Entanglement of Bipartite Quantum Systems driven by Repeated Interactions

We consider a non-interacting bipartite quantum system $\mathcal H_S^A\otimes\mathcal H_S^B$ undergoing repeated quantum interactions with an environment modeled by a chain of independant quantum systems interacting one after the other with the bipartite system. The interactions are made so that the pieces of environment interact first with $\mathcal H_S^A$ and then with $\mathcal H_S^B$. Even though the bipartite systems are not interacting, the interactions with the environment create an entanglement. We show that, in the limit of short interaction times, the environment creates an effective interaction Hamiltonian between the two systems. This interaction Hamiltonian is explicitly computed and we show that it keeps track of the order of the successive interactions with $\mathcal H_S^A$ and $\mathcal H_S^B$. Particular physical models are studied, where the evolution of the entanglement can be explicitly computed. We also show the property of return of equilibrium and thermalization for a family of examples.