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arXiv:1307.1990 · Condensed Matter Physics

Self-consistent approach for Bose-condensed atoms in optical lattices

Yukalov

Published Jan 1, 2013• 4 citations
arXivHTMLPDF

Abstract

Bose atoms in optical lattices are considered at low temperatures and weak interactions, when Bose-Einstein condensate is formed.A self-consistent approach, based on the use of a representative statistical ensemble, is employed, ensuring a gapless spectrum of collective excitations and the validity of conservation laws.In order to show that the approach is applicable to both weak and tight binding, the problem is treated in the Bloch as well as in the Wannier representations.Both these ways result in similar expressions that are compared for the self-consistent Hartree-Fock-Bogolubov approximation.A convenient general formula for the superfluid fraction of atoms in an optical lattice is derived.

DOI
10.5488/cmp.16.23002

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Bose atoms in optical lattices are considered at low temperatures and weak interactions, when Bose-Einstein condensate is formed.A self-consistent approach, based on the use of a representative statistical ensemble, is employed, ensuring a gapless spectrum of collective excitations and the validity of conservation laws.In order to show that the approach is applicable to both weak and tight binding, the problem is treated in the Bloch as well as in the Wannier representations.Both these ways result in similar expressions that are compared for the self-consistent Hartree-Fock-Bogolubov approximation.A convenient general formula for the superfluid fraction of atoms in an optical lattice is derived.
A theoretical study of interacting bosons in a periodic optical lattice is presented. Instead of the commonly used tight-binding approach (applicable near the Mott-insulating regime of the phase diagram), the present work starts from the exact single-particle states of bosons in a cubic optical lattice, satisfying the Mathieu equation, an approach that can be particularly useful at large boson fillings. The effects of short-range interactions are incorporated using a self-consistent Hartree-Fock approximation, and predictions for experimental observables such as the superfluid transition temperature, condensate fraction, and boson momentum distribution are presented.
The necessity of accurately taking into account the existence of nonequivalent operator representations associated with canonical transformations is discussed. It is demonstrated that Bose systems in the presence of the Bose-Einstein condensate and without it correspond to different Fock spaces, orthogonal to each other. A composite representation for the field operators is constructed allowing for a self-consistent description of Bose-condensed systems. Equations of motion are derived from the given Hamiltonian, which guarantees the validity of conservation laws and thermodynamic self-consistency. At the same time, the particle spectrum obtained either from diagonalizing this Hamiltonian or from linearizing the field-operator equations of motion has no gap. The condition of the condensate existence assures the absence of the gap in the spectrum, irrespectively to the approximation involved. The suggested self-consistent theory is both conserving and gapless.
A practical finite-temperature theory is developed for the superfluid regime of a weakly interacting Bose gas in an optical lattice with additional harmonic confinement. We derive an extended Bose-Hubbard model that is valid for shallow lattices and when excited bands are occupied. Using the Hartree-Fock-Bogoliubov-Popov mean-field approach, and applying local-density and coarse-grained envelope approximations, we arrive at a theory that can be numerically implemented accurately and efficiently. We present results for a three-dimensional system, characterizing the importance of the features of the extended Bose-Hubbard model and compare against other theoretical results and show an improved agreement with experimental data.
Abstract When interactions between particles are strong, at low temperature, these particles can form self-organized quantum crystals, and when the particles interact weakly, periodic structures can be imposed by external fields, e.g. by optical lattices. These opposite cases usually are treated separately, dealing either with quantum crystals or with optical lattices. Here the unified theory is developed for arbitrary particle interaction strength, treating in the frame of the same model both the limiting cases of quantum crystals and optical lattices, as well as the states intermediate between these two limits. Bose particles are considered, hence at low temperature in optical lattices Bose–Einstein condensation can happen, while it seems to be prohibited in ideal quantum crystals that do not contain mesoscopic regions of disorder, such as dislocations and grain boundaries.