electric and magnetic interaction between quantum dots … · electric and magnetic interaction...

Electric and Magnetic Interaction

between Quantum Dots and Light

A dissertation

submitted to the Niels Bohr Institute

at the University of Copenhagen

in partial fulllment of the requirements

for the degree of

philosophiae doctor

Petru Tighineanu

February 12, 2015

Electric and Magnetic Interaction

between Quantum Dots and Light

To my parents

iii

Preface

The research presented in this thesis was conducted from January 2012 to February 2015 in the

Quantum Photonics Group at the Niels Bohr Institute, University of Copenhagen, under the

supervision of professor Peter Lodahl. First and foremost, thank you Peter for your priceless

encouragement and support throughout my PhD project, and for giving me the fantastic possi-

bility to be part of this exciting research environment. Your professional insight and experience

have helped me immensely to develop my set of skills and abilities, and to dene my scientic

personality. Your feedback on papers and reports conferred a completely new dimension to my

understanding of science and I am deeply grateful for that.

I feel greatly privileged to have been co-supervised by Søren Stobbe, whose prominent ex-

citonic heart always motivated me to knock on his oce door and discuss ideas from science

and beyond. Søren's excellent ability of intertwining deep scientic knowledge with subtle and

elegant humor rendered our discussions productive and fun at the same time. One of the most

exciting terms describing my work, the quantum banana, was a product of our discussions and

I would like to thank Søren for that.

The collaboration with Anders Søndberg Sørensen was a fantastic experience. His deep

scientic insight and the ability to explain complex processes in simple terms have been of great

educational value. Our meetings were a gigantic burst of energy and motivation for myself.

The lengthy discussions about classical and quantum physics, and the association between a

quantum dot and a bent wire, were an extraordinary enjoyment. Thank you Anders for sharing

your knowledge and expertise with so much enthusiasm.

The results presented in this thesis would have not been possible to achieve without the

contribution of other group members. I greatly beneted from the contribution of Raphaël

Daveau to the work presented in Chapters 3 and 4, in particular his support in the lab and his

insight into the interaction between quantum dots and phonons. Tau Lehmann, Kristian Høeg

Madsen and Inah Yeo had a substantial contribution to the results from Chapter 3 through their

outstanding knowledge and expertise in the optics lab. I would like to thank the people who

played an important role in improving my thesis by proofreading it: Sahand Mahmoodian, Immo

Søllner, Søren Stobbe and Leonardo Midolo.

I arrived in this group with no previous lab experience and I am therefore forever indebted

to every single group member who introduced me into the world of experimental physics. Alisa

Javadi and David Garcia, thanks for introducing me the ow-cryo setup and for dropping down

v

in the lab countless times to help me. Immo Söllner, thanks for your insightful lab advices and

for being patient with me, especially at the beginning when I was playing with re. Literally!

Tom Bienaimé, you helped me build my very rst optical setup and I am deeply grateful for

that. Tau Lehmann, Kristian Høeg Madsen and Inah Yeo, thanks so much for introducing me

the dry-cryo setup and for helping me with the measurements. Gabija Kirsanske and Tommaso

Pregnolato, your micrometer-precise skills of manipulating tweezers were a huge help, thank you!

The foundation of the exciton gang was a milestone event that lead to an unforgettable golden

age of measurements, results and discussions. To this end, I would like to thank the members

of the gang Raphaël Daveau, Gabija Kirsanske, Miguel Carro and Tommaso Pregnolato for

their enthusiastic contribution. Also, I had the pleasure to supervise Raphaël and Miguel as

Master students. The countless hours spent in the lab catching single photons and discussing

quantum-dot physics were a great source of enjoyment.

The fantastic atmosphere present in the group helped me connect with the people not only

on a professional but also on a personal level. I became good friends with Immo Söllner, Kristian

Høeg Madsen, Alisa Javadi, Marta Arcari, Sahand Mahmoodian and many others. I would like

to thank Alisa for the many get togethers with so much fun, in particular playing table tennis

and chess, and for the bike and shing trips in and around the city. The many trips to the bio

canteen with Gabi and the related discussions about dogs were extremely enjoyable. The trip

to Rome with Kristian was an unforgettable exciton-polariton brainstorm. The Friday beers

were memorable events in which lots of joy, excitement and laughs were shared. To this end, I

would like to send my warmest regards to all the aforementioned people as well as Soe Lindskov

Hansen and Haitham El-Ella.

I would like to send my deep gratitude to my parents for their unconditional help and support.

Learning from their wisdom has been the main propeller of my accomplishments.

Petru Tighineanu

February 12, 2015

vi

Abstract

The present thesis reports research on the optical properties of quantum dots by developing

new theories and conducting optical measurements. We demonstrate experimentally single-

photon superradiance in interface-uctuation quantum dots by recording the temporal decay

dynamics in conjunction with second-order correlation measurements and a theoretical model.

We measure an oscillator strength of up to 96±0.8 and an average quantum eciency of (94.8±3.0)%. This enhanced light-matter coupling is known as the giant oscillator strength of quantum

dots, which is shown to be equivalent to superradiance. We argue that there is ample room

for improving the oscillator strength with prospects for approaching the ultra-strong-coupling

regime of cavity quantum electrodynamics with optical photons. These outstanding gures of

merit render interface-uctuation quantum dots excellent candidates for use in cavity quantum

electrodynamics and quantum-information science.

We investigate exciton localization in droplet-epitaxy quantum dots by conducting spectral

and time-resolved measurements. We nd small excitons despite the large physical size of droplet-

epitaxy quantum dots, which is attributed to material inter-diusion during the growth process.

The small size of excitons leads to a small oscillator strength of about 10. These ndings are cross-

checked by an analysis of the phonon-broadened spectra revealing a small exciton wavefunction.

We conclude that engineering large excitons with giant oscillator strength remains a future

challenge for the droplet-epitaxy technique.

A multipolar theory of spontaneous emission from quantum dots is developed to explain the

recent observation that In(Ga)As quantum dots break the dipole theory. The analysis yields

a large mesoscopic moment, which contains magnetic-dipole and electric-quadrupole contribu-

tions and may compete with the dipole moment in light-matter interactions. A theory for the

quantum-dot wavefunctions is developed showing that the mesoscopic moment originates from

distortions in the underlying crystal lattice. The resulting quantum-mechanical current den-

sity is curved leading to light-matter interaction of both electric and magnetic character. Our

study demonstrates that In(Ga)As quantum dots lack parity symmetry and, as consequence,

can be employed for locally probing the parity symmetry of complex photonic nanostructures.

This opens the prospect for interfacing quantum dots with optical metamaterials for tailoring

light-matter interaction at the single-electron and single-photon level.

vii

Resumé

Denne PhD-afhandling beskriver forskning i de optiske egenskaber af kvantepunkter, herun-

der udvikling af nye teorier og optiske eksperimenter. Vi demonstrerer eksperimentelt enkelt-

foton-superradians i grænselagsuktuationskvantepunkter ved hjælp af målinger af den tidslige

henfaldsdynamik og anden-ordens korrelationsmålinger, som sammenstilles med en teoretisk

model. Vi måler en oscillatorstyrke på op til 96 ± 0.8 og en gennemsnitlig kvanteeektivitet

på (94.8± 3.0)%. Denne forøgede lys-stof vekselvirkning er kendt som giant oscillator strength-

eekten for kvantepunkter og vi viser, at den er ækvivalent med superradians. Vi argumenterer

for, at der er mulighed for en betydelig forøgelse af oscillatorstyrken, hvilket kunne muliggøre

det ultrastærkt koblede regime af kavitetskvanteelektrodynamik med optiske fotoner. Disse be-

mærkelsesværdige egenskaber betyder, at grænselagsuktuationskvantepunkter er har stort po-

tentiale indenfor kavitetskvanteelektrodynamik og kvanteinformationsvidenskab.

Vi undersøger excitonlokalisering i dråbeepitaksikvantepunkter ved hjælp af spektrale og

tidsopløste målinger. Vi nder, at excitonerne er små, på trods af dråbeepitaksikvantepunkternes

relativt store størrelse, hvilket tilskrives interdiusion under dyrkningsprocessen. Excitonernes

lille størrelse fører til en oscillatorstyrke på omkring 10. Disse konklusioner underbygges af en

analyse af de fonon-forbredte spektre, som afslører små excitonbølgefunktioner. Vi konkluderer,

at demonstrationen af store excitoner med store oscillatorstyrker forbliver en fremtidig udfordring

for dråbeepitaksiteknikken.

En multipolteori for spontan emission fra kvantepunkter udvikles og anvendes til at forklare

den nylige observation, at dipolteori bryder sammen for In(Ga)As kvantepunkter. Analysen

viser, at kvantepunkter har et stort mesoskopisk moment, som indeholder magnetisk dipol- og

elektrisk quadrupol-bidrag, der kan indgå på lige fod med dipolmomentet i lys-stof vekselvirknin-

gen. En teori for kvantepunkters bølgefunktioner udvikles, og den viser, at det mesoskopiske

moment har sin oprindelse i forskydninger i det underliggende krystalgitter. Den resulterende

kvantemekaniske strømtæthed er kurvet og fører til en lys-stof vekselvirkning af både elektrisk og

magnetisk karakter. Dette arbejde viser, at In(Ga)As kvantepunkter ikke har paritetssymmetri,

og deraf følger, at de er følsomme for paritetssymmetrien af komplekse fotoniske nanostrukturer.

Dette åbner nye perspektiver for at forbinde kvantepunkter med optiske metamaterialer for at

skræddersy lys-stof vekselvirkningen på enkelt-elektron- og enkelte-foton-niveau.

ix

List of Publications

The work conducted in the present Ph.D.-project has resulted in the following publications:

Journal Publications

1. P. Tighineanu, M. L. Andersen, A. S. Sørensen, S. Stobbe and P. Lodahl, Probing Electric

and Magnetic Vacuum Fluctuations with Quantum Dots, Physical Review Letters 113,

043601 (2014).

2. P. Tighineanu, R. Daveau, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, Decay Dynamics

and Exciton Localization in Large GaAs Quantum Dots Grown by Droplet Epitaxy, Physical

Review B 88, 155320 (2013).

3. P. Tighineanu, A. S. Sørensen, S. Stobbe and P. Lodahl, Unraveling the Mesoscopic Char-

acter of Quantum Dots in Nanophotonics, arXiv:1409.0032, submitted to Physical Review

Letters (2014).

4. P. Tighineanu, R. Daveau, Tau B. Lehmann, H. E. Beere, D. A. Ritchie, P. Lodahl and

S. Stobbe, Single-Photon Dicke Superradiance from a Quantum Dot, submitted to Nature

Physics (2015).

Conference Contributions

1. P. Tighineanu, S. Stobbe and P. Lodahl, Forging the Flow of the Quantum-Mechanical

Current in Quantum Dots, Proceedings of the "Nonlinear Optics and Excitation Kinetics

in Semiconductors" conference, Bremen, Germany (2014).

2. R. Daveau, P. Tighineanu, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, Optical Proper-

ties of Large GaAs Quantum Dots Grown by Droplet Epitaxy, Proceedings of the "Nonlinear

Optics and Excitation Kinetics in Semiconductors" conference, Bremen, Germany (2014).

3. P. Tighineanu, A. S. Sørensen, S. Stobbe and P. Lodahl, Probing Electric and Magnetic

Vacuum Fluctuations with Quantum Dots, "Nonlinear Quantum Optics" workshop, Leiden,

the Netherlands (2014).

xi

4. P. Tighineanu, S. Stobbe and P. Lodahl, Accessing the Magnetic Dipole and Electric

Quadrupole of Quantum Dots with Light, Proceedings of the "CLEO 2014" conference,

San Jose, United States of America (2014).

5. P. Tighineanu, R. Daveau, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, Assessing the

Quality of Quantum Dots by Time-Resolved Spectroscopy, Proceedings of the "Optics of

Excitons in Conned Systems" conference, Rome, Italy (2013).

xii

Contents

Preface iii

Abstract vii

Resumé viii

List of publications x

1 Introduction 1

2 Fundamental Properties of Semiconductor Quantum Dots 5

2.1 Quantum mechanics of semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 From a huge multi-body system to a single-particle problem . . . . . . . . 6

2.1.2 Band structure of III-V semiconductors . . . . . . . . . . . . . . . . . . . 8

2.2 Basic structural, electronic and optical properties of quantum dots . . . . . . . . 12

2.2.1 Electronic models of quantum dots. Eective-mass theory . . . . . . . . . 13

2.2.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Excitons. Weak- and strong-connement regimes . . . . . . . . . . . . . . 18

2.2.4 Heavy-hole excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.5 Light-hole excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Density of states of conned systems . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 The electromagnetic quantum-vacuum eld . . . . . . . . . . . . . . . . . . . . . 25

2.5 Fundamental light-matter interaction with quantum dots . . . . . . . . . . . . . 27

2.5.1 Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2 The dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.3 Decay dynamics of quantum dots . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Single-Photon Dicke Superradiance from a Quantum Dot 35

3.1 Theory of single-photon superradiance from quantum dots . . . . . . . . . . . . . 38

3.1.1 Strong-connement regime . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Weak-connement regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xiii

CONTENTS

3.1.3 Relation between the giant oscillator strength of quantum dots and single-

photon Dicke superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Sample and experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Deterministic preparation of superradiant excitons . . . . . . . . . . . . . . . . . 44

3.4 Previous work on the giant oscillator strength of quantum dots . . . . . . . . . . 46

3.5 Extracting the impact of nonradiative processes . . . . . . . . . . . . . . . . . . . 46

3.6 Experimental demonstration of single-photon superradiance . . . . . . . . . . . . 48

3.7 Microscopic insight into the exciton wavefunction . . . . . . . . . . . . . . . . . . 51

3.8 Results on all measured quantum dots . . . . . . . . . . . . . . . . . . . . . . . . 52

3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Decay dynamics and Exciton Localization in Large GaAs Quantum Dots Grown by

Droplet Epitaxy 55

4.1 Sample growth and experimental procedure . . . . . . . . . . . . . . . . . . . . . 56

4.2 Spectral measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Oscillator strength and quantum eciency . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Temperature dependence of the eective transition strength . . . . . . . . . . . . 65

4.5 Acoustic-phonon broadening and exciton size . . . . . . . . . . . . . . . . . . . . 69

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Multipolar Theory of Spontaneous Emission from Quantum Dots 73

5.1 Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.1 Zeroth order: electric-dipole moment . . . . . . . . . . . . . . . . . . . . . 76

5.1.2 First order: electric-quadrupole and magnetic-dipole moments . . . . . . 77

5.1.3 Second-order: electric-octupole and magnetic-quadrupole moments . . . . 78

5.1.4 Summary of the multipole transition moments . . . . . . . . . . . . . . . 79

5.2 Origin dependence of the multipole transition moments . . . . . . . . . . . . . . 80

5.3 Radiative decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 Green's Tensor and derivatives in the vicinity of an Interface . . . . . . . . . . . 83

5.4.1 Homogeneous part of the Green tensor . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Scattering part of the Green tensor . . . . . . . . . . . . . . . . . . . . . . 85

5.5 Origin (in)dependence of the radiative decay rate . . . . . . . . . . . . . . . . . . 89

5.5.1 Spontaneous decay in a homogeneous medium . . . . . . . . . . . . . . . 90

5.5.2 Spontaneous decay in an arbitrary environment . . . . . . . . . . . . . . . 91

5.6 Decay dynamics of In(Ga)As quantum dots in the vicinity of an interface . . . . 91

5.6.1 Zeroth-order contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6.2 First-order contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.6.3 Second-order contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xiv

CONTENTS

6 Unraveling the Mesoscopic Character of Quantum Dots in Nanophotonics 99

6.1 Microscopic model for mesoscopic quantum dots . . . . . . . . . . . . . . . . . . 101

6.2 The quantum-mechanical current density . . . . . . . . . . . . . . . . . . . . . . 106

6.3 Breakdown of the dipole theory at nanoscale proximity to a dielectric interface . 107

6.4 Lattice-distortion eects beyond the multipolar theory . . . . . . . . . . . . . . . 112

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Probing Electric and Magnetic Vacuum Fluctuations with Quantum Dots 115

7.1 Electric and magnetic light-matter interaction . . . . . . . . . . . . . . . . . . . . 117

7.2 Probing the parity symmetry of nanophotonic environments . . . . . . . . . . . . 120

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8 Conclusion & Outlook 125

Appendices 129

A Operator Matrices for the Theory of Invariants 131

B Length and Velocity Representation 133

C Evaluation of the First-Order Mesoscopic Moment Λzx 135

D Evaluation of the Second-Order Mesoscopic Moment Ωzzx 137

E The Unit-Cell Dipole Approximation 139

F Quantum Dots as Building Blocks for Quantum Metamaterials 141

F.1 Polarizability of split-ring resonators . . . . . . . . . . . . . . . . . . . . . . . . . 142

F.2 Polarizability of quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

F.3 Quantum metamaterial with quantum dots . . . . . . . . . . . . . . . . . . . . . 145

Bibliography 149

xv

Chapter 1

Introduction

The remarkable clarity and beauty of classical physics led to a nearly complete and unques-

tionable mechanical model of the universe at the beginning of last century. According to Lord

Kelvin's famous speech in 1900 [1], two "little clouds" were contaminating the awless and clear

sky of physics, namely the inability to detect the ether and the ultraviolet catastrophe. The two

bothersome clouds precipitated the spectacular development of the two main pillars of modern

physics, general relativity and quantum mechanics, which immensely deepened our understand-

ing of the universe. The little clouds eradicated the complacency characterizing classical physics

because they were nothing else than fundamental limitations to a classical understanding of the

universe.

The mystery hidden behind those clouds propelled the development of science and technology

over the past and present centuries. The advent of quantum mechanics drastically changed our

perception of nature by conferring a wave-particle duality to light and matter [2]. The theory

of relativity, on the other hand, intertwines space and time in a four-dimensional universe [3].

The two revolutionary theories gave birth to the fascinating eld of quantum electrodynamics

explaining the complexity of vacuum, which consists of virtual particles popping in and out of

existence as allowed by Heisenberg's uncertainty relation [4]. The quantum vacuum not only

triggers spontaneous emission from quantum emitters [5], but also mediates interactions within

the emitter and perturbs the energy levels in an eect known as the Lamb shift [6]. The capability

of tailoring the density of vacuum uctuations lead to the discovery of the bizarre yet fascinating

Casimir force [7]. These breakthroughs cemented our understanding of the quantum world and

established quantum optics as a new and exciting research eld, in which concepts such as hidden

variables [8] and non-locality [9] sparked philosophical contemplations among physicists. The

subsequent demonstration of vacuum Rabi oscillations [10] provided the experimental evidence

of quantum entanglement as a fundamental property of quantum systems.

Quantum mechanics formulated the necessary ingredients for the development of solid-state

physics in the middle of the last century. This eld has undergone an extraordinary technological

and scientic revolution ever since. The discovery of new and fascinating phenomena such as

the quantum Hall eect [11], which has helped measure fundamental physical constants with

1

Chapter 1. Introduction

unprecedented precision, and the giant magnetoresistance [12], which has immensely increased

the information density that can be stored in modern electronics, had a direct inuence not

only on the economic and industrial progress but also on the life of each of us. New materials

with unique mechanical, optical and thermo-electric properties, such as carbon nanotubes [13]

and graphene [14], had a monumental impact on other branches of science as well as on the

global market. The transistor was a milestone discovery [15] that led to an exponential increase

in the density of logical gates in integrated circuits known as the Moore's law [16]. This has

dramatically enhanced the impact of digital electronics in practically every segment of the world's

economy [17].

Notwithstanding this extraordinary progress, the state-of-the-art circuits process informa-

tion according to the laws of classical physics. This represents a fundamental limitation to

simulating and understanding quantum systems. Indispensable quantum phenomena, such as

high-temperature superconductivity [18], are poorly understood owing to the intrinsic mismatch

between a multi-body quantum system and a classical simulator. Realizing the ultimate com-

puter, which deals with quantum states, is a fascinating emerging eld. So far, entanglement

and coherence of up to 14 quantum bits has been demonstrated [19] with promising prospects

for exploiting the property of quantum parallelism on a large-scale device.

At the intersection between quantum optics and solid-state physics, the eld of quantum

photonics has unfolded over the past years striving to combine the expertise developed for atoms

and the scalability demonstrated by solid-state systems. To this end, quantum dots provide

the essential link between light and matter degrees of freedom in an environment that may be

integrated monolithically into photonic devices. These nanometer-size purposefully engineered

impurities combine the atomic-like discrete spectra and excellent single-photon purity with the

large light-matter interaction strength inherent to solid-state systems. The ability to tailor the

density of vacuum uctuations around quantum dots has resulted in tremendous progress in

manipulating single quantum-dot excitations over the past decade. Strong coupling between a

quantum dot and a cavity [20] and near-unity coupling to a photonic-crystal-waveguide mode [21]

are a few out of many promising practical realizations for ecient manipulation of quantum

bits [22].

The atomic-like properties of quantum dots are supplemented by a myriad of new eects ow-

ing to their solid-state nature. For instance, vibrations of the underlying crystal lattice, known

as phonons, may decohere the light-matter interaction [23] or couple non-resonant quantum-dot

excitations to an optical cavity [24]. Similarly, the mesoscopic ensemble of the nuclei composing

the quantum dot can be used to tailor the hyperne interaction with the electron and is of high

relevance for spin-based quantum-information science [25]. A recent surprising discovery [26]

demonstrated that quantum dots may break the dipole approximation, which has been uncrit-

ically employed in the eld of quantum optics so far. These realizations underline the complex

yet fascinating nature of solid-state quantum emitters with potentially numerous eects yet to

be unraveled.

The very aim of the present thesis is to deepen our understanding of quantum dots and

their interaction with light. The underlying electric and magnetic oscillations compose the light

2

eld on an equal footing as is known from Maxwell's equations. The interaction with matter

is, however, only accomplished by the electric-eld component of light owing to the small size

of conventional quantum emitters. In the present work we strive for overcoming the limitations

inherent to conventional emitters by tailoring the coupling of quantum dots to both the electric-

and magnetic-eld components of the quantum vacuum. This is possible because the size, shape

and material composition of solid-state emitters can be accurately engineered. To this end, we

envision the possibility to engineer the "ideal" quantum emitter with the desired built-in electric

and magnetic sensing capabilities. Such quantum emitters would have complete control over

the interaction of light in terms of the radiative decay rate, direction of polarization or angular

distribution, which is one of the holy grails in the eld of nanophotonics.

Quantum dots greatly benet from their multi-body nature with an enhanced light-matter

interaction strength compared to atomic emitters. This renders them promising candidates for

improving the eciency of single-photons sources, solar cells and nano-lasers, to name a few

important practical realizations. Commonly employed quantum dots have, however, an upper

limit for the interaction strength with light, regardless of their size and shape. It has been

therefore a long-sought goal in quantum photonics to develop solid-state emitters with no such

upper limit. In the present work we demonstrate that monolayer-uctuation quantum dots [27]

can be used to enhance the interaction strength with light far beyond that of conventional

quantum dots. This remarkably large interaction strength is caused by the superradiant nature

of monolayer-uctuation quantum dots, which may be of great interest for fundamental science

and technology alike. In particular, such rapid radiative decays will likely exceed all dephasing

mechanisms resulting in highly coherent ying quantum bits, of high relevance for their use in

quantum-information science. The large enhancement of spontaneous emission envisions novel

possibilities for integrating such quantum emitters with super-bright optoelectronic devices. New

and so far largely unexplored solid-state quantum-electrodynamics regimes involving energy non-

conserving virtual processes, such as the ultra-strong coupling between light and matter, may

become within reach at optical frequencies for the rst time.

The aforementioned enhanced coupling to the light eld is nothing else than an increased

interaction between the quantum dot and the electric-eld component of the quantum vacuum.

This is because, according to the dipole theory, quantum emitters are completely blind to the

magnetic-eld component of light. The recent experimental demonstration that the dipole theory

may break in self-assembled In(Ga)As quantum dots motivated us to develop a self-consistent

multipolar theory of spontaneous emission from quantum dots. We nd that In(Ga)As quantum

dots are sensitive to the magnetic eld of light on dipole-allowed transitions. As a consequence,

quantum dots can no longer be treated as point-like entities and have prominent mesoscopic

properties. We pinpoint the microscopic mechanism governing the mesoscopic nature of quan-

tum dots by developing a theory for the quantum-mechanical wavefunctions. We show that the

underlying lattice distortion generates curved quantum-mechanical currents owing over meso-

scopic length scales inside the quantum dot. The resulting quantum-dot wavefunctions break

parity symmetry and are therefore excellent sensors of the parity of the surrounding photonic

nanostructure. Both fundamental science and quantum technologies may greatly benet from

3

Chapter 1. Introduction

these ndings. For instance, novel photonic environments could be designed to match the curved

current-density pattern of the quantum dot. Sensitivity to magnetic elds has been long sought

in nanophotonics, and quantum dots may be employed as non-invasive magnetic probes operat-

ing at the single-electron single-photon level. The curved quantum current density can curiously

be considered the quantum version of split-ring resonators that are often employed as building

blocks of optical metamaterials [28]. This opens the prospect for the realization of a quantum-dot

based quantum metamaterial combining the fascinating phenomena inherent to classical meta-

materials, such as negative index of refraction, super-lensing and cloaking, with single-photon

nonlinearities and non-classical statistics of light pertaining to the quantum world.

The outline of the present thesis is as follows. Chapter 2 introduces the indispensable in-

gredients required for describing the light-matter interaction with quantum dots. We show that

quantum dots can be modeled in a remarkably simple fashion despite their complex multi-body

nature. Fundamental quantities such as the oscillator strength and the local density of optical

states, which govern the process of spontaneous emission, are introduced.

The experimental demonstration of single-photon superradiance from a quantum dot is pre-

sented in Chapter 3. The strong and weak quantum-connement regimes are discussed at length,

and the mathematical equivalence between the giant oscillator strength and single-photon super-

radiance is pinpointed accordingly. We show that time-resolved spectroscopy is a powerful tool

not only for unambiguously extracting the impact of radiative processes, but also as a mean to

obtain deep insight into the microscopic characteristics of the quantum-dot wavefunctions.

Chapter 4 presents an extensive study of the optical properties of quantum dots grown by a

novel technique, droplet epitaxy, which promises to deliver high-quality quantum dots with no

built-in strain and related adverse eects. We perform an analysis of radiative and nonradiative

processes and show that droplet-epitaxy quantum dots are described by a model for strongly-

conned excitons.

A multipolar theory describing the spontaneous emission from quantum dots is developed in

Chapter 5. The dependence of the multipolar moments on the origin of the coordinate system

and the corresponding impact on the decay rate is discussed at length. The mesoscopic moments

having a large contribution to the light-matter interaction strength are identied through simple

and intuitive parity-symmetry arguments.

The microscopic theory pinpointing the origin of the mesoscopic character of quantum dots

is presented in Chapter 6. A simple extension of the eective-mass theory is developed and the

resulting wavefunctions inherit the structural asymmetry of the underlying crystal lattice. We

compute the quantum-mechanical current density owing through the quantum dot and obtain

excellent agreement with experimental data in a Drexhage-type geometry.

The large circular current density confers magnetic sensitivity to quantum dots as explained

in Chapter 7. As a consequence, quantum dots probe electric and magnetic eld simultaneously

and are therefore fundamentally dierent than atoms. The asymmetry inherent to the quantum-

mechanical wavefunctions can be exploited to sense the parity symmetry of complex photonic

nanostructures.

4

Chapter 2

Fundamental Properties of

Semiconductor Quantum Dots

The central topic of the present thesis is the study of the interaction between semiconductor QDs

and the electromagnetic vacuum eld. As such, the purpose of this chapter is to lay the theo-

retical foundations for the rest of the thesis. Quantum dots are semiconductor heterostructures

composed of thousands of atoms, thereby forming a complicated multi-body system. The beauty

of such a system is hidden in the powerful approximations that can simplify the problem im-

mensely leading to remarkably simple and intuitive results. The electromagnetic vacuum eld, on

the other hand, can be accurately engineered for tailoring the spontaneous-emission process from

QDs. Combined with detailed experimental investigations, a deep and complex microscopic un-

derstanding can be acquired, which is of crucial importance for the further development of elds

such as quantum photonics, nano-optics and scalable solid-state quantum-information science.

Quantum dots are three-dimensional crystalline blocks of one semiconductor material (e.g.,

InAs) embedded in a matrix of another material (e.g., GaAs). Since they are extended over a

few nanometers, comparable to the de Broglie wavelength of the electrons, QDs require a full

quantum-mechanical treatment. The principles of solid-state physics, which were developed in

the middle of the last century, lie at the heart of this description. We therefore discuss the central

topics and approximations of quantum mechanics in crystalline materials before presenting the

concept of a nanostructure and, in particular, of a QD. The density of states is an important

concept for understanding the interaction between a conned system and light, which is why

we are treating it in a separate section. Spontaneous emission is nothing but the interaction

between a QD excitation and the electromagnetic vacuum eld. The latter can be accurately

tailored to enhance or suppress this interaction via the so-called Purcell eect [5], or even to

bring this interaction in the strong light-matter coupling regime for studying cavity-quantum-

electrodynamics (CQED) eects in a solid-state platform, which are discussed towards the end

of the chapter. Fundamental quantities, such as the oscillator strength and the local density

of optical states, which govern the spontaneous-emission process, are introduced. Thus, this

5

Chapter 2. Fundamental Properties of Semiconductor Quantum Dots

chapter presents the three primordial ingredients for the present thesis: quantum dots, the

electromagnetic eld, and the light-matter interaction with QDs.

2.1 Quantum mechanics of semiconductors

2.1.1 From a huge multi-body system to a single-particle problem

In this section we outline the mathematical apparatus that is indispensable for understanding

a solid-state environment. We are following the treatment from Ref. [29]. A crystal can be

conceptually regarded as an innitely extended physical system, which is formed by periodically

translating a single unit cell until it lls the entire space. The fundamental building block of

a crystal, the unit cell, is the smallest entity that contains all the symmetry and structural

information required for building up the crystal. It is made up of positively charged nuclei

arranged in a well-dened geometric conguration, and of electrons, which surround the nuclei

and are potentially able to move freely. For understanding this physical system, the Schrödinger

equation of the crystal

HΨ = EΨ (2.1)

has to be solved, where

Ψ = Ψ(r1, r2, ..., rn,R1,R2, ...,RN ) (2.2)

is the wavefunction of the crystal which depends on the coordinates of all electrons ri and nuclei

Rj . The Hamilton operator reads

H =∑i

(− ~2

2m0∆i

)︸︷︷︸

kinetic energy of electrons

+∑j

(− ~2

2Mj∆j

)︸︷︷︸

kinetic energy of nuclei

+1

2

∑i

∑j

i6=j

e2

4πε0rij

︸︷︷︸potential energy of electron interaction

+ U(R1,R2, ...,RN )︸︷︷︸potential energy of nuclei interaction

+ M(r1, r2, ..., rn,R1,R2, ...,RN )︸︷︷︸potential energy of interaction between electrons and nuclei

,

(2.3)

where ε0 is the vacuum permittivity, m0 the electron mass, Mi the mass of i-th nucleus, rij

the absolute distance between electron i and j, ∆i the Laplace operator corresponding to the

i-th electron, and ∆j the Laplace operator corresponding to the j-th nucleus. The number of

unknowns in Eq. (2.1) is determined by the number of particles, which is of the order of 1023

within 1 cm3 of matter. It is, therefore, nearly impossible to solve such a problem exactly without

introducing further assumptions.

The dierent time scales at which electrons and nuclei move can be used to decouple their

motion within the so-called Born-Oppenheimer approximation. More specically, the kinetic

energy of electrons and nuclei is about the same in thermal equilibrium. Since electrons possess

a much smaller mass, they are faster by about two orders of magnitude. As a consequence,

the electronic distribution is formed instantaneously for a certain nuclear distribution and the

nuclear coordinates can be taken as free parameters Ri = Ri0. Thus, the nuclei do not move

6

Quantum mechanics of semiconductors

and form an ideal three-dimensional lattice. The kinetic energy of nuclei vanishes (second term

of Eq. (2.3)), and the potential energy of interaction between nuclei U (fourth term in Eq. (2.3))

becomes a constant and can be removed by changing the energy-scale reference. The simplied

Hamiltonian then takes the form

H =∑i

(− ~2

2m0∆i

)+

1

2

∑i

∑j

i6=j

e2

4πε0rij+ V (r1, r2, ..., rn,R10,R20, ...,RN0). (2.4)

Only the valence electrons are potentially able to move through the crystal and we therefore

merge the other electrons with the nucleus they belong to into a positively charged ion. As a

consequence, the indices i and j in Eq. (2.4) run only over the valence electrons. Despite the

considerable simplications, this equation still cannot be solved owing to the high number of

unknowns. We have to invoke the single-electron approximation, which decouples the electron-

electron interaction by assuming that a given electron moves through an averaged potential

created by all the other electrons, so that the electron interaction term can be written as a single

sum1

2

∑i

∑j

i6=j

e2

4πε0rij=∑i

Gi(ri), (2.5)

where Gi(ri) is the potential energy of the i-th electron in the potential created by all the other

electrons. Analogously, M(r1, r2, ..., rn,R10,R20, ...,RN0) =∑iMi(ri). These omitted eects

can be, in principle, included later on as a perturbation (electron-electron scattering). Now, the

Schrödinger equation reads[∑i

(− ~2

2m0∆i

)+∑i

Vi(ri)

]Ψe = EΨe, (2.6)

where Vi(ri) = Gi(ri) + Mi(ri), and Ψe is the wavefunction of valence electrons but in the

following we drop the index for convenience. The Hamiltonian can be nally written as H =∑i Hi, where Hi is the Hamiltonian of the i-th electron, and the multi-electron problem can be

reduced via the Ansatz Ψe(r1, ..., rn) =∏i Ψi(ri). Thus, Eq. (2.6) can be written as a system of

n equations, each depending on the coordinate of a single electron[− ~2

2m0∆i +Gi(ri) + Vi(ri)

]Ψi(ri) = EiΨi(ri). (2.7)

We have arrived at the single-electron Schrödinger equation. Even though it depends on a single

particle, it does have remarkable success in describing semiconductors. The reason is related to

the fact that the electron-electron scattering is normally reduced due to the so-called exchange-

correlation potential [30].

Equation (2.7) can be nally tackled because it discards the coupling between a given electron

and all the other particles forming the crystal. Given the periodic nature of the lattice, the

underlying potential V (r) is also periodic, which leads to the fundamental property that any

observable quantity must have the same periodicity. We assume that the crystal has N unit cells

7


Unit-cellfunction

Envelope

Bloch function

Figure 2.1: Visualization of the real part of a one-dimensional Bloch function. It consists of a

unit-cell function with the lattice periodicity modulated by an envelope.

and employ periodic boundary conditions ∗ The periodicity of the charge distribution ρ(r) ∝|Ψ(r)|2 = |Ψ(r + Rl)|2, where Rl is any translation vector of the lattice, implies that the electron

wavefunction obeys the Bloch theorem [31]

Ψ(k, r) = eikruk(r), (2.8)

where ki = 2πmiNi

and i = x, y, z, N is the total number of unit cells and m an integer with

m = −N/2, ..., N/2− 1. The electron wavefunction can thus be written as a product of a Bloch

function uk(r), which mimics the structure and symmetry of the underlying crystal potential,

and an envelope function eikr carrying information about the momentum of the electron ~k. An

example of how such a Bloch function may look like is illustrated in Fig. 2.1. Equation (2.8)

is extremely important for the rest of the thesis because the wavefunction of a QD (and of a

conned system in general) can be expressed in a very similar fashion, which renders powerful

simplications in practical calculations. Despite the apparent simplicity of Eq. (2.8), the Bloch

function uk(r) cannot be expressed analytically due to the complexity of the crystal potential.

It is at the heart of current research eorts using concepts from density functional theory to

evaluate the Bloch functions numerically [32]. There are, however, more established empirical

methods to determine the contribution of the Bloch functions, which is used in the powerful

eective-mass and k.p theories, as will be seen later. Finally, we emphasize that the Bloch

theorem is valid for virtually any periodic media, such as photonic crystals, which tailor the ow

of light similarly to the way crystals tailor the ow of electrons [33].

2.1.2 Band structure of III-V semiconductors

The solutions to the single-electron problem in Eq. (2.7) are the eigenvectors Ψ and eigenvalues

E for the given wavevector k. While computing Ψ is a complicated problem and is not discussed

∗In the limit of large N , the type of boundary conditions does not really matter. Periodic boundary conditions

are just mathematically convenient [31].

8


kx

ky(a) (b)

UX

WK

L

ΓΛΔΣ

kz

kyky

(c)

Ga

As

Figure 2.2: (a) Illustration of the rst Brillouin zone for a two-dimensional hexagonal lattice. (b)

A zincblende unit cell exemplied on GaAs. (c) The rst Brillouin zone of a zincblende structure.

here, nding the eigenenergies is a somewhat simpler task because the contribution of the Bloch

functions to the electron energy can be taken from experiments. The resulting dispersion relation

E = E(k) governs the electronic and optical properties of the material and is therefore an

important concept in semiconductor physics. As shown in the previous section, the wavevector

k of the electron takes a nite number of values and is bounded by

− πai≤ ki <

π

ai, i = x, y, z. (2.9)

Any value of k beyond this so-called rst Brillouin zone is redundant since it is physically

identical to k−G, where G is any vector of the reciprocal (or k-) lattice. The Brillouin zone

is dened as the region in k-space, which is closer to a given reference lattice point than to

any other, as visualized in Fig. 2.2(a) for a two-dimensional hexagonal lattice. In the present

thesis we are dealing with III-V semiconductors like indium arsenide (InAs), gallium arsenide

(GaAs) and aluminum arsenide (AlAs), which belong to the zincblende structure [34] and are

part of the face-centered cubic space group, see Fig. 2.2(b). The rst Brillouin zone of GaAs is

illustrated in Fig. 2.2(c), where labels are assigned to points and directions of high symmetry.

The symmetry points normally correspond to local minima or maxima in the dispersion E(k)

and are of fundamental importance for the absorption and emission of light from semiconductors.

In a crystal, a large number of atoms are brought in close proximity and each of the former

atomic orbital splits into an entire energy band. It is very common that band minima and

maxima are located at high symmetry points in reciprocal space (see Fig. 2.2(c)), where the

energy is quadratic versus k

E(k) =~2

2(k− kext)

←→M−1(k− kext) + V. (2.10)

Here, kext is the wavevector corresponding to the energy minimum/maximum in reciprocal space,←→M is the eective-mass matrix and V an arbitrary energy oset. Diagonalization of

←→M leads to

Ek =~2

2

[(kx − kext,x)2

mx+

(ky − kext,y)2

my+

(kz − kext,z)2

mz

]+ V. (2.11)

9


k

E

Eg

e

hhlh

so

(a) (b)

Figure 2.3: (a) Band structure of GaAs along the high-symmetry directions in reciprocal

space [35]. The region in the band structure relevant for optical measurements (shaded cir-

cle) is sketched in detail in (b); 'e', 'hh', 'lh' and 'so' correspond to the electron, heavy-hole,

light-hole and split-o bands, respectively; 'Eg' denotes the band gap.

10


The resulting band structure of GaAs is plotted in Fig. 2.3, where a multitude of bands can be

noticed. Only a few are, however, relevant for optics: the valence band(s), which are full at low

temperatures, and the conduction band(s), which are empty. GaAs and most of AlGaAs/InGaAs

alloys are direct-gap semiconductors with the relevant bands situated at the Γ point where

kext = 0 as sketched in Fig. 2.3. Even though the Bloch functions at the Γ point are generally

unknown, knowledge about their symmetry properties provides remarkable simplications in

practical calculations. The conduction band stems from the atomic s orbital and inherits its

spherical symmetry, while the three valence bands (heavy hole, light hole and split o) stem

from the three degenerate atomic p orbitals. Due to the spin-orbit interaction, only two valence

bands remain degenerate while the split-o band is shifted downwards in energy and plays a

negligible role in optical experiments, which is why we do not discuss it further. In terms of the

total angular momentum and its projection |j, jz〉, the heavy- and light-hole Bloch functions at

the Γ point can be written as [36]

uhh ≡ |3/2, 3/2〉 = − 1√2

(ux + iuy) ,

uhh ≡ |3/2,−3/2〉 =1√2

(ux − iuy) ,

ulh ≡ |3/2, 1/2〉 = − 1√6

(ux + iuy − 2uz) ,

ulh ≡ |3/2,−1/2〉 =1√6

(ux − iuy + 2uz) ,

(2.12)

where ui and ui denote spin-up and spin-down functions, and ui inherits the symmetry of the

atomic pi orbital. The coordinate system (x, y, z) in the above equation is chosen such that the

wavevector k of the electron points in the z-direction. For k pointing in another direction, the

above relations would have to be redened.

Another important parameter in the interpretation of the band structure is the eective

mass of an energy band. This concept lays the foundation of the simple yet powerful eective-

mass theory for semiconductor nanostructures, where the microscopic information about the

crystal potential is merged into an eective-mass parameter that simplies analyses tremendously.

Calculating the eective mass of a band can be done by plugging the Bloch solution of Eq. (2.8)

into the single-electron Schrödinger equation, Eq. (2.7), and doing perturbation theory [34, 37].

As a result, the energy can be written in the vicinity of the Γ point as

En(k) = En(0) +~2k2

2m0+

~2

m20

∑i6=n

|〈un(0) |k · p|ui(0)〉|2

En(0)− Ei(0)

= En(0) +~2k2

2meff,

(2.13)

where the sum runs over all the bands, n labels the band of interest and p is the momentum

operator. An electron in a crystal has a mass meff dierent from a free electron m0 due to

the coupling of the electronic states in dierent bands via k · p. The coupling elements are

normally inferred from absorption measurements. The eective mass of the energy bands of most

11


(a) (b)

GaAs WL InAsGaAs

InAs

1Å

20 nm

GaAs

Ener

gy

Position

910 911 912 913 914 9150.0

0.5

1.0

QD 3

QD 2

Nor

mal

ized

inte

nsity

Wavelength (nm)

QD 1

1.362 1.360 1.358 1.356

Energy (eV)(c)

Figure 2.4: Basic structural, electronic and optical properties of QDs. (a) Sketch of a single InAs

QD sitting on top of an InAs wetting layer (WL) and grown on a GaAs substrate. Illustration

from Ref. [40]. (b) Band diagram of QDs in the single-particle picture and eective-mass ap-

proximation. In a typical optical excitation, an electron-hole pair is created by the absorption

of a photon inside the wetting layer or the GaAs matrix. The pair would then relax via phonon

processes to the ground state of the QD and subsequently recombine radiatively by emitting a

single photon. (c) Emission spectrum of three self-assembled InAs QDs. Data from [41].

semiconductors has been thoroughly studied and a comprehensive compilation can be found in

Ref. [38]. This has been a major step forward towards the understanding of semiconductors

and lays the foundation for a formal description of semiconductor nanostructures, as seen in the

following.

2.2 Basic structural, electronic and optical properties of quan-

tum dots

The advent of modern nanotechnology has paved the way for the realization of complex semicon-

ductor heterostructures. The nanostructures investigated in the present thesis are quantum dots,

which represent three-dimensional nanosopic clusters of one material (e.g., InAs) embedded in a

host material with a larger band gap (e.g., GaAs). Quantum dots bring the high single-photon

purity of atoms [39] to a solid-state platform, which can be combined with mature semiconductor

processing techniques to tailor and scale their properties.

Many dierent classes of QD systems have been studied but the most commonly employed

are the so-called self-assembled In(Ga)As QDs grown in a GaAs matrix [42], as illustrated in

Fig. 2.5(a). They are grown by high-precision epitaxial methods under ultra-high-vacuum con-

ditions to minimize structural defects and impurities [43], which are ubiquitous in a solid-state

environment. The self-assembly growth relies on the 7 % lattice-constant mismatch between InAs

and GaAs to grow a thin (12 nm) wetting layer of InAs before the stored elastic energy is so

large that the strain relaxes and QDs are formed at random positions. The QDs are subsequently

capped by a thin layer of GaAs, which is partially shown in Fig. 2.5(a), to prevent oxidation and

12

Basic structural, electronic and optical properties of quantum dots

saturate the surface states. The size of QDs is in the few-nanometer range: a height of 35 nm

and an in-plane size of 1530 nm are usually found [44]. Due to quantum connement, quantized

states are formed in the QDs, see a sketch of the resulting band diagram in Fig. 2.5(b). We have

depicted one single valence band because in a QD only the heavy-hole band is relevant in optical

processes; a rigorous justication is given in the next section. Normally one or two quantized

states are formed in the conduction and valence bands before the continuum density of states of

the wetting layer sets in. If the surrounding material is excited optically, the created electrons

and holes can be captured by the QD. The quantized energy structure of the latter results in the

generation of a one-photon Fock state as depicted in Fig. 2.5(b). The fermionic nature of the

electron-hole pair results in strong Coulomb and exchange interactions, which, in turn, induce

an anharmonic electronic spectrum, thereby justifying the excellent single-photon purity of QDs

observed experimentally. The random self-assembled growth process results in QDs with various

sizes, shapes and material composition, which leads to a broad inhomogenous emission spectrum.

A typical example of a photo-luminescence spectrum is shown in Fig. 2.5(c), where each narrow

spectral feature corresponds to the emission of a single self-assembled QD.

Aside from self-assembled QDs, we extensively study two other QD systems in the present

thesis: interface-uctuation QDs [27] and droplet-epitaxy QDs [45]. Despite being grown with

dierent techniques, all these classes of QDs share most of the electronic and optical properties

described above. In the following we learn how to describe QDs using the concepts developed

for semiconductors, which are presented in Sec. 2.1.

2.2.1 Electronic models of quantum dots. Eective-mass theory

One of the most important properties of semiconductors, the translational symmetry induced by

the periodicity of the crystal lattice, does not hold for nanostructures. Consequently, the Bloch

theorem is no longer valid and cannot be used to describe the QD wavefunctions. A typical

semiconductor QD has somewhere between 10 to 100 thousand atoms, which constitutes a huge

multi-body system and developing electronic-structure methods is therefore an extremely chal-

lenging task. The most accurate theoretical models are the so-called ab-initio approaches using

concepts from density functional theory [46], where each atom is described individually within

a complete atomistic framework. Such methods have proven exceedingly useful for describing

molecules [47] and periodic systems of up to several hundred atoms [48] but are computation-

ally infeasible for larger structures. They do, however, provide reliable parameters for the more

practical semi-empirical methods, which are described in the following.

The bandstructure models developed so far for QDs rely on empirical parameters, which

quantify certain properties of the complicated crystal potential (e.g., the eective mass). There

are two classes of commonly employed models: atomistic theories, such as the empirical pseu-

dopotential theory [49], which simulate the contribution of every single atom comprising the QD,

and continuum approaches, such as the multiband k · p theory [37], which discard the QD atom-

istic nature and consider only the macroscopic potential. Atomistic models successfully address

the structure and symmetry of the mesoscopic QD potential as well as the underlying crystal

13


symmetry but suer from a limited generality and a high computational eort approaching ab-

initio methods [50]. The most feasible and mature electronic-structure models are the continuum

approaches, where the entire atomistic nature is merged into a couple of empirical parameters,

and only a "macroscopic" Schrödinger equation needs to be solved treating the mesoscopic po-

tential of the QD. An excellent example constitutes the 8 × 8 k · p theory [51], where the QD

potential and strain couple the 8 relevant bands (4 bands with a two-fold spin degeneracy, see

Fig. 2.3) and yield a system of 8 coupled dierential equations, which are solved on a modern

computer with relative ease.

The most commonly employed continuum theory is the envelope function theory or k · ptheory, which has been developed by Bastard [52] and uses the periodic Bloch functions uΓ(r)

as a complete and orthogonal set to expand the QD wavefunction Ψ(r) [53]

Ψ(r) =∑n

ψn(r)un,Γ(r), (2.14)

where the index n runs over all the bands in the semiconductor and ψn(r) are the expansion

coecients, also called slowly varying envelopes. Since there is an innite number of bands in

the solid, the sum has to be truncated in practice. There are 4 relevant bands governing the

optical properties of III-V semiconductors as explained in Sec. 2.1. As a consequence, three main

approaches are used to truncate the sum in Eq. (2.14), namely in a 8× 8 (all the four bands are

coupled), 6× 6 (the three valence bands) and 4× 4 band (the heavy- and light-holes) approach.

For instance, the 6 × 6 k · p theory yields solutions for the valence band, while the conduction

band is treated separately in an eective-mass fashion, as explained in the following paragraph.

The k · p theory has had remarkable success in modeling quantum wells. One shortcoming of the

Ansatz of Eq. (2.14) is that the expansion is performed over a set of functions that is complete and

orthogonal in bulk but not in the particular nanostructure. A rst-principles theory developed

by Burt [54] and Foreman [55] addresses this issue but has been largely ignored because it shows

little discrepancy with the formalism developed by Bastard.

A particular case of k · p theory is the eective-mass approximation that will be extensively

used in this work. We rst present the theory before discussing its physical justication. The

theory assumes that the bands, which are exact solutions in bulk, interact little with one an-

other so that the eigenstates of the nanostructure retain their bulk periodicity. Formally, this

corresponds to one single term in Eq. (2.14), so that a quantized eigenstate in every band can

be written as

Ψj(r) = ψj(r)uj,Γ(r), (2.15)

where j = e,hh, lh, so belongs to either of the four bands. The time-dependent wavefunction

of this eigenstate, Ψj(r, t) = Ψj(r)e−i(Ej/~)t, is dierentiated with respect to time and, using the

parabolic dispersion relation at the Γ point in Eq. (2.11), yields a Schrödinger-type equation [31],

whose time-independent part reads

Ejψj(r) = − ~2

2meff,j∆ψj(r) + Vj(r)ψj(r), (2.16)

14


Energy

Position

Eff-massapprox.

CB

VB

Figure 2.5: Physical interpretation of the eective-mass approximation. The complicated poten-

tial energy of the crystal (left) is merged into an eective-mass parameter (right).

where we have assumed that the eective mass is isotropic, which is a good approximation for III-

V semiconductors. The remarkable aspect about this eective-mass Schrödinger equation is that

the complicated unit-cell potential prole is merged into the eective-mass meff , a parameter

that can be accurately inferred from experiments. The potential energy V (r) now contains

only the smooth mesoscopic potential of the QD, as illustrated in Fig. 2.5. This particle-in-

a-box problem can be solved either analytically or numerically using the standard techniques

of quantum mechanics, which massively simplies the problem. There is need for one more

justication. In Eq. (2.16) V depends on r, which contradicts the single-electron approximation

by destroying the periodicity of the potential. However, if the potential is a smooth spatial

function, i.e., if its changes are small compared to the kinetic energy term over the scale of a

unit cell and over 2π |k|, then the material is called locally crystalline, and the eective-mass

approximation holds [31].

Even though QDs are complex three-dimensional structures, the eective-mass approximation

describes their properties remarkably well. This is because the rst valence-band eigenstate

is heavy-hole like with a negligible light-hole component for most QD systems. This can be

understood qualitatively as follows. The QDs that are presently studied have a small aspect

ratio, i.e., a height that is much smaller than the in-plane extension [44, 56, 57]. It turns out

that in the limit of vanishing in-plane connement, which is equivalent to a quantum well, the

heavy-hole band decouples from the light-hole band at the Γ point and is energetically closest

to the conduction band [58]. The splitting is of the order of 10 meV, which is suciently large

to confer a heavy-hole character to the rst quantized hole state of the quantum well. The

presence of a nite but small in-plane connement, as in the case of QDs with small aspect ratio,

15


induces a small light-hole contribution of less than 10 % and can be neglected for most practical

purposes [51, 59]. In this regard it is important to note that the eective-mass approximation is

only justied for QDs with small aspect ratio; other shapes may result in signicant heavy- and

light-hole mixing, and their electronic and optical properties would be altered. For instance, in

a spherical QD the heavy- and light-hole eigenstates remain degenerate because a sphere has a

higher symmetry than any crystal [60].

In addition to the aforementioned arguments, the eective-mass approximation is the most

widely employed QD theory due to its simplicity and intuitive nature. More complicated theories

have had a weak connection to experimental studies despite signicant theoretical eort. The

reason resides in the complexity of QDs, where optical spectroscopy is often incompatible with

studies of the exact geometry and material composition of QDs. In the present thesis we are

therefore employing the two-band eective-mass theory to describe the electronic properties of

QDs: we consider only the electron and the heavy-hole bands. There are QD systems, such as

self-assembled QDs, in which not only quantum connement but also strain plays an important

role in the electronic structure, as explained in the following.

2.2.2 Strain

There are three semiconductor systems investigated in the present thesis: Al(Ga)As, GaAs

and In(Ga)As. It so happens that the former two have the same lattice constant [38] and

the properties of the corresponding QDs are only governed by quantum connement. InAs

has, however, a 7 % larger lattice constant and, if grown on GaAs, the atoms are imposed to

accommodate to the lattice structure of the substrate, see the compressive-strain situation in

Fig. 2.6. Consequently, the atom i shifts from the equilibrium position ri to the non-equilibrium

position r′i, and the displacement vector u quanties this shift

u(ri) = r′i − ri. (2.17)

Since the atoms are away from equilibrium, internal forces tend to restore the equilibrium and

there is a certain elastic energy density E = E[u(r)] stored inside the QD. E is normally of the

order of tenths to hundredths of meV/nm3, i.e., of the same order as the quantization energy in

QDs, which is why strain plays an important role in the electronic structure of strained QDs.

The energy density E contains a dilatation component Es, which alters the band gap by

changing the volume of the unit cell without modifying the symmetry, and a distortion component

Ed, which lifts degeneracies in the valence band by lowering the symmetry of the unit cell. The

main inuence of compressive strain is to lower the light-hole band with respect to the heavy-

hole band, as illustrated in Fig. 2.7. As a consequence, in self-assembled QDs both quantum

connement and strain confer a negligible role to the light-hole band, thereby justifying the

eective-mass theory once again. It is worth mentioning the recent breakthrough of Huo et

al. [61], which managed to apply tensile strain to a QD (see Fig. 2.6) and to create a light-hole

ground state.

In a QD, the energy density E varies from unit cell to unit cell and, therefore, the band

structure is position-dependent as depicted in Fig. 2.8. In general, the distribution of strain

16


Material 2Compressive strain

SubstrateFStrain

Material 1Tensile strain

z

Figure 2.6: Illustration of compressive and tensile strain.

k

E E

k

EgE'g

e

lhhh

CompressiveStrain e

hh

lh

Figure 2.7: Qualitative visualization of the inuence of compressive strain on the band structure

of a semiconductor.

17


x

Eg

x

E

E'g

E

CompressiveStrain

e

hh+lh hh

lh

e

Figure 2.8: Qualitative visualization of the inuence of compressive strain on the band diagram

of a QD.

within a QD can be calculated with continuum as well as atomistic approaches [62]. Continuum

models are fully compatible with the eective-mass theory because they employ the continuum

elasticity theory [63], where the QD is treated as a continuous mesoscopic system and its atomistic

nature is discarded. The contribution of strain is simply included as a separate potential-energy

term in Eq. (2.16), i.e., V (r) = Vconfinement(r)+Vstrain(r), and is calculated by solving the central

equation of continuum elasticity theory Navier's equation

∇ ·(←→C ∇u

)= 0, (2.18)

where←→C is the elastic stiness tensor and is a material parameter, which is well documented for

III-V semiconductors. A thorough introduction to continuum elasticity theory can be found in

Ref. [63] and its application to QDs in Ref. [58]. The predictive power of such theories is, however,

limited because the strain distribution depends on many parameters like material composition,

amount of crystalline defects, which vary from QD to QD and are generally unknown. The

inuence of strain can be calculated more exactly for quantum wells, where the clean crystalline

growth provides a well-dened physical problem.

2.2.3 Excitons. Weak- and strong-connement regimes

All the results we have arrived at so far are a consequence of the single-electron Schrödinger

equation, which is able to explain a remarkably large class of eects in crystals as well as in

nanostructures. In photonics, the central physical process is the absorption and emission of

light, which is normally triggered by the creation or recombination of an electron-hole pair. While

electrons and holes can be described individually within the single-electron approximation, they

can also interact with one another because they possess charge and half-integer spin. In QDs,

the interaction between electrons and holes is further enhanced with respect to bulk because

they are squeezed together in a small region of space of a few nanometers. This electron-hole

bound state constitutes a fundamental quasi-particle, the exciton, which governs the optical

18


properties of a large class of semiconductor structures including QDs. Being a two-body system,

the description of the exciton goes beyond the single-electron approximation but is very similar

to the formalism we have presented so far, if the single-particle wavefunction Ψe/h is replaced

by the exciton wavefunction ΨX(re, rh). In a QD, ΨX can be expanded in the single-particle

electron and hole wavefunctions:

ΨX(re, rh) =∑n,m

Cn,mΨn(re)Ψm(rh), (2.19)

where Ψn corresponds to the n-th eigenstate of the QD. In the eective-mass approximation,

Ψn(r) = uΓ(r)ψn(r), where uΓ(r) is the periodic Bloch function evaluated at k = 0 and ψn(r)

the slowly varying envelope subject to the single-particle eective-mass Schrödinger equation.

In the following we drop the index Γ in the Bloch function for simplicity. Equation (2.19) can

therefore be written as

ΨX(re, rh) = ue(re)uh(rh)∑n,m

Cn,mψe,n(re)ψh,m(rh) = ue(re)uh(rh)ψX(re, rh), (2.20)

where χ(re, rh) is the slowly varying envelope of the exciton subject to the two-body eective-

mass Schrödinger equation(p2e

2me+

p2h

2mh+ Ve(re) + Vh(rh)− e2

4πε0εr |re − rh|

)ψX(re, rh) = EψX(re, rh). (2.21)

Here, εr is the background dielectric constant and E the eigenenergy of the exciton. In bulk, the

attraction between the electron and the hole results in a spatial separation between them known

as the exciton Bohr radius a0. Since the Coulomb energy EC scales inversely with the QD size

EC ∝ L−1, the Coulomb and exchange interactions in a QD are enhanced compared to bulk.

These processes confer a non-trivial ne structure to QDs, as explained in the next section.

Despite the enhanced Coulomb processes, the spatial motion and distribution of the exciton is

not only determined by Coulomb connement but also by quantum connement. It is well-known

from quantum-mechanics textbooks that the quantum-connement energy scales as † L−2 [31].

As a consequence, the exciton motion can be found in two regimes:

(i) The strong-connement regime, in which the QD size L is smaller than the exciton Bohr

radius a0 [60] and quantum connement dominates Coulomb connement. The latter can then be

treated as a vanishingly small perturbation and, in the rst approximation, neglected completely.

As a consequence, the electron and hole move independently of each other as non-interacting

particles and the exciton slowly varying envelope χ can be written as a product of the individual

electron and hole wavefunctions, i.e.,

ψX(re, rh) = ψe(re)ψh(rh). (2.22)

The single-particle wavefunctions ψe,h(r) can then be computed individually with the single-

particle eective-mass Schrödinger equation, thereby substantially simplifying the problem. Most

†More precisely, the connement energy scales as L−2 only for a potential well with innite barriers [31]. In

practice it scales as L−n with n ∈ (1; 2).

19


of the semiconductor QDs studied so far belong to the strong-connement regime and, despite

being a complex multi-body system, can be modelled within the single-particle approximation

remarkably well.

(ii) The weak-connement regime, in which L a0 and the electron-hole motion is strongly

correlated with a negligible role from the QD boundary. Here, Eq. (2.21) cannot be simplied

further and has to be solved as a two-body problem. Achieving this regime has been a long-

sought goal in quantum photonics because such QDs couple giantly to light, as is experimentally

demonstrated in Chapter 3.

Excitonic eects have a prominent role in determining the QD energy structure because they

couple the bare single-particle eigenstates, as seen in the following.

2.2.4 Heavy-hole excitons

An electron in the conduction band and a heavy hole in the valence band constitute the funda-

mental quasi-particle studied throughout the present thesis: the heavy-hole exciton. Combining

the electron contribution with a spin of ±1/2 with the heavy-hole contribution with a projected

angular momentum of ±3/2, see Eq. (2.12), yields four possible excitonic congurations: two

optically bright excitons with jz = ±1 and two optically dark with jz = ±2. Their optical

brightness can be checked explicitly by evaluating the dipole moment

µ =e

m0〈0 |p|ΨX〉 , (2.23)

where |0〉 denotes the vacuum state. The underlying fermionic nature of the excitons leads to an

exchange-type interaction between the the electron and hole, which couples and splits these four

bare excitonic eigenstates. Understanding their energy structure in an important prerequisite

for performing and interpreting spectroscopic analyses on QDs, as explained in Sec. 2.5. In this

section we present the formalism that can be used to provide such an understanding.

Using standard semiconductor-physics textbooks [64], it can be shown that the energy of the

exchange interaction is proportional to [65]

Eexchange ∝∫ ∫

dr1dr2Ψ∗X(re = r1, rh = r2)1

|r1 − r2|ΨX(re = r2, rh = r1). (2.24)

The integration is normally divided in two parts leading to a short-range contribution where

the electron and hole are in the same unit cell, and a long-range interaction where the particles

are in dierent cells. The latter has little eect on the energy structure [65] and we therefore

discuss only the former. The main role of the short-range interaction is to split the bright and

dark states in energy. This can be understood by using the short-range interaction Hamiltonian,

which is derived using the theory of invariants and contains an electron with spin se and a hole

with spin jh [66]

Hshort = −∑

i=x,y,z

(aijh,ise,i + bij

3h,ise,i

), (2.25)

where a and b are QD parameters. Evaluating the operators s and j on the projected angular

momentum bases (|+1〉 , |−1〉 , |+2〉 , |−2〉) is discussed in detail in Ref. [67] and is outlined in

20


Appendix A. The resulting matrix representation of the Hamiltonian reads

Hheavy-hole =1

2

+δ0 +δ1 0 0

+δ1 +δ0 0 0

0 0 −δ0 +δ2

0 0 +δ2 −δ0

, (2.26)

where δ0 = 1.5(az + 2.25bz), δ1 = 0.75(bx − by), and δ2 = 0.75(bx + by). Since the matrix is

block diagonal, bright and dark excitons do not mix with each other but are split by δ0. Bright

excitons, however, mix with each other and are split by δ1, as are dark excitons by δ2, and the

new eigenstates are symmetric and antisymmetric combinations of the bare states, i.e.,

|ΨX1〉 =1√2

(|−1〉+ |1〉) ,

|ΨX2〉 =1√2

(|−1〉 − |1〉) ,

|ΨX3〉 =1√2

(|−2〉+ |2〉) ,

|ΨX4〉 =1√2

(|−2〉 − |2〉) ,

(2.27)

where |b〉 and |d〉 denote bright and dark states, respectively. The transition dipole moment of

the dressed states can be computed with the help of Eqs. (2.12) and (2.23) yielding

µX1 = −iΠey,

µX2 = Πex,

µX3 = µX4 = 0,

(2.28)

where Π = 〈ux |px|ue〉 and we have omitted the slowly varying envelopes for simplicity since

they do not carry any information about polarization. The two bright states are orthogonally

polarized along the x = [1, 1, 0] and y = [1,−1, 0] crystallographic directions, respectively.

The splitting between the two bright states is of the order of tens of µeV [68] and is mostly

determined by the QD asymmetry. For in-plane symmetric QDs, bx = by and the two bright

eigenstates are degenerate. Such a scenario is hardly ever encountered in practice because it

can be shown with more exact atomistic models [68] that even perfectly symmetric QDs have a

lower crystallographic symmetry leading to the eigenstates of Eq. (2.27). The two parameters bx

and by depend on the QD wavefunctions and can be tuned by external elds, which change the

distribution of the wavefunctions within the QD [69]. Bright excitons are higher in energy than

dark excitons by several hundred µeV [68] and the corresponding energy structure is illustrated

in Fig. 2.9. These properties of the heavy-hole exciton will be used throughout the present thesis,

both in spectroscopic analyses and in theoretical calculations.

2.2.5 Light-hole excitons

The light-hole exciton is a fundamental quasi-particle formed by an electron in the conduction

band and a light hole in the valence band. Despite the fact that most of the QD studies have

21


Energy structure Dipole moment

Figure 2.9: Energy structure and the corresponding orientation of the dipole moment of a heavy-

hole exciton.

dealt with heavy-hole excitons so far, light-hole excitons conned in single QDs are becoming

experimentally accessible [61] and provide a new platform for the exploration of QDs in quantum

technologies. We therefore present their ne structure but note that this section is not required

for understanding the rest of the thesis.

The light-hole excitons contain a hole with j = 1/2 and an electron with s = 1/2, and form

4 possible congurations(|0jz=1/2,sz=−1/2〉 , |0jz=−1/2,sz=1/2〉 , |1〉 , |−1〉

)≡ (|0〉 , |0〉 , |1〉 , |−1〉).

It can be checked using Eq. (2.12) that, unlike heavy-hole excitons, all these bare states are

optically active. The Hamiltonian in Eq. (2.25) can be evaluated in this basis set with the help

of Appendix A yielding

Hlight−hole =1

2

+∆0 +∆1 0 0

+∆1 +∆0 0 0

0 0 −∆0 +∆2

0 0 +∆2 −∆0

, (2.29)

where ∆0 = 0.5(a3 + 0.125b3), ∆1 = −(a1 +a2 + 2.5b1 + 2.5b2) and ∆2 = (a2−a1) + 2.5(b2− b1).

The matrix is block diagonal and excitons with mj = 0 do not couple to excitons with |mj | = 1

but are split in energy by ∆0. Excitons with the same |mj | are coupled by ∆1 and ∆2, and the

dressed eigenstates constitute symmetric and antisymmetric superpositions of the bare states

|ΨX1〉 =1√2

(|0〉+ |0〉) ,

|ΨX2〉 =1√2

(|0〉 − |0〉) ,

|ΨX3〉 =1√2

(|−1〉+ |1〉) ,

|ΨX4〉 =1√2

(|−1〉 − |1〉) .

(2.30)

Despite being mathematically analogous, light-hole excitons have a dierent polarization spec-

22

Density of states of conned systems

Energy structure Dipole moment

Figure 2.10: Energy structure and the corresponding orientation of the dipole moment of a

light-hole exciton.

trum. The dipole moment of the 4 states is evaluated using Eq. (2.12) yielding

µX1 =2√3

Πez,

µX2 = 0,

µX3 = −i 1√3

Πey,

µX4 =1√3

Πex.

(2.31)

The light-hole excitons have electric dipoles oriented along all three Cartesian directions x, y and

z = [0, 0, 1]. The z-polarized exciton has a dipole moment twice larger than the other two bright

excitons, and is normally higher in energy by several hundred µeV [61] as shown schematically in

Fig. 2.10. The x- and y-polarized states inherit the properties of the bright heavy-hole excitons

but have a smaller dipole moment.

2.3 Density of states of conned systems

The density of electronic states (DES) is an important quantity that governs the process of

absorption and emission of light in semiconductor nanostructures. Quantum dots are often

grown on a wetting layer or a quantum well, which represents a combination of zero- and two-

dimensional systems. They therefore exhibit a non-trivial energy structure with a certain spectral

density of available states. In the following we derive the DES of an arbitrary n-dimensional

system and then analyze particular cases.

The DES is dened as the number of available electronic states per unit volume and energy.

According to Fermi's Golden Rule, which is discussed in Sec. 2.5, the absorption of light is

directly proportional to DES. We consider one single band with an isotropic eective mass meff

23


but the extension to multiple bands is done straightforwardly in an additive fashion

E =~2k2

2meff. (2.32)

Since the reciprocal lattice is discrete, the wavevector of the electron can take only discrete

values separated equidistantly by 2πL , where L is the length of the one dimensional system. In n

dimensions the elementary volume of the reciprocal lattice Vk,n equals

δVk,n =(2π)n

Vn, (2.33)

where Vn is the volume of the structure. The number of states gn(k) normalized to the volume

of both the real and reciprocal lattices is just the inverse of δVk,n divided by Vn, i.e., gn(k) =

1/(δVk,nVn)

gn(k) =1

(2π)n. (2.34)

The quantity gn(k) is the so-called density of states in k-space and is a constant. To relate it to

the relevant DES in energy, the dispersion relation between the k and energy spaces needs to be

used, see Eq. (2.32), which implies that the iso-energy surface is an n-dimensional sphere. We take

into account spin degeneracy (i.e., a factor of 2) and that the surface of a n-dimensional sphere

equals 2πn/2kn−1

Γ(n/2) , where Γ denotes the Gamma-function. Hence, the DES in energy multiplied

by an innitesimal energy interval dE must be equal to the DES in k-space multiplied by an

innitesimal volume in k-space dnk, i.e.,

gn(E)dE = 2gn(k)dnk = 21

(2π)n2π

n2 kn−1

Γ(n2

) (dk

dE

)dE. (2.35)

By carrying out the calculation one arrives at

gn(E) =2

Γ(n2

) (meff

2π~2

)n2

(E − E0)n2−1. (2.36)

The DES in 0, 1, 2 and 3 dimensions is evaluated as

g0(E) = 2δ(E − E0)

g1(E) =

√2meff

π~1√

E − E0

g2(E) =meff

π~2

g3(E) =

√2m

3/2eff

π2~3

√E − E0.

(2.37)

Figure 2.11 visualizes the DES in dierent dimensions. As the dimensionality of the system

is lowered, the DES becomes more conned in energy. In a zero-dimensional system, which

corresponds to a QD, the DES is a sequence of delta functions, thereby justifying the spectrally

narrow emission lines from QDs in Fig. 2.3(c). The DES can be measured eciently in absorption

measurements. An excellent example is shown in Ref. [70], where the stepwise DES of a quantum

well is observed experimentally and modelled theoretically remarkably well. The advantage of

the spectrally narrow DES in QDs is that the oscillator strength of an entire band is merged into

a single narrow line, which induces a large spectral interaction between QDs and light [71].

24

The electromagnetic quantum-vacuum eld

DES

Energy

n=2n=3 n=1 n=0

Figure 2.11: Illustration of the DES in 0, 1, 2 and 3 dimensions.

2.4 The electromagnetic quantum-vacuum eld

Spontaneous emission is the central physical process studied in the present thesis. An excited

emitter such as a QD may interact with the electromagnetic quantum-vacuum eld and sponta-

neously emit a photon. While the properties of the quantum emitter have been investigated in

detail in the previous section, here we present the physical properties of light, which comprises

electric and magnetic elds that oscillate in time and space. We rst describe their classical prop-

erties before generalizing them into a quantum-mechanical framework. Maxwell's equations are

the fundamental relations that govern the behavior and propagation of electromagnetic elds [72]

∇×E(r, t) = −∂B(r, t)

∂t,

∇ ·B(r, t) = 0,

∇×H(r, t) =∂D(r, t)

∂t+ j(r, t),

∇ ·D(r, t) = ρ(r, t),

(2.38)

where E, B, D, H denote the electric eld, magnetic induction, electric displacement and mag-

netic eld, respectively, and j and ρ the current and charge densities, respectively. These identi-

ties are complemented by the constitutive equations relating D and B to E and H through the

electric and magnetic properties of the material, which we assume to be isotropic

D(r, t) = ε0ε(r)E(r, t),

B(r, t) = µ0µ(r)H(r, t),(2.39)

where ε0 and µ0 (ε and µ) the vacuum (relative) electric permittivity and magnetic permeability,

respectively. We henceforth assume non-magnetic media at optical frequencies and set µ(r) = 1.

The electromagnetic eld generated by a quantum emitter propagates according to the rules

established by Maxwell's equations. Combining the latter with the constitutive relations yields

25


the fundamental equation governing the propagation of the electromagnetic eld in space and

time the wave equation

∇×∇×E(r, t) +ε(r)

c20

∂2E(r, t)

∂t2= −µ0j(r, t), (2.40)

subject to the condition ∇ × E = 0. Here, c0 = (ε0µ0)−1/2 is the vacuum speed of light and

we have assumed charge-free environments ρ = 0. An optically linear medium does not couple

dierent frequency components and the elds can be expanded in time-harmonic modes via

E(r, t) = E(r)e−iωt and similarly for j. The spatial prole E(r) satises the Helmholtz equation,

which is obtained by plugging the modes into the wave equation in Eq. (2.40)

∇×∇×E(r)− ε(r)k20E(r) = iωµ0j(r), (2.41)

where k0 = ω/c0 is the wavevector of light in vacuum. This equation contains a potentially

complex spatial distribution of a source current on the right-hand side, which generates the eld

prole from the left-hand side. A powerful method of solving such an inhomogeneous dierential

equation is to use the Green tensor of the system←→G (r, r′), which is dened by the electric eld

at the eld point r generated by an electric dipole located at r′ [72]. The Green tensor satises

the Helmholtz equation similarly to the electric eld E(r) but with a point-like source replacing

the complex current distribution j(r)

∇×∇×←→G (r, r′)− ε(r)k2

0

←→G (r, r′) =

←→I δ(r− r′), (2.42)

where←→I is the unit tensor. The Green function is the impulse response of the system and,

according to the convolution theorem, can be used to calculate the radiation E(r) of an arbitrary

current distribution j(r) via

E(r) = iωµ0

∫dr′←→G (r, r′)j(r′). (2.43)

The Green tensor is solely a property of the electromagnetic environment and is determined

by the distribution of the permittivity ε(r). The physical meaning of←→G can be grasped by

considering a radiating point dipole at r0, i.e., j(r) = −iωpδ(r − r0), where p is the electric

dipole moment

E(r) = ω2µ0←→G (r, r0)p. (2.44)

We can interpret the Green tensor as a propagator for the electric eld from r0 to r in an

environment determined by ε(r).

The concept of the Green function will be extensively used in the present thesis when studying

the light-matter interaction. Closed-form expressions for the Green tensor exist only for a few

specic environments, such as a stratied medium [73]. It is often convenient to separate←→G (r, r′)

into a homogeneous-medium component←→GH(r, r′), which normally coincides with the medium

where the quantum emitter is located, and a scattering component←→GS(r, r′), which provides the

response of the inhomogeneous character of the environment. The homogeneous solution can be

26

Fundamental light-matter interaction with quantum dots

expressed in a closed form via [74]

←→GH(r, r′) =

(←→I +

∇∇k2H

)eikHR

4πR, (2.45)

where kH = k0√εH is the wavevector in the background medium with a relative permittivity

εH , and R = |R| = |r− r′| is the relative distance.It is well known from classical electrodynamics that the electric E and magnetic B elds can

be recast in terms of the scalar φ and vector A potentials. From ∇ ·B = 0 it follows that the

magnetic induction can be written as the curl of the vector potential because ∇ · (∇×) ≡ 0.

Then it follows from Maxwell's equations that

B = ∇×A,

E = −∂A

∂t−∇φ.

(2.46)

Even though these relations uniquely determine E and B, they are not sucient to specify the

potentials uniquely. This is why a choice of gauge needs to be adopted where a further constraint

xes the uniqueness of A and φ. It is often preferred to deal with the potentials rather than the

physically more meaningful electric and magnetic elds, as is shown in the next section.

The spontaneous-emission process is triggered by the ground state of the electromagnetic eld,

formally denoted as the electromagnetic vacuum eld. In a quantum-mechanical picture, the

vacuum eld has a nite energy of ~ω/2 and can be explained by quantizing the electromagnetic

eld [75]. The electric eld becomes an operator

E(r) = i∑l

√~ωl2ε0

[alfl(r) + a†l f

∗l (r)

], (2.47)

where a†l and al are the creation and annihilation operators of a photon in the normal eld mode

fl(r), which satises the Helmholtz equation and obeys the normalization condition∫drε(r)f∗m(r) · fn(r) = δmn. (2.48)

The normal modes fi(r) are eigensolutions of the Helmholtz equation and any eld E(r) can be

expanded into them. The magnetic-eld operator can be recast from Eq. (2.47) using Maxwell's

equations. The quantization procedure confers a non-zero variance to the ground (or vacuum)

state of the electromagnetic eld |0〉, which is identically zero in the classical description. Such a

vacuum state can be depicted as virtual photons that exist in brief moments of time, as allowed

by the uncertainty principle of quantum mechanics. Spontaneous emission is triggered by the

interaction of an excited emitter with these virtual photons belonging to the vacuum state |0〉resulting in an emitted photon in the mode l, i.e., |1l〉, as shown in the following section.

2.5 Fundamental light-matter interaction with quantum dots

The interaction between light and matter is an ubiquitous process in nature and lies at the

heart of modern light sources and detectors, such as light-emitting diodes, lasers, photodiodes

27


and solar cells. Eciently interfacing solid-state quantum emitters with light plays a paramount

role in quantum-information science and is the subject of intense fundamental research in cavity

quantum electrodynamics [22]. In the present section we lay the fundamental as well as the

experimentally relevant aspects describing the interaction between QDs and light, which represent

the starting point for the research carrier out in this work.

2.5.1 Spontaneous emission

The process of photon emission governs the conversion of a stationary qubit to a ying qubit and

its understanding is of immense signicance for the realization of ecient light-matter interfaces.

The interaction between an exciton in a QD with the electromagnetic vacuum eld is the central

process studied in the present thesis. An important aspect of this process is the temporal

dynamics of the exciton-to-photon conversion, which is governed by the light-matter interaction

strength. The latter was considered to be solely an intrinsic property of the emitter before the

work of Purcell in 1946 [5], which showed that the spatial density of the optical modes can

tailor this interaction. Purcell found that the spontaneous-emission rate of an emitter in a cavity

was increased by a factor of FP = 3Qλ2/4π2V , where Q and V are the quality factor and

the mode volume of the cavity, respectively, and λ is the wavelength of light. This prediction

was experimentally veried by Drexhage in 1970 [76], where he found that the spontaneous-

emission rate can be both enhanced and suppressed when the distance between an emitter and

a metal surface is varied. Nowadays, modern fabrication techniques enable accurate tailoring

of the nanophotonic environment surrounding QDs, which in turn allows to carefully optimize

the light-matter coupling strength. Fundamental as well as applied studies of QDs in photonic-

crystal cavities [77] and waveguides [21], micropillar cavities [20] and nanowires [78] constitute a

few out of many hot research directions that are followed at the moment.

The QD-light interaction can be found in two fundamental regimes, namely the strong- and

weak-coupling regimes, which are distinguished based on the coupling strength g [72]

g =

√~ω

2ε0V

µ

~, (2.49)

where µ is the QD dipole matrix element from the excited to the ground state. For g Γrad,

where Γrad is the spontaneous-emission rate into the cavity, the interaction is in the strong-

coupling regime, and light and matter degrees of freedom become entangled. This leads to a

non-Markovian time decay of the emitter because the emitted photon is likely to be re-absorbed

a number of times before it leaks out of the cavity. These so-called vacuum Rabi oscillations

were experimentally demonstrated nearly 20 years ago at microwave frequencies [79], which has

been followed by a plethora of similar studies in other systems including QDs in solid-state

cavities [80].

The regime in which the cavity enhances the density of optical states with a subsequent

Markovian dynamics is the weak-coupling regime, i.e., g Γrad, which is the most widely

studied type of light-matter interaction and is main focus of the present thesis. In this regime,

the exciton population of the QD decays exponentially with the rate Γrad subject to Fermi's

28


Golden Rule [31]

Γrad(ω) =2π

~2

∑f

∣∣∣⟨f ∣∣∣Hint

∣∣∣ i⟩∣∣∣2 δ(ω − ωif ), (2.50)

where Hint is the light-matter interaction Hamiltonian triggering a transition from the initial |i〉to the nal |f〉 state. In this work we consider the minimal coupling interaction Hamiltonian [81]

between a particle with charge e and mass m0, and the eld described by the vector A and scalar

φ potentials

Hint = − e

2m0

(p · A + A · p− eA · A

). (2.51)

Another commonly used interaction Hamiltonian is the multipolar formalism, where the Hamil-

tonian is expressed in terms of the physically meaningful electric E and magnetic B elds. It

can be shown [81] that the two Hamiltonians give the same result for processes subject to energy

conservation such as spontaneous emission. In Chapter 5 we are showing this explicitly when

developing a multipolar theory of light-matter interaction with QDs.

The nonlinear term A · A can be neglected for the weak elds with low energies triggering

spontaneous emission at optical frequencies. We employ the generalized Coulomb gauge in which

the scalar potential vanishes φ = 0 [82]

∇ ·[ε(r)A(r)

]= 0, (2.52)

which renders the commutator [A, p] = ∇ · A, in general, dierent than zero. In this work

we consider spatial variations of the dielectric constant over length scales much smaller than

the QD spatial extent and we may treat ε(r) constant [82], thereby yielding for the interaction

Hamiltonian

Hint ' −e

m0A(r) · p. (2.53)

The vector-potential operator can be recast from Eq. (2.47) using Eq. (2.46)

A(r) =∑l

√~

2ε0ωl

[alfl(r) + a†l f

∗l (r)

]. (2.54)

We approximate the QD with a two-level system with the initial state |i〉 = |e〉 ⊗ |0〉, where |e〉denotes the excited electronic state and |0〉 the vacuum state of the electromagnetic eld, and

|f〉 = |g〉 ⊗ |1f 〉 the nal state with one excitation in the eld mode f and the emitter in the

ground state |g〉. We note that only the term containing a† yields a non-zero contribution to

Γrad and Eq. (2.50) can be written as

Γrad(ω) =πe2

ε0~m20

∑f

∣∣∣∣∣∑l

1√ωl〈g| f∗l (r) · p |e〉〈1f | a†l |0〉

∣∣∣∣∣2

δ(ω − ωf ). (2.55)

The eld matrix element yields 〈1f | a†l |0〉 = δlf and we arrive at the important result

Γrad(ω) =πe2

ε0~m20

∑l

1

ωl|〈g| f∗l (r) · p |e〉|2 δ(ω − ωl), (2.56)

29


which is the starting point for the formalisms developed in the present thesis. The meaning of

the sum∑l is that the emitter can decay over the entire 4π solid angle at a xed frequency

ωl. This is the most general expression describing the decay rate of a two-level emitter in the

weak-coupling regime. The expression is beyond the textbook dipole approximation because the

distribution of the eld over the emitter is fully taken into account. While providing interesting

theoretical challenges, Γrad is also a physical observable and, thus, an experimentally accessible

quantity. Novel and exciting information about both the QD and the eld can be inferred from

comparing theory with experiment, as shown by the research in the present thesis.

2.5.2 The dipole approximation

The standard-textbook approach to evaluating Eq. (2.56) is to assume that the electromagnetic

eld varies slowly over the spatial extent of the QD fl(r) ≈ fl(r0), where r0 is the center of the

QD. The resulting dipole approximation is excellent for most quantum emitters because they

are much smaller than the wavelength of light. Under this assumption, Fermi's Golden Rule,

Eq. (2.56), reads

Γrad(ω) =π |µ|2

ε0~ep ·

[∑l

1

ωlf∗(r0)f(r0)δ(ω − ωl)

]· e∗p, (2.57)

where ep is the unit vector pointing along the direction of the dipole moment µ, which is given

by Eq. (2.23). It can be shown [72, 83] that the term in square brackets is proportional to the

imaginary part of the Green tensor evaluated at the origin

Im[←→G (r0, r0)

]=πc202

∑l

1

ωlf∗l (r0)fl(r0)δ(ω − ωl), (2.58)

which yields for the decay rate

Γrad(ω) =2µ0 |µ|2

~Im[ep ·←→G (r0, r0) · e∗p

]. (2.59)

In the dipole approximation, the light-matter interaction strength is governed by two fundamental

quantities: the dipole moment, which is an intrinsic property of the emitter, and the projected

imaginary part of the Green tensor, which is purely a property of the electromagnetic eld.

Tailoring the spontaneous-emission process can therefore be done either at the emitter level using

novel growth techniques of QDs or at the eld level by structuring the environment surrounding

the QDs.

It is often useful to recast the emitter and eld properties in terms of quantities with a more

intuitive physical meaning, namely the oscillator strength f and the local density of optical

states (LDOS) ρ(r0, ω, ep). The oscillator strength is a dimensionless quantity dened as the

ratio between the radiative rate of the QD in a homogeneous environment and the emission rate

of a classical harmonic oscillator, and is related to the dipole moment via

f =m0

e2~ω|µ|2 . (2.60)

30


The oscillator strength of atoms is of the order of 1 and of QDs about 10 [84], which makes QDs

ecient light-matter interfaces. The LDOS is dened as the number of electromagnetic modes

per unit energy and volume that the emitter can decay into, and is related to the projected Green

tensor via

ρ(r0, ω, ep) =2ω

πc20Im[ep ·←→G (r0, r0) · e∗p

]. (2.61)

The LDOS can be both enhanced and suppressed with respect to a homogeneous medium, as

shown by Drexhage 40 years ago. Placing the emitter in the antinode of a cavity may substantially

enhance the LDOS and, thus, the light-matter coupling strength, whereas in a photonic band

gap the spontaneous-emission process would be strongly suppressed because there are no modes

the emitter can decay into.

In a homogeneous environment, Eq. (2.59) can be evaluated analytically by taking the limit

R→ 0 in the homogeneous part of the Green tensor in Eq. (5.46), which yields

Im[←→G (r0, r0)

]=

ωn

6πc0

←→I , (2.62)

where n = ε2 is the refractive index of the background medium. The LDOS scales linearly with

the wavevector of light kB because larger wavevectors correspond to an increased number of

propagating eld modes inside the medium. This yields the textbook expression for the decay

rate in a homogeneous medium

Γhomrad =

µ0ωn

3π~c0|µ|2 . (2.63)

Finally we note that only the imaginary part of the Green tensor contributes to the decay

rate because energy dissipation is described by the part of the response function, which is π/2

out of phase with the driving eld, as is known from linear response theory. The real part is

proportional to a self-energy term, also known as the Lamb shift, which shifts the frequency of

the QD exciton [85].

2.5.3 Decay dynamics of quantum dots

So far we have described the spontaneous-emission process of QDs as the recombination of

a two-level system. In reality, however, the level structure of QDs is more complicated, as

presented in Sec. 2.2: there are 4 excited states comprising 2 bright and 2 dark excitons. These

states are coupled by spin-ip processes because the constituent electron or hole may ip spin.

The complexity of the problem is further enhanced by the omnipresent nonradiative processes,

such as defect traps in the vicinity of the QD [86], which provide alternative pathways for

the recombination of the exciton. All in all, the experimentally measured decay dynamics of

QDs intertwines radiative, nonradiative and spin-ip process and is more complex than a single

decaying exponential. It is of high experimental signicance to extract all these rates from

measurement because they provide important information about the optical quality of QDs,

namely the oscillator strength, which is proportional to the radiative rate, and the quantum

eciency η, which quanties the probability that an exciton captured by the QD recombines

31


Figure 2.12: Level scheme describing the population transfer of the exciton in a QD, which has

a direct impact on the bright-exciton decay dynamics. The bright exciton |b〉 can decay either

radiatively (Γrad) or nonradiatively (Γnrad) and can interact with its dark counterpart |d〉 via thespin-ip rate (Γsf).

radiatively

η =Γhom

rad

Γhomrad + Γnrad

, (2.64)

where Γnrad is the nonradiative decay rate and Γhomrad denotes the radiative rate in a homogeneous

medium. In the following we present a method that can unambiguously extract these quantities

from optical measurements and that will be extensively used in the present work [57, 84].

The picture involving the 5 coupled QD levels (4 excited states and one ground state) can be

simplied by noting that spin-ip processes are strongly inhibited in QDs as compared to bulk

or quantum wells because the energy levels are quantized, which makes it dicult for charge

carriers to simultaneously ip spin and full energy conservation. We can therefore retain only

rst-order spin-ip processes and neglect double or higher-order processes. This leads to a picture

in which the coupling between bright-bright and dark-dark excitons is removed because it is a

second order process, i.e., it involves two spin-ip processes. As a consequence, only bright-dark

excitons are coupled by the spin-ip rate Γsf and the former ve-level picture is reduced to the

three-level scheme depicted in Fig. 2.12.

Bright excitons |b〉 may decay both radiatively with the rate Γrad or nonradiatively with

Γnrad. Additionally, bright excitons may ip spin and become dark with the rate Γsf . Dark

excitons |d〉 do not decay radiatively owing to optical selection rules presented in Sec. 2.2 but

they may decay nonradiatively with Γnrad and ip spin with Γsf . In the previous statements

we have implicitly carried out two further assumptions. First, the bright-dark and dark-bright

spin-ip rates are assumed to be the same because the spin-ip process is phonon mediated owing

to the energy-conservation requirement. As a consequence, the thermal energy kBT ≈ 1 meV

at a temperature of T = 10 K that the experiments in the present work are carried out at, is

much larger than the bright-dark energy splitting δ0 ≈ 100µeV. Second, the nonradiative rates

for bright and dark excitons are assumed to be the same due to their small energy splitting, as

demonstrated experimentally in Ref. [87].

The decay dynamics of the bright exciton are governed by the rate equations of the coupled

32


three-level system, which read(ρB

ρD

)=

(−Γrad − Γnrad − Γsf Γsf

Γsf −Γnrad − Γsf

)(ρB

ρD

), (2.65)

where ρ denotes the probability to occupy the corresponding level and the dot indicates the

time derivative. Under the realistic assumption that spin ip-processes are much slower than the

radiative decay rate, i.e., Γsf Γrad, Eq. (2.65) is solved yielding for the temporal decay of the

bright state

ρB(t) = ρB(0)e−(Γrad+Γnrad)t +Γsf

ΓradρD(0)e−(Γnrad+Γsf )t. (2.66)

The bright exciton exhibits a biexponential decay with the fast rate ΓF = Γrad + Γnrad and the

slow rate ΓS = Γnrad + Γsf . Consequently, by tting the measured decay curves with f(τ) =

AF e−ΓF τ+ASe

−ΓSτ+C, where τ is the time delay with respect to the start of the excitation pulse

and C is the background level, which is determined by the measured dark-count rate and after-

pulsing probability of the detector, the radiative and nonradiative rates can be unambiguously

extracted via

Γrad = ΓF − ΓS , (2.67)

Γnrad = ΓS −ASAF

ρB(0)

ρD(0)(ΓF − ΓS) , (2.68)

Γsf =ASAF

ρB(0)

ρD(0)(ΓF − ΓS) . (2.69)

The standard excitation scheme performed in this work is above-band excitation, where charge

carriers are photo-generated in the matrix surrounding the QD. The trapping process of these

carriers by the QD has a random character implying that bright and dark excitons are prepared

with equal likelihoodρB(0)

ρD(0)' 1. (2.70)

In our experiments, the decay dynamics is recorded by selecting a single spectral line and sending

it to the a single-photon detector for time-resolved measurements. In order to quantify how well

the t reproduces the experimental data we dene the weighted residual Wk as

Wk =ρM (tk)− ρF (tk)√

ρM (tk), (2.71)

where ρM is the measured data, ρF represents the tted value, and the discreetness of the time-

delay axis is denoted by the subscript k. The biexponential decay is tted to the acquired data

using a least-squares approach where the collapsed residual χ2R = 1

N−p∑Nk=1W

2k is minimized,

N being the total number of time bins and p the number of adjustable parameters in the model.

This self-consistent procedure allows to extract the oscillator strength f and quantum e-

ciency η from optical measurements. It is important to underline that, in general, the extracted

radiative rate Γrad does not coincide with the homogeneous-medium quantity Γhomrad because QDs

are often located close to dielectric-dielectric or dielectric-air interfaces, which may modify the

33


projected LDOS at the QD position. It is therefore important to determine the LDOS contribu-

tion in every experiment to correctly evaluate the oscillator strength and quantum eciency.

In conclusion, we stress that spin ip is the key process making the aforementioned scheme

work. In the present thesis we study three dierent classes of QDs and we nd the biexponential

model to reproduce the decay dynamics of the bright exciton remarkably well.

2.6 Summary

In this chapter we have presented the important physical processes describing light-matter in-

teraction with QDs. We have learned that QDs constitute a complex multibody system but

can be treated remarkably well with the simple and intuitive eective-mass formalism. We have

presented the energy structure and optical selection rules of QDs using the theory of invariants

within a two-body excitonic picture. Finally, the interaction of these energy states with the

electromagnetic quantum vacuum results in a biexponential decay of the bright excitons. The

radiative decay rate of QDs can be accurately modelled with Fermi's Golden Rule and, in the

dipole approximation, is governed by the product of the oscillator strength, an emitter prop-

erty, and the LDOS, an intrinsic eld property. These theoretical and experimental prerequisites

represent the starting point for the research carried out in the present thesis.

34

Chapter 3

Single-Photon Dicke

Superradiance from a Quantum

Dot

The interaction between quantum emitters and the uctuating electromagnetic vacuum eld

has been a central physical process propelling the remarkable advent of quantum optics and

quantum electrodynamics over the past and present centuries. The surprising and unexpected

discovery of the Lamb shift 70 years ago [6] conferred a new understanding to "empty space",

which consists of virtual photons popping in and out of existence as allowed by the uncertainty

principle. These virtual excitations are not only instrumental in the process of spontaneous emis-

sion from emitters but also mediate self-interactions within the emitter and perturb the energy

of light-matter-interaction processes. These fascinating ndings served both as a new motiva-

tion and platform for enhancing the interaction between emitters and the quantum vacuum in

order to reveal new phenomena inherent to the quantum world. This unavoidably led to the

emergence of quantum optics as a novel and exciting research eld and, in particular, to the

realization of optical cavities, which are capable of strongly enhancing the magnitude of vacuum

uctuations [88]. The subsequent demonstration of strong coupling between an emitter and the

cavity [10] has been the hallmark of fundamental CQED studies in a variety of physical systems

including Rydberg atoms [79], superconducting circuits [89], cavity quantum optomechanics [90]

and semiconductor quantum dots [91]. The Lamb shift is another fundamental property of the in-

teraction that can be tailored by the density of vacuum uctuations as demonstrated in Ref. [92].

These extraordinary breakthroughs have given birth to the eld of quantum-information process-

ing, which develops ecient quantum algorithms based on the concept of quantum parallelism,

which promises to exponentially outperform "classical" state-of-the-art silicon-based processors.

Quantum technologies have greatly advanced over the past decade; a promising example concerns

photonic nanostructures, which have proven useful in meeting the steep requirements for scal-

able quantum circuits, where, e.g., one-dimensional photonic waveguides enable ecient photonic

35

Chapter 3. Single-Photon Dicke Superradiance from a Quantum Dot

switches [93] and single-photon sources [21].

Another approach to enhancing the interaction with the quantum vacuum concerns tailoring

the dipole moment of quantum objects, which is the foremost capability of solid-state emitters.

In this regard, the eld of superconducting circuit QED greatly benets from the ability of care-

fully engineering the geometry of the emitters and of the cavities, which has helped extending the

quantum-optics toolbox towards the ultrastrong coupling regime [94] with a prominent contribu-

tion from energy non-conserving processes. At optical frequencies, however, the advances have

been more modest owing to the extraordinary challenge of engineering emitters and cavities with

nanometer accuracy. An approach widely used in atomic physics relies on a collective enhance-

ment of the light-matter interaction known as superradiance. As was pointed out by Dicke more

than half a century ago [95], N coherently excited atoms decay N times faster compared to an

incoherent ensemble, which is the very telltale of superradiance. So far, superradiance has been

studied in ensemble of atoms [96], ions [97], nuclei [98], Bose-Einstein condensates [99], and su-

perconducting circuits [100]. One of the most remarkable practical realizations of superradiance

is a superradiant laser with less than one intracavity photon [101]. Rather than relying on intra-

cavity photons to store phase coherence, the superradiant laser relies on collective superradiant

eects in the atomic cloud to store coherence.

A dierent yet intimately related N -fold emission speedup may occur at the single-photon

level, if a single excitation is distributed coherently and symmetrically in an ensemble of N emit-

ters [95] rather than localized in a single emitter. Coined "the greatest radiation anomaly" by

Dicke himself, single-photon superradiance is currently a hot topic in theoretical [102, 103] and

experimental [98] physics, and is central to schemes for robust quantum communication [104] and

quantum memories [105]. Previous experiments have studied ensembles with harmonic spectra

(and, thus, equidistant energy levels), where the absorption of a single laser pulse generates multi-

ple excitations. As such, single-photon superradiant states cannot be prepared deterministically

with a harmonic spectrum, cf. Fig. 3.1(a). Mutual (e.g., Coulomb) interactions between the

emitters are needed to create an anharmonic spectrum and this can be achieved with a spatially

conned ensemble.

In the present chapter we report measurements on a Dicke-superradiant and anharmonic

single-photon source in which internal Coulomb interactions are so strong that superradiant

quasiparticles can be deterministically prepared. The quasiparticles are single excitons consist-

ing of spatially correlated electron-hole pairs weakly conned to quantum dots of gallium arsenide

as shown in Fig. 3.1(b). Connement to subwavelength dimensions is ideal because in this limit,

i.e., the Dicke regime, the constructive cooperativity amongst the emitters is maximized while

it is reduced by destructive interference in larger ensembles, compare with Fig. 3.1(a). This

favorable combination of properties enables demonstrating for the rst time single-photon Dicke

superradiance (SPDS) as well as its deterministic and robust preparation. The studied solid-state

single-photon source has near-unity quantum eciency and an intrinsic radiative spontaneous

decay beyond 10 GHz. Specically, we measure a superradiant enhancement factor of up to

5.5 leading to a "giant" oscillator strength of 96.4 ± 0.8. Furthermore, we nd the quantum

dots to exhibit an average quantum eciency of (94.8 ± 3.0)%, which is the highest ever re-

36

(a)

_ +

(b)

... ...

e h

Figure 3.1: Single-photon superradiance in (a) atomic physics and (b) semiconductor quantum

dots. (a) A single electronic excitation, the atomic dipole, is distributed among the ensemble.

Since the latter is larger than the optical wavelength, the excitation is shared with a spatially

varying phase with both constructive and destructive cooperativity, as indicated by the striped

swirling arrow. The absorption of the ensemble is linear as denoted by the harmonic level

structure in the green circle. (b) A single excitation, the exciton, is distributed coherently

within the quantum dot. Constructive cooperativity is ensured by the optically small size of the

quantum dot. The spectrum is anharmonic, i.e., the energy ~ωXX of a biexciton, |XX〉, is lessthan the energy ~ωX of a single exciton, |X〉.

ported for quantum dots. Specically, it is higher than that of small (80− 93 %) [84] and large

(33 − 60 %) [106] self-assembled In(Ga)As quantum dots, and of droplet-epitaxy quantum dots

(70− 78 %) [57]. Our ndings show that the superradiant enhancement factor can, in principle,

be orders-of-magnitude larger, if the quantum-dot size and material composition are accurately

engineered. As such, coherent single-photon sources operating at terahertz clock speeds may be

realizable, with prospects for exploring riveting quantum-optical regimes, such as the ultrastrong

coupling between matter and the quantum vacuum, which is, at present, beyond reach at op-

tical frequencies. The increased emission speed may help addressing outstanding challenges in

quantum photonics, such as phonon-, charge-, and spin-induced photon dephasing [24, 107]. Our

results underline the extraordinary potential of semiconductor quantum dots for becoming the

bright and indistinguishable single-photon source of choice for quantum-information science [22].

37


3.1 Theory of single-photon superradiance from quantum

dots

In the present section we make a formal connection between the Gedankenexperiment of Dicke

describing single-photon superradiance [95] and SPDS in a semiconductor QD. We show that the

long-sought giant oscillator strength of quantum dots (GOSQD) and SPDS are two physically

equivalent phenomena.

If two atoms are placed much closer than one wavelength apart yet much farther apart than

the atomic size, the spontaneous-emission dynamics of a shared single electronic excitation diers

dramatically from the individual-atom case. The strongest enhancement of the light-matter

interaction occurs in the symmetrical state |Ψs〉

|Ψs〉 =1√2

(|e〉1 |g〉2 + |e〉2 |g〉1) , (3.1)

where e and g denote the excited and ground electronic states, and i = 1, 2 the index of the

atom. It is straightforward to check that this state has a dipole moment√

2 as large as the dipole

moment of an individual atom yielding a decay rate twice as fast. Importantly, a superradiant

state can be inferred from the entanglement of the underlying (quasi-) particles as reected in

Eq. (3.1). Equation (3.1) can be extended to cover the N -atom case via

|Ψs,N 〉 =1√N

∑j

|g〉1 |g〉2 ... |e〉j ... |g〉N , (3.2)

which decays to the ground state with an enhancement factor of N . The fundamental lim-

itation to harvest such eects in atomic physics is related to the large size of the atomic

cloud compared to the wavelength of light, in which case the constructive cooperativity may

be lost. Quantitatively, the absorption of a single photon with the wavevector k yields the state

(1/√N)∑j exp(ik · rj) |g〉1 |g〉2 ... |e〉j ... |g〉N [108], whose superradiant properties are washed

out by the phase-factor exponent. The deterministic preparation of the state from Eq. (3.2) is

impeded by the spectral harmonicity of the atomic ensemble. These impediments can be elimi-

nated in large semiconductor QDs, where a superradiant state that couples giantly to light can

be prepared deterministically, as shown in the following.

Quantum dots are well suited for enhancing the light-matter interaction strength due to their

multi-atomic nature. The property of the emitter governing the magnitude of the interaction

with light is the oscillator strength, which is proportional to the square of the dipole moment, cf.

Eq. (2.60). In conventional QDs, connement dominates and overwhelms the Coulomb attraction

between the electron and the hole. This corresponds to the strong-connement regime, in which

superradiant eects are inhibited by strong connement eects. It is the opposite case, the weak-

connement regime, in which the electron-hole motion is correlated and may form a superradiant

quantum state.

38

Theory of single-photon superradiance from quantum dots

(a) (b)

Figure 3.2: Superradiance with single QDs. (a) In small QDs, such as self-assembled In(Ga)As

QDs, the motion of electrons and holes is governed by quantum-connement eects and is com-

pletely uncorrelated, which limits the light-matter interaction strength. (b) In large interface-

uctuation QDs, the electron-hole motion is dominated by their mutual attraction in the plane

of the QD. The mean electron-hole separation (∼ exciton Bohr radius and is marked with gray)

is smaller than the exciton wavefunction (marked with violet) and leads to a strong superradiant

behavior of the ground-state exciton.

3.1.1 Strong-connement regime

In this limit, the QD is smaller than the exciton Bohr radius. The electron and the hole

do not "see" each other and "feel" only the conning barriers, see the sketch in Fig. 3.2(a).

Since quantum-connement eects are stronger than the Coulomb attraction, the latter can

be neglected resulting in no spatial correlations and a separable exciton wavefunction ΨX(r),

which can be written as a product of the individual electron Ψe and hole Ψh wavefunctions, i.e.

ΨX(r) = 〈r|e〉 = Ψe(r)Ψh(r). This lack of entanglement in the electron-hole motion implies that

no superradiant enhancement can be achieved, and that the oscillator strength is not aected by

the QD size in the strong-connement regime∗. The dipole moment of the x-polarized exciton is

µ =e

m0〈Ψh |px|Ψe〉 x '

e

m0pcv 〈ψh|ψe〉 x, (3.3)

where pcv = V −1UC

∫UC

d3ru∗xpxue is the interband Bloch matrix element with VUC being the

unit-cell volume. In the above equation we have exploited the slow variation of the envelopes ψ

over one unit cell. Inserting the dipole moment into Eq. (2.60) yields the oscillator strength of

strongly-conned excitons

f =Eg~ω|〈ψh(r)|ψe(r)〉|2 , (3.4)

where Eg is the Kane energy, an experimentally accessible and well-documented quantity. The

oscillator strength of small QDs has therefore an upper limit of fmax = Eg/~ω amounting to 17.4

for GaAs QDs at a wavelength of 750 nm. In other words, uncorrelated excitons cannot surpass

this limit and in the following we are referring to fmax as the limit for uncorrelated excitons.

∗Actually, the oscillator strength may depend weakly on the QD size because electrons and holes have dierent

connement energies and eective masses, which aects their overlap. This is, however, a small eect.

39


3.1.2 Weak-connement regime

In this regime, the QD is larger than the exciton Bohr radius and the excitons are conned

by the electron-hole mutual electrostatic interaction. Since the interaction couples the electron

with the hole, their motion is spatially entangled. A consequence of this correlation is that the

oscillator strength is proportional to the volume of the QD and may attain values much larger

than fmax. This GOSQD eect was rst considered by Hanamura [109] and is known as excitonic

enhancement of light-matter interaction. The latter has been studied in a range of solid-state

systems including impurities in bulk semiconductors [110] and quantum wells [111]. The unique

feature of the GOSQD is that it occurs for a single quantum state leading to single-photon

superradiance, as shown in the following.

Our experiments concern investigations of single GaAs interface-uctuation QDs embedded

in Al0.33Ga0.67As as presented in Fig. 3.2(b). Bound excitonic states are created by intentionally

engineered monolayer uctuations in a quantum well leading to weak in-plane quantum con-

nement, a technique that was pioneered by Gammon et al. [27]. The quantum-well thickness

(∼ 4 nm) is smaller than the bulk exciton Bohr radius a0 = 11.2 nm and leads to strong con-

nement along the QD height. Exciton enhancement is achieved only within the plane, where

the quantum-dot wavefunction is extended beyond the exciton Bohr radius. We model the stud-

ied QDs as being cylindrically symmetric with a slowly varying envelope that is separable into

in-plane ψX and out-of-plane φ(z) components. Due to strong connement, the electron-hole

motion is uncorrelated out of the plane and, thus, their wavefunction is separable in indepen-

dent components φ(z) = φe(z)φh(z). We therefore obtain for the exciton wavefunction in the

eective-mass approximation

ΨX(R, r, re, rh) = ψX(R, r)φh(zh)φe(ze)ux(rh)ue(re), (3.5)

where R = (mere +mhrh)/(me +mh) and r = re − rh the center-of-mass and relative in-plane

excitonic coordinates, and me and mh are the electron and hole eective masses, respectively.

The unit-cell Bloch functions contribute to the Kane energy and do not play an important role

in our study. The out of plane envelopes φe,h can be accurately computed because the QD

thickness is known precisely and amounts to Lz = 4.3 nm but they play no role for the GOSQD

eect, which is governed by the in-plane excitonic envelope ψX(R, r). To see this, we rst make

some realistic assumptions and consider a symmetric in-plane parabolic quantum connement, in

which case the excitonic envelope separates into a center-of-mass χCM(R) and a relative-motion

χr(r) components [83]

ψX(R, r) = χCM(R)χr(r), (3.6)

χCM(R) =

√2

π

1

βe−|R|

2/β2

, (3.7)

χr(r) =

√2

π

1

aQWe−|r|/aQW , (3.8)

where aQW is the exciton Bohr radius in the quantum well and β the in-plane HWHM of the

exciton wavefunction. Equations (3.73.8) are solutions to a dierential equation describing the

40

Theory of single-photon superradiance from quantum dots

e h e h

e h

Figure 3.3: Illustrative interpretation of ψX in Eq. (3.10). The excitonic enhancement of light-

matter interaction may be regarded as a generalization of single-photon superradiance: the

exciton is a symmetric superposition of dierent spatial positions of the excitation φX within the

QD.

two-dimensional excitonic hydrogen. The motion in a two-dimensional system is dierent than in

three dimensions, which is why the gure of merit characterizing the electron-hole separation in

interface-uctuation QDs is aQW rather than the bulk quantity a0. For a perfect two-dimensional

system, aQW = a0/2 ' 5.6 nm leading to a binding energy four times as large as in bulk. The

structure investigated in this work is, however, not a perfect two-dimensional system because

the exciton wavefunction has a non-zero thickness. As argued in Refs. [112, 113], the binding

energy of an exciton in a 4-nm thick quantum well is only twice larger than in bulk. We therefore

consider a value of the two-dimensional Bohr radius aQW ' a0/√

2 ≈ 8 nm. For β > aQW, the

mean distance between the electron-hole pair (≈ 2aQW) is smaller than the QD size (≈ 2β)

and forms the prerequisite for the GOSQD and single-photon superradiance, as shown in the

following.

3.1.3 Relation between the giant oscillator strength of quantum dots and

single-photon Dicke superradiance

The connection to the single-photon Dicke superradiance can be made by noting that, if β > aQW,

the center-of-mass motion can be written as a convolution between a function ca(R) capturing

the dynamics on the scale of aQW and a function cs(R) responsible for the coherent superradiant

enhancement, i.e.,

χCM(R) = ca(R) ∗ cs(R) =

∫d2Pcs(P)ca(R−P) ≈

∑n

c(Rn)ca(R−Rn), (3.9)

where the last step involves switching the integral to a sum and cs equals cs times the discretiza-

tion area. Consequently, the slowly varying excitonic envelope reads

ψX(R, r) =∑n

c(Rn)φX(R−Rn, r), (3.10)

where n runs over the unit cells of the atomic lattice constituting the QD. The internal exciton

dynamics is governed by φX, which has a spatial extent of the order of the Bohr radius (∼ 8 nm)

and is smaller than the QD. The exciton in Eq. (3.10) is therefore in a spatial superposition

of excitations corresponding to dierent positions of φX as illustrated in Fig. 3.3. An exciton

41


20 30 400

5

10

15

QD Diameter, 2L (nm)

Sup

erra

dian

t enh

ance

men

t

of th

e os

cilla

tor

stre

ngth

Figure 3.4: Superradiant enhancement of the oscillator strength for an interface-uctuation QD

with respect to the strong-connement limit fmax = 17.4.

governed by Eq. (3.10) has been long sought in solid-state quantum optics [56] because it has

been predicted to lead to GOSQD. The analysis shows that GOSQD is a generalization of SPDS,

compare Eq. (3.10) with Eq. (3.2), and the two eects are equivalent if c is constant throughout

the QD. The constructive cooperativity is ensured by the constant phase of c that is found for

the lowest-energy exciton state due to its s-like symmetry. In contrast, all optical experiments

so far concerned larger ensembles where additional phase factors reduce the cooperativity [108].

For parabolic in-plane connement we obtain the following expressions for ca and cs

ca(R) =1

πa20

e−2|R|2/a2QW , (3.11)

cs(R) = πe−|R|2/ξ2 , (3.12)

where ξ2 = β2−a2QW ≈ β2 for β aQW. In the following we quantify the expected superradiant

increase in the oscillator strength and, consequently, in the spontaneous-emission rate.

According to Fermi's Golden Rule, the probability of photon emission is proportional to the

excitonic charge density |〈0 |px|ΨX(R, r = 0, re, rh)〉|2. The relative motion is taken to be zero,

r = 0, because the exciton can recombine radiatively only if the electron and hole are found

at the same spatial position. After performing the standard procedure of merging the unit-cell

Bloch functions into the Bloch matrix element pcv = V −1UC 〈ux |px|ue〉UC, where the subscript

UC denotes integration over an unit cell, we obtain the following expression for the oscillator

strength (compare with Eq. (3.4))

f =Eg~ω

χr(0) |〈0|χCM(R)〉|2 |〈ψh(z)|ψe(z)〉|2 , (3.13)

where the rst (second) inner product on the right-hand side of the equation denotes a two-

dimensional (one-dimensional) integration over R (z). We dene the radius of the QD L =√

2β

42

Sample and experimental setup

as argued in Ref. [83] and, with the help of Eqs. (3.63.8), arrive at the following superradiant

enhancement S of the oscillator strength

S =f

fmax=

(√2L

aQW

)2

|〈ψh|ψe〉|2 . (3.14)

The electron and hole wavefunctions in the growth directions can be accurately calculated and

we nd that |〈ψh|ψe〉|2 ≈ 0.96 for the interface-uctuation QDs from the present study. We plot

the resulting superradiant enhancement of the oscillator strength in Fig. 3.4. It scales with the

QD area and is a dramatic eect; for realistic QD diameters of 35 nm, the light-matter interaction

strength exceeds the upper limit of strongly conned excitons by an order of magnitude.

3.2 Sample and experimental setup

The sample used in our experiment was grown on a GaAs (001) wafer following the procedure

developed by Gammon et al. [27]. The GaAs interface-uctuation quantum dots were created by

random monolayer uctuations in the GaAs quantum-well thickness. The GaAs quantum well is

surrounded by 5-nm-thick Al0.33Ga0.66As layers in order to obtain a high-quality interface, and

followed by a 100-nm-thick Al0.8Ga0.2As. The detailed structure is presented in Fig. 3.5(a). A

zirconia solid-immersion lens shaped as half a sphere with a radius of 1 mm and refractive index

of 2.18 was placed on top of the sample to improve the collection eciency.

There are several types of optical measurements performed in this study: spectral and time-

resolved measurements, and second-order correlation measurements. All of them are carried out

in a closed-cycle cryogen-free cryostat as sketched in Fig. 3.5(b). The sample holder is mounted

on piezoelectric nanopositioning translation stages. For all experiments, the sample is cooled to

a temperature of 7 K. After exiting the single-mode polarization-maintaining (PM) ber, the

excitation beam generated by a picosecond pulsed Ti:Sapph laser is collimated to a diameter of

2 mm. Then, it passes through a thin-lm linear polarizer and a 90:10 (transmitted:reected)

beam splitter before being focused on the sample through a microscope objective with a numer-

ical aperture of 0.85. The spatial resolution of the objective was measured to be 1.1µm2 at a

wavelength of 633 nm. The excitation laser is tuned to a wavelength of about 750 nm correspond-

ing either to resonant excitation of continuum states in the quantum well or to 2s-shell excitation

of the quantum dot. The photoluminescence of the investigated ground-state excitons is located

around 752 nm. The emission is collimated by the same microscope objective and ltered from

the excitation laser by the perpendicularly-oriented thin-lm linear polarizer, see Fig. 3.5(b).

The beam is then coupled into a PM ber and guided towards the detection setup.

Spectral measurements are performed by sending the emission to a spectrometer with a groove

density and spectral resolution of 1200 mm−1 and 25 pm, respectively, and subsequently detected

by a charge-coupled device (CCD), see Fig. 3.5(c). After the grating, a mirror can be ipped to

direct the emission to an avalanche photo-diode (APD) with a time resolution of 60 ps for time-

resolved measurements. For correlation measurements, a setup with a higher throughput is used,

cf. Fig. 3.5(d). The emission is rst ltered by a grating with a groove density of 1200 mm−1

43


Spectrometer

Flip mirror CCDAPD Grating

APD

APD

CryostatPiezo-stagesHeater

Microscopeobjective

Vaccum window

For powerstabilization

Detection

PM ber

PM berPolarizers

Photodiode Excitation

1 mm

10 nm

100 nm

5 nm(3.7 ± 0.3) nm5 nm

100 nm

Zr02 SIL

GaAs cap

Al0.8Ga0.2As

Al0.8Ga0.2As

Al0.33Ga0.67As

GaAs substrate

Al0.33Ga0.67AsGaAs QDs

(a) (b)

(c) (d)

Figure 3.5: (a) Cutaway prole of the investigated sample (not to scale). Lattice-matched GaAs

quantum dots are formed by random uctuations in the GaAs quantum-well thickness. A zirconia

solid-immersion lens enhances the collection eciency of the setup. (b) Sketch of the optical setup

around the cryostat. Optical excitation and collection are performed in a cross-polarized scheme

to discriminate between the photoluminescence and specular laser reection. (c) The emission

is sent through a spectrometer before being detected by a CCD for spectral measurements and

an APD for time-resolved measurements. (d) For correlation measurements, the quantum dot

emission line is spectrally ltered before being directed onto a Hanbury-Brown-Twiss (HBT)

setup.

before being coupled back into a single-mode PM ber and directed towards a beam splitter.

The grating setup has a spectral resolution of 50 pm. After the beam splitter, two APDs detect

coincident counts.

3.3 Deterministic preparation of superradiant excitons

It is essential to understand the energy-level structure of the studied conned system to identify

proper excitation conditions of superradiant excitons. This is because the 1s manifold contains

bright and dark excitons, which are prepared with random probability for above-band excitation,

44

Deterministic preparation of superradiant excitons

X

c

1s

2s3s2s

3s

1s

Quasi-continuum

Quantum dot exciton manifold

c

exci

ton

man

ifold

Energy (eV)(a) (b)

Figure 3.6: Deterministic preparation of single bright superradiant excitons. (a)

Photoluminescence-excitation spectrum obtained by integrating the emission of the 1s transi-

tion while scanning the excitation wavelength. It features a quasi-continuum band of states

followed by a sequence of quantum-dot states labeled as 1s, 2s and 3s. (b) Two excitation

schemes are used in our study. Pumping in the quasi-continuum band at the wavelength "C"

results in preparation of carriers with random spin and formation of an equal bright- and dark-

exciton population, which is important for extracting the impact of nonradiative processes. For

2s excitation, the spin is preserved and the bright exciton is prepared deterministically.

as explained in Sec. 2.4. For deterministic preparation of superradiant excitons, only the bright

states must be prepared.

The spectrum of states is probed using photoluminescence-excitation spectroscopy as dis-

played in Fig. 3.6(a), which shows a quasi-continuum band of QD states hybridized with quantum-

well resonances followed by the exciton manifold. We identify the 1s, 2s and 3s excitonic states

that are denoted according to the two-dimensional hydrogen atom. Note, the recombination of

excitons with dierent symmetry, such as p, d, etc., is optically forbidden. Key features of the

spectrum are summarized in Fig. 3.6(b). The measurement was carried out in continuous-wave

mode below the saturation power of the 1s exciton. The laser was scanned stepwise from around

752 nm down to 735 nm, where the QD and quantum-well resonances are present.

The deterministic preparation of superradiant excitons in the 1s state is achieved by applying

excitation through the 2s exciton state, cf. Fig. 3.6, with a pulsed laser. This is feasible since

the decay cascade from 2s to 1s is spin-conserving [114] so spin-dark states are not populated.

Deterministic excitation occurs when applying sucient optical power (300 nW) to saturate the

emission from the 1s state. Aside from deterministic excitation, a further advantage for pumping

in the 2s state is that the environment of the QD is not polluted by phonons and charges that may

raise the eective temperature at the QD position and couple the exciton levels thermally. This

requirement is particularly stringent for interface-uctuation QDs owing to the close proximity

of the exciton states of only a few meV induced by the weak quantum connement.

Despite the important advantages inherent to the 2s pumping, the oscillator strength cannot

45


be extracted in such a fashion because spin-dark excitons are not populated. This is conrmed by

the single-exponential character of the exciton decay, which implies that the impact of radiative

and nonradiative processes cannot be separated. We therefore use above-band excitation in

the quasi-continuum band of the quantum well (the wavelength labeled "C" in Fig. 3.6), which

prepares bright and dark states with equal probability, to extract the impact of nonradiative

processes. We note that for "C"-excitation, the radiative decay rate is found to depend on

excitation power, which is attributed to the presence of undesired thermal processes induced

by the local phonon bath, which is created by the relaxation of the (many) charge carriers

from the quantum well. This excitation scheme is therefore only used to extract the decay rate

of nonradiative processes occurring in the QD, while the oscillator strength is probed through

the 2s-excitation. We conrmed experimentally that nonradiative processes do not depend on

excitation conditions.

3.4 Previous work on the giant oscillator strength of quan-

tum dots

Previous searches for the GOSQD eect were inspired by the prediction of Andreani et al. [56]

that QDs in the GOSQD regime may enable reaching the strong-coupling regime of cavity quan-

tum electrodynamics. In some works [20, 115], the oscillator strength was estimated from the

vacuum Rabi splitting in the strong-coupling regime of cavity quantum electrodynamics. Such

estimates are inaccurate because multiple quantum dots may couple to the cavity even when they

are o resonance due to a (non-Dicke, non-single-photon) collective coupling of multiple quan-

tum dots to the cavity mediated by phonons [24, 116]. In other works, the oscillator strength

was estimated from absorption experiments [71] but in such experiments the inuence of other

emitters cannot be ruled out. The oscillator strength has also been estimated from time-resolved

measurements [117, 118] but, as pointed out above and also noted in Ref. [117], the nonradiative

and radiative processes must be measured independently. It is also crucial to extract properly the

radiative decay rate for a homogeneous medium because the local density of optical states may be

modied signicantly in photonic nanostructures even by the presence of nearby planar surfaces.

The importance of properly accounting for these eects was highlighted in recent results on large

InGaAs QDs: in Ref. [118], the total decay rate was used to estimate an oscillator strength of

∼50 but later measurements showed that non-radiative processes were very signicant and that

the oscillator strength was ∼5 times smaller [106], i.e., below the GOSQD regime.

3.5 Extracting the impact of nonradiative processes

Extracting the impact of nonradiative process in the decay of the exciton can be done by ex-

ploiting the ne structure of the 1s exciton manifold as explained in Sec. 2.4. To ensure an equal

preparation probability of bright and dark excitons ρB(0)/ρD(0) ∼ 1, the 1s manifold is excited

with a pulsed laser at the "C" wavelength in the quantum-well quasi-continuum. If, for some

46

Extracting the impact of nonradiative processes

0 . 1 1 1 0 1 0 01 0 0

1 0 1

1 0 2

1 0 3

0 . 0

0 . 5

1 . 0 ( d )

X

Norm

alized

Inten

sity

( c )

( a ) 0 . 1 P s a t

0 . 0

0 . 5

1 . 0 P s a t

X X

X

W a v e l e n g t h ( n m )

X X

( b )

7 . 5 P s a t

Integ

rated

Inten

sity (c

ounts

)

P o w e r ( n W )7 5 0 7 5 2 7 5 4 7 5 6

0 . 0

0 . 5

1 . 0

Figure 3.7: Spectral measurements for "C"-excitation. (a) Measured photoluminescence spec-

trum at 10 % of the exciton saturation power Psat = 20 nW. Only the exciton is observed. (b)

At saturation of the exciton, the biexciton is visible as a small peak. (c) Signicantly above the

exciton saturation (7.5Psat), the spectrum acquires further narrow peaks on top of a continu-

ous background, which indicates multi-excitonic features. (d) The exciton is distinguished from

biexcitons by their power-law dependence on excitation power P : the ts yield P 0.86 and P 2.01

for the exciton and biexciton, respectively.

reason (e.g., spin-conserving cascade to the ground-state exciton) ρB(0)/ρD(0) > 1, we would

actually be overestimating Γnrad because of Eq. (2.68). This means that we are estimating a

lower bound to the oscillator strength and quantum eciency.

The excitation power is set signicantly below the exciton saturation P ≈ 0.1Psat to ensure

that only the exciton is prepared. In this regime, the spectrum is dominated by the exciton line,

cf. Fig. 3.7, with a linewidth limited by the spectrometer (25 pm). At saturation, the spectrum

remains as clean but the biexciton line becomes discernible. Signicantly above saturation, both

the exciton and the biexciton lines are saturated and the spectrum features spectrally continuous

multibody emissions. The nature of the exciton and biexciton lines is conrmed by power series

measurements as shown in Fig. 3.7(c), where the exciton line is tted with a Lorentzian function

and subsequently integrated whereas the biexciton line is integrated directly owing to its irregular

spectral shape. The spectral broadening of the biexciton line is related to multibody eects

between the exciton and the free carriers populating the quantum well [57, 117].

The decay dynamics of the exciton was recorded by sending the corresponding line from

Fig. 3.7(a) to an avalanche photo-diode. The acquired data is tted by the biexponential model

presented in Sec. 2.4, which yields the fast rate ΓCF = ΓC

rad + Γnrad + Γsf and the extracted

parameters are outlined in Fig. 3.8. Here, the superscript "C" denotes quantities related for

47


1 0 3

1 0 4

1 0 5

0 2 4 6 8 1 0 1 2- 5

0

5

Γf = 7 . 6 4 n s - 1

Γs = 0 . 4 9 n s - 1

A F / A S = 2 1 . 0 3χ2 = 1 . 1 3

Inten

sity (c

ounts

)No

rmaliz

edres

iduals

T i m e d e l a y ( n s )

Figure 3.8: Time-resolved decay of the exciton (black dots) under "C"-excitation. The ne-

structure model yields an excellent biexponential t (yellow line) with the extracted parameters

indicated accordingly. The instrument response of the detector is indicated by the green line.

"C"-excitation only. We obtain a nonradiative rate Γnrad = 0.19 ns−1, and a spin-ip rate Γsf =

0.31 ns−1, quantities that do not depend on excitation conditions as conrmed experimentally

by measuring an excitation-independent slow rate. As mentioned before, ΓCrad is not related to

the oscillator strength because more than two levels participate in the exciton dynamics. The

rates obtained here are used to unambiguously extract the oscillator strength in the following.

3.6 Experimental demonstration of single-photon superradi-

ance

The experimental signature of the GOSQD is spontaneous emission of single photons with an

intrinsic (i.e., homogeneous-medium) radiative emission rate enhanced beyond the upper limit

for uncorrelated excitons. The 1s bright state is excited deterministically through the 2s shell

and we nd a clean spectrum below and at saturation, cf. Fig. 3.9. The excitation eciency

is diminished owing to the smaller absorption cross-section of the QD resonance. We nd a

saturation power of 300 nW, which is a factor of 15 higher than for "C"-excitation. The time-

resolved measurement is performed at P = 0.1Psat and we nd the decay to be close to single

48

Experimental demonstration of single-photon superradiance

0 . 0

0 . 5

1 . 0 l a s e r

Norm

alized

Inten

sity

( c )

( a ) 0 . 2 P s a t

0 . 0

0 . 5

1 . 0 P s a t

X


X X

( b )

5 . 5 P s a t

7 4 6 7 4 8 7 5 0 7 5 2 7 5 4 7 5 6 7 5 80 . 0

0 . 5

1 . 0

Figure 3.9: Spectral measurements for 2s-shell excitation (a) below, (b) at and (c) above the

exciton saturation power Psat = 300 nW. The exciton line exhibits a spectral behavior similar to

the "C"-excitation.

exponential†. The radiative rate of the exciton is Γrad = ΓF − Γnrad − Γsf = 8.4 ns−1, where

ΓF is the fast rate extracted from Fig. 3.10(a). Importantly, the radiative rate cannot be used

to directly compute the oscillator strength because it is not a homogeneous-medium quantity

owing to the layered structure of the sample, cf. Fig. 3.5(a). We calculate an LDOS [73] of 0.95,

which is normalized to the LDOS in homogeneous Al0.8Ga0.2As. We nd an oscillator strength

of f = 72.7± 0.8, which is enhanced far beyond the upper limit of f = 17.4 for an uncorrelated

exciton, cf. Fig. 3.10(a). This is a direct signature of exciton superradiance.

To conrm the single-photon nature of the emission, we measure the second-order correlation

function g(2)(τ) dened as [75]

g(2)(τ) =

⟨a†(t)a†(t+ τ)a(t+ τ)a(t)

⟩〈a†(t)a(t)〉2

, (3.15)

which determines the statistical character of the intensity uctuations. Here, τ denotes the

time delay between two photons. The function g(2)(τ) determines the probability of detecting

a photon at time t = τ subject to the condition that a photon was detected at t = 0. An

ideal single-photon source exhibits g(2)(0) = 0 but in practice any value below 0.5 is direct

experimental evidence of single photons. The value of 0.5 is related to the fact that a Fock

state |n〉 exhibits g(2)(0) = 1 − 1/n, and higher-order Fock states n ≥ 2 have g(2)(0) > 0.5.

Note, classical statistics of light-intensity uctuations obey g(2)(0) ≥ 1. Figure 3.10(b) shows

the second-order correlation function obtained in an HBT experiment. The data are tted by a

†Specically, the ratio of the slow-to-fast amplitude AS/AF is 10 times smaller than for "C"-excitation.

49


Figure 3.10: Experimental demonstration of single-photon superradiance from a quantum dot.

(a) Time-resolved decay (black points) of the 1s exciton obtained under 2s-resonant excitation.

The t to the theoretical model is indicated by the yellow line. We take into account the

impact of nonradiative processes presented in the previous section and extract a radiative decay

rate of 8.4 ns−1 (red line), which is deeply in the superradiant regime (green area). (b) HBT

measurement of the emitted photons showing g(2)(0) = 0.13, which demonstrates the single-

photon character of the emission. (c) Long-time-scale HBT measurement where each coincidence

peak has been numerically integrated. No blinking of the emission is observed.

sum of biexponentially decaying functions, and g(2)(0) is dened by the ratio between the energy

contained in the central peak around τ = 0 and in the adjacent peaks τ = mTlaser, where m ∈ Zand Tlaser = 12.6 ns−1 is the repetition period of the laser. We nd a zero-time correlation of

g(2)(0) = 0.13, demonstrating the single-photon nature of the emitted light. In conjunction with

the measured enhanced oscillator strength for a spatially conned exciton, this is the unequivocal

demonstration of SPDS in a QD.

Solid-state quantum light sources often suer from blinking of the emission, in which the QD

randomly switches to a dark state and does not emit light [119]. This may happen, if a charge

defect in the vicinity of the QD traps the electron or hole composing the exciton, thereby pre-

venting the radiative recombination. This decreases the radiative eciency of the single-photon

source and could be detrimental for practical applications. For epitaxially grown QDs, blinking

normally occurs within nanosecond-to-microsecond time scales with a corresponding bunching

in the QD second-order correlation function g(2)(τ) over such time scales. By numerically in-

tegrating each peak in the HBT correlation data we obtain the long-time-scale plot shown in

Fig. 3.10(c). No bunching eects are observed, which shows that this single-photon source is free

from blinking on a time scale of at least 10µs.

This chapter only presents the properties of one interface-uctuation QD but we have in

fact measured the oscillator strength of 9 dierent QDs and found them all to be superradiant

with an average oscillator strength of f = 76.2 ± 10.8. Remarkably, we have measured an

oscillator strength up to f = 96.4 corresponding to an intrinsic radiative rate beyond 10 GHz.

50

Microscopic insight into the exciton wavefunction

z (nm)

Ene

rgy

(eV

)

-4 0 4 -10 0 10

0

-10

10

xe (nm)

x h (n

m)

2

3

4

min

max

(a)

(b) (c)

Figure 3.11: Acquiring microscopic information about the exciton wavefunction from time-

resolved measurements. (a) Sketch of the interface-uctuation GaAs QD embedde in an AlGaAs

matrix (not to scale). The two-dimensional electron-hole pair (exciton) is coherently spread over

the spatial extent of the QD (green area). Exciton enhancement is achieved within the plane

(grey spiraling arrow), while out-of-plane cooperative eects are destroyed by the close proximity

of the GaAs-AlGaAs potential barrier. (b) Band diagram along the QD height and the corre-

sponding quantum-mechanical wavefunctions of the exciton. The material parameters used in

the calculation are taken from Ref. [38]. (c) Plot of the in-plane exciton density |ψX(xe, xh, 0, 0)|2.

Such a highly superradiant QD can deliver a radiative ux of single photons equivalent to more

than ve conventional QDs. The superradiant enhancement of the light-matter coupling in QDs

can potentially be orders-of-magnitude larger than the experiments reported here. This can be

achieved in yet larger excitons at millikelvin temperatures and is discussed in the Outlook section

of the present thesis.

3.7 Microscopic insight into the exciton wavefunction

The presented experimental ndings provide insightful information not only about macroscopic

properties such as the oscillator strength and quantum eciency, but also about the nanoscopic

structure of the quantum-mechanical wavefunctions of the QD. This opens the possibility of

engineering the QD size, shape and composition for accurately tailoring the superradiant behavior

of excitons.

The out-of-plane uncorrelated electron φe(z) and hole φh(z) wavefunctions are computed

51


with a tunneling resonance technique [31] and are plotted in Fig. 3.11(b) for the investigated

QD. While the microscopic structure of the out-of-plane wavefunctions can be accurately com-

puted because the quantum-well thickness is known with monolayer precision (see Sec. 3.2), the

in-plane geometry is generally unknown because the uctuations of the quantum-well thickness

are spatially random. The nanoscopic information is then inferred from the superradiant en-

hancement of spontaneous emission S, where it can be shown (see Sec. 3.1) that the QD radius

L is related to S via

L =aQW√

2

√S

|〈φh|φe〉|. (3.16)

From the measured value S ' 4.3 an in-plane diameter 2L ' 24 nm is obtained. The resulting

wavefunction |φX(xe, xh, 0, 0)|2 is plotted in Fig. 3.11(c), where a strong correlation between

the electron and hole position is observed within the QD, which gives rise to the superradiant

emission. The exciton wavefunction is spread over 90 thousand atoms in a collective quantum

state sharing a single excitation and exhibiting constructive cooperativity. Our results emphasize

that optical spectroscopy is a robust, non-invasive way of acquiring profound insight into the

nanoscopic wavefunctions of quantum emitters.

3.8 Results on all measured quantum dots

The measurement results for all studied QDs are presented in Tab. 3.1. The fastest measured

radiative decay rate belongs to QD3 and amounts to 11.1 ns−1, which corresponds to an oscillator

strength of 96.4. All studied QDs have a giant oscillator strength with an average value of

76.2, which constitutes an average superradiant enhancement of 4.4 compared to the limit for

uncorrelated excitons. The average quantum eciency is 94.8 %.

Quantum dot Γrad (ns−1) Γnrad (ns−1) Γsf (ns−1) f η (%)

QD1 8.41 0.19 0.31 72.7 97.9

QD2 8.35 0.41 0.033 72.2 95.6

QD3 11.1 1.33 0.15 96.4 89.8

QD4 10.5 0.42 0.046 90.5 96.4

QD5 7.64 0.94 0.10 66.2 89.5

QD6 9.66 0.34 0.008 83.6 96.7

QD7 7.13 0.30 0.013 61.7 96.1

QD8 8.13 0.37 0.015 70.4 95.8

QD9 8.34 0.46 0.10 72.1 95.0

Table 3.1: Data extracted from time-resolved measurements on all measured quantum dots:

radiative decay rate Γrad, non-radiative decay rate Γnrad, spin-ip rate Γsf , oscillator strength f ,

and quantum eciency η. QD1: data presented in most of the chapter, QD2: data of the PLE,

QD3: largest oscillator strength.

52

Summary

3.9 Summary

In the present chapter we have demonstrated single-photon Dicke superradiance from a quantum

dot. We have studied a single interface-uctuation QD with weakly conned excitons. One

single excitation, the exciton, is distributed over a large collective quantum state and couples

giantly to the electric eld of the quantum vacuum. This eect is also known as the giant

oscillator strength of quantum dots and we have shown that it is equivalent to superradiance.

The outstanding gures of merit characterizing interface-uctuation QDs render them promising

single-photon sources for cavity quantum electrodynamics and quantum-information processing.

53

Chapter 4

Decay dynamics and Exciton

Localization in Large GaAs

Quantum Dots Grown by Droplet

Epitaxy

The research carried out in the present chapter investigates the possibility of achieving super-

radiant enhancement of spontaneous emission in solid-state quantum emitters grown by a novel

technique, droplet epitaxy, which is capable of carefully engineering the size, shape and com-

position of quantum dots. The content of this chapter is partially adapted from Ref. [57].

As explained in the previous chapter, engineering QDs with giant oscillator strength is not

straightforwardly accomplished in practice because the QDs must have a uniform potential pro-

le over length scales larger than the exciton Bohr radius. For instance, the commonly employed

In(Ga)As/GaAs QDs suer from inhomogeneous strain and alloy composition, which create lo-

calized potential minima thereby impeding the coherent distribution of the ground-state exciton

over length scales comparable to the measured physical size of the QDs. Another fundamental is-

sue is the large value of the exciton Bohr radius, which attains 48 nm in InAs, as compared to only

11 nm in GaAs [34]. Employing a modied LDOS near a semiconductor-air interface revealed a

small oscillator strength of large In(Ga)As/GaAs QDs corresponding to strongly-conned charge

carriers [106]. The rst experimental demonstration of single-photon superradiance and giant

oscillator strength was presented in Chapter 2 for interface-uctuation GaAs QDs [120].

Droplet epitaxy [45, 121125] is a powerful emerging growth technique, which is capable of

growing QDs with an optical quality (i.e., narrow linewidths) approaching that of self-assembled

In(Ga)As QDs [126]. Droplet-epitaxy QDs are strain free because they are embedded in lattice-

matched AlGaAs barriers [45] with two important advantages. First, no strain energy is stored

in the QDs, which otherwise may degrade the homogeneity of the potential prole, and second,

55

Chapter 4. Decay dynamics and Exciton Localization in Large GaAs Quantum Dots Grown by Droplet

Epitaxy

strain-related structural defects are avoided. Growing physically large QDs, which is the main

prerequisite for the giant-oscillator-strength eect, is the central capability of the droplet-epitaxy

technique. A further advantage pertains to the growth of QDs with a very low surface density

(a few QDs per µm2), which enables their individual control and manipulation. Finally, QDs

grown by droplet epitaxy are promising for use in the visible spectrum where Si-based detectors

attain maximum eciency.

Despite these important advantages, droplet epitaxy is a relatively new technology and the

droplet-epitaxy QDs lack detailed and systematic studies of their optical properties. In partic-

ular, their oscillator strength and quantum eciency have not been studied. Being part of a

solid-state system, QDs are prone to growth imperfections and, hence, to nonradiative decay

channels, e.g., via carrier trapping by QD surface states [86]. Unfortunately, little attention is

being drawn in the literature to nonradiative decay and it is often implicitly assumed that the

QDs decay purely radiatively. Nonradiative processes degrade the ability of QDs to generate

single photons on demand, which is an important goal in the eld of quantum-information pro-

cessing. By controllably modifying the LDOS, it has been shown recently that self-assembled

In(Ga)As QDs possess non-negligible nonradiative contribution with quantum eciency between

80 % and 95 % [84]. Large In(Ga)As QDs were found to exhibit a quantum eciency of only 30

to 60 % [106]. Near-unity quantum eciency has been found so far in interface-uctuation QDs

only [120], as presented in Chapter 3.

In the present chapter we perform for the rst time a systematic study of the decay dynamics

of large QDs grown by droplet epitaxy and measure the oscillator strength and quantum eciency.

We present a detailed analysis of three individual QDs. Surprisingly, the oscillator strength

reveals that the excitons are in the strong-connement regime despite the large size of droplet-

epitaxy QDs. The small exciton size is cross-checked by quantitatively analyzing the phonon-

broadened spectra. Our results are in qualitative agreement with the work of Rol et al. [127]

for GaN QDs, where a similar analysis revealed that the excitons are smaller than the QD

size. The extracted quantum eciency (70 to 80 %) turns out to be lower than that of small

In(Ga)As QDs [84] yet larger than the quantum eciency of large In(Ga)As QDs [106]. Our

work conrms that nonradiative processes in semiconductor QDs have a profound impact on

their optical properties. We also show that some QDs exhibit a pronounced reduction in their

eective transition strength and quantum eciency with temperature, which we attribute to

coupling to excited states of the QD.

4.1 Sample growth and experimental procedure

The sample used in our experiment was grown on n-type GaAs (001) wafer. After thermal removal

of surface oxides, 0.1µm GaAs, 10 nm AlAs, 0.94µm GaAs, and 50 nm Al0.3Ga0.7As layers was

grown successively at 580 C. Thereafter, GaAs QDs were grown by droplet epitaxy according

to the following procedure. At a substrate temperature of 300 C, Ga atoms were injected onto

the surface at a vacuum level of 10−10 Torr. The amount of Ga is equivalent to the Ga content

in two GaAs monolayers. After injection of As and subsequent crystallization, a 20 nm-thick

56

Sample growth and experimental procedure

0.94 μm

GaAs droplets

Al0.3Ga0.7As

n-type GaAs (001) Substrate

3 nm

AlAs

GaAs

GaAs

10 nm

100 nm GaAs

155 nm

(a)

Figure 4.1: Sample structure and layout. (a) Schematic of the cutaway prole of our sample

(not to scale). (b) Scanning electron micrograph data of the uncapped reference sample.

Al0.3Ga0.7As layer was grown by migration-enhanced epitaxy, a technique used for growing high-

quality heterointerfaces at low temperatures [128]. The temperature was then raised back to

580 C and an additional 85 nm Al0.3Ga0.7As layer and a 3 nm GaAs cap were successively grown.

The sample was annealed at 850 C for 240 s in N2-atmosphere to improve the optical properties

of the QDs [129]. A sketch of the cutaway prole of our sample is depicted in Fig. 4.1(a),

while Fig. 4.1(b) shows a scanning-electron-microscope (SEM) image of a sample grown under

identical conditions but uncapped, which revealed a QD density of 67 µm−2. Atomic-force-

microscopy (AFM) studies showed that the uncapped dots are lens-shaped and asymmetric

in-plane with a major diameter of (82.4± 7.6) nm, a minor diameter of (54.4± 12.8) nm and a

height of (25.2± 8.8) nm but intermixing during overgrowth might change their size [129]. In

fact, we show in Sections 4.3 and 4.5 that ground-state excitons are strongly conned, which

represents direct evidence that signicant interdiusion during annealing reduces the eective

size of droplet-epitaxy QDs.

For optical measurements, the sample was placed in a liquid helium ow cryostat at 10 K

unless stated otherwise. A pulsed supercontinuum white-light source was spectrally ltered by

an acousto-optic modulator at a wavelength of 632 nm and was focused on the sample from

the top to a spot size of about 1.4 µm2 through a microscope objective with NA = 0.6. The

wavelength corresponds to above-band excitation of the QDs. The emission from the QDs was

collected by the same microscope objective. The cryostat was mounted on translation stages

to control the excitation and collection spot with an accuracy of 100 nm. The emission was

spatially ltered by a circular aperture with a diameter of 75µm and was subsequently dispersed

by a monochromator with a spectral resolution of 50 pm. The ltered light was sent either to a

charge-coupled device (CCD) for spectral measurements or to an avalanche photodiode (APD)

for time-resolved measurements.

57


Epitaxy

7 1 0 7 1 5 7 2 0 7 2 5 7 3 00

2

4

6

8

7 1 5 7 2 0 7 2 5 7 3 0 7 1 5 7 2 0 7 2 5 7 3 0

0 . 1 1 1 0 1 0 01 0 1

1 0 2

1 0 3

1 0 4

0 . 1 1 1 0 1 0 0 0 . 1 1 1 0 1 0 0

( f )( e )( d )

( c )( b )

Inten

sity (a

rb. un

its)


( a )

1 1 4 2 W / c m 22 2 8 5 W / c m 23 9 2 7 W / c m 2

5 7 1 W / c m 2

2 8 6 W / c m 2

1 4 3 W / c m 2

7 1 W / c m 2

3 6 W / c m 2

1 . 7 4 1 . 7 3 1 . 7 2 1 . 7 1 1 . 7 0E n e r g y ( e V )


1 . 7 3 1 . 7 2 1 . 7 1 1 . 7 0Q D CQ D BQ D A

E n e r g y ( e V )

X X X L O X X X X X XT


1 . 7 3 1 . 7 2 1 . 7 1 1 . 7 0E n e r g y ( e V )

2 . 0 7 ± 0 . 0 6

Inten

sity (a

rb. un

its)

P o w e r ( µ W )

1 . 0 7 ± 0 . 0 6

P o w e r ( µ W )

1 . 0 5 ± 0 . 0 7

1 . 2 7 ± 0 . 0 6 0 . 7 7 ± 0 . 0 2

1 . 5 7 ± 0 . 0 5

P o w e r ( µ W )

Figure 4.2: Spectral measurements on droplet-epitaxy QDs. (a-c) Spectra at dierent excitation

power densities for the three QDs discussed in this work. The exciton, trion and biexciton lines

are labelled as X, T and XX, respectively. For QD A, an LO-phonon replica is observed. (d-

f) Integrated intensity as a function of pumping power for the X (blue upward triangles) and XX

lines (green downward triangles) along with the corresponding polynomial ts.

4.2 Spectral measurements

The optical properties of the QDs are investigated by means of above-band optical excitation,

where electron-hole pairs are photoexcited in the Al0.3Ga0.7As matrix in which the QDs are

embedded. Due to the low areal density, the spectrum normally consists of individual lines

at low average excitation power densities of 143 W/cm2 or below, which corresponds to the

recombination of the ground-state exciton in the QD (further denoted as the X or exciton line,

see Fig. 4.2,(a) through (c)). For all three QDs, the integrated PL intensity of the X lines

is approximately linear with excitation power, cf. Fig. 4.2(d)-(f) as expected for excitons. The

integrated intensity is calculated as follows. The X line is tted with a Lorentzian and integrated.

The exciton line saturates at a pumping intensity of about 286 W/cm2, which corresponds to

the onset of the biexciton as discussed later. The spectral behavior of QD B is dierent because

two emission lines arise at low pumping powers. We identify them as an exciton and a trion via

time-resolved measurements, which is consistent with previous investigations of droplet-epitaxy

58

Spectral measurements

GaAs QDs [130133]. In particular, the exciton decays bi-exponentially (see Sec. 4.3) and the

trion single exponentially because the trion manifold does not have a dark state [65].

The emission linewidth of QDs B and C is limited by the resolution of the spectrometer

(50 pm equivalent to 120µeV at a wavelength of 720 nm), which is clear evidence of single QD

emission. The line belonging to QD A is, however, relatively broad and is found to be of the

order of 260 µeV after deconvolving it with the instrument response function, which is much

broader than the radiative linewidth of several µeV of the excitonic transition in single QDs.

This broadening is mainly caused by two factors. First, a noticeable broadening can be induced

by the ne-structure splitting [68] of the bright exciton [134]. Second, the broadening of the

exciton line is associated with spectral diusion induced by a time-uctuating quantum-conned

Stark eect [135] related to charging and discharging of trap defects [136] in the QD vicinity.

This scenario is plausible, given the low-temperature growth of the droplet-epitaxy QDs, which

can aect the quality of their crystalline structure.

At higher excitation powers (286 W/cm2 and above) we observe a second line, which is red-

shifted by 24 meV with respect to the exciton line. From power-series measurements (see

Fig. 4.2), the slope of the integrated intensity (the raw data were integrated directly due to

the broad and irregular shape of the line) is signicantly larger than that of the X-line, which

suggests a biexciton-to-exciton recombination. For biexcitons, the PL intensity is expected to be

quadratic with excitation power. For QD A the data show good agreement but for QDs B and

C the slope of the second peak is found to be superlinear but less than two. We have therefore

performed lifetime measurements on these two peaks to conrm the biexcitonic origin of the

second peak. We obtain a total decay rate for the exciton (secondary) line lying in the range

1.62.1 ns−1 (3.64.3 ns−1) for all three QDs. In the limit of slow spin-ip processes [137], the

biexciton is twice as fast as the exciton because it has two possible radiative decay channels, i.e.,

it can decay to either of the bright states. Similarly, the biexciton is expected to decay twice as

fast nonradiatively because any of its charge carriers is prone to nonradiative loss, as explained

in the following. We assume that one type of charge carriers (e.g., holes) have the largest nonra-

diative decay, and that the hole decays with the same rate from an exciton or biexciton. Then,

in the rst approximation, an exciton |↑⇓〉 decays to |↑ 0〉, while a biexciton |↑↓⇑⇓〉 can decay

either to |↑↓⇑ 0〉 or to |↑↓ 0 ⇓〉, where the single arrow denotes the electron spin and the double

arrow the hole spin. As a consequence, the total decay rate of the biexciton is expected to be

γXX = 2γX,RAD + 2γX,NRAD = 2γX , where γX is the total decay rate of the exciton. The time-

resolved measurements conrm that the second line is due to biexciton recombination (further

denoted as the XX line).

The XX saturation intensity corresponds to the onset of multi-particle recombination because

more than four single-particles (2 electrons + 2 holes) are stored in the QD. An increasing number

of spectral lines on top of a continuous background appear at yet larger excitation intensities.

Additionally, the XX line exhibits a peculiar feature: it is spectrally broadened and time-resolved

measurements show that the low-energy sideband decays rst. This behavior was previously

observed for GaAs interface-uctuation QDs at large excitation powers and was attributed to

Coulomb interaction between the biexciton and charge carriers present at higher lying states in

59


Epitaxy

CB, Γ

VB, Γ

1.94 eV

640 nm

~1.72 eV

~720 nm

92 meV

51 meV

Al0.3Ga0.7AsAl0.15Ga0.85As

Al0.3Ga0.7As

Figure 4.3: Approximated band diagram (solid black lines) of the QDs under the assumption

of constant Al content within the QD. Charge carriers are generated inside the Al0.3Ga0.7As

matrix and are subsequently trapped by the Al0.15Ga0.85As QDs before recombining radiatively

around a wavelength of 720 nm. The dotted line is a qualitative sketch of the actual potential

prole whose smooth spatial dependence is a consequence of alloy inhomogeneities within the

QD, thereby rendering the spatial extent of the ground-state exciton smaller than the size of the

QD (see text for details).

the surrounding quantum well [117]. Our scenario is in fact very similar, the main dierence

being that the higher lying states belong to the same QD and not to a quantum well.

Aside from these common features, each QD has its own spectral repertoire. An interesting

example is the line of QD A at a wavelength of 728 nm. Its energy distance to the exciton peak

is 33.4 meV, which suggests an optical phonon replica since the bulk GaAs LO (TO) phonon

energy is 36.6 meV (33.2 meV). By tting the emission spectrum with a Lorentzian function and

deconvolving with the point-spread function of the setup we obtain a FWHM of 430µeV. At this

particular excitation power (far beyond saturation), the X line is 436µeV broad. We therefore

attribute this red-shifted spectral emission to the optical phonon replica of the exciton.

To conclude this section, we calculate the band structure of the QDs. The only information

taken from experiment is the emission frequency of the exciton. The QDs emit in a wave-

length range between 700 and 740 nm and the band gap of GaAs (Al0.3Ga0.7As) is about 820 nm

(640 nm) at low temperature [38]. Given the fact that connement eects are supposed to be

small due to the large size of the QDs, we conclude that the growth has resulted in a substantial

interdiusion between the AlGaAs matrix and the GaAs QDs. We thus expect the conduction-

and valence-band potential proles to follow the intermixing prole, as is qualitatively sketched

in Fig. 4.3. Unfortunately, the explicit spatial dependence of the latter is unknown; in order to

provide a quantitative picture of the average interdiusion magnitude, we assume for now that

the potential prole is constant. This is a drastic assumption and is just meant to provide a

simplied picture of the band structure and, therefore, its consequences should be treated with

care (in fact, we show later that the potential prole does exhibit a spatial dependence). Simi-

60

Oscillator strength and quantum eciency

larly, by virtue of the previous arguments, we believe that connement eects are smaller than

the involved energy scales of the band diagram and we neglect them to simplify the discussion.

In the eective-mass approximation, we write down the energy position in eV of the conduction

Ec,Γ and valence Ev,Γ bands of AlxGa1−xAs at the Γ point in reciprocal space [38, 138]

Ec,Γ(x) = 2.979 + 0.765x+ 0.305x2, (4.1)

Ev,Γ(x) = 1.460− 0.509x. (4.2)

By solving for Ec,Γ − Ev,Γ = EPL, where EPL is the emission energy, we obtain an average

Al content of 15.3 % for a wavelength of 720 nm. This yields a total connement energy of

51 meV for holes and 92 meV for electrons. The corresponding band diagram, which includes the

aforementioned simplications, is sketched in Fig. 4.3. We underline that the rst valence-band

eigenstate is expected to be heavy-hole like due to the relatively small aspect ratio of the QDs [61],

see Sec. 4.1 and the discussion in Sec. 2.2. Intermixing during growth does not signicantly alter

the aspect ratio because it is approximately isotropic.

4.3 Oscillator strength and quantum eciency

Spectral measurements provide important insight to the level structure of QDs, as we have

seen in the previous section. However, phenomena with a lifetime signicantly shorter than a

few hundred milliseconds are averaged out and therefore not resolved. For instance, spontaneous

emission, spin-ip processes, phonon scattering, etc., are processes that occur somewhere between

picosecond to microsecond time scales. Time-resolved measurements of the PL signal have the

capability of providing rich information about such processes. The gure of merit quantifying

the coupling of an emitter to light is the oscillator strength f , which is dened in Sec. 2.5 and is

proportional to the radiative decay rate in a homogeneous medium Γhomrad via [83]

f =6πm0ε0c

30

e2nω20

Γhomrad , (4.3)

where n is the refractive index of the host material (i.e., Al0.3Ga0.7As), ω0 and c0 constitute

the frequency and speed of light, respectively, ε0 the vacuum permittivity, m0 the electron

mass, and e the elementary charge. In other words, the oscillator strength of the QD can be

obtained by measuring the radiative decay rate of the ground-state exciton. However, the latter

is not a straightforward task because the measured decay rate Γ is the sum of the radiative rate

Γrad and all nonradiative recombination channels Γnrad. In general, nonradiative processes are

omnipresent in solid-state systems even at low temperatures and, therefore, cannot be neglected.

In this section we extract the oscillator strength and quantum eciency of droplet-epitaxy QDs

by employing the exciton ne structure and bi-exponential decay dynamics presented in Sec. 2.5.

If the excitation intensity is below the onset of the biexciton line (see Fig. 4.2), only one

electron and one hole are captured by the QD. These carriers undergo phonon-scattering pro-

cesses on a time scale of the order of picoseconds (several orders of magnitude faster than the

radiative recombination) before ending up in the QD ground state. The exciton captured by the

61


Epitaxy

10-2

10-1

100

-303

0 1 2 3 4 5-303

XXXFitXbBackgroundXXbBackground

Inte

nsity

bDar

b.bu

nits

)W

eigh

ted

Res

idua

ls

TimebDelaybDns)

3.68bns-1

0.6bns-1

2.07bns-1

Figure 4.4: Time-resolved decay dynamics of the exciton (blue) and biexciton (green) of QD

A along with the corresponding weighted residuals marked by blue circles and green squares,

respectively. The red solid line indicates the t while the dashed/dotted lines denote the back-

ground level. The data are tted ∼0.5 ns later than the beginning of the decay due to lling

eects (see text).

QD ends up either being bright or dark with equal likelihood because above-band excitation is

performed [139]. As such, the bright exciton exhibits a biexponential decay with the fast rate

ΓF = Γrad + Γnrad and the slow rate ΓS = Γnrad + Γsf . The extracted radiative rate Γrad does

generally not equal the homogeneous radiative rate Γhomrad because the emitter is not placed in

an innite homogeneous medium. Therefore, we calculate the normalized LDOS at the position

of the emitter [73] for the layered structure outlined in Fig. 4.1(a) and obtain a value of 1.05.

The excitonic and biexcitonic decays of QD A, both below saturation, are plotted in Fig. 4.4(b)

along with the corresponding ts.

The exciton exhibits a biexponential behavior, which is conrmed by the low χ2R (see Ta-

ble 4.1; it is important to emphasize that a single exponent severely underts all the decay curves

at 10 K) with the fast rate of the order of 2 ns−1 and the slow rate about three times smaller, as

can be seen in Fig. 4.4(b). We extract an oscillator strength of around 9 and a quantum eciency

between 69 to 79 % (see Table 4.1). Even though the quantum eciency of droplet-epitaxy QDs

is found to be lower than that of small InAs QDs, their optical quality is signicantly higher

than that of large InAs QDs whose quantum eciency ranges between 30 and 60 % [106]. This

may be due to the lack of strain-related eects in GaAs QDs, which makes droplet epitaxy a

growth technique potentially capable of delivering QDs with very high optical quality suitable

for quantum-information applications. We attribute the less-than-unity quantum eciency to

the low-temperature growth of the capping layer, see Sec. 4.1.

It is commonly stated that a key advantage of QDs relies in their oscillator strength, which

is about one order of magnitude larger than that of atomic emitters. However, it is important to

62

Oscillator strength and quantum eciency

QD A QD B QD C

ΓF (ns−1) 2.07 1.61 2.00

ΓS (ns−1) 0.60 0.34 0.62

AS/AF × 10−3 10 50 23

χ2R 1.1 1.05 1.02

Γrad (ns−1) 1.47 1.27 1.38

Γnrad (ns−1) 0.58 0.28 0.59

Γsf (µs−1) 14.4 57.7 30.4

oscillator strength 9.4 8.2 9.0

quantum eciency (%) 70.1 78.1 69.0

|〈ψh|ψe〉|2 (%) 56.5 49.1 53.5

Table 4.1: Quantities extracted from the exciton decay.

underline that the oscillator strength depends on the QD size. Only for QDs smaller than the ex-

citon Bohr radius a0 (further denoted as `small QDs') does the oscillator strength become almost

independent of the QD size [83, 109, 140]. In the dipole- and eective-mass approximations, the

oscillator strength of small QDs is given by

f =EP~ω0|〈ψh| ψe〉|2 , (4.4)

where EP is the Kane energy and ψe (ψh) is the electron (hole) slowly-varying envelope function.

This so-called `strong connement regime' has an upper bound for the oscillator strength of

fmax = EP /~ω0, which amounts to 16.7 for a GaAs QD at an emission wavelength of 720 nm,

where we have used a GaAs value of 28.8 eV for the Kane energy [38].

On the other hand, QDs whose linear size L is larger than a0 (further denoted as `large QDs')

exhibit an enhanced light-matter interaction. For example, the oscillator strength of a spherical

QD is given by [83] fsph = fmax×√π (L/a0)

3, and scales with the number of unit cells the exciton

spreads itself across. The oscillator strength in weakly conned systems can become signicantly

larger than fmax if L > a0. This behavior of large QDs was coined the giant oscillator strength,

and its physical reason is related to the superradiant nature of the ground-state exciton, which

distributes itself coherently over a much larger volume than it otherwise does in small QDs or

bulk [109, 120].

According to the AFM data, the QDs have an in-plane radius of 3040 nm and a height of

25 nm before capping and annealing. Given the fact that all the dimensions are weakly conned,

we compare the QD to a sphere with the same volume and obtain an expected oscillator strength

beyond 900, which is two orders of magnitude larger than the observed oscillator strengths of

about 10 that are listed in Table 4.1. This value is within the strong connement limit, which

is direct evidence that the excitons are strongly conned in the droplet-epitaxy QDs. In other

words, it appears that the eective size of the QDs is diminished by the capping and annealing

63


Epitaxy

processes. This assumption is supported by the emission wavelength (see the discussion in

Sec. 4.2), which is substantially smaller than the GaAs bandgap, thereby suggesting considerable

alloy inhomogeneities in the QDs and, hence, a reduction of the ground-state exciton coherence

volume. Our results clearly underline the importance of a growth technique that induces as

little intermixing as possible between the QD and the surrounding matrix in order to obtain

enhanced light-matter interaction. With the help of Eq. (4.4) we calculate the electron-hole

overlap integral |〈ψh| ψe〉|2 to range between 0.490.57, which is comparable but smaller than

that of self-assembled In(Ga)As QDs (0.620.77) [84].

Let us return to the decay dynamics of the biexciton, as shown in Fig. 4.4. Surprisingly,

a single-exponential does not t the curve for all the investigated QDs. We attribute this to

spectral pollution of the biexciton line by LA phonons stemming from the exciton line [141

143]. The photo-luminescence signal IAPD at the emission frequency of the biexciton within a

frequency range determined by the resolution of the setup (50 pm) is given by

IAPD = AXXe−ΓXXt + CLAAF e

−ΓF t + CLAASe−ΓSt + C, (4.5)

where CLA is the integrated coupling coecient of the zero-phonon line to the modes that overlap

spectrally with the biexciton emission. It is clear from Eq. (4.5) that the decay curve is expected

to be triple exponential with the fastest rate ΓXX corresponding to the biexciton decay rate.

We have tted the biexciton curve with a triple exponential, see the red solid line in Fig. 4.4.

The extracted fast rate ΓXX = 3.68 ns−1 roughly equals 2ΓF of the bright exciton (see Table 4.1),

in good agreement with the theoretical considerations in Sec. 4.2. The middle and slow rates

are found to be 2.2 ns−1 and 0.44 ns−1, respectively, and reproduce quite accurately the fast and

slow rates of the exciton line (noteworthy, the biexciton line was recorded above the exciton

saturation, which is why the decay rates of the exciton at this elevated power might be dierent

than the ones given in Table 4.1), which brings further evidence of a phonon-mediated emission

of the ground-state exciton overlapping spectrally with the biexciton.

To conclude this section, we emphasize that although the ground state is clearly strongly

conned, the excited states do not necessarily have to be so. We have already shown that

it is very likely that the droplet-epitaxy QDs are characterized by a smooth spatial potential,

which follows the alloy-intermixing prole implying that excited states become less conned

(see Fig. 4.3). In general, there is no obvious correlation between the connement of ground-

state excitons and the QD size, if the potential prole is not uniform within the QD. Despite

the strong connement of the ground state, the droplet-epitaxy QDs can be considered `large'

because they contain a large number of excited states, which is supported by two independent

experimental ndings. First, time-resolved measurements of the exciton line above saturation

show pronounced lling eects (i.e., there is a substantial time interval between the excitation

pulse and the actual PL decay), which is characteristic to large QDs [106, 117]. Second, the

eective transition strength is diminished with increasing temperature, which is direct evidence

of nearby excited states (several meV away), as is shown in the following section.

64

Temperature dependence of the eective transition strength

4.4 Temperature dependence of the eective transition strength

Due to three-dimensional connement, QDs have discrete energy levels. At low excitation powers

and temperatures, only the ground state is relevant because the rst excited eigenstate is situated

at much higher energies than the thermal energy kBT . This picture is justied for small QDs

where connement eects are signicant but it may no longer be valid in large QDs where the

energy dierence between the eigenstates may become comparable to the thermal energy; for

example, kBT = 4.3 meV at 50 K. Thermal population of excited states leads to a modication

of the three-level scheme from Fig. 2.12 and, thus, of the exciton dynamics. If a single excitation

is thermally shared by several eigenstates, the eective transition strength becomes temperature-

dependent and does not coincide with the oscillator strength of the ground-state exciton [144].

Generally, the oscillator strength is a property of two energy levels and quanties the emission

rate of light. When an exciton is shared among many energy levels (as, e.g., in the case of a

quantum well at nite temperatures), the radiative decay rate of the system can no longer be

used to extract the oscillator strength. In this context, the eective transition strength becomes a

more relevant quantity and determines the light emission rate [144]. In the following, we present

a study of the temperature properties of droplet-epitaxy QDs.

Some of the studied QDs, in particular QD A, exhibit a striking reduction in the eective

transition strength with increasing temperature. In this section we show that this behavior

is caused by the large size of droplet-epitaxy QDs. Figure 4.5(a) displays the acquired PL

spectrum of QD A in a temperature range from 10 to 60 K below the exciton saturation. A

pronounced redshift of the excitonic line is observed due to the well-known band-gap shrinkage

with temperature. At 60 K the PL signal is quenched due to the onset of nonradiative processes

at elevated temperatures. A narrow line, blueshifted by 700µeV with respect to the exciton

line, appears with increasing temperature. It cannot be an excited state because it would have

been thermally populated at energies 4kBT ≈ 700µeV corresponding to T ≈ 2 K. Time-resolved

measurements revealed that it decays identically to the exciton line for all temperatures. This

is consistent with the behavior of a charged exciton (a trion), which is expected to decay with

roughly the same rate as the neutral exciton. Henceforth we turn our attention to the exciton

line.

QD A reveals a pronounced dependence of the decay dynamics on temperature, see Fig. 4.5(b).

As the temperature is increased, the bright exciton decays slower up to 50 K. Interestingly, the

decay curves become single-exponential at temperatures higher than 40 K. This is a consequence

of Γnrad becoming comparable to Γrad, whereby the biexponential decay is masked by the mea-

surement noise and the curves become single-exponential. More formally: in order to resolve a

biexponential decay, the PL signal of the fast component must decay before the amplitude of the

slow component becomes smaller than the background noise ∆ABG, i.e., AF e−ΓF t ≤ ASe

−ΓSt

and ∆ABG < ASe−ΓSt. This gives the important condition for experimentally observing the

slow decay component

Γnrad

Γrad.

ln(

AS∆ABG

)ln(AFAS

) . (4.6)

65


Epitaxy

710 715 720 7250

1

2

3

4

5

6

7

0 1 2 3 410l2

10l1

100

0V020V04

0V5

1V0

1V5

2V0

3

4

5

6

7

8

9

10

10 20 30 40 50 6020

30

40

50

60

70ddb

dcb

dbb

dab

60K

50K

40K

30K

20K

Inte

nsi

ty5d

arb

V5un

itsb

Wavelength5dnmb

10K

1V74 1V73 1V72Energy5deVb

50K40K30K20K10K

Inte

nsi

ty5d

arb

V5un

itsb

Time5Delay5dnsb

Effe

ctiv

e5T

ran

sitio

nS

tre

ng

thW

F

Temperature5dKb

Qu

an

tum

5Effi

cie

ncy

Wd.

b

Dec

ay5

rate

Wdn

sl1b

Figure 4.5: Spectra and decay dynamics of excitons captured by QD A at various temperatures

for an excitation power density of 286 W cm−2, which is below saturation of the exciton line.

(a) Spectra recorded within a temperature range of 1060 K. (b) Time-resolved decay of the

exciton line from 10 K to 50 K showing an increase in the exciton lifetime with temperature.

(c) Temperature dependence of the fast, radiative, nonradiative, and spin-ip rates of the exciton,

as well as (d) the eective transition strength and quantum eciency. The black dashed line ts

the spin-ip rates with a linear function passing through the origin. The dotted lines provide

guides to the eye.

In the limit of noiseless measurements, the slow component can always be detected, whereas if

the noise equals the slow component amplitude, the biexponential decay cannot be resolved at all

and the curve appears single exponential. It is also clear that a longer integration time τ of the

decay curves enables resolving the biexponential decay better because the PL signal scales with

τ and the measurement noise with√τ . In our experiment, AS/∆ABG ≈ 60 and AF /AS ≈ 20

at a temperature of 40 K, which yield a limit of Γnrad/Γrad . 1.4. Indeed, at 40 K Γnrad/Γrad

is about 1.4, cf. Fig. 4.5(c), while at 50 K the biexponential model overts the decay curve,

which is direct evidence of the low quantum eciency of the transition. This is conrmed by

the luminescence quenching in the spectrum, see Fig. 4.5(a). Henceforth only the biexponential

curves are discussed.

66

Temperature dependence of the eective transition strength

Figure 4.6: Extension of the three-level scheme from Fig. 2.12 at elevated temperatures. In

large QDs the ground-state bright exciton may be thermally activated to a higher energy (hot)

state |h〉 (Γ∗ph = Γph × B, where B is the Boltzmann factor). The hot state can decay back to

the bright state via Γph; furthermore, |h〉 may decay nonradiatively via Γh.

The sudden threefold drop in the radiative decay rate with temperature (see Fig. 4.5(c) and

(d)) may appear puzzling since the rate is expected to be independent of temperature at low

temperatures [145]. A quantum well for instance does decay slower with increasing temperature

owing to thermal excitation of excitons away from the Brillouin zone center rendering them

optically dark [146, 147]. A similar eect was predicted theoretically for QDs and attributed

to thermal population of excited states [148] but is not expected to occur in small QDs at low

temperatures by virtue of the zero-dimensional density of states. In large QDs, however, this

eect may become possible due to the small spacing between the energy states. We elaborate

on this in the next paragraphs. A reduction in the decay rate was observed for self-assembled

In(Ga)As QDs in a similar temperature range and was attributed to carrier redistribution among

dierent QDs via the wetting layer [149151]. Such a mechanism is unlikely to occur in droplet-

epitaxy QDs due to the lack of a wetting layer, and carrier redistribution among dierent QDs can

be safely neglected because the thermal energy is much smaller than the connement potential

of charge carriers in the present experiment, see Fig. 4.3.

We discuss the physical mechanism governing the decrease in the bright-exciton transition

strength qualitatively and return to a more formal discussion later. Due to the large size of

droplet-epitaxy QDs, excited states with small oscillator strength may be thermally activated. If

the hole populates the rst excited state |2h〉 (this is more likely because the eective mass of holes

is larger than of the electrons) and the electron is in the ground state |1e〉, the recombination

of such an exciton is parity forbidden. This is consistent with the fact that we do not see

excited states in the PL spectrum. The in-plane symmetry of the QDs results in two closely

spaced optically inactive excited states (they would be degenerate in case of perfect rotational

symmetry). A single excitation may therefore be shared between a parity-bright and two parity-

dark states. In the limit kBT ∆Ehb, where ∆Ehb is the energy dierence between the ground

and excited states, the exciton populates the bright state with a probability of 1/3 resulting in

a three-fold decrease of the eective transition strength.

67


Epitaxy

We denote the excited eigenstates as hot states |h〉 and the modied level scheme is sketched

in Fig. 4.6. We rst analyze the implications of a single hot state before generalizing the results.

The hot state decays to the bright state via the phonon-mediated rate Γph and becomes pop-

ulated from the bright state via Γ∗ph = Γph × e−∆Ehb/kBT . Additionally, the hot state decays

nonradiatively via Γh. We include only the bright exciton of the hot states in our model to

simplify the discussion, however, the implications of the dark state will also be addressed. We

therefore set the spin-ip rate Γsf = 0 and solve the rate equations analytically. For the realis-

tic assumption that Γph is the fastest rate in the system (phonon scattering is of the order of

picoseconds), the decay of the bright state takes the form

ρb(t) =1

1 + B[Bρb(0)− ρh(0)] e−Γph(1+B)t

+1

1 + B[ρb(0) + ρh(0)] e−(Γrad+Γnrad+BΓh)t/(1+B),

(4.7)

where B = e−∆Ehb/kBT is the Boltzmann factor. The rst term accounts for the build-up of

the excitonic population on a phonon scattering time scale. The population decay and thus the

experimentally accessible fast rate is given by the second term

ΓF =Γrad + Γnrad + BΓh

1 + B= Γ∗rad + Γ∗nrad.

(4.8)

where the asterisk denotes temperature-dependent quantities, and Γh is merged into Γnrad. The

fast decay rate of the bright state decreases up to a factor of two in the limit of large temperatures.

More generally, N parity-dark states decrease the eective transition strength F by a factor

of N + 1, i.e., F = f × Γhom∗rad /Γhom

rad = f/(N + 1). As a consequence, the almost threefold

decrease observed for F (see Fig. 4.5(d)) suggests the presence of two parity-dark states. At

50 K the fast decay rate is further reduced, which suggests that the eective transition strength

continues to decrease and interaction with more excited states becomes feasible. The energy

dierence between the bright and parity-dark states ∆Ehb is of the order of several meV, i.e.,

comparable to the thermal energy in the investigated temperature range. In the present study it

is unfortunately not possible to accurately quantify ∆Ehb because parameters such as the energy

dierence between the hot states, their nonradiative decay rates, etc., are unknown.

It is well known that at elevated temperatures nonradiative decay channels become increas-

ingly important [152154] and this is reected in Fig. 4.5(c), where the nonradiative decay rate

increases by about 50 %. This has direct impact on the quantum eciency, which diminishes

from 70 % to 40 %, see Fig. 4.5(d). Interestingly, our data show that it is incorrect to associate

a decrease in the fast decay rate with a reduction of nonradiative processes.

The spin-ip rate in droplet-epitaxy QDs is similar to self-assembled In(Ga)As QDs and

amounts to several tens of µs−1 at 10 K (cf. Table 4.1). Spin ip is a phonon-mediated process

as discussed in Sec. 4.3 and, therefore, depends on the number of available phonons NB given by

the Bose-Einstein distribution

NB =1

eδ0/kBT − 1, (4.9)

68

Acoustic-phonon broadening and exciton size

where δ0 is the energy splitting between bright and dark excitons. In our experiment kBT δ0

or, equivalently, NB 1, and the spin-ip rate can be written as Γsf ' Γ0NB ≈ kBT/δ0 × Γ0,

where Γ0 is the spin-ip rate at 0 K. By tting the data with a linear function passing through

the origin we obtain a good agreement, as seen in Fig. 4.5(c). We extract a slope of Γ0/δ0 ≈15 ns−1eV−1, and for typical values of δ0 ≈ 200µeV [68] obtain a zero-temperature spin-ip rate

of Γ0 ≈ 3 µs−1.

4.5 Acoustic-phonon broadening and exciton size

In Sec. 4.3 it was shown that ground-state excitons are strongly conned in droplet-epitaxy

QDs. This conclusion was based on the small oscillator strength extracted from time-resolved

measurements. In this section, we bring further evidence of strong connement of charge carriers

by analyzing the phonon sidebands in the emission spectra of droplet-epitaxy QDs. In particular,

we employ the independent-boson theory to model our experimental results, and we nd that

the electron and hole wavefunctions are smaller than the exciton Bohr radius, in good agreement

with time-resolved measurements. The model implemented in this section has been employed to

investigate the electron-phonon interaction [20, 141, 155], and is used here as a tool to quantify

the size of the electron and hole wavefunctions.

We consider a two-level system coupled to an acoustic phonon bath. The Hamiltonian of this

coupled exciton-phonon system reads

H = E0c†c+

∑k

~ωk

(b†kbk +

1

2

)+ c†c

∑k

Mk(b†k + bk), (4.10)

where c† and b†k (c and bk) are the creation (annihilation) operators of the exciton (with energy

E0) and the phonon (with momentum ~k), respectively. The last term in Eq. (4.10) denotes

the interaction Hamiltonian, where Mk is the electron-phonon interaction matrix element. The

phonon bath represents a continuous set of modes with momentum ~k, and each mode has a

probability Mk of interacting with the two-level system. The exciton has a nite lifetime given

by the radiative decay rate Γrad. In order to compute Mk, we follow a number of assumptions:

(1) The deformation-potential coupling to longitudinal-acoustic (LA) phonon modes is the

dominant electron-phonon interaction term and, therefore, interactions with transverse acoustic

modes and the piezoelectric coupling are neglected [156].

(2) We consider bulk phonons only, i.e., the QD couples to vibrational modes of the sur-

rounding material, Al0.3Ga0.7As. This assumption is justied by the small impedance mismatch

between the QD material and the surrounding matrix.

(3) The LA phonon dispersion relation is linear in the relevant energy range ωk = cs|k|, wherecs is the speed of sound in the crystal and is taken to be isotropic (averaged over all directions).

This is a good assumption since the dispersion of LA phonons becomes nonlinear towards the

edge of the Brillouin zone, which corresponds to harmonic modes with spatial oscillations beyond

the size of QDs.

69


Epitaxy

1.71 1.715 1.72Energy (eV)

QD C

10 K

20 K

30 K

1.73 1.735 1.740

200

400

600

800

Energy (eV)

Inte

nsity

(ar

b. u

nits

) QD A

10 K

20 K

30 K 40 K

Figure 4.7: Experimental data (colored circles) along with the ts (solid black lines) for QD A

and QD C at dierent temperatures. All data were recorded under the same conditions with an

excitation power density of 286 W cm−2.

(4) We employ the eective-mass approximation for charge carriers and work in the single-

particle picture where electrons and holes are independent entities. This is a good assumption

for at QDs and its justication is discussed in detail in Sec. 2.2.

Under these assumptions, the phonon matrix element reads [141]

Mk = Nk[De 〈ψe| eik·r |ψe〉 −Dg 〈ψg| eik·r |ψg〉

], (4.11)

where Nk =√~|k|/2dcsV , ψe (ψg) is the slowly-varying envelope function of the electron (hole),

and V is the quantization volume. The following constants are used for Al0.3Ga0.7As: the density

d = 4805 kg m−3, the speed of sound cs = 5396 m s−1 [157], and the deformation potentials

Dg = 5.6 eV and De = −11.5 eV [143]. We assume lens-shaped wavefunctions with a Gaussian

spatial prole

ψν(r) =1

π3/4σν,ρ√σν,z

e− ρ2

2σ2ν,ρ e− z2

2σ2ν,z , (4.12)

where ρ =√x2 + y2 is the in-plane radial coordinate, σ is the half-width at half maximum

(HWHM), and ν = e, g. The matrix element is evaluated to be

Mk = Nk[Dee

− 14 (σ2

e,ρk2ρ+σ2

e,zk2z) −Dge

− 14 (σ2

g,ρk2ρ+σ2

g,zk2z)]. (4.13)

The phonon contribution function Φ(t) gauges the temporal decay of the excitonic-polarization

coherence, and is derived directly from the interaction Hamiltonian in Eq. (4.10) [155]

Φ(t) =∑k

|Mk|2

(~ωk)2[i sin(ωkt) + [1− cos(ωkt)] (2nk + 1)] , (4.14)

where nk = (e~ωk/kBT−1)−1 is the thermal occupation function. For a large quantization volume,

the sum over k can be converted into an integral via∑

k →V

(2π)3

∫dk, so that Eq. (4.14) becomes

70

Acoustic-phonon broadening and exciton size

σe,ρ (nm) σg,ρ (nm) f

QD A 2.4 2.4 16.6

QD C 3.6 1.9 9.4

Table 4.2: Fitted sizes (HWHM) of the electron and hole wavefunctions and the resulting oscil-

lator strength f .

Φ(t) = C

∞∫0

kdk

1∫0

dy

∣∣∣∣Dee−σ

2e4 k

2(1−ξey2) −Dge−σ2g4 k

2(1−ξgy2)

∣∣∣∣2× [i sin(ωkt) + [1− cos(ωkt)] (2nk + 1)] ,

(4.15)

where C = 1/4π2dc3s~ and ξν = 1− σ2ν,z/σ

2ν,ρ. The integration over y is performed analytically,

and the integral over k is evaluated numerically. Finally, the emission spectrum is evaluated by

Fourier transforming the phonon-contribution function [155]

S(ω) =

+∞∫−∞

dte−i(ω−ω0−iΓrad/2)te−Φ(t), (4.16)

where ω0 is the emission frequency of the QD.

We t the acquired spectra with the independent-boson model using a least-square approach

so that the sum of the squared residuals is minimized. Following the observations from AFM

measurements, we x the ratio between the wavefunction height and radius σν,ρ = ασν,z with α =

3. This assumption is needed to avoid overtting the data. We thus have only two independent

tting parameters, namely the size of the hole and electron wavefunctions. For QD C, the

spectrum is tted at the highest recorded temperature (40 K) because the signal coming from the

phonon sidebands increases with temperature and enhances the accuracy of the tted parameters.

For QD A, the tting is performed at 30 K because at higher temperatures there is an additional

line in the vicinity of the exciton line, see Fig. 4.5(a), which renders the t dicult to realize. For

the data at lower temperatures we do not t but simply plot the evaluated emission spectrum.

Figure 4.7 shows the spectra of QDs A and C with very good agreement between theory and

experiment. The extracted sizes (HWHM of 24 nm) are well below the exciton Bohr radius

(11.2 nm). This independent analysis therefore conrms the observations from time-resolved

measurements, namely that ground-state excitons are strongly conned in droplet-epitaxy QDs.

We can give an estimate of the oscillator strength using Eq. (4.4), which agrees reasonably well

with experiment, compare Tables 4.1 and 4.2.

71


Epitaxy

4.6 Summary

In this chapter we have presented an extensive study of the optical properties and decay dy-

namics of large strain-free droplet-epitaxy GaAs QDs. From the measurements, we draw several

important conclusions:

(1) The droplet-epitaxy QDs exhibit an oscillator strength and quantum eciency of about

9 and 75 %, respectively.

(2) Ground-state excitons are strongly conned despite the large size of the droplet-epitaxy

QDs observed in AFM measurements. This is caused by material inter-diusion occurring be-

tween the QDs and the surrounding matrix, which creates a localized potential minimum that

traps carriers in a region of space smaller than the exciton Bohr radius. This physical picture

is supported by two independent analyses: the oscillator strength extracted from time-resolved

measurements, and the sizes of electron and hole wavefunctions obtained from the analysis of

the spectral phonon sidebands.

(3) For some QDs, the bright exciton is thermally activated to parity-dark eigenstates with

temperature. As a consequence, the radiative lifetime of bright excitons is substantially prolonged

and the eective transition strength decreases from 10 to 4 as the temperature is raised from

10 K to 40 K. Additionally, the nonradiative recombination rate is increased by almost a factor

of two in the same temperature range. Both aect the quantum eciency, which attains a value

of only 40 % at 40 K.

Our ndings show that droplet-epitaxy GaAs QDs, similarly to the commonly used self-

assembled In(Ga)As QDs, exhibit non-negligible nonradiative processes. This is likely due to the

low-temperature growth of the QDs and of the capping layer forming a crystalline structure of

low quality, which is not fully restored by thermal annealing. Although we have not found a giant

oscillator strength in these QDs, we believe that better growth techniques have the capability of

improving this aspect owing to the lack of strain in these structures.

Finally, we mention that the general conclusion that the actual exciton size can be signicantly

smaller than the QD size has also been reached for other material systems. By analyzing phonon-

broadened spectra, Rol et al. [127] found that the excitons conned in GaN/AlN QDs are much

smaller than the spatial extent of the QD. Stobbe et al. [106] extracted a small oscillator strength

of large In(Ga)As QDs of about 10 by controllably modifying the LDOS at the position of the

emitter. The latter work points to the same physical situation, namely that the induced material

inhomogeneities during growth create a non-uniform potential prole, which strongly connes

excitons. Engineering large QDs with large excitons and giant oscillator strength represents a

future challenge for the droplet-epitaxy growth technique.

72

Chapter 5

Multipolar Theory of

Spontaneous Emission from

Quantum Dots

The advent of quantum optics and cavity quantum electrodynamics (CQED) over the past cen-

tury has led to beautiful fundamental studies that revealed the quantum nature of light and

matter. The demonstration of the vacuum Rabi oscillations and subsequent entanglement be-

tween the eld and the emitter [10] was an extraordinary breakthrough that cemented our un-

derstanding of the "strange" world of quantum mechanics. Pioneered for Rydberg atoms at

microwave frequencies [79], fundamental CQED studies have been replicated in a vast range of

quantum systems, such as superconducting circuits [89], cavity quantum optomechanics [90] and

semiconductor quantum dots [91]. At the very heart of these studies lies the interaction between

a quantum emitter and the electromagnetic quantum-vacuum eld. The small size L of the

emitters compared to the wavelength of light λ lead to an extremely successful approximation

employed in quantum optics so far the dipole approximation. Its main consequence is that

emitters perceive only the magnitude of the electric eld and have been treated as dimensionless

entities (point dipoles) in practical calculations. The condition for the dipole-approximation

validity, kL 1, where k = 2π/λ is the wavevector of light, may, however, be compromised in

semiconductor quantum dots, which are grown by precise state-of-the-art epitaxial techniques

and attain mesoscopic sizes of 1030 nm [44]. It is straightforward to check that kL ≈ 0.5 for

quantum dots, where typical values for the wavelength in vacuum λ0 = 900 nm, refractive index

n = 3.42 and L = 20 nm have been used. The product kL may be further enhanced in the vicin-

ity of metal nanostructures, where additional propagating modes (surface plasmons) beyond the

light cone arise. The very rst experimental demonstration of the dipole-approximation break-

down has followed recently [26], where the spontaneous-emission process from quantum dots

placed near a metal interface showed pronounced deviations from the dipole theory. Aside from

the inherent mesoscopic size, the eect was found to be enhanced by the asymmetric nature of

73

Chapter 5. Multipolar Theory of Spontaneous Emission from Quantum Dots

the electron and hole wavefunctions. The eects beyond the dipole approximation were merged

into a phenomenologically dened quantum-dot parameter, the mesoscopic moment Λ, which,

combined with the well-understood dipole moment, describes the spontaneous-emission process

from quantum dots.

Despite the fundamentally novel experimental ndings, a comprehensive understanding of

the mesoscopic character of quantum dots, which may break the dipole approximation, has been

lacking so far, but is of crucial importance for the use of quantum dots in practical applications.

One reason for the lack of such an understanding is related to the fact that the microscopic

structure and symmetry of the QD wavefunctions obtained from the eective-mass theory can-

not explain the large mesoscopic moment Λ observed experimentally. Such a microscopic theory

is developed in the present thesis in Chapter 6, where it is shown that these eects are caused by

the inhomogeneous crystal structure of quantum dots. Another reason is connected to the lack of

a rigorous and well-established spontaneous-emission theory beyond the dipole approximation.

A couple of attempts have been realized [83, 158, 159] but they are all incapable of explaining the

experimental ndings from Ref. [26] because they allow contributions beyond the dipole approx-

imation only from the mesoscopic potential of quantum dots and discard their discrete nature.

It is the aim of the present chapter to develop such a theory of spontaneous emission in arbitrary

optical environments, which can be applied to any type of emitter in any eld provided that the

quantum-mechanical wavefunctions are known. We assume the weak-coupling regime between

light and matter but note that the theory can be readily generalized to the strong-coupling

regime, where the new multipolar terms renormalize the light-matter coupling strength. The

theory is applied to describe the spontaneous-emission process from quantum dots in Chapters 6

and 7, where it is shown that the inhomogeneous quantum-current distribution makes quantum

dots a remarkably ecient probe of electric and magnetic elds at optical frequencies.

In the present theory we choose to perform a multipolar expansion in the eld because it dras-

tically simplies practical calculations. We discuss every multipolar term in detail and underline

its physical interpretation. We then connect the multipolar moments to the radiative decay rate,

which is the desired physical observable to be computed and compared with experimental results.

A fundamental property of the multipolar expansion is the dependence of the multipolar terms

on the choice of the coordinate system. This has lead to signicant research eorts to determine

the optimum choice of the center of the coordinate system O [160, 161] because the radiative de-

cay rate was also found to be origin dependent. We nd, however, that by consistently collecting

the expansion orders in the radiative decay rate rather than in the multipolar moments, the rate

is remarkably robust against changes in O. These ndings are of paramount importance for the

physical justication of the multipolar expansion and its application to describe the spontaneous

emission with quantum dots. The central experiment discussed in the present thesis is the decay

dynamics of quantum dots in the vicinity of a dielectric-air or dielectric-metal interface. We

therefore study the properties of the elds in these layered structures towards the end of the

chapter. We nally apply the theory to semiconductor quantum dots and discuss the magnitude

of the multipolar eects. We note that a multipolar theory following analogous principles was

developed recently for plane-wave X-ray absorption by molecules [161].

74

Multipole expansion

5.1 Multipole expansion

In this section we perform a multipole expansion in the eld modes and show that the moments

resulting from the minimal-coupling Hamiltonian are physically equivalent to those resulting

from the more commonly used multipolar Hamiltonian. Let us recall Fermi's Golden Rule for

emission processes

Γ(ω) =π

ε0~∑l

1

ωl

∣∣∣∣〈0| em0f∗l (r) · p |ΨX〉

∣∣∣∣2 δ(ω − ωl). (5.1)

This expression can be written in terms of the imaginary part of the Green tensor, which we

generalize from Eq. (2.58) [83]

Im[←→G (r, r′)

]=πc202

∑l

1

ωlf∗l (r)fl(r

′)δ(ω − ωl), (5.2)

and the decay rate can be recast as

Γ(ω) =2µ0

~

∫ ∫d3rd3r′Im

[j(r) ·

←→G (r, r′) · j∗(r′)

]. (5.3)

Here, we have dened the quantum-mechanical current density j(r), which is an intrinsic property

of the emitter

j(r) =e

m0pΨX(r, r). (5.4)

This is the most general expression of light-matter interaction beyond the dipole approximation.

The integrand is a nonlocal function that intertwines matter and eld degrees of freedom and, in

general, is a six-dimensional integral. In Ref. [83] such an approach is used to develop a theory

of spontaneous emission beyond the dipole approximation from quantum dots, which, however,

is incomplete because only the mesoscopic quantum-dot potential is considered while potential

inhomogeneities at the crystal-lattice level are neglected. These approximations turn out to be

crude and fail to explain the surprising experimental ndings from Ref. [26]. Here we address this

issue and consider the structure and symmetry of the entire quantum-mechanical wavefunction.

We adopt a dierent route and perform a multipolar expansion in the eld modes f(r) because

the integral formulation oers limited physical insight and is often computationally infeasible.

The essential physics is normally captured by the rst few multipoles leading to a clear and

intuitive physical interpretation. First we rewrite Eq. (5.1) as

Γ(ω) =π

ε0~∑l

1

ωl|T0X |2 δ(ω − ωl), (5.5)

where the transition moment T0X is dened as

T0X =e

m0〈0| f∗l (r) · p |ΨX〉 . (5.6)

The starting point is the expansion of the normal mode fl in a Taylor series around a conveniently

chosen point r0. For simplicity, we discard the index l for the moment, and we assume r0 = 0

75


without loss of generality.∗ As such,

fi(r) = fi(0) + xj × ∂jfi(0) +1

2xjxk × ∂k∂jfi(0) + . . . , (5.7)

where the notation of implicit summation over repeated indices is used and ∂i ≡ ∂∂i. This is

substituted in Eq. (5.6) and the dierent orders are collected accordingly

T0X = T(0)0X + T

(1)0X + T

(2)0X + . . . (5.8)

It is important to underline that the dierent orders in T0X are not equivalent to the same orders

in Γ, as can be readily seen via

Γ = Γ(0) +Γ(1) +Γ(2) + . . . ∝(T

(0)0X + T

(1)0X + T

(2)0X + . . .

)×(T

(0),∗0X + T

(1),∗0X + T

(2),∗0X + . . .

). (5.9)

Here, we expand Γ up to the second order because the rst order vanishes in parity-symmetric

environments such as a homogeneous medium.

5.1.1 Zeroth order: electric-dipole moment

The electric-dipole term neglects the variation of the electromagnetic eld over the spatial extent

of the emitter

T(µ)0X =

e

m0〈0| f∗i (0)pi |ΨX〉 = f∗i (0) 〈0| µ(p)

i |ΨX〉 = f∗i (0)µi, (5.10)

where µi = 〈0| µ(p)i |ΨX〉 and µ(p)

i is the electric-dipole operator in the velocity representation

µ(p)i =

e

m0pi. (5.11)

By using the identities given in Appendix B, the matrix element of the electric-dipole moment

operator can be related to that in the more familiar length representation

〈0| pi |ΨX〉 = −iE0X

~〈0| µi |ΨX〉 , (5.12)

where we have introduced the electric-dipole moment in the length representation

µi = eri. (5.13)

The zeroth-order contribution therefore yields

T(0)0X = T

(µ)0X = f∗i (0) 〈0| µ(p)

i |ΨX〉 = −iE0X

~f∗i (0) 〈0| µi |ΨX〉 . (5.14)

∗A non-zero r0 can be straightforwardly included via xi → xi − x0,i.

76

Multipole expansion

5.1.2 First order: electric-quadrupole and magnetic-dipole moments

The rst-order contribution reads

T(1)0X =

e

m0∂jf∗i (0) 〈0|xj pi |ΨX〉

= ∂jf∗i (0)Λji,

(5.15)

where Λij = (e/m0) 〈0|xipj |ΨX〉 is the rst-order mesoscopic moment. T(1)0X can be written as a

sum of electric-quadrupole and magnetic-dipole contributions in the following fashion

T(1)0X =

e

2m0∂jf∗i (0) (〈0|xj pi + xipj |ΨX〉+ 〈0|xj pi − xipj |ΨX〉)

[xi,pj ]=i~δij=

e

2m0∂jf∗i (0) (〈0|xj pi + pjxi + i~δij |ΨX〉+ 〈0|xj pi − xipj |ΨX〉)

∇·f=0=

e

2m0∂jf∗i (0) (〈0|xj pi + pjxi |ΨX〉+ 〈0|xj pi − xipj |ΨX〉) .

(5.16)

The rst (second) term from the RHS denotes the matrix element of the electric-quadrupole

(magnetic-dipole) moment operator. First we address the electric-quadrupole contribution T(Q)0X

T(Q)0X =

1

2∂jf∗i (0) 〈0| Q(p)

ij |ΨX〉 , (5.17)

where Q(p)ij is the electric-quadrupole moment operator in the velocity representation

Q(p)ij =

e

m0(xipj + pixj) , (5.18)

which can be transformed to length representation with the help of the relations presented in

Appendix B

Qij = exixj , (5.19)

so that

T(Q)0X = − i

2

E0X

~∂jf∗i (0) 〈0| Qij |ΨX〉 . (5.20)

The magnetic-dipole contribution T(m)0X can be converted to the well-known canonical form

via

T(m)0X =

e

2m0∂jf∗i (0) 〈0|xj pi − xipj |ΨX〉

=e

2m0∂jf∗i (0) 〈0|xmpn(δmjδni − δmiδnj) |ΨX〉

=e

2m0∂jf∗i (0) 〈0|xmpnεljiεlmn |ΨX〉

=e

2m0∂jf∗i (0)εljiel 〈0| εkmnxmpnek |ΨX〉

=e

2m0∂jf∗i (0)εjilel 〈0| εmnkxmpnek |ΨX〉

=e

2m0[∇× f∗(0)] · 〈0| r× p |ΨX〉 ,

(5.21)

77


where εijk is the Levi-Civita tensor. The magnetic-dipole transition moment therefore takes the

form

T(m)0X = [∇× f∗(0)] · 〈0| m |ΨX〉 , (5.22)

where

m =e

2m0r× p (5.23)

is the magnetic-dipole operator.

Altogether, the rst-order transition moments consist of electric-quadrupole and magnetic-

dipole contributions

T(1)0X = T

(Q)0X + T

(m)0X . (5.24)

5.1.3 Second-order: electric-octupole and magnetic-quadrupole moments

The second order correction to the transition moment reads

T(2)0X =

e

2m0∂j∂kf

∗i (0) 〈0|xkxj pi |ΨX〉

= ∂j∂kf∗i (0)Ωkji

(5.25)

where Ωijk = (e/2m0) 〈0|xixj pk |ΨX〉 is the second-order mesoscopic moment. T(2)0X can be

rewritten in terms of electric-octupole and magnetic-quadrupole contributions

T(2)0X =

e

6m0∂j∂kf

∗i (0) (〈0|xkxj pi + xkpjxi + pkxjxi |ΨX〉

+ 〈0| 2xkxj pi − xkpjxi − pkxjxi |ΨX〉 ) .(5.26)

The rst (second) term from the RHS denotes the matrix element of the electric-octupole

(magnetic-quadrupole) operators. Explicitly, the electric-octupole transition moment reads

T(O)0X =

1

6∂j∂kf

∗i (0) 〈0| O(p)

ijk |ΨX〉 , (5.27)

where the electric-octupole operator in the velocity representation has been dened

O(p)ijk =

e

m0(xkxj pi + xkpjxi + pkxjxi) (5.28)

and can be converted to the length representation with the help of the identities presented in

Appendix B

Oijk = exixjxk. (5.29)

The corresponding electric-octupole transition moment takes the form

T(O)0X = − i

6

E0X

~∂j∂kf

∗i (0) 〈0| Oijk |ΨX〉 . (5.30)

78

Multipole expansion

Table 5.1: Overview of the dierent contributions to the multipole expansion of T0X up to second

order.

Order Overall Electric Magnetic

0 T(0)0X = µif

∗i (0) T

(µ)0X =

⟨µ(p)i

⟩f∗i (0)

1 T(1)0X = Λji∂jf

∗i (0) T

(Q)0X = 1

2

⟨Q

(p)ij

⟩∂jf∗i (0) T

(m)0X = 〈m〉 · [∇× f∗(0)]

2 T(2)0X = Ωkji∂j∂kf

∗i (0) T

(O)0X = 1

6

⟨O

(p)ijk

⟩∂j∂kf

∗i (0) T

(M)0X = 1

2

⟨Mij

⟩∂j [∇× f∗(0)]i

The magnetic-quadrupole transition moment T(M)0X can be brought to a canonical form as follows

T(M)0X =

e

6m0∂j∂kf

∗i (0) 〈0| 2xkxj pi − xkpjxi − pkxjxi |ΨX〉

=e

6m0∂j∂kf

∗i (0) [〈0|xkxj pi − xkpjxi |ΨX〉 − 〈0|xkxj pi − pkxjxi |ΨX〉]

∇·f=0=

e

6m0∂j∂kf

∗i (0) [〈0|xk (xj pi − pjxi) |ΨX〉 − 〈0| (xkpi − pkxi)xj |ΨX〉]

5.21=

e

6m0∂j [∇× f∗(0)]i 〈0|xj (r× p)i + (r× p)i xj |ΨX〉 ,

(5.31)

whereupon

T(M)0X =

1

2∂j [∇× f∗(0)]i 〈0| Mij |ΨX〉 , (5.32)

where

Mij =e

3m0[xj (r× p)i + (r× p)i xj ] (5.33)

is the magnetic-quadrupole operator.

Altogether, the second-order correction to the transition moment consists of electric-octupole

and magnetic-quadrupole contributions

T(2)0X = T

(O)0X + T

(M)0X . (5.34)

5.1.4 Summary of the multipole transition moments

The multipole expansion of the transition moment up to the second order results in ve dierent

contributions

T0X = T(0)0X + T

(1)0X + T

(2)0X + ...

= T(µ)0X + T

(Q)0X + T

(m)0X + T

(O)0X + T

(M)0X + ...,

(5.35)

which are summarized in Table 5.1 and sketched in Fig. 5.1. The zeroth order has only electric-

dipole contributions, while higher orders include terms of both electric and magnetic nature.

79


=+_ +

+_

_+ + + ...

Figure 5.1: Physical interpretation of the multipole expansion. The interaction between a poten-

tially complex current density j(r) and the eld E(r) is decomposed into a linear superposition

of multipoles and is weighted by the magnitude of the multipoles, which are intrinsic properties

of the current density j(r).

5.2 Origin dependence of the multipole transition moments

In this section we give explicit proof of the dependence of the multipole transition moments on

the choice of the origin of the coordinate system O. It turns out that only the dipole transition

moment is independent of O while all the higher transition moments are origin-dependent.

It is obvious that the dipole transition moment, µi = (e/m0) 〈0| pi |ΨX〉, is origin-independent.In the following, we show the origin-dependence for the rst- and second-order mesoscopic mo-

ments. Upon a shift of O to O + a, we obtain for←→Λ

Λij(O + a) =e

m0〈0| (xi − ai)pj |ΨX〉 =

e

m0[〈0|xipj |ΨX〉 − ai 〈0| pj |ΨX〉] = Λij(O)− aiµj ,

(5.36)

while the second-order mesoscopic moment transforms as

Ωijk(O + a) =e

2m0〈0| (xi − ai)(xj − aj)pk |ΨX〉

=e

2m0[〈0|xixj pk |ΨX〉 − ai 〈0|xj pk |ΨX〉 − aj 〈0|xipk |ΨX〉+ aiaj 〈0| pk |ΨX〉]

= Ωijk(O)− aiΛjk − ajΛik + aiajµk.

(5.37)

In the case of a purely mesoscopic emitter, i.e., if µ = 0, then←→Λ is origin-independent. Analogous

reasoning applies to Ω, if µ = 0 and←→Λ =

←→0 .

The origin dependence of the mesoscopic transition moments is a fundamental property of

the multipole expansion. Only the dipole transition moment is independent of the choice of

the origin of the coordinate system. All the higher-order moments are origin-dependent and,

therefore, their physical meaning should be treated with care. In particular, they should not be

regarded as intrinsic property of the emitter as long as the origin of the coordinate system O is

not rigorously dened.

80

Radiative decay rate

5.3 Radiative decay rate

In this section we derive the main quantity of interest, namely the radiative decay rate of an

emitter in an arbitrary optical environment. According to Eq. (5.38), the connection between

the various transition moments and the radiative decay rate is

Γ(ω) =π

ε0~∑l

1

ωl|T0X |2l δ(ω − ωl)

=π

ε0~∑l

1

ωl

(T

(0)0X + T

(1)0X + T

(2)0X + ...

)l

(T

(0)0X + T

(1)0X + T

(2)0X + ...

)∗lδ(ω − ωl).

(5.38)

Here, we expand Γ up to the second order because the rst order alone vanishes in many photonic

congurations

Γ(ω) ≈ Γ(0)(ω) + Γ(1)(ω) + Γ(2)(ω). (5.39)

The zeroth order in Γ is the contribution from the dipole nature of the emitter,

Γ(0)(ω) =π

ε0~∑l

1

ωl

∣∣∣T (0)0X

∣∣∣2lδ(ω − ωl)

5.2=

2µ0

~ImµiGij(0,0)µ∗j .

(5.40)

Within the dipole approximation, Γ(ω) = Γ(0)(ω) and can be calculated by multiplying the

squared absolute value of the dipole moment of the emitter with the projected Green tensor at

the position r0 of the dipole. The dissipation rate Γ(0) is a self-interference eect, where the

dipole moment probes the environment and interferes back with itself.

The rst-order contribution to Γ reads

Γ(1)(ω) =π

ε0~∑l

1

ωl2Re

[T

(1)0XT

(0),∗0X

]lδ(ω − ωl)

=2µ0e

2

~m20

2Re [〈0|xkpi |ΨX〉〈ΨX| pj |0〉] ∂kImGij(r,0)|r=0

=2µ0

~2Re

[Λkiµ

∗j

]∂kImGij(r,0)|r=0 ,

(5.41)

and is proportional to the gradient of the imaginary part of the Green tensor at the position of

the emitter. In homogeneous media Γ(1) ≡ 0 because ∂kImGHij (r,0)

∣∣r=0≡ 0, where

←→G H is

the homogeneous part of the Green tensor, as derived in Sec. 5.4. The rst-order contribution is

non-zero only in nanophotonic environments that violate parity symmetry.

Γ(1) can be interpreted as an interference process between the dipolar µi = 〈0| pi |ΨX〉 andmultipolar Λij = 〈0|xipj |ΨX〉 nature of the emitter. Importantly, Γ(1) 6= 0 only if both the QD

wavefunctions and the electromagnetic environment violate parity symmetry [162], see Chapter 7

for more details. Then, the second-order contribution Γ(2) is the next correction to the dipole

approximation and is more dicult to full because the criterion for the breakdown of the DA

becomes k2L2 6 1. The dipole approximation is therefore protected by parity symmetry to rst

order.

81


= + + + ...+

Figure 5.2: Physical interpretation of the spontaneous-emission rate when decomposed into

the constituent orders. The nonlocal interaction between the points r and r′ within the cur-

rent density of the emitter is converted into a local interaction between the dierent multipoles

characterizing the emitter.

Semiconductor QDs have built-in asymmetries and←→Λ is generally non-zero, as is extensively

investigated in Sec. 5.6 and Chapter 6. The rst-order contribution, Γ(1), can take both positive

and negative values depending on the orientation of the emitter with respect to the nanophotonic

environment. This makes mesoscopic QDs an ideal platform for both enhancing and suppressing

light-matter interaction at the nanoscale, as is demonstrated experimentally in Ref. [26].

The second-order contribution to the radiative decay rate is given by

Γ(2)(ω) =π

ε0~∑l

1

ωl

Re[T

(2)0XT

(0),∗0X

]+ T

(1)0XT

(1),∗0X

lδ(ω − ωl)

=2µ0

~Im[

Re[Ωlkiµ

∗j

]∂k∂l + ΛkiΛ

∗lj∂k∂

′l

]Gij(r, r

′)|r=r′=0

,

(5.42)

and is proportional to the second-order derivative of the imaginary part of the Green tensor.

The rst term from the right-hand side is a result of an interference between µ and←→Ω and is

generally non-zero even for high-symmetry emitters such as atoms because some of the involved

operators have the same parity. In contrast, the second term from the right-hand side vanishes

for parity-symmetric emitters owing to the orthogonality of the underlying µ- and←→Λ -operators.

The resulting contributions are sketched in Fig. 5.2.

Finally, we introduce the normalized decay rate ΓN of an emitter as the ratio between the

decay rate in the given nanophotonic geometry and the decay rate in a homogeneous medium

ΓH

ΓN =Γ(0) + Γ(1) + Γ(2)

Γ(0)H + Γ

(2)H

. (5.43)

The normalized decay rate is often the preferred computable quantity because, unlike Γ, is

normalized to the value of the dipole moment, whose magnitude may be hard to evaluate self-

consistently. The Green tensor can be decomposed into a homogeneous part, which emulates the

behavior in a homogeneous medium, and a scattering part, which quanties only the scattered

electromagnetic waves by the surrounding inhomogeneous medium, as explained in Sec. 2.4. The

decay rate Γ(ω) can therefore be written as a sum of a homogeneous part ΓH(ω) and a scattering

82

Green's Tensor and derivatives in the vicinity of an Interface

Figure 5.3: Conguration of the problem. The emitter, which is running a quantum-mechanical

current density j(r), is located in the medium characterized by the refractive index n1 at a

distance z0 to the interface with a medium characterized by n2.

part ΓS(ω), and the normalized decay rate takes the form

ΓN = 1 +Γ

(0)S + Γ

(1)S + Γ

(2)S

Γ(0)H + Γ

(2)H

. (5.44)

In conclusion, we emphasize that this theoretical framework can be used to study the spontaneous-

emission process in any kind of emitters (atoms or molecules in X-ray spectroscopy, QDs in

nanophotonic environments, etc.), provided that their relevant wavefunctions are known. A

number of entries in←→G and its derivatives often vanish, as do certain entries in the mesoscopic

moments for symmetry reasons. The light-matter interaction beyond the dipole approximation

can therefore be easily and accurately quantied using a small number of parameters.

5.4 Green's Tensor and derivatives in the vicinity of an In-

terface

The three contributions to the decay rate, Γ ≈ Γ(0) + Γ(1) + Γ(2), are governed by products

between an emitter property (e.g., electric and magnetic dipoles or quadrupoles) and a eld

property, which is given in terms of the Green tensor and its derivatives. The emitter multipoles

are solely a property of the underlying quantum-mechanical wavefunctions; the properties of the

QD multipoles are presented in Sec. 5.6. On the other hand, the eld properties are determined

by the spatial distribution of the dielectric permittivity ε(r). Since the present thesis is largely

concerned with describing the spontaneous-emission process of QDs in the vicinity of an interface,

see Chapters 6 and 7, in this section we derive the Green tensor and its derivatives for such a

nanophotonic geometry, where we assume that the emitter is located at a distance z = z0 above

the interface, see Fig. 5.3.

As presented in Sec. 2.4, the Green tensor←→G can be written as a sum of a homogeneous term←→

GH , which is the solution to the wave equation in a homogeneous medium hosting the emitter

and characterized by n1 =√ε1µ1, and a scattering term

←→GS , which accounts for the scattered

83


waves by the inhomogeneous photonic structure, i.e.,

←→G =

←→GH +

←→GS . (5.45)

We rst present the contribution of the homogeneous part before returning to the scattered

contribution from the plane interface.

5.4.1 Homogeneous part of the Green tensor

In a homogeneous medium characterized by ε1 and µ1, the dyadic Green function←→GH takes the

form [72, p. 30]

←→GH(r, r′) =

[←→1 +

1

k21

∇∇]

e(ik1R)

4πR

=

[←→1 +

ik1R− 1

k21R

2

←→1 +

3− 3ik1R− k21R

2

k21R

4RR

]e(ik1R)

4πR,

(5.46)

where R = |R| ≡ |r− r′| is the relative distance between r and r′, and k21 = ω2ε1µ1. In general,

←→GH diverges when evaluated at r = r′ = r0 but the imaginary part is bounded and well-behaved,

cf. Ref. [72, p. 239]. By taking the limit of Im←→GHas R→ 0, one obtains

Im←→GH(r0, r0) =

k1

6π

←→1 . (5.47)

The Cartesian derivatives ∂x, ∂y and ∂z of←→GH are taken by writing them in a spherical coordinate

system

∂x = sin(θ) cos(φ)∂R +1

Rcos(θ) cos(φ)∂θ +

1

R

sin(φ)

sin(θ)∂φ,

∂y = sin(θ) sin(φ)∂R +1

Rcos(θ) sin(φ)∂θ +

1

R

cos(φ)

sin(θ)∂φ,

∂z = cos(θ)∂R −1

Rsin(θ)∂θ.

(5.48)

It is clear that, rst, only the operator ∂R yields a non-zero contribution, and second, that

∂x = ∂y = ∂z due to the spherical symmetry of←→GH . Moreover, it is apparent from the denition

of←→GH that

∂i = −∂′i, (5.49)

where i = x, y, z. Inserting Eq. (5.48) into Eq. (5.46) and taking the limit R→ 0 yields

∂i Im←→GH(r, r′)

∣∣∣∣r0

=←→0 . (5.50)

84


The second-order derivatives are found in a similar fashion

∂x∂x Im←→GH(r, r′)

∣∣∣∣r0

= − k31

30π

1 0 0

0 2 0

0 0 2

,

∂x∂y Im←→GH(r, r′)

∣∣∣∣r0

= +k3

1

60π

0 1 0

1 0 0

0 0 0

,

(5.51)

and all the remaining derivatives can be recast by taking advantage of the spherical symmetry

of the geometry. The compact notation of the second-order derivatives reads

∂i∂jImGHmn(r0, r0) = − [2δijδmn(2− δim) + (1− δij)(δimδjn + δinδjm)]

k31

60π. (5.52)

Derivatives with respect to r′ can be written with the help of Eq. (5.49).

To conclude, we discuss the implications of the results. The rst-order derivatives of←→GH

vanish implying that Γ(1) = 0 in homogeneous media, and the rst non-zero term beyond the

dipole approximation is Γ(2), which is proportional to the second-order derivatives of←→GH . As a

consequence, it is more dicult to break the dipole approximation in homogeneous media, since

the condition of the dipole-approximation breakdown is k21L

2 6 1 instead of the usual k1L 6 1.

This property pertains to any parity-symmetric environment as is shown in Chapter 7.

5.4.2 Scattering part of the Green tensor

Let us assume the geometry presented in Fig. 5.3, where the emitter is placed at a distance z0

above the interface. The scattering part of the Green tensor←→GS in the upper half-space equals the

reected Green tensor from the interface. Since the reection coecient depends on the incident

angle of the wave, it is convenient to perform a two-dimensional in-plane Fourier transform

of the dyadic Green function (also called the angular-spectrum representation) and apply the

Fresnel reection coecient to every Fourier component individually. The reection coecient

is polarization dependent and it is convenient to split←→GS into a sum of s- and p-polarized

components. For compact notation, we consider the normalized Green tensor←→GS =

←→GS/k1 with

normalized wavevectors kx =√

1− k2y − k2

z . The scattered part of the Green tensor can then be

written as [72]

←→GS(r, r′) =

i

8π2

∫∫ ∞−∞

dkxdky

[←→M s

ref +←→M p

ref

]ei[kx(x−x′)+ky(y−y′)+kz1 (z+z′)],

←→M s

ref =rs(kx, ky)

kz1(k2x + k2

y)

k2y −kxky 0

−kxky k2x 0

0 0 0

,

←→M p

ref =−rp(kx, ky)

(k2x + k2

y)

k2xkz1 kxkykz1 kx(k2

x + k2y)

kxkykz1 k2ykz1 ky(k2

x + k2y)

−kx(k2x + k2

y) −ky(k2x + k2

y) −(k2x + k2

y)2/kz1

,

(5.53)

85


where the Fresnel reection coecients are dened below. It is convenient to switch to a cylin-

drical coordinate system in the xy and kxky planes

r = (ρ cosφ, ρ sinφ, z − z′) = (x− x′, y − y′, z − z′),

k = (kρ cos kφ, kρ sin kφ, kz) = (kx, ky, kz),(5.54)

whereupon←→GS can be recast as

←→GS(r, r′) =

i

8π2

∫ ∞0

dkρkρei[kz1 (z+z′)]

∫ 2π

0

dkφ

[←→M s

ref +←→M p

ref

]eikρρ[cos kφ cosφ+sin kφ sinφ],

←→M s

ref =rs(kρ)

kz1

sin2 kφ − sin kφ cos kφ 0

− sin kφ cos kφ sin2 kφ 0

0 0 0

,

←→M p

ref =−rp(kρ)

k21

kz1 cos2 kφ kz1 sin kφ cos kφ kρ cos kφ

kz1 sin kφ cos kφ kz1 sin2 kφ kρ sin kφ

−kρ cos kφ −kρ sin kφ −k2ρ/kz1

,

(5.55)

where kz1 is xed by the dispersion relation kz1(kρ) =√

1− k2ρ. The Fresnel coecients read

rs(kρ) =µ2kz1 − µ1kz2µ2kz1 + µ1kz2

,

rp(kρ) =ε2kz1 − ε1kz2ε2kz1 + ε1kz2

.

(5.56)

The integral over kφ can be evaluated analytically with the help of

Jn(kρρ) =i−n

2π

∫ 2π

0

dkφ cos(nkφ)eikρρ cos kφ , (5.57)

where Jn is the Bessel function of the rst kind and n-th order. The resulting expression reads

←→GS(r, r

′) =

i

8π

∫ ∞0

dkρkρ[←→M s

ref +←→M p

ref

]ei[kz1 (z+z′)],

←→M sref =

rs(kρ)

k2z1

J0(kρρ) + J2(kρρ) cos 2φ J2(kρρ) sin 2φ 0

J2(kρρ) sin 2φ J0(kρρ)− J2(kρρ) cos 2φ 0

0 0 0

,

←→M pref = −rp(kρ)kz1

J0(kρρ)− J2(kρρ) cos 2φ −J2(kρρ) sin 2φ −2i

kρkz1

J1(kρρ) cosφ

−J2(kρρ) sin 2φ J0(kρρ) + J2(kρρ) cos 2φ −2ikρkz1

J1(kρρ) sinφ

2ikρkz1

J1(kρρ) cosφ 2ikρkz1

J1(kρρ) sinφ 2(kρkz1

)2J0(kρρ)

.

(5.58)

This is the nal expression for the scattering part of the Green tensor for an emitter-interface

problem. Although it does not have analytic solutions, it is written in closed form involving a

one-dimensional integral, which has to be evaluated numerically.

In the following we evaluate←→GS and its derivatives at the origin r = r′ = r0. We consider

the integrand←→GSkρ dened as

←→GS(r, r′) =

i

8π

∫ ∞0

dkρkρ←→GSkρ(ρ, φ)e2ikz1z0 , (5.59)

86


Distance to interface, z0(nm)

Nor

mal

ized

Gre

en`s

func

tion

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3GaAs-Air

GaAs-Ag


Nor

mal

ized

Gre

en`s

func

tion

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3GaAs-Air

GaAs-Ag

Figure 5.4: Imaginary part of the Green tensor evaluated at the origin and normalized to the

homogeneous contribution for an emitter (a) parallel and (b) perpendicular to the interface. Two

cases of interest are taken: GaAs-air and GaAs-silver interfaces.

and take the limit ρ→ 0 yielding for the integrand

←→GSkρ(r0, r0) =

1

kz1

rs − kz1rp 0 0

0 rs − kz1rp 0

0 2k2ρr

p

. (5.60)

The Green tensor at (r0, r0) has only diagonal components and, due to the in-plane cylindrical

symmetry, only two of them have a distinct functional dependence Gxx = Gyy and Gzz, cor-

responding to a dipole moment parallel and perpendicular to the interface, respectively. The

resulting normalized imaginary part obtained after evaluating Eq. (5.59) numerically and adding

the homogeneous part is depicted in Fig. 5.4 for two cases: GaAs-air and GaAs-silver interfaces.

The plotted quantity is equivalent to the normalized decay rate within the dipole approxima-

tion. The following parameters are used: vacuum wavelength λ0 = 1000 nm, n1 = 3.42 and

nAg = 0.2 + 7i.

The two dependencies are out of phase because the reected eld acquires an additional

π phase shift upon reection from the metal. The LDOS, which is proportional to the Green

function at the origin, is considerably enhanced as the emitter approaches the metal because new

near-eld decay channels arise, namely surface-plasmons polaritons (SPPs) at distances below

∼100 nm, and ohmic-lossy modes inside the metal below ∼20 nm. Coupling to these near-eld

excitations occurs for in-plane k-vectors larger than the wavevector of light k1 in GaAs. It

is straightforward to separate the dierent decay channels in the angular-spectrum formalism.

The contribution from propagating photons is found by integrating Eq. (5.59) from 0 to 1, the

surface plasmons are found between 1 and 2kSPP−1, and ohmic-lossy from 2kSPP−1 to innity,

where kSPP =√ε2/(ε1 + ε2) is the propagation wavevector for surface-plasmon polaritons. The

resulting contributions are plotted in Fig. 5.5. A perpendicular dipole couples much stronger to

surface plasmons than an in-plane dipole owing to the well-known polarization and dispersion

of surface plasmons [163, p. 25]. The weak coupling to surface plasmons by a parallel dipole

87



Nor

mal

ized

Con

trib

utio

n

0 100 200 300 400 5000

1

2

3PhotonsSPPsOhmic Losses


Nor

mal

ized

Con

trib

utio

n

0 100 200 300 400 5000

1

2

3PhotonsSPPsOhmic Losses

Figure 5.5: Decomposition of the normalized imaginary part of the Green tensor for an emitter

(a) parallel and (b) perpendicular to a metal interface.

represents an intrinsic limitation for the use of semiconductor QDs in plasmon-based devices.

The mesoscopic character of QDs can, however, couple to the strong perpendicular eld and

substantially enhance or suppress the coupling to surface plasmons depending on the orientation

and symmetry of the exciton wavefunction [26, 162].

The derivatives of←→GS at (r0, r0) are evaluated by dierentiating Eq. (5.58) and subsequently

taking the limit ρ→ 0. In cylindrical coordinates, the Cartesian derivatives read

∂x = cosφ∂ρ −1

ρsinφ∂φ,

∂y = sinφ∂ρ +1

ρcosφ∂φ,

∂z = ∂z,

(5.61)

thereby yielding for the rst-order derivatives

∂x←→GSkρ(r, r

′)

∣∣∣∣r0

= k2ρr

p

0 0 1

0 0 0

−1 0 0

,

∂y←→GSkρ(r, r

′)

∣∣∣∣r0

= k2ρr

p

0 0 0

0 0 1

0 −1 0

,

∂z←→GSkρ(r, r

′)

∣∣∣∣r0

=

k2z1r

p − rs 0 0

0 k2z1r

p − rs 0

0 0 −2k2ρr

p

.

(5.62)

The rst-order derivatives of←→GSkρ are generally non-zero and rst-order processes may be allowed,

in contrast to a homogeneous medium.

As for←→GS(r0, r0), the entries of the rst-order derivatives of

←→GS inherit the two functional

dependences of a parallel dipole (compare, e.g., ∂z←→GSxx and

←→GSxx) and a perpendicular dipole

88

Origin (in)dependence of the radiative decay rate

(compare, e.g., ∂z←→GSzz and

←→GSzz). Interestingly, the x- and y-derivatives are generally non-zero

and may result in a signicant contribution for QDs, since they couple to the in-plane size of

QDs (∼2030 nm), which is normally much larger than the height (∼25 nm). These in-plane

derivatives inherit the properties of the strong z-polarized plasmon eld.

The second-order derivatives←→GS at (r0, r0) are evaluated by dierentiating Eq. (5.58) twice

and taking the limit ρ→ 0

∂x∂x←→GSkρ(r, r

′)

∣∣∣∣r0

=k2ρ

4kz1

3k2z1r

p − rs 0 0

0 k2z1r

p − 3rs 0

0 0 −4k2ρr

p

,

∂y∂y←→GSkρ(r, r

′)

∣∣∣∣r0

=k2ρ

4kz1

k2z1r

p − 3rs 0 0

0 3k2z1r

p − rs 0

0 0 −4k2ρr

p

,

∂z∂z←→GSkρ(r, r

′)

∣∣∣∣r0

= kz1

k2z1r

p − rs 0 0

0 k2z1r

p − rs 0

0 0 −2k2ρr

p

,

∂x∂y←→GSkρ(r, r

′)

∣∣∣∣r0

=k2ρ

4kz1

(k2z1r

p + rs)0 1 0

1 0 0

0 0 0

,

∂x∂z←→GSkρ(r, r

′)

∣∣∣∣r0

= k2ρkz1r

p

0 0 1

0 0 0

−1 0 0

,

∂y∂z←→GSkρ(r, r

′)

∣∣∣∣r0

= k2ρkz1r

p

0 0 0

0 0 1

0 −1 0

.

(5.63)

Finally, the derivatives with respect to r′ can be calculated with the help of

∂′x = −∂x,

∂′y = −∂y,

∂′z = ∂z,

(5.64)

as can be noted from Eq. (5.58).

In conclusion, we note that all the above one-dimensional integrals have a pole at kρ = k1 and

are easily evaluated on a computer. Combined with the analytic solutions to the homogeneous

part of←→G , it allows an easy and straightforward calculation of the radiative decay rate beyond

the dipole approximation.

5.5 Origin (in)dependence of the radiative decay rate

In Sec. 5.2 it is shown that the multipole transition moments change upon a shift of the origin

of the coordinate system from O to O + a. The relevant quantity of interest, the radiative

89


decay rate, is, however, robust to such a shift, if the orders in the decay rate (rather than in the

multipolar moments) are collected consistently. We rst address the spontaneous decay of an

emitter in a homogeneous medium before discussing a general nanophotonic environment.

5.5.1 Spontaneous decay in a homogeneous medium

In a homogeneous medium, the Green tensor is spatially invariant and does not change upon a

shift of the coordinate system. The zeroth-order contribution, Γ(0), is therefore origin-independent

because it contains only the electric-dipole contribution. The rst-order correction, Γ(1), trans-

forms as

Γ(1)H (O + a)− Γ

(1)H (O) = −4µ0

~akRe

[µiµ∗j

]∂kImGij(r,0)|r=0 .

The rst-order derivatives of the homogeneous part of the Green tensor vanish as proved in

Sec. 5.4 leading to

Γ(1)H (O + a)− Γ

(1)H (O) ≡ 0. (5.65)

The second-order contribution

Γ(2)H (ω) =

2µ0

~Im[

Re(Ωlkiµ

∗j

)∂k∂l + ΛkiΛ

∗lj∂k∂

′l

]Gij(r, r

′)|r=r′=0

changes when O shifts to O + a as

Γ(2)H (O + a)− Γ

(2)H (O) = Im

[−akRe

(Λliµ

∗j

)− alRe

(Λkiµ

∗j

)+ alakRe

(µiµ∗j

)]∂k∂lGij(r,0)|r=0

+ Im

[−akΛ∗ljµi − alΛkiµ

∗j + alakµiµ

∗j

]∂k∂

′lGij(r, r

′)∣∣r=r′=0

.

By noting that Gij(r, r′) = Gji(r

′, r) [74], we rearrange the above equation to obtain

Γ(2)H (O + a)− Γ

(2)H (O) = Im

[−2akRe

(Λliµ

∗j

)+ alakRe

(µiµ∗j

)]∂k∂lGij(r,0)|r=0

+ Im

[−2akRe (Λ∗liµj) + alakµiµ

∗j

]∂k∂

′lGij(r, r

′)|r=r′=0

.

(5.66)

This expression vanishes for the homogeneous part of←→G , as can be checked with the help of

Eqs. (5.49) and (5.51)

Γ(2)H (O + a)− Γ

(2)H (O) ≡ 0. (5.67)

It can be shown that the property of origin independence holds up to an arbitrary expansion

order in Γ [161]. The radiative decay rate is therefore independent of the choice of the origin of

the coordinate system in a homogeneous medium

ΓH(O + a)− ΓH(O) ≡ 0. (5.68)

This important result lies at the heart of practical calculations employing the multipole expan-

sion.

90

Decay dynamics of In(Ga)As quantum dots in the vicinity of an interface

5.5.2 Spontaneous decay in an arbitrary environment

The property of origin independence cannot be generalized to an arbitrary environment because,

in general, the Green tensor is spatially dependent. This implies that the LDOS is also spatially

dependent and all the orders (including the dipole contribution) are dependent on the origin of

the coordinate system. The expansion point O then has to be dened such that the multipole

expansion converges fastest. It can be shown that O coincides with the largest transition den-

sity [160], which corresponds to the excitonic density for QDs. The center-of-mass coordinate of

the exciton contains the highest excitonic density and we therefore obtain for O

O =mere +mhrhme +mh

, (5.69)

where me and mh (re and rh) are the eective-masses (central coordinates) of electrons and

holes, respectively.

In conclusion, we have been able to successfully address the fundamental problem of the ori-

gin dependence of the multipole expansion in two steps. First, collecting consistently the orders

in Γ rather than in the multipole moments results in origin independence for the homogeneous-

medium contribution in the spontaneous decay. In the second step we have rigorously dened

a natural choice for the origin of the coordinate system O, which is relevant in inhomogeneous

nanophotonic environments. We have therefore justied formally the use of the multipole ex-

pansion for modelling accurately and eciently the spontaneous-emission process.

5.6 Decay dynamics of In(Ga)As quantum dots in the vicin-

ity of an interface

Quantum dots are mesoscopic entities extended over tens of nanometers and they may break

the dipole approximation as argued in the introduction of the present chapter. In this sec-

tion we employ the developed multipolar theory to provide a profound understanding of the

spontaneous-emission process from QDs. In particular, we are interested in determining the

multipolar moments that may yield a signicant contribution to the light-matter interaction and

compete with the electric-dipole contribution. The rst-order mesoscopic moment←→Λ contains

9 entries and the second-order mesoscopic moment←→Ω has 27 entries. Many of them, however,

vanish or are negligible for symmetry reasons, and only a few capture the essential physics lead-

ing to a simple and intuitive interpretation of the light-matter interaction. For concreteness we

consider the spontaneous decay of a QD placed at a distance z = z0 above a silver interface.

Such a conguration was realized experimentally in Ref. [26], where QDs showed pronounced

deviations from the dipole theory. In Chapter 6 we present experimental data demonstrating

the breakdown of the dipole approximation for QDs placed in the vicinity of a dielectric-air

interface [164].

We assume the QDs to be lens shaped with in-plane cylindrical symmetry but with no well-

dened parity symmetry in the growth direction, in good agreement with the shape of self-

assembled In(Ga)As QDs [44]. We note that this analysis is not bound to this particular QD

91


shape and is also valid for pyramidal or in-plane elliptical QDs. The exciton state in In(Ga)As

QDs is found in the strong-connement regime [84] and, as argued in Sec. 2.2, we employ the

single-particle eective-mass approximation to model the electron Ψe and hole Ψhh wavefunctions

Ψe(r) = us(r)ψe(r),

Ψhh(r) = uhh(r)ψhh(r),(5.70)

where us (uhh) is the conduction-band (valence-band) Bloch function at the Γ-point in k-space,

and ψ(r) is the slowly varying envelope subject to the eective-mass Schrödinger equation. Due

to the exchange interaction, the two bright excitons are linearly polarized along x = [1, 1, 0] and

y = [1,−1, 0] as derived in Sec. 2.2. We consider one of the bright excitons uhh = ux, where ux

is the valence-band Bloch function with odd symmetry along x, but note that the properties of

the y-polarized exciton are derived analogously.

5.6.1 Zeroth-order contribution

The zeroth-order contribution to the radiative decay rate stems from the electric-dipole character

of the QD and reads (cf. Eq. (5.40))

Γ(0) =2µ0e

2

~m20

Im 〈0| pi |ΨX〉Gii(r0, r0) 〈ΨX| pj |0〉 . (5.71)

The transition moment can be simplied as follows

〈0| pi |ΨX〉 = 〈uxψhh| pi |usψe〉 ≈∫

d3rψ∗hhψeu∗xpius

≈N∑q=1

ψ∗hh(Rq)ψe(Rq)

∫U.C.

d3ru∗x(r)pius(r) ≈ 〈ψhh|ψe〉 pcv,(5.72)

where pcv = 1VU.C.

〈ux| px |us〉U.C. is the Bloch transition matrix element and is evaluated within

one single unit cell. The rst approximation in the above equation assumes that the momentum

operator acting on the electron slowly varying envelope has a negligible contribution because ψe

varies much slower than ue. We then have neglected the spatial variation of the slowly varying

envelopes over one unit cell and have separated the integral into a sum over all the unit cells

comprising the QD. Finally, we have converted the sum back into an integral involving the slowly

varying envelopes. We thus obtain for the rate

Γ(0) =2µ0e

2

~m20

|pcv|2 |〈ψhh|ψe〉|2 ImGxx(r0, r0). (5.73)

The imaginary part of Gxx can be recast with the help of Eqs. (5.46), (5.59) and (5.60)

ImGxx(r0, r0) = ImGHxx(r0, r0) + ImGSxx(r0, r0)

=k1

6π+k1

8πRe

[∫ ∞0

dkρkρkz1

(rs − kz1rp)

].

(5.74)

92


ux ψhh Ψhh us ψe Ψe

x -1 1 -1 1 1 1

y 1 1 1 1 1 1

z 1 0 0 1 0 0

Table 5.2: Symmetries of the electron and hole wavefunctions for a lens-shaped QD. `1' denotes

even parity, `-1' odd parity and '0` no parity.

5.6.2 First-order contribution

We recall Γ(1) from Eq. (5.41)

Γ(1) =2µ0

~02Re

[Λkiµ

∗j

]∂kImGij(r, r0)|r=r0

. (5.75)

The only non-zero component of the dipole transition moment is µx = 〈0| px |ΨX〉 because we

consider the x-polarized exciton. The sum over j therefore collapses yielding

Γ(1) =2µ0

~02Re [Λkiµ

∗x] ∂kImGix(r, r0)|r=r0

. (5.76)

In the following, we investigate the rst-order mesoscopic moment←→Λ

Λki =e

m0〈0| (xk − x0,k)pi |ΨX〉 =

e

m0〈uxψhh| (xk − x0,k)pi |usψe〉 . (5.77)

The choice of x0 and y0 is provided naturally by the cylindrical symmetry of the QD. Due to

a lack of parity symmetry of the QD wavefunctions in the z-direction, z0 cannot be chosen by

symmetry and we dene it as the z-component of the exciton center-of-mass coordinate as argued

in Eq. (5.69).

The 9 entries in←→Λ can be reduced to 2 non-zero entries using the symmetry properties of

the underlying wavefunctions. The valence-band Bloch function ux inherits the symmetry of the

px orbital and therefore exhibits odd parity ("-1") in the x-direction and even parity ("+1")

in y and z. The conduction-band Bloch function ue inherits the spherical symmetry of the s-

orbital and therefore contains even parity in all directions. Since the eective-mass theory is an

envelope-function formalism, the slowly varying envelopes ψ inherit the symmetry of the QD.

Table 5.2 summarizes these considerations. Applying parity-symmetry arguments, we nd that

only Λxz and Λzx contain non-zero entries and thus

←→Λ =

0 0 Λxz

0 0 0

Λzx 0 0

. (5.78)

This yields for the rst-order contribution

Γ(1) =2µ0

~2(Re [Λxzµ

∗x] ∂xImGzx(r, r0)|r0 + Re [Λzxµ

∗x] ∂zImGxx(r, r0)|r0

). (5.79)

93


Distance to interface, z0 (nm)

Qu

antu

m-d

ot

dec

ay r

ate

(ns-1

)

Silver

GaAs

Silver

GaAs

z

z

(a)

(b)

Figure 5.6: Experimental demonstration of the breakdown of the dipole approximation taken

from Ref. [26]. The decay rate of quantum dots close to a metal interface was measured for (a)

direct and (b) inverted QDs relative to the interface. The black dashed line denotes the dipole

theory, the triangles the data points and the colored solid lines the t.

To understand which of the two non-zero mesoscopic moments may be relevant and compete

with the electric-dipole contribution, we analyze their functional dependence and compare it to

the experiment performed in Ref. [26]. In the experiment, the QD spontaneous-emission rate

was found to be inhibited with respect to the dipole theory for the geometry illustrated by the

inset of Fig. 5.6(a). In contrast, the inverted structure shown in Fig. 5.6(b) showed an increase

in the rate; we note that the QDs from the direct and inverted structures would exhibit the same

rates, if they behaved as point dipoles, as shown by the dashed line corresponding to the dipole

theory. This breakdown of the dipole approximation was found to be caused by the mesoscopic

moment Λxz and we explain the reason in the following.

The two mesoscopic moments provide dierent contributions to the light-matter interaction

strength as shown in Fig. 5.7. If compared to Fig. 5.6, it follows that Λzx does not reproduce

the functional dependence observed in experiment. The contribution of Λzx aects mostly the

"phase" of the oscillations, a behavior that was not observed in experiment. This implies that

Λzx Λxz and in the following we give a qualitative explanation for this. The mesoscopic

moment Λzx = (e/m0) 〈ψh |zpx|ψe〉 scales with the height of QDs because of the z-operator in

the matrix element. In contrast, Λxz = (e/m0) 〈ψh |xpz|ψe〉 scales with the in-plane size, which

is much larger than the height for In(Ga)As QDs. As a consequence, Λzx is expected to yield

negligible contribution to the decay rate and this is what we show quantitatively in Appendix C.

Only Λxz may potentially attain large values and compete with the electric-dipole contribution

94


0 100 200 300 400 5000

1

2

3

Distance to interface, z (nm)

Nor

mal

ized

dec

ay r

ate Direct

DAInverted

0

0 100 200 300 400 5000

1

2

3

Distance to interface, z (nm)

Nor

mal

ized

dec

ay r

ate Direct

DAInverted

0

Figure 5.7: Normalized decay rate for (left) Λxz and (right) Λzx contribution. The size of the

mesoscopic moments Λij/µx is taken to be 10 nm, which is of the order of the size of In(Ga)As

QDs. "DA" denotes the dipole approximation.

Γ(0) and we may write

←→Λ '

0 0 Λ

0 0 0

0 0 0

, (5.80)

where Λ = Λxz. In the experiment from Ref. [26] a value of Λ/µ ' 10 nm was measured, which

is surprisingly large and provides an additional degree of freedom for the light-matter interaction

with QDs, in addition to the dipole moment. Evaluating Λ/µ self-consistently and understanding

the microscopic origin for the large mesoscopic moment is not straightforward and we return in

Chapter 6 to accomplish this task. We nally obtain for the rst-order contribution

Γ(1) =2µ0

~2Re [Λxzµ

∗x] ∂xImGzx(r, r0)|r0 . (5.81)

5.6.3 Second-order contribution

As shown in Sec. 5.3, the second-order correction to the decay rate is

Γ(2) =2µ0

~Im[

Re[Ωlkiµ

∗j

]∂k∂l + ΛkiΛ

∗lj∂k∂

′l

]Gij(r, r

′)|r=r′=r0

.

In the following we analyze the symmetry properties of the second-order mesoscopic moment

Ωijk. With the help of Table 5.2, it can be shown that only 7 entries out of 27 do not vanish:

Ωxxx, Ωxzz, Ωzxz = Ωxzz, Ωyyx, Ωyyz, Ωzzx and Ωzzz. The contribution of the entries Ωyyz and

Ωzzz vanishes in the proximity of an interface because the corresponding derivatives of the Green

tensor are zero, see Eq. 5.63. In Appendix D we show that the contribution of the mesoscopic

moment Ωzzx is negligible because it couples to the height of QDs, which is small. Analogous

reasoning applies to Ωxzz. Only Ωxxx and Ωyyx therefore may yield non-negligible contribution

95


0 100 200 3000.5

1

1.5

2


Nor

mal

ized

dec

ay r

ate

DirectDAInverted

0 100 200 3000.5

1

1.5

2


Nor

mal

ized

dec

ay r

ate

DirectDAInverted

Figure 5.8: Contribution of the second-order mesoscopic moments (left) Ωxxx and (right) Ωyyx

to the decay rate. Note that the direct and inverted structures yield the same contribution.

to the decay rate, and the second-order correction reads

Γ(2) =2µ0

~

Re [Ωxxxµ∗x] ∂x∂x ImGxx(r, r0)|r0 + Re [Ωyyxµ

∗x] ∂y∂y ImGxx(r, r0)|r0 +

+ Re [Ωxzzµ∗x] ∂x∂z ImGzx(r, r0)|r0 − |Λxz|

2∂x∂x ImGzz(r, r0)|r0 ]

(5.82)

All three second-order mesoscopic moments contribute with the same sign for the direct and

inverted structures. Due to the fact that we are considering relatively large wavelengths of

∼1000 nm corresponding to about ∼300 nm in GaAs, the second-order contribution Γ(2) is sub-

stantially reduced relative to the rst-order contribution as shown in the following. This may,

however, not hold for QDs operating at shorter wavelengths.

In the following, we estimate the impact of the second-order mesoscopic moments on the

decay rate of QDs near a silver interface. From the cylindrical symmetry of the QDs it follows

that Ωxxx = Ωyyx and are scaled by the in-plane size of QDs. Their contribution is, however,

not identical since they couple to dierent eld components

Ωxxxµx

=Ωyyxµx

=

⟨uxψhh

∣∣x2px∣∣usψe⟩

〈ψhh|ψe〉 pcvUCDA≈

⟨ψhh

∣∣x2∣∣ψe⟩

〈ψhh|ψe〉, (5.83)

where in the last step the term x2 was pulled outside the Bloch matrix element along with

the slowly-varying envelopes because the corresponding term stems from the expansion of the

eld, which may be considered constant over one unit cell. The justication of this unit-cell

dipole approximation (UCDA) is presented in Appendix E, where it is shown that UCDA is

excellent for most practical purposes owing to the small size of unit cells of ∼ 0.5 nm. The

quantitative estimations performed in Chapter 6 yield an in-plane extent HWHM for the QD

wavefunctions of about 15 nm, which corresponds to a value of the second-order mesoscopic

moments of Ωxxx/µx = Ωyyx/µx ∼ 225 nm2. These moments have a negligible contribution to

light-matter interaction as shown in Fig. 5.8 because k21 is small at a wavelength of 1 µm and the

mesoscopic terms are not large enough to compensate this. We therefore neglect the contribution

from Ωxxx and Ωyyx in the present thesis.

96


0 100 200 3000

1

2

3


Nor

mal

ized

con

trib

utio

n

Normalized ΓDA(ω)Normalized Γ(ω)Radiation ModesSPPs

0 100 200 3000

1

2

3


Nor

mal

ized

con

trib

utio

n

Normalized ΓDA(ω)Normalized Γ(ω)Radiation ModesSPPs

Figure 5.9: Decomposition of QD decay mechanisms in front of a silver mirror for the (left) direct

and (right) inverted structures. The mesoscopic moment Λ contributes mainly to the excitation

of surface plasmons with large eld gradients, while the coupling to radiation modes is largely

unaected.

The nal expression for the decay rate, which takes into account the aforementioned assump-

tions and justications, reads

Γ(ω) =2µ0

~

(|µx|2 ImGxx(r0, r0) + 2Re [Λxzµ

∗x] ∂xImGzx(r, r0)|r0

− |Λxz|2 ∂x∂x ImGzz(r, r0)|r0 ) .(5.84)

This is a general expression for the light-matter interaction between QDs and light and can be

applied to arbitrary nanophotonic environments. Every term in the above equation is analyzed

in detail in Chapter 7, where it is shown that QDs are the very rst known quantum emitters

that can probe simultaneously electric and magnetic elds at optical frequencies.

Finally, we note that decomposing the decay of the QD into dierent types of excitations can

be done using the angular-spectrum representation analogously to the dipole theory presented in

Sec. 5.4. It is apparent from Fig. 5.9 that the coupling to surface plasmons (SPPs) is very dierent

for the direct and inverted structures. This is because the dipole-moment µ and mesoscopic-

moment Λ operators have dierent parity along the z-direction and their interference changes

from destructive for the direct structure to constructive for the inverted geometry. The coupling

to surface plasmons βpl , which is dened as the ratio between the coupling rate to plasmons

divided by the coupling to all the available optical modes

βpl =ΓSPPs(ω)

ΓQD(ω)=

ΓSPPs(ω)

Γphotons(ω) + ΓSPPs(ω) + Γlosses(ω), (5.85)

can therefore be tuned by the interference between the dipole and mesoscopic moment of the QD

Γ(1), see Fig. 5.10. We therefore conclude that QDs have an additional optical degree of freedom,

the mesoscopic moment Λ, which breaks the dipole nature of QDs and may be used not only

to tailor the light-matter interaction strength but also to tune other emission-related properties

such as the radiation pattern or the polarization of the spontaneously emitted photons.

97


0 100 200 3000

20

40

60

80

100


Cou

plin

g to

SP

Ps,

βpl

(%

)

DADirectInverted

Figure 5.10: Eciency of coupling to plasmons βpl for mesoscopic QDs in front of a silver

mirror. Destructive (constructive) interference between µ and Λ results in suppressed (enhanced)

coupling to surface-plasmon polaritons for the direct (inverted) structure.

5.7 Summary

In the present chapter we have developed a comprehensive theory describing the process of spon-

taneous emission of light from quantum emitters beyond the dipole approximation. We have

performed a multipolar expansion in the eld and have collected the multipolar contributions up

to the second order in the radiative decay rate. A fundamental characteristic of the multipolar

expansion is the dependence of the multipolar moments on the choice of the origin of the coordi-

nate system, which may compromise the practical utility of the expansion. We have shown that

by carefully collecting the orders in the decay rate rather than transition moments, and by rigor-

ously dening the origin of the coordinate system, the origin dependence is not an issue and the

multipolar moments have a well-dened physical meaning. We have used the developed theory to

describe the spontaneous-emission process from self-assembled In(Ga)As and have found that the

QDs are mesoscopic entities possessing a large mesoscopic moment Λ, which may compete with

the dipole moment µ in light-matter interactions. The resulting expression for the decay rate is

simple and intuitive and can be applied to describe the spontaneous-emission process in arbitrary

nanophotonic environments. There are two questions left to be answered. First, the microscopic

origin of the large mesoscopic moment Λ has been unclear so far and we devote Chapter 6 to

answer this question. Second, the implications of the dipole-approximation breakdown on the

fundamental nature of the spontaneous-emission process is discussed in Chapter 7.

98

Chapter 6

Unraveling the Mesoscopic

Character of Quantum Dots in

Nanophotonics

The central physical process studied in the elds of quantum optics, cavity-quantum electrody-

namics, nano-optics and nanophotonics is the spontaneous emission of nonclassical light from

quantum emitters. State-of-the-art fabrication techniques enable the realization of advanced

photonic nanostructures such as photonic-crystal cavities and waveguides [22], or plasmonic

nanoantennas [165], which accurately and eciently tailor the density of optical states allowing

for the desired spontaneous-emission decay time [166], state of polarization [167] and direction of

propagation [21] of the spontaneously-emitted photons. The ability to control the spontaneous-

emission process provides envisioning prospects for the realization of ecient on-chip quantum-

information protocols interfacing stationary and ying quantum bits, as well as developing a new

understanding of fundamental CQED phenomena happening in a solid-state platform, such as

the Lamb shift [6] and other largely unexplored energy non-conserving processes [168]. So far,

at the heart of the light-matter-interaction studies has been the dipole approximation, which

has become a standard approximation used in quantum-optics textbooks [75]. According to the

dipole theory, the variation of the electromagnetic eld over the extent of the emitter is neglected

completely, which renders emitters to appear dimensionless entities when interacting with light.

The enormous success of the dipole theory in practical experiments has resulted in photonic

environments that have been engineered to target solely the dipole moment of emitters for tai-

loring the coupling to light in the desired fashion. For instance, a point-like emitter may couple

to an optically large plasmonic nanoantenna [167], where the interference among the multipolar

moments of the antenna may result in a highly directional emission of the single photons. Sim-

ilarly, the highly directional LDOS for in-plane oriented dipoles in photonic-crystal waveguides

leads to emission of single photons into the waveguide mode with near-unity probability [21].

Notwithstanding this extraordinary theoretical and experimental progress, the supremacy of the

99

Chapter 6. Unraveling the Mesoscopic Character of Quantum Dots in Nanophotonics

xyz(a) (b)

xyz

Figure 6.1: Unraveling the mesoscopic character of QDs in the vicinity of a GaAs-air interface.

The presence of the interface breaks the parity symmetry of the environment in the z-direction.

Since reections occur at the interface (the circular white arrow), the imaginary part of the elec-

tric eld E(r) generated by the electric-dipole component, which triggers spontaneous emission,

inherits this lack of symmetry and is curved (indicated by the green arrow). (a) In the dipole

approximation, the QD microscopic current j(r) (brown arrow) perceives only the parallel com-

ponent but not the out-of-plane component (the "curvature") of the electromagnetic eld at its

position. (b) In In(Ga)As self-assembled QDs, the current density ows along a curved path that

resembles the shape of the eld environment thereby exchanging energy more eciently with it.

As a consequence, the spontaneous-emission decay rate is enhanced and the photons (red arrows)

are emitted at a faster rate compared to the case in (a).

dipole approximation was recently challenged by the observation that self-assembled In(Ga)As

QDs do not follow the dipole theory when positioned in front of a silver mirror [26]. As presented

in Chapter 5, the breakdown of the dipole approximation is explained by a phenomenologically

dened mesoscopic moment, which may compete with the dipole contribution in light-matter

interactions, but no microscopic understanding of the eect has been established so far. Such an

understanding is, however, highly relevant for the use of QDs in the aforementioned studies and

applications.

In the present chapter we develop a microscopic theory of the QD wavefunctions that provides

physical insight into the mesoscopic character of QDs and has resulted in the publication of

Ref. [164]. Previous theories of light-matter interaction beyond the dipole approximation have

investigated mesoscopic eects at the level of the QD spatial extent and symmetry and have

discarded their atomistic nature because the unit cells are small compared to the wavelength

of light [83, 169173]. These approaches fail to explain the large mesoscopic moment observed

experimentally and the fundamental principle conferring mesoscopic properties to QDs has been

therefore unknown. In the present work we nd that, surprisingly, the atomistic nature plays

a crucial role and explains the mesoscopic character of QDs. We show that the mesoscopic

moment originates from structural inhomogeneities at the crystal-lattice level, which generate

100

Microscopic model for mesoscopic quantum dots

large circular quantum-mechanical current densities owing inside the QD over mesoscopic length

scales. The inhomogeneities are related to the change in the periodicity of the underlying crystal

lattice of the QD, which, in turn, is caused by the lattice-mismatched growth of self-assembled

QDs. Since Bloch functions with dierent periodicities cannot remain in phase throughout the

QD, this necessarily leads to a phase gradient and a resulting quantum-mechanical current in

the growth direction of the QD, which gives rise to the mesoscopic moment. Our ndings enrich

the understanding of the QD spontaneous-emission process, and can be immediately used for

engineering complex nanophotonic environments that maximize the coupling to the current-

density pattern of the QD. Such an example is shown in Fig. 6.1 in the vicinity of a dielectric-air

interface, where the electromagnetic eld is curved due to the lack of parity symmetry of the

environment in the z-direction. Matching this eld shape with a curved current density that

is running through self-assembled QDs, see Fig. 6.1(b), results in an enhanced light-matter

interaction strength compared to QDs obeying the dipole approximation shown in Fig. 6.1(a).

Our ndings are supported by experimental data showing an increased radiative decay rate

of QDs in the vicinity of an air interface compared to the dipole theory. By applying the

developed multipolar theory of spontaneous emission to the experimental results, we extract a

surprisingly strong variation of the mesoscopic moment across the inhomogeneously broadened

emission spectrum of QDs. Our ndings provide a new optical degree of freedom of QDs that

can be used in state-of-the-art CQED experiments as well as in more complex atomistic models

that take into account distortions of the nanoscopic lattice of QDs.

6.1 Microscopic model for mesoscopic quantum dots

The central quantity describing the optical transition from the excited state Ψe to the ground

state Ψg of a QD is the dipole moment µ = (e/m0) 〈Ψg|p|Ψe〉. We consider the x-polarized

exciton µ = µx, as sketched in Fig. 6.1, where x is the Cartesian unit vector. Until recently,

µ was the only QD property used to describe the interaction with light. Recent experimental

studies of spontaneous emission from QDs at nanoscale proximity to a mirror revealed strong

deviations from the dipole theory, which have been accounted so far by the phenomenologically

dened mesoscopic moment Λ = (e/m0) 〈Ψg|xpz|Ψe〉. Combined with the microscopically well-

understood dipole moment, this quantity accounts for the interaction with light caused by the

extended mesoscopic nature of QDs. In Ref. [26], a large value of Λ/µ ' 10 nm was measured and

approaches k−1GaAs = λ/(2πnGaAs), where nGaAs is the refractive index of GaAs. In the following

we show that such a large value cannot be obtained using the standard eective-mass theory,

and that a more extended description is required.

As presented in Sec. 5.6, the standard textbook approach for evaluating the transition dipole

moment µ is to assume that the envelope function ψ varies slowly over a unit cell so that µ

can be written as a product of the Bloch matrix element pcv and a three-dimensional overlap

integral between the envelope functions, i.e., µ = (e/m0) 〈uxψg|px|ueψe〉 ≈ (e/m0)pcv 〈ψg|ψe〉,where pcv = V −1

UC

∫UC

d3ru∗xpxue is given by an integral over the unit cell with VUC being the

unit-cell volume. In other words, the transition dipole moment is primarily a unit-cell eect and

101


is marginally aected by the envelope functions, as their overlap is normally close to unity [84].

Importantly, the large mesoscopic strength Λ/µ observed experimentally cannot be reproduced

by a similar calculation, which leads to

Λ =e

m0[〈ψg |x|ψe〉〈ux |pz|ue〉UC + 〈ψg|ψe〉〈ux |xpz|ue〉UC

+ 〈ψg |xpz|ψe〉〈ux|ue〉UC + 〈ψg |pz|ψe〉〈ux |x|ue〉UC] ,(6.1)

where 〈〉UC ≡ V −1UC

∫UC

d3r denotes integration over a unit cell. The rst three contributions

vanish for symmetry reasons. The fourth contribution is vanishingly small and does not scale

with the QD size: for Gaussian envelopes allowing for realistic mutual displacements of 12 nm

between the electron and the hole in the growth direction (note that the integral vanishes in

the absence of such a displacement) we estimate Λ/µ ∼ 10−4nm. This suggests that the large

mesoscopic strength Λ/µ ∼ 1020 nm observed experimentally cannot be explained solely by the

envelope wavefunctions. In the following we show that structural gradients at the nanoscopic

crystal-lattice length scale can explain the eect.

It is often assumed that solid-state emitters have a homogeneous chemical composition, which

renders substantial simplications in the computation of the wavefunctions. In particular, the

homogeneity justies the use of bulk-material Bloch functions, and only the slowly varying en-

velopes describe the properties of the nanostructure. This assumption works excellently for

quantum wells and lattice-matched QDs, where the structures are either strain free or pseudo-

morphically grown on the substrate material. As a result, the wavefunctions are conned to a

chemically homogeneous region of space. InAs QDs are grown by self-assembly induced by strain

relaxation, a violent process that unavoidably leads to the generation of chemical gradients at

the crystal-lattice level. In particular, large lattice-constant shifts are observed in the growth di-

rection of QDs [44, 174]. This limits the applicability of the standard envelope-function theories

and, in particular, of the eective-mass formalism. A complete theory encompassing the spatial

position and symmetry of every single atom comprising the QD would generally be required.

Remarkably, the essential physics of the mesoscopic light-matter interaction can be captured

by only a minor extension of the eective-mass theory. We assume that the lattice periodicity

changes at a certain position z = zT along the QD height by an amount ∆al = 110 pm at a

central value al = 605 pm as found experimentally in Ref. 174, see Fig. 6.2(a). This corresponds

to a relative lattice-constant shift of 18%, which is strain induced and is substantially larger

than the lattice-constant mismatch between InAs and GaAs of 7%. We note that, in general, the

lattice periodicity changes twice: rst it is expanded at the QD base (GaAs-In(Ga)As transition)

before being shrunk back at the QD tip (In(Ga)As-GaAs transition). Since the exciton is spa-

tially conned near the tip where the indium concentration is highest [44], we only consider the

second transition region. The Bloch functions change periodicity as well, cf. Fig. 6.2(b), and we

model this by expanding them in a Fourier series with a position-dependent lattice wavevector

102


hhe

(a) (c) 1

-1

(b)Bloch function

0

Figure 6.2: Sketch illustrating the microscopic model for mesoscopic QDs. (a) The atomic

lattice inside the QD is assumed to change periodicity at the position z = zT . (b) Sketch of how

the Bloch function u2x of the atomic lattice varies spatially inside the QD. (c) Illustration of the

matrix elements 〈px〉 ≡ 〈ux|px|ue〉 and 〈pz〉 ≡ 〈ux|pz|ue〉 for the three colored unit cells in (a).

The symmetry of the integrand is broken in the transition region around z = zT giving rise to

pronounced mesoscopic eects.

kl(z)

ux(r) =∑m

am(y, z) sin[mkl(z)x]

ue(r) =∑n

bn(y, z) cos[nkl(z)x].(6.2)

This Ansatz ensures opposite parity of the conduction- and valence-band Bloch functions along

x. Furthermore, we implicitly assume the shape of the Bloch functions to remain the same,

and only their periodicity to vary spatially. Now we return to the evaluation of the mesoscopic

moment and separate the slowly- and rapidly-varying contributions as

Λ =e

m0

N∑q=1

ψ∗g(Rq)Xqψe(Rq)

∫UC

d3ru∗x(r)pzue(r), (6.3)

where Rq denotes the position of the q-th unit cell and N is the total number of unit cells in the

QD. In a homogeneous region of the QD (the blue unit cell in Fig. 6.2(a)) the unit-cell integrand

of Eq. (6.3) is odd in x- and z-directions, cf. Fig. 6.2(c), which leads to a vanishing integral.

However, in the transition region around z = zT strong gradients are present, which destroy

the parity of the integrand (see the pink and green unit cells in Fig. 6.2(a,c)) and generate a

substantial contribution to Λ.

In the following, we calculate the mesoscopic moment Λ and show that its magnitude is

sensitive to the QD geometry. With the Ansatz in Eq. (6.2) we rst compute the dipole Bloch

matrix element pcv = 〈px〉UC

pcv =1

VUC

∫UC

d3ru∗xpxus

=i~VUC

∑n,m

∫UC

d3ra∗m(r)bn(r) sin[mkl(z)x]nkl(z) sin[nkl(z)x]

=i~VUC

∑n

∫UC

d3ra∗n(r)bn(r)nkl(z) sin2[nkl(z)x].

(6.4)

103


Since Λ contains the z-polarized Bloch matrix element 〈ux| pz |us〉, we evaluate it with the help

of Eq. (6.3)∫UC

d3ru∗x(r)pzue(r) = −i~∑m,n

∫UC

d3ra∗m(r) sin[mkl(z)x]∂zbn(r) cos[nkl(x)]

' i~∑m,n

∫UC

d3ra∗m(r)bn(r)∂kl(z)

∂zsin[mkl(z)x]nx sin[nkl(z)x],

(6.5)

where we have assumed that only the periodicity of the Bloch functions changes while the func-

tions a(r) and b(r) are independent of z. With this in mind, the mesoscopic moment Λ can be

written as

Λ =

N∑q=1

ψ∗hh(Rq)Xqψe(Rq)∑m,n

∫UC

d3ra∗m(r)bn(r)∂kl(z)

∂zsin[mkl(z)x]n(x+Xq) sin[nkl(z)x].

We assume that kl is varying linearly over several lattice constants as shown qualitatively in

Refs. [44, 174], and we can therefore pull it in front of the integral over the unit cell. The term

containing x vanishes because it renders the integral odd yielding for the mesoscopic moment

Λ =

N∑q=1

ψ∗hh(Rq)X2qψe(Rq)

∂kl(z)

∂z

∣∣∣∣z=Zq

∑m,n

∫UC

d3ra∗m(r)bn(r) sin[mkl(z)x]n sin[nkl(z)x]

'Ncells∑q=1

ψ∗hh(Rq)X2qψe(Rq)

1

kl

∂kl∂z

∣∣∣∣z=Zq

∑m,n

∫UC

d3ra∗m(r)bn(r) sin[mkl(z)x]nkl sin[nkl(z)x]

' 1

kl

⟨ψhh(r)

∣∣∣∣x2 ∂kl(z)∂z

∣∣∣∣ψe(r)

⟩pcv.

(6.6)

The resulting expression for the mesoscopic strength Λ/µ reads

Λ

µ=

1

kl

⟨ψg(r)

∣∣x2 [∂zkl(z)]∣∣ψe(r)

⟩〈ψg(r)|ψe(r)〉

. (6.7)

We have thus been able to express a crystal-lattice eect in terms of the slowly varying envelope

functions. The mesoscopic strength scales quadratically with the in-plane size of the QD, Λ/µ ∼L2r, because the term

⟨ψg∣∣x2 [∂zkl(z)]

∣∣ψe⟩ contains the variance of the exciton wavefunction in

the x-direction. Moreover, it increases with decreasing QD height, Λ/µ ∼ L−1z , since in shallow

QDs the relative importance of the lattice-constant transition region is increased.

Equation (6.7) is the most general expression for Λ/µ that can be simplied in order to

obtain an intuitive analytical expression. In this regard, we assume a sharp transition in the

lattice constant ∂zkl = ∆klδ(z − zT ) for simplicity. This approximation is excellent because

∂zkl is multiplied by a slowly varying integrand. Throughout this paper we consider zT = 0,

which coincides with the center of the QD wavefunctions, for the following reason. According to

Ref. [44], the shift happens at the tip of the QD, which turns out to be close to the position where

the QD wavefunctions are localized since the indium concentration is highest here. We therefore

expect zT to coincide with the region of high excitonic density. We have explicitly checked that

our results are robust to small shifts of zT of ±1 nm that may occur in practice. A quantitative

justication requires the knowledge of the distribution of both the material composition and the

104


5 10 15 20 250

20

40

60

80

In−plane QD radius (nm)

|Λ/µ

| (nm

)

Figure 6.3: The mesoscopic strength as a function of the in-plane size of the QD for three xed

QD heights.

local lattice parameter throughout the QD, parameters which are dicult to measure and are

generally unknown. For the particular case of in-plane rotationally symmetric Gaussian slowly

varying envelopes we obtain the following analytic expression for the mesoscopic strength

Λ

µ= −∆al

al

√1 + ξz

4π

σ2r

σz, (6.8)

where σz is the height (HWHM) of the electron envelope, σr the QD radius, ∆al/al the relative

lattice-constant shift and ξz ≈ 5 is the ratio between the electron and hole eective masses [34].

We plot the mesoscopic strength as a function of the in-plane radius for three xed heights in

Fig. 6.3. The largest mesoscopic strengths are achieved by shallow and wide (disk-shaped) QDs.

For instance, taking a relative extreme case of a height of 2σz = 2 nm and a radius of σr = 30 nm

yields a mesoscopic strength as large as Λ/µ = 120 nm, which is an order of magnitude larger

than the values observed in experiments so far. Such QDs would constitute a mesoscopic entity

in which mesoscopic eects may dominate the light-matter interaction strength. For instance, a

QD with Λ/µ = 120 nm placed in front of a silver mirror would exhibit a Purcell factor that is

nearly 100 times larger than the case of a point-dipole source. Aside from this, such QDs may also

be extremely ecient at interfacing both electric and magnetic degrees of freedom in structures

that conserve parity symmetry, such as photonic-crystal cavities and waveguides, owing to the

substantial increase of second-order light-matter-interaction processes that are weak for current

In(Ga)As QDs. Note, Λ/µ has units of nanometers and its physical relevance can be assessed

only in conjunction with the magnitude of the k-vector of the corresponding optical mode(s). In

other words, k × Λ/µ is the relevant gure of merit characterizing the strength of light-matter

interaction beyond the dipole approximation.

105


-5 0 5

-2

0

2

0

1

-5 0 5 -5 0 5-10-15 10 15

0 10 20 30-10-20-30

-2

0

2

(a) (b) (c)

(d)

x (nm)

z (n

m)

z (n

m)

Figure 6.4: The quantum-mechanical current density J(r) running through the QD for various

QD geometries. (a) Homogeneous crystal lattice where the current ow is uniform and points in

the direction of the dipole moment. (b) Inhomogeneous lattice for a QD radius of 5 nm giving

rise to a non-uniform current ow following a curved path. The QD height is 2σz = 4 nm. (c),

(d) Same as (b) but for QD radii of 10 and 20 nm, respectively. Both the length of the arrows

and the color scale indicate the magnitude of the ow and the direction of the arrows indicates

the pointwise direction of the ow. The dashed white line sketches the position and orientation

of the QD.

6.2 The quantum-mechanical current density

Knowledge about the quantum-mechanical wavefunctions allows computing the current density

jQD(r) owing through the QD. We dene the latter by comparing the interaction Hamiltonian

Hint = (e/m0)A · p, where A is the vector potential, to the classical particle-eld interaction

Hamiltonian Hint = A(r) · j(r) [72]. The quantum-mechanical current density can therefore be

written as jQD(r) = (e/m0)Ψg(r)pΨe(r) or

jQD(r) =e

m0

[Ψ∗g(r)pxΨe(r)x + Ψ∗g(r)pzΨe(r)z

]. (6.9)

The current density jQD(r) = JQD(r)p(r) is modulated by the Bloch element p(r) = ux(r)pxue(r)

but for simplicity we neglect it and in the following discuss only the relevant and physically

meaningful slowly varying component JQD(r), which can be written with the help of Eq. (6.6)

as

JQD(r) =e

m0ψ∗g(r)ψe(r)

(x + x

1

kl

∂kl∂z

z

). (6.10)

A sharp lattice-constant transition is a good simplifying assumption for evaluating Λ because

the properties of kl(z) are integrated out. The current density JQD(r) is, however, sensitive

to the exact spatial dependence of the lattice-constant shift. In the following we assume that

106

Breakdown of the dipole theory at nanoscale proximity to a dielectric interface

most of the transition happens over two lattice constants as shown experimentally in Ref. [44].

Slowly varying Gaussian envelopes are used to model the QD wavefunctions ψg and ψe. In

QDs with a homogeneous crystal lattice ∂kl/∂z = 0 and thus vanishing mesoscopic moment

Λ = 0, the current density ows only along the direction of the dipole moment because there

are no gradients in the z-direction and the second term from the right-hand side of Eq. (6.9)

vanishes (see Fig. 6.4(a)). The presence of lattice inhomogeneities changes the ow dramatically

because strong gradients in the z-direction arise. The current density ows along a curved path

as illustrated in Figs. 6.4(b-d), conferring pronounced mesoscopic properties to QDs. The wider

the QD is, the sharper the transverse oscillations of the current are and the larger Λ/µ is. This

eect oers the possibility to enhance (diminish) the light-matter interaction by placing QDs

in environments where the electric vacuum eld exhibits gradients with the same (opposite)

sign, see also Fig. 6.1. We underline that the current density is an intrinsic property of QDs

and does not depend on the nanophotonic environment surrounding the QD. The current has a

spatial curvature that can be decomposed into a curl-free component, which probes the electric

eld in light-matter interactions, and a circular component, which probes the magnetic eld

of the electromagnetic quantum vacuum. The ability to eciently probe magnetic elds at

optical frequencies has been a long-sought goal in nanophotonics, and our ndings show that self-

assembled In(Ga)As QDs are the very rst quantum emitters that are not "blind" to the magnetic

eld of light. This topic is explored in detail in Chapter 7. The curved current density of QDs

opens new opportunities for designing ecient light-matter interfaces that exploit mesoscopic

eects to enhance the interaction with light. Aside from the local light-matter coupling strength,

other degrees of freedom could be potentially tailored by exploiting the mesoscopic interaction,

such as the photon-emission directionality or polarization.

6.3 Breakdown of the dipole theory at nanoscale proximity

to a dielectric interface

Deviations from the dipole theory have been observed in the proximity of a metal interface [26]

and were attributed to the strong plasmonic gradients that eciently probed the mesoscopic

character of QDs. In the present section we demonstrate deviations from the dipole theory in

the vicinity of an air interface, which is a weakly conning dielectric structure as pictured in

Fig. 6.1. Our ndings show that the mesoscopic strength Λ/µ of QDs is so large that even

dielectric environments may be used to tailor the multipolar radiation from QDs, which opens a

new and potentially unexplored dimension in the eld of cavity quantum electrodynamics.

Previous experiments reported the measurements of spontaneous-emission decay rates for

ensembles of QDs that are placed at dierent distances to a GaAs-air interface [84, 86]. Figure 6.5

displays the data that were used to reliably extract the dipole moment by exploiting the data

points recorded at distances above 75 nm [84]. A systematic deviation from the dipole theory

was found at distances below ∼ 75 nm, whose origin has been unclear and was speculated to be a

result of enhanced loss processes at the etched interfaces. Here we show that the deviations can

107


0 100 200 300

0.9

1

1.1

1.2

1.3

1.4

Dec

ay r

ate

(ns−

1 )

0 100 200 300

0.9

1

1.1

1.2

1.3

1.4

z0

GaAs

Air

0 100 200 300

0.9

1

1.1

1.2

1.3

1.4

Dec

ay r

ate

(ns−

1 )

0 100 200 300

0.9

1

1.1

1.2

1.3

1.4

0 100 200 300

0.9

1

1.1

1.2

1.3

1.4

Distance to interface, z0

(nm)

Dec

ay r

ate

(ns−

1 )

0 100 200 300

0.9

1

1.1

1.2

1.3

1.4


(nm)

(a) (b)

(c) (d)

(e) (f)

E=1170 meV E=1186 meV

E=1204 meV E=1216 meV

E=1252 meV E=1272 meV

Figure 6.5: Observation of deviations from the dipole theory for QDs near an interface. (a)-(f)

Measured decay rates versus distance z0 to the GaAs-air interface (data points) at six dierent

energies E across the inhomogeneously broadened emission spectrum. The dipole (multipolar)

theory is indicated by the black dashed (blue solid) line. A refractive index n = 3.5 of GaAs was

used. The inset in (a) is a schematic illustrating the sample geometry.

be explained by the contributions from the mesoscopic moment Λ to the light-matter interaction

strength. We use the multipolar theory developed in Chapter 5 to analyze the experimental data

shown in Fig. 6.5(a). The decay rate beyond the dipole approximation can be decomposed into

108


ExpTheory

1.15 1.2 1.25 1.35

10

15

20

25

Energy (eV)

Λ/µ

(nm

)

(a) (b)

1.15 1.2 1.25 1.30

5

10

Energy (eV)L z

(c)

Height (nm)

Figure 6.6: Microscopic insight into the mesoscopic strength of QDs. (a) Extracted mesoscopic

strength Λ/µ over the emission spectrum of QDs (red squares) along with the prediction of the

theoretical model (blue dashed line) assuming that the QDs have a xed in-plane size and only

the height varies (see text for details). (b) Spectral dependence of the QD height as predicted by

the theoretical model, which agrees well with the atomic-force microscopy measurements from

(c). The data in (c) was published in Ref. [86].

three decay channels Γ = Γ(0) + Γ(1) + Γ(2) with

Γ(0)(r0) = Cµ2ImGxx(r0, r0)

Γ(1)(r0) = 2CΛµ ∂xImGzx(r, 0)|z=z0Γ(2)(r0) = CΛ2 ∂x∂

′xImGzz(r, r

′)|z=z′=z0 ,

(6.11)

where C = 2µ0/~. We set Λ and µ as free parameters and t the experimental data with the

resulting dependences plotted in Fig. 6.5 for all the emission energies. It should be mentioned

that a data point observed at a distance of 20 nm from the GaAs-air interface is not shown

in Fig. 6.5 and is omitted from the analysis because it shows a much higher decay rate and

lower photoluminescence intensity, which is likely to be caused by nonradiative tunneling of the

QD charge carriers to surface states [86]. A phenomenological distance-independent loss rate

is added to Γ to account for intrinsic nonradiative decay channels within the QD but is found

to be negligibly small in the present analysis. This procedure is used independently for every

emission energy resulting in the data points in Fig. 6.6(a). The dipole-theory t is performed

by setting Λ = 0 and excluding the rst six data points from the analysis. For completeness, we

plot distance-dependent decay rates for various mesoscopic strengths in Fig. 6.7. It can be seen

that both the amplitude and the phase of the oscillations are substantially aected at distances

less than ∼ 100 nm from the interface, which allows extracting the mesoscopic strength from

the experimental data. It is interesting to note that the eect is more dramatic for QDs ipped

upside down for which Λ/µ < 0 as indicated by the red curves in Fig. 6.7.

The multipolar theory quantitatively reproduces the functional dependence observed in the

experiment for all emission energies, see Fig. 6.5. The extracted mesoscopic strength Λ/µ in-

creases with emission energy and varies from 10 to 23 nm over the inhomogeneously broadened

emission spectrum, cf. Fig. 6.6(a), which reects a pronounced dependence on QD size. The

increase in Λ/µ with energy is successfully explained by our microscopic QD theory, which

is presented in the following. We use Eq. (6.7) to model the spectral dependence of Λ/µ in

109


0 100 200 3000.7

0.8

0.9

1

1.1


(nm)

Nor

mal

ized

dec

ay r

ate

z0

GaAs

Air

10 nm 0 nm-10 nm-25 nm

25 nm

Figure 6.7: Calculated spontaneous-emission decay rates as a function of distance to a GaAs-air

interface for various mesoscopic strengths. The vanishing mesoscopic strength corresponds to

the dipole theory. The decay rates are normalized to the decay rate in homogeneous GaAs. A

refractive index n = 3.5 of GaAs and an emission wavelength of λ = 1µm were employed.

Fig. 6.6(a), where we employ several assumptions. First, only the height of QDs is assumed to

vary across the spectrum while the in-plane size remains constant. This assumption is supported

by studies of size and shape performed on self-assembled QDs [42], where a small relative distri-

bution of the in-plane QD size is observed. Second, we consider the inhomogeneously broadened

spectrum to be caused only by the random distribution of the size (and consequently of the

quantization/emission energy) of QDs. Other parameters such as strain distribution or chemical

composition are considered constant over the emission spectrum. Third, we assume that only

the height of the QDs contributes to the quantization energy, since atomic-force microscopy mea-

surements show that the height of self-assembled In(Ga)As is generally much smaller than the

in-plane size [44]. Fourth, we assume disk-shaped wavefunctions that can be decomposed into

in-plane Φ and out-of-plane φ components, i.e., ψ(r) = Φ(x, y)φ(z). With this, Eq. (6.7) can be

rewritten asΛ

µ= −∆al

al

⟨Φg∣∣x2∣∣Φe⟩

〈Φg|Φe〉φg(zT )φe(zT )

〈φg|φe〉. (6.12)

The rst term from the right-hand side denotes the relative change in the lattice constant while

the second and third terms contain the dependence on the in-plane QD size and QD height,

respectively. We nd the functional dependence, f , between the third term and the quantization

energy, E − E0, using a nite-potential-well model, where E is the emission energy and E0 the

bulk band gap of the QD material. We therefore obtain

Λ

µ= S × f(E − E0), (6.13)

where S = −(∆al/al)⟨Φg∣∣x2∣∣Φe⟩ / 〈Φg|Φe〉. The trend of the experimental data from Fig. 6.6(a),

i.e., that Λ/µ increases with energy, agrees very well with our model (see the theory curve in the

same gure), if the in-plane QD size is constant across the emission spectrum and only the height

110


varies. This behavior has been reported in the literature in studies of QD size and shape using

similar growth conditions as used here [42]. We note that if the QDs had a constant aspect ratio,

the mesoscopic strength would be predicted to decrease with energy because the eect depends

stronger on the in-plane QD size, cf. Eq. (6.8). Some degree of correlation between height and

width has been observed [175], but our study suggests that such correlations are small in our

sample. We also stress that our study deals with the in-plane size of the QD wavefunctions,

which generally can be dierent than the QD size measured by surface-prole techniques. We

use S and E0 as tting parameters and obtain a bulk band gap E0 = 1.13 eV of the QD, which

yields quantization energies ranging from 40 meV up to 140 meV across the emission spectrum.

The resulting curve in Fig. 6.6(a) agrees well with the experimental results. By mapping the

quantization energy to the QD size, we are able to extract a QD height that varies from 11 nm

to 3 nm across the inhomogeneously broadened spectrum, cf. Fig. 6.6(b), which agrees well with

the values obtained from atomic-force microscopy measurements presented in Fig. 6.6(c). We

conclude that QDs with larger emission energy have larger mesoscopic strengths because they are

shallow so that a large part of the excitonic wavefunction is aected by the lattice inhomogeneity.

The increase in the spontaneous-emission rate compared to the dipole theory for small dis-

tances to the interface is consistent with the behavior near a silver interface [26] and can be

understood with the help of Fig. 6.1. The nanophotonic environment breaks parity symmetry

and the vacuum eld is curved at the position of the QDs. This curvature is probed by the

inhomogeneous quantum-mechanical current density of QDs through the mesoscopic moment Λ

leading to enhanced light-matter interaction. Note that in the vicinity of a metal interface the

decay rate is diminished rather than enhanced because the plasmonic eld exhibits opposite cur-

vature. We exemplify such as scenario for the coupling to the plasmonic eld of a silver nanowire

with a radius of 20 nm and a refractive index nAg = 0.2+7i, which exhibits larger eld gradients

and mesoscopic eects are strongly enhanced, see Fig. 6.8. By matching the plasmonic eld to

the QD current pattern, the light-matter interaction can be drastically improved. The congura-

tion in Fig. 6.8(a) exhibits a substantial coupling enhancement to surface plasmons compared to

a point dipole (from 75 % to 90 %), cf. Fig. 6.8(c). In contrast, the interaction is diminished by

ipping the QD, cf. Fig. 6.8(b), because the ows of the QD current and of the environment are

spatially orthogonal. In other words, µ and Λ interfere constructively in (a) and destructively

in (b). These ndings open the prospect of the realization of ecient nanophotonic designs for

harvesting the mesoscopic nature of QDs.

111


0 20

0

10

20

-20x (nm)

0

10

20

(a)

β pl

z0

z0 (nm)20100

0

0.2

0.4

0.6

0.8

1 (c)

0

1

(b)

z (n

m)

Figure 6.8: Quantum dots coupled to surface plasmons of a silver nanowire. The curved

quantum-mechanical current density of QDs probes the complex eld prole of the nanowire

as exemplied in (a). Since the eld matches the curvature of the QD current, the coupling

eciency βpl to surface plasmons is enhanced, red curve in (c), beyond that of a point dipole,

dashed curve in (c). (b) A QD ipped upside down exchanges energy less eciently with the

plasmonic eld as shown in (c) by the blue line. In (a)-(b), both the length of the arrows and

the color scale denote the magnitude of the vector eld.

6.4 Lattice-distortion eects beyond the multipolar theory

As shown earlier, the lattice distortion generates an inhomogeneous quantum-mechanical current

density owing through QDs. This eect results in two multipolar moments being signicant

for QDs: the dipole moment µ and the mesoscopic moment Λ. While the multipolar theory

of spontaneous emission developed in the present thesis readily accounts for these eects, the

question arises of how these eects can be incorporated in an exact formalism that does not

perform the multipolar expansion but rather treats the full nonlocal character of the light-matter

interaction beyond the dipole approximation from Eq. (5.3), which we rewrite for convenience

Γ(ω) =2µ0

~

∫ ∫d3rd3r′Im

[j(r) ·

←→G (r, r′) · j∗(r′)

].

Such a nonlocal formalism was developed earlier [83] but does not account for the lattice dis-

tortion occurring in the QD and treats only the mesoscopic QD potential as a source for the

dipole-approximation breakdown. Generalizing this formalism can be done using the presently

developed theory that evaluates the current distribution owing through the QD. Inserting

Eq. (6.10) into the above equation leads to a generalized and exact-to-all-orders expression for

112

Summary

the spontaneous-emission rate beyond the dipole approximation

Γ(ω) =2µ0 |pcv|2

~

[∫ ∫d3rd3r′Im [Jx(r)Gxx(r, r′)J∗x(r′)]

+2Re

(∫ ∫d3rd3r′Jx(r)ImGxz(r, r

′)J∗z (r′)

)+

∫ ∫d3rd3r′Im [Jz(r)Gzz(r, r

′)J∗z (r′)]

], (6.14)

where Jx(r) = x · JQD(r) = (e/m0)ψ∗g(r)ψe(r) and Jz(r) = z · JQD(r) = xk−1l (∂kl/∂z)Jx are

the x- and z-projections of the QD current density JQD(r), respectively. The rst term from the

right-hand side is identical to the expression derived in Ref. [83] and its zeroth-order expansion

contains the electric-dipole contribution Γ(0). The second an third terms are generated by the

transverse oscillations of the current density JQD(r) and contain the rst Γ(1) and second Γ(2)

order contributions, respectively. Due to the odd x-operator present in the transverse current

density Jz(r), which is a consequence of the transverse ow changing direction along x, the

second and third terms vanish in the dipole approximation and only the rst terms survives.

Equation (6.14) is the most general expression for the spontaneous emission of light from QDs

beyond the dipole approximation and should be preferred over the multipolar theory when the

gure of merit k × Λ/µ > 1, i.e., when the multipolar expansion diverges. Such a scenario is

however hard to achieve in practice because even the largest QDs grown so far are still small

compared to the wavelength of light λ/2π.

6.5 Summary

We have developed a novel microscopic model that successfully explains the large mesoscopic

strengths of In(Ga)As QDs observed experimentally. We nd the eect to be governed by the

lack of symmetry of the nanoscopic crystal lattice and scaled by the extended mesoscopic size of

the QD. The microscopic current density oscillates along a non-trivial curved path and can be

expressed as a superposition between the electric dipole and the mesoscopic moment of the QD.

This mesoscopic current is generated at the unit-cell level in analogy to the generation of currents

in macroscopic systems. Our work deepens the physical understanding of semiconductor QDs

and we therefore expect it to be of signicance for the active elds of solid-state quantum elec-

trodynamics and quantum-information processing, where ecient quantum interfaces between

QDs and light are exploited.

113

Chapter 7

Probing Electric and Magnetic

Vacuum Fluctuations with

Quantum Dots

Spontaneous emission is a fundamental physical process, which plays an essential role in nature

as the main source of optical radiation, and in applications as the principal source of articial

illumination. Be it the radiation from the sun or the indoor lighting, spontaneous emission

plays a paramount role in our lives and in the life of living organisms. At the heart of the

spontaneous-emission process lays the uctuating electromagnetic vacuum eld, which perturbs

quantum emitters and triggers their radiative decay. The electric and magnetic elds comprising

the vacuum eld are intimately connected through Maxwell's equations and do not exist without

one another. Despite being equally important for generating the electromagnetic eld, there is

a fundamental built-in asymmetry in their interaction with matter. Specically, the magnetic

force acting on a charged particle moving with velocity v is v/c times smaller than the electric

force. Only in environments where charges move extremely fast, such as charged plasmas [176],

do magnetic interactions become important.

The aforementioned asymmetry is inherited by the spontaneous emission of light from quan-

tum emitters, where magnetic and other higher-order multipole eld components do not play

normally a role. This is because the variation of the electromagnetic eld is negligible over the

spatial extent of most quantum emitters, which has rendered the dipole approximation a highly

successful approximation in quantum electrodynamics. Nevertheless, magnetic-dipole (MD) and

electric-quadrupole (EQ) transitions are well known in atomic physics and can be accessed with

light despite being much weaker [177179], since they have dierent selection rules than electric-

dipole (ED) transitions [158, 159, 180]. Self-assembled QDs are fundamentally dierent and the

dipole approximation may not apply to QDs even on dipole-allowed transitions, as was observed

experimentally in Refs. [26, 162] and discussed in Chapters 5 and 6. The asymmetry of the QD

wavefunctions originating from a lack of mirror-reection symmetry (parity symmetry) of the

115

Chapter 7. Probing Electric and Magnetic Vacuum Fluctuations with Quantum Dots

Figure 7.1: Schematic of the physical system studied in the present chapter. The QD has

three decay channels in the proximity of a metal interface: radiative ΓRAD, coupling to surface

plasmons ΓSP and ohmic losses heating the metal ΓLS. The electron (blue) and hole (red)

wavefunctions illustrate the built-in asymmetry of the QD. The Cartesian coordinate system is

indicated accordingly.

QD connement potential breaks the usual selection rules applicable in atomic physics leading

to a curved quantum-mechanical current density owing through QDs with both ED and mul-

tipolar contributions on the same transition. For atoms a related but very weak asymmetry

is induced by the electroweak interaction and has been used to probe the standard model of

particle physics [181]. In contrast, the parity violation is very strong for QDs due to their asym-

metric structure and, therefore, they may be exploited as a probe of the parity of the photonic

nanostructure or the nature of the multipolar quantum-vacuum uctuations.

In the present chapter, we show that the commonly used self-assembled In(Ga)As QDs are

sensitive to both electric and magnetic elds, which is a consequence of large circular quantum-

mechanical current densities running through QDs [164], and has resulted in the publication of

Ref. [162]. The multipolar eects explained by our theory are relevant and important in many

nanophotonic congurations. A current hot topic in nanophotonics exploits the role of non-

locality of the dielectric response in plasmonics [182, 183]. Here we study a dierent non-local

phenomenon by accounting for the spatial extent and symmetry of QDs and their interaction

with the complex eld proles found in nanostructures of importance for photon emission. The

eect is particularly pronounced, if the nanophotonic environment violates parity symmetry. For

concreteness we consider the QD spontaneous emission for two experimentally realistic nanopho-

tonic structures: a semiconductor-metal plane interface and a plasmonic nanowire, see Fig. 7.1

for a sketch of the investigated geometry. We note that our results apply to self-assembled QDs

while, e.g., spherical nanocrystals would not possess the required symmetry. Our study demon-

strates that single QDs can be employed for locally probing complex photonic nanostructures

116

Electric and magnetic light-matter interaction

that tailor both the electric and magnetic eld [184, 185]. Sensitivity to the magnetic eld has

been a long-sought goal in nanophotonics, and has been achieved so far only by scanning near-

eld spectroscopy [186] where the disturbance of the electromagnetic eld prole by the applied

near-eld probe can be an issue. The nanometer-size of single QDs enables non-invasive probing

that operates at the single-electron single-photon level. Furthermore, the multipolar coupling

of QDs can potentially be exploited for enhancing the light-matter interaction with immediate

applications to quantum light sources for quantum-information processing [22].

7.1 Electric and magnetic light-matter interaction

In this section we show that QDs probe electric and magnetic elds simultaneously and eciently.

The present analysis concentrates on the rate of spontaneous emission Γ of the QD because it

is a direct and experimentally relevant measure of the light-matter coupling strength [57, 166].

Another interesting property would be the emission pattern of the QD that can be modied

and tailored by the interference of dipolar and multipolar contributions [179]. In contrast, the

multipolar eects discussed here do not aect the QD interaction with phonons, an essential

dephasing mechanism, since the phonon interaction depends mainly on the QD volume rather

than symmetry. It should be mentioned that the mesoscopic terms do not inuence the photon

statistics of the source, i.e., the excellent single-photon purity observed for QD sources prevails

also under conditions where mesoscopic contributions are signicant.

As the starting point, we use the multipolar theory of spontaneous emission developed in

Chapter 5, where it is shown that QDs have two degrees of freedom while interacting with light,

namely the dipole moment µ = µx and the mesoscopic moment←→Λ = Λxz. One single parameter,

Λ = 〈Ψg |xpz|Ψe〉, describes the light-matter interaction beyond the dipole approximation. The

ratio |Λ/µ| quanties the mesoscopic strength of the QD and was measured to vary between

10 nm and 20 nm over the emission spectrum of standard self-assembled In(Ga)As QDs [26, 164].

A conservative value of 10 nm will be used throughout the present analysis. The interaction

with light can be either suppressed or enhanced by the mesoscopic moment Λ depending on

the properties of the environment of the QD. This eect is illustrated in Fig. 7.3(a), where the

emission rate of a QD in the proximity of a silver interface is shown (at an emission wavelength

of 1000 nm and with the refractive indices of GaAs nGaAs = 3.42 and of silver nAg = 0.2 + 7i). A

QD and a point dipole exhibit dierent functional dependencies to the metal interface because

the former couples also to eld gradients while the latter does not, as is shown in the following.

Note that, unlike QDs, atomic wavefunctions possess parity symmetry so that µ and←→Λ never

contribute simultaneously.

As argued in Chapter 5, we expand the decay rate up to the second order Γ ' Γ(0) +

Γ(1) + Γ(2). In the proximity of metals, the QD can decay into propagating photons with the

rate ΓRAD, propagating surface plasmons (ΓPL), or ohmic-lossy modes in the metal (ΓLS) [187],

see Fig. 7.1. The former coupling to radiative modes is essentially not aected by multipolar

eects since the responsible electromagnetic eld varies weakly in space, i.e., ΓRAD ≈ Γ(0)RAD. In

contrast, the plasmon eld varies strongly and therefore multipolar eects inuence the excitation

117


+ _

(b)

+ _(b)

+ _(c)

+ _+_

+ _+_(d)

Figure 7.2: Electric and magnetic light-matter interaction with mesoscopic QDs. (a) Light-

matter interaction processes governing Γ(0), where the ED interacts with the radiation modes of

the electric vacuum ERADx and the guided surface plasmon modes Ex. (b) Processes governing

the ED-MD interference. The light emitted by the ED µx interacts with the MD my and creates

a magnetic eld. The physical picture of Γ(1Q) is conceptually analogous and is presented in (c).

(d) Processes governing Γ(2) with pure MD and EQ contributions. The EQ Qxz couples to the

gradient of the electric vacuum.

rate of plasmons. The lossy modes are proportional to the imaginary part of the dielectric

permittivity [188] and raise the entropy of the system [189]. The coupling to lossy modes is

normally negligible for distances larger than ∼ 20 nm from the metal and we do not discuss them

further. We thus obtain the three light-matter interaction channels for mesoscopic QDs

Γ(0) = A |µ|2 ImGxx(0, 0) = ΓRAD + Γ(0)PL,

Γ(1) = 2ARe [Λµ∗] ∂xImGzx(r, 0)|r=0 ≈ Γ(1)PL,

Γ(2) = A |Λ|2 ∂x∂′xImGzz(r, r′)|r=r′=0 ≈ Γ

(2)PL,

(7.1)

where A = 2µ0/~. Each order has a clear physical meaning as explained below, where we

exemplify a semiconductor-silver interface as sketched in the inset of Fig. 7.3(a).

The zeroth-order rate, Γ(0), is the well-known ED contribution, and is given as a product of

a eld term, ImGxx, which is proportional to the (electric) LDOS, and a QD term, |µ|2, whichis proportional to the (electric) oscillator strength [72]. Here, a microscopic polarization in the

x-direction couples to the x-polarized electric eld, which probes the environment and interferes

118

Electric and magnetic light-matter interaction

z0

Silver

0 100 200 3000

1

2

3

DistanceGtoGtheGinterface,Gz0G(nm)

Nor

mal

ized

Gde

cayG

rate (a)

DistanceGtoGtheGinterface,Gz0G(nm)0 50 100 150

-1

0

1

2

z0

(b)

Gra

dien

tGofGG

G(nm

-1)

0G 50 100 150DistanceGtoGtheGinterface,Gz0 (nm)

-0.04-0.04

0

-0.02

0.02 (c)

z0

Figure 7.3: Decay dynamics of QDs near a silver interface. All the rates are normalized to the

decay rate in homogeneous GaAs. (a) Decay rate for the direct (inverted) QD orientation marked

by blue (orange) lines. The black dashed line denotes the dipole theory. (b) Decomposition of

the decay rates according to the expansion order. The ohmic losses are indicated by the dotted

black line. (c) The ED-MD and ED-EQ Green tensor probed by mesoscopic QDs and normalized

to ImGxx(0, 0) in homogeneous GaAs.

back with itself. The resulting eld excitation propagates away from the QD in the form of free

photons or surface plasmons, see Fig. 7.2(b). In the proximity of an interface, Γ(0) has the well-

known Drexhage dependence [76], see Fig. 7.3(a,b), where the red-violet color gradient indicates

that the coupling to the plasmonic eld becomes dominant at distances smaller than ∼ 50 nm.

The higher-order corrections to Γ depend on the mesoscopic moment Λ, which is responsible

for the non-local interaction with light. Γ(1) is a rst-order process and is negligible, if the

gure of merit G(1) ≡ |Γ(1)|/Γ(0) ≈ k × 2 |Λ/µ| is much smaller than unity. For In(Ga)As QDs,

G(1) ' 0.44 shows that the light-matter interaction beyond the dipole approximation can be

strong. The magnitude of such eects is determined by the eld gradients of the particular

photonic nanostructure and we compute them in the next paragraph. Γ(2) is a second-order

process and contains pure MD and EQ contributions as sketched in Fig. 7.2(d). For QDs, the

important quantity is G(2) ≡ |Γ(2)|/Γ(0) ≈ k2 |Λ/µ|2 ' 0.05, which is negligible. Note that the

dipole approximation is more robust for atoms and other high-symmetry emitters, since the rst

non-vanishing contribution is Γ(2), which has a weight of (kLQD)2 with respect to Γ(0).

In the following we discuss the rst-order contribution, Γ(1), in quantitative terms. The

mesoscopic moment Λ contains MD and EQ contributions, as can be seen from

Λxz∂xel,z(0) = iωmybl,y(0) +Qxz [∂xel,z(0) + ∂zel,x(0)] , (7.2)

where my ≡ m = (e/2m0) 〈Ψg |xpz − zpx|Ψe〉 the MD, Qxz ≡ Q = (e/2m0) 〈Ψg |xpz + pxz|Ψe〉the EQ of the QD, see Chapter 5 for details, and e and b are the electric- and magnetic-eld

modes, respectively. The two moments are equal, i.e., m = Q = Λ/2, but they couple to dierent

eld components and thus their contribution can be tailored independently. As a consequence,

Γ(1) intertwines the ED, MD and EQ characters of the QD with the following physical inter-

pretation. The ED couples to the x-polarized electric eld, which probes the environment and

interferes back with the MD and EQ components, see Fig. 7.2(b,c). The resulting eld excita-

tion propagates away in the form of surface plasmons. Note that Γ(1) 6= 0 only if both the QD

119


wavefunctions and the electromagnetic environment violate parity symmetry. This is because a

parity-symmetric electronic potential cannot be both µ- and←→Λ -allowed, and a parity-symmetric

environment contains either even or odd electromagnetic modes. The ED is an even operator and

would couple only to the even modes, while Λ corresponds to an odd operator and would couple

to the odd modes inducing no mutual interference between µ and Λ and a vanishing Γ(1). The

rst-order contribution can both enhance and suppress the light-matter interaction depending

on whether the light emitted by the ED µ interferes constructively or destructively with the

mesoscopic moment Λ. This can be seen in Fig. 7.3(a), where by ipping the QD orientation

Λ changes sign and, hence, Γ(1) changes from suppressing to enhancing the decay rate. The

multipolar contribution to Γ(1) is

Γ(1) = Γ(1m) + Γ(1Q)

= Aωmyµ∗ReByx(0, 0) +AQxzµ

∗ImQxz(0, 0),(7.3)

where we dene the ED-MD Green tensor Byx(0, 0) = −iω−1 [∂xGzx(r, 0)− ∂zGxx(r, 0)]r=0, the

ED-EQ Green tensor Qxz(0, 0) = [∂xGzx(r, 0) + ∂zGxx(r, 0)]r=0, and assume Λµ∗ to be real,

which holds in the eective-mass approximation in the absence of applied magnetic elds. Equa-

tion (7.3) shows that QDs access the magnetic and electric-quadrupole vacuum elds, similarly to

the way dipoles probe the electric component of the vacuum, and is demonstrated in Fig. 7.3(c),

where the contribution of ReByx(0, 0) and ImQxz(0, 0) is shown. The two components of the

Green tensor vary over length scales of tens of nanometers, which is comparable to the QD

size [44] and demonstrates that QDs are ecient probes of electric and magnetic elds on the

same electronic transition. In the following, we show that this novel property of QDs can be

used to probe the parity symmetry of complex nanophotonic environments.

7.2 Probing the parity symmetry of nanophotonic environ-

ments

Quantum dots interact with light as spatially extended objects and are therefore capable of

probing not only the electric-eld magnitude at their position but also eld variations. This is

the basic property allowing to use QDs for probing the complex nature of the electromagnetic

vacuum uctuations. If placed in an unknown nanophotonic structure, the spontaneous-emission

rate of the QD is generally given by ΓN ≈ Γ(0)N + Γ

(1)N . By ipping the QD orientation, which is

a feasible experimental procedure that can be done by etching away the substrate [26], the ED

contribution is the same but the rst-order term has opposite symmetry and changes sign, i.e.,

ΓH ≈ Γ(0)H +Γ

(1)H = Γ

(0)N −Γ

(1)N . As a consequence, both the projected Green tensor ImGxx(0, 0)

and the spatial gradient ∂xImGzx(0, 0) can be unambiguously extracted, cf. Eq. (7.1). While

the former corresponds to the electric-eld strength generated by an ED at the position of the

emitter, the latter describes the electric-eld gradient generated by the same ED. We exemplify

this aspect by investigating the interaction between QDs and surface plasmons in the proximity of

a silver nanowire (radius ρ = 30 nm), which is capable of collecting most of the QD emission into

120

Probing the parity symmetry of nanophotonic environments

R0sxnm)0 10 20 30 40

02468

10

02468

10

Silver

R0

Silver

R0

xa)

xd)

Nor

mal

ized

sde

cays

rate

Pro

ject

edsG

reen

Msste

nsor

02468

10

02468

10

0 10 20 30 40

xc)

xf)

rz

10

15

20

25

30

0 10 20-10-2010

15

20

25

30D

ista

nces

tosn

ano

wire

sxnm

)

xb)

xe)

R0sxnm)zsxnm)

Min

Max

Figure 7.4: Probing eld gradients with mesoscopic QDs. (a) For an axially-oriented dipole,

Γ(1) enhances (suppresses) the light-matter interaction for the conguration marked by orange

(blue). The dashed line is the prediction of the dipole theory. (b) Vector plot of the plasmonic

eld generated by the ED of the QD situated 20 nm away from the nanowire. Both the length of

the arrows and the color scale denote the eld magnitude. The white arrow inside the QD shows

the ow of the quantum-mechanical current inside the QD. (c) The eld projections probed by

the QD can be extracted by subtracting the decay curves in (a). The gray-shaded area is the

region where nonradiative losses are dominant. (d) For a radially-oriented dipole, Γ(1) = 0 and

the QD behaves as an electric dipole.

a single propagating eld mode, an important goal in the eld of quantum photonics [22, 188].

We nd the nanowire to support a single strongly conned plasmon mode with G(1) = kSP ×2 |Λ/µ| = 0.76 [72, 188] leading to stronger eld gradients than for the plane silver interface. The

contribution of Γ(2) is again negligible since G(2) = 0.14. The coupling to radiation and lossy

modes is modelled as a point-dipole in the simple quasi-static approximation [188, 190], which

gives excellent agreement with the full electrodynamic computation [191]. The Green tensor of

the plasmon eld acquires a particularly simple form [191] and for the geometry presented in

Fig. 7.4(a) the relevant rates read

Γ(0)PL

Γ(0)GaAs

= C |fz(0)|2 (7.4)

Γ(1)PL

Γ(0)GaAs

= 2CΛ

µRe [∂zf∗r (0)] fz(0) , (7.5)

where C = 3πc0ε0/nk20vg, vg is the group velocity of the guided mode and the decay rates have

been normalized to the decay rate in homogeneous GaAs. Equations (7.47.5) contain the two

eld components, which can be probed by QDs in spontaneous-emission experiments, as shown

in the following.

In order to acquire an understanding of the sensing capability of QDs, we analyze the prop-

121


erties of the surface-plasmon eld. The normal eld mode of the nanowire mode is [188]

f = N(−Er(r), 0, iEz(r)

)eikSPz, (7.6)

where Er and Ez are real positive quantities and N a normalization constant. Thus there are two

congurations in which the plasmon density of optical states is non-zero, namely for an axially

and radially oriented dipole, see the inset of Fig. 7.4(a,c), and in the following we study the elds

probed by QDs in these congurations. If the dipole moment is oriented axially, the rst-order

contribution acquires the simple form Γ(1) ∝ −(Λ/µ)kSPErEz and has about the same magnitude

as Γ(0). Consequently, the coupling to surface plasmons is suppressed completely when Λ and µ

are in phase (Λ/µ > 0, depicted with blue in Fig. 7.4(a)) and enhanced by a factor of two when

they are π out of phase (Λ/µ < 0, depicted with orange). Using the aforementioned procedure of

recording ΓN and ΓH from Fig. 7.4(a), QDs can be used to probe the magnitude and curvature

of the complex plasmonic eld plotted in Fig. 7.4(b). At the center of the QD, the eld is

completely polarized along the z-direction and the point-dipole character of the QD therefore

probes the local density of states via ImGzz(0, 0). Additionally, the eld exhibits a curvature

meaning that the radially-polarized eld varies over the QD despite the fact that its mean

value is zero. This radially-polarized axial gradient, ∂zImGrz(r, 0), is probed by the extended

mesoscopic character of the QD. Both elds exhibit a monotonic increase as the QD approaches

the nanowire and are plotted in Fig. 7.4(c). The axial gradient is multiplied by the in-plane QD

size (LQD = 20 nm)[44] to show the eld variation over the QD spatial extent. It is interesting to

note that the eld ImGrz exhibits a large variation over the QD that is comparable to the probed

eld itself ImGzz. This example shows the "ease" of breaking the dipole approximation with

QDs in nanophotonic structures. We nd that most of the contribution to Γ(1)PL stems from the

EQ nature of the QD, in contrast to the silver interface, where the MD and EQ contributions are

of comparable magnitude. These examples show that, even though the MD and EQ moments are

equal in magnitude, their individual contribution to the light-matter interaction can be tailored

by correspondingly engineering the nanophotonic environment. In this sense, QDs are promising

light emitters for embedment in optical metamaterials, whose practical realization has become

technologically feasible over the past years.

In the second conguration, the dipole moment is oriented radially (see Fig. 7.4(d)) and

the rst-order contribution Γ(1) vanishes because the environment is parity-symmetric along

the QD height, see the inset of Fig. 7.4(d). Consequently, the dipole approximation is a very

good assumption for this conguration, as seen in Fig. 7.4(d) that the prediction of the two

theories are very close. For a better understanding, we plot the in-phase component of the

electric eld generated by the dipole character of the QD in Fig. 7.4(e). The mesoscopic moment

Λ ≡ Λrz would couple to the radial gradient of the z-polarized eld but the latter vanishes in

this conguration owing to the aforementioned parity symmetry. There are two eld gradients

that do not vanish, the z-derivative of the z-polarized eld and the r-derivative of the r-polarized

eld. However, they are not sensed by QDs because they couple to other mesoscopic moments

(Λzz and Λrr, respectively), which vanish for In(Ga)As QDs. Therefore, a radial QD probes only

the (electric) local density of optical states as illustrated in Fig. 7.4(f).

122

Summary

7.3 Summary

We have shown that the commonly employed In(Ga)As QDs are capable of strongly interacting

with the multipolar quantum vacuum on dipole-allowed transitions. This striking behavior is

triggered by the lack of parity-symmetry of the electronic wavefunctions in the growth direction,

a feature that is absent in atomic physics because atoms have parity symmetry. The rst-order

expansion term Γ(1) can be comparable in magnitude to the dipole rate Γ(0) in nanophotonic

structures. This eect can be exploited to use QDs as a probe of the local eld environment

revealing not only information about the eld itself but also about its gradients. Furthermore,

by engineering the nanophotonic environment it is possible to selectively access the MD or EQ

nature of the QD and, thereby, to tailor the multipolar radiation of semiconductor QDs. We

have exemplied this for metal nanostructures but any strongly- or rapidly-varying optical modes

would produce deviations from the dipole approximation and we therefore expect this work to

be of signicance not only for plasmon-based devices [192] and photovoltaics [193], but also for

the active eld of photonic-crystal cavities and waveguides, where QDs have been described as

dipole emitters so far.

123

Chapter 8

Conclusion & Outlook

The present thesis has explored fundamental aspects of the optical properties of quantum dots.

The size, shape and material composition of quantum dots can be accurately engineered, which

oers precise control over their properties, in particular over the interaction with light. As a

consequence of their mesoscopic nature, quantum dots are fundamentally dierent from atomic

emitters in many respects.

One such example concerns the strength of the light-matter coupling, which can be tuned

over orders of magnitude depending on the quantum-dot size and geometry. This has been

demonstrated in the present work by measuring single-photon superradiance in a monolayer-

uctuation quantum dot. The tens of thousands of atoms comprising the quantum dot oscillate

in unison leading to a "giant" light-matter coupling. The latter is a manifestation not only of

the mesoscopic size of quantum dots but also of the underlying fermionic nature of the quantum-

dot excitation, which builds strong spatial correlations within the quantum dot. This eect is

expected to be of signicance for applications beneting from a large light-matter interaction at

the nanoscale, such as cavity quantum electrodynamics exploring strong coupling between light

and matter, one-dimensional waveguides where quantum dots induce strong interactions between

individual photons, as well as solid-state optoelectronic devices including solar cells, nano-lasers

and light-emitting diodes. An extraordinary challenge in quantum photonics pertains to the

creation of coherent quantum bits, and monolayer-uctuation quantum dots may provide a solu-

tion to the challenge owing to their fast radiative decay, which is likely to outspeed decoherence

processes. To this end, exploiting these quantum dots as an ecient interface between stationary

and ying quantum bits is a particularly attractive prospect for quantum-information science.

If the mesoscopic nature of quantum dots is combined with a lack of parity symmetry of

the underlying wavefunctions, the interaction with light can be further enriched. In particular,

we have shown that self-assembled In(Ga)As quantum dots are nanoscale probes of electric and

magnetic elds on a single optical transition, an eect unknown to the parity-symmetric atomic

emitters. This is caused by distortions at the level of the crystal lattice that forms the quantum

dot. The resulting quantum-mechanical current density owing through the quantum dot has

a pronounced circular character and explains the microscopic origin of the magnetic moment of

125

Chapter 8. Conclusion & Outlook

e h

e h

he

Excitonplocalization Currentpdensity

Self6assembledInOGa:AspQDs

Monolayer6fluctuationGaAspQDs

Droplet6epitaxyGaAspQDs

~15pnm

~4

pnm

~30pnm

~20pnm~

4pn

m~

4pn

m

OS:p10615QE:p806953

OS:p606100QE:p906993

OS:p8610QE:p706803

Figure 8.1: Synthesis of the electronic and optical properties of the three classes of quantum

dots explored in the present thesis. The red, blue and green envelopes denote the hole, electron

and exciton wavefunction, respectively. The pointwise direction and magnitude of the quantum-

mechanical current owing through quantum dots are denoted by the direction and length of

the white arrows, respectively. The oscillator strength (OS) and quantum eciency (QE) are

indicated accordingly.

quantum dots. Extensive eorts have been devoted to achieve sensitivity to the magnetic eld

of light. To this end, satisfactory results have been achieved so far only in optical metamaterials

and in the near eld of plasmonic structures. Quantum dots provide their quantum nature as an

extended degree of freedom, and the lack of ohmic loss as an intrinsic advantage for obtaining

magnetic sensitivity at optical frequencies.

Yet another complexity arising in solid-state environments pertains to the manifestation of

nonradiative processes. Defect impurities in the vicinity of quantum dots may trap one of the

charge carriers constituting the exciton and thus lower the eciency of radiative recombination.

We have measured an excellent near-unity quantum eciency of monolayer-uctuation quantum

dots, which is likely due to the ultra-clean growth procedure of these quantum dots. In contrast,

it was found that droplet-epitaxy quantum dots have a relatively modest quantum eciency of

about 75 %.

In the present thesis we have studied three dierent classes of quantum dots and we summarize

their properties in Fig. 8.1. Self-assembled In(Ga)As quantum dots have a prominent lattice-

constant shift inducing a curved quantum-mechanical current density with both electric and

magnetic character. The chemical gradient along the height of these quantum dots results in a

displacement of electrons and holes generating a static dipole moment. In monolayer-uctuation

126

quantum dots, the electron and the hole build spatial correlations within the extended exciton

wavefunction leading to a strongly and superradiantly enhanced light-matter interaction. The

large physical size of droplet-epitaxy quantum dots is not reected by the small spatial extent

of the exciton wavefunction. This is likely caused by material intermixing between the droplet-

epitaxy quantum dot and the surrounding matrix during the growth process.

The research carried out in the present study opens three major prospects for the further

development of science and technology. The rst prospect pertains to achieving unprecedented

enhancement of light-matter interaction at the nanoscale exploiting the eect of single-photon su-

perradiance. The superradiant enhancement can potentially be orders-of-magnitude larger than

the experiments reported here. This can be achieved in yet larger quantum dots and calculations

show [83] that a hundred-fold enhancement is realistically achievable in monolayer-uctuation

quantum dots with a radius of about 50 nm. This would correspond to a quantum dot that is

suciently large to build strong superradiant eects, yet suciently small to be in the Dicke

regime with constructive cooperativity in which the dipole approximation is valid. Due to the

weak quantum connement present in such large quantum dots, working at millikelvin temper-

atures would be crucial for preparing only the relevant 1s state of the exciton manifold. The

incorporation of such quantum dots in photonic structures beneting from large Purcell factors

such as nanocavities and waveguides may propel the generation speed of single photons to the

terahertz regime yielding remarkably large radiating powers of hundreds of nanowatts from a

single quantum emitter. The single photons emitted from a large quantum dot would potentially

be highly coherent, partly due to an intrinsically weaker coupling to nuclear spin noise [107]

and phonon dephasing [127] for large excitons, partly because the dephasing mechanisms present

in solid-state environments would be largely negligible compared to a radiative decay at sub-

picosecond time scales. Even faster radiative decays could be achieved in materials with large

Rydberg energies [194, 195]. This may allow studying energy non-conserving eects such as

the ultrastrong coupling regime of cavity quantum electrodynamics for the rst time at optical

frequencies. Another intriguing aspect is that the collective Lamb shift has been predicted to

be nite [108] without the renormalization schemes required in the quantum electrodynamics of

conventional emitters.

The second prospect is related to the inhomogeneous quantum-mechanical current density

owing through quantum dots. So far, only the dipole moment of In(Ga)As quantum dots has

been exploited in photonic nanostructures. We have shown, however, that the complex nature

of the quantum-mechanical current generates an additional optical degree of freedom, the meso-

scopic moment, which may play an important role in light-matter interactions. To this end, the

realization of nanostructures that tailor both the dipole and the mesoscopic nature of quantum

dots is a particularly attractive prospect that will likely lead to enhanced emission eciency. Such

a scenario can be readily realized in plasmonic systems with lack of parity symmetry and strong

eld gradients. For instance, the proposal for ultrastrong Purcell enhancement from Ref. [165]

tailors the out-of-plane dipole moment of quantum emitters but is incompatible with the in-

plane dipole of conventional quantum dots. The large mesoscopic moment of In(Ga)As quantum

dots may oer a solution by probing the strong out-of-plane eld gradients of the plasmonic

127

Chapter 8. Conclusion & Outlook

nanoparticle.

The third prospect concerns employing quantum dots as the building blocks of a quantum

metamaterial, a fundamentally new medium that combines the fascinating phenomena related

to classical metamaterials such as negative index of refraction, super-lensing and cloaking, with

the quantum nature of quantum dots that operate at the single-electron and single-photon level.

This is possible because the quantum-mechanical current owing through In(Ga)As quantum

dots is curved and resembles the current density running through split-ring resonators, the

building blocks of conventional metamaterials. An intrinsic limitation of split-ring resonators

is the ohmic loss of the underlying metal structure, which renders conventional metamaterials

extremely lossy and futile for practical applications. In particular, the largest measured propaga-

tion length of light in metamaterials is about 1 µm [196]. In contrast, quantum dots do not have

such a thermodynamically irreversible loss channel and could operate with signicantly reduced

propagation losses. A further prominent advantage of quantum dots is that the electric and

magnetic responses occur within the same optical resonance leading to an ideal spectral over-

lap. In Appendix F we elaborate on the possibility of designing a quantum-dot-based quantum

metamaterial and compare the relevant electric and magnetic gures of merit of quantum dots

and of split-ring resonators.

128

Appendices

129

Appendix A

Operator Matrices for the Theory

of Invariants

In the following we present the matrix representation of the angular-momentum operators σe of

the electron and jh of the hole composing the short-range exchange Hamiltonian. The matrices

σi are the well-known Pauli matrices because electrons have zero orbital angular momentum

σx =

(0 1

1 0

)σy =

(0 −ii 0

)σx =

(1 0

0 −1

). (A.1)

Heavy- and light-holes have a total angular momentum of j = 3/2 with four possible projec-

tions mj = +3/2,+1/2,−1/2,−3/2, which are taken as eigenbases of jh in the following. The

construction of these matrices is discussed in Ref. [197] and read

Jx =1

2

0√

3 0 0√3 0 2 0

0 2 0√

3

0 0√

3 0

Jy =i

2

0 −

√3 0 0√

3 0 −2 0

0 2 0 −√

3

0 0√

3 0

Jz =1

2

3 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −3

.

(A.2)

These matrices are used to evaluate the short-range exchange Hamiltonian Hshort in Sec. 2.2,

which yields the ne structure of heavy- and light-hole excitons.

131

Appendix B

Length and Velocity

Representation

Here, we show how the multipole moments in velocity representation can be converted to length

representation. We employ the following commutators:[ri, H0

]= i

~m0

pi, (B.1)

[rirj , H0

]= i

~m0

(pirj + ripj) , (B.2)

[rirjrk, H0

]= i

~m0

(pirjrk + ripjrk + rirj pk) , (B.3)

where H0 is the Hamiltonian in the absence of the electromagnetic perturbation. Next, we

evaluate the matrix element of the commutator of an operator A and H0

〈0| [A, H0] |ΨX〉 = 〈0| AH0−H0A |ΨX〉 = EX 〈0| A |ΨX〉−E0 〈0| A |ΨX〉 = E0X 〈0| A |ΨX〉 . (B.4)

We use these results to obtain

〈0| pi |ΨX〉 = −im~〈0| [ri, H0] |ΨX〉 = −iE0X

m

~〈0| ri |ΨX〉 , (B.5)

so that the electric-dipole transition becomes

〈0| µi |ΨX〉 =e

m0〈0| pi |ΨX〉 = −iE0X

~〈0| µi |ΨX〉 . (B.6)

Similarly, the electric-quadrupole and electric-octupole transition moments read

〈0| Q(p)ij |ΨX〉 = −iE0X

~〈0| Qij |ΨX〉 , (B.7)

〈0| O(p)ijk |ΨX〉 = −iE0X

~〈0| Oijk |ΨX〉 . (B.8)

133

Appendix C

Evaluation of the First-Order

Mesoscopic Moment Λzx

Here we give an estimate for the rst-order mesoscopic moment Λzx and show that it has a

negligible contribution to the light-matter interaction. Recall that

Λzx =e

m0〈Ψg| (z − z0)px |Ψe〉 ≈

e

m0pcv 〈ψg| (z − z0) |ψe〉 , (C.1)

where pcv is the Bloch matrix element. Λzx scales with the height of QDs, which amounts to

several nanometers and is much smaller than the in-plane size. This is a qualitative justication

for the negligible role of Λzx. In the following we provide a quantitative justication. The

relevant gure of merit for the magnitude of the mesoscopic moment is∣∣∣∣Λzxµ∣∣∣∣ =

∣∣∣∣ 〈ψg| (z − z0) |ψe〉〈ψg|ψe〉

∣∣∣∣ =

∣∣∣∣ 〈ψg| z |ψe〉〈ψg|ψe〉− z0

∣∣∣∣ . (C.2)

The QD is not symmetric in the z-direction and there is no predetermined choice for z0. Con-

ceptually, z0 should be dened such that the expansion in Γ converges fastest and corresponds

to the center-of-mass coordinate of the exciton as discussed in Sec. 5.5

z0 =ze +mrzg

1 +mr, (C.3)

where mr is the ratio of the eective masses of the hole and the electron. Now we are in a

position to estimate the magnitude of Λzx. Since this is an eect involving the slowly-varying

envelopes, we can make some realistic assumptions. We assume Gaussian wavefunctions for the

electron and hole with an out-of-plane HWHM of 2 nm for the electron and√ξ smaller for the

hole, where ξ = 5 is the ratio of their eective masses. Then, we plot |Λzx/µ| as a function of

the distance between the electron and hole wavefunctions, see Fig. C.1. For a realistic vertical

shift of 23 nm we obtain |Λzx/µ| ≈ 0.1 nm. Then, the relevant gure of merit for the breakdown

of the dipole approximation is 2k |Λzx/µ| ≈ 0.2% 1 and is negligible. This analysis provides

rigorous justication for neglecting Λzx both in this work and in Ref. 26.

135

Chapter C. Evaluation of the First-Order Mesoscopic Moment Λzx

0 1 2 30

0.05

0.1

0.15

0.2

Electron−hole vertical shift (nm)

|Λzx

/µ| (

nm)

Figure C.1: |Λzx/µ| as a function of the electron-hole vertical shift.

136

Appendix D

Evaluation of the Second-Order

Mesoscopic Moment Ωzzx

Here we show that Ωzzx has negligible contribution to light-matter interaction. We can write

Ωzzx =e

2m0〈Ψhh| zzpx |Ψe〉 ≈

e

2m0pcv 〈ψhh| zz |ψe〉 . (D.1)

The term 〈ψhh| zz |ψe〉 is reminiscent of the variance of the exciton wavefunction in the z-

direction, which determines the out-of-plane size of the exciton. Given the fact that InAs QDs are

at (∼25 nm) and wide (∼2030 nm), the contribution of this mesoscopic moment is negligible.

The relevant gure of merit for the breakdown of the dipole approximation is

k21

∣∣∣∣Ωzzxdx

∣∣∣∣ =

∣∣∣∣ 〈ψhh| zz |ψe〉〈ψhh|ψe〉

∣∣∣∣ 6 1. (D.2)

We consider the optimistic case of 〈ψhh| zz |ψe〉 ≈ 25 nm2, which yields k21 〈ψhh| zz |ψe〉 ≈ 1%.

This can be visualized in Fig. D.1(a), where Gaussian slowly varying envelopes are assumed

with σe,z = 5 nm and σh,z = 3 nm. The contribution to the decay rate is therefore negligible,

cf. Fig. D.1(b).

137

Chapter D. Evaluation of the Second-Order Mesoscopic Moment Ωzzx

0 1 2 3 4 50

5

10

15

20

Electron−Hole Vertical Shift (nm)

|Ωzz

x/| (

nm2 )

(a)

μ

0 100 200 3000

1

2

3


(nm)

Nor

mal

ized

dec

ay r

ate Direct

DAInverted

(b)

Figure D.1: (a) |Ωzzx/µ| as a function of the electron-hole vertical shift. (b) Normalized decay

rate for |Ωzzx/µ| = 25 nm2. Note that the contribution has the same sign for both direct and

inverted structures. "DA" denotes the dipole approximation in which Ωzzx = 0.

138

Appendix E

The Unit-Cell Dipole

Approximation

Here we justify the use of the UCDA for a QD positioned in front of an interface. If the interface

is of dielectric nature, the UCDA is justied because the QD does not couple to k-vectors larger

than k1, and k1LUC 1. Here, LUC ∼ 0.5 nm denotes the lattice constant of the QD material.

At a metal interface the situation is changed by the coupling to surface plasmons and ohmic

losses for which |k| /k1 > 1. As the QD approaches the metal, it can decay into a larger number

of available optical modes with enhanced spatial frequencies. The UCDA is therefore valid for

suciently large distances from the metal when the spatial frequencies do not resolve the size of

an unit cell.

Formally, the UCDA is performed by pulling Gij(r, r′) outside an integration over an unit-cell,

i.e., ∫∫ ∞−∞

d3rd3r′u∗x(r)pius(r)Im Gij(r, r′)ux(r′)p∗ju∗s(r′)

SDA≈

Ncells∑q=1

Ncells∑q′=1

Im Gij(Rq,Rq′)∫

UC

d3ru∗x(r)pius(r)

∫UC

d3rux(r)p∗ju∗s(r)

(E.1)

The UCDA is equivalent to Im Gij(Rq,Rq′) being slowly-varying over the extent of an unit

cell for all q and q′ belonging to the exciton wavefunction. This is a more stringent requirement

than in the dipole approximation, where it is sucient to check the variation of←→G at the center

of the emitter (r0, r0).

Since the breakdown of UCDA is equivalent to kLUC 6 1, we dene a threshold wavevector

kth below which the UCDA is justied. We let kthLUC ∼ 0.1. We then employ the angular-

spectrum representation for Im←→G

=∫

dkρIm←→Gkρ

and evaluate the ratio η between the

integrands

ηij(ρ, φ, z, z0) =

∫∞kth

dkρImGij,kρ(ρ, φ, z, z0)

∫∞0

dkρImGij,kρ(ρ, φ, z, z0)

(E.2)

139

Chapter E. The Unit-Cell Dipole Approximation

0 5 10 15 200

1

2

3

4

5


Rat

io, η

xx(r

0,r0)

(%)

z−z0 (nm)

ρ (n

m)

−5 0 5−20

−10

0

10

20

−10

−8

−6

−4

Figure E.1: (a) The ratio ηxx evaluated within the DA as a function of z0. (b) log10(ηxx)

calculated beyond the DA as a function of ρ and z for z0 = 10 nm and φ = 0.

The UCDA is justied if this ratio is negligible for all (ρ, φ, z). In Fig. E.1 we plot ηxx within

and beyond the dipole approximation (the results for ηzz are very similar). While within the

dipole approximation the UCDA is valid at distances larger than about 5 nm, beyond the dipole

approximation the UCDA is fullled at distances larger than about 10 nm because the exciton

wavefunction has a nite height, which is assumed to equal 10 nm for this calculation.

In conclusion, we have argued that the UCDA is justied at distances larger than 10 nm from

a metal interface for QDs of common size.

140

Appendix F

Quantum Dots as Building Blocks

for Quantum Metamaterials

In the present work we have shown that In(Ga)As QDs probe electric and magnetic vacuum

uctuations through their mesoscopic nature. This is possible because the QDs have a large

mesoscopic moment Λ = (e/m0) 〈Ψg |xpz|Ψe〉 that may compete with the dipole moment µ =

(e/m0) 〈Ψg |px|Ψe〉 in light-matter interactions. The relevant mesoscopic strength Λ/µ was mea-

sured to be between 10 and 20 nm [26, 164]. The mesoscopic moment contains magnetic-dipole

m = Λ/2 and electric-quadrupole Q = Λ/2 contributions.

The excellent electric and magnetic properties of QDs could potentially be exploited in a

quantum metamaterial (QMM) with QDs as building blocks. Such a QMM would intertwine

the classical properties of a conventional metamaterial with the quantum nature of QDs. In a

QMM, the QDs would be arranged in a subwavelength lattice in three dimensions to provide an

eective-medium response for light. The basic idea is to use the QDs as point scatterers for light,

and is a dierent approach than the spontaneous-emission experiments that the present thesis

has focused on. In a scattering picture, light creates a microscopic polarization inside the QD

providing an electric and magnetic response to the incident electromagnetic eld. For incident

intensities signicantly below saturation, the QD behaves as a classical scatterer and only confers

its quantum nature to the statistics of the scattered light. Close to saturation, however, the full

quantum-mechanical nature is revealed owing to the strongly nonlinear optical response. In the

following we present the basic properties of classical scatterers and analyze the relevant gures

of merit of split-ring resonators and of QDs.

The incident light creates a microscopic polarization inside the scatterer with an electric

dipole moment p =←→α EEE and a magnetic dipole moment m =←→α HHH, where←→α EE and←→α HH

are the electric and magnetic polarizability tensors, respectively. For simplicity we neglect the

electric-quadrupole contribution in the following. In the most general case, the electric dipole

moment can be induced by both electric and magnetic elds [198], and same holds for the

141

Chapter F. Quantum Dots as Building Blocks for Quantum Metamaterials

magnetic dipole moment, i.e., (p

m

)=←→α

(E

H

), (F.1)

where ←→α is a 6× 6 polarizability tensor consisting of four 3× 3 blocks

←→α =

(←→α EE←→α EH

←→α HE←→α HH

). (F.2)

The o-diagonal blocks describe the cross-coupling between the external electric (magnetic)

eld and the magnetic (electric) dipole moment of the scatterer. There are two fundamen-

tal constraints governing the polarizability tensors: reciprocity and energy conservation. The

reciprocity theorem for electromagnetic elds, which states that swapping source and detector

does not modify the detected eld, demands the following Onsager relations for the polarizabil-

ities [198]

←→α EE =←→α TEE,

←→α HH =←→α THH,

←→α HE = −←→α TEH. (F.3)

Energy conservation demands extinction to equal scattering leading to the following optical

theorem for the polarizability

←→α −1 =←→α −10 −

2

3k3i←→1 , (F.4)

where ←→α 0 is the electrostatic polarizability derived from RLC circuit theory. Since the polariz-

ability is the key quantity determining the optical response of point scatterers, we analyze the

polarizability of split-ring resonators and of QDs in the following.

F.1 Polarizability of split-ring resonators

We consider a widely employed class of split-ring resonators with the split along the x axis as

shown in Fig. F.2(a). The contents of this section are partially adapted from Ref. [198]. The

incident electric eld polarized along the x direction generates an electric dipole moment p =

αxxEEExx, and induces a circulating current leading to a magnetic dipole moment m = αyxHEExy.

Similar reasoning applies for a driving magnetic eld along y. The resulting polarizability tensor

therefore takes the form

←→α SRR =

αxxEE 0 0 0 αxyEH 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

αyxHE 0 0 0 αyyHH 0

0 0 0 0 0 0

. (F.5)

142

Polarizability of split-ring resonators

Taking a Lorentzian frequency dependence of the tensor elements, the static polarizability can

be written as [198]

←→α SRR0 = α(ω)

ηE 0 0 0 iηC 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

−iηC 0 0 0 ηH 0

0 0 0 0 0 0

, (F.6)

where ηE, ηC and ηH are dimensionless quantities. The prefactor α(ω) exhibits the Lorentzian

dependence

α(ω) =ω2

0

ω20 − ω2 − iωΓ

× V, (F.7)

where ω0 is the resonance frequency, Γ the damping rate and V the optical volume. This static

polarizability can be converted into a dynamic polarizability using Eq. (F.4). Here and in the

following we are using a unit system that leads to a clear and intuitive physical interpretation by

treating electric and magnetic elds on equal footing. This allows dening electric and magnetic

quantities sharing the same physical units, which enhances the transparency of the analysis. The

conversion to SI units is straightforward and can be found in Ref. [198]. The magneto-electric

coupling coecient ηC must full the following constraint enforced by losses Im(α(ω)) ≥ 0

|ηC| ≤√ηEηH, (F.8)

which holds for all magneto-electric scatterers including split-ring resonators and QDs. A

metamaterial can be created by arranging the split-ring resonators in a subwavelength three-

dimensional lattice, in which case the eective electric permittivity ε and magnetic permeability

µ are spatial averages of the polarizabilities.

The quantities characterizing the optical response of a point scatterer are the optical volume

V , the damping rate Γ, and the electric ηE, magnetic ηH and magneto-electric ηC coupling

coecients. In the following we consider the properties of a split ring made of gold that is resonant

at λ = 1.5µm with the geometry presented in Fig. F.2(a): width and height of 200 nm, thickness

of 30 nm and a gap of 90 nm. This yields a large optical volume of V ∼ 106nm3. The large

damping rate of gold Γ = 1.25 · 1014s−1 leads to a small quality factor of the resonance of about

5 and corresponds to a modest enhancement of the polarizabilities αEE = 4.6V , αHH = 2.1V and

αEH = 2.5V , at resonance. Here we have used the optical coupling constants ηE = 0.7, ηH = 0.3

and ηC = 0.4 as argued in Ref. [198]. This can lead to a strong electric and magnetic response

by tting a large number N of split-ring resonators within an optical volume of λ3

ε = 1 +NαEE

λ3, (F.9)

µ = 1 +NαHH

λ3. (F.10)

Simple estimations show that for N ∼ 100, ε and µ can readily achieve negative values simulta-

neously. In the following we present and compare the scattering properties of QDs.

143


~ 200 nm

~ 2

00 n

m(a)

e h

~ 20 nm

~ 4 nm

(b)(a)

z

x

Figure F.1: Split-ring resonators and QDs as electric and magnetic point scatterers for light. (a)

A single split-ring resonator with the electric-dipole moment p across the split and the magnetic-

dipole moment m induced by the circulating current j(r). (b) A single In(Ga)As QD resembles

a split-ring resonator in many respects owing to the curved quantum-mechanical current density.

The small optical volume V of QDs is compensated by a large quality factor Q.

F.2 Polarizability of quantum dots

The quantum-mechanical current running through QDs is qualitatively similar to that of split-

ring resonators as seen in Fig. F.2. As a consequence, the polarizability of QDs below saturation

takes the form of Eq. (F.6). The quantities characterizing the optical response of QDs, namely

V , Γ, ηE,H,C, are, however, fundamentally dierent than of split-ring resonators. While Γ is a

well documented quantity and is about 1 ns−1, the magneto-electric coecients and the optical

volume are not widely known and we calculate them in the following.

The optical volume characterizes the interaction strength between an emitter and light. As

a consequence, it is a function of the electric oscillator strength fEE of QDs and reads [72]

VQD =2e2fEE

4πεm0ω20

, (F.11)

where ε is the permittivity of the background material. In(Ga)As QDs have an oscillator strength

of about 15 [84] yielding VQD ' 0.18 nm3, in strong contrast to the physical volume of QDs

of about 700 nm3. The reason for this discrepancy is the strong-connement regime of charge

carriers in which Coulomb correlations are destroyed by quantum connement. As a consequence,

the optical volume of strongly conned excitons does not benet from the multi-body nature

of QDs and equals the volume of one single unit cell, the building block of the underlying

semiconductor. In other words, the oscillator strength of one unit cell is spatially spread over

the entire volume of the QD but overall is not enhanced by cooperative eects. These heuristic

144

Quantum metamaterial with quantum dots

arguments are reected in the volume of one unit cell VUC = a3UC ' 0.22 nm3, which roughly

equals VQD. Here, aUC ' 0.605 nm [38] denotes the lattice constant of InAs. This limitation of

strongly conned excitons can be overcome in the weak-connement regime and is discussed at

length in Chapter 3.

The electric coupling coecient ηE = 1 because the QD optical volume was expressed in

terms of the electric oscillator strength in Eq. (F.11). The magneto-electric ηC and magnetic ηH

coecients are related to ηE through the gure of merit characterizing the magnetic light-matter

interaction k0nm/µ = k0nΛ/2µ, where n is the refractive index of the background material

ηE = 1, ηC = k0nΛ

2µ, ηH =

(k0n

Λ

2µ

)2

. (F.12)

The mesoscopic strengths of Λ/µ ∼ 10− 20 nm yield ηC = 0.11− 0.22 and ηH = 0.01− 0.05 for

current In(Ga)As QDs. Even though the electric and magnetic responses are more balanced for

split-ring resonators, the magnetic response of QDs can be increased by tailoring their size and

shape as demonstrated in Section 6. Now that we have estimated the quantities governing the

optical response of QDs, we compare the polarizability of QDs and of split-ring resonators in the

following.

There is a strong mismatch between the optical volume of the two scatterers of

VSRR

VQD' 5 · 106. (F.13)

This pronounced discrepancy is partly due to the large physical size of split-ring resonators

compared to QDs, and partly due to the small optical volume of strongly conned excitons in the

QD. This mismatch is, however, largely compensated by the enhanced quality factor Q = 2Γ/ω0

of the QD optical transition compared to the lossy split-ring resonator, see Fig. F.2(b), yielding

QQD

QSRR' 2 · 105. (F.14)

As a consequence, QDs attain electric polarizabilities as large as αQDEE ∼ 2 ·105nm3 on resonance,

which is one order of magnitude smaller than of split-ring resonators. Quantum dots may,

however, approach or even exceed the eective optical response of metamaterials based on split-

ring resonators because they are smaller and can be packed with a higher density inside a unit

volume, as argued in the following.

F.3 Quantum metamaterial with quantum dots

It is well known that quantum emitters interact through dipole-dipole forces. These forces are at

the heart of collective processes in many quantum systems in which quantum emitters are situated

at nanoscale proximity of one another. The energy transferred from one emitter (further called

"donor") to another (further called "acceptor") may alter the decay rate and resonance frequency

of the donor. In this work we wish to estimate these eects between adjacent QDs in a QMM

145


−10 −5 0 5 10−200

0

200

400

600

Detuning (µeV)

ε QD

Re(ε)Im(ε)

(a)

−10 −5 0 5 10−1

0

1

2

3

Detuning (µeV)

µQ

D

Re(µ)Im(µ)

(b)

Figure F.2: (a) Electric and (b) magnetic response of a QD-based QMM. The following param-

eters have been used: Λ/µ = 20 nm, λ0 = 1µm.

to determine whether QDs can be treated as independent entities. The rate of energy transfer

ΓD→A between a donor and acceptor is [72]

ΓD→A

Γ=

(R0

R

)6

, (F.15)

where R is the distance between the emitters and R0 is the distance at which the rate of energy

transfer equals the decay rate Γ of the acceptor in the absence of the donor. It can be shown [72]

that R0 can be expressed as

R60 =

9c40κ2

8π

∫ ∞0

sD(ω)σA(ω)

n4(ω)ω4dω, (F.16)

where sD(ω) denotes the spectral overlap between the donor and the acceptor, σA(ω) the ab-

sorption cross-section of the acceptor, and the factor κ describes the relative orientation of the

dipoles and is given by

κ2 = [nA · nD − 3 (nR · nD) (nR · nA)]2. (F.17)

Evaluating R0 for QDs assuming perfect spectral overlap sD = 1 results in

R60 =

3c30κ2

2n3ω20Γ× VQD, (F.18)

and yields R0 ' 50 nm for QDs. To simplify the discussion, we consider such a distance between

adjacent QDs in the QMM, so that the approximation of non-interacting emitters holds reason-

ably well. Extending the description to cover interacting molecular arrays of QDs can be done

with the formalism presented in Ref. [198].

Simple estimations show that negative values for ε and µ can readily be achieved, if QDs are

placed in a subwavelength cubic lattice with the lattice constant of 50 nm, which leads to a QD

density of 8000µm−3. The electric and magnetic responses are plotted in Fig. F.2 showing strong

146

Quantum metamaterial with quantum dots

optical response at the QD transition frequency. The magnetic response is weaker for current

In(Ga)As QDs but may potentially be enhanced by tailoring the QD geometry during growth,

see Chapter 6 for details. Increasing Λ by a factor of 4 would render the magnetic and electric

couplings equal resulting in complete impedance match for the incident wave. Such a behavior

is highly demanded in optical cloaks [199], where light is following well-dened paths to hide an

object, and removing reections are key to successful cloaking.

Aside from electric and magnetic response, QDs have strong magneto-electric coupling ηC.

As a consequence, a QD-based QMM would exhibit structural chirality and circular dichroism.

Furthermore, since QDs do not have an irreversible loss channel similar to split-ring resonators,

they could operate with signicantly reduced losses in a metamaterial with a properly designed

structure. Placing a quantum emitter in such a low-loss medium may provide rich light-matter

interaction dynamics with a strong electric and magnetic character. So far we have presented the

optical response of QDs below saturation, i.e., when the induced polarization depends linearly on

external elds. The quantum nature of QDs, with the maximum capacity of one single excitation,

can be easily saturated leading to strongly nonlinear optical response. This may lead to new

and so far unexplored eects at the intersection between quantum optics, solid-state physics and

photonics. Quantifying these aspects is beyond the scope of this work and will be conducted

elsewhere. In the following we comment on the practical challenges of building a QD-based

QMM.

First and foremost, a QMM would require QDs placed at well-dened positions in a three-

dimensional structure, which is within reach of the currently employed growth techniques. In

particular, it has been demonstrated that In(Ga)As QDs grown in periodically arranged pits of

a pre-patterned substrate exhibit good optical properties [200]. This two-dimensional array of

QDs can be converted into a three-dimensional structure by vertical stacking several QD layers.

The strain eld stored in the vicinity of the QDs ensures spatially ordered growth also in the

third dimension. This technique has proven to yield QDs with excellent optical properties, such

as narrow linewidths, background-free single-photon emission and highly indistinguishable pho-

tons [201]. Despite these promising prospects, currently employed QDs suer from inhomogenous

broadening of their emission, which is induced by uctuations in their size and shape. Each QD

has its own spectral emission window destroying the spectral overlap among many such QDs

and impeding their collective contribution to the optical response. Solving this extraordinary

challenge would propel immensely the development of solid-state quantum technologies and, in

particular, would render the realization of a QD-based QMM experimentally accessible.

147

Bibliography

[1] Kelvin, L. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of

Science 2, 1 (1901).

[2] De Broglie, L. Foundations of Physics 1, 5 (1970).

[3] Einstein, A. Relativity: The special and general theory. Penguin, (1920).

[4] Mandelshtam, L. and Tamm, I. J. Phys. (USSR) 9, 122 (1945).

[5] Purcell, E. Phys. Rev. 69, 674 (1946).

[6] Lamb, W. E. and Retherford, R. C. Phys. Rev. 72, 241 (1947).

[7] Casimir, H. B. In Proc. K. Ned. Akad. Wet, volume 51, 150, (1948).

[8] Bell, J. S. Rev. Mod. Phys. 38, 447 (1966).

[9] Einstein, A., Podolsky, B., and Rosen, N. Phys. Rev. 47, 777 (1935).

[10] Raimond, J., Brune, M., and Haroche, S. Rev. Mod. Phys. 73, 565 (2001).

[11] von Klitzing, K. Rev. Mod. Phys. 58, 519 (1986).

[12] Baibich, M. N., Broto, J. M., Fert, A., Van Dau, F. N., Petro, F., Etienne, P., Creuzet,

G., Friederich, A., and Chazelas, J. Phys. Rev. Lett. 61, 2472 (1988).

[13] De Heer, W. A., Chatelain, A., and Ugarte, D. Science 270, 1179 (1995).

[14] Novoselov, K., Geim, A. K., Morozov, S., Jiang, D., Katsnelson, M., Grigorieva, I.,

Dubonos, S., and Firsov, A. Nature 438, 197 (2005).

[15] Shockley, W. Bell System Technical Journal 28, 435 (1949).

[16] Moore, G. E. Proc. IEEE 86, 82 (1998).

[17] Rauch, J. Atlantic Monthly 287, 35 (2001).

[18] Bednorz, J. G. and Müller, K. A. Zeitschrift für Physik B Condensed Matter 64, 189

(1986).

149

BIBLIOGRAPHY

[19] Monz, T., Schindler, P., Barreiro, J. T., Chwalla, M., Nigg, D., Coish, W. A., Harlander,

M., Hänsel, W., Hennrich, M., and Blatt, R. Phys. Rev. Lett. 106, 130506 (2011).

[20] Peter, E., Senellart, P., Martrou, D., Lemaître, A., Hours, J., Gérard, J., and Bloch, J.

Phys. Rev. Lett. 95, 067401 (2005).

[21] Arcari, M., Söllner, I., Javadi, A., Lindskov Hansen, S., Mahmoodian, S., Liu, J.,

Thyrrestrup, H., Lee, E. H., Song, J. D., Stobbe, S., and Lodahl, P. Phys. Rev. Lett.

113, 093603 (2014).

[22] Lodahl, P., Mahmoodian, S., and Stobbe, S. arXiv/1312.1079 (2013).

[23] Kaer, P., Lodahl, P., Jauho, A.-P., and Mork, J. Phys. Rev. B 87, 081308 (2013).

[24] Madsen, K. H., Kaer, P., Kreiner-Møller, A., Stobbe, S., Nysteen, A., Mørk, J., and Lodahl,

P. Phys. Rev. B 88, 045316 (2013).

[25] Chekhovich, E., Makhonin, M., Tartakovskii, A., Yacoby, A., Bluhm, H., Nowack, K., and

Vandersypen, L. Nat. Mat. 12, 494 (2013).

[26] Andersen, M. L., Stobbe, S., Sørensen, A. S., and Lodahl, P. Nat. Phys. 7, 215 (2011).

[27] Gammon, D., Snow, E., Shanabrook, B., Katzer, D., and Park, D. Phys. Rev. Lett. 76,

3005 (1996).

[28] Smith, D. R., Pendry, J. B., and Wiltshire, M. C. Science 305, 788 (2004).

[29] Shalimova, K. V. Semiconductor Physics (in Russian). Energoatomizdat., (1985).

[30] Rosencher, E. and Vinter, B. Optoelectronics. Cambridge University Press, (2002).

[31] Miller, D. A. B. Quantum Mechanics for Scientists and Engineers. Cambridge University

Press, (2007).

[32] Kohn, W., Becke, A. D., and Parr, R. G. J. Phys. Chem. 100, 12974 (1996).

[33] Joannopoulos, J. D., Meade, R. D., and Winn, J. N. Photonic Crystals. Molding the ow

of light. (1995).

[34] Cardona, M. and Peter, Y. Y. Fundamentals of semiconductors. Springer, (2005).

[35] Chelikowsky, J. R. and Cohen, M. L. Phys. Rev. B 14, 556 (1976).

[36] Coldren, L. A., Corzine, S. W., and Mashanovitch, M. L. Diode lasers and photonic

integrated circuits. First edition. John Wiley & Sons, (1995).

[37] Luttinger, J. and Kohn, W. Phys. Rev. 97, 869 (1955).

[38] Vurgaftman, I., Meyer, J., and Ram-Mohan, L. J. Appl. Phys. 89, 5815 (2001).

150

BIBLIOGRAPHY

[39] Michler, P., Kiraz, A., Becher, C., Schoenfeld, W., Petro, P., Zhang, L., Hu, E., and

Imamoglu, A. Science 290, 2282 (2000).

[40] Kristensen, P. T. Light-matter interaction in nano-structures materials. PhD thesis, De-

partment of Photonics Engineering at the Technical University of Denmark, (2009).

[41] Söllner, I., Mahmoodian, S., Javadi, A., and Lodahl, P. arXiv/1406.4295 (2014).

[42] Moison, J., Houzay, F., Barthe, F., Leprince, L., Andre, E., and Vatel, O. Appl. Phys.

Lett. 64, 196 (1994).

[43] Biasiol, G. and Heun, S. Phys. Rep. 500.

[44] Bruls, D., Vugs, J., Koenraad, P., Salemink, H., Wolter, J., Hopkinson, M., Skolnick, M.,

Long, F., and Gill, S. Appl. Phys. Lett. 81, 1708 (2002).

[45] Koguchi, N., Ishige, K., and Takahashi, S. Journal of Vacuum Science & Technology B:

Microelectronics and Nanometer Structures 11, 787 (1993).

[46] Kohn, W. and Sham, L. J. Phys. Rev. A 140, 1133 (1965).

[47] Puschnig, P., Berkebile, S., Fleming, A. J., Koller, G., Emtsev, K., Seyller, T., Riley, J. D.,

Ambrosch-Draxl, C., Netzer, F. P., and Ramsey, M. G. Science 326, 702 (2009).

[48] Jiang, H., Baranger, H. U., and Yang, W. Phys. Rev. B 68, 165337 (2003).

[49] Bester, G. J. Phys. Cond. Matt. 21, 023202 (2009).

[50] Majewski, J. A., Birner, S., Trellakis, A., Sabathil, M., and Vogl, P. Phys. Stat. Solidi (c)

1, 2003 (2004).

[51] Stier, O., Grundmann, M., and Bimberg, D. Phys. Rev. B 59, 5688 (1999).

[52] Bastard, G. Phys. Rev. B 25, 7584 (1982).

[53] Voon, L. C. L. Y. and Willatzen, M. The kp method: electronic properties of semiconduc-

tors. Springer, (2009).

[54] Burt, M. J. Cond. Matt. 11, 53 (1999).

[55] Foreman, B. A. Phys. Rev. B 48, 4964 (1993).

[56] Andreani, L. C., Panzarini, G., and Gérard, J.-M. Phys. Rev. B 60, 13276 (1999).

[57] Tighineanu, P., Daveau, R., Lee, E. H., Song, J. D., Stobbe, S., and Lodahl, P. Phys. Rev.

B 88, 155320 (2013).

[58] Harrison, P. Quantum wells, wires and dots: theoretical and computational physics of

semiconductor nanostructures. John Wiley & Sons, (2005).

151

BIBLIOGRAPHY

[59] Dreiser, J., Atatüre, M., Galland, C., Müller, T., Badolato, A., and Imamoglu, A. Phys.

Rev. B 77, 075317 (2008).

[60] Schmitt-Rink, S., Miller, D. A. B., and Chemla, D. S. Phys. Rev. B 35, 8113 (1987).

[61] Huo, Y., Witek, B., Kumar, S., Cardenas, J., Zhang, J., Akopian, N., Singh, R., Zallo, E.,

Grifone, R., Kriegner, D., Trotta, R., Ding, F., Stangl, J., Zwiller, V., Bester, G., Rastelli,

A., and Schmidt, O. G. Nat. Phys. (2013).

[62] Pryor, C. Phys. Rev. B 57, 7190 (1998).

[63] Landau, L. D. and Lifshitz, E. M. Elasticity Theory (in Russian). Nauka, (1987).

[64] Anselm, A. I. Introduction to semiconductor theory. Mir Publishers, (1981).

[65] Bayer, M., Ortner, G., Stern, O., Kuther, A., Gorbunov, A. A., Forchel, A., Hawrylak,

P., Fafard, S., Hinzer, K., Reinecke, T. L., Walck, S. N., Reithmaier, J. P., Klopf, F., and

Schäfer, F. Phys. Rev. B 65, 195315 (2002).

[66] van Kesteren, H. W., Cosman, E. C., van der Poel, W. A. J. A., and Foxon, C. T. Phys.

Rev. B 41, 5283 (1990).

[67] Trebin, H.-R., Rössler, U., and Ranvaud, R. Phys. Rev. B 20, 686 (1979).

[68] Bester, G., Nair, S., and Zunger, A. Phys. Rev. B 67, 161306 (2003).

[69] Ghali, M., Ohtani, K., Ohno, Y., and Ohno, H. Nat. Commun. 3, 661 (2012).

[70] Kuo, Y.-H., Lee, Y. K., Ge, Y., Ren, S., Roth, J. E., Kamins, T. I., Miller, D. A., and

Harris, J. S. Nature 437, 1334 (2005).

[71] Guest, J., Stievater, T., Li, X., Cheng, J., Steel, D., Gammon, D., Katzer, D., Park, D.,

Ell, C., Thränhardt, A., Khitrova, G., and Gibbs, H. M. Phys. Rev. B 65, 241310 (2002).

[72] Novotny, L. and Hecht, B. Principles of nano-optics. Cambridge university press, (2012).

[73] Paulus, M., Gay-Balmaz, P., and Martin, O. J. Phys. Rev. E 62, 5797 (2000).

[74] Tai, C.-T. Dyadic Green functions in electromagnetic theory, volume 272. IEEE press New

York, (1994).

[75] Gerry, C. and Knight, P. Introductory quantum optics. Cambridge university press, (2005).

[76] Drexhage, K. Journal of Luminescence 1, 693 (1970).

[77] Englund, D., Faraon, A., Fushman, I., Stoltz, N., Petro, P., and Vu£kovi¢, J. Nature 450,

857 (2007).

[78] Claudon, J., Bleuse, J., Malik, N. S., Bazin, M., Jarennou, P., Gregersen, N., Sauvan, C.,

Lalanne, P., and Gérard, J.-M. Nat. Photon. 4, 174 (2010).

152

BIBLIOGRAPHY

[79] Brune, M., Schmidt-Kaler, F., Maali, A., Dreyer, J., Hagley, E., Raimond, J., and Haroche,

S. Phys. Rev. Lett. 76, 1800 (1996).

[80] Hennessy, K., Badolato, A., Winger, M., Gerace, D., Atatüre, M., Gulde, S., Fält, S., Hu,

E. L., and Imamo§lu, A. Nature 445, 896 (2007).

[81] Craig, D. P. and Thirunamachandran, T. Molecular quantum electrodynamics: an intro-

duction to radiation-molecule interactions. Courier Dover Publications, (1998).

[82] Vats, N., John, S., and Busch, K. Phys. Rev. A 65, 043808 (2002).

[83] Stobbe, S., Kristensen, P. T., Mortensen, J. E., Hvam, J. M., Mørk, J., and Lodahl, P.

Phys. Rev. B 86, 085304 (2012).

[84] Johansen, J., Stobbe, S., Nikolaev, I. S., Lund-Hansen, T., Kristensen, P. T., Hvam, J. M.,

Vos, W. L., and Lodahl, P. Phys. Rev. B 77, 073303 (2008).

[85] Yao, P., Manga Rao, V., and Hughes, S. Laser & Photonics Reviews 4, 499 (2010).

[86] Stobbe, S., Johansen, J., Kristensen, P. T., Hvam, J. M., and Lodahl, P. Phys. Rev. B 80,

155307 (2009).

[87] Johansen, J., Julsgaard, B., Stobbe, S., Hvam, J. M., and Lodahl, P. Phys. Rev. B 81,

081304 (2010).

[88] Kuhr, S., Gleyzes, S., Guerlin, C., Bernu, J., Ho, U. B., Deléglise, S., Osnaghi, S., Brune,

M., Raimond, J.-M., Haroche, S., et al. Appl. Phys. Lett. 90, 164101 (2007).

[89] Johansson, J., Saito, S., Meno, T., Nakano, H., Ueda, M., Semba, K., and Takayanagi, H.

Phys. Rev. Lett. 96, 127006 (2006).

[90] Kippenberg, T. J. and Vahala, K. J. Science 321, 1172 (2008).

[91] Khitrova, G., Gibbs, H., Kira, M., Koch, S., and Scherer, A. Nat. Phys. 2, 81 (2006).

[92] Hoeppe, U., Wol, C., Küchenmeister, J., Niegemann, J., Drescher, M., Benner, H., and

Busch, K. Phys. Rev. Lett. 108, 043603 (2012).

[93] Tiecke, T., Thompson, J., de Leon, N., Liu, L., Vuleti¢, V., and Lukin, M. Nature 508,

241 (2014).

[94] Niemczyk, T., Deppe, F., Huebl, H., Menzel, E., Hocke, F., Schwarz, M., Garcia-Ripoll,

J., Zueco, D., Hümmer, T., Solano, E., et al. Nat. Phys. 6, 772 (2010).

[95] Dicke, R. Phys. Rev. 93, 99 (1954).

[96] Haroche, S. Rev. Mod. Phys. 85, 1083 (2013).

[97] DeVoe, R. and Brewer, R. Phys. Rev. Lett. 76, 2049 (1996).

153

BIBLIOGRAPHY

[98] Röhlsberger, R., Schlage, K., Sahoo, B., Couet, S., and Rüer, R. Science 328, 1248

(2010).

[99] Baumann, K., Guerlin, C., Brennecke, F., and Esslinger, T. Nature 464, 1301 (2010).

[100] Mlynek, J., Abdumalikov, A., Eichler, C., and Wallra, A. Nat. Commun. 5, 5186 (2014).

[101] Bohnet, J. G., Chen, Z., Weiner, J. M., Meiser, D., Holland, M. J., and Thompson, J. K.

Nature 484, 78 (2012).

[102] Scully, M. O., Fry, E. S., Ooi, C. R., and Wódkiewicz, K. Phys. Rev. Lett. 96, 010501

(2006).

[103] Svidzinsky, A. A., Yuan, L., and Scully, M. O. Phys. Rev. X 3, 041001 (2013).

[104] Duan, L.-M., Lukin, M., Cirac, J. I., and Zoller, P. Nature 414, 413 (2001).

[105] Hammerer, K., Sørensen, A. S., and Polzik, E. S. Rev. Mod. Phys. 82, 1041 (2010).

[106] Stobbe, S., Schlereth, T. W., Höing, S., Forchel, A., Hvam, J. M., and Lodahl, P. Phys.

Rev. B 82, 233302 (2010).

[107] Kuhlmann, A. V., Houel, J., Ludwig, A., Greuter, L., Reuter, D., Wieck, A. D., Poggio,

M., and Warburton, R. J. Nat. Phys. 9, 570 (2013).

[108] Scully, M. O. and Svidzinsky, A. A. Science 325, 1510 (2009).

[109] Hanamura, E. Phys. Rev. B 37, 1273 (1988).

[110] Rashba, E. and Gurgenishvili, G. Sov. Phys. Solid State 4, 759 (1962).

[111] Takagahara, T. and Hanamura, E. Phys. Rev. Lett. 56, 2533 (1986).

[112] Greene, R. L., Bajaj, K. K., and Phelps, D. E. Phys. Rev. B 29, 1807 (1984).

[113] Bastard, G., Mendez, E. E., Chang, L. L., and Esaki, L. Phys. Rev. B 26, 1974 (1982).

[114] Poem, E., Kodriano, Y., Tradonsky, C., Lindner, N. H., Gerardot, B. D., Petro, P. M.,

and Gershoni, D. Nat. Phys. 6, 993 (2010).

[115] Reithmaier, J. P., Löer, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L. V.,

Kulakovskii, V. D., Reinecke, T. L., and Forchel, A. Nature 432, 197 (2004).

[116] Diniz, I., Portolan, S., Ferreira, R., Gérard, J., Bertet, P., and Auèves, A. Phys. Rev. A

84, 063810 (2011).

[117] Hours, J., Senellart, P., Peter, E., Cavanna, A., and Bloch, J. Phys. Rev. B 71, 161306

(2005).

[118] Reitzenstein, S., Münch, S., Franeck, P., Rahimi-Iman, A., Löer, A., Höing, S.,

Worschech, L., and Forchel, A. Phys. Rev. Lett. 103, 127401 (2009).

154

BIBLIOGRAPHY

[119] Davanço, M., Hellberg, C. S., Ates, S., Badolato, A., and Srinivasan, K. Phys. Rev. B 89,

161303 (2014).

[120] Tighineanu, P., Daveau, R., Lehmann, T. B., Beere, H. E., Ritchie, D. A., Lodahl, P., and

Stobbe, S. submitted (2015).

[121] Watanabe, K., Koguchi, N., and Gotoh, Y. Jpn. J. Appl. Phys. 39, 79 (2000).

[122] Watanabe, K., Tsukamoto, S., Gotoh, Y., and Koguchi, N. Journal of Crystal Growth 227,

1073 (2001).

[123] Mantovani, V., Sanguinetti, S., Guzzi, M., Grilli, E., Gurioli, M., Watanabe, K., and

Koguchi, N. J. Appl. Phys. 96, 4416 (2004).

[124] Wang, Z. M., Holmes, K., Mazur, Y. I., Ramsey, K. A., and Salamo, G. J. Nanoscale

Research Letters 1, 57 (2006).

[125] Ha, S.-K., Bounouar, S., Song, J. D., Lim, J. Y., Donatini, F., Dang, L. S., Poizat, J. P.,

Kim, J. S., Choi, W. J., and Han, I. K. In AIP Conference Proceedings, volume 1399, 555,

(2011).

[126] Mano, T., Abbarchi, M., Kuroda, T., Mastrandrea, C. A., Vinattieri, A., Sanguinetti, S.,

Sakoda, K., and Gurioli, M. Nanotechnology 20, 395601 (2009).

[127] Rol, F., Founta, S., Mariette, H., Daudin, B., Dang, L. S., Bleuse, J., Peyrade, D., Gérard,

J.-M., and Gayral, B. Phys. Rev. B 75, 125306 (2007).

[128] Horikoshi, Y., Kawashima, M., and Yamaguchi, H. Jpn. J. Appl. Phys. 27, 169 (1988).

[129] Moon, P., Lee, J. D., Ha, S. K., Lee, E. H., Choi, W. J., Song, J. D., Kim, J. S., and Dang,

L. S. Phys. Status Solidi (RRL) 6, 445 (2012).

[130] Abbarchi, M., Mastrandrea, C. A., Kuroda, T., Mano, T., Sakoda, K., Koguchi, N., San-

guinetti, S., Vinattieri, A., and Gurioli, M. Phys. Rev. B 78, 125321 (2008).

[131] Kuroda, T., Sanguinetti, S., Gurioli, M., Watanabe, K., Minami, F., and Koguchi, N.

Phys. Rev. B 66, 121302 (2002).

[132] Kuroda, T., Belhadj, T., Abbarchi, M., Mastrandrea, C., Gurioli, M., Mano, T., Ikeda,

N., Sugimoto, Y., Asakawa, K., Koguchi, N., Sakoda, K., Urbaszek, B., Amand, T., and

Marie, X. Phys. Rev. B 79, 035330 (2009).

[133] Abbarchi, M., Mastrandrea, C., Kuroda, T., Vinattieri, A., Mano, T., Koguchi, N., Sakoda,

K., and Gurioli, M. Phys. Status Solidi (c) 6, 886 (2009).

[134] Abbarchi, M., Troiani, F., Mastrandrea, C., Goldoni, G., Kuroda, T., Mano, T., Sakoda,

K., Koguchi, N., Sanguinetti, S., and Vinattieri, A. Appl. Phys. Lett. 93, 162101 (2008).

155

BIBLIOGRAPHY

[135] Miller, D. A. B., Chemla, D. S., Damen, T. C., Gossard, A. C., Wiegmann, W., Wood,

T. H., and Burrus, C. A. Phys. Rev. Lett. 53, 2173 (1984).

[136] Berthelot, A., Favero, I., Cassabois, G., Voisin, C., Delalande, C., Roussignol, P., Ferreira,

R., and Gérard, J.-M. Nat. Phys. 2, 759 (2006).

[137] Narvaez, G. A., Bester, G., Franceschetti, A., and Zunger, A. Phys. Rev. B 74, 205422

(2006).

[138] Wei, S.-H. and Zunger, A. Appl. Phys. Lett. 72, 2011 (1998).

[139] Baylac, B., Marie, X., Amand, T., Brousseau, M., Barrau, J., and Shekun, Y. Surface

Science 326, 161 (1995).

[140] Einevoll, G. T. Phys. Rev. B 45, 3410 (1992).

[141] Besombes, L., Kheng, K., Marsal, L., and Mariette, H. Phys. Rev. B 63, 155307 (2001).

[142] Favero, I., Cassabois, G., Ferreira, R., Darson, D., Voisin, C., Tignon, J., Delalande, C.,

Bastard, G., Roussignol, P., and Gérard, J. M. Phys. Rev. B 68, 233301 (2003).

[143] Peter, E., Hours, J., Senellart, P., Vasanelli, A., Cavanna, A., Bloch, J., and Gérard, J. M.

Phys. Rev. B 69, 041307 (2004).

[144] Feldmann, J., Peter, G., Göbel, E., Dawson, P., Moore, K., Foxon, C., and Elliott, R.

Phys. Rev. Lett. 59, 2337 (1987).

[145] Fiore, A., Borri, P., Langbein, W., Hvam, J. M., Oesterle, U., Houdre, R., Stanley, R., and

Ilegems, M. Appl. Phys. Lett. 76, 3430 (2000).

[146] Butov, L. V., Imamoglu, A., Mintsev, A. V., Campman, K. L., and Gossard, A. C. Phys.

Rev. B 59, 1625 (1999).

[147] Koch, S. W., Kira, M., Khitrova, G., and Gibbs, H. M. Nature Materials 5, 523 (2006).

[148] Citrin, D. Superlattices and microstructures 13, 303 (1993).

[149] Wang, G., Fafard, S., Leonard, D., Bowers, J. E., Merz, J. L., and Petro, P. M. Appl.

Phys. Lett. 64, 2815 (1994).

[150] Yang, W., Lowe-Webb, R. R., Lee, H., and Sercel, P. C. Phys. Rev. B 56, 13314 (1997).

[151] Sanguinetti, S., Henini, M., Grassi Alessi, M., Capizzi, M., Frigeri, P., and Franchi, S.

Phys. Rev. B 60, 8276 (1999).

[152] Dai, Y., Fan, J., Chen, Y., Lin, R., Lee, S., and Lin, H. J. Appl. Phys. 82, 4489 (1997).

[153] Mukai, K., Ohtsuka, N., and Sugawara, M. Appl. Phys. Lett. 70, 2416 (1997).

[154] Wu, Y.-h., Arai, K., and Yao, T. Phys. Rev. B 53, R10485 (1996).

156

BIBLIOGRAPHY

[155] Mahan, G. Many Particle Physics. Physics of Solids and Liquids. Springer, (2000).

[156] Takagahara, T. Phys. Rev. B 60, 2638 (1999).

[157] Adachi, S., of Electrical Engineers, I., and Service), I. I. Properties of aluminium gallium

arsenide. EMIS datareviews series. INSPEC - The Institution of Electrical Engineers,

(1993).

[158] Zurita-Sánchez, J. R. and Novotny, L. JOSA B 19, 1355 (2002).

[159] Zurita-Sánchez, J. R. and Novotny, L. JOSA B 19, 2722 (2002).

[160] DeBeer George, S., Petrenko, T., and Neese, F. Inorganica Chimica Acta 361, 965 (2008).

[161] Bernadotte, S., Atkins, A. J., and Jacob, C. R. J. Chem. Phys. 137, 204106 (2012).

[162] Tighineanu, P., Andersen, M. L., Sørensen, A. S., Stobbe, S., and Lodahl, P. Phys. Rev.

Lett. 113, 043601 (2014).

[163] Maier, S. A. Plasmonics: fundamentals and applications. Springer Science+Business Me-

dia, (2007).

[164] Tighineanu, P., Sørensen, A. S., Stobbe, S., and Lodahl, P. arXiv/1409.0032 (2014).

[165] Chen, X.-W., Agio, M., and Sandoghdar, V. Phys. Rev. Lett. 108, 233001 (2012).

[166] Wang, Q., Stobbe, S., and Lodahl, P. Phys. Rev. Lett. 107, 167404 (2011).

[167] Curto, A. G., Taminiau, T. H., Volpe, G., Kreuzer, M. P., Quidant, R., and van Hulst,

N. F. Nat. Commun. 4, 1750 (2013).

[168] Sanchez-Burillo, E., Zueco, D., Garcia-Ripoll, J. J., and Martin-Moreno, L. Phys. Rev.

Lett. 113, 263604 (2014).

[169] Goupalov, S. V. Phys. Rev. B 68, 125311 (2003).

[170] Sugawara, M. Phys. Rev. B 51, 10743 (1995).

[171] Thränhardt, A., Ell, C., Khitrova, G., and Gibbs, H. M. Phys. Rev. B 65, 035327 (2002).

[172] Jun Ahn, K. and Knorr, A. Phys. Rev. B 68, 161307 (2003).

[173] Kristensen, P. T., Mortensen, J. E., Lodahl, P., and Stobbe, S. Phys. Rev. B 88, 205308

(2013).

[174] Eisele, H., Lenz, A., Heitz, R., Timm, R., Dähne, M., Temko, Y., Suzuki, T., and Jacobi,

K. J. Appl. Phys. 104, 124301 (2008).

[175] Ebiko, Y., Muto, S., Suzuki, D., Itoh, S., Shiramine, K., Haga, T., Nakata, Y., and

Yokoyama, N. Phys. Rev. Lett. 80, 2650 (1998).

157

BIBLIOGRAPHY

[176] Tatarakis, M., Watts, I., Beg, F., Clark, E., Dangor, A., Gopal, A., Haines, M., Norreys,

P., Wagner, U., Wei, M.-S., Zepf, M., and Krushelnick, K. Nature 415, 280 (2002).

[177] Noecker, M., Masterson, B., and Wieman, C. Phys. Rev. Lett. 61, 310 (1988).

[178] Rukhlenko, I. D., Handapangoda, D., Premaratne, M., Fedorov, A. V., Baranov, A. V.,

and Jagadish, C. Opt. Express 17, 17570 (2009).

[179] Taminiau, T. H., Karaveli, S., van Hulst, N. F., and Zia, R. Nat. Commun. 3, 979 (2012).

[180] Slepyan, G. Y., Magyarov, A., Maksimenko, S. A., and Homann, A. Phys. Rev. B 76,

195328 (2007).

[181] Wood, C., Bennett, S., Cho, D., Masterson, B., Roberts, J., Tanner, C., and Wieman, C.

Science 275, 1759 (1997).

[182] Ciracì, C., Hill, R., Mock, J., Urzhumov, Y., Fernández-Domínguez, A., Maier, S., Pendry,

J., Chilkoti, A., and Smith, D. Science 337, 1072 (2012).

[183] Toscano, G., Raza, S., Yan, W., Jeppesen, C., Xiao, S., Wubs, M., Jauho, A.-P., Bozhevol-

nyi, S. I., and Mortensen, N. A. Nanophotonics 2, 161 (2013).

[184] Soukoulis, C. M. and Wegener, M. Science 330, 1633 (2010).

[185] Pendry, J. B., Schurig, D., and Smith, D. R. Science 312, 1780 (2006).

[186] Burresi, M., Van Oosten, D., Kampfrath, T., Schoenmaker, H., Heideman, R., Leinse, A.,

and Kuipers, L. Science 326, 550 (2009).

[187] Kalkman, J., Gersen, H., Kuipers, L., and Polman, A. Phys. Rev. B 73, 075317 (2006).

[188] Chang, D. E., Sørensen, A. S., Hemmer, P. R., and Lukin, M. D. Phys. Rev. B 76, 035420

(2007).

[189] Landau, L. D., Lif²ic, E. M., Sykes, J. B., Bell, J. S., Kearsley, M., and Pitaevskii, L. P.

Electrodynamics of continuous media. Pergamon Press Oxford, (1960).

[190] Klimov, V. V. and Ducloy, M. Phys. Rev. A 69, 013812 (2004).

[191] Chen, Y., Nielsen, T. R., Gregersen, N., Lodahl, P., and Mørk, J. Phys. Rev. B 81, 125431

(2010).

[192] Schuller, J. A., Barnard, E. S., Cai, W., Jun, Y. C., White, J. S., and Brongersma, M. L.

Nat. Mat. 9, 193 (2010).

[193] Atwater, H. A. and Polman, A. Nat. Mat. 9, 205 (2010).

[194] Kazimierczuk, T., Fröhlich, D., Scheel, S., Stolz, H., and Bayer, M. Nature 514, 343 (2014).

158

BIBLIOGRAPHY

[195] Srivastava, A., Sidler, M., Allain, A. V., Lembke, D. S., Kis, A., and Imamoglu, A.

arXiv/1411.0025 (2014).

[196] García-Meca, C., Hurtado, J., Martí, J., Martínez, A., Dickson, W., and Zayats, A. V.

Phys. Rev. Lett. 106, 067402 (2011).

[197] Winkler, R. Spin-orbit coupling eects in two-dimensional electron and hole systems. Num-

ber 191. Springer Science & Business Media, (2003).

[198] Sersic, I., Tuambilangana, C., Kampfrath, T., and Koenderink, A. F. Phys. Rev. B 83,

245102 (2011).

[199] Schurig, D., Mock, J., Justice, B., Cummer, S. A., Pendry, J., Starr, A., and Smith, D.

Science 314, 977 (2006).

[200] Skiba-Szymanska, J., Jamil, A., Farrer, I., Ward, M. B., Nicoll, C. A., Ellis, D. J., Griths,

J. P., Anderson, D., Jones, G. A., Ritchie, D. A., and Shields, A. J. Nanotechnology 22,

065302 (2011).

[201] Jons, K., Atkinson, P., Muller, M., Heldmaier, M., Ulrich, S., Schmidt, O., and Michler,

P. Nano Lett. 13, 126 (2012).

159

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