A study on quantum gases: bosons in optical lattices and the
one-dimensional interacting Bose gas
- URL: http://arxiv.org/abs/2006.13100v2
- Date: Wed, 29 Jul 2020 16:45:58 GMT
- Title: A study on quantum gases: bosons in optical lattices and the
one-dimensional interacting Bose gas
- Authors: Felipe Taha Sant'Ana
- Abstract summary: Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model.
In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent lobes.
Motivated by this, we develop perturbative methods to solve the degeneracy-related problems.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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.
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