Driven-dissipative quantum mechanics on a lattice: Describing a fermionic reservoir with the master equation*

The possibility of simulating dissipative processes with digital circuits and quantum simulators, and using dissipation as a resource for state preparation, have created a renewed interest on the driven-dissipative many body problem.
Here, I address a tight binding model of electrons driven out-of-equilibrium by an electric field, where the system can exchange energy and number of particles with a fermionic reservoir. The problem is solved using the master equation formalism. I use the exact solution for the noninteracting driven-dissipative system to benchmark the accuracy of the results obtained with the master-equation, showing a very good quantitative agreement. Furthermore, to create a better link with the simulation of open quantum systems, I provide with the time dependent Kraus map that pumps the system to the non-equilibrium steady state. Generalizations of such a method to broken symmetry phases (like CDW) and their simulation on quantum computers are discussed too.