A semiclassical model for the non-equilibrium quantum transport of a many-electron Hamiltonian coupled to phonons

The semiclassical approach developed for non-equilibrium quantum transport in molecular junctions [J. Chem. Phys. 134, 164103 (2011)] is extended to include inelastic effects. The many-body Hamiltonian describing a quantum dot coupled to the non-interacting leads and to a phonon bath is mapped onto a classical model that preserves the fermionic character of electrons and the bosonic character of the phonons. Comparisons with a numerically exact real-time path integral approach generated for the non-equilibrium Holstein model reveal that a semiclassical treatment of inelastic transport provides a quantitative description of the dynamics at short times and at steady state for a wide range of electron-phonon couplings at different bias potentials. The approach opens up the possibility of treating inelastic tunnelling in complex models that remain far from the reach of fully quantum approaches - for example, when anharmonic effects play an essential role in the non-equilibrium dynamics.