Document [original]

Theory for strongly coupled quantum dot cavity

quantum electrodynamics

Photon statistics and phonon signatures in quantum light emission

———————————–

Vorgelegt von Diplom-Physiker

Alexander Carmele

aus Zell / Mosel

von der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

— Dr. rer. nat. —

genehmigte Dissertation

Promotionsausschuss

Vorsitzender : Prof. Dr. rer. nat. Michael Kneissl, TU Berlin

1. Gutachter : Prof. Dr. rer. nat. Andreas Knorr, TU Berlin

2. Gutachter : Prof. Dr. rer. nat. Torsten Meier, Universität Paderborn

Tag der wissenschaftlichen Aussprache: 30. November 2010

Berlin, 2011

D 83

Abstract

Advances in semiconductor technology constitute a new physical system to investigate

quantum optics in a solid state environment: a quantum dot strongly coupled to a single

cavity mode. In the strong coupling regime, the electron-photon coupling dominates over

dissipation processes and Rabi oscillations occur. This thesis focuses on the theoretical in-

vestigation of the strong coupling regime in the single-photon limit, in which few photons

interact with a single quantum dot and the combined electron and photon dynamics is

strongly correlated. Typical theoretical frameworks rely on factorization approaches, such

as the cluster expansion scheme. However, for strongly correlated dynamics, standard fac-

torization schemes fail and new theoretical frameworks become necessary. The work in

hand develops two novel methods within the equation of motion approach to solve the

complete quantum kinetics without a factorization of the electron-photon interaction, in-

cluding non-Markovian features, and hereby bridges this gap in the theoretical description

of semiconductor quantum optics.

First, for quantum dots with a fixed number of electrons, a mathematical induction method

for the combined electron, photon and phonon dynamics is introduced. This method is nu-

merically solvable up to an arbitrary accuracy. Surprisingly, for an initial thermal photon

distribution in the cavity the electron-phonon interaction leads to a counter-intuitive re-

duction in the intensity-intensity correlation function. The classical photon distribution

evolves into a nonclassical one. In addition to the well-known anti-crossing for zero de-

tuning, more anti-crossings and signatures for strong coupling are found for detunings of

a multiple of the LO-phonon energy below and above the band gap energy. Within this

LO-phonon cavity feeding, modified Rabi oscillations exhibit anharmonicities for elevated

temperatures. Assuming two electrons in the quantum dot, the polarization entanglement

of photons generated by a biexciton cascade is theoretically investigated in the weak and

strong coupling regime. Via an effective multi-phonon Hamiltonian approach, temperature

dependent relaxation rates from the QD to the wetting layer states are taken into account.

For elevated temperatures T≥120 K, these LO-phonon relaxation rates in GaAs result in a

vanishing degree of entanglement. This calculation is in agreement with recent experiments.

Focusing on quantum dots, in which the number of electrons and holes is not fixed due to

the presence of a wetting layer/carrier reservoir, many-particle correlations occur and are

treated within the photon-probability cluster expansion approach. This generalized cluster

expansion is a reliable theoretical framework to study the optical emission in the single

photon regime. Vacuum Rabi flopping is discussed. The oscillation amplitude of vacuum

Rabi flopping is interestingly determined by the number of electrons in the quantum dot.

Furthermore, the dynamics of an electrically pumped single quantum dot is described and

the full photon statistics of the quantum light emission for different pump rates is investi-

gated. Including semiconductor specific Pauli-blocking effects in the polarization dynamics,

a parameter study of an electrically driven single photon emitter is provided.

Abstrakt

Aktuelle Fortschritte in der Halbleitertechnologie etablieren ein neues physikalisches Sys-

tem, um quantenoptische Eigenschaften auf Nanoskalen zu untersuchen: ein Quanten-

punkt in einem Mikroresonator. Im Falle einer starken Kopplung übertrifft die Stärke der

Elektron-Licht Wechselwirkung die auftretenden Verluste und Rabi Oszillationen treten

auf. Diese Arbeit befasst sich mit der theoretischen Beschreibung solcher Systeme für den

Fall, dass wenige Photonen mit einem einzelnen Quantenpunkt wechselwirken und die Dy-

namik der Photonen und Elektronen stark korreliert ist. Typische theoretische Zugänge

basieren auf Faktorisierungsansätzen, die im Falle einer stark korrelierten Dynamik ihre

Gültigkeit einbüßen. Neue theoretische Zugänge sind zwingend erforderlich und werden

mit dieser Arbeit vorgestellt, die im Bewegungsgleichungsansatz eine Faktorisierung der

Elektron-Licht Wechselwirkung vermeiden und eine vollständige Beschreibung der Viel-

teilchenkinetik im Falle stark korrelierter Dynamiken gewährleisten, inklusive der Berück-

sichtigung von Gedächtniseffekten.

Für atomähnliche Quantenpunkte kann die Zahl der Elektronen und Löcher als konstant

angenommen werden. Mittels der vollständigen Induktion wird ein numerisch lösbares Mod-

ell eingeführt, das die ineinanderspielende Elektron-, Photon- und Phonondynamik bis auf

beliebige Genauigkeit auswertet. Überraschenderweise wird ein anfängliches thermisches

Lichtfeld im Resonator aufgrund der stark wechselwirkenden Phononen- und Photonendy-

namik in den Zustand eines nichtklassischen Lichtfeld gebracht. Neben dem üblichen spek-

tralen Anti-crossing, falls sich die Resonatormode in Resonanz mit der Bandlückenfrequenz

des Quantenpunktes befindet, treten weitere Anti-crossings auf, sobald die Kavitätsmode

von der Bandlückenfrequenz um ein ganzzahliges Vielfaches der LO-Phononenfrequenz ver-

stimmt ist. Diese von den Phononen herbeigeführte starke Kopplung führt zu anharmonis-

chen Rabi Oszillationen bei Raumtemperatur. In einem nächsten Schritt wird das induk-

tive Modell auf eine Biexziton-Kaskade in einem Quantenpunkt angewendet und für starke

und schwache Kopplung ein temperaturabhängiges Maß für Polarisationsverschränkung der

abgestrahlten Photonen bestimmt. Ab einer Temperatur von 120 K dominiert der Einfluss

der Halbleiterumgebung in GaAs und die Polarisationsverschränkung verschwindet. Dieses

Ergebnis stimmt mit aktuellen Messergebnissen überein.

In flachen Quantenpunkten ist die Zahl der Elektronen und der Löcher nicht mehr kon-

stant und die Vielteilchenkorrelationen der Ladungsträger können nicht mehr vernachläs-

sigt werden. Im Rahmen einer Photonwahrscheinlichkeitsentwicklung, die die starke Kor-

relation zwischen Elektronen und Photonen berücksichtigt, wird eine modifizierte Hartree-

Fock Faktorisierung eingeführt. Das erhöhte Pauli-Blocking in Halbleiterumgebungen führt

zu modizifierten Rabi Oszillationen, an deren Amplitude die Zahl der Ladungsträger im

Quantenpunkt bestimmt werden kann. Zudem wird die Photonenstatistik eines elektrisch

gepumpten Quantenpunktes berechnet und auf diese Weise eine Parameterstudie für elek-

trisch gepumpte Einzelphotonenemitter erstellt.

List of Publications

Regular articles

• M. Richter, A. Carmele, S. Butscher, N. Bücking, F. Milde, P. Kratzer, M. Scheffler,

and A. Knorr, “Two-Dimensional Electron Gases: Theory of Ultrafast Dynamics of

Electron-Phonon Interactions in Graphene, Surfaces and Quantum Wells“,

J. Appl. Phys. 105, 122409 (2009).

•A. Carmele, A. Knorr and M. Richter, “Photon statistics as a probe for exciton

correlations in coupled nanostructures”,

Phys. Rev. B 79, 035316 (2009).

• M. Richter, A. Carmele, A. Sitek and A. Knorr, "Few-photons model of the optical

emission of semiconductor quantum dots",

Phys. Rev. Lett. 103, 087407 (2009).

• M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim,

G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, "Theory of carrier and photon

dynamics in quantum dot light emitters",

Phys. Status Solidi B 247, 809 (2010).

•A. Carmele, M. Richter, W.W. Chow, and A. Knorr, "Antibunching of Thermal

Radiation by a Room-Temperature Phonon Bath: A Numerically Solvable Model for

a Strongly Interacting Light-Matter-Reservoir System",

Phys. Rev. Lett. 104, 156801 (2010).

•A. Carmele, F. Milde, M.-R. Dachner, M.B. Harouni, R. Roknizadeh, M. Richter,

and A. Knorr, "Formation dynamics of an entangled photon pair:

A temperature-dependent analysis",

Phys. Rev. B 81, 195319 (2010).

• Y. Su, M. Richter, A. Knorr, D. Bimberg, and A. Carmele, "Photon statistics of a

single quantum dot in a microcavity",

Phys. Status Solidi RRL 4, No. 10, 289 (2010)

• Y. Su, A. Carmele, M. Richter, K. Lüdge, E. Schöell, D. Bimberg, and A. Knorr,

“Theory of single quantum dot lasers: Pauli-blocking enhanced anti-bunching”,

Semicond. Sci. Technol. 25, accepted for publication (2010)

• J. Kabuss, A. Carmele, M. Richter, W.W. Chow, and A. Knorr, “Inductive Equation

of Motion Approach for a semiconductor QD-QED: Coherence induced Control of

Photon Statistics”,

Phys. Status Solidi B, accepted for publication (2010).

List of Conference Contributions

Invited talks

•A. Carmele, J. Kabuss, M. Richter, W. W. Chow, and A. Knorr, "Photon Statistics

and phonon signatures in the quantum light emission from semiconductor quantum

dots",

SPIE Photonics West, San Francisco, USA, January 2011

•A. Carmele, J. Kabuss, M. Richter, W. W. Chow, and A. Knorr, "Quantum dot

cavity-QED: Phonon induced antibunching of Thermal Radiation and cavity feed-

ing",

Physics of Quantum Electronics (PQE), Snowbird Utah, USA, January 2011

• Julia Kabuss, A. Carmele, M. Richter, W. W. Chow, and A. Knorr, "Theory of

coherence induced control of photon statistics",

Physics of Quantum Electronics (PQE), Snowbird Utah, USA, January 2011

• A. Knorr, A. Carmele, and J. Kabuss, "Theory of Light Scattering From Semicon-

ductor Quantum Dots",

International Workshop on Theoretical and Computational Nano-Photonics (TaCoNa),

Bad Honnef, Germany, November 2010

•A. Carmele, J. Kabuss, M. Richter, W. W. Chow, and A. Knorr, "Phonon signatures

in quantum light emission from semiconductor QDs",

Universität Magdeburg, Magdeburg, Germany, November 2010

• J. Kabuss, A. Carmele, M. Richter, W. W. Chow, and A. Knorr, "Microscopic

theory of the multi-phonon assisted Quantum Emission from a Semiconductor QD

in the high driving field limit",

University Paderborn, Paderborn, Germany, November 2010

•A. Carmele, M.-R. Dachner, J. Wolters, M. Richter, and A. Knorr "Quantum light

emission from Cavity enhanced LEDs",

Numerical Simulation of Optoelectronic Devices (NUSOD) at Georgia Tech, Atlanta,

USA, September 2010

•A. Carmele, M.-R. Dachner, F. Milde, M. Richter, and A. Knorr, "Photon statistics

and entanglement in quantum light emission from semiconductor quantum dots",

Colloqiuum talk at the University of Bremen, Bremen, Germany, November 2009

• A. Knorr, A. Carmele, J. Kabuss, M.-R. Dachner, and M. Richter, "Photon Statis-

tics and Entanglement in Phonon-assisted Quantum light emission from Semicon-

ductor Quantum Dots",

Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS), Paderborn,

Germany, August 2010

III

• A. Knorr, A. Carmele, M.-R. Dachner, J. Wolters, and M. Richter, "Theory of few

photon dynamics in electrically pumped light emitting quantum dot devices",

SPIE Photonics West, San Francisco, USA, January 2010

Poster and Talks

2010

•A. Carmele, M. Richter, W.W. Chow, and A. Knorr, "Antibunching of Thermal

Radiation by a Room temperature Phonon bath",

Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS), Paderborn,

Germany, August 2010 (Poster)

• M.-R. Dachner, A. Carmele, F. Milde, J. Wolter, M. Richter, and A. Knorr, "The-

ory of phonon-assisted relaxation in QD systems and how it affects photon entangle-

ment",

Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS), Paderborn,

Germany, August 2010 (Poster)

• J. Kabuss, A. Carmele, A. Knorr, and W.W. Chow, "Theory of externally controlled

photon-statistics for a cavity-coupled quantum dot",

Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS), Paderborn,

Germany, August 2010 (Poster)

•A. Carmele, M. Richter, A. Knorr, and W. W. Chow, "Antibunching of thermal

radiation by a room-temperature phonon bath: an exactly solvable model",

International Conference on the Physics of Semiconductors (ICPS), Seoul, South

Korea, July 2010 (Poster)

• M.-R. Dachner, A. Carmele, F. Milde, J. Wolters, M. Richter, and A. Knorr, "The-

ory of phonon-assisted relaxation and its impact on entanglement in QD systems",

Quantum Dot 2010, East Midlands Conference Centre, Nottingham, UK, April 2010

(Poster)

2009

•A. Carmele, A. Sitek, M. Richter, and A. Knorr "Theory of few photon dynamics

in light emitting quantum dot devices",

TaCoNa, Bad Honnef, Germany, November 2009 (talk)

• A. Sitek, A. Carmele, M. Richter, and A. Knorr, "Few photon dynamics in semi-

conductor quantum dot emitters",

Quantum Information Processing and Communication (QIPC), Rome, Italy, Septem-

ber 2009 (Poster)

•A. Carmele, A. Sitek, A. Knorr, and M. Richter, "Cluster expansion scheme for

few photon dynamics in semiconductor quantum dot emitters",

Optics of Excitons in confined Systems (OECS), Madrid, Spain, September 2010

(Poster)

• M. Richter, A. Carmele, T. Renger, and A. Knorr, "Optical Bloch equations for

light harvesting complexes: pump probe spectra and saturation dynamics at high

light intensity excitation",

Conference on Lasers and Opto-Electronics (CLEO-EUROPE), Munich, Germany,

June 2009 (Poster)

•A. Carmele, M. Richter, T. Renger, and A. Knorr, "Optical Bloch equation of light

harvesting nanostructures",

DPG, Dresden, Germany, February 2009 (Poster)

•A. Carmele, H.B. Harouni, M. Richter, and A. Knorr, "Photon statistics as a probe

of exciton correlation in Coulomb coupled quantum dots",

DPG, Dresden, Germany, February 2009 (Poster)

2008

• M. Richter, A. Carmele, T. Renger, and A. Knorr, "Excitation Energy transfer in

coupled quantum dots",

Workshop for Organo-Metallic Chemistry (OMC), Berlin, Germany, October 2008

(Poster)

•A. Carmele, M. Richter, and A. Knorr, "Theory of quantum light excitation of

Coulomb-coupled quantum dots",

Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS), Klink-Müritz,

Germany, August 2008 (Poster)

•A. Carmele, M. Richter, and A. Knorr, "Theory of Quantum Light Harvesting in

Coupled Quantum Dots",

DPG, Berlin, Germany, March (Poster)

Contents

List of Publications I

List of Conference contributions III

List of Figures IX

1 Introduction 2

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Quantum optics in the equation of motion approach: The factorization problem 6

2.1 Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Jaynes-Cummings solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Hierarchy problem and cluster expansion . . . . . . . . . . . . . . . . . . . . 9

2.4 Weakly driven Coulomb coupled quantum dots . . . . . . . . . . . . . . . . 14

2.4.1 Hamiltonian and equations of motion . . . . . . . . . . . . . . . . . . 15

2.4.2 Photon statistics as a probe for exciton correlations . . . . . . . . . . 19

3 Quantum dot cavity quantum electrodynamics 22

3.1 Mathematical induction approach . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.2 General equations of motion . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Benchmarking the induction model . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Independent boson model:LO-phonon satellite peaks . . . . . . . . . 28

3.2.2 Jaynes-Cummings model (1): Vacuum Rabi splitting . . . . . . . . . 30

3.2.3 Jaynes-Cummings model (2): Collapse and revivals . . . . . . . . . . 32

3.2.4 Jaynes-Cummings model (3): Analytical solution . . . . . . . . . . . 34

3.3 LO-phonon QD cavity-QED: LO-phonon assisted cavity feeding . . . . . . . 36

3.3.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.3 LO-phonon QD cavity-QED: Optical absorption spectrum . . . . . . 37

3.3.4 Phonon-induced Rabi frequency modification . . . . . . . . . . . . . 40

3.4 LO-phonon QD cavity-QED: Degradation of photon-statistics . . . . . . . . 42

3.4.1 Initial conditions and equations of motion . . . . . . . . . . . . . . . 42

3.4.2 LO-phonon induced anti-bunching . . . . . . . . . . . . . . . . . . . 43

3.5 LO-phonon QD cavity-QED: Biexciton cascade . . . . . . . . . . . . . . . . 49

3.5.1 Electron-electron interaction . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.2 Electron-photon interaction . . . . . . . . . . . . . . . . . . . . . . . 51

VII

Contents

3.5.3 Biexciton cascade: Equations of motion . . . . . . . . . . . . . . . . 53

3.5.4 Biexciton cascade: Dynamics in the strong coupling regime . . . . . 55

3.5.5 Biexciton cascade and entanglement . . . . . . . . . . . . . . . . . . 58

3.5.6 Biexciton cascade: Weak coupling regime . . . . . . . . . . . . . . . 59

4 Quantum dot - wetting layer cavity quantum electrodynamics 66

4.1 Standard cluster expansion (SCE) beyond the one-electron assumption (OEA) 67

4.1.1 Hartree-Fock approximation within the SCE . . . . . . . . . . . . . . 68

4.1.2 Modified equations of motion within the SCE . . . . . . . . . . . . . 69

4.2 Photon probability cluster expansion approach (PPCE) . . . . . . . . . . . 72

4.2.1 Photon probabilities expansion . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 PPCE and photon-statistics . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.3 PPCE and Hartree Fock factorization . . . . . . . . . . . . . . . . . 76

4.2.4 PPCE and modified equations of motion . . . . . . . . . . . . . . . . 78

4.3 Photon dynamics in the PPCE and SCE approach . . . . . . . . . . . . . . 79

4.3.1 Intensity-intensity correlations within the Hartree-Fock approximation 80

4.3.2 Pauli-blocking dependent Rabi oscillation amplitude . . . . . . . . . 83

4.4 PPCE and environment coupling . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.1 Electrical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.2 β-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.3 Cavity loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4.4 Quantum dot - wetting layer laser equations . . . . . . . . . . . . . . 91

4.5 Electrically pumped single photon emitter . . . . . . . . . . . . . . . . . . . 91

4.5.1 Laser dynamics of an electrically driven QD . . . . . . . . . . . . . . 92

4.5.2 Parameter studies of a single QD laser device . . . . . . . . . . . . . 96

4.5.3 Atom and QD-WL rate equations in the single-photon limit . . . . . 99

5 Conclusion and outlook 102

A Appendix i

A.1 Weakly driven QDs: equations of motion . . . . . . . . . . . . . . . . . . . . i

A.2 Numerical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

A.3 Biexciton cascade in the strong coupling regime: Equations of motions . . . iii

A.4 Biexciton cascade in the weak coupling regime: Equations of motions . . . . v

A.5 Derivation of the Hartree-Fock factorization . . . . . . . . . . . . . . . . . . vii

A.6 Photon probability picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

A.7 Jaynes-Cummings solution in the photon probability picture . . . . . . . . . x

A.8 Modified photon-assisted polarization (electron picture) . . . . . . . . . . . xii

A.9 Cavity loss in the photon probability picture . . . . . . . . . . . . . . . . . . xii

Bibliography xvi

Danksagung xxxi

VIII

List of Figures

1.1 Micropillar SEM [SDT+05]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Time-resolved emission scheme, calculated in the equation of motion ap-

proach [DMR+10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Jaynes-Cummings model solution . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 2nd order cluster expansion solution . . . . . . . . . . . . . . . . . . . . . . 10

2.4 3rd and 4th order cluster expansion solution . . . . . . . . . . . . . . . . . . 11

2.5 Cluster expansion solution for 50 Photons . . . . . . . . . . . . . . . . . . . 12

2.6 4th order modified cluster expansion solution . . . . . . . . . . . . . . . . . . 14

2.7 Hanbury Brown and Twiss Experiment . . . . . . . . . . . . . . . . . . . . . 15

2.8 Local (electron operators a(†)) and non-local (exciton operator B(†)) exci-

tation scheme of two Coulomb-coupled quantum dots . . . . . . . . . . . . . 16

2.9 Coulomb-interaction between two QD electrons. . . . . . . . . . . . . . . . . 17

2.10 Single-exciton and biexciton dynamics for stationary excitation with differ-

ent values for g(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.11 Exciton / biexciton density and percental exciton / biexciton density differ-

ences as a function of the excitation strength. . . . . . . . . . . . . . . . . . 20

3.1 GaAs/InAs-QD [MGJ01]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Scheme of the semiconductor quantum dot cavity-QED . . . . . . . . . . . . 23

3.3 Scheme of general equations of motion within the induction method without

classical pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Polarization dynamics within IBM and spectrum . . . . . . . . . . . . . . . 29

3.5 Temperature dependence of the LO-phonon QD spectrum . . . . . . . . . . 30

3.6 Rabi splitting in the QD-CQED . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7 QD-CQED induced anticrossing . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.8 Excited state dynamics for different photon-statistics . . . . . . . . . . . . . 33

3.9 Optical absorption spectrum of an LO-phonon QD-cavity system at 300 K. 38

3.10 Parameter plots for the Rabi splitting dependence . . . . . . . . . . . . . . . 39

3.11 Dynamics of the excited state density for different detunings at 3K and at

300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.12 Thermal light distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.13 Photon density dynamics and spectrum of a cavity mode in the thermal state 44

3.14 Mean phonon number of GaAs LO-phonon energy and g(2)(t, 0)-function . . 45

3.15 g(2)(t, 0) of a cavity mode in the thermal state for two temperatures of the

LO-phonon bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.16 Dynamics of the photon density and g(2)(t, 0) of a cavity mode in the Fock

state for two temperatures of the LO-phonon bath . . . . . . . . . . . . . . 47

List of Figures

3.17 Counting averages of g(2)(t, 0) at 3K and 300 K in case of Fock photons in

the cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.18 QD band structure with spin states |J, mji. . . . . . . . . . . . . . . . . . 50

3.19 Decay paths of the biexciton cascade . . . . . . . . . . . . . . . . . . . . . . 53

3.20 Electronic states in the biexciton dynamics . . . . . . . . . . . . . . . . . . 54

3.21 Biexciton dynamics in the strong coupling . . . . . . . . . . . . . . . . . . . 55

3.22 Photon dynamics in the biexciton cascade. . . . . . . . . . . . . . . . . . . . 56

3.23 Polarization coherence dynamics in the biexciton cascade . . . . . . . . . . . 57

3.24 Hanbury Brown and Twiss setup with polarisators . . . . . . . . . . . . . . 58

3.25 Set of equations of motion in the weak coupling regime . . . . . . . . . . . . 60

3.26 Scheme of the biexciton cascade dynamics in the weak coupling regime . . . 61

3.27 Concurrence in dependence of the fine structure splitting Vex . . . . . . . . 62

3.28 Experimental data from the group of Peter Michler [HUM+07]. . . . . . . . 63

3.29 LO-phonon assisted carrier relaxation in a higher-order Markovian process

for different temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.30 Temperature dependent degree of entanglement in GaAs . . . . . . . . . . . 64

4.1 Scheme of QD-WL cavity-QED . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 2nd order cluster expansion within the Hartree-Fock approximation . . . . . 71

4.3 4th order cluster expansion within the Hartree-Fock approximation . . . . . 71

4.4 Photon-statistics within the PPCE: thermal light and coherent light dynam-

ics for a two-level system, initially prepared in the excited state . . . . . . . 75

4.5 PPCE solution with and without Hartree-Fock contribution driven by cavity

photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Dynamics of the photon density g(1)(t, 0) and the intensity-intensity correla-

tion function g(2)(t, 0), calculated with the SCE and the PPCE in the OEA

and for a QD-WL system for N= 40 . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Dynamics of the photon density g(1)(t, 0) and the intensity-intensity correla-

tion function g(2)(t, 0), calculated with the SCE and the PPCE in the OEA

and for a QD-WL system for N= 2 . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Dynamics of the g(2)(t, 0)-function (solid, orange line) and the electron den-

sity for different initial conditions of the electronic system NE= 1+fe−fh

and initially two photons in the cavity . . . . . . . . . . . . . . . . . . . . . 83

4.9 g(2)(t, 0)-function oscillation amplitude dependence on the number of elec-

trons and holes in the QD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.10 Scheme of cavity losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.11 QD as a single photon emitter without electrical pumping . . . . . . . . . . 93

4.12 QD as a single photon emitter weakly pumped . . . . . . . . . . . . . . . . 94

4.13 QD as a single photon emitter transition regime . . . . . . . . . . . . . . . . 94

4.14 QD as a single photon emitter strongly pumped . . . . . . . . . . . . . . . . 95

4.15 The single photon emitter in steady state condition over different values of

electron-photon coupling M. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.16 The single photon emitter in steady state condition over different values of

cavity loss κ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.17 The single photon emitter in steady state condition over different values of

the inscattering rate Sin and for different detuning ∆. . . . . . . . . . . . . 99

List of Figures

4.18 Photon density g(1)(0) and g(2)(t, 0)-function in a steady state condition

over different values of the inscattering rate Sin in an atomic (OEA) and in

a QD-carrier reservoir system (CR) . . . . . . . . . . . . . . . . . . . . . . . 100

List of Figures

A sa recherche, l’écrivain progresse

laborieusement, tâtonne en aveugle, s’engage

dans des impasses, s’embourbe, repart — et, si

l’on veut à tout prix tirer un enseignement de

sa démarche, on dira que nous avançons

toujours sur des sables mouvants.

Claude Simon

1 Introduction

1.1 Motivation

Quantum optical properties pave the

Figure 1.1: Micropillar SEM [SDT+05].

way to new and spectacular techno-

logical advances by improving the ac-

curacy of measurements done so far

with stabilized lasers [Shi07], increas-

ing the degree of security in quantum

cryptography via the knowledge of

quantum states [UFT07] or exploit-

ing entanglement for quantum com-

puting [BPM+97]. In addition to an

atom-photon interface, ideal sources

for deterministic quantum light emis-

sion with tunable photon statistics

are semiconductor quantum dots (QDs) coupled to an optical microcavity, cf. Fig. 1.1

[STH+09; FMR+09]. These systems also exhibit signatures of strong coupling [RSL+04].

In this regime, the electron-photon interaction outrivals the dissipation processes and dis-

tinct quantum optical phenomena become accessible, such as vacuum Rabi oscillations or

the vacuum Rabi splitting [KGK+06; MKH08]. In this limit, the quantized nature of light is

of primary importance, and semiconductor research begins to shed new light on well-known

quantum optical effects, e.g., gigantic, experimentally robust resonances of vacuum Rabi

splitting of the second rung in the Jaynes-Cummings ladder [KK08; SKK08]. Conceptually

new approaches are introduced, such as implementing the coherent control of an exciton

two-level system (qubit) by means of a time-dependent electric interaction [VSSM+10]. For

weak coupling, a smooth transition from strong photon bunching to Poissonian statistics

in the output characteristics of semiconductor microcavity lasers is observed [GWJ08].

One of the most feasible features of solid state environments is the capability of position-

ing nanostructures such as quantum dots permanently in a microcavity [KKKG06]. The

size and geometry of the QD control the properties, e.g., the coupling strength and the

confinement energies [Sti01]. Those systems are designated candidates for future technolog-

ical applications, including single-photon emitters [LO05; MIM+00]. In optical quantum-

information processing, semiconductor quantum dots inside a microcavity have emerged

as promising sources of polarization-entangled photon pairs [TPT06]. However, the limits

of future technological applications depend on understanding of quantum many-body phe-

nomena in condensed matter environments such as electron-electron scattering or phonon-

induced dephasing [SCK01; MZ07; MR09; RCB+09; DMK05] , which are not present in

atom-photon systems.

Common non-Markovian quantum mechanical approaches for treating the electron-photon

interaction result in a hierarchy problem, if the Jaynes-Cummings-Model (JCM), which is

1 Introduction

valid only in the case of uncoupled two-level-systems interacting with a single cavity mode

[JC63], is exceeded. Hence, quantum optics in semiconductors, where Coulomb-interaction

and electron-phonon interaction take place, calls for different approaches such as the Born

approximation[VWW01] or the cluster-expansion [KK08]. Recent work incorporates sev-

eral aspects of many-body phenomena into the calculation of semiconductor quantum

dot cavity quantum electrodynamics (QD-CQED): Absorption spectra in the presence of

strong electron-phonon interaction have been studied with polaron operator techniques

[WRI02; MKH08].

The excitonic time dynamics in a perturba-

Figure 1.2: Time-resolved emission

scheme, calculated in the equation of

motion approach [DMR+10].

tion approach, based on the Lamb-Dicke ap-

proximation [LMC08] for LA-phonons, com-

parable to the second-order Born approxima-

tion, has been theoretically investigated. Of-

ten, many-particle interactions are taken into

account via an equation of motion approach

[GMK06; MTK07]. However, factorization tech-

niques become problematic in case of strongly

correlated interactions, such as the electron-

photon interaction in the strong coupling and

single-photon regime.

This thesis focuses on the theoretical description of semiconductor QD-CQED beyond

the JCM, including non-Markovian contributions and many-particle interactions such as

electron-phonon coupling on a microscopical level. The equation of motion approach is

chosen as an established technique to explain recent experimental results and to give in-

sight into the underlying physics. Two theoretical frameworks are introduced to provide

a simulation tool for parameter studies in the single-photon limit: a mathematical in-

duction approach and the photon-probability cluster expansion. In this regime, quantum

correlations are important, and fully quantized approaches become necessary, which take

into account the complete and strongly correlated quantum dynamics of the observables

of interest: the photon density, the intensity-intensity correlation g(2)(t, 0)-function, the

polarization and the electron density. The mathematical induction model solves the QD-

CQED dynamics including the electron-longitudinal optical (LO) phonon interaction up

to an arbitrary accuracy in case of a fixed number of electrons in the system but including

dissipation processes such as a cavity loss or a radiative decay. In the presence of a wetting

layer, the number of carriers, electrons and holes, is not fixed and a modified Hartree-Fock

factorization based on the photon-probability cluster expansion (PPCE) approach is intro-

duced to treat the occurring many-particle correlations.

1.2 Highlights

The mathematical induction approach allows the inclusion of the LO-phonon interaction

up to an arbitrary order. As a result, the LO-phonon assisted QD-CQED dynamics is

computed and a novel feature predicted: Antibunching of Thermal Radiation by a Room-

Temperature Phonon Bath [CRCK10]. Furthermore, LO-phonon assisted cavity feeding

leads to interesting anharmonic Rabi oscillations for elevated temperatures. In the weak

and strong coupling regime, the biexciton cascade in a QD is computed and for the first

1 Introduction

time a temperature dependent analysis of the formation dynamics of an entangled photon

pair is derived [CMD+10]. For wetting layer contributions in the QD-CQED, a few-photon

model of the optical emission of semiconductor quantum dots is introduced, the PPCE

[RCSK09a]. A modified Hartree-Fock factorization is derived and the photon statistics of

a single quantum dot in a microcavity is calculated [SRK+10; SCR+10], and hereby the

completion of a theory of few photon dynamics in electrically pumped light emitting QD

devices provided [CRD+10; DMR+10].

1.3 Structure

The thesis is divided into five chapters. The introduction in this chapter 1 contains the

motivation, the highlights and the outline of the thesis. The fundamental quantum optical

model to study cavity-QED, the Jaynes-Cummings model (JCM), is discussed in chapter

2. The JCM provides an analytical solution under the condition of an isolated two-level

system with one electron interacting with a single cavity mode without dissipation pro-

cesses. These conditions are not fulfilled in the case of semiconductor QD-CQED. In order

to consider further interactions, such as electron-phonon coupling, and losses due to envi-

ronmental coupling, the equation of motion approach is chosen, which allows the extension

of the JCM. However, this leads to the well-known hierarchy problem and factorization

approaches become necessary. In chapter 3, an equation of motion approach is introduced,

which does not rely on factorization and solves the hierarchy problem via a mathematical

induction model. The induction approach goes beyond the JCM by including the LO-

phonon interaction up to arbitrary accuracy and cavity losses. However, similiar to the

JCM, the induction model relies on the assumption of a fixed number of electrons in the

system. In chapter 4, a Hartree-Fock based perturbation approach is introduced and is

valid without assuming a fixed number of carriers. In particular, this is important in stud-

ies of a single-QD laser in the presence of wetting layer (WL) carriers, since the WL leads

to a modified spontaneous emission source term. In chapter 5, a conclusion and outlook is

given.

1 Introduction

2 Quantum optics in the equation of motion approach: The factorization problem

2 Quantum optics in the equation of motion

approach: The factorization problem

Theoretical approaches in semiconductor many-particle physics are often based on an equa-

tion of motion scheme in the formalism of second quantization [HK04]. The equations of

motion of observables are derived via commutating them with the full Hamiltonian of

the investigated system [KJHK99]. In contrast to typical master equation approaches,

the equation of motion approach does not rely on the Markovian approximation, i.e.,

the neglection of memory effects [Car99]. Therefore, to study quantum correlations of

higher order or ultrafast phenomena, the equation of motion approach is advantageous

and enables the investigation of novel quantum features, such as excitation induced de-

phasing [SCK01; SCK04], Coulomb scattering processes in quantum well and microcavity

systems [KKM01; KK08] or many-particle-induced non-Lorentzian lineshapes in semicon-

ductor nano-optics [FAD+02; MKH08].

The non-Markovian equation of motion approach results in the hierarchy problem. As an

example, the quantum correlations of second order couple to quantum correlations of the

third order [KJHK99]. The set of equations of motion is not closed, unless the quantum

correlations are factorized at a given level. A typical approach is the cluster expansion or

correlation expansion method [Fri96; GWLJ07; KK08]. Via a complete factorization of oc-

curring quantum correlations, the set of equations of motion is truncated and casted into

a closed set, which is numerically solvable. However, this cluster expansion approach is

problematic for systems, in which the quantum correlations are in the order of magnitude

of the corresponding mean value of the observable [RCSK09a]. A prominent example is

the single-photon emitter, where the fluctuations and mean value are of the same order of

magnitude.

In the following chapter, this factorization problem is discussed. A model Hamiltonian of a

two-level system strongly coupled to a single cavity mode is assumed. This simplest possible

fully quantized model of interest is exactly solvable [ENSM80]. The solution is called the

Jaynes-Cummings model (JCM) [JC63] and is taken as a benchmark for the solutions de-

rived in the equation of motion approach. The deviations from the exact Jaynes-Cummings

model are discussed.

2.1 Jaynes-Cummings Hamiltonian

The Hamiltonian of a single two-level system interacting with a single-mode field reads in

rotating-wave [HK04] and dipole approximation [SZ97]:

H=~ωva†

vav+~ωca†

cac+~ω0c†c−~M(a†

vacc†+a†

cavc),(2.1)

where ω0is the frequency of the quantized light field with bosonic creation (c†) and annihi-

lation operators (c). In the two-level system one electron is assumed, either in the ground

state (v) with energy ~ωvor in the excited state (c) with ~ωcthe excited-state energy.

2 Quantum optics in the equation of motion approach: The factorization problem

The fermionic creation (annhiliation) operator denote a(†)

c/v and Mis the electron-photon

coupling element.

Here, the coupling element is assumed to be

Figure 2.1: JCM Hamiltonian

real and includes the polarization of the pho-

ton and the dipole strength of the two-level

system, as well as the corresponding frequency

of the single-mode light field. The interaction

is depicted in Fig. 2.1. The two-level system is

driven by the quantized light field. If the elec-

tron is excited via a photon absorption from

the ground state to the excited state (a†

cav),

a photon from the quantized light field is an-

nihilated (c). If the electron relaxes from the

excited state to the ground state via spontaneous or induced emission (a†

vac), a photon

is created (c†). As long as the interaction is restricted to the electron-photon case, other

processes are not possible. Losses, such as cavity loss or radiative decay or pure dephas-

ing, are not considered. The zero point energy in the free-energy part of the photon-field

Pk~ωk/2is omitted for convenience [SZ97]. The zero-point energy sums frequencies with-

out an upper bound and is accordingly infinite, an awkward feature of the quantized light

field. Fortunately, usual measurements in quantum optics do not apply apparatus, which

would register a response proportional to the zero-point energy. They response only to the

change in the total energy of the electromagnetic field [Lou83].

2.2 Jaynes-Cummings solution

In the case of a two-level system with one electron interacting with a quantized light field,

the dynamics can be solved exactly. Several approaches are possible, e.g., the unitary time-

evolution operator method [SZ97]. If the excited and ground state energy are chosen as:

~ωc=~˜ω

2=−~ωv, Eq. (2.1) equals the Jaynes-Cummings Hamiltonian and reads:

H=~˜ω

2(a†

cac−a†

vav) + ~ω0c†c

|{z }

−~Ma†

vacc†+a†

cavc

|{z }

HII

.(2.2)

The dynamics of the system depends only on HII. It is convenient to describe the dynamics

in the interaction picture. The transformed interaction Hamiltonian HII reads:

Hint

II =ei

~HItHIIe−i

~HIt.(2.3)

Using the one-electron assumption, i.e., any combination of four electron operators becomes

zero in the expectation value and can be neglected (a†

ia†

jakal= 0 for i, j, k, l =v, c), and

assuming the two-level system to be in resonance with the light field, HII reads explicitely:

Hint

II =−~Ma†

vacc†+a†

cavc.(2.4)

2 Quantum optics in the equation of motion approach: The factorization problem

The time evolution of the system for a given state |Ψ(t)iis calculated via the unitary time

evolution operator and reads |Ψ(t)i=U(t)|Ψ(0)iwith |Ψ(0)ias the initial state and the

unitary time-evolution operator in the interaction picture:

U(t) = e−i

~Hint

II t.(2.5)

After evaluating the exponential function, again using the one-electron assumption, the

time-evolution operator reads:

U(t) = cos Mpc†c+ 1 ta†

cac+ cos M√c†c ta†

vav

−i

sin M√c†c+ 1 t

√c†c+ 1 c a†

cav−i c†sin M√c†c+ 1 t

√c†c+ 1 a†

vac.(2.6)

Now, the time evolution of a given state is easily calculated, bearing in mind, that cosine

and the square root are analytical functions and can be expressed in a Taylor series. As

an example, the electron is initially in the excited state with Nphotons in the system, i.e.

the state is |Ψ(0)i=|c, Ni. After applying the unitary time evolution operator, the time

evolution reads:

|Ψ(t)i=U(t)|c, Ni= cos M√N+ 1t|c, Ni−isin M√N+ 1t|v, N + 1i.(2.7)

The analytical solution results from a division of the Hilbert space in subspaces, consisting

of the only two occurring processes in the system: 1) a photon is absorbed (N→N−1)

and the electron moves into the excited state |vi → |ciand 2) a photon is emitted via

spontaneous or induced emission (N→N+1) and the electron moves back into the ground

state |ci → |vi. The time evolution of ground and excited state densities is calculated via

0 10 20 30 40 50

t [ps]

0.2

0.4

0.6

0.8

Electron density

Ground state

Excited state

Figure 2.2: Analytical solution of the JCM for a two-level system interacting with a quan-

tized light field. Rabi oscillations occur between the ground and excited state.

the probability amplitudes given in the time evolution of the system state with CN

hc, N|Ψ(t)iand CN

v=hv, N + 1|Ψ(t)i. The exact solution reads:

a†

vav=CN

v(t)2= sin2(ΩNt),a†

cac=CN

c(t)2= cos2(ΩNt),(2.8)

2 Quantum optics in the equation of motion approach: The factorization problem

with the quantum-mechanical Rabi frequency ΩN=M√N+ 1, cf. Fig. 2.2. The solu-

tion includes quantum optical properties of emission processes, e.g. spontaneous emission,

seen in the vacuum Rabi oscillations (N= 0) and experimentally observed in [NRS+94].

The solution enables one to implement photon statistics dependent Rabi oscillations by

calculating a superposition of JCM solutions [LP07; SZ97]:

a†

cac=

∞

n=0 |pn|2cos2(Ωnt),

where |pn|2is the probability to find nphotons in a given mode within a given photon mean

value. Although the JCM is a powerful tool, to implement quantum optical properties in the

emission and absorption dynamics, it is more or less limited to an isolated two-level system.

Including, for example, Coulomb or phonon induced dephasing processes microscopically,

the interaction Hamiltonian is more complex and the presented analytical solution Eq. (2.8)

is not valid.

2.3 Hierarchy problem and cluster expansion

In this section, a simple example of the hierarchy problem is presented, which occurs in

systems with many-particle quantum correlations. This equation of motion approach is

chosen to include non-Markovian many-particle features in the calculations. Starting from

the Hamiltonian Eq. (2.1), equations of motion for the observables of interest are derived

via the Heisenberg equation of motion:

−i~∂ta†

iaj= [ ˆ

H, a†

iaj].(2.9)

A single-mode quantized light field and one electron in a two-level system are considered,

as well as resonance between the light mode ω0and band gap frequency ωcv =ωc−ωv

assumed. The dynamics of the photon density, of the excited-, and ground state read:

∂ta†

cac= 2 ImhMa†

vacc†i,(2.10)

∂ta†

vav=−2Im hMa†

vacc†i=∂tc†c(2.11)

The equations of motion for the photon density and the ground state density are identical.

This is easy to understand: the JCM considers a closed system with particle conservation,

i.e., the number of excitations and photons (ν) is constant: ν=a†

cac+c†c=const.

Assuming an initially excited two-level system without photons in the cavity, ν= 1 is

valid for all times. The one-electron assumption is also valid for all times a†

cac+a†

vav=

1, which yields for the conservation of particles: ν= 1 = 1 −a†

vav(t) + c†c(t)→

a†

vav(t) = c†c(t). Of course, this is only valid for zero photons in the cavity and an

initially excited two-level system. The equations of motion of the densities couple to the

photon-assisted polarization:

∂ta†

vacc†=−iM a†

cac−iM a†

cacc†c−a†

vavc†c.(2.12)

2 Quantum optics in the equation of motion approach: The factorization problem

The hierarchy problem is now obvious: a quantity with nphoton operators couples to a

quantity with n+ 1 photon operators [GWLJ07]. If the one-electron assumption is aban-

doned, another hierarchy problem occurs within the electronic system, cf. Chap. 4. The

set of differential equations must be truncated to obtain a closed system with respect to

the electron-photon coupling M. Common approaches rely on the Born approximation

[RBSK07; VWW01] or the cluster expansion [FMWS97; KJHK99]. Here, the cluster ex-

pansion is applied to derive the truncated form for Eq. (2.12):

∂ta†

vacc†=−iM a†

cac−iM c†ca†

cac−a†

vav

−iM a†

cacc†cc−a†

vavc†cc.(2.13)

Pure, coherently driven quantities such as a†

vac,c†are neglected. They do not con-

tribute to the combined electron-photon dynamics without the inclusion of a classical

pump field [RCSK09a]. If the correction terms a†

iaic†ccare neglected, the set of dif-

0 10 20 30 40 50

t [ps]

-0.2

0.2

0.4

0.6

0.8

Electron density (excited state)

cluster-expansion 2nd order

Jaynes-Cummings-Model

Figure 2.3: Comparison of the second-order cluster-expansion (dashed, red line) in the

electron-photon coupling element with the exact solution of the JCM (solid, green line).

After 10 ps, the deviation of the exact solution becomes very large.

ferential equations [Eq. (2.10) -(2.11) and (2.13)] is closed and can be solved numerically

and analytically. With the same initial condition: c†c(0) = 0 and a†

cac(0) = 1, the

solution is plotted in Fig. 2.3. Again, Rabi-oscillations are visible for the JCM (solid, green

line). The second-order correlation expansion is a good approximation for the first pico

seconds. After 10 ps, the deviation becomes larger and the approximation fails completely.

Negative probabilities occur due to the factorization. Dissipation processes (cavity loss or

electronic dephasing) can prevent negativities but do not solve the underlying problem. In

conclusion, a second-order correlation expansion in the electron-light coupling element is

not sufficient to factorize the combined electron-photon dynamics correctly. Higher order

correlation terms in Eq. (2.13) are necessary. The dynamics of the correction terms are

derived via the same approach. The photon-density assisted electron density in the excited

state reads:

∂ta†

cacc†c= 2 ImhMa†

vacc†c†ci.(2.14)

2 Quantum optics in the equation of motion approach: The factorization problem

The correlation term is derived by:

∂ta†

cacc†cc=∂ta†

cacc†c−∂tha†

cacc†c+a†

cacc†c+a†

caccc†i.

After the neglecting coherently driven contributions, the equation of motion reads

∂ta†

cacc†cc= 2 ImMa†

vacc†c†cc+Ma†

vacc†cc†c+a†

cac.(2.15)

Using the one-electron assumption and Eq. (2.15), ∂ta†

cacc†cc=−∂ta†

vavc†ccis

shown. This identity is disturbing, since ground state and excited state dynamics differs in

quantum optical schemes. The ground state is driven by the higher-order photon-assisted

polarization and additionally by the lower-order photon-assisted polarization, whereas the

excited state couples only to the higher-order photon-assisted polarization. In the cluster

expansion approach, photon density-assisted ground and excited state lose this significant

difference in third order perturbation theory and are described via the same equation of

motion, only with different sign and initial conditions. Neglecting again the higher-order

0 10 20 30 40 50

t [ps]

0.2

0.4

0.6

0.8

Excited State Density

cluster expansion 3th order

Jaynes-Cummings Model

(a)

0 10 20 30 40 50

t [ps]

-5

-4

-3

-2

-1

Excited State Density

Jaynes-Cummings Model

cluster expansion 4th order

(b)

Figure 2.4: (a) Comparison of the third order cluster expansion and the JCM solution.

Although the approximation is quite good in the first 10 ps, the deviation becomes very

large at later times. (b) Solution of the 4th order cluster expansion (red, dashed line)

and the JCM solution (green, solid line). The solution obtained via the cluster expansion

diverges.

correlation term in Eq. (2.15), the set of equations of motion is closed and can be evaluated

numerically. The solution is plotted in Fig. 2.4(a). A truncation at the third order is prob-

lematic within a equation of motion approach. In Fig. 2.4(a), no negativities occur, but

the oscillation amplitude of the cluster expansion solution (dashed, red line) is too small.

Furthermore, the Rabi oscillation frequency is wrong due to the decreased amplitude. Note,

that the deviation from the JCM solution is bigger (difference of the minima: 0.3), com-

pared to the solution of the second order correlation expansion (−0.2). An alternating

convergence to the JCM is improbable. This becomes even clearer, when investigating the

4th order solution in the cluster expansion.

The higher-order photon-assisted polarization includes the intensity-intensity correlation

2 Quantum optics in the equation of motion approach: The factorization problem

implicitly, and after the factorization, explicitly. The dynamics in the fourth order of the

electron-photon coupling element reads:

∂ta†

vacc†c†c=−2i M a†

cacc†c−iM a†

cacc†c†c c −a†

vavc†c†c c ,(2.16)

∂tc†c†cc=−4Imh¯

Ma†

vacc†c†ci.(2.17)

The factorization implies more correction terms than in the lower order due to the different

possibilities to factorize singlett, dublett, and triplett expectation values:

∂ta†

vacc†c†cc=(2.18)

=−2iM a†

cacc†cc−iM c†c†c c ca†

cac−a†

vav

−2iMc†ca†

cacc†cc−a†

vavc†cc+ 4a†

vacc†cImhMa†

vacc†ci.

This equation yields unexpected terms, proportional to the photon density and results

even in a diverging dynamics. In Fig. 2.4(b), the solution of the 4th order cluster expan-

sion (red, dashed line) and the JCM solution (green, solid line) are plotted. Up to 10 ps,

the factorized solution reproduces the JCM. For longer times, the factorized solution de-

viates strongly from the JCM solution. In the 4th order the solution additionally loses the

periodicity property. The excited state density finally diverges.

The cluster expansion factorization approach leads to wrong results in the single-photon

limit: few emitters are correlated with few photons in the cavity.

For an ensemble of emitters, at least 10−50, or for a high number of photons (25−50), the

cluster expansion yields good results [SKK08; GWJ08; KK08]. In particular, the second

0 10 20

t [ps]

0.2

0.4

0.6

0.8

0 10 20

t [ps]

0.2

0.4

0.6

0.8

Excited State Density

2nd order

4th order

Figure 2.5: Solution of the 2nd and 4th order cluster expansion (red, dashed line) and the

JCM solution (green, solid line) for a high number of photons (N= 50).

2 Quantum optics in the equation of motion approach: The factorization problem

order factorization converges to the solution of the JCM model. In Fig. 2.5, the Jaynes-

Cummings model solution (green, solid line) is compared with the 2nd-order (upper panel)

and 4th-order (lower panel) solution, obtained via factorization in the cluster expansion

scheme for the initial condition of one excited two-level system with initially N= 50 pho-

tons in the cavity. The 2nd-order solution fits the JCM solution perfectly. The deviations

are vanishingly small, which proves the cluster expansion scheme to be well suited for a

situation, where one emitter interacts with a large number of photons. In other words, the

electron and photon dynamics are weakly correlated, since only one out of 50 photons is

absorbed or re-emitted. Assuming this weak correlation regime, the 2nd-order equations of

motion can be rewritten. Given the one electron approximation, Eq. (2.12) is written as:

∂ta†

vacc†=−iM a†

cac−iM c†c(2 a†

cac−1).

For a large photon number N,c†c=Ndominates over the product a†

cacc†c. In the

case of a single emitter, the photon density fluctuates ±1around N, whereas the product

of excited state and photon number oscillates between 0and 2N. The larger N, the closer

the photon-assisted polarization appears in the JCM limit. However, in the single-photon

limit, Nand a†

cacNare of the same order of magnitude and the deviations from the

JCM become large.

In Fig. 2.5 (lower panel), the solution of the 4th-order is compared with the JCM. Again,

the solution is closer to the JCM solution and shows a periodic behavior, which is lost in

the few photon limit, but the deviations are larger in comparison to the deviation between

the JCM solution and the 2nd-order solution. This suggests, that higher order correla-

tion expansion terms do not correspond and alter the dynamics artificially, introducing

anharmonic oscillation behavior seen for, e.g., t= 6 ps. The factorization scheme must be

adapted to the situation of weakly coupled electrons and photons. E.g., the cluster expan-

sion is done only within the photon field and correlation terms containing both fermion

and boson contributions such as a†

cacc†cc≈0are neglected:

a†

cacc†c†cc=a†

cacc†c†cc+ 2a†

cacc†cc†c+a†

cacc†c†ccc.(2.19)

Hence, correlations between the interacting fields are assumed to be small. Now, the cor-

rection term of 4th order reads:

∂ta†

vacc†c†cc=(2.20)

=−2iM a†

cacc†cc−iM c†c†c c ca†

cac−a†

vav

+ 2iMc†ca†

cacc†cc−a†

vavc†cc+ 4a†

vacc†cImhMa†

vacc†ci.

The Rabi oscillations, which are described exact within the JCM, are approximated to a

satisfying accuracy by Eq. (2.20). Fig. 2.6 shows no deviation between the approximated

solution, derived via this modified cluster expansion, and the exact solution from the

JCM. However, the suggested truncation scheme is not adaptable to the strong coupling

regime, for few photons and few emitters. In case of weak coupling, presumely without

cavity enhanced dynamics, the factorization schemes such as the cluster expansion or the

Born approximation work. In the next section, an example is given: Two Coulomb-coupled

quantum dots are excited by quantum light. The calculation does not assume a cavity.

2 Quantum optics in the equation of motion approach: The factorization problem

510 15

t [ps]

0.2

0.4

0.6

0.8

Excited State density

Excited State - 4.order

Excited state - JCM

Figure 2.6: The approximated Rabi oscillation via the modified cluster expansion in the

4th-order of the light-coupling element are in good accordance with the exact solution

given by the JCM .

Quantum light excitation of Coulomb-coupled semiconductor quantum dots is analyzed

for fixed light intensity but different photon statistics in the weak coupling regime.

2.4 Weakly driven Coulomb coupled quantum dots

Excitonic excitations in Coulomb-coupled nanostructures are of particular interest to un-

derstand many basic processes in many particle physics. Important examples are light har-

vesting complexes in coupled photosynthetical units [RM02] and nanoscale energy transfer

in metal structures as well as semiconducting nanocrystals [GLK07] or in polymer com-

posites [AMC+04]. A man made system for promising applications, such as entangled

photon processing [HPS07; SYA+06; ALP+06], are coupled semiconductor nanostructures

[SME+06], since their size, shape, composition and location can be controlled by modern

epitaxially growth techniques [BGL99] or chemical synthesis [DRVM+97]. Here, an anal-

ysis is performed of dipole-dipole coupled semiconductor quantum dots, providing typical

couplings and parameters, representative for many of the mentioned nanosystems.

Coulomb coupling between two QDs via dipole-dipole interaction leads to energy renor-

malisation of the ground-, exciton- and biexciton-state and Förster coupling. These in-

teractions transform the isolated two-level-systems into a multi-level-system [DAFK06].

Thus, the coupled quantum systems form new eigenstates and the individual systems can-

not be excited separately: excitons and many exciton states, such as biexcitons are formed

[UML+05]. The goal in this section is to consider the fully quantized electron-light in-

teraction in a multi-level system, expanding previous semi-classical investigations on the

nonlinear response of exciton-exciton correlations. The impact of photon statistics of the

incident light on the creation of excitons and biexcitons is investigated, using the pho-

ton statistics of the excitation light as an additional parameter to differentiate between

these many particle excitations and presenting hereby a theoretical approach to implement

photon-statistical properties [KK08] in coupled semiconductor structures.

Adressing the quantum-optical properties of the excitation light, the electron-light inter-

action is calculated on a microscopic level [AHZ97]. Self-consistently including photon-

2 Quantum optics in the equation of motion approach: The factorization problem

statistical and Coulomb-effects a measurable and strong impact of the light source on the

creation of biexcitons in comparison to the creation of single-excitons is predicted. Here,

it is focused on the excitation with

Correlator

Beam SplitterLight Source

Photon Detector

I(t+τ)

I(t)

Figure 2.7: Hanbury Brown and Twiss Experiment

thermal, coherent and squeezed light.

To characterize photon statistics of

the excitation light source, one typ-

ically uses the Hanbury Twiss and

Brown setup, which is illustrated in

Fig. 2.7. It consists of a light source,

a beam splitter, two photon detec-

tors and a correlator.[HT56] In this

setup one measures the second-order

correlation function of two intensities at measurement time t(detector 1) and t+τ(detector

2), defining τas the delay time:

g(2)(t, τ) = :I(t)I(t+τ) : 

I(t)I(t+τ),(2.21)

which represents the normalized deviation from a Poisson distribution [MW95]. Three typ-

ical cases for g(2) (i-iii) are discussed: (i) If there is no correlation between the emitted

photons, the g(2) of a Poisson distribution has the value 1. The probability to measure a

photon at the detector 2 is independent of the measurement result at detector 1. Exam-

ples for this photon statistics are laser fields well above threshold, representing coherent

(Glauber) states [Gla63]. (ii) If it is more probable to measure a photon at detector 2, after

detector 1 has measured a photon, the light field is described via a super-Poissonian statis-

tics, showing photon-bunching. An example is a thermal light field, which shows g(2) = 2.

An example is the Bose-Einstein-distribution of the black body radiation [MW95]. (iii) The

light field is in a sub-Poissonian state, if g(2) is smaller than 1, meaning it is less probable

to measure a photon at detector 2 after a positive measurement result at detector 1 is

registered. This effect is called anti-bunching. An example is squeezed light emitted by a

single-photon-emitter [BMJS05].

2.4.1 Hamiltonian and equations of motion

Starting in a local basis (see Fig. 2.8), the total Hamiltonian Hlocal =H0e+He−e+H0p+

He−pis expressed via local annihilation (creation) operators and includes the free energy

of the electronic H0eand photonic system H0p, the electron-electron interaction He−eand

the electron-light interaction He−p. The free energy part of the electronic system reads:

H0e=X

n~ωvna†

vnavn +~ωcna†

cnacn,(2.22)

The electronic states of Coulomb-coupled quantum dots are described via local annihilation

(creation) operators a(†)

in , where idenotes the electronic state and n= (1,2) the QD.

Each QD has one valence and one conduction band state in effective mass approximation

of each individual nanostructure [HK04], thus i= (v, c)with ~ωvn valence band and

~ωcn conduction band energy of quantum dot n. The number of electrons is conserved

2 Quantum optics in the equation of motion approach: The factorization problem

inside every quantum dot, i.e. tunneling between the quantum dots is prohibited. The

Figure 2.8: Local (electron operators a(†)) and non-local (exciton operator B(†)) excitation

scheme of two Coulomb-coupled quantum dots. Förster-Interaction transforms the coupled

two-level-systems into a four-level-system.

two-particle interaction Hamiltonian He−edescribes the Coulomb-interaction and can be

expressed in the local basis of creation and annihilation operator of the individual dots

[DAFK06; RRK08]:

He−e=VFa†

c1a†

v2ac2av1+V∗

Fa†

v1a†

c2av2ac1+Vcv a†

c1a†

v2av2ac1(2.23)

+Vvc a†

v1a†

c2ac2av1+Vcc a†

c1a†

c2ac2ac1+Vvv a†

v1a†

v2av2av1.

The parameters are chosen for Coulomb-coupled quantum dots (Egap = 1.4eV; VF=

1meV Förster interaction; Vcc =Vvc = 0.1meV for biexciton and single-exciton-shifts;

dvc = 0.3e nm for the dipole moment.) The Coulomb-interaction results in a ground

state shift Vvv, two monoexciton shifts V12

vc , V 21

cv and a biexciton shift Vcc. These shifts

correspond to energy renormalizations caused by electrostatic interactions. V12

Fdescribes a

non-radiative Coulomb energy transfer from one quantum dot to the other. H0eand He−e

can be written in terms of four two-electron operators by defining G=a†

v1a†

v2av2av1as

the ground state operator, P1

1=a†

v1a†

v2av2ac1and P1

2=a†

v1a†

v2ac2av1as single-polarisation

operators leading to an annihilation of a single-excitonic state and P2=a†

v1a†

v2ac2ac1as

the double-polarisation operator, leading to an annihilation of a biexcitonic state. In this

local two-electron basis [RAK+06], the Hamiltonian of the electronic system He=H0e+

He−ehas non-diagonal contributions due to the Förster-coupling VF. Thus, taking Förster-

coupling into account [UML+05], P1

1and P1

2are no longer eigenstates of the Hamiltonian.

To find the new eigenstates, the electron part of the Hamiltonian is diagonalized [RAK+06]

and the local basis illustrated in Fig. 2.8 is transformed into the non-local exciton basis:

in this case into a four-level-system. All Coulomb-induced energy shifts are included in the

definition of the new operators derived via diagonalization (ground state, monoexcitonic,

biexcitonic shifts and Förster-Interaction). New collective operators are formed: G(†)is the

ground state, B2(†)the biexciton annihilation (creation) operator, replacing two particle

operators a†

v1a†

v2ac2ac1and B1(†)

mthe exciton annihilation (creation) operator (m=1,2).

Within the non-local basis the electron operators (G(†), B1(†)

m, B2(†)) are eigenvectors of

2 Quantum optics in the equation of motion approach: The factorization problem

Figure 2.9: Coulomb-interaction between the electrons leads to energy renormalization of

the ground state Vvv, monoexcitonic Vvc and biexcitonic state Vcc, as well as to Förster-

coupling VF, corresponding to a non-radiative energy transfer between the quantum dots.

the electron part of the Hamiltonian with eigenvalues (~ωG,~ωm,~ωB). B1(†)

mrepresent a

superposition of localized polarisations [RAK+06]:

m=X

nP1

n.(2.24)

The coefficients vm

ndepend on the strength of the Förster-coupling, the energy of the

monoexcitonic energy shifts and on the band gap frequencies. In case of two coupled quan-

tum dots, they can be calculated analytically via the quantity ∆e=VF(B−~ω1)−1=

−VF(C−~ω2)−1:

1=−v2

2=−1 + ∆2

e−1

2∆e, v1

2=v2

1=1 + ∆2

e−1

2.(2.25)

The electron-light interaction (matrix element Mvc

kn) is also transformed into the new ba-

sis [RAK+06]. Starting from the local two-electron basis, He−preads in a rotating-wave

approximation [HK04]:

He−p=−X

k,n,m6=n

Mvc

kn G†P1

nc†

k+P1†

mP2c†

k+h.c. , (2.26)

where c(†)

kdenotes the Boson annihilation (creation) operator of the photon in the excita-

tion and dissipation mode with frequency ω=ck and m, n the quantum dot. Eq. (2.26)

incorporates radiative coupling between the dots, as well as local electron-light interaction.

Using Eq. (2.24) and introducing non-local matrix elements ( ¯

m,¯

m,2), He−pis expressed

via non-local electron operators. The non-local matrix elements are defined as [RAK+06]:

m=X

(vm

n)∗¯

Mvc

kn,¯

m,2=X

i,n6=i

n¯

Mvc

ki .(2.27)

2 Quantum optics in the equation of motion approach: The factorization problem

The new matrix elements include important characteristics of Förster-coupled systems,

e.g. a vanishing electron-light interaction (dark states) in the case of two identical systems

(quantum dots) [DAFK06]. Since the free energy Hamiltonian of the photonic system H0p

stays unchanged, the total Hamiltonian of the diagonalized system reads:

H=X

~ωmB1†

mB1

m+~ωBB2†B2+X

~ωc†

kck−~X

km ¯

mG†B1

m+¯

m,2B1†

mB2c†

k+h.c.

The ground state energy is chosen to be zero and ~ωmand ~ωBdenote the energy levels

of the four-level-system, illustrated in Fig. 2.8, with two exciton and one biexciton level.

The temporal evolution of the system is calculated via the Heisenberg equation of motion:

−i~∂tA= [H, A]for occurring many body expectation values. All correlations up to the

fourth order of the light coupling element ¯

mare included. Single-mode excitation is

assumed, i.e.: c(†)

k≡c(†)with ¯

m=¯

Mm. Also, to consider dissipation processes, the

electronic system is coupled to a photon reservoir, which causes a decay of the exciton

and biexciton states [MW95]. The decay rate is described via the Einstein-coefficient Γ =

(500ps)−1[DAFK06]. The equations for the ground state and excited state densities read:

∂tG†G= 2Γ B1†

1B1

1+B1†

2B1

2−2X

Im¯

MmG†B1

mc†,(2.28)

∂tB1†

mB1

m=−2Γ(B1†

mB1

m−B2†B2) + 2Im(¯

M∗

m,2B2†B1

mc+¯

MmG†B1

mc†),(2.29)

∂tB2†B2=−4ΓB2†B2+ 2 X

Im¯

Mm,2B1†

mB2c†.(2.30)

A Born-approximation in this order of electron-light coupling element leads to a photon

density driven inversion. Since the intensity of the driving field is expressed via the photon

density and is chosen to be equal for different light sources, differences in the photon statis-

tics do not enter in the present order of the electron-light coupling element. Eq. (A.2) and

(A.3) include spontaneous emission and couple to photon-assisted densities, which are given

in the App. A.1 with Eq. (A.4) - (A.5)]. A characteristic quantity of Förster-coupled nanos-

tructures is a two-photon polarisation G†B2c†c†, which enter in higher order photon-

assisted polarisation [see Eq. (A.7)]. On this level, also the photon statistics of the incident

radiation occurs via its characteristic g(2) value, including bunching or anti-bunching fea-

tures. In the fourth-order of the light-coupling element, the set of differential equations is

closed via the Born-Approximation [RBSK07]: B1†

1B1

1c†c†cc=B1†

1B1

1c†c†cc. Here,

the photon correlation c†c†ccis represented by g(2) =c†c†cc/c†c2[see Eq. (2.21)]:

replacing c†c†ccwith g(2)c†c2statistical properties are implemented via the choice of

2 Quantum optics in the equation of motion approach: The factorization problem

g(2). After these approximations and substitution and neglecting quantities, which couple

to fifth order in the light-coupling, hence Eq. (A.9) and (A.8) read:

∂tG†B1

nc†c†c=−iω1

n−ω−iΓG†B1

mc†c†c−2iX

M∗

lB1†

lB1

nc†c(2.31)

+ig(2)c†c2 ¯

M∗

nG†G−X

M∗

lB1†

lB1

n!+i¯

Mn,2G†B2c†c†,

∂tB1†

mB2c†c†c=−iω2−ω1

m−ωk−3iΓB1†

mB2c†c†c−2i¯

M∗

m,2B2†B2c†c

−ig(2)c†c2 ¯

M∗

m,2B2†B2−X

M∗

l,2B1†

mB1

l.!(2.32)

The set of differential equations is closed. In Eqs. (2.31) and (2.32), typical quantum

optical effects are visible like the spontaneous emission, depending in the fourth order only

on photon density-assisted exciton and biexciton densities. Also the photon-density driven

inversion, responsible for induced-absorption and emission processes, can be recognized,

second line in Eq. (2.32). Furthermore, characteristic for Förster-coupled nanostructures,

excitation transfer from, e.g. quantum dot l to quantum n influences the system dynamics.

2.4.2 Photon statistics as a probe for exciton correlations

The system of differential equations [Eq. (2.30) - (A.9)] is numerically solved. Initially, the

electronic system is in the ground state G†G= 1, all other quantities are zero. Light

sources with different g(2), but having an equal mean photon number c†care chosen

to discuss the g(2)-dependence of the excitation efficiency. The g(2)-function differs in the

case of τ= 0 for different statistics (thermal light: g(2) = 2, coherent light: g(2) = 1,

squeezed light: g(2) = 0.2) [UGA+07]. Since a single mode theory is derived, the station-

ary excitation of a constant flux of photons c†cbalances the loss of photons due to

spontaneous emission into the dissipative modes. The excitation strength is determined by

the dipole moment Mmand photon number. It is kept within the validity of the applied

Born approximation, which can be deduced by an independent evaluation of the JCM for

a single exciton, cf. Sec. 2.2. The excitation frequency is chosen to be resonant with one

ground-state to single-exciton-transition [see Fig. 2.8], which is approximately half of the

ground-state to biexciton-transition. Due to the Förster coupling, there is always an energy

splitting between the exciton frequencies. The ground-exciton transitions cannot be both

in resonance, cf. the absorption spectrum in Fig. 2.10(left,inset). Fig. 2.10(left), shows the

exciton density of the lowest optical transition, the photon flux is adiabatically switched

on. The optically generated exciton density built-up is plotted for three values of g(2). The

exciton density for squeezed and coherent excitation is higher than for thermal excitation.

Thus, a small value for g(2) is advantageous, if excitons are created. This occurs, since only

one photon is absorbed to create an exciton and a part of the bunched photons passes the

electronic system unused. In this case, it is more effective to excite with anti-bunched light,

when photons arrive successively. However, the quantitative influence of the photon statis-

tics remains small. This changes drastically for the exciton of higher excitonic complexes.

In Fig. 2.10(right), the biexciton density for different g(2) is plotted. Here, thermal excita-

tion is clearly advantageous for creating biexciton densities. The influence of the photon

2 Quantum optics in the equation of motion approach: The factorization problem

024

t [ns]

0.01

0.02

0.03

0.04

Exciton Density

-0.2 0 0.2

Detuning [meV]

Absorption

[arb.units]

0 2 4

t [ns]

0.0

0.5

1.0

1.5

2.0

Biexciton Density [x 108 ]

Figure 2.10: Left: Single-exciton dynamics for stationary excitation with different values for

g(2). (solid curve: g(2) = 1, coherent excitation; dashed curve: g(2) = 2, thermal excitation;

dotted curve: g(2) = 0.2, squeezed excitation). Right: Biexciton dynamics for stationary

excitation with different values for g(2). (solid curve: g(2) = 1, coherent excitation; dashed

curve: thermal excitation g(2) = 2; dotted curve: squeezed excitation g(2) = 0.2).

statistics has an opposite effect compared with excitonic excitations: Biexcitons creation

is a two-photon-process, thus more susceptible to photon-correlations. Compared to the

single-exciton dynamics in Fig. 2.10(left), where anti-bunched light is advantageous for the

creation of excitons, multi-photon-processes, such as biexciton generation, e.g. B2†Gcc,

favor bunched photons, i.e. simultaneous absorption of photon pairs. Therefore, biexcitonic

levels are populated stronger for thermal excitation. Next, the excitation-dependence of the

quantum optical excitation-scheme is studied. In Fig. 2.11(left), the exciton densities are

0500 1000

Photon Number [N]

0.0

5.0×10-3

1.0×10-2

1.5×10-2

2.0×10-2

Exciton Densities

0 500 1000

-2

∆B1 [%]

0500 1000

Photon Number [N]

2×10-10

4×10-10

6×10-10

Biexciton Densities

0 500 1000

-40

-20

∆B 2 [%]

Figure 2.11: Left: Exciton density and percental exciton density differences as a function

of the excitation strength. The coherently driven exciton-density (solid curve) is taken as

a benchmark (dashed curve: ∆B1

th for thermal excitation; dotted curve: ∆B1

sq for squeezed

excitation). Right: Biexciton density and percental biexciton density differences as a func-

tion of the excitation strength. The coherently driven biexciton-density (solid curve) is

taken as a benchmark (dashed curve: ∆B2

th for thermal excitation; dotted curve: ∆B2

sq for

squeezed excitation).

2 Quantum optics in the equation of motion approach: The factorization problem

plotted as a function of the excitation strength c†c=n(mean photon number). Also the

percental deviation from coherent (laser g(2) = 1) defined by:

∆B1

th/sq =B1†

mB1

mcoh −B1†

mB1

mth/sq

B1†

mB1

mcoh ·100 (2.33)

is shown. The exciton density deviation depends linearly on the mean photon number of

the photon flux. For weak pumping, the photon-statistical impact on the creation of single-

excitons can be neglected. The calculations confirm that for single-excitons different values

of g(2) are negligible, unless the excitation strength is high enough to populate many particle

correlations: G†B1†

mc†c†c,B1†

mB2c†c†cin the fourth order of the light-coupling element.

With increasing excitation strength, the difference becomes measurable in a typical pump-

probe experiment [HK04; AHZ97]. In this case, the theory predicts differences up to 3%

(not shown). For comparison, in Fig. 2.11(right), the more sensitive biexciton densities and

its deviation from coherent excitation (∆B2) is plotted:

∆B2

th/sq =B2†B2coh −B2†B2th/sq

B2†B2coh ·100.(2.34)

Here, photon statistics show a large impact: up to 40% deviation in populating the biexci-

ton density can be obtained, rather independently of the excitation strength. Furthermore,

a qualitative change cp. to Fig. 2.11(left) can be seen: thermal excitation populates biex-

citons stronger, thus the deviation is negative, in contrast to the case, when creation of

single-excitons is investigated and deviation is positive, showing an inversed behavior with

respect to exciton generation. With higher excitation strength only the absolute value of

the deviation increases, but not the relative.

The presented results suggest strong, qualitative differences of excitonic and biexcitonic be-

havior on the photon statistics of the incident light. The photon statistics takes advantage

of different aspects of the material excitation and may be used for possible optical device,

detecting absorption of different many particle species to measure the photon statistics

directly. However, the calculation were done in the stationary, weakly coupled regime. No

Rabi oscillations occur. In this limit, the Born factorization and the cluster expansion are

valid approaches for calculations as presented.

To go beyond the weak coupling regime, exact models are preferable. In the next chapter,

an induction model is introduced, which reproduces important exact solutions numerically

up to an arbitrary accuracy in the case of a fixed electron number, cf. Chap. 3. If the num-

ber of electrons is not constant, another approach is taken to reproduce the JCM in the

equation of motion approach by expanding the expectation values in photon probabilities,

cf. Chap. 4

3 Quantum dot cavity quantum electrodynamics

3.1 Mathematical induction approach

In this section, the QD cavity-QED within the

Figure 3.1: GaAs/InAs-QD [MGJ01].

equation of motion approach is described by

a mathematical induction approach. Based on

the factorization problems of the cluster ex-

pansion in the single-photon or strongly cor-

relation limit, cf. Chap. 2, a theoretical frame-

work is derived to treat strongly coupled quan-

tum correlations up to an arbitrary accuracy

in a non-Markovian approach. This novel ap-

proach allows one to include more interactions

without any factorization, e.g. the electron-

phonon interaction, or a classical pump field,

or Coulomb contribution in the case of two or

more electrons in the QD. If a set of general

equations of motion for the involved quantum

correlations is found, the induction method is

limited solely by numerical means, as long as

the number of electrons in the QD is fixed. In small and deep QDs, this is a good approx-

imation, since the WL electrons are energetically well separated from the QD states and

the QD is considered to be atom-like [SGB99a; Sti01]. For shallow QDs, the WL carriers

easily scatter into the QD states and a fixed number of electrons cannot be assumed, cf.

Chap. 4.

This section is organized as follows. First, the Hamiltonian is discussed in Sec. 3.1.1, in-

cluding QD specific properties. In Sec. 3.1.2, a general set of equations of motion is derived

and discussed, including all occurring quantum correlations. To benchmark this induction

approach, its results are compared with the JCM and the independent Boson solution in

Sec. 3.2. Then, the LO-phonon impact on the QD cavity-QED is evaluated with respect

to the absorption spectrum and LO-phonon assisted cavity feeding in Sec. 3.3 and the

remarkable feature, that a classical cavity field is transformed into a non-classical via the

LO-phonon interaction, i.e. LO-phonon induced anti-bunching in Sec. 3.4.

3.1.1 The Hamiltonian

Focusing on a QD strongly coupled to a microcavity within a semiconductor bulk mate-

rial, the Heisenberg equation of motion approach is chosen to calculate the full quantum

kinetics of quantum optical observables of interest. The QD is populated with one electron

and treated as a two level system with one valence band state with energy ~ωvand one

conduction band state with energy ~ωc, e.g., InAs/GaAs-QD with a base length of smaller

than 13 nm [SGB99a] or free-standing QD fabricated in the heterogeneous droplet epitaxy

3 Quantum dot cavity quantum electrodynamics

method [SMO+02]. For larger QD, where the one electron assumption is not valid anymore,

other approaches become necessary [RCSK09b]. The QD band gap energy ~ωcv := ~ωc−~ωv

and the wave functions are calculated in the effective mass approximation [HK04]. The QD

interacts with the classical pump field Ω(t), the effective single cavity mode ω0and with

LO-phonons of the semiconductor bulk material, cf. Fig 3.2. The LO-phonon frequency is

ωLO and a constant dispersion is assumed (Einstein model) [MZ07]. The total Hamiltonian

of the system reads in rotating-wave and dipole approximation [KK08; HK04]:

H=~ωva†

vav+~ωca†

cac−~Ω(t)(a†

vac+a†

cav)

+~ω0c†c−~M(a†

vacc†+a†

cavc)(3.1)

+~X

ωLOb†

qbq+a†

cacgc

qbq+gc∗

qb†

q+a†

vavgv

qbq+gv∗

qb†

q,

where a(†)denotes the fermionic annihilation (creation) operators for the electron in the

quantum dot and the bosonic annihilation (creation) operators c(†)for photons and b(†)

qfor

LO-phonons in the mode q. Electron and photons interact via the electron-light coupling-

element M, which depends on the geometry, size and position of the quantum dot in

the semiconductor microcavity [RSL+04; TPT06; Hoh10]. The electron-phonon interaction

is given with the exciton-phonon Fröhlich-coupling element g(c,v)

q[MZ07; KAK02]. The

modification of the bulk phonon modes due to the geometry and size of the microcavity

is assumed to be small and is therefore neglected [WLW09]. Due to the large band gap

frequency ωcv in comparison to the actual phonon energies in GaAs, non-diagonal electron-

phonon contribution are also not included. A semi-classical electron-light interaction is

Figure 3.2: Scheme of the semiconductor quantum dot cavity-QED. The quantum dot is

treated as a two-level system with one valence vand one conduction band state c. The

electron a(†)

v/c in the quantum dot interacts with LO-phonons b(†)

qfrom the semiconductor

bulk material via the electron-phonon coupling element g(v,c)

qand with the photons c(†)

of the cavity via the electron-photon coupling element M. A phenomenologically included

cavity loss κis considered, as well as a classical pump field Ω(t).

3 Quantum dot cavity quantum electrodynamics

included, assuming an excitation of the carriers in the QD by an external laser field.

The corresponding interaction strength is determined by Ω(t) = ME(t)/~with E(t) =

E(t) cos(ωlt), oscillating with ωland including a time dependent amplitude ˜

E(t).

In this investigation, all parameters are chosen for an InAs/GaAs-quantum dot in a GaAs

semiconductor bulk material, cf. App. A.2 for the list of numerical parameters.

3.1.2 General equations of motion

A general set of equations is derived with the Heisenberg equation of motion −i~∂tA=

[H, A], where Ais an arbitrary operator. For example, the excited state (a†

cac) dynamics

reads:

∂ta†

cac= 2 ImMa†

vacc†+ Ω(t)a†

vac.(3.2)

This equation couples to the photon-assisted polarization via the electron-photon coupling

element Mand to the polarization via the classical pump field Ω(t). The photon-assisted

polarization is given by:

∂ta†

vacc†=−iωcv −ω−iκ)a†

vacc†−iΩ(t)a†

cacc†−a†

vavc†(3.3)

−i M a†

cac−i M a†

cacc†c−a†

vavc†c

−iX

(gc

q−gv

q)a†

vacc†b−iX

(gc∗

q−gv∗

q)a†

vacc†b†.

To model a realistic microcavity, a cavity-loss is introduced phenomenologically, which is

assumed to be κ= 10 µeV [RSL+04]. The photon-assisted polarization couples again to

higher order photon-assisted quantities such as a†

cacc†c. Additionally, the LO-phonon

operators enter into the dynamics. The hierarchy problem is obvious. Here, a numerically

exactly solvabel model is introduced in order to calculate the LO-phonon quantum cavity-

QED for an initially fixed number of electrons in the QD [CRCK10].

In the first step, a general set of equations is derived via mathematical induction. It is not

necessary, to calculate all interactions at once. Applying the product rule for phonon- and

photon assisted quantities, the interaction adds up:

∂ta†

cacc†cb†

qbq=∂ta†

cacc†cb†

qbq+c†c∂ta†

cacb†

qbq.(3.4)

Therefore, the general equations of motion are derived separately for the phonon- and

photon-assisted quantities and added afterwards. Here, it is particularly convenient, since

the electron-photon coupling is non-diagonal, whereas the electron-phonon coupling is diag-

onal in the Hamiltonian. The derivation of a general set of equations of motion is straight-

forward, and proven via mathematical induction. To illustrate the derivation, an example is

given: the generalized equation of motion for photon-correlations. The equation of motion

3 Quantum dot cavity quantum electrodynamics

of the photon-density, or first-order photon correlation, and intensity-intensity correlation,

or second-order photon correlation read:

∂t(c†c) = iM a†

vacc†−a†

cavc,

∂t(c†c†cc) = 2 iM a†

vacc†c†c−a†

cavc†cc.

In this particularly simple example, the general equation of motion is easy to guess with p

an integer:

∂t(c†pcp) = p iM a†

vacc†pcp−1−a†

cavc†p−1cp.(3.5)

It follows from the commutator algebra [A, F(B)] = [A, B]F′(B)a convenient rule [CT99;

AHK99], which is applied to arbitrary powers of the photon-operators:

∂tc†p=∂tF(c†) = −iM[a†

vacc†+a†

cavc, F(c†)] = −iMa†

cav[c, c†]F′(c†) = −ipMa†

cavc†p−1,

and Eq. (3.5) is proven and is valid for the hermetian conjugate as well. In this illustrated

derivation, the combined electron-phonon and electron-photon dynamics are computed and

expressed in a general set of equations of motion. For convenience, some abbreviations are

introduced, taken into account the constant dispersion of the LO-phonons: ω(q) = ωLO

[KAK02; MZ07]. Since the free energy rotation is the same for every wave number q, it is

possible to write:

∂tX

bq|free =−iX

ωqbq=−i ωLO X

bq,(3.6)

defining:

b:= X

(gc

q−gv

q)bq.(3.7)

Now, only the number of phonon operators is of importance and the set of equations

is simplified for matters of clarity. In the one electron and two level assumption, only

three types of fermionic states are possible: Gis the ground, Ethe excited state and T

the transition from ground to excited state. All three of them can be photon(p, s)- and

phonon(m, n)-assisted with (p, s, m, and nintegers):

Gp,s

m,n := a†

vavc†pcs¯

b†m¯

bn,(3.8)

Ep,s

m,n := a†

cacc†pcs¯

b†m¯

bn,(3.9)

Tp,s

m,n := a†

vacc†pcs¯

b†m¯

bn.(3.10)

The Heisenberg equation of motion leads to a general set of equations of motion for different

orders in electron-photon and electron-phonon coupling elements by raising or lowering the

3 Quantum dot cavity quantum electrodynamics

indices p, s, m, n. An observable of interest to calculate the absorption is the microscopic

polarization. The dynamics are calculated via

∂tTp,s

m,n=(3.11)

=−i[ωcv −(p−s)ω0−(m−n)ωLO −i(p+s)κ−iγ]Tp,s

m,n

−ip MEp−1,s

m,n −iM(Ep,s+1

m,n −Gp,s+1

m,n )−iΩ(t)Ep,s

m,n−Gp,s

m,n

−iTp,s

m,n+1−iTp,s

m+1,n+i m gvTp,s

m−1,n−i n gcTp,s

m,n−1,

where gi=Pqgi

q(gc∗

q−gv∗

q)for i=v, c. A phenomenological pure dephasing γis included

and takes into account, e.g. longitudinal acoustical (LA) phonon interaction. A cavity loss

considers a finite cavity photon life time: κ. The microscopic polarization is driven via

spontaneous emission of photons / phonons, induced emission and absorption of photons

and modulated by LO-phonon assisted higher-order polarization terms. The classical pump

field is time-dependent, and can also be assumed as probe to calculate absorption spectra

in the linear optics regime, cf. Sec. 3.3. The photon- and phonon assisted ground and

excited state densities are source terms of the dynamics. The equation of the photon- and

phonon-assisted ground state reads:

∂tGp,s

m,n=(3.12)

=i[(m−n)ωLO + (p−s)ω0+i(p+s)κ]Gp,s

m,n

+i MTp+1,s

m,n −i MTs+1,p

n,m ∗+i s MTp,s−1

m,n −i p MTs,p−1

n,m ∗

+iΩ(t)(Tp,s

m,n−Ts,p

n,m∗) + i m gvGp,s

m−1,n−i n g∗

vGp,s

m,n−1

In the same way, spontaneous emission and absorption processes of LO-phonons are in-

cluded. The dynamics of the excited state is given by:

∂tEp,s

m,n=(3.13)

=i[(m−n)ωLO + (p−s)ω0+i(p+s)κ]Ep,s

m,n

−i MTp+1,s

m,n +i MTs+1,p

n,m ∗

−iΩ(t)(Tp,s

m,n−Ts,p

n,m∗) + i m gcEp,s

m−1,n−i n g∗

cEp,s

m,n−1.

The excited state and ground state density differ strongly: a typical quantum optical feature

[CSS75; SZ97]. The ground state couples to the higher and lower photon number state, but

the excited state only to the higher. This asymmetry is lost in the cluster expansion, where

the factorized excited and ground state density are determined via the same equations,

only with a reversed sign ∂ta†

cacc†cc=−∂ta†

vavc†cc, cf. Eq. (2.15). In this context, it is

worth noting, that purely photonic c†pcpor purely phononic expectation values ¯

b†p¯

bp

do not appear in Eq. (3.11) - (3.13), indicating the full quantum correlation of the combined

electron, LO-phonon and photon dynamics.

Since this set of equations is not closed in terms of n, m, p, s, many orders have to be

calculated to reach convergence. The number of orders depend on the initial conditions, as

well as the strength of the implemented parameters such as the coupling strengths of gv/c

and M. However, this set of equations of motion models the exact solution numerically to

an arbitrary accuracy, cf. Fig. 3.3.

3 Quantum dot cavity quantum electrodynamics

Figure 3.3: Scheme of general equations of motion within the induction method without

classical pumping.

3.1.3 Initial conditions

A wide range of initial conditions can be chosen to calculate the system dynamics. The

expectation values at t0= 0 are determined by the following system parameters: the

photon statistics and the mean number of photons in the cavity, the temperature of the

LO-phonon bath and the initial electronic state. In general, the initial conditions for a

cavity field prepared in a Fock or thermal state read:

Ep,s

n,m(t0) = ρE

0ρpt

pρph

nδm,n δp,s ,(3.14)

Gp,s

n,m(t0) = ρG

0ρpt

pρph

nδm,n δp,s ,

Tp,s

n,m(t0) = ρT

0ρpt

pρph

nδm,n δp,s ,

with ρE

0:= E0,0

0,0(t0) = a†

cac(t0),ρG

0:= G0,0

0,0(t0) = a†

vav(t0)and ρT

0:= T0,0

0,0(0) =

a†

vac(t0)denoting the probability to find the electron in the excited Eor in the ground

state Gat time t0= 0 or with an initially assumed transition probability T.

The photon statistics enters in ρpt

p:= c†pcp. The expectation value for p > 1differs

for coherent, thermal and non-classical states strongly [Lou83; CKR09]. Different initial

conditions in dependence on the chosen photon-statistics are discussed in Sec. 3.2.3.

Initially, the LO-phonon bath is in thermal equilibrium. Using Wick’s theorem [Mah90] and

including the q-sum of the different wave numbers of the LO-phonon operators, arbitrary

orders in the pure phonon expectation value read:

ρph

n:= ¯

b†m¯

bnδmn =n! X

q|gc

q−gv

q|2!n

¯nn

LO,(3.15)

3 Quantum dot cavity quantum electrodynamics

where the mean number of phonons in the LO-phonon mode ¯nLO is determined by the

Bose-Einstein distribution ¯nLO = [exp(β~ωLO)−1]−1and β= (kBT)−1, as long as the

temperature of the bath is fixed and the bath assumption is valid.

To benchmark the induction model, corresponding initial conditions are chosen to compare

the induction model results with two exact analytical solutions: for the electron-photon in-

teraction with the Jaynes-Cummings model (JCM) solution [JC63; SK93] and for electron-

phonon interaction with the independent boson model (IBM) [Mah90; KAK02].

3.2 Benchmarking the induction model

The mathematical induction approach is not an exact solution in the sense, that an an-

alytical formula describes the full quantum kinetics of the combined electron-photon and

LO-phonon dynamics. Analytical solutions, such as the Jaynes-Cummings model [JC63]

or the independent Boson model [Mah90] provide insights into the system dynamics with-

out further numerical evaluation. The mathematical induction model is a framework, that

generates exact solutions up to an arbitrary accuracy, such as the generating function

approach[AHK99]. To proof the accuracy of this numerical solution scheme, the JCM and

IBM solution are reproduced in this section.

3.2.1 Independent boson model:LO-phonon satellite peaks

In the independent boson model, the optical absorption of a two-level system interacting

with an LO-phonon bath at non-zero temperature can be calculated exactly [KAK02].

In case of LO-phonons in GaAs, the absorption spectrum shows a set of satellite peaks

at positions of a multiple of the LO-phonon energy ~ωLO = 36.4meV. The absorption

spectrum is calculated with Eq. (3.11) and setting the initial value of the polarization to

T0,0

0,0(0) =: ρT

0= 0.5, corresponding to an initial excitation via a δ-pulse, and the ground

state density is G0,0

0,0(0) =: ρG

0= 1.0. A pure dephasing of γp= 2.5ps−1is assumed.

Here, one can apply a phonon operator transformation, to consider an equilibrium with the

ground state (starting with an electron in the valence band, by replacing gc−→ gc−gvand

neglecting the coupling to the valence band state). An interaction with the cavity mode ω0

is not considered. The quantum optical coupling Mis set to zero. The equations of motion

are reduced to the electron-phonon dynamics. By Fourier-transforming the solution of the

time domain into the frequency domain, the absorption is calculated in case of a δ−pulse

with [KAK02]:

α(ω)∝Re"T0,0

0,0(ω)

ρT

0#.(3.16)

In Fig. 3.4(left), the dynamics of the imaginary part of the microscopic polarization T0,0

0,0

is depicted for the first 3ps. A delta-pulse excites the system initially, and the polarization

decays from its initial value due to the pure dephasing in the system, cf. Eq. (3.11). The

dynamics of the polarization exhibits quantum beats with multiples of the LO-phonon

frequency ωLO = 55.3ps−1. Here, the converging order is m, n ≥8.The main resonance

interferes with the satellites peaks to higher and lower energies, modulating the amplitude

of the polarization during the dephasing process. The strength of the modulation depends

3 Quantum dot cavity quantum electrodynamics

012 3

t [ps]

-0.4

-0.2

0.2

0.4

Im [<T >]

-108 -72 -36 0 36 72 108

Energy [meV]

Absorption α

0 0

Figure 3.4: Left: Dynamics of the imaginary part of the polarization. Phonon signatures are

visible in the modulated amplitude of the polarization. Right: The absorption spectrum (log

scale) of the quantum dot shows LO-phonon satellite peaks at multiples of the LO-phonon

energy 36.4meV. A temperature of 300K is chosen and converging order is m, n ≥8.

strongly on the temperature. Here, 300 K are chosen, which leads to multiple LO-phonon

satellite peaks left and right from the QD band gap frequency (shifted to zero plus polaron

shift) in the absorption spectrum, cf. Fig. 3.4(right). The LO-phonon satellite peaks are

broadened, since a pure dephasing is assumed. At exactly a multiple of the LO-phonon

frequency, the satellite peaks appear. Experimentally, LO-phonon peaks in the absorp-

tion spectra are found in measurements, using photoluminescence excitation spectroscopy

[SGB99b; HSB03; HMS+99]. The strength of the satellite peaks depends on the Huang-

Rhys factor of the system. For 0K, the peak heights of the different LO-phonon satellite

peaks follow from a Poissonian distribution [MK00]:

pi=e−FFi

i!(3.17)

for the i−th peak with the dimensionless Huang-Rhys factor: F=∆LO

ωLO , introducing the

Polaron shift: ∆LO =g2

eff

ωLO and g2

eff =Pq|gc

q−gv

q|2. This result is reproduced with the

mathematical induction method. For finite temperatures, the dependence of the satellite

peaks to the zero phonon line peaks is more complicate. The mean phonon number de-

pendence is introduced into the system, and thus induced emission, not only spontaneous

emission of LO-phonons must be included. With increasing temperature, the IBM predicts

higher satellite peaks. The induction method reproduces this result, also. In Fig. 3.5, the

temperature dependence of the peak heights of the LO-phonon satellite peaks is depicted.

For 0K, Eq. (3.17) is exact for the Stokes peaks. At this temperature, only the Stokes peak

appears on the higher energetic side, cf. Fig. 3.5. At elevated temperature, approximately

70 K, the first anti-Stokes peak appears in the absorption spectrum on the lower energetic

side, as well as the weak second Stokes-peak. The temperature dependency is weak due to

the high LO-phonon energy and thus, the small mean phonon number. Also, the effective

phonon coupling determines the peak height: g2

eff. In this value, the size of the quantum dot,

the band gap energy and the effective mass of the electron in the valence and conduction

3 Quantum dot cavity quantum electrodynamics

Figure 3.5: Temperature dependence of the LO-phonon satellite peaks. The quantum dot

band gap frequency including polaron shift is set to zero: ωcv −∆LO = 0.

band state as well as the shape of the QD enters. Hereby, the presented induction model

reproduces the important results of the LO-phonon independent Boson model.

3.2.2 Jaynes-Cummings model (1): Vacuum Rabi splitting

Restricting the system dynamics to the electron-photon interaction, the QD acts as a two-

level quantum emitter within a one mode cavity field [CTDRG89]. In a first step, the

absorption spectrum is derived via Eq. (3.16). Now, the electron-phonon coupling element

is set to zero (gv,c = 0) and, again, a small pure dephasing is assumed. The system is ini-

tially excited with a δ−pulse, resulting in an initial value for the polarization of ρT

0= 0.5.

No photons are inside the cavity and no further losses are included. Therefore, the sys-

tem dynamics is completely determined by two coupled harmonic oscillators, for which the

Jaynes-Cummings model gives the exact solution in the time domain, cf. Sec. 2.2.

Since photons and electrons are strongly coupled, new quasi-particles are formed: polari-

tons, giving rise to new eigen energies of the system. The derivation starts with the Jaynes-

Cummings model Hamiltonian, restricting Eq. (3.2) to the electron-photon interaction, i.e.

Eq. (2.2). The zero point of the energy is chosen to be in between the conduction and va-

lence band state energy: ωc=ωcv

2and ωv=−ωcv

2, cf. Sec 2.2. In the JCM, the dynamics is

divided into subspaces of the Hilbert space, depending on the initially fixed number of pho-

tons involved in the dynamics. The two types of states are: |ΨN

2i=|N, ciwith N-photons

in the cavity and an excited two-level system c, and |ΨN

1i=|N+ 1, viN+1-photons in

the cavity and a two-level system in the ground state v. The Hamiltonian applied to these

states leads to:

H|N+ 1, vi=~ω0(N+ 1) −ωcv|N+ 1, vi−~M√N+ 1|N, ci,(3.18)

H|N, ci=~ω0(N) + ωcv|N, ci−~M√N+ 1|N+ 1, vi.(3.19)

3 Quantum dot cavity quantum electrodynamics

The states |ΨN

1i,|ΨN

2iare obviously no eigenstates of the JCM -Hamiltonian. To calculate

the eigenvalues, each N-part of the Hamiltonian HNis diagonalized, defined by:

HN

ij := hΨN

1|H|ΨN

1i hΨN

1|H|ΨN

hΨN

2|H|ΨN

1i hΨN

2|H|ΨN

2i=~−ωcv/2 + (N+ 1)ω0−M√N+ 1

−M√N+ 1 ωcv/2 + Nω0.(3.20)

The eigenvalues are determined by the solution of the characteristic polynomial and read:

~ωN

+/−=~ω0(N+1

2)±~r(ωcv −ω0)2

4+M2(N+ 1),(3.21)

which results in the Jaynes-Cummings ladder with different rungs and Rabi splitting in

dependence of the photon number N[WRI02; SK93; SKK08]. The vacuum Rabi splitting

(N= 0) and in case of resonance (ωcv =ω0) has the value of ω0

+−ω0

−= 2 Mand is twice

the vacuum Rabi frequency. Due to the electron-photon interaction, the excited state en-

ergy splits up into a dressed state, leading to two resonance frequencies in the system. In

Fig. 3.6(left), the polarization dynamics are plotted. The pure dephasing leads to a decay

of the polarization. After 30 ps, the polariation vanishes. However, the electron-photon

interaction leads to an additional oscillation behavior and modulates the polarization in

dependence of the Rabi frequency of the system. This amplitude modulation is the signa-

ture of the Rabi splitting, depicted in Fig. 3.6(right). The QD band gap frequency is set

0 0.01 0.02 0.03

t [ns]

-0.4

-0.2

0.2

0.4

Im [<T >]

-0.002 0 0.002

Energy [meV]

Absorption α

0 0

Figure 3.6: Left: Dynamics of the imaginary part of the polarization. Photon signatures are

visible in the modulated amplitude of the polarization. Right: The absorption spectrum of

the quantum dot shows the vacuum Rabi splitting in case of N= 0 photons in the cavity

and resonance between the cavity mode and the QD band gap frequency.

to zero. Instead of one Lorentzian peak at ω= 0, two peaks appear, marking the polariton

frequencies in case of N= 0 photons in the cavity. From this kind of spectra, the electron-

photon coupling strength can be determined [KGK+06; WNIA92; YSH+04; MKH08].

In case of a detuning δbetween the QD resonance and the cavity mode, which exceeds the

coupling strength δ≫M, the M-contribution in the square root in Eq. (3.21) is neglibible

and the eigenenergies are again the QD band gap frequency and the cavity frequency, alone.

In this case, two peaks are visible in the spectrum. A large resonant peak of the QD and a

small cavity peak, only weakly driven by the initial delta-pulse. In the transition regime,

3 Quantum dot cavity quantum electrodynamics

when the detuning δ≈M, the polariton state starts to form, which leads close to the

resonance to an anti-crossing between the cavity mode and the QD band gap frequency.

In Fig. 3.7, the QD spectrum is plotted for different detunings (y-axis) of the cavity mode

Figure 3.7: Benchmark: anti-crossing for small detunings between the cavity mode and the

QD band gap frequency: pure dephasing γp= 0.002 fs−1, coupling strength M= 0.5ps−1

and no photons in the cavity N= 0. Detuning ωcv −ω0=δon y-axis, x-axis is the

absorption frequency, the anti-crossing is clearly visible.

from the QD band gap frequency (x-axis). The electron-coupling strength is chosen as

M= 0.5ps−1and no photons are initially in the cavity: N= 0. If the detuning is in the

order of magnitude larger than the electron-photon coupling strength, two peaks appear

in the spectrum, e.g. δ= 5.0fs−1. However, with decreasing detuning, the cavity peak

becomes larger, until the polariton state starts to form and a splitting appears, e.g. δ≈0.

Both, the cavity frequency and the QD band frequency changes. A typical anti-crossing

behavior is predicted by the exact model of Jaynes and Cummings and here, calculated

via the induction method visible in Fig. 3.7.

3.2.3 Jaynes-Cummings model (2): Collapse and revivals

In the JCM, also the time dynamics of the observables are calculated exactly and addition-

ally, different photon-statistics can be computed via a superposition of the general JCM

solution, cf. Sec 2.2. Hence, another benchmark for the induction model is the reproduc-

tion of these photon-statistics induced dynamics, such as the well-known phenomenon of

collapse and revivals in case of an initially coherent superposition of photon states in the

cavity [ENSM80; RWK87; NSME81]. The system is closed, as assumed in the JCM. No

dissipative processes, e.g. cavity losses or pure dephasing are considered: κ=γp= 0 µeV.

The QD is at t0= 0 in the excited state (E0,0

0,0(t0) = 1.0). In dependence on the initial

preparation of the cavity field, the Rabi oscillations differ strongly. The photon field can

3 Quantum dot cavity quantum electrodynamics

be prepared in the Fock state (a), thermal state (b) or in the coherent state (c).

The initial conditions for a photon field in the Fock state read:

ρpt

p=N!

(N−p)! for p≤N,

0for p > N, (3.22)

where Nis the number of photons in the cavity mode and pthe order of the photon

correlation, e.g. ρpt

2=c†c†cc=N(N−1) for p= 2. The expectation value is zero for all

p, which exceed N. In the Fock state, the mean photon number is exactly known, leading

to a complete uncertainty in the phase relation of the photons. In this perspective, the

Fock state is an extremely squeezed photon state and maximally non-classical [MW95]. In

Figure 3.8: The excited state dynamics for different photon-statistics of the cavity mode:

(a) Fock state N= 0, (b) thermal state N= 0.4and (c) coherent state N= 8, calcu-

lated within the induction model. In case of vanishing electron-phonon interaction, the

induction model reproduces the Jaynes-Cummings solution, not distinguishable in the plot

(converging order: p, s ≥60 for coherent and thermal states).

Fig. 3.8(a), the solution of the excited state dynamics (E0,0

0,0(t)) is plotted for a cavity

field prepared in the Fock state. Rabi oscillations are visible with a Rabi frequency of

ΩN=M√N+ 1. The quantized Rabi frequency includes the spontaneous emission of

photons, which is not included in the semi-classical Rabi frequency. Even if there are no

photons in the cavity, Rabi oscillations occur, as long as the electron is initially in the

excited state. In Fig. 3.8(a), Nis set 0to prove this remarkable result of the JCM [JC63;

SK93]. Vacuum Rabi oscillations are experimentally accessible in various experimental

3 Quantum dot cavity quantum electrodynamics

setups [BSKM+96; KRM+10]. The vacuum Rabi oscillation case is chosen here, to consider

the strongest possible quantum correlation between the QD and the cavity mode.

If the photon field is prepared in a thermal state, the mean photon number is calculated

with the Bose-Einstein statistics. The initial value is determined by Wick’s theorem. Like

in the case of the LO-phonons, cf. Sec. 3.1.3, the result reads:

ρpt

p=p! (N)p,(3.23)

with N= [exp(β~ω)−1]−1and β= (kBT)−1. In Fig. 3.8(b), irregular oscillations of the

excited state are observable (N= 0.1). A pattern is not visible and not to be expected.

Different Rabi frequencies interfere with each other. The induction model reproduces this

complicated oscillation pattern, predicted by the JCM [SZ97; SK93]. The thermal statistics

needs very high-orders in the electron-photon coupling element. The photon-correlations

increase fast and high numbers exceed easily typically numerical number formats. The

solution, depicted in Fig. 3.8(b), takes into account orders up to p= 60. In principle, the

solution cannot converge for very high N. Higher orders in the photon field are driven very

strongly and all orders exist with a large N, cf. Eq. (3.23). However, observables such as

the photon density or the g(2)(t, 0)-function, as well as the electron densities depend on

these higher order correlations only very weakly due to the increasing order in the electron-

photon coupling. Due to this fact, the solution converges and the limits are imposed only

by the numerical implementation and the computer facilities.

Finally, the photon field can be prepared in the coherent state. Glauber-states [Gla63] are

assumed. The corresponding expectation values read:

ρpt

p=X

m,n

Nm+n

2e−N

√m!n!hn|c†pcp|mi=Np.(3.24)

The remarkable phenomenon of periodic collapses and revivals are seen in Fig. 3.8(c) for

N= 8 [ENSM80]. Collapses and revivals are experimentally seen in (Ref. [RWK87]). This

phenomenon results from an interference between different orders in the photon-correlation

and is extremely sensible to deviations from the exact solution of the JCM [Rob03]. The

solution, depicted in Fig. 3.8(c), takes into account orders up to p≥60. If the dynamics

is calculated for longer times t≥20 ns, higher order in pneed to be calculated, since

higher orders of the quantum correlations enter into the observable dynamics. However,

the solution of the JCM (not distinguishable in the plot) is reproduced and proves that

the induction model contains the features of a coherently-driven JCM.

The results in Fig. 3.8(a),(b) and (c) are compared with the analytic and numerical solu-

tions of the JCM [SK93] and are found to be in complete agreement. Thus, equations (3.12)

- (3.13) reproduce the JCM in the equation of motion approach, if the electron-phonon

coupling is set to zero.

3.2.4 Jaynes-Cummings model (3): Analytical solution

Concluding the benchmark of the induction model regarding the electron-photon interac-

tion, an analytical solution of the excited state density, known from the JCM, is derived to

3 Quantum dot cavity quantum electrodynamics

clarify the induction model without a numerical evaluation. The vacuum Rabi oscillation

of the excited state density, depicted in Fig. 3.8(a), reads:

E0,0

0,0(t) = cos2(M t).

The Rabi splitting is proportional to the Rabi frequency, here the Rabi frequency is identi-

cal with the electron-light coupling element, since initially there are no photons in the cavity

N= 0. The equations of motion, needed to derive Eq. (3.25), read with 1 = a†

cac+a†

vav:

∂tE0,0

0,0= 2Im hMT1,0

0,0i(3.25)

∂tT1,0

0,0=−i ME0,0

0,0−i M(2E1,1

0,0−c†c)(3.26)

∂tc†c=−2Im hMT1,0

0,0i=−∂tE0,0

0,0.(3.27)

Now, the photon-assisted excited state density is not driven in case of vacuum Rabi os-

cillations, and the set of equations of motion is closed. For convenience, the equations

are solved for the difference between the photon denstiy and the excited state density

(W=c†c−a†

cac):

∂t(c†c−E0,0

0,0) = ∂tW=−4ImhMT1,0

0,0i(3.28)

∂tT1,0

0,0=i M(c†c−E0,0

0,0) = i M W (3.29)

→∂2

tW=−(2 M)2W. (3.30)

This simple differential equation is solved via a cosine ansatz, bearing in mind the initial

condition of W(0) = −1:

W(t) = c†c(t)−E0,0

0,0(t) = −cos(2 M t) = −cos2(Mt) + sin2(Mt),(3.31)

which gives the solution of the inversion, but also for the excited state dynamics. The

initial conditions verify: E0,0

0,0(t) = cos2(M t). The Rabi frequency is proportional to M,

proving, that the induction model includes the JCM solution analytically. However, higher

order in this truncation scheme are difficult to derive, and numerical evaluations become

necessary.

3 Quantum dot cavity quantum electrodynamics

3.3 LO-phonon QD cavity-QED: LO-phonon assisted cavity

feeding

Cavity quantum electrodynamics (CQED) studies the interaction between a single radiation-

field mode and a quantum emitter, here a semiconductor quantum dot (QD). The pursuit

of QD-CQED is motivated by the possibility to tailor the quantum emission properties,

e.g. the frequency and photon-statistics, within ultrasmall volumes [KGK+06; BSL+09].

Semiconductor technology paves the way for the design of monolithic structures, such as

micropillars, microdisks or photonic crystal nanocavities [MKB+00; KGK+06; BSL+09].

In these systems, typical cavity-QED phenomena are demonstrated, e.g. Vacuum Rabi

splitting and Purcell enhancement [LVT08], entanglement [TPT06; HPS07] and photon

antibunching [MKB+00; WKU+05]. These observations show that quantum information

tasks are achievable and technological applications are on the verge of realization like single

and entangled photon sources or low threshold nanolasers.

Recent studies of QD-CQED show intense cavity emission, and even lasing, when the cavity

mode is not in resonance with the QD. This cavity feeding is disussed widely in the litera-

ture and can be attributed to the solid-state environment of the QD [TKH+09; SDG+09].

Experimental and theoretical work confirm that longitudinal acoustical (LA) phonons play

a crucial role for small detunings (few meV) and quasi-excitonic transitions for larger detun-

ings [MKH08; Hoh10; WVT+09; TS10]. In case of large detunings, phonons are typically

not taken into account as a feeding channel.

In this section, the LO-phonon interaction comes into focus and is investigated within the

induction model approach to simulate the strong coupling signatures for an InAs/GaAs-

quantum dot in a microcavity for different detunings, in particular for high temperatures.

Strong coupling signatures become visible, if the cavity is one LO-phonon energy detuned

from the QD transition energy. A clear anti-crossing is visible for the Stokes position, the

anti-Stokes anti-crossing is weaker but still noticeable at 300 K.

The calculation is based on the general set of equations, cf. Eq. (3.11)-(3.13), derived by the

mathematical induction, cf. Sec. 3.1 with given initial conditions. To investigate the impact

of the LO-phonons, modified Rabi oscillations of the excited state density are plotted and

an analytical expression for the Rabi frequency is derived. First, the initial conditions need

to be discussed to set up the general set of equations of motion, adjusted to the initial

conditions.

3.3.1 Initial conditions

The absorption spectrum is calculated with Eq. (3.11) by including phenomenologically

a pure dephasing γ= 2 (ps)−1. The system is excited with a delta pulse Ωδ= Ω(t)δ(t),

creating an initial value of the polarization to T0,0

0,0(0) =: ρT

0= 0.5and the ground state

density is G0,0

0,0(0) =: ρG

0= 1.0. The delta pulse creates a polarization on a time scale, that

the electronic system and the LO-phonon bath remains in equilibrium at t= 0. Material

characteristics are implemented in the choice of initial conditions, here, in the calculation

of the normalized effective form factor |gc

q−gv

q|and the wave number (q) dependency,

as well as the electron-photon light coupling element M. The LO-phonons are initially in

equilibrium with the ground state density, which results in an operator transformation, i.e.

gv=gv∗= 0 and gc−→ gc−gv, cf. 3.2.1. The LO-phonon bath is assumed to be initially

3 Quantum dot cavity quantum electrodynamics

in thermal equilibrium, where the mean number of phonons in the LO-Mode ¯nLO is given

with the Bose-Einstein statistics, as long as the temperature of the bath is fixed. For t > 0,

the LO-phonon interaction is included beyond the bath assumption, since electrons and

phonons are not in equilibrium anymore and non-Markovian contributions are of interest.

3.3.2 Equations of Motion

The observable of interest to calculate the absorption is the microscopic polarization. The

dynamics is calculated via

∂tTp,s

m,n=−i[ωcv −(p−s)ω0−(m−n)ωLO −i(p+s)κ−iγ]Tp,s

m,n(3.32)

−ip M∗Ep−1,s

m,n −iM∗(Ep,s+1

m,n −Gp,s+1

m,n )

−iδp

0δs

0δm

0δn

0Ωδ−iTp,s

m,n+1−iTp,s

m+1,n−i n g2

eff Tp,s

m,n−1.

A phenomenological pure dephasing γis included and takes into account the LA-phonon

interaction. A cavity loss considers a finite photon life time: κ. The equations of the photon-

and phonon-assisted ground state and excited state read:

∂tGp,s

m,n=i[(m−n)ωLO + (p−s)ω0+i(p+s)κ]Gp,s

m,n(3.33)

+i MTp+1,s

m,n −i MTs+1,p

n,m ∗+i s MTp,s−1

m,n −i p M∗Ts,p−1

n,m ∗,

∂tEp,s

m,n=i[(m−n)ωLO + (p−s)ω0+i(p+s)κ]Ep,s

m,n(3.34)

−i MTp+1,s

m,n +i MTs+1,p

n,m ∗+i m g2

effEp,s

m−1,n−i n g2

effEp,s

m,n−1.

Via Eq. (3.32)-(3.34), the LO-phonon assisted cavity feeding can be calculated. Note, the

ground state density does not couple to the LO-phonons due to the initial conditions and

the applied operator transformation. The δ−pulse is included only in the polarization for

p=s=n=m= 0, which corresponds to the linear excitation regime.

3.3.3 LO-phonon QD cavity-QED: Optical absorption spectrum

In Fig. 3.9, the full absorption spectrum α(ω, ∆) of the LO-phonon QD-CQED is depicted

for a temperature of 300 K with ωas the Fourier transform frequency and the detuning

between the QD and the cavity mode: ∆ = ωcv −(ω0−∆LO). As band gap frequency

is 1.5eV chosen [Sti01]. Due to the polaron shift (∆LO =g2

eff

ωLO ), the main peak is shifted

compared to the case without LO-phonons. For zero detuning ∆ = 0, the cavity mode and

the band gap frequency are on resonance and an anti-crossing is visible, which depends

on the light coupling strength M, cf. Sec. 3.2.2. The anti-crossing is a signature for strong

coupling between the QD and the cavity mode [LVT09; VLT09]. In time domain, Rabi

oscillations occur (not shown). Their frequency depends linearly on the coupling strength.

The higher the Rabi frequency, the larger is the Rabi splitting of the QD peak. For minor

detunings, longitudinal acoustical (LA) phonon assisted cavity feeding occurs, but is not

considered in this investigation [TS10; Hoh10; SDG+09]. The mismatch between cavity

mode and QD frequency is compensated through LA-phonon emission or absorption. This

mechanism secures strong coupling features for small detunings, in order of typical longitu-

dinal acoustical phonon frequency (few meV). It is not incorporated here, since it is focused

on larger detunings, a typical longitudinal optical phonon frequency in GaAs (36,4meV).

3 Quantum dot cavity quantum electrodynamics

Figure 3.9: Optical absorption spectrum of an LO-phonon QD-cavity system at 300 K.

Three anti-crossings are visible: (1) in case of resonance between band gap frequency and

cavity mode ∆ = 0, which is dominated by the light coupling strength M, (2) in case of a

detuning of ∆ = −ωLO, the Stokes contribution, which depends strongly on the effective

phonon coupling strength and (3) in case of a detuning of ∆ = ωLO, the Anti-Stokes

contribution, only visible for temperatures higher than 70 K, strongly depending on the

mean phonon number.

If the cavity mode is detuned more than 10 meV from the band gap frequency, the Rabi

splitting vanishes [Hoh10]. The spectrum is reduced to the band gap resonance peak and

the cavity peak, cf. Sec. 3.2.2. But when the cavity mode matches an LO-phonon satellite

peak, the anti-crossing reappears. Strongly for the Stokes peak, at a detuning of ∆ = −ωLO,

weaker, but still visible at room temperature, the anti-Stokes contribution at ∆ = ωLO.

The anti-crossing at a detuning position depends strongly on the material parameter, on

the LO-phonon and light coupling strength. In this picture, the strong coupling signature

is transported via an LO-phonon emission (Stokes contribution) or an LO-phonon ab-

sorption (anti-Stokes). Since the mean LO-phonon number at room temperature does not

exceed ¯nLO <0.4for the given LO-phonon energy in GaAs, cf. Fig. 3.14, the anti-Stokes

contribution is small, in contrast to the Stokes-contribution, which is even strong at low

temperatures due to spontaneous emission of LO-phonons.

For different detunings, the Rabi oscillations have a different frequency. In the Jaynes-

Cummings model, which describes the main resonance without phonons, the detuning ap-

pears in a modified Rabi frequency, generalized as Ω = pM2(N+ 1) + ∆2=√M2+ ∆2

for vacuum Rabi flopping N= 0, cf. Sec. 3.2.2 and Eq. (3.21). The detuning leads to higher

Rabi frequency at the price of a reduced oscillation amplitude [SZ97]. Here, the LO-phonon

interaction introduces other resonance frequencies in the system and the Rabi frequency

cannot be described with the generalized formula given in the JCM.

3 Quantum dot cavity quantum electrodynamics

Figure 3.10: Parameter plots for a detuning of ∆ = −ωLO. Left: Rabi splitting for dif-

ferent electron-light coupling strength M(y-axis) for an effective phonon coupling of

geff = 75 ps−2. Right: Rabi splitting for different effective phonon coupling strengths geff

(y-axis) and an electron-light coupling of M= 0.75 ps−1.

By changing the numerical parameter, the dependence of the Rabi splitting on the cou-

pling strength can be studied. In Fig. 3.10(left), the dependence of the Rabi splitting on

the electron-photon strength Mis depicted for a fixed electron-phonon coupling strength.

The cavity and the QD are one LO-phonon energy detuned: ∆ = −ωLO and an effective

electron-phonon coupling strength of geff = 75 ps−2is chosen and in agreement with exper-

imental data [HBG+01]. For a vanishing coupling element M, the splitting also vanishes,

since no interaction can take place. For a coupling element M, which is larger than 1

2γp, the

splitting appears. For further increment of M, the splitting increases also. First linearly,

predicted by the JCM, but the deviation from the linear dependence is visible and caused

by the presence of LO-phonons and LO-phonon assisted cavity feeding. In Fig. 3.10(right).

The dependence of the Rabi splitting on the effective phonon-coupling strength geff is plot-

ted for a electron-light coupling element of M= 0.75 ps−1. If geff = 0, the cavity mode and

the QD are spectrally too far separated and since no feeding via LO-phonon can occur, no

strong coupling signature is visible. The Rabi splitting is zero. For increasing geff, the Rabi

splitting occurs fast and in a non-linear dependence.

Now, cavity feeding via LO-phonons drives the system, but with a decreased oscillator

strength due to the spectral separation. Interestingly, the Rabi splitting saturates slightly,

if only the effective phonon strength of the system is increased. This indicates a complicated

correlation between LO-phonon frequency, LO-phonon coupling strength, electron-photon

coupling and detuning, as well as the temperature dependent LO-phonon mean number.

An approximated functional dependence is derived in the next section via the Huang-Rhys

factor.

3 Quantum dot cavity quantum electrodynamics

3.3.4 Phonon-induced Rabi frequency modification

Now, three cases need to be discussed, as shown in Fig. 3.9 by investigating the vacuum

Rabi oscillation for given detuning and without external light field, i.e. Ωδ≡0. It is as-

sumed that the QD is initially in the excited state, ρE

0= 1 and ρT

0=ρG

0= 0. The equation

of motion set Eq. (3.32)-(3.34) still applies. In Fig. 3.11, the dynamics of the excited state

025 50 75

0.5

025 50 75

0.5

025 50 75

t [ps]

0.5

Excited state density

∆ = 0

∆ = ωLO

∆ = −ωLO

(1)

(2)

(3)

Figure 3.11: Dynamics of the excited state density for different detunings at 3K (black,

dotted line) and at 300 K (orange, solid line). (1) shows the Rabi oscillation for zero

detuning, which amplitude is modulated at 300 K due to the presence of LO-phonons.

(2) depicts the Rabi oscillation for a detuning of one LO-phonon frequency below the QD

frequency. The Rabi oscillation frequency is strongly modified and anharmonic oscillation

behavior is visible at 300 K. In case of a detuning of one LO-phonon frequency above the

QD frequency (3), no Rabi oscillation is visible at 3K. However, at 300 K LO-phonon cavity

feeding takes place and a Rabi oscillation occur, with a reduced oscillation amplitude.

density (hE0,0

0,0(t)) is depicted for different detunings and for 3K (dotted, black line) and

300 K (solid, orange line). Initially, the QD is in the excited state and no photon is in the

cavity.

If the cavity mode is in resonance with the QD band gap frequency (1), Rabi oscillations

occur with an amplitude of 1and a Rabi frequency equal to the JCM solution Ω1=M. At

300 K, a beating occurs, but does not change the Rabi frequency, instead the amplitude is

modulated in dependence on the effective phonon-coupling [CRCK10; AHK99]. This mod-

ulation cannot be seen in the absorption spectrum due to the strong pure dephasing. If the

cavity is detuned to the Stokes peak (2), the excited state oscillates also with an amplitude

of 1but with a modified Rabi frequency Ω2. The temperature dependence does not change

3 Quantum dot cavity quantum electrodynamics

the modified Rabi frequency at 300 K, which remains smaller than in the case of reso-

nance. Remarkably, the amplitude is not decreased although the cavity mode is detuned

from the band gap frequency of QD. In case of a detuning to the anti-Stokes peak (3), the

temperature dependence is very important and strong. At 3K, no oscillation occur. The

cavity feeding is not active for low temperature, since the anti-Stokes process depends on

the mean LO-phonon number, which is at 3K due to the high energy of 36.4meV negli-

gible. Due to the lack of cavity feeding, the excited state density oscillates very fast with

a negligible amplitude. However, at 300 K, cavity feeding occurs. The phonon occupation

number is increased to 0.3and an oscillation occurs with the same modified Rabi frequency

Ω3= Ω2, compared to the Stokes case, but with a decreased amplitude. The amplitude,

but not the Rabi frequency depends on the mean phonon number.

The modified Rabi frequency due to the LO-phonon interaction depends on the oscillator

strength at the given frequency. In the IBM [Mah90], the absorption spectrum of a QD

interacting with an LO-phonon bath can be calculated exactly. For 0K, the peak height-

s/oscillator strengths of the different LO-phonon satellite peaks follow from a Poissonian

distribution in comparison to the main resonance:

ξi=e−FFi

i!(3.35)

for the i−th peak with the dimensionless Huang-Rhys factor: F=∆LO

ωLO [MK00]. Since the

splitting depends on the peak height/oscillator strength, the splitting at the Stokes (2) and

anti-Stokes position (3) can be derived from the splitting (Ω1=M) in case of resonance

between QD band gap frequency and cavity mode (1) and is decreased via the Huang-Rhys

factor for i= 1 (first peak from the main resonance):

Ω2

Ω1≈ξ1=g2

eff

ω2

e−∆LO

ωLO −→ Ω2≈M g2

eff

ω2

e−∆LO

ωLO .(3.36)

This expression is not exact for elevated temperatures. The modified Rabi frequencies

are only proportional to the Huang-Rhys factor, since other contributions enter in the

electron dynamics, e.g. the detuned Rabi frequency from the QD band gap frequency,

and the temperature is 300 K. However, parameter studies, cf. Fig. 3.10, suggests an

approximately linear dependence of the anti-crossing at the Stokes and anti-Stokes position

on the electron-photon coupling Mand the effective phonon coupling g2

eff. The modulation

of the amplitude shows a more complex behavior: anharmonicities appear, cf. Fig. 3.11(2)

and (3). More frequencies enter the system dynamics at 300 K due to the LO-phonon

induced beating.

3 Quantum dot cavity quantum electrodynamics

3.4 LO-phonon QD cavity-QED: Degradation of

photon-statistics

Beyond spectral properties of the LO-phonon QD cavity-QED system, discussed in the last

section, with inherent cavity feeding due to the presence of LO-phonons, the time-dynamics

include other observables of interest. Next to the polarization, observables such as the pho-

ton density or the normalized intensity-intensity correlation function g(2)(t, 0)-function are

affected strongly by the semiconductor environment [UGA+07; MIM+00; AVB+09]. In gen-

eral, the semiconductor is assumed to destroy quantum correlation via enhanced dephasing

[HPS07]. In this section, the induction model is applied to the LO-phonon QD cavity-QED

and demonstrates an enhancement of quantum correlation, induced by phonons, leading

to an LO-phonon induced anti-bunching of the cavity field [CRCK10].

Photon-photon correlations are in focus of current research to exploit intrinsic features

of the quantized light field, such as photon-bunching (g(2)(t, 0)>1) for measurement tech-

niques [VSDS05; BO06; SBS06], coherence properties of laser emission above threshold

(g(2)(t, 0)=1) [Wie09; UGA+07; AVB+09], or anti-bunching (g(2)(t, 0)<1), e.g. single-photon

emission (g(2)(t, 0)=0) to control quantum information processes, quantum cryptography

protocols or measurement techniques [YKS+02; BUA+05; BSPY00; LST+09]. Since QDs

are grown and controlled in their optical properties [KGK+06; YSH+04] via growth tech-

niques, strain [SWL+09], lateral fields [VUH+07], etc., quantum emitter sources based

on semiconductor QDs are of great interest. To tailor the quantum light emission from

QDs, embedded in a semiconductor environment, many-particle contributions, such as LO-

phonons need to be taken into account. In particular in the single-photon limit, in which

the fluctuation around the mean value is in the same order of magnitude of the mean num-

ber itself, quantum correlations are strongly modulated by for example the LO-phonon

interaction. The single-photon limit has gained great importance recently. Unfortunately,

quantum correlations are highly sensible to factorization approaches, but at the same time,

those correlations are feasible for technological application, such as entanglement or anti-

bunching.

To simulate and investigate the complex system dynamics on a microscopical level, quan-

tum correlations must be treated as exact as possible. Therefore, the induction model is

applied to the single-photon limit to demonstrate the strength of this approach, to treat two

interactions simultaneously up to an arbitrary accuracy, including non-Markovian effects

and the impact of LO-phonons on the g(2)(t, 0)-function and the photon density. Hereby,

the induction model as a theoretical framework reveals possibilities to control the quantum

light emission via external parameters such as the temperature and proves the importance

of non-Markovian, non-factorized theoretical frameworks in the search for advantageous

properties of semiconductor quantum optics.

3.4.1 Initial conditions and equations of motion

At the starting point, the QD is brought via a short pulse in the excited state on a time

scale that the LO-phonon bath remains at t= 0 in equilibrium: ρE

0= 1. The cavity is

prepared either in a Fock or thermal light state. The initial conditions are calculated via

the method described in 3.1.3 with Eq. (3.14). The LO-phonon bath is assumed to be

initially in equilibrium and the mean number of phonons in the LO-Mode ¯nωLO is given

3 Quantum dot cavity quantum electrodynamics

with the Bose-Einstein statistics. Two situations are investigated: 3K and 300 K. No

external pumping is assumed, as well as for investigation purposes, losses are not included:

κ= 0 and γp= 0. The cavity mode (ω0) and the QD transition (ωcv) are in resonance

with respective to the polaronshift ωcv =ω0+ ∆LO. The LO-phonon bath is initially in

equilibrium with the ground state, cf. Sec. 3.3. In this case, the set of equations of motion

Eq. (3.11)-(3.13) are simplified. The QD transition dynamics read:

∂tTp+1,p

m,n =i(m−n)ωLOTp+1,p

m,n −i(p+ 1) MEp,p

m,n−ingeff Tp+1,p

m,n−1(3.37)

−iM(Ep+1,p+1

m,n −Gp+1,p+1

m,n )−iTp+1,p

m,n+1−iTp+1,p

m+1,n.

Due to the quantum optical dynamics, without external pumping, the annihilation and cre-

ation operators of the photon field remain in constant relation. Only transitions are driven

with one more or one less annihilation operator than the creation operator. In comparison

to Eq. (3.11), p, s is replaced by only p. Excited state and ground state dynamics have

always the same number of creation and annihilation operators. Non-diagonal contribution

are not driven in this setup.

∂tGp,p

m,n=i(m−n)ωLO Gp,p

m,n−2Im MTp+1,p

m,n +p MTp,p−1

m,n  (3.38)

∂tEp,p

m,n=i(m−n)ωLO Ep,p

m,n+ 2ImMTp+1,p

m,n +igeff(mEp,p

m−1,n−nEp,p

m,n−1).

With Eq. (3.37)-(3.38), the combined, and fully correlated electron, photon and phonon

dynamics can be evaluated for different photon-statistics and phonon initial values, e.g. the

temperature. The results reported here apply also to other materials, as long as the Rabi

frequency ΩN=M√N+ 1 with Nphotons in the cavity is in the order of magnitude

of the electron-phonon coupling strength g(c,v)

q, i.e. the strength of the electron-phonon

interaction is comparable with the strength of the electron-photon interaction:

ΩN

|gc

q−gv

q|≤100.(3.39)

3.4.2 LO-phonon induced anti-bunching

Since the induction model evaluates the set of equations up to an arbitrary accuracy, it is

now possible to study the impact of the LO-phonons on the quantum optical properties

of the cavity mode, such as the photon-statistics. The interplay between the phonon- and

photon interaction via the electronic system is most interesting in the regime, in which

neither of them is dominant, i.e. the Rabi frequency ΩNand the effective coupling strength

geff := qPq|gc

q−gv

q|2are in the same order of magnitude. In that case, none of the

interactions is negligible and surprising physics occur. As a first example, it is shown that

the electron-phonon interaction transforms a classical light state, such as a thermal light

state, into a non-classical one [CRCK10].

Starting with a cavity prepared in a thermal state with initial conditions, cf. Sec. 3.2.3,

and a mean photon number of N= 0.25. A thermal field at a frequency of the typical

transition frequency of an InAs/GaAs QD can hardly reach this mean photon number value

at reasonable temperatures. The initialization of the cavity mode can be intense tunable

narrow-band thermal sources, or thermal light beams used for imaging [BH84; GBBL04].

3 Quantum dot cavity quantum electrodynamics

Thus, the cavity mode is externally pumped and prepared artificially in a thermal state of

N= 0.25. Higher values for the mean photon number are possible, but results in a very

strong and photon dominated dynam-

00.5 11.5 2

Mean photon number N

0.05

0.1

0.15

0.2

0.25

0.3

Photon Probability pn

Figure 3.12: Thermal light distribution.

ics, for which the LO-phonon interac-

tion becomes negligible. Here, the single-

photon limit is investigated. The mean

photon number for the thermal light field

is chosen, that three or more photons

have a negligible probability. In Fig. 3.12,

the three-photon probability (p3) is plot-

ted for different mean photon numbers

of the thermal light field, calculated via

pn=Nn/(N+1)n+1 [SZ97]. Clearly, to

limit the dynamics in the single-photon

limit, i.e. p3≪1%, the mean photon number must be clearly below 0.3. The thermal mixed

state probability increases fast for higher mean photon numbers. Depicted in Fig. 3.12, a

higher mean photon number N > 0.6results in three-photon probabilities of about 5%,

which are not negligible and beyond the single-photon limit.

In Fig. 3.13(a), the dynamics of the photon density is plotted for different temperatures.

The QD is initially in the excited state and the photon density starts with N= 0.25 and

increases fast, since in the spontaneous emission process a Fock photon is created and the

QD electron relaxes into the ground state. The maximum value of the photon density is

N= 1.25. Due to the strong coupling between the cavity mode and the QD electron, the

photon is absorbed again. The thermal light field introduces many Rabi frequencies to the

system and an irregular oscillation pattern is the result. At 3K(dotted, black line), no

LO-phonon signatures are visible. The JCM solution for a thermal light field is reproduced

[SK93]. At 300 K(orangle, solid line), the oscillation pattern is changed. A modulation of

the Rabi amplitude appears. This beating frequency decreases the amplitude, in general.

At the right hand side, Fig. 3.13(b), the intensity spectrum is plotted, i.e. the Fourier

012

t [ns]

0.5

Photon density

60 90 120

Frequency [GHz]

Intensity spectrum

(a) (b)

Figure 3.13: (a) Photon density of a cavity mode in the thermal state for two temperatures

of the LO-phonon bath. At 3K (black, dotted line), the photon density fluctuates strongly.

At 300 K (orange, solid line), the fluctuation is decreased. (b) Fourier transform of the

photon density time trace. LO-phonon peaks appear at elevated temperatures.

3 Quantum dot cavity quantum electrodynamics

00.5 11.5 2

t [ns]

g2(t,0)

300K

0150 300 450

T [K]

0.3

0.6

nLO

(a) (b)

Figure 3.14: (a) Mean phonon number for different temperatures for GaAs LO-phonon

energy. (b) g(2)(t, 0) of a cavity mode in the thermal state for two temperatures of the LO-

phonon bath. At 3K (black, dotted line), g(2)(t, 0) fluctuates strongly. At 300 K (orange,

solid line), the fluctuation is decreased and leads to a lower average value of g(2)(t, 0) from

the bunching regime 1.1at 3K to 0.8at 300 K.

transform of the photon density dynamics for 3K and 300 K. At 3K (dotted, black

line), three peaks (75,105,130 GHz) indicate the multiple set of Rabi frequencies of the

thermal light field. The higher the mean photon number N, the more peaks appear and

the single peaks become more pronounced. All peaks at 3K can be contributed to the

electron-photon interaction, alone. An LO-phonon contribution is not visible. However, at

300 K (orange, solid line), additional peaks appear (around 50,65,90 Ghz). These peaks

originate from the LO-phonon interaction and act as local oscillator, modulating the pho-

ton and electron dynamics with different temperature dependent beating frequencies. In

particular, the peaks at 60 GHz and 87 GHz modulate the main oscillater frequency in

between at 75 GHz. The spacing between the LO-phonon assisted peaks can be expressed

via the mean LO-phonon number and the LO-phonon frequency, quantitatively n2

LOωLO.

For low temperatures T < 70 K, the mean phonon number is negligible, cf. Fig. 3.14(a).

However, for elevated temperature, e.g. room temperature, the mean phonon number is

around nLO = 0.3and the LO-phonons introduce a beating phenomenon into the sys-

tem dynamics. Remarkably, this quantum beating acts as a local oscillator, comparable

to a beam splitter and degradates the photon-statistics of the photon field. In contrast to

the photon density in Fig. 3.13(a), on which the LO-phonon interaction has only a minor

impact, the g(2)(t, 0)-function of the cavity field is strongly changed. The time trace of

the g(2)(t, 0)-function is depicted in Fig. 3.14(b) for 3K and 300 K. Initially, the cavity

field is in equilibrium and prepared in a thermal state. The g(2)(t, 0)-function is equal

to 2. After the dynamics starts, a Fock photon is emitted into the cavity mode and the

g(2)(t, 0)-function starts to fluctuate around 2for 3K (dotted, black line). The highest and

lowest value of the g(2)(t, 0)-function are far apart, but the mean value stays well within

the bunching regime with

¯g(2) := 1

TaZTa

dt g(2)(t, 0) = 1.1,(3.40)

3 Quantum dot cavity quantum electrodynamics

3K 300K

Figure 3.15: g(2)(t, 0) of a cavity mode in the thermal state for two temperatures of the LO-

phonon bath. At 3K (black, dotted line), the photon density and the g(2)(t, 0) fluctuates

strongly. At 300 K (orange, solid line), the fluctuation is decreased and leads to a lower

average value of g(2)(t, 0) from the bunching regime 1.1at 3K to 0.8anti-bunching at

300 K.

for a fixed average time Ta, e.g. Ta= 2 ns. The Fock photon does not change the ini-

tial photon-statistics of the cavity mode for low temperatures. For elevated temperatures

T > 70 K, e.g. 300 K (orange, solid line), the quantum beating leads to a modulated

g(2)(t, 0)-function and surprisingly, only the highest values of the g(2)(t, 0)-function are

decreased drastically. In consequence, the mean value of the g(2)(t, 0)-function drops for

300 K. The initial thermal cavity field is transformed into a non-classical one with a mean

value of ¯g(2) = 0.8, well within the anti-bunching regime. The LO-phonon interaction with

substantial mean phonon number has a strong impact on the quantum correlation of the

cavity field.

To illustrate this result, counting averages for the g(2)(t, 0)-function are plotted in Fig. 3.15

for 3K and 300 K. It is counted, up to a given time T(2ns), how many times the g(2)(t, 0)

function enters a given probability interval, e.g. between 0.25 and 0.3. For example, at

3K, the g(2)(t, 0)-function reaches only two times a value larger than 3, cf. Fig. 3.14, but

never at 300 K. This counting is done from the anti-bunching regime 0to the bunching

regime of 3and plotted via bars in Fig. 3.15. The remarkable result is now clearly visible.

The probability to measure a g(2)(t, 0)-value of larger than 1is drastically decreased at

300 K. It is much more likely to measure a value indicating anti-bunching (g(2) <1) in

comparison to the 3K case. Noteworthy, the probability to measure extreme anti-bunching

values (g(2) <0.2) is also decreased at 300 K. This can be explained by the degradation of

the Fock statistics.

To proof, that the transformation of the thermal light field is not caused by the LO-phonon

impact on the Fock photon, which is spontaneously emitted from the QD, the Fock case

is now investigated. The QD is initially in the excited state and the cavity contains one

Fock photon. In this preparation, the photon probability distribution pnis restricted to

two values, which are non-zero: p1and p2. Either there is one or two photons in the cavity,

depending on the state of the QD carrier. In Fig. 3.16, the dynamics of the photon density

and the intensity-intensity correlation function g(2)(t, 0) is plotted in case of a cavity mode

prepared in the Fock state with N= 1 and for two temperatures. At 3K (black, dotted

line), no LO-phonon impact is visible. The photon density oscillates with the Rabi fre-

quency ΩRabi =M√2between 1and 2. At elevated temperature, e.g. 300 K (orange, solid

3 Quantum dot cavity quantum electrodynamics

00.5 11.5

t [ns]

Photon density

00.5 11.5

t [ns]

0.2

0.4

0.6

g(2)(t,0)

Figure 3.16: Dynamics of the photon density and g(2)(t, 0) of a cavity mode in the Fock

state for two temperatures of the LO-phonon bath. At 3K (black, dotted line), the photon

density oscillates between 1and 2with the Rabi frequency. At 300 K (orange, solid line),

a phonon-induced quantum beating is visible, but the average value of g(2)(t, 0) is only

slightly changed and remains at approximately 0.4.

line), the quantum beating becomes visible and the oscillation amplitude of the photon

density and the g(2)(t, 0) is decreased. If the cavity mode is prepared in a Fock state, the

intensity-intensity correlation g(2)(t, 0)-function reads:

g(2)(t, 0) = 1 −1

c†c+(∆n)2

c†c2= 1 −1

c†c.(3.41)

Thus, for a system purely in the Fock state the standard deviation (∆n)2=c†cc†c−

c†c2is zero and the g(2)(t, 0)-function oscillates between 0.5for c†c= 2 or 0for

c†c= 1 [MW95]. This is shown in Fig. 3.16(right) with a temperature of 3K, at which

the LO-phonon interaction is negligible (dotted, black line). In contrast to the thermal

light, at 300 K the g(2)(t, 0) function is narrowed around a higher value. Thus, the stan-

dard deviation becomes non-zero due to a phonon-induced Fock state mixing and Eq. (3.41)

is not valid anymore. However, since the phonon occupation (¯nLO = 0.3) is small in com-

parison to the photon number (between 1and 2), the impact on the average value of g(2)

is small (from 0.35 to 0.39). The Fock state is only minorly transformed into a mixed state

[CRCK10].

Exactly treated, the electron-LO-phonon interaction results into a phonon-induced mixing

of Rabi frequencies and as a consequence into a quantum beating. Since the Fock state is

a maximally squeezed photon number state, the quantum beating just leads to a mixing

of different orders in the light coupling. A partial mixed state is created and it is more

probable to find slightly higher g(2)-values at elevated temperatures (300 K). However, the

photon-statistics itself is not changed. This is illustrated in Fig. 3.17. The counting averages

for the g(2)(t, 0) - function are plotted for 3K and 300 K in the Fock case. The distribution

is skewed to higher g(2)(t, 0)-function values, but the mean value is only slighty increased.

At 3K (left), a broad distribution from 0to 0.45 is visible. It is equally probable to find

ag(2)(t, 0)-function in between this intervall and the mean value ¯g(2) = 0.35 is well in the

single-photon limit. At 300 K (right), it is less probable to measure a g(2)(t, 0)-function

in the extreme single-photon limit g(2) <0.3, but the mean value remains at approxi-

mately the same value ¯g(2) = 0.39. This explains, why in the thermal case the extreme

3 Quantum dot cavity quantum electrodynamics

3K 300K

Figure 3.17: Counting averages of g(2)(t, 0) at 3K and 300 K in case of Fock photons in

the cavity. The distribution of probable g(2)-values is slightly skewed to higher values.

single-photon values of the g(2)(t, 0)-function are also decreased at 300 K. The emitted

Fock photon shifts the distribution slighty to higher values, but within the anti-bunching

regime.

The LO-phonon interaction has only a minor impact on a non-classical photon distribu-

tions, such as a cavity field prepared in a Fock state, but has strong impact on classical

photon distribution. As an example, a thermal cavity field was investigated. The com-

bined, and to arbitrary accuracy evaluated electron-, photon, and phonon dynamics result

in evolution of the thermal light field into a non-classical, anti-bunched distribution. The

counter-intuitive reduction of the g(2)(t, 0)-function from larger than 1to well below 1is

enforced by a strongly modified intensity-intensity correlation (c†c†cc) due to the LO-

phonon presence and is counter-intuitive because the photon field is made less bunched or

random via interaction with a bath. The antibunching mechanism appears similar to that

producing squeezed states with parametric amplification [SHY+85]. In this case, the local

oscillator is the LO phonon bath Fig. 3.13, where the energetically flat phonon disper-

sion imparts regularity in the beating of the different Rabi oscillations. The significantly

longer phonon wavelength compared to the microcavity length may be the reason for an

incoherent phonon bath to have the same effect as a coherent laser local oscillator: If in a

homodyne detection an incident light is mixed with a phase-matched laser local oscillator

at a beam splitter, photon-antibunching can be observed [TS90]. Hereby, the strength of

the induction model is proven in reproducing the Jaynes-Cummings solution, and being

able to include the LO-phonon interaction in a non-Markovian, non-factorizing approach.

The result occurs at high (TLO = 300K) temperatures, and is therefore interesting for

engineering applications involving nonclassical light. It is also relevant for understanding

photon antibunching experiments, when blackbody radiation from the cavity wall is con-

sidered.

This model offers a wide range of possible extensions, as more electrons, more levels, even

LA-phonons can be taken into account via the mathematical induction scheme and the

numerical evaluation techniques. In the next section, a biexciton cascade is studied, in the

weak coupling and in the strong coupling regime.

3 Quantum dot cavity quantum electrodynamics

3.5 LO-phonon QD cavity-QED: Biexciton cascade

In this section, the induction model is applied to the biexciton cascade of electrons in a

single QD, strongly coupled to a cavity mode. The dynamics of the biexciton cascades in-

cludes Coulomb-effects, such as energy renormalization and electron-hole exchange splitting

[Tak00; Tak93; NBFZ06], and is particularly important as a possible source for polariza-

tion entangled photons [BSPY00; BUM+04]. Entangled photons constitute an experimen-

tal realization of a qubit [THM00] and are technically feasible for quantum cryptography

[BBG+02], acceleration of quantum information processing [NC00] and even in measure-

ment setups [MSO+10]. Since QDs are promising solid state sources, and can be tailored to

a certain measure in their optical and electroncial properties [KKKG06], it is of great im-

portance to understand the underlying physical processes and quantum correlations within

the semiconductor environment. The induction model provides a theoretical framework to

study the combined photon-, phonon- and carrier interaction with additional semi-classical

electron-light interaction, or, with temperature dependent T1times due to LO-phonon as-

sisted QD-carrier-WL interaction.

To simulate the biexciton cascade, first the Coulomb interaction, including the electron-

hole exchange splitting, has to be considered [WSS+09]. Via an operator transformation,

done by diagonalization of the Coulomb-Hamiltonian, exciton operators are introduced

and the electron-photon interaction is expressed with these operators, cf. Sec. 3.5.1-3.5.2.

Within the exciton basis, a general set of equations of motion is derived. As an applica-

tion, the equations of motion are evaluated for the weak coupling regime and a temperature

dependent degree of entanglement is calculated, cf. Sec. 3.5.5 and 3.5.6.

3.5.1 Electron-electron interaction

The two electrons in the QD interact with each other via their Coulomb potential, an addi-

tional spin-depended, repulsive exchange interaction Vex is considered [WSS+09; Tak00].

In second quantization and after setting the ground energy to zero (~ωv↑+~ωv↓+Vvv = 0),

the Coulomb-part of the Hamiltonian reads:

H|Coulomb =~ωc↑a†

c↑ac↑+~ωc↓a†

c↓ac↓+Vvc (a†

c↓a†

v↓av↓ac↓+a†

c↑a†

v↑av↑ac↑)

+Vcc a†

c↑a†

c↓ac↓ac↑+Vex (a†

c↑a†

v↑av↓ac↓+a†

c↓a†

v↓av↑ac↑)(3.42)

where ↑,↓denote the spin-up, spin-down electron in ground vor excited state c. Here,

the spin state of the electron |J, mjiis given by the total angular momentum Jand the

spin projection mjin growth direction, i.e. |c↑i =|1/2,1/2iand |c↓i =|1/2,−1/2ifor

an electron in the excited state. In the ground state, the heavy hole (HH) spin state is

taken into account with |v↑i =|3/2,3/2iand |v↓i =|3/2,−3/2i. Light hole and split-off

spin states are energetically well below the HH and can be neglected here, cf. Fig. 3.18

[ERK+96; YC05; Sch04]. The Coulomb matrix elements Vcc and Vvc incorporating the

ground state energy, which is set zero for convenience. Dark state transitions are not

taken into account with ∆mj6=±1.Vcc is the biexciton shift, when both electrons are

in the excited state. Vvc is the monoexciton shift, when only one electron is excited,

whereas the other is in the ground state [RAK+06; DAFK06; CKR09]. From the spin-

orbit coupling originates an electron-hole exchange splitting (Vex), in which repulsive and

attraction forces induced by different spin-conformation give rise to the fine structure

3 Quantum dot cavity quantum electrodynamics

Figure 3.18: QD band structure with spin states |J, mji. Light hole (LH) and split-off (SO)

are energetically separated from the heavy hole state (HH), therefore the ground state spin

can be taken from the HH state.

splitting [WSS+09; Sch04]. Additionally, it is assumed that no spin preferences exist in

semiconductor QD [AKVP05], leading to the given Coulomb Hamiltonian, cf. Eq. (3.42).

Here, only four optically active states determine the system dynamics: |v↑v↓i,|v↑c↑i,

|v↓c↓i, and |c↑c↓i. The dark states are neglected, such as |v↓c↑i and |v↑c↓i. Four

operators can be introduced for convenience and read in the two-electron basis:

G(†)=a†

v↓a†

v↑av↑av↓, B =a†

v↓a†

v↑ac↑ac↓(3.43)

1=a†

v↓a†

v↑av↑ac↑, P1

2=a†

v↑a†

v↓av↓ac↓,(3.44)

where Eq. (3.43) are the ground state Gand the biexciton state operator B, whereas in

Eq. (3.44) the polarization operators P1

1/2are denoted. Via diagonalization, the Hamilto-

nian is transformed into a basis, in which the Hamiltonian with respect to the Coulomb

contributions is diagonal [CKR09; RAK+06]. New eigenvalues are the result. Commuting

ground and biexciton state operator (Eq. (3.43)) with the Coulomb operator show, they

are already eigenvectors of the Hamiltonian. However, the polarization operators are not

eigenvectors and the diagonalization lead to new exciton operators:

X†

m=vm

1P1

1+vm

2P1

2,(3.45)

with m:= H/V replacing 1/2as polarization operator index. Below, H/V denotes either a

horizontal or a vertical polarization. Here, the polarization is not specified. The coefficients

iare given by:

1=−vV

2=−∆e

p1 + ∆2

, vH

2=vV

1=1

p1 + ∆2

,∆e=Vex

~ω↑−~ωH

,(3.46)

3 Quantum dot cavity quantum electrodynamics

with the new eigenvalue ωH/V given by, for simplicity Vvc = 0:

ωH/V =ωc↓+ωc↑

2±r(ωc↓−ωc↑)2

4+|Vex|2.(3.47)

Within the diagonalized basis, the electron operators (G(†), X(†)

H, X(†)

V, B(†)) are eigenvec-

tors of the electron part of the Hamiltonian with eigenvalues (~ωG= 0,~ωH,~ωV,~ωB).

The transformed Hamiltonian reads:

H|Coulomb =~ωHX†

HXH+~ωVX†

VXV+~ωBB†B , (3.48)

with XH/V and B, the exciton and biexciton annihiliation operator, respectively. The

calculation for more complex Coulomb-Hamiltonians with more contributions, or more

states, and particle is straightforward [RAK+06]. First, new eigenvectors are calculated

and the coefficients are derived. The diagonalized Hamiltonian, e.g. Eq. (3.48), is used to

rewrite any other interaction in the system, such as the electron-photon interaction.

3.5.2 Electron-photon interaction

The electron-photon interaction Hamiltonian of a semiconductor QD is usually expressed in

terms of a dipole coupling between the interband polarization and the creation or annihili-

ation of a photon. cf. Eq. (3.2). The optical and electronic properties of the QD determine

the type of polarization and the wave number of the emitted quantum light. Independent

of the given confinement symmetry, electron-hole exchange interaction leads to a splitting

between dark and bright states and to a mixing of the dark and bright states, forming a

dark and bright doublet. Emission lines, involving only pure states, are circularly polarized,

whereas the mixed states result in emission lines showing linear polarization along the crys-

tal direction [SSR+05; Sch04]. The emission lines with linear polarization are produced by

a superposition of circularly polarized photons. In the rotating-wave approximation, this

interaction reads:

Hel−pt =−~X

Mka†

v↑ac↑c†

kσ++a†

v↓ac↓c†

kσ−+h.a., (3.49)

in which σ+, σ−denotes the polarization and kthe wave number of the photon. The

coupling element Mkσ depends on the wave number, as long as it is not specified, or

being transformed into an effective coupling, e.g. in a cavity. Here, a coupling strength

dependence on the photon polarization is neglected [AKVP05; HPS07]. To transform this

electron-photon interaction Hamiltonian into the exciton basis, the dipole interaction is

expressed in the two-electron basis by inserting a unity relation:

↑=a†

v↑av↑+a†

c↑ac↑,

↓=a†

v↓av↓+a†

c↓ac↓.(3.50)

The interaction Hamiltonian, cf. Eq. (3.49), reads after normal ordering and using the

two-electron assumption, as well as Pauli’s principle [RAK+06; CKR09]:

Hel−pt =−~X

kG†P1

1+P1†

2BMkc†

kσ++G†P1

2+P1†

1BMkc†

kσ−+h.a. (3.51)

3 Quantum dot cavity quantum electrodynamics

The basis transformation allows to express the electron-photon interaction in the new basis,

in which the electron operators are eigenvectors of the electronic part of the Hamiltonian,

cf. Eq. (3.45). This superposition of the exciton operators leads to a superposition of the

photon operators, as well. It is convenient to define new photon operators, e.g.:

c†

kH =vH

1c†

kσ++vV

1c†

kσ−, c†

kV =vV

1c†

kσ+−vH

1c†

kσ−,(3.52)

which fulfill the commutation relation as well, since the coefficient vH/V

1/2are taken from the

basis transformation. The Coulomb Hamiltonian determines the coefficients. If there is no

spin-preference in the system, which would break the symmetry between the four possible

transition in the two-electron case, and the exciton energies are degenerated (~ωc↑=~ωc↓),

thus energetically separated only due to the fine structure splitting (Vex), the electron-

photon interaction consists of only two photon operators and reads:

Hel−pt =−~X

MkG†XH+X†

HBc†

kH +MkG†XV−X†

VBc†

kV +h.a. (3.53)

with the new photon operators

c†

kH := 1

√2c†

kσ++c†

kσ−, c†

kV := 1

√2c†

kσ+−c†

kσ−.(3.54)

Furthermore, it can be assumed that the QD is placed inside a nanocavity, which supports

two different modes, one for the horizontal (ωH

0) and one for the vertical polarization (ωV

of the emitted photons. The energy difference between the vertical polarized and horizontal

polarized mode is in order of magnitude of µeV. Therefore, the coupling strength for both

modes is the same, i.e. MkH=MkV=Mand the simplified interaction Hamiltonian reads

finally:

Hel−pt =−~MG†XHc†

H+X†

HBc†

H+G†XVc†

V−X†

VBc†

V+h.a. (3.55)

Hereby, a cavity enhanced biexciton cascade can be studied. In Fig. 3.19, the interaction

scheme restricted to the electron-photon dynamics is depicted and illustrates Eq. (3.55).

Each cavity mode supports one relaxation path via the intermediate exciton state. A cavity

mode enhances only one polarization type [HHH+06]. The energy splitting of the interme-

diate exciton states depends on the excited state energy and on the exchange interaction:

δ=ωH−ωV=p|2Vex|2= 2|Vex|.(3.56)

If the intermediate exciton splitting is zero, i.e. the exciton energies are the same ωc↑=ωc↓

and no fine structure splitting in the system leads to an energy shift δ= 0, the four-level

system reduces to a three-level system. In general, the degeneracy of the exciton states is

lifted and four transition energies occur in the system: two between ground and exciton

states, and two between the exciton and biexciton. In this case, the cavity modes can only

be in resonance with one of this four, taken into account the polarization selection, cf.

Fig. 3.19. The biexciton cascade dynamics strongly depend on how close the cavity modes

and the QD transitions are in resonance, and furthermore, the degree of polarization en-

3 Quantum dot cavity quantum electrodynamics

Figure 3.19: Biexciton cascade scheme. The biexciton |Birelaxes via the intermediate

exciton states |XH/V ito the ground state |Gi. Two photons are emitted, depending on

the relaxation path. If the intermediate exciton states are energetically degenerated δ=

ωH−ωV= 0, the relaxation paths are indistinguishable and polarization entangled photons

are produced.

tanglement is determined by the fine structure splitting.

3.5.3 Biexciton cascade: Equations of motion

Now, the system dynamics can be calculated within an equation of motion approach.

Since the number of electrons is fixed, the induction model is applied. There are four elec-

tronic densities: the biexciton B†B, the intermediate exciton X†

HXH,X†

VXVand

the ground state density G†G, cf. Fig. 3.20. In consequence, there are four transitions:

the ground state to exciton transitions G†XH,G†XVand the exciton to biexciton

transitions X†

HB,X†

VB. New quantities are formed in this four-level system: a ground

state to biexciton transition G†Band the density-like quantity exciton to exciton tran-

sition X†

HXV, crucial for the degree of entanglement. All of these electronic states can

be photon-assisted in higher order by horizontal and vertical polarized photons. Follow-

ing abbreviations are convenient: Hm,n = (c†

H)m(cH)nand Vp,q = (c†

V)p(cV)q. Using the

Heisenberg equation of motion, the general set of equations of motion is derived and proven

via the induction method.

The dynamics of the biexciton density reads:

∂tB†BHm,nVp,q(3.57)

=i(m−n)ω0

H+ (p−q)ω0

V+iκ(m+n+p+q)B†BHm,nVp,q

−iMX†

HBHm+1,nVp,q+iMX†

VBHm,nVp+1,q

+iMB†XHHm,n+1Vp,q−iMB†XVHm,nVp,q+1.

The photon-assisted biexciton density couples to the photon-assisted polarization between

the intermediate exciton and the biexciton state, cf. equations in App. A.3. Phenomeno-

logically, the cavity loss is introduced κ, cf. Sec. 3.1.2. Since the two-electron assumption

holds, the biexciton couples only to energetic lower states, such as the intermediate exci-

ton states via the corresponding transitions. Due to spontaneous emission processes, the

3 Quantum dot cavity quantum electrodynamics

Figure 3.20: Electronic states in the biexciton dynamics. Besides to the four electronic den-

sities: biexciton B†B, intermediate exciton X†

HXH,X†

VXVand ground state density

G†G, and the four transitions between them, the ground to biexciton state polarization

G†Band the exciton-exciton transition X†

HXVmix the horizontal |Bi → |XHi → |Gi

and vertical cascade path |Bi → |XVi → |Gi.

exciton densities are driven stronger by the biexciton transition than by the ground state

transition. The exciton-exciton transition (X†

HXV) has a significant impact on the exciton

dynamics, if the two intermediate exciton levels are energetically close. The equation of

motion reads:

∂tX†

VXHHm,nVp,q(3.58)

=i(m−n)ω0

H+ (p−q)ω0

V+ωV−ωH+iκ(m+n+p+q)X†

VXHHm,nVp,q

−iMG†XHHm,nVp+1,q+iM p B†XHHm,nVp−1,q+iMB†XHHm,nVp,q+1

+iMX†

VGHm,n+1Vp,q+iM n X†

VBHm,n−1Vp,q+iMX†

VBHm+1,nVp,q

This exchange of excitation energy is of great importance for the degree of entanglement,

cf. Sec. 3.5.5. The horizontal polarized photon density can be converted completely into

vertical polarized photons, if the energy splitting between the intermediate exciton levels

is smaller than the cavity loss, i.e. some µeV. The conversion is driven by the ground to

exciton state transitions. The ground state is populated by all relaxation processes as the

energetic lowest electronic state and reads:

∂tG†GHm,nVp,q(3.59)

=i(m−n)ω0

H+ (p−q)ω0

V+iκ(m+n+p+q)G†GHm,nVp,q

−iM mX†

HGHm−1,nVp,q−iMX†

HGHm,n+1Vp,q−iM pX†

VGHm,nVp−1,q

−iMX†

VGHm,nVp,q+1+iM nG†XHHm,n−1Vp,q+iMG†XHHm+1,nVp,q

+iM qG†XVHm,nVp,q−1+iMG†XVHm,nVp+1,q.

3 Quantum dot cavity quantum electrodynamics

050 100

t [ps]

0.5

Probability

050 100

t [ps]

0.5

Probability

(b)(a)

Figure 3.21: Biexciton dynamics in the strong coupling for degenerated exciton energies

δ= 0 (a) and for a detuning δ= 10 µeV. (a): The biexciton density (blue line) decays

and the exciton densities (red) build up and are not distinguishable, until both biexciton

and exciton densities are decayed completely and the system enters the ground state state

(green). (b): The exciton energies are distinguishable (red and orange) and are differently

strong driven.

Eq. (3.57) - (3.59) describe the cavity enhanced biexciton cascade and form a complete

set with the equations in App. A.3. More interactions can be considered straightforwardly,

such as classical pumping of the biexciton density, or the LO-phonon interaction. However,

this set of equations of motion contains already interesting features and can be studied in

the strong and weak coupling regime.

3.5.4 Biexciton cascade: Dynamics in the strong coupling regime

To study the biexciton cascade dynamics, initial values have to be chosen, e.g. a popu-

lated biexciton density with B†B(0) = 1 and no photons in the cavity: c†

HcH(0) =

c†

VcV(0) = 0. The coupling strength is set to M= 100 µeV and a high-Q cavity is as-

sumed κ= 20 µeV. Phonon- or Coulomb induced pure dephasing is not included, and the

biexciton, and mono-exciton shift is neglected.

In Fig. 3.21(a) and (b), the dynamics of the electronic densities are plotted without dis-

sipation processes: κ= 0. In the case of complete resonance between the excitonic levels

and the cavity mode (ωH=ωV=ω0

V=ω0

H), the four-level system reduces to a three

level system with a degenerated intermediate exciton level with δ= 0. In Fig. 3.21(a),

the resonant case is plotted. The biexciton density (blue line) decays in dependence of the

assumed electron-photon coupling strength and the exciton densities (red line) are built

up. Both exciton densities are driven equally and are not distinguishable in this plot. With

increasing exciton densities, the ground state density (green line) also builds up due to

spontaneous emission processes. At 20 ps, the ground state reaches its maximum value

and no excitation is in the system, but two photons with horizontal and vertical polar-

ization have been emitted into the cavity mode. Since strong coupling is assumed, the

exciton and biexciton density is driven by the cavity photons and the oscillation starts

3 Quantum dot cavity quantum electrodynamics

00.15 0.3

0.5

1H1,1

V1,1

00.15 0.3

t [ns]

0.5

Photon densities

H2,2

V2,2

Figure 3.22: Photon density (H1,1, V 1,1) and intensity-intensity correlation (H2,2, V 2,2)

dynamics in the biexciton cascade with a detuning of δ= 10 µeV and a cavity loss of

κ= 10 µeV. Horizontal and vertical driven photon quantities exhibit different oscillation

patterns.

after approximately 40 ps again. In Fig. 3.21(b), a detuning between the exciton levels of

10 µeV is assumed. The same initial values are valid. Due to the detuning in the system,

between cavity mode and exciton densities, the dynamics of the exciton densities differ

(red and orange line). The Rabi oscillation of the biexciton and ground state is irregular.

The amplitude is modulated. Eventually, the value of B†B= 1 is reached, but after a

longer time (not shown).

Experimentally accessible are the transmitted photon densities and intensity-intensity cor-

relation from the cavity mode. The photon density appears in the set of equations Eq. (3.57)

- (3.59) not independently, only in correlation with transitions and electron densities. This

is a typical feature of a strongly coupled system, since factorization cannot be applied

and the full system dynamics depends on the correlation between the electron and photon

system. However, separate equations of motion of the pure photon dynamics are easily de-

rived, using the same induction method. The photon densities for the number of horizontal

and vertical polarized photons read:

∂tHm,m(3.60)

=−2mκHm,m−2mIm hMG†XHHm,m−1V0,0+MX†

HBHm,m−1V0,0i,

∂tVp,p(3.61)

=−2pκHp,p−2pIm hMG†XVH0,0Vp,p−1−MX†

VBH0,0Vp,p−1i.

For m=p= 1, these equations determine the photon density dynamics for horizontal

or vertical polarized photons, whereas for m=p= 2, the intensity-intensity correlation

is computed. In Fig. 3.22, the dynamics of the photon correlations is plotted. The initial

values are chosen as in Fig. 3.21(b) with a fine structure splitting of 10 µeV. Horizontal

3 Quantum dot cavity quantum electrodynamics

00.15 0.3

t [ns]

-0.3

0.3

Re [ <H2,0 V0,2>]

00.15 0.3

t [ns]

-0.3

0.3

Im [ <H2,0 V0,2>]

Figure 3.23: Polarization coherence dynamics (H2,0V0,2) in the biexciton cascade. This

quantity determines the degree of entanglement. The higher the oscillation frequency, the

smaller the degree of entanglement.

and vertical cavity mode are in resonance with the ground to exciton state transition, re-

spectively: ωH=ω0

H6=ωV=ω0

V. Initially, no photon is in the cavity and the biexciton

is populated. The photon densities H1,1(blue, dotted line) and V1,1(orange, solid line)

start at zero and increase equally fast to 1, indicating that via the biexciton cascade one

photon with horizontal and vertical polarization is generated, cf. Fig. 3.22(upper panel).

Due to the small cavity loss, Rabi oscillations occur. The detuning between the inter-

mediate exciton levels leads to a deviation between the oscillation behavior. The photon

density in the vertical polarized mode differs from 30 ps strongly from the oscillation of

the horizontal mode. The exciton-exciton transition X†

VXHenforces a mixing of the ex-

citation energy between the two cavity modes, indicated with photon densities above 1.

After 150 ps, the cavity loss dominates the dynamics and the oscillations become similar

again. In the lower panel of Fig. 3.22, the intensity-intensity correlation for the horizon-

tal mode H2,2(blue, dotted line) and the vertical mode V2,2(orange, solid line) is

plotted. The intensity-intensity correlation determines the g(2)(t, 0)-function of the cavity

mode, i.e. the photon-statitics. Again, the fine structure splitting leads to strong deviations

between the horizontal and vertical mode in an irregular pattern. Since the biexciton is

initially populated, it is highy unprobable that two horizontal or vertical polarized pho-

tons are generated. Therefore, the intensity-intensity correlation stays below 1and marks

ag(2)(t, 0)-function in the anti-bunching regime.

In terms of polarization entanglement, a coherence between the horizontal and vertical

polarized photons, is of great importance. The equation of motion reads:

∂tHm,0V0,m(3.62)

=i m (ω0

H−ω0

V) + 2iκHm,0V0,m−i m MX†

HGHm−1,0V0,m

+imM G†XVHm,0V0,m−1+B†XHHm−1,0V0,m−X†

VBHm,0V0,m−1

This polarization coherence oscillates with the assumed detuning between the cavity modes,

which are resonant with the ground to exciton transition of the QD system. In Fig. 3.23,

the dynamics of the real (left) and imaginary part (right) of this polarization coherence

3 Quantum dot cavity quantum electrodynamics

Figure 3.24: Hanbury Brown and Twiss setup. Intensity-intensity correlation in dependence

on polarizers are measured. The beam splitter is non-polarizing. After correlation, the

quantum state tomography elements are experimentally reconstructed, and the degree of

entanglement is determined.

is plotted. The oscillation depends on the number of photons, on the coupling strength

between electrons and photons and, in particular, on the detuning between the horizontal

and vertical mode. Note, the oscillation is around zero and thus, cancels time-integrated

out. Here, the mean value of the real part is less than 0.002 and for the imaginary part

0.0002. But the mean value determines the degree of entanglement.

3.5.5 Biexciton cascade and entanglement

Among different proposals [EOSI04; FAT+04], very promising solid-state sources for po-

larization entangled photon pairs are semiconductor QDs, since a single energy level in

a QD is saturated by two electrons (or holes) with opposite spins due to the exclusion

principle [BSPY00]. Entanglement in its simplest form is a non-separable superposition

of joint quantum states, that show non-local quantum correlations [Bel64]. The degree of

entanglement can be expressed with several quantities, such as the fidelity [Jos94], the

von-Neumann entropy [BDSW96] or the concurrence [Woo98]. Here, the concurrence is

taken as a measure for the degree of entanglement, which is experimentally accessible via

the quantum-state tomography [JKMW01; KK10]. In a Hanbury Brown and Twiss setup

[HT56], intensity-intensity correlations Hm,mVp,pare measured, for m+p= 2. Here,

an experimental setup is considered, cf. Fig. 3.24, where the distance within the two light

paths to the detector is appropriately adjusted to compensate the time difference between

the two photon emission processes in the biexciton-cascade [SYA+06]. This enhances the

probability to detect the two photons at the same time. In the quantum-state tomography

[JKMW01], time-integrated measurements are done:

ρHH := 1

TZT

dtH2,2(t), ρVV := 1

TZT

dtV2,2(t)(3.63)

ρHV = (ρVH)∗:= 1

TZT

dtH2,0V0,2(t),(3.64)

3 Quantum dot cavity quantum electrodynamics

where Tis chosen large enough to consider all possible arrival times, e.g. 1−2ns. The

time-integrated intensity-intensity correlations (ρV V , ρHH , ρHV ) determine the degree of

the entanglement of the emitted photon pair. Neglecting other detuning sources but the

fine structure splitting (δ≥0), the biexciton decays via two relaxation paths: either

B†B→X†

HXH→G†Gand H2,2>0or B†B→X†

VXV→G†Gand

V2,2>0. If the fine structure splitting is small (δ≤10 µeV), the two relaxation

path are not distinguishable due to the exciton-exciton transition: B†B→X†

HXH→

X†

VXH→X†

VXV→G†Gand H2,0V0,2>0. As a consequence, only four combi-

nation of the intensity-intensity correlation are non-zero and the photon density matrix in

the polarization sub-space (|HHi,|HV i,|V Hiand |V V i) reads:

ρpt := 





ρHH 0 0 ρHV

0 0 0 0

ρV H 0 0 ρV V





.(3.65)

Now, the concurrence is defined via the eigenvalues λi(i=1,2,3,4) of the polarization sub-

space density matrix with:

C(ρpt) := max {λ1−λ2−λ3−λ4,0},(3.66)

where the order of the eigenvalue is defined as: λ1≥λ2≥λ3≥λ4[SH06]. As a consequence,

the concurrence has a positive value and is smaller than 1due to the normalized character

of the density matrix with Tr ρpt= 1. Here with Eq. (3.65), the concurrence can directly

be calculated and expressed in dependence on the polarization coherence ρHV and reads:

C= 2|ρHV |.(3.67)

For a degenerated intermediate exciton level, the horizontal and vertical decay path are

not distinguishable. Therefore, the intensity-intensity correlation reads: ρHH =ρV V = 0.5.

The polarization coherence reaches its maximum value with ρV H = 0.5and the concur-

rence shows complete entanglement: C= 1.

Via Eq. (3.60) - (3.62), the intensity-intensity correlations are calculated for different pa-

rameter sets. In the strong and weak coupling regime, the degree of entanglement can be

discussed. In case of strong coupling, the induction method needs to be applied. An impor-

tant feature, the loss of entanglement in case of a strong fine structure splitting δ≈10 µeV,

is computable also in the weak coupling regime. This is done in the next section, as the

set of equations of motion reduces considerably.

3.5.6 Biexciton cascade: Weak coupling regime

In the weak coupling regime, the losses overrule the electron-photon coupling element:

M≪κ. As a consequence, the photons leave the system faster than a possible reabsorp-

tion from the electronic system may occur. Rabi oscillations are prevented and induced

absorption and emission processes are negligible. The spontaneous emission dominates the

system dynamics. The equations of motion are truncated to the pure cascade scheme, e.g.

quantities like B†BHm,nVp,qfor m, n, p, q 6= 0 or G†GHm,nVp,qand a†

vavHm,nVp,q

3 Quantum dot cavity quantum electrodynamics

Figure 3.25: Set of equation of motion in the weak coupling regime. No induced absorption

or emission processes occur. The dark blue quantities represent densities, which do not con-

tribute to the entanglement, whereas the dark orange and red quantities directly generate

a crossing of the different paths in the light orange boxes and are crucial to entanglement.

for m, n, p, q > 1vanish. E.g., the biexciton density cannot be photon-assisted and the

equation of motion reads:

∂tB†B=−2Γ B†B+ 2Im hMX†

HBH1,0−MX†

VBV 1,0i,(3.68)

a radiative decay Γ = Γrad =25 ps−1into non-cavity modes is assumed and corresponds

to a T1-time, incorporated in the Weisskopf-Wigner theory [SZ97; CKR09; NBFZ06]. The

cavity modes are in resonance with the ground-exciton transition: ω0

H=ωHand ω0

V=ωV

with the photon life time of κ= 10 µeV. The other equations of motion, truncated to the

weak coupling regime, are listed in the appendix, cf. App. A.4.

In Fig. 3.25, the involving quantities in the cascade scheme are depicted. The color of the

boxes indicate the importance of the quantity in generating a polarization entangled photon

pair. The dark blue quantities represent densities, which do not contribute to the entangle-

ment, whereas the dark orange and red quantities directly generate a crossing of the dif-

ferent paths in the light orange boxes and are crucial to entanglement. The dynamics start

with a populated biexciton B†B, which decays via the photon-assisted exciton-biexciton

polarization. The emitted photon has either a horizontal or vertical polarization. Therefore,

two decay paths are possible via the photon-density assisted intermediate exciton densities

X†

VXVH0,0V1,1and X†

HXHH1,1V0,0. However, the photon-assisted exciton-biexciton

polarization couples additionally to the photon-assisted ground-biexciton state polariza-

tion G†BH2,0V0,0or G†BH0,0V2,0. This characteristic quantity in a four-level system

leads to a decay path mixing. This can be seen in Fig. 3.25. The ground-biexciton transi-

tion couples to ground-exciton transitions, assisted from horizontal and vertical polarized

photons, such as G†XVH2,0V0,1or G†XHH0,1V2,0. In fact, these polarization-mixed

ground-exciton transitions drive the polarization coherence ρHV , which determines the de-

gree of entanglement in Eq. (3.67).

Furthermore, the intermediate exciton states also couples to the exciton-exciton transi-

3 Quantum dot cavity quantum electrodynamics

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500

Occupation probabilities

Time [ps]

〈B†B〉

104 • 〈X†

H XH a†

H aH〉

105 • ρHH

-1e-05

-5e-06

5e-06

1e-05

0 200 400 600 800 1000

Two-photon matrix ρ

Time [ps]

Re[ρVH] at Vex = 1 µeV

Im[ρVH] at Vex = 1 µeV

Re[ρVH] at Vex = 10 µeV

Im[ρVH] at Vex = 10 µeV

Figure 3.26: Biexciton cascade dynamics in the weak coupling regime. Left: The biexciton

decays (solid line). The intermediate exciton density builds up (dashed line) and decays,

while the intensity-intensity correlation is built up (dashed, dotted line). The biexciton

cascade is completed. Right: The dynamics of the off polarization coherence for different

fine structure splitting Vex.

tion via an exchange of a photon with respective polarization X†

VXHH0,1V1,0. This

exciton-exciton transitions drives also the polarization-mixed ground-exciton transitions.

The smaller the detuning between the exciton states, the less the exciton-exciton transition

oscillates and the stronger the polarization coherence is driven. But if the detuning is large

δ≥10 mueV, the decay path do not overlap and the biexciton decay is dominated by

the emission of either two horizontal or vertical polarized photons without a polarization-

mixed coherence, e.g. B†B→X†

HXHH1,1V0,0→ρHH. In this case, the entanglement

does not exist and the decay path can be reconstructed afterwards by the quantum state

tomography results.

In Fig. 3.26, the biexciton cascade dynamics in the weak coupling regime is depicted. On

the left, the biexciton decays (solid line), the intermediate exciton density builds up (dashed

line), and the first photon is emitted. After 50 ps, the biexciton density is completely re-

laxed into the exciton density and due to the losses in the system, the exciton density starts

to relax to the ground state via the emission of a second photon. The intensity-intensity

correlation increases and reaches its maximum, when the exiton density is zero. The cavity

loss leads to a decay of the intensity-intensity correlation and after around 500 ps, the

system dynamics stops. Two photons are emitted, either horizontal or vertical polarized

or entangled. In Fig. 3.26(right), the dynamics of the polarization coherence is plotted

for different detunings δbetween the intermediate exciton levels. If the detuning is small

δ= 1 µeV (dashed line), the polarization coherence is density-like and the oscillation fre-

quency is small. But for higher detuning values, e.g. Vex = 10 µeV, the oscillation frequency

is high and the difference to the intensity-intensity correlations ρHH and ρV V is big. The

oscillation frequency is determined by the detuning, cf. Eq. (3.62), of the horizontal and

vertical cavity mode.

Quantum-state tomography reconstructs the polarization density matrix via averaged mea-

surements, cf. Eq. (3.64). In Fig. 3.27(left), the time-integrated intensity-intensity corre-

lation ρHH and polarization coherence ρHV over the time is plotted. This illustrates the

importance of the detuning and the oscillation frequency of the polarization coherence.

3 Quantum dot cavity quantum electrodynamics

0.1

0.2

0.3

0.4

0.5

0 500 1000 1500

Integrated two-photon matrix ρ

Time [ps]

ρHH

ρVH at Vex = 1 µeV

ρVH at Vex = 10 µeV

Figure 3.27: Left: Time-integrated intensity-intensity correlations. Right: The concurrence

in dependence of the fine structure splitting Vex =δ

The intensity-intensity correlation is a higher order density and does not oscillate. Time-

integrated saturates ρHH (blue) at the value 0.5. The values in the quantum-state to-

mography are normalized, that the trace of ρpt equals 1. In the biexciton cascade, ρV V

and ρHH reaches time-integrated and normalized always 0.5. In contrast to the intensity-

intensity correlation (orange and red line), the polarization coherence ρHV oscillates with

2δ= 4|Vex|. If the oscillation frequency is high, compared to the duration of the biexciton

cascade, time-integration leads to a vanishing value for ρHV . For a Vex = 1 µeV (orange

line), the time-integrated value of ρHV is still close to the value of ρHH. However, the

oscillation frequency is around 5ns−1and the value is lowered after 0.4ns due to the

oscillation. More pronounced for a Vex = 10 µeV (red line), the oscillation takes place on

a time scale of ps and consequently the time-averaged value increases and decreases fast

and remains at a low value of around 0.01. In Fig. 3.27(right), the degree of entanglement

in dependence of the fine structure splitting is depicted. The cavity mode and the ground-

exciton transitions are kept in resonance. If Vex 6= 0, the cavity modes are detuned from

each other: ω0

H=ωH6=ω0

V=ωV. Consequently, the maximum entanglement is generated

for a vanishing fine structure splitting. Any detuning leads to a loss of entanglement. For

Vex ≥20 µeV, the entanglement vanishes completely [CMD+10].

The fine structure splitting of the intermediate exciton states imposes a severe restriction

to the generation of polarization entangled photon pairs [STS+02; SFP+02; USM+03]. An-

other restriction comes into play, if the QD carriers interact with the surrounding semicon-

ductor host material, e.g. with the carriers in the wetting layer or with the bulk phonons.

However, in a pure relaxation process, such as a biexciton cascade, the pure dephasing

contributions may be neglegible [HPS07]. Therefore the immediate attack of occurring

transitions by longitudinal acoustical and optical phonons have no impact on the degree of

entanglemenent, at least in the weak coupling regime. In Fig. 3.28, experimental data from

Hafenkramp et.al [HUM+07] is presented to illustrate the minor impact of pure dephasing

processes on the degree of entanglement. Pure dephasing increases considerably from 4K

to 30 K [KAK02], but the concurrence is only minorly decreased. Further temperature

increment leads to a vanishing signal. Hafenkramp et al. contributes this temperature ef-

fect to a loss of carriers to the wetting layer. Next, this effect is taken into account via

an LO-phonon coupling between WL and QD carrier states and is in agreement with the

3 Quantum dot cavity quantum electrodynamics

Figure 3.28: Experimental data from the Group of Peter Michler [HUM+07]. The con-

currence is measured in a quantum tomography setup for 4K and 30 K. The degree of

entanglement is weakly decreased for 30 K from 0.55 to 0.41, since pure dephasing processes

have no impact on the entanglement and WL carrier losses are negligible at temperatures

well below 70 K.

presented data in Fig. 3.28. The impact of the WL - QD carrier interaction depends on

the geometry, size and energy configuration of the QD. If the QD energy levels are far

separated from the WL energy level, the QD is atom-like and the two-electron assumption

holds [Sti01]. Yet, it may be feasible to pump the QD electrically with the ultimate goal

to fabricate an entangled photon source on demand in a LED structure [SSF+10]. In this

case, the WL-QD interaction cannot be neglected.

Depending on the QD size, the heavy hole state (HH) is typically separated between one

and two LO phonon energies from the WL band edge, cf. Fig. 3.29(left). To connect the

QD holes with the WL effectively, at least two LO-phonon processes have to be taken into

account. Single phonon processes cannot contribute to the relaxation process. WL elec-

trons are at least three LO-phonon energy separated from the QD electron state. Within

the weak density regime, Coulomb interaction within the WL is negligible [DMR+10]. The

microscopic relaxation rates Γware derived in an effective Hamiltonian approach, based on

a higher-order Markovian process. The whole two LO-phonon process is energy conserving:

ǫW L

vkres −ǫQD

v= 2~ωLO, cf. Fig. 3.29(left). However, in the transition to the intermediate

carrier state at ǫW L

vk, the energy conservation is always violated, since the hole state at

kis always less than one LO-phonon energy separated from ǫW L

vkres . Processes, which are

energy conserving, but consists of energy conservation violating subprocesses, are called

higher-order Markovian process [DMR+10]. The probability amplitude of the subprocesses

depend strongly on what extent the energy conservation is met.

The LO-phonon assisted relaxation rates contribute to the overall T1-time in the system,

which now reads Γ = Γr+ Γwand consists of the LO-phonon induced relaxation rate

Γwand the radiative decay Γr. In contrast to Γr, the LO-phonon relaxation rates depend

strongly on the temperature via the Bose-Einstein distribution function. For a two LO-

phonon process, the temperature dependence corresponds to the temperature dependence

3 Quantum dot cavity quantum electrodynamics

|k|

εQD

kres

gq1

gq2

kkres

Energy

εWL

vkres

εWL

Phonons

!ωLO

120

160

120

Figure 3.29: Left: LO-phonon assisted carrier relaxation in a higher-order Markovian pro-

cess. Right: T1-time in dependence on the temperature. With increasing temperature, the

LO-phonon assisted scattering rate increases due to the increasing number of phonons.

V [µeV]

Temperature [K]

Concurrence

-10

-5

10 50

100

110

0.2

0.4

0.6

0.8

Figure 3.30: With increasing temperature, the LO-phonon assisted scattering rate increases

due to the increasing number of phonons and the degree of entanglement is decreased.

Until for temperatures beyond T= 110 K, the entanglement is entirely lost due to the WL

induced damping, even for the ideal situation of degenerated exciton levels.

of a Bose-Einstein distribution function to the power of two. In Fig. 3.29(right), the relax-

ation rate over the temperature is plotted. From a temperature of T= 70 K and higher, the

relaxation rate exceeds the radiative decay rate Γr= 40 ns−1and becomes the dominant

damping and dephasing process in the system [CMD+10].

The temperature dependent T1-times of the LO-phonon assisted WL carrier QD carrier

scattering allow a temperature dependent analysis of the quality of entanglement of the

biexciton cascade in a semiconductor QD, embedded in GaAs bulk material. In Fig. 3.30,

the degree of entanglement (concurrence) is plotted for different exchange splitting strength

and for different temperatures, concerning the LO-phonon damping rates. First, it is shown

3 Quantum dot cavity quantum electrodynamics

how the entanglement is lost with increasing Vex. The FWHM of the concurrence depends

on the chosen Coulomb parameters. When temperature effects of the WL states are taken

into account, the concurrence is spoiled even in the ideal situation of degenerated interme-

diate exciton levels with Vex = 0. For low temperatures T < 70 K, the concurrence remains

unaffected by the WL-induced dephasing rate Γw, since the scattering times are well above

1ns, thus on a larger time scale than the whole biexciton dynamics. With increasing tem-

perature, the LO-phonon damping becomes the dominant process in the biexciton cascade.

Until for temperatures beyond T= 110 K, the entanglement is entirely lost due to the

WL induced damping, even for the ideal situation of degenerated exciton levels. Since in a

pure relaxation case, i.e. a biexciton cascade in the weak coupling regime, the pure dephas-

ing of longitudinal acoustical phonons is negligible, phonons imposes only via LO-phonon

assisted scattering processes a restriction to the generation of polarization entangled pho-

ton pairs, in particular in electrically driven LED structures [SSF+10]. Depending on the

host material of the QD, WL induced damping rates reduces the degree of entanglement,

e.g. in GaAs at a temperature of T= 100 Kin the weak coupling regime. Further in-

vestigations, in particular via the induction method in the strong coupling regime, with

inherent LO-phonon coupling to the QD exciton may reveal other probabilities. E.g. the

LO-phonon sidebands may enforces an erasing of the which-path information, similiar to

the modulation of the exciton and biexciton energies via external applied electrical fields

[CG09].

4 Quantum dot - wetting layer cavity quantum electrodynamics

4 Quantum dot - wetting layer cavity quantum

electrodynamics

The fundamental model to study quantum optical properties of a two-level system interact-

ing with a single-cavity mode is the Jaynes-Cummings model (JCM), cf. Sec. 2.2. The JCM

provides an analytical solution under three conditions: (i) the one-electron assumption, i.e.

expectation values such as ha†

1a†

2a3a4iare zero; (ii) only the electron-photon interaction

is considered and (iii) a closed system is assumed without pumping mechanisms or losses.

To place this theory into an experimental context, one may consider a stream of velocity-

selected excited Rydberg atoms pass through a microwave cavity at a rate that allows only

one atom to be in the cavity at any time [SK93]. Recent progress in nanotechnology pro-

vides a new fundamental scheme to study quantum optical phenomena, a semiconductor

QD in a microcavity [STH+09; FMR+09].

In Chap. 3, a mathematical induction method is introduced to include the LO-phonon

interaction, cavity loss and radiative decay and electrical pumping into the strongly cou-

pled electron-photon interaction. However, a fixed number of electrons is the fundamental

assumption for the induction model and two conditions for the JCM are not met: (ii)

and (iii). Considering a QD in a carrier reservoir - wetting layer (WL) system, all three

conditions of the JCM are not fulfilled, cf. Fig. 4.1. The model system to illustrate the pro-

posed photon probability cluster expansion (PPCE) is described by the electron-photon

Hamiltonian H=H0+Hel−pt [MW95] and illustrated in Fig. 4.1(b):

H=~ω0c†c+~ωca†

cac+~ωva†

vav−~M(a†

vacc†+a†

cavc).(4.1)

Figure 4.1: Scheme of the model system: a) a quantum dot is embedded inside a wetting

layer leading to confined QD states coupled to a wetting layer continuum, b) the emission

of the confined states couples to a photon mode in a cavity.

4 Quantum dot - wetting layer cavity quantum electrodynamics

For simplicity, only one valence band state vand conduction band state cis considered

with the energies ~ωvand ~ωc, respectively. Here, the cavity mode is assumed to be in reso-

nance with the QD band gap frequency: ω0=ωc−ωv. The off-diagonal coupling matrix M

denotes the electron-photon interaction strength. Additionally, the two level quantum dot

is assumed to be embedded in an electronically occupied carrier reservoir - wetting layer.

In particular, the one electron assumption is not valid and for the emission of non-classical

light, many-particle correlations for the dynamics of electrons and holes confined in the

QDs become important. In particular, electrical pump mechanisms are in focus of research

and feasible for future technological applications, such as optical quantum-information pro-

cessing or quantum cryptography [YKS+02; LO05].

For describing the quantum dynamics of semiconductor light emitters involving weak light-

matter coupling and large photon numbers, an expansion involving mean field quantities

and their fluctuations (often called cluster or correlation expansion) provides a well con-

trolled theoretical scheme to treat many particle systems of electrons, phonons and pho-

tons [Fri96; AFK05; KK06; GWLJ07]. However, for semiconductor emitters operated in

the limit of few photons [MIM+00] and few electronic levels, such as quantum dot-wetting

layer (QD-WL) systems [GWLJ07], the cluster expansion breaks down, since it is based

on the assumption that fluctuations of mean field quantities have minor influence, cf.

Sec. 2.3. With the rise of quantum information and the search for proper solid state emit-

ters, QD-WL systems for single and entangled photons are in the focus of current research

[YKS+02; GWLJ07; CKR09].

In contrast to Chap. 3, it is now important to treat the strongly coupled electron-photon in-

teraction within a many-particle perturbation approach, regarding the carrier fluctuations

in a QD-WL system, in which a fixed number of electrons cannot be assumed. Therefore,

there is an urgent need for the extension of the standard cluster expansion (SCE). Such an

extension: the photon probability cluster expansion (PPCE) is developed in this chapter.

The PPCE is a reliable approach for few photon dynamics in many body electron systems.

The strength of this method is to keep the accurate results of the SCE for large photon

numbers [KK06; GWLJ07], and, additionally, to include the strong coupling limit for high

quality cavities [RSL+04].

This chapter is organized as following: first, the standard cluster expansion is illustrated

and the Hartree-Fock factorization introduced, cf. Sec. 4.1. The intensity-intensity corre-

lation g(2)(t, 0)-function is computed within the SCE for a small number of photons and

emitters, cf. Sec. 4.1.1, and problems are discussed. In Sec. 4.2, the PPCE is derived and

PPCE and SCE solutions are compared in the single-photon limit. Within the PPCE, the

environment coupling of the QD to non-lasing modes, cavity losses, pure dephasing and

electrical pumping are discussed in Sec. 4.4. Finally, parameter studies of a single-QD laser

in the single-photon and lasing regime are presented, cf. Sec. 4.5.

4.1 Standard cluster expansion (SCE) beyond the

one-electron assumption (OEA)

A significant difference between semiconductor and atom physics is the number of carriers,

which are involved in the system dynamics [Mah90]. In atom physics, the number of car-

riers is fixed and in most cases, only the lowest unoccupied molecule orbital (LUMO) and

the highest occupied molecule orbital (HOMO) need to be considered, populated with one

4 Quantum dot - wetting layer cavity quantum electrodynamics

electron either in the excited or in the ground state. The one-electron assumption (OEA) is

assumed to be valid in most of the discussed cases [Car99; SZ97]. If not the OEA, at least

the ground state density is correlated to the excited state density. However, in semicon-

ductors, the number of carriers, electrons and holes, is high and even for confined system,

e.g. QDs, the OEA cannot be regarded as always valid. For shallow QD, which are ener-

getically not far separated from the carrier reservoir of the semiconductor, the OEA is not

a good approximation [BGWJ06]. More important, the number of carriers in the excited

and in the ground state are not correlated, since electrons and holes have different effective

masses and in case of InAs/GaAs QDs holes are energetically closer to the WL than their

electrons and pumping rates differ [Sti01].

In systems without a fixed number of electrons, i.e. expectation values like a†

1a†

2a3a4

are not vanishing, a hierarchy problem occurs. Quantities with two electronic operators

a†

1a4couple to expectation values with four operators a†

1a†

2a3a4, those to six operator

expectation values a†

1a†

2a†

5a6a3a4etc [DWRK10; SCK04]. To cast the set of equations

into a closed and solvable system of differential equations, the expectation values need to

be factorized to eliminate the coupling to higher orders in the electronic system. The lowest

order of factorization is a mean-field theory, which describes the many-particle dynamics

within a single-particle interaction [Fri96].

In this section, the conventional Hartree-Fock approximation is applied to the QD cavity-

QED case. The inclusion of many-particle contributions to the electron-photon dynamics

leads to a modified spontaneous emission source term. However, the Hartree-Fock contri-

butions do not solve the underlying problem of the SCE, regarding the factorization of a

strongly correlated system, such as the QD cavity-QED case, cf. Sec. 2.3. However, exper-

imental results can be explained and the calculations are valid for many emitter dynamics

within the cavity, or for a high number of photons.

4.1.1 Hartree-Fock approximation within the SCE

Considering a shallow QD, with many particle contribution, the equation of motion of the

polarization changes. Beyond the OEA, the microscopic photon-assisted polarization reads:

∂ta†

vacc†c=−iM a†

cac−a†

ca†

vavac−iM c†cna†

cac−a†

vavo

−iM na†

cacc†cc−a†

vavc†cco.(4.2)

Compared with Eq. (2.13), a significant difference is introduced in to the dynamics of the

photon-assisted polarization. The spontaneous emission source term is driven now via two

expectation values: a†

cacand a†

ca†

vavac. The second term vanishes in the OEA, since

the electron cannot be in both states at the same time. Due to the presence of the WL,

the OEA loses the validity and quantities with four electronic operators have to be taken

into account, even without including a wetting layer - quantum dot carrier interaction in

the Hamiltonian explicitely.

To close the set of differential equations, one needs to factorize the expectation values at a

given level, e.g. on a Hartree-Fock level: a†

ia†

jakal≈a†

iala†

jak−a†

iaka†

jal. This

factorization rule is derived via the approximation, that the electron-electron correlations

can be expressed as a general canonical statistical operator (GCSO), including only single

particle contributions [Fri96; FS90]. That is: the single particle dynamics is described in a

4 Quantum dot - wetting layer cavity quantum electrodynamics

mean field, induced by the other particles. The approximation is done in the choice of the

GCSO,[FS90] in which the observables of interest are included, e.g. ρHF:

a†

ia†

jakal=tr(a†

ia†

jakalρ)≈tr(a†

ia†

jakalρHF ).(4.3)

After introducing a unitary matrix φto conveniently express the single particle contribution

GCSO with new operators [see App. A.5 for details]:

ρHF =1

Ze−PnλD

nn d†

ndn, d(†)

n=X

φ(∗)

in a(†)

i,(4.4)

inheriting the fermionic character. The partition function is calculated with the complete

set of eigenfunctions of the new operators and reads:

Z= (1 + e−λD

11 )(1 + e−λD

22 )···(1 + e−λD

NN ) = Πk(1 + e−λD

kk ).(4.5)

Now, the GCSO is explicitly given and one can calculate on a Hartree-Fock level the

expectation value of the electron-electron correlations:

a†

ia†

jakal≈X

abcd

φaiφbjφ∗

kcφ∗

ldtr(ρHF d†

ad†

bdcdd).(4.6)

Only two contributions remain, if the trace is evaluated, since the states are orthogonal.

The expectation value of the four electron operator quantity yields:

a†

ia†

jakal≈X

cd "φdie−λD

cc φ∗

(1 + e−λD

cc )#"φcje−λD

dd φ∗

(1 + e−λD

dd )#−"φdje−λD

cc φ∗

(1 + e−λD

cc )#"φcie−λD

dd φ∗

(1 + e−λD

dd )#

=a†

iala†

jak−a†

iaka†

jal.(4.7)

The calculation is straightforward, after choosing the Hartree-Fock GCSO [FS90].

4.1.2 Modified equations of motion within the SCE

Now, the equation of the photon-assisted polarization is closed on the electronic single-

particle level:

∂ta†

vacc†=−iM a†

cac1−a†

vav−iM c†cha†

cac−a†

vavi

−iM ha†

cacc†cc−a†

vavc†cci.(4.8)

The source term of spontaneous emission is changed from being proportional to the excited

state density only a†

cac, to the product of ground and excited state a†

cac1−a†

vav

as a result of the interaction between WL carriers and the carriers inside the QD [BGWJ06;

GWLJ07]. To calculate the g(2)(t, 0)-function, higher-order photon-assisted polarizations

must be computed, considering photon-density assisted ground and excited state densities.

4 Quantum dot - wetting layer cavity quantum electrodynamics

They are not changed in comparison to the derivation within the OEA, cf. Sec. 2.3 and

Eq. (2.15). The higher-order photon-assisted polarization reads within the SCE:

∂ta†

vacc†c†cc=(4.9)

=−2iM ha†

cacc†cc(1 −a†

vav)−a†

vavc†cca†

caci+ 4a†

vacc†cImhMa†

vacc†ci

−iM c†c†c c cha†

cac−a†

vavi−2i˜

Mc†cha†

cacc†cc−a†

vavc†cci.

Again, the Pauli-blocking term for the spontaneous emission is changed and now photon-

assisted. The higher the order in the electron-photon coupling element, the more correction

terms originate from the cluster expansion method. Those corrections lead to interesting

terms in Eq. (4.9), such as the typical correlation expansion correction term ∝ |a†

vacc†c|2.

The Hartree-Fock factorization in the higher-order photon-assisted polarization includes

additionally a photon density assisted Pauli-blocking, cf. first line in Eq. (4.9)). But in the

limit of few photons, this quantity is small in comparison to other contributions and is

often neglected [GWLJ07]. Together with the equations in Sec. 2.3, a closed set of differ-

ential equations is derived and can be solved for different initial conditions of the electron

and the photon system.

In Fig. 4.2(left), the Hartree-Fock contributions of the carrier-carrier interaction in the

second order of the electron-light coupling element is compared with the calculation for

systems, in which the OEA is valid. The excited state is assumed to be initially 1and

no photons are in the cavity. In the SCE within the OEA (dashed, red line), the vacuum

Rabi oscillations lead to a negative probability for the electron density in the conduction

band state, cf. Sec. 2.3. Including the enhanced Pauli-blocking due to the implicit WL in-

teraction, the negativities are getting smaller, remain 10% large (green, solid line). Within

the first oscillation (up to 250 fs), the deviation between the OEA and the QD-WL case

is small and negligible. For longer times, a smaller Rabi frequency is observable for the

QD-WL case, but the solution is not valid for longer times [RCSK09a]. The same initial

values for the electron system is chosen in Fig. 4.2(right), but with three Fock photons

in the cavity p3= 1, i.e. c†c†cc(0) = 6 and c†c= 3. In addition to the vacuum Rabi

oscillation, the electronic system is now driven by the cavity photons. Therefore, the Rabi

frequency is higher. The dynamics of the OEA (dashed, red line) and QD-WL case (green,

solid line) do not deviate much (percentage range). The Rabi frequency is the same and

the negativities, still present, are decreased due to the higher number of photons, corre-

sponding to a weaker correlation between electron and photon system, cf. Chap. 2. Note,

the influence of the Hartree-Fock modified spontaneous emission processes is smaller and

vanishes for a high number of photons.

Since negativities still occur, even within the Hartree-Fock factorization, causing a de-

creased oscillation amplitude, it is necessary to go beyond second order and investigate

within the SCE approach the dynamics up to the fourth order. At this level, it is also

possible to compute the g(2)(t, 0)-function. The intensity-intensity correlation c†c†ccis

driven by the photon-density assisted polarization in Eq. (4.9) and is the crucial quantity

to determine the quantum light statistics of the emitted light. In Fig. 4.3, the Hartree-Fock

contributions of the carrier-carrier interaction is compared to the calculations for systems,

in which the OEA is valid, now up to the 4th-order in the electron-photon interaction.

The initial conditions for the electron system is not changed and the excited state assumed

4 Quantum dot - wetting layer cavity quantum electrodynamics

00.5 11.5 2

t [ps]

-0.2

0.2

0.4

0.6

0.8

Excited State Density

00.5 11.5 2

t [ps]

0.2

0.4

0.6

0.8

Excited State Density

Figure 4.2: Dynamics of the excited state density. Left: For the case of a initially occupied

conduction band, a†

cac(0) = 1 and zero photons in the cavity, the Hartree-Fock factor-

ization (solid, green line) results in decreased negativity in comparison to the calculation,

in which carrier-carrier interaction is neglected (dashed, red line). Right: With the same

electronic initial conditions but with c†c= 6, to show, that for increasing photon number

inside the cavity, first, the negativity is reduced and the difference between Hartree-Fock

case and one-electron assumption becomes small.

00.5 11.5 2

t [ps]

0.2

0.4

0.6

0.8

Excited State Density

00.5 11.5 2

t [ps]

0.2

0.4

0.6

0.8

Excited State Density

Figure 4.3: Dynamics of the excited state with higher contributions. Left: For the case of

a initially occupied conduction band, a†

cac(0) = 1 and zero photons in the cavity, the

Hartree-Fock factorization (solid, green line) results in a decreased oscillation amplitude,

it is now positive and does not oscillate between 0 and 1, like expected for the OEA case

(dashed, red line). The HF does not better this result, but worsened it, by decreasing the

amplitude further (solid, green line). Right: With the same electronic initial conditions but

with p3= 1. The deviation from the JCM is decreased.

4 Quantum dot - wetting layer cavity quantum electrodynamics

to be initially populated and for (left panel), the cavity is empty p0= 1. The dynam-

ics of the OEA (dashed, red line) and the QD-WL case (solid, green line) differ strongly.

Negativities do not occur, but the oscillation amplitude is much too small compared to

the known result of the JCM solution, cf. Sec. 2.2. The enhanced Pauli-blocking decreases

the amplitude further as well as the Rabi frequency. In Fig. 4.3(right), the dynamics are

investigated with three photons in the cavity. Still, the OEA and the QD-WL do not differ

much. With a higher number of photons, the solution converges to the JCM solution and

the carrier-carrier interaction do not lead to a different dynamics.

The calculations show, the break down of the cluster expansion approach is not compen-

sated by the inclusion of many particle contributions in the Hartree-Fock regime. The JCM

solution is reproduced only for very high numbers of photons or emitters. For few photons

and few emitters, the cluster expansion breaks down. There is the urgent need, to introduce

a theoretical framework for the Hartree-Fock factorization, which incorporates the JCM as

well as the possibility to factorize the electronic field within a mean field theory.

4.2 Photon probability cluster expansion approach (PPCE)

In this section, the standard cluster expansion (SCE) is generalized to the limit of few

photons in systems with few quantum confined electronic levels such as semiconductor

QD-WL systems, cf. Fig. 4.1(a). So far, the SCE of a†

ca†

vavacprovides access only to

large photon number dynamics or in weak coupling-short time limit (~M·t≪1), where

Mis the electron-photon coupling constant, cf. Eq. (4.1). In strong correlated dynamics,

the SCE becomes problematic, since the dynamics of relevant observables are not connected

directly to the photon number state. Examples include quantities such as a†

cacc†c, which

include contribution from photon states with one and more photons, similarily the quan-

tity ha†

cacc†c†cci, which includes contributions from two and more photons. However, it is

clear that this kind of expansion is not suitable, if correlations with a distinct number of

photons are of crucial importance.

A theoretical scheme, which attacks this problem for a single photon mode, but is easy to

extend to multiple photon modes, is presented in this section. This new approach allows

to treat few photon dynamics in many particle systems such as QD-WL systems with few

electronic levels, that contribute to the emission. This section is organized as follows: First,

the photon probability expansion is derived, cf. Sec. 4.2.1, expressing relevant observables

in terms of photon probabilities. In this probability picture, the JCM solution can be ana-

lytically re-derived and the inclusion of photon-statistics is discussed, cf. Sec. 4.2.2. Then,

a modified Hartree-Fock factorization rule is introduced in Sec. 4.2.3 and the modified

equations of motion are investigated, cf. Sec. 4.2.4.

4.2.1 Photon probabilities expansion

Important quantities to characterize a quantum optical field are the normalized intensity-

intensity correlation function g(2)(t, τ = 0) = hc†c†cci

hc†ci2and the photon intensity g(1)(τ=

0) = hc†ci[MW95]. Both quantities can be measured and they are directly connected

to the photon number hc†ciand intensity-intensity expectation value hc†c†cci. The idea

to circumvent the problems characteristic for the SCE (cf. Sec. 4.1) is to formulate all

quantities in terms of nphoton probabilities pn=h|nihn|i, where |nidenotes the Fock-

4 Quantum dot - wetting layer cavity quantum electrodynamics

state of nphotons in the system. This idea is already succesfully applied in JCM, but

needs to be generalized to the many electron case. The PPCE introduced now combines

the advantages of the exact JCM-solution and the standard cluster expansion. Rewriting

the observables by introducing a complete set of eigenfunctions, thus inserting an identity

of the photon subspace:

n|nihn|.(4.10)

The photon density reads:

c†c=c†c

=X

nc†c|nihn|=X

n|nihn|.=X

n pn.(4.11)

To calculate the photon density, one must solve an equation of motion for the photon

probability. The whole description of the combined electron and photon dynamics are

transfered onto the photon probabilities, which is comparable to the situation in the JCM.

In the JCM, the equations of motion are solved exactly for a single-photon subspace, in

which a photon is emitted or absorped from the state |ni, cf. Sec. 2.2. So, instead of

calculating the whole photon hierarchy:

c†c→c†c†cc→c†c†c†ccc→ ··· (4.12)

One calculates different photon probability subspaces. In this way, the calculation is orga-

nized and can be truncated safely, since it is easily to show, that for fixed initial conditions

(without environment coupling), certain photon probabilities are not driven. Therefore, it

exists an Nfrom which on the photon probability pnis zero for all times, if no external

pump mechanism is included.

The equation of motion for the pnis derived via the Hamiltonian in Eq. (4.1) with the

Heisenberg equation of motion:

∂tpn=−2√nImhM|nihn−1|a†

vaci+ 2 √n+ 1 Im hM|n+ 1ihn|a†

vaci.(4.13)

To calculate the photon probability, the photon-assisted polarization enters the system

dynamics. Here, the single-electron case is considered, assuming quantities like a†

ia†

jakal

to be zero for all time and combination of i, j, k, l ={v, c}.

∂ta†

vac|n+ 1ihn|=−i M √n+ 1 a†

cac|nihn|−a†

vav|n+ 1ihn+ 1|.(4.14)

This form of the photon-assisted polarization includes the induced absorption and emission

processes, as well as the spontaneous emission. In Eq. (4.14) a new form of the phase-filling

factor is derived. The spontaneous emission and induced emission is included in the quantity

a†

cac|nihn|, whereas the induced absorption is included in a†

vav|n+ 1ihn+ 1|. To see

4 Quantum dot - wetting layer cavity quantum electrodynamics

that, one can transform Eq. (4.14) back into the photon operator picture. Details are found

in App. A.6. The photon-assisted valence and conduction band state densities read:

∂ta†

vav|nihn|=−2√nIm Ma†

vac|nihn−1|(4.15)

∂ta†

cac|nihn|= 2 √n+ 1 Im Ma†

vac|n+ 1ihn|.(4.16)

In this expansion, the photon-assisted valence and conduction band density dynamics differ

strongly. This can be attributed to the rotating-wave approximation, already done in the

Hamiltonian. This results from the density matrix description, and that the valence state is

driven by a lower order of the photon-assisted polarization. Instead of the electron-picture,

one can formulate the same equations in the electron-hole picture, taking as photon-assisted

electron density fe

n=a†

cac|nihn|and as photon-assisted hole density fh

n=|nihn|−

a†

vav|nihn|. In this picture, the photon-probability reamains the same, but the photon-

assisted polarization reads now:

∂ta†

vac|n+ 1ihn|= +i M √n+ 1 (pn+1 −fh

n+1)−i M √n+ 1 fe

n.(4.17)

Quantum optical absorption and emission processes are related to fh

n=pn−|nihn|a†

vav

and fe

n=|nihn|a†

cacfor holes and electron densities assisted by nphotons. Finally, the

electron and hole densities fe/h

nare given by:

∂tfe/h

n= 2 √n+ 1 Im Ma†

vac|n+ 1ihn|.(4.18)

In the electron-hole picture, the dynamics of the photon-assisted electron and hole dynam-

ics obey the same equations of motion.

To close the hierarchy of equations of motion, a†

ia†

jakal|nhn|= 0 is used and is strictly

valid only for the single electron case, since two electrons are annihilated. Eq. (4.13) and

Eqs. (4.17)-(4.18)) reproduce the JCM, numerically and analytically. This is an important

benchmark of the new PPCE, if the model is limited to one electron in the quantum dot,

see App. A.7.

4.2.2 PPCE and photon-statistics

Via the choice of initial conditions for the cavity field, the JCM calculates the dynamics

for different photon-statistics, such as thermal light and coherent light, cf. Sec. 2.2. In the

PPCE, the expectation values are expressed with the statistical properties, which is one of

the advantages of this approach. For example, the photon density is calculated with:

c†c=X

n|nihn|=X

n pn.(4.19)

The whole information of the quantum correlations are incorporated in pn. Different light

fields are characterized via the deviations around the mean photon number. Even with a

given mean photon number, e.g., the standard deviation can be different. Important ex-

amples are thermal and coherent distributions, cf. Sec. 3.2.3.

4 Quantum dot - wetting layer cavity quantum electrodynamics

0510 15 20

0.2

0.4

0.6

0.8

Excited State Density

0510 15 20

t [ps]

0.2

0.4

0.6

0.8

Thermal light n=0.3

Coherent light n=8.0

Figure 4.4: Photon-statistics within the PPCE: thermal light and coherent light dynamics

for a two-level system, initially prepared in the excited state. The solution of the JCM

is reproduced. Superposition of Rabi oscillation lead to a complex oscillation pattern for

thermal light, and to the phenomenon of collapse and revival for coherent light.

The mean photon number of a thermal light field is ¯n=c†cwith a probability distribu-

tion:

pn=¯nn

(¯n+ 1)n+1 .(4.20)

To check this approach and the right choice of the statistical operator, the mean photon

number is calculated:

c†c=X

n pn=

∞

n=0

n¯nn

(¯n+ 1)n+1 =¯n

¯n+ 1 ¯n

¯n+ 1−11

¯n+ 1 X

n¯n

¯n+ 1n

It is convenient to abbreviate q=¯n

¯n+1 <1:

c†c=q

¯n+ 1 X

n qn−1=q

¯n+ 1∂q"X

qn#=q

¯n+ 1∂q1

1−q(4.21)

¯n+ 1 1

1−q2

=¯n

(¯n+ 1)2"1

1−¯n

¯n+1 #2

=¯n

(¯n+ 1)2[¯n+ 1]2= ¯n.

The thermal light cavity dynamics are initialized with the given pn, also reproducing the

JCM. Another example is the coherent state, with a probability distribution, taken into

account the Poissonian statistics of the coherent light field:

pn=¯nn

n!e−¯n,(4.22)

4 Quantum dot - wetting layer cavity quantum electrodynamics

here ¯n=|α|2corresponds to the mean photon number of the Glauber states. Again, the

expectation value is calculated:

c†c=e−¯n

∞

n=1

n¯nn

n!=e−¯n

∞

n=1

¯nn

(n−1)! = ¯n e−¯n

∞

n=1

¯nn−1

(n−1)!,(4.23)

now the summation index changed into m=n−1:

c†c=¯n

e¯n

∞

m=0

¯nm

m!=¯n

e¯ne¯n= ¯n. (4.24)

In Fig. 4.4, numerical evaluations of thermally and coherently prepared cavity dynamics

are plotted. The JCM solutions from Sec. 2.2 is reproduced within the PPCE approach.

Depending on the mean photon number, higher order of the n-photon probabilities need

to be taken into account.

However, the PPCE approach is not limited to the restrictions of the JCM, since it is now

possible to incorporate, e.g., enhanced Pauli-blocking due to carriers scatter in and out of

the quantum dot and violating the OEA.

4.2.3 PPCE and Hartree Fock factorization

In this section, a factorization rule for the photon-assisted electron-electron correlation is

derived. After expanding the photon system in terms of photon-probabilities pn=|nihn|,

the equation of motion of the photon-assisted polarization beyond the OEA reads:

∂ta†

vac|n+ 1ihn|(4.25)

=−i M √n+ 1 a†

cac|nihn|+i M √n+ 1 a†

ca†

vavac|nihn|

+i M √n+ 1 a†

vav|n+ 1ihn+ 1|−i M √n+ 1 a†

va†

cacav|n+ 1ihn+ 1|.

In Eq. (4.25), electron-electron correlations enter via e.g. a†

va†

cacav|nihn|. To close the

set of differential equations, one needs to factorize electron correlations expectation values.

For example, in the lowest order, one describes a two-particle quantity via the dynamics

of electron single particle quantities. This is a mean field theory, in which a single carrier

is influenced by a mean field. This mean field is the first approximation to include carrier-

carrier interaction, here on a Hartree-Fock level. The usual Hartree-Fock approximation is

not applicable here, since the photon contributions are neither part of the one electron op-

erator pair nor of the other. The Hartree-Fock factorization needs to be modified according

to the photon dynamics, and photon expansion technique. Starting with the expectation

value, which needs to be factorized, one derives, given a state with n1···nNelectrons and

mphotons:

|nihn|a†

ia†

jakal=X

m,{ni}hn1...nN, m| |nihn|a†

ia†

jakalρ|n1...nN, mi

≈X

{ni}hn1...nN|a†

ia†

jakalhn|ρ|ni|n1...nNi.(4.26)

4 Quantum dot - wetting layer cavity quantum electrodynamics

In the last line, the approximation is applied, that the photon state |nidoes not include

any electronic contributions, for all times. In other words, the electron and photon system

on each photon-probability pnis in a quasi-equilibrium. Note, this is not equivalent to

the Born factorization. The statistical operator does not factorize. The GCSO includes

every contribution, in dependence on the choice of the observable set. Here, the GCSO is

projected onto the different photon-probabilities, resulting in the PPCE - GCSO:

ρpn =1

pnhn|ρ|ni.(4.27)

With pnin the denominator, the PPCE-GCSO is trace conserving. To illustrate this, an

example is given. A simple case is considered, in which the electron ρel and photon system

ρpt =Pmpm|mihm|statistical operator are factorizing completely:

ρpn =1

pnhn|ρ|ni=1

pnhn|ρpt ⊗ρel|ni=1

pnhn|X

pm|mihm|⊗ρel|ni(4.28)

pnX

pmhn|mihm|ρel|ni=pn

pnhn|niρel =ρel.

The Hartree-Fock GCSO now has a different form, assuming the electron-electron correla-

tions are describable on a single-particle basis, and this, for every photon-probability, the

GCSO reads:

ρ≈ρHF := e−Pij λij a†

iaj

Z→ρpn =hn|ρ|ni

pn≈1

Ze−Pij λn

ij a†

iaj:= σn

el.(4.29)

With this statistical operator (σn

el), the usual Hartree-Fock factorization, cf. App. A.5, is

calculated straightforward. Therefore, one introduces a pn/pnto realize this normalized

GCSO in Eq. (4.26) to yield:

|nihn|a†

ia†

jakal(4.30)

≈pnX

{ni}hn1, ...nN|a†

ia†

jakalσn

el|n1...nNi

=pnX

cd "φdie−λn,D

cc φ∗

(1 + e−λn,D

cc )#"φcje−λn,D

dd φ∗

(1 + e−λn,D

dd )#−



φdje−λn,D

cc φ∗

(1 + e−λn,D

cc )

"φcie−λn,D

dd φ∗

(1 + e−λn,D

dd )#,

with λn,D

ii as the matrix element of the diagonalized matrix for the n-th photon-probability.

One calculates now with the same equilibrium assumption and introduced GCSO the single

electron quantity, to identify the contributions:

|nihn|a†

ial≈X

{ni}

pnhn1...nN|a†

ialσn

el|n1...nNi=X

d"pn

φdie−λn,D

cc φ∗

(1 + e−λn,D

cc )#.(4.31)

4 Quantum dot - wetting layer cavity quantum electrodynamics

Now, in Eq.(4.30), one can identify the modified Hartree-Fock rule, by inserting again

pn/pnto take into account the pncontribution to the single particle quantity in Eq. (4.31):

|nihn|a†

ia†

jakal(4.32)

≈1

pnX

cd "pn

φdie−λn,D

cc φ∗

(1 + e−λn,D

cc )#"pn

φcje−λn,D

dd φ∗

(1 + e−λn,D

dd )#−

pn

φdje−λn,D

cc φ∗

(1 + e−λn,D

cc )

"pn

φcie−λn,D

dd φ∗

(1 + e−λn,D

dd )#

pnn|nihn|a†

ial|nihn|a†

jak−|nihn|a†

iak|nihn|a†

jalo,

which defines the factorization rule, taken into account semiconductor many-particle con-

tributions, resulting from the independent hole and electron dynamics with valence and

conduction band state filling. The factorization rule originates from the assumption that

the photon and electron dynamics on every photon number is approximately described as

in equilibrium and thus with the statistical operator shown in Eq.(4.29). The GCSO does

not imply, that the total photon and electron dynamics are in equilibrium, which would

lead to a factorized statistical operator ρ=ρpt ⊗ρel and result in a Born factorization rule:

|nihn|a†

ia†

jakal≈|nihn|a†

ia†

jakal.(4.33)

This is only valid in case of a weak coupling, or a weak correlation between the photon and

electron dynamics, e.g. if the photons are treated as a bath [VWW01]. This assumption is

not valid in the case of a single QD coupled to single cavity mode in the strong coupling

limit, interesting for the deterministic single photon emission on demand [MKB+00].

4.2.4 PPCE and modified equations of motion

The equation of motion, describing the combined and strongly coupled electron-photon

dynamics, are modified due to the many-particle contributions in contrast to the equa-

tions, given within the mathematical induction approach in Chap. 3. Since in shallow

QD electrons and holes in the wetting layer interact with the carriers inside the QD, the

polarization dynamics of the QD is modified. New contributions occur in comparison to

the photon-assisted polarization in the one-electron assumption, Eq. (4.32), namely in

the braces. They lead to an enhanced Pauli-blocking. They decrease the impact of the

photon-assisted electron densities. In the electron-hole picture, [for the electron picture, cf.

App. A.8], the equation reads:

∂ta†

vac|n+ 1ihn|=−i M√n+ 1 "fh

nfe

pn−pn+1 −fh

n+1pn+1 −fe

n+1

pn+1 #.(4.34)

The electron densities and the photon densities stay unchanged. The set of equations of

motion can be evaluated numerically for given initial conditions.

To investigate the impact of the modified spontaneous emission and induced absorption

and emission processes, the dynamics of the electron densities are depicted in Fig. 4.5 in the

OEA (dashed, red line) and including the WL Pauli-blocking (solid, green line) the photon

probabilities are p0= 0.4, p1= 0.6. In Fig. 4.5(left), the electron densities are set initially

fe=fh= 1. Since higher order photon-assisted polarization are driven with the photon-

4 Quantum dot - wetting layer cavity quantum electrodynamics

t [ps]

0.2

0.4

0.6

0.8

012 3 4 5

t [ps]

0.2

0.3

0.4

0.5

Figure 4.5: Dynamics of fedriven by cavity photons. Left: For the case of a initially

occupied conduction band state, fe=fh= 1 and p0= 0.4, p1= 0.6photons in the cavity,

the Hartree-Fock factorization (solid, green line) results after 1ps in a different dynamics,

compared to the OEA (dashed, red line). Right: With fe=fh= 0.5, the enhanced

Pauli-blocking is clearly visible. The oscillation amplitude is decreased, if the Hartree-Fock

factorization is included (green, solid line).

probabilities, the Pauli-blocking modifies the dynamics and decreases the Rabi frequency.

Higher order photon-probabilities lead to a collapse and revival similiar dynamics, which is

enhanced due to the Pauli-blocking. In Fig. 4.5(right), the electron densities are set initially

fe=fh= 0.5and the picture does not change but the amplitude. The electron density

does not oscillate between 0.2and the initial value 0.7, like in the case, in which the OEA

(dashed, red line) is valid. The amplitude is decreased due to the interplay between hole

and electron density with the independent carrier interaction of the WL. As an additional

important result, the Rabi frequency is also different. The Pauli blocking decreases the Rabi

frequency (solid, green line) although the amplitude decreased. This is clearly a situation,

which goes beyond the JCM, cf. Sec. 2.2. In the OEA, a decreased amplitude lead to a

higher Rabi frequency, e.g. in case of a detuning, cf. Sec. 3.2.2. Here, the amplitude is

decreased as well as the Rabi frequency. After 2ps, the maximum of the dynamics are

clearly shifted. If more than one photon probabilities are inequal to zero in the beginning,

the Pauli-blocking factors modify the dynamics due to the denominator in Eq. (4.17).

In contrast to the JCM, the strength of the vacuum Rabi flopping is reduced in dependence

on how strong the electron and hole density deviates from the OEA fh=fe. This implies

that the amplitude of the Rabi flops might be used as a measure for the number of electrons

in the actual quantum dot. To connect this result to an easily accessible quantity, the g(2)-

function is calculated in the next section and the dynamics of the SCE and the PPCE are

compared.

4.3 Photon dynamics in the PPCE and SCE approach

The best benchmark for the new PPCE scheme is the case of vacuum Rabi flops, where

quantum fluctuations dominate the dynamics. The PPCE provides physical reasonable re-

sults as opposed to the SCE method: it guarantees positivity of the electronic density, cf.

Sec. 4.2.4. In particular, the exact solution of the JCM is recovered within the OEA for

4 Quantum dot - wetting layer cavity quantum electrodynamics

arbitrary initial conditions, cf. Sec. 4.2. In this section, the differences between the SCE

and PPCE are discussed within and beyond the validity of the OEA, focusing on the pho-

ton dynamics. The photon dynamics determine the easily accessible intensity correlation

g(2)(t, 0)-function and the photon density. Both depend strongly on the electron dynam-

ics discussed in the previously section. In Sec. 4.3.1, the differences within the OEA are

discussed, in which the PPCE-solution is identical with the JCM solution and is used as

a benchmark to determine to what extent the SCE is a good approximation in a QD-WL

system. Here, the investigation focuses on the g(2)(t, 0)-function. In Sec. 4.3.2, the results

of the PPCE are applied to extract the actual number of electrons in the QD from the

Rabi oscillation amplitude of the g(2)(t, 0)-function.

4.3.1 Intensity-intensity correlations within the Hartree-Fock approximation

The photon density, identical to the g(1)(t, τ = 0), and the g(2)(t, 0)-function, are very

interesting quantities for technological applications, and are well-known in atomic quantum

optics, in which the OEA is valid and results are known. To calculate the second-order

correlation g(2)(t, 0)-function within the SCE, one has to determine the correction quantity

c†c†ccc, which is defined as the difference between the full correlation c†c†ccand the

sum of all possible factorizations, cf. Sec. 2.3:

c†c†ccc=c†c†cc−2(c†c†cccc+c†cccc†c+c†ccc†cc+c†ccc†ccc)

−c†c†ccccc−cccc†cc†c−c†c†cccc−c†cc†ccccc

which simplifies in case of Fock states in the cavity to

c†c†ccc=c†c†cc−2c†cc2.(4.35)

Other correlations are not driven and have an initial value of zero. In this case, the photon

density is equal to the correlation: c†c=c†cc. This is not valid, if coherent states are

assumed within the cavity, i.e. Glauber states, which have non-vanishing singlet expectation

values: c†[Gla63]. The g(2)(t, 0)-function is calculated in the SCE approach with:

g(2)(t, τ = 0) = 2c†c2+c†c†ccc

c†c2= 2 + c†c†ccc

c†c2.(4.36)

Since the g(2)-function depends on the preparation of the cavity field, the correction term

depends on the initial conditions and has to be calculated with:

c†c†cc(0) −2c†c2(0) = c†c†ccc(0) = (g(2)(0) −2) c†c2(0).(4.37)

Now, with a given set of initial conditions the equations of motion are evaluated and

the differences between the SCE and PPCE can be discussed with the OEA: a†

cac=

1−a†

vav, which leads to the specific relation within the SCE, already mentioned and

discussed: a†

cacc†cc=−a†

vavc†cc.

For a high number of emitters, or a high number of photons inside the cavity, the SCE

and the PPCE converge, since for a high number of photons the SCE becomes a very

good approximation technique and is well established. In Fig. 4.6(left), the dynamics of

4 Quantum dot - wetting layer cavity quantum electrodynamics

0 0.3 0.6

40.2

40.4

40.6

40.8

g(1)(t,0)

0 0.3 0.6

t [ps]

0.98

1.02

g(2)(t,0)

0 0.3 0.6

39.9

40.1

40.2

40.3

40.4

g(1)(t,0)

0 0.3 0.6

t [ps]

0.975

0.98

0.985

0.99

0.995

g(2)(t,0)

Figure 4.6: Dynamics of the photon density g(1)(t, 0) and the intensity-intensity correlation

function g(2)(t, 0), calculated with the SCE (dashed, orange line) and the PPCE (solid,

green line) in the one-electron approximation (left) and for a QD-WL system (right). The

QD is initially excited and inside the cavity are g(1)(0) = 40 photons. The solutions differ

not much, but the Hartree-Fock factorization within the SCE results in a too high Rabi

frequency. The enhanced Pauli-blocking leads to a decreased oscillation amplitude for the

oscillation of the photon density and to a decreased photon-statistics.

the photon density and the g(2)(t, 0)-function is depicted for a high number of photons

initially inside the cavity and within the OEA. In this case, the SCE (dashed, orange

line) and the PPCE (solid, green line) converge. The higher the number of photons, the

better the SCE approximates the system, and the closer the SCE reproduces the JCM

solution, here derived within the PPCE. The photon density oscillates maximally between

40 and 41 with one QD in the cavity. The QD is initially in the excited state. However,

the Rabi frequency of the SCE solution deviates from the value of the exact model. The

SCE Rabi frequency is higher, approximately 1/3-of the JCM Rabi frequency, given by

Ω = M√N+ 1 = M√41. This would be visibile in a calculated spectrum, but does not

change the photon-statistics of the quantum light emission from the QD, which is almost 1

for both solutions. The deviation is in a percentage-range, which is small, compared to the

value of 1. The mean value for the SCE is 1.0, the mean value of the PPCE almost 0.98.

For a even higher number, the mean value of the PPCE slowly converges g(2)(t, 0) →1,

since the correlation function can be calculated for a Fock state with:

g(2)(t, 0) = 1 −1

c†c(t),(4.38)

which gives g(2) = 0.975 for N= 40, which is exactly the numerical value, visible in

Fig. 4.6(left). From about N= 100, the solution become the same, within less than one

percent. In Fig. 4.6(right), the initial conditions for the photon system are the same, but

the electronic system is in the state fe= 0.8and fh= 0.4, to emphasize the QD-WL prop-

erties for typical initial conditions, beyond the OEA. Again, the SCE (dashed, orange line)

and the PPCE solutions (solid, green line) are compared. The difference is small due to the

high number of photons. One photon more or less does not change the electron-photon in-

teraction much. However, the Hartree-Fock factorization does not solve the problem of a too

high Rabi frequency in the SCE solution (dashed, orange line). Interestingly, the enhanced

4 Quantum dot - wetting layer cavity quantum electrodynamics

00.5 11.5 2

1.2

1.4

1.6

1.8

g(1)(t,0)

00.5 11.5 2

t [ps]

-4

-3

-2

-1

g(2)(t,0)

00.5 11.5 2

1.5

1.6

1.7

1.8

1.9

g(1)(t,0)

00.5 11.5 2

t [ps]

-0.5

0.5

g(2)(t,0)

Figure 4.7: Dynamics of the photon density g(1)(t, 0) and the intensity-intensity correlation

function g(2)(t, 0), calculated with the SCE (dashed, orange line) and the PPCE (solid,

green line) in the one-electron approximation (left) and for a QD-WL system (right). The

QD is initially in the ground state. The Pauli-blocking leads to a decreased oscillation

amplitude for both solution, PPCE and SCE. In the SCE solution, negativities occur in

the g(2)(t, 0)-function oscillation.

Pauli-blocking leads to a decreased oscillation amplitude of the photon density, which here

is emphasized by the choice of the initial conditions. Moreover, the photon statistics is de-

creased also, in particular for the SCE solution. Without the Hartree-Fock Pauli-blocking

term, the photon-statistics oscillates between 1.02 and 0.98, with the Pauli-blocking term in

the spontaneous and induced emission, the g(2)-function oscillates only between 0.995 and

0.987. The SCE solution is closer to the PPCE solution now. The Pauli-blocking from the

many-particle carrier-carrier interaction leads to decreased g(2)(t, 0)-function. The funda-

mental problem with this theoretical approach cannot be solved and becomes more obvious

and problematic, if the QD is initially in the ground state.

In Fig. 4.7(left), the photon density and intensity-intensity correlation function is depicted

for a QD initially in the ground state and with two photons inside the cavity. The SCE

shows for the photon density (dashed, orange line) a strong deviation from the JCM solu-

tion. The oscillation amplitude is decreased, but now, the photon density does not reach 1

and does not create a full excitation within the system in contrast to the PPCE solution

(solid, green line), which correspond to the expected behavior. The decreased amplitude

leads now for the intensity-intensity correlation function to a obvious failure of the SCE,

since the positively defined correlation function becomes negative and oscillates between

0.5, which is the expected maximum, and −3, which exceeds the minimum 0of the JCM

solution. Even without a benchmark, this failure leads to the conclusion, that within the

OEA, the SCE is valid only for a high number of emitters, whereas the PPCE provides a

theoretical framework to rely on. In Fig. 4.7(right), the QD-WL case is investigated with

a QD initially in fe= 0.4and fh= 0.2. Two photons are inside the cavity. The Pauli-

blocking factor leads to the decrease in the oscillation amplitude, now for both, the PPCE

and the SCE solution with the result, that the negativities in the g(2)(t, 0)-function become

smaller for the SCE. The PPCE secures the positivity of the g(2)(t, 0)-function, as long as

the limit of the Hartree-Fock approximation is valid. The comparison now favoures clearly

4 Quantum dot - wetting layer cavity quantum electrodynamics

0510

0.5

1fe

g(2)(0)

0510

t [ps]

0.5

g(2)(t) , fe(t)

0510

0.5

0510

t [ps]

0.5

(a)

(b)

Figure 4.8: Dynamics of the g(2)(t, 0)-function (solid, orange line) and the electron density

(dashed, blue line) for different initial conditions of the electronic system NE= 1+fe−fh

and initially two photons in the cavity. (a) NE= 0: No dynamics for the g(2)(t, 0)-function,

since no electrons in the QD; (b) NE= 1: Cavity dynamics lead to strong Rabi oscillations

in the one-electron limit fh=fe; (c) NE= 1.5: the number of electron and holes are

increased, the oscillation amplitude is smaller due to Pauli-blocking; and (d) NE= 2: Full

Pauli-blocking prevent cavity dynamics.

in the single-photon limit, with few emitters and few photons, the theoretical approach of

the PPCE.

4.3.2 Pauli-blocking dependent Rabi oscillation amplitude

The Pauli blocking term occurring in the QD-WL system does not lead to strong deviations

from the JCM solution for short times. The impact of the Pauli-blocking terms depends

strongly on the number of electrons and holes in the QD, without environment coupling,

e.g. electrical pumping. Since the g(2)(t, 0)-function is an easy accessible observable and

the time evolution is, in principle, measurable, the amplitude can be used as a measure

for the number of electrons and holes in the actual QD for the QD-WL system. In those

systems, the oscillation amplitude differs for different initial conditions in the electronic

system strongly.

Focusing on the specific QD-WL properties, in Fig. 4.8, the dynamics of the electron density

(dashed, blue line) and the g(2)(t, 0)-function (solid, orange line) is depicted for different

initial conditions in the electronic field, the number of electrons and holes in the QD with

NE= 1 −fh+fe=a†

vav+a†

cac. The cavity has two photons in the beginning, i.e.

p2= 1 and the photon-statistics is anti-bunched:

g(2)(t= 0,0) = Pnn(n−1)pn

(Pnnpn)2=2

(2)2=1

2.(4.39)

Via the choice of different initial values for the total number of electrons and holes in

the QD, the impact of the Pauli-blocking on the oscillation behavior can be investigated.

4 Quantum dot - wetting layer cavity quantum electrodynamics

00.5 11.5 2

NE = 1 + f e - f h

0.2

0.4

g(2)(t,0)-amplitude

Figure 4.9: The PPCE solution for semiconductor quantum dots. The oscillation amplitude

of the first two Rabi flops of the g(2) for PPCE is plotted over the number of electrons in

the quantum dot, with p1= 1.0(black/straight) and p2= 1.0(red/dashed).

In Fig. 4.8(a), there is no electron in the QD and no dynamics takes place. The photon-

statistics is constantly g(2) = 0.5and the electron density fe= 0. In Fig. 4.8(b), the

total number is increased to NE= 1 by setting fh=fe= 0, or assuming a†

vav= 1.

Dynamics take place, the electron density oscillates strongly with an amplitude of nearly 1,

and the g(2)(t, 0)-function follows with an oscillation amplitude of about 0.3. For a higher

total number of electrons and holes, NE= 1.5, or a†

cac= 0.5and a†

vav= 1, the

oscillation amplitude is decreased again, cf. Fig. 4.8(c). The photon cannot be absorbed,

since the excited and ground state are Pauli-blocked. Due to the Hartree-Fock terms,

this Pauli-blocking is enhanced: a specific phenomenon of the many-particle interaction

in semiconductor cavity-QED. In Fig. 4.8(d), the Pauli-blocking is full and cannot be

enhanced. Excited and ground state are populated. The photon cannot be absorbed and

no further photon can be emitted. The g(2)(t, 0)-function is constant and 0.5. The total

number of electrons and holes determines the oscillation amplitude of the g(2)(t, 0)-function

and of the electron and hole density. This corresponds to the impact of the photon numbers.

If the photon number is high in the cavity, the oscillation amplitude of the g(2)(t, 0)-function

is small. For electrons and holes, the oscillation amplitude depends on the Pauli-blocking

in the system.

This is illustrated more clearly in Fig. 4.9. The number of electrons and holes in the

quantum dot is plotted as a function of the oscillation amplitude maximum of the g(2)(t, 0)-

function, which is directly connected to the maximum of vacuum Rabi flopping. It is

seen that a maximum is achieved, if the total number equals one: NE= 1, i.e. fe=fh,

corresponding to the one-electron assumption. This behaviour can be attributed to the

Pauli-blocking terms fn

e·fn

h, because they drive the emission and are enhanced in the

one electron case. In conclusion: A modified cluster expansion scheme is introduced to

describe semiconductor quantum dot - wetting layer quantum optical devices in the strong

coupling limit involving few photons and few dominant electronic states. This photon-

probability-cluster expansion (PPCE), reproduces the JCM as a well-known benchmark in

quantum electrodynamics, which the standard cluster expansion (SCE) cannot reproduce.

4 Quantum dot - wetting layer cavity quantum electrodynamics

To illustrate its strength, the new set of equations are applied to the case of vacuum Rabi

flopping, where quantum fluctuations are dominant. For the QD-WL system, it is shown

that the amplitude of the vacuum Rabi flopping and the maximum of g(2)(t, 0) depend on

the numbers of electrons and holes pumped into the actual quantum dot device. This is a

first important prediction from the PPCE. The next step is to introduce the environment

coupling to the PPCE, e.g. cavity loss and electrical pumping mechanism.

4.4 PPCE and environment coupling

Semiconductor QDs in this thesis are assumed to be self-assembled QDs, grown with the

Stranski-Krastanoff method [GSB95]. The optical and electrical properties are strongly in-

fluenced by the surrounding material, the bulk material with the phonons, and the wetting

layer, and or the carrier reservoir [Sti01]. The environment coupling includes dephasing pro-

cesses as well as pumping processes. In the following, two important examples are given: the

cavity loss, calculated within the PPCE and the electrical pumping. Hereby, the PPCE is

introduced as a theoretical framework, pronouncingly able to include these coupling mech-

anism microscopically [SRK+10; RCSK09a; CRD+10; DMR+10]. For application studies,

it is highly desirable to take into account the parameter dependencies and to introduce

a theoretical framework, in which the environmental coupling mechanisms can be studied

and their impact on the emission properties can be predicted.

In particular, carrier inscattering are of interest for technological applications, cf. Sec. 4.4.1,

emission into non-lasing modes expressed with β−factor in Sec. 4.4.2, and cavity loss in

Sec. 4.4.3. This section focuses on these three environment couplings to discuss the tem-

poral dynamic of the single photon device.

4.4.1 Electrical pumping

It is highly desirable to drive devices electrically, instead optical pumping [LST+09]. To

minitiuarize the device, injection current, carrier scattering, and quantum light emission

is the ideal choice [BAS+06]. In a QD, embedded in a carrier reservoir, which is embed-

ded in the semiconductor bulk material, carrier injection can be experimentally realized

[YKS+02]. This is one of the great advantages of semiconductor physics, and it is the goal

to exploit the advantages and at the same time to overcome the known problems with

semiconductor nanostructure based devices, such as the high dephasing due to phonons or

Coulomb scattering [BLS+02] .

In an electrically pumped QD structure carriers are injected into the bulk material, relax

into the WL / carrier reservoir and from there into the optically active (3D confined) QD

states [PRS+07]. There are two basic scattering channels: i) relaxing carriers that translate

their energy to other carriers via Coulomb interaction [SCK01] and ii) carriers relax by

emission of phonons [DMR+10]. The first case is important at high carrier densities, at

low carrier density regime it is focused on the electron-phonon interaction, using the effec-

tive Hamiltonian approach, cf. Sec. 3.5.6. Phonons and carriers, external electrons, in the

carrier reservoir are treated as bath. It is assumed that cavity mode photons and electrons

are uncorrelated to the inscattering electrons and holes. As a result, the density matrix ρ

of the full system can be expressed at any time as ρ(t)≈ρsys(t)⊗ρbath(t)with a system

part ρsys(t)describing the quantum dots low energy electron states and the cavity mode

4 Quantum dot - wetting layer cavity quantum electrodynamics

and with ρbath(t)describing all other electronic states and bosonic baths.

Therefore for any electronic states of the bath i,jand for the QD electronic states kand l,

the factorization rules read h|nihn|a†

ia†

kalaji≈h|nihn|a†

kaliha†

iaji. The in-scattering rates

Sin and the out-scattering rates Sout for electrons and holes are calculated via this factor-

ization and a corresponding interaction Hamiltonian between the QD and WL electronic

states, including blocking and heating self-consistently [KMR+10a]. Here, the in-scattering

and out-scattering is taken as a parameter. The electron and hole dynamics restricted to

the electrical pump mechanism reads:

∂tfe

n|pump =Sin

e(pn−fe

n)−Sout

efe

n(4.40)

∂tfh

n|pump =Sin

h(pn−fh

n)−Sout

hfh

n(4.41)

So only an index nis added to f·

nand 1is replaced by pncompared to the SCE inscattering

terms [SRK+10; CRD+10]. Note, that Sin and Sout do not depend on the photon manifold:

Such a dependence will appear if the correlations with the photons is also considered for

the inscattering electronic states or cross terms between the inscattering mechanisms and

the electron-photon interaction. Whether the pump processes are phonon- or Coulomb

assisted, the scattering rates Sdiffer and include the specific properties of the interaction

between electron and holes [DMR+10]. Furthermore, the external pumping introduces a

contribution to the dephasing of the microscopic polarization:

∂tD|n+ 1ihn|a†

vacE|pump =−1

2(Sin

e+Sin

h+Sout

e+Sout

h)h|n+ 1ihn|a†

vaci(4.42)

=−γc

p|n+ 1ihn|a†

vac

due to the out scattering from the QD ground state to other states. γc

pis the pure dephasing

contribution of the in- and out-scattering of carriers. In particular in the strong pumping

regime, the scattering induced dephasing term is important.

4.4.2 β-factor

To simulate a realistic laser dynamics inside the cavity, one needs to take into account the

emission into non-lasing modes [CKI94], leading to a loss of excitation of the electronic

system without a contribution to the photon density. To derive this loss, an interaction

Hamiltonian with the non-resonant, non-lasing modes is considered:

Hel-nl =−~

i=1

Mkia†

vacc†

ki+h.c.,(4.43)

in which k0is the cavity mode and all other kiare leading to a loss, i.e. are not strongly

coupled to the QD inside the cavity [YTC00; Car99]. The completeness relation is fulfilled

with n:= nk0and nk1···nkN, counting the photons in the lasing and non-lasing mode,

respectively:

nk1

···X

nkN

|n, nk1···nkNihn, nk1···nkN|=X

n,{nk}|n, {nk}ihn, {nk}| =

.(4.44)

4 Quantum dot - wetting layer cavity quantum electrodynamics

With the completeness relation of the modes, it is convenient to trace out the non-lasing

modes to study the laser dynamics of the system, for instance the dynamics of the photon-

assisted electron and hole densities:

n,{nk}|n, {nk}ihn, {nk}| =X

n|nihn|⊗

{nk},(4.45)

where ncounts the photons in the lasing modes and {nk}the photons in the non-lasing

modes. The interaction with the non-lasing modes result in a decay of the electron and hole

densities. Excitation is lost, but no photons in the cavity mode are created. The dynamics

of the photon-density assisted electron and hole density reads:

∂tfe/h

n,{nq}(4.46)

= 2 √n+ 1 ImhMa†

vac|n+ 1,{nk}ihn|,{nk}i

+ 2

i=1 pnki+ 1 ImhMkia†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|i

The non-dynamical contributions of the non-lasing modes is sumed up to study the laser

dynamics in the cavity:

{nq}

∂tfe/h

n,{nq}=∂tfe/h

n(4.47)

= 2 √n+ 1 Im hMa†

vac|n+ 1ihn|i

+ 2

i=1 X

{nki}pnki+ 1 ImhMkia†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|i.

Now, the dynamics of the photon-assisted polarization of the non-lasing must be investi-

gated, since in this equation the spontaneous emission enters the dynamics via the electron

density. The process of spontaneous emission is crucial for the β-factor, here the sponta-

neous emission into the non-lasing mode, which is not strongly coupled to the electron

transition in the QD:

∂ta†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|(4.48)

=−i(ωcv −ωki+iγp)a†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|

−i Mkipnki+ 1 fe

n,{nk}−i Mkipnki+ 1 (pn,nk1···nki+1···nkN−fh

n,nk1···nki+1···nkN),

where the pure dephasing γpis included, e.g. due to longitudinal acoustical phonon inter-

action of the QD carriers with the bulk phonons. Non-lasing modes do not contribute to

the photon density inside the cavity. In consequence, quantities such as a†

vav|nkihnk|=

0,a†

cac|nkihnk|= 0 vanish for k6=k0and nk≥1, and the photon probability distribution

reduces to

c†

kicki=X

nki=0

nki|nkiihnki|= 0.(4.49)

4 Quantum dot - wetting layer cavity quantum electrodynamics

It is valid for all time: pn,nk1···nki+1···nkN=0 = fh

n,nk1···nki+1···nkN, since no occupation in the

non-lasing modes is possible. The validity of this assumption depends on the quality factor

of the cavity to support only one mode [Vah03]. No feedback of non-lasing mode photons

is possible and thus, no induced processes. Therefore, the polarization dynamics can be

solved in the adiabatic limit, bearing in mind that higher order photon-assisted processes

and strong coupling interaction mechanism do not enter into this equation [SRK+10], and

the time dependence is omitted, as well:

a†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|=Mkipnki+ 1

ωcv −ωki+iγp

n,{nk}(4.50)

=Mkipnki+ 1 ωcv −ωki−iγp

(ωcv −ωki)2+γ2

n,{nk}.

Now, the result in Eq. (4.50) enters in Eq. (4.47):

∂tfe/h

n(4.51)

= 2 √n+ 1 ImhMa†

vac|n+ 1ihn|i−2γp

i=1 X

{nk}

|Mki|2(nki+ 1)

(ωcv −ωki)2+γ2

n,{nki}.

The vanishing non-lasing mode density needs to be taken into account: nki= 0 and,

assuming the vanishing contribution of non-lasing mode photons to the system dynamics,

the sum over the non-lasing modes can be computed:

∂tfe/h

n= 2 √n+ 1 Im hMa†

vac|n+ 1ihn|i−1

τnl

n,(4.52)

in which the rate of the spontaneous emission into the non-lasing mode is defined as:

τnl

= 2 γp

i=1

|Mki|2

(ωcv −ωki)2+γ2

.(4.53)

Since βdefines the fraction of the total spontaneous emission 1

τsp into the lasing-mode 1

τl,

one may write:

β=

τl

τsp

τl

τl+1

τnl −→ 1

τnl

=1

β−11

τl

=1−β

βτl

=1−β

τsp

.(4.54)

The total rate of spontaneous emission depends on the microcavity and its quality-factor,

and is enhanced by the Purcell effect and can be assumed in the order of τsp = 50 ps

[RSL+04]. The β-factor is in the following used as an input-parameter.

The coupling to the β-factor is changed in the QD-carrier reservoir scheme, since the

4 Quantum dot - wetting layer cavity quantum electrodynamics

source term of the spontaneous emission is modified, Sec. 4.2.4. The derivation is similar,

but Eq. (4.50) reads in the QD-WL case:

a†

vac|n, nk1···nki+ 1 ···nkNihn, {nk}|(4.55)

=Mkipnki+ 1 ωcv −ωki−iγp

(ωcv −ωki)2+γ2

pa†

cava†

vac|n, {nk}ihn, {nk}|,

with the consequence after the modified Hartree-Fock factorization, that the electron/hole

dynamics is changed to:

∂tfe/h

n=−1−β

τsp

nfh

+ 2 √n+ 1 Im hMa†

vac|n+ 1ihn|i.(4.56)

In Eq. (4.56), the environment is included in the modified decay factor of the electron

and hole densities and is a remarkable improvement to previous theoretical approaches in

the few photon and few emitter dynamics, discussing semiconductor specific properties of

quantum light emitter.

4.4.3 Cavity loss

To exploit the properties of the emitted

quantum light, e.g. to excite another quan-

Figure 4.10: Scheme of cavity loss.

tum system in a coupled quantum sys-

tem setup [KK08; Car93], a cavity loss

needs to be considered [Car99]. In par-

ticular for single-photon emission, the

cavity loss parameter is of great impor-

tance, since one photon needs to be emit-

ted at a time, before a second photon is

created via induced emission processes.

Focusing on repetition rate, the cavity

loss is the crucial factor, e.g. how fast

the system enters the steady state. Theoretically, cavity loss results from a tunnel Hamil-

tonian, in which dissipative photon modes qcouple via the mirror with the cavity photons,

cf. Fig. 4.10 [YTC00]. These photon states outside of the cavity are coupled to the cav-

ity modes [Car99]. The coupling of the cavity modes to the external modes is decribed

by an additional part in the Hamilton operator [VWW01], which reads together with the

homogenous part of the external photons:

Hpt−pt =~X

ωqd†

qdq+~X

Gq(c†+c)(d†

q+dq),(4.57)

The probability of the tunneling process from the cavity into the environment is determined

by the strength of the coupling element Gqof the mode q. The cavity loss via external

photon modes is described by dq,d†

q. Only contributions within the rotating-wave approx-

4 Quantum dot - wetting layer cavity quantum electrodynamics

imation are considered. The dynamics, calculated only for the photon-photon interaction

reads for an arbitrary QD operator Aassisted by |nihm|cavity photons:

∂tA|nihm||pt−pt =i(n−m)ω0A|nihm|(4.58)

+iX

Gq√n+ 1A|n+ 1ihm|dq+√nA|n−1ihm|d†

q

+iX

Gq√m+ 1A|nihm+ 1|d†

q+√mA|nihm−1|dq.

Applying a rotating frame for convenience, for details App. A.9, the dissipative photon

mode assisted quantities are calculated, to derive the set of equations up to the second

order in the tunneling matrix couling element. Following as an example is given:

Contributions proportional to d(†)

qd(†)

q′are neglected within the bath assumption, since

only dissipation is taken into account [See App. A.9 for details]. The bath photon do not

couple back into the cavity. The mean photon number is too low in the bath to consider

a tunnel process in the reversed direction. The equation is solved by integrating formally

and applying the Markovian approximation and assuming memory effects are negligible.

This is a good approximation in system-bath processes with coupling strength suggesting

interaction times higher than ps-scale. The solution reads:

A|nihm|d†

qR(t)|pt−pt ≈iGq√n+ 1A|n+ 1ihm|Rei(ω0−ωq)tZ∞

dτe−i(ω0−ωq)τ.(4.60)

If the lower limit is not extended from 0to −∞, the integral is the Heitler-Zeta function

ζ(ω0−ωq), consisting of the delta-function and of the Cauchy-principal value integral.

It must be checked, to what extent the Cauchy-principal vaule integral contributes to

the dynamics, normally an energy renormalization occurs, which can be included into the

definition of ω0[Car99; SZ97]. The solution is plugged into the Eq. (4.59) to derive the

cavity-loss for the general quantity A|nihm|, by defining:

κ:= X

q|Gq|2ζ(ω0−ωq),(4.61)

the PPCE formulation of the Markovian cavity loss, including energy renormalization is

derived and can be expressed as:

In the PPCE formalism, the dynamics of the photon loss describes the loss and gain of

photons of the different photon probabilities self-consistently. The dynamics of the pho-

ton probabilities are easier in the electron-photon interaction, but more complicated for

the photon-loss, since inscattering from higher order photon probabilities need to be con-

sidered. Therefore, the cavity loss appears more complicated than in the picture of the

c†, c-operator picture. An example is given in the App. A.9. The photon-probability ap-

proach opens up further insight into the combined electron-photon dynamics, even in the

4 Quantum dot - wetting layer cavity quantum electrodynamics

case of the Markovian-derived cavity loss. The few-electron approach, i.e. the need to fac-

torize in the electronic system, does not enter in this derivation of the cavity loss, since

the cavity loss results solely from the photon-photon interaction.

4.4.4 Quantum dot - wetting layer laser equations

After discussing important processes, that couple the QD-WL cavity QED dynamics to

the environment, the results are summarized and concluded with following equations of

motion. Starting with the probability to have nphotons in the system, it is obtained:

∂tpn=−n2κpn+ (n+ 1)2κpn+1 (4.63)

−2√nImhMa†

vac|nihn−1|i+ 2√n+ 1Im hMa†

vac|n+ 1ihn|i,

including now the cavity loss process. The impact of the environment coupling on the

dynamics of the transistion is larger, since electronic and photonic observables are included:

∂ta†

vac|n+ 1ihn|=(4.64)

=−i(ωcv −ω0−iγ)a†

vac|n+ 1ihn|

−κ(2n+ 1)a†

vac|n+ 1ihn|+κp(n+ 1)(n+ 2)a†

vac|n+ 2ihn+ 1|

−i M√n+ 1 "fe

nfh

pn−pn+1 −fh

n+1pn+1 −fe

n+1

pn+1 #,

where the different pure dephasing contributions are included in γ(pump process and

contributions originating from the interaction with LA phonons). The non-lasing modes do

not change the strongly coupled microscopic polarization dynamics of the cavity photon

assisted transition. The β-factor enters only in the dynamics of the electron and hole

densities, as well as the pump mechanism:

∂tfe/h

n=−1−β

τsp

nfh

+ 2 √n+ 1 ImhMa†

vac|n+ 1ihn|i(4.65)

−2κhn fe/h

n−(n+ 1)fe/h

n+1i+Se/h

in (pn−fe/h

n)−Se/h

out fe/h

These equations together constitute the theoretical framework of the single photon emitter

realized by a single quantum dot. Parameter dependent quantum light emission can now

be studied and specific semiconductor properties be discussed. Conveniently, the full infor-

mation about the quantum light is self-consistently included in the pndistribution, which

is time resolved accessible.

4.5 Electrically pumped single photon emitter

Due to the environment, the dynamics of the QD carriers is strongly damped. Pure de-

phasing, cavity loss and the spontaneous emission into non-lasing modes lead to dephasing

and typical strong coupling signatures like Rabi oscillations are not seen. Furthermore,

4 Quantum dot - wetting layer cavity quantum electrodynamics

the pump mechanism leads to an additional dephasing and the dynamics is additionally

damped via carrier relaxation and scattering from the WL to the QD states. First, the

laser dynamics of an electrically driven QD is discussed, cf. Sec. 4.5.1, in different pumping

regimes, from weak to strong pumping. This leads to different cavity field photon-statistics

from the anti-bunching to the coherent regime. In Sec. 4.5.2, parameter dependent quan-

tum light emission is studied, i.e. the dependence of the g(2)(t, 0)-function on the electron-

photon coupling strength, the cavity loss and detuning between the cavity mode ω0and

the QD transition energy ωcv. Finally, the atomic rate equation model is compared with

the semiconductor model in Sec. 4.5.3. It is shown, that enhanced Pauli-blocking in semi-

conductor quantum light devices is advantageous for single-photon emission.

4.5.1 Laser dynamics of an electrically driven QD

In Fig. 4.11, the dynamics of the carrier and photon related observables are depicted with-

out an electrical pump mechanism. The QD is populated initially with an electron density

of fe= 1 and a hole density of fh= 0.7. One photon is initially in the cavity. The

photon probability distribution reads: p0= 0, p1= 1, pn= 0 for n≥1. A cavity loss of

κ= 0.018 meV and a pure dephasing of γp= 0.1meV takes the environment coupling into

account. The cavity has a β-factor of 0.9and a Purcell enhanced spontaneous emission

rate of τsp = 50 ps and the QD transition frequency ωcv is in resonance with the cavity

mode ω0[RSL+04]. These parameters are chosen for all calculations in this section with a

electron-photon coupling strength of M= 0.22 meV [MIM+00].

On the left of Fig. 4.11, the carrier dynamics of the QD is depicted. The electron den-

sity fe(blue, solid line) shows oscillation as a feedback photon phenomenon of strong

coupling. The electron density is driven by the microscopic polarization a†

vac(orange,

dotted line). The polarization shows the oscillations of the phase filling factor or inver-

sion a†

cac−a†

vav, including the process of spontaneous emission [SZ97]. Here, the pure

dephasing attacks the polarization dynamics and stops the oscillation after 35 ps. In conse-

quence, the Rabi oscillation of the electron density stops at the same time. The polarization

is damped to zero and after the polarization dynamics comes to a stop, the electron density

is damped only via the cavity loss and the spontaneous emission into non-lasing modes.

The right side of Fig. 4.11 shows the dynamics of the photon observables. The photon

density c†c(red, solid line) is one at t= 0 due to the initial conditions. Since the QD

is initially populated, the exciton recombines and an additional photon is created. The

photon density rises. Due to the cavity loss, the photon occupation is decreased before the

photon density reaches 2. The photon density decays to zero in dependence on the cavity

quality factor. The oscillation of the photon density stops after 35 ps, also. The intensity-

intensity correlation g(2)(t, 0)-function (green, dotted line) shows longer oscillation due to

the cavity loss mechanism. The oscillation is expected to peak at around 0.5, since single

photon interact in the cavity. Anti-bunching is visible. For the time scale of the decaying

electron density, single photons are emitted out of the cavity until the whole dynamics

stops. Without pumping mechanism, the stationary state is zero for the observables and

for the polarization. However, if the system is pumped, a non-trivial stationary state is

reached with values inequal to zero.

Now, a stationary electrical pump mechanism is considered:Sin/out

e/h 6= 0. Initially, the QD

4 Quantum dot - wetting layer cavity quantum electrodynamics

050 100

t [ps]

-0.4

-0.2

0.2

0.4

0.6

0.8

Probability

rhovc

050 100

t [ps]

0.5

1.5

g(1)(0,t)

g(2)(0,t)

Figure 4.11: QD as a single photon emitter, using the parameters M= 0.22 meV, γp=

0.1meV and κ= 0.018 meV without electrical pumping. Dynamics show Rabi oscillation.

is unoccupied. A typical, low carrier density regime (8×109(cm)−2) value of thermal oc-

cupation probability is around fe≈0.001 at 70 K. Therefore, the initial conditions are

chosen with fe/h(0) = 0 and no photons are assumed in the cavity: p0(0) = 1, pn(0) = 0

for n≥1. In the following, holes and electrons are equally pumped: Se

in =Sh

in =Sin and

out =Sh

out =Sout, and three pump regimes are investigated: weak, transient and strong

pumping strength. These regimes correspond to a transition from single-photon emission

(weak) to lasing well above threshold (strong).

In Fig. 4.12(left), a weak electrical pumping drives the QD dynamics: Sin = 4ns−1and

the out-scattering is Sout = 0.1Sin. With increasing pump duration, an electron and a hole

density is built up in the QD (solid, blue line). After 10 ps, electrons and holes start to

recombine. The photon density (dotted, green line) rises. Emission processes are slowed

down by the dephasing in the system and the cavity loss leads to a loss of photons as

well. After 100 ps, the g(2)(t, 0)-function reaches a stationary value of approximately 0.3,

within the single-photon limit. The single photon emitter now operates in this stationary

state. Single photons are emitted out of the cavity. Excitons are created and recombine in

balance. Within the PPCE approach, the full information about the quantized light field

is easily obtained via the photon probability distribution [SRK+10; RCSK09a; MW95]. In

Fig. 4.12(right), the photon probability distribution in the stationary limit is depicted. A

Fock distribution is visible. Only two photon probabilities are inequal to zero. Rabi oscilla-

tions lead to an increase of p1. Prepared in a pure Fock state, the cavity field oscillates only

between two photon probabilities, if only one emitter, e.g. QD, is placed inside the cavity.

Here, the stationary value has a maximum of p0, but enough probability p1to actually

emit a photon from the cavity. A single photon source needs a small g(2)(t, 0)-function with

a high probability of a photon emission: c†c. In Fig. 4.12, both conditions are fulfilled.

4 Quantum dot - wetting layer cavity quantum electrodynamics

Figure 4.12: QD as a single photon emitter, using the parameters M= 0.22 meV, γp=

0.1meV and κ= 0.018 meV, weakly pumped:Sin = 4ns−1and the out-scattering is Sout =

0.1Sin.

The photon probability resolved dynamics and results are important to understand quan-

tum light correlation on a fundamental level, e.g. the importance of p2to find a strongly

anti-bunched cavity field. Physically, this corresponds to negligible induced emission in the

cavity. The electron and hole density is not built up fastly enough. An induced emission

process cannot take place. The next interesting regime is the transient regime for medium

pumping.

In the transient regime, the pump mechanism is fast enough to repopulate the QD before

the cavity photon is reabsorbed and creates an exciton. In Fig. 4.13(left), the observ-

ables of the system is shifted to a higher stationary value due to the stronger pumping:

Sin = 0.1ps−1and the out-scattering is Sout = 0.1Sin. The electron density (solid, blue

line) is built up faster. The inscattering is stronger and the electron density saturates on

Figure 4.13: QD as a single photon emitter, using the parameters M= 0.22 meV, γp=

0.1meV and κ= 0.018 meV, pumped in the transition regime.

4 Quantum dot - wetting layer cavity quantum electrodynamics

Figure 4.14: QD as a single photon emitter, using the parameters M= 0.22 meV, γp=

0.1meV and κ= 0.018 meV. The electrical pumping is strong. The QD starts to enter the

lasing regime.

a higher stationary value. Due to the higher electron and hole density, more excitons can

recombine on a shorter time scale and a higher photon number is reached with a stationary

value of approximately 0.1. The stationary value is reached faster due to the faster inscat-

tering processes and the additional dephasing, which is introduced in the photon-assisted

polarization. The g(2)(t, 0)-function saturates at a value of 0.55, above the single-photon

level. To understand the transient regime in more detail, the photon probability distribu-

tion is plotted on the right hand side of Fig. 4.13. Here, clearly visible, is the condition of

a single photon emitter not fulfilled. It is highly probable to find a second photon in the

cavity with around 7%. If the goal is a single QD laser, the opposite is to provide, or a

stronger pumping.

A lot of effort is aiming at a single QD laser, a very efficient laser device, operating above

threshold to provide a coherent quantum light emission [MKZ+00]. Conditions for a single

QD laser is a high-Q cavity with a strong electron-light coupling, exceeding the losses to

provide induced emission processes. A β-factor close to 1is necessary, for maximally quan-

tum exploit efficiency, as well as a fast pump mechanism, e.g. electrical pump mechanism

via a carrier reservoir [KMR+10b]. In Fig. 4.14(left), the dynamics of a single QD laser

is plotted. The increased pump rates Sin = 0.33ps−1and the out-scattering Sout = 0.1Sin

enforces the electron density (solid, blue line) to saturated very fast within only 5ps. After

10 ps, electron and photon density cross and the photon density saturates at a stationary

value of approximately 2. On approximately the same time scale, the g(2)(t, 0)-function

reaches the stationary value close to 1, indicating a coherent cavity light field.

Specific for this pumping regime is the order, in which the observables enter into the sta-

tionary regime. For weak and medium pumping, the photon density saturates before the

g(2)(t, 0)-function. But in the strong regime, the g(2)(t, 0)-function saturates fast and before

the photon density at 60 ps. Coherence is reached and cannot be more enforced as it is at

4 Quantum dot - wetting layer cavity quantum electrodynamics

that point of pumping. In this regime, the Hartree-Fock approximation begins to become

problematic. For high carrier densities, higher order correlation between electrons and holes

need to be taken into account. For much higher pumping rates beyond Sin = 0.33ps−1, the

calculation need to be cross checked with higher order calculations, since the density grow

to higher values than 0.5and correlations cannot be neglected. Additionally, an adiabatic

switch-on of the pumping mechanism need to be implemented for much higher pumping

rates, since for very strong pumping rates, the instantaneous pumping of the carrier states

in the QD may lead to numerical artefacts. It does not correspond to the slowly increased

injection current into the bulk material, or the tunneling of carriers directly into the carrier

reservor for the QD. In Fig. 4.14(right), the photon probability distribution is depicted.

A Poisson distribution is visible with the maximum at 2photons, the stationary value,

reached in the time dynamics. Up to 6photon become probable in the cavity, which con-

verts the initially Fock like squeezed cavity field into a Poisson-distributed coherent field

with a fluctuation around the mean photon number of √¯n, leading to laser emission.

Note that in the plots above Fig. 4.12-4.14, signatures of Rabi oscillation are not visible.

Since the QD is initially unpopulated and the losses are too strong, the electron density

does not start to oscillate. The theoretical approach is used in the next section to investi-

gate parameter dependent photon-statistics of the emitted quantum light.

4.5.2 Parameter studies of a single QD laser device

For device optimization, parameter studies are a valuable tool to frame efficient operating

points. In the following, the photon-statistics is calculated for different parameter set. A

chosen parameter, e.g. the cavity loss, is varied, whereas the other parameters are held con-

stant, to investigate the interplay between the different physical processes of gain, emission,

absorption, dephasing and damping. The system is started and evaluated until a stationary

state is reached. The corresponding values are plotted for the varied parameter. Since the

time the stationary state is reached and the values differ strongly, the order of equations

of motion need to be adjusted to the chosen parameter set. This is done by investigating

the photon probabilities |nihn|to find an Nmax, so that pn= 0 for n≥Nmax. This

Nmax depends on the parameter set. However, for every given initial conditions and pa-

rameter an Nmax can be found, bearing in mind that, this Nmax is dynamically located

and used, controlling the validity of the calculation. Note, the theoretical results provides

mean probabilities. So the quantities presented here, represent mean values after enough

repetitions of the experiments. Though data of single experiments can not be analysed

with this methods, in this case quantum jump theories are of particular importance.

The electron-light coupling strength Mis crucial, since the time scale of the emission and

absorption processes depends on it. The QD interacts with the cavity photons via M.

The coupling matrix element includes the dipole moment of the QD between valence and

conduction band state, as well as the photon mode and the Purcell factor, determined by

the qualtiy factor of the microcavity [YTC00]. Although κdepends also on the cavity pa-

rameter, the electron - light coupling is independently varied from κ, considering different

dipole moments of the QD in a comparable cavity environment. The dipole moments de-

pends on the size, on the geometry and of the material of the QD, given the respective wave

functions [Sti01]. In Fig. 4.15, the g(2)(t, 0)-function (red line), the photon density (green

line) and the electron density (blue line) are plotted for different electron-light coupling

4 Quantum dot - wetting layer cavity quantum electrodynamics

012 3

M [1/ps]

0.0

0.2

0.4

0.6

0.8

1.0

probability

g(1)(0)

g(2)(0)

Figure 4.15: The single photon emitter in steady state condition over different values of

electron-photon coupling M. Using the parameters κ= 1/(4.3ps),γp= 1/(10ps) and the

scattering rates Se

in = 1/(4ps),Se

out = 1/(40ps),Sh

in = 1/(25ps),Sh

out = 1/(8.3ps), and

β= 1.

strength. A cavity loss of κ= 1/(4.3ps), a pure dephasing of γp= 1/(10ps)are chosen and

scattering rates of Se

in = 1/(4ps),Se

out = 1/(40ps),Sh

in = 1/(25ps),Sh

out = 1/(8.3ps). A

β= 1 is assumed. This parameter set determines the range of values for M, which result

in a change of the stationary value. Beyond this range, Mcan be varied without changing

the stationary state value, since other processes prevent an efficient photon emission, e.g.

the chosen in- and out-scattering. If the inscattering of electrons is much slower than the

electron-photon coupling strength, the in-scattering is the crucial parameter and a higher

value of the electron-photon interaction does not change the quantum light emission. This

is visible in Fig. 4.15. After a value of M= 0.5ps−1, the photon density and the electron

density do not change much for higher values for M. The electron-photon coupling strength

exceeds the inscattering rate too much and loses the impact on the stationary value. Only

the g(2)(t, 0)-function is rising slightly afterwards due to the fact, that higher correlations

in the photon field are driven more efficiently with a higher Mvia spontaneous emission

processes. The electron density fe(blue line) is decreased for stronger electron-photon

coupling: excitons in the QD recombine faster with an increased M. In consequence, the

photon density (green line) rises with increasing M, until the cavity loss comes into play

and stops the continously increment of photons in the cavity, as well as the inscattering

rates of electron and holes. For this set of parameter, the g(2)(t, 0)-function is increasing,

but remains in the single-photon limit, i.e. below a value of 0.5. With given in-scattering

and losses, a single QD laser could not be realized, even with a very high dipole moment.

Other processes prevent the increment of g(2)(t, 0)-function to 1.

Another important parameter is the cavity loss with κ=ω0/Q, defining Qas the quality

factor of the cavity, cf. 3.1.1. On the one hand, it is highly desirable to have an efficient

out-coupling of photons from the cavity, to produce a fast triggered-single-photon source

[LO05]. On the other hand, a very high-Qcavity is desirable with strong electron-light cou-

pling, a high Purcell enhancement to create photons in the desired mode. Hence, one needs

a low and a high cavity loss at the same time. The impact of the cavity loss parameter is

4 Quantum dot - wetting layer cavity quantum electrodynamics

0.5 11.5

κ [1/ps]

0.0

0.2

0.4

0.6

0.8

1.0

probability

g(1)(0)

g(2)(0)

Figure 4.16: The single photon emitter in steady state condition over different values of

cavity loss κ. Using the parameters M= 1/(2ps),γp= 1/(10ps) and the scattering rates

in = 1/(4ps),Se

out = 1/(40ps),Sh

in = 1/(25ps),Sh

out = 1/(8.3ps), and βis assumed to be

depicted in Fig. 4.16. Obviously, for a high κ, the photon density (green line) is decreased

fast and is vanishingly small for κbeyond 1ps−1. Also, the g(2)(0)-function (red line) is

decreased due to a damping of higher order correlation in the photon field: the higher the

photon correlation, the higher the losses, cf. Eq. (4.63). Although the photon-statistics of

the cavity field remains in the anti-bunching regime, the mean photon number produced

by the single photon device is for high κtoo small, to operate in an efficient way in terms

of a single-photon device. Since κdoes not increase the g(2)(t, 0)-function, or changes the

photon-statistics, a very small κis advantageous for single photon emission on demand.

The g(2)(t, 0)-function decreases only as a result of the vanishing photons in the system.

A quantum correlation effect is not visible and is not expected. The electron density (blue

line) is not changed at all. A small increase of the stationary value of the electron density

is still visible and can be explained with an additional damping of the photon-assisted

densities. For high values of κ, the emission process takes longer and the electron density

fecan be filled slightly longer with the result of an increment in the stationary value of

the electron density.

Concluding the parameter studies, the photon density and the g(2)(t, 0)-function are inves-

tigated for different pump rates and for different detunings ∆between the QD transition

frequency and the cavity mode in units of ps−1. The β-factor is assumed to be 0.9. Obvi-

ously, the detuning enforces a transition from the strong coupling into the weak regime.

In Fig. 4.17(left), the stationary value of the photon density g(1)(0) for different detuning

and increasing inscattering is plotted. For a detuning ∆ = 1 ps−1(blue line), in order

of magnitude of the electron-photon coupling strength M= 0.5ps−1, the mean photon

number depends approximately linearly on the pump rate. With increasing pump rate, the

mean photon number rises also, outrivaling the cavity loss. For a stronger detuning (red

and orange line) with ∆ = 2 ps−1and ∆ = 3 ps−1, the detuning effect is stronger than

the electron-photon coupling and the mean photon number increases also, but slower. For

an increasing detuning, the emission of photons is less probable, until the emission stops

ultimately, if the detuning is too big and no further feeding mechanisms are considered

4 Quantum dot - wetting layer cavity quantum electrodynamics

0 0.2 0.4 0.6

Sin[ps-1]

g(1)(0)

0 0.2 0.4 0.6

Sin [ps-1]

0.5

1.5

g(2)(0)

∆=1

∆=2

∆=3

Figure 4.17: The single photon emitter in steady state condition over different values of

the inscattering rate Sin and for different detuning ∆between the cavity mode and the

QD transition energy in units of ps−1. Using the parameters M= 1/(2ps),γp= 1/(10ps)

and βis assumed to be 0.9.

[MKH08; Hoh10; WVT+09; TS10]. On the right hand side of Fig. 4.17, the g(2)(t, 0)-

function in dependence on the pump rate and on the detuning is plotted. In contrast to

the photon density, the detuning has a surprising impact on the g(2)(t, 0)-function. For a

small detuning (blue line), the g(2)(t, 0)-function rises and saturates at a stationary value

of approximately 1. The QD-carrier reservoir system operates in the laser limit well above

threshold. A Poisson-distribution is obtained. But for a higher detuning a remarkable effect

is visible. The g(2)(t, 0)-function does not progress to 1, but overshoots into the bunching

regime, reaches a maximum value and progresses then back to 1for high pump rates. This

effect is widely discussed in recent experimental and theoretical work and is attributed to

background noise [NKI+10], to a biexciton contribution[RGGJ10] or, like in this case, to a

detuning of the QD-cavity resonance. In the Hartree-Fock limit, higher pump rates become

problematic. The unfactorized correlation between electron-and holes must be negligible.

Here, the investigation is focused on the single-photon limit, not on the laser limit. To

investigate the laser dynamics of a single QD laser in the high pumping limit, it is neces-

sary to go beyond the Hartree-Fock factorization in the similar manner like the SCE. The

possibility of the PPCE to include correlation of higher order is one of its strength and

is in preparation. Remarkable, the bunching transition dynamics is observable within the

Hartree-Fock limit and within the given parameter set.

4.5.3 Atom and QD-WL rate equations in the single-photon limit

The main difference between an atomic model and a QD-carrier reservoir (WL) model stems

from the validity of the one-electron assumption (OEA), which is typically valid in atomic

system. In atomic systems electron and holes are correlated, e.g. via a†

cac+a†

vav= 1

[BY99]. In the atomic limit, the Hartree-Fock is not necessary. Quantities like a†

ca†

vavac

vanish and the spontaneous emission is only driven by the electron density fe=a†

cac.

4 Quantum dot - wetting layer cavity quantum electrodynamics

0.001 0.01 0.1

Sin[ps-1]

0.5

1.5

g(1)(0)

OEA

0.001 0.01 0.1

Sin[ps-1]

0.5

1.5

g(2)(0)

Figure 4.18: Photon density g(1)(0) and g(2)(t, 0)-function in a steady state condition over

different values of the inscattering rate Sin in an atomic (OEA) and in a QD-carrier reservoir

system (CR): ∆ = 0.05 ps−1,κ= 0.03 ps−1M= 1/(5ps),γp= 0.05 ps−1and βis assumed

to be 0.98 and Sout = 0.1Sin.

In consequence, the photon-assisted polarization is strongly modified in comparison to

Eq. (4.64) and reads:

∂ta†

vac|n+ 1ihn|= (−i∆−γp)a†

vac|n+ 1ihn|−iM√n+ 1(fe

n−pn+1 +fh

n+1).(4.66)

The laser rate equations are derived by assuming the polarization dynamics to be stationary

on the emission process time scale. Eq. (4.66) is solved in the adiabatic limit. The time-

derivative is assumed to be zero. With the given equation for the polarization, using the

one-electron assumption and applying second order Born-approximation: fe

n≈fepn, the

PPCE equations Eq. (4.63) - Eq. (4.65) reduce to the well-known laser rate equation for

the photon density [Hak94]:

∂tc†c=−2κc†c+β

τsp hc†c+ 1ife(4.67)

with c†c=Pnn pnand to close the set of equations, one obtains for the electron

dynamics with the expression of the microscopic polarization and in second order Born:

∂tfe=−1

τsp

fe−β

τsp

fec†c+Sin(1 −fe)−Soutfe.(4.68)

Given a constant pump rate, dephasing and cavity losses, the observable enter eventu-

ally in the stationary regime, in which the photon density and the g(2)(t, 0)-function can

be compared. In Fig. 4.18, the stationary values of the photon density g(1)(0) (left) and

100

4 Quantum dot - wetting layer cavity quantum electrodynamics

the intensity-intensity correlaton g(2)(t, 0)-function (right) is plotted for different in- and

outscattering Sin/out. The parameter set is given in the caption of Fig. 4.18. The cavity dy-

namics are evaluated for a atomic system (orange, dashed line, OEA) and for a QD-carrier

reservoir system (green, solid line,CR). Comparing the stationary values of the photon

density in Fig. 4.18(left), between the atomic and QD-CR-system, the difference is small.

With increasing pump rate, the mean photon number rises gradually. The higher the pump

rate, the faster the mean photon number rises. The atomic system shows a slightly faster

output of photons from a pump rate of Sin = 0.005 ps−1until Sin = 0.2ps−1. At this

value, both output curves start to converge again. However, this small difference indicates

a remarkable feature of the QD-CR system, concerning single-photon emission on demand.

Intensity-intensity correlation are surpressed without the lack of photons. The photon den-

sity is almost the same, but not the g(2)(t, 0)-function, cf. Fig. 4.18(right). For the atomic

system and with given parameters, the g(2)(t, 0)-function starts in the anti-bunching regime

at a value of 0.75 and rises gradually with increasing pump rate to the coherent regime

around 1. This indicates, the transition into the laser regime. However, with a similar out-

put curve, the QD-CR system shows a wide range of strongly anti-bunched quantum light

emission. Even for a pump rate of Sin = 0.01 ps−1, the device operates in the single-photon

regime, below a value of 0.5due to the fact, that the semiconductor specific property, uncor-

related electron and hole densities, lead to an enhanced Pauli-blocking, which in turn leads

to a decreased emission rate into the laser mode. This enhanced Pauli-blocking originates

from the spontaneous emission term: fe

nfh

n/pn, which is in the atomic case only propor-

tional to fe. In typical experimental data [NKI+10], the value of the g(2)(t, 0)-function in a

steady state condition is much lower than theoretical calculations predict. Those numeri-

cal simulations are based on the one-electron assumption and do not include the enhanced

Pauli-blocking due to the presence of the wetting layer. Typical experimental values of

the g(2)(t, 0)-function are well-below 0.5and thus in the single-photon limit. Within the

PPCE, this strong anti-bunching is enforced by the modified spontaneous emission term,

cf. Fig. 4.18 and agrees better with experimental data [NKI+10].

Being disadvantageous in other regimes (lasing), the single photon emission is enhanced

in this case and only modelled correctly within the PPCE-approach. This proves the im-

portance of microscopical models, which take into account carrier-carrier scattering to

investigate interesting operating conditions for technological applications, and to reveal

advantageous properties within a semiconductor environment.

101

Alles Wissen und alles Vermehren unseres

Wissens endet nicht mit einem Schlusspunkt,

sondern mit einem Fragezeichen.

Hermann Hesse

5 Conclusion and outlook

This thesis focuses on the single-photon regime of the semiconductor quantum dot-cavity

quantum electrodynamics (QD-CQED) beyond the Jaynes-Cummings model, including

non-Markovian contributions and many-particle interactions such as electron-phonon cou-

pling or enhanced Pauli-blocking on a microscopic level. Two theoretical frameworks are

introduced to provide a simulation tool for parameter studies of single-photon devices: (i)

the mathematical induction model solves the QD-CQED including the electron-LO phonon

interaction up to an arbitrary accuracy in case of a fixed number of electrons in the quan-

tum dot; and (ii) in the presence of a carrier reservoir, the photon-probability cluster

expansion (PPCE) is devoloped and introduces a modified Hartree-Fock factorization.

Within the mathematical induction model, the conversion of a thermal into a non-classical

cavity field with sub-Poissonian statistics has been shown and LO-phonon assisted cav-

ity feeding with additional anti-crossing signatures is discussed. Present anharmonicities

in the Rabi oscillations is attributed to the mixing of Rabi frequencies for elevated tem-

peratures. The modified Rabi frequency is expressed using the Huang-Rhys factor. An

analytical derivation with amplitude modulation is a next step to simplify the LO-phonon

QD-CQED via an effective Hamiltonian approach.

The biexciton cascade is calculated in the strong and weak coupling regime. The polariza-

tion entanglement of the generated photons is theoretically investigated. Via an effective

multi-phonon Hamiltonian approach, a temperature dependent analysis of the degree of

entanglement has been performed. For elevated temperatures (T≥120 K) in GaAs, the

polarization entanglement vanishes due to the presence of the wetting layer. Further in-

vestigations, in particular via the induction method in the strong coupling regime and of

LO-phonon coupling to the QD excitons may reveal other probabilities for an enhanced

degree of entanglement. For example, the LO-phonon sidebands may enforce an erasing

of the which-path information, similiar to the modulation of the exciton and biexciton

energies via external applied electrical fields.

Within the PPCE, an electrically driven single-photon emitter has been studied, includ-

ing enhanced Pauli-blocking in a modified spontaneous emission source term. This many-

particle specific Pauli-blocking is advantageous for generating single photons in a wide

range of the electrical pump strength in comparison to atom-based single photon emitter

devices. To calculate a single-QD laser operating well above threshold, the investigation

needs to go beyond the introduced modified Hartree-Fock limit. As a next step, correction

terms will be included into the calculation to study threshold behavior in the g(2)(t, 0)-

function. Additionally, the electron-phonon interaction has to be included to motivate the

phenomologically introduced pure dephasing microscopically.

102

5 Conclusion and outlook

103

A Appendix

A.1 Weakly driven QDs: equations of motion

In Förster-coupled nanostructures exciton transfer occurs. The equation of motion reads:

∂tB1†

mB1

n=−i(ωn−ωm−2iΓ)B1†

mB1

n(A.1)

−i¯

M∗

m,2B2†B1

nc−¯

Mn,2B1†

mB2c†+¯

MmG†B1

nc†−¯

M∗

nB1†

mGc.

The electronic densities in Eq. (2.28) - (2.30) couple to photon-assisted polarisations. The

photon-assisted polarisations in the second-order of electron-light coupling read:

∂tG†B1

mc†=−i(ωm−ω−iΓ) G†B1

mc†+i¯

M∗

mG†Gc†c+¯

Mm,2G†B2c†c†

−iX

M∗

lB1†

lB1

mc†c+B1†

lB1

m,(A.2)

∂tB1†

mB2c†=−i(ωB−ωm−ω−3iΓ)B1†

mB2c†+iX

M∗

l,2B1†

mB1

lc†c

−i¯

M∗

m,2B2†B2c†c+¯

M∗

m,2B2†B2−i¯

MmG†B2c†c†.(A.3)

This quantity depends strongly on the energy splitting, thus on the Coulomb-parameter.

It oscillates for strong Förster-coupling and is driven by the photon-assisted polarisations,

which couple to photon-assisted exciton-densities:

∂tG†G c†c= 2Γ B1†

1B1

1c†c+B1†

2B1

2c†c−2X

Im¯

MmG†B1

mc†

−2X

Im¯

MmG†B1

mc†c†c,(A.4)

∂tB2†B2c†c=−4Γ B2†B2c†c+ 2 X

Im¯

Mm,2B1†

mB2c†c†c,(A.5)

∂tB1†

mB1

nc†c=−i(ω1

n−ω1

m−2iΓ) B1†

mB1

nc†c

−i¯

MmG†B1

nc†c†c−¯

M∗

nB1†

mGc†cc

−i¯

M∗

m,2B2†B1

nc†cc−¯

Mn,2B1†

mB2c†c†c

−i¯

M∗

m,2B2†B1

nc−¯

Mn,2B1†

mB2c†.(A.6)

A Appendix

These photon-assisted exciton densities decay with the Einstein-coefficient Γ, exactly like

the exciton densities. The third order in the electron-light coupling element ¯

Mmcouples

to the fourth order, which as well includes the two-photon polarisation:

∂tG†B2c†c†=−i(ωB−2ω−2iΓ)G†B2c†c†+iX

M∗

m,2G†B1

mc†c†c(A.7)

−iX

M∗

m2B1†

mB2c†+B1†

mB2c†c†c.

The two-photon polarisation opens two additional channels to excite a biexciton in the

coupled system. Due to this quantity, which influences the quantitative results strongly, a

simple analytical solution cannot be derived.

∂tG†B1

nc†c†c=−iω1

n−ω−iΓG†B1

mc†c†c−2iX

M∗

lB1†

lB1

nc†c

−iX

M∗

lB1†

lB1

nc†c†cc−i¯

M∗

nG†Gc†c†cc

+i¯

Mn,2G†B2c†c†c†c+i¯

Mn,2G†B2c†c†,(A.8)

∂tB1†

mB2c†c†c=−iω2−ω1

m−ω−3iΓB1†

mB2c†c†c−2i¯

M∗

m,2B2†B2c†c

−i¯

M∗

m,2B2†B2c†c†cc+iX

M∗

l,2B1†

mB1

lc†c†cc

−i¯

MmG†B2c†c†c†c.(A.9)

Now, the set of equations of motions is given to calculate two Coulomb coupled QDs in

the weak coupling regime.

A.2 Numerical parameters

Parameter Symbol Value

electron mass m05.6856800 fs2eV nm−2

electron effective mass me0.043m0[R¨

02]

hole effective mass mh0.450m0[R¨

02]

LO phonon energy ~ωLO 36.4 meV [R¨

02]

QD band gap ~ωcv 1.5 eV

Electron confinement energy ~ωel 50 meV

Hole confinement energy ~ωh25 meV

QD diameter l=q~2ln2

m∗

em0~ωel 15,77 nm

High frequency dielectric constant ǫ∞10.9

Static dielectric constant ǫstat 12.53

A Appendix

A.3 Biexciton cascade in the strong coupling regime:

Equations of motions

Phenomenologically, the cavity loss is introduced κ, cf. Sec. 3.1.2. Since the two-electron

assumption holds, the biexciton couples only to energetic lower states, such as the inter-

mediate exciton states via the corresponding transitions:

∂tX†

HBHm,nVp,q(A.10)

=i(m−n)ω0

H+ (p−q)ω0

V+ωH−ωB+iκ(m+n+p+q)X†

HBHm,nVp,q

−iMG†BHm+1,nVp,q−iM mB†BHm−1,nVp,q

−iMB†BHm,n+1Vp,q+iMX†

HXHHm,n+1Vp,q−iMX†

HXVHm,n+1Vp,q+1

∂tX†

VBHm,nVp,q(A.11)

=i(m−n)ω0

H+ (p−q)ω0

V+ωV−ωB+iκ(m+n+p+q)X†

VBHm,nVp,q

−iMG†BHm,nVp+1,q+iM pB†BHm,nVp−1,q

+iMB†BHm,nVp,q+1+iMX†

VXHHm,n+1Vp,q−iMX†

VXVHm,nVp,q+1

In the photon-assisted exciton-biexciton transition is driven via the spontaneous emission

due to the biexciton relaxation. Either a horizontal or vertical photon is emitted. Note,

the frequency ωH/V

0originates from the cavity frequency and must not be in resonance

with transition energy ωH/V −ωB. The spontaneous emission process is crucial in the weak

coupling regime. In the strong coupled dynamics, the exciton-biexciton transition is also

driven by the photon-assisted exciton density:

∂tX†

HXHHm,nVp,q(A.12)

=i(m−n)ω0

H+ (p−q)ω0

V+iκ(m+n+p+q)X†

HXHHm,nVp,q

−iMG†XHHm+1,nVp,q−iM m B†XHHm−1,nVp,q−iMB†XHHm,n+1Vp,q

+iMX†

HGHm,n+1Vp,q+iM n X†

HBHm,n−1Vp,q+iMX†

HBHm+1,nVp,q

∂tX†

VXVHm,nVp,q(A.13)

=i(m−n)ω0

H+ (p−q)ω0

V+iκ(m+n+p+q)X†

VXVHm,nVp,q

−iMG†XVVp+1,q+iM p B†XVHm,nVp−1,q+iMB†XVVp,q+1

+iMG†XVHm,nVp,q+1−iM q X†

VBHm,nVp,q−1−iMX†

VBHm,nVp+1,q.

The photon-assisted intermediate exciton densities couple to the higher electronic levels

via the transition to the biexciton and to the lower electronic level via the transition to

the ground state. Due to spontaneous emission processes, the exciton densities are driven

stronger by the biexciton transition than by the ground state transition. The exciton-

exciton transition (X†

HXV) has a significant impact on the exciton dynamics, if the two

intermediate exciton levels are energetically close. This exchange of excitation energy is

iii

A Appendix

of great importance for the degree of entanglement, since in this quantity the relaxation

paths are crossing over.

∂tX†

VXHHm,nVp,q(A.14)

=i(m−n)ω0

H+ (p−q)ω0

V+ωV−ωH+iκ(m+n+p+q)X†

VXHHm,nVp,q

−iMG†XHHm,nVp+1,q+iM p B†XHHm,nVp−1,q+iMB†XHHm,nVp,q+1

+iMX†

VGHm,n+1Vp,q+iM n X†

VBHm,n−1Vp,q+iMX†

VBHm+1,nVp,q

The smaller the fine structure splitting, the more is this exciton-exciton transition density

like and is driven strongly by the photon-assisted polarization. Due to this quantity, it

is possible that vertical polarized photons are emitted in a biexciton cascade, which is

driven only by horizontal polarized photons. The horizontal polarized photon density can

be converted completely into vertical polarized photons, if the energy splitting between the

intermediate exciton levels is smaller than the cavity loss, i.e. some µeV. The conversion is

driven by the ground to exciton state transitions:

∂tG†XHHm,nVp,q(A.15)

=i(m−n)ω0

H+ (p−q)ω0

V−ωH+iκ(m+n+p+q)G†XHHm,nVp,q

−iM m X†

HXHHm−1,nVp,q−iMX†

HXHHm,n+1Vp,q

−iM p X†

VXHHm,nVp−1,q+iMG†GHm,n+1Vp,q+iM n G†BHm,n−1Vp,q

−iMX†

VXHHm,nVp,q+1+iMG†BHm+1,nVp,q

∂tG†XVHm,nVp,q(A.16)

=i(m−n)ω0

H+ (p−q)ω0

V−ωV+iκ(m+n+p+q)G†XVHm,nVp,q

−iMX†

HXVHm,n+1Vp,q−iM p X†

VXVHm,nVp−1,q

−iM m X†

HXVHm−1,nVp,q−iMX†

VXVHm,nVp,q+1+iMG†GHm,nVp,q+1

−iM q G†BHm,nVp,q−1−iMG†BHm,nVp+1,q

The ground to exciton transition is formally equivalent to the exciton to biexciton transi-

tion. It includes spontaneous emission processes, induced absorption and emission via the

phase filling factor, or photon-assisted inversion [SZ97]. Two new quantities appear, the

photon-assisted ground state density and the ground to biexciton state transition (G†B),

which is another crucial and characteristic quantity in the four-level exciton dynamics.

∂tG†BHm,nVp,q(A.17)

=i(m−n)ω0

H+ (p−q)ω0

V−ωB+iκ(m+n+p+q)G†BHm,nVp,q

−iM mX†

HBHm−1,nVp,q−iMX†

HBHm,n+1Vp,q−iMG†XVHm,nVp,q+1

−iM pX†

VBHm,nVp−1,q−iMX†

VBHm,nVp,q+1+iMG†XHHm,n+1Vp,q

The ground to biexciton transition does not couple the biexciton to the ground state

density, but the exciton-biexciton to the ground to exciton transitions and is formally

A Appendix

equivalent to the exciton-exciton transition. Due to this transition, the relaxation path can

also interfere. Finally, the ground state density reads:

∂tG†GHm,nVp,q(A.18)

=i(m−n)ω0

H+ (p−q)ω0

V+iκ(m+n+p+q)G†GHm,nVp,q

−iM mX†

HGHm−1,nVp,q−iMX†

HGHm,n+1Vp,q−iM pX†

VGHm,nVp−1,q

−iMX†

VGHm,nVp,q+1+iM nG†XHHm,n−1Vp,q+iMG†XHHm+1,nVp,q

+iM qG†XVHm,nVp,q−1+iMG†XVHm,nVp+1,q

The ground state is populated by all relaxation processes as the energetic lowest electronic

state. The ground state to horizontal exciton state transition reads:

∂tG†XHHm,nVp,q(A.19)

=i(m−n)ω0

H+ (p−q)ω0

V−ωH+iκ(m+n+p+q)G†XHHm,nVp,q

−iM m X†

HXHHm−1,nVp,q−iMX†

HXHHm,n+1Vp,q

−iM p X†

VXHHm,nVp−1,q+iMG†GHm,n+1Vp,q+iM n G†BHm,n−1Vp,q

−iMX†

VXHHm,nVp,q+1+iMG†BHm+1,nVp,q.

The ground to exciton transition is formally equivalent to the exciton to biexciton transition

A.4 Biexciton cascade in the weak coupling regime:

Equations of motions

The equation of motion of a biexciton cascade in the weak coupling regime differ from the

strong copuling regime in the number of photon operators, which are taken into account.

The temporal evolution of the driving terms in Eq. (3.62) is given by

∂tG†XHc†

Vc†

VcH=i−ωH+ 2ωV

0−ωH

0+iΓ + 3iκG†XHc†

Vc†

VcH(A.20)

−2iMX†

VXHc†

VcH+iMG†Bc†

Vc†

V

and

∂tX†

VG c†

VcHcH=iωV+ωV

0−2ωH

0+iΓ + 3iκX†

VG c†

VcHcH(A.21)

+ 2iMX†

VXHc†

VcH+iMB†G cHcH.

The driving terms of the two-photon density matrix in turn couple to combined exciton-

and photon coherences X†

VXHc†

VcHand to the direct decay channel from B†Bto G†G

emitting two photons with the same polarization G†Bc†

Vc†

V. Crucial for entangling the two

decay paths is the exciton-exciton transition, assisted by a photon coherence:

∂tX†

VXHc†

VcH=iωV−ωH+ωV

0−ωH

0+ 2iΓ + 2iκX†

VXHc†

VcH

−iM G†XHc†

Vc†

VcH+iM B†XHcH

+iMX†

VG c†

VcHcH+iMX†

VBc†

V.(A.22)

A Appendix

In this equation, the two paths interfere. The influence in the two-particle correlation

X†

VXHc†

VcHincreases the degree of entanglement as this term couples back to the driving

terms of ρV H , Eq. (A.20) and (A.21). Here again the resonance condition of the frequencies

is essential (ωV−ωH=ωV

0−ωH

0=δ): A high detuning δwill diminish the contribution

of Eq. (A.22) to the cascade and both paths cannot interfere. The other characteristic

and important quantity in the two-electron biexciton-cascade situation are the two-photon

polarizations

∂tG†Bc†

Vc†

V=i−ωB+ 2ωV

0+ 2iΓ + 2iκG†Bc†

Vc†

V(A.23)

+iM G†XHc†

Vc†

VcH−iM G†XVc†

Vc†

VcV−2iM X†

VBc†

V

and

∂tB†G cHcH=iωB−2ωH

0+ 2iΓ + 2iκB†G cHcH(A.24)

−iM X†

HG c†

HcHcH+iM X†

VG c†

VcHcH+ 2iM B†XHcH.

Each path in the cascade has one biexciton-to-ground state transition like G†Bc†

Vc†

V. Its

dynamics couples the biexciton-to-exciton transition X†

VBc†

Vwith both exciton-to-ground

state transitions G†Xi. Remarkably, the origin of the entanglement is directly visible,

since a quantity of a different path enters in Eq. (A.23): G†XHc†

Vc†

VcH. Here again, the

two paths interfere. For maximum entanglement the contributions of the different paths

G†XHand G†XVto the expectation values should be equally weighted. The photon-

assisted biexciton-to-exciton transition enters in the two-photon polarization and drives

this quantity via the biexciton decay:

∂tB†XHcH=iωB−ωH−ωH

0+ 3iΓ + iκB†XHcH+iMB†B(A.25)

−iM X†

HXHc†

HcH+iM X†

VXHc†

VcH+iM B†G cHcH,

∂tX†

VBc†

V=i−ωB+ωV+ωV

0+ 3iΓ + iκX†

VBc†

V+iM B†B(A.26)

−iM G†Bc†

Vc†

V+iM X†

VXHc†

VcH−iM X†

VXVc†

VcV.

The occurring biexciton as well as the intermediate exciton-photon densities are driven by

the biexciton-exciton transition X†

iBc†

i:

∂tX†

HXHc†

HcH=−(2Γ + 2κ)X†

HXHc†

HcH(A.27)

−2ImMX†

HBc†

H+MX†

HG c†

HcHcH,

∂tX†

VX†

Vc†

VcV=−(2Γ + 2κ)X†

VX†

Vc†

VcV(A.28)

+ 2 ImMX†

VBc†

V−MX†

VG c†

VcVcV.

In the visualization of the complex interplay, it is a bottom-to-top trail followed through

the cascade, starting with the concurrence determining ρV H. The biexciton B†Bas the

A Appendix

top element of the scheme decays via the Hor the Vintermediate exciton-to-ground-state

path

∂tB†B=−4ΓB†B+ 2 Im MX†

HBc†

H−MX†

VBc†

V.(A.29)

To complete the set of equation, two higher-order photon-assisted exciton-to-ground state

transitions of the direct and thus not entangled path are necessary:

∂tG†XHc†

Hc†

HcH=i−ωH+ωH

0+iΓ + 3iκG†XHc†

Hc†

HcH(A.30)

−2iM X†

HXHc†

HcH+iM G†Bc†

Hc†

H,

∂tG†XVc†

Vc†

VcV=i−ωV+ωV

0+iΓ + 3iκG†XVc†

Vc†

VcV(A.31)

−2iM X†

VXVc†

VcV−iM G†Bc†

Vc†

V.

With these polarization Eq. (A.30-A.31), the diagonal elements i=H, V of the density

matrix of the polarization subspace are given, too:

∂tc†

ic†

icici=−4κc†

ic†

icici−4ImMG†X†

ic†

ici.(A.32)

A.5 Derivation of the Hartree-Fock factorization

To close the set of differential equations, one needs to factorize the expectation values at a

given level, e.g. on a Hartree-Fock level: a†

ia†

jakal≈a†

iala†

jak−a†

iaka†

jal. This

factorization rule is derived via the approximation, that the electron-electron correlations

can be expressed as a general canonical statistical operator (GCSO), including only single

particle contributions.[Fri96; FS90] That is: the single particle dynamics is described in a

mean field, induced by the other particles. The GCSO reads:

ρ≈ρHF =1

Ze−Pij λij a†

iaj, Z =tr(e−Pij λij a†

iaj),(A.33)

with the partition function as a normalization to secure tr(ρ) = 1 and a(†)

ifermionic

operators with [a†

i, aj]+=δij and a†

ia†

j=−a†

ja†

i. Now, one introduces a unitary matrix φ

to diagonalize, so that λD=φλφ∗, which exist as long as HHF =Pij λij a†

iajconsists of

observables, i.e. HHF is hermitian. The matrix element reads:

λij =X

φ∗

inλD

nmφmj =X

φ∗

inλD

nnφnj.(A.34)

The GCSO can now be written in this new basis and is diagonal:

ρHF =1

Ze−PnλD

nn d†

ndn,(A.35)

with the new operators

d(†)

n=X

φ(∗)

in a(†)

i,(A.36)

vii

A Appendix

inheriting the fermionic character. The partition function can be calculated, given a com-

plete set of eigenfunction of the new operators:

|ΨNi:= |n1, n2, n3, ..., nNi=d†

1d†

2···d†

N|0i=−|n2, n1, n3, ..., nNi.(A.37)

With this definition, one can calculate with Pithe number of necessary anti-commutations

to bring the state ito the front:

d†

idi|ΨNi=d†

idi(−1)Pi|ni, n1, ..., ni−1, ni+1, ..., nNi=ni(−1)Pi2|ΨNi=ni|ΨNi.(A.38)

The trace is the summation of every state, which can be occupied ni= 1 or be unoccupied

ni= 0, taken into account Pauli’s principle. The trace of the diagonalized operator reads:

Z=X

{ni}hΨN|e−PnλD

nn d†

ndn|ΨNi=

n1=0

n2=0 ···

nN=0hΨN|e−PnλD

nn d†

ndn|ΨNi(A.39)

n1=0

n2=0 ···

nN=0hΨN|e−PN−1

n=1 λD

nn d†

ndn|ΨNie−λD

NN nN

n1=0

n2=0 ···

nN=0

e−λD

11 n1e−λD

22 n2···e−λD

NN nN

= (1 + e−λD

11 )(1 + e−λD

22 )···(1 + e−λD

NN ) = Πk(1 + e−λD

kk ).

Now, the GCSO is explicitly given and one can calculate on a Hartree-Fock level the

expectation value of the electron-electron correlations. The approximation is done in the

choice of the GCSO, in which the observables of interest are included, here single particle

contribution.

a†

ia†

jakal=tr(a†

ia†

jakalρ)≈X

abcd

φaiφbjφ∗

kcφ∗

ldtr(ρHF d†

ad†

bdcdd).(A.40)

The trace must now be taken into account, for that, one investigates:

d†

ad†

bdcdd|nd, nc, ..., nNi=√ncndhnc, nd, ...nN|d†

ad†

b|nd−1, nc−1, ..., nNi(A.41)

Summing over bresults only in two contributions, since the states are orthogonal:

d†

ad†

bdcdd|ΨNi=√ndncd†

a|nd−1, nc, ..., nNiδb,c (A.42)

+nd√ncd†

a|nd, nc−1, ..., nNiδb,d

=ndnc|ΨNi(δa,dδb,c −δa,cδb,d).

viii

A Appendix

Now, the expectation value can be calculated:

a†

ia†

jakal≈X

{ni}X

abcd

φaiφbjφ∗

kcφ∗

ldhΨN|ρHF d†

ad†

bdcdd|ΨNi(A.43)

cd X

{ni}

ncndhΨN|ρHF |ΨNiφdiφcjφ∗

kcφ∗

ld −φciφdjφ∗

kcφ∗

ld

nc,nd=0 "nce−λD

ccnc

(1 + e−λD

cc )#"nde−λD

ddnd

(1 + e−λD

dd )#φdiφcjφ∗

kcφ∗

ld −φciφdjφ∗

kcφ∗

ld

cd "φdie−λD

cc φ∗

(1 + e−λD

cc )#"φcje−λD

dd φ∗

(1 + e−λD

dd )#−"φdje−λD

cc φ∗

(1 + e−λD

cc )#"φcie−λD

dd φ∗

(1 + e−λD

dd )#

=a†

iala†

jak−a†

iaka†

jal.

After choosing the Hartree-Fock GCSO, the calculation leads to automatically to the fac-

torization.

A.6 Photon probability picture

In Eq. (4.14) a new form of the phase-filling factor is derived. The spontaneous emission and

induced emission is included in the quantity a†

cac|nihn|, whereas the induced absorption

is included in a†

vav|n+ 1ihn+ 1|. To see that, one can transform Eq. (4.14) back into

the photon operator picture. E.g., one sums over nafter multiplying by √n+ 1, keeping

in mind the annihilation and creation operator relations:

c†|ni=√n+ 1 |n+ 1i ≡ hn|c=hn+ 1|√n+ 1,(A.44)

c|ni=√n|n−1i ≡ hn|c†=hn−1|√n. (A.45)

With that, one transforms the equation of the photon-assisted polarization part by part

back:

√n+ 1a†

vac|n+ 1ihn|=X

na†

vacc†|nihn|=a†

vacc† X

n|nihn|!=a†

vacc†.

The inhomogeneity in Eq. (4.14), are already multiplied by √n+ 1, so we have:

(n+ 1)a†

cac|nihn|=X

na†

cac|nihn|+na†

cac|nihn|(A.46)

=a†

cac+X

na†

cacc†c|nihn|=a†

cac+a†

cacc†c

and

(n+ 1)a†

vav|n+ 1ihn+ 1|=X

na†

vavc†|nihn|c=a†

vavc†c.(A.47)

A Appendix

We come up with the photon-assisted polarization, in the second order of the electron-light

coupling in its known form:

∂ta†

vacc†=−i(ωcv −ω0)a†

vacc†−i M a†

cac+a†

cacc†c−a†

vavc†c,(A.48)

and now the spontaneous emission term, proportional to the excited state density, and

induced absorption and emission in the phase-filling factor. Remarkable, in dependence

on the factor one multiplies the equation, one automatically derives different order in the

electron-photon coupling element. As an example, we investigate the photon probability

|nihn|:

n|nihn|=c†c(A.49)

n(n−1) |nihn|=c†c†cc.(A.50)

This works for every expectation value. Thus, one needs for the whole dynamics in arbri-

trary orders only four equations of motions.

A.7 Jaynes-Cummings solution in the photon probability

picture

In the one-electron assumption without losses and other interactions, the exact solution is

given by the JCM. In case of the vacuum Rabi oscillation, i.e. zero photons in the cavity and

a initially excited two-level system, the photon density reads: hc†ci(t) = sin2(M t).[JC63]

We start by showing that the PPCE reproduces this solution. In the one-electron case, the

photon-assisted polarization reads for ωcv −ω0= 0:

∂th|n+ 1ihn|a†

vaci= +i M √n+ 1(pn+1 −fh

n+1)−i M √n+ 1fe

n.(A.51)

Here, we use the electron-hole picture. Note, the n-quantity considers spontaneous emission,

since the dynamics couple to a lower photon probability in comparison to the photon-

assisted polarization and in contrast to the n+1-quantities. The equation of motion for the

photon-assisted electron and hole densities couple again to the photon-assisted polarization:

∂tfe/h

n= 2√n+ 1Im(M|n+ 1ihn|a†

vac).(A.52)

For fixed initial conditions, without environment coupling, the set of equations [Eq. (A.51)

and A.52] can be solved analytically, e.g. in the case of vacuum Rabi oscillation. The initial

conditions are: fe

0=fh

0= 1, p0= 1, p1= 0. This solution is obtained by differentiating Eq.

(A.52) for n= 0 and (4.13) for n= 1 with respect to time:

∂2

tfe/h

0= 2Im(M ∂t|1ih0|a†

vac) = −∂2

tp1.(A.53)

A Appendix

This is valid, as long, as higher order photon assisted polarization are not driven, i.e.

a†

vac|n+ 1ihn|for n > 0. Investing the next higher photon assisted polarization n= 1:

∂t|2ih1|a†

vac= +i M √2(p2−fh

2)−i M √2fe

1,(A.54)

which is initially zero. To start the dynamics, one of the inhomogeneties must be inequal

to zero at least at one time. fe

1is initially zero and couples back to |2ih1|a†

vacand

cannot start the dynamics; the same for fh

2, which couples to a even higher photon assisted

polarization and is initially zero as well. At last, p2needs to be investigated. Since there

is initially zero photons in the cavity, and there is only one excitation in the system, it

is physically impossible to come up with a probability for two photons, in the rotating

wave assumption and considering a band gap of the quantum dot of at least order of

magnitude eV . Time and energy uncertainty forbids an existence of an photon longer than

a tenth of a femtosecond. But it is also mathematically clear, p2couples to the photon-

assisted polarization of |3ih2|a†

vac, which gives clearly no contribution, and couples back

to |2ih1|a†

vac. As long as initially p2is zero, higher photon assisted polarization can be

neglected and Eq. (A.53) is valid and the photon assisted polarization of n= 0 reads:

∂t|1ih0|a†

vac= +i M (p1−fe

0),(A.55)

since fh

1is not driven and initially zero, it couples to Eq. (A.54). Obviously, it is now

convenient, to investigate the dynamics of the difference between the electron density of

n0and the photon probability of n= 1, this

∂2

t(fe

0−p1) = 2Im(M ∂t|1ih0|a†

vac)−∂2

tp1(A.56)

= 4Im(M ∂t|1ih0|a†

vac)

After inserting the photon-assisted polarization for n= 0 from Eq. (A.51), the set of

equations of motions is closed, since all other quantities in higher photon manifolds are

not driven with given initial conditions:

∂2

t(fe

0−p1) = −(2M)2(fe

0−p1).(A.57)

The solution reads fe

0(t)−p1(t) = cos (2M t) = cos2(M t)−sin2(M t). Since fe

0(t= 0) =

1, one identifies fe

0(t) = cos2(M t)and p1(t) = sin2(M t). With the initial conditions, the

photon density depends only on p1and p0and reads:

hc†ci(t) = = 0 ·p0+ 1 ·p1=p1(t) = sin2(M t),(A.58)

the JCM solution is reproduced. The oscillation frequency depends on the number of

photons in the cavity and the off-diagonal light coupling Mvc, forming together the Rabi

frequency [JC63]. Thus, the JCM is naturally contained in the PPCE-approach.[RCSK09a]

Another benchmark is to include the photon-statistics via initial conditions like in the JCM,

which is done in the next subsection.

A Appendix

A.8 Modified photon-assisted polarization (electron picture)

The photon-assisted polarization reads in the electron picture:

∂ta†

vac|n+ 1ihn|(A.59)

=−i(ωcv −ω0)a†

vac|n+ 1ihn|

+i M √n+ 1 a†

vav|n+ 1ihn+ 1|1−1

pn+1a†

cac|n+ 1ihn+ 1|

−i M√n+ 1 a†

cac|nihn|1−1

pna†

vav|nihn|.

A.9 Cavity loss in the photon probability picture

The coupling of the cavity modes to the external modes is decribed by an additional part

in the Hamiltonoperator,[VWW01] which reads together with the homogenous part of the

external photons:

Hpt−pt =~X

ωqd†

qdq+~X

Gq(c†+c)(d†

q+dq),(A.60)

The probability of the tunneling process from the cavity into the environment is determined

by the strength of the coupling element Gqof the mode q. For the cavity loss external pho-

ton modes described by dq,d†

q. Only contributions within the rotating-wave approximation

are considered. The dynamics, calculated only for the photon-photon interaction reads for

an arbitrary QD operator Aassisted by |nihm|cavity photons:

∂tA|nihm||pt−pt =i(n−m)ω0A|nihm|(A.61)

+iX

Gq√n+ 1A|n+ 1ihm|dq+√nA|n−1ihm|d†

q

+iX

Gq√m+ 1A|nihm+ 1|d†

q+√mA|nihm−1|dq.

Now, a rotating frame is used:

A|nihm|=A|nihm|Rei(m−n)ω0t(A.62)

A|nihm|dq=A|nihm|dqRei((m−n)ω0−ωq)t(A.63)

xii

A Appendix

Now, the free part is changed into a phase-factor:

∂tA|nihm|R|pt−pt =iX

Gq√n+ 1A|n+ 1ihm|dqRei(ω0−ωq)t(A.64)

+iX

Gq√nA|n−1ihm|d†

qRe−i(ω0−ωq)t

+iX

Gq√m+ 1A|nihm+ 1|d†

qRe−i(ω0−ωq)t

+iX

Gq√mA|nihm−1|dqRei(ω0−ωq)t.

Now, the dissipative photon mode assisted quantities must be calculated, to derive the set

of equation up to the second order in the tunneling matrix couling element. They read:

∂tA|nihm|dqR|pt−pt =iX

q′

Gq′√nA|n−1ihm|d†

q′dqRe−i(ω0−ωq′)t(A.65)

−iX

q′

Gq′√m+ 1A|nihm+ 1|Re−i(ω0−ωq′)tδq,q′

−iX

Gq√m+ 1A|nihm+ 1|d†

q′dqRe−i(ω0−ωq′)t

≈ −iGq√m+ 1A|nihm+ 1|Re−i(ω0−ωq)t.

Contributions proportional to d(†)

qd(†)

q′are neglected within the bath assumption, since

only dissipative are taken into account. The bath photon do not couple back into the

cavity. The mean photon number is too low in the bath to consider a tunnel process in

the reversed direction. The phase-relation is the same for the inhomogeneities, since this

contributions reflects one part of the two part density matrix process for the cavity loss,

the other one reads:

∂tA|nihm|d†

qR|pt−pt ≈iGq√n+ 1A|n+ 1ihm|Rei(ω0−ωq)t.(A.66)

Now, the equation is solved by integrating formally and applying the Markovian approxi-

mation, assuming memory effects are negligible, which is a good approximation in system-

bath processes, with coupling strength suggesting interaction time higher than ps-scale.

The tunnel coupling strength is weak enough and the solution reads:

A|nihm|d†

qR(t)|pt−pt ≈iGq√n+ 1A|n+ 1ihm|RZ∞

dτ ei(ω0−ωq)(t−τ)(A.67)

=iGq√n+ 1A|n+ 1ihm|Rei(ω0−ωq)tZ∞

dτ e−i(ω0−ωq)τ.

If the lower limit is not extended from 0to −∞, the integral is the Heitler-Zeta function

ζ(ω0−ωq), consisting of the delta-function and of the Cauchy-principal value integral.

It must be checked, to what extent the Cauchy-principal vaule integral contributes to

the dynamics, normally a energy renormalization occurs, which can be included into the

xiii

A Appendix

definition of ω0. In Eq. (A.65), several combinations of dissipative photon mode assisted

quantities need to be calculated like in Eq. (4.60). They read:

A|n+ 1ihm|dqR=−i Gq√m+ 1A|n+ 1ihm+ 1|Re−i(ω0−ωq)tζ(ω0−ωq)(A.68)

A|n−1ihm|d†

qR=i Gq√nA|nihm|Rei(ω0−ωq)tζ(ω0−ωq)

A|nihm−1|dqR=−i Gq√mA|nihm|Re−i(ω0−ωq)tζ(ω0−ωq)

A|nihm+ 1|d†

qR=i Gq√nA|n+ 1ihm+ 1|Rei(ω0−ωq)tζ(ω0−ωq)

These solutions can be plug into the Eq. (A.65) to derive the cavity-loss for the general

quantity A|nihm|, by defining:

κ:= X

q|Gq|2ζ(ω0−ωq),(A.69)

the PPCE formulation of the Markovian cavity loss, including energy renormalization is

derived and can be expressed as:

∂t|nihm|AR|ph−ph =−(m+n)κ|nihm|A

+2√m+ 1√n+ 1κ|n+ 1ihm+ 1|A.(A.70)

In the PPCE formalism, the dynamics of the photon loss describes the loss and gain of

photons of the different photon probabilities self-consistently. The dynamics of the pho-

ton probabilities are easier in the electron-photon interaction, but more complicated for

the photon-loss, since inscattering from higher order photon probabilities need to be con-

sidered. Therefore, the cavity loss appears more complicated than in the picture of the

c†, c-operators. To give an example, one chooses in Eq. (4.62) the photon-assisted transi-

tion. The equation reads, only for the photon-photon interaction:

∂ta†

vac|n+ 1ihn|R=−(2n+ 1)κa†

vac|n+ 1ihn|R+ 2√n+ 1√n+ 2κa†

vac|n+ 2ihn+ 1|R.

To transform Eq. (A.71) into the operator formalism, one multiplies, e.g. with √n+ 1, and

sums over n:

√n+ 1∂ta†

vac|n+ 1ihn|R=X

∂ta†

vac|nihn|R=∂ta†

vacc†R

−X

√n+ 1(2n+ 1)κa†

vac|n+ 1ihn|R=−κa†

vacc†R−2κa†

vacc†c†c

2√n+ 1√n+ 2κa†

vac|n+ 2ihn+ 1|R= 2κa†

vacc†c†cR.

Only with the inscattering of photons from higher photon-probabilities in case of losses,

the usual cavity loss can be derived, since Eq. (A.71) reads now:

∂ta†

vacc†R=−κa†

vacc†R.(A.71)

The last equation has been derived to clarify the processes, which do not appear in the

operator-formalism, that the photon-probability approach opens up further insight into

xiv

A Appendix

the combined electron-photon dynamics, even in the case of the Markovian-derived cavity

loss. The few-electron approach, i.e. the need to factorize in the electronic system, does

not enter in this derivation of the cavity loss, since the cavity loss results solely from the

photon-photon interaction.

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Danksagung

Ich danke Prof. Andreas Knorr für sein Vertrauen, seine Unterstützung und Ermutigungen

und dafür, meinen verschiedenen Projekten so viel Aufmerksamkeit gewidmet zu haben,

dass sie in dieser kurzen Zeit gelingen konnten. Ich danke in diesem Sinne ebenfalls Dr.

Marten Richter, der mir mit Rat und Tat bei allen Problemen zur Seite gestanden hat

und ohne den diese vorliegende Arbeit ebenfalls gar nicht möglich gewesen wäre, sowie der

gesamten Arbeitsgruppe Knorr für die freundliche, interessante und inspirierende Arbeits-

atmosphäre.

Mein Dank geht auch an Prof. Torsten Meier für die Übernahme des Zweitgutachters

sowie an Prof. Michael Kneissl, dass er den Prüfungsvorsitz übernommen hat.

Ich danke Prof. Weng W. Chow für die zahlreichen fruchtbaren Diskussion hier in Berlin

und in Albuquerque, wo er zusammen mit Prof. Andreas Knorr Julia Kabuss und mir einen

einmonatigen Forschungsaufenthalt ermöglicht hat.

Für das Korrekturlesen und die unermüdliche Bereitschaft, wissenschaftliche Diskussionen

zu führen, seien Marten Richter, Frank Milde, Carsten Weber, Matthias-Rene Dachner,

Mario Schoth, Yumian Su, Christina Pöltl und vor allem Julia Kabuss gedankt. Unzählige

Male habt ihr mir geholfen und euch für meine Fragen Zeit genommen. In diesem Sinne

danke ich auch Jeong-Eun Kim und Ermin Malic für ihre Freundlichkeit und Verlässlichkeit.

Matthias-Rene Dachner danke ich für die temperaturabhängigen T1-Zeiten für das Ver-

schränkungsprojekt und zusammen mit Alexander Wilms für die Kaffeemaschine, deren

Kaffee mich über so manche müde Stunde gerettet hat. An Mario Schoth geht mein Dank

dafür, dass er das NLPF-Projekt so erfolgreich übernommen hat.

Meinen Eltern möchte ich herzlich danken, dass sie mir dieses wunderbare Studium mit

seinen Irrungen und Wirrungen ermöglicht haben und nicht ein einziges Mal an meiner

wissenschaftlichen Ernsthaftigkeit und meinem Wissensdurst zweifelten. Und vor allen an-

deren danke ich meiner lieben Julia, dafür, dass sie so ist, wie sie ist, für dieses gemeinsame

Leben, das ich mir gar nicht schöner vorstellen kann.

xxxi