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We study the spatial correlations of strongly interacting bosons in a ground state, confined in a two-dimensional square and a three-dimensional cubic lattice. Using the combined Bogoliubov method and the quantum rotor approach, we map the Hamiltonian of strongly interacting bosons onto U(1) phase action in order to calculate the atom-atom correlations' decay along the principal axis and a diagonal of the lattice-plane direction as a function of distance. Lower tunneling rates lead to quicker decays of the correlations, whose character becomes exponential. Finally, correlation functions allow us to calculate quantities that are directly bound to experimental outcomes, namely time-of-flight absorption images and resulting visibility. Our results contain all the characteristic features present in experimental data (transition from Mott insulating blob to superfluid peaks, etc.), emphasizing the usability of the proposed approach.

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We study a model of n-layer high-temperature cuprates of homologous series like HgBa 2 Ca n−1 Cu n O 2+2n+␦ to explain the dependence of the critical temperature T c ͑n͒ on the number n of Cu-O planes in the elementary cell. Focusing on the description of the high-temperature superconducting system in terms of the collective phase variables, we have considered a semimicroscopic anisotropic three-dimensional vector XY model of stacked copper-oxide layers with adjustable parameters representing microscopic in-plane and out-of-plane phase stiffnesses. The model captures the layered composition and block structure along the c axis of homologous series. Implementing the spherical closure relation for vector variables we have approximately solved the phase XY model with the help of the transfer matrix method and calculated T c ͑n͒ for arbitrary block size n, elucidating the role of the c-axis anisotropy and its influence on the critical temperature. Furthermore, we accommodate inhomogeneous charge distribution among planes characterized by the charge imbalance coefficient R being the function of number of layers n. By making a physically justified assumption regarding the doping dependence of the microscopic phase stiffnesses, we have calculated the values of the parameter R as a function of block size n in reasonable agreement with the nuclear magnetic resonance data of the carrier distribution in multilayered high-T c cuprates.