Bose condensation of interacting gases in traps with and without optical lattice

We discuss effects of particle interaction on Bose condensation in inhomogeneous traps with and without optical lattice. Interaction pushes normal particles away from the condensate droplet, which is located in the center of the trap, towards the periphery of the trap where the trapping potential is large. In the end, the remaining normal particles are squeezed to a quasi-2D shell around the condensate droplet thus changing the effective dimensionality of the system. In the absence of the optical lattice the index in the temperature dependence of the condensate density at the later stages of the process is close to 2 with a weak dependence on the number of trapped particles. In the presence of the lattice inside the trap this index acquires a strong dependence on the number of particles inside the trap and gradually falls from a 3D to a 2D value with an increase in the number of particles. This change in index is explained by the lattice-driven spread of the condensate droplet and the localization of the narrow band particles by the trap potential.