Quantum Dissipative Systems

The following sections are included:

• Preface

• Acknowledgements

• Preface to the First Edition

• Contents

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

The following sections are included:

• Diverse limited approaches: a brief survey
  • Classical Langevin equation
  • New schemes of quantization
  • Traditional system–plus–reservoir methods
    • Quantum-mechanical master equations for weak coupling
    • Operator Langevin equations for weak coupling
    • Quantum and quasiclassical Langevin equation
    • Phenomenological methods
  • Stochastic dynamics in Hilbert space

• System–plus–reservoir models
  • Harmonic oscillator bath with linear coupling
    • The Hamiltonian of the global system
    • The road to the classical generalized Langevin equation
    • Phenomenological modeling
    • Ohmic and frequency-dependent damping
    • Rubin model
  • The Spin–Boson model
    • Truncated two-state system: trapped flux in a rf SQUID
    • The model Hamiltonian
  • Microscopic models
    • Acoustic polaron: one-phonon and two-phonon coupling
    • Optical polaron
    • Interaction with fermions (normal and superconducting)
    • Superconducting tunnel junction
  • Charging and environmental effects in tunnel junctions
    • The global system for single electron tunneling
    • Resistor, inductor and transmission lines
    • Charging effects in Josephson junctions
  • Nonlinear quantum environments

• Imaginary–time path integrals and statistical mechanics
  • The density matrix: general concepts
  • Effective action and equilibrium density matrix
    • Open system with bilinear coupling to a harmonic reservoir
    • State-dependent memory-friction
    • Spin-boson model
    • Acoustic polaron and defect: one-phonon coupling
    • Acoustic polaron: two-phonon coupling
    • Tunneling between surfaces: one-phonon coupling
    • Optical polaron
    • Heavy particle in a metal
    • Heavy particle in a superconductor
    • Effective action for a Josephson junction
  • Partition function of the open system
    • General path integral expression
    • Semiclassical approximation
    • Partition function of the damped harmonic oscillator
    • Functional measure in Fourier space
    • Partition function of the damped harmonic oscillator revisited
  • Quantum statistical expectation values in phase space
    • Generalized Weyl correspondence
    • Generalized Wigner function and expectation values

• Real–time path integrals and dynamics
  • Feynman–Vernon method for a product initial state
  • Decoherence and friction
  • 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

Up to this point, we have used functional integral methods in order to find the exact formal solution for the dynamics of a damped quantum system. Now we move to new grounds and consider the properties of simple exactly solvable linear models. Then we discuss a variational method which brings us into the position to treat nonlinear systems in the quantum regime in much the same way as linear systems. We conclude this part with a discussion of the quantum decoherence problem.

In the second part, I considered the exactly solvable case of linear dissipative quantum systems, the thermodynamic variational approach useful for nonlinear quantum systems, and the quantum decoherence problem. We now turn to a study of the frequent situation in which a metastable state is separated from the outside region by a free energy barrier. The decay of a metastable state plays a central role in many scientific areas including low-temperature physics, nuclear physics, chemical kinetics and transport in biomolecules. At high temperatures, the system escapes from the metastable well predominantly over the barrier by thermal activation. At zero temperature on the other hand, the system is in the localized ground state of the metastable well and can escape only by quantum mechanical tunneling through the barrier. We shall focus the discussion on the influence of friction on the decay process. The treatment will be based on an effective method which allows to study this problem in a unified manner for temperatures ranging from T = 0 up to the classical regime.

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 clockwise motion. The simplest model involving quantum coherence is a two-state system. The problem of a quantum 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 strongly affected by the surrounding medium. We will see that the coupling can lead to qualitative changes in the behaviors: the environment-induced fluctuations can destroy quantum coherence and can even lead to a phase transition to a state in which quantum tunneling is quenched.

A quantum Brownian particle in a multi-well potential coupled to a dissipative environment is archetypal for many problems in physics and chemistry. Examples include superionic conductors, atoms on surfaces, and interstitials in dielectrics and metals. Also the current-voltage characteristics of a Josephson junction, and charge transport in a quantum wire hindered by impurity scattering are described by this model. At high temperatures, the particle moves by incoherent tunneling or thermally activated transitions between nearest-neighbor states. As the temperature is lowered, coherent tunneling between many states leading to constructive or destructive interference may become significant. The model has an interesting duality symmetry between the weak-binding and strong-binding representation. In view of the broad area of applications, the understanding of quantum transport in multi-state systems is a central issue.

The following sections are included:

• Bibliography

• Index