Solvable Model of a Mixture of Bose-Einstein Condensates
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Solvable Model of a Mixture of Bose-Einstein Condensates
Shachar Klaiman, ∗ Alexej I. Streltsov, † and Ofir E. Alon ‡ Theoretische Chemie, Physikalisch–Chemisches Institut, Universit¨at Heidelberg,Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Department of Physics, University of Haifa at Oranim, Tivon 36006, Israel (Dated: October 11, 2018)
Abstract
A mixture of two kinds of identical bosons held in a harmonic potential and interacting by har-monic particle-particle interactions is discussed. This is an exactly-solvable model of a mixture oftwo trapped Bose-Einstein condensates which allows us to examine analytically various properties.Generalizing the treatment in [Cohen and Lee, J. Math. Phys. , 3105 (1985)], closed formexpressions for the ground-state energy, wave-function, and lowest-order densities are obtainedand analyzed for attractive and repulsive intra-species and inter-species particle-particle interac-tions. A particular mean-field solution of the corresponding Gross-Pitaevskii theory is also foundanalytically. This allows us to compare properties of the mixture at the exact, many-body andmean-field levels, both for finite systems and at the limit of an infinite number of particles. Wehereby prove that the exact ground-state energy and lowest-order intra-species and inter-speciesdensities converge at the infinite-particle limit (when the products of the number of particles timesthe intra-species and inter-species interaction strengths are held fixed) to the results of the Gross-Pitaevskii theory for the mixture. Finally and on the other end, the separability of the mixture’scenter-of-mass coordinate is used to show that the Gross-Pitaevskii theory for mixtures is unableto describe the variance of many-particle operators in the mixture, even in the infinite-particlelimit. Our analytical results show that many-body correlations exist in a mixture of Bose-Einsteincondensates made of any number of particles. Implications are briefly discussed. PACS numbers: 03.75.Mn, 03.75.Hh, 03.65.-w ∗ [email protected] † [email protected] ‡ ofi[email protected] . INTRODUCTION Mixtures of Bose-Einstein condensates, following their first experimental realizations inultra-cold atoms [1–4], have been immensely explored theoretically for their ground stateand excitations, statics and out-of-equilibrium dynamics, at zero and finite temperatures,when miscible or immiscible, and in traps of various shapes and topologies, see, e.g., [5–52],representing in our viewing how fascinating and rich these quantum systems are. Method-ologically, bosonic mixtures have been treated by a variety of many-body approaches andnumerical tools, as well as within Gross-Pitaevskii, mean-field theory. Whereas the Gross-Pitaevskii theory is often employed, only the use of many-body theory can actually ensurewhen the mean-field theory provides an adequate description and when it is not. Obviouslyand quite generally, a many-body description is more demanding than a mean-field descrip-tion. In this respect, having an exactly-solvable many-body model at hand is, of course, theideal situation.In this work we present such an exactly-solvable model of a mixture of two trappedBose-Einstein condensates. Explicitly, we consider two kinds of identical bosons held in aharmonic potential and interacting by harmonic particle-particle interactions, or, briefly, theharmonic-interaction model (HIM) for bosonic mixtures. We emphasize that the HIM modelfor single-species bosons is well known and has provided ample many-body, exact results aswell as means to benchmark numerical investigations [53–58]. Similarly, the HIM model forfermions has been employed, see, e.g., [54, 59, 60]. We would also like to mention the useof an analytical treatment based on coupled harmonic oscillators for vibrations [61], whichare distinguishable degrees of freedom.The mixture’s HIM model to be derived below allows us to examine analytically variousproperties, in particular the ground-state energy, wave-function, and lowest-order intra-species and inter-species densities, and how they depend on the number of particles andinteractions in the mixture. A plethora of exact, many-body results is reported and analyzed.In addition, we also find analytically the ground-state solution of the Gross-Pitaevskiitheory for the mixture’s HIM. This enables us to compare properties of the mixture atthe exact, many-body and mean-field levels, both for finite mixtures and in the limit ofan infinite number of particles. The later topic, i.e., when the many-body and mean-fielddescriptions of a Bose-Einstein condensate coincide, is of much interest for single-species2osons [62–67], but has yet to be addressed for mixtures. We hereby prove that the exactground-state energy and lowest-order intra-species and inter-species densities converge atthe infinite-particle limit to the results of the Gross-Pitaevskii theory for the mixture.Last but not least, the separability of the mixture’s center-of-mass coordinate is used toshow that the Gross-Pitaevskii theory for mixtures can deviate strongly when computingthe variance of many-particle operators in the mixture, even in the infinite-particle limit.Unlike the variance of operators of a single particle, which is a basic notion in any quantummechanics textbook [68], the variance of operators of many-particle systems is more involved[66, 67], also see [69] in this context. Our analytical results show that many-body correlationsexist in a trapped mixture of Bose-Einstein condensates consisting of any number of particles . II. THE HARMONIC-INTERACTION MODEL FOR MIXTURES
Consider a mixture of two kinds of identical bosons which we denote A and B . Thebosons are trapped in a three-dimensional isotropic harmonic potential and interact betweenthem via harmonic particle-particle interactions. We focus in this work on what might beconsidered the simplest case, the symmetric mixture. This is a mixture consisting of M bosons of type A and an equal number of M bosons of type B , all having the same mass(taken below to be 1) and trapped in the same harmonic potential (of frequency ω ). The totalnumber of particles is denoted by N = 2 M . Furthermore, the two intra-species interactionsare alike (denoted by λ ). The inter-species interaction is donated by λ . Positive valuesof λ and λ mean attractive particle-particle interactions whereas negative values implyrepulsive interactions. Of course, the intra-species interactions may be repulsive and theinter-species attractive, and vice versa.The mixture’s Hamiltonian is then given by ( ~ = 1)ˆ H ( x , . . . , x M , y , . . . , y M ) = M X j =1 (cid:20)(cid:18) − ∂ ∂ x j + 12 ω x j (cid:19) + (cid:18) − ∂ ∂ y j + 12 ω y j (cid:19)(cid:21) ++ λ M X ≤ j 2, 1, and 1, respectively.Let us examine the frequencies (7) and their dependence on the particle-particle interac-tions more closely. The frequency of all intra-species relative coordinates Q , . . . , Q M − ,Ω rel , depends on both the intra-species λ and inter-species λ interactions, yet only throughtheir sum λ + λ . The frequency Ω AB of the intra-species relative coordinate Q N − dependson λ only. The frequency of the center-of-mass degree of freedom Q N , which is equal tothe trap frequency ω , naturally does not depend on either. When λ = 0, i.e., when theintra-species interactions vanish, we are still dealing with a mixture described by three fre-quencies, solely due to the inter-species interaction which couples all N particles together.On the other hand, when λ + λ = 0, we get Ω rel = ω and the mixture is now describedby two frequencies only. We will analyze this case further below. When the intra- andinter-species interactions are equal, λ = λ , the frequency of the relative coordinate be-tween the center-of-masses of the A and B species degenerates to that of the other relativecoordinates, i.e., Ω AB = Ω rel , and the standard, single-species HIM model of N bosons isrecovered. Finally, in the limiting case when λ = 0, we have Ω AB = ω and, as one wouldexpect, we find two independent species each described by the HIM model for M bosons.5he frequencies (7) must be positive in order for a bound solution to exist. This dictatesbounds on both the intra-species λ and inter-species λ interactions which are:Ω AB = ω + 2 N λ > ⇒ λ > − ω N , Ω rel = ω + N ( λ + λ ) > ⇒ λ > − λ + ω N . (8)The meaning of these bounds are as follows: The inter-species interaction λ is boundedfrom below, irrespective of the intra-species interaction λ , otherwise the mixture cannot betrapped in the harmonic potential. On the other hand, the intra-species interaction λ islimited by the chosen inter-species interaction λ .We can now proceed and prescribe the normalized ground-state wave-functionΨ( Q , . . . , Q N ) = (cid:18) Ω rel π (cid:19) N − (cid:18) Ω AB π (cid:19) (cid:16) ωπ (cid:17) e − ( Ω rel P N − k =1 Q k +Ω AB Q N − + ω Q N ) , (9)along with the ground-state eigen-energy E = 32 [( N − rel + Ω AB + ω ] = 32 h ( N − p ω + N ( λ + λ ) + p ω + 2 N λ + ω i . (10)To express the wave-function with respect to the original spatial coordinates we use therelations (5) and findΨ( x , . . . , x M , y , . . . , y M ) = (cid:18) Ω rel π (cid:19) N − (cid:18) Ω AB π (cid:19) (cid:16) ωπ (cid:17) × (11) × e − α P Mj =1 x j − β P M ≤ j The HIM model for a mixture of two Bose-Einstein condensates presented above admitsa wealth of properties that can all be studied analytically. We present in this section a6ather detailed account of the ground-state energy, intra- and inter-species densities, a mean-field solution, and the variance of center-of-mass operators in the mixture. Side by sidethe plethora of closed-form expressions and results, a guiding line of our exploration areproperties of finite mixtures as well as at the infinite-particle limit (to be defined preciselybelow), and how the many-body and mean-field solutions are related to each other. A. Ground-state energy It is instructive and interesting to analyze the ground-state energy (10) in several cases.As mentioned above, for λ = λ we recover the ground-state energy of the single-speciesHIM model [53]. Keeping λ and λ fixed and increasing the number of particles N theenergy to leading order in N scales like N , i.e. adding more particles increases further themixture’s energy. This situation applies as long as the bounds (8) are not reached, that iswhen the mixture is predominantly attractive, λ + λ > λ > λ + λ = 0, i.e., when the intra- and inter-speciesinteractions are exactly opposite in sign. In this case, as we touched upon above, Ω rel degenerates to ω , and the energy (10) reduces to E | λ + λ =0 = 32 h ( N − ω + p ω + 2 N λ i (13)and seen to depend only on the inter-species interaction λ . Now, the ground-state energy(13) is linear to leading order in the number of particles N .Using the bounds for λ and λ in (8), we obtain that the mixture’s energy is boundfrom below by E > ω , which is obtained for Ω rel → + and Ω AB → + . This means thatall relative degrees of freedom are marginally bound, and essentially only the center-of-massdegree of freedom is bound in the harmonic trap.We now move to analyze the mixture’s energy in the so-called infinite-particle limit.To this end, we introduce the intra-species Λ = λ ( M − 1) and inter-species Λ = λ M interaction parameters. In the infinite-particle limit the interaction parameters Λ and Λ are kept fixed, while the number of particles is increased. Hence, the interaction strengths λ and λ diminish accordingly. The two interaction parameters Λ and Λ appear naturallyin the mean-field treatment discussed below, and would facilitate the comparison betweenthe exact, many-body and mean-field solutions of the mixture.7eeping Λ and Λ constant, the energy per particle in the limit of an infinite number ofparticles reads lim N →∞ EN = 32 p ω + 2(Λ + Λ ) . (14)We see that the limit of the energy per particle depends on the sum of intra- and inter-speciesinteraction parameters Λ + Λ only. Finally, in the particular case when the interactionparameters are fixed and opposite in sign, i.e., Λ + Λ = 0, we find in the infinite-particlelimit lim N →∞ E | Λ +Λ =0 N = 32 ω. (15)The meaning of (15) is that the energy per particle of the interacting system becomes thatof the non-interacting system; A situation that cannot occur in the single-species system.Here, the intra- and inter-species interaction parameters ‘cancel’ each other in the infinite-particle limit, at least as far as the energy per particle in concerned. We will return to thispeculiar situation below when analyzing the density and mean-field solution of the mixture. B. Intra- and inter-species densities We start from the N -particle density of the mixture | Ψ( x , . . . , x M , y , . . . , y M ) | = (cid:18) Ω rel π (cid:19) N − (cid:18) Ω AB π (cid:19) (cid:16) ωπ (cid:17) × (16) × e − α P Mj =1 x j − β P M ≤ j 1) coordinates andcorresponding constants C M − and D M − . To simplify the relation between the constantsin F M and F M − , it is useful to take appropriate linear combinations. The final result reads˜ C M − = ˜ C M − ( β + ˜ C M ) α + ˜ C M , ˜ C M = (2 C M − D M ) = − γ,D M − = D M − ( β + D M ) α + D M , D M = γ, (25)where the values of ˜ C M and D M are obtained when we equate the auxiliary function F M and the N -body density (16).We can now write the recurrence relation connecting the function F j with 2 j variablesand constants C j and D j to the function F j − with 2( j − 1) variables and constants C j − and D j − . The equality looks like (24) and need not be pasted here. What is important isthe recurrence relation between the corresponding constants which is given by˜ C j − = ˜ C j − ( β + ˜ C j ) α + ˜ C j , ˜ C j = (2 C j − D j ) ,D j − = D j − ( β + D j ) α + D j . (26)In turn, the recursion ends with the ‘lowest-order’ auxiliary function F ( x , y ; α, β, C , D ) = e − ( α + C )( x + y )+2( D − C ) x · y . (27)As we shall see, the constants C and D are required to evaluate the intra- and inter-speciesdensities of the mixture (18) and (17).Interestingly and importantly, the recurrence relations (26) for the parameters ˜ C j and D j appearing in the double-Gaussian integrations of the mixture’s auxiliary function have10xactly the same structure as the recurrence relation emerging in the single-Gaussian inte-grations of the HIM system [53]. Thus, we use this result directly and write for the finalresult of the respective solutions˜ C j = − α + ( α − β )(1 + jη − )1 + ( j + 1) η − , D j = − α + ( α − β )(1 + jη + )1 + ( j + 1) η + ,η ± = ( α − β ) − ( α ± γ )( M + 1)( α ± γ ) − M ( α − β ) , (28)where the initial conditions in (25) have been used [70].We can now proceed to express explicitly the intra- and inter-species lowest-order densi-ties. Because the auxiliary function F M , together with the initial conditions ˜ C M = − γ and D M = γ , is proportional to the N -particle density (16), the function F is proportional tothe intra-species two-body density (17). Thus we readily have ρ AB ( x , y ) = M (cid:20) ( α + C ) − ( D − C ) π (cid:21) e − ( α + C )( x + y )+2( D − C ) x · y , = M (cid:20) ( α + C ) − ( D − C ) π (cid:21) e − [( α + C )+( D − C )]( x − y ) e − [( α + C ) − ( D − C )]( x + y ) , (29)for the two-body density and, upon one additional integration, ρ A ( x ) = M " ( α + D )( α + ˜ C ) π ( α + C ) e − ( α + D α + ˜ C α + C x ,ρ B ( y ) = M " ( α + D )( α + ˜ C ) π ( α + C ) e − ( α + D α + ˜ C α + C y , (30)for the one-body densities. We see that the two-body density can be viewed as an ellipsoid inthe x – y coordinates, see the second line of (29), whereas the one-body densities are isotropicGaussians in x and y coordinates.To complete the computation of the densities we are left to determine ˜ C and D as afunction of the mixture’s frequencies Ω rel , Ω AB , and ω and the number of particles in eachspecies M . Using (12) we find α − β = Ω rel , α + γ = ( M − rel + Ω AB M , α − γ = ( M − rel + ωM = ⇒ η + = Ω rel − Ω AB ( M + 1)Ω AB − Ω rel , η − = Ω rel − ω ( M + 1) ω − Ω rel . (31)11rom which we obtain the ingredients α + ˜ C = Ω rel M ω ( M − ω + Ω rel ,α + D = Ω rel M Ω AB ( M − AB + Ω rel ,α + C = ( α + ˜ C ) + ( α + D )2 = Ω rel M [2( M − ω Ω AB + ( ω + Ω AB )Ω rel ]2[( M − ω + Ω rel ][( M − AB + Ω rel ] , (32)and combinations thereof2( D − C ) = ( α + D ) − ( α + ˜ C ) = M Ω rel (Ω AB − ω )[( M − ω + Ω rel ][( M − AB + Ω rel ] , ( α + C ) − ( D − C ) = ( α + D )( α + ˜ C ) = Ω rel M ω Ω AB [( M − ω + Ω rel ][( M − AB + Ω rel ] , ( α + D )( α + ˜ C )( α + C ) = 2 α + D + α + ˜ C = Ω rel M ω Ω AB M − ω Ω AB + Ω rel ( ω + Ω AB ) (33)entering the expressions (29) and (30). We have now computed explicitly and analyticallythe lowest-order densities of the mixture.With analytical expressions for the densities one can examine various situations. We wishto elaborate on two. The two-body density couples the A and B species as soon as the inter-spices interaction, λ , is present, see (29). This is because λ = 0 leads to Ω AB = ω , see (7),which implies ( D − C ) = 0, see the first line of (33). Furthermore, from the aspect ratio ofthe ellipsoid in the x – y coordinates, ( D − C ) > D − C ) < D − C ) is positive (negative) when Ω AB is bigger (smaller) than ω ,i.e., when λ is positive (negative). To remind, λ > λ < and Λ fixed, we havefor the mixtue’s frequencies in the infinite-particle limit lim M →∞ Ω rel = p ω + 2(Λ + Λ )12nd lim M →∞ Ω AB = √ ω + 4Λ . Now, the ingredients (32) entering the densities satisfyin the limit of an infinite number of particles lim M →∞ ( α + C ) = lim M →∞ ( α + D ) =lim M →∞ ( α + e C ) = p ω + 2(Λ + Λ ), and similarly in (33). Consequently, in the infinite-particle limit the expressions for the densities, see (18) and (17), depend on the interactionparameters only and simplifylim M →∞ ρ A ( x ) M = p ω + 2(Λ + Λ ) π ! e − √ ω +2(Λ +Λ ) x , lim M →∞ ρ B ( y ) M = p ω + 2(Λ + Λ ) π ! e − √ ω +2(Λ +Λ ) y , (34)and lim M →∞ ρ AB ( x , y ) M = p ω + 2(Λ + Λ ) π ! e − √ ω +2(Λ +Λ )( x + y ) . (35)In particular, because lim M →∞ ( D − C ) = 0 the inter-species two-body density is seen tofactorize to a product of two one-body densities, independent of the magnitude of the inter-species interaction parameter Λ . We would come back to this point in the following section,when the mean-field solution of the mixture’s HIM model is to be derived and analyzed. C. Mean-field (Gross-Pitaevskii) solution At the other end of the exact, many-body solution of the mixture’s HIM model liesthe Gross-Pitaevskii, mean-field solution. In the mean-field theory the many-particle wave-function is approximated as a product state, where all the bosons of the A species lies inthe one and the same orbital, and all the bosons of the B species similarly lie in one orbitalwhich is generally different than the A species one. In our case of a symmetric mixture(1), excluding a solution where demixing of the two species occurs in the ground state, themean-field ansatz for the mixture is the product wave-function where, due to symmetrybetween the A and B species, each of the species occupies the same spatial functionΦ GP ( x , . . . , x M , y , . . . , y M ) = M Y j =1 φ ( x j ) M Y k =1 φ ( y k ) . (36)13he Gross-Pitaevskii energy functional of the mixture thus simplifies and reads ε GP = 12 " Z d x φ ∗ ( x ) (cid:18) − ∂ ∂ x + 12 ω x (cid:19) φ ( x ) + Λ Z d x d x ′ | φ ( x ) | | φ ( x ′ ) | ( x − x ′ ) ++ Z d y φ ∗ ( y ) (cid:18) − ∂ ∂ y + 12 ω y (cid:19) φ ( y ) + Λ Z d y d y ′ | φ ( y ) | | φ ( y ′ ) | ( y − y ′ ) ++ Λ Z d x d y | φ ( x ) | | φ ( y ) | ( x − y ) == Z d r φ ∗ ( r ) (cid:18) − ∂ ∂ r + 12 ω r (cid:19) φ ( r ) + 12 (Λ + Λ ) Z d r d r ′ | φ ( r ) | | φ ( r ′ ) | ( r − r ′ ) , (37)where r represents the coordinate of any of the particles. To remind, ε GP is the total mean-field energy of the mixture divided by the number of particle N = 2 M . Consequently andside by side, the two coupled Gross-Pitaevskii equations of the mixture degenerate to one (cid:26) − ∂ ∂ r + 12 ω r + (Λ + Λ ) Z d r ′ | φ ( r ′ ) | ( r − r ′ ) (cid:27) φ ( r ) = µφ ( r ) , (38)where µ is the chemical potential of each species. The solution of (38) follows exactlythe same way as for the HIM problem [53], and is now briefly followed for completeness.Expanding the interaction term in (38) we find (cid:26) − ∂ ∂ r + 12 (cid:2) ω + 2(Λ + Λ ) (cid:3) r (cid:27) φ ( r ) = (cid:20) µ − (Λ + Λ ) Z d r ′ | φ ( r ′ ) | r ′ (cid:21) φ ( r ) . (39)The solution of (39) is the Gaussian function. φ ( r ) = p ω + 2(Λ + Λ ) π ! e − √ ω +2(Λ +Λ ) r == p Ω rel − λ π ! e − √ Ω rel − λ r . (40)Since the orbital (40) is an even function, there is no linear in r term in (39). We can nowevaluate the integral R d r ′ | φ ( r ′ ) | r ′ = √ ω +2(Λ +Λ ) in (39) and determine the chemicalpotential µ = 32 p ω + 2(Λ + Λ ) + 34 Λ + Λ p ω + 2(Λ + Λ ) = 34 2 ω + 3(Λ + Λ ) p ω + 2(Λ + Λ ) . (41)Indeed, for λ = λ the single-species HIM chemical potential [53] is found. With this, themean-field energy is given by ε GP = (cid:26) µ − 12 (Λ + Λ ) Z d r d r ′ | φ ( r ) | | φ ( r ′ ) | ( r − r ′ ) (cid:27) == 32 p ω + 2(Λ + Λ ) = 32 q Ω rel − λ , (42)14here R d r d r ′ | φ ( r ) | | φ ( r ′ ) | ( r − r ′ ) = √ ω +2(Λ +Λ ) is used. In the specific case whereΛ + Λ = 0, the mean-field energy becomes that of the non-interacting system. Thisreminds the properties of the many-body energy discussed above, see (13) and (15) andassociated text.We can now compare the exact, many-body energy E and the mean-field energy perparticle ε GP . Of course, the many-body energy is always lower, for repulsive as well as forattractive particle-particle interactions, than the mean-field energy because of the variationalprinciple. In the infinite-particle limit we find when Λ and Λ are held constant thatlim N →∞ EN = ε GP , (43)which establishes the connection between the exact energy and mean-field (Gross-Pitaevskii)energy per particle in this limit for the mixture.We now discuss the one-body density. From (40) we have ρ GP ( r ) = | φ GP ( r ) | = p ω + 2(Λ + Λ ) π ! e − √ ω +2(Λ +Λ ) r == p Ω rel − λ π ! e − √ Ω rel − λ r . (44)From the mean-field solution we see that the density narrows for overall attractive interac-tions and broadens for repulsive ones. To be more precise, the density narrows for Λ +Λ > + Λ < 0. Within the mean-field theory this is the manifestation ofthe self-consistency used to solve the non-linear Schr¨odinger (Gross-Pitaevskii) equationand find the orbital (40). Furthermore, from (44) and using (36) we can write for thelowest-order densities of the mixture within the mean-field theory, ρ GPA ( x ) = M ρ GP ( x ), ρ GPB ( y ) = M ρ GP ( y ), and ρ GPAB ( x , y ) = M ρ GP ( x ) ρ GP ( y ). As expected, for mixtures with afinite number of particles, the mean-field densities differ from their many-body counterparts(30) and (29). For instance, in contrast to the two-body inter-species density ρ AB ( x , y ), themean-field density ρ GPAB ( x , y ) is always factorized to a product of two one-body densities.Finally, in the limit of an infinite number of particles the many-body (35) and (34) and theGross-Pitaevskii densities coincide in the sense that what one might have suspected holdstrue, namely lim M →∞ ρ A ( x ) M = ρ GP ( x ) , lim M →∞ ρ B ( y ) M = ρ GP ( y ) , (45)15nd lim M →∞ ρ AB ( x , y ) M = ρ GP ( x ) ρ GP ( y ) . (46)With (43), (45), and (46), we have proved and established that the many-body energy anddensities of the mixture coincide with their mean-field values in the infinite-particle limit.This constitutes a generalization for mixtures of Bose-Einstein condensates, at least withinthe mixture’s HIM model, of what is known in the literature for single-species trappedBose-Einstein condensates [62, 63]. D. Center-of-mass variance and uncertainty product So far we have discussed the energy and densities of the mixture, and have established therelations in the infinite-particle limit in which the exact solution and the Gross-Pitaevskiitheory coincide. As has recently been shown for single-species Bose-Einstein condensates, thevariance of many-particle operators can deviate strongly when the many-body and mean-field treatments are compared [66, 67]. These quantities have recently been found to besensitive to correlations in single-species Bose-Einstein condensates even in the infinite-particle limit. The many-body and mean-field analytical solutions of the mixture allow usto examine directly the variances of the center-of-mass position and momentum operatorsof the mixture, as well as their uncertainty product. We shall keep the discussion concise.Consider the center-of-mass position and momentum operators of the mixtureˆ R CM = 1 N M X j =1 ( x j + y j ) = 1 √ N Q N , ˆ P CM = M X j =1 (cid:18) i ∂∂ x j + 1 i ∂∂ y j (cid:19) = √ N i ∂∂ Q N , [ ˆ R CM , ˆ P CM ] = i , ∀ N, (47)where i (and analogously below) is a shorthand symbol for i in each of the three Cartesiancomponents. ˆ R CM and ˆ P CM are one-particle operators in the sense that they are linearwith respect to all particles in the mixture. To compute their variances we need also theoperators squares, ˆ R CM and ˆ P CM , which are two-particle operators. These would lead todifferences between the many-body and mean-field results; see for an extended discussion ofthe matter in the case of single-species Bose-Einstein condensates [66, 67].We can now compute the variances of the center-of-mass position and momentum oper-16tors, ∆ R CM = h Ψ | ˆ R CM | Ψ i − h Ψ | ˆ R CM | Ψ i = 1 N × ω , ∆ P CM = h Ψ | ˆ P CM | Ψ i − h Ψ | ˆ P CM | Ψ i = N × ω , ∀ N, (48)which are straightforwardly obtained due to the separability of the wave-function (9). Forcomparison, the corresponding expressions at the Gross-Pitaevskii level are computed fromthe mean-field wave-function (36) and given by∆ R CM ,GP = h Φ GP | ˆ R CM | Φ GP i − h Φ GP | ˆ R CM | Φ GP i = 1 N × p ω + 2(Λ + Λ ) , ∆ P CM ,GP = h Φ GP | ˆ P CM | Φ GP i − h Φ GP | ˆ P CM | Φ GP i = N × p ω + 2(Λ + Λ )2 , ∀ N. (49)Comparing (48) and (49) we see how the self-consistency of the Gross-Pitaevskii orbtial(40), whose shape depends on the sum of the intra- and inter-species interaction parametersΛ + Λ , alters the values of the mean-field variances in comparison with the exact result.This is an interesting result for the mixture which is best expressed by the respective ratios∆ R CM ,GP ∆ R CM = 1 q ω (Λ + Λ ) , ∆ P CM ,GP ∆ P CM = r ω (Λ + Λ ) , ∀ N, (50)in which the total number of particles does not appear. In particular for predominantlyattractive mixtures, Λ + Λ > 0, since the sum Λ + Λ is unbound from above the ratiobetween the mean-field and exact center-of-mass position variances (50) can be as smallas one wishes. However, the ratio of the center-of-mass momentum variances (50) is then,inevitably, very large.We can now discuss the uncertainty product. Here, because the trap and all interactionsare harmonic, the center-of-mass wave-function is a Gaussian and the mean-field orbital is aGaussian as well, although being dressed by the particle-particle interactions. Consequently,∆ R CM ∆ P CM = 14 , ∆ R CM ,GP ∆ P CM ,GP = 14 , ∀ N, (51)irrespective to the interactions between the particles. The comparison between the exactand mean-field solutions in terms of the uncertainty product might erroneously imply thatthere are no correlations in the mixture, especially in the limit of an infinite number ofparticles. Here, the correlations which survive the infinite-particle limit can for instance be17een in the ratios of the variances (50). Although not discussed in [53], this generalizes thesituation for the single-species HIM to the mixture’s HIM model.A final note. The result (51) raises the useful question what happens for general intra-and inter-species interactions. Is this equivalence between the center-of-mass uncertaintyproducts at the exact and mean-field levels lifted? The answer goes beyond the solution ofthe HIM model for mixtures but, together with the related issue of an explicit constructionof the center-of-mass separability in the generic case, is presented for completeness in theAppendix. IV. CONCLUDING REMARKS We have presented in this work a solvable quantum model for a mixture of two trappedBose-Einstein condensates. The model consists of two kinds of identical bosons, A and B , held in a harmonic potential and interacting by harmonic particle-particle interactions,namely, the harmonic-interaction model for mixtures. We have concentrated in the presentinvestigation on the case of a symmetric mixture in which the number of bosons and interac-tions within species A are the same as for species B . By generalizing the treatment of Cohenand Lee [53] to mixtures, closed form expressions for the ground-state energy, wave-function,and lowest-order intra- and inter-species densities are obtained. These quantities are ana-lyzed analytically as a function of the number of particles and the intra- and inter-speciesinteractions.Aside from the many-body solution, we have also obtained analytically the Gross-Pitaevskii solution for the ground-state, as far as demixing of the two Bose-Einstein con-densates is excluded. This has allowed us to compare the exact, many-body and mean-fieldsolutions for any number of particles. In particular, keeping the products of the num-ber of particles times the intra-species and inter-species interaction strengths fixed, whileincreasing the number of particles in the mixture, we were able to prove that the exactground-state energy and lowest-order intra-species and inter-species densities converge atthe infinite-particle limit to the results of the Gross-Pitaevskii theory for the mixture. Thisholds as a particular generalization for bosonic mixtures of the known literature results forsingle-species bosons [62, 63]. On the other end, the separability of the mixture’s center-of-mass coordinate is used to show that the result for the variance of many-particle operators in18he mixture obtained within Gross-Pitaevskii theory for mixtures can deviate substantiallyfrom the exact, many-body result, even in the limit of an infinite number of particles. Thepresent analytical results hence show that many-body correlations exist in the ground stateof a trapped mixture of Bose-Einstein condensates made of any number of particles, thusgeneralizing our recent result for single-species bosons [66].As an outlook we mention the asymmetric mixture, out-of-equilibrium dynamics, andmore, all within the harmonic-interaction model for mixtures. In particular, an impuritymade of one or two particles would be interesting to solve, once the techniques used above areadapted to asymmetric integrations. Last but not least, we hope that the present analyticalresults and proofs obtained in the limit of an infinite number of particles for a particularsolvable model would stimulate generalizations for mixtures with short-range interactions,in as much as single-species bosons were offered mathematically rigorous results in this limitin the literature. ACKNOWLEDGEMENTS This research was supported by the Israel Science Foundation (Grant No. 600/15). 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