Quantum Dissipative Systems

The following sections are included:

• Preface

• Preface to the Second Edition

• 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 quanta1 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. Other experimental tools are, e. g., spin relaxation experiments, studies of the absorptive and dispersive acoustic behavior] and investigations of transport properties. In such experiments, the system's response gives information about the dynamical behavior of the spontaneous fluctuations. Theoretically, the response is rigorously described in terms of time correlation functions. Therefore, time correlation functions are the center of interest in theoretical studies of the relaxation of non-equilibrium systems.

The following sections are included:

• Diverse limited approaches: a brief survey
  • Langevin equation for a damped classical system
  • 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
    • Quasiclassical Langevin equation
    • Ohmic and frequency–dependent damping
    • Rubin model
  • The Spin–Boson model
    • The model Hamiltonian
    • Josephson two-state systems: flux and charge qubit
  • 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 tunneling: 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
    • Electromagnetic environment
  • 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 Wagner 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
  • Stochastic unraveling of influence functionals
  • Brief summary and outlook

Up to this point, we have employed functional integral methods in order to set out exact formal expressions which describe the thermodynamics and dynamics of damped quantum systems. Now we move to new grounds and consider the physical properties of exactly solvable damped linear systems, i.e. the damped quantum oscillator and quantum Brownian particle. Subsequently, we discuss a variational method which makes it possible for us to treat nonlinear damped quantum systems in much the same way as linear systems. We conclude this part with a brief general discussion of quantum decoherence.

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 tunneling clockwise forth and back between two different localized states. In reality, such a system is strongly affected by the surroundings. We will see that the coupling can lead to qualitative changes in the behaviours: 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 forward or backward from well to well by incoherent tunneling or thermally activated transitions. As the temperature is lowered, coherent tunneling across many wells may become significant, and the competing different tunneling paths may interfere constructively or destructively. The model has a far reaching 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-well systems is a central issue.

The following sections are included:

• Bibliography

• Index