We propose a model that includes itinerant and localized states to study Bose–Einstein condensation of ultracold atoms in optical lattices (Bose–Anderson model). It is found that the original itinerant and localized states intermix to give rise to a new energy band structure with two quasiparticle energy bands. We have computed the critical temperature Tc of the Bose–Einstein condensation of the quasiparticles in the Bose–Anderson model using our newly developed numerical algorithm and found that Tc increases as na3 (the number density times the lattice constant cubed) increases according to the power law Tc≈18.93(na3)0.59 nK for na3<0.125 and according to the linear relation Tc≈8.75+10.53na3 nK for 1.25<na3<12.5 for the given model parameters. With the self-consistent equations for the condensation fractions obtained within the Bogoliubov mean-field approximation, the effects of the on-site repulsion U on the quasiparticle condensation are investigated. We have found that, for values up to several times the zeroth-order critical temperature, U enhances the zeroth-order condensation fraction at intermediate temperatures and effectively raises the critical temperature, while it slightly suppresses the zeroth-order condensation fraction at very low temperatures.
Bose-Einstein condensation and superfluidity of a dilute Bose gas in a random potential