First and second sound in a highly elongated Fermi gas at unitarity

We consider a Fermi gas at unitarity trapped by a highly elongated harmonic potential and solve the equations of two fluid hydrodynamics at finite temperature. The propagation of sound waves as well as the discretized solutions in the presence of weak axial trapping are considered. The relevant thermodynamic functions entering the hydrodynamic equations are discussed in the superfluid and normal regimes in terms of universal scaling functions. Both first sound and second sound solutions are calculated as a function of temperature and the role of the superfluid density is explicitly pointed out. The density fluctuations in the second sound wave are found to be large enough to be measured as a consequence of the finite thermal expansion coefficient of the gas. Emphasis is given to the comparison with recent experimental data.

Figure 1

Equation of state $\mu /k_{B}T$ vs $T/T_F$. The blue dash-dotted line corresponds to the phonon contribution to thermodynamics (Sec. 3c). The red solid circles correspond to the experiment data in higher $T$ regime, while the black solid line to the virial expansion in classical limit (Sec. 3d). The green arrow indicates the critical point $T_c/T_F=0.167\left(13\right)$. The inset on the upper right corner is an amplification in the lower $T$ regime.

Figure 2

Universal scaling functions $f_n$ and $f_p$ as a function of the dimensionless variable $\mu /k_{B}T$. See Fig. 1 for the notation.

Figure 3

Entropy and specific heats per particle in uniform matter, evaluated using Eqs. (14, 15, 16). In the lower panel, the red square-guided line corresponds to $C_p/N{k}_B$; the black full-circle-guided line to $C_v/N{k}_B$; the inset is for the specific heats in a large temperature interval. The vertical green line indicates the critical temperature.

Figure 4

1D entropy ${\overline{s}}_1/k_B$ and specific heats ${\overline{c}}_1/k_B$ evaluated using Eqs. (22) and (24, 25, 26). In the lower panel, the red dashed line corresponds to ${\overline{c}}_{p1}/k_B$; the black dash-dotted line to ${\overline{c}}_{v1}/k_B$. The vertical green line indicates the critical temperature.

Figure 5

1D first and second sound velocity in units of $v_F^{1\mathrm{D}}$. Upper branch (first sound) the blue dashed line corresponds to Eq. (49); the experiment data (red dots) in the pink shaded region are taken from [26]. Lower branch (second sound): the theoretical lines calculated using Eq. (60) with two different models for the superfluid density. The red dashed line corresponds to the phenomenological ansatz: $n_s/n=1-{(T/T_c)}^4$ while the black dash-dotted line to the choice: $n_s/n={(1-T/T_c)}^{2/3}$. The blue dots are the experimental data from [26]. The shaded area indicates the uncertainty range of experimental data. The vertical green line indicates the critical temperature. At low temperature the 1D second sound velocity is expected to vanish like $\sqrt{T}$.

Figure 6

Frequency for the $k=2$ (upper panel) and $k=3$ (lower panel) first sound collective frequencies. Experiment data are from [24, 25]. The green lines are the theoretical predictions based on Eqs. (52) and (53) using the equation of state of the unitary (solid) and ideal (dashed) Fermi gas. The thin horizontal dashed lines mark the zero-$T$ superfluid limit (50) and the classical hydrodynamic limit (51), respectively. The red dash-dot vertical lines in (a) and (b) indicate the critical temperature. In this figure and Fig. 7 the Fermi temperature corresponds to the definition $T_F^{\mathrm{trap}}={\left(3N\right)}^{1/3}\hslash {\overline{\omega }}_{ho}/k_B$ introduced in the text.

Figure 7

Equilibrium profiles (upper figure) and density oscillations for the $k=2$ (middle figure) and the $k=3$ (lower figure) first sound collective modes at different temperatures from [25]. See Fig. 6 for the definition of the Fermi temperature.

Figure 8

Ratio (61) between the relative density and temperature fluctuations calculated for 1D second sound. The vertical green line indicates the critical temperature.

Figure 9

Uniform superfluid density; the red dashed line corresponds to the phenomenological ansatz, $n_s/n=1-{(T/T_c)}^4$ while the black dash-dotted line to the choice, $n_s/n={(1-T/T_c)}^{2/3}$. The blue dots are the experimental data [26]. The shaded area indicates the uncertainty range of experimental data.

Figure 10

Frequency for the lowest discretized second sound mode in an axially trapped configuration with ${\omega }_z\ll {\omega }_{\perp }$. See Fig. 5 for the notations. The vertical green line indicates the critical temperature.