Classical and quantum transport from generalized Landauer-Büttiker equations. II. Time-dependent resonant tunneling

A generalization of the Landauer-Büttiker picture of transport applicable to time-dependent problems in the presence of dissipation is presented. Starting from the Schrödinger equation in the Keldysh formalism under appropriate boundary conditions and using a linear-response approximation, we find that an ideal voltage probe is described by a generalized Landauer-Büttiker equation (GLBE). Thus, the GLBE may be considered as a particular case of the Keldysh kinetic equation. The transmission probabilities become retarded functions of the elapsed time, and are evaluated in terms of the retarded and advanced Green’s functions. The interpretation of the mathematical formalism is emphasized. The GLBE is applied to get the time-dependent dissipative resonant tunneling by considering a resonant dot coupled with (a) a dephasing process and (b) a finite side probe. For case (a), we find that the resonant-tunneling conductance G is associated with an inductance L≊${\mathrm{\tau }}_{\mathit{q}}$/G which accounts for the response in the low-frequency regime; G depends on the dissipation inside the resonant region, while ${\mathrm{\tau }}_{\mathit{q}}$, being twice the natural lifetime of the resonant state, does not. For case (b), we find an additional delay in the response accounting for the ‘‘inertia’’ of the side probe. This result clarifies the concepts of ideal voltage probe and voltage source as applied in the case of time-dependent transport.