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Condensed Matter Physics

Bose condensation in (random) traps

Jaeck, Pulé, Zagrebnov

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.5488/cmp.12.4.559

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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.
We study the perfect Bose gas in random external potentials and show that there is generalized Bose-Einstein condensation in the random eigenstates 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. Our method is based on the derivation of an explicit formula for the occupation measure in the one-body kinetic-energy eigenstates which describes the repartition of particles among these non-random states. This technique can be adapted to re-examine the properties of the perfect Bose gas in the presence of weak (scaled) non-random potentials, for which we establish similar results.
In a previous paper we established that for the perfect Bose gas and the mean-field Bose gas with an external random or weak potential, whenever there is generalized Bose-Einstein condensation in the eigenstates of the single particle Hamiltonian, there is also generalized condensation in the kinetic energy states. In these cases Bose-Einstein condensation is produced or enhanced by the external potential. In the present paper we establish a criterion for the absence of condensation in single kinetic energy states and prove that this criterion is satisfied for a class of random potentials and weak potentials. This means that the condensate is spread over an infinite number of states with low kinetic energy without any of them being macroscopically occupied.
In a previous paper we established that for the perfect Bose gas and the mean-field Bose gas with an external random or weak potential, whenever there is generalized Bose–Einstein condensation in the eigenstates of the single particle Hamiltonian, there is also generalized condensation in the kinetic-energy states. In these cases Bose–Einstein condensation is produced or enhanced by the external potential. In the present paper we establish a criterion for the absence of condensation in single kinetic-energy states and prove that this criterion is satisfied for a class of random potentials and weak potentials. This means that the condensate is spread over an infinite number of states with low kinetic-energy without any of them being macroscopically occupied.
We investigate the phase transition exhibited by the Bose gas in systems which are not translation-invariant. Though it has been known since the sixties that Bose-Einstein condensation (BEC) cannot occur in translation invariant systems for dimension 1 or 2, one can nevertheless enhance this phase transition in low-dimensional Bose gases by the addition of suitable external potentials (thus losing translation invariance in the process). However, the resulting condensate is then found to be in localised states, while BEC is usually understood to be the macroscopic occupation of extended kinetic eigenstates. It is therefore not clear whether the phase transition obtained by means of localisation is of the same nature as the one related to the usual concept of BEC. In this thesis, we consider two classes of localised systems. The first one is a family of random systems, where the Bose gas is contained in a disordered medium, which is modelled by a random external potential. Our second model consists of weak (scaled) external potentials. We first recall necessary conditions on these external potentials to enhance condensation in the localised states. We then show under very general assumptions that in these models, BEC in the usual sense occurs also, in a generalised sense. This means that the particles condense on kinetic eigenstates with arbitrary small energy. For the non-interacting Bose gas, we can moreover show that the densities of both condensates are actually equal. Next, we investigate BEC on a finer scale, asking whether one can obtain condensation in a single kinetic eigenstate. We show that in spite of the existence of a phase transition, and the occurence of generalised BEC, no condensation exists in any single kinetic eigenstate. In particular, the so-called "ground-state condensation" does not occur in these localised systems. Finally, we establish a possible generalisation of the Bogoliubov c-number approximation to take into account the very specific properties of BEC in the presence of localisation, and discuss how to interpret the result of the corresponding variational problem.
We investigate the validity of the Bogoliubov c-number approximation in the case of interacting Bose-gas in a homogeneous random media. To take into account the possible occurence of type III generalized Bose-Einstein condensation (i.e. the occurrence of condensation in an infinitesimal band of low kinetic energy modes without macroscopic occupation of any of them) we generalize the c-number substitution procedure to this band of modes with low momentum. We show that, as in the case of the one-mode condensation for translation-invariant interacting systems, this procedure has no effect on the exact value of the pressure in the thermodynamic limit, assuming that the c-numbers are chosen according to a suitable variational principle. We then discuss the relation between these c-numbers and the (total) density of the condensate.
We investigate the validity of the Bogoliubov c-number approximation in the case of interacting Bose-gas in a homogeneous random media. To take into account the possible occurence of type III generalized Bose–Einstein condensation (i.e., the occurrence of condensation in an infinitesimal band of low kinetic energy modes without macroscopic occupation of any of them) we generalize the c-number substitution procedure to this band of modes with low momentum. We show that, as in the case of the one-mode condensation for translation-invariant interacting systems, this procedure has no effect on the exact value of the pressure in the thermodynamic limit, assuming that the c-numbers are chosen according to a suitable variational principle. We then discuss the relation between these c-numbers and the (total) density of the condensate.