Superfluid density and condensate fraction in the BCS-BEC crossover regime at finite temperatures

The superfluid density is a fundamental quantity describing the response to a rotation as well as in two-fluid collisional hydrodynamics. We present extensive calculations of the superfluid density ${\rho }_s$ in the BCS-BEC crossover regime of a uniform superfluid Fermi gas at finite temperatures. We include strong-coupling or fluctuation effects on these quantities within a Gaussian approximation. We also incorporate the same fluctuation effects into the BCS single-particle excitations described by the superfluid order parameter $\Delta$ and Fermi chemical potential $\mu$, using the Nozières–Schmitt-Rink approximation. This treatment is shown to be necessary for consistent treatment of ${\rho }_s$ over the entire BCS-BEC crossover. We also calculate the condensate fraction $N_c$ as a function of the temperature, a quantity which is quite different from the superfluid density ${\rho }_s$. We show that the mean-field expression for the condensate fraction $N_c$ is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ${\rho }_s$ and $N_c$ depend on temperature, from the weak-coupling BCS region to the BEC region of tightly bound Cooper pair molecules. In a companion paper [Phys. Rev. A 74, 063626 (2006)], we derive an equivalent expression for ${\rho }_s$ from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.