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Physical review. B, Condensed matter

Resonantly enhanced quantum decay: A time-dependent Wentzel-Kramers-Brillouin approach

Matthew P. A. Fisher

Published Jan 1, 1988• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.37.75

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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.
This paper considers the tunneling out of a metastable state at T=0T=0T=0 of a system whose classical equation of motion is, in Fourier-transformed form, $K(\ensuremath{\omega})q(\ensuremath{\omega})=\ensuremath{-}[\frac{\ensuremath{\partial}V(q)}{\ensuremath{\partial}q}](\ensuremath{\omega})$ where V(q)V(q)V(q) is a conservative potential and $K(\ensuremath{\omega})$ represents the effects of arbitrary linear dissipative and/or reactive elements. It is shown that, provided a few commonly satisfied conditions obtain, there is a simple prescription for writing down the imaginary-time effective action functional which determines the tunneling rate in the Wentzel-Kramers-Brillouin limit; namely, it contains the usual term in V(q)V(q)V(q), plus a term of the form $(\frac{1}{2\ensuremath{\pi}}) \ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\frac{1}{2} K(\ensuremath{-}i|\ensuremath{\omega}|){|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})|}^{2}d\ensuremath{\omega}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})$ is the Fourier transform of the imaginary-time trajectory. Previously obtained results are special cases of this prescription. Applications are made to the case of "anomalous" dissipation (rate of dissipation proportional to the squared velocity of the momentum conjugate to the tunneling variable), to the "mixed" case (relaxation by collisions subject to a conservation law), and to more realistic models of a rf superconducting quantum-interference device.
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.
A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.
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.
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).