We propose a pairing-based method for cooling an atomic Fermi gas. A three component (labels 1, 2, 3) mixture of Fermions is considered where the components 1 and 2 interact and, for instance, form pairs whereas the component 3 is in the normal state. For cooling, the components 2 and 3 are coupled by an electromagnetic field. Since the quasiparticle distributions in the paired and in the normal states are different, the coupling leads to cooling of the normal state even when initially $T_{paired}geq T_{normal}$ (notation $T_Sgeq T_N$). The cooling efficiency is given by the pairing energy and by the linewidth of the coupling field. No superfluidity is required: any type of pairing, or other phenomenon that produces a suitable spectral density, is sufficient. In principle, the paired state could be cooled as well but this requires $T_N<T_S$. The method has a conceptual analogy to cooling based on superconductor -- normal metal (SN) tunneling junctions. Main differences arise from the exact momentum conservation in the case of the field-matter coupling vs. non-conservation of momentum in the solid state tunneling process. Moreover, the role of processes that relax the energy conservation requirement in the tunneling, e.g. thermal fluctuations of an external reservoir, is now played by the linewidth of the field. The proposed method should be experimentally feasible due to its close connection to RF-spectroscopy of ultracold gases which is already in use.
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