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Physical Review A

Quantum decay into a continuum at weak bias

Peter Hänggi A, Ulrich Weiß, Peter S. Riseborough

Published Dec 1, 1986• 15 citations
Publisher

Abstract

We consider the quantum decay from the locally stable ground state of a one-dimensional metastable potential. We consider the case where the metastable minimum is almost degenerate with the vacuum level. In this case, the quantum decay probability is influenced by backscattering from the continuum states; a feature which is not accounted for in standard WKB theory. We evaluate the decay rates, from the ground state and the higher-resonance states, by the use of complex-time path-integral methods. The decay rate from the low-lying states of the metastable well is characterized by two quasi-zero-modes and a transmission factor, which approaches zero proportional to the square root of the potential drop. Moreover, we present a useful set of rules for the complex-time path-integral phase factors. These rules considerably simplify the calculation of decay rates (i.e., the pole condition of the Fourier-transformed Green's function).

DOI
10.1103/physreva.34.4558

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We consider the quantum decay from the locally stable ground state of a one-dimensional metastable potential. We consider the case where the metastable minimum is almost degenerate with the vacuum level. In this case, the quantum decay probability is influenced by backscattering from the continuum states; a feature which is not accounted for in standard WKB theory. We evaluate the decay rates, from the ground state and the higher-resonance states, by the use of complex-time path-integral methods. The decay rate from the low-lying states of the metastable well is characterized by two quasi-zero-modes and a transmission factor, which approaches zero proportional to the square root of the potential drop. Moreover, we present a useful set of rules for the complex-time path-integral phase factors. These rules considerably simplify the calculation of decay rates (i.e., the pole condition of the Fourier-transformed Green's function).
The WKB approach to quantum tunneling from a metastable well is generalized to include a weak oscillatory force. Both one-dimensional decay and multidimensional decay, appropriate for a system coupled to a dissipative environment, are considered. In both cases, general expressions are derived for the enhancement of the decay rate linear in the applied force. The enhancement is expressed exclusively in terms of the instanton trajectories of the unperturbed system. For a cubic potential with no damping, the decay rate exhibits sharp resonances when the applied frequency coincides with the energy spacing between the ground and excited states in the metastable well. In the strong-damping limit, this resonance structure is completely suppressed.
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 Wentzel–Kramers–Brillouin and bounce (path integral) answers for the rate of escape from the ground state of a metastable potential well are both reduced to an expression involving only functionals of the potential, without the need for solving any auxiliary problems. The results are extended to low-lying excited states, and they are applied to the classic cubic parabola and other model potentials. A numerical calculation is done of the escape rate from the ground state of the cubic parabolic well for a variety of barrier heights, and is found to compare well with the WKB result.
Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert WWW function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation ΓT=TtransΓT = T_{\text{trans}}ΓT=Ttrans​ between the decay rate ΓΓΓ, oscillation period TTT, and transmission probability TtransT_{\text{trans}}Ttrans​, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.
The periodic instanton method is used to study the decay rates of metastable ground state and excited states of the cubic potential. The imaginary part of the energy is calculated through the standard procedure in the path-integral scheme. A formula of the decay rate valid for the entire region of energy is obtained. This formula provides a linkage between classical thermal activation at high temperatures and purely quantum tunneling at zero temperature. It is shown that in the low energy limit this more general result reduces exactly to the vacuum result. The temperature dependence of the decay rate agrees with earlier works in the literature.
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.