The loading of a Bose-Einstein condensate into a deep two-dimensional optical lattice provides a unique way to study one-dimensional Bose gases: the strong radial confinement freezes any motion in two dimensions, and for deep enough lattices, the system can be seen as an array of independent I D tubers. For our experimental parameters, the ID gas is predicted to be in an intermediate regime between the Tonks-Girardeau and the Thomas-Fermi regimes. We performed experiments showing that some long range phase coherence is present in this regime. We investigated correlation properties of these gases by studying their collective oscillations. In addition, we investigated the ID Mott transition by adiabatically loading the 1D gases into a 1D optical lattice.
We study the limit of large on-site repulsion of the one-dimensional Bose-Hubbard model at low densities, and derive a strong-coupling effective Hamiltonian. By taking the lattice parameter to zero, the Hamiltonian becomes a continuum model of fermions with attractive interactions. The leading corrections to the internal energy of a hard-core-boson (Tonks) gas as well as the (finite temperature) pair correlations of a strongly interacting Bose gas are calculated. We explore the possibility of realizing, in an optical lattice, a Luttinger liquid with stronger density correlations than the Tonks gas. A quantum phase transition to a charge-density-wave Mott insulator is also discussed.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal
k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of
N interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for
N particles at finite interaction, the Tonks-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 interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.