In this paper, the vortices number and the critical atoms number are calculated. Calculations are performed for ultracold trapped Bose gas in two-dimensional deep optical lattice. Hellmann–Feynman theorem is used here to calculate the expectation values of the angular momentum and thermal average of square radius (in situ size). The above-mentioned parameters are calculated from the thermodynamic grand potential [Formula: see text]. The latter is calculated using a conventional semi-classical approximation. This thermodynamic grand potential allows us to investigate the impacts of the finite size and inter-atomic interaction effects. The obtained result for the vortices number has a peak with increase in the rate of rotation. While the critical atoms number has increase in nature with increase in the critical temperature. These parameters show a major dependence on the inter-atomic interaction, depth of optical lattice and number of particles.
We show that in contrast with contact interacting gases, an optical lattice changes drastically the stability properties of a dipolar condensate, inducing a crossover from dipolar destabilization to dipolar stabilization for increasing lattice depths. Performing stability measurements on a
52Cr Bose-Einstein condensate in an interaction-dominated regime, repulsive dipolar interaction balances negative scattering lengths down to $\ensuremath{-}17$ Bohr radii. Our findings are in excellent agreement with mean-field calculations, revealing the important destabilizing role played by intersite dipolar interactions in deep lattices.