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

• Autor(a) principal: Sant'Ana, Felipe Taha
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/76/76134/tde-29062020-150004/
• topic: Bose-Hubbard model; Gás de Bose interagente; Gás de Tonks-Girardeau; Interacting Bose gas; Modelo de Bose-Hubbard; Optical lattices; Quantum phase transition; Redes ópticas; Tonks-Girardeau gas; Transição quântica de fase

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, such as the on-site interaction between particles and the hopping parameter. Differently from classical phase transitions, the Mott-insulatorsuperfluid transition can happen even at zero temperature, driven by quantum fluctuations, thus characterizing a quantum phase transition. For the homogeneous system, we can approximate the particle excitations as a mean-field over time, thus providing a local Hamiltonian, which makes possible the evaluation of physical properties from a single lattice site. From the Landau theory of second-order phase transitions, it is possible to expand the thermodynamic potential in a power series in terms of the order parameter, giving rise to the Mott-insulator-superfluid phase diagram. As the condensate density goes from a finite value to a vanishing one when the system transits from superfluid to a Mott insulator, it can be considered as the order parameter of the system. In the vicinity of the phase boundary, it is possible to consider the hopping term as a perturbation, since it contains the order parameter. Thence, one can apply perturbation theory in order to calculate important physical quantities, such as the condensate density. However, due to degeneracies that happen to exist between every two adjacent Mott lobes, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition due to the vanishing of the order parameter in a point of the phase diagram where no transition occurs. Motivated by such a misleading calculation, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Firstly, we develop a degenerate perturbative method based on Brillouin-Wigner perturbation theory to tackle the zero-temperature case. Afterwards, we develop another degenerate perturbative method based on a projection operator formalism to deal with the finite-temperature regime. Both methods have the common feature of separating the Hilbert subspace where the degeneracies are contained in from the complementary one. Therefore, such a separation of the Hilbert subspaces fixes the degeneracy-related problems and provides us a framework to obtain physically consistent results for the condensate density near the phase boundary. 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 an universal k4 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. From the known solutions of the system composed of two particles, we are able to acquire important knowledge about the manifestation of the interaction, e.g., the cusp condition that implies the vanishing of the many-body wave function whenever two particles meet. Then, we investigate the system constituted of N interacting particles in the strongly interacting limit, also known as Tonks-Girardeau gas. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas, an effect known as fermionization,. Because of the difficulty in analytically solving the system for N particles at finite interaction, the Tonks-Girardeau regime provides, through the fermionization of the bosons, a favorable scenario to probe the Tan´s contact. Therefore, within such a regime, we are able to provide an analytical formula for the Tan´s contact in terms of the single-particle orbitals of the harmonic oscillator. Furthermore, we analyze the scaling properties of the Tan´s contact in terms of the number of particles 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.