Recently there has been renewed interest in second sound in superfluid Bose and Fermi gases. By using two-fluid hydrodynamic theory, we review the density response $chi_{nn}(bq,omega)$ of these systems as a tool to identify second sound in experiment
s based on density probes. Our work generalizes the well-known studies of the dynamic structure factor $S(bq,omega)$ in superfluid $^4$He in the critical region. We show that, in the unitary limit of uniform superfluid Fermi gases, the relative weight of second vs. first sound in the compressibility sum rule is given by the Landau--Placzek ratio $lpequiv (bar{c}_p-bar{c}_v)/bar{c}_v$ for all temperatures below $T_c$. In contrast to superfluid $^4$He, $lp$ is much larger in strongly interacting Fermi gases, being already of order unity for $T sim 0.8 T_c$, thereby providing promising opportunities to excite second sound with density probes. The relative weights of first and second sound are quite different in $S(bq,omega)$ (measured in pulse propagation studies) as compared to $mathrm{Im}chi_{nn}(bq,omega)$ (measured in two-photon Bragg scattering). We show that first and second sound in $S(bq,omega)$ in a strongly interacting Bose-condensed gas are similar to those in a Fermi gas at unitarity. However, in a weakly interacting Bose gas, first and second sound are mainly uncoupled oscillations of the thermal cloud and condensate, respectively, and second sound has most of the spectral weight in $S(bq,omega)$. We also discuss the behaviour of the superfluid and normal fluid velocity fields involved in first and second sound.
We examine the superfluid and collapse instabilities of a quasi two-dimensional gas of dipolar fermions aligned by an orientable external field. It is shown that the interplay between the anisotropy of the dipolar interaction, the geometry of the sys
tem, and the p-wave symmetry of the superfluid order parameter means that the effective interaction for pairing can be made very large without the system collapsing. This leads to a broad region in the phase diagram where the system forms a stable superfluid. Analyzing the superfluid transition at finite temperatures, we calculate the Berezinskii--Kosterlitz--Thouless temperature as a function of the dipole angle.
We develop a variational approach to calculate the density response function at finite temperatures of the lowest-lying two-fluid modes in a trapped two-component Fermi superfluid close to a Feshbach resonance. The out-of-phase oscillations, which ar
e the analogue in trapped gases of second sound in uniform superfluids, have so far not been observed in cold-atom experiments. At unitarity, we show that these modes are observable at finite temperatures via two-photon Bragg scattering, whose spectrum is related to the imaginary part of density response function. This provides direct evidence for superfluidity and a promising way to test microscopic results for thermodynamics at unitarity.
We present the results of a variational calculation of the frequencies of the low-lying Landau two-fluid hydrodynamic modes in a trapped Fermi superfluid gas at unitarity. Landaus two-fluid hydrodynamics is expected to be the correct theory of Fermi
superfluids at finite temperatures close to unitarity, where strong interactions give rise to collisional hydrodynamics. Two-fluid hydrodynamics predicts the existence of in-phase modes in which the superfluid and normal fluid components oscillate together, as well as out-of-phase modes where the two components move against each other. We prove that at unitarity, the dipole and breathing in-phase modes are locally isentropic. Their frequencies are independent of temperature and are the same above and below the superfluid transition. The out-of-phase modes, in contrast, are strongly dependent on temperature and hence, can be used to test the thermodynamic properties and superfluid density of a Fermi gas at unitarity. We give numerical results for the frequencies of these new modes as function of temperature in an isotropic trap at unitarity.
