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Physical Review A

Influence-functional theory of metastability in a dissipative quantum system

W. Zwerger

Published Mar 1, 1985• 28 citations
Publisher

Abstract

The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as ${T}^{n+1}$ for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.

DOI
10.1103/physreva.31.1745

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The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as \ensuremathωn{\ensuremath{\omega}}^{n}\ensuremathωn. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
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The tunneling process in dissipative quantum systems can be described in terms of a Feynman functional integral. The quasiclassical approximation entails the evaluation of solutions of the Euler-Lagrange equations in imaginary time and diagonalizing the small-amplitude excursions about the extremal trajectories. The latter process is equivalent to evaluating all the eigenvalues of integro-differential eigenvalue equations. We show that the exact eigenvalues are related to the phase shifts and derive a Hulth\'en-type of variational principle for the phase shifts appropriate for the above-described eigenvalue equation. We illustrate the method by applying it to the problem of the decay of a metastable state coupled to an Ohmic heat bath at T=0. We compare the accuracy with the Born approximation, rigorous results, and numerical evaluations.
In this paper the so-called Feynman-Vernon influence functional theory has been generalized in order to include the possibility of dealing with initial conditions of the system plus environment other than the factorizable one. It has been shown that the new influence functional can now be written in terms of paths which represent the coupling of quantum-statistical to quantum-dynamical effects. Once the thermal paths are integrated out of this new influence functional, a generalization of the original Feynman-Vernon expression is obtained.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
Finite temperatures are incorporated into the calculation of the tunneling rate in the presence of linear dissipation. A theory of finite-temperature tunneling is introduced which is based on the density matrix which provides an approximate description of the metastable state. Semiclassical functional integral methods are used to calculate the density matrix and the Wentzel-Kramers-Brillouin approximation is used to derive the tunnel current from the density matrix. Dissipation appropriate to superconducting quantum-interference devices is introduced into the calculation by the use of the existing model of a heat bath consisting of a prescribed distribution of harmonic oscillators linearly coupled to the tunneling variable. The finite-temperature tunnel escape rate obtained is of the form \ensuremath{\Gamma}=A exp(-S/\ensuremath{\Elzxh}). S is determined from the solution of a nonlinear integro-differential equation. An existing numerical technique was modified to achieve this. The quantity A is approximately evaluated.