Precursor phenomena of nucleations of quantized vortices in the presence of a uniformly moving obstacle in Bose-Einstein condensates

We investigate excitations and fluctuations of Bose-Einstein condensates in a two-dimensional torus with a uniformly moving Gaussian potential by solving the Gross-Pitaevskii equation and the Bogoliubov equation. The energy gap $\Delta$ between the current-flowing metastable state (that reduces to the ground state for sufficiently slowly-moving potential) and the first excited state vanishes when the moving velocity $v$ of the potential approaches a critical velocity v_c(>0). We find a scaling law $\Delta \propto (1-|v|/v_c)^{1/4}$, which implies that a characteristic time scale diverges toward the critical velocity. Near the critical velocity, we show that low-energy local density fluctuations are enhanced. These behaviors can be regarded as precursor phenomena of the vortex nucleation.