Quantum Kinetic Theory (eBook)

Preface to the Second Edition 5
Preface to the First Edition 7
Contents 10
Symbols 16
1 Introduction 18
1.1 Correlated Many-Particle Systems 19
1.2 Thermodynamic Properties of Correlated Systems 26
1.3 Ultrafast Nonequilibrium Phenomena 29
1.3.1 Dynamics of Isolated Systems 30
1.3.2 Interaction of Matter with Short Laser Pulses 31
1.3.3 Overview of Relaxation Processes 35
1.4 The Boltzmann Equation--Successes and Failure 37
1.4.1 An Elementary Introduction to the Boltzmann Equation 37
1.4.2 Unphysical Ultrafast Relaxation in Charged Particle Systems 40
1.5 Improved Theoretical Concepts 41
1.5.1 Outline of this Book 43
1.6 Problems 44
2 The Method of Reduced Density Operators 45
2.1 Density Operator. Von Neumann Equation 45
2.2 BBGKY-Hierarchy 48
2.2.1 Reduced Density Operators. Equations of Motion 48
2.2.2 Conservation Laws 53
2.3 Basic Representations of the Hierarchy 57
2.3.1 Coordinate Representation 57
2.3.2 Wigner Representation 59
2.3.3 Classical Limit and Quantum Corrections 61
2.3.4 Spatially Homogeneous Systems. Momentum Representation 62
2.4 Multi-component and Multi-band Systems 66
2.4.1 Bloch Representation of the Hierarchy 68
2.4.2 Remarks on General Properties of the BBGKY-Hierarchy 71
2.5 Correlations in Many-Particle Systems 72
2.5.1 BBGKY-Hierarchy for Correlation Operators 72
2.5.2 Energy Conservation Condition in Terms of Correlation Operators 74
2.6 Decoupling of the BBGKY-Hierarchy 75
2.6.1 Correlation Effects 78
2.6.2 *Selfenergy Effects 81
2.7 Relation to Equilibrium Correlation Functions 83
2.8 Problems 85
3 *Correlations Due to the Spin Statistics 86
3.1 (Anti-)Symmetrization of the Density Operators 88
3.2 Exchange and Phase Space Filling Effects 89
3.3 (Anti-)Symmetrization of the First and Second Hierarchy Equations 92
3.4 (Anti-)Symmetrization of the Third Hierarchy Equation 93
3.4.1 (Anti-)Symmetrization of the Selfenergy Terms 94
3.4.2 Energy Conservation with Spin Statistics 95
3.5 Problems 97
4 Mean--Field Approximation. Quantum Vlasov Equation. Collective Effects 98
4.1 Linearization of the Quantum Vlasov Equation. Dielectric Function 100
4.2 Collective Plasma Excitations (Plasmons) 107
4.3 Plasma Instabilities 110
4.4 Examples: Plasmons in Quantum Systems 113
4.4.1 One-Dimensional Quantum Plasmas 114
4.4.2 Plasmons in 2D and 3D Quantum Systems 121
4.5 *Quasilinear Theory for Classical and Quantum Systems 124
4.6 Numerical Solutions of the Nonlinear Quantum Vlasov Equation 128
4.7 *Kinetic Equations for Carrier--Plasmon Interaction 129
4.8 Problem 132
5 Correlations and Their Dynamics 133
5.1 Hierarchy of Relaxation Processes. Time Scale Separation 134
5.2 Correlation Buildup. Correlation Time Approximation 137
6 Correlation Dynamics and Non-Markovian Effects 140
6.1 Solution for g12 in Second Born Approximation 141
6.2 Non-Markovian Quantum Landau Equation 143
6.3 Markov Limit 145
6.4 Non-Markovian Quantum Landau Equation with Exchange Renormalization and Time-Dependent Fields 147
6.5 Problems 152
7 Non-Markovian Kinetic Equations with Selfenergy 153
7.1 *Selfenergy in Density Operator Approach 154
7.2 Renormalized Binary Correlation Operator 158
7.3 Non-Markovian Quantum Landau Equation with Selfenergy 160
7.3.1 Properties of the Landau Equation. Memory Effects 161
7.3.2 Dynamics of Physical Observables. Energy Conservation 166
7.3.3 Markov Limit and Corrections. Retardation Expansion 170
7.3.4 *Approximations for the Selfenergy 175
7.4 *Discussion of the Selfenergy Concept. Relation to Green Functions Results 178
7.5 Problem 181
8 Properties of the Quantum Kinetic Equation 182
8.1 Markovian Dynamics of Macroscopic Observables 183
8.2 Irreversibility. H-Theorem. Equilibrium Solution of the Markovian Kinetic Equation 184
8.3 Equilibrium Correlations 187
8.4 Non-Markovian Dynamics of Macroscopic Observables 188
8.5 Total Energy Conservation in Non-Markovian Kinetics 190
8.6 H-Theorem in Non-Markovian Kinetics 191
8.7 Problems 193
9 Strong Coupling Effects. Ladder (T-Matrix) Approximation 194
9.1 Generalized Binary Collision Approximation 195
9.2 *Selfenergy in Ladder (T-Matrix) Approximation 196
9.3 Correlation Operator in Binary Collision Approximation 198
9.3.1 Propagators and Scattering Quantities 198
9.3.2 Møller Operators and T-Operators 201
9.3.3 Correlation Operator in Binary Collision Approximation 203
9.3.4 *Gradient Expansion of g12 and Physical Observables 206
9.3.5 *Recovery of the Generalized Kadanoff-Baym Ansatz 209
9.4 Collision Integral with Memory Effects 210
9.5 Kinetic Equation in First Order Gradient Expansion 213
9.6 Numerical Results and Discussion 217
9.6.1 Markovian T-Matrix Scattering Rates 218
9.6.2 Summary and Comments on the T-Matrix Approximation 220
9.6.3 Numerical Results for Lattice Systems 220
10 *Random Phase Approximation 222
10.1 Generalized Polarization Approximation: Selfenergy 223
10.2 Dynamical Screening in Nonequilibrium 227
10.3 Non-Markovian Balescu-Lenard Equation 232
10.3.1 Properties of the Non-Markovian Balescu-Lenard Equation. Markov Limit 233
10.3.2 Correlation Energy in RPA 237
10.3.3 Short-Time Behavior: Screening Buildup 238
10.4 Problem 240
11 *Dynamically Screened Ladder Approximation 241
11.1 Generalized Screened Ladder Approximation. Selfenergy 242
11.1.1 Limiting Cases of the Screened Ladder Approximation 245
11.2 Gould--DeWitt Approximation 246
12 Charged Many-Particle Systems in Electromagnetic Fields. Generalized Bloch Equations 247
12.1 Field-Matter Interaction 248
12.2 Field Effects on the Distribution and the Propagators 252
12.3 Interaction of Optical Fields with Multiband Systems 259
12.4 Bloch Representation of the First Hierarchy Equation 262
12.5 *Bloch Representation of the Solution g12(t) 269
12.6 *Correlation Operator, Non-Markovian Collision Integral and Selfenergy in an Electromagnetic Field 274
12.7 *Non-Markovian Bloch Equations Beyond the Static Born Approximation 277
12.8 Problem 280
13 *Nonequilibrium Green Functions Approach to Field-Matter Dynamics 281
13.1 Introduction 282
13.2 Basic Concepts of Relativistic Quantum Electrodynamics 283
13.2.1 Field Operators of the Maxwell Field 284
13.2.2 Relativistic Field Operators for Fermions 284
13.2.3 Statistical Description in Nonequilibrium 286
13.2.4 Green Functions for Photons and Charge Carriers 288
13.3 Relativistic Keldysh-Kadanoff-Baym Equations for Particles and Photons 292
13.4 Approximations for the Selfenergies 294
13.4.1 Expansion in Terms of G0 and D 295
13.4.2 Expansion in Terms of G and D 298
13.4.3 Adiabatic Approximation for the Electromagnetic Field 298
13.5 Nonrelativistic Keldysh-Kadanoff-Baym Equations 300
13.5.1 Nonrelativistic Limit. Pauli Equation 300
13.5.2 Green Functions for Carriers, Photons and Plasmons 302
13.5.3 Keldysh-Kadanoff-Baym Equations for Carriers, Plasmons and Photons 305
13.6 Particle Keldysh-Kadanoff-Baym Equations. Properties and Approximations 305
13.6.1 Approximations for the Selfenergy 307
13.6.2 Properties of the Keldysh-Kadanoff-Baym Equations 309
13.6.3 Numerical Results 311
13.7 Interband KBE 314
13.7.1 Two-Time Semiconductor Bloch Equations 315
13.7.2 Illustration: NEGF-Simulation for Laser Excitation of Electrons in a Harmonic Oscillator 317
13.7.3 Numerical Results for Ultrafast Relaxation of Femtosecond-Laser Excited Semiconductors 320
13.7.4 Computing Optical Absorption Via Solution of the Interband KBE 323
13.8 Nonequilibrium KBE-Approach to Equilibrium Response Properties 326
13.8.1 Response Properties in Lowest Order (Linear Response) 327
13.8.2 Relation Between the Two-Particle Kernel ? and the KBE-Selfenergy ? 328
13.8.3 Interband Approach to Plasma Oscillations of the Correlated Electron Gas 329
13.8.4 Optical Absorption of Atoms and Molecules. Electronic Double Excitations 331
13.9 Kinetic Equations for Single-Time Functions. Comparison to Density Operators 333
13.10 Build Up of Dynamical Screening 338
13.10.1 Theoretical Approaches to the Screening Dynamics 338
13.10.2 Femtosecond Buildup of the RPA Dielectric Function 339
13.10.3 Selfconsistent Solution of the KBE in RPA 341
13.10.4 Experimental Results. Outlook 341
13.11 Problems 343
14 Conclusion 344
Appendix AUsed Mathematical Formulas 346
Appendix BWigner Representationof the BBGKY-Hierarchy 350
Appendix CEquations of Motion for Binaryand Ternary Correlations 354
Appendix DProperties of the FreePropagators U0 and U0± 361
Appendix ERetardation Expansion 365
Appendix FNumerical Solution of Quantum KineticEquations 371
Appendix GSolutions to Problems 382
References 391
Index 407