Variational theory of two-fluid hydrodynamic modes 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. Landau’s 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, a feature due as much to the harmonic trapping potential as to the thermodynamic properties at unitarity. 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 modes as function of temperature in an isotropic trap at unitarity.

Figure 1

(Color online) Superfluid density fraction in a uniform Fermi gas at unitarity as a function of temperature. The different theoretical predictions are discussed in the text.

Figure 2

(Color online) Profiles of several thermodynamic functions at temperature $T=0.20T_F$, where $T_F$ is the Fermi temperature of a trapped ideal Fermi gas. Here we have used the results given by the NSR-type Gaussian fluctuation theory [12]. The superfluid transition temperature is $T_c\simeq 0.27T_F$. The adiabatic compressibility is only needed in the superfluid region of the trap.

Figure 3

(Color online) Out-of-phase breathing mode at unitarity as a function of temperature for an isotropic trap using an improved NSR calculation for the equation of state [12]. The frequencies obtained using the fitted NSR ${\rho }_s$ data and the scaled BCS mean-field data are given by the solid and dashed lines, respectively (see Fig. 1). The arrow indicates the superfluid transition temperature, $T_c\simeq 0.27T_F$.

Figure 4

(Color online) Frequency of the out-of-phase dipole mode at unitarity in an isotropic trap as a function of temperature. See caption of Fig. 3.

Figure 5

(Color online) Comparison of an LDA calculation of the frequency of the out-of-phase breathing mode with a calculation using the Bogoliubov–de Gennes equations. Here the LDA result (solid line) is obtained by using the BCS mean-field theory for a Fermi gas at unitarity. The solid squares are calculated using the self-consistent Bogoliubov–de Gennes mean-field equations.