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A quantum system interacting with a dilute gas experiences irreversibledynamics. The corresponding master equation can be derived within two differentapproaches: The fully quantum description in the low-density limit and thesemiclassical collision model, where the motion of gas particles is classicalwhereas their internal degrees of freedom are quantum. The two approaches havebeen extensively studied in the literature, but their predictions have not beencompared. This is mainly due to the fact that the low-density limit isextensively studied for mathematical physics purposes, whereas the collisionmodels have been essentially developed for quantum information tasks such as atractable description of the open quantum dynamics. Here we develop and for thefirst time compare both approaches for a spin system interacting with a gas ofspin particles. Using some approximations, we explicitly find the correspondingmaster equations including the Lamb shifts and the dissipators. The low densitylimit in the Born approximation for fast particles is shown to be equivalent tothe semiclassical collision model in the stroboscopic approximation. We revealthat both approaches give exactly the same master equation if the gastemperature is high enough. This allows to interchangeably use complicatedcalculations in the low density limit and rather simple calculations in thecollision model.

Quantum master equations for a system interacting with quantum gas in the low density limit and for the semiclassical collision model