High-Frequency Sound in a Unitary Fermi Gas

We present an experimental and theoretical study of the phonon mode in a unitary Fermi gas. Using two-photon Bragg spectroscopy, we measure excitation spectra at a momentum of approximately half the Fermi momentum, both above and below the superfluid critical temperature $T_{\mathrm{c}}$. Below $T_{\mathrm{c}}$, the dominant excitation is the Bogoliubov-Anderson (BA) phonon mode, driven by gradients in the phase of the superfluid order parameter. The temperature dependence of the BA phonon is consistent with a theoretical model based on the quasiparticle random phase approximation in which the dominant damping mechanism is via collisions with thermally excited quasiparticles. As the temperature is increased above $T_{\mathrm{c}}$, the phonon evolves into a strongly damped collisional mode, accompanied by an abrupt increase in spectral width. Our study reveals strong similarities between sound propagation in the unitary Fermi gas and bosonic liquid helium.

Figure 1

Excitation spectra for a unitary Fermi gas for a range of temperatures $T/T_{\mathrm{F}}$ at $k/k_{\mathrm{F}}=0.55\pm 0.03$. (a) Filled circles show the experimental data; solid curves and shaded areas are fits to Eq. (2), convolved with a Gaussian instrumental broadening function as described in the text. (b) Calculated excitation spectra using the QRPA theory, also convolved with the instrumental broadening function. Red solid lines indicate the superfluid transition temperature $T_c$. All spectra are presented in units of $(4\pi {\epsilon }_{\mathrm{r}}{E}_{\mathrm{F}}{)}^{-1}$.

Figure 2

(a) Mode frequency as a function of $T/T_{\mathrm{c}}$. Blue circles are experimental data, the red dash-dotted line is the hydrodynamic sound frequency [43], and the grey solid line is the QRPA prediction [34]. Note that $T_{\mathrm{c}}$ is indicated by the green dashed line. (b) Mode frequency as a function of momentum at two temperatures, below and above $T_c$, respectively. Straight lines are linear fits that cross the origin, confirming the dispersion is linear within experimental resolution. (c) Zoomed-in plots of two experimental spectra (points and solid lines) in the normal phase at two different temperatures along with the corresponding QRPA spectra (dashed and dash-dotted shaded curves). Error bars include uncertainties in the density measurement used to determine $E_{\mathrm{F}}$ and the standard error in the mean of the measured data points.

Figure 3

(a) Normalized peak amplitude of the phonon mode as a function of temperature. Blue circles are experimental data, and the gray solid line is the QRPA calculation. (b) Blue points show the FWHM of the experimental spectra determined by fitting Eq. (2). The grey solid line is the damping predicted by the QRPA theory; blue and red dash-dotted lines are hydrodynamic predictions from Eq. (3) using the classical result for the shear viscosity [46] and for $\eta$ obtained from a $\mathcal{T}$-matrix calculation [47]. (c) Quality factor ${\omega }_0/\mathrm{\Gamma }$ of the sound mode. Blue circles are the experimental data; the grey solid line is the QRPA theory. Brown stars and dashed lines in panels (b) and (c) show scaled results for liquid helium measured at a wave vector of $0.4{\AA }^{-1}$ [36].