Effect of dimensional crossover in a dipolar supersolid

We investigate the dimensional crossover in a dipolar Bose-Einstein condensate (BEC) from 3D to 2D, highlighting how anisotropic long-range dipolar interaction is affected by the reduced dimensionality. Starting from the 3D supersolid phase, a coexistence of superfluid and solid phases, we explore how it evolves with reduced dimensionality. The dynamical evolution is modeled using the mean-field Gross-Pitaevskii equation with beyond mean-field quantum fluctuation corrections. By tuning the trap aspect ratio, we analyze how this crossover impacts supersolid stability and potential transitions to other phases, i.e., crystal or superfluid. We also consider the presence of vortices, unique to superfluids, to study their effects on the dimensional crossover.