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Physical Review A

Transition-state theory for tunneling in dissipative media

Eli Pollak

Published Jun 1, 1986• 86 citations
Publisher

Abstract

The recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.

DOI
10.1103/physreva.33.4244

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The recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.
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%.
As a prelude to a discussion of tunneling in the presence of dissipation, we discuss the damped harmonic oscillator. We show that for a harmonic-oscillator potential, the Caldeira-Leggett Hamiltonian can be reduced to a sum of independent harmonic-oscillator Hamiltonians by a normal-mode transformation. We then discuss tunneling through a parabolic barrier and explain the apparently contradictory results in the literature. The decay of a metastable state is then discussed. We argue that much of the published work on tunneling in dissipative systems has assumed unphysical initial conditions. We calculate the tunneling out of a metastable state, on the assumption that the environment is initially at zero temperature, and find that quantum tunneling is enhanced by dissipation.
The effect of a low-frequency dissipative medium on the tunneling rate of a particle trapped in a metastable potential is investigated with the aid of a frozen-bath sudden approximation. The sudden theory is formulated for arbitrary time-dependent friction and for all temperatures at which the escape process is dominated by tunneling. The validity criterion of the theory is only that the bath frequency spectrum be substantially lower than the system frequency. It is applicable both in the weak- and strong-damping limits. The sudden theory is applied to tunneling in a cubic potential and a piecewise harmonic potential where the barrier frequency differs from the well frequency. In contrast to the ImF method, the sudden theory, when valid, can provide estimates of tunneling rates from well-defined excited resonance states and is not limited to estimates of only thermal rates. We find that, if the barrier is thin relative to the well, dissipation can serve to enhance tunneling rates from excited states.
The transition rate for a two-state system interacting with a bosonic heat bath, from the initial state `up' at time t=0 to `down' at time t>0, was derived formally in the seminal paper [12] by Leggett, Chakravarty, Dorsey, Fisher, Garg and Zwerger. The expression was obtained for small tunneling parameter of the two-level system using time-dependent perturbation theory to second order. It contains all orders in the parameter of coupling between the system and the bath. We give a rigorous derivation of the transition rate, using a quantum resonance approach.
We develop a quantal cumulant dynamics method for the quantum tunneling in dissipative environment. Reduced equations of motion of classical and quantal cumulant variables without bath degrees of freedom are derived. We observed suppression of the tunneling that depends on the sign of a friction constant for an Ohmic approximation and on the magnitude of a bath frequency for a single bath mode approximation. A possible mechanism of the suppression is explored by analyzing an effective quantal potential of the tunneling path.