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arXiv:cond-mat/0509323 · Physical Review A

Sound propagation in elongated superfluid fermionic clouds

P. Capuzzi, Patrizia Vignolo, F. Federici, Monica Tosi

Published Feb 27, 2006• 54 citations
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Abstract

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.

DOI
10.1103/physreva.73.021603

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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 3∕5\sqrt{3∕5}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.
We use hydrodynamic equations to study the formation of Faraday waves in a superfluid Fermi gas at zero temperature confined in a strongly elongated cigar-shaped trap. First, we treat the role of the radial density profile in the limit of an infinite cylindrical geometry and analytically evaluate the wavelength of the Faraday pattern. The effect of the axial confinement is fully taken into account in the numerical solution of hydrodynamic equations, and shows that the infinite cylinder geometry provides a very good description of the phenomena.
This thesis is devoted to the study of the hydrodynamic behavior of the unitary Fermi gas trapped by a highly elongated harmonic potential. Propagation of sound is one of the most exciting features exhibited by interacting many-body systems. It provides crucial information on the dynamic behavior of the system as well as on key thermodynamic quantities. The propagation of sound is particularly interesting in superfluids where two-fluid hydrodynamic theory predicts the occurrence of two different sounds: first sound, where the normal and superfluid component oscillate in phase, and second sound, where the two components oscillate with opposite phase. In the thesis, we investigate the propagation of sound waves of the unitary Fermi gas in a cylindrical geometry by solving the equations of two-fluid hydrodynamics in the `1D' scenario at finite temperature. The relevant thermodynamic functions entering the hydrodynamic equations are discussed in the superfluid and normal regimes in terms of universal scaling functions. Both the 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, which is the strategy used in a recent experiment carried out at Innsbruck where second sound was detected in the unitary Fermi gas. We also provide an 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.
We evaluate the transition from zero-sound to first-sound behaviour with increasing collisionality in the propagation of density waves through an ultracold gaseous mixture of fermionic atoms in the normal state confined inside a cigar-shaped harmonic trap. We study for this purpose the evolution of the one-body distribution functions associated with a density perturbation generated in the central region of the cloud, as obtained by numerical solution of the Vlasov–Landau equations. We examine a variety of trap anisotropies and of repulsive or attractive interaction strengths between the components of the mixture. The speed of propagation of the density disturbance is found to decrease in both cases as the magnitude of the coupling strength is increased. The results are compared with the values of the speed of zero sound and of first sound, as obtained analytically from the limit of vanishing collisionality and from linearized hydrodynamics. The main effects of the quasi-one-dimensional confinement are the stabilization of zero-sound excitations in the attractive regime before collapse and the lowering of the hydrodynamic sound velocity by a factor relative to three-dimensional behaviour.
Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which was recently derived by G. Bertaina and coworkers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of the temperature and the form of superfluid density. We show that the density fluctuations in second-sound modes, due to their coupling to first-sound modes, are large enough to be measured in current experimental setups such as that exploited by M. K. Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second-sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far has remained elusive.
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.
We provide an overview of the recent theoretical and experimental advances in the study of second sound in ultracold atomic gases. Starting from the Landau two fluid hydrodynamic equations we develop the theory of first and second sound in various configurations characterized by different geometries and quantum statistics. These include the weakly interacting 3D Bose gas, the strongly interacting Fermi gas at unitarity in the presence of highly elongated traps and the dilute 2D Bose gas, characterized by the Berezinskii-Kosterlitz-Thouless transition. An explicit comparison with the propagation of second sound in liquid Helium is carried out to elucidate the main analogies and differences. We also make an explicit comparison with the available experimental data and point out the crucial role played by the superfluid density in determining the temperature dependence of the second sound speed.