Convergence of bipartite open quantum systems stabilized by reservoir engineering

We study a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, quantum system-a strategy known as quantum reservoir engineering. We provide sufficient conditions for convergence of the considered Lindblad equations; our setting accommodates the case where steady-states are not unique but rather supported on a given subspace of the underlying Hilbert space. We apply our result to a Lindblad master equation modeling engineered multi-photon emission and absorption processes, a setting that received considerable attention in recent years due to its potential applications for the stabilization of so-called cat qubits.