Quantum master equation for a system interacting with a spin gas environment in the low-density limit and for the semiclassical collision model

A quantum system interacting with a dilute gas experiences the irreversible dynamics. The corresponding master equation can be derived within two different approaches: the fully quantum description in the low-density limit and the semiclassical collision model, where the motion of gas particles is classical whereas their internal degrees of freedom are quantum. The two approaches have been extensively studied in the literature but their predictions have not been compared. This is mainly due to the fact that the low-density limit is extensively studied for mathematical physics purposes, whereas the collision models have been essentially developed for quantum information tasks as a tractable description of the open quantum dynamics. Here we develop and for the first time compare both approaches for a spin system interacting with a gas of spin particles. Using some approximations, we explicitly find the corresponding master equations including the Lamb shifts and the dissipators. The low density limit in the Born approximation for fast particles is shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. We reveal that the both approaches give exactly the same master equation if the gas temperature is high enough. This allows to interchangeably use complicated calculations in the low density limit and rather simple calculations in the collision model.