Can Three-Body Recombination Purify a Quantum Gas?

Three-body recombination in quantum gases is traditionally associated with heating, but it was recently found that it can also cool the gas. We show that in a partially condensed three-dimensional homogeneous Bose gas three-body loss could even purify the sample, that is, reduce the entropy per particle and increase the condensed fraction $\eta$. We predict that the evolution of $\eta$ under continuous three-body loss can, depending on small changes in the initial conditions, exhibit two qualitatively different behaviors—if it is initially above a certain critical value, $\eta$ increases further, whereas clouds with lower initial $\eta$ evolve towards a thermal gas. These dynamical effects should be observable under realistic experimental conditions.

Figure 1

Microscopic dynamics of an ideal homogeneous Bose gas with three-body loss. (a) Saturation-driven cooling. Loss of atoms from a saturated thermal cloud (at a rate ${\mathrm{\Gamma }}_{\mathrm{th}}{N}_{\mathrm{th}}$) induces a flow of zero-energy atoms (${\dot{N}}_t$) from the BEC to the thermal gas, which lowers the temperature. Direct loss of the BEC atoms (${\mathrm{\Gamma }}_0{N}_0$) has no effect on the temperature. (b) Three-body loss processes. The rates of three-body collisions between different numbers of BEC (blue) and thermal (orange) atoms involve different combinatorial terms, reflecting the boson bunching that occurs in a thermal cloud but not in a BEC. Normalized by the appropriate powers of BEC and thermal densities, the relative rates of the processes (i)–(iv) are, respectively, $1/(3!)$, $1/(2!)$, 1, and 1. This preferential loss of thermal atoms can lead to purification of the gas.

Figure 2

Three-body cooling and purification of an ideal partially condensed homogeneous Bose gas. The purification coefficient $\mathcal{P}$ (see text), which determines whether the gas purifies ($\mathcal{P}>1$) or cools without purifying ($0<\mathcal{P}<1$), depends only on the condensed fraction $\eta$. The critical value ${\eta }^{*}=0.76$ (indicated by the red dot) defines a “bifurcation point” for the evolution of the cloud. As indicated by the arrows, for $\eta <{\eta }^{*}$ the condensed fraction keeps dropping, but for $\eta >{\eta }^{*}$ the gas self-purifies and $\eta \to 1$. The horizontal dashed line, $\mathcal{P}=3/5$, corresponds to the result for one-body loss, which cannot purify the gas. Inset: evolution of $\eta$ for different initial conditions; here $N_{\mathrm{i}}$ is the initial atom number.

Figure 3

Cooling and purification in a weakly interacting gas. The purification coefficient $\mathcal{P}$ now depends on the condensed fraction $\eta$ and the gas parameter $n{a}^3$. The evolution of the system is described by trajectories that flow either to $\eta =1$ or to $\eta =0$, depending on which side of the critical trajectory ${\eta }^{*}(n{a}^3)$ (red line) they are. On this log-log graph the slope of the trajectories is given by $\mathcal{P}-1$ [see Eq. (3)] and the background shading indicates whether instantaneously the gas purifies (green), cools without purifying (blue), or heats (white). In the regime below the dashed line, which corresponds to $k_{B}T=2gn$, the dynamics could deviate from our results due to phononic nature of the thermal excitations.