We describe weakly interacting atomic or molecular bosons trapped in harmonically confined optical lattices, which exhibit a wedding cake structure consisting of insulating (Mott ) shells. We show that superfluid regions emerge between Mott shells as a result of fluctuations due to finite hopping and find that the order parameter equation in the superfluid regions is not of the standard Gross- Pitaevskii type except near the insulator- to- superfluid boundaries. We obtain the excitation spectra in the Mott and superfluid regions, and we show that the superfluid shells possess low- energy sound modes with spatially dependent sound velocity described by a local index of refraction directly related to the local superfluid density. Furthermore, we discuss the Berezinskii- Kosterlitz- Thouless transition and vortex- antivortex pairs in thin ( wide ) superfluid shells ( rings ) limited by three- ( two- ) dimensional Mott regions. Last, we propose an experiment to detect superfluid shells using Laguerre- Gaussian beams and Bragg spectroscopy.
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