Limits of sympathetic cooling of fermions by zero temperature bosons due to particle losses

Abstract

It has been suggested by Timmermans [Phys. Rev. Lett. {\bf 87}, 240403 (2001)] that loss of fermions in a degenerate system causes strong heating. We address the fundamental limit imposed by this loss on the temperature that may be obtained by sympathetic cooling of fermions by bosons. Both a quantum Boltzmann equation and a quantum Boltzmann \emph{master} equation are used to study the evolution of the occupation number distribution. It is shown that, in the thermodynamic limit, the Fermi gas cools to a minimal temperature $k_{\rm B}T/μ\propto(γ_{\rm loss}/γ_{\rm coll})^{0.44}$, where $γ_{\rm loss}$ is a constant loss rate, $γ_{\rm coll}$ is the bare fermion--boson collision rate not including the reduction due to Fermi statistics, and $μ\sim k_{\rm B}T_{\rm F}$ is the chemical potential. It is demonstrated that, beyond the thermodynamic limit, the discrete nature of the momentum spectrum of the system can block cooling. The unusual non-thermal nature of the number distribution is illustrated from several points of view: the Fermi surface is distorted, and in the region of zero momentum the number distribution can descend to values significantly less than unity. Our model explicitly depends on a constant evaporation rate, the value of which can strongly affect the minimum temperature.