Bose-Einstein condensation of particles with spin

One of possible ground states and low-lying collective modes of Bose-Einstein condensate (BEC) of atoms with arbitrary spin in a magnetic field is studied using Bogoliubov's model for weakly interacting Bose gas. The equation for the vectorial order parameter, valid at temperatures T→0 , is der...

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Дата:	2007
Автори:	Peletminskii, A.S., Peletminskii, S.V.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
Назва видання:	Вопросы атомной науки и техники
Теми:	Physics of quantum liquids
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/111058
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Bose-Einstein condensation of particles with spin / A.S. Peletminskii, S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 413-417. — Бібліогр.: 22 назв. — англ.

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spelling	irk-123456789-1110582017-01-08T03:04:29Z Bose-Einstein condensation of particles with spin Peletminskii, A.S. Peletminskii, S.V. Physics of quantum liquids One of possible ground states and low-lying collective modes of Bose-Einstein condensate (BEC) of atoms with arbitrary spin in a magnetic field is studied using Bogoliubov's model for weakly interacting Bose gas. The equation for the vectorial order parameter, valid at temperatures T→0 , is derived and its specific solution is found. This solution corresponds to the formation of BEC of atoms with a definite spin projection onto magnetic field. We obtain also the necessary condition for thermodynamic stability of such a condensate and the explicit expressions for low-lying collective modes and magnetization. На основі моделі Боголюбова слабко неідеального бозе-газу вивчено один із можливих основних станів і колективні збудження бозе-ейнштейнівського конденсату (БЕК) атомів із довільним цілим спіном у магнітному полі. Отримано рівняння для векторного параметра порядку, справедливе при температурах T→0, та знайдено його частковий розв'язок. Цей розв'язок відповідає утворенню БЕК атомів із визначеною проекцією спіну на магнітне поле. Одержано також необхідну умову термодинамічної стійкості такого конденсату та вирази для спектрів елементарних збуджень і намагніченості. На основе модели Боголюбова слабовзаимодействующего бозе-газа изучены одно из возможных основных состояний и низколежащие коллективные возбуждения бозе-эйнштейновского конденсата (БЭК) атомов с произвольным целым спином в магнитном поле. Получено уравнение для векторного параметра порядка, справедливое при температурах T→0, и найдено его частное решение. Это решение соответствует образованию БЭК атомов с определенной проекцией спина на направление магнитного поля. Найдены также необходимое условие термодинамической устойчивости такого конденсата и выражения для спектров элементарных возбуждений и намагниченности. 2007 Article Bose-Einstein condensation of particles with spin / A.S. Peletminskii, S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 413-417. — Бібліогр.: 22 назв. — англ. 1562-6016 PACS: 12.20.-m, 13.40.-f, 13.60-Hb, 13.88.+e http://dspace.nbuv.gov.ua/handle/123456789/111058 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Physics of quantum liquids Physics of quantum liquids
spellingShingle	Physics of quantum liquids Physics of quantum liquids Peletminskii, A.S. Peletminskii, S.V. Bose-Einstein condensation of particles with spin Вопросы атомной науки и техники
description	One of possible ground states and low-lying collective modes of Bose-Einstein condensate (BEC) of atoms with arbitrary spin in a magnetic field is studied using Bogoliubov's model for weakly interacting Bose gas. The equation for the vectorial order parameter, valid at temperatures T→0 , is derived and its specific solution is found. This solution corresponds to the formation of BEC of atoms with a definite spin projection onto magnetic field. We obtain also the necessary condition for thermodynamic stability of such a condensate and the explicit expressions for low-lying collective modes and magnetization.
format	Article
author	Peletminskii, A.S. Peletminskii, S.V.
author_facet	Peletminskii, A.S. Peletminskii, S.V.
author_sort	Peletminskii, A.S.
title	Bose-Einstein condensation of particles with spin
title_short	Bose-Einstein condensation of particles with spin
title_full	Bose-Einstein condensation of particles with spin
title_fullStr	Bose-Einstein condensation of particles with spin
title_full_unstemmed	Bose-Einstein condensation of particles with spin
title_sort	bose-einstein condensation of particles with spin
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2007
topic_facet	Physics of quantum liquids
url	http://dspace.nbuv.gov.ua/handle/123456789/111058
citation_txt	Bose-Einstein condensation of particles with spin / A.S. Peletminskii, S.V. Peletminskii, Yu.V. Slyusarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 413-417. — Бібліогр.: 22 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT peletminskiias boseeinsteincondensationofparticleswithspin AT peletminskiisv boseeinsteincondensationofparticleswithspin
first_indexed	2025-07-08T01:33:37Z
last_indexed	2025-07-08T01:33:37Z
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fulltext	BOSE-EINSTEIN CONDENSATION OF PARTICLES WITH SPIN A.S. Peletminskii, S.V. Peletminskii, and Yu.V. Slyusarenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; e-mail: spelet@kipt.kharkov.ua One of possible ground states and low-lying collective modes of Bose-Einstein condensate (BEC) of atoms with arbitrary spin in a magnetic field is studied using Bogoliubov's model for weakly interacting Bose gas. The equation for the vectorial order parameter, valid at temperatures T , is derived and its specific solution is found. This solution corresponds to the formation of BEC of atoms with a definite spin projection onto magnetic field. We ob- tain also the necessary condition for thermodynamic stability of such a condensate and the explicit expressions for low-lying collective modes and magnetization. 0→ PACS: 12.20.-m, 13.40.-f, 13.60-Hb, 13.88.+e 1. INTRODUCTION After the first remarkable experiments concerning the observation of BEC in dilute gases of alkali atoms such as 87Rb [1], 23Na [2], and 7Li [3] the interest to this phenomenon has revived [4,5]. Later, BEC has been also obtained in other atomic species: atomic hydrogen [6], metastable 4He [7], and 41K [8]. The experimental observation of BEC has become possible due to devel- opment of laser cooling and trapping techniques [9]. The carried out experiments have proved many predic- tions of the micro-scopic theory for weakly interacting Bose gas, which originates from the pioneering work of Bogoliubov [10]. Bogoliubov's theory has become al- most the first theory in which it was necessary to move essentially from the methods of standard perturbative approach while describing the interaction effects. How- ever, this theory, in its original formulation, did not take into account the internal degrees of freedom of atoms. The spin degrees of freedom have been taken into ac- count for a weakly interacting Bose gas (spinor BEC) in [10-19]. The realization of optical trapping of atomic con- densate [20] has stimulated theoretical interest to spinor BEC. Bose condensation in a weakly interacting gas of bosonic atoms has been theoretically studied by many authors both for spin-1 [12-17] and spin-2 [18,19] bos- ons. These investigations are based on the effective in- teraction Hamiltonians of two bosons in which the in- teraction is characterized by a definite number of inter- action constants s-wave scattering lengths. The number of scattering lengths is determined by the total spin of two interacting bosons taking into account the symme- try properties of their wave function. For example, in case of spin-1 atoms the interaction Hamiltonian con- tains two interaction constants [12-17], in case of spin-2 atoms there are three interaction constants [18,19]. Thus, as the spin value of the atoms grows, the number of constants which characterize the interaction of two bosons is increased under phenomenological descrip- tion. Note that in the mentioned effective Hamiltonians it is difficult to interpret the physical nature of the iso- lated term of non-relativistic interaction not associated with neither potential nor spin-exchange interactions (see e.g. [18]). In this paper we study a weakly interacting Bose gas of particles with arbitrary integer spin S in a magnetic field (see also [11]). We start from the interaction Ham- iltonian for two spin-S bosons. This Hamiltonian is specified by two functions, which describe the potential and spin-exchange interactions of spin-S atoms. Ac- cording to general rules of quantum mechanics we pass from the pairwise interaction of two bosons to the stan- dard expression for binary interaction of arbitrary num- ber of bosons in the second quantization representation. By solving the problem of multichannel scattering for the considered Hamiltonian we could find, in principle, all scattering lengths in terms of the functions character- izing the potential and spin-exchange interactions. Thereby, it would be possible to obtain the Hamilto- nians analogous to the above mentioned effective inter- action Hamiltonians (see e.g. [18]). However, the Ham- iltonian of the present paper gives a possibility to re- strict ourselves by two interaction constants even in the case of arbitrary spin while studying the ground state, stability, and excitations in a weakly interacting gas in the presence of BEC. 2. SEPARATION OF A CONDENSATE To describe the system with a spontaneously broken symmetry we address to the method of quasiaverages [21,22]. According to this method the Gibbs statistical operator is modified so that it possesses the symmetry of the degenerate state. This modification is usually done by introducing the infinitesimal "source" ( 0 ), which has the symmetry of phase under con- sideration into the Gibbs exponent. Then the average value of any physical quantity A is defined as F̂ν →ν ˆ ,ˆˆlimlimˆ AwA V ν ∞→→ν =>< Tr 0 (1) where the Gibbs statistical operator w has the form νˆ ( )( )FNHw ˆˆˆexpˆ ν+µ−β−Ω= νν . (2) Here ,/T1=β are the reciprocal temperature and chemical potential respectively, and H are the sys- tem Hamiltonian and the particle number operator. The thermodynamic potential being a function of ther- modynamic variables is found from the µ N̂,ˆ νΩ µβ, PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 413-417. 413 normalization condition . Notice that the lim- its in (1) are not permutable. 1=νŵTr Consider a gas of condensed bosonic atoms with spin The origination of a condensate is accompanied by the gauge symmetry breaking and, therefore, in order to remove this kind of degeneracy we should choose the "source" in (2) such that [ (N is the generator of phase transformation), .S F̂ν 0≠ν ]ˆ,ˆ Nw ˆ ( )∫ + ααα ψ+ψν=ν )(ˆ)(ˆˆ xxxdF 3 + , where are the creation and annihilation operators with index taking values (the sum- mation over the repeated indices is assumed). Then that corresponds to the formation of a condensate of atoms with momenta . The quantity represents the order parameter usually called the condensate wave function (V is the volume of the system). )(ˆ),(ˆ xx αα ψψ α 21) / <=> −Vx 12 +S 1 <− 21 a/ 0 ~(ˆ >ψ< αα 0=p =α V â Ψ >α0ˆ Note that the method of quasiaverages and the spa- tial correlation decay principle allow to justify the re- placement of creation and annihilation operators with by c -numbers, 0=p ,ˆ,ˆ αα + αα ΨΨ→ VVaa 00 [21-22] (the condensate separation procedure). The basic statement of the method of quasiaverages applied to the description of BEC consists in the follow- ing [21-22]: the Gibbs statistical operator is replaced by ( )( ))(ˆ)(ˆ)(exp)(ˆ Ψµ−Ψβ−ΨΩ=Ψ NHw , (3) where is found from the following equation: },{ αα ΨΨ≡Ψ . )( 0= Ψ∂ ΨΩ∂ (4) → 3. THE GROUND STATE OF SPIN-S BEC In this Section we study the ground state properties of spin-S BEC in a magnetic field. In doing so, we start from the Hamiltonian which deter- mines the Gibbs statistical operator (3). This Hamilto- nian is given by , where ,ˆˆˆ NH µ−=H ep HH ˆˆ +0HH ˆˆ += [ ]∑ =ε−δµ−ε= βαβαβ + α 1 2 1110 2M p aa p,ˆ)(ˆˆ hSH , ( )∑ βα + β + α++δ−= 1234 4321432131 2 1 aaaaU V ˆˆˆˆˆ ,pH , (5) ( )∑ ργβραγ + β + α++δ−= 1234 4321432131 2 1 aaaaJ V ˆˆˆˆˆ , SSeH , where are the spin matrices, U ( p ) are the Fourier transforms of the am- plitudes of potential and spin-exchange interactions respectively, and , where g is the Bohr magneton, and is an external magnetic field. For our next calculations it is convenient to introduce the so- called ladder operators S . Then their non- zero matrix elements in the representation where S is a diagonal matrix, < , have the form αβS 1p − ),( 13p )( 13pJ 313 p= Sg /Hh = H yx SiS ˆˆˆ ±=± >=α′α \|ˆ\| zS ẑ α′ααδ ,)()( 11 +αα−+>= SS .)()( 11 −αα−+>= SS 0=p α0̂ ˆ =w Ψµ ζ ˆ \|ˆ\|1 α+α< +S \|ˆ\|1 α−α< −S (6) Now we separate the components a in the Hamiltonian (the replacement of a by c -numbers, α0̂ αα Ψ→ Va0ˆ αpˆ αΨ ) and keep the terms only up to second order in a . We omit the higher order terms, since they should be taken into account only when examining the interaction between quasiparticles, which we will introduce in the next section. As a result the Hamil- tonian takes the form . The explicit expression for which contains only c -numbers reads (2)H )0 Ĥ+Hˆ (≈ ,)(0H ( ) ( ) ( ) 22* 0 )ˆ( 2 0)( 2 0 ΨΨ+ΨΨ= SJU V H ,ˆ ΨΨµ−ΨΨ− Sh (7) where . (8) ˆˆ, βαβααα ΨΨ=ΨΨΨΨ=ΨΨ SS The explicit form for will be written in the next section. )(ˆ 2H Next, using the normalization condition Tr we immediately find the density of thermodynamic po- tential in the main approximation ( 0T ) of the model for weakly interacting Bose gas, ,1 VT /Ω=ω ( ) ( ) .ˆ)ˆ()( * Ψ−ΨΨ−ΨΨ+ΨΨ=ω ShS 22 2 0 2 0 JU αΨTherefore, Eq.(5) for takes the form ( ) ααµ ΨΨ−Ψ )(0 U Ψ (9) ( ) 00 =Ψ+ΨΨΨ− βαββαβ hSSS )ˆ( J . If to introduce the normalized spin functions ,α ,αα ζ=Ψ n * ζζ αα where is the condensate den- sity and then the latter equation is written as ααΨΨ= n ,1= ( ) ( ) .0)ˆ(0 0 =+− − βαββαβ αα ζζζζ ζµζ hSSSnJ nU (10) Its solution , which is an eigenfunction of S is of the form )(m αζ (mζ= ,z ,)ˆ( ))( mm zS αβαβζ .)( m m αα δ=ζ (11) Assuming the vector directed along z -axis, and taking into account that S is a diago- nal matrix, whereas have no diagonal matrix ele- ments in the considered representation of spin matrices, one finds from Eq. (3.8) h ),,( h00=h ẑ ±Ŝ ( ) ( ) . 00 2JmU mh n + +µ = (12) 414 Formulae (11), (12) result in the following expression for the density of equilibrium thermodynamic potential: ( ) ( ) ( ) . 002 1 2 2 JmU mh + +µ −=ω (13) We are now in a position to study the stability of the possible ground states (11). In the considered approxi- mation the thermodynamic potential of normal state is zero (the order parameter vanishes). Therefore, for stability of the studied ground state the density of ther- modynamic potential must be negative, and, consequently, according to (13), we can write the nec- essary condition of thermodynamic stability, αΨ 0<ω ( ) ( ) .000 2 >+ JmU (14) = JH Let us find now such spin projections m , which corre- spond to minimum of the potential (13). For simplicity, we study the case of h (or sufficiently weak h ). Then 0= ( ) ( ) . )( 0 002 2 2 < + µ −=ω JmU (15) As it can be easily seen that in contrast to the usual Bo- goliubov theory, where U (the necessary condi- tion of stability), the case U is also permissible. Therefore, we have the following three situations: 00 >)( 0 <)( 0 1) U 0 . In this case the requirement (14) is automatically fulfilled. The density of thermody- namic potential (15) has a minimum at m in which We call this case as antifer- romagnetic ordering. ,)( 00 > (/ 022 Uµ 0 >)(J ). ,0= −=ω 2) U 0J but such that the requirement (14) is fulfilled. The minimum of ω (see (15)) is reached for where ,)( 00 > minm 0 <)( (±= cm ),1− [ ] 1,)0(/)0( c 2/1 +≤−= SmJUmc (16) 2 (the square brackets denote an integer part). This case corresponds to ferromagnetic ordering. 3) U 0 but again such that Here the minimum of is given by the spin projections m , where m is also defined by (16) but with U . This case also corresponds to ferromagnetic ordering. ,)( 00 < ( )02 >+ Jm 0 >)(J .0 min ( )0U ω )( 1+±= cm c ,)( 00 < αp̂ 00 >)(J ≠ [ ].)ˆ +p h.caˆ()ˆ ΨΨ+ p SS )ˆˆ( −+ − SS ± ˆ ,ˆ α + α pa ++ − mp [ ],−1ˆˆ + − + +1 mm aa pp +mp [ ],ˆ 11 + + + mm ap 4. LOW-LYING COLLECTIVE MODES Here we shall obtain the excitation spectra for spin- BEC employing the well-known diagonalization procedure (Bogoliubov’s transformations [10]) for the Hamiltonian quadratic in creation and annihila- tion operators. S −u v The part of the spin-exchange interaction Hamilto- nian (see (5)), which is quadratic in a ( 0p ) has the form +ΨΨ ∑ + p ppSS aa ˆˆˆˆ)(ˆ 0(2) e (17) ( ) ,ˆ()ˆˆ()ˆˆ(∑ +ΨΨ+ + − + p pp SSp aaaJ 2 1 where we have used the notations (8). Taking into ac- count that )ˆˆ(ˆ −+ += SSSx 2 1 , ˆ −= i y 2 S and employing (6) for nonzero matrix elements of S as well as the explicit expression for the condensate wave function ,)( m m n αα δ=Ψ one finds ˆ)ˆˆˆ()ˆ( * ∑ α + α=ΨΨ pppSS anmaa =ΨΨ ++ − + maanmaa ppp SS ˆˆ)ˆˆ)(ˆˆ( 2 ˆˆ + +− + −− ++ 112 mmmm aaSS n pp =ΨΨ ++ maanmaa ppp SS ˆˆ)ˆˆ)(ˆˆ( * 2 ˆˆˆ 2 11 2 − + −− ++ mmmm aSaaS n ppp where ( ) ( )11 +−+= mmSSSm . As a result the "Hamiltonian" (17) takes the form ( ) ( )[ ]++++−= ∑ + + + + − + − p pppppp 1111 110 mmmmmm aamaamaamnmJ ˆˆˆˆˆˆ)(ˆ (2) eH [ ]++++α+ ∑ ∑ α − ++ − + α + α , ˆˆˆˆˆˆ)(ˆˆ)( p pppppp p pp p mmmmmm aaaaaamJ n aanmJ 2 2 0 2 ( )[ ]ˆˆˆˆˆˆˆˆ)( 11 2 11 2 11112 + + +− + −−−+− + +− + −− ++++ ∑ mmmmmmmmmmmm aaSaaSaaaaSSJ n pppppppp p p . Here the summation index in the second term takes all the values of spin projections except m , m , and . We have separated these three spin projections and written them as the first term of the above "Hamil- tonian". In a similar manner assuming the vector directed along -axis, , and eliminating the chemical potential α ,(0 1− 1+m h z ),h0=h µ by using (12), we can find the explicit expressions for and thereby the total Hamiltonian quadratic in creation and annihilation operators: )2() ˆ, pHH (ˆ 2 0 ),(ˆ)(ˆˆˆ )()()()( 112222 +−++= α mmm HHHH , (18) where ( ) ( )[ ] ,ˆˆ)(ˆ , )( ∑ α α + αα α−+α−−ε= p pp aamhmnmJp 02H ( ) [ ] ++ε= ∑ + p ppp mmmp aagm ˆˆ)(ˆ )(2H [ ]∑ − + − + ++ p ppppp mmmmm aaaag ˆˆˆˆ)( 2 1 , (19) 415 [ ]∑ + + +β+−ε=+− p ppp 11 2 11 mmmp aahmm ˆˆ)(),(ˆ )(H +[ ]∑ −−−β++ε+ p ppp 11 mmmp aah ˆˆ)( [ ].)( 1111∑ +−− + +− + − ++ p ppppp mmmmm aaaaα The introduced quantities α , )(pm )(pmβ , are given by )(pmg mmm SSJ n −=α )()( pp 2 , mnJSJ n mm )()()( 0 2 2 +=β pp , (20) ( ) γω ( ).)()()( 2 ppp JmUngm += Now we are in a position to carry out the diagonali- zation of the total Hamiltonian (18). In this connection let us note that the "Hamiltonians" (19) contain the crea- tion and annihilation operators with not overlapping sets of indices , , , (α ). Therefore, we can perform their diagonalization inde- pendently. The evidence of this statement is also associ- ated with the fact that (18) can be considered as the Hamiltonian of the system consisting of four kinds (α ) of noninteracting particles. α 1 1−m m 1+m mm ,1±≠ ±mm,, The "Hamiltonian" has already a diagonal form with the following spectrum: )(ˆ 2 αH ( ) ( ) ( .)(, α−+α−−ε=ω α mhmnmJpm 0p ) 1 (21) To carry out the diagonalization of we introduce the creation and anni- hilation operators b ( ), ),(ˆ )( 112 +− mmH p ˆ σ+m ±=σ ( ) ( ) ,ˆˆˆ ,, + σ−−σσ+σσ+ += mmmmm bvbua ppp pp ( ) ( ) σ−−σ + σ+σ + σ+ += mmmmm bvbua ppp pp ˆˆˆ * , * , in terms of which this "Hamiltonian" has the diagonal form, ( ) ,)( 0 , , )2( Ebbm mmm +=+ + + +∑ σσ σ σωσ pp p pH where and E are the excitation spectra and the ground state energy respectively. In order that the introduced operators meet the canonical commutation relations, the functions u , must obey the relationships: )(, pσωm 0 )(, pσm )(, pσmv 122 =− σσ \|)(\|\|)(\| ,, pp mm vu , ( ) ( ) ( ) ( ) 0=−−− σ−σσ−σ pppp ,,,, mmmm uvvu . Simple mathematical manipulations (see e.g. [22]) result to the following expression for low-lying collective modes: ±σ−      −σ=ω σ hJJnmm )()()(, pp 2 10 (22) , )( ))()(( 2 1 2 22 2 1              +−+ε+ε± mnJ mSSnJpp p p moreover, ( ) ( ) ( ) ( )( ) , 2 ,, 2 , σσ σ γωα α mmm m mu −− = pp p p ( ) ( ) ( ) ( )( )22 σσ σσ σ γ−ω−α γ−ω = ,, ,, , mmm mm mv pp p p , where ( ) hmpm σ−β+ε=γ σσ p, (23) (here )(pσβm ,mu depends on the product m ). In fact, the functions , do not depend on be- cause, as it can be easily shown, the quantity σ )(pσ σ )(, pσmv σ σ − ,mp,m is independent of . The sign plus before the square root in (22) corresponds (for σ ) to the wave, which propagates in one direction, whereas the sign minus corresponds (for ) to the wave propagating in opposite direction. Notice that the ob- tained spectra as well as (21) contain only the amplitude of spin-exchange interaction and does not depend on the amplitude of potential interaction. σ 1− 1= =σ When (the antiferromagnetic ordering), the excitation spectrum (22) takes the form 0=m ( ) hSSnJpp ±+ε+ε=ω 12 )()( pp . In this case for h and p we have 0= 0→ cpp =ω )( , ( )10 2 += SSJ M n )(c . In ferromagnetic case (when m ) the excitation spectrum is of the form S= hSnJ SnJ p σ−σ+σ−+ε=ω )()( )( )( 01 2 p p . In a similar manner we can perform the diagonaliza- tion of and thereby to obtain another spec- trum of excitations. This spectrum depends on the am- plitudes of spin-exchange and potential interactions, ( )m)(ˆ 2H ))()(()( ppp JmUnppm 22 2 +ε+ε=ω . (24) The corresponding functions u and v can be eas- ily found and have the form, pm pm )( )( )( p p p mp mp mu ωε ω+ε = 2 , )( )( )( p p p mp pm m ωε ε−ω = 2 v . When the excitation spectrum (24) coincides with those, obtained Bogoliubov [10]. At small the spectrum has the phonon behavior, 0=)(pJ p ( ) ,cpm =ω p ))()(( 00 2JmU M n +=c , In this formula and in (24) we have chosen the arithme- tic value of the square root. The magnetization is defined as , where is the Bohr magneton, is the spin operator in the second quantization representation and the Gibbs statistical operator w is given by (3). Up to the second order in a this magnetization is of the form SM ˆ)(ˆ Ψ= wg Tr g ˆ Ŝ )(ˆ Ψ αp ∑ ≠ ++= 0 22 )()\|)(\|\|)((\| p pppM mmm fvugmgVm ))()(()\|)(\|\|)((\| 11 0 21,21, pppp p −+ ≠ +++ ∑ mmmm ffvugm 416 ∑ ∑∑ ≠ ±≠≠ −+ +++ 0 1,0 11 )())()(( pp ppp mm mm fgffg α αα , where we have taken into account the fact that the func- tions , do not depend on , , . The fun- ctions , , , represent the boson distribution functions of quasiparticles with chemi- cal potential µ and excitation spectra , , , respectively (see (24), (22), (21)). )(, pσmu )p 1+= mv )(pmf f )p ,−ωm )(, pσmv )p −mu )(p1+ mf 0 )p ,αωm σ )p ((1−mv m = (,1mω (1 )()( pp 11 += mu )(p1− )(pαf )(p (mω In conclusion, we have studied BEC of atoms with ar- bitrary spin in a magnetic field on the basis of the model for weakly interacting Bose gas. We have derived the equation, which determines the ground state of spin-S BEC at T and found its specific solution. This solu- tion corresponds to the formation of BEC of spin-S atoms with a definite spin projection m that also holds for an ideal Bose gas [11]. The explicit expression for thermody- namic potential as a function of chemical potential and spin projection has been obtained. It generalizes the ther- modynamic potential for weakly interacting Bose gas to the case when both potential and spin-exchange interac- tions act between bosons. The thermodynamic stability of the obtained ground state has been studied and the spin projections which give a minimum of thermodynamic potential have been found. These projections are deter- mined by the integral part of the ratio of the potential and spin-exchange interaction amplitudes. The expressions for low-lying collective modes corresponding to the ground state (11) as well as the magnetization have been obtained. Notice that Eq. (10) for the order parameter has also other solutions different from (11). The goal of our present re- search is to seek such solutions. 0→ REFERENCES 1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell //Science. 1995, v. 269, p. 198-201. 2. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle //Phys. Rev. Lett. 1995, v. 75, p. 3969-3973. 3. C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet //Phys. Rev. Lett. 1995, v. 75, p. 1687-1690. 4. C.J. Pethick, H. Smith. Bose-Einstein condensation in dilute gases. Cambridge: "Cambridge University Press", 2002, 402 p. 5. L. Pitaevskii, S. Stringari. Bose-Einstein condensation. New York: "Oxford University Press", 2003, 382 p. 6. D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, T.J. Greytak //Phys. Rev. Lett. 1998, v. 81, p. 3811-3814. 7. F. Pereira dos Santos, J. Lonard, J. Wang, C.J. Barrelet, F. Perales, E. Rasel, C.S. Unikrishnan, M. Leduc, C. Cohen-Tannoudji //Phys. Rev. Lett. 2001, v. 86, p. 3459-3462. 8. G. Modugno, G. Ferrari, G. Roati, R.J. Brecha, A. Simoni, M. Inguscio //Science. 2001, v. 294, p. 1320-1322. 9. S. Chu // Usp. Fiz. Nauk. 1999, v. 169, p. 274-291; C.N. Cohen-Tannoudji //Usp. Fiz. Nauk. 1999, v. 169, p. 292-304; W.D. Phillips // Usp. Fiz. Nauk. 1999, v. 169, p. 305-322 (in Russian). 10. N.N. Bogoliubov //J. Phys. (USSR). 1947, v. 11, p. 23- 32. 11. A.I. Akhiezer, S.V. Peletminskii, Yu.V. Slyusarenko //JETP. 1998, v. 86, p. 501-506. 12. T. Ohmi, K. Machida //J. Phys. Soc. Japan. 1998, v. 67, p. 1822-1825. 13. T.-L. Ho //Phys. Rev. Lett. 1998, v. 81, p. 742-745. 14. M. Ueda //Phys. Rev. A. 2000, v. 63, p. 013601-1- 013601-4. 15. C.K. Law, H. Pu, N.P. Bigelow //Phys. Rev. Lett. 1998, v. 81, p. 5257-5261. 16. M. Koashi, M. Ueda //Phys. Rev. Lett. 2000, v. 84, p. 1066-1069. 17. T.-L. Ho, S.-K. Yip //Phys. Rev. Lett. 2000, v. 84, p. 4031-4034. 18. M. Ueda, M. Koashi //Phys. Rev. A. 2002, v. 65, p. 063602-1-063602-22. 19. J.-P. Martikainen, K.-A. Suominen //J. Phys. B: At. Mol. Opt. Phys. 2001, v. 34, p. 4091-4101. 20. D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, W. Ketterle //Phys. Rev. Lett. 1998, v. 80, p. 2027-2030. 21. N.N. Bogoliubov. Lectures on quantum statistics, vol. 2. Quasiaverages. New York: "Gordon and Breach", 1970, 231 p. 22. A.I. Akhiezer, S.V. Peletminskii. Methods of statistical physics. Oxford: "Pergamon Press", 1981, 450 p. БОЗЕ-ЭЙНШТЕЙНОВСКАЯ КОНДЕНСАЦИЯ ЧАСТИЦ СО СПИНОМ А.С. Пелетминский, С.В. Пелетминский, Ю.В. Слюсаренко На основе модели Боголюбова слабовзаимодействующего бозе-газа изучены одно из возможных основных состоя- ний и низколежащие коллективные возбуждения бозе-эйнштейновского конденсата (БЭК) атомов с произвольным целым спином в магнитном поле. Получено уравнение для векторного параметра порядка, справедливое при темпера- турах T , и найдено его частное решение. Это решение соответствует образованию БЭК атомов с определенной проекцией спина на направление магнитного поля. Найдены также необходимое условие термодинамической устойчи- вости такого конденсата и выражения для спектров элементарных возбуждений и намагниченности. 0→ БОЗЕ-ЕЙНШТЕЙНІВСЬКА КОНДЕНСАЦІЯ ЧАСТИНОК ЗІ СПІНОМ О.С. Пелетмінський, С.В. Пелетмінський, Ю.В. Слюсаренко На основі моделі Боголюбова слабко неідеального бозе-газу вивчено один із можливих основних станів і колективні збудження бозе-ейнштейнівського конденсату (БЕК) атомів із довільним цілим спіном у магнітному полі. Отримано рів- няння для векторного параметра порядку, справедливе при температурах T , та знайдено його частковий розв'язок. Цей розв'язок відповідає утворенню БЕК атомів із визначеною проекцією спіну на магнітне поле. Одержано також необ- хідну умову термодинамічної стійкості такого конденсату та вирази для спектрів елементарних збуджень і намагніченості. 0→ 417

Bose-Einstein condensation of particles with spin

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