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Springer eBooks

Basic Mean-Field Theory for Bose-Einstein Condensates

P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González

Published Dec 29, 2007• 8 citations
Publisher
DOI
10.1007/978-3-540-73591-5_1

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Since the nowadays growing interest in Bose-Einstein condensates (BEC) due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, its many-body quantum bases and equations are elementarily presented emphasizing on the order parameter, pseudo-potential and mean field approaches, while emphasizing the common places with the Density Functional Theory, like the integral normalization of density or the chemical potential, as the main vehicles of both theories of matter (Putz, 2011d, 2011e).
This chapter introduces basic notions about Bose-Einstein condensates (BECs). One first studies ideal Bose gases, putting an emphasis on the role of the trapping geometry and of the dimensionality. One also introduces the correlation functions characterizing coherence. The second section deals with weakly-interacting BECs, which are described, in a mean-field approach, by the so-called Gross-Pitaevskii equation. This equation is derived and applied to a variety of experimentally relevant situations. The third section deals with beyond-mean-field effects that appear when the interactions are stronger, and introduces in particular the Bogolubov approximation. The last section is devoted to the study of BECs in double-well potentials, a situation inspired from condensed matter systems where beyond-mean-field effects can appear quite easily, and which remains simple enough to be studied theoretically in details.
There is a wide-spread belief in the literature on Bose-Einstein condensation of interacting atoms that all variants of mean-field theory incorrectly describe the condensation phase transition, exhibiting, instead of the necessary second-order transition, a first-order transition, even for weakly interacting Bose gas. In the present paper, it is shown that a self-consistent mean-field approach is the sole mean-field theory that provides the correct second-order condensation transition for Bose systems with atomic interactions of arbitrary strength, whether weak or strong.