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We consider theoretically the problem of an artificial gauge potentialapplied to a cold atomic system of interacting neutral bosons in atight-binding optical lattice. Using the Bose-Hubbard model, we show that aneffective magnetic field leads to superfluid phases with simultaneous spatialorder, which we analyze using Bogoliubov theory. This gives a consistentexpansion in terms of quantum and thermal fluctuations, in which the lowestorder gives a Gross-Pitaevskii equation determining the condensateconfiguration. We apply an analysis based on the magnetic symmetry group toshow how the spatial structure of this configuration depends on commensurationbetween the magnetic field and the lattice. Higher orders describe thequasiparticle excitations, whose spectrum combines the intricacy the Hofstadterbutterfly with the characteristic features of the superfluid phase. We use thedepletion of the condensate to determine the range of validity of ourapproximations and also to find an estimate for the onset of the Mott insulatorphase. Our theory provides concrete experimental predictions, for bothtime-of-flight imagery and Bragg spectroscopy.

Bogoliubov theory of interacting bosons on a lattice in a synthetic magnetic field