Coupling phenomena of quantum dots to photonic crystal modes

Lehrstuhl fur experimentelle Halbleiterphysik E24

Walter Schottky Institut, Zentrum fur Nanotechnologie und Nanomaterialien

Technische Universitat Munchen

Semiconductor Quantum Optics with TailoredPhotonic Nanostructures

Arne Laucht

Vollstandiger Abdruck der von der Fakultat fur Physik der Technischen Universitat

Munchen zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Dr. H. Friedrich

Prufer der Dissertation:

1. Univ.-Prof. J. J. Finley, Ph.D.

2. Univ.-Prof. Dr. A. Holleitner

Die Dissertation wurde am 10.05.2011 bei der Technischen Universitat Munchen

eingereicht und durch die Fakultat fur Physik am 30.05.2011 angenommen.

1. Auflage Juni 2011

Copyright 2011 by

Verein zur Förderung des Walter Schottky Instituts der Technischen Universität

München e.V., Am Coulombwall 4, 85748 Garching.

Alle Rechte vorbehalten. Dieses Werk ist urheberrechtlich geschützt. Die

Vervielfältigung des Buches oder von Teilen daraus ist nur in den Grenzen der

geltenden gesetzlichen Bestimmungen zulässig und grundsätzlich vergütungspflichtig.

Titelbild: Calculated spectral function of a strongly coupled double-quantum dot - cavity system as a function of detuning. Background: Scanning electron microscope image of a photonic crystal L3 cavity. Upper left: Photoluminescence spectrum of a strongly coupled quantum dot - cavity system for increasing lattice temperature. Lower right: Temporal photoluminescence emission spectrum of a quantum dot and a non-resonant cavity mode. Druck: Printy Digitaldruck, München (http://www.printy.de)

ISBN: 978-3-941650-29-9

“Physics is, hopefully, simple.

Physicists are not.”

Edward Teller (1908-2003)

- To my lovely wife -

Abstract

This thesis describes detailed investigations of the effects of photonic nanostructures

on the light emission properties of self-assembled InGaAs quantum dots. Nanoscale

optical cavities and waveguides are employed to enhance the interaction between light

and matter, i.e. photons and excitons, up to the point where optical non-linearities

appear at the quantum (single photon) level. Such non-linearities are an essential

component for the realization of hardware for photon based quantum computing since

they can be used for the creation and detection of non-classical states of light and

may open the way to new genres of quantum optoelectronic devices such as optical

modulators and optical transistors.

For single semiconductor quantum dots in photonic crystal nanocavities we investi-

gate the coupling between excitonic transitions and the highly localized mode of the

optical cavity. We explore the non-resonant coupling mechanisms which allow excitons

to couple to the cavity mode, even when they are not spectrally in resonance. This

effect is not observed for atomic cavity quantum electrodynamics experiments and its

origin is traced to phonon-assisted scattering for small detunings (∆E <∼ 5 meV)

and a multi-exciton-based, Auger-like process for larger detunings (∆E >∼ 5 meV).

For quantum dots in high-Q cavities we observe the coherent coupling between exci-

ton and cavity mode in the strong coupling regime of light-matter interaction, probe

the influence of pure dephasing on the coherent interaction at high excitation levels

and high lattice temperatures, and examine the coupling of two spatially separated

quantum dots via the exchange of real and virtual photons mediated by the cavity

mode. Furthermore, we study the spontaneous emission properties of quantum dots

in photonic crystal waveguide structures, estimate the fraction of all photons emit-

ted into the propagating waveguide mode, and demonstrate the on-chip generation of

single photon emission into the waveguide.

i

Abstract

The results obtained during the course of this thesis contribute significantly to the

understanding of coupling phenomena between excitons in self-assembled quantum

dots and optical modes of tailored photonic nanostructures realized on the basis of

two-dimensional photonic crystals. While we highlight the potential for advanced

applications in the direction of quantum optics and quantum computation, we also

identify some of the challenges which will need to be overcome on the way.

ii

Zusammenfassung

Diese Dissertation untersucht die Effekte von photonischen Nanostrukturen auf die

optischen Emissionseigenschaften von selbstorganisierten InGaAs Quantenpunkten.

Optische Resonatoren und Wellenleiter werden verwendet um die Wechselwirkung

zwischen Licht und Materie, d.h. Photonen und Exzitonen, zu verstarken, bis sich

optische Nichtlinearitaten auf dem Quantumlevel einzelner Photonen bilden. Diese

Nichtlinearitaten sind eine essentielle Komponente fur die Realisierung von optischen

Quantencomputern, da sie sowohl als optischen Modulatoren und optische Transi-

storen, als auch fur die Erzeugung und Detektion von nicht-klassischen Quanten-

zustanden des Lichts verwendet werden konnen.

In Proben mit einzelnen Quantenpunkten in Photonischen Kristall Nanoresonatoren

untersuchen wir die Kopplung zwischen exzitonischen Ubergangen und der optis-

chen Mode des Resonators. Wir erforschen die nicht-resonanten Kopplungsmechanis-

men die es dem Exziton erlauben an die photonische Mode zu koppeln selbst wenn

beide spektral nicht in Resonanz sind. Dieser Effekt wird nicht in Experimenten

der Quantenelektrodynamik mit Atomen in Resonatoren beobachtet. Fur kleine Ver-

stimmungen (∆E <∼ 5 meV) konnen wir diesen Prozess auf Phonon unterstutzte

Prozesse und bei großen Verstimmungen (∆E >∼ 5 meV) auf einen Auger-ahnlichen

Zerfall von Multiexzitonen zuruckfuhren. Bei Quantenpunkten in Resonatoren mit ho-

hen Qualitatsfaktoren beobachten wir die koharente Kopplung zwischen Exziton und

Resonatormode im Regime der starken Kopplung der Licht-Materie Wechselwirkung,

untersuchen den Einfluss von phasenzerstorenden Prozessen bei hohen Anregungsleis-

tungen und hohen Kristalltemperaturen, und inspizieren die Kopplung von zwei sep-

araten Quantenpunkten uber den durch die Resonatormode vermittelten Austausch

von reellen und virtuellen Photonen. Des Weiteren untersuchen wir die Eigenschaften

von Quantenpunkten in Photonischen Kristall Wellenleitern bezuglich der spontanen

iii

Zusammenfassung

Emission, schatzen den Anteil aller Photonen ab der in die sich ausbreitende Wellen-

leitermode emittiert wird und demonstrieren die Emission einzelner Photonen in den

Wellenleiter.

Die Ergebnisse die wir im Laufe dieser Dissertation erhalten haben, tragen maßge-

blich zum Verstandnis von Kopplungsphanomenen zwischen Exzitonen in selbstorgan-

isierten Quantenpunkten und den optischen Moden von Photonischen Kristall Nanos-

trukturen bei. Wahrend wir das Potential fur komplexere Anwendungen in Rich-

tung Quantenoptik und Quanteninformationsverarbeitung aufzeigen, indentifizieren

wir auch ein paar Aufgaben die auf dem Weg dorthin noch gelost werden mussen.

iv

Contents

Contents

Abstract i

Zusammenfassung iii

1. Motivation: Let the Photons Interact 1

2. Quantum Dots and Photonic Crystal Structures 5

2.1. Quantum Dot - Cavity Interaction . . . . . . . . . . . . . . . . . . . 7

2.1.1. Cavity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2. Weak Coupling Regime . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3. Strong Coupling Regime . . . . . . . . . . . . . . . . . . . . . 13

2.2. Photonic Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1. Basic Properties of Photonic Crystals . . . . . . . . . . . . . . 18

2.2.2. Photonic Crystal L3 Microcavities . . . . . . . . . . . . . . . . 22

2.2.3. Photonic Crystal W1 Waveguide . . . . . . . . . . . . . . . . 24

3. Sample Design and Experimental Techniques 27

3.1. Design of Electrically Tunable Samples . . . . . . . . . . . . . . . . . 29

3.1.1. Vertical Layer Design . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2. Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.3. Electrical Contacts . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2. Design of Undoped Photonic Crystal Samples . . . . . . . . . . . . . 35

3.2.1. Layer Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2. Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 35

v

Contents

3.3. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1. Micro-Photoluminescence Setup . . . . . . . . . . . . . . . . . 40

3.3.2. Photo- and Electroluminescence . . . . . . . . . . . . . . . . . 42

3.3.3. Time-Correlated Single Photon Counting . . . . . . . . . . . . 42

3.3.4. Photocurrent Measurement . . . . . . . . . . . . . . . . . . . 44

4. Characterization of Electrically Tunable Samples 47

4.1. p-i-n Diode Characteristic . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2. Photoluminescence vs. Photocurrent . . . . . . . . . . . . . . . . . . 52

4.2.1. Electric Field Effects . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.2. L3 Cavity Investigation . . . . . . . . . . . . . . . . . . . . . . 54

4.3. Cavity Resonant Excitation . . . . . . . . . . . . . . . . . . . . . . . 55

4.4. Quantum Dot Emission Energy Tuning . . . . . . . . . . . . . . . . . 57

4.4.1. Quantum Confined Stark-Effect . . . . . . . . . . . . . . . . . 57

4.4.2. Tuning Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3. Frequency Limit . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5. Cavity Emission Energy Tuning . . . . . . . . . . . . . . . . . . . . . 64

5. Non-Resonant Quantum Dot - Cavity Coupling 67

5.1. Phonon-Assisted Transitions . . . . . . . . . . . . . . . . . . . . . . . 70

5.1.1. Theory of Phonon-Assisted Transitions . . . . . . . . . . . . . 70

5.1.2. Experimental Evidence for Phonon-Assisted Transitions . . . . 74

5.2. Feeding from Multi-Excitonic Transitions . . . . . . . . . . . . . . . . 82

5.2.1. Model of Multi-Excitonic Feeding . . . . . . . . . . . . . . . . 82

5.2.2. Experimental Evidence for Multi-Excitonic Feeding . . . . . . 85

5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6. Strongly Coupled Quantum Dot - Cavity Systems 95

6.1. Theory of Quantum Dot - Cavity Coupling . . . . . . . . . . . . . . . 97

6.1.1. Emission Spectrum of a N-Quantum Dot - Cavity System . . . 97

6.1.2. Special Case: Single-Quantum Dot - Cavity System . . . . . . 100

6.2. The Role of Dephasing in Strongly Coupled Quantum Dot - Cavity

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2.1. Electrically Tunable, Strongly Coupled Quantum Dot - Cavity

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.2. Dephasing by Dynamical Fluctuations . . . . . . . . . . . . . 105

vi

Contents

6.2.3. Phonon Induced Dephasing . . . . . . . . . . . . . . . . . . . 107

6.3. Mutual Coupling of two Quantum Dots via an Optical Nanocavity Mode110

6.3.1. Experimental Realization . . . . . . . . . . . . . . . . . . . . . 111

6.3.2. Analysis of the Eigenvectors . . . . . . . . . . . . . . . . . . . 116

6.3.3. Coupling via an Optical ”V-System” . . . . . . . . . . . . . . 118

6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7. Spontaneous Emission Properties of Quantum Dots in Photonic Crystal

Waveguides 121

7.1. Quantum Dot Ensemble - Photonic Crystal Waveguide Systems . . . 123

7.2. Single Quantum Dot - Photonic Crystal Waveguide Systems . . . . . 131

7.2.1. Photonic Crystal Waveguides with Direct Optical Access . . . 131

7.2.2. Photonic Crystal Waveguide Bandstructure and Transmission

Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2.3. Determination of β-Factors . . . . . . . . . . . . . . . . . . . 134

7.2.4. Comparison of the Emission Channels . . . . . . . . . . . . . 138

7.2.5. Single Photon Emission into the Photonic Crystal Waveguide

Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3. Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 144

8. Outlook: On-Chip Quantum Optics 147

8.1. Coherent Coupling of Quantum Dot - Microcavity Systems via Waveg-

uides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.2. Creation and Investigation of Entangled Photons On-Chip . . . . . . 149

8.3. Recommendations for the Future . . . . . . . . . . . . . . . . . . . . 150

A. Strong Coupling Data 153

B. Mutual Coupling of two Quantum Dots - Alternative Interpretations 168

C. Python Codes for Quantum Dot - Cavity Systems 171

C.1. Calculation of the Emission Spectrum of an N-Quantum Dot - Cavity

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

C.2. Fitting of the Emission Spectrum of an N-Quantum Dot - Cavity System174

vii

Contents

D. Cavity versus Dot Emission in Strongly Coupled Quantum Dots–Cavity

Systems 179

D.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

D.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

D.3. One Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

D.4. More than one Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . 189

D.4.1. Two Emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

D.4.2. Three Emitters and Beyond . . . . . . . . . . . . . . . . . . . 195

D.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

. Bibliography 199

. List of Publications 225

. Acknowledgements 229

viii

All that glitters may not be gold, but at least it

contains free electrons.John D. Baernal

Chapter 1Motivation: Let the Photons Interact

Commonly available light sources such as LEDs, incandescent lamps and fluores-

cent tubes produce electromagnetic radiation with predominantly classical proper-

ties. [Lou83] This manifests itself in the appearance of wave phenomena such as re-

fraction and diffraction or the fact that an optical detector will always measure a

constant average optical intensity. Only when one strongly attenuates the intensity

of the stream of light further and further, does one obtain evidence for individual

quanta of light, the photons.

The properties of light can be characterized by taking a closer look at the statistics

of the detected photons (for experimental details see sect. 3.3.3). One pertinent

question that arises is: If we detect a photon at the time t1, what is the probability to

detect another photon at time t2? For a coherent light source, such as a laser beam,

the probability to detect a photon at time t2 is independent of the time difference

τ = t2 − t1. The number of light quanta detected per unit time follows a Poissonian

distribution, and the second order correlation function g(2)Laser(τ) = 1 is independent

of τ . [Lou83] Expressed verbally, this means that the probability to detect a second

photon is completely uncorrelated to the detection of the first photon.

However, if we consider a classical light source such as a light bulb or the sun,

we will notice a different behaviour. The intensity fluctuations of these “chaotic”

light sources typically exhibit intensity fluctuations with maxima where photons tend

to bunch together and minima with fewer photon counts. This results in a photon

1

Motivation: Let the Photons Interact

correlation function that is > 1 for timescales short compared to the coherence time

of the light source: g(2)Chaotic(0) > 1. Such a light source is then said to exhibit photon

bunching corresponding to a super-Poissonian distribution of photons. [Lou83]

In addition to classical light fields such as coherent and chaotic (incoherent) light,

non-classical light can be generated by special types of individual quantum emitters.

The second order correlation of such non-classical fields will exhibit sub-Poissonian

statistics which manifests itself as photon antibunching : g(2)Non−Classical(0) < 1. [Lou83]

This means that the probability to detect a second photon shortly after detection

of the first photon is smaller than the probability to detect it at later times. The

origin of this effect comes about due to the nature of isolated, light emitting quantum

systems that are only able to emit one photon at a time after which they must be

re-excited before a second photon can be generated. The optical transitions of a single

excited atom, molecule or quantum dot are examples of such non-classical sources of

light. [Kim77, San01]

For some applications such as interconnects between classical and quantum infor-

mation processing systems, it is desirable to create non-classical light from classi-

cal sources. In order to do that we need to employ processes that are sensitive to

the photon number, favouring single photons over multiple photons. This is only

possible when the light matter interaction becomes so strong such that optical non-

linearities produce measurable changes of the optical response at the single photon

level. In order to obtain non-classical light with photon antibunching, individual

photons should effectively “repel” each other. However, due to their Bosonic nature

photons do not directly interact with one another and the interaction has, there-

fore, to be mediated indirectly. [Har10] Mediators can be all optical emitters such as

atoms, [Tho92, Wal06, Sch08a, Sch08b, Kub08] quantum dot excitons, [Rei04, Yos04,

Pet05, Eng07a, Far08a, Far10b] colour centers, [Par06, San10, Eng10] or supercon-

ducting circuits. [Wal04, Fin08, Fin09, Fil10]

The strength of such an optical non-linearity depends on the strength of the inter-

action between light and matter, or more precisely photon and emitter. This is again

dependent on two major factors: The electric dipole matrix element of the emitter

transition and the strength of the electromagnetic field at the position of the emitter

(see sect. 2.1). Whilst the first factor is a property of the emitter itself, the second

can be influenced by placing the emitter inside an optical cavity where the photon

is confined to a small volume for a long time. When the rate at which energy is

2

exchanged between the photon field and the emitter exceeds the decoherence rate of

the system, the coupling will lead to a mode splitting in analogy to the physics of two

coupled classical oscillators (see sect. 6). During this mode splitting the energies of

the emitter and the cavity mode are altered by the presence of a single photon, which

is effectively a strong optical non-linearity at the single photon level. Applications

include the design of optical modulators or single photon transistors which can be

used for all-optical computing, [Eng07a, Eng09] but also the conversion of classical

into non-classical states of light. [Far08a, Far10b]

The Scope of this Thesis

The scope of this thesis is to investigate the light-matter coupling between semicon-

ductor quantum dots and nanoscale cavities and waveguides, realized using photonic

crystals. This is done in 8 chapters with the following contents:

• Chapter 2 describes investigations of the basic properties of quantum dots and

photonic crystal structures, and discusses how quantum dots interact with optical

cavities.

• In Chapter 3 we present the design and fabrication of the electrically tunable

photonic crystal cavity samples and the photonic crystal waveguide samples. Fur-

thermore, we discuss the experimental setup and techniques which were used to in-

vestigate these samples.

• Chapter 4 characterizes the electrically tunable photonic crystal cavity samples

and demonstrates how these structures can be used not only for electric tuning of the

quantum dot emission energy, but also for photocurrent investigations.

• In Chapter 5 we investigate the non-resonant coupling between quantum dots and

photonic crystal cavity modes via phonon emission and absorption for small spectral

detunings, and via an Auger-type process for multi-excitons at large spectral detun-

ings.

• Chapter 6 presents investigations on a strongly coupled, electrically tunable quan-

tum dot – photonic crystal cavity system, and demonstrates how excitation- and

temperature-induced dephasing affects the coherent interaction and how two individ-

3

Motivation: Let the Photons Interact

ual quantum dots can be coherently coupled directly or indirectly via the cavity mode.

• Chapter 7 examines the spontaneous emission dynamics of quantum dot ensembles

and single quantum dots inside photonic crystal waveguides.

• Finally, in Chapter 8 we present an outlook on future experiments for the integra-

tion of photonic crystal cavities and waveguides in more advanced on-chip quantum

optics experiments.

4

physics, Science that deals with the structure of

matter and the interactions between the

fundamental constituents of the observable universe.Encyclopædia Britannica

Chapter 2Quantum Dots and Photonic Crystal Structures

A fundamental requirement for our investigations of cavity quantum electrodynamics

effects is the incorporation of quantum emitters in optical nanocavities. In our sam-

ples we use self-assembled semiconductor quantum dots as emitters since they have

suitable properties for our experiments: They possess a discrete emission spectrum

which does not change over time since they are embedded in a solid state mate-

rial. Furthermore, fabrication is well established and their fundamental electronic

and optical properties are well understood and have been discussed elaborately else-

where. [Adl96, Bay98, Bim99, Fin01b, Fin01a, Kre01, Ger01, Mic03, Sab04, Kre05,

Kre06, Kro08, Hof08, Mul09, Kan09a, Abb09] In this thesis we will, therefore, not

discuss the basic properties of quantum dots. For the understanding of the results

in chapters 4, 5, 6, and 7 it is, however, important to keep in mind, that a quantum

dot can be considered as a quantum emitter with an atom-like electronic structure

consisting of different shells, while being responsive to its solid state environment.

The use of self-assembled quantum dots and their optical and electronic prop-

erties in optical semiconductor physics has led to many different applications in-

cluding quantum dot lasers, [Ara82, Asr96] photo-detectors, [Sti01] single photon

sources [Mic00, San01] with high degrees of quantum indistinguishability, [San02]

sources of entangled photon pairs, [Ako06, Shi07, Dou10], optically non-linear systems

for optical modulators at the single photon level, [Eng07a, Eng09] and the conversion

of classical into non-classical states of light. [Far08a, Far10b] Many of these appli-

5

Quantum Dots and Photonic Crystal Structures

cations already make use of an optical cavity and quantum electrodynamics effects

to increase the efficiency of the device or, more fundamentally, to create the desired

effect at all. The underlying physics of the interaction between quantum dot and

cavity is the topic of this chapter and will be discussed in the following sections.

In the first part of this chapter (sect. 2.1) we introduce the theoretical background of

the interaction between an emitter and a cavity and discuss the physics of the light-

matter interaction in the weak and the strong coupling regimes. We focus on the

Purcell effect [Pur46] in the weak coupling regime (sect. 2.1.2), and on the formation

of dressed states in the strong coupling regime (sect. 2.1.3), respectively.

In sect. 2.2 we introduce the concept of photonic crystal nanostructures and their

implementation in cavity quantum electrodynamics experiments. Firstly we review

the principle characteristics of the photonic crystals and present the key concepts of

photonic band structure and photonic bandgaps in sect. 2.2.1. We then introduce the

L3 type of photonic crystal microcavities (sect. 2.2.2) and discuss their properties,

including their influence on the spontaneous emission enhancement of the embedded

emitters. In addition to photonic crystal microcavities where light is spatially con-

fined, we also discuss photonic crystal W1 waveguides (sect. 2.2.3) which can be used

for efficient guiding of light over a moderate energy range.

6

2.1 Quantum Dot - Cavity Interaction

2.1. Quantum Dot - Cavity Interaction

In our optical studies we use photonic crystal defect nanostructures as optical res-

onators, which will be discussed in sect. 2.2.2. The following section will introduce

the basic theoretical understanding of the interaction between cavities and an incor-

porated quantum emitter.

2.1.1. Cavity Theory

Photon Density of States

In a three dimensional isotropic photonic environment of refractive index n, the pho-

ton density of states g3D(ω) is defined as [Fox06]

g3D(ω) =n3ω2

π2c3, (2.1)

with c denoting the vacuum speed of light. By spatially confining a photon in all

three dimensions, for example via a cavity, the spectral density of states is strongly

modified. We assume a cubic box cavity, which consists of perfectly reflecting walls as

illustrated in fig. 2.1(a). This resonator only supports modes with wavelengths that

have electric field nodes at the walls, which leads to the following condition for the

supported photon wavevectors kph:

kph =

kxkykz

= π

n/am/b

l/c

, (2.2)

where a, b and c are the length, height and width of the cubic cavity and n, m

and l are the number of nodes minus one for each supported mode. Therefore, the

supported photon frequencies ωn,m,l form a discrete series defined by

ωn,m,l =1√ε0µ0

|kph| =1√ε0µ0

((nπa

)2

+(mπb

)2

+

(lπ

c

)2) 1

2

. (2.3)

We have to keep in mind that the density of photon modes is only a δ-function when

the boundaries of the resonator are perfectly reflecting. In the case of non-perfectly

reflecting walls of the box, photons escape from the cavity leading, therefore, to a

reduction of field intensity inside the cavity over time.

7


a

b

c

n=1

n=2

(a) (b)

R N

R N

RN

N

2

3

(1-R)N

(1-R)R N

(1-R)R N

(1-R)RN2

3

n

Lcav

Mirror Mirror

Figure 2.1.: (a) Schematic illustration of a cubic resonator with perfectly reflectingboundaries. The first two eigenmodes for each spatial dimension areindicated. (b) Planar cavity with length Lcav consisting of two mirrorswith reflectivity R according to a Fabry-Perot resonator (adapted from[Ang08]).

Photon Lifetime in a Cavity

To illustrate the influence of the finite photon lifetime on the cavity mode spectrum, we

consider a Fabry-Perot cavity with mirrors of reflectivity R as depicted in fig. 2.1(b).

The time t it takes for a photon to propagate along one cavity length is

t =nLcavc

, (2.4)

with n being the refractive index inside the resonator and c being the vacuum

speed of light. If initially N photons are in the cavity mode, after one reflection at

the mirrors R ·N photons remain in the cavity and (1−R) ·N photons escape from

the cavity. An additional reflection leads to R2N remaining photons with another

(1−R) ·RN photons escaping from the cavity. In general, the number of photons N

remaining in a Fabry-Perot resonator with mirror reflectivity R as a function of time

can be described as differential equation

8


dN

dt=

∆N

nLcav/c=c(1−R)

nLcavN, (2.5)

which has the solution

⇒ N(t) = N0exp(−t/τph) with τph =nLcav

c(1−R). (2.6)

We can relate the photon lifetime τph of equation (2.6) to the Lorentzian linewidth

∆ω of the cavity mode via Heisenberg’s energy-time uncertainty relationship

~ ·∆ω · τph ≈ ~⇒ ∆ω ≈ 1

τph=c(1−R)

nLcav. (2.7)

This indicates that we can probe the lifetime of a photon inside a cavity by measur-

ing the corresponding mode emission linewidth. We can now define the mode quality

factor Q as a figure of merit for the photon lifetime inside a cavity, where

Q = ω0 · τph =ω0

∆ω. (2.8)

Quantum Dot - Cavity Coupling

When an emitter like a quantum dot is placed inside a cavity, it couples to the electric

field in the cavity at the position of the emitter. The strength of this coupling depends

on basic parameters of the cavity and emitter. These properties have been extensively

discussed in the literature, [Kre05, Hen06a, Hof08, Eng08, Kan09a, Far09, dVa10a]

and in the case of a single exciton (1X ≡ 1e−+1h+) in a quantum dot placed into the

cavity in spectral and spatial resonance with a cavity eigenmode, the relative strength

of the emitter - cavity-interaction is mainly determined by:

1. The incoherent interactions of the cavity mode λc

(e.g. photon loss rate Γc = 1/τph)

2. The incoherent interactions of the emitter λd

(e.g. non-resonant spontaneous emission decay rate Γd = 1/τ1X)

3. The emitter-photon coupling parameter g0.

In chapter 6 we will show how the relative strength of the emitter - cavity interaction

is dependent on these parameters. At this point, we focus on the fact that depending

9


on the strength of coupling g0 compared to the incoherent interactions, the system

can be distributed into two different regimes of operation. Depending on the value of

the Rabi splitting Ω =√g2

0 − (λd − λc)2/4 (see sect. 6.1.2), we distinguish between

two physically different interaction regimes:

• Ω is completely imaginary: weak coupling regime

• Ω has a real part: strong coupling regime.

In the weak coupling regime the emission of a photon by the emitter into the cavity

is an irreversible process. After some time the photon will escape from the cavity.

This is based on the same principle as in free-space spontaneous emission with a mod-

ified emission rate due to the change in the photonic density of states inside the cavity.

In contrast, in the strong coupling regime the emitter-photon interaction is coherent

and energy can be periodically exchanged between the cavity field and the emitter.

As we will show later in sect. 6.2, in experiments it is not always straightforward to

differentiate whether a system operates in the weak or the strong coupling regimes

of the light-matter interaction. Depending on the relative importance of decoherence

due to pure dephasing, incoherent pumping, and decay, strong coupling often appears

”in disguise”. [Lau08]

At low temperatures and low excitation rates the photon loss rate of the cavity Γc

is the dominant source of decoherence in the system (see sect. 6.2). Thus, we have

to compare g0 and Γc to estimate in which regime a quantum dot is interacting with

the cavity. The emitter-cavity coupling g0 is defined by the interaction energy ∆E

between the exciton and the vacuum field via

~g0 = ∆E, (2.9)

with ∆E defined as

∆E = |µ12Evac|, (2.10)

where µ12 ≡ −e〈1|x|2〉 is the electric dipole matrix element of the emitter transi-

tion and Evac =(

~ω2ε0V0

)1/2

is the magnitude of the vacuum field with mode volume

V0. [Fox06] By combining equations (2.9) and (2.10) we obtain

g0 =

(µ2

12ω

2ε0~V0

) 12

. (2.11)

10


From this equation we see that the emitter photon coupling rate g0 is determined

by the dipole moment µ12, the angular frequency ω and the mode volume V0 of the

cavity. We can now derive a mathematical expression to define the criterion whether

the system operates in the weak or strong coupling regimes, respectively. We do this

by comparing equation (2.11) with the photon loss rate of the cavity mode Γc, since

we assume that the cavity loss rate is the dominant loss mechanism. We then obtain

g0 =

(µ2

12ω

2ε0~V0

) 12

Γc =ω

Q, (2.12)

which can be rewritten as

Q(

2ε0~ωV0

µ212

). (2.13)

We see that we need cavities with a high Q/V0 ratio to reach the strong coupling

regime of the light-matter interaction.

2.1.2. Weak Coupling Regime

When the emitter-cavity coupling constant g0 is smaller than the rate at which pro-

cesses destroy the coherence of the interacting emitter - cavity system (e.g. photon

loss rate of the cavity mode Γc), the interaction between the emitter and the cavity

is in the weak coupling regime and photons leak out of the cavity over timescales

faster than the characteristic interaction time between the emitter and the cavity.

Therefore, emission of a photon by the emitter into the cavity is entirely irreversible

like in free space. [Joa95] The influence of the cavity in the weak coupling regime

is considered as a weak perturbation. Thus, by applying perturbation theory, the

transition rate is given by Fermi’s golden rule to be [Fox06]

Γ1→2 =2π

~〈|E(r)e · 〈2|d|1〉|2〉ρ(~ω), (2.14)

where E(r) is the amplitude of the vacuum field, e the vacuum field polarization

direction, 〈2|d|1〉 the dipole matrix element between the initial state |1〉 and the final

state |2〉, and ρ(~ω) the effective photon density of states at the transition energy ~ω.

The only parameter which is modified by tuning an emitter spectrally through the

cavity mode is the photon density of states ρ. To give an idea of ρ for a 0D cavity

mode as in our case, we can assume in first approximation the density of states as

11


a square Heavyside function with a spectral width ∆ω. This leads to an effective

photonic mode density

ρcav(ωcav) ≈1 mode / volume

frequency interval=

1Vcav

∆ω=

Q

ωcavVcavwith Q =

ωcav∆ω

. (2.15)

It can be shown that by considering the exact Lorentzian lineshape instead of the

Heavyside we get an additional factor of 1/π. [Fox06] By orienting the emitter dipole

moment parallel to the electric field in the cavity we obtain an extra factor of 3, [Fox06]

since we do not consider averaging over all possible dipole orientations. This leads to

the photon density of states of an ideally positioned emitter inside a 0D cavity:

ρcav(ωcav) =3

π· Q

ωcavVcav. (2.16)

In comparison, the free photon density of states is

ρfree(ω) =ω2

π2c3. (2.17)

We now introduce the Purcell factor FP , [Pur46] a dimensionless parameter that is

defined as the maximum ratio of the spontaneous emission rate in the cavity γcav

compared to the same emitter in free space. The Purcell factor is

FP =γcavγfree

=τfreeτcav

=ρcavρfree

. (2.18)

By inserting equations (2.16) and (2.17) into equation (2.18) and by replacing c/ω

with (λ/n)/2π we obtain

FP =ρcav(ω)

ρfree(ω)=

3π· QωVcavω2

π2c3

= 3π · c3

ω3· QVcav

=3(λ/n)3

4π2·(

Q

Vcav

). (2.19)

From this equation we see that high Q-factors and low mode volumes are essential to

obtain large Purcell enhancements. This expression for the Purcell factor describes

the change in emission properties for ideal conditions when the emitter is placed

precisely at an antinode of the cavity field and its transition frequency is resonant

with the cavity mode. Usually we have the following deviations:

1. The emitter can be located outside an antinode of the cavity field. This reduces

12


the spatial coupling since the coupling strength is proportional to the amplitude

of the cavity field at the position of the emitter.

2. The emission frequency of the emitter can be detuned from the cavity eigenfre-

quency. The coupling strength is determined by the deviation from the central

frequency of the Lorentzian lineshape cavity mode.

3. The orientation of the emitter dipole differs from the ideal case of being parallel

to the electric field of the cavity mode.

If we take all these three deviations into account, we obtain an extended expression

for the change in spontaneous emission of an emitter inside a cavity compared to an

emitter in an isotropic photonic environment [Fox06]:

γcavγfree

=τfreeτcav

= FP ·∣∣∣∣E(re)

Emax

∣∣∣∣2︸︷︷︸spatial coupling

· ∆ω2cav

4(ωe − ωcav)2 + ∆ω2cav︸︷︷︸

spectral coupling

·∣∣∣∣e〈2| − er|1〉|〈2| − er|1〉|

∣∣∣∣2︸︷︷︸dipole coupling

, (2.20)

where re is the position of the emitter, Emax the maximum field amplitude in the

cavity, ∆ωcav the full width at half maximum of the cavity mode and e the vacuum

field polarization direction.

From equation (2.20) it is immediately apparent how we can directly influence the

emission rate of an emitter by spectrally tuning it into and out of resonance with a

cavity eigenmode, by varying the field strength of the cavity mode at the position of

the emitter and by changing its dipole orientation.

2.1.3. Strong Coupling Regime

In 1963, Edwin Jaynes and Fred Cummings were the first to analyze the interac-

tion between a two level system and a resonant cavity mode using a fully quantum

model. [Jay63] They developed the Jaynes-Cummings model which describes the in-

teraction of a two-level atom system with a single quantized mode of an optical cavity.

In the case of undisturbed states of an uncoupled resonant system with a single

emitter of angular frequency ω in the cavity, we can describe the quantum state of

the coupled system Ψ via the quantum state of the emitter ψ and the number of

photons n. We write this as

Ψ = |ψ;n〉. (2.21)

13


The ground state of the system is, therefore, the state with the emitter in the ground

state |g〉 and no photon in the cavity:

Ψ0 = |g; 0〉. (2.22)

The energy of the ground state Ψ0 is (1/2)~ω due to the zero-point energy of the cavity

vacuum field. We now assume that we do not have any emitter cavity interaction. In

this case, the first excited state has an energy of (3/2)~ω and consists of either an

excited emitter with no photon in the cavity |e; 0〉 or the emitter in the ground state

and a single photon in the cavity |g; 1〉. In analogy, the energy of the n-th excited

state is (n + 1/2)~ω and corresponds to |e;n − 1〉 or |g;n〉. All excited states are

doubly degenerate.

If we take the emitter-cavity interaction into account this degeneracy is lifted due

to the electric-dipole interaction that couples the electronic and photonic states of the

system. The first excited state then becomes a non-degenerate doublet with energies

[Fox06]

E±1 =3

2~ω ± ~g0. (2.23)

The wavefunctions for the first excited state can be written as

Ψ±1 =1√2

(|g; 1〉 ∓ |e; 0〉) . (2.24)

In general, the energies of the n-th excited doublet states are

E±n =

(n+

1

2

)~ω ±√n~g0, (2.25)

with the corresponding wavefunctions

Ψ±n =1√2

(|g;n〉 ∓ |e;n− 1〉) . (2.26)

Therefore each doublet has a splitting of

∆En = 2√n~g0. (2.27)

The whole spectrum of resulting coupled emitter-cavity field states is schematically

illustrated in fig. 2.2 and is termed the Jaynes-Cummings ladder. The mixed emitter-

14


DressedEmitter-Cavity

w

wQuantumField Energy

h-

h-

2hg0-

2hg0-

nhg0-2

2

-Ψ1

+Ψ1

-Ψ2

+Ψ2

+Ψn

-Ψn

Ψ0

Ψ1

Ψ2

Ψn

Figure 2.2.: The Jaynes-Cummings “ladder” describes the states of a coupled emitter-cavity system with a coupling constant of g0. The quantum states of thecavity field are split up due to the strong interaction with the emitter and“dressed states” are formed.

photon states are called dressed states. In the case of n = 1 we obtain the so called

vacuum Rabi splitting ∆Evac with

∆Evac ≡ E+1 − E−1 = 2~g0 =

(2µ2

12~ωε0V0

) 12

, (2.28)

where ω is the angular frequency and V0 the mode volume.

A classical analogy for the n = 1 mode splitting is the system of two coupled

classical oscillators, with ωcav and ωemi being the natural angular eigenfrequencies

15


of the uncoupled oscillators (in our case the cavity and the emitter) and with g0

as coupling strength. The eigenfrequencies ω± of the coupled system can then be

calculated to be [Fox06]

ω± =ωcav + ωemi

2±√g2

0 +

(ωcav − ωQD

4

)2

. (2.29)

In the case of resonant emission into the cavity we obtain

ω± = ω ± g0 with ω ≡ ωcav = ωemi, (2.30)

which is the same result as the Jaynes Cummings model for n = 1. The frequency

characteristic as described by equation (2.29) is schematically illustrated in fig. 2.3(a)

and (b) for an emitter tuned into and out of resonance with a cavity mode. We

observe an anticrossing of the two peaks split by ∆E = 2g0 at resonance. This

comes about due to the strong (coherent) coupling of exciton and photon, which

leads to the formation of so-called exciton polaritons, which are half exciton and half

photon. [dVa10a]

We have to keep in mind that this quasi-classical picture is only valid for n = 1.

Higher order excitations can only be explained in a completely quantum mechanical

way. In particular, the vacuum field is a purely quantum concept without any classical

analogue. In this thesis we will limit the discussions to the strong coupling regime for

n = 1, since higher excited states have not been observed in our quantum dot - cavity

systems. However, ref. [Fin08] reports the observation of strong coupling in the n > 1

regime for a superconducting qubit - microwave resonator system and ref. [Kas10] in

a quantum dot - micropillar cavity system.

16


Detuning

Ene

rgy

Mode

QD

0

ΔE

Energy

Inte

nsit

y

(a) (b)

QD

-Mod

e D

etun

ing

QD Mode

Figure 2.3.: (a) Schematic illustration of the emission energies (solid lines) of twoexciton polariton branches in the strong coupling regime. The changefrom dot to mode characteristic and vice versa within each branch isindicated by the color scheme. Dashed lines correspond to the uncoupledmode and quantum dot emission energies. The energy splitting ∆E = 2g0

is illustrated by the orange arrow. (b) Simulated luminescence signal ofan exciton anticrossing with a cavity mode shown in a waterfall plot.

17


2.2. Photonic Crystal Structures

In the previous section, we showed how optical cavities can be used to enhance the

strength of the light-matter interaction by resonant recirculation of energy. In this

section, we introduce the optical cavities that we use in our experiments and briefly

explore their key characteristics. The size of these cavities is typically in the range of a

few micrometers down to hundreds of nanometers. As such, they are usually referred

to as microcavities. In recent years, several types of microcavities have been developed,

each of which use various methods to confine the light, mainly in combination with

total internal reflection. For instance, micropillar cavities employ distributed Bragg

reflectors to trap the light in vertical direction while applying total internal reflection

for the lateral confinement. [Pre07] Microdisks and microtoroids, however, entirely

use total internal reflection to trap the light in the form of the so-called whispering

gallery modes around their peripheries. [Pet05, Arm03] However, probably the most

widespread type of the optical microresonators employed in the optical regime with

semiconductor materials are photonic crystal defect microcavities. [Joh87, Joa95] In

our experiments, we employ photonic crystal microcavities for studying cavity quan-

tum electrodynamics effects. Therefore, in the following we will describe the basic

properties of photonic crystals (sect. 2.2.1) and then will present one specific cavity,

namely the photonic crystal L3 microcavity (sect. 2.2.2), that combines sufficiently

high Q and low mode volume to probe light-matter couplings with single dots in the

weak and strong coupling regimes of the light-matter interaction.

However, for experiments where photons are guided between two photonic crystal

L3 microcavities, we need to couple the emission of the dots to propagating optical

modes, rather than localized modes in a microcavity. Thus, we will also explore the

properties of photonic crystal defect waveguides in sect. 2.2.3. [Mea94, Dor08a, Hue07,

Sai10]

2.2.1. Basic Properties of Photonic Crystals

Photonic crystals are structures in which a periodic modulation of dielectric con-

stants is applied over one, two, or all three spatial dimensions of the structure. [Joa08]

This periodic arrangement of dielectric constants can be regarded as an optical ana-

logue to the periodic Coulomb potential of the semiconductor materials. [Kit02] This

conventional optics-electronics analogy suggests that the optical properties of the

18

2.2 Photonic Crystal Structures

photonic crystals should be comparable to the electronic properties of the semicon-

ductors. [Yab87, Joa95] As a particular case of resemblance, it has been shown that

similar to the presence of electronic bandgaps in semiconductor materials, a pho-

tonic bandgap can be realized in photonic crystals which prohibits the propagation

of photons with specific energy ranges through the structure. [Joa08] Photonic crys-

(a) 1-D (c) 3-D(b) 2-D

x

yz

periodic periodicperiodicpe

riodic

perio

dic

periodic

Figure 2.4.: Simple examples of (a) one-, (b) two-, and (c) three-dimensional photoniccrystals. The different colors represent materials with different dielectricconstants. The defining feature of a photonic crystal is the periodicity ofdielectric material along one or more axes. Adapted from ref. [Joa08].

tal structures can be categorized based on the number of dimensions in which they

are periodic. One-dimensional photonic crystals have been realized since many years

as Bragg reflectors commonly used in high reflectivity mirrors and anti-reflection

coatings. These structures consist of multiple layers of alternating materials with

different dielectric constants [see schematic in fig. 2.4(a)]. They reflect the light in a

specific range of wavelengths and, therefore, confine it to a two-dimensional subspace

[y-z plane in fig. 2.4(a)]. An example for an one-dimensional photonic crystal is a

micropost cavity which utilizes two Bragg reflectors to confine the light in vertical

direction for cavity quantum electrodynamics applications. [Pel02] In contrast to the

one-dimensional photonic crystals which can only reflect the light within a limited

range of incidence angles, two-dimensional photonic crystals with their periodicity

plane being parallel to the x-y plane [see schematic in fig. 2.4(b)] can completely

prohibit light propagating in the plane of the periodic modulation of the dielectric

function. In the case of two-dimensional photonic crystal slabs, in which total internal

reflection is employed for the out-of-plane confinement, the required periodicity can

be realized by patterning a periodic set of air holes into a dielectric membrane (see

19


section 3.2.2). Here, the photonic crystal structure is formed by a periodic hexagonal

pattern of air holes (with dielectric constant of εAir = 1.0) in GaAs material (with

εGaAs = 11.7). [Kan09a] Nonetheless, two-dimensional photonic crystals exhibit spec-

trally wide bandgaps only for certain polarizations of the light in the structure. In fact,

a complete bandgap for all spatial incidence angles and all polarizations should only be

present in three-dimensional photonic crystals [see schematic in fig. 2.4(c)]. However,

due to the complicated fabrication procedure, three-dimensional photonic crystals are

rarely used in cavity quantum electrodynamics experiments. [Aok08] A sophisticated

fabrication method for three-dimensional photonic crystals was recently demonstrated

whereby several layers of one-dimensional photonic crystals were stacked on top of

each other by micromanipulation. [Aok03] In order to establish periodicity in the third

dimension, each layer is then rotated by 90 relative to its neighbouring layers.

In general, the periodicity of the photonic crystals is quantified by the lattice con-

stants of the structure, defined in the same way as in solid state physics, i.e. the

constant distances between constitutive unit cells of the crystal. In the case of two-

dimensional hexagonal photonic crystals in which the two primitive lattice vectors

(a1 and a2) are of the same length, a unique lattice constant a, can be defined for

the whole structure as presented in fig. 2.5(a). Since the mechanism of the photonic

crystals is based on the diffraction and interference phenomena, the relevant lattice

constants need to be of the same order as the wavelength of the propagating light

in the structure. This results in a typical lattice constant value between 200 nm to

500 nm for GaAs photonic crystals working at visible or near-infrared wavelengths.

Due to the periodicity, the dispersion relation of light in photonic crystal structures

is substantially different from the uniform media and appears in the form of photonic

band structures as depicted in fig. 2.5(b). Here, the dispersion relation of the trans-

verse electric (TE)1 polarized light for a two-dimensional hexagonal photonic crystal

slab consisting of air holes in a dielectric membrane is calculated up to four bands

using the plane-wave expansion method as explained in more detail in ref. [Hau09].

The dielectric membrane has a refractive index of nGaAs = 3.5, thickness of h = 0.6a,

and air holes radius of r = 0.32a. The frequencies are normalized to the lattice con-

stant a (with λ denoting the free space wavelength of light) and the wave vectors are

specified along the path which contains the highly symmetric points of the Brillouin

zone (Γ, M and K) as shown in fig. 2.5(c). Due to index guiding of photonic crystal

1Here, TE polarization implies an in-plane electric field.

20


Fre

quency

(a/

)λ

ar

1 2

b2b1

Γ K

M

ex

ey

kx

ky

aaLightcone

M K

TE band structure(a)

(c)

(b)

Bandgap

Figure 2.5.: (a) Bravais lattice of a two-dimensional hexagonal photonic crystal struc-ture with lattice constant a, hole radius r, and primitive lattice vectorsa1 and a2. The lattice unit cell is depicted by a hexagon. (b) Corre-sponding photonic band structure for the given parameters r = 0.32a,h = 0.6a, nhole = 1.0, and nGaAs = 3.5 for TE polarized light. Adaptedfrom ref. [Hau09]. (c) Corresponding reciprocal lattice with reciprocalvectors b1 and b2. The first and irreducible Brillouin zones are depictedas a hexagon and a wedge, respectively.

slabs for out-of-plane direction, the guided modes should lay outside of the so-called

light cone. Inside of the light cone, there is a continuum of leaky modes for which

photons are not guided by total internal reflection in the slab waveguide. These are

indicated by the dark blue regions in fig. 2.5(b). Considering this photonic band

structure, one observes a frequency region in which no band is existent and, there-

fore, no guided photon can propagate through the structure [red area in fig. 2.5(b)].

This key characteristic of photonic crystals is called photonic bandgap and results in

a reduced photonic density of states and, therefore, suppression of the spontaneous

emission rate of the emitter if it lies within the bandgap region. [Yab87] Practically,

this can be employed to spatially redistribute the emission of the light and, thus, to

enhance the external quantum efficiency of the emitter that is embedded in such a

structure. [Str06a, Fra08, Kan08b]

21


2.2.2. Photonic Crystal L3 Microcavities

In this section we give an explanation why point defects introduced into photonic

crystals can be used to realize microcavities and, therefore, for quantum dot - cavity

systems. Any perturbation in the periodicity of photonic crystals can modify their

photonic band structure and the corresponding density of photonic states. In general,

introducing a defect into the periodic lattice of the photonic crystal results in the

formation of localized modes. With proper engineering of the defects, it is possible to

create photonic crystal cavities with localized modes whose frequencies lie within the

photonic bandgap of the structure. In two-dimensional photonic crystal slabs, this

can be realized by omitting air holes at the position of the cavity. In particular, a

so-called L3 cavity can be created by omitting three holes in a row in a hexagonal

photonic crystal structure, as depicted in fig. 2.6(a). In order to achieve a high quality

factor and low mode volume microcavity, Akahane et al. have proposed an optimized

design of the L3 cavities. [Aka03] The modification is carried out by shifting the first

and third longitudinal side holes of the cavity outwards, as shown in fig. 2.6(a). This

leads to a smooth confinement of the electric field inside the cavity and reduces the

leaky modes which couple into the continuum of the free space modes. [Hau09]

Fig. 2.6(b) shows the calculated band structure of a L3 cavity consisting of air

holes with radius r = 0.32a (a the lattice constant) in a dielectric membrane with a

refractive index of nGaAs = 3.5 and thickness of h = 0.6a for TE polarized light. The

setback of the side holes is set to the ideal value of ∆a = 0.17a and ∆c = 0.15a which

gives the highest quality factor according to ref. [Hau09]. When we compare these

calculations to the band structure of the equivalent intact photonic crystal structure

presented in fig. 2.5(b), we expect six almost dispersionless modes within the bandgap

of the photonic crystal. The lack of dispersion indicates that these modes are localized

and stem from the L3 cavity defect. The fundamental mode with the lowest frequency

denoted as M1 is well separated from the other modes and exhibits the highest cavity

quality factor among all modes. Simulations demonstrate a theoretical quality factor

of Q ≈ 225,000 for such a cavity mode. [Hau09] However, the experimental quality

factors of our L3 cavities typically lie in the range of Q ≈ 3000-12000 for the fun-

damental cavity mode. The discrepancy is mainly due to the fabrication disorders

and residual absorption in GaAs (compared to Si) which can drastically change the

quality factor of the cavity. The highest measured quality factor in GaAs photonic

crystal cavities reported in literature has a value of Q = 25,000 and is reported by

22


Fre

quency

(a/

)λex

y

Defect modesin bandgap

Lightcone

M K

TE band structure

(a)

(c)

(b)

Lightcone

Defect modes

in bandgap

Fre

quen

cy (

a/λ)

M K

TE band structure(a) (b)

(c)

min

max

a ae

M6

M1

M2-5

| |2

cc

Figure 2.6.: (a) Schematic of an optimized L3 photonic crystal cavity. The positionof the side holes without setback is sketched by red circles. (b) TE bandstructure of the L3 cavity with nGaAs = 3.5, r = 0.32a, h = 0.6a, andsetback of ∆a = 0.17a and ∆a = 0.15a with six modes M1 - M6 in thebandgap. (c) Distribution of the electric field intensity |E|2 of the L3cavity modes for r = 0.28a. (b) and (c) adapted from ref. [Hau09].

Faraon et al. [Far08a]

The distribution of the electric field intensity |E(re)|2 for the L3 cavity modes with

r = 0.28a is presented in fig. 2.6(c). The modes are indexed using the nomenclature

M1 (the fundamental mode) to M6 according to the increment of their frequencies,

and demonstrate a strong localization within the defect region of the cavity.

According to the spatial term∣∣∣E(re)Emax

∣∣∣2 in equation (2.20) of the previous section, the

field distribution in the cavity can substantially influence the spontaneous emission

rate of the emitters. In order to achieve the strongest spatial coupling, the emitter

must be placed exactly at the position of the maximum field intensity in the cavity,

which has a width of ∼120 nm. [Tho09] Moreover, the field distribution influences

the spontaneous emission rate since it determines the mode volume of the cavity.

According to equation (2.19) higher emission rates can be realized in cavities with

smaller mode volumes. The fundamental mode volume of the L3 microcavities is

calculated to be V ≈ 0.92(λ/n)3, [Hau09] with λ as the free space wavelength of the

23


light and in a dielectric material with refractive index of n.

The high quality factor and the low mode volume of the L3 cavities make them

ideal candidates for cavity quantum electrodynamics experiments. For a more detailed

discussion about the photonic crystal L3 microcavities, which we use in this thesis,

we refer the reader to refs. [Hau09], [Kan09b] and [Kre05].

2.2.3. Photonic Crystal W1 Waveguide

As discussed in the previous section, the introduction of a point defect into the pe-

riodic lattice of the photonic crystal results in the formation of localized modes in

the photonic crystal bandgap. In addition to point defects, which are used for pho-

tonic crystal cavities, one can introduce line defects into the photonic crystal slab, i.e.

omitting a whole row of air holes, to create a waveguide structure which can guide

light. Here, we discuss why waveguides are very suitable for the transmission of light,

whilst we describe the experimental investigation of quantum dots coupled to a W1

waveguide in chapter 7.

In fig. 2.7(a) we present a schematic illustration of a W1 photonic crystal waveguide

consisting of a photonic crystal with a single omitted row of air holes along which

it guides light. In order to calculate the band structure of the photonic crystal W1

w

(b)(a)

ey

ex

ky

kx

KK’M

‚

ex

ye

KK’M

‚No defect Line defect

W1 waveguide Brillouin zone

Figure 2.7.: (a) Schematic of a photonic crystal W1 waveguide. Omitted air holesmarked as dashed black circles. (b) Construction of the new Brillouinzone edge (K′ point) by projection of M onto the the nearest-neighbourdirection [see fig. 2.5(c)] The resulting Brillouin zone is marked with theblue arrow on the right panel. Adapted from ref. [Dor08a].

24


TE band structure

Lightcone

Slab band

continuum

(b)

‚

‚M K

No defect(a)

Γ

KM

Line defect

Slow lightregion

Lightcone

Bandgap

Figure 2.8.: (a) TE band structure of a 2-D hexagonal photonic crystal with no defectsand its first and its irreducible Brillouin zones (see fig. 2.5). Adapted fromref. [Hau09]. (b) TE band structure of a photonic crystal W1 waveguide(r/a = 0.3, h/a = 0.6) within its 1-D Brillouin zone [see fig. 2.7(b)]. Thelight grey area is the slab band continuum. The dark blue area indicatesthe light cone. The red, green and blue line are the waveguide modes.The orange dashed box marks the slow light region of the blue waveguidemode. Adapted from ref. [Dor08a].

waveguide, we must correct the Brillouin zone as depicted in fig. 2.7(b). As the

introduction of the line defect breaks the translational symmetry along the y direction

[see fig. 2.7(a)], ky is no longer a good mode index. Since kz is also not a good quantum

number, the new Brillouin zone is one-dimensional and lies along the Γ-K direction

corresponding to the nearest-neighbour direction. However, K itself is not the edge

of the new Brillouin zone. Instead, it is a point K′ that lies at a distance πa

from Γ.

The construction of K′ and the new Brillouin zone (blue arrow) is also presented in

fig. 2.7(b). [Hue07]

The calculated TE band structure of a photonic crystal W1 waveguide with r/a =

0.3, h/a = 0.6 is presented in fig. 2.8(b). For comparison we display in fig. 2.8(a)

the TE band structure of a perfectly periodic photonic crystal, as already shown

in fig. 2.5(b). The continuous bands are indicated by a light grey area (“slab band

continuum”) in fig. 2.8(b). These bands are not confined to the line-defect but extend

infinitely in the x-y plane. 2 The light cone, in which the states are not confined by

total internal reflection along the vertical direction (“leaky modes”), is shaded dark

2For a detailed discussion about the origin of the slab band continuum we refer to ref. [Dor08a].

25


blue. We can identify six guided modes in the band structure which are also depicted

in fig. 2.8(b). When we compare this band structure with the band structure of a

defect-free photonic crystal [see fig. 2.8(a)] we notice, that the three lowest modes

(black curves), which are located below the lower slab band continuum, do not lie

within the photonic bandgap of the structure. Therefore, they must be purely index-

guided, both in vertical direction and in-plane by the substrate-air interface at the

air hole sidewalls. [Fra06] In contrast, the guided modes [red, green and blue curve

presented in fig. 2.8(b)] between the slab band continuum (light grey areas), do lie

within the photonic bandgap and we call them waveguide modes. Since these modes

lie sufficiently isolated within the photonic bandgap, light which is coupled into these

modes cannot couple to the slab band continuum of the photonic crystal and is guided

along the line defect. Therefore, photonic crystals with a line defect can be used to

guide light very efficiently. [Sto00] The transmission band of such a waveguide depends

on the bandwidth of the waveguide modes. For example, the blue waveguide mode

has a transmission band between a/λ = 0.26 and 0.28, which corresponds to an energy

(wavelength) range of ∼ 1.2− 1.3 eV (950− 1030 nm) for a common lattice constant

a of 270 nm used for our samples. [Sai10] Therefore, we expect low propagation losses

and a high transmission coefficient at this energy (wavelength) range.

In the region which we marked with an orange dashed box in fig. 2.8(b) the blue

waveguide mode has a flat band structure and, therefore, a small group velocity vg =

dω/dk. Hence, this region of the mode is termed the slow light mode. For this region

large Purcell factors are expected due to the strong enhancement of the spectral mode

density. [Rao07a] As a result, the spontaneous emission of an emitter coupled to the

slow light mode is expected to be strongly enhanced. [Hug04, Lun08, Dew10, Thy10]

The high transmission efficiency and the large transmission bandwidth make pho-

tonic crystal waveguides promising candidates to couple two photonic crystal L3 cavi-

ties and, therefore, for quantum optics experiments moving towards the goal of photon

based distributed quantum computation. [Kim08] Due to their slow light modes and

the enhanced spontaneous emission defect waveguides are also suitable for quantum

electrodynamics experiments. In chapter 7 we will present experimental results con-

ducted on photonic crystal W1 waveguides. For a more detailed discussion about

photonic crystal waveguides, especially the simulation of the band structure, the ori-

gin of the slab band continuum and the properties of the waveguide modes we refer

the reader to refs. [Hue07], [Dor08a] and [Sai10].

26

A scientist can discover a new star, but he cannot

make one. He would have to ask an engineer to do

that.

Gordon L. Glegg (Engineer)

Chapter 3Sample Design and Experimental Techniques

Technological advances in material and optical technologies over the past century

were necessary to allow the fabrication and characterization of solid-state microcav-

ities with active emitters. The list includes the invention of the scanning electron

microscope by Ernst Ruska in 1931, [Kno32] the development of the laser by Charles

H. Townes in 1954, [Sch58] the development of highly transmissive optical fibers by

Charles K. Kao in 1966, [Kao66] and the invention of the CCD camera by Willard

Boyle and George E. Smith in 1969, [Boy70] to name just a few. The impact of these

achievements of mankind were recognized by Nobel prizes in 1986, 1964, 2009 and

2009, respectively. The omnipresence of highly sophisticated technical equipment,

without which the fabrication of structures with a size of a few 100 nm or the de-

tection of single photons with a temporal resolution of 100 ps would not be possible,

will become clear as we present the fabrication process and the measurement setup

in this chapter.

In the first part of this chapter (sect. 3.1) we describe the design of the electrically

tunable photonic crystal samples investigated and discuss the methods used for their

fabrication. Essentially, they consist of a p-i-n diode structure formed on a n-doped

GaAs substrate with quantum dots incorporated in the intrinsic region (sect. 3.1.1). In

some regions, the p-i-n layer is underetched to form a photonic crystal slab into which

we introduce a defect microcavity. The fabrication process will be briefly discussed

27

Sample Design and Experimental Techniques

in sects. 3.1.2 and 3.1.3. A more detailed description can be found in ref. [Hof08].

We continue by presenting the design and fabrication of our samples, fabricated on

undoped GaAs substrates in sect. 3.2. We employed the electron beam resist ZEP that

has a higher resistance against Ar-Cl2 reactive ion etching such that the previously

used Si3N4 hardmask [Hof08] can be abolished. Instead, the patterned resist is used

as a softmask to fabricate photonic crystal cavities and photonic crystal waveguides

(see sect. 3.2.2).

In the final part of the chapter (sect. 3.3), we will then introduce the experimen-

tal setup (sect. 3.3.1) and the principle experimental measurement techniques used in

this thesis, namely photo- and electroluminescence emission spectroscopy (sect. 3.3.2),

time-correlated emission spectroscopy (sect. 3.3.3) and photocurrent absorption spec-

troscopy (sect. 3.3.4).

28

3.1 Design of Electrically Tunable Samples

3.1. Design of Electrically Tunable Samples

As discussed in sect. 2.1.3, in order to investigate strong interactions between exci-

tonic transitions of quantum dots and optical cavity modes, we need high Q-factor,

low mode volume microcavities. In our case, the microcavities are realized via L3

defects introduced into photonic crystal slab structures, which provide maximum ex-

perimental Q-factors of ∼ 12,000 and calculated mode volumes of ∼ 0.92 (λ/n)3

(sect. 2.2.2). The photonic crystal slabs are realized by a free standing air-GaAs-air

membrane into which we incorporate semiconductor self-assembled quantum dots as

emitters. By making use of the quantum confined Stark-effect (see sect. 4.4.1), we

can electrically tune the quantum dot emission energy relative to the mode emission

energy by applying static electric fields up to F ∼ 100 kV/cm at the position of the

quantum dots. This is realized using the p-i-n diode structure with the quantum dots

located in the intrinsic region.

3.1.1. Vertical Layer Design

The layer structure of the electrically tunable samples investigated in this thesis,

including the self-assembled quantum dots, was grown by molecular beam epitaxy.

Fig. 3.1 depicts a schematic of the layer sequence deposited with the nominal ma-

terial composition and dopant profile indicated. The samples were grown on a n-

doped GaAs substrate by depositing 500 nm of n-Al0.80Ga0.20As. This serves as a

sacrificial layer for wet chemical etching using a buffered hydrofluoric (HF) acid solu-

tion. [Wu85, Mer79] Since n-AlxGa1−xAs oxidizes quickly under ambient conditions

for x > 0.87, [Cho94] and high selectivity in the wet etching process is required, the

nominal Aluminium content was chosen to be ∼ 80 %. On top of the sacrificial layer a

180 nm thick p-i-n GaAs structure was grown, into which the photonic crystal struc-

ture will be patterned. Both, the 35 nm p- and n-doped regions are subdivided into

high and low doping concentration layers, with the higher concentrations closer to the

contacts to shorten the depletion length. [Sze85] In the middle of the 110 nm thick

intrinsic GaAs region, low-density self-assembled In0.4Ga0.6As quantum dots were in-

troduced. For a detailed description of the quantum dot growth studies we refer the

reader to refs. [Hof08, Kan09a, Mul09].

29


180 n

m500 n

m

1.0

E+

18

0.0

E+

00

2.5

E+

19

-2.0

E+

18

-1.0

E+

18

-3(c

m)

In Ga As QDs0.4 0.6

n-Al Ga As0.80 0.20

n-GaAs Substrate

Air

p-GaAs

n-GaAs

i-GaAs with

10 nm

15 nm

110 nm

20 nm

500 nm

25 nm

Doping ProfileMaterial Structure

e /x ey

ez

Figure 3.1.: Vertical sample design in growth direction showing material compositionand dopant profile. Adapted from ref. [Hof08].

3.1.2. Lithography

Fig. 3.2 shows schematically the entire fabrication process, starting from the grown

layer structure in panel (a). The first step is to deposit ∼ 85 nm of Si3N4 on the top

surface via plasma enhanced chemical vapor deposition (PECVD) [fig. 3.2(b)]. On

top of this, ∼ 120 nm of Polymethylmethacrylate (PMMA) is spun coated [fig. 3.2(c)].

This polymer serves as a positive electron beam resist. The PMMA layer is used to

define the photonic crystal hole pattern via e-beam lithography [fig. 3.2(d)], which is

then transferred into the Si3N4 layer by CF4 electron cyclotron resonance reactive ion

30


18

0 n

m5

00

nm

In Ga As QD’sx 1-x

n-Al Ga As80 20

n-GaAs Substrate

Air

p-GaAs

n-GaAs

i-GaAs with

85 nm Si N3 4

120 nm PMMA

CF RIE etching4

e-beam lithography+

develop PMMA Ar-Cl RIE etching2

120 nm PMMA

CF RIE etching4

Top- and backcontact +

etching of mesaHF etching

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.2.: Photonic crystal fabrication process: (a) basic MBE-grown layer struc-ture, (b)-(e) deposition and patterning of a Si3N4 mask by e-beam lithog-raphy and RIE etching with CF4, (f) etching of the photonic crystal holesby Ar-Cl2 RIE etching, (g) removing residual Si3N4 mask, (h) applica-tion of electrical contacts and (i) underetching with hydrofluoric acid.Adapted from ref. [Hof08].

31


etching (ECR-RIE) [fig. 3.2(e)]. The Si3N4 layer is used as mask for the subsequent Ar-

Cl2 RIE, which then defines the holes of the photonic crystal slab structure [fig. 3.2(f)].

The remaining Si3N4 layer is removed by CF4 ECR-RIE [fig. 3.2(g)]. GeAuNiAu top-

and back-contacts are then fabricated using photo lithography and metal evaporation.

In order to electrically separate the individual diodes on the chip mesas are formed

by optical lithography [fig. 3.2(h)] and subsequent wet etching. The last step is the

underetching process of the sacrificial n-Al0.80Ga0.20As layer with hydrofluoric acid to

create a freestanding slab structure [fig. 3.2(i)]. For more informations regarding the

fabrication process we refer the reader to ref. [Hof08].

Fig. 3.3(a) shows a schematic view of the final sample design with a L3 photonic

crystal microcavity. A perspective cross sectional view is illustrated in fig. 3.3(b)

with a schematic quantum dot, which is ideally incorporated at the center of the

microcavity. In fig. 3.3(c) an optical microscope image is depicted, which shows the

top contact and the mesa structure together with the dimensions. In contrast to the

schematic picture in figs. 3.3(a) and (b), we have actually an array of 5× 5 photonic

crystal structures within a 200×200 µm2 window in the top contact. 1 A 250×125 µm2

gold panel at the bottom of the window is used for gluing gold wires to the sample

as contacts. The mesa structure has a total size of 300× 400 µm2 which results in a

total area of 0.12 mm2. In fig. 3.3(d) we show the height profile across the diode, with

a mesa height of ∼ 100 nm and a top contact height of ∼ 355 nm. Three scanning

electron microscope (SEM) images of a L3 photonic crystal microcavity at different

magnifications are shown in fig. 3.3(e).

3.1.3. Electrical Contacts

We typically process a whole array of diodes on a wafer as illustrated schematically in

fig. 3.4 (a). The diodes share a common back contact and each diode has a separate

top contact. In combination with the mesa structure, this allows us to locally control

the electric field under the top contact of a single diode. After cleaving the processed

wafer, the back contact of a sample piece is glued with silver epoxy onto an Al2O3

sample carrier with gold contacts. The top contacts of the sample and the sample

carrier are connected with Au bonding wires. Since we want to avoid damaging

the photonic crystal slabs and prevent annealing of the contacts, both arising from

the ultrasonic bonding process, the bonding wires are attached with silver epoxy.

1For better visibility we depict schematic views with only one photonic crystal structure.

32


N2

(b)(a)

exey

ez

h = 355 nm1

(d)

h =110 nm2

(c)

200 µm

20

0 µ

m

250 µm

35

0 µ

m

PhC

40

0 µ

m

300 µm

4µm

(e)

2µm

Top contact

Back gate:n-GaAs

i-GaAs

n-GaAs substrate

Photonic crystal

Quantum Dot

n-Al Ga As0.8 0.2

Top gate:p-GaAs

Back contact

Figure 3.3.: (a) Schematic view of the electrical contacted photonic crystal sampledesign with L3 microcavity. (b) Cross sectional view of the sample in thecutplane through the longitudinal L3 microcavity axis. (c) Microscopeoptical image of the top contact and mesa structure with dimensions.(d) Height profile across the top sample surface. (e) Scanning electronmicroscope images of a L3 photonic crystal microcavity. [Hau09]

33


(a)

(c)

L3

(b) Bonding wireCleaved sample

Al Ocarrier

2 3

Au contact

Accesswire

Backcontact

Figure 3.4.: (a) Schematic view of an electrical contacted photonic crystal 3×3 samplearray with L3 microcavities. [Hau09] (b) Schematic top view with aninserted optical image of the cleaved sample on the Al2O3 sample carrierwith some Au bonding- and Cu access-wires attached. [Hau09] (c) Photoof contacted sample on carrier.

34

3.2 Design of Undoped Photonic Crystal Samples

Electrical contact is then realized by connecting the outer contacts of the sample

carrier with copper access wires [see fig. 3.4(b) and (c)], which is done by soldering.

It is important to short-circuit the wires when the sample undergoes temperature

changes, since the devices have a high likelihood to be damaged, probably due to the

accumulation of charges (during cool down), leading to a fatal breakthrough. With no

wires connected we did not observe that the diodes were damaged during temperature

changes.

3.2. Design of Undoped Photonic Crystal Samples

As mentioned at the beginning of this chapter, we also fabricated photonic crystal

structures using a different e-beam resist (ZEP) that has a better resistance in the Ar-

Cl2 RIE etching step, thus allowing us to abolish the Si3N4 hardmask. This fabrication

process was employed to fabricate samples with L3 cavities (see sect. 2.2.2), but also

with photonic crystal waveguides (see sect. 2.2.3 and chapter 7).

In the following sections we will present the layer structure of these samples (sect. 3.2.1)

and explain the necessary fabrication steps to create the investigated nanostructures

(sect. 3.2.2).

3.2.1. Layer Structure

Fig. 3.5 shows the entire fabrication process schematically, starting from the grown

layer sequence with the respective material compositions in fig. 3.5(a). The layer

structure of the sample is grown by molecular beam epitaxy. As for the doped struc-

tures, the growth begins with 500 nm of Al0.80Ga0.20As on an intrinsic GaAs substrate

(300 nm undoped GaAs buffer layer). Again this serves as sacrificial layer for wet

chemical etching using hydrofluoric acid. [Wu85, Mer79] On top of the sacrificial layer

a 180 nm thick intrinsic GaAs layer is grown, into which the photonic crystal struc-

ture will be patterned. In the middle of this undoped GaAs region, self-assembled

low density In0.5Ga0.5As quantum dots are introduced.

3.2.2. Sample Fabrication

The first step is to spin coat approximately 260 nm of ZEP on the sample [see

fig. 3.5(b)] which serves as a positive electron beam resist. The ZEP layer is used

35


to write the photonic crystal hole pattern via electron beam (e-beam) lithography

[see fig. 3.5(c)]. After the ZEP layer is developed [see fig. 3.5(d)] it is used as a mask

for the subsequent Ar-Cl2 reactive ion etching (RIE), which transfers the holes of the

photonic crystal structure into the GaAs [see fig. 3.5(e)]. After the Ar-Cl2 RIE the

remaining ZEP layer is removed with organic solvents [also fig. 3.5(e)]. The last step

is the selective removal of the sacrificial Al0.80Ga0.20As layer with hydrofluoric acid to

create a freestanding slab structure [see fig. 3.5(f)].

GaAs Substrate

Al Ga As0.80 0.20

GaAs with

InGaAs QDs180

500

300

nm

(a) (b)e-beam Resist Spin Coating

260 nm ZEP

(c)e-beam

Lithography

(d)

Developing ZEP

(e)

Ar-Cl RIE Etching2

plusResist Removal

(f)

HF Etching

GaAs Buffer

Figure 3.5.: Photonic crystal fabrication process: (a) basic MBE-grown layer struc-ture, (b)-(d) spin-coating and patterning of the resist by e-beam lithog-raphy, (e) etching of the photonic crystal holes by Ar-Cl2 RIE etchingplus resist removal and (f) underetching with hydrofluoric acid. Adaptedfrom ref. [Mul09].

36


Photonic Crystal L3 Cavity

In fig. 3.6(a) we present a schematic view of the sample design with a L3 photonic

crystal microcavity. A magnified cross sectional view is presented in fig. 3.6(b) with

a schematic quantum dot layer in the middle of the free-standing membrane (red).

Ideally one of these quantum dots is incorporated at the centre of the microcavity.

In fig. 3.6(c) a scanning electron microscope (SEM) image is presented, showing a

hexagonal photonic crystal structure with four (much larger) etch holes surrounding

it. The aim of these holes is to facilitate a proper underetching of the photonic

Quantum Dots

1 µm 200 nm

(a)

(c) (d)

(b)

Figure 3.6.: (a) Schematic view of the photonic crystal sample design with a L3 micro-cavity. (b) Cross sectional view of the sample in the cutplane through thelongitudinal L3 microcavity axis. (c) Scanning electron microscope imageof a photonic crystal with a L3 cavity at the centre. (d) Magnified view ofthe L3 cavity presented in (c). The little bright spots are most probablyquantum dots on the sample surface, which were grown for atomic forcemicroscopy studies. (a) and (b) are adapted from ref. [Mul09].

37


crystal. Furthermore, they allow us to find the photonic crystal structures more

easily with an optical microscope. Finally, in fig. 3.6(d) we show a SEM image with

higher magnification of the same photonic crystal, clearly showing the L3 cavity in

the centre. The small bright spots are most probably uncapped quantum dots on

the sample surface, which were grown for atomic force microscopy studies. Quantum

dots in the middle of the slab membrane, however, cannot be seen in SEM images.

We fabricated an array of 6 × 6 photonic crystals, similar to the one displayed in

fig. 3.6(c), within an area of 300× 300 µm2.

Photonic Crystal Waveguide

The fabrication of a photonic crystal waveguide contains one additional step as com-

pared to a photonic crystal L3 cavity. Since we want to use the side detection and

excitation capability of our setup (as depicted in fig. 7.5) to check if the photonic

crystal waveguide transmits light, it has to be cleaved along the GaAs easy cleavage

axis [110], perpendicular to the row of omitted air holes. In fig. 3.7(a) we depict a

SEM image of a waveguide after processing steps fig. 3.5(b)-(e) showing optical in-

and out-couplers at both ends of the waveguide. The purpose of these structures is

to enable more efficient in- and out-scattering of light into and from the waveguide

and they were designed according to ref. [Far08b]. The two rectangles directly next to

the photonic crystal structure serve for orientation purposes and have no influence on

the photonic properties of the waveguide. When the waveguide has been cleaved into

two pieces, as we show in fig. 3.7(b), and underetched by means of hydrofluoric acid

[see fig. 3.5(e)] we are able to use the second microscope objective to directly detect

light from the end of the waveguide or focus light into the waveguide [see fig. 7.5].

We can cleave a photonic crystal waveguide perpendicular to the omitted row of air

holes with fairly high precision when it is oriented along the [110] crystal axes of the

GaAs substrate. This can be seen in the SEM image in fig. 3.7(c), which is recorded

with higher magnification. The underetched region is also visible.

38


1 µm

(a)

(c)

(c)

(b)

10 µm

1 µm

Figure 3.7.: SEM pictures of a W1 waveguide before cleaving (a), after cleaving (b)and a magnification of the cleaved facet (c).

39


3.3. Experimental techniques

In this section we will introduce the principle measurement techniques used in this

thesis, namely:

• Photo- and electroluminescence spectroscopy

• Time-correlated emission spectroscopy

• Photocurrent absorption spectroscopy

We will now explain each technique in more detail together with the measurement

setups developed in the laboratory for their implementation.

3.3.1. Micro-Photoluminescence Setup

For our optical studies we use the setup schematically depicted in fig. 3.8. Light from

a Titanium-Sapphire (Ti:Sa) laser is coupled into a single mode optical fibre and

guided to the setup (1). A neutral density power wheel (2) attenuates the laser power

to the desired intensity. The laser light is then directed into a beamsplitter cube

(3) with 50% transmission and 50% reflection, where the intensity of the transmitted

light is measured by an optical powermeter (4) to have a reference for the incident

intensity on the sample. The reflected light is directed onto the sample through a

×100 magnifying microscope objective (5) with a numerical aperture of NA = 0.5

and a long working distance of 12 mm. The objective can be positioned in all three

spatial dimensions with a piezo-stage (6) to allow a precise targeting of the beam.

The size of the excitation spot determines the spatial resolution of the system. It was

measured to be 1.3 µm at a wavelength of λ = 815 nm (see sect. 7.1), which is very

close to the Abbe diffraction limit of the objective. To observe the exact position of

the laser spot on the sample, an imaging CCD (7) is used together with a white light

source (8). The sample is placed inside a liquid helium flow cryostat with a top window

allowing optical access. The luminescence signal from the sample is then collected via

the same microscope objective and transmitted by the beam splitter on a separate

optical system, which guides the light through a longpass filter (9) to the entrance

slit of a Triax 550 imaging spectrometer (10). The light is dispersed by gratings with

either 150, 600 or 1200 lines/mm and detected by a liquid nitrogen cooled Si-CCD

detector (11) with a 1340 × 100 array of 20 µm sized pixels. The side exit of the

40

3.3 Experimental techniques

Cryostat

Sample

x-y Stage

x100

Piezo xyz - Stage

LiquidHe

BeamSplitter50:50

Laser

Single ModeOptical Fiber

PowerWheel

Excitation

Piezo xyz - Stage

Powermeter

DetectionFlipMirror

ImagingCCD

WhiteLight Source

92:8Pellicle

LongpassFilter

0.Order

1.Order

TurretwithGratings

Mirror

Entrance Slitwith Shutter

LN2

CooledCCD

Voltage SourceLock InAmplifier

VoltageSource

Photocurrent

Triax 550 spectrometer

Time-Correlated

FlipMirror

(1)(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

SideExitSlit

(12)

Vacuum

Figure 3.8.: Schematic illustration of the micro-photoluminescence setup. Adaptedfrom ref. [Hau09].

spectrometer (12) can be used to guide the filtered light out of the spectrometer

and onto additional detectors as it is done for the time-correlated measurements (see

sect. 3.3.3).

41


3.3.2. Photo- and Electroluminescence

To study non-destructively the properties of semiconductor quantum dots and their in-

teraction with an optical cavity mode we perform luminescence spectroscopy. Fig. 3.9

shows the basic principle of photoluminescence, starting with an unexcited semicon-

ductor quantum dot in fig. 3.9(a). When exciting above the wetting layer (EWL ∼1.44 eV at 10K) with a laser, charge carriers are generated in the vicinity of the quan-

tum dot as illustrated in fig. 3.9(b1). Alternatively, this can be realized by carrier

injection from electric contacts as in fig. 3.9(b2). The charge carriers are captured into

the quantum dot (τcapture ∼ 3 ps) and relax subsequently by phonon mediated pro-

cesses (τrelax ∼ 10 ps) to the energetically lowest unoccupied states, [Tru05, Ohn96]

as depicted in fig. 3.9(c). The carriers recombine radiatively after a typical spon-

taneous recombination time τrad ∼ 1 ns [Bim99, Mic03] and generate photons of

distinctive energies. The whole process fig. 3.9(a)-(c) involving fig. 3.9(b1) is called

photo-luminescence, while the process depicted schematically in fig. 3.9(b2) is called

electro-luminescence. By analyzing the emitted luminescence we can probe the elec-

tronic structure of the quantum dot.

Laser

(b1) Optical excitation (b2) Electrical excitation

Carrierinjectionfrom thecontacts

(c) Relaxation and recombination

Emissionl~1mm

(a) Unexcited QD

Figure 3.9.: Schematic illustration of optical spectroscopy of a single quantum dot: (a)unexcited quantum dot. (b1) creating electron hole pairs by laser excita-tion or (b2) by injection from the leads and capturing into the quantumdot. (c) Radiative recombination of few exciton states populating thequantum dot. Adapted from ref. [Hof08].

3.3.3. Time-Correlated Single Photon Counting

As presented in fig. 3.8, we can guide the spectrally filtered light to a time-correlated

part of our setup via the side exit (12) of the spectrometer by flipping a mirror in front

42


of the CCD (11). Thus, we are able to perform time-correlated single photon counting

experiments either for spontaneous emission lifetime or autocorrelation measurements

as depicted in fig. 3.10.

Spontaneous Emission Lifetime

For spontaneous emission lifetime measurements, the light directed to the side exit

of the spectrometer (12) is coupled into a multi mode optical fibre and guided to a

single photon avalanche diode (SPAD) with a time resolution of 350 ps as depicted in

fig. 3.10(a). In time-correlated measurements, we use a pulsed Ti:Sa laser with a pulse

width of 6 ps and a repetition frequency of 80 MHz (≡ 12.5 ns). In order to determine

the decay lifetime, the detected photons are timed with respect to a reference signal

from a fast photodiode inside the laser. This is done using a TimeHarp card, [Pic11]

which creates a time difference histogram from many repeats of the same experiment

as depicted in fig. 3.10(a). By measuring the reflected laser pulses directly (i.e. the

spectrometer is driven to the wavelength of the laser), we obtain the instrument

response function (IRF), presented in blue. The characteristic form is affected by

time-jitter in the electronic signal from the photodiode inside the laser, the SPAD

0

1

g(2

) (t)

Cryostat

Sample

x-y Stage

x100

Piezo xyz - Stage

LiquidHe

BeamSplitter50:50

LA

SE

R

Single ModeOptical Fibre

PowerWheel

Excitation

Piezo xyz - Stage

Powermeter

DetectionFlipMirror

ImagingCCD

WhiteLight Source

92:8Pellicle

LongpassFilter

0.Order

1.Order

TurretwithGratings

Mirror

Entrance Slitwith Shutter

LNCooledCCD

2

Lock InAmplifier

VoltageSource

Triax 550 Spectrometer

Time Harp Card

PhotonsCounted

Start Stop

OutputSignal

Optical SplitFibre

Time-CorrelatedFlipMirror

(1)(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

SideExitSlit

(12)

Vacuum

x50

Pie

zo

xyz

- S

tage

Pie

zo

xyz

- S

tag

e

(13)

(14)FlipMirror

Imaging CCDplus

White Light Source

(16)

Multi ModeOptical Fibre

xyz-Stage

T-CubeMirror

(15)

To spectrometeror

from LASER

from µ-PL setupMulti ModeOptical Fibre


To spectrometer

(18)

SPAD

SPAD

Autocorrelation

Fro

mS

pect

rom

ete

r

SPAD

Time Harp Card

Reference Signalfrom Laser

PhotonsCounted

Start Stop

OutputSignal

Time

PL

Laser Beam(IRF)

MeasurementHistogram

ExponentialFit


Time-Correlated

Fro

mS

pect

rom

ete

r

Decay Lifetime(a) (b)

Hanbury-BrownTwiss Setup

Figure 3.10.: Schematic illustration of the time-correlated part of the micro-photoluminescence setup for (a) lifetime measurements and (b) autocor-relation measurements (Hanbury-Brown Twiss Setup). Adapted fromref. [Hau09].

43


and the TimeHarp card. When we deconvolute the recorded histogram of photon

arrival times with the IRF, we can enhance the temporal resolution to a maximum of

∼100 ps. [Hau09]

Autocorrelation

If we want to measure the second order photon correlation function g(2)(τ) (auto-

correlation), e.g. from a single photon source, the “time-correlated” part of our

setup is modified to a Hanbury-Brown Twiss system [see fig. 3.10(b)]. In contrast

to spontaneous emission lifetime measurements the spectrally filtered photolumines-

cence signal leaving the side exit of the spectrometer (12) is coupled into a 50:50

beamsplitter spliced fibre connected to two different SPADs with similar time reso-

lution of ∼750 ps. To obtain the autocorrelation function the detected photons from

one SPAD are timed with respect to the detected photons from the other one, which is

again performed by the TimeHarp card. As the split fibre divides the signal into two

equal parts and photons are fundamental, indivisible particles, the resulting g(2)(τ),

depicted in fig. 3.10(b), shows an expected dip at a delay time τ = 0 when investi-

gating a single photon source. For an ideal single photon source the probability of

detecting a photon in both SPADs at the same time is zero. For more information on

photon-correlation functions we refer the reader to ref. [Lou83].

3.3.4. Photocurrent Measurement

The sample design of the p-i-n photonic crystal structure provides us with the unique

possibility to perform photocurrent investigations on photonic crystal structures.

However, the magnitude of the generated photocurrent is typically of the order of

a few picoamperes, especially for small applied electric fields, since the majority of

photo-generated carriers does not tunnel out of the quantum dots, but instead re-

combines radiatively. In addition, the leakage current of the diodes is of the same

magnitude and is superimposed to the photo-generated current. Therefore, we use a

lock-in amplifier to measure photocurrent in the picoampere regime with high signal

to noise ratio. The lock-in amplifier distinguishes between current generated by the

excitation laser and current arising from other sources. The electric circuit of the

lock-in amplifier is depicted in the ”photocurrent” box in fig. 3.11. A chopper is used

to modulate the excitation intensity and, therefore, the photogenerated current at a

given frequency. The lock-in amplifier only amplifies and detects the current which

44


Photocurrent

ChopperWheel

LaserBeam

Lock-in Amplifier

LaserBeam

ChopperWheel Reference Signal

+-+-

Sample

Electric circuit

Lock In

VoltageSource

Figure 3.11.: Schematic illustration of the photocurrent part of the micro-photoluminescence setup. Adapted from ref. [Hau09].

is modulated at that frequency. Using this technique allows us to easily measure

photocurrents with amplitudes down to < 500 fA.

45

Electrical force is defined as something which causes

motion of electrical charge; an electrical charge is

something which exerts electric force.

Arthur Eddington

Chapter 4Characterization of Electrically Tunable Samples

For the investigations of cavity quantum electrodynamics phenomena on single quan-

tum dots with high quality photonic crystal nanocavities, the spectral detuning be-

tween the quantum dot emission line and the cavity mode is crucial. Even very small

spectral detunings can change the quantum dot cavity coupling significantly. In the

weak coupling regime the spectral dependence of the emission rate is governed by a

Lorentzian lineshape, as discussed in sects. 2.1.2 and 5.1. If the cavity has a relatively

high quality factor of Q > 5000 at E = 1240 meV a spectral detuning of less than

∆E < 0.2 meV is sufficient to significantly change the quantum dot emission life-

time. In particular, in the strong coupling regime with Q > 10000 (see sect. 2.1.3 and

chapter 6), the probability of coherent energy transfer between coupled excitonic and

photonic states is severely hindered by a spectral mismatch for detunings much larger

than the cavity linewidth. [Yos04, Rei04] Therefore, precise spectral tuning mecha-

nisms are needed for either the cavity mode or the quantum dot emission energy.

Common methods for adjusting the detuning include irreversible methods such as

digital etching, [Hen05, Bad05, Dal05, Dal10] local oxidation [Hen06b, Kon09, Lee09,

Kir11] and atomic layer deposition, [Kir11] and reversible methods such as changing

the lattice temperature, [Yos04, Rei04, Eng07a, Nom10] and condensation of inert

gases at the surface. [Mos05, Str06b, Hen07] While the first methods can be used

for a coarse adjustment, they are impractical for cavity quantum electrodynamics

experiments where the detuning needs to be varied repeatedly. Although the latter

47

Characterization of Electrically Tunable Samples

methods are reversible and can be used in-situ in the cryogenic environment, they

are slow, rendering them impractical for future single-quantum dot devices. How-

ever, in the last years an electrical tuning method was designed which employs the

quantum confined Stark effect to shift the emission energy of the quantum dot exci-

tons. [Fin04, Hof07, Hof08, Kis08, Eng09, Far10a] This method is completely reversible

and was shown to work up to frequencies of 150 MHz. [Far10a]

In this chapter we discuss the characteristic properties of our electrically tunable

samples. We will start by investigating current-voltage characteristics at different

temperatures and under different illumination powers in sect. 4.1. We will then discuss

the influence of the applied electric field on photoluminescence and photocurrent,

respectively, and investigate L3 cavities with these techniques to show that we have

a local electrical access to the nanocavity in sect. 4.2. Furthermore, the technique

of cavity resonant excitation is explained and its advantages for our experiments are

discussed in sect. 4.3.

Following this, the electrical tuning mechanism of the exciton emission energy due to

the quantum confined Stark-effect is discussed in sect. 4.4. This allows us to precisely

control the exciton-cavity detuning via the bias voltage. Furthermore, we investigate

the response of the system to alternating bias voltages by taking photoluminescence

spectra of single dot exciton emission lines.

In the last part of this chapter (sect. 4.5) we will introduce nitrogen deposition as

a method to reversibly tune the nanocavity mode emission energy.

48

4.1 p-i-n Diode Characteristic

4.1. p-i-n Diode Characteristic

We begin by discussing the electrical characteristics of our p-i-n devices with and

without illumination. Fig. 4.1(a) shows the growth layer sequence of the sample with

p- and n-doped layers of GaAs sandwiching a layer of nominally undoped intrinsic

GaAs into which we incorporate the self-assembled quantum dot layer. Fig. 4.1(b) de-

picts a schematic of the corresponding band diagram with tilted bands in the intrinsic

region. The resulting electric field distribution is illustrated in fig. 4.1(c), where the

electric field in the intrinsic region is constant. When Ohmic contacts are established

to the n- and p-doped regions of the diode, an external bias voltage Vapp can be ap-

plied to the sample to influence the electric field in the intrinsic region. The static

electric field Fi in this region and, therefore, at the position of the quantum dots, is

then approximately given by

Fi(Vapp) =Vbi − Vapp

di, (4.1)

where Vbi is the built-in potential [Sze85] of the diode p-i-n structure and di is the

nominal thickness of the intrinsic GaAs region.

Fig. 4.2(a) shows typical current-voltage (I-V) characteristics of our p-i-n diode

sample in the dark at different temperatures between 307 K and 16 K. The measured

current was normalized to the mesa area of 0.12 mm2 to obtain the current density.

With decreasing temperature, the onset voltage in forward bias moves from ∼ 0.0 V

to ∼ 0.9 V. The leakage current at −4 V bias is reduced from ∼ 200 nA/mm2

down to < 1 nA/mm2 at 16 K. Currents < 10 pA cannot be resolved by our

HP4142B DC-Source/Monitor, which leads to the large noise in the low current regime

(< 0.1 nA/mm2). Comparing the I-V characteristics at 16 K and 117 K, we observe

in the range from −4 V to 1 V no difference in the leakage current and only a small

shift in the onset voltage. This means we can perform temperature dependent studies

in this range without affecting the electric field properties of the quantum confined

Stark-effect (see sect. 4.4.1). Considering the threshold of the current in the I-V

characteristics at low temperatures, the built-in voltage can be determined to be

Vbi ∼ 0.9 V. We note that the onset voltage of our photodiodes is typically ∼ 0.9 V

instead of the ∼ 1.5 V expected for a GaAs p-i-n structure. This discrepancy prob-

ably arises due to the low doping level of the GaAs n-contact (−2 · 1018 cm−3) and

Al0.8Ga0.1As sacrificial layers. This low doping level was experimentally found to be

49


pi

55nm

110nm

n

EF

x,y

z

QDs

ip ni

E

z

(a)

(b)

(c)

z

El. F

ield

F Fi

Figure 4.1.: (a) Schematic image of the grown p-i-n layer sequence with incorporatedquantum dot layer. (b) Schematic band diagram with Fermi level EF ofthe samples. (c) Electric field distribution with constant electric field inthe intrinsic region. [Hau09]

required to obtain high-Q cavity modes.

In fig. 4.2(b) we compare I-V characteristics of the same diode after different num-

bers of cooldown 1 cycles at low temperatures of ∼ 16 K. The characteristics are

identical within the experimental error, indicating that there is no deleterious phys-

ical degradation induced by the cooldown procedure. 2 This was found to be the

case so long as the electric contact wires were short-circuited during cooldown. Con-

1Decrease sample temperature from room-temperature to ∼ 16 K.2Reference [Hof08] reported a degradation of the diode characteristic induced by the cooldown

procedure with an older sample design.

50

4.1 p-i-n Diode Characteristic

-4 -3 -2 -1 0 1 20.01

0.1

1

10

100

1000

10000

Power(W/cm²)

1400 380 80 20 5 1 dark

Cu

rre

nt-

de

nsi

ty(n

A/m

m2)

Applied Voltage (V)

0.1 1 10 100 10001

10

100

1000

10000

Cu

rre

nt-

de

nsity -1

V(n

A/m

m2)

Illumination Power (W/cm²)

m = 0.98 +/- 0.02

-4 -3 -2 -1 0 1 20.01

0.1

1

10

100

1000

10000 307 K 286 K 245 K 181 K 117 K 16 K

Cu

rre

nt-

de

nsity

(nA

/mm

2)

Applied Voltage (V)

(a)

(c) (d)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.01

0.1

1

10

100

1000

10000

1 cooldown 5 cooldowns 10 cooldowns

Cu

rre

nt-

de

nsity

(nA

/mm

2)

Applied Voltage (V)

(b)

080930

Figure 4.2.: Diode characteristic: (a) IV characteristic at different temperatures. (b)IV characteristic after several cooldown cycles. (c) IV characteristic at ∼16 K for different illumination intensities at 1550 meV excitation energy.(d) Current-density at −1 V backward bias plotted over illuminationintensity. [Hau09]

tact wires, which are not short-circuited during temperature changes lead to charge

accumulations which might destroy the diode.

Fig. 4.2(c) compares I-V characteristics at ∼ 16 K of the same diode illuminated at

different intensities by a titanium sapphire laser in continuous wave mode, emitting

above the GaAs bandgap (1550 meV). The illumination power was increased from

∼ 1 W/cm2 to ∼ 1400 W/cm2 in a ∼ 1 µm2 spot on the sample. For all illumination

powers, we observe a minimum in photocurrent at ∼ 0.9−1.0 V. From equation (4.1)

we see that at this voltage the built-in field is canceled and, therefore, photogenerated

carriers do not move out of the intrinsic region, reducing the measured photocurrent.

51


When decreasing the applied voltage Vapp at a fixed illumination intensity, we have

an increase in photocurrent since the electric field Fi in the intrinsic region is rising

according to equation (4.1). The increase in Fi drives the carriers out of the intrinsic

region towards the electric contacts, increasing the amplitude of the measured pho-

tocurrent. When further increasing Fi, by decreasing Vapp, we observe a saturation

of the photocurrent, since all photogenerated carriers are extracted by the electric

contacts.

The plot in fig. 4.2(d) shows the current density at Vbias = −1 V for different il-

lumination powers from ∼ 1 W/cm2 to ∼ 1400 W/cm2 in a ∼ 1 µm2 spot on the

sample together with a linear fit. We observe a linear dependence of the photocur-

rent with illumination power, indicating that over the measured range the generation

of electron-hole pairs and the extraction efficiency stays constant when exciting in

resonance with the GaAs absorption at 1550 meV.

4.2. Photoluminescence vs. Photocurrent

In this section we explain in detail the influence of the electric field Fi on the photo-

luminescence and photocurrent signal.

4.2.1. Electric Field Effects

Photogenerated electrons and holes in the quantum dot relax to the lowest unoccu-

pied orbital states over picosecond timescales. [Hei97] They recombine subsequently

with a typical lifetime of ∼ 1 ns [Hof04] and emit a photon as illustrated in fig. 4.3(a).

If we neglect thermal loss of carriers from quantum dots, 3 as well as non-radiative

recombination, the radiative decay is the only possible decay channel for the exciton

when no electric field is present. Therefore, all photogenerated excitons in the quan-

tum dot will recombine and generate light. However, when an electric field is present,

a non-radiative decay channel is available since the conduction and valence bands of

the quantum dot are tilted as illustrated in fig. 4.3(b). This tilt leads to a finite prob-

ability for electrons and holes to tunnel out of the quantum dot [Fli05, Gri06, Ang08]

and drift under the influence of the electric field to the electric contacts, where they

are detected as an optically induced current.

3The thermal energy at T ∼ 16 K (∼ 1.5 meV) is much smaller than the confinement (∼ 20 meV).

52

4.2 Photoluminescence vs. Photocurrent

GaAs GaAsQDWL

--

+

z

E

GaAs WL QD GaAs

--

+

(a) (b)

Fi

Figure 4.3.: Schematic of decay dynamics in a quantum dot at low temperatures:(a) Optical recombination without electric field. (b) When applying anelectric field Fi, carriers can tunnel out of the quantum dot and generatephotocurrent. [Hau09]

Optical recombination and photocurrent are two competing processes. An electron-

hole pair which recombines radiatively cannot be detected as photocurrent and vice

versa. In the absence of an electric field, the conduction and valence bands are not

tilted resulting in a quasi impenetrable barrier. Thus, the photoluminescence signal

is strong and no photocurrent is generated. In contrast, for large electric fields,

the bands are strongly tilted resulting in a very thin tunneling barrier. Since the

tunneling probability depends exponentially on the barrier width, [Gri06, Ang08] the

tunneling time is very short for large electric fields. Therefore, the photoluminescence

signal quenches and photogenerated electron-hole pairs tunnel out of the quantum dot

and generate photocurrent. Detailed discussions with experimental results of those

competing processes can be found in refs. [Gri06, Ang08, Hof08].

53


4.2.2. L3 Cavity Investigation

We use the technique of photoluminescence and photocurrent spectroscopy, as de-

scribed above and in sect. 3.3, to probe the properties of our photonic crystal nanocav-

ities. In the following experiment we investigate an optimized L3 cavity in a hexagonal

lattice with r/a = 0.28, where the first outer holes are shifted by ∆a = 0.15 lattice pe-

riods (see sect. 2.2.2). The excitation energy of the titanium sapphire laser focused on

the center of the L3 cavity is swept, while detecting at the energy of the fundamental

mode (M1) with a single photon detector through the side exit of our spectrometer

(see sect. 3.3.1). This special photoluminescence measurement technique is called

photoluminescence excitation (PLE). While sweeping the excitation energy, we also

measure the photocurrent. However, one drawback of photoluminescence excitation

measurements is that we cannot probe the excitation of the fundamental mode, since

we are detecting on it.

Fig. 4.4(a) shows the photoluminescence excitation intensity on a logarithmic scale

at the energy of the fundamental cavity mode (EM1 = 1215 meV) plotted versus

the excitation energy while keeping Vapp = 0.5 V fixed. 4 In fig. 4.4(b) we show

the corresponding detected photocurrent on a logarithmic scale plotted versus the

excitation energy. The photocurrent is normalized with respect to the excitation

power. We clearly observe four peaks (marked in orange) in the photoluminescence

excitation signal and the corresponding peaks in the photocurrent measurement. The

photocurrent background signal is decreasing with decreasing excitation energy, since

states which are at lower energy are less likely to tunnel out compared to higher

energetic states (see fig. 4.3).

If we compare the results of figs. 4.4(a) and (b) with our simulation results shown

in figs. 2.6 and 4.4(c), we can identify five modes of the L3 cavity. The modes M4

and M5 are too close together to resolve them. Therefore, we only see four peaks

in photoluminescence. The measurements presented in this section show that we are

able to establish a direct electrical contact to the photonic crystal nanocavity and that

the etching process described in sect. 3.1.2 does not destroy the electric connection

within the p- and n-doped GaAs layers. This finding is crucial for the tuning of the

exciton energies presented in sect. 4.4 and the experiments presented in chapter 6.

4Vapp = 0.5 V resulted in the best signal to noise ratio for photocurrent.

54

4.3 Cavity Resonant Excitation

|E|² Cutplanes

max

0

(a) (c)

M2 M3

M5M4

M6

(b)M4/5M2 M3 M6

Excitation Energy (meV)

Figure 4.4.: Investigation of optimized L3 cavity modes. (a) Measured photolumines-cence excitation intensity at the first order cavity mode energy plottedversus excitation energy. (b) Measured photocurrent plotted versus exci-tation energy. The energy of the cavity modes are marked in orange andlabeled from M2 to M6. (c) Calculated electric field distribution of thecorresponding L3 cavity modes.

4.3. Cavity Resonant Excitation

According to refs. [Nom06, Kan09b], resonant excitation of single quantum dots via

a higher order cavity mode results in a ∼ 100-fold increase in the pumping efficiency.

Furthermore, they show that this excitation scheme leads to a selective excitation

of quantum dots coupled to the cavity. Fig. 4.5 compares the photoluminescence

spectrum under higher order cavity resonant excitation at 1292 meV with above GaAs

bandgap excitation at 1550 meV. The strongest peak is the 1st order cavity mode.

Furthermore, we observe single exciton lines emerging in the resonant case while in

above bandgap pumping we see other and weaker exciton lines although we measure

the photoluminescence signal at the same position on the sample. The reason for

these observations are that, when exciting resonantly in a cavity mode, quantum dots

55


1205 1210 1215 1220 1225

Cavity resonant excitation Non resonant excitation

PL

Inte

nsiit

y(a

rb.

units)

Energy (meV)

Cavity Mode

Exciton linesExciton lines

080409

Figure 4.5.: Photoluminescence spectra under cavity resonant excitation and underabove GaAs bandgap excitation. In the resonant case single quantum dotlines coupled to the cavity emerge out of the continuum. The intensity ofthe fundamental cavity mode increases for resonant excitation conditiondue to a more efficient pumping mechanism.

positioned at antinodes of the mode field pattern are strongly pumped while quantum

dots positioned at nodes experience no excitation. This leads to a selective, highly

efficient excitation of quantum dots coupled to the cavity via the mode field. [Kan09b]

These efficiently pumped quantum dots also emit into other cavity modes resulting

in a stronger mode emission than for the non resonant excitation (see chapter 5).

According to equations (2.9) and (2.10), we have a stronger exciton-mode coupling

at antinodes of the mode field pattern. As a result, the cavity resonant excitation

scheme is very suitable for “picking out” the quantum dot transitions which have a

good chance of showing strong coupling behavior with a cavity mode. In addition, the

high pumping efficiency allows us to investigate quantum dot - cavity systems at low

excitation intensities. This is crucial to minimize disturbing effects such as screening

of the electric field due to free photogenerated carriers or heating of the sample by

the laser.

56

4.4 Quantum Dot Emission Energy Tuning

4.4. Quantum Dot Emission Energy Tuning

We have designed the samples introduced in sect. 3.1, such that we can make use of the

quantum confined Stark-effect to shift the recombination energy of excitons Eexciton

relative to the fixed emission energy of a photonic crystal slab nanocavity mode. 5 To

create a controllable electric field Fi at the position of the self-assembled quantum

dots, we incorporate them in the middle of the intrinsic region of a p-i-n diode (see

fig. 4.1), which is connected to a voltage source. Thus, by changing the applied bias

voltage Vapp we can directly influence Eexciton. Compared to other tuning mechanisms,

like temperature tuning [Rei04, Yos04, Pet05, Pre07] or gas deposition [Mos05, Hen07],

the quantum confined Stark-effect is more attractive since it is reversible and can be

used to switch the exciton-cavity coupling over much faster time timescales.

4.4.1. Quantum Confined Stark-Effect

An exciton is a bound state of an electron and a quasi particle, called hole. It interacts

with an externally applied electric field which gives rise to numerous effects such as the

generation of photocurrent when illuminating the sample, symmetry lowering, [Guo04,

Ima02] or a shift in the emission energy of the exciton, called quantum confined Stark-

effect. [Mil84, Fin04]

A cross sectional view of a single In0.4Ga0.6As self-assembled quantum dot, incor-

porated in the intrinsic region of a p-i-n diode as in our sample design, is sketched in

fig. 4.6(a), together with cartoon illustrations of the localized electron (hole) wave-

function in blue (red). Our quantum dots are grown by the Stranski-Krastanow

method [Str39] and show a gradient in both alloy concentration and lateral size along

the growth direction. This leads to a deeper potential well at the top compared to

the bottom of the quantum dot. Because of their 5-7 times larger effective mass com-

pared to electrons in InGaAs, [Fry00] holes tend to occupy the more confined upper

region inside the quantum dot while electrons stay at the lower region. This leads to

the formation of an intrinsic dipole p as illustrated in fig. 4.6, which is present even

without applying an external electric field. The typical values for this permanent

excitonic dipole are of |p| = −8 · 10−29 ± 0.5 · 10−29Cm. [Fin01b, Kre05]

A change in an externally applied electric field can change the separation d [see

5The mode emission remains fixed due to the negligible variation of the refractive index of theGaAs matrix since we are spectrally far away from the GaAs bandedge.

57


-+ In

concentrationp

p-doped GaAs

n-doped GaAs

x,yz

(a)

+-

(b)

yx

d

i-GaAs

InGaAs

Figure 4.6.: (a) Schematic illustration of In0.4Ga0.6As self-assembled single quantumdot in a p-i-n diode with In concentration gradient and shape variationalong the growth axis as used in our sample design. Schematically illus-trated in blue (red) is the localized electron (hole) wavefunction togetherwith the corresponding dipole moment p (green). (b) Schematic crosssectional top view, again with electron (hole) wavefunction and dipolemoment.

fig. 4.6(a)] between the carrier wavefunctions by either pulling them further apart

or by pushing them together. Since the change in dipole moment p = e · d, with

e = −1.6 · 10−19 C, is very small, it is possible to perform a Taylor expansion of p as

a function of the electric field Fi in the intrinsic region and neglect all terms higher

than first order:

p(Fi) = p1(Fi) + p0 = β · Fi + p0, (4.2)

with β being the polarizability 6 of the exciton. The electric dipole p interacts

with the electric field Fi, therefore, it is possible to change the total energy of the

electron-hole pair Eexciton = Ee + Eh as described by ref. [Jac66]

Eexciton(Fi) = p(Fi) · Fi + E0 = p0 · Fi + β · |Fi|2 + E0, (4.3)

6A detailed discussion on the dependence of β from the quantum dot size can be found in [Ang08].

58


where E0 is the non-field-dependent total energy of the exciton. Equation (4.3)

shows that when changing the electric field Fi, the emission energy of an exciton

in a quantum dot changes in a quadratic way. Fig. 4.7(a) depicts schematically the

dependence of the exciton recombination energy with respect to the electric field Fi

in the intrinsic region of the p-i-n diode. The electron and hole spatial distributions

are illustrated in (b) for the five different electric fields marked in (a). Points 2 , 3

and 4 are in the small dipole regime, since the separation of electron and hole is

smaller than the size of the quantum dot. 1 and 5 correspond to the large dipole

regime where the separation of the carriers is close to the size of the quantum dot

and the dipole moment becomes field independent.

Electric Field F 0

Reco

mbin

atio

n E

nerg

y

43

1

0 Bias Voltage V

5

(a) (b)

bias

i

- +3 -+

p4

-+ p2 F=01

5

FiFi

i-

+p Fimax

-

+p Fimax

small dipolelargedipole

largedipole

2

Figure 4.7.: (a) Schematic of exciton recombination energy in a quantum dot plottedversus electric field Fi in the intrinsic region and applied bias voltage.Small and large dipole regimes are illustrated. The yellow region high-lights the regime in which we can tune our sample. (b) Schematic illus-tration of electron and hole spatial distribution in a quantum dot at fivedifferent electric fields Fi, as indicated in (a).

In detail:

2 : Here, |Fi| = 0 in the intrinsic region. To achieve this condition we need

to apply forward bias to cancel the built-in field, pointing from the n-doped

layer to the p-doped layer. This is a situation similar to a quantum dot in

59


undoped bulk material, where the hole favors an upper position in the dot and

the electron favors a lower position as explained previously.

3 , 4 : The case of Fi pointing from the p-doped to the n-doped layer cannot be

reached with our sample design due to the onset of current flow, which prevents

hole and electron to move more into the middle of the quantum dot where they

would cancel the electric dipole moment. Hence, we cannot tune the exciton

recombination energy over the maximum value.

1 : Hole and electron get pulled apart further, such that the hole is located

at the very top of the quantum dot and the electron at the very bottom. This

configuration cannot be reached with our sample design because electron and

hole will tunnel out of the quantum dot beforehand.

5 : Hole and electron get pushed to the very bottom of the quantum dot and

the very top, respectively. This is the reversed configuration of 1 and can also

not be reached in our samples.

4.4.2. Tuning Range

As discussed in the last section, we expect the emission energy of an exciton in

a quantum dot to shift parabolically with the applied electric field Fi. To avoid

breakthrough of the p-i-n diode at large backward bias and current flow in forward

bias, we are limited to a bias voltage range from ∼ −4 V to ∼ 0.9 V. Fig. 4.8(a) shows

typical photoluminescence spectra recorded when exciting resonantly in a high order

L3 cavity mode at 1345 meV with a continuous wave laser. The spectra are recorded

as a function of increasing electric field Fi (pointing from n-doped layer to p-doped

layer) from the top of the graph to the bottom as indicated by the orange arrow. The

single quantum dot emission lines shift to lower energy when increasing the electric

field in backward bias, as expected from our discussions in fig. 4.7(a). 7

Fig. 4.8(b) shows a more detailed analysis of the emission line marked by black

arrows in (a), with the extracted peak position plotted as a function of the applied

voltage Vapp. Fig. 4.8(c) shows the differential shift of the peak energy δmeV/δV of

the emission line. For Vapp > 0.9 V (blue region) we do not observe a shift in the

extracted peak position, while the maximum differential shift is at Vapp ∼ 0.5 V. When

7The kink at Vapp = 0 V arises when the voltage source switches from positive to negative polarity.

60


(a) (b)

(c)

(d)

3.5 meV

δ

i

1221

1222

1223

1224

1216 1218 1220 1222 1224

PL

Inte

nsity

(arb

.units)

Energy (meV)

Energ

y(m

eV

)

0

1

2

dm

eV

/dV

-2 -1 0 110

-1

100

101

102

103

104

(nA

/mm

²)C

urr

ent

Density

Bias Voltage (V)

080912

F

Figure 4.8.: Electric tuning of exciton emission lines using the quantum confinedStark-effect: (a) Waterfall plot of photoluminescence spectra when chang-ing Vapp. Analysis of marked exciton line: (b) Energetic peak position,(c) differential peak position shift and (d) I-V characteristic of the illu-minated sample.

further decreasing the applied voltage, the differential shift gradually reduces and for

Vapp < −1.4 V (green region) we do not observe any energy shift. If we look at the

diode characteristic in fig. 4.8(d), recorded under the same illumination conditions

as the photoluminescence spectra in (a), we see that the regime with current flowing

in forward direction corresponds to the regime where the exciton emission lines stop

shifting. When switching to forward bias the electric field Fi acting on the quantum

dot is zero for all Vapp > 0.9 V since the electric field is screened by the current flow.

In the voltage range from −1.4 V< Vapp < 0.9 V the electric field acts on the quantum

dot and the exciton energy shifts gradually over a range of 3.5 meV to lower energy,

while the differential shift is gradually reducing. If we examine the I-V characteristic,

we see a continuous increase in backward current which explains the reduction of

61


the differential shift. 8 The discrepancy between the theoretically predicted quantum

confined Stark effect (see equ. 4.3) and the experimentally observed shift is probably

due to screening by photogenerated charges that, in combination with a high access

resistance, prevent the field from increasing linearly with the applied bias voltage.

For Vapp < −1.4 V the photo-induced leakage current at the excitation spot is large

enough to prevent the electric field at the position of the quantum dots becoming

larger. [Hof08]

In summary, by employing the quantum confined Stark-effect we have a reversible

tuning mechanism to control the dot-cavity detuning precisely over a range of ∼ 4

meV. We can adjust the bias voltages in the range from ∼ 0.9 V to ∼ -1.4 V which

corresponds to a maximum electric field of Fmax = 200 kV/cm which is reached at

Vapp = −1.4 V. According to ref. [Fin04] the system should be in the small dipole

regime for applied voltages as indicated in fig. 4.7(a) by the yellow region. We have

to consider that for our sample design equation 4.1 is not valid for all Vapp due to

current onset and screening effects, leading to a non-linear dependence of |Fi| on Vapp.

4.4.3. Frequency Limit

In the last section we have shown that by applying a voltage Vapp we are able to tune

the emission of exciton lines over a range of ∼ 4 meV. So far, we just considered

a static Vapp to shift the exciton emission. To investigate the characteristics of our

sample under alternating voltage we connected the sample to a frequency generator

and applied a rectangular voltage with an amplitude of 0.2 V. Thus, a single exciton

emission line should switch between two emission energies at the frequency of the

applied rectangular voltage as illustrated in fig. 4.9(a). By taking photoluminescence

spectra with a long integration time of 120 seconds we average over the emitted

spectrum during this time and, therefore, we expect to observe two peaks, one for

Vapp = 0.2 V and one for Vapp = −0.2 V.

Fig. 4.9(b) shows photoluminescence spectra of a single exciton line with long in-

tegration times for different frequencies f of the rectangular alternating Vapp in a

waterfall plot. We used the cavity resonant excitation scheme with P ∼ 2µW to ob-

tain a good signal to noise ratio and detected the exciton line at the spatial position

of the L3 nanocavity. For f = 1 Hz and f = 10 Hz we can clearly resolve two different

peaks, separated by ∼ 0.5 meV, since the electric field Fi in the intrinsic region of

8Note the log scale.

62


1253.0 1253.5 1254.0 Energy (meV)

1 Hz 10 Hz 100 Hz 1 kHz

PL

Inte

nsity

(arb

.units)

(a)

Electric Field F 0

Re

com

bin

atio

n E

ne

rgy 0 Bias Voltage Vapp

ΔV

ΔE

080730

(b)

V

t

app

Figure 4.9.: Frequency characteristic of the quantum confined Stark shift: (a)Schematic of exciton recombination energy change ∆E induced by achange in the applied bias voltage ∆Vapp. (b) Waterfall plot of photo-luminescence spectra of a single exciton line with long integration timefor different switching frequencies.

the p-i-n diode follows Vapp and oscillates between two values Fmin and Fmax. When

increasing the frequency to f = 100 Hz, the photoluminescence intensity of the peaks

reduces and they broaden. This indicates that the electric field Fi cannot completely

follow the change in Vapp. When further increasing f to 1 kHz, we observe only one line

of the exciton, showing a stronger intensity, in between the line positions from lower

frequencies. This means that although Vapp alternates at 1 kHz, the electric field at

the position of the quantum dots stays constant at an average value Favg ∼ Fmin+Fmax2

,

leading to a single emission peak.

We can use this frequency dependence to determine the electric access resistance

Raccess to the nanocavity. Our sample is basically a RC circuit, where the p- and

n-doped GaAs layers can be considered as capacitor plates and the intrinsic region as

insulator. 9 The capacitance Cdiode of the p-i-n diode can be calculated as

Cdiode = ε0 · εr ·A

d= 8.85 · 10−12 As

Vm· 13.1 · 400 · 300µm2

110nm= 12.7 nF,

where ε0 is the electric field constant, εr is the relative dielectric constant of GaAs,

A is the mesa area and d is the thickness of the intrinsic region. The time constant τRC

9We neglect the capacitance of the electric access wires.

63


is defined as the product of the circuit resistance R and the circuit capacitance C and

gives the time required to charge the capacitor, through a resistance R, to ∼ 63.2%

of full charge. [Jac66, Sze85] From our measurements in fig. 4.9(b) we determine the

cutoff frequency to fC ∼ 100 Hz for excitons in quantum dots near the cavity. The

time constant τRC can be calculated from the cutoff frequency fC with

τRC = Raccess · Cdiode =1

2π fC= 1.59 · 10−3 s, (4.4)

and this allows us to determine the access resistance R to the cavity:

⇒ R =τRCCdiode

= 12.5 MΩ. (4.5)

The high resistance in the MΩ regime explains the relatively low cutoff frequency

of ∼ 100 Hz compared to other p-i-n designs, switching in the GHz regime with an

access resistance Raccess < 10 Ω. [Put94]

One reason for the large resistance could be the etching processes, which might

significantly increase the specific resistance of the doped GaAs layers due to surface

roughness and defects. 10 However, since we could not detect single exciton lines

under alternating voltage with large enough signal to noise ratio on the bulk, due to

the missing photonic environment, we were not able to verify this theory.

4.5. Cavity Emission Energy Tuning

In sect. 4.4 we have shown that exciton emission lines can be spectrally tuned via

the quantum confined Stark-effect. However, from our discussions in sect. 4.2.1 we

know that changing the applied voltage has an influence on the the exciton decay

dynamics and, hence, on the measured photoluminescence signal. In order to inves-

tigate exciton-mode detuning dependent phenomena it is required to have a tuning

mechanism which has no intrinsic effect on the decay dynamics.

A well established technique for cavity mode tuning is the condensation of inert

gases such as xenon or nitrogen on the photonic crystal structure at low tempera-

tures. [Mos05, Str06b] After cooling the sample down in a liquid helium flow cryostat

to a temperature of approximately 15 K, small amounts of gaseous nitrogen are in-

10One way to overcome this issue is to fabricate samples with gold contacts which reach directlytowards the nanocavity, e.g. by gold strips on top of the waveguide of an A1 nanocavity.

64

4.5 Cavity Emission Energy Tuning

jected into the vacuum of the cryostat via a capillary tube which ends in the vicinity

of the sample. When the nitrogen gas gets in contact with the cold sample surface it

freezes and coats the surface of the photonic crystal with a thin nitrogen film. Thus,

the r/a ratio is decreased and the cavity mode shifts to lower energies with a precision

of < 0.05 meV per tuning step and up to 10 meV in total. [Kan09b]

Nitrogen deposition has, in contrast to other techniques, like e.g. digital etching

[Hen05, Bad05, Dal05, Hof08], the advantage that it is reversible. When heating the

sample above ∼ 30 K the condensed nitrogen evaporates, thus, when cooling it down

again to 15 K the mode is shifted back to its original position.

65

Jeder Mensch hat in der Nahe und in der Ferne

gewisse ortliche Einzelheiten, die ihn anziehen, die

ihm, seinem Charakter nach, um des ersten

Eindrucks, gewisser Umstande, der Gewohnheit

willen, besonders lieb und aufregend sind.

Johann Wolfgang von Goethe

Chapter 5Non-Resonant Quantum Dot - Cavity Coupling

The atom-like properties of semiconductor quantum dots make them ideal candidates

for cavity quantum electrodynamics experiments in the solid-state. [Vah03] Over the

past ten years quantum dot - cavity systems have been used for many different appli-

cations including the efficient and deterministic generation of single photons with a

high degree of quantum indistinguishability, [San02] devices that exploit single photon

quantum non-linearities, [Eng07a] and ultra low threshold nanolasers. [Str06a] Whilst

these systems exhibit many effects already known from atomic cavity quantum elec-

trodynamics experiments, the solid-state environment leads to a number of surprising

results. For example, photoluminescence studies of nanocavities containing a few

quantum dot emitters typically exhibit intense emission from the cavity even though

dot and cavity mode are spectrally detuned. This can be seen in the photolumines-

cence spectrum presented in fig. 5.1 that shows a typical result for such structures.

It is important to understand the origin of this non-resonant feeding mechanism in

order to be able to exploit it, since it mediates the principle gain mechanism in few

quantum dot lasers, [Str06a] or develop strategies to suppress it for the construction

of efficient single photon sources. [Pre07, Kan08b]

For small spectral dot - cavity detunings (|∆E| < 5 meV), acoustic phonon me-

diated dot - cavity coupling has been shown to feed such non-resonant cavity emis-

sion. [Wil02, Xue08, Mil08, Hoh09, Suf09, Ate09, Ota09b, Hoh10, Maj10, Kae10] How-

ever, even when the discrete quantum dot transitions and the mode are very strongly

67

Non-Resonant Quantum Dot - Cavity Coupling

Figure 5.1.: Photoluminescence spectrum of a single quantum dot - cavity system withbright emission from the non-resonant cavity mode.

detuned from one another by |∆E| > 5 meV a mechanism exists by which the quan-

tum dots non-resonantly couple to the cavity mode under conditions of strong optical

pumping. [Hen07, Pre07, Kan08b, Win09, Taw09] For these large detunings, acoustic

phonon mediated coupling becomes ineffective and the work presented in this chap-

ter shows that the cavity mode emission stems from optical transitions between ex-

cited multi-exciton states and an energetically lower quasi-continuum of multi-exciton

states. [Kan08b, Win09, Cha09, Yam09, Lau10b, Lau10a, Ill10]

In this chapter, we present investigations of the mechanisms mediating non-resonant

photon feeding phenomena from a single quantum dot coupling to the mode of a two-

dimensional photonic crystal nanocavity. We present investigations of the phonon-

assisted emission process in sect. 5.1, starting by introducing the theory of cavity-

enhanced photon emission from excitons coupled to a phonon bath in sect. 5.1.1.

Within the framework of the independent boson model we find that such transitions

can be very efficient, even for relatively large exciton - cavity detunings of several

meVs. [Hoh09, Hoh10] Furthermore, we predict a strong detuning asymmetry for the

exciton lifetime that vanishes for elevated lattice temperature. [Hoh09] These theoret-

ical predictions are corroborated by experiments in sect. 5.1.2. For a strongly coupled

system of a single exciton and the cavity mode, we track the detuning-dependent

photoluminescence intensity of the exciton-polariton peaks at different lattice tem-

peratures. At low temperatures we observe a clear asymmetry in the emission inten-

sity depending on whether the exciton is at higher or lower energy than the cavity

68

mode. At high temperatures this asymmetry vanishes when the probabilities to emit

or absorb a phonon become comparable. Furthermore, we conduct time-resolved

measurements on a weakly coupled exciton - cavity system and observe an enhanced

spontaneous emission rate in the presence of the mode. This enhancement extends

over a much larger spectral range than would be expected from the measured linewidth

of the cavity mode. By fitting the theoretical model to our experimental data, we are

able to explain this enhancement with phonon-assisted emission processes.

We then continue to investigate the feeding of a non-resonant cavity mode by quan-

tum dot multi-excitons in sect. 5.2. We explain the theoretic model which allows a

multi-exciton to emit at the energy of the cavity mode due to Purcell enhanced de-

cay into a multi-exciton final state continuum (sect. 5.2.1). [Kan08b, Win09] We

continue to present measurements for large detunings |∆E| > 5 meV for which

phonon-assisted processes are inefficient (sect. 5.2.2). The presented photon cross-

correlation, [Kan08b] photoluminescence intensity saturation spectroscopy, and time-

resolved photoluminescence saturation spectroscopy measurements unambiguously

prove the multi-excitonic origin of the non-resonant cavity mode emission.

69


5.1. Phonon-Assisted Transitions

5.1.1. Theory of Phonon-Assisted Transitions

We start by considering the theory of a system where a single exciton, confined in

a semiconductor quantum dot, interacts with a single mode of a microcavity. The

theory was developed by Ulrich Hohenester from the Karl-Franzens-Universitat Graz

and adapted by him to describe our systems.

Under appropriate conditions, we can describe the system as a genuine two-level

system, where |0〉 denotes the cavity mode and |1〉 the exciton state. To account

for the coupling of the exciton to the phonon environment, we introduce an effective

Hamiltonian (we set ~ = 1)

Heff = H0 + H ′ + Hph + Hep −iΓ

2, (5.1)

where, H0 = ∆|1〉〈1| describes the uncoupled two-level system. ∆ is the energy de-

tuning between exciton and cavity and we have set the cavity mode energy to the zero

of our energy scale. Throughout this section we employ the rotating-wave approxima-

tion.

H ′ = g (|1〉〈0|+ |0〉〈1|) describes the electromagnetic coupling between the exciton

and the cavity, which depends on the exciton dipole moment and the field distribu-

tion of the cavity mode. [And99] The phonons and the exciton-phonon coupling are

described by the independent boson Hamiltonian [Mah81, Kru02, Hoh07]

Hph + Hep =∑λ

ωλa†λaλ +

∑λ

gλ

(aλ + a†λ

)|1〉〈1| , (5.2)

with λ denoting the different phonon modes with energy ωλ. aλ and a†λ are the phonon

field operators which obey the usual equal-time commutation relations for bosons. gλ

are the exciton-phonon matrix elements, which depend on the material parameters

of the host semiconductor and the exciton wavefunction. [Kru02, Hoh07] Equ. (5.2)

describes a coupling where the exciton deforms the surrounding lattice and a polaron

is formed, but no transitions to other quantum dot or cavity states are induced.

Finally, Γ = κ|0〉〈0|+γ|1〉〈1| in equ. 5.1 accounts for the radiative decay of the cavity

and the quantum dot states. 1

1We can use the Γ description in a situation where the decay irretrievably brings the system to

70

5.1 Phonon-Assisted Transitions

The quantum dot state and the cavity mode become coupled via H ′, and new

polariton modes are formed, which have partial exciton and partial cavity mode char-

acter. [Wal95, Rei04, Hen07] As we will show in the following, the phonon-mediated

exciton-cavity coupling opens a new decay channel, which is efficient also for relatively

large detunings. Our starting point is the master equation in the Born approxima-

tion [Wal95, Wil02]

∂ρ(t)

∂t≈ −

∫ t

0

trph

[H ′(t),[H ′(τ),ρ(t)⊗ ρph] ] dτ , (5.3)

where we use an interaction representation according to Hib = H0 + Hph + Hep. In

equ. (5.3), we treat the exciton - cavity coupling H ′ as a perturbation and describe

the exciton-phonon coupling without further approximation. We suppose that the

system is initially in the exciton state ρ = |1〉〈1| and the phonons are in thermal

equilibrium. From equ. (5.3), we then obtain for the increase of population in the

cavity

ρ00 ≈ g2ρ11

∫ ∞0

ei∆(τ−t)trph 〈1| ρphe−iHibτ |1〉 dτ + c.c. . (5.4)

This expression is of the form ρ00 = Γ10ρ11 and, thus, can be associated with a

scattering process. The memory kernel accounts for the buildup of a polaron state,

and is governed by the correlation function

C(τ) = exp−∑

λ

(gλωλ

)2 [(nλ + 1)e−iωλτ + nλe

iωλτ] (5.5)

This correlation function also determines pure phonon dephasing and the spectral

lineshape in semiconductor quantum dots. [Bor01, Kru02] nλ is the occupation number

of the phonons in thermal equilibrium. Then,

Γ10 = 2g2<e∫ ∞

0

ei∆τC(τ) dτ (5.6)

gives the scattering rate for a phonon-mediated transition from the exciton to the

cavity mode. In a similar fashion, we can also determine the scattering rate Γ01

for the reverse process, although this back-scattering is usually inefficient due to the

a state that is decoupled from |0〉 and |1〉, such as the empty quantum dot and a propagatingphoton. When the system can be re-excited, e.g., under continuous-wave pumping conditions,we should use a more complete master equation approach.

71


strong losses κ of the cavity.

For a quantum dot embedded in a bulk material, the correlation function of

equ. (5.5) describes the time evolution of the interband polarization, when at time

zero the quantum dot is prepared in a superposition between ground and excited

state. Subsequent to this preparation, a polaron is formed and the excess energy

is emitted as a phonon wavepacket. [Hoh07] Similarly, the phonon-assisted exciton -

cavity scattering rate can be interpreted as a process where the exciton first “tries” to

decay to the cavity, and forms a superposition between |0〉 and |1〉. It takes a time of

the order 1/∆ for the system to “perceive” that this transition is energetically forbid-

den, and to be transferred back to the initial state. If during this correlation buildup

the superposition state is subject to phonon scatterings or photon emission through

leaky modes, a real scattering occurs where the energy mismatch is compensated by

the environmental degrees of freedom.

Fig. 5.2(a) shows the calculated phonon-assisted scattering rate of equ. (5.6) as a

function of detuning, and for a range of different lattice temperatures. We observe

that at low temperatures the scattering is only efficient for positive (blue) detun-

ings, i.e., when ωQD ≥ ωcav. The corresponding scattering is associated with phonon

emission. With increasing temperature phonon absorption becomes more important,

and the scattering rate becomes increasingly symmetric with respect to detuning. To

estimate the importance of such phonon scatterings under realistic conditions, we

consider the situation where initially the quantum dot is populated and subsequently

decays through cavity and phonon couplings. We fully account for the electromag-

netic coupling H ′ between exciton and cavity, and describe the radiative quantum

dot and cavity losses through Γ. The phonon scatterings are modeled in a master-

equation approach of Lindblad form. [Wal95, Lau08, Lau09c] We use typical values

for state of the art cavity quantum electrodynamics experiments, with the quality

factor being Q = 10000 (κ = ~ωQD/Q, where ~ωQD ≈ 1.3 eV is the exciton transi-

tion energy), the exciton decay time ~/γ = 5 ns, and the exciton - cavity coupling

g = 60µeV. [Hen07, Kan08a] In the inset of fig. 5.2(b) we present the decay transients

calculated for the detunings 0.2 meV (red line), 0.5 meV (black line) and 1 meV (blue

line). For the smallest detuning (0.2 meV) we observe oscillations of the quantum

dot population at early times, which are a clear signature that the system operates in

the strong coupling regime. In all cases we observe a mono-exponential decay at later

times, which is associated with the radiative decay and phonon scatterings. Panel

72


0

1

2

3

4

5

Sca

tte

rin

g r

ate

(1

/ns)

(a)

-15 -10 -5 0 5 100

1

2

3

4

5

Detuning (meV)

SE

de

ca

y tim

e (

ns)

(b)

w/o phonons

T = 2 K

T = 20 K

T = 40 K

0 50 10010

-1

100

Time (ps)

QD population

Figure 5.2.: (a) Phonon-assisted scattering rate from exciton to cavity as a functionof exciton - cavity detuning, and for different temperatures. We use ma-terial parameters for GaAs and consider deformation potential and piezoelectric interactions with acoustic phonons. For the electron and holewavefunctions we assume Gaussians, with full-width of half maxima of6 and 3.5 nm along the transversal and growth directions, respectively.For details of our model see ref. [Hoh07]. (b) Full lines - spontaneousemission lifetime for the quantum dot exciton as a function of tempera-ture. We use Q = 10000, ~/γ = 5 ns, and g = 60µeV. The symbols showresults of our quantum-kinetic simulations, in good agreement with therate equation model for the range of parameters investigated. The insetshows the calculated decay transients for detunings of 0.2 (red line), 0.5(black line), and 1 meV (blue line), in order of decreasing decay constant,and for T = 2 K.

73


(b) of fig. 5.2 shows the extracted decay times for different detunings and tempera-

tures. At low temperatures we find that the decay is strongly enhanced at positive

detunings, which is due to processes in which phonons are emitted. This can be most

clearly seen by comparing the calculations with the thick gray line that represents

the results of a simulation where phonon scatterings have been artificially neglected.

Upon increasing the lattice temperature from ∼ 2 − 40 K phonon absorption gains

importance, due to the thermal occupation of the relevant phonon modes, and the

strong detuning asymmetry in the scattering rate is significantly weakened. Similar

results are also obtained for smaller Q values, where the dot - cavity system operates

firmly in the weak coupling regime of light-matter interaction. Thus, the results of the

calculations presented in fig. 5.2 are quite general and do not depend on the system

operating in the strong or weak coupling regimes of the light-matter interaction.

To check the results of our simulations we additionally performed quantum-kinetic

simulations for the coupled cavity-dot system in presence of phonon dephasing. The

approach was based on a density matrix description, including all one- and two-

phonon assisted density matrices, and a surrogate Hamiltonian with a finite number

of representative phonon modes. [For03, Hoh04, Hoh07] The symbols in fig. 5.2(b)

show the results of these more sophisticated simulations. Throughout, we observe

an almost perfect agreement between the full simulations and the much simpler rate-

equation approach, 2 demonstrating that equ. (5.6) incorporates the pertinent physical

mechanisms of phonon-assisted exciton - cavity scatterings. Thus, we conclude that

phonon mediated scattering into the cavity mode should: (i) enhance the bandwidth

over which the spontaneous emission lifetime is influenced by the proximal cavity

mode, and (ii) give rise to an asymmetry in the ∆ω dependence of the spontaneous

emission lifetime that should be observable for lattice temperatures ≤ 40 K. In the

next section, we will present experimental investigations which verify our conclusions.

5.1.2. Experimental Evidence for Phonon-Assisted Transitions

In this section we will present the result of photoluminescence and time-resolved mea-

surements performed on single quantum dot - cavity systems. The results obtained

prove the expectations we obtained from the calculations in the previous section,

2The small discrepancies between the quantum-kinetic and rate-equation simulations are attributedto the too small number of about hundred phonon modes in our simulation, which induce artificialphonon revivals in the simulated decay transients.

74


namely the asymmetry in the coupling strength for positive and negative detunings

at low lattice temperatures, and an enhanced coupling due to acoustic phonons for

detunings 1 meV < ∆E < 5 meV.

Strong Coupling Regime

In fig. 5.3 we present optical investigations performed on a strongly coupled, electri-

cally tunable quantum dot - cavity sample (see chapters 3, 4, and 6). 3 We show

in fig.5.3(a) a false-color plot of the emission spectra for a lattice temperature of

T = 23 K. The peak positions of the exciton and the cavity mode are indicated

with black circles. The exciton is tuned through resonance with the cavity mode

(Ecav = 1207.25 meV) as we change the bias voltage due to the quantum confined

Stark effect (see sect. 4.4). We observe a clear anticrossing in the photoluminescence

spectra for V23K = −0.21 V, with a vacuum Rabi splitting 2g ∼ 120 µeV which is

a clear sign of strong light-matter coupling as discussed in sect. 2.1.3 and chapter 6.

In fig. 5.3(b), we plot photoluminescence spectra of the same system at a higher

lattice temperature of T = 52 K. At this temperature, pure dephasing due to in-

teraction with acoustic phonons broadens the two polariton peaks and reduces the

coherent interaction between exciton and cavity mode (see sect. 6.2). As a result,

the anticrossing at resonance (V52K = 0.35 V) is not as clearly visible as for the low

temperature measurements. For these two temperatures and two intermediate tem-

peratures (T = 31 K and T = 42 K) we extracted the peak area of the two polariton

peaks and normalized it to the total luminescence intensity of both polariton peaks.

The results of these measurements are plotted in fig. 5.3(c) as a function of detuning

from resonance ∆E =√

(Epeak1 − Epeak2)2 − (2g)2. The region of the data presented

in fig.5.3(c) with higher intensity corresponds to the cavity-like emission peak and re-

gion with lower intensity corresponds to the exciton-like emission peak. In resonance

(∆E = 0 meV), the exciton and the cavity mode enter the strong coupling regime and

a polariton is formed. Here, both peaks have equal exciton- and photon-like character

and their emission intensity is equal (see sect. 2.1.3 and chapter 6). We now focus

the attention on the relative intensities of the exciton- and cavity-like peaks when

the system is detuned from resonance. When the quantum dot is at higher energy

than the cavity mode (∆E is positive) it can generate a photon in the cavity mode

3In this section only a few experiments on samples operating in the strong coupling regime of thelight-matter interaction are presented, while the focus of chapter 6 lies entirely on this topic.

75


- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 40 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

Q D - l i k e

Q D a t h i g h e r e n e r g y

( c )Normalized Peak Area

2 3 K 3 1 K 4 2 K 5 2 K 2 3 K m i r r o r

D e t u n i n g ( m e V )

Q D a t l o w e r e n e r g y

C a v i t y - l i k e

0 . 2 0 . 3 0 . 4 0 . 5

1 2 0 6 . 4

1 2 0 6 . 8

1 2 0 7 . 2

( b )

B i a s V o l t a g e ( V )

Energ

y (me

V)

- 0 . 3 - 0 . 2 - 0 . 1

1 2 0 6 . 8

1 2 0 7 . 2

1 2 0 7 . 6( a )

T = 5 2 K

B i a s V o l t a g e ( V )

Energ

y (me

V)

T = 2 3 K

V 2 3 K = - 0 . 2 1 V

V 5 2 K = 0 . 3 5 V

Figure 5.3.: (a) and (b) Photoluminescence spectra of a strongly coupled dot - cavitysystem as a function of bias voltage (false-color plot) for (a) T=23 K and(b) T=52 K. The black circles indicate the extracted peak positions. (c)Extracted, normalized peak area of the two polariton lines for differenttemperatures as a function of detuning between exciton and cavity mode.

by a phonon mediated emission process as shown above in sect. 5.1.1. [Hoh09, Hoh10]

For this case we do not observe a significant difference in intensities as we raise the

temperature since phonon emission is independent of the lattice temperature. In

strong contrast, when the quantum dot is detuned to lower energy than the cavity

mode (∆E is negative), the normalized intensity of the exciton-like polariton branch

is clearly highest for the lowest temperatures investigated (T = 23 K - blue curve).

Upon increasing the temperatures its relative emission intensity decreases, and ex-

hibits the minimum value at the highest temperature investigated (T = 53 K - red

76


curve). These observations can be explained by the strong temperature dependence of

the phonon-mediated scattering into the cavity-like polariton branch. [Hoh09, Hoh10]

Only for temperatures for which kBT Ephonon, is the phonon absorption rate suf-

ficiently high such that scattering of the exciton into the cavity mode via a phonon

absorption process can take place. These observations are fully consistent with our

expectation from sect. 5.1.1 and recent publications. [Hoh09, Suf09, Hoh10, Ota09b]

The phonon-mediated, Purcell-enhanced photon feeding of the cavity mode is active

for small quantum dot - cavity detunings and shows a clear asymmetry for positive

and negative detunings at low temperatures.

Weak Coupling Regime

We continue by investigating the spontaneous emission dynamics of two different sin-

gle quantum dot - cavity systems, as a function of the exciton - cavity mode detuning

∆ω. We employ the technique of nitrogen deposition (see sect. 4.5) to systematically

shift the cavity mode and vary ∆ω. [Mos05, Hof08, Kan09a] We performed control

measurements to check that this tuning method has no direct influence on the intrinsic

spontaneous emission dynamics of the quantum dot itself. Fig. 5.4(a) shows a series of

photoluminescence spectra recorded from quantum dot - cavity system I as a function

of successive nitrogen deposition steps. 4 A number of single exciton-like transitions

were observed in the vicinity of the quantum dot s-shell interband transition and were

identified by their linear power dependence. We focused on one such prominent single

exciton transition, labeled 1X1 on fig. 5.4(a). For each nitrogen detuning step we

observed a clear shift of the mode emission towards lower energies and away from the

emission of 1X1 up to a maximum detuning of ~∆ω ∼ 11 meV. For these different

detunings we performed time-resolved photoluminescence measurements to investi-

gate the influence of the mode on the spontaneous emission dynamics of the exciton

as a function of ~∆ω. In fig. 5.4(b) we plot three representative decay transients

recorded whilst detecting on 1X1. The solid lines on fig. 5.4(b) are mono-exponential

fits to the time resolved data convoluted with the instrument response function of the

detection system. All transients show a clear monoexponential decay with a decay

time that becomes systematically longer as the cavity mode is detuned, precisely as

4The photoluminescence spectra were recorded with relatively strong excitation levels (∼10 W/cm2) in order to clearly observe the cavity mode, whilst the time resolved measurementspresented later were performed with much weaker excitation power (≤ 1 W/cm2) in order toavoid state filling effects.

77


1 3 20

1 3 21

1 3 22

1 3 23

1 3 24

1 3 25

PL Intensity (arb. units)

( c )E n e r g y ( m e V )

1 X 2


- 1 . 0 0 m e V

+ 0 . 1 1 m e V

N 2 tunin

g

M o d e

+ 1 . 0 8 m e V

0 2 4 6 8 1 0

( d )

T i m e ( n s )

+ 1 . 0 8 m e V + 0 . 1 1 m e V - 1 . 0 0 m e V

0 2 4 6 8 1 0

( b ) - 1 . 4 8 m e V - 4 . 0 8 m e V - 1 1 . 1 m e V

1 3 00

x 4 . 2

( a )

1 2 88

1 2 92

1 2 96

PL In

tensit

y (arb

. unit

s)

1 X 1M o d e

- 1 1 . 1 m e V

- 4 . 0 8 m e V

- 1 . 4 8 m e V

PL In

tensit

y (arb

. unit

s)

Figure 5.4.: (a) Photoluminescence spectra of quantum dot - cavity system I, withQ = 2900 and T = 15 K. The mode was spectrally tuned away from thequantum dot s-shell to lower energies by adsorption of molecular nitrogen.(b) Decay transients of the indicated quantum dot state for three differentdot - cavity detunings. (c) Photoluminescence spectra of quantum dot -cavity system II, with Q = 3200 and T = 18 K. The mode was spectrallytuned through a quantum dot transition to lower energies by adsorptionof molecular nitrogen. (d) Decay transients of the quantum dot transitionfor three different quantum dot - cavity detunings.

78


expected due to the Purcell effect. [Pur46] However, as we will show below the cavity

mode exerts an influence on the decay lifetime for detunings far beyond what would

be expected due to the Purcell effect.

Fig. 5.4(c) shows a series of representative photoluminescence spectra recorded from

quantum dot - cavity system II. In this case, we conducted measurements for positive

and negative detunings where the quantum dot is at lower and higher energies than

the mode (−2 meV ≤ ∆ω ≤ 2 meV), respectively. However, the absolute range of

detunings attainable was smaller as compared with system I. We excited in resonance

with a higher energy cavity mode at 1433 meV and an excitation power density far

below the saturation of the exciton (∼ 1 W/cm2). [Kan09b] Three representative de-

cay transients are plotted in fig. 5.4(d). In contrast to system I, we observe a clear

biexponential decay for all detunings, with both fast (τf ) and slow (τs) decay com-

ponents. These two lifetimes can be attributed to the two fine structure components

of the neutral exciton that couple with a different strength to the linearly polarized

cavity mode. 5 A comparison of the three decay transients presented in fig. 2(d) re-

veals that both τf and slow τs are minimum close to zero detuning (green curve), as

expected due to the Purcell effect. [Pur46] The other two decay transients selected for

presentation in fig. 5.4(b) were recorded at ~∆ω = +1 meV (red curve) and −1 meV

(blue curve), respectively and exhibit markedly differing lifetimes despite the similar

magnitude of the detuning. This observation already hints at an asymmetry in the

lifetimes for positive and negative detunings.

In fig. 5.5(a) we summarize all measured lifetimes for quantum dot - cavity system

I as a function of ∆ω. We performed simulations using the experimentally measured

quality factor Q = 2900 and g = 45 µeV. When the phonon assisted dot - cavity

coupling is artificially neglected (black dotted line) the experimental data (squares)

cannot be reproduced and, furthermore, the dip in τ(∆ω) is much narrower than is

observed experimentally. However, when coupling to a phonon bath with a tempera-

ture of 15 K is switched on (solid cyan line), the calculation accounts very well for the

measured data. Similarly, in fig. 5.5(b) we summarize the information extracted from

the decay transients of quantum dot - cavity system II. We plot the measured lifetimes

τs and τf with circles and squares, respectively. The gray traces are the data points for

5This is inevitable for these systems since the L3 cavity axis was not intentionally aligned withthe [110] crystal axis and the neutral exciton consists of a linearly polarized doublet with a finestructure splitting below our spectral resolution of 60 µeV.

79


- 2 - 1 0 1 2 0 . 00 . 10 . 20 . 30 . 40 . 5

1

2

3

4

5

N o p h o n o n s 1 0 K 2 0 K 3 0 K

D o t o n l o we n e r g y s i d e

D o t o n h i g he n e r g y s i d e

( b )

Lifetime (ns)

D e t u n i n g ( m e V )

( a )

0 2 4 6 8 1 0 1 20

1

2

3

4

1 5 K

Lifeti

me (n

s)

Figure 5.5.: (a) Extracted decay times of quantum dot - cavity system I as a functionof detuning. The squares correspond to the 1X1 state. The black dottedline and the cyan solid line are theoretical decay times obtained fromcalculations with the phonon coupling turned off and a phonon bath at15 K temperature, respectively. (b) Extracted decay times of quantumdot - cavity system II as a function of detuning. The squares and circlescorrespond to the fast and the slow decay component, respectively. Thegray traces are shown for better comparison, and display the data for neg-ative detunings mirrored on the positive axis. The lines are calculationsof the system without phonon coupling and for different temperatures(10 K, 20 K, 30 K). Note that the measured lifetimes in resonance arelimited by the temporal resolution of the detection system of ∼ 150 psafter deconvolution.

80


negative detunings mirrored onto the positive axis to facilitate a direct comparison.

For the slow decay component, we clearly observe a pronounced asymmetry in the

data as a function of detuning, with the measured lifetimes being generally shorter for

positive detunings as compared to negative detunings. The fast decay component is

more symmetric with respect to detunings but also reveals a dip with a width that can

only be accounted for when phonon assisted scattering into the cavity is included. To

see this, we calculated the lifetimes without coupling to phonons (black dotted line),

and with coupling to a phonon bath with a temperature of 10 K (blue solid line),

20 K (green solid line) and 30 K (red solid line). We used Q = 3200 and g = 60 µeV

as obtained from the optical measurements. Clearly, coupling to the phonon bath is

needed to correctly represent the dip in the measured lifetime, underlining the role of

phonons mediating dot-cavity coupling at small detunings. As for the slow compo-

nent, we consider a level scheme with two fine structure components of the exciton,

where g = 10 µeV for the weakly coupled component. In fig. 5.5(b) we present results

of simulations with coupling to a phonon bath at 30 K (red solid line) and without

phonon scatterings (black dotted line). Whilst the overall agreement with experiment

is not entirely satisfactory, our calculations qualitatively reproduce the asymmetric

detuning dependence in good agreement with experiment.

In summary, in this section we have shown photoluminescence intensity measure-

ments and time-resolved photoluminescence measurements that unambiguously demon-

strate phonon-mediated feeding of the cavity mode in our single quantum dot - cavity

systems. While this process can explain the cavity emission for small detunings to

the quantum dot states |∆E| < 5 meV, there is an additional mechanism related to

multi-excitonic transitions which we will investigate in the next section, sect. 5.2.

81


5.2. Feeding from Multi-Excitonic Transitions

The phonon-assisted process presented in the last section (sect. 5.1), explains the cav-

ity emission for small detunings to the quantum dot states |∆E| < 5 meV. However,

emission from the cavity mode can still be seen at large detunings when the cavity

mode is detuned by |∆E| > 15 meV. This is because of an additional mechanism

related to multi-excitonic transitions in strongly excited quantum dots. We introduce

this mechanism in sect. 5.2.1 and compare its predictions with the experiments in

sect. 5.2.2.

5.2.1. Model of Multi-Excitonic Feeding

The experimental evidence presented below in sect. 5.2.2 indicates that the mecha-

nism which leads to strongly non-resonant emission of photons into the cavity mode is

related to multi-excitons. Before presenting our experimental results, we outline the

mechanism responsible to qualitatively illustrate what we would expect in experiment.

In fig. 5.6(a) we plot a schematic energy level diagram for a quantum dot - cavity

system (redrawn from ref. [Win09]). The thick black lines, labeled with n = 0, n = 1,

n = 2, and n = 3 denote the lowest energy discrete configurations of n excitons in the

quantum dot. The corresponding optical transitions are labeled with X0, 2X0, and

3X0, respectively. In a more realistic picture, additional discrete energy levels have

to be considered as well, since the particles can be distributed amongst the different

orbital states of the quantum dot. Each of these different charge configurations has

a specific energy that is determined by the competition between the orbital kinetic

energy and the attractive or repulsive Coulomb interactions between confined parti-

cles. [Kar04] We schematically denote these states as thin black lines in fig. 5.6(a). At

higher excitation levels transitions between this plethora of states will merge into a

quasi-continuum and generate a broad quantum dot background at higher excitation

levels. [Dek98] Furthermore, due to the possibility for charge carriers to occupy states

in the wetting layer, there is also the possibility for the occurrence of an Auger-type

process. Here, an electron or hole can scatter into the two-dimensional wetting layer

continuum. [Kar04] Such continuum excited few particle states are indicated as the

gray background in fig. 5.6(a). In fig. 5.6(b) we plot schematics of the occupancy of

the electron and hole energy levels in the quantum dot for the lowest energy state of

the n = 3 charge configuration. In addition to the initial state we indicate the several

82

5.2 Feeding from Multi-Excitonic Transitions

Lowest

energy state

Different

occupational

configurations

Hybridization with

2D continuum

3X0

2X0

En

erg

y

X0

n=

3n

=2

n=

1n

=0

(a) (b)

Figure 5.6.: (a) Schematic energy-level diagram (n ≤ 3) of a neutral quantum dot(drawn after ref. [Win09]). (b) Schematic diagram of the occupancy ofthe electron and hole energy levels in the quantum dot for the lowestenergy state of the n = 3 charge configuration as initial state and thedifferent possible final states of the n = 2 charge configuration after thedecay of one exciton.

different possible final states of the n = 2 charge configuration after the radiative

decay of one exciton. Corresponding to the energetic states indicated in fig. 5.6(a)

these are (i) the lowest energy state where both excitons occupy the s-shell level, (ii)

an example for a different occupational configuration where one electron-hole pair is

in the s-shell and one pair is in the p-shell, and (iii) an example for an Auger-type

process where one hole is lifted into the two-dimensional wetting layer continuum.

We now assume that the energy of the cavity mode is smaller than that of a single

exciton and regard the excitation and decay cascade of the quantum dot as depicted

schematically in fig. 5.7. The quantum dot is non-resonantly excited from the ground

state (e.g. with a laser pulse) and then decays until it is occupied by three excitons

(n = 3) in the energetically most favorable configuration. The next exciton can then

decay into a continuum of final states. Some of the states within this continuum will

83


En

erg

y

n=

3n

=2

n=

1n

=0

cavity mode

cavity mode

1X

Figure 5.7.: Excitation and decay cascade of the quantum dot. Multi-exciton transi-tions can emit photons into the cavity mode by decaying into an alterna-tive, higher energy final state.

be spectrally in resonance with the cavity mode and their Purcell-enhanced emission

can efficiently feed photons into the mode. [Kan08b, Win09] The charges will then

quickly relax into the most favorable configuration of the n = 2 occupancy state by

emission of phonons. The system can now decay in a very similar manner from the

n = 2 state to the n = 1 state and, crucially, it can again emit a photon into the

cavity mode. Only during the last transition from the n = 1 state to the n = 0 state

can a photon at the energy of the cavity mode not appear, a direct consequence of

the absence of a continuum of final states.

The two-dimensional continuum of states of the wetting layer is crucial for the ob-

servation of non-resonant mode emission. Similar to experiments performed with sin-

gle self-assembled InGaAs quantum dots, experiments with single site-controlled InP

quantum dots exhibit mode emission via phonon-assisted processes for small detun-

ings. [Dal10, Cal10] However, when the detuning to the discrete quantum dot states is

increased, the emission intensity of the cavity mode rapidly decreases. [Dal10, Cal10]

This absence of mode emission for large detunings (|∆E| > 5 meV) can be related

to the growth process of the site-controlled quantum dots and, more specifically, the

absence of a wetting layer for this type of nanostructures.

84


With the above presented model the puzzling observations seen in experiments can

be explained:

- Light emitted from the cavity mode obeys Poissonian statistics. [Hen07, Pre07,

Kan08b, Win09]

- Cross-correlation experiments with the exciton line of a quantum dot show a

clear anti-correlation. [Hen07, Pre07, Kan08b, Win09]

- The emission dynamics (i.e. intensity and decay time) measured on the emission

line of the cavity mode are insensitive to small detunings. [Kan08b, Hof08,

Lau09b]

In the next section we will present a number of different experiments which un-

equivocally relate the origin of the mode emission to multi-excitonic quantum dot

transitions and prove the model presented in this section.

5.2.2. Experimental Evidence for Multi-Excitonic Feeding

Whilst phonon-mediated feeding of the cavity mode is effective only for small detun-

ings (|∆E| < 5 meV) an additional mechanism must exist that non-resonantly feeds

photons into the cavity mode for much larger detunings. [Hen07, Pre07, Kan08b,

Win09] In fig. 5.8(a) we plot the photoluminescence spectrum of a quantum dot -

cavity system where the cavity mode is strongly detuned to lower energy than the

quantum dot s-shell excitonic emission (∆E ∼ 15 meV). Although no discrete quan-

tum dot transitions are present in the spectral vicinity of the cavity mode, pronounced

photoluminescence emission is still observed from the cavity mode. For this system

we conduct photon-cross-correlation measurements between the cavity mode [marked

with the blue shaded region on fig. 5.8(a)] and a quantum dot single exciton transition

[marked with the red shaded region on fig. 5.8(a)], which are detuned from another

by ∆E ∼ 18 meV. The single exciton transition was clearly identified from its linear

power dependence (I ∝ Pm with m = 0.95 - data not shown), and the excitation

power for the experiment was chosen just below saturation of the single exciton line.

Fig. 5.8(b) shows the cross-correlation histogram as a function of the time τ between

two detection events. Here, τ > 0 corresponds to detection of a photon from the ex-

citon after detection of a photon from the cavity mode. At zero time delay (τ = 0 ns)

we observe a clear dip in the histogram indicating that cavity mode photons are not

85


Figure 5.8.: (a) Photoluminescence spectrum of a quantum dot - cavity system withthe cavity mode (marked in blue) detuned to ∆E ∼ 18 meV lower energythan the single exciton transition (marked in red). (b) Cross-correlationhistogram between the cavity mode and the single exciton. Here, τ > 0corresponds to detection of a photon from the single exciton upon de-tection of a photon from the cavity mode. The red line is a fit to thedata.

emitted simultaneously with photons from the exciton transition. We can fit the

measured data using the formula

g(2)(τ) = 1− ρ2e(−|τ |/t±m), (5.7)

86


where ρ represents the degree of anti-correlation between the two emission lines,

and t±m is the anti-bunching time constant of the decay for positive and negative τ ,

respectively. [Zwi04, Ate09] In our case t±m will correspond to the decay time of the

single exciton and the decay time which can be measured on the cavity mode. 6 [Hen07]

In a final step, equation 5.7 convoluted with a Gaussian of width 750 ps, was fitted to

the data to obtain the degree of anti-correlation ρ and g(2)cavity−X(0). The deconvoluted

result of the fit is plotted as a red line in fig. 5.8(b). We obtain ρ = 95 %, g(2)cavity−X(0) =

0.19 before and g(2)cavity−X(0) = 0.11 after deconvolution, and time constants of t+m =

1.2 ns and t−m = 5.2 ns. The high value of ρ proves the very high degree of anti-

correlated emission from the cavity mode and the investigated single exciton line,

and evidences that this cavity mode is predominantly fed by only this one quantum

dot. [Hen07, Kan08b, Win09] The asymmetry of the dip is a result of the different

lifetimes of the cavity mode and the exciton. [Hen07, Kan08b, Win09] The extracted

value of t+m = 1.2 ns corresponds well to the lifetime we normally measure on the

cavity mode, while the time constant t−m = 5.2 ns corresponds to the lifetime of an

excitonic transition inside the photonic bandgap. [Lau09b] Whilst this measurement

proves that the origin of the cavity mode emission is indeed due to the quantum dot, it

does not allow us to readily draw any conclusion about the mechanism that mediates

such non-resonant coupling.

In fig. 5.9(a) we show photoluminescence spectra of a different quantum dot - cavity

system as a function of the continuous-wave equivalent power of the excitation pulses

used in our experiments. The power is increased from 10 nW for the lowermost

spectrum to 200 nW for the uppermost spectrum. At low excitation levels we observe

emission from the s-shells of two quantum dots (labeled QD1 and QD2) at λs−shell1 =

940−945 nm and λs−shell2 = 962−967 nm, respectively. Measurements obtained with

mode-resonant excitation (data not shown) demonstrated that QD1 is spatially well

coupled to the electric field maximum of the investigated cavity mode, whilst QD2

is less strongly coupled. [Nom06, Kan09b] At higher excitation levels, we observe the

emergence of p-shell emission from QD1 and QD2 at λp−shell1 = 930 − 935 nm and

λp−shell2 = 943 − 948 nm, respectively. The fundamental mode of the L3 cavity is

observed at λcav = 954 nm, spectrally detuned from any of the discrete quantum dot

transitions. At low excitation powers (< 30 nW) the cavity mode emission is hardly

6The decay time measured on the cavity mode does not correspond to the cavity loss rate. Ratherit is the decay time of the non-resonant states which emit into the cavity mode as shown in thischapter.

87


9 3 0 9 4 0 9 5 0 9 6 0

(

(

!# !#

##

PL In

tensit

y (arb

. unit

s)

%$

##

1 0 1 1 0 2 1 0 3 1 0 4

%

!#!#


'$$ &"

!#!#

Figure 5.9.: (a) Power dependent photoluminescence spectra recorded from a quantumdot cavity system with two quantum dots, excitation level increasing frombottom to top. (b) Integrated photoluminescence intensity for the cavitymode (black stars), selected single exciton transitions from the s-shellof QD1 (labeled X1 - red circles) and from the s-shell of QD2 (labeledX2 - cyan rectangles). At higher power additional emission is observedfrom the p-shell multi-excitons of QD1 (blue triangles) and QD2 (greendiamonds).

visible. In contrast, it dominates the spectrum at higher excitation powers, its onset

occurring at an excitation level that is comparable to that needed to observe emission

from the p-shell of QD1. This is already a first indication that feeding of the cavity

mode is mediated by multi-exciton quantum dot transitions.

For the transitions marked in fig. 5.9(a), we extracted the integrated photolumi-

nescence intensity and plot the power dependence in fig. 5.9(b). The intensities of

X1 (red circles) and X2 (cyan squares) exhibit a nearly linear power law dependence

88


I ∝ Pm with exponents mX1 = 0.91 ± 0.05 and mX2 = 0.95 ± 0.04, respectively.

These observations are strong evidence that these lines possess single exciton char-

acter. [Fin01b, Abb09] The intensities of both transitions saturate at a power of

∼ 100 nW as the dot is pumped out of the single excitation regime. 7 The p-shell

emission intensity increases superlinearly with excitation power exhibiting exponents

of mp−shell1 = 1.53 ± 0.07 and mp−shell2 = 1.42 ± 0.07, respectively. 8 Furthermore,

these transitions saturate at a much higher laser power (> 1 µW), compared to the

single exciton emission. In comparison with the s-shell and p-shell emission, the cav-

ity mode intensity also increases superlinearly with an exponent of mcav = 1.68±0.07

(black stars) and saturates at an excitation power comparable to that observed for

the quantum dot p-shell transitions. The similar emission behaviour of the cavity

mode and the multi-excitons strongly suggests that the cavity mode emission is fed

by the multi-exciton transitions of the quantum dot.

Consequently, we conducted similar power dependent measurements for various dif-

ferent quantum dot - cavity systems and for a range of quantum dot - cavity mode

detunings. A selection of six representative spectra recorded from different quantum

dot - cavity systems is presented in fig. 5.10(a). The data is presented on a relative

wavelength scale with respect to the cavity mode emission. For the uppermost spec-

trum, the s-shell of the quantum dot is on the short wavelength side of the mode (blue

detuned), but for the lowermost spectrum it is on the long wavelength side of the mode

(red detuned). In the latter case, the mode is in resonance with the p-shell emission

of the investigated quantum dot. We extract the exponent of the power dependent

mode photoluminescence intensity [see fig. 5.9(b)] and plot it as a function of detun-

ing to the quantum dot s-shell and p-shell in fig. 5.10(b) and (c), respectively. When

the cavity mode is detuned from the s-shell, the mode intensity exhibits a superlinear

power dependence with typical exponents in the range mcav = 1.2 − 1.6, similar to

our findings for the exponents of multi-exciton transitions. Only when the mode is in

resonance with the s-shell emission [−5 < ∆λ < 1 nm in fig. 5.10(b)], in a range where

coupling via acoustic phonons dominates (see sect. 5.1), do we find that the exponents

decrease to mcav = 1.0 − 1.2. This is a characteristic signature of photon emission

7The increase of the intensity of X1 for higher excitation powers is explained by the appearanceof a background under very strong pumping. The unexpected reduction of the intensity of X2,however, is attributed to the long lifetime (τ ∼ 13 ns) emitting deep inside the photonic bandgap,comparable to the laser repetition period.

8For the photoluminescence intensity of the p-shell transitions of QD1 and QD2 we integrated overthe p-shell emission band since individual transitions could not be well resolved.

89


Figure 5.10.: (a) Selected photoluminescence spectra of six different quantum dot -cavity systems. (b),(c) The exponents of the extracted cavity modeemission intensity obtained from power dependent photoluminescencemeasurements of different quantum dot - cavity systems plotted againstthe detuning relative to the s-shell (b) and p-shell (c).

related to single excitons. In fig. 5.10(c), we plot the measured exponents of the mode

emission as a function of p-shell detuning. Provided that the mode is in resonance

with the p-shell transitions or at lower energies [p-shell blue detuned, ∆λ < 2 nm in

fig. 5.10 (c)], we observe a superlinear behaviour with exponents between mcav = 1.0

and mcav = 1.7. For these detunings p-shell states have excess energy and can effec-

tively feed the cavity mode. However, when the p-shell transitions are red detuned

(∆λ > 2 nm), the extracted exponents show values of mcav > 2.0 with the highest

value being mcav = 3.9 measured at a detuning of ∆λ ∼ +32 nm. Here, feeding occurs

90


from higher quantum dot transitions and most likely from quantum dot-wetting layer

transitions. In summary, we observe strong indication for multi-exciton feeding of

the mode with additional contributions of single exciton feeding for small detunings.

This behaviour is in excellent agreement with the model introduced in sect. 5.2.1 and

the phonon-mediated processes presented in sect. 5.1.

An unambiguous proof for the multi-exciton feeding of the cavity mode is obtained

from time-resolved photoluminescence measurements when the quantum dot - cavity

system is excited above saturation of the s-shell levels (P = 250 nW). [Cha09] In

fig. 5.11(a) we plot the time-resolved emission intensity of a quantum dot - cavity

system as a function of wavelength. Emission from the p-shell transitions of QD1

[λp−shell1 = 930−935 nm - indicated by the blue color in fig. 5.9] and QD2 [λp−shell2 =

943− 948 nm - indicated by the green color in fig. 5.9] occurs rapidly after the arrival

of the laser excitation pulse and decays within a few nanoseconds. 9 Emission from

the s-shell transitions of QD1 [λs−shell1 = 940−945 nm - indicated by the red color in

fig. 5.9] and QD2 [λs−shell2 = 962− 967 nm - indicated by the cyan color in fig. 5.9] is

temporally delayed, such that the maximum intensity is not reached until ∼ 4− 6 ns

after excitation. At that time the population in the dot has already decayed towards

the single exciton level. Emission prior to the arrival of the excitation laser pulse at

0 ns originates from the previous excitation cycle, 12.5 ns earlier. The emission from

the cavity mode at λcav = 954 nm occurs rapidly after arrival of the excitation pulse

and decays quickly within ∼ 2 ns.

The spectra plotted in fig. 5.11(b) show again the time evolution of the entire

quantum dot - cavity system. Here, we integrated the measured photoluminescence

signal over ∆t = 1 ns time intervals and present the resulting spectra in fig. 5.11(b)

for the time bins from 0 − 1 ns, 2 − 3 ns, 4 − 5 ns, 6 − 7 ns, and 8 − 9 ns, from

bottom to top. For the first time interval, emission from the p-shell states of QD1

and QD2 and from the cavity mode dominates the spectrum. However, the intensity

of this emission decreases rapidly and vanishes almost completely by 5 ns after the

excitation. The dominating peaks of the spectrum are now the s-shell emission of

QD1 and QD2, whilst hardly any signal from the according p-shells or the cavity

9Here, the lifetime of all transitions is lengthened due to the two dimensional photonicbandgap. [Kan07, Kan08a]

10The spectrally broad mode emission at early times is an artifact of plotting and not a physicaleffect. Here, the linear colour scale was chosen such that features of low intensity are well visibleand the intensity above a certain threshold is plotted with the same colour.

91


0123456789

PL Intensity (norm.)PL Intensity (norm.)

Time (

ns)

0 . 0

0 . 5

1 . 0

X 1 X 2

X 1 X 2

( b )

( a )

9 3 0 9 3 5 9 4 0 9 4 5 9 5 0 9 5 5 9 6 0 9 6 5

8 - 9 n s

6 - 7 n s

4 - 5 n s

2 - 3 n s

PL In

tensit

y (arb

. unit

s)

0 - 1 n s

W a v e l e n g t h ( n m )0 1 2 3 4 5 6 7 8 9

0 . 0

0 . 5

1 . 0

T i m e ( n s )

I R F m o d e b k g r d p - s h e l l 1

( d )

m o d eb k g r dp - s h e l l 1 ( c )

Figure 5.11.: (a) False colour contour plot of the time-resolved photoluminescence in-tensity of a quantum dot - cavity system as a function of emission wave-length. 10(b) photoluminescence spectra at different time delays afterthe laser pulse, each integrated over 1 ns. (c),(d) Extracted, normalizedphotoluminescence intensity for (c) X1 (red circles) and X2 (cyan rect-angles), and (d) cavity mode (black stars), p-shell 1 (blue triangles), andp-shell 1 background (dark red diamonds) as a function of time delayafter the laser pulse. The IRF is plotted as gray solid line.

mode is observed. For a more quantitative comparison, we plot in fig. 5.11(c) the

integrated, normalized photoluminescence intensity of X1 (red circles) and X2 (cyan

rectangles). This is compared with the integrated photoluminescence intensity of the

mode (black stars), p-shell 1 (blue triangles) and p-shell background emission (dark

red diamonds) in fig. 5.11(d). The solid gray line is the instrument response function

of our experimental setup, measured with the spectral detection window tuned to the

laser wavelength. It serves as reference for the time when the laser pulse excites the

92


0 2 4 6 8

( a )

d e l a y

T i m e ( n s )

1 Xb a c k g r o u n d

0 2 4 6 8

d e l a y

( b ) 2 X

0 2 4 6 8

d e l a y

( c ) p - s h e l l

0 2 4 6 8

1 0 05 03 01 0

5 . 03 . 01 . 00 . 50 . 30 . 1

( d ) m o d e

Figure 5.12.: (a-d) Time-resolved photoluminescence intensity (logarithmic scale) forincreasing excitation power (from bottom to top) for (a) the single ex-citon, (b) the biexciton of QD1, (c) a p-shell state, and (d) the cavitymode. The black arrows indicate the temporal position of maximumphotoluminescence emission from the states.

sample and allows us to determine the zero point of the time axis. While emission from

the cavity mode, p-shell states and p-shell background [Dek98] occurs immediately

after the laser pulse, the emission from the single excitons X1 and X2 is delayed and is

temporally completely uncorrelated with the mode emission. It is also interesting to

take a closer look at the decay times. The cavity mode decays (τmode = 1.4± 0.1 ns)

even faster than the selected discrete p-shell state (τp−shell1 = 2.2±0.1 ns), but slower

than the p-shell background between the discrete emission lines (τbkgrd = 0.8±0.1 ns).

Thus, feeding of the cavity mode apparently occurs from many different multi-exciton

states as well as from the broad background which is present at higher excitation

levels. [Dek98]

Finally, in fig. 5.12(a-d) we present time-resolved measurements of the same quan-

tum dot - cavity system at increasing levels of the cw equivalent excitation power

(from bottom to top). The different panels show the decay transients measured at

the wavelength of a single exciton transition (a), the biexciton (b), a transition in the

p-shell (c) of QD1, and the cavity mode (d). The exciton is the longest-living state

and its emission shifts rapidly to later times as the excitation power is increased (the

small black arrows mark the temporal delay), since the population in the dot has

93


to decay to the single excitation level first. The prompt emission observed at short

times for high powers (labeled background) originates from the broadband emission

that is always present for very strong pumping. [Dek98] For the biexciton the delay

due to state filling effects is not as strongly pronounced as for the exciton, and for

the p-shell state it is again smaller, as expected. However, all three time resolved -

excitation power plots exhibit a comparable contribution from the broad background

at the highest powers investigated. Again, we can relate these emission characteristics

to the emission from the cavity mode. We observe that the mode emits more quickly

than all the other states and no delay at high powers can be recognized. There is no

visible correlation with the single exciton and the biexciton state for this detuning.

Most of the emission appears over timescales that are even faster than that of the

p-shell, showing a good temporal accordance to the background emission.

5.3. Summary

In this chapter we have shown a combined theoretical and experimental study on

the origin of the non-resonant cavity mode emission in photonic crystal nanocavity

samples with self-assembled InGaAs quantum dots. We presented a theoretical model

for the Purcell-enhanced, phonon-assisted emission of photons into the cavity mode

in sect. 5.1.1 which we corroborated by performing detuning-dependent intensity and

lifetime measurements in the strong and the weak coupling regimes of the light-matter

interaction (sect. 5.1.2). This emission process is efficient for small detunings (|∆E| <5 meV), while for larger detunings (|∆E| > 5 meV) the dominant source for mode

emission originates from the feeding from multi-excitonic states. We discussed a

more realistic energy level diagram of a self-assembled quantum dot and explained

the origin of mode emission due to the Purcell-enhanced decay of a multi-exciton

into a hybridized wetting layer - quantum dot continuum in sect. 5.2.1. This model

was then substantiated by photon cross-correlation, power-dependent intensity and

time-resolved saturation spectroscopy measurements in sect. 5.2.2.

94

The interaction of a single dipole with a

monochromatic radiation field presents an important

problem in electrodynamics. It is an unrealistic

problem in the sense that experiments are not done

with single atoms or single-mode fields.

Leslie Allen and Joseph H. Eberly

Chapter 6Strongly Coupled Quantum Dot - Cavity Systems

Spontaneous emission is entirely fundamental and is often regarded as an inherent

property of an excited atom, molecule or quantum dot. However, this view somewhat

overlooks the fact that it is not only a property of the emitter, but of the com-

bined emitter - vacuum system. [Har89] The irreversibility of spontaneous emission

comes about due to the infinite number of vacuum states available to the emitted

photon. If the photonic environment is modified, for instance by placing the emit-

ter within a cavity, spontaneous emission can be inhibited, enhanced or even made

to become reversible. [Dal92, Khi06] As discussed in chapter 2 the coupling between

light (photons) and matter (excitons) is classified into the weak coupling regime with

irreversible emission of photons and the strong coupling regime where the interac-

tion strength overcomes the losses, leading to the formation of exciton-polaritons.

The border between these two regimes was discussed mostly in theoretical work so

far. [Khi06, Lau08, Lau09c] Pioneering, experimental work has only proven the possi-

bility to obtain such a strong coupling in different semiconductor microcavities such

as micropillars, [Rei04] photonic crystals, [Yos04, Mos05] and microdiscs, [Pet05]

and more recent experiments have shown non-linear effects in the strong coupling

regime. [Pre07, Hen07, Eng07a, Far08a, Win08] Detailed investigations that experi-

mentally address the form of the emission spectrum as the degree of pumping and/or

incoherent interactions become important are lacking. This is because traditional

tuning mechanisms between exciton and cavity mode cannot be used for systematic

95

Strongly Coupled Quantum Dot - Cavity Systems

studies since they can only be employed at low temperatures (inert gas deposition)

or at no fixed temperature at all (temperature tuning).

In this chapter we use the electrically active single dot nanocavities (chapter 4)

to conduct detailed investigations on the strong coupling between a semiconductor

quantum dot and a photonic crystal microcavity. We introduce the theoretical for-

malism of quantum dot - cavity coupling in sect. 6.1 by deriving the spectral function

for the general case of N quantum dots coupled to one cavity mode in sect. 6.1.1,

and the special case of one quantum dot coupled to one cavity mode (sect. 6.1.2).

We then present detailed power and temperature dependent investigations (sect. 6.2),

and apply the theoretical model to fit the emission spectra in order to probe the

transition from the strong coupling to the weak coupling regime. In sect. 6.2.1 we

introduce our electrically tunable quantum dot - photonic crystal samples operating

in the strong coupling regime of the light-matter interaction. With these samples

we can increase the level of incoherent excitation power to monitor the influence of

dephasing induced by dynamical fluctuations (sect. 6.2.2) and the lattice temperature

to examine the influence of phonon induced dephasing (sect. 6.2.3) on the polariton

spectrum. In sect. 6.3 we employ a similar system to investigate the coupling of two

quantum dots via the optical mode of a photonic crystal cavity. After presenting the

experimental realization of the measurements we use the theoretical model to fit the

emission spectra in sect. 6.3.1, substantiating that both quantum dots are coherently

coupled to the same cavity mode. By analyzing the eigenvectors of the eigenstates of

the coupled system we demonstrate the possibility to couple the two quantum dots

indirectly via an optical “V-system” (sect. 6.3.2). Sect. 6.3.3 then discusses the ad-

vantages of this indirect coupling, namely the possibility to circumvent the largest

source of incoherence in the system; the loss of photons from the cavity mode.

96

6.1 Theory of Quantum Dot - Cavity Coupling

6.1. Theory of Quantum Dot - Cavity Coupling

6.1.1. Emission Spectrum of a N-Quantum Dot - Cavity System

We begin this chapter by describing the theory of the emission spectrum of N in-

dependent quantum dots coupled to the same mode of one photonic crystal cav-

ity. The model was developed by Jose Maria Villas-Boas 1 and is based on the

formalism introduced by Howard J. Carmichael et al. [Car89] and Fabrice Laussy et

al. [Lau08, Lau09c] We start from the following Hamiltonian:

H =∑i

[~ωi2σiz + ~gi(a†σi− + σi+a)

]+ ~ωca†a (6.1)

where σi+, σi− and σiz are the pseudospin operators for the two level system consisting

of the ith–quantum dot ground state |0〉 and a single exciton |Xi〉 state, ωi is the

exciton frequency, a† and a are the creation and destruction operators of photons in

the cavity mode with frequency ωc, and gi is the dipole coupling between cavity mode

and exciton of the ith–quantum dot. The incoherent loss and gain (pumping) of the

dot-cavity system is then included in the master equation of the Lindblad form

dρ

dt= − ı

~[H,ρ] + L(ρ) (6.2)

with the super-operator

L(ρ) =∑i

[Γi2

(2σi−ρσi+ − σi+σi−ρ− ρσi+σi−)

+Pi2

(2σi+ρσi− − σi−σi+ρ− ρσi−σi+)

+γφi2

(2σizρσiz − ρ)

]+

Γc2

(2aρa† − a†aρ− ρa†a)

+Pc2

(2a†ρa− aa†ρ− ρaa†), (6.3)

1Current address: Instituto de Fısica, Universidade Federal de Uberlandia, 38400-902 Uberlandia,MG, Brazil.

97


where Γi is the exciton decay rate of the ith–quantum dot, Pi is the rate in which

excitons are created by a continuous-wave pump laser in the ith–quantum dot, γφi is

the pure dephasing rate of the exciton in the ith–quantum dot, included to better

describe effects originating from high excitation powers or high temperatures (see

sect. 6.2), Γc is the cavity loss and Pc is the incoherent pumping of the cavity mode.

The origin of this non-resonant pumping was discussed in chapter 5.

To calculate the spectral function we use a formalism described in ref. [Car89].

Using the Wiener-Khintchine theorem, the spectral function of the exciton emission

of the ith–quantum dot is given by [Scu97]

Si(ω) =1

πlimt→∞

Re

∫ ∞0

dτe−(Γr−ıω)τ 〈σi+(t)σi−(t+ τ)〉, (6.4)

and the cavity mode emission is given by

Sc(ω) =1

πlimt→∞

Re

∫ ∞0

dτe−(Γr−ıω)τ 〈a†(t)a(t+ τ)〉, (6.5)

where the term Γr = 30 µeV (half-width) was added to take into account the finite

spectral resolution of our Triax550 imaging spectrometer with 1200 lines/mm grat-

ing. [Ebe77] Here, we have also assumed that the system achieves a steady state for

long times. In both cases we need to compute the correlation function of first order for

different times, which in the absence of coupling to a reservoir is relatively straight

forward, as one needs only to find the unitary evolution operator describing time

evolution. In general, in the presence of coupling to a reservoir, finding multi-time

correlations is more complicated. However, when the reservoir can be treated in the

Markovian approximation, as in our case here, the two-time correlation functions can

be computed using the quantum regression theorem as described in ref. [Car89]. The

quantum regression theorem can be stated in terms of the time evolution of single-

time correlation functions. For a set of operators Oj(t), the master equation given by

equ. 6.2 can be shown to yield for any initial condition

∂

∂t〈Oi(t)〉 =

∑j

Mij(t)〈Oj(t)〉, (6.6)

where the single time expectation value of a given operator O(t) can be computed by

〈O(t)〉 = Tr[O(t)ρ(0)] = Tr[O(0)ρ(t)], (6.7)

98


where ρ(t) is the density operator fulfilling the master equation given by equ. 6.2.

The quantum regression theorem then asserts that the two-time correlation function

is governed by the same set of equations as in equ. 6.6:

∂

∂τ〈Ol(t)Oj(t+ τ)〉 =

∑j

Mij(t)〈Ol(t)Oj(t+ τ)〉. (6.8)

To compute the two-time correlation functions 〈σi+(t)σi−(t+ τ)〉 and 〈a†(t)a(t+ τ)〉we need to know the appropriate single-time correlation functions 〈σi−(t)〉 and 〈a(t)〉.From the master equation (equ. 6.2), we can then derive the Liouvillian equations for

the single time expectation value of a given operator, which can be separated in two

sets of differential equations:

˙〈a〉 = − (ıωc + γc) 〈a〉 − ı∑i

gl〈σi−〉

˙〈σj−〉 = ıgj〈aσjz〉 − (ıωj + γj) 〈σj−〉 (6.9)

and

˙〈a†a〉 = −ı∑i

gi[〈a†σi−〉 − 〈σi+a〉

]− Γc〈a†a〉+ Pc〈aa†〉

˙〈σj+σj−〉 = ıgj[〈a†σj−〉 − 〈σj+a〉

]+ Pj〈σj−σj+〉 − Γj〈σj+σj−〉

˙〈a†σj−〉 = ıgj〈a†aσjz〉+ ı∑i

gi〈σi+σj−〉 − (ı∆jc + γj + γc) 〈a†σj−〉

˙〈σj+a〉 = −ıgj〈a†aσjz〉 − ı∑i

gi〈σi−σj+〉+ (ı∆jc − γj − γc) 〈σj+a〉

˙〈σj+σk−〉 = ıgk〈σj+aσkz 〉 − ıgj〈σk−a†σjz〉+ (ı∆jk + γj + γk) 〈σj+σk−〉. (6.10)

In these equations ˙〈O〉 represents the time derivative of the expectation value of the

operator O (d〈O〉/dt), γc = (Γc − Pc)/2, γi = γφi + (Γi + Pi)/2, ∆ic = ωi − ωc,

and ∆ij = ωi − ωj. Notice that in equ. 6.9 and 6.10 we have higher order terms

〈aσjz〉, 〈a†aσjz〉, 〈a†aσjz〉 and 〈σj+aσkz 〉 which can be approximated by −〈a〉, −〈a†a〉,−〈a†a〉 and −〈σj+a〉, respectively, as the other terms are only important for strong

excitation of the system. To compute the two-time correlation function we need now

to solve equ. 6.9 for a single-time correlation. The initial condition for the two-time

99


correlation functions 〈σi+(t)σi−(t+ τ)〉 and 〈a†(t)a(t+ τ)〉 is then given by the solution

of equ. 6.10 (see example in the next section).

6.1.2. Special Case: Single-Quantum Dot - Cavity System

Assuming that most of the light collected in our setup escapes the system through the

leakage of the cavity mode and using the Wiener-Khintchine theorem, the spectral

function of the collected light is given by equ. 6.5. For a system with a single quantum

dot inside the cavity the Liouvillian equations for a single-time correlation function

(equ. 6.9) become

∂

∂t

(〈a〉〈σ−〉

)= M

(〈a〉〈σ−〉

), (6.11)

where

M =

(− (ıωc + γc) −ıg−ıg − (ıωd + γd)

), (6.12)

with γc = (Γc − Pc)/2, γd = γφd + (Γd + Pd)/2. For the two-time correlation we have

to solve then

∂

∂τ

(˙〈a†(t)a(t+ τ)〉˙〈a†(t)σ−(t+ τ)〉

)= M

(〈a†(t)a(t+ τ)〉〈a†(t)σ−(t+ τ)〉

), (6.13)

with initial condition (τ = 0) given by the solution of

∂

∂t

〈a†a〉〈σ+σ−〉〈a†σ−〉〈σ+a〉

=

−(Γc − Pc) 0 −ıg ıg

0 −(Γd + Pd) ıg −ıg−ıg ıg −ı∆− γ 0

ıg −ıg 0 ı∆− γ

〈a†a〉〈σ+σ−〉〈a†σ−〉〈σ+a〉

+

Pc

Pd

0

0

,

(6.14)

where ∆ = ωd − ωc, and γ = γc + γd. To arrive in this equation from equ. 6.10 we

have used 〈σ+σ−〉 = 1 + 〈σ−σ+〉 and 〈a†a〉 = 1 − 〈aa†〉. Equ. 6.13 can be solved

analytically, and the resulting spectral function (equ. 6.5) can be written as

S(ω) =1

πRe

[A+

ıλ+ − ıω + Γr+

A−ıλ− − ıω + Γr

], (6.15)

100


where

λ± =λc + λd

2± Ω (6.16)

is the emission frequency with effective Rabi splitting

Ω =√g2 − (λd − λc)2/4, (6.17)

and

A± =1

2Ω

[(ω ± ıλd − λc

2

)〈a†a〉(t)± g〈a†σ−〉(t)

]. (6.18)

Here we have set λd = ıωd + γd, λc = ıωc + γc.

For a continuously pumped system similar to that used in our experiments, we can

assume that the system achieves a steady state for long times (t→∞). In this case

〈a†a〉(t) and 〈a†σ−〉(t) can be obtained by setting the derivative in equ. 6.14 to zero

to obtain

〈a†a〉ss =2γg2(Pc + Pd) + Pc(Γd + Pd)(∆

2 + γ2)

2γg2(Γc − Pc + Γd + Pd) + (Γc − Pc)(Γd + Pd)(∆2 + γ2), (6.19)

and

〈a†σ−〉ss =g(ΓcPd − ΓdPc − 2PcPd)(∆ + ıγ)

2γg2(Γc − Pc + Γd + Pd) + (Γc − Pc)(Γd + Pd)(∆2 + γ2). (6.20)

We can now use equ. 6.15 to fit the experimental results using an Levenberg-

Marquardt algorithm as it is done in sect. 6.2.

101


6.2. The Role of Dephasing in Strongly Coupled

Quantum Dot - Cavity Systems

As discussed in chapter 2 two distinct regimes of light-matter interaction exist in

cavity-quantum electrodynamics; weak coupling in which excited emitters decay ir-

reversibly and strong coupling, where a coherent exchange of energy occurs between

the emitter and a single mode of the quantized electromagnetic field. [And99, Khi06]

The strong coupling regime is of particular interest since it directly provides optical

non-linearities at the single photon level, properties that may be used to construct

single photon transistors [Ber06a, Cha07] or even mediate coherent coupling between

spatially separated quantum systems. [Bou00, Ima99] For individual semiconductor

quantum dots embedded within solid-state microcavities strong coupling was first

observed a few years ago, [Rei04, Yos04] the key signature being an anticrossing be-

tween an exciton (ωx) and the cavity mode (ωc) as they are brought into resonance

(δ = ωx−ωc = 0). [And99, Khi06] Surprisingly, recent theoretical work has suggested

that an anticrossing in the emission spectrum is not an unambiguous signature of

strong coupling, [Lau08, Lau09c] rather its appearance depends on the balance be-

tween the incoherent pumping rates of the excitonic (Px) and photonic (Pc) sub sys-

tems. 2 The appearance of an anticrossing then depends on whether the steady quan-

tum state of the system is more photon-like or exciton-like [Lau08, Lau09c] and the

relative importance of pure dephasing. Clearly, probing the nature of the light-matter

coupling in such systems calls for a method to measure the emission spectrum as a

function of detuning δ under well defined experimental conditions of optical pumping

and lattice temperature. Comparison of the results with theory can then be used

to extract quantitative information on the pure dephasing rate and the underlying

mechanisms. To date δ has been predominantly tuned by varying the lattice tem-

perature, [Rei04, Yos04, Pet05, Pre07, Eng07a] or by condensing inert gases into the

cavity at low temperatures. [Mos05, Hen07, Win08] Varying the lattice temperature

simultaneously affects the exciton dephasing rate [Bor05] and δ, greatly complicating

the comparison of experiment with theory. Thus, it was not possible to disentangle

the influence on the line width of changing the lattice temperature from an influence

on the line width due to a change of the dot cavity detuning. Similarly, inert gas

2The dot is pumped by continuous laser excitation into the excited state continuum and the cavityfield by other dots that non-resonantly emit photons into the resonator (see chapter 5).

102

6.2 The Role of Dephasing in Strongly Coupled Quantum Dot - CavitySystems

deposition can only be used for T ≤ 25 K [Mos05] making detailed temperature de-

pendent studies impossible. In the following section (sect. 6.2.1) we introduce our

electrically tunable quantum dot - photonic crystal samples operating in the strong

coupling regime of the light-matter interaction. They offer the advantage that the

detuning between exciton and cavity mode can be adjusted by simply varying the

applied bias voltage without fiddling with the properties of quantum dot and cavity.

6.2.1. Electrically Tunable, Strongly Coupled Quantum Dot -

Cavity Systems

Photoluminescence spectra were recorded from one of the electrically tunable nanocav-

ities, discussed in chapters 3 and 4, using confocal microscopy as a function of the

applied bias voltage Vapp. The excitation laser was tuned to the third order cavity

mode (M3) in order to selectively excite only dots spatially coupled to the funda-

mental mode (see sect. 4.3). [Nom06, Kan09b] Using the quantum confined Stark

effect (see sect. 4.4), one of the quantum dot single exciton transitions can be shifted

through the cavity mode (~ωc = 1207.1 ± 0.1 meV) for a gate voltage where car-

rier tunneling escape from the dots is unimportant and the emission intensity is field

independent. [Lau09b]

A detailed high resolution voltage sweep of this situation is presented in fig. 6.1(a)

for T = 18 K and Pexc = 25 W/cm2. A clear anticrossing is observed between

the exciton and cavity mode as they are electrically tuned into resonance. To extract

quantitative information on the evolution of the spectral function S(ω) and dephasing

of the quantum dot polaritons we simulated the spectral function of the dot-cavity

system using the theoretical model described in sect. 6.1. The open blue circles

correspond to the experimental data whilst the solid black lines are fits using the

theoretical model. Fig. 6.1(b) shows a magnified plot of three of these spectra to

demonstrate the excellent agreement between the measured and calculated spectral

functions.

Using the spectral function equ. 6.15 we globally fit the entire set of δ−dependent

curves presented in fig. 6.1 using an Levenberg-Marquardt algorithm. The best fit

was obtained for ~g = 59 µeV (which was then held constant for all fittings), ~Γx =

0.2 µeV, ~Px = 0.5 µeV, ~Γc = 68.0 µeV, ~Pc = 4.5 µeV, and ~γφx = 19.9 µeV.

The pure dephasing rate is much larger than the spontaneous emission rate or the

incoherent pumping rate of the exciton. This already highlights that pure dephasing

103

Strongly Coupled Quantum Dot - Cavity SystemsPL

Inten

sity (a

rb. un

its)

1 20 6

. 6

1 20 6

. 8

1 20 7

. 0

1 20 7

. 2

1 20 7

. 4

1 20 7

. 6 - 0 .2 0 . 0 0 . 2

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

V app

δ

δ

PL Intensity (cts/s)

δ

Figure 6.1.: (a) Waterfall plot of exciton-cavity anticrossing as Vapp is changedfrom −0.05 V to −0.40 V, corresponding to detunings ranging fromδ = +450 µeV to δ = −260 µeV. The open circles correspond to experi-mental data whilst the solid lines are fits to the theory. Selected curvesclose to resonance (δ = 0 µeV) are presented in (b) showing the excellentagreement between the observed and calculated spectral functions.

plays a significant role even for the low excitation power densities (∼ 25 W/cm2) and

temperatures (T = 18 K) investigated.

104


6.2.2. Dephasing by Dynamical Fluctuations

Being able to reproduce the measured spectra from theory with excellent agreement,

and to extract the different parameters from the fits, allows us to systematically

modify the experimental conditions and monitor the influence on the pumping, decay

and dephasing rates. First, we conducted similar measurements as presented in fig. 6.1

for different excitation power densities. The normalized spectra recorded close to

zero detuning (δ ∼ 0 µeV) are plotted in fig. 6.2(a) with excitation power densities

ranging from 6 - 400 W/cm2. 3 As in fig. 6.1, the open blue circles correspond to the

experimental data points whilst the solid black lines are fits. We observe a splitting at

low excitation powers into two polariton peaks. These peaks clearly broaden and are

not well resolved for δ ∼ 0 µeV at high excitation powers. It is important to note that

this line broadening does not reflect a simple sum of the pure dephasing, decay and

pumping rates. Rather, the various parameters contribute differently to the steady

state solution of 〈a†a〉ss and 〈a†σ−〉ss that appear in equ. 6.15. Thus, as in fig. 6.1 a

global fit to the entire set of δ−dependent data is required for each excitation power

to extract the different parameters from the fits.

The parameters corresponding to the fits of these data sets are summarized in

fig. 6.2(b). The green (open circles) and red (open squares) curves show the quantum

dot (Px) and cavity mode (Pc) pumping rates, respectively. Both exhibit a clear linear

and monotonous increase as expected since Px and Pc should scale with the excitation

power density. Similarly, the black (filled circles) and the blue (filled squares) curves

show the exciton decay (Γx) and cavity loss rates (Γc), respectively. These were kept

constant since they depend, respectively, on the dipole moment of the dot and the

radiation losses of the cavity, but not the excitation power. Furthermore, ~Γx =

0.2 µeV corresponds to a radiative lifetime of τ = 3 ns, in good agreement with

time resolved measurements performed on similar quantum dots emitting into a two-

dimensional photonic bandgap [Kan07] and ~Γc = 68 µeV corresponds to a cavity Q

of ∼ 17600, close to the measured value of Q > 11500 for this mode. Bearing in mind

the limited spectral resolution of the setup this is good agreement. Thus, the values of

~Px, ~Pc, ~Γx and ~Γc extracted from the fits have physically meaningful values and

exhibit precisely the expected dependence on excitation power. These observations

strongly support the general validity of our model.

3The complete detuning-dependent sets of data including the theoretical fits are plotted in Ap-pendix A in figs. A.1 - A.7.

105


!

- 0 .6

- 0 .4

- 0 .2 0 . 0 0 . 2 0 . 4 0 . 6

"

"

"

"

"

"

"

0 . 1

1

1 02 0

4 0

6 0

8 0

φγ

Γ

Γ

!"

1 0 1 0 0 0

4 0

8 0

1 2 0

1 6 0

!

!

Figure 6.2.: (a) Spectra of the exciton polaritons at resonance as a function of excita-tion power. The circles correspond to experimental data, while the solidlines are fits to the theory. In (b) the fitting parameters are displayedas they change with increasing excitation power, and the diagram in (c)shows the theoretically expected Rabi splitting at resonance (filled blacksquares) and the observed position of the peaks in the photoluminescencespectra (open blue circles).

The only additional parameter obtained from fitting our data is the dephasing rate

~γφx , as shown in fig. 6.2(b) by the magenta filled triangles. ~γφx does not vary for low

excitation powers <50 W/cm2, remaining close to ~γφx ∼ 20 µeV, but increases rapidly

106


for higher excitation powers. This observation is a clear evidence for the presence of

excitation induced dephasing, as was recently reported for excitons in self-assembled

quantum dots. [Ber06b, Fav07] Here, spectral wandering induced by fluctuations in

the charge status of the quantum dot environment was shown to result in a linear

temperature dependence and a characteristic power dependence, very similar to the

data for ~γφx presented in fig. 6.2(b). Physically, the power induced broadening arises

from a transition between a regime where motional narrowing of the fluctuating en-

vironment takes place (low power), to one where the dephasing is dominated by the

fluctuating quantum dot environment. [Ber06b] Given the proximity of the investi-

gated quantum dot to the etched surfaces (<∼ 100 nm) of the photonic crystal it is,

perhaps, hardly surprising that excitation induced dephasing plays a significant role.

In fig. 6.2(c) we plot the effective vacuum Rabi splitting (2Ω - equ. 6.17) as a

function of excitation power density. The open blue circles correspond to the result of

fitting two Lorentzian peaks to the experimental spectra as was done in refs. [Yos04,

Rei04], whilst the filled black squares are the values obtained from the theoretical

fits. The simple analysis would indicate that the vacuum Rabi frequency reduces at

higher excitation levels, whereas the full theory shows that 2Ω is largely unaffected by

excitation. This discrepancy between a simple and full analysis was already predicted

in refs. [Lau08, Lau09c]. Our results confirm that strong coupling often appears “in

disguise”of a single peak, depending on the relative importance of decoherence due

to pure dephasing, incoherent pumping and decay. From the high power spectrum

(400 W/cm2) in fig. 6.2(a), the simple treatment would indicate that the system is

not longer in the strong coupling regime whilst our theoretical analysis indicates that

the system remains in the strong coupling regime, albeit with significant broadening

due to pure dephasing.

6.2.3. Phonon Induced Dephasing

Besides excitation induced dephasing, another major source of decoherence in quan-

tum dots is due to coupling to the crystal lattice. Therefore, we investigated the

spectrum of the quantum dot-polaritons as a function of detuning and temperature.

Examples of the recorded spectra close to δ ∼ 0 µeV are plotted in fig. 6.3(a) for tem-

peratures ranging from 17.5 - 48.1 K. 4 The excitation power density used for these

4The complete detuning-dependent sets of data including the theoretical fits are plotted in Ap-pendix A in figs. A.8 - A.14.

107


measurements was fixed at 10 W/cm2 in the low power regime of fig. 6.2. Clearly we

resolve the two polariton peaks for T < 30 K but they broaden rapidly between 30 -

40 K and merge into a single peak at higher temperatures (blue circles). Again, we

used our model to fit the entire δ−dependent data at each fixed temperature (black

lines) and the resulting parameters are summarized in fig. 6.3(b). In contrast to the

power dependent measurements discussed above (see fig. 6.2) the temperature is not

expected to influence either the decay (Γx, Γc) or pumping (Px, Pc) rates of quantum

dot and cavity mode. This expectation is confirmed by the results of our fitting, where

Γx/c were kept constant and Px/c remain constant as T varies. The only fit parameter

that varies strongly with the temperature is the pure dephasing rate ~γφx [filled ma-

genta triangles - fig. 6.3(b)] that increases linearly ~γφx = ~γφ0 +α0T for temperatures

below 30 K with a temperature coefficient α0 ∼ 1.1 µeV/K, and more rapidly for

higher temperatures. The linear temperature dependence of the dephasing rate at

low temperatures is strong evidence for decoherence mediated by coupling to acous-

tic phonons [Bes01, Bay02, Fav03, Fav07] for which temperature coefficients of the

zero-phonon exciton transition lie in the range α0 = 0.04− 4 µeV/K. [Urb04, Bor05]

Finally in fig. 6.3(c), we plot the experimentally observed polariton peak splitting

obtained using the simple model (open blue circles) and from our theory (filled black

squares). The effective vacuum Rabi splitting is insensitive to the temperature for

T < 40 K and reduces slightly at higher temperature. As discussed earlier, the

effective vacuum Rabi splitting differs strongly from the simple analysis of the peak

splitting in all cases except conditions of weak pumping and zero temperature. This

observation underscores the need for a full theoretical description of the emission

spectrum and its variation with detuning in order to unambiguously determine if the

system operates in the strong or weak coupling regime of the light-matter interaction.

108


( c )

( a )

E n e r g y ( m e V )

- 0 .6

- 0 .4

- 0 .2 0 . 0 0 . 2 0 . 4 0 . 6

4 8 . 1 K

4 3 . 2 K

3 8 . 8 K

3 2 . 9 K

2 7 . 3 K

2 3 . 2 K

1 7 . 5 K

0

2 0

4 0

6 0

8 0

1 0 0

φγx

P x Γx

P c

Γc

2 0 3 0 4 0 5 0 0

4 0

8 0

1 2 0

1 6 0

T e m p e r a t u r e ( K )

t h e o r y

s i m p l e a n a l y s i s

( b )

Figure 6.3.: (a) Spectra of the exciton polaritons at resonance as a function of tem-perature. The circles correspond to experimental data, while the solidlines are fits to the theory. In (b) the fitting parameters are plotted as afunction of temperature, and the diagram in (c) shows the theoreticallyexpected Rabi splitting at resonance (filled black squares) and the ob-served position of the peaks in the photoluminescence spectra (open bluecircles).

109


6.3. Mutual Coupling of two Quantum Dots via an

Optical Nanocavity Mode

Cavity quantum electrodynamics experiments using semiconductor quantum dots

have recently attracted much interest in the solid-state quantum optics commu-

nity. [Khi06, Shi07] Much progress has been made with a number of spectacular

demonstrations, including efficient generation of non-classical light, [San02] the ob-

servation and investigation of strong coupling phenomena [Rei04, Yos04, Pet05, Rei06,

Lau09a] and, excitingly, the possibility to observe and exploit quantum optical non-

linearities. [Eng07a, Ger09a] These developments are all ingredients needed for the

realization of a solid-state all-optical quantum network [Cir97] where quantum mem-

ory elements are coupled via single light quanta. In 1999 Imamoglu et al. [Ima99]

proposed that two spatially separated electron spins in quantum dots could be co-

herently coupled via a common optical cavity field. However, it was only in the

past seven years that the strong coupling regime was reached for a single quan-

tum dot [Rei04, Yos04, Pet05] and, to the best of our knowledge, only one obser-

vation of two dots coherently interacting with a common cavity mode has been pub-

lished. [Rei06] This would provide a new way to entangle spatially separated quantum

emitters via the electromagnetic quantum vacuum.

In this section, we present experimental and theoretical investigations of a system

of two spatially separated quantum dots strongly coupled to the same high-Q photonic

crystal nanocavity mode. In sect. 6.3.1 we identify the two different quantum dots

via their different voltage dependent shifts when we tune their emission energies via

the quantum confined Stark effect (see sect. 6.3.1). [War00, Fin04, Lau09b] Further-

more, we use the same effect to tune their emission energies into resonance with each

other and through resonance with a cavity mode, which is energetically close to their

crossing point. The photoluminescence data shows a characteristic triple peak during

the double anticrossing which is an unambiguous signature for the coherent coupling

of the three quantum states. 5 Previously, the authors of ref. [Rei06] presented a

double dot micropillar system operating in the strong coupling regime but did not

analyze the spectral function of the system and its dependence on detuning. Here,

we obtain new information by theoretically modeling the spectral function of our sys-

5The triple peak observed here is different from the one observed in ref. [Hen07]. It originates fromthe coherent coupling of three quantum states, not from charge fluctuations in the occupation ofthe quantum dot levels.

110

6.3 Mutual Coupling of two Quantum Dots via an Optical NanocavityMode

tem and fitting the experimental data using the model introduced in sect. 6.1. By

fitting this model to our data we extract the contributions of each quantum state to

the three branches of the double anti-crossing in the spectral emission in sect. 6.3.2.

This comparison clearly indicates that coherent coupling is established between the

two spatially separated quantum dots and, moreover, allows us to identify a situation

where the two quantum dot states have the same detuning from the common cavity

mode. The strongly coupled double dot - cavity system then forms a V-like three

level quantum system [Scu97] [see fig. 6.4(b) (inset)] where entanglement between

the two different quantum dot excitons is established by virtual photon emission and

absorption, without populating the cavity mode. This would allow to circumvent the

dominant source of decoherence in solid-state cavity quantum electrodynamics, pho-

ton loss from the cavity as shown in sect. 6.3.3. Since it is technologically challenging

to further enhance the Q-factor of GaAs based photonic crystal nanocavities, [Mic07]

this would provide a route towards creating cavity mediated entanglement of the two

quantum dot excitons on the basis of currently existing solid-state technology. In this

case photon loss from the cavity, which determines the mode Q-factor, is no longer

important for the cavity mediated coupling of the two quantum dots.

6.3.1. Experimental Realization

The sample investigated had the same design and characteristics as already introduced

in sect. 3.1, chapter 4 and sect. 6.2. Photoluminescence spectra from the nanocavities

were recorded using confocal microscopy as a function of Vapp. The excitation laser

was tuned to the higher energy cavity mode M2 in order to spatially select only dots

coupled to the fundamental mode. [Nom06, Kan09b]

In fig. 6.4(a) we present photoluminescence data recorded as a function of Vapp as

an overview of the investigated situation. Besides the emission of the fundamental

cavity mode that is not influenced by Vapp, we observe a number of quantum dot

emission lines that clearly shift as Vapp varies, due to the quantum confined Stark

effect (see sect. 4.4). Two classes of lines are observed, each exhibiting a markedly

different voltage dependence. We attribute the two classes of lines to two different

quantum dots with different size and In-Ga compositional profiles. This leads to

intrinsically different distributions of the electron and hole wavefunctions [Bar00,

War00] and, consequently, different polarizabilities of the exciton transitions. As a

result the different quantum dot exciton lines can be electrically tuned relative to

111


- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 61 2 1 4

1 2 1 5

1 2 1 6

1 2 1 7

1 2 1 8

1 2 1 9

( a )

A p p l i e d B i a s V o l t a g e V a p p ( V )

Energ

y (me

V)

Q D 2Q D 1

M o d eM o d e

- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6

∆g 2g 1

Q D 2Q D 1 ( b )

c a v i t y m o d e

Q D 2

Q D 1

Figure 6.4.: (a) Photoluminescence spectra of the double dot-cavity system as a func-tion of bias voltage (false-colour plot). (b) Peak positions of cavity mode(black circles), QD1 (red squares) and QD2 (blue diamonds) for differentbias voltages, emphasizing the different DC Stark shifts of QD1 and QD2.(inset) Schematic of the energy levels when QD1 and QD2 are in reso-nance and both are detuned from the cavity mode, forming a V-system(indicated by the gray ellipse at Vapp = 0.45 V).

one another. Fig. 6.4(b) shows the extracted peak positions of all lines shown in

fig. 6.4(a) for better comparison. The black circles, red squares and blue diamonds

correspond to emission from the cavity mode, QD1 and QD2, respectively. Two of

112


these exciton lines from different quantum dots (labeled QD1 and QD2) cross at an

energy of ~ωX = 1217.8 ± 0.1 meV, very close to the energy of the cavity mode

(~ωc = 1217.6 ± 0.1 meV). Both these transitions can be assigned to single excitons

due to the observed linear power dependence of their emission intensity (data not

shown). By changing Vapp, the QD1 and QD2 excitonic transitions can be shifted

through the cavity mode and are resonant at almost the same voltage. We note that

at this value of Vapp the carrier tunneling escape rate from the dots is negligible and

the emission intensity is field independent. [Lau09b]

The system described above allows us to investigate the cavity mediated coupling

between QD1 and QD2 in two different ways. Depending on Vapp, the states of the

dots can be simultaneously tuned into resonance with the cavity mode where coherent

coupling occurs via the cavity field. Alternatively, they can be in resonance with each

other but detuned from the cavity mode to form a degenerate V-system [see fig. 6.4 (b)

- inset]. Most importantly, in the latter case dephasing due to incoherent loss of

photons from the cavity, the dominant source of dephasing, can be circumvented as

discussed in sect. 6.3.3.

We investigate the described system experimentally and theoretically in fig.6.5.

Panel (a) shows high-resolution photoluminescence spectra plotted in a false-color

plot recorded as a function of Vapp. Whilst the cavity mode is unaffected by the

electric field, the two quantum dot excitons shift into resonance with the mode at

Vapp ∼ 0.4 V due to the quantum confined Stark effect and into resonance with each

other at Vapp ∼ 0.5 V. During the double anticrossing, when both excitons and the

cavity mode are tuned into mutual resonance, we observe a triple peak feature which

is an unambiguous sign for the coherent coupling of three quantum states.

In order to describe the system of two quantum dots and one cavity mode theo-

retically we use the model presented in sect. 6.1.1 for i = 1,2, i.e. two independent

excitons coupled to a common cavity mode. Now, the Hamiltonian becomes:

H =2∑i=1

[~ωi2σiz + ~gi(a†σi− + σi+a)

]+ ~ωca†a (6.21)

with the formalism used in sect. 6.1.1. The emission eigenfrequency is obtained by

solving the Liouvillian equations for the single time expectation value, similar to

113


1 2 1 7 . 0

1 2 1 7 . 5

1 2 1 8 . 0

1 2 1 8 . 5 ( d )( a ) E x p e r i m e n t

1 2 1 7 . 0

1 2 1 7 . 5

1 2 1 8 . 0

1 2 1 8 . 5 ( b ) T h e o r y

Energ

y (me

V)

1 2 1 7 . 4 1 2 1 7 . 6 1 2 1 7 . 8



λ2

λ1

λ0

( c )

0 . 3 0 . 4 0 . 5

1 2 1 7 . 5

1 2 1 8 . 0E i g e n v a l u e s

A p p l i e d B i a s V o l t a g e V a p p ( V )

Figure 6.5.: (a) High-resolution photoluminescence spectra of the investigated systemas function of Vapp (false-colour plot). (b) Calculated spectral function ofthe same system. Parameters were obtained from best-fits. (c) Calculatedeigenvalues λ0, λ1 and λ2 (solid black lines), clearly showing the threebranches of the double anticrossing and the triple peak in resonance. Thepredicted anticrossing of the two quantum dots when they are detunedfrom the mode is magnified in the inset. For the gray shaded area, wecompare in (d) the observed (open circles) and the fitted, theoretical(black solid lines) spectral functions, showing the good agreement.

114


ref. [Car89]:

ı∂

∂t

〈a〉〈σ1−〉〈σ2−〉

=

ωc g1 g2

g1 ω1 0

g2 0 ω2

〈a〉〈σ1

−〉〈σ2−〉

(6.22)

where ωc = ωc − ıγc and ωi = ωi − ıγi, with γc = (Γc − Pc)/2, and γi = γφi + (Γi +

Pi)/2. From the eigenstate of the emission eigenfrequency we can obtain the degree of

mixture of each peak in the spectrum, i.e. the strength of the contribution of cavity

mode, QD1 exciton and QD2 exciton to each individual eigenstate (see sect. 6.3.2).

Using the spectral function S(ω) for two quantum dots and one cavity mode (see

equ. 6.5 in sect. 6.1.1) we globally fit the entire set of spectra generated as a function

of Vapp using a Levenberg-Marquardt algorithm in the same manner as reported in

sect. 6.3.1. The best fit was obtained for ~g1 = 44 µeV and ~g2 = 51 µeV, ~ΓQD1 =

0.1 µeV and ~ΓQD2 = 0.8 µeV, ~PQD1 = 1.5 µeV and ~PQD2 = 1.9 µeV, ~γφQD1 =

20 µeV and ~γφQD2 = 9.8 µeV, ~Γc = 147 µeV, and ~Pc = 5.7 µeV.

The calculated spectral function presented in fig. 6.5(b), shows very good quantita-

tive agreement with the measured data. The spectra close to resonance are plotted in

fig. 6.5(d) to directly compare the experimental data (open symbols) and the calcu-

lated spectral function (black solid lines). The blue arrows mark the position of the

third (middle) peak in resonance which remains visible for all detunings, and is very

well reproduced by our calculations with three coupled quantum states. The compar-

ison with the experimental spectrum, which shows the triple peak structure, strongly

supports our conclusion that we observe two excitons from two independent quantum

dots. 6 In fig. 6.5(c) we plot the calculated eigenvalues of the matrix in equ. 6.22 (i.e.

the peak positions) λ0, λ1 and λ2 (black curves). The data clearly indicate the exact

evolution of the three branches as function of Vapp. Besides the double anticrossing in

resonance with the cavity mode, our calculations indicate that an anticrossing must

exist for the two quantum dots when they are tuned into exact resonance with one

other, but not with the mode (see magnified inset). Here, the splitting was numeri-

cally determined to be ∼ 10 µeV. This energy is below the spectral resolution of our

setup and cannot, therefore, be directly resolved by our experiments. 7 Finally, we

6The possibility that the observed spectral signature could be produced by (i) two different singleexciton transitions of the same quantum dot, and (ii) by two different quantum dots, one weaklyand one strongly coupled to the cavity is investigated in app. B.

7For large detunings of the quantum dots from the cavity mode (∆ >> gi,γi,γc) it is also possibleto adiabatically eliminate 〈a〉 from equ. 6.22, which gives us an effective coupling between thequantum dot excitons geff = g1g2/∆ ∼ 7.2 µeV due to virtual photon transitions. Since this

115


note that we have observed similar double anticrossings in two other samples (data

not shown).

6.3.2. Analysis of the Eigenvectors

The good agreement between experiment and theory (see fig. 6.5) lends support to

our model and allows us to extract additional information about the coupling between

the cavity mode and the excitons. The calculation of the eigenvectors for each of the

eigenstates reveals the respective contributions of cavity mode, QD1 and QD2 to the

three different branches of our system. We plot the results in fig. 6.6, where (a), (b)

and (c) correspond to the eigenstates λ0, λ1 and λ2, respectively. The different curves

correspond to the normalized admixture of QD1 (red dotted), QD2 (blue dashed) and

cavity mode (green solid) to the quantum states of the coupled system. Starting with

λ0 fig. 6.6(a) we can trace the curves for increasing Vapp to monitor the evolution from

an almost pure QD1-like state to the mode-like state with only a weak contribution

of QD2 close to resonance.

The eigenstate λ1 [fig. 6.6(b)] is initially QD2-like, becomes a mixture of all three

states for Vapp ∼ 0.4 V, is mainly QD1-like for Vapp ∼ 0.45 V, and becomes a mixture

of QD1 and QD2 only for Vapp ∼ 0.49 V. For large Vapp it remains QD2-like since

the system is strongly detuned. For this eigenvalue we can readily see where coupling

between QD1 and QD2 can occur. When both excitons are in resonance with the

cavity mode we obtain a coupling between the two excitons, but the dissipation is

governed by the photon lifetime in the cavity effectively limiting the coherent inter-

action. However, when QD1 and QD2 are tuned into resonance with each other, but

not in resonance with the cavity mode (Vapp ∼ 0.49 V), the system can be seen as

a V-type system as depicted schematically in fig. 6.4(b) - inset. The coupling occurs

via a Raman-type transition as proposed by Imamoglu et al. in ref. [Ima99]. Here,

incoherent losses of the cavity mode, that dominate the dissipation, are only of mi-

nor importance. This increases the coherence of the system and somewhat relaxes

the stringent criteria demanding high Q-factors of the cavity mode, needed for the

coupling of the two quantum dots as discussed in sect. 6.3.3.

The last state λ2 [fig. 6.6(c)] starts mode-like, becomes a strongly mixed state of

condition is not entirely fulfilled for our system (∆ ∼ 303 µeV), the numerically extracted split-ting of ∼ 10.3 µeV does not exactly correspond to the splitting calculated from the adiabaticapproximation ∼ 11.2 µeV.

116


Figure 6.6.: Calculated eigenvectors of the investigated system for the eigenvalues λ0

(a), λ1 (b) and λ2 (c). The plotted curves show the contributions of QD1(red dotted lines), QD2 (blue dashed lines), and the cavity mode (greensolid lines) to the individual states for different detunings as a functionof Vapp.

mode and QD2 for Vapp ∼ 0.4 V, is QD2-like for Vapp ∼ 0.45 V, and becomes a

mixture of QD1 and QD2 for Vapp ∼ 0.49 V before it ends QD1-like. For this state,

we can also see the coupling of QD1 and QD2 when they are not in resonance with

the mode, due to the mixed character.

117


6.3.3. Coupling via an Optical ”V-System”

We will now take a closer look at the coupling of the two quantum dots via the optical

V-system and the influence of the cavity mode. To do this we employ the theory

introduced in sect. 6.1.1 and numerically calculate the splitting of the two exciton-

like eigenstates for parameters that reflect the achievable range in experiments. The

result is presented in fig. 6.7(a) as a function of detuning to the cavity mode and as

a function of Q-factor of the cavity mode. The position of the quantum dot - cavity

system, investigated in sect. 6.3.1, in the parameter space of our figure is marked with

the red circle (∆ = 300 meV, Q = 8300). For these values we obtain a splitting of

∼ 10.3 µeV for the numerical solution.

In fig. 6.7(b) and (c) we extract the splitting of the two exciton-like eigenstates as

a function of Q-factor for a detuning of ∆ = 0 µeV and ∆ = 300 µeV, respectively.

In the first case, the two excitons are in exact resonance with the cavity mode. The

Q-factor of our quantum dot - cavity system investigated in sect. 6.3.1 is indicated by

the gray dotted line. For this Q-factor of Q = 8300 we calculate a splitting of 61 µeV.

For decreasing Q-factors, the splitting decreases until it drops to 0 µeV at a Q-factor

of ∼ 4000. This is the set of parameters for which the system leaves the regime of

coherent coupling and enters the weak coupling regime. [Lau09c, dVa09]

A detuning of ∆ = 300 µeV corresponds to the detuning of our quantum dot -

cavity system investigated in sect. 6.3.1. The corresponding Q-factor of Q = 8300

is indicated with the gray dotted line. If we now increase or decrease the Q-factor

of the cavity mode, we will move along the abscissa of the figure and observe the

influence of the Q-factor on the splitting of the two exciton-like eigenstates when

the two quantum dot transitions are coupled via the V-type system. Remarkably,

the splitting stays at values > 5 µeV for Q-factors as low as 2000. We can see

that the Q-factor of the cavity mode is only of minor importance, and to obtain

coupling between the two quantum dots high Q-factors of the cavity mode are not

absolutely necessary. For a system where the quantum dots are subject only to

little dephasing this is still sufficient to obtain coherent coupling between the two

quantum dots. Low-temperature high resolution laser spectroscopy experiments on

quantum dots were able to observe excitonic linewidths as small as 1 − 2 µeV in

transmission. [Kro06, Ger09b]

118


5 . 05 . 0

1 0 . 02 0

5 0

5 . 0

- 1 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 02 0 0 0

3 0 0 0

4 0 0 0

5 0 0 0

6 0 0 0

7 0 0 0

8 0 0 0

9 0 0 0

1 0 0 0 0

∆

0 . 04 . 08 . 01 21 62 02 42 83 23 64 04 44 85 25 66 06 4

2 00 0

4 00 0

6 00 0

8 00 0

1 00 0

0

01 02 03 04 05 06 07 0

∆

2 00 0

4 00 0

6 00 0

8 00 0

1 00 0

0 024681 0

∆

Figure 6.7.: (a) Calculated splitting of the two quantum dots in the optical V-system(numerical solution). The red circle corresponds to the experimentalparameters (∆ = 300 meV, Q = 8300) of the system investigated insect. 6.3.1. (b) and (c) Calculated splitting as function of Q-factor for adetuning of (a) ∆ = 0 µeV and (b) ∆ = 300 µeV.

119


6.4. Summary

In this chapter we started by introducing a theoretical model to calculate the emission

spectrum of N quantum dots coupled to a single cavity mode (sect. 6.1). This model

was then refined for the special case of a single-quantum dot - cavity system.

In sect. 6.2 we presented a combined experimental and theoretical study of the

emission spectrum of quantum dot exciton polaritons in electrically tunable single

dot nanocavities. Such devices provided us with the unique capability to vary the

dot-cavity detuning in-situ and probe the emission spectrum under well controlled

conditions of lattice temperature and incoherent excitation level. The results showed

that the observation of a double-peak in the emission spectrum is not an unequivocal

signature of strong coupling and by comparing our results with theory we extracted

the effective vacuum Rabi splitting, the pure dephasing rate and their dependence on

the incoherent optical pumping power and lattice temperature. Our results confirmed

that strong coupling often appears in disguise of a single peak, depending on the exact

set of experimental parameters.

Finally, in the last section sect. 6.3 we presented an experimental and theoretical

study of a system where two quantum dots are coherently coupled via an optical

cavity mode. Coupling was established by electrically tuning both quantum dots into

mutual resonance and into resonance with the cavity mode, or by tuning them in

resonance with each other but detuned from the mode. We pointed out that the

latter configuration offers the advantage that photon loss from the cavity can be

circumvented, leaving the system in a state of coherent superposition for a longer

time and relaxing the stringent criteria for having extremely high mode Q-factors.

120

Light thinks it travels faster than anything but it is

wrong. No matter how fast light travels it finds the

darkness has always got there first, and is waiting

for it.Terry Pratchett

Chapter 7Spontaneous Emission Properties of Quantum Dots in

Photonic Crystal Waveguides

In contrast to point defects in photonic crystals that form localized optical modes,

waveguides such as line defects allow light to be guided on a chip. Such elements may

play an important role for on-chip photonic, and quantum optics applications. [Hug07,

Eng07b, Far08b, Yao10] They have been used to route single photons from a cavity-

coupled single quantum dot, [Eng07b, Yao09, Yao10] but also to control the direc-

tion and efficiency of quantum dot spontaneous emission. [Hug04, Lec07, Rao07a,

Rao07b, Rao08, Lun08, Pat09b, Dew10, Thy10, Yao10] These properties, combined

with the inherent non-classical nature of the light emission of individual quantum

dots [Mic00, San01, Kan08a] open up the possibility for the realization of highly effi-

cient single photon sources for on-chip quantum optics experiments.

In this chapter we present experimental studies of the emission characteristics of

self-assembled InGaAs quantum dots integrated inside photonic crystal waveguides.

In sect. 7.1 we conduct measurements on an ensemble of quantum dots. We compare

the emission properties normal to the sample surface for quantum dots in (i) the bulk

GaAs, (ii) the photonic crystal membrane and (iii) the photonic crystal waveguide

region. These measurements show an up to ∼ 21× enhancement of the photolumines-

cence intensity for quantum dots spatially and spectrally coupled to the waveguide

121

Spontaneous Emission Properties of Quantum Dots in Photonic CrystalWaveguides

mode. We trace this enhancement to a combination of angular redistribution of the

emission and Purcell enhanced emission into the waveguide mode with subsequent

scattering.

In sect. 7.2 we continue to investigate structures containing single self-assembled

quantum dots. We present our sample design which allows direct optical access to

photons emitted into the waveguide mode via the cleaved facet (sect. 7.2.1). Employ-

ing this geometry and a tunable laser source allows us to measure the transmission

band of a waveguide and compare it with bandstructure calculations (sect. 7.2.2). We

then compare the emission intensities of single quantum dots in the vertical direction,

perpendicular to the sample surface, and in the lateral direction into the third pho-

tonic crystal waveguide mode (sect. 7.2.3). This enables us to calculate the β-factor

of an individual quantum dot from the emitted intensities. We find β = 64 % and

compare this value to the β-factor obtained from lifetime measurements. In sect. 7.2.4

we compare the two different emission channels. For a less well-coupled quantum dot

(β = 12 %), we observe very similar photoluminescence spectra, power-dependences

and decay dynamics (τ ∼ 6.3 ns) for both detection geometries. Finally, second order

photon autocorrelation g(2)(τ) measurements prove the single photon character of the

quantum dot emission for a well coupled quantum dot (β > 90 %) (sect. 7.2.5). The

broad bandwidth of the photonic crystal waveguide mode allows the design of on-chip

quantum optics experiments without the need to tune the quantum dot transition

into resonance with a high-Q photonic crystal cavity mode.

122

7.1 Quantum Dot Ensemble - Photonic Crystal Waveguide Systems

7.1. Quantum Dot Ensemble - Photonic Crystal

Waveguide Systems

We start our investigations on W1 photonic crystal waveguides as discussed in chap-

ter 2. The basis material used is undoped and has a high quantum dot density

(> 50 µm−2). In fig. 7.1(a) and (b) we present a scanning electron microscope image

of the investigated samples. The photonic structure has a total length of 45 µm, a slab

thickness of h = 180 nm, a photonic crystal lattice constant of a = 270 nm, and an

air hole radius to lattice constant ratio of r/a = 0.34. 1 We obtained a spatially and

spectrally resolved photoluminescence map of this particular structure by scanning

the laser spot over the surface of the sample and recording photoluminescence spectra

at every position in a confocal detection geometry. 2 A typical representative result

of this scan is presented in fig. 7.1(c) where we have integrated the detected photo-

luminescence over the spectral range of the quantum dots (1298 meV to 1340 meV)

and plotted the resulting intensity in a false color representation. The contours of the

photonic structure can be easily recognized in the photoluminescence map and we ob-

serve a clear luminescence enhancement on the photonic structure with the maximum

intensity detected at the position of the photonic crystal waveguide being a factor of

∼ 21× higher than on the unpatterned substrate.

We now investigate the origin of this photoluminescence enhancement. To do

this, we performed photonic bandstructure calculations using the software package

RSoft. [RSo08] We use the appropriate parameters for GaAs of nGaAs = 3.5 and geo-

metric parameters of h/a = 0.6667 and r/a = 0.34, corresponding to the investigated

W1 photonic crystal waveguide shown in the scanning electron microscope image in

the inset of fig. 7.2(a). The result of this simulation is plotted in fig. 7.2(a) where

we plot the normalized frequency of the photonic bands as a function of k-vector on

the path from the Γ point to the K ′ point. [Joh00, Dor08b] We present the photonic

crystal waveguide modes as blue solid lines, the slab waveguide modes as light gray re-

gions, and the lossy region above the light cone is shaded in dark gray. 3 We calculate

1The two rectangles directly next to the photonic crystal structure serve for orientation purposesduring optical measurements and have no influence on the photonic properties of the waveguide.

2The size of the excitation laser spot on the sample was determined to have a FWHM ∼ 1.3 µm,while the size of the detection spot has a diameter of FWHM ∼ 6.0 µm.

3The region above the light cone corresponds to the energy-wavevector combinations for whichphotons are not confined to the slab by total internal reflection, i.e. they can leave the waveguidein vertical directions.

123


8 . 0 0 0 E + 0 6

7 . 3 8 5 E + 0 78 . 0 0 0 E + 0 7

Figure 7.1.: (a) and (b) Scanning electron microscope image of the investigated W1waveguide - quantum dot-ensemble system from the top. (c) Spatially-resolved scan of the photoluminescence signal performed with excitationand detection from the top and integrated over the 1298 - 1340 meVspectral range. The area of high photoluminescence intensity correspondsto the waveguide region.

the guided part of the lowest energy waveguide mode WM1 to be at a normalized fre-

quency of a/λ = 0.262−0.274 which corresponds to an energy of E = 1203−1259 meV.

For small wavevectors k < 0.28 this mode overlaps with the region above the light

cone. Photons of these wavevector-energy combinations can, therefore, leave the

sample in vertical directions. This unguided waveguide mode spans the normalized

frequency range from a/λ = 0.275 − 0.308 (E = 1260 − 1413 meV). The second

waveguide mode WM2 extends from a/λ = 0.285 − 0.296 (E = 1309 − 1359 meV)

with an additional unguided region at a/λ = 0.291− 0.296 (E = 1338− 1359 meV).

The third waveguide mode WM3 is at much higher energies of a/λ = 0.334 − 0.337

(E = 1534− 1548 meV) (data not shown).

For comparison we plot two examples of recorded photoluminescence spectra in

124


0 . 50

0 . 45

0 . 40

0 . 35

0 . 30

0 . 25

0 . 20

0 . 15

0 . 2 6

0 . 2 7

0 . 2 8

0 . 2 9

0 . 3 0

0 . 3 1

0 . 3 2

W M 2

( a ) N

ormaliz

ed Fr

eque

ncy a

/λ( b )

W a v e v e c t o r K ’Γ

L i g h t c o n er / a = 0 . 3 4

W M 11 2 0 0

1 2 5 0

1 3 0 0

1 3 5 0

1 4 0 0

1 4 5 0

Q u a n t u m D o t si n W a v e g u i d e

x 2 1

P L I n t e n s i t y ( a r b . u n i t s )

Energy (meV)

x 5

Q u a n t u m D o t si n B u l k M a t e r i a l

Figure 7.2.: (a) Photonic bandstructure calculations for a W1 waveguide with r/a =0.34 and h/a = 0.6667. The blue lines correspond to the photonicwaveguide modes and the light gray region to the slab waveguide modes.The dark gray region indicates the region above the light cone. (inset)Scanning electron microscope image of the investigated photonic crystalwaveguide structure. (b) Comparison of the photoluminescence signalmeasured on the ensemble of quantum dots in the unprocessed bulk ma-terial (black line) and on the ensemble of quantum dots in the waveguideregion (red line).

fig. 7.2(b). The red line corresponds to a spectrum recorded directly on the photonic

crystal waveguide, while the black line corresponds to a spectrum recorded next to

the photonic crystal waveguide on the unprocessed GaAs bulk material (note the ×5

magnified scale). We clearly observe a strong enhancement of the photoluminescence

intensity in the spectral range between 1300 meV and 1325 meV. For these exci-

tation conditions, we detect the maximum signal of ∼ 78000 cts/s at an energy of

∼ 1317 meV, ∼ 21× stronger than the ∼ 3700 cts/s obtained from the unprocessed

bulk material. We notice that the energy range over which we observe photolumines-

125


Figure 7.3.: (a) Spatial image of the reflection of the laser spot on the sample surface.(b) Spatial image of the photoluminescence signal of the waveguide modenormal to the sample surface (filtered with a 10nm bandpass filter at theenergy of the mode). The dotted white lines indicate the outline of thephotonic crystal membrane.

cence enhancement coincides very well with the flat part of the dispersion relation of

the second energy mode, but also with the energy range of the unguided parts of the

first and the second waveguide mode as shown in fig. 7.2(a).

The observation of a higher emission amplitude in the direction normal to the

photonic crystal membrane may seem puzzling at first. In ideal structures we would

expect the Purcell enhanced emission to be efficiently guided away from the excitation

spot along the waveguide, resulting in a decrease of emission detected in the vertical

direction. However, disorder will lead to a scattering of the guided light. Scattering

in the in-plane direction is inhibited due to the photonic bandgap of the surrounding

photonic crystal, but scattering into modes above the light cone that can readily

escape from the slab is still possible. [Kur05, Pat09a, Pat10b] Furthermore, the finite

length of the photonic crystal waveguide structure also enables the quantum dots

to emit in vertical directions due to Fabry-Perot effects. [Fus08] In our structures,

possibly the most important reason for emission in vertical directions is the presence

of non-guided waveguide modes at the same energy range. Quantum dots can emit

light directly into the leaky modes, or propagating photons from the guided mode can

be scattered into these modes due to disorder in the crystal. [Kur05, Pat09a, Pat10b]

In order to support these suggestions we performed additional measurements with

a Si-based, Peltier-cooled CCD camera for spatial imaging. In fig. 7.3(a) we present

126


an image taken of the laser excitation spot on the sample surface at an energy of

1521 meV. The red (blue) curve corresponds to the cross section along the x- (y-)

direction at the position indicated by the red (blue) line. From these plots, we can

extract the full-width-half-maximum (FWHM) of the laser spot to be ∼ 1.3 µm in

both the x- and the y-direction. Fig. 7.3(b) shows a similar image of the photolu-

minescence emitted by the photonic crystal waveguide mode in the vertical direc-

tion. In order to record this image, we mounted a 10 nm-bandpass filter, centered

at the energy of the strongest photoluminescence enhancement, in front of the cam-

era. While the photoluminescence signal recorded along the y-axis, perpendicular to

the waveguide direction, exhibits a similar width as the reflection of the laser spot

FWHM = 1.4 µm, the signal along the waveguide direction (x-axis) is much broader

with FWHM = 5.1 µm. This observation strongly supports the assertion, that light

traveling along the waveguide is scattered into vertical directions by the processes

mentioned above. Since our detection spot has a size of FWHM ∼ 6.0 µm, we

collect most of this scattered light and observe, therefore, not only photolumines-

cence emitted into the vertical direction, but also photoluminescence emitted into the

guided modes and subsequently scattered into the vertical direction.

We check for Purcell-enhanced emission by performing a spatially-resolved scan

of the time-resolved photoluminescence signal emitted at the peak of the amplitude

(1317 meV). In order to do this we scan the excitation spot over the surface of the

sample in 1 µm-steps and record the decay transients at every position over a 61x16

matrix. The decay transients are then fitted taking into account the instrument

response function of the experimental setup. The fitting algorithm first tries to fit a

biexponential decay but automatically switches to a monoexponential decay when one

of the fitting parameters is unrealistic. 4 The result of the fitting routine is plotted in

fig. 7.4. The different panels of fig. 7.4 show the amplitude Aslow and decay time τslow

of the slow decay component and Afast and τfast of the fast decay component. When

the decay transient is fitted monoexponentially, the extracted amplitude and decay

time are plotted in the panels for the slow decay component and the corresponding

pixel in the panels for the fast decay component are blackened.

We can clearly recognize the outline of the photonic crystal waveguide structure

4A biexponential decay is regarded as not being able to describe the fitted decay transient properlywhen (i) one of the amplitudes is negative, (ii) the two lifetimes differ by less than 20%, (iii) oneof the lifetimes is shorter than 50 ps, or (iv) one of the amplitudes is more than 25× larger thanthe other one.

127


A

B

C

C

B

A

Figure 7.4.: Spatially-resolved scan of the time-resolved photoluminescence signal.The different panels show the fitted amplitude Aslow and decay time τslowof the slow decay component and the amplitude Afast and decay timeτfast of the fast decay component. A , B and C present examples ofdecay transients of the spatially-resolved scan recorded at the position A

the bulk material, B the photonic crystal membrane, and C the photoniccrystal waveguide.

from the data in fig. 7.4. On the unprocessed bulk material the decay transients are

globally well described by monoexponential decays, leading to the large blackened

area around the photonic crystal membrane structure (see position A on fig. 7.4).

We present a typical example of one of these decay transients in the panel of fig. 7.4,

marked by A . The amplitude and the decay time of the monoexponential decay are

fitted to be Abulk = 111 ± 14 Gcts/s2 and τbulk = 0.60 ± 0.1 ns, which results in an

intensity of Ibulk = Abulk · τbulk = 67 ± 14 cts/s. The extracted parameters of the

decays at the different positions are summarized in Tab. 7.1.

When we move from the bulk material onto the photonic crystal membrane, but

not on the W1 waveguide, the decay transients become clearly biexponential. A

representative transient is plotted in fig. 7.4 and marked by B . Here, we observe a

very slow decay component due to the effect of the photonic bandgap and an additional

faster decay component probably due to quantum dots located in proximity to the

etched holes of the photonic crystal. The extracted parameters are summarized in

Tab. 7.1. We observe an enhancement in the total intensity of ∼ 4.2× compared

to the intensity of the quantum dot ensemble in the bulk material, which we relate

128


Position τ (ns) A (Gcts/s2) I (cts/s) Itotal (cts/s)

A Bulk 0.60± 0.1 111± 14 67± 14 67± 14

B Photonic Crystal Membrane0.81± 0.1 106± 12 86± 14

282± 323.92± 0.1 40± 4 196± 18

C Photonic Crystal Waveguide0.42± 0.1 2048± 87 860± 208

1116± 2341.74± 0.16 147± 7 256± 26

Table 7.1.: Overview on the extracted decay times and amplitudes for A the bulkmaterial, B the photonic crystal membrane, and C the photonic crystalwaveguide.

to a more efficient collection of the emitted photons due to angular redistribution of

emission. [Fuj05, Kan08a]

Next, we focus on the waveguide region. 5 We observe a highly enhanced amplitude

for the fast as well as for the slow decay component, as can be seen in the example

transient in fig. 7.4 marked by C . The extracted parameters are summarized in

Tab. 7.1. With a total intensity of 1116 ± 234 cts/s the photoluminescence signal

detected from the photonic crystal waveguide is a factor of ∼ 4.0× stronger than

on the photonic crystal membrane. This enhancement corresponds to the spatial

influence of the waveguide mode, but the origin is not obvious. In an experiment

with pulsed excitation where the lifetime of the emitter is much shorter than the

time between two successive laser pulses, the emitted intensity is independent of the

lifetime itself. When compared to the lifetime of quantum dots in the unprocessed

bulk material, the average lifetime of the quantum dot transitions is lengthened to

∼ 4− 5 ns due to the photonic bandgap. [Kan07, Kan08a] However it is still a factor

of ∼ 3× shorter than the ∼ 12.5 ns between two successive laser pulses. We would,

therefore, only expect a marginal increase in intensity due to a shortening of the

lifetime. We believe that the enhanced intensity is related to a process similar to the

non-resonant feeding of photonic crystal cavity modes discussed in detail in sect. 5.2,

where multi-excitons emit at an energy where the photonic density of states is high

as is the case in resonance with the waveguide mode. However, in order to verify this

speculation additional power-dependent and time-resolved measurements would need

to be conducted. Finally, compared to the emission from quantum dots in the bulk

5The oscillations in Afast in fig. 7.4 are an artifact of the slightly rotated waveguide structure andthe 1 µm steps during the scan.

129


material, the measured intensity of 1116±234 cts/s from the waveguide is a factor of

∼ 17× higher in good agreement to the emission enhancement of ∼ 21× extracted

from figs. 7.1 and 7.2.

In this section we were able to demonstrate a ∼ 21× higher emission intensity for

quantum dots spectrally and spatially in resonance with the photonic crystal waveg-

uide mode compared to quantum dots located in the unpatterned bulk material. By

comparing the emission dynamics for the different photonic environments we could

trace this emission enhancement to a combination of angular redistribution of emission

and Purcell enhanced emission of resonant and, probably also, non-resonant quantum

dot transitions into the waveguide mode (similar to the results presented in chap-

ter 5). In the next section we turn our attention to low quantum dot density photonic

crystal waveguide structures that allow us to probe single quantum dots coupled to

the photonic crystal waveguide mode.

130

7.2 Single Quantum Dot - Photonic Crystal Waveguide Systems

7.2. Single Quantum Dot - Photonic Crystal

Waveguide Systems

7.2.1. Photonic Crystal Waveguides with Direct Optical Access

The sample employed for our single dot studies is very similar to that used for our

ensemble studies above. The two main differences are, (i) the much lower quantum dot

density of ρsingle−QD < 1 µm−2 which allows us to selectively excite single quantum

dots, and (ii) the cleaved facet which runs perpendicular through the waveguide and

allows for collection of light directly propagating in the waveguide mode. Cleaving of

the sample was done before we created the free-standing membrane in the wet-etching

step with hydrofluoric acid (see sect. 3.2). In fig. 7.5(a) and (b) we present scanning

electron microscope images of the investigated sample. Fig. 7.5(a) shows a plan view

image, clearly showing the W1 photonic crystal waveguide and the cleaved edge at

which it ends. The cleaved edge runs almost perpendicular through the omitted row

of air holes. This can be quite easily realized when orienting the photonic structure

and the cleaved edge along the [110] crystal axes of the GaAs substrate. The image

also shows the underetched region of the waveguide which extends ∼ 0.6 µm into the

bulk material. In fig. 7.5(b) we present an image of the cleaved edge of the sample,

taken at a 45 angle. We can identify the relatively smooth surface of the cleaved

facet and also the free-standing membrane.

For the following experiments we always excited the sample from the top (i.e.

perpendicular to the sample surface) using a 100× microscope objective with NA =

0.50. The photoluminescence signal, however, was either detected from the top using

the same objective, or from the cleaved end of the photonic crystal waveguide (i.e.

perpendicular to the cleaved edge) using a 50× microscope objective with NA = 0.42.

A schematic of the excitation and detection scheme is shown in fig. 7.5(c).

7.2.2. Photonic Crystal Waveguide Bandstructure and

Transmission Band

Before we start with detailed investigations on these samples, we determine the spec-

tral properties of the guided waveguide mode. To do this, we conduct photonic

bandstructure calculations using the software package RSoft, [RSo08] and appropri-

ate parameters for GaAs (nGaAs = 3.5) and geometric parameters for the investigated

131


!

Figure 7.5.: (a) and (b) Scanning electron microscope images of the W1 waveguide(a) from the top, and (b) from the side. (c) Schematic of the excitationand detection geometry.

W1 photonic crystal waveguide (h/a = 0.6667 and r/a = 0.25). This is shown in

the scanning electron microscope image in the inset of fig. 7.6(a). The result of

this simulation is plotted in fig. 7.6(a) where we plot the normalized frequency of

the photonic bands as a function of k-vector on the path from the Γ point to the

K ′ point. [Joh00, Dor08b] We draw the guided photonic waveguide modes as blue

132


0 . 2 0 . 3 0 . 4 0 . 50 . 2 20 . 2 40 . 2 60 . 2 80 . 3 00 . 3 20 . 3 4

W M 3W M 2

( b )( a )No

rmaliz

ed fre

quen

cy a/λ

W a v e v e c t o r K ’Γ

L i g h tc o n e

S l a b b a n dc o n t i n u u m

W M 11 0 0 01 1 0 01 2 0 01 3 0 01 4 0 01 5 0 01 6 0 0

Energy (meV)

( a r b . u n i t s )T r a n s m i s s i o n

G a A sr / a = 0 . 2 5

Figure 7.6.: (a) Photonic bandstructure calculations for a W1 waveguide with r/a =0.25 and h/a = 0.6667. The blue lines correspond to the photonic waveg-uide modes and the light gray region to the slab waveguide modes. Thedark gray region indicates the light cone. (inset) SEM image of the inves-tigated photonic crystal waveguide structure. (b) Spectral transmissionof the waveguide for illumination from the top at the inner end of thewaveguide and detection from the side at the cleaved end. The red (lightgray) shaded region marks the spectral region of quantum dot emission(bulk GaAs).

solid lines, the slab waveguides modes as light gray region, and the light cone as

dark gray region. 6 We calculate the lowest energy waveguide mode WM1 to span

the normalized frequency range a/λ = 0.245 − 0.266, corresponding to an energy of

E = 1125− 1221 meV. The second waveguide mode WM2 is at a/λ = 0.260− 0.272

(E = 1194− 1249 meV) and the third waveguide mode WM3 at a/λ = 0.288− 0.296

(E = 1322− 1359 meV). 7

In order to verify the calculations and to check that the quantum dot emission

is spectrally in resonance with the guided mode of the photonic crystal waveguide,

we measure the spectral transmission band. To do this, we focus the laser beam

6The region above the light cone corresponds to the energy-wavevector combinations for whichphotons are not confined to the slab by total internal reflection, i.e. they can leave the waveguidein vertical directions.

7Due to the large unit cell used for the simulations, the third waveguide mode was hidden withinfolded bands for r/a = 0.25. We, therefore, linearly extrapolated its position from six calculationsperformed for r/a = 0.29− 0.34.

133


at the inner end of the photonic crystal waveguide within the body of the photonic

crystal. Laser light is scattered into the waveguide and the transmitted intensity can

be detected from the cleaved facet as the laser wavelength is scanned. Fig. 7.6(b)

shows the spectrum of the transmitted light recorded using this method. The open

circles correspond to the experimental data points while the blue solid line is a moving

7-point average. The transmission band is centered at 1323 meV, with a width of

55 meV. This is in fairly good accord with the simulations conducted in fig. 7.6(a).

The gray shaded region in fig. 7.6(b) indicates the spectral range over which we expect

absorption from the bulk GaAs, while the red shaded region indicates the range

for which we observe photoluminescence emission from the quantum dots, clearly

showing that they are located well inside the transmission band of the photonic crystal

waveguide.

7.2.3. Determination of β-Factors

When a semiconductor quantum dot is placed inside the photonic crystal waveguide,

the exciton can decay via different radiative and non-radiative channels with indi-

vidual decay rates. The β-factor is then defined as the ratio of the radiative rate of

photons emitted into a specific mode of the waveguide to the total decay rate of the

exciton, and serves as a figure of merit to quantify the coupling strength to the pho-

tonic crystal waveguide mode. [Lec07, Rao07a, Lun08] The fraction of light emitted

into the waveguide mode is then

βΓ =ΓWG

ΓWG + Γrad + Γnr, (7.1)

where ΓWG is the radiative decay rate of the exciton into the photonic crystal waveg-

uide mode, Γrad is the radiative decay rate into non-guided (leaky) modes, and Γnr is

the non-radiative decay rate. In experiments, only the total decay rate Γtot is imme-

diately accessible. The exact values of the individual contributions of ΓWG, Γrad and

Γnr to Γtot can only be determined via more complex experiments. [Joh08a, Kan08b,

Thy10] However, to the best of our knowledge there is no report of the simultaneous

measurement of ΓWG, Γrad and Γnr for a specific quantum dot exciton, which leaves

the determination of the β-factor subject to estimations. In recent publications Γrad

and/or Γnr were estimated from reference values obtained for similar quantum dots

or the same quantum dot detuned from cavity mode at a different lattice tempera-

134


ture. [Lun08, Thy10]

A quantum dot in a cleaved waveguide offers the possibility for an alternative esti-

mation of the β-factor. Here, the intensities obtained for the two different detection

geometries can be directly compared. Top detection of the emission will provide infor-

mation about the radiative emission into non-guided modes, while side detection will

supply information about the radiative emission into the photonic crystal waveguide

mode. Therefore, we can alternatively define βI based on the ratio of the intensity

detected from the cleaved facet of the waveguide and the intensity detected in both

geometries as follows:

βI =Iside

Iside + Itop + Inr. (7.2)

Here, Iside is the intensity detected from the cleaved facet of the waveguide, Itop is

the intensity detected normal to the sample surface, and Inr is the intensity lost by

non-radiative processes. Although Inr is still unknown and needs to be estimated, we

obtain values for both Iside and Itop from experiments, and can determine the β-factor

of a quantum dot exciton by intensity measurements instead of measuring the decay

dynamics.

In fig. 7.7 we present spatial photoluminescence maps of the photonic crystal waveg-

uide region obtained by scanning the excitation spot over the sample surface and

recording spectra at every position. In fig. 7.7(a) and (b) we plot the photolumines-

cence signal integrated over the spectral range Eint = 1328− 1333 meV for detection

from the top and from the side, respectively. The white outline in both plots indicates

the position of the photonic crystal membrane structure and the cleaved edge of the

sample. For both detection geometries we observe clear emission of the same quantum

dot located in the photonic crystal waveguide region, dedgeQD ∼ 8.0± 0.5 µm away from

the cleaved facet. A comparison of the two plots shows, that under identical nominal

excitation conditions the quantum dot is much brighter in side geometry than in top

geometry [note the 2× enhanced scale in fig. 7.7(a)].

The photoluminescence spectra of the quantum dot s-shell and p-shell are plotted in

fig. 7.7(c) and (d), respectively. While the excitation power density is chosen very low

(P = 3 W/cm2) to record the spectra of the s-shell, it is increased to P = 13 W/cm2

to record the spectra of the p-shell. The spectra recorded for side (top) detection are

plotted as red (black) lines. A comparison of the two detection geometries indicates

that the intensity of both the s-shell and p-shell of the quantum dot is higher for the

135


#

&

# # # #

$

1 3 2 6 1 3 2 9 1 3 3 20

5 0

1 0 0

PL In

tensity

(cts/

s)

! %

" "

1 3 4 7 1 3 5 0 1 3 5 3

# "

"

Figure 7.7.: Comparison of the photoluminescence intensity for top and side detec-tion. Spatially-resolved photoluminescence scan performed with (a) ex-citation and detection from the top, and (b) excitation from the top anddetection from the cleaved facet of the waveguide. The photoluminescenceintensity is integrated over the spectral range Eint = 1328 − 1333 meV.(c) Photoluminescence spectra of the quantum dot s-shell recorded withdetection from the top (black line) and from the side (red line). (d)Corresponding photoluminescence spectra of the quantum dot p-shell athigher excitation power.

side detection geometry. While in a high-Q microcavity with spectrally narrow modes

only a small energy range can be well-coupled to the cavity mode, [Lau09b, Hoh09]

the broad bandwidth of the photonic crystal waveguide mode allows transitions of a

136


1 1 0 1 0 0

1 0

1 0 0

1 0 0 0

" %

! %

PL

Inten

sity (c

ts/s)

$

$ " " # 0 1 2 3 4 5 6

0 . 1

1τ % !

!


Figure 7.8.: (a) Comparison of the photoluminescence intensity of a single excitontransition for side (red squares) and top (black circles) detection as afunction of excitation power. (b) Time-resolved photoluminescence in-tensity of the single exciton transition measured in side detection. Thered curve is a fit to the data. The instrument response function of thesetup is displayed as gray line.

wider energy range to efficiently emit into the waveguide. [Rao07a]

For the transition marked with the gray area in fig. 7.7(c), we measure and ex-

tract the power-dependent photoluminescence intensity and plot it in fig. 7.8(a) for

side detection (red squares) and top detection (black circles). For both detection

geometries we observe a power law increase I ∝ Pm with mside = 0.98 ± 0.02 and

mtop = 1.01 ± 0.04 very close to unity, attesting the single exciton character of the

investigated line. [Fin01b, Abb09] Throughout the linear regime, we notice a ∼ 3.5×higher intensity for side detection than for top detection (marked with the gray lines).

Furthermore, we observe a very similar saturation behaviour for both detection geome-

tries. Since we use identical nominal excitation conditions this is hardly surprising.

However, at very high excitation powers (P > 10 W/cm2) the photoluminescence

intensity measured in side detection seems to increase further, indicating the ap-

pearance of a background. This background could be related to a process similar

to the highly non-resonant feeding of cavity modes from multi-exciton states (see

sect. 5.2), [Kan08b, Win09, Lau10b] albeit over a much larger energy range due to

the large bandwidth of the waveguide mode.

137


We now use the ratio of intensities in the linear regime of fig. 7.8(a) to estimate

βI of the system. If we assume that non-radiative processes contribute, at most,

in a similar manner as radiative emission into non-guided modes, 8 [Joh08a, Kan08a,

Thy10] and if we neglect the different detection efficiencies for top and side detection, 9

we can estimate the β-factor of the quantum dot emission to be βI = IsideIside+Itop+Inr

=3.5

3.5+1+1= 64 %. This value is in good agreement with theoretical predictions and

values measured from other groups for photonic crystal waveguide modes. [Rao07a,

Lec07, Thy10]

We check the validity of our assumptions and approximations by conducting time-

resolved measurements of the emission dynamics of this exciton state. In fig. 7.8(b)

we plot the decay transient measured in side detection geometry. We will show

in sect. 7.2.4 that we measure the same decay time independent on the detection

geometry. The decay was fitted with a monoexponential function convoluted with

the instrument response function (IRF) of the system [shown in gray in fig. 7.8(b)].

We obtain a decay time of τ = 1.1 ± 0.1 ns, which corresponds to a decay rate

of Γtot = 0.9 ± 0.1 ns−1. The intrinsic decay rate of this quantum dot transition

without the presence of the mode consists of contributions from non-radiative pro-

cesses and decay into non-guided modes Γint = Γrad + Γnr. Using equ. 7.1 and

βI = 64 % we can calculate Γint = Γtot(1 − β) = 0.33 ns−1 or τint = 3.0 ns. Since

this value is rather short for the lifetime of a single exciton transition in the photonic

bandgap, [Kan08b, Thy10, Lau10b] we believe that our estimation of the β-factor is

on the conservative side and the real β-factor is probably larger. The reason for this

can be an overestimation of non-radiative processes, an unequal detection efficiency,

or loss of photons inside the waveguide due to scattering. [Kur05]

7.2.4. Comparison of the Emission Channels

In this section we compare the single photon spontaneous emission characteristics

of individual quantum dots for the two different emission geometries. In fig. 7.9 we

8This is a worst case approximation based on measurements in ref. [Kan07] which indicate thatnonradiative recombination is negligible for similar systems.

9The detection path for top detection consisted of a 100x objective with NA=0.50, a 50:50 beam-splitter and a multimode fibre to the spectrometer. The detection path for side detection consistedof a 50x objective with NA=0.42 and a multimode fibre to the same spectrometer. While the50:50 beamsplitter is taken into account in the analysis, an exact determination of the detectionefficiencies was not possible because of too many unknown factors as e.g. the farfield emissionpattern of the waveguide.

138


!! !!

!

!

!!

!!!

Emiss

ion In

tensity

(arb.

log.

scale)

"

- 1 . 0- 0 . 5- 0 . 10 . 40 . 8

Ratio of Emission Intensities (log. units)

- 1 . 0- 0 . 8- 0 . 5- 0 . 3- 0 . 1

- 1 . 1- 0 . 8- 0 . 4- 0 . 10 . 2

Figure 7.9.: Comparison of the photoluminescence intensity for top and side de-tection. (a) Spatially-resolved photoluminescence scans performed withexcitation and detection from the top (left panel), and excitation fromthe top and detection from the cleaved facet of the waveguide (middlepanel). The photoluminescence intensity is integrated over the spec-tral range Eint = 1240 − 1476 meV. The right panel shows the ratioof the detected intensities in logarithmic units. (b) and (c) Same mea-surement as in (a) but integrated over the limited spectral range of (b)Eint = 1288− 1295 meV and (c) Eint = 1305− 1312 meV.

present spatial photoluminescence maps of the photonic crystal waveguide region ob-

tained by scanning the excitation spot over the sample surface and recording spectra

at every position. Spatially-resolved contour plots of the photoluminescence signal in-

tegrated over the spectral range Eint = 1240− 1476 meV are presented in fig. 7.9(a),

including photoluminescence from the wetting layer and the quantum dots. We per-

form the measurement in the left panel with excitation and detection from the top and

the measurement in the middle panel with excitation from the top and detection at

the cleaved end of the photonic crystal waveguide. Integration over this energy range

allows us to locate the position of the photonic crystal waveguide and the cleaved facet

on the maps (indicated by the white outline). In the rightmost panel we present the

calculated ratio of the emission intensities obtained by dividing the intensity detected

in the side geometry by the intensity detected in the top geometry. Before plotting it

we take the logarithm to the base of 10 in order to emphasize ratios above and below

unity, corresponding to stronger lateral / vertical emission, respectively. We observe

139


a general trend to slightly higher intensities in side detection (red color) at positions

closer to the cleaved facet of the waveguide, probably due to the proximity of the

excitation spot to the outcoupling facet.

While integration over a wide spectral range enables us to locate the photonic

crystal waveguides, integration over a narrow spectral window allows us to pinpoint

the position of individual self-assembled quantum dots. In fig. 7.9(b) we integrate

the same dataset as in fig. 7.9(a) over the limited spectral range Eint = 1288 −1295 meV. Again, the left panel shows the signal detected from the top, and the

middle panel the signal detected from the side. One of the quantum dots (labeled

QD1), which is dedgeQD1=11.3±0.5 µm away from the cleaved facet, exhibits weak out-

of-plane emission but strong in-plane emission. This is evidence for good coupling

between the quantum dot and the photonic crystal waveguide mode. The calculated

ratio of intensities shows ∼ 100.82 = 6.6× higher intensity detected in side geometry

than in top geometry, averaged over this spectral window. In analogy to the procedure

outlined in sect. 7.2.3 we can estimate the β-factor of the quantum dot emission to be

as large as βQD1 = IsideIside+Itop+Inr

= 6.66.6+1+1

= 77 %, which is in good agreement with

refs. [Rao07a, Lec07, Thy10].

In fig. 7.9(c) we integrate the same dataset as in fig. 7.9(a) and (b) over the spectral

window Eint = 1305−1312 meV. We observe a very similar photoluminescence pattern

for the quantum dots inside the photonic crystal waveguide region for both detection

geometries (top detection in the left panel, side detection in the middle panel). We

will now focus on the quantum dot labeled QD2 which is dedgeQD2=8.8±0.5 µm away

from the cleaved facet. The calculated ratio of intensities (right panel) indicates a

factor of ∼ 10−0.57 = 0.27× difference in intensities, which corresponds to a lower

intensity detected in side geometry than in top geometry, averaged over this spectral

window. For this quantum dot we calculate a β-factor of βQD2 = 0.270.27+1+1

= 12 %.

This quantum dot is less well coupled to the photonic crystal waveguide mode than

QD1, which we attribute to a misalignment of QD2 to the electric field maxima of

the photonic crystal waveguide mode. [Eng07b] However, QD2 offers the advantage

that we obtain similar count rates for both detection geometries. We will use this to

directly compare the spontaneous emission characteristics for top and side detection.

In fig. 7.10(a) we present photoluminescence spectra recorded from QD2, measured

with detection from the top (black solid line) and from the cleaved facet of the photonic

crystal waveguide (red solid line). The spectra are very similar for both detection

140


0 . 00 . 20 . 40 . 60 . 81 . 01 . 2

1 3 0 8 1 3 1 2 1 3 1 6 1 3 2 0

PL Intensity (kcts/s)

$

PL In

tensit

y (kc

ts/s)

!

1 0 0 1 0 0 0

1

1 0

%

#!!"

!%

!

0 2 4 6 8 1 0 1 2

2

4

68 !!!

τ % τ!%

PL In

tensit

y (cts

/s)

0 2 4 6 8 1 0 1 2

1

2

3 !!


Figure 7.10.: (a) Photoluminescence spectra of QD2 recorded with detection from

the top (black line) and from the side (red line). (b) Power-dependentphotoluminescence intensity of the single exciton line of QD2 at EX =1311.1 meV with detection from the top (black circles) and from the side(red squares). (c) and (d) Time-resolved photoluminescence intensitymeasurements recorded at the single exciton line of QD2 at EQD2

X =1311.1 meV with detection from (c) the top, and (d) the side.

geometries, however, the absolute intensities (at identical excitation conditions) are

different. For QD2 the absolute intensity detected from the end of the photonic crystal

waveguide is a factor 4-6× lower than compared to the situation detecting from the

top, in good agreement with the factor of 0.27× obtained in the paragraph above.

141


We continue investigating the emission properties by conducting measurements on

the single exciton line of QD2 at EQD2X = 1311.1 meV (indicated by the gray shaded

area in fig. 7.10(a)) since it is well separated from other emission lines and exhibits

a comparatively high intensity for top as well as for side detection. Fig. 7.10(b)

shows the integrated photoluminescence intensity of this line as a function of laser

excitation power for top (black circles) and side detection (red squares). At low

powers, we observe a power law increase of the intensity (I ∝ Pm) with exponents

mQD2top = 0.88 ± 0.04 and mQD2

side = 0.94 ± 0.03, identical to within the experimental

error, for both detection geometries. The approximately linear increase hints on the

single excitonic nature of the investigated line. [Fin01b, Abb09] As expected, the

saturation behaviour of this transition is independent of the detection geometry and

occurs at an excitation power of PQD2sat ∼ 150 nW (indicated by the dotted gray line).

In fig. 7.10(c) and (d) we present the time-resolved photoluminescence intensity

measured on the single exciton line of QD2 for top and side detection, respectively.

The fitted spontaneous emission decay times τQD2top = 6.3 ± 0.3 ns and τQD2

side = 6.4 ±0.3 ns are almost equal for both detection geometries. However, they are much longer

than that of a quantum dot coupled to a photonic crystal cavity mode and more similar

to the value of a quantum dot emitting into the photonic bandgap. [Kan08a, Lau09b]

We attribute this to the imperfect spatial coupling between QD2 and the electric field

maxima of the waveguide mode. In addition, the lower photonic density of states

at the spectral midpoint of the photonic crystal waveguide mode compared to the

high photonic density of states in the slow light regime, close to the band edges,

most likely plays a relevant role. [Lun08] From the measured β-factor of this quantum

dot βQD2 = 12% and the lifetime of this single exciton state τQD2 = 6.35 ns, we

can estimate the intrinsic lifetime τQD2int for the case that it was not coupled to the

photonic crystal waveguide mode in analogy to sect. 7.2.3. The estimated intrinsic

lifetime τQD2int = 7.2 ns corresponds to a realistic value for a quantum dot emitting into

the photonic bandgap and agrees well with values from literature. [Kan08a, Lun08]

7.2.5. Single Photon Emission into the Photonic Crystal

Waveguide Mode

In the final section of this chapter, we demonstrate the single photon emission char-

acter of a quantum dot coupled to the photonic crystal waveguide. We compare the

photoluminescence spectra for the two different detection geometries in fig. 7.11(a)

142


Figure 7.11.: (a) Photoluminescence spectra of a quantum dot coupled to the pho-tonic crystal waveguide recorded with detection from the side (blackline) and from the top (red line). (b) Power-dependent photolumines-cence intensity of the single exciton line at EX = 1362.9 meV withdetection from the top (black circles) and from the side (red squares).(c) Corresponding time-resolved photoluminescence intensity measure-ment with detection from the side. (d) Corresponding autocorrelationmeasurement g(2)(τ) with detection from the side.

where the spectrum detected from the top is plotted as red line (note the ×10 en-

hanced scale) and detection from the side is plotted as black line. For the line marked

with the gray shaded region, we conduct power-dependent measurements and plot

the result in fig. 7.11(b) for both top (red squares) and side (black circles) detection.

For both detection geometries we observe a slightly sublinear power law dependence

I ∝ Pm with mtop = 0.73 ± 0.04 and mside = 0.81 ± 0.09, providing evidence that

143


the transition line investigated has single exciton character. [Fin01b, Abb09] While

the power-dependence and the saturation behaviour is very similar for both detection

geometries, the absolute detected intensity for side detection is ∼ 50× higher than for

top detection. With equ. 7.2 we can estimate the β-factor of this quantum dot to be

as high as βI ∼ 96 %, attesting to the excellent coupling of the quantum dot to the

waveguide mode. Further support for this conclusion is provided by the time-resolved

measurement presented in fig. 7.11(c). The black circles correspond to the measured

decay transient, while the red line is a fit to the data taking into account the instru-

ment response function (IRF) of the detection and excitation systems (gray line). For

the specific transition under study we measure a lifetime of τ = 0.87± 0.15 ns. With

equ. 7.1 and typical intrinsic lifetimes of uncoupled quantum dots emitting into the

two-dimensional photonic bandgap of τint ∼ 5 − 20 ns, [Kan07, Kan08a, Lun08] this

short lifetime indicates that the β-factor is in the range 85 % < βΓ < 96 %, in good

agreement with the βI = 96 % estimated above.

Finally, we present a g(2)(τ) measurement recorded with side detection in fig. 7.11(d)

using pulsed excitation. The peak at τ = 0 is, clearly, much weaker than the adjacent

peaks and from comparing the areas of the peaks we estimate g(2)(0) = 0.27. This

value is significantly below the value of 0.50, proving that almost all of the detected

light originates from the investigated single exciton transition and that we observe

clean single photon emission. The results presented in fig. 7.11 correspond, there-

fore, to a highly directed and efficient single photon emitter with β > 85 % and a

maximum emission rate f > 1 GHz. Photons are emitted directly into an on-chip

photonic waveguide providing significant flexibility for on-chip quantum optics exper-

iments. Compared to geometries where the quantum dot is resonantly coupled to

a cavity mode, in this system there is no need to spectrally tune the quantum dot

into resonance and, thus, efficient single photon emission can be realized over a wide

bandwidth. This opens up perspectives for quantum information experiments with

wavelength division multiplexing capabilities. [Tre09]

7.3. Summary and Conclusion

In summary, we have presented investigations on the emission properties of self-

assembled quantum dots inside photonic crystal waveguides. For experiments in-

144

7.3 Summary and Conclusion

volving quantum dot ensembles we observed an enhanced spontaneous emission with

a ∼ 21× increase in photoluminescence for out-of-plane emission which we related

to a combination of angular redistribution of emission and a Purcell-related effect in

combination with subsequent scattering. For experiments involving single quantum

dots we demonstrated a method to estimate the β-factors of single quantum dots

from intensity measurements and compared the results to estimations from lifetime

measurements. When comparing the signal obtained in the two different detection

geometries we observed the same spectral features for detection perpendicular to the

sample surface and for detection at the cleaved end of the waveguide, albeit with

a different intensity depending mainly on the coupling strength between quantum

dot and waveguide mode. Furthermore, both detection geometries show very similar

power dependences and lifetimes. For a different single quantum dot we estimate a β-

factor of β > 85% from intensity and lifetime measurements, and demonstrate single

photon emission with g(2)(0) = 0.27, making these structures prospective candidates

for on-chip quantum-optics, -communication and -information experiments.

145

New ideas pass through three periods:

• It can’t be done.

• It probably can be done, but it’s not worth doing.

• I knew it was a good idea all along!

Arthur C. Clarke

Chapter 8Outlook: On-Chip Quantum Optics

In the last years a number of experimental and theoretical developments have led to-

wards advances in creating the basic components of quantum information processing

devices. One essential element for photonic based implementations of quantum in-

formation processing is a source of single indistinguishable photons. Applications for

such a photon source include quantum teleportation, [Ben93, Vai94, Bou97, Bos98,

Mar03, Joh08b, Dyn09] quantum computation with linear optics, [Kni01] and sev-

eral schemes for quantum cryptography. [Nie00, Kim08] In experiments, such pho-

ton sources have been realized and demonstrated in a variety of systems including

semiconductor quantum dots, whose efficiency and quantum indistinguishability can

be drastically improved in combination with a microcavity. [Ben00, Ste06, Ako06,

Joh08b, Dou10] A second major component is a device for efficiently transferring “in-

formation” between spatially separated nodes of a quantum network. [Kim08] Such a

network would enable to store and manipulate information in quantum dots while the

information is transfered between nodes via photons which interact coherently with

the quantum memory. [Yao05, Kim08]. In sects. 8.1 and 8.2 we present two basic

components for quantum information processing which are within reach in the next

few years.

147

Outlook: On-Chip Quantum Optics

8.1. Coherent Coupling of Quantum Dot - Microcavity

Systems via Waveguides

In quantum information processing scaling is a major challenge. Preparation and

manipulation of a single quantum bit (qubit) has already been demonstrated for

excitons and spins in quantum dots, [Zre02, Ata06, Kim10, Pre10, dlG10] however

for applications the number of qubits must be increased. A promising route towards

the coherent coupling of several qubits is the realization by means of quantum dot -

microcavity systems which are coupled via photonic crystal waveguides. This would

allow the integration of all components for quantum information processing on-chip

and would mean a big step forward towards quantum information processing. A

schematic of a possible device for the coupling of 2 quantum dot - cavity systems is

displayed in figure 8.1. It consists of two electrically tunable quantum dot - photonic

V1

V2

Waveguide

Electrically TunableSingle Quantum DotMicrocavities

QuantumDot 1

QuantumDot 2

Figure 8.1.: Schematic of electrically tunable single QD L3 cavity systems coherentlycoupled on-chip via a W1 waveguide. Adapted from ref. [Hau10].

crystal L3 cavity systems [Rei08, Hof08, Kan10, Far10a] which are coupled via a

photonic crystal W1 waveguide (see sects. 2.2). Quantum dot 1 and quantum dot 2

(see fig. 8.1) can be independently tuned to emit at the same energy and, furthermore,

to be in resonance with their surrounding cavities. When both quantum dots are

148

8.2 Creation and Investigation of Entangled Photons On-Chip

strongly coupled to the cavities, it should be possible to coherently exchange photons

between the two quantum dots via the waveguide. [Eng07b] This would provide a

fundamental and potentially scalable method to exchange quantum “information”

between the quantum dots via photons.

8.2. Creation and Investigation of Entangled Photons

On-Chip

Quantum dots and photonic components as photonic crystal cavities and waveguides

can also be used to create and investigate indistinguishable photon pairs on-chip.

Recently, it has been shown that photons from two separate quantum dots can emit

indistinguishable pairs of photons when they are tuned into resonance. [Ben09, Fla10,

Pat10a] In order to verify if the photons are indistinguishable, a Hong-Ou-Mandel

interferometer [Hon87] is employed which is best implemented on-chip [Mat09] to

avoid photon loss due to out-coupling of the photons from the chip. In figure 8.2 we

present the schematic of an on-chip Hong-Ou-Mandel interferometer. Two electrically

V1

V2

WaveguideElectrically TunableSingle Quantum DotMicrocavities

QuantumDots

GND

50:50 Beamsplitter

Hong-Ou-Mandel Interferometer

Figure 8.2.: Schematic of an on-chip Hong-Ou-Mandel interferometer. Adapted fromref. [Hau10].

tunable quantum dot - cavity systems emit photons into two waveguides which guide

149


the photons to different sides of a 50:50 beamsplitter where the photons will interfere.

If both quantum dot - cavity systems emit photons of the same properties (e.g. energy)

into the waveguide, the two photons are indistinguishable and will always exit the

beam splitter together at the same side. [Hon87, Ben09] Furthermore, such a device

can also be used to create entangled pairs of photons when a post-selection mechanism

is applied. [San02, Fat04]

8.3. Recommendations for the Future

By introducing two sample designs suitable for the coherent exchange of information

and the on-chip creation of indistinguishable photons in the last two sections, we

have pointed out the goals and directions for the, hopefully, not too distant future.

However, since these devices will require further developments in terms of fabrication

as well as measurement techniques, we conclude this section by presenting, more

specific ideas for the next steps in the near future.

- Photonic crystal fabrication: Due to several changes in the cleanroom equip-

ment, the recipe for the fabrication of photonic crystal structures needs to be

optimized again. This is crucial for the fabrication of photonic crystals with

high quality in a reproducible manner. The major optimizations necessary are

the determination of the exact dose factors and the correction of the proxim-

ity effect during the electron beam lithography, and the determination of the

exact parameters for the reactive ion etching. [Kre05, Hof08, Eng08] Further-

more, the exploration of different cavity designs such as photonic heterostruc-

tures, [Son05, Asa06] nanobeam resonators [For97, Deo09] and waveguide cou-

pled cavities [Hue07, Eng07b, Dor08a] may provide access to new ultra high-Q

regimes of cavity quantum electrodynamics, whilst simultaneously being com-

patible with requirements for on-chip quantum optics experiments.

- Quantum dot material : High quality quantum dots with a distinctive electronic

structure, narrow transition linewidths and large dipole matrix elements will be

essential to the implementation and interpretation of all experiments. While

atomic force microscope images of some of the older quantum dot structures

show non ideal, elongated quantum dot shapes, with the resulting strong mix-

ing of exciton states via the electron hole exchange interaction. [Bay00] More

150

8.3 Recommendations for the Future

recently grown wafer material, using a lower growth temperature (540 C cf.

590 C) and wafer rotation during growth, seems to indicate that the quantum

dots have a round shape. First characterizations of these quantum dots also

indicate better optical properties, however detailed characterizations, also in

connection with photonic structures, will be required to reveal their full poten-

tial.

- Non-resonant coupling : While the non-resonant coupling investigated in chap-

ter 5 might be helpful for some applications, for the continuation of this project

it is unwanted since it leads to an incoherent, Poissonian background that blurs

the clear signatures of non-classical light emission. While the prevention of

phonon-mediated processes is not straight-forward, it might be of assistance to

use resonant excitation schemes to get rid of the multi-exciton related mode

feeding. [Pre07, Fla09] Here, we suggest that resonant florescence experiments

could be performed to cleanly excite the system. Such resonant measurements

could be performed, for example, by exciting the system via a waveguide whilst

detecting the light along the direction normal to the waveguide slab or vice

versa. [Mul07]

- Cavity quantum electrodynamics experiments : For the implementation of cav-

ity quantum electrodynamics experiments with semiconductor quantum dots in

photonic crystal cavities, all of the points mentioned above will be helpful. A

better fabrication leads to higher Q-factors, better quantum dots to less intrin-

sic dephasing and higher coupling constants, and resonant excitation schemes to

less charge fluctuations in the vicinity of the quantum and coherent excitation

of the system. This will increase the quality of the quantum dot-cavity system

and allows to unravel non-linear features of the Jaynes–Cummings ladder which

will otherwise be lost due to the strong dephasing. [dVa09]

- Photonic crystal waveguides : A disadvantage of the waveguide structures pre-

sented and investigated in chapter 7 is the fact that the quantum dots are in

resonance with a higher order waveguide mode (WM2 or WM3) and not with

the fundamental mode (WM1). The properties of the fundamental mode are

better in a way that the slow light regime with stronger light-matter interaction

is significantly more pronounced. Furthermore, there are no other waveguide

modes in the leaky region of the k-space at the same energy. Hence, the WM1

151


mode would be expected to exhibit much better coupling to the quantum dots

and weaker scattering related losses. [Kur05]

152

In this age people are experiencing a delight, the

tremendous delight that you can guess how nature

will work in a new situation never seen before.Richard Feynman

Appendix AStrong Coupling Data

This appendix chapter presents data of the strongly coupled quantum dot - cavity

systems introduced in chapter 6. We show the experimental data and the fits obtained

from theory for the power-dependent and temperature-dependent investigations per-

formed in section 6.2.

153

Strong Coupling Data

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

Figure A.1.: Experimental data (blue circles) and fits (black lines) for P = 6 W/cm2

and T = 18 K. The parameters extracted from the fits are: ~γc = 68 µeV,~Pc = 0.78 µeV, ~γφx = 18.2 µeV, ~Γx = 0.2 µeV, ~Px = 0.1 µeV, and~g = 59 µeV.

154

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)



155


1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)



156

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)



157


1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

Figure A.5.: Experimental data (blue circles) and fits (black lines) for P =100 W/cm2 and T = 18 K. The parameters extracted from the fitsare: ~γc = 68 µeV, ~Pc = 8.5 µeV, ~γφx = 41.8 µeV, ~Γx = 0.2 µeV,~Px = 1.0 µeV, and ~g = 59 µeV.

158

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

Figure A.6.: Experimental data (blue circles) and fits (black lines) for P =200 W/cm2 and T = 18 K. The parameters extracted from the fits are:~γc = 68 µeV, ~Pc = 13.2 µeV, ~γφx = 66.8 µeV, ~Γx = 0.2 µeV,~Px = 1.6 µeV, and ~g = 59 µeV.

159


1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)

Figure A.7.: Experimental data (blue circles) and fits (black lines) for P =400 W/cm2 and T = 18 K. The parameters extracted from the fits are:~γc = 68 µeV, ~Pc = 16.4 µeV, ~γφx = 71.8 µeV, ~Γx = 0.2 µeV,~Px = 4.4 µeV, and ~g = 59 µeV.

160

1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


T = 1 7 . 5 K

1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


Figure A.8.: Experimental data (blue circles) and fits (black lines) for T = 17.5 Kand P = 10 W/cm2. The parameters extracted from the fits are: ~γc =68 µeV, ~Pc = 11.6 µeV, ~γφx = 10.7 µeV, ~Γx = 0.2 µeV, ~Px = 2.8 µeV,and ~g = 59 µeV.

161


1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


T = 2 3 . 2 K

1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


Figure A.9.: Experimental data (blue circles) and fits (black lines) for T = 23.2 Kand P = 10 W/cm2. The parameters extracted from the fits are: ~γc =68 µeV, ~Pc = 11.4 µeV, ~γφx = 16.3 µeV, ~Γx = 0.2 µeV, ~Px = 3.6 µeV,and ~g = 59 µeV.

162

1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


T = 2 7 . 3 K

1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


Figure A.10.: Experimental data (blue circles) and fits (black lines) for T = 27.3 Kand P = 10 W/cm2. The parameters extracted from the fits are: ~γc =68 µeV, ~Pc = 13.0 µeV, ~γφx = 18.5 µeV, ~Γx = 0.2 µeV, ~Px =3.7 µeV, and ~g = 59 µeV.

163


1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)


T = 3 2 . 9 K

1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6 1 2 0 7 . 8

PL In

tensit

y (arb

. unit

s)



164

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)


T = 3 8 . 8 K

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)



165


1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)


T = 4 3 . 2 K

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4 1 2 0 7 . 6

PL In

tensit

y (arb

. unit

s)



166

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4

PL In

tensit

y (arb

. unit

s)


T = 4 8 . 1 K

1 2 0 6 . 6 1 2 0 6 . 8 1 2 0 7 . 0 1 2 0 7 . 2 1 2 0 7 . 4

PL In

tensit

y (arb

. unit

s)



167

Truth and clarity are complementary.

Niels Bohr

Appendix BMutual Coupling of two Quantum Dots - Alternative

Interpretations

In this section we investigate the possibility of ambiguity in the interpretation of our

data from sect. 6.3. We investigate the possibility that the observed spectral signature

of the double anticrossing could be produced by (i) two different single exciton tran-

sitions of the same quantum dot, and (ii) by two different quantum dots, one weakly

and one strongly coupled to the cavity. In fig. B.1, we compare the experimental data

with calculations obtained for these two different, alternative models.

In fig. B.1(a) we present spectra obtained from fitting the experimental data (blue

circles) using the same fitting procedure as in sect. 6.3.1, but assuming that the states

QD1 and QD2 cannot coexist at the same time (red solid lines). This would be the case

for two different states of the same quantum dot, e.g. exciton and charged exciton.

While the overall agreement between the calculated spectra and the experimental data

is good, 1 we find clear derivations between the experimental data and the theory close

to resonance. In this situation the resulting spectral function does not exhibit a triple

peak, rather it is the sum of the spectra of two independent quantum systems (as

described in sects. 6.1.2 and 6.2) S(ω) = (S1(ω) + S2(ω)) and, therefore, shows only

a double peak close to resonance. We mark the significant difference between the

1When the two quantum dots and the cavity mode are detuned from each other, the coupling is veryweak. In this case, both models, the genuine model from sect. 6.3.1 and the model investigatedhere, will be equivalent.

168

0 . 30

0 . 35

0 . 40

0 . 45

0 . 50

0 . 55

0 . 60

1 2 1

7 . 21 2 1

7 . 41 2 1

7 . 61 2 1

7 . 81 2 1

8 . 0

!

"

0 . 30

0 . 35

0 . 40

0 . 45

0 . 50

0 . 55

0 . 60

1 2 1 7 . 0

1 2 1 7 . 5

1 2 1 8 . 0

1 2 1 8 . 5

Energ

y (me

V)

1 2 17 . 2

1 2 17 . 4

1 2 17 . 6

1 2 17 . 8

1 2 18 . 0

PL In

tensit

y (arb

. unit

s)

Figure B.1.: Resonant spectra (Vapp = 0.41 − 0.46 V) obtained from fitting of analternative theoretical model to the experimental data (blue circles), as-suming that (a) the two quantum dot states cannot coexist (e.g. neutralexciton and charged exciton of the same quantum dot), and (c) QD2 isonly weakly coupled to the cavity mode with a fixed ~g2 = 20 µeV. (b)and (d) Contour plots (false color) of the calculated spectra of the samedata for a wider range showing the qualitative evolution of the peaksduring the crossing.

169

Mutual Coupling of two Quantum Dots - Alternative Interpretations

experimental data and the calculated spectral functions with the gray shaded area on

the figure. Fig. B.1(b) shows the same calculations in a contour plot presentation.

In this plot, the third peak in resonance is, of course, also missing. This observation

and the difference in the observed Stark shifts (see sect. 6.3.1) excludes this possible

explanation.

In fig. B.1(c) we present calculations obtained, employing the very same model as

in sect. 6.3.1, but assuming that QD2 is only weakly coupled to the cavity mode with

a fixed ~g2 = 20 µeV (green solid lines), and compare it with the experimental data

(blue circles). The quantitative agreement between calculations and experiment is

clearly unsatisfactory. The qualitative agreement can be evaluated from the contour

plot in fig. B.1(d). Since QD2 is only weakly coupled to the cavity mode, no exciton

polaritons form and its spectral position is not affected by the presence of the cavity

mode. It simply crosses the cavity mode on the path dictated by the quantum confined

Stark shift. Furthermore, in resonance with the cavity mode the emission of QD2

is enhanced due to the Purcell effect, which is a clear signature of weak coupling.

However, this situation does not show any similarity to the spectral function observed

in the experiments and plotted in fig. 6.5(a).

Of the three presented models, only the genuine model employed in sect. 6.3 is

able to satisfactorily fit the experimental data and represent all features observed in

the spectral function of the system. We, therefore, conclude that the quantum dot -

cavity system investigated in our experiments, consists of two independent quantum

dots which are both strongly coupled to one common cavity mode.

170

Debugging is twice as hard as writing the code in

the first place. Therefore, if you write the code as

cleverly as possible, you are, by definition, not smart

enough to debug it.

Brian W. Kernighan

Appendix CPython Codes for Quantum Dot - Cavity Systems

C.1. Calculation of the Emission Spectrum of an

N-Quantum Dot - Cavity System

#---------------------------------------------------------------------------------#

# Jose Maria Villas-Boas (WSI - 2009) #

# This program calculates the emission spectrum of one or more QDs in a Cavity#

# Requirements: #

# numpy for matrix operation (www.scipy.org) #

# scipy for interpolation, integral and fitting (www.scipy.org) #

#---------------------------------------------------------------------------------#

# initializing functions for fitting

from scipy.integrate import simps

from scipy.optimize import leastsq

from scipy import interpolate

from numpy.linalg import inv, det, eig

from numpy import *

import time

#-------------------------------------------------------------

#Function to find index of matrix for a value of x (to reduce the matrix to region of interest)

def findi(x,v):

i=0; f0=v[0]-x;

while (v[i]-x)*f0 > 0.0:

i=i+1

return i-1;

#-------------------------------------------------------------

#-------------------------------------------------------------

#Sorting the eigenvalues and eigenvectors (for windows version)

def sorteig(E,Vec):

ne=size(E);

for i in range(ne):

for j in range(i+1,ne):

if(E[i] > E[j]):

Ena=E[i]

171

Python Codes for Quantum Dot - Cavity Systems

E[i]=E[j]

E[j]=Ena

for iv in range(ne):

vau=Vec[iv,i]

Vec[iv,i]=Vec[iv,j]

Vec[iv,j]=vau;

return E,Vec;

#-------------------------------------------------------------

#-------------------------------------------------------------

I=complex(0,1); # just a complex number!

#-------------------------------------------------------------

#-------------------------------------------------------------

# Matrix for the spectral funcion on the basis: 00, 01, 02, ..., 10, 11, 12, ...

def Matrix_M(En,g,G,P,Gp):

y = Gp+(G+P)/2; # total exciton dephasing

y[0] = (G[0]-P[0])/2; # total cavity dephasing

Pe = G+P; # effective pumping

Pe[0] = G[0]-P[0]; # effective pumping of cavity

#---------------------------------------------------------

n=size(g); n2=n*n;

M = zeros((n2,n2),dtype=complex);

im=-1;

for i in range(n):

for j in range(n):

im+=1; jm=-1;

for k in range(n):

for l in range(n):

jm+=1;

if i==k:

if j == 0 and l != 0: M[im,jm]=g[l];

if l == 0 and j != 0: M[im,jm]=g[j];

if j==l:

if i == 0 and k != 0: M[im,jm]=-g[k];

if k == 0 and i != 0: M[im,jm]=-g[i];

if im == jm and k != l: M[im,jm]+=-(En[k]-En[l])-I*(y[k]+y[l]);

if im == jm and k == l: M[im,jm]+=-I*(Pe[k]);

return -I*M;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Matrix for the spectral funcion on the basis: <a>, <S0>, <S1>, ...

def Matrix_H(En,g,G,P,Gp):

y = Gp+(G+P)/2; # total exciton dephasing

y[0] = (G[0]-P[0])/2; # total cavity dephasing

#---------------------------------------------------------

n=size(g);

H = zeros((n,n),dtype=complex);

for i in range(n):

for j in range(n):

if i==j: H[i,j]=En[i]-I*y[i];

if i==0 and j != 0: H[i,j]=g[j];

if j==0 and i != 0: H[i,j]=g[i];

return -I*H;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Steady state of Matrix_M for using to calculate the spectral function

def steady(En,g,G,P,Gp): # Numerical from solving a system of linear equ.

n=size(En);

#---------------------------------------------------------

M = Matrix_M(En,g,G,P,Gp);

X = zeros(n*n,dtype=complex);

ix=0;

for i in range(n):

X[ix]=-P[i];

172

C.1 Calculation of the Emission Spectrum of an N-Quantum Dot -Cavity System

ix+=n+1;

ns=linalg.solve(M,X)

return ns;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Spectral Function for the cavity and dots

# Numerical solution

def spectral_numerical(w,En,g,G,P,Gp,E_res,s=0):

H = Matrix_H(En,g,G,P,Gp); n=size(En); n2=n*n;

ns=steady(En,g,G,P,Gp);

Xs = zeros(n,dtype=complex);

for i in range(n):

Xs[i]=ns[i+s*n];

L,S =eig(H);

Sm=inv(S);

Xa=dot(Sm,Xs);

Xb=0.0;

for i in range(n):

Xi=-Xa[i]/(I*w-E_res+L[i]);

Xb=Xb+S[s,i]*Xi

Sw=real(Xb)/(Xs[s]*pi);

sw=real(Sw)

return sw;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Change only after this line

#-------------------------------------------------------------

# cavity dot1 dot2 dot3 Parameters

g = r_[ 0.0, 0.044,0.051]; #coupling strength g

Gp= r_[ 0.0, 0.020, 0.0098]; #Gamma_puredephasing

G = r_[ 0.1, 0.0001, 0.0008]; #Gamma

P = r_[0.0057, 0.0015, 0.0019]; #Pumping

Ei= r_[1217.6, 1217.0, 1217.25]; #Initial energy in the plot

Ef= r_[1217.6, 1218.5, 1218.3]; #Final energy in the plot

Norm=1.0;

#-------------------------------------------------------------

E_res=0.018; # resolution of the experimental results (HWHM)

wi = 1216.6; wf=1218.6; nw=501; # region of interest (emission frequency) and number of points for the fit

wv = linspace(wi,wf,nw); # create a vector with the frequencies.

#-------------------------------------------------------------

n=size(g); nf=nw;

En0 = zeros((nf,n),’d’);

for i in range(n):

En0[:,i] = linspace(Ei[i], Ef[i],nf); # Cavity mode spectrum

V = linspace(-1,1,nw);

#-------------------------------------------------------------

Env = zeros((nf,n+1),’d’); Env[:,0]=V;

Vec = zeros((nf,n+1,n),’d’);

Sw = zeros((nw,nf+1,n),’d’); Sw[:,0,0]=wv;

for s in range(n):

Vec[:,0,s]=V;

Sw[:,0,s]=wv;

for i in range(nf):

En=0.0

for s in range(n):

En = r_[En,En0[i,s]]

En=delete(En,[0])

H=I*Matrix_H(En,g,G,P,Gp);

L,S = eig(H); L,S = sorteig(L,S)

Env[i,1:]=real(L);

for s in range(n):

Sw[:,i+1,s] = G[s]*Norm*spectral_numerical(wv,En,g,G,P,Gp,E_res,s=s)

Vec[i,1:,s] = real(S[:,s]*conj(S[:,s]));

#-------------------------------------------------------------

savetxt(’Eigenvalues.dat’,Env, fmt="%.5f"); #Save Eigenvalues of the Eigenstates

173


for s in range(n):

savetxt(’Vec_’+str(s)+’.dat’,Vec[:,:,s], fmt="%.5f"); #Save Eigenvectors of the N Eigenstates

savetxt(’Theory_’+str(s)+’.dat’,Sw[:,:,s], fmt="%.5f"); #Save Emission Spectra of the N Eigenstates

# ==================================================================================================

C.2. Fitting of the Emission Spectrum of an

N-Quantum Dot - Cavity System

#---------------------------------------------------------------------------------#

# Jose Maria Villas-Boas (WSI - 2009) #

# This program fits the emission spectrum of one or more QDs in a Cavity #

# Requirements: #

# numpy for matrix operation (www.scipy.org) #

# scipy for interpolation, integral and fitting (www.scipy.org) #

#---------------------------------------------------------------------------------#

# initializing functions for fitting

from scipy.integrate import simps

from scipy.optimize import leastsq

from scipy import interpolate

import string

#-------------------------------------------------------------

# Function to find index of matrix for a value of x (to reduce the matrix to region of interest)

def findi(x,v):

i=0; f0=v[0]-x;

while (v[i]-x)*f0 > 0.0:

i=i+1

return i-1;

#-------------------------------------------------------------

#---------------------------------------

def readArray(filename, skipchar = ’#’):

"""

PURPOSE: read an array from a file. Skip empty lines or lines

starting with the comment delimiter (defaulted to ’#’).

OUTPUT: a float numpy array

EXAMPLE: >>> data = readArray("test.dat")

>>> x = data[:,0] # first column

>>> y = data[:,1] # second column

"""

myfile = open(filename, "r")

contents = myfile.readlines()

myfile.close()

data = []

for line in contents:

stripped_line = string.lstrip(line)

if (len(stripped_line) != 0):

if (stripped_line[0] != skipchar):

items = string.split(stripped_line)

data.append(map(float, items))

a = array(data)

(Nrow,Ncol) = a.shape

if ((Nrow == 1) or (Ncol == 1)): a = ravel(a)

return(a)

#---------------------------------------

#-------------------------------------------------------------

I=complex(0,1); # just a complex number!

174

C.2 Fitting of the Emission Spectrum of an N-Quantum Dot - CavitySystem

#-------------------------------------------------------------

#-------------------------------------------------------------

# Matrix for the spectral funcion

# (not used, only for numerical solution for checking)

def Dmatrix3(wc,wd,g,Gd,Gc,Pd,Pc,Gp):

yd = Gp+(Pd+Gd)/2; # total exciton dephasing

yc = (Gc-Pc)/2; # total cavity dephasing

#---------------------------------------------------------

n2=3;

Tr = zeros((n2,n2),dtype=complex);

Tr[0,0]=-I*wc-yc; Tr[0,1]=-I*g[0]; Tr[0,2]=-I*g[1]

Tr[1,0]=-I*g[0]; Tr[1,1]=-I*wd[0]-yd[0]; Tr[1,2]=0.0

Tr[2,0]=-I*g[1]; Tr[2,1]=0.0; Tr[2,2]=-I*wd[1]-yd[1]

return Tr;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Matrix for the averages operators <a*a>,<SpSm>,<a*Sm>,<Spa>

# used for numerical solution of initial condition

def Dmatrix4(wc,wd,g,Gd,Gc,Pd,Pc,Gp):

n=9; D=wd-wc; D10=wd[0]-wd[1]



Tr = zeros((n,n),dtype=complex);

Tr[0,0]=-Gc+Pc; Tr[0,1]=0.0; Tr[0,2]=-I*g[0]; Tr[0,3]= I*g[0]; Tr[0,5]=-I*g[1]; Tr[0,6]= I*g[1];

Tr[1,0]=0.0; Tr[1,1]=-Gd[0]-Pd[0]; Tr[1,2]= I*g[0]; Tr[1,3]=-I*g[0];

Tr[2,0]=-I*g[0]; Tr[2,1]= I*g[0]; Tr[2,2]=-I*D[0]-yc-yd[0]; Tr[2,3]=0.0; Tr[2,8]= I*g[1];

Tr[3,0]= I*g[0]; Tr[3,1]=-I*g[0]; Tr[3,2]=0.0; Tr[3,3]=I*D[0]-yc-yd[0]; Tr[3,7]=-I*g[1];

Tr[4,0]=0.0; Tr[4,4]=-Gd[1]-Pd[1]; Tr[4,5]= I*g[1]; Tr[4,6]=-I*g[1];

Tr[5,0]=-I*g[1]; Tr[5,4]= I*g[1]; Tr[5,5]=-I*D[1]-yc-yd[1]; Tr[5,6]=0.0; Tr[5,7]= I*g[0];

Tr[6,0]= I*g[1]; Tr[6,4]=-I*g[1]; Tr[6,5]=0.0; Tr[6,6]=I*D[1]-yc-yd[1]; Tr[6,8]=-I*g[0];

Tr[7,3]=-I*g[1]; Tr[7,5]= I*g[0]; Tr[7,7]=I*D10-yd[0]-yd[1]; Tr[7,8]=0.0;

Tr[8,2]= I*g[1]; Tr[8,6]=-I*g[0]; Tr[8,7]=0.0; Tr[8,8]=-I*D10-yd[0]-yd[1];

return Tr;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Steady state of Dmatrix4 for using to calculate the spectral function

def steady(wc,wd,g,Gd,Gc,Pd,Pc,Gp): # Numerical from solving a system of linear equ.

#---------------------------------------------------------

X = zeros(9,dtype=complex)

X[0]=-Pc; X[1]=-Pd[0]; X[4]=-Pd[1]

Tr= Dmatrix4(wc,wd,g,Gd,Gc,Pd,Pc,Gp)

ns=linalg.solve(Tr,X)

return ns;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Function to compute the spectral fnction numerically

# (Only used for 2 or more QDs or to check analytical solution)

def spec(Tr, w, X, Gres): ## dX/dt=Tr*X X=exp(Tr)*X(0) or ...

n=size(Tr[:,0]);

L,S =eig(Tr);

Sm=inv(S);

Xa=dot(Sm,X);

Xb=0.0;

for i in range(n):

Xi=-Xa[i]/(I*w-Gres+L[i]);

Xb=Xb+S[0,i]*Xi

Sw=real(Xb)/(X[0]*pi);

return Sw;

#---------------------------------------------------------------------------

175


#-------------------------------------------------------------

# Spectral Function for the cavity

# Numerical solution

def spectralc_numerical(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res):

Tr=Dmatrix3(wc,wd,g,Gd,Gc,Pd,Pc,Gp);

ns=steady(wc,wd,g,Gd,Gc,Pd,Pc,Gp)

Xs = zeros(3,dtype=complex); Xs[0]=ns[0]; Xs[1]=ns[2]; Xs[2]=ns[5];

sw=real(spec(Tr, w, Xs, E_res))

return sw;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Spectral Function for the cavity

# Analytical solution

def spectral_analitical(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res):



yt = yd + yc;

Ec=wc-I*yc;

Ed=wd-I*yd;

d=(Ed-Ec)/2;

Em=(Ed+Ec)/2;

R=sqrt(g**2+d**2);

#---------------------------------------------------------

Dw=wd-wc; # <a*a>,<SpSm>,<a*Sm>,<Spa>

Nsden=2*yt*g**2*(Gc-Pc+Gd+Pd)+(Gc-Pc)*(Gd+Pd)*(Dw**2+yt**2)

ApAm=(2*yt*g**2*(Pc+Pd)+Pc*(Gd+Pd)*(Dw**2+yt**2))/Nsden

# SpSm=(2*yt*g**2*(Pc+Pd)+Pd*(Gc-Pc)*(Dw**2+yt**2))/Nsden

ApSm=g*(Gc*Pd-Gd*Pc-2*Pc*Pd)*(Dw+I*yt)/Nsden

# SpAm=g*(Gc*Pd-Gd*Pc-2*Pc*Pd)*(Dw-I*yt)/Nsden

#---------------------------------------------------------

A=((R+d)*ApAm-g*ApSm)/(2*R)

B=((R-d)*ApAm+g*ApSm)/(2*R)

sf=real(-I*A/(Em-R-w-I*E_res)-I*B/(Em+R-w-I*E_res))/(ApAm*pi) #

return sf;

#-------------------------------------------------------------

#-------------------------------------------------------------

def spectralc_analitical2(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res):

sw=0.5*(spectral_analitical(w,wc,wd[0],g[0],Gd[0],Gc,Pd[0],Pc,Gp[0],E_res)+

spectral_analitical(w,wc,wd[1],g[1],Gd[1],Gc,Pd[1],Pc,Gp[1],E_res))

return sw;

#-------------------------------------------------------------

#-------------------------------------------------------------

def spectralc(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res):

# sw=spectralc_analitical2(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res);

sw=spectralc_numerical(w,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res);

return sw;

#-------------------------------------------------------------

#-------------------------------------------------------------

def residuals(p,EMat,E_res,Par,G_Par,L_Par):

if any(p<0.0001): # or (p[2]>0.0004): # forcing parameters to be positive for physical reasons!

return 10.0;

err = 0.0; # inicialize the erreituwy

N_EF=size(EMat[0,:])-1;

jp=0;

for pa in G_Par:

Par[pa]=p[jp];

jp=jp+1;

for i in range(N_EF):

for pa in L_Par:

Par[pa]=p[jp];

jp=jp+1;

wc = Par[’wc’]; Pc = Par[’Pc’]; Gc = Par[’Gc’];

wd = r_[Par[’wd0’],Par[’wd1’]];

Pd = r_[Par[’Pd0’],Par[’Pd1’]]; # Exciton pumping rate

176

C.2 Fitting of the Emission Spectrum of an N-Quantum Dot - CavitySystem

Gd = r_[Par[’Gd0’],Par[’Gd1’]]; # Exciton decay rate

Gp = r_[Par[’Gp0’],Par[’Gp1’]]; # Exciton pure dephasing

g = r_[Par[’g0’],Par[’g1’]]; # Exciton-cavity coupling

Swt=Par[’Norm’]*spectralc(EMat[:,0],wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res)

err = r_[err,Swt-EMat[:,i+1]];

err=delete(err,[0]); # delete first value! 0

return err

#-------------------------------------------------------------

#-------------------------------------------------------------

# Function to initialize the initial guess for fittings according to what we decide to vary

def initial_guess(f_peaks,G_Par,L_Par,Par0):

#-------------------------------------------------------------

Par=Par0.copy();

a2 = readArray(f_peaks); N_EF=size(a2[:,0]);

#-------------------------------------------------------------

p0=0.0;

for pa in G_Par:

p0=r_[p0,Par[pa]];


Par[’wc’]=a2[i,0]; Par[’wd0’]=a2[i,1]; Par[’wd1’]=a2[i,2];

for pa in L_Par:

p0=r_[p0,Par[pa]];

p0=delete(p0,[0])

return p0;

#-------------------------------------------------------------

#-------------------------------------------------------------

# Function to fit the experimental data!

# Read the file, select the region of interest give by the vector wv

def fit_exp(f_data,wv,E_res,p_guess,Par0,G_Par,L_Par,interpolation=False):

#-------------------------------------------------------------

a1 = readArray(f_data); # Experimental results

nw = size(wv);

i1 = findi(wv[0],a1[:,0]); i2 = findi(wv[nw-1],a1[:,0]);

EMat = a1[i1:i2:sign(i2-i1),:] # Emission

N_EF=size(EMat[0,:])-1;


ymin=min(EMat[:,i+1]);

EMat[:,i+1]=(EMat[:,i+1]-ymin);

ymax=EMat[:,1::].max(); # Nonalization of individual curves

EMat[:,1::]=EMat[:,1::]/ymax;

#-------------------------------------------------------------

if (interpolation): # decide if interpolate exp. date (might give better result, but it is time consuming)

EMat_aux=EMat;

EMat = zeros((nw,N_EF+1),’d’); EMat[:,0]=wv


tck = interpolate.splrep(EMat_aux[:,0],EMat_aux[:,i+1])

EMat[:,i+1] = interpolate.splev(wv,tck)

#-------------------------------------------------------------

Par=Par0.copy();

# Do the fittings and return the best parameters that fits the data

pbest = leastsq(residuals,p_guess,args=(EMat,E_res,Par,G_Par,L_Par),ftol=1.4e-12,xtol=1.4e-12);

p_best=pbest[0];

#-------------------------------------------------------------

npar = 14;

TMat = zeros((nw,N_EF+1),’d’); TMat[:,0]=wv;

bpar = zeros((N_EF,npar),’d’);

#-------------------------------------------------------------

jp=0;

for pa in G_Par:

Par[pa]=p_best[jp];

jp=jp+1;


for pa in L_Par:

Par[pa]=p_best[jp];

jp=jp+1;

177








TMat[:,i+1]=Par[’Norm’]*spectralc(wv,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res)

bpar[i,:]=r_[Par[’wc’],Par[’wd0’],Par[’wd1’],Par[’Gc’],Par[’Pc’],Par[’Gp0’],Par[’Gd0’],Par[’Pd0’],Par[’g0’],

Par[’Gp1’],Par[’Gd1’],Par[’Pd1’],Par[’g1’],Par[’Norm’]]

#-------------------------------------------------------------

return bpar,TMat,EMat; # returns the best parameters with the experimental data and fitted data

#-------------------------------------------------------------

#-------------------------------------------------------------

E_res=0.018; # resolution of the experimental results

wi = 1216.5; wf=1219.0; nw=501; # region of interest (emission frequency) and number of points for the fit

wv = linspace(wi,wf,nw); # create a vector with the frequencies.

#-------------------------------------------------------------

# Initial guess for the rates and parameters

Par0=’wc’:0.0,’wd0’:0.0,’wd1’:0.0,’Pc’:0.00554,’Gc’:0.14079,’Pd0’:0.00299,’Gd0’:0.00035,’Gp0’:0.026,’g0’:0.050,

’Pd1’:0.00292,’Gd1’:0.00035,’Gp1’:0.016,’g1’:0.044,’Norm’:0.3

# Defining which parameters vary locally or globally!

G_Par=(’Pc’,’Gc’,’Gp1’,’Gp0’,’Gd0’,’Gd1’,’Pd0’,’Pd1’,’g0’,’g1’);

L_Par=(’wc’,’wd0’,’wd1’,’Norm’);

#-------------------------------------------------------------

Par=Par0.copy();







Swt=spectralc(wv,wc,wd,g,Gd,Gc,Pd,Pc,Gp,E_res)

#-------------------------------------------------------------

data = ’PLV32new.DAT’ # File with spectra. 1st column: energy, other columns:spectra

f_peaks = ’PLV32newpeakpos2.DAT’ # File with estimates of peak positions. n-th column: E_n

#-------------------------------------------------------------

p_guess=initial_guess(f_peaks,G_Par,L_Par,Par0)

bpar,TMat,EMat = fit_exp(f_data,wv,E_res,p_guess,Par0,G_Par,L_Par,interpolation=False) # fitting the data

#-------------------------------------------------------------

savetxt(’Parameters’+’.dat’,bpar, fmt="%.5f")

savetxt(’Theory’+’.dat’,TMat, fmt="%.5f")

savetxt(’Exp’+’.dat’,EMat, fmt="%.5f")

#-------------------------------------------------------------

178

Science has ’explained’ nothing; the more we know

the more fantastic the world becomes and the

profounder the surrounding darkness.

Aldous Huxley

Appendix DCavity versus Dot Emission in Strongly Coupled

Quantum Dots–Cavity Systems

In this chapter we discuss the spectral lineshapes ofN quantum dots in strong coupling

with the single mode of a microcavity. Nontrivial features are brought by detuning the

emitters or probing the direct exciton emission spectrum. We describe dark states,

quantum non-linearities, emission dips and interferences and show how these various

effects may coexist, giving rise to highly peculiar lineshapes. The calculations are

based on the theoretical model developed by Jose Maria Villas-Boas and introduced

in sect. 6.1 in the linear regime, and on the theoretical model developed by Fabrice

Laussy and Elena del Valle in refs. [Lau09c] and [dVa09] in the non-linear regime.

D.1. Introduction

The coherent coupling of a semiconductor quantum dot exciton to the optical mode

of a microcavity has been intensely investigated over the last years in cavity quantum

electrodynamics experiments [Yos04, Rei04, Pet05, Rei06, Hen07, Eng07a, Far08a,

Kis08, Win08, Lau09b, Lau09a, Mun09, Rei09, Ota09a, Suf09, Tho09, Nom10, Kas10,

Lau10c] and theory. [And99, Cui06, Lau08, Ino08, Nae08, Auf08, Lau09c, dVa09,

Yam09, Hug09, Auf09, Ave09, Ric09, Ver09c, Ver09a, Tar10, Kae10, Hoh10, Rit10,

Pod10] In some of these works, the experimental spectral function of the strongly

179

Cavity versus Dot Emission in Strongly Coupled Quantum Dots–CavitySystems

coupled quantum dot–cavity system was directly compared to a theoretical model,

and the agreement is excellent (see chapter 6). [Lau08, Lau09a, Mun09, Lau10c] It

was assumed that most of the light escapes the system via the radiation pattern of the

cavity mode, and the experimental spectra were compared to the spectral function

calculated from the cavity occupation. This detection geometry is known in atomic

cavity quantum electrodynamics as “end emission” or “forward emission”. [Yam99] In

atomic systems, a negligible fraction of the light escapes the cavity through the cavity

mode, that is to say, the cavity photon lifetime is so long such as to be considered

infinite. Light is then detected in the so-called “side-emission” geometry, where the

radiation pattern of the emitter is directly probed. With microcavities, the situation

is reversed: the cavity mode is often measured with an emitter that has a much longer

lifetime than the cavity photon lifetime. In the spontaneous emission regime of a sys-

tem operating in the strong-coupling regime of the light-matter coupling, this makes

measurements of the Rabi doublet in the photoluminescence more difficult, unless

some cavity feeding makes the quantum state of the system photon-like. This arises,

since changing the nature of the excitation then becomes equivalent to changing the

channel of detection. [Lau08] In the nonlinear regime, this also hinders manifestations

of the Jaynes–Cummings ladder. All the transitions between its rungs have the same

intensity in the exciton emission. In the cavity emission, however, the photon has

two paths to be emitted, one with the dot in its ground state, the other with the dot

in its excited states. [dVa09] These two paths interfere destructively when the initial

and final states are out of phase, which is the case for two out of the four possible

transitions in the Jaynes–Cummings ladder. On the other hand, these two paths

interfere constructively when the initial and final states are in phase, or, up to a pho-

ton, indistinguishable. In the dressed-state picture, the cavity photon to be emitted

decouples from the polaritons and carries away little information from the coupled

system, being more like a cavity photon as the number of excitations becomes higher.

On the other hand, the dot photon does not decouple from the system, regardless the

number of excitations: the dot cannot de-excite without fundamentally altering the

state of the entire system. As a result, the dot photon carries more information about

the coupled system. Summarizing, the dot is essentially a quantum emitter whereas

the cavity is essentially a classical emitter.

It is, therefore, interesting to develop techniques to directly probe the dot emission

using an experimental geometry that excludes light arising from the cavity. The

180

D.1 Introduction

ratio R of photons emitted by the dot compared with the cavity depends on the

populations of the dot (cavity), n1 (na) and their rate of emission, γ1 (γa):

R =n1γ1

naγa. (D.1)

This ratio is typically small since—apart from the fact that 0 ≤ n1 ≤ 1 whereas na is

unbounded—in typical experiments, γ1 γa. One should, therefore, take advantage

of the detection geometry, to detect light in a solid angle where the cavity does

not emit. In a photonic crystal, one could filter out the areas of most intense cavity

emission by selective collection of the farfield emission. The gain is however negligible,

the cavity emission being reduced by a factor of about 2 as compared to the dot

emission 1 when one needs several order of magnitude increase to compensate R. The

direct dot emission is therefore more easily accessible in a micropillar geometry, [Rei10]

where one can detect on the side of the structure. The feasibility of such experiments

has already been demonstrated. [San06]

In this chapter, we will not focus on the practical question of how to separately

detect the dot and cavity emission and rather focus entirely the differences that are

observed when probing the strong-coupling physics in the cavity and the dot emis-

sion. As the possibility of coupling strongly N > 1 dots to the (one) microcav-

ity mode [Yeo98, dVa07, Ver09b, dVa10b, Pod10] is starting to emerge experimen-

tally, [Rei06, Gal10, Lau10c] we will consider the general case of many emitters si-

multaneously coupling to the cavity field.. We address both the linear and non-linear

regimes, with incoherent and continuous pumping as the scheme of excitation, which

is a favoured way of probing the system experimentally. The effects we will discuss

are not limited to the case of quantum dots in a microcavity. Systems with many

identical quantum emitters are equally well described by our formalism. Examples

include atomic ensembles coupled to a high finesse Fabry Perot cavity, [Tho92, Wal06]

superconducting qubits coupled to a microwave resonator [Wal04, Fin08, Fin09, Fil10]

or an ensemble of colour centers in diamond. [Par06, San10, Eng10]

1We estimated the ratio of cavity to dot emission by comparing the farfield emission pattern of a L3cavity mode obtained from FDTD simulations with the emission of an isotropic emitter. Fromassuming collection for different numerical apertures or blocking of these numerical apertures wecould estimate a maximum inhibition of the cavity emission compared to the dot emission of afactor ∼ 2.5.

181


D.2. Model

The Hamiltonian for N independent excitons (in different quantum dots) coupled to

a common cavity mode reads

H =N∑j=1

[ωjσ

†jσj + gj(a

†σj + σ†ja)]

+ ωaa†a, (D.2)

where σ†j , σj are the pseudospin operators for the excitonic two level systems consisting

of ground state |0〉 and a single exciton |Xj〉 state of the jth–quantum dot. ωj is the

exciton frequency, a† and a are the creation and destruction operators of photons

in the cavity mode with frequency ωa, and gj describes the strength of the dipole

coupling between cavity mode and exciton of the jth–quantum dot. The incoherent

loss and gain (pumping) of the dot-cavity system is included in a master equation of

the Lindblad form dρdt

= −i[H,ρ] + L(ρ), where:

L(ρ) =N∑j=1

[γj2

(2σjρσ†j − σ†jσjρ− ρσ†jσj)

+Pj2

(2σ†jρσj − σjσ†jρ− ρσjσ†j)]

+γa2

(2aρa† − a†aρ− ρa†a)

+Pa2

(2a†ρa− aa†ρ− ρaa†) . (D.3)

Here, γj is the jth exciton decay rate, Pj is the rate at which excitons are created

by a continuous wave pump laser in the jth quantum dot, γa is the cavity loss and

Pa is the incoherent pumping of the cavity. Pumping of the cavity from non-resonant

quantum dots was investigated in detail in chapter 5 and is a phenomena observed by

many different groups. [Pre07, Hen07, Kan08b, Win09, Suf09, Hoh09, Cha09, Ota09a,

Lau10b] It has been shown how the effective quantum state realized in the system

under the interplay of the two types of pumping (cavity and exciton) imparts a strong

influence on the lineshape, [Lau08, Lau09c, dVa09] which, in the linear regime, is

a counterpart of the different channels of emission. As shown in sect. 6.2, a pure

dephasing rate of the exciton in the jth–quantum dot could be included to account

for effects originating from high excitation powers or high temperatures. [Lau09a]

However, such effects have a tendency to merge together the closely spaced peaks

182

D.3 One Emitter

arising from higher rungs of the Jaynes–Cummings ladder, [Gon10] and we neglect it

here for the sake of simplicity.

Assuming that the system achieves a steady state for long times and employing the

Wiener-Khintchine theorem, we calculate the spectral function of our quantum dot-

exciton systems [Scu97] when photon emission occurs via the normalized radiation

pattern of the cavity

Sa(ω) =1

naπlimt→∞

Re

∫ ∞0

dτe−(Γr−iω)τ 〈a†(t)a(t+ τ)〉, (D.4)

and via the normalized radiation pattern of the jth quantum dot:

Sj(ω) =1

njπlimt→∞

Re

∫ ∞0

dτe−(Γr−iω)τ 〈σ†j(t)σj(t+ τ)〉. (D.5)

We included in the expression the term Γr that takes into account the finite spectral

resolution of a monochromator, [Ebe77] which is typically ≈ 18 µeV (half-width) for

a good monochromator by today standards. The qualitative effect of this term is to

broaden the peaks and blur the features. Therefore, in the following we shall assume

it is zero (case of a perfect detector).

Using the quantum regression theorem, [Car89] the emission eigenfrequency is ob-

tained by solving the Liouvillian equations for the single time expectation value, and

has been amply detailed elsewhere. [dVa10a] Assuming that the emission energy of

different quantum dots change differently with respect to a control parameter, which

is the case for electrically tuning quantum dots, we can bring two or more quantum

dots at resonance at the same time. This can be simply modeled by an effective con-

trol parameter such as ωj = αjVcontrol, where αj gives the different slopes of emission

frequency with the control parameter Vcontrol. The emission frequency of the cavity

mode is assumed not to be affected by this control parameter.

D.3. One Emitter

We revisit the simplest and most popular case of one quantum dot strongly coupled

to the cavity mode. This was the system previously studied in refs. [Lau09c, dVa09].

We compare the emission spectra of the cavity and dot emission in fig. D.1, where we

plot the emission spectra from the radiation channel of the exciton S1(ω), from the

183


ω/g

Sa(ω)

−10

−5

0

5

10

ω/g

S1(ω)

−10

−5

0

5

10

Control parameter (arb. units)

ω/g

λ0

λ1

Eigenenergies

−10

−5

0

5

10

−1 −0.5 0 0.5 1

Figure D.1.: One dot in a cavity. Emission spectrum from the radiation channel ofthe exciton S1(ω), from the radiation channel of the cavity mode Sa(ω),and eigenstates of the system. Parameters: γa/g = 0.5, γ1/g = 0.1,P1/g = 10−3 and Pa = P1.

radiation channel of the cavity mode Sa(ω), and the eigenstates. Close to resonance,

both the exciton and the cavity mode emit into both radiation channels and the ra-

diation patterns are very similar [see also fig. D.2(a)]. Far away from resonance, the

spectra look different, the excitonic channel S1(ω) being dominated by the emission

of the exciton and the cavity channel Sa(ω) being dominated by the cavity emission.

However, a difference in the relative emission strengths cannot be directly attributed

to a preference in the radiation channel. It is also dependent on the experimental pa-

rameters, on the specific pumping rate of the exciton, the number of other transitions

184

D.3 One Emitter

- 4 - 2 0 2 40.0

0.2

0.4

0.6

0.8

1.0 (a)

- 4 - 2 0 2 40.0

0.2

0.4

0.6

0.8 (c)

- 4 - 2 0 2 40.0

0.1

0.2

0.3

0.4

0.5(b)

- 4 - 2 0 2 40.0

0.5

1.0

1.5 (d)

Figure D.2.: Cavity (solid blue) and dot (dotted purple) emission spectra for onedot in a cavity as a function of pumping power. In the spontaneousemission regime (a), there are no qualitative differences between the twotypes of spectrum. Increasing pumping (b–d) shows markedly differentbehaviours in the two channels of emission. Whereas the cavity spectrumdoes not present a particularly rich phenomenology (collapse of the Rabidoublet), the dot emission displays more characteristic lineshapes. Sideelbows are formed [outlined with arrows in (b)] and strong deviationsfrom Lorentzian lines are obtained even when the spectrum has only onepeak. Parameters: γa/g = 0.5, γ1/g = 0.1, Pa = 0 and P1/g = 10−3, 0.5,1 and 2 from left to right panel.

or quantum dots feeding the cavity mode, and of course on the different coupling

terms. In the solid state environment the exciton is not only directly coupled to the

cavity mode, but also via the phonon bath. [Hoh09, Hoh10, Kae10, Hug11]

Upon increasing the pumping level, one reaches the non-linear regime and climbs

the Jaynes–Cummings ladder. [dVa09] Considering cavity emission at resonance, in a

typical system where the coupling strength is not much larger than the decay rates,

this transition results in an apparent collapse of the Rabi doublet into a single line. If

the number of photons is high enough, this line narrows as a consequence of the cavity

entering the lasing regime. This transition, displayed in solid blue in fig. D.2, has been

recently reported experimentally. [Nom10] Its counterpart in the dot emission (purple

dotted line) exhibits richer qualitative features in the non-linear regime. [dVa09] This

185


manifests by the elbows in fig. D.2(b-d) [indicated by arrows in (b)] arising from the

transitions |n,±〉 → |n − 1,∓〉, that are suppressed in the cavity emission. We have

used the standard notation for the eigenstates of the Jaynes–Cummings Hamiltonian

with |n,±〉 the state with n excitation of higher (+) and lower (−) energy. In the lasing

regime, (d), the dot emission is strongly non-Lorentzian and exhibits a characteristic

lineshape reminiscent of the one-atom laser. [dVa10d]

A simple and convenient way to evidence Jaynes–Cummings non-linearities is to

bring the system out of resonance. In this way, transitions that are otherwise closely

packed together, can be separated and resolved in the photoluminescence spectrum.

This is shown in fig. D.3 for the same system as previously but detuning the two bare

emitters from each other by ∆ = ωa − ω1. The Rabi doublet at resonance turns into

a triplet at small detunings. This is because one of the transitions |2,±〉 → |1,±〉—that are too broad and too close from the linear transitions |1,±〉 → |vacuum〉 at

resonance—can be distinguished at a small detuning, as shown in the lower panel (a-

b).

Transitions between dressed states provide a faithful mapping to the exact system

dynamics when the system operates in the very strong coupling regime, such that

dressed states are well defined and do not overlap appreciably with each other. In the

case where they overlap, say because the splitting between dressed states is small or

because their broadening is large, interferences between the states become important.

The luminescence can indeed be decomposed into a sum of Lorentzian emissions from

the dressed states with dispersive corrections arising from each dressed states driving

or being driven by the others. [dVa09] These interferences can become particularly

strong and complex when the system is brought towards the classical regime, that

is, with a lot of excitations. In this case, many dressed states enter the collective

dynamics, and their overlap as well as mutual disturbance are thus much stronger.

This effect is also better seen at nonzero detuning. Although the interference grows

with pumping even at resonance, it tends to be canceled by symmetry: dressed states

from both sides of the origin (set at the bare cavity emission) equilibrate each other.

However, with detuning, imbalance magnifies the interferences with striking conse-

quences. This is shown for instance in fig. D.4 that reproduces fig. D.2 (for the dot

emission only) but at detuning ∆ = 2g. The figure shows how, as pumping is in-

creased, an emission dip develops at the origin as the lines broaden. It starts to be

particularly visible in panel (b) whereas, at low excitation [panel (a)], one merely sees

186

D.3 One Emitter

- 3 - 2 - 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5(a)

- 3 - 2 - 1 0 1 2 3

(b)

Figure D.3.: Cavity spectrum of one dot in a cavity as a function of detuning. Lowerpanels, decomposition of the spectrum in its dressed states emissionlines (|1,±〉 → |vacuum〉 in dashed green and |2,±〉 → |1,±〉 in dot-ted red), showing how (a) Jaynes–Cummings transitions are hindered atresonances but (b) are revealed out of resonance. Parameters are thesame as in fig. D.2 but with the detunings ∆/g = 0, 0.6, 1 and 2 andP1/g = 0.4.

the exciton, detuned, favouring its own mode of radiation, just as in the linear case

(see fig. D.1). Note how, at resonance, in fig. D.2, this interference is masked, being

essentially canceled by symmetry.

The physical meaning of this dip can be assigned, in this case, to the cavity entering

the lasing regime at this frequency and, therefore, depleting excitations from the

quantum dot. In fig. D.5, we show a decomposition of fig. D.4(d) into the sum of

dressed states emission (thick solid purple) and the sum of interferences between

the dressed states (dashed brown). The observed photoluminescence spectrum is the

sum of these two contributions, obtained from a mathematical decomposition of the

G(1)(t,τ) = 〈a†(t)a(t + τ)〉 into terms that give rise to Lorentzian lines (identifying

the dressed states) and to dispersive interferences when the dressed state emissions

187


- 5 0 5 10

0.05

0.10

0.15

0.20

- 10 - 5 0 5 10

- 10 - 5 0 5 10 - 10 - 5 0 5 10

(a) (b)

(c) (d)

0.05

0.10

0.15

0.20

Figure D.4.: Dot emission spectrum of one dot in a cavity, out of resonance, as afunction of pumping. The Rabi doublet in the spontaneous emissionregime (a) gives rise, with increasing pumping (b–d), to an emissiondip: the incoherent excitation is coherently scattered to the cavity whichenters the lasing regime. This is better seen for the case of nonzerodetuning. Parameters are the same as in fig. D.2 but for ∆ = 2g.

overlap. [dVa09] The final result can be adequately described by the dressed states

only whenever interferences between them are negligible, that is, when they do not

overlap. This is the case at low excitation and in the very strong coupling regime,

when g γa,γ1 and the splitting to broadening ratio of all transitions is large. In

this case, a kinetic theory that computes mean occupation of the dressed states with

rate (Boltzmann) equations is adequate. [Pod10] Otherwise, a full master equation

approach is required, although it quickly becomes intractable. As the intensity is

further increased, interferences result in the breakdown of the dressed state picture

when the system acquires some macroscopic coherence. Analytical investigations show

that the emission dip corresponds to a coherent scattering peak from the dot to the

cavity, which, in some approximations, becomes a Dirac δ function, that describes

Rayleigh scattering. [dVa10d]

188

D.4 More than one Emitter

-10 -5 0 5 10

-0.05

0.00

0.05

0.10

0.15

0.20

Figure D.5.: Detail of fig. D.4(d). The interference that arises as the system enterslasing is best seen in the detuned system because dressed states (solidpurple) are then of a markedly different character, see the exciton linebroadened by pumping, right, and the narrow line of the cavity thatenters lasing (origin). The interferences between dressed states (dottedbrown), which added to the dressed state emission provide the photolu-minescence spectrum (filled purple), strongly modify the results of kinetictheory applied to the dressed states.

D.4. More than one Emitter

The dynamics of strongly-coupled and strongly dissipative systems containing several

emitters extremely interesting and rich as new paths of coherence flow between the

dressed states are opened by pumping and decay. This can give rises to new peaks

(of emission or absorption) not accounted for by the dressed state picture. [dVa10c]

At low pumping, when all N -quantum dots excitons are exactly at resonance with

the cavity mode, the eigenvalues and eigenstates of the system (neglecting incoherent

loss, and setting the origin of the energy scale to ωa) have a simple and well known

form [Man95]: two are split in energy by λN0

= ±Ω where Ω =√∑N

j=1 g2j , while the

other N − 1 are degenerated and equal to 0. The corresponding eigenstates are:

|λN0〉 =

1√2

(N∑j=1

gjΩ|0,Xj〉 ± |1,0〉

), (D.6a)

|λj−1〉 =1

Ωj

(g1|0,Xj〉 − gj|1,0〉

), (D.6b)

with Ωj =√g2

1 + g2j , with j ranging from 2 toN for the degenerated eigenstates (D.6b).

189


From this solution we can see that only the |λN0〉 states have a contribution from the

cavity photons. The cavity mode does not contribute to the other states |λj−1〉 which

are called, for this reason, dark states. They consequently cannot be probed in the

cavity spectrum Sa(ω), but they can be very well seen in the excitonic radiation

channel or a mixture of all radiation channels. These superpositions also give rise to

the phenomenon of sub- and superradiance, first reported by Dicke. [Dic54] This is

essentially a classical effect, that is also observed with vibrating strings. Recently,

such configurations have been analyzed in the microcavity quantum electrodynamics

context by Temnov and Woggon, [Tem09] who studied the photon statistics, and by

Poddubny et al., [Pod10] who studied the photoluminescence lineshapes. The latter

authors found that this classical regime is particularly fragile to incoherent pump-

ing since dark states, being also excited by pumping, act as a long-lived reservoir

for bright states from higher manifolds. They also analyzed the regime of very high

excitations, when the system is (or is going towards) lasing. They observe in such

a case that, due to the predominance of the Dicke states that are the most highly

degenerated, the cavity spectrum is either oddly or evenly peaked depending on the

parity of the number of strongly coupled dots, which is a strong manifestation in a

readily measured observable of the underlying microscopic configuration. In the fol-

lowing, we address particular cases of larger than one, but still small number (given

the difficulty in coupling larger assemblies) of dots and show how, in the linear regime,

photoluminescence spectra vary greatly in a qualitative way because of the contribu-

tion or suppression of the dark states. We confirm that these states are quickly spoiled

with increasing pumping [Pod10] and give rise to non-linear quantum features, that

also manifest in strikingly different ways depending on the radiation channel that is

probed.

D.4.1. Two Emitters

In the case of two emitters coupled to a single cavity mode, the eigenfrequencies in

the linear regime can be obtained by solving the eigenvalue problem given by the

following equation:

i∂

∂t

〈a〉〈σ1−〉〈σ2−〉

=

ωa g1 g2

g1 ω1 0

g2 0 ω2

〈a〉〈σ1

−〉〈σ2−〉

(D.7)

190



ω/g

Sa(ω)

−10

−5

0

5

10

−1 −0.5 0 0.5 1

ω/g

S1(ω)

−10

−5

0

5

10

ω/g

S2(ω)

−10

−5

0

5

10


ω/g

λ0

λ1

λ2

Eigenenergies

−10

−5

0

5

10

−1 −0.5 0 0.5 1

Probability

|λ0〉

0

0.5

1

|1, 0〉|0, X1〉|0, X2〉

Probability

|λ1〉

0

0.5

1

|1, 0〉|0, X1〉|0, X2〉

Probability

|λ2〉

0

0.5

1

|1, 0〉|0, X1〉|0, X2〉

Figure D.6.: Emission spectrum from the radiation channel of the first exciton S1(ω),the second exciton S2(ω), the cavity mode Sa(ω), and the eigenstates fora strongly coupled system of two excitons and the cavity mode. Theeigenvectors of the three eigenstates |λ2〉, |λ1〉, and |λ0〉 of the coupledsystem display the contributions of the individual quantum states |1,0〉,|0,X1〉, and |0,X2〉 to the specific eigenstate. Parameters: same as infig. D.1 with identical pumping of the dots.

where ωa = ωa−iΓa/2 and ωj = ωj−iΓj/2, with Γa = γa−Pa, and Γj = γj+Pj. From

the eigenstate of the emission eigenfrequency we can obtain the degree of mixture of

each peak in the spectrum, i.e., the strength of the contribution of the cavity mode,

QD1 exciton and QD2 exciton to each individual eigenstate.

In fig. D.6, we investigate a system of two excitons in different quantum dots simul-

taneously coupled to one cavity mode. We compare the emission spectra obtained

191


ω/g

Sa(ω)

−10

−5

0

5

10


ω/g

λ0

λ1

λ2

Eigenenergies

−10

−5

0

5

10

−1 −0.5 0 0.5 1

Figure D.7.: Emission spectrum from the radiation channel of the cavity mode Sa(ω),and eigenstates for a strongly coupled system of two excitons and thecavity mode where the two excitons cross out of resonance from thecavity mode. Parameters: same as in fig. D.6.

via the radiation channel of the first exciton S1(ω), the second exciton S2(ω) and

the cavity mode Sa(ω). In the spontaneous emission regime, while all three radiation

channels exhibit a markedly different emission spectrum, the most striking difference

can be found in Sa(ω) where one of the emission lines vanishes completely. This oc-

curs when subradiance sets in, and follows for the case of two emitters an analysis of

the eigenvectors similar to that in sect.6.3.2. The plot of the eigenvectors in fig. D.6

presents the contributions of the three quantum states |1,0〉, |0,X1〉, and |0,X2〉 to

the three eigenstates |λ2〉, |λ1〉, and |λ0〉 (as marked in the plot of the eigenstates) of

the coupled system. While |λ2〉 and |λ0〉 have contributions from all three quantum

states at resonance, for the eigenstate |λ1〉 the contribution from |1,0〉 goes to zero

due to destructive interference.

A measurement of this kind would also be possible for the system just described,

192


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

(a)

(b)

(c)

Figure D.8.: Spectral shapes in the cavity emission when two dots are strongly cou-pled to the cavity mode. (a) With increasing excitation (solid) the emis-sion dip effect becomes visible in the cavity emission. (b) It is betterseen slightly out resonance since it can now “feed” on the line previ-ously canceled by subradiance. (c) Such interferences are strong evenin systems that do not exhibit spontaneous emission features (such asRabi splitting), although this still requires strong-coupling. Parameters:g1 = g2 = g (setting the unit), γa = g, γ1 = γ2 = 0.1g, Pa = 0, then:(a) ∆1 = ∆2 = 0, P1 = 10−4g with (dotted) P2 = 10−4g and (solid)P2 = 0.1g. (b) Same as (a) but for γa = 2g, P1 = 10−3g, P2 = 0, ∆1 = 0and (solid) ∆2 = 0.1g or (dashed) ∆2 = 0. (c) Same as (a) but forγa = 5g, ∆1 = 0 with (dotted) ∆2 = 0 or (solid) ∆2 = 0.1g.

but when the two exciton lines anticross out of resonance from the cavity mode,

similar to that discussed in sect. 6.3. In fig. D.7, we plot the emission spectrum from

193


the radiation channel of the cavity mode Sa(ω), and the eigenstates of such a system.

Probing the cavity emission, one of the emission lines vanishes, similar to the case

in fig. D.6, but this time when the two excitons are crossing out of resonance from

the cavity mode. In this situation the eigenvalues and eigenstates have also a simple

form, λ1 = ∆, λ20

= ∆/2±√

∆2 + 4(g21 + g2

2)/2, with ∆ = ω1 − ωa being the mutual

detuning from the cavity mode. In this case, the eigenstates are:

|λ20〉 =

1√(λ2

0)2 + Ω2

2

(g1|0,X1〉+ g2|0,X2〉 − λ2

0|1,0〉

),

|λ1〉 =1

Ω2

(g1|0,X2〉 − g2|0,X1〉

). (D.8)

Again, since the contribution from the cavity mode to this particular eigenstate (|λ1〉)goes to zero, it cannot be probed by the cavity radiation. The plot of all three

eigenstates, however, clearly shows the anticrossing behaviour of the two exciton-like

states when they come in resonance detuned from the cavity mode. [Lau10c]. The

disappearance of the second peak during the anticrossing of the excitons brings a clear

signature of a collective strong coupling with the two dots and that this is probed in

the cavity radiation channel only.

This interference in the linear regime persists in the non-linear regime where it

turns into the interference related to coherence buildup in the system, as was the case

with one dot in the cavity, see figs. D.4-D.5. Such interferences are, however, now

directly accessible through the cavity spectrum, whereas they were previously only to

be seen in the dot emission, which is technically more challenging. This is shown in

fig. D.8, at resonance (a) and out of resonance (b-c). The sharp line near the origin in

panels (b) and (c) is the exciton line that appears suddenly as subradiance cancellation

is destroyed by going out of resonance, see fig. D.6. It results from the interplay of

subradiance and detuning, studied in ref. [Ave09], and the method outlined there, in

the linear regime, indeed reproduces such spectral features. In the non-linear regime,

this line also suffers from the dip carved by the cavity, where it is sharply located.

This effect is robust regardless of the broadening of the cavity, i.e., with and without

observation of the Rabi doublet. Here again, detuning is paramount in revealing the

underlying physics, as seen in (b-c) where the resonant case is superimposed as dotted

line: a very small dip at ω = 0 is hardly visible at resonance [similar to the solid line

in panel (a)], in contrast to the detuned case that produces a significant discontinuity

194


ω/g

Sa(ω)

−10

−5

0

5

10


ω/g

λ0

λ1

λ2

λ3

λ4

λ5

Eigenenergies

−10

−5

0

5

10

−1 −0.5 0 0.5 1

Figure D.9.: Emission spectrum from the radiation channel of the cavity mode Sa(ω)and the eigenstates of a strongly coupled system of five quantum dotexcitons and the cavity mode. Parameters: same as in fig. D.6.

in the spectral shape.

D.4.2. Three Emitters and Beyond

In the linear regime, the interference in the cavity emission due to settling of dark

states simply scales in the expected way with the number of dots, by the very nature

of linearity. As many lines vanish as there are corresponding strongly coupled dots

(minus one). As an example, we simulate the spectral shapes of a system of five

identical quantum emitters coupled to the same cavity mode and plot in fig. D.9

the emission spectrum from the radiation channel of the cavity mode Sa(ω) and the

corresponding eigenvalues. At resonance, the cavity radiation channel shows only

the anticrossing of two of the eigenstates of the coupled system, albeit with a larger

splitting corresponding to 2Ω, as described before in equ. (D.6a).

195


Figure D.10.: Spectral shapes in the cavity emission when three dots are stronglycoupled to the cavity mode for low pumping (dashed line) and highpumping (solid line). This case brings together most features describedabove. At resonance (dotted), although the emission dip is seen atω = 0, and also the peaks from higher rungs, these are faint and wouldbe difficult to observe with finite detector resolution or counting noise.Detuning the system, (solid), destroys the dark state and gives riseto a sharp line nearby the origin. This one is then strongly affectedby the emission dip. Finally, the peaks from non-linear transitionsbecome much more visible. Inset, zoom around the origin. Parameters:g1 = g2 = g3 = 1, γa/g = 0.75, γ1 = γ2 = γ3 = 0.004, P1 = P2 = 10−6g,P3 = 0.01g, Pa = 0 and (dotted) ∆/g = 0 or (solid) ∆/g = 0.1.

In the non-linear regime, the situations discussed previously can be reproduced

and even brought together, with a higher degree of complexity, by the very nature of

non-linearities. A typical example is shown in fig. D.10 for three emitters, featuring

the cavity emission at resonance (thick dotted) and slightly out of resonance (thin

solid). In both cases the emission dip is seen at ω = 0. In the former case, mainly

the Rabi doublet, now at ±√

3g, is visible, with less clearly resolved peaks from the

transitions of multiply-excited states. Both these features would be lost by finite

spectral resolution of the detector. A small detuning breaks the subradiance just like

in the linear case and results in the dot eigenstate showing up strongly as a very sharp

peak. This also displays the emission dip, like in the case with two dots, as seen in the

inset which is a zoom around the origin. This case therefore brings together the two

types of strong interferences that manifest in this system: subradiance and emission

dip. Note that detuning, which unravels them, also makes much more prominent the

peaks that arise from quantum non-linearities, producing a neat quadruplet, a result

196

D.5 Conclusions

we also put forward for the case of one emitter in strong coupling with the cavity.

D.5. Conclusions

We theoretically investigated the spectral form of the emission from N quantum dots

strongly coupled to a single mode of a microcavity. The emission spectra obtained

from the two different radiation channels offered by the cavity and direct quantum

dot emission were compared, both in the linear and non-linear regime, in and out

of resonance. In the spontaneous emission regime, dark states forming at resonance

result in a vanishing of spectral lines in the cavity emission only. Similar darken-

ings are not observed for the QD emission. As the level of excitation is increased and

multiple-photon effects become important, dressed states enter the picture. These are

difficult to observe in state-of-the-art experiments where splitting to broadening ratio

does not allow them to be clearly resolved. Although strong-coupling is maximum at

resonance, we find that the study of detuned systems can provide a route to reveal the

quantum non-linear features. Moreover, here too dot emission behaves qualitatively

differently from cavity emission. As excitation is further increased and a large number

of cavity photons is generated (the system enters lasing), another interference due to

onset of coherence takes place. This manifests itself as an emission dip that results

from coherent and elastic scattering between the modes. A plethora of phenomena

remain to be observed in these systems. One can access it either by detecting direct

dot emission—which is technically difficult—or by considering strong-coupling involv-

ing more than one emitter. In both cases, detuning is a powerful tool to unravel this

new physics, since, although optimum, strong-coupling is balanced and/or canceled

at resonance.

197

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List of Publications

1. K. Muller, G. Reithmaier, V. Jovanov, A. Bechthold, A. Laucht, H. J. Krenner,

M. Bichler, G. Abstreiter, and J. J. Finley. Single Exciton Energy Pinning in

Quantum Dot Molecules. In course of preparation for Applied Physics Letters.

2. A. Laucht, T. Gunthner, S. Putz, N. Hauke, R. Saive, S. Frederick, M. Bichler,

M.-C. Amann, A. W. Holleitner, M. Kaniber, and J. J. Finley. Spontaneous

Emission Properties of Quantum Dots in a Photonic Crystal Waveguide. In

course of preparation for New Journal of Physics.

3. A. Laucht, S. Putz, T. Gunthner, N. Hauke, R. Saive, S. Frederick, M. Bichler,

M.-C. Amann, A. W. Holleitner, M. Kaniber, and J. J. Finley. Quantum Dot

- Photonic Crystal Waveguide System as an Efficient On-chip Source of Single

Photons. In course of preparation for Nano Letters.

4. F. P. Laussy, E. del Valle, M. Schrapp, A. Laucht, and J. J. Finley. Evidencing

quantum nonlinearities in strongly dissipative cavity QED systems. In course of

preparation for Phys. Stat. Sol. C.

5. F. P. Laussy, E. del Valle, M. Schrapp, A. Laucht, and J. J. Finley. Evidenc-

ing quantum nonlinearities in strongly dissipative cavity QED systems. Under

consideration at Phys. Rev. Lett.

6. F. P. Laussy, A. Laucht, E. del Valle, J. J. Finley, and J. M. Villas-Boas. Lumi-

nescence spectra of quantum dots in microcavities. III. Multiple quantum dots.

Under consideration at Phys. Rev. B.

7. N. Hauke, S. Lichtmannecker, T. Zabel, F. P. Laussy, M. Kaniber, A. Laucht,

G. Abstreiter, J. J. Finley, D. Bougeard, and Y. Arakawa. Investigation of

225

Emitter-Cavity Coupling in two-dimensional Photonic Crystals with embedded

Germanium Islands. Under consideration at Phys. Rev. B.

8. F. Jahnke. Quantum optics with semiconductor nanostructures. C. Tejedor,

E. del Valle, F. P. Laussy, A. Gonzalez-Tudela, A. Laucht, M. Kaniber and J.

J. Finley. Luminescence spectra of quantum dots in microcavities. Woodhead

Publishing (2011). In print.

9. A. Laucht, N. Hauke, A. Neumann, T. Gunthner, F. Hofbauer, A. Mohtashami,

K. Muller, G. Bohm, M. Bichler, M.-C. Amann, M. Kaniber, and J. J. Finley.

Non-resonant feeding of photonic crystal nanocavity modes by quantum dots. J.

of Appl. Phys. 109, 102404 (2011).

10. A. Laucht, J. M. Villas-Boas, S. Stobbe, N. Hauke, F. Hofbauer, G. Bohm, P.

Lodahl, M.-C. Amann, M. Kaniber, and J. J. Finley. Mutual Coupling of two

Semiconductor Quantum Dots via an Optical Nanocavity. Phys. Rev. B 82,

075305 (2010).

11. A. Laucht, M. Kaniber, A. Mohtashami, N. Hauke, M. Bichler, and J. J. Fin-

ley. Temporal Monitoring of Non-resonant Feeding of Semiconductor Nanocavity

Modes by Quantum Dot Multiexciton Transitions. Phys. Rev. B 81, 241302(R)

(2010).

12. N. Hauke, T. Zabel, K. Muller, M. Kaniber, A. Laucht, Arakawa, and J. J. Fin-

ley. Enhanced photoluminescence emission from two-dimensional silicon pho-

tonic crystal nanocavities. New Journal of Physics 12, 053005 (2010).

13. S. Volk, F. J. R. Schulein, F. Knall, A. Wixforth, H. J. Krenner, A. Laucht,

J. J. Finley, J. Riikonen, M. Mattila, M. Sopanen, H. Lipsanen, J. He, T. A.

Truong, H. Kim, and P. Petroff. Recent progress towards acoustically mediated

carrier injection into individual nanostructures for single photon generation.

Proc. SPIE 7610, 76100J (2010).

14. U. Hohenester, A. Laucht, M. Kaniber, N. Hauke, Andre Neumann, A. Mo-

htashami, M. Seliger, M. Bichler, and J. J. Finley. Phonon-assisted transitions

from quantum dot excitons to cavity photons. Phys. Rev. B 80, 201311(R)

(2009).

226

15. F. J. R. Schulein, A. Laucht, J. Riikonen, J. Toivonen, M. Sopanen, H. Lipsanen,

J. J. Finley, A. Wixforth, and H. J. Krenner. Cascaded exciton emission of an

individual strain-induced quantum dot. Appl. Phys. Lett. 95, 083122 (2009).

16. A. Laucht, N. Hauke, J. M. Villas-Boas, F. Hofbauer, M. Kaniber, G. Bohm, and

J. J. Finley. Dephasing of Exciton Polaritons in Photoexcited InGaAs Quantum

Dots in GaAs Nanocavities., Phys. Rev. Lett. 103, 087405 (2009).

17. A. Laucht, F. Hofbauer, N. Hauke, J. Angele, S. Stobbe, M. Kaniber, G. Bohm,

P. Lodahl, and J. J. Finley. Electrical control of spontaneous emission and

strong coupling for a single quantum dot., New Journal of Physics 11, 023034

(2009).

18. M. Kaniber, A. Neumann, A. Laucht, M. F. Huck, M. Bichler, M.-C. Amann,

and J. J. Finley. Efficient and selective cavity-resonant excitation for single

photon generation. New Journal of Physics 11, 013031 (2009).

19. M. Kaniber, A. Laucht, A. Neumann, F. Hofbauer, J. Angele, M. Bichler, M.-

C. Amann, and J. J. Finley. Observation of non resonant coupling of single

quantum dots to photonic crystal nanocavity modes., Conference on Lasers and

Electro-Optics, CLEO (2008).

20. M. Kaniber, A. Laucht, A. Neumann, M. Bichler, M.-C. Amann, and J. J.

Finley. Tunable single quantum dot nanocavities for cavity QED experiments.,

J. Phys.: Condens. Matter 20, 454209 (2008).

21. M. Kaniber, A. Laucht, A. Neumann, J. M. Villas-Boas, M. Bichler, M.-C.

Amann, and J. J. Finley. Investigation of the nonresonant dot-cavity coupling

in two-dimensional photonic crystal nanocavities., Phys. Rev. B 77, 161303(R)

(2008).

22. M. Kaniber, A. Laucht, T. Hurlimann, M. Bichler, R. Meyer, M.-C. Amann,

and J. J. Finley. Highly efficient single-photon emission from single quantum

dots within a two-dimensional photonic band-gap., Phys. Rev. B 77, 073312

(2008).

23. M. Kaniber, A. Kress, A. Laucht, M. Bichler, R. Meyer, M.-C. Amann, and J.

J. Finley. Efficient spatial redistribution of quantum dot spontaneous emission

from two-dimensional photonic crystals., Appl. Phys. Lett. 91, 061106 (2007).

227

Art is “I”; science is “we”.

Claude Bernard

Acknowledgements

A lot of people contributed in numerous ways to the successful completion of my

doctoral thesis. I want to sincerely thank all of you for every little bit! You really

helped me a lot and I learned so much over the last years! Science can simply not

be considered as the effort of an individual, but always needs to be considered as

teamwork!

I would like to thank Jonathan J. Finley for all the different aspects of being the

supervisor of my doctoral thesis from offering me a project to signing my PhD cer-

tificate (hopefully)! Thank you for paying my salary for 3.75 years, all your ideas,

your enthusiasm, being such an unreachable paragon in giving talks, not taking any

offense in my sometimes very direct manner, all the corrections, the letters of ref-

erence, and letting me fly to so many nice conferences! Your always happily given

support was crucial for the success of the project! I am glad, that I will be able to con-

tinue working with you in the future and I am looking forward to seeing you in Sydney!

I would like to thank Gerhard Abstreiter for being the motivating leader of the whole

E24 family. Thank you for promoting our work all around the world, for supplying

reference letters for my postdoc applications and also for all the financial benefits of

working within your chair!

I would like to thank Michael Kaniber for being a great supervisor and colleague

over the last years! Thank you for all the training I got during my Master’s thesis

which helped me a lot during my doctoral thesis, proof-reading and correcting all my

manuscripts, the discussions, your ideas, and not to forget your nice single-dot sample

229

Acknowledgements

which I finally destroyed (although some parts might still be alive...)!

I would like to thank Felix Hofbauer, whose project I continued, for all the prepara-

tory work, his setup and, of course, the great electrically-tunable, strong coupling

samples! Your brilliant work in the cleanroom allowed me to focus on the optical

measurements and do so many nice experiments!

At this point I would also like to thank Jakob Angele who did his Diploma the-

sis with Felix for his preparatory work in the lab! And thank you for starting at

McKinsey so that I could get the PhD position on this project!

I would like to thank Jose M. Villas-Boas for the excellent theoretical description

of our experimental data! I enjoyed working with you and going to conferences with

you! I think our collaboration was always very fruitful! I remember the coffee breaks

at the QD2010 in Nottingham where we made the figures for our last paper. Without

your theoretical interpretation and your fitting skills the results would not even be

half as interesting!

I would like to thank my 413

Dipl./MSc. students Norman Hauke, Abbas Mo-

htashami, Thomas Gunthner, Simon Putz and 13

Rebecca Saive for all the great work

and their motivation and contributions! Without your hard work and input it would

have simply not been possible to obtain so many nice results! I am glad that four

of you have started your PhD already and the last one of you is at least thinking

about it. I take that as an indication that you did not always feel miserable under

my supervision.

In particular, I would like to thank Norman for mounting the new CCD onto the

spectrometer while I was on vacation, staying at the WSI as a colleague and good

friend, and being my best man! I would like to thank Thomas for his great motivation

at work, always high spirits and his stupid jokes! I would like to thank Simon for all

the work late at night, finally finding a bright and fast dot, and being my chauffeur

to Zurich!

I would like to thank Fabrice P. Laussy, Elena del Valle and Michael Schrapp for

the fresh theoretical support and all the fantastic new ideas! It is a pity that we do

not have the right samples and equipment to do all the corresponding measurements!

230

Acknowledgements

I would like to thank Ulrich Hohenester from the Karl-Franzens-Universitat Graz

for his theoretical description of the phonon-mediated exciton-cavity coupling. It was

a pleasure working with you!

I would like to thank Max Bichler, Gerhard Bohm and, more recently, Gregor

Koblmuller and Kai Muller for the MBE growth of the samples! Thank you for

wafers and quantum dots!

I would like to thank Markus-Christian Amann and Ralf Meyer for the manage-

ment and maintenance of the cleanroom of the Walter Schottky Insitut! Many thanks

also to Linda Mora, Sepp Grottenthaler, Edith Sckopke, Elke Thiel and Gabi Riedl for

their support inside the cleanroom, and Tobias Grundl, Simeon Katz, and Christian

Grasse for their support with the RIE system!

For the introduction to and the support with the ELine and the FIB, I would like

to especially thank Peter Weiser who set everything up and keeps it running, but

also Sonja Matich!

I would like to thank Irmgard Neuner, Daniela Huber, Veronika Enter, Liane Lin-

der, Sonja Kraus, and Regina Morinaga for their invaluable help in administrative

concerns!

I would like to thank Hubert Riedl, Max Bichler, Ade Ziegltrum, Michi Fischer and

Sepp Grottenthaler for their help and support regarding technical problems!

I would like to thank Hannes Seitz, Wolfgang Bendak, Fritz Sedlmeir, and Bernhard

Kratzer for their great work in the workshop! My special thanks go to Hannes who

was always able to figure out my hand-drawn sketches and only mildly complained

when I ordered 100+ posts for the optical setup!

I would like to thank Claudia Paulus and Jose A. Garrido for running the upper

technology in the Walter Schottky Institut!

231

Acknowledgements

I would like to thank my numerous office mates Dominic Dorfner, Jakob Angele,

Tobias Hurlimann, Søren Stobbe, Norman Hauke, Kai Muller, Andre Neumann, Malte

Huck, Hagen Langhuth, Abbas Mohtashami, Linus Gisslen, Simon Frederick, Thomas

Gunthner, Stefan Lichtmannecker, Watcharapong Paosangthong, Daniel Rudolph,

Gregor Bracher, Gunther Reithmaier, and Alex Bechthold for sharing an office with

me, their help in work-related things but also for the distractions from work!

I would like to thank my fellow doctoral candidates Emily Clark, Michael Kaniber,

Dominik Heiss, Vase Jovanov, Flo Klotz, Norman Hauke, Kai Muller, Gregor Bracher,

Daniel Rudolph, Stefan Lichtmannecker, and Gunther Reithmaier for “suffering” with

me!

I would like to thank Michael Kaniber, Jonathan J. Finley, Jakob Angele, Felix Hof-

bauer, Thomas Zabel, Emily Clark, Jose M. Villas-Boas, Vase Jovanov, Flo Klotz,

Heike Schwager, Hagen Langhuth, Norman Hauke, Thomas Zabel, Simon Frederick,

Fabrice Laussy, Elena del Valle, and Michael Schrapp for the great time at confer-

ences and winterschools!

I would like to thank the Quantum Dudes for all the fun things we did besides work

like kicker, BBQ, poker, go-kart, playing football, watching football, running, going

to concerts, movie nights, flat warming parties, bachelor parties, and so on. I am glad

that many of you have become true friends!

Finally, I would like to thank my wife Daniela for her understanding and continuous

patience, and my parents for their support during the years of my undergrad studies!

232

Acknowledgements

We gratefully acknowledge financial support of the Deutsche Forschungsgemein-

schaft via the Sonderforschungbereich 631, Teilprojekt B3, the German Excellence

Initiative via the“Nanosystems Initiative Munich” (NIM), the European Union via

SOLID in the seventh framework program, and the Bundesministerium fur Forschung

und Bildung via QuaHLRep. My special thanks go the the TUM Graduate School

that paid my research visit to Prof. Y. Yamamoto’s group at the Stanford University.

233

In der Schriftenreihe des Walter Schottky Instituts der Technischen Universität München sind bisher folgende Bände erschienen: Vol. 1 Cornelia Engel Si/SiGe basierende Phototransistoren 131 Seiten ISBN 3-932749-01-4

Vol. 7 Markus Sexl Verspannte und gitterrelaxierte In(GaAl)As Heterostrukturen 144 Seiten ISBN 3-932749-07-3

Vol. 2 Peter Schittenhelm Selbst-Organisation und Selbst-Ordnung in Si/SiGe-Heterostrukturen 151 Seiten ISBN 3-932749-02-2

Vol. 8 Christian Obermüller Photolumineszenszspektroskopie mit optischen Nahfeldmethoden an GaAs-Nanostrukturen 140 Seiten ISBN 3-932749-08-1

Vol. 3 Andreas Nutsch Selektive Epitaxie von (GaIn)(AsP) Schichtstrukturen 129 Seiten ISBN 3-932749-03-0

Vol. 9 Edilson Silveira Inelastische Lichtstreuung an niedrig-dimensionalen Halbleiterstrukturen 104 Seiten ISBN 3-932749-09-X

Vol. 4 Peter Baumgartner Optische und elektronische Eigenschaften lasergeschriebener GaAs-Nanostrukturen 180 Seiten ISBN 3-932749-04-9

Vol. 10 Eberhard Christian Rohrer Photoleitungs-Spektroskopie von Diamant 153 Seiten ISBN 3-932749-10-03

Vol. 5 Walter Franz Rieger Untersuchung der elektronischen und strukturellen Eigenschaften von GaNAlN und deren Legierungen 158 Seiten ISBN 3-932749-05-7

Vol. 11 Thomas Wimbauer Magnetische Resonanz-Untersuchungen an modernen Halbleitermaterialien 125 Seiten ISBN 3-932749-11-1

Vol. 6 Markus Hauser Oberflächenemittierende Laserdioden mit Mehrfachepitaxie 148 Seiten ISBN 3-932749-06-5

Vol. 12 Herbert Verhoeven Thermische Eigenschaften von CVD-Diamantschichten 154 Seiten ISBN 3-932749-12-X

Vol. 13 Hans-Christoph Ostendorf Trennung von Volumen- und Oberflächenrekombination in Silizium 128 Seiten ISBN 3-932749-13-8 Vol. 14 Martin Städele

Vol. 20 Christoph Martin Engelhardt Zyklotronresonanz zweidimensionaler Ladungsträgersysteme in Halbleitern, Effekte der Elektron-Elektron-Wechsel-wirkung und Lokalisierung 317 Seiten ISBN 3-932749-20-0

Dichtefunktionaltheorie mit exaktem Austausch für Halbleiter 202 Seiten ISBN 3-932749-14-6 Vol. 15 Helmut Angerer Herstellung von Gruppe III-Nitriden mit Molekularstrahlepitaxie 144 Seiten ISBN 3-932749-15-4 Vol. 16 Wolfgang Heller Spektroskopie einzelner Quantenpunkte in magnetischen und elektrischen Feldern 128 Seiten ISBN 3-932749-16-2 Vol. 17 Molela Moukara Pseudopotentiale mit exaktem Austausch 117 Seiten ISBN 3-932749-17-0 Vol. 18 Ralph Oberhuber Elektronische Struktur und Transport in verspannten Halbleiterschichtsystemen 110 Seiten ISBN 3-932749-18-9 Vol. 19 Reiner Pech High-Energy Boron-Implantation into Different Silicon Substrates 158 Seiten ISBN 3-932749-19-7

Vol. 21 Eduard Neufeld Erbium-dotierte Si/SiGe-Lichtemitter und -Wellenleiter 136 Seiten ISBN 3-932749-21-9 Vol. 22 Gert Schedelbeck Optische Eigenschaften von Halbleiter-nanostrukturen hergestellt durch Über- wachsen von Spaltflächen 154 Seiten ISBN 3-932749-22-7 Vol. 23 Jürgen Zimmer Optoelektronisches Verhalten von Dünn-schichtbauelementen aus amorphem und mikrokristallinem Silizium 171 Seiten ISBN 3-932749-23-5 Vol. 24 Berthold Schmidt Leistungsoptimierung abstimmbarer InGaAsP/InP Halbleiterlaser 85 Seiten ISBN 3-932749-24-3 Vol. 25 Jianhong Zhu Ordering of self-assembled Ge and SiGe nanostructures on vicinal Si surfaces 120 Seiten ISBN 3-932749-25-1

Vol. 26 Gerhard Groos Herstellung und Charakterisierung von Silizium-Nanostrukturen 168 Seiten ISBN 3-932749-26-X

Vol. 33 Martin Rother Elektronische Eigenschaften von Halbleiternanostrukturen hergestellt durch Überwachsen von Spaltflächen 196 Seiten ISBN 3-932749-33-2

Vol. 27 Uwe Hansen Theorie der Reaktionskinetik an Festkörperoberflächen 119 Seiten ISBN 3-932749-27-8

Vol. 34 Frank Findeis Optical spectroscopy on single self-assembled quantum dots 156 Seiten ISBN 3-932749-34-0

Vol. 28 Roman Dimitrov Herstellung und Charakterisierung von AlGaN/GaN-Transistoren 196 Seiten ISBN 3-932749-28-6 Vol. 29 Martin Eickhoff Piezowiderstandsmechanismen in Halbleitern mit großer Bandlücke 151 Seiten ISBN 3-932749-29-4 Vol. 30 Nikolai Wieser Ramanspektroskopie an Gruppe III-Nitriden 161 Seiten ISBN 3-932749-30-8 Vol. 31 Rainer Janssen Strukturelle und elektronische Eigenschaften amorpher Silizium-Suboxide275 Seiten ISBN 3-932749-31-6

Vol. 35 Markus Ortsiefer Langwellige Vertikalresonator-Laser-dioden im Materialsystem InGaAlAs/InP 152 Seiten ISBN 3-932749-35-9 Vol. 36 Roland Zeisel Optoelectronic properties of defects in diamond and AlGaN alloys 140 Seiten ISBN 3-932749-36-7 Vol. 37 Liwen Chu Inter- und Intraband Spektroskopie an selbstorganisierten In(Ga)As/GaAs Quantenpunkten 124 Seiten ISBN 3-932749-37-5 Vol. 38 Christian Alexander Miesner Intra-Valenzbandspektroskopie an SiGe-Nanostrukturen in Si 100 Seiten ISBN 3-932749-38-3

Vol. 32 Martin W. Bayerl Magnetic resonance investigations of group III-nitrides 155 Seiten ISBN 3-932749-32-4

Vol. 39 Szabolcs Kátai Investigation of the nucleation process of chemical vapour deposited diamond films 178 Seiten ISBN 3-932749-39-1

Vol. 40 Markus Arzberger Wachstum, Eigenschaften und An-wendungen selbstorganisierter InAs-Quantenpunkte 236 Seiten ISBN 3-932749-40-5

Vol. 46 Jan Schalwig Feldeffekt-Gassensoren und ihre Anwendung in Abgas-nachbehandlungssystemen 125 Seiten ISBN 3-932749-46-4

Vol. 41 Markus Oliver Markmann Optische Eigenschaften von Erbium in Si/Si1-xCx, Si/Si1-xGex und Si/SiOx Heterostrukturen 182 Seiten ISBN 3-932749-41-3

Vol. 47 Christopher Eisele Novel absorber structures for Si-based thin film solar cells 126 Seiten ISBN 3-932749-47-2

Vol. 42 Rainer Alexander Deutschmann Two dimensional electron systems in atomically precise periodic potential 210 Seiten ISBN 3-932749-42-1

Vol. 48 Stefan Hackenbuchner Elektronische Struktur von Halbleiter-Nanobauelementen im thermodynamischen Nichtgleichgewicht 213 Seiten ISBN 3-932749-48-0

Vol. 43 Uwe Karrer Schottky-Dioden auf Galliumnitrid: Eigenschaften und Anwendungen in der Sensorik 182 Seiten ISBN 3-932749-43-X

Vol. 49 Andreas Sticht Herstellung und Charakterisierung von dünnen Silizium/Siliziumoxid-Schichtsystemen 166 Seiten ISBN 3-932749-49-9

Vol. 44 Günther Anton Johann Vogg Epitaxial thin films of Si and Ge based Zintl phases and sheet polymers 169 Seiten ISBN 3-932749-44-8

Vol. 50 Giuseppe Scarpa Design and fabrication of Quantum Cascade Lasers 193 Seiten ISBN 3-932749-50-2

Vol. 45 Christian Strahberger Vertikaler Transport und extreme Magnetfelder in Halbleitern 167 Seiten ISBN 3-932749-45-6

Vol. 51 Jörg Frankenberger Optische Untersuchungen an zwei-dimensionalen Ladungsträgersystemen 158 Seiten ISBN 3-932749-51-0

Vol. 52 Doris Heinrich Wavelength selective optically induced charge storage in self-assembled semiconductor quantum dots 144 Seiten ISBN 3-932749-52-9

Vol. 58 Po-Wen Chiu Towards carbon nanotube-based molecular electronics 116 Seiten ISBN 3-932749-58-8

Vol. 53 Vol. 59 Nicolaus Ulbrich Tobias Graf Entwurf und Charakterisierung von Spin-spin interactions of localized Quanten-Kaskadenlasern und electronic states in semiconductors Quantenpunktkaskaden 194 Seiten 133 Seiten ISBN 3-932749-59-6 ISBN 3-932749-53-7 Vol. 54 Vol. 60 Lutz Carsten Görgens Stefan Klein Analyse stickstoffhaltiger III-V Halbleiter- Microcrystalline silicon prepared by hot Heterosysteme mit hochenergetischen wire CVD: preparation and characteri- schweren Ionen sation of material and solar cells 116 Seiten 157 Seiten ISBN 3-932749-54-5 ISBN 3-932749-60-X Vol. 55 Vol. 61 Andreas Janotta Markus Krach Doping, light-induced modification and Frequenzverdreifacher mit Anti-Seriellem biocompatibility of amorphous silicon Schottky-Varaktor für den Terahertz- suboxides bereich 180 Seiten 156 Seiten ISBN 3-932749-55-3 ISBN 3-932749-61-8 Vol. 56 Vol. 62 Sebastian Tobias Benedikt Gönnenwein Ralph Thomas Neuberger Two-dimensional electron gases and AlGaN/GaN-Heterostrukturen als ferromagnetic semiconductors: chemische Sensoren in korrosiven materials for spintronics Medien 198 Seiten 153 Seiten ISBN 3-932749-56-1 ISBN 3-932749-62-6 Vol. 57 Vol. 63 Evelin Beham Sonia Perna Photostromspektroskopie an einzelnen Wasserstoff-Passivierung von tri- Quantenpunkten kristallinem Silizium durch hydro- 186 Seiten genisiertes Siliziumnitrid ISBN 3-932749-57-X 136 Seiten ISBN 3-932749-63-4

Vol. 64 Vol. 71 Oliver Schumann Andreas Florian Kreß Einfluss von Stickstoff auf das Manipulation of the Light-Matter-Inter- Wachstum und die Eigenschaften action in Photonic Crystal Nanocavities von InAs-Quantenpunkten 185 Seiten 148 Seiten ISBN 3-932749-71-5 ISBN 3-932749-64-2 Vol. 65 Vol. 72 Gerhard Rösel Markus Grau Entwicklung und Charakterisierung von Molekularstrahlepitaktische Herstellung Typ-II-Heterostrukturen für die Abstimm- von antimonidischen Laserdioden für region in abstimmbaren Laserdioden die Gassensorik 101 Seiten 138 Seiten ISBN 3-932749-65-0 ISBN 3-932749-72-3 Vol. 66 Vol. 73 Angela Link Karin Buchholz Zweidimensionale Elektronen- und Löcher- Microprocessing of silicon on insulator Gase in GaN/AlGaN Heterostrukturen substrates and biofunctionalisation of 156 Seiten silicon dioxide surfaces for sensing ISBN 3-932749-66-9 applications in fluids 170 Seiten Vol. 67 ISBN 3-932749-73-1 Matthias Sabathil Opto-electronic and quantum transport Vol. 74 properties of semiconductor nanostructures Dominique Bougeard 156 Seiten Spektroskopische Charakterisierung von ISBN 3-932749-67-7 Germanium-Quantenpunkten in Silizium 154 Seiten Vol. 68 ISBN 3-932749-74-X Frank Fischer Growth and electronic properties of two- Vol. 75 dimensional systems on (110) oriented GaAs Jochen Bauer 139 Seiten Untersuchungen zum kontrollierten ISBN 3-932749-68-5 Wachstum von InAs-Nanostrukturen auf Spaltflächen Vol. 69 140 Seiten Robert Shau ISBN 3-932749-75-8 Langwellige oberflächenemittierende Laser- dioden mit hoher Ausgangsleistung und Vol. 76 Modulationsbandbreite Ingo Bormann 198 Seiten Intersubband Spektroskopie an Silizium- ISBN 3-932749-69-3 Germanium Quantenkaskadenstrukturen 124 Seiten Vol. 70 ISBN 3-932749-76-6 Andrea Baumer Structural and electronic properties of hydrosilylated silicon surfaces 163 Seiten ISBN 3-932749-70-7

Vol. 77 Vol. 84 Hubert Johannes Krenner Claudio Ronald Miskys Coherent quantum coupling of excitons New substrates for epitaxy of group III in single quantum dots and quantum nitride semiconductors: challenges and dot molecules potential 160 Seiten 207 Seiten ISBN 3-932749-77-4 ISBN 978-3-932749-84-1 Vol. 78 Vol. 85 Ulrich Rant Sebastian Friedrich Roth Electrical manipulation of DNA-layers n- and p-type transport in (110) GaAs on gold surfaces substrates, single- and double-cleave 249 Seiten structures ISBN 3-932749-78-2 138 Seiten ISBN 978-3-932749-85-8 Vol. 79 René Todt Vol. 86 Widely tunable laser diodes with Mario Gjukic distributed feedback Metal-induced crystallization of 152 Seiten silicon-germanium alloys ISBN 3-932749-79-0 309 Seiten ISBN 978-3-932749-86-5 Vol. 80 Miroslav Kroutvar Vol. 87 Charge and spin storage in quantum dots Tobias Zibold 150 Seiten Semiconductor based quantum ISBN 3-932749-80-4 information devices: Theory and

simulations Vol. 81 151 Seiten Markus Maute ISBN 978-3-932749-87-2 Mikromechanisch abstimmbare Laser-Dioden mit Vertikalresonator Vol. 88 170 Seiten Thomas Jacke ISBN 3-932749-81-2 Weit abstimmbare Laserdiode mit vertikal integriertem Mach-Zehnder- Vol. 82 Interferometer Frank Ertl 165 Seiten Anisotrope Quanten-Hall-Systeme, Vertikale ISBN 978-3-932749-88-9 Ultrakurzkanal- und Tunneltransistoren 170 Seiten Vol. 89 ISBN 3-932749-82-0 Nenad Ocelić Quantitative near-field phonon- Vol. 83 polariton spectroscopy Sebastian M. Luber 174 Seiten III-V semiconductor structures for biosensor ISBN 978-3-932749-89-6 and molecular electronics applications 212 Seiten ISBN 978-3-932749-83-4

Vol. 90 Vol. 96 Kenji Arinaga Stefan Ahlers Control and manipulation of DNA on Magnetic and electrical properties gold and its application for biosensing of epitaxial GeMn 111 Seiten 184 Seiten ISBN 978-3-932749-90-2 ISBN 978-3-932749-96-0 Vol. 91 Vol. 97 Hans-Gregor Hübl Emanuele Uccelli Coherent manipulation and electrical Guided self-assembly of InAs quantum detection of phosphorus donor spins dots arrays on (110) surfaces in silicon 172 Seiten 162 Seiten ISBN 978-3-932749-97-1 ISBN 978-3-932749-91-9 Vol. 92 Vol. 98 Andrea Friedrich Shavaji Dasgupta Quanten-Kaskaden-Laser ohne Growth optimization and characteri- Injektorbereiche zation of high mobility two-dimensional 140 Seiten electron systems in AlAs quantum wells ISBN 978-3-932749-92-6 152 Seiten ISBN 978-3-932749-98-8 Vol. 93 Vol. 99 Oliver Dier Werner Hofmann Das Materialsystem (AlGaIn) (AsSb): Eigen- InP-based long-wavelength VCSELs schaften und Eignung für GaSb-basierte and VCSEL arrays for high-speed Vertikalresonator-Laserdioden optical communication 174 Seiten 142 Seiten ISBN 978-3-932749-93-3 ISBN 978-3-932749-99-5 Vol. 94 Vol. 100 Georg Steinhoff Robert Lechner Group III-nitrides for bio- and electro- Silicon nanocrystal films for chemical sensors electronic applications 197 Seiten 227 Seiten ISBN 978-3-932749-94-0 ISBN 978-3-941650-00-8 Vol. 95 Vol. 101 Stefan Harrer Nebile Işık Next-generation nanoimprint lithography: Investigation of Landau level spin Innovative approaches towards improving reversal in (110) oriented p-type flexibility and resolution of nanofabrication GaAs quantum wells in the sub-15-nm region 114 Seiten 161 Seiten ISBN 978-3-941650-01-5 ISBN 978-3-932749-95-7

Vol. 102 Vol. 109 Andreas Florian Härtl Sebastian Strobel Novel concepts for biosensors using Nanoscale contacts to organic molecules diamond-based field effect transistors based on layered semiconductor 255 Seiten substrates ISBN 978-3-941650-02-2 140 Seiten ISBN 978-3-941650-09-1 Vol. 103 Felix Florian Georg Hofbauer Realization of electrically tunable single Vol. 110 quantum dot nanocavities Ying Xiang 160 Seiten Semiconductor nanowires and ISBN 978-3-941650-03-9 templates for electronic applications 152 Seiten Vol. 104 ISBN 978-3-941650-10-7 Dominic F. Dorfner Novel photonic biosensing based on Vol. 111 silicon nanostructures Michael Kaniber 169 Seiten Non-classical light generation in ISBN 978-3-941650-04-6 photonic crystal nanostructures 177 Seiten Vol. 105 ISBN 978-3-941650-11-4 Till Andlauer Optoelectronic and spin-related properties Vol. 112 of semiconductor nanostructures in Martin Hermann magnetic fields Epitaktische AlN-Schichten auf 157 Seiten Saphir und Diamant ISBN 978-3-941650-05-3 216 Seiten ISBN 978-3-941650-12-1 Vol. 106 Christoph Bihler Vol. 113 Magnetic semiconductors Dominik Heiss 190 Seiten Spin storage in quantum dot ensembles ISBN 978-3-941650-06-0 and single quantum dots 196 Seiten Vol. 107 ISBN 978-3-941650-13-8 Michael Huber Tunnel-Spektroskopie im Vol. 114 Quanten-Hall-Regime Tillmann Christoph Kubis 164 Seiten Quantum transport in semiconductor ISBN 978-3-941650-07-7 nanostructures 253 Seiten Vol. 108 ISBN 978-3-941650-14-5 Philipp Achatz Metal-insulator transition and super- Conductivity in heavily boron-doped diamond and related materials 151 Seiten ISBN 978-3-941650-08-4

Vol. 115 Vol. 122 Lucia Steinke Ilaria Zardo Magnetotransport of coupled quantum Growth and raman spectroscopy Hall edges in a bent quantum well studies of gold-free catalyzed semi- 194 Seiten conductor nanowires ISBN 978-3-941650-15-2 184 Seiten ISBN 978-3-941650-22-0 Vol. 116 Christian Lauer Vol. 123 Antimonid-basierte Vertikalresonator- Andre Rainer Stegner Laserdioden für Wellenlängen oberhalb 2 m Shallow dopants in nanostructured and 180 Seiten in isotopically engineered silicon ISBN 978-3-941650-16-9 185 Seiten ISBN 978-3-941650-23-7 Vol. 117 Simone Maria Kaniber Vol. 124 Optoelektronische Phänomene in hybriden Andreas J. Huber Schaltkreisen aus Kohlenstoffnanoröhren und Nanoscale surface-polariton spectros- dem Photosystem I copy by mid- and far-infrared near- 136 Seiten field microscopy ISBN 978-3-941650-17-6 144 Seiten ISBN 978-3-941650-24-4 Vol. 118 Martin Heiß Vol. 125 Growth and properties of low-dimensional Marco Andreas Höb III-V semiconductor nanowire hetero- Funktionalisierung von Gruppe structures IV-Halbleitern 172 Seiten 186 Seiten ISBN 978-3-941650-18-3 ISBN 978-3-941650-25-1 Vol. 119 Vol. 126 Sandro Francesco Tedde Daniel Claudio Pedone Design, fabrication and characterization of Nanopore analytics – electro-optical organic photodiodes for industrial and studies on single molecules medical applications 114 Seiten 277 Seiten ISBN 978-3-941650-26-8 ISBN 978-3-941650-19-0 Vol. 120 Vol. 127 Danche Spirkoska Jovanov Casimir Richard Simeon Katz Fundamental properties of self-catalyzed Multi-alloy structures for injectorless GaAs nanowires and related heterostructures Quantum Cascade Lasers 200 Seiten 131 Seiten ISBN 978-3-941650-20-6 ISBN 978-3-941650-27-5 Vol. 121 Vol. 128 Jürgen Sailer Barbara Annemarie Kathrin Baur Materials and devices for quantum Functionalization of group III-nitrides Information processing in Si/SiGe for biosensor applications 158 Seiten 215 Seiten ISBN 978-3-941650-21-3 ISBN 978-3-941650-28-2

Vol. 129 Arne Laucht Semiconductor quantum optics with tailored photonic nanostructures 232 Seiten ISBN 978-3-941650-29-9

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