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 ρₛ 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 Δ and Fermi chemical potential μ, using the Nozi\`eres--Schmitt-Rink approximation. This treatment is shown to be necessary for consistent treatment of ρₛ over the entire BCS-BEC crossover. We also calculate the condensate fraction Nc as a function of the temperature, a quantity which is quite different from the superfluid density ρₛ. We show that the mean-field expression for the condensate fraction Nc is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ρₛ and Nc 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 ρₛ from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.