Repeated Interaction Processes in the Continuous-Time Limit, Applied to Quadratic Fermionic Systems

We study a class of Lindblad equations on finite-dimensional Fermionic systems. The model is obtained as the continuous-time limit of a repeated interaction process between Fermionic systems with quadratic Hamiltonians, a setup already used by Platini and Karevski for the one-dimensional XY-model. We prove a necessary and sufficient condition for the convergence to a unique stationary state, which is similar to the Kalman criterion in control theory. Several examples are treated, including a spin chain with interactions at both ends.