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arXiv:cond-mat/0204439 · Physical Review Letters

Existence of Long-Range Order for Trapped Interacting Bosons

Uwe R. Fischer

Published Dec 31, 2002• 33 citations
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Abstract

We derive an inequality governing "long-range" order for a localized Bose-condensed state, relating the condensate fraction at a given temperature with effective curvature radius of the condensate and total particle number. For the specific example of a one-dimensional, harmonically trapped dilute Bose condensate, it is shown that the inequality gives an explicit upper bound for the Thomas-Fermi condensate size which may be tested in current experiments.

DOI
10.1103/physrevlett.89.280402

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We derive an inequality governing "long-range" order for a localized Bose-condensed state, relating the condensate fraction at a given temperature with effective curvature radius of the condensate and total particle number. For the specific example of a one-dimensional, harmonically trapped dilute Bose condensate, it is shown that the inequality gives an explicit upper bound for the Thomas-Fermi condensate size which may be tested in current experiments.
I discuss a Bogoliubov inequality for obtaining a rigorous bound on the maximal axial extension of inhomogeneous one-dimensional Bose-Einstein condensates. An explicit upper limit for the aspect ratio of a strongly elongated, harmonically trapped Thomas-Fermi condensate is derived. The question on the existence and dimension dependence of off-diagonal long-range order, for an infinitely extended system of scalar uncharged bosons, was conclusively answered by Hohenberg 1. He made use of the fact that in the thermodynamic limit of an infinite system of interacting particles of bare mass m, the relation (we set � = kB = 1 in what follows, where kB is the Boltzmann constant) Nk ≥ − 1 mT 2 k2 n0
Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order ${R}^{\ensuremath{-}4}\mathrm{ln}R$ and ${R}^{\ensuremath{-}4},$ where R is the number-dependent dimensionless condensate radius in units of the oscillator length $\sqrt{\ensuremath{\Elzxh}/m{\ensuremath{\omega}}_{0}}.$ The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order ${R}^{\ensuremath{-}4}.$
We calculate the shift in the critical temperature of Bose-Einstein condensation for a dilute Bose-Fermi mixture confined by a harmonic potential, to lowest order in both the Bose-Bose and Bose-Fermi coupling constans. The relativ importance of the effect on the critical temperature of the boson-fermion interactions is investigated as a function of the parameters of mixture. The possible relevance of the shift of the transition temperature in current experiments on trapped Bose-Fermi mixtures is discussed.
We calculate the shift in the critical temperature of Bose–Einstein condensation for a dilute Bose–Fermi mixture confined by a harmonic potential, to lowest order in both the Bose–Bose and Bose–Fermi coupling constants. The relative importance of the effect on the critical temperature of the boson–boson and boson–fermion interactions is investigated as a function of the parameters of the mixture. The possible relevance of the shift of the transition temperature in current experiments on trapped Bose–Fermi mixtures is discussed.
In this paper we explore the low-dimension long-range superfluid order in the trapped Bose-Einstein condensate driven by spin-orbit coupling at any finite temperature. With the help of the Bogoliubov inequality we give rigorous proofs in both one and two dimensions. For one dimension, no long-range superfluid order can be found, even if spin-orbit coupling exists. For two dimensions, we find that the long-range superfluid order cannot be ruled out in the presence of the unequal Rashba-Dresselhaus-type spin-orbit coupling. However, this long-range superfluid order disappears for the equal case, which has been achieved in recent experiments.