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We study interacting bosons in optical lattices in the weak-tunneling regimein systems that exhibit the coexistence of Mott-insulating and condensedphases. We discuss the nature of the condensed ground state in this regime andthe validity of the mean-field treatment thereof. We suggest two experimentalsignatures of condensate order in the system. (1) We analyze the hyperfineconfiguration of the system and propose a set of experimental parameters forobserving radio-frequency spectra that would demonstrate the existence of thecondensed phase between Mott-insulating phases. We derive the structure of thesignal from the condensate in a typical trapped system, taking into accountGoldstone excitations, and discuss its evolution as a function of temperature.(2) We study matter-wave interference patterns displayed by the system uponrelease from all confining potentials. We show that as the density profilesevolve very differently for the Mott-insulating phase and the condensed phase,they can be distinguished from one another when the two phases coexist.

Probing condensate order in deep optical lattices