Optical Two-Dimensional Fourier Transform
Spectroscopy of Semiconductors
byTianhao Zhang
M.S., University of Colorado, 2002
B.S., University of Science and Technology of China, 1997

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
2008This thesis entitled:
Optical Two-Dimensional Fourier Transform Spectroscopy of Semiconductors
written by Tianhao Zhang
has been approved for the Department of Physics

Prof. Steven T. Cundiff

Prof. Charles Rogers

Date

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

iii
Zhang, Tianhao (Ph.D., Physics)
Optical Two-Dimensional Fourier Transform Spectroscopy of Semiconductors
Thesis directed by Prof. Steven T. Cundiff

Optical two-dimensional (2D) Fourier transform spectroscopy (FTS) is implemented in the near-IR regime and employed for the study of exciton dynamics, manybody interactions, and disorders in semiconductors. As the optical analog of multidimensional nuclear magnetic resonance, 2D FTS is based on a highly enhanced transient
four-wave mixing (FWM) experiment. A FWM signal is generated by a non-collinear
three-pulse sequence and heterodyne-detected with a reference to provide both amplitude and phase. With active interferometric stabilization and scanning, the evolutionof FWM electric field is coherently tracked and presented on a 2D map of the absorption and emission frequencies. With capabilities that include disentangling congested
spectra, identifying resonant coupling, isolating coherent pathways, determining inhomogeneous broadening, and separating complex spectra into real and imaginary parts,
2D FTS is a powerful tool to resolve problems in traditional FWM spectroscopies.
In a typical 2D spectrum of semiconductor quantum wells, diagonal peaks arise
from exciton resonances and cross peaks represent their coupling, with features suchas cross peak strength and absorption of continuum dominated by many-body interactions. Based on the modified optical Bloch equations with phenomenological terms
including excitation-induced dephasing, excitation-induced shift, and local field effect,
numerical calculations are performed to reproduce these features and determine themicroscopic origin of many-body effects by comparing to the experimental amplitude
and real part spectra. The dependence of 2D spectra on the excitation polarization is
employed to further explore the many-body interactions. In comparison with microscopic calculations with contributions of Pauli blocking, Hartree–Fock approximation,

iv
and higher-order Coulomb correlations, it is found that exciton correlations play thedominant role in the case of cocircular-polarized excitation. With an alternative 2D
projection, Raman coherences between excited excitons are isolated. The experimental
and calculated 2D spectra in this form also demonstrate the similar result on exciton
correlations.

Dedication

This thesis is dedicated to my parents.

vi

Acknowledgements

First and foremost, I would like to thank my advisor, Prof. Steve Cundiff, for
his constant support, guide and encouragement throughout my graduate career. His
expertise and deep insight in research have greatly enlightened me and advanced mywork. His excellence and enthusiasm in science always motivate me to explore the nature
further.
I also have a great gratitude to Prof. Charles Rogers, as well as Dr. Rich Mirin,
Prof. David Jonas, and Prof. Dan Dessau on my thesis committee, for their invaluableadvices and supports.
I am deeply grateful for the inspirations and contributions from our theory collaborators, including Prof. Torsten Meier (Universität Paderborn), Prof. Peter Thomas and
Dr. Irina Kuznetsova (Philipps Universität, Marburg), as well as Prof. Shaul Mukamel
and Dr. Lijun Yang (University of California, Irvine). The efficient exchanges of ideas
and results with Irina and Lijun have been very beneficial and pleasant. I thank Dr. Rich
Mirin (NIST) and Prof. Duncan Steel (University of Michigan) for high quality quantum
well samples.
I appreciate the friendship and help from all the fellow Cundiff group members,
past and present. Particularly I thank all my postdoctoral co-workers, Dr. Camelia
Borca, Dr. Xiaoqin (Elaine) Li, and Dr. Alan Bristow, for their aids and encouragements. I am indebted to Ryan Smith and Alan Bristow for reading my thesis manuscript
and valuable suggestions on writing. I also thank Loree Kaleth for the administrative

vii
support.
JILA is an exceptional place for scientific research. I am grateful for the technicalassistance from David Alchenberger and Tom Foote of the instrument shop, as well as
Paul Beckingham, Terry Brown, and Mike Whitmore of the electronic shop in JILA.
I express my ultimate gratitude to my parents, for their lifelong love, care, andsupport.

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Contents

Chapter
1 Introduction

1

2 Coherent Optical Properties of Semiconductors

7

2.1

Excitons in semiconductors . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Semiconductor quantum wells . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3

Optical transitions in quantum wells . . . . . . . . . . . . . . . . . . . .

15

2.4

Relaxations of optical excitations . . . . . . . . . . . . . . . . . . . . . .

19

2.5

Nonlinear optical spectroscopies of semiconductors . . . . . . . . . . . .

21

2.6

Many-body interactions in semiconductors . . . . . . . . . . . . . . . . .

24

3 Principles of Optical Two-Dimensional Fourier Transform Spectroscopy

30

3.1

Development of IR and optical 2D FTS . . . . . . . . . . . . . . . . . .

31

3.2

Nonlinear response function . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.3

Double-sided Feynman diagrams . . . . . . . . . . . . . . . . . . . . . .

38

3.4

Two-dimensional Fourier transform spectroscopy . . . . . . . . . . . . .

43

3.5

Advantages of 2D FTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.5.1

Identification of resonant coupling . . . . . . . . . . . . . . . . .

49

3.5.2

Isolation of coherent pathways . . . . . . . . . . . . . . . . . . .

51

3.5.3

Other capabilities of 2D FTS . . . . . . . . . . . . . . . . . . . .

57

ix
4 2D FTS Experiments with Active Interferometric Stabilization

59

4.1

Implementations of 2D FTS with phase stability . . . . . . . . . . . . .

59

4.2

Four-wave mixing generation . . . . . . . . . . . . . . . . . . . . . . . .

62

4.3

Interferometric phase stabilization . . . . . . . . . . . . . . . . . . . . .

64

4.4

Fourier transform spectral interferometry . . . . . . . . . . . . . . . . .

68

4.5

Generating 2D Fourier transform spectra. . . . . . . . . . . . . . . . .

71

4.6

Primary experimental results . . . . . . . . . . . . . . . . . . . . . . . .

74

5 2D FTS Interpreted with Optical Bloch Equations

78

5.1

Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.2

Amplitude 2D spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.3

Full 2D spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.4

Determination of homogeneous and inhomogeneous broadening . . . . .

91

6 2D FTS Interpreted with Microscopic Semiconductor Theory

95

6.1

General formalism of the microscopic semiconductor theory . . . . . . .

96

6.2

One-dimensional tight-binding model . . . . . . . . . . . . . . . . . . . .

99

6.3

Excitation dependence of 2D spectra . . . . . . . . . . . . . . . . . . . . 101

6.4

2D spectra with cocircular-polarized excitation . . . . . . . . . . . . . . 104

6.5

2D spectra with cross-linear-polarized excitation . . . . . . . . . . . . . 108

7 Raman Coherences Revealed by Alternative 2D FTS

111

7.1

Coherent pathways contributing to the photon echo signal . . . . . . . . 112

7.2

Experimental SI (τ, ωT , ωt ) spectra . . . . . . . . . . . . . . . . . . . . . 115

7.3

Microscopic calculations of SI (τ, ωT , ωt ) spectra . . . . . . . . . . . . . . 118

Appendix
A 2D FTS of Double Quantum Wells

121

x
B 2D FTS of Exciton Continuum

127

Bibliography

130

xi

FiguresFigure2.1

Optical excitation in a direct bandgap semiconductor . . . . . . . . . . .

9

2.2

Exciton E(n, k) relation and the linear absorption . . . . . . . . . . . .

11

2.3

HH and LH valence sub-bands and the optical transitions between thevalence band and conduction band . . . . . . . . . . . . . . . . . . . . .

15

2.4

Exciton energy level scheme with doubly-excited states included . . . .

17

2.5

Experimental setup of three-pulse FWM generation with TI and TR signal detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

23

Experimental evidences for strong many-body interactions in semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.1

Excitation pulse sequence in a three-pulse nonlinear experiment . . . . .

36

3.2

Double-sided Feynman diagrams representing contributions to the phasematched directions kI , kII , and kIII . . . . . . . . . . . . . . . . . . . . .

3.3

40

Spatial arrangement and time ordering of excitation pulses in 2D FTSexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.4

Excitation pulse sequence for rephasing and non-phasing experiments

.

46

3.5

Coupling of resonances in different level systems revealed by 2D spectra

49

3.6

Isolating contributions of coherent pathways in 2D spectra . . . . . . . .

52

3.7

Isolating contributions of two-exciton coherent pathways in 2D spectra .

54

xii
3.8

Contributions of coherent pathways in 2D spectra and 1D spectrallyresolved FWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

55

Double-sided Feynman diagrams of coherent pathways contributing to
non-rephasing 2D spectra . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.1

Experimental setup for 2D FTS with active interferometric stabilization

62

4.2

Linear absorption of the GaAs/Al0.3 Ga0.7 As multiple quantum well sample and the excitation pulse spectrum . . . . . . . . . . . . . . . . . . .

4.3

63

Error signals from the excitation phase and reference phase stabilizationinterferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.4

He-Ne fringes during stepping and the flowchart of locking and stepping

66

4.5

Full information of FWM signal retrieved by Fourier transform spectralinterferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.6

Fluctuation of signal phase change as a function of delay τ. . . . . . .

72

4.7

SRDT measurement and the least-squared fit for global 2D phase . . . .

74

4.8

Excitation power dependence of TI FWM signal intensity . . . . . . . .

75

4.9

Amplitude 2D spectrum of rephasing pathway with a laser tuning above
LH exciton resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.1

Numerical calculations for cross peak strength in amplitude 2D spectrum

84

5.2

Numerical calculations for the amplitude 2D spectrum of HH exciton andcontinuum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Experimental amplitude and real part 2D spectra of rephasing and nonrephasing pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

85

87

Calculated amplitude and real part 2D spectra without any many-bodyeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.5

Calculated amplitude and real part 2D spectra with EID . . . . . . . . .

88

5.6

Calculated amplitude and real part 2D spectra with EIS

90. . . . . . . .

xiii
5.7

Simulated broadening in 2D spectra for a two-level system . . . . . . . .

5.8

Simulation result of diagonal and cross-diagonal linewidths as functionsof dephasing rate and inhomogeneous width . . . . . . . . . . . . . . . .

5.9

92

93

Excitation power dependence of homogeneous and inhomogeneous linewidths
for HH and LH excitons obtained from experimental 2D spectra. . . . .

94

6.1

One-dimensional tight-binding model for semiconductor quantum wells . 100

6.2

Excitation power dependence of TI FWM signal for different excitation
polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3

Experimental real part 2D spectra of colinear-polarized excitation withdifferent tuning and excitation power . . . . . . . . . . . . . . . . . . . . 103

6.4

Real part 2D spectra of colinear-polarized excitation at different tunings
calculated with microscopic theory . . . . . . . . . . . . . . . . . . . . . 104

6.5

Experimental real part 2D spectra with cocircular-polarized excitation inboth rephasing and non-rephasing pathways . . . . . . . . . . . . . . . . 105

6.6Real part 2D spectra of cocircular-polarized excitation calculated withmicroscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.7

Experimental and calculated amplitude 2D spectra with cross-linearpolarized excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.8

Exciton and biexciton peaks in power-dependent photoluminescence (PL)spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1

Contributions of coherent pathways in SI (ωτ , T, ωt ) and SI (τ, ωT , ωt ) 2D
projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2

Linear absorption of the GaAs/Al0.3 Ga0.7 As multiple quantum well sample and the excitation pulse spectrum . . . . . . . . . . . . . . . . . . . 115

7.3

Experimental amplitude SI (τ, ΩT , Ωt ) spectra with colinear- and cocircularpolarized excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xiv
7.4

Cross-sections at Ωt = ΩH and ΩL in experimental SI (τ, ΩT , Ωt ) spectrumwith colinear-polarized excitation . . . . . . . . . . . . . . . . . . . . . . 117

7.5

Calculated amplitude 2D spectra with colinear-polarized and cocircularpolarized excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.6

Calculated amplitude 2D spectra with colinear-polarized and cocircularpolarized excitations in the time-dependent Hartree–Fock approximation 119

A.1 Schematic energy level structure and optical transitions for double quantum wells with different barrier thickness. . . . . . . . . . . . . . . . . 122

A.2 Amplitude 2D spectra of the DQW sample with a barrier thickness of
10 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.3 Amplitude 2D spectra of the DQW sample with a barrier thickness of
1.7 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.4 Amplitude 2D spectra of the DQW sample (barrier thickness of 1.7 nm)with different waiting times . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.1 Experimental rephasing 2D spectra of exciton continuum . . . . . . . . 128

Chapter 1

Introduction

In direct bandgap semiconductors at low temperatures, the optical properties
near the fundamental gap are dominated by excitonic resonances, which correspond to
the formation of bound electron and hole pairs. The carrier dynamics and coupling insemiconductors have been extensively studied by ultrafast optical spectroscopic methods, primarily transient four-wave mixing (FWM) [1]. Various experiments, including
time-integrated, time-resolved, and spectrally-resolved FWM, have demonstrated thatthe many-body interactions among excitons and carriers dominate the nonlinear optical properties [2]. Phenomenological effects such as local field correction (LFC) [3],biexciton formation [4], excitation-induced dephasing (EID) [5], and excitation-induced
shift (EIS) [6] describe the many-body interactions. Although conventional FWM spectroscopies have revealed rich information about excitons and carriers, there are somedifficulties. Multiple electronic transitions can be simultaneously excited by broadband
ultrafast pulses, resulting in a complicated oscillatory temporal evolution. Whether it
is due to the quantum beats from resonant coupling, or the polarization interferenceof uncoupled oscillators often remains undetermined [7]. In addition, it is hard to isolate dynamical information from the mixed contributions of many quantum-mechanicalcoherent pathways in a one-dimensional spectrum. Moreover, there is an intrinsic ambiguity regarding the microscopic origins of many-body effects, as the manifestations of
LFC, EID, EIS, and biexcitons are similar in FWM magnitude measurements.

2
Optical two-dimensional (2D) Fourier transform spectroscopy (FTS) based on,
but surpassing, traditional FWM techniques is a promising tool to address these problems. As an optical analog of multidimensional nuclear magnetic resonance (NMR)
spectroscopy [8], 2D FTS has been implemented for probing the vibrational and electronic states of molecular systems in the infrared (IR) and optical regimes [9, 10, 11,
12, 13, 14, 15]. Compared to traditional one-dimensional or non-Fourier transform twodimensional spectroscopies, 2D FTS provides many advantages by coherently tracking
signal phase evolution and presenting the correlation in two time periods. Congested
spectra are spread out into two spectral dimensions, allowing identification of resonant
couplings and isolation of coherent pathways. The preservation of phase informationin 2D FTS leads to the separation of real and imaginary part spectra, making it possible to improve resolution of lineshapes and provide microscopic information unavailable
in magnitude measurements. In 2D FTS, inhomogeneous broadening of the ensemble
manifests itself as the diagonal elongation, leading to the separation of homogeneousand inhomogeneous linewidths. Inhomogeneous broadening can be determined to provide disorder information of the system. In contrast to NMR, optical 2D FTS provides
snapshots of the coherent dynamics on the timescale of femtosecond. Besides, phasematched directions can be employed in optical 2D FTS with non-collinear geometry,
resulting in the separation of the signal from incident pulses and spatial isolation ofcoherent pathways. By selecting the phase-matched direction, the temporal orderingof excitation pulses, and the Fourier-transform time variables, different 2D projections
can be produced for specific problems.
In 2D FTS experiments, excitation pulses are scanned with sub-optical-cycle step
size and the signal phase is tracked coherently. Therefore high phase stability andpulse positioning accuracy are critical, especially for experiments in the near-IR orvisible wavelength range. Various techniques have been developed to implement 2D FTSexperiment, including diffractive optics [16, 17, 18, 19], femtosecond pulse shaping [20,3
21, 22], and active phase stabilization [23, 24].
This thesis is focused on the implementation of 2D FTS and its applications in the
study of exciton dynamics, many-body interactions, and disorders in semiconductors.
The structure of this thesis is as follows. Chapter 2 is an introduction to the coherent
optical properties of semiconductors. The concept of excitons in semiconductors andsemiconductor nanostructures is presented, followed by discussions of the selection rules
for optical transitions, the relaxation processes after optical excitation, the coherent
optical spectroscopic techniques, as well as the experimental evidences for the dominanceof many-body interactions in semiconductors.
In Chapter 3, the development of IR and optical 2D FTS from multidimensional
NMR and the applications in the studies of vibrational and electronic states in molecularsystems are first reviewed. Then the nonlinear response function, third-order polarization and double-sided Feynman diagrams are introduced. The principles of 2D FTS arepresented and the advantageous features are discussed.
Chapter 4 covers the technical aspects of the 2D FTS experiments we have implemented. Details including FWM generation, the interferometric stabilization andscanning of delay, the retrieval of full signal information with Fourier transform spectralinterferometry in heterodyne detection, the formation of 2D spectra, and the experimental determination of the global phase in 2D spectra are reviewed. The basic featuresin an amplitude 2D spectrum from a typical experiment are discussed.
In Chapter 5, the density matrix formalism and the optical Bloch equations withphenomenological modifications are introduced. Numerical calculations are performed
to qualitatively reproduce the features in experimental data and demonstrate the dominance of many-body interactions. The calculated amplitude and real part spectra withdifferent types of many-body effects are compared to the experimental spectra.
In Chapter 6, the formalism of a microscopic semiconductor theory is first introduced, followed by a discussion on the polarization dependence of 2D spectra. Ex-

4perimental results of different polarization configurations are compared to microscopic
calculations based on a one-dimensional tight binding model. The contributions of Pauliblocking, Hartree–Fock approximation, and Coulomb correlations beyond the Hartree–
Fock limit are discussed.
An alternative form of 2D FTS that is able to isolate the Raman coherences
between heavy-hole and light-hole excitons is presented in Chapter 7. Experimental 2D
spectra with colinear and cocircular-polarized excitation are presented and compared tomicroscopic calculations with time-dependent Hartree–Fock approximation and beyond.
In addition, the 2D FTS approach is employed to study the electronic coupling
and tunneling in asymmetric double quantum wells of different barrier thickness. The
results are presented in Appendix A. The experimental study of exciton continuum
states with 2D FTS is discussed in Appendix B.

Publications and conference presentations in frame of the thesis

Journal publications[1] X. Li, T. Zhang, S. Mukamel, R. P. Mirin, and S. T. Cundiff, “Investigation of electronic coupling in semiconductor double quantum wells using optical two-dimensional
Fourier transform spectroscopy,” in preparation.
[2] L. Yang, T. Zhang, A. D. Bristow, S. T. Cundiff, and S. Mukamel, “Isolating excitonic
Raman coherence in semiconductors using two-dimensional correlation spectroscopy,”
submitted to J. Chem. Phys.
[3] I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, and S. T. Cundiff, “Determinationof homogeneous and inhomogeneous broadening of quantum-well excitons by 2DFTS:
An experiment-theory comparison,” submitted to Phys. Stat. Solid.

5[4] T. Zhang, I. Kuznetsova, T. Meier, X. Li, R. P. Mirin, P. Thomas, and S. T. Cundiff,“Polarization-dependent optical 2D Fourier transform spectroscopy of semiconductors,”
Proc. Natl. Acad. Sci. 104, 14227 (2007).
[5] I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, X. Li, R. P. Mirin, and S. T. Cundiff,“Signatures of many-particle correlations in two-dimensional Fourier-transform spectraof semiconductor nanostructures,” Sol. Stat. Comm. 142, 154 (2007).
[6] X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional Fourier transform spectroscopy,” Phys.
Rev. Lett. 96, 057406 (2006).
[7] C. N. Borca, T. Zhang, X. Li, and S. T. Cundiff, “Optical two-dimensional Fouriertransform spectroscopy of semiconductors,” Chem. Phys. Lett. 416, 311 (2005).
[8] T. Zhang, C. Borca, X. Li, and S. T. Cundiff, “Optical two-dimensional Fouriertransform spectroscopy with active interferometric stabilization,” Opt. Expr. 13, 7432
(2005).

Conference presentations[1] T. Zhang, L. Yang, A. D. Bristow, S. Mukamel, and S. T. Cundiff, “Exciton Raman coherence revealed in two-dimensional Fourier transform spectroscopy of semiconductors,” Quantum Electronics and Laser Science (QELS) Conference, San Jose, CA,
May 4-9, 2008.
[2] T. Zhang, A. D. Bristow, I. Kuznetsova, T. Meier, P. Thomas, and S. T. Cundiff, “Direct determination of exciton homogeneous and inhomogeneous linewidths insemiconductor quantum wells with two-dimensional Fourier transform spectroscopy,”
Quantum Electronics and Laser Science (QELS) Conference, San Jose, CA, May 4-9,
2008.

6
[3] T. Zhang, I. Kuznetsova, T. Meier, P. Thomas, R. P. Mirin, and S. T. Cundiff,“Polarization-dependent optical 2D Fourier transform spectroscopy of quantum wells,”
Montana Meeting on Fundamental Optical Processes in Semiconductors (FOPS), Big
Sky, MT, Jul. 23-27, 2007.
[4] T. Zhang, A. D. Bristow, S. T. Cundiff, L. Yang, and S. Mukamel, “Isolating excitonic Raman coherences using 2D Fourier transform spectroscopy,” Montana Meetingon Fundamental Optical Processes in Semiconductors (FOPS), Big Sky, MT, Jul. 23-27,
2007.
[5] T. Zhang, X. Li, S. T. Cundiff, R. P. Mirin, I. Kuznetsova, P. Thomas, and T.
Meier, “Experimental and theoretical studies of exciton correlations using optical twodimensional Fourier transform spectroscopy,” Quantum Electronics and Laser Science(QELS) Conference, Baltimore, MD, May 6-11, 2007.
[6] T. Zhang, X. Li, S. T. Cundiff, R. P. Mirin, and I. Kuznetsova, “Many-body interactions in semiconductors probed by optical two-dimensional Fourier transform spectroscopy,” American Physical Society March Meeting, Denver, CO, Mar. 5-9, 2007.
[7] T. Zhang, X. Li, S. T. Cundiff, R. P. Mirin, and I. Kuznetsova, “Polarized optical twodimensional Fourier transform spectroscopy of semiconductors,” The 15th International
Conference on Ultrafast Phenomena, Pacific Grove, CA, Jul. 31-Aug. 4, 2006.
[8] T. Zhang, X. Li, C. N. Borca, and S. T. Cundiff, “Probing the microscopic origin of many-body interactions in semiconductors using optical two-dimensional Fouriertransform spectroscopy,” Quantum Electronics and Laser Science (QELS) Conference,
Baltimore, MD, May 22-27, 2005.
[9] T. Zhang, C. N. Borca, X. Li, and S. T. Cundiff, “Optical two-dimensional Fouriertransform spectroscopy technique,” Colorado Meeting on Fundamental Optical Processes in Semiconductors (FOPS), Estes Park, CO, Aug. 8-13, 2004.

Chapter 2

Coherent Optical Properties of Semiconductors

The rapid development of ultrafast lasers has paved the way to the coherent regime
in semiconductors, in which the phase coherence established by optical excitation only
exists on the timescale of a picosecond. The coherent behavior of excitons, electrons,
and holes is critical to the understanding of many fundamental quantum mechanicalprocesses [1]. These processes have profound influences on ultrafast electronic and optoelectronic device applications. Besides, the knowledge of carrier dynamics is essential
for research activities recently emerging in coherent regime, such as quantum computing,coherent control, and ultrafast switching [25]. The carrier dynamics and coupling havebeen extensively studied by nonlinear optical spectroscopic tools, primarily transient
four-wave mixing (FWM) and pump-probe experiments. Unlike dilute atomic systems,
many-body interactions dominate the nonlinear response of semiconductors [26].
In Section 2.1, the concept of an exciton is introduced as the dominant featurefor optics near band edge in direct bandgap semiconductors, followed by discussionof exciton properties in bulk semiconductor and quantum wells (Section 2.2). Then
selection rules and optical excitations, including bound and unbound two-exciton states,
are discussed in Section 2.3. The relaxation processes of optical excitation at different
time scales is reviewed in Section 2.4, with a primary focus on the coherent regimeof semiconductors. Common experimental techniques including time-integrated, timeresolved, and spectrally-resolved FWM are discussed in Section 2.5. The results of

8many-body interactions in semiconductors probed with FWM experiments are reviewed
in Section 2.6, with a discussion on the limits of conventional techniques.

2.1

Excitons in semiconductors
The electronic levels of a semiconductor with a periodic lattice can be canonicallydescribed by an energy band structure in momentum space. For a semiconductor in the
ground state, the valence band is completely occupied by electrons and the conduction
band is empty. During the optical excitation of a semiconductor with direct bandgap
Eg , as shown in Fig. 2.1, a photon with sufficient energy (~ω > Eg ) promotes anelectron from a state (mostly p-like) in the valence band to a state (mostly s-like)in the conduction band. Simultaneously a vacancy is created in the valence band,
which is referred as a “hole”. Since the momentum of a photon can be neglectedcompared to that of electron or hole, the transition is essentially vertical (δk = 0).
The Coulomb interaction between the excited electron and hole can be treated as aperturbation for excitations high into the conduction band. However, for excitations
close to the fundamental bandgap, the attractive Coulomb force between an electron
and a hole results in the formation of a bound state, known as an “exciton”. The
exciton can be treated as a quasi-particle, with the unbound electron-hole pairs being
the continuum states. In general, excitons have a large oscillator strength and thus
dominate the optical properties of direct bandgap semiconductors near band edge. The
concept of quasi-particle is useful for describing basic excitation and linear response to
weak external perturbations. However, the “residual” part of the Coulomb force, i.e.
the interaction not accounted for in the formation of excitons, leads to the interaction
between excitons and induces a nonlinear response. Excitons are usually classified into
two types: Frenkel excitons of tightly-bound electron-hole pairs [27, 28], and Wannier
excitons of less highly-correlated electrons and holes [29]. Excitons in semiconductors

9E(k)Conduction
Bandelectron
Eg

=ω

k
0

hole

Valence
Band

Figure 2.1: Optical excitation in a direct bandgap semiconductor. A photon promotes
an electron from a valence state to a conduction state, simultaneously creating a “hole”
in the valence band.

are generally the Wannier type with large radii [30].
With a structure similar to a hydrogen atom formed by an electron and a proton,
an exciton presents a hydrogen-like energy level scheme. In the center-of-mass frame, thereduced mass of exciton is µ = me mh /(me + mh ), and the total mass is M = me + mh ,where me and mh are the respective electron and hole masses. The eigenvalues of theexciton energy are given simply by analogy with hydrogenic levels:
Ẽ(n) = −

ER
,
n2

(2.1)

where n is the principal quantum number of the exciton, and ER is the exciton Rydberg,
i.e. the binding energy of the lowest state. ER is determined by the reduced mass of
exciton, µ, and the background dielectric constant, :
ER =

~2
µe4
=
,
22 ~2
2µa2B

(2.2)

where aB = ~2 /µe2 is the exciton Bohr radius characterizing the space occupied by
an exciton. The total energy of the exciton in the n-th excited state is a sum of the

10bandgap, internal energy, and kinetic energy [2]:E(n, k) = Eg −ER~2+
|k|2 ,
n2
2M

(2.3)

where k = ke + kh is the center-of-mass wavevector of exciton.
Due to the small reduced mass and large dielectric constant in background, excitons differ significantly from hydrogen atoms. The exciton binding energy is much
weaker than that of the hydrogen atom and the distance between the electron and holeare quite large, according to Eqn. (2.2). Here are some typical numbers for excitons inbulk GaAs semiconductor: Eg = 1.519 eV, me = 0.067 m0 , mh = 0.51 m0 , where m0 is
the free electron mass,  = 12.4, aB = 11.0 nm, and ER = 4.2 meV [31]. As the lattice
constant of GaAs is 0.56 nm, the electron in an exciton is many lattice sites away fromthe hole, confirming that excitons in GaAs are the Wannier type [30].
Based on the bandgap picture of semiconductors, optical excitation and emission
can be treated as a two-particle process, i.e. an electron in the conduction band and ahole in the valence band are created simultaneously in photon absorption and they canrecombine radiatively to emit a photon. This treatment leads to a useful representationof the semiconductor system, known as the pair picture. In Fig. 2.2(a), where the exciton
total energy E(n, k) is plotted versus exciton center-of-mass wavevector k, the origin
corresponds to the crystal ground state, and excitons of n = 1, 2, 3, . . . and electronhole pairs are presented as excited states. An electron-hole pair is created along thephoton line following the equation E(k) = ~ck/n, where c is the speed of light and n is
the index of refraction. If the coupling between photons and excitons is included, thepair picture is then converted to the polariton picture. An anti-crossing feature occurs
where the photon line and the exciton state intersect, resulting in an upper and a lower
branch.
The pair picture helps explain the dominance of excitons in the optical responseof semiconductors. The center-of-mass momentum of an exciton obtained by absorbing11
E(n, k)(a)electron-hole
continuum

n=2

(b)

n=1

ER

α

n=1

E2
E1

Eg

Optical excitation:
=c
E= k
n

n=2

E1

E2 Eg

continuumEk0

Figure 2.2: (a) The exciton total energy E(n, k) versus the center-of-mass wavevectork (not to scale) and the photon line E(k) = ~ck/n; (b) A schematic figure of the linear
absorption spectrum, with strong absorbing bound exciton states below bandgap.

a photon in the visible wavelength range is negligibly small compared to the extent
of the first Brillouin zone, thus k ' 0 in Eqn. (2.3). The photon energy needed tocreate an exciton bound state is less than the energy gap between the top of the valence
and the bottom of the conduction band. In a schematic linear absorption spectrum
depicted in Fig. 2.2(b) [31], strong hydrogenic absorption resonances corresponding to
the 1s, 2s, . . . bound excitons appear below the bandgap Eg , followed by the continuum
absorption of electron-hole pairs. The continuum absorption approaches a finite value
when photon energy ~ω decreases to Eg , instead of going to zero with a square-root
law as the prediction of free-carrier theory. This difference is known as the Coulomb
enhancement of continuum absorption [31]. The presence of discrete absorption lines
below bandgap and enhanced absorption of electron-hole above bandgap is significantly
different from the free-carrier absorption predicted with no Coulomb interaction between
electrons and holes. Therefore the formation of excitons and electron-hole pairs by
Coulomb interaction in semiconductors influences the linear optical response near band

12edge prominently. The oscillator strength of absorption is determined by the overlappingof the electron and hole wavefunctions. Hence much larger absorption happens at boundexciton states rather than the unbound electron-hole pair states. The exciton oscillator
strength f (n) ∝ 1/n3 for the n-th state, therefore the discrete resonance absorption
decreases quickly with n. Experimentally, only the first a few lines are observablebecause of various broadening mechanisms, such as scattering of electron-hole pairs byphonons.

2.2

Semiconductor quantum wells
Two-dimensional heterostructures known as quantum wells can be readily fabri-cated with superior lattice quality by growth techniques such as molecular beam epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD). In quantum wells,a layer thinner than the extent of the envelope wavefunction of electrons and holes
is sandwiched between two barrier layers of another semiconductor with larger energy
bandgap. If the barrier layers have sufficient thickness and energy gap, carriers will
be confined in the low energy layer where they were created, but remain free to movearound within that layer. One of the most widely used quantum well system consists of
GaAs and Alx Ga1−x As as well and barrier material, respectively [31]. The bandgap of
Alx Ga1−x As is larger than that of GaAs by about 0.5 eV, depending on the Al concentration. GaAs and Alx Ga1−x As (0 < x < 0.45) are both direct-gap semiconductors ofthe zinc blende structure, with almost the same lattice constant [31]. The close lattice
match reduces strain during growth and thus makes structures of many periods possible.
The wavefunction and energy levels of excitons in a quantum well are alteredsignificantly due to the quantum confinement. The wavefunction in the growth (ẑ)
direction is compressed while it remains a plane wave in the in-plane (x̂-ŷ) direction.
The ẑ-direction component is the solution to the quantum mechanical problem of a

13particle in a box. With such modification to the system, the eigenvalues of the excitonic
energy in an ideal two-dimensional case are:Ẽ(n) = −and the Bohr radius is (n − 12 ) aB .

ER,(n − 21 )2(2.4)

In addition, the exciton oscillator strength isf (n) ∝ 1/(n − 1/2)3 for 2D excitons instead of 1/n3 for bulk. Thus the binding energy (oscillator strength) of 1s exciton in 2D case is four (eight) times that of the bulk
exciton. The confinement greatly increases the exciton binding energy and absorption
strength by improving the overlapping of electron and hole wavefunctions. Due to finite
well and barrier thickness, excitons in real quantum heterostructures are not completely
two-dimensional. A more realistic treatment of the confinement on excitonic states inquantum wells can be found in [32].
In GaAs quantum wells, while the electron states and wavefunctions in the conduction band are similar to those in bulk, the hole states and wavefunctions in thevalence band are more complicated, since the valence band at the center of the Brillouin zone (k = 0) is degenerate. For bulk GaAs with a cubic symmetry, the conduction
band at the Brillouin zone center has s-orbital character with a two-fold degeneracy
since the electron spin J = 12 . The valence band around k = 0 has p-orbital character,which can be described by wavefunctions with a total angular momentum J = 23 for theupper two sub-bands and J = 12 for the lower sub-band. Known as the split-off band,the lower valence band is separated by 0.34 eV from the upper band at k = 0 due tospin-orbit coupling. The two upper valence bands are referred as the heavy-hole (HH)
and light-hole (LH) sub-bands, with projection of the angular momentum along ẑ-axis
Jz equal to ± 23 and ± 12 , respectively.
The energy of valence bands has an anisotropic dependence on the direction of
the wavevector k. The dispersion relations for HH and LH sub-bands near the center14of the Brillouin zone are [31]:2~2 kxy~2 kz2− (γ1 + γ2 )
2m0
2m0
2
~2 kxy~2 kz2− (γ1 − γ2 )εLH = −(γ1 + 2γ2 )2m0
2m0

εHH = −(γ1 − 2γ2 )3,2
1for Jz = ± ,2
for Jz = ±

(2.5)
(2.6)

where kxy and kz are the respective in-plane (x̂-ŷ direction) and perpendicular (ẑ direction) component of the hole wavevector, in a frame defined by the corresponding
quantum well geometry. m0 is the free electron mass and γ1 and γ2 are the Luttinger
parameters with values of γ1 ' 6.9 and γ2 ' 2.4 for GaAs [31]. The HH and LH
sub-bands are degenerate at k = 0 in bulk GaAs. Consequently, the effective holemass is also anisotropic and different for the two sub-bands: mxy
HH = m0 /(γ1 + γ2 ),xyzmzHH = m0 /(γ1 − 2γ2 ), mxy
LH = m0 /(γ1 − γ2 ), and mLH = m0 /(γ1 + 2γ2 ), where mHHz
and mzHH (mxy
LH and mLH ) are the respective in-plane and perpendicular component of

the hole mass in HH (LH) sub-band. The values of hole mass in the two sub-bands arexyz
mzHH ' 0.5 m0 , mxy
HH ' 0.11 m0 , mLH ' 0.086 m0 , and mLH ' 0.23 m0 . In quantum

wells, the quantum confinement acts as a perturbation and lifts the degeneracy of theupper valence band, resulting in an energetic separation of the HH and LH sub-bands,
as depicted in Fig. 2.3(a). The energy splitting of the two sub-bands near k = 0 is
∼ 10 meV in a typical 10 nm thick GaAs quantum well. The splitting is determined by
the well thickness and increases as the thickness decreases. For optical excitation withpulse width larger than the splitting, excitons are formed by both HH and LH withelectrons, commonly referred as HH and LH excitons.

15
E(k)

(a)

(b)

J = 1/ 2

Conduction band
-1/2

+1/2

σ+

σ-

σ-

+1/2

+3/2

Eg

=ω
σ+
HH

J = 3/ 2
J z = ±3 / 2
J = 3/ 2LHk
HH sub-band

HH
LH
Jz =

-3/2

-1/2

Valence band
LH sub-band

J z = ±1 / 2
J = 1/ 2

Split-off band

Figure 2.3: (a) The heavy-hole (HH) (J = 23 , Jz = ± 32 ), light-hole (LH) (J = 23 , Jz =± 12 ), and split-off (J = 12 ) sub-bands in the valence band of GaAs quantum wells; (b)Optical transitions between the HH and LH valence bands and the conduction bandwith σ + and σ − circular-polarized light.

2.3

Optical transitions in quantum wells
A photon carries an angular momentum l of 1 in the dipole approximation. Dueto the conservation of angular momentum, optical transitions obey the selection rule of∆l = ±1 for incident light perpendicular to the well plane. As the total exciton angularmomentum contains contributions from the internal motion and envelope motion, theselection rule of transitions can be expressed as ∆lint + ∆lenv = ±1, where lint andlenv are the angular momentum of exciton internal and envelope motion, respectively.
For dipole-allowed transitions, the only possible situation is ∆lint = ±1 and ∆lenv = 0.
This leads to a Jz = ±1 transition that occurs between a p-like valence band and ans-like conduction band, resulting in an exciton with an s-like envelope function.
In the optical excitation of a quantum well, only the electron and hole states at thecenter of Brillouin zone need to be considered, due to the negligible photon momentum.
The upper valence band can be represented by a pair of Jz = ± 32 hole states for HH

16
sub-band and a pair of Jz = ± 21 hole states for LH sub-band, and the conduction bandby a pair of Jz = ± 12 electron states, as depicted in Fig. 2.3(b). With right circularpolarized (σ + ) light, a transition can occur between Jz = − 32 HH state and Jz = − 12electron state, or between Jz = − 21 LH state and Jz = 12 electron state, with respective
3

1

1 1

transition dipole moment µ(− 2 ,− 2 ) and µ(− 2 , 2 ) . Similarly, transitions between Jz = 32
3 1

HH state and Jz = 21 electron state with dipole µ( 2 , 2 ) , and between Jz = 21 LH state11

and Jz = − 12 electron state with dipole µ( 2 ,− 2 ) happen with left circular-polarized (σ − )light. The transition dipole moments for HH and LH excitons excited with left andright circular-polarized light can be expressed as [33]:311µ(− 2 ,− 2 ) = µ0 σ + = √ µ0 (x̂ + iŷ) ,
2
1 1
1
1
µ(− 2 , 2 ) = √ µ0 σ + = √ µ0 (x̂ + iŷ) ,
3
6
3 1
1
µ( 2 , 2 ) = µ0 σ − = √ µ0 (x̂ − iŷ) ,
2
1
1
1
1
µ( 2 ,− 2 ) = √ µ0 σ − = √ µ0 (x̂ − iŷ) ,
3
6

(2.7)
(2.8)
(2.9)
(2.10)

where µ0 is the transition dipole modulus of HH excitons, and x̂ and ŷ are unit vectors.
The transition dipole of LH excitons is √13 times that of HH excitons. The two transitions with either left or right circular-polarized light occur independently with eachother, therefore the system in each case can be treated as a level scheme containing
two separate two-level systems. For excitation with linear-polarized light, which is acombination of left and right circular-polarized light with equal intensity, there are two
degenerate transitions between the HH state of Jz = ± 32 in the valence band and the
conduction band, and two degenerate transitions between the LH state of Jz = ± 12 andthe conduction band [4]. These transitions are coupled through the common electronstate.
The “residual” Coulomb force between excitons, which is the part not accounted
for in the formation of excitons, leads to the interactions of excitons. As a result of the17interactions, bound or unbound two-exciton states can be produced. When two excitonsof opposite electron spins are close enough, they can form a bound state with loweredsystem energy, which is known as the biexciton [34]. On the other hand, two excitonsof the same electron spins close to each other have a repulsive interaction, thus form
an unbound two-exciton. There are two types of HH excitons: (− 23 , − 12 ) and ( 32 , 12 ),and two types of LH excitons: (− 21 , 12 ) and ( 12 , − 12 ) [33], labeled in the form of (Jzv , Jzc )by the hole state Jzv and electron state Jzc involved in the transition. Ten different
kinds of two-exciton states can be formed: three from a pair of single HH excitons,
labeled as fH , three from a pair of single LH excitons, fL , and four from a mixtureof one HH exciton and one LH exciton, fM [33]. The two-exciton (− 23 , − 12 )+( 32 , 12 ) isa bound state by two HH excitons, whereas (− 12 , 12 )+( 12 , − 12 ) is a bound state by two
LH excitons, and (− 23 , − 21 )+(− 21 , 12 ) and ( 23 , 12 )+( 12 , − 21 ) are two bound states of mixed
excitons. Unbound states include four two-excitons formed by two identical excitons: apair of (− 32 , − 21 ) HH excitons, a pair of ( 32 , 12 ) HH excitons, a pair of (− 12 , 21 ) LH excitons,
and a pair of ( 12 , − 12 ) LH excitons, as well as two mixed pairs: (− 23 , − 12 )+( 12 , − 21 ) and( 32 , 21 )+(− 12 , 21 ).

2 XLH
XHH & XLH

Two-exciton
states

2 XHH

2 XHH

2 XHH

σSingle-exciton
states

2 XLH
XHH & XLH

2 XLH
XHH & XLH

σ+

σ+

σ-

XLH

XLH

XHH

XHH

σ-

σ+

GroundTotal spin:

-2

-1

0

1

2

Figure 2.4: Exciton energy level scheme with doubly-excited states included and opticaltransitions with σ + and σ − circular-polarized light.

18
In the exciton picture of energy level scheme depicted in Fig. 2.4, two-exciton
states are included as doubly-excited states in addition to the ground state and two
singly-excited HH and LH exciton states with spin Jz = −1 and 1 [4, 33]. A twoexciton state is presented at an energy level equal to the sum of the individual exciton
transition energy plus an energetic shift [4]. For bound two-exciton states, the doublyexcited states are shifted lower by an amount equal to the biexciton binding energy,
whereas the doubly-excited states of unbound two-excitons are shifted higher by thescattering energy. From left to right, the two-exciton states with total spin -2, labeled
(−2)

as fH

(0)

, formed by a pair of identical ( 23 , 12 ) HH excitons, fH with total spin 0 by two(2)

HH excitons (− 23 , − 21 )+( 32 , 21 ), and fH with total spin 2 by two identical HH excitons(− 23 , − 12 ) are the three bottom levels of the doubly-excited states in Fig. 2.4. Mixed(−2)

two-exciton states fM

with total spin -2 formed by HH exciton ( 32 , 12 ) and LH exciton(0)( 12 , − 21 ), fM with total spin 0 by HH exciton (− 23 , − 12 ) and LH exciton ( 21 , − 12 ), and(2)

by HH exciton ( 32 , 21 ) and LH exciton (− 12 , 12 ), and fM with total spin 2 by HH exciton(− 23 , − 12 ) and LH exciton (− 21 , 12 ) are the middle levels of two-exciton states. Two(−2)

exciton states fL

with total spin -2 formed by a pair of identical ( 12 , − 21 ) LH excitons,

(0)

(2)

fL with total spin 0 by two LH excitons (− 12 , 12 )+( 12 , − 12 ), and fL with total spin 2 by
two identical LH excitons (− 12 , 12 ) are the three top levels of the doubly-excited states.
Optical transitions between singly-excited states and doubly-excited states follow
the same selection rule as that for the transitions between ground state and singlyexcited states. The dipole-allowed transitions with σ + and σ − circular-polarized lightare depicted in Fig. 2.4, which gives a complete exciton picture of the transitions between
valence band and conduction band with different circular-polarized light, including exciton interactions. In practice, different simplified models based on this picture are often
used. For example, if the two-exciton states are totally neglected, an exciton picturewith a ground state and four excitonic states are presented, equivalent to the electron
picture in Fig. 2.3(b).

19

2.4

Relaxations of optical excitations
In this section, we discuss the time scales of relaxation processes following opticalexcitations in semiconductors. After a semiconductor is excited from thermodynamic
equilibrium by an ultrafast pulse, several stages of relaxation occur until it returns tothermodynamic equilibrium. These processes include coherent relaxation, non-thermal
relaxation, hot-carrier relaxation, and isothermal relaxation, in four temporally overlapped regimes [2].
The coherent regime starts right after an excitation and occurs on a time scaleof picoseconds. The excitation creates a well-defined phase relation within the excitedensemble and with the incident field. Unlike atomic and molecular systems, scattering
processes are so fast in semiconductors that the phase coherence is destroyed withinpicoseconds. Therefore, the study of coherence phenomena in semiconductors was deferred until the generation and application of ultrafast laser pulse were widely available.
Many fundamental quantum mechanical processes take place in the coherent regime
and have profound influences on the behaviors of excitons, electrons, and holes in the
scattering.
Following the coherent regime, the distribution of excitons or free electron-holepairs is usually non-thermal, i.e., the distribution function can not be characterized bya temperature. Non-thermal scattering processes of excitons and carriers can also beprobed by optical spectroscopy. The non-thermal relaxation lasts several picoseconds.
Carrier or exciton scattering can redistribute the energy within the carrier or exciton
system and eventually leads to the hot-carrier regime with a thermalized distribution
characterized by a temperature. This temperature is higher than the lattice temperature and can be different for subsystems of electrons, holes and excitons. The timeof thermalization with lattice depends strongly on many factors, such as the interactions with phonons, thus the hot-carrier regime varies largely on timescale from several

20
picoseconds to hundreds of picoseconds. At the end of the hot-carrier regime, all carriers, phonons and excitons are in an equilibrium that can be described by the same
temperature as that of the lattice. In the isothermal relaxation, the excess electrons
and holes, or excitons recombine either radiatively or non-radiatively and return thesemiconductor back to thermodynamic equilibrium.
In this work, we are focused on the coherent spectroscopic properties of semiconductors with ultrafast optical excitations. The third-order polarization P (3) created by
the electromagnetic field of the excitation pulse is initially in phase with the field. The
decay of phase coherence happens immediately after the excitation as the result of coupling to the “bath”, in the Markovian limit. The characteristic time of phase relaxation,
T2 , is called the decoherence time, also known as the dephasing time. Processes that
destroy the phase coherence of the polarization include scattering with phonons, elastic
and inelastic exciton-exciton or exciton-electron scattering, as well as scattering at impurities and crystal defects (including interface roughness and alloy fluctuations) [35].
The coherence is lost ultimately in the recombination process, which can be describedby an exponential decay of the population with a lifetime T1 . Since the decoherence timeT2 describes the amplitude decay of polarization, we can derive a relation of T2 6 2T1 ,which gives the upper limit of T2 as 2T1 , in the case that no other phase destroying
process but recombination happens. In general, T2 is much smaller than 2T1 due tothe various scattering processes in semiconductors. To characterize these scattering
processes but excluding the recombination, a “pure” dephasing time T20 can be definedwith [35]:
1
1
1
=
+ 0 ,
T2
2T1 T2(2.11)

where T20 is usually small for the fast scattering processes.
Linewidth broadening of the exciton resonance is common in semiconductor bulk
and quantum wells. In the case of homogeneous broadening, all oscillators in the en-21semble have exactly the same resonant frequency. Therefore the polarization of every
oscillator and the ensemble decays at the same rate T2 . The linewidth of the Lorentzian
resonance (homogeneous linewidth) is Γh = ~/T2 , in energy units. In contrast, every
oscillator in an inhomogeneous broadened system has a slightly different eigenfrequency,
which follows a distribution within an interval of δω centered at the resonant frequency
ω0 . The polarization decay of the ensemble is faster than the decoherence of individual
oscillator. All oscillators in the ensemble are in-phase when it is excited by an optical field. A phase distribution between oscillators develops with time as the result ofdifferent eigenfrequencies, causing the cancellation of polarizations from individual oscillators. The dephasing of the ensemble can be characterized roughly by a time constantT2∗ ' 1/δω, which is shorter than the decoherence time T2 of individual oscillators.
The amplitude of the emitted field from an ensemble of inhomogeneously broadened
oscillators decays with T2∗ . Inhomogeneous broadening dominates in semiconductor alloys or quantum wells at low temperatures, where disorder due to composition or well
thickness fluctuations localizes excitons. In high quality bulk samples or at elevatedtemperatures, the homogeneous broadening overwhelms inhomogeneous broadening as
the result of inelastic scattering of free excitons with optic and acoustic phonons. In
nonlinear spectroscopic experiments such as transient FWM, it is possible to distinguish different mechanisms of broadening and obtain the decoherence time T2 , even in
the presence of inhomogeneous broadening.

2.5

Nonlinear optical spectroscopies of semiconductors
The study of the coherent dynamical response of semiconductors is only possible

with ultrafast spectroscopic techniques of picosecond or femtosecond time resolution.
Transient FWM and pump-probe experiments are the primary nonlinear spectroscopic
methods commonly used to probe the nonlinear properties of various systems. FWM

22measurements are sensitive to the coherent response, whereas pump-probe experiments
can study both coherent and population dynamics.
In a generalized form of transient FWM experiments, three successive incident
pulses of either different or the same frequency are used to excite a medium. The
first pulse, with a wavevector k1 , induces a polarization, i.e., a coherent superposition
between the ground and excited states in the medium. The second pulse of wavevector
k2 , arrives after a delay and converts the superposition into a population. The amplitudeof the population is maximal where the electric field of the second pulse interferesconstructively with the polarization created by the first pulse, and minimal where theyinterfere destructively, forming a dynamic amplitude grating with a spatial period ofk2 − k1 . The third pulse of wavevector k3 , arrives with a delay after k2 and is scatteredby the dynamic grating into the phase-matched direction ks = −k1 + k2 + k3 as the
FWM signal. The nonlinear FWM signal depends on the pulse ordering and interpulse
time delays. Dynamics of the excited system can be studied by varying pulse delays.
The time period between the first and second pulses reflects the coherent evolution ofpolarization generated by the first pulse, therefore it is called the evolution time τ . The
period between the second and third pulses corresponds to the population relaxation of
the excited states, usually named as the waiting time T . The time-domain behaviors
of FWM as a function of τ and T provide essential information on the dynamics ofpolarization and population relaxation. In a simplified configuration, FWM can begenerated with two pulses where the second pulse functions as both the second andthird pulses in a three-pulse experiment, thus the waiting time T is always zero.
In a simple form of pump-probe spectroscopy, the medium under investigation
is excited by a pump pulse and the induced changes are measured by another pulse,
the probe, with a delay relative to the pump. Linear responses of the probe, such as
absorption, reflectivity, Raman scattering, and luminescence are recorded and comparedto the case without a pump. The probe is usually much weaker than the pump to avoid

23

Laser
Delay

Up-conversion pulset
TR-FWM

Delay
Pulse 2

Lens

Lens

Nonlinear
Crystalk2
Pulse 3Sample
k3

Delay

k1

Pulse 1T

Slow
Detector

kFWM = – k1+ k2+ k3

τ

TI-FWM

Figure 2.5: Experimental setup of three-pulse FWM generation and time-integrated(TI) and time-resolved (TR) signal detection. The signal can be measured with a slow
detector for TI FWM or up-converted with a reference pulse in a nonlinear crystal for
TR FWM.

introducing additional nonlinearities.
A generic FWM experimental configuration is illustrated in Fig. 2.5, where thenonlinear signal is generated by three pulses with adjustable delays τ and T . The FWM
signal is commonly detected by a slow detector for the integral of signal intensity withvarying delay τ , called time-integrated (TI) FWM signal. TI FWM can be expressedas:Z +∞S TI (τ ) ∝|P (3) (t, τ )|2 dt ,(2.12)

0where t is the signal emission time, and P (3) (t, τ ) is the macroscopic third-order polarization. The signal can also be measured by up-conversion with a reference in anonlinear crystal to obtain the time-resolved (TR) FWM as a function of the real timet. TR FWM has the form:
Z +∞S TR (t, τ ) ∝|P (3) (t0 , τ )|2 |E(t − t0 )|2 dt0 ,(2.13)−∞

where E(t) is the electric field of the reference. For dilute atomic systems, the temporal

24
behavior of TI-FWM as a function of delay τ is same as that of TR FWM as a functionof the real time t. The signal spectrum can also be measured to obtain the spectrallyresolved (SR) FWM:Z +∞S SR (ωt , τ ) ∝|P (3) (t, τ )|2 eiωt t dt ,(2.14)−∞

where ωt is the emission frequency. Moreover, it is often desirable to obtain the full
information of the emitted signal including phase, for a complete characterization of thenonlinear signal. This can be accomplished by a technique known as Fourier transform
spectral interferometry [36]. The signal to be measured is collinearly combined with areference pulse, whose spectrum is well-characterized and overlapped with that of thesignal, to produce a spectral interferogram. Fourier transform analysis of the interferogram allows the determination of signal amplitude and phase in temporal and spectral
domains. This heterodyne detection technique is commonly used for the phase characterization of nonlinear signals. In Chapter 4, we will discuss the details of heterodyne
detection of the FWM signal implemented for experiments of two-dimensional Fouriertransform spectroscopy.
The exciton dynamics and coupling in semiconductor bulk and nanostructure materials have been extensively studied by nonlinear optical spectroscopic tools, primarily
FWM experiments. It has been revealed that the interactions among elementary opticalexcitations (excitons and electron-hole pairs) have profound influences on the coherent
nonlinear optical response. In the following section, we review the studies of exciton
many-body interactions in semiconductors.

2.6

Many-body interactions in semiconductors
The many-body problem in condensed-matter systems is a fundamental topic

with longstanding and continuing interest. Optically excited semiconductors provide

25
a convenient system for the study of many-body effects [26]. In the case of diluteatomic systems, the coherent phenomena are well understood since only independent
atoms need to be considered. A simple picture such as a two-level scheme is usuallyadequate to describe the system. In materials such as semiconductors, many-bodyinteractions make dramatic differences from the atomic limit and often dominate thecarrier behaviors. It is essential to study these effects for a better understanding of thefundamental quantum mechanical processes and the influence on the performance of
optoelectronic devices.
The manifestation of many-body interactions in the nonlinear response of semiconductors has been revealed by many experimental signatures. Exciton dephasing inquantum wells was studied by TI FWM experiments [37, 38] and interpreted on thebasis of a two-level model developed by Yajima and Taira [39]. The appearance of asignal for “negative” delay in a two-pulse TI FWM experiment [40] is the first demon-

(a)

(b)

Figure 2.6: (a) A strong signal for negative time delay in two-pulse TI FWM. Reproduced from K. Leo et al., Phys. Rev. Lett. 65, 1340-1343 (1990); (b) Delayed FWM
peaks in two-pulse TR FWM. Reproduced from D.-S. Kim et al., Phys. Rev. Lett. 69,
2725-2728 (1992).

26
stration that the nonlinear response of semiconductors is completely different from thatof non-interacting two-level systems. As depicted in Fig. 2.6(a), TI FWM from a highquality GaAs quantum well sample was measured at different lattice temperatures. The
TI FWM signal in the phase-matched direction 2k2 − k1 was displayed as a functionof pulse delay τ , which is considered positive (negative) when pulse k1 precedes (follows) pulse k2 . A signal on a time scale much longer than the laser pulse width wasobserved for negative time delay (τ < 0), with a time constant for the rise equal to halfof that for the decay. This surprising result is in qualitative contrast to the expectation of no signal for negative delay from a non-interacting two-level system. Negative
delay signals were also observed in experiments on dense systems including moleculariodine [41] and atomic potassium [42]. Further surprising results were found in TR
FWM experiments [43, 44]. As shown in Fig. 2.6(b), the peak of a TR signal appears
at a delayed time that is extremely long (as much as 10 times the pulse width). The
delay decreases quickly with increasing temperature and shows no dependence on pulse
delay τ , in dramatic contrast to the photon echo behavior of two-level atomic systems
with inhomogeneous broadening. The asymmetric temporal evolution and the fact thatthe signal delay decreases with temperature indicate the signal is not a photon echo.
Detailed analysis shows the delayed signal is from the diffraction of the polarization,rather than the diffraction of the field. The TI signal for negative time delay and thedelayed peak in TR signal demonstrate that the nonlinear response of semiconductors
differs completely from that of non-interacting two-level systems.
These behaviors were explained in terms of a local field effect resulted from
Coulomb interactions in semiconductors [3]. Excitons can sense not only the electricfield of the incident light, but also the field of all other excitons in the semiconductor,
thus the effective field needs to be corrected to reflect the change by excitation. In a
two-pulse experiment on a non-interacting two-level system, the FWM signal emitted
in the 2k2 − k1 direction is produced by an interaction between the polarization created27
by the first pulse, k1 , and the field of the second pulse, k2 , thus resulting in no signal
when they do not overlap at τ < 0. In semiconductors, however, polarizations createdby two pulses can interact with each other due to the interaction between incident fieldand excitons. Since polarizations persist for a duration of the dephasing time after theelectric field, they can overlap at a negative τ to produce a FWM signal. In addition,
the diffraction of polarization leads to a TR FWM signal that increases with time andpeaks at a delay determined by the dephasing time. That is why the signal in TR FWM
experiments presents a delayed peak that is not a photon echo.
The dephasing time of excitons measured by TI FWM experiments is influenced
by many-body interactions. The homogeneous linewidth displays a linear dependence
on exciton or free carrier densities created by pre-excitation [45]. The broadening by
exciton-free carrier interaction is about eight times bigger than that by exciton-exciton
interaction. Also the electron-exciton scattering is more efficient than hole-exciton scattering. Exciton interactions are stronger in quantum wells due to the reduced screening
in two dimensions [45]. The differential transmission spectrum at a relatively low exciton density is characterized primarily by broadening of the exciton resonance resulting
from exciton-exciton interaction [5]. The excitation-induced dephasing (EID), i.e., thedensity dependence of T2 , has a strong contribution to TI FWM signal excited bycolinear-polarized light but no contribution in cross-linear-polarized excitation. The dephasing rate depends linearly on density at low densities and saturates at higher. EID
allows the diffraction of polarization [46], therefore it is also a possible reason for thedelayed signal in TR FWM.
A strong biexcitonic contribution to the TR FWM signal from GaAs quantum
wells was observed [47].

Biexciton effects are closely related to problems such aspolarization-dependent properties of FWM [4], quantum beat behaviors, and TI FWM
signal at negative delay, in experiments including grating-induced biexcitonic FWM signals, biexcitonic quantum beats, and two-photon coherence. Excitation-induced shift

28(EIS) was observed in SR FWM and confirmed with spectrally-resolved differential
transmission [6]. EIS can result in a signal comparable to or stronger than that arisingfrom saturation, local fields, or EID in semiconductors.
Coupling between optically induced excitations was also addressed. Beating in
the FWM signal was observed as a result of the interference of excitons with slightly
different quantum confinement energy due to well width fluctuations [48]. To distinguish whether the beats from disordered quantum wells were quantum beats or electromagnetic interference, TR FWM [7] and SR FWM [49] experiments were performed.
Interference between the exciton resonance and unbound electron-hole pairs was also
observed [50]. The coupling of excitons and continuum states by many-body interactions was discovered in partially-non-degenerate FWM [51] and three-pulse FWM
measurements [52, 53]. These results explained the earlier observation that the FWM
signal from excitons decays anomalously fast when the excitation spectrum overlapped
continuum states [54].
These effects all resulted from the many-body interactions among excitons andcarriers in semiconductors. The theoretical description of these phenomena requires asophisticated model rather than an independent two-level scheme of atomic systems.
Phenomenological terms for local field correction, EID, EIS, and biexciton formation
are added in the optical Bloch equations of the two-level model to describe the experimental observations [6]. However, there is an intrinsic ambiguity regarding the
microscopic mechanism of many-body interactions underlying the nonlinear response.
The manifestations of local field, EID, EIS, and biexciton formation in FWM signal aresimilar, making it difficult to determine the microscopic origin of the interactions. The
ambiguities come from the fact that TI or TR FWM experiments only measure the intensity of the signal, thereby discarding the essential information encoded in the phase.
Furthermore, conventional FWM techniques have other limitations. Many electronicresonances could be excited simultaneously by broadband ultrafast pulses, leading to a

29complicated oscillatory temporal evolution. It is difficult to distinguish whether such oscillations arise from polarization interference of uncoupled oscillators or quantum beatsof coupled resonances. In addition, many quantum mechanical pathways with various
relaxation times contribute to the FWM signal. It is hard to separate different dynamical information from these mixed contributions. Two-dimensional Fourier transform
spectroscopy (2D FTS), a highly enhanced FWM technique that tracks coherent phaseevolution, could resolve the aforementioned problems and provide microscopic information about the interactions [55].

Chapter 3

Principles of Optical Two-Dimensional Fourier Transform
Spectroscopy

Optical two-dimensional (2D) Fourier transform spectroscopy (FTS) originatesconceptually from the revolutionary development of multidimensional nuclear magnetic
resonance (NMR) in the 1970s [8]. Compared to traditional one-dimensional or nonFourier transform two-dimensional spectroscopic approaches, 2D FTS provides many
advantageous features by coherently tracking phase evolution and correlating it in two
frequency dimensions. Congested spectra are spread out into two dimensions to allow
isolation of coherent pathways and identification of resonant couplings. Furthermore,
the preservation of phase information in 2D FTS enables the separation of real andimaginary part spectra, making it possible to improve resolution of lineshapes andprovide microscopic information unavailable in magnitude measurements. These powerful features make optical 2D FTS a very promising tool for probing the structureand dynamics of complex systems including liquids [9, 11, 15], chemical and biologicalmolecules [10, 12, 13, 14], and semiconductors [24, 56, 55, 57]. The experimental andtheoretical progress in 2D infrared (IR) and optical spectroscopy is presented in a recent
perspective by Hochstrasser [58], along with the challenges that can be addressed bythese advances.
In this chapter, the development of IR and optical 2D FTS from multidimensional NMR and the applications for vibrational and electronic dynamics of molecular

31
systems are first reviewed in Section 3.1. In the following section, the nonlinear response
function for optical excitation and the third-order polarization are discussed. Doublesided Feynman diagrams are introduced in Section 3.3 as a diagrammatic description ofquantum mechanical coherent pathways in Liouville space that contribute to the nonlinear polarization. The principles of 2D FTS are presented in Section 3.4, followed bya discussion of the advantageous features that 2D technique offers in Section 3.5.

3.1

Development of IR and optical 2D FTS
The extension of Fourier transform spectroscopy into multidimensions revolution-ized nuclear magnetic resonance spectroscopy ultimately by producing fundamentallynew types of spectra [8]. The first implementation of the Fourier transform approachto 2D magnetic resonance was demonstrated by the Ernst group in 1976 [59]. It wassoon recognized that 2D NMR spectra contain valuable molecular phase information notavailable in double resonance experiments [60]. In a 2D NMR spectrum that measures
the immediate response of excited spins, spin coupling between nuclei connected withchemical bonds provides information on bond lengths and bond angles, while spin fliptransfer can be used to determine distances between nuclei without bond connections.
In vibrational 2D NMR, cross peaks reveal the connectivity of adjacent bonds while thetime dependence of vibrational relaxation could map out longer range structures.
The radio-frequency 2D NMR was rapidly extended to 2D FT microwave spectroscopy [61, 62], 2D FT Raman spectra of magnetic sublevels in atomic vapors [63, 64],
and 2D IR spectroscopy [65, 66, 67]. In early 2D IR spectra, an outer product wasused to display correlation among a series of perturbed 1D optical spectra [66], without
information on spectral relaxation. The second-harmonic phase-matching map of a nonlinear crystal was measured in two dimensions of the sum and difference frequencies [68].
Tanimura and Mukamel proposed femtosecond vibrational 2D Raman spectroscopy [9],32which was followed by experimental implementations [11, 12, 69]. However, it was foundthat cascaded third-order processes were contributing to the the fifth-order response andmasking the real fifth-order process when Raman excitation was used [12]. Doubly vibrationally enhanced FWM was also used to generate 2D vibrational spectra [70]. Direct
excitation in IR was then utilized to obtain better results on vibrations [13, 14]. Similar to NMR, structural information about molecules can be obtained by studying thecoupling between the IR active vibrational modes. The anharmonic nuclear potential
and correlated fluctuations in the transition energies of two coupled molecular vibrations were studied by 2D IR spectroscopy [71, 72]. The absorptive correlation spectrum
was obtained by summing complementary spectra of two vibrational coherences that
oscillate with conjugate frequencies in the initial evolution time period [73].
2D electronic spectroscopy was introduced and investigated by Jonas and coworkers [10, 74, 75, 76, 77]. 2D electronic spectra of the dye IR144 in methanol were
measured at shorter wavelengths. Separate real absorptive and imaginary dispersive
spectra were obtained and compared to theoretical predictions [10, 76]. Electronic
couplings in the molecular complex of photosynthetic protein were also studied by 2D
FTS in the visible range [78, 79, 80]. The development of vibrational and electronic 2D
FTS was reviewed by Mukamel et al. [81, 82, 83, 84] and Jonas [85].
Dynamic hole burning was used in early 2D spectroscopic studies of vibrationaltransitions [86, 87]. A narrowband mid-IR pulse was scanned across a protein absorption spectrum and the intensity change of the protein carbonyl stretch band (amide Iband) showed up as cross peaks in a 2D representation, indicating coupled vibrationalmodes. Then femtosecond-pulse 2D IR spectroscopy was implemented with better time
resolution using heterodyne detection of photon echoes [13]. Fourier transform spectralinterferometry [68, 88, 89, 90], based on heterodyne detection, is utilized in most 2D
FTS experiments to obtain full information including amplitude and phase of the nonlinear signal. Time-domain heterodyne techniques for full signal characterization have33
been discussed [91, 92, 93], however they are less widely used since both time dimensions
have to be scanned.
The power of 2D FTS has not been extended to the studies of carrier dynamics and many-body effects in semiconductors until recently, although there were someattempts of two-dimensional spectroscopic approaches in primitive forms. To distinguish whether the beats from disordered quantum wells with a bimodal inhomogeneous
distribution were quantum beats or electromagnetic interference, a technique based ontime resolving the FWM was applied, in which the time-resolved signal was displayed
as a function of the delay between excitation pulses to construct a two-dimensional
plot [7, 94, 95]. A similar method based on spectrally resolving the FWM signal was
also developed [49]. Based on a model that the temporal positions of beat peaks forquantum beats and electromagnetic interference have different dependence on the pulse
delay, a conclusion was reached that the beats were truly quantum beats, meaning the
spatially separated exciton transitions were coupled [96]. This result was interpreted
by a calculation that includes both disorder and many-body effects [97]. Another formof non-Fourier transform two-dimensional technique is coherent excitation spectroscopy(CES) [98], an extension to partially-degenerate FWM [51, 99]. The FWM signal is generated with a first pulse of narrow bandwidth and a second pulse of broad bandwidthand spectrally resolved at different frequencies of the first pulse in two dimensions. CES
suggested that spatially localized excitons in disordered quantum wells are not coupled
to each other, in contradiction to the earlier conclusion. Two-dimensional luminescence
excitation spectra were developed for single dot spectroscopy, which displayed coupling
between resonances due to incoherent energy migration in naturally occurring quantum
dots in thin quantum wells [100, 101, 102, 103, 104]. As in the context of multidimensional NMR, “two-dimensional Fourier transform spectroscopy” specifically means that
coherent phase evolution is preserved in both dimensions so a 2D Fourier transformation
is feasible. The spectroscopic techniques mentioned in this paragraph simply display a34stack of data in two dimensions, therefore they are not considered as 2D FTS.
Although optical 2D FTS shares many similarities with 2D NMR in concepts,
there are fundamental differences between them [85]. First, the ultrafast pulses used in
optical 2D FTS contain far fewer cycles as compared to the radio-frequency pulses in
NMR, meaning molecules are frozen in place in the femtosecond pulse duration rather
than motionally averaged in NMR. Therefore ultrafast 2D FTS provides multidimensional snapshots of the fast chemical dynamics on extremely short timescale. In addition,
the wavelength of optical pulses is significantly smaller than the sample dimensions (thesize of coherent area in a sample is typically ∼ 100λ for near-IR excitations). In this case,the nonlinear electric field is emitted in specific phase-matched directions if the excitation pulses have a non-collinear geometry (see Section 3.2). Time-gating is commonlyused in collinearly excited NMR to separate the signal from strong excitation pulses.
In femtosecond experiments, however, the signal is often emitted temporally together
with excitation pulses, thus it cannot be separated by time-gating. If non-collinear excitation geometry is used, the signal can be detected background-free in phase-matcheddirections with no need for time-gating.
Furthermore, non-collinear excitation enables the spatial isolation of coherent
pathways. For a particular phase-matched direction, only the corresponding parts of
the nonlinear response are in resonance with excitation, thus the contributions of coherent pathways can be isolated in different directions. In NMR, phase cycling is used toobtain the signal from desired coherent pathways. The phase of excitation pulses needs
to be systematically varied to generate a nonlinear signal corresponding to a specificcoherence pathway from a collection of measured signals [8]. A 2D absorption spectrum
can be obtained by superposing equally weighted signals excited by two pulse sequenceswith oppositely signed coherence orders during the evolution period. An optical 2D
experiment with a collinear geometry, in which signals from all coherence pathways are
emitted in the same direction, was performed as analog of the simplest 2D NMR to ex-35amine the possible extension of 2D NMR into the optical regime [20]. In non-collinear
optical 2D FTS experiments, a specific phase-matched direction and temporal orderingof excitation pulses can be selected to isolate the coherent pathways of interest. The isolation of coherent pathways in 2D FTS is to be discussed in Section 3.5. A combinationof non-collinear pulse geometry for coherence pathway selection with the phase control
ability in current pulse-shaping technologies would allow optical 2D spectroscopy tohighlight more specific properties of the sample being excited.

3.2

Nonlinear response function
Microscopic information about electronic couplings and dynamics in a system isrepresented by the multidimensional nonlinear response function, when a sequence ofn excitation pulses is applied to the system to generate and maintain coherences andpopulations of the electronic states [81]. In a three-pulse experiment, excitation pulses
arrive at times τ1 , τ2 , and τ3 respectively, separated by a time delay of t1 = τ2 − τ1between the first two pulses, and delay t2 = τ3 − τ2 between the second and third pulses,
as shown in Fig. 3.1. The nonlinear third-order signal generated by the interaction withall three pulses is measured at time t, after a delay t3 from the last excitation pulse.1
The electric field of each pulse in a three-pulse sequence has the form:i(kj ·r−ωj t)−i(kj ·r−ωj t)
Ej (r, t) = [ ε++ ε−]êj ,
j (t)ej (t)e(3.1)

for the j-th pulse (j=1, 2, or 3) with wavevector kj , carrier frequency ωj , and unit po+ ∗−larization vector êj . ε+j (t) (εj = (εj ) ) is the positive (negative) frequency componentof the slow-varying pulse envelope. The pulse sequence can be expressed as the sum of
three electric fields:
E(r, t) =3X

Ej (r, t − τj ) .

(3.2)

j=1
1

For convenience, interpulse time delays are denoted as t1 , t2 , and t3 in Section 3.2 and 3.3, rather
than τ , T , and t elsewhere in this thesis.

36

k1

k2

k3
FWM

τ1

t1

τ2

τ '1

t2

τ '2

τ3

t3

t

Time

τ '3

Figure 3.1: Excitation pulse sequence in a three-pulse nonlinear experiment. Pulses
arrive at times τ1 , τ2 , and τ3 respectively, separated by a time delay of t1 = τ2 − τ1between the first two pulses, and delay t2 = τ3 − τ2 between the second and third
pulses. Nonlinear third-order signal is measured at time t, after a delay t3 from the last
excitation pulse.

The polarization induced by three excitation pulses occurs at third-order in theapplied fields [82]:P (3) (r, τ1 , τ2 , τ3 ) =Z ∞Z ∞Z ∞00

R(3) (τ10 , τ20 , τ30 )E1 (r, τ10 − τ1 )E2 (r, τ20 − τ2 )0

×E3 (r, τ30 − τ3 ) dτ10 dτ20 dτ30 ,(3.3)

where R(3) is the third-order time-dependent response function of the system, and τi0(i = 1, 2, or 3) is the integration variable associated with pulse i. The n-th order response
function R(n) can be expressed as a combination of n + 1 order correlation functions of
the dipole moment operator [82]:R(1) (t1 ) =ih[ µ(t1 ), µ(0) ]i ,~iR(2) (t1 , t2 ) = ( )2 h[ µ(t1 + t2 ), [ µ(t1 ), µ(0) ] ]i ,~i
R(3) (t1 , t2 , t3 ) = ( )3 h[ µ(t1 + t2 + t3 ), [ µ(t1 + t2 ), [ µ(t1 ), µ(0) ] ] ]i ,
~

(3.4)
(3.5)
(3.6)

and so forth. The time evolution of the dipole moment operator µ(t) is given by thesystem Hamiltonian in the absence of applied fields. In general, a sum-over-states
expression of R(n) can be derived for an arbitrary multilevel system [81].

37
In the limit of impulsive excitation, where the pulse envelope is much shorter
than both the dynamic time scale and the delays between pulses, the time integrations
in Eqn. (3.3) can be evaluated by approximating the pulses as delta functions. Aftereach electric field is plugged in, the third-order polarization has the following form [33]:± ± i(±k1 ±k2 ±k3 )·r −i(±ω1 ±ω2 ±ω3 )t
P (3) (r, t, t1 , t2 , t3 ) = R(3) (t1 , t2 , t3 ) ε±e
1 ε2 ε3 e× ei(±ω1 ±ω2 ±ω3 )t3 ei(±ω1 ±ω2 )t2 e±iω1 t1 ,(3.7)

where t1 , t2 , and t3 are pulse time delays. There are several possible contributions to thepolarization associated with different wavevectors ks = ±k1 ±k2 ±k3 . Each contribution
produces a distinct signal field in the corresponding ks direction after the polarization isplugged into the Maxwell’s equations, provided the sample size is larger than the signal
wavelength. This requirement is usually satisfied in optical experiments, thus the emission of optical fields produced by the polarization has directionality. Phase-matchingconditions together with the frequency dependence of the refraction index of the system
may favor some of these directions and suppress others. For a particular direction of
signal wavevector or combination of wavevectors, only the corresponding parts of thenonlinear response function are in resonance with the fields and contribute significantly
to the polarization, resulting in a resonant signal dominated by these contributions.
Among the eight possible signal wavevectors in a three-pulse excitation, only fourof them are independent as ks and −ks correspond to complex conjugated contributions.
The possible wavevector of the polarization is one of the following: kI = −k1 + k2 + k3 ,kII = k1 − k2 + k3 , kIII = k1 + k2 − k3 , and kIV = k1 + k2 + k3 , labeled with
subscript I, II, III, or IV. Each wavevector also corresponds to an emission frequency
ωI = −ω1 + ω2 + ω3 , ωII = ω1 − ω2 + ω3 , ωIII = ω1 + ω2 − ω3 , and ωIV = ω1 + ω2 + ω3 ,respectively. Consequently, Eqn. (3.7) can be recast to show the polarization as a sum

38of the four contributions with wavevectors kI , kII , kIII , and kIV :P (3) (r, t, t1 , t2 , t3 ) =IVX

Ps (t1 , t2 , t3 )ei(ks ·r−ωs t) + c.c. ,

(3.8)

s=I

where the polarization term Ps (t1 , t2 , t3 ) has the following expression for s=I, II, III, or
IV:
(3)

(3.9)

(3)

(3.10)(3)(3.11)(3)(3.12)+ + i(−ω1 +ω2 +ω3 )t3 i(−ω1 +ω2 )t2 −iω1 t1
PI (t1 , t2 , t3 ) = RI (t1 , t2 , t3 ) ε−ee,
1 ε2 ε3 e− + i(ω1 −ω2 +ω3 )t3 i(ω1 −ω2 )t2 iω1 t1
PII (t1 , t2 , t3 ) = RII (t1 , t2 , t3 ) ε+ee,
1 ε2 ε3 e+ − i(ω1 +ω2 −ω3 )t3 i(ω1 +ω2 )t2 iω1 t1
PIII (t1 , t2 , t3 ) = RIII (t1 , t2 , t3 ) ε+ee,
1 ε2 ε3 e+ + i(ω1 +ω2 +ω3 )t3 i(ω1 +ω2 )t2 iω1 t1ee
PIV (t1 , t2 , t3 ) = RIV (t1 , t2 , t3 ) ε+.
1 ε2 ε3 e

The last term, PIV , corresponds to a highly oscillatory polarization at approximately
the third harmonic of the incident field, thus it is neglected within the rotating-wave
approximation (RWA) [33]. The third-order polarization associated with each direction
can be obtained in the form of a sum-over-states expression with the help of double-sided
Feynman diagrams.

3.3

Double-sided Feynman diagrams
In perturbation theory, the evolution of density matrix elements can be conve-

niently tracked by double-sided Feynman diagrams. With the capability of carrying
the complete information including the time ordering and the choice of frequencies andtheir signs, the double-sided Feynman diagram is a tool for diagrammatically describing nonlinear interactions of light with matter. Each diagram corresponds to a coherent
pathway in the Liouville space.
There are some basic rules for double-sided Feynman diagrams [81]. The density
matrix operator is represented by two vertical lines, where the left line stands for the ket
and the right line for the bra, with time increasing vertically upward. Each incident light39
field is represented by a wiggly line with an arrow and the interaction of a field with thesystem is indicated by a vertex of the wiggly line and the left or right vertical line. The
density matrix state is changed by the interaction and a label below (above) the vertex
gives the state before (after) the interaction. A wiggly line pointing to the right meansa contribution of Ej exp(ikj · r − iωj t) to the polarization, while a line pointing to theleft means the conjugate contribution of Ej∗ exp(−ikj · r + iωj t). A wiggly line coming
towards the vertical lines means absorption, thus taking the system to or from a coherent
superposition. Conversely, a line going away represents photon emission, sending thesystem from higher energy back to lower energy state. The overall wavevector of adiagram is the sum of those of the individual fields with appropriate signs. The sign ofa diagram is (−1)n where n is the number of vertexes on the right vertical line (bra),since each interaction acting from the right in a commutator introduces a minus sign.
Each order in the perturbation of interaction is described by a basic diagram containing
one vertex on the left or the right vertical line, and n such diagrams are concatenatedvertically in time order to produce a diagram for the n-th order perturbation theory.
The n-th order perturbation usually consists of many double-sided Feynman diagramswith different wavevectors, frequencies and time ordering of the incident fields. From
the double-sided diagrams one can directly write down the mathematical form of thenonlinear response function with a few rules.
To illustrate how double-sided Feynman diagrams track the density matrix elements, we discuss a system containing a ground state |gi, singly-excited states |ei and|e0 i, and doubly-excited states |f i, just as a simplified model for the semiconductor
quantum well system with excited HH excitons, LH excitons and biexcitons in Chapter
2. As shown in Fig. 3.2, the double-sided Feynman diagrams representing the third-order
perturbation in a three-pulse excitation of such a level scheme (depicted in Fig. 3.2(d))
can be grouped by the phase-matched directions [82]. The diagrams for direction kI , kII ,and kIII are listed in box (a), (b), and (c), respectively. The diagrams for the direction

40(a) kI = -k1 + k2 + k3
g

g

g

k4

(1)

e'

g

k4

t3

e

k3

k2

e'

(3)

f

t3

e

t2

k3
e'

t1

-k1

g

k3

t2

e

k2

t1
g

t3

(2)

t2

e'

k4

g

g

g

g

t1
k2

-k1

g

g

e'

e'

-k1

(b) kII = k1 - k2 + k3
g

g

k4

k4

k4

t3

(4)

k3

t2

t3

g

e'

g

(7)
-k3

e'

g

g

g

f

t2

e

t1

k2
k1

g

e'

e'

f

k4

(8)

μ ef

t3
f

t2

e

t1

k2
g

g

(d)

g

t3

-k2

t1

k1

g

(c) kIII = k1 + k2 - k3
k4

t2
e

k1
g

t3

f

k3

t1

e

k1

(6)

t2
-k2

-k2

t1

e

e'
k3

(5)

k1

g

e

-k3

μ ge
g

g

Figure 3.2: Double-sided Feynman diagrams representing contributions to the phasematched directions kI = −k1 +k2 +k3 (a), kII = k1 −k2 +k3 (b), and kIII = k1 +k2 −k3(c) for a system containing a ground state, excited states |ei, and doubly-excited states|f i (d).

kIV are not presented since the third-order polarization corresponding to kIV vanishes
within RWA.
For phase-matched direction kI , the three double-sided Feynman diagrams contributing to the polarization are listed in box (a) of Fig. 3.2. Labeled as (1), (2), and41(3), these diagrams describe the respective process of ground-state bleaching (GSB),
excited-state emission (ESE), and excited-state absorption (ESA) [82]. In diagram (1),for example, the conjugated field k1 comes first and excites the system to state |e0 i.
The wiggly line of k1 is pointing up to the vertical line from the right hand side since itmakes a conjugate contribution of E1∗ exp(−ik1 · r + iω1 t) to the polarization. After adelay t1 , the second field k2 brings the system back to state |gi, as the k2 line is goingaway from the vertical line. The third field k3 arrives with a delay t2 after the second
field and excites the system again to state |ei. In the end, the system returns to groundstate by emitting a signal along kI at delay t3 after the third pulse. Similarly, diagram(2) and (3) describe other possible transitions with the same pulse time ordering forthe emission in direction kI . For emission along direction kII , there are also three contributions represented by diagram (4), (5), and (6) in box (b) of Fig. 3.2, where pulse
k2 has the conjugated field. Furthermore, two diagrams (7) and (8) in box (c) are thecontributions to the emission in direction kIII . Note that in the case of kI or kII , there
is only one diagram involving doubly-excited stated |f i, while in the case of kIII , both
diagrams include |f i.
Once all the possible Feynman diagrams for a particular phase-matched direction
are presented, it is straightforward to obtain the sum-over-states expression of the nonlinear response function from all the diagrams with an appropriate set of translation
(3)

(3)

(3)

rules. The third-order response function RI , RII , and RIII can be written as a sumof all contributions presented in Fig. 3.2(a), (b), and (c), respectively [33]:

42

i
(3)
RI (t1 , t2 , t3 ) = ( )3 [~X(µge0 · ê1 )(µe0 g · ê2 )(µeg · ê3 )(µge · ê4 )e(iωe0 g −Γe0 g )t1 e−Γgg t2 e−(iωeg +Γeg )t3
e,e0

+

X

(µge0 · ê1 )(µeg · ê2 )(µe0 g · ê3 )(µge · ê4 )e(iωe0 g −Γe0 g )t1 e−(iωee0 +Γee0 )t2 e−(iωeg +Γeg )t3

e,e0

−

X

(µge0 · ê1 )(µeg · ê2 )(µef · ê3 )(µe0 f · ê4 )e(iωe0 g −Γe0 g )t1 e−(iωee0 +Γee0 )t2 e−(iωf e0 +Γf e0 )t3 ] ,e,e0 , f(3.13)i(3)
RII (t1 , t2 , t3 ) = ( )3 [~X(µeg · ê1 )(µge · ê2 )(µe0 g · ê3 )(µge0 · ê4 )e−(iωeg +Γeg )t1 e−Γgg t2 e−(iωe0 g +Γe0 g )t3
e,e0

+

X

(µeg · ê1 )(µge0 · ê2 )(µe0 g · ê3 )(µge · ê4 )e−(iωeg +Γeg )t1 e−(iωee0 +Γee0 )t2 e−(iωeg +Γeg )t3

e,e0

−

X

(µeg · ê1 )(µge0 · ê2 )(µf e · ê3 )(µe0 f · ê4 )e−(iωeg +Γeg )t1 e−(iωee0 +Γee0 )t2 e−(iωf e0 +Γf e0 )t3 ] ,e,e0 , f(3.14)

i(3)
RIII (t1 , t2 , t3 ) = ( )3 [~X(µeg · ê1 )(µf e · ê2 )(µe0 f · ê3 )(µge0 · ê4 )e−(iωeg +Γeg )t1 e−(iωf g +Γf g )t2 e−(iωe0 g +Γe0 g )t3
e,e0 , f−X(µeg · ê1 )(µf e · ê2 )(µge0 · ê3 )(µe0 f · ê4 )e−(iωeg +Γeg )t1 e−(iωf g +Γf g )t2 e−(iωf e0 +Γf e0 )t3 ] ,e,e0 , f(3.15)
where µij , ωij , and Γij are the dipole moment, transition frequency and dephasing rateof the i → j (i, j = g, e, e0 , or f ) transition respectively. In Eqns. (3.13), (3.13), and(3.15), each line on the right-hand side of the equation corresponds to a double-sided
Feynman diagram in Fig. 3.2(a), (b), and (c) respectively.

43

3.4

Two-dimensional Fourier transform spectroscopy
The 2D FTS technique is based on a highly enhanced version of degenerate FWM

experiments, in which the coherent phase evolution of the emitted signal is tracked withthe delay of excitation pulses in sub-optical-cycle precision and the signal phase is obtained with heterodyne detection. The challenges of implementing 2D FTS are discussed
in detail in the next chapter. A typical box geometry used in 2D FTS experiments isshown in Fig. 3.3(a), where pulses are labeled by their geometric positions rather thantime ordering. Three excitation beams of same frequency propagate with wavevector
ka , kb , and kc arranged on three corners of a square before they are focused on a sample. The FWM electric field induced by the third-order polarization is emitted in thephase-matched direction ks = −ka + kb + kc . Fig. 3.3(c) shows the FWM and excitation beam spots on the output lens viewed from the sample side. In this square box
geometry configuration, the length of the polarization wavevector matches that of the

(a)

Focal
Lens

Pulse bkb

Pulse a

Focal
Lens

FWM:

ks= - ka+ kb+ kc

ka

Pulse cSample

kc

τ

T

(b)

ka

kb

Evolutionkc
Waiting(c)

Emission
FW M

τa

τ

τb

T

τc

t

T im e

kc

ks

ka

kb

Output Lensview

Figure 3.3: (a) The box geometry of excitation beams ka , kb , and kc . FWM is emitted
in the phase-matched direction ks = −ka + kb + kc ; (b) Time ordering of excitation
pulses, where the evolution time, waiting time, and emission time are labeled as τ , T ,and t, respectively; (c) FWM and excitation beam spots on the output lens, viewed
from the sample side.

44
incident fields, thus phase mismatch in the sample is minimized [76]. The time orderingof pulses is depicted in Fig. 3.3(b), where the delay t1 between the first and second
pulses is the evolution time τ , and the delay t2 between the second and third pulse is
the waiting time T (or waiting time in NMR). The delay between signal and the third
pulse corresponds to the signal emission time t, provided the origin of time is set to thecenter of the third pulse.
The FWM signal is measured with frequency-domain heterodyne detection to
obtain both spectral amplitude and phase. The electric field of the emitted signal isdetermined by the third-order polarization (in Gaussian units) [76]:E(τ, T, ωt ) =2π li ωt P (3) (τ, T, ωt ) ,n(ωt ) c(3.16)

which is a function of the evolution time τ , waiting time T , and emission frequency ωt .
Here l is the sample thickness, n(ωt ) is the refraction index of the sample, and c is the
speed of light in vacuum. Eqn. (3.16) is valid only when absorption and propagation
effects in the sample can be neglected. The pulse propagation effects in thick samples
can be treated with a suitable multidimensional frequency filter function [85].
In the standard technique of 2D FTS, the electric field of the emitted FWM signal
is presented in two frequency dimensions after Fourier transforms with respect to twotime variables: the evolution time τ and emission time t. Since the transform withrespect to t is implicit in frequency-domain heterodyne detection, a Fourier transformwith respect to τ produces the 2D spectrum:
Z +∞S(ωτ , T, ωt ) =E(τ, T, ωt ) eiωτ τ dτ ,(3.17)−∞

at a fixed value of T . The new dimension ωτ is called the absorption frequency as itcorresponds to the absorption process in excitation. The optical 2D FTS defined inthis manner is closest to the Correlated Spectroscopy (COSY) and Nuclear Overhauser
Effects Spectroscopy (NOESY) among many variations of multidimensional NMR spectroscopy [8]. In both COSY and NOESY, the “direct” frequency is produced in the45measurement of the emitted field and the “indirect” frequency is obtained from theFourier transform with respect to the scanning delay between the first two pulses. In
NOESY, the waiting time T between the second and third pulses is nonzero, allowing
population dynamics to occur. In contrast, there is no delay between the last two pulses(or a single pulse functions as two pulses) in COSY. Despite its frequency-domain appearance, 2D FTS is still considered as a time domain technique, as it relies completelyon the temporal control of the excitation pulse sequence [82].
A 2D Fourier transform spectrum SI (ωτ , T, ωt ) correlates the absorption frequencyωτ and the emission frequency ωt in a two-dimensional map by monitoring the coherent
phase evolution of the third-order polarization during the evolution period τ and emission period t, separated by a waiting time T . By extending the FWM signal traditionallymeasured in one-dimension to two dimensions with independent frequency variables,
congested spectra are disentangled and couplings between resonances are identified. In
addition, contributions from various quantum mechanical coherent pathways can beisolated and associated with peaks in 2D spectra. The power of 2D FTS comes fromthe fact that the phase evolution of the third-order polarization is coherently tracked
and correlated in the evolution and emission time periods, rather than from the useof Fourier transformation, which just facilitates the presentation of data in frequency
domain. In some early experiments on semiconductors, the complete FWM electric field
including phase was measured to obtain phase dynamics of the emitted signal [92, 105].
However, the crucial aspect of tracking signal phase as a function of excitation pulse
delay was not included. A closer analogy to 2D FTS is the coherent excitation spectroscopy (CES), which was developed and applied to the study of coupling between
disorder localized excitons [98].
To understand why coherent tracking of the signal phase is critical, we consider
the FWM signal that is emitted from a transition with frequency ωik due to excitationof a transition with frequency ωij by the first pulse. If the first pulse is the conjugated46
field, as in the case of kI experiments, the electric field of the signal can be written
as [56]:S(τ, t) = D(τ, T, t) µ2ij µ2ik ei(ωij τ −ωik t) ,(3.18)

where µij and µik are the dipole moments of the respective excitation and emission transitions. All the decay dynamics and any process between the second and third pulses
are lumped into the term D(τ, T, t). Eqn. (3.18) shows that phase evolution between thefirst two pulses results in a constant overall phase in the emitted field. Thus, by coherently tracking the phase of the signal as the delay between the excitation pulses is variedwith sub-cycle precision, one can determine the frequency of the excitation transition
simultaneously with that of the emission transition. The correlation between these two
frequencies can be conveniently displayed in a 2D map with a two-dimensional Fouriertransform that converts the time domain data into frequency domain. Coherently tracking signal phase with the time delay between excitation pulses and maintaining sub-cycle
precision of the delay is experimentally challenging. We achieved interferometric accuracy in delay scanning and stabilization in our 2D FTS experimental apparatus, which
will be discussed in detail in the next chapter.
The coherent phase evolution is directly related to the time ordering of excita-

Rephasing scan: τ ≡ τ b −τ a > 0

(a)

ka

kb

Non-rephasing scan: τ ≡ τ b −τ a < 0

(b)

kc

kb

ka

kc

FWM

τa

τ

τb

T

τc

t

Time

FWM

τb

τ

τa

T

τc

t

Time

Figure 3.4: (a) Excitation pulse sequence for a rephasing experiment, in which the
conjugated pulse a is scanned to earlier times while pulse b and c are fixed and separated
by waiting time T . τ is positive by definition; (b) Pulse sequence for a non-rephasing
experiment, in which pulse b is scanned to earlier times while pulse a and c are fixedand separated by waiting time T . τ is negative.

47
tion pulses. The phase evolution described by Eqn. (3.18) is actually produced by a“rephasing” pulse sequence in the kI pulse geometry. As shown in Fig. 3.4(a), the conjugated pulse a arrives first and creates a coherent superposition between the groundand excited state in the ensemble. Then the oscillators in the ensemble dephase withdifferent resonant frequencies in the evolution time period τ , provided the system is
inhomogeneously broadened. The third pulse introduces a conjugated phase evolution
in the emission time period t and causes the oscillators to reverse their phases, resulting
in a photon echo signal with a delay τ from the third pulse [106].2 Therefore we referto this time ordering as a “rephasing” excitation sequence. One advantage of the threepulse excitation is that the time ordering of the first two pulses can be interchanged.
If the pulse b comes earlier than the conjugated pulse a in the case of “non-rephasing”
pulse sequence shown in Fig. 3.4(b), the phase evolution in the evolution and emission
periods has the form of S(τ, t) = D(τ, T, t) µ2ij µ2ik e−i(ωij τ +ωik t) . There is no conjugationin phase evolution and thus a free polarization decay, instead of photon echo signal, is
produced in a system with inhomogeneous broadening. Delay τ is defined as the difference between the arriving times of pulse b and pulse a: τ = tb −ta , so τ is positive for therephasing pathway and negative for the non-rephasing pathway. As depicted in Fig. 3.4,
the conjugated pulse a is scanned in a rephasing experiment whereas pulse b and c are
fixed in time and separated by a waiting time T . The emitted FWM is measured whilepulse a is being scanned away from tb , the time-overlapping position with pulses b, toearlier times ta for a series of data with increasing τ . In a non-rephasing experiment,
however, the conjugated pulse a and pulse c are stationary as pulse b is scanned withnegatively increasing τ from ta to earlier times tb [24].
The sign of the absorption frequency ωτ is different for the rephasing and nonrephasing pathways. The emission frequency ωt is usually chosen to be positive, then ωτ2An enlightening illustration of spin echo can be found on the front cover of the November 1953
edition of P hysics T oday.

48
is negative in a rephasing experiment as a result of the conjugated phase evolutions in theevolution and emission time periods. The 2D spectra of the rephasing pathway appear in
the lower-right quadrant of the (ωτ , ωt ) plane, i.e., ωt > 0 and ωτ < 0. In a non-rephasing
experiment, ωτ and ωt have the same sign and the 2D spectra show up in the upperright quadrant. Comparison of spectra in rephasing and non-rephasing pathways gives
insight into the presence of inhomogeneity and allows isolation of different contributionsin the Liouville space [55, 73].
As discussed earlier in Section 3.1 of this chapter, one advantage of optical 2D
FTS over 2D NMR is that non-collinear excitation geometry can be used in optical 2D
experiments to allow the selection of coherent pathways from different phase-matcheddirections. Most 2D FTS experiments are performed in the photon echo geometry where
the FWM signal is emitted in direction kI , as shown in Fig. 3.3. The measured 2D
spectrum is correspondingly denoted as SI (ωτ , T, ωt ). Other spatial arrangements canalso be utilized to implement 2D spectroscopies that probe different coherent pathways
for additional information. For example, the SIII 2D spectra produced from FWM signal
emitted in the kIII direction are intrinsically sensitive to two-exciton correlations [107].
Two-exciton contributions are demonstrated in both frequency dimensions of the SIII
spectra, while they manifest themselves only along one frequency axis in other thirdorder nonlinear spectroscopies. Consequently, high two-exciton resolution unavailable inconventional 1D FWM and other 2D FTS experiments is achieved and the two-exciton
binding energy can be revealed even when it is smaller than the line broadening.

3.5

Advantages of 2D FTS
Being a new spectroscopic method that tracks coherent phase evolution and dis-

plays correlation in two frequency dimensions, 2D FTS presents many advantageous
features, such as identifying coupling of resonances with the appearance of cross peaks

49
and isolating quantum mechanical coherent pathways in Liouville space. These features
are discussed in the following subsections.

3.5.1

Identification of resonant coupling

In 2D spectra, resonances result in diagonal peaks and the coupling of resonances
is indicated by the appearance of off-diagonal peaks, allowing an intuitive way for theidentification of couplings. For a system with two independent transitions of frequency
ω1 and ω2 (two two-level systems) depicted in Fig. 3.5(a), there are two diagonal peaks at(ω1 , −ω1 ) and (ω2 , −ω2 ) in the corresponding 2D spectrum. A peak in the 2D spectrumis specified by the coordinates of emission frequency ωt and absorption frequency ωτ inthe form of (ωt , ωτ ). The rephasing pathway is shown here, therefore ωτ is negative andωt is positive. If the two transitions share the same ground state, as the “V” system
shown in Fig. 3.5(b), they are not independent any more. The coupling between the
two transitions is revealed directly in the 2D spectrum: two additional off-diagonal

(a)

(b)
ω1

(c)
ω1

ω2

− ω1

ω1

ω2

− ω1

ωτ

− ω1

ωτ
− ω2

ωτ
− ω2

ω1

ωt

ω2

ω2

− ω2

ω1

ωt

ω2

ω1

ωt

ω2

Figure 3.5: (a) A system of two independent two-level transitions and the corresponding
2D spectrum; (b) A 3-level system with shared ground state and the corresponding
2D spectrum; (c) A system of two independent two-level transitions with incoherent
population relaxation from higher exited state to lower state and the corresponding 2D
spectrum.

50
peaks appear at (ω1 , −ω2 ) and (ω2 , −ω1 ), besides two resonance peaks at (ω1 , −ω1 )and (ω2 , −ω2 ) on the diagonal. Therefore, a 2D spectrum can identify whether two
resonances are from two independent transitions (two two-level systems) or from two
coupled transitions (a three-level “V” system), while a one-dimensional spectrum suchas linear absorption or luminescence gives identical results for these two situations. It
is also possible to determine the coupling mechanism in 2D FTS. For example, the twotransitions in a two two-level system will not be independent if there is an incoherent
non-radiation relaxation happening from the higher excited state to the lower state, as
shown in Fig. 3.5(c). This coupling mechanism results in a 2D spectrum with only one
cross peak instead of two symmetric ones on both sides of the diagonal. The cross peak
appears at the lower-left corner with an absorption frequency (absolute value) greaterthan the emission frequency, indicating an incoherent population relaxation occurringfrom the higher excited state to the lower one. This incoherent process generally requires
a finite value of the waiting time T .
Furthermore, the interactions of two transitions in a two-two-level system or athree-level “V” system can be studied with 2D FTS, too. A realistic example is theinteractions between HH and LH excitons in quantum wells (as discussed in Chapter2), where a biexciton can form as the bound state of two interacting excitons, with atypical binding energy of 2∼4 meV. For the case of scattering between two excitons,
it can be treated as an unbound state. This system can be described equivalently as
a four-level system consisting of one ground state, two singly-excited states and one
doubly-excited state, where the doubly-excited state has an energy level equal to thesum of the transition energy of the two singly-excited states with a shift [4]. The doublyexcited state is shifted lower energetically by an amount equal to the biexciton binding
energy for bound two-exciton states or higher by the amount of the scattering energy for
unbound two-excitons. The interactions of transitions in such a system can be revealedby 2D spectra in a similar way as couplings. Only two diagonal peaks appear if the

51
two transitions are independent as in a two-two-level system; however, both peaks are
shifted along emission frequency axis, corresponding to the shift of the doubly-excitedstate. In the case of a three-level “V” system, two off-diagonal peaks also appear as
the result of coupling, and all four peaks are shifted along emission frequency axis in
the same way as in the case of a two-two-level system. We will find such a behavior in
the 2D spectrum excited with cross-linear-polarized light in Chapter 6. Moreover, theinteractions of HH and LH excitons with exciton continuum states (electron-hole pairs)
can also be reflected in 2D FTS. We will see examples of such spectra in experiments
and study the exciton interactions with the continuum numerically in Chapter 5.
Identifying resonance coupling is not a capability unique to 2D FTS. As discussed
in Section 3.1 of this chapter, some primitive two-dimensional non-Fourier-transform
techniques [7, 49] are also able to make this identification. However, only 2D spectra canprovide the complete information needed in an intuitive but general way. Besides, 2D
spectra also give further insight into more complicated cases of coupling, such as excitonexciton correlations and exciton-continuum interactions, which cannot be described insimple level schemes.

3.5.2

Isolation of coherent pathways

Another essential feature of 2D FTS is the ability to isolate contributions ofvarious coherent pathways in Liouville space. As an example, we consider a threelevel “V” excitonic system consisting of the ground state, excited HH exciton state|eH i and LH exciton state |eL i, with respective transition frequency ωH and ωL . Thisis a simplified picture of the exciton energy level scheme shown in Fig. 2.4, without
considering the two-exciton states. The double-sided Feynman diagrams representing
the coherent pathways for the SI 2D spectra can be derived from Fig. 3.2(a), where thesingly-excited state |ei and |e0 i in Fig. 3.2(d) can be either |eH i or |eL i. Therefore we
obtain four diagrams of type (1) and four of type (2). There is no type (3) diagram

52
since the doubly-excited level |f i is neglected. The difference between type (1) and type(2) is that during the waiting time period the system in pathways of type (2) is in the
excitonic state rather than the ground state as is the case for type (1).

(a)

g

g

g

(1a)

t

eH

k3

g

T

k2

t

k3

g

g

g

(2a)

T

-k1

k2

eL

g

g

g

g

t

eH

k3

k2

τ
g

t

k3

-k1

g

eH

g

g

g

g

τ

-k1

k3

t

eH

k2

eH
g

k2

-k1

g

g

g

-k1

k4

(2d)

k3

t

eL

T

τ
g

eL

g

T

k2

τ

eL
g

(2c)

T

k3

t

eL

T

k4

(1d) eL

T

(2b)

τ

-k1

k4

k4

k3

t

eH

g

k4

T

k2

τ

eH

g

g

g

k4

(1b) eL

τ

(1c)

g

k4

k4

k2

eL
g

-k1

τ
g

eH
g

-k1

(b)
− ωH

(1a)
(2a)

(1d)
(2d)

− ωL

(1c)
(2c)

(1b)
(2b)

ωτ

ωH

ωt

ωL

Figure 3.6: (a) Double-sided Feynman diagrams for heavy-hole and light-hole exciton
transitions; (b) Isolated contributions of coherent pathways to peaks in 2D spectra.

All eight possible coherent pathways are listed in Fig. 3.6(a), where (1a), (1b),(1c), and (1d) are type (1) diagrams, and (2a), (2b), (2c), and (2d) are type (2) [33].
Pathways (1a) and (2a) occur at the same absorption and emission frequency of ωH ,while (1b) and (2b) occur at both frequencies equal to ωL . For the other four pathways,
the absorption and emission frequencies are not equal: (1c) and (2c) have an absorption

53
frequency of ωL and an emission frequency of ωH , while (1d) and (2d) have an absorption
frequency of ωH and an emission frequency of ωL . In the corresponding (rephasing) 2D
spectrum in Fig. 3.6(b), the diagonal peak at (ωH , −ωH ) is from the contributions ofpathways (1a) and (2a), and the other diagonal peak at (ωL , −ωL ) is from pathways(1b) and (2b). The cross peak at (ωH , −ωL ) arises from pathways (1c) and (2c), whereas
the other cross peak at (ωL , −ωH ) is from (1d) and (2d). The pathways (1c) and (2c)are well isolated from (1a) and (2a) at emission frequency of ωH , and (1d) and (2d) are
separated from (1b) and (2b) at ωL . In an one-dimensional spectrally resolved FWM
curve, which is the vertical projection of the 2D spectrum, pathways (1c), (2c), (1a),and (2a) ((1d), (2d), (1b), and (2b)) are all mixed together and contribute to the HH
exciton (LH exciton) peak. Here, the advantage of spreading out a spectrum along thesecond dimension by the 2D technique is apparent.
In the following, we examine the contributions of two-exciton states to 2D spectra.
The six diagrams with all possible combinations of states |ei, |e0 i, and |f i derived fromdiagram (3) in Fig. 3.2(a) are listed in Fig. 3.7(a), where |fH i, |fL i, and |fM i standfor the state of two HH excitons, two LH excitons, and mixed HH and LH excitons,
respectively. There are four pathways leading to |fM i, one to |fH i, and one to |fL i.
Every diagram contains the contributions from both bound and unbound two-exciton
states. The corresponding (rephasing) 2D spectrum is depicted in Fig. 3.7(b), where
all peaks are elongated along ωt axis to demonstrate the effects of red-shift by boundtwo-excitons (indicated by a suffix “-” to the labels of diagrams) and blue-shift byunbound two-excitons (indicated by a suffix “+”). Determined by the absorption andemission frequencies, each coherent pathway can be associated with either a diagonal
peak or an off-diagonal peak. The absorption and emission frequency of pathway (3a)are approximately equal to ωH , thus it shows up as diagonal peak at (ωH , −ωH ), and thepathway (3b) appears as the other diagonal peak at (ωL , −ωL ). The off-diagonal peakat (ωH , −ωL ) arises as a mixture of pathway (3b0 ) and (3c), while the other off-diagonal

54

(a)

eH

eH

eH

(3a)

eH

eL

k4

k4
fH

k3
eH

t

(3a’) f

T

k3

τ

eH

k2
g

g

-k1

M

t

(3b)

T

k3

τ

eL

k2
g

g

eL

eL

eL

k4
fL

t

t

(3b’) f

T

k3

T

τ

eL

τ

k2

-k1

g

g

eH

eH

-k1

M

k2
g

g

-k1

k4

k4

(3c) f

eL

eL

k4

M

k3
eH

(3d)

T

k3

T

τ

eL

τ

k2
g

t

t

g

fM

k2

-k1

g

g

-k1

(b)
− ωH

(3a-) (3a+)(3a’-) (3a’+)(3d-) (3d+)− ωL(3b’-) (3b’+)(3c-) (3c+)(3b-) (3b+)

ωτ

ωH

ωt

ωL

Figure 3.7: (a) Double-sided Feynman diagrams involving bound and unbound twoexcitons; (b) Isolated contributions of coherent pathways to peaks in 2D spectra.

peak at (ωL , −ωH ) is due to pathway (3a0 ) and (3d). Similar to Fig. 3.6(b), different
coherent pathways are isolated along ωτ axis as diagonal and off-diagonal peaks at ωtequal to ωH and ωL .
Contributions from HH and LH excitons as well as two-excitons are all presented
in the total 2D spectrum in Fig. 3.8(a), which is a combination of Fig. 3.6(b) and
Fig. 3.7(b). We compare the total 2D spectrum with a 1D spectrally-resolved FWM
in Fig. 3.8(b). Besides contributions of single-exciton pathways (1a), (2a), (1c), and(2c) to the HH exciton peak (labeled as XHH ), and (1b), (2b), (1d), and (2d) to the

55

(a)

− ωH

(1a)
(3a-) (3a+)
(2a)

(1d)
(3a’-) (3a’+)(3d-) (3d+)
(2d)

− ωL

(1c)
(3b’-) (3b’+)(3c-) (3c+)
(2c)

(1b)
(3b-) (3b+)
(2b)

ωH

ωL

ωτ

SR-FWM(b)(3a-)(3b’-)(3c-)(3a+)(3b’+)(3c+)

XHH (1a)
(2a)
(1c)
(2c)

(1b)
(2b)
(1d)
(3a’-)(3a’+)
(2d)
(3d-)
(3d+)
(3b-)
(3b+)

XLH

ωt

Figure 3.8: Contributions of all coherent pathways including biexciton transitions in 2D
spectra (a) and 1D spectrally-resolved FWM (b).

LH exciton peak (XLH ), two-exciton pathways also contribute to the wings of peaks.
As indicated in Fig. 3.8(b), the bound two-exciton pathways (3a-), (3b0 −), and (3c-)
and unbound two-exciton pathways (3a+), (3b0 +), and (3c+) contribute to the lefthand side and right-hand side of the HH exciton peak (XHH ) respectively, whereas
pathway (3a0 −), (3d-), and (3b-) of bound two-exciton states and (3a0 +), (3d+), and(3b+) of unbound two-exciton states show up on the two sides of LH exciton peak
(XLH ). The typical two-exciton binding energy in GaAs semiconductor quantum wells
is ∼ 2 meV, while the energy separation between XHH and XLH is 7∼9 meV. In a small
spectral range about 10 meV, there are eight single-exciton contributions and twelve twoexciton contributions, resulting in a very congested 1D FWM spectrum. The presenceof exciton homogeneous and inhomogeneous broadening makes the case even worse. 2D

56
FTS demonstrates its power by spreading out the spectrum along the new ωτ dimension
and attributing peaks to different pathways.
The isolation of coherent pathways can be performed in different schemes bychoosing the pair of time periods to be correlated, the pulse sequence, and the spatial
arrangement of excitation pulses, thus providing a great flexibility for selecting thecontributions of interest. For example, an alternative 2D spectroscopy based on thesame kI geometry but with a different pair of time variables, T and t instead of τand t, for Fourier transformations could separate the pathways contributing to theRaman coherence between HH and LH excitons. The implementation of this new 2D
spectroscopy will be discussed in Chapter 7.
As another example, we inspect how coherent pathways can be isolated differently
by selecting the excitation pulse sequence. To illustrate this, the double-sided Feynman
diagrams for pathways contributing to the nonlinear signal from a 3-level “V” system aredepicted in Fig. 3.6(a) and Fig. 3.9(a), for rephasing and non-rephasing pulse sequences,respectively. In the rephasing case, there are eight pathways, where pathways (1a) and(2a), (1b) and (2b), (1c) and (2c), and (1d) and (2d) contribute to peaks (ωH , −ωH ),(ωL , −ωL ), (ωH , −ωL ), and (ωL , −ωH ), respectively, in the rephasing 2D spectrum in
Fig. 3.6(b). There are also eight pathways in the non-rephasing case. Pathways (n1a),(n2a), and (n1c) contribute to peak (ωH , ωH ), (n1b), (n2b), and (n1d) to peak (ωL , ωL ),(n2c) to peak (ωL , ωH ), and (n2d) to peak (ωH , ωL ) in the non-rephasing 2D spectrum
in Fig. 3.9(b). In this case pathways (n2c) and (n2d) are separated as cross peaks.
The pathways isolated in a different scheme for the non-rephasing pulse sequence canprovide supplemental information to rephasing pathways.

57

(a)

g

g

g

k4

g

g

k4

t

(n1a)

k3 (n2a)
eH

T

τ

eH

eH

t

k3

g

g

g

g

k4

t

k3 (n2c)

τ

τ

eH

g

g

g

eL

t

g

g

g

g

g

g

t

τ
g

g

g

g

τ

eL

eH

t

k3

eH

T

T
-k2
eL

-k2

τ

k1

g

g

ωL

(n2d)

(n1b)
(n2b)
(n1d)

ωH

(n1a)
(n2a)
(n1c)

(n2c)

(b)

T
-k2
eL

k3 (n2d)

k1
g

t

k4

T

τ

g

k1

(n1d)

k1

k1

τ

eL

k3
-k2

k4

-k2
eH

-k2

eL

k1
g

k3

eL

T

k3 (n2b)

T
eL

k4

(n1c)

t

(n1b)

k1

k1

g

k4

T
-k2
eH

-k2

g

k4

g

g

g

ωτ

ωH

ωt

ωL

Figure 3.9: (a) Double-sided Feynman diagrams for the non-rephasing pulse sequence;(b) Isolated contributions of coherent pathways to peaks in non-rephasing 2D spectra.

3.5.3

Other capabilities of 2D FTS

Besides the capabilities of identifying resonance coupling and isolating coherent
pathways, there are some more advantageous features in 2D FTS. One that is of particular importance for complicated spectra is the ability to represent inhomogeneous
broadening as an elongation along the diagonal, therefore inhomogeneous broadening
can be separated from homogeneous broadening to provide valuable information about
the fluctuations of oscillators in the ensemble. With inhomogeneous broadening represented on the diagonal, identification of coupling of resonances and isolation of pathways

58
are possible even in the presence of strong inhomogeneity.
Furthermore, the time resolution of 2D spectra is determined by excitation pulse
width and the frequency resolution is limited by pulse separation, therefore overcoming
the compromise between time and frequency resolutions imposed by the time-bandwidth
product [76]. Consequently, 2D FTS technique can capture the fast dynamics withfemtosecond time resolution in a broad spectral range covering all resonances of interest,
while every individual resonance is well characterized spectrally.
Another advantage of 2D spectra is the preservation of phase information. With
full signal information preserved, many conventional spectroscopic results can be produced as various projections of 2D spectra. For example, a vertical projection alongωτ axis provides the spectrally-resolved FWM field, from which the time-resolved signal can also be retrieved by an inverse Fourier transform with respect to ωt . The
absorption spectra can be obtained from the projection along ωt axis. Moreover, thespectrally-resolved differential transmission from a pump-probe measurement is equal
to the product of the real part 2D projection along ωτ axis and the probe electric field.
In general, a complex 2D spectrum with a correct global phase can be presented inseparate real and imaginary parts that are linked by the Kramers-Kronig relation. Thelineshape resolution is improved in the absorptive spectra by removing the dispersivelineshape with long wings that tends to blur peaks. In addition, the phase information
is of great value in distinguishing microscopic mechanisms for many-body interactions,where ambiguity arises due to the lack of information in conventional time or frequency
domain FWM experiments.
With all these advantageous features, 2D FTS technique is being adopted as apromising tool in the studies of exciton dynamics, many-body interactions, and disorders
in semiconductors. We will discuss the development of 2D FTS experiments in near-IR
wavelength for semiconductors in the next chapter.

Chapter 4

2D FTS Experiments with Active Interferometric Stabilization

Optical 2D FTS experiments are commonly implemented with impulsive timedomain techniques. The excitation pulses are scanned with sub-optical-cycle step size
and the corresponding change of signal phase is tracked coherently. Due to the natureof sensitive phase measurement, it is of critical importance to have high phase stability
and pulse positioning accuracy in 2D FTS experiments. The mechanical instability ofthe optical mounts and the positioning inaccuracy of the delay stages can introduce
significant fluctuations in beam direction and optical path length, resulting in degraded2D spectra with artifacts such as ghost peaks. Compared to 2D IR experiments, therequirement for phase accuracy is more demanding in near-IR or visible wavelength
range, as the relative phase fluctuation increases at shorter wavelengths. Therefore, theimplementation of 2D FTS is experimentally challenging, and a delicate apparatus withspecial designs to improve phase stability is generally essential.

4.1

Implementations of 2D FTS with phase stability
Various approaches have been developed to implement 2D FTS experiment withhigh phase stability and pulse positioning accuracy. One method is to maximize thestability of the apparatus passively and measure the phase delays [10, 76]. The measured
data are resampled to correct phase errors, however, it must be done with great care.

60
Passive phase stability for 2D experiments was accomplished with diffractive optics [16,17, 18, 19]. Three excitation pulses and a local oscillator are generated as the first-orderdiffraction from a diffractive optic and thus inherently phase stable. 2D IR spectra wereobtained [108] and the ultrafast hydrogen bond network dynamics in liquid water wasstudied [15]. Electronic couplings in the molecular complex of photonsynthetic protein
were studied with 2D spectra in the visible range [78, 80]. However, the diffractive
optics method suffers from limited delay range provided by movable glass wedges inIR spectroscopy, where the slow dephasing of vibrational states requires quite longtime scan. Adding delay with a translation stage destroys the inherent phase stability.
Although phase stability can be maintained when a delay stage is added in a way that
does not affect the difference in the relative delay between the first two pulses andbetween the third pulse and the local oscillator [108], the delays cannot be scanned
independently. Besides, the local oscillator passes through the excited spot on thesample in this geometry, resulting in undesirable dynamical modifications to systems
with strong excitation-induced effects, such as semiconductors.
Femtosecond pulse shaping techniques are applied to generate an inherently phasestable excitation pulse sequence for 2D FTS experiments [20, 22]. Pulse timing, phase,and waveform shape are well specified, with optical coherence among all pulses. A
collinear three-pulse train created by an acousto-optic pulse shaper with well controlled
and scannable interpulse delays and phases was used to excite 2D spectra of atomic
Rubidium vapor [21]. In this collinear geometry, the desired nonlinear coherent pathways are selected by phase cycling schemes [20] that are routinely performed in 2D
NMR, as discussed in Section 3.1. Diffraction based two-dimensional pulse shaping
is developed to produce multiple non-collinear phase-stable femtosecond pulses in aphase-matched FWM geometry [22]. Combining the waveform generation capabilityof multidimensional NMR with the wavevector specification and phase-matching feature of non-collinear optical spectroscopy, this technique accomplishes a new level of61
spatial and temporal control over excitation pulses and selectivity of coherence pathways. 2D spectra of prototype systems including Rubidium vapor and semiconductor
quantum well structures were produced [109, 110]. There are certain limitations inpulse-shaping techniques. For example, spurious pulses often overlap with main peaks
from an acousto-optic pulse shaper in time-domain and cause harmful amplitude modulation. Furthermore, the range of interpulse delays is constrained by the frequency
resolution to about tens of picoseconds. This restriction causes difficulties to experiments of longer time scan, such as population relaxation studies of semiconductors.
Similar to the approach of diffractive optics, unwanted dynamical modifications by the
local oscillator can also be a problem in pulse shaping techniques.
Active phase stabilization has been developed for conventional non-collinear coherent optical spectroscopy [23, 24]. The incorporated phase stability does not compromise the non-collinear configuration for phase-matching and the flexibility of control
over excitation pulses. We have implemented 2D FTS for the studies of semiconductors at near-visible wavelengths around 800 nm [24]. Two separate interferometers andfeedback loops are used to maintain superior stability of relative phase between thefirst two excitation pulses and between the third pulse and a reference pulse (local
oscillator). The delay between the first two pulses is scanned with interferometric accuracy. The measurement of the optical phase is accomplished by heterodyne detectionof the nonlinear signal with the reference. The 2D FTS experimental setup is depicted
in Fig. 4.1, where the FWM signal generation and heterodyne detection, the excitation pulse stabilization interferometer, and the reference stabilization interferometerare shown. Detailed technical aspects of the experiments are presented in the following
sections of this chapter.

62
Ti:sapphire
Laser
He-Ne
Laser

CMP
Pulse c

BS1

SpectrometerSM fiber
φ = 6 μm

DBS1
PZT2

BS4
Reference

BS2
DBS2

Cryostat

L3

Pulse bPZT
DriverD2

SM fiber
φ = 6 μm

PZT
DriverL2
Loop
FilterSampleL1

BS3

Reference stabilizationinterferometer

Pulse aLoop
FilterPZT1
D1

Excitation pulse stabilizationinterferometer

Figure 4.1: The experimental setup for 2D Fourier transform spectroscopy with active
interferometric stabilization. FWM is generated in a standard box geometry. The
delay between the first two excitation pulses is stabilized and scanned by a stabilizationinterferometer (enclosed in the right box). The phase of the reference is locked to that of
the third pulse by another stabilization interferometer (enclosed in the left box). CMP:
chirped mirror pair; BS1-4: beam splitters; DBS1-2: dichroic beam splitters; L1-3: focal
lens; D1-2: photodiode detectors; PZT1-2: piezoelectric transducers.

4.2

Four-wave mixing generation
The standard box geometry depicted in Fig. 3.3 is used for FWM generation in 2D

FTS experiments. Three excitation beams are produced from a Kerr-lens mode-locked
Ti:sapphire laser with a pulse duration of 100 fs and a repetition rate of 76 MHz. The
wavelength of all three beams is identical and tunable around 800 nm. The beams arearranged on the three corners of a square on a singlet lens (L1 in Fig. 4.1) of 20 cm focal
length to propagate with wavevectors ka , kb , and kc before focused on a sample. The
FWM signal generated by all three beams is emitted in the phase-matched direction−ka + kb + kc . With such a non-collinear three-pulse geometry, the nonlinear coherence

63
pathway can be selected with the phase-matched direction, temporal pulse ordering,and Fourier transform variables. The polarization orientations of all beams are alignedby polarizers and manipulated by half-wave plates or quarter-wave plates (not shownin Fig. 4.1) in front of the focusing lens L1, thus different polarization configuration of
excitation beams can be applied.
1 .5
E x c ita tio n p u ls e

1 .0

A b s o rb a n c e

0 .5
0 .5

L ig h t h o lee x c ito n

N o r m a liz e d In te n s ity

1 .0

H e a v y h o lee x c ito n

0 .0

0 .0
1 5 3 51 5 4 01 5 4 51 5 5 0P h o to n E n e rg y (m e V )Figure 4.2: Linear absorption of the GaAs/Al0.3 Ga0.7 As multiple-quantum well sample(solid line) and the excitation pulse spectrum (dash line).

A typical sample is a GaAs/Al0.3 Ga0.7 As multiple quantum well structure consisting of 10 periods 10 nm well and 10 nm barrier grown by molecular beam epitaxy.
The heterostructure is affixed to a wedged sapphire plate and the substrate is removed
by lapping and chemical etching. The sample is held below 10 K in a cold-finger cryostat
with continuous liquid-Helium flow. As depicted in Fig. 4.2, linear absorption spectrumof the sample displays prominent resonance peaks of HH and LH excitons with an energy
separation of about 6 meV. The excitation pulses have sufficient bandwidth to excite
both excitonic resonances and create unbound electron-hole pairs. The HH exciton has
a linewidth of ∼ 1.5 meV, due in part to inhomogeneous broadening from well-width
fluctuations. At the excitation densities of ∼ 1010 excitons/well/cm2 , the homogeneous
linewidth is ∼ 0.8 meV from a TI FWM measurement. The spectral positions of the64resonances are sensitive to strain arising from the difference in thermal coefficient ofexpansion between the sample and sapphire plate and the mechanical strain imposedby mounting the plate to cold-finger. Consequently, the excitonic peak positions varyslightly in spectra of samples from the same wafer.
As depicted in Fig. 4.1, the emitted FWM signal is collimated by a lens (L2)
identical to the focal lens before the sample. The signal is then combined on a beam
splitter (BS4) with a reference beam, which is split from the third beam before thesample and routed around the cryostat. With such a design, the reference does not passthrough the sample, thus undesired dynamical modification to the excitation is avoided.
The combined signal and reference are coupled into a single-mode fiber and transmitted
to a grating spectrograph, and the resulting spectral interferogram is recorded with a
CCD camera. The complex FWM spectrum, i.e. amplitude and phase, is retrieved fromthe interferogram [36].

4.3Interferometric phase stabilization
The delay τ between excitation pulses a and b is stabilized and scanned by aninterferometer formed with a separate Helium-Neon (He-Ne) CW laser, as shown in theright box in Fig. 4.1. The He-Ne laser follows the same optical path of the Ti:sapphire
beam between two dichroic beam splitters, DBS1 and DBS2, to form a folded Michelsoninterferometer with the beam paths of pulse a and b as the two arms. The second dichroic
beam splitter (DBS2) is common to both arms of the interferometer. This design relieson the fact that rotational motion of the dichroic is minimal. The recombined HeNe beam exiting the interferometer is detected with a silicon photodiode (D1). The
error signal monitoring the relative arm length is sent to a loop filter and fed backto a piezoelectric actuator (PZT1) mounted on the back of a mirror in one arm, whichcorrects for any fluctuation. The servo loop is enabled by computer control to stabilize τ65during data acquisition of the FWM signal and disabled to allow scanning τ . Fig. 4.3(a)presents the phase errors with unlocked and locked τ from the He-Ne interferometer in ascan. Fluctuations in the error signal drops to below 1% when the servo loop is enabled.
Taking into account the fact that the He-Ne beam passes the interferometer twice, thephase error between pulses a and b is reduced to less than 0.5% of one optical-cycle.
Thus the phase fluctuation is suppressed to below 0.01 π (peak-to-peak deviation) by
active stabilization.

(a)Reference Phase Error (π)

Unlocked

0.8
Excitation Phase Error (π)

(b)

0.6
0.4
0.2
0.0

Locked
-0.2

0

500

x50
1000
1500
Delay τ (fs)

2000

0.8

Unlocked

0.6
0.4
0.2
0.0

Locked
x20

-0.2
0

500

1000
1500
Delay τ (fs)2000

Figure 4.3: (a) Error signal from the excitation phase stabilization interferometer with
unlocked and locked delay τ ; (b) Error signal from the reference stabilization interferometer while reference phase is unlocked and locked. Each locked error signal is shownin smaller scale in inset.

After acquiring the nonlinear signal at a delay τ , the feedback loop is disabledand τ is increased by stepping a picomotor actuator with step size of ∼ 30 nm. The
fringes of He-Ne interferometer cycle through one period when the beam path lengthchanges by one-half of the He-Ne laser wavelength, 316.4 nm, or 1.06 fs in delay changeof τ . The He-Ne interference signal is monitored during the stepping of picomotor, andan overall offset is removed to make the fringes oscillate symmetrically around zero.
Once the signal crosses the zero level, the servo loop is enabled to lock delay τ atthe steepest slope of the fringe. The change of path length from one locking point toanother is exactly 316.4 nm, as the small error due to overshooting zero by a partial

66
step is eliminated by the servo loop after locking is established. Fig. 4.4(a) shows the
tracked intensity of He-Ne fringes during stepping, where the circled dots are the locking
positions of τ . The length of picomotor step is not uniform, thus the number of steps
between two signal acquisitions presents a distribution. As shown in Fig. 4.4(b), thedistribution indicates a scan of one-half He-Ne wavelength is completed in 10 steps mostof the time. The same process of locking, acquiring, and stepping is repeated for every
delay τ until all signal measurements are completed, as illustrated by the flowchart in
Fig. 4.4(c).
λHe-Ne /2(b) 600

0.6

Number of Occurrence

He-Ne Error Signal (V)

(a)

0.4
0.2
0.0
-0.2
-0.4

500
400
300
200
100

-0.6
10

(c)

20
30
40
50
Picomotor Steps

60

EnableLockingStart: n=1

6

7

8

9
10 11
Picomotor Steps

12

13

14

DisableLocking

AcquireInterferogram

Step
Picomotor
No

End

Yes

n>N?

n = n+1

Yesacross 0?

No

Figure 4.4: (a) The error signal from the He-Ne interferometer during locking andstepping (the circled dots are where delay τ is locked); (b) A histogram that presents
the distribution of the number of picomotor steps per measurement in a 2D experimentof 1200 measurements; (c) The whole process of phase locking, spectrum acquiring, andτ stepping in a 2D FTS experiment. N is the total number of measurements.

The reference derived from the third excitation beam is combined with the FWM
signal at a beam splitter (BS4) of 10% reflectivity. Glass wedges and a pair of singlet

67
lens identical to L1 and L2 are inserted into the reference path (not shown in Fig. 4.1)
to compensate for the delay and chirp introduced by the sample, cryostat windows andlenses. Fluctuation in the relative path length between reference and the third pulse are
significant due to the turbulent air flow around the cold cryostat used to hold and cool
the sample. Therefore it is necessary to actively stabilize the phase relative to that of
the third beam. As depicted in the left box in Fig. 4.1, a Mach-Zehnder interferometer isformed with the transmitted reference and the third beam. The two beams are focusedwith a 20 cm singlet focal lens (L3) onto to one end of a single-mode fiber (mode field
diameter 5.6 µm), where a spatial fringe pattern is formed with fringe spacing muchlarger than the fiber core. The resulting interferometric error signal is detected witha high gain photo detector (D2). The error signal is fed back to drive a piezoelectric
actuator (PZT2), which moves a mirror in the reference beam to compensate the path
length fluctuation. Fig. 4.3(b) shows the error signal from the reference stabilizationinterferometer, where the fluctuation of relative path length is reduced to less than 5%of one wavelength when the servo loop is enabled. The reference phase is locked to a
fixed value with peak-to-peak deviation below 0.1 π in the whole 2D data-taking time
(several hours). In a long period of time, the drift of relative path length can be quite
significant, as exacerbated by the presence of the cryostat. The phase drift is monitoredwith the PZT driving voltage and canceled by moving the glass wedge in the reference
used for delay compensation.
During long data acquisition time, the laser beam direction usually also needs
to be actively stabilized. The drift in horizontal and vertical directions are monitored
by a pair of quadrant detectors and stabilized by a turning mirror with piezoelectric
actuators in a feedback loop. The experimental setup is enclosed in a sturdy acrylic box
to minimize the influences of air flow, environmental noise and temperature variations.

68

4.4

Fourier transform spectral interferometry
The FWM signal is measured by heterodyne detection with the reference to obtainfull information of amplitude and phase [36, 111]. The spectral interferogram of the
FWM and reference is recorded by a 0.75 m imaging spectrograph and a CCD camera.
The thermo-electric cooled CCD has a 1024 × 256 array of 26 µm pixels and a dynamic
range of 16 bits. The collinear signal and reference beams are focused with a 10×
microscope objective into a single-mode fiber to ensure complete spatial mode-matchingfor interference. The fiber has a numeric aperture of 0.12 and a mode field diameter of
5.6 µm. A linear polarizer is placed in front of the fiber to ensure accurate polarizationalignment. The light exiting the fiber is coupled directly into the spectrograph entrance
slit and a spectral resolution of 0.1 nm is achieved. The FWM signal arrives after
the reference with a delay adjustable by the glass wedge in the reference. The delay,
denoted as τ0 , is set to a large enough value so that a spectral interferogram with dense
fringes is resulted. The maximal density of fringes is limited by the spectral resolution.
The intensity ratio of FWM signal to reference is set to 1/10 initially at τ = 0 and
the interferogram intensity is adjusted to utilize the full dynamic range of the CCD inspectral measurements.
The intensity of the spectral interferogram between FWM and reference can be
expressed as [36]:|E(ωt ) + E0 (ωt )|2 = |E(ωt )|2 + |E0 (ωt )|2 + 2Re[ E(ωt )E0∗ (ωt ) eiωt τ0 ] ,(4.1)

where E(ωt ) and E0 (ωt ) are the respective electric field of the signal and reference, andτ0 is the delay between the two fields. In Fig. 4.5(a), the power spectra of the FWM andreference, |E(ωt )|2 and |E0 (ωt )|2 , and their interferogram in a measurement are plotted,
where the reference bandwidth is broad enough to cover the whole signal spectrum.
There are about 40 interference fringes in the signal range, with a delay τ0 equal to
7.0 ps. The third term in the right-hand side of Eqn.(4.1) is the interferometric term

69with rapid oscillatory fringes, which can be separated by subtracting the power spectraof signal and reference from the interferogram. The net interferometric term is shownin Fig. 4.5(b).
700000(a)Spectral interfrogram

60000040000

Amplitude (a.u.)

30000
20000

300000200000

0
374
376
378
380
Emission Frequency (THz)

382

(b)
Net spectralinterfrogram

0-10000

374
376
378
380
Emission Frequency (THz)-20Spectral Amplitude (a.u.)

Spectrum (a.u.)

400000100000FWM Spectrum

0
372

372

500000

Reference

10000

10000

(c)

382

40

-10

0
Time (ps)

10

(d)

20

15

30

10

20

5

10

0

0
372

374
376
378
380
Emission Frequency (THz)Spectral phase (rad.)Spectra (a.u.)

50000

382

Figure 4.5: Full information of FWM signal including amplitude and phase is retrieved
by Fourier transform spectral interferometry. (a) The power spectra of the FWM andreference, and their spectral interferogram; (b) The net interferometric term obtainedby subtracting the power spectra of FWM and reference from the interferogram; (c)The inverse Fourier transform of the interferometric part contains two terms that are
time-reversed from each other in time-domain; (d) The retrieved spectral amplitude(solid line) and phase (dot line) of the FWM signal.

Since the net interferometric term, S(ωt ) = 2Re [ E(ωt )E0∗ (ωt ) e iωt τ0 ], is real, its
inverse Fourier transform contains two terms that are time-reversed from each otherand separated by 2 τ0 in time-domain:s(t) = F −1 [ S(ωt ) ] = f (t − τ0 ) + f (−t − τ0 ) ,(4.2)

where F −1 stands for the inverse Fourier transform, and the term f (t−τ0 ) and f (−t−τ0 )correspond to the peak of positive time and negative time in Fig. 4.5(c), respectively. In

70general, the transient FWM signal has a fast leading edge followed by a long decay tailin time-domain, and its time-domain interferogram with the reference should present asimilar shape with the addition of fringes. Only the term f (t − τ0 ) has the appropriate
behavior that satisfies the causality principle. In Fig. 4.5(c), the small feature aroundtime zero corresponds to the slow-varying and DC components in an imperfect spectralinterferogram. The delay of the interferogram from time zero, is equal to that between
the FWM and reference in spectral interferogram. With large enough τ0 , the termf (t − τ0 ) can be well isolated from f (−t − τ0 ) and the component around time zero. A
Fourier transform of f (t − τ ) recovers the complex spectral amplitude and phase of theFWM, which can be expressed in the following [36]:E(ωt ) =F[ Θ(t − t0 ) F −1 [S(ωt )] ] e−iωt τ0,
E0∗ (ωt )(4.3)

where Θ(t − t0 ) is the Heaviside step function rising from t0 to isolate f (t − τ0 ) out of
the inverse Fourier transform of S(ωt ), and F stands for the forward Fourier transform.
The complex exponential term e−iωt τ0 is multiplied to remove the linear phase associatedwith delay τ0 . Fig. 4.5(d) shows the retrieved FWM amplitude and phase in frequencydomain.
Strictly speaking, the phase measured is the difference between the signal phase,φ(ωt ), and reference phase, φ0 (ωt ). The reference has a flat phase as a transform-limited
pulse, which is accomplished with a chirped mirror pair (CMP in Fig. 4.1) to compensate
the material dispersion gained from the optics in the setup. However, the constant phaseof the reference is unknown in heterodyne detection, resulting in an unknown offset in
the signal phase. This offset can be determined by a separate pump-probe experiment,
as discussed in the next section.

71

4.5

Generating 2D Fourier transform spectra
The process of Fourier transform spectral interferometry is applied to every in-terferogram acquired while scanning delay τ . In general, τ is scanned with a step size
equal to one half of a He-Ne wavelength, or 1.06 fs in delay change. A few thousandsteps are usually needed to reach a delay of several picoseconds until the FWM signal
decays completely to the background level. For example, a scan of 8192 steps covers
a delay range of 8.68 ps, providing a spectral limit of 473.61 THz (Nyquist frequency)with a resolution of 0.12 THz for the absorption frequency ωτ after Fourier transform.
The 2D FTS experiment can be quite time consuming with a slow dephasing signal, such as that from a high-quality quantum well with a narrow excitonic linewidth.
In this case, undersampling with larger step size is favorable to speed up the measurement. A sampling frequency below that required by the Nyquist sampling theorem
would result in an aliasing problem [112]. However, if the spectral bandwidth of signal
is sufficiently narrow, the aliasing can be exploited with an appropriate undersamplingrate. For the FWM signal from excited HH and LH excitons in quantum wells, thebandwidth is within 10 THz (see Fig. 4.5). If τ is scanned in such a way that two full
He-Ne fringes are cycled in the error signal before each acquisition, undersampling isresulted as the Nyquist frequency of 236.81 THz is far below the upper-limit of the signal oscillation frequency, 380 THz. An image spectral peak appears around 98.5 THz,within a bandwidth of 10 THz. The real signal spectrum is recovered from the image
peak by folding it around the Nyquist frequency. Similarly, higher undersampling ratio
such as 4 or 8 can be used in experiments to significantly shorten the measurement
duration and reduce the long-term stabilization issues.
Before the Fourier transform along τ axis, we can verify the performance of phasestabilization and stepping in an experiment by analyzing the phase evolution with τ . In
addition to testing the entire apparatus, it also serves as an out-of-loop characterization

72
of the data in a 2D measurement, whereas the error signals presented in Fig. 4.3(a)and (b) are in-loop. According to Eqn. (3.18), the change of signal phase at a certain
emission frequency ωt is ∆φ = ωt ∆τ = 2πνt ∆τ , where the delay change between two
acquisitions of spectral interferograms is ∆τ = 1.06 fs. At the HH exciton resonant
frequency νt = 372.12 THz, the phase change is ∆φ = 0.79 π. This constant value issubtracted from the phase change of the retrieved FWM signal at 372.12 THz, leaving
a residual fluctuation shown in Fig. 4.6. Only the first half portion of a whole 2D FTS
measurement with adequate signal strength is shown. The experimental result of phasechange agrees with the expected value quite well, with a mean error of 0.02 π and astandard deviation of 0.06 π. These results are consistent with the phase stability ofthe excitation and reference, 0.01 π and 0.10 π (peak-to-peak deviation), respectively.
Note that this analysis is only valid if the signal is dominated by a diagonal peak. The
presence of a cross peak with the LH exciton resonance adds a systematic variation to
the phase, worsening the apparent fluctuations.

P h a s e C h a n g e (π)

0 .2
0 .1
0 .0

-0 .1
-0 .2
0

2 0 0

4 0 0
6 0 0
D e la y τ ( fs )8 0 0

1 0 0 0Figure 4.6: The fluctuation of signal phase change as a function of delay τ at the
HH resonant frequency 372.12 THz. The nominal value of 0.79 π has been subtracted,
leaving a mean error of 0.02 π and a standard deviation of 0.06 π.

73
All retrieved spectra form a 2D map of complex FWM electric field as a functionof the emission frequency ωt and delay τ . A Fourier transform with respect to τ producesthe 2D spectrum with emission frequency ωt and absorption frequency ωτ . The complex
2D spectrum can be represented in plots of real and imaginary parts if the global
constant phase is known. However, there is an ambiguity in the global 2D phase resulting
from the aforementioned offset in retrieved FWM phase that is not determined in Fouriertransform spectral interferometry. The global 2D phase can be determined with thespectrally-resolved differential transmission (SRDT) measured in an independent pumpprobe measurement. A tracer beam that comes from the fourth corner of the square onlens L1 in Fig. 4.1 and travels along the signal path is used as the probe, whereas thethird beam is treated as the pump. The beam power is carefully chosen to reproduce theexcitation conditions in 2D FTS experiments since excitation-induced effects dominate
the optical nonlinear response of semiconductors. Therefore the pump is set to a power
equal to the sum of the first two beams and the probe power is half of that. The delay
between pump and probe is identical to the waiting time T used in FWM generation.
The SRDT signal is obtained as the change in transmitted probe spectrum induced by
the pump. With identical transform-limited excitation pulses, the SRDT signal is givenby the real-part product of the 2D projection to ωt axis and the probe electric field [75]:Z ∞S2D (ωτ , T, ωt ) dωτ ] ,∆T pp (ωt , T ) ∝ Re [ E pr (ωt )

(4.4)

−∞

where E pr (ωt ) is the probe electric field, and ∆T pp (ωt , T ) is the SRDT spectrum, with adelay T between the pump and probe pulses. Equivalently, the integral in Eqn. (4.4) canbe substituted by E(0, T, ωt ), the electric field of FWM at τ = 0. In data processing,a constant phase rotation is applied to the retrieved FWM field at τ = 0 and its
product with the probe electric field is least-squared fit to the SRDT spectrum. The
optimized constant phase in the fit is the global phase of the 2D spectrum. In Fig. 4.7,
a typical pump-probe measurement of SRDT (dotted line) is shown with the optimized

74
fit by correcting the spectral phase of FWM at τ = 0 (solid line), where the maximal
mismatch is below 10%.

D iffe r e n tia l T r a n s m is s io n

0 .8

P u m p -P ro b e m e a s u re m e n t
M a tc h e d D T

0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
3 6 8

3 7 0

3 7 2

3 7 4

3 7 6

E m is s io n F r e q u e n c y ( T H z )Figure 4.7: A typical pump-probe measurement of SRDT (open circle) and its leastsquared fit by correcting the spectral phase of FWM at τ = 0 (dot line). The maximal
mismatch is below 10%.

4.6

Primary experimental results
Using the 2D FTS apparatus with active interferometric stabilization, we have

performed experiments to study the optical excitations in GaAs multiple quantum wells.
The dependence of TI FWM intensity on the power of excitation beams is measured
first to determine the excitation density for 2D FTS experiments. The TI FWM is taken
by lock-in detection at τ corresponding to the peak of the signal and T matching thatused in 2D measures. As depicted in Fig. 4.8, the TI intensity increases from the weakexcitation limit as the cube of the excitation power, until saturation starts at about
1 mW. Usually experiments are performed in the cubic range in order to compare withtheory in the χ(3) limit. An excitation power of 1.6 mW per beam is used to obtain 2D
spectra with good signal-to-noise ratio, with an excitation dependence close to cubic.

75
C o lin e a r - p o la r iz e d e x c ita tio n
C u b ic

T I-F W M

In te n s ity ( m V )1 0 0 0
1 0 0
1 0
1
0 .1
0 .0 10 .1
1
P o w e r p e r b e a m (m W )Figure 4.8: The dependence of TI FWM intensity on the power of excitation beams.
2D experiments were performed with an excitation power of 1.6 mW per beam.

The corresponding excitation density is on the order of 1010 excitons/well/cm2 .
The linear absorption of the sample described in Section 4.2 is shown in Fig. 4.9(a),along with the excitation pulse spectrum. The laser is tuned a few meV above the LH
exciton resonance in order to compensate the oscillator strength and excite HH and LH
excitons, and exciton continuum (unbound electron-hole pairs) simultaneously. Withan energy separation of ∼ 6 meV between HH and LH excitons, which is close to theexciton binding energy, the LH exciton is degenerate with the edge of the HH exciton
continuum. All three excitation beams have the same linear polarization orientation.
An amplitude 2D Fourier transform spectrum of rephasing pathway is shown in
Fig. 4.9(b). Diagonal peaks corresponding to HH and LH exciton resonances arise fromoscillations at the same frequency during absorption and emission time periods, whereas
off-diagonal peaks unveil that the oscillation frequency changes during or between thetwo periods as a result of coupling between the two resonances. With collinear-polarized
excitation, the HH and LH exciton resonances are coupled through common conduction

76
1.5
Absorbance

1.0
0.5

(b)

Excitation pulse

1.0
0.5

Light hole
exciton0.0

Absorption photon energy (meV)

0.0

Heavy hole
excitonNormal. Inten.

(a)Emission photon energy (meV)

Figure 4.9: (a) Linear absorption of the GaAs/Al0.3 Ga0.7 As multiple quantum well
sample (black) and the excitation pulse spectrum (red); (b) Amplitude 2D spectrum ofrephasing pathway with colinear-polarized excitation. The normalized spectral strength
is represented with 20 contour lines.

band states while different valence bands are involved. The dynamics of the coupledresonances can be modeled with the three-level “V” system discussed in Section 3.5,
without considering the doubly-excited states and polarization selection rules [4]. The
waiting time T is set to 100 fs to avoid other coherent orders when the excitation pulses
overlap temporally. There is no qualitative difference in separate experiments with
T = 0 (not shown).
In the amplitude 2D spectrum in Fig. 4.9(b), a distinct feature is that one crosspeak is the strongest among all peaks and the cross peaks are asymmetric in strength.
In addition, a vertical stripe shows up at the emission photon energy of either HH
or LH exciton, as a result of the absorption by continuum states. For a three-level

77
system, the relative strengths of the diagonal and cross peaks can be estimated from
Eqn. (3.18). The diagonal peaks should have amplitudes of µ4ij and µ4ik , while thecross peaks should be equal in strength with amplitude µ2ij µ2ik . If µij > µik , then
µ4ij > µ2ij µ2ik > µ4ik . Therefore, the cross peaks should never be stronger than the
diagonal peaks. This analysis ignores weighting due the finite bandwidth and tuning of
the excitation pulses, which can be taken into account by weighting the dipole moments,
and does not change the conclusion. Apparently, the asymmetry of cross peaks inexperimental data can not be interpreted with a simple approximation of three-level“V” system. Incoherent population relaxation from the upper excited state to the lower
one observed in molecular systems [78] can result in unequal cross peaks, however only
for a finite waiting time T . This relaxation process is not the case in our experiments, as
the asymmetry feature still remains at T = 0. The influences of many-body interactionsof excitons need to be considered. In the next chapter, a theoretical approach basedon the optical Bloch equations is introduced, with phenomenological modifications forsemiconductor systems. The manifestations of many-body effects in 2D spectra are thenstudied numerically and compared to experimental results.

Chapter 5

2D FTS Interpreted with Optical Bloch Equations

The optical Bloch equations (OBE) are commonly employed to interpret the coherent optical response of atomic systems, where the energy structure is well characterized by a few-level model [113]. However, this approach encounters difficulties when it
is applied to interacting materials, such as semiconductors, as the dominant many-bodyinteractions alters the coherent dynamics dramatically [26]. To account for that, optical Bloch equations are modified phenomenologically to incorporate many-body terms,
including excitation-induced dephasing (EID), excitation-induced shift (EIS), and local
field correction (LFC) [6].
In this chapter, the manifestation of many-body effects in 2D spectra is studied
with numerical calculations based on the modified optical Bloch equations (MOBE). The
theoretical approach of the density matrix formalism and the OBE for a two-level system
are first introduced in Section 5.1, followed by the modification with many-body terms.
The amplitude 2D spectra are calculated in Section 5.2 and the dominance of manybody effects is demonstrated. The cross peak strength and the presence of continuum
absorption in experimental data are represented qualitatively with the inclusion of EID.
Full 2D spectra with phase information are calculated in Section 5.3. The lineshapeof real part spectrum shows a strong dependence on different many-body mechanisms,
making it possible to distinguish the microscopic origin of many-body interactions. In
Section 5.4, the manifestation of inhomogeneous broadening as diagonal elongation in79
2D spectra is demonstrated with calculations. The inhomogeneity and dephasing rate
are determined from the diagonal and cross-diagonal linewidths of experimental spectra.
The excitation dependence of broadening for both HH and LH excitons is studied.

5.1

Optical Bloch equations
The quantum physics of a material system can be described by a wavefunction in

general, and the time evolution of the system and observable properties are determinedby the Schrödinger equation. It is often convenient to employ the density matrix formalism to obtain a statistical description of the quantum system. The density matrix
operator is defined as [81]:ρ(t) ≡ |ψ(t)ihψ(t)| ,(5.1)

where |ψ(t)i is the wavefunction of the quantum system. Using the eigenkets |ni of thesystem as the basis, the wavefunction can be expanded as:|ψ(t)i =X

cn (t) |ni ,

(5.2)

n

where cn (t) is the time-dependent coefficient for state n. The density matrix elementsρnm = cn (t)c∗m (t) describe the probability of the system in a certain physical status. The
diagonal element ρnn = |cn (t)|2 gives the probability of the system in energy eigenstaten, thus it is usually referred as the population density. The off-diagonal element ρnm(n 6= m) is the interstate coherence between states n and m, which means ρnm is nonzero
only if the system is in a coherent superposition of the two states.
The expectation value of any observable variable in the system can be expressedwith the density matrix operator. For example, the expectation value of the polarization
operator P is:
hP i = hψ(t)| P |ψ(t)i = T r( P ρ(t) ) ,

(5.3)

80
where T r stands for the trace of an operator, i.e., the sum of all diagonal elements. The
trace of the density matrix operator T r[ ρ(t) ] = 1, which is the normalization condition
for population density.
For a two-level system with ground state |1i and excited state |2i, the wavefunction has the form: |ψ(t)i = c1 (t)|1i + c2 (t)|2i. Consequently the density matrix of thesystem is:






∗
 c1 c1

c1 c∗2  ρ11 ρ12 
ρ=
=
 ,
∗
∗
ρ21 ρ22c2 c1 c2 c2(5.4)

where the diagonal element ρ11 = |c1 |2 (ρ22 = |c2 |2 ) is the population density in ground(excited) state, whereas ρ12 (ρ21 = ρ∗21 ) describes the coherence of the system and is proportional to the dipole moment µ12 of ground to excited state transition. An ensembleof independent quantum systems can be described by the mixed density matrix, which
is formed as a summation of the pure case density matrices weighted by the probabilityof each individual system.
From the Schrödinger equation of the wavefunction, one can derive the equationof motion for the density matrix in the following form:ρ̇ =1[ H, ρ ] .
i~

The Hamiltonian of a simple two-level system can be written as: ~ω1 V12 
H = H0 + V =  ,V21 ~ω2

(5.5)

(5.6)

where H0 is the Hamiltonian of free particles, with eigenenergy ~ω1 and ~ω2 for the twolevels. Potential V characterizes the interaction between the applied electric field E(t)and the system. In the dipole approximation of optical excitation, the field is coupled
to the system via the dipole transition moment, resulting in a potential with elements
V12 = V21 = −µ12 E(t). In general form, the equation of motion for density matrix
reads:
ρ̇ =1[ H0 + V, ρ ] − decays ,
i~

(5.7)

81
where “decays” terms are added by hand in this simplistic two-level theory. The relevant
decays for the system include spontaneous emission from the excited state to ground(with decay rate γ sp ), or to a “reservoir” level (with decay rate γ res ) if there is one,and the decay of coherence (with decay rate γ ph ). In the quantum theory with second quantization of the applied field, the decay term of spontaneous emission appears
intrinsically.
For a closed system, i.e. γ1res = γ2res = 0, the conservation of population gives therelation ρ11 + ρ22 = 1, or ρ̇11 = −ρ̇22 . Along with the property ρ21 = ρ∗12 , the density
matrix is completely determined by two equations following Eqn. (5.7):iµ12 E (ρ12 − ρ21 ) ,~iρ̇12 = −γ ph ρ12 + i ω0 ρ12 + µ12 E (ρ22 − ρ11 ) ,~ρ̇22 = −γ sp ρ22 +

(5.8)
(5.9)

where the population relaxation with rate γ sp due to decay of spontaneous emission
to ground state and the decay of coherence with rate γ ph have been added. µ12 is the
dipole moment of the transition between the ground and excited states, which has an
oscillation frequency ω0 = ω2 − ω1 . Commonly referred as Bloch equations, Eqn. (5.8)
and (5.9) were originally developed by F. Bloch for the description of nuclear magnetic
resonance [114] and later extended to problems in coherent optical excitation. In orderto account for the strong many-body effects in semiconductors, optical Bloch equations
are modified to include phenomenological terms, such as EID, EIS, and LFC [6]:ρ̇22 = − γ sp ρ22 +iµ12 (E + LP )(ρ12 − ρ21 ) ,~(5.10)

ρ̇12 = − (γ ph + γ 0 N ρ22 ) ρ12 + i(ω0 + ω 0 N ρ22 ) ρ12+iµ12 (E + LP )(ρ22 − ρ11 ) ,~(5.11)

where N is the number density of oscillators and P = N T r(µρ) is the polarization.
EID is included through the term γ 0 N ρ22 , which causes the dephasing rate to increase
with excitation if γ 0 > 0. Excitation-induced narrowing has been discovered in atomic

82
vapors [115], corresponding to a negative γ 0 . Similarly, EIS is added through ω 0 N ρ22 ,which indicates the resonant frequency increasing with excitation for positive ω 0 . The
local field correction is linear to the polarization, with strength controlled by parameter L. Eqn. (5.10) and (5.11) are called modified optical Bloch equations (MOBE).
The linear dependence of the dephasing rate, resonance frequency, and local field on
excitation density is a good approximation at the weak power limit. Note that the EID(γ 0 N ρ22 ρ12 ) and EIS (iω 0 N ρ22 ρ12 ) terms modify the interstate coherence differently witha phase shift introduced by the constant i in Eqn. (5.11). Therefore, phase-sensitive
measurements can directly address the distinction between different many-body effects
that intensity measurements can not [5, 116].
It is not practical to derive an analytical solution in a general form for the MOBE.
Perturbation calculation is usually employed to produce a solution to a certain order in
the incident field. For the common initial conditions of ρ22 (0) = 0 and ρ12 (0) = 0, theodd powers of population ρ22 and the even powers of polarization (interstate coherence)ρ12 vanish. Typically, the polarization is calculated to the third-order to represent thenonlinear response of the system in experiments such as FWM. However, it is difficult
to characterize excitation-dependent effects with the perturbation calculation, since itis only valid in the weak field limit. Neither the dephasing rate nor the oscillation frequency shows a dependence on the excitation population, even when the polarization is
calculated to the fifth-order [117]. There is an essential breakdown of the perturbation
theory when excitation-induced effects need to be considered. In order to model these
phenomena correctly, MOBE can be numerically solved with the spatial Fourier expansion of the incident fields in a paraxial approximation [117]. This numerical approachis not perturbative, but an approximation in that the spatial expansion of the density
matrix elements is truncated to a desired order.

83

5.2

Amplitude 2D spectra
Numerical calculations of 2D spectra based on MOBE were performed to study

the influences of many-body interactions [56, 55]. In this section, the effort is focusedon features in the experimental amplitude 2D spectrum in Fig. 4.9(b). As depicted
in Fig. 5.1(a), a three-level scheme is used to represent the ground state, excited HH
exciton state and LH exciton state in semiconductor quantum wells. The MOBE for thethree-level system can be derived in a way similar to that discussed in Section 5.1. The
nonlinear polarization is calculated to give a FWM signal as a function of evolution timeτ and emission time t. The FWM electric field is displayed as a map on two frequency
axes after Fourier transforms with respect to τ and t. The calculated linear absorption
spectrum is shown in Fig. 5.1(b), along with an excitation spectrum chosen to matchthat in Fig. 4.9(a).
The simulated amplitude 2D spectrum is shown in Fig. 5.1(c), where both frequency axes are normalized to the dephasing rate γ ph and in a rotating frame chosensuch that the HH exciton resonance has a frequency of 1. The diagonal peaks for the two
exciton resonances are reproduced, with cross peaks indicating the coupling through theshared ground state. The two cross peaks have the same strength, which is in between
those of the two diagonal peaks, confirming the simple analysis based on a three-levelsystem in Section 4.6. However, this result is quite different from the experimental
spectrum in Fig. 4.9(b). The distinct feature of asymmetric cross peak strength andone cross peak being the strongest could be reproduced with the inclusion of manybody effects. Specifically, this feature presents if EID or EIS dominates and there is adifference in how each resonance is affected. To verify this, we include EID that only
affects the HH exciton resonance in the calculation and obtain a spectrum in Fig. 5.1(d).
Apparently this calculation matches the experimental spectrum better. The diagonal
and off-diagonal peaks at the emission frequency of HH exciton are enhanced, leading

84

(a)

XLH
XHH

ω2

ωτ /γ ph

ω1

(b)

(c)

(d)

ωt /γ phFigure 5.1: Numerical calculations for cross peak strength in amplitude 2D spectrum.
(a) The three-level system representing HH and LH excitonic states and ground state;(b) Calculated linear absorption (solid line) and excitation pulse spectrum (dash line);
Amplitude 2D spectrum is calculated without any many-body interactions (c), or with
EID only affecting HH exciton resonance (d), which results in asymmetric cross peakstrength and one cross peak being the strongest. Amplitudes of two spectra are normalized individually and each is represented with 20 contour lines.

to asymmetric cross peaks and a cross peak with the strongest strength.
Another striking feature in Fig. 4.9(b) is the vertical “ridge” at the emission frequency of HH or LH exciton. These ridges are due to the absorption of continuum
states. A simple model of the continuum states treats them as a set of inhomogeneously
broadened two-level systems. We employ a single strong resonance to model the HH
exciton state and a set of 20 weaker resonances with higher dephasing rates for thecontinuum states, as depicted in Fig. 5.2(a). The calculated linear absorption and excitation spectrum used are shown in Fig. 5.2(b). A diagonal elongation feature, instead ofa vertical ridge, presents in the calculation without any many-body effect, as shown in

85
Fig. 5.2(c). If EID is included to affect the strong resonance only, a 2D spectrum thatis qualitatively similar to the experiment is obtained, as shown in Fig. 5.2(d). Since
this calculation only models the HH exciton and continuum states, only a single verticalridge appears, whereas in the experiment two vertical ridges show up, in association
with the HH exciton and LH exciton.

(a)continuum
(b)

XHH
ω1(1)

ω1( 2 )

ω1(19) ω1( 20 )ωτ /γ ph

ω1

(c)

(d)

ωt /γ phFigure 5.2: Numerical calculations for the amplitude 2D spectrum of HH exciton andcontinuum states. (a) The system is modeled by a single strong resonance for the HH
exciton state and a set of 20 weaker resonances for the continuum states; (b) Calculated
linear absorption (solid line) and excitation pulse spectrum (dash line); Amplitude 2D
spectrum is calculated without any many-body interactions (c), or with EID included
to affect the strong resonance only (d), which results in a vertical ridge feature. Eachamplitude spectrum is represented with 20 contour lines.

The goal of the phenomenological calculations is to qualitatively demonstrate theinfluence of many-body effects on 2D spectra, rather than reproducing experimental
data with full details. EID is selected as an instance of many-body effects and thecalculation is done without special parameter refinement.

86

5.3

Full 2D spectra
The dominant influences of many-body interactions on the amplitude 2D spectraof semiconductors have been demonstrated in the previous section. In general, however,different mechanisms of interactions have quite similar manifestations in the amplitude
spectrum, thus it is difficult to distinguish these mechanisms. With coherent phase
information available, full 2D spectra could give insight into the microscopic nature of
many-body interactions.
In simple terms, the imaginary part of a 2D spectrum measures the transientchange in the refractive index, whereas the real part gives the resonant absorption of aprobe field at emission frequency ωt , induced at excitation frequency ωτ [55]. Proper
decomposition of a complex 2D spectrum into real and imaginary parts relies on the determination of the global phase, which is made possible by comparing to the spectrallyresolved differential transmission signal from an independent pump-probe measurement,
as described in Section 4.5. In Fig. 5.3, the amplitude and real part 2D spectra of arephasing experiment with colinear-polarized excitation are shown as (c) and (d), respectively, along with the amplitude (a) and real part (b) spectra of the non-rephasing
pathway. The excitation spectrum is tuned to the middle of the HH and LH exciton resonances to reduce continuum density. Other experimental conditions are same as thosefor the spectrum in Fig. 4.9. The non-rephasing experiment was performed by scanning
pulse b, instead of scanning pulse a in rephasing measurements. Non-rephasing spectra
can provide complementary information to rephasing spectra, such as resolving inhomogeneity and the degree of correlation of inhomogeneity between coupled resonances [72].
In the rephasing amplitude spectrum of Fig. 5.3(c), the strength of cross peaks is still
asymmetric, but the lower-left cross peak is no longer the strongest due to the change oftuning. The relative strength of cross peaks to that of diagonal peaks in non-rephasing
spectra is weaker than that in rephasing spectra. This difference can be explained with

87

Real Part

Amplitude
(b)

(c)

(d)

Absorption photon energy (meV)(a)

Emission photon energy (meV)
Figure 5.3: Experimental amplitude (a) and real part (b) 2D spectra of non-rephasing
pathway, along with the amplitude (c) and real part (d) 2D spectra of rephasing pathway.
Both rephasing and non-rephasing spectra are normalized to the respective amplitude
and represented with 32 contour lines.

the coherent pathway analysis in Section 3.5. As shown in Fig. 3.9, each diagonal peakcorresponds to the contributions from three coherent pathways, whereas each cross peakis due to one pathway in non-rephasing spectra. In contrast, every diagonal or crosspeak in rephasing spectra arises from the contributions of two pathways.
Each peak in a real part spectrum presents a strong dispersive (derivative of apeak) feature, either perpendicular to the diagonal in rephasing spectra (Fig. 5.3(d)), or
along the diagonal in non-rephasing spectra (Fig. 5.3(b)). The dominance of dispersive

Absorption photon energy (a.u.)

88

Real Part

Amplitude(a)(b)
Emission photon energy (a.u.)

Absorption photon energy (a.u.)

Figure 5.4: Calculated amplitude (a) and real part (b) 2D spectra based on a three-levelsystem, without any many-body effects. Spectra are normalized to the amplitude andrepresented with 32 contour lines.

Real Part

Amplitude(a)(b)
Emission photon energy (a.u.)

Figure 5.5: Calculated amplitude (a) and real part (b) 2D spectra based on a three-levelsystem, with EID included. Spectra are normalized to the amplitude and represented
with 32 contour lines.

lineshapes qualitatively contradicts the prediction of simple optical Bloch equations. As
shown in Fig. 5.4, the amplitude (a) and real part (b) 2D spectra are the calculated resultof rephasing pathway for the three-level model depicted in Fig. 5.1(a). No many-bodyeffects are included in the calculation. The diagonal peaks display strong absorptive

89
lineshapes in the real part spectrum. The slightly unequal strength of cross peaks isdue to the different dephasing times used for HH and LH exciton resonances in thecalculation. With EID included, the calculation produces a real part spectrum shownin Fig. 5.5(b), where both diagonal and off-diagonal peaks are all absorptive. These
results are quite different from the measurements. However, the inclusion of EIS in
the calculation results in dominant dispersive lineshapes in the real part spectrum,
as shown in Fig. 5.6(d). For further comparison, amplitude and real part spectra ofnon-rephasing pathway are also calculated, which are displayed in Fig. 5.6(a) and (b),respectively. The calculation with EIS qualitatively matches the experimental spectrain both rephasing and non-rephasing pathways. Although the calculation is not aimed
to quantitatively determine the phenomenological parameters, they are in reasonableagreement with previous studies [6].
While the real part 2D spectrum presents a distinct change with different manybody effects, the amplitude with EID (Fig. 5.5(a)), or with EIS (Fig. 5.6(c)) has lineshape features similar to that without many-body effects (Fig. 5.4(a)). The phase,rather than the amplitude of 2D spectra, is sensitive to the mechanism of many-bodyeffects. With full signal information presented, 2D FTS approach is a powerful tool
for revealing the microscopic origin of many-body interactions in semiconductors. The
experimental spectra in Fig. 5.3 were taken at a relative high excitation power, thus
the peaks are primarily homogeneously broadened. Consequently no inhomogeneous
broadening was considered in all the numerical calculations. It is also possible to makea distinction between EID and EIS in the presence of inhomogeneous broadening, which
manifests itself as the diagonal elongation of peaks and does not alter the dispersive or
absorptive lineshape.
Although the calculation with EID results in a real part spectrum that does
not match the experiment, its contribution can not be simply ruled out. In fact, asa direct manifestation of the EID effect, the peak broadening in 2D spectra has a90

Absorption photon energy (a.u.)

(a)

(b)

Real Part

Amplitude(c)(d)
Emission photon energy (a.u.)

Figure 5.6: Calculated amplitude (a) and real part (b) 2D spectra of non-rephasing
pathway, and amplitude (c) and real part (d) 2D spectra of rephasing pathway basedon a three-level system, with EIS included. Both rephasing and non-rephasing spectra
are normalized to the respective amplitude and represented with 32 contour lines.

distinct dependence on excitation power. The power dependence of homogeneous andinhomogeneous linewidths will be discussed in the following section. In the Green’s
function approach of microscopic semiconductor theory [118], EID and EIS correspondto the renormalization of the imaginary and real parts of exciton self-energy, respectively,
thus they are both present and can not be treated as independent parameters. Possible
contributions of the renormalization include band-gap renormalization, screening andcorrelation terms beyond the Hartree–Fock approximation.

91

5.4

Determination of homogeneous and inhomogeneous broadening
The linewidth of optical transitions in semiconductors and other materials pro-vides rich information on the structure of electronic states and light-matter interactions.
In semiconductor quantum nanostructures at low temperatures, disorder spatially localizes excitons and leads to inhomogeneous broadening [1], whereas phonon induced
homogeneous broadening dominates at elevated temperatures. Various nonlinear optical approaches have been utilized to study the broadening in nanostructures, including
TI and TR FWM spectroscopies [1, 25]. TI FWM provides the homogeneous width in a
simple two-level system, but this is often not true in semiconductors with strong manybody coupling [51]. The temporal width of the photon echo in TR FWM spectroscopy
characterizes inhomogeneous broadening. However, echo peak is not well defined dueto the interplay between disorder and many-body interactions when disorder-induced
broadening is small. In addition, the echo peak can be distorted by beating when morethan one resonance is excited [119]. The retrieval of homogeneous width from TI FWM
signal with beating requires careful fits to a sophisticated model in general.
2D FTS can be used as a robust tool to determine exciton homogeneous andinhomogeneous linewidths directly, despite the presence of multiple resonances. To
demonstrate the manifestation of broadening in 2D spectra, numerical simulations were
performed based on the perturbation calculation for a simple two-level system [39].
Inhomogeneous broadening is introduced to the ensemble as a Gaussian distribution [39]:g(ω) = √1(ω − ω0 )2
exp [−],
2(δω)22π δω

(5.12)

where ω0 is the center of the resonance and δω is the inhomogeneous width in angularfrequency. The homogeneous width is proportional to the dephasing rate γ, as discussed in Section 2.4. 2D spectra are calculated with different δω and γ. In Fig. 5.7,
an amplitude 2D spectrum with γ of 0.3 ps−1 and no inhomogeneous broadening (a) is
compared to one with the same γ but an inhomogeneous width δE of 0.7 meV (b). Here

In
us
eo
e n in gog n
m de
ho roab

Absorption photon energy (meV)

92

(a)

(b)
Emission photon energy (meV)

Figure 5.7: Simulated broadening in 2D spectra for a two-level system. (a) Amplitude
2D spectrum with homogeneous width γ of 0.3 ps−1 and no inhomogeneous broadening;(b) Amplitude spectrum with homogeneous width γ of 0.3 ps−1 and inhomogeneous
width δE of 0.7 meV.

√
δE is the full-width at half-maximum (FWHM) in energy defined as δE = 2 2 ln 2 ~δωfor convenience. Clearly the inhomogeneous broadening elongates the peak along the
diagonal. The diagonal linewidth has a Voigt profile, which corresponds to the convolution of the Gaussian function and the Lorentzian distribution representing homogeneous
broadening. The inhomogeneous broadening can be deduced from the Voigt profile byempirical fit [120]. The cross-diagonal width Γ C (FWHM) and the deduced inhomogeneous width ΓI (FWHM) in 2D spectra are shown as functions of the dephasing rate γ
and inhomogeneous width δE in Fig. 5.8 [121]. From the linewidths along and acrossdiagonal direction, one can retrieve γ and δE directly.
We determine the dephasing rate and inhomogeneous broadening of HH and LH
excitons from experimental 2D spectra. The sample under study is a multiple quantum well structure containing four periods of 10 nm GaAs well and 10 nm Al0.3 Ga0.7 As
barrier cooled below 10 K. The amplitude 2D spectrum of rephasing pathway is used toobtain δE and γ. Real or imaginary part spectrum can also be employed for linewidth

93

(b)

3.0

ΓI (meV)ΓC (meV)

(a)

2.0
1.0
0
1.35

1.00

δE

0.50

( me
V)

0 0

0.50

1.00
-1 )

γ (ps

1.30

3.0
2.0
1.0
0
1.35

1.00

δE

0.50

( me
V)

0

0

0.50

1.00

1.30

-1 )

γ (ps

Figure 5.8: Simulation result of the cross-diagonal width ΓC (FWHM) (a) and the
deduced inhomogeneous width ΓI (FWHM) (b) in 2D spectra as functions of dephasingrate γ and inhomogeneous width δE.

analysis, with the advantage of resolution improvement when congested spectra arepresent. Fig. 5.9 shows the linewidths of HH and LH excitons from experiments of different excitation power within the weak excitation regime [121]. From the homogeneous
widths we find the dephasing time is 4.5 ps for HH exciton and 3.8 ps for LH exciton at
200 µW. The homogeneous width of HH or LH exciton increases with power, in good
agreement with early TI FWM experiments [45]. The inhomogeneous broadening of
HH and LH excitons at 200 µW are 0.54 meV and 0.40 meV, respectively. The larger
inhomogeneous width of the HH exciton is possibly due to the localization by disorder,
since the HH exciton is more likely localized with a lower mobility compared to LH
exciton. If the disorder is of short range compared to the exciton diameter, the smaller
inhomogeneous width of LH exciton can be the result of a more effective averaging
of inhomogeneity over LH excitons [122], as the Bohr radius of LH exciton is bigger
than that of the HH exciton. The inhomogeneous broadening also shows an excitation
dependence, which demands an understanding of the interplay between disorder andmany-body interactions. Interactions between HH exciton continuum and LH excitons
should be considered due to the spectral overlap, thus the broadening of LH exciton

0 .6

H H e x c i t o n 

0 .6

0 .5

L H e x c i t o n 

0 .5

I n h o m o g e n e o u s w i d t h  ( m e V )H o m o g e n e o u s w i d t h γ (p s - 1 )94

H H d e lt a E

0 .4

0 .4

L H d e lt a E
H H g a m m a
L H g a m m a

L H e x c ito n γ

0 .3

0 .3

H H e x c ito n γ

0 .2
2 0 0

4 0 0
6 0 0
E x c i t a t i o n p o w e r (µW )

0 .2
8 0 0

Figure 5.9: Excitation power dependence of homogeneous and inhomogeneous
linewidths for HH and LH excitons obtained from experimental 2D spectra.

is more complicated. The coupling of HH and LH excitons also affects the linewidthof both resonances. Such problems can be avoided by reducing the bandwidth of laser
pulses and exciting the HH exciton resonance only. It is possible to observe the differencein homogeneous broadening between bound and unbound excitons in 2D spectra. Compared to bound excitons, the decoherence of unbound excitons is faster due to stronger
scattering processes, thus results in a broader homogeneous linewidth. Consequently,
the cross-diagonal width on the higher energy side of the resonance peak is larger thanthat on the lower energy side, leading to an asymmetric elongation along the diagonal.
In addition to the linewidth study with 2D spectra in rephasing pathway, the nonrephasing spectra can also be employed for complementary information on broadening.
In microscopic calculations of 2D spectra with the inclusion of many-body interactions of
HH and LH excitons as well as biexcitons [119], inhomogeneous broadening demonstrates
different behaviors in non-rephasing and rephasing pathways, making it possible toseparate disorder-induced broadening. Experimental studies towards this goal are stillon the way.

Chapter 6

2D FTS Interpreted with Microscopic Semiconductor Theory

In the previous chapter, we reproduced the qualitative features of 2D spectra innumerical calculations based on the modified optical Bloch equations with the inclusionof phenomenological many-body effects. While phenomenological few-level models arewidely employed due to the simple formalisms and thus the relatively easy implementation in practice, there has been significant progress in the development of fundamental
microscopic theory for semiconductors [25, 118, 123, 124, 125, 126]. In microscopic theories, an optically-excited semiconductor is treated as a quantum many-body system
of interacting electrons and holes. Usually only the electrons in the lowest conduction
band and holes in the highest valence band are considered. The Coulomb interactions
profoundly influencing the coherent nonlinear response are taken into account in thispicture. Compared to a few-level model, the Coulomb interactions renormalize theelectron and hole energies and produce excitons. The interaction strength between theradiation field and the semiconductor is also renormalized [2]. Different microscopic approaches in terms of density matrices as well as non-equilibrium Green’s functions havebeen developed. Within the density matrix approach, the Coulomb interaction leads toan infinite hierarchy of higher order of density matrices, which can be truncated using
either correlation expansion or dynamics-controlled truncation scheme [124]. Within
the Green’s function approach, the problem of an infinite hierarchy is deferred to constructing the self-energy. EID and EIS in the phenomenological model correspond to the96
renormalization of the imaginary and real parts of the exciton self-energy, respectively,
therefore they are not independent parameters in the microscopic theory. The relation
between phenomenological few-level models and microscopic semiconductor theories has
been discussed [127].
With the rich information of carrier dynamics and many-body interactions preserved, 2D FTS provides a stringent test of theoretical models and gives insight into
the microscopic nature of exciton correlations in semiconductors. The 2D spectra of
excitonic resonances are sensitive to excitation conditions, which include the spectrum,
intensity, and polarization of the incident laser pulses. In the 2D FTS experiments discussed in the previous chapters, we have used colinear-polarized excitation pulses, whichis experimentally the most straightforward method but not the simplest for interpretation. The earlier studies also only used a single tuning and power of the incident pulses,
with the latter corresponding to a relatively high excitation density in the sample. In
this chapter, we study the dependence of 2D spectra on the excitation conditions, particularly the polarization of excitation beams. Cocircular-polarized excitation provides thesimplest case for isolating many-body effects as they are responsible for the coupling between the HH and LH exciton resonances. Cross-linear-polarized excitation suppresses
the exciton resonances and reveals contributions from biexcitons. Experimental results
show a strong dependence of 2D spectra on the polarization of excitation beams, and thedominant influence of many-body interactions in each case. These results are comparedto calculations based on the microscopic theory, with qualitative agreement.

6.1

General formalism of the microscopic semiconductor theory
For the numerical studies of 2D spectra, we use a general formalism of the mi-

croscopic semiconductor theory, which has been employed to investigate the effects ofcarrier correlations on the excitonic optical response of semiconductors [125]. As a real-

97
space approach, this formalism is particularly suitable for semiconductor systems withdisorders [25]. The Hamiltonian of the system can be written in the following form:
H = H0 + HC + HI ,(6.1)

where H0 is the free-particle Hamiltonian, HC is due to the Coulomb interaction, andHI is from the interaction of the system with the electric field of incident light.
The free-particle Hamiltonian reads:
H0 =

X

c
Tijc ac+i aj +

ij c

Xv
Tijv av+
i aj ,

(6.2)

ij v

where i and j are indices to real-space sites, and c (v) means the conduction (valence)v+
cv
band. ac+i (ai ) creates (destroys) an electron in conduction band at site i, and ai (ai )creates (destroys) a hole in valence band at site i. The diagonal terms of the T matrices
give the site electronic energies while the off-diagonal terms define the couplings betweensites.
The Coulomb Hamiltonian HC has the form:
HC =

0
0
1 X
0
0
v+ v
c
(aci + aci − avi + avi )Vij (ac+j aj − aj aj ) ,
2
0 0

(6.3)

ij cv c v

where Vij is the Coulomb potential between particles at site i and j. The dipole interaction of the system with a classical electric field E(t) is:
HI = −E(t) · P = −E(t) ·Xv cvc ∗ c+ v+(µvc
ij ai aj + (µij ) aj ai ) ,

(6.4)

ijvc

where P is the optical interband polarization, and µ is the dipole matrix for opticaltransition between valence and conduction bands.
The interband coherence, pv121 c2 = hav11 ac22 i, is characterized by the equation of

98
motion, with the total Hamiltonian plugged in, as follows:−i

X
X∂pv121 c2
c v 1 c2v v 1 c2
= −
T2j
p1j −
Ti1
pi2 + V12 pv121 c2
∂t
j
i
X
X
cc2
v 1 c2 ∗2 ∗ vv1(µv1j1 c )∗ fj2−(µvc− E(t) · [(µ12 ) −i2 ) fi1 ]
jc

+

X

+

X

iv

[Vk1 havkk av11 avkk + ac22 i − V2k hav11 avkk ac22 akvk + i]

kvk

[Vk1 hackk + av11 ackk ac22 i − V2k hav11 ackk + ac22 ackk i] .

(6.5)

kck
cc2
Here the electron and hole populations and intraband coherences are defined as fj2=
v1 v
v1 + vc2
hac+
j a2 i and f1i = ha1 ai i, respectively. The four-point functions represent the firststep of the infinite hierarchy of many-body correlations induced by Coulomb interaction.
For simplification, we consider the coherent limit in which dephasing processes
due to scattering with other quasi-particles are neglected. Besides, relatively weakexcitation is assumed and terms contributing to the optical response are only included
to the third-order in the optical field. Within these assumptions the electron and hole
populations and coherences are no longer independent but can be expressed by theinterband coherence as:X

c1 c2
pva2a c2 (pva1a c1 )∗ = f12
,

(6.6)

v 1 v2.
pv1a1 ca (pv2a2 ca )∗ = f12

(6.7)

ava

and
X
aca

The optical response up to χ(3) can be expressed with two transition-type quantiv1 vcc2
ties: the interband coherences pv121 c2 related to single-exciton and σ1234= hav11 av2 ac3 ac42 i

describing two-exciton excitations. In order to study the pure correlation effects beyond
the time-dependent Hartree–Fock approximation, the uncorrelated parts can be sepav1 vcc2
v1 vcc2
v1 c vc2
rated from the four-point quantities by defining: σ̄1234= σ1234− pv141 c2 pvc
23 + p13 p24 .

This procedure results in closed equations of motion for the single-exciton amplitude
v1 cvc2
v1 vcc2pvc≡ −σ̄1234
that completely determine the
12 and the two-exciton amplitude B1324

optical response within the coherent χ(3) -limit.

99
The equation of motion for p becomes:
−i

X
X
∂pvc
c vc
v vc
12
=−
T2j
p1j −
Ti1
pi2 + V12 pvc
12
∂t
j

+

X

i

0 0

0

0

0 0

0 0

v c vcv c vc(Va2 − Va1 − Vb2 + Vb1 )(pvbac )∗ [ pvb2c pvc
1a − pba p12 − Bba12 ]abv 0 c0∗
+ E(t) · [ (µvc
12 ) −

X

0

0 0

0

0

0 0

0

∗ vc ∗ vc
v c ∗ v c ∗ vc((µvc
1b ) (pab ) pa2 + (µb2 ) (pba ) p1a ) ] ,(6.8)

abv 0 c0

and the equation of motion for B is:0 0v c vcX∂Bba12c
v 0 c0 vc
v
v 0 c0 vc
c
v 0 c0 vc
v 0 c0 vc
=−
( T2i
Bba1i
+ Ti1
Bbai2
+ Tai
Bbi12
+ Tibv Bia12
)
−i
∂t
i

0 0

v c vc+ (Vba + Vb2 + V1a + V12 − Vb1 − Va2 ) Bba12
0

0

0 0

vc
v c vc− (Vba + V12 − Vb1 − Va2 ) pvc
1a pb2 + (V1a + Vb2 − Vb1 − Va2 ) pba p12 .

(6.9)
The total interband polarization P measured in an experiment is:
P=

X

vc
µvc
ij pij ,(6.10)

ijvc

from which the electric field of the emitted FWM signal can be obtained with Eqn. (3.16).

6.2

One-dimensional tight-binding modelTo keep the complexity of numerical computations within tractable limits, a one-dimensional tight-binding model is commonly used to simulate the semiconductor band
structure [125]. Quantitative agreements between calculations and experiments are notexpected with this model, however, it has qualitatively produced important signaturesin different types of experiments in the χ(3) -limit, such as pump-probe, transient FWM,
and coherent excitation spectroscopy [25, 34, 98, 125, 128, 129].
In the one-dimensional tight-binding model, it is assumed that N particles are
spatially localized to form a one-dimensional chain. The site positions are labeled as ia,

100
where i = 1, 2, . . . , N is the index of sites, and a is the spatial separation between two
neighbor sites, as depicted in Fig. 6.1 [130]. The energetic levels of the electron, HH, and
LH at site i are denoted as εci , εhi , and εli , respectively. The coupling between localized
electrons in two close neighbor sites i and j is described by Jijc , and that between HHs(LHs) is Jijh (Jijl ). The interband coherence between sites i and j is given by pij . µhi E(µli E) is the optical transition strength from the HH (LH) level to the electron level insite i, induced by an electric field E. The energy separation of the electron and HHlevels, E0 , and the band width due to the coupling, determine the energy gap Eg at theΓ-point for the given semiconductor material. The quantum well system is modeled with
the introduction of LH levels, which are below the HH levels with an energy splitting
provided by the linear absorption spectrum.

ε ic

i

J ijc

j

μ il E
μih E

E0

ε ih

pij
J ijh

ε il

J ijl

a

Figure 6.1: The one-dimensional tight-binding model for semiconductor quantum
wells [130].

In the tight-binding approximation, the Hamiltonian matrices T c and T v inh(l)Eqn. (6.2) are defined with diagonal elements Tiic = εci and Tiiv = εih(l)elements Tijc = Jijc and Tijv = Jij, off-diagonal

for |i − j| = 1, and zero for all other off-diagonal

elements. The Coulomb potential has the form [125]:
Vij = U0a,|i − j| a + a0

(6.11)

101
where a is the site spacing constant, a0 regularizes the potential in order to have a finite
exciton binding energy, and U0 characterizes the strength of Coulomb interactions.
Based on the one-dimensional tight-binding model, 2D spectra of different excitation polarizations were calculated with 40 sites [57]. Realistic material parameters, suchas effective masses, dipole matrix elements, resonance centers, and oscillator strengths
come from the experimental linear absorption spectrum. The experimental conditions,
including laser pulse width, tuning, excitation polarization and pulse ordering, are incorporated into the model. The dephasing times of excitons and biexcitons are taken as
phenomenological parameters and adjusted to match the experimental spectra. Unlike
the phenomenological few-level model discussed in Chapter 5, the many-body contributions are intrinsic in the microscopic theory, and their relative strengths cannot be
adjusted. Details of the microscopic theory and numerical calculations have been discussed in reference [130].

6.3

Excitation dependence of 2D spectra
In this section, we demonstrate the dependence of 2D spectra on the tuning andpower of the excitation pulses. First of all, the power dependence of FWM signal isstudied to determine the excitation density for 2D experiments. The TI FWM intensity
is measured as a function of the excitation pulse power, with τ corresponding to thepeak of the signal, T equal to 100 fs, and the laser tuned to the LH exciton resonance.
Similar to Fig. 4.8, the signal intensity is shown as a function of the pulse power for
excitations of colinear, cocircular, and cross-linear polarization in Fig. 6.2. The FWM
intensity in all cases increases approximately as the cube of the excitation power, until
saturation starts at about 1 mW, which corresponds to an excitation density about
1010 excitons/well/cm2 .
In Fig. 6.3, real part 2D spectra of the rephasing pathway with colinear-polarized

102
C o lin e a r - p o la r iz e d
C o c ir c u la r - p o la r iz e d
C r o s s - lin e a r - p o la r iz e d
C u b ic

In te n s ity ( m V )1 0 0

T I-F W M

1 0 0 00 .0 11 0
10 .1
0 .0 0 1
0 .0 0 0 1

0 .1
1
P o w e r p e r b e a m (m W )Figure 6.2: Excitation power dependence of TI FWM intensity for excitations of colinear(solid square), cocircular (solid circle), and cross-linear (hollow square) polarization.

excitation are depicted for four different laser tunings. For each tuning, the spectra at
two excitation powers, 0.8 mW and 1.6 mW, are shown. The linear absorption spectrum and respective excitation spectrum are displayed above each pair of spectra. The
lowest tuning corresponds to the peak of the laser spectrum coinciding with that of
the HH exciton resonance. From there it is increased to be halfway between the HH
and LH excitons, coincident with the LH exciton, and finally several meV above the
LH exciton, which puts it well into the continuum of unbound electron-hole pairs. The
laser bandwidth is sufficient so that both exciton resonances and unbound electron-holepairs are excited in all four cases. Clearly, the relative peak strengths vary with tuning,as a relatively straightforward effect caused by the changing excitation density due tospectral overlap.
All of the 2D spectra in Fig. 6.3 exhibit some common features. Similar to
the amplitude 2D spectra discussed in Section 4.6, two diagonal peaks appear at thephoton energies of the HH and LH excitons, with two off-diagonal peaks indicatingAbsorbance

103
2.0
1.0

-1545

1.0

Absorption Energy (meV)

-1550

0.5

-1555

0

-1545

-0.5

-1.0

-1550

-1555
1545

1550

1555

1545

1550

1555

1545

1550

Emission Energy (meV)

1555

1545

1550

1555

Figure 6.3: Experimental real part 2D spectra with colinear-polarized excitation, rephasing time ordering and T = 200 fs. For each tuning, the lower spectrum is taken at twice
the power of the upper one. The laser spectrum is overlaid on the linear absorption
spectrum above the respective pair of spectra [57].

the resonant coupling. Cross peaks are expected for colinear-polarized excitation as
the shared electronic state in the conduction band results in resonant coupling. For
higher laser tunings, a vertical stripe also arises at the emission photon energies of both
excitons, as a result of the excitation of unbound electron-hole pairs, or continuum.
With the calculations based on the modified optical Bloch equations in Chapter 5, wehave demonstrated that the continuum states manifest as a diagonal feature in 2D
spectra, rather than a vertical stripe, if they act as a set of inhomogeneously broadenedtransitions. The vertical stripes occur because free pairs strongly couple to the excitonic
resonances, resulting in a dominant signal at the excitons [51].
Microscopic calculations were performed by our collaborators in Philipps Universität, Marburg, for 2D spectra with colinear-polarized excitation at different tunings
corresponding to those used in Fig. 6.3 [57]. As shown in Fig. 6.4, the experimental

104
real part spectra are qualitatively reproduced, including the spectral structures, thedispersive lineshapes, and the tuning dependence. The good agreement indicates thatthe theory well captures the many-body processes. Since the theory is only valid in the
weak excitation limit, the dispersive feature in the experimental spectra for excitation
at the HH exciton and for higher intensity must be the result of effects beyond thirdorder in the incident electric field. The relative strengths of peaks depend strongly on

Absorbancethe dephasing rates of the excitonic resonances.

1.0
0.5
0

Absorption Energy (meV)

1.0

-1545

0.5

0

-1550

-0.5

-1555
1545

1550

1555

1545

1550

1555
1545
1550
Emission Energy (meV)

-1.0

1555

1545

1550

1555

Figure 6.4: Real part 2D spectra of colinear-polarized excitation calculated with themicroscopic semiconductor theory, at different tunings. T = 200 fs, and an inhomogeneous broadening of 0.7 meV is applied. The calculated linear absorption spectrum
overlaid with the laser spectrum is shown at the top. Calculations were performed bycollaborators [57].

6.4

2D spectra with cocircular-polarized excitation
Although colinear-polarized excitation is the most straightforward experimen-

tally, it is not optimal for revealing many-body contributions. Cocircular-polarized
excitation, on the other hand, provides an immediate and striking demonstration of thedominant role that many-body processes play in semiconductors. Based on the selection
rule of optical transitions between the HH and LH valence bands and the conduction
band, as depicted in Fig. 2.3(b), no coupling between HH and LH excitons would beAbsorbance

105
1.0

0

1565

1560

1555

Absorption Energy (meV)

1.0

1550

0.5

1545

T = 100 fs-1545

T = 2 ps

0

-0.5

-1550

-1.0

-1555

-1560

-1565
1545

1550

1555

1560

1565 1545 1550

1555

1560

1565

Emission Energy (meV)

Figure 6.5: Experimental real part 2D spectra with cocircular-polarized excitation. Bothrephasing (Bottom) and non-rephasing (Middle) are shown for T = 100 fs (Left) andT = 2 ps (Right). The linear absorption and laser spectrum are shown at the top [57].

expected for cocircular-polarized excitation. Previous FWM [131], spectrally-resolved
differential transmission [132], and coherent excitation spectroscopy [129] studies haveshown that indeed coupling does occur in cocircular-polarized excitation, attributing
it to many-body correlations. Experimental 2D spectra of both rephasing and nonrephasing pathways with cocircular-polarized excitation are shown in Fig. 6.5. A laser
tuning overlapping the LH exciton resonance is used, with T = 100 fs and 2 ps. The
clear appearance of cross peaks between the HH and LH excitons in both pathways,
as a strong evidence of the many-body coupling, confirms the conclusion of the earlier
studies.
To check the manifestation of many-body effects in 2D spectra, microscopic cal-106
culations were performed with different levels of Coulomb interactions. As shown in
Fig. 6.6, the real part 2D spectra with cocircular-polarized excitation are calculatedwith contributions of Pauli blocking only, within the Hartree–Fock approximation, andwith higher-order Coulomb correlations, respectively. Both rephasing and non-rephasing
pathways are shown, with T = 100 fs. If only Pauli blocking terms are included, theCoulomb interaction is neglected except for the part that contributes to exciton for-Absorbancemation, and the optical nonlinearity arises from the saturation of the resonances by

1.0
0.5
0

1570

Pauli blocking

Hartree-Fock

Full calculation

1565
1560
Absorption Energy (meV)

1.0

1555
0.5

1550
0

-1550
-0.5

-1555
-1.0

-1560
-1565
-1570
1550 1555 1560 1565 1570 1550 1555 1560 1565 1570 1550 1555 1560 1565 1570
Emission Energy (meV)

Figure 6.6: Real part 2D spectra of cocircular-polarized excitation calculated with themicroscopic theory by collaborators [57]. Both rephasing (Bottom) and non-rephasing(Middle) cases are shown with only Pauli blocking included (Left), within the Hartree–
Fock approximation (Center), and for the full calculation including all correlation terms(Right) for cocircular-polarized excitation with T = 100 fs. The calculated linear absorption and laser spectrum are shown at the top.

107
phase-space filling. In this case, no cross peaks occur, and the continuum states appear
on the diagonal. The two diagonal peaks from HH and LH excitons present a strong
absorptive feature in both rephasing and non-rephasing pathways. When the Coulombinteraction is included, but only in the Hartree–Fock approximation, weak cross peaks
appear, but the relative strengths of the cross peaks and diagonal peaks do not agreewith the experiment. In addition, the continuum states display cross peaks, but no vertical feature, as occurs in the experiment. Only the full calculation, including Coulomb
correlations beyond the Hartree–Fock approximation, provides good agreement withthe experiment. The relative strengths of peaks are reproduced, with the cross peakat the LH exciton absorption photon energy and HH exciton emission photon energy
dominating. Furthermore, the continuum exhibits vertical stripes at the exciton emission photon energies. Therefore, we conclude that the higher-order exciton correlations
beyond the Hartree–Fock limit play dominant roles in the nonlinear optical response of
semiconductors.
Enhancement of low-energy resonances can also occur because of incoherent relaxation. The incoherent relaxation can be studied by varying T , as it occurs when thesystem is in a population state [78]. In the 2D spectra of T = 100 fs and 2 ps shown
Fig. 6.5, there is some limited strengthening of the emission at the HH exciton emission
photon energy with T , thus the dominance of the HH exciton emission is not caused byincoherent relaxation. The microscopic theory is within the coherent limit, so incoherent relaxation is not considered. The good agreement provided by the coherent limit
further supports the conclusion that incoherent relaxation is not significant for spectra
taken with T of a few hundred femtoseconds.

108

6.5

2D spectra with cross-linear-polarized excitation
As the bound state of two interacting excitons, the biexciton has significant in-fluences on the optical properties of semiconductor quantum wells at low temperatures.
Prominent evidences include the presence of beats with a frequency corresponding to
the biexciton binding energy for “negative” delays [4] or mediated by strong inhomogeneous broadening [133]. With a small binding energy of a few meV in quantum wells,
biexcitons are often masked by inhomogeneous broadening due to the disorder in thesystem, making it difficult to isolate their contributions in FWM signal.

1.0

Absorption Energy (meV)

-1545
0.75

-1550
0.5

-1555

0.25

-1560

-1565

0

1545

1550

1555

1545 1550
1560 1565
Emission Energy (meV)

1555

1560

1565

Figure 6.7: Experimental (left) and calculated (right) amplitude 2D spectra with crosslinear-polarized excitation, where the polarization of the first beam is perpendicular tothat of the rest. T = 100 fs and a higher excitation power is used [57].

Biexcitonic effects are more apparent in experiments with cross-linear-polarized
excitation, where the polarization of the first beam is perpendicular to that of the rest,since many-body interactions arising from single exciton resonances are suppressed inthis case. Fig. 6.7(a) shows an experimental amplitude 2D spectrum of cross-linearpolarized excitation, with T = 100 fs and a higher excitation power because of the
reduced signal strength in this polarization configuration. Currently only the amplitude spectrum is available, since the spectrally-resolved differential transmission mea-

109
surement for the determination of the global phase in 2D spectra is no longer valid
for cross-linear excitation. Similar to the 2D spectrum of colinear-polarized excitation,
there are two diagonal peaks and two cross peaks in Fig. 6.7(a), however, all thesepeaks are dominated by a horizontal elongation. This feature is due to the arising of
biexciton peaks, which are red-shifted by about 2 meV compared to the single-exciton
peaks. The manifestation of biexcitons is verified by the calculated 2D spectrum forcross-linear-polarized excitation, as depicted in Fig. 6.7(b), where the horizontal elongation is not as significant as that in the experimental spectrum without optimizing
parameters in the calculation. The elongation of the peaks corresponds to the emission at the exciton-to-biexciton transition. This feature disappears in calculations thatneglect two-exciton states [33]. Biexciton coherence is formed by either the transition
from an exciton population or via a direct and coherent two-quantum transition fromthe ground state to the biexciton state. The two-quantum transition can be isolatedwith experiments where all three excitation pulses are phase-locked [33].

(b )

(a )

B ie x c ito n p e a k a r e a
1 0 0 0P e a k a re a (a .u .)P L S p e c tru m(a .u .)

3 0 0 02 0 0 01 0 0 01 0 0
1 0
E x c ito n p e a k a r e a10
1 5 4 01 5 4 51 5 5 0P h o to n e n e rg y (m e V )0 .1

1
P L e x c ita tio n p o w e r ( m W )1 0

Figure 6.8: Exciton and biexciton peaks in power-dependent photoluminescence (PL)spectra. (a) A PL spectrum (square and dot line) measured at excitation power of
1 mW is fitted with an exciton peak (dash-dot line) and a biexciton peak (dot line) of
the Lorentzian shape; (b) The peak areas of exciton (solid circle) and biexciton (opencircle) from the fits in (a) versus PL excitation power.

110
The formation of biexciton can also be observed in photoluminescence (PL) spectra. In the PL spectrum shown in Fig. 6.8(a), the peak at HH exciton energy is revealed
as the combination of an exciton peak and a biexciton peak by a double Lorentzian fit.
The peak height, peak center and linewidth can be obtained from the fit, which gives abiexciton linewidth equal to about twice that of the exciton. The power-dependent PL
measurement further confirms the biexciton formation. PL spectra were measured withvarying excitation power and the relative peak areas of exciton and biexciton contributions were extracted by the fit. Fig. 6.8(b) shows the exciton peak area has a linear
dependence on the excitation power, whereas the biexciton peak area increases quadratically before saturation. This power-dependence of the biexciton is in good agreementwith an early work on the thermodynamic evaluation of exciton and biexciton populations [134].

Chapter 7

Raman Coherences Revealed by Alternative 2D FTS

In general, a heterodyne-detected transient FWM signal is three-dimensional,
i.e. it is determined by time variables τ , T , and t in time-domain, thus its spectral
representation is a function of ωτ , ωT , and ωt generated by a three-dimensional Fouriertransformation. In the 2D FTS approach discussed in previous chapters, a spectrum
is displayed as a map on two frequencies, the absorption frequency ωτ and emission
frequency ωt , while the waiting time T is fixed. With a different scheme of Fouriertransform variables, an alternative approach can project the photon echo signal in theplane of ωT and ωt , with a constant τ . The spectrum projected in this way is denoted
as SI (τ, ωT , ωt ), whereas the conventional 2D FTS is referred as SI (ωτ , T, ωt ). In thischapter, we demonstrate that the new 2D projection can isolate Raman coherences andother many-body correlations that are mixed in the conventional 2D FTS, thus provides
complementary information to the latter one [135].
Raman coherences are of interest for the study of coherent processes in photosynthetic complexes [78, 136] and in semiconductors. In semiconductors, non-radiative
Raman coherence between HH and LH excitonic states was measured from the quantum
beats in a transient absorption experiment [137]. Further work studied the essential roleof exciton-exciton interactions in producing the signal [138]. Evidence for inter-valenceband coherences was also found by using the optical Stark effect [132]. Recently, timeintegrated FWM was measured in three-pulse experiments for the simultaneous de-112phasing of both the Raman and optical coherences [139], allowing the determination of
the correlation coefficient for the dephasing processes. The presence of non-radiative
Raman coherences is essential for effects including electromagnetically-induced transparency, lasing without inversion, and slow light [140].

7.1

Coherent pathways contributing to the photon echo signal
To learn how Raman coherences can be isolated by the alternative 2D approach,we compare the contributions of coherent pathways to peaks in the two 2D projections.

Within the rotating-wave approximation, three fundamental coherent path-ways contribute to the photon echo signal emitted in the phase-matched direction
kI = −k1 + k2 + k3 . These pathways, corresponding to the processes of ground-statebleaching (GSB), excited-state emission (ESE), and excited-state absorption (ESA) [82],can be respectively represented by the Feynman diagrams labeled with (1), (2), and (3)in Fig. 3.2(a). We first consider the coherent pathway contributions from HH and LH
excitons only, in this case only diagrams (1) and (2) are involved. For a three-levelsystem with HH and LH exciton states |eH i and |eL i, and ground state |gi, with atransition frequency ωH between |eH i and |gi and ωL between |eL i and |gi, there are
four diagrams of the GSB type and four of the ESE type. They are all depicted inFig. 3.6(a). The rephasing 2D spectrum of the SI (ωτ , T, ωt ) projection is illustrated
in Fig. 7.1(a), which is a replica of Fig. 3.6(b) to allow comparison between different
2D projections. Each diagonal peak or cross peak arises from the contributions of two
different coherent pathways. The pathways corresponding to the Raman coherences
between the two excited states, (2c) and (2d), mix up with (1c) and (1d), respectively,
and contribute to the two cross peaks. With such a degeneracy, interference of the two
types of coherent pathways forms, making it not possible to observe Raman coherences
alone. In general, one has to measure SI (ωτ , T, ωt ) spectra with varying waiting time T

113
and observes the oscillation of cross peak strength [78, 136]. This indirect approach forthe study of Raman coherences requires a large number of 2D spectra to be measured
and thus quite time consuming.

(a)

(b)

− ωH

(1a)
(2a)

(1d)
(2d)

(3a-) (3a+)(3a’-) (3a’+)(3d-) (3d+)

− ωL

(1c)
(2c)

(1b)
(2b)

(3b’-) (3b’+)(3c-) (3c+)(3b-) (3b+)

ωL

ωH

ωτ

ωH

ωt

(c)

0

ωH − ωLωL(d)ωL − ωH

ωT

ωt

(2d)

(1a) (1c)(2a)(1b) (1d)
(2b)

(2c)

ωH

(3d-) (3d+)(3a-) (3a+)(3b’-) (3b’+)(3a’-) (3a’+)(3b-) (3b+)(3c-) (3c+)

ωt

ωL

ωH

ωt

ωL

Figure 7.1: The schematic 2D spectrum arising from the contributions of type (1) andtype (2) coherent pathways, and that from type (3) pathway contributions are shown
respectively in (a) and (b) in the conventional SI (ωτ , T, ωt ) 2D projection, and in (c)and (d) in the alternative SI (τ, ωT , ωt ) projection.

In the SI (τ, ωT , ωt ) 2D projection, coherent pathways distribute among peaks
differently. As shown in Fig. 7.1(c), Raman coherence pathway (2c) contributes tothe side peak at ωt = ωH and ωT = ωH − ωL , whereas (2d) falls onto another side
peak at ωt = ωL and ωT = ωL − ωH , completely separated from other pathways. The

114
primary peak at ωt = ωH and ωT = 0 arises from three pathways: (1a), (2a), and (1c),whereas the other primary one is from (1b), (2b), and (1d). Similarly, the coherent
pathways involving two-exciton states also contribute to 2D spectra differently in the
new projection. As listed in Fig. 3.7(a), there are six such diagrams derived from thetype (3) diagram (ESA) in Fig. 3.2(a). The pathway (3a) leads to |fH i, the state of
two HH excitons, pathway (3b) to |fL i, that of two LH excitons, and pathways (3a0 ),(3b0 ), (3c), and (3d) to |fM i, the state of mixed HH and LH excitons. The 2D spectrumwith two-exciton contributions in the SI (ωτ , T, ωt ) projection is shown schematicallyin Fig. 7.1(b)(duplicated from Fig. 3.7(b)), along with the SI (τ, ωT , ωt ) spectrum in
Fig. 7.1(d), separated from the 2D spectra of single-exciton contributions in (a) and (c)for clarity. Each two-exciton state can have three different conditions: the bound state,unbound state, and bare two-exciton state (two correlated excitons without energy shiftarising from the correlation). Consequently, all peaks in the 2D spectra of (b) and (d)are elongated along ωt axis to reflect the red-shift by bound two-excitons and blueshift by unbound two-excitons. The pathways of bound and unbound two-excitons areindicated respectively by a suffix “-” and “+” to the labels of diagrams. Apparently,
two-exciton coherent pathways contribute differently in the two projections. Pathways(3c) and (3d) are isolated as the respective side peak at ωt = ωH , ωT = ωH − ωL ,and ωt = ωL , ωT = ωL − ωH in the SI (τ, ωT , ωt ) spectrum. The contribution from the
ESA pathways (type (3) diagrams) to Raman coherence has a minus sign, as comparedto that from ESE (type (2) diagrams), thus it reduces signal strength. However, thecontribution from two-exciton states in semiconductor quantum wells is small if theexcitation pulse is in near resonance with the single-exciton resonances. In this case,the two side peaks in the SI (τ, ωT , ωt ) projection are dominated by the ESE pathways.

115

7.2

Experimental SI (τ, ωT , ωt ) spectra
In this section, we demonstrate the experimental implementation of SI (τ, ωT , ωt )2D projection. The same box-geometry described in Chapter 4 is employed to obtain
photon echo signals in the kI phase-matched direction. The delay T between the second
and third excitation pulses is scanned with a step size of 26.67 fs by a translation stage,
whereas the delay τ is set to 0 and actively stabilized by the servo loop. The phaseof the reference relative to the third pulse is also actively stabilized for the heterodyne
detection. The SI (τ, ωT , ωt ) 2D spectrum is produced from the complex FWM electricfield retrieved from the spectral interferometry with the reference. The direct frequency,
ωt , is obtained from the spectral interferometry, and indirect frequency, ωT , from theFourier transform with respect to T that is performed numerically.

1 .5 0E x c ita tio n p u ls e

1 .0 0A b s o rb a n c e

1 .2 5
0 .7 5

1 .0 00 .7 5

0 .5 0L ig h t h o lee x c ito n

0 .5 00 .2 50 .2 50 .0 0N o r m a liz e d la s e r s p e c .

H e a v y h o lee x c ito n0 .0 01 5 4 51 5 5 01 5 5 51 5 6 0P h o to n e n e rg y (m e V )Figure 7.2: Linear absorption of the GaAs/Al0.3 Ga0.7 As multiple quantum well sample(solid line) and the excitation pulse spectrum (dash line).

A GaAs multiple quantum well sample with the same well structure as describedin Section 4.2 was used for the 2D measurements. In Fig. 7.2, the linear absorption
spectrum of the sample shows an energy separation of 8.4 meV between HH and LH
exciton resonances. The laser is tuned to the LH exciton resonance to compensate for

116

(a)

+15

(b)

+10

ΩT (meV)

+5
0
-5

-10
-15

1545

1550

1555

1545

1550

1555

Ωt (meV)Figure 7.3: Experimental amplitude SI (τ, ΩT , Ωt ) spectra measured in the photon echophase-matched direction with colinear-polarized (a) and cocircular-polarized (b) excitations. Both spectra are normalized to amplitude separately and represented with 100
contour lines at 10% of the full scale to enhance side peaks.

its weak oscillator strength. The excitation power for new experiments is comparable
to that used in conventional measurements.
The SI (τ, ωT , ωt ) 2D spectra were measured with colinear- and cocircular-polarized
excitations. The experimental results are shown in Fig. 7.3(a) and (b), where the twoaxes of the spectra are defined as Ωt = ~ωt and ΩT = ~ωT in photon energy. Bothspectra are plotted at 10% of the full scale to enhance the weak side peaks. The twoside peaks show up at (Ωt , ΩT ) = (1546.0, −8.4) and (1553.4, 8.4), well separated fromthe strong primary peaks at ΩT = 0. The appearance of side peaks manifests the coherent pathway analysis in the previous section. The lower side peak arises from pathways(2c) of Raman coherence and (3c±) of two-exciton contributions, whereas the upper
one from the pathways (2d) and (3d±). It is worth noting that the relative strength ofside peaks in the spectrum with cocircular-polarized excitation (Fig. 7.3(b)) is smaller

117
than that in the case of colinear-polarized excitation (Fig. 7.3(a)).
We note that “windowing” is applied to the time series of FWM signal withwaiting time T for the spectra in Fig. 7.3. The decay of Raman coherence with waitingtime T happens simultaneously with the population relaxation, whereas the latter lasts
much longer than the Raman coherence. Truncation at the end of the time series
with limited length causes severe “ringing” along ΩT after the Fourier transform. An
appropriate window function is usually needed to suppress the ripples without undue
linewidth broadening along transform direction [8]. The cross-sections along ΩT atΩt = ΩH and ΩL in the 2D spectrum with colinear-polarized excitation (Fig. 7.3(a))
are shown in Fig. 7.4(a) and (b), respectively. The weak side peaks are made visible
from the ripples when a Hanning window is applied, which is of the form h(T ) =0.5 + 0.5 cos(πT /T0 ). The Hanning window 1 and 2 in Fig. 7.4 correspond to a T0 equal
to 60% and 100% of the total time series length, respectively.

(a )0 .2 0N o W in d o w
H a n n in g 1
H a n n in g 2

N o W in d o w
H a n n in g 1
H a n n in g 2

A m p litu d e ( a .u .)0 .1 5A m p litu d e ( a .u .)(b )0 .1 00 .1 00 .0 50 .0 0
-1 5

-1 0

-5

0

ΩT ( m e V )

5

1 0

1 5

0 .0 50 .0 0
-1 5

-1 0

-5

0
5
ΩT ( m e V )

1 0

1 5

Figure 7.4: Cross-sections along ΩT at Ωt = ΩH (a) and ΩL (b) in experimental amplitude SI (τ, ΩT , Ωt ) spectrum with colinear-polarized excitation. The suppression ofripples by applying window is demonstrated.

118

7.3

Microscopic calculations of SI (τ, ωT , ωt ) spectra
To better understand the experimental observations, we compare to microscopic

calculations of the SI (τ, ωT , ωt ) spectra, which were performed by our collaborators
in University of California, Irvine [135]. The calculations are based on a multi-band
1D tight-binding model [33], which includes HH, LH excitons and their continuum
states in a tractable way and accounts for various features in time-integrated/timeresolved FWM [25] and 2D FTS [33, 107] qualitatively. The equations of motion are
truncated according to the dynamics-controlled truncation scheme [124, 141]. In thecalculations, material parameters are chosen to fit the HH and LH exciton resonanceenergies, oscillator strengths and energy splitting between them.

(b)ΩT (meV)(a)Ωt (meV)Figure 7.5: Calculated amplitude 2D spectra with colinear-polarized (a) and cocircularpolarized (b) excitations. Energy values along the Ωt axis are relative to the top of the
bandgap. Both spectra are normalized to amplitude separately and represented with 50
contour lines at 30% of the full scale.

119
Fig. 7.5 shows the calculated SI (τ, ΩT , Ωt ) 2D spectra of colinear-polarized andcocircular-polarized excitation in (a) and (b), respectively. Energy values along the Ωtaxis are relative to the top of the bandgap. The HH (LH) exciton resonance energy
is -37.9 meV (-29.5 meV). In each spectrum there are two Raman coherence peaks, one
centered at (Ωt , ΩT ) = (−37.9, −8.4), and the other at (Ωt , ΩT ) = (−29.5, 8.4). We findthat the relative strength of side peaks in (b) is smaller compared to that in (a). The
appearance of side peaks and the relative peak strength agree with the experimental
result in Fig. 7.3 qualitatively.

(b)ΩT (meV)(a)Ωt (meV)Figure 7.6: Calculated amplitude 2D spectra with colinear-polarized (a) and cocircularpolarized (b) excitations in the time-dependent Hartree–Fock approximation. Energy
values along the Ωt axis are relative to the top of the bandgap. Both spectra arenormalized to amplitude separately and represented with 50 contour lines at 30% of thefull scale.

In comparison, Fig. 7.6 shows the calculated 2D spectra with colinear-polarized
and cocircular-polarized excitations in the time-dependent Hartree–Fock (TDHF) ap-

120
proximation [33], i.e. without including correlated two-exciton states. There are still
two side peaks corresponding to Raman coherences in Fig. 7.6(a), meaning HH and LH
excitons are coupled in the case of colinear-polarized excitation, even within the TDHF
approximation. However, the side peaks disappear completely in the spectrum withcocircular-polarized excitation in Fig. 7.6(b), where the weak features close to the side
peak positions are from the continuum states. Therefore, there is no coupling between
HH and LH excitons in the TDHF limit, with cocircular-polarized excitation. The side
peaks presented in the full calculation (Fig. 7.5(b)) are from higher-order correlation
effects beyond the TDHF approximation. The alternative 2D projection of photon echo
signal, SI (τ, ΩT , Ωt ), provides additional microscopic information about the coupling of
excitons.

Appendix A

2D FTS of Double Quantum Wells

An asymmetric double quantum well (DQW) structure consists of two quantum wells of unequal thickness separated by a barrier layer. With electronic couplingstrength and tunneling rate between the two wells controlled conveniently by the barrier
thickness, the DQW structure is a unique model system for many intriguing dynamic
processes in semiconductors [2]. In practice, many optoelectronic devices employ DQW
structures, where the electronic coupling between inter-well transitions has significant
influences on the electro-optical properties [142].
We apply the approach of 2D FTS to DQW structures for the identification ofelectronic coupling and measurement of coupling strength [143]. Two different DQW
samples are studied, both consisting of 10 periods of alternating 8 nm and 9 nm GaAs
quantum wells. The thickness of the barrier, Al0.3 Ga0.7 As, is 10 nm in one sample(referred as sample A) and 1.7 nm in another (sample B). Based on numerical calculations of the single-particle eigenstates of the conduction band, HH, and LH valence
sub-bands [144], the energy level structure and possible optical transitions for the two
DQW samples are depicted schematically in Fig. A.1, where the electron, HH, and LH
states in the wide (narrow) well are labeled as E1 (E2), HH1 (HH2), and LH1 (LH2),
respectively. Four optical transitions are allowed in the DQW sample A, as shown in
Fig. A.1(a). In the order of increasing energy, these transitions are HH1→E1, HH2→E2,
LH1→E1, and LH2→E2. The corresponding exciton resonances observed experimen-

122
tally are denoted as XHH1→E1 , XHH2→E2 , XLH1→E1 , and XLH2→E2 , respectively. Witha barrier of 10 nm thick, electron and hole wavefunctions are localized in individual
wells, therefore no coupling is expected between exciton resonances in different wells.
In the DQW sample with a barrier thickness of 1.7 nm, the electron wavefunctions areextended in both wells and overlap with the HH wavefunctions, resulting in six possible transitions: HH1→E1, HH2→E1, HH1→E2, HH2→E2, LH1→E1, and LH2→E2.
The energetic splitting between the two HH states is so small that they are unresolvedin our experiments. The four resonances observed are tentatively labeled as XHH→E1 ,XHH→E2 , XLH1→E1 , and XLH2→E2 from the lower to higher energies. The accurate assignment of these resonances is difficult due to complications caused by valence band
mixing and strain effects.

(a)

(b)

Barrier

Conduction
band

Valence
band

E2

E1

HH1

Valence
band

HH2
LH1

9 nm

LH2
10 nm

Conduction
band

8 nm

E1

HH1

E2

HH2
LH1

9 nm

LH2
1.7 nm 8 nm

Figure A.1: Schematic energy level structure and optical transitions for the DQW sample with a barrier thickness of 10 nm (a) or 1.7 nm (b). The electron, HH, and LH
states in the wide (narrow) well are labeled as E1 (E2), HH1 (HH2), and LH1 (LH2),
respectively.

Next we discuss the manifestation of electronic coupling in 2D spectra. In the
DQW sample A, transitions from the HH and LH states to the electron state confinedin the same well are coupled, whereas no coupling is expected between transitions indifferent wells. Therefore, an amplitude 2D spectrum of the SI (ωτ , T, ωt ) type can beplotted schematically as Fig. A.2(a), where the red circles on the diagonal represent

123

Absorbance

1.0

0.5

0.5

Excitation pulse

1.5
1.0

1.0

0.5

0.5

0.0 0.0

0.0

Normal. Inten.

1.0

0.0

Absorption photon energy (meV)(d)
Excitation pulse

Normal. Inten.

Absorbance

(b)
1.5

⊗

⊗

CA

⊗
⊗
(a)

⊗

⊗
⊗

⊗
CB

(c)

CC

(e)

Emission photon energy (meV)

Figure A.2: (a) Expected amplitude 2D spectrum for the DQW sample A (barrier thickness of 10 nm); (b) The linear absorption of sample A (black line) and the excitation
pulse (red line) that is tuned to the middle of the lower two transitions; (c) The experimental amplitude 2D spectrum with the tuning in (b), corresponding to an area
enclosed by the red box in (a); (d) The linear absorption (black line) and the excitation
pulse (red line) that is tuned to the higher three transitions; (e) The experimental amplitude 2D spectrum with the tuning in (d), corresponding to an area enclosed by theblue box in (a). T = 6.67 ps for both 2D spectra.

the four exciton resonances, the off-diagonal green circles indicate the couplings due
to common electron states, and the circles with a cross inside mean no cross peaks
expected there. As depicted in Fig. A.2(b), the resonance peaks in the linear absorption
spectrum of sample A arise from the transitions HH1→E1, HH2→E2, LH1→E1, and
LH2→E2. Since the second and third transitions have quite close energy, they are notwell resolved in the absorption. Due to limited bandwidth of the laser pulse, not all
the four transitions can be excited simultaneously. The laser was first tuned to the
middle of the lower two transitions, as shown in Fig. A.2(b), leading to an amplitude
2D spectrum displayed in Fig. A.2(c). Only two diagonal peaks appear without any
cross peaks, confirming that these two resonances arise from the HH exciton transitions
localized in the two spatially-separated quantum wells. Then a different laser tuning
was used to cover the three transitions at higher energies, as shown in Fig. A.2(d). In

124
the resulted amplitude 2D spectrum in Fig. A.2(e), two cross peaks (CA and CB) arise
as the indication of coupling between transitions HH2→E2 and LH2→E2. It is worth
noting that the cross peak CB has the strongest strength, similar to the feature observedin the 2D spectra of ordinary multiple quantum wells. This feature is resulted from the
many-body interactions of excitons, as discussed in Chapter 5.
The appearance of cross peak CC is surprising for a large barrier thickness of
10 nm. The same cross peak is still present even when the waiting time is reduced
from 6.67 ps to 200 fs (spectrum not shown). This peak may arise from energy transfer,
although it is not clear whether energy transfer can happen effectively between excitons
localized in spatially-separated quantum wells. Possible transfer mechanisms include
dipole-dipole interaction, intrinsic structural inhomogeneity, Auger process, and twophoton absorption [143]. It is difficult to identify the exact mechanism based on themeasurements presented here. The fact that only one cross peak appears below thediagonal implies the mechanism of coupling between these two LH transitions may be
incoherent relaxation. If this is the case, the energy transfer or relaxation happens on
a time scale shorter than the laser pulse duration.
For the DQW sample B with a barrier thickness of 1.7 nm, four distinct resonances
show up in the linear absorption spectrum in Fig. A.3(b). These resonances are dueto the six possible transitions, HH1→E1, HH2→E1, HH1→E2, HH2→E2, LH1→E1,
and LH2→E2. The speculated amplitude 2D spectrum is shown in Fig. A.3(a), where
only coupling due to common electron states is considered to contribute to cross peaks.
Because of the difficulties in assigning the transitions, it is not clear if some of the crosspeaks would appear. These peaks are indicated by circles with a question mark inside.
As shown in Fig. A.3(b), the laser pulse was tuned to excite all the resonances and theresulted amplitude 2D spectrum is displayed in Fig. A.3(c). A checker-broad pattern isobserved, indicating that all the resonances are coupled with one another. This result
suggests that there are contributions from effects of valence band mixing, which are

125

Excitation pulse

0.5

1.0
0.5

0.0

0.0

Absorption photon energy (meV)

1.0

1.5

Normal. Inten.

Absorbance

(b)
2.0

?
?

?

?
?

(a)

?

(c)
Emission photon energy (meV)

Figure A.3: (a) Expected amplitude 2D spectrum for the DQW sample B (barrier
thickness of 1.7 nm); (b) The linear absorption of sample B (black line) and the laser
pulse (red line) that excites all the transitions; (c) The experimental amplitude 2D
spectrum with the tuning in (b), measured with T = 6.67 ps.

not included in the calculations of the single-particle eigenstates. Valence band mixing
would couple HH and LH states and account for the observed 2D spectrum.
The strength of cross peaks is observed to change with the waiting time. Fig. A.4
show the amplitude 2D spectra of sample B with T = 280 fs (a) and 400 fs (b), whichcorrespond to the respective maximal and minimal point in the scan of time-integrated
FWM signal with T . These changes are resulted from the interference between different
coherent pathways contributing to the cross peaks. As discussed in Chapter 7, the crosspeaks in the SI (ωτ , T, ωt ) type 2D spectra arise from two terms, referred as “ground statebleaching” and “excited state emission” [33]. The “excited state emission” terms, whenappearing at the cross-peak positions, include the Raman coherence. As the waitingtime T varies, the phase of the Raman coherence terms changes, leading to constructive
or destructive interference with the coherent pathways of the “ground state bleaching”

Absorption photon energy (meV)

126
(a)

(b)

Emission photon energy (meV)

Figure A.4: Experimental amplitude 2D spectra of sample B with different waitingtimes. (a) T = 280 fs and (b) T = 400 fs correspond to the respective maximal andminimal point in the scan of time-integrated FWM signal with T .

terms. Such oscillatory behavior of cross peak strength has been observed in the 2D
spectra of electronic transitions in photosynthetic systems [136].

Appendix B

2D FTS of Exciton Continuum

The carrier-carrier interactions in semiconductors have been demonstrated to havesignificant contributions to the coherent optical response [51, 145, 146, 147]. Unboundelectron-hole pairs, i.e. exciton continuum states, can be created with laser pulses
tuned above the exciton resonance. As observed in the SI (ωτ , T, ωt ) type 2D spectra,
a vertical stripe due to the absorption of the continuum states appears at the emission
photon energies of both HH and LH excitons. However, the continuum manifests as
a diagonal feature in the 2D spectra, rather than a vertical stripe, if it acts as a setof inhomogeneously broadened transitions, as shown by the calculations in Chapter 5.
The vertical stripes occur because unbound electron-hole pairs strongly couple to theexcitonic resonances [51], resulting in a dominant signal at the resonances.
In order to explore the structure of continuum states, 2D FTS experiments havebeen performed with excitation pulses tuned well above the exciton resonance. The
bulk exciton resonance appears at 1515 meV in a GaAs sample cooled to 10 K. Laser
pulses with a tuning of 1521 meV and an excitation power of 4.0 mW per beam isused to obtain dominant continuum states. The continuum is excited by the full pulse
bandwidth (FWHM ∼ 12 meV) and the coherence is expected to exist on a timescalecomparable to the pulse duration. Fig. B.1 depicts the experimental rephasing 2D
spectra of exciton continuum states with colinear-polarized excitation, where amplitude
and real part spectra with waiting time T of 50, 150, and 250 fs are shown. For different

128

Absorption photon energy (meV)

Amplitude

Real Part

(a1)

(a2)

(b1)

(b2)

(c1)

(c2)

Emission photon energy (meV)
Figure B.1: Experimental rephasing 2D spectra of exciton continuum. Amplitude (leftcolumn) and real part (right column) spectra with waiting time T of 50, 150, and
250 fs are shown in row (a), (b), and (c), respectively. All spectra were obtained withcolinear-polarized excitation.

values of T , a broad and almost round-shape peak appears in the amplitude plot, witha strongly-dispersive lineshape for the real part. The peak appears at a position withthe absorption and emission photon energy equal to -1521 and 1518 meV, respectively.
The red-shift from the diagonal along the emission axis is consistent with a weakening

129of the Coulomb interaction for higher k-states and possibly an increase in the dephasingrate [148]. The red-shift increases slightly with T , however, it remains small even at alarge T corresponding to the complete decoherence of the continuum states, implyingthat the incoherent scattering process may not dominate the spectra. Compared to
the rephasing 2D spectra, the non-rephasing spectra of continuum are much weaker(not shown), indicating the continuum acts like an inhomogeneously broadened system.
However, such a system would result in a diagonal feature in the rephasing spectra,
which is in contradiction to the observation. This mixed behavior may imply thatthe continuum states have an underlying inhomogeneous structure, but the states arecoupled together through many-body interactions.
While the amplitude 2D spectra are similar in the range of T discussed, the realpart spectra present a phase inversion from T = 50 fs to 250 fs. The phase oscillation only
occurs during the coherent lifetime of the population, therefore no phase inversion canbe observed with larger T . It is possible that the coherent intraband scattering processes
may result in such a phase inversion, although further investigations are needed to verify
this.
2D FTS experiments with cocircular-polarized excitation were also performed. In
the amplitude and real part spectra, only subtle differences can be observed from thecase of colinear-polarized excitation with the same T and pulse power. The weak dependence on polarization is not surprising for the continuum, since the spin-splitting of thecontinuum states is negligible compared to the range of k-states simultaneously excited.
In this regime, intraband electron-electron effects, rather than interband contributions,dominate over the initial coherence and relaxation.

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