Simulation of Gross-pitaevskii equation ~Vortex lattice formation in a rotating Bose-einstein condensate~

Akira Kato, Yuki Nakano, Kenichi Kasamatsu +1 more

We study the dynamics of vortex nucleation and lattice formation in a Bose--Einstein condensate in a rotating square optical lattice by numerical simulations of the Gross--Pitaevskii equation. Different dynamical regimes of vortex nucleation are found, depending on the depth and period of the optical lattice. We make an extensive comparison with the experiments by R. A. Williams et al. [Phys. Rev. Lett. 104, 050404 (2010)], especially focusing on the issues of the critical rotation frequency for the first vortex nucleation and the vortex number as a function of rotation frequency.

Makoto Tsubota, Kenichi Kasamatsu, Masahito Ueda

Satori Tsuzuki

This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.

Masashi Yasunaga, Makoto Tsubota

SB Prasad, T Bland, BC Mulkerin +2 more

The behavior of a harmonically trapped dipolar Bose-Einstein condensate with its dipole moments rotating at angular frequencies lower than the transverse harmonic trapping frequency is explored in the co-rotating frame. We obtain semi-analytical solutions for the stationary states in the Thomas-Fermi limit of the corresponding dipolar Gross-Pitaevskii equation and utilize linear stability analysis to elucidate a phase diagram for the dynamical stability of these stationary solutions with respect to collective modes. These results are verified via direct numerical simulations of the dipolar Gross-Pitaevskii equation, which demonstrate that dynamical instabilities of the co-rotating stationary solutions lead to the seeding of vortices that eventually relax into a triangular lattice configuration. Our results illustrate that rotation of the dipole polarization represents a new route to vortex formation in dipolar Bose-Einstein condensates.

Shih-Kuang Tung, Volker Schweikhard, Ian R. Coddington +2 more