I Introduction .................................................. 1
1 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
10 Introduction ............................................... 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 T0 ...................... 235
16.2 Beyond steepest descent below T0 ...................... 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
17.5 Concluding remarks .................................... 257
IV THE DISSIPATIVE TWO-STATE SYSTEM .......................... 259
18 Introduction ............................................... 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 ωc ...................................... 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/r2 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 К = 1/2 and
К = 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 ID 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 (σj)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 К 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 (σz)t for К = 1/2 ........ 377
21.6.3 The case К = 1/2 - к; coherent-incoherent
crossover ...................................... 379
21.6.4 Equilibrium σz autocorrelation function ........ 380
21.6.5 Equilibrium σx autocorrelation function ........ 385
21.6.6 Correlation functions in the Toulouse model .... 387
21.7 Long-time behaviour at T = 0 for К < 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 К = 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
23.1 Introduction .......................................... 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
К = 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 К = 1/2 .............. 439
25 Duality symmetry ........................................... 439
25.1 Duality for general spectral density .................. 440
25.1.1 The map between the ТВ 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 К = 1/2
and К = 2 ............................................. 446
25.2.1 Full counting statistics at К = 1/2 ............ 446
25.2.2 Full counting statistics at К = 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
ρ << 1 ........................................ 453
25.3.3 Supercurrent-voltage characteristics at
ρ = 1/2 ........................................ 454
25.3.4 Supercurrent-voltage characteristics at
ρ = 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 К ....... 459
25.5.1 Construction of the self-dual scaling
solution ....................................... 459
25.5.2 Supercurrent-voltage characteristics at T = 0
for arbitrary ρ ................................ 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
Bibliography .................................................. 483
Index ......................................................... 503
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