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arXiv:cond-mat/0409612 · Journal of Physics A Mathematical and General

Bose–Einstein condensation in geometrically deformed tubes

Pavel Exner, Valentin Anatol'evich Zagrebnov

Published Jun 8, 2005• 8 citations
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Abstract

We show that Bose-Einstein condensate can be created in quasi-one-dimensional systems in a purely geometrical way, namely by bending or other suitable deformation of a tube.
DOI
10.1088/0305-4470/38/25/l01

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Quasi-crystal structures of Bose-Einstein condensates

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Quasi-crystal structures of Bose-Einstein condensates which correspond to a Penrose tiling of the plane are discussed.

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We consider a quasi-one-dimensional (quasi-1D) system that is an optical hard-wall nanotube with axial harmonic potential. This quasi-1D system possesses a quasi-1D harmonic potential that consists of a 1D harmonic trap along the z axis and a two-dimensional box trap in the x-y plane. Within the framework of quantum statistical mechanics, we find that in a quasi-1D harmonic potential, the generalized Bose-Einstein condensation (GBEC) can be classified into the two types. The first type of GBEC (GBEC_{1}) corresponds to T_{m}<T_{c}. The second type of GBEC (GBEC_{2}) corresponds to T_{c}<T_{m}. Here, T_{c} is the critical temperature of the first-step condensation and T_{m} is the critical temperature of the second-step condensation. There is a competitive mechanism between normal Bose-Einstein condensation (NBEC) and GBEC. We shall show that the standard transition temperature T_{c} is much larger than the transition temperature T_{n} of NBEC. Therefore, the state of NBEC does not occur in a quasi-1D harmonic potential. We have proposed an analytical solution to the problem of GBEC of ideal atoms in the quasi-1D harmonic potential. The number of noncondensed atoms in the first-step condensation is characterized by a series of elliptic theta functions and the number of noncondensed atoms in the second-step condensation is characterized by a single q-digamma function. If r denotes the tube radius, then this quasi-1D system can show the quantum size effect. There is a critical tube radius r_{c}. (1) If r>r_{c}, then the mediate condensate fraction N_{m}/N=0; (2) if r>r_{c}, then the standard transition temperature T_{c}=0; (3) if r>r_{c}, then T_{m}>T_{c} and this is the state of GBEC_{2}. The results of numerical calculation of the analytical solution predict many new experimental phenomena in the GBEC of ideal atoms in the quasi-1D harmonic potential. In the thermodynamic limit, the analytical expressions of the three critical temperatures and the three condensate fractions are derived.