Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction

We study a superfluid Bose system with single-particle and pair condensates on the basis of a half-phenomenological theory of a Bose liquid not involving the weakness of interparticle interaction. The coupled equations describing the equilibrium state of such system are derived from the variational...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2013
Hauptverfasser:	Peletminskii, A.S., Peletminskii, S.V., Poluektov, Yu.M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2013
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/120798
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction / A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13603: 1–17. — Бібліогр.: 65 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-120798
record_format	dspace
spelling	irk-123456789-1207982017-06-14T03:05:09Z Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction Peletminskii, A.S. Peletminskii, S.V. Poluektov, Yu.M. We study a superfluid Bose system with single-particle and pair condensates on the basis of a half-phenomenological theory of a Bose liquid not involving the weakness of interparticle interaction. The coupled equations describing the equilibrium state of such system are derived from the variational principle for entropy. These equations are analyzed at zero temperature both analytically and numerically. It is shown that the fraction of particles in the single-particle and pair condensates essentially depends on the total density of the system. At densities attainable in condensates of alkali-metal atoms, almost all particles are in the single-particle condensate. The pair condensate fraction grows with increasing total density and becomes dominant. It is shown that at density of liquid helium, the single-particle condensate fraction is less than 10% that agrees with experimental data on inelastic neutron scattering, Monte Carlo calculations and other theoretical predictions. The ground state energy, pressure, and compressibility are found for the system under consideration. The spectrum of single-particle excitations is also analyzed. Вивчається надплинна бозе-система з одночастинковим та парним конденсатами на основi напiвфеноменологiчної теорiї бозе-рiдини, яка не припускає слабкостi мiжчастинкової взаємодiї. Iз варiацiйного принципу для ентропiї виведено систему рiвнянь, що описують рiвноважний стан такої системи. Цi рiвняння проаналiзовано у випадку нульової температури як аналiтично, так i чисельно. Показано, що частки одночастинкового та парного конденсатiв суттєво залежать вiд повної густини системи. При густинах, що досягаються у конденсатах атомiв лужних металiв, майже всi частинки знаходяться в одночастинковому конденсатi. Доведено, що при густинi рiдкого гелiю частка одночастинкового конденсату є меншою за 10%, що узгоджується з експериментальними даними з непружного нейтронного розсiювання, розрахунками методом Монте-Карло та iншими теоретичними передбаченнями. Знайдено вирази для енергiї основного стану, тиску, стисливостi системи, що вивчається. Проаналiзовано також спектр одночастин-кових збуджень. 2013 Article Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction / A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13603: 1–17. — Бібліогр.: 65 назв. — англ. 1607-324X PACS: 67.10.Fj, 67.25.D, 67.85.Jk DOI:10.5488/CMP.16.13603 arXiv:1303.5539 http://dspace.nbuv.gov.ua/handle/123456789/120798 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study a superfluid Bose system with single-particle and pair condensates on the basis of a half-phenomenological theory of a Bose liquid not involving the weakness of interparticle interaction. The coupled equations describing the equilibrium state of such system are derived from the variational principle for entropy. These equations are analyzed at zero temperature both analytically and numerically. It is shown that the fraction of particles in the single-particle and pair condensates essentially depends on the total density of the system. At densities attainable in condensates of alkali-metal atoms, almost all particles are in the single-particle condensate. The pair condensate fraction grows with increasing total density and becomes dominant. It is shown that at density of liquid helium, the single-particle condensate fraction is less than 10% that agrees with experimental data on inelastic neutron scattering, Monte Carlo calculations and other theoretical predictions. The ground state energy, pressure, and compressibility are found for the system under consideration. The spectrum of single-particle excitations is also analyzed.
format	Article
author	Peletminskii, A.S. Peletminskii, S.V. Poluektov, Yu.M.
spellingShingle	Peletminskii, A.S. Peletminskii, S.V. Poluektov, Yu.M. Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction Condensed Matter Physics
author_facet	Peletminskii, A.S. Peletminskii, S.V. Poluektov, Yu.M.
author_sort	Peletminskii, A.S.
title	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction
title_short	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction
title_full	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction
title_fullStr	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction
title_full_unstemmed	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction
title_sort	role of single-particle and pair condensates in bose systems with arbitrary intensity of interaction
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/120798
citation_txt	Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction / A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13603: 1–17. — Бібліогр.: 65 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT peletminskiias roleofsingleparticleandpaircondensatesinbosesystemswitharbitraryintensityofinteraction AT peletminskiisv roleofsingleparticleandpaircondensatesinbosesystemswitharbitraryintensityofinteraction AT poluektovyum roleofsingleparticleandpaircondensatesinbosesystemswitharbitraryintensityofinteraction
first_indexed	2025-07-08T18:35:41Z
last_indexed	2025-07-08T18:35:41Z
_version_	1837104882051448832
fulltext	Condensed Matter Physics, 2013, Vol. 16, No 1, 13603: 1–17 DOI: 10.5488/CMP.16.13603 http://www.icmp.lviv.ua/journal Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov Akhiezer Institute for Theoretical Physics, National Science Center Kharkiv Institute of Physics and Technology, 1 Akademichna St., 61108 Kharkiv, Ukraine Received April 27, 2012, in final form August 13, 2012 We study a superfluid Bose system with single-particle and pair condensates on the basis of a half-phenome- nological theory of a Bose liquid not involving the weakness of interparticle interaction. The coupled equations describing the equilibrium state of such system are derived from the variational principle for entropy. These equations are analyzed at zero temperature both analytically and numerically. It is shown that the fraction of particles in the single-particle and pair condensates essentially depends on the total density of the system. At densities attainable in condensates of alkali-metal atoms, almost all particles are in the single-particle conden- sate. The pair condensate fraction grows with an increasing total density and becomes dominant. It is shown that at density of liquid helium, the single-particle condensate fraction is less than 10%, which agrees with ex- perimental data on inelastic neutron scattering, Monte Carlo calculations and other theoretical predictions. The ground state energy, pressure, and compressibility are found for the system under consideration. The spectrum of single-particle excitations is also analyzed. Key words: superfluidity, Bose-Einstein condensation, single-particle and pair condensates, quasiparticles, excitation spectrum PACS: 67.10.Fj, 67.25.D, 67.85.Jk 1. Introduction The first experimental observations of Bose-Einstein condensation in dilute gases of alkali-metal atoms [1–3] have stimulated a great interest to this remarkable phenomenon manifested also in super- fluids and superconductors. However, in spite of a significant progress [4–8] in the study of Bose systems with condensate, their theory is far from being completed. Among the open theoretical problems wemen- tion the following ones: description of Bose systems with strong interaction, microscopic justification of observable excitation spectrum in a superfluid 4He, role of single-particle and pair condensates in the phenomenon of superfluidity, the existence of elementary excitations with activation energy along with sound excitations. As was first showed by Bogolyubov [9], the interparticle interaction essentially affects the behavior of a many-body Bose system at low temperature. In particular, in consequence of a weak interaction, the number of particles in a condensate at zero temperature is not equal to the total particle number in the system. This effect, usually referred to as depletion of a condensate, is caused by the presence of pair anomalous averages which are similar to Cooper correlations in superconductors. The pair corre- lated bosonic atoms form a pair condensate which, along with a single-particle condensate, specifies the superfluid density. The role of pair correlations in superfluid Bose systems has been studied by many au- thors [10–19]. Superfluidity has also been treated in terms of pair condensation only, in the total absence of a single-particle condensate [20–22]. Note that accounting for pair correlations in a Bose system can produce a gap in the spectrum of el- ementary excitations [11–14]. The existence of a gap essentially depends on the used approximation or truncated Hamiltonian [5, 8, 23, 24]. In particular, the well-known Hartree-Fock-Bogolyubov approxima- tion extended to bosons generates a gap in the spectrum. In virtue of the Hugenholtz-Pines theorem [25], © A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov, 2013 13603-1 http://dx.doi.org/10.5488/CMP.16.13603 http://www.icmp.lviv.ua/journal A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov according to which the spectrum should be gapless, this fact is usually considered as a defect of the men- tioned approximation. Therefore, some efforts have been done to reformulate the Hartree-Fock-Bogolyu- bov theory in order to remove a gap in the single-particle excitation spectrum (see [26] and references therein). However, such schemes involve additional assumptions or parameters which does not alow one to treat them as a rigorous solution of the problem. Note that Pines [27] has pointed out that the correctness of their theorem depends on the validity of expansions into a series of perturbation theory. Moreover, the Hugenholtz-Pines theorem cannot be applied to all models because it is valid for a specific truncated Hamiltonian used by the authors. The subsequent analysis of Bogolyubov’s 1/k2-theorem has showed that no general conclusion can be made concerning the excitation spectrum of a superfluid and long-wavelength density excitations are insensitive to U (1) symmetry breaking [28]. As was also stressed by Bogolyubov and Bogolyubov (jr.), the presence of correlated pairs in a Bose gas of interacting parti- cles leads to the fact that the spectrum consists of two branches — the branch with activation energy (or with a gap) and a phonon mode [29]. The similar form of spectrum for a superfluid 4He was also pro- posed within a phenomenological model [30]. Perhaps, one may consider that the existence of a gap in the spectrum of single-particle excitations does not contradict the general physical principles. Moreover, the possible existence of elementary excitations with activation energy along with sound excitations re- quires further theoretical and experimental investigations. The existence of a gap in the single-particle excitation spectrum has been recently discussed in terms of variational wave functions [31] and with- out the involvement of a conjecture of c-number representation of creation and annihilation operators with zero momentum [32]. Note that the c-number representation results in a reduction of Fock space, though one can prove that it correctly reproduces the pressure [33] and condensate density [34, 35] in thermodynamic limit. In a recent study [36], it has been shown that the restoration of the state with zero momentum to the Hilbert spacewith subsequent exact numerical diagonalization of the total single-mode Hamiltonian yields the excitation spectrum with a finite gap. The description of many-body systems that does not require the weakness of interparticle interaction can be formulated using the quasiparticle concept. The well-known example of such a description is the Fermi liquid theory proposed by Landau [37] and Silin [38]. This theory has been extended to the study of various superfluid states of Fermi systems by introducing pair anomalous averages [39–41]. The main advantage of the extended half-phenomenological approach is that it is valid for an arbitrary energy functional and does not restrict the intensity of interaction. It can be applied to a wide range of systems, including strongly interacting superfluid nuclear matter [42–44]. On microscopic level, such half-phenomenological description is equivalent to a self-consistent mean field theory [45]. Subsequently, the ideas of the extended Fermi liquid approach have also been disseminated to a Bose superfluid [41, 46] with a corresponding microscopic justification [14]. Note that the self-consistent mean field model is an effective zeroth-order approximation of quantum-field perturbative theory [47]. In the present paper, we study a superfluid Bose system with single-particle and pair condensates without any restriction on intensity of interaction and conjecture of c-number representation of creation and annihilation operators. To this end, we employ the formalism [41, 46] developed by us earlier. From the principle of maximum entropy, we present a brief derivation of the coupled equations describing the system at finite temperature. The obtained equations are analyzed in detail both analytically and numerically in case of zero temperature and contact interaction. It is shown that for systems of low density (dilute gases of alkali-metal atoms), the pair condensate fraction is negligibly small in comparison with the fraction of single-particle condensate. Such systems, as expected, arewell described by the Gross- Pitaevskii equation [4, 6, 7]. The role of pair condensate becomes determinative for dense systems. In particular, it is found that the single-particle condensate fraction in a superfluid 4He is less than 10% at zero temperature. This result is in a good agreement with experiments [48–51] on inelastic neutron scattering, with Monte Carlo calculations [52] and with theoretical approach that involves the calculation of single-particle density matrix expressed through the structure factor [53–55]. We also calculate the ground state energy, pressure, speed of sound for a system with two condensates. The single-particle excitation spectrum is analyzed. 13603-2 Single-particle and pair condensates in Bose systems 2. Formalism and basic equations Consider the basic principles that underlie the theory of superfluid Bose systems with single-particle and pair condensates (for details see the reviews [41, 46]). This theory is formulated in close analogy with extended Fermi-liquid approach to superfluids [39–41]. The state of a superfluid Bose system is specified by the condensate amplitudes, bp = Spρap , b∗ p = Spρa† p , (2.1) as well as by normal and anomalous single-particle density matrices, fpp′ = Spρa† p′ ap , gpp′ = Spρap′ ap , g † pp′ = Spρa† p′ a † p , (2.2) where creation and annihilation operators a† p, ap satisfy the usual Bose commutation relations and ρ is a statistical operator of the system which we define below by equation (2.3). Note that fpp′ = f ∗ p′p and gpp′ = gp′p. The condensate amplitudes (2.1) describe a single-particle condensate, while anomalous av- erages (2.2) characterize the pair correlations between particles and indicate the existence of a pair con- densate. We will approximate the statistical operator ρ by operator that contains a general quadratic form of ap, a† p as well as linear terms in creation and annihilation operators, ρ = exp(Z −F ), F = a† Aa + 1 2 ( aB a +a†B∗a† ) +a†C +C∗a. (2.3) Here, we have omitted the repeated summation indices bearing in mind that, e.g., a† Aa ≡ a† p App′ap′ or a†C ≡ a† pCp and Z is found from the normalization condition, Spρ = 1. The quantities App′ , Bpp′ , Cp are related to fpp′ , gpp′ , bp by equations (2.2), (2.1). The terms linear in creation and annihilation operators that stand in the exponent of ρ can be removed by the unitary transformation of a c-number shift (for details see references [41, 46, 56]): UapU † = ap +bp , Ua† pU † = a† p +bp . (2.4) Then, the statistical operator ̺ = UρU † will include only quadratic terms in a† p, ap as well as matrices App′ , Bpp′ , ̺= exp [ Z̃ −a† Aa − 1 2 ( aB a +a†B∗a† ) ] , Z̃ = Z +b∗Ab + 1 2 ( bBb +b∗B∗b∗) . (2.5) For this statistical operator, we have Sp̺ap = 0. Therefore, according to equations (2.4), one obtains Spρap = Sp̺(ap +bp) = bp , 0. Using the unitary transformation (2.4) and equations (2.2), it is easy to introduce the correlation functions f c pp′ and g c pp′ : fpp′ = Spρa† p′ ap = b∗ p′bp + f c pp′ , gpp′ = Spρap′ ap = bp′bp + g c pp′ , (2.6) where f c pp′ = Sp̺a† p′ ap , g c pp′ = Sp̺ap′ap . (2.7) Fromequations (2.7), which relate thematrices App′ ,Bpp′ to f c pp′ , g c pp′ , it follows [56] that ̺= ̺( f c , g c , g c†). Therefore, the entropy of a Bose system S =−Spρ lnρ =−Sp̺ ln̺ is a functional of correlation functions only, S = S( f c , g c , g c†). Moreover, it can be shown [41, 46] that the matrices App′ and Bpp′ represent the derivatives of entropy with respect to correlation functions: ∂S ∂ f c p′p = App′ , ∂S ∂g c p′p = 1 2 Bpp′ , ∂S ∂g c† p′p = 1 2 B† pp′ . (2.8) 13603-3 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov Taking into account equations (2.8), (2.7) we can see that these matrices are expressed through the av- erages of creation and annihilation operators. Therefore, they are also transformed under the global phase transformations of ap, a† p, so that statistical operator (2.5) is invariant. In addition, such choice of ρ (or ̺) is based on the fact that it satisfies the principle of spatial correlation weakening and Wick’s theorem applies to it [56]. The introduced statistical operator defined by equation (2.5) describes a su- perfluid many-body system of interacting particles in the language of free particles (or quasiparticles) with a modified dispersion law. It is worth stressing that the efficiency of such description is essentially determined by the choice of coefficients App′ and Bpp′ . As we will see below, these coefficients are chosen to satisfy the requirement of maximum entropy for fixed values of additive integrals of motion (or con- served quantities). In the language of a self-consistent mean field theory, this requirement is equivalent to the fact that the Hamiltonian of such approximation is the closest to the exact one [14]. Therefore, the statistical operator (2.5) and the maximum entropy principle give the most accurate description of the system within the quasiparticle approximation even if the interparticle interaction is not weak. A compact formulation of a theory under consideration is given in terms of a two-row matrix f̂ c that combines the correlation functions and a vector ψ̂, whose components are the condensate amplitudes, f̂ c = ( f c −g c g c† −1− f̃ c ) , ψ̂= ( b b∗ ) , (2.9) where tilde denotes the transposed matrix. Thermodynamic equilibrium of the system is determined by the maximum of entropy S =−Spρ lnρ for fixed values of conserved quantities, such as energy, momen- tum, and total particle number. Using the unitary transformation (2.4) and u − v transformations, one can show [41, 46] that the statistical operator (2.5) is reduced to a diagonal form ρ0. Then, for the entropy of the system S =−Spρ0 lnρ0, the following combinatorial expression is valid: S =−tr [ f0 ln f0 − (1+ f0) ln(1+ f0) ] , f0 = Spρ0a†a, which allows it to be expressed through the introduced two-row matrix [41, 46]: S( f̂ c ) =−ReTr f̂ c ln f̂ c . (2.10) Here, tr . . . is a trace over the momentum variables which specify a single-particle state, while Tr. . . is taken over the the two-row matrix as well as the momentum variables. The introduced entropy depends on correlation functions only and does not depend on the condensate amplitudes. The energy of a superfluid Bose system is a functional of correlation functions and condensate ampli- tudes, E = E ( f̂ c ,ψ̂). It is obtained by averaging a microscopic Hamiltonian, E ( f̂ c ,ψ̂) = SpρH ( a† p, ap ) . (2.11) In fact, the normal-ordered Hamiltonian H ( a† p, ap ) of a Bose system can take into account binary, triple, and higher-order interactions of particles. However, we will use the following second quantized Hamil- tonian with binary interparticle interaction, H ( a† p, ap ) = ∑ p p2 2m a† pap + 1 2V ∑ p1...p4 ν(p1 −p3)a† p1 a† p2 ap3 ap4δp1+p2,p3+p4 . (2.12) Now, we find the explicit form of the energy functional E ( f̂ c ,ψ̂) for the written Hamiltonian. Due to unitary transformation (2.4), the averaging given by equation (2.11) is reduced to E ( f̂ c ,ψ̂ ) = Sp̺H ( a† p +b∗ p , ap +bp ) . Since ̺ has a Gaussian form that involves both normal and anomalous pairs of second quantized opera- tors in the exponent, we can apply Wick’s theorem with non-vanishing pairwise normal and anomalous 13603-4 Single-particle and pair condensates in Bose systems averages [or contractions, see equations (2.7)]. Thus, we have [57] E ( f̂ c ,ψ̂) = E (ψ̂)+ ∑ p1p2 p2 1 2m f c p1p2 δp1,p2 + 1 V ∑ p1...p4 f c p1p2 [ ν(p1 −p2)+ν(p2 −p4) ] b∗ p3 bp4δp3+p2,p1+p4 + 1 2V ∑ p1 ...p4 f c p1p2 f c p3p4 [ν(p1 −p2)+ν(p2 −p3)]δp1+p3,p2+p4 + 1 2V ∑ p1 ...p4 ν(p2 −p3) [ g c p1p2 b∗ p3 b∗ p4 +h.c. ] δp1+p2,p3+p4 + 1 4V ∑ p1 ...p4 g∗c p1p2 g c p3p4 [ν(p1 −p3)+ν(p1 −p4)]δp1+p2,p3+p4 , (2.13) where the first term E (ψ̂) ≡ E (bp,b∗ p) is constructed from the condensate amplitudes bp, b∗ p only, E (ψ̂) = ∑ p p2 2m b∗ pbp + 1 2V ∑ p1...p4 ν(p1 −p3)b∗ p1 b∗ p2 bp3 bp4δp1+p2,p3+p4 . (2.14) It can be proved that E (ψ̂) is related to the total energy functional E ( f̂ c ,ψ̂) by the differential opera- tor [41, 46], E ( f̂ c ,ψ̂) = RE (ψ̂), where R = exp ( ∂ ∂b f c ∂ ∂b∗ + 1 2 ∂ ∂b g c ∂ ∂b + 1 2 ∂ ∂b∗ g c∗ ∂ ∂b∗ ) . In the expression for R, we have omitted the repeated summation indices, like in equations (2.3), (2.5). The total particle number, similar to entropy (2.10), can be expressed through the two-rowmatrix f̂ c . Like in quantum mechanics, in which the physical quantities correspond to operators, we assume that a physical quantity a is associated with a two-row matrix â. Then, the average of a physical quantity a is given by [41, 46] 〈a〉 ≡ tr f a = 1 2 ( Tr f̂ c â − tr a + ψ̂∗τ̂3âψ̂ ) , â = ( a 0 0 −ã ) , where τ̂3 is the Pauli matrix and tr f a ≡ ∑ p( f a)pp. Since τ̂3 is the generator of unitary gauge transfor- mations, it should be interpreted as the particle number operator. Therefore, according to the above formula, we have N ( f̂ c ,ψ̂ ) = tr f = 1 2 ( Tr f̂ c τ̂3 − tr1+ ψ̂∗ψ̂ ) , τ̂3 = ( 1 0 0 −1 ) , where 1≡ δpp′ . The calculation of the traces gives the following expression for the total particle number: N ( f̂ c ,ψ̂ ) = ∑ p ( f c p +b∗ pbp ) , f c pp ≡ f c p . (2.15) As we have already mentioned, equations that determine the equilibrium values of correlation func- tions and condensate amplitudes are obtained from the principle of maximum entropy for fixed values of the additive integrals of motion — energy, total particle number, and total momentum. However, below we will study the system at rest (we do not introduce the latter integral of motion). Then, the problem of conditional maximization of the entropy can be reduced to the problem of unconditional minimization of the following non-equilibrium thermodynamic potential: Ω ( f̂ c ,ψ̂ ) =−S ( f̂ c ) +β [ E ( f̂ c ,ψ̂ ) −µN ( f̂ c ,ψ̂ )] , (2.16) 13603-5 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov where β, βµ are the corresponding Lagrange multipliers (β = 1/T is the reciprocal temperature and µ is the chemical potential). The solution of the formulated variational problem gives the self-consistency equations [41, 46]: f̂ c = [expβ(ε̂− µ̂)−1]−1, (2.17) η̂−µψ̂= 0, (2.18) where ε̂= ( ε ∆ −∆∗ −ε̃ ) , µ̂= ( µ 0 0 −µ ) , η̂= ( η η∗ ) (2.19) and εpp′ = ∂E ( f̂ c ,ψ̂ ) ∂ f c p′p , ∆pp′ = 2 ∂E ( f̂ c ,ψ̂ ) ∂g c∗ p′p , ηp = ∂E ( f̂ c ,ψ̂ ) ∂b∗ p . (2.20) Equation (2.17) has a natural form — namely, it reflects the fact that the matrix f̂ c , which is constructed from the correlation functions, has a structure of the Bose distribution function if ε̂ is interpreted as the operator of quasiparticle energy. Note that ε̂ and f̂ c are Hermitian matrices in indefinite metrics intro- duced in references [41, 46]. Equation (2.18) has a structure similar to the stationary Gross-Pitaevskii equation without external potential. However, the principal difference is that the energy of the system depends now not only on the condensate amplitudes but also on correlation functions. The coupled equa- tions (2.17), (2.18) describe an inhomogeneous equilibrium state of a superfluid Bose system with single- particle and pair condensates if the energy functional E ( f̂ c ,ψ̂ ) is known. In conclusion of this section we would like to note that a similar approach has been developed to extend the theory for a normal Fermi liquid to superfluid states [39–41]. In this case 〈ap〉 = 〈a† p〉 = 0 and the system is described by one equation only, which has a structure of the Fermi distribution function [similar to equation (2.17)]. 3. Spatially homogeneous state In the case of a homogeneous system, the self-consistency equations (2.17)–(2.20) have a more simple form. In particular, the correlation functions and condensate amplitudes meet the following relations: f c pp′ = f c p δp,p′ , g c pp′ = g c pδp,−p′ , bp = b0δp,0 , (3.1) where f c p = f c∗ p and g c p = g c −p. These relations show that only particles with zero momentum and pairs of particles with total zero momentum are in a condensate. Then, in accordance with equations (2.20), we have εpp′ = εpδp,p′ , ∆pp′ =∆pδp,−p′ , (3.2) where εp = ∂E ( f̂ c ,ψ̂ ) ∂ f c p , ∆p = ∂E ( f̂ c ,ψ̂ ) ∂g c∗ p . (3.3) Let us transform equation (2.17) taking into account the above conditions for spatial homogeneity. To this end, we expand the 2×2 matrix βξ̂=β ( ε̂− µ̂ ) , entering this equation, in terms of the traceless Pauli matrices τ̂i and the identity matrix τ̂0: βξ̂=β ( ξ ∆ −∆∗ −ξ ) = ci τ̂i +c0τ̂0 , (3.4) where ξ= ε−µ. We have employed the fact that for a spatially homogeneous state the first relation from (3.2) yields ξ̃= ξ. Taking the trace of both sides of equation (3.4) we have c0 = 0 and, consequently, (ciτi )2 =β2E 2 p , E 2 p = ξ2 p −∆p∆ ∗ p . 13603-6 Single-particle and pair condensates in Bose systems Let n(x) = (ex −1)−1 be the Bose distribution function. Then, equation (2.17) can be written in the form f̂ c = n ( βξ̂ ) . Bearing in mind equation (3.4), we can find the following representation for this equation: f̂ c = 1 2 [ n ( βEp ) +n ( −βEp )] + 1 2Ep [ n ( βEp ) −n ( −βEp )] ξ̂. Finally, the comparison of the matrix elements of both sides of this equation yields, f c p =−1 2 + ξp 2Ep [ 1+2n ( βEp )] , g c p =− ∆p 2Ep [ 1+2n ( βEp )] , (3.5) where we have used equations (3.1), (3.2) and the evident property n(−x) =−1−n(x). Here, ξp = εp −µ , Ep = √ ξ2 p −\|∆p\|2 (3.6) and εp, ∆p are determined by the energy functional according to equations (3.3). In the case of a homogeneous system, equation (2.18) is reduced to ∂E ( f̂ c ,ψ̂ ) ∂b∗ 0 −µb0 = 0. (3.7) Equations (3.5)–(3.7) provide a complete description of a homogeneous Bose system with single-particle and pair condensates if the energy functional E ( f̂ c ,ψ̂ ) is given. The quantities b0 and ∆p should be interpreted as the order parameters associated with single-particle and pair condensates, respectively, and Ep as the quasiparticle energy. Let us now obtain the system of coupled equations using an explicit form of the energy functional given by equations (2.13), (2.14). In this case, equation (3.7) for the condensate amplitudes takes the form b0 [ n0ν(0)−µ ] + b0 V ∑ p f c p [ ν(0)+ν ( p )] + b∗ 0 V ∑ p g c pν(p) = 0, (3.8) where we have used equations (3.1) and introduced the single-particle condensate density n0 = b∗ 0 b0/V . Next, using equations (3.3), we can find the quantities εp and ∆p through f c p = f c∗ p and g c p = g c −p. The subsequent substitution of equilibrium correlation functions (3.5) gives equations for ξp = εp−µ and ∆p. We should also eliminate the correlation functions in equation (3.8). After some algebraic manipulations, we come to the desired system of coupled equations for determining ξp, ∆p, b0: ξp = p2 2m −µ+n0 [ ν(0)+ν(p) ] − 1 2V ∑ p′ [ ν(0)+ν ( p ′−p )] { 1− ξp′ Ep′ [ 1+2n ( βEp′ )] } , ∆p = 1 V ν(p)b2 0 − 1 2V ∑ p′ ν ( p+p ′) ∆p′ Ep′ [ 1+2n ( βEp′ )] , b0 ( 2n0ν(0)−ξ0 − b∗ 0 b0 ∆0 ) = 0. (3.9) Note that these equations, as well as equation (3.8), are invariant with respect to the following transfor- mations: b0 → b′ 0 = b0e iϕ, ξp → ξ′p = ξp, ∆p → ∆ ′ p = ∆pe2iϕ. The same system of the coupled equations has been also derived within other approaches [14, 58]. It has three types of solutions. The first one, with n0 = 0 and ∆p = 0, describes the state with no broken symmetry. In this normal state there is neither single-particle nor pair condensate. The second type of solution, with n0 = 0 and∆p , 0, breaksU (1) sym- metry and corresponds to the state with a pair condensate, which is similar to a condensate of Cooper pairs in the theory of superconductivity. This condensation has been studied by a number of authors [20– 22, 59]. Finally, the third kind of solution, with n0 , 0 and∆p , 0, characterizes the state with brokenU (1) symmetry containing both single-particle and pair condensates. Note that the derived equations have no solution of the type ∆p = 0, n0 , 0. 13603-7 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov The chemical potential is related to the total particle number. From equations (2.15), (3.5), we find that the total particle density can be written in the form n = n0 +npair +nquas , (3.10) where npair = 1 2V ∑ p ( ξp Ep −1 ) [ 1+2n ( βEp )] , (3.11) nquas = 1 V ∑ p n ( βEp ) . (3.12) Here, we have introduced the particle density in a pair condensate (pair condensate density) npair and quasiparticle density nquas. In the normal phase, i.e., in the absence of both condensates, the particle number coincides with the quasiparticle number, as in the normal Fermi liquid. In a superfluid phase, the particle number is always greater than quasiparticle number. 4. Zero temperature and contact interaction The coupled equations (3.9) can be further simplified in the case of zero temperature and contact interaction, ν(p) = ν= const. Indeed, in this case, there are no quasiparticle excitations, n ( βEp ) = 0, and all particles are in the single-particle or pair condensates. Moreover, due to contact interaction, ∆p does not depend on momentum, ∆p ≡∆, and the structure of equations allows us to consider b0 and ∆ as real and positive quantities. After simple transformations, equations (3.9) are reduced to ν V ∑ p ( ξp Ep −1 ) −µ+∆= 0, (4.1) ∆ ( 1+ ν 2V ∑ p 1 Ep ) −νn0 = 0, (4.2) where ξp = p2 2m +α, α= 2νn0 −∆. (4.3) The spectrum of elementary excitations [see equations (3.6)] takes the form Ep = √ ( p2 2m +α )2 −∆2. (4.4) As we see, it has a gap δ≡ Ep=0 given by δ= √ α2 −∆2 = √ 4νn0(νn0 −∆). (4.5) We will address the issue of single-particle excitation spectrum and its behavior in the region of small momenta herein below. Equations (4.1), (4.2) allow one to find n0 and ∆ as functions of chemical potential [or taking into account equations (3.10), (3.11) as functions of total density]. As we have already mentioned, besides the solution corresponding to the normal state, these equations describe the state with a pair condensate only as well as the state with both condensates. Equation (4.2) shows that the solution corresponding to the state with a pair condensate (n0 = 0, ∆ , 0) exists only in the case of attractive interaction, ν < 0. However, this state was found to be thermodynamically unstable [59]. Therefore, we will study the solution of equations (4.1), (4.2) that describes the state with both condensates (n0 , 0, ∆, 0). Moreover, the interaction is assumed to be repulsive, ν> 0. The total particle density at T = 0, according to equations (3.10), (3.11), becomes n = n0 +npair = n0 + 1 2V ∑ p ( ξp Ep −1 ) . (4.6) 13603-8 Single-particle and pair condensates in Bose systems The comparison of this formula with equation (4.1) yields the expression for the chemical potential, µ=∆−2ν(n0 −n). (4.7) From equation (3.5) we can also find the normal and anomalous correlation functions at T = 0: f c p = ξp −Ep 2Ep , g c p =− ∆ 2Ep . (4.8) The ground state energy E (0) is obtained by setting T = 0 in equations (2.13), (3.5). In addition, we should take into account the conditions for spatial homogeneity (3.1) and equations (4.6), (4.8). The ground state energy density E (0) = E (0)/V is found to be E (0) = ν(n−n0)2 +n∆− ∆ 2 2ν + 1 2V ∑ p ( Ep −ξp ) . (4.9) When deriving this result we have eliminated the terms ∑ p ( 1/Ep ) and p2/2m using equations (4.2), (4.3), respectively. Let us obtain an explicit expression for pressure P = −Ω/βV . Here, Ω is the equilibrium thermody- namic potential which is found by substituting the equilibrium values of the correlation functions and condensate amplitudes into equation (2.16). Using the fact that S = 0 at T = 0, one obtains P = µn −E (0) and, consequently, P = ν ( n2 −n2 0 ) + ∆ 2 2ν + 1 2V ∑ p ( ξp −Ep ) . (4.10) Here, as well as in equation (4.9), the quantity ξp is given by equation (4.3). From equation (4.10) we can conclude that the pressure is positive when ν> 0 that is one of the stability conditions. We now return to the coupled equations (4.1), (4.2). First, let us eliminate the chemical potential in equation (4.1) using equation (4.7). Then, we replace the summation by integration over the variable x = p2/2m in both equations. The integral that appears in equation (4.2) diverges at the upper limit. The physical reason for this divergence is that we have taken a delta-like contact potential with zero radius of interparticle interaction. In order to remove the divergence, we introduce a typical length scale r0 for the range of interaction. In subsequent numerical computations, we will assume that r0 coincides with the value of a repulsive core of the interaction potential, which is typically equal to a few angstroms. When integrating over the variable x having the dimension of energy, we will cut off the divergence by x0 = (ħk0)2/2m = ħ2/2mr 2 0 . In addition, for numerical analysis of these equations it is convenient to introduce the following dimensionless quantities: x̃ = x/x0, ∆̃=∆/x0, α̃=α/x0, ñ0 = n0r 3 0 , ñ = nr 3 0 and to consider α̃= 4gñ0 − ∆̃ [see equations (4.3)] as the sought quantity, instead of ñ0. Consequently, equations (4.1), (4.2) in dimensionless form are written as ñ = α̃+ ∆̃ 4g + 1 8π2 ∞ ∫ 0 dx̃ p x̃ [ x̃ + α̃ √ (x̃ + α̃)2 − ∆̃2 −1 ] , (4.11) ∆̃ α̃  1+ g 2π2 1 ∫ 0 dx̃ p x̃ √ (x̃ + α̃)2 − ∆̃2  = 1, (4.12) where g= νm p 2x0m ħ3 = νm ħ2r0 (4.13) is a dimensionless coupling constant and ν is related to s-wave scattering length a by the well-known expression ν= 4πħ2a/m. We see that the dimensionless coupling constant is determined by the ratio of the scattering length to the radius of the interaction potential, g= 4π a r0 . (4.14) 13603-9 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov Equation (4.11) reflects the fact that the total particle density is the sum of the particle densities in the single-particle and pair condensates (the first and second terms, respectively). The coupled equations (4.11), (4.12) are characterized by two free dimensionless parameters — the total particle density ñ and coupling constant g. Therefore, having specified these quantities, we can find α̃ and ∆̃ and thereby deter- mine the fractions of single-particle and pair condensates as well as other characteristics of the system. We now present the dimensionless expressions for the quasiparticle energy, pressure, and density of the ground state energy. According to equation (4.4), the single-particle excitation spectrum in terms of dimensionless quantities has the form Ẽp̃ = √ ( p̃ + α̃ )2 − ∆̃2 , (4.15) where Ẽp = Ep/x0 and p̃ = pr0/ħ. In order to find an appropriate expression for pressure, let us replace the summation by integration over the variable x = p2/2m in equation (4.10). The divergent integral, as above, is cut off by x0. Moreover, along with the introduced dimensionless quantities, we define the dimensionless value of pressure P̃ = P/P0 where P0 =ħ2/mr 5 0 . Hence, P̃ = g ( ñ2 − ñ2 0 ) + ∆̃ 2 8g + 1 16π2 J0 ( α̃,∆̃ ) , (4.16) where J0 ( α̃,∆̃ ) = 1 ∫ 0 dx̃ p x̃ [ x̃ + α̃− √ (x̃ + α̃)2 − ∆̃2 ] . In a similar manner, one obtains the ground state energy density in a dimensionless form, Ẽ (0) = 1 2 ñ∆̃− ∆̃ 2 8g +g(ñ − ñ0)2 − 1 16π2 J0 ( α̃,∆̃ ) , (4.17) where Ẽ (0) = E (0)/P0. In conclusion of this section, we estimate the value of P0 for a superfluid 4He. The interaction poten- tial of this system has a strong short-range repulsion whose radius (core) is r0 = 2.55 Å. The atomic mass of helium is m = 6.65 ·10−24 g. For these values, we find P0 ≈ 15.3 atm which is somewhat less than the value of crystallization pressure ≈ 25 atm. 5. Weak interaction Here, we study the asymptotic solution of equations (4.11), (4.12) in the case of a weak interaction. These equations contain two parameters g/π2 and ñg, whose values are assumed to be small, g π2 ≪ 1, ñg≪ 1. As one can see from equations (4.11), (4.12), α̃= ∆̃= 0 when g = 0 and, consequently, α̃≈ ∆̃≪ 1 at small g. Therefore, from equation (4.12), one obtains ∆̃= α̃ [ 1− g π2 (p 1+2α̃− p 2α̃ )] , with an accuracy of the terms of the order of α̃3/2: ∆̃= α̃ [ 1− g π2 ( 1− p 2α̃ )] . (5.1) Next, since α̃ and ∆̃ are small, we neglect the terms α̃2 and ∆̃ 2 in equation (4.11). Then, after its integration and subsequent substitution of equation (5.1), we come to the following equation for α̃: 4ñg= 2α̃− g π2 α̃+ 4 p 2 3π2 gα̃3/2. 13603-10 Single-particle and pair condensates in Bose systems Its approximate solution at small α̃ is α̃= 2ñg+ g π2 [ ñg− 8 3 ( ñg )3/2 ] . (5.2) After the substitution of equation (5.2) into (5.1) we can obtain ∆̃ with the same level of accuracy: ∆̃= 2ñg− g π2 [ ñg− 4 3 ( ñg )3/2 ] . (5.3) The particle density in a single-particle condensate is expressed through α̃ and ∆̃ as follows: ñ0 = ( α̃+ ∆̃ ) /4g [see equation (4.11)]. Therefore, the asymptotic solution given by equations (5.2), (5.3) allows us to find the particle number densities in both condensates: ñ0 = ñ − 1 3π2 ( ñg )3/2 , ñpair = 1 3π2 ( ñg )3/2 . (5.4) As we see, the terms ∼ ( ñg )3/2 account for the presence of a pair condensate. The next step is to calculate the ground state energy density and pressure in the case of a weak inter- action. Both quantities, according to equation (4.16), (4.17), are determined by the integral J0 ( α̃,∆̃ ) which at small α̃ and ∆̃ has the following asymptotic behavior: J0 ( α̃,∆̃ ) ≈ J0 (α̃) ≈ α̃2 − 16 p 2 15 α̃5/2 , or taking into account equation (5.2), J0 ( α̃,∆̃ ) ≈ J0 (α̃) ≈ 4 ( ñg )2 − 128 15 ( ñg )5/2 . (5.5) Therefore, using equations (5.2)–(5.4), we come to the following dimensionless expression for pressure: P̃ = ñ2g 2 − 1 4π2 ( ñg )2 + 4 5π2 ( ñg )5/2 . (5.6) It is also easy to find the pressure in dimensional form, P = νn2 2 { 1− 1 2π2 [ νm ħ2r0 − 16 5 (νm)3/2n1/2 ħ3 ]} . (5.7) Having obtained an explicit dependence of pressure on density, we can write down the speed of sound u: u2 = 1 m ∂P ∂n = νn m [ 1− νm 2π2ħ2r0 + 2(νm)3/2n1/2 π2ħ3 ] . (5.8) The second term in equations (5.7), (5.8) accounts for the finite range of interaction potential, while the third term is a correction responsible for the presence of a pair condensate. The ground state energy density (4.17) is determined by the same integral J0 ( α̃,∆̃ ) . Therefore, taking into account equations (5.3)– (5.5) we have Ẽ (0) = ñ2g 2 − 1 4π2 ( ñg )2 + 8 15π2 ( ñg )5/2 , (5.9) or in a dimensional form, E (0) = νn2 2 [ 1− νm 2π2ħ2r0 + 16 15π2 (mν)3/2n1/2 ħ3 ] . (5.10) The ground state energy E (0) = E (0)V can be expressed in terms of the scattering length a, E (0) = 2πaħ2N 2 mV [ 1− 2 π a r0 + 128 15 p π ( a3N V )1/2 ] . (5.11) 13603-11 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov This formula almost coincides with those obtained by Lee and Yang [60]. The difference is in the second term which takes into account the final range of interaction potential. However, the renormalization of scattering length a → a (1−2a/πr0) in equation (5.11) gives their original result. Finally, we note that α̃→ ∆̃ in the limit ν→ 0 and, consequently, a gap δ in the quasiparticle spectrum (4.4) tends to zero. In this case we come to the Bogolyubov spectrum [9], Ep = √ ( p2 2m )2 + p2 m νn0 . In the same limit, the density of a single-particle condensate tends to the total particle density (the pair condensate density tends to zero). 6. Small density and strong interaction Here, we study the system of small density without any restrictions on the value of interparticle in- teraction. To this end, it is convenient to introduce new dimensionless variables χ= ∆̃/α̃ and η= α̃/4ñg instead of α̃ and ∆̃. In virtue of their definition, these variables meet the inequalities 0 < χ,η < 1. Then, equations (4.11), (4.12) in terms of χ and η read η ( 1+χ ) + g π2 ( ñg )1/2 η3/2 J1 ( χ ) = 1, (6.1) 1 χ − g 2π2 J2 ( 4ñgη,4ñgηχ ) = 1, (6.2) where J1 ( χ ) = ∞ ∫ 0 dx p x [ x +1 √ (x +1)2 −χ2 −1 ] , (6.3) J2 ( 4ñgη,4ñgηχ ) = 1 ∫ 0 dx p x √ ( x +4ñgη )2 − ( 4ñgηχ )2 . (6.4) We remind the reader that the left-hand side of equation (6.1) is just the sum of the single-particle con- densate fraction n0/n and pair condensate fraction npair/n, respectively. Now, assuming that the density is small we do not make any restrictions on the value of interaction, ñg≪ 1, g π2 ( ñg )1/2 ≪ 1. Then, from (6.1)–(6.4), we find the equations for determining χ and η in the zeroth order in small param- eters: η ( 1+χ ) −1 = 0, 1 χ − g π2 −1 = 0, whence χ= 1 1+ ( g/π2 ) , η= 1+ ( g/π2 ) 2+ ( g/π2 ) . (6.5) The explicit expressions for χ and η give α̃ = 4gñη and ∆̃ = 4gñχη. Note that in this zero-order approx- imation, the total particle density coincides with the particle density in the single-particle condensate, n = n0 (there is no pair condensate). Let us now obtain the explicit form for the ground state energy and pressure. Both quantities, accord- ing to equations (4.16), (4.17), are determined by the integral J0 ( α̃,∆̃ ) , which at small density is written as follows: J0 ( α̃,∆̃ ) = J0 ( 4ñgη,4ñgηχ ) ≈ (4ñgηχ)2, 13603-12 Single-particle and pair condensates in Bose systems where χ and η are given by equations (6.5). It is easy to find that in the given approximation, the ground state energy density coincides with the pressure, P̃ = Ẽ (0) = ñ2g 2+ g /π2 , or in a dimensional form, P = E (0) = νr0(πħn)2 2r0(πħ)2 +νm . The obtained expression for pressure allows us to calculate the speed of sound: u2 = 1 m ∂P ∂n = νnr0(πħ)2 mr0(πħ)2 +νm2/2 . We see that u2 > 0 for repulsive interaction. This fact indicates the thermodynamic stability of the system. Next, from equation (4.5) we find the dimensionless expression for the energy gap in the single-particle excitation spectrum, δ̃= √ α̃2 − ∆̃2 = 4ñg √ g/π2 2+g/π2 . (6.6) In virtue of the small value of ñg, this gap remains small even at quite strong interaction. In conclusion of this section, let us calculate the pair condensate fraction [the second term in equation (6.1)]. In the leading non-vanishing approximation over the small parameter, it has the form npair n = g π2 ( ñg )1/2 η3/2 J1 ( χ ) . Taking into account that J1 ( χ ) ≈ (π/4)χ2, where χ is given by the first formula in equations (6.5), the pair condensate fraction is found to be ñpair ñ = π2 4 ñ1/2 ( g /π2 )3/2 [ 1+ ( g /π2 )]1/2 [ 2+ ( g /π2 )]3/2 . It is easy to show that the obtained function increases in the range of small values of g and decreases for “strong” interaction. Its maximum corresponds to the point g∗ ≈ 50.9 and does not depend on the total particle density in the approximation under consideration. This fact agrees with numerical results shown in figure 1. 7. Numerical results 7.1. Dilute gases Bose-Einstein condensation in dilute ultracold gases of alkali-metal atoms was first realized experi- mentally in 1995 [1–3]. The particle number density in the condensed atomic cloud is n ∼ 1013÷1015 cm−3. Then, the corresponding dimensionless density in equations (6.1), (6.2) is a small quantity, ñ ∼ 10−11 ÷ 10−9, where we have taken r0 ≈ 3 Å. The binary atomic interaction in such systems is usually approx- imated by the contact interaction potential expressed through the scattering length a. For example, for 87Rb and 23Na, the scattering lengths are equal to a ≈ 90a0 and a ≈ 19.1a0 , respectively, where a0 ≈ 0.53 Å is the Bohr radius [6]. From equation (4.14), we can find that the dimensionless coupling constant: g≈ 200 for 87Rb and g≈ 42 for 23Na. The numerical analysis of equations (6.1), (6.2) for a systemwith parameters of dilute gases of alkali-metal atoms are presented in figures 1, 2. Figure 1 shows the dependencies of the pair condensate fraction ñpair/ñ on the coupling constant g for the above shown values of the total particle density ñ (all quantities are dimensionless). The curves have nonmonotonous character. Moreover, their maximum is reached at g∗ ≈ 50 and practically does not depend on the total density. This result, as we have already seen in the previous section, follows 13603-13 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov Figure 1. (Color online) Pair condensate fraction ñpair/ñ as a function of the coupling constant g for various values of the total particle density ñ. Figure 2. (Color online) Energy gap δ̃ in the single- particle excitation spectrum as a function of the cou- pling constant g for various values of the total parti- cle density ñ. from the analytical study of equations (6.1), (6.2). We can conclude that the pair condensate fraction is several orders less than the single-particle condensate fraction and, consequently, the dilute systems are described by the Gross-Pitaevskii equation with a high level of accuracy. In addition, in the limit g→ 0, the pair condensate density also tends to zero and thereby ñ0 → ñ. Figure 2 presents the dimensionless energy gap δ̃ in the spectrum of single-particle excitations as a function of dimensionless coupling constant g for various values of the total particle density ñ. At small densities, the gap remains small even for large values of the coupling constant. The gap tends to zero in the limit g→ 0 which agrees with Bogolyubov’s result [9]. 7.2. Model of superfluid 4He Superfluid 4He is a strongly interacting system of sufficiently high density in which Bose-Einstein condensation is also manifested [48–51]. The density of a liquid helium at zero temperature and low pressure is about n ∼ 2.18 ·1022 cm−3. The atomic mass is m = 6.65 ·10−24 g. The interaction potential of helium has a strong short-range repulsion whose radius is r0 ≈ 2.55 Å, followed by a weak intermediate range attraction. Therefore, the dimensionless density ñ = nr 3 0 in equations (6.1), (6.2) is ñ ≈ 0.36. The scattering length a for 4He atoms varies from 46.1 Å to 100 Å depending on the interaction potential [61]. Hence, equation (4.14) shows that the dimensionless coupling constant g lies in the interval from 227 to 492. The numerical analysis of equations (6.1), (6.2) for a superfluid 4He are presented in figures 3, 4. Figure 3. Pair condensate fraction ñpair/ñ as a func- tion of the coupling constant g at the density of liquid 4He, ñ = 0.36. Figure 4. Energy gap δ̃ in the single-particle excita- tion spectrum as a function of the coupling constant g at the density of liquid 4He, ñ = 0.36. 13603-14 Single-particle and pair condensates in Bose systems Figure 3 illustrates the behavior of the pair condensate fraction depending on the coupling constant. We can see that it significantly exceeds the single-particle condensate fraction. For the above shown val- ues of the coupling constant, the single-particle condensate fraction is just about 5%÷10%. This agrees with the condensate fraction in a superfluid 4He measured in experiments [48–51] and found within Monte Carlo calculations [52] and theoretical predictions based on the computation of single-particle density matrix which is expressed through experimentally measured structure factor [53, 54] (see also reference [55] and references therein). Figure 4 presents the dependence of the dimensionless energy gap δ̃ in the single-particle excitation spectrum on g at helium density. For example, at g = 227 we have δ̃ ≈ 13.45 or in temperature units δ≈ 12.5 K. This value is somewhat greater than the value of the roton gap ∆rot ≈ 8.65 K and less than the maxon energy ∆max ≈ 14 K. The Landau gapless spectrum can be a result of superposition of excitations of various nature. Moreover, some experiments [62, 63] directed toward studying the excitations in a superfluid 4He also show the complex structure of Landau spectrum. 8. Discussion Within the quasiparticle approach based on the maximum entropy principle, we have given a deriva- tion of the coupled equations describing a superfluid Bose system with single-particle and pair conden- sates. These equations have been analyzed both analytically and numerically. In order to remove the di- vergences, we introduce a typical length scale r0 for the range of interaction potential. For dilute systems (alkali atomic gases), the pair condensate fraction is negligibly small at zero temperature, npair/n ∼ 10−6 and it grows with an increasing density n. For a superfluid 4He, the role of pair condensate at T = 0 is dominant because the single-particle condensate fraction n0/n is less than 10%,which agreeswith experi- mental data on neutron scattering in a superfluid helium [48–51], with Monte Carlo calculations [52], and with other theoretical predictions (see references [53–55] and references therein). We have also found other characteristics of the system such as pressure, ground state energy, speed of sound, and single- particle excitation spectrum. Note that the latter exhibits a gap. We believe that this fact is sufficiently justified because the used approximation introduces quasiparticles by the general way and respects the basic principles of statistical physics. The studied approach can also be used to derive equations of two- fluid hydrodynamics predicting the second sound wave [41, 46]. As we have already mentioned in the introduction, the analysis of Bogolyubov’s 1/k2 -theorem does not provide a general conclusion concern- ing the excitation spectrum of a superfluid, and long-wavelength density excitations are insensitive to the gauge symmetry breaking [28]. The latter fact is directly confirmed by the neutron scattering experiments in a superfluid 4He [62, 63]. The single-particle excitation spectrum exhibits a gap in the presence of a pair condensate because the pairs have the dissociation energy and, consequently, in addition to phonon branch of spectrum, there exists another branch corresponding to the excitation of pairs [29]. The same qualitative structure of the spectrum is obtained within the proposed approach. Note that the existence of a gap resembles the situation in a solid: if there is more than one atom per unit cell, both acoustic and optical branches appear. Moreover, as we know, a neutral Fermi superfluid is characterized by the single-particle excitations with a gap as well as gapless phonon mode. The separation of single-particle and collective excitations is probably less marked in a Bose system than in a Fermi system as a conse- quence of “hybridization” of these branches due to the presence of a single-particle condensate [64, 65]. Therefore, one can expect that the Landau spectrum is a result of superposition of excitations of various nature. It is worth stressing that the experiments [62, 63] directed toward studying the excitations in a superfluid 4He also show its complex structure. Acknowledgement We thank Dr. Igor Tanatarov for valuable discussions of numerical calculations. 13603-15 A.S. Peletminskii, S.V. Peletminskii, Yu.M. Poluektov References 1. Anderson M.H., Ensher J.R., Matthews M.R., Wieman C.E., Cornell E.A., Science, 1995, 269, 198; doi:10.1126/science.269.5221.198. 2. Davis K.B., Mewes M.-O., Andrews M.R., van Druten N.J., Durfee D.S., Kurn D.M., Ketterle W., Phys. Rev. Lett., 1995, 75, 3969; doi:10.1103/PhysRevLett.75.3969. 3. Bradley C.C., Sackett C.A., Tollett J.J., Hulet R.G., Phys. Rev. Lett., 1995, 75, 1687; doi:10.1103/PhysRevLett.75.1687. 4. Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S., Rev. Mod. Phys., 1999, 71, 463; doi:10.1103/RevModPhys.71.463. 5. Zagrebnov V.A., Bru J.B., Phys. Rep., 2001, 350, 291; doi:10.1016/S0370-1573(00)00132-0. 6. Pethick C.J., Smith H., Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2002. 7. Pitaevskii L., Stringari S., Bose-Einstein Condensation, Oxford Science Publications, Oxford, 2003. 8. Andersen J.O., Rev. Mod. Phys., 2004, 76, 599; doi:10.1103/RevModPhys.76.599. 9. Bogolyubov N.N., J. Phys. USSR, 1947, 11, 23. 10. Gross E.P., Ann. Phys. (New York), 1960, 9, 292; doi:10.1016/0003-4916(60)90033-6. 11. Gerardeau M., Arnowitt R., Phys. Rev., 1959, 113, 755; doi:10.1103/PhysRev.113.755. 12. Wentzel G., Phys. Rev., 1960, 120, 1572; doi:10.1103/PhysRev.120.1572. 13. Luban M., Phys. Rev., 1962, 128, 965; doi:10.1103/PhysRev.128.965. 14. Poluektov Yu.M., Low Temp. Phys., 2002, 28, 429; doi:10.1063/1.1491184 [Fiz. Nizk. Temp., 2002, 28, 604 (in Rus- sian)]. 15. Kobe D.H., Ann. Phys., 1968, 47, 15; doi:10.1016/0003-4916(68)90224-8. 16. Shevchenko S.I., Sov. J. Low Temp. Phys., 1985, 11, 183 [Fiz. Nizk. Temp., 1985, 11, 339 (in Russian)]. 17. Nepomnyaschii Yu.A., Pashitskii E.A., Sov. Phys.–JETP, 1990, 71, 98 [Zh. Eksp. Teor. Fiz., 1990, 98, 178 (in Russian)]. 18. Pashitskii E.A., Low Temp. Phys., 1999, 25, 81; doi:10.1063/1.593709; [Fiz. Nizk. Temp., 1999, 25, 115 (in Russian)]. 19. Pulé J.V., Zagrebnov V.A., Rev. Math. Phys., 2007, 19, 157; doi:10.1142/S0129055X07002924. 20. Valatin J.G., Butler D., Nuovo Cimento, 1958, 10, 37; doi:10.1007/BF02859603. 21. Evans W.A.B., Imry Y., Nuovo Cimento B, 1969, 63, 155; doi:10.1007/BF02711051. 22. Kondratenko P.S., Theor. Math. Phys., 1975, 22, 196; doi:10.1007/BF01036327 [Teor. Mat. Fiz., 1975 22, 278 (in Russian)]. 23. Hohenberg P.C., Martin P.C., Ann. Phys., 1965, 34, 291; doi:10.1016/0003-4916(65)90280-0. 24. Griffin A., Phys. Rev. B, 1996, 53, 9341; doi:10.1103/PhysRevB.53.9341. 25. Hugenholtz N.M., Pines D., Phys. Rev., 1959, 116, 489; doi:10.1103/PhysRev.116.489. 26. Yukalov V.I., Kleinert H., Phys. Rev. A, 2006, 73, 063612; doi:10.1103/PhysRevA.73.063612. 27. Pines D., The many-body problem, Benjamin, New York, 1961. 28. Wagner H., Z. Phys. A, 1966, 195, 273; doi:10.1007/BF01325630. 29. Bogoliubov N.N., Bogoliubov N.N. (Jr.), Introduction to Quantum Statistical Mechanics, Gordon and Breach, Lau- sanne, 1994. 30. Hastings R., Halley J.W., Phys. Rev. B, 1975, 12, 267; doi:10.1103/PhysRevB.12.267. 31. Sütő A., Szépfalusy P., Phys. Rev. A, 2008, 77, 023606; doi:10.1103/PhysRevA.77.023606. 32. Bobrov V.B., Trigger S.A., Yurin I.M., Phys. Lett. A, 2010, 374, 1938; doi:10.1016/j.physleta.2010.02.075. 33. Ginibre J., Commun. Math. Phys., 1968, 8, 26; doi:10.1007/BF01646422. 34. Lieb E.H., Seiringer R., Yngvason J., Phys. Rev. Lett., 2005, 94, 080401; doi:10.1103/PhysRevLett.94.080401. 35. Sütő A., Phys. Rev. Lett., 2005, 94, 080402; doi:10.1103/PhysRevLett.94.080402. 36. Ettouhami A.M., Prog. Theor. Phys., 2012, 127, 453; doi:10.1143/PTP.127.453. 37. Landau L.D., Sov. Phys.–JETP, 1956, 3, 920 [Zh. Eksp. Teor. Fiz., 1956, 30, 1058 (in Russian)]. 38. Silin V.P., Sov. Phys.–JETP, 1958, 6, 387 [Zh. Eksp. Teor. Fiz., 1957, 33, 495 (in Russian)]. 39. Krasil’nikov V.V., Peletminskij S.V., Yatsenko A.A., Physica A, 1990, 162, 513; doi:10.1016/0378-4371(90)90432-R. 40. Akhiezer A.I., Krasil’nikov V.V., Peletminskii S.V., Yatsenko A.A., Physics-Uspekhi, 1993, 36, 35; doi:10.1070/PU1993v036n02ABEH002127 [Usp. Fiz. Nauk, 1993, 163, 1 (in Russian); doi:10.3367/UFNr.0163.199302a.0001]. 41. Akhiezer A.I., Krasil’nikov V.V., Peletminskii S.V., Yatsenko A.A., Phys. Rep., 1994, 245, 1; doi:10.1016/0370-1573(94)90060-4. 42. Akhiezer A.I., Isaev A.A., Peletminskii S.V., Rekalo A.P., Yatsenko A.A., J. Exp. Theor. Phys., 1997, 85, 1; doi:10.1134/1.558307 [Zh. Eksp. Teor. Fiz., 1997, 112, 3 (in Russian)]. 43. Akhiezer A.I., Isayev A.A., Peletminsky S.V., Yatsenko A.A., Phys. Lett. B, 1999, 451, 430; doi:10.1016/S0370-2693(99)00241-5. 44. Akhiezer A.I., Isayev A.A., Peletminsky S.V., Yatsenko A.A., Phys. Rev. C, 2001, 63, 021304(R); doi:10.1103/PhysRevC.63.021304. 45. Poluektov Yu.M., Ukr. J. Phys., 2005, 50, 1303. 13603-16 http://dx.doi.org/10.1126/science.269.5221.198 http://dx.doi.org/10.1103/PhysRevLett.75.3969 http://dx.doi.org/10.1103/PhysRevLett.75.1687 http://dx.doi.org/10.1103/RevModPhys.71.463 http://dx.doi.org/10.1016/S0370-1573(00)00132-0 http://dx.doi.org/10.1103/RevModPhys.76.599 http://dx.doi.org/10.1016/0003-4916(60)90033-6 http://dx.doi.org/10.1103/PhysRev.113.755 http://dx.doi.org/10.1103/PhysRev.120.1572 http://dx.doi.org/10.1103/PhysRev.128.965 http://dx.doi.org/10.1063/1.1491184 http://dx.doi.org/10.1016/0003-4916(68)90224-8 http://dx.doi.org/10.1063/1.593709 http://dx.doi.org/10.1142/S0129055X07002924 http://dx.doi.org/10.1007/BF02859603 http://dx.doi.org/10.1007/BF02711051 http://dx.doi.org/10.1007/BF01036327 http://dx.doi.org/10.1016/0003-4916(65)90280-0 http://dx.doi.org/10.1103/PhysRevB.53.9341 http://dx.doi.org/10.1103/PhysRev.116.489 http://dx.doi.org/10.1103/PhysRevA.73.063612 http://dx.doi.org/10.1007/BF01325630 http://dx.doi.org/10.1103/PhysRevB.12.267 http://dx.doi.org/10.1103/PhysRevA.77.023606 http://dx.doi.org/10.1016/j.physleta.2010.02.075 http://dx.doi.org/10.1007/BF01646422 http://dx.doi.org/10.1103/PhysRevLett.94.080401 http://dx.doi.org/10.1103/PhysRevLett.94.080402 http://dx.doi.org/10.1143/PTP.127.453 http://dx.doi.org/10.1016/0378-4371(90)90432-R http://dx.doi.org/10.1070/PU1993v036n02ABEH002127 http://dx.doi.org/10.3367/UFNr.0163.199302a.0001 http://dx.doi.org/10.1016/0370-1573(94)90060-4 http://dx.doi.org/10.1134/1.558307 http://dx.doi.org/10.1016/S0370-2693(99)00241-5 http://dx.doi.org/10.1103/PhysRevC.63.021304 Single-particle and pair condensates in Bose systems 46. Krasil’nikov V.V., Peletminskii S.V., Phys. Part. Nucl., 1993, 24, 200 [Fiz. Elem. Chastits At. Yadra, 1993, 24, 463 (in Russian)]. 47. Poluektov Yu.M., Ukr. J. Phys., 2007, 52, 579. 48. Sosnick T.R., Snow W.M., Sokol P.E., Silver R.N., Europhys. Lett., 1989, 9, 707; doi:10.1209/0295-5075/9/7/016. 49. Bogoyavlenskii I.V., Karnatsevich L.V., Kozlov Sh.A., Puchkov A.V., Sov. J. Low Temp. Phys., 1990, 16, 77 [Fiz. Nizk. Temp., 1990, 16, 139 (in Russian)]. 50. Snow W.M., Sokol P.E., J. Low Temp. Phys., 1995, 101, 881; doi:10.1007/BF00754515. 51. Glyde H.R., Diallo S.O., Azuah R.T., Kirichek O., Taylor J.W., Phys. Rev. B, 2011, 83, 100507(R); doi:10.1103/PhysRevB.83.100507. 52. Moroni S., Boninsegni M., J. Low. Temp. Phys., 2004, 136, 129; doi:10.1023/B:JOLT.0000038518.10132.30. 53. Vakarchuk I.A., Theor. Math. Phys., 1985, 65, 1164; doi:10.1007/BF01017941 [Teor. Mat. Fiz., 1985 65, 285 (in Russian)]; 54. Vakarchuk I.A., Theor. Math. Phys., 1990, 82, 308; doi:10.1007/BF01029225 [Teor. Mat. Fiz., 1990 82, 438 (in Rus- sian)]. 55. Rovenchak A.A., Vakarchuk I.O., J. Phys. Stud., 2007, 11, 404. 56. Akhiezer A.I., Peletminskii S.V., Methods of Statistical Physics, Pergamon Press, Oxford, 1981. 57. Peletminskii A.S., Peletminskii S.V., Low Temp. Phys., 2010, 36, 693; doi:10.1063/1.3490834 [Fiz. Nizk. Temp., 2010, 36, 875 (in Russian)]. 58. Tolmachev V.V., Theory of a Bose Gas, Moscow University Press, Moscow, 1969 (in Russian). 59. Jeon G.S., Yin L., Rhee S.W., Thouless D.J., Phys. Rev. A, 2002, 66, 011603(R); doi:10.1103/PhysRevA.66.011603. 60. Lee T.D., Yang C.N., Phys. Rev., 1957, 105, 1119; doi:10.1103/PhysRev.105.1119. 61. Jamieson M.J., Dalgarno A., Kimura M., Phys. Rev. A, 1995, 51, 2626; doi:10.1103/PhysRevA.51.2626. 62. Blagoveshchenskii N.M., Bogoyavlenskii I.V., Karnatsevich L.V., Kozlov Zh.A., Kolobrodov V.G., PuchkovA.V., Sko- morokhov A.N., JETP Lett., 1993, 57, 428 [Pis’ma Zh. Eksp. Teor. Fiz., 1993, 57, 414 (in Russian)]. 63. Blagoveshchenskii N.M., Bogoyavlenskii I.V., Karnatsevich L.V., Kozlov Zh.A., Kolobrodov V.G., Priezzhev V.B., Puchkov A.V., Skomorokhov A.N., Yarunin V.S., Phys. Rev. B, 1994, 50, 16550; doi:10.1103/PhysRevB.50.16550. 64. Glyde H.R., Griffin A., Phys. Rev. Lett., 1990, 65, 1454; doi:10.1103/PhysRevLett.65.1454. 65. Glyde H.R., Phys. Rev. B, 1992, 45, 7321; doi:10.1103/PhysRevB.45.7321. Роль одночастинкового та парного конденсатiв у бозе-системах iз довiльною iнтенсивнiстю взаємодiї О.С. Пелетминський, С.В. Пелетминський, Ю.М. Полуектов Iнститут теоретичної фiзики iм. О.I. Ахiєзера, ННЦ ХФТI, вул. Академiчна, 1, 61108 Харкiв, Україна Вивчається надплинна бозе-система з одночастинковим та парним конденсатами на основi напiвфено- менологiчної теорiї бозе-рiдини, яка не припускає слабкостi мiжчастинкової взаємодiї. Iз варiацiйного принципу для ентропiї виведено систему рiвнянь, що описують рiвноважний стан такої системи. Цi рiв- няння проаналiзовано у випадку нульової температури як аналiтично, так i чисельно. Показано, що час- тки одночастинкового та парного конденсатiв суттєво залежать вiд повної густини системи. При густинах, що досягаються у конденсатах атомiв лужних металiв, майже всi частинки знаходяться в одночастинко- вому конденсатi. Доведено, що при густинi рiдкого гелiю частка одночастинкового конденсату є меншою за 10%, що узгоджується з експериментальними даними з непружного нейтронного розсiювання, розра- хунками методом Монте-Карло та iншими теоретичними передбаченнями. Знайдено вирази для енергiї основного стану, тиску, стисливостi системи, що вивчається. Проаналiзовано також спектр одночастин- кових збуджень. Ключовi слова: надплиннiсть, бозе-ейнштейнiвська конденсацiя, одночастинковий та парний конденсати, квазiчастинки, спектр збуджень 13603-17 http://dx.doi.org/10.1209/0295-5075/9/7/016 http://dx.doi.org/10.1007/BF00754515 http://dx.doi.org/10.1103/PhysRevB.83.100507 http://dx.doi.org/10.1023/B:JOLT.0000038518.10132.30 http://dx.doi.org/10.1007/BF01017941 http://dx.doi.org/10.1007/BF01029225 http://dx.doi.org/10.1063/1.3490834 http://dx.doi.org/10.1103/PhysRevA.66.011603 http://dx.doi.org/10.1103/PhysRev.105.1119 http://dx.doi.org/10.1103/PhysRevA.51.2626 http://dx.doi.org/10.1103/PhysRevB.50.16550 http://dx.doi.org/10.1103/PhysRevLett.65.1454 http://dx.doi.org/10.1103/PhysRevB.45.7321 Introduction Formalism and basic equations Spatially homogeneous state Zero temperature and contact interaction Weak interaction Small density and strong interaction Numerical results Dilute gases Model of superfluid 4He Discussion

Role of single-particle and pair condensates in Bose systems with arbitrary intensity of interaction

Institution

Ähnliche Einträge