theory of light propagation in nano-structured materials

Theory of Light Propagation in Nano-StructuredMaterials and Semiconductors

vorgelegt von Diplom-Physiker

Martin Schaarschmidtaus Berlin

Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Dr. rer. nat.

genehmigte Dissertation

Tag der wissenschaftlichen Aussprache: 3. Mai 2006

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. E. Sedlmayr

1. Gutachter: Prof. Dr. rer. nat. A. Knorr

2. Gutachter: Prof. Dr. rer. nat. H. Engel

Berlin 2006

D 83

5

Zusammenfassung

In dieser Arbeit wird eine Theorie präsentiert, die die Ausbreitung intensiver elektromagnetischerStrahlung in nano-strukturierten Materialien und Halbleitern beschreibt.

Ein Schwerpunkt ist die Modellbildung und Simulation der mikroskopischen Materialdynamikniederdimensionaler Halbleiterstrukturen, Volumenhalbleitern und Laser-induzierter Plasmen. Die ul-traschnelle mikroskopische Dynamik dieses elektronischen Vielteilchensystems unter dem Einflussvon Lichtfeldern und Streumechanismen wie Elektron-Elektron Wechselwirkung oder Wechselwir-kung mit Gitterschwingungen wird im Dichtematrixformalismus beschrieben. Diese Beschreibung inzweiter Quantisierung liefert die zeitlich nichtlokale und nichtlineare Antwort des Materialsystemsauf Erregung. Es wird gezeigt, dass die Laser-induzierten Plasmen in Quantenfilmen als neuartigeHalbleiterquellen für Terahertzstrahlung (Wellenlänge mm bis µm) genutzt werden können.

Der zweite Schwerpunkt der Arbeit ist die Simulation der Ausbreitung elektromagnetischer Wel-len in verschiedensten Systemen wie in photonischen Kristallen, in Glasfasern und Wellenleitern. DiePropagation in Volumenhalbleitern und Wellenleitern wird in Näherung einer langsam veränderlichenEinhüllenden (SVEA) und mittels der nichtlinearen Schrödingergleichung behandelt. Für komplexestrukturierte Systeme wie eindimensionale photonische Kristalle wird bei hoher Symmetrie ein Ma-trixtransferformalismus verwendet. Für beliebige strukturierte Systeme (die auch lokale Brechungender Symmetrie enthalten können) wird ein sehr flexibler Finite-Differenzen-Algorithmus zum Einsatzgebracht.

Die Kombination der mikroskopischen Materialdynamik und der Lichtausbreitung erlaubt die Be-rechnung von Reflexions- und Transmissionseigenschaften von nano-strukturierten Materialien nichtnur bei linearer Anregung sondern auch in Intensitätsbereichen, in denen die nichtlineare Licht-Materialkopplung dominiert und völlig neuartige Effekte auftreten. Als ein Modell-System kommtdabei der Bragg-resonante Vielfach-Quantenfilm zum Einsatz. An diesem eindimensionalen resonantabsorbierenden photonische Kristall werden neu auftretende nichtlineare Effekte studiert. Hierbei istvor allem die Möglichkeit zur Anwendung als schneller optischer Schalter (sub-Picosekunden) unddie Möglichkeit zum Einfang und zur Speicherung von Lichtpulsen zu nennen. Es wird weiter ge-zeigt, dass es unter gewissen Randbedingungen bei hohen Intensitäten zu solitonartiger Lichtpuls-ausbreitung in der ansonsten linear verbotenen photonischen Bandlücke des eindimensionalen photo-nischen Kristalls kommt. Die Ergebnisse werden mit aktuellen Messungen verglichen und durch diesebestätigt.

In Volumenhalbleitern und Wellenleitern wird die nah-resonante Propagation unterhalb der funda-mentalen Materialresonanz untersucht. Es wird gezeigt, dass es bei hohen Lichtintensintäten zu einermit linearer Optik nicht zu verstehenden Pulskompression und solitonartigen Lichtpulsausbreitungkommt. Durch Untersuchung der Propagationsdynamik (insbesondere der Phasendynamik) und Ver-gleich mit aktuellen Experimenten wird gezeigt, dass in diesem Regime eine Beschreibung durch diesonst übliche nichtlineare Schrödinger Gleichung nicht ausreicht und eine mikroskopische Material-beschreibung angewandt werden muss.

7

Abstract

This work presents a theory for the propagation of intense electromagnetic radiation in nano-structuresmaterials and semiconductors.

One main area is the modelling and simulation of the microscopic material dynamics of low di-mensional semiconductors, both structured and bulk, and of a laser induced plasma. The ultrafastmicroscopic dynamics of these electronic many particle systems under the influence of light fieldsand scattering mechanisms like electron-electron interaction or interaction with lattice vibrations isdescribed in the density matrix formalism. This description in second quantization yields the tempo-ral nonlocal and nonlinear response of the material to electromagnetic fields. Laser induced plasmain quantum wells will be shown to be a possible new semiconductor source for terahertz-emission(wavelength mm to -µm).

The second focus of interest of this work lies in the simulation of the propagation of electromagne-tic waves in different systems like photonic crystals, optical fibers and wave guides. The propagationin bulk semiconductors and waveguides is considered in both slowly varying envelope approximation(SVEA) and with the nonlinear Schrödinger equation. For complex systems like photonic band gapstructures with high symmetry a matrix-transfer formalism is applied. For arbitrary structured sys-tems (which may include local breaches of symmetry) a very flexible finite-differences algorithm isemployed.

The combination of microscopic material dynamics and light propagation enables the calculationof reflection and transmission properties of nano-structured materials not only in linear excitation butalso for high intensities where nonlinear light-matter coupling dominates and novel effects arise. Oneused model system are Bragg-resonant multiple quantum wells. On this one dimensional resonantabsorbing photonic crystal new nonlinear effects are studied. Some effect to mention are the possibleapplication as an ultrafast optical switch (sub picosecond) and the possibility to store light pulses.Furthermore it will be shown that (under certain boundary conditions) soliton-like pulse propagationin the otherwise forbidden linear photonic band gap of a one dimensional photonic crystal is pos-sible. The obtained analytical and numerical results will be compared with (and verified by) recentexperiments.

In bulk semiconductors and waveguides the near-resonant propagation below the fundamental ma-terial resonance is analyzed. It will be shown that for high light intensities a pulse compression andsoliton-like propagation is to be expected which cannot be understood in linear optics. By analyzingthe propagation dynamics (especially the phase dynamics) and comparison with new experimentaldata it will be shown that in this regime the usually applied nonlinear Schrödinger equation for offre-sonant propagation is not sufficient and a microscopic material description has to be applied to obtaincorrect results.

CONTENTS 9

Contents

Introduction 15Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

I Propagation of the Electromagnetic Field 17

1 Bulk Propagation 211.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 Splitting of Background Polarization and Dynamic Polarization . . . . . . . . . . . . 221.4 Background Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.1 Quasi-monochromatic Wave Packet / SVEA . . . . . . . . . . . . . . . . . . 231.4.2 Group Velocity and Group Velocity Dispersion . . . . . . . . . . . . . . . . 231.4.3 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 Dynamic Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.5.1 Kerr-Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Moving Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7 Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.8 Split-Step Fourier-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9 Effects of a Kerr-Type Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.9.1 Self-focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.9.2 Self-Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.9.3 Pulse Compression and Breakup . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Ordered Structures: Transfer-Matrix Method 352.1 Forward Backward Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Continuity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 Continuity of the Tangential Component . . . . . . . . . . . . . . . . . . . . 362.2.2 Second Condition of Continuity . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Solution for N Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Slowly Varying Envelope Approximation . . . . . . . . . . . . . . . . . . . . . . . 382.6 Range of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10 CONTENTS

3 Arbitrary Structures: FDTD 413.1 Discretization of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 FDTD Equations in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 FDTD Equations in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 Advantages and Disadvantages of the FDTD Method . . . . . . . . . . . . . . . . . 44

II Microscopic Material Theory 47

4 Density Matrix Formalism 514.1 Schrödinger-Picture and Heisenberg-Picture . . . . . . . . . . . . . . . . . . . . . . 514.2 Pure States and Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Hamiltonian 555.1 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Classical Hamilton-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Hamilton-Operator of a Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.1 Bloch-Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3.2 Classical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.2.1 Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.3 Coulomb-Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.3.1 2D Electron Gas - Perfect Confinement . . . . . . . . . . . . . . . 625.3.3.2 3D Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3.4 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3.5 Phonon-Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.6 Quantized Light Field (Photons) . . . . . . . . . . . . . . . . . . . . . . . . 635.3.7 Photon-Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.8 Photon-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Electron-Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4.1 Free Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4.2 Free Photons, Photon-Electron and Electron-Electron . . . . . . . . . . . . . 665.4.3 Electron-Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Hartree-Fock-Approximation 676.1 Example Hierarchy-Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Factorizing and Mean-Field-Approximation . . . . . . . . . . . . . . . . . . . . . . 686.3 Generalized Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Equations of Motion 717.1 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.1 Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.1.1.1 Bloch-Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.1.1.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.1.2 Semiconductor Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . 737.1.3 Parabolic Band Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 747.1.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

CONTENTS 11

7.1.5 Maxwell Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8 Wannier-Expansion 778.1 Wannier-Expansion of the Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788.3 Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.4 Validity of the Wannier-Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9 Rabi-Oscillation 819.1 Validity of the Wannier-Expansion for Highly Nonlinear Excitation . . . . . . . . . . 83

10 Adiabatic Following and First Order Memory Effects 8710.1 Adiabatic Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 Memory Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.3 Continuum Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

11 Boltzmann-Equation 9311.1 Markov-Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9411.2 Boltzmann-Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9511.3 Scattering Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12 Quantum Dot on Wetting Layer 9712.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9712.2 Coulomb-Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9912.3 Phonon Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012.5 Linear Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10112.6 Nonlinear Dynamics - Rabi-Oscillation . . . . . . . . . . . . . . . . . . . . . . . . 10212.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

13 THz-Luminescence 10513.1 Luminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

13.1.1 Spectral Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10513.1.2 Free Propagation Approximation . . . . . . . . . . . . . . . . . . . . . . . . 10613.1.3 Free-mode Expansion of the Field Operators . . . . . . . . . . . . . . . . . 10613.1.4 Stationary Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

13.2 THz-Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10713.2.1 Emission from Free Electron . . . . . . . . . . . . . . . . . . . . . . . . . . 10713.2.2 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

13.2.2.1 Effective Scattering Matrix . . . . . . . . . . . . . . . . . . . . . 10813.2.2.2 Plasmon-Polariton . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.2.2.3 Statistical Ensemble Average . . . . . . . . . . . . . . . . . . . . 11013.2.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

13.2.3 Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11213.2.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

12 CONTENTS

III Propagation in Nonlinear Materials 117

14 Multiple Quantum Wells 12114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.2 Linear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

14.2.1 Superradiant Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.2.2 Radiative Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12314.2.3 Photonic Band Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12514.2.4 Deviation from Bragg-Periodicity . . . . . . . . . . . . . . . . . . . . . . . 12514.2.5 Polariton-like Beating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

14.2.5.1 Comparison to the Experiment . . . . . . . . . . . . . . . . . . . 12814.3 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

14.3.1 Analytic Solutions for Long Structures . . . . . . . . . . . . . . . . . . . . 13014.3.1.1 Two Level System Equations of Motion . . . . . . . . . . . . . . 13014.3.1.2 First Order Wave Equation . . . . . . . . . . . . . . . . . . . . . 13014.3.1.3 Sine-Gordon-Equation . . . . . . . . . . . . . . . . . . . . . . . . 13114.3.1.4 Solutions of the Sine-Gordon-Equation . . . . . . . . . . . . . . . 13214.3.1.5 Validity of Derivation . . . . . . . . . . . . . . . . . . . . . . . . 133

14.3.2 Pauli-Blocking Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 13414.3.3 Adiabatic Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13414.3.4 Ultrafast Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13514.3.5 Suppression of Polariton-like Beating . . . . . . . . . . . . . . . . . . . . . 13614.3.6 Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13914.3.7 Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14114.3.8 Soliton-like Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

14.4 Rabi-Oscillation in Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . 14314.5 Light Capturing and Storage in MQW . . . . . . . . . . . . . . . . . . . . . . . . . 143

14.5.1 Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14514.5.1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 14514.5.1.2 Free Standing Wave . . . . . . . . . . . . . . . . . . . . . . . . . 14514.5.1.3 Standing Wave Inside the MQW . . . . . . . . . . . . . . . . . . 14614.5.1.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 14714.5.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

14.5.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14714.5.2.1 Arbitrary Stationary Solutions . . . . . . . . . . . . . . . . . . . . 14714.5.2.2 Trapping External Light Pulses . . . . . . . . . . . . . . . . . . . 14814.5.2.3 Stability of the Numerical Solution . . . . . . . . . . . . . . . . . 15314.5.2.4 Classes of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 153

14.5.3 Example of Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . 15514.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

15 Pulse Propagation in Bulk Semiconductors 15715.1 Resonant Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

15.1.1 Linear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15815.1.2 Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16015.1.3 Influence of Chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

15.2 Near-resonant Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

CONTENTS 13

15.2.1 Linear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16215.2.2 Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.2.3 Transversal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

15.2.3.1 Comparison with the Experiment . . . . . . . . . . . . . . . . . . 17315.2.4 Comparison of Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . 176

15.2.4.1 Semiconductor Bloch Equations . . . . . . . . . . . . . . . . . . 17715.2.4.2 Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . 17715.2.4.3 Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . 17715.2.4.4 Numerical Comparison . . . . . . . . . . . . . . . . . . . . . . . 17715.2.4.5 Linear Spectra and Pulse Propagation . . . . . . . . . . . . . . . . 17715.2.4.6 Nonlinear Pulse Propagation . . . . . . . . . . . . . . . . . . . . 17815.2.4.7 Instantaneous Frequency . . . . . . . . . . . . . . . . . . . . . . 17915.2.4.8 Validity of Model Systems . . . . . . . . . . . . . . . . . . . . . 187

Appendix 189

A Commutator Relations 191A.1 Basic Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

A.1.1 General Fermionic Commutators . . . . . . . . . . . . . . . . . . . . . . . . 191

B Markov-Approximation 193B.1 Markov-Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

C Equations of Motion for Generalized Coulomb Hamiltonians 195C.1 Generalized Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195C.2 Generalized Coulomb-Matrix-Element . . . . . . . . . . . . . . . . . . . . . . . . . 195

C.2.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196C.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

D Transfer-Matrix for N Quantum Wells 199

E Equations of Motion for Quantum Dot on Wetting Layer 201E.1 Quantum Dot Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202E.2 Wetting Layer Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202E.3 Quantum Dot - Wetting Layer quantities . . . . . . . . . . . . . . . . . . . . . . . . 203

F Instantaneous Frequency 205F.1 Refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205F.2 Instantaneous Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

F.2.1 Only linear contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207F.2.2 Only Kerr nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207F.2.3 Memory Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

G Material Parameters 209

14 CONTENTS

H Acronyms, Notations and Symbols 211H.1 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211H.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212H.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Bibliography 213

CONTENTS 15

Introduction

Motivation

In the last years there has been great progress in semiconductor epitaxy and the generation of ultrashortlaser pulses.

With the introduction of mode-coupled Ti:Sapphire-Lasers it is now possible to generate sub-picosecond laser pulses with intensities above 30 MW

cm2 . The pulse energy is high enough to inducehighly nonlinear material response in semiconductors. On the other hand the pulse is short enoughto enable experiments on timescales where quantum mechanical coherences play an important rolein the material response. Thus the collective coherent dynamics of semiconductor systems (e.g. lowdimension semiconductor structures) under nonlinear excitations can be investigated.

Improvements in semiconductor epitaxy allow for growth control at a level where it is possibleto produce layered structures of a few hundred quantum wells with such a precision that allows theconstruction of optical Bragg-reflectors. Furthermore the purity of the samples can be increased tolimit unwanted dephasing processes of the optical dipole density by impurity scattering to a levelwhere the observation of coherent phenomena is possible.

Due to these experimental possibilities it is of great interest to theoretically examine such systemsand structures to be able to interpret experiments and to gain insight into the fundamental mechanismsinvolved.

For possible applications it is interesting to study if effects known from bulk semiconductor (likeself-induced transparency [1]) or from structured dielectric materials (e.g. optical band gaps in pho-tonic crystals[2] and gap-solitons in Bragg-structured optical fibers [3]) are present in mesoscopicsemiconductor structures.

Structure of the Thesis

This work is structured in three parts. In part I the theory of light propagation is developed. Differentapproaches and algorithms are discussed: From slowly varying envelope approximation and nonlinearSchrödinger equation for bulk propagation to the matrix-transfer formalism for ordered structuresand the finite-differences time-domain algorithm for arbitrary geometries. In part II the microscopicmaterial dynamics of semiconductors and laser induced plasma are discussed in the frame work ofsecond quantization. Part III combines the material dynamics and light propagation to investigate thelinear and nonlinear properties of different bulk configurations and nano-structures.

16 CONTENTS

17

Part I

Propagation of the Electromagnetic Field

19

In this part the propagation of electromagnetic radiation is considered. In the first chapter propaga-tion equations for light propagating in bulk semiconductor are derived: The slowly varying envelopeapproximation (SVEA) still coupling to an microscopic model for the material response and the non-linear Schrödinger equation up to 3rd order which approximates the material response by a timelocal expansion of the polarization. In the second chapter the finite-differences time-domain algo-rithm (FDTD) is presented as a very versatile approach for solving Maxwell’s equation in arbitrarystructures and systems. In the last chapter the propagation in ordered structures is described withthe transfer-matrix method on the model system of Bragg-periodic multiple quantum wells. Both theSVEA propagation equation and the FDTD algorithm can be coupled to a microscopic model for thematerial response.

21

Chapter 1

Bulk Propagation

In a bulk semiconductor a wave equation can be derived from Maxwell’s equations. In contrast to vac-uum the material polarization enters the wave equation as a source term. With a few approximationsa propagation equation for an electromagnetic pulse can be derived (SVEA [1]) which can be coupledto a microscopic description of the polarization. To further simplify the theory the material responsecan be expanded locally in time resulting in the well known nonlinear Schrödinger equation (e.g. [4]).

1.1 Maxwell’s Equations

Maxwell’s equations in their most general form are given by [5]

∇×E = −∂tB , ∇ ·D = ρ ,∇ ·B = 0 , ∇×H = ∂t D+ j (1.1)

and the material equations (which couple the electric displacement field to the polarization and themagnetic field strength to the magnetization)

D = ε0E+P3D , H =1µ0

B−M (1.2)

with the following quantities

E electric field, M magnetization,D electric displacement field, ρ charge density,B magnetic flux density, j current density,H magnetic field strength, ε0 permittivity,

P3D polarization, µ0 permeability.

In this thesis only systems without macroscopic currents, charges, and magnetization are consid-ered:

ρ = 0 , (1.3)

j = 0 .

Maxwell’s equations now take the simplified form

∂tE = 1ε0µ0

∇×B− 1ε0

∂t P3D , ∇ ·B = 0 ,

∂tB = −∇×E , ∇ ·D = 0(1.4)

22 1. BULK PROPAGATION

with the polarization P3D acting as a source term. This set of equations can be directly numericallysolved as described in chapter 3 or, as in the following, a wave equation can be derived and thepropagation of light pulses considered.

1.2 Wave Equation

From eq. (1.4) follows the wave equation for the electric field

−∇(∇ ·E)+∆E = ε0µ0∂2t E+µ0∂2

t P3D , (1.5)

where the divergence of the electric field can be rewritten

− 1ε0

∇(∇ ·D−∇ ·P3D)+∆E = ε0µ0∂2t E+µ0∂2

t P3D

⇔ 1ε0

∇(∇ ·P3D)+∆E = ε0µ0∂2t E+µ0∂2

t P3D (1.6)

leading to the canonical form of the wave equation for the electrical field driven by the polarizationP3D with the phase velocity c0 = 1√ε0µ0

:

(∆− 1c2

0∂2

t )E = µ0∂2t P3D

dyn −1ε0

∇(∇ ·P3D) . (1.7)

The term ∇(∇ ·P3D) is neglected as part of the paraxial approximation [4] as used in section 15.2.3.The paraxial approximation considers transversal propagation effects in first order while retaining theoverall planar wave character of the wave. The resulting wave equation is

(∆− 1c2

0∂2

t )E = µ0∂2t P3D . (1.8)

1.3 Splitting of Background Polarization and Dynamic Polarization

In general the polarization P3D is nonlinear in E, is dependent on the frequency of the exciting field,and time-nonlocal (i.e. exhibiting memory effects). A theory is needed to describe the response ofthe material via the polarization to an electric field. The macroscopic polarization P3D is decomposedinto two contributions: the linear polarization of the background material P3D

back = ε0χbackE and thedynamic polarization P3D

dyn caused by the coherent dynamics of the semiconductor:

P3D = P3Dback +P3D

dyn . (1.9)

The wave equation now reads

(∆− 1c2

0∂2

t )E = µ0∂2t P3D

back +µ0∂2t P3D

dyn . (1.10)

1.4 Background Polarization

The background polarization P3Ddyn is time local and linear in the electric field E but depends on the

frequency of the exciting field. It is therefore possible to treat P3Ddyn in the frequency domain by means

of a Taylor expansion in the frequency ω. This expansion will lead to the the group velocity vg = ∂ω∂k

and group velocity dispersion (GVD) ∂2ω∂k2 .

1.4. BACKGROUND POLARIZATION 23

1.4.1 Quasi-monochromatic Wave Packet / SVEA

As propagating plane waves are a solution of eq. (1.8) a superposition of plane waves is now con-structed. Without loss of generality only one component of the field is considered (E = ∑i Eiei ).Furthermore the direction of propagation of this wave packet is assumed to be along the z-axis andonly propagation in one direction is assumed. This effectively neglects reflection and backscatteringwhich is a good assumption for bulk propagation. A wave packet can be written as

Ei(t,r) = F(x,y)1

2π

Z ∞

−∞Ê(ω−ω0,z)ei(kz−ωt) dω (1.11)

with Ê(ω−ω0,z) being the spectral distribution function of the plane wave superposition and F(x,y)the transversal pulse profile. Equation (1.11) can be rewritten as follows

Ei(t,r) = F(x,y)1

2π

Z ∞

−∞Ê(ω−ω0,z)ei(kz−ωt) dω (1.12)

= F(x,y)ei(kz−ω0t) 12π

Z ∞

−∞Ê(ω−ω0,z)e−i(ω−ω0)t dω (1.13)

and with the Fourier-transformation E(t,z) = 12π

R ∞−∞

Ê(ω−ω0)e−i(ω−ω0)t dω the field component Eibecomes

Ei(t,r) = F(x,y)Ei(t,z)ei(kz−ω0t) . (1.14)

It can be seen that the representation of the form (1.14) with a monochromatic planar wave and amodulating function in space and time (F(x,y)E(t,z)) is equivalent to the wave packet of equation(1.11).

For any expansion in terms of the frequency ω around ω0 to be valid, the wave packet has to benarrowly centered around the central frequency ω0 (and therefore around k(ω0)), i.e. quasi monochro-matic. This narrow spectral distribution function translates into a slowly varying modulating functionF(x,y)E(t,z). For perfect monochromatic waves the modulation function is constant. If the polariza-tion is expanded in ω it is also valid to perform a Slowly Varying Envelope Approximation (SVEA,e.g. [4, 6]) by neglecting derivatives of the slowly varying envelopes which is done in the followingcalculation.

1.4.2 Group Velocity and Group Velocity Dispersion

The equation for one component of the field

(∆− 1c2

0∂2

t )Ei(t,r) = µ0∂2t P3D

i,back +µ0∂2t P3D

i,dyn (1.15)

is transformed into the frequency domain via the Fourier-transformation G(ω) =R ∞−∞ G(t)eiωt dt :

(∆+1c2

0ω2)E = −µ0ω2P3D

back −µ0ω2P3Ddyn . (1.16)

The background polarization is written with the frequency-dependant susceptibilityP3D

back = ε0χback(ω)E resulting in

(∆+1c2

0ω2(1+χback(ω)))E = −µ0ω2P3D

dyn (1.17)


with the frequency dependent refractive index n(ω) =√

1+χback(ω) and subsequently the frequencydependent c = c0

n(ω) :

(∆+ω2

c2 )E = −µ0ω2P3Ddyn . (1.18)

Inserting the Fourier-transformed wave packet ansatz (1.14) for the field

E(x,y,z,ω−ω0) =

Z ∞

−∞Ei(t,r)eiωt dt (1.19)

=

Z ∞

−∞F(x,y)Ei(t,z)ei(kz−ω0t)eiωt dt

= F(x,y)eikzZ ∞

−∞E(t,z)ei(ω−ω0)t ddt

= F(x,y)eikz Ê(ω−ω0,z)

yields with the transversal Laplace-operator ∆t = ∆−∂2z

(∆t +∂2z +

ω2

c2(ω))F(x,y)eikz Ê(ω−ω0,z) = −µ0ω2P3D

dyn (1.20)

⇔ (eikz Ê(ω−ω0,z)∆t F(x,y)+F(x,y)∂2z eikz Ê(ω−ω0,z) (1.21)

+F(x,y)eikz Ê(ω−ω0,z)ω2

c2 ) = −µ0ω2P3Ddyn

⇔ (∆tF(x,y)F(x,y)

− k2 +2ik∂z

Ê(ω−ω0,z)Ê(ω−ω0,z)

+∂2

zÊ(ω−ω0,z)

Ê(ω−ω0,z)+

ω2

c2 ) (1.22)

= − 1

F(x,y) Ê(ω−ω0,z)eikzµ0ω2P3D

dyn .

For this calculation to yield the pure effects of the background polarization, it is necessary to assumethe dynamic polarization to be of perturbative nature and in first order to ignore it altogether for thetime being (as discussed in [4]). The differential equation can now be decoupled [4]

(∆tF(x,y)F(x,y)

+∂2

zÊ(ω−ω0,z)

Ê(ω−ω0,z)+

2ik∂zÊ(ω−ω0,z)

Ê(ω−ω0,z)− k2 +

ω2

c2

)= 0 , (1.23)

which leads to two equations

∆tF(x,y)+(ω2

c2 − k2)F(x,y) = 0 (1.24)

∂2z

Ê(ω−ω0,z)+2ik∂zÊ(ω−ω0,z)+(−k2 + k2) Ê(ω−ω0,z)) = 0 (1.25)

with the propagation constant k following from the eigenvalue problem (1.24). As the envelope func-tion Ê(ω−ω0,z) is slowly varying for spectrally narrow wave packets, the second derivative in z isneglected (as part of the SVEA), leading to the equations

∆tF(x,y)+(ω2

c2 − k2)F(x,y) = 0 (1.26)

2ik∂zÊ(ω−ω0,z)+(−k2 + k2) Ê(ω−ω0,z)) = 0 . (1.27)

1.4. BACKGROUND POLARIZATION 25

By assuming k to differ from k by a small ∆k

k2 − k2 = (k +∆k)2 − k2

= k2 +2k∆k +(∆k)2 − k2

= 2k(∆k +12

k− 12

k)+(∆k)2

= 2k(∆k + k− k)+(∆k)2

= 2k(k− k)+(∆k)2 ≈ 2k(k− k)

and neglecting (∆k)2, equation (1.27) can be transformed into

∂zÊ(ω−ω0,z)− i(k− k) Ê(ω−ω0,z)) = 0 . (1.28)

Now the frequency dependent propagation constant k(ω) is expanded in terms of ω in the vicinity ofω0 up to second order:

k ≈ k0 +(ω−ω0)dkdω

+12(ω−ω0)

2 d2kdω2 . (1.29)

The relevant parameters are the inverse group velocity 1vg

= dkdω and the inverse group velocity disper-

sion (GVD) β2 = d2kdω2 :

k =ω0

c+(ω−ω0)

1vg

+12(ω−ω0)

2β2 . (1.30)

Inserting (1.30) into equation (1.28) results in

∂zÊ(ω−ω0,z)− i(ω0

c +(ω−ω0)1vg

+ 12(ω−ω0)

2β2 − k0)Ê(ω−ω0,z) = 0 (1.31)

⇔ ∂zÊ(ω−ω0,z)− i((ω−ω0)

1vg

+ 12 (ω−ω0)

2β2)Ê(ω−ω0,z) = 0 . (1.32)

After applying the dispersion relation k2 = ω2

c2 , equation (1.26) reads

12ik0

∆tF(x,y) = 0 . (1.33)

Combining (1.32) and (1.33) and consistently taking into account again the previously neglected dy-namic polarization results in

∂zÊ(ω−ω0,z)F(x,y)− i((ω−ω0)

1vg

+12(ω−ω0)

2β2)Ê(ω−ω0,z)F(x,y)

+1

2ik0∆tF(x,y) Ê(ω−ω0,z) = − 1

2ik0e−ikzµ0ω2P3D

dyn (1.34)

⇔ (∂z − i(ω−ω0)1vg

+ i12(ω−ω0)

2β2

+1

2ik0∆t)F(x,y) Ê(ω−ω0,z) = − 1

2ik0e−ikzµ0k2c2P3D

dyn . (1.35)


At this stage, to be consistent, the remaining k has to be expanded in ω (up to first order)

k2c2P3Ddyn ≈ (

ω0

c)2c2P3D

dyn = ω20P3D

dyn .

In the time domain this is equivalent to the neglection of the derivatives of the envelope of the dynamicpolarization and is therefore justified in the context of the SVEA:

P3Di,dyn = Pi,dyn(t,x,y,z)ei(kz−ω0t) (1.36)

∂2t P3D

i,dyn = ei(kz−ω0t)(∂2t −2iω0∂t −ω2

0)Pi,dyn(t,x,y,z)

≈ −ω20ei(kz−ω0t)Pi,dyn(t,x,y,z) = −ω2

0P3Di,dyn . (1.37)

The equivalence can be seen by the Fourier-transform:

∂2t P3D

i,dyn → −ω20P3D

i,dyn

↓ ↓−ω2P3D

dyn → −ω20P3D

dyn .

Equation (1.35) now reads

(∂z − i(ω−ω0)1vg

+ i12(ω−ω0)

2β2

+1

2ik0∆t)F(x,y) ˆE(ω−ω0,z) = − 1

2ik0e−ikzµ0ω2

0P3Ddyn . (1.38)

This equation is transformed back into time space via the inverse Fourier-transformG(t) = 1

2πR ∞−∞ G(ω−ω0)e−i(ω−ω0)t d(ω):

(∂z +1vg

+ i12

β2 +1

2ik0∆t)F(x,y)Ei(t,z) = −eiω0t 1

2ik0e−ikzµ0ω2

0P3Ddyn . (1.39)

With (1.36) the wave equation for one component of the envelope functions reads

(∂z +1vg

∂t + i12

∂2t β2 +

12ik0

∆t)F(x,y)Ei(t,z) = − 12ik0

µ0ω20Pi,dyn(t,x,y,z)

= − ω0

2icε0Pi,dyn(t,x,y,z) , (1.40)

and the full vector equation for the envelopes E = ∑i Eiei:

(1

2ik0∆t +∂z +

1vg

∂t +i2

β2∂2t )E(t,x,y,z) = − ω0

2icε0Pdyn(t,x,y,z) . (1.41)

1.4.3 Absorption

In the previous subsection the susceptibility χback and therefore also the refractive index n have beenassumed to be real numbers which is not necessarily true. Assuming an imaginary contribution n =n+ iκ leads to a damping of the propagating wave while propagating (Beer’s Law):

ei(kx−ωt) = ei( ωnc0

z−ωt) (1.42)

= ei( ωn+iκc0

z−ωt) (1.43)

= e−κc zei( n

c z−ωt) . (1.44)

1.5. DYNAMIC POLARIZATION 27

This absorptive contribution of the background polarization is assumed to be independent from thefrequency and is incorporated by an absorption coefficient α

(1

2ik0∆t +∂z +

1vg

∂t +i2

β2∂2t +

α2

)E = − ω0

2icε0Pdyn . (1.45)

1.5 Dynamic Polarization

In general the dynamic polarization has to be calculated from a microscopic material theory (cp.Part II). However, where memory effects can be neglected the dynamic polarization can be expandedin E to derive a propagation equation.

1.5.1 Kerr-Nonlinearity

For pulses which propagate off-resonant (i.e. spectrally not overlapping resonances) memory effectscan be neglected in the dynamic polarization (adiabatic following, cp. chapter 10 and section 15.2)and P3D

dyn can be expanded in a Taylor series in E

P3Ddyn = ε0(χ1E+χ2EE+χ3E2E+ ...) , (1.46)

which leads to a classical description (opposed to the microscopic quantum mechanical descriptionpresented in part II) of the medium with classical material parameters:

χ1 is the linear susceptibility of the dynamic polarization responsible for the same effects as thebackground polarization (section 1.4), e.g. chromatic dispersion, and thus result in a constant renor-malization of the group velocity vg, the group velocity dispersion coefficient β2 and the absorptioncoefficient α.

χ2 is non-vanishing only in crystals without inversion centre and gives rise for quadratic effectslike second harmonic generation (SHG), the linear Stark-effect and the Pockels-effect.

χ3 generates 3rd order effects like self-focusing (section 1.9.1), self-phase modulation (SPM,section 1.9.2), the Kerr-effect and Four-Wave-Mixing. As the semiconductors investigated in thiswork all posses an inversion centre, χ2is omitted.

As the first order of the expansion results just in coefficient renormalization and the second ordervanishes, the third order of the expansion of P3D

dynis inserted into the wave equation (1.45):

(1

2ik0∆t +∂z +

1vg

∂t +i2

β2∂2t +

α2

)E = − ω0

2icε0ε0χ3|E|2E (1.47)

(1

2ik0∆t +∂z +

1vg

∂t +i2

β2∂2t +

α2− i

χ3ω0

2c|E|2

)E = 0 (1.48)

(1

2ik0∆t +∂z +

1vg

∂t +i2

β2∂2t +

α2− iγ|E|2

)E = 0 (1.49)

1.6 Moving Frame

Both the wave equation with general dynamic polarization (eq. 1.45) and the wave equation withKerr-nonlinearity (eq. 1.49) can be further simplified by bearing in mind the character of the solutions


which are propagating pulses. A new coordinate system can be introduced which is moving alongwith the propagating pulse (moving frame). The coordinate transformation is given by

η = t − zvg

(1.50)

ξ = z , (1.51)

where ζ is the space coordinate in propagation direction and η is called the local time of the pulse.The derivatives are transformed accordingly

∂t = ∂η (1.52)

∂z = − 1vg

∂η +∂ξ , (1.53)

and the wave equations in the moving frame and slowly varying envelope approximation take the form

(− i

2k0∆t +∂ξ +

i2

β2∂2η +

α2

)E =

iω0

2cε0Pdyn (1.54)

for the general dynamic polarization (eq. 1.45) and

(− i

2k0∆t +∂ξ +

i2

β2∂2η +

α2− iγ|E|2

)E = 0 (1.55)

for the Kerr-nonlinearity (eq. 1.49).

1.7 Nonlinear Schrödinger Equation

For a propagating of a transversal infinite plane wave (i.e. no transversal effects due to symmetry)equation (1.55) is called the nonlinear Schrödinger equation (NLSE, e.g. [4]) and is usually written inthe form of a propagation equation

∂ξE(ξ,η) = −α2

E(ξ,η)− i2

β2∂2ηE(ξ,η)+ iγ|E|2E(ξ,η) . (1.56)

The nonlinear Schrödinger equation has been studied intensively in the context of soliton propagation.It has been used to model successfully e.g. pulse interactions in waveguides [7], soliton dynamics inoptical lattices [8] and fibers [9], propagation in a hot plasma [10], filamentation of laser pulses in air[11] and propagation in Kerr-media[12]. The general nonlinear Schrödinger equation incorporates thefollowing contributions

∂ξE(ξ,η) = −α2 E(ξ,η) absorption (1.57)

− i2 β2∂2

ηE(ξ,η) dispersion

+iγ|E|2E(ξ,η) nonlinearity

+ i2k0

∆t E (possibly) transversal effects .

1.8. SPLIT-STEP FOURIER-ALGORITHM 29

1.8 Split-Step Fourier-Algorithm

The equations (1.55) and (1.54) normally do not lend themselves to analytic solutions and have tobe numerically solved. In case of (1.54) this has to be done self-consistently with the microscopicmaterial equations for the polarization. Both equations are of the form

∂z f = (A+ B) f (1.58)

with differential operators A and B. Without loss of generality only two operators are considered asthe results can easily be transfered for more operators. As equations (1.55) and (1.54) are nonlinearpartial differential equations, the operators may depend on f and do not generally commutate. Asimple discretization of the derivatives on the RHS of this equation is not an optimal way as thenumerical errors can not be controlled with justifiably effort. A more elegant way is to treat theterms in the equation individually in momentum- or frequency space. This concept is called split-stepFourier-method[4] and a widely used algorithm.

The nonlinear partial differential equation (1.58) is formally solved by

f (z+h, t) = eh(A+B) f (z, t) . (1.59)

For solution (1.59) to be valid, the operators A and B are required to be self-adjoint. For small step-sizes h the exponential function can be expand in h:

eh(A+B) = 1+hA+hB+O(h2) . (1.60)

Only the first order in h is considered.1 Inserting (1.60) into the formal solution (1.59) results in

f (z+h, t) ≈ f (z, t)+hA f (z, t)+hB f (z, t) . (1.61)

Expression (1.61) can easily be evaluated using a Fourier-transform for operators A ∝ ∂nt and B ∝ ∂n

x,y.For differential operators containing time derivatives a transformation into frequency-space (FTω) willbe used

A f (z, t) = (FT−1ω

Ã(ω) f (z,ω)FTω)

and for operators containing real-space derivatives transformations into transversal momentum (orwavenumber) space (FTq)

B f (z, t) = (FT−1q

˜B(q) f (z, t,q)FTq) .

Altogether one gets

f (z+h, t) ≈ f (z, t)+h(FT−1ω

Ã(ω) f (z,ω)FTω)+h(FT−1q

˜B(q) f (z, t,q)FTq) . (1.62)

Both Ã(ω) and ˜B(q) are now pure multiplicative operators which is the main advantage of thisapproach for numerical evaluation (together with the availability of highly optimized Fast-Fourier-algorithms). In (1.62) the operators A and B act independently on the function f (z, t). In the picture

1This is equivalent to ignoring the non-commutating nature of the operators: For two bounded non-commutating opera-tors the Baker-Campell-Hausdorff formula holds true: ehAehB = ehA+hB+ 1

2 h2[A,B]+ 112 h3[A−B,[A,B]]+.... Assuming the commu-

tator to vanish [A, B] = 0 is equivalent to neglecting terms in h2: ehAehB ≈ ehA+hB .


of a propagating pulse this translates into the following assumption: for sufficiently small propagationlength h the different effects (e.g. dispersion and nonlinearity) act independently on the propagatinglight pulse. In the numerical algorithm the light pulse propagates multiple times through a layer ofthe thickness h. First, the effect of the dispersion is computed and the resulting pulse is once morepropagated through the layer to evaluate the influence of the nonlinearity. Because of these multi-ple propagations through the same layer and the application of Fourier-Transforms this method iscalled split-step Fourier. As seen above this method is accurate to the first order in the step size h fornon-commutating operators.

Applying this method on equation (1.54) results in

E(ξ+h,η) = E(ξ,η)+i

2k0h∆t E(ξ,η)− i

2β2h∂2

ηE(ξ,η)− α2

hE(ξ,η)+hiω0

2cε0Pdyn (1.63)

= (1− αh2

)E(ξ,η)+FT−1q

i2k0

hi2q2 Ê(q,ξ,η)FTq (1.64)

−FT−1ω

i2

β2i2ω2h Ê(ξ,ω)FTω +hiω0

2cε0Pdyn

= (1− αh2

)E(ξ,η)− ih2k0

(FT−1

q q2 Ê(q,ξ,η)FTq

)(1.65)

+ihβ2

2

(FT−1

ω ω2 Ê(ξ,ω)FTω

)+

iω0h2cε0

Pdyn ,

which can easily be numerically implemented.

1.9 Effects of a Kerr-Type Nonlinearity

In section 1.4.2 it has been shown that the refractive index is

n(ω) =√

1+χback =

√

1+P

ε0E. (1.66)

With a Kerr-type nonlinearity P = ε0(χ1E + χ3|E|2E) as introduced in section 1.5.1 the refractiveindex gets

n(ω) =

√1+χ1 +χ3|E|2 . (1.67)

Equation (1.67) is now expanded in E yielding

n(E) ≈√

1+χ1 +χ3

2√

1+χ1|E|2 = n0 +n2|E|2 . (1.68)

1.9.1 Self-focusing

For positive χ3 (and therefore positive n2) the expansion (1.68) leads to an increase of the refractiveindex with increasing pulse intensity. Considering the transversal profile of a propagating pulse therefractive index gets higher at the centre of the pulse. This results in a waveguide-like structure(Fig. 1.1). The pulse is focused in this waveguide leading to a higher intensity in the pulse centerand an even higher change of refractive index. If not counteracted by higher order nonlinearities andabsorption this effect would lead to a focusing of the pulse to an infinite small transversal profile.

1.9. EFFECTS OF A KERR-TYPE NONLINEARITY 31

Figure 1.1: Self-induced waveguide leads to transversal profile compression. The shape of the refrac-tive index is exaggerated to show the principle.

1.9.2 Self-Phase Modulation

The refractive index expansion (1.68) also leads to a similar temporal effect described as Self-phasemodulation. The field (without transversal coordinates) can be written as

E(t,z) = E(t,z)ei(kz−ω0t) (1.69)

= E(t,z)ei( 1c ω0nz−ω0t) . (1.70)

E(t,z) is the pulse envelope and the time dependent phase φ is

φ(t) = (1c

ω0n(|E(t)|)z−ω0t) . (1.71)

With the intensity dependent refractive index, it gets

φ(t) =1c

ω0(n0 +n2E2)z−ω0t) . (1.72)

The instantaneous frequency is given by the time derivative of the phase

ω(t) = φ(t) =1c

ω0n2∂t E2z−ω0 . (1.73)

Assuming a Gaussian intensity

E2(t) = I(t) = I0e−t2

τ2

the instantaneous frequency gets

ω(t) = φ(t) = −2tτ2

1c

ω0n2I0e−t2

τ2 z−ω0 ∝ tI −ω0 . (1.74)

The instantaneous frequency is modulated around the central frequency ω0. Depending on the sign ofn2 the frequency of the first part of the pulse is decreased and of the last part increased or vice versa.The typical instantaneous frequency of a self-phase modulated Gaussian pulse is depicted in Fig. 1.2.


Figure 1.2: Self-phase modulation of a Gaussian pulse due to a Kerr-type nonlinearity. The instanta-neous frequency is S-shaped: ω(t) ∝ tI while the pulse envelope is unaltered by the chirp.

1.9.3 Pulse Compression and Breakup

The group velocity dispersion, which is a linear effect in E as was shown in section 1.4.2, does interactwith the self-phase modulation: The increase or decrease in frequency of parts of the pulse leads inconjunction with the GVD to different propagation velocities of the pulse components. This can leadto two effects depending on the sign of SPM and GVD:

If the leading part of the pulse is decelerated and the last part is accelerated the pulse is in effectcompressed in the temporal domain. The steepening of the pulse would lead to infinite pulse intensitiesand destruction of the material if not counteracted by higher order effects. These so called freak-wavesare not only known from electrodynamics but also from oceanic surface waves where they caused alot of attention in recent time[13]. A typical pulse evolution is shown in Fig. 1.3.

If however the leading part of the pulse is accelerated and the trailing part is decelerated theinterplay of GVD and SPM leads to a pulse breakup into two (or more) separated components. Atypical pulse breakup for near-resonant propagation in bulk GaAs[14] is shown in Fig. 1.4. The SPMleads also to a spectral splitting into several components corresponding to the temporal components.

1.9. EFFECTS OF A KERR-TYPE NONLINEARITY 33

Figure 1.3: Typical pulse steepening due to SPM and GVD. The pulse intensities were renormalizedfor better visibility and do not represent the original calculations.

7 8 9 10 11 12 13

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

Fiel

d [a

rb.u

nits

]

Time [ps]

0 2 4 6 8 10 12

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Fiel

d [a

rb.u

nits

]

Frequency [arb.units]

Figure 1.4: Pulse breakup due to SPM and GVD: Time resolved Fourier-analysis of a pulse afterpropagating off-resonant in bulk GaAs. The temporal splitting corresponds to the spectral splitting.Faster (earlier) temporal components correspond to blue spectral components and vice versa.

35

Chapter 2

Ordered Structures: Transfer-MatrixMethod

If a structure to be analyzed consists mainly of a passive background material and only of small,ordered regions of active material the most convenient approach is the transfer-matrix method: Insuch a setup in most places nothing “interesting” happens. A propagating wave only suffers a timedelay and a phase change while traveling through such an empty (i.e. passive background) space. It ismore convenient to only calculate these delays and phase changes between areas of interest instead ofe.g. discretizing the whole domain (as in the FDTD method, section 3). This calculation is known astransfer-matrix method and not only utilized in electrodynamics but is also frequently used in quantummechanics to calculate transmission- and reflection-coefficients for scattering at arbitrary potentials[15].

A structure of N parallel two-dimensional quantum-wells (QW) embedded into a passive back-ground material (Fig. 2.1) is considered. The QWs are equally spaced at a distance of l = λ

2 . Thewidth L of the QWs is small compared to l. In this quasi-one dimensional problem we are consideringlinear polarized light propagating perpendicular to the QWs as depicted in Fig. 2.1.

2.1 Forward Backward Splitting

To solve the wave equation (1.8) (with the background polarization incorporated into the refractiveindex c = c0

n )

(∆− 1c2 ∂2

t )E = µ0∂2t P3D

dyn , (2.1)

the electric field between the n-th and (n+1) -th QW is split into a forward propagating part and intoa backward propagating part [16, 17, 18, 19] (Fig. 2.1):

En(z, t) = E+n (t − z

c)+E−

n (t +zc) . (2.2)

In the space between the QWs the equations for both components decouple

(∆− 1c2 ∂2

t )E+n (t − z

c) = 0 = (∆− 1

c2 ∂2t )E

−n (t +

zc)

and are only coupled by the polarization at the places zn of the QWs

(∆− 1c2 ∂2

t )(E+n (t − zn

c)+E−

n (t +zn

c)) = µ0∂2

t P3Ddyn(zn, t).

36 2. ORDERED STRUCTURES: TRANSFER-MATRIX METHOD

E0

E0

E1

E1

EN

EN

+ + +

- - -

Ll

z2 zNz1

z

Figure 2.1: N parallel two-dimensional quantum-wells (QW) embedded into a passive backgroundmaterial. The light-field is split into forward and backward propagating fractions.

2.2 Continuity Conditions

To solve the wave equation in this setup two continuity conditions for the electric field at the placeszn of the QWs have to be derived.

2.2.1 Continuity of the Tangential Component

The tangential component of the electric field at the interface between two layers has to be continuous(Et(z+ ε) = Et(z− ε), e.g. [5]):

E+n (t − z

c)+E−

n (t +zc) = E+

n+1(t −zc)+E−

n+1(t +zc) . (2.3)

2.2.2 Second Condition of Continuity

The second condition of continuity emerges from an integration of the wave equation. This derivationis based on [16]. The wave equation (2.1) is integrated over an interval 2ε around a QW and the limitof ε → 0 is calculated:

limε→0

zn+εZ

zn−ε

(∂2z −

1c2 ∂2

t )(E+n (t − z

c)+E−

n (t +zc))dz = lim

ε→0

zn+εZ

zn−ε

1εc2 ∂2

t P3Ddyn(z, t)dz . (2.4)

2.3. SOLUTION FOR N QUANTUM WELLS 37

The individual terms of equation (2.4) are now considered separately:

limε→0

zn+εZ

zn−ε

∂2z

(E+

n (t − zc)+E−

n (t +zc))

dz

= limε→0

∂z

(E+

n (t − zc)+E−

n (t +zc))|zn+εzn−ε

= ∂z

(E+

n+1(t −zn

c)+E−

n+1(t +zn

c)−E+

n (t − zn

c)−E−

n (t +zn

c))

=1c

∂t

(−E+

n+1(t −zn

c)+E−

n+1(t +zn

c)+E+

n (t − zn

c)−E−

n (t +zn

c))

.

(2.5)

In the last step was used that for functions of the form f (t ± zc) the time derivative can be replaced by

the space derivative via ∂z = ∓ 1c ∂t . For the second term follows with (2.3)

limε→0

zn+εZ

zn−ε

− 1c2 ∂2

t (E+n (t − z

c)+E−

n (t +zc))dz

= ∂2t lim

ε→0

zn+εZ

zn−ε

− 1c2 (E+

n (t − zc)+E−

n (t +zc))dz = 0 .

(2.6)

As mentioned above, the width L of the QW is small compared to the spacing of l. Therefore theconfinement of the electron and hole wave functions can by approximated by a δ-distribution in z-direction. For the polarization follows:

P3Ddyn(z, t) = ∑

mP2D

m (t)Φme (z)Φm

h (z) ≈∑m

P2Dm (t)δ(z− zm) , (2.7)

and the RHS of (2.4) becomes

limε→0

zn+εZ

zn−ε

1εc2 ∂2

t P3Ddyn(z, t)dz

= limε→0

zn+εZ

zn−ε

1εc2 ∂2

t ∑m

P2Dm (t)δ(z− zm)dz

=1

εc2 ∂2t P2D

n (t) .

(2.8)

Reassembling (2.5), (2.6) and (2.8) and integrating over time yields the second condition of continuity:

−E+n+1(t −

zn

c)+E−

n+1(t +zn

c)+E+

n (t − zn

c)−E−

n (t +zc) =

1εc

∂tP2Dn (t) . (2.9)

2.3 Solution for N Quantum Wells

With the conditions of continuity (2.3) and (2.9) equations can be derived which relate the electricfield at an arbitrary quantum well in the structure with the incident light field. The light fields in allother quantum wells are eliminated from these equations.


The effect of the polarization of QW m (which is located at zM) at an arbitrary position z is retardeddue to the finite velocity of electromagnetic waves by z−zm

c :

P2Dm (z, t) = = P2D

m (t − z− zm

c) . (2.10)

The actual calculation is given in appendix D and for E+n (t − z

c) yields

E+n (t − z

c) = E+

0 (t − zc)− 1

2εc

n

∑m=1

∂t P2Dm (t − z− zm

c) . (2.11)

An analogous calculation for E−n (t + z

c) leads to

E−n (t +

zc) = E−

N (t +zc)− 1

2εc

N

∑m=n+1

∂tP2Dm (t +

z− zm

c) . (2.12)

With (D.17) and (D.18) the transmitted and reflected waves can be directly computed from the incidentwaves and the polarizations inside the quantum-wells.

2.4 Rotating Frame

Both the electric field and the polarization are assumed to consist of a fast rotating part (oscillatingwith the light frequency) and a slowly varying envelope (analogous to section1.4.1)

E(t,z) =12

e−iωgt E(t,z) (2.13)

P(t,z) = e−iωgt P(t,z) . (2.14)

Rewriting equations (D.17) and (D.18) in this rotating frame

12

e−iωgt E+n (t − z

c) =

12

e−iωgt E+0 (t − z

c)− 1

2εc

n

∑m=1

∂t e−iωgt P2Dm (t − z− zm

c) (2.15)

12

e−iωgt E−n (t +

zc) =

12

e−iωgt E−N (t +

zc)− 1

2εc

N

∑m=n+1

∂t e−iωgt P2Dm (t +

z− zm

c) (2.16)

leads to

E+n (t − z

c) = E+

0 (t − zc)− 1

εc

n

∑m=1

−iωgP2Dm (t − z− zm

c)+∂t P2D

m (t − z− zm

c)

E−n (t +

zc) = E−

N (t +zc)− 1

εc

N

∑m=n+1

−iωgP2Dm (t +

z− zm

c)+∂t P2D

m (t +z− zm

c) . (2.17)

2.5 Slowly Varying Envelope Approximation

If the envelopes E and P change slowly in time (compared to the carrier wave) the time derivativescan be neglected (slowly varying envelope approximation, SVEA):

E+n (t − z

c) = E+

0 (t − zc)+

iωg

εc

n

∑m=1

P2Dm (t − z− zm

c) (2.18)

E−n (t +

zc) = E−

N (t +zc)+

iωg

εc

N

∑m=n+1

P2Dm (t +

z− zm

c) . (2.19)

2.6. RANGE OF APPLICATION 39

If (and only if) the number of quantum wells is small (N < 10) the retardation of the polarization maybe neglected and the polarization becomes

P2Dm (t +

z− zm

c) ≈ φmP2D

m (t) (2.20)

with φm = eikzm = eiklm being a pure phase factor. This results in

E+n (t − z

c) = E+

0 (t − zc)+

iωg

εc

n

∑m=1

φmP2Dm (t)

E−n (t +

zc) = E−

N (t +zc)+

iωg

εc

N

∑m=n+1

φmP2Dm (t) . (2.21)

2.6 Range of Application

If the structure consists of only a small number of quantum wells equations (2.21) can be used fornumerical or even analytical calculations. If however, the structure is extended, equations (2.17) haveto be employed. For such a simulation to be numerically feasible, the distance l between the quantum-wells has to be constant throughout the structure to ensure constant time discretization for the wholespatial domain. For structures with disorder or complex defects the FDTD method (as described inchapter 3), even though more computational demanding, is better suited.

41

Chapter 3

Arbitrary Structures: FDTD

The Finite-Differences Time-Domain Algorithm (FDTD) solves the propagation of electromagneticradiation by directly discretizing Maxwell’s equation and therefore allows for the analysis of arbitrarystructures accurately up to the second order in the discretized quantities. The FDTD equations for onedimensional problems as used in chapter 14 are derived. Though not employed in part III the threedimensional problem is addressed for the sake of completeness.

3.1 Discretization of Maxwell’s Equations

In a configuration were the light propagates exclusively in z direction and is polarized in the x direction(e.g. planar waves propagating in a MQW, as discussed in chapter 14) the symmetries can be used toreduce Maxwell’s equations (1.4). The outer products simplify to

∇×B = −∂zByex

∇×E = ∂zExey .(3.1)

Therefore Maxwell’s equations for a one-dimensional problem read:

∂tE = − 1εµ0

∂zB− 1ε

∂t P3Ddyn

∂t B = −∂zE

∂yB = 0

∂xD = 0 .

(3.2)

Equations (3.2) can be handled numerically by directly discretizing space and time (∆z and ∆t )[20],

z = i∆z

t = n∆t (3.3)

i,n ∈ N .

Therefore all quantities depending on space or time are now denoted as

A(z, t) = A(i∆z,n∆t) → Ani . (3.4)

42 3. ARBITRARY STRUCTURES: FDTD

The time and space derivatives can be expressed as a central difference quotient (with the correspond-ing half-numbered indices symmetric around the central point of the derivative):

∂zA(z, t) → An+ 1

2i −A

n− 12

i∆t

(3.5)

∂t A(z, t) →An

i+ 12−An

i− 12

∆z. (3.6)

Maxwell’s equations become

1∆t

(Bn+ 1

2i+ 1

2−B

n− 12

i+ 12) = − 1

∆z(En

i+1 −Eni ) (3.7)

1∆t

(En+1i −En

i ) = − 1εµ0

1∆z

(Bn+ 1

2i+ 1

2−B

n+ 12

i− 12)− 1

ε1∆t

(Pn+1i −Pn

i ) . (3.8)

Note that the time and space indices of the magnetic flux density B are half numbers and of the electricfield E are integers. This choice will be motivated in section 3.2.

3.2 FDTD Equations in One Dimension

The equations (3.7) and (3.8) can be resorted for later times:

Bn+ 1

2i+ 1

2= B

n− 12

i+ 12− ∆t

∆z(En

i+1 −Eni ) (3.9)

En+1i = En

i −1

εµ0

∆t

∆z(Bn+ 1

2i+ 1

2−Bn+ 1

2i− 1

2)− 1

ε1∆t

(Pn+1i −Pn

i ) . (3.10)

(3.9) and (3.10) are the one dimensional FDTD-equations (finite-difference time-domain) for linearpolarized light. The field distributions at later times can now recursively be calculated from the fielddistributions at earlier times:

En,Bn− 12 → Bn+ 1

2

Bn+ 12 ,En → En+1 . (3.11)

Magnetic flux density B and electric field E are interleaved by half a discretization interval in time.This interleaved spatio-temporal alignment is known as Yee-grid [20] and B and E are updated al-ternately (the so called Leapfrog-algorithm as shown in Fig. 3.2). In (3.11) the polarization Pn+1 isrequired to calculate Bn+ 1

2 . This is especially a problem if the polarization is calculated dynamicallyfrom microscopic material equations. It has to be predicted appropriately. One possible way is toextrapolate Pn+1 with a linear regression algorithm. Another way is to calculate the polarization witha higher order algorithm like Runge-Kutta and to extrapolate only the electric field at the supportingpoints. The later approach is used in this work.

3.3 FDTD Equations in Three Dimensions

For inhomogeneous excitation or material response perpendicular to the propagation direction theFDTD-method has to be extended to two or three dimensions. A three dimensional Yee-Grid is em-ployed (Fig. 3.3).

3.3. FDTD EQUATIONS IN THREE DIMENSIONS 43

0 1 2 3 4 50.5 1.5 2.5 3.5 4.5

5

4

3

2

1

0

4.5

3.5

2.5

1.5

0.5

t=n∆t

z=i∆z

E:

H:

Figure 3.1: Interleaved discretization and update scheme of the one dimensional FDTD method.

Ey

Ex

Ez

HyEzEz

Hx

HzEx

Ey

Ey

Ex

x

y

z

(i,j,k)

Figure 3.2: Yee grid[21] for the three dimension FDTD algorithm.


The emerging equations are similar to those of the one dimensional case. However, coupledequations for all three spatial components of the fields have to be derived. Here the equation for thex-component of the electric field is exemplary shown:

Ex|n+ 1

2i+ 1

2 , j,k= Ex|

n− 12

i+ 12 , j,k

(3.12)

− 1εµ0

∆t

∆z(By|ni+ 1

2 , j− 12 ,k −By|ni+ 1

2 , j+ 12 ,k +Bz|ni+ 1

2 , j,k+ 12−Bz|ni+ 1

2 , j,k− 12)

−1ε

1∆t

(Px|n+ 12

i+ 12 , j,k

−Px|n−12

i+ 12 , j,k

) .

3.4 Absorbing Boundary Conditions

To limit the numerical demands of the simulations, it is convenient to restrict the size of the simulateddomain. If the fields outside the simulated domain are just set zero, which is equivalent to a metal-lic wall, all incident waves are reflected. To prevent this reflections one has to introduce absorbingboundary conditions (ABC’s) which simulate a propagation of the incident waves into infinite spaceoutside the simulated domain. Different algorithm implementing absorbing boundary conditions havebeen proposed[20]. For the one dimensional case a scheme known as Mur-ABC is appropriate. It em-ploys a simple subtraction of the analytic solution for propagating plane waves and therefore cancelsall reflections.

In a higher dimensional system the Mur-ABC cannot be used. The waves no longer arrive per-pendicular to the border of the simulated domain and the Mur-algorithm breaks down. For bordersparallel to the propagation direction it is sometimes opportune to use periodic boundary conditionswere the left side of a region is connected to the right side to form an infinite region reflecting in-herent symmetries of the structure studied [22]. One of the most successful ABCs for two or moredimensions is the Uniaxial Perfect Matched Layer (UPML) approach. It employs an artificial materialmodel which is absorbing incident light without reflection. This is achieved by a special anisotropicunidirectional tensor for the electric and magnetic susceptibility in the form of

ε = ε0εrs , µ = µ0µrs with s =

η−1 0 00 η 00 0 η

. (3.13)

In theory this leads to a perfect absorption even for thin layers. Keeping in mind the finite precision ofthe FDTD scheme the UPML layer consist of several grid points gradually leading from the simulateddomain to the absorbing material. Although the parameter η can be analytically determined, its valuefor best numerical results differs and has to be determined by trial and error.

3.5 Advantages and Disadvantages of the FDTD Method

The main advantage of the FDTD method is its flexibility and versatility. The FDTD technique al-lows the user to specify the material at all points within the computational domain. All materials arepossible and dielectrics, magnetic materials, etc. can be simply modeled without the need to resortto work-arounds or tricks to model these materials. In contrast to other methods like e.g. a matrix-transfer-method (section 2), it is possible to introduce disorder or defects effortlessly. It provides thefield distributions directly in time and space and therefore lends itself to providing animated displays

3.5. ADVANTAGES AND DISADVANTAGES OF THE FDTD METHOD 45

of the electromagnetic field movement through the model. This type of display is useful in under-standing what is going on in the model, and to help ensure that the model is working correctly. TheFDTD method is easy to understand and implement which is another big advantage. Also being acentral difference method it cancels out quadratic contribution from Taylor expansion exactly. It is,despite its appearance, exact in second order.

The FDTD method is computationally very demanding which is its main disadvantage. FDTD re-quires that the entire computational domain be gridded and the discretization must be small comparedto the smallest wavelength (at least 20 spatial steps per wavelength). This normally calls for verylarge computational domains, which result in very long solution times and high memory usage. Thetemporal discretization cannot be chosen independently from the spatial resolution - it has to be equalor smaller than the magical time step ∆t = ∆z

c [20]. Models with long, thin features, (like wires) aredifficult to model in FDTD because of the excessively large computational domain required. Thereforit is also not feasible to extract far field distributions directly out of the model. Also emerging fromthe restriction of the computational domain is the need for artificial boundary conditions to preventunwanted reflections.

47

Part II

Microscopic Material Theory

49

In this part the microscopic material theory of semiconductors (and, as a similar system, of laserinduced electron plasma) is developed. In the first chapter (chapter 4) a short introduction to thedensity matrix formalism is given. Then the relevant Hamiltonian for a semiconductor system ispresented in chapter 5 and the equations of motion (ch. 7) in Hartree-Fock-approximation (chapter6) are derived. To simplify the obtained equations and to enable analytic solutions an expansionin the excitonic base is performed (Wannier-expansion, chapter 8). In the context of the derivedoptical Bloch and semiconductor Bloch equations resonant Rabi-oscillations are discussed and thevalidity of the Wannier-expansion and the influence of dephasing-processes on these Rabi-oscillationsare analyzed (ch. 9). Analytic solutions for the near-resonant nonlinear material response in theadiabatic following limit are presented in chapter 10. The influence of higher order electron-electroninteraction is estimated in Markovian approximation resulting in a Boltzmann-equation (ch. 11). Thedynamics of a typical setup of a quantum dot laser (i.e. e coupled system of quantum dots and awetting-layer) is discussed in chapter 12. Finally the theory of photoluminescence is developed informalism of quantized light field as this effect is not included in the formalism of semi-classicalelectrodynamics. Photo-luminescence spectra for laser induced plasma and a semiconductor quantumwell are calculated in chapter 13.

51

Chapter 4

Density Matrix Formalism

In this chapter a short introduction to the underlying quantum mechanics and the density matrix for-malism used throughout this work is given. Different descriptions of quantum mechanics, as theSchrödinger picture and the Heisenberg picture (used in this work) are introduced and the differencesare pointed out. Furthermore the need of the concept of mixed states and the statistical operatoris motivated and the fundamental equations are quoted. The goal of this chapter is to give a shortintroduction/repetition of text-book knowledge about quantum mechanics found e.g. in [15].

4.1 Schrödinger-Picture and Heisenberg-Picture

The state ψ of a quantum mechanical system is described by a state vector |ψ〉 which lies in a Hilbert-space H . A physically observable quantity is represented by a Hermitian operator A : H → H witheigenvectors |ai〉 and eigenvalues αi. The result of a measurement of the observable A has to be aneigenvalue αi. If the state of the measured system is an eigenstate |ψ〉= |ai〉 the probability to measureαi is one, otherwise the probability pi is the square of the projection on the corresponding eigenstatepi = |〈ψ|ai〉|2. Accordingly the expectation value is defined as 〈A〉 = ∑i piαi = 〈ψ|A|ψ〉.

The expectation value is the main quantity of interest in this work as for example the expectationvalue of the operator of the microscopic coherence ρcv = a†

cav (which will be introduced in chapter5) enters Maxwell’s equations via the macroscopic polarization as a source term and therefore is anexperimentally accessible quantity. The goal of the theoretical framework discussed in this chapteris to derive equations of motion for expectation values. As can be seen from the definition of theexpectation value the total time derivative is given by

ddt〈A〉 =

ddt〈ψ|A|ψ〉 = (

ddt〈ψ|)A|ψ〉+ 〈ψ|( d

dtA)|ψ〉+ 〈ψ|A(

ddt|ψ〉) . (4.1)

In the usually considered Schrödinger-picture (denoted by the index S) of quantum mechanics the timederivative of the operator vanishes (besides an explicit time dependence) d

dt As = 0 and the dynamicsof the state vector is governed by Schrödinger’s equation

ddt|ψ〉S =

1i~

HS|ψ〉S . (4.2)

However, it is technically more convenient to introduce a time evolution propagator U(t, t0) = ei~

H(t)

which transfers the time dependence from the state vectors onto the operators. As U(t, t0) is an unitary

52 4. DENSITY MATRIX FORMALISM

operator, the expectation values do not change. This new picture is called Heisenberg-picture (denotedby the index H) and the equation of motion now reads

ddt〈A〉 = 〈ψH |(

ddt

AH)|ψH〉 . (4.3)

The equation of motion for the operator AH can be calculated from the Schrödinger-picture via thetime evolution propagator and reads

ddt

AH =1i~

[AH ,HH ] (4.4)

for vanishing explicit time dependence of AS. Reinserting this Heisenberg-equation of motion into eq.4.3 yields the universally valid (i.e. picture independent) Ehrenfest-theorem

ddt〈A〉 =

i~〈[A,H]〉 . (4.5)

A non-vanishing explicit time dependence of the operator A leads to an additional term yielding thecanonical form of the Ehrenfest-theorem:

ddt〈A〉 =

i~〈[A,H]〉+ 〈 ∂

∂tA〉 . (4.6)

4.2 Pure States and Mixed States

For the above definition of the expectation value to be valid one has to consider exactly one quantummechanical system and one has to know the exact state vector of the system. In the systems consideredin this work the exact state of the system is not necessarily known and in almost all cases a statisticalensemble of systems (many quantum dots or many excitons for example) is analyzed .

If the knowledge of the exact state vector of a system is uncertain or if an ensemble of systemscoupled to a bath of temperature T is considered the expectation value has to incorporate an addi-tional statistical averaging. This statistical average differs considerably from the quantum mechanicalaverage: The quantum mechanical uncertainty of the actual value measured is inherent and cannot bereduced. The statistical uncertainty however is only due to our insufficient knowledge of the systemand can theoretically be removed by exactly determining or preparating the state vector of the sys-tem. If the quantum mechanical state vector of a system is known it is denoted as to be in a “purestate” whereas if knowledge of the state vector is uncertain (or a statistical ensemble is considered)the system is said to be in a “mixed state”. The expectation value of an operator A is defined as

〈A〉 = ∑i

Pi〈ψi|A|ψi〉

if |ψn〉is a basis of the Hilbert-space and Pi the statistical probability of the mixed state to be inthe pure state |ψi〉. This can also be written as a trace over the statistical operator (or density matrixoperator) ρ

〈A〉 = ∑i

Pi〈ψi|A|ψi〉 = ∑i〈ψi|ρA|ψi〉 = tr(ρA) ,

which is defined as

ρ = ∑n

Pn|ψn〉〈ψn| . (4.7)

4.2. PURE STATES AND MIXED STATES 53

The equation of motion for the statistical operator in the Schrödinger picture can easily be calculatedfrom Schrödinger’s equation and reads

ddt

ρ =1i~

[ρ,H] . (4.8)

Equation (4.8) is called the von-Neumann-equation or Liouville-equation. The equation of motion forthe expectation value can now be written as

ddt〈A〉 = tr(

ddt

ρA)+ tr(ρddt

A)

= tr(1i~

[ρ,H]A)+ 〈 ∂∂t

A〉

=1i~

tr(ρ[H,A])+ 〈 ∂∂t

A〉

=i~〈[A,H]〉+ 〈 ∂

∂tA〉 , (4.9)

which is the well known, picture independent Ehrenfest-theorem (4.6) and identical to the result ofthe pure state calculation.

54 4. DENSITY MATRIX FORMALISM

55

Chapter 5

Hamiltonian

In the first part of this chapter the Hamiltonian-operator of a semiconductor including the electrons(and holes), classical light field, quantized light field (photons), lattice vibrations (phonons) andtheir various interactions and scatterings is derived in second quantization. In the second part theHamiltonian-operator for the somewhat similar system of a Laser induced electron plasma is given.

5.1 Confinement

Throughout this work systems are considered where the movement of electrons is restricted in one ormore dimensions (Fig. 5.1). For the following calculations, to be as general as possible, the dimensionof the confinement is left undefined. Spatial coordinates are decomposed in a contribution in theconfined direction and a distribution in the unconfined (free) direction

r = rc + r f . (5.1)

The electrons are assumed to be strongly localized in the confined direction preventing free motionin this direction. In the confined dimension Schrödinger’s equation can be solved resulting in discreteenergy eigenvalues Ei (which give rise to the subband splitting) and the corresponding eigenfunctionsφi(rc) in the confined dimension (Fig. 5.2). Consequently the wavenumber vector k of confinedelectrons has no contribution in the confined dimension. Wavenumber vectors of particles interactingwith the confined electrons are again decomposed in confined and free contributions

q = qc +q f . (5.2)

5.2 Classical Hamilton-Function

From the standard Lagrange-function (e.g.[23])

L =12 ∑

imir2

i (t)

︸︷︷︸free particles

+ε0

2

Z

d3r(E2(r, t)− c2B2(r, t))︸︷︷︸

electromagnetic field

(5.3)

+∑i

qi(ri(t) ·A(ri(t), t)−φ(ri(t), t))

︸︷︷︸particle-field interaction

(5.4)

56 5. HAMILTONIAN

bulk

quantum wire quantum dot

quantum well

Figure 5.1: Schematics of low-dimensional quantum structures. The free motion of the electrons ismore and more confined: Free motion in bulk material, two dimensional motion in quantum wells,one dimensional motion in a quantum wire and no free motion at all in a quantum dot. The possibledirections of free motion (q f ) are shown.

rc

V

?2(rc)

?1(rc)

?0(rc)

Figure 5.2: Discrete eigenfunctions in a confinement potential. The energetic differences between theeigenfunctions in the confined dimension gives rise to the subband splitting.

5.3. HAMILTON-OPERATOR OF A SEMICONDUCTOR 57

which describes multiple particles i at the positions ri with masses mi and charges qi in an electromag-netic field (E(r, t) and B(r, t)) with the vector potential A(r, t) and the scalar potential φ(r, t). TheLagrangian (5.3) gives rise to the minimal-coupling Hamiltonian in Coulomb gauge (A = AC, φ = 0)[24]:

H = ∑i

12mi

(pi −qiA(ri))2 +

12

14πε0

∑i, j

qiq j

|ri − r j|+

ε0

2

Z

d3r(E2T (r)+ c2B2(r) . (5.5)

The first term includes kinetic energy of the particles and particle-field interaction, the second termdescribes interaction of the particles and the last term the field energy (with the transversal electricfield ET ⊥ k).

5.3 Hamilton-Operator of a Semiconductor

This section follows the Born-Oppenheimer-Approach (e.g. [25]) in separating the multi particlesystem into the outer shell electrons and the background ion cores (i.e. the ions consisting of theatomic core and the bound electrons without the outer shell). As the mass of the atom is more thenfive orders of magnitude greater than the mass of the electron we assume the ions cannot follow themovement of the electrons and therefore are well localized around there equilibrium lattice positions.If the position of the ion is Taylor expanded around its equilibrium position it is stationary in firstorder and the electronic and ionic system decouple. The electrons travel in the constant periodicbackground potential of the crystal lattice. The second order of the expansion which corresponds toharmonic oscillations of the ion around its equilibrium position gives rise to the quasi-particle of thelattice vibration, the phonon.

5.3.1 Bloch-Electrons

According to Bloch’s theorem [26] the periodicity of the underlying potential is imposed on the wavefunction. This wave function can be decomposed into planar waves resulting in a description wherecollective electronic states called Bloch-electron can be described like free particles.

For the results to be general and independent from possible confinement of the electrons in somedimensions, the spatial coordinates are split into a free and a confined part

r = (r f ,rc) . (5.6)

All indices are compound indices consisting of band- , subband- and spin-index

i = (λi,σi,Si) . (5.7)

As mentioned above, in first order the ionic background is considered as a static periodic potential.According to Bloch’s Theorem [26] the electronic wave function ψ(r) can be decomposed into planewaves:

ψ(r) =1√Vf

∑i,k

φi(rc)eik·r f ui,k(r)ai,k , (5.8)

with φi(rc) being the wave function in direction of a possible confinement (cp. 5.1) and V f the volumeof a elementary cell in the free dimensions. ui,k(r) is the periodic Bloch-function with unrestrictedk in the free dimensions (it is later restricted to k ≈ 0 in the parabolic band approximation). In the

58 5. HAMILTONIAN

framework of second (or canonical) quantization (e.g. [6, 27]) the Fourier-coefficients are interpretedas field operators on a Fock-space and ai,k is called the annihilation operator of a Bloch electronin λi with the (reduced) wave vector k (i.e. ai,k applied onto the Fock-state of the Bloch-electron|i,k〉 produces the vacuum state ai,k|i,k〉 = |0〉 ). Bloch-electrons being fermionic yields an anti-commutator relation for the annihilation and creation operators (e.g. [15, 28]):

[ai,k,a†i′,k′ ]+ = ai,ka†

i′,k′ +a†i′,k′ai,k = δi,i′δkk′ . (5.9)

The expansion of the electronic wave functions results in the Hamilton-operator in second quantizationfor free Bloch-electrons:

HEl,0 = ∑i,k

εi,ka†i,kai,k . (5.10)

Due to the decoupling of the ionic and electronic system it is diagonal and simply a sum over thenumber operators ni,k = a†

i,kai,k and the corresponding single particle energies εi,k (which includethe confinement energy through the compound index i). If direct semiconductors are considered theenergy can be expanded into a Taylor-series around the Γ-point (i.e. around k = 0) yielding in secondorder

εi,k = εi,0 +~

2k2

2m?i

, (5.11)

which is called parabolic band approximation. For the parabolic dispersion to resemble the free parti-cle kinetic energy the effective mass

m?i = ~

2(d2εdk2

∣∣k=0)

−1 (5.12)

is introduced which can be both positive and negative. The effective mass can be derived from theband structure and enters the calculation as a material constant.

5.3.2 Classical Field

For the interaction of the electronic system with the classic electromagnetic field, the Hamiltonianin the form (5.5) (the A ·p representation) is inconvenient. By neglecting terms ∝ A2and neglectingdeviations of A on the scale of a elementary unit cell (i.e. performing a dipole approximation) the firstterm of (5.5) can be transformed [24, 29] into the semi-classical r ·E form with the transversal electricfield ET:

∑i(pi −qiAi(ri))

2 →Z

Vd3rP(r) ·ET(r) (5.13)

with the macroscopic polarization P(r) = erψ?(r)ψ(r). This polarization corresponds to the dynamicpolarization P3D

dyn of chapter 1. The electron wave function are decomposed into plane waves as above

ψ(r) =1√Vf

∑i,k

φi(rc)eik·r f ui,k(r)ai,k (5.14)

and transforming the polarization into

P(r) = ∑i, j,k,l

er1

Vfe−i(k−l)·r f u?

i,k(r)u j,l(r)a†i,ka j,lφ?

i (rc)φ j(rc) . (5.15)


The vector r is separated into a lattice vector R and an intra-cell vector r (as well as into contributionin free and confined dimension, cp. section 5.1):

r = r+ R = r f + rc + R f + Rc . (5.16)

Because of the periodicity of the Bloch function u one gets

P(r) = ∑i, j,k,l

e(r f + rc + R f + Rc)1

Vfe−i(k−l)·(r f +R f )u?

i,k(r f + rc)u j,l(r f + rc) (5.17)

×a†i,ka j,lφ?

i (rc + Rc)φ j(rc + Rc) .

As k and l are of the same order of magnitude the exponential function do only slightly change overan elementary cell. Further the confinement function φ is also assumed to change only slightly on thescale of an elementary cell

P(r) = ∑i, j,k,l

e(r f + rc + R f + Rc)1

Vfe−i(k−l)·R f u?

i,k(r f + rc)u j,l(r f + rc) (5.18)

×a†i,ka j,lφ?

i (Rc)φ j(Rc) .

Splitting (5.13) into an integral over the lattice vector R and an intra-cell vector r (EC denoting theintegration volume of an elementary cell) yields

Z

Vd3r P(r) ·ET(r) =

Z

Vd3R

1Ω0

Z

ECd3rP(r) ·ET(r) . (5.19)

ET is assumed to be nearly constant over an elementary cell:Z

Vd3r P(r) ·ET(r) =

Z

Vd3R

1Ω0

ET (R) ·Z

ECd3rP(r) (5.20)

=

Z

Vd3RET (R) · 〈P(r)〉 (5.21)

with the expectation-value of the polarization averaged over an elementary cell

〈P(r)〉 =1

Ω0

Z

ECd3rP(r) . (5.22)

(5.18) can now be inserted into (5.22) and with the orthogonality of the Bloch-functions u results in :

〈P(r)〉 =e

Vf Ω0∑

i, j,k,le−i(k−l)·R f a†

i,ka j,lφ?i (Rc)φ j(Rc) (5.23)

×(R f Ω0δ12 + RcΩ0δ12

+Z

ECd3ru?

i,k(r f + rc)r f u j,l(r f + rc)

+Z

ECd3ru?

i,k(r f + rc)rcu j,l(r f + rc)) .

Electrons in a direct semiconductor near the band edge, i.e. k ≈ 0 are considered and thereforeui,k ≈ ui,0:

〈P(r)〉 =e

Vf Ω0∑

i, j,k,le−i(k−l)·R f a†

i,ka j,lφ?i (Rc)φ j(Rc)((R f + Rc)Ω0δ12 (5.24)

+

Z

ECd3ru?

i,0(r f + rc)r f u j,0(r f + rc)

+Z

ECd3ru?

i,0(r f + rc)rcu j,0(r f + rc)) .

60 5. HAMILTONIAN

The two integrals are the dipole-matrix elements for the confined and free dimensions:

d f /ci, j = − e

Ω0

Z

ECd3ru?

i,0(r)r f /cu j,0(r) . (5.25)

Assuming spatial homogeneity of the polarization in the free direction (i.e. e−i(k−l)·R f = 1) the polar-ization can be written as

〈P(r)〉 =1

Vf∑

i, j,k,lφ?

i (Rc)φ j(Rc) (5.26)

×(−d fi, j −dc

i, j︸︷︷︸interband

+e(R f + Rc)δ12︸︷︷︸intraband

)a†i,ka j,l .

Finally the interaction Hamilton operator for Bloch electrons and a classical electromagnetic field is

HEl,EM = ∑i, j,k,l

φ?i (Rc)φ j(Rc)

Vf(−d f

i, j −dci, j︸︷︷︸

interband

+e(R f + Rc)δ12︸︷︷︸intraband

) ·ET (r)a†i,ka j,l . (5.27)

5.3.2.1 Quantum Well

For a given problem (5.27) can be simplified considerably. As an example a two dimensional quantumwell in the x,y-plane is considered: The width of the well is assumed to be zero ( φ?

i (Rc)φ j(Rc)Vf

≈ δ(z)),i.e. perfect 2d confinement. In this perfect confinement no polarization perpendicular to the well canbe exited. Only optical transitions are considered, i.e. no intraband (=transport) and no diagonaltransitions occur. The resulting Hamilton-operator under this assumptions is

HEl,EM = ∑i6= j,k

−d fi, j ·ET (r)a†

i,ka j,k (5.28)

〈P(r)〉 = δ(z) ∑i6= j,k

−d fi, ja

†i,ka j,k . (5.29)

5.3.3 Coulomb-Interaction

In this section the interaction Hamilton-operator between electrons due to Coulomb-interaction isderived. The interaction part of the classical Hamilton-function (5.5) is quantized with the electronictotal multi-particle wave function ψ and reads

HEL,El =12

14πε0

Z

d3rZ

d3r′ψ†(r)ψ†(r′)e2

|r− r′|ψ(r)ψ(r′) . (5.30)

The wave function is decomposed in planar waves (i.e. single particle wave functions in the frameworkof canonical quantization)

ψ(r) =1√Vf

∑i

φi(rc)eiki·r f ui,ki(r)ai,ki , (5.31)


where the ionic background is incorporated via the lattice periodic Bloch-functions u. The Hamilto-nian becomes

HEl,El =12 ∑

1,2,3,4

e2

4πε0

Z

d3rZ

d3r′e−i(k1−k3)·r f e−i(k2−k4)·r′ f (5.32)

× 1V 2

f

1|r− r′|u

?λ1

(r)u?λ2

(r′)uλ3(r)uλ4(r′)

×φ?σ1

(rc)φ?σ2

(r′c)φσ3(rc)φσ4(r′c)a

†1a†

2a3a4

=12 ∑

1234V 12

34 a†1a†

2a3a4 (5.33)

with the Coulomb-matrix element V 1234 . Note that as above compound indices i = (λi,σi,Si) are used.

The Coulomb matrix element can be further simplified by introducing relative ( r = r f −r′f ) and centre

of mass (r =r f +r′f

2 ) coordinates in the free dimensions

V 1234 =

e2

4πε0

Z

d3rZ

d3rZ

d3rc

Z

d3r′ce−i(k1−k3)·(r+ 12 r)e−i(k2−k4)·(r− 1

2 r) (5.34)

× 1V 2

f

1|r+ rc − r′c|

u?λ1

(r+12

r+ rc)u?λ2

(r− 12

r+ r′c)uλ3(r+12

r+ rc)uλ4(r−12

r+ r′c)

×φ?σ1

(rc)φ?σ2

(r′c)φσ3(rc)φσ4(r′c)

=e2

4πε0

Z

d3rZ

d3rZ

d3rc

Z

d3r′ce−i(k1−k3)·(r+ 12 r)e−i(k2−k4)·(r− 1

2 r) (5.35)

× 1V 2

f

1|r+ rc − r′c|

u?λ1

(r+12

r+ rc)u?λ2

(r− 12

r+ r′c)uλ3(r+12

r+ rc)uλ4(r−12

r+ r′c)

×φ?σ1

(rc)φ?σ2

(r′c)φσ3(rc)φσ4(r′c)

=e2

4πε0δλ1λ3 δλ2λ4 δk1+k2,k3+k4

1Vf

Z

d3rc

Z

d3r′cφ?σ1

(rc)φ?σ2

(r′c)φσ3(rc)φσ4(r′c) (5.36)

Z

d3re−i(k1−k3)·r

|r+ rc − r′c|.

The integralR

d3r e−i(k1−k3)·r|r+rc−r′c| has to be evaluated for a given confinement. This non-trivial calcula-

tion leads to the following results (were the momentum conservation given by δk1+k2,k3+k4 has beenincorporated)

HEl,El =12 ∑

1234kk′q f

V 1234 (q f )a

†1,k+q f

a†2,k′−q f

a3,k′a4,k (5.37)

V 1234 (q f ) = δλ1λ3 δλ2λ4 ∑

qc

V (q)F14(qc)F?23(qc)

V (q) =4πe2

Vf ε0εr(q, t)q2

Fi j(qc) =Z

drcφ?i (rc)eiqc·rc φ j(rc)

with the so called form factor Fi j(qc) describing the confinement. The indices 1,2,3,4, i, j are stillcompound indices incorporating different subbands. V (q) is the three dimensional Coulomb-matrixelement. Screening can be incorporated into the dielectric function εr(q, t).

62 5. HAMILTONIAN

5.3.3.1 2D Electron Gas - Perfect Confinement

For a perfect quantum well, i.e. a two dimensional electron gas with perfect confinement (δ(x)), theHamiltonian gets

H2DEl,El =

12 ∑

λ1,λ2,k,k′,q6=0

4πe2

2Aε0εr(q, t)1|q|a

†λ1,k+qa†

λ2,k′−qaλ2,k′aλ1,k . (5.38)

5.3.3.2 3D Electron Gas

For a non-confined three dimensional electron gas the resulting Hamiltonian is

H3DEl,El =

12 ∑

λ1,λ2,k,k′,q6=0

4πe2

V ε0εr(q, t)1

|q|2 a†λ1,k+qa†

λ2,k′−qaλ2,k′aλ1,k . (5.39)

5.3.4 Phonons

Despite the fact that electron-phonon coupling is not the main subject of this work, the derivations ofthe Hamiltonian is included here as the resulting equations are used in chapter 9 and chapter 12. Asmentioned above, the ionic and electronic systems decouple because of the mass difference betweenelectron and ion. The potential energy of the whole ionic system is the sum over the two-particlepotentials:

VIon = ∑i6= j

V (Ri −R j) . (5.40)

In a crystal structure points of minimal potential exist were the ions are situated (equilibrium posi-tions). The potential can be Taylor expanded around these minima. The place R jof the j-th ion is splitinto its equilibrium position R0

j and its elongation s j

R j = R0j + s j (5.41)

and the potential reads in second order

V ≈ V0 +12

m jω2js

2j . (5.42)

Now only nearest neighbour interactions (denoted as ∑i, j (nn) in the sums) are considered. This yieldsthe potential energy of the ionic system as

VIon ≈ ∑i, j (nn)

12

mω2(si − s j)2 . (5.43)

The corresponding Hamilton-operator of the ionic system is

H = ∑i

p2i

2m+

12

mω2 ∑i, j (nn)

(si − s j)2 , (5.44)

which is the well known problem of N coupled harmonic oscillators (e.g. [27]). It is solved byintroducing ladder operators which are interpreted as creation (b†) and annihilation (b) operators of the


quasi-particle “phonon”. They obey the bosonic commutation relation [b†i ,b j]− = b†

i b j − b jb†i = δi j.

The Hamilton-operator reads

HPh,0 = ∑µ,q

~ωµ,q(b†µ,qbµ,q +

12) . (5.45)

q is the wave-vector of the phonon and µ denotes the dispersion branch. If p ions are in a elementarycell 3 acoustic branches and 3p−3 optical branches exist. The branches can be further classified intolongitudinal and transversal modes. The longitudinal optical (LO) phonons normally have a weakwavenumber dependence at the vicinity of the Γ point ωLO(q) ≈ ω0

LO. The wavenumber dependenceof the longitudinal acoustic (LA) phonons is linear near the Γ point ωLA(q) ≈ vLA|q| with vLA beingthe sound propagation velocity.

5.3.5 Phonon-Electron Interaction

Considering the lowest order of interaction between phonons and electrons beyond the Born-Oppenheimerapproximation the respective Hamilton-operator can be derived via a phonon-mode expansion of theion movement as above. The Hamiltonian reads

HEl,Ph = ∑i, j,q,k,µ

gi jµ(q)a†i,k+q f

(bµ,q +b†µ,−q)a j,k (5.46)

with the electron-phonon matrix element

gi jµ(q) = g3Di jµ(q)Fi j(qc) . (5.47)

As in this work only polar crystals are considered the 3D electron-phonon matrix element g3Di jµ(q)

describing the coupling of electrons to optical phonons is the well known dipole-dipole or Fröhlich-coupling element (e.g. [27])

g3Di jµ(q) = gLO

i jµ (q) =

√e2~ωLO

2ε0V(

1εb

− 1εst

)1|q| . (5.48)

In this formula the dielectric functions enters in two limits: εst is the limit of εrfor low frequencies(static) and εbis the dielectric function at optical (i.e. high) frequencies. The form factor Fi j dependson the wave-function in the confined directions:

Fi j(qc) =

Z

drcφic(rc)

?φ jc(rc)eiqcrc . (5.49)

For the first subband in a quantum well with the width L it is [30]

F11(qz) =8π2sin(Lqz

2 )

Lqz(−4π2 +(Lqz)2). (5.50)

5.3.6 Quantized Light Field (Photons)

Starting point is the Hamilton density of the electromagnetic field in Coulomb gauge (see e.g. [31])

HEM =1

2ε0εr(Π2 + εrc2(∇×A)2) (5.51)

64 5. HAMILTONIAN

with the canonical momentum of the photon Π = dLdA = εεrA. The vector potential A is now decom-

posed in plane waves (in the quantization volume V ):

A(r) = ∑kσ

√~

2ε0ωkσVεkσ(eik·rbkσ + e−ik·rb†

kσ) (5.52)

= ∑kσ

√~

2ε0ωkσVεkσeik·r(ckσ + c†

−kσ) . (5.53)

εkσ = ε−kσ is the unity vector of the polarization direction σ and the wave number vector k. The cre-ation and annihilation operators c and c†of the quasi particle “photon” obey the bosonic commutationrules:

[ckσ,ck′σ′ ] = [c†kσ,c†

k′σ′ ] = 0 (5.54)

[ckσ,c†k′σ′ ] = δkk′δσσ′ . (5.55)

The free photonic part of the Hamilton operator takes the form

HPhot = ∑kσ

~ωkσ(c†kσckσ +

12) . (5.56)

5.3.7 Photon-Electron Interaction

Rewriting the interaction term in (5.5) yields

Hinteraction = ∑i

eAi(ri)2

2m+∑

i

qipi ·Ai(ri)

2m+∑

i

qiAi(ri) ·pi

2m.

Electron wave functions and vector potential can be expanded according to (5.8) and (5.52) transfer-ring the Hamilton operator in second quantization[32]:

HEl,Phot = ~∑ki j

Fkσi j (c†

−kσ + ckσ)a†i a j (5.57)

HA2 = ~∑i j

Ai ja†i a j .

HEl−phot describes the linear coupling and HA2 the nonlinear coupling. The linear coupling element is

Fkσi j =

√~

2ε0V ωk

em

εkσ ·kiδi j (5.58)

and in the nonlinear Hamiltonian Ai j containing photonic operators is

Ai j =e2

2m0∑

kk′σ

√~

2ε0V ωk

√~

2ε0Vωk′δk+k j,k′+ki(c

†−kσ + ckσ)(c†

−k′σ + ck′σ) . (5.59)

5.4. ELECTRON-PLASMA 65

5.3.8 Photon-Phonon Interaction

If, like in section 13.2.3, the interaction of phonons and photons have to be considered, it is convenientto diagonalize the resulting phonon-photon and electron-photon Hamiltonian and thereby introducingnew quasi-particles (phonon-polaritons) and operators mixing phononic and photonic operators. Thestarting point is the Hamilton-function (5.51) extended by the phononic-contributions [33]

H =1

2ε0εr(Π2 + εrc2(∇×A)2)+

ωplε0

2(Πpn −A)2 +

ω2TO

2ω2plε0

P2 (5.60)

with the TO-phonon-frequency ωTO and the plasma frequency ωpl =√

Ne2

Mε0where N is the effective

ion-density and M the ion-mass. Πpn = dLdA = 1

ω2plε0

P+A is the canonical momentum of the phonon.

Decomposing the electromagnetic field in planar waves as above yields the interaction-Hamiltonianas

HPh,Phot = ∑kσ

~

2

√ωTO

ε20ωkω2

pl(ckσ + c†

−kσ)(b−kσ +b†kσ) (5.61)

with the dispersion relation ωk = ck√εr

for the electromagnetic wave inside the material. The phonon-photon-Hamiltonian together with the linear electron-photon-Hamiltonian (Hel−phot 5.57) is diagonal-ized according to [33] resulting in

HAP = ~ ∑abkνµσρ

Fνkσab Sρ,kσ(αρ,kσ +α†

ρ,−kσ)a†νaaµb (5.62)

with the phonon-polariton-operators α† and α (containing both phononic and photonic operators ) andthe phonon-photon-interaction matrix-element

Sρ,kσ =ω2

TO +ω2ρ,kσ√

ω2TOω2

pl +(ω2ρ,kσ −ω2

TOω2pl)

2

√ωk

ωρ,kσ. (5.63)

5.4 Electron-Plasma

A laser induced electron plasma (as investigated in chapter 13.2.2 to describe the THz luminescence)consists of ionized atoms, i.e. the electrons and ions, and photons. As the mass of the ions is muchgreater than the mass of the electrons the ions are treated semi-classical while the electrons and pho-tons are treated in the framework of second quantization.

5.4.1 Free Electrons

The Hamiltonian for the free electrons is derived analogous to section 5.3.1 but with a strictly kinetic(i.e. parabolic) energy dispersion

HEl,0 = ∑k

εka†kak . (5.64)

66 5. HAMILTONIAN

5.4.2 Free Photons, Photon-Electron and Electron-Electron

The free photon contribution, the photon-electron interaction and the electron-electron interactionyield identical contributions to the Hamilton operator as in sections 5.3.6, 5.3.7 and 5.3.3 :

HPhot = ∑kσ

~ωkσ(c†kσckσ +

12) (5.65)

HEl,Phot = ~∑ki j

Fkσi j (c†

−kσ + ckσ)a†i a j (5.66)

HA2 = ~∑i j

Ai ja†i a j (5.67)

HEl,El =12 ∑

1234kk′q f

V 1234 (q f )a

†k+q f

a†k′−q f

ak′ak . (5.68)

5.4.3 Electron-Ion

Only the interaction between free electrons and ions is considered. The interaction between ionsas well as interaction between bound electrons and (different) ions are neglected. The electron-ionHamiltonian reads

HEl,Ion = −∑i j

Z e2

4πε0

1|Ri − r j|

(5.69)

with Ri being the positions of the ions and r j the positions of the free electrons. Decomposing theelectron wave-functions into plane waves yields the Hamiltonian in 2nd quantization:

HEl,Ion = ~∑i j

Ui− ja†i a j . (5.70)

67

Chapter 6

Hartree-Fock-Approximation

In this chapter it is shown how a hierarchy problem emerges in the attempt to calculate the equationsof motion for expectation values of observables via the Ehrenfest theorem (eq. 4.9) for many particleinteractions. Then the Hartree-Fock (or Mean-Field)-approximation is introduced to obtain a closedset of equations for the expectation value.

6.1 Example Hierarchy-Problem

To understand what the hierarchy problem is and how it arises, the Coulomb-interaction-Hamiltonian(eq. 5.37)

HEl,El =12 ∑

λ1,λ2k1,k2q6=0

Vqa†λ1,k1+qa†

λ2,k2−qaλ2,k2 aλ1,k1 (6.1)

is considered. The goal is to calculate the equations of motion for the expectation value of electroniccoherences

pα,βk = a†

α,kaβ,k , (6.2)

because this expectation values enters Maxwell’s equation as a source term via the dynamic polar-ization (cp. sections 5.3.2 and 1.1). The Ehrenfest-theorem (4.6) (or the Heisenberg-equation for theoperator itself, which is equivalent)

−i~∂t〈A〉 = 〈[H,A]−〉 (6.3)

68 6. HARTREE-FOCK-APPROXIMATION

yields for the electronic coherences:

−i~∂t〈a†α,kaβ,k〉 = 〈[1

2 ∑λ1,λ2k1,k2q6=0

Vqa†λ1,k1+qa†

λ2,k2−qaλ2,k2 aλ1,k1 ,a†α,kaβ,k]〉 (6.4)

= +12 ∑

λ2,k2q6=0

Vq〈a†α,k+qa†

λ2,k2−qaλ2,k2 aβ,k〉 (6.5)

+12 ∑

λ1,k1q6=0

Vq〈a†λ1,k1+qa†

α,k−qaβ,kaλ1,k1〉

−12 ∑

λ2,k2q6=0

Vq〈a†α,ka†

λ2,k2−qaλ2,k2 aβ,k−q〉

−12 ∑

λ1,k1q6=0

Vq〈a†λ1,k1+qa†

α,kaβ,k+qaλ1,k1〉 .

The commutator relations from appendix A have been utilized and the expectation values have beentransformed into normal ordering (i.e. 〈a†a†aa〉). As can bee seen the obtain equations of motion forthe quantities 〈a†

α,kaβ,k〉 are not a closed set of differential equations: They couple to four-operator

expectation values 〈a†aa†

bacad〉, i.e. higher order correlations. The equations of motion for the four-operator expectation values (which are explicitly given for this example in chapter 11 as eq. 11.3)couple to six-operator expectation values and so forth. Aside from some special cases it is generallyimpossible to derive a closed set of equations by following the hierarchy. This hierarchy problem isan inherent problem of the density matrix formalism for many particle systems and not limited to thediscussed example.

6.2 Factorizing and Mean-Field-Approximation

Apart from special cases (e.g. electron-phonon coupling problems in the independent boson model[34]) were the hierarchy problem can be solved analytically, some sort of truncation scheme has tobe used to obtain a closed set of equations. For some systems it arises naturally like in a quantumdot which can only be occupied by two electrons. In this case all expectation-values concerning morethan two electrons have to be zero.

In the general case however the hierarchy has to be truncated at one point or another. First thecorrelation expansion is introduced by which four-point expectation values are split into products oftwo point expectation values [6]:

〈a†aa†

bacad〉 = 〈a†aad〉〈a†

bac〉−〈a†aac〉〈a†

bad〉+ 〈a†aa†

bacad〉corr. . (6.6)

This is always possible by putting all possibly made errors into the correlation term 〈a†aa†

bacad〉corr..The correlation term is assumed to be small in comparison to the two operator expectation values. Byneglecting the correlation term the many-body problem is reduced to a pseudo one particle problem inthe mean field of the other particles (hence mean-field approximation). The Hartree-Fock (or mean-field)-approximation is a basic concept in semiconductor physics (e.g. [6, 27, 35, 36, 37, 38]).

The obtained solution for the above discussed example of Coulomb-interaction is given in ap-pendix C.

6.3. GENERALIZED FACTORIZATION 69

6.3 Generalized Factorization

The Hartree-Fock-factorization scheme (eq. 6.6) can be generalized for six-operator quantities:

〈a†aa†

ba†cadaea f 〉 = 〈a†

aad〉(〈a†ba f 〉〈a†

cae〉−〈a†bae〉〈a†

ca f 〉) (6.7)

− 〈a†aae〉(〈a†

ba f 〉〈a†cad〉−〈a†

bad〉〈a†ca f 〉)

+ 〈a†aa f 〉(〈a†

bae〉〈a†cad〉−〈a†

bad〉〈a†cae〉)

+ 〈a†aa†

ba†cadaea f 〉corr. .

The factorization of bosonic (denoted by operators b† and b) expectation values is done in an anal-ogous manner. Here even two-operator quantities can be factorized due to the bosonic nature of theparticles (i.e. no conservation of the particle number):

〈b†1b2〉 = 〈b†

1〉〈b2〉+ 〈b†1b2〉corr. .

In calculations involving interaction of bosonic fields like phonons or photons with the electronicsystem, mixed expectation values have to be factorized in the following manner:

〈b1a†2a3〉 = 〈b1〉〈a†

2a3〉+ 〈b1a†2a3〉corr. (6.8)

〈b†1b2a†

3a4〉 = 〈b†1〉〈b2a†

3a4〉corr. + 〈b2〉〈b†1a†

3a4〉corr.

+〈b†1〉〈b2〉〈a†

3a4〉+ 〈b†1b2〉corr.〈a†

3a4〉+ 〈b†1b2a†

3a4〉corr. .

70 6. HARTREE-FOCK-APPROXIMATION

71

Chapter 7

Equations of Motion

In this chapter the coherent dynamics of the electron system in a semiconductor is considered. Theequations of motion for the relevant observables coherence (or microscopic polarization) and electronoccupation (or density) are presented in the first part and the rotating wave approximation (RWA) isapplied in the second part.

7.1 Bloch Equations

7.1.1 Optical Bloch Equations

Many systems like a quantum dot, a simplified atom or even a semiconductor in first order Wannier-expansion (chapter 8) are described by a system with only a few (here two for simplicity) discreteenergy levels. The Hamilton-operator for such a two level system (TLS) is given by the sum over thefree energies (like the free Bloch-electron Hamilton-operator (5.10) but without k-dispersion) and thecorresponding electron-light interaction Hamilton-operator (analogous to eq.(5.28)):

HTLS = ∑i

εia†i ai, + ∑

i6= j−d f

i, j ·ET a†i a j . (7.1)

The (macroscopic) polarization is

〈P(r)〉 = δ(z)∑i6= j

−d fi, ja

†i a j . (7.2)

The equations of motion for the coherence pi j = 〈a†i a j〉 and occupation fi = 〈a†

i ai〉 are derived usingthe Heisenberg-equation (4.4). For two energy levels in the electron hole-image they read

p12 = −iω12 p12 + iΩ(1− f11 − f22) (7.3)

f11/22 = 2Im(Ω∗p12) (7.4)

with the Rabi-frequency Ω := d21ET~

and ω12 = ε1−ε2~

. Without further scattering processes the densityof electrons in level 2 and the density of holes in level 1 are equal. With p12 = p and f11 = f22 = fone gets

p = −iωp+ iΩ(1−2 f )

f = 2 Im(Ω∗p) . (7.5)

72 7. EQUATIONS OF MOTION

u

v

w

s

Figure 7.1: Bloch-sphere. The Bloch-vector s is precessing around the w (inversion) axis.

7.1.1.1 Bloch-Sphere

The Bloch equations are often discussed in the somewhat more graphical and intuitive image of thedynamics of a vector on the so called Bloch-sphere. The new quantities inversion w = 1−2 f , real partu and imaginary part −v of the coherence p = u− iv and renormalized field κE =− 1

2 Ω are introducedand the Bloch equations are written as

u = −ωv (7.6)

v = ωu+κEw (7.7)

w = −κEv . (7.8)

This set of equations can be interpreted as a torque equation

s = ΩF × s (7.9)

with the torque vector ΩF = (−κE,0,ω) and the Bloch-vector s = (u,v,w). As can be seen the normof s is constant. The possible orbit of s is called the Bloch-sphere (Fig. 7.1.1.1). s is rotating withthe angular velocity ω around the w axis (this free rotation can be eliminated by the rotating waveapproximation). Analogous to chapter 9 an inversion of a TLS with a 1π pules corresponds to a180 rotation of the Bloch-vector on the Bloch-sphere. The full inversion or Rabi-Flop induced bya 2π pulse rotates the Bloch-vector completely around the Bloch-sphere leaving it at its beginningposition.

7.1. BLOCH EQUATIONS 73

7.1.1.2 Conservation Laws

As f and p are elements of the density matrix ρ (cp. chapter 4)

ρ =

(f1 p12

p21 f2

)

certain conservation laws arise. ρ has to be a positive semidefinite Hermitian operator with tr(ρ) = 1which implies

f1 + f2 = 1 (7.10)

p12 = p?21 . (7.11)

From the equations of motion (7.5) another conservation law can be derived: :

|p|2 = f − f 2 . (7.12)

7.1.2 Semiconductor Bloch Equations

For modelling a semiconductor one has to take into account the k-dispersion (i.e. the band structure)and electron-electron interaction (Coulomb-interaction). The corresponding Hamiltonian is composedof the contributions (5.10),(5.28) and (5.37):

HSC = ∑i,k

εi,ka†i,kai,k

︸︷︷︸free Bloch

+ ∑i6= j,k

−d fi, j ·ET a†

i,ka j,k

︸︷︷︸electron-field interaction

(7.13)

+12 ∑

1234kk′q f

V 1234 (q f )a

†1,k+q f

a†2,k′−q f

a3,k′a4,k

︸︷︷︸Coulomb-interaction

with the macroscopic polarization

〈P(r)〉 = δ(z) ∑i6= j,k

−d fi, ja

†i,ka j,k . (7.14)

Two bands are considered: the highest occupied (valence) band v and the lowest unoccupied (conduc-tion) band c. The Hamilton-operator gets

HTB = ∑k

εc,ka†c,kav,k + εv,ka†

v,kac,k

︸︷︷︸free Bloch

(7.15)

+∑k−d f

c,v ·ET (r)a†c,kav,k −d f

v,c ·ET a†v,kac,k

︸︷︷︸electron-field interaction

+12 ∑

k,k′,q f 6=0V (q f )(a

†c,k+q f

a†v,k′−q f

av,k′ac,k +a†cv,k+q f

a†c,k′−q f

ac,k′av,k

+a†c,k+q f

a†c,k′−q f

ac,k′ac,k +a†v,k+q f

a†v,k′−q f

av,k′av,k)

︸︷︷︸Coulomb interaction

.


The equations of motion for the coherence pcvk = 〈a†

c,kav,k〉 = pk and occupation fc/v = 〈a†c/vac/v〉 are

derived using the Heisenberg-equation (4.4). For two energy levels in the electron hole image (i.e.felectron = fc, fhole = 1− fv) and Hartree-Fock-approximation (cp. section 6.6) they read

pk = −iωk pk︸︷︷︸free Bloch

(7.16)

+iΩ(1− f ek − f h

k )︸︷︷︸electron-field interaction

−ipk ∑q6=0

Vq

~(1− f e

k+q − f hk+q)

︸︷︷︸exchange self-energy

+i(1− f ek − f h

k ) ∑q6=0

Vq

~pk+q

︸︷︷︸excitonic contribution

and

f e/hk = 2Im(Ω? pk + pk ∑

q6=0

Vq

~p?

k+q) (7.17)

with the Rabi-frequency Ω := dcvET~

.

• The first term describes the free dynamics of the Bloch-electrons.

• The electron-field interaction describes optical interband transitions. The factor (1− f ek − f l

k)enforces the Pauli-blocking.

• The excitonic contribution accounts for the formation of the hydrogen-like excitons. i.e. exci-tonic modes and resonances.

• The exchange self-energy term causes a reduction in energy of the excitonic resonance.

7.1.3 Parabolic Band Approximation

For direct semiconductors and small wavenumber vectors k the energy of the Bloch-states can beTaylor expanded. The second order reads

εc,k = εg +~

2k2

2mc(7.18)

εv,k = 0− ~2k2

2mv(7.19)

~ωk = εv − εc = εg +~

2k2

2(

1m1

+1

m2) = εg +

~2k2

2mr(7.20)

with the band-gap energy εg and the reduced mass mr of the effective electron-masses mv and mc ofthe valence and conduction band (cp. section 5.3.1).

7.2. ROTATING WAVE APPROXIMATION 75

7.1.4 Damping

The Bloch equations can incorporate additional dephasing processes like e.g. electron-phonon-scattering(as covered in chapter 12), electron-electron-scattering (of higher order, cp. chapter 11), surfaceroughness or impurity scattering. These can be treated either dynamically or be included as a phe-nomenological damping. For resonant systems of many quantum wells, however all these processesare of minor importance compared to the radiative damping (cp. section 14.2.2 and chapter 11). For aself consistent damping of the Bloch-vector s the density-like quantities fk have to be damped twiceas much as the coherence like quantities pk, i.e. the terms −γ0 pk and −2γ0 fkhave to be added [1]. Forthe quantum wells considered in chapter 14 the measured dephasing constant is γ0 ≈ 0.5 meV ([39],appendix G).

7.1.5 Maxwell Bloch Equations

The coupled system of Bloch-equations (either (7.5) or (7.16) and (7.17)) and Maxwell’s equations(either their full form (1.1) or the wave equation (1.8)) are normally referred to as the MaxwellBloch Equations. This system of equations has to be computed self consistently as the electromag-netic field influences the carrier-dynamics through the Rabi-frequency and the carrier-dynamics entersMaxwell’s equations as a source-term via the dynamic polarization.

7.2 Rotating Wave Approximation

Analogous to 1.4.1 the rotating wave approximation can also be applied to the Bloch-equations. Theoptical field is expressed as a product of a fast oscillating part and a slowly varying envelope function:

Ω(t) = cos(ωlt +θ)Ω(t) (7.21)

=12(e−iωlt−iθ + eiωlt+iθ)Ω(t)

=12(e−i(ωg+∆)t−iθ + ei(ωg+∆)t+iθ)Ω(t)

with

ωl frequency of the carrier wave (laser)θ phase of the carrier wave

ωg frequency corresponding the band gap∆ optical detuning ∆ = ωl −ωg

Ω(t) slowly varying envelope function.

The second exponential function leads to a non-resonant term in the equation of motion. Foroptical detunings ∆ ωg this term can be neglected. This neglecting of non-resonant term is calledthe Rotating-Wave-Approximation (RWA) [6]. It is

Ω(t) ≈ 12

e−i(ωg+∆)t−iθΩ(t) (7.22)

=12

e−iωgte−i∆t e−iθΩ(t) . (7.23)

Now the quantity Ω is introduced, which includes phase and detuning:

Ω(t) ≈ 12

e−iωgtΩ(t) . (7.24)


For the coherence pk a similar ansatz is made:

pk(t) = e−iωgt pk(t) (7.25)

with pk(t) as slowly varying envelope of the coherence. The semiconductor Bloch equations read now

˙pk = −i~k2

2mrpk + i

12

Ω(1− f ek − f l

k)

−ipk ∑q6=0

Vq

~(1− f e

k+q − f lk+q)

+i(1− f ek − f l

k) ∑q6=0

Vq

~pk+q − γ0 pk (7.26)

and

f e/lk = 2 Im( 1

2 Ω? pk + pk ∑q6=0

Vq~

p?k+q)−2γ0 f e/h

k . (7.27)

77

Chapter 8

Wannier-Expansion

To reduce the numerical expenses of numerical simulations or to be able to derive analytical pre-dictions (see e.g. 15.2.4.3), the semiconductor Bloch equations are often considered in a first orderWannier-expansion. This results in a approximation by a two-level resonator and reproduces the linearabsorption spectra of the excitonic resonance.

8.1 Wannier-Expansion of the Polarization

For linear excitation the electron- and hole-densities can be neglected as almost no carriers are excited.The equation of motion for the coherence (beyond rotating wave approximation (7.16))

(i~∂t −Eg−~

2k2

2mr)pk = −(Ed1,2 + ∑

q6=kV|k−q|pq) (8.1)

is Fourier-transformed into real-space

(~(ω+ iγ0)−Eg −~

2∇2

2mr+V(r))p(r,ω) = −(E(ω)d1,2δ(r)L2) . (8.2)

The corresponding homogeneous differential equation has the form(−~

2∇2

2mr+V (r)

)p(r,ω) = E p(r,ω) (8.3)

with the Coulomb-potential V (r) and is called Wannier-equation[6]. As this equation is identical tothe eigenvalue problem of the hydrogen atom the solutions are well known (e.g. [15]). The arisingdiscrete eigen-energies ε can be interpreted as bound states between an electron and a hole due toCoulomb-interaction analogous to a hydrogen atom.

To solve the inhomogeneous differential equation (8.2) the polarization is expanded into solutionsof the Wannier-equation (8.3), i.e. into hydrogen-like wave functions:

p(r,ω) = ∑ν

bνψν(r) . (8.4)

The equation of motion is now

(~(ω+ iγ0)−Eg −~

2∇2

2mr+V (r))∑

νbνψν(r) = −(E(ω)d1,2δ(r)L2) . (8.5)

78 8. WANNIER-EXPANSION

By projecting onto the state ψµ the equation can be solved for the expansion coefficients

−E(ω)d1,2ψ?µ(r = 0)L2

~(ω+ iγ0)−Eg −Eµ= ∑

νbν

Z

ψ?ν(r)ψν(r)d2r (8.6)

= ∑ν

bνδµν (8.7)

= bµ . (8.8)

The coherence is

p(r,ω) = ∑ν

−E(ω)d1,2ψ?ν(r = 0)L2

~(ω+ iγ0)−Eg −Eνψν(r) (8.9)

and after back transformation into k-space

p(k,ω) = −∑ν

E(ω)d1,2ψ?ν(r = 0)

~(ω+ iγ0)−Eg −Eν

Z

ψν(r)eikr d2r . (8.10)

The macroscopic polarization is (cp. section 5.3.2)

P(ω) = ∑k

p(k,ω)d1,2 + p?(k,−ω)d1,2 (8.11)

= −∑ν,k

E(ω)d1,2ψ?ν(r = 0)

~(ω+ iγ0)−Eg −Eν

Z

ψν(r)eikrd d2r (8.12)

−∑ν,k

E?(ω)d1,2ψν(r = 0)

~(ω− iγ0)−Eg −Eν

Z

ψ?ν(r)e

−ikrd d2r .

which can be transformed into

P(ω) =−2∑ν|d1,2|2|ψν(r = 0)|2E(ω)

× (1

~(ω+ iγ0)−Eg −Eν− 1

~(ω− iγ0)+Eg +Eν) .

(8.13)

8.2 Susceptibility

With (8.13) the susceptibility can be computed via Beer’s Law [40] as

χ(ω) =P(ω)

E(ω)(8.14)

= −2∑ν|d1,2|2|ψν(r = 0)|2 × (8.15)

(1

~(ω+ iγ0)−Eg −Eν− 1

~(ω− iγ0)+Eg +Eν) .

Neglecting the non-resonant second term yields

χ(ω) = ∑ν

−2|d1,2|2|ψν(r = 0)|2~(ω+ iγ0)−Eg −Eν

. (8.16)

8.3. NONLINEAR DYNAMICS 79

-8 -4 0 4

abso

rptio

n [a

rb. u

nits

]

detuning [arb. units]

1s

2s3s..

continuum

Figure 8.1: Analytical absorption-spectrum of a single quantum well [6].

The imaginary part of (8.16) is a measure of the absorption

Imχ(ω) = ∑ν−2|d1,2|2|ψν(r = 0)|2 ~γ0

(~ω−Eg −Eν)2 +~2γ20

. (8.17)

The spectrum is a superposition of Lorentz-peaks - one for every eigenstate of the exciton. TheWannier-equation - like the hydrogen problem - has not only discrete solutions but also a continuumof states energetically above the band-gap. The width γ0 of the resonance results from interactions ofthe carriers with themselves (electron-electron scattering) and other systems (like radiative dampingor phonon scattering). For linear absorption resonant to the excitonic resonance the system can beapproximated by only the first term in the expansion, i.e. a two level system. The analytic solution forthe ground state and the first few exited states of the exciton is depicted in Fig. 8.1. The damping waschosen to illustrate the Lorentz-character of the excitonic line. For positive detunings (i.e. energiesabove the band edge) the continuum is visible.

8.3 Nonlinear Dynamics

The existence of discrete energies εi and eigenstates ψi, arising from the semiconductor Bloch equa-tion as shown in section 8.1, eq. (8.3), give rise to the introduction of a new quasi-particle: theWannier-exciton. While being a collective effect of the electronic system in a semiconductor, theexciton can be interpreted as a hydrogen like bound state of a Bloch-electron in the conduction bandand a hole in the valence band. To describe the exciton in second quantization creation and annihila-tion operators xi and x†

i in the excitonic state ψi with the energy εi are introduced. The free excitonHamiltonian is

HEx,0 = ∑i

εix†i xi (8.18)

with the exciton number operator nEx = x†i xi. Analogous to the considerations concerning Bloch-

electrons in chapter 5, section 5.3.2 the interaction Hamiltonian between excitons and the classical

80 8. WANNIER-EXPANSION

light field can be derived as

HEx,EM = ∑i6= j

−di, j ·Ex†i,x j (8.19)

with the dipole moment

di, j =Z

d3rψirψ j . (8.20)

The excitonic problem in Wannier expansion is now fully defined as the wave functions ψ i and eigen-energies εi are known. In most applications only the lowest two discrete energies are consideredresulting in a two level system (TLS) description of the semiconductor system.

8.4 Validity of the Wannier-Expansion

The Wannier-expansion is able to reproduce the linear absorption around the excitonic resonance. Itis not able to describe the continuum effects of a semiconductor as only the discrete eigenstates ofthe Wannier-equation (8.3) are considered for the expansion. For nonlinear excitation it fails evenfor excitation at the excitonic resonance as is shown in chapter 9.1. However it is possible to obtainnumerically fast results in resonant or below resonance (chapter 15.2.4) which can be used to estimatethe results which to anticipate from a full semiconductor Bloch calculation.

81

Chapter 9

Rabi-Oscillation

In this chapter analytic solutions for the undamped Bloch equations are given and the influence ofdephasing mechanisms on the dynamics of the system is analyzed. The validity of the Wannier-equations for high excitations is discussed.

Starting from the pumped Bloch equations (7.5, here written in a rotating frame relative to thelaser frequency)

ddt

p = i12

Ω(1−2 f ) (9.1)

ddt

f = ΩIm(p) (9.2)

the inversion w = 1− 2 f and, since ddt p is always imaginary, the imaginary part of the coherence

v = 2 Im(p) are introduced (similar to section 7.1.1.1). This leads to a new set of differential equations

w = −Ωv (9.3)

v = Ωw , (9.4)

which are solved by

v(t) = −sin(θ(t)−π) (9.5)

w(t) = −cos(θ(t)−π)

with

θ =Z t

−∞Ω(t ′)dt (9.6)

being the area under the Rabi-frequency (i.e. the external field) Ω. The phase factor π takes sensibleboundary conditions into account (i.e. both coherence and inversion zero before the exciting pulse).Going back to the original variables yields

Im(p) = −12

sin(θ(t)−π) (9.7)

f =12

+12

cos(θ(t)−π) . (9.8)

The final values for both the coherence and the inversion only depend on the area under the excitingpulse (area theorem [41]). They are independent from the actual shape or intensity of the pulse. For a

82 9. RABI-OSCILLATION

u

v

w

s

s

Figure 9.1: Bloch-sphere. When excited with a θ = 1π-pulse the Bloch-vector s is turned around 180.The system is fully inverted.

0

0.5

1

0 0.5 1time [arb.units]

polarizationdensity

Figure 9.2: Two level system excited with a θ = 1π-pulse. The system is fully inverted.

9.1. VALIDITY OF THE WANNIER-EXPANSION FOR HIGHLY NONLINEAR EXCITATION 83

0

0.5

1


polarizationdensity

Figure 9.3: Two level system excited with a θ = 2π-pulse. The system performs a full Rabi-oscillation.

0

0.5

1


polarizationdensity

Figure 9.4: Damped (γ0 = 0.002meV) two level system excited with a θ = 2π-pulse. The Rabi-oscillation is slightly damped.

pulse area of π the imaginary part of the coherence becomes 0 and the probability to find the system inthe excited state is 1, the system is completely inverted (Fig. 9.1 and 9.2). A pulse with an area of 2πwill leave the system in the initial state. This full rotation of the Bloch-vector on the Bloch-sphere (cp.7.1.1.1 and Fig. 7.1.1.1) is called a Rabi-oscillation or Rabi-flop (Fig. 9.3). A damping of the systemprevents the system from performing perfect Rabi-Oscillations or reaching full inversion. However,the dynamics of the system is still an oscillation. Fig. 9.4 and Fig. 9.5 show phenomenologicallydamped two level-system whereas Fig. 9.6 shows the simulation of the actual dynamics of a quantum-dot damped by interaction with phonons in the surrounding semiconductor.

9.1 Validity of the Wannier-Expansion for Highly Nonlinear Excitation

An excitonic resonance is often expanded in a Wannier basis. Taking only the first order into accountresults in an approximation of the semiconductor by a two level system (chapter 8). This approx-imation reproduces the absorption spectra of the excitonic resonance. But the nonlinear resonant


0

0.5

1


polarizationdensity

Figure 9.5: Damped (γ0 = 0.2meV) two level system excited with a θ = 2π-pulse. The Rabi-oscillationis considerably distorted.

0

0.5

1


polarizationdensity

Figure 9.6: Two level system excited with a θ = 2π-pulse. The system is damped by the dynamiccoupling to the phononic system of the surrounding semiconductor.

9.1. VALIDITY OF THE WANNIER-EXPANSION FOR HIGHLY NONLINEAR EXCITATION 85

0

0.001

0.002

0 0.5 1

k-sp

ace

inte

grat

ed [1

/a02 ]

time [arb.units]

polarizationdensity

Figure 9.7: Two band Hartree-Fock-model (section 7.1.2) of a semiconductor: When excited with aθ = 2π-pulse the system does not perform a full Rabi-oscillation.

0

0.001

0.002

0 0.5 1

k-sp

ace

inte

grat

ed [1

/a02 ]

time [arb.units]

polarizationdensity

Figure 9.8: Damped (γ0 = 0.55meV) two band Hartree-Fock model of a semiconductor, excited witha θ = 2π-pulse: highly distorted Rabi-oscillation without zero-crossings.

excitation of an exciton is not reproduced adequately. Being a collective effect of all continuum car-riers the exciton does not perform perfect Rabi-oscillations as a two level system would. Looking atthe k-space integrated coherence and carrier density the resulting dynamics is a superposition of theindividual Rabi-oscillation of every continuum carrier. Even for an undamped system (Fig. 9.7) theexcitation with a 2π pulse does not return the system into its original state. For a damped systemeven the zero-crossings vanish (Fig. 9.8). The influence of the continuum is the more pronounced thenearer the excitonic resonance is to the continuum as more continuum states are excited.

Recapitulating what was said above the Wannier-expansion is only valid for linear excitation andonly around the excitonic resonance. It does not reproduce nonlinear response correctly, especially ifthe resonance is close to the continuum.

87

Chapter 10

Adiabatic Following and First OrderMemory Effects

In this chapter the microscopic equations of motion for the polarization are considered in a regimewhere memory-effects can be neglected and an expansion of the polarization in the electric field ispossible. In the first section the theory of adiabatic following is presented which totally neglectsmemory effects. In the second section memory effects in first order are considered. Finally theinfluence of continuum-states on memory effects is discussed.

Through this expansion physical insight into e.g. the phase dynamics of near resonant propagatingpulses (as discussed in chapter 15, section 15.2) can be gained.

10.1 Adiabatic Following

The fundamental idea of adiabatic following is based on the dynamics of a damped oscillator. If theoscillator is driven by an external force with a frequency far below or above the resonance frequencyand for a time period much smaller than the relaxation time the motion of the oscillator just follows thedriving force. The classical example would be a pendulum which is elongated very slowly and veryslowly returned to the initial point. Instead of performing oscillations the pendulum just adiabaticallyfollows the elongating force.

As was shown in section 7.1.1.1 the equations of motion for a two level system (the optical Blochequations) can be written in the form of a torque-equation 7.9

s = ΩF × s , (10.1)

which describes the precession of the Bloch-vector s around the torque vector ΩF = (−κE,0,ω)(Fig. 10.1). For great detuning (i.e. great ∆) s and ΩF are nearly parallel and the Bloch-vectoradiabatically follows the torque vector.

The dynamics of the Bloch-vector in the limit of adiabatic following is now derived. Startingfrom the pumped and damped Bloch equations (7.5) (here written in a rotating frame relative to theresonance frequency )

ddt

p = −i∆p− i12

Ω(1−2 f )− γp

ddt

f = −2γ( f − feq)+ΩIm(p) (10.2)

88 10. ADIABATIC FOLLOWING AND FIRST ORDER MEMORY EFFECTS

u

v

w

s

WF

Figure 10.1: Bloch-sphere. Bloch-vector s is precessing around the torque vector ΩF .

10.1. ADIABATIC FOLLOWING 89

with γ = 1T2

being the damping constant. In first order pertubation theory the density is assumed tobe constant f (t) ≡ feq with feqbeing the equilibrium density. The equation for p can be formallyintegrated yielding

p =

Z t

−∞−i

12

Ω(t ′)(1−2 f (t ′))e−(γ+i∆)(t−t ′)dt ′ . (10.3)

A new relative time coordinate s = t − t ′ is introduced

p =

Z ∞

0−i

12

Ω(t − s)(1−2 f (t − s))e−(γ+i∆)sds . (10.4)

The Taylor-series of the term

12

Ω(t − s)(1−2 f (t − s)) =∞

∑n=0

(−1)nsn

n!dn

dtn (12

Ω(t)(1−2 f (t))) (10.5)

allows the evaluation of integral (10.4)

p =−i

γ+ i∆

∞

∑n=0

(−1)n

(γ+ i∆)ndn

dtn (12

Ω(t)(1−2 f (t))) . (10.6)

Only the first two terms in the expansion are considered

p =−i

γ+ i∆(12

Ω(1−2 f )− 1γ+ i∆

ddt

12

Ω(1−2 f )) . (10.7)

Now the time derivative is computed

ddt

(12

Ω(1−2 f ))) =ddt

(12

Ω−Ω f ) =12

ddt

Ω− fddt

Ω−Ωddt

f (10.8)

and the equation of motion for f is inserted (terms ∝ γ are neglected, adiabatic following):

ddt

(12

Ω(1−2 f ))) =12

ddt

Ω(1−2 f )−Ω2Im(p) . (10.9)

Inserting (10.9) into (10.7) yields

p =−i

γ+ i∆(12

Ω(1−2 f )− 1γ+ i∆

(12

ddt

Ω(1−2 f )−Ω2Im(p)) . (10.10)

Neglecting γ in respect to ∆ (adiabatic following)

p = Re(p)+ iIm(p) =−1∆

(12

Ω(1−2 f )− 1i∆

(12

ddt

Ω(1−2 f )−Ω2Im(p)) (10.11)

and separately taking into account the real part and the imaginary part

Re(p) = − 1∆

(12

Ω(1−2 f ) (10.12)

Im(p) =−1∆2 (

12

ddt

Ω(1−2 f )−Ω2Im(p)) (10.13)

=−1∆2

12

ddt Ω(1−2 f )

(1− 1∆2 Ω2)

(10.14)

=12

ddt Ω(1−2 f )(Ω2 −∆2)

. (10.15)


From this set of equations p and f can be computed [1]:

Re(p) = −12 Ω√

∆2 +( 12 Ω)2

(10.16)

Im(p) = − ∆√∆2 +( 1

2 Ω)2

12

ddt Ω

∆2 +( 12 Ω)2

(10.17)

f = (12− ∆

2√

∆2 +( 12Ω)2

) . (10.18)

Now the material response to an external field, i.e. the polarization P = d ∗ p+ cc = 2d Re(p) can bedirectly computed in the adiabatic following limit. Expanding the real part of p yields

P = − dΩ√∆2 + 1

4 Ω2≈−d

∆Ω+

d8∆3 Ω3 +O(Ω5) . (10.19)

Equation (10.19) contains no memory effects.

10.2 Memory Effects

The results of the first order perturbation theory for the density, eq. (10.18), are inserted into theequations of motion (10.2) to calculate the next order perturbation theory. The polarization can beformally integrated analogous to eq. (10.4) yielding

p(t) = iZ ∞

0dse−(γ+i∆)s(Ω(t − s)− 2

∆2 Ω3(t − s)) . (10.20)

The Taylor expansion of the field up to second order

Ω(t − s) = ∑n

∂nt Ω(t)

(−s)n

n!≈ Ω(t)− s∂tΩ(t) (10.21)

is inserted into the polarization :

p(t) = iZ ∞

0dse−(γ+i∆)s(Ω(t)− s∂tΩ(t)− 2

∆2 (Ω(t)− s∂tΩ(t))3) (10.22)

= iZ ∞

0dse−(γ+i∆)s(Ω− sΩ− 2

∆2 (Ω3 −3Ω2sΩ+3Ωs2Ω2 − s3Ω3)) (10.23)

= iZ ∞

0dse−(γ+i∆)sΩ− i

Z ∞

0dse−(γ+i∆)ssΩ− i

Z ∞

0dse−(γ+i∆)s 2

∆2 Ω3 (10.24)

+iZ ∞

0dse−(γ+i∆)s 2

∆2 3Ω2sΩ− iZ ∞

0dse−(γ+i∆)s 2

∆2 3Ωs2Ω2

+iZ ∞

0dse−(γ+i∆)s 2

∆2 s3Ω3) .

The field quantities Ω(t) in this equation do not depend on s and can be drawn out of the integrals

p(t) = iΩZ ∞

0dse−(γ+i∆)s − iΩ

Z ∞

0dse−(γ+i∆)ss− i

2ω2

gΩ3

Z ∞

0dse−(γ+i∆)s (10.25)

+i2

∆2 3Ω2ΩZ ∞

0dse−(γ+i∆)ss− i

2∆2 3ΩΩ2

Z ∞

0dse−(γ+i∆)ss2 + i

2ω2

0Ω3

Z ∞

0dse−(γ+i∆)ss3 .

10.3. CONTINUUM EFFECTS 91

The integrals are computed in the limit of neglectable damping (γ → 0)Z ∞

0dse−(γ+i∆)s =

i∆

Z ∞

0dse−(γ+i∆)ss = − 1

∆2Z ∞

0dse−(γ+i∆)ss2 = − 2i

∆3Z ∞

0dse−(γ+i∆)ss3 =

6∆4

resulting in

p(t) = iΩi∆− iΩ(− 1

∆2 )− i2

∆2 Ω3 i∆

+ i2

∆2 3Ω2Ω(− 1∆2 ) (10.26)

−i2

∆2 3ΩΩ2(− 2i∆3 )+ i

2∆2 Ω3 6

∆4

= − 1∆

Ω+i

∆2 Ω+2

∆3 Ω3 − 6i∆4 Ω2Ω− 12

∆5 ΩΩ2 +12i∆6 Ω3 . (10.27)

With P = d ∗ p+ cc = 2d Re(p) the polarization P gets

P = −2d∆

Ω+4d∆3 Ω3 − 24d

∆5 ΩΩ2 (10.28)

= −αΩ+βΩ3 − γΩΩ2 . (10.29)

Comparing the result to chapter 1 the α term corresponds to the linear refractive index, β correspondsto the Kerr-nonlinearity and γ is the first order memory contribution.

10.3 Continuum Effects

To determine the validity of the above calculations for a semiconductor the influence of the contin-uum states are now considered. In the semiconductor Bloch equations (7.16) and (7.17) the differentk-states decouple if the Coulomb-interaction is neglected to see only the continuum-effects. Thesemiconductor Bloch equations read under this assumption

ddt

pk = −i∆k pk − i12

Ω(1−2 fk)− γpk

ddt

fk = −2γ( fk − fk,eq)+ΩIm(pk) (10.30)

and are (except the k index) identical to eqns. (10.2) . The solution (10.27) can be transfered readingnow

pk(t) = − 1∆k

Ω+i

∆2k

Ω+2

∆3k

Ω3 − 6i∆4

kΩ2Ω− 12

∆5k

ΩΩ2 +12i∆6

kΩ3 . (10.31)

The polarization is (cp. eq. (7.14))

P = ∑k

d ∗ pk + cc = 2d ∑k

Re(pk) (10.32)

= 2d ∑k

(− 1∆k

Ω+2

∆3k

Ω3 − 12∆5

kΩΩ2) (10.33)

= −2Ωd ∑k

1∆k

+4Ω3d ∑k

1∆3

k−24ΩΩ2d ∑

k

1∆5

k. (10.34)


The consideration of continuum-states effectively results in the transformation 1∆l

0→ ∑k

1∆l

k. The in-

verse detuning between the light frequency and one resonance 1∆0

is superceded by the sum over thedifferent inverse detunings between the light frequency and the frequency of each continuum state.A numerical evaluation of the sums for bulk GaAs parameters (appendix G) yields the followingestimates:

∑k

1ωk

≈ 1ω0

→ refractive index

∑k

1ω3

k<

1ω3

0→ Kerr nonlinearity

∑k

1ω3

k 1

ω30

→ memory contribution .

The following conclusions can be drawn from this estimation: The refractive index is only slightlyinfluenced by the continuum states and the Kerr-nonlinearity is moderately weakened. The memorycontribution however is considerably suppressed by the continuum states. As the first order memorycontribution enters the coherence as a coherent superpositions of contributions from different k-states,

Pmemory ∝Z ∞

0

(

∑k

ei∆kss∂t Ω(t)

)ds , (10.35)

these contributions partially cancel each other out, leading to reduction of overall memory effects.

93

Chapter 11

Boltzmann-Equation

In this chapter the scattering rates of the electron-electron interaction in a quasi two dimensionalquantum well are considered. To derive the Quantum-Boltzmann-equation (e.g. [42, 37] ) one has togo beyond the Hartree-Fock-approximation (introduced in chapter 6). Beginning with the equation ofmotion for a two operator expectation value (6.4)

−i~∂t〈a†α,kaβ,k〉 = (εα,k − εβk)〈a

†α,kaβ,k〉+ ∑abcd V βk a

bc 〈a†α,ka†

aabac〉−V abcαk〈a†

aa†bacaβk〉 (11.1)

and performing the Hartree-Fock-factorization (6.6) yields

i~∂t〈a†α,kaβ,k〉 = (εα,k − εβk)〈a

†α,kaβ,k〉+

∑abc

V βk abc 〈a†

α,kac〉〈a†aab〉−V βk a

bc 〈a†α,kab〉〈a†

aac〉

−V abcαk〈a†

aaβ,k〉〈a†bac〉+V ab

cαk〈a†aac〉〈a†

baβ,k〉+V βk a

bc 〈a†α,ka†

aabac〉corr −V abcαk〈a†

aa†bacaβ,k〉corr . (11.2)

The equation of motion for the four-point expectation value couples to six point-entities:

i~∂t〈a†1a†

2a3a4〉 = −(ε1 + ε2 − ε3 − ε4)〈a†1a†

2a3a4〉+∑

abcV ab

c1 (〈a†aa†

ba†2aca3a4〉

−V abc2 〈a†

1a†aa†

baca3a4〉−V ab

c3 〈a†1a†

2a†caaaba4〉

+V abc4 〈a†

1a†2a†

ca3aaab〉)−∑

ab(V ab

21 〈a†aa†

ba3a4〉)

−∑ab

(V ab34 〈a†

1a†2aaab〉) , (11.3)

which are factorized according to (6.7). The hierarchy is truncated at this point by neglecting thesix-operator correlations. Combining eqns. (11.2) and (11.3) yields the equation of motion for the

94 11. BOLTZMANN-EQUATION

four-point correlation (the actual calculation is trivial but a Sisyphean challenge):

i~∂t〈a†1a†

2a3a4〉corr = −(ε1 + ε2 − ε3 − ε4)〈a†1a†

2a3a4〉corr + ∑abcd

(V abcd −V ba

cd )( (11.4)

−〈a†1aa〉〈a†

2ac〉〈a†aa4〉〈a†

ba3〉+〈a†

aa4〉〈a†ba3〉〈a†

2ab〉〈a†1aa〉

+δ2c〈a†aa4〉〈a†

ba3〉〈a†1aa〉

+δ1d〈a†aa4〉〈a†

ba3〉〈a†2ac〉

−δ3b〈a†1aa〉〈a†

2ab〉〈a†aa4〉

−δ4a〈a†1aa〉〈a†

2ab〉〈a†ba3〉

−δ2cδ1d〈a†aa4〉〈a†

ba3〉+δ3bδ4a〈a†

1aa〉〈a†2ab〉

+12

δ1d〈a†aa†

ba3a4〉corr〈a†2ac〉

+12

δ2c〈a†aa†

ba3a4〉corr〈a†1ad〉

+12

δ3c〈a†1a†

2aaab〉corr〈a†da4〉

+12

δ4d〈a†1a†

2aaab〉corr〈a†ca3〉

−12

δ1dδ2c〈a†aa†

ba3a4〉corr

+12

δ4dδ3c〈a†1a†

2aaab〉corr) .

In order to solve this equation only the free rotation and scattering terms where the equation of motioncouples to two-operator-coherences are considered. All higher terms (which result in an effectivescreening of the coulomb-interaction) are neglected. The products of two-operator-coherences can berewritten (utilizing the basic fermionic commutation relation) and the equation of motion reads

i~∂t〈a†1a†

2a3a4〉corr = −(ε1 + ε2 − ε3 − ε4)〈a†1a†

2a3a4〉corr + ∑abcd

(V abcd −V ba

cd )(

−〈a†aa4〉〈a†

ba3〉〈aca†2〉〈ada†

1〉+〈a4a†

a〉〈a3a†b〉〈a

†2ac〉〈a†

1ad〉) . (11.5)

11.1 Markov-Approximation

Equation 11.5 has the form of

i~∂t f (t) = −ε f (t)+C(t) , (11.6)

which can be formally integrated

f (t) =i~

Z t

−∞dt ′C(t)e

i~

ε(t−t ′) =i~

Z ∞

0dsC(t − s)e

i~

εs (11.7)

11.2. BOLTZMANN-EQUATION 95

with s = t−t ′. The Markov-approximation (cp. appendix B) is the negligence of the so called memorykernel ( C(t − s) ≈ C(t) ) of the integral. The physical consequence of the negligence of memory-effects is the strict selection of scattering processes which preserve the energy:

f (t) =i~

Z ∞

0dsC(t − s)e

i~

εs ≈ i~

C(t)Z ∞

0dse

i~

εs = iC(t)2πδ(ε) . (11.8)

Solutions of equation (11.5) can be inserted into (11.2) getting a closed set of equations for the two-operator coherences. After a few elementary manipulations the equation of motion for diagonal ele-ments of the density matrix (i.e. the electron densities) can be written as:

∂t〈a†1a1〉 =

π~

∑abc

δ(ε1 − εa − εb + εc)(V c1ab −V 1c

ab )(V bac1 −V ab

c1 ) (11.9)

×(〈a†aaa〉〈a†

bab〉(1−〈a†1a1〉)(1−〈a†

cac〉)−〈a†

1a1〉〈a†cac〉(1−〈a†

bab〉)(1−〈a†aaa〉)) .

11.2 Boltzmann-Equation

Using the notation fa = 〈a†aaa〉 and converting the equation into the canonical form yields

∂t f1 = (1− f1)π~

∑abc


ab )(V bac1 −V ab

c1 ) fa fb(1− fc) (11.10)

− f1π~

∑abc


ab )(V bac1 −V ab

c1 ) fc(1− fb)(1− fa)

= Γin(1− f1)−Γout f1 (11.11)

with the in- and out-scattering rates

Γin (1) =π~

∑abc


ab )(V bac1 −V ab

c1 ) fa fb(1− fc) (11.12)

Γout (1) =π~

∑abc


ab )(V bac1 −V ab

c1 ) fc(1− fb)(1− fa) . (11.13)

11.3 Scattering Rates

To elucidate the influence of electron-electron-scattering on the overall-dynamics of multiple quantumwells (as discussed in chapter 14) the dephasing rates are computed. In contrast to the calculations forthe electromagnetic field (chapter 3 and chapter 2) the QW can no longer be assumed as infinitely thinas vanishing form-factors Fi j(qc) =

R

drcφ?i (rc)eiqc·rc φ j(rc) in the matrix elements V ab

cd (5.37) wouldlead to vanishing scattering rates (as the Coulomb induced dephasing is a interference effect betweenthe different k-states which cancel out in this case [43]). It has been shown ([44]) that a QW with finitedepth can be well approximated by an infinitely deep QW with an effective well-width. To simplifythe calculation, this approximation is used (and only the lowest subband is considered). Numericalestimations show the damping constant γCoulomb to be in the order of 1-2 ps. This is consistent withexperimental measurements were the overall damping constant for a single QW was found to be0.55meV [39]. As the dynamics of a system is governed by the fastest process, the electron-electronscattering is of minor importance for the MQW as the superradiative coupling leads to dephasingtimes in the order of a few fs (as shown in chapter 14).

96 11. BOLTZMANN-EQUATION

97

Chapter 12

Quantum Dot on Wetting Layer

In this chapter, the optical properties of a self-organized quantum-dot are considered. On a semicon-ductor substrate a layer of atoms is deposited (wetting layer). The lattice constant mismatch betweenthe substrate and the deposited atoms gives rise to pyramid-like structures which minimize the totalenergy (Stranski-Kranstanov mode, Fig. 12.1). These structures on top of the wetting layer have adiameter of about 100 atoms. This system has attracted much interest in recent times as it is typicalfor a quantum dot laser where electrons are injected into the wetting layer and than scatter into thequantum dot states to be able to perform a laser-transition. The laser-transition in this kind of laser canbe influenced to some extent by the geometry of the quantum dots in contrast to conventional inter-band lasers where the laser-transition is determined by the material system. The microscopic theoryof the quantum dot laser is of interest because of two processes limiting the laser performance: Thetime scale of the scattering mechanisms from the wetting layer states into the lasing level limits theefficiency of the carrier injection into this state and consequently the efficiency of the whole laser. Arelated problem is the damping of the coherent dynamics of the lasing transition itself as it limits themodulation band width of the laser. The problem of damping has been studied on the optically inducedRabi-oscillations of quantum dots. Experiments[45, 46] show a damping which can be explained byelectron-phonon interaction[34] for high temperatures but not for low temperature experiments.

Aim of this chapter is to elucidate the question if interactions between quantum dot states andwetting layer states (as proposed in [47]) can cause a damping of Rabi-oscillations in the quantumdot. The system is modeled by a GaAs, multilevel, zero-dimensional quantum dot (two electron andtwo holes levels) for the self-organized pyramidal quantum dot and a two-dimensional quantum wellfor the wetting layer. The wetting layer and the quantum dot interact via Coulomb-interaction. Theenergy scheme of this coupled system is depicted in Fig. 12.2. Used as a quantum dot laser, theelectrons would be injected into the wetting layer and then cascade down from the wetting layer bandc into the upper quantum dot state D1 and further down into the ground state D2 to be able to performthe optical laser-transition D2−D3.

12.1 Hamiltonian

The Hamiltonian used for calculating the system dynamics includes the following contributions: freeelectrons in the quantum dot (HQD,0), free electrons in the wetting layer (HQW,0, QW for quantum-well), the phononic bath (HPH,0), the Coulomb interaction inside the wetting layer (HQW,QW ) andbetween wetting layer and the quantum-dot (HQW,QD) as well as the interaction of the quantum dot

98 12. QUANTUM DOT ON WETTING LAYER

Figure 12.1: Self-organized quantum dot in the shape of a pyramidal stump. Simulation by MichaelBlock (Technische Universität Berlin).

excitonic conduction band

QW conduction band

QD conduction excited state

QD conduction ground state

excitonic conduction band

QD conduction excited state

QD conduction ground state

QW conduction band

energy

electrons

holes

c

v

D1

D2

D3

D4

Figure 12.2: Energy scheme of the coupled quantum dot and quantum well (wetting layer) system.The quantum well and exciton bands have a k-dispersion perpendicular to the confinement.

12.2. COULOMB-SCATTERING PROCESSES 99

in the 1. excited state

e−h cap e−e cap e−h cap e−e rel e−h rel e−e rel2

AA B F GC D E

e−e cap

WL

WL

capture processesin the ground statecapture processes

relaxation processes

elec

tron

sho

lesbo

und

QD

−sta

tes

Figure 12.3: he seven possible fundamental electron-electron scattering processes [48, 49].

with the phononic system (HQD,PH) (cp. section 5.3):

H = HQW,0 +HQD,0 +HPH,0 +HQW,QW +HQW,QD +HQD,PH (12.1)

= ∑i,k

εi,ka†i,kai,k (12.2)

+ ∑I

εIa†I aI

+ ∑µ,q

~ωµ,q(b†µ,qbµ,q +

12)

+12 ∑

i, j,k,k′ ,q6=0

4πe2

2Aε0εr(q, t)1|q|a

†i,k+qa†

j,k′−qa j,k′ai,k

+12 ∑

1234V 12

34 (q f )a†1a†

2a3a4

+ ∑I,J,q,µ

gIJµ(q)a†I (bµ,q +b†

µ,−q)aJ .

Here, small letters (i, j, etc) denote quantum-well states in the valence (v) or conduction (c) bandwhereas capital letters (I,J, etc.) denote quantum-dot states in the ground and excited electron states(1 and 2) or hole states (3 and 4).

12.2 Coulomb-Scattering Processes

In the considered system, 28 different electron-electron scattering processes are possible. The numbercan be reduced by taking into account symmetries between electrons and holes as well as the timereversal of processes. This leaves seven fundamental processes depicted in Fig. 12.3. The scattering


times have been extensively discussed on the level of rate equations in the diploma thesis of ErminMalic[48]. The dominant processes have turned out to be the capturing into the upper quantum dotstate (capture process B in Fig. 12.3) and the relaxation from the upper into the lower quantum dotstate (relaxation process F in Fig. 12.3)[48, 49]. The matrix elements were calculated by assuming adisc-shaped quantum dot. The result[48] for the capturing process is

V D1k2k3k1

=1

A3/2

e2

ε

√~πmω

F(q)exp(− ~

2mω|q−k1|2) (12.3)

q = k2 −k3, (12.4)

F(q) :=1q

2

Lq+

1

Lq+ 4π2

Lq

− 2L6 (1− exp(−Lq))

(4π2

q(q2 + 4π2

L2 )

)2 (12.5)

and for the relaxation process

V D1k2D2k3

= −i1A

e2

4ε

√~

mωqF(q)(k1 −q)exp(− ~

4mωq2) . (12.6)

12.3 Phonon Matrix Element

Analogous to [50], the deformation-potential coupling to acoustic phonons is considered. To reducenumerical calculations, the form factors were chosen for a spherical quantum dot. The electron-phonon coupling matrix element reads[50]

gq12 = gq

11 −gq22 (12.7)

gqii =

√~q

2ρcVDi e−

q2~

4mω . (12.8)

12.4 Equations of Motion

In a first step, a simplified model with only two quantum dot levels (3 and 2) and only one elec-tron per quantum dot is considered. The electron-phonon interaction is considered in first order andthe phononic system is in thermodynamical equilibrium. Therefore the phonon matrix-element isgi j ∼ δi j. The Coulomb-interaction is assumed to be diagonal in the quantum well (i.e. no intra-bandpolarization) and polarizations between quantum dot hole states and quantum well electron states aswell as polarizations between quantum dot electron states and quantum well hole states are neglected.A rotating frame on the band gap frequency of the wetting-layer is introduced. A closed set of equa-tions in Hartree-Fock approximation (cp. chapter 6) for the following quantities is derived with theEhrenfest-theorem (eq. 4.9):

12.5. LINEAR ABSORPTION SPECTRA 101

0

2

4

6

8

10

12

-200 -150 -100 -50 0

-Imχ

detuning [meV]

Figure 12.4: Realistic absorption spectrum. The quantum dot resonance lies 120 meV below thewetting layer band-edge.

quantum dotcoherence p23 =< a†

2a3 >

occupation 2 f2 =< a†2a2 >


quantum wellinterband coherence pcv¸k =< a†

ckavk >

occupation conduction band fck =< a†ckack >

occupation valence band fvk =< a†vkavk >

QD-phononphonon assisted density-matrix Sq =< a†

2a3bq >c

phonon assisted density-matrix Tq =< a†2a3b†

−q >c

QD-QWdot-well polarization Uk =< a†

2avk >

dot-well polarization Vk =< a†3ack >

The equations are given explicitly in appendix E.

12.5 Linear Absorption Spectra

The calculated linear absorption spectrum is shown in Fig. 12.4. As the absorption line of the quantumdot is much narrower than the excitonic line of the wetting layer and both are separated by more than100 meV, the linewidth and spectral position of the quantum dot line have been artificially altered inFig. 12.5 for better illustration. In the spectrum the main features are visible: the band edge at 0 meVdetuning with the continuum states of the wetting layer at detunings greater than 0. Just a few meVbelow the band edge is the first excited excitonic state (2s). The excitonic resonance below 10 meV


0

0.2

0.4

0.6

0.8

1

1.2

-40 -30 -20 -10 0 10

-Imχ

detuning [meV]

Figure 12.5: Absorption spectrum artificially altered for better visibility. The continuum as wellas the 1s-and 2s-excitonic resonance of the wetting layer are visible. The quantum dot resonance(artificially broadened and shifted to -30 meV) features the typical zero phonon-line and phonon side-bands[50, 34].

has a width of 1 meV (the used damping constant of γQW = 0.55 meV was chosen in accordance with[39]). The line of the quantum-dot (which is shown in Fig. 12.5 at -30 meV but lies in reality at -120meV) consists of a narrow zero-phonon line (width about 0.0013 meV) and two phonon sidebands(shown here for 77K). At low temperatures it is easier to emit phonons than to absorb phonons (asthere are to few to absorb). This can be seen in the asymmetry of the phonon side bands. No evidenceof the coupling between wetting layer and quantum dot influencing the linewidth could be detected.

12.6 Nonlinear Dynamics - Rabi-Oscillation

To identify the different contributions to the damping of the coherent dynamics of the quantum dot anexcitation with a 3ps-, 3π-light pulse resonant to the quantum dot resonance was simulated. First aquantum dot without coupling to either the phononic system nor the wetting layer was simulated. Theundamped Rabi-oscillation of the polarization and occupation in the quantum dot is depicted in Fig.12.6. As expected from a 3π-pulse (cp. chapter 9), the polarization performs 3 flops and the density1.5 flops - resulting in a complete inversion of the system.

Next a quantum dot interacting with the phonon-bath is considered. The Rabi-oscillation isdamped considerably as can be seen in Fig. 12.7. The damping is consistent with the linear linebroadening and phonon side-bands as shown in Fig. 12.5.

The final calculation including the full wetting layer dynamics and Coulomb-coupling betweenquantum dot and wetting layer shows no influence on the Rabi-oscillation (Fig. 12.8). No significantdifferences between Fig. 12.7 and Fig. 12.8 are visible.

12.6. NONLINEAR DYNAMICS - RABI-OSCILLATION 103

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9time [ps]

Figure 12.6: Free Rabi-oscillation with a 3π-excitation. Coherence (red line) and occupation (blueline) perform Rabi-oscillations.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9time [ps]

Figure 12.7: Damped Rabi-oscillation due to electron-phonon-interaction. Coherence (red line) andoccupation (blue line) are shown.


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9time [ps]

Figure 12.8: Calculation of Rabi-oscillations including both electron-phonon interaction and quan-tum dot / quantum well interaction. Coherence (red line) and occupation (blue line) are shown. Noadditional effects compared to Fig. 12.7 are evident.

12.7 Discussion

In the wetting layer the different continuum states perform separate Rabi-oscillations at different fre-quencies resulting in an integrated carrier density that shows traces of Rabi-oscillations but is con-siderably damped (Fig. 9.7). For real parameters including dephasing processes in the continuum, itshows as modulations on top of the carrier dynamics (Fig. 9.8). It was hoped that these effects wouldinfluence the dynamics of the quantum dot which is Coulomb-interacting with the wetting layer. Nei-ther in the linear absorption spectrum nor in the nonlinear excitation were any signs of such influencefound.

However, as this calculation presents only a first approach, the theory leaves room for criticismand refinement:

• In the calculation only one electron and one hole state of the quantum-dot were considered.

• Not all possible scattering processes (Fig. 12.3) were incorporated.

• Polarizations between quantum dot hole states and quantum well electron states as well aspolarizations between quantum dot electron states and quantum well hole states were neglected.These polarizations could give rise to additional optical transitions.

• The Coulomb matrix elements and the phonon coupling matrix elements were calculated in-consistently. For the former, a spherical quantum dot was assumed, and a disc-shaped for thelatter. In a further step realistic wave functions of the quantum dot (which can be calculated bya k ·p-theory) should be used as it has been shown that the actual shape of the quantum dot isof importance [50, 22].

Recapitulating, it can be said that with the used model and assumptions an influence of the wettinglayer on the damping of Rabi-oscillations in the quantum well could not be verified.

105

Chapter 13

THz-Luminescence

To describe effects like photoluminescence and spontaneous emission which are not included in theformalism of semi-classical electrodynamics it is necessary to quantize the light field. This quanti-zation gives rise to the light quasi particles or photons as shown in section 5.3.6. In the first part ofthis chapter an universal expression for the photo-luminescence is derived. In the final part the THz-luminescence spectrum of free electrons, an interacting electron plasma and a semiconductor quantumwell is calculated[32, 51].

13.1 Luminescence

13.1.1 Spectral Intensity

The Wiener-Khintchine-Theorem connects the power-spectrum with the auto-correlation function.[52]The photo-emission is given by

IS(Ω,T ) =

Z ∞

−∞

Z ∞

−∞〈E(−)(r, t1)E(+)(r, t2)〉G(t1 −T)G(t2 −T )e−iΩt1 eiΩt2 dt1dt2 (13.1)

and the photon-flux (i.e. the intensity) which is measured by the photo-detector

S(Ω,T ) =IS(Ω,T )

∆T(13.2)

with the following entities

G(t) = envelope of the detector-function (13.3)

Ω = central frequency of the detection (13.4)

T = central time of the detection (13.5)

∆T = duration of the measurement (13.6)

E(−)(r, t) = part of the electric field rotating with e−iωqt (13.7)

E(+)(r, t) = part of the electric field rotating with eiωqt . (13.8)

106 13. THZ-LUMINESCENCE

13.1.2 Free Propagation Approximation

We are now assuming free propagation of the field in the eldirection. The separate space and timedependence can be combined as the field now obeys the wave-equation:

(rel , t) → (r− ct) . (13.9)

This allows for a change of the integration variable from time-like to space-like

dt → 1c

dr (13.10)

and a shift of the temporal auto-correlation to spatial auto-correlation:

(r− ct1/2) → (r1/2 − ct) (13.11)

t1 =r1 − cT

c(13.12)

t2 =r2 − cT

c(13.13)

IS(Ω,T ) =1c2

Z ∞

−∞

Z ∞

−∞〈E(−)

l (r1el ,T )E(+)l (r2el ,T )〉 (13.14)

×G(r1 − cT

c)G(

r2 − cTc

)e−iΩ r1−cTc eiΩ r2−cT

c dr1dr2 . (13.15)

This is strictly true only for an integration over the whole space (i.e. free propagation in the wholespace). For the detection of the photoluminescence signal far away from the investigated structure inthe far-field (13.14) should be a good approximation.

13.1.3 Free-mode Expansion of the Field Operators

Analogous to the expansion of the vector-potential (5.3.6) the field operators can be expanded infree-modes:

E(−)l (r1, t1) = ∑

q1

√~ωq1

2ε0e−iq1r1c†

q1,l(t1)eq1,l (13.16)

E(+)l (r2, t2) = ∑

q2

√~ωq2

2ε0e+iq2r2cq2,l(t2)eq2,l . (13.17)

Inserting this expansion into the expression for the photo emission 13.14 yields:

IS(Ω,T ) =1c2 ∑

q1,q2

~√ωq1ωq2

2ε0(eq1,l · eq2,l)〈c†

q1,l(T )cq2,l(T )〉 (13.18)

Z ∞

−∞

Z ∞

−∞G(

r1 − cTc

)G(r−cT

c)ei(q2−Ω

c )r2e−i(q1−Ωc )r1dr1dr2 . (13.19)

13.2. THZ-EMISSION 107

13.1.4 Stationary Limit

For long detection times the envelope G(t) of the detector-function approaches unity: G(t) = 1. Thephoto emission can be written as:

IS(Ω,T ) =1c2 ∑

q1,q2

~√ωq1ωq2

2ε0(eq1,l · eq2,l)〈c†

q1,l(T )cq2,l(T )〉 (13.20)

Z ∞

−∞ei(q2−Ω

c )r2dr2

Z ∞

−∞e−i(q1−Ω

c )dr1

=1c2 ∑

q1,q2

~√ωq1ωq2

2ε0(eq1,l · eq1,l)〈c†

q1,l(T )cq2,l(T )〉(2π)2 (13.21)

δq1,Ωcδq2,

Ωc

.

Evaluating the Kronecker-deltas and substituting Q = Ωc the photo emission reads

IS(Ω,T ) =2π2

~ωQ

c2ε0〈c†

Q,l(T )cQ,l(T )〉 . (13.22)

Considering the stationary case with long detection-times, the photo emission gets linear in T [53]and the photon-flux gets

S(Ω,T ) =d

dTIs(Ω,T ) (13.23)

=2π2

~ωQ

c2ε0

ddT

〈c†Q,l(T )cQ,l(T )〉 . (13.24)

13.2 THz-Emission

13.2.1 Emission from Free Electron

If an electron emits a photon the conservation of momentum has to be obeyed. For a free electron itreads

a = b+k (13.25)

with k being the photon momentum and a and b the electron momentum before and after emission.The total energy has also to be conserved:

~2a2

2m=

~2b2

2m+~ωk . (13.26)

Together one gets

~2a2

2m=

~2(a−k)2

2m+~ωk . (13.27)

And with the dispersion relation ωk = ck of the photon it reads

~2a2

2m=

~2(a−k)2

2m+~ck (13.28)

~2a2

2m=

~2(a2 +k2 −2a ·k)

2m+~ck (13.29)

0 =~k2

2m− ~ak

mcos θ+ ck . (13.30)


energy

wavenumber

?

photonemission

Figure 13.1: Free electrons cannot emit photons due to energy and momentum conservation.

For small photon momenta k2 can be neglected with respect to k2

0 = −~akm

cosθ+ ck . (13.31)

For this equation to hold true, cosθ has to be > 0. This leads to

~am

1 c . (13.32)

The velocity of the electron has to be greater than c to enable the electron to emit a photon. Fornon-relativistic electron velocities (which in this work are considered) this holds true more then ever.To calculate the photoluminescence scattering and interaction processes must be included to enablephoton emission processes.

13.2.2 Plasma

As seen above one has to include interaction or scattering mechanism to obtain photon emission. TheHamiltonian for a Laser-induced interacting electron plasma has been derived in 5.4. The mechanismwhich allows for luminescence to occur is electron-electron and electron-ion scattering 13.2.

13.2.2.1 Effective Scattering Matrix

In the limit of weak ionization the contribution of free-electron scattering is small and only the scatter-ing of free-electrons with bound electrons and atomic cores are considered. The Hartree-contributionof the many particle interaction can be written in the form

HU = ~∑i j

Ui− ja†i a j . (13.33)

This is done by only considering elastic scattering, i.e. scattering between two bound and two unboundelectrons. The contributions of the bound electrons are approximated by the expectation value of theelectron density[54]:

HEL,EL = ∑1234

V 1234 a†

1a†2a3a4 ≈ ∑

12νUν

12〈a†νaν〉a†

1a2 . (13.34)


energy

wavenumber

el-el or el-ionscattering

photonemission

Figure 13.2: In a laser induced interacting electron plasma the interaction between the electrons andwith the atomic cores allows for the momentum and energy conservation to be fulfilled. The plasmacan emit photons.

The same can be done for the free-electron atomic core scattering. Inserting the wave functions for thebound electrons (φν(r)) and the ionic cores (which are due to the semi-classical treatment essentiallyδ-distributions) yields the effective scattering matrix

Uq = ∑α

e−iqRα(∑µ〈a†

µαaµα〉Uµα,el−elq −Uα,el−core

q ) (13.35)

Uα,el−coreq =

e2

V ε0

Zq2 (13.36)

Uµα,el−elq =

e2

V ε0

1q2 Fν(q) (13.37)

Fν(q) =Z

|φ(r+Rν)|2e−iqrdr , (13.38)

where α is the index of atomic systems and ν the index of atomic shells. Only the electron-electroninteraction features the form factor F νas the atomic cores are treated semi-classically.

13.2.2.2 Plasmon-Polariton

The photonic system can be treated similar to the electronic system in 13.2.2.1 by substituting theelectron density operator by its expectation value. With the free electron density n0 and the plasma-frequency ω2

pl = n0e2

m0ε0the Hamiltonian for the free photons and the nonlinear electron-light interaction

reads:

Hphot +HA2 = ~∑kσ

ωkω2

pl

4ω2k(ckσ + c†

−kσ)(c−kσ + c†kσ)+~∑

kσωkc†

kσckσ . (13.39)


This Hamiltonian is diagonalized by means of a Bogoljubov-transformation:

B†kσ =

1

2ε14k

((1+√

εk)c†kσ +(

√εk −1)c−kσ) (13.40)

Bkσ =1

2ε14k

((1+√

εk)ckσ +(√

εk −1)c†−kσ) (13.41)

εk = 1+ω2

pl

ω2k

. (13.42)

The new quasi-particles denoted by the creation- and annihilation-operators B† and B are plasmon-polaritons which obey the canonical bosonic commutation-relations. The Hamiltonian now reads

Hphot +HA2 = ∑kσ

~ωk(B†kσBkσ +

12) (13.43)

with the dispersion relation ωk =√

εkωk, which is the well known[5] dispersion relation for transver-sal electromagnetic radiation in plasma. The analogous calculation for the linear electron light inter-action yields

Hel−phot = ∑i j

∑kσ

Fkσi j (B†

kσ +B−kσ)a†i a j (13.44)

Fkσi j = ε

1−4k Fkσ

i j . (13.45)

13.2.2.3 Statistical Ensemble Average

Deriving the equations of motion for the plasmon-polariton density 〈B†kσBkσ〉and subsequent quan-

tities up to three-operator-quantities (factorization according to 6.8) shows that up to this order noluminescence occurs. The first non-vanishing order is in F 2U2. Introducing an ensemble averageover all atoms in the plasma 〈·〉U and utilizing statistical results for density fluctuation in gases thescattering-matrix square results in

〈UqUq′〉U = δ−q,q′U0e−q2ζ2UqU−q (13.46)

with ζ being the correlation length depending on the average distance of ions and U0 ∝ ω0 being ameasure of the intensity of density fluctuations. The resulting system of equations of motions is

∂t〈B†kσBkσ〉 = −2∑

ab

1

ε14k

Im(F−kσ

ab 〈Bkσa†aab〉

)(13.47)

i~∂t〈Bkσa†cad〉 = ~(ωk + εd − εc)〈Bkσa†

cad〉+~∑

bFkσ

db 〈a†cab〉

+~∑q

(〈UqBkσa†cad−q〉−〈UqBkσa†

c+qad〉) (13.48)


ı~∂t〈〈UqBkσa†c+qad〉〉U =~(ωk + εd − εc+q)〈〈UqBkσa†

c+qad〉〉U (13.49)

+~∑i

Fkσdi 〈〈Uqa†

c+qai〉〉U

+~∑i

(〈〈Ud−iUqBkσa†

c+qai〉〉U −〈〈Ui−c−qUqBkσa†i ad〉〉U

)

ı~∂t〈〈Uqa†c+qab〉〉U =~(εb − εc+q)〈〈Uqa†

c+qab〉〉U (13.50)

+~〈〈U2q a†

c+qab+q〉〉U −~〈〈U2q a†

cab〉〉U

ı~∂t〈〈U2q Bkσa†

c+qad+q〉〉U =~(ωk + εd+q− εc+q)〈〈U2q Bkσa†

c+qad+q〉〉U (13.51)

+~∑i

Fkσd+qi〈〈U2

q a†c+qai〉〉U .

Utilizing the Markov-approximation (Appendix B) for the last three equations and considering a spa-tial homogeneous system the equations reduce to

∂t〈B†kσBkσ〉 = −2∑

ab

1

ε14k

Im(F−kσ

ab 〈Bkσa†aab〉

)

ı~∂t〈Bkσa†cad〉 =~(ωk + εd− εc)〈Bkσa†

cad〉 (13.52)

+~∑q

Fkσdc U2

q ( fc+qζ(εc − εc+q)ζ(ωk + εd − εc+q)

− fcζ(εc − εc+q)ζ(ωk + εd − εc+q))

+~∑q

Fkσd−q,c−qU2

q ( fc−qζ(εc−q − εc)ζ(ωk + εd−q − εc)

− fcζ(εc−q − εc)ζ(ωk + εd−q− εc))

−~∑q

Fkσdc U2

q ( fcζ(ωk + εd−q − εc)ζ(ωk + εd − εc)

+ fcζ(ωk + εd − εc+q)ζ(ωk + εd − εc))

+~∑q

Fkσd+q,c+qU2

q ( fc+qζ(ωk + εd+q − εc)ζ(ωk + εd+q− εc+q)

+ fc+qζ(ωk + εd− εc+q)ζ(ωk + εd+q − εc+q)) .

with fc = 〈a†cac〉 being the electron density. Applying the Markov-approximation (appendix B) once

more yields the final result for the plasmon-polariton density

∂t〈B†kσBkσ〉 = 2π|ζ(ωk)|2A2

kσ1

V 2 ∑cq

U2q ε2

q fc+qδ(ωk + εc − εc+q) (13.53)

with εq = εkσ ·q and A2kσ = ~e2

2ε0V ωkm20.

13.2.2.4 Results

The luminescence for a hydrogen-like gas is evaluated. Considering a weakly ionized gas the mainscattering contribution is electron-atom-interaction. The form factor F(q) in this situation is

F(q) =1

(1+q2a2

04 )2

. (13.54)


0 2 4 6 8 10ΩΩpl

[email protected]

itsD

Figure 13.3: Plasma luminescence at T = 300K and n0 = 10−5nm−3. The plasma frequency is fpl =0.9 THz. Different correlation length are shown: ζ = 30nm (solid line), ζ = 10nm (dashed line) andζ = 0.35nm (dash-dotted line)

The electron-distribution is assumed to be a Boltzmann-function

f (k) = Nke−~εkkT . (13.55)

The resulting luminescence spectra are shown in Figs. 13.3, 13.4, 13.5 and 13.6. For long correlationlength and for low temperatures the maximum of the spectrum is near the classic plasma frequency.For smaller correlations length (Fig. 13.3) and higher temperature (Figs. 13.4 and 13.6) the position ofthe maximum shifts to higher frequencies and the spectrum is broadened. For higher electron densitiesthe maximum shifts to lower frequencies and the spectrum gets narrower (Fig. 13.5).

13.2.3 Quantum Well

The procedure for obtaining the photon-density 〈c†c〉 is similar to section 13.2.2. The difference isthe scattering mechanism. In a quantum well not the electron-ion scattering but the electron-phononscattering enables the THz-luminescence (Fig. 13.7). Using the Hamiltonian from section 5.3 and theHeisenberg equation (4.4) the equation of motion for the photon-density and subsequent expectationvalues are derived and factorizing according to (6.8). Coherent photons〈c〉 are neglected as onlyspontaneous emission is considered. The phonons are assumed to be in thermal equilibrium with anexternal bath. Due to this bath-assumption all coherent phonons 〈b〉vanish and only phonon-densitiesnq = 〈b†

qbq〉 are non zero. As in section 13.2.2 the first non-vanishing order that causes luminescenceis in F2g2. The resulting system of equations is (note that in the second equation only one exemplarypart involving phonon emission is shown)

∂〈c†kσckσ〉 = −2 ∑

abνIm(Fνkσ

ab 〈bkσa†νaaνb〉 (13.56)


0 2 4 6 8 10ΩΩpl

[email protected]

itsD

Figure 13.4: Plasma luminescence at ζ = 3.5nm and n0 = 10−5nm−3. The plasma frequency is fpl =0.9 THz. Different temperatures are shown: T = 10000K (solid line), T = 2000K (dashed line) andT = 300K (dash-dotted line)

10-6 10-5 10-4

n0 @nm-3 D0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ωm

axΩ pl

DΩ

Ω pl

Figure 13.5: Plasma luminescence at ζ = 30nm and T = 300K. The frequency of the maximum (solidline) and the FWHM of the spectrum (dashed line) are shown over the electron density n0.


0 200 400 600 800 1000T@KD

1

2

3

4

5

6

7

8Ω

max

Ω plD

ΩΩ pl

Figure 13.6: Plasma luminescence at ζ = 3.5nm and n0 = 10−5nm−3. The frequency of the maximum(solid line) and the FWHM of the spectrum (dashed line) are shown over the temperature T .

energy

wavenumber

photonemission

phononemission

Figure 13.7: In a quantum well energy band the interaction with the pononic system allows for themomentum and energy conservation to e fulfilled. The Bloch-electrons can emit photons.


ı~∂t〈ckσa†νcaνd〉

∣∣partial =~(ωkσ + ενd − ενc)〈ckσa†

νcaνd〉 (13.57)

+~∑q|gq|2ζ(ωkσ + ενd−q‖ − ενc +ωl)nq

(Fνkσ

cd∗( f νν

cd−q‖ − f ννcd )ζ(ενd−q‖ − ενd +ωl)

+Fνkσc−q‖d−q‖

∗( f νν

c−q‖d−q‖ − f ννcd−q‖)ζ(ενc−q‖ − ενc +ωl)

−Fνkσcd

∗f ννcd ζ(ωkσ + ενd − ενc)

+Fνkσc−q‖d−q‖

∗f ννc−q‖d−q‖ζ(ωkσ + ενd−q‖ − ενc−q‖)

)

+~∑q|gq|2ζ(ωkσ + ενd− ενc+q‖ +ωl)nq

(Fνkσ

c+q‖d+q‖∗( f νν

c+q‖d+q‖ − f ννc+q‖d)ζ(ενd − ενd+q‖ +ωl)

+Fνkσcd

∗( f νν

c+q‖d − f ννcd )ζ(ενc − ενc+q‖ +ωl)

−Fνkσcd

∗f ννcd ζ(ωkσ + ενd − ενc)

+Fνkσc+q‖d+q‖

∗f ννc+q‖d+q‖ζ(ωkσ + ενd+q‖ − ενc+q‖)

)

with f νµcd = f ν

c (1 − f µd ) = 〈a†

νcaνc〉(1 − 〈a†µdaµd〉) being Pauli-blocking terms. Using the Markov-

approximation again leads to the final result for the photon density

∂t〈b†kσbkσ〉 = 2π ∑

cqν|gνν

q |2|ζ(ωkσ)|2|Fνkσqq |2 (13.58)

((1+nq) f νν

c+q‖c δ(ωkσ + ενc − ενc+q‖ +ωl)

+nq f ννc+q‖c δ(ωkσ + ενc − ενc+q‖ −ωl)

).

13.2.3.1 Results

The resulting luminescence spectra of a GaAs quantum well embedded in a AlGaAs bulk semicon-ductor is shown in Fig. 13.8.

The dispersion-relation of the AlGaAs-TO-phonons are depicted in Fig. 13.9. The TO-phononsfeature three dispersion-branches which are separated by phononic band gaps. These three branchesand the band-gaps are a result of the coupling induced anti-crossing of two GaAs-like and two AlAs-like dispersion-branches. As the interaction of the electronic system with the phononic system is theprime mechanism enabling the luminescence of the QW these phononic-band gaps are clearly visiblein the luminescence spectrum.


0 20 40 60 80 100E(ω) [meV]

S(ω

)[ar

b. u

nits

]

T=400K

T=50K

Figure 13.8: GaAs-AlGaAs quantum well luminescence at n0 = 1.2∗1013m−2. Different temperaturesranging from 50K to 400K are shown. Tho two gaps are due to the phonon-band gap of the coupledphononic systems of the GaAs quantum well and the AlGaAs bulk semiconductor.

kHΩL@arb. unitsD20

40

60

80

100

ÑΩ@meVD

Figure 13.9: Dispersion relation of the transversal optical phonons of Al1−xGaxAs. The anti-crossingof two GaAs-like branches and two AlAs-like branches causes the formation of three dispersion-branches with two separating phononic band gaps.

117

Part III

Propagation in Nonlinear Materials

119

In this final part the material dynamics (part II) and the light propagation (part I) are combined toinvestigate the linear and nonlinear properties of different semiconductor setups.

In the chapter 14 a multiple quantum well is considered as an example for an active photoniccrystal. The linear and nonlinear propagation properties are investigated analytically and numerically.Novel applications are proposed: ultrafast optical switching on a sub picosecond timescale (section14.3.4) and the utilization for stopping and storing of light pulses of arbitrary shape and intensity(section 14.5).

In chapter 15 the propagation of light pulses in a bulk semiconductor is analyzed for two regimes:The phase dynamics for resonant propagation is considered as it only recently has been experimentallypossible to fully characterize ultra short laser pulses including the phase. For near-resonant excitationthe validity of different theoretical approaches is evaluated as it is not a priori evident if theoriesusually applied for resonant or far off-resonant excitation are valid in this transition regime. Thenumerical calculations are compared to experimental data.

121

Chapter 14

Multiple Quantum Wells

14.1 Introduction

In this chapter an example for a semiconductor hetero-structure is considered: the multiple quantumwell (MQW). A MQW consists of N equally spaced (In,Ga)As quantum wells embedded in a dielectricGaAs-bulk semiconductor. The width of a single quantum well is about 8 nm. The spacing is chosento be d = λex/2nb with λex being the wavelength of the exciton-resonance of the quantum well excitonand nb the background refractive index. In GaAs this corresponds to an interwell spacing of d =113.7nm. The spacing is great enough to prevent direct Coulomb interaction between the quantumwells. Therefore the individual quantum wells couple only via the light field [55, 56, 57]. The structureis depicted in Fig 14.1.

Plane waves propagating perpendicular to the QWs are assumed. In the numerical simulations thereflected and transmitted signals are calculated by numerically solving the semiconductor Maxwell-Bloch equations (SMBE)[58, 6] (cp. section 7.1.2) using the Finite-Difference-Time-Domain method(FDTD, chapter 3) [20, 21]. The parameters (cp. appendix G ) used in the calculations are chosen toreflect typical experiments.

First the linear properties of a MQW are investigated: the linear spectrum is explained by theformation of a superradiant mode and Bragg-reflection and the influence of radiative damping is dis-cussed.

For stronger excitation an analytical solution for infinite MWQ-structures is presented giving riseto soliton-like propagation. For finite length structures the Pauli-nonlinearity is discussed and theeffect of adiabatic driving is explained and a possible application for ultrafast switching is devel-oped. The transition from linear polariton beating to suppression of this beating and finally to pulsecompression for high excitation is elucidated. The influence of the substrate of the sample and Rabi-oscillations is discussed. Finally the MQW is proposed as a device for stopping and storing arbitrarylight pulse.

14.2 Linear Optics

14.2.1 Superradiant Mode

In chapter 2 a formalism for solving the electromagnetic field inside a MQW has been developed. Byneglecting retardation effects of the pulse envelope in equation (2.21) an equation for the electromag-netic field at arbitrary positions inside the structure was derived:

122 14. MULTIPLE QUANTUM WELLS

N

It

I0

Ir

d =l

ex

2nb

detector

Figure 14.1: Bragg-resonant multiple quantum well.

E+n (t − z

c) = E+

0 (t − zc)+

iωg

εc

n

∑m=1

φmP2Dm (t) (14.1)

E−n (t +

zc) = E−

N (t +zc)+

iωg

εc

N

∑m=n+1

φmP2Dm (t) . (14.2)

For a spacing of d = λex/2nbthe phase factor φmbecomes real:

φm = φ = eikd = eikλex/2nb = −1 . (14.3)

The resulting field at the places of the quantum wells can now be written as

En(t) = (−1)n(E+0 (t)+ E−

N (t))+iωg

εc

N

∑m=1

(−1)n+mP2Dm (t) . (14.4)

For even and odd n it is

Eeven(t) = (E+0 (t)+ E−

N (t))+iωg

εc

N

∑m=1

(−1)mP2Dm (t) = −Eodd(t) . (14.5)

The field at the places of quantum wells alternates in sign but is otherwise identical throughout thestructure. Considering the equations of motion for the polarization (7.16), it can be noted that thesubstitution E →−E results in P →−P and therefore

P2Deven(t) = −P2D

odd(t) . (14.6)

This leads to the final equation

Eeven(t) = (E+0 (t)+ E−

N (t))+iωg

εc

N

∑m=1

P2Deven(t) (14.7)

= (E+0 (t)+ E−

N (t))+iωg

εcNP2D

even(t) . (14.8)

14.2. LINEAR OPTICS 123

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20 30 40

abso

rptio

n

detuning [meV]

N=1N=10N=20

Figure 14.2: Superradiant linewidth of an MQW with 1, 10 and 20 wells.

In the assumed circumstances of neglectable retardation inside the structure (i.e. only a small numberof quantum wells) all quantum wells behave collectively like one well with N-fold increased couplingto the electromagnetic field and therefore a N-fold decrease in radiative damping and correspondingdecrease in lifetime [18]. This behavior is known from clusters of atoms as superradiance: The atomsare within a distance smaller than the resonance wavelength. If one atom spontaneously emits aphoton the rest of the cluster is dragged along by stimulated emission. The lifetime/linewidth of sucha cluster is altered accordingly. In the case of a MQW the formation of such a superradiant mode ismade possible by the constant phase relation between the individual quantum wells. In Fig 14.2 thequalitative behavior of the resonance line of a MQW is depicted. The linewidth grows linear with thenumber of quantum wells while the lineshape stays Lorentzian.

14.2.2 Radiative Damping

It turns out that the radiative damping in a Bragg-resonant MQW is much stronger than any othermechanism of dephasing (e.g. section 11) [16]. Transforming the results of chapter 2 into frequencyspace

E+n (ω) = E+

0 (ω)− 12εc ∑n

m=1 iωφ?mP2Dm (ω) (14.9)

E−n (ω) = E−

N (ω)− 12εc ∑N

m=n+1 iωφmP2Dm (ω) (14.10)

and inserting them into the Wannier-expansion of the polarization (chapter 8) yields

P2Dn (ω) = χ(ω) ( φnE+

0 (ω)−φn 12εc

n

∑m=1

iωφ?nP2Dm (ω) (14.11)

+ φ?nE−N (ω)−φ?n 1

2εc

N

∑m=n+1

iωφnP2Dm (ω)) . (14.12)


Introducing a matrix of phase factors (analogous to [16])

Dnm =

φnφ?m for m ≤ nφ?nφm for m > n

= eikl l|n−m| (14.13)

and denoting the m-dependent entities as vectors , e.g.

φ =

φ1

φ2

...φN

, (14.14)

the polarization can be written as

P2D(ω) = (1+iωχ(ω)

2εcD)−1χ(ω)(φE+

0 (ω)+φ?E−N (ω)). (14.15)

By inserting (14.15) into (14.9) and (14.10)

E+N (ω) = E+

0 (ω)− iω2εc

φ?(1+iωχ(ω)


0 (ω)+φ?E−N (ω)) (14.16)

E−0 (ω) = E−

N (ω)− iω2εc

φ(1+iωχ(ω)


0 (ω)+φ?E−N (ω)) (14.17)

the absorption α(ω) can be computed. After a few basic transformations the absorption is obtained as

α(ω) =|E+

0 (ω)|2 −|E+N (ω)|2 −|E−

0 (ω)|2|E+

0 (ω)|2 (14.18)

=ωεc Im(χ(ω))

|1+ iωχ(ω)2εc |2

.

Inserting (8.16) and the ground state wave function of the exciton [6],

φ2D1s (0) =

√1

πa20(

13)3

, (14.19)

the absorption is

α(ω) =2γ0Γ

~2(ω−ωg)2 +(γ0 +Γ)2 (14.20)

with the radiative damping constant

Γ =9d2

1,2ωg

2εcπa20

. (14.21)

For one quantum well the radiative damping constant is Γ ≈ 0.06 meV and therefore one order ofmagnitude weaker than the measured dephasing constant γ0 ≈ 0.5 meV [39] (cp. appendix G) whichincludes e.g. phonon interaction and impurity scattering. But as was shown above (14.2.1) the cou-pling of a MQW to the electromagnetic field scales with N and therefore also Γ ∝ N. For N = 10 theradiative damping Γ is already in the same order of magnitude as γ0. For the experimentally exam-ined MQWs with 60 and 200 quantum wells, the radiative damping is the most important dephasingprocess as it is at least one order of magnitude faster than all other dephasing processes.


0

0.5

1

1.5

-30 -20 -10 0 10

trans

mis

sion

detuning [meV]

N=1N=10N=60

N=100N=200

Figure 14.3: Photonic band gap for different quantum well numbers. The absorption line is broadenedand squared out by superradiant coupling of the individual quantum wells.

14.2.3 Photonic Band Gap

For structures greater than a few 10 quantum wells the retardation of the pulse envelope becomesimportant. Due to multiple reflection and destructive interference inside the structure the linear ab-sorption is no longer Lorentzian in shape[59]. In the limit of large N, the reflectivity reaches unityand the square profile of an active photonic band gap is formed (Fig. 14.3) [19, 60, 18, 61]. Typicalbandwidths are in the order of 10-20 meV.

The resonant band gap itself is similar to a passive dielectric band gap of a Bragg-reflector causedby multiple reflection at the periodic surfaces and interference effects (analytic solutions are knownfrom atomic systems [62]). In the linear regime a MQW is a one dimensional photonic crystal[63, 59,64]. However, in contrast to passive dielectrics, the excitonic resonance which forms the band gap,can be directly influenced by the strength of the light field because of strong optical nonlinearities dueto Coulomb many body and other interaction effects, and Pauli-blocking. Such nonlinearities maylead to exciton saturation resulting in a breakdown of the band gap[58]. A MQW with many quantumwells spaced λ/2 is therefore called a Resonant Absorbing Bragg-Reflector (RABR).

14.2.4 Deviation from Bragg-Periodicity

As the photonic band gap is a result of interference between (multiple) reflected light inside the struc-ture, the phase relation of the incident and reflected light is of utmost importance. Therefore evensmall deviations from the Bragg-condition lead to a breakdown of the superradiant mode and there-fore to a breakdown of the photonic band gap (Fig. 14.4). This sensitivity on the exact fulfillment ofthe Bragg-condition is one of the biggest experimental challenges. The samples are usually grown ina wedged shape enabling the experimenter to tune the quantum well distance to Bragg-condition byaltering the spatial position of the incident laser (Fig. 14.5).


0

0.5

1

1.5

-30 -20 -10 0 10

trans

mis

sion

detuning [meV]

d=0.40λd=0.49λd=0.50λd=0.51λd=0.60λ

Figure 14.4: N=200, Breakdown of the photonic band gap due to derivation from Bragg-condition

Figure 14.5: Actual buildup of the samples used in the experiments [65]. The wedged shape of theMQW enables the tuning into Bragg-condition by altering the spatial position of the sample.


Theory Experiment

Figure 14.6: Propagation of low-intensity 560fs-pulses resonant to the exciton through the MQWstructure for Bragg-periodic and detuned interwell spacing. (a) Normalized transmitted intensities|E(t)|2 and (b) |E(∆λ)|2. (c), (d) Experiment on 1s-hh-exciton[61]: (c) Normalized cross-correlationtraces and (d) transmitted spectra. In experiment and theory, the lowest curves represent the inputpulses.

14.2.5 Polariton-like Beating

At low light intensities, the structure exhibits a temporal propagation beating of the transmitted pulseenvelope similar to the temporal polariton beating observed in bulk semiconductors[66] (cp. section15.1.1). This beating is strongly dependent on the optical dephasing time T2 which is dominated bythe radiative interwell coupling (cp. section 14.2.1).

Clearly observable is a pulse splitting into two distinct pulse components separated by approxi-mately 770 fs (Fig. 14.6). The full dip in between appears about 410 fs after the leading pulse peak.Due to the steep edges of the dip, both components exhibit considerable reshaping and narrowing withrespect to the input pulse envelope. The corresponding normalized spectra are shown in Fig. 14.6(b).Here, the pronounced dip in the spectrum transmitted through the sample is caused by the enhancedreflection from the Bragg-coupled QWs and minor absorption contributions.

The linear pulse splitting in the time domain can be explained by interference effects[67]: The ex-citonic resonance is vastly broadened due to superradiant coupling (section 14.2.1) inside the Bragg-structure. Cutting out this photonic band gap from the input pulse spectrum leaves two spectral wingspropagating freely in the dispersive structure. These two spectral components translate into a prop-agation beating in time with phase shifts of 180 degrees between the temporal pulse components(Fig. 14.7). Qualitatively, the effect is similar to the polariton propagation beating known from bulksemiconductors (cp. section 15.1.1)[66, 59]. However, one finds the temporal pulse envelope, espe-


0

1

2

3

4

0 1 2time [ps]

normalized field intensityphase

Figure 14.7: Linear polariton-like beating: between the two pulse components is a phase jump of 2πwhich is a clear signature of a beating phenomenon.

cially the amplitude ratio of the two pulse components, to be significantly different from the propa-gation beating exhibited by MQW structures with comparable length but with an arbitrary interwellspacing [68]. In those structures no photonic band gap is formed because of the lack of superradiantcoupling between the individual quantum wells.

Parameters that critically determine the observed propagation beating are the pulse length τ p, theoptical dephasing time T2, and the propagation distance Nd. A greater number of Bragg-periodicQWs N will result in a wider photonic band gap, exhibiting a smaller dephasing time T2. A shorterpulse length τp means a larger spectral width of the pulse. If τp is short with respect to T2/2 (theoptical dephasing time T2 is defined with respect to the spectral HWHM, whereas the pulse length τ pis defined with respect to the spectral FWHM), a significant amount of spectral components will betransmitted on both sides of the photonic band gap. The interference of these spectral wings duringpropagation through the dispersive structure will lead to a pronounced temporal beating with severalbeat periods. On the other side, if τp is long with respect to T2/2, almost the entire pulse spectrumwill overlap with the photonic band gap. The transmitted pulse will be damped exponentially, andsplit-off pulse components will be merely weakly pronounced.

14.2.5.1 Comparison to the Experiment

The influence of the optical dephasing time T2 on the pulse propagation characteristics can be ex-perimentally studied directly using the wedged design of the MQW sample (Fig. 14.5). Detuning thestructure from the Bragg-condition weakens the radiative interwell coupling, so that the fast superradi-ant response of the optical polarization is suppressed, the photonic band gap shrinks, and nonradiativecontributions become dominant. In effect, the T2-time can be tuned from approximately 400 fs forthe N = 60 MQW structure with Bragg-period (compare Ref. [69]) to a few picoseconds duration forthe decoupled wells. An upper limit of T2 ≈ 10 ps for the corresponding (In,Ga)As/GaAs single QWshas been determined from the radiative dephasing rate measured on a series of Bragg-structures withdifferent numbers of QWs N (compare Ref. [69]) and the nonradiative contributions deduced fromthe decay of the subradiant modes observed in degenerate four-wave mixing (DFWM) spectroscopy.Figures 14.6(c) and 14.6(d) show experimental results[61] of the low-intensity propagation of 560 fspulses resonant to the 1s-hh-exciton for increasing detuning from the Bragg-condition. The temporal

14.3. NONLINEAR OPTICS 129

cross-correlation traces are plotted in the left column, whereas the right column illustrates the trans-mitted spectra behind the sample. The lowest curves characterize the input pulse as a reference. Allcurves have been normalized for comparison. For Bragg-periodic arrangement nbd = 0.5 λex (uppercurves in Fig. 14.6), pulse splitting in time with two pulse components separated by a full coherent dipare found. In consequence of the enhanced radiative dephasing, the overall pulse envelope declinesrapidly after the leading pulse peak so that only one beat period emerges from the coupled QWs. Ina phase-sensitive XFROG-experiment the expected phase shift of about 180 degrees between the twopulse components can be verified. The observed spectral dip is caused by the asymmetrically broad-ened reflection from the superradiant mode. Basically the wings of the input spectrum matched to theresonance are transmitted through the structure. For the gradual reduction of the interwell distance tonbd = 0.49 λex (going from top to bottom in Fig. 14.6), the measured cross-correlation traces demon-strate considerably altered pulse shapes. Due to the increased T2-time, the later pulse componentsuccessively grows in intensity at the expense of the earlier and becomes dominant already for a slightdetuning to nbd = 0.497 λex. At nbd ≈ 0.494 λex and even more clearly at nbd = 0.49 λex, two fullbeat periods are visible with the three pulse peaks delayed by about 790 fs and 1290 fs with respect toone another. No further change of the pulse envelope for interwell distances below nbd = 0.49 λex isobserved. As the influence of radiative coupling can almost be neglected in this case, the characteris-tics are qualitatively similar to the pulse distortions found by Kim et al. [68] for subpicosecond pulsestransmitted through MQW samples with an arbitrary interwell spacing but with comparable length.The opposite detuning from the Bragg-condition to nbd > 0.5 λex showed an analogous propagationbehavior with the same strong dependence on the superradiant coupling.

The measured spectra plotted in Fig. 14.6(d) are characterized by an extinction line that is con-tinuously narrowing with decreasing interwell distance. The transmitted wings of the input spectrumgrow in intensity and change their amplitude ratios due to the asymmetric breakdown of the reflec-tion profile starting from the blue edge [70]. The considerably smaller width of the extinction line atnbd = 0.49 λex indicates the suppression of the enhanced radiative polarization decay, which resultsin an increased dephasing time T2 and substantial nonradiative contributions.

The results of numerical simulation are depicted in Figs. 14.6(a) and 14.6(b) both temporally andspectrally for a detuning from the Bragg-condition to nbd = 0.49 λex. The employed semiconductorMaxwell-Bloch theory is capable of describing the observed propagation beating and the dependenceon the radiative interwell coupling in excellent agreement with the experimental data: the change in theamplitude ratios of the temporal pulse components as well as the development of a second beat period.The predicted delays of the three pulse peaks with respect to one another amount to 780 fs and 1000 fs,which is confirmed by experimentally determined values. The extinction line, which dominates thespectra, drastically narrows with the detuning to nbd = 0.49 λex. Simultaneously, the transmittedspectral wings exhibit an asymmetric growth in intensity that is qualitatively similar to the spectralbehavior shown in Fig. 14.6(d). The theoretical investigation demonstrates the strong influence ofsuperradiant coupling on the characteristics of subpicosecond pulses propagating resonantly throughMQW Bragg-structures. Especially, it is possible to study the impact of a gradually varied dephasingtime T2 in a model system with defined pulse length τp and nearly constant propagation distance.

14.3 Nonlinear Optics

In this section the propagation of strong light pulses is investigated. As the excitonic resonance whichforms the band gap, can be directly influenced by the strength of the light field because of strongoptical nonlinearities due to Coulomb many body and other interaction effects, the behavior differs


considerably from linear propagation.

14.3.1 Analytic Solutions for Long Structures

For an infinite extended MQW it is possible to derive an analytic solution for nonlinear excitation. Thissolution predicts soliton-like propagation of strong light pulses inside this structure [1, 71] similar toSIT (self induced transparency) in bulk semiconductor . However, to derive this solution one has touse the Wannier expansion in the first order. As is shown in chapter 8 and section 9.1, this expansionis only partially valid for nonlinear resonant excitation of an excitonic resonance.

14.3.1.1 Two Level System Equations of Motion

The first order Wannier expansion corresponds to a two level system model for the excitonic resonance(cp. section 8). The equation of motion for the coherence and electro/hole density [1] (section 7.5)

˙p = i12

Ω(1− f e − f h) (14.22)

f e/l = 2Im(12

Ω? p) (14.23)

are solved by Rabi-oscillations as shown in chapter 9.

14.3.1.2 First Order Wave Equation

The electromagnetic wave is split in a slowly varying envelope (SVE) and a fast oscillating part aswell as into a forward and backward propagating part[17] (analogous to chapter 2):

E(z, t) = E+(z, t)ei(kz−ωt) +E−(z, t)e−i(kz−ωt) + c.c. (14.24)

Inserting this ansatz into the wave equation (1.8) and neglecting the second derivative of the slowlyvarying envelope yields

(ik∂zE+ +iωc2 ∂tE+)ei(kz−ωt) +(−ik∂zE−+

iωc2 ∂t E−)e−i(kz+ωt) =

12εc2 ∂2

t P . (14.25)

A time-average over ∆t 1ω is performed from which the envelope is not affected:

(∂zE+ +1c

∂tE+)eikz +(−∂zE−+1c

∂tE−)e−ikz = − i2ωεc

〈eiωt ∂2t P〉 . (14.26)

The polarization for the localized two level systems can be written as (cp. 5.3.2)

〈eiωt ∂2t P〉 = = −ω2 ∑

ja†

0(t,r j)a1(t,r j)d0,1δ(r− r j) . (14.27)

Inserting (14.27) into (14.26) and averaging over one wavelength results in:

1λ

Z

λdz(∂zE+ +

1c

∂tE+)e2ikz +(−∂zE− +1c

∂tE−)

=1λ

Z

λdz

i2ωεc

ω2 ∑j

a†0(t,z j)a1(t,z j)d0,1δ(r− z j)eikz .

(14.28)


The envelope is nearly constant over one wavelength, which leads to

−∂zE−+1c

∂tE− =iωσd0,1

2λεc ∑j

a†0(t,z j)a1(t,z j)eikzi δ(z− zi) (14.29)

with the spatial oscillator density σ and the step function

δ(z) =

1 for z ∈ [− 1

2 ,+ 12 ]

0 otherwise. (14.30)

A similar equation holds true for E+:

∂zE+ +1c

∂tE+ =iωσd0,1

2λεc ∑j

a†0(t,z j)a1(t,z j)e−ikzi δ(z− zi) . (14.31)

Introducing the coherence p = a†0a1 and the Rabi-frequency Ω = d

~E yields

(±c∂z +∂t)Ω± =iωσd2

0,1

2λε~∑

jpe∓ikzi δ(z− zi) . (14.32)

The quantum wells are spaced at λ/2 therefore the phase factor vanishes and one can perform a spatialaveraging over a domain with exactly one quantum well. With this averaging the MQW property of thestructure is lost and the equations can no longer be distinguished from bulk equations. It is thereforenot surprising that the calculation will arrive at the same results as a bulk calculation.

(±c∂z +∂t)Ω± =iωσd2

0,1

2λε~p . (14.33)

After a substitution similar to section 9 the propagation equation reads

(±c∂z +∂t)Ω± = τ−2c p (14.34)

with

τc =

√8πε~c

ω2σd20,1

. (14.35)

14.3.1.3 Sine-Gordon-Equation

Substituting the electrical field in 9.5 according to the transformations

Ω = Ω+ + Ω− (14.36)

Ω′ = Ω+ − Ω− (14.37)

and adding (subtracting) the two equations yields

c∂zΩ′ +∂tΩ = 2τ−2c p(z, t) (14.38)

c∂zΩ+∂tΩ′ = 0 . (14.39)


Figure 14.8: Soliton on the Scott Russell Aqueduct on the Union Canal, Heriot-Watt University,12 July 1995. [73]

The first time derivative of (14.38) and the first space derivative of (14.39) are

c∂z∂tΩ′ +∂2t Ω = 2τ−2

c ∂t p(z, t) (14.40)

c∂2z Ω+∂z∂tΩ′ = 0 . (14.41)

Inserting these derivatives into each other

−c2∂2z Ω+∂2

t Ω = 2τ−2c ∂t p(z, t) (14.42)

and integrating over time while introducing the pulse area θ (9.6) leads to

−c2∂2z θ(z, t)+∂2

t θ(z, t) = 2τ−2c p(z, t) (14.43)

where the polarization can be substituted by the known solution (9.5). The result

c2∂2z θ−∂2

t θ = 2τ−2c sinθ (14.44)

is the well known Sine-Gordon-equation (e.g. [1, 72]).

14.3.1.4 Solutions of the Sine-Gordon-Equation

Solutions of the Sine-Gordon-equation (14.44) were first observed by John Scott Russell in the UnionCanal in Edinburgh, Scotland in 1834 (Fig. 14.8). The wave he observed did not disperse for overa mile. These solitary waves or solitons are of great importance for modern communication as theinformation transmission in fiber-optics are nowadays completely based on solitons.

Now the simplest solitonic solution is presented. Solutions of higher order can be derived byBäcklund-transformation[72, 74]. One known solution is

θ(X ,T ) = 4arctan(eT−uX√

1−u2 ) (14.45)


0 5

10 15space 0

5

10

time 0 1 2field

Figure 14.9: Solitary wave as solution of the Sine-Gordon-Equation.

with u ∈ [0,1] being the relative propagation velocity. The time and space coordinates are renormal-ized:

T = t√

2τ−2c i (14.46)

X = z

√2τ−2

c

ci . (14.47)

Undoing all substitutions leads to the solution for the electrical field

Ω(z, t) = 2αcosh−1(α(t − uc

z)) (14.48)

with the constant α = i√

2τ2

c(1−u2). It has to be u > 1 for the solution to be real. The solution (14.48)

presents a solitary wave propagating with the velocity cu inside the structure (Fig. 14.9). The pulse

width, which can take every real value, depends on the velocity u. The pulse area of solution (14.48)is

θ(t) =

Z ∞

−∞dt ′2αcosh−1(α(t − u

cz)) = 2π (14.49)

and independent from α and u. 2π is exactly the pulse area which induces a full Rabi-oscillation inthe individual two level systems as shown in chapter 9 [6]. The derived solution (14.48) will be usedas input pulse in the numerical simulations of MQWs as it also is a good approximation of the actualpulse shape used in the experiment.

14.3.1.5 Validity of Derivation

In the derivation of the Sine-Gordon-Equation (14.44) three approximations were made which couldinvalidate the reached results for real world application: First, the excitonic resonance was expandedin a Wannier-basis and only the first order was taken into account. As was shown in chapter 8 the firstorder Wannier-expansion is not valid for strong resonant excitation. But the soliton solution (14.48)is predicted in exactly this high intensity regime (14.49). Second, the polarization is averaged overa period of the structure and thereby the character of the hetero-structure is lost. Due to the strict


phase relation of the QWs this approximation can be justified but would be, just as the superradiativecoupling (cp. section 14.2.4), extremely sensitive to the smallest deviation from the exact spacinginside the structure. Third, an infinite extended structure is assumed. As shown in section 14.5 theinjection of a light pulse into a MQW can be a problem.

In section 14.3.8 it is shown that short light pulses do indeed propagate like solitons. Howeverthis holds not true for long pulses. In Fig. 14.24 it is shown that long pulses are reflected and do notpropagate as predicted by this calculation as the spatial pulse length is in the order of the length of theMQW-structure.

14.3.2 Pauli-Blocking Nonlinearity

In the semiconductor Bloch equations (7.16) the Pauli-exclusion principle (no two identical fermionsmay occupy the same quantum state simultaneously) is enforced by the Pauli-blocking term (1−2 f k)in the term governing the coupling to the light field iΩ(1− 2 fk). Beginning with a system were theelectron density in the conduction band equals zero the Pauli-blocking term is (1−2 fk) = (1−2 ·0) =1. In this situation the incoming light is absorbed and electron density in the conduction band is buildup. The blocking term decreases and thereby decreases the probability of further light being absorbed.At the same time the probability of stimulated emission increases. At fk = 1

2 , i.e. half of the electronsare in the conduction band, the probabilities of absorption and stimulated emission are at balance.The blocking term gets (1 − 2 fk) = (1 − 2 · 1

2) = 0. The polarization is decoupled from the lightfield. In a MQW the superradiant mode would break down and the individual QWs decouple. Aweak pulse would be able to propagate such a structure much unhindered. This phenomenon is alsocalled bleaching of the resonance. For further rising fk the semiconductor reaches a regime werestimulated absorption dominates as the blocking term changes sign. For fkreaching 1 the blockingterm becomes (1− 2 fk) = (1− 2 · 1) = −1. No further light can be absorbed as the probability ofan electron transition into the conduction band is 0 (Pauli exclusion principle). The system is fullyinverted and the only allowed process is emission.

14.3.3 Adiabatic Driving

The effect of adiabatic driving [75] is somewhat similar to adiabatic following (cp. 10) but is in theresonant excitation regime. The time scale of the optical response in nano-optical systems near res-onance is limited by the response time of the structured material and the duration of the excitationpulse. If all intrinsic relaxation processes in the material are fast compared to the duration of theexcitation pulse, the optical pulse adiabatically drives the material variables and determines their tem-poral response.[1] So when the MQW is excited by a pulse with a temporal duration longer than theresponse time of the photonic lattice the polarization and density dynamics are adiabatically driven bythe pulse [76]. The lattice response time is given by the inverse HWHM of the band gap (about 100 fsfor 200 QWs) and must be much shorter than the applied pulse duration. In Fig. 14.10 the electron dy-namics inside different QWs during the propagation of a 1.6 ps pulse is depicted. Typical bandwidthsare on the order of 10-20 meV, compared to 1-2 meV for the spectral width of a switching pulse sothe requirements are met. As can be seen, the 2π- and 8π-pulses induce a temporal density dynamicswhich directly follows the pulse envelope (Fig. 14.15). For intensities greater than 5π (peak Rabi-frequency larger than the lattice band width) weak Rabi-oscillations occur in the nonlinear materialresponse and the dynamics of the density can no longer be adiabatically driven by the pulse envelope.During the action of the pulse, Pauli-blocking of the Coulomb renormalized field [58](section 14.3.2)reduces the strength of the excitonic resonance, thus weakening the radiative coupling between the


0

0.001

0.002

0.003

420420

elec

tron

dens

ity [1

/a02 ]

time [ps]

2 π (x4) 8 π

1.QW100.QW200.QW

Figure 14.10: Electron density in the 1st, 100th and last (200th) QW for different incident fieldstrengths. For nonlinear excitation, the system dynamics occur simultaneously with the pulse en-velope. For strong excitation (8π), the populations exhibit Rabi-oscillations.

QWs. In consequence, the photonic band gap collapses. After the pulse, the band gap recovers.

14.3.4 Ultrafast Switching

The effect of adiabatic driving (section 14.3.3) can be utilized for ultrafast switching of a secondpulse[75]: The influence of the induced carrier dynamics on the photonic band gap[77] is investigated.A pump-probe setup for cross-linearly polarized light pulses is simulated. A weak, spectrally broad(FWHM 14.4 meV = 126 fs) probe pulse experiences the nonlinear band gap dynamics induced bya strong pump pulse at different time delays. The investigated signal S(τ) is chosen to be the ratio ofthe energy of the reflected probe pulse (Ere f ) to the incident probe pulse (Eprobe) energy computed fordifferent pump-probe delays τ

S(τ) =

R |Ere f (t)|2R

|Eprobe(t)|2dt . (14.50)

A suppression of the photonic band gap by the pump induced density dynamics is shown by a drop inthe reflectivity of the weak probe pulse (enlarged transmission). In addition, different detunings of thepump pulse with respect to the resonance are investigated: excitation below (-8 meV), inside (0 meV)and above (14 meV) the photonic band gap (Fig. 14.22).

In Fig. 14.12(top), the integrated reflected probe intensities are shown as a function of the delayτ between pump and probe pulses. Pump pulses with the relatively large detuning -8 meV belowthe center of the band gap do not significantly influence the reflection of the sample. However, forexcitation inside the photonic band gap (0 meV), the reflectivity drops noticeably and recovers on thetime scale of the pump pulse due to the transient bleaching of the exciton resonance. For an excitation14 meV above the photonic band gap (near the semiconductor band edge), electron-hole populationsare excited due to the overlap of the pump pulse with the interband absorption spectrum. Due to thelack of the superradiant coupling of the unbound interband transitions, their phase coherence is lost


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-10 -5 0 5 10 15 20

elec

trica

l fie

ld [a

rb. u

nits

]

detuning [meV]

pump -8 meVpump 0 meVpump 14 meV

probe pulselinear reflection

Figure 14.11: Spectra of pump pulses located at different energies with respect to the excitonic reso-nance are used to investigate the influence of the induced carrier dynamics on a weak probe pulse.

and the process cannot be reversed during the switch off of the optical pulse. Therefore, the opticalband gap does not recover on the time scale of the pulse. One signature of the model is that forexcitation above the gap, the normalized reflected intensities are slightly greater during the buildupperiod of the pump pulse than with out pulse. This results from a short period of gain during the onsetof the pulse. The oscillations in the reflectivity visible at later times are due to beating between theinsufficiently numerically resolved continuum states and are purely artificial. The magnitude of thenonlinear band gap suppression of 15% depends on the intensity and shape of the pump pulse. Forstronger pulses (in the order of 10 π) complete suppression can be realized.

The obtained theoretical results are in full compliance with reflection measurements on MQWsamples [77] (Fig. 14.12, bottom). The experiments were conducted on a N = 200, In0.04Ga0.96As/GaAswedged MQW structure (DBR28) with a pump-probe setup in reflection geometry. The observed in-tegrated reflection is not influenced by excitation below the band gap. If excited in or above the bandgap, a significant drop in reflectivity is observed. This drop recovers instantaneously if the sample isexcited inside the band gap whereas it stays at a low level for excitation above the band gap. All theseeffects are reproduced in every spectral excitation regime by the theoretical approach (Fig. 14.12,top). As a potential application, the observed effect could be utilized for ultrafast optical switchingwith switching times down to the response time of the lattice (100 fs).

14.3.5 Suppression of Polariton-like Beating

From the resonant propagation of low-intensity pulses through MQWs with arbitrary interwell spac-ing d, it is known (section 14.2.5) that those structures may exhibit distortions on the transmittedpulse envelope similar to the temporal polariton beating observed in bulk semiconductors[66] (cp.section 15.1.1). These distortions were found to depend on the total thickness Nd and the dephas-ing time T2. Especially when exciting the structure with an intense pump pulse before the arrivalof the propagating probe pulse – thus generating an incoherent electron-hole density – the beatingcould effectively be suppressed. However, the situation is different if the transition from the linear to


1

0.8

1

0.8

-6 -4 -2 0 2 4 6 8

norm

aliz

ed re

flect

ion

of p

robe

pul

se

pump-probe-delay [ps]

theory

experiment

below gapinside gapabove gap

Figure 14.12: Integrated probe pulse reflection[75]: A pump pulse (1π) inside or above the photonicband gap suppresses the photonic band gap. The gap recovers on the time scale of the pump pulsewhen excited inside the photonic band gap. For excitation above the gap it does not recover fornanoseconds. Top: theoretical calculations. Bottom: Measurements[77] with a 4 µJ, 1.6 ps FWHMpulse.


the nonlinear excitation regime is investigated by an intensity increase of the propagating pulse itselfwhen excitation-induced correlations occur among coherently excited excitons[78].

In an intermediate intensity regime, the Pauli-blocking nonlinearity (cp. section 14.3.2) leads togradual suppression of the photonic band gap and vanishing of the linear propagation beating.

The semiconductor Maxwell-Bloch theory in Hartree-Fock-approximation for the Coulomb-interactionpredicts the observed pulse splitting and its gradual suppression for increasing pulse area up toΘ = 1.0 π in agreement with the experimental data (Fig. 14.13). Even the asymmetric spectral behav-ior with the growing red wing of the transmitted input spectrum is found in the experiment. For pulseareas beyond Θ = 1.0 π, the temporal beating phenomenon has completely vanished. However, theenvelopes of the propagated pulses plotted in Fig. 14.13(a) exhibit considerable pulse reshaping: Thepulse peak shifts to later times while the initial pulse component declines in intensity. At Θ = 2.0 π,the input pulse is compressed from 650 fs to about 450 fs duration. The associated spectrum reveals aslight red-shift and marginal asymmetric broadening.

Comparing experimental and calculated curves in Fig. 14.13, excellent agreement in the prop-agation behavior of the pulses with I = 28 MW/cm2 and Θ = 1.5 π, 12 MW/cm2 and 1.0 π, and3.9 MW/cm2 and 0.5 πis found. With 2.3 and 3.1, the ratios of these intensities roughly correspond tothe squared area ratios that amount to 2.25 and 4, respectively. Assuming an intensity of 12 MW/cm2

for a 1 π-pulse with τp = 560 fs, we can give an estimation for the transition dipole moment dc,v:Since the measured intensity I = 1

2

√ε0/µ0 E2

0 and the pulse area Θ ≈ (dc,v/~)E0 τp, the dipole mo-ment amounts to dc,v ≈ 3.9 e nm. This value is in full accordance with the dipole moment dc,v = 4.2 eVused for the simulations and literature values for GaAs ranging from dc,v ≈ 3 eV to 6 eV. [79].

The transition from the resonant linear to nonlinear pulse propagation can be explained by thePauli-blocking nonlinearity (section 14.3.2). For an input pulse with area Θ ≤ 1.0π, the reflectionfrom the effective photonic band gap together with the absorptive contributions cause a splitting ofthe transmitted spectrum into two portions. The corresponding Fourier-transforms yield a beatingphenomenon in time. Initially, the temporal duration of the pulse is longer than the response timeof the MQW Bragg-structure. Therefore the pulse-induced temporal dynamics are quasi-stationaryand the population follows the pulse envelope adiabatically. Due to the Pauli-blocking nonlinearity,further interband transitions are prevented. Thus, for an increasing pulse area towards Θ = 1.0π, thedip in the transmitted spectrum is filled up and the temporal beating phenomenon gradually vanishes.As Pauli-blocking leads to a decoupling of the QW polarization from the propagating light field (thelight interaction term + iΩn(1− f e,n

k − f h,nk ) ) is reduced for increasing carrier densities f e/h,n

k ), theeffective interwell coupling decreases, starting at the first QW. In consequence, the photonic band gapis suppressed but quickly recovers after the pulse transit because of radiative decay of the populationinversion [77]. Thus, for pulse areas Θ > 1.0π, the pulse transmitted through the structure will havea temporal shape and spectrum more similar to the input characteristics. The deviations, such as timedelay and shortening with respect to the incident pulse, can be explained as follows: During the propa-gation there is a temporally delayed interchange of energy between the pulse and the excitonic system,i.e., excitation and reemission (decay and reestablishment of the band gap). This interchange causesthe temporal pulse delay. As the breakdown and recovery of the band gap appears instantaneouslywith a higher-order nonlinearity, temporal steepening and shortening of the transmitted pulse is ob-served. Of course, many-body interaction in the semiconductor system will cause deviations fromthis idealized description. Despite the similarity of input and transmitted pulses, pure solitons such asSIT-solitons (known from the dynamics of two-level systems) or gap solitons are not expected in sucha semiconductor system. Nevertheless, Rabi-flopping of the carrier density (section 14.4), coherentnonlinear long-distance propagation, and a high degree of this so-called self induced transmission


ExperimentTheory

Figure 14.13: (a) and (b) Numerical simulation based on the semiconductor Maxwell-Bloch equa-tions: propagation of 600 fs pulses resonant to the 1s-hh-exciton through a N = 60 Bragg-periodicMQW structure for increasing intensities. The normalized transmitted intensities |E(t)|2 and|E(∆λ)|2 are plotted at the left and at the right, respectively, for increasing pulse area. (c) and (d)Experiment[61]: The normalized cross-correlation traces and transmitted spectra are shown at the leftand at the right, respectively. The high-frequency modulations in the spectra are due to the steppedstructure of the reflecting pulse-shaper mask and interferences in the CCD-camera of the spectrometer.In (a) and (b), the lowest curves represent the input pulses as a reference.

have been predicted [76, 80] and shown recently for bulk semiconductors. [81, 82]For highly nonlinear propagation with pulse area θ = 2π, the theory also predicts pulse compres-

sion from 650 fs for the input pulse to about 450 fs and the corresponding spectral broadening. Thepulse compression results from a beginning single Rabi-flop at this intensity.

14.3.6 Pulse Compression

The much stronger pulse compression, the temporal modulations, and above all the wide spectralbroadening found in the experiment for the highest pulse intensity of I = 66 MW/cm2 (Figs. 14.14(a)and (b)) cannot be reproduced by the theoretical calculations involving only the MQW. This deviationis is explained by the contribution of near-resonant self-phase modulation (SPM, cp. section 1.9.2)in the GaAs substrate material [83]. As is shown in chapter 15 the propagation through the substratematerial in transmission geometry results in the observed pulse compression. Experiments performedon bulk GaAs showed similar nonlinear effects, thus giving proof for the strong influence of SPM.


Theory Experiment

Figure 14.14: High-intensity propagation of 560 fs input pulses (short-dashed lines) resonant to the hh1s exciton through the Bragg-periodic N = 200 MQW structure (solid lines). (a) Normalized calcu-lated intensities |E(t)|2 and (b) |E(∆λ)|2. (c) Normalized cross-correlation traces and (d) transmittedspectra.


1

0 2 4 0 2 4 0 2 4

el. f

ield

[arb

. uni

ts]

time [ps]

input transmitted reflected

linear pulse2.0 π pulse8.0 π pulse

Figure 14.15: Reflected and transmitted pulse shapes for increasing intensity. With increasing inten-sity the transmission increases and envelope modulations due to Rabi-oscillations occur. All pulsesare normalized to the respective input pulses maxima.

14.3.7 Pulse Propagation

Here the propagation of spectrally narrow (FWHM 1.6 meV = 1.14 ps) Gaussian pulses with increas-ing intensities is investigated and the reflected and transmitted signal is calculated. To classify thestrength of the pulse-induced nonlinearity, one can compare the peak Rabi-frequency Ω0 = E0dcv/~

of the time dependent Rabi-frequency of the pulse Ω0e−t2

τ2 with the width ∆ of the photonic bandgap. For the investigated sample (N = 200 quantum wells), the width of the band gap is found tobe 15 meV. For comparison, a pulse with 3 meV peak Rabi-frequency corresponds to a pulse area

Θ =R ∞−∞ dtΩ0e

−t2

τ2 of 1π (resulting in full inversion of a non-interacting two level system). Figure14.15 shows the transmitted and reflected intensities for different pulse areas. Figure 14.15 (input)shows the normalized electric field envelope at the sample entrance as a function of time and is givenfor comparison with the transmitted (Fig. 14.15 transmitted) and the reflected fields (Fig. 14.15 re-flected). With increasing input pulse area, the transmitted pulse exhibits pulse shortening and developsan asymmetric shape. For large input areas (8π), the transmitted pulse resembles more or less the inputfield. The reflected signal is weakened with increasing pulse area and the trailing edge is flattened.

14.3.8 Soliton-like Propagation

For highly nonlinear excitation, signatures of self-induced transmission due to Rabi-flopping and adi-abatic following of the carrier density are found. Fig. 14.16 shows the soliton-like propagation of ashort (10fs) 2π-pulse. The pulse can propagate the structure without alteration in form or intensity.This is in sharp contrast to the linear behavior were the spectrally broad pulse would expose a pro-nounced beating between the spectral components propagating above and below the photonic bandgap. The electron density and polarization in the individual QW are performing nearly perfect Rabi-oscillations even in this simulation of a two band semiconductor. This soliton-like propagation in incompliance with the predictions made in section 14.3.1. Long pulses however are reflected by the


0

0.1

0.2

0.3

0.9 1 1.1 1.2

field

inte

nsity

[arb

. uni

ts]

time [ps]

input pulsetransmitted pulse

Figure 14.16: A short (10fs) intense 2π-pulse propagates the N = 200 MQW without alteration inform or intensity.

0

0.01

0.95 1 1.05

k-sp

ace

inte

grat

ed [1

/a0]

time [ps]

densitypolarization

Figure 14.17: Electron density and polarization inside one QW during the propagation of a short(10fs) intense 2π-pulse. The electronic system performs a Rabi-flop (chapter 9).

14.4. RABI-OSCILLATION IN QUANTUM WELLS 143

0

0.5

1

0 10 20 30 40

0 5 10 15 20

el. f

ield

[arb

. uni

ts]

t/τ

time [ps]

τ=500fs inputreflection

transmission

Figure 14.18: A long (500fs) intense 2π-pulse is reflected by the N = 200 MQW structure.

MQW-structure (Fig. 14.18). This behavior violates the predictions of section 14.3.1 due to the finitenumber of quantum wells.

14.4 Rabi-Oscillation in Quantum Wells

The experimental and theoretical results of highly nonlinear propagation through the Bragg-resonantMQW structure (e.g. Fig. 14.14 and Fig. 14.10) show pulse reshaping and compression due to asingle Rabi-cycle and adiabatic following of the carrier density. To highlight the difference betweenthe dynamics of two-level systems and the MQW photonic crystal, Fig. 14.19 illustrates the band gapdynamics and carrier-density Rabi-flopping in the semiconductor material. The densities 1

A ∑k f e/hk (t)

in the final QW of the Bragg-periodic MQW structure are plotted with respect to the correspondinginput and transmitted field envelopes. For an input pulse area of Θ = 2.0 π (Fig. 14.19(a)), the carrierdensity follows the input pulse instantaneously, similar to a single Rabi-cycle, which here, however,is due to breakdown and recovery of the photonic band gap (section 14.3.2 and 14.3.4)[77]. For anarea of Θ = 10 π (Fig. 14.19(b)), we observe on top of the band gap dynamics five distinct Rabi-oscillations of the carrier density. Clearly visible are several modulations of the transmitted pulseenvelope that reach their maxima shortly after the peak densities, i.e., shortly after the onset of thecoherent nonlinear reemission from the final QW. Multiple pulse breakup, which is well known fromcoherent nonlinear pulse propagation in optically thick bulk semiconductors [81], does not developduring propagation through N = 60 QWs. However, the results are in good agreement with pump-probe experiments performed on MQWs with arbitrary interwell spacing (compare Ref. [84]).

14.5 Light Capturing and Storage in MQW

In this section the coherent nonlinear propagation properties of a phonic crystal consisting of Bragg-resonant quantum wells (or MQW) are calculated beyond slowly varying envelope approximation.


0.000

0.002

0.004

0.006

0.008

0.010

Nor

m. F

ield

Env

elop

e |E

(t)|

Car

rier D

ensi

ty (a

0-2)

Time (ps)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.000

0.004

0.008

0.012

0.016

0.020

(b)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 14.19: Carrier densities in the 60th QW (thick solid lines) and normalized field envelopes ofthe resonant 650 fs input (thin dashed lines) and transmitted pulses (thin solid lines) calculated for theBragg-periodic MQW structure. (a) Θ = 2.0 π and (b) Θ = 10 π.

14.5. LIGHT CAPTURING AND STORAGE IN MQW 145

New phenomena at the transition regime between transparency and total reflection are shown andanalytic solutions are presented.

The theory in this section circumvents the limitations of section 14.3.1 by explicitly taking intoaccount a MQW with a finite number of quantum wells. The derivation presented here does howeverrely on the Wannier-expansion of the semiconductor Bloch equations in first order (chapter 8) with alllimitations mentioned there and in section 14.3.1.5. But as the results presented here predict solutionsin the linear regime, where the Wannier-expansion is valid, it is an justifiable approximation.

14.5.1 Analytic Solutions

In this section the theory of linear and nonlinear light propagation and capturing inside the structureis developed. Solutions of Maxwell’s equations are constructed: first (as a reminder) a standing wavein free space and second a standing wave inside the Bragg-structure. In the second case the nonlinearinteraction between material polarization and electro-magnetic field has to be treated self-consistently.

14.5.1.1 Equations of Motion

The excitonic resonance of the individual quantum wells are considered in first order Wannier-expansionand the equations of motion for population f = 〈a†

cac〉 and coherence p = 〈a†cav〉 are given in section

7.1.1 as:

p = −iωg p+ iΩ(1−2 f ) (14.51)

f = i(Ωp? −Ω?p) (14.52)

with the Rabi-frequency Ω = dcv~

E .

14.5.1.2 Free Standing Wave

A standing wave with its knots at a distance of d = nλl/2, n ∈ N, i.e.

E(z, t) = E(z, t)12

Re(e−i(ωlt−kz) + e−i(ωlt+kz)) (14.53)

= E(z, t)cos(ωl t)cos(kz) (14.54)

with k = 2πλl

, is a valid solution of the free wave equation (no material interaction, i.e. no polarization):

(∂2z −

1c2 ∂2

t )E = 0 . (14.55)

Insertion of (14.54) into (14.55) yields

cos(kz)∂2z E(z)−2ksin(kz)∂zE(z) = 0 (14.56)

under the assumption of a stationary solution (∂2t E = ∂t E = 0). This differential equation is satisfied

by

∂zE1(z) = 0 (14.57)

∂zE2(z) = sec(kz)2 . (14.58)

Both lead to a constant envelope of the standing wave over the complete space (i.e. in free space onlyinfinitely extended standing waves exist).


14.5.1.3 Standing Wave Inside the MQW

Now quantum wells at the positions of the knots in equation (14.54) are considered. As the electricfield vanishes at the knots, the equations of motion for the coherence (14.51) and the population(14.52) read

p = −iωg p (14.59)

f = 0 . (14.60)

The population f is constant and the coherence p is performing a free rotation in the complex planewith angular frequency ωg. The wave equation (1.8) including the background polarization throughthe refractive index c = c0

n and taking the dynamic material response into account reads

(∂2z −

1c2 ∂2

t )E =1

ε0εrc2 ∂2t Pdyn (14.61)

with the dielectric function εr of the background refractive index and dynamic polarization

Pdyn(z) = ∑n

Pnδ(z− zn) (14.62)

as a sum of the polarizations Pn caused by the quantum wells at the positions nz. Both the electromag-netic field and the polarization are rotating with the laser frequency ω l if the resonance frequency ofthe quantum wells equals the light frequency (ωg = ωl ):

P = 2dcvRe(p) = 2dcvRe( pe−iωlt) (14.63)

E = E(z, t)cos(kz)cos(ωt) (14.64)

=12(e−iωlt + eiωlt)cos(kz)E .

Considering Maxwell’s equation in a stationary case where

∂2t p|well = ∂t p|well = 0 (14.65)

and

∂t E|well = ∂2t E|well = 0 (14.66)

holds true at the places of the quantum wells and therefor at the knots of the standing wave. Underthese conditions the wave equation (14.61) for the envelope of the electric field E at the positions ofthe quantum wells becomes

∂zE(1+ ei2ωlt) =ω2

l dcv

2kε0εrc2 ( p+ p?ei2ωlt) . (14.67)

Neglecting the fast oscillating non-resonant terms ∝ ei2ωlt (rotating wave approximation) yields

∂zE = =ω2

l dcv

2kε0εrc2 p . (14.68)

Note that this equation is only defined at the positions of the quantum wells.The self-consistent solution for the field inside the structure is a standing wave (14.54) with a

perturbed envelope E(z) at the quantum wells according to equation (14.68). The solution is depictedin Fig. 14.20.


14.5.1.4 Boundary Conditions

For the solution (14.68) to be stationary an additional condition has to be met: even though infinitystanding waves outside the structure are formally valid solutions, in practice the field-envelope has tobe zero outside the structure to avoid infinity field energy:

E(z0) = = E(z0 +N/2∗λ) = 0 . (14.69)

As f and p are elements of the density matrix ρ of the TLS (cp. section 4), which has to be a positivesemidefinite hermitian operator with trρ = 1, a relation between this two quantities can be derived(eq. 7.12):

| p|2 = f − f 2 . (14.70)

This also limits (in conjunction with equation (14.68)) the gradient of the field envelope to

E ′max =

ω2l dcv

4kε0εrc2 . (14.71)

14.5.1.5 Conclusion

Any set of field distribution of the form (14.54), coherence and population obeying the conditions(14.68), (14.69), (14.70) and (14.71) are stationary solutions. In other words: standing waves withconstant envelope between the quantum wells and a change of this envelope inside the quantum wellsaccording to above conditions is trapped inside the structure without any radiative losses at all. Thestructure can be interpreted as a sequence of cavities or resonators. In the stationary situation the po-larization inside the quantum wells separating the cavities is such that it exactly compensate the differ-ence in amplitude inside the adjacent cavities. Therefor the electromagnetic fields inside the cavitiesare completely decoupled from each other (Fig.14.20). The shape and intensity of the pulse enve-lope is nearly arbitrary, i.e. unlike previous analytic solutions[85] employing forward-backward wavesplitting and slowly varying envelope approximation this solution beyond SVEA gives rise to station-ary solutions which do not necessarily obey the strict condition for zero-velocity solitons. Therefor amuch greater multitude of stationary solutions exist than previously thought.

14.5.2 Numerical Simulation

The material equations (14.51) and (14.52) are solved self-consistently together with Maxwell’s equa-tions utilizing a combined Runge-Kutta-FDTD (chapter 3) approach.

14.5.2.1 Arbitrary Stationary Solutions

As theoretically predicted a multitude of nearly arbitrary solutions exist. The system was preparedaccording to the conditions (14.54),(14.68), (14.69),(14.70) and (14.71). Due to numerical inaccu-racy in this preparation the solutions slightly changes in the first picosecond but remains stationarythereafter. The figures 14.21 are just examples of different possible field distribution.


Amplitude

Standing-

wave

Quantum wells

Figure 14.20: The standing waves between the quantum wells are decoupled by the polarizationsinside the quantum wells. The structure acts like decoupled cavities.

14.5.2.2 Trapping External Light Pulses

To demonstrate the possibility of actually preparing an stationary state the propagation of a 2π-sech-pulse for different pulse durations (Fig. 14.22) is investigated. The simulation shows that very shortpulses (τ = 20fs, Fig. 14.23) travel without alterations in shape or amplitude (as predicted in section14.3.1). This is in contrast to linear optics where the pulse is strongly distorted into polariton beatsduring propagation (e.g. Fig. 14.7). For the nonlinear excitation with a 2π-pulse the transmissionis almost exactly equal to the incident pulse and only a very small signal can be seen in reflection.This effect is similar to self-induced transparency (SIT) in unstructured atomic vapor [1] (cp. analyticcalculation in section 14.3.1).

The opposite extreme of long pulses (τ = 500fs, Fig. 14.24) corresponds to spectral excitationcompletely within the stop band. As one would expect from the linear spectrum, the propagation inthis spectral region is strongly suppressed. Most of the light is reflected back and only a small partis transmitted through the sample. This is in contrast to the SIT-like behavior of the τ = 20fs-pulse.To derive an analytical solution for the SIT-like propagation one has to make the assumption of aninfinite extended structure (section 14.3.1). This is a good assumption for the τ = 20fs-pulse whichis short in comparison to the length of the structure. The τ = 500fs-pulse however is overlapping thewhole structure and the assumption of infinite extension breaks down.

The intermediate situation of a pulse which has roughly the same spectral width as the band gap(τ = 106fs corresponding to a spectral FWHM of about 10meV, Fig. 14.25) whereas lying mostly inthe stop band shows a different behavior: As initial response the TLS structure immediately reflectsand transmits parts of the pulse. Additionally, after about 2ps time delay a second response is emittedin both reflection and transmission direction. Fig. 14.26 displays the dynamics of the upper leveloccupation of quantum wells at different positions. During pulse excitation all quantum wells are first


0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

0 10 20 30z [µm]

Figure 14.21: Trapped light pulse, initial field distribution (red), field distribution after 3ps (blue), finalfield distribution (t ≥ 10ps) (magenta), place of QWs (green), from left to right and top to bottom:asymmetric with change of sign, without change of sign, symmetric with side wings, cosh-shaped(as would be expected from a soliton), symmetric with change of sign, triangle, Gaussian, stronglymodulated.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

-40 -30 -20 -10 0 10 20 30 40

Tran

smis

sion

Detuning [meV]

linear Transmission20fs pulse

106fs pulse500fs pulse

Figure 14.22: Transmission T = |ETrans|2|Einput |2 of the structure and spectral form of input pulses.

0

0.5

1

0 10 20 30 40

0 0.2 0.4 0.6 0.8

el. f

ield

[arb

. uni

ts]

t/τ

time [ps]


transmission

Figure 14.23: 20fs-2π-input-pulse, reflected and transmitted pulse. The pulse propagated unhindered.

0

0.5

1

0 10 20 30 40

0 5 10 15 20

el. f

ield

[arb

. uni

ts]

t/τ

time [ps]


transmission

Figure 14.24: 500fs-2π-input-pulse, reflected and transmitted pulse. The pulse is reflected by thestructure.


0

0.5

1

0 10 20 30 40

0 1 2 3 4

el. f

ield

[arb

. uni

ts]

t/τ

time [ps]


transmission

Figure 14.25: 106fs-2π-input-pulse, reflected and transmitted pulse. Only part of the pulse is initiallyreflected and transmitted. The main energy is released about 2ps later.

0

0.5

1

0 10 20 30 40

0 1 2 3 4

popu

latio

n

t/τ

time [ps]

1. QW100. QW190. QW

Figure 14.26: Population inside the first, middle and last quantum well during excitation with a 106fs-2π-pulse.

completely inverted and subsequently coherently depopulated to some extend. The 100th quantumwell, lying in the center of the structure, is pumped into complete inversion and stays there for asubstantial time (1ps) until decaying back into its ground state. Quantum wells at the beginning andthe end of the structure first show a partial decay but subsequently a second increase of the population,that is not directly induced by the incident pulse, occurs. However, this decay is only partial andthe upper level occupation remains on a non-vanishing constant value. Also the polarization (notshown) does not vanish (but only oscillates with a constant amplitude), therefore one would stillexpect radiative decay. All in all, the system converges into a stationary solution in which the excitedstates and therefore energy (population) and coherent excitation (polarization) can be stored. Thetemporal dynamics of the whole process is depicted in Fig.14.27.

This is further illustrated in Fig. 14.28 showing the spatially resolved electric field, polarizationand upper state population for the stationary situation, i.e. few ps after pulse excitation. The electricfield forms an oscillating standing wave within the structure with a profile that has its maximum in the


0

0.5

1

0 100 200z [µm]

1.9 ps el. fieldQWs

populationpolarization

0

0.5

1

0 100 200z [µm]

2.8 ps el. fieldQWs


0

0.5

1

0 100 200z [µm]

3.8 ps el. fieldQWs


0

0.5

1

0 100 200z [µm]

4.8 ps el. fieldQWs


0

0.5

1

0 100 200z [µm]

6.3 ps el. fieldQWs


0

0.5

1

0 100 200z [µm]

20 ps el. fieldQWs


Figure 14.27: Space resolved sequence showing the structure at different times during excitation witha 106fs-2π-pulse.


0

0.2

0.4

0.6

100 110 120 130z [µm]

el. fieldQWs


abs(E’)

Figure 14.28: Spatial field, polarization and population while in stationary solution. Scaled derivativeof electric field (absolute value plotted) is identical to polarization as analytically predicted.

middle and declines towards the boundaries. A magnification of the electric field as given in Fig. 14.29finally helps to understand why a stationary solution exists and how it is preserved. First, one canobserve that the electric field is zero at the plane positions. Therefore the material equations are notdriven and remain stationary. However the polarization is non-vanishing and occurs in Maxwell’sequations as a driving term. In the balanced stationary condition it exactly cancels out propagatingfields from the surrounding which enter the quantum well, hence stabilizing the stationary solution.

In Fig. 14.28 the numerically observed electrical field envelope and its derivation is compared tothe numerically observed polarization. The observations are in perfect agreement with the theoreticalpredictions derived above.

14.5.2.3 Stability of the Numerical Solution

Numerical simulations show this solution to be very stable against perturbation with a second externallight pulse. Even a second pulse as strong as the input pulse does not destroy the stationary solutioninside the structure. This stability is of critical importance for a possible application for light (andinformation) storage.

14.5.2.4 Classes of Solutions

The input pulse shown in Fig. 14.25 is not the only one resulting in a stationary solution. Fig. 14.30shows the energy captured inside the structure over the pulse area and pulse width. As can be seenpulses along straight lines in the area-width plane are resulting in a stationary solutions. These are notlines of constant peak Rabi-frequency as those would be straight lines through the origin.


-0.05

0

0.05

99 100 101

el. f

ield

[arb

. uni

ts]

z [µm]

el. fieldQWs

Figure 14.29: Magnification of the structure while in stationary solution: standing wave betweenquantum wells.

Figure 14.30: Energy captured inside the structure over input pulse area and width. Pulses along astraight line lead to stationary solutions.


0

0.5

1

0 0.5 1z [µm]

el. fieldpolarizationpopulation

Figure 14.31: Analytic solution of electric field and polarization.

14.5.3 Example of Analytic Solution

The field

E(z, t) = cos(2πzλln

− π2)cos(ωl t)cos(

2πλl

z) (14.72)

is one solution which meets all conditions above and is similar to the numerically observed solution.The corresponding polarization is

p = −4πkε0εrc2

λlnω2l dcv

sin(2πzλln

− π2) . (14.73)

The population according to (14.70) is given by

f =12±√

14−|4πkε0εrc2

λlnω2l dcv

sin(2πzλln

− π2)|2 . (14.74)

Fig. 14.31 depicts both field and polarization.

14.5.4 Conclusions

It has been shown analytically and through numeric simulations that any set of field distribution of theform (14.54), coherence and population obeying the conditions (14.68), (14.69), (14.70) and (14.71)are stationary solutions in a MQW. Moreover it has been demonstrated that the preparation of suchcaptured pulses should be feasible.

157

Chapter 15

Pulse Propagation in BulkSemiconductors

In this chapter the pulse propagation in a bulk semiconductor is investigated. In the first part resonantpropagation (i.e. a pulse with center frequency at the frequency of the excitonic resonance) andespecially the phase characteristics of the light pulse is addressed and compared with experimentaldata[86]. Only recently the experimental techniques were developed to fully characterize the temporal,spectral and phase characteristics of ultra-short laser pulses. The differences of linear and nonlinearpropagation are worked out.

The second part addresses the near-resonant propagation. The spectral region near resonant belowthe semiconductor band edge is of particular interest for two reasons: 1. The transmission experimenton MQW samples, as theoretically described in chapter 14, are usually performed on MQWs epitax-ially grown on a GaAs substrate typically a few hundred micrometers thick. The light pulse has topropagate through the substrate after passing through the MQW. 2. Typically, the transparent regionbelow the band gap features a normal dispersion (i.e. dn(λ)

dλ > 0) and a negative Kerr-nonlinearity (cp.chapter 1). This regime is of fundamental interest since bright solitons are possible [4].

In the final part the suitability of three different theoretical models for the nonlinear propagationof very short, intense laser pulses near resonant below the band gap of GaAs bulk semiconductor iscompared [14]. The differences between the nonlinear Schrödinger equation (NLSE, section 1.7), theoptical Bloch equations (OBE, section 7.1.1) and the Maxwell semiconductor Bloch equations (SBE,section 7.1.2) are worked out and compared with the experiment[87].

For the theoretical description of the linear and nonlinear light propagation effects, Maxwell’sequations have been evaluated using the Slowly Varying Envelope Approximation (SVEA[88, 1],chapter 1). The dynamic dipole density, which is a source term in Maxwell’s equations, is calculatedusing the semiconductor Bloch equations in the Hartree-Fock limit [58] (section 7.1.2). This approachallows to reproduce the observed excitonic self-induced transmission and multiple pulse breakup. Thecombined semiconductor Maxwell-Bloch equations can be numerically evaluated for arbitrary shapes,strength, and duration of the input pulse.

158 15. PULSE PROPAGATION IN BULK SEMICONDUCTORS

Figure 15.1: Linear spectrum and refractive index. The propagating pulse will be split into twocomponents by the excitonic resonance. Calculations of refractive index and absorption employingthe semiconductor Bloch equations and the optical Bloch equations are shown.

15.1 Resonant Propagation

15.1.1 Linear Propagation

The linear propagation is governed by the linear absorption and refractive index as shown in Fig. 15.1.A pulse propagating resonant on the excitonic resonance is split into two spectral components, onespectrally below the resonance and one between the resonance and the continuum of the semicon-ductor. The interference between these two components translates into a temporal beating as shownin Fig. 15.2. As to be expected from a beating phenomenon sharp 1π phase jumps are between thetemporal components (Fig. 15.3)

This beating can also be understood in the image of a new quasi-particle: The propagating photonand the exciton constitute the new quasi-particle polariton. The dispersion-relation of the polaritonfeatures two branches arising from the anticrossing of light and excitonic dispersion. These twodispersion branches relate to the two spectral components and cause the temporal beating. In Fig. 15.4theoretical calculation are compared with the actual experiment [86]. To be able to compare the thecalculations with the experiment the asymmetry and chirp properties of the real input pulse were takeninto account in the calculations. The experimentally retrieved normalized intensity and phase in thetemporal and spectral domain are shown in Figs. 15.4(c) and 15.4(d), respectively. The lower curvescharacterize the input pulse as measured after propagation through the sample substrate, featuring aroughly single-sided exponential envelope and (according to the sign conventions) a slight positive

15.1. RESONANT PROPAGATION 159

Figure 15.2: Linear resonant propagation in bulk semiconductor: Beating between the two polaritonbranches.

Figure 15.3: Linear resonant propagation in bulk semiconductor: Beating between the two polaritonbranches results in sharp 1π phase jumps.


p p

Figure 15.4: (a) temporal and (b) spectral normalized intensity (dashed line) and phase (solid line) forthe input pulse (bottom) and after linear propagation through the sample (top). (c),(d) Results of thecorresponding experiment[86].

chirp caused by the employed Lorentzian bandpass filter in the experiment. As expected for a beatingphenomenon, the temporal phase shows a phase jump of almost π between both pulse components,which is equivalent to a change of sign of the field. Before and after the phase jump, the curve followsthe input pulse phase. The spectral phase distortion is a measure of the refractive index dispersionaround resonance, which could be used to accurately describe the material system. The polaritonbeating as well as the spectral phase are well reproduced by the theoretical calculations [Figs. 15.4(1)and 15.4(b)].

15.1.2 Nonlinear Propagation

In the nonlinear regime the effects of the Pauli-blocking nonlinearity (section 14.3.2) and adiabaticdriving (section 14.3.3) lead to pulse breakup and formation of individual soliton-like pulses (SIT,analytic derivation similar to 14.3.1). The individual pulses manifesting themselves clearly in thedevelopment of separate phase characteristics for each pulse component. In Fig. 15.5 the theoreticalresults are compared with the experiment[86]. As above the pulse was modeled to take into accountthe asymmetry and chirp properties of the real pulse.

For the propagating pulses with an area of 2π (middle curves), temporal reshaping, steepening,and breakup into two distinct pulse components occur. At 2.6π (upper curves), further steepeningand a third pulse component can be identified. In both cases no remarkable distortions in the spectraldomain is found. These temporal and spectral properties in addition to the high nonlinear transmissionindicate Rabi-flopping as expected for a growing pulse area. In contrast to the linear experiments, thenonlinear pulse breakup is accompanied by a temporal phase that essentially follows the quadraticinput phase [Fig. 15.5(a)and (c)]. Slight phase modulations of less than 0.2π occur in the experimentsynchronously with the temporal pulse breakup. In the spectral domain, the phase of the transmittedpulses roughly corresponds to the input pulse phase [Fig. 15.5(b) and (d)].

The theoretical predictions are well confirmed by the experiment. The excitation-dependent tem-

15.1. RESONANT PROPAGATION 161

p p

Figure 15.5: (a) temporal and (b) spectral normalized intensity (dashed line) and phase (solid line)for the input pulse (bottom) and after nonlinear propagation through the sample. (c),(d) Results of thecorresponding experiment. Pulse area is θ = 2.0 π (middle) and θ = 2.6 π (top).[86]

poral pulse breakup due to Rabi-flopping of the carrier density can be observed as well as a phase char-acteristic that essentially follows the input phase profile except for slight modulations [Fig. 15.5(c)].These phase modulations are explained with the coherent reemission of light during the Rabi-flopping.The emission is accelerated or decelerated depending on the field gradient, and the temporal phase iscorrespondingly compressed or stretched. The asymmetric spectra shown in Fig. 15.5(d) marginallychange with increasing pulse area. The respective spectral phases are nearly invariant.

15.1.3 Influence of Chirp

In experimental measurements one great problem is the control of the chirp of the used laser pulses.Chirp denotes a change of instantaneous frequency ω(t) over time. For a wave packet of the form (cp.section 1.4.1, eq. (1.14))

E(t,r) = E(t,r)eiφ(t) (15.1)

the instantaneous frequency is given by the time derivative of the phaseφ:

ω(t) = ∂tϕ.

For a ideal monochromatic pulse the the instantaneous frequency is a constant, but often it is lin-ear or (like in the above presented experiments) quadratic. To investigate the influence of chirpedinput pulses numerical calculations for different pulse chirp and propagation lengths are performed.The normalized intensity and phase characteristics of the input and transmitted pulses after nonlinearpropagation are depicted in Fig. 15.6. The pulses are modeled according to the experiment, i.e. aGaussian pulse shape with a FWHM of 400 fs and an pulse area of 2.6 π. For chirp-free input pulses


Figure 15.6: (a)–(c) Plots of temporal intensity (dashed line) and phase (solid line) for the 400fsGaussian input pulse (top) and after nonlinear propagation according to the experiment with a pulsearea θ = 2.6 π (bottom). (a) Chirp-free, (b) positive chirp, and (c) negative chirp.

[Fig. 15.6(a)], the pulse breaks up in the temporal domain and the output phase is essentially flat withslight modulations as described in the preceding paragraph. For chirped input pulses [Figs. 15.6(b)and 15.6(c)], the transmitted pulse shape is altered due to the different relative velocity of the pulsecomponents after breakup. Here, the output phase is given mainly by the quadratic input phase su-perimposed by the modulations already apparent in the chirp-free case. For a significantly longerpropagation distance, pulse breakup into well-separated soliton-like pulses occurs. The input chirpdetermines the relative delay of the individual pulses. Each pulse is characterized by a temporal phasethat features a flat section at the position of the pulse peak surrounded by the phase deviations in thecourse of Rabi-flopping. The development of separate phase characteristics without a defined phaserelation is in contrast to the linear polariton beating with phase jumps of π between the discrete pulsecomponents (compare Figs. 15.4(a) and 15.4(c)).

15.2 Near-resonant Propagation

In this section the near-resonant propagation of a pulse below the excitonic resonance is considered.This is a transition regime between the resonant propagation as described in section 15.1.1 and thefar off-resonant propagation, e.g. the propagation of communication lasers at half the frequency ofthe fundamental material resonance of optical fibers[9, 4]. As can be seen in Fig. 15.7 the parameterswere chosen for the pulse to be spectrally well separated from the resonance line by several linewidthsof the resonance (and several FWHM of the pulse).

15.2.1 Linear Propagation

As for the linear resonant propagation (section 15.1.1) the near-resonant linear propagation is gov-erned by the linear absorption and refractive index. As can be seen from Fig. 15.7 a pulse propagatingoff resonant below the excitonic resonance will suffers only slight distortions due to the frequency

15.2. NEAR-RESONANT PROPAGATION 163

0.01

0

3.55

3.56

3.57

-40 -30 -20 -10 0 10 20detuning [meV]

abso

rptio

nre

fract

ive

inde

x

pulse spectrumSBEOBE

Figure 15.7: Absorption spectra and refractive index of GaAs. The initial spectrum is well belowthe resonance. Results of calculations employing Semiconductor Bloch equations and Optical Blochequations are shown.

dependent refractive index: The pulse is broadened and a slight asymmetry of the pulse shape isintroduced while propagating (Fig. 15.8).

15.2.2 Nonlinear Propagation

For nonlinear propagation the dynamics is dominated by self-phase modulation (SPM, cp. section1.9.2). In Fig. 15.9 the temporal shape and the spectrum of a propagating pulse is shown for differentpropagation length up to 13 mm. The pulse is first compressed and spectral wings show up. Forgreater propagation length the pulse is splits up into two components both in time and frequency.Looking at the field instead of the intensity even reveals a third temporal component (Fig. 15.10).Fourier-transforms of selected, finite time intervals (denoted by different colors in Fig. 15.10) helpsto attribute the spectral components to the different temporally separated pulses in time domain. Eachwell separated spectral peak corresponds to a temporal peak (Fig. 15.11). Generally, blue componentscorrespond to faster temporal components.

To understand the spectral features the instantaneous frequency ω(t) is considered. As has beenshown in section 1.9.2 a Kerr-type nonlinearity results in an S-shaped modulation of the pulse enve-lope around the centre of the pulse as instantaneous frequency is

ω(t)−ωL ∝ −(t − t0)|Ω| . (15.2)

The near resonantly transmitted pulse shown in Fig. 15.12 exhibits these typical SPM features (cp.


0

1

-2 -1 0 1 2

norm

aliz

ed fi

eld

inte

nsity

time [ps]

inputafter 13mm

Figure 15.8: Linear near resonant propagation (z = 1.3cm): pulse is widened while propagating dueto the dispersion of the refractive index.

section 1.9.2): A considerable chirp (which is given by the slope of ω(t)) of the central part of thepulse and an decrease of frequency at the later pulse components. The pulse breakup is due to theGVD resulting in a deceleration of the later pulse components and an acceleration of the rising pulseedge. However the instantaneous frequency depicted in Fig. 15.12 shows deviations from the idealsolution (eq. 15.2) which originate from a nonlocal time retardation of close to resonance exciteddipoles[66] and is discussed in more detail in section 15.2.4.6 and appendix F. The characteristicsof the instantaneous frequency results in a chirped main pulse component with the same central fre-quency as the input pulse and a smaller less chirped pulse component with a central frequency below.The relatively low chirped, temporally less intensive second component dominates the spectrum aschirped main component is spectrally broadened. These qualitative results are also found experimen-tally (Fig. 15.13)[87].

15.2.3 Transversal Effects

In a linear dispersive medium, the evolution of an optical pulse in time and transverse space can beunderstood independently. The situation is fundamentally different if an ultrashort pulse propagatesthrough a dispersive medium that exhibits a Kerr-type nonlinearity. In this case, temporal (dispersive,SPM) and transverse spatial (diffractive, self-focusing) effects do not only occur simultaneously butbecome mutually coupled (section 1.9). This nonlinear spatio-temporal coupling has been scarcelyinvestigated to date. Much more effort was made to study short pulse propagation in fibers and one-dimensional waveguide structures, in which transverse confinement keeps the pulse intensity andallows to exclude diffractive effects. In these structures, the interplay of the focusing (γ ≥ 0) ordefocusing (γ ≤ 0) nonlinearity with the appropriate anomalous (β2 ≤ 0) or normal (β2 ≥ 0) groupvelocity dispersion leads to temporal pulse compression and the formation of soliton-like pulses [89](cp. chapter 1). For planar waveguides and bulk materials, the transverse confinement is removed, andnonlinear diffractive effects and spatio-temporal coupling apply. Here, the limitation to a treatmentin the time domain alone is not sufficient, especially if a defocusing nonlinearity without intrinsic


-2 0 2

norm

aliz

ed in

tens

ity

time [ps]

(a)

phas

e [π

]

-40 -20 0

norm

aliz

ed in

tens

ity

detuning [meV]

(b)

Figure 15.9: (a) Temporal compression (solid line) due to SPM and breakup into two components forgreater propagation distances. The phase (dotted line) does not jump between the main componentsfor long propagations. (b) Spectral splitting into two asymmetric components corresponding to thetwo temporal components.


9 10 11 12 13 14 15 16

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

Fiel

d [a

rb.u

nits

]

Time [ps]

Figure 15.10: SBE: Pulse form after nonlinear propagation. Individual pulse components are color-coded. The pulse components are not separated by zero-crossing of the envelope or phase jumps.

2 4 6 8 10 0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

Fiel

d [a

rb.u

nits

]


Figure 15.11: SBE: Spectrum of pulse and pulse components of Fig. 15.10. The faster componentscorresponds to the spectral components on blue side of the spectrum.


1

0

-1-4 -2 0 2 4

Time [ps]

1

0

-1-4 -2 0 2 4

Time [ps]

Figure 15.12: Field amplitude (red), phase φ(η) (green) and ddt φ(η) = ω(t) (blue) after nonlinear prop-

agation. The centers of the individual pulse components are marked on the instantaneous frequencyby black dots and the angle representing the chirp are marked by dashed lines.

self-guiding is considered.The theory applied so far assumes the propagation of planar waves. This is justified either for

propagation in waveguides or for illumination of the sample with a large laser spot, i.e. quasi-planarwaves. In the actual experiment however the sample is illuminated by a very focused laser to achievethe needed high intensities. For spot diameters in the order of a few microns the transversal effects ase.g. in equation 1.57 cannot be neglected. To elucidate the influence of these transversal effects thetransversal Laplace-operator in the wave equation is treated in paraxial approximation (section 1.2).This approximation retains the planar-wave character of the propagating wave while the transversalintensity profile is governed by the transversal Laplace-operator.

The temporal characteristics of the propagating pulse is similar to the pure planar-wave propaga-tion. As can be seen in Fig. 15.14, the pulse is initially compressed and then breaks up into severalpulse components. The transversal profile is shown in Fig. 15.15: Simultaneous to the temporal com-pression the transversal pulse profile is focused while propagating the semiconductor (self-focusing,section1.9.1). The profile maintains its transversal symmetry and does not develop side maxima. Theresulting overall pulse shape is depicted in Fig. 15.16.

Now the frequency and impulse distribution is considered in a strong nonlinear pulse (area about10π) 17meV below the band edge after a propagation of a few millimeters. The resulting pulse isdepicted in Fig. 15.17 and features considerable compression both in transversal space and time aswell as breakup into one main component and several smaller components.

A Fourier-transformation along the red line in Fig. 15.17 yields a similar result as in the planar-wave propagation. The spectrum (Fig. 15.18) features two spectral components below and above theoriginal spectral maximum of the input pulse. The spectral components correspond to the temporalcomponents with the fast temporal components at the blue side of the spectrum. The k-space distribu-tion (i.e. momentum) along the yellow line, as shown in Fig. 15.19, does not exhibit any qualitativechange while the pulse propagates in the bulk material. It only reflects the damping and losses along


Figure 15.13: Experimental data[87] of near-offresonant (23 meV below band gap) propagation. Theintensity of 540MW/cm2 correspond to a pulse area of about 4π. (a) As predicted the pulse is firstcompressed before breaking up into several components. (b) The spectral splitting and the enhance-ment of the red component are in accordance with the theoretical considerations.


0 1 2 3 4

field

[arb

. uni

ts]

time [ps] 0 1 2 3 4 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

Figure 15.14: Nonlinear pulse propagating at 17 meV below band edge with a pulse area of 3π. Tem-poral pulse profile of the input pulse (red) and while propagating (green). The propagation distancegrows from left to right and top to bottom. The pulse is initially compressed and then breaks up intoindividual pulse components.


0 5 10 15 20 25

field

[arb

. uni

ts]

transversal space [micron] 0 5 10 15 20 25 0 5 10 15 20 25

0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25

0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25

Figure 15.15: Nonlinear pulse propagating at 17meV below band edge with a pulse area of 3π.Transversal pulse profile of the input pulse (red) and while propagating (green). The propagationdistance grows from left to right and top to bottom. The pulse is focused while propagating.


timespace

field

Figure 15.16: Nonlinear pulse propagating at 17meV below band edge with a pulse area of 3π. Overallpulse shape while propagating the semiconductor.


Figure 15.17: 10π-pulse at 17meV below band edge after a few mm of propagation. The pulse isconsiderably compressed both in transversal space and time and is broken up into one main componentand several smaller components. The initial pulse form is shown translucent.

Figure 15.18: Frequency distribution of Fig. 15.17 along the red time-like line. The spectrum splitsinto three components corresponding to the temporal components.


Figure 15.19: k-vector (momentum) distribution of Fig. 15.17 along the yellow transversal-space-likeline. The spectrum does not change during propagation but is only damped. The initial transversalmomentum distribution is shown translucent.

its path.

15.2.3.1 Comparison with the Experiment

The theoretical predictions are now compared with the actual experiment performed at 9K. A 600 fslong and 34µm wide pulse with a central wavelength of 836 nm and an input intensity of 8 MW

cm2 is prop-agated through a 450µm bulk GaAs. For linear propagation the input pulse (depicted in Fig. 15.20)does not significantly differs from the transmitted pulse either in the calculation (Fig. 15.21) or theexperiment (Fig. 15.22). The pulse only suffers a small broadening as expected from a defocusingmedium.

For nonlinear propagation the theoretically predicted compression both in time and transver-sal space as well as the breakup in individual pulses (Fig. 15.23) are confirmed by the experiment(Fig. 15.24). A close inspection of the experimental data revealed side-maxima in the transversal di-rection of the transmitted pulse. These are not present in the calculation. The cause of these discrep-ancies could not be finally determined. They could be due to interference effects in the CCD-detectorof the experiment which are a commonly known problem. Another possibility is the paraxial approx-imation used in the calculations: The neglection of higher order transversal effects could lead to a thedeviations of the transversal profile.


Figure 15.20: Input pulse intensity. The transversal FWHM is about 34µm.

Figure 15.21: Pulse after 450µm linear near-resonant bulk propagation: The pulse is slightly broad-ened but no considerable change in overall pulse shape.


Figure 15.22: Experiment[90]: Pulse after 450µm linear near-resonant bulk propagation: No consid-erable change in overall pulse shape.

Figure 15.23: Pulse after 450µm nonlinear propagation: The pulse is compressed both in time and intransversal space. The pulse breaks up into two separate pulses.


Figure 15.24: Experiment[90]: Pulse after 450µm nonlinear propagation: The pulse compressed bothin time and in transversal space. The pulse breaks up into two separate pulses along the temporaldirection. Side-maxima along the transversal space axis are visible.

15.2.4 Comparison of Model Systems

In this section the validity of three different theoretical models for describing the material responseand the limits for the prediction of pulse spectra in transparent materials where the detuning betweenthe carrier frequency ωL and the typical material resonance frequency ωM (∆ = ωM −ωL) is on theorder of the applied Rabi-frequency ∆ ≤ ~Ω0 = d E0 (with d being a typical dipole moment of theoff-resonant transitions and E0 the maximal field envelope) are discussed[14]. As this is a transitionregime between the resonant propagation a far off-resonant propagation it is not inherently evidentwhich theoretical models are valid. Different levels of material equations to describe the polarizationP are considered: At the microscopic level of material description the semiconductor Bloch equations(SBE, section 7.1.2) which have been shown to describe the nonlinear properties of the optical Stark-effect for samples where propagation effects can be excluded [6, 91, 92] or dominate [93, 83, 88].After the SBE two reduced models from the SBE are discussed: the optical Bloch equations (OBE,section 7.1.1) and the nonlinear Schrödinger equation (NLSE, section 1.7), thus discussing a hierarchyof models for near off-resonant pulse propagation. The main disadvantage of the NLSE may bethat the nonlinear coupling may occur differently as described by a time local Kerr-effect (P(t) ∝|E(t)|2E(t) ), since it may be retarded and of higher order. The bulk semiconductor dielectric modelsystem is microscopically well studied [6] and clarifies the validity of different material nonlinearitiesin the transparent region, ranging from the Kerr-nonlinearity and simple Pauli-blocking (cp. section14.3.2) up to semiconductor Bloch equations (SBE) including many particle effects [93, 89, 87].Strong phase modulation and temporal pulse breakup effects close to the semiconductor band gaphas been found in 2d wave guides[83]. In contrast, the focus is here on spectral pulse features and asystematic comparison of a microscopic model and a correspondingly derived NLSE.


15.2.4.1 Semiconductor Bloch Equations

As the SBE (7.1.2) include the complete experimentally relevant nonlinearity (shown by comparisonto the experiments on the optical Stark-effect [6, 83, 94, 95]) it can be used for the description of nearresonant excitation as long as not too many real carriers are created. Thus, the SBE can be studied tojustify the NLSE (with third order nonlinearities) or reveal its limits.

15.2.4.2 Optical Bloch equations

Below the excitonic resonance the SBE can be reduced to the optical Bloch equations by an expansionin Wannier-states (i.e. hydrogen-like wave-functions) including their Pauli-blocking nonlinearity asshown in section 8. Further exciton-exciton interactions are shown to be of minor importance forwell below band edge excitation (optical Stark-effect [6]). The equations of motion are similar to theSBE but lack exciton-exciton interactions, however they keep a time retarded nonlinear polarizationto infinite order since they involve the equation of the occupation density.

15.2.4.3 Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation (NLSE, cp. chapter 1, eq. (1.57)) is a well established descriptionof pulse propagation in transparent media. The NLSE has been used to model successfully e.g. pulseinteractions in waveguides[7], soliton dynamics in optical lattices[8], fibers[9], wave guides[96] andwaveguide arrays[97], propagation in a hot plasma[10], propagation[98] and filamentation of laserpulses in air[11] and propagation in Kerr-media[12]. The cubic Kerr-nonlinearity is non-retarded andresults from an expansion of the full optical nonlinearity . The coefficient γ typically is used as a fitparameter since the involved material transitions cannot be discussed microscopically. The NLSE isbased on the expansion of the dipole density up to 3rd order in field (section 1.5.1) and 2nd order infrequency ω around ω0 (section 1.4.2). The expansion up to third order in the field can be directlyobtained in the limit of adiabatic following (cp. section 10) [1] and yields the Kerr coefficient γ ∝ 1

∆3

and the absorption coefficient. From the analytically calculated excitonic susceptibility χ(ω) (eqn.8.16) [6] the expansion of the wavenumber vector k(ω) = ω

c

√1−χ(ω) in ω can be calculated and

yields the inverse group velocity 1vg

= dkdω and the inverse group velocity dispersion (GVD) β2 = d2k

dω2 =

4√

2d2ωMa0c~γ2 .

15.2.4.4 Numerical Comparison

For the numerical evaluation standard GaAs material parameters (appendix G) have been used for allmaterial equations. Therefore, a fair comparison of the different model systems is possible. The OBEand the SBE are solved using a 4th order Runge-Kutta-algorithm, the NLSE is solved with a split-stepFourier-algorithm (section 1.8) and the reduced wave equation is solved in slowly varying envelopeapproximation (SVEA, section1.4.1) and forward-backward splitting[17].

15.2.4.5 Linear Spectra and Pulse Propagation

The numerical parameters are chosen to reflect the actual experiment[87]. For the initial Gaussianpulse, a FWHM of 600fs (corresponding to a spectral width of 5meV) and ∆ = −23meV is used.Linear absorption and refraction are depicted in Fig. 15.25. The excitonic resonance of the SBE isperfectly reproduced by the OBE. However, as expected, the electron-hole continuum is missed. Notethat only the spectral specifics at the pulse position are important for linear propagation. The NLSE


0.01

0

3.55

3.56

3.57

-40 -30 -20 -10 0 10 20detuning [meV]

abso

rptio

nre

fract

ive

inde

x

pulse spectrumSBEOBE

Figure 15.25: Absorption spectra and refractive index: The excitonic resonance of the SBE is repro-duced by the OBE. The initial spectrum is well below the band gap.

features (compared to SBE and OBE) a flat refractive index and absorption. Small deviations of therefractive index between SBE and OBE due to the influence of continuum states are noticeable. Linearpropagation of the near resonant pulse exhibits similar features in all three models: the pulse smoothlybroadens due to group velocity dispersion as it propagates while maintaining its spectral shape. Thepulse width of the propagated pulse is slightly underestimated over a distance of a few cm in the OBE-model. This is expected from the different value and gradient of the refractive index (Fig. 15.25) whichleads to a more pronounced dispersion of the linear pulse in the SBE-model. As the parameters of theNLSE are derived accordingly, the pulse width is nearly identical to the OBE. OBE and SBE producea slight asymmetric pulse in contrast to the perfect Gaussian of the NLSE. However no importantqualitative difference is observed and all three models are appropriate for linear propagation.

15.2.4.6 Nonlinear Pulse Propagation

Figures 15.27 (NLSE),15.28 (OBE) and 15.29 (SBE) depict the nonlinear propagation for the differ-ent models: (a) shows the normalized intensities |Ω|2 (solid line) and the phase φ (dotted line) in thetime domain (over η = t − z

c ) while the corresponding power spectra are shown in (b). In the bottomline the temporal input pulse shape and its spectrum are shown. From bottom to top the propagationdistance increases in equidistant steps up to about 1.3 cm. For a fair comparison of the different mod-els, a measure of the light intensity is introduced as the ratio of the spectral detuning ∆ and the initialpeak Rabi-frequency Ω0 = E0dcv

~: δ = Ω0

∆ . For each model δ is chosen to obtain clear pulse breakupsignatures (ration 1:4) in the different models: In the NLSE simulation it does not occur until δ ≈ 2


0

1

-4 -2 0 2 4

norm

aliz

ed fi

eld

inte

nsity

time [ps]

SBEOBE

NLSE

Figure 15.26: Linear near resonant propagation (z = 1.3cm): SBE and OBE lead to slight pulsereshaping. OBE and NLSE slightly underestimate the pulse width due to the lack of continuum states.

while in the OBE simulation it is observed for δ ≈ 1 and in the SBE simulation even for δ ≈ 12 . All

models exhibit at low propagation distances a temporal pulse compression and with increasing propa-gation length a temporal and spectral splitting (pulse breakup) due to self-phase modulation (SPM, cp.1.9.2)[4]. Since the time domain results (pulse breakup ratio) are fixed, the spectral deviations of theOBE and SBE in comparison to the NLSE show, where the simpler models may fail: For the NLSE,all results are perfectly symmetric in time and frequency (Fig. 15.27). For the OBE (Fig. 15.28) andSBE (Fig. 15.29) calculation, the spectral and temporal pulse shapes [93, 87, 83, 86] are asymmetricand therefore exhibits a qualitative difference to the NLSE.

A close inspection shows a third spectral component in the red for the SBE (Fig. 15.32) and inthe blue for the OBE (Fig. 15.31) simulation. Fourier-transform of selected, finite time intervals helpsto attribute the spectral components to the different temporally separated pulses in time domain forall models (Fig. 15.32, Fig 15.31 and Fig. 15.30). Each well separated spectral peak corresponds toa temporal peak. Generally, blue components correspond to faster temporal component. The spectralasymmetry for short propagation distances are qualitatively similar in OBE and SBE (i.e. the redcomponent is stronger than the blue one). For long propagations in the SBE model however the redcomponent gets more pronounced than the blue one. This is the opposite behavior in comparison toth OBE.

15.2.4.7 Instantaneous Frequency

To understand the origin of the spectral and temporal pulse breakup for the different models, a waveequation for the phase dynamics is considered: ∂ξφ(η,ξ) ∝ P

Ω . In the NLSE-simulation, Fig. 15.33,no strict phase jumps (zero-crossings of the field correspond to 1π phase jumps) between the temporalcomponents are observed. Here, ∆φ ∝ |Ω|2∆ξ, the induced phase follows the field envelope up toa large extent of the propagation length. Considering the instantaneous frequency of the pulse (cp.section 15.2.2), which is given by the time derivative of the phase ∂t φ(t) = ω(t) and, as shown in


-2 0 2

norm

aliz

ed in

tens

ity

time [ps]

(a)

phas

e [π

]

-40 -20 0

norm

aliz

ed in

tens

ity

detuning [meV]

(b)

Figure 15.27: NLSE, δ = Ω∆ ≈ 2: (a) In the temporal domain, the pulse (solid line) is symmetrically

compressed with subsequent breakup in symmetric components. The phase (dotted line) does notjump between the main components for long propagations. (b) The spectrum features two distinctsymmetric spectral components.


-2 0 2

norm

aliz

ed in

tens

ity

time [ps]

(a)

phas

e [π

]

-40 -20 0

norm

aliz

ed in

tens

ity

detuning [meV]

(b)

Figure 15.28: OBE,δ = Ω∆ ≈ 1: (a) Temporal compression (solid line) including a simultaneous pulse

splitting and a 1π-phase-jump (dotted line) between the first two main components. (b) Spectralsplitting into two main asymmetric components corresponding to the two main temporal components(a third is barely visible).


-2 0 2

norm

aliz

ed in

tens

ity

time [ps]

(a)

phas

e [π

]

-40 -20 0

norm

aliz

ed in

tens

ity

detuning [meV]

(b)

Figure 15.29: SBE δ = Ω∆ ≈ 1

2 : (a) temporal compression (solid line) due to SPM and breakup intotwo components for greater propagation distances. The phase (dotted line) does not jump betweenthe main components for long propagations. (b) Spectral splitting into two asymmetric componentscorresponding to the two temporal components. The asymmetry for long propagations is opposite incomparison to th OBE.


7 8 9 10 11

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

Fiel

d [a

rb.u

nits

]

Time [ps]

0 -2 -4 -6 -8 -10

0,0

0,2

0,4

0,6

0,8

Fiel

d [a

rb.u

nits

]


Figure 15.30: NLSE: Pulse form after nonlinear propagation. Individual pulse components are color-coded.

7 8 9 10 11 12 13

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

Fiel

d [a

rb.u

nits

]

Time [ps]

0 2 4 6 8 10 12

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Fiel

d [a

rb.u

nits

]


Figure 15.31: OBE: Pulse form after nonlinear propagation. Individual pulse components are color-coded. The first two pulse components are separated by a zero-crossing.


9 10 11 12 13 14 15 16

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

Fiel

d [a

rb.u

nits

]

Time [ps]

2 4 6 8 10 0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

Fiel

d [a

rb.u

nits

]


Figure 15.32: SBE: Pulse form after nonlinear propagation. Individual pulse components are color-coded. The pulse components are not separated by zero-crossing or phase jumps.

section 1.9, correlates to the polarization

ω(t) = ∂t φ(t) (15.3)

= ∂t(1c

ω0nz−ω0t) (15.4)

∂t(1c

ω0z

√1+

Pε0E

−ω0t) . (15.5)

For a Kerr-type nonlinearity, the polarization P =−αΩ+βΩ3 (section 1.5.1) leads to an instantaneousfrequency

ω(t) ∝ (t − t0)I(t)−ω0 (15.6)

with I(t) = |Ω|2 being the intensity of the pulse envelope (centered around t0). The NLSE simulationexhibits the typical SPM features (Fig. 15.33) to be expected from eq. (15.6) : A considerable chirpof the central part of the pulse and an increase/decrease of frequency at the rising and falling flank ofthe pulse. The pulse breakup is due to the GVD resulting in a deceleration of the falling pulse flankand an acceleration of the rising pulse flank. As the leading and trailing pulse components are rela-tively unchirped in respect to the central pulse component, their corresponding spectral components(with central frequencies above and below the central frequency of the input pulse) are narrower andtherefore dominate the pulse spectrum.

In contrast, in the OBE-simulation (Fig. 15.34) between the first two temporal pulse components(corresponding to the well separated red component in the spectra and first blue component) a zero-crossing and phase jumps of 1π is observed. This behavior can be traced back to the single res-onance approximation in the OBE-model: zero-crossing between pulse components and the cleardevelopment of π-phase jumps results from a nonlocal time retardation of close to resonance exciteddipoles[66]. Therefore, for the OBE ∆ϕ ∝

R

dt ′Ω(t ′)∆ξ, memory effects (time integral) are dominantand yield an asymmetric spectrum. If memory effects are taken into account in first order (as derivedin chapter 10) the polarization reads (eq. (10.29))

P = −αΩ+βΩ3 − γΩΩ2 (15.7)


1

0

-1-4 -2 0 2 4

Time [ps]

Figure 15.33: NLSE: Field amplitude (red), phase φ(t) (green) and ddt φ(t) (blue) after nonlinear prop-

agation. The centers of the individual pulse components are marked on the instantaneous frequencyby black dots and the angle representing the chirp are marked by dashed lines. The lesser chirpedpulse components with central frequencies above and below the incident light frequency ω0 dominatethe spectrum.

with an additional term proportional to Ω2. Expanding the resulting instantaneous frequency in termsof the I(t) yields (for the actual calculation see appendix F):

ω(t) ∝ (t −2t3)I− (t3 −2t5)I2 +O3(I) . (15.8)

The SPM like instantaneous frequency is superimposed by a term proportional t 3I due to memoryeffects. In effect both pulse components are chirped nearly identical around their respective centerfrequencies (Fig. 15.34). The intensity relation of the two developing spectral components reflect theintensity relation of the two pulse components in the time domain resulting in a more pronounced bluecomponent.

In contrast to the OBE, the temporal phase characteristics of the SBE (Fig. 15.35) are more likethe NLSE: There are no clear phase-jumps between the temporal components, however, signaturesof a phase jump remain at the position of the sample, where the temporal pulse breakup occurs(Fig. 15.29). Here, the interference between the reemissions of the different exciton states in theSBE-model prevents a strict phase jump as observed in the OBE-model, since well off-resonant ex-citonic states permit a adiabatic elimination of the field: ∆ϕ ∝ |Ω|2∆ξ and close to resonant states donot: ∆ϕ ∝

R

dt ′Ω(t ′)∆ξ.As shown in chapter 10 the influence of continuum states partially suppresses the memory effect

and weakens the Kerr-nonlinearity contribution. This leads to less pronounced and flatter featuresof the instantaneous frequency than in the OBE-model. The characteristics of the instantaneous fre-quency results in a chirped main pulse component with the same central frequency as the input pulseand a smaller less chirped pulse component with a central frequency below (Fig. 15.35). As in theNLSE the relatively low chirped, temporally less intensive second component dominates the spec-trum. Aspects of this mixed dynamics of OBE and NLSE which occurs for the full SBE-solution arefound experimentally (Fig. 15.13)[87].


1

0

-1-4 -2 0 2 4

Time [ps]

Figure 15.34: OBE: Field amplitude (red), phase φ(t) (green) and ddt φ(t) (blue) after nonlinear prop-

agation. The centers of the individual pulse components are marked on the instantaneous frequencyby black dots and the angle representing the chirp are marked by dashed lines. Very pronounced1π-phase-jump between the first and second component. The intensity of the equally chirped tem-poral pulse components with central frequencies above and below the incident light frequency ω0correspond to the spectral intensities of these pulse components.

1

0

-1-4 -2 0 2 4

Time [ps]

1

0

-1-4 -2 0 2 4

Time [ps]

Figure 15.35: SBE: Field amplitude (red), phase φ(t) (green) and ddt φ(t) (blue) after nonlinear propa-

gation. The centers of the individual pulse components are marked on the instantaneous frequency byblack dots and the angle representing the chirp are marked by dashed lines.


15.2.4.8 Validity of Model Systems

All three models are qualitatively valid for linear propagation, the OBE however slightly underesti-mates the linear dispersion. The linear spectra at the excitonic resonance are nearly identical for theSBE and OBE. The refractive index of the OBE-and SBE-model differ noticeably even at the excitonicresonance.

For nonlinear excitation all three models exhibit pulse compression but only the OBE and SBEresult in the distinctive shape observed in the experiment[87]. The OBE and NLSE underestimates thenonlinear coupling considerably as the pulse compression is only visible at higher intensities than inthe SBE-model which reaches agreement on the pulse intensity with the experiment[87]. The NLSEand SBE exhibit phase characteristics typical for pulse breakup while the OBE phase characteristicshas signatures of beating phenomena. Only the SBE-simulation is capable of correctly reproducingthe spectral features observed in the experiment.

The NLSE only considers nonlinearities up to the third order in the field but includes no memoryeffects. This is apparently not a got approximation for near-resonant highly nonlinear excitation. In thelimit δ ≥ 1 memory effects must be taken into account to describe the proper self-phase modulation.Therefor the The OBE-model is partly suitable for obtaining (numerically) quick results but doesnot reproduce the correct phase. It is known that for resonant excitation the continuum states areplaying an important role (cp. section 9.1). This holds true for near resonant excitation as well as theSBE-simulations shows features not included in the other models.

The conclusion therefore is: The NLSE-model is not suitable for strong near resonant excitationas it does not exhibit the asymmetric pulse shape seen in the experiment. The OBE-model is partlysuitable for obtaining (numerically) quick results in the time domain but does not reproduce the correctphase or spectrum. The best of the three models is the SBE.


Appendix

191

Appendix A

Commutator Relations

To derive the equations of motion for the density matrix of a quantum-mechanical system one has toevaluate the Heisenberg-equation 4.4 (as shown in chapter 4). The Heisenberg-equation incorporatescommutators between the operators of interest and the Hamiltonian-operator of the system. In mostcases these commutators can be reduced to a general form. In this appendix the most importantcommutators of this work are shown.

A.1 Basic Commutators

The basic commutators for fermionic creation and annihilation operators a†1 and a2 (where 1 and 2 are

compound indices incorporating wavenumber, spin, band-index etc.) are given by:

[a1,a†2]+ = a1a†

2 +a†2a1 = δ1,2

[a1,a2]+ = a1a2 +a2a1 = 0

[a†1,a

†2]+ = a†

1a†2 +a†

2a†1 = 0 . (A.1)

For bosonic creation and annihilation operators b†1 and b2 the anti-commutator [ , ]+ has to be substi-

tuted by the commutator [ , ]−:

[b1,b†2]− = b1b†

2 −b†2b1 = δ1,2

[b1,b2]− = b1b2 −b2b1 = 0

[b†1,b

†2]− = b†

1b†2 −b†

2b†1 = 0 . (A.2)

A.1.1 General Fermionic Commutators

From the basic fermionic commutators (A.1) general commutators quantities consisting of multipleoperators can be computed. The most commonly used are

[a†1a2,a

†3a4]− = δ2,3a†

1a4 −δ1,4a†3a2 (A.3)

[a†1a†

2a3a4,a†5a6]− = +δ3,5a†

1a†2a6a4 −δ6,2a†

1a†5a3a4 (A.4)

+δ4,5a†1a†

2a3a6 −δ1,6a†5a†

2a3a4

192 A. COMMUTATOR RELATIONS

[a†1a†

2a3a4,a†5a†

6a7a8]− = +δ2,7a†1a†

5a†6a3a4a8 −δ3,6a†

1a†2a†

5a4a7a8 (A.5)

+δ4,6a†1a†

2a†5a3a7a8 −δ2,8a†

1a†5a†

6a3a4a7

+δ4,5a†1a†

2a†6a3a7a8 −δ1,8a†

2a†5a†

6a3a4a7

+δ3,5a†1a†

2a†6a4a7a8 −δ1,7a†

2a†5a†

6a3a4a8

+δ1,8δ2,7a†5a†

6a3a4 −δ4,5δ3,6a†1a†

2a7a8

+δ1,7δ2,8a†5a†

6a3a4 −δ3,5δ4,6a†1a†

2a7a8 .

193

Appendix B

Markov-Approximation

B.1 Markov-Approximation

The Markov-approximation (e.g. [99]) is a technique to solve coupled differential equations by ne-glecting memory effects. If applied to equations of motions of a scattering problem this lack ofmemory corresponds to the strict enforcement of energy conservation which otherwise has not strictlyto be obeyed. We are considering a general system of two differential equations of the form

x(t) = iωxx(t)+ icxy(t) (B.1)

y(t) = iωyy(t)+ icyx(t) . (B.2)

The second equation is formally solved by

y(t) = y(t0)eiωyt +

Z t

t0icyx(t ′)eiωy(t−t ′)dt ′ . (B.3)

Assuming the initial time to be t0 = −∞ and the boundary condition to be y(t0) = y(−∞) = 0we get

y(t) = icyeiωytZ t

t0x(t ′)e−iωyt ′dt ′ . (B.4)

Substituting t ′ = t − s yields

y(t) = icyeiωytZ ∞

0x(t − s)e−iωy(t−s)ds . (B.5)

In this equation the memory kernel of the integral is neglected, i.e. the s-dependent part of thefunction x(t − s) is replaced by its free rotation

x(t − s) ≈ x(t) · e−iωxs (B.6)

resulting in

y(t) = icyeiωytZ ∞

0x(t) · e−iωxse−iωy(t−s)ds (B.7)

= icyx(t)Z ∞

0e−i(ωx−ωy)sds (B.8)

= cyx(t)ζ(ωx −ωy) (B.9)

194 B. MARKOV-APPROXIMATION

with the generalized delta function (or Heitler-Zeta-function) ζ(ω) = limγ→0i

ω+iγ = limγ→0γ+iω

ω2+γ2 =

πδ(ω)+ iPV ( 1ω) and PV being the Cauchy-principal value.

195

Appendix C

Equations of Motion for GeneralizedCoulomb Hamiltonians

In this chapter the equations of motion for a generalized two-particle interaction between identicalfermions will be derived. These equations are utilized in this work to describe the Coulomb-interactionof Bloch-electrons.

C.1 Generalized Hamiltonian

The generalized form of 5.37 is

H =12 ∑

1234V 12

34 a†1a†

2a3a4 (C.1)

where the indices 1,2,3,4 are compound indices consisting of all relevant quantum numbers.

C.2 Generalized Coulomb-Matrix-Element

The Coulomb-matrix-element is given by the Coulomb-potential and the wave functions of the initial(φ3 and φ4) and final states(φ1 and φ2) (cp. chapter 5.3.3):

V 1234 =

Z

dxZ

dx′φ1(x)∗φ2(x′)∗e2

|x− x′|φ4(x)φ3(x′) . (C.2)

196 C. EQUATIONS OF MOTION FOR GENERALIZED COULOMB HAMILTONIANS

V1 2

34

1

2

3

4

Figure C.1: Schematic two particle interaction mediated by the matrix-element V 1234 .

C.2.1 Symmetry

For this matrix element two fundamental symmetries can be shown. The first follows by exchangingx and x′:

V 1234 =

Z

dxZ

dx′φ1(x)∗φ2(x′)∗e2

|x− x′|φ4(x)φ3(x′) (C.3)

x↔x′=

Z

dx′Z

dxφ1(x′)∗φ2(x)∗e2

|x′− x|φ4(x′)φ3(x) (C.4)

=

Z

dxZ

dx′φ2(x)∗φ1(x′)∗e2

|x− x′|φ3(x)φ4(x′) (C.5)

= V 2143 . (C.6)

The second symmetries is obtained by considering the complex conjugated of the matrix element:

(V 1234 )? = (

Z

dxZ

dx′φ1(x)∗φ2(x′)∗e2

|x− x′|φ4(x)φ3(x′))? (C.7)

=Z

dxZ

dx′φ1(x)φ2(x′)e2

|x− x′|φ4(x)?φ3(x′)? (C.8)

=

Z

dxZ

dx′φ4(x)?φ3(x′)?e2

|x− x′|φ1(x)φ2(x′) (C.9)

= V 4321 = V 34

12 . (C.10)

C.3 Equations of Motion

Using the Heisenberg-equation (4.4) the equation of motion for a generic two operator expectationvalue follows as

∂t〈a†5a6〉 =

i~[a†

5a6,H]− (C.11)

=i~

∑123

V 2135 〈a†

2a†1a3a6〉−

i~

∑134

V 1643 〈a†

5a†1a3a4〉 . (C.12)

C.3. EQUATIONS OF MOTION 197

Performing a factorization according to 6.6 in Hartree-Fock-approximation yields

∂t〈a†5a6〉 =

i~

∑123

〈a†1a2〉(V 31

25 〈a†3a6〉+V 16

23 〈a†5a3〉) (C.13)

with the matrix element V 2234 = V 12

34 −V 2134 under utilization of all symmetries.

198 C. EQUATIONS OF MOTION FOR GENERALIZED COULOMB HAMILTONIANS

199

Appendix D

Transfer-Matrix for N Quantum Wells

The goal of this section is to derive equations which relate the electric field at an arbitrary quantumwell inside a multiple quantum well structure (Fig. 2.1) with the incident light field. The light fieldsat all other quantum wells are eliminated from the equation. The effect of the polarization of QW m(which is located at zM) at an arbitrary position z is retarded due to the finite velocity of electromag-netic waves by z−zm

c :

P2Dm (z, t) = P2D

m (t − z− zm

c) . (D.1)

The conditions of continuity (2.3) and (2.9) for n = 0 read

E+0 (t − z

c)+E−

0 (t +zc)−E+

1 (t − zc)−E−

1 (t +zc) = 0 (D.2)

−E+1 (t − z

c)+E−

1 (t +zc)+E+

0 (t − zc)−E−

0 (t +zc) = 0 (D.3)

and for n = 1

E+1 (t − z

c)+E−

1 (t +zc)−E+

2 (t − zc)−E−

2 (t +zc) = 0 (D.4)

−E+2 (t − z

c)+E−

2 (t +zc)+E+

1 (t − zc)−E−

1 (t +zc) =

1εc

∂tP2D1 (t − z− z1

c) . (D.5)

Adding (D.2) and (D.4) as well as (D.3) and (D.5) one gets

E+0 (t − z

c)+E−

0 (t +zc)−E+

2 (t − zc)−E−

2 (t +zc) = 0 (D.6)

−E+2 (t − z

c)+E−

2 (t +zc)+E+

0 (t − zc)−E−

0 (t +zc) =

1εc

∂tP2D1 (t − z− z1

c) . (D.7)

With the conditions of continuity for n = 2

E+2 (t − z

c)+E−

2 (t +zc)−E+

3 (t − zc)−E−

3 (t +zc) = 0 (D.8)

−E+3 (t − z

c)+E−

3 (t +zc)+E+

2 (t − zc)−E−

2 (t +zc) =

1εc

∂t P2D2 (t − z− z1

c) (D.9)

one gets in the same manner from (D.6) and (D.7) to

E+0 (t − z

c)+E−

0 (t +zc)−E+

3 (t − zc)−E−

3 (t +zc) = 0 (D.10)

−E+3 (t − z

c)+E−

3 (t +zc)+E+

0 (t − zc)−E−

0 (t +zc) =

1εc

2

∑m=1

∂tP2Dm (t − z− zm

c) . (D.11)

200 D. TRANSFER-MATRIX FOR N QUANTUM WELLS

This process can be repeated infinitely. This leads to the universal expression

E+0 (t − z

c)+E−

0 (t +zc)−E+

n (t − zc)−E−

n (t +zc) = 0 (D.12)

−E+n (t − z

c)+E−

n (t +zc)+E+

0 (t − zc)−E−

0 (t +zc) =

1εc

n

∑m=1


c) . (D.13)

Sorting the forward and backward propagating fields in (D.12) and (D.13) results in

E+0 (t − z

c)−E+

n (t − zc) = E−

n (t +zc)−E−

0 (t +zc) (D.14)

and

E+0 (t − z

c)−E+

n (t − zc)+E−

n (t +zc)−E−

0 (t +zc) =

1εc

n

∑m=1


c) . (D.15)

The backward propagating term in (D.15) can be eliminated with(D.14):

E+0 (t − z

c)−E+

n (t − zc) =

12εc

n

∑m=1


c) . (D.16)

Resorting (D.16) for E+n (t − z

c) yields

E+n (t − z

c) = E+

0 (t − zc)− 1

2εc

n

∑m=1


c) . (D.17)

An analogous calculation for E−n (t + z

c) leads to

E−n (t +

zc) = E−

N (t +zc)− 1

2εc

N

∑m=n+1

∂tP2Dm (t +

z− zm

c) . (D.18)

With (D.17) and (D.18) the transmitted and reflected waves can be directly computed from the incidentwaves and the polarizations inside the quantum wells.

201

Appendix E

Equations of Motion for Quantum Dot onWetting Layer

The quantities of interest are

quantum dotcoherence p23 =< a†

2a3 >



quantum wellinterband coherence pcv¸k =< a†

ckavk >

occupation conduction band fck =< a†ckack >

occupation valence band fvk =< a†vkavk >

QD-phononphonon assisted density-matrix Sq =< a†

2a3bq >c

phonon assisted density-matrix Tq =< a†2a3b†

−q >c

QD-QWdot-well polarization Uk =< a†

2avk >

dot-well polarization Vk =< a†3ack >

202 E. EQUATIONS OF MOTION FOR QUANTUM DOT ON WETTING LAYER

E.1 Quantum Dot Quantities

˙p23 = (−i∆QD − γQD) p12 + id23

2~E(1− f2 − f3)+

i~

∑q

gq12(Sq + Tq) (E.1)

+i~

∑k1k3

fck1Uk3W3,ck1vk3,ck1

f2 = −2γQD f2 +2Im(d23

2~Ep23)+

i~

∑k1k3

pcvk1U?k3

(W 2,vk1ck1 ,vk3

)? (E.2)

f3 = −2γQD f3 +2Im(d23

2~E? p23)+

i~

∑k1k3

p?cvk1

V ?k3

(W 3,ck1vk1 ,ck3

)? (E.3)

˙Sq = (−i∆QD − iωq − γQD)Sq +i~

g−q23 p23(nq +1) (E.4)

˙Tq = (−i∆QD + iωq − γQD)Tq +i~

g−q23 p23n−q (E.5)

W abcd = W ab

cd −W bacd (E.6)

with the quantum dot polarization P23 = d23 p23 and the equilibrium distribution of phononsnq = 1

e~vsound h

kBT −1(Bose-distribution).

E.2 Wetting Layer Quantities

˙pcvk = (−i~k2

2mr− γQW ) pcvk + i

dcv

2~E(1− fck − fvk) (E.7)

+ ipcvk ∑q6=0

Vq

~(− fck+q − fvk+q)+ i(1− fck − fvk) ∑

q6=0

Vq

~pcvk+q

+i~

∑k1

Vk1 fvk(W 3,ckvk,ck1

)? +i~

∑k2

Uk2 fvkW 2,vkvk,vk2

+i~

∑k1

V ?k fck!(W

3,ck1vk,ck1

)?

fck = 2Im(dcv

2~E? pcvk + pcvk ∑

q6=0

Vq

~p?

cvk+q)−2γQW fck (E.8)

+i~

∑k1

Vk1 p?cvk(W 3,ck

vk,ck1)? +

i~

∑k2

Uk2 p?cvkW 2,vk

ck,vk2

+i~

∑k1

Vk pcvk1W3,ck1vk1,ck

fck = 2Im(dcv

2~E? pcvk + pcvk ∑

q6=0

Vq

~p?

cvk+q)−2γQW fvk (E.9)

+i~

∑k1

U?k1

pcvk(W 2,vkck,vk1

)? +i~

∑k2

Vk2 pcvkW 3,ckvk,ck2

+i~

∑k1

Uk p?cvk1

W 2,vk1ck1,vk

E.3. QUANTUM DOT - WETTING LAYER QUANTITIES 203

with the wetting layer (quantum well) polarization: Pvc = dcv ∑k

pcv,k .

E.3 Quantum Dot - Wetting Layer quantities

˙Uk = i(− ~k2

2mh−ωc2)Uk +

i~

∑k1

pcvk1 fvk(W 2,vk1ck1,vk)? (E.10)

+i~

∑k1

fck1 p23(W3,ck1vk1,ck)? + γUUk

˙Vk = −i(~k2

2me+ωv3)Vk +

i~

∑k1

p?cvk1

fck(W 3,ck1vk1,ck)? (E.11)

+i~

∑k1

fvk1 p?23(W

2,vk1ck,vk1

)? + γVVk .

204 E. EQUATIONS OF MOTION FOR QUANTUM DOT ON WETTING LAYER

205

Appendix F

Instantaneous Frequency

In this appendix the instantaneous frequency of a Gaussian pulse under the influence of polarizationin the adiabatic following limit but incorporating first order memory effects is computed. In chapter 1it has been shown that the instantaneous frequency can be written as

ω(t) = ∂tφ(t) (F.1)

= ∂t(1c

ω0nz−ω0t) (F.2)

∂t(1c

ω0z

√1+

d P~ε0E

−ω0t) . (F.3)

In chapter 10 the polarization of a semiconductor expanded in the electric field Ω and incorporatingfirst order memory effects has be calculated to be

P = −αΩ+βΩ3 − γΩΩ2 . (F.4)

F.1 Refractive index

The polarization (F.4) is inserted in the refractive index in eq. (F.3)

n =

√1+

d P~ε0E

(F.5)

=

√

1+(−α+βΩ2 − γΩ2)d

ε0~

=

√1− dα

~ε0+

dβ~ε0

Ω2 − dγ~ε0

Ω2

and expanded in Ω

n = ∑n

∂nΩn(Ω)|Ω=0

Ωn

n!. (F.6)

206 F. INSTANTANEOUS FREQUENCY

With the derivatives

∂0Ωn(Ω)|Ω=0 =

√1− dα

~ε0− dγ

~ε0E2 (F.7)

∂1Ωn(Ω)|Ω=0 = 0 (F.8)

∂2Ωn(Ω)|Ω=0 =

dβ~ε0√

1− dα~ε0

− dγ~ε0

Ω2(F.9)

the second order expansion of the refractive index is

n ≈√

1− dα~ε0

− dγ~ε0

E2 +

dβ~ε0√

1− dα~ε0

− dγ~ε0

Ω2Ω2

= n0 +dβ

~ε0n0E2 = n0 +n2E2 . (F.10)

F.2 Instantaneous Frequency

A Gaussian pulse is considered

Ω(z, t) = Ω(z)e−t2

τ2 cos(ω0nc0

z−ω0t) (F.11)

with the phase

ϕ =ω0nc0

z−ω0t . (F.12)

With (F.10) the instantaneous frequency (F.3) gets

ω(t) =ω0zc0

∂t n−ω0 (F.13)

=ω0zc0

∂t(

√1− dα

~ε0− dγ

~ε0Ω2 +

dβ~ε0√

1− dα~ε0

− dγ~ε0

Ω2Ω2)−ω0 (F.14)

=ω0zc0

∂t(

√1− dα

~ε0− dγ

~ε0

4t2

τ4 e−2 t2τ2 +

dβ~ε0√

1− dα~ε0

− dγ~ε0

4t2

τ4 e−2 t2

τ2

e−2 t2

τ2 )−ω0 (F.15)

=ω0zc0

(− dγ

~ε0

8tτ4 e−2 t2

τ2 − dγ~ε0

4t2

τ4 e−2 t2

τ2 (−4 tτ2 )

2

√1− dα

~ε0− dγ

~ε0

4t2

τ4 e−2 t2

τ2

(F.16)

−dβ~ε0

(2 dγ~ε0

tτ4 e−2 t2

τ2 +2 dγ~ε0

4t3

τ4 (−2 1τ2 )e

−2 t2

τ2 )

2

√(1− dα

~ε0− dγ

~ε0

4t2

τ4 e−2 t2

τ2 )3

e−2 t2

τ2

+

dβ~ε0

(4 tτ2 )√

1− dα~ε0

− dγ~ε0

4t2

τ4 e−2 t2

τ2

e−2 t2

τ2 )−ω0 .

F.2. INSTANTANEOUS FREQUENCY 207

Figure F.1: Typical instantaneous frequency of a Gaussian pulse due to SPM.

To distinguish between the different contributions they are considered separately:

F.2.1 Only linear contributions

ωlinear(t) = −ω0 (F.17)

∝ const.

No influence due to linear contributions.

F.2.2 Only Kerr nonlinearity

ωKerr(t) =ω0zc0

dβ~ε0

4t

τ2 e−2 t2

τ2 (F.18)

=16d4ωlz

~3ω30ε0c0τ2

tI

∝ t · I

Classic SPM with S-shaped instantaneous frequency (Fig. F.1)

F.2.3 Memory Contributions

ωmemory(t) =ω0zc0

(− dγ

~ε0

8tτ4 e−2 t2

τ2 + dγ~ε0

4t2

τ4 e−2 t2

τ2 (4 tτ2 )

2

√1− dγ

~ε0

4t2

τ4 e−2 t2τ2

) (F.19)

=ω0zc0

(− dγ

~ε0

8tτ4 I + dγ

~ε0

4t2

τ4 I(4 tτ2 )

2√

1+ dγ~ε0

4t2

τ4 I)

Expanding the instantaneous frequency in terms of I yields (without coefficients):

ωmemory(t) ≈ (t −2t3)I − (t3 −2t5)I2 +O3(I) (F.20)

208 F. INSTANTANEOUS FREQUENCY

Figure F.2: Typical instantaneous frequency of a Gaussian pulse due to first order memory effects.

Figure F.3: The interplay of Kerr nonlinearity and first order memory effects can emphasised or dampthe pure Kerr induced instantaneous frequency.

The memory effects cause a mixing of terms proportional to t 3 into the first order in I of the instanta-neous frequency. An example is shown in Fig. F.2.

The combination of both Kerr nonlinearity and first order memory effects results in an interplaybetween both contributions which vary with the signs and magnitudes of the coefficients β and γ: TheKerr typical S-shape can be emphasised or damped as depicted in Fig. F.3.

209

Appendix G

Material Parameters

For calculations concerning GaAs bulk and InGaAs quantum wells the following parameters wereused in this work.

d1,2 dipole matrix-element of the quantum well excitondcv dipole matrix-element of the bulk excitonεr relative dielectric constant of GaAsωg band gap frequencyγ0 phenomenological damping constantme relative mass of Bloch-electrons in the conduction band of a InGaAs quantum wellmhh relative mass of heavy holes in the valence band of a InGaAs quantum wellσ oscillator density for TLS-simulationsL width of a quantum well.

The dipole moment d1,2 and the phenomenological damping γ0 are fitted to reproduce the referenceexperiment of C. Ell (Optical Sciences Center, University of Arizona, Tucson) [39] on the sampleDBR13 (Fig. G.1).

unit valued1,2 e nm 0.42dcv e nm 0.3εr 12.7ωg fs−1 2.309γ0 meV 0.55me m0 0.067mhh m0 0.377σ cm−2 4.1×1012

L nm 8.5

Table G.1: Material parameters for InGaAs Quantum wells

210 G. MATERIAL PARAMETERS

energy [meV]

ab

so

rpti

on

[1/n

m]

energy [meV]

ab

so

rpti

on

[1/n

m]

Figure G.1: Two level Hartree-Fock-simulation, 30 quantum wells: fitting to reference experiment([39] Fig. 1).

211

Appendix H

Acronyms, Notations and Symbols

H.1 Acronyms

Acronyms expansionABS absorbing boundary conditions

DFWM degenerate four-wave mixingFDTD fnite-difference time domain

FWHM full width at half maximumGaAs gallium-arsenid

hh heavy holeHWHM half width at half maximum

LA longitudinal accoustic phononLHS left hand side (equation)LO longitudinal optical

MQW multiple quantum wellNLSE nonlinear Schrödinger equationOBE optical Bloch equationsQD quantum dotQW quantum well

RABR resonant obsorbing Bragg reflectorRHS right hand side (equation)RWA rotating wave approximationSBE semiconductor Bloch equationSIT self-induced transmission / transparancySHG second harmonic generation

SMBE semiconductor Maxwell-Bloch equationsSVEA slowly varying envelop approximationTLS two level system

UPML uniaxial perfectly matched layerWL wetting layer

XFROG cross-correlation frequency-resolved optical gating

212 H. ACRONYMS, NOTATIONS AND SYMBOLS

H.2 NotationsNotation meaning

∂x∂∂x

f temporal derivative of fr scalarr vectorrc vector in confined dimensionr f vector in free dimensoon

H.3 Symbols

Symbol meaning

Ω Rabi-frequency Ω = dcvE~

F slowly varying envelope of FF Fourier-transform of Fr vectorrc vector in confined dimensionr f vector in free dimensoon

a,a† electron creation and annihilation operatorsb,b† phonon creation and annihilation operatorsc,c† photon creation and annihilation operatorsx,x† exciton creation and annihilation operators

BIBLIOGRAPHY 213

Bibliography

[1] L. Allen, J. H. Eberly. Optical resonance and two-level atoms. Dover Publ., New York, 1987.

[2] K. Busch, S. John. Photonic band gap formation in certain self-organizing systems. PhysicalReview E, 58(3):3896, 1998.

[3] B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, J. E. Sipe. Bragg grating solitons.Physical Review Letters, 76(10):1627, 1996.

[4] Govind P. Agrawal. Nonlinear Fiber Optics. Academic Press, Boston, 1989.

[5] John David Jackson. Classical Electrodynamics. John Wiley & Sons, New York, wydanie 3rd,1999.

[6] H. Haug, S. W. Koch. Quantum theory of the optical and electronic properties of semiconductors.World Scientific, Singapore, wydanie 3rd, 1994.

[7] J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison. Nonlinear optical beaminteraction in waveguide arrays. Physical Review Letters, 93:093903, 2004.

[8] Y. Kartashov, L.-C. Crasovan, A. S. Zelenina, V. A. Vysloukh, A. Sanpera, M. Lewenstein,L. Torner. Soliton eigenvalue control in optical lattices. Physical Review Letters, 93:143902,2004.

[9] A. Hasegawa. Generation of a train of soliton pulses by induced modulation instability in opticalfibers. Optics Letters, 9(7):288, 1984.

[10] P. K. Shukla, B. Eliasson. Localization of intense electromagnetic waves in a relativistically hotplasma. Physical Review Letters, 94:065002, 2005.

[11] K. D. Moll, A. K. Gaeta, G. Fibich. Self-similar optical wave collapse: observation of theTownes profile. Physical Review Letters, 90:203902, 2003.

[12] G. G. Luthers, A. C. Newell, J. V. Moloney, E. M. Wright. Short-pulse conical emission andspectral broadening in normally dispersive media. Optics Letters, 19(11):789, 1994.

[13] A. R. Osborne, M. Onorato, M. Serio. The nonlinear dynamics of rogue waves and holes indepp-water gravity wave trains. Physics Letters A, 275(5):386, 2000.

[14] M. Schaarschmidt, A. Knorr. Impact of off-resonant nonlinearities on pulse propagation indielectric bulk material - limits of the nonlinear Schrödinger equation. submitted to JOSA-B,2006.

214 BIBLIOGRAPHY

[15] C. Cohen-Tannoudji, B. Diu, F. Laloë. Quantenmechanik. Walter de Gruyter, Berlin, 1999.

[16] Tineke Warncke. Coherent dynamics of radiatively coupled quantum well excitons. VerlagGörich & Weiershäuser GmbH, Marburg, 1996. Dissertation.

[17] Tomobumi Mishina. Forward and backward self-consistent approach to coherent-light propaga-tion in stratified structures. Physical Review B, 54(19):13630, 1996.

[18] M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, K. Ploog. Collective effectsof excitons in multiple-quantum-well Bragg and anti-Bragg structures. Physical Review Letters,76(22):4199, 1996.

[19] T. Stroucken, A. Knorr, P. Thomas, S. W. Koch. Coherent dynamics of radiatively coupledquantum-well excitons. Physical Review B, 53(4):2026, 1996.

[20] Allen Taflove, Susan C. Hagness. Computational electrodynamics: the finite-difference time-domain method. Artech House Publishers, Boston, wydanie 2nd, 2000.

[21] K.S. Yee. Numerical solution of initial boundary value problems involving Maxwell’s equationsin isotropic media. IEEE Trans. Antennas and Propagation, 14:302–307, 1966.

[22] Kai Zimmermann. Zur Theorie der elektromagnetischen Kopplung in periodischenQuantenpunkt-Strukturen. Technische Universität Berlin, Berlin, 2003. Diplomarbeit.

[23] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg. Photons and Atoms. John Wiley & Sons,Chichester, wydanie 1rd, 1989.

[24] D. P. Craig, T. Thirunamachandran. Molecular quantum elecrodynamics: An Introduction toRadiation-Molecule Interactions. Academic Press, London, 1984.

[25] O. Madelung. Introduction to Solid-State Theory. Springer, 1996.

[26] F. Bloch. Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Physik, 52:555–600,1928.

[27] Hermann Haken. Quantenfeldtheorie des Festkörpers. Teubner, Stuttgart, 1973.

[28] A. S. Davydov. Quantum Mechanics. Elsevier Science, 1966.

[29] S. Mukamel. Principles of nonlinear optical spectroscopy. Oxford University Press, New York,1995.

[30] Stefan Butscher. Theory of Non-markovian Intersubband Dynamics of Semiconductor QuantumWells. Technische Universität Berlin, Berlin, 2004. Diplomarbeit.

[31] W. Vogel, D.-G. Welsch. Lectures on Quantum Optics. Akademie Verlag, Berlin, 1994.

[32] M. Richter, M. Schaarschmidt, A. Knorr, W. Hoyer, J. V. Moloney, E. M. Wright, M. Kira,S. W. Koch. Quantum theory of incoherent THz emission of an interacting electron-ion plasma.Physical Review A, 71(5):053819, 2004.

[33] J. J. Hopfield. Theory of the contribution of excitons to the complex dielectric constant ofcrystals. Phys. Rev., 112(5):1555, 1958.

BIBLIOGRAPHY 215

[34] J. Förstner, C. Weber, J. Dankwerts, A. Knorr. Phonon-assisted damping of Rabi oscillations insemiconductor quantum dots. Physical Review Letters, 91:127401, 2003.

[35] Peter Y. Yu, Manuel Cardona. Fundamentals of Semiconductors. Springer, wydanie second,1996.

[36] W.W. Chow, S.W. Koch. Semiconductor-Laser Fundamentals - Physics of the Gain Materials.Springer, 1999.

[37] Gerald D. Mahan. Many-Particle Physics. Plenum Press, 1981.

[38] N.W. Ashcroft, N.D. Mermin. Solid State Physics. Holt, Rinehart and Winston, Philadelphia,1976.

[39] C. Ell, J. Prineas, T. R. Nelson Jr., S. Park, H. M. Gibbs, G. Khitrova, S. W. Koch, R. Houdré.Influence of structural disorder and light coupling on excitonic response of semiconductor mi-crocavities. Physical Review Letters, 80(21):4795, 1998.

[40] W. Schäfer, M. Wegener. Semicoductor Optics and Transport Phenomena. Springer Verlag,Berlin, 2002.

[41] S. L. McCall, E. L. Hahn. Self-induced transparency by pulsed coherent light. Physical ReviewLetters, 18:908, 1967.

[42] Leo Kadanoff, Gordon Baym. Quantum Statistical Mechanics. Addison Wesley PublishingCompany, 1994.

[43] Andreas Knorr. Spatial effects in the Ultrafast Optics and Kinetics of Semiconductor. Philipps-Universität Marburg, Marburg, 1997. Habilitationsschrift.

[44] Inés Waldmüller. Intersubband Dynamics in Semiconductor Quantum Wells - Linear and Non-linear Response of Quantum Confined Electrons. Technische Universität Berlin, Berlin, 2005.Dissertation.

[45] A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, G. Abstreiter. Coherent properties of atwo-level system based on a quantum-dot photodiode. Nature, 418:612–614, 2002.

[46] T. Unold, K. Mueller, C. Linau, T. Elsaesser, A. D. Wieck. Optical control of excitons in a pairof quantum dots coupled by the dipole-dipole interaction. Physical Review Letters, 94:137404,2005.

[47] J. M. Villas-Bôas, S. E. Ulloa, A. O. Govorovl. Decoherence of Rabi oscillation in a singlequantum dot. Physical Review Letters, 94:057404, 2005.

[48] Ermin Malic. Dynamic Theory of Semiconductor Quantum Dot Lasers. Technische UniversitätBerlin, Berlin, 2005. Diplomarbeit.

[49] T. R. Nielsen, P. Gartner, F. Jahnke. Many-body theory of carrier capture and relaxation insemiconductor quantum dot lasers. Physical Review B, 69:235314, 2004.

[50] Carsten Weber. On the Theory of Electron-Phonon Coupling and the Optical Linewidth in Semi-conductor Quantum Dots. Technische Universität Berlin, Berlin, 2002. Diplomarbeit.

216 BIBLIOGRAPHY

[51] Marten Richter. Quantentheorie der Terahertz-Emission laserinduzierter Plasmen. TechnischeUniversität Berlin, Berlin, 2005. Diplomarbeit.

[52] M. O. Scully, M. S. Zubairy. Quantum Optics. Cambridge University Press, Cambridge, 1997.

[53] M. Kira, F. Jahnke, W. Hoyer, S. W. Koch. Quantum theory of spontaneous emission and coher-ent effects in semiconductor microstructures. Progress in Quantum Electronics, 23(6):189–279,1999.

[54] E. Fick, G. Sauermann. The Quantum Statistics of Dynamic Processes. Springer, Berlin, 1990.

[55] L. C. Andreani. Exciton-polaritons in superlattices. Physics Letters A, 192(1):99, 1994.

[56] D. S. Citrin. Material and optical approaches to exciton polaritons in multiple quantum wells:Formal results. Physical Review B, 50(8):5497, 1994.

[57] I. Waldmüller, J. Förstner, S. C. Lee SC, A. Knorr, M. Woerner, K. Reimann, R. A. Kaindl,T. Elsaesser, R. Hey, K. H. Ploog. Optical dephasing of coherent intersubband transitions in aquasi-two-dimensional electron gas. Physical Review B, 69:205307, 2004.

[58] M. Lindberg, S. W. Koch. Effective Bloch equations for semiconductors. Physical Review B,38(5):3342, 1988.

[59] J. Förstner, K. J. Ahn, J. Danckwerts, M. Schaarschmidt, I. Waldmüller, C. Weber, A. Knorr.Light propagation- and many-particle-induced non-Lorentzian lineshapes in semiconductornanooptics. Physica status solidi B, 234(1):155–165, 2002.

[60] J. Kuhl, M. Hübner, D. Ammerlahn, T. Strouken, B. Grote, S. Haas, S. W. Koch, G. Khitrova,H. M. Gibbs, R. Hey, K. Ploog. Superradiant exciton/light coupling in semiconductor het-erostructures - part ii: Experiments. Advances in Solid State Physics, 38:281, 1998.

[61] N. C. Nielsen, J. Kuhl, M. Schaarschmidt, J. Förstner, A. Knorr, S. W. Koch, G. Khitrova, H. M.Gibbs, H. Giessen. Linear and nonlinear pulse propagation in multiple-quantum-well photoniccrystal. Physical Review B, 70, 2004.

[62] I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, W. D: Phillips. Photonic band gaps in opticallattices. Physical Review A, 52(2):1394, 1995.

[63] T. Ikawa, K. Cho. Fate of the superradiant mode ina resonant Bragg reflector. Physical ReviewB, 66:085338, 2002.

[64] Leon Brillouin. Wave Propagation in Periodic Structures. Dover Publ., New York, 2003.

[65] N. C. Nielsen, J. Kuhl, M. Schaarschmidt, J. Förstner, A. Knorr, S. W. Koch, H. M. Gibbs,G. Khitrova, H. Giessen. Pulse propagation in Bragg-resonant multiple quantum wells: frompulse breakup to compression. Physica status solidi C, 5(1):1484–1487, 2003.

[66] D. Fröhlich, A. Kulik, B. Uebbing, A. Mysyrowicz, V. Langer, H. Stolz, W. von der Osten.Coherent propagation and quantum beats of quadrupole polaritons in Cu2O. Physical ReviewLetters, 67(17):2343, 1991.

[67] M. D. Crisp. Propagation of small-area pulses of coherent light through a resonant medium.Physical Review A, 1(6):1604, 1970.

BIBLIOGRAPHY 217

[68] D. S. Kim, J. Shah, D. A. B. Miller, T. C. Damen, A. Vinattieri. Femtosecond pulse distortion inGaAs quantum wells and its effect on pump-probe or four-wave-mixing experiments. PhysicalReview B, 50:18240, 1994.

[69] J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, S. W. Koch. Exciton-polaritoneigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well peri-odic structures. Physical Review B, 61(20):13863, 2000.

[70] D. Ammerlahn, J. Kuhl, B. Grote, S. W. Koch, G. Khitrova, H. Gibbs. Collective radiative decayof light- and heavy-hole exciton polaritons in multiple-quantum-well structure. Physical ReviewB, 62:7350, 2000.

[71] B. I. Mantsyzov, R. N. Kuz’min. Coherent interaction of light with a discrete periodic resonantmedium. Sov. Phys. JETP, 64(37), 1986.

[72] Manfredo P. do Carmo. Differential geometry of curves and surfaces. Vieweg Verlag, Braun-schweig, wydanie 3rd, 1993.

[73] Heriot-Watt University Dugald Duncan. http://www.ma.hw.ac.uk/solitons.

[74] Chaohao Gu. Soliton theory and its applications. Springer Verlag, Berlin, 1995.

[75] M. Schaarschmidt, J. Förstner, A. Knorr, J.P. Prineas, N. C. Nielsen, J. Kuhl, G. Khitrova, H. M.Gibbs, H. Giessen, S. W. Koch. Adiabatically driven electron dynamics in a resonant photonicband gap: Optical switching of a Bragg periodic semiconductor. Physical Review B, 70:233302,2005.

[76] R. Binder, S. W. Koch, M. Lindberg, N. Peyghambarian. Ultrafast adiabatic following in semi-conductors. Physical Review Letters, 65(7):899, 1990.

[77] J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, A. Knorr. Ultrafastac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductorquantum wells. Applied Physics Letters, 81(23):4332, 2002.

[78] J. Förstner, A. Knorr, S. Kuckenburg, T. Meier, S. W. Koch, H. Giessen, S. Linden, J. Kuhl. Non-linear polariton pulse propagation in bulk semiconductors. Physica status solidi B, 221(1):453–457, 2000.

[79] O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, F. X. Kärtner. Signatures of carrier-waveRabi flopping in GaAs. Physical Review Letters, 87(5):057401, 2001.

[80] S. W. Koch, A. Knorr, R. Binder, M. Lindberg. Microscopic theory of Rabi flopping, photon-echo, and resonant pulse-propagation in semiconductors. Physica status solidi B, 173:177, 1992.

[81] N. C. Nielsen, S. Linden, J. Kuhl, J. Förstner, A. Knorr, S. W. Koch, H. Giessen. Coherentnonlinear pulse propagation on a free-exciton resonance in a semiconductor. Physical Review B,64:24, 2001.

[82] H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grün, , C. Kling-shirn. Self-induced transmission on a free exciton resonance in a semiconductor. Physical Re-view Letters, 81(19):4260, 1992.

218 BIBLIOGRAPHY

[83] P. A. Harten, A. Knorr, J. P. Sokoloff, F. Brown de Colstoun, S. G. Lee, R. Jin, E. M. Wright,G. Khitrova, H. M. Gibbs, S. W. Koch, N. Peyghambarian. Propagation-induced escape fromadiabatic following in a semiconductor. Physical Review Letters, 69(5):852, 1992.

[84] A. Schülzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke1, H. M. Gibbs, G. Khitrova,N. Peyghambarian. Direct observation of excitonic Rabi oscillations in semiconductors. PhysicalReview Letters, 82(11):2346, 1999.

[85] J. P. Prineas W. N. Xiao, J. Y. Zhou. Storage of ultrashort optical pulses in a resonantly absorbingBragg reflector. optics Express, 11(24):3277–3283, 2003.

[86] N. C. Nielsen, T. Höner zu Siederdissen, J. Kuhl, M. Schaarschmidt, J. Förstner, A. Knorr,H. Giessen. Phase evolution of solitonlike optical pulses during excitonic Rabi flopping in asemiconductor. Physical Review Letters, 94(5):057406, 2005.

[87] T. Höner zu Siederdissen, N. C. Nielsen, J. Kuhl, H. Giessen. Influence of near-resonant self-phase modulation on pulse propagation in semiconductors. submitted to JOSA-B, 2005.

[88] A. Knorr, R. Binder, M. Lindberg, S. W. Koch. Theoretical study of resonant ultrashort-pulsepropagation in semiconductors. Physical Review A, 46(11):7179, 1992.

[89] P. Dumais, A. Villeneuve, J.S. Aitchison. Bright temporal solitonlike pulses in self-defocusingAlGaAs waveguides near 800 nm. Optics Letters, 21(4):260, 1996.

[90] N. C. Nielsen, T. Höner zu Siederdissen, J. Kuhl, M. Schaarschmidt, J. Förstner, A. Knorr,S. W. Koch, H. Giessen. Temporal and spatial pulse compression in a nonlinear defocussingmaterial. in Ultrafast Phenomena XIV, T. Kobayashi, T. Okada, T. Kobayashi, K. Nelson and S.de Silvestri, Springer Verlag, Heidelberg, 2005.

[91] S. W. Koch, N. Peyghambarian, M. Lindberg. Transient and steady-state optical nonlinearitiesin semiconductors. Journal of Physics C, 21(30):5229, 1988.

[92] C. Ell, J. F. Müller, K. El Sayed, H. Haug. Influence of many-body interactions on the excitonicoptical Stark effect. Physical Review Letters, 62(3):304, 1989.

[93] T. Höner zu Siederdissen, N. C. Nielsen, J. Kuhl, M. Schaarschmidt, J. Förstner, A. Knorr,G. Khitrova, H. M. Gibbs, S. W. Koch, H. Giessen. Transition between different coherentlight-matter interaction regimes analyzed by phase-resolved pulse propagation. Optics Letters,30(11):1384, 2005.

[94] J. D. Dow, D. Redfield. Electroabsorption in semiconductors: The excitonic absorption edge.Physical Review B, 1:3358, 1970.

[95] I. A. Merkulov, V. I. Perel. Effects of electron-hole interaction on electroabsorption in semicon-ductors. Physics Letters, 45(2):83, 1973.

[96] U. Bartuch, U. Peschel, Th. Gabler, R. Waldhäusl, H.-H. Hörhold. Experimental investigationsand numerical simulations of spatial solitons in planar polymer waveguides. Optics Communi-cations, 134:49–54, 1997.

[97] T. Pertsch, U. Peschel, F. Lederer. Discrete solitons in inhomogeneous waveguide arrays. Chaos,13:774, 2003.

BIBLIOGRAPHY 219

[98] M. Mlejnek, E. M. Wright, J. V. Moloney. Dynamic spatial replenishment of femtosecond pulsespropagating in air. Optics Letters, 23(5):382, 1998.

[99] Heitler. The Quantum Theory of Radiation. Oxford Clarendon Press, 1954.

220 BIBLIOGRAPHY

221

Danksagung

Es sei all jenen gedankt, die das Entstehen dieser Arbeit ermöglicht oder erleichtert haben.Insbesondere möchte ich meinem Doktorvater Prof. Dr. Andreas Knorr danken, der stets ein offenesOhr für Fragen hatte und mich durch viele konstruktive Diskussionen unterstützt hat.Der gesamten Arbeitsgruppe “Nichtlineare Optik und Quantenelektronik von Halbleitern” danke ichfür die außergewöhnlich gute Arbeitsatmosphäre und die stete Bereitschaft zu wissenschaftlichenStreitgesprächen, insbesondere Norbert Bücking und Stefan Butscher.Den folgenden Arbeitesgruppenmitgliedern möchte ich für die fruchtbare Zusammenarbeitdanken: Marten Richter (THz-Lumineszenz), Ermin Malic (Quantenpunkt-Wettinglayer Kopplung),Carsten Weber (Rabi-Oszillation in Quantenpunkten), Jens Förstner (MQW).Für die hervorragende Zusammenarbeit auf dem Gebiet der MQW- und Bulk-Propagation mit derexperimentellen Optics and Spectroscopy Group des Max-Planck-Institut für Festkörperforschung inStuttgart danke ich insbesondere Dr. Jürgen Kuhl, Tilman Höner zu Siederdissen undDr. Nils C. Nielsen (zur Zeit am Lawrence Berkeley National Laboratory).Für viele Anregungen und Diskussionen danke ich Prof. Dr. Stephan W. Koch (Philipps-UniversitätMarburg) und Prof. Dr. Harald Giessen (Universität Bonn / Universität Stuttgart).Der Person, ohne die jegliche wissenschaftliche Arbeit an unserem Institut jäh zum Erliegen kommenwürde möchte ich an dieser Stelle für seine (leider nicht oft genug gewürdigten) Leistungen Anerken-nung und Dank zukommen lassen: unserem System-Administrator Peter Orlowski.Nicht zuletzt möchte ich meine Familie und insbesondere Jessica für die Unterstützung danken.Abschließend möchte ich Prof. Dr. Erwin Sedlmayr für die bereitwillige Übernahme des Vorsitzesund Prof. Dr. Harald Engel für die Begutachtung meiner Arbeit unter schwierigen terminlichen Vor-aussetzungen danken.

theory of light propagation in nano-structured materials

Documents