Virtual Process in Time-dependent Tunneling

The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.