Transmission spectrum of a tunneling particle interacting with dynamical fields: real-time functional-integral approach
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 $ω$, an analytic expression of the transmission spectrum is obtained which covers from sudden tunneling ($ωT_0\ll 1$) to adiabatic one ($ωT_0\gg 1$), 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 $ηT_0\ll 1$ is found to obey a power law $\sim(eVT_0/\hbar)^{ηT_0S_0/2π\hbar}$, where $η$ is the friction coefficient and $S_0$ is the tunneling exponent in the absence of interaction.