Vortex dynamics in lattice Bose gases in a synthesized magnetic field with a random noise and a dissipation: Study by the stochastic Gross-Pitaevskii equation

In this paper, we investigate vortex dynamics in a two-dimensional Bose-Hubbard model coupled with a weak artificial magnetic field, a random white noise and a dissipation. Origin of the noise and dissipation is considered as thermal fluctuations of atoms that do not participate the Bose-Einstein condensation (BEC). Solving a stochastic Gross-Pitaevskii equation to this system, we show that the interplay of the magnetic field and the white noise generates vortices in the bulk of the BEC and stable steady states of vortices form after a transition period. We calculate the incompressible part of the kinetic-energy spectrum of the BEC. In the transition period, a Kolmogorov $k^{-5\mathrm{/}3}$ spectrum appears in the infrared regime with the wave number $k$, $k<{\zeta }^{-1}$, where $\zeta$ is the healing length, whereas in the ultraviolet region, $k>{\zeta }^{-1}$, the spectrum behaves as $k^{-3}$. On the other hand in the steady states, another scaling low appears. We find a relationship between the above mentioned kinetic-energy spectra and the velocity of vortices. By an inverse cascade, the large velocity of a few created vortices develops the Kolmogorov $k^{-5\mathrm{/}3}$ spectrum.