Excitons in Semiconductor Quantum Wells Studied Using
Two-Dimensional Coherent SpectroscopybyRohan Singh
B.Sc., Indian Institute of Technology Kharagpur, 2010
M.Sc., Indian Institute of Technology Kharagpur, 2010
M.Sc., University of Colorado Boulder, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2015This thesis entitled:
Excitons in Semiconductor Quantum Wells Studied Using Two-Dimensional Coherent
Spectroscopy
written by Rohan Singh
has been approved for the Department of Physics

Steven T. Cundiff

Markus B. Raschke

Date

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

Singh, Rohan (Ph.D., Physics)
Excitons in Semiconductor Quantum Wells Studied Using Two-Dimensional Coherent Spectroscopy
Thesis directed by Prof. Steven T. Cundiff

Correlated electron-hole pairs, or excitons, in semiconductor nanostructures have beenstudied extensively over the past few decades. The optical response of excitons is complicated due to inhomogeneous broadening, presence of multiple states, and exciton-exciton
interactions. In this work we bring new perspectives to exciton physics in semiconductor
quantum wells (QWs) through two-dimensional coherent spectroscopy (2DCS).
The effect of QW growth direction on the optical properties of excitons is exploredby studying (110)-oriented GaAs QWs. The homogeneous and inhomogeneous linewidthsof the heavy-hole exciton resonance are measured. By probing the optical nonlinear response for polarization along the in-plane crystal axes [11̄0] and [001], we measure different
homogeneous linewidths for the two orthogonal directions. This difference is found to be
due to anisotropic excitation-induced dephasing, caused by a crystal-axis-dependent absorption coefficient. The extrapolated zero-excitation density homogeneous linewidth exhibits
an activation-like temperature dependence.
Spectral diffusion of excitons in (001)-oriented QWs has been studied. We show thatthe spectral diffusion characteristics depend strongly on the sample temperature. Spectral
diffusion is generally assumed to follow the strong-redistribution approximation, partly because of lack of any evidence to the contrary. We find that this assumption is violated at
low sample temperatures for excitons in QWs; high-energy excitons preferentially relax due
to a negligible phonon population at low temperatures. The frequency-frequency correlation
function is measured through a numerical fitting procedure to quantify spectral diffusion forsample temperatures > 20 K.

iv
Exciton-exciton interactions affect the light-matter interactions in QWs significantly.
We present an intuitive and simple model for these interactions by treating excitons asinteracting bosons. We show that the polarization-dependent exciton dephasing rate in GaAs
quantum wells is due to the bosonic character of excitons. We fit slices from simulated spectrato those from the experimentally measured spectra and show that interference between two
different quantum mechanical pathways results in a slower dephasing rate for co-circularand co-linear polarization of optical excitation pulses. This interference does not exist forcross-linearly polarized excitation pulses resulting in a faster dephasing rate. Additionally,
we were able to separately quantify inter- and intra-mode interactions between excitons
through exciton-density-dependent measurements.

Dedication

This thesis is dedicated to my parents.

vi

Acknowledgements

First and foremost I would like to thank Steve Cundiff for giving me the opportunity
to work under his guidance. He has been an excellent advisor and a constant source ofinspiration.
I would also like to thank all the members of the Cundiff lab for being excellent labmates and making my life as a graduate student a lot of fun. I have learned a lot throughstimulating interactions with all of them. The members of the 2D team – Galan Moody,
Hebin Li, Takeshi Suzuki, Bo Sun, Gaël Nardin, Travis Autry, Bachana Lomsadze, Chris
Smallwood, and Diogo Almeida have been especially helpful. Soobong Choi, Andy Funk,and Martina Carsjens, along with Galan and Hebin, helped me immensely with the SEA
TADPOLE project. I am also grateful to have the opportunity to interact with and learn
from Eric Martin, Andy Almand-Hunter, Marc Aβmann, George Roumpos, Kai Wang, Matt
Day, Nick Finnigan, and Connor Awe.
I was fortunate to have had the opportunity to collaborate with many other researchgroups. I am thankful to Rich Mirin, Klaus Pierz and Mark Bieler for high quality quantum
well samples. Working and interacting with Mark Siemens, Marten Richter, Lazaro Pradilha,
Ilya Akimov, and Mack Kira has been a rewarding experience.
I would also like to acknowledge the amazing support staff at JILA especially Kim
Hagen and Hans Green in the machine shop, Carl Sauer in the electronics shop, J. R. Raithand Corey Keasling for computing help, and Agnieszka Lynch for administrative help.
Finally, I would like to thank my family and friends for their continued support andencouragement over the years.

Contents

Chapter
1 Introduction

1

2 Excitons in Semiconductor Quantum Wells

4

2.1

Semiconductor quantum well . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Bulk Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Quantum well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Exciton states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.1

Bulk excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.2

Quantum well excitons . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2.3

Polarization selection rules . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3

Decay processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.4

Inhomogeneous broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.5

Spectral Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6

Crystal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.7

Exciton-Exciton Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.8

Exciton as a boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.9

Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.9.1

(001)-Oriented GaAs quantum well . . . . . . . . . . . . . . . . . . .

25

2.9.2

(110)-Oriented GaAs quantum well . . . . . . . . . . . . . . . . . . .

25

2.2

viii
3 Two-Dimensional Coherent Spectroscopy
3.1

3.2

3.3

3.4

27

Transient four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.1.1

Three-pulse realization . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.1.2

Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.1.3

Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Two-dimensional coherent spectroscopy – theory . . . . . . . . . . . . . . . .

37

3.2.1

Rephasing one-quantum scan . . . . . . . . . . . . . . . . . . . . . .

37

3.2.2

Rephasing zero-quantum scan . . . . . . . . . . . . . . . . . . . . . .

43

3.2.3

Non-rephasing one-quantum scan . . . . . . . . . . . . . . . . . . . .

45

3.2.4

Correlation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.5

Two-quantum scan . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Two-dimensional coherent spectroscopy – experiment . . . . . . . . . . . . .

54

3.3.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.2

Obtaining 2D spectrum. . . . . . . . . . . . . . . . . . . . . . . . .

58

Two-dimensional coherent spectroscopy – previous studies . . . . . . . . . .

63

4 Excitons in (110)-Oriented GaAs Quantum Wells

66

4.1

Measuring linewidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.2

Excitation-density dependence . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.3

Population decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.4

Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5 Spectral Diffusion

76

5.1

Frequency-frequency correlation function . . . . . . . . . . . . . . . . . . . .

77

5.2

Validity of the strong redistribution approximation . . . . . . . . . . . . . .

79

5.2.1

Spectral diffusion from 2D spectra . . . . . . . . . . . . . . . . . . . .

79

5.2.2

Population redistribution . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.2.3

Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . .

83

ix
5.3

High temperature regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

5.3.1

Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.3.2

Analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6 Excitons as Interacting Bosons
6.1

6.2

6.3

90

Bosonic model for excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.1.1

Interactions between same-spin excitons . . . . . . . . . . . . . . . .

91

6.1.2

Interactions between opposite-spin excitons . . . . . . . . . . . . . . .

93

Polarization dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6.2.1

Linear and circular bases . . . . . . . . . . . . . . . . . . . . . . . . .

94

6.2.2

Co-circular polarization scheme . . . . . . . . . . . . . . . . . . . . .

98

6.2.3

Cross-linear polarization scheme . . . . . . . . . . . . . . . . . . . . . 101

6.2.4

Co-linear polarization scheme . . . . . . . . . . . . . . . . . . . . . . 102

6.2.5

Polarization dependent dephasing rate . . . . . . . . . . . . . . . . . 106

Excitation-density dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.1

Inter-mode interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3.2

Intra-mode interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3.3

Biexcitonic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 Conclusion

112

7.1

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Bibliography

115

Appendix
A Optical Bloch equations

127

x
B Numerical simulations

130

B.1 Phase-matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.2 Inhomogeneous broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.3 Phenomenological many-body effects . . . . . . . . . . . . . . . . . . . . . . 134
C Nominally Disallowed Transitions in (110)-oriented GaAs Quantum Well

136

D Population Evolution

138

xi

TablesTable
3.1

Phase cycling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.1

Parameters of Eq. (4.1) for the fits shown in Fig. 4.3. . . . . . . . . . . . . .

70

4.2

Parameters of Eq. (4.2) for the fits shown in Fig. 4.5. . . . . . . . . . . . . .

74

6.1

Contributions to the transient four-wave mixing signal from the Feynman
diagrams in Fig. 6.2 for the co-circular, cross-linear and co-linear polarization
schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Figures

Figure2.1

Band structure of bulk GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2Band structure of GaAs quantum well with Al0.3 Ga0.7 As barriers . . . . . . .

9

2.3

Exciton dispersion and absorption spectrum for bulk GaAs . . . . . . . . . .

11

2.4

Measured absorption spectrum for GaAs quantum well . . . . . . . . . . . .

13

2.5

Polarization selection rules for excitonic transitions . . . . . . . . . . . . . .

14

2.6

Width fluctuations in a quantum well and the localized exciton . . . . . . . .

17

2.7

Homogeneous vs. inhomogeneous broadening . . . . . . . . . . . . . . . . . .

18

2.8

Illustration of spectral diffusion . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.9

Band structure for (110)-oriented GaAs quantum well . . . . . . . . . . . . .

22

2.10 Polarization selection rules for the exciton and biexciton states . . . . . . . .

24

2.11 Measured absorption spectrum for (110)-oriented GaAs quantum well . . . .

26

3.1

SI time ordering, and non-collinear three-pulse four-wave mixing geometry .

30

3.2

Two-level system, and Feynman diagrams that contribute to signal for the SI
time ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

31

One-quantum rephasing 2D spectra for a homogeneously broadened two-levelsystem – absolute value and real and imaginary parts . . . . . . . . . . . . .

38

3.4

Effect of inhomogeneity on rephasing one-quantum absolute value 2D spectrum 39

3.5

Coherently coupled transition in a three-level V system . . . . . . . . . . . .

40

xiii
3.6

Feynman diagrams indicating coherent coupling in the three-level V system
for the rephasing time ordering . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.7

Rephasing one-quantum absolute value 2D spectrum for a three-level V system 42

3.8

Zero-quantum rephasing absolute value 2D spectrum for tow-level and threelevel V systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

44

Non-rephasing one-quantum time ordering and the Feynman diagrams that
contribute to the four-wave mixing signal . . . . . . . . . . . . . . . . . . . .

46

3.10 One-quantum non-rephasing 2D spectra for a homogeneously broadened twolevel system – absolute value and real and imaginary parts . . . . . . . . . .

47

3.11 One-quantum correlation 2D spectra for a homogeneously broadened two-levelsystem – absolute value and real and imaginary parts . . . . . . . . . . . . .

48

3.12 The SIII or double-quantum time ordering . . . . . . . . . . . . . . . . . . .

49

3.13 Three-level ladder system and the Feynman diagrams that contribute to signal
for the SIII time-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.14 Two-quantum 2D spectra for a homogeneously broadened two-level system –absolute value and real and imaginary parts . . . . . . . . . . . . . . . . . .

51

3.15 Six-level system arising from interaction between two three-level ladder systems 52
3.16 Feynman diagrams indicating coupling in the four-wave mixing signal for SIII
time-ordering and absolute value two-quantum non-rephasing spectrum for
six-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.17 Experimental setup for the two-dimensional coherent spectroscopy . . . . . .

56

3.18 Absolute value one-quantum 2D signal in time and frequency domains . . . .

61

3.19 Phase extraction using spectrally resolved pump-probe . . . . . . . . . . . .

62

3.20 Real and imaginary parts of the absolute value 2D signal shown in Fig. 3.18(b) 62
4.1

Real part and absolute value spectrum from (110)-oriented GaAs quantum
well at a temperature of 10 K . . . . . . . . . . . . . . . . . . . . . . . . . .

68

xiv
4.2

Experimental and best-fit slices from the spectrum in Fig. 4.1(b) for obtaining
the homogeneous and inhomogeneous linewidths . . . . . . . . . . . . . . . .

4.3

Incident photon density dependence of homogeneous linewidth for X and Y
polarization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

69

Rephasing zero-quantum 2D spectrum and vertical slice to measure the population time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

68

72

Temperature dependence of the zero-excitation-density homogeneous linewidthfor X and Y polarization schemes . . . . . . . . . . . . . . . . . . . . . . . .

73

5.1

Effect of the frequency-frequency correlation function on simulated 2D spectrum 78

5.2

Experimental 2D spectra showing spectral diffusion at 5 K . . . . . . . . . .

79

5.3

Experimental 2D spectra showing spectral diffusion at 25 K . . . . . . . . .

80

5.4

Population redistribution of excitons created at two energies and asymmetry
between them with the sample at 5K . . . . . . . . . . . . . . . . . . . . . .

5.5Population redistribution of excitons created at two energies and asymmetry
between them with the sample at 5K . . . . . . . . . . . . . . . . . . . . . .

5.6

87

Change in the total signal as a function of the waiting time for different sample
temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

86

Change in frequency-frequency correlation function with waiting time for different sample temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.9

84

Experimental and simulated diagonal and cross-diagonal slices for T = 0.2
and 26.6 ps for sample temperature of 25 K . . . . . . . . . . . . . . . . . .

5.8

83

Variation of asymmetry between relaxation and excitation processes with sample temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

82

88

Energy level scheme for excitons as harmonic oscillators with the same spin
and both ±1 spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

xv
6.2

Feynman diagrams denoting the dipole allowed quantum pathways that contribute to the TFWM signal for the energy level scheme shown in Fig. 6.1(b)

6.3

95

Absolute value and real part 2D spectra for the co-circular polarization scheme– measured and simulated . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

6.4

Experimental and simulated best-fit slices for the co-circular polarization scheme100

6.5

Absolute value 2D spectra for the co-circular polarization scheme – measured
and simulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.6

Experimental and simulated best-fit slices for the cross-linear polarization
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.7

Absolute value and real part 2D spectra for the co-linear polarization scheme– measured and simulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.8

Experimental and simulated best-fit slices for the co-linear polarization scheme105

6.9

Comparison of cross-diagonal slices through the exciton peak for the cocircular, cross-linear and co-linear polarization schemes. . . . . . . . . . . . . 106

6.10 Simulated time-integrated transient four-wave mixing signal for individual
parts and the total signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.11 Excitation density dependence of (a) the resonance energy and (b) the dephasing rate of the |Gi ↔ |1±i transitions . . . . . . . . . . . . . . . . . . . 109
6.12 Intra-mode exciton-exciton interaction terms as a function of the exciton density110
6.13 Biexciton binding energy and dephasing terms as a function of the exciton
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.1 Longitudinal “spatial” profile of the incident electric field and phase-matchingcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.2 Inhomogeneous broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.3 Numerical simulation for an inhomogeneously-broadened two-level system . . 133

xvi
B.4 Numerical simulation for an inhomogeneously-broadened two-level system including excitation-induced shift term . . . . . . . . . . . . . . . . . . . . . . 135
C.1 Rephasing one-quantum 2D spectrum with the heavy and light hole excitons,
coupling peaks and nominally disallowed transitions in (110)-oriented GaAs
quantum well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
D.1 Change in time-integrated transient four wave mixing signal with waiting time
for different sample temperatures . . . . . . . . . . . . . . . . . . . . . . . . 138

Chapter 1
Introduction

Several semiconductor nanostructures such as quantum wells (QWs), nanowires andquantum dots have been developed and studies over the last few decades. QWs are perhaps
the most widely studied of these nanostructures. The optical properties of these structures
has been the focus of many of these studies. The near-bandgap optical properties of QWs isdominated by excitons, which are correlated electron-hole pairs, for direct bandgap materials
at low temperatures. The interaction of exciton with light is complicated due to several
factors. The presence of multiple transitions, inhomogeneous broadening, exciton-exciton
interaction, etc. provide a plethora of interesting physics problems, which motivate continued
research in this field.
The optical excitation in QWs typically last for tens of picoseconds. Thus, ultrafast
spectroscopy techniques are ideal for probing the optical properties of excitons. One of themost common techniques is transient four-wave mixing (TFWM) [1]. The exciton coherence
lifetime was revealed through the decay of photon echo even in the presence of inhomogeneous
broadening [2]. Coupling between exciton states was indicated by the presence of quantum
beats in the TFWM signal [3]. Exciton-exciton interactions in the form of excitation-induceddephasing (EID) [4], excitation-induced shift (EIS) [5], and local field effects [6] were studied.
Although TFWM experiments have been instrumental in studying exciton physics, it has
a few shortcomings. Interpreting the correct behavior from oscillatory signal in the timedomain can be challenging [3]. Furthermore, any spectral dependence of the aforementioned

2
properties is averaged for an ensemble of oscillators.
Two-dimensional coherent spectroscopy (2DCS), which is an extension of the TFWM
technique, overcomes these shortcomings. 2DCS is motivated by a similar technique usedin nuclear magnetic resonance studies [7] and was developed in the visible [8] and infrared [9] regimes to study electronic and vibrational properties of molecules. Since then, thistechnique has also been used to study excitons in semiconductor nanostructures [10]. The
spectrally-resolved homogeneous response can be measured in the presence of inhomogeneity[11]. Coupling between multiple states appear as isolated peaks in a 2D spectrum [12]. The
measurement of the signal phase has revealed many-body effects such as EIS [13]. The
existence of multiple-exciton states have also been studied through 2DCS experiments.
This thesis brings new perspectives to exciton physics in semiconductor QWs using
2DCS. Although many exciton properties have been discussed qualitatively in previous 2DCS
experiments, few studies have focused on quantifying these effects, in part due to complicatedsimulation procedure [14]. A major focus of this work is on developing numerical fitting
methods that can be used to quantify these properties.
In Chapter 2, we will present a brief review of some of the basic concepts in semiconductor optics. We will introduce the concept of excitons and discuss some of its properties suchas dephasing rate, inhomogeneous broadening, etc. Specific properties of excitons, relevant tothis work, will be discussed. The general principles of 2DCS will be explained in Chapter 3.
We will discuss modeling the excitonic system through the optical Bloch equations (OBEs).
A simple theoretical framework, based on the double-sided Feynman diagrams, which canbe used to write down the TFWM signal will be presented. We will highlight the advantagesof the 2DCS technique through simulated 2D spectra for various model systems. We will
describe the experimental considerations and features of our implementation of 2DCS. The
polarization-dependent measurements of the exciton resonance in (110)-oriented GaAs QWs
will be presented in Chapter 4. The thermal broadening due to phonons will be discussed.
Our studies of the spectral diffusion of excitons will be presented in Chapter 5. The limita-

3
tions of the previous methods of analyzing spectral diffusion for low sample temperatures will
be highlighted. We will quantify spectral diffusion for high sample temperatures through a
numerical fitting procedure. Chapter 6 will discuss modeling excitons as interacting bosons.
This model will be used to explain the polarization-dependent dephasing rate of excitons.
We will separate the exciton-exciton interactions into inter- and intra-mode components. A
numerical fitting procedure will be demonstrated to quantify these interaction terms.
The derivation of the OBEs for a two-level system will be discussed in Appendix A.
Appendix B will discuss the numerical solution of the OBEs. Some preliminary measurements indicating the contribution of nominally disallowed transition to 2D signal will bepresented in Appendix C. The observation of non-monotonic exciton population decay rates
will be demonstrated in Appendix D.

Chapter 2
Excitons in Semiconductor Quantum Wells

The optical response of direct-gap semiconductors is dominated by excitons near thefundamental band gap at low temperatures.

The interaction of exciton with photons,

phonons, other excitons and free carriers affects the optical response of excitons significantly
resulting in very rich physics, which has been studied over the past several decades. In thiswork, we focus on the non-linear light-matter interactions in semiconductor heterostructures,
specifically direct band gap GaAs QWs. Before we can discuss the findings of this work, it
is imperative to provide a brief summary of the relevant exciton properties.
In this chapter, we will review the properties of excitons in semiconductor QWs thatare relevant to the work presented in this thesis. The band structure of direct band gap
semiconductors will be discussed in Section 2.1 followed by an introduction to excitons in
Section 2.2. In Section 2.3, we will discuss the processes that result in decay of excitations
in a semiconductor. Inhomogeneous broadening of the exciton resonance due to fluctuations
in QW width will be discussed in Section 2.4. After giving a general description of excitons,
we will discuss the specific properties of excitons that will be explored later on in this work.
We will also provide an overview of the previous works that have studied these properties.
Inhomogeneous broadening of the exciton resonance results in spectral diffusion of the exciton
resonance energy, which will be discussed in Section 2.5. The effect of crystal symmetry on
excitons will be discussed in Section 2.6. Exciton-exciton interactions significantly affect thelight-matter interaction in semiconductors. Some of these interactions will be discussed in5Section 2.7. Finally, we will introduce the concept of thinking of excitons as interacting
bosons in Section 2.8.
2.1

Semiconductor quantum well

2.1.1

Bulk Semiconductor

Semiconductors are a class of materials that have electronic properties intermediate
between those of an insulator and a metal. The electronic properties of a crystalline material
are generally determined by the electronic band structure of the material. Valence bandis the highest energy band, which is completely occupied by electrons. The next higherlying energy band is the conduction band. The electrical conductivity of a material isdetermined by the distribution of electrons in the conduction and valence bands. In a
metal, the conduction band is partially occupied. These high-energy electrons can be easily
scattered into unoccupied states on application of an external electric field resulting in thegood electrical conductivity of metals. In an insulator, the highest energy electron is in thevalence band, which is completely filled. As a result, there is no scope for the intra-band
scattering of electrons on the application of an external electric field and insulators do not
conduct electricity. The conduction and valence bands in an insulator are separated by anenergy gap, known as the band gap energy. The situation for a semiconductor is very similar
to that in an insulator. However, the band gap in case of a semiconductor (∼ 1 eV) is muchless than that in insulators. As a result, electrons can be excited from the valence band tothe conduction band by absorption of light in the visible and near infra-red region of theelectromagnetic spectrum, which results in many interesting properties of semiconductors.
The repeating pattern of the crystal structure in a semiconductor results in a periodic
potential
V0 (r) = V0 (r + Rn )(2.1)

where Rn = ni ai is a lattice vector, ai is a basis vector, and ni is an integer. As a result, an6electron in a crystalline lattice must satisfy the Bloch theorem [15]eik·Rn ψλ (k, r) = ψλ (k, r + Rn )(2.2)

where k is the crystal momentum, and ψλ is the electron wave function.

For a one
dimensional lattice with periodicity a, the first Brillouin zone is defined as the range − a2 , a2 .
We will only discuss the range of k in the first Brillouin zone. The Bloch theorem is satisfied
by the Bloch wave function1
ψλ (k, r) = √ eik·r uλ (k, r)
L3

(2.3)

where L is the length of each side of a cubic crystal, and uλ is the Bloch function, which
has the same periodicity as the periodic potential in Eq. (2.1). The plane wave part of the
Bloch wave function implies that the electronic states are delocalized. The electronic states
defined by the aforementioned equations include interactions of electrons with the periodic
lattice of nuclei. As a result, these states describe pseudo-particles, which are also called
electrons [16].
The band structure for GaAs is shown in Fig. 2.1, which shows a few electronic statesaround wavevector k = 0. A parabolic band dispersion is assumed, in the effective mass
approximation limit, around the band extrema at k = 0. The energy of the conduction bandand two valence sub-bands – heavy hole (HH) and light hole (LH) are shown as a functionof the crystal wave-vector k. The minimum of the conduction band and the maxima of
the valence sub-bands lie at k = 0, which makes GaAs a direct-gap semiconductor. The
difference in energy of the conduction band minimum and the valence band maxima is theband gap energy Eg . A photon with energy ~ω ≥ Eg can excite an electron from one of thevalence bands to the conduction band leaving behind a hole, which is a positively charged
pseudo-particle, in the valence band as shown in Fig. 2.1. The linear momentum of aphoton is negligible, compared to the crystal momentum at the edge of the first Brillouinzone, and is ignored. In other words, the excited electron and hole are assumed to have equalwave-vectors.

7
2.0

Energy (eV)CB
1.5

E = ħω

Eg
0.0

HH
LH

-0.5
-0.10 -0.05

0.00

0.05

0.10

-1

Wavevector (A )Figure 2.1: Band structure of bulk GaAs showing the conduction (CB) and valence subbands – heavy hole (HH), and light hole (LH). The direct band-gap Eg is indicated. A
photon with energy E = ~ω can excite an electron, indicated by the solid circle, from avalence band to the conduction band leaving behind a hole, indicated by the empty circle,
in the valence band. Note that the HH and LH bands are degenerate at k = 0. This figure
has been adapted from Ref. [17].

The conduction and valence bands have s- and p-orbital like wave functions, respectively, and, thus, an orbital angular momentum L = 0 and 1, respectively. We can define
the total angular momentum J = L + S for both the bands. The electron and hole have aspin angular momentum S = 12 . Thus, J = S = 21 in the conduction band. The conduction
band in Fig. 2.1 shows the degenerate electron states with Jz = Sz = ± 12 , where Sz is
the projection of S along the Z direction. We obtain J = 21 , 32 for the valence bands. The
J = 32 valence sub-bands are shown in Fig. 2.1. The HH and LH bands are also doubly
degenerate states with total angular momentum projections Jz = ± 32 and ± 21 , respectively.
The states with J = 12 constitute the split-off band; this band is separated from the HH and
LH sub-bands by a large energy difference and is not shown in Fig. 2.1.
The electron or hole effective mass at k = 0 is proportional to the radius of curvatureof the respective band, and is given by the relation [15]m∗i = ~2

 2 −1
∂ Ei
∂k 2

(2.4)
k=0

8
where the index i indicates either an electron (e) in the conduction band, or a heavy hole(hh), or light hole (lh) in the valence sub-bands. Ei indicates the energy in the same band.
The different curvatures of the conduction and valence bands results in different effective
masses for electron and holes; m∗e = 0.067m0 , m∗hh = 0.51m0 , and m∗lh = 0.082m0 in GaAs
for temperatures < 100 K [18]. m0 is the free electron rest mass. The group velocity of theelectron or hole at a particular k is proportional to the slope of the band, and is given by
[19]
vi =

1 ∂Ei
.
~ ∂k

(2.5)

Thus, vi = 0 at k = 0. This property is relevant for formation of excitons, as we will discussin Section 2.2.

2.1.2

Quantum well

A QW is formed when a thin layer of a low band gap semiconductor is sandwiched
between a higher band gap material, which acts as the barrier. These structures can be grownthrough molecular beam epitaxy or metal-organic chemical vapor deposition. Multiple layers
of QW and barriers are grown to increase the effective light-matter interaction length.
A commonly used QW for studying fundamental many-particle physics in the solidstate is GaAs QW with aluminum gallium arsenide Alx Ga1−x As barriers; x is the fraction of
Al. Figure 2.2(a) shows the relevant band gaps, at 0 K, for a GaAs QW with Alx Ga1−x As
barriers, for x = 0.3. The confinement potentials for the conduction and valence bands arealso indicated. The QW thickness is usually around 10 nm, which results in confinementof the electron and hole wave functions between the barriers. This confinement results indiscrete energy levels indicated by the labels CB, HH and LH for electron, heavy hole andlight hole states, respectively, in Fig. 2.2(a).
The electrons and holes, however, are free to move around in the plane of the QW,
resulting in the quasi-2D structure of QWs. The band structure for a GaAs QW is shown in9

QW width2.0

CB1.95 eV

Energy (eV)0.27 eVCB

1.52 eV

1.5

0.0

HH
LH

HH
LH
(a)

Al0.3Ga0.7As-0.5

0.16 eVGaAs

Al0.3Ga0.7As-0.10 -0.05

(b)

0.00

0.05

0.10

Wavevector (A-1)

Figure 2.2: (a) Sample structure of GaAs quantum well with Al0.3 Ga0.7 As barriers. GaAs andAl0.3 Ga0.7 As have band gaps of 1.52 and 1.95 eV, respectively. The confinement potentials
for the conduction and valence bands are 0.27 and 0.16 eV, respectively. The ground stateenergies of the conduction band electron (CB), heavy hole (HH), and light hole (LH) areindicated along with their wave functions. This figure has been adapted from [20]. (b)Modified band structure for GaAs quantum well showing the conduction (CB) and valence
sub-bands - heavy hole (HH) and light hole (LH). Notice the broken degeneracy for HH and
LH bands at peak of the valence sub-bands. This figure has be adapted from [17].

Fig. 2.2(b), where the wavevector is defined in the QW-plane. The usual labels are used forthe different states. Although this band structure looks very similar to the one for bulk GaAs
in Fig. 2.1, quantum confinement affects the electronic band structure in two major ways
– (1) the band gap is increased compared to bulk GaAs, and (2) the degeneracy between
HH and LH bands at k = 0 is lifted. In addition to these effects, lattice mismatch between
the QW and barrier materials introduces strain in the QW structure, which also affects theelectronic band structure. However, the lattice mismatch between GaAs and Al0.3 Ga0.7 As is
negligible and can be ignored.
In the next section, we will describe the exciton states in semiconductors, which is thefocus of this work.

10
2.2

Exciton states

2.2.1

Bulk excitons

We have discussed the creation of electron-hole pairs in a semiconductor by the absorption of light. A Coulomb attraction potential exists between the excited electrons andholes. This electron-hole pair interaction results in new eigenstates, which are the exciton
states. The exciton wave function can be separated into the relative and center-of-mass coordinates. The exciton comprises a positively charged hole and a negatively charged electron.
This situation is analogous to that in the hydrogen atom and the relative wave functionsof an exciton are similar to the wave functions of the hydrogen atom. One can also define
an exciton Bohr radius aB , which gives the mean distance between an electron-hole pair.
However, the center-of-mass wave functions of an exciton are delocalized, similar to thedelocalized electron and hole states given by Eq. (2.3).
An essential condition for the creation of an exciton is that the electron and hole shouldhave the same group velocity, which is given by Eq. (2.5). Otherwise, the electrons and holes
move apart and cannot form a bound state. For example, the electronic excitation shownin Fig. 2.1 creates electron and hole with opposite group velocities, and will not form an
exciton. As a result, optical creation of excitons only occurs ∼ k = 0 for band structures
similar to that shown in Fig. 2.1.
Instead of dealing with electrons and holes separately, it is possible, and convenient,to define exciton states. Figure 2.3(a) shows the dispersion relation for different exciton
states. The E(n, 0) = 0 point indicates the unexcited semiconductor, i.e. the valence bandis completely filled and the conduction band is completely unoccupied. This state can bethought of as the vacuum state in the exciton basis. The shaded electron-hole continuum
region starts at the band gap energy Eg at k = 0. The n = 1 exciton state is red-shiftedwith respect to the continuum state by the exciton binding energy ER , which is analogous
to the Rydberg energy for the hydrogen atom. A photon with energy ~ω = E1 can create

11
E(n,k)
electron-hole
continuum

n=2

n=1

α

n=1

ER

E2

n=2

E1

continuum

Optical excitation:

Eg

E=

c
k
n

(b)

E1

E2 Eg

E

(a)
0

k

Figure 2.3: (a) Dispersion of the exciton states. The n = 1 and n = 2 states have energies E1
and E2 , respectively, at k = 0. The exciton ground state energy E1 is less than the band-gap
energy Eg by the exciton ground state binding energy ER . The gray region indicates theexciton continuum states. (b) The absorption spectrum for bulk GaAs. The sharp peaks are
due to the quantized exciton states, while the broad-band absorption is due the continuum
states of the exciton. These figures have been adapted from [21].

an exciton in the n = 1 state. A small linear momentum is transferred by the photon to theexciton.
For bulk semiconductors, the energy of the nth exciton state is given by
En = Eg −

ER
.
n2

(2.6)

Similar to the hydrogen atom case, ER and aB are related as
ER =

~2
2µa2B

m∗ m∗

(2.7)
2

~
where µ = m∗e+mh∗ is the reduced mass of the exciton and aB = µe
2 with dielectric constanteh. m∗e and m∗h are the electron and hole effective masses, respectively. For the HH excitonin bulk GaAs, aB = 14nm and ER = 4.2 meV [22].
The discrete exciton states can be readily identified as the sharp resonances in theabsorption spectrum shown in Fig. 2.3(b). The broad band absorption feature is due to12absorption in the continuum states. The overlap integral between the electron and hole
wave functions determines the inter-band transition dipole moment and oscillator strength.
This overlap integral is significantly greater for the exciton states compared to the continuum
states, which results in a higher absorption coefficient for the exciton states. The oscillator
strength for the various exciton state fn ∝ n−3 [15] and results in a fast decrease in absorption
coefficient for higher energy exciton states.
2.2.2

Quantum well excitons

Quantum confinement, in a QW, modifies the density of states in the continuum. The
general exciton properties discussed in Section 2.2.1 are still valid in QWs. However, there
are a few quantitative differences. The energy of the nth exciton state in a QW is given by
[15]
ER,QW
(2.8)
2
n − 12where the same definition is used as earlier and subscript QW is used to denote that the
En,QW = Eg,QW −quantities are for a QW. For infinite potential barriers, aB,QW = 12 aB and ER,QW = 4ER ,which is not the case in reality. For example, ER,QW ≈ 8.5 meV for the HH exciton in GaAs
QWs with Al0.3 Ga0.7 As barriers [23, 24]. Finally, the change in the oscillator strength for
−3
[15].
different exciton states follows the relation fn ∝ n − 12
Figure 2.4 shows the experimentally measured absorption spectrum for a four-period
10-nm-wide GaAs QW with 10-nm wide Al0.3 Ga0.7 As barriers. The absorption spectrum
comprises two peaks corresponding to absorption in the n = 1 state of the HH and LH
excitons. The two peaks are separated because the degeneracy between the HH and LH
bands is lifted due to quantum confinement as shown in Fig. 2.2(b).
2.2.3

Polarization selection rules

The angular momentum values for the electron and hole states were discussed in Section
2.1.1. Electrons are excited from the valence band to the conduction band by the absorption

13
0.6

Absorbance

HH
0.4

LH
0.2

0.0
1540

1545

1550

1555

Photon energy (meV)

Figure 2.4: Measured absorption spectrum for a four-period 10-nm-wide GaAs quantum
well with 10-nm-wide Al0.3 Ga0.7 As barriers showing the heavy hole (HH) and light hole(LH) peaks.

of photons. As the photon is absorbed, its momenta – angular and linear – are transferedto the electron. The linear momentum of a photon is negligible and can be ignored for allpractical purposes. A photon also has unit spin angular momentum – +1 and -1, respectively,for σ+ and σ− polarized light. The total angular momentum should be conserved during
inter-band transitions. This conservation involves the spin angular momentum of light, andresults in the polarization selection rules shown in Fig. 2.5.
Figure 2.5(a) shows all the optically allowed transitions of electrons from the valenceband to the conduction band. σ+ polarized light can excite electron with Jz = − 23 Jz = − 21from the HH (LH) band to the conduction band with Sz = − 21 Sz = + 21 . Alternatively, σ−polarized light can excite electron with Jz = + 23 Jz = + 21 from the HH (LH) band to theconduction band with Sz = + 21 Sz = − 21 . The HH and LH states are coupled through theelectron states. The Sz = − 12 electron state couples the Jz = − 23 HH state to the Jz = + 12
LH state. Similarly, the other HH and LH states are coupled through the Sz = + 12 electronstate.
Another, more convenient, way to understand the polarizations selection rules is in the

14
CB electron

−1/2

−1

+1/2

+1

LH exciton
HH exciton

σ+

σ−
σ+

σ−
HH
−3/2
LH
(a)

+3/2
+1/2

−1/2

(b)

G

Figure 2.5: (a) Polarization selection rules for excitons in the electron-hole basis. The labels
indicate the projection of the total angular momentum Jz along the growth direction for thedifferent electron and hole states. The spin of the photon (±1) is transfered to the electron
as it is excited from the conduction band to the valence band by circularly polarized light.
σ+ and σ− polarized light is indicated by solid and dashed arrows, respectively. (b) The
same transitions are shown in the exciton basis. The exciton states are labeled with spin ±1
indicating the circular polarization of light that was used to create the exciton.

exciton picture, as shown in Fig. 2.5(b). In the exciton picture, both HH and LH exciton
states are considered to be doubly degenerate state with ±1 spin and the exciton is createdwith a spin, which is equal to the photon spin. σ+ and σ− polarized light creates HH and
LH excitons with +1 and −1 spins, respectively.
Now that we have discussed excitons in QWs and how they are created through opticalexcitation, we would like to discuss the processes that lead to a decay of these excitations.
We will only discuss the decay processes relevant for excitons.
2.3

Decay processes
We will use the density matrix formalism to describe excitations of the excitonic system.

Such a formalism is especially useful for an ensemble of particles, as is the case for excitons.
A two-level system with ground and excited states, respectively, can be described using thefollowing density matrix ρgg ρge ρ=ρeg ρee(2.9)

15
where the indices e and g are used for the ground and excited states, respectively. Thus, thediagonal terms of the density matrix (ρgg and ρee ) denote the population in the ground andexcited states, respectively; tr(ρ) = 1 for a closed system. The two off-diagonal elements ofthe density matrix (ρge and ρeg ) are complex conjugates of each other and denote a coherent
superposition, or coherence, between the two states. The coherence has an associated phase,
which evolves with the frequency given by the energy difference between the two states. In
the case of excitons, we can assume the excited state to be one of the exciton states.
If the excited state has a lifetime T1 , the following equation describes the evolution of
the excited state population with time t− t

ρee (t) = ρee (0)e T1 .

(2.10)In the frequency domain, we obtain a Lorentzian line shape with width quantified by T11 . A
similar relation is generally used to describe evolution of the coherence between the groundand exciton states− tρeg (t) = ρeg (0)e−iωt e T2

(2.11)

where T2 is the dephasing time and the energy between the states is ~ω. The term e−iωt accounts for the phase evolution of the coherence. A decrease in population of the excited statealso results in decay of the coherence. In the absence of other coherence decay mechanisms,
the population and dephasing times are related as
T2 = 2T1(2.12)

in a closed system. If the decay of population results in the emission of photons, due toelectron-hole recombination in the case of excitons, the lifetime and dephasing times are
said to be radiatively limited. Bulk exciton population lifetime in GaAs is ∼ 1 ns [25]. The
exciton lifetime in QWs depends on the growth and sample quality. Radiatively limited
lifetimes of 10-30 ps have been reported for GaAs QWs [2, 26].

16
In addition to radiative decay, non-radiative decay is also possible in semiconductors.
For instance, trap sites due impurities or imperfections in crystal lattice can result in population decay in addition to that due to electron-hole recombination. Another important
non-radiative decay process is “pure dephasing”, which is a decay in coherence without any
corresponding population decay. Pure dephasing can also be understood as a loss of phasememory during the evolution of the coherence. Exciton-exciton scattering is an importantsource of pure dephasing in QWs. This effect is same as the phenomenological many-bodyeffect – excitation-induced dephasing (EID), which is described in Section 2.7. In general,the combined effect of all the pure dephasing mechanisms is taken into account through therelation
1
1
1
=
+ ∗
T2
2T1 T2(2.13)

where T2∗ is the effective pure-dephasing lifetime.
In order to obtain Eq. (2.13), an exponential coherence decay, similar to Eq. (2.11),
is assumed for the pure-dephasing processes. This assumption is known as the Markovian
approximation, which assumes that the phase-memory loss occurs instantaneously. As thediscussion on spectral diffusion in Section 2.5 will highlight, the Markovian approximation
is not valid for QW excitons. Spectral diffusion occurs due the presence of inhomogeneous
broadening in the exciton resonances, which will be discussed in the next section.
2.4

Inhomogeneous broadening
In our treatment of excitons, we have assumed a single energy for each state. Thiscondition would be true for perfectly smooth layers of the barrier and QW materials, whichcannot be realized in practice. Spatial variations in QW widths are introduced in the growth
process as indicated in Fig. 2.6. These variations result in different exciton resonancefrequency at different spatial locations. Thus, instead of a single, well-defined resonanceenergy, all the exciton states have an inhomogeneous distribution of resonance energies.

17This distribution of exciton energies results in different phase evolution of distinct frequency
components. The collective coherence is called the macroscopic polarization P , which isdefined by the relation
P = N tr(µρ)(2.14)

where N is the total number of oscillators and µ is the transition dipole moment matrix 0 µge µ=
(2.15)
µge 0
where the non-zero elements are complex conjugates of each other. Since an optical transition
involves two states, the diagonal elements of the dipole matrix are zero. In the presence of
inhomogeneity, the macroscopic polarization can decay much faster than the dephasing time
discussed in Section 2.3.
Due to the variations in the QW-width, areas of larger width can be surrounded bysmaller width regions. Such a situation results in a small in-plane confinement potential for
excitons in the areas with larger QW width since the exciton energy increases with decreasing
QW width. These areas are known as localization sites and can be as big as 10s of nanometersin size [27, 28]. A low energy exciton can be trapped in a localization site, as shown in Fig.
2.6, resulting in a localized exciton. An extreme limit of such a localization results inthe formation of natural quantum dots [29]. Thus, the treatment of excitons as perfectly
delocalized states is not strictly valid in the presence of inhomogeneous broadening. The© 2012, IEEE

Figure 2.6: Illustration of well width fluctuations in a quantum well sample. These fluctuations result in an inhomogeneous distribution of the resonance energies of the exciton state.
A localized exciton, which is trapped due to the width fluctuations is indicated. This figure
has been reproduced from Ref. [10].

Absorbance18

Photon energy

Figure 2.7: A broad absorption peak shown by the solid line can be be composed of several
narrower peaks such as shown by the dotted lines.

excitons with high resonance energy, however, are still delocalized. Localized and delocalized
excitons have distinct properties, such as exciton decay times [30]. Interface roughness also
introduces non-radiative decay channels.
As a result of the inhomogeneous broadening, the peaks in the absorption spectrum
shown in Fig. 2.4 result from several narrower peaks. This situation is depicted in Fig.
2.7. The peaks indicated by the dotted lines are the narrower resonances. We have useda Lorentzian distribution for these peaks, which is consistent with the exponential decayof excitations discussed in Section 2.3. The linewidth of the narrow peaks is the homogeneous linewidth. A collection of these spectrally separated narrow peaks results in theinhomogeneously broadened exciton resonance indicated by the solid line in Fig. 2.7. The
inhomogeneous distribution is usually a Gaussian function, as we have assumed here. The
linewidth of the Gaussian distribution gives the inhomogeneous linewidth of the system.
The width fluctuations, and, consequently, the inhomogeneous linewidth of excitons in QWs
depends significantly on growth conditions. Improved growth techniques have resulted in a
reduction in the inhomogeneous linewidth from a few meVs [27] to less than a meV [31].
Figure 2.7 depicts all the Lorentzian resonances with the same width, which is not19necessarily true for excitons in QWs. The localized excitons have a lower linewidth thandelocalized excitons. The linewidth of the localized excitons is nominally independent of
the resonance energy. The linewidth of the delocalized excitons, however, increases with anincrease in the resonance energy. The lowest resonant energy of a delocalized exciton is themobility edge, which usually lies at the center of the inhomogeneous distribution of exciton
resonance energies [32, 33].
We have provided a general overview of excitons in semiconductor QWs. In the following sections, we will introduce specific properties of QW excitons – spectral diffusion,
effect of crystal symmetry on excitonic properties, exciton-exciton interactions, and treating
excitons as interacting bosons, which will be explored in detail in rest of this work.
2.5

Spectral Diffusion
Due to the inhomogeneity in the exciton resonance energy, spatial migration of an

exciton, in the QW plane, results in a change in the resonance energy of the exciton. Thisphenomenon is spectral diffusion [34]. The migration of excitons involves interactions withphonons – the exciton energy increases and decreases on absorption and emission of phonons,
respectively. This effect is similar to the time-dependent frequency fluctuations seen in
atomic [35] and molecular [36, 37] systems. The time-dependent transition energy ω can be
defined as ω(T ) = hωi + δω(T ), where T is the time, hωi is the time-averaged value of thetransition energy and δω is the difference between the instantaneous and mean values of thetransition energy. Additionally, one can define a normalized frequency-frequency correlation
functionC(T ) =hδω(T )δω(0)ihδω 2 i(2.16)

where the average is an ensemble average, and hδω 2 i is the inhomogeneous width. The
Markovian approximation for pure dephasing, discussed in Section 2.3, assumes δ-function
correlated frequencies [38]. Spectral diffusion essentially violates this assumption and, thus,

20
(b)

Absorbance(a)

Excitation energy

Figure 2.8: Illustration of spectral diffusion. (a) Spectral diffusion in the strong redistribution
approximation. Red and blue arrows indicate a decrease and increase in the energy ofthe excitations. (b) Preferential relaxation results in violation of the strong redistribution
approximation.

is quantified through the decay in C(T ) as T increases in several systems, including QW
excitons [39, 40, 41].
The description of spectral diffusion presented above is valid only in the strong redistribution approximation, which assumes that spectral diffusion results in an equal redistributionof the transition energy of an excitation to higher and lower energies as indicated in Fig.
2.8(a). While this approximation is valid for atomic and molecular systems, which lack static
inhomogeneity, we will show in Chapter 5 that this assumption is not valid for spectral diffusion of excitons in QWs, especially for low (< 25 K) sample temperatures. This situation
is similar to the one shown in Fig. 2.8(b). Furthermore, we will also show that even for highsample temperatures, when the strong-redistribution approximation is valid, C(T ) does notdecay exponentially, which is generally assumed in the Gauss-Markov approximation [38, 40].
2.6

Crystal symmetry
Different semiconductor materials belong to different crystal structure categories and,thus, have different structural symmetry. A material’s structural symmetry affects its electronic properties significantly. The same holds true in the case of QWs. In addition to21changing material composition, it is possible to change the symmetry of a QW structure
depending on the growth direction. While the most commonly used GaAs QWs are grownalong the (001) crystal axis, GaAs QWs in other growth directions such as (110), (111), and(113) are also possible. All these QWs have different symmetries and their distinct opticalproperties.
In addition to (001)-oriented GaAs, in this work, we will discuss excitons in (110)oriented GaAs QWs, which have garnered significant interest recently for optical generationof spin currents [42] and terahertz radiation [43]. For a (110)-oriented GaAs QW, where the
QW itself is symmetric, the in-plane symmetry is reduced to C2v point group compared to
Td for bulk GaAs and D2d for an undoped (001)-oriented GaAs QW. The reduced symmetry
results in the different spin (angular momentum) states of the conduction (valence) bands
being split in k-space along the [11̄0] crystal axis while no splitting exists along the [001]crystal axis [44, 45], and, along with the k-linear terms of the transition dipole moment,
contributes to the generation of spin currents. The schematic band structures for the first
conduction (Sz = ± 1/2) and HH valence (Jz = ± 3/2) bands along the X and Y directions
are shown in Figs. 2.9(a) and 2.9(b), respectively. Sz and Jz are the projections, on the
Z-axis, of the spin of an electron and the total angular momentum of a HH, respectively. We
use a co-ordinate system with the X, Y and Z axes aligned along the [001], [11̄0] and [110]crystal axes, respectively, of the sample.
Very long spin relaxation times [47, 48] and generation of spin currents by opticalexcitation [42] in these nanostructures find applications in terahertz generation [43] andspintronics. In coherent control experiments, carrier dephasing rates influence the opticallygenerated currents in bulk GaAs [49] and (001)-oriented QWs [50]. A recent coherent control
study on (110)-oriented GaAs QWs revealed novel photocurrents that directly depend on
exciton polarization dephasing [51]. The contribution of excitons to spin currents [52] and theelectron spin relaxation time [53] were also inferred. However, not many works have studied
excitonic effects in these QWs. We will present measurements of the exciton dephasing times

22
EnergyEnergyS z =−1/2

S z =1/2_ 1/2
Sz = +
ky

kx

(a)

_ 3/2
Jz = +(b)

J z =3/2

J z =−3/2

Figure 2.9: An illustration of the energy levels for the conduction and HH valence bandsin (110)-oriented GaAs QWs, as a function of wave vector along [001] (kx ) and [11̄0] (ky )directions are shown in (a) and (b), respectively. The energy is shown for the conduction(HH valence) bands with spin and angular momentum states Sz = ±1/2 and Jz = ±3/2,
respectively. The figure has been adapted from Ref. [46].

for (110)-oriented GaAs QWs in Chapter 4 [46].
2.7

Exciton-Exciton Interactions
Our discussion on excitations in a semiconductors is only strictly valid for an unex-

cited sample. Excitations in a semiconductor affect its electronic properties significantly due
to several effects such as phase-space filling, screening, band gap renormalization, etc [16].
The semiconductor Bloch equations (SBEs), which are a modification of the more widelyused optical Bloch equations (OBEs), are used for a rigorous description of these effects[54]. However, a phenomenological treatment of these effects in terms of exciton-exciton interactions, dubbed many-body effects, is usually sufficient to gain a physical understandingof these effects [5]. These effects are especially relevant for non-linear light-matter interactions, which is the central theme of this work. Some of the prominent many-body effects are
excitation-induced dephasing (EID), excitation-induced shift (EIS), local field corrections,and higher order correlations such as biexcitons.
EID refers to an increase in the dephasing rate of the excitonic coherence with increase23
in the exciton density [55, 4]. This effect is analogous to the thermal collisional broadening
in case of atoms [56].
EIS is a complimentary effect due to which the exciton resonance energy changes withthe exciton density. Two competing effects contribute to EIS – (1) screening of the Coulombinteraction between electrons and holes results in a blue-shift in the exciton resonance due
to a decrease in the exciton binding energy, and (2) band gap renormalization results ina red-shift. These two effects almost exactly cancel each other out for excitons in bulk
semiconductors, resulting in no measurable EIS [57]. In case of excitons in QWs, however,the screening effect dominates and results in a blue-shift of the exciton resonance with anincrease in the exciton density [58, 59].
Local-field correction for excitons [60, 6] accounts for the fact that in addition to theexternal electric fields, the macroscopic polarization inside the sample also interacts with thesystem. The effect of local-field correction has been shown to be very similar to EIS [5].
Higher order correlations between two or more electron-hole pairs have significant effecton the non-linear optical response of QWs. Biexciton, which is a bound state of two excitons
with opposite spins, is an example of such a higher-order correlation [61]. Many othercorrelated states of electron-hole pairs have been identified in QWs recently [62, 63, 64, 65].
Presently, we will limit our discussion to that of biexcitons.
Effect of biexcitons can be included phenomenologically by adding a bound two-exciton
state to the energy level scheme, as shown in Fig. 2.10 [66, 67]. The biexciton state hasa energy that is less than twice the exciton state energy by the biexciton binding energy∆B . Two excitons with opposite spins form a biexciton, which is analogous to a hydrogenmolecule. As a result, biexciton is a spin 0 pseudo-particle. The polarization selection
rules for the exciton-biexciton system, in the basis of circularly polarized light, is shownin Fig. 2.10(a). Opposite circularly polarized photons are needed to create a biexciton.
An equivalent polarization selection rule, in the linear polarization basis, is shown in Fig.
2.10(b) [67].

24

Two-exciton∆B

Biexciton

σ+

σ−

X

Y

X

Y

One-excitonσ−σ+Ground(a)(b)

Figure 2.10: The ground, one-exciton, two-exciton and biexciton states are shown with polarization selection rules in the (a) circular and (b) linear bases. The energy of the biexciton
state is less than that of the two-exciton state by the the biexciton binding energy ∆B . σ+
and σ− polarized light is indicated by solid and dashed arrows, respectively, in (a). Orthogonal linear polarizations – Y and X are indicated by solid and dashed arrows, respectively,in (b).

2.8

Exciton as a boson
Excitons are spin 1 pseudo-particles. Although excitons are composed of two fermions– electrons and holes, it has been shown that excitons can be considered as bosons at low
excitation densities, where phase-space filling effects are minimal [68, 69, 70]. The bosoniccharacter of excitons is invoked to describe excitons in microcavities [71] and macroscopic
condensed phases of excitons [72]. Also, some effects of the bosonic character of excitons on
FWM signal has been discussed [73, 74].
Nevertheless the validity of modeling excitons as bosons has been under debate [75]. It
has been argued that the fermionic nature of electrons and holes masks the bosonic natureof excitons. Also, it has been claimed that the bosonic model of excitons does not take into
account the higher-order correlations, even at low excitation densities [76].
In Chapter 6 we will discuss the origin of FWM signal when excitons are modeled asinteracting bosons. We will show that such a model is consistent with FWM and 2DCS
experiments performed in the χ(3) -regime as argued by Shirane et al. [74]. Additionally,

25
modeling excitons as interacting bosons has several advantages. Firstly, we find that the polarization dependent exciton dephasing rate [77, 78, 79] is a direct consequence of the bosonicnature of excitons. Secondly, exciton-exciton interactions are included in a physically intuitive and straightforward way, which is significantly simpler than using either semiconductor
Bloch equations [54] or the phenomenological many-body effects [5]. The simplified modeling enables quantifying the exciton-exciton interactions. Finally, we will see that multiple
exciton modes are excited optically in a FWM experiment, and the exciton-exciton interactions can be separated into inter- and intra-mode interaction. Power dependent experiments
are used to separately quantify these interactions.
2.9

SamplesIn this work, we will discuss experimental findings obtained from a couple of different

QW samples. We will discuss the samples that we have used here.

2.9.1(001)-Oriented GaAs quantum well

The (001)-oriented GaAs QW has been used to study spectral diffusion and the bosonicmodel of excitons. Since similar samples have been studied extensively, we have used thissample to demonstrate new perspectives that we have obtained through 2DCS experiments.
This sample comprises a series of four uncoupled 10-nm-wide QWs with 10-nm-wide
Al0.3 Ga0.7 As barriers. The absorption spectrum in Fig. 2.4 is obtained from this sample. This
sample was grown and provided to us by Richard Mirin at National Institute of Standards
and Technology, Boulder, USA.

2.9.2

(110)-Oriented GaAs quantum well

As mentioned in Section 2.6, we have also studied excitons in (110)-oriented GaAs QWs.
These QWs were grown by Klaus Pierz’s group at Physikalisch-Technische Bundesanstalt,
Braunschweig, Germany.

26
Y
X

0.3

Absorbance (αL)

HH
0.2

LH
0.1

0.0

1550

1560

1570

1580

1590

Photon energy (meV)

Figure 2.11: Measured absorption spectrum for a 10 period 8-nm-wide (110)-oriented GaAs
quantum well with 8-nm-wide Al0.3 Ga0.7 As barriers. The HH and LH resonances are marked.
Note the different absorption coefficients for orthogonal linear polarizations – X and Y.

We have studied 10 period samples with 5 and 8 nm wide QWs separated by 8-nm-wide
Al0.3 Ga0.7 As barriers. Figure 2.11 shows the absorption spectra for the 8-nm-wide QW for X
and Y polarized light, which are aligned with the [001] and [11̄0] crystal axes, respectively.
The polarization dependent absorption arises from the broken symmetry in the QW plane
discussed in Section 2.6, and has been reported earlier [80]. Absorption at the HH exciton
resonance is higher for light polarized along the [11̄0] crystal axis. In case of the LH exciton
resonance, higher absorption coefficient is measured for light polarized along the [001] crystal
axis.
Now we are ready to discuss studies of excitons in QWs using 2DCS, which will bepresented in the next chapter following an overview of the experimental and theoreticalmethods.

Chapter 3
Two-Dimensional Coherent Spectroscopy

Various spectroscopic techniques are used to investigate a material’s properties throughlight-matter interaction. Electronic and structural properties can be probed by using lightfields of different wavelengths. The most simple techniques, such as linear absorption andphotoluminescence reveal information about the various transitions in a system. However,this information is not sufficient to describe more complicated systems with multiple resonances and/or inhomogeneity. This is especially true for semiconductor heterostructures.
Strong many-body interactions in semiconductors complicate things even further. Nonlinear spectroscopy is sensitive to many-body interactions because of higher excitation densities than linear techniques, and can be broadly classified into coherent and non-coherent
techniques. The difference between the two class of experiments is in the sensitivity, or lackthereof, to phase fluctuations of quantum mechanical superpositions between states that is
excited in the sample. Coherent spectroscopy techniques are sensitive to these fluctuations
and, thus, are more sensitive to many-body interactions. Furthermore, knowledge about thecoherent response of a system can be utilized for coherent control. For instance, chemical
reactions can be controlled through optical pulses [81, 82]. Coherent control of excitons andphotocurrents in QWs has also been studied [83, 84, 49].
Perhaps the most commonly used non-linear coherent spectroscopy technique is thetransient four-wave mixing (TFWM) experiment. Although, a variety of experimental implementations exist, a common characteristic of this class of experiments is that the signal

28
is measured as a function of delay(s) between multiple ultrafast laser pulses. A TFWM
experiment is ideal to study excitons in semiconductors, which have dynamics in the range
of a few to tens of picoseconds.
Two-dimensional coherent spectroscopy (2DCS) is a relatively new spectroscopic technique, which is inspired from two-dimensional Fourier transform nuclear magnetic resonancespectroscopy [7]. Since the first demonstration of 2DCS in the optical domain [8], the technique has been extended to the terahertz [85], infra-red [9, 86] and ultra-violet [87, 88]
frequency regimes. Recently, the technique has been extended to three frequency dimensions [89, 90]. 2DCS has been used extensively to study a variety of materials such as atomic
vapors [91, 92, 93, 94], molecules [9, 86], colloidal nanocrystals [95], energy transfer in lightharvesting complexes [96], semiconductor heterostructures [17, 62, 97], and Nitrogen vacancy
centers in diamond [98]. 2DCS is an extension of the TWFM experiment, with the additionof phase-sensitive detection and precise scanning of time delay(s). The detection of signal
phase has numerous advantages. 2DCS studies of excitons in QWs have been instrumental inunderstanding many-body interactions – the effect of many-body interactions on the signal
phase [13], existence of multiple exciton correlation states [62, 63, 64], limitations of meanfield theories through comparison with experiments [99] have been some of the importantdiscoveries.
In this chapter, we will discuss the principles of optical 2DCS. We will start off with a
discussion on the TFWM technique, since it is essentially a precursor to 2DCS experiments.
Next, we will provide an overview of the relevant experimental and theoretical details of
2DCS. Finally, we will review previous studies of QW excitons using 2DCS.
3.1

Transient four-wave mixing
TFWM is one of the most commonly used non-linear spectroscopy techniques. Strictly

speaking, the signal field is proportional to the cube of the incident optical electric field in a
TFWM experiment. Although, some TFWM signal would be radiated by a single incident

29
optical pulse, the non-linear signal would be buried in the much stronger linear signal.
As a result, two or three pulses are used so that the non-linear response can be isolatedusing frequency modulations for collinear and phase-matching for non-collinear geometries.
Furthermore, the delay between multiple pulses can be scanned to probe excitation dynamics.

3.1.1

Three-pulse realization

The two- and three- pulse implementations of TFWM are very similar; we will only
discuss the three-pulse experiments since that discussion can be easily extended to discuss
2DCS experiments. Additionally, we would limit our discussion to degenerate TFWM, inwhich all the excitation pulses have identical spectrum. One realization of three excitation
pulses, A, B and C, in a typical TFWM experiment is shown in Fig. 3.1(a). The delay
between consecutive excitation pulses are τ and T . The signal is radiated during time t after
pulse C is incident on the sample.
The electric field for each pulse is10
E = |Ei |[ei(ki ·r−ωi t ) + c.c.]
2

(3.1)

where |Ei | is the electric-field envelope, ki is the wavevector, and ωi is the center frequencyof pulse i. Time is indicated by t0 to distinguish it from the delay t in Fig. 3.1(a). Each
incident pulse acts once (either the conjugated or the non-conjugated part) and the radiatedsignal field is proportional to |EA ||EB ||EC |. One of the pulses is conjugated so that thesignal frequency is close to the frequency of the incident pulses. For instance, pulse A is the
conjugated pulse in Fig. 3.1(a), and is labeled A∗ . Thus, the radiated signal has a frequency
ωS = −ωA + ωB + ωC

(3.2)

where ωA , ωB , and ωC are the center frequencies of the incident pulses. The negative signfor ωA is used since A is the conjugated pulse. Since the conjugate pulse is the first pulsethat is incident on the sample, this pulse sequence is referred to as the SI time ordering.

30
ks
A*

B

C

TFWM

kA
Sample

kB
τ

T

t
kC
(b)

(a)

Figure 3.1: (a) Three excitation pulses, A, B, and C, in the SI time ordering in a transient
four-wave mixing (TFWM) experiment. (b) A typical three-pulse non-collinear geometry.
The excitation pulses are incident on the sample along the directions indicated by kA,B,C .
The TFWM signal is emitted along direction kS .

Several TFWM signals with different pulses acting as the conjugated pulse are radiatedby the sample. Usually one of these signals is isolated experimentally, which sets the pulsethat acts as the conjugated pulse. A non-collinear geometry of incident pulses and detectedsignal is shown in Fig. 3.1(b) for a three-pulse TFWM experiment. The excitation pulses
are incident along the directions given by kA , kB , and kC . The signal is detected in thephase-matching direction
kS = −kA + kB + kC .

(3.3)

The direction in which the TFWM signal is detected determines which pulse acts as the
conjugated pulse. The − sign for kA means that pulse A is conjugated.
3.1.2

Theoretical description

We will now discuss the origin of TFWM signal, from a physical perspective, for a
two-level system shown in Fig. 3.2(a) with the ground and excited states denoted as |0i
and |1i, respectively. The generation of TFWM signal is usually understood through theperturbative interactions of three light pulses with the sample. In such a description, Pulse A
creates a coherence between the |0i and |1i states. This coherence evolves with the frequency

31

Γ1
ω01
γ01

σ01

+

+

Signal 0 0

0 0

1 0

1 0

t

1 1

T

0 1

τ

C

0 0

B

0 1
0 0
(a)

(b)

A*

0 0
(c)

Figure 3.2: (a) A two-level system with ground |0i and excited |1i states. The resonancefrequency ω01 , dephasing rate γ01 , and inhomogeneous width σ01 of the |0i ↔ |1i transition
is indicated. Γ1 is the population decay rate of state |1i. (b) and (c) show the double-sided
Feynman diagrams that contribute to the TFWM signal. The pulse sequence is indicated in(b) and the relevant time delays are shown in (c). The + sign above the Feynman diagrams
indicates the signal from the two diagrams are radiated in phase.

ω01 and decays with the dephasing rate γ01 . The sample interacts with pulse B after delay
τ , which creates a spatial population grating depending on the relative phase between pulses
A and B. Since the system is in a population state, there is no phase evolution during
delay T . However, the grating decays due to population decay rate Γ1 during this timedelay. Pulse C is incident on the sample next; it puts the system in a coherent state again,
which radiates the TFWM signal during time t. Because pulse A is conjugated, the phaseevolution during τ is opposite to the phase evolution during t for the SI time ordering. As
a result, the decay of macroscopic polarization due to inhomogeneous broadening during τis undone during t and signal is emitted as a photon echo at t = τ . As a result, the SI timeordering is also referred to as the rephasing or photon-echo time ordering. Since the effectof inhomogeneity is essentially eliminated, such an experiment can be used to measure thehomogeneous dephasing rate even in the presence of inhomogeneous broadening.
The TFWM signal can be quantitatively calculated by solving the optical Bloch equations (OBEs). These are a set of coupled differential equations that relate the time evolution

32of the density matrix elements ρ00 ρ01 
ρ=.
ρ10 ρ11(3.4)

It is convenient to work in the rotating frame of the laser center-frequency and define thecoherence ρ01 in terms of a slowly varying component ρe01 such that
ρ01 = ρe01 eiωt .

(3.5)

ω is the center-frequency of the light. ρe10 and ρ10 are related through the complex conjugate
of Eq. (3.5).
The OBEs for a two level system in the rotating wave-approximation are:iρe˙ 01 = (−γ01 + i∆01 )e
ρ01 + µ01 E ∗ (2ρ11 − 1)2~iρ̇11 = −Γ1 ρ11 + µ01 (E ρe01 − E ∗ ρe10 )
2~

(3.6)

where µ01 is the transition dipole moment of |0i ↔ |1i transition, E = |E|ei(k·r−ωt) is thecomplex electric field of light, and ∆01 = ω01 −ω is the detuning of the transition energy withthe light frequency. In case of pulsed light, ω is the center frequency of the laser spectrum.
Only two equations are sufficient to completely define the density matrix evolution because
ρ10 = ρ∗01 and ρ00 + ρ11 = 1. These equations incorporate the light-matter interaction in the
dipole approximation and excitation decays are included phenomenologically. An outlineof the derivation from OBEs starting from the dipole approximation of the light-matterinteraction is presented in Appendix A. The OBEs can be extended, in a straightforwardmanner, to more complicated energy level schemes following a similar procedure.
In general, the OBEs need to be solved numerically. However, an analytical solution
can be obtained in the perturbative regime assuming δ-function pulses in time. This simplification implicitly assumes infinitely broad laser spectrum, which is not possible to realize
experimentally. Nevertheless, this assumption is reasonable if the resonance is significantly

33
narrower than the laser spectrum. If that is not the case, the spectral dependence of amplitude will not be correctly reproduced through the simplified simulations, and will needad-hoc adjustments. The detailed procedure for solving the OBEs using the aforementioned
assumptions has been discussed previously [100], and will not be discussed in this work.
We will take an alternate, shorthand approach to calculate the TFWM signal using
double-sided Feynman diagrams [38, 100]. In this method, the TFWM signal is separated
into several quantum pathways, each represented by a Feynman diagram. Figures 3.2(b)and 3.2(c) show the Feynman diagrams that contribute to the TFWM signal for the SI timeordering in the two-level system shown in Fig. 3.2(a). Interaction with each pulse and theradiation of TFWM signal is indicated by an arrow. The excitation pulses and the signal
are labeled in Fig. 3.2(b). The relevant time delays are shown in Fig. 3.2(c). Without going
into specific details, which are discussed elsewhere [100], the TFWM signal can be calculated
by following the rules/properties of the Feynman diagrams:• The numbers on each line indicate a specific density matrix element. For instant,the second row from the bottom in Fig. 3.2(b) indicates the density matrix element|0i h1| or ρ01 .
• The interaction with each pulse takes the system from one density matrix element
to another through optical transitions. For instance, pulse A takes the system from
ρ00 to ρ01 through the |0i ↔ |1i transition.
• For a particular Feynman diagram, it is assumed that, during a particular delay thesystem is in the state defined by the density matrix element during that delay as
indicated in the Feynman diagram.
• Since pulse A is conjugated, the arrow indicating it is opposite to those indicating
pulses B and C.
• An arrow pointing inwards creates an excitation in the system whereas an outward

34
pointing arrow results in relaxation.
• The system should be in a population state after the TFWM signal is radiated.
• The overall signal phase of a particular Feynman diagram is (−1)n where n is thenumber of interactions, including signal emission, on either the left or right side of
the Feynman diagram.
• All possible Feynman diagrams have to be included to correctly calculate the TFWM
signal.
Following these rules and properties, only two diagrams can be drawn for the SI time ordering
as shown in Figs. 3.2(b) and 3.2(c). Both these diagrams give rise to equal TFWM signals;
the total radiated signal field can be written, in the rotating frame, as
12

2

SSI ,2level (τ, T, t) = 2|E|3 µ01 4 e−i∆01 (τ −t) e−(γ01 τ +Γ1 T +γ01 t) e− 2 σ01 (τ −t) Θ(τ )Θ(T )Θ(t)(3.7)

where all the variables have been previously defined and Θ denotes the Heaviside theta
function to ensure the correct time ordering.
It is imperative to take a close look at the different factors in Eq. (3.7) in order tounderstand how it was obtained from the Feynman diagrams in Figs. 3.2(b) and 3.2(c):• Both the Feynman diagrams comprise two interactions each on the left and rightside, and, consequently, have the same phase. The factor of 2 accounts for theconstructive interference between these two diagrams.
• The factor |E|3 is obtained by assuming equal electric field amplitude for all the
excitation pulses, which is generally true for our experiment.
• There are four interactions involving the |0i ↔ |1i transition, which are incorporated
in the factor µ01 4 .

35• The first exponential term includes the oscillation of the signal with the resonancefrequency, which corresponds to the detuning frequency in the rotating-wave approximation. The opposite sign for delays τ and t is due to the rephasing timeordering.
• The second exponential includes the homogeneous decays. The dephasing rate isrelevant for delays τ and t because the system is in a coherent state during thoseintervals. During delay T the population decay rate is the relevant quantity since
the system is in a population state. The population decay with T is included evenfor the Feynman diagram in Fig. 3.2(b), although the system is in the |0i state, tocorrectly account for the decay in the population grating, which determines the rateof decay of TFWM signal.
• The last term accounts for the inhomogeneous broadening in the system that resultsin the signal being emitted as a photon echo at t = τ . A Gaussian lineshape forthe inhomogeneous broadening is assumed. Inhomogeneity does not contribute to
the decay during T when the system is in a population state, as long as no spectral
diffusion occurs.
The above theoretical description of the origin of TFWM signal will be used extensively
throughout this work to calculate 2D spectra. Before we get to the details of 2DCS, we will
provide an overview of typical variations of TFWM experiments that have been used to
study excitons in QWs. This will provide some context to the work presented later on andalso highlight the advantages of 2DCS vis-à-vis TFWM experiments.
3.1.3

Experimental studies

In a typical TFWM experiment, one of the time delays, τ or T , is varied and the signal
can either be resolved spectrally or temporally, or the time-integrated TFWM (TI-TFWM)
signal can be measured by a slow photo-detector. The two-pulse TFWM experiment is more36
widely used because of simpler experimental implementation. Theoretically, a two-pulse
experiment can be treated similar to the three-pulse experiment discussed in Section 3.1.2,
with pulses B and C incident on the sample simultaneously. In this case only delay τ can bescanned. We will now review some of the commonly used variations of the TFWM technique.
Photon echo spectroscopy:

This realization uses the SI or rephasing time-orderingand is most commonly used for its ability to extract the homogeneous response of a systemeven in the presence of inhomogeneity. Generally, the TI-TFWM is measured as a functionof delay τ . As discussed earlier the effect of inhomogeneity on the macroscopic polarizationis reversed. As a result, the homogeneous dephasing rate can be extracted, from the decayof TI-TFWM signal with delay τ , even in an inhomogeneously broadened spectrum. This
scheme can also reveal coherent coupling between different states through oscillations in the
TI-TFWM signal as a function of τ , although care needs to be taken to distinguish such
oscillations from polarization interference [3, 101]. An interesting result was the observationof finite and fast-decaying TI-TFWM signal for negative values of τ , which seemed to violatecausality and was shown to be due to local field effects [60, 6]. We will briefly discuss theorigin of this signal in Section 3.2.5 from the perspective of interactions.
Three-pulse photon echo peak shift (3PEPS) spectroscopy:

3PEPS exper-

iments have been used to study non-Markovian decay dynamics of coherences in various
systems [36, 37, 102], including QWs [40]. The photon-echo takes some time to build up,
and the TI-TFWM is maximum at some finite τ > 0, say τm . In this experiment, τm ismeasured as a function of delay T . In the absence of spectral diffusion, the value of τm does
not change with T . However, since the transition frequency for each oscillator changes withincreasing T due to spectral diffusion, the generation of a photon-echo is not perfect. As
a result, the system behaves more and more like a homogeneously broadened system withincreasing T , and τm decreases, which is used to quantify spectral diffusion under certain
approximations [39, 40].
With a brief background of non-linear spectroscopy in general and TFWM in par-37
ticular, we will now discuss the principles of 2DCS from the theoretical and experimental
perspectives.
3.2

Two-dimensional coherent spectroscopy – theory
2DCS is an extension of the time resolved TFWM technique, and requires phase-sensitive detection of signal and scanning of time delays. This poses several experimental
challenges that will be discussed in Section 3.3. However, the theoretical description for 2DCS
follows directly from that of TFWM discussed in Section 3.1.2; a numerical Fourier transformof the time-domain TFWM signal along delays τ or T and t results in a 2D spectrum.
Nevertheless, this seemingly straightforward extension results in significant advantages of
2DCS compared to TFWM techniques because the TFWM signal is unfolded on to two
frequency dimensions. Different kinds of 2D scans can be performed by changing the timeordering of excitation pulses and/or the time delay that is scanned. These variations yieldcomprehensive information about the system under study. Through detailed analysis, one
can quantify several physical properties of the system. However, the physical properties caneasily be understood qualitatively by just looking at different 2D spectra. Presently, we will
discuss the interpretation of different kinds of 2D spectra for various model systems that
include few energy levels.

3.2.1

Rephasing one-quantum scan

The first kind of 2D scan that we will discuss is the most obvious extension of thephoton-echo spectroscopy discussed in Section 3.1.3; the excitation pulses are incident on
the sample in the SI time-ordering [Fig. 3.1(a)] and the delay τ is scanned. The rephasing
one-quantum 2D spectrum is obtained by taking a Fourier transform of Eq. (3.7) along
delays τ and t. The name one-quantum is used because the system is in a superposition
state, which evolves at the frequency of the transition between the ground and excited states
during the scanned time delay τ . We will discuss this 2D spectrum for a two-level system

38
and a three-level V system.
Homogeneous broadening:First, we will consider a two-level system, as shownin Fig. 3.2(a), without any inhomogeneous broadening, i.e. σ01 = 0. The absolute value,
real part, and imaginary part of the 2D signal are shown in Figs. 3.3(a), 3.3(b), and 3.3(c),respectively. We use ∆01 = 5 meV, and γ01 = 0.1 meV. Γ1 does not affect these spectra since
T is kept constant for this scan.

0.4

0.6

0.8

−1

−0.5

0

0.5

Real part

Absolute value

Imaginary part

−7

C
di ro
ag ss
on al

−5

D
l
na

go
ia

Excitation energy (meV)

0.2

−3

(a)

(b)
3

5

7

3

(c)
5

7

3

5

7

Emission energy (meV)

Figure 3.3: (a) Absolute value, (b) real part, and (c) imaginary part rephasing one-quantum
2D spectra for a homogeneously broadened two-level system. The dashed line in each spectrum indicated the diagonal direction. The cross-diagonal direction is indicated by arrowsin (a). The negative value region is outlined in (b).

The horizontal energy axis is the ~ωt axis, denotes the photon emission energy, andcorresponds to the emission of the TFWM signal. This axis is labeled “emission energy”.
The vertical axis is labeled excitation energy and corresponds to ~ωτ . The excitation energy
axis is labeled with negative energies to indicate opposite phase evolution during delays
τ and t; it is assumed that the radiated TFWM signal has positive frequency. Equal theemission and excitation energies with equal magnitudes are indicated by the dashed diagonallines. The direction perpendicular to the diagonal direction is the cross-diagonal direction,
as indicated by arrows in Fig. 3.3(a).

39The absolute value spectrum in Fig. 3.3(a) shows a star-shaped peak on the diagonalline at the detuning energy used in the simulation. The real part spectrum in Fig. 3.3(b)has a positive peak with slight negative values in the wings, and is outlined. This feature
results from the discrete Fourier transform of the Heaviside theta functions, Θ(τ ) and Θ(t),in Eq. (3.7), and is referred to as the phase-twist in the spectrum [103]. This phase twist
also results in a broadening of the peak in the real part spectrum. Both these issues can be
resolved through the correlation spectrum [103], as will be shown in Section 3.2.4.
Effect of inhomogeneous broadening:

The 2D spectrum is modified significantlyin the presence of inhomogeneity. Figures 3.4(a) and 3.4(b) show rephasing one-quantum
absolute value spectra for σ01 = 0.2 and 1 meV, respectively. The other system parameters
are kept the same as earlier. The signal from different frequency groups appears along the
diagonal line resulting in an elongated peak in the 2D spectrum. This elongation increases
with an increase in the inhomogeneity in the sample.

Excitation energy (meV)

0.2

0.4

0.6

0.8

−3

−5

−7

(a)

(b)
3

5

7

3

5

7

Emission energy (meV)

Figure 3.4: Rephasing one-quantum spectrum with (a) σ01 = 0.2 meV and (b) σ01 = 1 meV.
Other parameters are the same as those used to simulate the spectra shown in Fig. 3.3.
Inhomogeneity results in an elongation of the peak along the diagonal direction.

The width of slices taken along the diagonal and cross-diagonal directions generally
correspond to the inhomogeneous and homogeneous linewidths. This situation is strictly40
true for extreme inhomogeneity, similar to the case in Fig. 3.4(b). For an intermediate case,as shown in Fig.3.4(a), the diagonal and cross-diagonal slices can be simultaneously fit tocomplex lineshapes to obtain the homogeneous and inhomogeneous linewidths [104]. We
made an assumption that the dephasing rate does not depend on the resonance energy toobtain Eq. (3.7). However, this assumption is not always valid, which results in varying crossdiagonal width across the inhomogeneous distribution. This spectral dependence can bemeasured using a single 2DCS experiment [11] as opposed to spectrally averaged informationthat is obtained from TI-TFWM experiments.
Coherent coupling:

2DCS is an ideal technique to study coherent coupling between

different eigenstates of a system. We will illustrate the effect of coupling on a rephasing onequantum 2D spectrum using a three-level V system shown in Fig. 3.5 with ground state |0i
and excited states |1i and |2i. The transitions |0i ↔ |1i and |0i ↔ |2i are the only opticallyallowed transitions. The excited states are, however, coupled through the ground state. As
was done previously, all the parameters for these transitions are indicated.
Γ2
Γ1
ω01 σ
01
γ01

σ02
ω02
γ02

Figure 3.5: Three-level system with the ground state |0i and two excited states |1i and |2i.
The resonance frequency ω, dephasing rate γ and inhomogeneity σ of the |0i ↔ |1i and|0i ↔ |2i transitions are indicated. The population decay rates Γ for the excited states arealso indicated.

Feynman diagrams similar to those in Fig. 3.2 can be written down with interactions
involving only one of |0i ↔ |1i and |0i ↔ |2i transitions. Each of these diagrams includes
interactions in both |0i ↔ |1i and |0i ↔ |2i transitions. In addition to these diagrams,
the Feynman diagrams shown in Fig. 3.6 also contribute to the TFWM signal. All these41
diagrams have the same phase and the total TFWM signal can be written ash
1 2
2
SSI ,3level (τ, T, t) =|E|3 2µ01 4 e−i∆01 (τ −t) e−(γ01 τ +Γ1 T +γ01 t) e− 2 σ01 (τ −t) + 2µ02 4 e−i∆02 (τ −t)1

2

2

e−(γ02 τ +Γ2 T +γ02 t) e− 2 σ02 (τ −t) + µ01 2 µ02 2 e−i(∆01 τ −∆02 t) e−(γ01 τ +γ02 t)− 12 (σ01 τ −σ02 t)2−Γ1 Ti∆21 T −γ12 T − 12 σ12 2 T 2
e
+
e
+e
e
e
1

(3.8)

2

µ01 2 µ02 2 e−i(∆02 τ −∆01 t) e−(γ02 τ +γ01 t) e− 2 (σ02 τ −σ01 t)

i
−Γ2 T
i∆12 T −γ12 T − 12 σ12 2 T 2e+eee
Θ(τ )Θ(T )Θ(t).

Signal
C

+

+

+

+

0 0

0 0

0 0

0 0

2 0

2 0

t

1 0

1 0

2 1

T

0 0

1 2

0 1

τ

0 2

0 2

0 0
0 1

B
A*

0 0

0 0

0 0

0 0

(a)

(b)

(c)

(d)

Figure 3.6: Feynman diagrams that contribute to the four-wave mixing signal due to coherent
coupling between the |0i ↔ |1i and |0i ↔ |2i transitions. The pulse sequence is indicatedin (a) and the relevant time delays are shown in (b). The signal due to all the Feynman
diagrams have the same phase, as indicated by the + sign above each diagram.
The first and second terms include signals from the Feynman diagrams similar to
those in Fig. 3.2 for the |0i ↔ |1i and |0i ↔ |2i transitions, respectively. The other two
terms include contribution from the Feynman diagrams shown in Fig. 3.6, and involve twointeractions each with the two transitions, which is accounted for by the factor µ01 2 µ02 2 .
The Feynman diagrams in Fig. 3.6 have different evolution frequencies during delays τ andt. The third term includes contributions from the Feynman diagrams shown in Figs. 3.6(a)and 3.6(b) while the signal due to the Feynman diagrams in Figs. 3.6(c) and 3.6(d) is
incorporated in the fourth term. These terms include new parameters to correctly describe
the change in the TFWM signal with T . These parameters are defined in Sec. 3.2.2 alongwith the discussion of change in TFWM signal with T .

42

Excitation energy (meV)

0.2

0.4

0.6

0.8

−3

−5

−7

3

5

7

Emission energy (meV)

Figure 3.7: Rephasing one-quantum absolute value 2D spectrum for the three-level system
in Fig. 3.5. The off-diagonal peaks are circled, and indicate coherent coupling between
states |1i and |2i. Note the tilt of the off-diagonal peaks with respect to the diagonal line,
as indicated by the lines through the peak, due to different inhomogeneity of the |0i ↔ |1i
and |0i ↔ |2i transitions.

Figure 3.7 shows the simulated rephasing one-quantum absolute value 2D spectrum
obtained from Eq. (3.8) using the following parameter values: ∆01 = 4 meV, ∆02 = 6 meV,
γ01 = γ02 = 0.1 meV, σ01 = 0.2 meV, σ02 = 0.25 meV, and µ02 /µ01 = 0.95. There are several
key features of this spectrum, which we would like to highlight:• The diagonal peaks result from the first two terms in Eq. (3.8), and results from alllight-matter interactions involving only a single transition.
• Coherent coupling between states |1i and |2i results in the circled off-diagonal peaks.
Feynman diagrams in Figs. 3.6(a) and 3.6(b) result in the above-diagonal peak,which is circled by a solid line. The Feynman diagrams in Figs. 3.6(c) and 3.6(d)give rise to the peak below the diagonal line, which is circled by a dashed line.
• The peak strength of the coupling peaks are equal to the geometric mean of the peakstrength of the peaks on the diagonal.

43• The solid line through the off-diagonal peaks indicate the tilt of these peaks withrespect to the diagonal line due to the different, but correlated inhomogeneous
linewidths of the |0i ↔ |1i and |0i ↔ |2i transitions. Uncorrelated inhomogeneitiesof the two transitions will result in circular, instead of elongated, peaks.
The appearance of off-diagonal coupling peaks makes 2DCS extremely useful to study
such systems. While coherent coupling can be interpreted from beating in TI-TFWM with
τ , one needs to be careful to distinguish between quantum and polarization beats [3, 101].
Furthermore, we can identify the correlation between inhomogeneities of different states
through the peak shape from such a 2D spectrum. Next we will discuss a 2D scan which
also uses the SI time-ordering, but involves scanning time delay T instead of τ .
3.2.2

Rephasing zero-quantum scan

As mentioned earlier, for a two-level system, a spatial population grating exists during
delay T , and there is not phase evolution during this time delay. Thus, information about the
population decay times can be obtained by measuring TFWM signal as delay T is scanned.
However, a Raman-like non-radiative coherence between two excited states exists during T
as shown in the Feynman diagrams in Figs. 3.6(b) and (d) for a three-level system. The
TFWM signal phase evolves with T at a frequency that is much smaller compared to thetransition frequency and this scan is known as zero-quantum scan. We will look at typical
spectra for this scan for both two- and three-level systems.
Population decay rate:

A rephasing zero-quantum absolute value 2D spectrum for

an inhomogeneously broadened system is shown in Fig. 3.8(a). This spectrum was obtainedby taking a numerical Fourier transform of Eq. (3.7) along time delays T and t. The systemparameters were the same as those used to obtain the rephasing one-quantum spectrum
shown in Fig. 3.4(b). The population decay rate Γ1 = 0.2 meV was used, which satisfies Eq.
(2.12) and assumes no pure dephasing processes.

44
The vertical axis is labeled mixing energy and corresponds to ωT ; zero mixing energy
is denoted by the dashed line. Since the system is in a population state during T , the single
peak appears on the zero mixing energy line. The peak is elongated along the emission
energy axis due to the inhomogeneous broadening. The peak-width along the mixing energy
axis, as indicated by the vertical line, gives the population decay rate.

Mixing energy (meV)

0.2

2

0.4

0.6

0.8

(b)

(a)

0

−2

3

5

7

3

5

7

Emission energy (meV)

Figure 3.8: Zero-quantum rephasing absolute value 2D spectrum for (a) two-level system
shown in Fig. 3.1(b) and (b) three-level V system shown in Fig. 3.5. The elongation ofpeaks along the emission energy axis is due to inhomogeneity. The width of the vertical slice
indicated gives the population decay rate of the exciton. The circled peaks with non-zero
mixing energy in (b) result from coherent coupling between states |1i and |2i.

Coherent coupling:

Similar to the one-quantum spectrum, coherent coupling be-

tween states has apparent signature in zero-quantum scan as shown by the spectrum in Fig.
3.8(b). The system parameters are the same as the ones used to obtain the spectrum in
Fig. 3.7. The population decay rates Γ1 = Γ2 = 0.2 meV were used consistent with no puredephasing processes.
The most interesting feature in this spectrum is the presence of the circled peaks at
non-zero mixing energy obtained due to the coherent coupling between states |1i and |2i.
The Feynman diagrams in Figs. 3.6(b) and 3.6(d) contribute to the peaks circled with

45
solid and dashed lines, respectively. In these Feynman diagrams, a non-radiative coherence
between states |1i and |2i exists, which results in coupling peaks at mixing energies equal
to the difference in the transition energies of |0i ↔ |1i and |0i ↔ |2i transitions.
Further insight into the coupling is gained through the terms that are underlined in
Eq. (3.8), which result in the coupling peaks. We should first define the new parametersused. ∆21 = −∆12 = ω02 − ω01 = 2 meV give the resonance energies of the non-radiative
coherences between states |1i and |2i, γ12 = γ01 + γ02 = 0.2 meV is the dephasing rateof this coherence, and σ12 = |σ01 − σ02 | = 0.05 meV is the corresponding inhomogeneity,
assuming correlated inhomogeneity between states |1i and |2i. It is worth noting that both
γ12 and σ12 contribute to the signal decay with T resulting in coupling peaks that appear
circular, instead of elongated. Furthermore, only a single Feynman diagram contributes
to each coupling peak compared to three Feynman diagrams that contribute to the peaks
with zero mixing energy, which results in the significant difference in the amplitudes ofthese peaks. Finally, we would like to note that while the coupling peaks in a one-quantum
spectrum might appear due to population transfer, during T , between the different states,
the coupling peaks in zero-quantum only show up in the presence of coherent coupling.
All the 2DCS experimental results presented in this work have been obtained for the
SI time ordering. However, for the sake of completeness, we will discuss other time orderings
and the corresponding 2D spectra.

3.2.3

Non-rephasing one-quantum scan

The non-rephasing one-quantum scan is similar to its rephasing version. The only
difference is that pulse B is incident on the sample first followed by pulse A, as shown in Fig.
3.9(a). Since the conjugate pulse is the second pulse that is incident on the sample, this timeordering is referred as the SII time ordering. The phase evolution during τ and t are thesame, which prevents the formation of a photon-echo even in the presence of inhomogeneity.
The Feynman diagrams that contribute to the TFWM signal in the two-level system

46
A*

B

C

FWM

Signal
C
A*

τ

T

t

(a)

B

+

+

0 0

0 0

1 0

1 0

t

0 0

1 1

T

1 0

1 0

τ

0 0

0 0

(b)

(c)

Figure 3.9: (a) The non-rephasing one-quantum time ordering. (b) The Feynman diagrams
that contribute to the four-wave mixing signal for this time-ordering. The pulse sequence isindicated in (b) and the relevant time delays are shown in (c).

shown in Fig. 3.2(a) for this time ordering are shown in Figs. 3.9(b) and 3.9(c). The
corresponding TFWM signal can be written as1

2

2

SSII ,2level (τ, T, t) = 2|E|3 µ01 4 ei∆01 (τ +t) e−(γ01 τ +Γ1 T +γ01 t) e− 2 σ01 (τ +t) Θ(τ )Θ(T )Θ(t)(3.9)

where all the variables have their usual meaning. Equations (3.9) and (3.7) are almost
identical with a couple of differences – (1) the first exponential denotes similar phase evolution
during delays τ and t, and (2) (τ + t)2 in the last exponential accounts for the fact the signal
is not emitted as a photon-echo even in the presence of inhomogeneity.
Figures 3.10(a), 3.10(b), and 3.10(c) show the absolute value, real part and imaginary
part non-rephasing one-quantum 2D spectrum, respectively for a two-level system in theabsence of inhomogeneous broadening. The system parameters are the same as used forthe spectra in Fig. 3.3. Equations (3.9) and (3.7) are essentially identical for σ01 = 0; theonly difference is that positive frequency during both delays τ and t. This is reflected in
the identical peak shapes in 2D spectra in Figs. 3.3 and 3.10. An important distinction,
however, is that the excitation energy axis has positive values for the non-rephasing spectra.
The one-quantum 2D spectra for SI and SII time orderings differ significantly in thepresence of inhomogeneity. Due to the lack of rephasing, the peaks are uniformly broadened
along the diagonal and cross-diagonal directions as a result of inhomogeneous broadening.

47

Excitation energy (meV)

0.2

0.4

0.6

0.8

−1

Absolute value

−0.5

0

0.5

Real part

Imaginary part

7

5

3

(a)
3

5

7

(b)
3

5

7

(c)
3

5

7

Emission energy (meV)

Figure 3.10: (a) Absolute value, (b) real part, and (c) imaginary part one-quantum nonrephasing 2D spectra for a homogeneously broadened two-level system. The negative value
region is outlined in (b).

Coupling peaks, for the three-level system in Fig. 3.5, appear in a non-rephasing
spectrum too, similar to those for the rephasing spectrum in Fig. 3.7. Since, both rephasing
and non-rephasing one-quantum spectra give similar information, the non-rephasing spectra
are generally used to measure correlation spectrum, which we will discuss next.

3.2.4

Correlation spectrum

Both the rephasing and non-rephasing one-quantum spectrum show effects of phasetwist due to the Heaviside theta functions in Eqs. (3.7) and (3.9). This phase-twist results inthe broadening of the peak in the real part spectrum. It also results in the negative values in
the real part spectrum, although the time domain signal is strictly positive if ∆01 = 0. These
anomalies can be resolved by adding the rephasing and non-rephasing spectra to obtain thecorrelation spectrum. Mathematically a one-quantum correlation spectrum is related to therephasing and non-rephasing spectra as
SCor,2level (ωτ , T, ωt ) = SSI ,2level (|ωτ |, T, ωt ) + SSII ,2level (ωτ , T, ωt )(3.10)

48
where all the signals are written in the 2D frequency domain.

0.2

0.4

0.6

0.8

−1

Excitation energy (meV)

Absolute value

−0.5

0

0.5

Real part

Imaginary part

7

5

3

(a)
3

5

7

(b)
3

5

7

(c)
3

5

7

Emission energy (meV)

Figure 3.11: (a) Absolute value, (b) real part, and (c) imaginary part one-quantum correlation 2D spectra for a homogeneously broadened two-level system. Note that the peak in(b) has only positive values and is narrower compared to the one-quantum rephasing andnon-rephasing spectra shown in Figs. 3.3(b) and 3.10(b), respectively.

The correlation spectra obtained from the 2D spectra in Figs. 3.3 and 3.10 are shownin Fig. 3.11. The real part spectrum in Fig. 3.11(b) does not have any negative value
and is also much narrower than the peak in Figs. 3.3(b) and 3.10(b). As a result, measuring a correlation spectrum is especially useful in molecular systems where several partiallyoverlapping resonances can be more easily identified due to the narrow peaks [103].

3.2.5

Two-quantum scanFinally, we will discuss TFWM signal originating when the excitation pulses are incident on the sample in the SIII time-ordering, shown in Fig. 3.12, in which the conjugate
pulse A is the last pulse that is incident on the sample. These scans are equivalent to thenegative τ scans in two-pulse TFWM experiments mentioned in Section 3.1.3. While it was
initially explained using the local field effects [60, 6], we will describe the origin of TFWM
signal for this time ordering from the perspective of coherent interactions between multiple

49
excitations.
Single resonance:

We need a three-level ladder system, as shown in Fig. 3.13(a),to obtain signal for this time ordering. This level scheme is different than the V system inFig. 3.5 in that the transition |1i ↔ |2i is optically allowed while |0i ↔ |2i is not. Also, thistransition has a resonance energy that is close to that of |0i ↔ |1i transition. The Feynman
diagrams that contribute to the TFWM signal are shown in Fig. 3.13(b) and (c). Both these
diagrams indicate that the system is in the non-radiative |0i ↔ |2i coherence during T . This
coherence cannot be created in a two-level system, thus resulting in the absence of TFWM
signal for such a system.
B

A*

C

τ

T

FWM

t

Figure 3.12: The SIII or double-quantum time ordering with the conjugate pulse A∗ incident
last on the sample. Note that the phase evolution during T is twice as fast as that during τand t.

The |0i ↔ |2i coherence during delay T evolves at a rate that is faster and close totwice the rate of evolution of the |0i ↔ |1i coherence during delay τ , as indicated in Fig.
3.12. Since the delay T is scanned, this scan is called a two-quantum scan. The two Feynman
diagrams in Fig. 3.13 have opposite signs, which results in destructive interference between
the TFWM signal from these diagrams. As a result, the total TFWM signal would be zero
if the transitions |0i ↔ |1i and |1i ↔ |2i are identical. This symmetry can be broken if
the resonance energy, the dephasing rate, and/or the dipole moment of |1i ↔ |2i transition
is different from those for the |0i ↔ |1i transition, which gives rise to finite TFWM signal.
In the energy-level scheme in Fig. 3.13(a), the resonance energy and dephasing rate of the|1i ↔ |2i transition is greater than those of the |0i ↔ |1i transition by ω 0 and γ 0 , respectively.
It is important to realize that state here |2i may indicate correlation between two excitations

50
in state |1i. The correlation exists due to interactions between multiple excitations. Thus,
including state |2i models the interactions between multiple excitations [63, 94].

ω01+ω'
γ01+γ'
ω01
γ01

Signal
σ01
A*
σ01

C
B

(a)

+

−

0 0

1 1

1 0

2 1

t

2 0

2 0

T

1 0

1 0

τ

0 0

0 0

(b)

(c)

Figure 3.13: (a) Three level ladder system. (b) and (c) shown the Feynman diagrams that
contribute to the signal in the SIII time-ordering. The pulse sequence is indicated in (b)and the relevant time delays are shown in (c). The two diagrams destructively interfere
indicated by the + and − signs above the Feynman diagrams in (b) and (c), respectively.
The broken symmetry between the |0i ↔ |1i and |1i ↔ |2i transitions, due to different
resonance energies and/or dephasing rates, results in a finite four-wave mixing signal.

The TFWM signal arising from the Feynman diagrams in Fig. 3.13 can be written as
1

2

SSIII ,3level (τ, T, t) =|E|3 µ01 2 µ12 2 ei(∆01 τ +∆02 T ) e−(γ01 τ +γ02 T ) e− 2 (σ01 τ +σ02 T +σ01 t)ei∆01 t e−γ01 t − ei∆12 t e−γ12 t Θ(τ )Θ(T )Θ(t)(3.11)

where the usual meaning of variables is used along with subscripts 01, 02 and 12 denoting
transitions |0i ↔ |1i , |0i ↔ |2i , and |1i ↔ |2i, respectively. The parameters of these threetransitions are related:ω12 = ω01 + ω 0

ω02 = ω01 + ω12 = 2ω01 + ω 0

γ12 = γ01 + γ 0

γ02 = γ01 + γ12 = 2γ01 + γ 0(3.12)

σ12 = σ01

σ02 = σ01 + σ12 = 2σ02µ12 = µ01

∆02 = ω02 − 2ω = 2∆01 + ω 0

The relations in Eq. (3.12) can be used to simplify Eq. (3.11) and obtain
1

2

2

0

0

SSIII ,3level (τ, T, t) =|E|3 µ01 4 ei∆01 (τ +2T +t) e−γ01 (τ +2T +t) e− 2 σ01 (τ +2T +t) e(iω −γ )Thi001 − e(iω −γ )t Θ(τ )Θ(T )Θ(t).

(3.13)

51
The first and second terms in Eq. (3.13) correspond to the Feynman diagrams shown in Fig.
3.13(b) and 3.13(c), respectively. It is apparent from Eq. (3.13) that the TFWM signal will
be non-zero if and only if ω 0 and/or γ 0 are non-zero.
0.2

0.4

0.6

0.8

−1

Absolute value

−0.5

0

0.5

Real part

Imaginary partTwo−quantum excitation energy (meV)

14

12

10

8

6

(a)
3

5

7

(b)
3

5

7

(c)
3

5

7

Emission energy (meV)

Figure 3.14: (a) Absolute value, (b) real part, and (c) imaginary part two-quantum 2D
spectra for a homogeneously broadened two-level system. The peak in (b) comprises positive
and negative peaks originating from Feynman diagrams in Fig. 3.13(b) and (c), respectively.

Figures 3.14(a), 3.14(b), and 3.14(c) show the simulated two-quantum absolute value,
real and imaginary parts 2D spectra, respectively, for a homogeneously broadened system
by taking a numerical Fourier transform of Eq. (3.13) along delays T and t. Similar toearlier values, we chose γ01 = 0.1 meV and ∆01 = 5 meV. We set the symmetry breaking
parameters as: γ 0 = 0.01 meV and ω 0 = 0.1 meV. The vertical axis for these spectra iscalled the two-quantum excitation energy and corresponds to ωT . ωT = 2ωt is indicated by52
a dashed diagonal line. The absolute value spectrum in Fig. 3.14(a) shows a single peak on
the diagonal line corresponding to ωt = ∆01 . The real part spectrum in Fig. 3.14(b) reveals
that the peak in the absolute value spectrum arises from two peaks that are out of phaseand very close to each other. The positive and negative peaks in Fig. 3.14(b) result from thefirst and second terms in Eq. (3.13), respectively, and are shifted along the emission energy
axis by ω 0 . Next we will discuss how the presence of multiple coupled resonances is revealed
in non-rephasing two-quantum spectrum.
Coherent coupling:

The coupled states in the three-level V system shown in Fig.

3.5 does not result in any TFWM signal for the SIII time-ordering due to the same reason
that results in no signal from a two-level system. Thus, we will discuss coupling between
multiple resonances through the six-level system in Fig. 3.15. The various transitions andtheir parameters are very similar to those used previously, and is indicated in the figure. A
similar system has been used to model interactions in various excitonic systems [105, 106].
The doubly excited states are now labeled |11i, |12i, and |22i. The states |11i and |22i
denote two correlated excitations in the states |1i and |2i, respectively. Similarly, the state|12i represents excitations in states |1i and |2i, which are correlated.

ω02+ω'
ω01+ω'
γ01+γ'

γ02+γ'σ01

σ02

ω01+ω'
γ01+γ'σ01

ω02+ω'
γ02+γ'
σ02

ω02

ω01
γ01

γ02
σ01

σ02

Figure 3.15: The six-level system originating from interactions between two three-level laddersystems similar to one shown in Fig. 3.13(a). The resonance energy ω, dephasing rate γ,
and inhomogeneity σ are indicated for each transition.

53
As a result, the coherent coupling between states |1i and |2i involves the state |12i
and contributes to the TFWM signal through Feynman diagrams shown in Figs. 3.16(a) –(d). These diagrams include the first and second interactions at energies close to ~ω01 and~ω02 , respectively. An alternate set of Feynman diagrams exist with these two interactions
swapped. These two sets of diagrams are identical for τ = 0; we will make this assumption
for the sake of simplicity. In addition to the terms involving the |12i state, other Feynman
diagrams similar to those in Fig. 3.13 also contribute to the TFWM signal, which can be
written as

h
1
2
2
0
0
0
0
SSIII ,6level (T, t) =|E|3 e(iω −γ )T 1 − e(iω −γ )t µ01 4 e(i∆01 −γ01 )(2T +t) e− 2 σ01 (2T +t) +1

2

2µ02 4 e(i∆02 −γ02 )(2T +t) e− 2 σ02 (2T +t) + 2µ01 2 µ02 2 ei(∆01 +∆02 )Tn12e−(γ01 +γ02 )T e(i∆01 −γ01 )t e− 2 [(σ01 +σ02 )T +σ01 t] + e(i∆02 −γ02 )toi
12
Θ(τ )Θ(T )Θ(t).
e− 2 [(σ01 +σ02 )T +σ02 t](3.14)The non-rephasing two-quantum absolute value 2D spectrum corresponding to the
TFWM signal in Eq. (3.14) is shown in Fig. 3.16(e). We have used ∆01 = 4 meV, ∆02 = 6
meV, γ01 = γ02 = 0.1 meV, ω 0 = 0.1 meV, γ 0 = 0.01 meV, and µ02 /µ01 = 0.95. The first
two terms in the square brackets in Eq. (3.14) result in the two peaks on the diagonal line.
The rest of the terms give rise to the circled coupling peaks, which appear shifted fromthe diagonal. Feynman diagrams in Figs. 3.16(a) and 3.16(b) result in the peak circled
with dashed line and those in Figs. 3.16(c) and 3.16(d) give rise to the peak circled withsolid line. Both coupling peaks have two-quantum excitation energy = ∆01 + ∆02 + ω 0 ≈∆01 + ∆02 . These peaks are more intense than the peaks on the diagonal because more
Feynman diagrams contribute to these peaks.
It is important to note that for the SIII time-ordering the system is never in a population state. As a result, the signal originates entirely from interaction and/or coherent
coupling between states. Thus, the two-quantum scans are extremely useful for studying
coherent interactions.

54

Signal
A*
C
B

0.2

−

0

0

2

2

1

0

12

2

t

12

0

12

0

T

1

0

1

0

τ

0

0

0

0

(a)

(b)

+

−

0

0

1

1

2

0

12

1

12

0

12

0

1

0

1

0

0

0

0

0

(c)

0.4

0.6

0.8

(e)
14

Two−quantum excitation energy (meV)

+

12

10

8

6

(d)
3

5

7

Emission energy (meV)

Figure 3.16: (a) – (d) Feynman diagrams for the coupling terms in the SIII time-ordering
for the six-level system shown in Fig. 3.15. The pulse sequence is indicated in (a) and therelevant time delays are shown in (b). The + and - sign above the diagrams indicates thesignal phase from the corresponding diagram. These diagrams show the first interaction
involving the |0i ↔ |1i transition followed by the |1i ↔ |12i. An equivalent set of diagramswith the first and second interactions involving the |0i ↔ |2i and |2i ↔ |12i transitions,
respectively, is not shown. (e) Two-quantum non-rephasing absolute value spectrum. The
coupling terms appear as the circled off-diagonal peaks.

Now that we have discussed the different types of 2D scans and spectra from thetheoretical perspective, we will discuss the experimental aspects of 2DCS.
3.3

Two-dimensional coherent spectroscopy – experiment
We have shown how the TFWM technique is similar to 2DCS. The experimental im-

plementations are also similar, in that even in a 2DCS experiment the TFWM signal is55
detected as the time delay between a pair of pulses is scanned. However, 2DCS requires
phase-stabilized measurement of signal electric field to enable taking a Fourier transform,
which is a significant challenge for optical frequencies. Since stabilities up to a percent of the
optical wavelength are needed, the experimental challenge increases for smaller wavelengthsof light. Various experimental techniques have been developed to achieve phase-stability
many of which rely on passive phase stabilization thorough dispersive optics [107, 108] andoptical birefringence in the ultra-violet regime [88] and detection of signal field in the frequency domain. A similar method involving single-shot measurement of 2D spectrum has
also been implemented [109]. A major disadvantage of these techniques is limited time delays
that can be scanned. This disadvantage is countered by phase modulating collinear excitation beams and detecting photoluminescence [93] or photocurrent in case of semiconductors[110]. These experiments involve scanning two time delays through mechanical stage motion,
which results in long scan durations. Additionally, since signal is detected as photocurrent,
only heterostructures in a diode structure can be studied using this method. Since we want
to study bare quantum wells with coherence lasting more than 10 ps, we use active stabilization of pulse delays is used [111]. The signal is detected on a spectrometer for different
pulse delays.

3.3.1

Experimental setup

The experimental schematic is shown in Fig. 3.17, which shows four beam paths, in thebox geometry, that are focused on the sample, which is kept in a cryostat, using lens f1 . The
three excitation pulses go through three of these paths as shown in Fig. 3.17. The TFWM
signal, which is emitted in the phase-matched direction given by Eq. (3.3), is collected by
the collimating lens f2 . This direction is the same as that a pulse along the fourth pathwould follow. The fourth pulse is, however, routed around the sample while the experimentis executing, and is called the “Reference” pulse. The TFWM signal is interfered withthis pulse and detected on a spectrometer through spectral interferometry, which enables

56
the detection of signal electric field [112]. The frequency resolution of the spectrometer islimited by the detection camera at ∼ 40 µeV, half-width at half maximum (HWHM). This
resolution can be improved by magnifying the image detected by the camera, although it
results in decrease in efficiency due to reflection.
Sample incryostatf2

Spectrometer
TFWM

f1
A*
B

BS
ce
en
r
e
f
Re

C
τ

BS

T

Figure 3.17: The experimental setup for two-dimensional coherent spectroscopy used in thiswork. The three excitation pulses are incident on the sample in the SI time-ordering and arelabeled A∗ , B and C. Lenses f1 and f2 are used to focus the excitation pulses and collimate
the emitted transient four-wave mixing signal (TFWM), respectively. The reference pulse isrouted around the sample, and interfered with the signal in a spectrometer. This figure has
been adapted from Ref. [113].

Before we discuss the detection of signal in detail, it is imperative to mention how thepulses in the box-geometry are generated and phase stabilized. These pulses are generated
from a mode-locked Titanium:Sapphire laser, which has a repetition rate of 76 MHz. The
laser can be tuned in the wavelength range of 750 – 920 nm. The transform limited pulse
duration can be set in the range of 100 – 200 fs. A single pulse from this laser is split into fourparts, and aligned in the box-geometry, by a set of three actively phase stabilized Michelsoninterferometers dubbed the JILA multidimensional optical nonlinear spectrometer or JILAMONSTR. Each pulse passes through a mechanincal stage, which is moved to scan the delay
between different pulses. A continuous-wave HeNe laser co-propagates with the laser pulses,
and is retro-reflected by a dichroic mirror at the exit of JILA-MONSTR, resulting in nested
Michelson interferometers. These retro-reflected beams are interfered on photo-diodes. A
mirror in each interferometer is mounted on a piezoelectric transducer. Thus, the path
length of one arm of each interferometer can be adjusted by changing the drive voltage for

57
the transducer. The pulse delays are actively stabilized by stabilizing the voltage due to theinterference from HeNe laser on the photo-diodes by driving the piezoelectric transducers
through servo-loops. This system is used to step the delay between pulses with a λ/100
stability and accuracy [111]. Depending on the pulse being scanned, time delays of 300 ps –1 ns can be achieved. More technical details on the JILA-MONSTR and active stabilization
can be found elsewhere [111, 100].
Since the Reference pulse goes through different optics after exiting the JILA-MONSTR,
another phase lock loop is used to stabilize it with respect to one of the excitation pulses
coming passing through the sample to ensure no phase jitter in detection of the TFWM
signal.
The polarization and average energy of the pulses coming out of the JILA-MONSTR
are set using a half-wave plate followed by a linear polarizer in the path of each pulse.
Circularly polarized pulses can be obtained by adding quarter-wave plates after the linear
polarizers. The wave-plates and polarizers give us complete control over the polarizationof the excitation pulses and the detected signal. In general, we define the polarization
scheme used in a 2DCS experiment by a series of four alphabets/symbols, which indicate
the polarization of the three excitation pulses, and the detected signal. For the SI timeordering, the four alphabets/symbols denote the polarization of pulses A, B and C followedby that of the detected signal. Orthogonal linear polarizations – horizontal and vertical withrespect to the optics table is denoted by letters X and Y, respectively. Similarly orthogonal
circular polarizations – right-circular (σ+ ) and left-circular (σ− ) polarizations are denoted by
the symbols + and −, respectively. For instance, the polarizations scheme + − +− implies
pulses A and C with σ+ polarization and B and the signal with σ− polarization. Some of thepolarization schemes that are commonly used are – collinear (XXXX or YYYY), cross-linear(XYYX or YXXY), and co-circular (+ + ++ or − − −−) As we will see later, this versatility
is extremely useful.
In addition to the polarizers and wave-plates, we also have a liquid crystal variable58
retarder in the beam path of pulses A and B. These variable retarders are used to eliminate
noise due to scatter in the detected signal through phase cycling, which will be discussed in
Section 3.3.2.
3.3.2

Obtaining 2D spectrum

We have outlined the experimental design and setup. Next we will discuss how a 2D
spectrum is obtained from experimentally measured data, including tricks to reduce noiseand detection of the signal phase.
Electric field detection:

Optical electric fields in the visible range oscillate at a

few hundred terahertz, which cannot be detected by electronics directly. We use spectralinterferometry to measure the electric field of the signal [112]. The signal and reference
pulses propagate in the same direction with a time delay between them. The frequency
domain interference (I) between a reference pulse and the signal is detected asI(ω) = |ER (ω) + ES (ω)|2(3.15)
2

2= |ER (ω)| + |ES (ω)|+ ER∗ (ω)ES (ω) + ER (ω)ES∗ (ω)

where ER and ES are the reference and signal complex electric fields, respectively. We canFourier transform the measured spectrum to time-domain, isolate one of the interferenceterms, and, through inverse Fourier transform, obtain the interference term
Sint (ω) = ER∗ (ω)ES (ω) = |ER (ω)||ES (ω)|ei[ω δt +{φS (ω)−φR (ω)}](3.16)

where δt is the time delay between signal and reference. φR and φS are the spectral phasesof the reference pulse and the signal, respectively. |ER | is obtained through the reference
spectrum while δt can be measured from the Fourier transform of Eq. (3.15). Thus, Eq.
(3.16) results in the signal field ES (ω) under the assumption that φR (ω) = 0. The signal
electric field spectrum ES (τ, ωt ) is measured as a function of the time-delay that is scanned.1
We shifted to denoting the frequency as ωt to relate to the emission energy axis in a 2D1

The actual implementation is slightly different, as we will discuss later.

59
spectrum. A numerical Fourier transform of the measured signal along delay τ is taken to
obtain a 2D spectrum ES (ωτ , ωt ).
Phase cycling:

As mentioned in Section 3.1.1, phase-matching condition is used

to isolate the TFWM signal from the linear response of the sample. However, a perfectisolation of the signal is an idealization; some scatter from excitation pulses is also emitted,
in the phase-matched direction, along with the TFWM signal. This scatter can be reduced
significantly through phase cycling [111].
The basic idea behind phase cycling is that the TFWM signal is measured for different
phases of the incident pulses, which also modifies the detected signal. The liquid crystals
are used to delay pulses A and B so as to modify their phase on the sample. Four spectra
are recorded for each time delay. These spectra are processed to eliminate the noise due toscattering.

1
2
3
4

A

B

C

S

+
+

+
+
-

+
+
+
+

+
+
-

Table 3.1: Phase cycling procedure

The phase cycling process is outlined in Table 3.1. Each row indicates the phase of
excitation pulses (A, B and C) and the TFWM signal (S) for one of the four recorded
spectrum. + and − signs are used to indicate electric fields with π phase shift between
them. The phase of the TFWM signal depends on the phase of the excitation pulses. The
phase-cycled signal is then calculated using the relation
1
S = (S1 − S2 + S3 − S4 )4(3.17)

where the subscripts denote the row numbers in Table 3.1. This procedure removes thescatter from each of the pulses.

60
2D spectrum:

Although we mentioned earlier that ES (ω) is measured for eachtime delay, that is not how the analysis is implemented. It is more efficient to record theinterference term in Eq. (3.16) in the time domain, given by its Fourier transform Sint (t),which also enables us to deconvolve the artificial broadening due to the finite resolution of
the spectrometer [114].
The time-domain interference term is recorded as a function of the scanned time delay
τ to obtain Sint (τ, t). An example of this signal is shown in Fig. 3.18(a), which was experimentally measured from a GaAs QW. The real time axis is obtained through a Fouriertransform of the frequency axis obtained from the spectrometer. The dashed line indicatest = 0 and the value of real time axis at this point gives the delay δt between the signal andthe reference pulse. This interference term Sint (τ, t) is isolated by taking time cuts in thetime-domain; the two solid lines indicate the region that is selected. The photon-echo likebehavior of the signal is apparent, which allows us to take a time cut such as the one shownin Fig. 3.18(a). This time-cut reduces the noise significantly and results in a much cleaner
spectrum than the one that would result without it. The reduced noise also allows us toscan up to larger values of τ , which results in better resolution in 2D spectrum.
After performing the processing mentioned above, we obtain the phase-cycled, twodimensional time-domain interference term Sint (τ, t). By taking a Fourier transform along
both τ and t, we end up with the interference term in two-dimensions
Sint (ωτ , ωt ) = |ER (ωt )||ES (ωτ , ωt )|eiφS (ωτ ,ωt ) .

(3.18)

Equation (3.18) is obtained after removing the spectral phase due to δt and under theassumption φR = 0. A 2D spectrum ES (ωτ , ωt ) is obtained by dividing Eq. (3.18) by|ER (ωt )|, which is measured by the spectrometer directly. Figure 3.18(b) shows the absolutevalue spectrum |ES (ωτ , ωt )| obtained from the time-domain signal shown in Fig. 3.18(a).
Signal phase:In the discussion presented above we have implied that the signal

phase is correctly measured. This signal phase is, in general, incorrect because of the ran-

61
0.2

0.4

0.6

0.8
−1542

Excitation energy (meV)30

t=0τ (ps)

20

10

0
0

10

20

30

Real time (ps)
(a)

−1544

−1546

40

(b)

1542

1544

1546

Emission energy (meV)

Figure 3.18: (a) Experimentally obtained absolute value 2D signal in the time domain fromthe heavy hole exciton resonance in GaAs/Al0.3 Ga0.7 As quantum well. The dashed line in (a)indicates t = 0. The solid lines in (a) indicate the time cuts applied to reduce the detected
scatter; the signal outside the two lines is set to 0. (b) 2D spectrum in the frequency domain
by taking Fourier transform of the signal shown in (a) along τ and t.

dom phase shifts introduced when the delay between pulses is locked, and can be corrected
through an all-optical technique [115]. Another method to obtain the correct signal phaseis though a complimentary experiment – spectrally resolved pump-probe (SRPP) signal,
which is analogous to a TFWM experiment. Under the right conditions the SRPP signal isequivalent to the TFWM signal [116].
SP P (ωt ) = Re ES (ωt )|τ =0 eiφ0(3.19)

where SP P (ωt ) is the spectrally resolved pump-probe signal, ES (ωt ) is the measured complex
signal field, and φ0 is the phase offset is added to satisfy the equality. The experimentallymeasured spectra from the 2DCS and SRPP experiments are matched by varying φ0 . The
result of such a procedure for the absolute value spectrum in Fig. 3.18(b) is illustrated inFig. 3.19, where the SRPP signal is shown as black dots and the equivalent 2DCS signal
is plotted as the red line. The measured φ0 is then used to obtain the correctly-phased 2D

62
spectrum
ES,cor (ωτ , ωt ) = ES (ωτ , ωt )eiφ0 .

(3.20)

Figures 3.20(a) and 3.20(b) plot the real and imaginary parts of the absolute value 2DSpectrally resolved pump-probe

spectrum in Fig. 3.18(b).
2D
PP

0.6
0.4
0.2
0
−0.2
−0.4
−0.6
372

373

374

375

Emission frequency (THz)

Figure 3.19: The real part of the 2D signal for τ = 0, shown in red line, is adjusted to matchthe measured spectrally resolved pump-probe signal (PP) shown as black dots.

−1

−0.5

0

0.5

Excitation energy (meV)

−1542

−1544

−1546

(a)
1542

(b)
1544

1546

1542

1544

1546

Emission energy (meV)

Figure 3.20: (a) Real and (b) imaginary part of the absolute value spectrum shown in Fig.
3.18(b).

However, there are certain disadvantages of using this method to find the signal phase.

63
These disadvantages arise from some of the following conditions, which must be satisfied for
Eq. (3.19) to be true, and, consequently, this method to work:• The system must be in a real population state during T . For instance, pulses A and
B with orthogonal polarizations create a spin grating, which is not a real population
grating. As a result, this technique will not work for such a polarization scheme.
• The delay between the pump and probe pulses should be set equal to T in the 2DCS
experiment.
• The average power of the pump should equal the sum of the average powers of pulses
A and B. The average power of the probe should equal the average power of pulse
C.
• The measured signal phase is very sensitive to the pump-probe overlap and theoverlap of excitation pulses in the 2DCS experiment.
The last two conditions are unique to semiconductors because of excitation induced
effects such as EID and EIS. These conditions somewhat limit the applicability of this technique to find the signal phase. Despite these limitations, SRPP is applied extensively to
extract the correct signal phase in this work.
3.4

Two-dimensional coherent spectroscopy – previous studies
In this chapter, we have discussed the experimental and theoretical aspects of 2DCS.

Although this discussion was carried out for model systems, similar spectra are observed in
QWs. In addition to the kinds of spectra discussed several modifications are made dependingon the system and the experimental objective. We will discuss some of these modifications,which are relevant to this work, which will be discussed now.
Polarization dependence:

We discussed the exciton and biexciton polarization

selection rules in Section 2.7. As a result of those selection rules, the TFWM signal emitted

64
by an excitonic system can vary significantly as demonstrated by Bristow et al. [117]:• The biexciton signal appears red-shifted compared to the exciton peak along theemission energy axis.
• The exciton signal is reduced significantly for the cross-linear polarization scheme,
resulting in relatively stronger biexciton peak.
• The biexciton peak is not present for the co-circular polarization scheme, as is thecase in the spectrum in Fig. 3.18(b).
Since the biexciton signal appears as a separate peak, we will see that, it is possible
to obtain the parameters of the exciton ↔ biexciton resonance. Furthermore, frequency
dependent biexciton binding energy can also be easily measured [118].
Many-body effects:

It is worth noting that the real part of 2D spectrum for the

HH exciton resonance shown in Fig. 3.20(a) has a dispersive lineshape, i.e. it comprises comparable positive and negative parts with node on the diagonal line, which is not expectedfrom a two-level system. Such a peak-shape was explained through EIS, which is a phenomenological many-body effect resulting from exciton-exciton interactions [13]. In addition
to this, other many-body effects such as EID and local-field effects also modify the TFWM
signal [5], and, as a result, the 2D spectrum. 2DCS is ideal to identify the effect of these
phenomenological many-body effects, and can be simulated through numerical calculations,which are presented in Appendix B. Such numerical calculations are computationally intensive. In this work, however, we will discuss these many-body interactions from a different
perspective, similar to those discussed in Section 3.2.5. We will show that such a description
is much easier to simulate, and enable quantifying the strength of these interactions.
T -dependence:

We will present studies on the spectral diffusion of excitons in QWs

in this work. Previously spectral diffusion has been studied using other techniques such as
3PEPS spectroscopy [36, 40]. 2DCS is an excellent technique to study such a phenomenon,

65
and has been used to study spectral diffusion in molecular systems extensively [39, 119].
Spectral diffusion is measured by recording one-quantum 2DCS spectra for different valuesof T . These spectra directly depict the redistribution of resonance frequencies of the excited
population. Through these measurements we will highlight the limitations of the strongredistribution approximation, which is generally assumed for QW excitons. Additionally, it
can be used to accurately model exciton-phonon interactions.
In this chapter we presented a general discussion of 2DCS, and the several studies
relevant for the results that will be presented in this work. We will discuss the specificexperimental results obtained in the following chapters. These results will include studies
of excitons in (110)-oriented GaAs QWs, spectral diffusion of exciton in QWs and modeling
QW excitons as bosons. We will find that the properties of 2DCS that we have discussed inthis chapter, result in significant advantages for the aforementioned studies.

Chapter 4
Excitons in (110)-Oriented GaAs Quantum Wells

In this chapter, we report measurements of the homogeneous and inhomogeneous
linewidths of the HH exciton state in a (110)-oriented GaAs QW sample for excitation
polarized along X and Y directions in the plane of the QW. We have studied the 8-nm-wide
QWs, which we discussed in Section 2.9.2. We have already discussed the reduced symmetryfor (011)-oriented GaAs QWs, and some of the resulting properties in Section 2.6. We also
discussed the anisotropic absorption coefficients for two orthogonal linear polarizations in
Section 2.9.2. We find that while unequal absorption coefficients give rise to anisotropic EID
effects, and, thus, different homogeneous linewidths for excitons excited along the two polarization directions, the extrapolated zero-excitation density homogeneous linewidth is thesame for excitations along both directions. A study of the temperature dependence of thehomogeneous linewidth reveals an activation-like behavior of thermal broadening. We also
compare the results we obtain with those reported for (001)-oriented GaAs QWs to contrast
the exciton dephasing mechanisms in QWs grown along the two growth directions.
The sample was kept in a cold finger continuous flow helium cryostat. The sample
temperature was varied from 6 – 50 K. The excitation pulses are ∼ 120 fs long and are
obtained from a 76 MHz repetition rate mode-locked Ti:sapphire laser. All the excitation
beams are set to have, on an average, equal photon density per pulse. We present results forthe collinear polarization schemes with X and Y polarization. The X and Y axes coincidewith the [001] and [11̄0] crystal axes, respectively. Experiments were performed with the

67
excitation pulses in the rephasing time ordering.
4.1

Measuring linewidths
First we will present a typical 2D spectrum for these QWs and illustrate the mea-

surement of the homogeneous and inhomogeneous linewidths. Figure 4.1 shows the (a) realpart and (b) absolute value of a typical rephasing one-quantum 2D spectrum obtained forpolarization scheme Y with the sample at a temperature of 10 K. The laser spectrum had
a full width at half-maximum bandwidth of ∼ 16 meV and was tuned to be resonant with
the HH exciton resonance, as indicated by the dotted lines in Fig. 4.1. In Fig. 4.1(a), thecentral dispersive peak (Ex) on the diagonal line is the HH exciton resonance. The negative
absorptive peak (Bx) that is red-shifted relative to the exciton peak along the emission axis
is due to the HH bound biexciton state. The marked elongation of the HH exciton peakin the diagonal direction is because of large inhomogeneous broadening in the sample. The
dispersive lineshape of the exciton peak suggests that the signal is dominated by many-bodyeffects including EID and excitation-induced shift [13].
From an absolute value 2D spectrum, such as Fig. 4.1(b), the homogeneous andinhomogeneous linewidths are obtained by simultaneously fitting the diagonal and crossdiagonal slices to analytical lineshapes [104]. Figure 4.2 shows an example of the data
(dashed line) and fit curves (solid line) for slices from a 2D spectrum for polarization scheme
Y. The sample was kept at a temperature of 10 K and the average photon density of ∼ 5.4
x 1011 cm−2 per pulse was incident on it. There is a distinct difference between the crossdiagonal slice and the obtained fit on the negative energy wing of the main peak. This
discrepancy is due to the weak biexciton peak, which was not considered while performing
the fits. For the slices shown in Fig. 4.2, we obtained values of 0.105 and 1.9 meV for
the homogeneous and inhomogeneous linewidths, respectively. These, and the linewidths
reported later on, are the half width at half-maximum (HWHM) values of the Lorentzianand Gaussian distribution functions, which correspond to homogeneous and inhomogeneous

68

Excitation energy (meV)

−0.2

0

0.2 0.4 0.6

0.2

0.4

0.6

0.8

1

(a)
−1558

(b)

Ex

−1562

Bx
−1566

1558

1562

1566

1558

1562

1566

Emission energy (meV)

Figure 4.1: A typical (a) real part and (b) absolute value normalized 2D spectra of thecomplex rephasing signal for polarization scheme Y and sample temperature of 10 K. The
exciton (Ex) and biexciton (Bx) peaks are marked in (a). The dotted line at the bottom of
the figures shows the linear absorption spectrum.

linewidths, respectively, [104] and have been obtained after deconvolution of the signal withthe spectrometer response. The ratio of the linewidths suggest that inhomogeneity is thedominant broadening mechanism in (110)-oriented GaAs QWs. This result is expected froma rough growth front along the (110) direction for GaAs [120].
Cross−diagonal slice

Amplitude (arb. units)

Diagonal slice
1

(a)

(b)

0.5

0
−5

0

5

−5

0

5

Energy offset from peak (meV)

Figure 4.2: Slices (dashed line) and fits (solid line) along the (a) diagonal and (b) crossdiagonal directions of the normalized absolute value spectrum.

69
4.2

Excitation-density dependence
On repeating the above measurement for polarization scheme X while keeping the otherparameters the same, we measured a smaller homogeneous linewidth (0.086 meV) while theinhomogeneous linewidth was the same. We investigated the reason for unequal homogeneous linewidths for the two polarization schemes by varying the average photon density
incident on the sample. Figure 4.3 shows the variation of the homogeneous linewidth withthe average photon density per pulse incident on the sample kept at a temperature of 10 K,
along with a representative error bar obtained by repeating the experiment multiple times.
Over the range of photon density shown, the HH exciton state exhibits different homogeneous linewidths for X and Y polarization schemes. For both the polarization schemes, thehomogeneous linewidth increases with increasing photon density due to EID [55, 4], while

Homogeneous linewidth(meV)

the inhomogeneous linewidth is constant at ∼ 1.9 meV.
0.12

X
X (Fit)Y
Y (Fit)

0.10
0.08
0.06
0.04
0

1

2

3

4

5

6

Incident photon density
per pulse (10 11 cm −2)

Figure 4.3: Incident photon density dependence of homogeneous linewidth for X (triangles)
and Y (circles) polarization schemes. The linear fits to the data are shown by the solid lines.
A representative error bar, obtained by repeating the same experiment multiple times, isshown. The sample was kept at a temperature of 10 K.

For low photon density incident on the sample, the photon density dependence of the

70homogeneous linewidth can be approximated as [121]γ(φ) = γ0 + γEID ∗ φ(4.1)

where γ0 is the zero-excitation density homogeneous linewidth, γEID is the rate of increaseof homogeneous linewidth with average photon density per pulse φ incident on the sample.
The solid lines in Fig. 4.3 are the fits to the data using Eq. (4.1). The fit parameters listed inTable 4.1 show that while the value of γ0 is the same (within the errors obtained by assuming
constant error in homogeneous linewidth over the range of photon density), different valuesof γEID are obtained for the two polarization schemes.
Polarization scheme

γ0 (µeV)γEID (10−17 eVcm2 )X
Y

41 ± 242 ± 28.1 ± 0.6
12.1 ± 0.5

Table 4.1: Parameters of Eq. (4.1) for the fits shown in Fig. 4.3.

In Eq. (4.1), we use the average photon density per pulse incident on the sample rather
than the exciton density in the QWs to quantify the effect of EID on the homogeneous
linewidth. This is valid because, for relatively low photon density, the excitation density is
directly proportional to the incident photon density. From the absorption spectra shown in
Fig. 2.11, we estimate that the absorption of the HH exciton transition is ∼ 1.3 times larger
for Y polarization compared to X polarization. The difference in the exciton density forthe same photon density incident on the sample results in different γEID values for the twopolarization schemes. From the ratio of γEID for the two polarization schemes, we estimate
an excitation density ratio of 1.5 ± 0.1 for polarization scheme Y to that for X. This ratio
is in reasonable agreement with the estimate from the absorption curves and supports ouranalysis that anisotropic EID results in different homogeneous linewidth of the HH exciton
transition polarized along X and Y axes.
For an electronic transition, both the absorption coefficient and the radiatively limited
homogeneous linewidth are directly proportional to the square of the transition dipole mo-71ment [15]. Hence, if radiative decay was the only process contributing to γ0 , from unequal
absorption coefficients for the HH exciton transitions polarized along X and Y axes, we would
expect different γ0 for the two polarizations. Equal values of γ0 for both the polarizations
suggest that it is not radiatively limited.
4.3

Population decay rate
We measure the population decay times from a rephasing zero-quantum 2D spectrum.

Figure 4.4(a) shows such a 2D spectrum, for Y polarization scheme with the sample at atemperature of 10 K, where we plot the normalized signal intensity. When only a single
resonance is excited, as is the case presently, the system is in a population state during T .
Hence the linewidth in the mixing energy direction gives the population linewidth. We fit aslice from the zero-quantum 2DFT spectrum, indicated by the dashed line in Fig. 4.4(a), toa double Lorentzian. The linewidth of the narrower Lorentzian gives the population decayrate. The slice (dashed line) and the best-fit curve (solid line) are shown in Fig. 4.4(b). We
measure a population linewidth of ∼ 10 µeV at a temperature of 10 K. We would like topoint out that we did not measure different population linewidths for X and Y polarization
schemes. However, due to ∼ 2 µeV uncertainty in the population linewidths measured, wecannot confirm the presence of non-radiative population decay pathways. In the absence ofpure dephasing mechanisms, the measured population linewidth would result in a homogeneous linewidth of ∼5 µeV for both X and Y polarization schemes, according to Eq. (2.12).
Since measured γ0 is roughly 8 times this value, we can unambiguously conclude that puredephasing processes contribute significantly to the measured γ0 . Exciton-phonon interaction is an important mechanism that contributes to the homogeneous linewidth in QWs.
We investigate this interaction through the temperature dependence of the homogeneous
linewidth.

72

0.2 0.4 0.6 0.8
1Intensity slice(arb. units)

Mixing energy
(meV)

0.5

0

−0.5
1558

(a)

1562

1566

0.5

0
−0.4 −0.2

Emission energy (meV)

(b)

0

0.2

0.4

Mixing energy (meV)

Figure 4.4: A rephasing zero-quantum 2DFT spectrum for Y polarization scheme is shownin (a). The spectrum plots signal intensity. The slice (dashed line) at the center of theinhomogeneous distribution of the spectrum in (a) is shown in (b). The solid line in (b)shows the best-fit double Lorentzian curve.

4.4

Temperature dependence
We repeated the power dependence study presented earlier for different sample tem-

peratures varying from 6 – 50 K. We plot the values of γ0 for X and Y polarization schemes
for different temperatures in Fig. 4.5. For both the polarization schemes, nearly equal values of γ0 are obtained over the temperature range studied as expected from the discussionof the anisotropic EID effect earlier. Additionally, we find that γ0 increases with the sample temperature for both the polarization schemes. As the lattice temperature increases,
exciton-phonon interactions increase resulting in a faster dephasing of excitons. Since, thephonon population follows the Bose - Einstein distribution, we can describe the thermal
broadening as
γ0 (Ts ) = γ0 (0) + γ

∗


e

Eph
kB Ts

−1
−1

(4.2)

where we have written γ0 as a function of the sample temperature Ts , γ ∗ is the amplitudeof the thermal dephasing term, Eph is the energy of the dominant phonon mode responsible
for the dephasing and kB is the Boltzmann constant. In Eq. (4.2) we have considered thecontribution of only a single phonon mode to the pure dephasing of excitons. Equation (4.2)
is similar to the one used by Lee et al. [122] for (001)-oriented GaAs QWs. The difference

73
here is that we have not used the term that is linear in Ts , kept Eph as an adjustableparameter and not considered impurity scattering effects. The best-fit curves are shown in
Fig. 4.5 by a solid (dashed) line for polarization scheme X (Y). The values obtained forthe fit parameters, which are equal for the two polarization schemes (within the errors), arelisted in Table 4.2.
0.30

X
X (Fit)Y
Y (Fit)γ 0 (meV)

0.25
0.20
0.15
0.10
0.05
10

20

30

40

50

Temperature (K)Figure 4.5: Sample temperature dependence of γ0 for X (triangles) and Y (circles) polarization schemes. The fit, as per Eq. (4.2), is represented by the solid (dashed) line for X(Y) polarization scheme. The error bars represent the statistical errors only and additional
systematic error might also be present. Note that for a few temperatures the data points for
X and Y polarization schemes overlap.

We find the zero-excitation density homogeneous linewidth extrapolated to 0 K, γ0 (0),is ∼ 34 µeV (corresponding to a dephasing time of ∼ 20 ps) and phonons with energy of ∼ 3
meV are predominantly responsible for the faster dephasing of the HH exciton polarization
at high temperatures. Usually Eph corresponds to a transition to a real state which, in thiscase, could be delocalized exciton states centered ∼ 3 meV above the line-center of the HH
exciton inhomogeneous distribution. This activation would be an incoherent process that
we can probe by varying the time delay between the second and third pulses [113]. We
do not see any activation peak in these complimentary experiments (spectra not shown),
which suggests that the thermal broadening is due to an elastic exciton-phonon scattering
mediated virtual activation process. However, an activation peak too weak to be seen and/or

74

Polarization scheme

γ0 (0) (µeV)γ ∗ (meV)

Eph (meV)X
Y

34 ± 235 ± 40.24 ± 0.03
0.22 ± 0.08

3.1 ± 0.3
2.9 ± 0.7

Table 4.2: Parameters of Eq. (4.2) for the fits shown in Fig. 4.5.

activation to a dark state due to a change in the in-plane momentum of excitons cannot be
ruled out. Equal values of γ ∗ and Eph for both the polarization schemes suggest that theactivation mechanism is isotropic for exciton transitions polarized along X and Y directions.
Compared with the population times reported earlier, the values of γ0 (0) obtained
for the two polarizations indicate that the HH exciton transition is dominated by puredephasing processes even at low temperatures. We attribute this dephasing to acoustic
phonon assisted tunneling of excitons between localization sites,[123] which exists due todisorder in the sample. This tunneling is important at low temperatures and does not show
any activation-like behavior if it is accompanied with the emission of phonons.
We would like to point out that studies of temperature dependence of homogeneous
linewidth of HH excitons in (001)-oriented GaAs QWs have shown [122, 124, 125, 126] that at
low temperatures (< 50 K) the linewidth increased linearly with temperature. This behavior
would be replicated by Eq. (4.2) for Eph  kB Ts . Thus, we conclude that for (001)-oriented
GaAs QWs Eph would be lesser than what we measure for the (110)-oriented QWs. We would
also like to note that for (001)-oriented GaAs QWs with well widths ranging from 5 – 13
nm, low temperature dephasing times of 1 – 12 ps have been reported [124, 55, 126, 127, 31].
The homogeneous linewidth of excitons is sensitive to the inhomogeneity. As a consequence,
the results in the publications cited above show quite large variation in dephasing times
although all those results are for samples with relatively small inhomogeneous linewidth(HWHM < 0.5 meV). While the dephasing time we report here for (110)-oriented QWs
is slightly longer than these values, it is smaller than the value of ∼ 70 ps reported for(001)-oriented sample with significantly greater inhomogeneity [2].

75
In conclusion, we have measured the linewidths of the HH exciton resonance in (110)oriented GaAs QWs. The measurements reveal that inhomogeneous broadening is the dominant broadening mechanism. In addition to reporting the dephasing time in a QW with
reduced symmetry such as (110)-oriented GaAs QWs for the first time, we have also discussed important phenomena that affect the dephasing time. We show that anisotropic
dipole moments along X and Y crystal axes, through EID, result in different homogeneous
linewidths for exciton transitions polarized along the two axes. We find that the increasein homogeneous linewidth with temperature is isotropic along X and Y axes and shows anactivation-like behavior, and that the homogeneous linewidth, even at low temperatures, isdominated by pure dephasing processes.
Another interesting feature of these QWs, which we have not discussed, is the presenceof finite absorption between the HH and LH peaks in the absorption spectrum shown in Fig.
2.11. The absorption in this spectral region is probably due to the transition between thesecond HH state and the electron ground state. We observe peaks due to this nominallydisallowed transition in some preliminary studies, which are discussed in Appendix C.

Chapter 5
Spectral Diffusion

We introduced the phenomenon of spectral diffusion, in which the resonance energy ofthe excitation changes over time, in Section 2.5. We also defined the normalized frequencyfrequency correlation function, which we will be referred to as the correlation function fromhereon, which is generally used to quantify spectral diffusion. Several experiments including
3PEPS [36, 37, 40, 102] and various observables in 2DCS [39, 119, 128] are used to study
the spectral diffusion in various systems. Regardless of the technique, the final goal is tomeasure the correlation function, which is approximated through these experiments.
As mentioned in Section 2.5, the description of spectral diffusion through the correlation
function is only valid in the case of the strong redistribution approximation. In this regime
both increase and decrease of energy of the excitation are equally likely[129]. We have studied
the spectral diffusion of excitons in QWs. In general, the strong redistribution approximation
is assumed to be valid for excitons in QWs [130, 40, 41]. Furthermore, it is assumed that thecorrelation function decays exponentially with time; this is the Gauss-Markov approximation[38, 40].
In this chapter, we will discuss theoretical description of spectral diffusion in the strong
redistribution approximation and its effect on 2D spectra through simulations. Next, we will
present experimental results that show that the strong redistribution approximation is not
valid for excitons in QWs for low sample temperatures. Finally, we will quantify spectral
diffusion, for high sample temperatures, by measuring the correlation function through a

77
numerical fitting procedure. We show that even in the limit of strong redistribution approximation, the correlation function decay is non-exponential, which violates the Gauss-Markov
approximation.
5.1

Frequency-frequency correlation function
The most common way of quantifying spectral diffusion is through the normalized thecorrelation function C(T ) defined in Eq. (2.16). Usually C(T ) is not measured directly,
but is inferred through other experiments. One such experiment is the 3PEPS spectroscopy,
which we discussed in 3.1.3. The decrease in the photon echo peak delay τm corresponds toa decrease in the value of the correlation function. It has been shown that the two values
are proportional to each other, at least for large time delays [36, 37]. However quantitative
analysis is based on assumptions of functional form of C(T ) [39]; an exponential decay is
assumed in the Gauss-Markov approximation [38, 40]. Thus, a direct measurement of thecorrelation function is useful for quantifying spectral diffusion.
The TFWM signal given by Eq. (3.7) is modified due to the correlation function as
[39]
1

2

2

2

SSI ,SD (τ, T, t) = 2|E|3 µ01 4 e−i∆01 (τ −t) e−(γ01 τ +Γ1 T +γ01 t) e− 2 σ01 (τ +t −2Cτ t) Θ(τ )Θ(T )Θ(t) (5.1)
where all the parameters have the usual meaning. Equation (5.1) is exactly the same as Eq.
(3.7) for C = 1.
The effect of change in the correlation function is apparent in 2D spectrum. Figures
5.1(a) and 5.1(b) illustrate this effect for C = 1 and C = 0, respectively. We used γ01 = 0.1
meV and σ01 = 0.4 meV. It is apparent that the the peak in 2D spectrum is no longer
elongated along the diagonal line for C = 0. The excitation and emission energies are
perfectly correlated for C = 1 resulting in the peak that is elongated along the diagonal
direction in Fig. 5.1(a). For C = 0 there is no correlation between the excitation andemission frequencies, resulting in a peak that looks homogeneously broadened. The visible

78

Excitation energy (meV)

0.2

0.4

0.6

0.8

C=1

C=0

−3

−5

−7

(a)
3

(b)
5

7

3

5

7

Emission energy (meV)

Figure 5.1: Effect of the frequency-frequency correlation function C on 2D spectrum for an
inhomogeneously broadened resonance with (a) C = 1 and (b) C = 0.

effect of correlation function on 2D spectrum makes 2DCS a powerful technique to study
spectral diffusion. A number of observables such as peak ellipticity [119], peak slope [128],and several others, as summarized by Roberts et al. [39] have been used as a proxy for thecorrelation function. In general, the change in the shape of the 2D peak is measured as afunction of the waiting time T to quantify spectral diffusion through C(T ). Nevertheless,these observables only approximate the correlation function and not measure them directly.
In this chapter, we would show that C(T ) can be obtained directly though a numerical fitting
procedure. However, before we discuss those measurements, we present measurements thattest the validity of the strong redistribution approximation for excitons in QWs. This is an
important exercise because the correlation function does not have a physical meaning if thestrong redistribution approximation is not valid.

79
5.2

Validity of the strong redistribution approximation

5.2.1

Spectral diffusion from 2D spectra

Spectral diffusion is measured by taking a series of rephasing one-quantum 2D spectra
for different values of T . Since the system is in a population state during delay T , the change
in 2D peak shape with T directly gives the redistribution of exciton energy. We have studied
the spectral diffusion of HH excitons in (001)-oriented GaAs QW sample, which is describedin Section 2.9.1. We can directly observe the effect of spectral diffusion in experimentallymeasured 2D spectra.

Excitation energy (meV)

0.2

0.4

0.6

0.8

0

0.03

T = 0.2 ps

0.06

0.09

T = 26.6 ps

−1542

−1544

−1546

−1548

(a)
1542

(b)
1544

1546

1548

1542

1544

1546

1548

Emission energy (meV)

Figure 5.2: Rephasing one-quantum absolute value 2D spectra with (a) T = 0.2 ps and (b)T = 26.6 ps with the sample at a temperature of 5 K. We will compare the redistribution of
the population of excitons created with energies indicated by the two horizontal lines. Notethe asymmetric peak in (b).

Figures 5.2(a) and 5.2(b) show the HH exicton peak in rephasing one-quantum absolutevalue 2D spectra taken with T = 0.2 and 26.6 ps, respectively, with the sample kept at atemperature of 5 K. The peak in Fig. 5.2(a) is elongated along the diagonal line because
the excitation and emission energies are correlated across the inhomogeneous distribution.
In other words, excitons created with a particular energy radiate TFWM signal at the same80
energy. The peak in Fig. 5.2(b), however, broadens due to redistribution of the exciton
energy. This broadening, however, is different than in the simulation shown in Fig. 5.1(b).
The difference arises because of preferential relaxation of excitons that were created at ahigher energy, as denoted by the dash-dot line, to lower localized states compared to theexcitation of low energy excitons to higher energies.
The high energy excitons are delocalized, and can easily be trapped in localizationsites through emission of phonons. The low energy excitons are localized, and need toabsorb phonons to tunnel through the localization barrier. Since, there isn’t a significant
phonon population at 5 K, the excitation of low energy excitons to higher energies is notfavorable. Emission of phonons, however, does not need a phonon population. Thus, for
low temperatures phonon emission is more likely than phonon absorption, resulting in thepreferential relaxation of excitons and the resulting asymmetric peak in Fig. 5.2(b). We
further confirm this claim through experiment performed at higher temperatures.

0.2

0.4

0.6

0.8

0

0.03

0.06

0.09

Absorption energy (meV)−1540

T = 0.2 ps

T = 26.6 ps

−1542

−1544

−1546

(a)

−1548
1540

(b)
1542

1544

1546

1548

1542

1544

1546

1548

Emission energy (meV)

Figure 5.3: 2D spectra with (a) T = 0.2 ps and (b) T = 26.6 ps with the sample at
a temperature of 25K. We will compare the redistribution of the population of excitons
created with energies indicated by the two horizontal lines.

Figure 5.3 shows 2D spectra which are the same as those in Fig. 5.2 but were measured with the sample at a temperature of 25 K, which results in significant differences

81
between them. Firstly, the peak in Fig. 5.3(a) is broader due to increase in the homogeneous linewidth with temperature. But more importantly, the peak in Fig. 5.3(b) is almost
perfectly symmetric. This observation is consistent with our explanation of the asymmetric peak in Fig. 5.2(b); increased phonon population, due to higher sample temperature,
increases the likelihood of increase in exciton energy due to absorption of phonons, which
results in a symmetrically broadened peak.

5.2.2

Population redistribution

Now we will analyze the asymmetry between the exciton relaxation and excitation
processes through the evolution of the population of the excitons created with the excitation energies indicated by the horizontal lines in Figs. 5.2 and 5.3. These energies areoffset by ±0.3 meV from the excitation energy corresponding to the peak maximum. If thestrong redistribution approximation is satisfied, the population redistribution for both these
excitation energies should be identical. Conversely, different population redistribution of
excitons created with these excitation energies would suggest that the strong redistribution
approximation is not valid.
Figures 5.4(a) and 5.4(b) show the population redistribution of excitons created at
excitation energies denoted by the dot-dash and solid lines, respectively, in Fig. 5.2. The
spectrally-resolved change in the normalized population distribution1 is plotted for both theexcitation energies. Mathematically, the plotted quantity is
Sdif f,ωτ0 (T, ωt ) = SN (ωτ0 , T, ωt ) − SN (ωτ0 , 0.2 ps, ωt )(5.2)

where Sdif f,ωτ0 (T, ωt ) is the population redistribution spectrum, SN is the normalized 2D
spectrum, and ωτ0 is the excitation energy for which the population redistribution is calculated. The emission energy equal to ωτ0 is indicated by the vertical lines in both the spectra
1Normalized population distribution is obtained by selecting a peak area and dividing by the average
amplitude of the peak for each T . This procedure eliminates the effect of overall population decay with T ,and isolates the population redistribution.

82
shown in Fig. 5.4.
Population redistribution
High excitation energy
Low excitation energy
−3

−2

−1

0

1

−2

−1

0

Asymmetry
−3

1

−2

−1

0

0

2

1

Waiting time T (ps)

50
40
30
20
10

(a)

(b)

(c)

03
1542

1544

1546

1548

1542

1544

1546

Emission energy (meV)

1548 −4−2

4

Energy offset (meV)

Figure 5.4: Population redistribution for (a) high excitation energy and (b) low excitationenergy slices indicated in the 2D spectra in Fig. 5.2. The excitation energy is denoted by
the vertical lines in (a) and (b). (c) The difference of the plots in (a) and (b). This plot
highlights the asymmetry between the relaxation and excitation processes.

The plots in Figs. 5.4(a) and 5.4(b) look different in that the decrease and increasein population is significantly higher in the former spectrum. To highlight the asymmetry
between these two plots, we take a difference of the two plots, which is shown in Fig. 5.4(c).
One of the plots is rotated about the emission energy of the peak before calculating thedifference and the horizontal axis is plotted as the energy offset from the emission energy ofthe peak. We would expect this plot to have values close to zero if the strong redistribution
approximation valid.
The same set of plots are shown in Fig. 5.5 for the 25 K data shown in Fig. 5.3. We
can clearly see that the plots shown in Figs. 5.5(a) and 5.5(b) are more symmetric thantheir counterparts in Fig. 5.4. This is further corroborated by the plot in Fig. 5.5(c), which
has smaller values compared to the plot in Fig. 5.4(c).

83Population redistribution
High excitation energy
Low excitation energy−1.5−1 −0.5

0

−1.5

0.5

−1

−0.5

0

0.5

Asymmetry
−1.5

−1

−0.5

0

−2

0

2

Waiting time T (ps)

50
40
30
20
10

(a)

03
1540

(b)
1542

1544

1546

1548

(c)
1542

1544

1546

Emission energy (meV)−44Energy offset (meV)

Figure 5.5: Population redistribution for (a) high excitation energy and (b) low excitationenergy slices indicated in the 2D spectra in Fig. 5.3. The excitation energy is denoted by
the vertical lines in (a) and (b). (c) The difference of the plots in (a) and (b). This plot
highlights the asymmetry between the relaxation and excitation processes.

5.2.3

Temperature dependence

The analysis presented in Section 5.2.2 is helpful to qualitatively highlight the changes
in the spectral diffusion characteristics with the sample temperatures. Even so, it merely
stresses the features that were already apparent from the 2D spectra and the accompanying discussion in Section 5.2.1. Now we will use spectra similar to those in Figs. 5.4(c)and 5.5(c) to compare the validity of the strong redistribution approximation for different
temperatures. We quantify the asymmetry between the exciton relaxation and excitation
processes, for a given temperature, by taking the sum of the absolute value of all the points
in the “Asymmetry” spectrum, which yields zero for symmetric population redistribution.
This asymmetry, as a function of sample temperature, is shown in Fig. 5.6. The asymmetry
values have been normalized by dividing by the value for 5 K.
We observe a fast decrease in the asymmetry parameter with an increase in the sample
temperature. Furthermore, we find that, for temperatures higher than 20 K, the asymmetry

84

Asymmetry (a. u.)

1.0

0.8

0.6

0.4

0.2
0

20

40

60

80

Temperature (K)Figure 5.6: Asymmetry between the relaxation and excitation processes as a function ofsample temperature. The asymmetry value is obtained by summing the absolute value of all
points in the plots similar to those shown in Figs. 5.4(c) and 5.5(c) for each temperature.

value is basically independent of the sample temperature. Ideally, the normalized asymmetry
quantity should go from 1 to 0 as the temperature is increased. We observe that this value
only decays to ∼ 0.3 ± 0.1. We attribute this anomaly to experimental noise in the measured
spectra, which is accentuated on taking the difference between spectra. Nevertheless, fromthe graph in Fig. 5.6, we infer that the strong redistribution approximation is valid fortemperatures above 20 K. Thus, spectral diffusion can be quantified using the correlation
function for temperatures higher than 20 K, as will be demonstrated next.
5.3

High temperature regime
We have already discussed the effect of the change in correlation function on 2D spec-

trum in Section 5.1. Now we will see how to use that effect to measure the correlation
function from experimental data.

85
5.3.1

Fitting procedure

In Section 3.2, we discussed extensively the procedure to simulate 2D spectrum fromtime-domain TFWM signal, which is given by Eq. (5.1) including the correlation function C.
Now we will match the simulated and experimental spectra to obtain the best-fit parameters
through a numerical fitting procedure. Before, we describe the fitting procedure, we rewrite
Eq. (5.1), in a more useful form, as
1

2

2

2

SSI ,SD (τ, T, t) = A(T )e−γ01 (τ +t) e− 2 σ01 (τ +t −2C(T )τ t) Θ(τ )Θ(T )Θ(t)(5.3)

where A is the amplitude term, which includes electric fields, dipole moments, and the signal
decay with T . We show the dependence of the amplitude and correlation function on waitingtime T explicitly. The variable parameters in Eq. (5.3) are – A(T ), γ01 , σ01 , and C(T ). The
exciton resonance energy is directly measured from 2D spectrum, and is not varied during thefitting procedure; thus, we do not show that term in Eq. (5.3). In order to obtain meaningfulvalue of A(T ), normalized signal is used in the fitting procedure. This normalized signal isrelated to the one given by Eq. (5.3) as

Snorm (τ, T, t) = "Z ∞Z ∞−∞

SSI ,SD (τ, T, t)

SSI ,SD (τ, T, t)
dt dτ
A(T )
−∞

#
2 1/2

.

(5.4)

Diagonal and cross-diagonal slices from the simulated and measured 2D spectra arematched, by minimizing the sum of square of the difference between simulated and measured slices, to obtain the values of A, γ01 , and σ01 . This procedure was followed for 2D
spectrum with smallest T , which in the present case was 0.2 ps. We set C = 1 for the first
spectrum. γ01 , and σ01 are kept fixed for subsequent 2D spectra; A and C are the only
variable parameters.

86
5.3.2

Analysis results

The plots in Fig. 5.7 show slices from the experimental and simulated spectra. The
simulated spectra used the parameters obtained from the fitting procedure described in
Section 5.3.1. The diagonal and cross-diagonal slices are shown in the left and right columns,
respectively. The top and bottom rows show slices from spectra measured with T = 0.2 and26.6 ps, respectively. An excellent match between the measured and simulated slices isobtained.
Diagonal

Cross−diagonal

1

Absolute value spectrum slices

T = 0.2 ps

0.5

(a)

(b)

0
0.12

T = 26.6 ps

0.06

(c)

(d)

0
1541

1544

1547 1541

1544

1547

Emission energy (meV)

Figure 5.7: (a) Diagonal and (b) cross-diagonal slices from experimental (solid) and simulated (dashed) spectrum for T = 0.2 ps. (c) Diagonal and (d) cross-diagonal slices fromexperimental (solid) and simulated (dashed) spectrum for T = 26.6 ps. The simulated spectrum is obtained using the parameters obtained through the fitting procedure. The sample
temperature was 25 K.

The fitting procedure is repeated for several sample temperatures from 5 – 80 K.
The measured values of the correlation function as a function of the waiting time for a
few temperatures are shown in Fig. 5.8 in a semi-log plot. 2D spectra were measured for87
T = 0.2 − 50.9 ps for each temperature; a small subset of these points are shown in Fig.
5.8 to avoid the noisy points with C < 0.1. As expected, the correlation function decays
with increasing waiting time and the rate of decay increases with sample temperature. Aninteresting finding is that the decay of the correlation function is not exponential with thewaiting time. This deviation from exponential decay is most apparent for 20 K and becomes
progressively less apparent as the sample temperature is increased. We conclude that eventhough strong-redistribution approximation is valid for temperatures higher than 20 K, theGauss-Markov approximation is not valid.
1

Frequency-frequency
correlation funtion

70 K
50 K
30 K
20 K

0.1
0

10

20

30

40

Waiting time T (ps)Figure 5.8: Decrease in the normalized frequency-frequency correlation function with waitingtime for different sample temperatures on a semi-log scale.

Next we present the measured values for the amplitude term A as a function of the
waiting time for various sample temperatures, which are shown in Fig. 5.9. The amplitude
term denotes the total exciton population. There are several interesting features of the evolution of signal amplitude with the waiting time. Firstly, all the curves are non-exponential.
This suggests that our treatment of population decay in Chapter 3 is not valid for excitons.
Even more surprising is the non-monotonic population evolution for 50 and 70 K – there
is a sharp fall for small T followed first by an increase and then a slow decay. This non-88monotonic behavior is reproduced consistently for all the sample temperatures above 50 K,
and suggests an increase in population with time.
70 K
50 K
30 K
20 K

Amplitude (a.u.)

100

10
0

10

20

30

40

50

Waiting time T (ps)Figure 5.9: Variation in total signal strength with the waiting time for various sample
temperatures. Notice the non-monotonic behavior for 50 K and 70 K.

The non-monotonic evolution is not an artifact of our fitting procedure, and is reproduced from rephasing zero-quantum scans, which are discussed in Appendix D. We do notunderstand this unexpected increase in signal strength. Exciton spin-flip [131] and creationof excitons from electron-hole pairs created in the continuum are a couple of probable causes
for this unexpected behavior.
In this chapter we have shown that the usual description of spectral diffusion basedon the correlation function and its exponential decay does not work for excitons in semiconductor QWs. At low temperatures, the strong redistribution approximation is not valid,
which means spectral diffusion cannot be described through the correlation function. Thisapproximation is satisfied for high (> 20 K) temperatures. For these temperatures the correlation function is measured through a numerical fitting procedure, which, to the best of ourknowledge, has been demonstrated for the first time. The decay of this correlation function
is non-exponential, which violates the Gauss-Markov approximation usually used to model

89
spectral diffusion. In addition to the novel properties of spectral diffusion of excitons in QWs,
our results also show interesting, non-monotonic evolution of TFWM signal with the waitingtime for temperatures > 50 K. Further work is needed to explain this phenomenon.

Chapter 6
Excitons as Interacting Bosons

One of the puzzles in early TFWM experiments was the dependence of signal decayrate on the polarization of the excitation pulses. It was observed that the TFWM signal
decays faster for the cross-linear polarization scheme than for the co-circular and co-linear
polarization schemes [77, 78, 79]. The contribution of disorder-mediated coupling [132] andmulti-exciton correlations through bound and unbound two-exciton states to the TFWM
signal [133] have been previously used to explain this phenomenon. However, these models
cannot reproduce the correct TFWM signal phase [13].
Exciton physics in QWs is generally modeled using a two-level system, as has been
done so far in this work. Many-body effects can be added phenomenologically to the OBEs
describing such a system and solved numerically [5]. A more rigorous description of excitons
is possible through the semiconductor Bloch equations [54]. Both methods of including interexcitonic interactions are computationally intensive, which makes it impractical to use thesemethods to quantify exciton-exciton interaction strengths.
In this chapter, we discuss modeling excitons as interacting bosons. First we will
introduce the theoretical framework; we will show that exciton-exciton interactions can beseparated into intra- and inter-mode interactions. Next we will show that the polarization
dependent dephasing rate as well as EIS-modified signal phase [13] are a direct consequence ofthe bosonic nature of excitons. Due to the significant reduction in computational complexity,
we can fit simulation results to experimental data and quantify exciton-exciton interactions.

91
Also, we highlight how the signal in a non-linear optics experiment, such as TFWM, isaffected by the bosonic character of the system being studied. After establishing the validityof this model, we will separate and quantify the intra- and inter-mode interactions throughexciton-density-dependent measurements.
6.1

Bosonic model for excitons

6.1.1

Interactions between same-spin excitons

For the sake of simplicity, we consider excitons with the same spin. We describe excitons
as a harmonic ladder of states. Inter-exciton interactions are introduced through anharmonic
terms in the Hamiltonian [134, 135]. The Hamiltonian for multiple exciton modes can be
written as
Htot = ~NXi=1

ωi ni +NX

e i ni (ni − 1) +
∆

i=1

N
X

e ij ni nj
∆

(6.1)

i,j=1, i6=jwhere i, j denote exciton modes, N is the total number of available modes, ~ωi is the energy
e i and ∆e ij areof an isolated exciton in mode i, ni is occupation number for mode i, and ∆the intra- and inter-mode interaction energies, respectively. The multiple modes could be
due to different in-plane exciton wavevector or the different states due to the inhomogeneity.
Assuming constant intra- and inter-mode interaction energies of 21 ∆ and ∆0 , respectively,
Eq. (6.1) can be simplified to obtain the Hamiltonian for a single exciton mode1
Hi = ~ωi ni + ∆ni (ni − 1) + ∆0 Nex ni2where Nex =

N
P

(6.2)

nj is approximately the total number of excitons. Under the mean-field

j=1; j6=iapproximation, we consider the system to be defined by the Hamiltonian in Eq. (6.2), which
we can rewrite by explicitly removing the mode-index i1
H = (~ω0 + ∆0 Nex )n + ∆n(n − 1)
2
1
= ~ωn + ∆n(n − 1)
2

(6.3)

92
where ~ω0 is the resonance energy of a non-interacting exciton. The density dependenttransition energy of the |0i ↔ |1i transition is~ω = ~ω0 + ∆0 Nex .

(6.4)

The energy scheme shown in Fig. 6.1(a) represents the Hamiltonian given by Eq. (6.3)
showing the ground |0i, single exciton |1i, and two-exciton |2i states. The higher lying
states are not shown because they do not contribute to the TFWM signal in the χ(3) regime,
and are therefore not ordinarily needed if pulse energies are kept sufficiently low during anexperiment. In addition to the energy shift, exciton-exciton interactions result in increased
dephasing rates, which are added phenomenologically. Intra-mode interactions result in adifference of ξ in the dephasing rates between the |0i ↔ |1i and |1i ↔ |2i transitions. Similar
to the relation in Eq. (6.4), inter-mode interactions results in an exciton-density dependentdephasing rate of the |0i ↔ |1i transition
γ = γ0 + ξ 0 Nex(6.5)

where γ0 is the dephasing rate of the |0i ↔ |1i transition for an unexcited sample. The
aforementioned energy shift terms (∆ and ∆0 ) and dephasing rate terms (ξ and ξ 0 ) areequivalent of the many-body terms EIS and EID, respectively [134]. The obvious differenceis that we have now separated the EIS and EID terms into inter- and intra-mode interactions.
Later on in this chapter, we will quantify all of these terms.
The transition dipole moments for the |0i ↔ |1i and |1i ↔ |2i are µ1 and µ2 , respectively. In general, these are related asµ2 =where the factor of√√

2µ1 − µ0(6.6)

2 results from the bosonic nature and µ0 accounts for phase-spacefilling (PSF) effect due to the fermionic nature of electrons and holes. From experimental√
measurements, we find that µ0 ≈ 0. Thus, we ignore the PSF effect and set µ2 = 2µ1 .

93
∆
∆B

∆
µ2

µ2
γ+ξ

γ+ξ

µ2
µ1

γ+ξ

γ+ξB
µ1

µ1

γ

γ

(a)

(b)

Figure 6.1: Energy level scheme for excitons as harmonic oscillators for excitons with (a)the same spin and (b) both ±1 spins. (a) Shows the ground |0i, single exciton |1i, andtwo-exciton |2i states where the excitons are implicitly assumed to have the same spin.
(b) Shows the ground |0i, single exciton |1±i, two-exciton |2±i and biexciton |Bxi states.
Optical transitions are indicated by arrows. The solid and dashed arrows indicate transitions
excited by σ+ and σ− polarized light, respectively, in (b). The transition dipole moments(µ1 and µ2 ) and√dephasing rates (γ, γ + ξ and γ + ξB ) are indicated next to the arrows.
Note that µ2 = 2µ1 . The dashed line indicates the energy of the two-exciton states in theabsence of inter-exciton interactions. The same spin two-exciton states, |2i in (a) and |2±i
in (b), are blue shifted by interaction energy ∆. The bound biexciton state, |Bxi in (b), is
red shifted by binding energy ∆B and states

6.1.2

Interactions between opposite-spin excitons

We have described the same-spin exciton as bosons. Now we extend this model toinclude excitons of opposite spins and biexcitons, which are bound opposite-spin exciton
pairs. The combined Hamiltonian for excitons with both ±1 spins, based on Eq. (6.3), canbe written as
H = ~ω1 X
ni (ni − 1) + ∆B n+ n−
ni + ∆
2 i=±
i=±

X

(6.7)

where the index i is now used to denote exciton-spin and ∆B is the biexciton binding energy.
This Hamiltonian is represented by the energy level scheme shown in Fig. 6.1(b). As in Fig.
6.1(a), we only show the energy levels up to the two-exciton states. The optical transitions
that are possible using σ+ and σ− polarized lights are indicated by solid and dashed arrows,
respectively. The spin of the exciton states is explicitly indicated. In addition to the binding

94
energy, the interactions between opposite-spin excitons results in the change in dephasingrate of the |1±i ↔ |Bxi transitions with respect to that of the |0i ↔ |1±i transitions by ξB .
6.2

Polarization dependence
In the previous section, we presented a model to treat excitons as interacting bosons.

Now we will test the validity of this model by using it to analyze polarization-dependent
2DCS measurements. We have performed experiments for the co-circular, cross-linear andco-linear polarization schemes. All the transition shown in Fig. 6.1 are accessible throughcircularly polarized light. Thus, we need to be able to model the TFWM signal for linearlypolarized light using the circularly-polarized transitions shown in Fig. 6.1. We will discuss
this procedure next.

6.2.1

Linear and circular bases

We begin by listing the Feynman diagrams for all possible polarization schemes in thecircular basis. These Feynman diagrams are labeled I – VII in Fig. 6.2. The first interaction
in all these diagrams are with σ+ polarized light. The polarization of each pulse and theradiated signal is indicated next to the arrows in each Feynman diagram. Feynman diagrams
I – IV involve only the |0i↔|1±i transitions. Feynman diagram V includes contributions
from |0i↔|1+i and |1+i↔|2+i transitions. Another identical set of Feynman diagrams exist
where the σ+ polarized pulses are replaced with σ− polarized pulses and vice-versa, whichhave not been shown. Since these diagrams essentially include interactions identical to thosein the diagrams shown in Fig. 6.2, we will use the labels I0 – VII0 for these diagrams. We will
discuss the origin of TFWM signal for the co-circular, cross-linear and co-linear polarization
schemes using these fourteen Feynman diagrams.
To calculate the FWM signal for linearly polarized excitation pulses, we use the rela-

95
σ+

0

0

1+

0

0

0

σ+

0

σ+

σ+
σ+

1+

0

σ+

0

0

1+

0

1+

1+

0

σ−
σ+
σ−

1+

0

(I)
σ+

0

0

σ+

(II)
1+

1+

2+

1+

σ+

1+

1+

σ+

0

1+

0

0

0

0

1−

0

0

0

0

1+

0

0

σ−

σ+
σ−
σ+

0

0

1−

0

1−

1+

0

1+

0

0

(III)
σ−

1+

1+

Bx

1+

σ−

1+

1+

σ+

0

1+

0

0

σ+

(V)

σ+

σ+

(IV)
σ−

1+

1+

Bx

1+

σ+

1−

1+

σ−

0

1+

0

0

σ+

(VI)

σ+

(VII)

Figure 6.2: Feynman diagrams (I – VII) denoting the dipole allowed quantum pathways that
contribute to TFWM signal for the energy level scheme shown in Fig. 6.1(b). All possible
excitation pulse polarizations in the circular basis have been considered with σ+ polarizationof the first pulse. Feynman diagrams (I – IV) involve the transitions between the ground andone-exciton levels only, while Feynman diagrams (V – VII) also involve transitions between
one-exciton and two-exciton levels. Another identical set of diagrams can be written down
by swapping the σ+ and σ− polarized pulses, which are not shown. These diagrams will bereferred to by numbers I0 – VII0 indicating the corresponding diagram that has been shown.

tionship between linear and circular polarizations:1σ+ = √ (X + iY )21
σ− = √ (X − iY )
2

(6.8)

where σ± denote orthogonal circular polarizations and X and Y denote orthogonal linear
polarizations. The relations in Eq. (6.8) can be re-written as
1
X = √ (σ+ + σ− )2iY = − √ (σ+ − σ− ) .
2

(6.9)

We can write the contribution of each Feynman diagram to the TFWM signal foreach polarization scheme using the relations in Eqs. (6.8) and (6.9). First, we consider
only the strength, phase and the polarization of the TFWM signal, which are collectively

96
Feynman diagram

Co-circularIµ1 4 σ+I00IIµ1 4 σ+

II0

0

III

0

III0

0

IV

0

IV0

0

V

−(µ1 µ2 )2 σ+

V0

0

VI

0

VI0

0

VII

0

VII0

0

Cross-linear!
1√ µ1 4 σ +−2 2!
1√ µ1 4 σ −−2 2!
1√ µ1 4 σ +−2 2!
1√ µ1 4 σ −−2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ+
2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ+
2 !21√ (µ1 µ2 )2 σ+
2 2!
1√ (µ1 µ2 )2 σ−
2 2 !
1√ µ1 4 σ −−2 2!
1√ µ1 4 σ +−2 2!
1√ µ1 4 σ −−2 2!
1√ µ1 4 σ +−2 2Co-linear!
1√ µ1 4 σ+
2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ+
2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ+
2 2!
1√ µ1 4 σ−
2 2!
1√ µ1 4 σ+
2 2!
1√ (µ1 µ2 )2 σ+−
2 2!
1√ (µ1 µ2 )2 σ−
−
2 2 !
1
√ µ1 4 σ−−
2 2!
1√ µ1 4 σ+−
2 2!
1√ µ1 4 σ−−
2 2!
1√ µ1 4 σ+−
2 2

Table 6.1: Contributions to the transient four-wave mixing signal from the Feynman diagrams in Fig. 6.2 for the co-circular, cross-linear and co-linear polarization schemes.

shown for each Feynman diagram for different polarization schemes. We have consideredσ+ polarization for the co-circular polarization scheme. For the cross-linear and co-linear

97
polarization schemes, we have assumed XYYX and XXXX polarizations, respectively.
In addition to the contributions of each Feynman diagram listed in Table 6.1, we
need to include the excitation and emission frequencies for each diagram. We note that all
the Feynman diagrams have the same excitation energy ~ω corresponding to the transitionenergy of the |0i ↔ |1±i transitions. Also, all the diagrams decay with the same dephasingrate γ during delay τ . However, based on the emission energies and dephasing rates, theFeynman diagrams can be separated into the following groups:(1) The diagrams I – IV and I0 – IV0 with the emission energy equal to the excitationenergy ~ω. These diagrams decay with dephasing rate γ during delay t.
(2) The diagrams V and V0 , which have the emission energy of ~ω + ∆ corresponding
to the |1±i ↔ |2±i transitions. These diagrams decay with dephasing rate γ + ξduring delay t.
(3) The remaining diagrams with the emission energy of ~ω − ∆B corresponding to the|1±i ↔ |Bxi transitions. These diagrams decay with dephasing rate γ + ξB during
delay t.
Now we have all the information needed to write down the TFWM signal for each√polarization scheme. We use the relation µ2 = 2µ1 and those in Eq. (6.9) to obtain thefollowing TFWM signals:

1 2
2SCoCir (τ, t) =ACoCir e−γ(τ +t) e− 2 σ (τ −t) 1 − e(i∆−ξ)t(6.10)


1 2
1
2
2
SXLin (τ, t) = e−γ(τ +t) A1XLin e− 2 σ (τ −t) e(i∆−ξ)t −A2XLin e− 2 (στ −σB t) e(−i∆B −ξB )t (6.11)hi−γ(τ +t)− 12 σ 2 (τ −t)2(i∆−ξ)t− 12 (στ −σB t)2 (−i∆B −ξB )t
SCoLin (τ, t) = e
A1CoLin e
2−e− A2CoLin ee(6.12)
where the subscripts CoCir, XLin and CoLin are used for the co-circular, cross-linear andco-linear polarization schemes, respectively. These signals are written as a function of delays

98t and τ since we will only consider one-quantum rephasing spectra. Signal amplitudes aredenoted by A; different amplitudes are used for the different polarization schemes. Separate
exciton and biexciton signal amplitudes are used. The parameters σ and σB describe theinhomogeneous broadening for the |0i ↔ |1±i and |1±i ↔ |Bxi transitions, respectively.
Equal inhomogeneities for the |0i ↔ |1±i and |1±i ↔ |2±i transitions are assumed. All theother parameters are shown in Fig. 6.1 and explained earlier. We do not show the exciton
resonance energy explicitly.
We have laid down the theoretical background to understand the dependence of the
TFWM signal on the polarization scheme. The physical consequence of the above equations
will be discussed in detail with the experimental results, which will be discussed next. We
have measured rephasing one-quantum 2D spectra for the (001)-oriented GaAs QWs described in Section 2.9.1. The experiments were performed with ∼ 200 fs pulses generated
by a mode-locked Ti:sapphire oscillator. The delay between pulses B and C was kept constant at 300 fs to ensure correct time-ordering. On average an exciton density of ∼ 1010
cm−2 per pulse per QW was excited. The sample was kept at a temperature of 10 K in asample-in-vapor flow cryostat.

6.2.2

Co-circular polarization scheme

We will discuss the experimental results for the co-circular polarization scheme first.
The |Bxi state does not contribute to the signal for this polarization scheme [117]. A
measured absolute value 2D spectrum for co-circular polarization scheme is shown in Fig.
6.3(a), which comprises a single peak labeled P1. Figure 6.3(c) shows the real part of thespectrum shown in Fig. 6.3(a); the signal phase is obtained through the SRPP technique
discussed in Section 3.3.2. The peak in Fig. 6.3(c) has a dispersive lineshape, which has
been previously attributed to EIS through numerical simulations [13].
We simulate the 2D spectra from the TFWM signal in Eq. (6.10). It should be notedthat for ∆, ξ = 0 both the terms in the parenthesis cancel each other exactly and there is no−1541

Experiment
0.5

0.2

0.5

0.8

−0.2

0.2

0.6

P1

(a)

−1541

−1547

0.8

(b)

−0.2

0.2

0.6

−1547

−1544

Excitation energy (meV)

−1544

0.2

99

Simulation

(d)

(c)
1541

1544

1547

1541

1544

1547

Emission energy (meV)

Figure 6.3: Measured (a) absolute value and (c) real part 2D spectra for the co-circularpolarization scheme. The corresponding simulated spectra using best-fit parameter values
are shown in (b), and (d), respectively. The peak is labeled P1 in (a). The excitation
spectrum is shown as the solid line in (c).

TFWM signal, as expected from a perfectly bosonic system [136]. If the above equality is notsatisfied, however, this cancellation is not perfect and a TFWM signal results. Specifically,for ∆ > 0, the real part of the signal comprises a positive and a negative peak shifted along
the emission energy axis. The interference between these peaks results in a peak shapesimilar to the dispersive peak in Fig. 6.3(c) for small ∆ (< γ). Such a lineshape is not
obtained for a two-level system, which yields a single positive peak, as shown in Section
3.2.1.
We perform a fitting procedure similar to the one outlined in Section 5.3.1. The

Absolute value spectrum
slicesDiagonal
1

Cross−diagonal(a)Horizontal
(b)

100

Vertical
(c)

(d)

0.5

0
1541

1544

1547 1541

1544

1547 1541

1544

1547 −1547

Emission Energy (meV)

Cross−diagonal−1541

Excitation
Energy (meV)Horizontal(e)Real part spectrum
slices−1544

Vertical
(f)

(g)

0.5

0

−0.5
1541

1544

1547 1541

1544

Emission Energy (meV)

1547 −1547

−1544

−1541

Excitation
Energy (meV)

Figure 6.4: Experimental and simulation results for the slices used in the fitting procedure forthe co-circular polarization scheme are shown in solid black and dashed red lines, respectively.
The slices are taken through the maximum of the peak P1 shown in Fig. 6.3(a). The top
and bottom rows shows the slices taken from the absolute value (Fig. 6.3(a)) and real part(Fig. 6.3(c)) spectra, respectively. The particular slice is indicated by the label above eachplot.

simulated absolute value and real part spectra, using the best-fit parameter values, areshown in Figs. 6.3(b) and 6.3(d), respectively. We obtain an excellent match between themeasured and simulated spectra for the following best-fit parameters: σ = 383 ± 2 µeV,γ = 191 ± 3 µeV, ∆ = 13 ± 10 µeV, and ξ = 6 ± 6 µeV. The good agreement between thesimulated and measured spectra is also apparent in the slices used in the fitting procedure,
which are shown in Fig. 6.4. The experiment and fitting procedure was repeated five times
and we report the statistical standard deviation in the parameters values as the error. As

101discussed earlier, the dispersive lineshape in Fig. 6.3(c) is a consequence of ∆ > 0. The
measured dephasing rate γ is significantly different than that obtained by fitting the diagonal
and cross-diagonal slices of absolute value spectrum to lineshapes obtained by considering
exciton as a two-level system (102 ± 1 µeV) [104].

6.2.3

Cross-linear polarization scheme

We will now discuss results from 2DCS experiment performed with the cross-linear
polarization scheme. Figure 6.5(a) shows the measured absolute value 2D spectrum for thispolarization scheme. We see two distinct peaks in the spectrum – P2 and P3. The peak P3
is red-shifted, along the emission energy axis, with respect to peak P2 due to the biexciton
binding energy. The relevant TFWM signal is given by Eq. (6.11); the first and second
terms of the equation result in the peaks P2 and P3, respectively. The peak P3 is tilted
with respect to the diagonal line, as indicated by the solid black line through it. This tilt
is due to energy dependent biexciton binding energy and can be accounted for by different
inhomogeneous widths for the |0i ↔ |1±i and |1±i ↔ |Bxi transitions [118].

−1541

0.5

−1544

0.2

Simulation
0.8

(a)
1541

0.2

0.5

0.8

P2

P3
−1547

Excitation energy (meV)

Experiment

(b)
1544

1547

1541

1544

1547

Emission energy (meV)

Figure 6.5: (a) Measured absolute value 2D spectrum for the cross-linear polarization scheme.
(b) The corresponding simulated spectra using best-fit parameter values.The different peaks
are labeled P2 and P3.

102
Although Eq. (6.11) results in two peaks, it does not indicate the correct relative phasebetween the two peaks. Thus, we add an ad-hoc phase term eiφ to Eq. (6.11) and obtain


1 2
1
2
2
SXLin (τ, t) = e−γ(τ +t) A1XLin e− 2 σ (τ −t) e(i∆−ξ)t −A2XLin eiφ e− 2 (στ −σB t) e(−i∆B −ξB )t .
(6.13)
We match the slices from the 2D spectra simulated using Eq. (6.13) to the slices from theexperimentally measured spectra. The variable parameters for this fitting procedure are –A1XLin , A2XLin , φ, ∆B , and ξB . The other parameters are fixed to those obtained for theco-circular polarization scheme. Different inhomogeneous widths during delays τ and t resultin a peak that is tilted with respect to the diagonal line. Thus, the relative tilt between thetwo peaks gives the ratio between σ/σB , which is used to fix σB = 467 ± 5 µeV. The best-fit
parameter values are – ∆B = 2.099 ± 0.016 meV, and ξB = 12 ± 8 µeV. The simulated
spectrum from the parameter values obtained through the fitting procedure is shown in Fig.
6.5(b). The excellent match between theory and experiment is also highlighted by the slices
shown in Fig. 6.6.
The horizontal slices in Figs. 6.6(c) and 6.6(g) indicate a weak peak between the two
prominent peaks. Our model does not include that peak, and results in the incorrect phasebetween peaks P2 and P3, which is compensated by the ad-hoc phase term to some extent.

6.2.4

Co-linear polarization scheme

Next we will discuss at the experimental results obtained for the co-linear polarization
scheme. The absolute value and real part 2D spectra for this scheme are shown in Figs.
6.7(a) and 6.7(c). The spectrum exhibits two peaks, which are labeled P4 and P5 in Fig.
6.7(a). Although the positions of these peaks are similar to those of P2 and P3 in Fig.
6.5(a), the relative strengths of the two peaks are very different. Also, the real part of peakP4 in Fig. 6.7(c) has a distinctive dispersive character similar to that in Fig. 6.3(c) for theco-circular polarization scheme.

Diagonal

Cross-diagonal

Horizontal

103

Vertical

1(b)

Slices throughpeak P2

(a)

(c)

(d)

0.5

0
1541

1544

1547 1541

1544

1547 1541

1544

1547 −1547

Emission Energy (meV)Diagonal

Slices throughpeak P3

1

Cross-diagonal
(e)

−1544

−1541

Excitation
Energy (meV)Horizontal

Vertical
(g)

(f)

(h)

0.5

0
1541

1544

1547 1541

1544

1547 1541

Emission Energy (meV)

1544

1547 −1547

−1544

−1541

Excitation
Energy (meV)

Figure 6.6: Experimental and simulation results for the slices used in the fitting procedure
for the cross-linear polarization scheme are shown in solid black and dashed red lines, respectively. The slices are taken through the maximum of the peaks P2 and P3 in Fig. 6.5(a).
The top and bottom rows shows the slices taken for peaks P2 and P3, respectively. The
particular slice is indicated by the label above each plot.

The first and second terms, in the square brackets, in Eq. (6.12) give rise to peaks
P4 and P5, similar to the situation for the cross-linear polarization scheme. The first ofthese terms looks similar to the TFWM signal in Eq. (6.10) for the co-circular polarization
scheme. A key difference, however, is a ratio of two between the two parts of this term. As aresult, the real part of the 2D spectrum obtained from Eq. (6.12) does not give a dispersive
peak. This anomaly probably arises due to our simplistic treatment of density dependent
resonance energies and dephasing rates given by Eqs. (6.4) and (6.5). Specifically, thedensity-dependent terms give rise to additional signal terms, which need to be calculated

104
numerically [5]. The model we have presented clearly does not capture this feature andresults in the aforementioned anomaly. Since we find that the peak shape for the co-linear
and co-circular polarizations are the same, we modify the first term in Eq. (6.12) to be
similar to that in Eq. (6.10). The modified TFWM signal for the co-linear polarization is
h
i

1 2
1
2
2
SCoLin (τ, t) = e−γ(τ +t) A1CoLin e− 2 σ (τ −t) 1 − e(i∆−ξ)t − A2CoLin e− 2 (στ −σB t) e(−i∆B −ξB )t .
(6.14)−1541

Experiment
0.5

0.8

0.2

0.5

0.8

−0.2

0.2

0.6

P4

(a)

−1541

−1547

P5
(b)

−0.2

0.2

0.6

−1547

−1544

Excitation energy (meV)

−1544

0.2

Simulation

(c)
1541

(d)
1544

1547

1541

1544

1547

Emission energy (meV)

Figure 6.7: Measured (a) absolute value and (c) real part 2D spectra for the co-linear
polarization scheme. The corresponding simulated spectra using best-fit parameter values
are shown in (b), and (d), respectively. The peaks are labeled P4 and P5 in (a).

We perform the fitting procedure using slices from 2D spectrum simulated from Eq.
(6.14). The only free parameters are the amplitude terms A1CoLin and A2CoLin since all the105other parameters were obtained from the co-circular and cross-linear spectra. Once again we
obtain a very good match between theory and experiment, which is highlighted in the slices
shown in Fig. 6.8. Only slices through peak P4 are shown because peak P5 is much weaker.

Absolute value spectrum
slicesDiagonal
1

Cross−diagonal(a)Horizontal(b)Vertical
(c)

(d)

0.5

0
1541

1544

1547 1541

1544

1547 1541

1544

1547 −1547

Emission Energy (meV)

Cross−diagonal−1541

Excitation
Energy (meV)Horizontal(e)Real part spectrum
slices−1544

Vertical
(f)

(g)

0.5

0

−0.5
1541

1544

1547 1541

1544

Emission Energy (meV)

1547 −1547

−1544

−1541

Excitation
Energy (meV)

Figure 6.8: Experimental and simulation results for the slices used in the fitting procedure forthe co-linear polarization scheme are shown in solid black and dashed red lines, respectively.
The slices are taken through the maximum of the peak P4 shown in Fig. 6.7(a). The top
and bottom rows shows the slices taken from the absolute value (Fig. 6.7(a)) and real part(Fig. 6.7(c)) spectra, respectively. The particular slice is indicated by the label above eachplot.

The polarization dependent measurements that we have presented so far, highlight theeffectiveness of the bosonic model. Despite its certain limitations, we can reproduce theexperimentally obtained spectrum using a self consistent set of parameters, which were obtained though a fitting procedure. Next we will discuss the polarization-dependent dephasing

106
rate of excitons in these QWs measured from TFWM experiments [77, 78, 79].

6.2.5

Polarization dependent dephasing rate

The polarization dependent dephasing rate can be clearly seen in the cross-diagonal
absolute value slices through peaks P1, P2 and P4 as shown in Fig. 6.9 as the markers.
We find that while the peak-widths along the cross-diagonal direction for co-circular andco-linear polarization schemes are identical, they are significantly smaller than that for the
cross-linear polarization scheme. This observation is similar to those previously reported[77, 78, 79]. Figure 6.9 also shows the cross-diagonal slices from the simulated spectra
as lines. We can clearly reproduce the lineshapes measured for the different polarization
schemes although the relevant linewidths are obtained only from 2D spectra measured for
co-circular polarization scheme.
1

Co−cirCross-diagonal slice normalized amplitude (a.u.)

X−lin
0.8

Co−lin

0.6

0.4

0.2

0
−1

−0.5

0

0.5

1

Energy offset from peak (meV)

Figure 6.9: The cross-diagonal slices in measured and simulated spectra for the co-circular(Co-cir), cross-linear (X-lin) and co-linear (Co-lin) polarization schemes are shown by markers and lines, respectively. Note the broader cross-diagonal slice for cross-linear polarizationcompared to those for co-circular and co-linear polarizations. The asymmetric lineshape forcross-linear polarization is due to the wing of the biexciton peak.

107Time−integrated FWM (a.u)

100

1st term
2nd term
Total
10−1

10−2

10−3
0

2

4

6

8

10

τ (ps)

Figure 6.10: Simulated time-integrated transient four wave mixing signal for the first andsecond terms and the total signal in Eq. (6.10). The signals have been rescaled to havecomparable values.

The difference in the lineshapes observed for different polarization schemes can be understood through the interference, or lack thereof, of signals from different quantum pathways
that contribute to the peak appearing on the diagonal line. The spectral proximity of signal
due to the different quantum pathways for co-circular polarization – including or excluding
the |1±i↔|2±i transitions – leads to nearly complete destructive interference at the wingsof P1, along the horizontal direction, in Fig. 6.3(a). This interference results in a total nonlinear signal with significantly narrower width compared to the individual quantum pathway
contributions. To highlight this point, we plot the time-integrated TFWM signal intensities
for each quantum pathway (individual terms in Eq. (6.10)) as well as the total signal in Fig.
6.10 1 . It is apparent that independently, the TFWM signal intensity from each quantum
pathway is similar and decays at a fast rate (given by γ) compared to the total signal. The
slower total signal decay rate results in a narrower peak in the frequency domain. However,for the cross-linear polarization scheme, quantum pathways involving only the |Gi↔|1±itransitions do not contribute to the signal at peak P2; the aforementioned interference goes1

We use the example of the time-integrated FWM signal to compare with the previous experimental
results. However, the following discussion is valid for the decay of the signal during both delays – τ and t.

108
away, resulting in a much broader peak. Thus, the decay rate of total signal can be significantly different than decay rates of the individual transitions that constitute the signal.
This effect is a manifestation of the bosonic character of excitons and has not been previous
highlighted although scattering states of unbound two-exciton states have been considered[137, 138, 133, 105, 106] to obtain energy-level schemes similar to the one shown in Fig.
6.1(b). The scattering state was either considered to have the same polarization selection
rule as the biexciton state and was ignored for co-circular excitation [137, 138, 133] or the
relation µ1 = µ2 was assumed [105, 106], which does not give the dispersive peak observedin Fig. 6.3(c).
6.3

Excitation-density dependence
In Section 6.2 we illustrated the validity of the bosonic model of excitons throughpolarization dependent experiments. We were also able to quantify the exciton-exciton
interaction terms. While introducing this model in Section 6.1.1, we distinguished between
the inter- and intra-mode interactions. Now we will quantify these interactions throughexciton-density-dependent measurements.

6.3.1

Inter-mode interactions

The exciton-exciton interactions between different modes results in an exciton-density
dependent resonance energy of the |0i ↔ |1±i transitions. The exciton-density dependenceof these parameters is linear and given by Eqs. (6.4) and (6.5). Measurements of ~ω and γ
as a function of the exciton density will be used to obtain the inter-mode interaction terms∆0 and ξ 0 .
The transition energy of the |0i ↔ |1±i can be directly measured from the peak
position in the absolute value spectrum for the co-circular polarization. The measured values
as a function of the exciton density is shown in Fig. 6.11(a) as the markers. The statisticalerror bars are obtained by repeating the measurement for each power thrice. The figure also

Homogeneous linewidthγ (meV)

Resonance energyħω (meV)1545.0MeasuredFit

1544.91544.81544.71544.6

Measured
Fit

0.30
0.25
0.20
0.15
0.10

0

2

4

6

8

Exciton density per QW (1010 cm−2)

(a)

109

0.35

0

(b)

2

4

6

8

Exciton density per QW (1010 cm−2)

Figure 6.11: The excitation density dependence of (a) the resonance energy and (b) thedephasing rate of the |Gi ↔ |1±i transitions. The statistical error bars are obtained byrepeating measurements thrice. The linear fit to the data is indicated by the solid red line.

shows the linear-fit to the data as the red line. We obtain ~ω0 = 1544.687 ± 0.005 meV and∆0 ≈ 0.53 µeV/exciton for a spot diameter of ∼ 50 microns.
We repeat the fitting procedure for different exciton densities to obtain the measured
γ as shown in Fig. 6.11(b). Once again we observe that γ increases linearly with the exciton
density, as expected from Eq. (6.5). The linear fit to the data is indicated by the red line in
Fig. 6.11(b). We measure γ0 = 0.103 ± 0.003 meV and ξ 0 ≈ 0.55 µeV/exciton. We find that∆0 and ξ 0 are approximately equal. Approximate values of ∆0 and ξ 0 are quoted because theexciton density indicated in the plots in Fig. 6.11 are approximate.

6.3.2

Intra-mode interactions

It follows from the discussion in Section 6.1.1 that the intra-mode interaction terms
should be independent of the exciton density. The measured values of the intra-mode interaction terms ∆ and ξ are plotted in Figs. 6.12(a) and 6.12(b), respectively. As expected, thevalues of these parameters, within the error bars, do not change with the exciton density.
We obtain ∆ = 19 ± 3 µeV and ξ = 6 ± 1 µeV by taking a weighted mean over the values
measured over the entire exciton-density range studied. We would like to emphasize that
although the values of ∆ and ξ are independent of the excitation density, the resonanceIntra-mode dephasing
term ξ (meV)Intra-mode energy
shift ∆ (meV)

0.06
0.04
0.02
0.00
0

(a)

110

0.03

0.08

2

4

6

0.01
0.00
0

8

Exciton density per QW (1010 cm−2)

0.02

(b)

2

4

6

8

Exciton density per QW (1010 cm−2)

Figure 6.12: The intra-mode exciton-exciton (a) interaction energy and (b) dephasing term
as a function of the exciton density. Both parameters are independent of the exciton density.
The statistical error bars are obtained by repeating measurements thrice.

energy and dephasing rate during delay t is different than that during delay τ . As a result,these parameters indicate EIS and EID, respectively.

6.3.3

Biexcitonic interactions

We also performed the exciton-density-dependence experiment for the cross-linear polarization scheme. We repeated the fitting procedure mentioned in Section 6.2.3 for each
exciton density. We plot the biexciton interaction energies ∆B and ξB in Figs. 6.13(a) and6.13(b), respectively.
We find that the biexciton binding energy increases with the exciton density. This
trend is similar to the trend measured, for significantly higher exciton densities, by AlmandHunter et al. [65]. Interestingly the rate of increase in the biexciton binding energy isapproximately the same as the rate of increase in the transition energy of the |0i ↔ |1±itransitions indicated in Fig. 6.11(a). The upshot is that the energy of the |Bxi state isindependent of the exciton density in the range of exciton densities probed.
The difference between the dephasing rates of the |1±i ↔ |Bxi and |0i ↔ |1±i transitions decreases, and changes sign, with an increase in the exciton density, as shown in Fig.
6.13(b). This behavior is not expected from the model that we have presented. Coinci-

Biexciton dephasing
interaction term ξB (meV)

Biexciton binding
energy ∆B (meV)

2.25
2.20
2.15
2.10
2.05
0

(a)

2

4

6

8

Exciton density per QW (1010 cm−2)

111

0.10
0.05
0.00
−0.05
−0.10
0

(b)

2

4

6

8

Exciton density per QW (1010 cm−2)

Figure 6.13: The biexciton (a) binding energy and (b) dephasing term as a function of theexciton density. Both the values are nominally independent of the exciton density. The
statistical error bars are obtained by repeating measurements thrice.

dentally, this decrease in dephasing rate almost cancels the increase in the dephasing rateof the |0i ↔ |1±i transition shown in Fig. 6.11(b), which results in the dephasing rate of
the |1±i ↔ |Bxi transition that is independent of the exciton density. In addition to theinteraction between opposite-spin excitons, other higher-order correlations also contributeto the formation of biexctions [139]. This simplification could be a possible reason for theunexpected exciton density dependence of the interactions between opposite-spin excitons.
In this chapter, we have used 2DCS experiments to highlight the bosonic nature of QW
excitons and their effect on the radiated non-linear signal. The previous measurements of polarization dependent exciton dephasing rate is a natural consequence of this bosonic nature.
Exciton-exciton interactions are included in a physically intuitive and straightforward way
using the model of interacting bosons. Also, we have shown that the observed decay rate ofmeasured non-linear signal can be different from the dephasing rate of individual transitions
that contribute to the signal for a boson-like system. In addition to providing important
insight into exciton physics in QWs, these results also highlight the effects of a bosonictransition in non-linear optics experiments. Finally, we were able to separate and quantify inter- and intra-mode interactions between excitons through exciton-density-dependent
experiments.

Chapter 7
Conclusion

We have explored various aspects of exciton physics in semiconductor QWs through
2DCS. The effect of structural symmetry on QW excitons was studied in (110)-oriented GaAs
QWs. Properties of excitons such as spectral diffusion and exciton-exciton interactions,which have been previously studied extensively, were presented in a new light.
The polarization dependent dephasing rate of excitons in (110)-oriented QWs wasfound to be due to the anisotropic absorption coefficient in these structures. We found thatthe exciton dephasing is dominated by non-radiative processes. The thermal broadening
revealed an activation-like behavior.
We demonstrated that the strong-redistribution approximation is not valid for the
low sample temperatures although it is generally used to analyze spectral diffusion. Other
methods such as 3PEPS are not sensitive enough to reveal this behavior. We have attributed
this behavior to the lack of exciton population at low temperatures.
An important aspect of this work is the development of a numerical fitting procedure
to analyze the measured 2D spectra quantitatively. The normalized frequency-frequency
correlation function was quantified, for high sample temperatures, using this procedure.
Although this correlation function is generally assumed to follow an exponential decay, ourresults show that this assumption is not valid.
A simple and physically-intuitive model of treating excitons as interacting bosons is
introduced. We separated the exciton-exciton interactions into inter- and intra-mode com-

113
ponents. The numerical fitting procedure was used to quantify these interactions. The
significantly simpler modeling of exciton-exciton interactions, compared to the previously
used models, was critical for quantifying these interactions. We were able to explain thepolarization-dependent exciton dephasing rates in these samples using the bosonic model.
7.1

OutlookIn this work we have revealed several novel properties of excitons in semiconductor QWs

using 2DCS. A detailed investigation into some of these effects is currently being pursued.
Some of the specific ongoing efforts are:
• We have proved that the strong-redistribution approximation is not valid in thecase of excitons. We are collaborating with Marten Richter (Technische Universität
Berlin) to develop a theoretical model of spectral diffusion. A series of exciton states
are considered based on the spatial fluctuations in the QW width. Initial results
reproduce the experimental observations qualitatively. Further work is being done
to obtain a quantitative agreement between the experiment and theory.
• In Section 6.2.3, we noted that our modeling of the biexciton state is probably too
simplistic. Specifically, the horizontal slices in Figs. 6.6(c) and 6.6(g) indicate an additional, albeit weak, peak. This peak suggests the existence of a bound vibrational
state of the biexciton. Further modeling is required to confirm this hypothesis.
• We have also developed a numerical simulation method to solve the modified OBEs,
which is presented in Appendix B. Our aim is to compare the effects due to EIS andlocal-field corrections. We are working on the correctly incorporate the local-fieldcorrections in order to compare these effects.
• Preliminary results indicating the contribution of nominally disallowed exciton transition in (110)-oriented GaAs QWs to the 2D spectrum are presented in Appendix

114
D. Further experiments are needed to study the properties of this state along with
its coupling with the other states. This exciton state might play a crucial for theobserved spectral dependence of photocurrent direction in these samples [45].
• Preliminary experiments have also revealed the non-monotonic exciton-population
evolution for sample temperatures > 40 K. This result is pretty interesting, andprobably results from thermal activation of bright excitons to some high-energy
dark exciton states. More experimental and theoretical studies are needed to probe
this behavior.
Furthermore, the numerical fitting technique developed in this work can be applied toother systems. This procedure would be especially useful to quantify spectral diffusion using
2DCS since several observables are used to approximate the frequency-frequency correlation
function [39]. Also, the bosonic model of excitons, and the modeling procedure can beextended to other systems such as exciton-polaritons [140] and plasmons [141]. The simplified
simulation and fitting procedure is, in principle, very general and can be applied to any
system where the system response can be modeled through OBEs.

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Appendix A
Optical Bloch equations

In the discussion of FWM theory in Section 3.1.2, we had directly written down the
OBEs for a two-level system in the rotating wave approximation. Here, we will derive theOBEs for a two-level system starting from the light-matter interaction in the dipole approximation. A similar procedure can be followed to obtain the OBEs for a more complicatedsystem. The derivation presented here is based on the discussion by Boyd [142]. We will
derive these equations for the two-level system shown in Fig. 3.2(a).
The total Hamiltonian of the system can be written as
H = H0 + Hint

(A.1)

where H0 is the unperturbed Hamiltonian and Hint is the light-matter interaction. For a
two-level system, these Hamiltonians can be represented by 2x2 matrices:




0 
0
−µ01 E 
 0
H0 = 
Hint = 
0 ~ω01−µ01 E0(A.2)

where Hint is written in the dipole approximation for light-matter interaction and E is theelectric field of light given by1
E = |E| ei(k·r−ωt) + e−i(k·r−ωt)
2

(A.3)

where |E|, k, and ω are the amplitude, wavevector, and frequency of the optical electricfield, respectively, and r is a position vector.

128
The time-evolution of the density matrix is given byiρ̇ = − [H, ρ]
~

(A.4)

where ρ is the density matrix for the two-level system and the square brackets indicate thecommutator operation. Equation (A.4) can be written explicitly in the matrix form




 


µ01 E   ρ00 ρ01   ρ00 ρ01   0µ01 E i  0 ρ̇00 ρ̇01  = − 
−

 .
~
µ01 E ~ω01ρ̇10 ρ̇11
ρ10 ρ11ρ10 ρ11µ01 E ~ω01
(A.5)
Equation (A.5) yields the following set of equations:iρ̇00 = − µ01 E (ρ10 − ρ01 )~iρ̇01 = − [µ01 E (ρ11 − ρ00 ) − ~ω01 ρ01 ]~iρ̇10 = − [µ01 E (ρ00 − ρ11 ) + ~ω01 ρ10 ]~iρ̇11 = − µ01 E (ρ01 − ρ10 )
~

(A.6)

It is apparent from Eqs. (A.6) that ρ̇01 and ρ̇10 are complex conjugates of each other. Also,ρ̇00 = −ρ̇11 , which follows from ρ00 + ρ11 = 1. Thus, we can completely describe the timeevolution of the system through the time-evolution of ρ01 and ρ11 . It should be noted that
we have not included any decay terms in Eqs. (A.6). The relevant equations with the decay
terms can be written asiρ̇01 = (−γ01 + iω01 )ρ01 − µ01 E (2ρ11 − 1)~iρ̇11 = −Γ1 ρ11 − µ01 E (ρ01 − ρ10 ).
~Rotating-wave approximation:(A.7)

Equations (A.7) are sufficient to describe thetime-evolution of density matrix elements. However, it is often convenient to go to therotating-wave approximation, in which the fast oscillating coherence term is described by
the product of a slowly varying term with an oscillatory term.
ρ01 = ρe01 eiω01 t

ρ10 = ρe10 e−iω10 t

(A.8)

129
where ρe01 and ρe10 are the slowly-varying terms. Also, we can redefine the optical electricfield E in a more useful form as
1
E = (Ee−iωt + E ∗ eiωt )
2

(A.9)

where E = |E|eik·r .
By substituting Eqs. (A.8) and (A.9) in Eqs. (A.7) and ignoring terms that oscillatewith frequency 2ω, we obtainiρe˙ 01 = (−γ01 + i∆01 )e
ρ01 − µ01 E∗ (2ρ11 − 1)2~iρ̇11 = −Γ1 ρ11 − µ01 (E ρe01 − E ∗ ρe10 ).
~Equations (A.10) are the same as Eq. (3.6) in Section 3.1.2.

(A.10)

Appendix B
Numerical simulations

Many-body effects, such as EIS, EID, and local field effects can be added phenomenologically to the OBEs. The perturbative solution does not work for these modified OBEs,
which need to be solved numerically [143]. The signature of the many-body effects, such as
EIS and EID, on 2D spectrum have been observed previously, although without considering
inhomogeneous broadening [13]. However, the impact of local field effects on 2DCS experiments have not been studied. In order to explore this phenomenon, we have developed anumerical method to solve the OBEs, which will be discussed next.
We will only consider a two-level system, although the procedure is general and can beextended to more complicated systems easily. We start by ignoring the many-body effects, tokeep things simple. The OBEs for such a system was derived in Appendix A, and is given by
Eq. (A.7). For a given time-dependent electric field, these coupled differential equations canbe easily solved on a computer. The time-dependent electric field can be modified dependingon the time delays. For a rephasing one-quantum spectrum, we change the delay τ and solve
the OBEs for each value of τ . This procedure evaluates the total coherence, up to all orders,
including the linear response and all possible TFWM signals. In order to isolate a particular
TFWM signal we need to implement a signal-selection procedure that is analogous to theexperimental phase-matching condition.

131
B.1

Phase-matching
The basic idea for phase-matching in the simulation is basically the same as that ina non-collinear experiment. We tag the individual pulses by distinct “wavevectors”. The
OBEs are solved for a series of “spatial” positions. We take a Fourier transform of the signal
along this spatial dimension and isolate the signal in the “k-space”. We consider only asingle spatial axis to reduce the computational complexity. We define the electric field due
to a single pulse as02i(t −tj ) h1−i{ω(t0 −tj )−kj z}−i{ω(t0 −tj )−kj z}
2tp 2
Ej (t ) = |E| e
+e
e
2
0

(B.1)

where tj is the time of incidence of pulse j with wavevector kj , tp is the pulse duration, ωis the center frequency of the light field, and z is the spatial dimension. We denote time by
t0 to distinguish it from the time delay t in a 2DCS experiment. The total electric field isthe sum of the electric fields from all the incident pulses. The oscillating part of the electricfield in Eq. (B.1) slows down the computational process significantly. Thus, we set ω = 0;
the transition energy is defined as the detuning from ω. This simplification is equivalent to
the rotating-wave approximation discussed in Appendix A.

Signal amplitude (a.u.)Light-field amplitude (a. u.)

104
1
0.8
0.6
0.4
0.2
0
−2

(a)

−1

0

1

Position z (a. u.)

Signal
102

100

10−2
−10

2

(b)

−5

0

5

10

Wavevector k (a. u.)

Figure B.1: (a) The longitudinal “spatial” amplitude profile of the incident electric field. (b)The transient four-wave mixing signal amplitude as a function of the wavevector.

132
The spatial positions considered in the simulation are shown in Fig. B.1(a); we use 101
spatial points. We have used a Gaussian spatial profile so that the electric field amplitude
peaks at z = 0 and decays smoothly to zero. Using such an amplitude profile ensures that
the Fourier transform is well-behaved. The OBEs are solved as a function of two time delays
τ and t for each of these spatial points to obtain the radiated signal S(τ, t, z) from themacroscopic polarization, which in turn is related to the coherence ρ01 as indicated in Eq.
(2.14). This procedure is repeated four times with different phases of the incident pulses
to implement a phase-cycling routine similar to the one described in Section 3.3.2. The
individual spectra are processed to obtain the third-order polarization S (3) (τ, t, z). We takea Fourier transform of this signal along the position axis to obtain S (3) (τ, t, k). An exampleof the integrated third-order signal is shown in Fig. B.1(b) as a function of wavevector k.
There are several peaks in this plot. Each peak corresponds to a different phase-matchingdirection. We use kA = −4, kB = 1, and kC = −1, which gives kS = 4 when pulse A is
conjugated, as labeled in the plot. We select the region k = 4 ± 1 in S (3) (τ, t, k) and set thesignal at all other k values to zero. This signal is Fourier transformed back to the spatial
domain to obtain Scut (τ, t, z). Finally, we obtain the TFWM signal SSI (τ, t) = Scut (τ, t, 0).
The corresponding 2D spectrum is obtained by taking a two-dimensional Fourier transformof SSI (τ, t).
B.2

Inhomogeneous broadening
The procedure mentioned in the previous section is for a single oscillator. In case of in-homogeneous broadening, the same procedure has to be repeated for oscillators with different
resonance energies as indicated in Fig. B.2 and different amplitudes αi indicating different
number of oscillators at each frequency position. The amplitude terms αi are normalized
such that the sum Σ αi = 1. This normalization ensures that, for a given inhomogeneous
linewidth, the total signal is independent of the number of oscillators chosen in the simulations. The total signal for the inhomogeneous distribution is obtained by taking the sum

133of the signal from all the frequency components after weighting the signal from a particularfrequency component i by the amplitude αi .

Signal intensity (a. u.)

0.15

0.1

0.05

0
−2

−1

0

1

2

Resonance energy offset (meV)

Figure B.2: The inhomogeneous distribution used for the simulations.

The 2D spectra obtained for the inhomogeneous broadening indicated in Fig. B.2 areshown in Fig. B.3. As expected the peak is elongated along the diagonal line. The signal
phase is similar to that obtained for the homogeneous case through the analytical procedure,as shown in Fig. 3.3.

Excitation energy (meV)

10

20

30

−20

Absolute value

0

20

−20

Real part

020

Imaginary part

−7

−5

−3

(a)

(b)
3

5

7

(c)
3

5

7

3

5

7

Emission energy (meV)

Figure B.3: The (a) absolute value, (b) real part, and (c) imaginary part of the rephasing
one-quantum 2D spectrum for a two-level system.

134
B.3

Phenomenological many-body effects
The motivation for developing the numerical procedure to solve the OBEs was toinclude the phenomenological many-body effects, which we will discuss next. The transitionenergy, dephasing rate and the effective electric field are modified by these interaction terms
terms [143].
γ01 = γ0 + N γ 0 ρ11
ω01 = ω0 + N ω 0 ρ11(B.2)

E = E0 + L P
where γ0 and ω0 are the dephasing rate and resonance frequency, respectively, for an unexcited sample, N is the total number of oscillators, γ 0 and ω 0 are the EID and EIS parameters,
respectively, E0 is the electric field of the incident light, L is the local-field parameter, andP is the macroscopic polarization given by Eq. (2.14). In case of inhomogeneous broadening, the local-field term is, generally, neglected because the macroscopic polarization decays
quickly.
The modified OBEs are solved according to the procedure mentioned above to calculate
the TFWM signal. An example of such a situation is shown in Fig. B.4 where we have only
included the EIS term, i.e. γ 0 = L = 0. We obtain a dispersive peak in the real part
spectrum shown in Fig. B.4(b). This feature is similar to the experimentally measured
signal phase [13].
Including the many-body terms in the absence of inhomogeneous broadening is straightforward. In the presence of inhomogeneous broadening, however, there are several ways toinclude the many-body terms. The complication arises because separate density matrices
are defined for each frequency element. By testing the different methods, we concluded that
EID and EIS are correctly incorporated as
γ01,i = γ0 + Ai N γ 0 ρ11,i(B.3)
0

ω01,i = ω0 + Ai N ω ρ11,i

135

Excitation energy (meV)

20 40 60 80 100 120−100

Absolute value

0

100

−100

Real part

0100

Imaginary part

−7

−5

−3

(a)

(b)
3

5

7

(c)
3

5

7

3

5

7

Emission energy (meV)

Figure B.4: The (a) absolute value, (b) real part, and (c) imaginary part of the rephasing onequantum 2D spectrum for an inhomogenoeusly-broadened two-level system. Phenomenological excitation-induced shift is included. Notice the increase in amplitude compared to thespectra in Fig. B.3 and the dispersive peak in (b).

where index i is used to indicate a frequency element. Consequently, the effect of the manybody terms is different for the different frequency groups. If we extend this treatment to the
local-field effect, the effective electric field experienced by each frequency group is different,
and is given by
Ei = E0 + L Pi .

(B.4)

This treatment of the local-field implies that it cannot be ignored even for inhomogeneously
broadened systems. We are still working on correctly including the local-field correction and,thus, do not show any results for the same.

Appendix C
Nominally Disallowed Transitions in (110)-oriented GaAs Quantum Well

We have mentioned a few effects of the reduced symmetry of the (110)-oriented GaAs
QWs in Section 2.6. Also, we discussed the resulting anisotropic exciton-dephasing rates in
Chapter 4. Another consequence of the reduced symmetry is the mixing of the HH and LH
valence bands, which results in the HH band having some LH characteristics, and vice-versa.
As a result, the polarization selection rules discussed in Section 2.2.3, are not strictly valid.
The effect of this band mixing on photocurrents generated in these QWs has been studied
both experimentally [144] and theoretically [45]. We find that another consequence of thisband mixing is that an excitonic transition that should be disallowed, is optically active.
Figure C.1 shows a 2D spectrum spanning both the HH and LH exciton resonances;the excitation-laser spectrum (blue) and the absorption spectrum (red) are shown in the
top panel. This spectrum was obtained with the sample at a temperature of 10 K and
the collinear polarization scheme with Y polarization. The spectrum comprises five distinct
peaks, which are labeled A – E. The HH and LH peaks, which are labeled A and B, respectively, appear on the diagonal line. The peaks labeled C and D indicate coupling between
the HH and LH excitons. Similar to the cross peaks in the simulated spectrum shown in
Fig. 3.7, these peaks are titled with respect to the diagonal line due to different inhomogeneous linewidths of the HH and LH exciton resonances. The origin of these coupling peaks
is similar to those observed in the (001)-oriented GaAs QWs [99]. The presence of the peaklabeled E, however, is unexpected; a similar peak does not show up for the (001)-oriented

137

1

0.3

0.5

0.15
0

Excitation energy (meV)0

Absorption
spectrum

Laser
spectrum

GaAs QW.

1

A
−1560

C
0.8

−1570

0.6

E
−1580

B

0.4
0.2

D
−1590
1560

1570

1580

1590

Emission energy (meV)

Figure C.1: Rephasing one-quantum spectrum spanning both the heavy and light hole excitons in (110)-oriented quantum well is shown in the bottom panel. Sample temperature was
10 K and co-linear polarization scheme with Y polarized pulses was used. The heavy holeand light hole exciton peaks on the diagonal are labeled A and C, respectively. The couplingpeaks between these exciton states are labeled C and D. Note the tilt of these peaks similar
to those shown in Fig. 3.7. Another peak E appears which should be nominally disallowed.
The top panel shows the laser spectrum in blue and the absorption spectrum in red.

We hypothesize that the presence of this peak indicates the presence of excitons formed
with the second HH band and the first electron state. This exciton transition should not beallowed through the dipole selection rules, but results from the HH and LH band-mixing. A
diagonal peak due to this exciton resonance should be present, but is too weak due to thesmall dipole moment of this transition. The resonance energy of this peak is in qualitative
agreement with numerical calculations [45]. Although this exciton resonance is weak, it canhave significant effect on the optically generated currents in these samples. A more detailed
study into the origin and effects of this resonance is needed, which is beyond the scope ofthe current work.

Appendix D
Population Evolution

In Section 5.3.2, we presented interesting evolution of signal strength as a function of
the waiting time T , as shown in Fig. 5.9. The amplitude term in Eq. (5.3) gives the TFWM
signal strength. We measured this amplitude term through the fitting procedure discussed
in Section 5.3.1. To ensure that the signal evolution is not an artifact of the fitting procedureor an incorrect model, we performed rephasing zero-quantum scan to directly measure theevolution of signal strength with the waiting time. These experiments were performed forseveral sample temperatures. The time-integrated TFWM signal as a function of the waiting

TI-TFWM intensity

1

20 K
30 K
50 K
80 K

0.1

0

20

40

60

80

Waiting time T (ps)Figure D.1: Evolution of time-integrated transient four wave mixing signal with the waitingtime for different sample temperatures. Note the similarities between these plots and thoseshown in Fig. 5.9.

139
time are shown in Fig. D.1 for several sample temperatures. Although 2DCS experiments
were performed, we plot the time-integrated signal intensity to compare with the plots shownin Fig. 5.9. The measurements were taken for much longer values of delay T . We only show
a section of it to highlight the behavior for short waiting times.
The similarity between the plots in Figs. 5.9 and D.1 is apparent; the non-exponential
evolution along with the non-monotonic signal evolution for temperatures > 50 K are seen
in both figures. We estimate an exciton population decay time in the range of ∼ 100 − 300
ps over the temperature range studied. This population decay time is measured from theslow, exponential decay of the signal for T > 50 ps.
While these results confirm the results obtained from the spectral diffusion experiment,
they do not shed any light on the possible reason for the non-monotonic behavior for highsample temperatures.