Dynamical Density Fluctuations of Superfluids near the Critical Velocity

We propose a stability criterion of superfluids in condensed Bose-Einstein systems, which incorporates the spectral function or the autocorrelation function of the local density. Within the Gross-Pitaevskii-Bogoliubov theory, we demonstrate the validity of our criterion for the soliton-emission instability, with use of explicit forms of zero modes of the Bogoliubov equation and a dynamical scaling near the saddle-node bifurcation. We also show that the criterion is applicable to the Landau phonon instability and the Landau roton instability within the single-mode approximation.

Figure 1

Scaling relation near the saddle-node bifurcation for dimensions $d=1$, 2, and 3. The scaled spectral functions $F(x=0,\tilde{\omega }=\omega |V-V_c{|}^{-1/2})={\omega }^{2-d}I(x=0,\omega )$ near the critical current are shown as a function of scaled frequency $\tilde{\omega }$. The critical velocity $V_c$ in dimensionless form is given by $V_c=0.049753\dots$ for $U_{\mathrm{ex}}(x)=U_0\delta (x)$ with $U_0=10$. Circle (red), square (blue), and triangle (green) points are data for $V=0.04975$, 0.0497, and 0.049, respectively.