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 ρ 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 ∆ and Fermi chemical potential µ, using the Nozières and Schmitt-Rink (NSR) approximation. This treatment is shown to be necessary for consistent treatment of ρ 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 ρ 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 ρ 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 by the authors (cond-mat/0609187), we derive an equivalent expression for ρ s from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.

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We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozières and Schmitt-Rink, we use this definition of s to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap ⌬, the chemical potential , and s all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density n ϵ n − s naturally separates into a sum of contributions from Fermi BCS quasiparticles ͑ n F ͒ and Bose collective modes ͑ n B ͒. The expression for n F is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution n B is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of s using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.

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