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Acta Physica Sinica

Influence of nonlinear coupling on quantum decay rate of metastable dissipative systems

Bi Lei, Jing-Dong Bao, 北京师范大学物理系，北京 100875

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We develop the fast-Fourier-transformation path-integral approach to investigate the quantum decay of a nonlinear dissipative system. The action of the bounce trajectory, i.e. the exponential factor of decay rate, is obtained. In the case of the nonlinear coupling f(x)=tanh［λ(x-xb)］ between the system and its environment, we find that the nonlinear coupling suppresses the decay rate. In contrast to the usual linear coupling, the action will not abide by the law SB=a［1-b(T/Tc)2］ and the crossover temperature rebounds, which means that the system steps into the quantum tunneling region at a higher temperature.

DOI
10.7498/aps.56.1919

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We develop the fast-Fourier-transformation path-integral approach to investigate the quantum decay of a nonlinear dissipative system. The action of the bounce trajectory, i.e. the exponential factor of decay rate, is obtained. In the case of the nonlinear coupling f(x)=tanh［λ(x-xb)］ between the system and its environment, we find that the nonlinear coupling suppresses the decay rate. In contrast to the usual linear coupling, the action will not abide by the law SB=a［1-b(T/Tc)2］ and the crossover temperature rebounds, which means that the system steps into the quantum tunneling region at a higher temperature.
The coupling between system and reservoir is considered to be linear in the coordinates of the bath but nonlinear in the system's coordinate. A dissipative threshold is observed at finite temperatures due to nonlinear dissipation. The quantum decay rate of a metastable state including higher-order expanded terms of the coupling form function is proposed, which can be strongly decreased at finite temperatures when the quantum dissipative threshold is added to the saddle point of the potential.
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 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 ${\ensuremath{\omega}}^{n}$. 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.
A quantum-activated rate process in the presence of a nonlinear dissipative environment is studied with a Feynman path-integral technique. The nonlinear dissipation arises from specific nonlinear couplings between the reaction coordinate and a many-dimensional bath of effective harmonic oscillators. Such coupling results in a nonlinear influence functional for the effective Feynman path integral along the reaction coordinate. Large nonlinear effects on the rate constant are found at low temperatures.