First and second sound of a unitary Fermi gas in highly elongated harmonic traps

Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which was recently derived by G. Bertaina and coworkers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of the temperature and the form of superfluid density. We show that the density fluctuations in second-sound modes, due to their coupling to first-sound modes, are large enough to be measured in current experimental setups such as that exploited by M. K. Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second-sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far has remained elusive.

Figure 1

(a) One-dimensional universal scaling functions $f_p\left(x\right),f_n\left(x\right)$, and $d{f}_n\left(x\right)/dx$ as a function of the dimensionless variable $x=\mu /\left(k_{B}T\right)$. (b) One-dimensional entropy $\overline{s}=s/\left(n{k}_B\right)$ and specific heat per particle ${\overline{c}}_v=c_v/\left(n{k}_B\right)$ as a function of $x=\mu /\left(k_{B}T\right)$. Vertical gray lines indicate the critical threshold for superfluidity, $x_c\simeq 2.49$ [28].

Figure 2

Landau-Placzek parameter ${\epsilon }_{\mathrm{LP}}$ as a function of $x=\mu /\left(k_{B}T\right)$ in a highly elongated unitary Fermi gas.

Figure 3

(a) Temperature dependence of the first-sound mode frequencies. (b) Enlarged view for the fourth first-sound mode frequency. The dashed (red) line shows the result with the ansatz $u_a\left(z\right)=A_3{z}^3+A_{1}z$, obtained earlier by Hou, Pitaevskii, and Stringari [12]. Vertical gray lines show the critical temperature, $T_c\simeq 0.223T_F$.

Figure 4

(a) Temperature dependence of the second-sound mode frequencies with $N_p=24$. Mode frequencies vanish right at the superfluid phase transition. However, our numerical calculations become less accurate slightly below the transition and cannot produce correctly the zero frequency exactly at $T_c$. (b) Enlarged view for the lowest second-sound mode frequency. The mode frequency converges with an increasing number of variational ansätze $N_p$. The result with $N_p=1$ [dot-dashed (green) line] corresponds to a constant displacement field $u_e$ [12]. The vertical gray lines show the critical temperature.

Figure 5

Temperature dependence of the full two-fluid hydrodynamic mode frequencies [(blue) circles]. For comparison, we show also the mode frequencies of the decoupled first and second sounds, by solid black and dashed (red) lines, respectively. The vertical gray line indicates the critical temperature.

Figure 6

Blowup of the full two-fluid hydrodynamic mode frequencies [(blue) circles], near (a) the $k=3$ first-sound mode and (b) the lowest second-sound modes. In (a), experimental data from the Innsbruck experiment [9] are shown by filled (brown) squares with error bars.

Figure 7

(a) Density distribution (solid line) and superfluid density distribution (dashed line) at $T=0.18T_F$, in units of the peak linear density of an ideal Fermi gas at the trap center $\left(n_F\right)$. (b) Density fluctuations of the $k=2$ and $k=3$ first-sound modes (thin lines) and of the three lowest second-sound modes (thick lines), at the frequencies indicated in Fig. 6 by A, B, and C. The amplitude of the second-sound density fluctuations is about one-third that of the first-sound density fluctuations.

Figure 8

Sensitivity of the lowest second-sound mode on the superfluid density. The critical temperature $T_c\simeq 0.223T_F$.