Second sound in the crossover from the Bose-Einstein condensate to the Bardeen-Cooper-Schrieffer superfluid

Abstract

Second sound is an entropy wave which propagates in the superfluid component of a quantum liquid. Because it is an entropy wave, it probes the thermodynamic properties of the quantum liquid. Here, we study second sound propagation for a large range of interaction strengths within the crossover between a Bose-Einstein condensate (BEC) and the Bardeen-Cooper-Schrieffer (BCS) superfluid, extending previous work at unitarity. In particular, we investigate the strongly-interacting regime where currently theoretical predictions only exist in terms of an interpolation in the crossover. Working with a quantum gas of ultracold fermionic ⁶Li atoms with tunable interactions, we show that the second sound speed varies only slightly in the crossover regime. By varying the excitation procedure, we gain deeper insight on sound propagation. We compare our measurement results with classical-field simulations, which help with the interpretation of our experiments.

Fig. 1. Sound excitation in a trapped superfluid Fermi gas in the vicinity of the BEC-BCS crossover.

a Set-up: A focussed, intensity-modulated, blue-detuned laser beam excites sound waves in the cigar-shaped atom cloud. b Two different modulation sequences of the laser intensity. Purple dashed line: step excitation. Green solid line: heat pulse. The time t is given in units of the axial trapping period 2π/ω_x. c Sketch of a bimodal density distribution of a trapped BEC (purple line) at y = z = 0. At the center of the trap a blue detuned beam produces a dimple in the potential. Modulating the beam intensity produces first sound waves (red arrows) and second sound (orange arrows) waves. Second sound reduces the local density of the cloud, while for first sound a density peak emerges. The thin black line shows the profile of the unperturbed cloud. Please note that the crests and troughs of the waves are shown in an exaggerated way for better visibility. d The false color plot shows the measured local change in the density $\mathrm{\Delta }\overline{n}\left(x,t\right)$ as a function of axial position x and time t. Here, ${\left(k_{\mathrm{F}}a\right)}^{-1}=\left(1.61\pm 0.05\right)$ at B = 735 G and T/T_c = (0.80 ± 0.15). After excitation, two wave packets (bright traces, marked with red arrows) propagate with first sound velocity u₁ towards the edges of the cloud. The excitation method predominantly excites first sound. Second sound is present as well but is barely discernible here. e Propagation of first sound waves (bright traces, marked with red arrow) and second sound waves (dark traces, marked with orange arrows) after excitation with sinusoidal pulse of (b). All other settings are the same as in (d). f Simulated sound propagation for the same parameters as in (e). The orange arrows mark the propagating second sound and the red arrows the first sound, respectively.

Fig. 2. Second sound velocity u₂ as a function of interaction strength.

The purple circles depict measured data for temperatures in the range T = 105−230 nK which corresponds to T/T_c = 0.66−0.84 (see Supplementary Note 1). The error bars are due to statistical uncertainties. The brown and blue solid line show hydrodynamic predictions for the BEC and BCS regime at T = 0.75T_c, respectively (see Supplementary Note 3). The shaded areas mark the second sound velocity in the temperature range of the experiments. The blue dash-dotted line shows a theoretical prediction of second sound in the crossover for a homogeneous gas at T/T_c = 0.75. It interpolates between the results from hydrodynamic theory in the BEC and BCS regime. The green squares are results of our numerical c-field simulations which are consistent with both, analytic and experimental results. For comparison we also show the second sound velocity on the resonance measured in ref. at the temperatures T/T_c = 0.65 (blue triangle), T/T_c = 0.75 (brown triangle), and T/T_c = 0.85 (red triangle).

Fig. 3. Comparing signal strength of first and second sound.

a Sound excitation experiment at ${\left(k_{\mathrm{F}}a\right)}^{-1}=\left(1.61\pm 0.05\right)$ and at a temperature of T/T_c = (0.80 ± 0.15). In contrast to Fig. 1d, first sound (red arrows) and second sound (orange arrows) are now visible simultaneously. For tω_x/2π < 0.15 first and second sound waves overlap and therefore cannot be distinguished from each other. b shows $\mathrm{\Delta }\overline{n}$ for tω_x/2π = 0.43. We fit the center position of each of the two sound waves using a Gaussian function (solid line).

Fig. 4. Sound excitation with different modulation sequences.

a, c $\mathrm{\Delta }\overline{n}\left(x,t\right)$ data at ω_ex = 0.61ω_r, ΔU = 0.3U₀ and ${\left(k_{\mathrm{F}}a\right)}^{-1}=\left(1.61\pm 0.05\right)$ for a modulation of 1.5 cycles and of 1 cycle, respectively. The excitation pulse excites both, first and second sound waves. b, d $\mathrm{\Delta }\overline{n}\left(x,t\right)$ from numerical c-field simulations. Top row: False color images of $\mathrm{\Delta }\overline{n}\left(x,t\right)$. First and second sound waves are marked with red and orange arrows, respectively. The inset in (d) is an enlargement, showing how the calculated bright first sound wave (black arrow) is damped when it crosses the second sound wave (orange arrow). Mid row: Shown is $\mathrm{\Delta }\overline{n}$ for tω_x/2π = 0 (a, b) and for tω_x/2π = 0.15 (c, d). Bottom row: Applied excitation scheme.