  We investigate the long-range phase coherence of homogeneous and trapped Bose
gases as a function of the geometry of the trap, the temperature, and the
mean-field interactions in the weakly interacting limit. We explicitly take
into account the (quasi)condensate depletion due to quantum and thermal
fluctuations, i.e., we include the effects of both phase and density
fluctuations. In particular, we determine the phase diagram of the gas by
calculating the off-diagonal one-particle density matrix and discuss the
various crossovers that occur in this phase diagram and the feasibility of
their experimental observation in trapped Bose gases.
