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Acta Physica Polonica A

Soluble Model of Interacting Bosons Trapped in Harmonic Potential: Quality of Bogoliubov Approximation

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

Published Oct 1, 2001• 1 citation
Publisher

Abstract

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

DOI
10.12693/aphyspola.100.485

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In treating system of bosons localized in a trapping potential, having a macroscopic number N0 of them condensing at the lowest single-particle level v0, Bogoliubov approximation is to replace the creation/annihilation operators [Formula: see text] of the state v0 by [Formula: see text]. We show that this approximation is justified if the inter-particle potential is repulsive in the sense specified. In fact, we show, by using [Formula: see text], that [Formula: see text] is effectively of the order [Formula: see text] under the condition stated.

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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
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.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with FFF the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of FFF and in this range the deviation of WWW from the Bogoliubov value is small. There is, however, a transition point FT{F}_{T}FT​, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT{F}_{T}FT​. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
cs, Po l i sh Ac ademy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and b Center for Theo reti cal Physi cs, Po l ish Aca demy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and ( Recei ved Novem ber 15, 2001)W e rev iew a pr ogress i n un derstandi ng of statisti cal prop erties of a quantu m degenerate Bose gas.W e show how the Ma xw ell demon ensem ble helps to compute Ûuctuations of the Bose{ Eins tein condensate of an ideal Bose gas according to the micro canoni cal ensemble.T hen, we review a metho d of measuring these Ûuctuations.U sing a solubl e mo del of interacting Bose gas w e also stress the imp ortance of higher-order correlation functions.Finall y, w e review our no vel computational metho d of studying an interacting Bose gas near its critical temp erature.
ct iv it y to B ose C on d en sation R .M ic na s In st i tute of P hysi cs, A. Mi cki ewi cz Uni versi ty Um ultowsk a 85, 61-61 4 Po zna¥, Pol an d ( Recei ved December 17, 2001)W e out li ne a mi cro scopic approac h to the sup ercon ducti ng Ûuctuation s and pairing correlations in the attracti ve H ubbard mo del ab ove T c , using the functional integral metho d.A crosso ver from BC S sup erconducti vit y to Bose condensatio n of pref ormed pairs is studied by constructing the appropriate Gin zburg { La ndau functional s.T he di ˜erences bet ween the lattice and the continuum mo dels are discussed.T he case of quasi -tw o-dimensional sup erconductors as w ell as the mo del with non-local pairin g interaction are also examined.T he e˜ects of Gaussian Ûuctuation s of the order parameter are analyzed in the T -matrix approach, the self-consistent H artree approach and Ùnally w ithin the Ginzburg{Lan dau theory .Similari tie s with a paramagnon theory of itineran t-electron magnetism are p ointed out.
In treating system of bosons localized in a trapping potential, having a macroscopic number N0 of them condensing at the lowest single-particle level v0, Bogoliubov approximation is to replace the creation/annihilation operators [Formula: see text] of the state v0 by [Formula: see text]. We show that this approximation is justified if the inter-particle potential is repulsive in the sense specified. In fact, we show, by using [Formula: see text], that [Formula: see text] is effectively of the order [Formula: see text] under the condition stated.