Numerical model of the Gross-Pitaevskii equation for rotating Bose-Einstein condensates using smoothed-particle hydrodynamics

TL;DR

This paper introduces a new numerical scheme using smoothed-particle hydrodynamics to simulate vortex lattice formation in rotating Bose-Einstein condensates, accurately reproducing experimental geometric patterns and vortex dynamics.

Contribution

The study develops a novel SPH-based numerical method for solving the Gross-Pitaevskii equation in rotating BECs, enabling realistic vortex lattice simulations.

Findings

Simulations qualitatively match experimental vortex lattice patterns.

The method captures vortex formation and phase-vortex correspondence.

Stable rotation and surface instabilities are accurately modeled.

Abstract

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 (GP) 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 4th-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 the 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.