Time-coarse-grained dynamics of open quantum systems: The system-plus-reservoir approach and effective models

In this work, we study temporal coarse-graining as a basis to derive accurate effective models for open quantum systems. In particular, we demonstrate that the coarse-graining time scale, when treated as a physical parameter, determines the validity of various commonly employed approximations such as the RWA and the Markov approximation. With a suitable choice of the coarse-graining time scale, we systemically derive the Lindblad dynamics without assuming the all-time factorizability of the full density matrix. Deviations from the Lindblad dynamics are quantified analytically and then compared to numerical results obtained from the full system-plus-reservoir master equation. We further compare the system-plus-reservoir approach to direct time-coarse-graining of the Lindblad master equation and to naïve time-averaging of the dynamical variables, emphasizing on the sources of their differences as well as the corresponding physical interpretations. Finally, we address the initial condition problem of time-coarse-grained dynamics and argue that the choice of the initial condition depends on one's interpretation of the physical nature of time-coarse-graining.