PHYSICAL REVIEW A 73, 013622 共2006兲

Exact solution of two bosons in a trap potential: Transition to fragmentation Shachar Klaiman and Nimrod Moiseyev Department of Chemistry and Minerva Center for Nonlinear Physics of Complex Systems, Technion-Israel Institute of Technology, Haifa 32000, Israel

Lorenz S. Cederbaum Theoretische Chemie, Univerdität Heidelberg, D-69120 Heidelberg, Germany 共Received 25 July 2005; published 24 January 2006兲 Fragmentation of a two-boson system in an external potential with contact interaction 共i.e., ␦-function interaction兲 is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method 共Gross-Pitaevskii equation兲 and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the bosonboson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate. DOI: 10.1103/PhysRevA.73.013622

PACS number共s兲: 03.75.Hh, 03.65.Ge, 03.75.Nt

I. INTRODUCTION

The remarkable ability to control the interaction strength in Bose-Einstein condensation 共BEC兲 experiments over the past years 关1,2兴 has stimulated extensive theoretical and experimental research on the effect of the interaction on the condensate’s characteristics, see, for examples, Refs. 关3,4兴. Recently, several theoretical studies considered fragmentation of condensates due to a change of the interaction strength or a change of the external trap 关5–7兴. Fragmentation of condensates was initially argued against by Nozières 关8兴 on the basis of the additional exchange term in the macroscopic energy of the fragmented condensate. As discussed by Spekkens and Sipe, the argument holds only in the case where the fragmented wave functions overlap extensively and is not generally applicable 关5,9兴. The mean-field approximation as implemented by the Gross-Pitaevskii equation 共GPE兲 has been the main theoretical tool used to explain many experiments, see Refs. 关10,11兴, and references within. Fragmentation of condensates, however, cannot be described using the GPE and, therefore, requires a different approach. Some of the approaches used to describe the fragmentation of condensates are based on a restricted variational treatment 关5兴, and more recently on a more general mean-field approach named the best mean-field 共BMF兲 关12兴. In light of the rising interest in the fragmentation of condensates it seems appropriate to solve a model problem exactly and compare the results to the different mean-field approaches. The model chosen in our work consists of two bosons with repulsive contact interaction inside an external potential. Other aspects related to condensates have been modeled similarly by others, see, for example, Ref. 关13兴. We emphasize that the model chosen is in no way a condensate but rather a microscopic system which we are able to solve exactly. Hopefully, one will be able to learn more about the mesoscopic system from the results given in this paper. Furthermore, there is currently much interest in the problem of 1050-2947/2006/73共1兲/013622共7兲/$23.00

few-boson systems as such and the exact solution of twoboson systems add to the understanding of these problems, see, e.g., 关14–16兴. We solve the two-boson model numerically and compare the results to those obtained by the meanfield the GPE and BMF approaches. The article is organized as follows. In Sec. II we present the problems and solution in performing a full variational calculation. In Sec. III we account for the different mean-field approaches used and in Sec. IV we compare the results of the different methods. Finally in Sec. V the main conclusions are drawn. II. THE HAMILTONIAN AND ITS EVALUATION

The time-independent Schrödinger equation for two bosons in an external trap with contact interaction reads

冋兺 冉 2

i=1

−

册

冊

1 ⳵2 + V共xi兲 + a0␦共x2 − x1兲 ⌿共x1,x2兲 2 ⳵ x i2

= E⌿共x1,x2兲,

共1兲

where a length scale L has been chosen for convenience so that the energy unit ប2 / ML2 = 1 and M is the mass of the boson. Consequently, the coordinates xi , i = 1, 2, are dimensionless quantities. The system described is purely one dimensional. For brevity we will consider only repulsive interaction 共i.e. a0 ⬎ 0兲. The external potential chosen for our model is 共see Fig. 1兲: V共x兲 =

冦

冉

冊

冧

x2 2 − 0.8 e−0.1x 兩x兩 ⬍ 15 2 . 兩x兩 艌 15 ⬁

共2兲

This potential, which has been used recently in the study of fragmentation 关7兴, is a modified version of a potential that has been used before as a test-case model for new theorems and computational methods 关17,18兴. Also recently, a similar

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well for the results to converge. In our calculations this is manifested as a problem of finding a consistent method for selecting and increasing the size of our basis systematically. The method with which the above-mentioned problem was overcome will be described below. The two-particle 共i.e., two-dimensional兲 basis set needed can be extremely large and, therefore, the calculation is very time consuming numerically. We construct the wave function ⌿共x1 , x2兲 as follows: ⌿共x1,x2兲 =

兺 Cmn⌽mn共x1,x2兲.

共5兲

n艌m

Utilizing the symmetry of bosons with respect to permutation we express ⌽共x1 , x2兲 as FIG. 1. 共Color online兲 The external potential used in our model. Notice the different regions defined by the potential: The region in between the potential barriers 共marked I兲 and the region beyond the potential barriers 共marked II兲.

potential has been manufactured using wire sculpturing in atom chips 关19兴. In this external potential the particles can be localized either “inside” the potential well between the two potential barriers 关marked 共I兲 in Fig. 1兴, or “outside” beyond the potential barriers 关marked 共II兲 in Fig. 1兴. We define two possible states of the system: The condensed state and the fragmented state. The definition of a condensed state was given by Penrose and Onsager 关20兴 through the existence of a macroscopically 共with respect to the number of particles兲 large eigenvalue of the first-order-reduced density matrix 共DM兲. In analogy 关11兴, the fragmented state is defined as a state for which more than one macroscopically large eigenvalue of this matrix exists. The DM, sometimes referred to as the one-particle density matrix, is given by 共for the case of two bosons兲 关21,22兴: ⌫共x1,x⬘1兲 = 2

冕

⌿*共x1,x2兲⌿共x⬘1,x2兲dx2 .

共3兲

The eigenfunctions ␸l of this matrix are called natural orbitals and can be determined from

冕

⌫共x1,x⬘1兲␸l共x⬘1兲dx⬘1 = ␭l␸l共x1兲.

共4兲

The eigenvalues ␭l 艌 0 represent the occupation of the natural orbital ␸l, and 兺l␭l = N. From now on we assume that the eigenvalues are ordered such that ␭1 ⬎ ␭2 ⬎ ¯ ⬎ ␭l. Similarly ⌫共x2 , x⬘2兲 can be calculated but one can easily show that if the wave function is symmetric with respect to permutation the natural orbitals of both ⌫共x2 , x⬘2兲 and ⌫共x1 , x⬘1兲 are the same. Finally, the condensed state in our two-boson system is one for which ␭1 ⬇ 2 and the fragmented state will ideally have ␭1 ⬇ ␭2 ⬇ 1. Performing a full variational calculation 共FVC兲 of the Hamiltonian of two particles interacting by a contact interaction can be a numerically cumbersome task indeed. This is mainly due to the fact that a ␦ function interaction occurs only along the line where x1 = x2 and, therefore, the grid used must be chosen such that this line is sampled sufficiently

⌽mn共x1,x2兲 =

冦

m = n ␾n共x1兲␾n共x2兲, m⫽n

1

冑2 关␾m共x1兲␾n共x2兲 + ␾m共x2兲␾n共x1兲兴,

冧

共6兲

where the single-particle basis functions ␾共x兲 are the solutions of the one-boson Hamiltonian given by

冉

−

冊

1 ⳵2 + V共x兲 ␾l共x兲 = ␧l␾l共x兲. 2 ⳵ x2

共7兲

The Hamiltonian in 共1兲 with the external potential 共2兲 is invariant under the following symmetry operations: Pˆ1 ⬅ 兵x1 ↔ x2其,

共8兲

Pˆ2 ⬅ 兵x1 ↔ − x2其.

共9兲

The ground state is “gerade” and the representative Hamiltonian matrix is constructed through the use of ⌽mn which are a product of either two even functions or two odd functions, i.e., the quantum numbers m and n in Eq. 共6兲 are both even or odd. A substantial contraction of the basis set is achieved by using criteria based on the first-order perturbation theory in order to select the basis functions that will be used in the diagonalization of the Hamiltonian 共1兲. The first-order correction to the wave function ␾0共x1兲␾0共x2兲 is 共1兲 = ⌽00

兺

Wmn⌽mn共x1,x2兲,

共10兲

m艌n,m⫽0

where Wmn =

具⌽mn共x1,x2兲兩␦共x2 − x1兲兩⌽00共x1,x2兲典 , 2␧0 − ␧m − ␧n

共11兲

and 2␧0 is the ground-state energy of two non interacting bosons. In Fig. 2, we portray the absolute value of these expansion coefficients with respect to different quantum numbers. The super index, k = 1, 2, 3, …, in Fig. 2 is ordered such that 共n,m兲 = 共0,1兲,共0,2兲, ¯ ,共0,N兲,共1,1兲, 共1,2兲, ¯ ,共1,N兲, ¯ ,共N,N兲,

共12兲

where N is the highest quantum number chosen for the one

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Eq. 共1兲, one is left with the energy functional, EGP = 2

再冕

␾*hˆ␾ dx +

a0 2

冕

冎

兩␾兩4dx ,

共14兲

where hˆ is the single-particle Hamiltonian 关see Eq. 共7兲兴. Minimizing the energy with respect to ␾, one finds a singleparticle nonlinear Schrödinger equation, i.e., the well known GPE, 关hˆ共x兲 + a0兩␾共x兲兩2兴␾共x兲 = ␮␾共x兲,

共15兲

where ␮ is the chemical potential and is related to the energy per particle by

FIG. 2. The absolute value of the expansion coefficients in firstorder perturbation theory 关see Eqs. 共10兲 and 共11兲兴. k is a super index enumerating the different particle quantum number pairs 共m , n兲. The nonvanishing coefficients are localized around specific configurations that relate to states of the noninteracting bosons 关the central trap potential-well-plus-barriers in Fig. 1 supports a single bound state, a few resonance states, and a 共quasi兲 continuum兴. See Sec. II for further discussion.

particle basis set. It turns out that the values of Wmn peak at a specific k. The k’s corresponding to these peaks are those for which both quantum numbers m and n belong to basis functions that are localized in between the two potential barriers, i.e., the bound state and the “resonance” states of the central potential in Fig. 1. The energies of these one-particle states are indicated in Fig. 2. The important contributions to Wmn come from values of k located at and around these peaks. The latter correspond to 共m , n兲 pairs where m belongs to a localized basis function while n belongs to a delocalized function which, physically, describes the “continuum” states of the central potential in Fig. 1. One can now choose the basis set ⌽mn by initially selecting those k’s which correspond to peaks in Wmn and then choose an energy up to which all delocalized basis functions will also be used to construct the configuration ⌽mn 关see Eq. 共6兲兴 such that one of the basis functions is either a resonance or the bound state of the central potential and the other is a delocalized basis function. This criterium for selecting the basis functions provides a very efficient and rapid convergence of the calculation. The discussion of the results of the full variational calculations and the analysis of the eigenvalues of the density matrix will be presented in Sec. IV.

EGP a0 =␮− 2 2

⬁

兩␾共x兲兩4dx.

共16兲

−⬁

A known characteristic of the GPE is the bifurcation of the energy 共or the chemical potential兲 under different conditions. Examples for such bifurcations can be found in 关23–25兴. A further discussion of this issue will be given in the next section. In this paper we also consider a more general mean-field approach developed recently, the BMF. More specifically, we consider a private case of the BMF, the MF共2兲. We shall present here only the final results as the details of the derivation are given elsewhere 关6,12兴. In the MF共2兲 approach one assumes that n1 bosons occupy the orbital ␾1 while n2 = N − n1 occupy the orbital ␾2, where N is the total number of bosons in the system. For the case of only two bosons the only possible choice for n1 and n2 is n1 = n2 = N / 2 = 1. The multiparticle wave function for the case of two bosons thus takes on the simple appearance ⌿共x1,x2兲 =

1

冑2 关␾1共x1兲␾2共x2兲 + ␾1共x2兲␾2共x1兲兴.

共17兲

The energy expectation value of the Hamiltonian discussed above 关Eq. 共1兲兴 then reads E MF共2兲 =

再冕

␾*1hˆ␾1dx +

冕

␾*2hˆ␾2dx + 2a0

冕

冎

兩␾1兩2兩␾2兩2dx . 共18兲

Minimizing the energy with respect to ␾1 and ␾2 we are left with the two coupled nonlinear equations 关hˆ共x兲 + 2a0兩␾2共x兲兩2兴␾1共x兲 = ␮11␾1共x兲 + ␮12␾2共x兲, 共19兲 关hˆ共x兲 + 2a0兩␾1共x兲兩2兴␾2共x兲 = ␮22␾2共x兲 + ␮21␾1共x兲, 共20兲

III. MEAN-FIELD SOLUTIONS

The GP approach for the case of a “spinless” gas at zero temperature 关10,11兴 approximates the multiparticle wave function as a product of identical single-boson orbitals. For the case of two bosons the wave function reads ⌿共x1,x2兲 = ␾共x1兲␾共x2兲.

冕

共13兲

The expectation value of the energy is given by, EGP ˆ 兩⌿共x , x 兲典, and inserting the Hamiltonian in = 具⌿共x1 , x2兲兩H 1 2

where ␮11 and ␮22 are the chemical potentials of ␾1 and ␾2, respectively, and ␮12␮21 are the Lagrange multipliers responsible for the orthogonality condition on ␾1 and ␾2. In the BMF approach the n1 and n2 are obtained variationally. Consequentially, the GP ansatz can be obtained as a special case of the BMF 共n1 = 2, n2 = 0兲. The BMF energy as a function of the boson-boson interaction strength a0 can thus be the GP energy up to some value of a0 and the MF共2兲 energy for larger values.

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FIG. 3. 共Color online兲 The system’s energy for different values of the interaction strength 共a0兲 is depicted for different methods of calculation. The numerically exact energy 关solid 共black兲 line兴, the two branches of the GPE 关dash-dot line 共blue兲兴 for the upper branch 关dash-dot line 共violet兲兴 for the lower branch, and the MF共2兲 energy 关dashed 共red兲 line兴. Note the transition to fragmentation and the respective critical interaction strengths marked in the figure for the exact and BMF results. The bifurcation point of the GP energy is also indicated.

In practice, the ground-state energy and wave function of the GPE and the MF共2兲 were calculated by relaxation of an initial wave function in imaginary time ␶ = −it 关26兴:

⳵ ␾共x兲 ˆ ␾共x兲. =−H ⳵␶

共21兲

Propagating in imaginary time the initial wave function 共renormalized in every time step兲 will relax to the solution ˆ , i.e., the ground state. The with the smallest eigenvalue of H results were verified by an adiabatic iterative procedure where the Hamiltonian is diagonalized in every step using the solution of the preceding iteration where the solution is chosen such that the total energy of the system is minimized. Unlike the propagation algorithm mentioned above this procedure requires a very large number of iterations to converge. The upper branch mentioned before which is, in a sense, an excited state of the GPE can be found with the diagonalization method mentioned above using an overlap condition rather than an energy condition in order to select the solution to be inserted in the following iteration. A discussion of the results obtained will be given in the following section. IV. COMPARISON OF EXACT AND MEAN-FIELD RESULTS

The results of the numerically exact, the GPE and of the MF共2兲 calculations, are depicted in Fig. 3 for different values of the interaction strength parameter a0. It is clear from the results that increasing the interaction strength increases the error of the GPE result. At infinite strength the GP energy diverges. The error originates from the underlying particular mean-field approximation which does not account, by defi-

FIG. 4. 共Color online兲 The density 关兩⌿共x1 , x2兲兩2兴 as a function of the dimensionless positions x1 and x2 of the two particles of the FVC and GPE calculation for no interaction, i.e., a0 = 0, and for an interaction strength of a0 = 1. Note the dynamical correlation effect on the wave function in the exact calculation and the lack of it in the standard mean-field approach.

nition, for the dynamical correlation between the two bosons. The effect of the dynamical correlation is not only apparent in the energy of the system but also in the structure of the wave functions. Let us first discuss the results up to a0 ⲏ 1. The GPE describes an isotropic potential field which includes both the external potential and the repulsive interaction. This is an approximation of the true field in which the repulsive interaction is only along the line in space where x1 = x2 关see Eq. 共1兲兴. Therefore, as the interaction increases the wave function that is obtained from the GPE will remain isotropic, simply expanding as it is pushed outwards by the repulsive interaction, see the second row in Fig. 4. The exact wave function, on the other hand, exhibits the correct spatial behavior of the interaction and, as can be expected from the repulsive nature of the interaction, the particles tend to stretch in one direction as the interaction is increased 共see the first row in Fig. 4兲. So far we have shown that the dynamical correlation between the particles has an increasing effect on the GPE energy as the interaction strength grows. Still the qualitative description of the system using the GPE is favorable considering the simplicity of the GPE especially for large systems where exact solutions are intractable. This qualitative description will hold, however, only for interaction strengths prior to a critical interaction strength where the fragmentation of the system occurs. The results presented in Fig. 3 show that the behavior of the energy as a function of the interaction strength changes once a certain critical interaction strength is reached. This critical interaction strength varies between the different methods used and each method will initially be discussed separately. The exact energy increases monotonically up to cr at which the energy the critical interaction strength a0共exact兲 becomes nearly constant with any further increase of the interaction strength. We now return to the definitions given in

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FIG. 5. 共Color online兲 The first two eigenvalues of the exact density matrix, ␭1 共solid兲 and ␭2 共dashed兲, as a function of the interaction strength a0. After a critical interaction strength the two eigenvalues become degenerate. See Sec. IV for further discussion. Note that according to the BMF results, ␭1 is equal to 2共␭2 = 0兲 up cr cr to a0 ⬍ a0共BMF兲 and ␭1 = ␭2 = 1 for a0 ⬎ a0共BMF兲 . In the GP theory ␭1 = 2 for all values of a0.

Sec. II for the condensed and fragmented states and consider the two largest eigenvalues of the density matrix. These are depicted in Fig. 5 as a function of the interaction strength. Indeed, one can now infer that the critical interaction strength represents the transition from the condensed state to cr the fragmented state of the system. For a0 ⬍ a0共exact兲 obviously only ␭1 is substantial and the state is condensed, while cr for a0 ⬎ a0共exact兲 we find ␭1 ⬇ ␭2 ⬇ 1 and the state is fragmented. What happens after fragmentation is easily understood by comparing the wave function of the condensed state, shown in Fig. 4, with that of the fragmented state depicted in Fig. 6. In the condensed state both particles are located in between the two potential barriers 共marked I in Fig. 1兲, while in the fragmented state one particle lives in between the barriers while the other lives beyond the barriers 共marked II in Fig. 1兲. Now it is clear why the energy is almost constant above the critical interaction strength as the particles hardly interact if they are spatially separated. In optical lattices, where usually many Bosons are involved, the transition from the superfluid to the Mott insulator phase is very similar in its nature to the fragmentation phenomenon described above for the case of two bosons 关27兴. It is important to keep in mind that although we use here the same definitions for the condensed and fragmented states used in the study of many particle condensates, the system described in this work is in no way a condensate. We do, however, hope that one can learn more about the mesoscopic system from the analysis of the microscopic system which in the case presented here, i.e., a comparison between an exact solution to the approximated mean-field solution, cannot be done for the mesoscopic systems. The GP ansatz, by definition, cannot describe fragmentation. There is only one natural orbital—which is identical with the GP orbital ␾共x兲—and the respective eigenvalue of the density matrix is ␭ = 2. Nevertheless, one can define a

FIG. 6. 共Color online兲 The two particle wave functions ⌿共x1 , x2兲 as a function of the dimensionless particle coordinates x1 and x2 of the 共a兲 exact and 共b兲 GP calculations for an interaction strength a0 = 1.5 after fragmentation has taken place in the exact calculation. In 共a兲 one particle is located in between the potential barriers 共i.e., x = 0兲 and the other outside the potential barriers while in 共b兲 the particles are found in the same location. Note that the result of the BMF is essentially identical to the exact one for the interaction strength used. cr critical interaction strength a0共GPE兲 as the interaction strength for which the energy bifurcates into two branches. The upper branch represents the energy of a state where both particles remain in between the barriers while the lower branch represents a state in which both particles possess a probability to be outside of the barriers. Of course, the latter state is also condensed. We mention that condensed and fragmented states have different physical properties like, e.g., momentum distribution. Evidently up to a0 ⬇ 1 the GP formalism is able to qualitatively describe our model well. For larger values of a0, the bifurcation can even tell us that some phenomena in which the particles can go beyond the potential barriers may occur. But this strongly contradicts the physics of the fragmented state. Within the GP formalism, the multiparticle wave function is a product of the same single particle orbital for each of the particles in the system. Surpassing the critical interaction strength, the orbital represents a particle that has some probability to be in between the potential barriers and some probability to be beyond the barriers. Therefore, there is always a probability of finding all particles in the same place, i.e., either both are between the barriers or both are beyond the barriers. The particles can still “feel” each other even after fragmentation has occurred and this is why the energy of the lower branch of the GPE does not approach a constant after the bifurcation. This is not consistent with the exact results. The above discussion is further substantiated in Fig. 6 where the exact wave function and that corresponding to the lower branch of the GPE are portrayed for an interaction strength greater than the critical interaction strength. Let us now turn to the results obtained by the BMF. According to this approach the energy is given by the GP energy for those values of a0 where this energy is lower than the MF共2兲 energy and by the MF共2兲 energy otherwise 共see Fig. 3兲. When the MF共2兲 energy becomes lower than that of the GPE, the configuration in which two orbitals are occupied is energetically favorable to the occupation of one and

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the same orbital, i.e., the GP orbital. The intersection between the energies of the MF共2兲 and of the GPE defines a cr new critical interaction strength a0共BMF兲 . One can see that the critical interaction strength of the intersection is fairly close to the critical interaction strength where the bifurcation of cr the GPE occurs. For a0 ⬎ a0共exact兲 the energy calculated by the MF共2兲 is very accurate and compares very well with that of the numerically exact calculation. In addition, the energy hardly varies with the interaction strength as is to be expected of the fragmented two-boson state. What makes the MF共2兲 such a good approximation? The exact wave function can be reconstructed using the natural orbitals of the density matrix as is easily verified by substituting into Eq. 共4兲, according to ⌿共x1,x2兲 =

1

冑2 兺l dl␸l共x1兲␸l共x2兲,

共22兲

where ␭l = d2l . Above the critical interaction strength there are only two degenerate states in the spectral representation of the density matrix each having an eigenvalue of 1 共i.e., ␭1 = ␭2 = 1兲. One can, therefore, construct the wave function from linear combinations of the two degenerate natural orbitals. Taking 关␸1共x兲 + ␸2共x兲兴 ⬅ ␾1 and 关␸1共x兲 − ␸2共x兲兴 ⬅ ␾2, one can construct the wave function such that, ⌿共x1,x2兲 =

1

冑2 关␾1共x1兲␾2共x2兲 + ␾2共x1兲␾1共x2兲兴.

共23兲

It is easy to verify that Eq. 共23兲 is equivalent to Eq. 共22兲 by putting d1 = −d2 = 1. The wave function in Eq. 共23兲 has exactly the same structure as the wave function used in the MF共2兲 calculation, i.e., Eq. 共17兲. This is why the MF共2兲 calculation describes the fragmentation of the two particles so well. It has the same form as the exact wave function after fragmentation occurs. This form of the wave function is precisely what reflects the difference between the GP on the one hand and the exact and MF共2兲 on the other hand in describing the fragmentation phenomenon. The wave functions described in Eqs. 共23兲 and 共17兲 are entangled states. This essentially means that once a particle is measured either in between the barriers 共area I in Fig. 1兲 or beyond the barriers 共area II in Fig. 1兲 the other particle can be located in the complementary area only. The GP wave function, on the contrary, does not describe an entangled state and both particles, apparently, can be found in the same place.

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V. CONCLUDING REMARKS

The effect of changing the interaction 共repulsive兲 between two bosons in an external trap has been studied using a full variational treatment. The results were compared to those obtained from the approximate mean-field approaches, the GPE and the BMF. At weaker interaction strengths 共a0 ⬍ acr 0 兲, the GPE cannot account for the dynamical correlation between the two bosons which results in an error in the calculated energy. This limitation of the GPE although significant quantitatively, does not affect the qualitative description of the system for weak interaction strengths. Once the interaction strength reaches a critical value 共a0 = acr 0 兲, fragmentation occurs and the GPE, although capable of predicting a change in the system, can no longer describe the system neither quantitatively nor qualitatively. In order to continue and describe the system within the mean-field approximation one must relax some of the GPE assumptions and construct a more general mean-field formalism. This is precisely the role of the BMF. Indeed, the BMF can describe the system in a qualitative and even in a quantitative way very well. The BMF, however, cannot compensate for the dynamical correlation and cannot offer any improvement of the description cr 兲 from prior to the critical interaction strength 共a0 = a0共BMF兲 which on the MF共2兲 energy becomes lower than that of the GPE. At present, exact solutions of trapped many boson systems are still intractable and, therefore, the use of mean-field approaches with which sound approximate solutions of such systems can be found, are indispensable. To describe fragmentation of the system requires a more general mean-field approach, i.e., the BMF. In systems containing more than two bosons, when the interaction strength is sufficiently strong one may find fermionization of the system 关28兴 and in optical lattices a transition from superfluidity to a Mott insulator 关29兴, all of which can be described by the BMF approach.

ACKNOWLEDGMENTS

The authors acknowledge useful discussions with Ido Gilary and Alexej I. Streltsov. This work was supported in part by the Israel Science Foundation 共Grant No. 1152/04兲 and by the Fund of promotion of research at the Technion. The support of the DIP is also gratefully acknowledged.

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