Broad vs. narrow Fano-Feshbach resonances in the BCS-BEC crossover with trapped Fermi atoms

With reference to the broad and narrow Fano-Feshbach resonances of Li-6 at about 822G and 543 G, we show that for the broad resonance a molecular coupled-channel calculation can be mapped with excellent accuracy onto an effective single-channel problem with a contact interaction. This occurs for a wide enough range of the magnetic field that the full BCS-BEC crossover can be realized with a typical trap. For the narrow resonance, the mapping onto a single-channel problem and the realization of the BCS-BEC crossover are restricted to too narrow a range of the magnetic field to obtain them in practice. A general criterion is also formulated for deciding whether the BCS-BEC crossover can be exhausted within the single-channel model for any specific Fano-Feshbach resonance. In this way, the BCS-BEC crossover for Fermi atoms with the broad resonance is placed on the same footing as the corresponding crossover for different physical systems.