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Physical Review Letters

Probabilities for Quantum Tunneling through a Barrier with Linear Passive Dissipation

A. Widom, Terrence D. Clark

Published Jan 11, 1982• 70 citations
Publisher

Abstract

The transmission coefficient for passage through a barrier with a parabolic maximum is computed rigorously when linear passive dissipation is present. The exact result for energies above and below the barrier height can be expressed in terms of the Coleman renormalized one-bounce time. The quasiclassical (WKB) approximation is recovered for energies significantly below the barrier height where "friction" enhances the tunneling probability.

DOI
10.1103/physrevlett.48.63

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The transmission coefficient for passage through a barrier with a parabolic maximum is computed rigorously when linear passive dissipation is present. The exact result for energies above and below the barrier height can be expressed in terms of the Coleman renormalized one-bounce time. The quasiclassical (WKB) approximation is recovered for energies significantly below the barrier height where "friction" enhances the tunneling probability.
A phenomenological approach to dissipative tunneling, originally put forth by Pollak [Phys. Rev. A 33, 4244 (1986)], is discussed and contrasted with the bounce formalism. This phenomenological approach is based on the WKB theory for parabolic barrier tunneling. It is quite simple and readily evaluated. We study the results for the leading exponential dissipative decay rate in metastable potentials and contrast them with previous instanton-type (bounce-technique) calculations. In doing so, we compare the zero-temperature decay rate in various metastable potentials, different dissipation mechanisms, and also consider the rate enhancement at very low temperatures. One finds between the two methods unexpected good agreement at weak to moderate dissipation strength; the accuracy decreases with increasing dissipation strength and increasing deviation from a harmonic barrier shape.
Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.
A quantum system which can tunnel, at T=0T=0T=0, out of a metastable state and whose interaction with its environment is adequately described in the classically accessible region by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor $\mathrm{exp}[\ensuremath{-}\frac{A\ensuremath{\eta}{(\ensuremath{\Delta}q)}^{2}}{\ensuremath{\hbar}}]$, where $\ensuremath{\Delta}q$ is the "distance under the barrier" and AAA is a numerical factor which is generally of order unity.