Universal thermodynamics of a strongly interacting Fermi gas: theory versus experiment

Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal thermodynamic behaviour. This is notoriously difficult to understand theoretically because of the absence of a small interaction parameter. Here we present a systematic comparison of theoretical predictions from different quantum many-body theories with recent experimental data of Nascimbene et. al. (arXiv:0911.0747v1). Our comparisons have no adjustable parameters, either theoretically or experimentally. A simple Gaussian pair fluctuation theory is shown to give the best quantitative agreement in the superfluid state below threshold. In the normal state, we also calculate the equation of state by using a quantum cluster expansion theory and explore in detail its applicability to low temperatures. Using the accurate experimental result for the thermodynamic function $S(T)$, we determine the temperature $T$ of a trapped Fermi gas at unitarity as a function of a non-interacting temperature $T_{i}$ which can be obtained by an adiabatic sweep to the free gas limit. By analyzing the recent experimental data, we find a normal-superfluid transition temperature $(T/T_{F})_{c}=0.19\pm0.02$ or $(T_{i}/T_{F})_{c}=0.16\pm0.02$ in a harmonic trap, where $T_{F}$ is the Fermi temperature for a trapped ideal, non-interacting Fermi gas.