Collisionless Sound in a Uniform Two-Dimensional Bose Gas

Using linear response theory within the random phase approximation, we investigate the propagation of sound in a uniform two dimensional (2D) Bose gas in the collisionless regime. We show that the sudden removal of a static density perturbation produces a damped oscillatory behavior revealing that sound can propagate also in the absence of collisions, due to mean-field interaction effects. We provide explicit results for the sound velocity and damping as a function of temperature, pointing out the crucial role played by Landau damping. We support our predictions by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii equation. The results are consistent with the recent experimental observation of sound in a weakly interacting 2D Bose gas both below and above the superfluid Berezinskii-Kosterlitz-Thouless transition.

Figure 1

Dimensionless isothermal compressibility as a function of temperature for $g_{2\mathrm{D}}=0.1{\hslash }^2/m$. The blue solid line and the red squares correspond to ${\kappa }_T$ calculated with the RPA expression (3) and with SGPE, respectively. The error bars of the SGPE data represent the statistical uncertainty of the average over different noise realizations. The black dots are the results obtained from the universal relations of Refs. [25, 26].

Figure 2

Upper panel: Imaginary part of the inverse frequency weighted response function, ${\chi }^{\prime \prime }(u)/u$, calculated in the RPA at $T=0.15T^{*}=1.2T_c$, with $g_{2\mathrm{D}}=0.16{\hslash }^2/m$, as a function of the dimensionless frequency $u=(\omega /k)\sqrt{m/(2k_{B}T)}$. Lower panel: Fourier transform (4), as a function of the dimensionless time $\tilde{t}=kt\sqrt{2k_{B}T/m}$. The blue solid and green dashed lines correspond to the interacting and ideal gas, respectively. The red dotted line is the fit based on Eq. (5).

Figure 3

Sound velocity in units of $c_0=\sqrt{g_{2\mathrm{D}}n/m}$ for $g_{2\mathrm{D}}=0.1{\hslash }^2/m$. The blue solid line is the sound velocity $c=\tilde{\omega }/k$ extracted from the Fourier transform Eq. (4) of ${\chi }^{\prime \prime }/\omega$ calculated in RPA, while the blue dashed line is the isothermal sound velocity $c_T=\sqrt{1/(mn{\kappa }_T)}$, within the same theory. Solid and open squares represent the sound speed extracted from real time simulations and the isothermal sound velocity, respectively, both obtained with SGPE. The error bars arise from statistical fluctuations; below $T_c$ they are of the same order of the marker size.

Figure 4

Upper panel shows the sound velocity calculated for $g_{2\mathrm{D}}=0.16{\hslash }^2/m$. Black circles: experimental data of Ref. [11]; blue solid line: RPA; red squares: SGPE; green dashed line: second sound predicted by Landau’s two-fluids hydrodynamics [12, 38]. Lower panel shows the quality factor $Q=2\tilde{\omega }/\mathrm{\Gamma }$. The blue solid line is $Q$ evaluated from the Fourier transform $\mathcal{F}$ in RPA; black circles are experimental data [11]; red squares are the results of SGPE simulations, obtained by averaging over simulations done at different values of excited frequency (see inset). The inset shows $Q$ as a function of frequency at different values of $T$ from SGPE; from top to bottom: $T/T_c=0.29$, 0.52, 0.75, 1.02. The error bars of the SGPE data in both panels represent the statistical deviations due to different noise realizations.