Decoherence in Optically Excited Semiconductors: a Perspective from Non-Equilibrium Green Functions

Decoherence in Optically Excited Semiconductors: a perspective from non-equilibrium Green functions

Kuljit Singh Virk

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

Decoherence in Optically Excited Semiconductors: a perspective from non-equilibrium Green functions

Kuljit Singh Virk Doctor of Philosophy Graduate Department of Physics University of Toronto 2010

Decoherence is central to our understanding of the transition from the quantum to the classical world. It is also a way of probing the dynamics of interacting many-body systems. Photoexcited semiconductors are such systems in which the transient dynamics can be studied in considerable detail experimentally. Recent advances in spectroscopy of semiconductors provide powerful tools to explore many-body physics in new regimes.

An appropriate theoretical framework is necessary to describe new physical eﬀects now accessible for observation. We present a possible approach in this thesis, and discuss results of its application to an experimentally relevant scenario.

The major portion of this thesis is devoted to a formalism for the multi-dimensional

Fourier spectroscopy of semiconductors. A perturbative treatment of the electromagnetic

field is used to derive a closed set of differential equations for the multi-particle correlation functions, which take into account the many-body effects up to third order in the field.

A diagrammatic method is developed, in which we retain all features of the double-sided

Feynman diagrams for bookkeeping the excitation scenario, and complement them by allowing for the description of interactions.

We apply the formalism to study decoherence between the states of optically excited excitons embedded in an electron gas, and compare it with the decoherence between these states and the ground state. We derive a dynamical equation for the two-time

ii correlation functions of excitons, and compare it with the corresponding equation for the interband polarization. It is argued, and veriﬁed by numerical calculation, that

the decay of Raman coherence depends sensitively on how diﬀerently the superimposed exciton states interact with the electron gas, and that it can be much slower than the decay of interband polarization.

We also present a new numerical approach based on the length gauge for modeling the time-dependent laser-semiconductor interaction. The interaction in the length gauge involves the position operator for electrons, as opposed to the momentum operator in the velocity gauge. The approach is free of the unphysical divergences that arise in the velocity gauge. It is invariant under local gauge symmetry of the Bloch functions, and can handle arbitrary electronic structure and temporal dependence of the ﬁelds.

iii Acknowledgements

I am indebted to the exceptional teachers and researchers from whom I learned throughout my university education. I am thankful to my supervisor John Sipe for suggesting many challenging problems for my thesis. John has always encouraged me to pursue my own ideas, and he provided much insight and feedback to guide me towards interesting and rich directions in research. My development as a physicist has been enriched by the broad range of topics of deep interest to him. His enthusiasm for research, and the time he devotes to his students, make him a great supervisor.

I started my research path as an undergraduate in Prof. Jeff Young’s laboratory at UBC, and the remarkable balance of theory and experiment that exists there gave me a broad and rich perspective of physics. I have always drawn inspiration from Jeff’s creative approach to physics, and I learned from him how much fun physics is at many different levels. I thank him for his continuing support and encouragement in pursuing my own research.

I have also beneﬁted from numerous discussions and debates with Fred Nastos, Ali Na- jmaie, and Eugene Sherman. They happily gave me their time as I learned the basics while starting my life as a graduate student. I also thank Julien Rioux for providing the band structure plot shown in Chapter 1 of the thesis.

I had the privilege to spend a period of two weeks in the research group of Prof. Steven Cundiﬀ (JILA, University of Colorado at Boulder). I am grateful to him for his hos- pitality, and I thank Dr. Alan Bristow for his time and patience in explaining me the experimental side of two-dimensional Fourier spectroscopy of semiconductors.

The constant love and support from my wife, Sumandeep, has been crucial in the com- pletion of this work. She always lifts my spirits, and keeps me going through the ups and downs of research. I also thank her for freeing me from various responsibilities so that I could pursue my research. Finally, I thank my parents for their love, encouragement, and unwavering support in all my pursuits.

iv Contents

1 Introduction 1

1.1 Background ...... 1

1.2 Perspective ...... 2

1.2.1 Experiment ...... 3

1.2.2 Theory...... 6

1.3 ThesisOverview...... 11

1.3.1 Brief introduction to non-equilibrium Green functions...... 11

1.3.2 Summaryandorganization...... 15

2 Semiconductor Optics in Length Gauge 20

2.1 Background ...... 21

2.2 NumericalMethod ...... 24

2.2.1 Position matrix elements on a lattice in the discrete Brillouin zone 24

2.2.2 Implementationissues ...... 28

2.2.3 Numericaltime-evolution...... 30

2.3 IllustrationandDiscussion ...... 34

2.3.1 Quantumwellbandstructure ...... 35

2.3.2 Comparison of length and velocity gauges ...... 36

2.4 Summary ...... 40

v 3 Multidimensional Fourier Spectroscopy: Formalism 42

3.1 TheoreticalBackground ...... 43

3.1.1 Electrodynamics ...... 43

3.1.2 Basis states and Hamiltonian in length gauge ...... 50

3.1.3 Greenfunctions...... 52

3.1.4 The eﬀective two-particle interaction ...... 56

3.2 ResponseFunctionsFramework ...... 59

3.2.1 The susceptibility expansion ...... 59

3.2.2 Hierarchy of correlation functions and its approximate termination 62

3.2.3 Equationsofmotion ...... 68

3.3 Application ...... 81

3.3.1 Externalandeﬀectiveﬁelds ...... 81

3.3.2 Analysisviadiagrams ...... 83

3.3.3 Signal and two-dimensional spectrum ...... 91

3.4 Summary ...... 96

4 Multidimensional Fourier Spectroscopy: Exciton Decoherence 98

4.1 Background ...... 101

4.2 RelationshiptoExperiment ...... 105

4.3 Equationsofmotion ...... 108

4.4 Modeleﬀectiveinteractions ...... 110

4.5 Sources ...... 119

4.6 Dynamics ...... 126

4.7 NumericalMethod ...... 136

4.7.1 Electrongas...... 136

4.7.2 Numerical time-stepping ...... 137

4.8 Results...... 139

4.9 Summary ...... 150

vi 5 Conclusion 152

A Eﬀective two-particle interaction: Details 159

B Derivation of Integral Bethe-Salpeter Equations 161

C Transforming between T (j) and X(j) 163

(2) D Derivation of EOM for Xn 165

E Diagram Rule 4 167

F Two-time Approximation 168

G Derivation of EOM 171

G.1 Derivationoftwo-particleEOM ...... 171

G.2 Derivationofsingle-particleEOM ...... 172

G.3 Expressions for interaction matrix components ...... 176

Bibliography 177 List of Tables

viii List of Figures

1.1 Schematic illustration of electron-hole excitations ...... 3

1.2 Schematic illustration of a four-wave mixing (FWM) setup...... 6

1.3 Structure of dynamics controlled truncation ...... 8

1.4 Modiﬁcation of DCT in cluster expansion and Green function methods . 12

1.5 Keldyshcontour...... 14

2.1 Discretized momentum space ...... 25

2.2 Interblock and intrablock matrix elements ...... 26

2.3 Linkoperators...... 27

2.4 Plaquetteoperator ...... 27

2.5 Illustration of operators W x(k) and W y(k) ...... 33

2.6 Energy bands in the presence of Pöschl-Teller potential ...... 37

2.7 Comparison of density in the velocity and the length gauge calculations . 39

2.8 Conduction band populations for ﬁeld frequencies far below the mid gap 40

3.1 Graphical symbols for constructing diagrams ...... 56

3.2 TheBSEforfourpointfunction ...... 57

3.3 Diagrams for 4-point eﬀective interaction I(2) ...... 57

3.4 Bethe-Salpeter equations for six and eight point functions...... 61

3.5 Illustration of the hierarchy problem in many body physics...... 64

3.6 Contributions of two-particle correlations to the two-particleEOM. . . . 78

ix 3.7 Examples of pair evolution induced by three pulses ...... 89

3.8 Example diagram of a term sensitive to the composite nature of excitons 89

3.9 Source diagrams for the generation of Raman coherence ...... 91

3.10 Diagrams contributing to the biexciton amplitude ...... 91

3.11 Diagrams that contribute to the third order signal ...... 94

4.1 Diagrams of the interactions in 2-point Bethe-Salpeter equation . . . . . 111

4.2 Exchange process between an exciton and the electron gas due to vertex correctionstoself-energy...... 115

4.3 Source diagrams that transfer interband polarization to Raman coherence amongexcitonstates ...... 122

4.4 Typical (τ) ...... 135 Bnn 4.5 Two-time grid used in the computation of exciton correlation functions . 139

4.6 Exciton energies vs. electron gas density ...... 140

4.7 Plot of interband polarizations for 2s and 2p excitonstates ...... 142

4.8 Plots of Raman coherence between 2s and 2p excitonstates...... 143

4.9 Plot of a phenomenological measure for two-particle correlations . . . . . 144

4.10 Generalized rates for decay of interband and Raman coherence ...... 146

4.11 Two-dimensional Fourier spectroscopy plots for lower density ...... 148

4.12 Two-dimensional Fourier spectroscopy plots for higher density ...... 149

4.13 Proﬁles of the peaks in the two-dimensional Fourier spectroscopy plots . 150 Chapter 1

Introduction

1.1 Background

Photoexcitation of semiconductors by ultrafast laser pulses accesses a transient regime in which the most fundamental quantum processes in solids can be observed [1, 2]. The transient regime begins with a coherent superposition of the semiconductor ground state and the excited states, and ends with its destruction via decoherence due to scattering. In recent years, pulsewidths have reached down to attoseconds, and remarkable progress has occurred in the control and detection of the phase of laser pulses [2, 3]. This has enabled researchers to explore the previously inaccessible regime of dynamics in which coherent superpositions, the formation of quasi-particles, and the evolution of their distribution functions can be observed. This thesis focuses on the ﬁrst of these phenomena, and addresses some theoretical and computational challenges related to ultrafast dynamics in semiconductor systems. The ground state of a semiconductor is approximated by fully occupied single particle energy bands below the fundamental gap, and completely empty bands above it [4] (see left side of Fig. 1.1). This constitutes a semiconductor vacuum. The simplest excitation above the ground state consists of an electron promoted from a valence band to the conduction band. This excitation leaves a positively charged vacancy in the valence band, or a hole, and thus creates a two-particle state consisting of an electron-hole pair called an exciton [5, 6, 7]. The Coulomb interaction between the electron and the hole leads to a set of discrete and continuum ﬁnal states of the pair. The former, below the fundamental gap of a direct gap semiconductor, behave like bound states of hydrogen

1 1.2 Perspective 2

atoms in many ways, but at a far lower energy scale (15 meV or less in contrast to 13 eV). The latter are often approximated by the conduction bands, due to the much larger kinetic energy of the pair than its Coulomb potential energy. This two-particle picture of excitations close to the fundamental gap is shown schematically on the right in Fig. 1.1. More complex excitations such as trions (charged excitons), and biexcitons (molecules of excitons), can also be created.

Linear interaction of light with a semiconductor, at photon energies close to the fundamental gap, creates a superposition between the ground state and an exciton state. It has a natural frequency associated with the energy of the state above the ground state energy, and is often termed interband polarization. This superposition between the ground state and a state in the two-particle Hilbert space is the only one accessible in a linear response. The nonlinear response consists of a plethora of effects, the simplest of which are photon echoes arising from the inhomogeneous broadening of the continuum states, and coherent control phenomena arising from the interference of different pathways between initial and final states. These are found already within the independent particle approximation. More complex effects involve Coulomb interactions, which may be viewed as correlating different linear excitations, or the same linear excitation to different orders. Among these, and the main focus of this thesis, are the transfer of interband polarization to superpositions among the two-particle states, the so-called Raman coherences [8]. However, unlike the Raman coherences studied previously [9, 10, 11, 8], in this thesis we consider Raman coherences that arise when the initial state is not a semiconductor vacuum, but contains a finite density of carriers. This involves a different mechanism for the emergence of these coherences, and also puts decoherence into an unconventional regime.

1.2 Perspective

There are many kinds of decoherence processes that have been studied in semiconductors and their microstructures. In recent years, signiﬁcant attention has been given to the application of quantum dots to quantum information processing. Research in this ﬁeld has produced many studies of decoherence processes in semiconductors. There has also been interest in using precisely controlled doping of semiconductors by magnetic impurities to 1.2 Perspective 3

Figure 1.1: Band structure of a direct gap semiconductor (GaAs), (left), and a schematic picture of optical excitations close to the fundamental gap (right). The optical excitation of an electron to the conduction band is represented as an electron-hole excitation into the two-particle continuum. Bound states are shown as lines below the continuum. make quantum bits. Thus decoherence of spin states of electrons bound to donors, e.g. phosphorous atoms in silicon, has also been investigated [12, 13, 14, 15, 16, 17, 18, 19]. We do not touch upon these topics further in this thesis, and instead focus on decoherence of optically excited states in semiconductors, which arises mainly due to carrier-carrier interaction.

1.2.1 Experiment

Decoherence mediated by carriers has been studied actively in the past twenty years, but primarily within the context of interband polarization. The interband polarization, which is a superposition of the ground state with the excited state, experiences scattering by electrons, and phonons. The earliest experiment by Becker et al. [20] measured the photon echo amplitude as a function of delay, and extracted decay rates for various carrier densities. The results showed carrier-carrier interaction, rather than carrier-phonon interaction, as the main decoherence mechanism. The interband polarization involves states of very diﬀerent properties - that of the presence of an exciton and no exciton. In the language of decoherence theory, the states either scatter an environment particle or do not scatter at all, so that substantial decoherence occurs with any scattering event. As the carrier-carrier scattering occurs at a much faster timescale, it is perhaps 1.2 Perspective 4

not surprising that carriers dominate the decay of interband polarization.

Following the work of Becker et al., there have been numerous theoretical and experimental studies of coherence in semiconductors [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. A multitude of techniques have been applied to study transient phenomena at shorter and shorter timescales. Among these, a prominent experimental method has been four-wave mixing (FWM) spectroscopy [2, 1, 36, 3], which can be viewed as radiation generated by third order nonlinear polarization in the semiconductor medium. In these experiments the interference of two pulses creates a transient grating in the sample, which diﬀracts either part of the second pulse or another third pulse into a background free direction (see Fig. 1.2). The two pulses that generate the grating do not have to be overlapping in time - only the polarization induced by the earliest pulse need survive until the arrival of the second pulse. Thus the signal in these experiments is directly related to the interband polarization, and is free of the inhomogeneous broadening that leads to a much faster vanishing of macroscopic polarization.

Furthermore, these experiments also exhibit clear signatures of interactions. In a two pulse experiment on non-interacting or dilute media, signals in some directions other than those of the incident pulses depend on the order in which the pulses arrive. On the other hand, local field effects in the presence of interactions allow generation of signals for reverse orders, and with delays that are much longer than the pulsewidths [29, 37, 38, 39, 40]. Other signatures include biexcitonic resonances [41, 42, 43, 44, 45, 46, 47, 48], electromagnetically induced transparency [49, 50, 51, 52, 53, 54, 55], and dynamical Stark shift [56, 29, 49]. Effects of carrier density and phonon coupling on decoherence have largely been confined to theoretical work [57, 58, 59, 60, 52, 53, 47, 61, 62, 55, 63, 64]. This list of effects, and investigations into them, is by far not an exhaustive one.

Four-wave mixing experiments typically measure the intensity of the signal, which still carries information on the interference facilitated by coherence in the medium due to the nonlinear response. Recently, a new technique called two-dimensional Fourier spectroscopy (TDFS) has been introduced [65, 3]. It enhances FWM with the measurement of the phase of the signal, as well as the correlation of this phase with that of the exciting optical pulses [65, 3]. As will be shown in detail in the following chapters, a full knowledge of phase and magnitude of the signal allows one to construct the underlying two-particle correlation functions. Furthermore, this can be performed over experimentally controllable time-windows, and with high resolution [3]. The constructed functions 1.2 Perspective 5 can then be Fourier transformed, and various excitations and coherences can be separated spectrally, and spatially via selecting appropriate directions for the exciting pulses. This unprecedented control over excitations, and extraction of information from the signal, leads to a much more detailed picture of ultrafast dynamics than possible previously.

The new information that a comparison between theory and these experiments brings is valuable in at least two specific areas in semiconductor physics. One of these is the coherence between excited many-body states, which appears in the radiated electromagnetic wave at a frequency corresponding to the energy difference between the two superimposed states. In a FWM signal, this contribution is difficult to separate from the rest of the signal. In a TDFS experiment, the complex electron-hole correlation function can be inferred and plotted in a two-dimensional frequency plane; the excited coherence appears as a peak in this plot. The extra degrees of freedom in the comparison of theory and experiment allow a conclusive detection of these coherences [66], in contrast to the past studies based on traditional FWM [8].

Another use of the new technique is to probe correlated many-body states in semiconductors. In most experimental scenarios to date, initial carrier densities are often negligible compared to those induced by the optical excitation. This neglect is a good assumption because, due to the large energy difference between the excited states and the ground state in these experiments, correlations in the semiconductor vacuum do not affect significantly the subsequent dynamics of electrons. Therefore, in the past studies, decoherence effects arose mainly from the interaction of electrons with phonons and with excitation induced charge density.

However, semiconductors also oﬀer great versatility in creating strongly correlated states other than the semiconductor vacuum. Of particular relevance to this thesis are the states with signiﬁcant carrier densities prior to excitation, which may be formed by doping with impurities, or by optically injecting an electron-hole plasma. As shown in subsequent chapters, many important properties of electronic correlations in the initial state can be directly related to parameters of the excitation sequence and qualitative aspects of the two-dimensional signals. Thus TDFS opens many new possibilities in investigating correlated systems, in or out of equilibrium, with sub-picosecond resolution.

We now turn to the current theoretical methods that are used in describing the phenomena captured by experimental semiconductor optics. We pinpoint the limitations of these methods in exploring new regimes with TDFS, and in turn motivate the techniques 1.2 Perspective 6 developed in this thesis.

Figure 1.2: Schematic illustration of a four-wave mixing (FWM) setup. The signal arrives in a background free direction.

1.2.2 Theory

Hartree-Fock initial states

As mentioned above, correlations in the semiconductor vacuum are not directly relevant to the excited state dynamics probed in a majority of experimental studies. Therefore, the simplest ground state is assumed as the starting point of most theoretical approaches in the ﬁeld. This is the Hartree-Fock (HF) ground state, consisting of the Slater determinant of ﬁlled valence states of the semiconductor.

The key obstacle in the theory of interacting electronic systems is the infinite hierarchy of correlation functions that are coupled by Coulomb interaction. Axt and Stahl [67] recognized that if the initial state is a HF ground state, then this hierarchy can be truncated rigorously by expanding the correlation functions to a chosen order in the optical field. A straightforward analysis of the hierarchy of equations reveals that there is a one-to-one correspondence between the order in the field, and the number of electron and hole annihilation operators in the correlation function. This led to the celebrated method of dynamics controlled truncation (DCT), which has since been used extensively to study the nonlinear response of semiconductors in the ultrafast regime. 1.2 Perspective 7

The DCT approach, however, cannot capture any decoherence effects in an all electronic system1. This can be understood from Figure 1.3, which shows schematically the re- (n) lationship between correlation functions retained in this approach. The symbol ρm in the figure represents a correlation function of order m in the electric field containing n electron/hole annihilation operators. The arrows describe the flow of information in the coupled system of equations; the variable at the tail acts as a source term for the variable at the tip of the arrow.

In DCT, the HF ground state excludes all correlations below the diagonal in Figure 1.3, and only the couplings corresponding to the solid red arrows are retained. Thus self-energy eﬀects from inter-particle scattering do not enter dynamically in the system of equations. Since these eﬀects are entirely responsible for decoherence in a purely electronic system, the evolution of the system remains fully coherent. Furthermore, Victor et. al. [68] prove that in the DCT formulation of such systems, all correlation functions can be factorized into a product of transition amplitudes. This generalizes the perfectly coherent single particle evolution, in which the density matrix can be factorized into the product of a wavefunction with its complex conjugate, to perfectly coherent evolution of multi-particle correlation functions.

That the self energy eﬀects responsible for decoherence are missed in DCT is made explicit by Kwong and Binder in their derivation of DCT from the non-equilibrium Green function formalism [69]. Kwong et. al. translate the truncation scheme of DCT into the vanishing of contribution by classes of diagrams that contain at least one propagator proportional to the carrier density. Such propagators are identiﬁed as oriented backwards in time along a contour, as explained in Section 1.3 below. A related approach by Maialle and Sham [70] involves constructing the χ(3) response from electron and hole Green functions, dressed by Coulomb interactions. While providing a physical picture of the basic interaction processes, it has been developed only as far as the assumptions of DCT, with diagram rules that allow for only bare Coulomb interaction, assume an initial state of zero density, and stay within a two-band model of the semiconductor. Decoherence is only introduced phenomenologically in the dynamical equation for interband polarization.

In summary, the DCT approach of starting from the HF ground state and expanding correlations to a ﬁnite order in the electric ﬁeld cannot capture decoherence in a purely electronic system. Decoherence in such systems at least requires either a correlated initial

1Note that the dephasing eﬀects of a phonon bath can be included within DCT. 1.2 Perspective 8

state with ﬁnite carrier densities, or a non-perturbative treatment of the electric ﬁeld. In the latter scenario, decoherence is excitation induced and is therefore called excitation induced dephasing (EID). This thesis focuses on the former, and views decoherence as a probe of the correlated initial state.

Figure 1.3: Schematic diagram showing the structure of the dynamics controlled trun- (j) cation (DCT) method. Only the ρm on and above the diagonal (shown in blue) are included in the DCT equations.

General initial states

For states other than the HF ground state, the order of the electric field no longer implies truncation of the hierarchy of correlation functions. In the pictorial representation of Figure 1.3, this means that each column continues indefinitely as opposed to being terminated at the diagonal. Furthermore, the rest of the couplings, represented by the dotted lines, must also be included. Their inclusion sets up feedback in the system of equations, and allows all self-energy effects to appear dynamically. In practice, the columns must be terminated at some point and this can best be done on physical grounds by assuming that correlations involving a large number of electrons are dominated by their factorization into correlations involving a smaller number of electrons.

In semiconductor optics, the most widely used density matrix approach that goes beyond DCT is the method of cluster expansion introduced by Kira and Koch [71]. Cluster 1.2 Perspective 9 expansions aim to be a complete description of the quantum dynamics of electrons, phonons, and photons in optically excited semiconductors. The method is formulated using single time correlation functions involving all three types of quantum mechanical excitations in the system. As it includes all correlations that are implied by the equations of motion, including those from the initial state, cluster expansion is in principle capable of treating arbitrary initial states. In order to capture EID effects, expansion in the electric field. is also abandoned. Thus in cluster expansions, the diagram of Figure 1.3 is reduced to a single column, and all the couplings are included (see Figure 1.4a). The result is a large set of coupled nonlinear equations, which are local in time. Limitations of cluster expansion arise from the difficulty of analyzing and solving these equations in practice.

In practice, cluster expansions end up being purely numerical techniques. Their method- ology involves studying the behaviour of numerical calculations when certain terms are removed or new terms are added to the system of equations. The physical interpretation of these terms then provides insight into the role of the corresponding physical mechanisms. It is most often used in conjunction with experiments, where data is interpreted by comparison with numerical calculations to identify the dominant physical processes.

The computational limitations restrict these calculations to one-dimensional systems, and systems in higher dimensions can only be studied by appropriate reduction of the coupled equations to single dimension. Limitations of computer modeling further restrict the capability of cluster expansions. To account for an initial state other than the HF vacuum, initial correlation functions are needed in the coupled equations as parameters. In practice, they are computed by subjecting a HF ground state to an optical excitation that would lead to the formation of a quasi-equilibrium state that is close to the one desired. Naturally, the same numerical limitations that are encountered in computing the dynamics also restrict the class of initial states that can be studied in this method.

Diﬃculties also arise in treating dynamical screening in density matrix based approaches. Screening in these approaches resides in the couplings between correlation functions the equations of which are local in time. Within transport theories in the past [72, 73], as well as in cluster expansions at present, it is now understood that screening at the level of the random phase approximation [7] in single-particle correlation functions is captured by self-consistent treatment of certain two-particle correlations. Similarly, the screened interaction at the two particle level must arise from three-particle correlations. However, 1.2 Perspective 10

analysis becomes formidable already beyond the single-particle equations, and it remains difficult to ascertain the nature of the screened interaction for excitons, bi-excitons and the like in cluster expansion method. Nonetheless, cluster expansions have been highly successful in comparison with experiment, and as an ab-initio approach to semiconductor dynamics. Advances in computational science will make it more and more useful. However, it provides little insight analytically. It is also difficult to connect cluster expansion with other phenomenological approaches. Since the dynamical screening arises only in the course of numerical integration, it does not allow a clear identification of terms of interest to decoherence in a purely electronic system. It also does not allow one to develop an intuition for optically probing the properties of an arbitrary correlated many-body system. As we wish to study decoherence due to correlated initial states, we seek a complementary technique that emphasizes analysis over numerics. The approach we develop is based on non-equilibrium Green functions, and focuses on deriving effective models that relate physical properties of the correlated initial state to the radiated signal of a multipulse excitation such as in a TDFS experiment. Leaving the details to Section 1.3, and Chapter 3, we note the important features of our approach. We retain the expansion in the electric field, and remove restrictions of the Axt-Stahl theorem by taking into account the correlations below the diagonal in Figure 1.3 as well as all the couplings. The picture now changes to the one shown in Figure 1.4b. The approach may be seen as the most general order-by-order formulation in the electric field, and clearly identifies the relationship to the description of the optical response in terms of a susceptibility expansion. This is particularly suitable to TDFS experiments, which identify a finite order response function of the semiconductor system. Furthermore, the technique is applicable to any system for which an appropriate self- energy can be identified. This is the only restriction on the class of initial states. Models of self-energy and dynamical screening for a correlated system can be derived within the Green function theory itself. Furthermore, the screened interaction is employed as the fundamental interaction in construction of the framework. Therefore, even though the screening must be approximated, it is applied consistently to all quasi-particles in the system. The use of Green functions also allows us to derive many existing phenomenological approaches from this method. In addition, by the work of Kwong and Binder, our 1.3 Thesis Overview 11

approach reduces to DCT in the limit of a HF ground state as the initial state. Another advantage of using Green functions is the convenient diagrammatic representation. The present framework extends the double sided Feynman diagrams of nonlinear optics to include the depiction of Coulomb interactions. The new diagrams are valuable in relating the optical excitation sequence to the dominant many body processes in the electronic system. We expect them to be very useful in design and interpretation of future TDFS experiments.

We remark that Green functions have been applied to various problems in semiconductor optics in the past. The work by Schäfer et al. [74], based on a Green function formalism developed from functional differentiation, is a benchmark in this field. But in separating the even and odd orders of the field it relies on the assumption of a slowly varying single pulse, and is not aimed at extracting a χ(n) response. Excitation induced dephasing is studied using non-equilibrium Green functions by Jahnke and Koch [75]. In that work, the dynamics of interband polarization is studied at the level of Born approximation. Thus it does not address non-trivial two-particle effects in general, and Raman coherence of excitons in particular. Nonetheless, the works by Schäfer et al. and Jahnke et al. are important precursors to the work in this thesis.

1.3 Thesis Overview

1.3.1 Brief introduction to non-equilibrium Green functions

In the foregoing, we have mentioned non-equilibrium Green functions without an explicit deﬁnition. We now turn to a discussion of how Green functions arise naturally in one strategy for attacking the many-body problems described above. Since the method is well- known in the literature [76, 77, 78, 79, 80, 81, 82], we focus our attention on specializing it to the systems we address in this thesis.

The theoretical object corresponding to an experimental measurement is the expectation value of a Hermitian operator. In the case of optical excitation and probing of semiconductors, within the semiclassical approximation, it is the single particle operators, such as the current and charge density, that are of primary interest. In a ﬁeld theoretic description, we associate ﬁeld operators ψ (r, t) and ψ† (r, t) that annihilate or create an 1.3 Thesis Overview 12

(a) Cluster Expansions (b) Approach taken in this thesis

Figure 1.4: Diagrams showing how the DCT picture (Figure 1.3) is modiﬁed in the cluster expansion and the non-equilibrium Green function approach of this thesis. All (j) ρm participate, and hence are shown in blue.

electron at position r and which in general evolve with time t. A single particle operator may be expanded in terms of the ﬁeld operators as,

Oˆ (t) = dr dr′ O (r′, r, t) ψ† (r′, t) ψ (r, t) , (1.1) ˆ ˆ

where O (r′, r, t) are the matrix elements of Oˆ in position basis, and all operators are in the Heisenberg picture. Thus a calculation of the expectation value Oˆ(t) turns

into a calculation of the expectation values ψ† (r, t) ψ (r′, t) . The dynamicsD E of the

latter follows from the Heisenberg equations of motion for ψ and ψ†. In the absence of interactions, they reduce to the familiar semiconductor Bloch equations [6, 83, 84]. In the presence of interactions, the dynamics is coupled to the dynamics of expectation values containing an arbitrarily large number of ﬁeld operators. In one strategy, a coupled nonlinear system of equations is obtained with an equation of motion for each expectation value. This is the approach taken in density matrix formulations, in which the equations are local in time.

In the strategy of Green functions, we attach fundamental signiﬁcance to the evolution 1.3 Thesis Overview 13

of ψ (r, t) and ψ† (r, t), and consider more general expectation values of the form

Φ φˆ (1) φˆ (2) φˆ (3) φˆ (4) Φ , (1.2) 0 ··· 0 D E ˆ where φ (1) represents either ψ (r 1, t1) or ψ† (r1, t1). To see why this is useful we note that the expectation value of any operator that acts on the system’s Hilbert space is given by

Oˆ(t) = Φ † (t, t ) Oˆ (t, t ) Φ , (1.3) 0 U 0 U 0 0 D E D E

where t0 is an initial time chosen to precede any dynamics of interest, Φ0 is the state | i of the system at this time, and (t, t ) is the unitary evolution operator. In terms of the U 0 full Hamiltonian for the system and its interaction with external ﬁelds,

′ i t 1 t t (t, t ) = 1 dt′ H(t′) dt′ dt′′ H(t′)H(t′′) .... (1.4) 0 ~ ~2 U − ˆt0 − ˆt0 ˆt0 It is often written in a compact form as

i t (t, t ) = exp dt′ H(t′) , (1.5) U 0 T+ −~ ˆ t0 where orders operators in the expanded exponential such that operators at earlier T+ times appear to the right of those at later times. Extending (1.3) to H(t), and using (1.4-1.5) we see that the right hand side of Eq. (1.3) is a sum of expectation values of the form (1.2).

The interpretation of (1.3) is an important conceptual step in the construction of Green functions. The operator (t, t ) acts on the ket Φ and yields the state of the system U 0 | 0i at time t, which includes the eﬀects of all interactions the system undergoes until that time. The operator Oˆ acts on this state, and yields in general a new ket in the Hilbert

space of the system. This ket is then evolved back to the initial time t0 by the evolution

operator † (t, t ). The overlap of the resulting ket with Φ yields the expectation value U 0 | 0i of Oˆ. The backward evolution operator, † (t, t ), is given by conjugating and reversing U 0 the order of operators in (1.5) so that the operators at earlier times are placed to the left of those at later times. Mixing the two directions in time is clearly cumbersome, and instead it is more convenient to introduce a contour along the real axis which runs from

t0 to t, and then back to t0 (see Fig. 1.5). Both types of time orderings can then be viewed as putting operators that appear earlier along the contour to the right of those 1.3 Thesis Overview 14

Figure 1.5: Illustration of the Keldysh contour. The pulse indicates a possible excitation

that may occur between t0 and t. that appear later. Representing this by , where “C” stands for contour, and using TC (1.5) in (1.3), we obtain

i ′ ′ ˆ ~ C dt H(t ) C Oe− ´ Oˆ(t) = T (1.6) D i dt′ H(t′) E e− ~ C D E TC ´ D E To this point we have kept the initial time t and the state Φ arbitrary. A judicious 0 | 0i choice is to pick t = , and let Φ be a Hartree-Fock state, which may subjected 0 −∞ | 0i to correlations at t> to form other kinds of correlated ground states. Hartree-Fock −∞ states are useful because they admit Wick decomposition [76] in which (1.2) may be written as a sum of products of simpler expectation values2,

G (rt; r′t′) i Φ ψ (r, t) ψ† (r′, t′) Φ . (1.7) ≡ − 0 TC 0

This expression defines a single particle Green function. Thus instead of equations local in time for infinitely many expectation values, we obtain an equation nonlocal in time but for a single expectation value, ∂ i G rt; r′t′ H (r,t) G rt; r′t′ dr′′dt′′Σ rt; r′′t′′ G r′′t′′; r′t′ (1.8) ∂t − 0 − ˆ = δ r r′ δ t t′ . − − We have written this so-called Dyson equation using the time variables t and t′ that lie on the contour introduced above. The Hamiltonian, H0 (r, t), is a single particle operator that includes the kinetic energy of the electron, and its interaction with the electromagnetic fields. The self-energy, Σ(rt; r′′t′′), introduces corrections to this dynamics, which

2 Wick decomposition places all operators with the property φˆ (1) Φ0 =0 to the right of the rest. For Hartree Fock states such operators correspond to annihilation operators| i for empty orbitals, and creation operators for the ﬁlled orbitals. 1.3 Thesis Overview 15

in the models we adopt arise from Coulomb interactions among electrons. For the choice of Φ mentioned above, it follows that Σ may be considered a functional of G itself. | 0i Thus, in principle, (1.8) is a closed equation for G. In practice, calculation of Σ [G] is prohibitively complex, and can only be done approximately.

Note that the coupling of laser and semiconductor is contained entirely in H0. Therefore, the probe of multi-particle complexes occurs only through Σ [G] . Thus Σ [G] is the link between optics and the dynamics of excitons, biexcitons, and the like. Naturally, the approximations made while constructing Σ [G] are central to any Green function based theory of semiconductor optics.

1.3.2 Summary and organization

In this thesis, we develop a general theoretical approach for constructing the χ(n) response within the framework of nonequilibrium Green functions. We neglect phonons and restrict ourselves solely to interacting charged particles in the presence of a lattice potential. The fundamental entity here is the single particle Green function G, which can be used to calculate the charge-current density that is the source of the signal in a TDFS experiment. In this approach, which is not restricted to an uncorrelated initial state, the expansion of the response in terms of susceptibilities χ(n) is accomplished by developing an expansion (1) (1) (1) (1) for changes ∆G to G in terms of quantities Xn , ∆G = X + X + X + , where 1 2 3 ··· each term involves a higher power of the full incident ﬁeld.

The inevitable hierarchy that arises in many-body physics here manifests itself in the (1) (j) coupling of the Xn to higher correlations Xn with j > 1 (the superscript j identifies (j) the number of annihilation operators as in the ρn of 1.2.2). Different dynamical models are associated with different approximations of this hierarchy, corresponding to approximations to vertex functions or, equivalently, with sets of diagrams kept in the expansion of the self-energy. Thus the physics of any adopted approximations can be identified.

In practice, the hierarchy must be truncated. We develop a truncation scheme particularly suited to the perturbative response characteristic of TDFS. This is essential because, to treat decoherence eﬀects properly, we must go beyond treatments of the type of Schäfer et al. [74] that include only screened Hartree-Fock theory, extended to the ladder approximation of a Bethe-Salpeter equation to account for excitons. Truncation may also violate several properties of the full theory. An important fundamental property 1.3 Thesis Overview 16

of exact vertex functions is that they are functionals only of the single particle Green functions. The truncation scheme at first deviates from this property to include an explicit dependence on multi-particle correlation functions as well. The result is that the (j) quantities Xn satisfy a finite set of linear differential equations that couple them to each other and to source terms built from the renormalized correlation functions. The above fundamental property then holds for the vertex functions in the optically excited state as long as it holds for those in the initial state.

Initial states of the kind considered here are certainly relevant experimentally, and may also be controlled to a large extent in terms of the distribution function of carriers. The scattering of these carriers from the optically excited quantum state presents the main mechanism of decoherence. A description of the decoherence phenomenon in this setting, and developing a link to experimental observations of the considered eﬀects, are two of the main challenges addressed in the present work. The former is addressed using the framework of non-equilibrium Green functions, and the latter using the recently developed technique of two-dimensional Fourier spectroscopy of semiconductors [85, 86, 87].

A new perspective on decoherence arises naturally from this work. In many treatments, decoherence arises in a system interacting with a reservoir, and appears formally when the reservoir is traced over to yield a reduced density operator that then characterizes the results of measurements on the system. The approach presented here is necessarily much more general since, due to the indistinguishability of electrons, formal factorization of the Hilbert space into system and reservoir cannot be made. Here decoherence as a phenomenon can be associated with the fact that relevant experiments are sensitive only (1) (j) to Xn , and not to the higher order Xn , j > 1. Experimental consequences of these (j) (1) higher Xn arise only insofar as they affect the dynamics of Xn . While this parallels the simpler situation where there is a well-defined system and reservoir, and experimental consequences of the reservoir arise only insofar as the reservoir affects the dynamics of the reduced density operator of the system, it offers a broader view and allows for a richer perspective on what are phenomenologically called “decoherence effects.”

We apply our approach to study the decay of Raman coherence of exciton states in the background of an electron gas, and demonstrate step-by step the application of the above formalism to a case of experimental interest. The motivation for studying this problem is three-fold. First, excitons are arguably the most experimentally accessible many-body 1.3 Thesis Overview 17

states. Second, exciton states represent the simplest many-body excitations that can be used as a concrete example for testing any formalism. Third, it is an opportunity to study decoherence in a system with the aforementioned lack of system-bath separation.

We show that, at least within a physically motivated set of approximations, decoherence effects in this scenario can be brought into direct comparison with the conventional system-bath models without sacrificing the indistinguishability. We also derive diagrams and analytical expressions for the transfer of interband coherence to Raman coherence. It is shown analytically and numerically that the decay of Raman coherence depends sensitively on how differently the corresponding exciton states interact with the electron gas. Furthermore, the contribution of the composite nature of the exciton is clearly iden- tifiable in the dynamical equations derived. In a calculation, we study the decoherence between 2s and 2p exciton states.

We now turn to a computational problem in semiconductor optics that is also addressed in this thesis. Although most of the experimental work to date has focused on optical frequencies, close to the fundamental gap, new effects may emerge when low frequency excitations are possible, particularly in the burgeoning terahertz (1012 Hz) regime. These excitations can also occur in biased heterostructures, in optical resonators where the undepleted pump approximation is violated, or when the adiabatic intraband motion of carriers probes the Berry curvature (for definition, see Chapter 2 Eq. (2.10) or [88, 89, 90]). To perform detailed studies of such a wide range of phenomena, and to make experimentally observable predictions, it is desirable to have a unified numerical approach that is applicable for electromagnetic fields from very low to very high frequencies.

The interaction between semiconductors and light has largely been studied within the so- called velocity gauge, which follows directly from the minimum coupling principle. While theoretically sound, the velocity gauge is not an ideal choice in numerical studies for several reasons. First, due to the fact that electric and magnetic fields are gauge invariant, it is better to have the electromagnetic field enter the equations directly; the velocity gauge couples light and matter via the vector potential A(t). Second, in frequency space the vector potential is related to the uniform electric field via A(ω)= iE(ω)/ω, − which means that it formally diverges at zero frequency in the presence of finite E(ω) at ω = 0. At low frequencies, a numerical calculation in the velocity gauge will amplify round-off errors and thus introduce unphysical effects. For example, derivation of the first order optical susceptibility in the velocity gauge via perturbation theory yields a 1.3 Thesis Overview 18

1 2 result with three terms, one of which diverges as ω− , and another as ω− [91]. These divergences are not real in unexcited semiconductors, where the Fermi energy lies in the 1 band gap of the unexcited crystal. The prefactor of the ω− term vanishes due to the 2 time reversal symmetry, and that of the ω− term vanishes due to the effective mass sum rule. However, in a numerical integration of the Schrodinger equation the cancellations of terms that eliminate the effects of these divergences will inevitably never be exact. The problem also plagues calculations beyond perturbation theory, since an infinite number of bands is required to ensure complete cancellations, and it survives in a fully quantum electrodynamic calculation since the same sum rules are required for removing divergences there as well.

A remedy to this problem is provided by the length-gauge, which requires the calculation of intraband components of the position operator in the basis of Bloch states. The intraband components involve a diﬀerentiation of Bloch functions, which generally contain arbitrary phases when calculated by numerical diagonalization. Therefore a method is desired that circumvents these diﬃculties. It should provide a well-behaved numerical procedure that allows us to study coherent dynamics of semiconductors induced by arbitrary optical excitations, and take into account the topological properties of band structure. We develop a method that meets these criteria.

The Berry phase formulation of polarization and its numerical implementation, e.g. the work by Vanderbilt and Resta [92, 93], introduces the main concepts that we use in our approach. More recent work by Souza et al. [94] on dynamics of polarization also addresses the same physical question, but uses a different technique that requires maximal localiza- tion of Wannier functions (performed variationally) before dynamical computations can be done. This is invaluable for visualizing the Wannier functions and for computations extending over the entire Brillouin zone, but it is also susceptible to problems in numerical optimization. Recent work on the quantum Hall effects, anomalous Hall effects, and wavepacket dynamics has also stressed the importance of the intraband components of the position operator. Of most relevance to us is the wavepacket dynamics in semiconductor materials [95, 96]. Here we note the work by Sundaram et al. [95] on a single band case, and Culcer et al. [96] on the two-band case. The analytical results therein provide significant insights into the effects of Berry curvature on the coherent intraband motion of charge carriers. However, being semiclassical and restricted to intraband motion, they are not directly applicable to the ultrafast phenomena in which we are interested. 1.3 Thesis Overview 19

The approach we have developed is oblivious to the random phases of the Bloch states, and thus free of the time consuming optimization step. It also directly lends itself to a two-time Green function approach for studying many body physics. To the best of our knowledge, ours is the ﬁrst attempt at developing a formally complete numerical approach to the semiconductor-laser interaction within the length gauge that applies to the entire range of frequencies of interest. We also employ a unitarity preserving technique for integration of the dynamical equations. This prevents growth of errors over time and allows for calculations over a longer period of time than is possible with other popular but non-unitary schemes, such as Runge-Kutta. However, our methods are best suited to the problems where computations are restricted within a subset of the Brillouin zone. This includes the familiar regime of excitation near the band-gap of a direct gap semiconductor, the focus of much of traditional semiconductor optics.

To highlight the benefits of the length gauge over the velocity gauge, we show how the effects of artificial low-frequency divergence in the velocity gauge are avoided. The usefulness of the length gauge in isolating the intraband motion and studying the effects of Berry curvature become clear in the course of the presentation.

The thesis is organized as follows. In Chapter 2, we present our computational method for studying semiconductor optics within the length gauge. The contents of this chapter have been published in Physical Review B 76, 035213 (2007). In Chapter 3, we describe our formalism for multidimensional Fourier spectroscopy based on non-equilibrium Green functions. This chapter focuses only on the formalism, with a short digression into concrete illustrations at the end. An application of the formalism is presented in Chapter 4. There we discuss the excitation and decay of Raman coherences among exciton states, and the relationship between these coherences to those between the ground state and the respective exciton states. Chapters 3 and 4 are published in Physical Review B 80, 165318 (2009), and Physical Review B 80, 165319 (2009) respectively. Chapter 2

Semiconductor Optics in Length Gauge

Solving the Schrödinger equation for the density matrix directly in time domain is central to any study of transients, and the situation is no different in laser-semiconductor interaction. However, as discussed in Chapter 1, numerical calculations based on the velocity gauge are often unreliable at frequencies below the fundamental gap, and away from resonances in the semiconductor. In this chapter, we turn to a solution of this problem, and develop a numerical method based on the length gauge. As discussed below, the length gauge also presents several challenges due to the infinite extent of the crystal. A numerically well-defined operator of translations in the momentum space must be constructed in order to implement a calculation based on the length gauge. The main issue in this implementation is the phase of the Bloch functions, which may vary arbitrarily as a function of the continuous crystal momentum variable k. While there is a random contribution to this phase arising from the band structure calculation itself, there may also be a physical contribution arising from the Berry curvature [92, 88]. Thus the goal is to develop a scheme that keeps the latter intact, while removes the effects of the former.

This chapter is organized into three sections. In Section 2.1 we provide background on the issues related to the velocity and the length gauge. In Section 2.2 we ﬁrst represent the position operator on a discrete lattice in a manner directly applicable to numerical calculation, and then discuss its implementation. We then describe our algorithm for using this numerical procedure in the time-dependent Schrödinger equation for the density matrix. In Section 2.3 we apply these methods to the single electron dynamics in a quantum well. We compute the density matrix in both the velocity and the length gauge. The calculations demonstrate that the results in the velocity gauge at low frequencies

20 2.1 Background 21

are unphysical, while those in the length gauge agree with expected behaviour. Our calculations are done for a generic type I quantum well with two bound states, and an eight band Kane model for the underlying bulk material.

2.1 Background

The interaction of light with matter is described using the minimum coupling principle. Let A (t) be the vector potential describing the electromagnetic ﬁeld and p be the momentum of an electron. We represent the mass of an electron by m and the charge by e (note that in this convention, e is a negative number). Then the Hamiltonian for this particle in the presence of A (t) is 1 Hvel = (p eA (t))2 + V, (2.1) 2m − where V is the potential energy depending on the spatial coordinates of the particle. The vector potential also has spatial dependence in general, which we will neglect in the so-called dipole approximation. This approximation holds if the wavelength of the electromagnetic ﬁeld is long compared to the typical length scales associated with electronic correlations. This is often a good approximation since typical wavelengths are much longer than the typical length scales governing the dynamics of electrons. We will discuss these scales further in the following Chapter, where we consider excitations in quantum wells. In this thesis we also take A (t) to be classical, but the discussion and results of this chapter hold equally well if A (t) is a quantum-mechanical operator.

The Hamiltonian (2.1) is written using the velocity gauge, in which the interaction of light with electron takes the form, e Hvel = A (t) p. int −m · The dependence on A (t) leads to numerical problems at low frequencies due to the apparent divergence of A (ω) as ω 0 (see Chapter 1 and references therein). → In order to switch from the vector potential to the electromagnetic ﬁelds, we perform a gauge transformation on the wavefunction, φvel(r, t), of the electron. We deﬁne

vel ieA(t) r/~ φ(r, t) = φ (r, t)e · , 2.1 Background 22

substitute it in the Schrödinger equation for φvel, and obtain the Schrödinger equation for φ (r, t),

∂ p2 i~ φ (r, t) = + V eE (t) r φ (r, t) . (2.2) ∂t 2m − · The function E(t) is the electric ﬁeld,

dA (t) E(t) = , − dt and (2.2) is written in the so called length gauge. Generalizing (2.2) to the motion of electrons in a crystal, the Hamiltonian in this gauge may be written in the form

H = H er E(t), (2.3) 0 − ·

where H0 is the full Hamiltonian of the crystal. When expanded in terms of the Bloch states, the second-quantized interaction Hamiltonian (H H ) takes the form − 0

H = eE(t) dkr (k)a† (k)a (k), (2.4) int − · ˆ nm n m n X

where an† (k) and an(k) are the creation and annihilation operators for excitation into band n with crystal momentum k. Adams and Blount [90] showed that the matrix

elements of the position operator between Bloch states nk and mk′ should be taken | i | i as δ(k k′)r (k), where − nm ∂ r (k) = iδ + ξ (k), (2.5) nm nm ∂k nm and ∂ ξ (k) = i u∗ (k; r) u (k; r)dr, (2.6) nm ˆ n ∂k m is the optical transition matrix [90, 97]. Here u (k; x)= x u (k) is the periodic part n h | n i of the Bloch functions, i.e un(k; x + R)= un(k; x) where R is any lattice vector. These functions are normalized according to dr un∗ (k, r)um(k; r) = δnm, ˆ Ωcell where Ωcell is the volume of the unit cell of the crystal. The deﬁnition (2.5) is an explicit construction of r as a generator of parallel translations in the space of Bloch functions [98]. To see this more explicitly, consider the space of all Bloch kets nk , which identiﬁes | i 2.1 Background 23

a set of inﬁnite dimensional Hilbert spaces, one for each k. A ket f in this space is | i given by a vector-valued function f(k), specifying the projection of f on each of the | i basis vectors,

f = dk nk f (k). (2.7) | i ˆ | i n n BZ X When we apply I ia r on this ket, the resulting ket is described by the functions, − · ∂f (k) f (k; a) = f (k)+ a n i a ξ (k)f (k). (2.8) n n · ∂k − · nm m m X This represents a translation of fn(k) where the first two terms on the left hand side give the ordinary displacement by an infinitesimal vector a, while the third represents the geometric phase correction. If these infinitesimal translations are ordered into a closed loop, the resulting phase correction is given by the loop integral of ξ(k), and is non-trivial in general.

This non-triviality is the price that we must pay to avoid the undesirable features of the velocity gauge. To appreciate this point, we consider the matrix elements of ξ(k). Let E (k) be the energy eigenvalue of Bloch state nk . Then in general the elements n | i between non-degenerate states are related to the velocity matrix by ~v (k) ξ (k)= nm , (2.9) nm i(E (k) E (k)) n − m while no such relationship exists between components within degenerate subspaces. They are not zero, however, because of the commutation of diﬀerent Cartesian components of r. The matrix elements between states n and m that belong to a degenerate subspace, , are related to the rest of the matrix elements by[97] S Ω (k) ξ i ξ ξ = i ξ ξ . (2.10) nm ≡ ∇ × nm − nj × jn np × pm j p/ X∈S X∈S This sum rule reﬂects the fact that all three Cartesian components of the position operator

commute with each other, and deﬁnes a (possibly non-abelian) gauge ﬁeld, ξnm (k). The

matrix elements ξnm(k) between degenerate states are constrained only by the above equation, and their arbitrariness is due to the gauge symmetry of Bloch states. On the

other hand, Ωnm (k) is gauge-invariant and carries physical meaning; within semiclassical treatment it aﬀects motion of carriers in momentum space, analogous to the way in which

magnetic ﬁeld aﬀects the motion of charges in real space. Ωnm (k) is known as the Berry curvature. 2.2 Numerical Method 24

The gauge symmetry of Bloch states comes from the fact that we are free to make gauge transformations on the Bloch functions in the form

u (k; x) u (k; x)W ∗ (k) (2.11) n 7→ m nm m X where W (k) is a unitary matrix that acts as a phase for non-degenerate bands and mixes degenerate states at k. This makes the question of differentiability of Bloch states subtle, as the class of such transformations surely include those that are non-differentiable. However, for most materials, it is possible to define a gauge for the Bloch functions such that differentiability exists almost everywhere. Sets of measure zero excluded from this condition are lines and points of degeneracy. Materials excluded from this possibility of differentiation posses Brillouin zone with special topology[99]. Semiconductor based materials that we are interested here do not generally belong to this class. So we will assume that differentiability exists almost everywhere.

Nonetheless, the definition (2.6) for calculating the gauge fields is inapplicable in practice because systems with more than two or three bands are necessarily solved using numerical diagonalization, which picks an arbitrary gauge independently at different k points. Existence of differentiability almost everywhere makes it possible to deal with random phases in a way such that (2.6) may be used; it does not give us a method for doing so. In this chapter we develop such a method. Although we discuss how to handle self energies in the case of an interacting particle problem in this framework, the essential points are best illustrated within the coherent dynamics of single particle semiconductor Bloch equations. The present chapter therefore focuses on this regime.

2.2 Numerical Method

2.2.1 Position matrix elements on a lattice in the discrete Bril- louin zone

As shown in Fig. 2.1, we discretize the relevant subset of the Brillouin zone by using a rectangular mesh. The domain is given by k [ K ,K ] with λ as the vector between µ ∈ − µ µ µ nearest neighbours in direction µ. Around this domain is a buﬀer zone that is necessary to handle boundary conditions consistently with the dynamical calculation. At each k 2.2 Numerical Method 25

point, the single particle Hamiltonian matrix is diagonalized to obtain the electronic structure.

Let us say that a convention is adopted to label eigenstates at each k using integers in the order of increasing eigenvalues. We deﬁne two eigenvectors, j and n, to be disconnected if the corresponding eigenvalues Ej(k) and En(k) never cross; otherwise we call them connected. We deﬁne the elements of any operator to be interblock if they are between disconnected states and intrablock if they are within connected states. Figure 2.2 illustrates this schematically.

The interblock elements of the position operator are related to the velocity matrix elements by Eq. (2.9). Here we change the notation from Eqs. (2.5) and (2.6) by deﬁning the elements of ξµ(k) within connected subspaces to be zero, where µ denotes the Carte- sian component. We will use the symbol ϑµ(k) to denote the matrix containing the intrablock elements. To determine these elements, we note that the periodic envelope functions of Bloch states at two points are related to each other by[97]

u (k + λ ) = u (k) [δ iλ ξ (k)] gµ (k)+ O λ 2 , (2.12) | m µ i | n i nl − µ · nl lm | µ| nl X

Figure 2.1: Discretized momentum space. The shaded rectangle represents the space of interest, while the region between this rectangle and the boundaries is the buﬀer zone used for numerical consistency. 2.2 Numerical Method 26

Figure 2.2: Schematic illustration of interblock and intrablock matrix elements.

where the unitary matrix gµ(k) accounts for geometric phases that may be necessary to maintain single-valuedness in the Brillouin zone. The matrix elements of gµ(k) are deﬁned to be nonzero only between the connected states, and are related to the intrablock elements of Blount’s deﬁnition via

k+λµ gµ(k) = exp i ϑµ(k)dk . (2.13) − ˆ k A convenient choice for the integration path is a straight line linking the neighbouring points on a lattice. For this reason these operators are also termed link operators or simply links. They define fields that live not on the lattice points but in the space between them. Such a view offers intuitive rules when constructing lattice definitions of the operations of r in more complex situations. It is helpful to visualize gµ(k) as an

arrow with its tail at k and the tip at k + λµ (we have adopted the convention that the argument k corresponds to the location of the tail, see Fig. 2.3). It acts on the vectors f(k) (see Eqs. 2.7 and 2.8) in the space at its tip, and expresses them in terms of the basis of the space at its tail. Reversal of an arrow corresponds to the conjugate transpose of the associated link. This immediately gives the prescription for constructing a parallel transport operator along a speciﬁed path: concatenate links along a path γ that starts

at k′ and ends at k on the lattice.

Crucial properties of Eq. (2.13) must be respected in order that the discrete theory reduces to the continuum one in the appropriate limit[100]. In particular, (a) we must have

µ µ g− (k + λµ) = g †(k), (2.14) 2.2 Numerical Method 27

µ Figure 2.3: Link operators g (k0) originating from point k = k0 on a rectangular grid in the BZ.

Figure 2.4: Plaquette operator Zxy corresponding to the smallest loop that begins and ends at k0 on a rectangular grid in the BZ. and (b) under an arbitrary gauge transformation of the form (2.11), gµ(k) must transform according to

µ µ g (k) W (k)g (k)W †(k + λ ). (2.15) 7→ µ If these two properties are respected, then the plaquette operator deﬁned by an ordered product of links along a closed path (see Fig. 2.4),

µν µ ν µ ν Z (k) = g (k)g (k + λµ)g− (k + λµ + λν )g− (k + λν), (2.16) satisﬁes

1 µν a− [Z (k) δ ] (2.17) mn − mn = i (∂ ϑν (k) ∂ ϑµ (k)) m [ϑµ(k), ϑν(k)] n + O(a), µ mn − ν mn −h | i where a is the area of loop. Recall that the right hand side of this equation is equal to [ξµ(k),ξν(k)] by Eq. (2.10).

It remains to ﬁnd a method to construct the link matrices gµ(k) numerically while preserving properties (2.14) and (2.15). The scheme we introduce makes use only of a given 2.2 Numerical Method 28

set of Bloch functions and the associated eigenvalues. We begin by deﬁning the overlap matrix that gives basis transformations associated with translations,

Sµ (k) u (k) u (k + λ ) . (2.18) lm ≡ h l | m µ i In the case of inﬁnite number of bands it is easily seen that Sµ(k) is unitary. However a computational scheme must truncate the number of bands, and therefore break the unitarity of Sµ(k). The deviation from this unitarity measures the projection of Bloch

states at k + λµ onto the states discarded at k. Unitarity is important to keep the Hamiltonian Hermitian, and we construct an eﬀective unitary overlap matrix by ﬁrst performing a singular value decomposition (SVD) of Sµ(k)[101],

µ µ µ µ S (k)= Q1 (k)Σ (k)Q2†(k), (2.19)

µ µ where the matrix Σ contains singular values, and Qj are unitary matrices that eﬀect the basis transformation. The deviation of the singular values from unity is a measure of the incompleteness of the computational subspace. Together, they can be used to construct the polar decomposition of Sµ(k)= Qµ(k)M µ(k), where

µ µ µ Q (k) = Q1 (k)Q2†(k), (2.20) plays the role of the angle of a complex number and M µ(k) its magnitude. Thus Qµ(k) is the transformation matrix between the Bloch functions at points k and k + λµ, and provides a unitary replacement for Sµ(k). At points away from degeneracies Qµ(k) becomes a phase factor eiθ(k), as can be seen from k p perturbation expansion. ·

2.2.2 Implementation issues

In this section we discuss the calculation of these matrix elements, and the slight modifi- cations necessary to ensure that the numerical algorithm is well-behaved. The first step in a numerical implementation is to examine the connectedness of the different energy bands. The simplest approach would be to compute the difference in energies for each combination of the bands and mark the bands as connected if the difference is below a user-specified tolerance. Here care must be taken to define a mesh that includes degenerate points on the lattice or those close to them, although, as discussed later, this will not not be a restriction in a dynamical calculation. 2.2 Numerical Method 29

To handle degeneracies, we must construct one more matrix, and this can be motivated as follows. Consider an arbitrary degenerate point. An expansion of Bloch functions from this point to its neighbours will require exact diagonalization of the k p perturbation · in the degenerate subspace. Considering this expansion in all directions makes it clear that in general a diﬀerent diagonalization is required for each neighbour. Thus it is not possible to assign links in the form of phases since each neighbour requires a diﬀerent basis at the degeneracy point. The problem can be circumvented if the links are matrices within the space of the two or more degenerate bands.

We construct a matrix ∆ starting from identity such that once two bands n and m are identiﬁed as connected, ∆nm and ∆mn are also set equal to unity. In addition, matrix elements between bands that are indirectly connected, in the sense that they can be reached by continuously moving through any number of bands, are also set equal to unity. This procedure always results in a block diagonal form for ∆, such that bands belonging to diﬀerent blocks are separated entirely from each other.

Alternately one might consider all the Bloch functions at k, and not just the degenerate

ones, to be members of a single set, putting ∆mn = 1 for all n and m. This is allowed since physical results would not depend on the grouping of basis functions. However, mixing bands that are clearly separated by a ﬁnite gap would only make the problem unnecessarily opaque since, as we will see below, direct interband transitions between these bands would then appear in parallel transport of the density matrix.

Returning to the problem at hand, once the matrix ∆ is constructed the desired matrices ξµ(k) and gµ(k) are obtained by ﬁrst performing the element-wise multiplication, ¯µ µ Snm(k) = ∆nmSnm(k). This prevents mixing of disconnected bands in the SVD proce- ¯µ dure. After singular value decomposition of Snm(k) (see Eqs. (2.19),(2.20)), we set

gµ(k) = Qµ(k).

By construction the links are unitary and their matrix elements are non-zero only between the connected states. Their gauge transformation properties are easily veriﬁed by making

use of the unitarity of Q1 and Q2, and the real valuedness of singular values. Incidentally, we can also compute ξµ(k) from the S¯µ(k) by the formula

µ i µ µ µ µ ξ (k) = A (k)g †(k) g (k)A †(k) , (2.21) 2 λ − | µ| where A (k)= Sµ (k) S¯µ (k) is an interblock matrix. This equation is constructed nm nm − nm 2.2 Numerical Method 30

from the ﬁrst order expansion (2.12), which in the matrix form reads

Aµ (k) = δ iλ ξ (k) gµ (k), jm jl − µ · jl lm where the band j is disconnected from the band m. The symmetrization in Eq. (2.21) is done to enforce exact Hermiticity. The relation of ξjl(k) to the velocity matrix elements thus holds to ﬁrst order in the lattice spacing. The formula (2.21) is useful when an analytical expression for the Hamiltonian is not available, as is the case in ab-initio band structure calculations.

2.2.3 Numerical time-evolution

The dynamics of single particle operators is speciﬁed by the density matrix ρmn(k, k′, t)=

a† (k′, t)a (k, t) , where denotes the expectation value in the ground state. In n m 0 h·i equilibrium the periodicity of the electronic density makes this matrix diagonal when k is restricted to lie within a BZ. From Eq. (2.5), it follows that the Heisenberg equations

for an† (k) and am(k) for diﬀerent k become decoupled in the dipolar limit, and thus

ρ(k, k′, t) reduces to its momentum diagonal part, which we denote as ρ(k, t). The equation of motion for ρ(k, t) is

∂ e ∂ ie + E(t) ρ(k, t) [E(t) ϑ(k), ρ(k, t)] ∂t ~ · ∂k − ~ · i = [H (k) eE(t) ξ(k), ρ(k, t)] . (2.22) −~ 0 − · In the presence of non-uniform ﬁelds or interactions with phonons, the discrete translational symmetry is broken and the terms oﬀ-diagonal in k will evolve to non-zero values.

We construct an integrator of arbitrary order by expanding in powers of ∆t the exact evolution operator,

i t+∆t U(t + ∆t, t) = exp H(t′)dt′ , T −~ ˆ t where H(t) is the full Hamiltonian[102, 103, 104]. Here we apply a second order integrator, U = U + O(∆t3), where

U(t + ∆t, t) = exp [ i∆tH(t, ∆t)] , − ∆t d H(t) = H(t)+ H(t). (2.23) 2 dt 2.2 Numerical Method 31

In this ordinary matrix exponential, we split the Hamiltonian into the interblock and

intrablock parts as H(t) = Hter(t)+ Htra(t), where Hter(t) and Htra(t) are direct sum of their counterparts at each k, 1 e ∆t dE H (k, t) = H (k) E(t)+ ξ(k), ter ~ 0 − ~ 2 dt · e ∆t dE ∂ H (k, t) = E(t)+ i + ϑ(k) . tra −~ 2 dt · ∂k Then the Baker-Campbell-Hausdorﬀ identity implies that ∆t ∆t U(t + ∆t, t) U t, U (t, ∆t)U t, , (2.24) ≈ tra 2 ter tra 2 where the symbol indicates that terms of the order ∆t3 have been ignored. We con- ≈ struct U exp [ i∆tH (t, ∆t)] by numerical diagonalization in the space of bands at ter ≡ − ter each k point. Since the interblock operators at diﬀerent k commute, the resulting operators can be multiplied in any order to construct the interblock propagator. This property can be directly exploited for parallel implementation. It also introduces no theoretical error; numerical noise or a possible ill-conditioned matrix are the only sources of errors.

To construct Utra (t, ∆t), we approach the problem on a uniform grid of k points and split the motion in each orthogonal direction. We then perform translations by transforming to a smooth gauge in the given direction. This is allowed because the gauge fields are specified only up to a gradient, and a gauge choice that makes them orthogonal to a given plane is always possible in a proper subset of the BZ. We define a gauge transformation operator W α(k) that brings all points on a string in coincidence with the first point. Thus at a point k =( K + n λ , K + n λ ), where n and n are integers, − x x x − y y y x y W α(k) = gα(k λ )gα(k 2λ )gα(k 3λ ) ...gα(k n λ ), (2.25) − α − α − α − α α where the operators are ordered from left to right in the increasing order of m, as shown in Fig. 2.5. In the continuum limit, this expression is a direct analog of a time-ordered operator and solves the differential equation ∂ W α(k) = iW α(k)ϑα(k), (2.26) ∂kα − along the concatenation path. Note that ϑα and W α may not commute in general, and thus we cannot write W α as an exponential of the line integral of ϑα. Either of the two representations (2.25-2.26) shows that after making the basis transformation

α α ρ(k) W (k)ρ(k)W †(k), 7→ 2.2 Numerical Method 32

the links gα become equal to the identity operator, which corresponds to a gauge ﬁeld perpendicular to the direction α. The intrablock propagator, Utra (t, ∆t) can be split as

Utra(t, ∆t) = Uα(t, ∆t), (2.27) α=1 Y so that it is correct only to the ﬁrst order in time step. Due to the symmetry of the composition in Eq. (2.24), terms proportional to ∆t2 cancel and the overall error remains third order. The action of the unitary operators for each of the D dimensions is a translation with the above gauge condition and a transformation back to the original gauge:

α α α α Uα(t, ∆t) = W †(k)ϕα W (k)ρ(k, t)W †(k); t, ∆t W (k). (2.28)

The map ϕ [ρ; t, ∆t] implements the translation ρ(k, t) ρ (k δk (t), t) , where the 7→ − α shift in the wave-vector, e∆t ∆t dE(t) δk(t) = E(t)+ , ~ 2 dt is the average crystal momentum injected by the ﬁeld over the given time interval.

The translation of ρ(k, t) is achieved by interpolating using the Dirichlet kernel [105], a procedure that can be implemented using fast Fourier transforms along each grid line. A shortcoming of the Dirichlet kernel is the Gibbs phenomenon, which can be overcome by using a large number of grid points. Over the thousands of time steps required, we found that other interpolating kernels (such as Fejer [105]) and shape preserving interpolation schemes [106] excessively smooth the density matrix to the extent that it becomes uniform in k space. Thus we chose to perform simulations with the Dirichlet kernel, and we suppress the Gibbs phenomenon by using a ﬁne sampling of k space.

It is also necessary that the grid be padded with a border that deﬁnes a region where the transition matrix elements are set equal to zero. This is because the propagation of excitations to the domain edges would inevitably contradict the periodic boundary conditions assumed implicitly by the Fourier series solution. Unequal values of density matrix elements at the opposite edges of the domain behave like step discontinuities under the propagation algorithm, and thus cause long tails of oscillations that resemble a sinc function. Our numerical experiments suggest that a pad of about half the size of the grid is needed, with a taper that ramps down in about 3% of the pad. 2.2 Numerical Method 33

Figure 2.5: Operators W x(k) and W y(k) that bring the operators at point k in coincidence with the ﬁrst point on the respective grid lines.

Here it is also useful to broaden our definition of connectedness. Bands that are closer in energy than the maximum frequency resolution of the calculation are effectively degenerate, and therefore may be considered connected in the algorithm for computing links. As well, two bands that anticross can lead to large variations in the optical matrix elements that approach delta functions as the anticrossings become smaller and smaller. These sharp features are another source of Gibbs phenomenon, and can be avoided by considering such anticrossed bands to be connected as well. Note, however, that the Gibbs phenomenon is a result of using a discrete mesh to represent continuum quantities. Therefore, its effect reduces monotonically as sampling in k-space becomes finer. On the other hand, the unphysical effects of velocity gauge arise from the spectrum of the vector potential itself, and a non-trivial change in the method of calculation would be necessary in order to remove them.

We end this section by discussing the generalization of the foregoing to Kadanoﬀ-Baym equations in the context of a dipole Hamiltonian acting as an external perturbation. The derivation and discussion of these equations is in Chapter 3. The Keldysh matrix bb′ G (kt, k′t′), where b and b′ label branch indices on the Keldysh contour obeys the 2.3 Illustration and Discussion 34

equation of motion

∂ ∂ bb′ i~ + eE(t) i + ϑ(k) G (kt, k′t′) = δ ′ δ(t t′)δ(k k′)I ∂t · ∂k bb − − bb′ +(H (k) eE(t) ξ(k)) G (kt, k′t′) 0 − · + dt′′dk′′Σ(kt, k′′t′′)G(k′′t′′, k′t′), ˆ and a conjugate one for the second set of arguments. Here Σ is the self energy, and summation over the branch and band indices is implied. It is easy to generalize Eqs. (2.27) and (2.28) to handle propagation along k and k′ separately. Convolutions in the variable k require either a common basis throughout the lattice, or a complete specification of the four-point Coulomb vertex including the phases of the states involved in a scattering process. The latter is inefficient, particularly when we can ignore the dependence of Coulomb interaction on band indices, or when Coulomb interaction does not create or annihilate quasi-particles. On the other hand, all points can be expressed in the basis at the gamma point by first constructing the overlap matrices, T (k) u (0) u (k) , nm ≡ h n | m i and then performing the SVD to extract the basis transformation matrices as shown in Eqs. (2.19)-(2.20). This transformation allows the use of Fourier transforms to calculate convolutions. For example, the instantaneous Hartree-Fock correction to Eq. (2.22) is given by

H (k, t) = W †(k) V (k k′) W (k′)ρ(k′, t)W †(k′) W (k), HF − k X iR k = W †(k) V (R)ρ(R, t)e− · W (k), R X where ρ(R) is the FFT of the term in square brackets in the ﬁrst equality, V (R) is the Coulomb potential in real space, and W (k) are the unitary transformation matrices.

2.3 Illustration and Discussion

In this section we present numerical studies to illustrate the methods outlined above. These are carried out for electronic structure corresponding to a generic type I quantum well, which is brieﬂy discussed below. We emphasize that this system acts as a testbed for making general points about velocity and length gauges in a numerical setting. The validity of our arguments is unaﬀected by details of the electronic structure. 2.3 Illustration and Discussion 35

2.3.1 Quantum well band structure

The band structure is calculated for a quantum well of thickness 80 Å, and with bulk parameters corresponding to GaAs [107]. The bulk Hamiltonian corresponds to the eight band Kane model [107]. Staying within the envelope function approximation, we replace k by i∂ in the bulk Hamiltonian and model the z-dependent band edges by a z − z symmetric Pöschl-Teller potential[108] ~2 Λ 2 Vll(z) = 2 sech (z/z0) , −2Mllz0

where the subscript l is a label for the bulk band and Mll is the mass appearing in the ~2 2 term kz /2Mll term on the diagonal. The total number of bound states, Nb, available in this potential is N Λ 1/2. The matrix elements of the operators kˆ = i∂ and b ≤ − z − z kˆ2 = ∂2, are available in literature; we used the expressions derived by Zuniga et al. z − z [109]. Ignoring coupling to the continuum states, the eigenfunctions of the quantum-well Hamiltonian are expanded as follows:

8 Nb ik r Ψ(k; r) = cjl(k)ϕj(z)ul(k; r)e · , j=1 Xl=1 X where ϕj(z) is an eigenfunction for the Pöschl-Teller Hamiltonian, and normalized as

dzϕ∗(z)ϕ (z) = δ . ˆ j l jl

We arrange the coeﬃcients cjl(k) into an 8Nb dimensional column vector at each k and diagonalize the Hamiltonian.

Figure (2.6) shows the band structure resulting from this diagonalization with Nb = 2

and Λ= Nb +0.1. The value for Λ was chosen so that the top state does not lie too close to the top of the well, since then the band structure can become very sensitive to the parameters. The bands shown in the left panel are clustered around the heavy and light hole bands of the bulk, while the ones around the split-oﬀ bands are not shown. The four conduction bands are shown in the right panel. Each set of bulk bands is repeated twice and shifted by approximately the conﬁnement energy of the band. The neglect of the continuum states leads to a violation of the canonical commutation relation [rµ,rν]=0, and thus to a violation of the sum rule in Eq. (2.10). As discussed in relation to Eqs. (2.16)-(2.17), this sum rule on a discrete grid takes the form

m (Zµν 1 i [ξµ,ξν]) n 0, (2.29) h | − − | i ≈ 2.3 Illustration and Discussion 36

where m and n belong to a set of connected bands. The symbol “ ” is used to indicate ≈ that the left hand side is expected to be zero only to the second order in grid spacing, and when an infinite number of bands are used. Several different measures of the violation of this relation can be used, and the most suitable may depend on the kinds of quantities that are being calculated. Perhaps the most important consequence of the non-commutativity is the error introduced in the expectation values of operators that involves integration over the Brillouin zone. We characterize the error in the sum rule by the largest eigenvalue at each k, and take our measure of the sum rule violation to be the average value of this error normalized to the largest eigenvalue of √ξ ξ. As expected the sum rule (2.29) is · satisfied with greater accuracy as the grid spacing is made smaller. The grid chosen for the numerical studies here, discussed in the next section, gives 0.1 for this measure. As noted previously, band truncation also causes deviation of the singular values in Eq. (2.19) from

unity. We ﬁnd that bands lying near the middle of the spectrum of H0 correspond to 6 average deviations of less than 10− , so that the subspace can be considered complete for them. On the other hand, the bands near the maximum of the spectrum may deviate as much as 0.05 on average.

2.3.2 Comparison of length and velocity gauges

In this section we discuss calculations using the quantum well band structure described above with an external electric ﬁeld. In the velocity gauge, the equation of motion (2.22) takes the form ∂ i ie ρ(k, t) = [H (k), ρ(k, t)] [A(t) v(k), ρ(k, t)] , (2.30) ∂t −~ 0 − ~ · t A(t) = E(t′)dt′, (2.31) − ˆ0 where v(k) is the velocity matrix. In both types of calculations the ﬁeld at t = 0 is set equal to zero. To illustrate the source of the problems in velocity gauge, we write (2.30) in the frequency domain,

1 e v(k) E(ω′) ρ(k,ω) = [H (k), ρ(k,ω)]+ dω′ · , ρ(k,ω ω′) , (2.32) ~ω 0 ~ω ˆ iω − ′

In this non-perturbative expression, the iω′ in the denominator of the second commutator leads to false poles with prefactors that would vanish only when all sum rules obeyed by 2.3 Illustration and Discussion 37

−0.779 −0.7604 0.7915

−0.78 0.7899 −0.7606

−0.781 Energy (eV)

−0.7608 0.7879

−0.782

−0.783 −0.761 0.7859 −5 0 5 −5 0 5 −5 0 5 −3 −3 −3 k[100](1/A˚) x 10 k[100](1/A˚) x 10 k[100](1/A˚) x 10

Figure 2.6: Energy bands corresponding to the bands of heavy and light hole character. The left panel shows valence bands and the right panel shows the conduction bands 2.3 Illustration and Discussion 38

the velocity matrix are satisfied. This in general requires an extremely large (in principle infinite) number of bands, only a few of which contain significant carrier population. When the fields contain frequencies only above the gap, the true resonances of the system are dominant while the false resonances are suppressed by negligible spectral weight of the field in the low frequency regime. We refer the reader to Aversa et al.[91] for detail.

Three calculations are presented for frequencies below the gap. The results for frequencies above the gap match exactly within the machine errors in the two gauges. The frequencies in the presented results are: ∆E = (Egap +0.1)/2 eV, ∆E/2, and ∆E/4, and the pulse widths are 40 fs. Lower frequencies are completely beyond straightforward description within the velocity gauge, since the vector potential takes a non-zero constant value after the pulse vanishes. In the case of sub-gap excitations, we explicitly checked that the 20 spectral overlap with energy above the gap is less than 10− , the numerical noise ﬂoor of the pulse. Time steps of 0.02 fs were found adequate, in the sense that smaller time steps cause no visible change in the results. The grid of k points is a rectangle centered at k = (0, 0) of size 0.07 2π/a in each of the two directions (a =5.655 Å is the × latt latt lattice constant of GaAs). The electric ﬁeld used in the calculation has an amplitude of 4 10− V/Å and is linearly polarized along the [100] direction. A pad of half the size of the grid is present on both edges in this direction only. As the translational motion is only along [100], it is reasonable to compare the densities along grid lines in this direction.

Figure (2.7) shows the mean density along the line where the density achieves its maximum. We have plotted the logarithm of total populations in the four conduction bands resulting from both the length and velocity gauge calculations in each panel. The values are normalized to the maximum in each case. The velocity gauge consistently gives larger values of the densities during the time the pulse is on, and the diﬀerence is larger for lower frequencies. In fact, below the mid gap frequencies, the densities calculated by velocity gauge are almost two orders of magnitude larger than those in length gauge. In a calculation that also includes Coulomb interactions among carriers, this diﬀerence would nonphysically amplify the interaction energies.

Furthermore the density for (Egap +0.1)/8 is much higher than (Egap +0.1)/4 in the velocity gauge, as shown in Figure 2.8. In the length gauge, however, they are virtually equal. This is a positive outcome for the length gauge, since for such low frequencies only virtual population exists in the conduction bands, and therefore there is no reason 2.3 Illustration and Discussion 39

0 0

−2 −2 ] c −4 −4

−6 −6 Log[n

−8 −8

−10 −10 0 200 0 200

0 0

−2 −2 ] c −4 −4

−6 −6 Log[n

−8 −8

−10 −10 0 200 0 200

0 0

−2 −2 ] c −4 −4

−6 −6 Log[n

−8 −8

−10 −10 0 200 0 200 Time (fs) Time (fs)

Figure 2.7: Log (nc)for the velocity gauge (solid lines), and the length gauge (dashed lines), where nc is the population in conduction bands (see text). The frequencies (in units of eV) from top to bottom are: (Egap + 0.1)/2, and (Egap + 0.1)/4, (Egap + 0.1)/8. The left column shows calculation without the split-oﬀ bands included in the dynamics, and the right column shows results for all 16 bands. The values are normalized to maximum in each plot (logarithm base is 10). 2.4 Summary 40

1 1 (E +0.1)/4 (E +0.1)/4 gap gap 0.9 (E +0.1)/8 0.9 (E +0.1)/8 gap gap

0.8 0.8

0.7 0.7

0.6 0.6 c c

n 0.5 n 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 50 100 150 0 50 100 150 Time [fs] Time [fs]

Figure 2.8: Conduction band populations for ﬁeld frequencies far below the mid gap, in velocity (left panel) and length (right panel) gauges. The densities are normalized to the maximum in each plot. The dashed lines correspond to (Egap +0.1)/8, and the solid lines to (Egap +0.1)/4.

to expect a large diﬀerence in the two cases when the ﬁeld amplitude is kept constant.

2.4 Summary

We have presented a method to numerically solve semiconductor Bloch equations in the length gauge. While in principle any gauge leads to the same physics, our motivation for developing this method is the inadequacy of the velocity gauge in numerical computations, especially when intense low frequency fields are present. The main difficulty in the length gauge arises from the existence of gauge fields in the momentum space. Within perturbation theory, the issues in both gauges can be resolved by the use of exact sum rules when deriving the analytical form for response functions. In the velocity gauge, the simplest such sum rule is the effective mass sum rule [110, 111]. In the length gauge, the sum rules arise from the constraints imposed by the commutativity of the Cartesian components of the position operator on the commutator of its inter- and intraband components [90, 111, 91, 112].

For non-perturbative calculations, the velocity gauge is inappropriate, while the length 2.4 Summary 41 gauge requires that the gauge fields be taken into account to all orders. The direct numerical calculation of these fields is impossible due to the independence of basis at different k points. This freedom in choice of basis also prevents a direct differentiation of operators such as the density matrix in k space, a step that must be performed to construct the complete position operator. We implement this step by supplementing the chosen lattice with directed links, and then computing the basis transformation (termed link operators) between the vector space at the tip of the link and that at its tail. We then concatenate the unitary link operators along each line of a rectangular grid of points, a step that rotates the gauge fields into the plane orthogonal to the grid line. This converts covariant differentiation into ordinary differentiation and leads to translation in momentum space, which is done by FFT-based interpolation.

We compared the method to a velocity gauge calculation for a model quantum well with a Pöschl-Teller conﬁnement potential. We found that as the ﬁeld frequencies are lowered below the gap frequencies, the velocity gauge results in larger populations in the conduction bands at intermediate times. These can be up to two orders of magnitude larger than in the length gauge for frequencies below the mid-gap. At frequencies much smaller than the band gap frequency, the magnitude of virtual excitations is found to be independent of frequency in the length gauge, as expected. Chapter 3

Multidimensional Fourier Spectroscopy: Formalism

In this chapter we turn to the many-body description of laser-semiconductor interaction. We describe a framework based on non-equilibrium Green functions that is suitable for studying multi-pulse excitations of interacting multi-particle systems. As discussed in Chapter 1, the most recent and powerful technique is two-dimensional Fourier spectroscopy, from which detailed information about coherences can be obtained via optical excitation by three pulses. The purpose of the present framework is to develop a general description of such experiments, which can be extended to arbitrary sophistication of many-body eﬀects. In particular, treatment of many-body states other than the Hartree- Fock ground state, as well as states that are in a quasi-equilibrium is possible in principle within this approach. Though the techniques developed can be applied to other materials, for concreteness we focus on two-dimensional Fourier transform spectroscopy of semiconductors.

This chapter is organized as follows. In Section 3.1, we begin with a discussion of the electrodynamics of optical excitation of a semiconductor quantum well, and the generation of a signal in the far ﬁeld. We derive the necessary formulas that relate the propagating signal to the time-dependent charge and current densities inside the well. The calculation of these two quantities is the main task addressed in the framework that follows. In the remainder of Section 3.1, we transform the Hamiltonian from the velocity gauge to the length gauge, and express it in terms of nominal single particle basis states. We discuss how we specialize the existing general formalism of non-equilibrium Green functions to

42 3.1 Theoretical Background 43

describe the nonlinear optical response in TDFS.

As mentioned in Chapter 1, our approach consists of expanding the changes ∆G to Green (1) (1) (1) (1) function G in in terms of quantities Xn , ∆G = X + X + X + , where each term 1 2 3 ··· involves a higher power of the full incident ﬁeld. In section 3.2 we discuss the expansion of ∆G and address the hierarchy problem of multi-particle correlations. We end the (j) section with a derivation of the equations of motion for Xn .

Finally, in Section 3.3, we develop a diagram method as a general bookeeping device. We then illustrate the main points of the formalism by determining the TDFS signal (leaving all details to the following chapter) for speciﬁc pulse sequence leading to coherence among exciton states. We also extend the diagram method as a tool for building approximate solutions, which is useful at least in speciﬁc cases.

3.1 Theoretical Background

3.1.1 Electrodynamics

In an n-dimensional Fourier spectroscopy experiment there are up to n pulses incident on the target material, which for the moment we take to be a single quantum well embedded in a semiconductor with a background relative dielectric constant ε that is real and positive. The pulses are not necessarily distinct since the measured response may involve one or more pulses to higher order. We describe these pulses classically, and write the electric ﬁelds associated with them as E1(r, t), E2(r, t),..., and so on. The total incident ﬁeld on the quantum well, with its growth axis taken to be the z axis and centered at z =0, is then l

Einc(r, t)= Ej(r, t), (3.1) j=1 X where l n. Our goal is to calculate the response of the expectation values of the charge ≤ density operator and the current density operator, ρ(r, t) and J(r, t) respectively; h i h i these will be perturbed from their equilibrium values in the neighborhood of the quantum well. These expectation values are then taken, in the usual semiclassical manner, as the

source of the generated signal Ed(r, t). The problem is simpliﬁed because the thickness of the quantum well is much less than the wavelength of light, and the variation of the expectation value of electromagnetic ﬁeld in the plane of the quantum well, which arises 3.1 Theoretical Background 44

because the pulses will not all be normally incident on the well, will be over distances large compared to the thickness of the well. Within these approximations, we argue that we can calculate ρ¯(r, t) and J¯(r, t) at a given (x y ), where the overbar indicates an h i o o average over the lattice spacing, by calculating the response of a quantum well to a

ﬁeld described by a nominal uniform vector potential Anom(t). That nominal vector

potential takes the value that an effective vector potential actually takes at (xoyo), where the effective vector potential is essentially uniform along the growth direction of the quantum well and includes contributions from the incident fields and the transverse field from the quantum well itself.

If we take Fourier transforms with respect to both time and the x and y coordinates, writing for example dω iωt E (r, t)= E (r,ω)e− , d ˆ 2π d where dk ik R E (r,ω)= E (k, z; ω)e · d ˆ (2π)2 d

with k =(kx,ky) and R =(x, y), we have

1 E (k, z; ω)= iω− (k, z z′; ω) J(k, z′; ω) dz′, (3.2) d ˆ G − h i where

2 iω˜ iw(z z′) (k, z z′; ω) = θ(z z′)(sˆsˆ +p ˆ+pˆ+)e − (3.3) G − 2ǫ0w − 2 iω˜ iw(z z′) + θ(z′ z)(sˆsˆ +p ˆ pˆ )e− − 2ǫ0w − − − zˆzˆ δ(z z′), −ǫ0ε −

with ǫ the permittivity of free space, ω˜ = ω/c, w = √ω˜2ε k2 (Imw 0, with Rew 0 0 − ≥ ≥ if Imw = 0), where k = k , θ(z) is the step function (θ(z)=0 or 1 as z < 0 or z > 0) | | and the unit vectors sˆ and pˆ are given by ± sˆ ≡ kˆ×zˆ, k zˆ∓ wkˆ pˆ , ± ≡ ω˜√ε

where kˆ = k/k. This follows directly from the solution of the Maxwell equations[113]. Far from the source but within the semiconductor, say for z > 0, if we ﬁx rˆ and let 3.1 Theoretical Background 45

r we find, →∞ iω˜√εr iµ0ω e E (rˆr, ω) γ J(k , z′; ω) dz′ (3.4) d ∼ 4π r · ˆ h 0 i where µ is the permeability of free space, k =ω ˜√ε (rˆ zˆ(ˆz · rˆ)), and γ = sˆsˆ +p ˆ pˆ 0 o − + + evaluated at k = k0 [114]. Alternatively, if the quantum well is close to a semic onductor interface with air or another material, the Fresnel coefficients can be applied to the s- and p- polarized components of the field (3.2) to find the field outside the semiconductor, and then a corresponding asymptotic result can be deduced. Hence once J(r, t) is h i determined the radiation signal is easily found. We address the problem of calculating J(r, t) from a many-body framework using non-equilibrium Green functions. h i In the absence of any perturbing field, the standard Hamiltonian of many-body physics is

(t) (3.5) H 1 ~ † ~ = ψ(r) ψ(r) dr 2m ˆ i ∇ i ∇ + v (r)ψ†(r)ψ(r)dr + ψ†(r)ψ†(r′)v(r − r′)ψ(r′)ψ(r)drdr′. ˆ 0 ˆ

Here v(r) is the Coulomb interaction,

e2 v(r) = , (3.6) 4πǫ0εr

and v0(r) is the periodic potential energy due to the lattice structure, as modulated by any variations in chemical composition that, for example, create the quantum well; the spin-orbit interaction could be easily included, but has not been written here for simplicity. The simplest way to generalize that Hamiltonian to include the eﬀect of an

incident field Einc(r, t) on the system, when that incident field is treated classically, is to describe the field by potentials and include them as time dependent terms in the Hamiltonian. Since we are dealing with pulses of light, the simplest gauge is the so-called

radiation gauge, where no scalar potential is introduced for Einc(r, t) but we write simply E (r, t)= ∂A (r, t)/∂t, describing both magnetic and electric ﬁelds in terms of only inc − inc a vector potential. The inclusion of this vector potential in the Hamiltonian leads to 3.1 Theoretical Background 46

(t) (3.7) H 1 ~ † ~ = eA (r, t) ψ(r) eA (r, t) ψ(r) dr 2m ˆ i ∇ − inc i ∇ − inc + v (r)ψ†(r)ψ(r)dr + ψ†(r)ψ†(r′)v(r − r′)ψ(r′)ψ(r)drdr′. ˆ 0 ˆ In describing the response to the incident field, however, it would be worrying to neglect the transverse component of the field due to the electrons, even if that component can be neglected in equilibrium calculations. One reason is that, since the incident field is a transverse field, it would seem natural to take into account the transverse field from the driven charge-current densities themselves. A second reason is more specific: It is through the interaction of the charge-current density with its own transverse field (the so-called radiation reaction) that the system responds to the fact that it is radiating energy to infinity [115]. Hence taking into account the effects of the transverse field of the charge-current densities on those densities themselves is essential for maintaining energy conservation.

At the level of the semiclassical description used here, the field radiated to infinity follows from the expectation value of the current density (3.2). Thus in calculating the dynamics of the charge-current densities it is appropriate to include the expectation value of the transverse field generated by the charge-current densities as an additional driving field,

1 E (k, z; ω)= iω− (k, z z′; ω) J(k, z′; ω) , T ˆ GT − h i where (k, z; ω)= (k, z; ω) (k, z; ω), (3.8) GT G − GL

with L(k, z; ω) = limω/k˜ 0 (k, z; ω). Again using the radiation gauge by writing G → G E (r,t)= ∂A (r, t)/∂t, we introduce an eﬀective vector potential T − T

Aeff (r, t)= Ainc(r, t)+ AT (r, t), (3.9)

and take as our time-dependent Hamiltonian

(t) (3.10) H 1 ~ † ~ = eA (r, t) ψ(r) eA (r, t) ψ(r) dr 2m ˆ i ∇ − eff i ∇ − eff + v (r)ψ†(r)ψ(r)dr + ψ†(r)ψ†(r′)v(r − r′)ψ(r′)ψ(r)drdr′. ˆ 0 ˆ 3.1 Theoretical Background 47

The strategy is to use the Hamiltonian (3.10) to solve for ρ(r, t) and J(r, t) in terms h i h i of A (r, t); then using (3.9) and writing A (r, t) in terms of J(r, t) , one can in eff T h i principle find ρ(r, t) and J(r, t) self-consistently in terms of A (r, t) and hence in h i h i inc terms of Einc(r, t). Using (3.2) the signal field generated is then identified. In practice the problem simplifies because of the different length scales involved. The

variation of Einc(r, t) (and thus Ainc(r, t)) is on the order of the wavelength of light λ, which is much greater than the thickness of the quantum well, typical exciton radii, and

of course the lattice constant a. To identify the length scale over which ET (r, t) varies, note that the expectation value J(k, z; ω) will have k components with k on the order h i 1 1 1 of λ− and on the order of a− λ− . For large k/ω˜ the transverse component of the ≫ electric field becomes negligible, as can be seen from evaluating the terms in (3.8); and 1 in any case the large transverse components k a− ω˜√ε lead to evanescent fields ≃ ≫ that do not carry energy to infinity, and hence are not responsible for radiation reaction 1 and not of primary interest. Thus we need only consider components k λ− , and for ≃ such components we find

(k, z z′; ω) (3.11) GT − (k; ω) ≃ GT i ω˜2 k2 ik w ik sˆsˆ + + zˆzˆ + − kˆkˆ ≡ 2ǫ w εw ε ε 0 for z z′ λ. | − | ≪ from (3.8). So ET (r, t), like Einc(r, t), is essentially uniform over the quantum well and varies slowly, over a range on the order of λ, as x and y vary. This range is larger than the typical exciton radii that will characterize the correlation lengths of the electron ﬁeld operators. This suggests the following strategy: (a) To

determine the response in the neighborhood of each (xoyo), replace Eeff (r, t) and hence

Aeff (r, t) by a nominal uniform eﬀective ﬁeld and potential Enom(t) and Anom(t), but

with values equal to the actual Eeff (xo,yo, 0, t) and Aeff (xo,yo, 0, t), and then (b) solve for the ρ(r, t) and J(r,t) that follow from the Hamiltonian (3.5), and call them h i h i ρo(r, t) and J o(r,t) . As functions of x and y these will be uniform except for variations h i h i on the order of the lattice constant. We denote the components that are uniform over x and y by ρ¯(r, t) and J¯(r,t) ; they are functionals of A (t) and hence of E (t), h i nom nom ρ¯(r, t) = [E (t)] , h i Fρ nom J¯(r, t) = J [E (t)] . F nom

3.1 Theoretical Background 48

of the system in the neighborhood of each (xoyo), we take

ρ¯ (r, t) = [E (x, y, 0, t)] , (3.12) h eff i Fρ eff J¯ (r, t) = J [E (x, y, 0, t)] , eff F eff

where ρ¯ (r, t) and J¯ (r, t) denote the actual ρ(r, t) and J(r, t) , within our h eff i eff h i h i approximations, when averaged in the xy plane over distances on the order of the lattice constant. This allows for ρ¯ (r, t) and J¯ (r, t) to adiabatically follow the slow h eff i eff variation of the eﬀective ﬁeld as it varies on distances on the order of λ in the plane of the quantum well.

The averaging involved in constructing ρ¯ (k, z; ω) and J¯ (k, z; ω) means that h eff i eff there will be components k in ρ(k, z; ω) and J(k, z; ω) that will be lost in ρ¯(k, z; ω) h i h i h i 1 and J¯(k, z; ω) . But since those components will have k a− they will not contribute ≃ to the signal ﬁeld that can propagate to inﬁnity, and in place of (3.2) and (3.4) we can write respectively

1 E (k, z; ω)= iω− (k, z z′; ω) J¯ (k, z′; ω) dz′ (3.13) d ˆ G − eff and iω˜√εr iµ0ω e E (rˆr, ω) γ J¯ (k , z′; ω) dz′ (3.14) d ∼ 4π r · ˆ eff o for evaluating the signal ﬁeld away from the quantum well. Note that even a few lattice spacings away from the quantum well (3.13) will essentially agree with (3.2). In (3.12) we put

Eeff (x, y, 0, t)= Einc(x, y, 0, t)+ ET (x, y, 0, t), (3.15)

(cf. (3.9)), and using (3.11) we have

1 E (k, 0; ω)= iω− (k; ω) (k; ω), (3.16) T GT ·J where

(k; ω)= J¯ (k, z′; ω) dz′. J ˆ eff

Since ET (k, 0; ω) in fact only involves the total integral over z of J¯eff (k, z; ω) , the consistent solution of (3.12,3.15,3.16) is feasible, at least within a perturbation scheme. This approach can be generalized to deal with multi well structures, if eﬀects such as Coulomb drag are neglected. That is, while fully taking into account the many-body 3.1 Theoretical Background 49

eﬀects within each well, we assume that we can treat the interaction between wells at the mean ﬁeld level, for both the longitudinal and transverse components of the electro-

magnetic ﬁeld. Then, if the wells are all identical with their centers located by zm, we take

ρ¯ (x, y, z + z , t) = [E (x, y, z , t)] , (3.17) h eff m i Fρ eff m J¯ (x, y, z + z , t) = J [E (x, y, z , t)] , eff m F eff m where and J are the functionals introduced above for a single quantum well (cf. Fρ F (3.12)); the eﬀective ﬁeld at each well is then

Eeff (x, y, zm, t)= Einc(x, y, zm, t)+ Ewells(x, y, zm, t) (3.18)

(cf. (3.15)), where now

1 E (k, z ; ω) = iω− (k; ω) (k; ω) (3.19) wells m GT ·Jm 1 +iω− (k, z z ′ ; ω) ′ (k; ω), G m − m ·Jm m′=m X6 (cf. (3.16)) with

m(k; ω)= J¯eff (k, z′; ω) dz′. (3.20) J ˆwell m

The full appears in (3.19) for m′ = m because it carries both the transverse and G 6 the longitudinal ﬁeld from well m′ at the position of well m. The integral originally appearing in (3.2) has been replaced by the sum in (3.19) because, except for the Dirac

delta function term in (3.3) that does not contribute if z and z′ are in diﬀerent wells, 1 (k, z z′; ω) varies slowly as z and z′ vary over a quantum well for the k λ− that G − ≃ appear. This also allows us to write our signal ﬁeld as

1 E (k, z; ω)= iω− (k, z z ; ω) (k; ω) (3.21) d G − m ·Jm m X for planes z at least a few lattice constants away from any quantum well.

Equations (3.17,3.18,3.19,3.20,3.21) allow us to determine the signal field and its dependence on the incident field E (r, t) in a TDFS experiment once the functionals and inc Fρ J are identified. That is the main task of this chapter. F 3.1 Theoretical Background 50

3.1.2 Basis states and Hamiltonian in length gauge

The Hamiltonian (3.10) is expressed in the velocity gauge. As the calculations involving this Hamiltonian are often done numerically, it is convenient to use the length gauge for reasons discussed in Chapter 2. By performing the transformation

ieAnom(t) r/~ ψ(r, t) ψ(r, t)e · , 7→ and substituting E(t) = ∂A /∂t, (3.10) is rendered in the form of length-gauge for − nom the electromagnetic coupling,

(t)= + (t)+ ψ†(r)ψ†(r′)v(r − r′)ψ(r′)ψ(r)drdr′, H H0 Hext ˆ where ~2 2 = − ψ†(r) ψ(r)dr + v (r)ψ†(r)ψ(r)dr, H0 2m ˆ ∇ ˆ 0 and

(t)= eE(t) ψ†(r)rψ(r)dr Hext − · ˆ describes the driving of by external electromagnetic field within the dipole approximation. The Hamiltonian defines a static eigenvalue equation H0 ~2 2 + v (ρ, z) ϕ(ρ, z)= Eϕ(ρ, z), −2m∇ 0 where ρ = (x, y) in which the potential possesses the property of two-dimensional periodicity in ρ and confinement in z. Thus the in-plane wavevector, which we denote by k = (kx,ky), remains a good quantum number. The discrete levels at each k include both the periodicity and the confinement effects (more physically, a set of bands for each transverse state of the well), and we label them with the symbol ζ. Thus the basis functions are ik ρ r ζ, k = u (k; ρ, z)e · , (3.22) h | i ζ

and we denote the corresponding energies by ~ωζ(k), which allows us to expand the electron ﬁeld operator as

ik ρ ψ(ρ, z)= aζ (k)uζ(k; ρ, z)e · , k Xζ

where the eigenfunctions uζ(k; ρ, z) are appropriately normalized. 3.1 Theoretical Background 51

Turning now to the coupling with the classical electromagnetic ﬁeld that occurs through (t), we expand the coupling in the basis, (3.22), to ﬁnd Hext

(t)= ~φ ′ (k, t)a† (k)a ′ (k). (3.23) Hext ζζ ζ ζ k ′ Xζζ The matrix elements φζζ′ follow from Eqs. (2.4-2.6), e φ ′ (k, t)= E(t) ξ ′ (k), ζζ −~ · ζζ

for ζ = ζ′, and 6 e φ = E(t) (i k ϑ ) . ζζ ~ · ∇ − ζζ Finally, the Coulomb interaction in the above basis takes the form ~ (t) (t)= V (k , k ; k , k )a† (k )a† (k )a (k )a (k ). H − Hext 2 ζ1ζ4ζ2ζ3 1 4 2 3 ζ1 1 ζ4 4 ζ2 2 ζ3 3 X It is convenient to write the matrix elements of the Coulomb potential as

Vζ1ζ4ζ2ζ3 (k1, k4; k2, k3) (3.24) 1 ll′ = v(q) δ (k k q l) δ (k k q l′) F (k , k ; k , k ), ~ 1 − 3 − − 2 − 4 − − ζ1ζ4ζ2ζ3 1 4 2 3 q ′ X Xll where v(q) is the two-dimensional Fourier transform of the Coulomb potential, e2 v(q)= . 2ǫ0εq

The vectors l, l′ that are summed over are reciprocal lattice vectors of the 2D lattice, and ll′ the matrix elements Fζ1ζ4ζ2ζ3 (k1, k4; k2, k3) are given by the integral

′ ′ ll q z z′ l l F (k , k ; k , k ) = dz dz′ e− | − |F (k k ; z)F (k k ; z′), (3.25) ζ1ζ4ζ2ζ3 1 4 2 3 ˆ ˆ ζ1ζ3 1 3 ζ4ζ2 4 2

l il ρ ′ Fζζ′ (kk′; z) = dρ uζ∗(k; ρ, z)e− · uζ (k′; ρ, z). (3.26) ˆcell

In (3.24), the terms l =0 or l′ =0 arise from the conservation of total crystal momentum, 6 6 which is only within a reciprocal lattice vector. Only for l =0 and l′ =0 do they behave in the form expected in free space. We refrain from imposing such restrictions here as they have no bearing on the subsequent analysis, except for additional rules in the diagrammatic method to be described at the end of the chapter.

Collecting all of the above notation and approximations, the full Hamiltonian can now be speciﬁed as 3.1 Theoretical Background 52

1 ~− (t) H + H (t) H ≡ ext 1 = ω (k)a† (k)a (k)+ V (k , k ; k , k )a† (k )a† (k )a (k )a (k ) ζ ζ ζ 2 ζ1ζ4ζ2ζ3 1 4 2 3 ζ1 1 ζ4 4 ζ2 2 ζ3 3 k X X ′ ′ + φζζ (k,t)aζ† (k)aζ (k), (3.27) ′ kXζζ Although the contribution of phonons could be easily added, it is not included at this stage to avoid unnecessary complications in the formalism. Even without it we can address situations with intermediate carrier densities at low temperature, where the Coulomb interaction is the dominant decoherence mechanism.

3.1.3 Green functions

If there are multiple quantum wells in the sample, we follow the strategy of Section 3.1.1 to take into account their interactions. For each quantum well, we calculate the charge and current densities, ρ¯(r, t) and J¯(r, t) respectively, as functionals of the h i ﬁeld E(t) appearing in (3.23) using the method of Green functions. The single-particle Green function is deﬁned as

G(12) = i a(1)a†(2) . (3.28) − TC

Here, and in the following discussion, we follow the standard practice of using numbers to denote a set of arguments not shown explicitly; thus 1=(k1, t1,ζ1) where t1 is time variable on the Keldysh contour. If we perform a transformation from the basis function to real space, then analogously 1=(r1, t1). To denote equal time limits, we deﬁne + + + the symbol 1 = (k1+ , t1 ,ζ1+ ), where t1 stands for inﬁnitesimal advancement of time variable on the Keldysh contour. The symbol orders the operators on the contour, TC and the meaning of O for an arbitrary operator, O, is through a trace with a statistical h i operator ρ (not to be confused with the single particle density matrix),

1 iS[Φ] b O = Tr Oρe− . (3.29) h i ZC The factor ZC is the general generating functionalb

iS[Φ] Z [Φ] = Tr ρe− , (3.30) C TC b 3.1 Theoretical Background 53

and

S[Φ] = dτ drH[a†, a]+ Φ(12)a†(1)a(2) (3.31) ˆC ˆ ˆC is deﬁned for an arbitrary bilinear coupling via the bare two-time external potential Φ(12). In the physical limit, Φ(12) will depend on a single time and is identiﬁed with

φζζ′ (k, t) of (3.23) above, as discussed in section 3.3. From the deﬁnitions of the charge and current densities, we see that

ρ¯(r, t) = ieG(rt; rt+) (3.32) h i − ~ e + + J¯(r, t) = lim rG(rt; r′t ) r′ G(rt; r′t ) . (3.33) 2m r r′ − → ∇ − ∇ Thus the self-consistent calculation determines G(12) via the Hamiltonian (3.27). To ﬁ- nally determine the signal in a TDFS experiment, the quantities ρ¯ (r, t) and J¯ (r, t) h eff i eff of (3.12) are then calculated (see Section 3.1.1), and treated as the source of the signal. The rest of the chapter is devoted to developing the formal machinery to determine G(12) in the presence of many-body interactions.

Prior to the introduction of the pulse sequences we assume that the system is in a quasi- equilibrium state that evolves on a timescale much longer than the one associated with the optical excitation; any photoluminescence that would appear is neglected. The

Green function GQ associated with this state leads to vanishing signal at the respective frequencies. We handle the subsequent optical excitation by writing the full Green function as

G(12) = GQ(12)+ ∆G(12). (3.34)

where ∆G(12) describes the effects of optical excitation, and will lead to nonvanishing J¯(r, t) , and hence to a signal field. While the defining equation (3.28) introduces G(12) as a functional of the driving field Φ, G(12) = G(12; Φ), it is convenient to introduce a

self-consistent ﬁeld UHartree,

U (1, 2) = Φ(12) i V (13; 42)G(43), (3.35) Hartree − ˆ that represents the dressing of the external driving ﬁeld by the medium of charges interacting via the bare Coulomb interaction. Here the Coulomb interaction has also been re-written with implied delta functions in time,

V (14;23) = V (k , k ; k , k )δ(t t+)δ(t+ t )δ(t t ), (3.36) ζ1ζ4ζ2ζ3 1 4 2 3 1 − 3 2 − 4 1 − 2 3.1 Theoretical Background 54

to establish a uniform notation with four-point eﬀective interactions below. The change in this self-consistent ﬁeld due to the pulses of light is given by

U(12) U (12) UQ(12), (3.37) ≡ Hartree − where the self-consistent ﬁeld in the (quasi-)equilibrium states is just

UQ(12) = i V (13; 42)GQ(43). (3.38) − ˆ The Dyson equation the Green function satisﬁes can then be written in the form

G(12) = G0(12) + G0(11′)UQ(1′1′′)G(1′′2) (3.39)

+G0(11′)U(1′1′′)G(1′′2) + G0(11′)Σ(1′1′′)G(1′′2), introducing a self-energy Σ(1′1′′). With this equation in hand, the strategy now is to treat the basic field U as the effective field, such that Φ = Φ[U]. Therefore when written as a solution to (3.39), G is functionally dependent on U rather than on Φ, or in other words G(12; U)= G(12; Φ[U]). Similarly the self-energy is also taken as a functional of U.

Increasing the effective field from U(34) to U(34)+δU(34) leads to a change in the Green function from G(12) to G(12) + δG(12), where for infinitesimally small δU(34) we have δG(12) = P (14; 23)δU(34), and the functional derivative

δG(12) P (14; 23) (3.40) ≡ δU(34) is evaluated at U(34). Hence in the limit of a weak eﬀective ﬁeld we have

δG(12) δ2G(12) ∆G(12) = U(34) + U(34)U(56) (3.41) δU(34) δU(34)δU(56) Q Q δ3G(12) + U(34)U(56)U(78) δU(34)δU(56)δU(78) Q + ··· = P (14; 23)U(34) where the subscript Q indicates that the functional derivative is evaluated in the quasi- 3.1 Theoretical Background 55

equilibrium state, with U =0, and we have written

δG(12) P (14;23) = (3.42) δU(34) Q δ2G(12) + U(56) δU(34)δU(56) Q δ3G(12) + U(56)U(78) δU(34)δU(56)δU(78) Q + ··· Despite the fact that one then has to extract an expression for ∆G in terms of Φ, the driving ﬁeld, at the end of the calculation, we will see that this approach simpliﬁes the analysis.

Since the use of the expansion (3.41) for ∆G(12) in the expressions (3.32,3.33) leads to expansions for ρ¯(r, t) and J¯(r, t) , ultimately in powers of the driving field, that h i expansion constitutes the Green function form of the usual expansion in nonlinear optics of the response of the system in increasing powers of the electric field. In a TDFS experiment the signal in the background-free direction will be from the third term in (3.42) when it is used in (3.41). All three incident pulses will contribute to all the U(ij) and P , of course, but the signals due to the different combinations of the pulses will propagate in different directions. For example, one contribution to the signal will result from each of the U(34), U(56) and U(78) corresponding to a different pulse, and it will propagate in the four-wave mixing direction. Thus in general a contribution to the signal can be identified as pulse the non-equilibrium response to a probe of a system excited by pump pulses. The full signal is a sum over all 27 possibilities labelling the three pulses as probe or pump pulses.

In doped semiconductors, the quasi-equilibrium state may consist of interacting electrons, in equilibrium with the lattice. This is a true equilibrium situation, where the response of the system depends only on diﬀerences in time. In terms of Wigner variables, τ = t t , 1 − 2 and t = (t1 + t2) /2, GQ depends only on τ. We may also allow greater ﬂexibility by considering an unexcited semiconductor that is excited optically to produce a non-thermal

distribution of carriers. In this scenario GQ(t, τ) depends on t, and this dependence is slowly varying with respect to the dynamics induced by the subsequent TDFS excitation. This more complex scenario can be handled by a multiple-scale expansion in the variable t. 3.1 Theoretical Background 56

3.1.4 The eﬀective two-particle interaction

Central to this approach are the properties of P (14; 23), which we identify here. For the derivation of the integral equations that arise, we refer to the reader to one of a number of standard references on Green function theory[116, 117, 83, 118, 74, 119, 120, 121]. The quantity P (14; 23) satisﬁes the Bethe-Saltpeter equation (BSE),

(2) P (14;23) = G(13)G(42) + G(11′)G(2′2)I (1′6;2′5)P (54; 63). (3.43) ˆ

This equation follows from the deﬁnition (3.40) of P if the eﬀective two-particle interaction, I(2), is written as

(2) δΣ(1′2′) I (1′6;2′5) = . (3.44) δG(56)

The validity of (3.43) is based upon the usual assumption that the self energy can be expressed as a functional of G alone. Using the diagrammatic notation indicated in Figure 3.1, the BSE equation is sketched in Figure 3.2. Since I(2) is one of the basic objects that will be used in the rest of the chapter, we brieﬂy discuss it here using the diagrams shown in Figure 3.3; in Appendix A we work out the detailed algebraic form that is captured by those diagrams.

Figure 3.1: Graphical symbols for constructing diagrams. Note that P (14;23) = X(2)(14; 23) . 3.1 Theoretical Background 57

Figure 3.2: The BSE for four point function

Figure 3.3: Diagrams for I(2)

The first diagram indicates the basic interaction between electrons in inter-particle scattering, and corresponds to an effective Coulomb interaction W (14; 23) with a vertex correction Γ(14; 23). The effective interaction is defined as the solution of the integral equation

W (14;23) = V (14; 23) i V (15; 63)P (66′;55′)W (5′4;26′). (3.45) − ˆ In its diagrammatic form, the incoming and outgoing lines are attached to W as shown in Figure 1. The vertex correction corrects P (14; 23) from its simplest approximation,

P (0)(14;23) = G(13)G(42), (3.46) such that the exact P (14;23) = G(11′)G(2′2)Γ(1′4;2′3), and 1 δG− (12) Γ(14;23) = = δ(13)δ(24) + T (12′;21′)G(1′3)G(42′), (3.47) − δU(34) where we have introduced a T -matrix as the solution of the integral equation

(2) (2) T (14;23) = I (14;23) + I (12′;21′)G(1′5)G(62′)T (54; 63), (3.48)

The second and third diagrams of Figure 3.3 play a crucial role in the decoherence process that occurs due to the scattering of an electron from a correlated pair, and we will turn to them in section 3.2.3. The fourth and fifth arise entirely from δW/δG, and are a precursor for a new scattering channel. Insight into their nature follows from using an expression for the self-energy in terms of the T -matrix (or Γ), that avoids explicit reference to the inverse Green function. From the definition (3.39) of the self-energy, and those of the vertex correction (3.47) and T -matrix (3.48), we find

Σ(12) = i W (13;3′1′)G(1′1′′)Γ(1′′3′;23) (3.49) ˆ

= i W (13; 21′)G(1′3) + i W (13;3′1′)G(1′1′′)T (1′′2′′;23′′)G(3′′3)G(3′2′′) ˆ ˆ 3.1 Theoretical Background 58

Iterating the coupled equations (3.44,3.48,3.49) it becomes evident that this new channel involves a "horizontal" ladder series in the self-energy and represents repeated interaction of the single particle with another particle in the system. By crossing symmetry[117, 119] in the exact formulation, this channel can simply be represented by the same equation as (3.48) but with the effective interaction now acting on 1, 3 in T (14; 23). While optical excitation creates coherence only between the ground state and an excited state, correlations such as those underlying dynamical interactions transfer this coherence to coherence between excited states. As shown in Chapter 4, the fourth and fifth diagrams in Fig. 3.3 are the kind that describe one such transfer mechanism. The coherences thus generated, sometimes referred to as Raman coherences [8], decay mainly under the influence of the first three diagrams. However, the fourth and fifth diagrams also make further contributions to the dynamics when there is an electron-hole plasma at reasonably high density regimes, but below the Mott transition point such that bound electron-hole states still exist. In this case, they significantly affect the nature of two-particle interaction.

Finally, the last two diagrams arise from the six point function formed by the variational derivative of the T -matrix itself. The six-point function is reducible, and therefore generates diagrams in Fig. 3.3 that are combinations of the two and four point functions. In particular, note that the set of diagrams is asymmetric in the sense that diagrams that would be obtained by ﬂipping them upside down are not shown explicitly. These diagrams do occur, and are contributed precisely by these last two terms. In an exact formulation these are guaranteed to occur due to the equivalence of the two self-energy formulas,

+ + Σ(11′)G(1′2) = iW (13;3′ 1′)P (1′3′ ) (3.50)

+ + G(11′)Σ(1′2) = iP (13′;2′ )W (2′3;3′ 2). (3.51)

Their equality implies that the particle number, and energy are conserved, which is an important property to build into approximations. Therefore in an approximate treatment, which in general picks a finite number of vertex diagrams, the symmetric diagrams can be included “by hand” and excluded from any approximations for the six point function. This corresponds to including all diagrams of the same topology (i.e. those which differ only by relabelling external vertices). The remaining contribution of δT/δG is then to used generate scattering channels of topologies beyond those already included in the (approximate) T -matrix being differentiated. 3.2 Response Functions Framework 59

In general, the vertex correction contains all two-particle reducible graphs, and represents an inﬁnite number of scattering channels. However, in a given regime of density, temperature, and optical frequencies, only a few will generally be dominant. Thus any approximation would necessarily pick a particular analytical form for Γ [W,G,P,...] as a starting point of the self-consistent calculation. We now turn to ours.

3.2 Response Functions Framework

3.2.1 The susceptibility expansion

Returning to the expression (3.34) for G(12) and the expansion (3.41) for its deviation ∆G(12) from quasi-equilibrium, we can write

∆G(12) = ∆G1 + ∆G2 + ∆G3 + . . . (3.52) = X(1)(12) X(1)(12). − Q (1) In anticipation of some future notation G(12) and GQ(12) are written as X (12) and (1) XQ (12) respectively. For any function, O, we deﬁne δnO ∆O = U(1′′1′)U(2′′2′) U(n′′n′). (3.53) n ˆ δU(1 1 )δU(2 2 ) δU(n n ) ··· ′′ ′ ′′ ′ ··· ′′ ′ Q The quantities ∆Gn are of fundamental interest in predicting experimental results. Since they are also central to the formalism, we use a separate symbol for them,

X(1)(12) ∆G (12) . (3.54) n ≡ n The meaning of the superscript will become clear below. In the absence of Coulomb interactions, from (3.43) we see that P (14;23) = G(13)G(24), and constructing the (1) expressions for the X1...3(12) leads to

(1) X1 (12) = GQ(13)U(34)GQ(42), (3.55) (1) (1) (1) X2 (12) = GQ(13)U(34)X1 (42) + X1 (13)U(34)GQ(42), (3.56) (1) (1) (1) (1) X3 (12) = X1 (13)U(34)X1 (42) + GQ(13)U(34)X2 (42) (3.57) (1) +X2 (13)U(34)GQ(42).

(1) (1) (1) That is, the set of three dynamical variables X1 , X2 , X3 is closed and this system of equations fully captures the dynamics of thisn non-interactingo system. In this limit, the 3.2 Response Functions Framework 60

coherence between the ground state and a particle-hole pair is preserved and uniquely determines all higher order correlations. This is not true when Coulomb interactions are present, which can build non-trivial correlations among the particles. Then the (1) (2) equations for the variables Xn involve quantities Xn (14; 23) that become dynamical variables themselves, where in general for j > 1 we put

(j) (j) Xn (1a1′ ...aj′ 1; 2a1 ...aj 1) ∆Xn (1a1′ ...aj′ 1; 2a1 ...aj 1) (3.58) − − ≡ − − n (j) δ X (1a1′ ...aj′ 1; 2a1 ...aj 1) = − − U(1′′1′) U(n′′n′) ˆ δU(1 1 ) ...δU(n n ) · · · ′′ ′ ′′ ′ !Q (j+n) = X U(1′′1′) U(n′′n′), ˆ Q · · ·

(j+1) (j+1) and where for j > 0 it is useful to deﬁne X and XQ as respectively the 2j-point correlation functions

j (j+1) δ G(12) X (1a′ ...a′ ;2a ...a ) , (3.59) 1 j 1 j ≡ δU(a a ) δU(a a ) 1 1′ ··· j j′ evaluated at U =0, and their values at U =0 as 6 j (j+1) δ G(12) XQ (1a1′ ...aj′ ;2a1...aj) . (3.60) ≡ δU(a a′ ) δU(a a′ ) 1 1 ··· j j Q (j) (j) Again, since ∆Xn are central to the formalism, we use a separate symbol, Xn , for them. (j) n (j) (j) Thus Xn is generally the contribution of order (U ) to the deviation X X , and O − Q X(j) X(j) ∆X(j) = X(j), (3.61) − Q ≡ n n X of which (3.52) is the special case for j =1.

(2) In the presence of Coulomb interactions the Xn can no longer be written in terms of the (1) (1) (1) variables X1 , X2 , X3 , and they acquire a dynamics beyond that implied by simple factorizationn (3.46) of Po(14; 32) as a product of Green functions. The BSE and the (2) (2) (3) discussion of I above implies that the Xn are coupled to the Xn , and so on. Thus, (j) in the presence of interactions, the set of dynamical variables expands to include Xn for all j =1, ... . ∞ At ﬁrst sight it might seem better to work directly with the X(j), since what is required for (4) (j) comparison with experiment is simply XQ (see (3.42,3.53,3.61)). The equations for X (j′) consist of the products of X for j′ < j, and an interaction of the same order, j, which builds non-trivial correlations and couples them to X(k) for k > j. The factorization 3.2 Response Functions Framework 61

follows from the lengthy but straightforward integral equations derived in Appendix C for up to j = 4, and shown diagrammatically in Figure 3.4. In these diagrams, the effective interactions I(n) are defined as n 1 (n) δ − Σ(11′) I (12′....n′;1′2...n)= , n> 1. (3.62) δG(22 ) δG(nn ) ′ ··· ′ of which the I(2) defined earlier (3.44) is a special case. The equations for the six (j = 3) and eight (j = 4) point correlation functions have a simple combinatorial structure, in which the correlation function is factorized into all possible combinations of the lower order correlation functions (Fig. 3.4). Diagrams separated into unconnected components represent the independent evolution of these components. In some of the diagrams, these components are brought into interaction either by a 4, 6, or 8 point effective interaction I(2), I(3), or I(4). They arise out of Coulomb interaction, and thus form an essential part of the many-body physics.

(a)

(b)

Figure 3.4: The BSE’s for six (X(3)) and eight point (X(4)) functions. All permutations are shown explicitly. We use a to represent the set (a, a′) etc.

(4) The exact solution of these integral equations yields XQ , from which ∆G can be obtained by integration. However, such a calculation is prohibitive as it includes immensely complex non-perturbative eﬀects arising from the summation of inﬁnite subsets of dia- (j) grams. So we abandon this approach in favor of focusing on the Xn , which provide the 3.2 Response Functions Framework 62

essential information for comparison with experiment in an alternative fashion. As we show below in section 3.2.3, and in a following chapter, it is possible to derive differential (j) equations for the Xn at a chosen level of approximation. Analytically, these equations of motion (EOM) offer insight by explicitly identifying the rates of different processes. Nu- merically, the self-consistent solution of these equations, via time stepping, allows many non-perturbative effects to be included automatically. The formulation of the problem (j) is reduced to a dynamical interplay of the deviations Xn (see (3.61)), where the rules of the dynamics are specified in the interactions that depend on the quasi-equilibrium correlation functions.

We devote the following sections of the chapter to these equations of motion.

3.2.2 Hierarchy of correlation functions and its approximate termination

(1) A ﬁrst step is the description of the hierarchy coupling the Xn of interest (3.52) to the (2) (2) (3) Xn , and the Xn to the Xn , and so on; the identiﬁcation of approximations that can be made to truncate the hierarchy; and description of the consequences of this truncation on the properties of the correlation functions being studied here. That is the goal of this section.

To discuss the hierarchy economically, it is convenient to extend the deﬁnition (3.48) of T (14; 23) to terms involving higher order derivatives,

n 2 (n) δ − T (14′′;1′3′′) T (14′′2′ ...n′;1′3′′2 ...n)= , n 2. (3.63) δU(22′) ...δU(nn′) ≥ With this term in hand we will be able to identify the non-factorizable parts of correlation functions. To illustrate this we use a schematic notation in which we suppress the arguments of the functions, and discuss the contribution T (n) makes to X(n). For n =2, the function T (2) corresponds to the one-particle irreducible amputated diagrams[121, 119, 120] comprising X(2), i.e.,

GGT (2)GG = X(2) GG. − Relationships of this type are explained in Appendix C. Diﬀerentiating with respect to U, we obtain GGT (3)GG = X(3) 2X(2)G 4X(2)GGGT (2), − − 3.2 Response Functions Framework 63

where the numerical factors indicate the number of terms of the same topology. The second term on the right hand side is unconnected, involving independent propagation of single- and two-particle correlation functions. The third term is single-particle reducible since only one leg of X(2) is connected to T (2). From the graphical six-point BSE (Figure 3.4) it is clear that X(3) is a sum of these two terms and single-particle irreducible terms. Therefore T (3) must equal single-particle irreducible contributions to X(3) with four legs amputated. Similarly, T (4) consists of analogous diagrams for X(4) and so on. Expressing the correlation functions using T (n) has the advantage that the trivial eﬀects of independent propagation, which are strictly determined by lower order correlation functions already at a given particle order, are removed. Thus these functionals bring in the fully interacting components at each order j.

Through the dependence (3.49) of the self-energy on T (2), the functional I(2) contains a contribution from the variational derivative of T (2) via (3.62), and this derivative is related to T (3) via (3.63). This particular term is the sum of two contributions that can be represented schematically by the following diagrams

(64)

where the dots represent external vertices, and the solid circles represent the T (n) as indicated. The number of lines and dots on the shaded circles equals the number of arguments for the corresponding T (n), and therefore identify its superscript n. These two contributions to I(2) both contain the functional T (3). When these diagrams are differentiated again to obtain I(3), they will bring in a contribution of T (4), and thus the hierarchy shown in Fig. 3.5 ensues. Since T (n) is related to X(n) by n incoming and outgoing quasi-particles, the effective interaction I(n) can be viewed as becoming dependent on X(n+1). This effective interaction in the equation for X(n) couples it to X(n+1) and results in a parallel hierarchy for X(n); the explicit conversion between T (n) and X(n) is shown in Appendix D up to n =4.

The exact solution has the property that the self-energy, Σ, is consistent with all T (n), which requires summation of all scattering channels. This means, in particular, that if the functional derivative of Σ with respect to G is substituted in the equation (3.48) for T (2), then its solution is self-consistent with the T (2) matrix that Σ explicitly depends on 3.2 Response Functions Framework 64 via (3.49). The hierarchy necessitates that this hold not just for T (2), but also for T (n) on which the self energy depends implicitly. Anything short of this, and therefore any practical iterative method of calculation, will break this consistency[122, 123].

It is instructive to consider how the consistency is broken in the well-known Kadanoﬀ- Baym approach[116, 118]. As an example, consider the T -matrix for the Bethe-Goldstone equation[118], which is an example of a conserving approximation,

(2) (2) Tapprox = V + V GGTapprox. (3.65)

(2) The approximate Σ is written as Σapprox = TapproxG. The latter is then diﬀerentiated to obtain an algebraic expression,

δT (2) I(2) = T (2) + approx G, (3.66) approx approx δG which when substituted in the BSE provides a two-particle correlation function that obeys (2) conservation laws. However, the functional Iapprox is clearly not equal to the kernel, V , in (3.65), and therefore the complete set of single and two particle equations is not self- (2) consistent. It also means that when Iapprox is used in the BSE, Papprox (14; 23) contains scattering in only two pairs of coordinates, namely 12 and 34. Nonetheless, the diagrams that contribute to Σapprox ensure that it is still a functional derivative of some functional, and this is a suﬃcient condition for macroscopic conservation to hold.

In view of the underlying lattice, we do not expect momentum conservation to hold in solids. The particle number conservation for G, and the conservation of the total energy, requires only that the approximate T (2) obey T (2)(14;23) = T (2)(41; 32), which can be easily guaranteed by symmetrizing the integral equation for T (2). These are conditions

Figure 3.5: Hierarchy of I(n) and T (n) in exact form (top left) and cut at eight point level (top right). The arrows point to the function that needs as input the function at its tail. The dashed arrow is the link that is broken in terminating the hierarchy. 3.2 Response Functions Framework 65

(A) and (B) of the classic Kadanoff-Baym paper[118], and if P is generated from such a G via functional differentiation (3.40), then it inherits these conservation laws from G. These translate into sum rules for conductivities and other quantities that describe linear response. On the other hand, including scattering between all pairs of external vertices of the correlation functions, as opposed to just two for Papprox, has important consequences for the decoherence of many-body states (see also Chapter 4). We therefore follow the latter route below, and will see that it means losing the ability to make a general statement about sum rules for two, and higher particle correlations. To proceed we will restrict ourselves to perturbative expansion in the field up to order 3, i.e. three pulse experiments. In the exact eight-point equation, Figure 3.4b, the effective interaction I(4) will depend on T (5) via the diagram, (67)

This diagram is at most contracted with three field-dressed correlation functions, and when they are all placed on the T (5), there remains one vertex exposed on it, which together with the vertex at the bottom right corner gives a two-point function that contributes to the self-energy of third order in the field. It is clear from this diagram that at least one pair of the external vertices of T (5) is already closed on itself, and connected to the rest of the self-energy diagram by an interaction line. The closure and the accom- (5 k) panying contraction with the field generates a field-dressed functional, ∆Tk − , which th (5 k) describes the k order deviation of T − from its quasi-equilibrium value. To convince oneself that this is correct, one need only write the self-energy expression (suppressing the arguments) as, Σ = WG + WGGGT , (cf (3.49)) and then expand each side of the equation to third order in field by expanding each quantity as in (3.52). Then one sees (2) that the contribution [WGGG]Q ∆T3 in this expansion, and the expression corresponding to placing all three field lines on T (5) in the above diagram, have the same form. (5) (2) Thus we identify the T connected to three field lines with ∆T3 , and in general we (n k) th (n) identify ∆Tk − with a k order field-induced deviation in T . Examining the diagrammatic equations in Figure 3.4, and the Dyson equation, we see that they depend (2...4) (2...3) (2) explicitly only on the field-dressed functionals ∆T1 , ∆T2 , and ∆T3 , and the (2) quasi-equilibrium functional TQ . Equivalently we can state this dependence in terms of (2) (2...4) (2...3) the effective interaction IQ , and the field-induced deviations in it, ∆I1 , ∆I2 , and 3.2 Response Functions Framework 66

(2) ∆I3 . From the above analysis, we see that each term in each of the BSE’s (Figs. 3.4a-3.4b) (j) (j) can be written as a product of Xn and ∆In , or the field induced deviations in the correlation functions and effective interactions respectively. The resulting equations are (j) still exact to all orders in Coulomb interaction so long as exact expression for ∆In is (j) known. In general such an expression would be dependent upon all Xn , and would not be possible to specify in practice. Therefore, at this point, we introduce an approximation strategy, or phenomenology, into the formalism by demanding that a model for I(2), I(3), and I(4) be specified in a way that these functionals are written as functions of the correlations X(1...3) and W . Note that none of these quantities has been forced to be at quasi-equilibrium. We thus have

I(j) = I(j) W, X(1),X(2),X(3) . (3.68) The arguments of the function I(j) correspond to subgraphs consisting of a possibly infinite number of diagrams constructed out of G and W in the exact I(j). We treat these subgraphs as independent objects in the model I(j). To write the (model) field induced (j) deviations ∆In , we need the field-induced deviations in the screened interaction, W . Using (3.45), it can be written in terms of X(j), and W via

δW δX(2) = W W = WX(3)W. (3.69) δU δU We let δnW (14; 23) ∆Wn(14;23) = U(1¯1¯′) ...U(¯nn¯′), (3.70) δU(1¯1¯′) ...δU(¯nn¯′)

(j) which allows us to write ∆In as follows,

∂I(j) ∂I(j) ∆I(j) = ∆W + X(k) (3.71) 1 ∂W 1 ∂X(k) 1 Q ∂I(j) ∂I(j) ∂2I(j) ∆I(j) = ∆W + X(k) + X(k)X(l) (3.72) 2 ∂W 2 ∂X(k) 2 ∂X(k)∂X(l) 1 1 Q Q Q ∂I(j) ∂I(j) + ∆W X(k), ∂W 1 · ∂X(k) 1 Q Q and so on for larger n. We bring to the reader’s attention the change in the symbol for derivatives, and its associated meaning. These derivatives are taken by treating the arguments of the function in (3.68) as independent, and a derivative with respect to one 3.2 Response Functions Framework 67

argument is taken by holding all other constant. As a result they do not remove G lines implicit in graphs for X(2),X(3),.... In contrast, functional derivatives in (3.62) differentiate all arguments with respect to G and generate new diagrams including those (j) that are topologically distinct from the ones differentiated. Thus the above ∆In do not couple to an infinite number of correlation functions; they couple only to the ones (j) on which they explicitly depend. Furthermore, in this scheme ∆In are not related to each other by a chain of functional derivatives. In other words, while a functional derivative of I(j) with respect to G results in I(j+1), it does not hold in general for the phenomenological I(j). In particular we have implicitly set I(5) = 0, and therefore manifestly break the hierarchy, i.e. δI(j) I(j+1) = . (3.73) 6 δG Substitution of these effective interactions in the BSE’s forms a coupled set of integral equations, instead of functional-integro-differential equations of the exact theory. Their (j) solution yields a set of correlation functions Xn , which we now treat as fundamental; equations such as (3.41) then hold only approximately.n o Therefore in this approach the Kadanoff-Baym method for proving that the conservation laws that hold for G also hold for X(2), X(3), and so on is inapplicable. Furthermore, in the exact theory, all correlation functions can be computed by functional differentiation of G, and this is equivalent to the statement that the self energy can be expressed as a functional only of G. Clearly, the correlation functions calculated in the above scheme are no longer connected to G via functional derivatives, and this in turn implies that the self energy must be considered a functional of all the correlation functions upon which the effective interactions depend. Since one can still take quasi-equilibrium

self energy to be a functional of GQ alone, it acquires additional functional dependence on the deviations. In order to maintain the structure of the self energy implied by the Dyson equation, we write

(1) (2) Σ = Σ GQ, Xn , Xn , (3.74)

(1) (j) so that it would produce the exact Xn if the exact ∆ In were used. As discussed in (2) (3) the next section, the equations of motion for Xn and Xn show that both these can be (j) (1) considered functionals of WQ, XQ , and Xm . Therefore, if the correlation functions at

quasi-equilibrium can all be considered functionals of GQ, then

(1) Σ = Σ GQ, Xn = Σ [G] . (3.75) 3.2 Response Functions Framework 68

Thus, when restricted to the solutions of the equations of motion below, Σ continues to be a functional of the non-equilibrium, but approximate G. This ensures that the non- equilibrium G calculated from this self-energy will give back the same self-energy. Thus the optical response, which only involves G explicitly, can be obtained self-consistently with the self-energy of the charges in the system.

(j) It remains to determine which models for ∆In are admissible, and if one can follow a (2) recipe to construct them. A formula for the simplest case, ∆I1 , follows from the BSE and is discussed below in Section 3.2.3.2 (see (3.121)), but higher orders in the ﬁeld are dictated by the particular system of charges to which the formalism is applied. However, we list some general rules to take into consideration. First, an important rule is the symmetry,

I(2) (14;23) = I(2) (41; 32) ,

(2) which is then inherited by the functions Xn . This symmetry implies that the two approximate equations of motion for G(12) for coordinates 1 and 2 are exact conjugates of each other, and the particle number conservation follows when the equal time limit is taken[116]. We remark that this condition can be ensured by inspection of an approximate model for I(2) and symmetrizing it so that channels that are conjugates in the exact theory do remain conjugate in the approximate theory. Secondly, the approximate model must be able to capture the transient coherence and its subsequent loss. This coherence is non-local in the sense that it can be communicated across a time-lag via within the quasi-particle phase breaking time. In such a case, it is also necessary that the dynamic interaction, W , acts on all pairs of a correlation function symmetrically. That is, for a given correlation function X(j), each pair of one incoming and one outgoing line linked by W be equally weighted with the interaction among the outgoing and incoming lines. This is in contrast to solving the BSE in a single channel of ladder series by letting I(2) = W in (3.43). The explicit demonstration is deferred to Chapter 4.

3.2.3 Equations of motion

In this section, we derive diﬀerential equations in the time variables of the deviations (j) (j) Xn in the correlation functions X . The general strategy is to begin with an integral equation for X(j) (Figs. 3.2, and 3.4a-3.4b), apply a diﬀerential operator to it, and 3.2 Response Functions Framework 69

expand each function, O, in the resulting equation as OQ +∆O. The differential equation (j) (j) for Xn then follows from the differential operator acting on ∆X , and the remaining terms are classified as either couplings or sources.

To show the strategy, we ﬁrst outline the derivation for single, and two particle correlation (1) functions, X = G. We start with the general Dyson equation, writing Σ = ΣQ + ∆Σ,

1 G− UQ ΣQ G = 1+(U + ∆Σ) G, 0 − − and use the quasi-equilibrium Dyson equation for GQ to obtain

1 GQ− (11′) G (1′2) = δ (12)+ [U (11′)+∆Σ(11′)] G (1′2) .

Expanding G on both sides as GQ + ∆G,

1 GQ− (11′) ∆G (1′2) = [U (11′)+∆Σ(11′)] [GQ (1′2)+∆G (1′2)] . (3.76)

1 The inverse function GQ− is a diﬀerential operator written explicitly as,

1 ∂ G− (11′) = i H (1) δ (11′) UQ (11′) ΣQ (11′) , Q ∂t − − − 1 which yields (3.76) as a differential equation. The deviations ∆G and ∆Σ are expanded up to the desired order in the field, and by matching the terms of like orders we get the (1) equation of motion for the correlation functions, X1...3 (12) in the first argument. Thus to the first order in U,

1 (1) G− (11′) X (1′2) ∆Σ (11′) GQ (1′2) = U (11′) GQ (1′2) . (3.77) Q 1 − 1 (1) (2) The self energy deviation ∆Σ1 couples X1 to X1 , and the equations for these two correlations have to be solved self-consistently. We will omit the equations for the latter (j) until the next section. Once a solution is obtained for X1 it can be substituted into the right hand side of the equation following from (3.76) that is second order in U,

1 (1) (1) G− (11′) X (1′2) ∆Σ (11′) GQ (1′2) = [U (11′) + ∆Σ (11′)] X (1′2) . (3.78) Q 2 − 2 1 1 In both (3.77) and (3.78), the left hand side consists of deviations of the same order in 1 the ﬁeld as the one being diﬀerentiated by application of GQ− , while the right hand side consists strictly of the deviations of lower orders. The left hand generates coupled linear 3.2 Response Functions Framework 70

(j) dynamics of Xn , and the right hand side acts as a driving force, and we term the latter the source terms. Similarly, at the third order,

1 (1) (1) G− (11′) X (1′2) ∆Σ (11′) GQ (1′2) = [U (11′) + ∆Σ (11′)] X (1′2) (3.79) Q 3 − 3 1 2 (1) +∆Σ2 (11′) X1 (1′2) .

(2...3) The nature of the couplings to Xn in these equations depends upon the approximation for the self-energy, which depends on the models picked for I(j). We will elaborate on these couplings in the subsection below, and discuss the similar derivation for higher particle orders below.

The next equation in the hierarchy is the four-point BSE for X(2) = P , and for which we write four equations in the form,

1 (2) G− U ∆Σ P = G + GI P, Q − − where the operator on the left hand side acts on one of the four arguments of P . This provides diﬀerential equations for evolution in each of the four time arguments of P . Again, expanding quantities on both sides as their quasi-equilibrium value plus a deviation, and making use of the quasi-equilibrium BSE, we obtain

1 (2) (2) (2) G− U ∆Σ ∆P = ∆G + ∆GI P + G∆I P + GI ∆P. (3.80) Q − − (2) Expanding in the order of the field then yields equations for Xn , whose detailed form depends on I(j), and it is left to the subsections below. Similarly, the equations for (3...4) 1 ∆X can be written using the BSE in 3.4b and applying G− to each external vertex. 1 When the operator G− is acting on a quasi-particle diagram, it simply removes it. If it acts on a vertex of X(2...3), then the BSE for that correlation function is used to expand the term so that it contains a quasi-particle line at the vertex. This results in an equation for the time derivative of X(j), which is then expanded about the quasi-equilibrium point (3) as shown above to obtain an equation of motion for the deviations, Xn . While the above equations for ∆G and ∆P are derived without diagrams, the different contributions with proper counting are best determined using the diagrams where all permutations of the verticies and connections of the different components are explicit. This is the task to which we turn in the subsection below.

(1) (j) As we have seen in the case of Xn , the expansion yields equations of motion for Xn (k) consisting of terms that are linear combinations of Xn , which are the same order in 3.2 Response Functions Framework 71

(j) (j) the field, and products of Xp Xn p, which are of the lower order in the field. Thus one − (j) starts by solving for X1 , which are driven by U only, and then uses these correlations (j) to solve for X2 and so on. Thus the equations for a given order form a linear system which are driven by the non-linear interactions of correlation functions of lower order. The linearity arises from the perturbative treatment in the field amplitude. The driving terms would amalgamate with the homogeneous terms in a non-perturbative treatment leading to a full solution of the non-equilibrium many body problem. Such a treatment is not attempted here.

Before presenting the results it is helpful to visualise the system of equations in abstract form. This also breaks down the derivation and presentation of the results into smaller components. The coeﬃcients and source terms introduced below will be given in detail in the following two subsections. To ﬁrst order in U,

∂ (1) (11;1) (1) (12;1) (2) (1;1) i X1 (12) = 1 X1 + 1 X1 + 1 , (3.81) ∂t1 M M S ∂ (2) (21;1) (1) (22;1) (2) (23;1) (3) (2;1) i X1 (14;23) = 1 X1 + 1 X1 + 1 X1 + 1 , (3.82) ∂t1 M M M S ∂ (3) (31;1) (1) (32;1) (2) (33;1) (3) (3;1) i X1 (146;235) = 1 X1 + 1 X1 + 1 X1 + 1 , (3.83) ∂t1 M M M S

(jj′;l) (n) (j;l) where the n describe coupling between the X , and the terms n are the sources. M 1 S Here jj′ in the superscript (left of semicolon) represent the particle order of the correlation functions, l stands for the time argument for which the differential equation is written, and n is the order in the external field. Here we have explicitly written the equations for just one time variable. For each X(j), there are 2j time variables and therefore 2j such first order differential equations defined on the Keldysh contour. The couplings for M one of the times can be obtained from another by changing the appropriate arguments, and is done almost trivially from the self-energy diagrams. We will derive explicit forms for for only t1, and the relation to rest of the couplings will be made clear in the M (3) process. In the first equation, there is no explicit appearance of X1 , since the self- energy depends only on G and P . The third equation is where truncation is invoked to keep the system exact only up to the eight-point correlation functions. In the context of the previous section, we have explicitly neglected the dependence of all I(j) on X(4). 3.2 Response Functions Framework 72

To second order we have

∂ (1) (11;1) (1) (12;1) (2) (1;1) i X2 (12) = 2 X2 + 2 X2 + 2 , (3.84) ∂t1 M M S ∂ (2) (21;1) (1) (22;1) (2) (2;1) i X2 (14;23) = 2 X2 + 2 X2 + 2 , (3.85) ∂t1 M M S and to third order, there are only the single particle Dyson equations,

∂ (1) (11;1) (1) (1;1) i X3 (12) = 3 X3 + 3 , (3.86) ∂t1 M S whose solution determines the polarization signal that would be observed. The sum of the (j) coupling terms in each of the above diﬀerential equations map the Xn to itself. We thus refer to these terms collectively as dynamical maps. While leavingn mosto of the details to Appendices, we provide an outline of the derivation of the maps and sources below, and focus on putting the results in the context of two-dimensional Fourier spectroscopy.

3.2.3.1 Dynamical maps

(ij;l) In this section, we identify the expressions for the couplings, n . In particular, we M will use the diagrammatic BSE’s to ﬁnd the contributions to the ∆Σn terms in (3.77)- (2) (3.79) above, as well as the analogous terms in the equations for Xn . The expansion for ∆Σ follows immediately from Eqs. (B.4-B.6) of Appendix B, where the equation th for the n order derivative of Σ is contracted with n factors of U to obtain ∆Σn. In Figs. 3.2, 3.4a, and 3.4b we have placed these terms on the left hand side where all X(j) functions are connected to the right of eﬀective interaction blobs. Each of these is contracted with a U on its free pair of vertices. Thus these diagrams in Figs. 3.2,

3.4a, and 3.4b yield ∆Σ1, ∆Σ2, and ∆Σ3 respectively. While this prescription provides (j) general expressions for ∆Σn, we still need expressions for ∆I for BSE’s for higher order correlation functions. One strategy is to leave the equations in the form of (3.77-3.80) above, and then directly substitute the model effective interactions. However, the general expressions (3.44), (3.49), and (3.62) can be exploited further to identify contributions of different types of interactions. The extra effort provides insight into how ∆Σ, and ∆I(j) (j) must couple different Xn in any model for effective interactions. Models for effective interactions would inevitably include only a few scattering channels. In the following we also discuss the relative importance, in various quasi-equilibrium states, of different scattering channels appearing explicitly in the general expressions. 3.2 Response Functions Framework 73

(j) To proceed we note that terms such as ∆In in higher order correlation functions can be obtained by expanding the corresponding BSE’s in a form analogous to (3.61). The (j) couplings originate from terms containing a single Xn multiplied by a function that is evaluated in the quasi-equilibrium state. In the diagrammatic form all n ﬁelds are placed (j) (j n) on the same correlation box, X , which convert it into the deviation Xn − .

For the single particle correlation functions, we have

(11;1) (1) (1) (1) (11;1) (1) X = (11′)X (1′2) + (12 1′2′)+ (12 1′2′) X (1′2′)(3.87) Mn n H n W | K | n (12;1) (2) (12;1) (2) X = (12 1′4′;2′3′) X (1′4′;2′3′). (3.88) Mn n K | n Here we have introduced several new quantities,

(11′) = H(1)δ(11′) + ΣQ(11′), (3.89) H (1) (12 1′2′) = i WQ(13;3′1′)ΓQ(2′3′;2′′3)GQ(2′′2), (3.90) W | ˆ

(11;1) ∂Γ(1′′3′;2′′3) (12 1′2′) = i WQ(13;3′3′′)GQ(3′′1′′) GQ(2′′2), (3.91) K | ˆ ∂X(1)(1 2 ) ′ ′ Q (12;1) ∂Γ(1′′5′;2′′5) (12 1′4′;2′3′) = i WQ(15;5′3′′)GQ(3′′1′′) GQ(2′′2) K | ˆ ∂X(2)(1 4 ;2 3 ) ′ ′ ′ ′ Q (12;1) + (12 1′4′;2′3′), (3.92) Ke | which depend only on the quasi-equilibrium state. If Coulomb assisted band-to-band (12;l) transitions are ignored, the kernel e exists only at even orders in the ﬁeld, and K corresponds to the variation of the dynamical susceptibility due to the optical excitation. The deviation in the dynamical potential has been given in equations (3.69)-(3.70). A part of it contributes to the source terms, and the part that contributes to the dynamical map is

(2) ∆W (14; 23) = iWQ(12′;1′3)X (1′4′;2′3′)WQ(3′4;24′) (3.93) n |map − n and substituting this in the self-energy expression we obtain

(12;1) (12 1′4′;2′3′) = WQ(12′;1′7)WQ(3′5;64′)GQ(77′)ΓQ(7′6;25). (3.94) Ke | ˆ

(1j;2) Similarly, the maps n are obtained from the adjoint Dyson equation, and it produces M (11;2) (1) (1) (2) (11;2) (1) X = X (12) (2′2) + (12 1′2′)+ (12 1′2′) X (1′2′) (3.95) Mn n n H W | K | n (12;2) (2) (12;2) (2) X = (12 1′4′;2′3′) X (1′4′;2′3′). (3.96) Mn n K | n 3.2 Response Functions Framework 74

The diﬀerent components are

(2) (12 1′2′) = i GQ(11′′)ΓQ(1′′3′;1′3)WQ(2′3;3′2), (3.97) W | ˆ

(11;2) ∂Γ(1′′3′;2′′3) (12 1′2′) = i GQ(11′′) GQ(2′′3′′)WQ(3′′3;3′2), (3.98) K | ˆ ∂X(1)(1 2 ) ′ ′ Q (12;2) ∂Γ(1′′5′;2′′5) (12 1′4′;2′3′) = i G(11′′) GQ(2′′3′′)WQ(3′′5;5′2) K | ˆ ∂X(1)(1 4 ;2 3 ) ′ ′ ′ ′ Q (12;1) + (12 1′4′;2′3′). (3.99) Ke | (2) For Xn , the integral form of the equation, without source terms, is derived in equation 1 (D.3) of Appendix D. Applying the inverse Green function, GQ− , to this equation at each of the four arguments generates the four evolution equations corresponding to the (jj′;l) maps n . From (D.3), we can now identify the components of the dynamical maps M as follows,

(22;1) (2) (2) (2) (2) X = (11′)X (1′4;23) + GQ(2′′2)I (12′;2′′1′)X (1′4;2′3) Mn n H n Q n (22;1) (2) + (14; 23 1′4′;2′3′)X (1′4′;2′3′), (3.100) K | n (21;1) (1) (21;1) (1) X = δ(22′)δ(1′4) + (14; 23 1′2′) X (1′2′), (3.101) Mn n K | n (23;1) (3) (23;1) (3) X = (14; 23 1′4′6′;2′3′5′)X (1′4′6′;2′3′5′). (3.102) Mn n K | n (23;1) Due to termination of the hierarchy, the map n is set equal to zero for n > N 2 M − for O(U N ) calculation. The three kernels (ij;1) are given by the following expressions, K (2)r (22;1) 1 ∂I (1′′4′′;23′′) (14; 23 1′4′;2′3′) = G− (11′′) ΓQ(3′′4;4′′3) (3.103) K | Q ∂X(2)(1 4 ;2 3 ) ′ ′ ′ ′ Q e + 22(14; 23 1′4′;2′3′) K | (2)r (23;1) 1 ∂I (1′′4′′;23′′) (14; 23 1′4′6′;2′3′5′) = G− (11′′) ΓQ(3′′4;4′′3), (3.104) K | Q ∂X(3)(1 4 6 ;2 3 5 ) ′ ′ ′ ′ ′ ′ Q (21;1) 1 (14; 23 1′2′) = G− (11′)GQ(2′′2)ΓQ(2′4;2′′3) (3.105) K | Q 1 0 (2) G− (11′′) P I (1′′4′;21′)GQ(4′4′′)ΓQ(2′4;4′3) − Q Q Q 1 0 (2) G− (11′′) P I (1′′2′;23′′)GQ(3′′3′)ΓQ(3′4;1′3) − Q Q Q (2)r 1 ∂I (1′′4′′;23′′) +G− (11′′) . Q ∂X(1)(1 2 ) ′ ′ Q The function I(2)r is the reducible interaction,

(2)r (0) (2) (0) I (14;23) = P (12′;21′)T (1′4′;2′3′)P (3′4;4′3), (3.106) 3.2 Response Functions Framework 75

and where P (0) has been deﬁned earlier in (3.46). The partial derivative of I(2)r with respect to X(2) appears in (3.103), which is ultimately substituted in (3.100). There, (2) (2)r in the last term, it multiplies Xn , and a yields contribution of the latter to ∆In via (2) (2)r (3.72). This contribution of Xn to ∆In is shown diagrammatically in Figs. 3.6a and 3.6c. The middle terms in (3.105) for (21;1) are necessary to cancel the over counting K done by the use of I(2)r, and they are obtained by replacing the quasiparticle arrows (1) attached to one of the four vertices by Xn . Their diagrammatic form is shown in Fig. 3.6b. The kernel (23;1) is obtained by setting the two terms with six-point correlation K (2) (3) functions in I to Xn . In contrast to diagrams in Fig. 3.3, Fig. 3.6a explicitly shows I(2) symmetrically, and the substitution of the resulting two-particle correlation back into the self-energy would ensure that particle number conservation holds.

′ (jj ;2) 1 The map n is obtained by acting on the argument 2 in (D.3) by G− . We obtain M Q (22;2) (2) (2) (2) (2) X = X (14;2′3) (2′2) + GQ(11′′)I (1′′2′;21′)X (1′4;2′3) Mn n n H Q n (22;2) (2) + (14; 23 1′4′;2′3′)X (1′4′;2′3′), (3.107) K | n (21;2) (1) (21;1) (1) X = δ(11′)δ(2′3) + (14; 23 1′2′) X (1′2′), (3.108) Mn n K | n (23;2) (3) (23;1) (3) X = (14; 23 1′4′6′;2′3′5′)X (1′4′6′;2′3′5′). (3.109) Mn n K | n Here the kernels (ij;2) are given by K (2)r (22;2) ∂I (14′′;2′′3′′) 1 (14; 23 1′4′;2′3′) = G− (2′′2)ΓQ(3′′4;4′′3) (3.110) K | ∂X(2)(1 4 ;2 3 ) Q ′ ′ ′ ′ Q e + 22(14; 23 1′4′;2′3′) K (2)r | (23;2) ∂I (14′′;2′′3′′) 1 (14; 23 1′4′6′;2′3′5′) = G− (2′′2)ΓQ(3′′4;4′′3), (3.111) K | ∂X(3)(1 4 6 ;2 3 5 ) Q ′ ′ ′ ′ ′ ′ Q (21;2) 1 (14; 23 1′2′) = G− (2′2)GQ(11′′)ΓQ(1′′4;1′3) (3.112) K | Q 0 (2) 1 P I (14′′;2′′1′)G− (2′′2)GQ(4′4′′)ΓQ(2′4;4′3) − Q Q Q 0 (2) 1 P I (12′;2′′3′′)G− (2′′2)GQ(3′′3′)ΓQ(3′4;1′3) − Q Q Q (2)r ∂I (1′′4′′;23′′) 1 + G− (2′′2). ∂X(1)(1 2 ) Q ′ ′ Q The maps, or equations for the variables 3 and 4, are obtained by exploiting the symmetry condition, X(2)(14;23) = X(2)(41; 32). Diagrammatically, this amounts to ﬂipping the diagrams left to right and then upside down.

The diagrams shown in Figures 3.6a contribute to (3.100,3.107), while those in 3.6b contribute to (3.101,3.108). We now brieﬂy discuss their connection to decoherence. These 3.2 Response Functions Framework 76

diagrams originate from those for I(2) shown in Fig. 3.3, and discussed briefly at the end of Sec. 3.1.4. The kernel (22;l), and the subtracted terms in (21;l) are shown dia- K K 1 grammatically in Figures 3.6a and 3.6b respectively, but without the application of GQ− . Figure 3.6a shows three classes of diagrams. The top line shows retarded interaction in the particle-hole channel in such a way that a full correlation develops between the times this retarded interaction travels. The term particle-hole channel is meant to describe a pair of particles anti-parallel in time. Thus these diagrams in a sense measure the state of the correlation both at a creation and annihilation, or, heuristically, the “bra” and “ket” state of the two-particle density matrix. The retarded interaction in fact contains a susceptibility bubble that represents the interaction of this composite particle with an external electron, and traced over this external particle. Taking into account the propagators in the polarization bubble, we can also view this term as a three-particle Green function traced over two of its arguments to form a four-point correlation function. In the context of transport problems, similar diagrams involving three-particle functions are also found for treating many body correlations for plasmas, and in calculations of dielectric functions[119, 121]. There the necessity for the three particle function arises from a density high enough that three particle scattering is important. Here they are important even for a low density system in order to capture the subtle decoherence effects consistently. Appropriate excitation could put the system in a superposition of two states, and this superposition would undergo decoherence due to interactions. Because these interactions are dynamic, coherence can exist across a time delay and therefore between the incoming and outgoing particles of a temporally extended correlation function. Including all these diagrams on the same footing as the ladder series has dramatic effects on the calculation of the decoherence of excitons, as discussed in Chapter 4. We also remark that decoherence in the picture developed so far arises from partially tracing over correlation functions in the higher particle Hilbert spaces. The bubble thus represents a kind of effective bath, and interacting with it the two-particle correlation can undergo decoherence.

The first two diagrams in the second line of Fig. 3.6a show the interaction in a particle- hole channel, which starts at W . This interaction is renormalized to produce the dynamic Coulomb interaction, and vertex corrections are inserted on both ends to account for self- interaction of the two different interacting particles. The two diagrams thus show the field induced effects on these vertices. From the self-energy expressions, it follows that the 3.2 Response Functions Framework 77

(2) Xn (14; 23) joins with the interaction line at 3, 4, and therefore must satisfy ζ3 = ζ4. The direct contribution of this term to the evolution of excitons thus occurs as field-induced changes in the particle-particle vertex. Their direct role in decoherence will appear at higher orders in Coulomb interaction. Finally, the rest of the four diagrams arise from the field-induced changes in the dynamical susceptibility. If Coulomb assisted transitions across the gap can be neglected, then these diagrams do not contribute directly in the horizontal particle-hole channel in the dynamical map. They will contribute if a coherence is present to flip the band type of electron as it travels from the bottom to top of the diagram. Thus these diagrams will only appear in the source terms when the calculation is restricted to the horizontal channel. 3.2 Response Functions Framework 78

∂I(2)r (2) (a) Terms comprising ∂X(2) Xn . Expressions for these diagrams fol- Q low from Feynman rules in Fig. 3.1. Cut the external lines at corner l to obtain the kernel (22;l). K

r (1) (b) Subtracted terms, included in ∂I(2) /∂X(2)Xn . The shaded boxes and − (1) triangles are TQ and ΓQ respectively. Double lines represent Xn . See text after Eq. (3.105)

∂I(2)r (2) (c) Terms comprising ∂X(2) Xn at even orders due to ﬁeld induced Q e (2) variation in dynamic potential, and contributing to Xn . K22 Figure 3.6: Contributions of two-particle correlations to the two-particle EOM 3.2 Response Functions Framework 79

3.2.3.2 Source terms

Here we give the source terms at all three orders in the field in the integral form. They consist of the effective field U, and corrections arising from the optically induced correla- (j;l) tion functions beyond the Hartree one. The sources n have already been identified in S the equations of motion corresponding to time derivative in the argument l. For brevity, (j) 1 we will instead give explicit expressions for functions n , which when acted on by GQ− (j;l) S on the argument l yield n , S

(j;l) 1 (j) (1 ...l...j;1′ ...l′ ...j′) = G− (ll′′) (1 ...l′′ ...j;1′ ...l′ ...j′) ,(3.113) Sn Q Sn (j;l′) (j) 1 (1 ...l...j;1′ ...l′ ...j′) = (1 ...l...j;1′ ...l′′ ...j′) G− (l′′l′) .(3.114) Sn Sn Q

The explicit presence of GQ(ll′′) and GQ(l′′l) in the expressions below makes this operation trivial. Furthermore, this explicit presence also means that the source terms below form the leading term in a series solution to the dynamical equations (3.81-3.86).

We have for the ﬁrst order,

(1) (12) = GQ(11′)U(11′)GQ(1′2), (3.115) S1 which just describes propagation of free electron and a valence-band hole generated by the external field. At the second order, there start to appear local field corrections beyond the Hartree level. We denote this modification to U by

(1) U n (12) = U(12)δn1 + ∆Σn(12), (3.116) where ∆Σn is the deviation in self-energy that describes all possible interactions among (j) the pairs generated by n pulses, and includes contributions from Xm for m < n only (see discussion at the beginning of 3.2.3.1). In general,

(1) ∆Σn(12) = WQ(13;3′1′) Xn (1′1′′)ΓQ(1′′3′;23)+ GQ(1′1′′)∆Γn(1′′3′;23) , for odd n, and where ∆Γn is the corresponding deviation in the vertex function. For even n, we also have to include

(2) ∆W2n(14;23) = iWQ(15; 63)X2n (66′;55′)WQ(5′4;26′) − n 1 − (2) iWQ(15; 63) X2n (66′;55′)∆W2(n j)(5′4;26′), − − j=1 X 3.2 Response Functions Framework 80

where we have used (3.69). Thus we write the second-order source as

(1) (1) (1) (12) = GQ(11′)U (1′1′′)GQ(1′′2′)U (2′2′′)GQ(2′′2). (3.117) S2 1 1 Similarly, the third order source (1) has the analogous form, S3 (1) (1) (1) (12) = GQ(11′)U (1′1′′)GQ(1′′2′)U (2′2′′)GQ(2′′2) S3 2 1 (1) (1) +GQ(11′)U 1 (1′1′′)GQ(1′′2′)U 2 (2′2′′)GQ(2′′2) (1) (1) (1) +GQ(11′)U 1 (1′1′′)GQ(1′′3′)U 1 (3′3′′)GQ(3′′2′)U 1 (2′2′′)GQ(2′′2). (3.118)

(2) Next we consider n , which gives rise to the formation of excitons and the coherence S among their states. To the ﬁrst order

(2) (1) (1) (14;23) = (11′)GQ(2′2) + (2′2)GQ(11′) ΓQ(1′4;2′3). (3.119) S1 S1 S1 n o To obtain (2), we contract the eight-point diagrams in Fig. 3.4 with two ﬁeld lines, not S2 connecting two separate correlation functions with a ﬁeld line; this rule is explained in the next section. Proceeding in this manner we get the expression,

(2)(14; 23) (3.120) S2 (1) (1) = GQ(11′)U 1 (1′3′)GQ(3′1′′)GQ(2′′4′)U 1 (4′2′)GQ(2′2)ΓQ(1′′4;2′′3) (1) (2) (2) +GQ(11′)U 1 (1′1′′)IQ (1′′4′;2′3′)X1 (3′4;4′3)GQ(2′2) (2) (2) (1) +GQ(11′)IQ (1′4′;2′3′)X1 (3′4′′;4′3)U 1 (4′′2′′)GQ(2′′2) (1) (2) (2) +GQ(11′)U 1 (1′1′′)∆I1 (1′′4′;2′3′)XQ (3′4;4′3) GQ(2′2) (2) (1) (2) +GQ(11′)∆I1 (1′4′;2′′3′)U 1 (2′′2′)XQ (3′4;4′3) GQ(2′2).

(2) Here ∆I1 is the O(U) variation induced in the eﬀective interaction, and it can be related explicitly to the inverse functionals of quasi-equilibrium BSE, assuming they exist,

(2) 1 (2) 1 ∆I1 (14;23) = PQ− (12′;21′)X1 (1′4′;2′3′)PQ− (3′4;4′3) (3.121) 0 1 0 0 1 P − (12′;21′)P (1′4′;2′3′) P − (3′4;4′3). − Q 1 Q (2) (2) Note that the term ∆I1 X1 is not present in (3.120) because it implicitly contains a partial summation that forces it to be a second order deviation in Γ, and it has been included in the dynamical map above. It is now clear that the sources can be expressed algebraically in a compact form, and can be understood as successive corrections to the Hartree renormalization of the driving ﬁeld. 3.3 Application 81

We now come back to the point raised earlier (Eq. (3.75)) that Σ may be considered a functional of G only. Since all couplings are linear, they remain functionals of GQ so long

as the quasi-equilibrium problem is formulated to ensure that ΣQ = ΣQ [GQ]. Therefore,

when the coupled equations are solved, the solutions are functionals of GQ and whatever functions the sources depend on. Consider now the solution of the ﬁrst order equations (3) driven by the sources in (4.34), and (3.117) above as well as 1 not shown here explicitly. (1) (1) S The latter is a function of GQ, 1 , and 2 , and by (3.117) it may be considered a S S (j) function of GQ and U alone. Therefore all X1 can be considered functionals of GQ and U. When the sources for the second-order equations are constructed, they are always (j) (j) expressible in terms of X1 , and similarly the third order sources in terms of X1 and (j) (j) X2 . Consequently, all Xn can be considered functionals of GQ and U. Assuming that the dependence of ∆G on U is invertible, they may in turn be considered functionals of the non-equilibrium G alone.

3.3 Application

3.3.1 External and eﬀective ﬁelds

The bare source that drives interband coherence is taken within the dipole approximation, and thus is of the form,

Φ (12) = φ (k , t ) δ(t t )δ (k k ) δ , (3.122) cv cv 1 1 1 − 2 1 − 2 σ1σ2 For convenience we have written the band indices corresponding the arguments of Φ as subscripts. The subscript c corresponds to the conduction band, and v to the valence 1 iωt band. Here φ (k, t ) = e~− ξ (k) E (t) e− is the effective amplitude for making cv 1 − · the inter-band transition, and the wave-vector is the same for initial and final state due to negligible momentum of the optical field: k1 = k2 = k. The positive frequency iωt component of the electric field E (t) e− is the macroscopic field inside the medium of quantum well with a dielectric constant different from unity, and with a center frequency ω. The discussion below refers to fields inside the well (see Appendix 3.1.1 to obtain fields outside the well).

This particular identiﬁcation of bands is useful only for normal insulators and semiconductors, where there is an energy gap between the ground state and the lowest excited 3.3 Application 82 state. In the presence of Coulomb interactions, this imposes restrictions on the solutions of the Dyson equation and thus the single particle self energy. Within the framework of Green functions, the Dyson equation for the system in thermodynamic equilibrium takes the form,

[E H (r)] G(r, r′; E) dr′′ Σ(r, r′′; E) G(r′′, r; E) = δ (r r′) . (3.123) − 0 − ˆ − In general, the solution to this equation consists of isolated, and perhaps also a continuous, set of poles in the complex plane. If no poles exist in a finite interval on the real axis (or close to it), then that interval may be identified with an energy gap. The solutions with isolated poles close to the real axis, and to the right of this gap then define a set of conduction states in which a quasiparticle can freely propagate over a time interval corresponding to the imaginary part of the pole. Similarly isolated poles to the left of the gap in the complex plane define valence states that are all filled by quasiparticles in the ground state. The formal manipulations of the preceding sections are independent of these considerations, but the restrictions like these are necessary steps in actual calculations. Without any interaction induced symmetry breaking, the quasiparticle wavefunctions will also obey the lattice symmetry, and a crystal momentum can be associated with them. In this sense the description of the semiconductor via energy bands survives. Clearly these single particle concepts remain useful only at timescales shorter than the lifetime set by the imaginary self energy. Assuming the many-body system to be a semiconductor, the effective field, U, is given by

Uζ1ζ2 (k, t1; k, t2) = Φcv (k, t1; k, t2) δζ1cδζ2v δG (43) Vζ1ζ3;ζ4ζ2 (13; 42) φcvk (t3′ ) δ(t3′ t4′ ), − ˆ δΦcv(3′4′) − where we have not fixed the band subscripts on U, since in general, the second term on right hand side may couple the inter-band transitions to each other and to the intraband motion. This term represents the contribution made by the change in the Hartree energy of the electron gas due to the creation of electron-hole pairs. We first consider the modification of the inter-band transitions between the conduction and the valence bands, δG (43) Ucv (k, t1; k, t2) Φcv (k, t1; k, t2) = Vcζ3;ζ4v(13; 42) φcvk (t3′ ) δ(t3′ t4′ ). − − ˆ δΦcv(3′4′) −

By (3.24-3.26), only the Umklapp processes contribute to the matrix element Vcζ3;ζ4v(13; 42) close to k = 0 (Γ point). This is so because in this region the cell-periodic functions 3.3 Application 83

uζ (k; ρ, z) can be taken approximately equal to uζ (0; ρ, z), where the latter belonging to diﬀerent bands are orthogonal. If all relevant dynamics is concentrated in a small enough region around the Γ point, then the matrix elements of interest involve only uζ (0; ρ, z) to a good approximation. In essence the particular Umklapp processes that allow Coulomb interaction to drive inter-band transitions can resonate with optical excitation close to the band-edge in general. However, neglecting that is a good approximation for most problems of interest, such as exciton dynamics. This also holds for the intraband terms,

Ucc and Uvv because in the response function δG/δU, they require either an Umklapp process at the 34 coordinate of V , or a band-transition other than the one driven by the field,. Since we only consider Coulomb interaction, the Umklapp process is also the only choice for the latter that is consistent with the Hamiltonian. Therefore, we assume that the dielectric function arising from (3.37), which connects the bare field with the Maxwell field, is unity at band-gap frequencies and diagonal in band indices. More generally, transitions among subbands, which can be low in energy, can couple to the low energy excitations of the charge density and the functions Ucv and Φcv will become substantially different.

3.3.2 Analysis via diagrams

(j) In the foregoing we have obtained differential equations for Xn that consist of dynamical maps and source terms driving them. Formally, a series solution to these equations be- (j) gins with an integral of the source terms, which corresponds to n in Section 3.2.3. This S approximate solution neglects the couplings. The effect of the couplings can be treated by iteration. The iteration generates a series of terms that represent propagation at the external vertices, and interactions among those vertices in all possible ways. A contribution to the solution can be represented diagrammatically to depict the physical processes it entails. Without interactions, a finite number of diagrams, namely, the double sided Feynman diagrams, capture the full solution. With interactions, the set of such diagrams is infinite, and it is not practical to represent the entire solution diagrammatically. It is practical, however, to use a diagrammatic form to represent the simplest term in (j) the series, the n . Diagrams provide an intuitive yet rigorous method for understanding S (j) the sources that drive the dynamics of a particular Xn . They also provide an extremely useful pedagogical tool to build approximate solutions, and identify their qualitative aspects. Furthermore, since the sources are those components in the equations that relate 3.3 Application 84

directly to the experimental parameters for pulses, they allow one to depict an experimental scenario as well. In particular, different subsets of diagrams are distinguished as dominant under different excitation conditions corresponding to pulse sequences and choice of observation directions. Recalling the relation in (3.32), the branch indices and particle species lying on the two contours at observation time t are also fixed by the experiment. So a connection with experiment can be made directly using the diagrams. In the rest of this section we develop a set of rules to construct diagrams, which can be applied to either sources or solutions. Note that we have already encountered a set of diagrams in the previous sections. Those diagrams are general, but not explicit in the contour branch indices, pulse sequences, and absorption/emission of the electromagnetic fields.

In developing the diagrams, we assume that the rotating wave approximation holds so that we are close to a resonance in the system and the optical cycle is much shorter than the timescales in which we are interested. The diagrams are represented on Keldysh contours oriented vertically with time increasing upwards. The left contour is positive, and the right is negative, as shown in Figure 3.7. We represent the self-consistent potential, U, by a directed dashed line, where the line entering the diagram represents the unconjugated field that is absorbed, and the line leaving the diagram is the conjugated field that is emitted by the system. The temporal location of these lines clearly exhibits the pulse sequence and time delays. The observed field emitted by the polarization is shown as a small arrow pointing out of the horizontal line placed at the observation time. The quasi-particle lines can either lie on the contours, parallel or anti-parallel to its direction, or can run from one contour to the other. In either case, if the arrow points against the contour direction, the field operators in the expectation value are reversed. They thus carry factors of density.

(j) We now discuss rules for constructing these diagrams for the source terms n in the (j;l) S form in which GQ lines terminate at all external vertices. The n may be obtained S by removing the appropriate GQ. Using the integral equations derived above and the expression (3.32) for the density matrix, it can be veriﬁed that admissible diagrams can be drawn via the following recipe.

1. Draw the positive and negative contours, and place n electric-field vertices at desired points on the contour, with the arrow pointing outwards for fields that are conjugated, and inwards for the unconjugated fields. 3.3 Application 85

(j) 2. On the contour, mark the points 1, 2,...,j corresponding to the arguments of n S function being represented. Fill the diagram via the following rules.

3. Insert a diagram between the contours using either Figure 3.1 to convert the mathematical expression into a diagram, or directly from the BSE for X(j+n) (Figures 3.2 and 3.4). Attach the n external field lines to the diagrams, with a pair of vertices (other than those in rule 2) for each line. At the electric field vertex, a conduction electron should leave and valence electron enter the vertex if the field line is incoming, and the opposite if the line is outgoing.

4. The pair to which an external ﬁeld line connects must belong to the same correlation under partial contraction. If a pulse fully contracts a diagram, i.e., it produces (1) Xn;cv, then it is allowed to connect two otherwise separate parts of a diagram. The proof of this rule is in Appendix E.

5. A correlation function can be connected to rest of the diagram only via eﬀective interaction or a ﬁeld line, and up to only two vertices may be attached to an interaction. This rule follows by examining (B.2-B.3) and noting the placement of arguments in δG/δU relative to δΣ/δG.

6. A completed diagram with all three electric ﬁeld lines attached must not be in a separated form. This is because the ﬁnal diagram represents a term in the expansion of single-particle Green function. The internal vertices are not placed on the contours but in the space between them.

7. A diagram containing all three pulses must have a conduction electron (band index c) arriving at a point on the positive contour and a valence electron (band index v) departing from a point on the negative contour. The two points should be at equal real time. Note that this rule is a specialization of the formalism to semiconductors excited close to the gap frequencies.

8. At this point, a diagram contains exactly one eﬀective interaction blob since all BSE’s are linear in the eﬀective interaction. More interactions are placed only when iterating to build an approximate solution.

This produces a set of diagrams that contain products of correlation functions, eﬀective interactions, and the self-consistent ﬁeld. Besides the above rules to construct a diagram 3.3 Application 86 from a mathematical expression, we also need to specify rules for the reverse process of converting a diagram into a mathematical expression. The diagrams are evaluated, and source diagrams are distinguished, by the following rules:

(j) (j k) (a) For each X connected directly to k instances of U, write Xk − .

(2) (3) (b) For four, six, and eight point blobs, write I (11′;21′′), I (11′2′;21′′2′′), and (4) I (11′2′3′;21′′2′′3′′) respectively as given by Fig. 3.1.

for each line leaving Uvc(k, t) regardless of which contour the vertex lies on. Con- serve the total quasi-momentum at the ﬁeld vertices as a consequence of the dipole approximation. Again, this rule is also a specialization to semiconductors excited close to the gap frequencies.

(d) Any vertex that is not on the ﬁnal observation time, and not connected to the external ﬁelds is an internal vertex and must be summed over time, branch indices, particle species, and momenta.

(e) Some internal summations involving I(j) will be of the form where both arguments (1) (j) of Xn are connected to the same I . To each diagram containing m functions, (1) (1) (1) (j) m Xn1 , Xn2 ,... Xn , connected to I in this form, associate n n , and p m ≡ i=1 i ≡ j m. Collect all diagrams that have the same n and p, and substitute for all of − P (j) (p) them a single diagram with I replaced by ∆In . It is always possible to do this if all possible diagrams are constructed with proper counting such that the diagrams (j) (1) are identical if the I and Xni identiﬁed above are removed. The reason for this is simple: these diagrams result from derivatives of I(p).

(f) Set the remaining correlations to their quasi-equilibrium values.

(j) (g) If a diagram corresponds to a source term, n , then it must not have any com- S n (n) ponent that is O(U ), and its external vertices must all be connected to XQ or

GQ. The latter can be ensured by iterating BSE of a correlation function as de-

scribed below Eq. (3.80). Remove the GQ line from the contribution to EOM for the corresponding vertex.

(n) (h) Among the remaining XQ , there will be up to j connected directly to j pulses. (n 1) Express these as X1 − . 3.3 Application 87

(j) (j) (i) Using the model for I , substitute the model expressions for ∆In as given by (3.70-3.72).

In a chemical picture, where the focus is on excitons, biexcitons, and the like, certain correlation functions are identiﬁed with particles of new species. It is often desirable, and quite simple, to determine whether a diagram is proportional to the density of a particular species. We note that any quasi-particle line directed against the contour is proportional to the density of that quasi-particle species. For any correlation function, if traveling along the contour, one encounters a creation point before the annihilation point, then either (a) the diagram is proportional to the density of quasi-particle related to the two points, or (b) if all constituents of a composite particle species are annihilated before they are created, the diagram is proportional to the density of species in question. We illustrate the conversion of a diagram to a mathematical expression by discussing some example diagrams. The construction of source diagrams will be illustrated using a concrete example in section 3.3.3 below.

(1) The diagrams we discuss now are some of the contributions to X3;cv (12) for the case of three pulses a, b, c, and the observation direction corresponding to ω ω +ω . Depending a− b c on the contour on which the fields are placed, there are eight combinations. In the DCT formalism, only the diagrams where the first pulse is placed on the left contribute, due to the electron-hole vacuum as the ground state; this is essentially the content of the Axt- Stahl theorem[67]. In the diagrams where the first pulse is on the right the quasi-particle propagators at the first pulse will always be anti-contour ordered, and would thus be given by the removal amplitudes. Consequently these diagrams contribute only when the quasi-equilibrium state contains finite quasi-particle densities. The quasiparticle lines are shown in Figure 3.7a-3.7b for non-interacting electrons. The physical process can be read off from these diagrams as follows. In the present discussion, we define the term pair to mean a conduction electron and valence hole. The first pulse creates a pair, the second annihilates, and the third creates a pair again. Due to the full connectedness of the diagram, this occurs at the same quasi-momentum state, and thus represents coherent driving of an electron between the valence and conduction bands. Since the quasi-particle lines are all parallel to the contour, the resulting amplitude is proportional to vacancy of the electron states. In Figure 3.7b, the first pulse is placed on the negative contour, and thus the quasi-particle lines leaving this vertex are anti- contour ordered. The result is a diagram proportional to the occupation of electrons. In 3.3 Application 88

the solution to the Schrödinger equation, the two diagrams add to form a contribution proportional to the Pauli blocking factor, f f . For pulses of finite width, a summation v − c must be performed for different locations of the field lines. In the case of non-overlapping pulses, the structure of the diagram does not change, and thus only quantitative changes occur. The result is then an expression of the form

i(ω1 ω4)t dω e− − U (k,ω ω )U (k,ω ω )U (k,ω ω ) ˆ j cv 1 − 2 cv 3 − 4 vc 3 − 2 j=1,4 Y AQ (k ω )AQ ( k ω )AQ (k ω )AQ ( k ω ) × ;cc | 1 ;vv − | − 2 ;cc | 3 ;hh − | − 4 (1 n (ω ))(1 n (ω ))(1 n ( ω ))(1 n ( ω )). × − F 1 − F 3 − F − 2 − F − 4

Here the functions AQ;ζζ are spectral functions for electrons in band ζ, and nF is the Fermi distribution function. If pulses are signiﬁcantly overlapping, then new diagrams would result when the order of ﬁeld lines changes.

Next we consider the same two diagrams when the Coulomb interaction is present. Fig- ures 3.7c-3.7d show the case where correlated pairs are generated, but they evolve independently from each other. This diagram would be needed to describe an ideal gas of excitons generated via optical excitation. Note that in both correlation boxes, the vertices connected to the ﬁelds occur ﬁrst. These correlation functions treat the exciton as

elementary,[124, 119, 121] and correspond to the exchange-correlation part of av† acac†av , 1 where the pair creation is placed to the right of pair annihilation . Similarly, placing the ﬁrst pulse on the negative contour, as shown in the right hand side Figure, corresponds

to the exchange-correlation part of ac†avav† ac . On the other hand, placing the second pulse on the positive contour brings the composite nature of the exciton into the process, see Figure 3.8. This is so because the third pulse produces a correlated pair that prop- agates into a state from which the second pulse had removed an electron, but not the valence hole. It can be veriﬁed, using a formal expansion over exact eigenstates, that this correlation function is indeed described by a subset of states in the N 1 electron Hilbert − space of the N-electron system. A symmetrical contribution comes from the states in the N +1-electron Hilbert space that results if the second pulse had annihilated a valence hole instead of an electron from a pair. On the other hand, if the outgoing ﬁeld is the last

pulse, then it is easily shown that only the correlation functions of the type av† acac†av

and ac†avav† ac are involved.

1Note that the operators in the expectation value may have diﬀerent real-time arguments. 3.3 Application 89

(a) (b)

Figure 3.7: Pair evolution induced by three pulses, where the pairs are either uncorrelated (top) or correlated (bottom).

Figure 3.8: Correlated pair evolution induced by three pulses. The third pulse sees the composite nature of excitons.

The diagrams introduced above are for two-point correlation functions at O(U 3). We now briefly discuss some diagrams for four-point functions that are O(U 2). Diagrams of this kind are central to the study of multi-particle correlations, and they contribute to the source term (2) (14; 23) of Eq. (3.120). With the choice of band and Keldysh indices, S2 many different topologies are possible for (2) (14; 23), of which two are shown in Figures S2 3.9 and 3.10. Figure 3.9 shows diagrams with one pulse on each side of the contour. Heuristically speaking, it thus affects both the “bra” and “ket” side of the correlation 3.3 Application 90

functions, and leads to coherences among two-particle excited states. In contrast, the diagrams in Fig. 3.10 contain both pulses on the same side, and they therefore generate coherence between the ground state and a multi-particle excited state - a biexciton state in this case.

In Fig. 3.9 we show two diagrams contributing to (2) , which under appropriate ex- S2;cvvc citation frequencies generate Raman coherence between exciton states. The diagram (2) 3.9a couples X1;ζv;ζ′c on the “bra” side of the exciton correlation, and directly to the self-consistent field U on the “ket” side, where the interaction blob is taken at quasi- equilibrium. Thus this diagram corresponds to the second term of (3.120). The dia- (2) gram 3.9b couples the second pulse to effective interaction, which generates ∆I1 , and corresponds to the second last term of (3.120). It can be verified that the first term of (3.120) does not contribute to (2) in the absence of interband coherence in the S2;cvvc quasi-equilibrium state. We will discuss these diagrams and their behaviour at length in Chapter 4.

We now turn to Figure 3.10, which shows two contributions to the biexciton source (2) 2;cc;vv. The coherence between a biexciton state and the ground state is a four-point S (2) function, a biexciton amplitude, which is described by the correlation function Xcc;vv. The amplitude is given by creation of two conduction electrons and two valence holes, as expected for a coherence between two excitons and the ground state. Its source term is also given by (3.120), but where in contrast to Raman coherence, the ﬁrst term is now a (2) leading order term (shown in Fig. 3.10a). The term containing X1 is shown in 3.10b, (2) where the eﬀective interaction blob is IQvv;cc. These diagrams can be used to build a fully many-body description of the optical generation of biexcitons as a function of pulse parameters, as well as the quasi-equilibrium state.

Thus we see that the contour diagrams are a useful bookkeeping device that help us understand the relationship of various correlation functions to the pulse sequences and time delays, which constitute experimental input. 3.3 Application 91

(a) (b)

Figure 3.9: Diagrams for (2) (14; 23), which lead to Raman coherence between two S2cvvc exciton states.

(a) (b)

Figure 3.10: Diagrams contributing to the biexciton amplitude (2) (14; 23). S2cc;vv

3.3.3 Signal and two-dimensional spectrum

In this section we discuss the third order source term, which directly drives the detected signal. We also identify the two-dimensional Fourier spectrum which follows from this signal and can be compared with the one obtained experimentally. The source term is discussed by making several simplifying approximations as our goal is illustration. The calculation of ﬁrst and second order correlations that contribute to this source will be taken up in Chapter 4, and they are assumed to be known here.

We let the function φcv (k, t1) in (3.122) be

iωt φ (k, t) = uke− δ (t t ) δ , (3.124) cv − d σ1σ2

where uk is a complex number, and the delta function is an idealization of a pulse with temporal width smaller than the timescale of the dynamics. From (3.118), the source 3.3 Application 92

(1) driving the correlation X3 (12) in its ﬁrst time argument is,

(1) (1) (1) (12) = GQ(11′)U (1′1′′)GQ(1′′2′)U (2′2′′)GQ(2′′2) (3.125) S3;cv 2 1 (1) (1) +GQ(11′)U 1 (1′1′′)GQ(1′2′)U 2 (2′2′′)GQ(2′′2) (1) (1) (1) +GQ(11′)U 1 (1′1′′)GQ(1′′3′)U 1 (3′3′′)GQ(3′′2′)U 1 (2′2′′)GQ(2′′2),

(1) where U n , given by (3.116), include the self-energy corrections to Ucv beyond Hartree. As expected, the source term contains correlations of the first and second order, and accounts for their interference via quasi-particle propagation between them. This source term has two distinct types of contributions. The first two terms originate from the two-particle correlations arising from (any) two of the pulses, which are linked to the electron-hole correlation from the third pulse by a quasi-particle. That is, the two types of correlations (each of order lower than three) interfere within the phase breaking time of the electron gas. In terms of the BSE of Figure 3.4b, these terms are a sum of all those terms on the right hand side that contain a X(3) block. The third term arises from the rest of the two diagrams on the RHS, which contain products of only four-point functions. This term describes the interference of independent single particle correlations due to the three pulses. The functions GQ account for propagation as well as memory effects if there is also a time delay between the pulses.

By removing GQ (11′) or GQ(2′′2) from (3.125) we obtain the source term for the dif- (1)+ ferential equation (3.86). In general, the dynamical evolution of X3;cvk− (12) must be taken fully into account so as to treat the correlations of each individual pulse on an (1)+ equal footing. However, X3;cvk− (12) is driven also by the exciton-coherence, unlike the inter-band polarizations of the first two pulses. The next chapter is devoted to a calculation demonstrating that, under experimentally relevant conditions, the contribution of exciton-coherence can last much longer than the inter-band coherence. In such cases, the (1) dynamical evolution of X3 has the overall effect of convolving a sharp response function with the above source term. Neglecting all couplings other than the self-coupling, (1) ΣQ (11′) X3;cv (1′2), which restores the removed GQ, an approximation for the solution is given by the integrated source term. Further corrections can be made on the diagram by allowing interactions between the external vertices 1, and 2. Thus, as alluded to in the previous section, the source diagrams can be used to solve the equations directly. We will see a specific example below.

Figure 3.11 shows two diﬀerent contributions to (1) . The diagrams show an identical S3;cv 3.3 Application 93

four-point correlation formed by the first two pulses as indicated by the inner-most dashed box. The shaded ellipse in both diagrams stands for a sum of different interaction blobs as identified below. However, the diagrams differ entirely in terms of what it is with which the third pulse interacts. In diagram 3.11a the third pulse interacts with a fully (1) + correlated second order source U 2 . The line GQvv− directed into the pulse accounts for the effect of hole lifetime in linking this source across a time-delay to the pulse. By comparing with (3.125), and considering also the case when the interaction blob is absent, the reader may easily verify that this diagram originates from the last two terms. As shown on the

diagram, the part of the diagram other than the factor GQ;cc and Ucv from the third pulse (1) may be replaced by X2;vv, or the field induced hole population. Analogous diagram with (1) X2;cc also exists and originates from the first and the last terms of (3.125). These contributions of diagram 3.11a account for the effect of Pauli blocking due to the first two pulses on the driving of the system by the third pulse. This can be seen by (1) writing the source term for the equal time limit for Xcv (12). We write the source term as a function of the time delays, τ, tb, and td as shown in Figure 3.11. It is convenient to switch to the Wigner representation such that a two time function f(t, t′) is written as (1) f((t + t′)/2, t t′). Then, letting S (t ,τ,t ) represent (12) as a function of the three − d b S3 variables τ, tb, and td, e

S (td,τ,tb) (3.126) dk (1) τ (1) τ = uk X k t + t + , τ X k t + t + , τ ˆ 4π2 2;vv | d b 2 − 2;cc | d b 2 − +S (t ,τ,th ) , i X d eb e

where t = td + tb + τ, and SX stands for the difference between (3.125) and the first term. Thus the Pauli blocking term will map to excitations of the conduction and valence band populations, and the Fourier transform of this function, (t , τ) (ω, Ω), will naturally d → contain peaks at (0,ω k), and (0, ω k). vv − cc Diagram 3.11b contains contributions from entirely different ways of assigning pulses to (2) U j depending on the details of effective interaction, and leads to richer possibilities. If the interaction is such that a continuous chain of quasi-particle lines links all three pulses, then this diagram corresponds to the third term (3.125). The topology is then like diagram 3.11a and a contribution exists without the interaction as well. When such a continuous chain does not exist, then the diagram corresponds to the first two terms (1) (3.125), but now the source U 2 is formed by the first and third pulse. It interacts with 3.3 Application 94

(2) the fully correlated source U 1 of the second pulse. The interaction in the latter case is not optional, as it is in diagram 3.11a, because its absence would result in a separated form in violation of Rule 6 above. Depending on the topology inside blob, its connec- (2) (1) (2) tion to the ﬁrst two pulses contributes either a ∆I2 or the products GQX1 ∆I1 PQ, (1) (2) (2) and GQX1 ∆IQ X1 . All these terms also follow if one performs a formal expansion of (3.43) up to second order in the ﬁeld by making substitutions as in (3.52), and in strict application of the formalism they are an application of Rule (e) in the previous section.

(a) (b)

+ Figure 3.11: Diagrams for X3;cv− (tt). Intermediate state (before last pulse arrives) is a hole population on the left and exciton on the right and - both these are resonant only at second order in the ﬁeld in a perturbation expansion.

As noted above, the resulting expression for the diagram would normally be substituted in the diﬀerential equations but it is reasonable to consider an iterative solution instead. We consider building such an iterative solution to Dyson equation by starting from this diagram. The Dyson equation explicitly leads to interactions between the external vertices, 12, and since they correspond to conduction and valence band respectively, there will be a contribution from the diagrams that form bound exciton states. It is shown in (1) (1) Chapter 4 that there are no interaction free (i.e. X1;cc (13) X1;vv (42)) contributions to this diagram. From (3.43) it then follows that allowing all possible interactions among + σ the four vertices would lead to the function X − − (14; 23), where σ = , and the 3, 4 2;cvvc ± coordinates are coupled into the third pulse. Thus we obtain the integral form of the 3.3 Application 95

(1) excitonic contribution to the solution X3;cvk,

+ + + + + 11 X −−− 14;1 4 U 44 . (3.127) SX ≈ 2;cvvc cv (2) Note that the equation is exactly in the form of (3.41) when we replace P by X2;cvvc. + + + Since the third pulse precedes the measurement time, it follows that X2;cvvc−−− (14;1 4 )= + + + + X2;cvvc− − (14;1 4 ), where the latter can be identiﬁed with the exciton correlation (Chapter 4). Thus the third pulse beats against the two-time exciton correlation. Using the Wigner variables to deﬁne a function ˜ via, P

+ + + + i(ωnt1 ωmt2) X − − 12;1 2 = ((t + t ) /2, t t ) e− − , 2;cvvc Pnm 1 2 1 − 2 we write the approximate solution, (3.127),e as a function of pulse parameters,

11+ S (t ,τ,t ) (3.128) SX ≈ X d b τ iωnm(t +t ) iωnτ/2 θ (τ) θ (t ) ϕ (k) u t + t + , τ e− d b e− . ≡ d n mPnm d b 2 n X In (3.128) θ ( ) is the step function, ϕ (k) eis the wavefunction of the exciton state n in · n momentum space, and um is the projection of the transition amplitude onto the exciton state m. Apart from a shift t , the source S (t ,τ,t ) maps directly to ˜ in this impulse b d b P response scheme. Since the evolution of X3;cv is linear, this direct mapping extends to (1) that quantity, although convolution with the pulse will make it diﬃcult to relate X3;cv and in a straightforward way. We let P + ∞ ∞ e (ω, Ω) = dτ dt (t, τ) eiωt+iΩτ , P ˆ0 ˆ0 P which then yields thee Fourier transform of the sourcee term,

ω ωn + ωm i(ω ωnm)t S (ω, Ω) = ϕ (k) u ω ω , Ω e− − b(3.129). X n mPnm − nm − 2 − 2 nm X e The shift ωnm in the ﬁrst argument represents the fact that it is conjugate to the evolution of the correlation function for a ﬁxed delay. This is also seen from the diagram where td maps to the time at which the correlation is two-particle. The third pulse directly couples to from right, and leaves vertices on the left side of the correlation to evolve Pnm iωnτ freely. The evolution of these vertices along the contour produces the factor e− as it evolves. Taking the Fourier transform yields this frequency shift for Ω.

(1) Equation (4.10) is an approximate solution to X3 , which is a response to the Maxwell ﬁeld. The radiated ﬁeld that is detected is given by (3.14-3.16), which are written in 3.4 Summary 96

terms of the expectation value of the current density. In the length gauge the current density operator, e~ J(r, t) = ψ† (r, t) ψ (r, t) ψ† (r, t) ψ (r, t) , (3.130) im ∇ − ∇ can be re-written in the Heisenberg picture as, ie J(r, t) = ψ† (r, t) [ (t), rˆ] ψ (r, t) , (3.131) − ~ H due to the commutativity of Coulomb interaction with the position operator, rˆ. The desired expectation value from the above equation is then e J = Tr [ (t), rˆ] G(11+) , (3.132) h i −~ H where we have switched to the Schrödinger picture for operators inside the trace. The O(U 3) contribution to the current density within the two-band model employed here is

J (3) (q, t) (3.133) D E dk = eδ (q k + k k ) ω (k) rˆ (k) X(1) (kt; kt+)+ c.c. , − − 1 2 − 3 ˆ 4π2 cv vc 3;cv n o where ki are wavevectors of the three optical ﬁelds. The summation in (3.133) is essen-

tially given by SX (4.10) apart from the additional factor of kinetic energy of the electron and hole. This expression is substituted into (3.14) to obtain the TDFS signal in the far ﬁeld.

3.4 Summary

In this chapter, we have presented a framework suitable for applications to multi-pulse optical excitations of semiconductors based on non-equilibrium Green functions. In particular, we speciﬁcally addressed multidimensional Fourier spectroscopy experiments. Given a model of the many body system in a quasi-equilibrium state, the formalism applies functional diﬀerentiation to determine the optically induced variations in the single particle Green function by computing the respective variations in the self-energy. The model itself may be formulated within the framework of non-equilibrium Green functions.

We discussed the challenges posed by the hierarchy problem, and used a truncation scheme that leads to a self-consistent solution for up to n-order correlation functions; it 3.4 Summary 97

includes the eﬀects of all scattering channels involving n particles. This scheme is appropriate for a consistent treatment of decoherence processes, since all scattering processes at a given level of scattering are present by construction.

Using this truncation scheme we obtained a set of coupled, linear, diﬀerential equations that consist of dynamical maps, and driving terms (source terms). We identiﬁed and discussed the physical interpretation of both types of contributions to the equations of motion. Furthermore, the source terms are also conveniently described by diagrams on the Keldysh contour, which turn out to be a natural extension of the double sided Feyn- man diagrams used in more phenomenological treatments. We showed simple examples of such diagrams. While the full solution to the equations of motion found here is not feasible at present, they are valuable as a starting point to model a particular experimental situation. A solvable approximation to the multi-particle correlation functions is necessary in order to obtain solutions. An illustration of this is made in the following chapter.

The class of problems where this method is the most appropriate is optically induced dynamics of a multi-particle system that contains finite density of quasiparticles. Thus within the field of semiconductor optics, doped (optically or impurity) quantum wells constitute an obvious system for testing this theory. In addition, the formalism may be extended to include the decoherence effects of the excitation process itself. This excitation induced dephasing is important in the proper interpretation of experimental data since even when ultrashort pulses are tuned to excitations below the fundamental gap, they usually have enough bandwidth to generate free carriers. Chapter 4

Multidimensional Fourier Spectroscopy: Exciton Decoherence

In the previous chapter, we presented a formalism that uses non-equilibrium Green functions to model three pulse excitation of semiconductors. The formalism is general enough to handle a variety of initial states, including a quasi-equilibrium distribution of carriers established prior to the excitation. We also developed a diagram method that is useful in analyzing the many body interactions and optical excitation on an equal footing. In this chapter we complement the abstract nature of Chapter 3 by applying the formalism to a specific problem, the decoherence of exciton states in a background electron gas. We focus on systems in which the hole and exciton densities prior to optical excitation densities are vanishing. The motivation for studying this problem is three-fold. First, excitons are arguably the most experimentally accessible many-body states. Second, exciton states represent the simplest many-body excitations that can be used as a concrete example for testing any formalism. Third, their interaction with an electron gas presents a new regime of decoherence, where the usual system-bath separation does not exist owing to the indistinguishability of the electron comprising the exciton from that in the gas. In this chapter we show that, at least within a physically motivated set of approximations, decoherence effects in this scenario can be brought into direct comparison with the conventional system-bath models while respecting the indistinguishability. In studying the dynamics of optically excited excitons, the specific property of interest is Raman coherence[8]. Raman coherence refers to the coherence between excited states of

98 99

the semiconductor. Being a coherence between excited states, it is not directly injected by linear optical excitation, in contrast to interband polarization, which is a coherence between the ground state and an excited state. Raman coherences can be dominated by the exchange and correlation in the multi-particle states, and can thus be fundamentally diﬀerent from the mere beating of interband coherences. The formalism that we apply in the present chapter is built to handle precisely this exchange-correlation part, and we quantify the extent to which the exchange and correlation dominate. Nonetheless it is the beating of interband coherences that gives rise to a driving term, which leads to the rise of coherences in multi-particle correlations. The ﬁrst major result of the present chapter is the microscopic description of one such mechanism of this transfer of coherence.

The mechanism in question is driven by the dynamical response of the electron gas, more precisely the density-density correlation. This response leads to a non-factorizable correlation among two interband polarization amplitudes. At the same time, the continuum of excitations in the electron gas, consisting of plasmons and pair excitations, lead to a “measurement” of this coherence. The same dynamical response thus also leads to the diminishing of the coherence it helps build. Yet the two processes do not cancel. While the appearance of this coherence originates from the strength of the multipole moments of the two exciton states, the decoherence depends sensitively on how differently the two multipoles interact with the electron gas. Incidentally, this is also the origin of decoherence in traditional models where different system states scatter a bath state into two different states with vanishing overlap. The reasoning applies also to the interband coherence, which is a superposition of the presence and the absence of an exciton. In terms of scattering, the two states are distinguished by any scattering event. The second major result of the chapter is an explicit analytical and numerical demonstration of these facts. We find that exciton states with similar momentum space profiles tend to have Raman coherences that far outlive their respective coherences with the ground state.

Furthermore, an exciton is a composite particle, consisting of an electron and a hole. In the limit of low density of the surrounding electron gas, the composite nature of the exciton can largely be ignored. However, as the density rises, the electron and hole, besides being bound to each other, can individually interact with the electrons in the gas. This interaction includes not only the Coulomb interaction but also the exchange interaction, since the electrons in the gas can be exchanged with that bound to the hole. It then becomes necessary to take into account these new eﬀects in describing the 100 dynamics of excitons embedded in an electron gas.

This brings us to the third major result of the chapter. We find that the dynamical equations for the excitons involve two classes of functions: those that treat exciton as indivisible and thus are sensitive only to its multipole moments, and those that treat exciton as a composite particle and take into account the individual propagation of the electron and the hole. The latter are connected to excitations of the N-particle many body system in the N 1-particle Hilbert space. We find that for densities with our choice ± of material parameters, the composite nature of the exciton makes little contribution. Therefore we ignore it in our analysis, but its appearance in the equations will be useful for future analyses in different parameter regimes.

In the calculations described below, we first study the exciton levels in the presence of static screening. The screening originates from the conduction band electrons that are assumed to be present in the samples via doping, and we take the hole and exciton densities to be equal to zero in the initial quasi-equilibrium state. The screening is controlled by a single parameter, which is calculated self consistently for a fixed density of the electron gas. For at least three bound exciton states to exist, we find that a density lower than an electron per exciton radius is required. We study the decoherence between the 2s and 2p exciton states, neglecting spin as well as anisotropy of energy bands.

This chapter is organized as follows. In Section 4.1, we discuss interband and Raman coherences formally in the Green function perspective. In Section 4.2, we relate the experimentally measurable signal to the theoretical quantity that is central to Raman coherence. In Sections 4.3 and 4.4 we discuss a set of approximations that are necessary for identifying the purely two-particle eﬀects from the general three particle Bethe Salpeter equation. We then discuss, in Section 4.5, the sources originating from the laser pulses, that drive the single and two-particle correlations, and the signal that can be compared with the experimental observation. In Section 4.6, we write the dynamical equations in a form convenient for analysis and numerical calculation. We discuss physical properties of the solutions of those equations analytically, and compare them with conventional system-bath models. The latter provides insight into the physics the conventional system-bath models fail to capture in the type of decoherence problems discussed here, and how the present approach succeeds at the same task. The details of derivations are left to the Appendix. In Section 4.7, we discuss our numerical technique for solving the two-time non-Markov equations, as well as the model for electron gas we 4.1 Background 101

have used in performing calculations. In Section 4.8 we present the results of calculations, and discuss them in the context of our analysis of the dynamical model. In Appendix F, we discuss our approximation method to restrict four-point functions to depend only on two time variables. In Appendix, G.1 we discuss the derivation of the equation of motion for exciton correlation function, and in Appendix G.2 we discuss a similar derivation for interband polarization.

4.1 Background

In this section, we discuss various types of coherences and the issues related to them on formal grounds. In particular, we begin with a traditional wavefunction perspective to identify the two main types of coherences that are accessible optically in semiconductors. We then recast them in the Green function language and point out the important physical aspects of the dynamics of these coherences that can be handled naturally within our formalism.

Consider an optical excitation of a quantum well close to its bandgap energy with a pulse so weak that the dominant eﬀect is the generation of an interband polarization. In the independent particle approximation we can describe the state of the system as

ψ(t) Φ + α k (t) cv, k , | i∝| 0i cv | i k X where Φ is the ground state of the system and cv, k is a state with a conduction | 0i | i electron and a valence hole. The amplitudes αcvk (t) are proportional to the dipole matrix elements and the electric ﬁeld. In a perfectly coherent system, the unitary evolution of

this state is fully captured by the amplitudes αcvk (t). To step beyond this regime, we write the corresponding density matrix,

̺(t) Φ Φ + (α k(t) cv, k Φ α∗ (t) Φ cv, k ) (4.1) ∝ | 0ih 0| cv | ih 0| − cvk | 0ih | k X ′ + αcvk (t) αcv∗ k (t) cv, k cv, k′ . ′ | ih | Xkk Here we interpret the second term as a coherence between the ground state and an excited state containing one electron-hole pair. It arises from the superposition between existence and non-existence of the electron-hole pair. The third term describes another 4.1 Background 102

superposition, in which both states involved contain an electron-hole pair but in diﬀerent states of the two-particle Hilbert space. These are often referred to as Raman coherences [8]. In other words, the second term is a coherence between two diﬀerent Hilbert spaces (namely zero and two-particle), while the third term is a coherence within a single Hilbert space.

We can abstract from this perspective a more general expression for ̺ (t) in the form,

̺(t) Φ Φ + ̺ (t)+ ̺ (t) . (4.2) ∝| 0ih 0| 1 2

Elements of ̺1 (t) are coherences between the ground state and the excited states (inter-

band coherences), and those of ̺2 (t) are coherences among the excited states (intraband coherences). While their relationship is ﬁxed by (4.1) within the non-interacting particle picture, their evolution can be dramatically diﬀerent when interactions are taken into account. When interactions are present, of course, the very idea of a "single particle state" loses its validity. Nonetheless, at short enough timescales the concepts of quasiparticle bands and crystal momentum retain an approximate validity. This "quasiparticle regime" is discussed, within our approach, in Section 3.3. It is an underlying assumption in the present calculation that this quasiparticle regime extends to picosecond timescales.

We now turn to a Green function description of the scenario of (4.1). Very generally, the single and two-particle Green functions are deﬁned as

G (12) = i a(1)a†(2) , (4.3) − TC G(14;23) = a(1)a(4)a† (3)a†(2) , (4.4) − TC

in the (standard) notation of Chapter 3; a and a† are respectively annihilation and creation operators, and indices such as 1 can be taken to label a space and time point, or alternately an index of a nominal band and a crystal momentum, together with a time. A usual decomposition of G(14; 23) is

(2) 1 G(14;23) = G(12)G(43) X (14′;23′) ε− (3′4;4′3), (4.5) − where the first term alone is the Hartree approximation, and the second is a correction that accounts for exchange and correlation; the second term is written in notation of 1 Chapter 3. Here ε− (3′4;4′3) is the inverse dielectric function that transforms the response to Maxwell field into the response to the external field. In a semiconductor, once the conduction (c) and valence (v) bands have been identified, the interband coherences 4.1 Background 103

(αcvk) arising in the simple description (4.1,4.2) are described in the Green function formalism by the equal time limit of the component

G (12) = i a (1) a† (2) , cv − TC c v of the Green function (4.3). Here we explicitly display the bands associated with the indices (c with 1 and v with 2), and the indices themselves only indicate a crystal momentum and time. The quasi-equilibrium background state is subjected to the condition

Gcv (12) = 0, which means that no coherence exists in the system prior to optical excitation. The two-particle propagation involved in exciton states is handled by a component of the four-point Green function (4.4), in which one electron resides in the conduction and one in valence band,

G (14;23) = a (1)a (4)a†(3)a† (2) . cvvc − TC c v c v This Green function describes the propagation of two electrons, added to the system at times t2 and t3, and subsequent removal at times t1 and t2 respectively. The equal time limit (t = t t, t = t = t+) describes the intraband coherences that are given by 2 1 ≡ 4 3 1 ′ αcvk(t)αcv∗ k (t) in the simple description (4.1,4.2). More generally,

G 14;1+4+ = G (11+)G (44+) X(2) 14;1+4+ , (4.6) cvvc cv vc − cvvc + where Gcv(11 ) describes the interband coherence. Here we have set the components 1 ε− (3′4;4′3) entering (4.5) equal to unity. This approximation neglects the matrix ζ3′ v;ζ4′ c elements of Coulomb interaction across the fundamental gap, as well as the eﬀects on the response due to the motion of carriers perpendicular to the plane of the quantum well. In the equal time limit, and with interactions turned oﬀ,

G k t, k t; k t+, k t+ = G (k t, k t+)G (k t, k t+) G (k t, k t+)G (k t, k t+). cvvc a d b c cv a b vc d c − cc a c vv d b (4.7)

′ Comparing (4.7) with the product αcvk(t)αcv∗ k (t) and recalling (4.1), it becomes clear that a wavefunction description is possible only if the carrier density can be ignored so that the exchange interaction among electrons is negligible. The second term in the last equation represents the eﬀect of indistinguishability of electrons that prevents one from from treating an electron-hole pair as a closed system in the presence of electron gas, even with all interactions turned oﬀ. If the Coulomb interactions are allowed but G (k t, k t+) 0, then they bind electron and hole into an exciton that can be treated cc a c ≈ as a particle in the semiconductor vacuum. 4.1 Background 104

Further interactions also screen the Coulomb interaction, via the response of the medium surrounding the exciton. If that response is such that the screening is eﬀectively static, then G (k t, k t; k t+, k t+) G (k t, k t+)G (k t, k t+) continues to hold, but cvvc a d b c ≈ cv a b vc d c the exciton states associated with the poles of the two-particle Green function will be modiﬁed. If the dynamic nature of the response is taken into account, then correlations between the screening medium and the exciton will contribute to X(2), thus destroying a wavefunction description of the state of the system. So we see that in general there are two reasons why it is impossible to maintain (4.1) as the state of the system: the presence of exchange interaction, and the presence of interaction with a surrounding medium including electrons.

In following the excitation of the system by optical fields, we track the deviations induced in the original, quasi-equilibrium X(2), which in Chapter 3 are identified by the functions (2) Xcvvc; they will contribute significantly to Gcvvc. At frequencies corresponding to the exciton states, the optically generated densities may indeed be small, and the dominant correction will arise mainly from dynamical interactions. The purely exchange term will be important if the coupling between exciton dynamics and the optically injected electron and hole densities is large.

The dynamical interactions will drive the system towards a quasi-equilibrium state involving excited electron-hole pairs, the electron gas, and other quasiparticles, such as phonons, that may be important. Since it is a good approximation to neglect any interband transitions via Coulomb interaction, this quasi-equilibrium state will be devoid of any interband polarization, i.e. Gcv 0. Thus as the system moves from an initial → (2) excitation of interband coherence to quasi-equilibrium, the functions Xcvvc entirely take over the contribution to the two-particle states of the system. Furthermore, when bound (2) excitons can be formed, the dynamics of Xcvvc is much richer than the dynamics of Gcv alone. This is because the decay of Gcv is sensitive to any scattering event involving (2) the excited pair, while the decay of Xcvvc is sensitive to how diﬀerent are the scattering properties of the two superimposed states. This point becomes explicit in the equations of motion we introduce below for the two objects, which can be compared on an equal footing.

(2) Thus it is natural to study the decoherence of excitons using only the function Xcvvc, which is free of the short-lived contributions of interband polarization to the total two-particle Green functions. As functions of four arguments, these are too large for computation, 4.2 Relationship to Experiment 105

and describe far more eﬀects than we aim to capture in this work. In Section 4.4 we will describe a set of approximations that restrict these functions to describe excitons in a way similar to quasiparticles, while being sensitive to their composite nature as well.

(2) We also remark that both the rise of coherence in Xcvvc associated with the optical excitation, and the subsequent decoherence, are mediated by dynamical interactions. Neglecting phonons, as we do in this chapter, these dynamical interactions in turn require that the density-density correlation in the many body electronic system is non-zero, i.e.,

(2) + + X rt, r′t′; rt , r′t′ = nˆ (r, t)ˆn (r′, t′) nˆ (r, t) nˆ (r′, t′) = 0, h i−h ih i 6 where nˆ (r, t) is the the electron density operator. In the original quasi-equilibrium state, we associate this fluctuation with the quasiparticle density in the partially filled conduction band, and treat the filled valence bands as inert. Fluctuations in the semiconductor vacuum must involve the formation of virtual electron-hole pairs with energy close to the fundamental gap, and therefore they are very short lived. Only the much longer timescale of fluctuations of the gas formed in the conduction band is of relevance in this work. The role of dynamical interactions in both coherence and decoherence will also be seen explicitly below when comparing the dynamical map for X(2) with the source terms driving it.

4.2 Relationship to Experiment

The main quantity studied in this chapter is the two-particle correlation function for excitons, and the main property of this function that we focus on is Raman coherence. In this section we discuss how this quantity is related to experimentally measurable signals. For concreteness we pick the exciton states 2s and 2p, which lie within a few meV of the conduction band edge for the electron density involved. The 2s state is excited by single photon absorption while the 2p state is excited by two-photon absorption. We assume that the first two pulses in the excitation sequence are coincident and predominantly excite these states. This restriction on excitations holds for optical frequencies that resonate with excitation from the ground state to exciton states. At much lower frequencies, field driven transitions from exciton to higher-particle correlations may become important, but they can be safely ignored here. We are not interested in the dynamics with respect to the delay between the first and the second pulse, and have thus taken the 4.2 Relationship to Experiment 106

pulses to be overlapping in time. This minimizes the eﬀects of ﬁrst order deviations in multi-particle correlations, such that the coherence between two-particle states is driven

mainly by interband polarization only. We let the third pulse arrive a time td after the first two pulses, and we let τ be the time after the third pulse when the signal is detected. The process is illustrated diagrammatically in Fig. 3.11. As discussed in Section 3.3.3, the electromagnetic signal corresponding to TDFS for this scenario is given by the current density (3.133), dk J (q t) = eδ (q k + k k ) ω (k) ξ (k) X(1) kt; kt+ + c.c., (4.8) h 3 | i − − 1 2 − 3 ˆ 4π2 cv vc 3;cv ~ where ξvc (k) is the matrix element of the position operator, ωcv (k)=(Ec (k) Ev (k)) / (1) − where Ec and Ev are the conduction and valence band energies, and X3;cv is the third order deviation (in the effective field) in the single particle Green function. In the following

we set ωcv (k) equal to the fundamental gap frequency, ωg, and neglect the small con- (1) tribution arising from its dependence on k. We estimate X3;cv by its source terms alone and ignore the convolution eﬀects of the decoherence causing terms in this interband polarization.

There are two contributions to the third-order interband polarization, as shown in Fig. 3.11. The ﬁrst arises from the Pauli blocking due to the optically injected carriers, and the second due to the Raman coherence of exciton states. It is the second contribution that is of primary interest to us in this chapter, and it appears in a diﬀerent region of two-dimensional spectrum than the former. Thus we study only this contribution. To indicate that only the exciton contribution is included, we put the superscript “ex” on

J 3. Based on the above approximations, we obtain,

J ex (q; t , τ) = eδ (q k + k k ) ω (4.9) h 3 d i − − 1 2 − 3 g τ iωnmt iωnτ/2 d t + , τ e− d e− + c.c. · mnPnm d 2 n X e Here the vector coeﬃcients dmn are given by the projection of optical transition matrix

elements onto the exciton states. The arguments td and τ correspond to the delays between the first two and the third pulse, and between the third pulse and the time of measurement respectively. The function ˜ (t, τ) is the exciton correlation function P of second order in the effective field. It is the exchange-correlation part of the exciton Green function that gives an amplitude for the system with an exciton removed from 4.2 Relationship to Experiment 107

state m at time td to evolve into the state with an exciton removed from level n at time (2) td + τ/2. It is also a restricted form of the more general correlation functions, X2;cvvc, of the formalism of Chapter 3.1 When the radiated electromagnetic ﬁeld is calculated from this current density, there are additional eﬀects arising from the propagation inside the well and radiation reaction, and their treatment is discussed at length in Section 3.1.1. We do not consider them in this chapter.

Due to the dependence of the right hand side of (4.9) on t and τ, J ex also becomes d h 3 i a function of these two variables. The two-dimensional Fourier transform of the signal with respect to the two time delays is then given by

J ex (q; ω, Ω) = eδ (q k + k k ) ω (0) h 3 i − − 1 2 − 3 cv ω ω + ω d ω ω , Ω n m (4.10) · mnPnm − nm − 2 − 2 nm X e where ˜ (ω, Ω) is the two-dimensional Fourier transform of ˜ (t, τ). Thus the signal Pnm Pnm can also be viewed as the transfer of Raman coherence back to interband coherence, which radiates at optical frequencies.

The two-dimensional spectrum in (4.10) is a sum of the matrix elements ˜ (ω, Ω), and Pnm in the numerical calculations the dominant matrix elements are those between the 2s and 2p states. The different components ˜ (ω, Ω) appear as peaks in different parts Pnm of the two-dimensional spectrum. Furthermore, we show in Section 4.5 that, at least under the assumption of isotropic bands, ˜ (ω, Ω) and ˜ (ω, Ω), lead to radiation Pnm Pmn in different directions. Therefore when presenting results in Section 4.8, we plot the functions ˜ (ω, Ω). Pnm The pulse sequence is such that only the beating of the interband polarizations appears in the two-dimensional spectrum. However each interband polarization also emits a signal. Though it may be difficult to detect, it is nonetheless related directly to the projection of (1) + X1;cv(11 ) onto the exciton basis. In the equations below, we write this O(U) deviation in single particle density matrix as

ρ (1) iX(1) (11+). (4.11) ≡ − 1;cv 1In the following sections we discuss the set of physical and mathematical approximations within which this correlation function alone captures the entire TDFS signal generated by Raman coherence among exciton states. 4.3 Equations of motion 108

4.3 Equations of motion

The formal structure of the equations of motion is given in Eqs. (3.81-3.85), which we now apply to the present problem. While the formalism in Chapter 3 identiﬁes many- body contributions in full generality to the third order in the driving ﬁeld, in a particular application only a few of those contributions may be of interest. In this section, we identify the relevant contributions based on the fact that the density of electron gas is relatively small.

It is clear from the discussion in the previous section that the results of measurements in the parameter regime of interest are sensitive primarily to the single and two-particle (1) (2) correlations, X1 and X2 , and we approximate the remaining functions as slaved to these (1) two. The signal is related to X3 , the calculation of which pertaining to problems like the present is discussed in Section 3.3.3. As was also shown in that section, the Keldysh (2)+ + component X2 − − is the one that contributes to the signal of interest. The equations (1) (2) for the two functions, X1 and X2 , are reproduced here for easy reference,

∂ (1) (11;j) (1) (12;j) (2) (1;j) i X1 (12) = 1 X1 + 1 X1 + 1 , (4.12) ∂tj M M S ∂ (2) (21;j) (1) (22;j) (2) (2;j) i X2 (14;23) = 2 X2 + 2 X2 + 2 . (4.13) ∂tj M M S

(2) The first equation contains coupling to the function X1 . Within the low density regime, (1) we show in Appendix G.2 that this function can be approximated by a product of X1 (2) and XQcvvc, and can therefore be subsumed into the first term (see (G.4-G.5)). The second (2) (1) equation couples X2 to the functions X2 , which in the two-band model we adopt are (1) identified with electron and hole densities. This coupling, along with Eq. (3.84) for X2 , describes effects of excitation induced carrier density on the dynamics of excitons. At strictly second order in the electric field, one can verify that the coupling term (21;1j) M2 has one higher factor of electron density compared to (22;j). This is due to the vanishing M2 hole and exciton densities in the background state 2.

However, we remark that the coupling (21;j) also describes contributions to excitation M2 induced dephasing (EID), which corresponds to a partial resummation to all orders in the ﬁeld. This is not the topic of the thesis, and we argue in Section 4.6 that EID makes

2 The diagrams that contribute to this coupling inevitably contain at least one GQ line from negative to the positive contour, which as discussed in Chapter 3, is proportional to density. 4.3 Equations of motion 109

only a quantitative difference in the type of decoherence studied. We also identify the coherences that are affected significantly by EID.

Therefore, we need to construct the couplings (11;j) and (22;j), in accordance with M1 M2 the general expressions outlined in Section 3.2.3. The coupling (11;j) is given by (3.87) M1 and (3.89-3.91), and (22;j) by (3.100,3.103,3.106,3.107, and 3.110). We will focus on M2 the derivation of (22;j); the couplings (11;j) and (12;j) can be obtained from the M2 M1 M1 same derivation by demanding consistency in the approximations to many-body physics in (4.12) and (4.13); explicit steps are also given in Appendix G.2.

To proceed we write (3.100) in the expanded form using (3.89),

(22;1) (2) (2) X = H(1)δ(11′)X (1′4;23) (4.14) M2 2 2 (2) (2) + ΣQ(11′)δ(22′)+ GQ(2′′2)IQ (12′;2′′1′) X2 (1′4;2′3) h (22;1) (2) i + (14; 23 1′4′;2′3′)X (1′4′;2′3′), K | 2 where the quantities on the right hand side are defined in Chapter 3. The above equation, and the expression for (22;1) given by (3.103), shows that the only effective interaction K (2) (2) needed is I , specified as a function of WQ and X . In the next subsection, we construct a model for this interaction. The most general form for this four-point function is displayed diagrammatically in Figures 3.6a-3.6c, and the derivation in Sec. 4.4 identifies the relevant contributions from these diagrams.

While we have reduced the computational eﬀort to two functions, a separate equation of motion (EOM) governs each time argument of these functions. This results in two (1) (2) equations for X1 , and four equations for X2 . Furthermore, the Keldysh matrix of the former has four components, while that for the latter has sixteen. In Appendix F, we introduce a two-time approximation based on treating an exciton as approximately a (2) quasiparticle. A special structure for the Keldysh matrix in this limit reduces X2 to a set of six functions, which is further decoupled to a set of three functions when there is a vanishing exciton density at equilibrium. Of these functions, two that we label as ≶ P are sensitive only to exciton as an indivisible particle. The other four functions, labelled

ˆ±, and ˇ± probe the composite nature of electron-hole pair, and therefore contribute P P (1) signiﬁcantly at high densities. A corresponding approximation reduces X1 to a single equation for interband polarization. The two-time approximation is not restricted to the low density regime, and can be introduced more generally as a re-arrangement of the summation of self-energy diagrams as discussed in the next section. 4.4 Model eﬀective interactions 110

In the subsequent sections we will identify the source terms diagrammatically, and then substitute them, along with the model eﬀective interaction, into (4.13). The resulting equation is discussed analytically and put in a form resembling a conventional master equation.

4.4 Model eﬀective interactions

We now outline the derivation of the effective interaction I(2). We obtain this function by restricting the three-time functions such that they are a product of fully interacting two-particle propagation, and an independent propagation of the remaining quasiparticle after one of them is destroyed. This approximation yields an effective two particle interaction I(2). We also neglect optically induced carrier densities in comparison to the background electron gas, and this simplifies the interaction to just a function of W and X(2) alone. The approximation employed is also convenient in computation, since it closes the dynamical equations within a set of two-time functions. The neglected physics is the retarded interaction between the destroyed particle and the remaining quasiparticle, and is discussed at the end of this section.

Of particular use will be four-point functions of two time variables, deﬁned as the limit

+ + F (tt′) lim lim lim lim F (14;23) = F (14;1 4 ). (4.15) ′ + + ≡ t1 t t2 t t2 t t3 t → → → 1 → 4 The limits in (4.15) apply only to the value of the time variable; the two branch indices of the Keldysh contour are still arbitrary. As shown in Appendix F, the branch indices yield a 4 4 matrix in the Keldysh space with six diﬀerent functions that have a clear × physical interpretation. We denote the Keldysh matrix for functions F (14; 23) as F .

The approximation method to accomplish the reduction from three-time to two-timeb BSE is elaborated in Appendix F. It depends on the ansatz,

z iG(tt′′)σ G(t′′t′) t > t′′ > t′ G(tt′) = , (4.16)  z iG(tt′′)σ G(t′′t′) t < t′′ < t′ −b b b where G is the 2 2 matrix in Keldyshb spaceb [125] for G (12) . This is a semigroup × approximation for the fully interacting Green functions. In the following we apply (4.16) b to the three-time Bethe-Salpeter equation, and reduce it to an equation with two-time 4.4 Model eﬀective interactions 111

Figure 4.1: Diagrams for the right hand side of (4.18).

quantities in accordance with (4.15). The result yields an eﬀective interaction that is consistent with the BSE, and the three to two time reduction.

When applying the two-time approximation to (4.14), the semigroup approximation is applied several times, and care must be taken to avoid double counting. Instead, it is simpler to apply the approximation to the exact BSE, and then identify the terms in (4.14). In this method, double counting is automatically avoided, and the last three terms of (4.14) are all captured by a single two-time function denoted below as J. We 1 take the BSE, (3.43), and let G− act on each of its four external vertices. Setting the time limits as shown in (4.15), the result of acting on argument “1” of X(2)(14;1+4+) is

∂ (2) + (2) + + i X (14; 24 ) H(1)X (1′4;1 4 ) (4.17) ∂t1 + − t2=t1 + + + (2) (2) + + + = Σ(11′)P (1′4;1 4 )+ G(2′1 )I (12′′;2′1′′)X (1′′4;2′′4 )+ G(41 )δ(14 ).

The last term makes this equation inhomogeneous, and it will drop out in the final form of the equations. Thus we will omit it below. An important aspect of the derivation is to treat the self energy terms at the same footing as I(2), so that they are subsumed into a single effective interaction driven by X(2) explicitly. Also, we will consider only the Hartree-Fock contribution to Σ. The reason for this is that we will perform a resummation of diagrams for I(2) after making the two-time approximations, and the resulting diagrams will be explicitly of first order in the dynamically screened two-particle interaction. Higher level approximations will consist of picking higher order diagrams in both Σ and I(2), which correspond to certain kinds of vertex corrections. In the present simpler description, we neglect these corrections. In the limit of a low density of optically injected carriers, this equation decouples from G, and yields a simpler effective interaction that acts like a two-time potential in the equation for X(2). This potential is the function J mentioned earlier. We note that the decoupling from G is the justification for dropping the coupling (21;j) in (4.13) above. M2 In the exact equation (4.17), the two interaction terms Σ and I(2), have the diagrammatic 4.4 Model effective interactions 112

representation shown in Fig. 4.1. In this diagram we have written X(2) as GGΓ (3.46- 3.47), in order to carry out the operations described above for the two-time limit. To proceed, we take the ﬁrst diagram,

with the time indicated explicitly and denote it by K for brevity. The diagram involves + the three time function Γ(t′t4; t′′t4 ), where we have already used the two-time limit to

set the external times equal. The interaction line at t′ is a natural choice at which to + split the diagram into two time functions. By cutting the propagator Gvv(t′′t1 ) at t′ and using (4.16), we obtain

t′ + K dt′σ(t′t ) dt′′ (4.18) ≈ − ˆ 1 ˆ + + iW (t1t′)Gcc(t1t1′ )Gcc(t′t1′ )Γ(t1′ t4; t′′t4 )Gvv(t′′t′)Gvv(t′t1 ).

The choice of overall sign is fixed by the order of the real times, which we will consider in detail in the following sections. Here we have taken into account the different signs in + (4.16) by defining σ(tt′) to be +1 if t′ > t , and 1 otherwise. We will proceed assuming 1 − t1 > t′ > t′′, and restore σ in the final result. At this stage of the approximation we have neglected diagrams where interaction can be placed such that the above cutting is not possible. Thus partial summations beyond those defining the exciton states via static interactions, and corrected by retaining two-time graphs, are assumed to be negligible.

Finally, we perform the integral over t′′ and set

(2) + + K dt′ iW (t t′)G (t t′ )G (t′ t′)X (t′t ; t′t )G (t′t ), (4.19) ≈ − ˆ 1 cc 1 1 cc 1 4 4 vv 1

where we have used the relation between Γ and X(2) to write

t′ (2) + + X (t′t ; t′t ) dt′′ G (t′t′ )Γ(t′ t ; t′′t )G (t′′t′). 4 4 ≈ ˆ cc 1 1 4 4 vv This relation holds at the same level of approximation as (4.18), because in the diagrammatic expansion of the above X(2), we also neglect two-particle interactions that cut across the line at time t. 4.4 Model eﬀective interactions 113

The second diagram in Fig. 4.1 is handled in a similar manner since it has the same structure at the point where the two-time limit is applied. We now substitute the two approximated diagrams into the right hand side of (4.17) and obtain,

(20)

In this equation, the ﬁrst equality shows the application of (4.16), where the minus sign is the result of the fact that the quasiparticle lines at the top are valence electrons directed backwards in time. The second equality employs the relations P = GGTGG + GG = GGΓ, so that the end result is in terms of P and the binary interaction W only. Here we have implicitly made use of the fact that the external vertices at the top of the diagram correspond to valence electron. We are concerned with deriving the dynamical map at the second order in the driving ﬁeld, which means that each component of the diagram must conserve the total number of particles; barring exchange processes that link conduction and valence states, we assume that the propagator cut by the dashed line at the top of the diagram is a valence hole, or a valence electron directed backwards in time.

The diagrams of Fig. 4.1 also have counterparts where the interaction occurs between the valence and conduction electron,

(21)

In this equation, the semigroup approximation is applied in the lower branch, or to the conduction electron. We have assumed this to be directed forward in time so that only 4.4 Model eﬀective interactions 114

the interaction of valence hole and conduction electron plays a role here, and the overall sign is positive. We have neglected the eﬀects arising from holes in the conduction band interacting with the valence electrons and holes. These eﬀects will be negligible at low densities.

(2) + By using the formula G(11′)Σ(1′2) = iX (13; 23 )W (32) (see (3.51)), and the same (2) methods above, we generate the two analogous contributions to ∂X (14; 23) /∂t2. Thus (2) + + when diﬀerentiating X (14;1 4 ) with respect to t1, we add the derivatives with respect to the 1 and 1+ variables. Combining these, we arrive at an eﬀective two-particle two-time interaction,

J W, X(2) (14;1+4+) (4.22)

(2) + (2) + + = iW (14′;41′)X (1′4;1 4′) iW (14;4′1′)X (1′4′;1 4 ) − + (2) + + + (2) +iW (41′;1 4′)X (14′;1′4 ) iW (4′1′;1 4 )X (14;1′4′), − which can be considered a function of W and P . When the components of this equation are expanded as in (3.61), there are contributions to both the dynamical maps and the source terms. Here we are concerned only with the dynamical maps, and deﬁne,

(2) JQ J WQ,X , ≡ Q h (2) i J J WQ, X . 2 ≡ 2 h i We must now restore the sign σ introduced in (4.18). Since we decouple the BSE from (1) Xn , we let the G lines inserted in accordance with (4.16) be evaluated at GQ so that they (2) (2) (2) are absorbed by JQ in the expression JQX2;cvvc and XQ in the term J2XQ . The dynamical map (22;1) + (22;2) can now be identiﬁed using (4.17), and (4.22) can written as M2 M2 (22;1)X(2) + (22;2)X(2) (14;1+4+) (4.23) M2 2cvvc M2 2cvvc = hH (1) H (1+) X(2) (14;1i+4+) cc − vv 2cvvc + + (2) + + + σ(11′′)JQcvvc(11′′ ;1 1′′) X2cvvc(1′′4;1′′ 4 ) + + (2) + + +J2cvvc(11′′;1 1′′ ) σ(1′′4)XQcvvc(1′′ 4;1′′4 ) , h i (22;j) where in the last two terms, σ(11′′) σ(t t′′). The couplings for j =3, 4 can be ≡ 1 1 M2 constructed by ensuring the symmetries discussed in Section 3.2.2, which we use in Sec. 4.6 below. 4.4 Model eﬀective interactions 115

Figure 4.2: These diagrams arise from leading order vertex corrections to self-energy, and exchange an electron in the bound pair with one in a virtual excitation in the background gas.

The vertex corrections to the self-energy, which we neglected above, can also be included within the two-time approximation. The result of including the leading vertex corrections in the self energy are shown in Fig. (4.2). These diagrams represent the process where

a virtual electron hole pair in the conduction band forms at the time, t′. The electron in this pair is exchanged with the electron bound to the valence hole in X(2) while the electron from X(2) annihilates the conduction hole at time t. Applying the diagram rules of Chapter 3, the eﬀect of these diagrams is to add the exchange corrections to the screened interaction W such that

W (14; 23) W (14; 23) GQ (3′′4′) GQ (2′4′′) GQ(41′′)W (14′;2′3)W (1′′4′′;23′′), 7→ − in (4.22) (summation over repeated arguments implied). Naturally, the contribution of the additional term exists only at high densities where the exchange process can be signiﬁcant. In any case, at low frequencies, it is dominated by the plasma excitations contained in W , which contribute a large response. In the calculations below, we neglect this particular contribution of exchange process, but remark that the subsequent analysis remains valid even if it is included.

The dynamical interaction W (14; 23) is a two-time function in its most general form (see (3.36) and (3.45)). Its Keldysh matrix has the same properties as G, and it is convenient < + + > ++ to use the functions W (tt′)= W −− (14; 23), and W (tt′)= W − −(14; 23) in place of the four-point functions. Below we use these functions evaluated at equilibrium, and in this case they depend only on one time variable, the diﬀerence t t′. We write them − ≶ as WQ (τ), and also introduce their Fourier transforms

+ ∞ W ≶ (ω) = dτ W ≶ (τ) eiωτ . Q ˆ Q −∞ 4.4 Model eﬀective interactions 116

(2) < By exploiting the structure of two time matrix X , and the relationship iW (t′t) = (2) > iW (tt′), we can verify that J has the same matrix structure as X and therefore a >,< c matrix with components , ˆ±, and ˇ± can be deﬁned, as discussed in Appendix F. J bJ J c This provides a natural deﬁnition for the retarded and advanced two-particle interaction, since it follows that the linear combination,

r σασ′γ (tt′) i αγJ (tt′), (4.24) J ≡ − α,γ= X± of the Keldysh components of J vanishes for t < t′, while

a σασ′γ (tt′) i σσ′J (tt′), (4.25) J ≡ α,γ= X±

vanishes for t > t′. These definitions are used in Sec. 4.6, where we analyze the full (2) equations for X2 and discuss their solutions. Comparing (4.22) with the diagrams in Figures 3.6a-3.6c, we see that the contributions retained correspond to the top row of Fig. 3.6a. Diagrams in the bottom row of that figure either vanish due to vanishing hole density, or contribute with one higher factor of electron density than those in the top row. Diagrams of Fig. 3.6c will contribute to excitation induced carrier density effects, mainly the EID. We have neglected this as discussed in the previous section, and we also comment in Sec. 4.6 on the relevance of this effect to our calculations. Our two-time approximation eliminates diagrams in the top row of Fig. 3.6b, as the semigroup approximation allows us to subsume the field dressed quasiparticle lines in these diagrams into J or P in the two-time BSE. In other words, these diagrams are a property of three-time BSE only. Diagrams of Figs. 3.6b-3.6c also involve three-particle effects, and we now turn to our arguments for neglecting them.

The leading diagrams neglected in the two-time approximation are of the form,

These diagrams are proportional to one higher power of density of the electron gas than the ones retained. This is so because they involve a ﬁnite time interval over which a 4.4 Model eﬀective interactions 117

virtual electron-hole pair must propagate and interact with the two correlated electrons. Such four particle complexes may become important in high density plasmas, but not at small densities in the present parameter regime. Furthermore, including them will not introduce new poles in the two-particle spectrum; they will only adjust the spectrum quantitatively in the complex domain. It should be noted that including these diagrams does not necessarily break the two-time structure. They can be decomposed further into a product of two-time correlations, but doing so transforms a single expression for three-time effective interaction to a sum over infinite two-time effective interactions in the BSE. Nonetheless, when a finite number of these can be identified as capturing the desired physics, the two-time approximation becomes useful a computational tool.

We emphasize the difference between our approach and that of Bornath et al.[124], who also employ the semigroup properties up to second order in their perturbation expansion and identify a two-particle self energy by re-grouping the expressions. Their expressions for the two-particle self-energy, which are analogous to J in (4.19), depend only on the single particle propagators. Their procedure must be infinitely repeated to identify the contribution of the two-particle propagator. Nonetheless the expressions obtained are the kind that are useful in discussing high density plasmas where collisions among electrons dominate the physics, rather than the self-consistent propagation of a bound pair. On the other hand, in the ansatz (4.19), the contribution of a pair of conduction and valence electrons to the self-energy is assumed to be dominated by the two-particle rather than single particle properties of the pair. This is done to include self-consistently the back-action of an exciton on itself via its interaction with the electron gas. Such a back-action makes an important qualitative contribution to the decoherence phenomenon as discussed below. Besides the contribution of the exciton, there are also contributions of two-particle functions where both particles belong to the same band. We take these to be only perturbative, since their effect does not change the dynamics qualitatively.

(1) (2) We conclude with the explicit form of dynamical equations for X1 and X2 under our approximations for the couplings. In Eq. (4.12) it is the the “+ ” Keldysh component − 4.4 Model eﬀective interactions 118

of interest to us. The derivation in Appendix G.2 shows that for j =1, 2,

∂ (1)+ i X1;cv − (12) (4.26) ∂t1 (1;1)+ = −(12) S1;cv s (1)+ s (1)+ + + H (1)+Σ (1) X − (12) iV (14′;2′3′) X − (3′4′) G − (2′2) cc Q;cc 1;cv − 1;cv Qvv +σσ+ iW (14′;41′) − Q (2)+σ σ (1)σσ + (2)+σ σ (1)σσ + X − (1′4′;24) X 44 X − (1′4′;24) X 4′4′ · Q;cvvc 1;cv − Q;cvvc 1;cv n o ∂ (1)+ i X1;cv − (12) (4.27) − ∂t2 (1;1)+ = −(12) S2;cv s (1)+ + (1)+ s + H (2)+Σ (2) X − (12) iG − (11′) X − (3′4′) V (1′4′;23′) vv Q;vv 1;cv − Qcc 1;cv σ σ iW −− (4′2′;24) − Q (2)+σ σ (1)σσ + (2)+σ σ (1)σσ + X − (14;2′4′) X 4′4′ X − (14;2′4′) X 44 . · Q;cvvc 1;cv − Q;cvvc 1;cv n o s In the above equations ΣQ is the singular component of the Hartree-Fock self energy, in which interaction corresponds to the static Coulomb interaction, V s.

The derivation of the couplings for the exciton correlation in the above text yields (22;j) M2 for j = 1, 2 while the equations for j = 3, 4 follow by conjugation. We here write only the former explicitly. In the two time approximation,

∂ ∂ lim i + X(2) (14; 24+) (4.28) + 2cvvc t2 t ∂t1 ∂t2 → 1 = (2;1) (2;2) S2;cvvc − S2;cvvc H (1) H (1+) X(2) (14;1+4+) cc − vv 2cvvc + + (2) + + + σ(11′′)JQcvvc(11′′ ;1 1′′)X2cvvc(1′′4;1′′ 4 ) + + (2) + + +J2cvvc(11′′;1 1′′ )σ(1′′4)XQcvvc(1′′ 4;1′′4 ).

Note that in contrast to (4.26) and (4.27), the singular self energy does not explicitly appear here. It is subsumed into the function J, and would be re-introduced when J is decomposed into instantaneous and purely retarded terms (see Section 4.6). It remains to ﬁnd the source terms (1;j) and (2;j) . We now turn to these. S2;cv S2;cvvc 4.5 Sources 119

4.5 Sources

(j) (j;l) The source terms, p , from which p can be determined as shown in Chapter 3, S S originate from the application of the optical pulses to the system, and the physical eﬀects captured by them. From Chapter 3 the interband polarization is driven by the optical dipole interaction projected onto the exciton states. We represent an exciton state by the ket n, q , which stands for the exciton state with quantum numbers n, and total | i momentum ~q. By deﬁning α = m /(m m ), and α = m /(m m ), momenta for c v v − c v c v − c conduction and valence electron in the pair are ~k = ~ (k α q), and ~k = ~ (k α q) c − v v − c respectively such that ~k is equal to the reduced mass times relative velocity of the pair. Letting k , k represent the direct product of conduction and valence electron states, | c vi we write the eigenfunctions in momentum space as,

ϕq (k) = k α q, k α q n, q . (4.29) n h − v − c | i It is often convenient to express various functions below in the basis n, q instead of | i k , k . We write the projection of two point functions, f (kt; kt′), onto the exciton | c vi cv basis as, dk f (tt′) = f (kt; kt′) ϕ∗ (k) . (4.30) qn ˆ 4π2 cv qn

(1) (1) For deviation of order j in U, such as Xj;cv, we write Xj;qn etc. Similarly, exploiting the conservation of total momentum of the electron-hole pair, we write the exciton correlation (2) (2) function Xcvvc (14; 23) in momentum space as Xcvvc (k + q, t1; k′ q, t4 k, t2; k′, t3). Its (j;l) − | projection onto ϕ , which also applies to n;cvvc, is deﬁned as qn S

(2) Xqnm (t1, t4; t2t3) (4.31)

dk dk′ (2) = ϕ∗ (k + α q) X (k + q, t ; k′ q, t k, t ; k′, t ) ϕq (k′ + α q) , ˆ 4π2 ˆ 4π2 qn c cvvc 1 − 4| 2 3 m v

Turning to the optical excitation, for states with the parity opposite to the states at the top of the valence band, the source driving ﬁrst order interband polarization can be characterized by its projection onto the exciton states,

dk + U (t) = δ U kt; kt ϕ∗ (k) (4.32) qn q ˆ 4π2 cv qn + e iωt U kt; kt = E (t) e− ξ (k) ϕq (k) . (4.33) cv −~ cv n 4.5 Sources 120

Parity forbidden states are excited by two-photon absorption [126],

+ e i j i2ω′t ij U kt; kt = ′ (t) ′ (t) e− η , cv −~E E n

ij where ηn are the Cartesian tensor components of the two-photon transition projected onto the exciton state n. The center frequencies ω and 2ω′ are assumed to overlap with the excitation energy of exciton states with respect to the ground state. Only excitons with a zero total momentum, q = 0, are generated owing to the neglect of the photon momentum. The source terms (1;j) that drive ﬁrst order deviation of the Green function S1 (4.26-4.27) are essentially these eﬀective energies:

(1;1) (12) = U (1) G (12) , (4.34) S1;cv cv vv (1;2) (12) = G (12) U (2) . (4.35) S1;cv cc cv

(2) The source term (tt′) is related to the optical field only via the effect of the field S2;cvvc on the single particle propagators. More precisely, the source term of second order in the field arises from the products of the first order correlations as shown in (3.120). We are interested in the “lesser” component, and by using (3.116), it is given by

(2)+ + − −(14; 23) (4.36) S2;cvvc = X1;cv(11′)X1;cv(2′2)ΓQ;vvcc(1′4; 2′3)

+GQ;cc(11′)X1;cv(2′2)

(2) ∆I (1′2′′; 2′1′′)PQ (1′′4; 2′′3) + IQ (1′2′′; 2′1′′)X (1′′4; 2′′3) ·  1;cvcc cvvc ;cζcζ 1;ζvζc   Xζ  +G (2 2)X (11 )  Q;vv ′ 1;cv ′ 

(2) ∆I (1′2′′; 2′1′′)PQ (1′′4; 2′′3) + IQ (1′2′′; 2′1′′)X (1′′4; 2′′3) . ·  1;vvvc cvvc ;vζvζ 1;ζvζc   Xζ  There is an implied correspondence between the branch indices explicit on the left hand side and the external vertices on the right hand side. It follows from the lack of interband coherence in equilibrium that the ﬁrst term vanishes. The second and the third term both form a contribution to (2). However, in (2;l), the second contributes to diﬀerentiation in S S 1 while the third to 2 to the lowest order in the dynamical interaction. We will describe only the second term below and show that the third can be related to it by appropriate 4.5 Sources 121

relabelling of the second. Thus in the notation of Chapter 3 we have,

(2;1) (t ,t ) (4.37) S2;cvvc 1 2

(2) = X (2 2) ∆I (12 ; 2 1 )PQ (1 4; 2 3) + IQ (12 ; 2 1 )X (1 4; 2 3) 1;cv ′  1;cvcc ′′ ′ ′′ cvvc ′′ ′′ ;cζcζ ′′ ′ ′′ 1;ζvζc ′′ ′′   Xζ 

(2;2) (t ,t) (4.38) S2;cvvc 1 2

(2) = X (11 ) ∆I (1 2 ; 21 )PQ (1 4; 2 3) + IQ (1 2 ; 21 )X (1 4; 2 3) , 1;cv ′  1;vvvc ′ ′′ ′′ cvvc ′′ ′′ ;vζvζ ′ ′′ ′′ 1;ζvζc ′′ ′′   Xζ  We consider diagrams of the lowest order in the dynamical interaction. In the four left sides of the panels in Figure 4.3 we show the four diagrams that contribute to (2;1) ; S2;cvvc the diagrams for (2;2) are constructed by switching G and G lines on the positive S2;cvvc cc vv contour. In each of the four panels, the left hand side shows the exact diagram in which + the arrows represent Ucv (kt; kt ) generating pairs. The pair is in general correlated as represented by the box that the two lines following the pulse enter. In these diagrams both the “bra” and “ket” side of the diagram is linked to a field line, since there is no population of excitons in the equilibrium state. This immediately tells us that coherence in the two particle states is necessarily of the second order in the field in this scenario. In contrast, in a finite pre-existing population of excitons, diagrams where the two field lines lie on the same contour would also survive. They will represent the response of the exciton population to the second order in the driving field.

Finally, we see that in accordance with the rule that only fully connected diagrams are allowed, the two correlated pairs interact with each other. At this lowest order there are two interaction lines because all diagrams with a single such line are accounted for in the deﬁnition of W ≶ (see also Chapter 3). Since they are on the opposite sides of the Keldysh contour, the interaction must be W ≶ = W rχ≶W a, where χ is the longitudinal susceptibility, and therefore is a purely dynamic response. This is in agreement with the discussion following (4.7) where the buildup of correlations is attributed to mainly to the response of the medium. The source term will vanish identically in the zero quasiparticle density, or more generally, in the absence of dynamical and exchange eﬀects. This is so + because as discussed earlier, the “amplitudes” Gcv (11 ) in this case fully describe the system with a perfectly coherent evolution.

In these diagrams, the pulse connected to a two-particle correlation yields the functions 4.5 Sources 122

(2) Figure 4.3: Source diagrams for 2 : top row corresponds to ∆I1 terms and the bottom to (2) S X1 terms. For each diagram, switch c and v lines on the positive branch for the diagrams to complete the set. Note that the interaction vertex and the interband polarization are placed at equal times, and the separation is shown only for clarity. The approximation is discussed in the text. 4.5 Sources 123

(1) (1) X1;cv (2′2) on the positive branch and X1;vc (3′′3′) on the negative branch by definition. Therefore a full solution of the Dyson equation varied to first order in the field can be inserted to compute the diagrams. However, it is useful to analyze these diagrams using the approximate semigroup ansatz, which in this case is also equivalent to using the generalized Kadanoff-Baym ansatz [82]. In this approximation, only the advanced Green

function contributes since the time-ordering, t2 > t2′ is ﬁxed by the zero hole density of the equilibrium state, and thus

(1) a X1;cv(2′2) = ρcv(2′)GQvv (2′2) , (4.39) where for simplicity we set ρ (1) = iG (11+). In the four right-hand panels of Fig. cv − cv 4.3, we show the result of applying this approximation to the diagrams in the left-hand panels. We have represented ρcv and ρvc in the right-hand panels by squares. Note that there is no interaction placed between the GQcc and GQvv to make another correlated pair on the right hand side diagrams3.

The diagrams in the right-hand panels connect the interband coherence with the memory (2) of the system and show more clearly how the coherence in X2;cvvc arises. To see this in < detail, we begin by noting that by deﬁnition Gcv can be written as

< G (tt′) = iTr a† (t′, t)a (t, t )̺Q (t , t) cv vU cU 0 U 0 = iTr a† (t′, t) a ̺(t) . vU { c } where ̺Q is the many-body statistical operator describing the equilibrium background state, and Z is the normalization defined in (3.30). The are exact evolution operators, C U and we use the cyclic property of the trace to obtain the second line. The term ac̺(t) represents the removal of a conduction electron from the state as it evolves from the initial time t0 to the time t. It is clear from Figure 4.1 that t′ > t, and therefore the modified state evolves from t to t′, and at the latter time a valence electron is added to it. The sufficient condition for the trace to be non-zero is that coherence between the conduction and the valence band exist in ̺(t). Beyond time t, the state evolves with an extra valence hole, or a missing valence electron. This is described precisely by the function GQvv (t′t), as the interaction with the field has occurred before time t. Now the phase breaking

3Doing so would be inconsistent with the left hand side because, in the full diagram, the pair of propagators is GcvGcc or GvvGvc etc. It would also be inconsistent because it would modify the eﬀective interaction in such a way that it would include both pulses, which is a second order deviation that should be in the dynamical map and not the source term. 4.5 Sources 124

time of the quasiparticle propagator GQvv is given by the on-shell inverse imaginary self- energy, ΣQvv, and it can be much longer than the decoherence time of the interband ℑ (1) ′ polarization. Therefore the amplitude X1;cv (2′2) exists for t2 beyond t2 + tdec, where tdec is the time-span over which ρcv vanishes. Since the two-particle correlation forms from the interaction of single-particle correlations, it becomes clear that coherence in the two-particle function can form within the phase breaking time of the quasiparticle once an interband polarization has formed. This is essentially the content of the propagators linking the two interaction lines in the right-hand panels of Fig. 4.1. Once a summation over momentum of the propagators is performed, the inhomogeneous broadening will further restrict the temporal spread of the coherence. The interaction lines themselves control the maximum delay between the interband polarizations that is allowed such that (2) coherence can be transferred to X2;cvvc. Therefore the time delay between the two optical < pulses, which is an experimental parameter, is equal to τ in WQ (τ), and its variation leads to a probe of that function.

As mentioned earlier, the 4 4 matrix (in contour indices) corresponding to X(2) takes × a special form, with six diﬀerent components. We use these components, and (4.30- (2;1) 4.31) to write source term 2;cvvc in the exciton bases. As we always use (4.39) in (1) S constructing X1;cv, the subscript used for indicating the order in U is always 2 when 4 (2;1) appearing explicitly. For brevity we will omit this subscript . Thus we will write 2;cvvc (2;1) S as cvvc , and its projection onto the exciton states as S

′ (2;1) i(ωqnt ωmt ) ∞ ∞ ∞ > qnm (t, t′) = e− − dτa dτb dτ ′ Q;qmm ( τ ′) (4.40) S ˆ0 ˆ0 ˆ0 P − c ′ ′ (τ , τ , τ ′) ρq ′ (t τ ) ρ∗ ′ (t′ τ τ ′) , Rqnm;n m a b n − a qm − b − ′ (2;2) i(ωnt ωmt ) ∞ ∞ ∞ > qnm (t, t′) = e− − dτa dτb dτ ′ Q;qmm ( τ ′) (4.41) S ˆ0 ˆ0 ˆ0 P − v ′ ′ (τ , τ , τ ′) ρq ′ (t τ ) ρ∗ ′ (t′ τ τ ′) , Rqnm;n m a b n − a qm − b − where the “response function” is

η ′ ′ (τ , τ , τ ′) (4.42) Rqnm;n m a b < > = dq′ W (q q′, τ ′) W (q′, τ + τ ′ + τ τ ) ˆ Q − Q b − a ηη ζζ′ ′ fnn′ (q q′, q′ τa) (1 2δζζ ) fmm∗′ (q q′, q′ τb) . − | ′ − − | Xζζ 4However, we will continue using “Q” explicitly when a function is to be evaluated at equilibrium. 4.5 Sources 125

≶ The correlation functions of the bath are represented by WQ and can be interpreted as absorption and emission of the electron gas excitations, where a total momentum of q is gained by the pair via the two interactions. The factor 1 2δ ′ accounts for the change − ζζ in sign of the interaction when conduction electron and a valence hole are involved. The functions f ′ ( , , ) are deﬁned as nn · · ·

ζζ′ dk f ′ (q q′, q′, τ) = ϕ∗ (k + α (q q′)) ϕ0 ′ (k α ′ q′) (4.43) nn − ˆ 4π2 qn ζ − n − ζ GQ (k α q, τ) GQ (k α q′, τ) , cc − v vv − c − where the propagators in (4.43) represent the evolution of the pair for a time delay, τ, between the two scattering events.

In the two-time plane (t, t′), the direction of macroscopic evolution of the system occurs along lines of constant τ = t t′. Along this line the propagation is driven by the − difference of the two source terms, which vanishes for q = q′ = 0. Thus any non-zero momentum exchange necessarily occurs in the driving terms. From (4.43) we also see that the distribution of the state in momentum space strongly “filters” the memory effects of the quasiparticles. States that are tightly bound would generally produce less smearing of the interband polarization than those that are loosely bound, and therefore their driving terms will more closely follow the direct beating of the interband polarizations. Higher energy states will restrict the total number of excitations of the electron gas contributing to this source to fewer numbers, and will therefore have longer lasting driving term but it will have a lower amplitude. Furthermore, we see the competing effects of density whereby the larger density yields a larger source term, but lowers its duration at the same time. These observations are confirmed in the numerical calculations presented below.

When we restrict the above expressions to the isotropic model, the angular integrations over q′ suppress transitions among states of different parity. Thus the elements q ′ ′ R nm;n m are nonzero only if the difference in parity between n and n′ is the same as that between m and m′. In particular, this implies that if the directions for fields that excite the two states n, q and m, q are kˆ and kˆ , then the coherence driven by and will | i | i n m Snm Smn radiate in directions kˆ kˆ + kˆ and kˆ kˆ + kˆ respectively; the vector kˆ is the n − m 3 m − n 3 3 direction of the third field amplitude that generates the signal. This clean separation is an advantage of the two-dimensional Fourier spectroscopy. 4.6 Dynamics 126

4.6 Dynamics

We now return to equations (4.26), (4.27) and (4.28), and substitute into them the source expressions (4.34, 4.35, 4.40, 4.41) derived above. These are the dynamical equations we wish to solve.

We will ﬁrst obtain a physically intuitive form for (4.28). In Appendix G.1 we deﬁne a set of six functions for the two-time approximation to the components of Keldysh matrix < of four-point functions. We show that within the two time limit, the function (t, t′) P (2)+ + + + (2)+ + + + from this set corresponds to Xcvvc − − (14;1 4 ), and thus also to Xcvvc − − (14;1 4 ). The derivation in the Appendix results in the following equation,

∂ < < i (tt′) (tt′) ∂tP − Heff P < > = Xvv(t′t) Xcc(tt′) tmax− r < < a Q (tt′′) (t′′t′)+ Q (tt′′) (t′′t′) − ˆtmin J P J P tmin < > > < i (tt′′) (t′′t′)+ (tt′′) (t′′t′) dt′′ − ˆ JQ P JQ P −∞ tmax r < < a + (tt′′) Q (t′′t′)+ (tt′′) Q (t′′t′) ˆtmin J P J P tmin < > > < +i (tt′′) (t′′t′)+ (tt′′) (t′′t′) dt′′ ˆ J PQ J PQ −∞ tmin + + +i ˆ−(tt′′) ˇ (t′′t′)+ ˆ (tt′′) ˇ−(t′′t′) dt′′ ˆ JQ P JQ P −∞ tmin + + i ˆ−(tt′′) ˇ (t′′t′)+ ˆ (tt′′) ˇ−(t′′t′) dt′′. (4.44) − ˆ J PQ J PQ −∞ Here is an eﬀective Hamiltonian that deﬁnes the nominal exciton energies calculated Heff self-consistently within a model for the equilibrium electron gas (see Section 4.7). The row vector J (tt′) has been discussed in Section G. The precise form of all its components, ≶ , ˆ±, ˇ±, is derived in Appendix G.3. The equation applies to both above and below J J J the time diagonal t = t′ where tmin = min (t, t′), and tmax = max (t, t′). The equation for < derivative with respect to the second argument, ∂ (tt′)/∂t′, is obtained by using the P < < relation (t, t′) = †(t′, t). We will explicitly treat only the region t > t′, since the P P other half of the time-plane is related by Hermitian conjugation and time-reversal.

Equation (4.44) is not restricted to vanishing density of holes and excitons in the equilibrium state, but we take this limit by setting < and < equal to zero. We also omit the PQ JQ 4.6 Dynamics 127

ﬁrst term on the right hand side, which involves single particle Green functions, and the last term which involves ˆ functions, etc. The former will cancel out when the equations P are subtracted to obtain dynamics parallel to the time-diagonal in the two-time plane. The latter is one order of density higher than others, and so is small in the regime where at least two exciton states are stable. We drop all those terms containing < and < JQ PQ because they are proportional to the equilibrium density of excitons, which is zero. Note

that in keeping with the approximation to neglect the terms involving ˇQ, ˆQ we have J J dropped them from the retarded functions as well, and thus

r > (tt′) i (tt′)θ (t t′) , JQ ≈ − JQ − r > (tt′) i (tt′)θ (t t′) . PQ ≈ − PQ − Substitution of these expressions (and their conjugates for the advanced functions) into (4.44) yields

∂ < < i (tt′) (tt′) (4.45) ∂tP −Heff P t′ t (2;1) < > > < = tt′ + i (tt′′) (t′′t′) dt′′ i (tt′′) (t′′t′) dt′′ S ˆ J PQ − ˆ JQ P −∞ −∞ ∂ < < i (tt′) (tt′) eff (4.46) − ∂t′ P − P H t t′ (2;2) > < < > = tt′ i (tt′′) (t′′t′) dt′′ + i (tt′′) (t′′t′) dt′′. S − ˆ PQ J ˆ J PQ −∞ −∞ When the expressions for ≶, given in Appendix G.3, are substituted in (4.45) and J (4.46), we obtain,

∂ ∂ + < (t , t )+ i , < (t , t ) (4.47) ∂t ∂t Pqnm 1 2 Heff Pqnm 1 2 1 2 = qnm (t1, t2)+ S t2 < dt′′ ′ (t t′′) ′ (q t′′ t ) ˆ −Pqnm 1 Bm m | − 2 −∞ dq′ < + ′ ′ (q, q′ t t′′, t′′ t ) ′ ′ ′ (t t′′) ˆ 4π2 Cnm;n m | 1 − − 2 Pq n m 1 t1 < dt′′ ′ (q t t′′) ′ (t′′t ) ˆ −Bnn | 1 − Pqn m 2 −∞ dq′ < + ′ ′ (q, q′ t′′ t , t t′′) ′ ′ ′ (t′′t ) . ˆ 4π2 Cnm;n m | − 2 1 − Pq n m 2 4.6 Dynamics 128

The source term follows from the diﬀerence of (4.40) and (4.41), and the addition of the conjugate terms for the rest of the two arguments,

(2;1) (2;2) (2;1) (2;2) q (t, t′) = (t, t′) (t, t′)+ ∗ (t′, t) ∗ (t′, t) . S nm Sqnm − Sqnm Sqmn − Sqmn We have written the expression for the interaction as a superoperator, , that maps C matrices to matrices,

< > ′ ′ (q, q′ τ, τ ′) = iW (q q′ τ)A ′ (q, q′)A ′ (q′, q) (τ ′) (4.48) Cnm;n m | Q − | nn m m PQ;qmm and as a matrix related to via B C

′ (q τ) = dq′ ′ (q′, q τ, τ) . (4.49) Bmm | ˆ Cjj;m m | − j X The matrix A is given by

A (q, q′) O (α q, α q′) O (α q, α q′), (4.50) nm ≡ nm c c − nm v v with

i(q q′) rˆ dk O (q, q′) = n, q e− − · m, q′ = ϕ∗ (k + q)ϕ ′ (k + q′). nm ˆ 4π2 q,n q m D E

The leading contribution to A (for small q q′) is the dipole or the quadrupole matrix − elements of the exciton.

The dynamics of Raman coherence in < is captured by and operators on the right P C B hand side of (4.47). The action of leads to decoherence of each exciton state with B respect to the ground state, since it is sensitive only to one index of <. That this term P leads to decay follows from the overall sign of . The action of is with the opposite B C sign and is sensitive to both indices of <. Its effect is to subtract away a “mean” P decoherence so that the combination of and is sensitive only to difference between B C the decoherence of the two states with respect to the ground state. This is an explicit analytical verification of the physically motivated discussion of Section 4.1.

The physical process that determines A (or ) is the scattering of the surrounding gas by C the electron-hole pair. In a calculation, the transfer of momentum to the pair as a result of this scattering appears through virtual or real transitions among the internal states of the exciton. The transfer of momentum via electron or hole corresponds to Onm(αcq, αcq′) or

Onm(αvq, αvq′) respectively. A quasiparticle interacting with the exciton sees a neutral 4.6 Dynamics 129

particle, unless the scattering process is sensitive to the composite nature of the exciton. As shown in G.3, this matrix arises naturally in the derivation where contributions from the composite functions are also retained.

We note that when (4.49) is substituted into (4.47), a conservation law follows,

∂ < dq (t, t) = dq q (tt) . ∂t ˆ Pqnn ˆ S nn n n X X Physically, this is a consequence of ascribing all the interband transitions to the optical excitation, so that the Coulomb interaction composing the dynamical map cannot create or annihilate excitons. Thus the set of approximations employed in deriving this map respect this property of the Hamiltonian exactly.

The single particle equation, written for the ρqn(t) deﬁned in (4.30), obeys an equation that involves only the matrix (see Appendix G.2), B d ∞ + iωq ρq (t) = Uq (t)+ dτ (q τ) ρq (t τ) . (4.51) dt n n n ˆ Bnm | m − 0 Equations (4.47) and (4.51) are the main results of this chapter. The rest of this section is devoted to their analysis, and determining the main properties of their solutions.

A deﬁning property of the map , which is expected to hold at any level of approximation, C can be taken to be its relationship (4.49) to and the single particle equation (4.51). B This is so because all those diagrams in the equation where the interaction acts only on one side of the function < can be closed on the other side, using a X line to form P 1;cv a contribution to the equation for interband polarization. Thus these diagrams generate self-coupling in the equation for X . Diagrams that contribute to contain at least two 1;cv C optical excitations already, and therefore can only contribute to sources at third or higher order, or they can appear as cross-coupling in the full set of second order equations.

We now consider the generation of excitons with a finite total momentum, q. While (4.32) strictly restricts the interband polarization to q = 0, the interactions of these polarizations via pair-excitations can result in a transfer of momentum from the electron gas to the exciton. While the momentum transferred will have an average value of zero, 2 (2) q will in general be finite so that qnm (tt′) will have a finite spread in momentum h i S ≶ 1 space. As the KMS relations [127, 121, 120] imply that W (q,ω) V (q) ε− (q,ω), this ∝ ℑ Q spread decays at least as 1/q. It is further suppressed by the form factors, which at large 3 4 q decay as q− for s states and q− for p states, as implied by the analytical expressions 4.6 Dynamics 130

for the ideal two-dimensional hydrogen atom. Therefore large momentum transfers are (2) signiﬁcantly suppressed, and we expect that the spread of qnm in momentum space is S only on the order of the inverse exciton radius. We use this result to reduce the numerical eﬀort in solving for correlation functions by replacing all quantities by their average over q.

We also assume that the eﬀective mass approximation (EMA) applies within this region, and that band anisotropy can be neglected. In this case, the internal and external motion of excitons decouples and wavefunctions become separable in the relative and < total momentum variables. The correlation functions (t, t′) can then depend on Pqnm 2 2 q via the center of mass kinetic energy of the exciton, ~Ωq = ~ q /2M. Furthermore,

within the EMA, the matrix elements Anm (q, q′) also depend only on the diﬀerence of momenta, i.e. the momentum transferred and not how fast the exciton moves in initial and ﬁnal states. Thus the superoperator has the simpler form C

> ′ ′ (q, q′ τ, τ ′) = ′ ′ (q q′ τ) (τ ′) , Cmn;m n | Cmn;m n − | PQ;qnn where

< ′ ′ (q τ) iW (q τ) A ′ (q) A ′ (q) . Cmn;m n | ≡ Q | mm n n

> Apart from the factor (t′′ t ), the integral over q′ in (4.47) becomes a convolution. PQ;qnn − 2 By switching to the interaction picture with respect to Ωq, the main contribution of ′′ > i(Ωq Ωq′ )(t1 t ) (t′′ t ) to the integrand in (4.47) is a factor e − − . This can be taken PQ;qnn − 2 to be unity to a good approximation, since the time diﬀerence t t′′ for which there is 1 − signiﬁcant contribution is restricted by the inverse bandwidth of the kernel. The kinetic

energy transfer Ωq Ωq′ will be large only for large values of q′, which are expected to lie | − | ′′ i Ωq Ω ′ (t1 t ) in the pair-excitation continuum and thus have a large bandwidth. Thus e ( − q ) − will deviate from only slightly from unity. In the preliminary study here, we neglect this deviation and introduce the exciton correlations traced over the total momentum states,

< dq < (t, t′) = (t, t′) , Pnm ˆ 4π2 Pqnm and also deﬁne (t, t′) as the result of a similar integration of q (t, t′). Snm S nm Integrating over q in (4.47) and neglecting the kinetic energy of the exciton motion, we 4.6 Dynamics 131 get

∂ ∂ + < (t , t )+ i [ , < (t , t )] (t , t ) (4.52) ∂t ∂t Pnm 1 2 H Pnm 1 2 − Snm 1 2 1 2 t2 < < = dt′′ ′ ′ (t t′′, t′′ t ) ′ ′ (t t′′) ′ (t t′′) ′ (t′′ t ) ˆ Cnm;n m 1 − − 2 Pn m 1 − Pnm 1 Bm m − 2 −∞ t1 < < + dt′′ ′ ′ (t′′ t , t t′′) ′ ′ (t′′t ) ′ (t t′′) ′ (t′′t ) . ˆ Cnm;n m − 2 1 − Pn m 2 − Bnn 1 − Pn m 2 −∞

Similarly, we deﬁne ρn (t) be equal to ρqn summed over all q, and obtain,

d ∞ + iω ρ (t) = U (t)+ dτ (τ) ρ (t τ) . (4.53) dt n n n ˆ Bnm m − 0 Here and represent the integration of (4.48) and (4.49) over the momentum argu- C B ments, while the relation between and in the new form is C B

′ (τ) = ′ ( τ, τ) . (4.54) Bmm Cjj;m m − j X Note that due to the assumption of isotropic bands, the elements ′ ′ are nonzero Cnm;n m only if the diﬀerence in parity between n and n′ is the same as that between m and m′.

Equation (4.53) has the formal solution

t ρn (t) = dt′ ψn (t t′) Un (t′) , (4.55) ˆ0 − where ψn (t) is the solution obtained by replacing Un (t) by δ(t). We write the Laplace ˆ transform of a function f(t) as f(z), and obtain the Laplace transform of ψn (t) in the form,

1 ψˆ (z) = zI ˆ(z) − , − B h i where I is the identity matrix of the same dimension as . In the case of 2s and 2p B superposition, the use of isotropic model, and the neglect of coupling to states of principal quantum number n = 2, the matrix (τ) becomes diagonal. In the following sections, 6 B we discuss the calculation of (τ), but it is convenient to use the result obtained there B in the present analysis. As shown in Figure 4.4 the shapes of functions (τ) are close Bnn γnt to damped exponentials, and thus may be approximate as the functions b e− , where − n γn lie in the right half complex plane. Then the Laplace transform may be inverted to 4.6 Dynamics 132 obtain,

t ′ γnt /2 γn ρ (t) = dt′ e− cosh (β t′)+ sinh (β t′) U (t t′) , (4.56) n ˆ n β n n − 0 n 2 γn βn bn. ≡ r 4 −

This simple formula is useful in explaining the behaviour of ρn (t) determined by numerical calculation.

When short pulses of less then 100 fs are used, the functions multiplying U (t t′) in n − (4.56) may be taken out of the integral, and evaluated at t′ = t. Thus we expect two rates of decay that are mixed together after convolution with the optical pulse. The temporal behaviour naturally approaches Markovian at times beyond few time constants of the faster decay rate but only if β vanishes. The system becomes manifestly non- ℑ n Markov for large values of bn, which corresponds to large coupling to the dynamical environment. Similarly a small γ also results in non-Markov behaviour, which is due | n| to the coupling to fewer modes of the environment and therefore a slower rate at which the phase information is lost.

We now turn to the two particle equation (4.52). At first, for purely pedagogical reasons, let us consider only the diagonal terms in both and so that the integrand in (4.52) C B takes the form 1 ∞ dτ dq iW <(q,τ) A (q) A (q) 2 < (t,t τ)+ < (t τ,t) −2 ˆ ˆ Q { mm − nn } Pnm − Pnm − 0 1 ∞ dτ dq iW <(q,τ) A2 (q) A2 (q) < (t,t τ) < (t τ,t)(4.57). −2 ˆ ˆ Q mm − nn Pnm − − Pnm − 0 < < By the relation † (tt′) = (t′t), the first term plays the dominant role in decay of P P , while the second contributes to oscillations. Analogous formula for (4.53) reads, Pnm 1 ∞ dτ dq iW <(q, τ)A2 (q) ρ (t τ) . (4.58) −2 ˆ ˆ Q nn n − 0 Thus the rate of decoherence of each state with the ground state depends on how strongly the state scatters the surrounding quasiparticles. On the other hand, the factor A (q) mm − Ann(q) in (4.57) shows that their decoherence with respect to each other depends on how differently they scatter the quasiparticles. Therefore states that decohere at a very fast timescale may continue to be coherent with each other if their spatial profiles are such that their multipole moments are similar. Similarly, if the rates are vastly different, 4.6 Dynamics 133 the mutual decoherence will be only slightly less than the faster decay rate of interband polarizations. Note that in both cases one or both interband polarizations can completely vanish while a substantial mutual coherence remains. This is verified in the numerical calculations below.

In general, the off-diagonal terms in are also important. However, they involve overlaps C between eigenfunctions of different energies, which tend to be smaller. Furthermore, due < to the dominance of low frequencies in WQ (τ), the contribution of off-diagonal terms is also small on average when (4.52) is integrated. This is confirmed in our numerical calculations below.

We now turn to the conditions that determine how far the system is from Markov dynamics. It is obvious from the integrals in (4.52) that the width of (τ, τ ′) in its first C argument, τ, controls how close the dynamical system is to being Markovian. This width is in turn controlled by the matrix elements Anm (q), which depend on the nature of < the exciton states involved in scattering, and the function WQ (q, τ), which connects the static properties of these states to dynamics. The matrix elements Anm (q) can be expanded in an infinite sum over the multipole moments of the exciton. The profile for Anm (q) is broader and centered at larger momentum for more tightly bound states than for loosely bound states. The dielectric function generally has sharper peaks in the frequency domain at smaller q owing to the presence of the plasmon branch at low momenta. At higher frequencies where there is a continuum of particle-hole excitations, < the correlation functions WQ (q,ω) acquire a broad frequency distribution. As a result, virtual excitations among lower energy, more tightly bound states, will have broader spectra, or in other, words will be closer to Markov than the higher energy states.

Furthermore, only very fast quasiparticles of the background gas will significantly scatter from tightly bound states. If such a quasiparticle has a velocity uq, then it will typically interact with the state of size a for a time a/u , and for it to be effective in causing ∼ q decoherence the quantity a/u must be significant. It follows then that the lower lying |C| q states, besides being less affected by memory terms, also have slower decoherence rates 2 due to their smaller size. In two-dimensions the (n +1/2)− dependence of energy levels enhances the size difference between the lowest and the second lowest states of excitons, which makes a pronounced difference in their decoherence behavior as well.

It is worth noting that the integral limits in (4.52) force the domain of integration to be the horizontal and vertical lines whose one endpoint is the point where the solution is 4.6 Dynamics 134

sought (see Fig. 4.5). In the more general equations (4.44), this point lies on the same lines but there is also an additional contribution from the line connecting the point to

the line, t1 = t2. The additional integration accounts for excitation induced eﬀects in the evolution of the exciton removal amplitudes. In other words, it describes the deviation in the evolution of the exciton population, and is zero by assumption of vanishing hole density in the background state. In the exact treatment, but still keeping the initial hole density zero, this contribution will account for the excitation induced dephasing (EID) and relaxation eﬀects. Its absence here allows us to solve (4.52) with explicit propagation only along t =(t + t ) /2, which then implicitly leads to dependence on τ = t t via 1 2 1 − 2 the memory integrals only.

The neglected EID can be accounted for by a resummation, where the bubble diagrams comprising the density response function are corrected by the induced carrier density. The main effect of this will be the much broader spectral response, because the rate of change of the density is of the order of the pulsewidth, and hence sub-picosecond. Due to the fact that the equilibrium gas interacts very weakly with the 1s exciton state, the latter is expected to decohere from the semiconductor vacuum mainly via EID. This is roughly due to the fact that the equilibrium density response is sharply suppressed at large energy, which also severely limits the momentum transfer. All these effects act against 1s decoherence. On the other hand, EID may provide enough spectral range to couple into the large energy transfers, and thus allows the intermediate transitions from 1s to all the states lying above it. This coupling may be the dominant source of decoherence of 1s state. However, the 2s and higher states are dephased quickly by the equilibrium gas alone, and EID is therefore expected to affect them only quantitatively. Below we perform calculations for 2s and 2p states only.

We conclude this section by drawing comparison of our approach and its results to the conventional system-bath models that would treat the electron gas as distinguishable from the exciton. The two-time correlation function in our model is in direct contrast to single-time density matrices used in system-bath models. Diﬀerent contour orderings of the former generate correlation functions describing diﬀerent types of excitations in the multi-electron system. In (4.44) we see that the exciton correlation function < P is coupled to the correlation functions ˆ± and ˇ± the spectrum of which corresponds P P to excitations in the N 1 particle Hilbert space. These correlations are related to ± the composite nature of the exciton. While this arises naturally in the present model, 4.6 Dynamics 135

6 −ℜ B 2p,2p −ℑ B 2p,2p

0 −0.5 0 0.5 1 1.5 2 2.5 3 t [ps]

Figure 4.4: Typical (τ) Bnn these diﬀerent types of excitations would have to be introduced phenomenologically in a system-bath model that treats exciton as distinguishable from the rest of the electronic system. The present model is also in a form in which a pre-existing population of excitons can be taken into account self-consistently.

Another important diﬀerence between the present model and the system-bath models is the presence of source terms in (4.47). These source terms result from treating exciton as part of a fully quantum mechanical multi-particle system, as is clear from their relation to the eﬀective interactions and the single particle self-energy. Also, the sources terms and the couplings ultimately originate from a single equation: the Dyson equation. On the other hand, a conventional system-bath model describes only the evolution of a density matrix, the preparation of which is outside the realm of the model. Such a treatment cannot account for the simultaneous generation and dephasing of excitons as is done in the present model. Given the relatively long timescales of decoherence compared to the ultrashort pulses in systems like the one being studied, treating the generation and dynamics of excitons at the same footing is indeed important. 4.7 Numerical Method 136

4.7 Numerical Method

We now turn to the numerical calculation of (4.52) and (4.53). To proceed we need to < compute the superopertors (τ, τ ′), which depend on W (q, τ) as shown in (4.48). The C Q < KMS relations [120, 127, 121] yield the Fourier transform of WQ (q, τ) in the form,

1 2V (q) ε− (q,ω) iW < (q,ω) = ℑ , Q − eβω 1 − In this section we ﬁrst discuss the model used in computing relevant properties of the 1 electron gas, including the inverse dielectric function ε− (q,ω). We then describe our time-stepping scheme for solving the dynamical equations.

4.7.1 Electron gas

1 We model ε− (q,ω) by setting its imaginary part equal to that given by the Lindhard formula. At very small q, the plasmon branch appears and leads to very sharp peaks that we handle by replacing ε (q,ω) by the single plasmon pole model for small q,

2 1 ωpl 1 1 ε− (q,ω) = , 2 2 2 2 ℑ 2ωq "(ω + ωq) + γq − (ω ωq) + γq # − where ωq is the energy of plasmon at wavenumber q, γq is its decay rate. The real part can be calculated by Kramers-Kronig relation, and its zero frequency limit relates the static screening to plasmon dispersion at small q, i.e. q ω2 = ω2 (q) 1+ γ2. q pl κ − q Here κ is the screening parameter, and we took the nominal dielectric constant to be 13. Independent of the temperature, the RPA calculation will produce γ 0 in the limit q → q 0, so that the above formula makes sense as ω2 0 in two-dimensions in the same → pl → limit. We set temperature equal to 10 K in calculations.

Note that the single plasmon pole approximation is made only in a very small region of (ω, q) plane. The dielectric function over the rest of this plane is dominated by what would correspond to incoherent electron-hole excitations in a degenerate electron gas. In 1 a degenerate gas, ε− (q,ω) is non-zero only over a sharply deﬁned region bounded by ℑ 1 two parabolas [7]. In a non-degenerate gas, ε− (q,ω) varies smoothly over the entire ℑ 4.7 Numerical Method 137

(ω, q) plane [128]. Furthermore, pair excitations lead to continuously distributed poles 1 in ε− (q,ω), which makes RPA a poor approximation in the degenerate case. In the ℑ non-degenerate gas, hole excitations in the Fermi sphere are absent because the chemical potential is negative. Now each electron line directed backward in time generates a 2 factor η = ncλT , where nc is the areal density of the gas, and λT is the thermal de- Broglie wavelength. Since this parameter is small in the non-degenerate case, it becomes an expansion parameter for vertex corrections, which correspond to ηm for m> 1. This justiﬁes the use of RPA for a non-degenerate gas in the low-density regime, which is also the regime for the present calculation. Note however that a degenerate gas corresponds to a large η, which means that vertex corrections can become increasingly important. In this case RPA is valid only in the high-density regime, where screened interaction replaces η as the small expansion parameter.

With the above model for the dynamical potential, we computed γq self-consistently with the quasiparticle energy shifts, which are equal to the real part of the retarded self-energy. We took the latter within the GW approximation, as is consistent with the assumptions of Sec. 4.4,

r i r r Σ (k,ω) = dq dω′W (q,ω′)G (k q,ω ω′). 8π3 ˆ ˆ − −

This energy shift is decomposed into two parts, the larger static part, Σs(k), and the smaller shift, Σ (k)= Σr(k, E(k)) Σ (k), which arises from the dynamical self energy. d ℜ − s Within the GW approximation, and at densities and temperatures involved, we found that Σ Σ , and could therefore be neglected. These calculations were performed by d ≪ s ignoring the quasiparticle broadening in the integrals. The results support this neglect because the on-shell imaginary self-energy, Σr(k, E(k)), is almost an order of magnitude ℑ smaller than γq for small k, where the energy shifts are the largest.

4.7.2 Numerical time-stepping

We obtained the numerical solution of (4.53) by using the fourth order backward differentiation formula (BDF) [129]. We applied this to both the time-stepping and the memory integral. While other quadrature schemes (trapezoidal and Simpson rule as well as second order BDF) also produced results that agreed with the fourth order BDF, they required smaller time steps and became unstable after fewer time steps than the fourth order BDF. 4.7 Numerical Method 138

We used the same quadrature scheme for the numerical solution of (4.52) in the two- dimensional time plane. To describe the numerical algorithm brieﬂy, we begin by making a change of variables to T = (t + t′) /2 and τ = t t′ and denote a two-time function, − , in terms of these variables ˜ (t, τ). The two-dimensional time plane was divided into F F lines (referred to as fronts hereafter) along which τ varied at a ﬁxed T . The memory integrals in (4.53) are such that only the information to the left of the front is necessary to propagate the solution. A square grid in the (t, t′) plane was used, and the above scheme partitions it into two mutually exclusive subgrids - one containing the diagonal t = t′ and the one that does not. Propagation was performed by alternating the propagation of the fronts on each subgrid as depicted in Fig 4.5.

It is necessary to restrict the maximum value τ of τ to compute a solution with max | | reasonable memory requirements. We chose τmax by trial and error, and thus obtained solution in a strip of ﬁnite width centered at t = t′. Due to the ﬁnite memory time of the kernels, the strip can be much longer than its width. Thus a grid large enough to contain the width can be translated parallel to τ =0 line when computing the full dynamics. We increased τmax from one memory width until the solution converged, which occurred for

τmax larger than about ﬁve memory widths. Smaller values τmax introduced divergence in the tail of the graphs. This is due to the buildup of errors at the edge of strip, which propagate via the memory integrals back to the time-diagonal.

Finally, we performed the two-dimensional Fourier transforms by fast Fourier transform (FFT) on the function ˜< (t, τ). We applied an envelope function 1 τ /τ to pre- P −| | max vent spurious oscillations for FFT along the anti-diagonals in the two-time plane. This modiﬁcation preserved the shape of the plots but removed extremely fast oscillations due to the Gibbs phenomenon arising from a step-like behaviour at the edges of the strip. 4.8 Results 139

Figure 4.5: Two-time grid used in computation of (4.52). The black and blue dots show the complementary subgrids. The shaded rectangle is domain of memory integral

contributing to the point (t, t′) at its top right corner. The thick dashed line indicates the “front” on which all points are calculated by linear multistep method from the previous front of the same subgrid.

4.8 Results

Figure 4.6 shows the behaviour of the energies of exciton states as a function of the density. The lowest lying 1s state is far below and is not shown. The p and s states at low density merge towards their ideal values (n +1/2)2 Ryd, while the effect of static screening with increasing density is to make the p states rise in energy. All states merge into the continuum as density is increased, where the continuum itself is lowered by the Hartree-Fock self energy. The lines in the figure show the effect of only the static self- energy correction, while the green dots show the results of a few calculations done with the dynamical self-energy included. Clearly the dynamical effects on these energies are too small and were neglected in calculating the kernels.

The energy of the 1s state lies below 3 Ryd (1 Ryd 5.23 meV) for densities below − ≈ 3 2 2 10− a− . Here a is the Bohr radius in the zero screening limit using the DC dielectric × 0 0 constant of GaAs. Its value is 10.58 nm. We perform calculations with the 2s and 2p states in the model without spin splitting. The energy diagram shows that the continuum 4.8 Results 140

3 2 is lowered signiﬁcantly at 2 10− a− , and above this density the wavefunctions of n =2 × 0 states start to be dominated by mixing from the scattering states.

−0.05 3p

−0.1 2p

−0.15

2s −0.2 [Ryd] n,l E

−0.25

−0.3

−0.35

0.5 1 1.5 2 2.5 2 3 3.5 4 4.5 5 n [1/a ] −3 c 0 x 10

Figure 4.6: Exciton energies vs. electron gas density nc. Calculation includes static Hartree-Fock correction. Red lines for p states and blue lines for s states. The green dots are from the dynamical corrections.

We note that the energy difference between the continuum and the 2p state is approx- 3 2 imately 0.1 Ryd at n = 2 10− a− , so that a pulse width of greater than 2 ps is c × 0 required to keep the magnitude of excitations into the conduction band low. Since the density in the conduction band is proportional to the intensity, while the interband polarizations, are proportional to the amplitude, the effects of density are not significant for much shorter pulses so long as the field is weak enough. However, this restriction can be relaxed further by pulse shaping, and by shifting the center frequency of the pulse deep into the band gap. In the latter case, the ratio of the intensity at the conduction band-edge to the amplitude at the 2p pulse decays rapidly.

We now discuss the results of solving the equation of motion (4.53) via the numerical procedure described above. Figure 4.7 shows the coherence, ρn (t), between the ground state and the 2s and 2p exciton states. Graphs are shown at two diﬀerent pulsewidths and two diﬀerent densities. On the left are graphs for pulsewidth of 0.0625 ps, and on the 3 2 right the pulse width is 0.5 ps. The top row corresponds to the density of nc1 = 10− a0− 9 2 (9 10 cm− ), and the bottom row shows the results for twice this density (called n × c2 4.8 Results 141

hereafter). The pulse envelope function is superimposed as dashed line but it is not to scale.

In both cases the the coherence remains for the same duration of approximately 16 ps - much longer than the pulse. The transient eﬀects of the pulse remain for longer than the pulsewidth due to memory. The memory, or the temporal width of (τ) , is Bnm approximately 1 ps for both densities. The response to the shorter pulse shows three regimes of decay.

The ﬁrst regime following the peak is marked by an approximately linear drop in ρn (t). The slopes of the curves in this regime are approximately 1.5 and 2.5 at n and n − − c1 c2 respectively. The third regime, or the tail end of ρn (t) is predominantly exponential.

The exponential decay rates for nc1 and nc2 do not diﬀer much, and thus the decoherence rate grows sublinearly with density despite the linear growth of the amplitude of Bnm with density. The intermediate regime is not described by a simple power law or an exponential decay, since it is aﬀected by all the decay rates.

< We now turn to the results obtained by solving for the exciton correlation (t, t′) P via (4.52), and shown in Figure 4.8 along the line t = t′ in the two-time plane. The solid curves show the matrix elements < (t, t), while the dashed lines show the driving Pnm terms, (t, t), along the same line in two-time plane. The interband coherence has almost S the same behaviour as the driving term, as expected. By comparing the left and right columns, we note that the amplitude of scales approximately linearly with density. S This is consistent with the fact that the source is driven by the dynamic part of the potential, and two factors of this scale as the ﬁrst power of density.

The top row in Fig. 4.8 shows < (t, t), which is the coherence between the 2s and P2p,2s 2p. Clearly, the decoherence is weak enough that < (t, t) remains signiﬁcant for time P2p,2s much longer than the duration of the interband coherence and the driving term. The approximate exponential decay following the peak, and the slowly decaying tail of the curves, indicates the presence of several diﬀerent rates in the dynamics of < (t, t). P2s,2p 4.8 Results 142

t = 0.0625 ps t = 0.5 ps p p

ρ ρ 2s 1.4 2s ρ ρ 2 2p 2p pulse 1.2 pulse

1 1.5

0.8

1 (t)| [arb. units] (t)| [arb. units] 0.6 n n ρ ρ | |

0.4 0.5 0.2

0 0 0 5 10 15 20 0 5 10 15 20 time [ps] time [ps]

t = 0.0625 ps t = 0.5 ps p p

ρ ρ 2s 2s 2 ρ ρ 2p 1 2p pulse pulse

1.5 0.8

0.6 1 (t)| [arb. units] (t)| [arb. units] n n ρ ρ | | 0.4

0.5 0.2

0 0 0 5 10 15 20 0 5 10 15 20 time [ps] time [ps]

Figure 4.7: Interband polarizations for the 2s and 2p states at two diﬀerent pulse widths. 3 2 3 2 Densities are n = 10− a− (top) and n =2 10− a− (bottom) c 0 c × 0

To investigate further, we introduce time-dependent rates, d γ (t) = log ρ (t) , n −dt | n | 0 d R (t) = log ρ (t)ρ∗ (t) , nm −dt | n m | d R (t) = log < (t, t) , nm −dt |Pnm | such that these functions would be constant for a purely exponential decay or Markov 0 behaviour. The function Rnm(t) is essentially the sum of decoherence rates for the two 4.8 Results 143

interband polarization amplitudes involved, and thus it represents the decay rate of the

non-interacting approximation to the Green function (4.7). The function Rnm (t) isolates the rates associated with the terms beyond the free Green function. Based on these rates a phenomenological measure of correlation can be introduced, in analogy to that

tp = 0.0625 ps tp = 0.0625 ps 0.09 0.25 |S2s,2p| |S2s,2p| 0.08 |P2s,2p| |P2s,2p| |ρ2sρ2p| |ρ2sρ2p| 0.2 0.07 ] 0.06 ] s s t t i i 0.15 0.05 b un b un r r a a [ [ p 0.04 p 2 2 s s 0.1 2 2 P 0.03 P

0.02 0.05

0.01

0 0 0 50 100 150 200 0 20 40 60 80 time [ps] time [ps]

tp = 0.0625 ps tp = 0.0625 ps 0.3 0.7

0.25 0.6

0.5 0.2 ] ] s s t t 0.4 i i 0.15 b un b un r r 0.3 a a [ [ n 0.1 n , , n n P P 0.2

0.05 0.1 S2s,2s S2s,2s S2p,2p S2p,2p 0 0 P2s,2s P2s,2s P2p,2p P2p,2p −0.05 −0.1 0 50 100 150 200 0 20 40 60 80 time [ps] time [ps]

Figure 4.8: Coherence (top) between the 2s and 2p states of < (t, t) at densities n = P c 3 2 3 2 10− a− (left) and n = 2 10− a− (right). The bottom row shows the populations 0 c × 0 of the respective levels. The source terms are superimposed as dashed curves, and its duration is about the same as interband polarization. 4.8 Results 144

−0.93 −0.85

−0.935 −0.9 −0.94 −0.95 −0.945 ) ) −1 t t −0.95 ( ( s s 2 2 , , −0.955 −1.05 p p 2 2 Q Q −0.96 −1.1 −0.965 −1.15 −0.97 −1.2 −0.975

−0.98 −1.25 10 20 30 40 50 60 10 20 30 40 50 60 time [ps] time [ps]

Figure 4.9: Phenomenological parameter corresponding to the plots in Fig. 4.8 (den- Q 3 2 3 2 sities n = 10− a− (left) and n =2 10− a− (right)). c 0 c × 0 by Ferrio et. al[8]. We let (t) be such that, Qnm R (t) = γ (t)+ γ (t)+2 (t) γ (t) γ (t). (4.59) nm n m Qnm n m p (2) A large magnitude of (t) corresponds to large contribution of the correlation Xcvvc Qnm to G in (4.6). A positive value of (t) indicates a stronger decoherence among the cvvc Qnm exciton states compared to that of interband polarization. On the other hand, a value of (t) close to 1 indicates that R (t) is approximately the square of the diﬀerence Qnm − nm of the decay rates of interband polarization. Thus summarizes the relationship Qnm between the decoherence between excited states, and the decoherence of these states with the ground states. Figure 4.9 shows (t) for the two calculations, and its values Qnm are indeed close to 1. The detailed behaviour of (t) is more easily understood from − Qnm 0 that of Rnm and Rnm, to which we now turn. Figure 4.10 shows the plots for these two functions based on the calculations presented 0 above. It is immediately clear that the Rnm (t) is an order of magnitude less than Rnm (t). The decoherence rates rise sub-linearly with respect to density as is evident by comparing 0 0 the rates in the free evolution regime for both R and R. The oscillations in Rnm(t) can

be understood as bath-induced eﬀects in the dynamics of ρn (t). The simple analytical

solution (4.56) shows that at larger densities, bn increases and eventually makes the 4.8 Results 145

eﬀective shift in the rates, β , complex. Since solutions include both β and β , the n n − n oscillations continue to exist in ρ (t) . | n | Turning now to the function R, we note that it has several rates, including very low ones that dominate the tail of the coherence. The latter would be superseded by other processes that begin to appear on timescale of tens of picoseconds. There is a steep transition between the decay rate of the intermediate regime that follows the pulsed excitation, and the one in the long time limit. The rates in each regime vary by only a few percent, where there are plateaus in the function Rnm (t) at both densities. One could associate a short-ranged Markov behaviour in these regimes. The sharp transition is partly due to the slow oscillation in the real and imaginary parts of < (t, t); the Pnm period of this oscillation is similar to the time over which the slow decay takes over from the fast decay of Rnm (t).

Finally we discuss the two-dimensional plots of the signal that results after interaction with a third pulse. As mentioned in Section 4.2, we take the first two pulses to be coincident (tb = 0 in (3.128)), and performed the Fourier transform of the numerically solved function ˜< (t, τ) as discussed in Section 4.7 above. Recall from Section 4.2 that P the two-dimensional Fourier transform is applied to the function ˜< (t + τ/2, τ), where P d td is the time between the first two and the third pulse. The variable ω is conjugate to the time td and Ω to τ. The variable τ maps directly to the difference time, so that for a

ﬁxed td, the Fourier transform from τ to Ω is related to the spectrum of the exciton levels.

The spectrum evolves as the variable td changes such that the point (td, Ω) represents the spectrum for which td is the time at which electron-hole pair begins propagation. Since the time of measurement follows the third pulse, the Fourier transforms are computed only in the region t t , or for τ > 0. Therefore the functions ˜< (t, τ) describe the 1 ≥ 2 Pnm overlap at time t + τ/2 between state with exciton in level n removed and the state from which an exciton in level m is removed a time τ earlier.

Figures 4.11 and 4.12 show the two-dimensional Fourier transforms at densities nc1 and nc2 respectively. In the left-hand panels are the driving terms, and in the right-hand panels are the solutions. While only the solutions constitute experimental predictions, we have included the driving terms to identify the features of the solutions originating from them, as opposed to the dynamical map. This would clarify the properties that < Pnm inherit from interband polarization, and what eﬀects the dynamical map has on them. 4.8 Results 146

As discussed in Section 4.2, the contributions of and are separated spatially P2s,2p P2p,2s at the detector. The ﬁgures show these contributions in separate windows.

The peaks are in proximity of the expected locations, (Ω,ω)=(ωn,ωnm), with a slight overall shift along the Ω axis, and the shift is almost equal in both the source and the solution plots. Thus they are largely due to the oscillations in the source term, and appear in as a result of being driven by the source terms. That the dynamical map Pnm makes negligible contribution to the shifts is also consistent with the fact that dynamical

1.032 1.9

1.8 1.0315 1.7

1.6 1.031 ) )

t t 1.5 ( ( s s 2 2 , 1.0305 , 1.4 p p 2 2 0 0

R R 1.3 1.03 1.2

1.1 1.0295 1

1.029 0.9 10 20 30 40 50 60 10 20 30 40 50 60 time [ps] time [ps]

0.14 0.3

0.12 0.25 0.1

0.08 0.2 ) )

t 0.06 t

( ( 0.15 s s 2 2 , 0.04 , p p 2 2 0.1 R 0.02 R

0 0.05

−0.02 0 −0.04

−0.06 −0.05 0 20 40 60 80 100 10 20 30 40 50 60 70 time [ps] time [ps]

< 3 2 Figure 4.10: Comparison of d log f/dt where f = ρ , , at densities n = 10− a− n P nm c 0 3 2 (left) and n =2 10− a− (right). c × 0 4.8 Results 147

corrections show little effect in the exciton energy levels of Fig. 4.6. We define the peak location along this axis by the first moment of the absolute value squared of the corresponding function. The shifts at higher density are slightly larger, and the peak for shifts in the positive direction while the peak for shifts to the negative P2p,2s P2s,2p direction and by a smaller amount. The spread in the Ω direction in both driving terms and solution is also similar. This is due to the fact that excitation induced changes in the exciton spectrum show up as deviations in the propagation of the exciton population. As was discussed at the end of Section 4.6, these effects occur either in the presence of initial exciton population or if excitation induced effects are somehow taken into account.

Another aspect common to both the and is the asymmetry in the magnitude of Snm Pnm the functions; and are smaller than and respectively. This result P2s,2p S2s,2p P2p,2s S2p,2s is more naturally understood from the perspective of than . The function P S P2s,2p corresponds to removing a 2p exciton first, which leaves the 2s state to propagate for the time delay τ. The overlap is taken with a state from which the 2s exciton is removed, which in the presence of 2s 2p coherence is predominantly p-like. Since 2s state is − already symmetric, evolution for time τ has little effect on its small overlap with a p-like state. On the other hand, in the function the 2s exciton is removed first and thus P2p,2s the p-like remaining state is driven to a more symmetric form via interactions with the electron gas. The resulting state has higher overlap with the s-like state that arises from removing a 2p exciton. Thus indeed, we expect the to be of lower magnitude than P2s,2p 2p,2s when the Fourier transforms are performed only with τ 0. The source terms P ζζ′ ≥ S contain this property through the functions fnn′ (q, q′, τ) that describe how faithfully the

exciton state n′ evolves into n after time τ and momentum exchanges q and q′.

Finally, the most obvious effect of the dynamical map is that the spread of the peaks along ω axis is greatly reduced from the source to the solution term. This reduction is much greater at lower density than it is at higher density, as expected. The non-Markov behaviour can also be seen from the profile of the peak along ω for the Wigner function, ˜ (ω, Ω). A pure exponential decay of Markov behaviour would have ω-dependence of P 1 the form γ2 +(ω ω )2 − that is symmetric about the center frequency. The profiles − 0 of the solutions ˜ at the two densities are shown in Figure 4.13. The curves are P2p,2s generally asymmetric around their respective peaks. The curves close to the center of the two-dimensional peak are more symmetric, and therefore correspond to oscillators that evolve via smaller memory effect. 4.8 Results 148

S 2p,2s P 2p,2s 0.1 0.25 0.6 0.6 0.09

0.5 0.08 0.5 0.2 0.4 0.07 0.4

] 0.06 ]

V 0.3 V 0.3 0.15 e e m m

[ 0.05 [

ω 0.2 ω 0.2 0.04 0.1 0.1 0.1 0.03 0 0 0.02 0.05

−0.1 0.01 −0.1

−1.8 −1.6 −1.4 −1.2 −1 −1.8 −1.6 −1.4 −1.2 −1 Ω[meV] Ω[meV]

S 2s ,2p P 2s ,2p

0.4 0.4 0.2

0.06 0.18 0.2 0.2 0.16 0.05 0 0 0.14

] 0.04 ] V V 0.12 e e m m [ −0.2 [ −0.2 0.1 ω 0.03 ω 0.08 −0.4 −0.4 0.02 0.06

0.04 −0.6 0.01 −0.6 0.02

−2 −1.5 −1 −2 −1.5 −1 Ω[meV] Ω[meV]

Figure 4.11: Two dimensional plots showing absolute value of , and at density S2p,2s P2p,2s 3 2 nc = 10− a0− . 4.8 Results 149

S 2p,2s P 2p,2s

0.8 0.8 0.9 0.9 0.7 0.7 0.8 0.8 0.6 0.6 0.7 0.7 0.5 0.5 0.6 0.6 ] ]

V 0.4 V 0.4 e e

m 0.5 m 0.5 [ [

ω 0.3 ω 0.3 0.4 0.4 0.2 0.2 0.3 0.3 0.1 0.1 0.2 0.2 0 0 0.1 0.1 −0.1 −0.1

−1.2 −1 −0.8 −0.6 −0.4 −1.2 −1 −0.8 −0.6 −0.4 Ω[meV] Ω[meV]

S 2s ,2p P 2s ,2p

0.8 0.8 0.2 0.2 0.7 0.7

0 0.6 0 0.6

] 0.5 ] V −0.2 V −0.2 0.5 e e m m [ 0.4 [ ω ω 0.4 −0.4 −0.4 0.3 0.3

−0.6 0.2 −0.6 0.2

0.1 0.1 −0.8 −0.8

−1.6 −1.4 −1.2 −1 −0.8 −0.6 −1.6 −1.4 −1.2 −1 −0.8 −0.6 Ω[meV] Ω[meV]

Figure 4.12: Two dimensional plots showing absolute value of , and at density S2p,2s P2p,2s 3 2 n =2 10− a− c × 0 4.9 Summary 150

0.35 1

0.3 0.8 0.25

s s 0.6 2 0.2 2 , , p p 2 2 ˜ ˜ P 0.15 P 0.4

0.1 0.2 0.05

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Ω[meV] Ω[meV]

3 3 Figure 4.13: Proﬁles of the peaks along Ω at both densities (10− on the left and 2 10− × on the right).

4.9 Summary

In this chapter we applied the formalism developed in Chapter 3 to study the decoherence of excitons and interband polarization. The discussion is centered on the dynamics of excitons and how it can be accessed in detail using a three-pulse optical excitation. We have shown how the interband coherence is transferred to the exciton correlation function, which can dominate the Raman coherence. This dominance is conﬁrmed in the particular case of 2s and 2p exciton states that we considered. We remark that the signal generated by the third order polarization can be viewed as the transfer of Raman coherence back to interband coherence. While the composite nature of the excitons is not relevant for decoherence in the parameter regime considered, we have presented the dynamical equations that allow for its treatment in more general problems.

Within a set of approximations for two-particle interactions, we derived equations of motion that allowed us to close these equations at the level of two-time functions. We also showed how these equations relate to those for the interband polarization at the same level of approximation. The main analytical result, Eq. (4.47) for the correlation functions of exciton states, may be thought of as a two-time generalization of a master equation. The equation for the evolution of interband polarization, Eq. (4.51), is related to (4.47) in 4.9 Summary 151 a physically transparent way. While the decoherence rate of an exciton depends on the difference between the spatial properties of the two superimposed states, the interband polarization is unaffected by any such difference because it addresses superposition with respect to the ground state.

It is interesting that Eq. (4.47) and its interpretation also connect to the conventional system-bath interaction. This holds even though no distinction between the system and the bath was made in the derivation. The present model, however, includes several features that a conventional system-bath model would not contain. We have seen that the two-time correlation function, , is central to our model as opposed to the single-time P density matrices used in system-bath models. This correlation function, which treats an exciton as indivisible particle, is coupled to those correlations that are related to the composite nature of the exciton. This arises naturally in the present model, and it can only be introduced phenomenologically in a system-bath model that treated exciton as distinguishable from the rest of the electronic system.

The source terms that drive the exciton correlations in (4.47) also arise naturally in our model, and they would be out of the scope of any system-bath model based on conventional master equations. The source terms allow us to treat the generation and dephasing of excitons on an equal footing. Due to the similarity of timescales for both processes, such a treatment is indeed important.

We also showed that a three pulse excitation can, within the convolution eﬀects of the third optical pulse, directly probe the two-time exciton correlations. That this is the case is important from a theoretical perspective because it leads to a much deeper comparison of theory and experiment than was possible in the past. In addition, it may provide guidance in approximations and calculations. We also obtained plots of two-dimensional Fourier spectroscopy from our calculation of the two-time functions. These plots are the main link between experiment and theory. Chapter 5

Conclusion

In this thesis, we have presented various new theoretical and computational methods, which we hope will be useful in understanding decoherence processes in the transient regime of optically induced dynamics in semiconductors. We have constructed a formalism for capturing the decoherence phenomena using the framework of nonequilibrium Green functions. We applied this formalism to decoherence of excitons embedded in an electron gas. The indistinguishability of system and bath particles in this experimentally relevant scenario brings decoherence into an unconventional regime.

In Chapter 2, we presented a new method to numerically solve semiconductor Bloch equations in the length gauge. While in principle any gauge leads to the same physics, our motivation for developing this method is the inadequacy of the velocity gauge in numerical computations, especially when low frequency fields are present. The main difficulty in the length gauge arises from the existence of gauge fields in the momentum space. Their direct numerical calculation is impossible due to the independence of basis at different k points. This freedom in choice of basis also prevents a direct differentiation of operators such as density matrix in k space, a step that must be performed to construct the complete position operator. We implement this step by supplementing the chosen lattice with directed links, and then computing the basis transformation (termed link operators) between the vector space at the tip of the link and that at its tail. We then concatenate the unitary link operators along each line of a rectangular grid of points, a step that rotates the gauge fields into the plane orthogonal to the grid line. This converts covariant differentiation into ordinary differentiation and leads to translation in momentum space, which is done by FFT-based interpolation. Although the presentation

152 5. Conclusion 153 in Chapter 2 focused on single particle dynamics, the method lends itself directly to numerical solutions of Kadanoﬀ-Baym equations.

We compared the method to a velocity gauge calculation for a quantum well with a Pöschl-Teller conﬁnement potential. We found that as the ﬁeld frequencies are lowered below the gap frequencies, the velocity gauge results in larger populations in the conduction bands at intermediate times. These can be up to two orders of magnitude larger than in the length gauge for frequencies below the mid-gap. At frequencies much smaller than the band gap frequency, the magnitude of virtual excitations is found to be independent of frequency in the length gauge, as expected.

In Chapter 3, we presented a framework suitable for applications to multi-pulse optical excitations of semiconductors, and in particular, multidimensional Fourier spectroscopy experiments. A calculation in this framework begins with a model of the many body system in a quasi-equilibrium state, where the important quasi-particle species and the scattering processes are taken into account by vertex corrections to the self energy. The formalism then applies functional diﬀerentiation to determine variations in the single particle Green function up to third order, by computing the respective variations in self-energy.

This straightforward and physically motivated method requires a suﬃciently complex machinery to be stated in an explicit form. We have discussed the challenges posed by the hierarchy of many-body correlation functions, and the form in which it appears in the current context. A possible truncation method was discussed and applied; it is analogous to the use of non-interacting two-particle correlation functions in the GW approximation [130, 120, 119]. However, the truncation used here is a self-consistent solution for up to n-order correlation functions that includes the eﬀects of all scattering channels involving n particles. This allows for a consistent treatment of decoherence processes, since all scattering processes at a given level of scattering are present by construction.

The basic structure of the formalism consists of a set of dynamical variables that are deviations of correlation functions of the quasi-equilibrium state. These deviations then evolve according to linear dynamical maps, which form one part of driven diﬀerential equations. The dynamical maps are given by kernels, which correspond to correlation functions and screened interactions of the quasi-equilibrium. Thus a consistent way of including necessary resummations is built into the formalism. We showed the explicit dynamical maps for up to four-point functions, which already includes decoherence ef- 5. Conclusion 154 fects of all two-particle interactions on, for instance, the exciton and biexciton correlation functions. These general equations, when specialized to particular cases, produce eﬀec- tive models that are appropriate to study decoherence by way of many-body correlation functions.

The second part of these equations are the source terms, the simplest of which is just the optical field. At a given order in the field, deviations of all lower orders combine to form local field corrections to this source term, and eventually form a link between the observed signal and multi-particle states in the system. We have identified and discussed the physical interpretation of these terms.

The source terms are also conveniently described by diagrams on the Keldysh contour, which turn out to be a natural extension of the double sided Feynman diagrams used in more phenomenological treatments. While keeping the same bookkeeping of optical excitations, they allow for a description of the interaction processes as well. Their char- acterization of source terms is completed by a set of rules that translate a diagram to a mathematical expression for a source, and vice versa. The arrangement of three pulses, and physical parameters of the quasi-equilibrium state such as densities, temperatures, and chemical potentials, can then be combined to identify the dominant sources. By the same token, they can be used to identify appropriate pulse sequences for suppressing or enhancing certain superpositions. We mentioned only simple examples of this, as the detailed investigation must be done case by case.

The full solution to the set of equations derived is not numerically feasible at present. However, the purpose of this formalism is to act as a starting point to model a particular experimental situation, with a solvable approximation to the multi-particle correlation functions. Within simple parametrization of the screened interaction, which is a reasonable expansion parameter as opposed to the bare interaction, the kernels of the dynamical maps can be constructed approximately. A way to perform this at the level of two-time correlation functions for up to two particle correlation functions has already been developed [124]. We have shown that it also leads to a two-time generalization of a Lindblad map for excitons, where the kernel to the lowest order is proportional to screened interaction and density-density correlation of the quasi-equilibrium state.

In Chapter 4, we discussed an application of the formalism to expound its main features, and to demonstrate the qualitatively diﬀerent nature of coherences in optical excitation of the semiconductors. The discussion in Chapter 4 focused on the dynamics of excitons 5. Conclusion 155

and how it can be accessed in detail using a three-pulse optical excitation. We argued that the coherence between exciton levels is a coherence within the two-particle Hilbert space and is analogous to coherence in more conventional models. On the other hand, the interband polarization is a coherence between the ground state of the semiconductor and the two-particle Hilbert space. We derived equations of motion within a set of approximations that allow us to close these equations at the level of two-time functions. We then related these equations to those for the interband polarization by applying the same level of approximation.

The main analytical result found in Chapter 4 is an equation (Eq. (4.47)) for the evolution of a two-time exciton correlation function, which may be thought of as a two-time generalization of a master equation. The equation (Eq. (4.51)) governing interband polarization was found to be related to this equation in a physically transparent way. There is a reduction in the decoherence rate of an exciton if the two superimposed states have similar spatial properties. The decoherence of interband polarization is unaﬀected by this property, because it addresses superposition with respect to the ground state.

Equation (4.47) and its interpretation presented in Chapter 4 also connects directly to the conventional system-bath interaction, where decoherence results from the diminishing overlap of the two “final states” of the bath in an interaction process with the two superimposed states. However, the derivation of (4.47) makes no distinction between the system and the bath. The charge configurations characterize the different states of the “system” or the electron-hole pair, and the approximation that leads only to the appearance of A(q) in that equation replaces these charge configurations by the multipole moments. Thus the ability of “bath” of quasiparticles to detect a multipole moment of charge distribution maps directly to the decoherence of interband polarization, and their ability to distinguish the two moments, maps directly to the mutual decoherence of the two states.

In relating the dynamical equations to experimental scenarios, we found that a three pulse excitation can, within the convolution eﬀects of the third optical pulse, directly probe the two-time exciton correlations. This illustrates a powerful aspect of the two-dimensional Fourier spectroscopy. A reasonably direct access to the full temporal behaviour of the two time functions allows one to compare theory and experiments in a way that allows various approximations and paradigms for describing these complex systems to be tested.

The numerical calculations conﬁrm the main points of the analysis presented. In particu- 5. Conclusion 156 lar, the exciton decoherence rates were found to be much smaller than those of interband polarization. We also obtained plots of two-dimensional Fourier spectroscopy from our calculation of the two-time functions. These plots are the main link between experiment and theory. Their main qualitative features were explained, and the slow decoherence rates were clearly identiﬁed by comparing the plots for the solution to those for the driving terms.

We briefly discuss some possible directions for future research based on this thesis. The computational method of Chapter 2 can be easily incorporated into any numerical sim- ulation of semiconductor dynamics. It would be interesting to explore the effects of non-trivial Berry curvature on the dynamics. In relation to the multi-pulse excitation, frequencies in the terahertz (1012 Hz) regime may be used for transferring either among the exciton states, or from these states to those of higher-particle order such as biexcitons. The intraband components of the position operator play an important role in describing the dipole transitions among these states. In multi-band models that are sophisticated enough to contain interesting topological effects, our method would be useful in exploring transport in transient regime optically. Its incorporation in numerical simulations would also be useful in correct interpretation of data in experiments involving sub-gap frequency excitations of semiconductors. A particular example is the fully dynamical treatment of coherent control phenomena where one of the exciting fields is at half the gap frequency.

The formalism presented in Chapter 3 is suﬃciently general to serve as a basis for developing phenomenological models of laser-semiconductor interaction. It provides a ﬁrmer grounding of these models in a microscopic description. Even within the realm of approximations and computational techniques employed in Chapter 4, several more interesting problems can be tackled.

An interesting set of experiments are the “pre-pulse” experiments, in which carrier distributions far from equilibrium are created optically. Here a coherent control scheme may be used to control the momentum space distribution of these distributions. An approach similar to the one presented for an equilibrium electron gas can be used to study the evolution of far-from equilibrium states. These states will have interesting eﬀects on both the spectrum of excitations as well as decoherence mechanisms. More generally, such a technique enables us to explore response functions of nonequilibrium many body systems. In this respect the dependence of susceptibility on the pulse delays, as was shown in Chapter 4, may be exploited to suggest experiments from which to construct 5. Conclusion 157 these susceptibilities.

It would also be interesting to explore the effects of excitation induced dephasing (EID). Short duration of pulses in the ultrafast regime inevitably generates carrier densities, which may be a major source of decoherence. Interactions among excitons lead to higher order correlations, whose net effect on the exciton correlations may also appear as decoherence. Therefore, understanding EID is very important in the analysis of experimental data. It would be valuable to understand the qualitative effects EID has on the two- dimensional plots. A possible approach to tackle this complex problem is to first consider a simple, yet experimentally relevant, scenario. For example, one may study the model in which three or higher particle correlations can be neglected. The resulting system will be dominated by the coupled dynamics of excitons and carrier densities. This coupling may be decomposed further into two couplings. One would describe the contribution of induced densities to the dynamical screening. It may be modeled with an equation like (4.47) in which the susceptibility evaluated at equilibrium is corrected up to second order in the effective field. Formation of excitons from the induced densities will require addition of new terms.

In the presence of high magnetic ﬁeld a quantum hall state may be formed in the two- dimensional electron gas. Multi-pulse excitation can probe the interior of the state at an ultrafast timescale, as opposed to the edge states in d.c. transport experiments. Here structured light, such as Laguerre-Gaussian beams could be used to couple into the angular momentum of cyclotron orbits of the electron. Using our formalism, models for the dynamics of magneto-excitons, spin-waves, magneto-plasmons, and other optically accessible phenomena can be constructed. Thus new experiments may be suggested that will allow us to complement the traditional picture of these systems built within the context of transport theory.

Finally, the theoretical framework developed in this thesis, and the experimental technique of multidimensional Fourier spectroscopy can be applied to many other types of materials. It is straightforward to extend these techniques for studying the role of coherence in organic materials. As a particular example, we refer to recent investigations of photosynthesis [131] using the two-dimensional Fourier spectroscopy. Insights gained from these studies may also be useful in developing technological applications such as eﬃcient solar cells.

Thus decoherence can be studied using multidimensional Fourier spectroscopy in a large 5. Conclusion 158 variety of systems. We hope that the general approach presented in this thesis will provide a unifying perspective of this phenomenon. Appendix A

Eﬀective two-particle interaction: Details

An explicit expression for I(2) follows from the functional derivative of (3.49) with respect to G, yielding

(2) I (14;23) = i W (15;5′3)Γ(45′;25) ˆ

δW (15;5′6) δΓ(1′6;25) +i G(11′)Γ(1′6;25) + i W (15;5′6)G(61′) . ˆ δG(34) ˆ δG(34)

From the formal solution of (3.45) we ﬁnd

δW (14; 23) δP (68; 57) = iW (15; 63) W (74; 28) δG(3′4′) − δG(3′4′) = iW (15; 63) G(5′5)Γ(68; 5′7) + G(66′)Γ(6′8;57) W (74; 28) − { } iW (15; 63) Γ(68; 57′)G(7′7) + Γ(68′;57)G(8′8) W (74; 28) − { } δT (6′8′;5′7′) iW (15; 63) G(5′5)G(66′) G(7′7)G(8′8) W (74; 28), − δG(3 4 ) ′ ′

159 A. Eﬀective two-particle interaction: Details 160 and substituting this in the equation for I(2) yields

I(2)(14; 23)

= i W (15;5′1′)Γ(1′5′;25) ˆ

+i W (14; 61′)G(11′′)G(66′)T (1′′6′;23)+ i W (15; 31′)G(11′′)G(5′5)T (1′′4;25′) ˆ ˆ

+ W (15;61˜ ′) G(5˜′5)Γ(68;˜ 5˜′7) + G(66′)Γ(6′8; 57)˜ W (75;5′8)G(1′1′′)Γ(1′′5′;25) ˆ + W (15;61˜ ′) Γ(68; 57˜ ′)G(7′7) + Γ(68′; 57)˜ G(8′8) W (75;5′8)G(1′1′′)Γ(1′′5′;25) ˆ

δT (6′8′; 5˜′7′) + W (15;61˜ ′) G(5˜′5)˜ G(66′) G(7′7)G(8′8) W (75;5′8)G(1′1′′)Γ(1′′5′;25) ˆ δG(3 4 ) ′ ′ δT (1′′6′;25′) +i W (15; 61′)G(11′′)G(5′5)G(66′) . ˆ δG(34)

The equation is understood most clearly in the form of diagrams shown in Section 3.1.3. Appendix B

Derivation of Integral Bethe-Salpeter Equations

We start with the Dyson equation

G = G0 + G0Σ˜G,

Σ˜ UQ + U +Σ. ≡

1 1 1 Using the identity δG = GδG− G, where G− = G− Σ˜, the three functional deriva- − 0 − tives of G follow. The ﬁrst derivative is1, ˜ (2) δG(12) δΣ(1′2′) X 1a′; 2a = = G(11′) G(2′2), (B.1) δUaa′ δUaa′ the second derivative is,

(3) δ δG(12) X 1b′a′; 2ba = (B.2) δUbb′ δUaa′ δG(11′) δG(2′2) δΣ(1˜ ′2′) = G(2′2) + G(11′) δU ′ δU ′ δU ′ bb bb aa 2 δ Σ(1˜ ′2′) +G(11′)G(2′2) , δUbb′ δUaa′

1 ′ For brevity U(aa ) is written as Uaa′ in this appendix.

161 B. Derivation of Integral Bethe-Salpeter Equations 162 and the third derivative is,

3 (4) δ G(12) X 1c′b′a′; 2cba = (B.3) δUcc′ δUbb′ δUaa′ 3 δ Σ(1˜ ′2′) = G(11′)G(2′2) δUcc′ δUbb′ δUaa′ δG(11 ) δG(2 2) δG(11 ) δG(2 2) δΣ(1˜ 2 ) ′ ′ + ′ ′ ′ ′ δU ′ δU ′ δU ′ δU ′ δU ′ bb cc cc bb aa 2 δG(11′) δG(2′2) δ Σ(1˜ ′2′) + G(2′2) + G(11′) δU ′ δU ′ δU ′ δU ′ cc cc bb aa 2 δG(11′) δG(2′2) δ Σ(1˜ ′2′) + G(2′2) + G(11′) δU ′ δU ′ δU ′ δU ′ bb bb cc aa 2 2 δ G(11′) δ G(2′2) δΣ(1˜ ′2′) + G(2′2) + G(11′) . δU ′ δU ′ δU ′ δU ′ δU ′ cc bb cc bb aa Next we obtain expressions for the ﬁrst three derivatives of the self-energy:

δΣ(12)˜ δΣ(12) δG(34) = δ(1a)δ(2′a′)+ (B.4) δUaa′ δG(34) δUaa′

δ2Σ(12)˜ δΣ(12) δ2G(34) δ2Σ(12) δG(56) δG(34) = + (B.5) δUbb′ δUaa′ δG(34) δUbb′ δUaa′ δG(56)δG(34) δUbb′ δUaa′

δ3Σ(12)˜ δ3Σ(12) δG(78) δG(56) δG(34) δΣ(12) δ3G(34) = + (B.6) δUcc′ δUbb′ δUaa′ δG(78)δG(56)δG(34) δUcc′ δUbb′ δUaa′ δG(34) δUcc′ δUbb′ δUaa′ δ2Σ(12) δG(56) δ2G(34) δG(34) δ2G(56) δG(56) δ2G(34) + + + . δG(56)δG(34) δU ′ δU ′ δU ′ δU ′ δU ′ δU ′ δU ′ δU ′ δU ′ cc bb aa aa cc bb bb cc aa These substituted in (B.1)-(B.2) result in equations written entirely in terms of G and effective interaction constructed by removing G from the self-energy graphs. The effective interaction is defined in (3.62). Appendix C

Transforming between T (j) and X(j)

In this Appendix we derive some useful relations to convert a typical vertex function to correlation functions. The starting expression is based on (3.40) and (3.48), and written here schematically,

T = I + I [GG + IGG + IGGIGG + . . .] I = I + IX(2)I.

In eﬀective interactions, this expression is joined by four lines, either G or connecting to the correlation functions X(j). In either case, one can use the vertex function Γ to explicitly multiply T by G,

GGTGG = P GG. − When connected to one or more X(j), the derivative of T with respect to G, absorbs the connected correlation in deﬁning the derivative of T with respect to U via the chain rule. We need only up to two derivatives in our formalism. They are δT = I(3)X(2) + I(3)X(2)X(2)I(2) + I(2)X(2)X(2)I(3) + I(2)X(3)I(2), δU δ2T = I(4)X(2)X(2) + I(3)X(3) + I(2)X(3)I(2) + I(2)X(4)I(2) δUδU + I(3)X(2)X(2)I(2) + I(4)X(2)X(2)X(2)I(2) + I(2)X(2)X(2)X(2)I(4) + I(3)X(3)X(2)I(2) + I(3)X(2)X(3)I(2) , where the indicate that the expression is to be symmetrized by taking all permuta- {·} tions with I(j) always on the outside. The functionals δT/δU and δ2T/δUδU generate integral kernels for X(3) and X(4) in the sense that they are amputated diagrams of the

163 C. Transforming between T (j) and X(j) 164 corresponding correlation functions obtained by cutting correlation functions joined to eﬀective interactions. To see this write δT δP GG GG = PG GP PGTG GPTG GGTPG GGTGP. δU δU − − − − − − Each of the terms being subtracted are either explicitly unconnected, or they contain a P joined to a fully connected four-point function by just one leg. Since P begins at GG, this again contains an unconnected piece implicitly and is therefore not fully connected. In this manner all explicitly and implicitly unconnected diagrams are subtracted and the remaining 6-point function, after amputating, represents the connected kernel for X(3). Appendix D

(2) Derivation of EOM for Xn

(2) Here we derive the integral form of the equation of motion for Xn , which is converted to diﬀerential form in the text. It is convenient to begin with the BSE (3.43). Suppressing the arguments of all the functions involved, we write

P = P 0 + P 0I(2)P,

where P 0 = GG is the non-interacting correlation function. We now substitute P = (2) (2) (2) (2) 0 0 0 PQ + Xn , I = IQ + ∆In , and P = PQ + Pn , where

0 (1) (1) P X GQ + GQX . n ≡ n n Using the relationship,

P = GG + GGTGG, (D.1)

the BSE now takes the form

(2) 0 (2) (2) 0 0 (2) (2) X = P I X + P ΓQ + P ∆′I PQ + , n Q Q n n Q n Sn (2) where ∆′In consists only of n-th order correlation functions, and all combinations of correlation functions of order lower than n in the ﬁeld are subsumed into the source (2) term, n , which we write in integral form without the superscript l. Substituting (D.1) S evaluated at quasi-equilibrium in the third term,

(2) 0 (2) (2) 0 0 (2) 0 (2) 0 (2) X = P Γ X + P + P I + P δ′ I P ΓQ + . n Q Q n n n Q Q n Q Sn h i

165 (2) D. Derivation of EOM for Xn 166

The third term in the above equation can be re-written using the variation of P 0I(2) with only n-order correlation functions. To do this we write P 0 = P 0 + P 0 P 0 in the term Q Q n − n multiplying square brackets. Thus

0 (2) 0 (2) 0 (2)r 0 (2) 0 P I + P ∆′I P ΓQ = ∆′I P I P ΓQ. n Q Q n Q n − Q Q n h i h i On the right hand side we have introduced the two-particle reducible interaction

(2)r 0 (2) 0 I (14;23) = P (12′;21′)T (1′4′;2′3′)P (3′4;4′3), (D.2) which is diagrammatically the same as I(2) in Fig. 3.3, but the extra quasi-particle lines convert each T into P P 0 (see also Figure (3.6a)). There are also many terms arising − from the products of lower order correlation functions, which we will pick up in the section on sources. These terms describe the contribution of local ﬁeld corrections to the driving of the deviations. We now have the following integral equation with arguments of the functions restored,

(2) 0 0 (2) (2) Xn (14;23) = Pn (14;23) + PQ (14′;23′)IQ (3′2′;4′1′)Xn (1′4;2′3) (D.3) (2)r 0 (2) 0 + ∆′I (14′;23′) P I (12′;21′)P (1′4′;2′3′) ΓQ(3′4;4′3) n − Q Q n + h(2)(14; 23) i Sn The interpretation of the terms is the same as in the original equation. The first is the field induced changes in the free-particle propagation, while the second defines an effective two-particle potential. The third term represents field-induced changes in this potential. (1) (2) It consists of two terms, where the first describes these changes by Xn and Xn , and the (2) second removes those effects due to free-particle propagation that are contained by Xn but already included by the first two terms in the equation. Appendix E

Diagram Rule 4

In this section we derive the diagram rules 4 and 5 in Section 3.3.

(j) Rule 4 : From the definition (3.58) of Xn ,we note that it corresponds to differentiating X(j) n times with respect to the external two-point fields. The differentiation converts each constituent X(i) into X(i+1), where the two new arguments of the correlation function occur in pair and belong to the same function U. Thus each correlation is promoted to X(i+1), and then the new pair of arguments is contracted with an external field line. Since this is done separately for each correlation, no two separate correlation functions are allowed to be connected to the same U. This can also be interpreted as a rule to prevent over-counting. Two individual components connected by a field line form a contribution of the same field order to a lower particle order correlation function. Therefore they have been included in the equation for the corresponding function already.

The exception to this rule for the pulse that fully contracts a diagram is due to the fact that this contraction corresponds to lowering of the particle order of correlation, rather than a deviation in it. This pulse produces a two-point function after two arguments of (2) (1) the correlation functions evolving to that point are contracted. It takes Xn Xn+1 via (2) → the products of the form Xn (1a;2a′)U(a′a). From Fig. 3.4, it becomes clear that a, a′ can exist on separate correlation functions. The rule is also equivalent to the statement that U occurs explicitly only in the equation for G, and is implicit in equations for all X(2...) via G [U].

167 Appendix F

Two-time Approximation

We use (4.15) to deﬁne the two-time exciton Green function as,

G (t , t ) G (14;1+4+), (F.1) X 1 4 ≡ cvvc such that the electrons are created and destroyed in pairs. The branch indices produce

a 4 4 matrix G (tt′) in the notation deﬁned below (4.15) in the text. Arranging this × X matrix such that the branch indices of arguments, 1 and 2, vary along rows, and those b of 3 and 4 along columns, it follows that for t > t′, all rows of GX (tt′) are identical, and given by, b > + < G(tt′) (tt′) ˆ−(tt′) ˆ (tt′) (tt′) . (F.2) ≡ G G G G h i For t < t′, all columns are identical and given by,

< (tt′) G +  ˇ (tt′)  G(tt′) = G . (F.3)  ˇ−(tt′)   G   >   (tt′)   G    Here we have deﬁned six new functions. The lesser and greater functions,

> (tt′) = a† (t)a (t)a†(t′)a (t′) , G − v c c v < (tt′) = a†(t′)a (t′)a† (t)a (t) , G − c v v c have equal time operators placed together and thus they are sensitive only to the atomic nature of the exciton (see I for deﬁnition of ). The next two functions account for the h·i

168 F. Two-time Approximation 169

composite nature of the exciton,

ˆ−(tt′) = a†(t′)a (t)a† (t)a (t′) , (F.4) G c c v v + ˆ (tt′) = a (t′)a† (t)a (t)a†(t′) , (F.5) G v v c c ˇ−(tt′) = a† (t)a (t′)a†(t′)a (t) , (F.6) G v v c c + ˇ (tt′) = a (t)a†(t′)a (t′)a† (t) . (F.7) G c c v v The superscripts on these functions do not stand for the contour index. Rather, they ± signify the fact that ˆ± ( ˇ±) represent the annihilation of the electron-hole pair at time G G t (t′) in a state that evolves from N 1 particle state at time t′ (t). This is in contrast ± to the functions ≷ the Lehmann representations of which are described by states only G within the N-particle Hilbert space. Exploiting the equal time anticommutation relation

ac†(t), av(t) =0, it can be veriﬁed that the equalities between Keldysh components of (2) (2) G carry over to X as well. Thus the matrix X has identical rows,

> ˆ ˆ+ < X (tt′) = (tt′) −(ttc′) (tt′) (tt′) , (F.8) P P P P h i for t > t′, and identical columns

< (tt′) P +  ˇ (tt′)  X (tt′) = P , (F.9)  ˇ−(tt′)   P   >   (tt′)   P    for t < t′. As discussed in the text, we make further approximations to close the Bethe-Salpeter equations within the space of two-time functions. To formulate them mathematically, we begin with the semigroup property obeyed by non-interacting Green functions,

r r r G0(tt′) = iG0(tt′′)G0(t′′t′), (F.10) a a a G (tt′) = iG (tt′′)G (t′′t′). (F.11) 0 − 0 0 Note that no integration is performed over the time variables, but all other degrees of freedom are summed using the matrix notation.The two time limit is constructed by constructing the analogue of semigroup property of the ideal single particle Green functions, which holds approximately for the full Green functions[124, 132, 133],

r r r G (tt′) iG (tt′′)G (t′′t′), (F.12) ≈ a a a G (tt′) iG (tt′′)G (t′′t′). (F.13) ≈ − 170

The “ ” here denotes the neglect of all those diagrams where the time t′′ lies inside at ≈ least one self-energy insertion. This is seen from the diagrammatic expansion of Dyson

equation, in which we start with G0, pick a time t′, and apply the properties (F.10) and

(F.11). Then we dress each of the two G0 with self-energy insertions, and reject all those diagrams that cannot be cut at t′ by only cutting a propagator.

Substituting the approximate equations (F.12) and (F.13) in the Dyson equation in turn implies,

r < < iG (tt′′)G (t′′t′) t > t′′ > t′ G (tt′) . ≈  < a iG (tt′′)G (t′′t′) t < t′′ < t′ − Collecting these functions in the matrix form, and transforming back to the contour indices, we get

z iG(tt′′)σ G(t′′t′) t > t′′ > t′ G(tt′) = .  z iG(tt′′)σ G(t′′t′) t < t′′ < t′ −b b b  b b Appendix G

Derivation of EOM

G.1 Derivation of two-particle EOM

(2) In this appendix, we show how to write the JQX2 part of (4.23) using the Keldysh (2) matrix representation, with the special structure (F.8-F.9). The J2XQ term is obtained analogously.

To proceed, we write the interaction part of the BSE as

+ + + + + + I(tt′; t t′ ) = J Q(tt′′; t t′′ )ZP (t′′t′; t′′ t′ ) dt′′, ˆ b 1b b b  1  where Z = − .  1   −    b  1      To further simplify this, we divide the integration over t′′ into three regions,

tmax (1) + + + + + + Icvvc(tt′; t t′ ) = J Q;cvvc(tt′′; t t′′ )ZP (t′′t′; t′′ t′ ) dt′′, ˆtmin b(2) + + ∞ b + + b b + + Icvvc(tt′; t t′ ) = J Q;cvvc(tt′′; t t′′ )ZP (t′′t′; t′′ t′ ) dt′′, ˆtmax tmin b(3) + + b + + b b + + I (tt′; t t′ ) = J Q (tt′′; t t′′ )ZP (t′′t′; t′′ t′ ) dt′′. cvvc ˆ ;cvvc −∞ b b b b

171 G.2 Derivation of single-particle EOM 172

where tmin = min(t, t′) and tmax = max(t, t′). For t > t′, we obtain t (1) r Icvvc(tt′) = Q;cvvc(tt′′)P(t′′t′) dt′′, ˆt′ J (2) Icvvc(tt′) = 0, t′ (3) I (tt′) = J Q(tt′′)ZP(t′′t′) dt′′. cvvc ˆ −∞ Here I(j) are the row vectors of the same form as bP with the corresponding elements. From the properties of it also follows that J (3)σγσ′ γ′ σσ′ (tt′) = 0. IQ;cvvc σσ′ < X< In the equation for , it is that contributes. In the two regimes, t > t′ and t < t′, P I it is t < r < (tt′) = dt′′ Q (tt′′) (t′′t′) I ˆt′ J P t′ +i J Q(tt′′)ZP(t′′t′) dt′′, t>t′, ˆ −∞ t′ < ba (tt′) = dt′′ J Q(tt′′) (t′′t′) I ˆt P t +i J Q(tt′′)ZP(t′′t′) dt′′, t > < +i (tt′′) (t′′t′)+ (tt′′) (t′′t′) dt′′ ˆ JQ P JQ P −∞ tmin + + i ˆ−(tt′′) ˇ (t′′t′)+ ˆ (tt′′) ˇ−(t′′t′) dt′′, − ˆ JQ P JQ P −∞ where tmin = min (t, t′) and tmax = max (t, t′). Similar steps lead to an expression, in (2) terms of the Keldysh components, for the term J2XQ in (4.23). After restoring the (2) σ(11′′) and σ(1′′4) in (4.23) as discussed just above that equation, and adding JQX2 and (2) J2XQ terms, we obtain (4.44) in Section 4.6.

G.2 Derivation of single-particle EOM

We now derive an expression for the interband polarization by using the semigroup approximation, which is the same as generalized Kadanoﬀ Baym ansatz (GKBA) in this G.2 Derivation of single-particle EOM 173 case. We will consider the dynamically screened Hartree-Fock self-energy with a vertex correction arising from electron-hole interaction. From the Dyson equation, the “+ ” − Keldysh components of the couplings in (4.12) are

(11;1) (1) s + X = Σ (k, t ) X − (k t t ) , (G.2) M1 1 Qcc 1 1;cv | 1 2 (12;1) (2) +σσ+ +σ; σ + + X = iW (q t t′)X − k, k′, q tt′; t t′ (G.3) M1 1 Q | − 1;cvvv | +σσ+ +σ; σ + + +iW (q t t′)X − k, k′, q tt′; t t′ . Q | − 1;ccvc | We obtained these by separating the self-energy into the static part Σs and a purely dynamical part Σ, and we have indicated the ﬁrst order variation in the self energy by superscript “1”. The above formula gives (12;1) (4.12), since it is coupled to the M1 functions X1;cvvv and X1;ccvc.

We now employ the GKBA[84, 82] in the form,

+ + + a r + X − (12) = X− (12) = X − (1)G (12) G (12)X − (2), 1;cv 1;cv 1;cv vv − cc 1;cv ++ + a r + X (12) = X−− (12) = X − (1)G (12) G (12)X − (2). 1;cv 1;cv 1;cv vv − cc 1;cv

Note that all O(U) components of the Keldysh matrix for Gcv are equal, and this is due to the anticommutation relation a k, a k = a k, a† = 0. We apply GKBA to { v c } v ck approximate X1;cvvv, and X1;ccvc functions by a productn ofo particle conserving exciton correlation X1;cvvc and the interband polarization as follows,

(2)+σ σ (2)+σ σ σσ X − (k, k′, q tt′; tt′) = X − (k, k′, q tt′; tt′)X (k′, t′; k′t′) (G.4) 1;cvvv | − Q;cvvc | 1;cv (2)+σ σ (2)+σ σ σσ X − (k, k′, q tt′; tt′) = X − (k, k′, q tt′; tt′)X (k′ + q, t′; k′ + q, t′).(G.5) 1;ccvc | Q;cvvc | 1;cv This approximation is most easily understood diagrammatically by directly performing vertex corrections corresponding to electron-hole interaction in the Hartree-Fock self energy.

Combining (G.3) and (G.2) with their adjoint, imposing the GKBA, setting t1 = t2 = t G.2 Derivation of single-particle EOM 174

+ and writing Xcv−(tt)= iρcv(t) for brevity, we get

∂ + s + iω k ρ (k, t) iU kt; kt f k f k Σ (k, t) ρ (k, t) (G.6) ∂t c cv − cv { v − c } − Qcc cv dk′ s = f k f k V (k k′) ρ (k′, t) { v − c } ˆ 4π2 − cv

dq +σσ+ +σ + dt′ iW (q t t′) S (t t′) ˆ ˆ 4π2 Q | − −

dkdk′ (2)+σ σ (2)+σ σ X − (k, k′, q tt′; tt′)+ X − (k, k′, q tt′; tt′) · ˆ 16π4 1;cvvv | 1;ccvc | n o dq σ σ σ dt′ iW −− (q t′ t) S− (t t′) − ˆ ˆ 4π2 Q | − −

dkdk′ (2)+σ σ (2)+σ σ X − (k q, k′, q tt′; tt′)+ X − (k q, k′, q tt′; tt′) . · ˆ 16π4 1;cvvv − | 1;ccvc − | n o σσ′ The integration domain for integral over t′ is restricted by the function S (t) deﬁned as

′ 1 δσσ′ t =0 Sσσ (t) = − . 1 t =0  6  This restriction allows us to explicitly write the contribution of V s and Σs, the singular parts of screened interaction, in (G.6). We have not yet substituted (G.4) and (G.5) in (G.6), but will do so when projecting this equation onto the exciton basis using (4.30). The condition of zero exciton density in equilibrium when applied to the last two terms in (G.6) allows only the positive contour index, i.e. σ =+. With σ = there is always − > at least one occurrence of GQvv, which vanishes. To project (G.6) onto the exciton basis, we substitute (G.4) and (G.5) and integrate them over k and k′ as follows,

dk dk′ ++ + ++ + ϕ∗ (k) X − (k, k′, q tt′; tt′)+ X − (k, k′, q tt′; tt′) ˆ 4π2 n ˆ 4π2 1cvvv | 1ccvc |

> dk = (tt′) ρ (t′) ϕ∗ (k) ϕ (k + α q) − PQjj m ˆ 4π2 n j c jm X dk′ ϕ∗ (k′ α q) ϕ (k′) ϕ∗ (k′ α q) ϕ (k′ + q) · ˆ 4π2 j − v m − j − v m > = O (0, α q) A (q) (tt′) ρ (t′) , nj c jm PQjj m jm X where A (q) = A(0, q) (see discussion below (4.50) in text). To obtain the last line we G.2 Derivation of single-particle EOM 175

have used the relation αc =1+ αv in the integral over k′,

dk′ dk′ ϕ∗ (k′ α q) ϕ (k′ + q) = ϕ∗ (k′) ϕ (k′ +(1+ α ) q) ˆ 4π2 j − v m ˆ 4π2 j m v = Ojm(0, αcq).

The integrals in the last term of (G.6) are handled similarly, obtaining

dk ++ + ++ + ϕ∗ (k) dk′ X − (k q, k′, q tt′; tt′)+ X − (k q, k′, q tt′; tt′) ˆ 4π2 n ˆ 1cvvv − | 1ccvc − |

> dk = (tt′) ρ (t′) ϕ∗ (k) ϕ (k α q) − PQjj m ˆ 4π2 n j − v jm X dk′ ϕ∗ (k′ α q) ϕ (k′) ϕ∗ (k′ α q) ϕ (k′ + q) · ˆ 4π2 j − v m − j − v m > = O (0, α q) A (q) (tt′) ρ (t′) . nj − v PQjj m jm X The diﬀerential equation now takes the form

∂ + iω ρ (t) U (t) ∂t n n − n ∞ dq = dτ iW < (q, τ) O (0, α q) O (0, α q) A (q) > (τ) ρ (t τ) ˆ ˆ 4π2 Q { nj c − nj v } jm PQjj m − jm 0 X ∞ dq = dτ iW < (q, τ) A (q) A (q) > (τ) ρ (t τ) ˆ ˆ 4π2 Q − nj jm PQjj m − m 0 ( j ) X X ∞ = dτ (τ) ρ (t τ) . ˆ Bnm m − m 0 X

Note that the integration domain does not conﬂict with the condition t′ = t in (G.6) 6 because only W < appears in the integrand, and therefore no instantaneous term (in the contour sense) contributes to the integral.

In the last equality in the above equation, we have used the deﬁnition (4.49) of the matrix (τ), while on the left hand side we have combined the coulomb interaction B terms and the band energies into the exciton energy levels, ~ωn. The projection of + iU (kt; kt ) f k f k on the exciton states is denoted by U (t). Thus the single cv { v − c } n particle equation takes the form shown in (4.47) or (4.52). G.3 Expressions for interaction matrix components 176

G.3 Expressions for interaction matrix components

After several algebraic steps starting from the expression (4.22) deﬁned on the Keldysh

contour, the expressions for the components of J q are as follows,

> dq′ > > (tt′) = i W (q q′ tt′)A (q, q′) ′ (tt′)A (q′, q) Jq − ˆ 4π2 − | Pq

< dq′ < < (tt′) = i W (q q′ tt′)A (q, q′) ′ (tt′)A (q′, q) . Jq − ˆ 4π2 − | Pq While we mainly work with these components in the text, we also give expressions for the rest for completeness,

dq′ < ˆ− ′ q q′ ′ q q′ ˆ− ′ ′ q′ q (tt ) = i 2 W ( tt )A ( , ) q q (tt )A ( ) J − ˆ 4π − | P − 1 < > [iW (q q′ tt′) iW (q q′ tt′)] F (q′) ˆ−′ (tt′)A (q′, q) −2 − | − − | Pq + dq′ > + ˆ (tt′) = i W (q q′ tt′)A (q, q′) ˆ ′ (tt′)A (q′, q) Jq − ˆ 4π2 − | Pq 1 > < + [iW (q q′ tt′) iW (q q′ tt′)] F (q, q′) ˆ ′ (tt′)A (q′, q) −2 − | − − | Pq + dq′ < + ˇ (tt′) = i W (q q′ tt′)A (q, q′) ˇ ′ (tt′)A (q′, q) Jq ˆ 4π2 − | Pq 1 < > + + [iW (q q′ tt′) iW (q q′ tt′)] A (q, q′) ˇ ′ (tt′)F (q′, q) 2 − | − − | Pq dq′ > ˇ− (tt′) = i W (q q′ tt′)A (q, q′) ˇ−′ (tt′)A (q′, q) Jq ˆ 4π2 − | Pq 1 > < + iW (q q′ tt′) iW (q q′ tt′) A (q, q′) ˇ−′ (tt′)F (q′, q) 2 − | − Q − | Pq Note that only the matrix A appears in expressions for ≶, which are sensitive only J to the multipole moments of the exciton. On the other hand, the composite functions always contain a contribution from F ,

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