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Journal of Physics A Mathematical and General

A model with coexistence of two kinds of Bose condensation

J.-B. Bru, Valentin Anatol'evich Zagrebnov

Published Jan 7, 2000• 27 citations
Publisher

Abstract

We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.

DOI
10.1088/0305-4470/33/3/302

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We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.
It is shown that the condensation in the Bogoliubov Weakly Imperfect Bose{Gas (WIBG) may appear in two stages.If the interaction is such that the pressure of the WIBG does not coincide with the pressure of the Perfect Bose{Gas (PBG), then the WIBG may manifest two kinds of condensations: a non{conventional condensation in the zero{mode due to the interaction (the rst stage) and a conventional (generalized of type I) Bose{Einstein condensation in modes next to the zero{mode due to the particle density saturation (the second stage).
We present a review of some published results concerning evolution of the notion of (conventional ) Bose-Einstein condensation (BEC) predicted in Ideal Bose-Gas (IBG) in 1925. The first step forward was the notion of the conventional generalised BEC à la van den Berg-Lewis-Pulé discovered in the IBG and later in certain models with interactions in 1980s. The next development was a discovery of non-conventional condensation of bosons that occurs only due to interaction. Finally we discuss an example of (non-ideal) the Weakly Imperfect Bose-Gas (WIBG), where the conventional and non-conventional condensates may coexist due to interaction.
We study a non-interacting (perfect) Bose-gas in random external potentials (traps).It is shown that a generalized Bose-Einstein condensation in the random eigenstates manifests if and only if the same occurs in the one-particle kinetic-energy eigenstates, which corresponds to the generalized condensation of the free Bose-gas.Moreover, we prove that the amounts of both condensate densities are equal.This statement is relevant for justification of the Bogoliubov approximation in the theory of disordered boson systems.