Resonant Light Scattering from Semiconductor Quantum Dots

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University of South Florida Scholar Commons Graduate eses and Dissertations Graduate School 11-18-2016 Resonant Light Scaering from Semiconductor Quantum Dots Kumarasiri Konthasinghe University of South Florida, [email protected] Follow this and additional works at: hp://scholarcommons.usf.edu/etd Part of the Optics Commons , and the Quantum Physics Commons is Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate eses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Scholar Commons Citation Konthasinghe, Kumarasiri, "Resonant Light Scaering from Semiconductor Quantum Dots" (2016). Graduate eses and Dissertations. hp://scholarcommons.usf.edu/etd/6527

Transcript of Resonant Light Scattering from Semiconductor Quantum Dots

Page 1: Resonant Light Scattering from Semiconductor Quantum Dots

University of South FloridaScholar Commons

Graduate Theses and Dissertations Graduate School

11-18-2016

Resonant Light Scattering from SemiconductorQuantum DotsKumarasiri KonthasingheUniversity of South Florida, [email protected]

Follow this and additional works at: http://scholarcommons.usf.edu/etd

Part of the Optics Commons, and the Quantum Physics Commons

This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion inGraduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please [email protected].

Scholar Commons CitationKonthasinghe, Kumarasiri, "Resonant Light Scattering from Semiconductor Quantum Dots" (2016). Graduate Theses andDissertations.http://scholarcommons.usf.edu/etd/6527

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Resonant Light Scattering from Semiconductor Quantum Dots

by

Kumarasiri Konthasinghe

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of PhilosophyDepartment of Physics

College of Arts and SciencesUniversity of South Florida

Major Professor: Andreas Muller, Ph.D.Martin Muschol, Ph.D.

Zhimin Shi, Ph.D.Jiangfeng Zhou, Ph.D.

Date of Approval:November 03, 2016

Keywords: Resonance fluorescence, Semiconductor quantum dots, Single photon sources,Correlation functions, Frequency comb, Laser induced fluorescence

Copyright c© 2016, Kumarasiri Konthasinghe

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DEDICATION

To my loving wife, Chamani.

Thank you for being by my side no matter where life takes us.

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ACKNOWLEDGMENTS

I would like to express my sincere appreciation and gratitude to all of the people who

supported and encouraged me throughout my doctoral journey. First, I would like to

express my heartfelt gratitude to my advisor, Dr. Andreas Muller, as without his men-

torship and support this dissertation would not have been possible. I am also grateful to

my committee members, Dr. Martin Muschol, Dr. Zhimin Shi, and Dr. Jiangfeng Zhou

for their time, and valuable feedback throughout this process. Additionally, I would like

to thank Dr. Denis Karaiskaj for his valuable time and advice during my candidacy. I

would also like to extend my gratitude to Dr. Kartik Srinivasan at the National Institute

of Standards and Technology (NIST), for the great opportunity to intern and whose advice

and encouragement has been immeasurable. I would like to thank James Christopher for

his assistance. I must also acknowledge the support and research experience gained from

working with my fellow graduate students over past several years. This research was made

possible in part by two grants from National Science Foundation (grant number:1254324),

and Defense Threat Reduction Agency (grant number:HDTRA12-1-0040). Lastly, I would

like to give thanks and appreciation to my family and friends for their love, support, and

encouragement.

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TABLE OF CONTENTS

LIST OF TABLES iv

LIST OF FIGURES v

LIST OF ABBREVIATIONS x

ABSTRACT xi

CHAPTER 1 INTRODUCTION 1

CHAPTER 2 MONOCHROMATIC RESONANT LIGHT SCATTERING FROMA QUANTUM DOT 4

2.1 Introduction 4

2.2 Semiconductor quantum dots: a general overview 5

2.3 Theoretical background 6

2.4 QD sample 7

2.5 Experimental setup 7

2.6 Excitation and emission spectrum 8

2.7 Effect of spectral diffusion on coherently scattered light 13

2.8 First-order coherence 16

2.9 Second-order correlation function g(2)(τ) 17

2.9.1 Principle 17

2.9.2 Hanbury-Brown and Twiss (HBT) setup 18

2.9.3 Characterization of light using g(2)(τ) 19

2.9.4 Experimental observation of g(2)(τ) 19

2.10 Summary and discussion 19

CHAPTER 3 RESONANT LIGHT SCATTERING FROM TWO REMOTE QUAN-TUM DOTS 21

3.1 Introduction 21

3.2 Experimental setup 22

3.3 Excitation and emission spectrum 23

3.4 One-photon interference 24

3.5 Two-photon interference 25

3.5.1 Mathematical description 25

3.5.2 Experimental observation 26

3.6 Theoretical analysis 27

3.7 Summary and discussion 32

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CHAPTER 4 RESONANT INTERACTION BETWEEN A TWO-LEVEL SYS-TEM AND A PERIODICALLY-PULSED LASER 33

4.1 Introduction 33

4.2 Theory 34

4.3 Simulations of spectrum and second-order correlation function 35

4.4 Experiment 37

4.5 Scattered light spectrum 37

4.5.1 Long pulse excitation 38

4.5.2 Short pulse excitation 39

4.6 Rabi oscillations 40

4.7 Second-order correlation functions 42

4.8 Summary and discussion 42

CHAPTER 5 LASER-INDUCED FLUORESCENCE FROM N+2 IONS GENER-

ATED BY A CORONA DISCHARGE IN AMBIENT AIR 44

5.1 Introduction 44

5.2 Experimental setup 45

5.2.1 Dye laser system 46

5.2.2 Excitation/detection methods 47

5.3 Spectrum of the corona discharge 47

5.3.1 Excitation spectrum of N+2 48

5.3.2 LIF at different pressures of N2 48

5.4 Time-resolved measurements of LIF 50

5.5 Estimation of the N+2 concentration in ambient air 51

5.5.1 Using fluorescence signal 52

5.5.2 Using electric current measurement 54

5.6 Summary and discussion 55

CHAPTER 6 CONCLUSIONS AND OUTLOOK 56

REFERENCES 58

APPENDICES 65

Appendix A Copyright permissions 66

Appendix B Theory of two-level system interacting with a near-resonantlaser 70

B.1 Hamiltonian and density operator 70

B.2 Transforming to a rotating frame and applying rotating wave approxi-mation 71

B.2.1 Rotating wave approximation 71

B.3 Evolution of two-level system and optical Bloch equations 72

B.3.1 Density operator and Liouville theorem 72

B.3.2 Inclusion of damping processes into Bloch equations via Mas-ter equation 73

B.3.2.1 Spontaneous emission 74

B.3.2.2 Pure dephasing 74

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B.3.3 Steady state solution to a two-level system 76B.3.4 Power spectrum of the scattered light 76B.3.5 Second-order correlation function 76

Appendix C Theoretical fringe visibilities of one-photon and two-photoninterference 78

C.1 Most general Stark shift model 78C.1.1 Numerical model 80C.1.2 Analytical model 83

Appendix D Interaction between a two-level system and a pulsed laser 87D.1 Theoretical background 87D.2 Calculation of the scattered light spectrum 89D.3 Coherently scattered light spectrum 93

D.3.1 Radiatively-broadened two-level system 93D.3.2 Radiatively-broadened two-level system in the presence of a

fluctuating environment 94D.4 Incoherently scattered light spectrum 95

D.4.1 Radiatively-broadened two-level system 95D.4.2 Radiatively-broadened two-level system in the presence of a

fluctuating environment 96D.5 Calculation of second-order correlation function 96D.6 Second-order correlation function in the presence of a fluctuating envi-

ronment 100D.7 Scattered light correlations under mode-locked laser frequency comb ex-

citation 100Appendix E List of publications 103

ABOUT THE AUTHOR End Page

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LIST OF TABLES

Table 3.1 Summary of one-photon interference visibilities 31

Table C.1 Summary of selection of reported Stark shift coefficients 79

Table C.2 Summary of coefficients from numerical analysis 82

Table C.3 Summary of coefficients from analytical model 86

Table C.4 Summary of one-photon interference visibilities for analytical and nu-merical analysis 86

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LIST OF FIGURES

Figure 2.1 (a) Typical energy level diagram of InAs/GaAs system showing s andp shells. (b) Filtered photoluminescence image of a low-density InAsQD sample. 6

Figure 2.2 (a) Schematics of experimental setup with the QD sample inside acryostat with optical access. An in situ lens collects the fluorescencelight, scattered perpendicular to the sample surface. The high reso-lution scanning Fabry-Perot interferometer along with single photoncounting module allow to analyze the scattered light spectrally. (b) Aphotograph of the sample, and excitation, detection optics. (c) InAsquantum dots in a planar microcavity containing alternating layers ofGaAs/AlAs Bragg mirrors. 8

Figure 2.3 (a) A photograph of the experimental setup showing orthogonal exci-tation/detection geometry. Polarizing beamsplitter (PBS) reflects thevertically polarized light. (b) Image of the closed-cycle cryostat whosebase temperature is 4 K. (c) Custom-made high-resolution Fabry-Perotinterferometers with finesse of 100-150. 9

Figure 2.4 (a) QD emission spectrum, recorded with a grating spectrometer un-der a weak resonant laser excitation. (b) Same data as in (a) but onlogarithmic scale. The pink shaded area corresponds to the phonon-broadband. 10

Figure 2.5 QD excitation spectra for two Rabi frequencies. Solid red line is a fitto the experimental data using Mollow’s theory, including the effect ofspectral diffusion. FWHM of the excitation spectrum increases as theRabi frequency is increased due to power broadening. 10

Figure 2.6 Intensity of scattered light as a function of time showing flickering (darkblue trace) that is inhibited when an additional weak auxiliary laser isintroduced (light green trace). The black trace was recorded with theauxiliary laser only. 11

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Figure 2.7 (a) Maps of scattered light intensity as a function of detection fre-quency (abscissas) and excitation frequency (ordinates relative to theQD transition frequency, for three values of Rabi frequency. (b) Theo-retical maps Eq. (2.5) corresponds to a radiatively broadened two-levelsystem subjects to spectral diffusion. 12

Figure 2.8 Power spectrum of the light scattered by the QD at exact resonance(∆ω = 0) represented on a linear (left) and logarithmic (right) ordinatescale, for a range of Rabi frequencies. Long-dashed (red), short-dashed(red) represent total, and incoherently scattered light intensity, respec-tively. The shaded (pink) area corresponds to the coherently scatteredlight. 14

Figure 2.9 Total scattered light intensity as a function of Rabi frequency. Thesolid-red line represents the corresponding theoretical trace when theeffect of spectral diffusion is included. 15

Figure 2.10 (a) Plot of the total scattered light intensity (solid red trace) and thefraction of coherently scattered light intensity (dashed red trace) onresonance, using the same parameters as those used in Fig. 2.8. (b).Same as (a), but for ∆ω/2π = 1 GHz. 15

Figure 2.11 (a) Measurement of mutual phase coherence between the coherentlyscattered light and a local oscillator (LO) by interferometry. (b) Inten-sity of light at the output of the beamsplitter in (a) as a function ofthe LO phase. 16

Figure 2.12 (a) Schematics of Hanbury-Brown and Twiss setup. (b) Second-ordercorrelation functions for light sources following different photon statis-tics. 18

Figure 2.13 Second-order correlation function of the light scattered by the QD,for three Rabi frequencies. The solid red traces correspond to thetheoretical second-order correlation function, convolved with the IRF(gray trace in the right most panel). 20

Figure 3.1 Experimental setup with the QD sample inside a cryostat with opticalaccess. An in situ lens separates the scattered light from QDL andQDR which is then recombined at a non-polarizing 50:50 beamsplitter(BS). The relative phase of the two waves is controlled with a piezo-electric actuator (PZT). Avalanche photon counting detectors (APD)record events at the beamsplitter outputs. A flip mount (FM) allows usto replace one of the QD signals with a reference local oscillator (LO).For polarization control, a half wave plate (HWP) is inserted into oneof the arms. 22

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Figure 3.2 (a) Image of the sample surface showing the two probed QDs. (b)Excitation spectra for QDL (blue dots) and QDR (red dots), for arange of Rabi frequencies. Solid lines are the corresponding theoreticaltraces obtained from the Mollow’s theory. The slight asymmetry of theexcitation spectra arises due to quadratic Stark shift. 24

Figure 3.3 (a) Spectra of the light scattered by QDL (red trace) and QDR (bluetrace) for three Rabi frequencies. (b) Corresponding theoretical spec-tra of the scattered light at ∆ω/2π = 0.3 GHz. Long-dashed andshort-dashed lines represent the intensity of the total and incoherentlyscattered light, respectively. The shaded area corresponds to the co-herently scattered light. 25

Figure 3.4 (a)-(b) One-photon interference between the light scattered by QDLand QDR for two Rabi frequencies. (c)-(d) One-photon interferencebetween the light scattered by QDR and local oscillator for the sameRabi frequencies. The dashed data lines correspond to the case whenthe polarization of one of the inputs is rotated by 90 deg. 26

Figure 3.5 Beamsplitter input (1,2) and output ports (2,4). Each input port con-tains one photon. 27

Figure 3.6 Schematics of the experimental setup to study HOM effect using lightscattered by two QDs. A half-wave plate is inserted to one of thebeamsplitter input ports to study cross correlations. 28

Figure 3.7 (a) Second-order correlation function for the light scattered by QDL(top) and QDR (bottom). (b) Second-order correlation function whenboth the QD signals are entering to the beamsplitter, for two Rabifrequencies. (c) Same as in (b) but with a perpendicularly polarizedinputs. 29

Figure 3.8 (a) Temporal flickering of the scattered light for a QD in the samesample as QDL and QDR. (b) Histogram of the signal in (a), plottedtogether with theoretical probability distributions that assume a purelylinear (solid purple line), and a purely quadratic (dashed orange line)Stark shift. The inset shows the corresponding excitation spectrumand a theoretical fit that includes a quadratic Stark shift. 30

Figure 4.1 (a) Spectrum of the frequency comb represented by Eq. (4.1). (b)Temporal profile of (a). 35

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Figure 4.2 (a) Applied field intensity in time domain. (b) Intensity of total (red),coherently (blue), and incoherently (green) scattered light, calculatedwith (solid lines) and without (dashed lines) spectral diffusion. (c)-(e)Calculated spectra of the scattered light for the applied field shown inFig. 4.2(a) for range of Rabi frequencies as indicated. Blue and redtraces correspond to the spectra calculated with and without spectraldiffusion, respectively. (f)-(h) Corresponding second-order correlationfunctions under the same conditions, for range of Rabi frequencies,calculated with (blue) and without (red) spectral diffusion. 36

Figure 4.3 (a) Laser spectrum for tp= 560 ps. (b) Corresponding second-order cor-relation function. (c) Experimental (blue) and theoretical (red) spectraof the light scattered by the QD for tp=560 ps, represented on a lin-ear (left) and logarithmic (right) ordinate scale, for a range of peakRabi frequencies. Black and green arrows indicate the positions of pri-mary and secondary sidebands, respectively. Shaded area representsthe incoherently scattered light. 38

Figure 4.4 (a) Laser spectrum for tp= 35 ps. (b) Zoomed-in view of (a). (c)Experimental (blue) and theoretical (red) spectra of the light scatteredby the QD for tp=35 ps for a range of peak Rabi frequencies. (d) Sameas (c) but for tp=80 ps. Shaded area represents incoherently scatteredlight. 40

Figure 4.5 (a) Intensity of total (red), coherently (blue) and incoherently (green)scattered light, measured (markers) and calculated (solid lines) fortp=560 ps. (b) Same as (b) but tp=80 ps. (c) Same as (a) but fortp=35 ps. 41

Figure 4.6 Experimental (blue) and theoretical (red) second-order correlation func-tions for (a) tp=560 ps, and (b) tp=35 ps. 42

Figure 5.1 (a) Schematics of the experimental setup. Corona discharge occursbetween positively biased Tungsten needle and a ground plate. Thecollected signal was sent to a PMT or a CCD for the analyzing. (b)N+

2 energy level diagram. When the band near 391.4 nm is excited,fluorescence is detected near 427.5 nm. 45

Figure 5.2 (a) Photograph of the experimental setup. Fluorescence signal wascollected in a direction perpendicular to both the excitation and theneedle axes. (b) Dye oscillator containing a flow cell, 3600 grooves/mmdiffraction grating, and two mirrors. 46

Figure 5.3 (a) Spectrum of the corona discharge at 1 Torr of N2. Emission linesdue to N+

2 ions dominate over those due to N2. (b) Same as (a), butfor 50 Torr of N2, in which N+

2 emission lines are no longer dominant. 48

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Figure 5.4 (a) LIF excitation spectrum of N+2 at p =4 Torr of N2. R and P

branches can be clearly identified. (b) Same as (a) but for p =140 Torrof N2 and with a better resolution (5 pm). 49

Figure 5.5 (a) Fluorescence decay for p=5 Torr (red) and p=8 Torr (green). Dashedlines (black) are the exponential fits to the decay curves, which are usedto extract the decay rate at each pressure. (b) Fluorescence decay rateversus pressure. Solid red line is a linear-fit to the data. 49

Figure 5.6 (a) N+2 LIF measurement in ambient air for on-line and off-line exci-

tation. Both N+2 fluorescence and N2 Raman signals are visible when

the laser is on-resonance. (b) Polarization properties of the collectedsignal. 50

Figure 5.7 (a) Delayed emission of N2. Therefore, time-gated detection is neces-sary to distinguish the fluorescence signal from the delayed emission ofN2. (b) Measured N+

2 ion concentration using electric measurement. 51

Figure 5.8 Linewidth extraction of individual transition, (ν ′′=0, N ′′=6)→(ν ′=0,N ′=7), at the center of the R-branch in Fig. 5.4(a). Solid-red line is aLorentzian fit to the experimental data. 53

Figure C.1 Probability density distribution of Stark shift at Ω/2π = 0.21 GHz. 82

Figure C.2 Probability density distribution of scattered light intensity at Ω/2π =0.21 GHz. Note that although the abscissas are denoted in arb. units,the function is normalized so that the area equals unity. 83

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LIST OF ABBREVIATIONS

QD Quantum Dot

LIF Laser Induced Fluorescence

FWHM Full Width at Half Maximum

SPCM Single Photon Counting Module

PMT Photo Multiplier Tube

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ABSTRACT

In this work, resonant laser spectroscopy has been utilized in two major projects –

resonance fluorescence measurements in solid-state quantum-confined nanostructures and

laser-induced fluorescence measurements in gases. The first project focuses on studying

resonant light-matter interactions in semiconductor quantum dots “artificial atoms” with

potential applications in quantum information science. Of primary interest is the under-

standing of fundamental processes and how they are affected by the solid-state matrix.

Unlike atoms, quantum dots are susceptible to a variety of environmental influences such

as phonon scattering and spectral diffusion. These interactions alter the desired properties

of the scattered light and hinder uses in certain single photon source applications. One

application of current interest is the use of quantum dots in quantum repeaters for which

two-photon interference is key. Motivated by such an application we have explored the

limits imposed by environmental effects on two quantum dots in the same sample, the

scattered light from which is being interfered. We find that both one-photon and two-

photon interference, although substantial, are affected in a variety of ways, in particular

by spectral diffusion. These observations are discussed and compared with a theoretical

model. We further investigated correlations in pulsed resonance fluorescence, and found

significant unexpected spectral and temporal deviations from those studied under continu-

ous wave excitation. Under these conditions, the scattered light exhibits Rabi oscillations

and photon anti-bunching, while maintaining a rich spectrum containing many spectral

features. These observations are discussed and compared with a theoretical model. In the

second project, the focus is on the investigation of the possibility of detecting N+2 ions

in air using laser induced fluorescence, with potential applications in detection of fissile

materials at a distance. A photon-counting analysis reveals that the fluorescence decay

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rate rapidly increases with increasing N2 pressure and thus limits the detection at elevated

pressures, in particular at atmospheric pressure. We show that time-gated detection can

be used to isolate N+2 fluorescence from delayed N2 emission. Based on the spontaneous

Raman signal from N2 simultaneously observed with N+2 fluorescence, we could estimate a

limit of detection in air of order 108 − 1010 cm3.

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CHAPTER 1

INTRODUCTION

Due to its capability for selective excitation of rovibrational transitions in atomic and

molecular species, resonant laser spectroscopy has attracted great interest during the past

decade as a powerful technique for a variety of applications in chemistry, molecular bi-

ology as well as in quantum optics. For example, resonant laser spectroscopy has been

commonly used for gas sensing [1], pollutant monitoring [2], and recently for implementing

coherent control in quantum bits [3]. In modern quantum information science, resonant

laser spectroscopy is strictly required for controlling the quantum phase coherence in qubits.

Semiconductor QDs are well suited for investigating quantum coherence in solids at optical

frequencies [4]. Most of the previous experiments typically were performed using photo-

luminescence in which incoherent relaxation is involved between excitation and emission.

Unlike QD photoluminescence, resonantly-excited QD fluorescence, or resonance fluores-

cence uniquely bears the distinctive features of the driven QD. In this context, resonant

light scattering in QDs has attracted significant interest during the past decade. Milestone

experiments include the measurement of the resonance fluorescence spectrum [5–7], photon

anti-bunching [6], cascaded photon emission [8], Rabi oscillations and their damping [9–11].

However, surprisingly, a majority of the previous measurements have been based on the

most basic light-matter interaction, i.e., the interaction between a two-level system and a

monochromatic resonant laser. Yet, even in this simple case, it has been difficult to obtain

a complete picture of the physics of resonant light scattering in QDs. For example, coher-

ent scattering, as opposed to incoherent scattering remains largely unexplored either due

to poor instrument resolution and/or issues associated with background laser scattering.

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In the first part of this work, a series of experiments was performed to understand

the spectral and statistical properties of the light scattered by QDs under different ex-

citation methods. In particular, I started with addressing a current issue, i.e., measur-

ing background-free high-resolution scattered light spectra under resonant excitation by a

monochromatic laser [12]. These measurements will be presented in chapter two. The mea-

sured spectra and second-order correlation functions are highly affected by environmental

effects such as spectral diffusion and phonon scattering.

It has been suggested that semiconductor QDs have a great potential for applications in

quantum information science. In particular, there is a recent interest in building a quantum

repeater based on semiconductor QDs. The basic requirement for such an application is to

be able to observe two-photon interference using light originated from different emitters.

Before combining many independent emitters, the most simple case is to study interference

between the light from two separate emitters. Motivated by such an application we have

explored the limits imposed by environmental effects on two QDs in the same sample, the

scattered light from which is being interfered [13]. Chapter 3 will discuss the experimental

approaches and the observation of one-photon and two-photon interference using the light

scattered from two nearby QDs.

The interaction between a two-level system and a monochromatic laser is a well-

understood topic in quantum optics and has been experimentally realized in different sys-

tems such as atoms, ions as well as semiconductor QDs. However, there is little knowledge

beyond this basic light-matter interaction. While monochromatic light-matter interac-

tions provide a testbed for studying dynamics of a two-level system under an external

field, pulsed-excitation is strictly required for generating deterministic or on-demand sin-

gle photons [14, 15]. In this context, studying the spectral and statistical properties under

pulsed-excitation will be an important step. Chapter 4 will present the correlations in

pulsed-resonance fluorescence under the excitation of stabilized periodically-pulsed laser

source [16, 17].

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In the second part of the study I will describe a more practical use of resonant laser

spectroscopy. Recently, it has been reported that ionizing radiation in the presence of

radioactive materials can be detected via prompt fluorescence[18, 19]. However, these

demonstrations are not suitable for day-time detection, where the background light contri-

bution is significant. LIF is of particular interest as it combines high chemical specificity

with low background and is potentially suited for stand-off sensing applications [1, 20].

Chapter 5 will present experimental approaches, for which LIF was used to determine the

limiting factors for the detection of N+2 ions generated by corona discharge in ambient air

[21].

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CHAPTER 2

MONOCHROMATIC RESONANT LIGHT SCATTERING FROM AQUANTUM DOT

2.1 Introduction

Recently, there has been a significant interest in measuring and collecting light emit-

ted by a QD under strictly resonant excitation. Unlike in non-resonant excitation such

as photoluminescence, the light emitted by a resonantly-excited QD bares improved co-

herent properties while exhibiting better photon indistinguishability [22, 23]. However,

true resonant excitation is always challenging as it is necessary to isolate the resonance

fluorescence signal from unwanted background laser scattering. This issue has been ad-

dressed by various methods in recent years. In one method QDs are embedded in a planar

microcavity, containing top/bottom distributed Bragg reflectors [5]. Here the cavity acts

as planar waveguide for an efficient resonant excitation/detection while allows to suppress

stray background light significantly. Another approach uses cross-polarized excitation and

detection [24]. So far, resonant fluorescence has been realized to investigate oscillatory

field correlations [5], the spectral Mollow triplet [6, 7, 22], and photon anti-bunching [6].

However, there has been little knowledge of the light elastically scattered by a QD, which

bares the inherent properties of the excitation laser.

This chapter will first give an overview of self-assembled semiconductor quantum dots.

Then it discusses experimental approaches to collect background-free scattered light. The

measurement of scattered light spectrum and second-order correlation function will be dis-

cussed. Finally, the impact of solid state environment on spectral and statistical properties

of the scattered light will be discussed and compared with Mollow’s theory.

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2.2 Semiconductor quantum dots: a general overview

QDs are nanometer-sized structures in which the motion of charge carriers (electrons,

holes) are confined in all directions. As a result of this confinement they exhibit discrete

energy levels similar to those of atoms and thus often referred to as “artificial atoms”. QDs

have been identified as promising candidates for applications in optoelectronic as well as

in quantum information science due to their high quantum yield and localized density of

states. Over the last few years, semiconductor industry has achieved a remarkable success.

A variety of growth mechanisms are available, but epitaxial based techniques, in particular,

molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD),

are widely used. The type of epitaxial growth is mainly determined by the interface energy

parameters and lattice mismatch. In general, a smaller band gap material is deposited

on a substrate with a larger band gap. For example, the energy level diagram of the

InAs/GaAs system is depicted in Fig. 2.1(a). Here GaAs has a larger band gap (1.424 eV)

compared to that of InAs (0.354 eV). This system is well-known in semiconductor industry

for fabricating high quality QDs using MBE based Stranski-Krastanov (SK) growth mode

[25]. In this method, as the growth process continues lattice mismatch between the two

materials (7% for GaAs/InAs system) leads to accumulate the strain induced energy inside

the system. Since this is thermodynamically unfavorable, the minimization of strain energy

is needed, which results in the formation of self-assembled three dimensional islands or QDs,

typically with a pyramidal or lens shape [Figure 2.1(b)]. Due to recent advancements in

semiconductor growth technology, size and thus the emission wavelength, and density of

the QDs can be controlled to some extent by adjusting parameters such as growth rate

and growth temperature. However, lack of position control of these QDs remains a key

problem, and hinders their use in some applications.

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- S

S

P

P

InA

s ba

nd g

ap

GaA

s ba

nd g

ap

+

(a) (b)

20 µm

Figure 2.1. (a) Typical energy level diagram of InAs/GaAs system showing s and p shells.(b) Filtered photoluminescence image of a low-density InAs QD sample.

2.3 Theoretical background

Here we consider a two-level system with natural resonance frequency ω0. The power

spectrum of the light scattered by such a system under the interaction with a monochro-

matic near-resonant laser with frequency ω is given by [26]

g(ν, ω, ω0) = 2π|α∞|2δ(ν − ω) + n∞κΩ2 (ν − ω)2 + Ω2/2 + κ2

|f(i(ν − ω))|2(2.1)

where ν is the emission frequency, and the only source of broadening is a decay of the

upper state to the lower state at a rate κ due to spontaneous emission. Ω is the Rabi

frequency, which is proportional to the square root of the excitation laser intensity and it

quantifies the interaction between the two-level system and an external field. The steady

state population inversion, n∞ and the steady state quantum mechanical expectation value

of the two-level coherence, α∞ are given by

n∞ =Ω2

κ+ 2γ

∆ω2 + (κ+ 2γ)(κ+ 2γ + 2Ω2/κ)/4(2.2)

and

α∞ =iΩ

4

κ+ 2γ + 2i∆ω

∆ω2 + (κ+ 2γ)(κ+ 2γ + 2Ω2/κ)/4(2.3)

6

Page 23: Resonant Light Scattering from Semiconductor Quantum Dots

laser detuning ∆ω = (ω−ω0) is inserted into Eq. (2.1) via f(s), a third-order polynomial,

which is defined as

f(s) = s3 + 2κs2 + [(Ω2 + (∆ω)2 + (5/4)κ2]s+ κ[1

2Ω2 + (∆ω)2 +

1

4κ2] (2.4)

A detailed derivation of the power spectrum and other related quantities will be provided in

Appendix B. The power spectrum consists of both coherently and incoherently scattered

light, and their relative intensities are solely governed by the Rabi frequency and the

detuning from the resonance. In particular, coherently scattered light, which appears as a

δ-function like peak in the scattered light spectrum, dominates under weak excitation. On

the other hand, when the Rabi frequency is large enough, the power spectrum is dominated

by incoherently scattered light or resonance fluorescence.

2.4 QD sample

Our sample, containing InAs QDs at the center of a planar optical microcavity, was

grown using a solid source VEECO Gen-II molecular beam epitaxy (MBE) system on a

semi-insulating GaAs (100) substrate with a 12-pair top and 20-pair bottom Al0.9Ga0.1As/GaAs

distributed Bragg reflector [Fig. 2.2(c)]. The substrate rotation was stopped during MBE

growth of InAs layers to obtain a QD density varying uniformly from 109 to 108/cm2 [27].

The dominant vertical cavity mode is centered around λ ≈ 925 nm.

2.5 Experimental setup

Figure 2.2(a) depicts schematics of our experimental setup. The QD sample was main-

tained around 4 K in a closed-cycle cryostat and the QD emission was collected by an in

situ high-numerical aperture aspheric lens. Orthogonal excitation/detection geometry was

used to discriminate resonantly scattered signal from background laser scattering. The col-

lected signal was focused into a single-mode fiber and then analyzed either using a grating

spectrometer with cooled charged-coupled device (CCD) camera or high-resolution scan-

7

Page 24: Resonant Light Scattering from Semiconductor Quantum Dots

Tunable laser

Scanning Fabry-Perotinterferometer

Single photon counting module (SPCM)

QD

4 K Cryostat

GaAsMirrors

InAs quantum dots(c)

(b)(a)

Figure 2.2. (a) Schematics of experimental setup with the QD sample inside a cryostatwith optical access. An in situ lens collects the fluorescence light, scattered perpendicularto the sample surface. The high resolution scanning Fabry-Perot interferometer alongwith single photon counting module allow to analyze the scattered light spectrally. (b) Aphotograph of the sample, and excitation, detection optics. (c) InAs quantum dots in aplanar microcavity containing alternating layers of GaAs/AlAs Bragg mirrors.

ning Fabry-Perot interferometer in conjunction with a single-photon counting detector. In

order to study the photon statistics of the scattered light, photon correlation measurements

were performed with a Hanbury-Brown and Twiss setup [28]. In this setup collected QD

signal was sent to a 50:50 beamsplitter, which with equal probability send photons to one

or the other single-photon detector, and detected photons were histogrammed based on

their arrival times (τ).

2.6 Excitation and emission spectrum

Figure 2.4(a) shows the power spectrum of the scattered light under weak resonant

excitation, recorded using a grating spectrometer with a 10 GHz resolution. Although

the main emission line is not resolved in this measurement, a broadband emission around

8

Page 25: Resonant Light Scattering from Semiconductor Quantum Dots

Chamber

PBS

Lens

To camera

Laser in

Fluorescence signal

(a)

Partial reector

(b)

(c)

FSR 4.2 GHz FSR 20 GHz

Figure 2.3. (a) A photograph of the experimental setup showing orthogonal excita-tion/detection geometry. Polarizing beamsplitter (PBS) reflects the vertically polarizedlight. (b) Image of the closed-cycle cryostat whose base temperature is 4 K. (c) Custom-made high-resolution Fabry-Perot interferometers with finesse of 100-150.

it can be clearly identified [Fig. 2.4(b)]. In fact, photoluminescence measurements [29],

four-wave mixing studies [30, 31] reveal that the origin of this broad band emission is due

to fast scattering processes with acoustic phonons. We further verify that the latter has a

≈ 5% contribution to the total scattered light intensity at 4 K. Therefore, as much as 95%

of the light may be scattered coherently, irrespective of Ω and ∆ω.

Figure 2.5 presents the excitation spectra, for both below and above the saturation

intensity, obtained by scanning the laser across the QD resonance frequency while collecting

the total scattered light. When the excitation power is small [Fig. 2.5(a)], the full width

at half-maximum (FWHM) of the resonance is about 0.7 GHz and further increases as the

excitation power is increased due to the power broadening [Fig. 2.5(b)]. Interestingly, the

measured FWHM is significantly larger than the radiative decay rate, reported earlier using

four-wave mixing [30], and photoluminescence measurements [31]. The additional apparent

broadening is due to spectral diffusion, a process by which the QD transition frequency

is randomly shifted during the measurement [32]. This shift is known to originate from a

9

Page 26: Resonant Light Scattering from Semiconductor Quantum Dots

924 922926

Inte

nsity

(arb

. uni

ts)

Emission wavelength (nm)

924 922926Emission wavelength (nm)

Log

inte

nsity

(a) (b)

10-1

101

103

105

Figure 2.4. (a) QD emission spectrum, recorded with a grating spectrometer under a weakresonant laser excitation. (b) Same data as in (a) but on logarithmic scale. The pinkshaded area corresponds to the phonon-broadband.

0 1 2-1-2 0 1 2-1-2

Inte

nsity

(arb

. uni

ts)

Relative excitation frequency (GHz)

Ω/2π=0.006 GHz

Ω/2π=0.35 GHz

Figure 2.5. QD excitation spectra for two Rabi frequencies. Solid red line is a fit tothe experimental data using Mollow’s theory, including the effect of spectral diffusion.FWHM of the excitation spectrum increases as the Rabi frequency is increased due topower broadening.

fluctuating charge environment of the QD that occurs on a time scale that is long compared

to the radiative decay process. The presence of spectral diffusion can be identified in

different forms. For example, in resonant scattering experiments, the effect of spectral

diffusion can be seen via flickering of the scattered light intensity (dark blue trace in Fig

2.6) at low laser intensities. It has been proposed that the time scale of the flickering can

be controlled using an auxiliary non-resonant laser (light green trace in Fig. 2.6), which

leads to neutralize the surrounding charges [33]. We also observed such a phenomenon in

our sample whenever a weak red laser, which does not generate any fluorescence itself, is

introduced (black trace in Fig. 2.6).

10

Page 27: Resonant Light Scattering from Semiconductor Quantum Dots

Inte

nsity

(arb

. uni

ts)

0 8642 10Time (s)

Figure 2.6. Intensity of scattered light as a function of time showing flickering (dark bluetrace) that is inhibited when an additional weak auxiliary laser is introduced (light greentrace). The black trace was recorded with the auxiliary laser only.

As given by Eq. (2.1), the power spectrum of the light scattered by a two-level system

contains both coherently and incoherently scattered light. Therefore, it should be, in

theory, possible to distinguish coherently and incoherently scattered light spectrally by

analyzing the scattered light with a resolution better than radiative decay rate (κ). Figure

2.7(a) shows maps of the scattered light intensity as a function of emission and excitation

frequency, for a range of Rabi frequencies. The latter was varied by varying the excitation

laser intensity. When Ω κ [leftmost panel in Fig. 2.7 (a)], the emission resonance

linewidth is much less than κ, as expected from Eq. (2.1). It is here limited by the

resolution of our Fabry-Perot interferometer (35 MHz). However, in theory, the latter

should be as narrow as the linewidth of the laser (∼1 MHz). On the other hand, when

the Rabi frequency is increased, resonance fluorescence starts to contribute, and eventually

dominates in the form of the Mollow triplet, consisting of three peaks with a FWHM that

is on the order of κ [middle and rightmost maps in Fig. 2.7 (a)].

The theoretically expected excitation-emission maps can be obtained through a careful

mathematical rearrangement of Eq. (2.1). The effect of spectral diffusion can be included

into Mollow’s power spectrum by integrating Eq. (2.1) over all possible random detunings

to obtain

11

Page 28: Resonant Light Scattering from Semiconductor Quantum Dots

0.0

0.5

0.0

-0.5

-1.0

-1.0 1.0 0.0-1.0 1.0 2.0

0.0

0.5

-0.5

-1.0

1.0

-1.5

0.0

0.5

-0.5

-1.0

1.0

0.0-1.0 1.0 2.0

0.0

0.5

0.0

-0.5

-1.0

-1.0 1.0 0.0-1.0 1.0 2.0

0.0

0.5

-0.5

-1.0

1.0

-1.5

0.0

0.5

-0.5

-1.0

1.0

0.0-1.0 1.0 2.0

(a) Experiment

(b) Theory (radiative decay and spectral diusion)

Ω/2π=0.006 GHz Ω/2π=0.3 GHz Ω/2π=0.98 GHz

Lase

r det

unin

g, ∆ω/2π

(GH

z)La

ser d

etun

ing,

∆ω/2π

(GH

z)

Relative frequency (GHz) Relative frequency (GHz) Relative frequency (GHz)In

tens

ity (a

rb. u

nits

)

Figure 2.7. (a) Maps of scattered light intensity as a function of detection frequency (abscis-sas) and excitation frequency (ordinates relative to the QD transition frequency, for threevalues of Rabi frequency. (b) Theoretical maps Eq. (2.5) corresponds to a radiativelybroadened two-level system subjects to spectral diffusion.

I(ν, ω, ω0) ∝∫

g(ν, ω, ω′0)e−(ω

′0−ω0)2/2σ2

dω′0 (2.5)

where we assume a Gaussian distribution of QD transition frequencies due to spectral

diffusion with a FWHM s ≈ 2.355σ. Furthermore, to account for finite apparatus resolu-

tion we replace the δ-function in Eq. (2.1) with a normalized Lorentzian with a FWHM

equal to the resolution of our Fabry-Perot interferometer (35 MHz). The theoretically

expected excitation-emission maps are shown in Fig. 2.7(b). We assume radiative decay

rate, κ/2π=180 MHz, which is consistent with typical radiative life times [τ = 1/(2π ×

180 MHz) = 0.9 ns] for InAs QDs. Moreover, FWHM of the Gaussian distributed QD

transition frequencies, s/2π = 0.7 GHz, was chosen to agree with the measured excitation

12

Page 29: Resonant Light Scattering from Semiconductor Quantum Dots

linewidth in Fig. 2.5(a). As seen in Fig. 2.7, a faithful agreement can be found when the

effect of spectral diffusion is included. By comparison with similar theoretical maps as in

Fig. 2.7(b), but with the effect of pure dephasing, we further verify that pure depahsing

does not play a major role in our measurement temperature.

For a quantitative analysis, we now consider the case where the QD is resonant with the

excitation laser (∆ω = 0). The resulting scattered light spectra are shown in Fig. 2.8, both

on linear (left) and logarithmic (right) ordinate scales, for range of Rabi frequencies. The

theoretical traces were obtained using Eq. (2.5) with the same parameters as in Fig. 2.7(b),

and only a common scale factor for all traces was used. Long-dashed (red) and short-dashed

(red) lines represent the total and the incoherently scattered light, respectively. The shaded

(red) area corresponds to the coherently scattered light. As can be seen in Fig. 2.8, there

is an excellent agreement between the experimental and theoretical traces. However, when

the Rabi frequency is large enough [bottom left trace (black) in Fig. 2.8], the central peak

deviates from the theory, showing elevated intensity for coherently scattered light. We think

that this extra light may originate from other neighboring detuned QDs. Nonetheless, we

estimate that 90% of the light is scattered off from the probed QD when Ω/2π = 0.49 GHz.

We also extracted the total scattered light intensity by taking the area under each

emission spectrum similar to those shown in Fig. 2.8, and the results are shown as a function

of Rabi frequency in Fig. 2.9. Solid red trace represents the corresponding theoretical curve

generated using Mollow’s theory, including the effect of spectral diffusion. κ and s were

chosen to have the same values as in Fig. 2.8. Saturation nature of the total intensity at

lager Rabi frequencies is clearly visible.

2.7 Effect of spectral diffusion on coherently scattered light

Spectral diffusion significantly alters the spectral and temporal characteristics of the

scattering process compared to the ideal two-level system. Figure 2.10 shows the total

normalized scattered light intensity as a function of Rabi frequency for the same param-

eters used in Fig. 2.8. Correspondingly, the dashed lines of the same color represent the

13

Page 30: Resonant Light Scattering from Semiconductor Quantum Dots

0 1-1 0 1-10.1

1

10

0.1

1

10

0.1

1

10

0.01

0.1

1

0.01

0.1

1

0

10

20

30

0

5

10

15

0

4

8

12

0

0.4

0.8

1.2

1.6

0

0.02

0.04

0.06

Ω/2π =0.49 GHz

Ω/2π =1.34 GHz

Ω/2π =0.007 GHz

Ω/2π =0.04 GHz

Ω/2π =0.3 GHz

Relative frequency (GHz)

Scat

tere

d lig

ht in

tens

ity (a

rb. u

nits

)

ExperimentTheory: Total scatt. lightTheory: Incoh. scatt. lightCoh. scatt. light

Figure 2.8. Power spectrum of the light scattered by the QD at exact resonance (∆ω =0) represented on a linear (left) and logarithmic (right) ordinate scale, for a range ofRabi frequencies. Long-dashed (red), short-dashed (red) represent total, and incoherentlyscattered light intensity, respectively. The shaded (pink) area corresponds to the coherentlyscattered light.

14

Page 31: Resonant Light Scattering from Semiconductor Quantum Dots

Tota

l sca

tt. i

nten

sity

(arb

. uni

ts)

0 1.51.00.5Rabi frequency (GHz)

Figure 2.9. Total scattered light intensity as a function of Rabi frequency. The solid-redline represents the corresponding theoretical trace when the effect of spectral diffusion isincluded.

fraction of coherently scattered light. As evident from Fig. 2.10(a), spectral diffusion leads

to enhance the fraction of coherently scattered light as it is an common feature of any

inhomogeneous broadening. It is also important to note that at large values of Ω, more

light is actually scattered coherently with the laser off resonance than with the laser at

exact resonance [Figure 2.10(b)].

Rabi frequency (GHz)

Scat

t. lig

ht in

tens

ity (a

rb. u

nits

)

0 0.4 0.8 1.2

0.5

0

1frac. coherently scatt. light

total scatt. light

Rabi frequency (GHz)0 0.4 0.8 1.2

(b)(a)

0.5

0

1total scatt. light

frac. coherentlyscatt. light

no spectral di.spectral di.

∆ω/2π=0 ∆ω/2π=1 GHz

Figure 2.10. (a) Plot of the total scattered light intensity (solid red trace) and the fractionof coherently scattered light intensity (dashed red trace) on resonance, using the sameparameters as those used in Fig. 2.8. (b). Same as (a), but for ∆ω/2π = 1 GHz.

15

Page 32: Resonant Light Scattering from Semiconductor Quantum Dots

2.8 First-order coherence

First-order coherence is a measure of the phase correlations of a light field. These

correlations are readily available in the fringe contrast of the interference pattern that can

be obtained using a Michelson interferometer. In such an experiment, interference signal is

detected using a photo multiplier tube (PMT) or a photo diode while scanning the delay

time τ between the two light fields. If τ is larger than the coherence time τcoh of the

impinging light, the resulting interference visibility reaches to zero regardless of the nature

of light source.

Tunablelaser

Single photoncounting detector

PZT

QD(a) (b)

Inte

nsity

(arb

. uni

ts)

LO phase shift0 2π 4π

LO Scatt. light

LO+scatt. light

Figure 2.11. (a) Measurement of mutual phase coherence between the coherently scatteredlight and a local oscillator (LO) by interferometry. (b) Intensity of light at the output ofthe beamsplitter in (a) as a function of the LO phase.

In our experiment, we made use of above idea to study the phase relationship between

excitation laser and scattered light signal using the setup shown in Fig. 2.11(a). Here

we combined QD signal with a part of the excitation laser (local oscillator (LO)) at a

50:50 beamsplitter. Additionally, a piezoelectric transducer (PZT) was used to vary the

path length (phase) of the LO signal with respect to that of the QD signal. Furthermore,

intensities of the two fields were adjusted so that the registered count rate on the detector

by each field is approximately the same. This guaranteed that the interfering fields have

nearly the same average intensity, and thus we can expect to have optimum fringe contrast.

16

Page 33: Resonant Light Scattering from Semiconductor Quantum Dots

Figure 2.11(b) displays the fringe contrast as a function of LO phase shift. The reduced

fringe visibility (≈ 40%) is observed mainly due to the spectral diffusion, which causes

large fluctuations in photon flux at the beamsplitter. We calculate that the latter leads

to reduce the fringe visibility by a factor of order κ/s ≈ 0.25. A detailed explanation on

reduced fringe visibility will be provided in chapter 3 as well as in Appendix C.

2.9 Second-order correlation function g(2)(τ)

As mentioned in the previous section, first-order coherence provides phase correlations

of two light sources. How ever, phase information alone is not enough to characterize a

given light source. For example, a laser beam and light from a thermal source can have

the same spectral properties, but, obviously, their intensity distributions are different.

Since first order correlation function is not sensitive to these information, the use of higher

order correlation functions is necessary. In the following section, I will summarize the

principle behind second-order correlation function and also typical experimental approach

to measure it in a lab.

2.9.1 Principle

In contrast to the first-order correlation function, the second-order correlation function

measures the probability of detecting a photon at time t+τ , when a photon was detected

at time t. From a mathematical point of view, g(2)(τ) is expressed in terms of photon

creation and annihilation operators a†(t), a(t) respectively, as [34],

g(2)(τ) =〈a†(t)a†(t+ τ)a(t+ τ)a(t)〉

〈a†(t)a(t)〉2, (2.6)

where τ is the delay time between two signals. In principle, measuring g(2)(τ) seems

straightforward using a photodetector. However, in practice most photodetectors have

a “deadtime”, during which the detector can not detect any photons. This deadtime is

typically on the order of 100 ns. Therefore, a single detector is not sufficient to measure

17

Page 34: Resonant Light Scattering from Semiconductor Quantum Dots

arrival times of the incoming photons within a certain time window less than 100 ns. This

issue can be overcome by using two photodetectors and a nonpolarizing 50:50 beamsplitter

much like in HBT setup [28].

2.9.2 Hanbury-Brown and Twiss (HBT) setup

Figure 2.12(a) shows a typical HBT setup containing two photodetectors and a 50:50

nonpolarizing beamsplitter. The beamsplitter divides incoming photons into two parts with

equal probability and sends them towards the detectors. An additional delay is usually

introduced to the path of the second APD to collect negative correlations. Once APD1

detects a photon, a counter starts, and it stops as soon as APD2 detects another photon.

The time interval between “start” and “stop”, τ , is repeatedly measured for many events.

Then g(2)(τ) can be directly obtained by histogramming these time intervals. Since the

setup consists of two detectors, APD2 is still active even though APD1 is in its “dead”

state. However, the time jitter (resolution) intrinsic through the detectors still imposes

some limitations to the setup. The detectors we used in our experiments have 100-400 ps

resolution. For example, PerkinElmer SPCM-AQRH-14-FC single photon counting module

has a deadtime of 32 ns and a resolution of 350 ps near 825 nm.

startstop

APD2

APD150:50 beam splitter

Correlationelectronics

(a) (b)

g(2)(τ)

τ0

1.0

2.0Thermal light

Antibunched lightCoherent lightτ

Figure 2.12. (a) Schematics of Hanbury-Brown and Twiss setup. (b) Second-order corre-lation functions for light sources following different photon statistics.

18

Page 35: Resonant Light Scattering from Semiconductor Quantum Dots

2.9.3 Characterization of light using g(2)(τ)

Depending on the shape of the second-order correlation function, the nature of the

light source can be determined. Figure 2.12(b) illustrates three possible outcomes of a

HBT experiment. For example, a classical light source such as one produces thermal light

generates photons in bunches, and thus g(2)(0) > g(2)(τ). In contrast to thermal light,a

laser emits photons randomly and thus second-order correlation function is independent of

time delay. These observations can be easily explained both classically as well as quantum

mechanically. However, photon antibunching can not be explained classically and it is a

purely quantum mechanical effect.

2.9.4 Experimental observation of g(2)(τ)

The measured normalized second-order correlation functions, together with the theory,

for the QD studied here are shown in Fig. 2.13, for three Rabi frequencies. For the

theoretical traces, the effect of spectral diffusion was included for a better comparison with

the experimentally observed traces. Furthermore, in order to account for the finite time

resolution of the detector, the theoretical second-order correlation function was convolved

with the instrument response function (IRF) of the detector [6]. Photon ant-bunching can

be clearly seen near (τ=0) even at larger Rabi frequencies, as expected from the theory.

In addition, Rabi oscillations, the time domain analog of the Mollow side-bands, are seen

(rightmost panel of Fig. 2.13). It is worth noting that it is the IRF, but not the background

laser scattering light, which leads to raise the dip at τ = 0.

2.10 Summary and discussion

We have measured background-free high resolution scattered light spectra from a res-

onantly driven QD, which can be reproduced theoretically by taking the effect of spectral

diffusion into account. Spectral diffusion reshapes the scattered light spectra in different

ways. Primarily, it increases the fraction of coherently scattered light via inhomogeneous

broadening. As a secondary effect, the shape of the sidebands has slightly deviated from

19

Page 36: Resonant Light Scattering from Semiconductor Quantum Dots

IRF

Correlation time τ (ns)

g(2) (τ

)

0

0.8

0.4

1.2

-4 0 4 -4 0 4 -4 0 4

Ω/2π=0.02 GHz Ω/2π=0.09 GHz Ω/2π=0.64 GHz

Figure 2.13. Second-order correlation function of the light scattered by the QD, for threeRabi frequencies. The solid red traces correspond to the theoretical second-order correla-tion function, convolved with the IRF (gray trace in the right most panel).

Lorentzian and has now become triangular. This is a result of summing up Mollow triplets

with different detunings. The single emission nature of QD is clearly evidenced by mea-

sured second-order correlation functions, which are also highly affected by inhomogeneous

broadening, and partially due to the limited time resolution of the detectors. The observed

mutual phase coherence between the LO and QD signal under weak excitation regime

along with high photon antibunching may be of use for generating reliable single photons

for future quantum computation.

20

Page 37: Resonant Light Scattering from Semiconductor Quantum Dots

CHAPTER 3

RESONANT LIGHT SCATTERING FROM TWO REMOTE QUANTUMDOTS

3.1 Introduction

Interference plays a key role in all optical phenomena, but its presence is not always ob-

vious. For example, interference can never be achieved using incoherent light sources such

as ordinary light bulbs due to the absence of a fixed phase relationship between the light

emitted by individual bulbs. In 1801, Thomas Young first observed interference fringes on

a screen in his well known double-slit experiment using a laser and two slits. Here two

slits act as two point sources having identical spectral properties and a well defined mutual

phase relationship. Recently, interference of light scattered from independent emitters has

received great deal of attention due to its possible applications for quantum networking,

in particular, quantum teleportation [35] and entanglement swapping [36, 37]. The basic

requirement for these applications is to be able to observe interference of light from inde-

pendent emitters. Rather than working with complex system involving many independent

emitters, the most simple scenario is to observe interference of light using two separate

emitters. In this regard, as an analogy to Young’s experiment, Eichmann et al. first ob-

served fringes on a screen positioned at a fixed distance from the light scattered by two

localized atoms driven by a weak laser field due to one-photon interference[38]. In addi-

tion to well-known one-photon interference, two-photon interference is also possible with

single photons having identical spatial, temporal properties and polarizations. In seminal

work, Hong et al. showed that when such photon pairs impinge on a 50:50 beamsplitter,

they exit together due to the interference of probability amplitudes [39]. In this context,

studying correlations between the light scattered by two QDs on the same sample will be

21

Page 38: Resonant Light Scattering from Semiconductor Quantum Dots

an important step towards realizing scalable correlated single photon devices. However,

due to the self-assembled nature and interactions with the underlying substrate, two QDs

may not have the same spectral and emission properties. These imperfections will intro-

duce additional challenges. Here we explore the possibility of obtaining one-photon and

two-photon interference effects from the light scattered by two separate QDs on the same

sample.

This chapter is organized as follows. It will first introduce the experimental setup with

two probed QDs. Then it will present properties of the selected QDs such as linewidth and

resonance frequency. Next it discusses how to overcome some of the challenges to achieve

one-photon and two-photon interference effects. At the end of the chapter, a theoretical

model, based on most general Stark shift will be built to explain reduced fringe visibilities.

3.2 Experimental setup

BS

QDL

QDR

SampleLaser

Lens

Correlationelectronics

Cryostat (4 K)

APD1

APD2

PZT

FM

LO

Figure 3.1. Experimental setup with the QD sample inside a cryostat with optical access.An in situ lens separates the scattered light from QDL and QDR which is then recombinedat a non-polarizing 50:50 beamsplitter (BS). The relative phase of the two waves is con-trolled with a piezoelectric actuator (PZT). Avalanche photon counting detectors (APD)record events at the beamsplitter outputs. A flip mount (FM) allows us to replace one ofthe QD signals with a reference local oscillator (LO). For polarization control, a half waveplate (HWP) is inserted into one of the arms.

22

Page 39: Resonant Light Scattering from Semiconductor Quantum Dots

The experimental setup is displayed schematically in Fig. 3.1. The same QD sample

and the excitation/detection geometry were used as in the previous chapter. However,

here we probe two nearby QDs (labeled “QDL”, “QDR”), which are grown in the same

sample. Both QDs were excited simultaneously by a near-resonant monochromatic laser

at a wavelength ≈ 925 nm. The light scattered from each QD was spatially separated,

recombined at a beamsplitter, and then analyzed by two single photon counting detectors

and a photon counting module.

3.3 Excitation and emission spectrum

As can be seen in an image of the sample surface [Fig. 3.2(a)], the two QDs are

separated by about 40 µm and are well isolated from any other background scatters. The

excitation spectra shown in Fig. 3.2(b) further reveal that the resonance frequencies of

the two probed QDs differ by about 0.6 GHz. Note that the overlap of the two excitation

spectra is incomplete, even at larger excitation powers [middle and rightmost panels of

Fig. 3.2(b)]. The theoretical traces [solid blue and solid red in Fig. 3.2(b)] were obtained

by plotting the steady state population n∞, of the excited state of the QD. This slight

asymmetry, observed in some of the recorded excitation spectra [red trace in the rightmost

panel of Fig. 3.2(b)] can be explained using quadratic Stark shifts, as discussed at the end

of this chapter.

The emission characteristics of the two QDs are shown in Fig. 3.3. Data was recorded

using a high resolution (35 MHz) scanning Fabry-Perot interferometer (free-spectral range

4.2 GHz) in conjunction with a singe photon counting detector. Even though the overlap

of the excitation spectra in Fig. 3.2(b) is poor, the spectra of the light scattered by QDL

and QDR can be identical when the laser detuning (∆ω = ω−ω0) is suitably chosen. Here

we have selected the laser frequency so that it satisfies the condition ∆ωQDL = −∆ωQDR

when ΩQDL = ΩQDR. Because of this perfect overlap of the emission spectra, it may be

possible to observe interference fringes.

23

Page 40: Resonant Light Scattering from Semiconductor Quantum Dots

0-2 -1 1 0-2 -1 1

Ω/2π=0.12 GHz

Ω/2π=0.19 GHz

20 µm

QDL QDRlaser

QDL

QDR

0-2 -1 1

Ω/2π=0.46 GHz

Relative laser frequency (GHz)

(a)

(b)

Inte

nsity

(arb

. uni

ts)

Figure 3.2. (a) Image of the sample surface showing the two probed QDs. (b) Excitationspectra for QDL (blue dots) and QDR (red dots), for a range of Rabi frequencies. Solidlines are the corresponding theoretical traces obtained from the Mollow’s theory. The slightasymmetry of the excitation spectra arises due to quadratic Stark shift.

3.4 One-photon interference

One photon interference was studied by recording the field-field correlations via the

light intensity at the output of the beamsplitter in Fig. 3.1. A piezoelectric actuator was

used to vary the relative path length traveled by the light scattered by QDR. Figure 3.4

displays the one-photon interference results, for three Rabi frequencies. Each experimental

trace (solid blue lines) represents the resulting fringe contrast, obtained as the difference

between the intensities at the beamsplitter output divided by their sum. When interfering

the signals from the two QDs [Figs. 3.4(a)-(c)], the observed fringe contrasts are as large

as 20%. However, when QDL signal is replaced with the local oscillator signal [Figs.

3.4(d)-(f)], the resulting fringe contrasts increased to 50%. It is worth noting that in

photoluminescence experiments, which lack a well-defined phase relationship between the

incident laser and the emitted light, no such interference is ever possible. It is the coherent

part of the scattered light, illustrated as the shaded area in Fig. 3.3, that gives rise to the

interference fringes.

24

Page 41: Resonant Light Scattering from Semiconductor Quantum Dots

Relative detection frequency (GHz)0-1 1 0-1 1 0-1 1

Ω/2π=0.46 GHz

Ω/2π=0.21 GHz

Ω/2π=0.15 GHz

Scat

tere

d lig

ht in

tens

ity

Relative detection frequency (GHz)0-1 1 0-1 10-1 1

Coh. scatt. light

(a) (b)Ω/2π=0.46 GHz

Ω/2π=0.21 GHz

Ω/2π=0.15 GHz

Figure 3.3. (a) Spectra of the light scattered by QDL (red trace) and QDR (blue trace)for three Rabi frequencies. (b) Corresponding theoretical spectra of the scattered lightat ∆ω/2π = 0.3 GHz. Long-dashed and short-dashed lines represent the intensity of thetotal and incoherently scattered light, respectively. The shaded area corresponds to thecoherently scattered light.

3.5 Two-photon interference

3.5.1 Mathematical description

If we consider the case where one photon is in the input port 1 and another photon is

in the input port 2 (Fig. 3.5), the initial state of the two photons can be written as |1112〉.

The beamsplitter input-output relations read [40],

a†1 =− ra†3 + ta†4,

a†2 =ta†3 + ra†4,

(3.1)

with r2 +t2 = 1 where a†i (i=1,2,3,4) is the photon creation operator in the ith beamsplitter

port. The initial state of the two photons can be written in terms of vacuum state |0〉 as

follows.

|1112〉 =a†1a†2 |0〉

=(−ra†3 + ta†4)(ta†3 + ra†4) |0〉

=(t2 − r2) |1314〉 −√

2rt |2304〉+√

2rt |0324〉

(3.2)

25

Page 42: Resonant Light Scattering from Semiconductor Quantum Dots

Frin

ge

con

tras

t

phase shift

QDL/QDR QDL/LO

0 4π2π 0 4π2π

-0.4

0.4 0.2

-0.2 0

-0.4

0.4 0.2

-0.2 0

(b)

(c)

(d)

(a)

Ω/2π= 0.46 GHz

Ω/2π= 0.15 GHz

Figure 3.4. (a)-(b) One-photon interference between the light scattered by QDL and QDRfor two Rabi frequencies. (c)-(d) One-photon interference between the light scattered byQDR and local oscillator for the same Rabi frequencies. The dashed data lines correspondto the case when the polarization of one of the inputs is rotated by 90 deg.

For a 50:50 beamsplitter (i.e r=t=1/√

2), the first term is zero by virtue of the destructive

interference of the corresponding two-photon probability amplitudes and Eq. (3.2) reduces

to

|1112〉 = − 1√2|2304〉+

1√2|0324〉 . (3.3)

Therefore, when two photons with identical wave packets (i.e. with identical polarization,

frequency, spatial-temporal mode structure) impinging on a 50:50 beamsplitter, they al-

ways exit together. This phenomena was first discovered by Hong-Ou and Mandel using

parametrically down converted photons [39] and known as Hong-Ou-Mandel effect.

3.5.2 Experimental observation

The experimental setup shown in Fig. 3.6 can be used to study the second-order corre-

lation function of individual QDs, as well as Hong-Ou-Mandel effect from the light emitted

by both QDs. The former was recorded by histogramming photon arrival times at the

output ports of the beamsplitter when one of the input ports is blocked at a time. As

26

Page 43: Resonant Light Scattering from Semiconductor Quantum Dots

1

2

3

4

Figure 3.5. Beamsplitter input (1,2) and output ports (2,4). Each input port contains onephoton.

can be seen in Fig. 3.7(a), light scattered by both quantum dots exhibits photon anti-

bunching, as expected. The latter can be studied via a similar measurement, but with

both the input ports are opened. Here one can investigate photon correlations for both

parallel [Fig. 3.7(b)] and perpendicular [Fig. 3.7(c)] polarizations. A half-wave plate was

inserted into one of the beamsplitter input ports to rotate the polarization with respect to

the other. The raw interference visibility is large as 44%, but it is limited here by g(2)|| (0),

which is increased from zero mainly due to the detector’s finite resolution function. The

detector’s instrument response function, shown in Fig. 2.13, has been convolved with the

theoretically expected g(2)(τ) to obtain the solid lines in Fig. 3.7 (a).

3.6 Theoretical analysis

Although we expect a higher fringe contrast at low Rabi frequencies due to larger

fraction of coherently scattered light, the observed fringe contrasts significantly deviate

from unity even when one of the QD signal is replaced with a part of the excitation

laser (LO). Hence, there should be some underlying limitations which govern the visibility

of interference fringes. We performed a statistical analysis to explain the reduced fringe

visibilities based on the effect of spectral diffusion, which causes to fluctuate the photon flux

at a beamsplitter. In resonance fluorescence experiments, the effect of spectral diffusion can

27

Page 44: Resonant Light Scattering from Semiconductor Quantum Dots

APD1

QDRsignal

APD2

QDL signal

start

stop

Correlation electronics

Figure 3.6. Schematics of the experimental setup to study HOM effect using light scatteredby two QDs. A half-wave plate is inserted to one of the beamsplitter input ports to studycross correlations.

be directly visible in the form of fluctuations of the scattered light intensity at very low

laser intensities. These fluctuations originate due to random Stark shifts, of magnitude

δStark. The effect of Stark shift can be included into the total scattered light intensity

represented by Eq. (2.2)

Itot = n∞ =Ω2/4

(∆ω + δStark)2 + κ2/4 + Ω2/2(3.4)

which is the scattered light intensity integrated over all detection frequencies. In other

words it is the population inversion of the two-level system at times long compared to the

quantum evolution of the system, but short compared to the spectral diffusion time scale.

Similarly, the coherent portion of the scattered light is given by

Icoh = |α∞|2 =Ω2

4

(∆ω + δStark)2 + κ2/4((∆ω + δStark)2 + κ2/4 + Ω2/2

)2 (3.5)

where α∞ is the steady-state coherence of the two-level system as defined in [26]. The shift

in resonance frequency due to an external electric field has been studied extensively for

InAs QDs [41–43]. For an in-plane applied electric field, a quadratic Stark shift is typically

reported [42, 43] whereas a linear Stark shift is found to dominate when the electric field

28

Page 45: Resonant Light Scattering from Semiconductor Quantum Dots

0 10-5 5

g(2)(

τ)

0

1.2

0.8

0.4

0

1.2

0.8

0.4

Ω/2π=0.12 GHz

Ω/2π=0.12 GHz

QD L

QD R

0

1.2

0.8

0.4

0 10-5 5Correlation time τ (ns)

Ω/2π=0.12 GHz

g(2)(

τ)

Ω/2π=0.12 GHz

Ω/2π=0.15 GHz0

1.2

0.8

0.4

Ω/2π = 0.15 GHz

g(2)(

τ)g(

2)(τ)

(a) (b) (c)

g ||(2)

(τ)

g ||(2)

(τ)

0

1.2

0.8

0.4

0

1.2

0.8

0.4

0 10-5 5

Figure 3.7. (a) Second-order correlation function for the light scattered by QDL (top)and QDR (bottom). (b) Second-order correlation function when both the QD signals areentering to the beamsplitter, for two Rabi frequencies. (c) Same as in (b) but with aperpendicularly polarized inputs.

is oriented perpendicular to the sample plane [41]. In our measurements QD flickering is

influenced by an external non-resonant pump laser focused at a point laterally displaced

by up to several tens of micrometers [12]. Thus we may conclude that in-plane fields are

present and we can expect a quadratic Stark shift, in addition to a linear term due to a

permanent electric dipole moment. In general we have, δStark(V ) ≈ c1V + c2V2. If we

assume that the electrical potential V at the location of the QD is fluctuating around

its mean value (set to zero without loss of generality) following a normal distribution,

PV (V ) = e−V2/2σ2

/σ√

2π, then the average value of the intensity can be written as

〈Itot(∆ω)〉 =

∫g(V )PV (V )dV (3.6)

where g(V ) = n∞(δStark(V )). By comparison of this theoretical expression with the ob-

served excitation spectrum, shown in Fig. 3.8(b), it is seen that the asymmetry of this

spectrum, also visible in Fig. 3.2 for QDL and QDR, is the result of the quadratic term in

the Stark shift.

29

Page 46: Resonant Light Scattering from Semiconductor Quantum Dots

Inte

nsi

ty (a

rb. u

nit

s)

0 400 800Time (s) Scatt. light intensity (arb. units)

0 0.04 0.08 0.12

0.04

0.08

0 0

20

40

60(a)

Pro

bab

ility

den

sity

(b)

-0.5 0-1 0.5-1.5Rel. laser freq. (GHz)

Scat

t. lig

ht I

nten

sity

Figure 3.8. (a) Temporal flickering of the scattered light for a QD in the same sample asQDL and QDR. (b) Histogram of the signal in (a), plotted together with theoretical prob-ability distributions that assume a purely linear (solid purple line), and a purely quadratic(dashed orange line) Stark shift. The inset shows the corresponding excitation spectrumand a theoretical fit that includes a quadratic Stark shift.

In addition, the distribution of the intensity of the scattered light, PI(I), is given by

PI(I) = PV (g−1(I))∣∣∣ ddIg−1(I)

∣∣∣ (3.7)

where g−1 is the inverse of the function g. PI(I) is obtained experimentally as the his-

togram of the time trace in Fig. 3.8(a), shown in Fig. 3.8(b). The histogram closely follows

the expected theoretical expression plotted as a solid red line in Fig. 3.8(b). With the

distribution function, PI(I), we can calculate the theoretically expected fringe contrast

for the data in Fig. 3.3. The one-photon fringe visibility, V(1), is given by the normalized

first-order correlation function of the two fields at the beamsplitter, as [44]

V(1) =⟨U∗1U2

⟩/√〈I1〉〈I2〉 (3.8)

where Uk = Ake−iωt (k = 1, 2) are the amplitudes of the electric field of the two waves

entering the beamsplitter. The corresponding average intensities are given by Ik =⟨U∗kUk

⟩.

When one of the fields is the scattered light from a QD and the other is a local oscillator

30

Page 47: Resonant Light Scattering from Semiconductor Quantum Dots

with the same average intensity, then

V(1)QD,LO =

⟨√Icoh

⟩√⟨Itot

⟩ =

∫|α∞(δStark(V ))|PV (V )dV√∫n∞(δStark(V ))PV (V )dV

(3.9)

It is important to note that in the numerator only Icoh is taken into account as there is

no first-order interference from incoherently scattered light. Similarly, if the two fields

correspond to the scattered light from QDL and QDR, respectively, with mutually un-

correlated intensity fluctuations, then V(1)QDL,QDR =

⟨√Icoh

⟩2/⟨Itot

⟩. Using experimentally

obtained values for ∆ω and values for c1, c2, and σ that best match our excitation spectra

of Fig. 3.2(b) and the statistical distribution of Fig. 3.8(b), we obtain, V(1)QDL,LO = 0.72

and V(1)QDL,QDR = 0.52 when Ω/2π = 0.23 GHz.

Theoretical visibilities of the second-order interference experiment of Fig. 3.7 can be

calculated in a similar approach, starting from an expression for the second-order correla-

tion function written in terms of the autocorrelation functions of the constituent signals

[45]

g(2)⊥ (τ) =

I2Lg

(2)L (τ) + I2

Rg(2)R (τ) + 2ILIR

(IL + IR)2(3.10)

and

g(2)‖ (τ) =

I2Lg

(2)L (τ) + I2

Rg(2)R (τ)

(IL + IR)2(3.11)

where IL (IR) is the intensity of the light scattered from QDL (QDR) and g(2)L (τ) (g

(2)R (τ))

the corresponding second order autocorrelation function. Time averages can be calculated

in the same manner as the first-order correlation function, as discussed in appendix B.

Table 3.1. Summary of one-photon interference visibilities

Experimental TheoreticalΩ/2π (GHz) QD-LO QD-QD QD-LO QD-QD

0.15 0.29 0.16 0.77 0.590.21 0.44 0.19 0.74 0.550.46 0.51 0.15 0.60 0.36

31

Page 48: Resonant Light Scattering from Semiconductor Quantum Dots

3.7 Summary and discussion

Here we have studied one-photon and two-photon interference using the light scattered

by two spatially separated QDs on the same sample under the simultaneous excitation of

a monochromatic near-resonant laser. Unlike in photoluminescence experiments, resonant

excitation allows to have identical emission spectra for the two probed QDs despite their

unequal resonance frequencies, when the laser detuning is suitably chosen. One-photon in-

terference is possible due to coherently scattered light, which dominates at weak excitation

regime (Ω κ). Although we expect to have unity fringe visibility under these condition,

the measured fringe visibilities are highly deviated from unity. This “apparent” reduction

in fringe visibility is primarily due to spectral diffusion, which leads to vary the photon

flux at the input of the beamsplitter. The two-photon interference visibility, on the other

hand, is expected to be unity for any Ω, but is also reduced in the presence of spectral

diffusion. For large Ω, two-photon visibility is further affected by the finite time resolu-

tion of the detectors, which makes the value of g(2)(0) seems larger due to the reduction

in width of the antibunching dip. The presented theoretical model relies on a number of

assumptions. In particular, it ignores anisotropic field fluctuations and any contributions

due to interferometer misalignments. It also neglects the incoherent contribution due to

phonon scattering (5% at 4 K). In general, therefore, the observations of this study strongly

suggest the importance of controlling spectral diffusion for future applications.

32

Page 49: Resonant Light Scattering from Semiconductor Quantum Dots

CHAPTER 4

RESONANT INTERACTION BETWEEN A TWO-LEVEL SYSTEM ANDA PERIODICALLY-PULSED LASER

4.1 Introduction

The interaction between a two-level system with an monochromatic near-resonant laser

is a well studied topic in quantum optics. However, on the other hand pulsed excitation

plays a key role in quantum information science, in particular for generating on demand

photons [46], and for quantum state preparation [47]. Rabi oscillations [48–50] and their

damping [51] have been studied under pulsed excitation in great detail. However, the

spectral properties of the scattered light have so far primarily been studied for the case

of monochromatic excitation. Although, bichromatic resonant light scattering has been

studied in atomic beams [52–56] and more recently in semiconductor QDs [57], to the

best of our knowledge, the measurement of the spectrum of the light scattered by a two-

level system under strong pulsed excitation has, surprisingly, not been measured. Earlier

theoretical work has also focused more specifically on the interactions involving a pulse

train [58, 59]. Resonance fluorescence under high-frequency modulation has also been

studied theoretically [55] as well as experimentally [60, 61].

Here we study the correlations of the light scattered by a quantum dot under pulsed-

excitation. A mathematical model will be presented at the beginning of the chapter to

describe the interaction between a two-level system and a periodically-pulsed laser or a

frequency comb. Then it will discuss the experimental observation of spectrum and second-

order correlation function under long and short pulse excitations. Pulse area dependent

Rabi oscillations will also be presented. Finally, these observations will be compared with

the theoretical model.

33

Page 50: Resonant Light Scattering from Semiconductor Quantum Dots

4.2 Theory

A general formalism for calculating spectrum and the second-order correlation function

of the scattered light under multichromatic excitation was proposed using the matrix trun-

cation method [16]. In this section, I will summarize the major definitions that are useful

to understand the interaction between the QD and a frequency comb.

We describe the incoming pulse train by its constituent monochromatic waves, so that

the applied electric field can be written as

E(t) =1

2e−iωst

p∑n=−p

Ene−inδt + c.c. (4.1)

where ωs(n=0) is the central frequency, and En is the amplitude of the nth mode which

can be related to its Rabi frequency Ωn = µhEn. Here µ represents the magnitude of the

transition dipole moment. The frequency difference δ (in rad/s) between adjacent waves

coincides with 2π times the pulse repetition rate (in Hz).

In general the temporal envelope of the incoming field is given by the function

E(t) =∞∑n=0

Es(t−2πn

δ) (4.2)

with

Es = E0 sech(1.76t/tp) (4.3)

where tp is the FWHM of the temporal intensity profile of a single pulse. Mode-locked laser

oscillators commonly produce such pulse trains. For such a pulse train, the time-bandwidth

product is tp∆νp ≈ 0.315, where ∆νp is the FWHM of the pulse’s power spectrum in Hz.

We further assume that the cavity is stable over the measurement duration, and thus the

applied electric field can be described by

E(t) = E(t)cos(ωst) (4.4)

34

Page 51: Resonant Light Scattering from Semiconductor Quantum Dots

Time (ns)

(b)(a)

|E|2

2π/δ

t

Frequency (GHz)

δ/2π

0 2-2 0 10 20

∆ν p p

Figure 4.1. (a) Spectrum of the frequency comb represented by Eq. (4.1). (b) Temporalprofile of (a).

comparing Eqs. (4.1) and (4.4), En can be written as

En =δ

∫ π/δ

−π/δEs(t)einδtdt (4.5)

with φn=0. The laser’s power spectrum is then given by

Slaser(ν) ∝ 2π

p∑n=−p

|En|2δD(ν − ωs − nδ) (4.6)

where ν is the detection frequency. Furthermore, peak Rabi frequency ΩR = |µE0|/h

quantifies the strength of the interaction between QD and the laser pulse. In addition,

input pulse area, which describes maxima or minima of the population of the two-level

system after passage of the pulse, is often used in pulse excitation experiments, and defined

as

θ =µ

h

∫ ∞−∞Es(t)dt = ΩR

πtp1.76

(4.7)

A full description of the theoretical model will be discussed in Appendix D.

4.3 Simulations of spectrum and second-order correlation function

In general, one can use the above theoretical model to study the scattered light spec-

trum and second-order correlation function for different pulse widths and laser repetition

rates. This will provide a elegant way to understand the fine details of the interaction

35

Page 52: Resonant Light Scattering from Semiconductor Quantum Dots

d

e

(g)

(h)

(f )

Figure 4.2. (a) Applied field intensity in time domain. (b) Intensity of total (red), coherently(blue), and incoherently (green) scattered light, calculated with (solid lines) and without(dashed lines) spectral diffusion. (c)-(e) Calculated spectra of the scattered light for theapplied field shown in Fig. 4.2(a) for range of Rabi frequencies as indicated. Blue and redtraces correspond to the spectra calculated with and without spectral diffusion, respectively.(f)-(h) Corresponding second-order correlation functions under the same conditions, forrange of Rabi frequencies, calculated with (blue) and without (red) spectral diffusion.

process under different excitation regimes. For example, one can expect to have different

characteristics of the scattered for light laser repetition rates under and above the radiative

decay rate of the two-level system being studied [16]. Figure 4.2 illustrates the simulated

results for a specific case where δ/2π=100 MHz and tp=800 ps. Figure 4.2(a) represents

the electric field intensity as a function of time. Obviously, these pulses are separated by

36

Page 53: Resonant Light Scattering from Semiconductor Quantum Dots

100 MHz in the frequency domain. As shown in Fig. 4.2 (b), frequency comb approach

allows us to calculate the intensity of total, coherently, and incoherently scattered light as a

function of input pulse area with and without the effect of spectral diffusion. Clearly, each

component exhibits pulse area dependent Rabi oscillations with different phase. We also

studied the scattered light spectra for range of laser repetition rates and pulse durations.

Figure 4.2(c)-(e) shows the simulated spectra for 100 MHz repetition rate and 800 ps pulse

duration for range of peak Rabi frequencies. Even though one might expect to have a

broader version of Mollow triplet, actually spectrum has a rich structure containing many

number of spectral features. When the Rabi frequency is small (Fig. 4.2(c)), most of the

light is coherently scattered in the form of δ- function like sharp peaks. As the Rabi fre-

quency is increased, spectrum evolves very similar to those studied under monochromatic

excitation. Sidebands appear and they are moving away from the central frequency as the

Rabi frequency is further increased. As can be seen in Fig. 4.2(e) in addition to primary

sidebands, secondary sidebands also visible at larger Rabi frequencies. This is one of the

major deviations from the Mollow triplet.

4.4 Experiment

In the present study, we used the same QD sample and the excitation/detection geome-

try discussed in the previous chapters. Now the excitation source is a passively mode-locked

Ti:sapphire laser with a fixed repetition rate δ/2π=188 MHz. The pulse width, tp, was

controlled by intracavity etalons of different thickness. The scattered light was analyzed

using a Fbry-Perot interferometer with a resolution of 20 MHz along with single photon

counting detector. In order to study the photon statistics of the scattered light, a typical

HBT setup was used.

4.5 Scattered light spectrum

It turns out from the theory that the spectral properties of the scattered light is strongly

depend on the bandwidth of the excitation laser pulses. Therefore, we studied the scattered

37

Page 54: Resonant Light Scattering from Semiconductor Quantum Dots

light spectrum for three different bandwidths (∆νp=0.56 GHz, 3.9 GHz, and 9 GHz), but

with the same repetition rate. In the following subsections, I will summarize the results

under long and short pulse excitation.

4.5.1 Long pulse excitation

Linear Logarithmic

(a) (b)

(c)

Time τ (ns)0 20 3010 40 50

Frequency (GHz)

∆νp=0.56 GHz

Frequency (GHz)Frequency (GHz)

|E|2 (a

rb. u

nit

s)

g(2

) (τ) (

arb.

uni

ts)

θ=1.6π

10-1

103

101

10-1

103

101

10-1

103

101

tp=560 ps

ΩR

2π=0.8 GHz

θ=3.8π

ΩR

2π=1.9 GHz

θ=6.0π

ΩR

2π=3.0 GHz

-4 -2 0 2 4

Scat

t. lig

ht in

tens

ity (a

rb. u

nits

)

-4 -2 0 2 4 -4 -2 0 2 4

Figure 4.3. (a) Laser spectrum for tp= 560 ps. (b) Corresponding second-order correlationfunction. (c) Experimental (blue) and theoretical (red) spectra of the light scattered by theQD for tp=560 ps, represented on a linear (left) and logarithmic (right) ordinate scale, fora range of peak Rabi frequencies. Black and green arrows indicate the positions of primaryand secondary sidebands, respectively. Shaded area represents the incoherently scatteredlight.

38

Page 55: Resonant Light Scattering from Semiconductor Quantum Dots

Figure 4.3(a) displays the laser spectrum for 560 ps pulse width recorded using a Fabry-

Perot interferometer and an oscilloscope. By applying active feedback to the laser cavity

length we maintained the laser repetition rate fixed and thus these spectra are actually

frequency combs. In the time domain, laser pulses are separated by the inverse of the

frequency-comb teeth separation, as seen in the second-order correlation measurements of

Fig. 4.3(b). Figure 4.3(c) summarizes the high resolution scattered light spectra under

long pulsed excitation, for a range of Rabi frequencies, i.e., laser intensities. Logarithmic

representation helps to identify the subtle features in the spectra. As can be seen in

Fig. 4.3(c), the measured spectra evolve similar to those studied under monochromatic

excitation. This is not surprising since the spectral diffusion broadened bandwidth of InAs

QD is on the order of 1 GHz, and in this case it matches with that of the frequency comb,

which contains only a few modes. Specifically, dominant sidebands can be seen at larger

Rabi frequencies near ωs ± ΩR and they are moving away from the central frequency as

the Rabi frequency is further increased. However, in addition to primary sidebands (black

arrows) secondary sidebands (green arrows) are also visible, which is a major deviation

from Mollow’s theory where only two sidebands are visible at high laser intensities.

4.5.2 Short pulse excitation

When the pulse duration is short (∆νp > 1 GHz), frequency comb contains large number

of modes as seen in the Fig. 4.4(a). A close-up view of Fig. 4.4(a) is shown in Fig. 4.4(b)

in which comb lines are under-sampled. We measured the scattered light spectra under

the excitation of short pulses tp=35 ps (∆νp=9 GHz), and the corresponding scattered

light spectra are shown in Fig. 4.4(c). Clearly, coherently scattered light, which appears

as sharp peaks in the scattered light spectrum, dominates under weak excitation regime

[upper panel of Fig. 4.4(c)]. On the other hand, the intensity of incoherently scattered

light (shaded area) starts from a minimum, reaches to a maximum and then decreases

again as the Rabi frequency is increased. Sidebands are no longer visible even for larger

Rabi frequencies. Under π-pulse excitation almost all light is scattered incoherently (center

39

Page 56: Resonant Light Scattering from Semiconductor Quantum Dots

-4 -2 420

Scat

t. lig

ht in

tens

ity (a

rb. u

nits

)

Frequency (GHz)

|E|2 (a

rb. u

nits

)(c)

Frequency (GHz) Frequency (GHz)(d)

(a) (b)

-4 -2 420Frequency (GHz)

∆νp=9 GHztp=35 ps

θ=0.4π

ΩR

2π =3 GHz

θ=π

ΩR

2π =8 GHz

θ=1.5π

ΩR

2π =12 GHz

θ=0.4π

ΩR

2π =1.25 GHz

θ=0.7π

ΩR

2π =2.5 GHz

θ=1.6π

ΩR

2π =5.5 GHz

tp=35 ps (∆νp=9 GHz) tp=80 ps (∆νp=3.9 GHz)

-4 -2 0 2 4-10 -5 0 5 10

Figure 4.4. (a) Laser spectrum for tp= 35 ps. (b) Zoomed-in view of (a). (c) Experimental(blue) and theoretical (red) spectra of the light scattered by the QD for tp=35 ps for arange of peak Rabi frequencies. (d) Same as (c) but for tp=80 ps. Shaded area representsincoherently scattered light.

panel of Fig. 4.4(c)), as expected from the theory. A similar evolution was observed under

the excitation of intermediate bandwidth ∆νp=3.9 GHz (tp=80 ps)as seen in Fig. 4.4(d).

4.6 Rabi oscillations

Rabi oscillations play a vital role in resonant interaction of radiation with two-level

systems and have been actively studied in different quantum systems, including supercon-

40

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0

50

100

150

0 2 4

Pulse area θ/π

Scat

t. ra

te (M

Hz)

6

Total scatt. rateIncoh. scatt. rateCoh. scatt. rate

(c)

0

50

100

150

0 2 4

Scat

t. ra

te (M

Hz)

(a) (b)

Pulse area θ/π

1 2

50

100

00

Scat

t. ra

te (M

Hz)

Figure 4.5. (a) Intensity of total (red), coherently (blue) and incoherently (green) scatteredlight, measured (markers) and calculated (solid lines) for tp=560 ps. (b) Same as (b) buttp=80 ps. (c) Same as (a) but for tp=35 ps.

ducting qubits and semiconductor QDs. However, so far most of the studies to investigate

Rabi oscillations have been performed by means of measuring total scattered light inten-

sity. As discussed below, frequency comb approach allows us to study Rabi oscillations of

total, coherently and incoherently scattered light. Maintaining phase coherence between

subsequent laser pulses during the measurement duration, i.e., keeping the laser cavity

length fixed, ensures that the coherently scattered light appears in the form of sharp peaks

in the scattered light spectra, and is therefore easily separated from the incoherently scat-

tered light. Here we use numerical low pass filtering of the raw spectra to first remove

the coherently scattered light component, the remainder being the incoherent component.

Figure 4.5 displays the evolution of coherently (blue), incoherently (green) and total (red)

spectrally- integrated intensities as a function of input pulse area for the three bandwidths

studied. Each component exhibits pulse area dependent Rabi oscillations but with differ-

ent phases as predicted by the theory. When the pulse width is large, incomplete Rabi

rotations are visible [Fig. 4.5(a)] and eventually they become more complete as the laser

pulse width is decreased as seen in Fig. 4.5(c).

41

Page 58: Resonant Light Scattering from Semiconductor Quantum Dots

4.7 Second-order correlation functions

To understand the photon statistics of the scattered light under pulsed excitation, we

also measured the second order correlation function for two different bandwidths, for a

range of Rabi frequencies. This data is displayed in Fig. 4.6. When the pulse duration is

large [Fig. 4.6(a)], second-order correlation functions actually resemble those found under

bichromatic excitation [52, 53, 57]. This is perhaps not surprising, given that in this case

only a few modes of the laser interact with the QD. With a much shorter pulse width,

on the other hand, subsequent pulses become well-separated and pulse-integrated photon

antibunching is pronounced at low excitation power. In that regime, the correlations are

similar to those measured under non-resonant pumping. However, as the Rabi frequency

is increased the second-order correlation function changes and photon antibunching is pro-

nounced only near τ = 0 [bottom panel of Fig. 4.6(b)]. Solid red lines represent simulated

traces using the theory given in [16].

(a) (b)

-20 -10 0 10 20 -20 -10 0 10 20Time τ (ns)Time τ (ns)

0

0.5

1.0

0

0.5

1.00

0.5

1.0

0

0.5

1.0

g(2) (τ

)

g(2) (τ

)

tp=35 ps (∆νp=9 GHz)tp=560 ps (∆νp=0.56 GHz)

θ=1.5π

ΩR

2π=0.75 GHzθ=0.38π

ΩR

2π=0.3 GHz

θ=4π

ΩR

2π=2 GHzθ=1.62π

ΩR

2π=13 GHz

Figure 4.6. Experimental (blue) and theoretical (red) second-order correlation functionsfor (a) tp=560 ps, and (b) tp=35 ps.

4.8 Summary and discussion

In this part of the study, we have measured the spectrum and the second-order cor-

relation function of the light scattered from an InAs QD interacting with a periodically

42

Page 59: Resonant Light Scattering from Semiconductor Quantum Dots

pulsed laser. For longer pulse widths multiple sidebands are observed, moving away from

the central frequency with increasing excitation power. As the pulse width is decreased,

the formation of Rabi oscillations is visualized via the extracted intensity of the total, co-

herently and incoherently scattered light. At low excitation power, the light is scattered

mostly coherently, implying phase coherence between the laser and the scattered wave

packet. As such, the resulting emission becomes a highly periodic single photon emission

with potential use in single photon metrology, where minimal “jitter” between photons is

important [17].

43

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CHAPTER 5

LASER-INDUCED FLUORESCENCE FROM N+2 IONS GENERATED BY

A CORONA DISCHARGE IN AMBIENT AIR

5.1 Introduction

Recent improvements in laser performance and photon counting technology opened the

possibility of investigating a wide rage of potential applications, such as stand off detection

of fissile materials using LIF. It has been suggested that positive ions generated in the

presence of radioactive materials can survive for minutes before recombining or forming

radiolytic compounds such as NOx or O3 [62]. N+2 cation is of interest as an indicator

for the presence of ionizing materials due to its strong absorption band in the near-ultra

violet regime. However, although the basic spectroscopy of N+2 ions is well-known [63, 64],

previous measurements have all been performed under controlled conditions and direct LIF

measurement of N+2 in ambient air has been lack. For example, LIF measurements based

on B 2Σ+u −X 2Σ+

g transition of N+2 have been reported at low pressures (on the order of

1 Torr) using electric discharge [65, 66], electron beams [67, 68], and synchrotron radiation

[69]. More recently, cavity ring-down absorption spectroscopy has been used to detect N+2

ions in an atmospheric pressure nitrogen glow discharge with an estimated detection of

7×1010 cm−3 [70].

In this chapter, I will discuss the measurement of LIF from N+2 ions via the B 2Σ+

u −X2Σ+

g band system in the near-ultraviolet. Spectral and temporal properties of the LIF

signal will be presented in great detail. Furthermore, the limitations for the detection of

N+2 in ambient air will be discussed. Finally, N+

2 number density will be estimated using

two independent methods.

44

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Figure 5.1. (a) Schematics of the experimental setup. Corona discharge occurs betweenpositively biased Tungsten needle and a ground plate. The collected signal was sent toa PMT or a CCD for the analyzing. (b) N+

2 energy level diagram. When the band near391.4 nm is excited, fluorescence is detected near 427.5 nm.

5.2 Experimental setup

Our experimental setup is illustrated in Fig. 5.1 (a), where the electric discharge takes

place inside the metal chamber. The chamber can either be evacuated and filled with any

other gas (operating at pressures between ≈ 1 and 760 Torr) or opened to ambient air. This

allows us to perform pressure dependent measurements as well as ambient measurements.

In order to generate N+2 ions, we applied a high DC voltage between a positively biased

tungsten needle and a ground plate, placed nearly 2 cm away from the tip of the needle.

The generated ions were excited by a dye laser, and the fluorescence signal was collected

perpendicular to both excitation and needle axes, as shown in Fig. 5.1 (a). A detailed

description of the dye laser, excitation and detection system are provided in the following

subsections.

45

Page 62: Resonant Light Scattering from Semiconductor Quantum Dots

Chamber

YAG laser

Beam splitter

Dye oscillator

Cylindrical lens

(a) (b)

Back mirror

Lens

Tuning mirror

Dye amplier

Diraction grating

Figure 5.2. (a) Photograph of the experimental setup. Fluorescence signal was collectedin a direction perpendicular to both the excitation and the needle axes. (b) Dye oscillatorcontaining a flow cell, 3600 grooves/mm diffraction grating, and two mirrors.

5.2.1 Dye laser system

The excitation source was a frequency-tripled YAG (6 ns pulse duration, 10 Hz rep-

etition period) pumped dye laser with two stages (a dye oscillator and a dye amplifier).

The dye circulator contained exalite 389 and exalite 398 dye in a 50/50, 10 Molar mix-

ture in P-Dioxane. The dye oscillator consists of a flow cell, two mirrors and a 3600

grooves/mm diffraction grating, arranged in Littman-Matclaf configuration to get a very

narrow linewidth [71]. The dye oscillator itself produced 0.5 mJ per pulse, and can be

continuously tunable from 388 nm to 398 nm. A dye amplifier system was also introduced

to enhance the output power. In order to facilitate the amplification process, the pump

laser path length was adjusted such that it reached to the flow cell before the oscillator

output beam, as shown is the photograph in Fig. 5.2 (a) After passing though the dye am-

plifier, laser typically provided a 5 mJ energy per pulse near 391 nm with a time-averaged

bandwidth of few pm. Moreover a LabView controlled motor was attached to the tuning

mirror to automate the tuning process with fairly even steps. The laser frequency was

monitored using a wavemeter (Burleigh WA-5500) with an accuracy of 2 pm.

46

Page 63: Resonant Light Scattering from Semiconductor Quantum Dots

5.2.2 Excitation/detection methods

The dye laser was focused into the discharge region, typically 1 mm away from the

needle tip, using a lens with 45 mm focal length. The polarization of the excitation laser

was set along the needle axis to maximize the fluorescence signal, collected perpendicular

to both the excitation and the needle axes. The LIF signal was collected using a lens and

a multimode optical fiber (400 µm diameter, numerical aperture NA=0.39). For some of

the measurements a multimode fiber with a smaller core diameter (62.5 µm) was used. The

diameter and the focal length of the collection lens were 25 mm and 45 mm, respectively.

The collection numerical aperture was ≈ 0.15. The collected LIF signal was then filtered

with a 405 nm long-pass filter to block out any contribution from direct laser scattering and

analyzed either with a photomultiplier tube (Hamamatsu R928) equipped with a bandpass

filter around 425 nm or with an intensified charge coupled device (ICCD) detector (Andor

iStar DH734 Gen II) fitted to a grating spectrometer (Andor Shamrock 303i, 2400 line/mm

grating). The ICCD and spectrometer allowed to record time-gated emission spectra with

a temporal window as small as 2 ns.

5.3 Spectrum of the corona discharge

The spectrum of the corona discharge itself is shown in Fig. 5.3(a), recorded using a

CCD and a grating spectrometer at 4 Torr of N2. Equally spaced emission bands due to

N2 were identified in addition to the dominant N+2 emission bands. In particular, a band

near 427.5 nm is due the spontaneous emission of N+2 ions. Since the spontaneous emission

rate is proportional to the number of ions available, intensity of this band can be used as a

measure of the N+2 ions generated during the discharge. A similar measurement, recorded

at 50 Torr of N2 [Fig. 5.3(b)], revealed that N2 bands are dominated over N+2 bands. This

trend continued to grow at larger pressures of N2, and N+2 bands were barely visible at 300

Torr of N2. This suggested that the available N+2 ion concentration decreases drastically

with the increase of N2 pressure. Therefore, in order to check the feasibility of detecting

LIF of N+2 ions, we first performed the measurements at low pressure of N2.

47

Page 64: Resonant Light Scattering from Semiconductor Quantum Dots

1 Torr

1N(0

,0)

2P(2

,5)

1N(0

,1)

2P(1

,4)

2P(0

,3)

Wavelength (nm)

Inte

nsity

(arb

. uni

ts)

Wavelength (nm)

Inte

nsity

(arb

. uni

ts) 50 Torr

2P(2

,6) 2P

(2,6

)

2P(0

,3)

2P(1

,4)

2P(2

,5)

2P(1

,5)

2P(3

,7)

390 400 410 420 430 390 400 410 420 430

Figure 5.3. (a) Spectrum of the corona discharge at 1 Torr of N2. Emission lines due toN+

2 ions dominate over those due to N2. (b) Same as (a), but for 50 Torr of N2, in whichN+

2 emission lines are no longer dominant.

5.3.1 Excitation spectrum of N+2

Since the measured fluorescence intensity is directly proportional to the absorption

cross section of the excited molecules, LIF provides a direct way to reproduce excitation

spectrum. The excitation spectrum of (ν′′

= 0 )→(ν′

= 0) branch of N+2 ions at 4 Torr of

N2 is shown in Fig. 5.4(a). It was recorded by changing the laser’s wavelength from 389.5

nm to 391.5 nm in 10 pm steps. In order to obtain this spectrum, three oscilloscope traces

(discharge only, laser only, both laser and discharge) were recorded at each wavelength

and the net effect from the LIF signal was calculated. As can be seen in Fig. 5.4(a), at

this resolution, the R-branch (∆N = 1) is fully resolved into rotational transitions but

the P-branch (∆N = −1) is not. Where N is the rotational quantum number. A similar

measurement performed at 140 Torr of N2 with 5 pm step size, is shown in Fig. 5.4(b). At

this pressure of N2, discharge is not a glow discharge, but a streamer discharge. In contrast

to Fig. 5.4(a), center of the R-branch in Fig. 5.4(b) has now well-resolved, and has shifted

towards the lower wavelengths, indicating a higher gas temperature.

5.3.2 LIF at different pressures of N2

The temporal decay of the fluorescence signal was studied at different pressures of N2

with the help of a PMT equipped with a SPCM. In particular, here SPCM was used to

48

Page 65: Resonant Light Scattering from Semiconductor Quantum Dots

(a) (b)

389.5 390.5 391.5Wavelength (nm)

389.5 390.5 391.5Wavelength (nm)

Inte

nsity

(arb

. uni

ts)

Inte

nsity

(arb

. uni

ts)4 Torr 140 Torr

R-branch

p-branch

Figure 5.4. (a) LIF excitation spectrum of N+2 at p =4 Torr of N2. R and P branches can be

clearly identified. (b) Same as (a) but for p =140 Torr of N2 and with a better resolution(5 pm).

2 4 6 8 10 0 20 40 60 80 100 2

6

10

14

Time (ns) Pressure (Torr)

(a) (b)

Inte

nsity

(arb

. uni

ts)

Dec

ay ra

te 1

07 s-1

Figure 5.5. (a) Fluorescence decay for p=5 Torr (red) and p=8 Torr (green). Dashed lines(black) are the exponential fits to the decay curves, which are used to extract the decayrate at each pressure. (b) Fluorescence decay rate versus pressure. Solid red line is alinear-fit to the data.

achieve a better time resolution ∼ 100 ps. We fixed the laser excitation at 391.42 nm and

histogrammed the collected fluorescence photons based on their arrival time. The collected

data was then fitted with an exponential function to extract the decay rate at each pressure

as shown in Fig. 5.5(a). Even at the lowest pressure (≈ 3.5 Torr), the measured decay

time is already significantly shorter than the radiative decay time of 62 ns. As can be seen

in Fig. 5.5(b), it decreases further as the pressure is increased and is roughly equal to the

laser pulse duration at p ≈ 12 Torr. Thus, the fluorescence is rapidly quenched with a

decay rate that can be estimated by extrapolation to γd ≈ 1010/s. Rapid increase of the

decay rate with pressure limits the detection at elevated pressures of N2.

49

Page 66: Resonant Light Scattering from Semiconductor Quantum Dots

5.4 Time-resolved measurements of LIF

Spectral properties of the collected signal were analyzed using a gated ICCD (0.4 nm

resolution) equipped with a 405 nm long pass filter to remove the background laser scatter-

ing. The data was collected with a 6 ns gate width and no delay with respect to the laser

pulse arrival to exclude photons not generated by the LIF process. We study the signal

under different combinations of laser, discharge for both on and off resonance (392 nm)

laser excitation, as shown in Fig. 5.6(a). When the discharge is off, and the laser is tuned

to a resonance with a transition in the first negative band of N+2 (λlaser = 391.41 nm), the

only feature seen is at 430.7 nm due to N2 Raman scattering, not fluorescence. It is shown

clearly in the blue (top) trace of Fig. 5.6(a). When corona discharge is also turned on,

two additional peaks near 426 nm and 427.5 nm, corresponding to the R-branch and P-

branch of N+2 LIF, respectively, arise [red trace of Fig. 5.6(a)]. These two additional peaks

disappear for off resonance laser excitation while a red-shift of the spontaneous Raman

signal of N2 is observed, as seen in the green (second from bottom) trace of Fig. 5.6(a).

The discharge itself produces a light violet glow, but does not have any spectral features

measurable on the nanosecond time scale, as displayed in the black (bottom) trace in Fig.

5.6(a). In order to further verify that the emitted light near 427 nm is indeed associated

N2+ LIF N2 Raman

Wavelength (nm) Wavelength (nm)420 432428424 420 432428424

Inte

nsity

(arb

. uni

ts)

Inte

nsity

(arb

. uni

ts)

(a) (b)

Unpolarized

Discharge only

Discharge+ laser (392.00 nm)

Discharge+ laser (391.42 nm)

Laser only (391.42 nm)

Figure 5.6. (a) N+2 LIF measurement in ambient air for on-line and off-line excitation.

Both N+2 fluorescence and N2 Raman signals are visible when the laser is on-resonance.

(b) Polarization properties of the collected signal.

with N+2 LIF, we have also measured its polarization, relative to the laser polarization.

We observed that the intensity of the emitted light near 427 nm dropped to half of its

50

Page 67: Resonant Light Scattering from Semiconductor Quantum Dots

un-polarized intensity with the introduction of the polarizer. This rules out other origins

for the observed signal at 427 nm, such as resonance Raman scattering. Also, we studied

the collected signal with different gating delays, durations and observed the second positive

bands of N2 for delayed detection. Therefore, time- gating capability of is mandatory to

differentiate N+2 LIF from N2 fluorescence.

0 2 3 410

2

4

Conc

entr

atio

n (1

09 cm

-3)

Wavelength (nm)420 432428424

Current (µA)

Inte

nsity

(arb

. uni

ts)

2P(0

,4)

2P(1

,5)

2P(2

,6)

tw=134 nstd=9 ns

tw=9 nstd=0 ns

(a) (b)

Figure 5.7. (a) Delayed emission of N2. Therefore, time-gated detection is necessary todistinguish the fluorescence signal from the delayed emission of N2. (b) Measured N+

2 ionconcentration using electric measurement.

5.5 Estimation of the N+2 concentration in ambient air

In order to estimate the N+2 ion concentration in ambient air we made use of the

spontaneous Raman scattering signal (∆ν = 2331 cm−1) simultaneously observed with the

LIF signal. This approach is convenient as the measured ion concentration is independent of

the geometric excitation parameters and photon collection efficiencies. We performed this

estimation using the information extracted via (ν ′′=0, N ′′=6)→(ν ′=0, N ′=7) transition of

N+2 near the center of the R-branch at 390.508 nm. Figure 5.7(b) shows the estimated N+

2

ion concentration as a function of the current measured between the needle and the ground

plate. 36% uncertainty is expected in the estimated concentration due to the uncertainties

of the parameters involved such as extrapolated decay rate, excitation linewidth and gas

temperature. We further estimated the positive ion concentration as ≈1011 cm−3 using the

drift velocity approach.

51

Page 68: Resonant Light Scattering from Semiconductor Quantum Dots

5.5.1 Using fluorescence signal

In order to estimate the N+2 ion concentration, we made use of N2 spontaneous Raman

scattering signal simultaneously observed with the LIF signal. This approach allows us to

estimate concentration measurement independent of geometric factors such as collection

solid angle, collection efficiencies. The differential cross section for the spontaneous Raman

signal, (dσ/dΩ)RS,N2=4.3(10)−31 cm2/sr for a laser wavelength of λlaser =488 nm, is well-

documented [72].

The ratio of N+2 ion concentration, NN+

2, and N2 concentration, NN2 , can be written as [73]

NN+2

NN2

=SLIF,N+

2

SRS,N2

[8πλR3λL

] (dσdΩ

)RS,N2[

σa4π

Aeγd

] (5.1)

where,

SLIF,N+2

-magnitude of LIF signal from N+2

SRS,N2-spontaneous Raman emission from N2

λL-wavelength of the excitation laser

λR-wavelength of the fluorescence signal

σa-absorption cross section for N+2

Ae-Einstein A-coefficient for the emission

γd-LIF decay rate

According to our definition, λL = 391.4 nm and λR = 428 nm. For this analysis, we are

particularly interested in (ν ′′=0, N ′′=6)→(ν ′=0, N ′=7) transition near the center of the

R-branch at λa=390.508 nm (wavelengths quoted throughout are in vacuum). Einstein

coefficient for the emission via (ν ′ = 0 → ν ′′ = 1) transition has been calculated as

Ae = 3.71(10)6/s [74]. The decay rate of the LIF signal at ambient pressure can be found

as, γd ≈ 1010/s, by extrapolating the decay rates measured at low N2 pressures. We also

need to calculate the absorption cross section for the above selected rotational line. In

general absorption cross section can be written as [75]

52

Page 69: Resonant Light Scattering from Semiconductor Quantum Dots

σa =1

4(g′/g′′)λ2

a

Aa

∆ωeρI ≈ 2.2(10)−15cm2 (5.2)

where

Aa-Einstein A-coefficient for the absorption

g′(g′′)-degeneracy of the upper (lower) states

∆ωe-excitation linewidth of the specific transition

ρI-fractional population of the initial vibrational-rotational state

Wavelength (nm)

Inte

nsity

(arb

. uni

ts)

390.48 390.54390.52390.50

Figure 5.8. Linewidth extraction of individual transition, (ν ′′=0, N ′′=6)→(ν ′=0, N ′=7),at the center of the R-branch in Fig. 5.4(a). Solid-red line is a Lorentzian fit to theexperimental data.

For (ν ′′=0, N ′′=6)→(ν ′=0, N ′=7) transition, Aa = 1.14(107)/s [74], and g′/g′′ =15/13.

Moreover, the excitation linewidth in ambient air, ∆ωe/2π = 30 GHz (∆λ ≈ 15 pm), has

been calculated using a Lorentzian fit to the fluorescence data as shown in the Fig.5.8.

Finally, ρI can be calculated via the Boltzmann fractions [76],

ρI =hcB

kT(2j + 1)e−Bj(j+1) hc

kT (5.3)

where

h-Planck’s constant

53

Page 70: Resonant Light Scattering from Semiconductor Quantum Dots

c-speed of light

B-rotational constant

j-rotational quantum number

k-Boltzmann’s constant

T -gas temperature

Furthermore, B can be calculated from the relation B = h8π2cIe

as 200 m−1. Where Ie is

the moment of inertia of the molecule. Assuming that the temperature of the air inside the

chamber is near 300 K. By substituting these quantities into the Eq. (5.2), one can obtain

an estimation for the absorption cross section, σa ≈ 2.2(10)−15cm2. Using Eq. (5.1) with

the numbers given above and taking the account of the λ4 dependence of Raman scattering

cross section, we findNN+

2

NN2

≈ 1.4(10)−10SLIF,N+

2

SRS,N2

. (5.4)

Therefore, Eq. (5.4) can be used to calculate the N+2 ions per unit volume in the chamber

when the intensity ratio between N+2 fluorescence signal and N2 spontaneous Raman signal

is known. This ratio is readily available in the traces shown in Figure. We are also inter-

ested in the uncertainty associated with Eq. (5.4) due to uncertainties in gas temperature,

excitation linewidth and extrapolated decay rate.

5.5.2 Using electric current measurement

We also estimate the positive ion concentration inside the chamber, N+, using electric

current via drift velocity, vD, approach. The concentration of positive ions can be approx-

imated as N+ = I/AvDe

. Where A is the cross-sectional area of the ion beam between needle

and the ground plate, and I is current through the ground plate. Obviously, e represents

the electronic charge. We also use the drift velocities of positive ions in air or N2 that have

been measured as a function of pressure and electric field [77–79]. For our measurements,

applied voltages are typically 5-6 kV at ambient pressure. Therefore, once the current

through the ground plate is known, above formula can be used to calculate positive ion

concentration for a particular current. We obtain N+ ≈ I × 6(10)10, with I in µA. Figure

54

Page 71: Resonant Light Scattering from Semiconductor Quantum Dots

5.7 (b) shows both the N+2 ion concentration and the positive ion concentration as a func-

tion of measured current through the ground plate. The former has been obtained from

the measured LIF data as discussed previously, and the latter has been inferred from the

electrical measurement. These two concentrations differ by several orders of magnitude,

as expected. It can be understood that N+ includes the contributions from many other

ions such as N+, N+3 , N+

4 , O+, O+2 , and NO+ which may be created during the corona

discharge.

5.6 Summary and discussion

We have measured LIF of N+2 ions generated by corona discharge in ambient air. The

rapid increase of the LIF decay rate with pressure makes it difficult to detect the fluores-

cence signal at ambient air. The measured N+2 ion concentration is consistent with the

electric measurement and this is a reasonable estimate considering that it can be done

with OH [80]. We believe that it may be possible to detect N+2 ions with lower concentra-

tions, 106 cm−3 using LIF with increased laser power and collection efficiency along with

improved signal to noise ratio.

55

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CHAPTER 6

CONCLUSIONS AND OUTLOOK

The main goal of this study was to investigate the importance of the solid state envi-

ronment on the optical properties of a resonantly driven QD. Embedding QDs in a planar

microcavity along with orthogonal excitation/detection approach provided an efficient way

to collect the scattered light while suppressing background laser scattering significantly.

The measured background-free high resolution scattered light spectra compared with the-

ory show that QD environment plays an major role. In particular, spectral diffusion leads

to broaden the excitation spectra and results in a larger fraction of coherently scattered

light even at intermediate Rabi frequencies. Additionally, this scattered light fluctuate

randomly on a time scale longer than the QD upper state life time, but shorter than the

measurement time. Regardless of these imperfections, we experimentally observed one-

photon and two-photon interference with relatively high fringe visibilities from the light

scattered by two nearby QDs in the same sample. The measured fringe visibilities were

highly affected by spectral diffusion, which induced variations in photon flux at the beam-

splitter, and can be explained using a most general Stark shift based model. Furthermore,

the interaction between a two-level system and a periodically pulsed laser provided a deep

insight into resonant light-matter interactions beyond the existing knowledge. Under these

conditions, scattered light spectrum contains many number of spectral features that are

not visible in the conventional Mollow triplet. Among other interesting observations, pulse

area dependent Rabi oscillations were observed for each component (total, incoherently, co-

herently) of the scattered light with different phase. Further investigations will be required

to understand the precise origin of this observation.

56

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Taken together, this work resulted in a number of key findings. Solid-state environ-

ment significantly modifies the optical properties of a resonantly-driven QD and strongly

limit their use in practical applications, in particular those involving many different QDs.

Therefore, it is important to find a way to stabilize QD linewidth. One possibility might

be to implement an active stabilization mechanism based on a secondary measurement

made on a neighboring QD. It may help to overcome some of the limitations introduced

by spectral diffusion. However, on the other hand, phonon scattering still prevents their

use at elevated temperatures, which will be the major challenge for the realization of QD

based devices at room temperature.

57

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REFERENCES

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APPENDICES

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Appendix A Copyright permissions

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Appendix A (Continued)

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Appendix A (Continued)

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Appendix A (Continued)

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Appendix B Theory of two-level system interacting with a near-resonant laser

The system we consider here is a two-level system with ground state |0〉 and excited

state |1〉, with energies hωa and hωb, respectively. For such a system, the basic quantum

mechanical operators can be defined as,

a = |0〉 〈1|

a† = |1〉 〈0|

aa† = |0〉 〈0|

a†a = |1〉 〈1|

(B.1)

B.1 Hamiltonian and density operator

The Hamiltonian of the system can be written as H = H0 + HI . Where H0 is the

Hamiltonian of the unperturbed system and HI represents the interaction Hamiltonian. In

the dipole approximation, the interaction Hamiltonian takes the form, HI = −µE(|0〉 〈1|+

|1〉 〈0|). Therefore we have

H = hωb |1〉 〈1|+ hωa |0〉 〈0| − µE(|0〉 〈1|+ |1〉 〈0|) (B.2)

where µ is the transition dipole moment of the system. At a given time t, wave function

of the system,Ψ(t), is a linear combination of the two states and can be written as Ψ(t) =

a+(t) |0〉+ a−(t) |1〉. Then Ψ(t) obeys Schrodinger equation, ih ∂∂tΨ(t) = HΨ(t).

ih∂

∂tΨ(t) = [hωb |1〉 〈1|+ hωa |0〉 〈0| − µE(|0〉 〈1|+ |1〉 〈0|)]Ψ(t)

by expanding the above equation in |0〉 , |1〉 basis, two coupled equations can be obtained.

ih∂

∂ta+(t) = hωba+(t)− µE0

2(eiωt + e−iωt)a−(t) (B.3)

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Appendix B (Continued)

ih∂

∂ta−(t) = hωaa−(t)− µE0

2(eiωt + e−iωt)a+(t) (B.4)

B.2 Transforming to a rotating frame and applying rotating wave approxima-

tion

Let’s transform equations B.3, B.4 into a frame, rotating with angular frequency ω2 ,

using

b+(t) = eiωt/2a+(t)

b−(t) = eiωt/2a−(t)

It can easily be shown that, equations B.3, B.4 have the new forms in the rotating frame

ih∂

∂tb+(t) = (hωa − hω/2)b+(t)− µE0

2(e2iωt + 1)b−(t) (B.5)

ih∂

∂tb−(t) = (hωa + hω/2)b−(t)− µE0

2(1 + e2iωt)b+(t) (B.6)

B.2.1 Rotating wave approximation

We will be interested in the case in which the two-level system is nearly resonant with

the incident field. Therefore, all the non-resonant terms (rapidly oscillating terms) in Eqs.

(B.5) and (B.6) can be neglected. This is known as the rotating wave approximation. Thus

new equations can be written as

ih∂

∂tb+(t) = (hωa − hω/2)b+(t)− µE0

2b−(t) (B.7)

ih∂

∂tb−(t) = (hωa + hω/2)b−(t)− µE0

2b+(t) (B.8)

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Appendix B (Continued)

The corresponding new Hamiltonian, HRWA which describes Eqs. (B.7) and (B.8) has the

form

HRWA =

hωb − hω2 −µE0

2h

−µE0

2h hωa + hω2

After shifting the energy of the system and introducing the Rabi frequency, Ω = µE0

h , we

can define a time-independent Hamiltonian H′.

H′

= h

−∆ω −µE0

2h

−µE0

2h 0

where ∆ω = (ω − ω0) is the laser detuning, i.e., the difference between the resonance

frequency and the laser’s frequency. Note that for the convenience, H will be used instead

of H′, in the rest of this article.

B.3 Evolution of two-level system and optical Bloch equations

Once we know the total Hamiltonian of the system, time evolution can be easily achieved

using the density operator and the Liouville equation.

B.3.1 Density operator and Liouville theorem

Density operator for a two-level system, ρ = |Ψ〉 〈Ψ|, can be written as the following

form

ρ(t) = ρ00(t) |0〉 〈0|+ ρ01(t) |0〉 〈1|+ ρ10(t) |1〉 〈0|+ ρ11(t) |1〉 〈1| (B.9)

where ρaa(t), ρab(t), ρba(t) and ρbb(t) represent the populations and the coherences of the

two-level system.

The evolution of an ideal two-level system, i.e, in the absence of any damping processes, is

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Appendix B (Continued)

governed by “Liouville equation” and written as,

ihdρ

dt= [H, ρ] (B.10)

where

H = −h∆ω |1〉 〈1| − hΩ

2|1〉 〈0| − hΩ

2|0〉 〈1|

by expanding Eq. (B.10) in the basis |0〉 , |1〉, we get four simultaneous equations,

often known as “optical Bloch equations”.

ihdρ11

dt=hΩ

2ρab −

2ρba

ihdρ00

dt=h∆ωρab −

2(ρbb − ρaa)

ihdρ01

dt=− h∆ωρba +

2(ρbb − ρaa)

ihdρ10

dt=− hΩ

2ρab +

2ρba

(B.11)

B.3.2 Inclusion of damping processes into Bloch equations via Master equa-

tion

In general, real systems always suffer from damping processes such as radiative (sponta-

neous emission) and non-radiative (pure dephasing) excited state relaxations. Contribution

from such processes can be included into optical Bloch equations via Master equation. The

Master equation for a dissipative system (often called as “Lindblad equation”) can be

written as [81]

d

dtρ = − i

h[H, ρ] +

∑k

(LkρL†k −

1

2LkL

†kρ−

1

2ρLkL

†k) (B.12)

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Appendix B (Continued)

the Lindbladian term, L[ρ] =∑

k(LkρL†k−

12LkL

†kρ−

12 ρLkL

†k) can be calculated for partic-

ular decay channels. Here Lk =√

Γji |i〉 〈j| is the quantum jump operator which makes a

quantum jump from state |i〉 to state |j〉 with a probability Γij . The complete Lindbladian

term for a given system is calculated by taking the sum over all the possible decay channels

k = ij.

B.3.2.1 Spontaneous emission

In order to calculate the Lindbladian term of a two-level system due to radiative decay,

we consider the quantum jump from state |1〉 to state |0〉 as the only decay channel. if

Γ10 = κ, then the Lindbladian, L[ρ], reads:

L[ρ] =κ

2[2 |1〉 〈0| ρ |1〉 〈0| − |1〉 〈0| 0〉 〈1| ρ− ρ| |1〉 〈0| 0〉 |1〉]

This can be simplified to get

L[ρ] = κρ11 |0〉 〈0| − κρ11 |1〉 〈1| −κ

2ρ10 |1〉 〈0| −

κ

2ρ01 |0〉 〈1| (B.13)

B.3.2.2 Pure dephasing

The corresponding Lindbladian for the Pure dephasing of a two-level system can be

evaluated via two decay channels, i.e., from state |1〉 to itself (i = 1, j = 1) and from state

|0〉 to itself (i = 0, j = 0). By assuming both decay channels have equal jumping probabil-

ities (Γ11 = Γ00 = γ), one can write the resultant Lindbadian for the pure dephasing.

L[ρ]pd = γ[|0〉 〈0| ρ |0〉 〈0| − 1

2|0〉 〈0| 0〉 〈0| ρ− 1

2ρ |0〉 〈0| 0〉 〈0|] + γ[|1〉 〈1| ρ |1〉 〈1|

− 1

2|1〉 〈1| 1〉 〈1| ρ− 1

2ρ |1〉 〈1| 1〉 〈1|]

74

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Appendix B (Continued)

It can be easily shown that the two terms in the above equations are equal. After simpli-

fication,

L[ρ]pd = −γ[ρ10 |1〉 〈0|+ ρ01 |0〉 〈1|] (B.14)

combining Eqs. [B.11, B.13, B.14], we can rewrite optical Bloch equations for a dissipative

two-level system in the following form

d

dtR(t) = M.R(t) (B.15)

where

M =

−κ −iΩ/2 iΩ/2 0

−iΩ/2 −κ/2− γ + i∆ω 0 iΩ/2

iΩ/2 0 −κ/2− γ − i∆ω −iΩ/2

κ iΩ/2 −iΩ/2 0

(B.16)

and

R(t) =

n(t)

α(t)

α∗(t)

m(t)

(B.17)

Here, n(t) = Trρ(t)a†a, α(t) = Trρ(t)a, α∗(t) = Trρ(t)a† and m(t) = Trρ(t)aa†.

κ, γ are the radiative and non-radiative decay rates of the two-level system. Note that laser

detuning, ∆ω = ωL − ω0, is also included into Bloch equations to study the non-resonant

excitations.

75

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Appendix B (Continued)

B.3.3 Steady state solution to a two-level system

Equation (B.15) has steady-state solution as t→∞

n∞ =Ω2

κ+ 2γ

∆ω2 + (κ+ 2γ)(κ+ 2γ + 2Ω2/κ)/4(B.18)

and

α∞ =iΩ

4

κ+ 2γ + 2i∆ω

∆ω2 + (κ+ 2γ)(κ+ 2γ + 2Ω2/κ)/4(B.19)

B.3.4 Power spectrum of the scattered light

The power spectrum of the scattered light can be calculated using the Fourier transform

of the first order correlation function 〈a†(t)a(t+τ)〉. This has been discussed in great detail

by Mollow [26], and the result is

g(ν) = 2π|α∞|2δ(ν − ω) + n∞κΩ2((ν − ω)2 + 1

2Ω2 + κ2

|f(i(ν − ω))|2) (B.20)

where ν is the emission frequency, and the only source of broadening is a decay of the

upper state to the lower state at a rate κ due to spontaneous emission. The third-order

polynomial, f(s), is defined as

f(s) = s3 + 2κs2 + (Ω2 + (∆ω)2 + (5/4)κ2)s+ κ(1

2Ω2 + (∆ω)2 +

1

4κ2)

B.3.5 Second-order correlation function

In general, second-order correlation function is used to study the photon statistics of

the scattered light. In the steady state, the normalized second-order correlation function

g(2)(τ) can be calculated via two-time correlation function 〈a†(t)a†(t+τ)a(t+τ)a(t)〉. This

76

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Appendix B (Continued)

can be simplified using quantum regression theorem, and the result is [82]

g(2)(τ) = 1− (cosµτ +3κ

4µsinµτ)e−3κτ/4 (B.21)

where µ =√

Ω2 + ∆ω2 − κ2

16

77

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Appendix C Theoretical fringe visibilities of one-photon and two-photon inter-ference

Note to reader:From “Field-Field and Photon-Photon Correlations of Light Scattered

by Two Remote Two-Level InAs Quantum Dots on the Same Substrate”, by K. Konthas-

inghe et.al., Phys. Rev. Lett. 109, 267402 (2012). Copyright 2016 by Copyright Holder.

Reprinted with permission.

C.1 Most general Stark shift model

In chapter 3, a statistical model is used to estimate the effect of spectral diffusion on

one-photon and two-photon interference visibilities from two quantum dots. The basis of

this model is the assumption that random changes in the QD resonance frequencies are the

result of random changes in the electrostatic environment of each QD. The random shifts

in QD resonance frequency in turn cause a flickering of the scattered light intensity and

thus variations in photon flux at a beamsplitter. This variation in photon flux causes an

“apparent” reduction in interference visibilities. In order to obtain analytical solutions to

this model we have assumed that the shift in resonance frequency is the result of normally-

distributed fluctuations in the electrical potential, V , at the location of the QD, namely

that,

δStark(V ) = c1V + c2V2

PV (V ) = e−V2/2σ2

/σ√

2π (C.1)

This model is a simplification because, in general, it is the electric field that can ran-

domly fluctuate in each spatial direction independently [83]. Accordingly, the most general

expression for the Stark shift, ignoring larger than second-order terms, is given by

δStark = c1xFx + c2xF2x

+c1yFy + c2yF2y + c1zFz + c2zF

2z (C.2)

78

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Appendix C (Continued)

Table C.1. Summary of selection of reported Stark shift coefficients

Linear Quadratic

GHz/(kV/cm) GHz/(kV/cm)2

z-direction -15.5 [84], +12.5 [85] -0.125 [84], -0.085 [86]x, y-direction -1.0 [87], +0.75 [88]

where Fi (i = x, y, z) is the component of the electric field along the i-direction and c1x,

c2x, c1y, c2y, c1z, c2z are constants.

The quantum-confined Stark effect has been studied extensively for InAs QDs. In

typical experimental studies of this effect, a static electric field is applied along a cho-

sen direction and the resulting QD resonance frequency shift is measured. In Table C.1

we summarize Stark shift coefficients reported in the literature which are based on such

measurements.

In agreement with the known shape of self-assembled QDs (pyramidal with few nm in

height, and ∼10-20 nm in lateral extent), it is found that a permanent dipole moment is

present in the growth (z) direction only. Therefore, when applying an electric field along z,

the Stark shift is dominated by a linear term for weak field strength, whereas a quadratic

term associated with the polarizability of the QD is only significant for a strong applied

field in the z direction. In the lateral (x, y) direction on the other hand, a quadratic Stark

shift is systematically observed due to a much larger lateral polarizability.

In order to simplify Eq. (C.2) for the present situation it is necessary to have an estimate

for the expected random field fluctuations that might be present in our sample. Since we

know that the typical spectral-diffusion-broadened excitation linewidth of an InAs is on

the order of 1 GHz at low temperature, we can estimate from Ref. [84] a corresponding

electric field fluctuation of 0.07 kV/cm (along the z direction). For an electric field of this

magnitude in the growth direction, it is clear from Table I that the linear term dominates.

79

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Appendix C (Continued)

Therefore, we can neglect c1x, c1y, c2z in Eq. (C.2) yielding

δStark = c2xF2x + c2yF

2y + c1zFz (C.3)

Note that we have assumed that the electric field fluctuates around zero. In general, an

internal field might be present such that fluctuations occur around a finite field other than

zero. This offset can be absorbed into a constant Stark shift and as long as the internal

field is not too large compared to the magnitude of its fluctuations the above estimates are

still valid.

C.1.1 Numerical model

In order to calculate physical observables relevant to our main experiment, in particular

the one-photon and two-photon interference visibilities for two QDs, it will be necessary to

calculate the expectation value of quantities such as the intensity of the light scattered by a

QD in the presence of the electric field fluctuations discussed above. Since the distribution

of intensity fluctuations is also available experimentally [Fig. 3.8(b)], we will also want to

obtain a theoretical expression for it. However, in general each electric field component in

Eq. (C.3) is fluctuating independently. In this case we can not obtain an analytical form

for a distribution function and we resort here to a numerical stochastical evaluation. For

this evaluation we make the following assumptions:

1. Fluctuations of Fx, Fy, and Fz are random and independent

2. Fx, Fy, and Fz follow normal distributions with zero mean value and identical stan-

dard deviation, σ.

3. c2x=c2y

80

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Appendix C (Continued)

For each random realization of the field components obtained from a pseudo-random num-

ber generator, we then calculate the intensity of the scattered light, according to [26] as

Itot = n∞ =Ω2/4

(∆ω + δStark)2 + κ2/4 + Ω2/2(C.4)

where δStark is given by Eq. (C.3). The numerical expectation value of this quantity, as a

function of detuning, namely, 〈Itot〉(∆ω), represents the QD excitation spectrum obtained

by scanning the laser over the QD resonance and recording the total scattered light [Fig.

3.2(b)]. The distribution function of Itot is obtained numerically by building a histogram.

There are three input parameters to this numerical model. These are: σ, c2x = c2y, and

c1z, as well as a scale factor than relates Itot to the experimentally measured scattered light

intensity. The linear (c1z) and quadratic (c2x and c2y) coefficients have a quite different

effect on the excitation spectrum, 〈Itot〉(∆ω). In particular, the quadratic coefficients

cause an asymmetry of the excitation spectrum. In the experimental spectra of Fig. 3.2(b)

this asymmetry is slightly visible, but only at larger Rabi frequencies. Therefore, we can

conclude that the linear term must dominate the quadratic term. This is consistent with

the data of Table C.1 of coefficients from the literature. In fact, according to Suppl. Table

C.1, even if the electric field would be 1 kV/cm (20 times larger than the value estimated for

the magnitude of the electric field fluctuations in our sample), the linear term would still be

one order of magnitude larger than the quadratic term. Consequently, we begin by choosing

the linear coefficient, c1z, and then adjusting the quadratic coefficient c2x = c2y in order to

obtain a best fit of the excitation spectrum. It is reasonable to assume that the magnitude

of the electric field fluctuations, and therefore σ, will depend on the laser intensity: in our

experiment the laser irradiates a large area around each QD and increasing its intensity

should cause an increase in σ, which we thus allow to change with Rabi frequency in our

model. Best fit values are summarized in Suppl. Table C.1.1. We estimate the fitting errors

for the linear (c1z) and quadratic (c1x and c1y) coefficients to be ±0.5 GHz/(kV/cm) and

81

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Appendix C (Continued)

Table C.2. Summary of coefficients from numerical analysis

Ω/2π c1z c2x c2y σGHz GHz/(kV/cm) GHz/(kV/cm)2 GHz/(kV/cm)2 (kV/cm)

0.15 9 -2 -2 0.050.21 9 -2 -2 0.060.46 9 -6 -6 0.09

±1 GHz/(kV/cm)2, respectively. The disproportionally larger uncertainty of the quadratic

coefficient is a reflection of the fact that the linear term provides the dominant contribution.

Figure C.1.1 further illustrates the distribution of Stark shifts at Ω/2π=0.21 GHz. The

0 1 2-2 -1

0.8

0

0.2

0.4

0.6

Stark shift (GHz)

Prob

abili

ty d

ensi

ty

Figure C.1. Probability density distribution of Stark shift at Ω/2π = 0.21 GHz.

corresponding intensity distribution is shown in Fig. C.1.1.

As discussed in chapter3, the fringe visibility of the first order interference of light

scattered by a single QD and a local oscillator is obtained as,

V(1)QD,LO =

⟨√Icoh

⟩√⟨Itot

⟩ (C.5)

82

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Appendix C (Continued)

Similarly, the fringe visibility of the first order interference between light scattered by two

QDs is,

V(1)QD,QD =

⟨√Icoh

⟩2⟨Itot

⟩ (C.6)

These visibilities were calculated using the best fit parameters at different intensities and

the results are shown in Table C.1.2. As can be seen by comparison with the visibilities

0

10

00.120.080.04

20

Prob

abili

ty d

ensi

ty

Scatt. light intensity (arb. units)

Figure C.2. Probability density distribution of scattered light intensity at Ω/2π =0.21 GHz. Note that although the abscissas are denoted in arb. units, the function isnormalized so that the area equals unity.

shown in chapter 3, the results are very similar.

C.1.2 Analytical model

In chapter 3, a simplified model of the Stark shift has been used for the purpose of

obtaining analytical solutions [Eq. (C.1)]. As discussed above, the linear Stark shift term

is in general found to dominate the quadratic term. Therefore, the numerical model above

and the analytical model of Eq. (C.1) should yield similar results. Using our simplified

model, the total intensity of the scattered light can be written as,

Itot = n∞(V ) =Ω2/4

(∆ω + c1V + c2V 2)2 + κ2/4 + Ω2/2(C.7)

83

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Appendix C (Continued)

We can then use substitution of variables to find the corresponding probability density

function via PI(I) = PV (g−1(I))∣∣∣ ddI g−1(I)

∣∣∣, where g−1(I) is the inverse function of g(V ) =

n∞(V ). However, complications arise due to the non-monotonic nature of g(V ). Therefore,

it is necessary to consider two cases, with c2 =0, ∆ω=0 and c1=0, ∆ω=0. Then, we can

use the following formula to get the corresponding distribution for the above two cases [89].

Because, g−1(I) has two branches −|g−1(I)| and +|g−1(I)|, we can write

PI(I) =2∑j=1

PV (g−1j (I))

d

dIg−1j (I) (C.8)

where g−11,2(I) =±|g−1(I)|.

Then,

PI(I) = [PV (|g−1(I)|) + PV (−|g−1(I)|)] ddIg−1(I) (C.9)

Because P (V ) is an even function, the two terms in the above equation gives the same

result. This introduces an additional factor of 2 into the calculations.

It is quite difficult to get an analytical expression for the intensity distribution when

both c1 and c2 are present. Therefore, we consider two separate cases to calculate PI(I).

Also, we assume that ∆ω=0 .

Case 1: c2 = 0

Then,

I =Ω2/4

(c1V )2 + κ2/4 + Ω2/2(C.10)

V =(Ω2/4I − κ2/4− Ω2/2)1/2

c1(C.11)

for two variables, I and V , PV (V )dV = PI(I)dI holds.

84

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Appendix C (Continued)

PI(I) = PV (V )dV

dI(C.12)

Therefore,

PI(I) = (Ω2/8√

2π)e−(Ω2/4I−κ2/4−Ω2/2)/2σ2c21

c1I2σ(Ω2/4I − κ2/4− Ω2/2)1/2(C.13)

Case 2 : c1 = 0

I =Ω2/4

(c2V 2)2 + κ2/4 + Ω2/2(C.14)

Then,

V =(Ω2/4I − κ2/4− Ω2/2)1/4

√c2

(C.15)

dV

dI=

Ω2

16√c2I

2(Ω2/4I − κ2/4− Ω2/2)3/4(C.16)

PI(I) = PV (V )dV

dI

PI(I) = (Ω2/16√

2√π)

e−(Ω2/4I−κ2/4−Ω2/2)1/2/2σ2c2

√c2I

2σ(Ω2/4I − κ2/4− Ω2/2)3/4(C.17)

As described above, I(V ) is a non-monotonic function and we need to introduce extra

factor of 2 to get the correct probability distribution. Therefore, following are the final

intensity probability distribution functions that we used to fit with our experimental data

shown in Fig. 3.8.

85

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Appendix C (Continued)

When quadratic term is zero,

PI(I) = (Ω2/4√

2π)e−(Ω2/4I−κ2/4−Ω2/2)/2σ2c21

c1I2σ(Ω2/4I − κ2/4− Ω2/2)1/2(C.18)

when linear term is zero,

PI(I) = (Ω2/8√

2π)e−(Ω2/4I−κ2/4−Ω2/2)1/2/2σ2c2

√c2σI

2(Ω2/4I − κ2/4− Ω2/2)3/4(C.19)

In Table C.1.2 summarizes the corresponding fitting parameters for the analytical

model. Also, Table C.1.2 shows the comparison of the calculated fringe visibilities for

the two models.

Table C.3. Summary of coefficients from analytical model

Ω/2π c1 c2 σGHz GHz/(kV/cm) GHz/(kV/cm)2 (kV/cm)

0.15 9.5 -3.5 0.0500.21 9.5 -3.2 0.0550.46 9.5 -12 0.076

Table C.4. Summary of one-photon interference visibilities for analytical and numericalanalysis

Analytical NumericalΩ/2π (GHz) QD-LO QD-QD QD-LO QD-QD

0.15 0.70 0.49 0.73 0.530.21 0.68 0.46 0.69 0.480.46 0.63 0.39 0.62 0.38

86

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Appendix D Interaction between a two-level system and a pulsed laser

Note to reader:From “Resonant Light Scattering of a Laser Frequency Comb by a Quan-

tum Dot”, by K. Konthasinghe et.al., Phys. Rev. A 90, 023810 (2014). Copyright 2016

by Copyright Holder. Reprinted with permission.

D.1 Theoretical background

We consider a QD with ground state |0〉 and excited state |1〉 (transition frequency

ω0), for which the relevant atomic operators are S+ = |1〉〈0|, S− = |0〉〈1|, and 2Sz =

|1〉〈1| − |0〉〈0|. Under pulsed laser excitation, the dynamics of the expectation values of

these operators are usually obtained by numerically integrating the optical Bloch equations

in which the Rabi frequency is taken to be proportional to the temporal envelope of the

(single) pulse’s electric field. Here we are concerned specifically with a rigorous description

of the experimental measurement which results from the time-averaged acquisition of the

scattered light correlations under excitation by millions of single pulses. In order to capture

such time-averaged effects, including coherent light scattering, we describe the incoming

pulse-train by its constituent monochromatic waves, so that the applied electric field is

written as

E(t) =1

2e−iωst

p∑n=−p

Ene−i(nδt+φn) + c.c. (D.1)

with time-independent coefficients En. The central frequency is ωs (n = 0), and the

frequency difference δ (in rad/s) between adjacent waves coincides with 2π times the pulse

repetition rate (in Hz).

Ficek et al. have analyzed the fluorescence spectrum of a two-level atom driven by

a modulated field in which the central component’s frequency coincides with the atom’s

natural transition frequency, and broadening is determined by spontaneous emission [?

]. We extend their treatment to the calculation of the power spectrum and second order

correlation functions with detuning and spectral diffusion, and investigate the scattered

87

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Appendix D (Continued)

light correlations for the case when the multichromatic field is that of a short pulsed laser

excitation. The optical Bloch equations then read,

dX(t)

dt= A(t)X(t) + v (D.2)

where X(t) = (X1(t), X2(t), X3(t)) with X1(t) = 〈S−(t)〉 = 〈S−(t)〉ei(ωst+Ψ), X2(t) =

〈S+(t)〉 = 〈S+(t)〉e−i(ωst+Ψ), X3(t) = 〈Sz(t)〉 = 〈Sz(t)〉,

A(t) =

−1

2Γ− i∆ 0 Ω(t)

0 −12Γ + i∆ Ω∗(t)

−Ω∗(t)/2 −Ω(t)/2 −Γ

(D.3)

and v = (0, 0,−Γ/2). Here the rotating-wave approximation was made and the time-

dependent (complex) quantity Ω(t) is defined as

Ω(t) =µ

h

p∑n=−p

Ene−i(nδt+φn). (D.4)

The radiative decay rate is denoted by Γ, and ∆ = ω0 − ωs is the detuning from exact

resonance. The magnitude, µ, of the transition dipole moment is known to be on the order

of 10 Debye for typical InAs QDs [? ]. Ψ is an arbitrary phase, chosen here to be π/2.

In what follows we begin by describing the method for calculating scattered light spectra

under multichromatic excitation (section D.2), with separate evaluation of coherent (section

D.3) and incoherent (section D.4) contributions with and without the effect of spectral

diffusion. We proceed with the calculation of second order correlations in the absence

(section D.5) and presence (section D.6) of spectral diffusion. In section D.7 results are

then presented for the specific case in which the multichromatic field is produced by a

periodically-pulsed laser oscillator.

88

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Appendix D (Continued)

D.2 Calculation of the scattered light spectrum

The power spectrum, S(t, ν), of the light scattered by a two-state quantum system is

given by the Fourier transform of the two-time correlation function of the dipole operators

[26] as

S(t, ν) = Γ

∫ ∞−∞

eiντ 〈S+(t)S−(t+ τ)〉dτ. (D.5)

Spectral measurements are typically performed on a timescale much longer than the ra-

diative decay, but also much longer than the phase coherence time (the reciprocal of the

linewidth) of the constituent waves of a real frequency comb that Eq. (D.1) is idealizing.

Therefore, the experimental spectrum must be obtained as,

S(ν) = limT→∞

δ

∫ T+2π/δ

TS(t, ν)dt, (D.6)

i.e., steady-state is assumed and an average over a period 2π/δ is performed. With the help

of the quantum regression theorem, it can be seen that the two-time correlation functions,

Y1(t, τ) = 〈S+(t)S−(t+ τ)〉 − 〈S+(t)〉〈S−(t+ τ)〉,

Y2(t, τ) = 〈S+(t)S+(t+ τ)〉 − 〈S+(t)〉〈S+(t+ τ)〉,

Y3(t, τ) = 〈S+(t)Sz(t+ τ)〉 − 〈S+(t)〉〈Sz(t+ τ)〉,

(D.7)

satisfy the same equations of motion as 〈S−(t+τ)〉, 〈S+(t+τ)〉, and 〈Sz(t+τ)〉, respectively,

namely Eq. (D.2), with v = 0, d/dt→ d/dτ , and t→ t+ τ [54], i.e.

dY(t, τ)

dτ= A(t+ τ)Y(t, τ). (D.8)

Thus the power spectrum of the scattered light [Eq. (D.5)] can be computed if Y1(t, τ)

is known. Note that Y1(t, τ) represents the incoherently scattered radiation since the

coherent contribution, 〈S+(t)〉〈S−(t + τ)〉, has been subtracted from the full correlation

89

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Appendix D (Continued)

function 〈S+(t)S−(t + τ)〉. Equation (D.8) can be solved with the help of the harmonic

expansion

Yj(t, τ) =∞∑

l=−∞Y

(l)j (t, τ)eilδ(t+τ) (D.9)

with slowly-varying coefficients Y(l)j (t, τ), which transforms the problem into an infinite set

of equations, written, after a Laplace transform, as

zY(l)

1 (z)− Y (l)1 (t, 0) = −(Γ/2 + i∆ + ilδ)Y

(l)1 (z)

+

p∑n=−p

ΩnY(l+n)

3 (z),(D.10)

zY(l)

2 (z)− Y (l)2 (t, 0) = −(Γ/2− i∆ + ilδ)Y

(l)2 (z)

+

p∑n=−p

Ω∗nY(l−n)

3 (z),(D.11)

and

zY(l)

3 (z)− Y (l)3 (t, 0) = −(Γ + ilδ)Y

(l)3 (z)

− 1

2

p∑n=−p

(Ω∗nY

(l−n)1 (z) + ΩnY

(l+n)2 (z)

),

(D.12)

where

Ωn = µEne−iφ/h. (D.13)

90

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Appendix D (Continued)

By substitution one then obtains a recursive relation for Y(l)

3 (z), written as

(z + Γ + ilδ)Y(l)

3 (z)+

1

2

∑n,m

(Ω∗nΩm

z + Γ/2 + i∆ + i(l − n)δY

(l−n+m)3 (z) +

ΩnΩ∗mz + Γ/2− i∆ + i(l + n)δ

Y(l+n−m)

3 (z)

)

= Y(l)

3 (t, 0)− 1

2

∑n

(Ω∗nY

(l−n)1 (t, 0)

z + Γ/2 + i∆ + i(l − n)δ+

ΩnY(l+n)

2 (t, 0)

z + Γ/2− i∆ + i(l + n)δ

)(D.14)

where Y(l)j (t, 0) are the initial values of the correlations’ expansion coefficients. The initial

values of the correlations Yj(t, 0) are given by setting τ = 0 in Eq. (D.7). Since the

calculation of the correlation function in Eq. (D.6) is at steady-state, we can replace

Yj(t, 0) with their steady-state values Y ssj (t, 0), which are computed as

Y ss1 (t, 0) =

1

2+Xss

3 (t)−Xss1 (t)Xss

2 (t), (D.15)

Y ss2 (t, 0) = −Xss

2 (t)Xss2 (t), (D.16)

and

Y ss3 (t, 0) = −

(1

2+Xss

3 (t)

)Xss

2 (t), (D.17)

where the superscript “ss” denotes steady-state. Using the expansion

Xssj (t) =

∞∑l=−∞

X(l)j eilδt (D.18)

with time-independent coefficients X(l)j (steady-state) we then obtain

Y(l)

1 (t, 0) =1

2δl,0 +X

(l)3 −

∑k

X(l−k)1 X

(k)2 , (D.19)

91

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Appendix D (Continued)

Y(l)

2 (t, 0) = −∑k

X(l−k)2 X

(k)2 , (D.20)

and

Y(l)

3 (t, 0) = −∑k

(1

2δk,0 +X

(k)3

)X

(l−k)2 . (D.21)

The coefficients X(l)j can be found by inserting Eq. (D.18) into Eq. (D.2) with dX/dt→ 0

(steady-state) which yields the set of equations

(Γ/2 + i∆ + ilδ)X(l)1 =

p∑n=−p

ΩnX(l+n)3 , (D.22)

(Γ/2− i∆ + ilδ)X(l)2 =

p∑n=−p

Ω∗nX(l−n)3 , (D.23)

and

(Γ + ilδ)X(l)3 =

−Γ

2δl,0−

1

2

p∑n=−p

(Ω∗nX

(l−n)1 + ΩnX

(l+n)2

).

(D.24)

After substitution of Eq. (D.22) and Eq. (D.23) into Eq. (D.24) one then obtains

(Γ + ilδ)X(l)3 +

1

2

∑n,m

(Ω∗nΩm

z + Γ/2 + i∆ + i(l − n)δX

(l−n+m)3 +

ΩnΩ∗mz + Γ/2− i∆ + i(l + n)δ

X(l+n−m)3

)= −Γ

2δ0,l.

(D.25)

Once the input field is specified via the set of coefficients Ωn−p≤n≤p in Eq. (D.13), the

calculation of the time-averaged scattered light spectrum [Eq. (D.6)] then begins with the

computation of X(l)3 using Eq. (D.25). This can be done, e.g., using the matrix methods

described in Ref. [55], in which the harmonic expansions are truncated. With X(l)3 known,

92

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Appendix D (Continued)

X(l)1 and X

(l)2 can be computed from Eq. (D.22) and Eq. (D.23), respectively. This then

allows the calculation of the initial values of the correlations’ expansion coefficients using

Eq. (D.19), Eq. (D.20), and Eq. (D.21), which can be used to solve Eq. (D.14) for Y(l)

3 (z).

Finally, Y(l)

1 (z) can be computed using Eq. (D.10).

D.3 Coherently scattered light spectrum

D.3.1 Radiatively-broadened two-level system

The light scattered coherently by a radiatively-broadened two-state quantum system

has its origins in the equilibrium oscillations of the two-time dipole correlation function.

Its spectrum is calculated as the Fourier transform of 〈S+(t)〉〈S−(t+ τ)〉 as

Scoh(ν) =

Γ limT→∞

δ

∫ T+2π/δ

T

∫ ∞−∞

eiντ 〈S+(t)〉〈S−(t+ τ)〉dτdt(D.26)

and therefore by

Scoh(ν) =

Γδ

∫ 2π/δ

0

∫ ∞−∞

ei(ν−ωs)τXss2 (t)Xss

1 (t+ τ)dτdt

(D.27)

or

Scoh(ν) =

Γδ

∫ 2π/δ

0

∫ ∞−∞

ei(ν−ωs+lδ)τ∑k,l

X(k)2 X

(l)1 ei(k+l)δtdτdt

(D.28)

93

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Appendix D (Continued)

and since X(−l)2 = X

∗(l)1 ,

Scoh(ν) = 2πΓ∑l

|X(l)1 |

2δD(ν − ωs + lδ), (D.29)

where δD(x) denotes the Dirac delta-function. The frequency-integrated rate of coherently

scattered photons is then given by

γcoh = Γ∑l

|X(l)1 |

2, (D.30)

while the total rate of scattered photons is given by

γtot = Γ

(1

2+X

(0)3

). (D.31)

D.3.2 Radiatively-broadened two-level system in the presence of a fluctuating

environment

In any real system, particularly in solids, the resonance frequency, ω0, of the two-level

system will fluctuate with time. The timescale on which this fluctuation occurs is typically

much slower than any other relevant timescales, except for the data acquisition time.

Therefore, such effects can be included by simply averaging the spectra obtained for an

ideal, radiatively-broadened two-level system, over a distribution of resonance frequencies.

In the case of InAs QDs, it is well-known that random Stark shifts due to fluctuations

of the charge density in the surrounding solid matrix are responsible for such spectral

diffusion. Although in general the distribution of resulting detunings may be non-trivial

[13] it is often sufficient to approximate it with a normal distribution [12]. The resulting

spectrum, Scoh(ν), is then given by

Scoh(ν) = 2πΓ∑l

|X(l)1 |2δD(ν − ωs + lδ), (D.32)

94

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Appendix D (Continued)

where

|X(l)1 |2 =

1

σ√

∫ ∞−∞|X(l)

1 |2e−(ω′

0−ω0)2/2σ2dω′0. (D.33)

Here the Gaussian distribution of resonance frequencies has a full width at half maximum

(in Hz) of s/2π ≈ 2.355σ/2π which is typically ≈ 1 GHz for InAs QDs. Likewise, we obtain

the frequency-integrated rate of coherently scattered photons in the presence of spectral

diffusion as

γcoh = Γ∑l

|X(l)1 |2, (D.34)

while the total rate of scattered photons in the presence of spectral diffusion is given by

γtot = Γ

(1

2+X

(0)3

), (D.35)

where

X(0)3 =

1

σ√

∫ ∞−∞

X(0)3 e−(ω′

0−ω0)2/2σ2dω′0. (D.36)

D.4 Incoherently scattered light spectrum

D.4.1 Radiatively-broadened two-level system

The spectrum of the light scattered incoherently by an ideal two-level quantum system

is given by

Sinc(ν) =

Γ limT→∞

δ

∫ T+2π/δ

T

∫ ∞−∞

Y1(t, τ)ei(ν−ωs)τdτdt,

(D.37)

or

Sinc(ν) = 2ΓRe(Y

(0)1 (z)|z=−i(ν−ωs)

), (D.38)

where the quantity Y(0)

1 (z) is obtained from Eq. (D.10) with l = 0.

95

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Appendix D (Continued)

D.4.2 Radiatively-broadened two-level system in the presence of a fluctuating

environment

When spectral diffusion is present, we obtain the scattered light spectrum as

Sinc(ν) = 2ΓRe

(Y

(0)1 (z)|z=−i(ν−ωs)

), (D.39)

where

Y(0)

1 (z) =1

σ√

∫ ∞−∞

Y(0)

1 (z)e−(ω′0−ω0)2/2σ2

dω′0. (D.40)

with the same parameter σ as for the case of coherently scattered light.

D.5 Calculation of second-order correlation function

The unnormalized time-averaged second-order correlation function of the scattered light

is given by

G(2)(τ) =

δ

2πlimT→∞

∫ T+2π/δ

T〈S+(t)S+(t+ τ)S−(t+ τ)S−(t)〉dt.

(D.41)

With the help of the quantum regression theorem it can be seen that the two-time corre-

lation functions

Z1(t, τ) = 〈S+(t)S−(t+ τ)S−(t)〉

−〈S+(t)S−(t)〉〈S−(t+ τ)〉,

Z2(t, τ) = 〈S+(t)S+(t+ τ)S−(t)〉

−〈S+(t)S−(t)〉〈S+(t+ τ)〉,

(D.42)

and

Z3(t, τ) = 〈S+(t)Sz(t+ τ)S−(t)〉

−〈S+(t)S−(t)〉〈Sz(t+ τ)〉,(D.43)

96

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Appendix D (Continued)

satisfy the same equations of motion as 〈S−(t+τ)〉, 〈S+(t+τ)〉, and 〈Sz(t+τ)〉, respectively,

namely Eq. (D.2), with v = 0, d/dt→ d/dτ , and t→ t+ τ , i.e.

dZ(t, τ)

dτ= A(t+ τ)Z(t, τ). (D.44)

Thus G(2)(τ) can be computed once Z3(t, τ) is known. Equation (D.44) can be solved

similarly to Eq. (D.8) with the help of the harmonic expansion

Zj(t, τ) =∞∑

l=−∞Z

(l)j (t, τ)eilδ(t+τ) (D.45)

with slowly-varying coefficients Z(l)j (t, τ). Note that in the final evaluation process the

sum in Eq. (D.45) will be truncated to some large integer so that an arbitrary degree of

precision can be achieved. This expansion transforms the problem into one involving an

infinite set of equations, written, after a Laplace transform, as

zZ(l)1 (z)− Z(l)

1 (t, 0) = −(Γ/2 + i∆ + ilδ)Z(l)1 (z)

+

p∑n=−p

ΩnZ(l+n)3 (z),

(D.46)

zZ(l)2 (z)− Z(l)

2 (t, 0) = −(Γ/2− i∆ + ilδ)Z(l)2 (z)

+

p∑n=−p

Ω∗nZ(l−n)3 (z),

(D.47)

and

zZ(l)3 (z)− Z(l)

3 (t, 0) = −(Γ + ilδ)Z(l)3 (z)

− 1

2

p∑n=−p

(Ω∗nZ

(l−n)1 (z) + ΩnZ

(l+n)2 (z)

).

(D.48)

97

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Appendix D (Continued)

After solving for Z(l)1 (z) in Eq. (D.46) and for Z

(l)2 (z) in Eq. (D.47) and substituting the

results into Eq. (D.48), one obtains a recursive relation for Z(l)3 (z), written as

(z + Γ + ilδ)Z(l)3 (z)+

1

2

∑n,m

(Ω∗nΩm

z + Γ/2 + i∆ + i(l − n)δZ

(l−n+m)3 (z) +

ΩnΩ∗mz + Γ/2− i∆ + i(l + n)δ

Z(l+n−m)3 (z)

)

= Z(l)3 (t, 0)− 1

2

∑n

(Ω∗nZ

(l−n)1 (t, 0)

z + Γ/2 + i∆ + i(l − n)δ+

ΩnZ(l+n)2 (t, 0)

z + Γ/2− i∆ + i(l + n)δ

)(D.49)

where Z(l)j (t, 0) are the initial values of the correlations’ expansion coefficients. The initial

values of the correlations Zj(t, 0) are given by setting τ = 0 in Eq. (D.42) and Eq. (D.43).

Since the calculation of G(2)(τ) is at steady-state, we can replace Zj(t, 0) with their steady-

state values Zssj (t, 0), which are computed as

Zss1 (t, 0) = −(Xss

3 (t) + 1/2)Xss1 (t), (D.50)

Zss2 (t, 0) = −(Xss

3 (t) + 1/2)Xss2 (t), (D.51)

Zss3 (t, 0) = −(Xss

3 (t) + 1/2)2. (D.52)

where the superscript “ss” denotes steady-state. Using again the expansion

Xssj (t) =

∞∑l=−∞

X(l)j eilδt (D.53)

with time-independent coefficients X(l)j (steady-state) we then obtain

Z(l)1 (t, 0) = −1

2X

(l)1 −

∑k

X(k)1 X

(l−k)3 , (D.54)

98

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Appendix D (Continued)

Z(l)2 (t, 0) = −1

2X

(l)2 −

∑k

X(k)2 X

(l−k)3 , (D.55)

Z(l)3 (t, 0) = −1

4δl,0 −X

(l)3 −

∑k

X(k)3 X

(l−k)3 . (D.56)

The relevant second-order correlation function (τ > 0) is then given as,

G(2)(τ) = F−12ReZ(0)3 (z)|z=−iν′

+∑k

X(k)3 X

(−k)3 eikδτ +X

(0)3 +

1

4,

(D.57)

where F−1 denotes the inverse Fourier transform with respect to ν ′, defined as,

F−1f(ν ′) =1

∫ ∞−∞

f(ν ′)e−iν′τdν ′. (D.58)

The quantity usually measured in an experiment is proportional to the normalized second-

order correlation function of the scattered light defined as

g(2)(τ) = G(2)(τ)/N, (D.59)

where N is a normalization factor given by

N =δ

2πlimT→∞

∫ T+2π/δ

T〈S+(t)S−(t)〉2dt. (D.60)

This normalization factor can be computed as

N =1

4+X

(0)3 +

∑k

X(k)3 X

(−k)3 . (D.61)

Thus for obtaining the desired normalized second-order correlation function [Eq. (D.59)],

Eq. (D.49) must first be solved for Z(l)3 (z). For this calculation the initial values of the

99

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Appendix D (Continued)

second-order correlation expansion function coefficients must be first obtained using Eqs.

(D.54), (D.55) and (D.56) using the Xj obtained before with Eqs. (D.22), (D.23), (D.24)

and (D.25).

D.6 Second-order correlation function in the presence of a fluctuating envi-

ronment

In the presence of a slowly-varying, normally-distributed random detuning, as in sec-

tions D.4.2 and D.3.2, we must further perform an average of the second order correlation

functions over these detunings as

g(2)av (τ) = G(2)(τ)/N, (D.62)

where

G(2)(τ) =1

σ√

∫ ∞−∞

G(2)(τ)e−(ω′0−ω0)2/2σ2

dω′0 (D.63)

and

N =1

σ√

∫ ∞−∞

Ne−(ω′0−ω0)2/2σ2

dω′0. (D.64)

D.7 Scattered light correlations under mode-locked laser frequency comb ex-

citation

We now numerically evaluate the first and second order correlations of light scattered

by a QD for the specific case for which the incoming field’s temporal envelope is given by

the function

E(t) =

∞∑n=0

Es(t−2πn

δ), (D.65)

with

Es(t) = E0sech(1.76t/tp), (D.66)

100

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Appendix D (Continued)

where tp is the full width at half-maximum of the temporal intensity profile of a single

pulse. Mode-locked laser oscillators commonly produce such pulse trains, for which the

time-bandwidth product is tp∆νp ≈0.315, where ∆νp is the FWHM, in Hz, of the pulse’s

power spectrum. We will further assume that the laser cavity has been sufficiently stabilized

so that over the measurement duration we can describe the applied field by

E(t) = E(t)cos(ωst), (D.67)

and thus in the notation of Eq. (D.1), φn = 0 and

En =δ

∫ π/δ

−π/δEs(t)einδtdt. (D.68)

The laser’s power spectrum is then given by

Slaser(ν) ∝ 2π

p∑n=−p

|En|2δD(ν − ωs − nδ). (D.69)

Note that the detection frequency ν is denoted in circular measure. Thus, together with

the detuning, ∆, the pulse width tp and the (temporal) peak Rabi frequency ΩR = µE0/h =∑n Ωn quantify the strength of the interaction between the QD and the laser pulse. The

input field is specified via the set of coefficients Ωn−p≤n≤p [Eq. (D.13)]. Because we

are concerned here with the time-averaged interaction involving a large number of pulses,

the magnitude of the pulse repetition period, 2π/δ, relative to the QD radiative lifetime,

2π/Γ, is another determinant factor for the magnitude of the average interaction strength.

In particular we must distinguish whether before arrival of a subsequent pulse the QD has

returned to its ground state (δ < Γ) or not. We define the input pulse area, θ, in the usual

way, as

θ =µ

h

∫ ∞−∞Es(t)dt = ΩR

πtp1.76

(D.70)

101

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Appendix D (Continued)

to describe maxima (θ = π, 3π, 5π, etc.) or minima (θ = 2π, 4π, 6π, etc.) of the population

of the two-level system after passage of the pulse. Throughout we will assume that the

radiative decay rate of the QD is Γ/2π=200 MHz, consistent with the radiative lifetime of

∼1 ns typically measured for InAs QDs [30].

102

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Appendix E List of publications

Journal articles

(1) K. Konthasinghe, J. G. Velez, A. J. Hopkins, M. Peiris, L. T. M. Profeta, Y.

Nieves, and A. Muller, “Fast Photothermal Canard Dynamics in High-Finesse Fabry-Perot

Microcavities”, Under Review.

(2) M. Peiris, K. Konthasinghe, and A. Muller, “Franson Interference Generated by a

Two-Level System”, Under Review.

(3) K. Konthasinghe, M. Peiris, B. Petrak, Y. Yu, Z. C. Niu, and A. Muller, “Correla-

tions in Pulsed Resonance Fluorescence”, Opt. Lett. 40, 1846 (2015).

(4) M. Peiris, B. Petrak, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Two-Color

Photon Correlations of the Light Scattered by a Quantum Dot”, Phys. Rev. B 91, 195125

(2015).

(5) K. Konthasinghe, K. Fitzmorris, M. Peiris, A. J. Hopkins, B. Petrak, D. K. Killinger,

and A. Muller, “Laser-Induced Fluorescence from N+2 Ions Generated by a Corona Dis-

charge in Ambient Air”, Appl. Spectrosc. 69, 1042 (2015).

(6) K. Konthasinghe, M. Peiris, and A. Muller, “Resonant Light Scattering of a Laser

Frequency Comb by a Quantum Dot”, Phys. Rev. A 90, 023810 (2014).

(7) M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic Resonant

Light Scattering from a Quantum Dot”, Phys. Rev. B 89, 155305 (2014).

(8) K. Konthasinghe, M. Peiris, Y. Yu, M. F. Li, J. F. He, L. J. Wang, H. Q. Ni, Z.

C. Niu, C. K. Shih, and A. Muller, “Field-Field and Photon-Photon Correlations of Light

Scattered by Two Remote Two-Level InAs Quantum Dots on the Same Substrate”, Phys.

Rev. Lett. 109, 267402 (2012).

(9) G. Zhao, Y. Zhang, D. G. Deppe, K. Konthasinghe, and A. Muller, “Buried Het-

erostructure Vertical-Cavity Surface-Emitting Laser with Semiconductor Mirrors”, Appl.

Phys. Lett. 101, 101103 (2012).

103

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Appendix E (Continued)

(10) K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J.

Wang, H. Q. Ni, Z. C. Niu, and A. Muller, “Coherent versus Incoherent Light Scattering

from a Quantum Dot”, Phys. Rev. B. 85, 235315 (2012).

(11) B. Petrak, K. Konthasinghe, S. Perez, and A. Muller, “Feedback-Controlled Laser

Fabrication of Micromirror Substrates”, Rev. Sci. Instrum. 82, 123112 (2011).

Conference proceedings

(1) K. Konthasinghe, M. Peiris, B. Petrak, Y. Yu, Z. C. Niu, and A. Muller, “Reso-

nance Fluorescence Spectrum from a Quantum Dot Driven by a Periodically-Pulsed Laser”,

Poster presentation at CLEO at San Jose, CA, JW2A.26 (2015).

(2) K. Konthasinghe, M. Peiris, B. Petrak, Y. Yu, Z. C. Niu, and A. Muller, “Resonance

Fluorescence Spectrum from a Quantum Dot Driven by a Periodically-Pulsed Laser”, Oral

presentation at APS March Meeting at San Antonio, TX, D37.00004 (2015).

(3) K. Konthasinghe, K. Fitzmorris, M. Peiris, A. J. Hopkins, B. Petrak, D. K. Killinger,

and A. Muller, “Laser-Induced Fluorescence from N+2 Ions Generated by a Corona Dis-

charge in Air”, Poster presentation at SciX at Reno, NV, 6583 (2014).

(4) K. Konthasinghe, K. Fitzmorris, M. Peiris, A. J. Hopkins, B. Petrak, D. K. Killinger,

and A. Muller, “Laser-Induced Fluorescence from N+2 Ions Generated by a Corona Dis-

charge in Air”, Poster presentation at DTRA at Springfield, VA, 6583 (2014).

(5) K. Konthasinghe, M. Peiris, Y. Yu, M. F. Li, J. F. He, L. J. Wang, H. Q. Ni, Z.

C. Niu, C. K. Shih, and A. Muller, “Field-Field and Photon-Photon Correlations of Light

Scattered by Two Remote Two-Level InAs Quantum Dots on the Same Substrate”, Poster

presentation at CLEO at San Jose, CA, JTh2A.75 (2013).

(6) K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J.

Wang, H. Q. Ni, Z. C. Niu, and A. Muller, “Coherent versus Incoherent Light Scattering

from a Quantum Dot”, Oral presentation at CLEO at San Jose, CA, session A5 (2012).

104

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Appendix E (Continued)

(7) K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J.

Wang, H. Q. Ni, Z. C. Niu, and A. Muller, “Coherent versus Incoherent Light Scattering

from a Quantum Dot”, Oral presentation at APS March Meeting at Boston, MA session

Y5 (2012).

105

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ABOUT THE AUTHOR

Kumarasiri Konthasinghe was born and raised in Alawwa, Sri Lanka. He attended

University of Peradeniya as an undergraduate where he majored in Physics. He Joined

the University of South Florida Physics department in the Fall of 2010 and the Solid State

Quantum Optics Lab the following Summer. He received Masters (M.S.) degree in Applied

Physics in 2013. In the Fall of 2015, he was an intern at National Institute of Standards

and Technology (NIST), Gaithersburg to study nano-positioning of semiconductor quantum

dots. He has published nearly ten peer-reviewed journals mainly studying resonant light-

matter interactions in epitaxially-grown semiconductor quantum dots.