Selected Topics:

Thermoelectric heat engine

Thermodynamics and Information

Self-propelled particle

Vesicles in flow

Membrane Adhesion

Hidden degree of freedom


 

Quantum Dissipative Systems: Contents - Prof. Dr. Ulrich Weiß

Quantum dissipative systems

1 Introduction

 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS

2 Diverse limited approaches: a brief survey
2.1 Classical Langevin equation 5
2.2 New schemes of quantization 7
2.3 Traditional system-plus-reservoir methods 7
2.3.1 Quantum-mechanical master equations for weak coupling 8
2.3.2 Operator Langevin equations for weak coupling 11
2.3.3 Quantum and quasiclassical Langevin equation 13
2.3.4 Phenomenological methods 14
2.4 Stochastic dynamics in Hilbert space 15
 
3 System-plus-reservoir models
3.1 Harmonic oscillator bath with linear coupling 18
3.1.1 The Hamiltonian of the global system 18
3.1.2 The road to the classical generalized Langevin equation 20
3.1.3 Phenomenological modeling 22
3.1.4 Ohmic and frequency-dependent damping 24
3.1.5 Rubin model 27
3.2 The Spin-Boson model 28
3.2.1 Truncated two-state system: trapped flux in a rf SQUID 28
3.2.2 The model Hamiltonian 30
3.3 Microscopic models 32
3.3.1 Acoustic polaron: one-phonon and two-phonon coupling 34
3.3.2 Optical polaron 36
3.3.3 Interaction with fermions (normal and superconducting) 38
3.3.4 Superconducting tunnel junction 41
3.4 Charging and environmental effects in tunnel junctions 42
3.4.1 The global system for single electron tunneling 43
3.4.2 Resistor, inductor and transmission lines 47
3.4.3 Charging effects in Josephson junctions 48
3.5 Nonlinear quantum environments 49
 
4 Imaginary-time path integrals
4.1 The density matrix: general concepts 52
4.2 Effective action and equilibrium density matrix 54
4.2.1 Open system with bilinear coupling to a harmonic reservoir 56
4.2.2 State-dependent memory-friction 60
4.2.3 Spin-boson model 61
4.2.4 Acoustic polaron and defect: one-phonon coupling 62
4.2.5 Acoustic polaron: two-phonon coupling 67
4.2.6 Tunneling between surfaces: one-phonon coupling 70
4.2.7 Optical polaron 71
4.2.8 Heavy particle in a metal 72
4.2.9 Heavy particle in a superconductor 79
4.2.10 Effective action for a Josephson junction 81
4.3 Partition function of the open system 88
4.3.1 General path integral expression 88
4.3.2 Semiclassical approximation 88
4.3.3 Partition function of the damped harmonic oscillator 90
4.3.4 Functional measure in Fourier space 91
4.3.5 Partition function of the damped harmonic oscillator revisited 92
4.4 Quantum statistical expectation values in phase space 94
4.4.1 Generalized Weyl correspondence 95
4.4.2 Generalized Wigner function and expectation values 97
 
5 Real-time path integrals and dynamics
5.1 Feynman-Vernon method for a product initial state 101
5.2 Decoherence and friction 104
5.3 General initial states and preparation function 107
5.4 Complex-time path integral for the propagating function 108
5.5 Real-time path integral for the propagating function 112
5.5.1 Extremal paths 116
5.5.2 Classical limit 117
5.5.3 Semiclassical limit: quasiclassical Langevin equation 117
5.6 Brief summary and outlook 120

 II FEW SIMPLE APPLICATIONS

6 Damped harmonic oscillator
6.1 Fluctuation-dissipation theorem 122
6.2 Stochastic modeling 125
6.3 Susceptibility for Ohmic friction and Drude model 127
6.4 The position autocorrelation function 128
6.4.1 Ohmic damping 130
6.4.2 Algebraic spectral density 131
6.5 Partition function and density of states 133
6.5.1 Partition function and ground-state energy 133
6.5.2 Density of states 134
6.6 Mean square of position and momentum 136
6.6.1 General expressions for colored noise 136
6.6.2 Strict Ohmic case 137
6.6.3 Ohmic friction with Drude regularization 138
6.7 The equilibrium density matrix 140
 
7 Quantum Brownian motion
7.1 Spectral density and damping coefficient 144
7.2 Displacement correlation and response function 145
7.3 Ohmic damping 147
7.4 Frequency-dependent damping 149
 
8 The thermodynamic variational approach
8.1 Centroid and the effective classical potential 154
8.2 Varlational method 156
8.2.1 Variational method for the free energy 157
8.2.2 Varlational method for the effective classical potential 157
8.2.3 Variational perturbation theory 161
8.2.4 Expectation values in coordinate and phase space 162
 
9 Suppression of quantum coherence
9.1 Nondynamical versus dynamical environment 165
9.2 Suppression of transversal and longitudinal interferences 166
9.3 Localized bath modes and universal decoherence 168
9.3.1 A model with localized bath modes 168
9.3.2 Statistical average of paths 170
9.3.3 Ballistic motion 171
9.3.4 Diffusive motion 172

 III QUANTUM STATISTICAL DECAY

10 Introduction
 
11 Classical rate theory: a brief overview
11.1 Classical transition state theory 178
11.2 Moderate-to-strong-damping regime 179
11.3 Strong damping regime 181
11.4 Weak-damping regime 182
 
12 Quantum rate theory: basic methods
12.1 Formal rate expressions in terms of flux operator 185
12.2 Quantum transition state theory 186
12.3 Semiclassical limit 187
12.4 Quantum tunneling regime 190
12.5 Free energy method 192
12.6 Centroid method 196
 
13 Multidimensional quantum rate theory
 
14 Crossover from thermal to quantum decay
14.1 Normal mode analysis at the barrier top 201
14.2 Turnover theory for activated rate processes 203
14.3 The crossover temperature 206
 
15 Thermally activated decay
15.1 Rate formula above the crossover regime 208
15.2 Quantum corrections 211
15.3 Multidimensional quantum transition state theory 212
 
16 The crossover region
16.1 Beyond steepest descent above To 217
16.2 Beyond steepest descent below To 218
16.3 The scaling region 222
 
17 Dissipative quantum tunneling
17.1 The quantum rate formula 224
17.2 Thermal enhancement of macroscopic quantum tunneling 227
17.3 Analytic results for strong Ohmic dissipation 229
17.3.1 Cubic metastable potential 229
17.3.2 Tilted cosine washboard potential 232
17.4 Concluding remarks 239

 IV THE DISSIPATIVE TWO-STATE SYSTEM

18 Introduction
18.1 Truncation of the double-well to the two-state systern 243
18.1.1 Adiabatic renormalization 243
18.1.2 Renormalized tunnel matrix element 244
18.1.3 Polaron transformation 249
18.2 Pair interaction in the charge picture 249
18.2.1 Analytic expression for any s and arbitrary cutoff 249
18.2.2 Ohmic dissipation and universality limit 251
 
19 Thermodynamics
19.1 Partition function and specific heat 252
19.1.1 Exact formal expression for the partition function 252
19.1.2 Static susceptibility and specific heat 245
19.1.3 The self-energy method 255
19.1.4 The limit of high temperatures 256
19.1.5 Noninteracting-kink-pair approximation 257
19.1.6 Weak-damping limit 258
19.1.7 The self-energy method revisited: partial resummation 260
19.2 Ohmic dissipation 261
19.2.1 General results 261
19.2.2 The special case K = 1/2 263
19.3 Non-Ohmic spectral densities 267
19.3.1 The sub-Ohmic case 267
19.3.2 The super-Ohmic case 268
19.4 Relation between the Ohmic TSS and the Kondo model 269
19.4.1 Anisotropic Kondo model 269
19.4.2 Resonance level model 272
19.5 Equivalence of the Ohmic TSS with the 1/(r^2) Ising model 272
 
20 Electron transfer and incoherent tunneling
20.1 Electron transfer 274
20.1.1 Adiabatic bath 275
20.1.2 Marcus theory for electron transfer 278
20.2 Incoherent tunneling in the nonadiabatic regime 281
20.2.1 General expressions for the nonadiabatic rate 282
20.2.2 Probability for energy exchange: general results 283
20.2.3 The probability function at T = 0 286
20.2.4 Crossover from quantum-mechanical to classical behavior 287
20.2.5 The Ohmic case 291
20.2.6 Exact nonadiabatic rates for K = 1/2 and K = 1 293
20.2.7 The sub-Ohmic case (0 < s < 1) 294
20.2.8 The super-Ohmic case (s > 1) 295
20.2.9 Incoherent defect tunneling in metals 298
20.3 Single charge tunneling 301
20.3.1 Weak-tunneling regime 301
20.3.2 The current-voltage characteristics 304
20.3.3 Weak tunneling of 1D interacting electrons 306
20.3.4 Tunneling of Cooper pairs 307
20.3.5 Tunneling of quasiparticles 308
 
21 Two-state dynamics
21.1 Initial preparation and relevant expectation values 309
21.1.1 Product initial state 309
21.1.2 Thermal initial state 312
21.2 Exact formal expressions for the system dynamics 313
21.2.1 Sojourns and blips 313
21.2.2 Conditional propagating functions 316
21.2.3 Populations and coherence 318
21.2.4 Correlation and response function of the populations 320
21.2.5 Correlation and response function of the coherences 321
21.3 Generalized exact master equation and integral relations 323
21.3.1 Irreducible kernels and self-energies 324
21.3.2 Dynamical theory of incoherent tunneling 326
21.4 The noninteracting-blip approximation (NIBA) 327
21.4.1 Symmetric Ohmic system in the scaling limit 330
21.4.2 Moderate-to-high temperatures for weak Ohmic damping 333
21.4.3 The super-Ohmic case 339
21.5 Weak-coupling theory beyond the NIBA for a biased system 342
21.5.1 The one-phonon self-energy 343
21.5.2 Populations and coherences (super-Ohmic and Ohmic) 345
21.6 The interacting-blip chain approximation 347
21.7 Ohmic dissipation with K at and near 1: exact results 349
21.7.1 Grand-canonical sums of collapsed blips and sojourns 350
21.7.2 The population for K = 1/2 351
21.7.3 The case K = 1/2; coherent-incoherent crossover 353
21.7.4 Equilibrium sigma_z autocorrelation function 354
21.7.5 Equilibrium sigma_x autocorrelation function 358
21.7.6 Correlation functions in the Toulouse model 361
21.8 Long-time behavior at T = 0 for K < 1: general discussion 362
21.8.1 The populations 362
21.8.2 The population correlations and generalized Shiba relation 363
21.8.3 The coherence correlation function 366
21.9 Thermodynamics from Dynamics 366
 
22 The driven two-state system
22.1 Time-dependent external fields 369
22.1.1 Diagonal and off-diagonal driving 369
22.1.2 Exact formal solution 370
22.1.3 Linear response 372
22.1.4 The case of Ohmic damping with K = 1/2 373
22.2 Markovian regime 373
22.3 High-frequency regime 374
22.4 Quantum stochastic resonance 378
22.5 Driving-induced symmetry breaking 380

 V THE DISSIPATIVE MULTI-STATE SYSTEM

23 Quantum Brownian particle in a cosine potential
23.1 Introduction 381
23.2 Weak- and tight-binding representation 382
 
24 Multi-state dynamics
24.1 Initial preparation 383
24.1.1 Product initial state 383
24.1.2 Thermal initial state 384
24.2 Exact formal expressions for the system dynamics 385
24.2.1 Product initial state 387
24.2.2 Thermal initial state 389
24.3 Mobility and Diffusion 391
24.3.1 Exact series expressions for transport coefficients 391
24.3.2 Incoherent nearest-neighbor tunneling dynamics 392
24.3.3 Einstein relation 394
24.4 The Ohmic case 395
24.4.1 Perturbative regime 395
24.4.2 Weak-damping limit 396
24.4.3 The exactly solvable case K = 1/2 398
24.5 The effects of a thermal initial state 399
24.5.1 Mean position and variance 399
24.5.2 Linear response 400
24.5.3 The exactly solvable case K = 1/2 402
 
25 Duality symmetry
25.1 Duality for general spectral density 403
25.1.1 The map between the TB and WB Hamiltonian 404
25.1.2 Frequency-dependent linear mobility 406
25.1.3 Nonlinear static mobility 408
25.1.4 Charge-phase duality in a Josephson junction 409
25.2 Self-duality in the Ohmic scaling limit 411
25.3 Exact scaling function at T = 0 for all K 414
25.3.1 Construction of the self-dual scaling solution 414
25.3.2 Connection with Seiberg-Witten theory 417
25.3.3 Special limits 418
25.4 Low-temperature enhancement of the nonlinear mobility 419
25.5 The sub- and super-Ohmic case 420
 
26 Tunneling of charge in a Luttinger liquid
26.1 Weak and strong barrier model 422
26.2 Equivalence with quantum Brownian motion 423
26.3 Self-duality in quantum impurity scattering 425
26.4 Nonequilibrium dc current noise 426
 
  Bibliography 427
 
  Index 445