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Physical review. B, Condensed matter

Quantum decay in a dissipative system

Lit-Deh Chang, Sudip Chakravarty

Published Jan 1, 1984• 85 citations
Publisher

Abstract

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%.

DOI
10.1103/physrevb.29.130

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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.
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.
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.
The decay time, τ, of a dissipative object out of the metastable minimum of a (cubic) potential due to quantum noise is determined by a numerical investigation of the quasiclassical Langevin equation. In the zero temperature limit, and for moderate to large damping (γ\gtrsimω o ), we find that σ=ln (ω o τ/2π) can be approximated by a σ≃κ·γΔV/\hbarω o 2 where the constant κ is κ≃3.20. This is in good agreement with an analytic treatment of the Langevin equation, but about a factor three smaller than the result obtained by Caldeira and Leggett by the instanton method.
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.