Transmission spectrum of a tunneling particle interacting with dynamical fields: real-time functional-integral approach

Abstract

A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency $\omega$, an analytic expression of the transmission spectrum is obtained which covers from sudden tunneling ($\omega T_0\ll 1$) to adiabatic one ($\omega T_0\gg 1$), where $T_0$ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with Ohmic spectrum, the differential tunneling conductance at low bias voltage $V$ and for $\eta T_0\ll 1$ is found to obey a power law $\sim (eV{T}_0/\hslash {)}^{\eta T_0{S}_0/2\pi \hslash }$, where $\eta$ is the friction coefficient and $S_0$ is the tunneling exponent in the absence of interaction.

RevTeX3.0, 18 pages, two figures (not included) are available upon request from the author. Accepted for publication in Phys. Rev. B15