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The nature of Bose-Einstein condensation enhanced by localisation

Thomas Jaeck

Published Sep 6, 2010• 0 citations
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Abstract

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.

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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 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 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 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.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.