Generic transport formula for a system driven by Markovian reservoirs

We present a generic, compact formula for the current flowing in interacting and non-interacting systems which are driven out-of-equilibrium by biased reservoirs described by Lindblad jump operators. We show that, in the limit of high temperature and chemical potential, our formula is equivalent to the well-known Meir-Wingreen formula, which describes the current flowing through a system connected to fermionic baths, therefore bridging the gap between the two formalisms. Our formulation gives a systematic way to address the transport properties of correlated systems strongly driven out of equilibrium. As an illustration, we provide explicit calculations of the current in three cases : {\it i)} a single-site impurity {\it ii)} a free fermionic chain {\it iii)} a fermionic chain with loss/gain terms along the chain. In this last case, we find that the current across the system has the same behavior for loss or gain terms and depends on the loss/gain rate in a non-monotonic way.