Sound propagation in elongated superfluid fermionic clouds

Abstract

We use hydrodynamic equations to study sound propagation in a superfluid Fermi gas inside a strongly elongated cigar-shaped trap, with main attention to the transition from the BCS to the unitary regime. We treat first the role of the radial density profile in the quasi-onedimensional limit 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 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 set-up such as that exploited by M.R. Andrews et al. [Phys. Rev. Lett. 79, 553 (1997)] for an atomic Bose-Einstein condensate.Strong evidence for a superfluid state in ultracold Fermi gases near a Feshbach resonance has come from experimental studies using as probes ballistic expansion [1,2,3,4,5], collective modes [6,7], RF spectroscopy [8], and the generation of quantized vortices [9]. Since the BCS weak-coupling limit and the unitary strongcoupling limit are characterized by the same collectivemode frequencies and expansion dynamics [10], attention has been drawn to the study of sound-wave propagation as allowing a clear identification of these two regimes. In particular Ho [11] and Heiselberg [12] have evaluated the speed of first and second sound in a homogeneous Fermi superfluid as functions of the coupling regime. At low temperature the first sound velocity is given by u 1 = v F (1 + (2/π)hk F a)/3 ≃ v F / √ 3 in the dilute BCS limit and by u 1 ≃ 0.37v F in the unitary limit.Here v F and k F are the Fermi velocity and wavenumber, while a is the s-wave scattering length.The main purpose of this Letter is to evaluate the propagation of density perturbations in a superfluid Fermi gas as a function of coupling strength in an experimentally realizable set-up. This simulates the set-up used in the experiments of Andrews et al.[13] on the propagation of sound pulses along the axis of an elongated cloud of Bose-Einstein condensed 23 Na atoms. In their experiments a density perturbation is generated by turning on a laser at the center of the trap and two pulses propagate in opposite directions along the trap axis. We describe the dynamics of density fluctuations in the superfluid by hydrodynamic equations, namely the continuity equation ∂ t n + ∇ · (nv) = 0(1) and the Euler equationfor the time-dependent density profile n(r, t) and velocity field v(r, t) of the trapped gas. The equation of state enters through the density-dependent chemical potential µ(n), and V (r) describes the external potentials.The laser beam is simulated in our study by a timeindependent effective potential U (z) = U 0 e −z 2 /w 2 having amplitude U 0 and width w, which is turned on at the center of the trap. The Q1D configuration. We discuss first the quasionedimensional (Q1D) case corresponding to V (r) = mω 2 ⊥ r 2 ⊥ /2, where ω...

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