Bose-Einstein condensation in a minimal inhomogeneous system

We study the effects of repulsive interaction and disorder on bosons in a two-site Bose-Hubbard system, which provides a simple model of the dirty boson problem. By comparison with exact numerical results, we demonstrate how a straightforward application of the Bogoliubov approximation fails even to deliver a qualitatively correct picture. It wrongly predicts an increase of the condensate depletion due to disorder. We show that, in the presence of disorder, the noncommutative character of the condensate operator has to be retained for a correct description of the system. While these corrections to a simple meanfield Bogoliubov theory are usually small in extended systems, we show that care has to be taken when modeling small inhomogeneous Bose systems.