Dual-Space Analysis of Vortex Configurations and Rotational Energy in a 2D Optical Lattice

We investigate the effects of a stationary two-dimensional periodic optical lattice on a rotating Bose--Einstein condensate confined by an isotropic harmonic trap. Within the mean-field description, the condensate is modeled by the two-dimensional Gross--Pitaevskii equation in the rotating frame and solved numerically using an imaginary-time split-step Crank--Nicolson method to obtain stationary vortex states. We examine how the lattice depth, lattice wave vector, particle number, and angular velocity influence the vortex configurations, real-space density, momentum-space density, and energetics of the condensate. We find that the stationary periodic lattice significantly modifies vortex nucleation and spatial organization: vortices emerge progressively with increasing angular velocity and, for sufficiently strong lattice strength, undergo rearrangement and merging. Importantly, changes in the vortex configuration are directly reflected in the momentum-space density, demonstrating that vortex nucleation and rearrangement produce pronounced modifications of the condensate momentum distribution. We further analyze the energy per particle and rotational energy as functions of the angular velocity and find that the presence of the optical lattice substantially alters the rotational response of the condensate. In particular, the rotational energy is progressively reduced at higher angular velocities as the lattice strength is increased. These results demonstrate how a stationary periodic optical lattice modifies vortex formation, momentum-space structure, and rotational energetics of a rotating trapped Bose--Einstein condensate.