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Recently, we presented a new numerical scheme for vortex lattice formation ina rotating Bose-Einstein condensate (BEC) using smoothed particle hydrodynamics(SPH) with an explicit time-integrating scheme; our SPH scheme could reproducethe vortex lattices and their formation processes in rotatingquasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In thisstudy, we have successfully demonstrated a simulation of rotating 3D BECstrapped in a 3D harmonic potential forming "cigar-shaped" condensates. We havefound that our scheme can reproduce the following typical behaviors of rotating3D BECs observed in the literature: (i) the characteristic shape of the latticeformed in the cross-section at the origin and its formation process, (ii) thestable existence of vortex lines along the vertical axis after reaching thesteady state, (iii) a "cookie-cutter" shape, with a similar lattice shapeobserved wherever we cut the condensate in a certain range in the verticaldirection, (iv) the bending of vortex lines when approaching the inner edges ofthe condensate, and (v) the formation of vortex lattices by vortices enteringfrom outside the condensate. Therefore, we further validated our scheme bysimulating rotating 3D BECs.

Three-dimensional analysis of vortex-lattice formation in rotating Bose-Einstein condensates using smoothed-particle hydrodynamics