First and Second Sound in a Compressible 3D Bose Fluid

Timon A. Hilker, Lena H. Dogra, Christoph Eigen, Jake A. P. Glidden, Robert P. Smith, and Zoran Hadzibabic

Phys. Rev. Lett. 128, 223601 – Published 2 June 2022

Abstract

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.

Received 31 December 2021
Accepted 12 April 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.223601

Physics Subject Headings (PhySH)

Superfluidity

Bose gases Bose-Einstein condensates Quantum fluids & solids Superfluids

Feshbach resonance

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsFluid Dynamics

Authors & Affiliations

Timon A. Hilker ^1,*,†, Lena H. Dogra ¹, Christoph Eigen ¹, Jake A. P. Glidden ¹, Robert P. Smith ², and Zoran Hadzibabic ¹

¹Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
²Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

^*Present address: Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany.
^†timon.hilker@mpq.mpg.de

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Vol. 128, Iss. 22 — 3 June 2022

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Figure 1
First and second sound in dilute Bose gases. (a) Mode structure and speeds of sound in a weakly interacting Bose gas based on the two-fluid model. Both modes involve motion of both fluids ( $v_{s}$ , $v_{n}$ ), but for $k_{B} T ≫ g n_{s}$ , where $g$ is the interaction strength and the first (second) sound is mainly an oscillation of the normal (superfluid) component. (b) In a box-trapped gas of ${}^{39}K$ atoms, we excite the $k_{L} = π / L$ mode of both sounds by sinusoidal forcing using a magnetic gradient $\nabla B$ . To attain hydrodynamic conditions, we reduce the mean free path $ℓ_{mfp}$ below the box length $L$ by tuning $g$ .
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Figure 2
Observation of first and second sound. (a) Center-of-mass velocity $v$ versus shaking time $t$ at several frequencies $ω / (2 π)$ reaching quasisteady state after 200 ms. (b) $| σ (ω) |^{2}$ , where $| σ (ω) |$ is proportional to the amplitude of the total current $N v (t)$ . The two peaks correspond to the two sound modes. We show spectra for different $N$ at approximately constant $T = 97 (3) nK$ (measured at $t = 200 ms$ ). The solid lines are fits using the sum of two resonances [20]. The diamonds correspond to the data shown in (a). (c) First (red) and second (blue) sound speeds, normalized by the Bogoliubov speed $c_{0} (N)$ , versus $T / T_{c}$ . The diamonds correspond to data shown in (b). The solid lines are the fit-free predictions of the two-fluid model [Eq. (2)] at fixed $T = 97 nK$ and varying $N$ , with $n_{s}$ calculated using Popov mean-field theory. Near $T_{c}$ mean-field theory is not valid, which we represent by fading of the lines. The dotted lines show $c_{I}$ of a noninteracting gas and $c_{II}$ calculated by equating $n_{s}$ with condensate density $n_{BEC}$ , and calculating $n_{BEC}$ in the ideal-gas approximation.
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Figure 3
Microscopic structure of the two sound modes. Here $T / T_{c} = 0.77 (3)$ , corresponding to a superfluid fraction of 43(3)%. (a) Evolution of $n_{k}$ on the second-sound resonance at 14 Hz [see Fig. 2]. The motion of the superfluid at low $k$ is visually clear, and from the wings of the distribution one can also extract the normal-gas velocity $v_{n}$ . (b) $k$ -resolved phase $ϕ$ (with respect to the phase of the drive) of the $n_{k} (t)$ oscillations. Here the low- and high- $k$ oscillations are almost completely out of phase, as predicted for second sound. (c) Extracted $v_{n}$ (orange) and $v_{s}$ (green); $v_{s}$ is larger and in phase with the drive (see phasor in the inset) as expected for a compressible superfluid. (d)–(f) Analogous analysis to (a)–(c), now for the first-sound resonance at 40 Hz; here $ϕ$ is close to 0 for all $k$ and $v_{n}$ is larger than $v_{s}$ .
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Figure 4
From collisionless dynamics to hydrodynamic sound in a normal gas. (a) For scattering lengths from $a = 10$ $a_{0}$ to 1000 $a_{0}$ , we measure the COM velocity $v$ for a free oscillation after shaking the cloud for 3 (gray circles) and 3.25 (brown circles) periods at 55 Hz. (b) Measured speed of sound $c_{I}$ normalized to the speed $c_{I}^{(HF)}$ predicted for dissipationless hydrodynamics. (c) Damping per period. In both (b) and (c), the data for $L = 70 μ m$ (purple squares) coalesce with the data taken using an additional box geometry ( $L = 50 μ m$ , red circles), when plotted against the inverse Knudsen number $L / ℓ_{mfp} \sim n a^{2} / k_{L}$ . The dashed lines show theoretical predictions to linear order in $ℓ_{mfp} / L$ , while the solid lines show the results of the full hydrodynamic calculation (see text and the Supplemental Material [20]). The latter captures the observed drop of the normalized sound speed below unity. The relative damping also agrees with this fit-free theory at $L / ℓ_{mfp} > 3$ , while for $L / ℓ_{mfp} \to 0$ it decreases but remains nonzero (see enlargement in the inset).
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