Markovian Repeated Interaction Quantum Systems

We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{\omega_n}\circ\cdots\circ\mathcal{L}_{\omega_1}(\rho_{\omega_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(\omega_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_\omega$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_\omega$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.