Sound Propagation in a Uniform Superfluid Two-Dimensional Bose Gas

In superfluid systems several sound modes can be excited, such as, for example, first and second sound in liquid helium. Here, we excite running and standing waves in a uniform two-dimensional Bose gas and we characterize the propagation of sound in both the superfluid and normal regimes. In the superfluid phase, the measured speed of sound is in good agreement with the prediction of a two-fluid hydrodynamic model, and the weak damping is well explained by the scattering with thermal excitations. In the normal phase we observe a stronger damping, which we attribute to a departure from hydrodynamic behavior.

Figure 1

Experimental protocol and observation of propagating waves. (a) Absorption image of the cloud perturbed by a local additional potential. The excitation is delimited by the horizontal dashed line and depletes the atomic density by a factor around $1/3$. (b) Example of time evolution of the variation of the density profile $n_{2\mathrm{D}}$ with respect to its spatial mean value (integrated along $x$) obtained after abruptly removing the additional potential. For this example $T/T_c=0.37(12)$ and $n_{2\mathrm{D}}=29(3)\mu {\mathrm{m}}^{-2}$. The position of the dip is fitted by a triangle function (black solid line) which gives, $c=1.49(3)\mathrm{mm}/\mathrm{s}$.

Figure 2

Time evolution of the normalized amplitude of the lowest-energy mode for (a) $T/T_c=0.21(11)$, (b) $T/T_c=0.95(5)$, (c) $T/T_c=1.38(18)$. The solid line is a fit of an exponentially damped sinusoidal oscillation. For (b) and (c) graphs, each data point is the average of three measurements and the error bars represent the associated standard deviation. In (a) each point corresponds to a single measurement.

Figure 3

Speed of sound and quality factor. (a) Measured speed of sound $c$ normalized to $c_B$. The vertical dashed line shows the position of the critical point. The solid line shows the result from the two-fluid hydrodynamic model applied to the 2D Bose gas [12]. A fit to the data points below $T_c$ by this hydrodynamic model with a free multiplicative factor shows that the measurements are globally 3% above the theoretical prediction. This could correspond to a 6% systematic error in the calibration of $n_{2\mathrm{D}}$ used to determine $c_B\propto n_{2\mathrm{D}}^{1/2}$. Our estimated uncertainty on $n_{2\mathrm{D}}$ is on the order of 11% (see Ref. [18]) and our measurements are thus compatible with the predicted value of the speed of second sound $c_{\mathrm{HD}}^{(2)}$. (b) Quality factor $Q=2\omega /\mathrm{\Gamma }$ of the lowest-energy mode. The solid line is the prediction for Landau damping [30] (continued as a dashed line for $T>T_c$). For both graphs, the error bars represent the statistical uncertainty extracted from the fitting procedures used to determine $c$, $\mathrm{\Gamma }$ and $T/T_c$.

Figure 4

Observation of standing waves in the box potential. Contribution of the three lowest-energy modes to the amplitude of the density modulation: $j=1$ (circles), $j=2$ (squares), $j=3$ (diamonds). The solid lines are Lorentzian fits. The two insets show the resonance frequencies ${\nu }_j$ and the full widths at half maximum ${\mathrm{\Gamma }}_j$ resulting from these fits. The solid lines in the insets are linear fit to the data and the shaded areas represent the uncertainty on the fitted slope. From the slope $c/(2L_y)$ of the fit to the resonance frequencies, we find $c=1.90(9)\mathrm{mm}/\mathrm{s}$. For this specific experiment, the length of the cloud is $L_y=57(1)\mu \mathrm{m}$ and the degree of degeneracy is $T/T_c=0.41(7)$.