Repeated Interaction Approach to Dissipation-driven Non-equilibrium Open Quantum Systems: Open Quantum Walks and Open Quantum Brownian Motion

Microscopic model-based approaches play a crucial role in non-equilibrium statistical mechanics. Traditionally, in quantum statistical mechanics one starts from the microscopic Hamiltonian of the system, bath, and system-bath interaction. After performing typical approximations, such as Born, Markov, and rotating wave, one obtains a quantum master equation for the reduced density matrix of the open system. Repeated interaction is a systematic and mathematically rigorous approach to the description of the dynamics and thermodynamics of non-equilibrium quantum systems. In the continuous time limit, together with the weak-coupling assumption, this approach naturally leads to a quantum master equation in Gorini-Kosakowski-Sudarshan-Lindblad form. Here, we extend the repeated interaction formalism to the description of open quantum walks and open quantum Brownian motion, which are examples of quantum stochastic dynamical systems purely driven by the dissipative interaction with an environment. We show that the repeated interaction approach provides a natural tool for the investigation and simulation of dynamical and thermodynamical properties of these non-equilibrium dissipation driven systems.