Short-range correlations and entropy in ultracold-atom Fermi gases

We relate short-range correlations in ultracold-atom Fermi gases to the entropy of the system over the entire temperature, T, vs coupling strength, -1/k(F)a, plane. In the low-temperature limit the entropy is dominated by phonon excitations and the correlations increase as T-4. In the Bose-Einstein condensate (BEC) limit, we calculate a boson model within the Bogoliubov approximation to show explicitly how phonons enhance the fermion correlations. In the high-temperature limit, we show from the virial expansion that the correlations decrease as 1/T. The correlations therefore reach a maximum at a finite-temperature. We infer the general structure of the isentropes of the Fermi gas in the (T,-1/k(F)a) plane, and the temperature dependence of the correlations in the unitary, BEC, and BCS limits. Our results compare well with measurements of the correlations via photoassociation experiments at higher temperatures.