Quantum Dissipative Systems

The following sections are included:

• Dedication

• Preface

• Contents

Quantum-statistical mechanics is a very rich and checkered field. It is the theory dealing with the dynamical behavior of spontaneous quantal fluctuations.

When probing dynamical processes in complex many-body systems, one usually employs an external force which drives the system slightly or far away from equilibrium and then measures the time-dependent response to this force. The standard experimental methods are quasielastic and inelastic scattering of light, electrons, or neutrons off a sample, and the system's dynamics is analyzed from the line shapes of the corresponding spectra.

The following sections are included:

• Traditional approaches to open quantum systems
  • Classical Langevin equation
  • New schemes of quantization
  • System–plus–reservoir methods
    • Generalized master equations
    • Generalized q–number Langevin equations
    • Quantum and quasiclassical Langevin equation
    • Phenomenological methods

• System–plus–reservoir models
  • Thermal reservoir with linear response
    • The Hamiltonian of the composite system
    • Phenomenological modelling
    • Ohmic and frequency–dependent damping
  • Microscopic models
    • Optical and acoustic polarons
    • Interaction with fermions in the normal and superconducting state
    • Superconducting tunnel junction

• Imaginary–time path integrals and statistical mechanics
  • Equilibrium density matrix of the open system
    • Linear response model of Caldeira and Leggett
    • Optical polarons
    • Acoustic polarons, interstitials, and defects
    • Heavy particle in normally conducting and superconducting environment
    • Effective action of a Josephson junction
  • Partition function of the open system
    • Semiclassical limit
    • The effective classical potential
    • Variational method

• Real–time path integrals and dynamics
  • Feynman–Vernon method for a product initial state
  • General initial states and preparation function
  • Complex–time path integral for the propagating function
  • Real–time path integral for the propagating function
    • Extremal paths
    • Classical limit
    • Semiclassical limit: quasiclassical Langevin equation
  • Brief summary and outlook

The following sections are included:

• Damped harmonic oscillator
  • Stochastic modelling
  • Fluctuation–dissipation theorem
  • The coordinate autocorrelation function
  • Ohmic friction
  • Mean square of coordinate and momentum
  • The equilibrium density matrix

• Quantum Brownian motion
  • Spectral density and damping coefficient
  • Displacement correlations and response function
  • Ohmic damping
  • Frequency–dependent damping

In part II, I considered examples of exactly solvable linear dissipative quantum systems. We now leave that simple class of models and consider the situation in which a metastable state is separated from a continuum or quasi-continuum of states by a free energy barrier. The problem of escape from a metastable state plays a central role in many scientific areas including low-temperature physics, nuclear physics, chemical kinetics and transport in biomolecules. At sufficiently high temperatures, the system escapes predominantly over the barrier by thermal activation. On the other hand, at zero temperature, the system can only decay by quantum-mechanical tunneling from the ground state in the metastable well. Our emphasis will be put on answering the question “What is the influence of dissipation on the decay process ?”. The discussion of the posed problem will focus on an effective method which enables the study in the range from T = 0 up to the classical regime in an unified approach.

One of the most interesting aspects of quantum theory is the phenomenon of constructive and destructive interference. The phase coherence between different quantum-mechanical states may lead to spectacular oscillation effects. The simplest model permitting coherent motion is a two-state system. The problem of a quantum-mechanical system whose state is effectively confined to a two-dimensional Hilbert space is often encountered in physics and chemistry. For instance, imagine a quantum-mechanical particle resonating or fluctuating between two different localized states. Such system is often coupled to the surrounding medium. We will see that the coupling can lead to qualitative changes in the behaviors; that is, the fluctuations in the environment can destroy quantum coherence and can even lead to a phase transition in which quantum tunneling is quenched.

A quantum particle moving in a multistable potential and coupled to environmental modes is the archetype model for many problems in physics and chemistry including superionic conductors, atoms on surfaces, and interstitials in dielectrics and metals. Even quantum effects in the current-voltage characteristics of a small Josephson junction are described by this model. The computation of transport properties is the most important issue for multi-state quantum systems. At high temperatures, these are determined by incoherent tunneling transitions between nearest neighbor states. As the temperature is lowered, coherent tunneling transitions between many states of the system leading to constructive and destructive interference may become increasingly important. The understanding of the outgrowth of these effects is still a challenging problem, and we shall put the emphasis here on the related questions.

The following sections are included:

• Bibliography

• Index