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 the 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 \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}${\mathit{T}}_{0}$\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}${\mathit{T}}_{0}$\ensuremath{\gg}1), where ${\mathit{T}}_{0}$ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}${\mathit{T}}_{0}$\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(${\mathit{eVT}}_{0}$/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and ${\mathit{S}}_{0}$ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.