Quantum Dissipative Systems (Third Edition) (Modern Condensed Matter Physics)

✍ Scribed by Ulrich Weiss

• Publisher: Wspc
• Year: 2008
• Tongue: English
• Leaves: 527
• Edition: 3
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
Preface to the Second Edition
Acknowledgements
Preface to the First Edition
1 Introduction
I GENERAL THEORY OF OPEN QUANTUM SYSTEMS
2 Diverse limited approaches: a brief survey
2.1 Langevin equation for a damped classical system
2.2 New schemes of quantization
2.3 Traditional system-plus-reservoir methods
2.3.1 Quantum-mechanical master equations for weak coupling
2.3.2 Operator Langevin equations for weak coupling
2.3.3 Quantum and quasiclassical Langevin equation
2.3.4 Phenomenological methods
2.4 Stochastic dynamics in Hilbert space
3 System-plus-reservoir models
3.1 Harmonic oscillator bath with linear coupling
3.1.1 The Hamiltonian of the global system
3.1.2 The road to the classical generalized Langevin equation
3.1.3 Phenomenological modeling
3.1.4 Quasiclassical Langevin equation
3.1.5 Ohmic and frequency-dependent damping
3.1.6 Rubin model
3.2 The Spin-Boson model
3.2.1 The model Hamiltonian
3.2.2 Josephson two-state systems: flux and charge qubit
3.3 Microscopic models
3.3.1 Acoustic polaron: one-phonon and two-phonon coupling
3.3.2 Optical polaron
3.3.3 Interaction with fermions (normal and superconducting)
3.3.4 Superconducting tunnel junction
3.4 Charging and environmental effects in tunnel junctions
3.4.1 The global system  or single electron tunneling
3.4.2 Resistor, inductor and transmission lines
3.4.3 Charging effects in Josephson junctions
3.5 Nonlinear quantum environments
4 Imaginary-time path integrals
4.1 The density matrix: general concepts
4.2 Effective action and equilibrium density matrix
4.2.1 Open system with bilinear coupling to a harmonic reservoir
4.2.2 State-dependent memory-friction
4.2.3 Spin-boson model
4.2.4 Acoustic polaron and defect tunneling: one-phonon coupling
4.2.5 Acoustic polaron: two-phonon coupling
4.2.6 Tunneling between surfaces: one-phonon coupling
4.2.7 Optical polaron
4.2.8 Heavy particle in a metal
4.2.9 Heavy particle in a superconductor
4.2.10 Effective action for a Josephson junction
4.2.11 Electromagnetic environment
4.3 Partition function of the open system
4.3.1 General path integral expression
4.3.2 Semiclassical approximation
4.3.3 Partition function of the damped harmonic oscillator
4.3.4 Functional measure in Fourier space
4.3.5 Partition function of the damped harmonic oscillator revisited
4.4Quantum statistical expectation values in phase space
4.4.1 Generalized Weyl correspondence
4.4.2 Generalized Wigner function and expectation values
5 Real-time path integrals and dynamics
5.1 Feynman-Vernon method for a product initial state
5.2 Decoherence and friction
5.3 General initial states and preparation function
5.4 Complex-time path integral for the propagating function
5 5 Real-time path integral for the propagating function
5.5.1 Extremal paths
5.5.2 Classical limit
5.5.3 Semiclassical limit: quasiclassical Langevin equation
5.6 Stochastic unraveling of influence functionals
5.7 Brief summary and outlook
II FEW SIMPLE APPLICATIONS
6 Damped harmonic oscillator
6.1 Fluctuation-dissipation theorem
6.2 Stochastic modeling
6.3 Susceptibility for Ohmic friction and Drude damping
6.3.1 Strict Ohmic friction
6.3.2 Drude damping
6.4 The position autocorrelation function
6.4.1 Ohmic damping
6.4.2 Algebraic spectral density
6.5 Partition function, internal energy and density of states
6.5.1 Partition function and internal energy
6.5.2 Spectral density of states
6.6 Mean square of position and momentum
6.6.1 General expressions for coloured noise
6.6.2 Strict Ohmic case
6.6.3 Ohmic friction with Drude regularization
6.7 Equilibrium density matrix
6.7.1 Purity
7 Quantum Brownian free motion
7.1 Spectral density. damping function and mass renormalization
7.2 Displacement correlation and response function
7.3 Ohmicdamping
7.4 Frequency-dependent damping
7.4.1 Response function and mobility
7.4.2 Mean square displacement
8 The thermodynamic variational approach
8.1 Centroid and the effective classical potential
8.1.1 Centroid
8.1.2 The effective classical potential
8.2 Variational method
8.2.1 Variational method for the free energy
8.2.2 Variational method for the effective classical potential
8.2.3 Variational perturbation theory
8.2.4 Expectation values in coordinate and phase space
9 Suppression of quantum coherence
9.1 Nondynamical versus dynamical environment
9.2 Suppression of transversal and longitudinal interferences
9.3 Localized bath modes and universal decoherence
9.3.1 A model with localized bath modes
9.3.2 Statistical average of paths
9.3.3 Ballistic motion
9.3.4 Diffusive motion
III QUANTUM STATISTICAL DECAY
10 Introduction
11 Classical rate theory: a brief overview
11.1 Classical transition state theory
11.2 Moderate-to-strong-damping regime
11.3 Strong damping regime
11.4 Weak-damping regime
1 2 Quantum rate theory: basic methods
12.1 Formal rate expressions in terms of flux operators
12.2 Quantum transition state theory
12.3 Semiclassical limit
12.4 Quantum tunneling regime
12.5 Free energy method
12.6 Centroid method
13 Multidimensional quantum rate theory
14 Crossover from thermal to quantum decay
14.1 Normal mode analysis at the barrier top
14.2 Turnover theory for activated rate processes
14.3 The crossover temperature
15 Thermally activated decay
15.1 Rate formula above the crossover regime
15.2 Quantum corrections in the preexponential factor
15.3 The quantum Smoluchowski equation approach
15.4 Multidimensional quantum transition state theory
16 The crossover region
16.1 Beyond steepest descent above To
16.2 Beyond steepest descent below To
16.3 The scaling region
17 Dissipative quantum tunneling
17.1 The quantum rate formula
17.2 Thermal enhancement of macroscopic quantum tunneling
17.3 Quantum decay in a cubic potential for Ohmic friction
17.3.1 Bounce action and quantum prefactor
17.3.2 Analytic results for strong Ohmic dissipation
17.4 Quantum decay in a tilted cosine washboard potential
17.5 Concluding remarks
IV THE DISSIPATIVE TWO-STATE SYSTEM
18 Introduction
18.1 Truncation of the double-well to the two-state system
18.1.1 Shifted oscillators and orthogonality catastrophe
18.1.2 Adiabatic renormalization
18.1.3 Renormalized tunnel matrix element
18.1.4 Polaron transformation
18.2 Pair interaction in the charge picture
18.2.1 Analytic expression for any s and arbitrary cutoff w,
18.2.2 Ohmic dissipation and universality limit
19 Thermodynamics
19.1 Partition function and specific heat
19.1.1 Exact formal expression for the partition function
19.1.2 Static susceptibility and specific heat
19.1.3 The self-energy method
19.1.4 The limit of high temperatures
19.1.5 Noninteracting-kink-pair approximation
19.1.6 Weak-damping limit
19.1.7 The self-energy method revisited: partial resummation
19.2 Ohmic dissipation
19.2.1 General results
19.2.2 The special case K = f
19.3 Non-Ohmic spectral densities
19.3.1 The sub-ohmic case
19.3.2 The super-ohmic case
19.4 Relation between the Ohmic TSS and the Kondo model
19.4.1 Anisotropic Kondo model
19.4.2 Resonance level model
19.5 Equivalence of the Ohmic TSS with the l/r2 Ising model
20 Electron transfer and incoherent tunneling
20.1 Electron transfer
20.1.1 Adiabatic bath
20.1.2 Marcus theory for electron transfer
20.2 Incoherent tunneling in the nonadiabatic regime
20.2.1 General expressions for the nonadiabatic rate
20.2.2 Probability for energy exchange: general results
20.2.3 The spectral probability density for absorption at T = 0
20.2.4 Crossover from quantum-mechanical to classical behaviour
20.2.5 The Ohmic case
20.2.6 Exact nonadiabatic rates for K = l / 2 and K = 1
20.2.7 The sub-ohmic case (0 < s < 1)
20.2.8 The super-ohmic case ( s > 1)
20.2.9 Incoherent defect tunneling in metals
20.3 Single charge tunneling
20.3.1 Weak-tunneling regime
20.3.2 The current-voltage characteristics
20.3.3 Weak tunneling of 1D interacting electrons
20.3.4 Tunneling of Cooper pairs
20.3.5 Tunneling of quasiparticles
21 Two-state dynamics
21.1 Initial preparation, expectation values, and correlations
21.1.1 Product initial state
21.1.2 Thermal initial state
21.2 Exact formal expressions for the system dynamics
21.2.1 Sojourns and blips
21.2.2 Conditional propagating functions
21.2.3 The expectation values (0, ) t ( j = z, y, z )
21.2.4 Correlation and response function of the populations
21.2.5 Correlation and response function of the coherences
21.2.6 Generalized exact master equation and integral relations
21.3 The noninteracting-blip approximation (NIBA)
21.3.1 Symmetric Ohmic system in the scaling limit
21.3.2 Weak Ohmic damping and moderate-to-high temperature
21.3.3 The super-ohmic case
21.4 Weak-coupling theory beyond the NIBA for a biased system
21.4.1 The one-boson self-energy .
21.4.2 Populations and coherences (super-ohmic and Ohmic)
21.5 The interacting-blip chain approximation
21.6 Ohmic dissipation with K at and near f : exact results
21.6.1 Grand-canonical sums of collapsed blips and sojourns
21.6.2 The expectation value ( u ~ ) ~ for K = f
21.6.3 The case K = f - K ; coherent-incoherent crossover
21.6.4 Equilibrium gz autocorrelation function
21.6.5 Equilibrium oZ autocorrelation function
21.6.6 Correlation functions in the Toulouse model
21.7 Long-time behaviour at T = 0 for K < 1: general discussion
21.7.1 The populations
21.7.2 The population correlations and generalized Shiba relation
21.7.3 The coherence correlation function
21.8 From weak to strong tunneling: relaxation and decoherence
21.8.1 Incoherent tunneling beyond the nonadiabatic limit
21.8.2 Decoherence at zero temperature: analytic results
21.9 Thermodynamics from dynamics
22 The driven two-state system
22.1 Time-dependent external fields
22.1.1 Diagonal and off-diagonal driving
22.1.2 Exact formal solution
22.1.3 Linear response
22.1.4 The Ohmic case with Kondo parameter K =
22.2 Markovian regime
22.3 High-frequency regime
22.4 Quantum stochastic resonance
22.5 Driving-induced symmetry breaking
V THE DISSIPATIVE MULTI-STATE SYSTEM
23 Quantum Brownian particle in a washboard potential
23.1 Introduction
23.2 Weak- and tight-binding representation
24 Multi-state dynamics
24 1 Quantum transport and quantum-statistical fluctuations
24.1.1 Product initial state
24.1.2 Characteristic function of moments and cumulants
24.1.3 Thermal initial state and correlation functions
24.2 Poissonian quantum transport
24.2.1 Dynamics by incoherent nearest-neighbour tunneling moves
24.2.2 The general case
24.3 Exact formal expressions for the system dynamics
24.3 1 Product initial state
24.3.2 Thermal initial state
24.4 Mobility and Diffusion
24.4.1 Exact formal series expressions for transport coefficients
24.4.2 Einstein relation
24.5 The Ohmic case
24.5.1 Weak-tunneling regime
24.5.2 Weak-damping limit
24.6 Exact solution in the Ohmic scaling limit at K =
24.6.1 Current and mobility
24.6.2 Diffusion and skewness
24.7 The effects of a thermal initial state
24.7.1 Mean position and variance
24.7.2 Linear response
24.7.3 The exactly solvable case K = 5
25 Duality symmetry
25.1 Duality for general spectral density
25.1.1 The map between the TB and WB Hamiltonian
25.1.2 Frequency-dependent linear mobility
25.1.3 Nonlinear static mobility
25.2 Self-duality in the exactly solvable cases K = f and K = 2
25.2.1 Full counting statistics at K = $
25.2.2 Full counting statistics at K = 2
25.3 Duality and supercurrent in Josephson junctions
25.3.1 Charge-phase duality
25.3.2 Supercurrent-voltage characteristics for p << 1
25.3.3 Supercurrent-voltage characteristics at p = f .
25.3.4 Supercurrent-voltage characteristics at p = 2
25.4 Self-duality in the Ohmic scaling limit
25.4.1 Linear mobility at finite T
25.4.2 Nonlinear mobility at T = 0
25.5 Exact scaling function at T = 0 for arbitrary K
25.5.1 Construction of the self-dual scaling solution
25.5.2 Supercurrent-voltage characteristics at T = 0 for arbitrary p
25.5.3 Connection with Seiberg-Witten theory
25.5.4 Special limits
25.6 Full counting statistics at zero temperature
25.7 Low temperature behaviour of the characteristic function
25.8 The sub- and super-ohmic case
26 Charge transport in quantum impurity systems
26.1 Generic models for transmission of charge through barriers
26.1.1 The Tomonaga-Luttinger liquid
26.1.2 Transport through a single weak barrier
26.1.3 Transport through a single strong barrier
26.1.4 Coherent conductor in an Ohmic environment
26.1.5 Equivalence with quantum transport in a washboard potential
26.2 Self-duality between weak and strong tunneling
26.3 Full counting statistics
26.3.1 Charge transport at low T for arbitrary g
26.3.2 Full counting statistics at g = and general temperature
Bibliography
Index