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arXiv:cond-mat/0004469

Exact versus mean-field description of the Bose-Einstein condensate: a model study

Magdalena A. Załuska–Kotur, Mariusz Gajda, Arkadiusz Orłowski, Jan Mostowski

Published Apr 27, 2000• 4 citations
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Abstract

We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.

DOI
10.48550/arxiv.cond-mat/0004469

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We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
g B oson s T ra p p ed in H ar m on ic P ote ntial : Qu alit y of B ogoliu b o v Ap p ro x im ation M .A .Z a¤u sk a-K otu r , M. Ga j da and J. Mo sto w sk i In st yt ut Fi zy ki , Po l i sh A cadem y of Sciences and Col l ege of Science Al .Lo tni k §w 32/ 46, 02-668 W arszawa, Po l and ( Recei ved Ju ly 26, 2 001)W e st udy a syste m of tra pp ed bosoni c partic les interactin g by mo del harmonic forces.O ur mo del allo w s for a detailed examination of the notion of an order parameter (a condensate w ave function ).By decomp osing a single particle density matrix into coherent eigenmo des w e study an e˜ect of interaction on the condensate.W e show that su£ciently strong interactions cause that the condensate disapp ears even if the whole system is in its lowest energy state.I n the second part of our pap er we discuss the v alidity of the Bogoliub ov approximation by comparing its prediction s w ith results inferred from the exactly soluble mo del.In particular we examine an energy spectrum , occupation , and Ûuctuation s of the condensate.W e conclude that Bogoliu bov approach gives a quite accurate descripti on of the system in the limit of w eak interactions.PAC S numb ers: 03
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We study an exactly solvable system of trapped bosonic particles interacting by model harmonic forces. The model allows for a detailed examination of the order parameter (condensate wavefunction) as well as a concept of the off-diagonal and diagonal order. We analyse the effect of interactions on the condensate and show that sufficiently strong interactions (attractive or repulsive) lead to the destruction of the condensate. In the thermodynamic limit this destruction has a critical character. It is shown that the existence of the coherent state of bosons is related to the existence of two length scales determined by one- and two-particle reduced density matrices. The condensate can exist only if the two length scales are of the same order. Interactions, both repulsive and attractive, change their relative size which may lead to destruction of coherence in the system and depletion of the condensate. We suggest that this scenario is model independent.
The mean ground-state occupation number and condensate fluctuations of interacting and noninteracting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground-state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $〈{\ensuremath{\delta}}^{2}{N}_{0}〉\ensuremath{\sim}N,$ while the fluctuations based on Bogoliubov theory behave as N4/3.{N}^{4/3}.N4/3.
The density of states of a Bose-condensed gas confined in a harmonic trap is investigated. The predictions of Bogoliubov theory are compared with those of Hartree-Fock theory and of the hydrodynamic model. We show that the Hartree-Fock scheme provides an excellent description of the excitation spectrum in a wide range of energy, revealing a major role played by single-particle excitations in these confined systems. The crossover from the hydrodynamic regime, holding at low energies, to the independent-particle regime is explicitly explored by studying the frequency of the surface mode as a function of their angular momentum. The applicability of the semiclassical approximation for the excited states is also discussed. We show that the semiclassical approach provides simple and accurate formulas for the density of states and the quantum depletion of the condensate.