Dependence of Repeated Interaction Asymptotic States on Environment

Consider a quantum system $\cS$ that interacts sequentially with a chain (environment) of identical probes ${\cal C} = \cP+\cP+...$, with each interaction governed by a fixed interaction time $\tau$ and operator $V$. It is known how to construct the asymptotic state (large times) if the initial states of $\cP$ belong to a class of so-called reference states. We generalize the analysis to a broader class of initial states, including the physically important situation of pure states. This is done by a simple modification to the effective dynamics generator.