Interacting bosons in an optical lattice

Abstract

Key words optical lattice, quantum phase transition, Mott insulator, functional integrals PACS 05.30. Jp,03.75.Hh,03.75.Lm Several models of a strongly interacting Bose gas in an optical lattice are studied within the functionalintegral approach. The one-dimensional Bose gas is briefly discussed. Then the Bose-Einstein condensate and the Mott insulator of a three-dimensional Bose gas are described in mean-field approximation, and the corresponding phase diagrams are evaluated. Other characteristic quantities, like the spectrum of quasiparticle excitations and the static structure factor, are obtained from Gaussian fluctuations around the mean-field solutions. We discuss the role of quantum and thermal fluctuations, and determine the behavior of physical quantities in terms of density and temperature of the Bose gas. In particular, we study the dilute limit, where the mean-field equation becomes the Gross-Pitaevskii equation. This allows us to extend the GrossPitaevskii equation to the dense regime by introducing renormalized parameters in the latter.

Ultracold bosons with short-range interaction in regular optical lattices

“…Since the local chemical potential µ = µ − varies from site to site but the occupation numbers n within the plateau do not, the local compressibility (89) vanishes. The density profiles can be also obtained within the local density approximation [4,246,247] replacing the chemical potential µ in Eq. (231) by µ = µ − .…”

Cold bosons in optical lattices

“…One evident condition is the normalization of the statistical operator, 18) where1 F is the unity operator in F (ψ). The Hamiltonian energy operatorĤ [ψ], which is a functional of ψ and ψ † , defines the internal energy 19) which is another statistical condition. The total number of particles N is given by the average 20) of the number-of-particle operatorN The statistical operatorρ of an equilibrium system is defined as the minimizer of the information functional [55] I , (2.24) where the trace is over F (ψ) and…”

Dynamical mean-field theory for the Bose-Hubbard model

“…1,6,7 For weakly interacting bosons for which the fluctuations around the mean-field state are small, the Bogolubov theory or the Gross-Pitaevskii equation applies but both fail to predict the SF-MI transition. 8,9 Beyond the mean field level, methods that can tackle the strong correlations have been applied 3,5,9,10,11,12 , including the Gutzwiller approach 13,14 , Bethe ansatz 15 , time-dependent variational principle method 16 , slave boson approach 17,18,19 , the strong coupling expansion 20,21,22,23 , variational method based on mean field theory 24 , and the effective action approach 24,25 , etc. Recently the cavity method based on the Bethe lattice is also applied to BHM 26 .…”