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arXiv:1310.3559 · Physical Review A

Probing the critical exponent of the superfluid fraction in a strongly interacting Fermi gas

Hui Hu, Xia-Ji Liu

Published Nov 26, 2013• 1 citation
arXivHTMLPDF

Abstract

We theoretically investigate the critical behavior of a second-sound mode in a harmonically trapped ultracold atomic Fermi gas with resonant interactions. Near the superfluid phase transition with critical temperature ${T}_{c}$, the frequency or the sound velocity of the second-sound mode crucially depends on the critical exponent $\ensuremath{\beta}$ of the superfluid fraction. In an isotropic harmonic trap, we predict that the mode frequency diverges like ${(1\ensuremath{-}T/{T}_{c})}^{\ensuremath{\beta}\ensuremath{-}1/2}$ when $\ensuremath{\beta}<1/2$. In a highly elongated trap, the speed of the second sound reduces by a factor of $1/\sqrt{2\ensuremath{\beta}+1}$ from that in a homogeneous three-dimensional superfluid. Our prediction could readily be tested by measurements of second-sound wave propagation in a setup, such as that exploited by Sidorenkov et al. [Nature (London) 498, 78 (2013)] for resonantly interacting lithium-6 atoms, once the experimental precision is improved.

DOI
10.1103/physreva.88.053635

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Superfluidity is a macroscopic quantum phenomenon, which shows up below a critical temperature and leads to a peculiar behavior of matter, with frictionless flow, the formation of quantized vortices, and the quenching of the moment of inertia being intriguing examples. A remarkable explanation for many phenomena exhibited by a superfluid at finite temperature can be given in terms of a two-fluid mixture comprised of a normal component that behaves like a usual fluid and a superfluid component with zero viscosity and zero entropy. Important examples of superfluid systems are liquid helium and neutron stars. More recently, ultracold atomic gases have emerged as new superfluid systems with unprecedented possibilities to control interactions and external confinement. Here we report the first observation of `second sound' in an ultracold Fermi gas with resonant interactions. Second sound is a striking manifestation of the two-component nature of a superfluid and corresponds to an entropy wave, where the superfluid and the non-superfluid components oscillate in opposite phase, different from ordinary sound (`first sound'), where they oscillate in phase. The speed of second sound depends explicitly on the value of the superfluid fraction, a quantity sensitive to the spectrum of elementary excitations. Our measurements allow us to extract the temperature dependence of the superfluid fraction, which in strongly interacting quantum gases has been an inaccessible quantity so far.
Abstract We briefly review the research on second sound in ultracold atomic physics, with emphasis on strongly interacting unitary Fermi gases with infinitely large s-wave scattering length. Second sound is a smoking-gun feature of superfluidity in any quantum superfluids. The observation and characterization of second sound in ultracold quantum gases have been a long-standing challenge, and in recent years, there are rapid developments due to the experimental realization of a uniform box-trap potential. The purpose of this review is to present a brief historical account of the key research activities on second sound over the past two decades. We summarize the initial theoretical works that reveal the characteristics of second sound in a unitary Fermi gas, and introduce its first observation in a highly elongated harmonic trap. We then discuss the most recent measurement on second sound attenuation in a uniform setup, which may open a new era to understand quantum transport near quantum criticality in the strongly interacting regime. The observation of second sound in homogeneous weakly interacting Bose condensates in both two and three dimensions are also briefly introduced.
The two-fluid model is fundamental for the description of superfluidity. In the nearly incompressible liquid regime, it successfully describes first and second sound, corresponding, respectively, to density and entropy waves, in both liquid helium and unitary Fermi gases. Here, we study the two sounds in the opposite regime of a highly compressible fluid, using an ultracold ^{39}K Bose gas in a three-dimensional box trap. We excite the longest-wavelength mode of our homogeneous gas, and observe two distinct resonant oscillations below the critical temperature, of which only one persists above it. In a microscopic mode-structure analysis, we find agreement with the hydrodynamic theory, where first and second sound involve density oscillations dominated by, respectively, thermal and condensed atoms. Varying the interaction strength, we explore the crossover from hydrodynamic to collisionless behavior in a normal gas.
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 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.
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.