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Influence of phonons on light emission and propagationin semiconductor nano optics

———————————–

Vorgelegt von Diplom-Physiker

Matthias-Rene Dachneraus Ueckermunde

von der Fakultat II – Mathematik und Naturwissenschaftender Technischen Universitat Berlin

zur Erlangung des akademischen GradesDoktor der Naturwissenschaften

— Dr. rer. nat. —Genehmigte Dissertation

Promotionsausschuss

Vorsitzende/r: Prof. Dr. rer. nat. Stephan Reitzenstein, TU Berlin1. Gutachter: Prof. Dr. rer. nat. Andreas Knorr, TU Berlin2. Gutachter: Priv.-Doz. Dr. rer. nat. Uwe Bandelow, WIAS

Tag der wissenschaftlichen Aussprache: 23. Oktober 2013

———————————–Berlin, 2013

D 83

To Science

Abstract

In this work the optical properties of semiconductor quantum dots (QDs) with regard to theirapplication in optical amplifiers are investigated. Quantum dot semiconductor optical amplifiers(SOAs) can be operated as non-linear optical devices. In this regime the propagation of an opticalpulse through an SOA may result in a reshaping of the pulse determined by the interplay of effectssuch as saturable gain and loss, external pumping and dispersion.

In a solid state environment, phonons introduce losses, but can be also responsible for thecarrier capture and thus for the pumping of the QDs. In this thesis, the pure dephasing, decoherencewithout a change of the levels’ occupations, is investigated on the basis of the independent bosonmodel. The influence of phonons on the shape of linear optical spectra with respect to the wavefunctions as well as the phonon dispersion relation is examined. Therefore, the resulting spectra ofmicroscopically calculated wave functions as well as simple model wave functions are compared.The impact of the wave function parameters size and internal dipole moment, i.e. displacementbetween the centers of mass of electron and hole wave function, are investigated for the typicalcoupling mechanisms deformation potential coupling, piezoelectric coupling for acoustic phononsand polar coupling to optical phonons. For optical spectra of QDs the phonon dispersion relation isimportant. Due to the spatial confinement, higher phonon wavenumbers gain importance, thus thetypical Debye and Einstein assumptions are compared to more realistic model dispersion relations.

Next, the phonon assisted carrier capturing is discussed. The capture process is described viaan effective multiphonon Hamiltonian, that is perturbatively derived within a projection operatorformalism. This leads to a rate equation with temperature dependent scattering rates.

The optical response of QDs is determined by the carrier dynamics. To calculate the dynamics,the non-resonant QD optical Bloch equations are investigated. With the aim of an analyticalexpression for the adiabatic response, the microscopical polarization is described by a perturbativeseries for the regimes: conservative (negligible losses), slow pumping (dephasing much strongerthan relaxation processes) and fast pumping (dephasing and relaxation of the same magnitude).

Finally, an electric field propagating through a QD medium is described using the micro-scopically derived adiabatic response. The propagation is formulated in terms of a generalizedGinzburg-Landau equation with parameters on a microscopic footing for the conservative as wellas the dissipative regime. Those wave equations are investigated for the existence of (dissipative)solitons.

ZusammenfassungDie optischen Eigenschaften von Halbleiter Quantenpunkten, speziell in Hinblick auf Anwendungenin optischen Verstarkern, sollen in dieser Arbeit naher beleuchtet werden. Auf Quantenpunktenbasierte Optische Verstarker konnen als nichtlineare Bauelemente betrieben werden. In diesemRegime findet eine Umformung der optischen Pulse durch sattigbaren Gewinn und Verlust, externenPumpen sowie Dispersion.

In Festkorpern werden durch Phononen Verluste erzeugt. Jedoch konnen Phononen auch beimEinfangen der Ladungstrager in den Quantenpunkt eine Rolle spielen und sind somit auch furdas Pumpen verantwortlich. In dieser Arbeit wird pure Dephasierung, also die Dekoharenz ohneVeranderung der Besetzung der Zustande, mittels des Modells unabhangiger Bosonen (independentboson model) untersucht. Der Einfluss von Phononen auf die Form linearer optischer Spektren wirdin Hinblick auf den Einfluss der Wellenfunktionen als auch der Dispersion der Phononen naherbetrachtet. Dazu werden, auf Grundlage von mikroskopischen Wellenfunktionen als auch einfachenModellwellenfunktionen, Spektren berechnet und verglichen. Die Auswirkung verschiedener Wel-lenfunktionsgroßen und intrinsischer Dipolmomente, d.h.sprich Abstanden zwischen den Elektron-und Loch-Wellenfunktionsschwerpunkten, wird fur die typischen WechselwirkungsmechanismenDeformationspotentialkopplung und Piezoelektrische Kopplung an akustische Phononen, sowiepolare Kopplung an optische Phononen untersucht. In optischen Quantenpunktspektren ist diePhononendispersionsrelation von Bedeutung. Durch die raumliche Einschrankung spielen hoherePhononwellenzahlen eine großere Rolle. Die typischen Naherungen der Phonondispersion fur kleineWellenzahlen nach Einstein und Debye werden mit realistischeren Naherungen verglichen.

Als Nachstes wird der von Phononen assistierte Einfang von Ladungstragern besprochen. DerEinfangprozess wird mit einem effektiven Hamiltonoperator beschrieben, der storungstheoretischinnerhalb eines Projektionsoperatorformalismus hergeleitet wird. Dies fuhrt zu einer Ratengleichungmit temperaturabhangigen Streuraten.

Die optische Antwort der Quantenpunkte wird durch ihre Ladungstragerdynamik bestimmt.Zur Berechnung dieser Dynamik werden die nicht-resonanten optischen Blochgleichungen ver-wendet. Mit dem Ziel die adiabatische optische Antwort analytisch zu beschreiben, werden dieBlochgleichungen storungstheoretisch, in den verschiedenen Bereichen: konservatives Regime(vernachlassigbare Verluste), Regime des langsamen Pumpens (signifikante Dephasierung, vielschwachere Relaxation), sowie Regime des schnellen Pumpens (Dephasierung und Relaxation sindin der gleichen Großenordnung), entwickelt.

Schließlich wird die Ausbreitung optischer Pulse durch ein Quantenpunktmedium mit Hilfe derzuvor mikroskopisch berechneten adiabatischen optischen Antwort beschrieben. Die Ausbreitungist als generalisierte Ginzburg-Landau Gleichung mit mikroskopische hergeleiteten Parametern imkonservativen, als auch im dissipativen Regime. Diese Wellengleichungen werden auf die Existenzvon dissipativen Solitonen untersucht.

Contents

1 Prologue 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Dramatis personæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Electronic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Electric fields, Maxwell’s equations, Wave equations . . . . . . . . . . . 15

1.3 Interactions – full systems’ Hamiltonian and equations of motion . . . . . . . . . 191.3.1 Electron–light system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Electron–phonon system . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Phonon influence on carrier dynamics in QDs 232.1 Linear absorption and luminescence spectra . . . . . . . . . . . . . . . . . . . . 23

2.1.1 Independent Boson model . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Diagonal coupling due to various interaction mechanisms . . . . . . . . 312.1.3 Numerical investigation of optical spectra . . . . . . . . . . . . . . . . . 38

2.2 Coupling to carrier reservoir – multi-phonon relaxation . . . . . . . . . . . . . . 412.2.1 The phonon bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2.2 Effective multi-phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . 412.2.3 Multiphonon relaxation rates . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Non-linear optical response of quantum dots 553.1 Conservative quantum dot Bloch equations . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Exact solutions of the optical Bloch equations . . . . . . . . . . . . . . . 573.1.2 Perturbative approach for non-resonant excitation . . . . . . . . . . . . . 59

3.2 Dissipation in QD Bloch equations in relaxation time approximation . . . . . . . 623.2.1 Perturbative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.2 Saturable absorber – adiabatic Ansatz for the dissipative Bloch equations 73

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Non-linear wave equations describing QD devices – The search for dissipative solitons 774.1 Introduction to non-linear wave equations and solitons . . . . . . . . . . . . . . 774.2 Wave equation – slowly varying envelope approximation . . . . . . . . . . . . . 79

CONTENTS

4.2.1 Pulse broadening due to group velocity dispersion . . . . . . . . . . . . 814.3 Generalized complex Ginzburg-Landau type equations . . . . . . . . . . . . . . 83

4.3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Conservative cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.3 Dissipative cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.4 Transition from dissipative to the conservative regime . . . . . . . . . . 97

5 Conclusion 99

A Prefactors of the piezoelectric coupling 101

B Induction proof of the Bloch equation’s perturbation treatment 103

2

Was einmal gedacht wurde, kann nicht mehr zuruckgenommen werden.Friedrich Durrenmatt, Die Physiker

1Prologue

1.1 Introduction

Our modern society is based on the transfer of information. Optical data transmission is thebackbone of long-range transmission and makes its way to shorter ranges as faster transmissionrates are required. In electronics the speed of information processing is limited, since the electrons’mass introduces resistance that is even enhanced by the skin effect, that pushes the electric currentto a conductor’s surface for higher frequencies [Mit10]. For photons these specific solid staterestrictions do not apply. However, the realization of fast optical devices has to overcome otherlimitations. In particular, optical pulses in a medium suffer dispersion, i.e. temporal broadeningafter a propagated distance due to different phase velocities of the frequencies the pulse is composedof. This limits the interval between two optical bits and thus the data transmission rate. Therefore,a deep understanding of propagation of optical pulses and the interaction with the host medium isof crucial importance to improve data transmission. In fiber optics the concept of optical solitonswas embraced, dealing specifically with this problem. Here the dispersion in the fiber is exploited,as it cancels the non-linear optical response of the material leading to stable optical pulse shapesover long distances [Mit10].

The fundamental concept of a soliton was introduced when John Scott Russell observed awater wave in a channel in 1834 that did not change in shape or velocity over a distance of 2-3 km.Korteweg and De Vries presented their famous equation for the wave propagation in a shallowwater channel and the corresponding “solitary wave” solution in 1895 [KV95]. In 1965 Zabuskyand Kruskal investigated waves in a collisionless plasma, which can be also described via theKorteweg-DeVries equation, and coined the name “soliton” for numerically found solutions, thatdid not change their shape during propagation and showed elastic scattering [ZK65]. Gardner et.

1.2. DRAMATIS PERSONÆ

al. discovered in 1967 an analytical method to obtain such soliton solutions including interaction[GGK+67]. The inverse scattering theory, which can be used to obtain soliton solutions for variousnon-linear equations is based on this work [ZS74; NML+84]. In 1973 Hasegawa and Tappertdescribed the light in fibers via the non-linear Schrodinger equation and proposed the existence ofoptical solitons in dielectric materials [HT73].

Despite that, over the years many non-linear equations having soliton solutions were intensivelystudied. The term soliton is restricted to conservative (closed, integrable) systems and in manypractical applications dissipation or coupling to the environment is not negligible. However, also indissipative systems localized structures can occur, leading to the concept of the “dissipative soliton”[AA05a; AA08a]. In the same way as solitons they are stable localized structures in a variety ofphysical, biological or chemical systems.

This work will shed some light on a host material that consists of semiconductor quantumdots (QDs) providing the basis for dissipative optical solitons. In QDs the electrons are spatiallyconfined below their DeBroglie wavelength and thus have discrete energy levels, rendering them ahighly efficient optical material for lasers or optical amplifiers [BGL99; Gru06].

The thesis is divided into five chapters. This first chapter introduces or revisits the nomenclatureand general properties of electric fields, carriers in semiconductor quantum dots, phonons, as wellas the respective interactions. The second chapter is focused on the dissipative influence of phononson the dynamics of electrons. Specifically, Sec. 2.1 shows the influence of pure dephasing onoptical spectra, whereas Sec. 2.2 focuses on the phonon assisted coupling to an electronic reservoir.The electron dynamics are described in a rate equation limit via the QD Bloch equations. InChap. 3 we explore the Bloch equations in various limits to yield analytical expressions for theQDs’ optical response. This non-linear optical response is then used in wave equations in Chap. 4,that consequently describe the propagation of optical pulses through QD media. The properties ofthose wave equations are investigated especially in connection with soliton solutions.

1.2 Dramatis personæ

This work describes the dynamics of electrons in semiconductor quantum dots, the influence ofphonons on them as well as the properties and evolution of optical fields interacting with this solidstate system. In this section the basics of the framework shall be briefly introduced. Section 1.2.1discusses wave functions and energies of the eigenstates of the electronic system, Sec. 1.2.2comments on the phonon modes and the interaction mechanisms to the electronic system and inSec. 1.2.3 the wave equations for the electric field are treated and linked with the electronic system.

We describe the electronic dynamics within the framework of second quantization. In practicethis means the systems wave functions obtain operator character and a commutation relation:

[ψ†(~r, t), ψ(~r′, t)]± = δ(~r − ~r′)

ψ denotes the full wave function of the electronic (anti-commutator) or bosonic system (commutator).Those operators are creation and annihilation operators that act on the vacuum state. This wave

4

CHAPTER 1. PROLOGUE

function operator is decomposed into the eigenstates of the systems where the coefficients ci are infact operators.

ψ =∑i

ciφi,

with the eigenfunctions of the systems Hamiltonian φi. In the next section we discuss the eigen-functions. The operators of the new systems are constructed by “sandwiching” the first quantizationoperators with wave function creators and annihilators.

1.2.1 Electronic Hamiltonian

The foundation of microscopic calculations presented in this work are the electronic wave functions.They contain the information of the material and the geometry of the structure and thus define thestrength of coupling elements and time scales of dynamics. We deal with heterostructures that arecomposed of different semiconductor materials, such that electrons are confined in a potential dueto different band gaps (cf. Fig. 1.1a) [Gru06].

The (self-organized) fabrication of such a structure starts with the bulk medium, where electronsare not confined below their DeBroglie wavelength and thus have defined momenta and correspond-ing energies in a quasi-continuous bandstructure. On top another material is deposited and due toa mismatch of the lattice constants stress is generated. Depending on the materials and growingconditions first a few monolayer thick film, the wetting layer (WL) is formed before the stress isalleviated by forming small islands, the quantum dots (QDs) [BGL99]. Typical QDs are flat andhave a typical diameter of 10-20 nm.

Due to the three-dimensional confinement, electrons in QDs have discrete energies and are thuscalled “artificial atoms”. They have advantageous optical properties as a distinguished and control-lable emission wavelength and a high quantum yield [Bim05]. Furthermore, they are prosperouscandidates for single and entangled photon sources [BSP+00; SYA+06; Bim08; SWO+10]. Wettinglayer and bulk have quasi-continuous energies that can be used as a carrier reservoir for the QDs[FCA+01; MUB+02; GBDT+08; WDM+09; Wil13].

Effective mass and envelope approximation

There are several microscopic approaches to calculate wave functions of semiconductor heterostruc-tures as 8~k ·~p theory [SWL+09; SWB07; SBL+03; WSB06] or tight binding calculations [SMS+11;SLS+11] beginning with the position of every atom.

It is however of advantage to get an analytical hold of the wave functions. This is the purpose ofthe envelope function approximation [HK04; YC05] for heterostructures. Here, the overall latticepotential is approximated by a microscopic, periodic lattice Vg with an additional slowly varyingenvelope potential U , (cf. Fig. 1.1b). The full systems Hamiltonian reads:

H |ψ〉 = (H0 + Vg + U) |ψ〉 = ε |ψ〉 . (1.1)

5


z

E

x, y

InAs

GaA

s

GaA

s

InGaA

s

GaAs InAs GaAs

V

x, y

Figure 1.1: a) Sketch of the energy landscape of a QD heterostructure (z denotes the growth direction).A lower potential confines carriers. b) Such a potential can be approximated via the envelope functionapproximation, where the whole structure is described by a fully periodic lattice potential with an additionalenvelope potential modeling the nano-structure.

H0 is the quasi-kinetic energy part of the electron Hamiltonian and ε the eigenenergy of theeigenstate |ψ〉. Without envelope potential U there are the eigenstates |i, λ〉 with eigenenergies Eλisatisfying Bloch’s theorem.

(H0 + Vg) |i, λ〉 = Eλi |i, λ〉 , |i, λ〉 ∧= 1√Vuλ~ki

(~r)ei~ki·~r, uλ~ki(~r + ~R) = uλ~ki

(~r). (1.2)

Here, λ describes the bands and i is the intraband index, denoting the energies within the bands. 1

Since |i, λ〉 are a complete orthonormal system |ψ〉 can be decomposed with coefficients cλi .

|ψ〉 =∑λ

∑i

cλi |i, λ〉 , (1.3)

⟨j, λ′

∣∣H∣∣ψ⟩ = cλ′j E

λ′j +

∑λ

∑i

cλi⟨j, λ′

∣∣U ∣∣i, λ⟩ = εcλ′j . (1.4)

The envelope potential changes much slower than the atomic potentials, thus we can assume it to beconstant over the range of a unit cell U(~R+ ~r) ≈ U(~R).

⟨j, λ∣∣U ∣∣i, λ′⟩ ≈ 1

N

∑n

U(~Rn)e−i(~kj−~ki)·~Rn 1

Vuc

∫Vuc

d~r uλ~kj∗(~r)uλ

′

~ki(~r)ei(

~ki−~kj)·~r, (1.5)

1In the typical III-V semiconductors there are four relevant bands: one conduction band and the three valence bandslight hole, heavy hole and split-off bands. We will however restrict ourselves to the heavy holes, since there are energeticoffsets between the valence bands in semiconductor-nano structures [YC05].

6

CHAPTER 1. PROLOGUE

with the typical approximation of small wave numbers uλ′~k (~r) ≈ uλ′0 (~r) (i.e. the resulting wave func-tions should have a larger spatial extent) the integral over the unit cell: 1

Vuc

∫Vucd~r uλ0

∗(~r)uλ

′0 (~r) =

δλ,λ′ yields band diagonality. This means within this approximation there are no band mixingeffects for the wave functions.

⟨j, λ∣∣U ∣∣i, λ′⟩ ≈ 1

N

∑n

U(~Rn)e−i(~kj−~ki)·~Rnδλ,λ′ . (1.6)

The next typical approximation is the effective mass approximation, where the band structureof the host material is expanded for small wave numbers yielding a parabolic behavior:

Eλj = Eλ0 +~2~k2

j

2m∗λ. (1.7)

m∗λ is an effective, band and material depending mass. Equation (1.4) is multiplied with ei~kj ·~r andthe sum over all ~kj is taken, then it reads

∑~kj

cλ~kj

~2~k2j

2m∗λei~kj ·~r +

∑~ki

cλ~ki

1

N

∑n

U(~Rn)∑~kj

e−i~kj ·(~Rn−~r)ei~ki·~Rnδλ,λ′ = (ε− Eλ0 )

∑~kj

cλ~kjei~kj ·~r.

(1.8)

This is a Schrodinger equation for the envelope wave functions φ =∑

~kcλ~kei~k·~r, where only the

envelope is the potential. (−~2∆

2m∗λ+ U(~r)

)φλ(~r) = Eλφλ(~r). (1.9)

The Eigenfunctions φλ yield the coefficients for the full wave function |ψ〉 by the definition inEq. (1.3)

ψλ(~r) =1√V

∑i

uλ~ki(~r)cλi e

i~ki·~r ≈ uλ0(~r)φλ(~r), (1.10)

ελ = Eλ0 + Eλ. (1.11)

The wave equation of a semiconductor heterostructure is composed of the eigenfunction of theenvelope Schrodinger equation and a Bloch factor. The eigenenergies are the ones from the envelopeproblem plus the band edge.

The following section presents the specific wave functions for this work.

7


Wave functions of the system

To describe the system, a convenient model potential U has to be chosen. For the sake of simplicitythis is done independently for QD and WL states, yielding an unintended wave function overlap.This can be corrected by the orthogonalized plane waves technique [Nie05; Mal08; Wil13].

A typical choice of the model potential describes lens shaped QDs via an in-plane harmonicoscillator potential and an one dimensional potential well in growth direction[WHF+96; Nie05;Mal08].

The lowest state φs is the product of the two dimensional harmonic oscillator ground stateϕλs (x, y) and the ground state of an one dimensional infinite well ξλ0 (z). This wave function has nonodes and is thus called s state in analogy to the hydrogen atom:

φλs (~r) = ϕλs (x, y)ξλ0 (z), (1.12)

ϕλs (x, y) =

√mλω

λdot

~πe−

mλωλdot

2~ (x2+y2), (1.13)

ξ0(z) =

√2L cos(πz/L), |z| ≤ L/2,

0, else,(1.14)

Eλ0 =~2ωλdot +

~2π2

2mλL2+ EInAs. (1.15)

L describes the height of the QDs with an effective width of the infinite quantum well. Alternativelythe x-y-part is written as

ϕλs (x, y) = e− x2+y2

2(lλdot)2 ln 2

, (1.16)

with lλdot =√

~ ln 2mλω

λdot

being the full width at half maximum of the probability density. This size can

be measured [MBM+03] or calculated by e.g. 8 band ~k · ~p calculations [SGB99; SWB07].To describe the next higher states, the first excited state of the harmonic oscillator is used:

ϕp = x√

2

(mλω

λdot

π~

) 12

e−mλω

λdot

2~ (x2+y2).

Those, so called, p states are two fold degenerated, one node in x direction, one in y direction,where x, y have to be chosen according to the crystal symmetry.

The wetting layer is described as one dimensional potential well in z direction, in the x-y-plane

8

CHAPTER 1. PROLOGUE

the states are described as plane waves:

φl = ϕl(x, y)ξl(z), l 6= 0, (1.17)

ϕl(x, y) = ei(~kxl x+~kyl y), (1.18)

ξl(z) =

√2Lz

cosπz/Lz, |z| ≤ Lz/2,0, else,

(1.19)

Eλl =~2(kxl

2 + kyl2)

2mλ+

~2π2

2mλL2z

+ EInGaAs. (1.20)

Lz denotes the effective width of the WL. Although multiple subbands with continuous energiesexist, we restrict ourselves to the lowest subband.

a)

E

z

GaA

s

InGaA

s

InAs

E0

EInGaAs

EInAs

Ei

b)

∆εc

∆εv

E

Ec0

Ev0

Γ ~kx,y

Egap

Figure 1.2: Energies of the QD–WL system (for a InAs/GaAs example) in space (a) and reciprocal space(b). The QD states have discrete energies, whereas the WL states are a continuum. The binding energy ∆Eis a characteristic quantity, especially for capture processes.

An important quantity for the phonon assisted relaxation (cf. Sec. 2.2) is the energetic distancebetween band edge of the WL and the QD energies, the binding energy: ∆ελ (cf. Fig. 1.2b). Withthe presented wave functions there is the following dependence

∆ελ = EλInGaAs − EλInAs +~2π2(L2/L2

z − 1)

2mλL2− ~2 ln 2

2mλ

(lλdot)2 . (1.21)

It depends on the size of the nanostructures as well as on the material parameters: band edges andeffective masses. For electrons, the binding energy is larger than for holes, due to the greater hole

9


mass. Further, smaller QDs have a smaller binding energy [SGB99] due to their larger ground stateenergy. However, since model wave functions are always an approximation, we will use the bindingenergy as a free parameter, with the dependencies above in mind.

For the calculation of optical spectra in Sec. 2.1 we will use an even simpler approach to theQD states, where the wave functions are Gaussian and isometric in all three spatial directions[WRR+08].

1.2.2 Phonon Hamiltonian

Having discussed the eigenstates of the electronic system we turn next to the quantized latticevibrations, the phonons. In solid state systems the dynamics of the lattice vibrations, the phonons,are of importance. The ions of the periodic lattice Vg(~r) =

∑n v(~r − ~Rn) (cf. Eq. (1.2.1)) are

in motion, described as coupled harmonic oscillators around the positions ~Rn = ~R(0)n + ~un. The

displacements ~un can be described via a continuous displacement field ~u(~r). To describe phononsthe system of oscillators is diagonalized leading to a system of uncoupled oscillators. The phononsare described by creation and annihilation operators b†~q, b~q with eigenenergies ~ω~q that depend onthe lattice symmetry. Figure 1.3 shows the dispersion relations of phonons in GaAs and AlN. Thereare two categories of branches with qualitatively different dispersions: acoustic and optical phonons.For acoustic phonons, ions in one unit cell oscillate in phase, which leads to a linear dispersionat the Γ point (q = 0) [Czy00]. For optical phonons the ions of an unit cell oscillate in oppositedirections leading to a non-vanishing dispersion at q = 0. The small displacements are describedvia phonon operators by the relation [Hak73; Czy00]

~u(R(0)n ) =

∑~q

~u~q(R(0)n ) =

1√N

∑~q

∑κ

√~

2Mnωκ(~q)~eκph(~q)ei~q·~R(0)

n (b~q,κ + b†−~q,κ), (1.22)

with Mn being the mass of the ions, N the number of unit cells, κ denotes the dispersion branches,ωκ(~q) is the dispersion relation of the respective modes. Finally, ~eκph(~q) denotes the unit vector ofthe phonon polarization.

The next important part of the Hamiltonian is the electron-phonon interaction. On the followingpages we want to recap the most important of the various interaction mechanisms.2

Deformation potential coupling

The deformation potential coupling directly takes into account the energetic changes of the electronicband structure for the displaced lattice. This short derivation follows Ref. [YC05]. The latticepotential is decomposed into the resting lattice and a correction from the displacement:

v(~r − ~R) = v(~r − ~R0 − ~u) ≈ v(~r − ~R0)−∇v(~r − ~R0) · ~u.

2A compilation of properties and derivations of those and other mechanisms can be found in Ref. [Hau70]

10

CHAPTER 1. PROLOGUEPh

onon

ener

gy[m

eV]

05

101520253035404550

Phon

onen

ergy

[meV

]0

102030405060708090

100110120

Figure 1.3: Dispersion relation of GaAs [GGP+91] and AlN [SBMP+99] and visualization of dispersionrelation approximations. Acoustic phonons start linearly at the Γ point (~q = 0) and in Debye approximation(not shown) they are linearly treated over the whole Brillouin zone. As an improvement we use a sinusoidaldescription as the approximation. Optical phonons have a constant value near the Γ point (~q = 0) and inEinstein approximation they are assumed to have a constant dispersion over the whole Brillouin zone. Weapproximate the dispersion by a cosine. The high symmetry points of the Brillouin zone are ~qΓ = (0, 0, 0), i.e.the center, the edges of the Zincblende Brillouin zone ~qK = 3π

2a (1, 1, 0), ~qX = 2πa (0, 1, 0), ~qL = πa (1, 1, 1)

and the edges of a Wurtzit Brillouin zone ~qK = 4π3a (1, 0, 0), ~qM = π√

3a(√

3, 1, 0), and ~qA = πc (0, 0, 1). The

Brillouin zone high symmetry points for various lattices can be found in [SC10]. For all approximations weuse an isotropic description, i.e. assuming the Brillouin zone to be spherical.

Since the lattice potential dictates the electronic energy, the interaction Hamiltonian is expressed interms of the electronic Hamiltonian.

He-ion =∂He

∂ ~R(0)~u ≈

∑λ′j

∑λi

⟨λ′j

∣∣∣∣ ∂Eλi∂ ~R(0)

~u

∣∣∣∣λi⟩ a†λ′jaλiThe energetic change δEλi with respect to the lattice displacement is typically accessed via a volumedilatation δV [YC05]

δEλi = DλiδV

V.

with the volume deformation potential Dλi.3 To relate this with the phonons, i.e. lattice displace-ment, linear deformation theory is used. The displacement field is described via a deformationmatrix dij :

ui(~R) =∑j

dijRj .

3Note that in the literature there is a, for non-experts confusing, amount of deformation potentials. For the couplingto LA phonons, typically the volume deformation potential is used.

11


For example dij = δxiδij would be stretching of the solid along the x-axis up to the double width.For a linear theory, i.e. dij = const., the definition can be inverted and divided into symmetric andasymmetric parts.

dij =∂ui∂Rj

= 0.5

[(∂ui∂Rj

+∂uj∂Ri

)+

(∂ui∂Rj

− ∂uj∂Ri

)]= sij + fij (1.23)

The strain tensor sij describes deformation, while the asymmetric fij describes rotations and doesnot change electronic energies. For small deformations the volume dilatation can be expressed asthe trace of the strain tensor:

δV

V= sxx + syy + szz + sxxsyy + syyszz + szzsxx + sxxsyyszz ≈ sxx + syy + szz.

The displacement field is expressed with phonon operators (cf. Eq. (1.22)):

sij =1√N

∑~q

0.5(qju0i + qiu

0j )∑κ

√~

2Mnωκ(~q)~eκph(~q)ei~q·

~R(0)n (b~q,κ + b†−~q,κ), (1.24)

∑i

sii =∑~q

~q · ~u(~q).

To be consistent and have a spatially constant strain tensor, this approximation is only valid inthe long wavelength limit ~q → 0. This yields the energy change per lattice displacement due todeformation potential coupling

δEλi = Dλi~q · ~u⇒∂Eλi

∂ ~R= Dλi~q,

Hel-ph∣∣DF = Dλi

∑~q

⟨λ′j∣∣~q · ~u∣∣λi⟩ a†λ′jaλi,

and thus the interaction Hamiltonian can be written formally

Hel-ph∣∣DF =

∑~q,κ

∑λij

g~q,κji a†λjaλi(b~q,κ + b†−~q,κ),

with the deformation potential coupling element:

g(~q)κ =

√~

2%V ωκ(~q)

(Dλ

∫d3r|ϕλ|2ei~q·~r

)~q · ~eκph(~q). (1.25)

Since ~q · ~eLA(~q) = q and ~q · ~eTA(~q) = 0 there is only coupling to longitudinal acoustic phonons.

12

CHAPTER 1. PROLOGUE

Piezoelectric coupling

In the case of piezoelectric coupling the displacement induced strain sij creates an electric field ~E.This effect is strong in polar material and is accounted by an electro-mechanical tensor emech. Theelectric displacement field from Maxwell’s equations reads[Mah72; YC05]

Dk =∑i,j

emechkij sji + ε0εrEk,

with the relative permittivity of the material εr (excluding piezoelectric effects) which is assumedto be isometric. This means a deformation of the i − j plane results in an electric field in the kdirection. εr is composed of the response from the electrons, which is commonly assumed to be thehigh frequency permittivity ε∞ and of the permittivity from the ions, which is the static permittivitywithout the electronic contribution εstat − ε∞ thus εr = εstat [Mah72].4,5

Zincblende

Wurtzite

Zincblende

Wurtzite

Wurtzite

Zincblende

Zincblende

Wurtzite

Figure 1.4: Visualization of the strain tensor and the piezoelectric fields created by deformation.

Using the source equation from Maxwell’s equations ∇ · ~D = 0 in Fourier domain yields therelation between the strain to a longitudinal electric field that is created:∑

k

qkDk = 0 =∑k

∑ij

qkemk,ijsij(~q) + ε0εr

∑k

qkEk(~q).

4Actually, one has to use the static relative permittivity without piezoelectric contributions.5Sometimes the literature states εr = ε∞ [YC05], which would neglect the influence of the ions on the permittivity.

13


To incorporate the energy into the Hamiltonian the induced electric field is taken into account viaits scalar potential φ:

Ek(~q) = −iqkφ(~q) ⇒ φ(~q) = i

∑k

∑ij qke

mk,ijsij(~q)

ε0εr∑

k q2k

, φ(~r) =∑~q

ei~q·~rφ(~q).

This potential leads to the electron-phonon interaction Hel-ph = −e0φ(~r). Symmetry propertiesallow the 3× 3× 3 tensor emk,ij to be described as a 3× 6 tensor as well as the strain tensor as avector with 6 elements.6

The phonon operators enter again via the strain tensor (cf. Eq. (1.24)):

Hel-ph = −|e0|∑mn

〈m|φ(~r)|n〉a†man

= −i∑~q

0.5|e|εε0q2

∑k

∑ij

qkemechk,ij qie

j~q

︸︷︷︸

Fκ(~q)

(~

2NV ρω

)1/2

〈m|ei~q·~r|n〉(b†−~q,κ + b~q,κ

)a†man.

The quantity Fκ(~q) does not depend on the value of ~q, but on the angle and is thus influenced bythe crystal symmetry. For a Zincblende system one number defines the piezoelectric interactione14 = e25 = e36, i.e. the induced electric field is transverse to the strain. The coupling in this casereads: ∑

k

∑ij

qkek,ijqiej~q

= 2(qxqyez + qxqze

y + qyqzex).

In a Wurtzit system there are three piezoelectric effects e31 = e32, e15 = e24, e33. Deformationof the c-plane (x− y plane ) leading to a field along the (0001) (i.e. z) direction; deformation alongthe (0001) direction leading also to a field along (0001); and an askew deformation leading to fieldsin the c-plane.

Polar coupling to optical phonons

Optical phonons, i.e. lattice vibrations where the ions in an unit cell do not oscillate in the samedirection, create polarization fields, that change the energy of the electronic states. The derivation issimilar to the piezoelectric coupling, because the polarization between the displaced ions creates anelectric field. In contrast to the coupling mechanisms discussed before, detailed derivations can befound in nearly every solid state text book (e.g. [Czy00; Hak73; YC05; Mah72]). The Hamiltonianreads

Hel-ph =∑m,n

∑~q

g~qmn

(b†−~q,LO + b~q,LO

)a†man,

6The convention is: 1 = xx, 2 = yy, 3 = zz, 4 = yz = zy, 5 = xz = zx, 6 = xy = yx and for the firstcoordinate 1 = x, 2 = y, 3 = z.

14

CHAPTER 1. PROLOGUE

with the Frohlich coupling element

g~qmn = −i

√~ωLO2V

·

√e2

ε0

(1

ε∞− 1

εstat

)1

q

⟨m∣∣∣ei~q·~r

∣∣∣n⟩ . (1.26)

A widespread formulation of this coupling element uses the Frohlich polaron constant α as amaterial parameter [Frö54]:7

g~qmn =

√1

ε04π

((~ωLO)3

2µ

) 14 √

4π~√α

1√V

1

q

⟨m∣∣∣ei~q·~r

∣∣∣n⟩ ; α = e2

√µ

2~3ωLO

(1

ε∞− 1

εstat

).

with the reduced mass µ = mhmemh+me

.

Form factor

All the phonon coupling Hamiltonians consist of the form factor⟨j∣∣ei~q·~r∣∣i⟩. For the QD states in

envelope function approximation (cf. Sec. 1.2.1) this means:⟨j∣∣∣ei~q·~r∣∣∣i⟩ =

∫d3rφ∗λjjφλiiu

λj0

∗uλi0 e

ei~q·~r

=

∫d3Rφ∗λjj(

~R)φλii(~R)ei~q·~R

∫Vez

d3ruλj0

∗(~r)uλi0 (~r)ei~q·~r︸︷︷︸

≈δλiλj

.

For long wavelength compared to the size of the unit cell, the phonon coupling is band diagonal.The form factor has typical Fourier transform properties, i.e. the smaller the extent of the wave

functions is, the higher are the wave numbers that contribute.

1.2.3 Electric fields, Maxwell’s equations, Wave equations

To investigate the dynamics of electromagnetic waves, Maxwell’s equations are always a goodstarting point.

∇ · ~E = %micr, ∇ · ~B = 0, (1.27a)

∇× ~B = ∂t ~E +~jmicr, ∇× ~E = −∂t ~B. (1.27b)

We start with the microscopic Maxwell’s equations, with %micr including all charges, i.e. electronsand atoms. However, assuming that in the ground state all charges are bound, we will only calculatethe microscopic dynamics of the electrons. The wave functions of the QD electrons reads:

Ψ(†)(~r) =∑i

∑l

ψ(∗)l (~r − ~ri)a(†)

i,l ,

7The original definition is in cgs units, the prefactor in red converts the expression to SI units.

15


with the states l of the ith quantum dot and a(†)i,l being the electronic annihilation (creation) operators

as introduced in Sec. 1.2.1.The electronic charge density of the electrons is defined by

%micr = −e0

⟨ψ†ψ

⟩= −e0

∑ij

∑kl

ϕ∗k(~r − ~rj)ϕl(~r − ~ri)⟨a†j,kai,l

⟩. (1.28)

The QD wave functions are assumed to have no overlap between different QDs, thus we introducea Kronecker delta δij . The overall charge is obtained by spatial integration and yields a result∼ δkl. In a neutral solid, those diagonal terms are canceled out by the charge of the ions thus thenon-diagonal contributions are left ∼ δkl = (1− δkl), because the wave functions are orthonormal.The residual part is used to define the macroscopic polarization with ∇ · ~P = %micr. Thus thisquantum mechanical charge expression has to be integrated.

~P (~r) = −e0

∑kl

∫ ~r

−∞d~r′∑i

ϕ∗k(~r′ − ~ri)ϕl(~r′ − ~ri)

⟨a†i,kai,l

⟩δkl

= −e0

∑kl

∑i

((~r − ~ri)ϕ∗k(~r − ~ri)ϕl(~r − ~ri)

−∫ r

−∞dr′(~r − ~ri)

∂

∂~rϕ∗k(~r

′ − ~ri)ϕl(~r′ − ~ri))⟨

a†i,kai,l

⟩δkl. (1.29)

The correction∇ϕ∗k(~r′ − ~ri)ϕl(~r′ − ~ri) is neglected, and therefore the polarization yields:

~P = −e∑kl

∑i

(~r − ~ri)ϕ∗k(~r − ~ri)ϕl(~r − ~ri)⟨a†kal

⟩(~ri).

Since the spatial dependence is microscopic and the QDs have an extent below the spatial resolutionof optical fields, thus the polarization of the whole QD is concentrated in one point and can beapproximated by a delta distribution.

−eϕ∗k(~r−~ri)ϕl(~r−~ri)(~r−~ri) ≈ ~dkl(~ri)δ(~r−~ri); ~dkl(~ri) = −e∫d~rϕ∗k(~r−~ri)ϕl(~r−~ri)(~r−~ri)

For practical reasons we pool N quantum dots in a volume V at position ~R together and assumethey have similar attributes and behave alike

~P (~R) =∑kl

~dkl(~R)⟨a†kal

⟩(~R)

∑~ri∈~R

δ(~r − ~ri)

︸︷︷︸=NV

=:nQD

,

16

CHAPTER 1. PROLOGUE

with the quantum dot density nQD.8 The microscopic current can be treated similar. From this weget the macroscopic Maxwell’s equations

∇ · ~D = %free, ∇ · ~B = 0, (1.30a)

∇× ~H = ∂t ~D +~jfree, ∇× ~E = −∂t ~B. (1.30b)

The material properties are included in

~D = ε0 ~E + ~P = ε ~E; ~P (ω) = ε0χ(ω) ~E(ω); ε = ε0 (1 + χ) ,

~H =1

µ0

~B − ~M =1

µ~B; ~M = χm ~H; µ = µ0 (1 + χm) .

We only consider the common case without free charges or currents %free = jfree = 0, whichyields the wave equation:

∆ ~E − 1

c2

d2

dt2~E = ∇(∇ · ~E) + µ0

d2

dt2~P . (1.31)

The part (∇ · ~E) = (∇ · ~P ) is neglected because we assume transversality of the electric field, thatis not spoiled by the polarization.

∆ ~E − 1

c2

d2

dt2~E = µ0

d2

dt2~P . (1.32)

This is the common wave equation for optical fields, which is able to describe the various phenomenadiscussed in this work. The spatial composition of ~P (~r) defines the optical modes of the system.The behavior of ~P in linear optics is presented in Sec. 2.1, a non-linear description is found inChap. 3.

Chirped pulses

In non-linear optics chirped pulses, i.e. pulses with a time dependent frequency can be of importance.Typically, we decompose an electric field E into a slow envelope and a carrier wave (cf. left panelof Fig. 1.5):

~E = ~Eeiωlt + ~E∗e−iωlt. (1.33)

However, this carrier frequency can time dependent (cf. middle and right upper panel of Fig. 1.5).In the most simple case this time dependence is linear:

ωl(t) = ωl(t0) + κ(t− t0). (1.34)

The time dependent part κ is called chirp and for rising frequency κ > 0 it is called upchirp,whereas a falling frequency κ < 0 is denoted downchirp. In this work we will make use of rotating

8The 1V

contribution stems from the discrete sum.

17


Time

field Eenvelope E

|E|Im(E)

|E|Im(E)

Figure 1.5: Left panel: temporal profile of an optical pulse is decomposed into carrier frequency and theslow envelope. Upper panels: a chirped pulse has a time dependent frequency, rising frequencies (left) arecalled upchirp; falling frequencies (right) downchirp. Lower panels: The absolute value and imaginary part(corresponds to phase) of the envelopes of the chirped pulses.

frames, where all quantities are rotated by the carrier frequency of the electric field. For chirpedpulses this rotating frame is done with the constant frequency ωl(t0), where t0 is the position ofthe maximum of the pulse. The lower panels of Fig. 1.5 show the resulting envelope of an up-and down-chirp. The chirp adds a complex behavior to the envelope E = |E|eiκt. The absolutevalue of the envelope is independent of the chirp, while the imaginary part represents the phaseIm(E) = |E| sin(κt) and is showing rising frequencies for positive and negative times. Upchirpand downchirp can be discriminated by the sign of the phase. We will handle with chirped pulsesin Chapters 3 and 4.

Electric field Hamiltonian

The electric field enters also in the Hamiltonian for the interaction with electrons. The derivation canbe thoroughly done via the combined Maxwell–Schrodinger Lagrangian for the electro-magneticpotentials [Mil10]. From there to the widely used electric field coupling Hamiltonian that resemblesthe energy of a dipole in an electric field [HK04]:

Hel-field =∑kl

~dkl · ~Ea†kal. (1.35)

The dipole coupling element is defined in the same way as above.This approach is called semi-classical, since the electric field is not quantized. Quantum optics

with quantum dots [Jah12] is very interesting field of research, but out of scope of this work. Forthe description of the excitation of a quantum system the semi-classical is mostly sufficient.

18

CHAPTER 1. PROLOGUE

1.3 Interactions – full systems’ Hamiltonian and equations of motion

The full Hamiltonian of the system consists of

H = Hel →∑i

εia†iai electronic eigenenergies, cf. Sec. 1.2.1

+Hph →∑~q,κ

~ω~q,κb†~q,κb~q,κ phonon eigenenergies, i.e. phonon dispersion

+Hel-ph →∑ij

∑~q,κ

g~q,κij a†iaj

(b~q,κ + b†−~q,κ

)electron-phonon interaction

+Hel-light. →∑ij

~dij · ~Ea†iaj semi-classical interaction with an optical field

Using this Hamiltonian, dynamical equations for the quantum mechanical values can be derived viaHeisenberg equations of motion.

i~∂tO = [O,H]. (1.36)

1.3.1 Electron–light system

For the electron light system H = Hel + Hel-light the Heisenberg equation of motion approachresults in the Bloch equations

i~∂

∂ta†iaj = (εj − εi)

~ωji

a†iaj +∑k

~E · (~djka†iak − ~dkia†kaj), (1.37)

which is a closed system of coupled equations. In our scope, dij is non-diagonal. For two levelsystems, polarizations (i 6= j) couple to densities (i = j) via the electric field and vice versa. Thoseequations are deeply discussed in Chap. 3. In Chap. 2 we will also make use of them in a linearresponse theory.

1.3.2 Electron–phonon system

The equations for the electron-phonon system H = Hel +Hel-ph +Hph are rather similar to theBloch equations on first sight, but due to the quantized nature of the phonons, the hierarchy problemoccurs:

i~∂ta†iaj = ~ωjia†iaj +∑k

(g~q,κjk a†iak − g

~q,κki a

†kaj)

(b~q,κ + b†−~q,κ

). (1.38)

19

1.3. INTERACTIONS – FULL SYSTEMS’ HAMILTONIAN AND EQUATIONS OF MOTION

The equation couples to phonon assisted operators of the kind a†kajb(†)−~q,κ whose dynamics would

have to be calculated and couple to even higher order terms.

i~∂t(a†iajb~q,κ) = (~ωji+~ω~q,κ)a†iajb~q,κ+∑k,κ′

(g~q′,κ′

jk a†iak−g~q′,κ′

ki a†kaj)(b~q′,κ′b~q,κ + b†−~q′,κ′b~q,κ

)+∑kl

g−~q,κlk

(a†iakδlj − a

†ia†l ajak

). (1.39)

That means the dynamics are described by an infinite system of coupled equations. The hierar-chy can be broken by factorization of higher order terms, thus neglecting correlations. Typicalapproximations are

• the one electron assumption, where a†ia†kajal vanishes exactly; An alternative treatment of the

term is the Hartree-Fock factorization:⟨a†ia†kajal

⟩=⟨a†ial

⟩⟨a†kaj

⟩−⟨a†iaj

⟩⟨a†kal

⟩,

which is a mean field approach.

• factorization of electrons and phonons⟨a†kajb

†−~q′,κ′b~q,κ

⟩=⟨a†kaj

⟩⟨b†−~q′,κ′b~q,κ

⟩• bath approximation for the phonons

⟨b†−~q′,κ′b~q,κ

⟩=⟨b†~q,κb~q,κ

⟩δ−~q′~qδκ,κ′ ,

⟨b~q,κb~q,κ

⟩= 0.

We will now use those approximations to derive a rate equation for the densities in second order Born-Markovian approximation. The dynamics equation is written with the definitions ρij =

⟨a†iaj

⟩and n~q,κ =

⟨b†~q,κb~q,κ

⟩as

i~∂t⟨a†iajb~q,κ

⟩ei(ωji+ω~q,κ)t =

∑k

(g−~q,κ

′

jk ρik(n~q,κ + 1

)− g−~q,κ

′

ki ρkjn~q,κ

−∑l

g−~q,κ′

lk (ρikρlj + ρijρlk)

)ei(ωji+ω~q,κ)t, (1.40)

which can be formally solved. To describe relaxation phenomena, it is convenient to neglectpolarizations (non-diagonal terms) on the right side, since they represent spatial inhomogeneities.

i~⟨a†iajb~q,κ

⟩=

∫ t

−∞dt′(g−~q,κ

′

ji ρii(n~q,κ + 1

)− g−~q,κ

′

ji ρjjn~q,κ

)e−i(ωji+ω~q,κ)(t−t′)

≈ g−~q,κ′

ji

(ρii(n~q,κ + 1

)− ρjj − ρjjn~q,κ

) ∫ t

−∞dt′e−i(ωji+ω~q,κ)(t−t′)︸︷︷︸

πδ(ωji+ω~q,κ)+iP 1ωji+ω~q,κ

(1.41)

20

CHAPTER 1. PROLOGUE

Inserting this into Eq. (1.38) yields a closed equation, e.g. for the occupation of level i:

i~∂tρii =∑~q,κ

∑l

Im(g~q,κil

⟨a†ialb~q,κ

⟩)−

∑~q,κ

∑l

Im(g~q,κli

⟨a†l aib~q,κ

⟩)

= − iπ

~∑~q,κ

∑l

|g~q,κil |2(ρii(1− ρll)

(n~q,κ + 1

)− ρll(1− ρii)n~q,κ

)δ(ωli + ω~q,κ)

+iπ

~∑~q,κ

∑lk

|g~q,κli |2(ρll(1− ρii)

(n~q,κ + 1

)− ρii(1− ρll)n~q,κ

)δ(ωil + ω~q,κ). (1.42)

This rate equation is the Fermi golden rule. Both lines consist of an out-scattering part (∼ −ρii)and an in-scattering part (∼ ρll). The many electron contributions create blocking terms (1− ρ)that inhibit scattering into occupied states. The delta distributions make the scattering processesenergy conserving. The upper line can only be non-zero for ωli < 0, i.e. εi > εl, correspondinglythe out-scattering from i is proportional to phonon emission (the emission term can be identifiedby the +1 due to spontaneous emission). The lower line represents cases, where εl > εi, thus thein-scattering is related to the phonon emission.

In this work we will deal with the independent boson model (IBM) in Sec. 2.1.1, with onlydiagonal coupling to phonons g~q,κki ∼ δki, which has an exact solution for the polarization in linearoptics, i.e. every order of the many particle problem is included in the solution. In Sec. 2.2 wederive an effective higher order electron-phonon Hamiltonian where the non-diagonal couplingplays the major role, and the hierarchy can be interrupted sooner.

21

1.3. INTERACTIONS – FULL SYSTEMS’ HAMILTONIAN AND EQUATIONS OF MOTION

22

Neither a borrower nor a lender be; For loan oft loses both itself and friendWilliam Shakespeare, Hamlet

2Phonon influence on carrier dynamics in QDs

In this work the optical response of QDs plays a major role. This response is determined by themicroscopic dynamics of the carriers in the system. In the semiconductor environment phonons,the quantized lattice vibrations, have a significant influence on the carrier dynamics. This chapterdescribes the impact of electron-phonon interaction on the dynamics and resulting effects. Onone hand interaction with phonons yields decoherence, i.e. the loss of microscopic transitionsprobabilities, which can be directly seen in optical spectra. Section 2.1 discusses how luminescenceor absorption spectra are influenced by the electron-phonon interaction and discusses the impactof phonon modes, carrier wave functions and phonon coupling strength. Here, calculations canbe easily compared to experiments. On the other hand the phonon system takes excess energy ofrelaxing carriers (or gives energy to excite carriers in the opposite process). Section 2.2 discusseshow carriers from the reservoir are captured, i.e. relax into the QD states, via interaction withoptical phonons. For that aim we present an effective multi-phonon Hamiltonian derived via aprojection operator technique. This capture process and the dependence on temperature and carrierreservoir density are described.

2.1 Linear absorption and luminescence spectra

Optical spectra are a traditional and versatile way to obtain information about the structure andproperties of matter. One of the most simple approaches are absorption spectra. They are commonlydescribed by the linear1 response of matter to optical light fields defined via the absorption coefficient

1In general optical response is a non-linear function of the optical field. Typically, weak probing fields are used, thatcan be approximated to not excite the system and thus yields only a linear response.

2.1. LINEAR ABSORPTION AND LUMINESCENCE SPECTRA

α(ω) from Beer-Lambert law |E(z, ω)|2 = |E(0, ω)|2e−α(ω)z . We consider an optical fieldE(ω, z) = E0(ω)eikz−α

2z traveling through a material with linear response P (ω) = ε0χ(ω)E(ω).

The typical wave equation Eq. (1.32) in frequency domain reads

∆E(ω) +ω2

c2E(ω) = − ω2

ε0c2P (ω) = −ω

2

c2χ(ω)E(ω).

By inserting the ansatz mentioned above, the wave number k as well as for the absorption α arerelated to the susceptibility χ:

0 = (ik − α

2)2 +

ω2

c2(1 + χ(ω)) .

Expressions for k and α are derived via splitting the equation into real and imaginary part

α(ω) =ω2

c2Im(χ(ω))

1

k(ω), k2 = +

α2

4+ω2

c2(1 + Re(χ(ω))) ,

⇒ α2 = 2ω2

c2(1 + Re(χ))

[√1 +

Im(χ)2

(1 + Re(χ))2− 1

]. (2.1)

To understand this further it is convenient to investigate the typical behavior of susceptibilities. InSec. 1.2.3 we showed the polarization is composed of all microscopic transition probabilities.

ε0χ(ω)E(ω) = P (ω) =∑ij

dijρij(ω) + c.c. (2.2)

The most simple approximation for semiconductors treat every transition as independent two levelsystem and investigate their linear response. Here, indirect transitions are neglected, i.e. every pointin the band structure is a free two level system with the dynamics in frequency domain (assumingconstant occupations):

−iωρij(ω) = −i(ωi − ωj − iγ)ρij(ω) + idijE(ω)

~(ρjj − ρii), (2.3)

P =∑ij

|dij |2(ρjj − ρii)~

(ω − (ωj − ωi)− iγ

(ω − (ωj − ωi))2 + γ2− ω + (ωj − ωi)− iγ

(ω + (ωj − ωi))2 + γ2

)︸︷︷︸

χ(ω)

E.

(2.4)

The imaginary part contains Lorentzian peaks at the systems resonances, while the real part hasdiscontinuities and decays only slowly. The dipole elements determine the strength and include thedensity of states. ρjj − ρii is the inversion and decides whether the response absorbs or enhancesincoming light. Figure 2.1 shows the susceptibilities of a two level system as well as of the GaAsband structure, both in the ground state.

24

CHAPTER 2. PHONON INFLUENCE ON CARRIER DYNAMICS IN QDS

0

1 1.2 1.4 1.6 1.8 2Energy [eV]

Re(χ)

Im(χ)

(a) Two level system

-15-10-505

1015202530

0 1 2 3 4 5 6 7Energy [eV]

band gapRe(χ)

Im(χ)

(b) GaAs

Figure 2.1: Susceptibilities of a two level system and of the semiconductor GaAs (data from Ref. [Ref]) Thereal part represents the refractive index and describes the scattering of light, whereas the imaginary partdetermines absorption/gain. For a bulk system the susceptibilities add up, since the real part decreases onlyslowly away from the resonance there is a significant contribution below the band gap.

Our primary interest is the response from QDs, with transition frequencies below the band gapof the host bulk semiconductor. Thus, the response is divided χ = χactive + χbg. The imaginarypart of the response of the background material is off-resonant, when investigating the activematerial, and does not contribute, whereas the real part response of the background almost alwaysgives a contribution and is larger then real part response of the smaller active system. Hence,Im(χ) ≈ Im(χactive) and Re(χ) ≈ Re(χbg). Typically, (1 + Re(χbg)) Im(χactive), then fromEq. (2.1) follows the absorption coefficient can be calculated via

α(ω) ≈ ω

cIm(χactive(ω))

1√1 + Re(χbg(ω))

. (2.5)

The wave number of the traveling light field is independent of absorption contributions:2

k(ω) ≈ ω

c

√1 + Re(χbg(ω)). (2.6)

The linear optical response to an electric field is derived from the polarization χ(ω) = P (ω)E(ω) , which

is obtained from the dynamics of the microscopic polarization.The temporal behavior of the polarization does not only have a signature in absorption spectra,

but also in luminescence spectra. Generally, to describe luminescence one needs a quantum optical2The wave number describes the refractive behavior of the material. The optical properties are often described in

terms of the refractive index and the wave length. For comparison with the literature we list here some useful relations:

λ =2πc

ω;

∂λ

∂ω= −λ

ω; k(ω) =

ω

cn(ω).

25


treatment, since essentially spontaneous emission dominates this effect. The luminescence spectrumis [SZ97]:

S(r, ω) =1

πRe

∫ ∞0

dτ⟨E(−)(r, t)E(+)(r, t+ τ)

⟩eiωτ , (2.7)

with the quantum optical electric field operators E(+), E(−), that describe the emission. Thespectrum is the inverse Fourier transform with respect to τ . The expectation value from the fieldoperators relates to the dynamics of the microscopic polarizations⟨

E(−)(r, t)E(+)(r, t+ τ)⟩

= I0 〈σ+(t)σ−(t+ τ)〉 . (2.8)

The polarization σ+ = a†jai defines the initial condition at t and the evolution of σ− = a†iaj isresponsible for the spectrum. For a two level system, that is prepared in the pure state |2〉〈2| thiswould mean the spectrum is defined by the temporal evolution of σ− = a†2a1. As we will see,absorption and luminescence spectra show the same behavior, but with mirrored frequency.

Next, we will show absorption/luminescence spectra of a QD system with phonon influence bycalculating the polarization dynamics under the influence of a phonon bath.

2.1.1 Independent Boson model

QDs interacting with phonons are commonly modeled by the so called “Independent BosonModel”(IBM), which describes the polarization of an electronic two level system without change ofthe occupation under interaction with a bath of phonons [Mah90]. Within this approximations themodel has an exact solution. The model is widely used for systems with discrete states like QDs (asin this work) [KAK02; KAK+05; Web08] or molecules [RAK+06]. This section briefly derives thewell known solution of the IBM,3 while Sec. 2.1.2 shows the properties of the phonon interactionin QDs with the help of the IBM solutions and Sec. 2.1.3 applies this to real systems and comparesto experimental data.

Start with the Hamiltonian from Sec. 1.3:

H =∑i

εia†iai +

∑q

~ωqb†qbq +∑ij

Vija†iaj , (2.9)

Vij =∑q

gqij(bq + b†−q). (2.10)

The Heisenberg equation of motion approach yields the dynamics equation for the microscopicpolarization:

i~∂ta†kal = (εl − εk)a†kal +∑i

(Vlia

†kai − Vika

†ial

)(2.11)

3The derivation of the IBM solution can be done in a more general way as presented here using propagators [Muk95],however we restrict ourselves to a simple derivation stressing the most important assumptions.

26


The equation can be formally solved by integration of the polarization in a rotating frame pkl(t) =

a†kal(t)ei(ωl−ωk)t.

pkl = pkl(0)− i

~

∫ t

0dt1∑i

(Vli(t1)ei(ωl−ωi)t1pki(t1)− Vik(t1)ei(ωi−ωk)t1pil(t1)

)(2.12)

Here, we restrict ourselves to diagonal coupling Vij ∼ δij . Further, to take into account initialconditions later on, we have to investigate the effect on the statistical operator ρ.

pkl(t)ρ = pkl(0)ρ− i

~

∫ t

0dt1 (Vll(t1)− Vkk(t1)) pkl(t1)ρ. (2.13)

The equations for all polarizations are decoupled. We define the effective coupling element Vlk =Vll − Vkk which is composed of the diagonal coupling elements of the interacting states. Thedynamic equation for the polarization can be formally solved via a Dyson series, which means theiterative replacement of pkl on the right side with the formal solution. This results in a sum overevery order of V . Since the phonons are assumed to be a thermalized bath, odd orders of V vanish.

〈pkl〉 (t) = 〈p(0)〉∞∑n=0

(− i

~

)n ∫ t

0dt1

∫ t1

0dt2 · · ·

∫ t2n−1

0dt2n〈Vlk(t1)Vlk(t2) . . . Vlk(t2n)〉

= 〈pkl(0)〉∞∑n=0

p(n)kl (t) (2.14)

For this simple problem with diagonal coupling only, we can use the solid state variant of Wick’stheorem [Mah90], where

p(n) =1

n!

(p(1))n.

In conclusion, using Wick’s theorem leads exactly to p(t) = p(0)ep(1)(t). For a more complex

problem, one would use the cumulant expansion [Ric07].To get grasp of the experimentally accessible absorption, the polarization and the exciting light

field have to be processed χ(ω) = P (ω)E(ω) . For practical use the interaction of electrons with the light

field on the one hand and all other interactions (in this work: phonons) on the other are temporallyseparated [Web08]. In order to do so the light field is modeled as a delta peak E(t) = E0δ(t)preparing the electronic system, before the other interactions kick in at t = 0.4

p(t ≤ 0) = iE0θ(t), p(t > 0) = p(0)ep(1)(t). (2.15)

This leads to the following expression for the macroscopic polarization:

P (t) =∑ij

dijTr(a†iajρ

)= d12e

i(ω2−ω1)tp(0)ep(1)(t) = iE0d12e

i(ω2−ω1)tep(1)(t).

4This starting condition can be expressed in a more elegant way using a description based on propagators.

27


Thus, the absorption can be calculated from the polarization dynamics via

α(ω) ∼ ω

cRe( F(ep

(1)(t))︸︷︷︸=f((ω2−ω1)−ω)

).

In the same way the luminescence is described by

S(ω) ∼ Re( F−1(ep(1)(t)︸︷︷︸

=f((ω2−ω1)+ω)

)),

according to Eq. (2.7). We see that luminescence and absorption have nearly the same behavior,but with mirrored frequencies around the transition energies ~(ω2 − ω1).

For the first order term one needs to evaluate:

〈V (t1)V (t2)〉 =∑qq′

g~q,κg~q′,κ⟨

(b~q(t1) + b†−~q(t1))(b~q′(t2) + b†−~q′(t2))⟩. (2.16)

In order two work with the two-time operators we need their temporal evolution. The dynamics canbe derived again via Heisenberg equation of motion. Here it is important to note, that the evolutionequation for the phonon operators is fed by electronic occupations, which leads to the problem, thateven if the system is in its ground state, the phonon dynamics would be driven. This problem isovercome by the Weyl transformation [Web08] or using the electron-hole picture. However, we areusing the bath approximation for the phonons via assuming the coupling in the phonon equations ofmotion vanishes anyways.

i~∂tb~q,κ = ~ω~q,κb~q,κb~q,κ ⇒ b~q,κ(t) = b~q,κ(0)e−iω~q,κt, b†~q,κ(t) = b†~q,κ(0)eiω~q,κt. (2.17)

Additionally we insert a phenomenological decay γ of the phonon dynamics.5 Temperature depen-dent decay rates can be found for example in Refs. [BKS94; RRB06]. The dynamics of the phononsis inserted and the time integral can be solved:

〈V (t1)V (t2)〉 ≈∑q

|g~q,κ|2⟨b~q,κb

†~q,κ

⟩e(−iω~q,κ−γ)(t1−t2) +

⟨b†~q,κb~q,κ

⟩e(iω~q,κ−γ)(t1−t2)

.

(2.18)

The temporal integrations now include only exponential functions and are thus easy to solve:∫ t

0dt1

∫ t1

0dt2e

(±iω~q,κ−γ)(t1−t2) = −∫ t

0dt1

1

±iω~q,κ − γ(1− e(±iω~q,κ−γ)t1)

= − t

±iω~q,κ − γ+

1

(±iω~q,κ − γ)2(e(±iω~q,κ−γ)t − 1). (2.19)

5This decay represents phonon-phonon interaction, where a phonon decays into phonons of other modes. How suchinfluence is incorporated microscopically can be found in [SZ97] as “Heisenberg-Langevin” approach.

28


This yields the solutions for the cumulant of the phonon dynamics

p(1)(t) = −∑q

|gq|2〈bqb†q〉

[t

iωq + γ+

1

(iωq + γ)2(e−iωqt−γt − 1)

]+〈b†qbq〉

[− t

iωq − γ+

1

(iωq − γ)2(eiωqt−γt − 1)

]. (2.20)

The solution can be decomposed into various parts with different meaning:

p(t) = p(0) exp i(−ωg −∆1)t−∆2t+R(t)− S1 − iS2 (2.21)

with the two contributions linear to time

∆1 = −∑~qκ

|g~qκ|2

~2

ω~qκω2~qκ + γ2

~qκ

Polaron shift, (2.22a)

∆2 = −∑~qκ

|g~qκ|2

~2

γ~qκω2~qκ + γ2

~qκ

(2n~qκ + 1) Zero-phonon line broadening. (2.22b)

Those terms shift the spectrum (a) or broaden the else delta shaped zero phonon line temperaturedependent (b), respectively. However the latter term is absent, if the phonon decay γ is not takeninto account. The next term is the only one with a complex temporal behavior and depending onthe phonon occupation and thus responsible for the shape of phonon side bands:

R(t) = −∑~qκ

|g~qκ|2

~2

[(n~qκ + 1)

(iω~qκ + γ~qκ)2e−(iω~qκ+γ~qκ)t +

n~qκ(iω~qκ − γ~qκ)2

e(iω~qκ−γ~qκ)t

]. (2.22c)

The part proportional to n~qκ + 1 represents phonon emission, while the other term corresponds tophonon absorption. Its meaning gets clearer by investigation of its role in the polarization, for thatthe exponential is expanded into a series [MK00]:

p(ω − ωg −∆1) ∼ F(eR(t)

)=∞∑n=0

1

n!F (R(t)n) .

The counter n has the meaning of the number of interactions with the phonon system. This can benicely seen for optical phonons in Einstein approximation ω~q,LO = ω0,LO in the zero temperaturelimit n~qκ = 0:

R(t)n|LO = R(0)ne−(inω~qκ+nγ~qκ)t,

F(R(t)ne−∆2t

)= R(0)n

1√2π

i(ω − nω~qκ

)+ ∆2 + nγ~qκ

(ω − nω~qκ)2 + (∆2 + nγ~qκ)2.

29


The spectrum in this case consists of Lorentzian lines (or delta shaped lines for the γ → 0 case,which is consistent with zero temperature) with the distance of an LO phonon energy, starting withthe zero phonon line at ωg + ∆1. For the γ → 0 case they have a size ratio of

p(nωLO)

p(0)=

1

n!R(0)n; T = 0 K,

which means they are Poisson distributed with the mean value of R(0), i.e. the highest line in thespectrum is the order n closest toR(0). This so called Huang-Rhys factorR(0) [HR50; MK00] is ameasure for the coupling strength and easily accessible in experiments. The last terms of Eq. (2.21)represent the Huang-Rhys factor R(t = 0) = S1 + iS2 and ensure the correct starting condition.

S1 =∑~qκ

|g~qκ|2

~2

ω2~qκ − γ

2~qκ

(ω2~qκ + γ2

~qκ)2(2n~qκ + 1) Huang-Rhys factor, (2.22d)

S2 =∑~qκ

|g~qκ|2

~2

2ω~qκγ~qκ(ω2~qκ + γ2

~qκ)2. (2.22e)

Often the IBM solution is presented/used without the phonon decay rate γ.

p(1)(t) =∑q

|gq|2

~2ω2q

(nq + 1)

(e−iωqt − 1

)+ nq

(eiωqt − 1

)+ iωqt

. (2.23)

Especially for this case it is convenient to define a spectral density [Muk95; RAK+06]: 6

Jκ(ω) =∑~q

|g~qκ|2

~2ω2δ(ω − ωκ(~q)) =

V

23π3

∫d~q|g~qκ|2

~2ω2δ(ω − ωκ(~q)). (2.24)

This quantity links the coupling strength for every phonon wave vector ~q to a frequency in thespectrum depending on the phonon mode’s dispersion relation. The spectral density contains allinformation but the temperature (respective the phonon occupation). In the T = 0 K limit R(t) canbe directly expressed as the Fourier transform of the spectral density:

R(t) =∑κ

∫dω Jκ(ω)e−(iω)t = F−1(J(−ω)). (2.25)

That means in the low temperature limit the spectrum is defined by the spectral density.

p(ω) ∼ F(ei(−ωg−∆1)t−S1eF

−1(f(ω)))

= e−S1F(eF−1(Jκ(ω−ωg−∆1))

)ω′=ω−ωg−∆1

= e−S1F

(∑i

1

i!(F−1(Jκ(ω′)))i

)≈ e−S1δ(ω′) + e−S1Jκ(ω′). (2.26)

6Note, that in the literature the power of ω in the definition can vary.

30


The absorption spectrum is approximative the delta like zero phonon line plus the spectral densityfrom the phonons (one-phonon contribution). For optical phonons that means again JLO(ω) =S1δ(ω − ωLO). For acoustic phonons it is a continuous band showing the coupling strengthdepending on the phonon energy. We will dive deeper into the shape and meaning of the spectraldensity on the next pages.

In the high temperature limit nqκ 1 the spectrum becomes symmetric around the zero phononline p(ω′) = p(−ω′).

2.1.2 Diagonal coupling due to various interaction mechanisms

In this part we will investigate the spectral density (Eq. (2.24)) caused by deformation potential andpiezoelectric coupling to acoustic phonons, as well as polar coupling to optical phonons. Thosecoupling mechanisms are introduced in Sec. 1.2.2. One major influence to the spectral density arethe QD’s wave functions. Here, we choose Gaussian QD model wave functions

ϕv(~r) =

√1

a2vπ

3

e− (~r−~r0)2

2a2v , ϕc(~r) =

√1

a2cπ

3

e− ~r2

2a2c . (2.27)

Where av(ac) denote the extent of the valence band (conduction band) wave functions. We introducethe distance of the centers of mass of valence and conduction band wave functions ~r0 and thusdescribes a dipole moment of the QD. Those model wave functions can be further generalized toan anisotropic form. The advantage of the chosen simple wave functions is the possible analyticaltreatment which displays fundamental properties of the coupling.

Deformation potential coupling of acoustic phonons According to Sec. 1.2.2 the effectivedeformation potential coupling element for Gaussian wave functions reads

g(~q)LA =

√~q2

2%V ωLA(~q)

(Dve

−a2v~q

2

4 ei~q·~r0 −Dce−a

2c~q

2

4

). (2.28)

For the spectral density (Eq. (2.24)) we need the absolute square of this function

|g(|~q|)LA|2DF :=~q2

2%V ωLA(~q)

(D2ve−a

2v~q

2

2 +D2ce−a

2c~q

2

2 − 2DvDce− (a2

c+a2v)~q2

4 cos(|~q||~r0| cos θq)

).

(2.29)

Here, we can see that the coupling is induced by the energy change due to deformation in valenceand conduction band and a mixed term that depends on the distance of the localized conductionand valence electron density. The cos θ describes, the part of the phonon wave that travels betweenelectron and hole wave function. For long wavelength the difference to the case ~r0 = 0 is notsignificant. However, for small QDs larger wave numbers q are important. This is due to theFourier character of the coupling element. Later on we will assume isotropic dispersion relations

31


ωLA(~q) = ωLA(q) and do not investigate the angular dependence of phonons or light, thus we areonly interested in the angular average of this quantity:

|g(q)LA|2DF :=1

4π

∫ 2π

0dφ

∫ π

0dθ sin θ|g(~q)LA|2

=~

2%V

q

ωLA(q)q

(D2ve−a

2v~q

2

2 +D2ce−a

2c~q

2

2 − 2DvDce− (a2

c+a2v)~q2

4sin q|~r0|q|~r0|

).

This already contains the most important properties of the typical phonon interaction strength.

0

4.0× 10−5

8.0× 10−5

1.2× 10−4

1.6× 10−4

2.0× 10−4

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]

ac = 2.5, av = 2.0ac = 5.0, av = 4.0ac = 10.0, av = 8.0ac = 20.0, av = 16.0

0

2.0× 10−5

4.0× 10−5

6.0× 10−5

8.0× 10−5

1.0× 10−4

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]

ac = 5.0, r0 = 0.0ac = 5.0, r0 = 1.5ac = 5.0, r0 = 3.0ac = 5.0, r0 = 6.0ac = 5.0, r0 = 12.0

Figure 2.2: Electron LA phonon interaction strength via deformation potential coupling. Left panel:influence of the wave function size on the coupling strength. Small QDs have a large extent in the q domain;due to the prefactor linear in q such couplings have a higher interaction strength. The right panel shows theinfluence of a built in dipole due to a distance between the wave functions, which has a significant influencefor q ≈ π

r0.

In Debye approximation ωLA(q) = cLAq the q dependence of the prefactor is linear. Without lossof generality, assume av ≤ ac (for the other case the indices c, v have to be switched). The qdependence can be written as

|g(q)LA|2DF ∼ D2vqe−a

2v2q2

(1− Dc

Dve−

a2c−a

2v

4q2

)2

+ 2Dc

Dvqe−

a2c4q2

(1− sin qr0

qr0

)The behavior for different QD wave function sizes and distances between the wave functions isshow in Fig. 2.2. In general on has the properties:

• For q = 0 (displacement of the whole solid) the coupling vanishes

• For large q the coupling decays as qe−a2

2q2

, a = min(av, ac), i.e. the smallest parts of thewave functions define the short wave length behavior.

32


• For medium q there is one or two maxima (in the case r0 = 0) depending on the size ofwave functions and deformation potential in valence and conduction band. The two maximacase (as presented in Ref. [Web08]) can occur if the deformation potentials have the samesign sgn(Dv) = sgn(Dc), av < ac, and |Dc| > |Dv| (or av > ac and |Dc| < |Dv|). Forequally sized wave functions a = av = ac the maximum is at qmax = 1

a . For unequal sizesthis maximum shifts towards zero. The size of the maximum |g(qmax)|2 is proportional to 1

a ,i.e. smaller QDs have a larger phonon interaction.

• A distance r0 between the wave functions introduces a correction, that can lead to additionaloscillations in a range dictated by the extent of the larger wave function. The influence ofthe correction is large for q ≈ π

r0< 1

a , which means the correction is important if the wavefunctions are strongly confined and separated, which can occur in a polar material.

Piezoelectric coupling to acoustic phonons The other interaction mechanism with acousticphonons covered in this work is the piezoelectric coupling, cf. Sec. 1.2.2.

gκ(~q) =

√~

2%V ωκ(~q)

(e−

a2v~q

2

4 ei~q·~r0 − e−a2c~q

2

4

)Fκ(~q), (2.30)

|gκ(~q)|2PZ =~F2

κ(~q)

2%V ωκ(~q)

(e−

a2v~q

2

2 + e−a2c~q

2

2 − 2e−(a2v+a2

c)~q2

4 cos(q|~r0| cos θ~q)

). (2.31)

Here, the coupling strength is defined by the piezoelectric factors Fκ, which depend on the orienta-tion but not on the absolute value of ~q. This angular dependence is different for different latticesymmetries, thus the typical lattice symmetries Zincblende and Wurtzit are two independent cases.An angular dependence of the part containing the wave function is introduced by a non-vanishing~r0 (for the chosen spherical wave functions). Thus, the coupling can be divided in a part where thewave function influence is angular independent and the angular dependent part.

|gκ(q)|2PZ∣∣isometric =

~F2κ

2%V

q

ωκ(q)

1

q

(e−

a2vq

2

2 + e−a2cq

2

2 − 2e−(a2v+a2

c)q2

4

), (2.32)

|gκ(~q)|2PZ∣∣angular =

~F2κ(~q)

2%V

q

ωκ(q)

1

q2e−

(a2v+a2

c)~q2

4(1− cos(q|~r0| cos θ~q)

). (2.33)

F2κ denotes the angular average of the piezoelectric coupling factor. First, we investigate the part of

the coupling where only the piezoelectric factors carry the angular dependence (corresponding tothe r0 = 0 case). The angular average will later on turn them into numbers that define the couplingstrength to the different modes. The QD size dependence of the piezoelectric coupling is shown inFig. 2.3. Here, the following general properties are valid:

• For equal sizes of the wave functions, the coupling vanishes

• For q = 0 the coupling vanishes

33


0

6.0× 10−5

1.2× 10−4

1.8× 10−4

2.4× 10−4

3.0× 10−4

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]


0

1.2× 10−3

2.4× 10−3

3.6× 10−3

4.8× 10−3

6.0× 10−3

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]


Figure 2.3: Piezoelectric coupling strength for different QD wave functions. Left panel: wave functionswith a larger spatial extent, have a maximum at small q and decay fast. Due to the 1/q prefactor, smallwave functions have a small coupling strength. The right panel shows the influence of the dipole moment(distance between electron and hole center of mass) on the coupling strength. A dipole moment enhances thepiezoelectric phonon coupling and can lead to oscillations for high q.

• For large q the coupling decays as 1q e−a

4q2

, a = min(av, ac), i.e. the smallest parts of thewave functions define the short wave length behavior.

• The coupling has one maximum between zero and 2√

ln(a2c/a

2v)

a2c−a2

v(which is where the maximum

of the terms in brackets without 1/q dependence occurs

• For a global scaling av ∼ ascale, ac ∼ ascale the position of the maximum has a dependenceqmax ∼ 1

ascale, i.e. for smaller QDs the maximum shifts to larger q. Subsequently the value of

the maximum becomes smaller for smaller QDs, |g(qmax)|2 ∼ ascale.

The angular dependent part, which is introduced by a non-vanishing dipole moment r0, is rathercomplicated, first the growth direction (defining the direction of the dipole) of the QD in respectto the lattice symmetry has to be taken into account, second the result is different for every modeand every crystal symmetry, then the result of angular averaging are composed from products ofpolynomials times sine and cosine functions of qr0. The results are shown in App. A. Since it isnot as easy to show general properties with those analytical results, we want to show numericallythe typical behavior in Fig. 2.3b. Generally the term can enhance the phonon coupling for higher q.

mixed contributions to LA phonon coupling For |gLA(~q)|2 there is also the mixed deforma-tion/piezoelectric contribution. In the Zincblende case these contributions vanish at the case withisotropic wave functions, as discussed here, when taking the angular average by integrating overthe in plane angle

∫ 2π0 dφFZ = 0 (valid for in plane symmetric wave functions). It also vanishes

also in case of Wurtzit structure.

34


Frohlich coupling to LO phonons The last coupling mechanism we cover is polar coupling toLO phonons.

gLO(~q) = −i

√~ωLO(~q)

2V

√e2

ε0

(1

ε∞− 1

εstat

)1

q

(e−

a2v~q

2

4 ei~q·~r0 − e−a2c~q

2

4

), (2.34)

|gLO(~q)|2 =~ωLO(~q)

2V

e2

ε0

(1

ε∞− 1

εstat

)1

q2

(e−

a2v~q

2

2 + e−a2c~q

2

2 − 2e−(a2v+a2

c)~q2

4 cos(q|~r0| cos θ~q)

).

(2.35)

Compared to the cases before, the investigation is rather simple, since there are neither differentcoupling strength to valence and conduction band, nor any angular dependent prefactors. Hence, itis easy to carry out the angular average. Since the spectral density is a ~q integration (cf. Eq. (2.24))it depends on the coupling strength times q2 (spherical coordinates). For the acoustic phonon casesthis q2 is compensated by ω−1(q), in the LO case it is convenient to define q2 as part of the couplingstrength:

q2|gLO(q)|2 =~ωLO(q)

2V· e

2

ε0

(1

ε∞− 1

εstat

)(e−

a2vq

2

2 + e−a2cq

2

2 − 2e−(a2v+a2

c)q2

4sin(qr0)

qr0

).

(2.36)

As before the expression can be split into the part for r0 = 0 and the correction for taking intoaccount the dipole moment of the wave functions. The following properties hold and are visualizedin Fig. 2.4:

• The coupling vanishes for q = 0 as well as for equally sized wave functions

• Also as before, the r0 independent part has one maximum at qmax = 2√

ln(a2c/a

2v)

a2c−a2

v. The

strength of the maximum is independent of the QD size. The width of the coupling scalesroughly as 1

a . Since for the calculation of the spectral density the area of the coupling strengthmatters, small QDs will show stronger LO phonon signatures.

• The correction term for non-vanishing r0 enhances the coupling and introduces a coupling,even if the sizes of the wave functions are equal

Influence of the phonon dispersion

After inspecting the wave number dependent interaction strength we turn next to the dispersionrelation in order to calculate the spectral density as a function of the phonon frequency. Typicallythe acoustic phonon dispersion relation is described within Debye approximation ωκ(q) = cκq,whereas optical phonons are described in Einstein approximation ωLO(q) = ωLO. The problem is,that those approximations only hold for small wave numbers (cf. Fig. 1.3 and especially for smallQDs relevant wave numbers can be easily higher than those approximations are reasonable). We

35


0

5.0× 10−4

1.0× 10−3

1.5× 10−3

2.0× 10−3

2.5× 10−3

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]


0

1.8× 10−2

3.6× 10−2

5.4× 10−2

7.2× 10−2

9.0× 10−2

0 0.3 0.6 0.9 1.2 1.5 1.8q [nm−1]


Figure 2.4: LO phonon coupling strength for QD wave functions with different sizes ac, av and dipolemoments r0. The maximum of the coupling strength does not depend on the wave function extent, but on theposition and thus the area. The right panel shows an enhancement of the coupling strength due to a dipolemoment r0.

improve those approximations by describing acoustic phonons with a sinusoidal dispersion relationand optical phonons with a cosine dispersion. An advantage of those approximations is that bothare invertible. We restrict ourselves to isotropic dispersion relations.7

Jκ(ω) =∑q

|gqκ|2

~2ω2δ(ω − ωκ(q)) =

V

2π2

∫ ∞−∞

dq1

∂qωκ(q)

q2|gqκ|2

~2ω2δ(ω−1

κ (ω)− q). (2.37)

For acoustic Phonons in Debye approximation ω−1κ (ω) = ω

cκthe spectral densities read (here

we only show the deformation potential coupling; piezoelectric coupling behaves analogously):

Jκ(ω) =1

%

1

4π2

1

~c4κ

ω

cκ

(D2ve−a

2vω

2

2c2κ +D2ce−a

2cω

2

2c2κ − 2DvDce− (a2

c+a2v)ω2

4c2κ

sin ωcκ|~r0|

ωcκ|~r0|

).

The spectral density in Debye approximation shows qualitatively the same behavior as the couplingstrength |g(q)|2 had in the q domain. The most simple improvement is to take into account asinusoidal dispersion ωκ(q) = ωmax

κ sin(alattq), with alatt = π/2/qmax and ωmaxκ is the dispersion

at the Brillouin zone edge qmax, e.g. at the K,X,M point or a respective mean value. To be moreconsistent with the Debye approximation for small q, ωmax

κ = 2cκqmax/π is a good choice, ratherthan the actual value. In the following expression you need to insert the wave number as a functionof the frequency q(ω) = arcsin

(ωωmaxκ

)1alatt

Jκ(ω) =1

2π2

1

~%1

alatt

q(ω)4

ω4√ωmaxκ

2 − ω2ω

(D2ve−a

2vq

2

2 +D2ce−a

2cq

2

2 − 2DvDce− (a2

c+a2v)q2

4sin qr0

qr0

).

7For LO phonons in nitride systems this is no good approximation.

36


In comparison to the Debye approximation this changes the spectral density especially for largerenergies, cf. Fig. 2.5. For very small QDs there can even occur a discontinuity at ω = ωmax

κ due tothe divergent phonon density of states at the band edge.

050

100150200250300

0 2 4 6 8 10 12 14 16 18energy [meV]

28 30 32 34 36 38 40050100150200250

energy [meV]

05

1015202530354045

0 10 20 30 40 50 60energy [meV]

85 90 95 100 105 110 115 1200510152025

energy [meV]

QD with ac = 1.2 nmin Debye approximationQD with ac = 3 nmin Debye approximation

QD with ac = 0.35 nmin Debye approximationQD with ac = 1 nmin Debye approximation

Figure 2.5: Comparison of spectral densities using different approximations for the phonon dispersion. Theupper figures show InAs/GaAs QDs, while the lower ones depict a GaN/AlN system. The left pictures showthe longitudinal acoustic phonons’ spectral density. For larger QDs Debye and sinusoidal description of thedispersion coincide, while for smaller QDs they behave differently, especially at the Brillouin zone edge. Theright panels compare the optical phonon spectral density. In Einstein approximation there is only a delta lineat the Γ point LO energy, while with the cosine description the peak gets an asymmetric shape. For smallerQDs the maximum shifts to lower energies.

Optical phonons in Einstein approximation consist of the q integration over the interactionstrength times a delta function at the LO energy:

Jκ(ω) =V

2π2

(∫ ∞−∞

dqq2|g~qκ|2)

1

~2ω2LOδ(ω − ωLO). (2.38)

That is, QD geometry and wave functions enter in this limit only via the total coupling strengthcompared to the zero phonon line, i.e. the Huang-Rhys factor Eq. (2.22d). Introducing the qdependence of the dispersion is here even more important. We do this by approximating it witha cosine ω(q) =

ωΓLO+ωX

LO2 +

ωΓLO−ω

XLO

2 · cosπ qqX

= ω0 + Ω0 cos(2alattq) with the inverse q(ω) =

37


12alatt

arccos(ω−ω0

Ω

). Thus

Jκ(ω) =1

4π2

e2

ε0

(1

ε∞− 1

εstat

)1

2alatt

√Ω2

0 − (ω − ω0)2

1

~ω(e−

a2vq

2

2 + e−a2cq

2

2 − 2e−(a2v+a2

c)q2

4sin(qr0)

qr0

).

Now also frequencies below the LO gamma point frequency contribute to the spectrum and themaximum is shifted towards lower energies. As the maximum of the coupling strength shifts tolarger q for smaller QDs, the frequency shift is also larger for smaller QDs, cf. Fig. 2.5.

2.1.3 Numerical investigation of optical spectra

Spectra can be numerically evaluated for any wave function. Thus using realistic 8k · p wavefunctions comparison to experimental spectra. Real quantum dots do not match the wave functionapproximations made in the previous section. Hence, for comparing theoretical models to measuredspectra it is convenient to use more realistic wave functions. However, the insights from the previoussection will help to understand the following observations:

InAs/GaAs QDs

In Reference [SDW+11] we compared the experimental electro-luminescence spectrum of a singleQD emitter [LST+09] with simulations within the IBM framework. Figure 2.6 shows the respectivedata. The simulations were done using the presented Gaussian wave functions as well as wavefunctions microscopically calculated via 8~k · ~p theory [SWB07]. For the spectrum of the acousticphonons we could see the above discussed properties of a spectrum, e.g. the asymmetry of thespectrum for low temperatures due to spontaneous phonon emission. An interesting finding isthat for larger q the coupling via Gaussian wave functions underestimates the coupling strengthcompared to the experiment and 8~k · ~p wave functions. That means the ansatz with Gaussianwave functions underestimates the spatial confinement. Further, we could observe that the spectraldensity calculated with 8~k · ~p wave functions did not vanish at the edge of the Brillouin zone, thusthe Debye approximation is not good for InAs/GaAs QDs. However, since the acoustic phononinteraction is rather weak in this material system, the experiment could not make a statement aboutbehavior at the Brillouin zone edge (at about 12 meV) of the spectrum. Piezoelectric coupling toacoustic phonons plays no rule here. For the LO part of the spectrum the experimental data showsa broad and asymmetric line incompatible to the usual Einstein approximation. As discussed above,using a cosine dispersion relation leads to such a line shape. Further, taking into account a finitephonon lifetime further broadens the spectrum to match the experimental data.

38


-10 -5 0 5 10

Lum

ines

cenc

e(a

.u.)

Energy (meV)-40 -38 -36 -34 -32 -30

Lum

ines

cenc

e(a

.u.)

Energy (meV)

T = 15KexperimentGaussian wave function

8k · p wave function

experimentinfinite phonon lifetime

5 ps phonon lifetime

Figure 2.6: Spectra of InAs/GaAs single QD emitter as in Ref. [SDW+11]. Comparison of experimentaldata with simulations. In the left panel acoustic phonon sidebands give a prediction of the wave functionsextent. Gaussian wave functions cannot describe the behavior as good as microscopically calculated wavefunctions, since a Gaussian description underestimates the spatial confinement. The right panel shows theLO phonon peak is asymmetric and broadened, which can be described by a cosine dispersion relation andan additional 5 ps phonon lifetime.

GaN/AlN QDs

Nitride based QDs are a promising basis for the development of light emitters for a broad spectralregion from visible to UV. Carriers in nitride hetero-structures are strongly confined making themcandidates for room temperature devices for quantum information applications. In those systemspiezoelectric and pyroelectric (temperature dependence of the dipoles) effects have a significantinfluence, because the lattice lacks inversion symmetry. This leads to a strong piezoelectric electron-phonon interaction on one hand and a large build-in dipole moment in the QDs on the other.Additionally, nitride QDs and their wave functions are rather small compared to the arsenide system.All in all, knowing the rules of thumb from above, this leads to a strong electron-phonon interactionin nitride QDs. In Ref. [OHR+12] we investigate the cathodoluminescence spectrum of GaN/AlNQDs. The data is shown in Fig. 2.7. Especially compared to the arsenide system the acousticphonon sidebands are huge as is the broadening of the zero phonon line. In the nitride system thepiezoelectric coupling to acoustic phonons is in the same order of magnitude as the deformationpotential coupling. Such a strong signature is the ideal basis for comparison to simulations withdifferent wave functions. Simulations with 8~k · ~p wave functions and Gaussian wave functionsunderline the influence of the intrinsic dipole in nitride quantum dots. The comparison of theexperimental spectra for different QDs yielded a broader spectrum, i.e. stronger acoustic phononinteraction for QDs with lower emission energy, i.e. larger QDs [Ost12]. As we saw in Sec. 2.1.2this can be attributed to the enhanced piezoelectric coupling for QDs with a larger intrinsic dipolemoment. For possible applications, where it is mostly favorable to have small coupling to phonons,this means it is important to find the QD right size since too small QDs have strong deformation

39


-40 -20 0 20 400.00

0.25

0.50

0.75

1.00

0 20 40 60 80 100 120 1404.245

4.250

4.255

4.260

ZPL

posi

tion

(eV)

T (K)

5K35K65K95K127K140K

CLintensity(arb.un.)

Relativeenergy,meV

Figure 2.7: Spectra of nitride QDs from cathodoluminescence. In the left panel experimental cathodolu-minescence spectra of a single QD for different temperatures are shown. The spectra are normalized withrespect to the area beneath them, since the area corresponds to the total emitted power. The right figure isFig. 2 from Ref. [OHR+12] and shows calculated spectra for various temperatures as well as the respectivespectral density decomposed into the different coupling mechanisms. The lower right panel compares thecalculated spectrum to the measured data.

potential coupling as too large QDs have a strong piezoelectric coupling.A thorough investigation of the optical phonons remains to be done in the material system.

40


2.2 Coupling to carrier reservoir – multi-phonon relaxation

Phonons play a significant role in intra-band and inter-subband transitions in semiconductors dueto the possibility of exchanging momentum and energy. The relaxation process can be attacked viaHeisenberg equation of motion approach and in Born-and Markovian approximation leading to firstorder relaxation processes as described in Sec. 1.3.

2.2.1 The phonon bottleneck

In a QD–WL system the description of capturing and relaxation via LO phonons suffers a problem:QD energies as well as phonon energies are discrete, which gives rise to an expected inefficientscattering mechanism, due to the required energy conservation from Markovian approximation (asin Fermi’s golden rule, cf. Sec. 1.3). This creates the necessity of taking into account higher orderprocesses, to allow for the violation of energy conservation on short timescales. There is a longhistory of experiments trying to confirm this so called “phonon bottleneck” [HBG+01; UNS+01],because other scattering channels as Coulomb scattering are efficient in relaxations [Wil13; Maj12].On the other side theories try to include mechanisms, that circumvent the strict energy conservation,e.g. by non-Markovian approaches as higher order density matrix formalism [FWD+03] or Green’sfunctions approaches using the polaron description [LNS+06; SNG+05]. Those approaches solvethe full dynamics of all quantities and need a high numerical effort. A perturbative multi-phonontreatment is another approach to the problem [MUB+02; IS94; IS92]. Such an approach can beconsidered as higher order Markovian, i.e. higher numbers of one-phonon interactions that violateenergy conservation are included implicitly, leading to an energy conserving process. The advantageof such an approach is that it leads to a rate equation description, which is often desired for furtheruse as the calculation of QD based devices [KDW+11] or for more complex systems with moreinteraction as the quantum optical treatment of the biexciton cascade where the interaction withphonons influences the degree of entanglement of the photons [CMD+10].

In this section we present a perturbative multi-phonon approach based on a projection operatortechnique. Within this approach it is easily possible to take into account many electron contributions,which is vital for solid state system. In the end we present the resulting rate equation and discussthe corresponding multi-phonon rates.

2.2.2 Effective multi-phonon Hamiltonian

We present an approach where an effective multi-phonon Hamiltonian is derived. Effective Hamil-tonians are often used to reduce the complexity of a problem. The approach we use is based on amulti photon approach from [Fai87, p. 158ff], and delivers the possibility to treat relaxations on arate equation limit.

The system is divided into a resonant subsystem where all (many particle) states have the sameenergy and a non-resonant subsystem consisting of all other states. Within a perturbation theorythe influence of the non-resonant subsystem is taken into account implicitly for transitions within

41

2.2. COUPLING TO CARRIER RESERVOIR – MULTI-PHONON RELAXATION

the resonant subsystem. The Hamiltonian of the full system H ′ is split into a free part H ′0 and theinteraction part V , which has the role of a perturbation:

H ′ |Ψ〉 = E |Ψ〉 , (2.39)

H ′ = H ′0 + V . (2.40)

Additionally we need a constant energy W which has to be assigned later on and decides on thekind of perturbation theory. 8

(H ′ −W · Id) |ψ〉 = H |ψ〉 = E |ψ〉 ; |ψ〉 = ei~W ·t |Ψ〉 , (2.41)

H = H0 + V ; H0 = H ′0 −W · Id. (2.42)

The eigenstates of H0, |i〉 are divided into resonant and non-resonant ones to define the twoprojection operators

P =∑

i|i〉〈i| (2.43)

Q = Id− P =∑

i|i〉〈i| . (2.44)

where∑

i sums up resonant states,∑

i the rest. For the application to the solid state system |i〉 willbe a product state of electronic and phonon states.

The projection operators are applied to Eq. (2.41) using the projector properties 1 = P +Qand P 2 = P .

EP |ψ〉 =: E |ψ〉P = PHP |ψ〉+ PHQ |ψ〉 =: HPP |ψ〉P + HPQ |ψ〉Q , (2.45)

EQ |ψ〉 = HQP |ψ〉P + HQQ |ψ〉Q ⇔ |ψ〉Q = (E · Id− HQQ)−1HQP |ψ〉P . (2.46)

The non resonant states |ψ〉Q are eliminated by inserting Eq. (2.46) into Eq. (2.45). Since theprojectors commute with H0 and PQ = 0 the interaction between the subsystems is solely theperturbation part HQP = VQP . The inverse operator is decomposed via a Neumann series:

(Id− O)−1 =

∞∑n=0

On, (2.47)

for a general operator O. The identity holds if∑∞

n=0 On converges, especially for ||O|| < 1 [Wer05;

RS80]. Thus, Eq. (2.45) reads:

E |ψ〉P = HPP |ψ〉P − VPQ(HQQ − E)−1VQP |ψ〉P=(HPP − VPQH−1

QQ(1− EH−1QQ)−1VQP

)|ψ〉P

=

(HPP − VPQH−1

QQ

∞∑n=0

(H−1QQ

)nVQPE

n

)|ψ〉P . (2.48)

8Such a constant can also be introduced in the perturbative treatment of the Schrodinger equations, its choice decidesfor example between Schrodinger and Brillouin–Wigner perturbation series.[Nol06]

42


HQQ still contains a free and a perturbation part, which is expanded in another Neumann series.

H−1QQ = (QH0Q+QV Q)−1 =

∞∑n=0

(−1)nH−10 (VQQH

−10 )n with H0 = QH0Q. (2.49)

Inserting into Eq. (2.48) leads to:

E |ψ〉P =

HPP − VPQ∞∑n′=0

( ∞∑n=0

(−1)nH−10 (VQQH

−10 )n

)(n′+1)

VQPEn′

︸︷︷︸

=:A

|ψ〉P . (2.50)

A is nearly an effective Hamiltonian except for the self consistent dependence on its eigenvalue E.To make it a Hamilton operator En is iteratively replaced by An (valid due to En |ψP 〉 = An |ψP 〉).Subsequently, the terms can be ordered by their order of V . This is, where we assign W in a way,that HPP = VPP . Else all higher terms would contain orders of H0.

HPP = VPP ⇔ PH0P = 0⇔W =⟨ψ∣∣∣PH ′0P ∣∣∣ψ⟩ . (2.51)

By construction all transitions within the resonant subsystem are energy conserving.Eventually, the effective Hamiltonian is explicitly given by

Heff = VPP − VPQH−10 VQP

+ VPQH−10 VQQH

−10 VQP − VPQH−2

0 VQP VPP

− VPQH−10 VQQH

−10 VQqH

−10 VQP + VPQH

−10 VQQH

−20 VQP VPP + VPQH

−20 VQQH

−10 VQP VPP

+ VPQH−20 VQP VPQH

−10 VQP − VPQ

(H−1

0

)3VQP V

2PP

· · · .(2.52)

The free energy part of the effective Hamiltonian is contained in W/~:

H ′0 |ψ〉P = W |ψ〉P ⇒ H ′eff = H ′0 + Heff. (2.53)

The first order term just represents the interaction from the initial system restricted to the resonantsubsystem. Higher order terms represent interactions within the resonant system, assisted by thenon-resonant states. This can be seen with the aid of the second order:

−VPQ(QH ′0Q−W )−1VQP =∑

i,jPV |i〉〈i| (H ′0 −W )−1 |j〉〈j| V P

=∑

i(⟨i∣∣∣H ′0∣∣∣i⟩−W )−1PV |i〉〈i| V P. (2.54)

43


This describes an interaction within the resonant subsystem, assisted by the non resonant states |i〉via an implicit transition with an weighting term

(⟨i∣∣∣H ′0∣∣∣i⟩−W)−1

that is inverse proportionalto the energetic difference of the resonant states and the energy of |i〉. This is an energy conservinginteraction composed of two transitions, that do not conserve energy. This can be seen as a higherorder Markovian process. Another point of view is to call |i〉√

〈i|H′0|i〉−Wa virtual state. Figure 2.8

shows some processes from the effective two-phonon Hamiltonian we will explicitly derive in aminute. The interpretation of higher order terms follows analogously.

: resonant states

resonant states

:

virtual states

intermediate states

a)

b)

Figure 2.8: Two phonon relaxation between two states with energetic difference of 2~ωLO. a) processes in aresonant two level system: The relaxation takes place via non-resonant states with a non-matching numberof phonons. Such states are typically called virtual. The two phonon transition is composed of one diagonalinteraction and a non-diagonal. b) a system with an additional continuum of non-resonant states. Here, therelaxation includes intermediate transitions into those states and is such composed of two non-diagonalinteractions.

44


Application to the electron–phonon system

The next step is to apply this general technique to the specific QD/WL electron and phonon problem.The electron phonon Hamiltonian reads 9

H ′0 =∑i

εia†iai + ~ω

∑q

b†qbq, (2.55a)

V =∑ij

∑q

gqija†iaj(bq + b†−q). (2.55b)

Now it is important to define the resonant states. The system states are composed of the electronicstates |ε〉 and the phonon states |n〉 each being many particle states.

|ψ〉 =∑ε,n

|ε, n〉 . (2.56)

We define one state |ε0, n0〉 to be the reference for the energy conservation. This is the state whereQD level is occupied and there is a phonon number state |n0〉. Everything has to be done for ageneral initial condition, where when the electron is occupied there are N0 Phonons in the systemand all electrons in the WL have an overall energy of ε0WL. The phonon number is lowered by n fora transition into other electronic states |εr〉 under energy conservation within an n phonon process.

W = ε0 + ε0WL + ~ωN0 = εr + ε0WL + ~ωNr, (2.57)

N0 =∑q

⟨n0

∣∣∣b†qbq∣∣∣n0

⟩, Nr =

∑q

⟨nr

∣∣∣b†qbq∣∣∣nr⟩ . (2.58)

The effective Hamiltonian shall only operate on the QD states and the electronic states being integertimes the phonon energy away with energies εr = ε0 + n~ωLO. Thus the projector P containsall states fulfilling this energy condition. We want to deal with a system, where the one phonontransition is energetically forbidden. Thus there are no resonant states for which a transition PV Pis allowed, which means VPP vanishes. We will directly turn to the two phonon case.

The effective two phonon Hamiltonian, already written in Eq. (2.54) yields for the actualinteraction Hamiltonian Eq. (2.55)

PH(2)P =∑

ψ

∑ij

∑q±

∑kl

∑q′±′

gqijgq′

kl(⟨ψ∣∣∣H ′0∣∣∣ψ⟩−W )−1Pa†iajb

±q |ψ〉〈ψ| a

†kalb

±′q′ P

with b+q = b†−q and b−q = bq. The energy of all states in P is W . The operators left and right of theprojector lead to the conditions⟨

ψ∣∣∣H ′0∣∣∣ψ⟩ = W − εl + εk ±′ ~ωLO = W − εi + εj ∓ ~ωLO. (2.59)

9Note, that we use the electron-hole picture, where conduction and valence band are symmetric. Later on we willrestrict ourselves to one band, when calculating the rates, the derivation here is, however, done in general.

45


This introduces an important energy relation. Now, the inverse term is not depending on theactual initial condition P anymore. The sum over the non-resonant states can be removed via∑

ψ |ψ〉〈ψ| = Q = Id− P .

PH(2)P =∑ij

∑q±

∑kl

∑q′±′

gqijgq′

kl(−εl + εk ±′ ~ωLO)−1Pa†iajb±q a†kalb

±′q′ P

−∑ij

∑q±

∑kl

∑q′±′

gqijgq′

kl(−εl + εk ±′ ~ωLO)−1Pa†iajb±q Pa

†kalb

±′q′ P.

The second line removes possible resonant one phonon processes. Assuming there are no electronicstates resonant to a single phonon transition, it vanishes as stated above. Hence, in normal orderingthe effective Hamiltonian reads 10

H(2) =∑il

∑q±

∑kj

∑q′±′

gqijgq′

kl

(a†ialδkj − a

†ia†kajal

)b±q b±′q′

−εl + εk ±′ ~ωLO. (2.60)

This effective Hamiltonian contains one and two electron transitions with interaction with twophonons. The one particle term represents the transition from level l to level i composed of thetransitions l → k and k → i without taking into account possible occupations of the level k.The effectiveness of the transitions is weighted by how close the intermediate level is to a singlephonon resonance. In terms of Fig. 2.8 it contains virtual and intermediate transitions. Thoseparts correspond to a one electron assumption. This level is widely used for the multiphonon QDrelaxation [MUB+02] or more general as a higher order Fermi golden rule. The two electron termbrings in the many body properties of the system. For the intermediate transitions the occupationof the intermediate levels has in general to be taken into account. For a one electron assumption,i.e. a completely empty WL, the term vanishes. Since we are interested in an effective one particlephonon interaction the term is treated via the Hartree-Fock factorization [Czy00]:

a†ia†kajal =

⟨a†kaj

⟩a†ial +

⟨a†ial

⟩a†kaj −

⟨a†iaj

⟩a†kal −

⟨a†kal

⟩a†iaj . (2.61)

Note, that this term contains the fermionic property a†ia†kajal ∼ (1− δik)(1− δjl) = δikδjl. We

will explicitly use those inverse Kronecker deltas for bookkeeping reasons. The first two termsdescribe one particle operators in the mean field of other one-particle operators, the last two terms,called exchange interaction, make the difference to classical mean field methods. Putting this intoEq. (2.60) one can re-index the sums to get the two contributions:

H(2)∣∣∣two particle

= −∑ij

∑q±

∑kl

∑q′±′

gqijgq′

kl

(⟨a†kaj

⟩a†ial

)δkiδjl

−εl + εk ±′ ~ωLO

(b±q b±′q′ − b

±′q′ b±q

)

+∑ij

∑q±

∑kl

∑q′±′

gqkjgq′

il

(⟨a†kaj

⟩a†ial

)δkiδjl

−εk + εj ∓ ~ωLO

(b±q b±′q′ − b

±′q′ b±q

). (2.62)

10Note that the denominator is equal to (−εi + εj ∓ ~ωLO), according to Eq. (2.59).

46


The first term has the same weighting as the one particle process in Eq. (2.60), but now includesthe occupation of the intermediate levels. Note, that in terms of Fig. 2.8 those processes are onlyintermediate processes (b). The exchange term however is weighted by the energetic difference ofthe resonant or the intermediate levels, respectively. This Hamiltonian is restricted to only operateon the interesting resonant electronic states via tracing out non-resonant states. This only leavesoperators acting on resonant energies as operators.

The next step is to explicitly treat the±,±′ sums, i.e. the single phonon absorption and emissionprocesses, which leads to two general cases:

• First case is both phonon operators are creators/annihilators ± = ±′. Those parts of theHamiltonian are the transition terms, where two phonons are absorbed/emitted. Here, theoverall phonon number changes by two via the transitions (cf. Fig. 2.8). From Eq. (2.59) weget the energy relation for the states:

(εl − εk)− (εi − εj) = ±2~ωLO (2.63)

Since l, i are resonant electronic states the possible energetic differences are (εl − εi) ∈−2~ωLO, 0, 2~ωLO. Thus, the assisting states need to obey (εk − εj) ∈ 0,±2~ωLO, 0,respectively. In the two phonon emission/absorption case the many particle correctionEq. (2.62) vanishes because [b±q , b

±q ] = 0. Hence, the two phonon transition part of the

effective Hamiltonian is the one electron contribution and reads:

H(2)∣∣∣±=±′

=∑ilk

∑qq′±

gqikgq′

kl

−εl + εk ± ~ωLOa†ialb

±q b±q′ . (2.64)

The energy conservation Eq. (2.59) results in the constraint i 6= l, i.e. only non-diagonalinteraction. The vanishing many particle contribution Eq. (2.62) can be understood as follows:There are two parts canceling each other; one part acts as a Pauli blocking correction to theone electron part, i.e. 1−

⟨a†kak

⟩, whereas the other part is proportional to the occupation

of the intermediate level, i.e. an electron from the intermediate level scatters into the QDand then the intermediate level is filled again. Comparing this to Fig. 2.8b both processesrepresent different time ordering of the left and the right side.

• In the second case ± = ∓′ there is one phonon creation as well as one annihilation, whichmeans the overall phonon number is not changed in the process. The energy relation Eq. (2.59)then reads:

(εl − εi)− (εk − εj) = 0. (2.65)

For the case of a diagonal interaction i = l (i.e. the electronic state does not change) itsimply means the carrier is virtually/intermediately occupying a non-resonant level and thenjumps back to its original state, or for the exchange part that diagonal interactions of twoelectrons compensate each other. A transition εl− εi = ±~ωLO would be compensated by anopposite two phonon transition of other states. We neglect this, since it would couple to fast

47


oscillating terms in the reservoir. Further, the intermediate resonance⟨a†kaj

⟩for εk = εj

can be assumed to be diagonal⟨a†kak

⟩δkj for homogeneous/isotropic systems[Ric07].

H(2)∣∣∣±=∓′

=∑il

∑±

∑k

∑qq′

gqikgq′

kl

b±q b∓q′ −⟨a†kak

⟩δkiδkl

(∓δqq′

)−εl + εk ∓ ~ωLO

a†ial. (2.66)

Eventually, we obtain an effective two phonon Hamiltonian with effective coupling elements. Wewill use only processes in one band since inter-band have weak contribution due to their largeenergy separation on one hand and band diagonal phonon coupling elements on the other.11 Weassume one QD state ε0 and a quasi-continuum εr in the carrier reservoir.

H(2)el-ph = g00a

†0a0 + grra

†rar

+∑q′q′′

gq′q′′

00 a†0a0b†−q′bq′′ +

∑q′q′′

gq′q′′rr a†rarb

†−q′bq′′

+∑q′q′′

gq′q′′

0r a†0arb†−q′b

†−q′′ +

∑q′q′′

gq′q′′

r0 a†ra0bq′bq′′ . (2.67)

With the six effective electron-two-phonon coupling elements, that include densities from the carrierreservoir, which we denote fk =

⟨a†kak

⟩and treat as equilibrium distribution. The first line does

not include phonon operators, thus those terms renormalize the electronic energies temperatureindependent:

gii =∑q

gqiig−qii

~ωLO+∑kk 6=i

∑q

(gqikg

−qki

(fk

−εi + εk − ~ωLO+

1− fk−εi + εk + ~ωLO

)), (2.68a)

This are processes, where a phonon is spontaneously emitted and absorbed again and thus create anew quasi-particle, the polaron. The first term is the polaron shift from the pure diagonal interactionwe already know from Sec. 2.1.1 (Eq. (2.22a)). The next represents an intermediate transition,where the relaxation from k to i and emission of the phonon is the first process, thus it is proportionalto the density of the level k. This process is most effective if εk > εi, i.e. at coupling to a higherenergy level. Then there is the process, where the electron is excited from i to k first under emissionof a phonon, proportional to the blocking of level k. This process is effective, if k has a lowerenergy than i.

The next effective coupling elements are diagonal in the electronic operators and do not changethe phonon number. They also represent polarons. This would correspond to re normalization ofboth, electrons and phonons, especially they lead to temperature dependent energy shifts of the

11Note that due to the exchange terms however densities of the other band would contribute.

48


electronic levels.

gqq′

ii =∑kk 6=i

gqikgq′

ki

−εi + εk − ~ωLO+

gq′

ikgqki

−εi + εk + ~ωLO. (2.68b)

Those two term arise from intermediate transitions, where the first favors intermediate levels aboveεk > εi and the second favors intermediate levels below.

Finally there are the relaxation/excitation matrix elements. The two phonon emission with thecapturing of the carrier from the WL.

gqq′

0r =∑kk 6=r

gq0kgq′

kr

−εr + εk + ~ωLO. (2.68c)

The first two contributions are direct transitions under the additional diagonal emission of a phonon.The last process covered is the two phonon absorption with excitation of the QD electron into

the WL.

gqq′

r0 =∑kk 6=r

gqrkgq′

k0

−ε0 + εk − ~ωLO. (2.68d)

This process is the complex conjugate of the two phonon emission. Next, we will use the effectiveHamiltonian to derive rate equations for the QD occupations.

2.2.3 Multiphonon relaxation rates

Using the effective Hamiltonian Eq. (2.67), we derive Heisenberg equations of motion for the QDoccupations

⟨a†0a0

⟩. The derivation of the dynamics equations is made just as with the conventional

Hamiltonian, cf. Sec. 1.3. The dynamic equation for the QD occupation in one band follows fromthe commutator with Hel +Hph +H

(2)el-ph and reads

i~∂ta†0a0 =∑q′q′′

∑r

gq′q′′

0r a†0arb†−q′b

†−q′′ −

∑q′q′′

∑r

gq′q′′

r0 a†ra0bq′bq′′ . (2.69)

49


The dynamics couple to transitions assisted by two phonon operators. The equations of motion forthose higher order terms read:

i~∂ta†0arb†−q1b

†−q2 = (εr+grr−ε0−g00−2~ωLO)a†0arb

†−q1b

†−q2−

∑q′,q′′

gq′q′′

00 a†0arb†−q′bq′′b

†−q1b

†−q2

+∑q′,q′′

gq′q′′rr a†0arb

†−q1b

†−q2b

†−q′bq′′−

∑q′,q′′

∑r′

gq′q′′

r′r′ a†0a†r′arar′

(−b†−q′b

†−q2δq′′,−q1 − b

†−q′b

†−q1δq′′−q2

)+∑q′,q′′

∑r′

gq′q′′

r′0

(a†0a0 − a

†r′ar

)b†−q1b

†−q2b−q′bq′′

+∑q′,q′′

∑r′

(gq′q′′

r′0 +gq′′q′

r′0 )(a†0a†r′ara0−a

†r′ar)

(b†−q1bq′δ−q2,q′′ + b†−q2bq′δ−q1,q′′ + δ−q1,q′δ−q2,q′′

).

The free part not only consist of the bare energy of the electrons and phonons but also of thepolaron shifts of the levels. The next two terms resemble also the two-phonon assisted transition,with additional assistance of the phonon number operator, thus they lead to a further, temperaturedependent, polaronic energy shift. The next part is also a polaronic energy shift, assisted by theoccupations of the electronic levels. Taking into account the polaronic terms however would needthe calculation of higher orders in the hierarchy. The last two contributions are the desired couplingsto electron and phonon densities.

At this point we stop the expansion and try to close the system of equations with the methods weintroduced in Sec. 1.3. The energy renormalizations are included in effective level energies εr, ε0 andare no longer in the focus of our investigation. The scattering will happen self consistently betweenstates that obey the energy conservation. Those terms do not directly contribute to scattering rates.

Now for the relevant part we apply Born factorization to separate electron and phonon dynamics,the four operator electron values are factorized via Hartree-Fock

⟨a†0a†r′ara0

⟩=⟨a†0a0

⟩⟨a†r′ar

⟩−⟨

a†0ar⟩⟨

a†r′a0

⟩, but the last part is omitted, rendering it effectively only a Hartree (classical

mean field) factorization. The four operator phonon terms are factorized assuming a thermalizedbath

⟨b†−q1b

†−q2b−q′bq′′

⟩=⟨b†−q1bq′′

⟩⟨b†−q2b−q′

⟩+⟨b†−q2bq′′

⟩⟨b†−q1b−q′

⟩and

⟨b†−q1bq′′

⟩=

nq1δ−q1,q′′ , where n denotes the Bose distribution. Finally, integration and Markovian approxima-tion yields⟨

a†0arb†−q1b

†−q2

⟩= −i

π

~δ(εr−ε0−2~ωLO)

((g−q1−q2r0 + g−q2−q1r0 )

(⟨a†0a0

⟩−⟨a†rar

⟩)(nq1nq2)

+(g−q1−q2r′0 + g−q2−q1r′0 )(⟨a†0a0

⟩⟨a†rar

⟩−⟨a†rar

⟩) (nq1 + nq2 + 1)

).

The rate equation for the QD occupation follows directly as

~∂tρ00 =∑q1q2

∑r

2π

~δ(εr − ε0 − 2~ωLO)gq1q20r (g−q1−q2r0 + g−q2−q1r0 )

((1− ρ00)ρrr (nq1 + 1) (nq2 + 1)− ρ00(1− ρrr) (nq1nq2)) (2.70)

50


Since the energy of the resonant states of the reservoir is set and the phonon distribution does notdepend on the phonon wave number in Einstein approximation, a rate independent of the phonon andelectron densities can be defined. This leads to a simple equation consisting of in- and out-scatteringrates:

∂tρ00 = Γρrr(n+ 1)2︸︷︷︸Sin

(1− ρ00)− Γ(1− ρrr)n2︸︷︷︸Sout

ρ00 (2.71)

The rates define the long time limit (steady state) of the QD occupation ρ∞, i.e. the equilibriumoccupation. Since the rates depend on the carrier distribution in the reservoir, pumping of thereservoir directly affects this steady state. A convenient description is the relaxation time picture,that is also used in Chap. 3.

∂tρ00 = −Γrelax(ρ00 − ρ∞); Γrelax = Sin + Sout, ρ∞ =

Sin

Sin + Sout. (2.72)

An non-equilibrium occupation is changed with the rate Γrelax until it reaches the equilibriumdefined by the pumping rates. The order of magnitude of Γrelax is defined by the system dependentΓ, which is multiplied with n2 +ρrr(2n+1) which has as rather simple behavior on the temperatureand the carrier density in the reservoir. Γ depends mainly on the binding energy of the QD levels,i.e. the energetic distance between QD state and WL band edge. The equilibrium occupation doesnot depend on Γ, but is defined via the occupation in the resonant level and the phonon distribution.We assume the WL carrier density to be thermalized, i.e. following a Fermi-Dirac distribution.

ρrr =1

1 + eβ (εr − µ)

with β = kBT and the chemical potential µ which is determined by the carrier density in the WLnWL. For a two dimensional reservoir this can be expressed analytically[CK99]:

µe/h =1

β

(exp

(β

~2π

2me/hnWL − 2ρ00nQD

)− 1

)2ρ00nQD presents a correction taking into account the carriers that are captured on average with theQD density nQD and the average occupation of a QD level ρ00. The carrier density in the reservoiris affected by pumping, e.g. via an applied current or optical excitation. The strength of Γ doesdepend on the wave functions of the system on one hand and on the QD binding energy, i.e. theenergetic distance between QD and band edge of the WL continuum on the other, cf. Sec. 1.2.1.Fig. 2.9 shows the dependence of the rates on this binding energy. For energies below ~ωLO ourassumptions of no possible one phonon processes fails, above 2~ωLO energy conservation is notpossible anymore.

The in- and out-scattering is determined by the pumping strength, which affects the carrierdensity. Figure 2.10 shows that for no carriers in the reservoir there are no in-scattering processes,but out-scattering is non-vanishing. This is in contrast to rates based on Coulomb interaction,where for an empty reservoir also the out-scattering vanishes [Wil13]. At high pumping strength,

51


020406080

100

40 45 50 55 60 65 70

Γp

s−1

∆ meV~ωLO 2~ωLO

Figure 2.9: Rate Γ over binding energy in units ~ωLO for InAs/GaAs QD material parameters. Near∆ = ~ωLO the rate is large due to many continuum states near a one phonon transition, for larger bindingenergies the number of such states is lowered, thus the rate decreases.

00.020.040.060.080.1

0.12

0 1 · 1012 2 · 1012 3 · 1012 4 · 1012

00.511.52

Sou

t/Γ

Sin/Γ

WL carrier density [cm−2]

SinSout

300 K200 K100 K10 K

Figure 2.10: In- and out-scattering rate as a function of the carrier density in the carrier reservoir. The inscattering depends mainly on the occupation of the resonant level. For low temperatures it resembles a Fermifunction, which is broadened for higher temperatures. Additionally higher temperatures enhance the rates.The out-scattering starts at a non-vanishing contribution proportional to n2 for low carrier densities anddecreases for higher carrier densities, due to the blocking in the resonant reservoir state.

however the resonant levels are completely occupied, hence the in-scattering saturates and the out-scattering vanishes. The most interesting influence is the temperature, which affects the occupationin the resonant states, but more over the phonon density. In Fig. 2.11 the temperature influence isshown. For low temperatures there are no phonons, hence only the in-scattering, proportional to thespontaneous phonon emission ∼ (n+ 1) is present. For higher temperatures the phonon density ismonotonously growing.

The rate equations derived in this chapter can be used to introduce losses or pumping terms inthe optical Bloch equations that are the topic of the next chapter.

52


0

0.2

0 50 100 150 200 250 300 350 400

Sin/Γ,S

out/

Γ

Temperature [K]

SinSout

×0.11012 cm−2

1011 cm−2

1010 cm−2

1009 cm−2

0 cm−2

Figure 2.11: Two phonon rates as a function of the temperature. For high carrier densities the in scattering isdecreased due to the decreasing occupation of the resonant level and increases for even higher temperaturesdue to the increasing phonon occupation. For lower carrier densities the in scattering rate is only increasing.The out scattering rates follow the temperature behavior of n2.

53


54

Greift nur hinein ins volle Menschenleben!Ein jeder lebt’s, nicht vielen ist’s bekannt,Und wo Ihr’s packt, da ist’s interessant.

Johann Wolfgang von Goethe, Faust

3Non-linear optical response of quantum dots

As described in Sec. 1.2.1, quantum dots (QDs) can be described by a discrete level system, similarto atomic systems, hence in the most simple approximation as a two level system or taking intoaccount higher states in the most simple approximation two uncoupled two level systems. Suchtwo level systems coupled to an electric field semi-classically can described via the so called Blochequations [AE87]. The Bloch equations are linear differential equations for the electronic quantities,but non-linear in the electric field, which leads to a non-linear optical response and hence a non-linear system in combination with the wave equation. This chapter deals with various approachesto describe the response, which is then used in Chap. 4 to yield wave equations that are capableof describing the propagation of optical pulses through QD media. Section 3.1 deals with theconservative Bloch equations. In Sec. 3.1.1 its well known analytic limits for pulses resonant to thetwo level system as well as non-resonant constant excitations are revisited, while Sec. 3.1.2 presentsa perturbative approach for non-resonant pulses. In Sec. 3.2 dissipative Bloch equations accordingto the losses and pumping introduced in Chap. 2 are discussed. Here, Sec. 3.2.1 generalizes theperturbative description for the dissipative case. The chapter ends with the description of the Blochequations in the strong dissipative regime in Sec. 3.2.2.

3.1 Conservative quantum dot Bloch equations

The optical response of a QD is defined by the microscopic polarization. Hence, its dynamics has tobe calculated. Using the electronic and semi-classical electric field Hamiltonian H = Hel +Hel-lightin the Heisenberg equation of motion approach (cf. Sec. 1.3) leads to the well known optical Bloch

3.1. CONSERVATIVE QUANTUM DOT BLOCH EQUATIONS

equations [AE87] (cf. Sec. 1.3) for the microscopic polarizations1

i~∂

∂tρ12 = (ε2 − ε1)

~ω21

ρ12 + ~d21 · ~E(ρ11 − ρ22), (3.1a)

i~∂

∂tρ21 = −~ω21ρ21 − ~d12 · ~Eu3, (3.1b)

as well as the equations for the occupations

i~∂

∂tρ11 = ( ~E · ~d12ρ12 − ~E · ~d21ρ21), (3.1c)

i~∂

∂tρ22 = −( ~E · ~d12ρ12 − ~E · ~d21ρ21). (3.1d)

The occupations of the levels are from now on described via the inversion u3 = ρ22 − ρ11 whichis one when the electron is in the upper level and −1, if the lower level is fully occupied. It is

Figure 3.1: Two level system of the states 1 and 2. The level occupations are observed via the numberoperators ρ11 = 〈a†1a1〉, ρ22 = 〈a†2a2〉. The transition probabilities between the levels are described via themicroscopic polarization ρ12 = 〈a†1a2 and its complex conjugate.

convenient to transform them into a rotating frame with the frequency of the external electric fieldωl. The field is decomposed into a slow envelope times the oscillation E = Eeiωlt +E∗e−iωlt andthe microscopic polarizations are treated in the same way ρ12 = ρ12e

iωlt + ρ∗21e−iωlt:

∂tρ12 = −i∆ωρ12 + iΩu3, ∂tu3 = −4Im(Ω∗ρ12), (3.2)

with Ω(t) = d12~ E(t) being the so called Rabi field and the important system parameter ∆ω =

ω21 − ω labels the detuning between the two level system and the frequency of the light. Here, weadditionally applied the rotating wave approximation, where occurring terms with the frequency2ωl in the equation for the inversion are neglected. Also neglected is the dynamics of ρ21, since itis governed by the oscillation −∆ω − 2ωl.2,3 Without any further interactions the phase space is

1Throughout this chapter quantities with a tilde (ρ12) represent the full quantities, while those without a tilde (ρ12)will represent the slowly varying envelope of ρ12.

2Note, that for the discrimination between near resonant and far resonant contributions one needs to constitute whichlevel has the higher energy.

3Subsequently the macroscopic polarization (cf. Sec. 1.2.3) P = Peiωlt + P ∗e−iωlt is given by P = · · · d12ρ12.

56

CHAPTER 3. NON-LINEAR OPTICAL RESPONSE OF QUANTUM DOTS

conserved, i.e. all solutions (2Re(ρ12), 2Im(ρ12), u3) (the so called Bloch vector) lie on a sphereof radius |ρ12|2 + 1

4u23, which is a conserved quantity. Those equations can be solved numerically.

However, in the two limits of resonant and non-resonant excitation, with completely differentdynamics, analytical solutions can be obtained.

3.1.1 Exact solutions of the optical Bloch equations

To recap the exact limits of the conservative Bloch equations we follow Ref. [AE87]. In the regimeof resonant excitation ∆ω = 0 this system is exactly solvable for real pulse envelopes

∂tρ12 = iΩu3, ∂tu3 = −4Im(ρ12)Ω. (3.3)

Their solutions depend on the pulse area

ρ12 = i sin

(∫ t

−∞Ω(t′)dt′

), u3 = − cos

(∫ t

−∞Ω(t′)dt′

).

The shown solutions represent the typical initial conditions of no polarization ρ12(−∞) = ρ(0)12 = 0

and the electron being in the lower level u3(−∞) = u(0)3 = −1. The state of the system after

an excitation does not depend on the shape of the envelope, but only on its integral. Figure 3.2ashows the solution to Eq. (3.3) for various pulse areas. The initial state of the electronic system isonly restored for pulse areas with a multiple of 2π. The optical response in this case results in thesine-Gordon equation briefly discussed in Sec. 4.1.

For continuous wave excitation Ω(t) = Ω0 well known so called Rabi oscillations with the Rabifrequency Ω0 occur:

ρ12 = i sin Ω0t, u3 = − cos Ω0t.

In the case of such a stationary field the Bloch equations can be analytically solved also for anon-vanishing detuning [AE87]. We start by writing Eqs. 3.2 in a matrix representation

∂

∂t

ρ12

ρ21

u3

(t) = i

−∆ω 0 Ω0 ∆ω −Ω∗

2Ω∗ −2Ω 0

ρ12

ρ21

u3

(t).

The dynamics of this equation can be calculated by diagonalizing the time independent matrixwhich has the eigenvalues 0,±

√∆ω2 + 4|Ω|2. The non-zero terms correspond to a renormalized

Rabi frequency. In order to solve the system, one needs to know the unitary transformations Tleading to the diagonalization [AE87]:0 0 0

0√

∆ω2 + 4|Ω|2 0

0 0 −√

∆ω2 + 4|Ω|2

= T−1

−∆ω 0 Ω0 ∆ω −Ω∗

2Ω∗ −2Ω 0

T.

57


Since the field is time independent, knowing the transformation matrices yields the solution:ρ12

ρ21

u3

(t) = T

1 0 0

0 ei√

∆ω2+4|Ω|2t 0

0 0 e−i√

∆ω2+4|Ω|2t

T−1

ρ12

ρ21

u3

(0).

Using the typical starting condition u3 = −1, ρ12 = 0 results in the exact solution:

ρ12 =Ω(

i√

∆ω2 + 4|Ω|2 sin(t√

∆ω2 + 4|Ω|2)

+ ∆ω(

1− cos(t√

∆ω2 + 4|Ω|2)))

∆ω2 + 4|Ω|2,

u3(t) = −∆ω2 + 4|Ω|2 cos

(t√

∆ω2 + 4|Ω|2)

∆ω2 + 4|Ω|2.

All quantities oscillate with the renormalized Rabi frequency√

∆ω2 + 4|Ω|2. Inversion and real

-0.50

0.5

-1 -0.5 0 0.5 1Time [ps]

Im(ρ12) 0.5× u3 field Ω

4π

2π

π

π/2

π/4

(a) Pulsed excitation on resonance for different pulse areas

-0.50

0.5

-1 -0.5 0 0.5 1Time [ps]

∆ω2Ω

= 0

∆ω2Ω

= −0.25

∆ω2Ω

= −1

∆ω2Ω

= −4

Re(ρ12) Im(ρ12) 0.5× u3

∆ω2Ω

= −16

(b) Non-resonant continuous excitation for different detun-ings.

Figure 3.2: Exact solutions of the optical Bloch equations for a) resonant pulsed optical fields with differentpulse areas and b) continuous optical fields for different detunings. The final state of the system is determinedsolely by the area of the interacting pulse. For areas that are a multiple of 2π, the final state recovers the initialstate. For a temporally constant field (b) polarization and inversion are oscillations with the (renormalized)Rabi frequency. With higher detuning the effective frequency of the oscillations rises, whereas the amplitudeand the mean value decreases.

part of the polarization are in phase, while the imaginary part is shifted by π/2. The importantquantity characterizing the behavior is the detuning in relation to the excitation strength x = 2|Ω|/∆ω.Figure 3.2(b) shows the dynamics depending on the detuning. The inversion oscillates around the

58


value −(1 + x2

)−1with an amplitude x2(1 + x2)−1.The real part of the polarization has a meanas well as an amplitude of 0.5x

(1 + x2

)−1,while the imaginary part oscillates around zero with anamplitude 0.5x

(1 + x2

)−1/2.There are certain properties of the non-resonant response to pulses we can see already here.

For weak fields the oscillations grow faster, as the amplitude becomes smaller. For this case|2Ω| < |∆ω| the mean values can be described as geometric series of the field:

ρ12 = 0.5

∞∑n=0

(−1)n(

2Ω

∆ω

)2n+1

, u3 =

∞∑n=0

(−1)n(

2Ω

∆ω

)2n

. (3.4a)

The mean value of the polarization is composed of odd orders of the field, whereas the inversionconsists of even orders. The series are alternating and every order becomes smaller, hence everyorder presents a correction for the order before.

3.1.2 Perturbative approach for non-resonant excitation

In the last section we have revisited the well known (e.g. from [AE87]) solutions for arbitrarypulse widths and no detuning as well as for temporally constant pulses with arbitrary detuning.Here, we present a perturbative approach for the cases in between. Perturbative treatment of theBloch equations is an approach, where the number of interactions with the field is counted andthus convenient for the description of four-wave-mixing techniques [DMR+10; MRH+10; Muk95;DWE+12]. For our approach the constraint is that the pulse is slow, or spectrally thin in comparisonto the detuning: ∂tΩ

Ω ∆ω.4 We start by integrating the Bloch equations (3.2) formally:

ρ12(t) = i

∫ t

−∞dt′ei∆ω(t′−t)Ω(t′)u3(t′), (3.5a)

u3 = u(0)3 − 4

∫ t

−∞dt′Im(Ω∗(t′)ρ12(t′)). (3.5b)

Equation (3.5a) shall be expanded with respect to the detuning, which can be done via multipleintegration by parts, or a treatment in Fourier space:

ρ12(ω) = iF(e−i∆ωt)F(Ω(t)u3(t)),

F(e−i∆ωt) =i

ω + ∆ω

ω<∆ω=

i

∆ω

∞∑n=0

(− ω

∆ω

)n≈ i

∆ω− iω

∆ω2.

This expansion is the better in the lower orders the smaller ω is compared to the detuning ∆ω.Thus, this is a good description for spectrally thin pulses Ω(∆ω) Ω(0), which vanish for larger

4 ∂tΩΩ∼ 1

Tis inversely proportional to the width of a pulse T , e.g. for a Gaussian ∂tΩ

Ω= 1

TtT

and tT

can beconsidered a constant.

59


-1 -0.5 0 0.5 1Time [ps]

-0.04∆ω

-0.02 0 0.02 0.04Energy [eV]

Re(ρ12) Im(ρ12) field Ω

Figure 3.3: Visualization of the slowly envelope approximation in Fourier space by the exactly solvedpolarization for different pulse widths. Spectrally broad pulses (upper figure) excite the systems resonance at∆ω and thus create oscillations with the detuning frequency. For longer pulses, i.e. spectrally thin ones, theresonance is not as strongly excited, but the polarization follows the field adiabatically and the oscillations intime domain vanish.

ω (cf. Fig. 3.3), i.e. ω∆ω 1. Back in time domain this results in a series of differential operators,

with the aforementioned constraint ∂tΩ ∆ωΩ:

ρ12(t) = − 1

∆ω

∞∑n=0

(− i∂t

∆ω

)nΩ(t)u3(t)

∂tΩ∆ωΩ≈ 1

∆ω

(Ω(t)u3(t) + i

1

∆ω∂tΩ(t)u3(t)

).

(3.6)

The polarization follows the electric field adiabatically, this can be thought of oscillations of thepolarization being averaged out.

However, the dynamics of the polarization and the inversion are still coupled. In order to arriveat a set of uncoupled differential equations an expansion in the order of the electric field is applied:

ρ =∑i

ρ(i); ρ(i) ∼ |Ω|i; u3 =∑i

u(i)3 ; u

(i)3 ∼ |Ω|

i.

Equations (3.5a) then couple a quantity of the order n+ 1 to the complete dynamics of the electricfield times a quantity of the order n.

ρ(n+1)12 (t) = i

∫ t

−∞dt′ei∆ω(t′−t)Ω(t′)u

(n)3 (t′), (3.7a)

u(n+1)3 = −4

∫ t

−∞dt′Im(Ω∗(t′)ρ

(n)12 (t′)). (3.7b)

60


Now the dynamics can be treated iteratively, assuming that the zeroth order is constant, i.e. noother interactions than the field, and inserting this value into the first order:

u(0)3 = const., ρ

(0)12 = 0, (3.8)

ρ(1)12 (t) ≈ u

(0)3

∆ω

(Ω(t) + i

1

∆ω∂tΩ(t)

). (3.9)

Here, the first order inversion u(1)3 vanishes due to the starting condition ρ(0)

12 = 0, which leads tothe property that the inversion exists only in even orders and the polarization only in odd orders (fortwo level systems without permanent dipoles). Subsequently, the second order inversion can bederived from the first order polarization, third order polarization from the second order inversionand so on. This procedure can be taken to arbitrary orders and written in general:

u3(t) =

[ ∞∑n=0

(−1)n1

∆ω2n

(2n)!

(n!)2|Ω|2n

]u

(0)3 , (3.10a)

ρ12(t) =

[ ∞∑n=0

(−1)n1

∆ω2n+1

(2n)!

(n!)2

(Ω(t)|Ω|2n(t) + i

1

∆ω∂t′ |Ω|2n(t′)Ω(t′)

)]u

(0)3 . (3.10b)

We proof this via mathematical induction in App. B. Figure 3.4 compares the perturbative solutionup to n = 2 to the numerically exact solution. For larger fields the perturbative description isnot accurate in low orders, since saturating effects are missing. For too short pulses oscillationsof the polarization reside in the system which cannot be described due to the slowly varyingenvelope approximation in Eq. 3.6, where we omitted oscillating behavior of the inversion. Theoscillating behavior of the exact non-resonant solution is not reproduced, but the mean values can beapproximated: The exact solution of the Bloch equations for a detuned steady field (cf. Eqs. (3.4))cannot be reproduced since it relies on the oscillations of the quantities, that are not present in theperturbative, adiabatic treatment. However the presented solution for pulses is close to the meanvalue of the steady field solution.

For further use, i.e. Chap. 4, we omit the time derivative consistent with the slowly varyingenvelope approximation.5 We will use the solution up to fifth order, namely up to n = 2:

ρ12(t) ≈[

1

∆ωΩ(t)− 2

∆ω3Ω(t)|Ω|2(t) +

6

∆ω5Ω(t)|Ω|4(t)

]u

(0)3 . (3.11)

The non-linear response presented here corresponds to the Kerr effect and leads to the NonlinearSchrodinger equation, if taken into account as optical response to a wave equation. This equation isdiscussed in detail in Sec. 4.33.

5For real pulse amplitudes this also omits the imaginary part of the polarization. In contrast to the real part it has asmall amplitude that even weakens with slower pulses.

61

3.2. DISSIPATION IN QD BLOCH EQUATIONS IN RELAXATION TIME APPROXIMATION

0

0.4

-0.5 0 0.50102030405060

Pola

rizat

ion

Elec

tric

field

Time (ps)

a)

0

0.4

-0.5 0 0.50102030405060

Pola

rizat

ion

Elec

tric

field

Time (ps)

b)

0

0.4

-0.5 0 0.50102030405060

Pola

rizat

ion

Elec

tric

field

Time (ps)

c)

Re(ρ12) pert.Im(ρ12) pert.

Re(ρ12) fullIm(ρ12) full

electric field

Figure 3.4: Perturbative solution of the optical Bloch equations up to fifth order in the field comparedto numerical exact solution. The perturbative approximation does not exhibit oscillations but follows thepulse adiabatically. a) A slow pulse with too large amplitude excites such that a low order perturbativedescription is not sufficient. b) The excitation stems from a slow pulse with small amplitude, the perturbativedescription matches the full solutions. c) The exciting pulse is too short, such that the slowly varying envelopeapproximation fails and persistent oscillations occur.

3.2 Dissipation in QD Bloch equations in relaxation time approxima-tion

The last section covered conservative Bloch equations. However, the aim of this work is to dealwith dissipative wave equations derived from QD dynamics (cf. Sec. 4.3.3). There are two ways toturn on dissipation:

i) input and output of carriers (relaxation/capturing) due to interaction with a carrier reservoiras presented in Sec. 2.2. We will refer to this process as pumping.

ii) On the other hand there is loss due to decoherence, i.e. loss of quantum mechanical infor-mation. Such decoherence is caused by capture and relaxation processes as in i), but alsowithout change of occupations as so called pure dephasing as discussed in Sec. 2.1.1.

As in the previous section one can derive the equations of motion from the Hamiltonian

H = Hel + Hel-field

via Heisenberg’s equation of motion i~ ∂∂tO = [O,H]. This time we additionally use a phenomeno-

logical treatment of the diagonal interaction with a phonon bath Hph,d like in Sec. 2.1.1 as well asphonon mediated interaction with carrier reservoirs Hph,nd as in Sec. 2.2, to introduce relaxation

62


and dephasing rates.

i~∂

∂tρ12 = (ε2 − ε1)ρ12 − ~d21 · ~Eu3 − i~γρ12, (3.12a)

i~∂

∂tρ11 = ( ~E · ~d12ρ12 − ~E · ~d21ρ21)− i~Γ1 (ρ11 − ρ∞11) , (3.12b)

i~∂

∂tρ22 = −( ~E · ~d12ρ12 − ~E · ~d21ρ21)− i~Γ2 (ρ22 − ρ∞22) . (3.12c)

The polarization dephases with a rate γ, and the densities in the states 1, 2 are driven to their steadystate occupations ρ∞ with rates Γ1/2 due to the interaction with a respective reservoir. Those ratesor time constants are calculated in Sec. 2.2. To create similar equations as in the conservative casein Sec. 3.1 we have to use the inversion as well as the total carrier density and thus relaxation ratesin center of mass and relative notation.

Γ =1

2(Γ1 + Γ2) , ∆Γ =

1

2(Γ1 − Γ2) . (3.13)

In comparison an additional equation for the total carrier number u4 = ρ11 + ρ22 shows up. Thisyields the dissipative Bloch equations, where neither energy, nor occupation is conserved.

ρ12 = −i∆ωρ12 + iΩu3 − γρ12, Ω =d12

~E, (3.14a)

u3 = −4Im(Ω∗ρ12)− Γ(u3 − u∞3 )−∆Γ(u4 − u∞4 ), ∆ω = ω21 − ω, (3.14b)u4 = −∆Γ(u3 − u∞3 )− Γ(u4 − u∞4 ). (3.14c)

Here, the phase space is not conserved, but in the long time limit the Bloch sphere (2Re(ρ12), 2Im(ρ12), u3)will shrink to one point (0, 0, u∞3 ). We will concentrate on ∆Γ = 0, i.e. the symmetric case whereboth levels have similar relaxation rates from the reservoir, which leads to a quasi carrier numberconservation.

ρ12 = −i∆ωρ12 + iΩu3 − γρ12, (3.15a)u3 = −4Im(Ω∗ρ12)− Γ(u3 − u∞3 ) (3.15b)

Note that this approximation corresponds to the usual treatment in atomic optics, where Γ corre-sponds rather to non-radiative recombination or direct pumping by another optical field, than to acoupling to a carrier reservoir. Hence we deal with the well known Bloch equations in relaxationtime approximation [AE87].

3.2.1 Perturbative approach

As in the conservative case in Sec. 3.1 we can solve the Bloch equations formally and expandin orders of the field. The approach is indeed the same, but the steps are more elaborate in the

63


dissipative case due to additional terms. Here, we do not give a general formula, but explicitlyderive the solution up to the fifth order. Equations (3.15) are integrated formally.

ρ12(t) = i

∫ t

−∞dt′ei(∆ω−iγ)(t′−t)Ω(t′)u3(t′), (3.16a)

u3 = u∞3 − 4

∫ t

−∞dt′eΓ(t′−t) (Im(Ω∗(t′)ρ12(t′))

). (3.16b)

As in Eq. (3.6), we use the approximation of a slowly varying pulse envelope. This time the linearspectra of the pulse are not delta like, but also broadened by γ. Now, the prerequisite of the slowlyvarying envelope is ∂tΩ

Ω √

∆ω2 + γ2 which is, as before, the spectral distance from the twolevel system’s resonance, which is now shifted on the imaginary axis by γ, cf. Fig. 3.5. By applyingthe approximation we get:

ρ12(t) = − 1

∆ω − iγ

∞∑n=0

(− i∂t

∆ω − iγ

)nΩ(t)u3(t)

≈ ∆ω + iγ

∆ω2 + γ2

(Ω(t)u3(t) + i

∆ω + iγ

∆ω2 + γ2∂tΩ(t)u3(t)

)=

1

a+

[(∆ω + iγ) Ω(t)u3(t) +

1

a+(ia− − 2∆ωγ) ∂tΩ(t)u3(t)

], (3.17)

with a± = ∆ω2 ± γ2.As is Sec. 3.1 we choose a perturbative approach by iteratively solving the polarization in orders

of the electric field:

u(0)3 = const. = u∞3 , (3.18)

ρ(1)12 (t) =

u(0)3

a+

[(∆ω + iγ) Ω(t) +

1

a+(ia− − 2∆ωγ) ∂tΩ(t)

], (3.19)

u(2)3 = −4

∫ t

−∞dt′eΓ(t′−t)

(Im(Ω∗(t′)ρ

(1)12 (t′))

). (3.20)

The difference to the conservative case in Sec. 3.1 is that now real and imaginary part of thepolarization are both dependent on the field and its derivative. Hence, the second order inversionconsists of two terms: an integral over Ω∂tΩ

∗, as in the conservative case and additionally theintegral over |Ω|2. Going up to higher orders, every term in the inversion leads to two in thepolarization. In the conservative case, Eq. (3.10), half of the polarization terms (∼ ∂tΩ) affectedthe inversion, thus polarization of every order always consisted of two parts, inversion of everyorder of one term (cf. Eq. (3.10)). Now the inversion has the same number of terms as the previouspolarization leading to 2n/2 (2(n+1)/2) terms for the nth (n+ 1st) order inversion (polarization).Accordingly, we will not give a general solution in this section, but derive the solution explicitly up

64


0

γ

√∆ω2 + γ2

0 ∆ω√

∆ω2 + γ2

Im(ω

)

Re(ω)

γ

γ/∆ω= -0

γ/∆ω= -0.125

-1 -0.5 0 0.5 1Time [ps]

γ/∆ω= -2

-0.04-0.02 0 0.02 0.04Energy [eV]

Re(ρ12) Im(ρ12) field Ω

Figure 3.5: Visualization of the approximation Eq. (3.17) in Fourier space. The resonance of the two levelsystem is expanded around the center frequency of the pulse ω = 0. a) the resonance of the two level systemis shifted by γ on the imaginary axis. For the quality of the approximation the ω is compared to the distanceof the resonance, which is a circle with the radius

√∆ω2 + γ2. b) a short (spectrally broad) pulse excites

the system. In the upper panel there is no dephasing such that the resonance of the system is excited andoscillations occur in the time domain. With a small dephasing (middle panel) the same pulse results in aweaker (and broadened) excitation of the system and a more significant adiabatic excitation, in the timedomain the oscillations are damped out. For strong dephasing (lower panel) the pulse excites the systemadiabatically and exists only on the timescale of the pulse.

to the fifth order. We use again Eq. (B.5): Ω∂tΩ∗ = 1

2∂t|Ω|2 − i|Ω|2∂tϕ, yielding the second order

inversion explicitly

u(2)3 = −4

u(0)3

a+

(γ

∫ t

−∞dt′eΓ(t′−t)|Ω|2 +

a−2a+

∫ t

−∞dt′eΓ(t′−t)∂t′ |Ω|2

+2∆ωγ

a+

∫ t

−∞dt′eΓ(t′−t)|Ω|2∂t′ϕ(t′)

). (3.21)

In comparison to the conservative case an additional memory integralM (1)(t) =∫ t−∞ dt

′eΓ(t′−t)|Ω|2

enters proportional to γ and has the property that ∂tM (1)(t) = |Ω|2(t) − ΓM (1)(t). This termrepresents the damped pulse energy. The second integral can be treated via integration by parts:∫ t

−∞dt′eΓ(t′−t)∂t′ |Ω|2 = |Ω|2 − Γ

∫ t

−∞dt′eΓ(t′−t)|Ω|2 = |Ω|2 − ΓM (1)(t) = ∂tM

(1)(t).

(3.22)

Luckily, this type of memory has the general property that integration and differentiation interchange,which will be advantageous for higher orders:∫ t

−∞dt′eΓ(t′−t)∂t′f(t′) = f(t)− Γ

∫ t

−∞dt′eΓ(t′−t)f(t′) = ∂t

∫ t

−∞dt′eΓ(t′−t)f(t′). (3.23)

65


For the further procedure we will neglect the chirp term ∂tϕ, in consistence to the slowly varyingenvelope approximation. This leads to a compact description of the second order inversion:

u(2)3 = −4

u(0)3

a+

(a−a+

1

2∂tM

(1) + γM (1)

). (3.24)

Note, that this ∂tM corresponds to the adiabatic following behavior from the conservative solutionEq. (3.10a). Hence, explicitly written, the inversion now consists of an adiabatically following term,as well as a memory term, that includes the relaxation via the rate Γ:

u(2)3 = −4

u(0)3

a+

(a−a+

|Ω|2

2+

(γ − Γ

2

a−a+

)M (1)

). (3.25)

The third order polarization is acquired straight forward, inserting the second order inversionEq. (3.24) into Eq. (3.17) leads to the compact representation:

ρ(3)12 = −4

u(0)3

a2+

[(∆ω + iγ)

(a−2a+

Ω(t)∂tM(1) + γΩ(t)M (1)(t)

)+

(ia−a+− 2

∆ωγ

a+

)(a−2a+

∂tΩ(t)∂tM(1) + γ∂tΩ(t)M (1)

)]. (3.26)

To distinguish the adiabatic and memory contributions, the derivatives ∂tM (1) = |Ω|2 − ΓM areexplicitly applied, leading to:

ρ(3)12 = −4

u(0)3

a2+

[∆ω

(a−2a+

(1 + 2

γΓ

a+

)− 2

γ2

a+

)+ i

a−2a+

(3γ − 2Γ

a−2a+

)]Ω(t)|Ω|2

+

(ia−a+− 2

∆ωγ

a+

)[a−2a+

∂tΩ(t)|Ω|2 +

(γ − Γ

a−2a+

)M (1)(t)∂tΩ(t)

]+

[∆ω + 2Γ

∆ωγ

a++ i

(γ − Γ

a−a+

)](γ − Γ

a−2a+

)M (1)(t)Ω(t)

. (3.27)

The third order polarization is composed of four components: an adiabatically following part, apart following the derivative of the field, terms depending on the memory as well as a mixtureof memory and derivative terms. Every component contributes to real and imaginary part of thepolarization.

By means of those lowest orders we can describe already the influence of dephasing on relaxationrates. As in Sec. 3.1 the higher orders are a corrective for the perturbation theory. Third order willbe of major importance in Sec. 4.3.3. The effect of dephasing can be investigated in the two limitsof fast and slow pumping.

66


0.000.040.080.120.16

Re(ρ

12)

γ/∆ω = 0

perturb. solutionadiabatic part

derivative partmemory part Γ = 0

γ/∆ω = 0.25 γ/∆ω = 1 γ/∆ω = 4

×10

γ/∆ω = 16

×100

-0.08

-0.04

0.00

Im(ρ

12)

-1.00

-0.80

-0.60

-0.40

-0.5 0.0 0.5

u3

-0.5 0.0 0.5 -0.5 0.0 0.5Time [ps]

-0.5 0.0 0.5 -0.5 0.0 0.5

Figure 3.6: Components of inversion, and polarization, Eqs. (3.28), up to second or third order, respectively,for a Gaussian pulse in the slow pumping limit Γ = 0. The solid line is the perturbative description, whereasthe shaded area represents the full numerical solution of the Bloch equations. The first column represents theconservative case, cf. Sec. 3.1.2: For vanishing dephasing the real part of the polarization as well as theinversion follow the pulse adiabatically, while the imaginary part of the polarization follows the derivativeof the pulse. With increasing dephasing, the influence of memory terms (dash-dotted lines) increases untilthe dephasing reaches the value of the detuning. For higher dephasing the leading prefactor a−1

+ affectsthe overall response: The dephasing destroys the coherence of the polarization and as such diminishes theinteraction with the electric field.

Slow pumping limit Γ→ 0. Here,M (1) becomes the pulse energy∫ t−∞ dt

′|Ω|2, whose derivativeis just the absolute square of the pulse |Ω|2. In this case the inversion does not return to its steadystate after the pulse passed through. Similar to the conservative case its value after the pulse dependson the pulse’s energy. This influence is large if the dephasing rate is faster, since the polarizationvanishes and cannot assist the decay of the inversion. The influence of dephasing is shown in

67


Fig. 3.6. The slow pumping limit of the polarization ρ12 = ρ(1)12 + ρ

(3)12 is:

Re(ρ12) ≈ u(0)3

a+∆ω

[(1− 2∆ω2

a2+

|Ω|2 + 6γ2

a2+

|Ω|2 − 4γ

a+M (1)(t)

)Ω(t)

−2γ

a+

(1− 6

1

a+

a−a+|Ω|2 − 4

γ

a+M (1)

)∂tΩ(t)

], (3.28a)

Im(ρ12) ≈ u(0)3

a+

[(1− 6a−

a2+

|Ω|2 − 4γ

a+M (1)(t)

)(γΩ(t) +

a−a+∂tΩ(t)

)], (3.28b)

u3 = u∞3

[1− 4

a+

(a−a+

|Ω|2

2+ γM (1)

)]. (3.28c)

We can see for non-vanishing γ (specifically γ > a−a+

γ2

∆ω2 >1−γ1+γ ) the inversion is dominated by

the memory contributions, which has its maximum at γ = ∆ω (because γa+

is maximal there). Theimaginary part of the polarization is dominated by the adiabatic behavior for larger γ also due to thesame relation. Since the behavior can be mostly characterized with the parameter γ

∆ω the effectivedephasing is influenced by the detuning. Figure 3.6 also shows that, as stated before, the approachgets better with increasing γ, since the slowly varying envelope approximation gets better. Theconservative limit (cf. Eq. (3.11)) is reproduced for vanishing dephasing γ → 0.

68


0.000.040.080.120.16

Re(ρ

12)

γ/∆ω = 0

perturb. solutionadiabatic part

derivative partmemory part Γ = γ

2

γ/∆ω = 0.25 γ/∆ω = 1 γ/∆ω = 4

×10

γ/∆ω = 16

×100

-0.08

-0.04

0.00

Im(ρ

12)

-1.00

-0.80

-0.5 0.0 0.5

u3

-0.5 0.0 0.5 -0.5 0.0 0.5Time [ps]

-0.5 0.0 0.5

×10

-0.5 0.0 0.5

×100

Figure 3.7: Components of inversion, and polarization, Eqs. (3.30), up to second or third order, respectively,for a Gaussian pulse in the fast pumping limit Γ = 0.5× γ. The solid line is the perturbative description,whereas the shaded area represents the full numerical behavior of the Bloch equations. The first columnrepresents the conservative case, cf. Sec. 3.1.2: For vanishing dephasing the real part of the polarizationas well as the inversion follow the pulse adiabatically, while the imaginary part of the polarization followsthe derivative of the pulse. The adiabatic description of the polarization approximates the full solution well,getting better with increasing dephasing. The inversion is described by the memory terms, that also showadiabatic behavior, cf. Fig. 3.7.

Fast pumping ∂t|Ω|2 Γ|Ω|2 Now the relaxation rate of the inversion is much faster, then thepulse duration, and the memory integrals can be approximated by the intensity:

M (n) = Γ−1|Ω|2n − Γ−1∂tM(n) ≈ Γ−1|Ω(t)|2n. (3.29)

Polarization and inversion in the fast pumping limit consist only of adiabatic terms:

ρ12 =u

(0)3

a+

[(∆ω + iγ)

(1− 4

a+

γ

Γ|Ω|2

)Ω(t) +

(ia−a+− 2

∆ωγ

a+

)(1− 12

γ

Γ|Ω|2

)∂tΩ(t)

],

(3.30a)

u3 = u∞3

[1− 4

a+

γ

Γ|Ω|2

]. (3.30b)

69


The strength of the higher order terms is given by the ratio γΓ . Since relaxation (Γ) and dephasing

rate (γ) are correlated γ = 2Γ + γpure (cf. Sec. 2.2.3), in the fast pumping limit the dephasingis completely defined by the relaxation rates γ ≈ 2Γ. The fast pumping limit up to second/thirdorder is shown in Fig. 3.7 for different values of dephasing. All quantities show clear adiabaticbehavior. As in the slow pumping case the real part polarization in monotonously decreasing withincreasing dephasing and the imaginary part has the maximal strength at γ = ∆ω because γ

a+is

maximal there. Also as in the slow pumping case the approximate description gets better withstronger dephasing. The main difference is the now adiabatic dispersion, which is clear, since in thefast pumping limit the influence of the memory becomes adiabatic.

Higher order corrections. To calculate the fourth order inversion the next step is again multiply-ing Eq. (3.26) by Ω and bringing the derivative in front, then write the memory terms of the nextorder using

Ω∗∂tΩM(1)0 =

1

2|Ω|2M (1)

0 +1

2|Ω|2∂tM (1)

0 − i|Ω|2M (1)0 ∂tϕ. (3.31)

Again the time dependence of a pulse’s phase ∂tϕ is neglected, leading to the fourth order inversionin the compact form:

u(4)3 = (−4)2u

(0)3

a2+

[γ

(a−2a+

N(2)1 + γN

(2)0

)+a−a+

1

2

(a−2a+

∂tN(2)1 +

a−2a+

N(2)2 + γ∂tN

(2)0 + γN

(2)1

)], (3.32)

with the generalized memory integrals

N(i)j,k =

∫ t

−∞dt′eΓ(t′−t)|Ω|2∂jtN

(i−1)k ,

containing derivatives of lower order generalized memory integrals. For i = 1 it is just the memoryintegral we used before N (1) = M (1). As we will see shortly they can be expressed as a sumof memory integrals without derivatives. Here, especially N (2)

j =∫ t−∞ dt

′eΓ(t′−t)|Ω|2∂jtM (1) isvalid. For further use the explicit form can be acquired via extracting the derivatives in the memoryintegrals:

N(2)0 = N

(2)0 , ∂tN

(2)0 = |Ω|2M (1) − ΓN

(2)0 ,

N(2)1 = M (2) − ΓN

(2)0 , ∂tN

(2)1 =

(|Ω|4 − ΓM (2)

)− Γ

(|Ω|2M (1) − ΓN

(2)0

),

N(2)2 =

1

2∂tM

(2) − ΓN(2)1 =

1

2

(|Ω|4 − 3ΓM (2)

)+ Γ2N

(2)0 .

The fourth order inversion consists of adiabatic following terms ∼ |Ω|4, memory of the pulse’senergy ∼M (1) and the corresponding square ∼M (2) as well as generalized memory ∼ N (2)

0 . It

70


reads explicitly written:

u(4)3 = (−4)2u

(0)3

a2+

a−2a+

[a−2a+

3

2|Ω|4 +

(2γ − a−

2a+

5

2Γ

)M (2)

+

(γ − a−

2a+Γ

)|Ω|2M (1) +

(2a+

a−γ2 − 3Γγ + 2

a−2a+

Γ2

)N

(2)0

]. (3.33)

Higher orders are proportional to a−(n+1)/2+ , which means for increasing γ the higher orders become

less important. The polarization of fifth order is acquired by inserting Eq. (3.33) and its derivative

∂tu(4)3 = (−4)2u

(0)3

a2+

a−2a+

[a−2a+

6|Ω|3∂t|Ω|+(

2γ − a−2a+

5

2Γ

)(|Ω|4 − ΓM (2)

)+

(γ − a−

2a+Γ

)(M (1)|Ω|∂t|Ω|+ |Ω|4 − Γ|Ω|2M (1)

)+

(2a+

a−γ2 − 3Γγ + 2

a−2a+

Γ2

)(|Ω|2M (1) − ΓN

(2)0

)]. (3.34)

into Eq. (3.17) to yield:

ρ(5)12 = (−4)2u

(0)3

a3+

a−2a+

(∆ω + iγ)

[a−2a+

3

2|Ω|4Ω +

(2γ − a−

2a+

5

2Γ

)M (2)Ω

+

(γ − a−

2a+Γ

)|Ω|2ΩM (1) +

(2a+

a−γ2 − 3Γγ + 2

a−2a+

Γ2

)N

(2)0 Ω

]+

ia− − 2∆ωγ

a+

[a−2a+

15

2|Ω|4∂tΩ +

(3γ − a−

2a+

7

2Γ

)|Ω|4Ω

−(

2γ − a−2a+

5

2Γ

)ΓM (2)Ω−

(γ − a−

2a+Γ

)(Γ|Ω|2ΩM (1)

)+

(2a+

a−γ2 − 3Γγ + 2

a−2a+

Γ2

)(|Ω|2ΩM (1) − ΓN

(2)0 Ω

)+

(2γ − a−

2a+

5

2Γ

)M (2)∂tΩ

+2

(γ − a−

2a+Γ

)|Ω|2M (1)∂tΩ +

(2a+

a−γ2 − 3Γγ + 2

a−2a+

Γ2

)N

(2)0 ∂tΩ

]. (3.35)

To describe non-linear response for Chap. 4 we are interested in the adiabatic terms. As for thethird order those terms give the main contribution to the solution of the polarization. We refrainfrom describing every term in the lengthy expression, but show, as for the third order, the slow and

71


the fast pumping limit of Eq. (3.35). In the slow pumping limit Γ→ 0 the polarization reads:

ρ(5)12 = (−4)2u

(0)3

a3+

a−2a+

(∆ω + iγ)

a−2a+

3

2|Ω|4Ω +

ia− − 2∆ωγ

a+3γ|Ω|4Ω

+ (∆ω + iγ)[2γM (2)Ω +γ|Ω|2ΩM (1) +

(2a+

a−γ2

)N

(2)0 Ω

]+

ia− − 2∆ωγ

a+

(2a+

a−γ2

)(|Ω|2ΩM (1)

). (3.36)

Whereas in the fast pumping limit the memory terms follow again adiabatically M (n) = Γ−1|Ω|2nas will the generalized memories N (2) = Γ−1M (2) = Γ−2|Ω|4.

ρ(5)12 = (−4)2u

(0)3

a3+

(∆ω + iγ)γ2

Γ2|Ω|4Ω. (3.37)

For use in Chap. 4 we will use the adiabatic parts of Eq. (3.35), that includes the adiabatic part ofthe slow pumping limit, as well as relaxation rate contributions, but not the fast pumping limit. Thereal part reads:

Re(ρ12) = ∆ωu∞3a+

1− 4

(a−2a+

(1 + 2

γΓ

a+

)− 2

γ2

a+

)|Ω|2

a++

+(4)2 a−2a+

[a−2a+

(3

2+ 7

1

a+Γγ

)− 6

γ2

a+

]|Ω|4

a2+

Ω, (3.38a)

whereas the imaginary part is:

Im(ρ12) = γu∞3a+

1− 4

a−2a+

(3− 2

Γ

γ

a−2a+

)|Ω|2

a++

+(4)2

(a−2a+

)2 [15

2− 7

Γ

γ

a−2a+

]|Ω|4

a2+

Ω. (3.38b)

72


3.2.2 Saturable absorber – adiabatic Ansatz for the dissipative Bloch equations

Another possible treatment of the dissipative Bloch equations Eq. (3.15) is similar to the casepresented in the last section, but uses the adiabatic following of the polarization and inversion rightfrom the start, i.e. (∂tρ)/ρ ∼ ∆ω γ,Γ. In this case the Bloch equations are in the quasi-steadystate [AE87].

0!

= ρ12 = −i∆ωρ12 + iΩu3 − γρ12, (3.39a)

0!

= u3 = −4Im(Ω∗ρ12)− Γ(u3 − u∞3 ). (3.39b)

The steady state solution in dependence on the field Ω are given by:

ρ12 =Ωu3

∆ω − iγ= (∆ω + iγ)

Ωu3

∆ω2 + γ2, (3.40a)

u3 = u∞3 −4

ΓIm(Ωρ12). (3.40b)

Indeed Eq. (3.40a) resembles the zeroth order of the slowly varying envelope approximation ofEq. (3.17) and Eq. (3.40b) is the corresponding inversion in the fast pumping limit (cf. Eq. (3.16b)).That means the description from Sec. 3.2.1 is just between this regime and the conservative case.Those coupled equations can be consistently solved to yield

u3 =

(1 + 4

γ

Γ

|Ω|2

∆ω2 + γ2

)−1

u∞3 , (3.41)

ρ12 =∆ω + iγ

∆ω2 + γ2

(1 + 4

γ

Γ

|Ω|2

∆ω2 + γ2

)−1

u∞3 Ω. (3.42)

The behavior can be boiled down to two parameters a = ∆ω+iγ∆ω2+γ2u

∞3 and b = 4 γΓ

1∆ω2+γ2 .

u3

u∞3=

1

1 + b|Ω|2, ρ12 =

a

1 + b|Ω|2Ω. (3.43)

The description is again adiabatically with intensity dependent coefficients. In contrast to Sec. 3.2.1all orders of th field are contained. The polarization has a maximum at |Ω| =

√b−1. The inversion

decays towards zero. This response describe a saturable absorber/gain depending on the sign ofthe steady state inversion. For small optical fields, |Ω|2 b−1 the response is linear, whereas forhigher fields it saturates and vanishes, cf. Fig. 3.8. We will use the non-linear response for waveequations in Sec. 4.3.3.

73


a2

√1b

u∞3

0√

1b |Ω|

polarizationinversion

-1-0.75-0.5

-0.250

00.40.81.2

γ/∆ω = 0.5

Γ = γ2

-1-0.75-0.5

-0.250

00.40.81.2

γ/∆ω = 3

-1-0.75-0.5

-0.250

-1 -0.5 0 0.5 100.40.81.2

Time [ps]

γ/∆ω = 3

×30

Re(ρ12)

Im(ρ12)

u3

Ω

Figure 3.8: Left panel: behavior of the saturable absorber non-linearity as a function of the field amplitude.For small intensities the polarization grows linearly, until it decreases again after a saturation intensity. Theinversion decreases with increasing field strength. Right panels: behavior of the quantities for a light pulse,the shaded areas represent the numerically exact solution. For too slow rates (upper panel) the exact solutionis asymmetric, i.e. memory effects are important. For fast enough rates (middle panel) the saturable absorbertreatment agrees with the exact solution: the polarization is excited for small fields and diminished for thelarger parts of the pulse, for the decreasing flank of the pulse the situations is the same. For small pulseamplitudes, i.e. where saturation does not occur, polarization and inversion follow the pulse adiabatically.

Polynomial limit

In the regime without saturation, b|Ω|2 1, the solution can be treated via a Taylor expansion orgeometric series, respectively, and becomes comparable to the perturbative treatment:

a

1 + b|Ω|2≈ a

∞∑k=0

(−b)k |Ω|2k

= u∞3∆ω + iγ

a+

∞∑k=0

(−4

γ

Γ

1

a+

)k|Ω|2k, (3.44)

74


which contains the perturbative solutions in the fast pumping limit we presented in the last section,cf. Eqs. (3.30a) and (3.37). The value of the polarization and inversion is determined by the ratioof the dephasing γ and relaxation Γ. For the assumption of vanishing pure dephasing this ratio isconstant γ/Γ = 2, so we can now investigate the conservative limit of those equations, b = 2

∆ω2 .6The difference between the conservative solution Eq. (3.10) and the series in Eq. (3.44) in order nis bn − 1

∆ω2n2n!

(n!)2 = 1∆ω2n

(2n − 2n!

(n!)2

), which is small for low orders.

3.3 Conclusion

In this chapter we started with the Bloch equations with and without dephasing and relaxation rates.The Bloch equations are a coupled systems of differential equations with a non-linear dependenceon the optical field. We showed derivations for the polarization in the different cases to use them inthe following Chap. 4 as non-linearity. To summarize, we used the following limits for a closeddescription:

• The conservative limit (Sec. 3.1.2) for negligible dephasing and relaxation yielded a pertur-bation series of arbitrary order (Eqs. (3.10)).

• With dephasing and relaxation rates (Sec. 3.2.1) also a perturbative approach for slow fieldswas possible. Within this treatment there were two limits yielding an adiabatic polarization:

– Slow pumping limit, with a negligible relaxation rate Γ

– In the fast pumping limit, where we assumed a relaxation rate that is faster than thepulse length Γ ∂t|Ω|2

|Ω|2

In both limits for higher dephasing rates the behavior of the polarization became moreadiabatic and thus the shown description more accurate.

• Saturable absorber treatment (Sec. 3.2.2) for fast relaxation rates (fast pumping limit) resultsin a non-perturbative expression, which can be expanded for small fields to result in the fastpumping limit of the perturbative approach.

6This limit seems of course inconsistent since we assumed fast rates, however it shows the quality of the perturbativeapproach, which is a description between conservative and fast pumping limit.

75

3.3. CONCLUSION

76

O eines Pulses Dauer nur Allwissenheit!Friedrich Schiller, Don Carlos

4Non-linear wave equations describing QD devices

– The search for dissipative solitons

In the previous chapter the material response of QDs to pulsed excitation was studied. We progressin this chapter by including the back-action of the QD response to the external pulse and investigatehow optical pulses can be shaped, when propagating through QD media. Therefore, the generalwave equation for pulses in semiconductors is briefly derived in Sec. 4.2. In the next sectionthe non-linear response from Chap. 3 is taken into account. This leads to well known non-linearequations, whose general features are discussed in Sec. 4.3. Further, those equations are known tosupport stable solutions, so called solitary waves and solitons, in the conservative as well as thedissipative regime. First, analytical solutions in the conservative regime are discussed, and finally,in Sec. 4.3.3 the formation of solitons in the dissipative regime is explored.

4.1 Introduction to non-linear wave equations and solitons

In this work we restricted ourselves to wave equations of the generalized Ginzburg-Landau type. Ofcourse there are many other non-linear equations with soliton solutions and interesting properties,we want to review briefly in this section.

Historically, the soliton story begins in 1834, when John Scott Russell observed a single waterwave, later called a solitary wave, that was constant in shape and velocity over a long distance in anEnglish channel. The theoretical description of the phenomenon followed in 1895 by Korteweg and

4.1. INTRODUCTION TO NON-LINEAR WAVE EQUATIONS AND SOLITONS

De Vries [KV95] describing the small elevation η on the plane water surface with the depth l:

∂η

∂t=

3

2

√g

l

∂

∂x

(1

2η2 +

2

3αη +

1

3σ∂2η

∂2x

). (4.1)

with α being an arbitrary parameter and σ = 13 l

2 − T lρg depending on the water depth as well as the

surface tension T and the density ρ. They show the solitary wave solution

η = h sech2

(x

√h

4σ

).

Nowadays the Korteweg-deVries (KdV) equation is probably the most known non-linear waveequation. The term soliton was coined in 1965 by Zabusky and Kruskal, when they numericallysolved the KdV equation to describe a collisionless plasma and found the solitons were left un-changed after collisions with each other [ZK65]. The next big event in the history of solitons wasthe inverse scattering theory by Gardner et al in 1967 [GGK+67], which provides a general methodto analytically derive the fundamental and higher order solutions of the KdV equation. Higher ordersolutions can be interpreted as multiple solitons scattering with each other. The inverse scatteringmethod can be also used to obtain soliton solutions of various non-linear wave equations as thenon-linear Schrodinger equation (NLSE) (which is featured in Sec. 4.3.2) as well as the sine-Gordonequation which we will shortly discuss below.

The NLSE plays a major role for the description of optical pulses in a fiber [Has90; Mit10].Here, the non-linear response, also referred to as the intensity dependent refractive index, modulatesthe phase of an optical pulse. As predicted by Hasegawa and Tappert in 1973 this self-phasemodulation in combination with the dispersion in fibers can lead to the existence of solitons.

The last well known non-linear equation, which was also found to have soliton solutions in thissection is the sine-Gordon equation. This equation is especially noteworthy for the description ofthe resonant response from discrete level systems, as QDs:

In Sec. 3.1.1 we showed, that the resonant response of QDs to an optical field is the sine of thepulse area, which leads just to the sine-Gordon equation:

(∂z − k′z∂t)︸︷︷︸∂ξ

E =d21ω

2l

2kzε0c2sin

d12

~

∫ τ

−∞dt′E(t′)︸︷︷︸

Θ

, ξ = z − k′z−1t; τ = t (4.2)

⇒ ∂ξ∂τΘ =|d21|2

4~ε0csin Θ. (4.3)

This is a conservative wave equation for the pulse area. To find the fundamental soliton solutionthe coordinate frame ξ′ = z − vsolitont; τ = t is useful. Here, the velocity of the soliton solutionenters as a free parameter. Those coordinates yield the equation

∂2τΘ =

(1

vsoliton− k′z

)−1 |d21|2

4~ε0c︸︷︷︸T−2

sin Θ. (4.4)

78

CHAPTER 4. THE SEARCH FOR DISSIPATIVE SOLITONS

This can be easily solved to yield:

Θ = 4 tan−1 eτ/T ⇒ E =1

Tsech

(t− z/vsoliton)

T(4.5)

This is a soliton solution where the electric field is a hyperbolic secant with a pulse area of 2π. As inthe NLSE case the height and the width of the pulse are inversely proportional. In the sine-Gordon

case additionally the relative velocity vsoliton = k′−1z

(1 + k′z

T 2|d21|24~ε0c

)−1depends on the width of the

pulse. Short pulses have a slow velocity whereas for long pulses the velocity approaches the groupvelocity. The phenomenon is called self induced transparency and was also investigated on theinfluence of relaxation and dephasing rates on the propagation [MH69]. The sine-Gordon equationis well studied and a description of the fundamental and higher order solutions can be found in[Man10]. In Ref. [ACM97] a generalized sine-Gordon equation with dissipation is investigated fordifferent properties in the conservative and the dissipative regime and the transition between theregimes.

The soliton concept can be generalized to dissipative systems, i.e. open systems that allowan energy exchange with the environment [AA05a; AA08a]. The field of dissipative solitonscovers the description physical, chemical and biological systems. One category, interesting in thesemiconductor optics context, but not covered by this thesis, are cavity solitons [Lug03]. Here, theactive material inside a cavity is pumped and has to offer to bistable operation, i.e. two differentlasing intensities. The solitons are then spatial spots with a higher intensity on a darker background.The formation of those solitons is a two dimensional spatial problem, the velocity can be influencedby external fields and they offer a rich interaction behavior [BMP+07; GSS+02; PGA+08; PBM+05].

4.2 Wave equation – slowly varying envelope approximation

To describe the propagation of optical pulses a wave equation for a full electric field ~E was derivedfrom Maxwell’s equations in Sec. 1.2.3. To obtain a wave equation for optical pulses with a slowlyvarying envelope, we start with the typical wave equation Eq. (1.32):

∆ ~E − 1

c2

d2

dt2~E = µ0

d2

dt2~P,

with the macroscopic polarization ~P . As stated in Sec. 1.2.3 the polarization connects the resultsfrom Chap. 3 to the propagation of electric fields since it is the sum of all microscopic polarizations,which form the response of the QD medium to the external field. Both, the electric field ~E andpolarization ~P are divided into a slowly varying envelope ~E, ~P and a fast wave part with the carrierfrequency of the light ωl and a corresponding wave vector ~k.~E(t) = ~E(t)e−i~k·~r+iωlt + c.c., ~P (t) = ~P (t)e−i~k·~r+iωlt + c.c., (4.6a)~E(ω) = ~E(ω − ωl)e−i~k·~r + ~E∗(ω + ωl)e

i~k·~r, ~P (ω) = ~P (ω − ωl)e−i~k·~r + ~P ∗(ω + ωl)ei~k·~r.(4.6b)

79

4.2. WAVE EQUATION – SLOWLY VARYING ENVELOPE APPROXIMATION

The slow envelopes ~E, ~P are complex vectors. In this work we will restrict ourselves to one-dimensional propagation in z direction, hence we define ~k = kz~ez without loss of generality.

The wave equation of the envelope E can be written in frequency domain1:(∆x,y + ∂2

z − 2ikz∂z − k2z +

ω2

c2

)~E(∆ω) = −µ0ω

2 ~P (∆ω), ∆ω = ω − ωl. (4.7)

The macroscopic polarization is now split into the part of the passive background medium aswell as the active QD medium ~P = ~Pbg + ~PQD. Where ~Pbg is assumed to be a linear response:~Pbg(ω) = ε0χ(ω) ~E(ω).2As already described in Sec 2.1 χ is a complex number with the real partbeing the index of refraction and the imaginary part corresponding to the absorption.

(∆x,y + ∂2

z − 2ikz∂z − k2z +

ω2

c2(1 + Re(χ(ω)))

)~E(∆ω) = −µ0ω

2 ~PQD(∆ω)− iω2Im(χ)

c2E.

(4.8)

The term K(ω) = ωc

√1 + Re(χ(ω)) represents the dispersion of the background material (cf.

Sec. 2.1). The envelope of a propagating pulse is influenced by the difference of kz and K. Ingeneral kz from the ansatz in Eq. 4.6a can be any function of ωl (with the constraint of obeyingboundary conditions). Here, we have the freedom to choose the pulse’s dispersion relation, whichaffects the solutions for envelopes. We choose kz to be the materials dispersion at the frequencyof the light kz = K(ωl) (Compare to Eq. (2.1) with vanishing Im(χ)) and expand the materialsdispersion K(ω) around ωl:

K(ω) ≈ kz + k′z∆ω +1

2k′′z∆ω2.

This ansatz is inserted into Eq. (4.8) and higher orders than ∆ω2 are neglected. Thus, a waveequation is obtained, where the dispersion of the background medium is included in the derivative,while the active medium enters as a non-linear source term.(

∆x,y + ∂2z − 2ikz∂z + 2kz

∂kz∂ωl

∆ω +

(∂kz∂ωl

)2

∆ω2 + kz∂2kz∂ω2

l

∆ω2

)~E(∆ω)

= −µ0ω2 ~PQD(∆ω)− i

ω2Im(χ)

c2~E(∆ω). (4.9)

Next, we divide the whole equation by 2kz ,3 and assume the term 12kz

(∂kz∂ω

)2 can be neglected.4

1This equation only represents the part proportional to e−i~k·~r , since it is linearly independent of ei~k·~r .2Note that for the Fourier transforms of the envelopes this means, that although the quantities are functions of ∆ω

due to the rotating frame, the connecting susceptibility is still a function of ω, i.e. ~Pbg(∆ω) = ε0χ(ω) ~E(∆ω), becauseit is defined for the full quantities, that are linear superpositions of the envelopes shifted by ±ωl, cf. Eq. (4.6b).

3which is of course only possible for kz 6= 0, which is reasonable for semiconductors below the band gap, cf.Fig. 2.1.

4Alternatively one can define k′′ = 1kz∂ωl (kz∂ωlkz) instead of k′′ = ∂2

ωlkz to arrive at the same wave equation.

80


Finally, the back transformation to the time domain is made in relation to ∆ω which yields thewave equation for the pulse envelope(

1

2kz∆x,y +

1

2kz∂2z − i∂z + ik′z∂t −

1

2k′′z∂

2t

)~E = −

ω2l

2kzc2ε0~PQD − iαE, (4.10)

with the absorption coefficient α = ωl2c

Im(χ)√1+Re(χ)

.

Next, the “Slowly varying envelope” approximation is applied by neglecting ∂zE kzE andthus the second derivative after z.

The spatial behavior perpendicular to the propagating direction can be calculated for specificgeometries of wave guides or cavities. For the phenomenon of cavity solitons the Laplacian ∆x,y isimportant. However, as we constrict ourselves to a one dimensional problem it can be neglected.

This wave equation is typically written in the co-moving frame τ = t−k′z, ξ = z, which meansthe observer at some point z takes into account the group velocity k′−1 for the time dependenceof the propagating pulse. Then the derivatives in those new coordinates are ∂ξ = ∂z + k′∂t and∂τ = ∂t, which leads to the typical partial differential equation with a first order derivative in thepropagation coordinate ξ and a second order derivative in the progression coordinate τ :(

i∂ξ −1

2k′′z∂

2τ

)~E = −

ω2l

2c2ε0~PQD − iαE. (4.11)

4.2.1 Pulse broadening due to group velocity dispersion

A prominent term in Eq. (4.11) is the group velocity dispersion (GVD) k′′ which leads to a broad-ening of the optical pulse due to different phase velocities of the frequency components of the pulse.The regime where k′′ > 0 is called normal dispersion, while k′′ < 0 is referred as anomalousdispersion [Has90].5 The pulse broadening effect of the GVD can be seen with from the waveequation without a non-linear response:

0 = i∂E

∂ξ− k′′

2

∂2E

∂τ2. (4.12)

This equation can be formally solved in frequency domain for known initial conditions via

E(ξ, ω) = E(0, ω)e−i2k′′ω2ξ, (4.13)

E(ξ, τ) =1

2π

∫dωE(ξ, ω)eiωτ . (4.14)

In particular, the effect of broadening due to GVD can be nicely shown on a Gaussian pulse since itstays Gaussian under Fourier transform and Eq. (4.13) is also of Gaussian shape. Since the product

5Sometimes the refractive index n(ω) = cωk(ω) is used to describe those regimes with ∂ωn(ω) > 0 being the

normal dispersion. Both descriptions are related via k′′ = 1c

(2∂ωn+ ω∂2

ωn).

81

4.2. WAVE EQUATION – SLOWLY VARYING ENVELOPE APPROXIMATION

of two Gaussian pulses results in a Gaussian, the broadening is fully analytically accessible. Westart with a Gaussian pulse with an inverse width of γ0 and a chirp κ0:6

E(0, τ) = e−(γ0−iκ0)τ2, (4.15)

E(0, ω) =√π (γ0 − iκ0)−

12 e−

14

(γ0−iκ0)−1ω2. (4.16)

The Fourier transform has an inverse width of γ = 14

γ0

γ20+κ2

0and a “chirp” of κ = −1

4κ0

γ20+κ2

0.

A chirp does not affect the width in time, but broadens the width in frequency domain. Afterthe propagation of a length ξ the pulse is still Gaussian, but with different properties due to theadditionally introduced “chirp” k′′

2 ξ from Eq. (4.13).

E(ξ, ω) =√π (γ0 − iκ0)−

12 e−

14

(γ0−iκ0)−1ω2e−

i2k′′ω2ξ, (4.17)

E(ξ, τ) =(1 + 2i (γ0 − iκ0) k′′ξ

)−1/2e− (γ2

0+κ20)

γ0+i(κ0+2(γ20+κ2

0)k′′ξ)τ2

. (4.18)

In the frequency domain the inverse width is still γ = 14

γ0

γ20+κ2

0, but κ is now modified to−1

4κ0

γ20+κ2

0+

k′′ξ2 , which yields a new width γ(ξ) in time domain depending on the propagated distance:

γ0

γ(ξ)= 1 + 2|k′′LDκ0|((1− ξ/LD)2 − 1); LD = − κ0

2(γ2

0 + κ20

)k′′. (4.19)

The distance dependent chirp, i.e. the imaginary part of the exponential, of the pulse reads:

κ(ξ) =κ0(1− ξ/LD)(γ2

0 + κ20)

γ20 + κ2

0 (1− ξ/LD)2 . (4.20)

With the dispersion length LD [Kno11],7 which is the maximal range for a possible compression ofa pulse, in the case κ0 < 0 (down-chirp) for normal dispersion k′′ > 0 (or in the opposite case withup-chirp and anomalous dispersion). After LD the “chirp” in the frequency domain κ vanishes. Inthis case a pulse recovers its starting width after a distance of 2LD, because the absolute value ofthe chirp is restored (cf. 4.1). Without a starting chirp the broadening of a Gaussian pulse after adistance ξ reads:

γ0

γ(ξ)= 1 + (2γ0k

′′ξ)2 (4.21)

The broadening depends on the square of the initial width and the distance traveled, hence groupvelocity dispersion is especially important for ultra short pulses and long distances.

6A chirp is the time dependent frequency of a pulse and briefly discussed in Sec. 1.2.3.7In a more technical context, the term dispersion length is reserved for the distance until a non-chirped pulse has

broadened by a factor of√

2 [Mit10], namely L′D = 2γ0k′′ for a Gaussian.

82


4.3 Generalized complex Ginzburg-Landau type equations

The wave equation Eq. (4.11) derived in the last section has the structure of a so called generalizedcomplex Ginzburg-Landau equation. In Sec. 4.3.1 we briefly comment on some general properties,then the non-linearities derived from the semiconductor QDs in Chap. 3 are introduced and theresulting wave equations are discussed. In Sec. 4.3.2 conservative wave equations in general andespecially the non-linear Schrodinger equation are treated. Section 4.3.3 addresses dissipative waveequations, especially the complex Ginzburg-Landau equation and the saturable absorber equation.The transition from dissipative to the conservative regime is presented in Sec. 4.3.4.

4.3.1 General properties

The generalized Ginzburg-Landau equation reads [Ros05]:

0 = i∂E

∂ξ− k′′

2

∂2E

∂τ2+ f(|E|2)E (4.22)

with the non-linearity f(|E|2)E which relates to the polarization of the QDs (active material)presented in Chap. 3. This implies restrictions to the cases that can be described with this waveequation:

i) the polarization has to vanish for vanishing fields, i.e. we cannot describe cases with significantmemory or dependence on the pulse area or the derivative of a pulse. The description can beused for adiabatic following hence for non-resonant interaction.

ii) f shall only depend on the absolute value of the field, which is the case for the QD dynamicsdiscussed in Chap. 3, due to the rotating wave approximation and no permanent dipoles.

Now we want to show some general properties of solutions to Eq. (4.22). First, we consider a formalsolution E0(τ, ξ) to show its invariances.

It is easy to see that Eq. (4.22) is invariant to translations and constant phases, i.e. if E0(τ, ξ) isa solution of Eq. (4.22) then E(τ, ξ) = E0(τ − τ0, ξ)e

iϕ0 is also a solution.

Galilean invariance

The next general invariance we want to discuss is the one to constant velocities within the co-movingframe. Inserting the ansatz E(τ, ξ) = E0(τ −Aξ, ξ)eiB′ξ+iC′τ into Eq. (4.22) leads to

−iA∂τE0 + i∂ξE0 −B′E0 −k′′

2

(∂2E0

∂2τ+ 2iC ′

∂E0

∂τ− C ′2E0

)= −f(|E0|2)E0, (4.23)

which leaves the following equation for the parameters of the moving solution:

−i(A+ k′′C ′

) ∂E0

∂τ−(B′ − k′′

2C ′2)E0 = 0. (4.24)

83

4.3. GENERALIZED COMPLEX GINZBURG-LANDAU TYPE EQUATIONS

The latter equation is fulfilled for C ′ = − Ak′′ and B′ = k′′

2 C′2 = A2

2k′′ . A pulse with a velocity A

relative to the group velocity and a phase linear in τ and ξ E(τ, ξ) = E0(τ −Aξ, ξ)eiA2ξ−2Aτ

2k′′ isalso a solution of the generalized complex Ginzburg-Landau equation. This means if we want tofind a solution it is only necessary to find the solution with the relative velocity A = 0 to knowa whole family of solutions. We also see, that the starting condition E(τ, 0) = E0(τ, 0)e−i 2Aτ

2k′′

which is a pulse with a center frequency other than ωl, leads to a velocity differing from the groupvelocity. Hence, two pulses with a different detuning will have a mutual distance depending on thedistance they traveled.

Scaling: normalized equations

A common technique to discuss wave equations is to switch into a normalized description [Has90].ξ and τ coordinate can be rescaled ξ′ = 1

λξ, τ′ = 1√

k′′λτ thus the non-linearity is rescaled and an

equivalence class of non-linearities can be described via the same equation

i∂

∂ξ′E − 1

2

∂2

∂τ ′2E = −λf(|E|2)E.

Which yields for a field ψ ∼ E with f(|ψ|2) = λf(|E|2) the same solutions in the scaled system.This works well in a limited number of non-linearities like the non-linear Schrodinger equation.Normalized equations typically normalize the leading term of f to 1. Correspondingly the problemis described by one parameter less than in the unnormalized description. Thus, the parameterspace is reduced. In Sec. 4.3.3 we present the microscopically derived non-linearities and show therespective normalized equations.

Energy flux

From Eq. (4.22) follows the continuity equation for the pulse energy [AA05b]

∂ξ|E|2 + ∂τik′′

2(E∗∂τE − E∂τE∗) = −Imf

(|E|2

)|E|2. (4.25)

This introduces the energy flux j = ik′′

2 (E∗∂τE − E∂τE∗) as well as the energy generating densityPgen = −Imf

(|E|2

)|E|2. The energy flux is a measure for the deformation of a pulse. A positive

(negative) sign means parts of the pulse are transported towards the negative (positive) direction ofthe τ axis. That is for a Gaussian pulse exposed to group velocity dispersion (cf. Eq. (4.18)):

j(τ, ξ) = −2|E|2k′′2(γ2

0 + κ20

)κ0 (1− ξ/LD)

γ20 + κ2

0 (1− ξ/LD)2 τ = −2|E|2κ(ξ). (4.26)

Figure 4.1 shows the energy flux of a Gaussian pulse with GVD for various distances. For ξ < LDthe pulse’s energy is transported towards the center, leading to a narrowing of the pulse, whereasfor ξ > LD the energy is transported to the wings, which broadens the pulse. As we can see, the

84


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Puls

eam

plitu

de/e

nerg

yflu

x

τ

amplitude energy flux

τ τ

0LD0.2LD0.4LD0.6LD0.8LD

1LD


2LD


3LD

Figure 4.1: The energy flux (dashed lines) of a chirped Gaussian pulse (solid lines) under GVD. The arrowsindicate the direction of the flux. For distances below the dispersion length energy is transported to the centerof the pulse, which leads to a narrowing. After the dispersion length the energy is transported to the wingswhich leads to broadening.

energy generation is depending on the intensity, while the flux is sensitive to the phase of a pulse.For the case with an unchirped Gaussian pulse the energy flux always broadens the pulse:

j(τ, ξ) = −|E|2 ξ1

4(γ0k′′)2 + ξ2

τ. (4.27)

In the regime of dissipative wave equations, Pgen is a non-vanishing function of the intensity,and an important quantity: for stable solutions it has to be in equilibrium with the energy flux j.We will discuss this further in Sec. 4.3.3.

Dispersion and Dissipation

Since E is a complex field it is convenient to decompose it into its absolute value and its phaseE = qeiϕ′ which leads to equations for the real and imaginary part of Eq. (4.22) (restrictingourselves to k′′ ∈ R).

85


Inserting the ansatz into Eq. (4.22), the real part of the equation corresponds to the dispersion:

0 = −q∂ϕ∂ξ− k′′

2

[∂2q

∂τ2− q

(∂ϕ

∂τ

)2]

+ Ref(q2)q. (4.28a)

It is mainly the propagation equation for the phase and is driven by the real part of the polarization.The imaginary part directly includes the propagation of the absolute value and contains the imaginarypart of the polarization, which we already saw corresponding to absorption/luminescence in Sec. 2.1.Thus the imaginary part of Eq. (4.22) resembles the continuity equation (4.25):

0 =∂q

∂ξ− k′′

2

[2∂ϕ

∂τ

∂q

∂τ+ q

∂2ϕ

∂τ2

]q−1 ∂

∂τ

(q2 ∂ϕ

∂τ

) + Imf(q2)q. (4.28b)

Both equations are coupled via the dispersion. In the next sections we show some conservativeand dissipative non-linearities f(q2) and solutions to the corresponding wave equations.

4.3.2 Conservative cases

In the conservative regime there is no energy consumption or generation, i.e. Im(f(q2)) = 0. Wedefine fR = Ref . In this section we want to derive the most simple localized solutions q(τ), the socalled fundamental solutions, which have one maximum. Higher order solutions that are able todescribe situations with multiple pulses and where solitons can collide etc. are not topic of thisthesis and can be learned from the literature [NML+84; Has90; Man10].

General approach to solitary solutions

There are many approaches to find the fundamental solitary solution for different specific non-linearequations, that work with similar schemes [KV95; Has90; Kno11]. Here, we show the generalizedprocedure for an arbitrary conservative non-linearity f(q2) and apply it on the 3-5 non-linearSchrodinger equation (NLSE). We search for the solitary pulse solution E0(τ, ξ) = q(τ)eiϕ(τ,ξ),i.e. a pulse that does not change its shape during propagation ∂ξq = 0. According to Sec. 4.3.1we can derive any solution with a constant velocity from this solution. In the conservative caseEqs. (4.28) read

0 = −q∂ξϕ−k′′

2

[∂2q

∂τ2− q (∂τϕ)2

]+ fR(q2)q, (4.29a)

0 = −k′′

2q−1 ∂

∂τ

(q2∂τϕ

)⇐ (∂τϕ = 0 ∧ q 6= 0) . (4.29b)

From the second equation we can see, that the phase has to be constant in time ϕ(ξ) for a non-moving8 solitary pulse. Specifically, this means there is no energy flux for a stable conservative

8Remember that non-moving means stationary in τ within the co-moving frame, i.e. in the t, z coordinate systemthis is a pulse, that moves with the group velocity k′−1.

86


solution. The first equation yields the pulse shape, with the definition C0 = −2/k′′∂ξϕ andmultiplication with ∂τq it reads:

0 = −k′′

4

∂

∂τ

(∂

∂τq

)2

+k′′

4C0∂q2

∂τ+

1

2

∂

∂τF (q2); ∂xF (x) = f(x), (4.30)

which can be integrated to yield:(∂

∂τq

)2

= C0q2 +

2

k′′F (q2) + C1. (4.31)

For a pulsed solution the integration constant has to be C1 = − 2k′′F (0), for the boundary condition

q = 0. This equation defines the maximum of q as a function of the free parameter C0 connectingthe phase ∂ξϕ and the amplitude q of the solution. That means a general solution of a conservativegeneralized Ginzburg-Landau equation we have to solve the integral for

±∫ q(τ)

q0

dq′1√

C0q′2 − 2

k′′

(F (q′2)− F (0)

) =

∫ τ

τ0

dτ = τ − τ0. (4.32)

The left side is a line integral where q is the curve, i.e. we search for the curve q(τ) on which theintegrand yields an integration linear in time (cf. Fig. 4.2). The sign of the integrals depends on the

0

qmax

τ0

2

4

6

8

10

12

14

-20-15-10 -5 0 5 10 15 20

q

Time (norm. coordinates)Figure 4.2: Left panel: Visualization of the approach described in the text. The color map represents theintegrand of the left hand side of Eq. (4.32) (or specifically Eq. (4.34) for the solution of the 3-5NLSE). Thesolution is obtained as the line integral that results in a linear growth with the time coordinate τ . Right panel:Solutions to the 3-5NLSE (the color code only discriminates different solutions). There is a continuum ofsolutions for every initial height with a corresponding width.

sign of ∂τq. Hence, four cases would have to be taken into account τ ≶ τ0 and q ≶ q0. Thus, it isconvenient to choose q0 = q(τ0) to be the maximum of the pulsed solution, given by:

C0q20 =

2

k′′(F (q0

2)− F (0)).

The solution is symmetric around τ0. Note that this is an equation for q20 but not q0, however only

the positive solution q0 is needed.

87


Application: cubic-quintic non-linear Schrodinger equation

In Sec. 3.1.2 we derived the perturbative response of the conservative Bloch equations (Eq. (3.2)).This resulted in a non-linear response that leads to the non-linear Schrodinger equation (NLSE).

i∂E

∂ξ− k′′

2

∂2E

∂τ2= −a|E|2E + b|E|4E. (a, b ∈ R) (4.33)

The NLSE is one of the best known conservative, non-linear wave equations. In particular it isfamous for describing the propagation of pulses through optical fibers [Has90; Mit10]. Mostlythe cubic NLSE is discussed in the literature. Here, we want to focus on its, also well known,generalization with an additional quintic term. The left side of Eq. (4.33) describes the broadeningof a pulse due to GVD, while the (cubic) non-linearity on the right side, which is sometimesdescribed as intensity dependent refractive index, affects the phase depending on the pulse amplitudeϕ = ϕ0 + aq2ξ. This is called self-phase modulation [Mit10] and is comparable to a chirp. Thequintic non-linearity is a saturation for the cubic part. Solitary solutions are pulse shapes for whichthe GVD chirp and the chirp from the non-linearity compensate each other. To find the fundamentalsolution to Eq. (4.33) with the general method we introduced above this we have to assign thenon-linearity f(q2) with:

f(x) = −ax+ bx2; F (x) = −a2x2 +

b

3x3; C1 = 0,

∂

∂τq = ±q

√C0 −Aq2 +Bq4; A =

a

k′′, B =

2b

3k′′.

As described in Eq. (4.3.2), the maximum of the solution q0 obeysC0 = −Bq40 +q2

0A. The solutionof the integral yields

τ − τ0 =

∫ q

q0

dq′

q′√C0 −Aq′2 +Bq′4

= − 1

2√C0

log

(2√C0

√−Aq2 +Bq4 + C0 −Aq2 + 2C0

−Aq2 + 2q2C0/q20

). (4.34)

This equation has to be resolved after q as a function of τ , starting with exponentiating the wholeexpression.

e−2√C0(τ−τ0) =

1

q2

(2q0

√−Bq2

0 +A√−A(q2 − q2

0) +B(q4 − q20)−Aq2 + 2(Aq2

0 −Bq40))

A− 2Bq20

.

(4.35)

Obviously, the left hand side of this expression is a monotonously falling function. Since we searchfor a symmetric, pulsed solution we add the rising flank e2

√C0(τ−τ0) and resolve the sum after q.

88


This leads to

cosh(

2√C0(τ − τ0)

)=

(2Aq20 − 2Bq4

0 −Aq2)

A− 2Bq20

1

q2. (4.36)

Finally, the fundamental soliton solution for the 3-5-NLSE can be obtained as

q2 =2q2

0

(a− 2

3bq20

)a+

(a− 4

3bq20

)cosh

(2q0/√k′′√a− 2

3bq20(τ − τ0)

) , (4.37)

ϕ = −1

2k′′C0z = −

(a− 2

3bq2

0

)q2

0

2ξ.

This is a continuous family of solutions, where every member has a maximum q0 and a correspondingwidth proportional to 1

q0.

The number of fundamental solutions is even larger taking into account the phase, translationaland Galilean invariance of the NLSE, which means the general fundamental solution:

q(ξ, τ) =

√√√√ 2q20

(a− 2

3bq20

)a+

(a− 4

3bq20

)cosh

(2q0/√k′′√a− 2

3bq20(τ − vξ − τ0)

) , (4.38)

ϕ(ξ, τ) =v2ξ − 2vτ

2k′′+ ϕ0 −

(a− 2

3bq2

0

)q2

0

2ξ.

Note, that the more commonly used limit of the cubic NLSE is of course included for b→ 0

q2 =2q2

0

1 + cosh(

2q0/√k′′√at) = q2

0 sech2

(q0

√a

k′′t

). (4.39)

4.3.3 Dissipative cases

When dissipative non-linearities occur in the wave equation, typically no analytical solution orclosed form expression exists. However there are stable solutions which can be found numerically. Inthis subsection we want to cover solitary waves in the dissipative regime. They are called dissipativeor auto solitons in analogy to the conservative case [AA05b; Ros05]. The most important differenceis that there is no continuum of solutions, but discrete ones playing the role of fixed points forthe dynamics. Additionally, there are also periodically changing solutions, corresponding to limitcycles [ASCT01; Akh98] depending on the parameter regime. Further, the scattering properties ofdissipative solitons show a wealth of variants, e.g. the soliton-typical preservation of the pulses,but also annihilation [ACM97], or the formation of stable bound states [BMA00; VKR01].

To find stable solutions the continuity equation Eq. (4.28b) is of importance:

0 = q∂q

∂ξ− k′′

2

∂

∂τ

(q2∂ϕ

∂τ

)+ Imf(q2)q2.

89


For stable pulsed solutions (∂ξq = 0) the energy flux j = k′′

2

(q2 ∂ϕ

∂τ

)has to balance the energy

generation Pgen = Imf(q2)q2,9 while the energy generation and drain has to cancel out over thewhole regime, i.e.

∫∞−∞ dτPgen(τ) has to vanish, this means it has to yield positive and negative

areas of the same size [AA08b]. Thus it is convenient to investigate the roots of Imf(q2)q2, whichare also homogeneous solutions (also known as continuous wave solution), i.e. q(τ) = const..Physically those points are where gain and absorption just compensate each other and gain changesto absorption with increasing intensity (stable fixed point) or the other way around (unstable fixedpoint). One root is always the solution q = 0 which should be stable for the existence of pulsedsolutions.

-8

-6

-4

-2

0

2

4

6

8

-3 -2 -1 0 1 2Time τ (arb. units)

Im(q2) = 0

Im(q2) = 0

-0.004-0.003-0.002-0.001

00.0010.002

-2.4 -2.1 -1.8 -1.5 -1.2

intensitypower generationenergy flux

Figure 4.3: Schematic of a stable solution in the dissipative regime. The horizontal lines show the roots ofthe energy generating function Pgen. At the points where the intensity crosses those levels the response isalternated between gain and absorption. The areas between Pgen and 0 visualize the generation/absorption ofenergy, for a stable solution the positive and negative areas have to be of the same size, while the energy fluxhas to balance energy generation and drain. The inset is a magnification of the graph for the temporal rangeit is shown.

The following sections will look in more detail on different special equations and their solutions,as well as regimes where they exist and how they can be numerically obtained.

9With the equilibrium condition ∂τ j = Pgen.

90


Complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation is the big brother of the non-linear Schrodinger equationEq. (4.33) from the last section. For the complex Ginzburg-Landau equation every real parametersof the NLSE become complex numbers, i.e. a→ a+ iε and b→ b− iµ. The Ginzburg-Landauequation originates from the description of second order phase transitions in hydrodynamics andsuperconductivity and is nowadays one of the most investigated non-linear equations in physics[AK02]. With the non-linear response from Sec. 3.2.1 we also obtain a complex Ginzburg-Landauequation describing the propagation of an optical pulse through a medium consisting of two levelsystems. That is the parameters of the non-linearity are microscopically derived from the dissipativeBloch equations in relaxation time approximation under the non-resonant excitation with a weakand slow optical pulse.

From the last paragraph we learned, that in order to obtain stable solutions, the response ofthe system has to yield regimes of absorption and gain depending on the amplitude of the pulse.However, this behavior cannot be achieved by the adiabatic, non-linear response of one two levelsystem since it shows either gain or absorption, independent on the intensity of the optical pulse.That is for a stable solution as shown in Fig. 4.3 we need the response of two two level systems aswell as an additional linear absorption. QDs typically consist of more than one electronic transition.We take into account s shell and p shell excitation in a regime, where the s shell exhibits gain, whilethe p shell absorbs. It is our aim to find soliton solutions for QD parameters from Sec. 3.2.1.

i∂E

∂ξ− k′′

2

∂2E

∂τ2= −iαE − (a+ iε)|E|2E + (b− iµ)|E|4E. (4.40)

We can obtain this equation with the results from Sec. 3.1.2 with the parameters in Table 4.1.All QD contributions show the common prefactor cpre =

ω2l

2c2ε0

|d12|2~

u∞3a+

stemming from the waveequation (4.11) and the common prefactors of every order of the polarization.

Note that often in studies there is also a complex GVD k′′ → k′′+ iβ, with the spectral filteringβ. Taking this into account can change a lot to the particular properties of the solutions. In thiswork, we will not discuss this parameter.

Normalized notation: Typically, non-linear equations are depicted in their normalized form, tomake general statements (cf. Sec. 4.3.1). For the cGLE this means the amplitude is renormalized tolet the renormalized a = 1 and τ to renormalize k′′ to 1.

i∂ψ

∂ξ′− D

2

∂2ψ

∂τ ′2= −iαψ − (1 + iε)|ψ|2ψ + (b− iµ)|ψ|4ψ. (4.41)

Homogeneous solutions of the cGLE: As noted before it is of advantage to know the homoge-neous or continuous wave solutions q(τ) = const. of a dissipative wave equation since they markthe intensities, where gain and absorption alternate and thus define the energy generating densityPgen. According to Eq. (4.28b) a homogeneous solution follows:

0 = ∂ξq + Imf(q2)q. (4.42)

91


Coefficient adiabatic part (slow pumping limit) fast pumping limitlinear absorption α→ γcpre γcpre

Kerr coefficient a→ ∆ω4|d12|2

~2a+

(a−2a+

(1 + 2

γΓ

a+

)− 2

γ2

a+

)cpre, ∆ω

4|d12|2

~2a+

γ

Γcpre

Cubic gain ε→ γ4|d12|2

~2a+

a−2a+

(3− 2

Γ

γ

a−2a+

)cpre, γ

4|d12|2

~2a+

γ

Γcpre

quintic phase b→ ∆ω

(4|d12|2

~2a+

)2a−2a+

[a−2a+

(3

2+ 7

Γγ

a+

)− 6

γ2

a+

]cpre, ∆ω

(4|d12|2

~2a+

)2γ2

Γ2cpre

quintic absorption µ→ γ

(4|d12|2

~2a+

)2(a−2a+

)2 [15

2− 7

Γ

γ

a−2a+

]cpre, γ

(4|d12|2

~2a+

)2γ2

Γ2cpre

Table 4.1: Parameters of the complex Ginzburg Landau equation derived with the perturbative approach inSec. 3.1.2. All coefficients have the common prefactor cpre =

ω2l

2c2ε0

|d12|2~

u∞3

a+.

One steady state solution is qS = q0 = 0. For the non-trivial steady state homogeneous solutionsqS of the cGLE one has to find the roots of

0!

= Imf(q2S) = α+ εq2

S + µq4S . (4.43)

Which yields to solutions for the square of the absolute value of the field:

q2S = q2

± = −1

2

ε

µ

(1±

√1− 4

αµ

ε2

). (4.44)

For the existence of such solutions the term has to be real and positive which means

• sgn(ε) = − sgn(µ): a non-linear gain has to be compensated by a non-linear absorption,because at least in one case the term in brackets is positive.

• 0 < αµε2

for both solutions being positive. That means α and µ have the same sign. Absorptionand gain alternate with the orders.

• Subsequently 4αµε2< 1 ⇔ 2|α|

|ε| <|ε|

2|µ| ⇔|ε|

2|µ| >√

αµ has to be valid to yield real values.

This implies for the coefficients if |α| > |ε|2 then |µ| < |ε|

2 and vice versa, i.e. they have to bemonotonous (compared to ε/2)!

Depending on the parameters there are two non-trivial solutions or none, which is the typical behaviorof a saddle-node-bifurcation.10 The homogeneous solutions depending on the systems parametersare plotted in Fig. 4.4. The stability of the solutions can be observed for small perturbations around

10The normal form of the saddle-node bifurcation reads ∂tx = r+x2, which also describes our case since Eq. (4.43)resembles this quadratic behavior for q2.

92


− ε2µ

q2 ±

0

√αµ

0√

αµ

Figure 4.4: Homogeneous solutions of the complex Ginzburg Landau equation depending on the strength ofthird and fifth order dissipative non-linearity. In the shaded area no real solutions exist; instead the real partof the complex solution is plotted. This is the behavior of a saddle-node bifurcation.

the steady state solutions qS : q(ξ) = qS + δ(ξ), with the deviation δ ∼ e−λξ growing or vanishingduring propagation. A solution is stable if those perturbations are damped out if the ansatz isinserted into Eq. (4.42), i.e. 0 < λ = 1

δ Imf((qS + δ)2

)(qS + δ). For the linear stability analysis

the non-linearity is expanded linearly in δ around qS .

λ ≈ 1

δImf

(q2S

)qS︸︷︷︸

=011

+Imf(q2S

)+ 2q2

SImf ′(q2S

).

The zero solution is stable for 0 < λ = Imf(0) = α, i.e. for linear absorption (α > 0).Consequently, for the existence of the two non-trivial homogeneous solutions the third order isnon-linear gain (ε < 0) and fifth order non-linear absorption (µ > 0). The stability of the non-trivial

solutions follows from λ/(2q2±)

= f ′(q2±) = ε+ 2µq2

±!> 0, where inserting Eq. (4.44) yields:

0 > ∓√

1− 4αµ

ε2.

Thus q+ is always stable, while q− is always unstable.

Pulsed solutions To find a pulsed solution, first of all a solution has to exist. Existence of varioussolutions with different properties is done elsewhere [ASCT01; SCAA+97; AASC96; AA08b;AA05b]. Secondly, to find a specific solution the initial conditions have to be in the proximityof it. For example, since the zero solution is a stable attractor, an initially too small pulse wouldrather evolve into the zero solution (i.e. vanish) than into a stable pulsed solution. Hence, to findreasonable initial conditions we take a close look at homogeneous solutions and thus on the energygenerating density.

11The term vanishes for qS = 0 as well as for qS = q± since Imf(q2±) = 0.

93


-20 -10 0 10 20 30

|E|

Time

starting condition #0starting condition #1starting condition #2starting condition #3starting condition #4starting condition #5starting condition #6starting condition #7starting condition #8

zero solutionsoliton solution

Pul

se E

nerg

y

travelled distance

starting condition #0starting condition #1starting condition #2starting condition #3starting condition #4starting condition #5starting condition #6starting condition #7starting condition #8

Figure 4.5: Pulsed and zero solution to the cGLE for various initial conditions. Horizontal lines depict thehomogeneous solutions. Initial conditions below the unstable E− level evolve into the zero solution whilethe initial conditions above lead to a unique pulsed solution.

Numerically we find solutions with a Fourier split step algorithm, where the discrete advancingstep ∆ξ is done in the Fourier domain for the temporal derivative and in time domain for thenon-linear part. As used for the introduction of the GVD (cf. Eq. (4.13)) the derivative part ofthe wave equation has the exact solution in Fourier domain. Hence, for the propagation fromξ = ∆ξ × j to ξ = ∆ξ × (j + 1) the field reads:

E(j + 1, ω) = E(j, ω)ei2k′′ω2dξ,

E(j + 1, ω) = E(j, ω) + E(j, ω)(e

i2k′′ω2dξ − 1

). (4.45)

The non-linear part can be solved via the Euler or a Runge-Kutta method [PTV92; PTV+96]:

E(j + 1, τ) = E(j, τ) + ∆ξf(|E(j, τ)|2)E(j, τ). (4.46)

Thus the combined propagated field reads

E(j + 1, τ) = E(j, τ) + ∆ξf(|E(j, τ)|2)E(j, τ) + F−1(F (E(j, τ))

(e

i2k′′ω2dξ − 1

)).

(4.47)

Figure 4.5 shows various initial conditions evolving into those two stable solutions. Initialconditions with intensities below the E− level are completely absorbed during propagation andstrive towards the zero solution. For initial conditions with higher intensities, the propagation leadsto a transformation into the same solution for all cases. The form of this solution depends on thesystem parameters. In the right panel of Fig. 4.5 the pulse energy as a function of the traveleddistance is plotted. One can see that the energy of the stable solutions is approached. The fixedpulse energy is the necessary condition for the arrival at the stable solution.

94


Saturable absorber non-linearity

In Sec. 3.2.2 we showed the fast pumping limit of the Bloch equations leading to the saturableabsorber description of the response. As discussed before, to obtain stable solutions we need totake into account two two level systems: the s states of the QD that offer gain as well as the p statesthat are responsible for a non-linear absorption. This yields the wave equation that is intenselydiscussed in Ref. [Ros05]:

i∂E

∂ξ− k′′

2

∂2E

∂τ2= −α′E − a

1 + ba|E|2E +

g

1 + bg|E|2E. (4.48)

Analog to finding the solution to the cGLE we start by investigating the homogeneous solutions toEq. (4.48):

0 = Imf(q2) = α′ +a

1 + baq2− g

1 + bgq2,

q2± =

((g − α′)ba − (a+ α′)bg

2α′babg

)(1±

√4

(−α′ + g − a)α′babg(g − α′)ba − (a+ α′)bg)2

+ 1

).

Formally one can see an analogue to the homogeneous solutions of the cGLE:

q2± = −1

2

ε

µ

(1±

√1− 4

αµ

ε2

),

α = α′ − g + a, ε = −(gba − abg) + (ba + bg)α′, µ = α′babg.

In the same way as for the cGLE there are parameter restrictions for the existence of real, positivesolutions:

• sgn ((g − α′)ba − (a+ α′)bg) = sgn(α′),

• (−α′ + g − a)α′babg < 0,

• 0 < 4(−α′+g−a)α′babg

((g−α′)ba−(a+α′)bg)2 + 1.

While the homogeneous solutions have the same qualitative behavior as those of the cGLE, thestability of those solutions is quite different. The stability of the zero solutions is ensured for

0 < α′+ a− g = α. The non-trivial solutions are stable for f ′(q2) = − aba(1+baq2)2 +

gbg(1+bgq2)2

!> 0:

0 <ε

α′

[α′

ε(ba + bg)(α

′ − α) + α−(

1− α′(ba + bg)

ε

)1

2

ε2

µ

].

95


05

1015202530

-20 -10 0 10 20 30 40

|E|V/µ

m

τ [ps]

|E+|

|E−|

z = 0µmz = 18µmz = 36µmz = 54µmz = 108µmz = 468µmz = 828µm

Figure 4.6: Optical pulse propagating through QD material described in the saturable absorber (fast pumping)limit. The pulse excites the homogeneous solutions piecewise: initial conditions below E− lead to the zerosolutions, whereas initial conditions above lead to the E+ solution. The dispersion is weak, such instead of apulse shape two fronts are created leading to a rectangular structure.

Table 4.2: Material parameters used in numerical simulations in the saturable absorber description

Parameter Value Parameter ValueNQD 5× 1013 cm2 ∆ωs −9.12× 10−2 fs−1

α 4.00× 10−4 nm−1 ∆ωp 6.08× 10−2 fs−1

k′′l 2.91× 10−3 fs2/nm γs 2.03 ps−1

nl 3.40 nm−1 Γs 1.38 ps−1

ds12 0.440e0 nm γp 3.03 ps−1

dp12 0.658e0 nm Γp 1.43 ps−1

Pulsed solutions For QD parameters (cf. Tab. 4.2) we find that an initial pulse evolves into arectangular structure [DPE+12], cf. Fig. 4.6. The reason for that is a small dispersion such thata piecewise evolution to the homogeneous fixed point levels occurs, i.e. parts of the pulse withamplitudes between E+ and E− are enhanced up to E+, while amplitudes below E− are absorbed.The resulting structure is no pulse solution in that sense, but two fronts.12 Thus, in contrast to pulsedsolutions there is no defined width of the localized structure, but it depends on the initial conditions.

For example an initially Gaussian pulse E(τ) = E0e− τ2

T2 would result in a rectangular structurewith a width of ∆τ = 2T

√2 ln(E0/E−). In conclusion, a QD waveguide that is pumped such

that the s shell exhibits gain and the p shell exhibits absorption, can act as a device that transformsoptical pulses into a solitary rectangular shape. Of course, we have to note that the rectangularsolution is only an approximate solution which cannot occur in its full beauty in reality since thesteady state limit of the Bloch equations breaks down for the fast change of the electric field. The

12A front is the border between two homogeneously excited regions and of special interest in pattern formation. Manyequations with pulsed solutions also support front solutions.

96


formation of stable rectangular pulses could be of importance in data transmission, where a herepresented amplifier would transform any pulse shape into a binary signal ideal for return-to-zeroon/off keying data transmission schemes, i.e. such a device would amplify the signal, but not thenoise.

4.3.4 Transition from dissipative to the conservative regime

In Chap. 3 the previously introduced non-linearities were derived from the Bloch equations withand without losses leading to conservative and dissipative wave equations, respectively. Here, wewant to shed some light on this transition. Clearly, the dephasing rate γ can be tuned to switchbetween the conservative and dissipative regime.

A simple way to investigate this transition is to introduce a linear scaling factor λ for thedissipative parameters in Eq. (4.41).

i∂ψ

∂ξ′− D

2

∂2ψ

∂τ ′2= −iαλψ − (1 + iελ)|ψ|2ψ + (b− iµλ)|ψ|4ψ. (4.49)

For λ = 0 an initial pulse, that is the solution to this NLSE is unchanged. Turning on the dissipationparameter this initial condition evolves into a solution to the cGLE (cf. Fig. 4.7). However, ina linear scaling approach the homogeneous solutions are not changed (cf. Eq. (4.44)), thus theresulting solutions are similar in height and width.

100120140160180200220240260280

0 0.05 0.1 0.15 0.2 0.2599.12

138.47

177.81

217.16

256.50

Puls

een

ergy

traveled distance (normalized units)

λ = 0

λ = 10−3

λ = 10−2

λ = 10−1

λ = 1

Figure 4.7: The evolution of an initial pulse for different values of λ and thus in a conservative or moredissipative regime. The pulse energy stays constant for the conservative case λ = 0. For the dissipative casesthe pulse energy is nearly the same, but the length scale on which the equilibrium is reach differs strongly.

97


98

Wir stehen selbst enttauscht und sehn betroffenDen Vorhang zu und alle Fragen offen.

Bertolt Brecht, Der gute Mensch von Sezuan

5Conclusion

In this work we described semiconductor QDs and their semiclassical optical properties. Chapter 2showed the influence of phonons to optical spectra and to the occupation of QDs. The influenceof the carrier wave functions, especially size and intrinsic dipole moment on various couplingmechanisms was discussed. Here, the importance of taking into account more realistic phonondispersions for small QDs was stressed. Further, a phonon mediated coupling between QD statesand a carrier reservoir was described via a rate equation approach. Therefore, a projection operatorbased formalism was used to derive an effective multiphonon Hamiltonian. With an equationof motion approach rate equations with temperature and reservoir carrier dependent rates wereacquired.

The equations of motion for QDs in the rate equations limit were treated in Chap. 3 with theaim to derive analytical expressions. In a perturbative approach expressions for the polarization,that adiabatically follows the electric field were achieved for the three regimes of negligible losses,the presence of dephasing only, as well as for relaxation and dephasing.

Finally, in Chap. 4 the QDs’ optical response was used as a non-linearity in wave equations.Those wave equations are of the type of a generalized Ginzburg-Landau equation. The non-linearwave equations were investigated with respect to (dissipative) soliton solutions and the operatingregime for pulse shaping semiconductor optical amplifiers was proposed.

APrefactors of the piezoelectric coupling

In Sec. 1.2.2 we introduce the piezoelectric coupling to acoustic phonons. Here, we want to showthe angular dependent prefactor for Zincblende and Wurtzit systems.

The angle dependent Piezoelectric factors for Zincblende

FZκ (~q) =

2e14e

q2εsε0(qxqye

zκ + qyqze

xκ + qzqxe

yκ) ,

or for Wurtzit systems (note that here the choice of the wave functions distance comes also intoplay. every thing shown here is valid for growth in 0001 direction)

FWκ (~q) =

e

q2εsε0

(e15(q2

x + q2y)e

zκ + e33q

2zezκ + (e15 + e31)qz(qxe

xκ + qye

yκ)).

In contrast to the deformation potential coupling both band have the same coupling strength, thusonly the difference of the wave functions leads to a coupling. To calculate the angular average onehas to insert ~q = q~eLA, and the unit vectors in spherical coordinates.

Longitudinal mode ~eLA =

sin θ cosφsin θ sinφ

cos θ

, first transversal mode ~eTA1 =

cos θ cosφcos θ sinφ− sin θ

,

second transversal mode ~eTA2 =

− sinφcosφ

0

FZ

LA(φ, θ) =6e14e

εsε0

(sin2 θ cosφ sinφ cos θ

),

FZTA1(φ, θ) =

2e14e

εsε0

(− sin2 θ + 2 cos2 θ

)sin θ cosφ sinφ,

FZTA2(φ, θ) =

2e14e

εsε0

(− sin2 φ+ cos2 φ

)sin θ cos θ,

averaged (FZ

LA)2

=1

4π

4e214e

2

ε2sε

20

12π

35(FZ

TA1)2

=1

4π

4e214e

2

ε2sε

20

23π

210,

(FZ

TA2)2

=1

4π

4e214e

2

ε2sε

20

π

6,

For Wurtzite

FWLA(φ, θ) =

e

εsε0

(e33 cos2 θ + (2e15 + e31) sin2 θ

)cos θ.

FWTA1(φ, θ) =

e

εsε0

(−e15 sin2 θ + (e15 + e31 − e33) cos2 θ

)sin θ.

FWTA2(φ, θ) = 0.

averaged (FW

LA)2

=1

4π

e2

ε2sε

20

(4

105

(8 (2e15 + e31)2 + 12 (2e15 + e31) e33 + 15e2

33

)π

)(FW

TA1)2

=1

4π

e2

ε2sε

20

(8

105

(19e2

15 + 3 (e31 − e33)2 + 2e15 (−e31 + e33))π

)(FW

TA2)2

= 0.

(FW

LA)2

=e2

ε2sε

20

1

105

(32e2

15 + 8e231 + 15e2

33 + 32e15e31 + 24e15e33 + 12e31e33

)(FW

TA1)2

=e2

ε2sε

20

1

105

(38e2

15 + 6e231 + 6e2

33 − 4e15e31 + 4e15e33 − 12e31e33

)

102

BInduction proof of the Bloch equation’s

perturbation treatment

In Sec. 3.1.2 an infinite series for the polarization and the inversion of the quantum dots is derived.Here is the proof via the mathematical induction method. For the proof we need Eqs. (3.7):

ρ(n+1)12 (t) = i

∫ t

−∞dt′ei∆ω(t′−t)Ω(t′)u

(n)3 (t′), (B.1a)

u(n+1)3 = −4

∫ t

−∞dt′Im(Ω∗(t′)ρ

(n)12 (t′)). (B.1b)

Assumeu

(2n)3 = (−1)n

4n

∆ω2n

Γ(n+ 1/2)√πn!

|Ω|2nu(0)3 , (B.2)

for n ≥ 0 with the Γ function with the property xΓ(x) = Γ(x + 1) and Γ(1/2) =√π. This

definition is true for n = 0. Then, using Eq. (B.1a) the polarization of order 2n+ 1 reads

ρ(2n+1)12 = (−1)n

4n

∆ω2n+1

Γ(n+ 1/2)√πn!

u(0)3

(Ω∗(t)|Ω|2n(t) + i

1

∆ω∂t′ |Ω|2n(t′)Ω∗(t′)

). (B.3)

This yields the next order inversion via Eq. (B.1b)

u3(2(n+1))(t) = (−1)(n+1) 4n+1

∆ω2n+1

Γ(n+ 1/2)√πn!

u(0)3 Im

(i

∆ωΩ(t)∂t|Ω|2n(t)Ω∗(t)

), (B.4)

where the derivative can be brought in front of the equation via

Ω∂tΩ∗|Ω|2n =

2n+ 1

2n+ 2∂t|Ω|2n+2 − i|Ω|2n+2∂tϕ; Ω = |Ω|eiϕ. (B.5)

Thus the inversion of order 2(n+ 1) is

u(2(n+1))3 (t) = (−1)(n+1) 4n+1

∆ω2(n+1)

Γ(n+ 1 + 1/2)!√π(n+ 1)!

u(0)3 |Ω|

2(n+1), (B.6)

which is the proof of the general form assumed in Eq. (B.2). Using Γ(n + 1/2) =√π (2n)!n!4n

[BSM+00] we can write:

u3(t) =

[ ∞∑n=0

(−1)n1

∆ω2n

(2n)!

(n!)2|Ω|2n

]u

(0)3 , (B.7)

ρ12(t) =

[ ∞∑n=0

(−1)n1

∆ω2n+1

(2n)!

(n!)2

(Ω∗(t)|Ω|2n(t) + i

1

∆ω∂t′ |Ω|2n(t′)Ω∗(t′)

)]u

(0)3 . (B.8)

104

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Acknowledgments

Zu erst mochte ich naturlich Prof. Dr. Andreas Knorr fur die Betreuung meiner Promotion danken,speziell fur die Moglichkeit eigenstandig forschen zu konnen, sowie die regelmaßigen Treffen.Desweiteren gilt mein Dank Prof. Dr. Harald Engel, fur die Hilfestellungen bei allen Fragen zurnichtlinearen Dynamik der Wellengleichungen. Dr. Uwe Bandelow mochte ich fur das Erstellendes Zweitgutachtens danken. Prof. Dr. Stephan Reitzenstein danke ich fur die Ubernahme desPrufungsvorsitzes.

Mein herzlichster Dank geht an die gesamte Arbeitsgruppe Knorr fur die einladende Atmosphareund insbesondere an meine langjahrigen Wegbegleiter Dr. Alexander Carmele, Dr. Thi Uyen-KhanhDang, Julia Kabuß, Christopher Kohler, Dr. Frank Milde, Dr. Ermin Malic, Dr. Marten Richter,Dr. Felix Schlosser, Mario Schoth, Franz Schulze, und Dr. Carsten Weber. Ihr wart wundervolleNachbarn, Zimmergenossen, Gesprachspartner (sowohl uber Physik als auch uber den weltlichenRest), Konferenzmitfahrer, Gesellschaft beim Essen, und Gnuplot- und Inkscapelehrer (sei es durchFragen oder Tricks).

Aus den anderen ITP Gruppen gilt mein Dank den “GRK”lern. Im Speziellen Stefan Fruhner, Dr.Christian Otto, Dr. Niels Majer, fur die vielen Interessanten Gesprache uber Dynamik, Bifurkationenund Quantenpunktlaser sowie die Fahrradtour(en).

Bei Sandra Kuhn, Marten Richter, Christopher Kohler und besonders bei Frank Milde mochteich mich fur das Korrekturlesen bedanken. Den (ehemaligen) Mitgliedern der AG Bimberg Dr.Irina Ostapenko, Dr. Sven Rodt, Dr. Christian Kindel, Dr. Erik Stock, Gerald Honig und Dr. AndrejSchliwa danke ich fur die angenehme und produktive Zusammenarbeit bei den Einzelquantenpunkt-spektren. Dank gilt auch meinen fleißigen Studierenden Maria Schuld, Anton Bruch und AndreHolzbecher fur die vielen Fragen die sie gestellt haben und den Details, die sie aufgebracht haben.Auch mochte ich der administrativen Seite des Instituts, speziell Kristina Ludwig, Peter Orlowskiund Julia Eckert fur die stete Unterstutzung danken.

Zu guter letzt danke ich der Deutschen Forschungsgemeinschaft, durch die meine Forschunginnerhalb des SFB 787 sowie des GRK 1558 finanziert wurde.

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