Asymptotics of repeated interaction quantum systems

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A quantum system interacts in a successive way with elements of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between and. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ--periodic in time and does not depend on the initial state. If the reference state is chosen so that and are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.