ / 2 ∑
i
( σ i
z ρσ i
z − ρ ) (7.0.4)
with σ i
z = σ i
11 − σ i
00 and ca vit y decay
D ph = κ/ 2(2 bρb † − b † bρ − ρb † b ) , (7.0.5)
see Fig. 7.1 (a). All con tributions to the master equation except D de and D pd commute with J 2
and are th us total spin preserving , (Fig. 7.1 (b)).
Subradiance is a collectiv e effect and the collectiv e processes in the sys tem, namely the ca v-
it y – t wo-lev el system in teraction and the external semi-classical driv e lead to suc h collectiv e
b eha vior. Ho w ev er these tw o pro cesses do not couple differen t Dic ke subspaces, th us only con-
sidering these t w o pro cesses w ould confine the system to the sup erradian t subspace, as stated
ab o v e. The individualization in tro duced through e.g. individual sp on taneous emission couples
the differen t Dic k e subspaces but leads to individualization. It w as found in Section 5.5.3 that
t w o-level systems deca ying through individual sp on taneous emission are accurately describ ed
without an y quan tum coherences/off-diagonal elements (Eq. (5.5.17)). If the t w o-lev el systems
b eha v e completely individual there will also b e no subradian t b eha vior in the system. Th us in
order to observ e gen uine Dic k e subradian t b eha vior an in terpla y b et w een collectivit y and indi-
vidualization is necessary .
In v estigating the p opulation in all the Dic k e basis states requires the computation of the whole
densit y matrix, esp ecially the off-diagonal elemen ts are imp ortan t (compare to the discussion
in Chapter 4, Section 5.6 and the explicit expressions of the Dic k e state projectors in App endix
B.4). Th us the information ab out the Dic k e state o ccupations is not accessible to a simple rate
equation theory .
Being able to practically treat collectiv e and individual pro cesses sim ultaneously while giving
access to the full densit y matrix is exactly the strength of the p erm utation symmetric metho d.
The n umerical results of this c hapter could not ha v e b een computed using a brute force solution,
ev en the use of an adaptiv e step width Runge-Kutta together with the p erm utation symmetric

7.1. Dic k e mo del ph ysics 93
Figure 7.1 – Illustrating the op en Dic k e mo del (a) Sc hematic represen tation of the
system. (b) Dick e states for N = 4 . The lo w est state in eac h l subspace is dark – the low est
state in the sup erradian t l max = N / 2 subspace is the ground state. The in teractions are
depicted: Hamiltonian part (purple,thic k), dissipators D de and D pd (blac k,thin) and dark
state cascade (orange,curv ed). Dashed lines indicate the additional states for N = 5 (with
differen t v alues of m, l ).
metho d w ould mak e the presen t discussion next to imp ossible. The com bination of fast time
scales through sup erradian t effects and slo w time scales in troduced through the subradiant ef-
fects mak es direct time in tegration v ery unfa v orable. Th us the presen t discussion pro vides an
excellen t testing ground for the steady state solv ers av ailable through PsiQuaSP .
The c hapter is organized as follo ws: First in Section 7.1 a short review of the ph ysics of the Dic k e
mo del is giv en. In Section 7.2 the rate equation theory will b e deriv ed and the emergence of
the bistable b eha vior will b e explained. F urthermore the connections to optical bistabilit y and
co op erativ e resonance fluorescence will b e explained. In Section 7.3 the sup erradian t to subradi-
an t phase transition is in tro duced and its manifestation in exp erimen tally accessible quan tities
is discussed. Thereafter in Section 7.4 the ground state relaxation prop erties of the system are
in v estigated, whic h leads to the formation of dark state cascades. Finally in Section 7.5 the
findings of this c hapter are summarized and implications and deduced researc h questions are
discussed.
7.1 Dic k e mo del ph ysics
The op en (and closed) system Dic k e mo del has b een a w ork horse in quan tum optics and b ey ond
for decades [25, 43, 18, 19, 23, 158, 159, 160, 85, 161, 162, 163, 22, 164, 165, 166, 167, 17, 168,
169, 78, 102]. Not only is the Dic k e mo del the theoretical foundation for lasers as discussed in
the previous c hapter but it also explains effects suc h as optical bistabilit y , co op erativ e resonance
flourescence, sup er- and subradiance. These effects are connected to theoretical concepts and
applications suc h as quan tum light generation, (m ultipartite) en tanglemen t and phase transi-
tions.
In recen t y ears sup erradiance has attracted a lot of in terest [161, 22, 168, 169, 170] since it is
an exp erimen tally accessible effect of truly collectiv e b eha vior and it w as b eliev ed for a long
time that there is an in trinsic connection betw een sup erradiance and en tanglemen t: The su-
p erradian t subspace is spanned b y the symmetric N particle generalizations of the Bell states,
these states are en tangled and are called W -states in the en tanglemen t comm unit y [171, 172].
Ho w ever it w as found recen tly that there is in fact no connection b et w een superradiance and
en tanglemen t, ev en though the pro cess liv es in a space whose basis states are en tangled states
the whole densit y matrix is separable [168]. Generally a mixed N -partite system is said to b e
separable (en tangled) if it (cannot) b e written as a sum o v er direct pro ducts of p ossibly mixed

94 The op en Dic k e mo del
single system densit y matrices [80, 81]
ρ = ∑
n
p n ρ (1)
n ⊗ ρ (2)
n ⊗ · · · ⊗ ρ ( N )
n . (7.1.1)
Sup erradiance has b een demonstrated in a large v ariet y of material platforms since it is a
sp on taneous pro cess that forms by itself: Preparing N t w o-lev el quan tum emitters in a fully
excited state results in a sp on taneous, highly directed sup erradian t burst pro vi ded that the
t w o-level emi tters in teract collectiv ely to their electromagnetic surrounding. This can either b e
ac hiev ed by confining a free space ensem ble of t w o-lev el quan tum emitters in as spatial v olume
of ( λ/ 2) 3 , with λ b eing the w a v elength of the emitted light, or b y placing the emitters in a bad
ca vit y ( κ ≫ g ). In fact the effect also o ccurs in classical acoustic, mec hanical systems and can
explained b y phase matc hing [173].
Con trary to sup erradiance, subradiance is b eing inv estigated for its prosp ects to store quan tum
information: As seen in Chapter 4 some of the subradian t states are dark and th us provide
excellen t candidates to store a quan tum state or rather provide long liv ed coheren t excitations
[169, 170]. Ho w ev er since the subradian t states do not or only w eakly couple to the external
radiation field and therefore also generally need a long time to b e p opulated this phenomenon
has remained more elusiv e.
The Hamiltonian Eq. (7.0.1) already is form ulated in the rotating-w a v e appro ximation (R W A).
This implies that it is only v alid for g ≲ 0 . 1 √ ω ω 1 . Including the non-R W A terms results in the
full Dic k e Hamiltonian
H = ℏ ω b † b + ℏ ω 1 J 11 + ℏ g ( J 10 + J 01 )( b † + b ) , (7.1.2)
where also the rotating frame is no longer applicable 1 . F or ultra-strong coupling this Hamiltonian
undergo es a quan tum phase transition at the critical coupling strength 4 N g 2 = ω ω 1 : The
Hamiltonian gap closes and the ground state of the system c hanges to a state with non-zero
mean photon n um b er. Therefore this phase transition is often called the sup erradian t phase
transition [18, 85]. Ho wev er there is an activ e debate where and if it is actually p ossible to
observ e this phase transition in quan tum optical setups [105, 106].
In this c hapter/thesis only mo derate coupling strengths are considered and therefore it is p ossible
to treat the Dic k e Hamiltonian in the R W A, lik e in Eq. (7.0.1).
7.2 Bistable effects in quan tum optics
As stated ab o v e the op en Dic k e mo del giv es birth to effects lik e optical bistabilit y and co op erativ e
resonance fluorescence. These t w o setups are closely related to the system studied in this c hapter.
This section starts with a rate equation/lo w order cluster expansion theory deriv ation, starting
from the master equation Eq. (7.0.2). This will serv e as an in tro duction to the notion of
bistabilit y in quan tum optics. How ev er the fo cus of the discussion here lies on where the rate
equation theory breaks do wn, where it pro duces results that are in ob vious con tradiction to
the full quan tum theory . It is this regime w ere quan tum correlations are exp ected b e strong
and should dominate the b eha vior of the system [14]. Afterw ards a short review of co op erativ e
resonance fluorescence and optical bistabilit y is giv en and the similarities and differences to the
in v estigated effects in this chapter are discussed.
The deriv ation of the rate equation theory starts from the time deriv ates of the exp ectation
v alues of the elemen tary p olarization op erators b , J 01 and the collectiv e excited state pro jector
1 Unless one wishes to w ork with an explicitly time dep enden t Hamiltonian.

7.2. Bistable effects in quan tum optics 95
J 11 2 . These exp ectation v alues couple to the higher order correlations ⟨ bJ 11 ⟩ and ⟨ bJ 10 ⟩ , whic h
again can b e factorized
⟨ bJ 11 ⟩ → ⟨ b ⟩⟨ J 11 ⟩ , ⟨ bJ 10 ⟩→⟨ b ⟩⟨ J 10 ⟩ , (7.2.1)
to giv e a closed set of equations of motion. On resonance ( ∆ 0 = ∆ 1 = 0 ) one can derive an
equation that relates the input p o w er ∝ E 2 to the mean, t w o-lev el system excitation n um b er
n = ⟨ J 11 ⟩ ϵ = − n
2 n − N (1 − 2 C (2 n − N ) + 1
2 C 2 (2 n − N ) 2 ) , (7.2.2)
with
ϵ = 4 E 2
γ ( γ + δ ) , C = 4 g 2
κ ( γ + δ ) . (7.2.3)
The parameter C is often called co op erativit y [174]. If it is large enough the system b eha v es
bistable. The in teresting part in Eq. (7.2.2) is that the righ t hand side is a rational function,
including a second order p olynomial and a first order p ole at n = N / 2 , whereas the left hand
side is a constan t that is fixed b y external parameters. Th us for a giv en external field strength E
there ma y b e m ultiple solutions to this equation. This observ ation serv es as the theoretical origin
of the phenomenon called optical bi- and m ultistabilit y [21, 175] and has attracted considerable
in terest o ver the last couple of decades: Bistabilit y and m ultistabilit y imply that dep ending on
the initial state the system ev olv es in to differen t steady states. In other w ords the steady state
of the system con tains information ab out the initial state whic h has spark ed vivid d iscussions
around the question of whether this could b e used to store quan tum information. This phe-
nomenon is closely related to or rather the origin of the notion of dissipativ e phase transitions
(see b elo w) [73]. Also the bistable b eha vior can b e exploited to build all optical transistor and
transmission devices or rather optical logical devices [176, 177]. The field of all-optical logical
devices is curren tly particularly relev an t for telecomm unication applications as it is believed
that suc h devices could result in an urgen tly needed reduction of the energy consumption of the
in ternet [178].
P articularly striking is the observ ation that the original master equation alw a ys has a unique,
defined steady state. The fact that b oth the ca vit y and the t wo-lev el systems are sub ject to
(individual) sp on taneous emission is a sufficien t criterion for a unique steady state [69, 72]. Th us
already at this lev el, without ha ving computed an ything, there is an apparent con tradiction b e-
t w een rate equation theory and the full quantum solution, as opp osed to laser theory .
The apparen t con tradiction b et w een bistable rate equation prediction and unique steady state
of the quan tum solution and the actual exp erimen tal observ ations of bistabilities w as one of the
ma jor researc h questions cen tered around bistable systems in quan tum optics. This problem w as
resolv ed when it w as realized that in these systems the Liouvillian gap can b ecome v ery small
and that it ma y ev en close in the thermo dynamic, infinite system size limit [179, 97, 180]. Th us
in the infinite system size limit the con tradiction is lifted, ho w ever in the small or mesoscopic
range in v estigated here this contradiction prev ails.
In Fig. 7.2 the rate equation theory exp ectation is compared to the n um b er state distributions
from the full quan tum solution of PsiQuaSP . The rate equation predicts the mean v alues of
the distribution, ho w ev er a distribution alwa ys has a unique mean v alue. In th e lo w and high
driving limit the rate equation accurately predicts the mean v alue ho wev er at the bistable, phase
transition p oin t the rate equation theory fails, at least in the small system limit: In the bistable
2 Here the p olarizations ⟨ b ⟩ and ⟨ J 10 ⟩ hav e to b e/can b e included since the semiclassical optical pump driv es
these p olarizations, contrary to the laser example of the previous c hapter. There all these p olarizations are strictly
zero. The semi-classical pump breaks breaks the U (1) symmetry , that leads to the ∝ N 2 scaling of the t w o-lev el
system degrees of freedom, see Chapter 9 for details.

96 The op en Dic k e mo del
Figure 7.2 – Rate equation vs full quan tum solution: The distributions are the full
quan tum solutions computed b y PsiQuaSP and the red line represen ts the rate equation
solution. F ar a w a y from the switc hing, bistable range the rate equation predictions are
somewhat accurate, ho w ever in the range 5 − 20 ps − 1 the rate equation theory clearly fails.
range the rate equation actually has three solutions but it can b e sho wn that the "middle" solu-
tion is alw a ys unstable, therefore only the upp er and lo w er most solutions represent the actual
bistabilit y . A t the p oint where the distribution of the full quan tum solution suddenly switc hes
– the phase transition p oin t – it actually b ears tw o maxima eac h in the photon n um b er states
as w ell as in the t wo-lev el system n um b er states. The doubly p eaked photon distribution has
b een iden tified to b e closely related to bistability: In the large system limit the p ositions of
these maxima con v erge tow ards the t w o stable solutions of the rate equation theory and the
p osition of the minim um b et w een the t w o maxima con v erges tow ards the unstable rate equation
solution [181]. This is highly reminiscen t of the observ ation in quan tum field theory that the
classical tra jectory of a particle is just the quan tum tra jectory of highest probabilit y [182]. Here
there are no tra jectories – only steady states, but still the stable classical solution coincides
with the quan tum solution of highest probabilit y and the unstable solution corresp onds to the
lo cal minim um – the extremal p oints of the distribution con v erge to w ards the classical results
and the curv ature of the distribution at these extremal p oin ts determines the stabilit y of the
corresp onding classical solution 3 .
The related exp erimen tal observ ations of bistabilties in these systems can b e understo o d b y
realizing that the t ypical time scale in whic h the system is measured can b ecome short to the
steady state con v ergence time and thus the exp erimen tal observ ations of bistabilities are in fact
transien t effects. The actual time dep endence of the bistable b eha vior in an exp erimen t has only
b een measured and calculated recen tly [183, 184].
The system defined b y Eqs. (7.0.2) and (7.2.2) is closely related to absorptiv e optical bistabilit y
and co op erativ e resonance fluorescence. Historically these t w o phenomena w ere the first ph ysical
effects that allo w ed to study co op erative/collectiv e effects in the op en Dic k e mo del theoretically
3 Here a discrete probabilit y distribution is considered, whic h ob viously has no curv ature. But going to larger
system sizes the discrete distribution can b e appro ximated b y a con tin uous distribution, whic h in fact w ould b e
the phase space form ulation of the problem. This is also similar to solid state ph ysics where discrete sums o v er
man y summands are appro ximated b y con tinuous functions.

7.2. Bistable effects in quan tum optics 97
as w ell as exp erimen tally . The discussion of these effects w ere also the main driving force of the
dev elopmen t of the quantum optical phase space methods in the 1970’s and 80’s that wer e origi-
nally dev elop ed around laser theory , but needed to b e impro v ed to describ e the strong collectiv e
effects in these systems 4 . The phase space metho ds also serv e as the man y emitter coun terpart
of the p erm utation symmetric metho d introduced in Chapter 5.
The notion of optical bistabilit y and co op erativ e resonance fluorescence w as coined b y the sem-
inal pap ers b y Bonifacio and Lugiato [21, 175]. They deriv ed a semi-classical rate equation
theory based on ph ysical in tuition rather than a microscopic Hamiltonian and observ ed bistable
b eha vior b et w een differen t branches – a collectiv e and an individual branc h.
In the bad ca vit y (short photon lifetime, κ ≫ g ) limit the setup considered in this c hapter
is equal to co op erativ e resonance fluorescence setup: In the bad ca vit y limit it is possible to
adiabatically eliminate the ca vit y mo de whic h results in the master equation (on resonance
∆ 0 = ∆ 1 = 0 ) [19, 97]
˙ ρ = + iE [ ρ, J 10 + J 01 ] + g 2
κ (2 J 01 ρJ 10 − J 10 J 01 ρ − ρJ 10 J 01 )
+ γ
2 ∑
i
(2 σ i
01 ρσ i
10 − σ i
11 ρ − ρσ i
11 ) . (7.2.4)
Here E is still the coheren t emitter driving strength, g is the ca vit y coupling strength, κ is the
ca vit y photon decay rate and γ is the individual deca y rate of the t w o-lev el systems. This is a
master equation that describ es t w o comp eting ph ysical deca y pro cesses: collectiv e and individ-
ual deca y . F or lo w driving strengths the system b eha v es collective and for high driving strength
the system b eha v es individual. In the region b etw een these t w o branc hes, rate equation theory
predicts a bistable b eha vior. In absorptive optical bistabilit y usually the ca vit y is driv en b y an
external laser instead of the t w o-lev el systems. Here usually the t w o-lev el systems are adiabat-
ically eliminated, which corresponds to the go o d ca vit y limit 5 . This system also splits in t w o
branc hes for lo w and high driving strengths with a bistable switching region predicted b y rate
equation theory . Generally these t w o setups are v ery similar and w ere in vestigated in the con-
text of the men tioned bi-, m ultistabiltiy , co op erativity , squeezing and quantum ligh t generation
(an ti-bunc hing). Since the phase space metho ds allo w analytic treatmen ts of the steady state
prop erties almost all of these w orks fo cus on steady state b eha vior and almost no w ork has b een
done on explicit transien t dynamics. F urthermore, ev en though these systems w ere studied in
the con text of collectiv e b eha vior there are – to the b est of m y kno wledge – no prior studies
in v estigating the p opulation of the v arious subradian t Dic ke states in these systems. There ha v e
b een J 2 , total spin preserving studies that essen tially studied the sup erradian t effects alb eit not
using this nomenclature [19, 23, 159, 160].
4 In the early da ys of the phase space metho ds it w as customary to expand the densit y matrix diagonal ly
in photonic and atomic coheren t states, see e.g. Ref. [19]. This how ever leads to a non-positive semidefinite
diffusion matrix in the asso ciated F okk er-Planc k equations and th us sometimes to unph ysical behavior. This w as
resolv ed b y expanding the densit y matrix in a non-diagonal w ay in the coheren t state, whic h is named generalized
or p ositiv e P represen tation [104, 4, 32]. F orm a mo dern p ersp ectiv e it seems hardly surprising that a system
that b ears strong quan tum correlations, enco ded in the off-diagonal elements of the densit y matrix, needs a full
expansion of the densit y matrix in order to giv e ph ysically meaningful results.
5 There are some w orks that treat optical bistabilit y without goo d or bad ca vit y limit e.g. Ref. [185]. These
w orks rely on the p ositiv e P represen tation in the high emitter n um ber limit, whic h essen tially amoun ts to a w eak
quan tum correlation limit. In this chapter the strong correlation limit is in vestigated.

98 The op en Dic k e mo del
7.3 Sup erradian t to subradian t phase transition
In this section the p opulation of subradiant states in the system through individual deca y and
pure dephasing pro cesses is in v estigated – b oth pro cesses do not conserv e the total spin. Coun-
terin tuitiv ely , the ca vit y lifetime determines the p opulation of the subradian t states, even though
the ca vit y decay Liouvillian does not break the J 2 symmetry and th us do es not couple differen t
Dic k e subspaces: Increasing the ca vit y lifetime c hanges the nature of the non-equilibrium phase
transition discussed in the previous section. In the bad ca vit y , co op erativ e resonance fluores-
cence limit the subradian t states are alw a ys suppressed b y quan tum coherence, essen tially the
quan tum correlations are pushing the system in to the sup erradian t subspace. Increasing the
ca vit y lifetime completely changes this behavior, it leads to an amplification of these subradian t
states ab o v e the phase transition. The individual sp on taneous emission pro cess is a necessary
requiremen t for the p opulation of subradian t Dic ke states, ho w ev er the asso ciated deca y rate has
no influence on the p opulation b eha vior of the subradian t states in realistic parameter regimes.
The coherences of the subradian t states are only formed through the ca vit y degrees of freedom.
The observ ed effect is th us a cavit y assisted generation of subradian t quan tum coherences.
This effect is accompanied b y clear c hanges in exp erimen tally accessible signatures. F urther-
more the system is found to exhibit gen uine m ulti-partite entanglemen t b elo w the transition.
Switc hing off the external driving, the subsequen t relaxation in to the ground state forms a long-
liv ed cascade of dark Dic k e states. This results in a simple, deterministic proto col for dark state
preparation with p opulations close to unit y under the influence of dephasing. This could b e
useful for storing quan tum information. The results presen ted here w ere published in M. Gegg,
A. Carmele, A. Knorr, M. Ric h ter arXiv:1705.02889 (2017).
As explained in Chapter 4 the total spin l ( l + 1) is the eigen v alue of the Casimir op erator
J 2 = ( J 10 J 01 + J 01 J 10 ) / 2 + J z , (7.3.1)
with J z = 1 / 2 ∑ i σ i
z . The J 2 and J z eigen v alues determine the coupling strength of the m ulti
TLS (Dic k e) state to an optical mo de, the collective dipole transition element. This coupling
strength distinguishes b et w een sup erradiance and subradiance. F or sup erradian t states the
dip ole elemen t scales sup erlinear in N , while for subradian t states the scaling is sublinear in
N and some subradian t states are dark [28]. Dark means that the dip ole transition elemen t
v anishes, meaning these states cannot deca y e.g. by creating a ca vit y photon. Ho w ever this
is only true for collectiv e, J 2 symmetric in teractions, these states still deca y in to other states
via the individual, J 2 symm try breaking deca y and dephasing pro cesses D de and D pd , c.f. Fig.
7.1 (b). Generally the spin preserving con tributions in the master equation generate quan-
tum correlations leading to collectiv e TLS b eha viour (suc h as sup er- and subradiance) and the
nonpreserving terms destro y correlations leading to individualization. Ho w ev er only the spin
nonpreserving con tributions in tro duce coupling b et w een sup erradian t and subradian t states,
th us in order to prepare subradian t states an in terpla y of collectivit y and individualization is
necessary . Throughout this discussion l max = N / 2 will refer to the sup erradian t subspace and
l min = 0 , 1 / 2 will refer to the most subradian t subspace.
As stated ab o v e, in the bad cavit y limit ( κ ≫ g ) Eq. (7.0.2) corresp onds to the co op erative
resonance fluorescence setup [19, 23]. The system exhibits a non-equilibrium phase transition
for increasing E for b oth total spin preserving and nonpreserving setups [19]. In the follo wing
the b eha vior of the system at mo derate ca vit y lifetimes κ ∼ g will b e in v estigated, the in terme-
diate range b et w een go o d and bad ca vit y limit. The system then represents the in termediate
regime b et w een co op erative resonance fluorescence and absorptiv e optical bistabilit y setup [176]
(instead of driving the TLS, in optical bistabilit y the ca vit y is driv en, opp osed to Fig. 7.1 (a)).

7.3. Sup erradian t to subradian t phase transition 99
7.3.1 Collectivit y measure
In v estigating sup er- and subradian t states requires a suitable measure. Unfortunately computing
the resp ectiv e Dic k e state p opulations is not sufficien t for in v estigating collectiv e effects, if
dephasing is presen t: Dick e states | l , m ⟩ are eigenstates of J 2 and J z with corresp onding quan tum
n um b ers l ( l + 1) , 0 ≤ l ≤ N / 2 and | m | ≤ l . l max = N / 2 defines the sup erradian t subspace and
l min = 0 , 1 / 2 defines the (most) subradian t subspace, see Fig. 7.1 (b). As an example consider
the N = 4 Dic k e states: The sup erradian t subspace consists of fiv e states and there are three l
subspaces in total with quan tum n umbers l = 2 , 1 , 0 . As explained in Section 5.6 and App endix
B.4 the Dic k e state pro jectors can b e expressed in the symmetrized Liouville basis sates ˆ
P [ n, k , l ] :
The sup erradian t subspace reads
| 2 , − 2 ⟩⟨ 2 , − 2 | = 1
1 ˆ
P [0 , 0 , 0] ,
| 2 , − 1 ⟩⟨ 2 , − 1 | = 1
4 ( ˆ
P [1 , 0 , 0] + ˆ
P [0 , 1 , 1]) ,
| 2 , 0 ⟩⟨ 2 , 0 | = 1
6 ( ˆ
P [2 , 0 , 0] + ˆ
P [1 , 1 , 1] + ˆ
P [0 , 2 , 2]) ,
| 2 , 1 ⟩⟨ 2 , 1 | = 1
4 ( ˆ
P [3 , 0 , 0] + ˆ
P [2 , 1 , 1]) ,
| 2 , 2 ⟩⟨ 2 , 2 | = 1
1 ˆ
P [4 , 0 , 0] . (7.3.2)
The other t w o subspaces are given b y
D 1 | 1 , − 1 ⟩⟨ 1 , − 1 | = 1
4 (3 ˆ
P [1 , 0 , 0] − ˆ
P [0 , 1 , 1]) ,
D 1 | 1 , 0 ⟩⟨ 1 , 0 | = 1
2 ( ˆ
P [2 , 0 , 0] − ˆ
P [0 , 2 , 2]) ,
D 1 | 1 , 1 ⟩⟨ 1 , 1 | = 1
4 (3 ˆ
P [3 , 0 , 0] − ˆ
P [2 , 1 , 1]) ,
D 0 | 0 , 0 ⟩⟨ 0 , 0 | = 1
6 (2 ˆ
P [2 , 0 , 0] + 2 ˆ
P [1 , 1 , 1] − ˆ
P [0 , 2 , 2]) . (7.3.3)
D 0 and D 1 are the degeneracies of the resp ectiv e Dic k e subspaces giv en b y Eq. (4.1.18). Since
the pro cesses in the master equation do not discriminate b et w een the differen t degenerate Dic k e
states, this degeneracy just en ters as a prefactor here.
Calculating the p opulation of the Dic k e states then amoun ts to computing the expectation v alues
tr [ | l , m ⟩⟨ l , m | ρ ] = ⟨ | l , m ⟩⟨ l , m | ⟩ ≡ p ( l , m ) (7.3.4)
Th us expressed in the p erm utation symmetric densit y matrix elements P [ n, k , l ] these p opula-
tions are generally giv en b y an expression like
p ( l , m ) = a 0 ( l , m ) P [ n, 0 , 0] ± a 1 ( l , m ) P [ n − 1 , 1 , 1] . . . , (7.3.5)
with n = m + N / 2 , where the prefactors a k ( l , m ) are rational n um b ers dep ending on the t w o
quan tum n um b ers l and m .
In the presence of dephasing the elemen ts P [ n, k , k ] for k  = 0 (represen ting quan tum correla-
tions/coherences) exp erience dephasing. If the dephasing is strong enough it will completely
suppress the quan tum correlations, i.e. P [ n, k , k ]=0 for k  = 0 . This situation then corresp onds
to a completely incoheren t mixture of TLS o ccupations. F or v arying n um b ers of TLS, P [ n, 0 , 0]

100 The op en Dic k e mo del
distributions allo w a large v ariet y of p opulations in sup er- and subradian t states ev en if quan-
tum coherences are absen t. Generally – when spin non-conserving terms are included – the
sup erradian t subspace p opulation decreases, since for large N the sup erradiant subspace is v ery
small compared to the full Hilb ert space ( N + 1 vs. 2 N ). Ho w ev er without quan tum coherences
( P [ n, k , k ] , k  = 0 ) in the individual TLS basis the lab el sup er- and subradiance b ecomes mean-
ingless, since the quan tum coherences are the signatures of the collectivit y of the Dic k e states
and reflect the redistribution of oscillator strength through collectiv e effects. In other w ords,
in general the study of quan tum coheren t effects b et w een distinct quantum emitters requires
explicitly lo oking at the offdiagonal elemen ts of the densit y matrix in the basis of the individual
quan tum emitters.
Th us – in the op en Dic k e mo del – P [ n − k , k , k ] are the key quan tities that distinguish a sup er-
or subradian t state from a classical, incoherent mixture of TLS population ( P [ i, j, k ]=0 for
j, k  = 0 ). Or in other w ords the P [ n − k , k , k ] distinguish b et w een collective and individual
b eha vior. The deca y pro cess D de and the pure dephasing D pd act individually on ev ery TLS
and th us destro y the collectivity , resulting in incoheren t mixtures.
T o quan tify the effect of collectivit y and distinguish b et w een collective and individual behavior
w e in tro duce the ratio b et w een the full Dick e subspace p opulation and its incoheren t part [20]
R ( l ) = ∑ m p ( l , m )
∑ m a 0 ( l , m ) P [ m + N / 2 , 0 , 0] , (7.3.6)
as a collectivit y measure. P [ i, k , k ] ∼ 0 , k  = 0 results in R ( l )=1 : The influence of quan tum
correlations b et w een the individual TLS on the subspace p opulation is zero or negligible – the
TLS act individual ly . P [ i, k , k ]  = 0 , k  = 0 results in R ( l ) < 1 / R ( l ) > 1 : Quantum corre lations
suppr ess / incr e ase the resp ectiv e subspace o ccupation – the TLS act c ol le ctively . R ( l ) pro vides
a realit y c hec k, since in an y exp eriment dephasing is presen t and isolated Dic ke subspaces (or
states) will lik ely not o ccur.
In the en tire c hapter the parameters for the individual deca y lifetime and the emitter ca vity
coupling are fixed at γ = 1 . 0 ns − 1 and g = 3 . 3 me V. A t optical frequencies this is w ell outside
the ultra-strong coupling regime, the parameters are w ell inside the v alidit y range of the R W A
and the presen ted effect is not related to the superradiant phase transition kno wn from the
closed Dic k e mo del [85].
There are t w o J 2 symmetry breaking effects in this setup: The individual sp on taneous deca y
D de ( ρ ) and the pure dephasing D pd ( ρ ) . First only the influence of the sp ontaneous deca y is
in v estigated and the influence of pure dephasing is inv estigated later. Including small pure de-
phasing preserv es all effects.
7.3.2 Nature of the phase transition
In the steady state the most basic feature of the nonequilibrium phase transition is the c hange
from the ground state to a half excited TLS state with increasing external driving field (Fig. 7.3
(a)). In the bad ca vit y limit this b eha vior is w ell studied in the con text of co op erativ e resonance
fluorescence [19]. Increasing the ca vit y qualit y (decreasing the ratio b et w een ca vit y deca y rate
and TLS-ca vit y coupling strength κ/g ) mak es the transition sharp er but the o v erall effect do es
not c hange m uch. Contrary a drastic c hange is seen in the in the b eha vior of the collectivit y mea-
sure for the sup erradian t subspace R ( l max = N / 2) , Fig. 7.3 (b). While in the bad ca vit y limit
the sup erradian t subspace p opulation is alw a ys increased b y collectiv e effects ( R ( l max ) ≥ 1 ), an
increased suppression ( R ( l max ) < 1 ) of the sup erradian t subspace o ccurs for increasing ca vit y
lifetime/qualit y . This is accompanied b y a drastic increase of coheren t ca vit y photons b elo w and

7.3. Sup erradian t to subradian t phase transition 101
Figure 7.3 – Lea ving the bad ca vit y limit: V ariation of the external pumping strength
for differen t ratios κ/g : (a) the normalized TLS excitation n um b er n/ N = ⟨ J 11 ⟩ / N , (b) the
relativ e sup erradian t subspace o ccupation R ( l max = N / 2) , (c) the ca vit y output rate κm =
κ ⟨ b † b ⟩ and (d) the photonic second order correlation function g (2) (0) : Drastic qualitativ e
c hange for κ/g approac hing unity .
a pronounced bunc hing at mo derate photon n um b ers ab o v e the phase transition (Fig. 7.3 (c)
and (d)). The maxim um in the second order photon correlation function indicates the transi-
tion p oin t from increased to suppressed sup erradiant subspace occupation. Please note that the
ca vit y decay does not lead to an effectiv e dephasing/individualization con tribution for the TLS
(it preserv es J 2 symmetry), th us the p opulation of subradian t states through differen t ca vit y
lifetimes is a highly non trivial effect.
Ab o v e the phase transition collectivit y fav ors the most subradian t subspace l min : The dep en-
dence of R ( l max ) on the n um b er of TLS N , Fig. 7.4 (a), sho ws a gro wing collectiv e change in
p opulation of the sup erradian t subspace for increasing N . Both the increase b elo w the phase
transition and the suppression ab o v e the phase transition increase. In Fig. 7.4 (b) the ra-
tio R ( l max − 2) is plotted whic h exists only for N ≥ 4 – it switc hes from complete collectiv e
suppression b elo w to collectiv e increase ab o v e the transition. How ev er the collectiv e increase
in p opulation decreases for increasing N . A t first glance it seems that the collectiv e increase
in subradian t subspaces drops for higher emitter n umbers. This is ho w ev er not the case: F or
N = 4 , 5 there are three differen t l subspaces: l max , l max − 1 and l max − 2 . Th us for N = 4 , 5 the
subspace l max − 2 corresp onds to the most subradian t subspace i.e. N = 4 : l min = 4 / 2 − 2 = 0
and N = 5 : l min = 1 / 2 . In these t w o cases the collectiv e increase in p opulation is strongest.
F or larger N subspaces with smaller l exist, e.g. N =6: l min = l max − 3 . Lo oking at R (0)
(only defined for ev en N , alwa ys corresp onds to the most subradian t subspace), Fig. 7.4 (c),
it app ears that the increase due to collectiv e effects in fact increases with N , ho w ev er alw a ys
only in the most subradian t subspace. Th us the collectiv e increase is alwa ys most pronounced in
the most subradian t subspace ( l min ) ab o v e the phase transition. Remarkably , b elo w the phase
transition the subradian t subspaces are completely suppressed, c.f. Figs. 7.4 (b), (c). In the bad
ca vit y limit this corresp onds to the collectiv e branc h kno wn from co op erative resonance fluores-
cence (explained ab o v e). This discussion sho ws that this collectiv e branc h can b e attributed to
sup erradian t b eha vior since subradian t subspaces are completely suppressed. In an incoheren t

102 The op en Dic k e mo del
Figure 7.4 – Increasing the system size: Relativ e Dic k e subspace o ccupation for v arying
N : (a) the sup erradian t subspace l = N / 2 , (b) l = N / 2 − 2 , (c) l = 0 . These states ha v e no
in teractions due to the Hamiltonian. They only couple to states with l > 0 through deca y
and dephasing. (d) Absolute o ccupation in the sup erradiant subspace: Approaching zero
ab o v e the phase transition for N → ∞ , ev en without correlations.
setting, lik e for example thermal distributions, all these curv es w ould b e strictly fixed at one, no
increase and no suppression. This means that subradian t states are in fact p opulated but the
p opulations are suc h that all quan tum coheren t effects completely cancel eac h other.
The total o ccupation in the sup erradian t subspace go es to zero ab ov e the phase transition for
N → ∞ , Fig. 7.4 (d). Naiv ely w e could asso ciate this with subradiance. Ho w ev er for E → ∞
the TLS are in a completely incoheren t, equipartitioned state [186]. This can b e seen from the
fact that for high pump rates the R ( l ) curves of all subspaces approac h unit y , whic h corresponds
to a completely incoheren t mixture. In this limit the sup erradian t subspace is only dep opulated
since this subspace b ecomes v ery small compared to the full Hilb ert (Liouville) space for large
N . This is clearly not a collectiv e effect. This illustrates that (in the steady state) it is imp os-
sible to distinguish b et w een collectiv e and individual b eha vior b y using Dic ke state occupations
alone, whic h serv es as a motiv ation for the collectivit y measure. Ho w ev er by looking at b oth
the absolute and relativ e p opulations it is p ossible to conclude that in the go o d ca vit y and large
N limit the system c hanges from a predominan tly sup erradian t to a predominan tly subradian t
phase at the transition, where b oth phases exhibit genuine collectiv e b eha vior. This constitutes
the main result of this discussion.
In Fig. 7.5 the scaling of exp erimen tally more accessible quan tities with the n um b er of individual
TLS N is presen ted: The normalized TLS excitation dev elops a kink for increasing N , indicating
a second-order transition, Fig. 7.5 (a). The normalized Lio villian gap | λ 1 | /γ , whic h corresp onds
to the slo w est time scale in the system to reac h steady state (see Section 3.4), decreases around
the phase transition for increasing N , Fig. 7.5 (b). It migh t ev en v anish for N → ∞ , creating
a second steady state. The in traca vit y mean photon n um b er sho ws the formation of a lo cal
minim um at the transition and an increase in the p eak in tensit y , Fig. 7.5 (c). Also bunching
( g (2) (0) > 1 ) increases for increasing N , Fig. 7.5 (d). Ov erall the transition b ecomes sharp er
and more pronounced for increasing N and decreasing κ/g , since these parameters increase the
system size. This displa ys a t ypical prop ert y of phase transitions, whic h are w ell defined only

7.3. Sup erradian t to subradian t phase transition 103
Figure 7.5 – Exp erimen tal signatures for v arying N : (a) the normalized TLS excita-
tion n um b er n/ N = ⟨ J 11 ⟩ / N , (b) the renormalized Liouvillian gap | λ 1 | /γ , (c) the rescaled
in traca vity photon n um b er m/ N = ⟨ b † b ⟩ / N and (d) the second order correlation g (2) (0) .
in the thermo dynamic limit (infinite system size) and blur for small system sizes [19, 84, 187].
The presence of individual sp on taneous deca y D de ( ρ ) is a necessary condition for the p opulation
of subradian t states, as it is the only parameter in the presen t discussion that breaks the J 2 sym-
metry . Therefore one w ould in tuitiv ely expect a strong dep endence of the p opulation b eha vior
of the subradian t states on the sp on taneous deca y parameter γ . This ho w ev er is not the case,
see Fig. 7.6: Both the normalized mean excitation as w ell as the relativ e subradiant subspace
p opulation R (1 / 2) hardly dep end on γ at all in realistic parameter ranges. Only in the limit of
(unrealistically) short lifetimes γ ∼ 10 ps there is an observ able dep endence in the bad ca vit y
limit and ev en in that limit the qualitativ e b eha vior of the relativ e p opulation R (1 / 2) do es not
c hange at all. In the mo derate ca vit y quality range there is practically no detectable influence
of this parameter. Please note that if the sp on taneous deca y rate γ w ould b e artificially set to
zero, the relativ e subradian t subspace occupation R (1 / 2) would strictly remain zero at all driv-
ing strengths. The reason wh y a stronger individualization leads to a p opulation of subradian t
states at higher pump p o w ers is due to the fact that the subradian t states are excited states
and the pump needs to o v ercome the sp on taneous deca y in order to b e able to p opulate these
excited states.
This means that the only pro cess in the master equation that enables p opulation in subradian t
states has no influence on the related subradian t coherences. The only parameter in the system
that determines the subradian t coherences is the ca vit y qualit y κ/g . This means that the present
pro cess is a ca vit y assisted generation of subradian t coherences, ev en though b oth ca vit y related
parameters g and κ do not break the J 2 symmetry and therefore do not couple differen t Dic k e
subspaces. F urthermore for ev en N the most subradian t states are all dark (asso ciated to R (0) ,
see Fig. 7.4 (c)), there are no brigh t states in this subspace, meaning that the en tire subspace
do es not couple to the ca vit y at all. But still the ca vit y qualit y factor determines its relativ e
p opulation.

104 The op en Dic k e mo del
Figure 7.6 – Dep endence on the sp on taneous deca y rate: N = 5 (a) and (b) the
normalized TLS excitation n um b er n/ N = ⟨ J 11 ⟩ / N . (c) and (d) the relativ e most subradian t
subspace o ccupation R (1 / 2) . (a) and (c) are in the bad ca vit y limit – (b) and (d) ha v e a
mo derate ca vit y qualit y . In the t ypical range of lifetimes of quan tum emitters 0 . 1 − 10 ns
there is no visible effect at all.
7.3.3 Robustness test and en tanglemen t prop erties
So far all results w ere presen ted without including pure deph asing. No w the robustness of the
collectiv e effects at the phase transition against pure dephasing is in v estigated: In Fig. 7.7 (a)
the collectiv e b eha vior of the relativ e Dick e subspace p opulation is reduced for increasing δ .
Ho w ever the effect of clear distinction of sup erradian t state b elo w and subradian t state ab o v e
phase transition is preserv ed for δ ∼ γ . The general trend of the total Dic k e subspace o ccupation
is not affected b y pure dephasing, as in Fig. 7.4 (d). This finding is only concerned with the
steady state prop erties in the system. The qualitativ e dynamical b eha vior remains the same, as
will b e seen in the next section.
In the spin preserving setup the TLS are en tangled via spin squeezing b elo w the phase transition
[22]. Spin squeezing is a concept originating from quan tum metrology , where it w as dev elop ed
around the idea that squeezed atomic coheren t states could b e used for measuremen t precision
b elo w the shot noise limit, but also has attracted a lot of atten tion as an en tanglement witness
[188, 189, 190, 191]. An en tanglemen t witness is just a qualitativ e measure whether a system
is excited (or not), it do es not pro vide a quan titativ e measure of the entanglemen t strength.
Here w e emplo y the spin squeezing inequalities introduced by T óth et al. that are explicitly
deriv ed as an en tanglement witness for man y t w o- (and m ulti-) lev el system setups [79, 80]. The
spin preserving case do es not con tain an y subradian t states/effects and cannot mo del the effects
of pure dephasing. The spin preserving and nonpreserving scenarios are t w o limits of the same
ph ysical system [192, 4]. Th us an in v estigation of en tanglemen t in our setup and its preserv ation
under dephasing is desirable: W e find that the spin squeezing inequalities (SSI) b y T óth et al.
detect en tanglemen t b elo w the phase transition for δ < γ , see Fig. 7.7 (b): The quan tit y plotted

7.4. Dark state cascades 105
Figure 7.7 – Robustness, en tanglemen t and dark state cascades (a) The ratio R ( l )
for N = 5 for l = l min , l max and v arying δ : The clear switc hing at the phase transition
surviv es for δ ∼ γ . (b) En tanglemen t via spin squeezing in equalities: en tanglemen t b elo w
the transition for δ < γ . (c) Driving the system to the maxim um subradiance point with
subsequen t relaxation to the ground state N = 5 , δ = 0 : A cascade of dark states is
generated. T otal dark state o ccupation close to unit y .
there corresp onds to one of the sev en inequalities deriv ed b y T óth et al.
⟨ J 2
y ⟩ + ⟨ J 2
z ⟩ − N
2 − ( N − 1) ( ∆ J x ) 2
  
=: A
≤ 0 , (7.3.7)
and the t w o-level systems are not i n a separable state as so on as the inequality is violated. A
is the quan tit y plotted in Fig. 7.7 (b). Hence the entanglemen t detected in the spin preserving
setup is still presen t for spin nonpreserving setups. F or more information on the spin squeezing
inequalities b y T óth et al. please refer to App endix C.
7.4 Dark state cascades
Sup er- and subradiance are concepts related to time ev olution and so far only steady state v alues
w ere presen ted: No w, the system is driv en to the steady state with maxim um R ( l min ) (see Fig.
7.4 (b) and (c)) and then the driving field is switc hed off. The system relaxes in to the ground
state and w e observ e that a cascade of dark states is generated, Fig. 7.7 (c): p (1 / 2 , − 1 / 2) and
p (3 / 2 , − 3 / 2) are the p opulations in the low est states of the smallest l = l min and in termediate
l max > l > l min subspace, c.f. Fig. 7.1 (b). Both states are dark. They are p opulated on
time scales of the in v erse TLS-photon coupling constant g − 1 , b ecause the higher energy brigh t
states of the asso ciated l subspaces deca y via the emission of ca vit y ph otons. This implies that
for ev en N the single, most subradian t state p opulation p (0 , 0) do es not exp erience fast initial
p opulation since there are no other, higher energy brigh t state in this l = 0 subspace, see Fig.
7.1 b). The ca vit y photons subsequently lea v e the ca vit y through the ca vit y deca y . After the
initial fast p opulation of the | l , − l ⟩ states d ue to the TLS ca vit y in teraction the dynamics are
go v erned by spontaneous emission. The o v erall dark state p opulation subsequen tly deca ys on

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