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

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 was 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 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.

Figure 1

Open dynamics of the system (large circle) with density operator $\varrho$ due to interaction with a diluted gas (small circles).

Figure 2

A gas particle with the initial momentum $\mathbf{p}$ and internal state $|j\rangle$ is scattered to the state with momentum $\mathbf{q}$ and internal state $|i\rangle$, whereas the system state is changed from $|l\rangle$ to $|k\rangle$. Operator $F$ defines the interaction between internal degrees of freedom of the gas particle and the system, potential $U\left(\mathbf{r}\right)$ determines the strength of the interaction.

Figure 3

Collision model with impact time $\tau$ and free propagation time $t_{\mathrm{free}}$. The system-particle Hamiltonian during the collision is $gF$.

Figure 4

The impact parameter $b$. The classical trajectories are approximated by straight lines for fast gas particles (left). The volume of particles with momentum $\mathbf{p}$ and impact parameter $b-(b+db)$ that reach the interaction region within time $t$, is $dV=2\pi bdb\times pt/m$ (right).