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Physica A Statistical Mechanics and its Applications

Decay of a metastable state: A variational approximation

E. Freidkin, Peter S. Riseborough, Peter Hänggi A

Published Apr 1, 1987• 4 citations
Publisher

Abstract

We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.

DOI
10.1016/0378-4371(87)90023-9

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Decay of a metastable state: A variational approach

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We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.
We evaluate the exponential suppression of the quantum tunneling rate due to the coupling with a thermal reservoir. The exponential dependence is calculated by a variational method, in which an approximate saddle point of the effective action is found. The effect of finite temperatures is accounted for. We calculate the exponential part of the rate for several potentials, some of which have not been considered previously. The method gives excellent agreement with numerically calculated values, where they exist. We also consider the effect of Ohmic damping mechanisms of Drude form (memory damping). This latter case may be of importance for experiments on rf superconducting quantum interference devices and Josephson junctions where frequency-dependent damping mechanisms have been observed. The trends exhibited by the variational approximation can be understood qualitatively on the basis of physical arguments.
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.
This chapter considers one important physical example of that kind, quantum dissipative decay of a metastable state. It demonstrates how to evaluate the quantum decay rate of a metastable state for a quantum particle in the presence of dissipation. In the presence of phase fluctuations, however, such current states become unstable and decay simply because the “particle” can escape from the local potential minimum either by thermal activation or by quantum tunneling. The chapter also considers sufficiently low temperatures in which case the decay is due to quantum tunnelling. Classical process of thermally activated decay of supercurrent in Josephson junctions was first demonstrated in 1974, and the results turned out to be in a good agreement with theoretical predictions. Later experiments allowed to quantitatively verify a number of important theoretical predictions, for example, ones concerning the effect of temperature and damping on the quantum decay rate.
In view of recent interest in the problem of macroscopic quantum tunneling in systems involving the Josephson effect, we present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment. Although we concentrate on a specific form of dissipation, as discussed by Caldeira and Leggett, we believe that such a numerical method can be extended to other forms of dissipation as well. Our method is based on the framework recently described by Caldeira and Leggett, and requires (a) a novel treatment of a nonlinear integro-differential equation and (b) an extension of the usual Fredholm scattering theory so as to be applicable to the present dissipative problem. We present explicit results for wide ranges of dissipation and estimate our error in the calculation of the exponent to be no larger than 0.1% and of the prefactor to be no larger than 2%.
The spin-bath induced decay of a quantum system out of a metastable state, which can tunnel and undergo thermal activation, is considered. The treatment is based on the c-number description of bath degrees of freedom and a canonical thermal distribution of the associated c-number variables. We show that, apart from the usual frequency prefactor and its dynamical modification in the spatial diffusion-limited regime, i.e., the Grote-Hynes factor, the rate coefficient contains two other factors. One is a quantum correction to the dynamical factor resulting from quantum transmission and reflection of the particle at the finite barrier. This term exhibits exponential rate enhancement at low temperature. The other one is a quantum corrected Arrhenius factor, which incorporates the effect of thermal saturation. At 0 K, dissipation due to tunneling multiplies the tunneling probability by a factor that gets exponentially damped by friction, implying that, at this temperature, the behavior of spin bath is almost identical to that of a harmonic bath.
The ground state of Bose-Einstein condensates with attractive particle interactions is metastable. One of the decay mechanisms of the condensate is a collapse by macroscopic quantum tunneling, which can be described by the bounce trajectory as a solution of the time-dependent Gross-Pitaevskii equation in imaginary time. For condensates with an electromagnetically induced gravity-like interaction, the bounce trajectory is computed with an extended variational approach using coupled Gaussian functions and is simulated numerically exact within the mean-field approach on a spacetime lattice. It is shown that the variational computations converge very rapidly to the numerically exact result with increasing number of Gaussians. The tunneling rate of the condensate is obtained from the classical action and additional parameters of the bounce trajectory. The converged variational and numerically exact results drastically improve by several orders of magnitude the decay rates obtained previously with a simple variational approach using a single Gaussian-type orbital for the condensate wave function.