Sympathetic cooling of trapped fermions by bosons in the presence of particle losses

We study the sympathetic cooling of a trapped Fermi gas interacting with an ideal Bose gas below the critical temperature of the Bose-Einstein condensation. We derive the quantum master equation, which describes the dynamics of the fermionic component, and postulating the thermal distribution for both gases we calculate analytically the rate at which fermions are cooled by the bosonic atoms. The particle losses constitute an important source of heating of the degenerate Fermi gas. We evaluate the rate of loss-induced heating and derive analytical results for the final temperature of fermions, which is limited in the presence of particle losses.