Quantum tunneling in the presence of an arbitrary linear dissipation mechanism

Abstract

This paper considers the tunneling out of a metastable state at $T=0$ 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 $V(q)$ 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 $V(q)$, 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.