Quantum dissipative systems / Ulrich Weiss.

Format:

Book

Author/Creator:

Weiss, U. (Ulrich)

Series:

Series in modern condensed matter physics ; v. 13.

Series in modern condensed matter physics ; v. 13

Language:

English

Subjects (All):

Quantum theory.

Mathematical physics.

Thermodynamics.

Quantum statistics.

Path integrals.

Physical Description:

xviii, 507 pages : illustrations ; 23 cm.

Edition:

Third edition.

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, [2008]

Summary:

Major advances in the quantum theory of macroscopic systems, in combination with stunning experimental achievements, have brightened the field and brought it to the attention of the general community in natural sciences. Today, working knowledge of dissipative quantum mechanics is an essential tool for many physicists. This book-originally published in 1990 and republished in 1999 as an enlarged second edition-delves much deeper than ever before into the fundamental concepts, methods, and applications of quantum dissipative systems, including the most recent developments.

In this third edition, 26 chapters from the second edition contain additional material and several chapters are completely rewritten. It deals with the phenomena and theory of decoherence, relaxation, and dissipation in quantum mechanics that arise from the interaction with the environment. In so doing, a general path integral description of equilibrium thermodynamics and nonequilibrium dynamics is developed. Book jacket.

Contents:

I General Theory of Open Quantum Systems 5

2 Diverse limited approaches: a brief survey 5

2.1 Langevin equation for a damped classical system 5

2.2 New schemes of quantization 7

2.3 Traditional system-plus-reservoir methods 8

2.3.1 Quantum-mechanical master equations for weak coupling 8

2.3.2 Operator Langevin equations for weak coupling 12

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 18

3.1 Harmonic oscillator bath with linear coupling 19

3.1.1 The Hamiltonian of the global system 19

3.1.2 The road to the classical generalized Langevin equation 21

3.1.3 Phenomenological modeling 24

3.1.4 Quasiclassical Langevin equation 25

3.1.5 Ohmic and frequency-dependent damping 27

3.1.6 Rubin model 30

3.2 The Spin-Boson model 31

3.2.1 The model Hamiltonian 31

3.2.2 Josephson two-state systems: flux and charge qubit 35

3.3 Microscopic models 38

3.3.1 Acoustic polaron: one-phonon and two-phonon coupling 40

3.3.2 Optical polaron 41

3.3.3 Interaction with fermions (normal and superconducting) 43

3.3.4 Superconducting tunnel junction 46

3.4 Charging and environmental effects in tunnel junctions 47

3.4.1 The global system for single electron tunneling 49

3.4.2 Resistor, inductor and transmission lines 53

3.4.3 Charging effects in Josephson junctions 54

3.5 Nonlinear quantum environments 55

4 Imaginary-time path integrals 57

4.1 The density matrix: general concepts 58

4.2 Effective action and equilibrium density matrix 62

4.2.1 Open system with bilinear coupling to a harmonic reservoir 63

4.2.2 State-dependent memory-friction 67

4.2.3 Spin-boson model 68

4.2.4 Acoustic polaron and defect tunneling: one-phonon coupling 69

4.2.5 Acoustic polaron: two-phonon coupling 75

4.2.6 Tunneling between surfaces: one-phonon coupling 77

4.2.7 Optical polaron 79

4.2.8 Heavy particle in a metal 80

4.2.9 Heavy particle in a superconductor 86

4.2.10 Effective action for a Josephson junction 88

4.2.11 Electromagnetic environment 95

4.3 Partition function of the open system 96

4.3.1 General path integral expression 96

4.3.2 Semiclassical approximation 97

4.3.3 Partition function of the damped harmonic oscillator 98

4.3.4 Functional measure in Fourier space 99

4.3.5 Partition function of the damped harmonic oscillator revisited 100

4.4 Quantum statistical expectation values in phase space 102

4.4.1 Generalized Weyl correspondence 103

4.4.2 Generalized Wigner function and expectation values 105

5 Real-time path integrals and dynamics 106

5.1 Feynman-Vernon method for a product initial state 108

5.2 Decoherence and friction 112

5.3 General initial states and preparation function 115

5.4 Complex-time path integral for the propagating function 116

5.5 Real-time path integral for the propagating function 120

5.5.1 Extremal paths 123

5.5.2 Classical limit 124

5.5.3 Semiclassical limit: quasiclassical Langevin equation 125

5.6 Stochastic unraveling of influence functionals 127

5.7 Brief summary and outlook 130

II Few Simple Applications 131

6 Damped harmonic oscillator 131

6.1 Fluctuation-dissipation theorem 132

6.2 Stochastic modeling 135

6.3 Susceptibility for Ohmic friction and Drude damping 138

6.3.1 Strict Ohmic friction 138

6.3.2 Drude damping 138

6.4 The position autocorrelation function 139

6.4.1 Ohmic damping 140

6.4.2 Algebraic spectral density 142

6.5 Partition function, internal energy and density of states 143

6.5.1 Partition function and internal energy 143

6.5.2 Spectral density of states 146

6.6 Mean square of position and momentum 147

6.6.1 General expressions for coloured noise 147

6.6.2 Strict Ohmic case 149

6.6.3 Ohmic friction with Drude regularization 150

6.7 Equilibrium density matrix 152

6.7.1 Purity 154

7 Quantum Brownian free motion 156

7.1 Spectral density, damping function and mass renormalization 157

7.2 Displacement correlation and response function 159

7.3 Ohmic damping 160

7.4 Frequency-dependent damping 163

7.4.1 Response function and mobility 163

7.4.2 Mean square displacement 165

8 The thermodynamic variational approach 167

8.1 Centroid and the effective classical potential 167

8.1.1 Centroid 167

8.1.2 The effective classical potential 169

8.2 Variational method 170

8.2.1 Variational method for the free energy 170

8.2.2 Variational method for the effective classical potential 171

8.2.3 Variational perturbation theory 174

8.2.4 Expectation values in coordinate and phase space 176

9 Suppression of quantum coherence 178

9.1 Nondynamical versus dynamical environment 179

9.2 Suppression of transversal and longitudinal interferences 180

9.3 Localized bath modes and universal decoherence 182

9.3.1 A model with localized bath modes 182

9.3.2 Statistical average of paths 184

9.3.3 Ballistic motion 185

9.3.4 Diffusive motion 186

III Quantum Statistical Decay 189

11 Classical rate theory: a brief overview 192

11.1 Classical transition state theory 192

11.2 Moderate-to-strong-damping regime 193

11.3 Strong damping regime 195

11.4 Weak-damping regime 197

12 Quantum rate theory: basic methods 199

12.1 Formal rate expressions in terms of flux operators 200

12.2 Quantum transition state theory 202

12.3 Semiclassical limit 203

12.4 Quantum tunneling regime 205

12.5 Free energy method 207

12.6 Centroid method 211

13 Multidimensional quantum rate theory 212

14 Crossover from thermal to quantum decay 216

14.1 Normal mode analysis at the barrier top 216

14.2 Turnover theory for activated rate processes 218

14.3 The crossover temperature 222

15 Thermally activated decay 223

15.1 Rate formula above the crossover regime 224

15.2 Quantum corrections in the preexponential factor 227

15.3 The quantum Smoluchowski equation approach 228

15.4 Multidimensional quantum transition state theory 230

16 The crossover region 233

16.1 Beyond steepest descent above T[subscript 0] 235

16.2 Beyond steepest descent below T[subscript 0] 236

16.3 The scaling region 239

17 Dissipative quantum tunneling 242

17.1 The quantum rate formula 242

17.2 Thermal enhancement of macroscopic quantum tunneling 245

17.3 Quantum decay in a cubic potential for Ohmic friction 246

17.3.1 Bounce action and quantum prefactor 247

17.3.2 Analytic results for strong Ohmic dissipation 248

17.4 Quantum decay in a tilted cosine washboard potential 250

IV The Dissipative Two-State System 259

18.1 Truncation of the double-well to the two-state system 261

18.1.1 Shifted oscillators and orthogonality catastrophe 261

18.1.2 Adiabatic renormalization 263

18.1.3 Renormalized tunnel matrix element 264

18.1.4 Polaron transformation 269

18.2 Pair interaction in the charge picture 269

18.2.1 Analytic expression for any s and arbitrary cutoff [omega subscript c] 269

18.2.2 Ohmic dissipation and universality limit 271

19 Thermodynamics 272

19.1 Partition function and specific heat 272

19.1.1 Exact formal expression for the partition function 272

19.1.2 Static susceptibility and specific heat 274

19.1.3 The self-energy method 275

19.1.4 The limit of high temperatures 277

19.1.5 Noninteracting-kink-pair approximation 277

19.1.6 Weak-damping limit 279

19.1.7 The self-energy method revisited: partial resummation 280

19.2 Ohmic dissipation 281

19.2.1 General results 282

19.2.2 The special case K = 1/2 283

19.3 Non-Ohmic spectral densities 288

19.3.1 The sub-Ohmic case 288

19.3.2 The super-Ohmic case 289

19.4 Relation between the Ohmic TSS and the Kondo model 290

19.4.1 Anisotropic Kondo model 290

19.4.2 Resonance level

model 292

19.5 Equivalence of the Ohmic TSS with the 1/r[superscript 2] Ising model 293

20 Electron transfer and incoherent tunneling 294

20.1 Electron transfer 295

20.1.1 Adiabatic bath 296

20.1.2 Marcus theory for electron transfer 298

20.2 Incoherent tunneling in the nonadiabatic regime 302

20.2.1 General expressions for the nonadiabatic rate 302

20.2.2 Probability for energy exchange: general results 304

20.2.3 The spectral probability density for absorption at T = 0 307

20.2.4 Crossover from quantum-mechanical to classical behaviour 308

20.2.5 The Ohmic case 312

20.2.6 Exact nonadiabatic rates for K = 1/2 and K = 1 314

20.2.7 The sub-Ohmic case (0 < s < 1) 315

20.2.8 The super-Ohmic case (s > 1) 317

20.2.9 Incoherent defect tunneling in metals 319

20.3 Single charge tunneling 322

20.3.1 Weak-tunneling regime 322

20.3.2 The current-voltage characteristics 326

20.3.3 Weak tunneling of 1D interacting electrons 328

20.3.4 Tunneling of Cooper pairs 330

20.3.5 Tunneling of quasiparticles 331

21 Two-state dynamics 333

21.1 Initial preparation, expectation values, and correlations 333

21.1.1 Product initial state 333

21.1.2 Thermal initial state 336

21.2 Exact formal expressions for the system dynamics 340

21.2.1 Sojourns and blips 340

21.2.2 Conditional propagating functions 343

21.2.3 The expectation values [left angle bracket sigma right angle bracket subscript t] (j = x, y, z) 344

21.2.4 Correlation and response function of the populations 346

21.2.5 Correlation and response function of the coherences 348

21.2.6 Generalized exact master equation and integral relations 349

21.3 The noninteracting-blip approximation (NIBA) 352

21.3.1 Symmetric Ohmic system in the scaling limit 355

21.3.2 Weak Ohmic damping and moderate-to-high temperature 359

21.3.3 The super-Ohmic case 365

21.4 Weak-coupling theory beyond the NIBA for a biased system 368

21.4.1 The one-boson self-energy 369

21.4.2 Populations and coherences (super-Ohmic and Ohmic) 371

21.5 The interacting-blip chain approximation 373

21.6 Ohmic dissipation with K at and near 1/2: exact results 376

21.6.1 Grand-canonical sums of collapsed blips and sojourns 376

21.6.2 The expectation value [left angle bracket sigma subscript z right angle bracket subscript t] for K = 1/2 377

21.6.3 The case K = 1/2 - [kappa]; coherent-incoherent crossover 379

21.6.4 Equilibrium [sigma subscript z] autocorrelation function 380

21.6.5 Equilibrium [sigma subscript x] autocorrelation function 385

21.6.6 Correlation functions in the Toulouse model 387

21.7 Long-time behaviour at T = 0 for K < 1: general discussion 388

21.7.1 The populations 389

21.7.2 The population correlations and generalized Shiba relation 389

21.7.3 The coherence correlation function 391

21.8 From weak to strong tunneling: relaxation and decoherence 392

21.8.1 Incoherent tunneling beyond the nonadiabatic limit 392

21.8.2 Decoherence at zero temperature: analytic results 395

21.9 Thermodynamics from dynamics 396

22 The driven two-state system 399

22.1 Time-dependent external fields 399

22.1.1 Diagonal and off-diagonal driving 399

22.1.2 Exact formal solution 400

22.1.3 Linear response 402

22.1.4 The Ohmic case with Kondo parameter K = 1/2 403

22.2 Markovian regime 403

22.3 High-frequency regime 404

22.4 Quantum stochastic resonance 407

22.5 Driving-induced symmetry breaking 409

V The Dissipative Multi-State System 411

23 Quantum Brownian particle in a washboard potential 411

23.2 Weak- and tight-binding representation 412

24 Multi-state dynamics 413

24.1 Quantum transport and quantum-statistical fluctuations 413

24.1.1 Product initial state 414

24.1.2 Characteristic function of moments and cumulants 414

24.1.3 Thermal initial state and correlation functions 415

24.2 Poissonian quantum transport 416

24.2.1 Dynamics by incoherent nearest-neighbour tunneling moves 416

24.2.2 The general case 418

24.3 Exact formal expressions for the system dynamics 419

24.3.1 Product initial state 421

24.3.2 Thermal initial state 423

24.4 Mobility and Diffusion 426

24.4.1 Exact formal series expressions for transport coefficients 426

24.4.2 Einstein relation 427

24.5 The Ohmic case 428

24.5.1 Weak-tunneling regime 429

24.5.2 Weak-damping limit 429

24.6 Exact solution in the Ohmic scaling limit at K = 1/2 431

24.6.1 Current and mobility 431

24.6.2 Diffusion and skewness 434

24.7 The effects of a thermal initial state 435

24.7.1 Mean position and variance 435

24.7.2 Linear response 436

24.7.3 The exactly solvable case K = 1/2 439

25 Duality symmetry 439

25.1 Duality for general spectral density 440

25.1.1 The map between the TB and WB Hamiltonian 440

25.1.2 Frequency-dependent linear mobility 443

25.1.3 Nonlinear static mobility 444

25.2 Self-duality in the exactly solvable cases K = 1/2 and K = 2 446

25.2.1 Full counting statistics at K = 1/2 446

25.2.2 Full counting statistics at K = 2 448

25.3 Duality and supercurrent in Josephson junctions 450

25.3.1 Charge-phase duality 450

25.3.2 Supercurrent-voltage characteristics for [rho] [double less-than sign] 1 453

25.3.3 Supercurrent-voltage characteristics at [rho] = 1/2 454

25.3.4 Supercurrent-voltage characteristics at [rho] = 2 454

25.4 Self-duality in the Ohmic scaling limit 455

25.4.1 Linear mobility at finite T 456

25.4.2 Nonlinear mobility at T = 0 457

25.5 Exact scaling function at T = 0 for arbitrary K 459

25.5.1 Construction of the self-dual scaling solution 459

25.5.2 Supercurrent-voltage characteristics at T = 0 for arbitrary [rho] 462

25.5.3 Connection with Seiberg-Witten theory 462

25.5.4 Special limits 463

25.6 Full counting statistics at zero temperature 464

25.7 Low temperature behaviour of the characteristic function 467

25.8 The sub- and super-Ohmic case 468

26 Charge transport in quantum impurity systems 470

26.1 Generic models for transmission of charge through barriers 471

26.1.1 The Tomonaga-Luttinger liquid 471

26.1.2 Transport through a single weak barrier 472

26.1.3 Transport through a single strong barrier 474

26.1.4 Coherent conductor in an Ohmic environment 476

26.1.5 Equivalence with quantum transport in a washboard potential 478

26.2 Self-duality between weak and strong tunneling 478

26.3 Full counting statistics 479

26.3.1 Charge transport at low T for arbitrary g 479

26.3.2 Full counting statistics at g = 1/2 and general temperature 482.

Notes:

Previous ed.: 1999.

Includes bibliographical references (pages 483-501) and index.

ISBN:

9789812791627

9812791620

OCLC:

227279980