Semi-classical theory of quantum stochastic resistors

We devise a semi-classical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We find a non-trivial dependence of the conductance of these systems on the chemical potential of the reservoirs. We then introduce a semi-classical approach that is in good agreement with the exact numerical solution and provides an intuitive and simpler interpretation of transport in quantum stochastic resistors. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states. We conclude by discussing the case of dissipative leads affected by dephasing, deriving the conditions for which they effectively behave as Markovian injectors of particles in the system.