Document [original]

Broadband ultrafast spectroscopy

on mixed–dimensional InAs/GaAs systems

vorgelegt von

Diplom–Physiker Mirco Kolarczik

ORCID: 0000-0001-8368-453X

von der Fakultät II — Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

— Dr. rer. nat. —

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. Stephan Reitzenstein (TU Berlin)

Gutachterin: Prof. Dr. rer. nat. Ulrike Woggon (TU Berlin)

Gutachter: Prof. Dr. rer. nat. Tobias Kampfrath (FU Berlin)

Gutachter: Dr. rer. nat. Jacek Kasprzak (Institut NÉEL,

Grenoble, Frankreich)

Tag der wissenschaftlichen Aussprache: 16. Januar 2019.

Berlin 2019

Daneben gibt es aber noch eine Reihe andrer Philosophen, die

die Möglichkeit einer Erkenntnis der Welt oder doch einer

erschöpfenden Erkenntnis bestreiten. […] Die schlagendste

Widerlegung dieser wie aller andern philosophischen Schrullen

ist die Praxis, nämlich das Experiment und die Industrie. Wenn

wir die Richtigkeit unsrer Auffassung eines Naturvorgangs

beweisen können, indem wir ihn selbst machen, ihn aus seinen

Bedingungen erzeugen, ihn obendrein unsern Zwecken

dienstbar werden lassen, so ist es mit dem Kantschen

unfaßbaren „Ding an sich“ zu Ende.

Friedrich Engels

Abstract

This thesis deals with light matter interaction and electron dynamics in coupled systems of nano-

structured III-V semiconductors, particularly with quantum dots embedded in quantum wells. Its

focus is on the role of dimensionality and the characterization of quantum dot states with respect

to their two- and many-level nature. The sample under investigation is a semiconductor optical

amplifier under standard operating conditions, in particular at room temperature.

An essential part of the work and a prerequisite of the experimental results is the re–

development of a setup for heterodyne detection of laser pulses. Compared to the established

pump–probe experiment, the duration of a single measurement was reduced by a factor of 100.

This allowed the introduction of simultaneous multi–power measurements that are used to in-

vestigate the non–linear response of the system. More than that, several novel techniques have

been developed: The Frequency Resolved Optical Short–pulse Characterization by Heterodyning

(FROSCH) for pulse shape analysis, the sideband pump–probe technique for low signal measure-

ments, and finally a method for two–dimensional coherent spectroscopy. All these techniques

can be used on the same setup without significant realignment. A detailed description is given in

part II.

In a first set of experiments we were able to observe Rabi oscillations of the quantum dot exciton

ground state at room temperature as a typical property of two–level systems. These observations

on a femtosecond timescale were enabled by the FROSCH technique when the ground state oscil-

lations caused a characteristic pulse shape modification.

On the other hand, the many–level characteristics dominated a large set of pump–probe mea-

surements presented in chapter 10. Under the assumption of a linear superposition of zero–

dimensional quantum dot states on the one hand and two- or three-dimensional continuum states

on the other hand, it was not possible to describe the electron dynamics. The introduction of

“crossed excitons” into the model allowed for a consistent explanation of the gain dynamics. These

crossed excitons are formed by Coulomb attraction of a zero–dimensional single carrier state and

another one of higher dimensionality. The crossed excitons affect the optical transitions as well as

the carrier diffusion in the quantum well.

The apparent contradiction in these findings with respect to the two- and many-level character-

istics is resolved in the final experimental part by two–dimensional coherent spectroscopy. This

technique uses white laser pulses and merges all technical approaches developed in the previous

parts. This allows the observation of the coherent coupling of states with large energy separation.

The reliability of the technique is proven by the extraction of dephasing times that are in excellent

agreement with literature values. In the two–dimensional spectra we find distinct signatures of

crossed excitons that can be investigated in detail by this novel technique.

Zusammenfassung

Diese Dissertation setzt sich mit der Licht–Materie–Wechselwirkung und der Elektronendynamik

in gekoppelten Systemen von nanostrukturierten III-V–Halbleitern am Beispiel in Quantenfilme

eingebetteter Quantenpunkte auseinander. Dabei wird insbesondere der Einfluss der Dimensionali-

tät untersucht und die Frage nach der Charakterisierung von Quantenpunktzuständen hinsichtlich

ihrer Eigenschaften als Zwei- bzw. Viel-Niveau-Systemen behandelt. Als Probe dient ein optischer

Halbleiter-Verstärker, der unter typischen Anwendungsbedingungen untersucht wird, also insbe-

sondere bei Raumtemperatur.

Wesentlicher Bestandteil der Arbeit und Grundlage der experimentellen Ergebnisse war die Wei-

terentwicklung eines Aufbaus zur heterodynen Detektion von Laserpulsen. Dabei konnte für die

etablierte Pump-Probe-Technik die nötige Messzeit um einen Faktor 100 verkürzt werden. Dies

ermöglichte auch die Einführung simultaner Messungen mit abgestuften Anregungsintensitäten

zur Untersuchung nichtlinearer Antworten des Systems. Darüber hinaus wurden mehrere neue

Techniken entwickelt: Die “Frequency Resolved Optical Short-pulse Characterization by Hetero-

dyning” (FROSCH) zur Analyse von Pulsformen, die Seitenband-Pump-Probe zur Messung sehr

kleiner Signale und zuletzt ein Aufbau zur zweidimensionalen kohärenten Spektroskopie. All die-

se Techniken sind ohne nennenswerte Umbauarbeiten am selben Versuchsstand nutzbar und im

Teil II der Dissertation detailiert beschrieben.

Zunächst konnten mit der FROSCH–Technik Rabi-Oszillationen des Exzitonen-Grundzustandes

als typische Eigenschaft eines Zwei-Niveau-Systems in Quantenpunkten bei Raumtemperatur

nachgewiesen werden. Die Beobachtung dieses Effekts auf der Zeitskala von Femtosekunden wur-

de durch die FROSCH-Technik ermöglicht. Das Oszillieren des Grundzustandes führt dabei zu einer

charakteristischen Pulsverformung.

Im Gegensatz dazu zeigte sich die Viel-Niveau-Charakteristik in einer Vielzahl von Pump-Probe-

Messungen, die in Abschnitt 10 präsentiert werden. Die erhobenen Daten konnten dabei nicht

im Rahmen einer linearen Überlagerung von nulldimensionalen Zuständen einerseits und zwei-

bzw. dreidimensionalen Zuständen andererseits erklärt werden. Durch die Einführung gekreuzter

Exzitonen (“crossed excitons”), die einen durch Coulomb-Weschselwirkung gebundenen Zustand

je eines null- und eines höherdimensionalen Einzelladungsträgerzustands darstellen, konnte eine

konsistente Erklärung für die beobachtete Verstärkungsdynamik geliefert werden. Dies gilt sowohl

in Hinblick auf die Anregung optischer Übergänge als auch bezüglich der Ladungsträgerdiffusion

im Quantenfilm.

Die Vereinigung dieser widersprüchlich anmutenden Ergebnisse hinsichtlich Zwei- oder Viel-

Niveau-Charakteristik erfolgt im letzten experimentellen Teil mit Hilfe der zweidimensionalen ko-

härenten Spektroskopie. Diese Technik vereint alle zuvor entwickelten technischen Ansätze und

ermöglicht durch den Einsatz von weißen Laserpulsen die Identifikation kohärenter Kopplungen

von Übergängen mit großem energetischen Abstand. Die Zuverlässigkeit der Technik wird durch

die Extraktion von Kohärenzzeiten der Quantenpunkte und den Abgleich mit Literaturdaten belegt.

Darüber hinaus weisen die zweidimensionalen Spektren deutliche Signaturen gekreuzter Exzito-

nen auf, womit erstmalig eine direkte Beobachtung dieser optischen Übergänge möglich wird.

iii

Contents

Acknowledgment vi

Introduction vii

I. Theoretical background 1

1. General concepts 2

1.1. The Fourier space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2. Electric field symbol conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3. Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Semiconductors 11

2.1. Carrier wavefunctions and their symmetry . . . . . . . . . . . . . . . . . . . . . . 11

2.2. Semiconductor classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3. Coupled semiconductor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. The theory of light–matter interaction 19

3.1. Matter acting on light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2. Light acting on matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3. Light interacting with matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

II. Experimental setup development 27

4. Established experimental concepts 28

4.1. Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2. Pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3. Fourier transform spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4. Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5. Multidimensional coherent spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 32

4.6. Pulse-shape analysis: FROG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5. New opportunities by fast data acquisition 37

5.1. Fast data acquisition approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2. The FROSCH technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3. Fast pump-probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6. Multiband detection for sideband pump–probe 44

6.1. Separation of multiple frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2. The sideband setup for low signals in Austin . . . . . . . . . . . . . . . . . . . . . 45

6.3. Modification of the Austin setup in Berlin . . . . . . . . . . . . . . . . . . . . . . . 47

Contents

6.4. Biexciton decay in colloidal PbS/CdS quantum dots . . . . . . . . . . . . . . . . . 49

6.5. Pathways for future development . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7. Two–dimensional coherent spectroscopy with white pulses 53

7.1. Concept of the heterodyne 2D coherent spectroscopy . . . . . . . . . . . . . . . . 53

7.2. Going digital: Changes enforced by the new HF2LI lock–in . . . . . . . . . . . . . 55

7.3. Pathways for future development . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

III. Investigation of subsystem coupling in the DWELL system 61

8. The sample 62

8.1. Sample structure and mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8.2. Amplified spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

9. Rabi Oscillations at room temperature 64

9.1. Pulse deformation caused by Rabi flops . . . . . . . . . . . . . . . . . . . . . . . . 64

9.2. Temporal vs. spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . 66

10.Evidence of Crossed Excitons in pump–probe experiments 68

10.1. Crossed Excitons revealed by gain excitation spectroscopy . . . . . . . . . . . . . 68

10.2. Carrier dynamics in a DWELL system . . . . . . . . . . . . . . . . . . . . . . . . . 71

11.Direct observation of Crossed Excitons in two-dimensional coherent spec-

troscopy 84

11.1. Alignment and measurement series . . . . . . . . . . . . . . . . . . . . . . . . . . 84

11.2. Crossed Exciton signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

11.3. Extraction of dephasing times from 2D spectra . . . . . . . . . . . . . . . . . . . . 88

IV. Summary 91

12.A summarizing outlook 92

12.1. The LabControl system: A versatile software concept for the optics lab . . . . . . 92

12.2. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A. LabControl XML file 99

Publications & Literature 108

Symbols 116

Abbreviations 118

Acknowledgment

Many people contributed to the work presented here. I would like to thank all of them and to

apologize to anyone who is not listed explicitly.

First of all, I would like to thank Ulrike Woggon, my principal investigator and supervisor. Her

trust and patience were the basis for many new ideas that took lots of time to evolve, but have

been finished succesfully.

During my membership in the Graduiertenkolleg 1558 (GRK 1558), Andreas Knorr acted as my

second supervisor: My cordial thanks for many discussions and a fruitful cooperation with his

entire group, especially with Sandra Kuhn.

The success of our work would have not been possible without the contribution of Nina

Owschimikow. As the team leader of the ultrafast spectroscopy sub–group she enabled the work

in the lab, deepened the insight into the physics, and guided the publication of our results. Widen-

ing our scope towards transitionmetal-dichalcogenides (TMDs) was one of her merits. More than

that, she became a real friend.

Many other colleagues contributed by discussions and hands–on assistance: Nicolai Grosse, who

recruited me for the group and brought in his profound knowledge of quamtum optics; Alexander

Achtstein contributed in many cases as an expert for nanocrystals; Yücel Kaptan, Bastian Herzog,

and Sophia Helmrich were the best lab mates one can imagine. And, finally, my student assistants

as well as bachelor and master students Mitra Pasche, Florian Buchholz, Kevin Thommes, and Aris

Simos brought new impulses into the group, everyone in her or his very own way.

Jenny Schwadtke, our secretary, deserves special thanks for shielding us from the administration

and for doing much more than her duty for us.

An invaluable stimulation of our experimental efforts was gained from the permanent interac-

tion with our colleagues of the Insitut für Theoretische Physik. Together with the group of Kathy

Lüdge and Eckehard Schöll and in particular with Benjamin Lingnau and Julian Korn, we developed

numerous experimental approaches and models for the description of carrier dynamics.

A highlight of my work was my research stay at the University of Texas (UT) at Austin with

the group of Elaine Li. The warmhearted welcome to her group and the collaboration during her

Humboldt stays in Berlin made this collaboration special, professionally as well as personally. I

owe special thanks to Akshay Singh, not only for introducing me to the Indian community of

Austin and to the “Crown and Anchor Pub”, but also for allowing me to contribute to his setup.

The work with him and Kha Tran granted me new insights into the TMD world and allowed first

applications of my ideas for multiband measurements.

Another special experience was my involvement in the PhotonX proposal for a Cluster of Ex-

cellence. The coordination of 25 principal investigators in the Berlin–Brandenburg area was chal-

lenging as well as exciting. Besides Ulrike Woggon, I would like to thank Stephan Reitzenstein and

Oliver Benson for this opportunity.

Finally, I’d like to express my gratitude to the institutions that enabled our research, namely

the Technische Universität Berlin (TU), the UT at Austin, the Deutsche Forschungsgemeinschaft

(DFG), the Sonderforschungsbereich 787 (SFB 787), and the GRK 1558. Not only their funding, but

also the countless people they represent were through all our work a solid foundation and reliable

backup.

Introduction

Over the past two decades the availability of mass data transfer had an enormous impact on the

lives of billions of people in the industrialized countries. Besides the achievements in solid state

physics and nanotechnology that enabled the miniaturization of processing units and data memo-

ries, the key technology of connecting the globe is the optical data transfer. The interface between

optics and electronics is still the hotspot of hardware development. Semiconductors are the ma-

terial system of choice to link both worlds. On established platforms—such as the III-V semicon-

ductors this thesis deals with—nanostructuring opens new avenues and generates a complexity

that is experimentally and theoretically challenging. On the other hand, promising new material

classes like the TMDs arise—and are nanostructured as well. Independent of the platform: Under-

standing interfaces and coupling mechanisms between subsystems is the path to future hardware

development.

Nowadays, hardware development does not exhaust the requirements of technology develop-

ment any more. “Big data” is one of the buzzwords of our days. The data processing structures do

not yet level the new abilities of modern hardware. Furthermore, the user interfaces for scientists

in the lab and the data handling and evaluation methods become a bottleneck for the efficient gen-

eration of ideas. While big companies like Google or Tencent address these problems using their

enormous manpower, comparatively small university labs like ours fall behind. From a societal

perspective, this means that publicly funded technology development is at risk.

This is the environment in which this thesis is settled. Before going into the physics, a look at

the experimental background is necessary. We inherited an established setup in 2011 on which

I collected the data for my diploma thesis. This heterodyne setup and its development are at the

core of the present thesis. Despite continuous data production, the original setup was not up–

to–date anymore: Single data point acquisition was on the timescale of seconds and there were

no functionalities for convenient alignment. The data transfer was digital, but via the outdated

General Purpose Interface Bus (GPIB) system which limited further development. In the course of

this thesis, the setup was improved tremendously, as will be described in detail in part II:

• Frequency Resolved Optical Short–pulse Characterization by Heterodyning (FROSCH) has

been developed for spectro-temporal pulse analysis, the More–Or–Less Characterization by

Heterodyning (MOLCH) became a valuable alignment tool.

• The traditional pump-probe (PP) data acquisition (DAQ) rate is today more than 1000 times

higher than five years ago, scanning a full curve about 100 times faster, while the data quality

has been significantly increased.

• While traditional PP requires signal levels of 10−2, the sideband PP has been developed for

the investigation of low signals in the range of 10−5without losing the phase resolution of

heterodyne detection.

• For traditional PP as well as FROSCH, the Simultaneous multi-power measurement (SMPM)

has been implemented to investigate pulse power dependencies with high reliability.

vii

Introduction

Figure 1:The logarithmic time scale visualizes the widespread number of events and circum-

stances this work deals with. The yellow flash in the millisecond time range indicates line noise

at 50 Hz and 100 Hz, respectively, posing an obstacle to every experiment based on electrical

signals. Important improvements of the experimental setup during this thesis are indicated by

dotted lines, notably leapfrogging the line noise range.

• The concepts of traditional PP, sideband PP, SMPM, and FROSCH have finally been

merged into a two-dimensional coherent spectroscopy (2DCS) setup that we refer to as

Supercontinuum–based Two–dimensional Observation of Radiation Coherence by Hetero-

dyning (STORCH). This setup also utilizes the broadband spectra of a fiber laser system

with supercontinuum generation, which enables the observation of state coupling over an

unusual wide energy range.

This development aimed to optimize the setup on every part of the timescale shown in fig. 1. The

FROSCH technique enabled us to have a look into the single cycles of our signals, corresponding

to a temporal resolution of about 100 as. This was a prerequisite for the results in chapter 9 that

required sub–pulse–duration time resolution not provided by PP experiments. The reduction of

lock–in integration times from the millisecond to the microsecond time range allowed the DAQ at

kilohertz rates and also reduced the susceptibility of the setup to acoustic noise and thermal drifts.

This fast DAQ approach discussed in chapter 5 enabled us to explore a much larger part of the

parameter space. The results in sections 10.1 and 10.2 are mainly based on this approach.

With FROSCH and STORCH, the concept of Fourier transform spectroscopy (FTS) has enriched

our methodological toolbox. This widened scope makes it necessary to emphasize the respective

terminology in this thesis. This is the motivation to start in chapter 1 with a detailed collection of

basic terms of Fourier analysis and a discussion of the frequency domain.

The sample under investigation is a semiconductor optical amplifier (SOA) with indium arsenide

(InAs) quantum dots (QDs) embedded in an indium gallium arsenide (InGaAs) quantum well (QW).

We nevertheless do not focus on technology development, but on the fundamental understanding

of the coupling mechanisms of these subsystems. The questions that arise under these circum-

stances are, for example: Does the QD preserve its characteristic properties as a discrete two level

system (2LS)? Is the coupled system a superposition of its constituents with defined transition

paths—or is the new system more than the sum of its parts? What channels interconnect the large

scale system in terms of excitation paths and carrier kinetics? And if there are new properties, can

we utilize them?

This thesis is a collection of answers to most of these questions. These answers have already

been published in various journal articles, but under varying circumstances. I would like to take

viii

the opportunity of this thesis to present the results in an overarching terminology and under con-

sideration of the development process. One important step to reach this goal is the introduction

of the four pulse scheme (4PS) in section 1.2.3. Although many graphs might seem familiar from

the former publications, almost everyone of them has been revised and adjusted to comply with

the symbols and theoretical background presented in the following chapters.

Part I.

Theoretical background

1. General concepts

1.1. The Fourier space

Understanding heterodyne experiments means understanding the frequency domain. Here and in

the following, any symbol νrefers to a frequency, while the analog symbol ωrefers to the angular

frequency with

ω= 2πν. (1.1)

For the sake of linguistic convenience, the term “angular” will often be omitted, the reader is asked

to decide on the basis of the symbols. In a similar manner, the photon energy

E=hν =~ω(1.2)

with the Planck constant hand the reduced Planck constant ~=h/(2π)will be used as an alter-

native representation of the frequency.

The one-dimensional Fourier transform (FT) Fof a function f:R→Cis defined as

F(ν) = F {f(t)}(ν) = Z∞

−∞

f(t′)e−i2πνt′dt′,(1.3a)

f(t) = F−1{F(ν)}(t) = Z∞

−∞

F(ν′)ei2πν′tdν′,(1.3b)

where a representation was chosen that omits prefactors and uses a notation in ν.18 From that

mathematical point of view, the Fourier transformation Fand the inverse Fourier transformation

F−1are structurally identical. All considerations in this section apply in both directions, although

from a technical point of view it might be easier to obtain sharp spectral lines in the lab than it is

to create short pulse durations. There is a reciprocal relation between pulse duration and spectral

width. This can be easily calculated for the important case of the Gaussian pulse with a spec-

trum FGauss(ν) = exp(−4 ln 2(ν/σ)2), where σdenotes the spectral full width at half maximum

(FWHM). The corresponding temporal signal is

fGauss(t) = σrπ

4ln 2 exp −π2σ2

4ln 2t2(1.4)

with a temporal FWHM of (2 ln 2)/(πσ). The time–bandwidth product for the Gaussian pulse is

therefore (2 ln 2)/π = 0.441, which is often used to approximate the minimal pulse duration for a

given spectrum or the necessary spectral width for a desired pulse duration.

In this calculation, a real function FGauss(ν)was chosen, which means a phase of zero for all

spectral components. This resulted in a real fGauss(t)centered at t= 0. A modification of the

spectral phase would have significant influence on the temporal signal: Adding a constant to the

spectral phase shifts the carrier wave temporally, while the envelope remains unchanged. A linear

tilt, on the other hand, shifts the temporal envelope. Finally, bending the phase function results

in a “chirped” pulse. The pulse duration is increased, the pulse is not transform–limited anymore.

1.1. The Fourier space

Figure 2:Three fundamental cases of FTs: a) A temporal Gaussian signal also has a Gaussian

spectrum. b) The exponential decay has a Lorentzian spectrum. c) A rectangular spectrum of

width eνresults in a temporal sinc-function with equally spaced zeros.

The pulse has a “positive chirp” if the components with long wavelengths arrive earlier than the

short wavelengths, which occurs upon normal dispersion in optical media.

Besides the Gaussian, there are two other fundamental Fourier transforms that will occur several

times. They are illustrated in fig. 2. In fig. 2 (b), the exponential temporal decay is shown, which

corresponds to a Lorentzian line in the spectrum. This case is important for optical transitions

(section 3.2.1) as well as for frequency filters (section 1.3.2). In fig. 2 (c), the FT correspondence of

the sinc and the rect function is shown. This correspondence is important for a spectral rectangle,

e.g. in the case of pulse amplitude shaping, but also for a temporal rectangle (section 7.3).

1.1.1. Products of signals: Convolution, cross–correlation, coherence

The multiplication of two monochrome signals, f(t) = exp(iωt)and g(t) = exp(ieωt), results in

f(t)g(t) = exp i(ω+eω)t(1.5)

and thereby creates a new mode ω+eω. The generation of new frequency bands will often be

caused by two or more original bands that mix in a non–linear element.

In interferometric measurements we will need to deal with products of more complex temporal

signals and their spectra, respectively. The convolution of two signals fand gis defined as

(f∗g) (t) = Z∞

−∞

f(t′)g(t−t′)dt′(1.6)

1. General concepts

and the well–known convolution theorem

F {(f∗g)(t)}(ω) = F {f(t)}(ω)·F {g(t)}(ω)(1.7)

says that its spectrum is the product of the spectra of the convoluted signals. In contrast to the

convolution, the cross–correlation is defined as

(f ? g) (t) = Z∞

−∞

f∗(t′)g(t+t′)dt′(1.8)

and its FT is given by

F {(f ? g)(t)}(ω) = F {f(t)}∗(ω)·F {g(t)}(ω).(1.9)

The mathematical cross–correlation leads directly to one of the central physical terms: the signal

coherence. The coherence of two signals fand gis given by

Cfg(t) = 

2 (f ? g) (t)

(f? f) (0) + (g ? g) (0).(1.10)

In the auto–correlation case f=g, the coherence is obviously Cff (0) = 1 and Cff (t > 0) ≤1.

One will usually observe a coherence decay when tincreases.

Signal convolutions are also very important for the FT of real world signals. As can be seen from

the definition of the FT in eq. (1.3a), infinite temporal information is required for a full FT. This

is not possible in general and sometimes not desired in particular. To restrict the Fourier analysis

to a certain range, a window function w(t)with w(t)→0for t→ ±∞ is applied. According to

eq. (1.7), the FT of f(t)w(t)is the convolution of F {f(t)}and F {w(t)}. In the case of a Gaussian

window the resulting transform

G(ω, t, ∆t) = 1

√2πZ∞

−∞

exp "−t′−t

∆t2#f(t′)e−iωt′(t)dt′(1.11)

is referred to as the Gabor transform (GT). The GT does only partially transfer information from

the time to the frequency domain and vice versa. ∆tis a tuning parameter that determines the

composition of spectral and temporal information, with G(ω, t, ∆t)→ F {f(t)}(ω)for ∆t→ ∞.

1.1.2. Composing signals: Modulation, repeating signals, discrete Fourier

transform

When two monochrome signals with different frequencies interfere, there is a periodic change be-

tween constructive and destructive interference. This oscillation of the signal envelope is called

beating. The beating is one example for the concept of signal modulations. In the time domain, a

modulation is the multiplication of a carrier signal fcar(t) = cos(ωt +ϕ)with a complex modula-

tion signal gmod =e

A0+e

Aexp(ieωt). The product of the real oscillations is now given by

fcar(t)gmod(t) = 1

2e

A0eiωt+iφ +e−iωt−iφ+e

Aei(ω+eω)t+e−i(ω−eω)t.(1.12)

In the frequency domain, the modulation is therefore represented by three components: the origi-

nal carrier band at ωand two sidebands, an upper one at ω+eωand a lower one at ω−eω. Without

a constant offset Re( e

A0), the carrier band would vanish. A signal can be modulated both, in am-

plitude and in phase. The particular type of the modulation is defined by the orientation of the

sidebands with respect to the main band. This is illustrated in fig. 3.

1.1. The Fourier space

Figure 3:Illustration of amplitude modulation (a) and phase modulation (b). In the upper row,

the signal is represented by a complex vector that varies temporally (shades of blue). This

variation can be approximated by a sideband (red arrow) that is added to the average value of

the oscillating signal. In the middle row, the contributions of the sidebands are displayed. The

lower row shows the resulting signals (red) in comparison to the main band (blue).

This example showed that a modulation of the time signal is represented by a replication of the

main band in form of a sideband. The concept works, of course, also the other way round: A repeti-

tion of the temporal signal causes a modulation of the frequency signal. This is especially the case

for pulsed lasers, where the time signal (the pulse) is not only duplicated, but repeated infinitely.

With a repetition rate bνrep, the corresponding spectrum is not only modulated by cos(ν/bνrep), but

frep(ν) = X

n∈N

cos ν

nbνrep =X

n∈N

δ(ν−nbνrep).(1.13)

The spectrum of a pulsed laser is therefore composed of sharp laser modes. Although a real world

laser does not provide δ-like modes, the mode spacing bνrep can exceed the mode width easily by a

factor of 106.19 This extreme reduction of the spectrum is at the core of heterodyne detection: All

this free spectral range in between the sharp laser modes can be used to encode information.

Mathematically very similar to the repeating laser are some considerations about the discrete FT.

The discrete sampling of a continuous signal will also result in an artificially repeating spectrum.

This effect is called aliasing. The second problem of discrete sampling is the impossibility of infinite

temporal information. In the most simple case, the measured signal will be a multiplication of the

actual signal and a rectangular window function. Also here, the measured spectrum will be a

convolution of the original signal and the applied sampling window according to eq. (1.6). These

facts result in some basic rules for discrete FT:

• Increasing the density of sampling points results in a wider spectral range due to the avoiding

of aliasing.

• An increased temporal range will improve the spectral resolution because the corresponding

window function spectrum is narrower.

1. General concepts

• It should be avoided to apply a rectangular window unless the sampled signal vanishes at

the border of the sampling window. As its FT is an extended sinc-function, the rectangular

window will result in a “ringing”. The spectrum can be improved at the expense of peak–

broadening by the numerical application of a window that vanishes at the border.

1.1.3. Spatial Fourier transform: The k-space

Now that we already touched on laser resonators, it is time for a brief discussion of space and

its Fourier correspondent, the k-space. The laser resonator is one particular representation of the

general particle in a box problem. A closed box of width L(in one dimension) restricts the allowed

spatial frequencies to wavelengths λwith nλ/2 = L(n∈N). The respective wave numbers are

kn=2π

λn

=π

Ln. (1.14)

The allowed spatial frequencies are therefore homogeneously distributed in the k-space.

Closely related to the particle in a box is the finite potential pot. This problem will become very

important under varying aspects. There is no conceptual difference between the treatment of light

and matter in this respect. In the case of light, the Helmholtz equation

∆ + ω2

c02n(r)E(r)= 0 (1.15)

can be used to calculate resonator modes or waveguide modes in a fiber based on the refractive

index landscape n(r). In the case of matter, we use the time–independent Schrödinger equation

Hψ(r) = −~

2m∆+V(r)ψ(r) = Eψ(r)(1.16)

to determine particle states in a potential landscape V(r). Depending on the symmetry of the

potential landscape, the coordinate system and a corresponding representation of the Laplace op-

erator ∆is chosen.

We restrict the discussion here to the textbook case of one dimension x, a particle obeying

eq. (1.16), and a finite potential pot of width L. This will be the typical approach to determine

quantum dot (QD) states. The potential is given by

V(x) = −V0:|x| ≤ L/2

0 : else.(1.17)

The problem can be solved in k-space using the ansatz ψ(x) = aiexp(kx) + ajexp(−kx). Here,

we need to distinguish between the different cases

ψ(r) = 





a1exp (koutx) + a2exp (−koutx) : x < −L/2

a3exp (kinx) + a4exp (−kinx) : |x| ≤ L/2

a5exp (koutx) + a6exp (−koutx) : x > L/2

(1.18)

with wave vectors outside and inside of the pot of

kout =√−2mE/~(1.19a)

kin =p−2m(E+V0)/~.(1.19b)

1.2. Electric field symbol conventions

The continuity of ψ(x)and ∂xψ(x)restricts the coefficients ai. Two cases need to be discussed.

Bound states exist for −V0< E < 0. Outside the potential pot, ψ(x)decays exponentially

(a2=a5= 0). Inside the pot, the solution is given by sine (a3=a4) and cosine (a3=−a4)

functions. Inside and outside wave vectors are related by

kout = tan kin

2kin.(1.20)

Both, eqs. (1.19) and (1.20), cannot be fulfilled for arbitrary energy values E. In the general case,

a numerical solution is necessary. In the limit case of V0→ ∞, representing the particle in a box,

the allowed energies are given by

En+V0=h

8m∗Ln2.(1.21)

For E>0, there are no more bound states. Although these states will still be calculated accord-

ing to the particle in a box model, e.g. using periodic boundary conditions, we will refer to those

states as continuum states. The continuum has an extension far beyond L, which makes the energy

separation between the single states (eq. (1.21)) negligible. There are still different wavevectors in-

side and outside the pot according to eq. (1.19), but now also eq. (1.19a) is imaginary. Continuum

states are extended states oscillating at a higher spatial frequency above the pot.

1.2. Electric field symbol conventions

1.2.1. Temporal and spectral representation

In the context of this thesis it will be unnecessary to consider electric fields as vectors. An electric

field is therefore a real scalar. For the description of oscillating fields, however, it is convenient to

define the complex electric field E(t). We will omit a special symbol for the real part of the electric

field and specify it by Re(E(t)) if necessary. The complex field spectrum E(ν)is defined as

E(ν) = F {E(t)}.(1.22)

The same symbol Ewill be used for both, temporal and spectral representation, thus it will be

depicted explicitly as E(t)or E(ν), respectively. In cases of a discrete spectral mode distribution,

E(ν)will be replaced by Eν.

1.2.2. Frequency ranges

Electrical frequencies will span an enormous range in the following. This has been already indi-

cated in fig. 1. From the highest frequencies in the THz range to switching frequencies at kHz to

thermal drifts of optical paths on the timescale of minutes: From the perspective of the field de-

scription there is no fundamental difference. My goal is to present an overarching description of

all these frequency ranges. A labeling convention will ease the identification of the functionality

of a certain frequency:

Light fields: ν.Light fields that will be found mostly in the range around 250 THz will be labeled

without an additional character on top.

Laser mode spacing: bν.Frequencies bνin the range of the laser repetition rate bνrep = 75.4 MHz

will be labeled with a hat. This includes modulation frequencies that are used to shift laser modes

by acousto-optic modulators (AOMs). As the subtraction of bνrep will often result in an almost

equivalent result, a prime will indicate a respective shift: bν′=bν−bνrep.

1. General concepts

Figure 4:The general four pulse scheme (4PS). The pulses are labeled A to D with corresponding

delays tij.

Slow modulation frequencies: eν.Switching and modulation frequencies in the range from 0 Hz

to few MHz will be labeled by a tilde.

Resulting signals may include all these components. In this case and under consideration of sec-

tion 1.1.2 it is important to have in mind that often ν+eν≈ ν.

1.2.3. The four pulse scheme

Different types of experiments will bring in different terminology. For example, in our publications

the terms “reference pulse”1–5,7,10,12 and “local oscillator”9,15 refer to one and the same pulse. The

usage depends on the experimental context.

One goal of this thesis is to give an overarching description of the experiments. Therefore I

decided to use a unified terminology to describe all heterodyne experiments. The pulse labeling

follows the terminology of two-dimensional coherent spectroscopy (2DCS).20 The pulse sequence

is shown in fig. 4. Four pulses will be sufficient to describe any experiment, they are labeled from

A to D. In this scheme, pulse D will correspond to the “reference pulse”. On the other hand, the

traditional 2DCS labeling of pulse delays τ,T, and tis unhandy. The τ, for example, will be

reserved for decay times within this thesis. Pulse delays are alternatively denoted tij where iand

jdepict the respective pulses (in lower case). Thus the traditional 2DCS notation corresponds to

this notation according to τ≡tab,T≡tbc, and t≡tcd.

1.3. Transfer functions

A transfer function (TF) is a mathematical expression that describes the modification of a signal

upon the passage of an object. There are numerous objects, e.g. semiconductor devices (samples)

or electrical frequency filters, that modify a passing signal. Therefor it will be useful to define a gen-

eral terminology for this work, although this terminology might deviate from the strict definition

of TFs in linear system engineering.21

1.3.1. Differential gain and phase

In our experiments, the phase of the signal waves is very important, so the signals will be described

by their complex field representation E(ν)rather than by their power. For a single element, a TF

is given by

Eout(ν) = TEin(ν) = GeiΦEin(ν)(1.23)

with the (field) gain G∈Rand the (field) phase shift Φ∈R. In general, Tcan depend on

numerous parameters, typically the signal frequency ν, the time tafter an excitation of the object,

1.3. Transfer functions

or the power Pof the excitation. If we compare the transfer functions for two parameter sets

A={tA, PA, . . .}an B={tB, PB, . . .}, we define

ln 10

10 ∆G(ν, A, B) + i∆Φ(ν, A, B) = ln Eout(ν, B)

Eout(ν, A)= ln TB(ν)Ein(ν)

TA(ν)Ein(ν)(1.24)

=ln "GB(ν)eiΦB(ν)

GA(ν)eiΦA(ν)#=ln GB(ν)

GA(ν)+i[ΦB(ν)−ΦA(ν)]

with the differential gain ∆G∈Rand the phase difference ∆Φ ∈R. As the exact input field

Ein(ν)is commonly unknown, the differential approach in eq. 1.24 is often the only quantitative

experimental access.

Especially for semiconductor devices it is appropriate to replace 1.23 by a description that takes

propagation into account. In this case we define analog to the Lambert–Beer law a

Eout(ν, x) = T(ν, x)Ein(ν, 0) = e(g(ν)+iϕ(ν))xEin(ν, 0) (1.25)

with the gain coefficient gand the phase coefficient φ. For a device of length Lthis means g=

(ln G)/L and φ= Φ/L. Equation 1.24 in this case becomes

ln 10

10 ∆G+i∆Φ = ([gB−gA] + i[φB−φA]) L(1.26)

which means that this representation eliminates the effect of propagation and the measured value

∆Glinearly corresponds to the material property gB−gA.

1.3.2. Frequency filters

In many cases a wide spectrum needs to be reduced to a narrow area of interest by electrical

filtering. Perfect spectral cuttoffs cannot be realized, of course, since they require infinite temporal

information. Based on resistors (R), coils (L), and capacitors (C), different configurations of RLC

circuits can be used for a variety of analogue filter types. In many applications, analogue filters are

replaced by digital ones nowadays, but the basic terms to describe filter properties are identical.

The most frequently used terms, cutoff frequency and roll-off, can be derived from the example of

a Butterworth filter.22 This class of filters is designed to provide flat spectral gain in the pass band.

The gain function is given by

Tn(bν) = "n

k=1 ibν

bνcut −exp i(2k+n−1)π

2n#−1

(1.27)

with cutoff frequency bνcut and the order nof the Butterworth filter.23 For n= 1 this becomes

T1(bν) = 1

bν

bνcut +1(1.28)

At the cutoff frequency the power gain drops to |T1(bνcut)|2= 0.5≈ −3 dB and is therefore

often referred to as the 3 dB frequency. In a RLC filter the order ndepicts the sum of numbers

of coils and capacitors, respectively. Often filters of even order n= 2mwith mpairs of coil and

capacitor are used. In this case the gain shows a roll-off for high frequencies, i.e. it decreases by

0.5n≈ −6 dB ·mas the frequency doubles according to eq. (1.27). Therefore specifications like

6 dB/octave are often used to characterize filters. It is also common to denote a time constant

τTC ≡bν−1

cut to visualize the delay of a changing signal that comes with the cutting of the high

frequencies. The actual filter settling time depends on the filter order (higher order means longer

settling time) and exceeds the time constant roughly by a factor of 5 for a first order filter.

1. General concepts

Digital low pass filter

A digital low–pass can be implemented in different ways. In the later used Zurich Instruments

(ZI) HF2LI lock–in, an exponential running average filter is used that resembles the properties of

eq. (1.28).24 A data point xnacquired with sampling time ∆tis added to the previous average value

¯xn−1

¯xn= ¯xn−1exp −∆t

τTC +xn1−exp −∆t

τTC .(1.29)

This method is elegant as it only needs a single memory and mimics the Butterworth filter well. In

section 7.3, however, we will discuss a case where this kind of filtering might pose an obstacle to

the ultimate exploitation of the frequency range in the future.

2. Semiconductors

2.1. Carrier wavefunctions and their symmetry

Before we start to discuss the coupling of semiconductor systems, some fundamental terms should

be reviewed. The basic statements of solid state physics are predominantly variations of the particle

in a box problem (section 1.1.3) under certain restrictions.

2.1.1. Extended states

In extended crystals, the atoms form a periodic lattice. The periodicity puts a boundary condition

to the wavefunctions of the electrons. The simple vacuum dispersion relation

Evac(k) = ~2

2m0

k2(2.1)

with wavevector kand electron rest mass m0is replaced by bands of allowed states. These states

are distributed homogeneously in k-space. Of particular interest are the the lowest energy states

of these bands, the so called band edge. At this energetic minimum, a Taylor expansion allows to

appoximate the dispersion relation E(k)by a parabola, which is again quite similar to eq. (2.1). In

correspondence, one defines the effective mass

m∗=~2∂2E

∂k2−1

(2.2)

of a carrier in the crystal based on the band curvature in k-space. Based on the homogeneous

distribution of states in the k-space and the band functions E(k), one can calculate a density of

states (DOS) D(E). The DOS is an important factor that determines the probability of optical

transitions.

Carriers occupy the low energy bands first up to the Fermi level, which is predominantly defined

by the available number of carriers in the neutrally charged crystal. The occupied bands are called

valence bands (VBs), the unoccupied bands of higher energy conduction bands (CBs). Materials are

classified as semiconductors if there is an energetic band gap between VB and CB of up to several

eV. Carrier transitions from CB to the VB correspond to photon energies from the infrared (IR)

to the ultraviolet (UV) range. This makes semiconductors the ideal interface between optics and

electronics.

To describe the carrier dynamics, it is useful to call occupied states in the CB “electrons” and

unoccupied states in the VB “holes”. The determination of states based on the particle in a box

model and periodic potentials has neglected the charge of these particles. In analogy to the hydro-

gen atom, electrons and holes can form bound states that are called excitons. These excitons can

be described as a new type of quasi–particle with a single wave function.25

2.1.2. Confined states and Dimensionality

The extension of a semiconductor can be confined in one or more dimensions. This has crucial

impact on the DOS. For each of the three dimensions, the crystal can either be extended, with a

2. Semiconductors

Figure 5:Density of states for the three confinement cases that will be important in the following.

continuuous DOS (section 2.1.1), or confined, with discrete energy levels (section 1.1.3). We call a

crystal n-dimensional, if it is extended in nof theses three dimensions. Three different cases will

occur in the following, they are illustrated in fig. 5: The three–dimensional (3D) or bulk material

has a D3D(E)∝√E. The two–dimensional (2D) case is referred to as quantum well (QW) and has

a stepwise increasing DOS. A semiconductor crystal with strong confinement in each direction, i.e.

the zero–dimensional (0D) case, is called QD. It has a discrete DOS which qualifies it to be called

an “artificial atom”. QDs will be the primary subject in the following.

A epitaxially grown QD will not be spherical, but rather a truncated pyramid.26 This is compa-

rable to the case of a strong confinement in zdirection and a moderate confinement in xand y

direction, respectively. According to eq. (1.21), the energy contribution of the zconfinement is

large, while that of the xy plane confinement is smaller. Typically, this leads to an exciton ground

state (GS) of the QD that is s-type in all direction. Above this GS, there are two states that are

p-type in one of the planar directions. This state is therefore twofold degenerate and referred to

as the first excited state (ES) of the QD.

2.1.3. Optical transitions

On a quantum mechanical level, the transition of carriers from one state to another upon optical

excitation can be described by first order perturbation theory. The Hamiltonian of the system is

split into a Hamiltonian b

H0of the unperturbed system, and a Hamiltonian b

H1that represents the

interaction of the external field and the system. The calculation results in Fermi’s Golden Rule

Γﬁ=2π

~D(Ef)⟨Ef|b

H1|Ei⟩2(2.3)

that describes the transition rate Γﬁfrom an initial state |Ei⟩to a final state |Ef⟩. This description

is actually not limited to optical transition, but describes any transition triggered by an external

perturbation.

In eq. (2.3), two factors contribute to the transition rate: Firstly, there is the DOS D(Ef)—without

a final state, there is no transition. Secondly, the transition rate is determined by the integral

2.2. Semiconductor classes

⟨Ef|b

H1|Ei⟩.Quantum confinement is an important technique to increase the overlap of the wave-

functions and to thereby maximize the integral. Additionally, this factor results in selection rules

depending on the respective symmetry of initial and final state under the particular perturbation.

Some transitions might be (dipole) forbidden. However, if the ideal system is modified, for instance

by the contribution of Coulomb interaction or by complicated potential landscapes, some selection

rules might be lifted. An exact theoretical description is extremely demanding.

2.2. Semiconductor classes

2.2.1. Epitaxially grown III-V Semiconductors

III-V semiconductors are basically composed of group III elements—aluminum (Al), gallium (Ga),

or indium (In)—and group V elements—nitrogen (N), phosphorus (P), arsenic (As), or antimony

(Sb). Their bandgap energies span almost the entire semiconducting range, from InAs at 0.35 eV

to AlN at 6.2 eV. By mixing of elements, for instance in InGaAs, the bandgap energy can be tuned.

III-V semiconductors are usually epitaxially grown. The standard techniques are molecular beam

epitaxy (MBE) and, more industrial, metalorganic vapor–phase epitaxy (MOVPE). In both cases,

new material is deposited on a substrate layer. The tunability of the band gap together with the

elaborate fabrication techniques make III-V semiconductors an important platform for a wide range

of applications.

Epitaxial growth requires that the lattice constant of the deposited material matches the lat-

tice constant of the substrate within a certain tolerance. A mismatch induces strain and thereby

increases the surface energy. The formation of islands can minimize the surface energy and be

therefore favorable. Depending on the surface energies of the materials and the lattice mismatch,

three different growth modes occur: The layer–by–layer growth (Frank–van der Merwe growth),

the formation of islands (Volmer–Weber growth), and the formation of islands on top of a single

layer (Stranski-Krastanow (SK) growth).26 This single layer is also called wetting layer (WL). For

example, InAs with a lattice constant of 0.606nm and GaAs with 0.565 nm at room temperature

have a lattice mismatch of 7.3 %. The formation of SK QDs is a self–organized process. This re-

sults in QDs of varying size and therefore varying transition energies. This energy distribution is

referred to as the inhomogeneous broadening of the QD ensemble. For typical QD ensembles, the

inhomogeneous broadening of the GS can be approximated by a Gaussian with a FWHM σGof

10 eV to 100 eV.26 A small inhomogeneous broadening is often desired to increase the DOS for a

desired wavelength.

Binding energies of excitons and exciton complexes in InAs/GaAs and its nanostructures are in

the range of <10 meV.27–29 Compared to the roomtemperature kB300 K ≈25 meV, these binding

energies are usually negligible and the quasiparticles therefore unstable. In our nanostructures, the

confinement energy is the dominant contribution to the exciton energy levels.

2.2.2. Colloidal PbS/CdS quantum dots

Although we are mostly dealing with III-V semiconductor, we will investigate colloidal PbS/CdS

core-shell30,31 QDs in a side–project. Unlike the InAs QDs, the PbS/CdS QDs are created in a wet–

chemical process.32 These QDs are also interesting for photonics and opto–electronics applications

as their emission wavelength is tunable from 900nm to 2000nm, covering all important telecommu-

nication windows33. In order for systems based on colloidal QDs to compete with the established

technology, e.g. In(Ga)As structures, data rates are required to lie in the Gbit/s range, correspond-

ing to one bit per 1 ns. An important aspect that needs consideration is the long lifetime of the

2. Semiconductors

exciton GS in colloidal QDs, exceeding this time by up to three orders of magnitude.33Fast data ma-

nipulation mechanisms therefore needs to be based on an efficient nonlinear process34,35 or the GS

lifetime has to be dramatically reduced. For all–optical signal conversion, the relevant processes

are in particular fast multi–exciton scattering events. A competing process is efficient broadband

intraband absorption, which has been observed in pump-probe (PP) experiments36. Recently, it

has been demonstrated that the effect of intra- and interband processes cancels at a specific wave-

length, making ultrafast switching operations based on nonlinear interaction possible.35

During the development process, spin–coating is a suitable fabrication technique for an easy

combination of waveguide structures and optically active colloidal QDs. One drawback of spin–

coating is the comparatively small overlap between the QD emission mode and the waveguide

mode.4The QDs are simply deposited on top of a chip with silicon nitride waveguides, while in

more sophisticated techniques an integration of the QDs into the waveguide is possible, which

increases the overlap. Silicon and silicon nitride (Si3N4) are important platforms in nanophotonics,

with the perspective of chip-sized devices based on a CMOS-compatible fabrication technology.37

Si3N4can be manufactured into high quality low-loss waveguides and resonators.38–40 Additionally,

a range of hybrid photonic devices based on silicon nitride structures has been demonstrated.41–46

In contrast to silicon, the material does not display large intrinsic nonlinearities or a substantial

free carrier absorption,47 and therefore allows us to observe the nonlinear response of the QDs

without waveguide–induced background.

The technical challenges that we will address in section 6.4 are in particular the low mode over-

lap and the fact that most studies on the ultrafast dynamics of similar samples have been performed

using amplified laser systems with kHz repetition rate. For the high repetition rates we use, tend-

ing towards 100MHz and thereby towards telecom applications, the long GS lifetime will have a

distinct effect on the observed effects.

2.2.3. Transitionmetal-Dichalcogenides

The material class of transitionmetal-dichalcogenides (TMDs) has attracted tremendous interest

of the semiconductor community since the crossover from an indirect to a direct bandgap in the

monolayer limit was demonstrated.48 TMDs have a chemical composition of the type MX2with a

transition metal ion M—molybdenum (Mo) or tungsten (W)—and two chalcogen ions X—sulfur (S),

selenium (Se), or tellurium (Te)—that form 2D layers. The bulk material is composed of stacked

layers held together by van der Waals interactions. Especially the compounds of Mo/W and S/Se

have been investigated by many groups, as their lowest exciton energies are in the range of 650 nm

to 850 nm, which is addressable by the popular Ti:sapphire laser. But also the investigation of

MoTe2with transition energies in the near infrared (NIR) has been intensified recently.16,49

In TMDs, very large binding energies have been observed. The exciton binding energy can

be larger than 300 meV, which is roughly two orders of magnitude more than in GaAs.50,51 Also

the trion binding energy is in the range of 30 meV.52,53 These high binding energies make these

quasiparticles interesting for opto–electronic applications at room temperature. More than that,

indirect excitons in TMD heterostructures are discussed in the literature,54 which is very similar

to the discussion of crossed excitons (CEs) later on (section 2.3.4 and chapters 10 and 11.

Simple monolayer TMD samples are prepared either by exfoliation or by chemical vapor de-

position. The optical excitation geometry is usually perpendicular to the TMD monolayer plane.

Although TMD monolayers absorb about 10 % of the incident cw radiation,55 the signals in the

non–linear experiments we perform are still on the level of 10−4. This made a modification of our

pump-probe (PP) detection scheme necessary that will be discussed in detail in section 6.2.

2.3. Coupled semiconductor systems

Parameter GaAs57 InAs58

E(0) [eV] 1.517 0.415

α[µeV K−1]550 276

β[K] 225 83

Table 1:Varshni parameters for GaAs and InAs.

2.3. Coupled semiconductor systems

2.3.1. Heterostructures and diodes

Heterostructures are a spatial combination of different semiconductors that are used to tailor a

spatial potential landscape. In epitaxially grown III-V semiconductors, this is achieved by a modi-

fication of the composition of the deposited material, for instance the ratio of In and Ga in InGaAs

crystals.26 But also TMDs are used to create heterostructures by stacking monolayers of different

types.56 Additional dopands can be used to shift the Fermi level inside the band gap either towards

the VB (p-doping) or the CB (n-doping). The creation of heterostructures is a key to realize new

technical applications. Besides transistors, the most important basic type of such heterostructures

are pin-diodes. These diodes are composed of an intrinsic (undoped, i) layer sandwiched between

ap- and an n-doped bulk material. In the contact region, a band bending due to carrier redistri-

bution equilibrates the Fermi level. The doped areas allow for carrier transport. Under forward

bias (+at p,−at n), carriers can be injected into the iregion. Under reverse bias (−at p,+at n),

carriers are extracted.

Electro–optical devices are often variations of pin-diodes with an optical active material in the

i-region. While there are also surface emitting devices (emitting in growth direction), we will deal

with edge emitters (emitting perpendicular to the growth direction). Semiconductor lasers, light

emitting diodes (LEDs), semiconductor optical amplifiers (SOAs), detectors, and solar–cells are in

principle variations of the same device—with totally different optimization demands, of course,

and emission will be only observed for direct band gaps. On one hand, an unbiased device acts as

a solar–cell, while detectors have usually a high reverse bias to extract newly generated carriers

efficiently from the active region. Under moderate forward bias, on the other hand, they start to

spontaneously emit photons. For higher forward bias above the lasing threshold, edge emitters

start lasing if no precautions are taken. To suppress lasing, SOAs have anti–reflection coated

waveguide facets. In the amplifying regime, the emission is referred to as amplified spontaneous

emission (ASE).

Electrical carrier injection always comes along with heat generation which affects the band gap

energy. The temperature dependence is well described by the empirical Varshni formula

E(T) = E(0) −αT2

T+β(2.4)

with phenomenological parameters αand βaccording to table 1 for the important cases of GaAs

and InAs.

2.3.2. The dot-in-a-well system

For the purpose of data transmission, the optical emission needs to fit the telecom windows around

wavelengths of 1.3µm (O-band) and 1.55 µm (C-band). These windows provide minimal absorption

in optical fibers that allows for long range optical data transfer. Tuning the emission wavelength of

2. Semiconductors

InAs QD to these transmission windows was a long development process.26,59 Besides the emission

wavelength, also the efficient carrier injection into the QDs was a development goal.59.

Investigating a single QD of the dot–in–a–well (DWELL) ensemble is quite challenging. In con-

trast to non–overgrown SK QDs, it is more difficult to identify a single dot in the ensemble.60 There

are also some other issues that need to be considered: On one hand, the optical excitation perpen-

dicular to the QD plain differs from that in–plane. On the other hand, the potential landscape of

the free–standing DWELL differs from that in the active region of the diode. The resulting states

might be affected by the modified environment.

2.3.3. Rate equation modeling

Rate equations are a simple yet powerful method to describe basic carrier kinetics in coupled sys-

tems. Rate equations represent in a simplified picture what Fermi’s Golden Rule (2.3) tries to

explain microscopically: There is an initial state N1, a final state N2, and transition rates Γij that

represents transitions from Njto Ni. The nature of the state is not important here, it acts as a “bin”

with population Ni. For these two states, the rate equation system is

N1= Γ11N1+ Γ21N2(2.5a)

N2= Γ12N1+ Γ22N2.(2.5b)

The rates Γii can be interpreted as decay (or feeding) rates coupling the system to its environment.

For more states, a matrix notation

N= Γ ·N(2.6)

is more appropriate. Just like eq. (2.3), the transition matrix Γmight depend on the number of

available final states. Especially in fermionic systems, where Pauli blocking needs to be consid-

ered, there will be a population–dependent transition rate Γ(N). This makes eq. (2.6) a nonlinear

differential equation (DEQ) which requires a numerical solution. In section 10.2.1, however, we

will even in the case of electrons discuss a linear rate equation model. In this case, for a static Γ,

eq. (2.6) can be solved by the determination of the eigenvalue and eigenvectors of Γ. The eigenval-

ues −τ−1

irepresent decay rates with the corresponding eigenvector

Niexp −t

τi.(2.7)

The numerical determination of eigenvalues and eigenvectors and the transformation of the prob-

lem to the according basis is much faster than the solution of the DEQ using methods like Runge–

Kutta. This allowed us to perform the calculations presented in section 10.2.1 on a standard desktop

PC.

In the extended 2D systems of QWs, we also will observe Brownian motion, i.e. diffusive pro-

cesses. These are described by a second order DEQ

∂t%(x, t) = D∂2

x%(x, t)(2.8)

for the density %of particles at a distance xfrom the origin with a diffusion constant D. This DEQ

has the solution

%(x, t) = N

√4πDt exp −x2

4Dt(2.9)

2.3. Coupled semiconductor systems

Figure 6:Illustration of Crossed Excitons. (a) An electron in the QD can either form a QD exciton

together with a hole that is also confined in the QD, or it can form a CE together with a hole

in the 2D continuum of the QW. (b) As both, QD exciton and CE state, share one carrier, there

is an intrinsic coherent coupling between these states that form a V system together with the

crystal ground state |0⟩.

for Nparticles in the system. In our case, the case x= 0 will be of particular interest, because

this will represent the QD in section 10.2.1. The density at this point is %(x= 0, t)∝1/√twith a

“decay constant”

τ(t) = −%(x= 0, t)

∂t%(x=0, t)∝t(2.10)

which is no constant, of course, but a linearly increasing value where t= 0 represents the δ-like

distribution according to eq. (2.9). This linear rise of the decay time characterizes the Brownian

motion in contrast to the first order linear DEQ characterized by eigenvectors according to eq. (2.7).

A representation of the Brownian motion in terms of rate equations requires a discretization

of the space. Every bin of the rate equation model represents a certain volume in space. A good

approximation of the diffusion behavior can be achieved by a random walk, which is characterized

by an equal probability to transit in any direction. Of course, bin sizes and bin geometry need to

be considered. An example will also be given in section 10.2.1.

2.3.4. Crossed Excitons

Excitons are bound states of electrons and holes, they are described analog to the hydrogen atom.61

The necessary Coulomb attraction will also be present if one carrier occupies a bound state (bound

with respect to the potential landscape), the other one a continuum state. The formation of a

hydrogen–like state is not forbidden by the different dimensionality of the states. In part III we will

find evidence for the existence of such states in the systems we investigate. Similar observations

can be found in the literature, and we decided to use the term crossed exciton (CE) in Ref. 2 and

derived it from the argumentation in Refs. 62, 63.

To evaluate the relevance of these anticipated states, several typical assumptions about excitons

need to be reviewed. First of all, it is necessary to leave the picture of photoluminescence (PL)

experiments. In these experiments, carriers are created with high excess energy and emission is

predominantly detected from the long–lived GS. Excitons need to be formed first and their binding

energy needs to exceed kBTto be detected in PL. In III-V semiconductors, this requires cryogenic

temperatures. In our experiments, however, we will deal with distinct non–equilibrium states. If

these states are occupied directly by optical excitation or from classical QD excitons, they might

2. Semiconductors

play a role at least on the ultrafast timescale. The optical excitability is the second aspect to be

reviewed: Within QDs, some transitions are forbidden by selection rules due to the wavefunction

symmetry. Such selection rules might be lifted under the formation of bound states from carriers

of different dimensionality and under the influence of Coulomb attraction. The third aspect is the

interaction with the surrounding environment. In the vacuum, a hydrogen atom either exists as

a bound state or it does not. In a semiconductor, scattering processes might keep a dissociated

exciton together, as the movement in the environment is rather diffusive than ballistic, as we will

discuss in section 10.2. Also effects like Anderson localization might come into play.

The theoretical handling of these states is challenging. Our observations will here stay on a

phenomenological level that is supposed to motivate a deeper investigation of the theory of CEs.

3. The theory of light–matter interaction

The scope of the theoretical description of light–matter interaction is set by the methods applied

and the samples investigated. While many fundamental properties can be derived in a fully classi-

cal theory of light as well as matter, the nature of QDs enforces a quantum–mechanical description

of matter. Such a semi–classical description, however, will be sufficient, as the experimental re-

sults presented in this thesis are obtained in the many–photon limit. Although the term “photon”

will be widely used in a phenomenological way and the heterodyne setup has been used to perform

experiments in the few–photon limit9, a detailed description of the quantization of the light will

be omitted.

The very fundamentals for the description of light–matter interaction can be found in Maxwell’s

equations:

∇·D(r, t) = ρ(r, t)≡0(3.1a)

∇·B(r, t) = 0 (3.1b)

∇×E(r, t) = −∂tB(r, t)(3.1c)

∇×H(r, t) = j(r, t) + ∂tD(r, t)≡∂tD(r, t).(3.1d)

Here it is already indicated by ≡that for our description of dielectric media the free charge–

carrier density ρand the free current density jcan be widely neglected. The interesting material

properties are contained in the electric displacement field Dand the magnetizing field H:

D(r, t) = ε0E(r, t) + P(r, t)(3.2a)

H(r, t) = 1

µ0

B(r,t)−M(r, t)≡1

µ0

B(r,t).(3.2b)

From eqs. (3.1) and (3.2) one can derive the wave equation

∆E(r, t)−1

∂2

tE(r, t) = µ0∂2

tP(r, t).(3.3)

3.1. Matter acting on light

It will be sufficient from now on to discuss the respective fields in one dimension. As the inves-

tigated semiconductors can be considered non–magnetizable (M(r, t)≡0), the description of

light–matter interaction shrinks down to the description of the relation of the electric field Eand

the polarization P. This relation is given by the electric susceptibility χ(E)that can be written as

a Taylor expansion

P=ε0χ(1)E+χ(2)EE +χ(3)EEE+. . ..(3.4)

The linear term χ(1) does not need a detailed discussion. Also the χ(2) contribution is of minor in-

terest, firstly, because χ(2) is suppressed in general by the crystal symmetry, and secondly, because

χ(2)-effects are not investigated. The χ(3)-term, however, will be of major importance. We will

here focus on four-wave mixing (FWM) as the most important non–linear effect in the following.

3. The theory of light–matter interaction

Figure 7:FWM in the box geometry. The original beams and the mixing bands can be filtered

spatially. Only the relevant mixing cases that incorporate exactly one conjugate field areshown.

3.1.1. Four wave mixing

FWM is very closely related to the four pulse scheme (4PS) that we will use to describe the exper-

iment in chapter 4. Three incident waves (A, B, and C) mix according to the χ(3)-nonlinearity of

the medium:

PFWM =ε0χ(3) E0,aeika·r+iωat+E∗

0,ae−ika·r−iωatE0,beikb·r+iωbt+E∗

0,be−ikb·r−iωbt

E0,ceikc·r+iωct+E∗

0,ce−ikc·r−iωct.

(3.5)

An illustration of a FWM experiment in the box geometry is given in fig. 7. There are eight different

contributions to PFWM which result in different fields obeying energy and momentum conserva-

tion. These resulting fields will be very important later on. To distinguish them, we will use the

notation

Ea¯

bc(t)∝ Ea(t)E∗

b(t)Ec(t) = E0,aE∗

0,bE0,cei(ka−kb+kc)r+i(ωa−ωb+ωc)t,(3.6)

so Ea¯

bc originates from pulses A, B, and C. The bar on top of the “b” indicates that the complex

conjugate wave has been used. Not all possible combinations are important to us, because all of

our incident pulses will be in the NIR range and the resulting field needs to be as well, in order to

be observed. For example, Eabc will oscillate at a frequency of ωa+ωb+ωc, which is out of our

spectral range. Relevant for us are all combinations that incorporate exactly one conjugate field.

These combinations are also displayed in fig. 7.

3.1.2. Frequency conversion

Every gain medium is limited to a certain frequency range. Although there are established gain

media from the Mid-IR to the UV range64 and free–electron lasers provide almost arbitrarily tun-

able coherent radiation, a typical optics lab needs to rely on frequency conversion to generate

desired frequencies from standard sources. In the visible (VIS) and UV the generation of higher

harmonics is a standard approach. However, the discussion of available methods is restricted to

the ones applied later on.

Ti:sapphire laser and parametric down–conversion

For 30 years now the Ti:sapphire laser has been the dominating solid state laser and has revolu-

tionized non–linear optics. Typical mode–locked Ti:sapphire lasers are built as free–space cavities

3.2. Light acting on matter

with a repetition rate around 80MHz. The emission is centered around 800 nm and is tunable from

600 nm to 1000 nm. For the investigation of transitions with longer wavelengths, an optical para-

metric oscillator (OPO) can be used for the frequency conversion. The OPO is basically a resonator

(with a length adjusted to the laser resonator length) with a non–linear crystal inside. In the non–

linear crystal, the pump photons of frequency νpare split into two photons of lower energy in

a parameteric downconversion process. The higher frequency photon νsis referred to as “signal”,

the lower one at νias “idler”, with νs+νi=νp. In one series of experiments (chapter 9) we will

use the signal photons created by an OPO.

Supercontinuum generation

The frequency range covered by the OPO is restricted by the pump pulse bandwidth due to the

defined photon–breakup in the phase–matched crystal. In highly non–linear devices that do not

fulfill such conditions, higher order mixing processes can lead to the generation of a so called

supercontinuum. The supercontinuum can cover more than one octave and is quasi–continuous,

but is still based on the resonator modes. The original laser modes have frequencies

νn=bνoﬀ +n·bνrep (3.7)

with an offset bνoﬀ and an equal mode spacing of bνrep. The generation of the supercontinuum can be

considered as a cascaded FWM process, and like in the FWM case only such processes are relevant

that incorporate one conjugate field. This, however, results in the cancellation of two of the bνoﬀ

and thereby forces all newly generated modes to also fulfill eq. (3.7).

3.2. Light acting on matter

For the understanding of the mechanisms of light–matter interaction, it is necessary to relate the

macroscopic parameters like χto microscopic processes on the level of single or few charge carriers.

In a first step, the oscillating light field is treated as an external and stiff boundary condition. It is

not modified by its action on the polarized matter.

3.2.1. The Hertzian dipole

In classical physics, a microscopic explanation of the optical susceptibility χis given by the

Hertzian dipole. We consider a single electron of charge eand mass m0. A displacement rof

this electron causes a proportional restoring force, so the electron movement under the influence

of the external field is that of a driven damped harmonic oscillator with the natural frequency ω0.

The inhomoegenous differential equation is

¨r+ 2γ˙r+ω2

0r=q

m0E0e−iωt(3.8)

and its homogeneous solution is

rhom(t) = Ae(−γ+iω0)t+c.c. (3.9)

under the assumption that γ≪ω0, which means that the reduction of the carrier frequency due

to the damping is neglected. Also the inhomogeneous solution is easily found:

rinh(t) = −1

ω2−ω2

0+2iωγ q

m0E0e−iωt.(3.10)

3. The theory of light–matter interaction

Figure 8:The Bloch sphere and the Bloch vector ρ.

By applying the near–resonant approximation ω2−ω2

0≈2ω0(ω−ω0), this solution can be written

in the more familiar form

rinh(t)≈ − 1

2ω0ω−ω0

(ω−ω0)2+γ2−i·γ

(ω−ω0)2+γ2q

m0E0e−iωt(3.11)

where the imaginary part of rinh(t)has the shape of a Lorentzian. The FWHM of this function is

2γ. As this is a property of the single oscillator, it is referred to as the homogeneous linewidth in

contrast to the inhomogeneous broadening, which is an ensemble property. For the interpretation

of two–dimensional spectra, the phase of the response function will be of particular interest later

on, so we calculate it from eq. (3.11) and find

ϕ(ω) = arctan −γ

ω−ω0.(3.12)

This describes the typical behavior of a driven harmonic oscillator: For ω→0, we find in–phase os-

cillation (ϕ→0), while counter–phase oscillation (ϕ→π) appears for ω≫ω0. At the resonance

there is a rapid phase change described by the derivative

dωϕ(ω)=1

1 + −γ

ω−ω02·γ

(ω−ω0)2=γ

(ω−ω0)2+γ2⇒d

dωϕ(ω0)=1

γ,(3.13)

so the phase slope at the resonance can be used to directly derive γ.

3.2.2. Optical Bloch equations

The classical description has revealed the fundamental mechanism that mediates the energy trans-

fer from one oscillator (the light field) to another (the dipole). It was also possible to derive the

spectral lineshape from the fact of a finite lifetime. But of course we know today that in the micro-

copic world we have to deal with quantized energy. Equations like eq. (3.10) that predict a linear

amplitude response to an E0increasing towards infinity must find their limitations in a quantum

world.

The quantum nature of matter must be taken into account, while the oscillatory behavior of the

light field can be further described in the previous classical way. In this semi–classical approach,

3.2. Light acting on matter

the Hamiltonian is composed of an unperturbed term which represents the energy stored in the

two level system (2LS) of eigenfrequency ω0and an interaction term. The temporal evolution

of the system is determined by the Schrödinger equation. An analysis yields the optical Bloch

equations25,65

˙u= (ω−ω0)v, (3.14a)

˙v=−(ω−ω0)u+ Ωw, (3.14b)

˙w=−Ωv(3.14c)

for an oscillatory electric field E(t) = E0exp(iωt)of frequency ωin the rotating wave approxi-

mation (RWA). The coupling to the electric field is given by the Rabi frequency

Ω = dE0

~.(3.15)

The pseudospin vector ρ= (u, v, w)Tis called the Bloch vector with |ρ| ≡ 1for an isolated

2LS. Its components uand vare the dispersive and absorptive components of the dipole moment,

respectively, while the component wis the inversion. The constant length of the Bloch vector

defines the Bloch sphere that is illustrated in fig. 8. It is important to have in mind that the Bloch

vector is defined under the RWA in a rotating frame according to the respective light frequency. If

the external field amplitude is reduced to zero, the temporal evolution is governed by the detuning

ω−ω0. In the resonant case ω=ω0, the Bloch vector would be static.

A fundamental property of the dynamics governed by eq. (3.14) are Rabi oscillations. Temporal

derivation of eq. (3.14b) and application of eqs. (3.14a) and (3.14c) yields

¨v=−(ω−ω0)2+ Ω2v(3.16)

which is the DEQ of a harmonic oscillation. For ρ(t= 0) = (−1,0,0)Tand under the condition

|ρ| ≡ 1, this results in

w(t) = −

(ω−ω0)2+ Ω2cos q(ω−ω0)2+Ω2t

(ω−ω0)2+Ω2.(3.17)

In the resonant case, the inversion oscillates between −1and 1at a frequency of Ω. With detuning,

the amplitude of the oscillation is reduced and the frequency increased to q(ω−ω0)2+ Ω2.

For the problems discussed in the following, we will use an average Bloch vector to describe the

state on an ensemble, with a Bloch vector length of |ρ| ≤ 1. In the real world, the coupling to the

environment will also cause a polarization decay as well as an inversion decay. These decays are

introduced phenomenologically to eq. (3.14) by decay constants:

˙u= (ω−ω0)v−u

,(3.18a)

˙v=−(ω−ω0)u−v

+Ωw, (3.18b)

˙w=−Ωv−w+ 1

.(3.18c)

These decay constant T1and T2will be crucial for the interpretation of our experiments. T1

describes the decay of the inversion, it represents the lifetime of a population. The decay of the

system polarization, on the other hand, is described by T2. This process is referred to as dephasing.

3. The theory of light–matter interaction

Figure 9:Illustration of a mechanical analog for the modification of a propagating light field. The

initial field from the laser is fed into the coupled pendula system by a weak coupling constant.

The coupled pendula exchange energy analog to the mutual energy exchange of light and matter

during Rabi oscillations. Similarly published in Ref. 1.

3.3. Light interacting with matter

3.3.1. Maxwell–Bloch equations

The Maxwell–Bloch approach is quite exactly what its name tells us: a combination of the Maxwell

equations (eq. (3.1)) and the Bloch equation (eq. (3.18)).66,67 While the Bloch equations determine

the inversion and polarization dynamics under the external perturbation, the Maxwell equations

are used to calculate the feedback of the polarization that modifies the electric field. Additionally,

a system environment can be modeled in the simple form of rate equations (section 2.3.3). Our

colleagues Kathy Lüdge, Julian Korn, and Benjamin Lingnau used a full model like this to calculate

the effect of Rabi oscillations induced by a laser pulse on this pulse itself:

∂tpj

m=−i∆ωj

mpj

m−iΩ

2ρj

e,m +ρj

h,m −1−1

m(3.19a)

∂tρj

b,m =−Im hΩpj

∗i−Wmρj

e,mρh,m +∂tρj

b,mcol (3.19b)

∂twb=J

e0−BSwewh−2NQD X

j,m

vmf(j)∂tρj

b,mcol (3.19c)

∂z+1

∂tE(z, t) = 1

iωγ

2εbgε0

P(z, t).(3.19d)

These equations are taken exactly from Ref. 1 and do not follow the nomenclature of this thesis,

of course. A brief discussion is worthwhile, because this equation system merges the numerical

approaches discussed so far. The first line, eq. (3.19a), describes the polarization. It is a complex

entity that represents both, uand v, in the form of p≡u+iv, with indices jrepresenting the

QD subensemble and m∈ {GS,ES}accounting for ground and excited state. The second line,

eq. (3.19b), describes the inversion ρ≡wwith b∈ {e,h}depicting electrons and holes. These

first two lines are basically the Bloch equation. The third line describes the population change of

the reservoir w. This reservoir wis coupled to the inversion ρby transition rates represented by

(∂tρj

b,m)|col. Finally, the last line, eq. (3.19d), is the Maxwell wave equation in the plane-wave and

the slowly-varying amplitude and phase approximation.67

3.3. Light interacting with matter

3.3.2. A mechanical analog

The full Maxwell–Bloch equations 3.19 are a powerful tool to evaluate light field and QD dynam-

ics. There is, however, only a very limited intuitive access. To illustrate the underlying physical

concepts, it is possible to reduce the system to an analogue mechanical toy model. We used this

model to understand the basic mechanisms of pulse deformation under the influence of Rabi os-

cillations.1The description of this toy model in Ref. 1 is in some ways misleading, although the

calculations are carefully performed and the conclusions will remain valid under the revision given

in the following paragraphs.

As a first step of simplification of the eq. (3.19) we neglect many aspects of the full model: The

QD ensemble is reduced to a single representative, only a single charge–carrier type is investigated

(pj

m→p,(ρj

e,m +ρj

h,m −1) →ρ), the reservoir is cut off (wb≡0), spontaneous recombination is

assumed to be infinitely slow (Wm≡0), propagation is ignored (neglect z), and the QD transition

energy shall match the light field central frequency (∆ωj

m≡0). This results in a simplified set of

equations:

∂tp=−iΩ

2ρ−1

p(3.20a)

∂tρ=−Im [Ωp∗](3.20b)

∂tE(t) = iωγ

2εbgε0

p. (3.20c)

From a mathematical point of view, the eqs. (3.20a) and (3.20b) are identical to the problem of two

coupled pendula as sketched in fig. 9. The equations of motion for the displacement x= (xp, xρ)T

are

∂2

tx+β

0(∂tx) + ω2+D−D

−D ω2+Dx=xext(t)

0.(3.21)

To evaluate the pulse deformation, it is necessary to look at eq. (3.20c). The pulse modification is

proportional to the polarization pendulum displacement. The resulting pulse would be the sum of

the original pulse (xext) and the contribution of the QD polarization. In Ref. 1, we reduced this

problem and negelcted the contribution of xext. This is done legitimately, as the single oscillator

model only represents an infinitesimal modification and the resulting modification still needs to be

calculated iteratively, i.e. by consideration of propagation. This, however, would leave the realm

of a toy model. Instead of choosing an arbitrary ratio for the summation of the original excitation

and the modification by polarization, we neglect the original pulse.

It is important to stress that the Rabi oscillations are not derived from this coupled system, but

rather constructed by the implementation of the coupling constant D. The mechanical analog

therefor resembles the dynamics of the system under Rabi oscillations, but does not replace the

derivation given in section 3.2.2. A much more sophisticated mechanical model can be found in

Ref. 68.

Part II.

Experimental setup development

4. Established experimental concepts

The sophisticated setups presented in the later chapters have not been build from scratch. There

are different types of experiments that have either been built by previous generations of scientists

in our lab or that inspired the development of new experimental configurations.

4.1. Heterodyne detection

The terms heterodyne and homodyne detection originate from radio frequency engineering, but

the concept has been used in IR and VIS optics for more than 25 years now.69 In both cases, a

signal wave of carrier frequency bνis superposed with a local oscillator (LO) (homodyne: bνLO ≈bν;

heterodyne: bνLO =bν) and the interfering signals are mixed by a non–linear element. In the optical

case, the LO and signal wave are laser modes and the non–linear element is simply the detector

(that is linear in power and quadratic in field).

The heterodyne concept is thus the concept of modulation and demodulation. Like for an ana-

log radio, the transmitter encodes information by modulating it onto a defined carrier band. The

receiver knows the band and demodulates the transmission accordingly. The very same is done

in the optical case: The set of laser modes is shifted by a defined frequency and demodulated later

on. The three frequency regimes introduced in section 1.2.2 become important for optical hetero-

dyning. The demodulation process is in fact twofold: The optical interference, which results in a

detectable beating, demodulates the light field oscillation from the THz (ν) to the MHz (bν) range.

The detector signal is subsequently demodulated to the kHz range (eν) which can be analyzed by

the computer. The signal quality of the optical interference is often improved by a balanced detec-

tion.70 This type of detection provides common mode rejection and suppresses all signals that do

not arise from the interference of the signal wave and the local oscillator.

4.1.1. Spectral shifting by acousto–optic modulators

Frequency shifts in the MHz range can be achieved by AOMs. An AOM can be understood both, in

the wave and in the particle picture. In the wave picture, the acoustic field in a medium creates a

Figure 10:Beams deflected by an AOM under consideration of momentum conservation. (a) Dis-

persion separates white light at single AOM frequency bν. (b) Application of two frequencies

bν1and bν2creates two beams from monochromatic light. (c) For a focused incident beam, the

overlap of beam cones can be spatially filtered in case b) (same applies for a)).

4.1. Heterodyne detection

periodic density modulation which comes along with a periodic refractive index modulation. This

periodic modulation acts as a Bragg grating where an incoming beam is diffracted. Besides the

transmitted beam, several orders of diffracted beams may appear. The particle picture, however,

is more intuitive: The AOM provides a field of phonons. Passing photons with frequency νin and

momentum ~kin may be scattered by a phonon of energy hbνphon and momentum ~kphon. Using

energy and momentum conservation, the resulting photon is defined by

hνout =hνin +hbνphon (4.1a)

~kout =~kin +~kphon.(4.1b)

Looking at the transmitted beam, the diffraction acts as an intensity modulation. The diffracted

beam, on the other hand, is frequency shifted. The AOM is in this case often referred to as an

acousto–optical frequency shifter (rather than a modulator). An amplitude modulation can also be

achieved by electro–optical modulators that utilize e.g. the Pockels effect. The distinct advantage

of AOMs is the easy selection of a single sideband by spatial filters due to eq. (4.1b).

This spatial selection becomes more complicated in the case of broadband laser pulses or if a

superposition of frequencies is applied to the AOM. Each combination of photon and phonon mo-

mentum can result in a different diffraction angle according to eq. (4.1b). These cases are illustrated

in fig. 10. In the white light experiments developed later on, the use of an aperture in the cone

overlap (fig. 10 (c)) of a focused incident beam was a useful trick to achieve the desired output.

4.1.2. Heterodyne detection with lock-in amplifiers

A lock-in amplifier is an adjustable band-pass filter that allows signal analysis in amplitude and

phase by homodyne detection. It consists of a signal demodulator of adjustable frequency and

subsequent low-pass filters. By a phase–locked loop, the demodulator can be locked to an external

reference frequency that is used as the local oscillator bνLO. Demodulation is performed by mul-

tiplication of the signal by cos(ωLOt)(denoted as Xor “in-phase” component) and sin(ωLOt)(Y

or “quadrature” component), respectively, so the signal is transformed to a rotating frame. In this

rotating frame, low-pass filters are applied that are adjustable by their filter order n(“slope”) and

their time constant ω−1

cut.

Lock–in amplifiers are nowadays available with frequency ranges up to several hundred MHz.

With slower lock–ins it is necessary to choose bνphon close enough to bνrep, so that the interference

of a mode with a neighboring mode at bν′

phon =bνphon −bνrep can be observed.

4.1.3. Heterodyne spectral interferometry

Although not used in our experiments, an advanced approach of heterodyne detection is worth a

brief discussion: the heterodyne spectral interferometry.71,72 This technique is not based on lock–

in detection. The modulation/demodulation concept is based solely on AOMs, while the low–pass

function is emulated by the comparatively slow charge–coupled device (CCD). After the sample,

the probe and the reference beam both enter a demodulation AOM set to the respective frequency

difference of the desired signal band. The angle between probe and reference beam is aligned to

the deflection angle of the first order of the AOM, so that the reference is deflected to the probe

beam and vice versa. The AOM power is set to 50:50 deflection ratio. The desired mode is thereby

shifted to 0 Hz and all other modes cancel out on the CCD.

This method has two advantages: A standard grating spectrometer can be used to detect a spec-

trally resolved signal and there is no frequency limitation by a limited bandwidth of the lock–in.

4. Established experimental concepts

Figure 11:The 4PS of a PP experiment. In this case, pulse A also acts as pulse B which prepares

a system population. Pulse D (“reference pulse”) is superposed with pulse C (“probe pulse”) to

maximize the heterodyne signal.

Drawbacks, on the other hand, are the more complex setup and alignment as well as the impos-

sibility of multi–band detection (chapter 6). Nowadays, lock–ins with far beyond 100MHz band-

width are available, so the entire relevant frequency range defined by the laser repetition rate is

covered. Also spectral analysis will in our case be done by Fourier transform spectroscopy (FTS)

(sections 4.3 and 5.2) rather than by grating spectrometers. Heterodyne spectral interferometry,

however, impressively demonstrates that the concept of heterodyne detection is not limited to

electronic filtering.

4.2. Pump-probe spectroscopy

In ultrafast spectroscopy, pump-probe (PP) experiments are used to reveal the temporal evolution

of a system state after an initial perturbation by a strong pump pulse. Of course, the system evolu-

tion is not accessible directly, but needs to be recovered from the temporal evolution of the sample

TF. A probe pulse with a defined time delay follows the pump pulse and passes under a modi-

fied TF. The differential gain and phase of the on and off case can be determined as described in

section 1.3.1. From that, one can recover the actual processes in the system.

The 4PS is shown in fig. 11. The probe pulse is represented by pulse C. The pump pulse that

generates a population modification in the sample is represented by a combination of pulse A and

pulse B. This description has multiple advantages and will become more clear in the discussion

of the 4PS of 2DCS in section 4.5.1. The pump probe delay is therefore tbc which is in this case

equivalent to tac.

4.2.1. By heterodyning

In heterodyne PP experiments, both, probe and reference pulse, are taken from the original laser

beam and shifted by a specific frequency marker: the pump by bνab, the probe by bνc. Pulse A and

B are completely identical in this case. The frequency marker makes the pulses which pass the

sample distinguishable even in collinear and co–polarized geometry. This is particularly useful

for samples in waveguides that restrict the experimental geometry. To distinguish the pulses, the

signal beam is interfered with the reference pulse D. The reference delay tcd is usually kept constant

at the signal maximum that defines the temporal overlap tcd = 0.

The data acquisition in the setup we inherited8was straightforward: After setting the pump–

probe delay tbc, the pump was switched on and off. For every state a lock–in value S(on)

cand S(oﬀ)

was acquired, respectively. According to 1.24 the differential gain ∆Gand phase difference ∆Φ

are given by

ln 10

10 ∆G(tbc) + i∆Φ(tbc) = ln 

S(on)

c(tbc)

S(oﬀ)

c+iϕ(on)(tbc)−ϕ(oﬀ).(4.2)

4.3. Fourier transform spectroscopy

4.2.2. Chopping setups

Heterodyne detection of a beat frequency due to frequency shifts as described above is not the only

pump-probe detection scheme using lock–in technique. In many cases an amplitude modulation

of the pump beam73 or both, pump and probe beam,53 is sufficient to retrieve the desired signal.

While mechanical chopping is limited to few kilohertz, AOMs are nearly unlimited in frequency

(up to several 10MHz) and allow for arbitrary modulation functions. Unlike heterodyne detection,

these techniques do not provide phase information as interference with a reference beam is omit-

ted. The major advantage, however, is superior sensitivity. While our standard heterodyne setup

(section 5.3) requires averaging below differential signals of 1 % (0.04 dB) the standard chopping

setups detected signals below 10−5(0.000 04 dB).

If pump and probe beam are not collinear, modulation of the pump at 2eνab is sufficient to gener-

ate a signal: The pump modulation creates a modulated excitation, this results in a modulated probe

signal. Since the pump beam can be spatially filtered and does not hit the detector, the modulation

imprinted on the probe is the only signal at 2eνab and can be detected easily by the lock-in amplifier.

In the 4PS the pump modulation is displayed as a superposition of two sidebands, namely pulse A

with frequency νa=νab −eνab and pulse B with νb=νab +eνab. Here it becomes evident why it

is useful to represent the pump pulse by two constituent pulses in the 4PS.

In collinear geometry or for very low signal levels this is more difficult: The pump beam cannot

be separated, its modulation is therefore also detected, and will dominate the lock-in signal. One

circumvents this problem by modulation of the probe beam at eνc. Instead of eνab the lock-in is

now locked to (eνab −eνc), i.e. one of the sidebands resulting from the modulation of the pump

imprinted to the already modulated probe signal. Neither pump nor probe alone carry this differ-

ence frequency, so their impact is filtered out by the lock-in. However, in this scheme there is

no qualitative difference between pump and probe anymore, both are modulated the same way.

This results in an undesired “probe-pump signal” that overlaps the pump-probe signal if both are

on the same intensity level. For this double-chop technique, pump power exceeding probe power

becomes crucial.

In sections 6.2 and 6.3 chopping techniques and frequency shift heterodyning will be applied in

a hybrid setup to combine the sensitivity of chopping setups and the advantage of phase resolution

and probe isolation of the frequency shift technique.

4.3. Fourier transform spectroscopy

Although one usually has prisms and gratings in mind when it comes to spectral analysis, a totally

different type of spectrometry shows superior performance in a wide range of applications: Fourier

transform spectroscopy (FTS). None of these commercial spectrometers has been used for this

thesis, but the basic idea of FTS was the seed for our development of the Frequency Resolved

Optical Short–pulse Characterization by Heterodyning (FROSCH) method (section 5.2) and the

elaborate theory of FTS18 has been and will be helpful to improve our approach.

The key idea of FTS is to obtain the spectral composition from a spatially resolved interference

signal by Fourier transformation. Broadband light sources and a Michelson interferometer are

used to create an interference pattern I(x)depending on an optical pathway difference x. For

monochromatic light of wavelength λ, a periodic sinusoidal signal is obtained. For a continuous

spectrum of light this periodicity allows to identify contributions of specific wavelengths. As

shown in section 1.1.2, the spectral resolution is here defined by the total range of x, while the

spectral range depends on the spatial sampling rate.

4. Established experimental concepts

Figure 12:The 4PS of FWM experiments. The traditional FWM experiments do not resolve the

frequency dependence, but rather scan the envelope of the signals.

The main advantages of FT spectrometers are high optical throughput, simultaneous broadband

measurements (multiplexing), and nevertheless high spectral resolution by only using a single

detector. Standard FT spectrometers cover pathway differences from 1m to 5 m. The limiting

factor for broadband measurements are the spectral limitations of beamsplitters, coatings, and

detector sensitivity. By replacing these components the spectral range can be extended.18

4.4. Four-wave mixing

The theoretical aspects of FWM have already been described in section 3.1.1. In PP spectroscopy,

carrier population dynamics are investigated and coherent effects are widely neglected. FWM

experiments, on the other side, focus on the polarization dynamics. The inherited setup we mostly

used for PP experiments has originally also been designed for FWM experiments. We will not

conduct FWM experiments in the original way. In our context, FWM is the fundamental ingredient

for 2DCS, so a brief review of this particular history will suffice.

Relevant for us are especially experiments in the self–diffraction configuration.74 In this configu-

ration, only two pulses are used instead of three. One of these two acts twice: It forms the transient

grating together with the second pulse and is also diffracted. The according 4PS is shown in fig. 12,

where the second pulse represents pulse B and pulse C as well. The heterodyne implementation of

FWM does not rely on spatial k-vector selection, but on frequency filtering.75 It is easy to see from

eq. (3.6) that the frequency identifies that band as well as the k-vector does. In the case illustrated

in fig. 12, the band of interest is bν¯acc, which oscillates at 2bνc−bνa.

The typical question in FWM experiments is the determination of the T2time of an optical

transition.70,76 Already in the early experiments, both delays, tab and tcd, have been scanned. A

scan of tcd is referred to as “time–resolved FWM”, while the signal integrated over tcd is called

“time–integrated FWM”. The delay between the two pulses, tab, reveals the coherence properties

of the material. The ability of the two delayed pulses to form a transient grating depends on the

ability of the material to maintain the coherence. This coherence decays with T2. If this decay

is measured over tab, this allows the direct determination of T2. For InAs quantum dots in SOAs

without carrier injection, T2decreases from 630 ps at cryogenic temperatures77 to 220 fs at room

temperature78.

4.5. Multidimensional coherent spectroscopy

Multi-dimensional coherent spectroscopy (MDCS) combines the concepts of FTS and FWM. The

observed signal is the very same as in FWM, but now it is not only “time–resolved”, but even

“phase–resolved”. The concept of MDCS originates from nuclear magnetic resonance (NMR).79

From this starting point in the radio frequency range it evolved to optical frequencies up to the

4.5. Multidimensional coherent spectroscopy

Figure 13:Excitation scheme for 2D spectroscopy. Four pulses (A to D) depicted by color interact

with two transitions ν1(solid Bloch vector) and ν2(open Bloch vector). Depending on the

respective delays, the resulting signals oscillates between the four extreme cases illustrated in

the grey Bloch spheres of pulse D. This oscillation enables us to identify the coupling by a 2D

FT.

VIS range. From a theoretical point of view, the concept of MDCS can be applied in an arbitrary

number of dimensions.80 In the following, however, the scope will be restricted to the 2D case.

4.5.1. Signal generation in 2DCS

In 2DCS three pulses (A – C) generate a fourth wave analog to the FWM case. The excitation

scheme is shown in fig. 13 and the typical explanation is as follows:81 Pulse A creates a coherence

in the medium. This coherence evolves during the time period tab (≡τin the 2DCS literature)

until pulse B arrives. Pulse B stores the phase of the initial coherence in a population state. The

population state decays during the period tbc (≡T) until pulse C arrives. This pulse generates a

coherence that radiates the signal during the time period tcd (≡t). The last pulse D interferes with

this radiation and thereby probes the result with phase resolution.

A detailed walk–through

The meaning of this description is illustrated in a case–by–case analysis in the Bloch sphere in

fig. 13. We assume two coupled transitions with eigen–frequencies ν1and ν2, respectively. In

a simplified case, we assume that pulses A and B are resonant to ν1, while pulse C is resonant

to ν2. It is very important to understand which part of the frequency information is encoded in

which delay and why it can be deciphered by FT. For this illustration we use π–pulses, but the

argumentation can be easily applied to the general case. As described in section 3.2.2, the Bloch

4. Established experimental concepts

vector is defined with respect to the driving field and preserves its orientation in the uv–plane

upon resonant excitation. However, there is not only one driving field in a three–pulse experiment.

Every pulse defines its own rotating frame and therefore its own Bloch vector, so the Bloch spheres

in fig. 13 are diplayed in the respective colors. The transition between two Bloch spheres depends

on the relative phase of the electric fields and thus on the respective delay. After the initial creation

of the coherence in the transition ν1by pulse A (red Bloch sphere), we can distinguish two extreme

cases. In the first case (upper row) the delay tab is chosen with respect to the eigen–frequency ν1

in a way that tabν1∈N. In this case, the second excitation is in–phase with the first excitation

and the Bloch vector is flipped to the north pole of the sphere. In the other case (lower row), the

delay tabν1+ 1/2∈Nand the second interaction has an inverse phase, so the Bloch vector is

flipped to the south pole. So the variation of the delay by half a wavelength results in different

scenarios, in this extreme case even in a change of the prepared population from w= 0 to w= 1.

The important thing is now, that the information about this change is encoded in the delay tab,

so a FT of any signal depending on this walong the delay tab will result in a Fourier component

unequal zero.

With this preparation of pure population, no phase dependent events are expected during the

period tbc. In many experiments in the literature the delay tbc ≡Tis therefore kept constant. If

there is a relaxation path from transition ν1to ν2, pulse C will find an initial population of this

transition depending on the phase scenario described above. Pulse C acts on the system analog

to pulse A: It creates a coherence (but now in the transition ν2) that radiates the FWM signal

during the period tcd, which is probed by pulse D with phase sensitivity. Here it is important to

understand that the result generated by pulse D depends in an oscillatory manner on two delays:

On one hand on tab with an oscillation frequency ν1(due to the initial state of pulse C, prepared

by pulse B), and on the other hand on tcd with a frequency of ν2. Both frequencies, ν1and ν2, can

be extracted by FT of the respective delay.

4.5.2. Presentation and interpretation of 2D spectra

Compared to the complexity of the 2DCS experimental concept, the presentation of extracted data

is quite intuitive. The two–dimensional FT transforms the oscillatory FWM patterns into a 2D

energy map. On the yaxis, the pump photon energy Epump is shown that corresponds to the

transformed delay tab. From the example process above, the association with a pump process can

be understood as the transition ν1is initially populated and therefore “pumped”. On the xaxis the

transformed delay tcd is shown as the probe photon energy Eprobe.

If two transitions ν1=ν2like in the example above participate in the process, an off-diagonal

contribution will indicate that coupling. In the case of a coherent coupling, the pump–probe pro-

cess works time–reversed as well, so both off-diagonal elements (hν1, hν2)and (hν2, hν1)will

appear on an equal level (if both transtitions are addressed by all pulses). Incoherent processes,

that may happen during the period tbc will result in off–diagonal unequal spots with the dominant

spot with Epump > Eprobe.

But besides coupling information, 2DCS also provides information if only a single transition

is involved. The homogeneous linewidth of the transition mediates a coupling between slightly

off–resonant external field components. In a simplistic model, the coupling can be derived

from eq. (3.11). The Lorentzian shape of this one–dimensional response function causes a two–

dimensional pattern that is illustrated in fig. 14(a). The shape is not circular as one might expect,

but shows a star pattern.

This pattern, however, can only be observed for a single transition or several transitions of the

exactly same transition energy. In most experiments, one observes an ensemble with an inhomo-

4.6. Pulse-shape analysis: FROG

Figure 14:Comparison of a 2D spectrum with (left) Lorentzian response x2+ 1y2+ 1−1

and (right) Gaussian response exp −x2+y2. While the 2D Gaussian is radial symmetric

due to x2+y2=r2, the Lorentzian is not and shows the typical star pattern.

SFG

Delay

Spectrometer

Ref.

tref

Figure 15:Concept of SFG FROG. If the beam–splitter (BS) is removed and an independent known

reference pulse (Ref.) is used instead, this becomes the XFROG configuration.

geneous broadening of transition energies. The resulting pattern is therefore the superposition of a

distribution of star patterns along the diagonal (ν1, ν2)with ν1=ν2. It is a major benefit of 2DCS

to distinguish between the inhomogeneous linewidth σ(diagonal section) and the homogeneous

linewidth γ(cross–diagonal section). The quantitative determination of linewidths depends on the

ratio of homogeneous an inhomogeneous linewidth. A calculation has been performed in ref. 82:

The case of the “inhomogeneous limit” with σ≫γ, which we assume for self–assembled QDs,

shows a cross–diagonal section identical to eq. (3.11). To extract T2from a 2D spectrum, the phase

slope along the cross–diagonal can be used according to eq. (3.13). This results in

T2=~

γ=~d

dEϕ(E0)(4.3)

with γin its photon energy representation.83

4.6. Pulse-shape analysis: FROG

FROG is a powerful non–linear technique for the analysis of femtosecond laser pulses. While

the first FROG utilized self–diffraction as a third–order non–linearity84, over the past 25 years

several other types have been demonstrated.85 Today the use second harmonic generation (SHG)86

4. Established experimental concepts

(or more general the SFG) FROG is very popular. It is the most sensitive configuration and the

formulae are very similar to the discussion of the FROSCH technique in section 5.2.1.

The scheme of the SFG FROG is displayed in fig. 15. The shown configuration is the standard

auto–correlation configuration. If an independent pulse is used as the reference pulse, the modified

scheme is referred to as the XFROG configuration. In the nonlinear crystal, the two incident pulses

generate the sum frequency under momentum conservation. The resulting sum frequency beam

analyzed by a spectrometer. The spectrogram is given by87

ISFG

XFROG(ω, tref) = Z∞

−∞

E(t)Eref(t−tref)e−iωtdt

,(4.4)

so for a FROG measurement for every reference delay tref a spectrum is acquired. This spectro–

temporal measurement allows the determination of the pulse frequency vs. time.

5. New opportunities by fast data

acquisition

From a technical point of view, the most impressive progress in our lab during the past years

has been reached due to a drastic reduction of lock–in integration times. This fast acquisition

brings several benefits to the established detection schemes, but is also a prerequisite for some

new experiment schemes. This section goes into the details of the new data acquisition scheme.

5.1. Fast data acquisition approach

In section 4.1 we have seen that heterodyne detection uses temporal integration to isolate a signal

from a noisy background. In an ideal lab, long integration times improve the signal–to–noise ratio

(SNR) by shrinking the bandpass window to the actual signal frequency (section 4.1.2). However,

there is no ideal lab and long integration times become a bad idea if the signal is unstable, especially

if it is subject to phase drifts of more than 180° as these will cancel out the signal upon integration.

For interferometric detection, this means that beam paths lengths of probe and LO must not change

by λ/2during integration. Vibrations of mirrors caused by acoustic noise therefore pose a serious

limitation to signal quality. To reduce the influence of the noisy lab environment, the optical table

is equipped with housing walls and covers. Figure 16 (a) shows typical noise spectra with and

without covers, respectively. The frequency range from 50 Hz to 5 kHz has a high noise level. The

noise reduction by the cover is efficient in the range of few kHz (fig. 16 (b)), but does not suppress

frequencies ≤200 kHz. In this range, with acoustic wavelengths of about 10 cm, resonator effects

inside the housing might play a role.

Under these external constraints, it is advisable to lift the data acquisition (DAQ) rate above this

range. Early tests showed that the necessary reduction of the lock–in time constant to 50 µs or

even 5µs has a minor negative impact on the SNR compared to acoustic and thermal drifts.

5.1.1. Acquiring data

With the old detection scheme, however, this frequency range was not accessible due to simple

speed limitations in digital data processing. In a digitalized world one tends to believe that ana-

logue signal transmission is outdated and can be replaced in general. This might be true in case of

well-defined tasks in a single apparatus, such as a personal computer or a digital lock-in amplifier.

However, this statement becomes questionable or even wrong when a large number of loosely

connected instruments is supposed to execute a task in a collective way. Without prepared data

highways, data exchange based on requests and answers is limited to repetition rates of less than a

kilohertz. If data from several instruments have to be documented simultaneously, serial requests

reduce this rate even more, while parallel requests are hard to synchronize.

As an answer to this challenge, DAQ cards have become the heart of all our setups. First of

all, they provide the setup clock defined by a single DAQ rate eνdaq. Signals are acquired by the

analog–to–digital converters (ADCs) of the DAQ card itself or by other instruments that receive a

trigger signal provided by the DAQ card. Typical DAQ rates are in the range of 10 kHz to 50 kHz. If

5. New opportunities by fast data acquisition

Figure 16:(a) Noise spectrum in the lab with and without table cover measured by a mobile phone.

(b) Damping induced by cover, calculated from the noise data in (a).

a slave instrument is not able to follow the trigger with its digital data processing, it is sometimes

advisable to use analog signal outputs that are linked to the ADCs of the DAQ card instead of

direct digital DAQ. Such an implementation is for example shown in fig. 19: The lock–in signal is

not acquired digitally, but transferred as an analog signal to the DAQ card. This scheme was used

for the old DSP7280 lock–in, but is not necessary any more for the new ZI HF2LI (fig. 28).

5.1.2. Rapid scanning and MOLCH

The fast data acquisition (fDAQ) in the kilohertz range contradicts the old concept of stepwise

setting delays and acquiring data. With fDAQ, it is inevitable to change to a rapid–scan approach.

Delay stages are moving at constant velocity and the fDAQ ensures a sufficiently dense set of data

points. The maximum velocity of the current stages is about v(max)

del = 15 mm/s. Two requirements

must be fulfilled to allow heterodyne measurements at this velocity: On the one hand, the DAQ

rate needs to be sufficiently high not to miss data features in between two sampling points. On

the other hand, the signal needs to be constant during the lock–in integration time. The lock–ins

used in our lab allow for minimal time constants (TCs) in the range of 1 µs, requiring the signal to

be constant for about 10 µs. A corresponding DAQ rate of e

νdaq = 100 kHz is easy to achieve. This

results in a scan range of 2v(max)

del /(ceνdaq) = 1 fs per data point. In the IR range, this equates λ/4.

With a phase rotation of 2π/λ and a slowly varying signal envelope, this value is at the limit of

what can be considered “constant”, but sufficient. With the present instruments, the stage velocity

is therefore not a limitation.

Scanning the reference delay tcd and tracking the lock–in signal S(tcd)has become a daily

routine in the lab. We refer to this experiment as More–Or–Less Characterization by Heterodyning

(MOLCH). More–Or–Less Characterization by Heterodyning (MOLCH) has been developed as a

reduced version of FROSCH. MOLCH omits the measurement of tcd on a sub–wavelength scale

and thus does not provide spectral information. It is used to check the slowly varying envelope,

so the typical display on the lab computer is |S(tcd)|(display of Xand Yis also possible). With

5.2. The FROSCH technique

Figure 17:The 4PS of a FROSCH measurement is quite simple. Pulse A and pulse B are omitted,

the measurement is a scan of the probe pulse C with the reference pulse D.

up to five scans per second in live–mode, MOLCH provides a real–time image of the heterodyne

signal during alignment. The finding of the temporal overlap tcd = 0 as well as the optimization

of the detector alignment have been simplified tremendously by the introduction of MOLCH.

5.1.3. Simultaneous multi-power measurements

In the traditional PP setup, the pump AOM power has been simply switched on and off. But when-

ever non–linear effects come into play, a more detailed power dependency becomes interesting.

With the old detection scheme, one could for example repeat the measurement with half the AOM

deflection power—but this would be ten minutes later and probably not very reliable. Here, fDAQ

opens a new avenue: The power sequence is not limited to on and off, arbitrary sequences of power

levels can be used. Instead of S(on) and S(oﬀ), a hole set of data like S(1.0), S(0.9), . . . , S(0.1), S(0)

is possible (where S(j)corresponds to a power level of P=j×Pmax), as long as such a sequence

period does not reach into the regime of acoustic noise. The diversification of power levels there-

fore implies an increased DAQ rate and a corresponding reduction of the lock–in time constant,

but this is possible in many cases without a significant loss of SNR.

The concept of Simultaneous multi-power measurement (SMPM) can be applied for the pump

AOM (detailed discussion in sections 5.3.1 and 10.2) as well as to the probe AOM (example in

section 9.2). For a quantitative interpretation, a calibration measurement of the deflected beam

power P(|EAOM|)over the driving electrical signal EAOM is necessary. A typical calibration curve

is shown in fig. 25 (a) and can be used as a look–up table to determine EAOM for a desired power

In most of the presented experiments, we used an analog SMPM implementation: The scaling in-

puts (“video input”) of the AOM drivers have been controlled via the analog output of the DAQ card

and is thereby inherently synchronized to the analog inputs of the DAQ card. One drawback of

this method is the scaling mechanism: While the frequency generator creates a continuous signal,

a 0 V input on the “video input” refers to a maximal suppression of this signal. This maximal sup-

pression, however, is not a complete annihilation of the signal, so a real “off” signal is not available.

This problem is overcome with a digital implementation of SMPM presented in section 7.2.3.

5.2. The FROSCH technique

Frequency Resolved Optical Short–pulse Characterization by Heterodyning (FROSCH) is a tech-

nique we developed to analyze pulse shapes. Instead of using non–linear elements like the FROG

technique (section 4.6), FROSCH is solely based on heterodyne detection. The pulse shape is cal-

culated from the lock–in signal of a scan of tcd

5. New opportunities by fast data acquisition

Figure 18:Minimal FROSCH setup. The incident pulse is split into a probe pulse C and the LO

pulse D. The reference delay tcd is scanned. The helium–neon laser is used to set up the complex

Michelson interferometer. Four real signals are acquired.

5.2.1. The FROSCH signal

The initial pulse is composed of discrete laser modes with complex amplitudes E(0)

ω. It is split into

the pulse Ejthat is supposed to be analyzed, and the LO pulse ELO:

ELO(t) = X

TLO(ω)E(0)

ωeiω t (5.1)

Ej(t) = Ej(t)eibω t =X

Tj(ω)E(0)

ωei(ω+bω)t,(5.2)

where b

Ej(t)includes the frequency shift bωintroduced by the AOM. The balanced detectors yield

an electric field Edet that is proportional to the light intensity:

Edet(t, tcd)∝Tdet(ω)1

2b

Ej(t)+ELO(t−tcd)2−b

Ej(t)−ELO(t−tcd)2

=Tdet(ω)b

Ej(t)E∗

LO(t−tcd) + c.c.(5.3a)

=Tdet(ω)X

ω,ω′Tj(ω)T∗

LO(ω′)E(0)

ωE(0)∗

ω′ei(ω−ω′+bω)t+c.c.

ωE(0)

ω2Tj(ω)T∗

LO(ω)eibω t +c.c..(5.3b)

In the last line we assumed that the comparatively slow detector electronics Tdet(ω)filter out all

frequencies with ω=ω′. At this point the major difference between FROSCH and FROG becomes

evident: In eq. (5.3b), the original pulse E(0) only contributes to the signal by its absolute value, not

by its phase. Even though there will be more arguments, the restriction imposed by the detector

electronics alone will make FROSCH insensitive to the phase of the initial pulse. FROG overcomes

this problem by the use of a spectrometer, i.e. an array of detectors. This allows FROG to include

the contributions of E(0)

ωE(0)∗

ω′for ω=ω′to its analysis which reveal the original phase. So be

aware: FROSCH is not suitable to characterize the original pulse, but it is an excellent tool to

determine the relative modifications introduced by Tj(ω)T∗

LO(ω).

5.2. The FROSCH technique

The detector signal is finally analyzed by the lock–in amplifier. The lock–in bandwidth is, of

course, much smaller than that of Tdet(ω)(we can simply assume an infinite TC), so we can ignore

Tdet(ω)and put eq. (5.3a) into the lock–in formula:

S(tcd) = Z∞

−∞ Edet(t, tcd)e−ibωtdt∝Z∞

−∞ b

Ej(t)E∗

LO(t−tcd)e−i∆ωtdt

=Z∞

−∞ Ej(t)E∗

LO(t−tcd) dt= (Ej?ELO) (tcd).(5.4)

The signal S(tcd)thus represents the cross–correlation of the two pulses according to eq. (1.8).

Utilizing this fact, we can generalize the insensitivity of FROSCH to the initial pulse phase: If both

pulses are exposed to the same additional transfer function T(ω) = G(ω) exp(iΦ(ω)) defined

according to eq. (1.23), we can use eq. (1.9) to determine the cross–correlation of the modified

fields:

E′

j?E′

LO(tcd) = F−1{(T(ω)Ej(ω))∗·(T(ω)ELO(ω))}

=F−1G2(ω)·E∗

j(ω)·ELO(ω)

=F−1G2(ω)∗(Ej?ELO) (tcd).(5.5)

Just like the initial pulse phase, a common phase change imposed by the TF in both arms cancels

out.

5.2.2. The complex-value interferometer

Precise position tracking is a crucial part of FROSCH measurements. A simple Michelson inter-

ferometer incorporating a Helium-Neon laser, a beamsplitter, two plane mirrors (one of them

mounted on the delay stage), and one detector has been used in Ref. 1 and provided sufficient

position control. However, there are two major drawbacks in this simple implementation: Firstly,

reference beam and control beam do not use the same optical element, so differences in motion of

these two elements will result in erroneous position values, in particular vibrations are observed.

Secondly, a single detector provides a sinusoidal signal. At extreme values this signal is symmetric

in direction, i.e. forward movement cannot be distinguished from backward movement.

Meanwhile we overcame both of these drawbacks. Using the same retro–reflector for the con-

trol beam as for the reference beam is a standard procedure.88 To obtain unambiguous directional

information, a second detector is necessary. It can be implemented as shown in fig. 18 in back-

ward direction that is often ignored although 50% of the intensity are back–reflected. Here, an

interferometric fact can be incorporated that usually is seen as an error source: Any real world

instrument with finite aperture size exhibits interference fringes in the detector plane,18 i.e. con-

structive and destructive interference alternate spatially. By a slight imperfection of alignment it

is easy to set one detector to a constructive fringe, while the other one is at a destructive fringe

for the same delay. This phase offset of ideally 90° is usually maintained for a sufficient distance

of some millimeters. If one detector is at its extreme value, the other one has a zero and provides

optimal directionality.

While we desire a circular signal in the complex plane, we will accept oval signals in most cases.

But these oval signals must be mapped to a circle. An easy way to do this is to rotate the oval

in a way that the long axis superposes the real axis and afterwards stretch the imaginary part to

5. New opportunities by fast data acquisition

Figure 19:Heterodyne pump–probe setup as it has been used in Ref. 3. Caution: Using the trans-

mitted AOM beam as local oscillator turned out to create artificial offsets. Similarly published

in Ref. 3.

retrieve the circle. There is an astonishingly simple trick how to determine the long axis from an

oval complex signal array S:

Aeiα =v

j=1 Sj−1

n n

i=1

Si!!2

.(5.6)

After the offset correction, the square turns opposite phase values to the same value, the oval is

transformed to an “egg” shape. The square root of the center of mass of this egg indicates the

position of the long axis.

5.3. Fast pump-probe

The general concept of traditional PP measurements has been described in section 4.2.1. The setup

in the state of Ref. 3 is sketched in fig. 19. In contrast to a FROSCH measurement, there is no need

to scan tcd, it is set to the maximum signal at tcd ≡0. The new rapid–scan approach can now be

applied to the pump–probe delay tbc. Like in the following experiments, tac will be the delay that

is controlled in the experiment. In the context of a PP experiment, this delay equals tbc.

It turned out that DAQ rates of eνdaq = 100 kHz with corresponding lock–in time constants of

≤2µsare possible. As discussed in section 5.1.2, this corresponds to a minimal temporal resolution

of 1 fs, which is way too high for a PP measurement with a pulse duration limited resolution of

5.3. Fast pump-probe

Figure 20:Example for processes that can be identified by SMPM. The excitation saturation causes

a non–linearly increasing peak height. TPA and subsequent relaxation from higher states

causes a carrier feeding on the timescale of tens of picoseconds. These resonant PP curves

are taken on the QD GS at low injection current (2 mA).

200 fs. This oversampling, however, allows to average the processed data S(on)/S(oﬀ) rather than

the original values S(on) and S(oﬀ), respectively. This reduces the susceptibility to common phase

drifts which are eliminated in the ratio.

In the setup shown in fig. 19, we still used the traditional scheme70 that uses the transmitted

beam of the AOM as reference pulse D. It turned out that this beam also carries a modulation at

the driving frequency bνc. We detected Sc= 0 with pulse C blocked and only with pulse D on the

detector. This results in artificial offsets, especially in PP experiments. With the later implemen-

tation (fig. 28) that splits pulse D from pulse C using a beam sampler right after the laser, such

offsets have been widely eliminated.

5.3.1. Advantages of SMPM

There might be many processes that contribute to the temporal evolution of a pump-probe trace.

Knowledge about the linearity of excitation response can be extremely useful to weight these

contributions. SMPM provides this check of linearity with a high degree of accuracy. In the context

of PP measurements, the observation of TPA and excitation saturation are typical phenomena to

be identified by SMPM.

Exemplary sets of SMPM differential gain curves are shown in fig. 20. The SMPM power levels

are linearly increasing, but the QD excitation at tbc = 0 is obviously not. This is a clear indicator

for the saturation of the QD population. An application and detailed discussion of this signature

will follow in section 10.2. The signature of TPA can also be easily identified: TPA creates carriers

in high energy bulk states that relax down to the probed spectral range on the timescale of tens

of picoseconds. As TPA scales with the square of power, the presence for high power and the

simultaneous absence for low power is an unambiguous indicator.

6. Multiband detection for sideband

pump–probe

So far, a single heterodyne frequency has been tracked. By tracking multiple frequencies in mul-

tiple demodulator units, one can overcome different limitations of the single lock–in setups pre-

sented in chapter 5. Similar usage of parallel lock–in amplifiers has been demonstrated for FWM89

as well as PP experiments90. However, the setups presented in the following add several new

aspects to this concept.

6.1. Separation of multiple frequencies

In our experiments, we use two methods to filter electrical signals. The final isolation of bands is

performed by a lock–in amplifier. Before that, we are able to filter the electrical detector signal by

high, low, and band pass filters. Although both methods are conceptually closely related, there are

some noteworthy differences.

6.1.1. Isolation of bands by lock–in amplifiers

For a single frequency of interest the requirementsto isolate the heterodyne frequency were easy to

fulfill: After leaving the range of acoustic and line noise affected frequencies towards the megahertz

range, there was merely white noise surrounding the desired peak. If two frequencies have to be

observed by two lock-ins, the situation is fairly different because crosstalk has to be suppressed.

The degree of isolation of two lock–in frequencies depends on the frequency spacing and the lock–

in TC. For a narrow spacing, a large TC can be used to isolate the bands at the cost of long settling

times resulting in reduced DAQ rates.

In modern digital lock–in amplifiers, the analog time signal is digitized after an initial amplifier

stage. Typical modern devices provide 14 bit resolution.24 This is usually not a limiting factor in

our case as the separation of signals with a level ratio of more than 1000 would require very large

TCs. The filtering in the lock–in amplifier is performed in the rotating frame of the reference signal:

The signal is demodulated and subsequently filtered. In this context, it is as easy to distinguish for

example 61 MHz from 60 MHz as it is to distinguish 1 MHz from 0 MHz. In the respective rotating

frame, both cases have a separation of 1MHz. The lock–in filtering therefore demands absolute

frequency separation.

6.1.2. Analog signal pre–conditioning

Let us now compare the lock–in filtering to the analog signal filtering, especially to the impor-

tant case of low pass filters. The low pass filter transfer function has already been discussed in

section 1.3.2. An expample of two signals is given in fig. 21 with an initial level ratio of 100 (red

curves). The strongest relative suppression is achieved in the range above the cut–off frequency

(linear range in the log–log graph). The suppression of one band becomes more efficient when the

ratio of the two frequencies is large. In contrast to the lock–in, the low–pass filtering demands

6.2. The sideband setup for low signals in Austin

Figure 21:Analog signal pre–conditioning by a low–pass filter. The heterodyne band at 2 MHz is

stronger than the sideband at 50 kHz by a factor of 100 on the detector (red curves). By a low–

pass filter, this ratio can be reduced to 2.5 which increases the dynamic range of the lock–in

amplifier.

relative frequency separation. This analog signal pre–conditioning will be one of the keys to de-

tect low signals in multi–band experiments. It allows us to use lower lock–in TCs and take data

much faster. In chapter 7, a band–pass filters will be applied. In that case, the argument of relative

frequency separation is not valid anymore, but it will still be useful to look at the transfer function

on a log scale.

6.2. The sideband setup for low signals in Austin

The main goal of my stay with the group of Professor Li at the University of Texas at Austin was

to introduce heterodyne detection to an existing pump-probe setup in order to filter out artifacts

efficiently. Our central research objective was a deeper understanding of exciton-trion dynam-

ics in TMDs upon polarization selective pumping and probing. In earlier experiments53 it was

possible to filter the pump even in collinear geometry since it was cross-polarized with respect

to the probe. For copolarized beams, however, probe-pump signals (section 4.2.2) obscured the

pump-probe signal.

The first straightforward approach of copying the existing Berlin detection scheme (section 5.3)

revealed its sensitivity limitations: For monolayer TMDs, signal levels with peak heights below

1 % were expected, which was impossible to retrieve as time–resolved signal. While heterodyne

detection distinguishes pump and probe qualitatively, the measurement of a signal modification

is not background-free and therefore lacks the sensitivity of chopping detection—loosely spoken,

it is easier to distinguish 0.01 from 0 than 1.01 from 1. Here, the idea of a hybrid setup was born:

A setup that relies on heterodyne detection and yet identifies signal modifications by continuous

modulation rather than by defined switching. The clue in this approach is that the continuous

6. Multiband detection for sideband pump–probe

Figure 22:The Austin setup is a hybrid of heterodyne detection and the double–chop approach.

A complicated analog frequency mixing scheme is necessary to create the modulation and to

lock the lock–in amplifiers to the respective bands.

modulation of a signal shows up as a sideband in Fourier space. Whenever the probe beam is

modified by a pump beam modulated at eν, the heterodyne signal oscillates at eν, i.e. sidebands

appear at bνc±eν. By detecting a sideband using a second lock–in, the signal modification can be

retrieved like it is illustrated in fig. 3.

The setup based on these considerations is shown in fig. 22. The titan-sapphire laser is a Griffin 5

from KMLabs with a repetition rate of bνrep = 91.078 MHz. The carrier frequencies of the AOMs are

bνab = 80 MHz for the pump and bνc= 91.638 MHz for the probe, respectively, so the heterodyne

frequency is at bν′

c= 560 kHz. To create the sideband, the pump carrier frequency is modulated

by a further frequency generator at eν= 480.00 kHz. The modulation scheme is quite complicated

and has been taken directly from the original double–chop scheme: The carrier signal is split, one

part is mixed with eνand afterwards recombined with the second part of the carrier signal. This

implementation therefore creates the signal

2(1 + sin(2πeνabt)) sin(2πbνabt),(6.1)

which in fact has some drawbacks that will be discussed in section 6.3.

Figure 22 shows the setup in transmission geometry, but it has been successfully tested in re-

flection geometry as well. In both cases, a microscope objective is used to focus pump and probe

6.3. Modification of the Austin setup in Berlin

Figure 23:Proof of principle data taken with the setup shown in fig. 22. The pump and probe

wavelengths are documented in the Austin lab book and therefore not available.

beam, respectively, onto a TMD flake that is mounted in a cryostat and kept at a temperature

of 15 K. The transmitted (or reflected) beam is superposed with the reference on a New Focus

2107-FC balanced photoreceiver. The modulation sideband imprinted on the probe beam by the

modulated pump beam is tiny compared to the heterodyne signal in general. To enable simultane-

ous detection, the low pass filter is set to the cut–off frequency 100 kHz, so the heterodyne signal

at bν′

c≈560 kHz is suppressed while the sideband at bν¯abc ≈80 kHz passes the filter with minor

loss. This pre–filtered signal is analyzed by two lock-ins from Stanford Research, a model SR844

for signal Scat the heterodyne frequency and a model SR830 for the sideband signal S¯abc.

Analog to eq. (4.2), the differential gain and phase are

ln 10

10 ∆G(tbc) + i∆Φ(tbc) = TcS¯abc(tbc)

T¯

abcSc

=G0exp (iϕ0)S¯abc(tbc)

.(6.2)

The TFs Tcand T¯abc take the differences in the signal propagation into account, especially the

differences in the electrical filtering. Although the lock–in signals Scand S¯abc have a defined

phase relation, the phase ϕ0is unknown. A calibration measurement is necessary to determine

ϕ0for the case ∆G > 0and ∆Φ = 0, i.e. a pure amplitude modulation. We used an additional

calibration AOM that is placed in the pulse C beam. During the measurement, this AOM is not

driven. For the calibration measurement, the pump beam is blocked and the pump AOM driving

voltage is applied to the calibration AOM. This scenario resembles a perfect amplitude modulation

of pulse C and the measured phase difference of Scand S¯abc can be used to compensate for ϕ0.

During my stay in Austin, we were able to demonstrate this novel approach and to acquire some

preliminary data during the testing process. An exemplary set of test data on a WSe2monolayer

sample is shown in fig. 23.

6.3. Modification of the Austin setup in Berlin

For the integration of the sideband PP scheme into the Berlin setup, several improvements have

been applied. Instead of two independent lock–ins, the use of the ZI HF2LI lock–in allows to skip

the complicated analog mixing. The necessary frequencies can be generated digitally. The setup

shown in fig. 24 is the state of the development that was used to acquire the data for Ref. 4. It still

incorporates independent AOM drivers, which will become unnecessary with the modifications

introduced in section 7.2.2.

6. Multiband detection for sideband pump–probe

Figure 24:This setup is a modification of the setup developed in Austin (fig. 22). Here, the fre-

quency generators provide internal modulation, so external mixing of AOM signals is not nec-

essary. Instead of an additional calibration AOM the probe AOM is used directly to generate a

calibration signal. Similarly published in Ref. 4.

6.4. Biexciton decay in colloidal PbS/CdS quantum dots

Figure 25:AOM characteristics: (a) The deflected power (solid blue) over the modulation volt-

age can be approximated by a parabola for low modulation levels. (b) The deviation from the

parabola results in the generation of higher harmonics. (c) The higher harmonics result in a tem-

poral signal that deviates from the ideal sinusoidal modulation for high modulation voltages.

Similarly published in Ref. 4.

Unlike for the Austin setup in eq. (6.1), the modulated pump AOM signal in this case is

sin(2πeνabt) sin(2πbνabt).(6.3)

The envelope of this oscillation is |sin(2πeνabt)|, which seems to be a non–ideal choice to create

a sinusoidal intensity modulation. If one takes into account the AOM deflection characteristics,

however, this modulation becomes reasonable. Figure 25 shows the deflected intensity over AOM

carrier level, i.e. the envelope of the modulation function. For low envelope levels this function

is parabolic, so the envelope of the intensity is |sin(2πeνabt)|2= (1 + sin (2π(2eνab)t)) /2. For

low deflection levels—which are likely for example for samples with low damage threshold—this

modulation shows superior shape compared to eq. (6.1). The mixing scheme in the original Austin

setup results in an intensity modulation with strong contribution of higher harmonics.

The calibration measurement has also been simplified. Instead of using an additional calibration

AOM, the artificial modulation is now directly applied to the probe AOM during the calibration

measurement. Because this is only a slight modulation on top of the standard carrier level, the

frequency is not doubled like at the minimum of the parabola. To achieve the same result, the

frequency doubling needs to be performed digitally.

6.4. Biexciton decay in colloidal PbS/CdS quantum dots

The investigation of PbS/CdS QDs spin–coated on silicon nitride waveguides37 was a collaboration

with colleagues from the University of Ghent. The spin–coating fabrication technique is ideal for

easy prototype preparation, but lacks the high mode overlap of waveguide and QDs achieved by

embedding the QDs into the waveguide. The resulting modulation depth is therefore comparatively

6. Multiband detection for sideband pump–probe

Figure 26:Variation of the probe photon energy with pump wavelength fixed at 1200 nm. Am-

plitude data in (a) and phase data in (b) are well described by a two–exponential fit (eq. (6.4)).

absorption (blue area) and emission (red area) spectra. Similarly published in Ref. 4.

Figure 27:Variation of the pump photon energy analog to the data shown in fig. 26. Similarly

published in Ref. 4.

6.4. Biexciton decay in colloidal PbS/CdS quantum dots

low which made this set of experiments an ideal test field for the sideband PP technique. A detailed

description of the sample and the results is given in ref. 4. These results are outside the context

of the investigation of SOAs discussed in part III. They are therefore presented here as a proof–

of–principle for the sideband PPtechnique. The experiments have been conducted primarily by

Christian Ulbrich and Bastian Herzog.

Figure 26(a) and (b) show representative traces of the time-resolved differential transmission in

amplitude and phase, respectively, detected by sideband PP spectroscopy with the pump close to

the center of the QD luminescence at 1200 nm, and the probe tuned from 1200 nm to 1600 nm, well

in the QD bandgap. The experimental data are shown as solid dots. Note that all PP traces are

shown on a logarithmic time scale. For early times, oscillations arising from acoustic noise are

visible and are not resolved for later times due to undersampling. The average laser power before

the waveguide was 100µW, with an incoupling efficiency of 10 % this equals 10 µW average power

or an energy of 150 fJ per pulse. As the repetition rate of the laser exceeds the decay rate of the

QD GS exciton of approximately 1µs by far, we create a quasi-constant GS exciton population.31

The transmission and refractive index changes induced by the pump pulse on the picosecond to

nanosecond time scale thus reflect not the dynamics of a single GS exciton, but rather the inter-

action of multiple exictations in the system. Throughout the whole investigated range of probe

wavelengths, we observe consistently a positive change of the differential transmission, caused

by state filling created by the pump pulse on the quasi-static exciton background. The amplitude

change is largest at a red detuning of about 50 nm to 75 nm from the pump pulse, consistent with

the Stark shift expected in this kind of QDs. In the differential phase response, we observe a change

in sign between 1200 nm and 1250 nm probe wavelength, indicative of the crossing of a resonance.

This supports the assumption that carriers are created within the GS manifold of the PbS/CdS QDs

rather than in an intraband process, as in this case no particular resonance feature is expected. The

data taken at 1550 nm and 1600 nm probe wavelength show a larger noise and artificially higher

amplitude, as the probe laser power had to be increased due to decreasing sensitivity of the de-

tector and an exceedingly small diffraction efficiency of the AOM at these long wavelengths. The

signature, however, is distinctly positive even at these wavelengths unlike for kHz excitation.35

This does not exclude the presence of an intraband absorption process, however, this process is

not modulated along with the pump modulation under our quasi-cw excitation conditions.

To extract more quantitative information, we fit the traces with a biexponential decay

y(tbc) = y0+A1exp(−tbc/τ1) + A2exp(−tbc/τ2),(6.4)

which describes the experimental curves well over three orders of magnitude in time. It is im-

portant to mention that amplitude and phase data are required. Using only the amplitude data, a

stretched exponential model would fit as well, not giving clear evidence of the presence of two

distinct processes. The data at 1550 nm and 1600 nm probe wavelength are not fitted because of

the signal-to-noise issues mentioned above. Also for the phase data taken at 1500 nm probe wave-

length, a meaningful fitting was not possible. The fits are shown in the figure as solid lines. The

fast and slow time constants are in the range of 30 ps to 40 ps, and 200 ps, respectively. We did not

observe a significant offset in the phase data, while in the amplitude the quasi-static background

amounted to up to one quarter of the total signal. The amplitude offset is displayed in fig. 26(c)

together with absorption and emission spectra and the wavelength settings of pump and probe

pulses.

Figure 27(a) and (b) show exemplary data obtained at pump wavelengths between 1200 nm and

950 nm for a fixed probe wavelength at the QD luminescence maximum at 1200 nm. Again, we

observe a consistently positive signal with a maximal amplitude in the response at a blue detuning

6. Multiband detection for sideband pump–probe

of 50 nm to 75 nm of the pump wavelength with respect to the probe wavelength. At shorter

pump wavelengths, the response decreases in amplitude, to reappear at wavelengths shorter than

1000 nm as the excited state continuum of the QDs is addressed. With the low number of carriers

created, also the phase response is low between 1200 nm and 1000 nm excitation wavelength. This

is also reflected by the variation in magnitude of the quasi-static amplitude offset y0displayed in

fig. 27(c). Biexponential fits of the pump-probe traces are shown in fig. 27(a) and (b) as solid lines.

For a range of phase traces, a meaningful fitting of the data was not possible due to an insufficient

SNR at arcsecond phase amplitudes.

6.5. Pathways for future development

Multiband detection promises several experiment configurations and improvements to existing

experiments that we have not yet implemented. To protect those ideas from being forgotten, I

would like to add them here as suggestions for later experiments:

Temporal overlap determination in PP experiments. So far, we always determined the tem-

poral overlap of PP experiments from the PP curve itself. This is not very accurate and might fail

totally, if the probed state is not addressed by the pump and the scattering processes are slow.

By additionally tracking the pump carrier band bνab it would be easy to determine the temporal

overlap exactly. Scanning the pump delay tbc over the static reference delay tcd ≡0resembles

a MOLCH measurement. The amplitude maximum locates the temporal overlap.

AC bias induced sidebands. One interesting option might be the generation of sidebands via

the electrical carrier injection. For example it would be interesting to combine the sideband and

the FROSCH technique. Using a bias tee, it would be possible to supply a constant bias as usual

and to combine it with an alternating voltage for sideband generation. In the sideband FROSCH

signal, only those light modes would contribute that experience a population modification in the

SOA upon a variation of the particular direct current (DC) level. With that experiment and under

comparison with the respective ASE change, it might be possible to calculate the absolute DOS

distribution.

7. Two–dimensional coherent spectroscopy

with white pulses

The observation of FWM signals was an important aspect of the original heterodyne setup,70 but it

has not been used in the setups presented in the previous chapters. This will change at this point.

Additional to the FWM observation, we have developed all the necessary “ingredients” to set up

an MDCS which is actually nothing else but a phase–resolved FWM measurement under variation

of multiple delays. These ingredients are:

• From the FROSCH concept we take the sub–wavelength resolution by interferometric posi-

tion control.

• SMPM plays an important role for maintenance of phase stability.

• Multiband detection reaches a new level when up to four different bands in two input signals

are observed simultaneously (up to six lock–in signals).

• We use a Toptica FemtoFiber pro SCIR fiber laser system with two independently tunable

outputs. The white light pulses provided by this system are used without intentional spectral

cutting, the spectral width of both pulses exceeds 200 nm.

The resulting 2DCS setup is presented in fig. 28. We refer to this method as Supercontinuum–based

Two–dimensional Observation of Radiation Coherence by Heterodyning (STORCH), where the

acronym is the German word for “stork”. Considering that a STORCH measurement will dispatch

up to six FROSCHs, this name seems highly suitable.

It is noteworthy that with the description giving in the following, the STORCH setup scheme in

fig. 28 replaces all former ones, as all the former experiments can be performed without hardware

changes.

7.1. Concept of the heterodyne 2D coherent spectroscopy

The preliminary goal of the development of this setup was to investigate SOAs under standard

operation conditions, particularly at room temperature and above. In this scenario, any coherent

signals will decay within hundreds of femtoseconds. The required scan widths of the delays will

therefore not exceed 1 mm.

To understand the signal generation of the heterodyne 2DCS one should have the sketch in fig. 13

in mind. This scheme remains valid, but some restrictions need to be considered here compared

to the general scheme. Our approach is so far restricted to two pulses that pass the sample, the

former probe and pump pulse, respectively. Just like in the classical FWM,70,77 we will use the

self–diffraction scheme given in fig. 12. Like in that scheme, pulse C will also act as pulse B,

but in general, the roles of the pulses can be swapped. The respective FWM bands will appear

at bν¯acc = 2bνc−bνaand bνac¯c = 2bνa−bνc. With respect to the white light nature of the laser it

is, however, advisable to concentrate on bν¯acc. This frequency represents pulse C that has been

diffracted by the transient grating induced by pulses A and C. For pulse C it is ensured that its

7. Two–dimensional coherent spectroscopy with white pulses

Figure 28:The STORCH setup for 2D coherent spectroscopy. The spots (α−δ) are equivalent with

respect to the characterizing FROSCH measurement in front of the sample. Similarly published

in Ref. 5.

7.2. Going digital: Changes enforced by the new HF2LI lock–in

constituent laser modes are present in the LO, as this has been split off from the same laser arm.

The other arm (pulse A) could create modes that are not present in the LO and therefore would not

be observable.

The calculation of the 2D spectrum requires the full scan of all combinations (tcd|tab). In the

present setup, the local oscillator delay tcd is swept rapidly (≈1 mm/s) back and forth while the

pump delay tab is scanned slowly (<1µm/s). In fact, every sweep of the reference resembles a

single FROSCH measurement with four SMPM. The time axis correction for the 2D scan is, however,

more complicated and will be explained in detail in section 7.2.3.

The STORCH measurements will require a comparatively detailed control of the used pulses.

The detection capabilities have therefor been extended by a second balanced detection “before the

sample”. The second arm of the beam–splitter that unites pulses A and C in front of the sample

is used to characterize the pulses that enter the sample by FROSCH. The necessary LO is split off

from the existing LO beam path by a beam sampler. A careful look at fig. 28 shows, that there is

something special about the beam path from spot αat the laser output to spot βon the beam–

splitter in front of the new balanced detection: The beam has not passed a single piece of glass.

The beam is either reflected by mirrors or by the glass surface of beam samplers. Therefore this

beam has been exposed to the minimal amount of dispersion and can be referred to as “the perfect

reference”. From the perspective of the detector, the spot βwhere it enters the final beam–splitter

is equivalent to spot γwhere the characterized beams enter: the same amount of glass is passed

subsequently which does not alter the FROSCH trace according to eq. (5.5). In the same way,

the spot δon the in–coupling lens of the sample is equivalent to spots γfrom the perspective

of the beam–splitter that unites pulses A and C. Thus, based on the relations α≡βand γ≡

δ, the FROSCH trace detected in the additional balanced detection is equivalent to a FROSCH

trace of the pulse at the in–coupling lens referenced with the very original laser output. This

FROSCH trace is used in particular to determine the chirp of pulses A and C with respect to the

LO. Using the 4fpulse shapers, a linear chirp can be compensated efficiently.91 Just like FROSCH,

STORCH is insensitive to the initial pulse phase. At the moment, we do not have the capabilities

to pre–characterize the original supercontinuum in amplitude and phase. We need to rely on the

sufficiency of its quality.

7.2. Going digital: Changes enforced by the new HF2LI lock–in

The previous description focused on the optics arrangement. But STORCH is much more a matter

of data acquisition and analysis. The heterodyne detection scheme in our lab changed tremen-

dously when we were able to purchase the Zurich Instruments (ZI) HF2LI lock–in and some of

its add–ons: The “Multi–frequency option”, the “AM/FM Modulation option”, and the “Real–time

option”.

7.2.1. Data acquisition and synchronization

With the HF2LI it is not longer necessary to acquire lock–in data via the DAQ board even at high

DAQ rates of up to 50 kHz. The HF2LI provides an efficient raw data exchange with the lab com-

puter where a server application makes processed data available. The DAQ board remains, how-

ever, still necessary to track the interferometer channels and it offers the opportunity to track

other analog signals in the future.

To synchronize the DAQ card and the HF2LI, the DAQ card operates as the master device and

generates a digital TTL trigger. The typical SMPM setting of the STORCH measurement has four

states. For reasons that will be discussed in section 7.2.3, it turned out to ease the experiment

7. Two–dimensional coherent spectroscopy with white pulses

Figure 29:The frequencies used in the 2DCS experiment and the filter transfer functions. The

ZI HF2LI has an internal 50 MHz low–pass filter (solid blue). The filter transfer function is

determined by directly connecting the lock–in frequency output to the signal input. An ad-

ditional band–bass filter (solid red) restricts the signal transmission from the detector to the

band of used frequencies. The color bars on top indicate the bandwidth of used instruments:

The AOMs restrict bνaand bνc, respectively, the PDB415C-AC detector covers the full frequency

range and replaced the Model 2117 detector.

control to use two digital triggers that provide four states in binary logic. So the two trigger

outputs of the DAQ board are connected to two digital inputs of the HF2LI.

Even though a synchronization of the data acquisition is ensured this way, the data processing

is still challenging. In general, large amounts of data are handled (typically at least 320 kB/s) that

need to be sorted, stored, and evaluated at run–time. One should avoid to create too many copies of

these data, which can easily happen in badly programmed Labview code. Labview data references

(defined memory space that is writable and readable from different access points) turned out to be

an efficient way to deal with this problem. A limited set of such data references are used to sort

the data: The lock–in data are requested from the HF2LI server application, but there is, of course,

a small latency compared to the DAQ card data points that do not need USB transfer. The first data

reference stores the newly acquired data channel by channel. In a next step, the program evaluates

how many full SMPM sets (all four data points in all channels) are available in the first stage. These

data are transferred from the first to the second data reference and used for the further evaluation.

7.2.2. Lock–in–driven AOMs and frequency choice

Following our traditional heterodyne detection scheme, the simultaneous tracking of bνc,bνa,bν¯acc,

and bνac¯c would require at least two reference inputs to lock to bνc−bνrep and bνa−bνrep. The ZI

HF2LI lock–in, however, does not provide more than two fast analog inputs that in our setup are

occupied by signal channels. The effective frequency range would therefore be restricted to less

than 2 MHz, which on the other hand would restrict the sampling rate and the according delay

stage speed.

To overcome this restriction, the ZI HF2LI itself is used as the frequency generator to drive the

AOMs. Locking to an external frequency is not necessary in this case, but the frequencies need

7.2. Going digital: Changes enforced by the new HF2LI lock–in

to be chosen within the AOM bandwidth (60 MHz to 100 MHz with a steep deflection efficiency

drop–off below 60 MHz). The demodulated frequency is now no longer the beating of a shifted

mode superposed with its neighboring mode ν′, but with its original mode νinstead. The New

Focus Model 2117 balanced detectors (bandwidth 10 MHz) used so far were replaced by Thorlabs

PDB415C-AC (bandwidth 100 MHz). As the ZI HF2LI is a 50 MHz lock–in (210 MS/s), this forces

us to operate the device beyond its specification. We chose bνc= 60 MHz and bνa= 65 MHz

as a trade-off and under consideration of the two FWM bands appearing at bν¯acc = 55 MHz and

bνac¯c = 70 MHz, respectively. This choice will favor the desired bν¯acc band.

Despite undersampling, the driving of the AOMs works well by using a 1.5 W amplifier (Becker

Nachrichtentechnik AMP5220031-T). On the detection side, however, both fast inputs are equipped

with an internal 50 MHz low–pass filter, so the chosen frequencies are already strongly suppressed.

To reduce the analog signal transmission to the used spectral window, an additional band–pass

filter (Minicircuits BBP-60+) is placed between detector and lock–in, that particularly suppresses

the residual signal of bνrep. The lock–in TF as well as the combined lock–in and band–pass filter TF

are shown in fig. 29.

As one can see, the STORCH setup works at the moment at the very technical limit and in some

respect beyond the specifications of the instruments. This implies that there is a wide range of

further quality gain whenever better instruments are available.

7.2.3. Phase stability: SMPM implemented by ZI HF2LI Real–Time option

The main challenge of MDCS is the stable tracking of the FWM signal with sub–wavelength res-

olution over the entire measurement. In the FROSCH setup it was sufficient to use a Michelson

interferometer as a time–reference because a single measurement does only take a few seconds and

the scan is linear. Thermal drifts of the beam pathways are therefore negligible. In 2D spectroscopy,

however, a single measurement takes several minutes. More than that, some kind of zig–zag pat-

tern is necessary to explore the two–dimensional time–area and a consistent data pattern requires

the reliable phase relation between nearby points. With respect to the passive stability in our lab,

simple Michelson interferometers are not sufficient.

The problem is solved by using pulses A and C itself as additional time–references that

are used to eliminate temporal drifts of the pulses. Similar approaches have recently also

been chosen by other groups.92 Of course, a time–reference needs to be constant over the en-

tire 2D pattern, which is not the case when pump and probe pulse are both switched on—

coherent as well as incoherent interactions of both pulses would inhibit the use of these sig-

nals. Here, the concept of SMPM becomes important: In an SMPM sequence of four points,

between two data points with both pulses on, we alternately switch one of the pulses off

to retrieve an undisturbed reference for the switched on pulse. For the power configura-

tion (Pa|Pc)(Pa/cdenote the normalized deflected power level at the respective AOM) we

use a repeating sequence {(1|1),(0|1),(1|1),(1|0)}to acquire a corresponding signal sequence

nS(1|1)

k, S(0|1)

k, S(1|1)

k, S(1|0)

ko(k∈ {(a,in),(c,in),(a,out),(c,out),¯acc}) at a switching rate of

20 kHz. This rate limits the TC of the lock–in to ≤5µsas the sequentially acquired values need to

be completely independent.

The analog implementation of SMPM has been described in section 5.3.1. The ZI HF2LI does

not provide an analog scaling input for the frequency outputs, so a fully digital setting of the out-

put levels driving the AOMs is required. The standard version of the ZI HF2LI does not provide

the functionality to switch the output signal level upon every data acquisition step. This requires

an instrument add–on, the “Real–time option”, which enables an onboard processor that is pro-

grammable and is able to execute a large number of operations without communication to the

7. Two–dimensional coherent spectroscopy with white pulses

Signal Acquired by Det. Freq. Demod. Trigger Description

S(1|1)

acc HF2LI after 55 MHz 2 1r, 1f FWM signal channel

S(0|1)

c,out HF2LI after 60 MHz 1 2r Undisturbed pulse C FROSCH

signal for tcd calculation

S(1|0)

a,out HF2LI after 65 MHz 4 2f Undisturbed pulse A FROSCH

signal for tab calculation

S(0|1)

c,in HF2LI before 60 MHz 3 2r Pulse C FROSCH signal be-

fore sample (for alignment)

S(1|0)

a,in HF2LI before 65 MHz 6 2f Pulse A FROSCH signal be-

fore sample (for alignment)

I(inter)

LO DAQ card interferometer all sin/cos signals from reference

stage interferometer

I(inter)

ac DAQ card interferometer all sin/cos signals from pump

stage interferometer

Table 2:Overview of the data channels. The lock–in channels analyze a certain combination of

input detector signals (Det.) and frequencies (Freq.). The demodulator occupation is restricted

by the “AM/FM Modulation option”. The last demodulator (number 5) is used to track Saa¯c,

which is not evaluated at the moment. A trigger “1r” refers to “trigger 1 rising”, “2f” to “trigger

2 falling”, and so on.

main computer. This makes these operations extremly fast. Typical latencies are in the range of

5 µs to 10 µs. The processor is programmed in C and the compiled program is uploaded to the

instrument.

The “Real–time option” is able to execute commands upon the occurrence of defined events. It

took us a while to understand that this event is not the digital trigger itself, but the acquisition of

a data point. The triggering has in fact two stages: The digital signal from the DAQ card triggers

the acquisition of a data point. Subsequently, this acquisition triggers the processor action. In

the first stage, the triggering of the acquisition, it is important to trigger only the demodulators

that are really necessary. For example, it does not make sense to acquire a data point S(1|0)

¯acc which

corresponds to a FWM signal with pulse C switched off. Here, the two trigger channels provided

by the DAQ card are important. The triggers rise and fall alternating at every DAQ card acquisition

event, which results in a sequence: 1r, 2r, 1f, 2f. Here, “1r” is for “trigger 1 rising”, for example.

The complete acquisition sequence is displayed in table 2.

7.2.4. Scanning procedure and Fourier transformation

As a first time base, the interferometer data are analyzed and every data set gets a corresponding

time stamp (t(0)

cd |t(0)

ab ). Based on this time basis we interpolate S(1|0)

1and S(0|1)

2for every FROSCH

scan, Fourier transform the interpolated trace, and derive temporal drifts of ∆tcd from S(1|0)

cby

comparing the phase of the spectral maximum with the respective values of the previous scan.

Taking ∆tcd into account, ∆tab is calculated in the same manner from S(0|1)

a. These drift values

are used to correct the time basis (t(0)

cd |t(0)

ab )to create the final time basis (tcd|tab).

After the correction of the time information, the one–dimensional chain of data is binned into a

two–dimensional matrix. The matrix size is chosen as a power of 2 (according to the FFT algorithm)

with a spatial bin width of <400 nm. For typical measurements, the matrix size is 512 ×512

7.3. Pathways for future development

Figure 30:Typical temporal data (real part). The data are presented in a rotating frame of 1200 nm.

The relevant data are restricted to a range of <1 ps around the pump–probe overlap. The FFT

of this particular data set is shown in fig. 43.

or 1024 ×1024. To improve the averaging, the data points are transformed to a rotating frame

according to the central wavelength before the binning. An example of the temporal matrix is

shown in fig. 30. The standard 2D FFT is applied to the temporal matrix to retrieve the spectra that

will be discussed in detail in chapter 11.

One final issue is the phase of the data. According to the coherent nature of the FWM generation,

one expects a fixed phase relation between the bands S¯acc,Sa, and Sc. Although this is true, it is

hard to determine. For example, not all spectral modes of pulse A might be detectable, as we will

discuss in section 11.1.1. In this case, the signal cannot be phase–normalized to this corresponding

mode. It turned out to be a reasonable solution to interpret

E2D(νa, νc) = F {S¯acc}(νa, νc)·F {Sc}(νc)

|F {Sc}(νc)|−1

(7.1)

as the 2D spectrum. The 2D spectrum phase is here displayed with respect to the phase of the re-

spective mode of pulse C. This makes sense, as the FWM beam is created when pulse C is diffracted

by the transient grating in the SOA.

7.3. Pathways for future development

Some limitations imposed by the current instruments have already been discussed. Another limit-

ing factor is, in general, the lock–in time constant. It requires from us to wait for several microsec-

onds to acquire a new data point, which results in DAQ rates of eνdaq = 20 kHz. Is it possible to

overcome this limitation? And to push eνdaq maybe even to the MHz range?

It probably is. The time–constant can be reduced to zero, if a temporal rect-function is used

instead of the Butterworth filter discussed in section 1.3.2. As we have seen in fig. 2 (c), this would

result in a broad spectral response. However, we can here make use of the fact that the heterodyne

bands are extremely narrow and equally spaced. The spectral sinc-function also has equally spaced

7. Two–dimensional coherent spectroscopy with white pulses

zeros—and nothing is better than a zero if you would like to filter a mode out. It is just necessary

to establish a proper relation between e

νdaq and the mode shift frequencies. Additionally, it is

even possible to adjust eνdaq to the basic sampling rate of the digital instrument. Let us assume a

instrument sampling rate of eνsamp = 500 MHz and a desired DAQ rate of eνdaq = 5 MHz. One

DAQ point will represent the FT of 100 original sampling points. We can now choose the AOM

frequencies as bνc= 12 eνdaq,bνa= 13 eνdaq, and additionally detect at bν¯acc = 11 eνdaq. This results in

exactly the same values as we used before. The demodulator at bνcwould (theoretically) perfectly

filter out the bands at bνaand bν¯acc because of the corresponding zeros in the sinc-function.

This acquisition would be 250 times faster than the present one. Of course, this requires an

integrated signal generation and analysis device which could fulfill these requirements. And it

would bring new challenges to the delay generation, to the data processing, and several other

aspects. From the perspective of physics, however, it would be possible to decrease the acquisition

time of a full 2D spectrum from 4 min today down to about 1 s. In this case, 2D spectroscopy could

even be used as an alignment tool.

Part III.

Investigation of subsystem coupling

in the DWELL system

8. The sample

In this chapter, the sample is introduced briefly and some of it basic properties are collected. These

are acquired by electrical and linear–optical methods. Several more detailed characteristics of the

SOA will be derived from the experiments in the following chapters.

8.1. Sample structure and mounting

The structure used in our experiments is an In(Ga)As-based SOA. The active region contains 15

layers of self-assembled QDs grown by molecular beam epitaxy with a nominal QD density of

ρQD = 1011 cm−2. The QDs are immersed in an InGaAs QW in a DWELL structure. The QD

layers are separated by 33-35 nm thick GaAs barriers to prevent vertical coupling. The active

region is enclosed between pand ndoped bulk GaAs (pin-structure). The lateral extension of the

active region is 2 µm, the length of the waveguide is 1.5 mm.

The SOA is mounted in a cryostat to enable cooling, although this has not been done throughout

this work to protect the SOA from damage. The SOA diode structure is bonded and contacted inside

the cryostat. This enables the application of an external bias Uand the injection of a current J.

At a forward bias of Uth = 855 mV, the injection threshold is reached. The J(U)characteristics

are displayed in fig. 45. In this work, only forward bias was applied. It has been varied from

0 V to 1.78 V, where a maximum injection current of 200mA was reached. There is no active

temperature control implemented, so the SOA is operated under standard operation conditions in

all experiments. A shrinking band edge upon carrier injection needs to be considered according

to eq. (2.4).

Light is coupled into and collected from the waveguide using aspheric lenses mounted outside

the cryostat. The coupling is adjusted by positioning of the lenses in x,y, and zdirection. Manually

operated micrometer screws were sufficient for alignment throughout the experiments presented

here. The waveguide of the shallow etched SOA provides gain guiding, i.e. the propagation of light

is favored in the area with high carrier injection. This results in a modification of the waveguide

properties upon variation of J. This does not pose a severeobstacle to our measurements. However,

the alignment will always be optimized for a particular injection current and will get slightly worse

when Jis changed. This will be discussed in section 11.1.

8.2. Amplified spontaneous emission

The ASE of the SOA is shown in fig. 31. At low injection, a first peak centered at 1281 nm (967 meV)

rises that can be attributed to the QD GS. It can be approximated by a Gaussian with a FWHM of

30 meV. From pump-probe experiments, the GS transparency current is determined to be J0=

5 mA at 1280 nm. ASE data show that at injection currents above 16J0= 80 mA the GS reaches

the maximal inversion and the GS luminescence intensity saturates. Above 40mA, a second peak

becomes visible that is attributed to the ES of the QDs.

Although the peak structure looks rather simple on the first sight, some features hint to a more

complicated nature of the energy structure. Figure 31 is taken from Ref. 11, where Benjamin

8.2. Amplified spontaneous emission

Figure 31:The measured ASE of the SOA (red, solid lines) compared to calculations (dashed gray)

performed by Benjamin Lingnau based on Maxwell–Bloch equations.11

Lingnau presented a high quality calculation of the ASE based on their Maxwell–Bloch model. Such

calculations are complicated, as different QD and QW states contribute to the dielectric function

and the resulting radiation is influenced by the DOS as well as the propagation behavior. In fig. 31,

a region between GS and ES is highlighted. The calculations always underestimate this particular

part of the spectrum. This might be a hint to the presence of a CE, as we will discuss in detail in

chapter 10. The ES cannot be approximated by a Gaussian as well as the GS. This might be due to

the proximity to the QW, but might also be influenced by the presence of a large number of CEs

in this energy range.

9. Rabi Oscillations at room temperature

The coherence time of self–assembled QDs has been extensively studied under various injection

and temperature conditions.78,93,94 At room temperature, the coherence time T2≤300 fs limits the

observation of coherent effects to a time range on the order of a single pulse length.78 In contrast

to low temperatures,77 it is therefore not possible to observe Rabi oscillation in a pump–probe

scheme at room temperature.

The modification of the shape of a single pulse caused by Rabi flops, however, has already been

predicted in the early days of laser physics.95 In parallel to our work, other groups also investigated

the influence of Rabi oscillations on the pulse shape experimentally,96 so this field still attracts

substantial interest. For data processing applications, also the interaction of pulses in the strong

coupling regime will be interesting.97,98

9.1. Pulse deformation caused by Rabi flops

From a technical point of view, the development of the FROSCH technique was a challenge on its

own, coming along with the change to the fast data acquistion paradigm. While the first physical

motivation was the pulse shape modification by ultrafast changes in the gain recovery by reservoir

feeding, discussions with Julian Korn and Benjamin Lingnau drew our attention to the impact of

Rabi oscillations.

In our lab, the field strength required to force a Rabi cycle below the pulse duration can only be

achieved by the Ti:sapphire laser combined with the OPO. The most well–defined 2LS is found at

the QD GS, so the OPO wavelength center was chosen resonant to the GS at 1280 nm. The OPO

provides high quality Gaussian pulses with a spectral FWHM of 15 nm, which eases the interpre-

tation of pulse shape modifications and the comparison to calculations. The pulse input power

measured in front of the SOA in–coupling lens was varied from 1 mW to 12 mW.

The effect of the Rabi oscillations is mainly determined by two experimental parameters: the

initial pulse power and the initial GS inversion. Any pulse modification caused by Rabi flops must

vanish when the power is lowered and the Rabi cycles exceed the pulse duration. On the other hand,

any effect should cancel out at zero inversion. An effective modification is only expected when the

majority of QDs contribute from the same initial state (inverted or not inverted) and thereby with

the same phase. To isolate the effect of Rabi oscillations from other pulse deformations, these two

parameters need to be varied.

In fig. 32 a current series is presented in columns (b-e) and in column (a) the vanishing of the

pulse shape modification effect for a low pulse intensity is demonstrated. The upper row contains

the amplitude values of the FROSCH signal: Without injection current, in column (b) a significant

deviation from the original Gaussian shape is observed. This “bump” disappears in the transparent

case in column (c) and reappears for the inverted system in (d). For very high injection currents

in column (e), the T2time is reduced and the pulse shape modification is less pronounced. The

middle row contains the corresponding phase data. A dip in the amplitude is accompanied by a

phase shift of π. This behavior is characteristic for any coupled oscillating system as it has been

described in section 3.3.2. The GT shown in the lower row will be discussed in section 9.2.

9.1. Pulse deformation caused by Rabi flops

Figure 32:Pulse shape modifications caused by Rabi oscillations. In the upper panel, amplitude

data are shown, with experimental data in black and theoretical calculations in red. The middle

panel shows the corresponding signal phase. The lower panel displays a GT of the experimental

data. The data show the expected behavior upon power and injection current variation: (a) For

low pulse power, the Rabi cycle exceeds the pulse duration, no modification is observed. (b)

For high power a pulse shape modification is observed in the case of a non–inverted system.

cancel out. (d) For high injection current with a majority of inverted QDs, the modification

reappears. (e) Further carrier injection reduces T2, the signature fades. Similarly published in

Ref. 1.

9. Rabi Oscillations at room temperature

Figure 33:Power dependence of the pulse shape modification induced by Rabi oscillations. (a)

The amplitude of the temporal signal on a logarithmic scale and (b) the corresponding phase

show typical phase shifts indicating reversed energy flow. (c–e) Also in the GT, the dip moves

to earlier times.

The theory values shown in the upper and middle row of fig. 32 as red solid curves are based

on calculations performed by Julian Korn and Benjamin Lingnau using Maxwell–Bloch equations

(section 3.3.1).1The calculations reproduce all the characteristic features and prove that the general

statements of the picture of coupled pendula1remain valid in an inhomogeneously broadened

ensemble and under consideration of propagation effects.1,99

In fig. 32 and Ref. 1 the power dependence is not discussed in detail. The original data series

included more power steps. A corresponding power series is presented in fig. 33. Dashed lines

indicate the phase shift positions. For increasing power, the flop events occur earlier, which is

exactly the expected behavior. For the highest input power of 12 mW, even a second phase shift

can be observed.

The quality of the phase data in figs. 32 and 33 is limited. In this very early experiment, an

additonal plane mirror mounted to the delay stage was used to set up the FROSCH Michelson

interferometer. The setup is described in Ref. 1. The velocity of the VT-80 delay stages varied about

a factor of 3 upon continous movement. The oscillatory movement induces a relative oscillation of

the plane mirror against the retro–reflector of the reference beam, which is read as an oscillation

of the phase. In later experiments, the plane mirror was removed and the interferometer beam is

now reflected by the same retro–reflector as the reference beam. This leads to much better phase

data, see for example in fig. 34.

9.2. Temporal vs. spectral representation

In fig. 32, a GT is displayed in the lower panel. It shows the temporal as well as the spectral dis-

tribution of the resulting pulse. A dip in the GT amplitude can be read as an indicator of Rabi

oscillations. However, there are some issues that need to be considered as drawbacks of this rep-

resentation. The GT is not unambiguous, it depends on the choice of the window function as it

has been shown in section 1.1.1. This also means that there is no defined position of the dip and

9.2. Temporal vs. spectral representation

Figure 34:A SMPM measurement of FROSCH traces spanning three orders of magnitude in pulse

input power.

therefore one cannot derive physical information from it. And finally, the shown amplitude repre-

sentation neglects the phase information of the GT, which is in fact hard to interpret. In conclusion,

the GT provides the interesting combination of spectral and temporal information, but is limited

in providing unchallengeable arguments.

Alternatively, it is also possible to restrict the discussion of FROSCH data either to the temporal

data (with the phase indicating qualitative changes as shown above) or to the pure spectral FT

data. An example for the spectral interpretation is shown in fig. 34: It is the demonstration of a

FROSCH SMPM measurement reproducing some of the results of Ref. 1. The data are acquired in

a single scan at an SOA injection current of J= 80 mA, i.e. at the GS inversion saturation. The

SMPM spans a power range of three orders of magnitude, from 14 µW to 14 mW, which—by the

way—impressively demonstrates the advantages of field based measurements. A temporal pulse

modification can be observed in fig. 34 (a). The interpretation of the modification, however, is

not as straightforward as in the data set of Ref. 1 due to a slightly different chirp of the input

pulses. The FFT of the FROSCH data shown in panel (b) is much more elucidating. Two effects

can be identified if one has in mind that the original pulse is slightly red–shifted with respect to

the GS: For low power pulses, the SOA amplification profile dominates the spectral position. With

increasing power, the resulting pulse shifts to the red, because the ratio between original power and

amplified spectrum increases. When the strong coupling regime is reached, the Rabi splitting65,100

sets in and becomes dominant for the maximum power of 14mW.

10. Evidence of Crossed Excitons in

pump–probe experiments

Crossed Excitons have been described as a bound state of carriers of mixed dimensionality in sec-

tion 2.3.4.62,63 In the DWELL system there are in general two types of CEs that might be observed

within the spectral window of our laser system: QD–bulk and QD–QW combinations. For both

of them we found experimental evidence, which motivated the further development of the experi-

ments following in chapter 11.

10.1. Crossed Excitons revealed by gain excitation

spectroscopy

The original question of the experimental sets presented in this chapter was: How exactly does the

QD couple to the QW? Can carriers be captured directly by the QD GS or is a cascading process

via the ES101 necessary? To clarify this, we chose the two-color pump-probe (2cPP) approach

with the pump tuned to the QW and the probe tuned to the GS. While the probe was fixed at a

wavelength of 1280 nm (0.969 eV), a series of curves is taken with the pump wavelength tuned from

1200 nm (1.033 eV) up to 1070 nm (1.159 eV) in steps of 10 nm. The injection current is varied for

each pump wavelength from 0 mA to 200 mA. The original plan was to compare this set of data to

an alternative case with the probe tuned to the ES. If the response of the ES would be substantially

faster than the response of the GS, this would be a proof of a cascading transfer mechanism.

A selection of PP traces is shown in fig. 35(a). Surprisingly, the response of the GS was already

faster than the temporal resolution of the experiment. Looking for a faster ES response in a second

measurement series was therefore futile. At this point, one of the tremendous benefits of the fast

data acquisition approach that we used here for the first time†came into play: the ability to acquire

a large number of traces. All the conclusions following in this section are not derived from a single

curve, but from the comparison of many curves.

10.1.1. Complexity reduction by linear fitting

The simultaneous handling of many curves enforces the reduction of the traces to appropriate

figures of merit characterizing their behavior. For the data in fig. 35(a), two properties of the

curves are of special interest: Firstly, the initial change of ∆G, and secondly, whether the traces

fall or rise in the subsequent time range. For low injection currents, the rising/falling behavior

is characterized by a linear fit in the temporal range of 1 ps to 5 ps. The fit slope is depicted in

fig. 35(b) as dots and stars. A change of sign indicates a qualitative change that can be attributed

†The actual setup deviated from that shown in fig. 19. In this very early stage, a mechanical chopper replaced the AOM

fast switching (this was due to a very slow amplifier in the inherited setup that was removed later on), thus SMPM

was not possible. A noteworthy fun fact: This highly important measurement series at the very end of a lab time

has almost been devastated by a server malfunction. Just in time we were able to emulate the lost control software

by a “Notfallprogramm” (emergency program). This was one experience that motivated the GIT system with local

and remote copies.

10.1. Crossed Excitons revealed by gain excitation spectroscopy

Figure 35:(a) A set of pump–probe traces at pump wavelengths around the QW band edge. (b)

The PP trace slope in the range of 1 ps to 5 ps (shown for clarity in the range 1ps to 20 ps)

revealing the QW band edge by a change of sign. (c) Gain excitation spectra (solid bold blue)

for varying injection currents. Similarly published in Ref. 2.

to the crossing of the band edge. Pumping above the QW band edge leads to a reservoir filling

and a subsequent carrier capture, leading to a rising differential gain in the QD GS in this early

time range. Pumping below the band edge only addresses QD related states, so the reservoir is not

filled and an immediate decay sets in. The exact determination of the QW band edge to 1164 nm

(EW= 1.065 eV) according to the zero crossing of the fit of the PP trace slopes is an important

step in the characterization of the SOA that could not be extracted from the ASE spectrum.

Besides the fit slope, also the fit intercept ∆G(0) provides important information. It represents

the differential gain excited immediately in the GS. As we were able to resolve this excitation spec-

trally by means of the large number of pump wavelengths, we referred to the analysis method as

gain excitation spectroscopy. The result is shown in fig. 35(c) for all injection currents∗, together

with the ASE and the QW band edge determined before. For low injection currents, there is a

positive excited gain, for high injection currents this changes to a gain depletion. The change of

sign, however, does not correspond to the GS transparency current of 5 mA but does rather occur

around 50 mA. Two peaks can be determined above the QW band edge especially for low injection

currents. These peaks remain visible for higher currents and show a well–explainable current de-

pendence: They shift to lower energies for high injection currents according to the sample heating

and the lower energy peak changes its sign at a lower current than the higher energy peak. Be-

low the band edge, a peak is visible that corresponds well to the ES ASE. ∆G(0)(0 mA) is slightly

shifted to higher energies with respect to the ASE at 200 mA, which excellently agrees with the

Varshni shift caused by the heating of the SOA under current injection. These dependencies are

an indicator for the high quality of the data.

Indicated in fig. 35 (a) and displayed in fig. 35 (c) as dotted lines, there is a second quantity that

shows a similar pattern: the minimum value ∆Gmin of the differential gain at the pump–probe

overlap. The minimal value can be determined by two factors. Firstly, a real gain depletion and

∗As the “Notfallprogramm” lacked automization, the planned measurement at 60mA has simply been forgotten in the

manual setting routine.

10. Evidence of Crossed Excitons in pump–probe experiments

Symbols Energy, eV

EG,eEE,eEW,eEB,e0.707 0.757 0.777 1.029

EG,hEGEWGEBG -0.262 0.969 1.019 1.039 1.291

EE,hEEEWE EBE -0.281 0.988 1.038 1.058 1.309

EW,hEGW EEW EW-0.288 0.995 1.045 1.065 1.317

EB,hEGB EEB EB-0.381 1.088 1.139 1.158 1.410

Table 3:First approximation of Crossed Exciton energies according to eq. (10.2).

a subsequent re–filling of the depleted states. Secondly, there might be also a contribution of

coherent interaction effects, often referred to as “coherent artifact”. The latter contribution cannot

be identified using a static reference delay tcd. The analysis of such contributions will be done in

the 2DCS experiments in chapter 11. Here, however, the similarity of the patterns of ∆G(0) and

∆Gmin is another indicator for the consistency of the measurements.

10.1.2. Crossed Excitons or Upper State?

The hypothesis of transitions from QD states to continuum states has already been extensively

discussed in my diploma thesis.102 An alternative explanation, however, would be the existence of

an “upper state” as it is assumed in several rate equation models.103,104 In this picture, the found

structure would be solely related to quantum dot states. In a first interpretation of the peak shapes

in fig. 35, this approach seemed plausible: The line width roughly mimics the inhomogeneous

broadening of the ASE. This contrasted our expectation of a continuous gain excitation, as the

involved bulk or QW states have a continuous DOS.

Calculations performed by Sandra Kuhn, however, convinced us to return to the interpretation

of bound–to–continuum transitions and CEs.105 These calculations are performed for intraband–

correlations of carriers and the exact calculation of the interband analog is much more complicated,

but one central argument applies also to this case: Although the DOS is continuous, the momentum

selction rules favor k= 0 and therefore the Γpoint. This results in a sharp peak in the absorption

spectrum. Of course, the resulting peak widths in fig. 35(c) are determined by the homogeneous

and inhomogeneous broadening as well as the pump and probe pulse spectral widths.

This does not negate the existence of an upper state. Quite in contrary, our interpretation pro-

vides an explanation for the formation of an upper state that is often useful to describe the electron

dynamics. The effect is actually the same, which will become more evident in section 10.2.

10.1.3. Approximation of Crossed Exciton energies

The calculation of exact CE energies is beyond the scope of our experimental work. A first ap-

proximation of the expected CE energies is based on a strongly simplified picture according to

section 2.1.2. We assume a fixed effective mass ratio for electrons and holes of

r=m∗

h/m∗

e= 2.7,(10.1)

neglecting any differences between the subsystems. This value is based on literature values for

InGaAs QWs of m∗

e= 0.071m0for electrons106 and m∗

h= 0.191m0for holes107, respectively.

We assume a very simple DOS with carrier energies derived from the transition energies of the

GS (EG= 0.969 eV), the ES (EE= 1.038 eV), the QW band edge (EW= 1.065 eV), and the

10.2. Carrier dynamics in a DWELL system

gallium arsenide (GaAs) bulk band edge (EB= 1.41 eV which corresponds to eq. (2.4) and table 1

at T= 328 K). Single carrier energies are calculated according to

Ei,h=−Ei

1+r=−Ei

3.7(10.2a)

Ei,e=Ei−Ei,h(10.2b)

and CE energies are given by the energy difference

Eij =Ei,e−Ej,h(10.3)

with i, j ∈ {B,W,E,G}. The respective energies and the resulting CE energies are summarized

in table 3. Eight CE energies are calculated for combinations of 0D and 2D/3D states. Six of them

are in the observation range of our presented gain excitation spectroscopy (only EBG and EBE

exceed this range). In general, all of them would be in the spectral range of the FemtoFiber pro

SCIR laser. A large spectral separation between pump and probe pulse, however, complicates the

alignment (the SOA is strongly absorptive for higher QW energies, even under maximum carrier

injection) and worsens the simultaneous in–coupling due to chromatic aberration.

These energies and this derivation will serve for the energetic localization of CEs throughout the

following chapters. Several aspects are completely neglected, e.g. the influence of semiconductor

doping on the relative shift of subsystem band edges. Moreover, as the two projects developed

independently, the calculations following in section 10.2 are partially based on different literature

values.3,108 This is, however, not very important. The real challenge of understanding the complex

system without feeding in parameters will be tackled in chapter 11.

Attributing Crossed Excitons to the gain excitation spectra

The CE energies are depicted in fig. 35(c). The two dominant peaks at energies above the QW

band edge can be attributed to CEGB and CEEB, respectively. In the context of our work, this is the

first experimental evidence of optical transitions of mixed dimensionality, in this case pumping an

electron from the GaAs VB to the QD CB in the non–inverted case. Although this might seem very

unlikely, one needs to consider that there is only a very limited set of allowed transitions in that

specific energy range that affect the QD immediately. Without electrical injection, no intra–band

absorption is possible, and even with injection current it would contribute only gain depletion.

Right below the band edge, a large number of possible transitions is found. The direct QD ES

is accompanied by CEWG, CEEW, and CEWE. This large number of potential configurations and

the rapidly changing spectral gain profile caused by the proximity of the QW band edge raise the

question, how purely the ASE ES peak can be attributed to the QD ES transition. If the above–

band–edge CEs have such an impact on the gain excitation spectra, do the below–band–edge CEs

not need to contribute in a similar way to the luminescence? Actually, this question will not be

answered in the following. But it needs to be kept in mind whenever thinking about the interpre-

tation of ASE spectra.

10.2. Carrier dynamics in a DWELL system

The existence of CEs will change the mechanisms of electron dynamics tremendously. The ques-

tion might not only be the one raised in the very beginning of section 10.1: Are the subsystems

coupled via a cascading process or via a direct capturing? The question is also: Are correlated

carriers captured simultaneously or do CEs offer an intermediate pathway that allows a stepwise

10. Evidence of Crossed Excitons in pump–probe experiments

Figure 36:Concept of the diffusion experiments. (a) All experiments are performed for low for-

ward bias below the carrier injection threshold. Carriers are created in the QDs, thermally

activated, and extracted by the band bending potential. (b) The GS ensemble is split in three

subensembles with respect to the probe pulse spectral width: A resonant subensemble, a lower

energy subensemble (“red subensemble”), and a higher energy subensemble (“blue subensem-

ble”). Similarly published in Ref. 3.

Parameter Resonant Off-resonant

Maximum power Pmax 1.35 mW 128 µW

Probe spectrum red limit 964 meV 965 meV

Probe spectrum blue limit 977 meV 975 meV

Pump spectrum red limit 951 meV 988 meV

Pump spectrum blue limit 998 meV 998 meV

Number of bias levels 18 22

Number of SMPM power levels 5 4

Total number of curves 90 88

Table 4:Parameters of the two experimental series, resonant and off-resonant, respectively. Spec-

trum limits denote energies at which the spectral density drops below 5 % of the maximum value.

The spectral distribution of pump pulses is illustrated in fig. 36 (b).

capturing—maybe even without breaking the carrier correlation? In this context we designed an

experimental setting that addresses two main aspects: How do CEs in general modify the inter–dot

diffusion and how do CEs in the GS spectral range contribute to the optical excitability?

The diffusion dynamics is directly related to the spatial localization of carriers around the QDs.

Considering CEs, there is a much larger number of localized states beyond the six exciton states

of GS and ES. The observation of localization requires isolated carriers. This excludes electrical

carrier injection that creates a random non–equilibrium distribution of carriers. The experiments

in this section are therefore performed at forward bias, but below the carrier injection threshold.

The corresponding potential scheme is shown in fig. 36 (a). This is the solar–cell configuration of

the pin–structure, so the results will be of particular interest for photovoltaic applications.109

The question of optical excitability governs the spectral choice of pump and probe pulses. These

considerations are sketched in fig. 36 (b). The probe is centered to the ASE GS maximum and defines

the “probe sub–ensemble”. The QDs with lower GS transition energy form the “red subensemble”,

those with higher energy the “blue subensemble”. The experiments comprise two series with re-

spect to the pump pulse spectrum: The resonant series addresses the probe subensemble directly.

10.2. Carrier dynamics in a DWELL system

Figure 37:Left: An exemplary set of data, taken from the resonant series at U= 750 mV. Right:

A plot of peak amplitude versus pump power (level visualized by dashed arrows) shows a satu-

ration well described by eq. (10.4). Similarly published in Ref. 3.

For the off–resonant series, the pump spectrum is tuned to the spectral range where CEGW, the

lowest energy CE, is expected. So the off–resonant series addresses the blue subensemble as well

as the CEGW of the probe subensemble. To identify the degree of saturation, which is necessary

to determine the ratio of GS and CE excitation, the SMPM technique is applied. In fig. 37 the ex-

citation saturation of GS pumping is shown for an exemplary set from the resonant series. With

linearly increasing pump power, the data show a clear saturation behavior that is well-described

by the function

∆G(sat)(P) = ∆G(max) 1−exp −ηG

∆G(max) P/Pmax (10.4)

with ∆G(max) = (11.8±0.3) dB,ηG/∆Gmax = 1.43 ±0.06 and the maximum power Pmax =

1.35 mW (18.0 pJ/pulse) for this particular experiment. If we compare fig. 37 to fig. 20, we do not

observe a significant contribution of TPA in fig. 37.

10.2.1. Numerical model

For a quantitative analysis of the data, we construct a rate equation model composed of the QD and

a surrounding 2D continuum. A direct comparison in Refs. 110 and 111 of QDs with surrounding

QW and such without even a wetting layer showed the decisive role of the QW in the inter-dot

coupling. The qualitative change of decay behavior upon variation of the extraction potential

Uextr(U) = Uth −U(10.5)

observed in fig. 38 (b) can be displayed in terms of an instantaneous time constant analog to

eq. (2.10):

τG(tbc) = −∆G(tbc)

∂tbc∆G(tbc).(10.6)

The instantaneous time constants for respective ranges of 15 ps are displayed in fig. 39 (a). It shows

a constant decay rate for high Uextr and a linear growth of the time constant for low extraction.

10. Evidence of Crossed Excitons in pump–probe experiments

Figure 38:Pump–probe data sets demonstrating carrier diffusion. Experimental data are shown

as blue dots, calculations based on rate equations according to section 10.2.1 as red dashed lines.

Light colors indicate high extraction potential (full band bending), dark colors low extraction

potential (compensated band bending). Bold numerals indicate the extraction potential Uextr.

The resonant series is shown in the left panel (a-c), the off–resonant series in the right panel

(d-f), respectively. Similarly published in Ref. 3.

Figure 39:Identification of diffusive carrier transport and implementation in the model. (a) In-

stantaneous decay constants τG(τ)according to eq. (10.6) for the resonant pump-probe series at

different extraction potentials, calculated from linear fits for 15 ps time ranges. The dashed (red)

lines represent linear fits of the limiting cases of zero and high extraction. (b) The transition

probability to other QDs is proportional to the number of QDs in area A, which is approximated

in eq. (10.17) as the area with bold (blue) contour. (c) Inward diffusion probability within one

sub-ensemble is proportional to the solid (red) fraction of the circle.

10.2. Carrier dynamics in a DWELL system

Figure 40:The basic scheme of states for the rate equation model. Each QD is represented by

three states: The GS (G), the ES (E), and an averaging CE state (C). The QW continuum is repre-

sented by a chain of bins (1. . . n), each of them representing a ring centered at the QD. Several

transitions, feeding, and extraction processes are implemented by time constants. Similarly

published in Ref. 3.

This linear growth is expected for diffusive processes. We assume, without knowing the particular

transfer mechanism, that the effective mathematical form of the carrier motion in 2D is the one

of a diffusion process (section 2.3.3). We thus model the QD as a three-level system, and for the

QW we assume an unstructured continuum in which we implement a random walk process for

the particles with Boltzmann-distributed thermal velocities.

We use a linear differential equation system according to eq. (2.6) and the description in sec-

tion 2.3.3. While this approach would not be sufficient to describe a QD-SOA in general, the

specific experimental conditions described above make it applicable: Here, every excitation is cre-

ated in the GS of a spatially isolated QD, for most quantum dots a single exciton or no exciton at

all is created (random biexcitons will be neglected). In this case, excitations are very dilute and

exciton-exciton interactions can be neglected. Most of all, the absence of other carriers allows to

treat fermions like bosons, without concerns about state filling. The calculations are carried out

using Scilab 5.5.1.

States

The fundamental scheme of the rate equation model and the allowed transitions between the states

are illustrated in fig. 40. The scheme is designed to take into account the spectral distribution of

the states within the inhomogeneously broadened QD ensemble, different behavior of paired and

unpaired carriers as well as different degrees of confinement. Instead of consecutive numbering

we will use a triplet (jkl)to identify these three features, respectively, and to address the elements

Njkl,ξ of N, which is nevertheless a column vector. The fourth index ξ∈ {res,oﬀ}for resonant

and off-resonant experiment will only appear if it is necessary to distinguish between the two

experimental series, and will be omitted otherwise for simplicity. For the elements of transition

matrix Γwe use the notation Γjkl

ικλ for the element in column (jkl)and row (ικλ), i.e. a transition

from (jkl)to (ικλ)for positive entries. All elements of Γnot mentioned explicitly below are zero.

10. Evidence of Crossed Excitons in pump–probe experiments

Confinement

By the first index j∈ {G,E,C,1, . . . , n}the quantum dot or 2D continuum state is identified.

The QD is represented by a three-level scheme, containing the QD GS (G), the QD ES (E), and a

generalized CE state (C). Note that the manifold of CE states available for equilibration contains

all possible CE states, while the only CE state energetically accessible for optical excitation is the

lowest state (CEGW, see fig. 36). The numbered states represent the two-dimensional continuum

into which the QD is embedded. In-plane radial symmetry allows to model the 2D continuum in a

1D representation: Each of the ncontinuum states represents a ring of width Rand outer radius

jR,j∈ {1, ..., n}, centered at the QD. The model for the continuum dynamics is explained in

detail in section 10.2.1. The different states of the system have different degeneracy: While the

sizes of GS N(max)

G= 1 and excited state N(max)

E= 2 are fixed, the size of the crossed exciton

state N(max)

Cis determined as a fit parameter. No specific size is attributed to continuum states, as

explained below.

Spectral ensemble

The second index k∈ {r,p,b}identifies the relative energy of the spectral sub-ensemble. Rela-

tive to the experimental probe pulse spectra, the QD ensemble is split into three sub-ensembles

(fig. 36(b), details in section 10.2.1). The central part of the QD ensemble is the probe sub-ensemble

(p), it represents a fraction of dpof the QDs. QDs with lower GS energies are represented by the

red sub-ensemble (r) with size dr, those with higher energies by the blue sub-ensemble (b) with

size db, with dr+dp+db≡1. For the QD GS ensemble we assume a Gaussian distribution for

the density of states:

D(E) = exp −ln 2 E−EG)

σG/22!(10.7)

centered at EGand a FWHM of σG(see. fig. 36 (b)).

Particle type

Finally, the particle type is given by the third index l∈ {x,e,h}for excitons (x), unpaired electrons

(e) or unpaired holes (h). Of course, unpaired carriers cannot occupy CE states, so the state j= C

does not exist for l∈ {e,h}.

Transitions

Every state (jkl)is placed at a specific position Ejkl on the energy scale. The GS energies of

excitons (j= G,l= x) are determined as weighted average of the Gaussian ensemble distribution

in eq. (10.7). We assume EEkx=EGkx+ 70 meV which can be read directly from the ASE spectra

(fig. 31) and

ECkx=EGkx+ 100 meV.(10.8)

Equation (10.8) reflects the estimation of a weighted average for all CEs based on the knowledge

that EGB and EEB exist above QW band edge (chapter 10) and the assumption that all other

CEs also provide states. All continuum states are located at EW= 1.061 eV. For electrons and

holes, ground and excited state energies (j∈ {G, E}) are set to fulfill the condition Ejke=

(rEjkx+EW)/(r+ 1) and Ejkh= (Ejkx+rEW)/(r+ 1), respectively.

10.2. Carrier dynamics in a DWELL system

In our model, the probability for upward transitions (Eικλ > Ejkl) is reduced by the Boltzmann

factor

Θjkl

ικλ = exp Ejkl −Eικλ

kBT(10.9)

with Boltzmann constant kBand temperature T= 300 K. Downward transitions are not modified

(Θjkl

ικλ ≡1for Eικλ ≤Ejkl).

For intra-dot transitions between the states G, E, and C, we assume a time constant of τQD =

2 ps.112,113 The matrix elements for intra-dot transitions are under consideration of degeneracy:

Γjlm

ιlm =N(max)

ιΘjlm

ιlm /τQD (10.10)

for j, ι ∈ {G,E,C}, j =ι.

Transitions are allowed between the QD and continuum state 1. An exciton transits between QD

state C and continuum state 1, so escape from the QD and capturing into the QD can be considered

as a stepwise process only applying to one of the constituent carriers at a time, similar to the carrier

capture via indirect excitons observed at elevated temperatures in Ref. 114. A state size N(max) is

not introduced explicitly for the continuum and therefore the degeneracy is included in the time

constants τesc for escape and τcap for capture, respectively. The matrix elements are given by

Γ1kx

Ckx= Θ1kx

Ckx/τcap,(10.11)

ΓCkx

1kx= ΘCkx

1kx/τesc.(10.12)

Since unpaired carriers l∈ {e,h}do not have a CE state, the continuum state 1 is coupled to states

G and E with the same time constants:

Γ1kl

jkl = Θ1kl

jkl N(max)

N(max)

G+N(max)

E!/τcap,(10.13)

Γjkl

1kl = Θjkl

1kl/τesc.(10.14)

for j∈ {G,E}.

Diffusion

In our model, we picture the lateral inter-dot carrier transfer as a consecutive process involving

thermal ejection from a QD, subsequent diffusion in the QW, and capture by a QD. Tunneling

between dots or the formation of QD molecules is unlikely at the given dot density, which corre-

sponds to an average inter-dot distance of 30 nm.115,116 The diffusion of carriers in the 2D contin-

uum and the transition from one spectral sub-ensemble to another is modeled in terms of simple

geometrical considerations illustrated in fig. 39(c,d). Carriers are treated as particles with masses

m∗

e= 0.041m0for electrons and m∗

h= 0.465m0for holes.108 The exciton mass is simply con-

sidered to be m∗

x=m∗

e+m∗

h. The particles move at thermal velocities given by the Boltzmann

distribution

Fl(v)dvdϕ=m∗

2πkBTexp −m∗

lv2

2kBTdvdϕ. (10.15)

Every carrier is assigned to its nearest QD. Other QDs are assumed to be distributed homoge-

neously in the 2D continuum according to nominal QD density ρQD. A carrier transits from one

QD to another if it is closer to a different QD after a motion of time interval dt(fig. 39 (b)). In this

moment, the distance to new and old nearest QD is equal, so the ring number jis conserved. The

10. Evidence of Crossed Excitons in pump–probe experiments

carrier starts at a distance rfrom its present QD in a direction given by angle ϕto a new position

at distance

r′=q(vdtsin ϕ)2+(r−vdtcos ϕ)2

≈pr2−2rvdtcos ϕ, (10.16)

where we used vdt≪r. The transition probability is equal to the number of QDs in the (red)

filled area A, which can be approximated as

A(r, v, ϕ) = |ϕ|(r′2−r2)+2r v dt|sin ϕ|

= 2 r v dt(|sin ϕ|−|ϕ|cos ϕ)(10.17)

with ϕ∈[−π, π]. By integrating over all directions and velocities, the transition matrix element

becomes

Γjkl

jκl =dκρQD Z∞

v=0 Zπ

φ=−π

dvdϕ Fl(v)A(r, v, ϕ)

=dκρQDs32 kBT

πm∗

r(10.18)

with k, κ ∈ {r,p,b}, k =κunder consideration of sub-ensemble size dk.

To derive the probability to move inward from ring j∈ {2, ..., n}to the next inner ring j−1

within the same sub-ensemble, we consider a carrier at distance Dfrom the ring border (fig. 39 (c)).

The probability to cross the border is given by p(v, D) = (arccos D

vdt)/π for D≤vdt,and

p(v, D)≡0otherwise. Integration over all velocities and points of the ring yields

Γjkl

(j−1)kl =2(j−1)

(2j−1)RdtZ∞

v=0

dvFl(v)ZR

D=0

dD p(v, D)

=2(j−1)

(2j−1)Rs2kBT

πm∗

(10.19)

where we used that the carrier distance from the QD is approximately constant at (j−1)R.

An analogous calculation yields the outward diffusion matrix element for j∈ {1, ..., (n−1)}.

Moving away from the nearest QD also increases the probability to reach the area of another QD.

Since outward movement becomes inward movement from the viewpoint of the new QD in this

case, we assume that a carrier that changes the sub-ensemble actually does not diffuse outward

any more, therefore we reduce the matrix element by the result from eq. (10.18):

Γjkl

(j+1)kl =2j

(2j−1)Rs2kBT

πm∗

l−X

κ=k

Γjkl

jκl.(10.20)

This matrix element also determines the necessary number of rings nthat is given by Γnkl

(n+1)kl ≤0.

Extraction and decay

So far, only off-diagonal elements of Γhave been discussed. All the processes described above con-

serve the number of particles in the system. Additionally, we included several decay processes into

the numerical model, which annihilate particles or dissociate excitons into unpaired carriers. First,

10.2. Carrier dynamics in a DWELL system

excitons in all states are subject to recombination determined by an average recombination time

constant τrec. Furthermore, excitons in the 2D continuum dissociate with a time constant τdis and

create an unpaired electron and hole. In the continuum states, also the extraction potential Uextr

extracts unpaired carriers from the system and single carriers from excitons with a time constant

τex, the latter meaning the annihilation of an exciton and simultaneous creation of an unpaired

carrier. Therefore we find another off-diagonal element for l∈ {e,h}that represents intrinsic and

potential-induced exciton dissociation, respectively, for continuum states j∈ {1, . . . , n}:

Γjkx

jkl =τ−1

dis +τ−1

ex (U).(10.21)

Thus, the diagonal-elements of exciton continuum states become

Γjkx

jkx=−

X

(ικλ)=(jkx)

Γjkx

ικλ

−τ−1

rec +τ−1

dis ,(10.22)

where the term in parentheses sums the corresponding off-diagonal elements for particle number

conservation and the last term compensates the double appearance of the first term of eq. (10.21)

in this sum (two unpaired carriers are created by dissociation, but only one exciton is annihilated).

For excitons l= x in quantum dot states j∈ {G,E,C}only recombination applies:

Γjkx

jkx=−

X

(ικλ)=(jkx)

Γjkx

ικλ

−τ−1

rec .(10.23)

Unpaired carriers l∈ {e,h}cannot recombine, so extraction is the only decay channel in the

continuum j∈ {1, . . . , n}

Γjkl

jkl =−

X

(ικλ)=(jkl)

Γjkl

ικλ

−τ−1

ex (U)(10.24)

and no decay channels apply for the quantum dot states j∈ {G,E,C}:

Γjkl

jkl =−

X

(ικλ)=(jkl)

Γjkl

ικλ

.(10.25)

Optical interaction

As a last step, optical pumping and probing have to be implemented, where pumping means setting

an initial state N(0)

ξUP =NξUP (t= 0) and probing means the derivation of a gain curve GξUP (t)

from the solution NξUP (t)of eq. (2.6) for every experimental series ξ∈ {res,oﬀ}, extraction

potential Uextr and optical pump power P. The calculation is based on two parameters, the optical

GS response ηGand the optical CEGW response ηC, respectively. Both parameters are allowed to

depend on U,ηCadditionally depends on P, to reproduce dependencies in experimental transition

probabilities. Table 5 summarizes the subsets of data used to determine ηGand ηC. If one of them

is not determined explicitly for a curve, it is interpolated from the parameters of other curves.

The optical GS response ηGgoverns the peak height in the resonant experiment series (fig. 37).

According to eq. (10.4), we determine ∆G(max)

Uand ηG,U /∆G(max)

Ufrom the SMPM power depen-

dence of the resonant experiment series for every Uextr. Since these fit parameters are strongly

interdependent, we will discuss the product ηG,U as the optical GS response.

10. Evidence of Crossed Excitons in pump–probe experiments

The model for the conversion of light to population follows the spectra shown in fig. 36(b). The

spectral limits of the probe sub-ensemble, ν(min)

Gp,ξ and ν(max)

Gp,ξ , are set where the spectral density of

the probe pulse drops below 5% of the maximum value (see table 4). From the measured pump

spectra we derive the spectral photon density Φξ(ν). According to this distribution, the initial

population is set to

N(0)

Gkx,ξUP =∆G(max)

UZν(max)

Gk,ξ

ν(min)

Gk,ξ 1−exp − ηG,U

∆G(max)

U!PΦξ(ν)!!D(hν)dν,(10.26)

N(0)

Ckx,ξUP =ηC,UP Zν(max)

Ck,ξ

ν(min)

Ck,ξ

PΦξ(ν)D(r+ 1)hν −EW

rdν.(10.27)

The differential gain for GS probing is calculated as a weighted sum of states from the solution

N(t)of eq. (2.6). Only carriers in the probe sub-ensemble contribute to the weighting vector

W, the marginal overlap with CEGW of the red sub-ensemble (fig. 36(b)) is neglected. Unpaired

carriers in state G switch a QD in the probe sub-ensemble from absorbing to transparency, this

contribution is set to WGpe =WGph = 1. Also crossed excitons in state C of the probe sub-

ensemble may switch the GS transition to transparency, if one carrier of the CE is in a QD GS state.

Here we assume that 80% of CEs are related to a conduction or valence band GS (20% related to

ES are not probed experimentally), so WCpx = 0.8. An exciton in state G switches a QD from

absorbing to amplifying, so it contributes with WGpx = 2. Considering the bias-dependent optical

GS response, the differential gain becomes

GξUP (t) = ηG,U (W·NξUP (t)) .(10.28)

For comparison with experimental data, all simulated curves are convoluted with a Gaussian of

500 fs FWHM to take temporal resolution into account. The model parameters are summarized

in table 5. They were fitted to the data by an iterative minimization of squares if not depicted

otherwise in the table.

10.2.2. Interpretation of the fit parameters

The numerical model now allows a more detailed discussion of the physical processes in the cou-

pled system. In the resonant case, the excitation decays in three stages. Within the first 3 ps, the

exciton distribution relaxes from the GS to a quasi thermal equilibrium (fig. 38 (a)). This proceeds

predominantly by intra–dot transitions to the ES and CE state governed by the time constant τQD.

In the intermediate time regime (3 ps to 300 ps) we observe a strong dependence of the population

decay on extraction potential. For strong extraction the decay becomes nearly single-exponential

since the return probability for excitons that escaped from QDs is strongly reduced. For low ex-

traction, the resulting decay rate decreases due to an increasing probability of re-capture from the

continuum. Finally, for tbc >300 ps, the excitation is spatially equilibrated over the entire system

and decays at a potential-dependent average rate. Nevertheless, we see a slight reduction of this

rate for later times, especially in the high extraction case. As experiment and model agree very

well in this aspect, this can be attributed to unpaired carriers that cannot recombine any more.

Generally, in our observations, the dynamics is well described by carriers relaxing and diffusing

as excitons, rather than as unpaired electrons and holes. In case of fast dissociation and diffu-

sion of unpaired carriers one would expect a strong dependence of the long time behavior on the

initial carrier density increasing with the SMPM power level, as re-pairing is necessary for recom-

bination. However, this is not observed. These findings are in contrast to results obtained in PL

10.2. Carrier dynamics in a DWELL system

Figure 41:Parameters with potential and power dependence. (a) Normalized optical GS response

ηG. (b) Optical CEGW response ηC, also normalized to the maximum of ηG. (c) Extraction

efficiency τ−1

ex . Efficiency reduction for high pump power can be attributed to compensating

fields caused by a large number of extracted carriers. Similarly published in Ref. 3.

10. Evidence of Crossed Excitons in pump–probe experiments

Parameter Value Source tmin tmax Lower limit Upper limit

GS ensemble center EG973.5 meV Both 0 ps ∞(971 meV) (974.5 meV)

GS ensemble width σG44.1 meV Both 0 ps ∞44.1 meV (55.8 meV)

Ring width R3 nm (Fixed)

CE state size N(max)

C25.9 Resonant 5 ps 15 ps (0)

Intra-dot time τQD 2 ps (Fixed)

Escape time τesc 16.5 ps (Ref. 3) 50 ps 200 ps (0 ps)

Capture time τcap 49.2 fs Both 0 ps 100 ps (0 ps)

Recombination time τrec 752 ps Both 0 ps ∞(0 ps)

Dissociation time τdis 2500 ps Both 0 ps ∞(0 ps) 2500 ps

Optical GS response ηGfig. 41(a) Resonant 0 ps 1.5 ps (0)

Optical CEGW resp. ηCfig. 41(b) Off-res. 0 ps 10 ps (0)

Extraction time τex fig. 41(c) Both 0 ps ∞(0 ps)

Table 5:Overview of model parameters. Except for the last three rows (explained in detail in

fig. 41), these parameters are used by all curves. The right hand side of the table depicts fitting

details: The fitting routine can be restricted to a certain subset of curves, to a time range tmin ≤

t≤tmax, and to upper and lower value limits. For values in parentheses, this limit has not been

reached. For τesc,τcap,R, and τdis, a detailed discussion is given in Ref. 3 (fig. 7).

spectroscopy117 with excitation into the bulk, where carriers were found to behave as independent

electrons and holes. Creating carriers with no excess energy in our case maintains the Coulomb

correlation, and leads to a dominating excitonic behavior.

The off-resonant experiment yields complementary information from the viewpoint of a

spectrally—and as well spatially—different QD sub-ensemble. An immediate response observed

for short times indicates absorption by CEGW, as it was intended by the choice of the off–resonant

spectral excitation range. Figure 38 (f) visualizes the relative contributions of QD GS and all CEs

according to eq. (10.28). At longer times, diffusing excitons arrive in the probed sub-ensemble and

start filling the GS in the intermediate time regime. The population maximum is reached after

137 ps for Uextr = 0 V. In the long time regime, the off-resonant experiment behaves like the

resonant one due to spatial equilibration.

A deeper insight into the nature of the contributing transitions is gained from the dependence

of the parameters presented in fig. 41 on the extraction potential Uextr and the pump power P. Of

particular interest is the comparison of the optical GS response ηG(fig. 41 (a)) and the optical CEGW

response ηC(fig. 41 (b)) obtained from the numerical fits to the data. Remarkably, these transitions

exhibit an opposite dependence on the extraction potential: While ηGis smallest for low Uextr

and increases continuously for increasing potential, the optical CEGW response ηCis largest for

compensated band bending and decreases for increasing Uextr. This is a strong indication that

GS and CEGW are indeed different states, as the natural polarization of the QD-QW CE is shown

in the literature62 to be perpendicular to the QD GS transition, hence the opposite response to a

polarizing electric field. The observed pump power dependence in fig. 41 (b) could be caused by an

excitation-power dependent linewidth broadening of the GS transition,118 resulting in an increased

spectral overlap of pump and probe pulse that becomes non-negligible compared to CEGW for high

power. The vertical offset thus can be attributed to unintended GS pumping.

The efficiency of carrier extraction is an important figure of merit for detectors and solar cells.

We quantify it for our system through the parameter τ−1

ex , which we determine for every curve

10.2. Carrier dynamics in a DWELL system

(fig. 41 (c)). The efficiency of extraction increases as the extraction potential aids the charge sepa-

ration. Strong band bending thus means strong extraction. Interestingly, the extraction efficiency

we observe shows at high Uextr (low forward bias) also a pump power dependence. For increasing

pump power, the data show a reduction of extraction efficiency. Two processes might contribute to

this behavior: On one hand, two-photon absorption occurs for high optical power that dominates

in the case of low electrical carrier injection (fig. 20), but is present also below threshold. The addi-

tional carriers relaxing from higher states would compensate the extraction partially. On the other

hand, and more important in our opinion, the larger number of created carriers for higher pump

powers will intrinsically compensate the band bending and screen the extraction potential.119

11. Direct observation of Crossed Excitons

in two-dimensional coherent

spectroscopy

The evidence of the existence of CEs collected by PP spectroscopy in chapter 10 seem convincing,

but some doubts may remain. For example, the separation of the exciton and CE transitions in

fig. 38 (f) does rely on the assumption that the exciton transition is not addressed by the pump

pulse. This is a simplification that neglects the homogeneous linewidth of the transitions and

several other aspects. As our knowledge on these aspects is limited so far, this simplification was

necessary to avoid unfounded speculation.

The primary purpose of the development of the STORCH setup (chapter 7) was to get access to

that information. We want to observe the complex interplay of the coupled states and the spectral

distribution of optical transitions under operating conditions of the SOA—if possible by white light

in a single experiment.

11.1. Alignment and measurement series

Although the fiber laser system provides a broad spectrum that covers much more than the nec-

essary spectral range, there are several restrictions that need to be considered. Firstly, coupling

white light into a wave guide will never be easy. Chromatic aberrations of the coupling optics

as well as the dispersion of the waveguide are hard to control or to compensate. Secondly, the

transmittance of the SOA changes upon variation of the injection current, so a universal align-

ment might be impossible. Thirdly, we are not able to measure the chirp of the pulses that leave

the laser system. Our FROSCH system is able to track the relative chirp accumulated on the table,

but the initial chirp remains unknown.

The alignment procedure is based on the maximization of the FWM band bν¯acc. Both pulses, A

and C, are coupled into the SOA at the temporal overlap tbc = 0. The pulse bands bνaand bνc,

respectively, are monitored before and after the SOA in rapid FROSCH measurements (two scans

per second). bν¯accis additionally monitored after the sample. These five parallel live–mode FROSCH

measurements are used for the alignment. The spectral composition of the FWM signal is imme-

diately revealed and the goal of the alignment is usually to create a spectrally broad FWM signal.

The FROSCH also reveals the relative chirp with respect to the original laser pulse (section 7.1) and

allows a compensation by the respective 4fpulse shaper. While the original idea was to minimize

the chirp that is detected before the sample, it turned out that a minimization of the output chirp

provided better results. In general, there is an enormous number of degrees of freedom for the

alignment. We finally focussed on a high and broad FWM level and a minimized output chirp.

11.1.1. Spectral visibility due to sample transmittance

We were not able to find a single alignment that fulfilled all the desires. This resulted in two

independently aligned current series: One that was aligned at J= 0 mA and the other one at

11.1. Alignment and measurement series

Figure 42:FROSCH spectra of S(0|1)

cbefore the sample (blue line) and after the sample (red line)

in relation to the SOA ASE (grey area). The data correspond to the 2D spectrum in fig. 44 at

J= 0 mA. The QD/QW states are highly absorptive, so the pulse is only transmitted in the

band gap spectral range.

J= 200 mA. In both cases, 2D spectra have been measured under variation of Jin the range from

0 mA to 200 mA, with decreasing spectrum quality upon increasing deviation from the alignment

value. Although a quantitative comparison of distant features in the 2D spectra is not possible yet,

the qualitative distribution of peaks appeared to be static and reliable as far as it is discussed in the

following.

In the case of J= 0 mA, the SOA states are highly absorptive and the transmittance spectrum

favors the band gap region. The corresponding FROSCH traces, before (blue) and after (red) the

SOA, are displayed for this series in fig. 42. The resulting 2D spectrum for J= 0 mA and U= 0 V is

presented in fig. 43 (detailed discussion in section 11.2). For its interpretation, it is very important

to have the transmittance spectrum of the SOA in mind: If the light carrier frequency νcis not

transmitted, there is no resulting signal, neither for the pulse band bνcnor for the FWM band

bν¯acc. Therefore the 2D spectrum is restricted on the emission axis to the range that is indicated

by the red curve in fig. 42. There is no corresponding restriction for the absorption axis. Even

though the pulse A might be completely absorbed during the propagation, it might have written

its information to pulse C (in the band bν¯acc) in the very first section of the SOA waveguide. So the

signal is visible as long as the corresponding light carrier mode νcis transmitted.

The series aligned at J= 200 mA focuses on the range of the QD GS and ES that is entirely

amplifying in this current range. For J < 10 mA, i.e. still above the GS transparency, this align-

ment did not provide FWM signals above the noise level. The 2D spectrum that corresponds to the

alignment current is displayed in fig. 44. Signatures of QD GS and ES are clearly visible along the

diagonal. Coupling between GS and ES is indicated by corresponding off–diagonal elements. The

coupling appears aligned to the cross–diagonal. This could be a hint to a volume conserving QD

growth, i.e. a reciprocal relation of lateral an vertical QD size (section 2.3.2). However, the quality

of the off–diagonal feature is not yet fully convincing. This could be clarified by an experiment

that reduces the respective spectra to the off–diagonal area.

11. Direct observation of Crossed Excitons in two-dimensional coherent spectroscopy

Figure 43:2D spectrum of the unbiased SOA (U= 0 V) measured and aligned at J= 0 mA. On

the emission axis, the signal is only visible where the DOS is low, in particular at energies below

the GS ensemble center. The dashed lines represent the expected CE energies of the GS, CEGW

(lower energy) and CEWG (higher energy), respectively.

11.1. Alignment and measurement series

Figure 44:2D spectrum of the SOA measured and aligned at an injection current of J= 200 mA.

GS and ES signatures are clearly visible along the diagonal. Off–diagonal areas above and below

the diagonal indicate coherent coupling between GS and ES.

11. Direct observation of Crossed Excitons in two-dimensional coherent spectroscopy

11.2. Crossed Exciton signatures

From the perspective of the pure QD theory, the most intriguing feature is the wide range of off–

diagonal signal in fig. 43. However, this observation is in excellent agreement with our previous

assumption of CEs. The spectrum for J= 0 mA resembles exactly the scenario of the PP experi-

ments in section 10.2. In the imaginary panel, a small square indicates the energy range that has

been addressed by the off–resonant PP experiment according to table 4. While it was still an open

question in the PP experiment whether there was a real spectral separation of pump and probe

pulse or not, the STORCH measurement provides the proof: There is coherent coupling between

these two energy ranges. And more than this, it provides a much more detailed insight into the

energetic structure because it covers the entire energy range.

At the moment, we are not able to fully explain the exact structure of the coupling mechanisms.

This would probably require calculations of the multi–body problem of an extended number of

carriers bound by the QD and for the SOA a Maxwell–Bloch approach to account for propagation

effects. Today, we restrain our interpretation to a phenomenological level to motivate future ap-

plications of the STORCH technique. The pattern of the off–diagonal area can be explained by the

effective mass ratio approach (section 10.1.3). Dashed lines in fig. 43 indicate the corresponding CE

distribution. The 2D spectrum shows excellent agreement with this prediction. STORCH might

therefore become a valuable tool to measure effective masses in the context of nanostructured

devices.

11.3. Extraction of dephasing times from 2D spectra

In both cases, fig. 43 and fig. 44, the diagonal peaks exhibit a change of sign upon crossing the

diagonal in the imaginary part, while the real part has a maximum. This corresponds to a phase

change of π, which is in excellent agreement with the theoretical prediction for a Lorentzian (sec-

tion 3.2.1). According to eq. (4.3), this allows us to extract the coherence time T2from the phase

slope of the cross–diagonal cut. This is easier than fitting complex Lorentzians to the 2D data,

because numerous features might overlap. The phase change, however, is dominated by the res-

onance at the diagonal, especially for large T2times that correspond to a steep phase slope. Be

aware that this method becomes erroneous for small T2times and large background signals.

To extract the T2times for the GS as well as the ES, respective energy ranges are set. In both

cases we assume a width of 40 meV. The GS area is centered at 0.97 eV, the ES area at 1.04 eV.

To get an average value for these areas, 20 homogeneously distributed cross–diagonal cuts are

analyzed. In a range of 2 meV, the phase slope is determined by linear fitting. The mean value and

the standard deviation of these fit parameters are plotted in fig. 45.

These mean values are in excellent agreement with the results in Ref. 78 on a similar DWELL

structure. This is a first proof of quality of the STORCH data with respect to the phase resolution.

For low carrier injection below the GS transparency, we observe a slight increase of the dephasing

time. Although the variation does not exceed the error range, this might be a hint that screening

effects and a modified formation of CEs play a role in case of the presence of only a few carriers. In

the high injection regime we find similar T2times for GS and ES. This contradicts the assumption

that the GS is more isolated from the continuum states than the ES. The coupling mechanisms

to the continuum seem to be more or less the same. From this perspective, there seems to be no

reason why the carrier transfer between QW and QD should require a cascading process.

11.3. Extraction of dephasing times from 2D spectra

Figure 45:Coherence time T2over voltage. The color indicates the measurement series, solid dots

represent GS data, open dots ES data. The UJ-characteristics of the SOA indicates different

regimes. For low injection current, only the GS is visible. A slight rise of the T2time for

increasing current below the transparency level might be caused by screening effects. In the

high injection regime, T2times for GS and ES do not differ significantly, indicating similar

coupling mechanisms to the environment.

Part IV.

Summary

12. A summarizing outlook

To begin a summary and a conclusion, let us start with a review of the three major sources of

progress. Firstly, there is the discovery of CEs that has extended the view on QDs embedded in a

continuum of states. Secondly, the improvement of the data acquisition and the development of

novel experimental configurations like FROSCH and STORCH has opened new avenues. Thirdly

and last—not least—the increased amount of data changed the way we analyze our data and how

we plan our next steps in the lab. This last point is essential to me and I would like to add a report

on software development in the following section, which can be seen as a conclusion on its own.

It concludes first steps gone so far and necessary next steps to improve the cycle of experiment

planning, data acquisition, data storage, data analysis, and repeated experiment planning.

12.1. The LabControl system: A versatile software concept for

the optics lab

During the course of my work, I spent a large amount of time on the development of a general

concept of control and storage software. The improvements to the measurement concept presented

in part II revealed that the bottleneck of our experimental work is no longer the pure alignment

and measurement duration, but the lacking ability to switch rapidly between experiments and

to cross–evaluate different experimental data sets. Although the work presented in this section

has not been finished—it has been postponed when the completion of the STORCH setup was of

highest priority—I would like to give a sketch of the basic ideas. Most of them have been worked

out in a test environment, but they have not yet been merged to a real system.

12.1.1. A little bit of dreaming

But let us start from the very end, from the picture of an ideal lab software. Think of a lab that

offers a multitude of experimental configurations as soon as it is once aligned. In an ideal lab, you

can easily switch between those configurations. You can use alignment tools whenever you need

a little readjustment. You switch between live–mode and measurement–mode as you please. And

as soon as you notice that an injection current dependence is much more interesting than the time

dependence, you just switch to the variable of choice—it is not much more than a click in your user

interface.

During your work on the optical table, the software supports you. All information stored on

your lab computer is also available on a hand-held device. It is no longer necessary to bow down

or stretch high to have a look at the lab computer monitor 5 m away—acquired data are available

on your tablet PC. And every parameter that can be set via the lab computer is also adjustable via

your tablet PC, of course. Two or more colleagues are working in the lab? No problem! Working

with multiple tablets or other access points is supported as a matter of course. It is just a question

of a few flexible access right limitations for security reasons.

After successfully taking your new data you would love to compare them to your previous data.

As the system has stored all your (and your colleagues) data including all and really all parameters,

12.1. The LabControl system: A versatile software concept for the optics lab

it is just a data base request to get the last years’ data with the matching wavelength range and

injection current setting. And as easy as the raw data, you get all your subsequent data evaluation.

Of course, some things on the table have changed, some mirrors and filters have been added—

but that is all documented in the software system. The computer simply compensates for the

respective transfer functions and this way it is easy to compare your new data to the old one even

quantitatively. So just go ahead and directly apply your prepared analysis routine. Why should

you wait? An immediate evaluation might bring up new ideas, right now, right while you are able

to acquire some complimentary data. And it is really astonishing how people did science before

that was possible, you might think once in a while…

12.1.2. Basic software concepts

But back to reality and to what has been implemented so far. As a first step towards this ideal

lab, it is necessary to set up a central control unit. This central unit is the LabControl system

that controls all the instruments attached to its computer. The centralization is necessary to avoid

conflicts between programs that run in parallel, because those conflicts often cause instrument

communication failures. The LabControl system takes care of all variable parameters. The present

state is always stored in an extended markup language (XML) file that literally represents the

entire parameter space. This parameter space can easily have more than hundred dimensions and

it would improve the long–time data reliability tremendously to always keep track of the entire

space, rather than writing some seemingly important parameters to a notebook.

Besides the LabControl system, there is a completely independent user interface which is called

LabCARS. It was conceptually and graphically inspired by the Library Access and Retrieval Sys-

tem (LCARS), the user interface of the Enterprise D in the famous TV show “Star Trek—The Next

Generation”. LabCARS connects to LabControl via TCP/IP and can therefore be executed on any

computer within the lab network. It is quite easy to set up a local WiFi network, but for the future

an integration into the university WiFi will be preferable. LabCARS is designed for mobile devices

and to be used with touchscreens. Any LabCARS instance has its own copy of the LabControl

XML file that is updated by the central system whenever necessary. The other way round, it is also

possible to modify the central system via LabCARS. Any modification is sent as a command to Lab-

Control which executes the command and delivers the updated state to the connected LabCARS

instances.

In the present state, National Instruments (NI) LabView (LV) has been chosen as a general soft-

ware platform to create both systems. While its capabilities in lab instrument interfacing are un-

paralleled, also mathematics and data processing provides a sufficient background to make use

of its orientation towards a haptic user interface and programming style. LabControl will also in

the future be based on LV, while for LabCARS a solution based on HTML and Javascript that is

executed in a web browser seems preferable. The interlink will, however, still be implemente via

the LabControl XML.

12.1.3. Object–oriented instrument and experiment programming

The desired flexibility is not compatible with a traditional, procedurally programmed experiment

flow. In contrary, it is necessary to execute a certain type of experiment, for instance a PP ex-

periment, with an arbitrary combination of compatible instruments (for example with another

delay stage). To achieve this flexibility, we need to change our programming paradigm to object–

oriented programming: Experiments as well as instruments are represented by objects and can be

combined freely within defined boundaries. The state of an object is defined by its properties and

12. A summarizing outlook

Figure 46:Scheme of a part of a heterodyne experiment split into basic elements. Some of these

basic elements can be controlled by the operator (red elements), some others have fixed prop-

erties (blue elements). Often used collections of elements can be handled as functional groups

(orange boxes). The real world instruments are represented by large collections of basic ele-

ments and functional groups (green boxes). Data are read at the memory nodes (M).

the object acts via its methods. This is quite exactly what we need to define a parameter space of

our entire lab system (where every object defines a subspace of the lab parameter space) and to

operate our lab based on these well–defined parameters.

LV offers a concept for object–oriented programming that is, however, quite limited and un-

handy on first sight: Objects do no represent the state of something, but rather a blueprint of

a state. It can be copied and modified without physical consequences. Within the LabControl

context, I modified the LV object approach by usage of the LabControl XML file and LV data ref-

erences. Object properties are no longer stored as variables within the object, but as entries in

the XML file. Additionally, the data reference approach uses a single data memory for an object.

Thereby it defines a state and not a blueprint anymore, although it is still virtual and not physical.

The LabControl XML file is constructed by a simple paradigm: Every object is represented by a

parent node. All object properties are stored as child nodes and the parameter value is stored as

the node’s text content. Any technical information that is necessary to handle the file and does

not change the parameter state is stored as an attribute. This means that the full parameter space

information is conserved even when all attributes are removed from the XML file.

An example of the LabControl XML file is given in appendix A. Figure 46 shows a typical experi-

mental setting with several instruments. The scheme is constructed of certain blocks that represent

software objects. Some of those object contain child objects, which corresponds to the typical tree

structure of XML files. The instruments are constructed in three layers:

Basic elements are for example electrical filters or modulators, small entities that are defined by

a very limited number of properties and that cannot be split into (meaningful) subentities. In

fig. 46, basic elements are displayed as red and blue boxes, respectively.

12.1. The LabControl system: A versatile software concept for the optics lab

Functional groups are groups of basic elements that can be interpreted as the minimal edition

of a certain functionality. In fig. 46 the functional groups are surrounded by an orange box.

One type of functional groups is the abstract “lock–in”, a combination of a demodulator and a

low–pass filter.

Physical instruments are surrounded by green boxes in fig. 46. This type of object represents a

specific device, its methods typically handle the control software provided by the manufacturer.

Experiment programming is less modular because an experiment is still a procedure. Experiments

are mainly based on functional groups: A sideband PP experiment, for instance, requires two

abstract lock–ins, one abstract detector, two abstract AOMs, and one abstract delay line. For the

experiment, it is not necessary to know which physical instrument is actually used. The experiment

relies on the fact that any modification it performs on the functional group will be translated by the

physical instrument according to its specific realization. In an sideband PPexperiment based on

the scheme in fig. 46, the operator needs to determine which one of the lock–in functional groups

is used for the sideband and which one for the main band. The LabControl system determines

an appropriate memory node (block “M” in the diagram) to acquire the lock–in functional group

output. The respective acquisition routine is, of course, determined in the physical instrument

object class.

12.1.4. Data storage and analysis

For data storage, the LabControl system uses the NI Technical Data Management Streaming

(TDMS) file format. This TDMS handling by LV is optimized to efficiently store continuous data

streams to a hard-drive. The data storage has three levels: the file, channel groups, and channels.

On every level, data can be labeled by user defined parameters. These parameters can later be used

to browse the data storage via the LV “data finder engine”. Typical labels in our experiments would

be the injection current or the pump laser wavelength. With well–labeled data it would be easy to

find all relevant data to compare, for instance in a defined wavelength range.

A key feature of future lab usability will be the accessibility of analysis tools. If the operator

is able to apply complex analysis routines as well as individual mathematical procedures from a

toolbox, the decision making during the experiment will be significantly improved. LabCARS is

supposed to provide this access to analysis tools. Like many other tasks, data analysis can be split

into a chain of sub–tasks. As an illustration, we can review the analysis routine that was used to

create fig. 35 (c) from raw PP data:

1. Calculate the derivative of the complex PP data.

2. Smooth the derivative to eliminate outliers.

3. Set tbc = 0 at the maximum of the smoothed derivative.

4. Determine the average value of tbc <−1 ps and subtract it from the raw data.

5. Perform a linear fit of the differential gain data in the range 1 ps ≤tbc ≤5 ps and determine

∆G(0).

6. Load the curves for all pump energies Ewith the same injection current Jand repeat the

procedure.

7. Plot ∆G(0) over E.

One goal of the LabCARS development is to compose such complex analysis tasks of fundamental

calculations and to make them available at lab runtime.

12. A summarizing outlook

Experiment Signal band b

νab

νbb

νctab tbc tcd

Traditional pump–probe bνcbνabνabνc0tac 0

Sideband pump–probe bν¯abc (bνab −eνab) (bνab +eνab)bνc0tac 0

FROSCH bνcbνctcd

Pump–FROSCH bνcbνabνabνc0tac tcd

STORCH bν¯acc bνabνcbνctac 0tcd

Table 6:The experiments developed in this thesis according to the A–C pulse scheme of 2DCS.

12.1.5. Transfer functions for quantitative measurements

If we look at a present FROSCH measurement, e.g. fig. 34, we are forced to display the amplitude

in units of V. It is the unit that the lock–in provides, but it is not the entity that we are actually

interested in. Interesting would be the field strength of the probe pulse inside the SOA, or at least

as close to the SOA output facet as possible. Between the output facet and the lock–in output,

there is also a chain of transfer functions:

1. Several lenses and mirrors, all with transmission/reflectance spectra T(ν)provided by the

manufacturer.

2. The detector with a spectral sensitivity Tdet(ν), also provided by the manufacturer. If the

local oscillator frequency is known, the relation between the light frequency νand the re-

sulting electrical eνis given.

3. The low and high pass filters with T(bν).

4. Even the coaxial cables connecting the instruments could be traced to determine signal re-

flections at impedance steps.

5. All the internal elements illustrated in fig. 46 also have their specific TFs.

The ultimate goal of the LabControl/LabCARS system is to image the entire setup, including every

mirror and every lens. This would allow to automatically calculate the original quantities, for

example the SOA output field strength from lock–in data. Such a quantitative tracing of the signal

can not only improve the interpretation of data, but would also be helpful in the lab to identify for

example defective components. In this context, it is very useful to think of the setup as a composite

of TFs, which was the original motivation to introduce this term for all linear processes already in

section 1.3.

12.2. Summary

12.2.1. Experiment development

The development of new types of heterodyne experiments was an essential part of this thesis. The

results are described in part II. With FROSCH, sideband PP, and STORCH there are three novel

experimental configurations. The speed of the existing traditional PP has been increased by a

factor of more than 100 and SMPM has been introduced for the detailed investigation of non–

linear behavior. Several minor tools like the MOLCH alignment tool improve the lab handling

significantly. All these experimental configurations are based on the very same hardware and can

be applied easily without realignment. The introduction of the four pulse scheme (4PS) unifies the

12.2. Summary

description of the different types of experiments. An overview of all experimental configurations

is given in table 6. Beyond the unification, the 4PS forms a linguistic link to the 2DCS literature.

The STORCH setup (chapter 7) for 2DCS is at the moment the highest level of development. It

merges all technical approaches developed before. After the preliminary FROSCH (section 5.2) and

fast PP experiments (section 5.3) it took more than two years and several new instruments to put

the setup in execution. The STORCH software analyzes a data stream of up to 16 real channels

at a data acquisition rate of eνdaq = 20 kHz. This allows to track temporal drifts and enables us

to perform the coherent experiment over a time range of several minutes without active phase

stabilization.

The technical basis of this experimental progress is the consequent exploitation of the entire fre-

quency range displayed in fig. 1. The attosecond time range has been entered by the introduction

of interferometric position control for sub–wavelength spatial resolution in the FROSCH configu-

ration. The pico- and nanosecond time range is reserved for the physics under investigation. This

range is limited by the laser repetition rate. The time range from 10 ps to 10 µs is used to perform

the optical heterodyning by integration in a lock–in amplifier. Subsequently, the fDAQ range from

10 µs to 100 ms completes the time range only accessible by machinery. However, also the human

time scales have been considered: The lab handling efficiency in the range of 100 ms to 1 day has

been significantly increased by several live–mode applications. Finally, a development basis has

been created in form of the LabControl system to store and categorize data on the time scale of

months and years, which is a requirement for big data analysis.

12.2.2. Investigation of coupled nanostructures of mixed dimensionality

The new experimental capabilities allowed us to investigate the electron dynamics in the SOA on

an extended scope. In a first experiment series, the 2LS nature of the QDs was demonstrated by

FROSCH measurements. A characteristic pulse shape modification caused by Rabi flops was ob-

served even at room temperature. The dependencies on injection current and pulse power were

fully consistent with the expectations for this type of pulse shape modification. A theoretical de-

scription was given on a low level using a mechanical analog, and on a higher level by detailed

calculations based on Maxwell–Bloch equations (calculations performed by Julian Korn and Ben-

jamin Lingnau).

Additional optical transitions attributed to CEs were first observed in gain excitation spec-

troscopy. This meta–analysis of a large number of PP experiments also allowed a precise deter-

mination of the QW band edge. Two distinct peaks in the gain excitation over energy appeared

above the QW band edge. These peaks can be attributed to CEs formed by an electron in the QD

GS or ES, respectively, and a hole at the GaAs bulk band edge.

In a second large set of PP experiments, the diffusion dynamics in the QW as well as the exci-

tation of the lowest energy CE have been investigated. The experiments were performed without

injection current, but under variation of the diode bias. This variation had significant influence on

the carrier extraction and accordingly on the decay rates of the PP traces. A total set of 178 PP

traces was described by a linear rate equation model that incorporated several excitation channels

and a detailed diffusion model. Fitting the model to the experimental data revealed that a large

bound state of CEs is necessary to describe the diffusion behavior. The excitation efficiency of

the QD GS and CEGW showed opposite bias dependence, which is an indicator for the qualitative

different nature of these states.

Finally, the new STORCH setup was used to investigate the entire QD/QW energy range simul-

taneously by white pulses. For two measurement series—one optimized for low, the other for high

injection current—the coherent coupling between the states and the coherence lifetime of GS and

12. A summarizing outlook

ES were investigated. The extracted coherence lifetimes are in excellent agreement with literature

values for similar samples, which proofs the quality of the newly developed method. Moreover,

the current dependence of T2has been observed. There was no significant difference between GS

and ES which hints to a similar coupling to the surrounding continuum. In the 2D spectra, we

found distinct off–diagonal contributions that are in very good agreement with the assumption

of the CEGW made before. These very first findings promise a much more detailed insight into

the nature of energetically widespread systems of coupled nanostructures by application of the

STORCH technique.

A. LabControl XML file

The LabControl XML file defines the interplay of LabView objects within the LabControl system.

A part of the LabControl class hierarchy is shown in fig. 47. Comments (red text) in the subsequent

XML code explain the structure of the XML tree. Both, fig. 47 and the code, represent snapshots

of the development process and do not represent a running system.

Figure 47:A part of the LabView class tree. Parent classes are shown on top, child classes that

inherit properties and methods from their parent are shown below.

1<?xml version="1.0" encoding="UTF-8" standalone="no" ?>

2<LC version="1.0.0.0">

3<!-- Begin of the library section. Holds information about publicly available classes

,→and their hierarchy -->

4<CLASS id="71C00000" label="LC XML element" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\LabControl XML element\

,→LabControl XML element.lvlibp" name="LabControl XML element">

5<CLASS id="71D00001" label="Parameter Set" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Parameter Set\Parameter

,→Set.lvlibp" name="Parameter Set">

6<CLASS id="71E00002" label="Experiment" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Experiment\Experiment.

,→lvlibp" name="Experiment">

7</CLASS>

A. LabControl XML file

8<CLASS id="71F00003" label="Functionality" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Functionality\

,→Functionality.lvlibp" name="Functionality">

9</CLASS>

10 <CLASS id="72000004" label="Mode" lib="C:\Users\Mirco\Documents\LabVIEW Data\

,→LabControl Development\Packed libraries\Basic\Parameter Set\Parameter Set.

,→lvlibp" name="Mode">

11 <CLASS id="72100005" label="FG off mode" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Functional group\

,→Functional group.lvlibp" name="FG off mode">

12 </CLASS>

13 <CLASS id="72200006" label="FG on mode" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Functional group\

→Functional group.lvlibp" name="FG on mode">

14 </CLASS>

15 </CLASS>

16 <CLASS id="72300007" label="Parameter Set with Default.lvclass" lib="C:\Users\

,→Mirco\Documents\LabVIEW Data\LabControl Development\Packed libraries\Basic

,→\Parameter Set with Default\Parameter Set with Default.lvlibp" name="

→Parameter Set with Default">

17 <CLASS id="72400008" label="Graph" lib="C:\Users\Mirco\Documents\LabVIEW Data\

,→LabControl Development\Packed libraries\Basic\Graph\Graph.lvlibp" name="

,→Graph">

18 </CLASS>

19 <CLASS id="72500009" label="Graph axis" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Graph axis\Graph axis.

,→lvlibp" name="Graph axis">

20 </CLASS>

21 </CLASS>

22 <CLASS id="7260000A" label="Sample.lvclass" lib="C:\Users\Mirco\Documents\LabVIEW

,→ Data\LabControl Development\Packed libraries\Basic\Sample\Sample.lvlibp"

,→name="Sample">

23 </CLASS>

24 <CLASS id="7270000B" label="Spatial object" lib="C:\Users\Mirco\Documents\LabVIEW

,→ Data\LabControl Development\Packed libraries\Basic\Spatial object\Spatial

,→ object.lvlibp" name="Spatial object">

25 <CLASS id="7280000C" label="Func. group" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Functional group\

,→Functional group.lvlibp" name="Functional group">

26 <CLASS id="7290000D" label="Frequency Generator" lib="C:\Users\Mirco\

,→Documents\LabVIEW Data\LabControl Development\Packed libraries\

,→Optional\Frequency generator\Frequency generator.lvlibp" name="

,→Frequency generator">

27 </CLASS>

28 <CLASS id="72A0000E" label="Lock-In" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Optional\Lock in\Lock in.

,→lvlibp" name="Lock in" refby="00000000">

29 </CLASS>

30 </CLASS>

31 <CLASS id="72B0000F" label="Instrument" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Instrument\Instrument.

,→lvlibp" name="Instrument">

32 <CLASS id="72C00010" label="National Instruments" lib="C:\Users\Mirco\

,→Documents\LabVIEW Data\LabControl Development\Packed libraries\

100

,→Optional\National Instruments\National Instruments.lvlibp" name="

,→National Instruments">

33 <CLASS id="72D00011" label="NI DAQmx device" lib="C:\Users\Mirco\Documents\

,→LabVIEW Data\LabControl Development\Packed libraries\Optional\NI

,→DAQmx device\NI DAQmx device.lvlibp" name="NI DAQmx device">

34 </CLASS>

35 </CLASS>

36 <CLASS id="72E00012" label="Zurich Instruments" lib="C:\Users\Mirco\Documents\

,→LabVIEW Data\LabControl Development\Packed libraries\Optional\Zurich

,→Instruments\Zurich Instruments.lvlibp" name="Zurich Instruments">

37 <CLASS id="72F00013" label="ZI HF2LI" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Optional\ZI HF2LI\ZI

,→HF2LI.lvlibp" name="ZI HF2LI">

38 </CLASS>

39 </CLASS>

40 </CLASS>

41 </CLASS>

42 </CLASS>

43 <CLASS id="73000014" label="Depot.lvclass" lib="C:\Users\Mirco\Documents\LabVIEW

→Data\LabControl Development\Packed libraries\Basic\Parameter Set\Parameter

,→Set.lvlibp" name="Depot">

44 </CLASS>

45 <CLASS id="73100015" label="Connector" lib="C:\Users\Mirco\Documents\LabVIEW Data\

,→LabControl Development\Packed libraries\Basic\Spatial object\Spatial object.

,→lvlibp" name="Connector">

46 <CLASS id="73200016" label="Input connector" lib="C:\Users\Mirco\Documents\

,→LabVIEW Data\LabControl Development\Packed libraries\Basic\Spatial object\

,→Spatial object.lvlibp" name="Input">

47 </CLASS>

48 <CLASS id="73300017" label="Output connector" lib="C:\Users\Mirco\Documents\

,→LabVIEW Data\LabControl Development\Packed libraries\Basic\Spatial object\

,→Spatial object.lvlibp" name="Output">

49 </CLASS>

50 </CLASS>

51 <CLASS id="73400018" label="Memory.lvclass" lib="C:\Users\Mirco\Documents\LabVIEW

,→Data\LabControl Development\Packed libraries\Basic\Spatial object\Spatial

,→object.lvlibp" name="Memory">

52 </CLASS>

53 <CLASS id="73500019" label="Space" lib="C:\Users\Mirco\Documents\LabVIEW Data\

,→LabControl Development\Packed libraries\Basic\Spatial object\Spatial object.

,→lvlibp" name="Space">

54 </CLASS>

55 </CLASS>

56

57 <DEP depot="Functionality" id="7480002C" name="Functionalities">

58 <!-- The depot node DEP can be filled by the user with classes inheriting from the

,→one defined in the "depot" attribute. The "id" value of any node is the

,→reference number of the corresponding LabView data reference. -->

59 <SET class="Functionality" created_by_user="1" id="73B0001F" name="0">SBLI

60 <!-- SET nodes are "Parameter Sets" that define a certain entity in the setup.

,→This one represents an object of class "Functionality". This class is used

,→to attribute a certain functionality to a "Functional group". The

,→functionality tag "SBLI" will be prepended to parameter names when they

101

A. LabControl XML file

,→are saved to the TDMS file. This says, e.g., that the specific "functional

,→ group" played the role of a sideband lock-in -->

61 <PAR datatype="80" id="73A0001E" label="Description" name="description">

62

63 <EL id="7390001D" name="0">0

64 <!-- To allow for array data types, the EL node represents an element of the

,→array and the text content "0" determines the index of the element in

,→the array. -->

65 <VAL id="7380001C" name="value">Sideband lock-in

66 <!-- Finally, the VAL node holds the value of the parameter element. In this

,→case, it is the description of the "Functionality" SET. -->

67 </VAL>

68 </EL>

69 </PAR>

70 <REF id="7360001A" name="class">

71 <!-- REF nodes are used to establish the cross-links within the XML file. While

,→the PAR node defines a data value, the REF node represents a reference.

,→-->

72 <VAL id="7370001B" name="0" options="/LC/CLASS//CLASS[@name='Functional group

→']/CLASS[name(.)='CLASS']" refID="72A0000E">/LC/CLASS[@name='LabControl

,→XML element']/CLASS[@name='Parameter Set']/CLASS[@name='Spatial object'

,→]/CLASS[@name='Functional group']/CLASS[@name='Lock in']

73 <!-- Also in this case, the VAL node holds the value of the reference, i.e.

,→the XPath string (an XML query language) to the referenced node/object.

,→Possible options are also defined as XPath in the "options" attribute.

,→The "id" attribute of the referenced node will change upon a restart

,→of LabControl, therefor the cross-link chosen by the user is saved as

,→XPath. However, LabControl will directly after a restart translate

,→this XPath to the corresponding "id" attribute and store it in the "

,→refID" attribute. So it is not necessary to execute the XPath query

,→every time the reference number is needed. -->

74 </VAL>

75 </REF>

76 </SET>

77 <SET class="Functionality" created_by_user="1" id="74100025" name="1">FUNCT1

78

79 </SET>

80 <SET class="Functionality" created_by_user="1" id="7470002B" name="2">FUNCT2

81

82 </SET>

83 </DEP>

84 <DEP depot="Instrument" id="82400108" name="Instruments">

85

86 <SET class="ZI HF2LI" created_by_user="1" id="81C00100" name="1">ZI HF2LI

87

88 <SPACE id="7490002D" name="spc">

89 <!-- Unlike the "Functionality" class, instruments define a space. Input and

,→output connectors are always located on the virtual surface of such sub-

,→spaces. In the far future, it might be useful to make "Instruments"

,→child nodes of "Lab" nodes. -->

90 </SPACE>

91 <IN id="818000FC" male="0" name="input1" refID="" space="0" type="BNC">

92 <!-- The IN node defines an input connector. This one is a male BNC connector

,→in space "0" (the outside world) and represents the "Input 1" of the

102

,→HF2LI. The IN node is very much like the VAL child node of a REF node (

,→see above): There is also a "refID" attribute that represents the output

,→node that acts as the signal source. -->

93 <OUT id="819000FD" male="0" name="out" space="7490002D" type="ZI HF2LI input">

94 <!-- The output corresponding to its parent input node. It is located inside

,→the HF2LI space ("space" attribute equals "id" attribute of the SPACE

,→node) and has a private class type that can only be used inside the

,→HF2LI. When connected, an input needs a unique source, while an output

,→can serve as a source for multiple other inputs. Therefore the IN

,→node has a "refNum" attribute while the OUT node does not. -->

95 </OUT>

96 </IN>

97 <IN id="81A000FE" male="0" name="input2" refID="" space="0" type="BNC">

98 <OUT id="81B000FF" male="0" name="out" space="7490002D" type="ZI HF2LI input">

99 </OUT>

100 </IN>

101 <!-- Memory nodes (MEM) for data readout are not implemented here, but they work

,→just like IN nodes and thereby probe a certain OUT node -->

102 <SET class="Frequency unit" id="77400058" name="frequnit">Frequency unit

103 <!-- Because the HF2LI is a rather complex instrument, it brings some private

,→classes (not found in the library section) to improve the structure.

,→This one comprises the six frequency sources of the HF2LI. -->

104 <SET class="Frequency generator" id="75000034" name="freqgen1">Frequency

,→generator

105 <SPACE id="74A0002E" name="spc">

106 </SPACE>

107 <MODES depot="" id="74B0002F" name="Modes" options="SET">

108 <!-- A frequency source might be switched "on" or "off" (note that "FG" is

,→not for "Frequency Generator", but for "Functional Group"). Other

,→modes might be defined if necessary. -->

109 <SET class="FG on mode" id="74C00030" name="on" short="?">FG on mode

110 </SET>

111 <SET class="FG off mode" id="74D00031" name="off" short="?">FG off mode

112 </SET>

113 </MODES>

114 <REF id="74E00032" name="functionality">

115 <!-- This is the attribution of the functionality. In fact, this solution of

,→attributing the functionality to a functional group directly is

,→deprecated and will be replaced. The attribution will be done in the

,→experiment class and the functionality will also store blueprint

,→settings that need to be set before the functional group can play a

,→certain role in an experiment. -->

116 <VAL id="74F00033" name="0" options="/LC/DEP[@name='Functionalities']/SET[

,→REF[@name='class']/VAL/@refID=/LC/CLASS//CLASS[@name='Frequency

,→generator']/@id][name(.)='SET'][@class=/LC/CLASS/descendant-or-

,→self::CLASS[@name='Functionality']/descendant-or-self::CLASS/@name]"

,→refID="">

117 </VAL>

118 </REF>

119 </SET>

120 <SET class="Frequency generator" id="7570003B" name="freqgen2">Frequency

,→generator

121

122 </SET>

103

A. LabControl XML file

123 <SET class="Frequency generator" id="75E00042" name="freqgen3">Frequency

,→generator

124

125 </SET>

126 <SET class="Frequency generator" id="76500049" name="freqgen4">Frequency

,→generator

127

128 </SET>

129 <SET class="Frequency generator" id="76C00050" name="freqgen5">Frequency

,→generator

130

131 </SET>

132 <SET class="Frequency generator" id="77300057" name="freqgen6">Frequency

→generator

133

134 </SET>

135 </SET>

136 <SET class="Demodulation unit" id="817000FB" name="demodunit">Demodulation unit

137 <SET class="Lock in" id="78F00073" name="demod0">Lock in

138 <SPACE id="77500059" name="spc">

139 </SPACE>

140 <PAR datatype="43" id="7780005C" label="Time constant" name="timeconstant"

,→unit="000000000001000000000000000000000000">

141 <EL id="7770005B" name="0">0

142 <VAL id="7760005A" name="value">3A83126F

143 </VAL>

144 </EL>

145 </PAR>

146 <PAR datatype="22" id="77B0005F" label="Slope" name="slope">

147 <OPTION id="77C00060" name="0000">

148 </OPTION>

149 <OPTION id="77D00061" name="0001">

150 </OPTION>

151 <OPTION id="77E00062" name="0002">

152 </OPTION>

153 <OPTION id="77F00063" name="0003">

154 </OPTION>

155 <OPTION id="78000064" name="0004">

156 </OPTION>

157 <OPTION id="78100065" name="0005">

158 </OPTION>

159 <OPTION id="78200066" name="0006">

160 </OPTION>

161 <OPTION id="78300067" name="0007">

162 </OPTION>

163 <EL id="77A0005E" name="0">0

164 <VAL id="7790005D" name="value" options="/LC/DEP[@name='Instruments']/SET[

,→@name='1']/SET[@name='demodunit']/SET[@name='demod0']/PAR[@name='

,→slope']/OPTION">0000

165 </VAL>

166 </EL>

167 </PAR>

168 <MODES depot="" id="78400068" name="Modes" options="SET">

169 <SET class="FG on mode" id="78500069" name="on" short="?">FG on mode

104

170 </SET>

171 <SET class="FG off mode" id="7860006A" name="off" short="?">FG off mode

172 </SET>

173 </MODES>

174 <REF id="7870006B" name="functionality">

175 <VAL id="7880006C" name="0" options="/LC/DEP[@name='Functionalities']/SET[

,→REF[@name='class']/VAL/@refID=/LC/CLASS//CLASS[@name='Lock in']/@id][

,→name(.)='SET'][@class=/LC/CLASS/descendant-or-self::CLASS[@name='

,→Functionality']/descendant-or-self::CLASS/@name]" refID="">

176 </VAL>

177 </REF>

178 <IN id="7890006D" name="sigout" refID="" space="77500059">

179 <OUT id="78A0006E" name="out" space="7490002D">

180 </OUT>

181 </IN>

182 <IN id="78B0006F" male="1" name="sigin" refID="" space="7490002D" type="ZI

,→HF2LI input">

183 <OUT id="78C00070" name="out" space="77500059">

184 </OUT>

185 </IN>

186 <IN id="78D00071" name="freqin" refID="" space="7490002D">

187 <OUT id="78E00072" name="out" space="77500059">

188 </OUT>

189 </IN>

190 </SET>

191 <SET class="Lock in" id="7AA0008E" name="demod1">Lock in

192

193 </SET>

194 <SET class="Lock in" id="7C5000A9" name="demod2">Lock in

195

196 </SET>

197 <SET class="Lock in" id="7E0000C4" name="demod3">Lock in

198

199 </SET>

200 <SET class="Lock in" id="7FB000DF" name="demod4">Lock in

201

202 </SET>

203 <SET class="Lock in" id="816000FA" name="demod5">Lock in

204

205 <SPACE id="7FC000E0" name="spc">

206 </SPACE>

207 <PAR datatype="43" id="7FF000E3" label="Time constant" name="timeconstant"

,→unit="000000000001000000000000000000000000">

208 <EL id="7FE000E2" name="0">0

209 <VAL id="7FD000E1" name="value">3A83126F

210 </VAL>

211 </EL>

212 </PAR>

213 <PAR datatype="22" id="802000E6" label="Slope" name="slope">

214 <OPTION id="803000E7" name="0000">

215 </OPTION>

216 <OPTION id="804000E8" name="0001">

217 </OPTION>

218 <OPTION id="805000E9" name="0002">

105

A. LabControl XML file

219 </OPTION>

220 <OPTION id="806000EA" name="0003">

221 </OPTION>

222 <OPTION id="807000EB" name="0004">

223 </OPTION>

224 <OPTION id="808000EC" name="0005">

225 </OPTION>

226 <OPTION id="809000ED" name="0006">

227 </OPTION>

228 <OPTION id="80A000EE" name="0007">

229 </OPTION>

230 <EL id="801000E5" name="0">0

231 <VAL id="800000E4" name="value" options="/LC/DEP[@name='Instruments']/SET[

→@name='1']/SET[@name='demodunit']/SET[@name='demod5']/PAR[@name='slope

,→']/OPTION">0000

232 </VAL>

233 </EL>

234 </PAR>

235 <MODES depot="" id="80B000EF" name="Modes" options="SET">

236 <SET class="FG on mode" id="80C000F0" name="on" short="?">FG on mode

237 </SET>

238 <SET class="FG off mode" id="80D000F1" name="off" short="?">FG off mode

239 </SET>

240 </MODES>

241 <REF id="80E000F2" name="functionality">

242 <VAL id="80F000F3" name="0" options="/LC/DEP[@name='Functionalities']/SET[REF

,→[@name='class']/VAL/@refID=/LC/CLASS//CLASS[@name='Lock in']/@id][name

,→(.)='SET'][@class=/LC/CLASS/descendant-or-self::CLASS[@name='

,→Functionality']/descendant-or-self::CLASS/@name]" refID="">

243 </VAL>

244 </REF>

245 <IN id="810000F4" name="sigout" refID="" space="7FC000E0">

246 <OUT id="811000F5" name="out" space="7490002D">

247 </OUT>

248 </IN>

249 <IN id="812000F6" male="1" name="sigin" refID="" space="7490002D" type="ZI

,→HF2LI input">

250 <OUT id="813000F7" name="out" space="7FC000E0">

251 </OUT>

252 </IN>

253 <IN id="814000F8" name="freqin" refID="" space="7490002D">

254 <OUT id="815000F9" name="out" space="7FC000E0">

255 </OUT>

256 </IN>

257 </SET>

258 </SET>

259 </SET>

260 <SET class="NI DAQmx device" created_by_user="1" id="82300107" name="2">NI DAQmx

,→device

261 <PAR datatype="80" id="82000104" label="Device" name="device">

262 <EL id="81F00103" name="0">0

263 <VAL id="81E00102" name="value" options="../../OPTION">

264 </VAL>

265 </EL>

106

266 <OPTION id="82100105" name="Dev1">

267 </OPTION>

268 <OPTION id="82200106" name="Dev2">

269 </OPTION>

270 </PAR>

271 <SPACE id="81D00101" name="spc">

272 </SPACE>

273 </SET>

274 </DEP>

275 <DEP depot="Experiment" id="82500109" name="Experiments">

276 </DEP>

277 <LIST name="VISA" ps="FALSE"/>

278 <LIST name="DAQ analog inputs" ps="FALSE"/>

279 <LIST name="DAQ analog outputs" ps="FALSE"/>

280 <LIST name="Measurement series" ps="FALSE"/>

281 </LC>

107

Publications & Literature

Publications related to this work (chronological)

1. Kolarczik, M., Owschimikow, N., Korn, J., Lingnau, B., Kaptan, Y. I., Bimberg, D., Schöll, E.,

Lüdge, K. & Woggon, U. Quantum coherence induces pulse shape modification in a semi-

conductor optical amplifier at room temperature. Nature Communications 4, 2953 (2013).

2. Owschimikow, N., Kolarczik, M., Kaptan, Y. I., Grosse, N. B. & Woggon, U. Crossed excitons in

a semiconductor nanostructure of mixed dimensionality. Applied Physics Letters 105, 101108

(2014).

3. Kolarczik, M., Owschimikow, N., Herzog, B., Buchholz, F., Kaptan, Y. I. & Woggon, U. Ex-

citon dynamics probe the energy structure of a quantum dot-in-a-well system: The role of

Coulomb attraction and dimensionality. Physical Review B 91, 235310 (2015).

4. Kolarczik, M., Ulbrich, C., Geiregat, P., Zhu, Y., Sagar, L. K., Singh, A., Herzog, B., Achtstein,

A. W., Li, X., van Thourhout, D., Hens, Z., Owschimikow, N. & Woggon, U. Sideband pump-

probe technique resolves nonlinear modulation response of PbS/CdS quantum dots on a

silicon nitride waveguide. APL Photonics 3, 016101 (2018).

5. Kolarczik, M., Thommes, K., Herzog, B., Helmrich, S., Owschimikow, N. & Woggon, U. Ultra-

fast photonics in coherently coupled III-V semiconductor nanostructures in Ultrafast Bandgap

Photonics III (ed Rafailov, M. K.) (SPIE, 2018), 17. isbn: 9781510617872. (2018).

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devices for dual-state operation. Applied Physics Letters 104, 261108 (2014).

8. Kaptan, Y. I., Röhm, A., Herzog, B., Lingnau, B., Schmeckebier, H., Arsenijević, D., Mikhe-

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N. & Lüdge, K. Stability of quantum-dot excited-state laser emission under simultaneous

ground-state perturbation. Applied Physics Letters 105, 191105 (2014).

9. Grosse, N. B., Owschimikow, N., Aust, R., Lingnau, B., Koltchanov, A., Kolarczik, M., Lüdge, K.

& Woggon, U. Pump-probe quantum state tomography in a semiconductor optical amplifier.

Optics Express 22, 32520–32525 (2014).

10. Herzog, B., Owschimikow, N., Schulze, J.-H., Rosales, R., Kaptan, Y. I., Kolarczik, M., Switaiski,

T., Strittmatter, A., Bimberg, D., Pohl, U. W. & Woggon, U. Fast gain and phase recovery

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N. Ultrafast gain recovery and large nonlinear optical response in submonolayer quantum

dots. Physical Review B 94, 014305 (2016).

12. Herzog, B., Lingnau, B., Kolarczik, M., Kaptan, Y. I., Bimberg, D., Maaßdorf, A., Pohl,

U. W., Rosales, R., Schulze, J.-H., Strittmatter, A., Weyers, M., Woggon, U., Lüdge, K. &

Owschimikow, N. Strong amplitude-phase coupling in submonolayer quantum dots. Applied

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13. Singh, A., Tran, K., Kolarczik, M., Seifert, J., Wang, Y., Hao, K., Pleskot, D., Gabor, N. M.,

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dots. Applied Physics Letters 110, 241102 (2017).

15. Böhm, F., Grosse, N. B., Kolarczik, M., Herzog, B., Achtstein, A., Owschimikow, N. & Wog-

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conductor opto-electronic devices in Quantum Nanophotonics (eds Dionne, J. A. & Lawrence,

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Kolarczik, M., Schöps, O., Bratschitsch, R., Woggon, U. & Owschimikow, N. Exciton–phonon

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115

Symbols

BMagnetic field 19

c0Vacuum speed of light 6

χElectric susceptibility 19

DElectric displacement field 19

∆GDifferential gain 9

∆Φ Differential phase 47

EElectric field 6

EGeneral energy 2

E2D 2D spectrum 59

ηExtraction efficiency 73

FFourier transformation 2

ΓTransition rate 12

γHomogeneous linewidth (typically with 2γFWHM of a Lorentzian) 21

HMagnetizing field 19

hThe Planck constant 2

~Reduced Planck constant h/(2π)2

JSOA injection current. 62

kBThe Boltzmann constant 13

MMagnetization 19

mMass 6

m0Electron rest mass 11

m∗Effective mass 11

m∗

eEffective electron mass 70

m∗

hEffective hole mass 70

m∗

xEffective exciton mass 77

nRefractive index 6

νLight frequency 2

bνGeneral mode shift frequency 7

bνrep Laser repetition rate 5

eνGeneral modulation frequency

eνdaq Data acquisition rate 37

ΩRabi frequency 23

ωAngular frequency ω= 2πν. 2

116

Symbols

PPolarization 19

PGeneral power 9

Pmax Maximum power in an SMPM. 39

rEffective mass ratio m∗

h/m∗

e70

ρBloch vector

vAbsorptive component of dipole moment 23

wInversion 23

σInhomogeneous linewidth (typically with σFWHM of a Gaussian) 2

Tthe temperature 77

T1Population lifetime 23

T2Coherence lifetime 23

tij General delay of pulse iand pulse j

tab Pulse delay in FWM experiments 32

t(0)

ab Uncorrected pulse delay in FWM experiments 58

tac Delay between the two pulses passing the sample. Can either be equivalent to tab or tbc. 30

tbc Pump–probe delay in pump–probe experiments 30

tcd Local oscillator delay 30

t(0)

cd Uncorrected local oscillator delay 58

τGeneral decay time

τTC Lock-in time constant 9

USOA bias. 62

Uth Injection threshold bias of the SOA. 62

117

Abbreviations

Notation Description Page

List

0D zero–dimensional 12

2D two–dimensional 12

2DCS two-dimensional coherent spectroscopy viii, 8

2LS two level system viii, 23

2cPP two-color pump-probe 68

3D three–dimensional 12

4PS four pulse scheme ix, 8, 20

ADC analog–to–digital converter 37

AOM acousto-optic modulator 7

ASE amplified spontaneous emission 15

CB conduction band 11

CCD charge–coupled device 29

CE crossed exciton 14

DAQ data acquisition vii, 37

DC direct current 52

DEQ differential equation 16

DFG Deutsche Forschungsgemeinschaft vi

DOS density of states 11

DWELL dot–in–a–well 16

ES excited state 12

fDAQ fast data acquisition 38

FROSCH Frequency Resolved Optical Short–pulse Charac-

terization by Heterodyning

ii, vii, 31

FT Fourier transform 2

FTS Fourier transform spectroscopy viii, 30

FWHM full width at half maximum 2

FWM four-wave mixing 19

GaAs gallium arsenide 71

GPIB General Purpose Interface Bus vii

118

Abbreviations

Notation Description Page

List

GRK 1558 Graduiertenkolleg 1558 vi

GS ground state 12

GT Gabor transform 4

InAs indium arsenide viii

InGaAs indium gallium arsenide viii

IR infrared 11

LED light emitting diode 15

LO local oscillator 28

LV LabView 93

MBE molecular beam epitaxy 13

MDCS multi-dimensional coherent spectroscopy 32

MOLCH More–Or–Less Characterization by Heterodyn-

ing

vii, 38

MOVPE metalorganic vapor–phase epitaxy 13

NI National Instruments 93

NIR near infrared 14

NMR nuclear magnetic resonance 32

OPO optical parametric oscillator 21

PL photoluminescence 17

PP pump-probe vii, 14

QD quantum dot viii, 6

QW quantum well viii, 12

RWA rotating wave approximation 23

SFB 787 Sonderforschungsbereich 787 vi

SHG second harmonic generation 35

SK Stranski-Krastanow 13

SMPM Simultaneous multi-power measurement vii, 39

SNR signal–to–noise ratio 37

SOA semiconductor optical amplifier viii, 15

STORCH Supercontinuum–based Two–dimensional Ob-

servation of Radiation Coherence by Heterodyn-

ing

viii

119

Abbreviations

Notation Description Page

List

TC time constant 38

TDMS Technical Data Management Streaming 95

TF transfer function 8

TMD transitionmetal-dichalcogenide vi, vii, 14

TU Technische Universität Berlin vi

UT University of Texas vi

UV ultraviolet 11

VB valence band 11

VIS visible 20

WL wetting layer 13

XML extended markup language 93

ZI Zurich Instruments 10

120

List of Figures

1. Logarithmic time scale from attoseconds to decades . . . . . . . . . . . . . . . . . viii

2. Three fundamental cases of Fourier transoforms . . . . . . . . . . . . . . . . . . . 3

3. Illustration of amplitude and phase modulation . . . . . . . . . . . . . . . . . . . . 5

4. The general four–pulse–scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5. Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6. Illustration of Crossed Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7. FWM in the box geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8. Bloch sphere and Bloch vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9. Mechanical analog of Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 24

10. AOM dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11. The pump–probe 4PS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12. The FWM 4PS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13. Excitation scheme for 2D spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 33

14. Lorentzian and Gaussian 2D spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 35

15. Concept of SFG FROG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16. Lab noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

17. The FROSCH 4PS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

18. Minimal FROSCH setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

19. Heterodyne pump–probe setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

20. Nonlinear processes identified by SMPM . . . . . . . . . . . . . . . . . . . . . . . 43

21. Analog signal pre–conditioning by a low–pass filter . . . . . . . . . . . . . . . . . 45

22. The sideband pump–probe setup in Austin . . . . . . . . . . . . . . . . . . . . . . 46

23. Sideband pump–probe on TMDs (proof of principle data) . . . . . . . . . . . . . . 47

24. The sideband pump–probe setup in Berlin . . . . . . . . . . . . . . . . . . . . . . 48

25. AOM characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

26. PbS sideband pump–probe: variation of probe photon energy . . . . . . . . . . . 50

27. PbS sideband pump–probe: variation of pump photon energy . . . . . . . . . . . 50

28. The STORCH setup for 2D coherent spectroscopy . . . . . . . . . . . . . . . . . . 54

29. Filter transfer functions for 2D coherent spectroscopy frequency selection . . . . 56

30. Temporal raw data of a 2D measurement . . . . . . . . . . . . . . . . . . . . . . . 59

31. The amplified spontaneous emission of the SOA . . . . . . . . . . . . . . . . . . . 63

32. Pulse shape modifications caused by Rabi oscillations . . . . . . . . . . . . . . . . 65

33. Power dependence of the pulse shape modification induced by Rabi oscillations . 66

121

List of Figures

34. FROSCH SMPM measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

35. Results of gain excitation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 69

36. Concept of the diffusion experiments . . . . . . . . . . . . . . . . . . . . . . . . . 72

37. Excitation saturation in pump–probe experiment observed by SMPM . . . . . . . 73

38. Carrier dynamics observed by pump–probe spectroscopy . . . . . . . . . . . . . . 74

39. Identification of diffusive carrier transport . . . . . . . . . . . . . . . . . . . . . . 74

40. Scheme of the diffusion rate equation model . . . . . . . . . . . . . . . . . . . . . 75

41. Diffusion model parameters with potential and power dependence . . . . . . . . . 81

42. FROSCH spectra of the transmitted pulses in a 2D experiment . . . . . . . . . . . 85

43. 2D spectrum of the unbiased SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

44. 2D spectrum of the SOA at high injection current . . . . . . . . . . . . . . . . . . 87

45. Coherence time T2over voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

46. Scheme of a heterodyne experiment split into basic elements . . . . . . . . . . . . 94

47. The LabView class tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

122