Sound propagation in elongated superfluid fermionic clouds

We use hydrodynamic equations to study sound propagation in a superfluid Fermi gas at zero temperature inside a strongly elongated cigar-shaped trap, with main attention to the transition from the BCS to the unitary regime. First, we treat the role of the radial density profile in the limit of a cylindrical geometry and then evaluate numerically the effect of the axial confinement in a configuration in which a hole is present in the gas density at the center of the trap. We find that in a strongly elongated trap the speed of sound in both the BCS and the unitary regime differs by a factor $\sqrt{3∕5}$ from that in a homogeneous three-dimensional superfluid. The predictions of the theory could be tested by measurements of sound-wave propagation in a setup such as that exploited by Andrews et al. [Phys. Rev. Lett. 79, 553 (1997)] for an atomic Bose-Einstein condensate.

Figure 1

Speed of sound $u_1$ in a cylindrical Fermi gas (in units of the Fermi velocity $v_{\mathrm{F}}$) as a function of $1∕k_{\mathrm{F}}\mid a\mid$ (on a logarithmic scale). The full and the dashed lines are the results from the equation of state, (7), and from the mean-field (MF) model, respectively. The dot-dashed lines mark the BCS (upper) and the unitary (lower) limits.

Figure 2

Integrated density profile $\tilde{n}$ (in arbitrary units) as a function of axial coordinate $z$ (in units of $a_{\perp }={[\hslash ∕\left(m{\omega }_{\perp }\right)]}^{1∕2}$) at different times starting from the bottom at $t=0$ and increasingly separated by $1.5∕{\omega }_{\perp }$. The left, middle, and right panel correspond to $a∕a_0=-80$, $-{10}^5$, and $-{10}^6$. For the sake of visibility the plotting range is smaller than the full extent of the density profile.

Figure 3

Sound velocity in a superfluid Fermi gas as a function of $1∕k_{\mathrm{F}}\mid a\mid$ (on a logarithmic scale). Circles and squares correspond to the average speed of perturbations propagating in a trap with $\lambda \simeq 0.06$ and $\lambda =0.01$. Empty and full symbols refer to $U_0∕\hslash {\omega }_z=0.8$ and $0.16$, respectively. The thick solid line is from Eq. (5), and the dot-dashed lines mark the BCS (upper) and unitary (lower) limits. The thin solid lines are guides to the eye.

Figure 4

Sound velocity in an elongated trap as a function of $1∕k_{\mathrm{F}}\mid a\mid$ (on a logarithmic scale). Circles and squares are for a superfluid at $T=0$ and for a normal Fermi gas at $T=0.2T_{\mathrm{F}}$, respectively. The error bars for the normal-gas results are from the standard deviation in the velocity distributions. The dot-dashed lines show the zero-sound, the BCS, and the unitary limits, from top to bottom.