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Bosonic atoms confined in optical lattices are described by the Bose-Hubbardmodel and can exist in two different phases, Mott insulator or superfluid,depending on the strength of the system parameters. In the vicinity of thephase boundary, there are degeneracies that occur between every two adjacentMott lobes. Because of this, nondegenerate perturbation theory fails to givemeaningful results for the condensate density: it predicts a phase transitionin a point of the phase diagram where no transition occurs. Motivated by this,we develop two different degenerate perturbative methods to solve thedegeneracy-related problems. Moreover, we study the one-dimensional repulsivelyinteracting Bose gas under harmonic confinement, with special attention to theasymptotic behavior of the momentum distribution, which is a universal $k^{-4}$decay characterized by the Tan's contact. The latter constitutes a directsignature of the short-range correlations in such an interacting system andprovides valuable insights about the role of the interparticle interactions. Weinvestigate the system constituted of $N$ interacting particles in the stronglyinteracting limit. In such a regime, the strong interparticle interaction makesthe bosons behave similarly to the ideal Fermi gas. Because of the difficultyin analytically solving the system for $N$ particles at finite interaction, theTonks-Girardeau regime provides a favorable scenario to probe the contact.Therefore, we are able to provide an analytical formula for the Tan's contact.Furthermore, we analyze the scaling properties of the Tan's contact in terms of$N$ in the high-temperature regime as well as in the strongly interactingregime. Finally, we compare our analytical calculations of the Tan's contact toquantum Monte Carlo simulations and discuss some fundamental differencesbetween the canonical and the grand-canonical ensembles.

A study on quantum gases: bosons in optical lattices and the one-dimensional interacting Bose gas