A semiclassical nonequilibrium Green’s function approach to electron transport in systems exhibiting electron–phonon couplings

We formulate a semiclassical theory for electron transport in open quantum systems with electron-phonon interactions adequate for situations when the system's phonon dynamics is comparable with the electron transport timescale. Starting from the Keldysh non-equilibrium Green's function formalism, we obtain equations of motion for the retarded and lesser electronic Green's functions, including contributions due to the phonon dynamics up to second order in the electron-phonon coupling strength. The resulting equations assume that the system's phonons follow classical time-local dynamics with delta-correlated noise. We apply our method to the study of the charging/discharging of a periodically driven quantum dot, and a three-level model for a single-electron pump, analyzing the signatures in the transient current, electron population, and process performance of the phonon dynamics. For these systems, we adopt the fluctuation-dissipation theorem and consider external harmonic driving of the phonon at frequencies comparable with the electron modulation, and different scenarios, varying electron-phonon coupling strength, coupling to the electron part of the system, and in-phase and anti-phase driving. Our results illustrate that our method provides an efficient protocol to describe the effects of nuclear motion in ultrafast transient phenomena.