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

TL;DR

This paper compares two approaches for deriving master equations of a quantum system interacting with a dilute gas, showing their equivalence under certain conditions and facilitating easier analysis of open quantum dynamics.

Contribution

It is the first comparison of the low-density limit and semiclassical collision model master equations for a spin system with a gas, highlighting their equivalence at high temperatures.

Findings

Both approaches yield identical master equations at high gas temperatures.

The low-density limit in the Born approximation is equivalent to the semiclassical collision model in the stroboscopic approximation.

The comparison enables interchangeable use of complex and simple calculation methods.

Abstract

A quantum system interacting with a dilute gas experiences 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 such 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 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.