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World Scientific Publishing Company, Incorporated
Quantum Dissipative Systems (Third Edition) / Edition 3

Quantum Dissipative Systems (Third Edition) / Edition 3

by Ulrich Weiss


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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.

Product Details

ISBN-13: 9789812791627
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 03/19/2008
Series: Series In Modern Condensed Matter Physics Series , #13
Edition description: 3rd ed.
Pages: 528
Product dimensions: 6.00(w) x 8.90(h) x 1.30(d)

Table of Contents

Introduction     1
General Theory of Open Quantum Systems     5
Diverse limited approaches: a brief survey     5
Langevin equation for a damped classical system     5
New schemes of quantization     7
Traditional system-plus-reservoir methods     8
Quantum-mechanical master equations for weak coupling     8
Operator Langevin equations for weak coupling     12
Quantum and quasiclassical Langevin equation     13
Phenomenological methods     14
Stochastic dynamics in Hilbert space     15
System-plus-reservoir models     18
Harmonic oscillator bath with linear coupling     19
The Hamiltonian of the global system     19
The road to the classical generalized Langevin equation     21
Phenomenological modeling     24
Quasiclassical Langevin equation     25
Ohmic and frequency-dependent damping     27
Rubin model     30
The Spin-Boson model     31
The model Hamiltonian     31
Josephson two-state systems: flux and charge qubit     35
Microscopic models     38
Acoustic polaron: one-phonon and two-phonon coupling     40
Optical polaron     41
Interaction with fermions (normal and superconducting)     43
Superconducting tunnel junction     46
Charging and environmental effects in tunnel junctions     47
The global system for single electron tunneling     49
Resistor, inductor and transmission lines     53
Charging effects in Josephson junctions     54
Nonlinear quantum environments     55
Imaginary-time path integrals     57
The density matrix: general concepts     58
Effective action and equilibrium density matrix     62
Open system with bilinear coupling to a harmonic reservoir     63
State-dependent memory-friction     67
Spin-boson model     68
Acoustic polaron and defect tunneling: one-phonon coupling     69
Acoustic polaron: two-phonon coupling     75
Tunneling between surfaces: one-phonon coupling     77
Optical polaron     79
Heavy particle in a metal     80
Heavy particle in a superconductor     86
Effective action for a Josephson junction     88
Electromagnetic environment     95
Partition function of the open system     96
General path integral expression      96
Semiclassical approximation     97
Partition function of the damped harmonic oscillator     98
Functional measure in Fourier space     99
Partition function of the damped harmonic oscillator revisited     100
Quantum statistical expectation values in phase space     102
Generalized Weyl correspondence     103
Generalized Wigner function and expectation values     105
Real-time path integrals and dynamics     106
Feynman-Vernon method for a product initial state     108
Decoherence and friction     112
General initial states and preparation function     115
Complex-time path integral for the propagating function     116
Real-time path integral for the propagating function     120
Extremal paths     123
Classical limit     124
Semiclassical limit: quasiclassical Langevin equation     125
Stochastic unraveling of influence functionals     127
Brief summary and outlook     130
Few Simple Applications     131
Damped harmonic oscillator     131
Fluctuation-dissipation theorem     132
Stochastic modeling     135
Susceptibility for Ohmic friction and Drude damping      138
Strict Ohmic friction     138
Drude damping     138
The position autocorrelation function     139
Ohmic damping     140
Algebraic spectral density     142
Partition function, internal energy and density of states     143
Partition function and internal energy     143
Spectral density of states     146
Mean square of position and momentum     147
General expressions for coloured noise     147
Strict Ohmic case     149
Ohmic friction with Drude regularization     150
Equilibrium density matrix     152
Purity     154
Quantum Brownian free motion     156
Spectral density, damping function and mass renormalization     157
Displacement correlation and response function     159
Ohmic damping     160
Frequency-dependent damping     163
Response function and mobility     163
Mean square displacement     165
The thermodynamic variational approach     167
Centroid and the effective classical potential     167
Centroid     167
The effective classical potential     169
Variational method      170
Variational method for the free energy     170
Variational method for the effective classical potential     171
Variational perturbation theory     174
Expectation values in coordinate and phase space     176
Suppression of quantum coherence     178
Nondynamical versus dynamical environment     179
Suppression of transversal and longitudinal interferences     180
Localized bath modes and universal decoherence     182
A model with localized bath modes     182
Statistical average of paths     184
Ballistic motion     185
Diffusive motion     186
Quantum Statistical Decay     189
Introduction     189
Classical rate theory: a brief overview     192
Classical transition state theory     192
Moderate-to-strong-damping regime     193
Strong damping regime     195
Weak-damping regime     197
Quantum rate theory: basic methods     199
Formal rate expressions in terms of flux operators     200
Quantum transition state theory     202
Semiclassical limit     203
Quantum tunneling regime     205
Free energy method      207
Centroid method     211
Multidimensional quantum rate theory     212
Crossover from thermal to quantum decay     216
Normal mode analysis at the barrier top     216
Turnover theory for activated rate processes     218
The crossover temperature     222
Thermally activated decay     223
Rate formula above the crossover regime     224
Quantum corrections in the preexponential factor     227
The quantum Smoluchowski equation approach     228
Multidimensional quantum transition state theory     230
The crossover region     233
Beyond steepest descent above T[subscript 0]     235
Beyond steepest descent below T[subscript 0]     236
The scaling region     239
Dissipative quantum tunneling     242
The quantum rate formula     242
Thermal enhancement of macroscopic quantum tunneling     245
Quantum decay in a cubic potential for Ohmic friction     246
Bounce action and quantum prefactor     247
Analytic results for strong Ohmic dissipation     248
Quantum decay in a tilted cosine washboard potential     250
Concluding remarks     257
The Dissipative Two-State System     259
Introduction     259
Truncation of the double-well to the two-state system     261
Shifted oscillators and orthogonality catastrophe     261
Adiabatic renormalization     263
Renormalized tunnel matrix element     264
Polaron transformation     269
Pair interaction in the charge picture     269
Analytic expression for any s and arbitrary cutoff [omega subscript c]     269
Ohmic dissipation and universality limit     271
Thermodynamics     272
Partition function and specific heat     272
Exact formal expression for the partition function     272
Static susceptibility and specific heat     274
The self-energy method     275
The limit of high temperatures     277
Noninteracting-kink-pair approximation     277
Weak-damping limit     279
The self-energy method revisited: partial resummation     280
Ohmic dissipation     281
General results     282
The special case K = 1/2     283
Non-Ohmic spectral densities     288
The sub-Ohmic case     288
The super-Ohmic case     289
Relation between the Ohmic TSS and the Kondo model     290
Anisotropic Kondo model     290
Resonance level model     292
Equivalence of the Ohmic TSS with the 1/r[superscript 2] Ising model     293
Electron transfer and incoherent tunneling     294
Electron transfer     295
Adiabatic bath     296
Marcus theory for electron transfer     298
Incoherent tunneling in the nonadiabatic regime     302
General expressions for the nonadiabatic rate     302
Probability for energy exchange: general results     304
The spectral probability density for absorption at T = 0     307
Crossover from quantum-mechanical to classical behaviour     308
The Ohmic case     312
Exact nonadiabatic rates for K = 1/2 and K = 1     314
The sub-Ohmic case (0 < s < 1)     315
The super-Ohmic case (s > 1)     317
Incoherent defect tunneling in metals     319
Single charge tunneling     322
Weak-tunneling regime     322
The current-voltage characteristics     326
Weak tunneling of 1D interacting electrons     328
Tunneling of Cooper pairs     330
Tunneling of quasiparticles     331
Two-state dynamics     333
Initial preparation, expectation values, and correlations     333
Product initial state     333
Thermal initial state     336
Exact formal expressions for the system dynamics     340
Sojourns and blips     340
Conditional propagating functions     343
The expectation values [left angle bracket sigma right angle bracket subscript t] (j = x, y, z)     344
Correlation and response function of the populations     346
Correlation and response function of the coherences     348
Generalized exact master equation and integral relations     349
The noninteracting-blip approximation (NIBA)     352
Symmetric Ohmic system in the scaling limit     355
Weak Ohmic damping and moderate-to-high temperature     359
The super-Ohmic case     365
Weak-coupling theory beyond the NIBA for a biased system     368
The one-boson self-energy     369
Populations and coherences (super-Ohmic and Ohmic)     371
The interacting-blip chain approximation     373
Ohmic dissipation with K at and near 1/2: exact results     376
Grand-canonical sums of collapsed blips and sojourns     376
The expectation value [left angle bracket sigma subscript z right angle bracket subscript t] for K = 1/2     377
The case K = 1/2 - [kappa]; coherent-incoherent crossover     379
Equilibrium [sigma subscript z] autocorrelation function     380
Equilibrium [sigma subscript x] autocorrelation function     385
Correlation functions in the Toulouse model     387
Long-time behaviour at T = 0 for K < 1: general discussion     388
The populations     389
The population correlations and generalized Shiba relation     389
The coherence correlation function     391
From weak to strong tunneling: relaxation and decoherence     392
Incoherent tunneling beyond the nonadiabatic limit     392
Decoherence at zero temperature: analytic results     395
Thermodynamics from dynamics     396
The driven two-state system     399
Time-dependent external fields     399
Diagonal and off-diagonal driving     399
Exact formal solution     400
Linear response     402
The Ohmic case with Kondo parameter K = 1/2     403
Markovian regime     403
High-frequency regime     404
Quantum stochastic resonance     407
Driving-induced symmetry breaking      409
The Dissipative Multi-State System     411
Quantum Brownian particle in a washboard potential     411
Introduction     411
Weak- and tight-binding representation     412
Multi-state dynamics     413
Quantum transport and quantum-statistical fluctuations     413
Product initial state     414
Characteristic function of moments and cumulants     414
Thermal initial state and correlation functions     415
Poissonian quantum transport     416
Dynamics by incoherent nearest-neighbour tunneling moves     416
The general case     418
Exact formal expressions for the system dynamics     419
Product initial state     421
Thermal initial state     423
Mobility and Diffusion     426
Exact formal series expressions for transport coefficients     426
Einstein relation     427
The Ohmic case     428
Weak-tunneling regime     429
Weak-damping limit     429
Exact solution in the Ohmic scaling limit at K = 1/2     431
Current and mobility     431
Diffusion and skewness     434
The effects of a thermal initial state      435
Mean position and variance     435
Linear response     436
The exactly solvable case K = 1/2     439
Duality symmetry     439
Duality for general spectral density     440
The map between the TB and WB Hamiltonian     440
Frequency-dependent linear mobility     443
Nonlinear static mobility     444
Self-duality in the exactly solvable cases K = 1/2 and K = 2     446
Full counting statistics at K = 1/2     446
Full counting statistics at K = 2     448
Duality and supercurrent in Josephson junctions     450
Charge-phase duality     450
Supercurrent-voltage characteristics for [rho] [double less-than sign] 1     453
Supercurrent-voltage characteristics at [rho] = 1/2     454
Supercurrent-voltage characteristics at [rho] = 2     454
Self-duality in the Ohmic scaling limit     455
Linear mobility at finite T     456
Nonlinear mobility at T = 0     457
Exact scaling function at T = 0 for arbitrary K     459
Construction of the self-dual scaling solution     459
Supercurrent-voltage characteristics at T = 0 for arbitrary [rho]     462
Connection with Seiberg-Witten theory      462
Special limits     463
Full counting statistics at zero temperature     464
Low temperature behaviour of the characteristic function     467
The sub- and super-Ohmic case     468
Charge transport in quantum impurity systems     470
Generic models for transmission of charge through barriers     471
The Tomonaga-Luttinger liquid     471
Transport through a single weak barrier     472
Transport through a single strong barrier     474
Coherent conductor in an Ohmic environment     476
Equivalence with quantum transport in a washboard potential     478
Self-duality between weak and strong tunneling     478
Full counting statistics     479
Charge transport at low T for arbitrary g     479
Full counting statistics at g = 1/2 and general temperature     482
Bibliography     483
Index     503

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