Collective Modes in a Unitary Fermi Gas across the Superfluid Phase Transition

We provide a joint theoretical and experimental investigation of the temperature dependence of the collective oscillations of first sound nature exhibited by a highly elongated harmonically trapped Fermi gas at unitarity, including the region below the critical temperature for superfluidity. Differently from the lowest axial breathing mode, the hydrodynamic frequencies of the higher-nodal excitations show a temperature dependence, which is calculated starting from Landau two-fluid theory and using the available experimental knowledge of the equation of state. The experimental results agree with high accuracy with the predictions of theory and provide the first evidence for the temperature dependence of the collective frequencies near the superfluid phase transition.

Figure 1

Probing a higher-order first sound longitudinal mode at the example of $k=2$. In (a), we illustrate the basic geometry of exciting the optically trapped cloud with a weak, power-modulated repulsive laser beam, which perpendicularly intersects the trapping beam. In (b), the experimental mode profiles (data points) are compared to theoretical curves based on the experimental EOS from Ref. 8 (solid lines) for two different temperatures. The corresponding cloud profiles in (c) are analyzed to extract the temperatures (see text). The solid lines show the density profiles obtained from the EOS [8] with $T/T_F=0.10$ and 0.45. The parameter $Z_{\mathrm{TF}}={\xi }^{1/4}{\omega }_z^{-1}\sqrt{2k_B{T}_F/m}$ represents the Thomas-Fermi radius of the zero-$T$ interacting gas.

Figure 2

Comparison between experimental and theoretical first sound frequencies of the $k=2$ mode. In (a) the experimental data are plotted versus the energy parameter $E/E_0$, while in (b) we use a temperature scale $T/T_F$. The theoretical curves (solid lines) are based on the EOS of Ref. 8. This EOS is also used to extract $T/T_F$ from the measured profile in two different ways: The filled symbols in (b) result from a direct conversion of $E/E_0$ to $T/T_F$, while the open symbols result from fitting the cloud profiles (see text). For comparison, we also show the mode frequencies (dashed curves) that would result from the EOS of the ideal Fermi gas. The thin horizontal dashed lines mark the zero-$T$ superfluid limit ($\omega /{\omega }_z=\sqrt{21/5}$) and the classical hydrodynamic limit ($\omega /{\omega }_z=\sqrt{19/5}$) according to Eqs. (2, 3), respectively. In (a) the dashed vertical line indicates the $T=0$ ground state with $E/E_0=\sqrt{\xi }=0.613(3)$, while the dash-dotted vertical lines in (a) and (b) indicate the critical energy $E_c/E_0=0.934(39)$ and temperature $T_c/T_F=0.223(15)$.