RDM of Cold Atom Gas Cederbaum

8/21/2019 RDM of Cold Atom Gas Cederbaum

1/17

Reduced density matrices and coherence of trapped interacting bosons

Kaspar Sakmann,*Alexej I. Streltsov, Ofir E. Alon, and Lorenz S. CederbaumTheoretische Chemie, Universitt Heidelberg, D-69120 Heidelberg, Germany

Received 22 February 2008; published 8 August 2008

The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are in-

vestigated numerically on a many-body level from first principles. Correlations in real space and momentumspace are treated. The coherence properties are analyzed. The results are obtained by solving the many-bodySchrdinger equation. It is shown in an example how many-body effects can be induced by the trap geometry.A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed alonga pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arisefrom the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals ofthe system are obtained and discussed. It is shown how the fragmentation of the condensate can be understoodin terms of its natural geminals. The many-body results are compared to those of mean-field theory. The bestsolution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined.In these limits the behavior of the correlation functions is explained within an analytical model.

DOI: 10.1103/PhysRevA.78.023615 PACS numbers: 03.75.Hh, 05.30.Jp, 03.65.w, 03.75.Nt

I. INTRODUCTION

The computation of correlation functions in interactingquantum many-body systems is a challenging problem ofcontemporary physics. Correlations between particles canexist in time, in real space, or in momentum space. Ofcourse, all combinations of the above three cases are pos-sible. Since the first experimental realization of Bose-Einstein condensatesBECsin ultracold atomic gases13,great experimental and theoretical progress has been made inthe determination of the coherence and the correlation func-tions of Bose-condensed systems. Over the years experi-ments have measured more and more accurately first-,second-, and to some extent even third-order correlations oftrapped BECs, see 411. Theoretically, the correlationfunctions of trapped interacting BECs have been investigatedin numerous works, see, e.g., Refs.1221. While analyticalapproaches from first principles are usually restricted to treathomogeneous gases without any trapping potential, numeri-cal methods can overcome this restriction. It is important tonote that the shape of the trapping potential can have a sub-stantial impact on the properties of the many-body system.This is particularly true for issues concerning condensation22 and fragmentation of Bose systems23. For example,the ground state of weakly interacting condensates in har-monic traps is almost fully condensed, while the ground state

of double-well potentials can be fragmented or condensed,depending on the height of the barrier, the number of par-ticles and the interaction strength2429. In this work weinvestigate first- and second-order correlations of trapped in-teracting condensates and their coherence properties depend-ing on the trap geometry from first principles. Our results are

obtained by solving the many-body Schrdinger equation ofthe interacting system numerically. From this many-body so-lution we extract the first- and second-order reduced densitymatrices which allow us to compute all real and momentumspace first- and second-order correlations and in particularthe fragmentation of the condensate. For illustration pur-poses we consider a stationary system in the ground state toshow how many-body effects can become dominant whenthe trap geometry is varied. As a numerical method to solvethe interacting many-body problem we use the recentlydeveloped multiconfigurational time-dependent Hartreemethod for bosons MCTDHB, which propagates a given

many-body state in time3032. By propagation in imagi-nary time it allows us to investigate the ground state andother stationary states. Alternatively, one can use the station-ary multiconfigurational Hartree method for bosons to com-pute these states 29. In order to identify true many-bodyeffects in the correlation functions, we compare our many-body results with those based on mean-field approaches.More specifically, we compute the energetically lowestmean-field solution of the same system for comparison. Thisis the best approximation to the true many-body wave func-tion within mean-field theory and, thereby, allows us to pin-point the limits of mean-field theory. A general method tocompute thisbest mean-fieldsolution has been developed in

our group 2528,33. For completeness we compare themany-body results also to the widely used Gross-Pitaevskiimean-field solution. In order to understand first- and second-order correlations in an intuitive way, we develop an analyti-cal mean-field model which explains the general structure ofour results in those regions where many-body effects can besafely neglected.

This paper is organized as follows. In Sec. II we reviewbasic facts about reduced density matrices, correlation func-tions, and coherence. In Sec. III we give a brief introductionto the numerical many-body and mean-field methods that weuse. In Sec. IV we introduce a generic one-dimensionalmodel system that we solve. In particular, we identify mean-

*Corresponding author; [email protected]@pci.uni-heidelberg.deofir.alon@[email protected]

PHYSICAL REVIEW A 78, 0236152008

1050-2947/2008/782/02361517 2008 The American Physical Society023615-1
http://dx.doi.org/10.1103/PhysRevA.78.023615http://dx.doi.org/10.1103/PhysRevA.78.023615


2/17

field and many-body regimes of this system. We explain thetransition from a condensed to a fragmented state in terms ofthe natural geminals of the system. In Secs. V and VI wepresent our many-body results for the first- and second-ordercorrelation functions and the coherence properties of themodel system. We compare our many-body results withthose obtained by using thebest mean-fieldand the Gross-

Pitaevskii approximation. The structure of the correlationfunctions and the coherence are explained within an analyti-

cal mean-field model in the limits where mean-field theory isapplicable.

II. BASIC DEFINITIONS

We consider a given wave functionr1 , . . . , rN; t ofNidentical spinless bosons with spatial coordinatesr iin D di-

mensions. The pth order reduced density matrix RDM34,35is defined by

pr1, .. . ,rpr1, ... ,rp;t=

N!

Np! r1, .. . ,rp,rp+1, .. . ,rN;t*r1, ... ,rp,rp+1, .. . ,rN;tdrp+1. . .drN, 1

where the wave function is assumed to be normalized

t t =1. Equivalent to the above equation 1,

pr1 , . . . , rp r1 , . . . ,rp ; t can be regarded as the kernel ofthepth order reduced density operator

p = N!

Np!TrNptt 2

in Hilbert space, where TrNp specifies taking the partialtrace over Np particles. Since the wave function is sym-metric in its coordinates, it does not matter over which par-ticles the trace is taken. In what follows, we add as anadditional subscript if a result is only valid for states ofa particular form.

The diagonal pr1 , . . . ,rp r1 , . . . ,rp ; t is the p-particleprobability distribution at time t multiplied by N! / Np!.The pth order RDM p can be expanded in its eigenfunc-tions, leading to the representation

pr1, .. . ,rpr1, ... ,rp;t

=i

nipti

pr1, .. . ,rp, tip*r1, .. . ,rp, t. 3

Here,niptdenotes theith eigenvalue of thepth order RDM

and ipr1 , . . . , rp , t the corresponding eigenfunction. The

eigenfunctions are known asnatural p functions and the ei-genvalues as natural occupations. For p =1 and p =2 theeigenfunctions are also known as natural orbitalsand natu-ral geminals, respectively. We order the eigenvalues ni

ptfor every p nonincreasingly, such that n1

pt denotes thelargest eigenvalue of thepth order RDM. The normalizationof the many-body wave function and Eqs.1and3put therestriction

i

nipt=

N!

Np! 4

on the eigenvalues of the pth order RDM. Thus the largesteigenvaluen 1

pt is bounded from above by35,36

n1p

t

N!

Np! . 5

Lower bounds on n1pt can be derived, relating RDMs of

different order37,38. In particular, for the casep =2 it canbe shown that37

n12t n1

1tn11t 1 . 6

It is a well-known fact that the natural orbitals of a sym-metricor antisymmetric function constitute a sufficientone-particle basis to expand and the eigenfunctions of theRDMs for all p 35. It is therefore possible to construct thenatural geminals in the basis of one-particle functionsspanned by the natural orbitals.

The determination of accurate bounds on eigenvalues ofRDMs is an active field of research 35,36 since it is pos-sible to express theexactenergy expectation value of a quan-tum system of identical particles interacting via two-bodyinteractions by an expression involving only the naturalgeminals, i

2r1 , r2 , t, and their occupations,ni2t. For a

general Hamiltonian

H=i=1

N

hri+ij

N

Wri rj , 7

consisting of one-body operatorshri and two-body opera-torsWri rj, the expectation value of the energyEcan beexpressed by making use of the time-dependent naturalgeminals i

2x1 ,x2 , t and following Refs. 35,36 throughthe equation

E=1

2

i

ni2t dr1dr2i2*r1,r2, thr1+hr2

N 1

+Wr1 r2i2r1,r2, t. 8Note that the many-body wave function does not appear ex-plicitly in Eq.8. We will not go any further into the detailsof these approaches to many-body physics and refer thereader to the literature35,36.

SAKMANN et al. PHYSICAL REVIEW A 78, 0236152008

023615-2


3/17

The natural orbitals also serve to define Bose-Einsteincondensation in interacting systems. According to Penroseand Onsager22, a system of identical bosons is said to becondensedif the largest eigenvalue of the first-order RDM isof the order of the number of particles in the system, n1

1

= ON. If more than one eigenvalue of the first-order RDMis of the order of the number of particles, the condensate is

said to be fragmented, according to Nozires and SaintJames23.

Equivalent to Eq. 1, the pth order RDM can be ex-pressed through field operators as

pr1, .. . ,rpr1, .. . ,rp;t

=t r1. . . rprp. . . r1t ,

9

where the Schrdinger field operators satisfy the usualbosonic commutation relations

r,

r=r r

,

r,

r

= 0. 10

The representation given in Eq.9shows that thepth orderRDM is identical to the pth order correlation function atequal times12,39.

In order to discuss correlations not only in real space, butalso in momentum space, we define the Fourier transform ofa function fr1 , . . . , rpofp D-dimensional coordinatesr iby

fk1, .. . ,kp= 1

2pD/2 dpreil=1p klrlfr1, .. . ,rp .

11

By applying the Fourier transform, Eq.11, to the coordi-nates r1 , . . . , rp and r1 , . . . ,rp of the natural p-functionsi

pr1 , . . . , rp , tand ipr1 , . . . , rp , tin Eq.3, one arrives

at the momentum space representation of p:

pk1, .. . ,kpk1, .. . ,kp;t

=i

nipti

pk1, .. . ,kp, tip*k1, .. . ,kp,t .

12

The diagonal pk1 , . . . ,kp k1 , . . . , kp ; t in momentumspace is thep-particle momentum distribution at timet, mul-tiplied by N! / Np!. It can be shown that the p-particle

momentum distribution at large momenta is dominated bycontributions ofpr1 , . . . , rp r1 , . . . , rp ; tclose to the diag-onal, i.e.,r i rifor i = 1 , . . . ,p. Similarly, thep-particle dis-tribution at low momenta is dominated by the behavior of

pr1 , . . . , rp r1 , . . . ,rp ; t on the off-diagonal at large dis-tances betweenr iand r i. See the Appendix for more details.

Apart from the p-particle distributions themselves, eitherin real space or in momentum space, it is also of great inter-est to compare thep-particle probabilities to their respectiveone-particle probabilities. Thereby, it becomes possible toidentify effects that are due to the quantum statistics of theparticles and the interaction between them. Thenormalizedpth order correlation functionat time tis defined by39

gpr1, .. . ,rp,r1, .. . ,rp;t=

pr1, .. . ,rpr1, .. . ,rp;t

i=1p 1riri;t1riri;t13

and is the key quantity in the definition of spatial coherence.Full spatial pth order coherence is obtained if

n

r1 , . . .rp r1

, . . . , rp

; tfactorizes for allnpinto a prod-uct ofone complex valued functionEr , t of the form

nr1, ... ,rnr1, .. . ,rn;t

=E*r1,t E*rn,tErn,t Er1, t. 14

In this case

gnr1, ... ,rn,r1, .. . ,rn;t= 1 15

for all np. Otherwise, the state t is only partiallycoherent. Full coherence in a system with a definite numberof particlesNcan only be obtained forp =1 40. However,when the particle numberNis large,pth order coherence canbe obtained up to correctionsO1 /N, at least forpN40.

The diagonal of the normalized pth order correlationfunction gpr1 , . . . , rp , r1 , . . . ,rp ; t gives a measurefor the degree of pth order coherence. For valuesgpr1 , . . . ,rp , r1 , . . . ,rp ; t11 the detection probabili-ties at positionsr1 , . . . , rp are correlatedanticorrelated.

Note that if Eq.14holds in real space, it must also holdin momentum space, as can be seen by Fourier transformingeach of the 2n variables in Eq.14. Futhermore, it is alsopossible to define the normalizedp th order correlation func-tion in momentum space by

gpk1, .. . ,kp,k1, ... ,kp;t=

pk1, .. . ,kpk1, .. . ,kp;t

i=1p 1kiki;t1kiki;t.

16

The diagonal of Eq. 16, gpk1 , . . . ,kp , k1 , . . . , kp ; t, ex-presses the tendency ofp momenta to be measured simulta-neously. For valuesgpk1 , . . . ,kp , k1 , . . . , kp ; t11thedetection probabilities of momentak1 , . . . ,kp are correlatedanticorrelated. The pth order momentum distribution

pk1 , . . . ,kp k1 , . . . ,kp ; t depends on the entire pth orderRDM pr1 , . . . ,rp r1 , . . . ,rp ; t; see the Appendix. Thus,gpk1 , . . . ,kp ,k1 , . . . ,kp ; t provides information aboutthe coherence of t which is not contained ingpr1 , . . . ,rp , r1 , . . . ,rp ; t. Of course, the reverse is equallytrue.

In Young double slit experiments using noninteractingbosonsg1r1 , r1 ; t0 =1 ensures the maximal fringe visibil-ity of an interference pattern. Here, t0 is the time of releasefrom the slits. See, e.g., Ref. 41, for an experiment usingBose-Einstein condensates. However, if interactions duringthe expansion behind the slit are not negligible, there is nosimple relation between the fringe visibility and the wavefunctionr1 , . . . , rN, t0at the time of release from the slits.In other words, the interaction between the particles canmodify the observed interference pattern4244.

In order to determine the degree of coherence of a givensystem, it is necessary to quantify how well Eq.14 is sat-isfied. A visualization of the degree of coherence is highly

REDUCED DENSITY MATRICES AND COHERENCE OF PHYSICAL REVIEW A 78, 0236152008

023615-3


4/17

desirable, as it helps to understand the coherence limitingfactors in an intuitive manner. Already for one-dimensionalsystems,D =1, the normalized first-order correlation functionat timet,g1r1 , r1 ; t, is a complex function of two variablesand cannot be visualized in a single plot. It is, therefore,necessary to consider quantities that sample parts ofgpr1 , . . . ,rp , r1 , . . . , rp ; t. In one-dimensional systems

g1r1 ,r1 ; t2 can be represented as a two-dimensional plotand gives a measure for the degree of first-order coherence.Similarly, g2r1 ,r2 , r1 , r2 ; t and g

2k1 , k2 , k1 , k2 ; t arereal and can be represented as two-dimensional plots, if D=1. In Secs. V and VI we will visualize the degree of first-and second-order coherence of a one-dimensional system,defined in Sec. IV, by means of g1r1 , r1 ; t

2,g2r1 , r2 ,r1 , r2 ; t and g

2k1 ,k2 , k1 , k2 ; t.

III. NUMERICAL METHODS

The main goal of this work is to investigate exactly thebehavior of first- and second-order correlation functions in

interacting many-body systems. This requires the computa-tion of the exact many-body wave function which is gener-ally a difficult problem to solve. In some cases, when thegeneral form of the wave function is known a priori, anexact solution can be obtained, either by solving transcen-dental equations or by exploiting mapping theorems, see e.g.,Refs.18,4553. However, in general it is necessary to solvethe full many-body Schrdinger equation numerically in anefficient way. In Sec. III A we give a brief account of thenumerical method MCTDHB to solve the interacting many-boson problem. In order to find out to which extent mean-field methods are applicable to bosonic interacting many-body systems, we compare our many-body results with those

based on mean-field approaches, namely the commonly usedGross-Pitaevskii mean field5456and thebest mean fieldBMF, which we describe briefly in Sec. III B. It is beyondthe scope of this work to explain either MCTDHB or BMF indetail and we refer the reader to Refs. 3032 and Refs.2528,33for more detailed explanations of MCTDHB andBMF, respectively.

A. The many-body wave function

The exact wave function of an interactingN-boson prob-lem can always be expanded in any complete set of perma-nents of Nparticles. Each of the permanents is constructedfrom a complete set of single-particle functions which arecommonly referred to asorbitals. Practical computations canneverbe carried out in complete basis sets and, therefore, itis crucial to cut the basis set carefully.

Our starting point is the Schrdinger picture field operator

r satisfying the usual bosonic commutation relations,Eqs.10. It is convenient to expand the field operator in acomplete set oftime-dependentorthonormal orbitals,

r=k

cktkr, t, 17

where the time-dependent annihilation and creation operatorsobey the usual commutation relations cktcj

t cjtckt

=kj for bosons at any time. Note that it is not necessary tospecify the shape of the orbitals at this point.

The many-body Hamiltonian7 is standardly written in

second quantized form asH = h+ W, namely

H =k,q

ckcqhkq+

1

2k,s,l,q

ckcs

clcqWksql, 18

where the matrix elements of the one-body Hamiltonianhrand two-body interaction potentialWr r are given by

hkqt= k*r,thrqr,tdr ,

Wksqlt= k*r,ts*r, tWr rqr,tlr, tdrdr. 19

Theansatzfor the many-body wave functiontin MCT-DHB is taken as a linear combination of time-dependent per-manents

t=n

Cntn1,n2, .. . ,nM;t ,

n1,n2, .. . ,nM;t

= 1

n1!n2! nM!c1

tn1c2tn2 cM

tnMvac ,

20

where n1 , n2 , . . . ,nM; t is assembled from the time-dependent orbitals above. The summation in Eq. 20 runs

over all N+M1

N permanents generated by distributing Nbosons over M orbitals. We collect the occupations in thevector n=n1 ,n2 , . . . , nM, where n1 +n2 + . . . + nM=N. Ofcourse, ifMgoes to infinity then theansatz20for the wavefunction becomes exact since the set of permanentsn1 , n2 , . . . , nM; t spans the complete N-particle Hilbertspace. In practical computations we have to restrict thenumber M of orbitals from which the permanentsn1 , n2 , . . . , nM; t are assembled. By substituting the many-body ansatz 20 into the action functional of the time-dependent Schrdinger equation, it is possible to derive acoupled set of equations of motion containing the coeffi-cients Cnt and the set of time-dependent orbitalskr , t.

The equations are obtained by requiring the stationarity ofthe action functional with respect to variations of the coeffi-cients Cnt andthe set of time-dependent orbitalskr , t.These coupled equations have to be solved simultaneously,leading to an efficient wave packet propagation method forbosons3032. At first sight it might seem to be an unnec-essary complication to allow the orbitalskr , t to dependon time. However, this additional degree of freedom allowsboth the basis of one-particle functionskr , tand the coef-ficients Cnt to be variationally optimal with respect to allparameters of the many-body Hamiltonian interactionstrength, number of particles, external potentialat any time.Note that this is fundamentally different from a multimode


023615-4


5/17

ansatzwith fixed orbitals, such as the Bose-Hubbard model,in which the quality of the chosen basis set may deteriorateas the system evolves in time.

In order to investigate stationary properties of Bose-Einstein condensates, we use a many-body relaxationmethod. By propagating a given initial guess in imaginarytime with MCTDHB, the system relaxes to the ground state,

which allows us to treat stationary systems as well. A neces-sary requirement for this procedure to work is that the initialguess has nonzero overlap with the ground state. The varia-tional principle ensures that the set of orbitalsk is varia-tionally optimal, in the sense that the lowest ground stateenergy within the Hilbert subspace of N bosons distributedoverMorbitals is obtained, see in this respect Ref.29. Wewill not go any further into the details of MCTDHB and referthe reader to the literature3032.

B. Best mean field

The exact many-body wave function of a bosonic system

ofNparticles can always be expanded in an infinite weightedsum over any complete set of permanents ofNparticles. Inmean-field theory the exact many-body wave function is ap-proximated by a single permanent. This single permanent isbuilt from a number MNof orthogonal orbitals in whichthe N bosons reside. In the field of Bose-Einstein conden-sates one particular mean field, the Gross-Pitaevskii GPmean-field, has proven to be very successful. In analogy tononinteracting BECs, in GP theory it is assumed that themany-body wave function is given by a single permanent inwhich all particles reside inone orbital, i.e., M=1. A mini-mization of the energy functional with the GPansatz wavefunction leads to the famous Gross-Pitaevskii equation

5456. The solution of the GP equation yields the singleorbital from which the GP mean-field permanent is con-structed.

However, it has been shown2528,33that the GP meanfield is not always the energetically lowest mean-field solu-tion. The assumption that all particles occupy the same or-bital is too restrictive. Especially in multiwell trapping ge-ometries the energetically lowest mean-field solution can befragmented2528,33, see also Sec. II.

In order to obtain the energetically lowest mean-field so-lution, it is necessary that theansatzfor the wave function isof the most general mean-field form. Due to the variationalprinciple, the minimization of the respective energy func-

tional with respect to all parameters of theansatzwave func-tion will then give thebestsolution within mean-field theory.It is therefore legitimate to call this mean-field solution thebest mean fieldBMF. A procedure to obtain the best mean-field BMF solution numerically has been developed re-cently2528,33.

In the best mean-field approach theansatz for the wavefunction is taken as a single permanent of N bosonsdistributed over M time-independent orthonormal orbitalskr:

=n1,n2, .. . ,nM. 21

Using this ansatz for the wave function, the energy func-

tional is minimized by a variation over the number of orbit-als M, the occupation numbers ni and the orbitals krthemselves 25,28. The variation leads to a set of couplednonlinear equations that have to be solved to obtain the BMFsolution. Thereby, the energetically most favorable perma-nent is selected to approximate the true many-body wavefunction. The GP mean field is contained in the BMFansatz

as can be seen by restricting the number of orbitals to M=1.

IV. A MODEL AND ITS PHYSICS

In order to examine correlation functions of Bose-condensed systems, we now turn to a specific example. Forsimplicity we work in one dimension,D =1, and henceforthwe substitute r =x and k = k. We will study the correlationfunctions of N=1000 repulsively interacting bosons in adouble-well trap at various barrier heights. The dynamics ofa similar system has been investigated recently in the contextof a dynamically raised barrier30. In order to isolate physi-

cal effects that are due to the trapping geometry and not dueto dynamical parameters such as the rate at which the barrieris raised, etc., we restrict our discussion to the ground state atdifferent barrier heights. The restriction to a stationary stateallows us to omit the time argument in all physical quantitiesfrom now on. Double-well systems have the interesting prop-erty that depending on the height of the barrier and/or theinteraction strength, the ground state undergoes a transitionfrom a single to a fragmented condensate24,26,28,29. Weshall show how this transition from a condensed state to afragmented condensate manifests itself in the correlationfunctions.

We work with a dimensionless Hamiltonian of the form

H=i=1

N

12

2

xi2+Vxi+ 0

ij

N

xixj , 22

to solve the stationary Schrdinger equation H=E. Allquantities in Eq.22 are dimensionless and the connectionto a dimensional Hamiltonian

H=i=1

N

22m

2

xi2+ V

xi+ 0ij

N

xixj 23

is made by the relationsxi =xi /L, where L is a length scale,

Vxi =mL2

2VLxi, 0 =

mL

20, x1 , . . . ,xN =x1L , . . . ,xNL,

andE=EmL

2

2 . As an external potential we choose a harmonictrap with an additional central barrier of variable height

Vxi =12xi

2 +Aexi2/22, whereA is the height of the potential

barrier and=2 a fixed width. For the strength of the dimen-sionless interparticle interaction we choose 0 =0.01. In thecomputations using MCTDHB we restrict the number of or-bitals toM=2, yielding a total ofN+M1N =1001 permanents.

A. Condensed state

We begin with a discussion of the ground state energy asa function of the barrier height. The ground state energy perparticle of the many-body solution, EMCTDHB /N blue line


023615-5


6/17

with circles, is shown in Fig.1 top. EMCTDHB /Nincreaseswith the height of the central barrier. The energy differencesper particle of the many-body and the BMF solution withrespect to the GP solution, EMCTDHB EGP /N blue linewith circles and EBMFEGP /N red line with triangles,are shown in the inset of Fig.1 top. The energy differenceEMCTDHB EGP /Nis negative, because the interacting sys-tem can lower its energy by depleting the condensate. At lowbarrier heights the GP mean field is the best mean field andthusEBMFEGP /N=0. A comparison of the energy scalesof Fig.1and its inset reveals that the energy of the many-body solution, the BMF solution, and the GP solution arevery close at all barrier heights. The nature of the many-body

ground state at different barrier heights varies neverthelessvery strongly, as we shall show below.

Figure 1 middle shows the occupations n11/N and

n21/Nof the first and second natural orbitals of the many-

body wave function as a function of the barrier height, com-puted with MCTDHB. The largest eigenvalue of the first-order RDM, n1

1, is only restricted by Eq. 5 and can

therefore take on any value between 0 and N. The dashedlines indicate these upper and lower bounds onn1

1. At lowbarrier heights only one natural orbital is significantly occu-pied. Therefore, we refer to the parameter range 0A13as the condensed regime, in accordance with the definition ofPenrose and Onsager 22. The occupation of the secondnatural orbital is due to the two-body interaction between theparticles. However, it remains below 1% for all values of thebarrier heightA13 and is even below 1 atA =0.

Sincen11 Nin the condensed regime, the upper and the

lower bounds, Eqs.5and 6, on the largest eigenvalue ofthe second-order RDM,n1

2, are almost identical. Therefore,n1

2 is constrained to take on a value very close toNN1.Consequently, there can be only one significantly occupied

natural geminal. This is confirmed in Fig.1 bottom, wherethe natural geminal occupations are shown as a function ofthe barrier height. For the purpose of describing first- andsecond-order correlations it is therefore legitimate to ap-proximate the many-body wave function in this regime by asingle permanent N, 0 in which all N bosons occupy thefirst natural orbital1

1x1.Note that the natural orbitals and the natural geminals are

generally complex functions. However, the ground statewave function is real and hence the natural orbitals and thenatural geminals are real functions. The first column of Fig.2shows the firstdashed redand the secondsolid bluenatu-ral orbitals of the many-body solution at barrier heights A

=0,13,19,24, from top to bottom. The first and the secondnatural orbitals are symmetric and antisymmetric about theorigin, respectively. AtA =0 the first natural orbital,1

1x1,takes on the shape of a broadened Gaussian, reflecting therepulsive interaction between the particles. The second natu-ral orbital,2

1x1, has a higher kinetic energy than the firstone due to the node at the center of the trap. Additionally, thesecond natural orbital forces the particles to occupy regionsof the trap where the trapping potential is higher. There is anenergy gap between the one-particle energies of the first andsecond natural orbital. The occupation of the second naturalorbital is therefore very small in the purely harmonic trap atthe chosen interaction strength.

As the barrier height is varied fromA =0 up toA =13, thenatural orbitals deform to fit the new shape of the externalpotential. The central peak of the first natural orbital splitsinto two maxima which become localized at positions x1=d/2, where d is the distance between the wells of theexternal potential.

At the center of the trap, where the barrier is raised, thefirst natural orbital develops a local minimum in order tominimize the potential energy. The second natural orbital onthe other hand has a node at the center of the trap at anybarrier height. Its maximum and minimum are localized atthe minima of the external potential. As the barrier is raised,the energy gap between the first two natural orbitals de-

0

2

4

6

8

10

12

14

1

0 5 10 15 20 25 30

E/N

MCTDHB

-1E-03

0E+00

0 5 10 15 20 25 30

0

20

40

60

80

100

0 5 10 15 20 25 30

n(1)

i

/N[%]

n(1)1

n(1)2

0

20

40

60

80

100

0 5 10 15 20 25 30

n(2)

i

/N(N-1)[%]

barrier height

n(2)1

n(2)2

n(2)3

FIG. 1. Color online Energy per particle, natural orbital, andnatural geminal occupations of the ground state ofN=1000 bosons

at 0 =0.01 in a harmonic trap with a central barrier. Shown is thedependence on the barrier height. Top: energy per particle E/Nofthe many-body solution. Inset: energy difference per particle be-tween the best mean-field and the GP solution,EBMFEGP /Ntri-angles, and between the many-body and GP solution, EMCTDHBEGP /Ncircles. Middle: the eigenvaluesn1

1 andn21 of the first-

order RDM 1x1 x1. The ground state fragments with increasingbarrier height. Bottom: the eigenvalues n1

2, n22, and n3

2 of thesecond-order RDM 2x1 ,x2 x1 ,x2. The dashed lines in themiddle and bottom panel indicate upper and lower bounds on thelargest eigenvalue of the first- and second-order RDMs. See text fordetails. The quantities shown are dimensionless.


023615-6


7/17

creases. However, the increase of the depletion of the con-densate from n2

1/N1at A =0 to n2

1/N1% at A =13

cannot be explained in this single particle picture. On asingle particle level the ground state would be fully con-densed at any finite barrier height, i.e., n2

1 =0. The reasonfor the observed increase in the depletion lies in the fact thatfor repulsively interacting many-boson systems in multiwellsetups it becomes energetically more favorable to fragment

as the barrier between the wells is raised 24,26,28,29,33.The increase in energy which results from the occupation ofnatural orbitals with a higher one-particle energy can be out-weighed by a decrease in interaction energy. This effect be-comes dominant at barrier heights above A =13, see Secs.IV B and IV C.

The second to fourth column of Fig.2showfrom left toright the first three natural geminalsi

2x1 ,x2 at the samebarrier heights as above. From the numerical many-bodysimulation we find that the natural geminals in the condensedregime are approximately given by symmetrized products ofthe natural orbitals,

12=2,0, 2

2=1,1, 32=0,2, 24

where m1 ,m2 denotes a state with m1 particles in the firstand m2 particles in the second natural orbital. Only the firstnatural geminal,1

2x1 ,x2, is significantly occupied in thecondensed regime. Due to the two-body interaction betweenthe particles there is a small occupation of the second and

third natural geminal. However, at low barrier heights theiroccupation is largely suppressed, due to the gap between thesingle particle energies of the first and second natural orbital.Since the geminals 2

2x1 ,x2 and 32x1 ,x2 contain the

second natural orbital in their expansion, see Eq.24, theiroccupation increases the total energy at low barrier heights.

Since n12 NN1 in the condensed regime, the only

substantially contributing natural geminal in the equation forthe energy expectation value, Eq. 8, is 1

2x1 ,x2. Theshape of 1

2x1 ,x2 is therefore particularly interesting. Ithas four maxima of similar height, located at positions x1=x2 =d/2 andx1 =x2 =d/2, see the second panel in thesecond row of Fig. 2. Since 1

2x1 ,x2 has peaks on the

diagonal at x1 =x2 =d/2, it contributes to both, the one-particle part and the interaction part of the energy, Eq.8.In contrast to the first natural geminal, 2

2x1 ,x2 and3

2x1 ,x2 both exhibit node lines going through the regionwhere the central barrier is raised. As the energy gap

11 h1

1 21 h2

1 between the single particle ener-gies of the natural orbitals1

1x1and 21x1decreases, so

do the energy gaps i2 hi

2 j2 hj

2, ij =1,2,3between the natural geminals 1

2x1 ,x2, 22x1 ,x2 and

23x1 ,x2. Similar to the discussion of the natural orbital

occupations above, this argument in terms of an energy gapdoes not explain the increase of the occupation of the secondand third natural geminal when the barrier is raised. Without

interactions the occupation numbers of all but the first natu-ral geminal would be exactly zero.

We shall demonstrate in Sec. IV C that fragmented statesallow the occupation of geminals that contribute very little tothe interaction energy as opposed to condensed states.Thereby, the system can lower its energy, once the barrier ishigh enough.

B. From condensation to fragmentation

At barrier heights 13A24, one finds that the occupa-tion of the second natural orbital n2

1/N increases continu-

ously from below 1% to almost 50%. The condensate frag-

ments in this regime according to the definition offragmented condensates23. In this regime we find numeri-cally that many permanents contribute to the wave functionand, therefore, we refer to the range of barrier heights 13A24 as the many-body regime. Along with the naturalorbital occupations the natural geminal occupations changeas well. Three natural geminals become occupied with in-creasing barrier height, see Fig.1.

In the many-body regime, the upper and lower bounds,Eqs. 5 and 6, on the largest eigenvalue of the second-order RDM,n1

2, no longer restrictn12 to a narrow region. In

fact, n12 takes on a value somewhere in between these

bounds.

FIG. 2. Color onlineNatural orbitals and geminals at differentbarrier heights. First column: the natural orbitals 1

1x1 dashedred lineand 2

1x1 solid blue lineof the many-body solution atdifferent barrier heights A =0,13,19,24, from top to bottom. Thetrapping potential is shown as a dashed-dotted black line in the firstcolumn. The state of the system changes from condensed to frag-mented betweenA =13 andA =24. Second to fourth columns: natu-ral geminals 1

2x1 ,x2, 22x1 ,x2, and 3

2x1 ,x2 from left toright at the same barrier heights as above. While the natural orbitalsremain qualitatively unchanged during the fragmentation transition,the natural geminals take on their final shapes only when the systembecomes fully fragmented. At low barrier heights only one naturalgeminal is occupied. At high barriers three natural geminals areoccupied, see Fig.1. The total energy is minimized by the occupa-

tion of a natural geminal that contributes practically nothing to theinteraction energy. See text for more details. The quantities shownare dimensionless.


023615-7


8/17

This onset of fragmentation manifests itself also in theBMF solution which jumps from a GP type permanentN, 0in the condensed regime to a fully fragmented solution of theformN/2 ,N/2. Note that already at barrier heightsA14this fragmented solution is lower in energy than a GP typepermanent. At this barrier height the many-body solution isonly slightly depleted, see Fig.1.

If we compare the natural orbitals in Fig. 2 at barrierheightA =19 with those atA =13, we note that they look verysimilar, apart from the fact that the peaks are slightly fartherapart, and the first natural orbital is closer to zero at thecenter of the trap. The energies of the first two natural orbit-als are almost degenerate and the total energy is minimizedby the occupation of both natural orbitals. Without interac-tions the system would remain in a condensed state, since thesingle particle energies of the first and the second naturalorbitals remain separated at any finite barrier height. Notethat in the absence of interactions the natural orbitals are the

eigenfunctions of h. However, as we noted in Sec. IV A, asystem of repulsively interacting bosons in multiwell trapscan lower its energy by occupying several natural orbitals,once the barrier is high enough2629,33. This is preciselythe reason for the observed onset of fragmentation.

In the many-body regime the natural geminals are nolonger symmetrized products of the natural orbitals. If wecompare 1

2x1 ,x2 in Fig. 2 at barrier heights A =13 andA =19, we see that the peaks on the diagonal at x1 =x2=d/2 decrease, while the peaks on the off-diagonal atx1=x2 =d/2 increase when the barrier is raised. The oppo-site is true for the third natural geminal3

2x1 ,x2: the off-diagonal maxima at x1 =x2 =d/2 have decreased, whilethe diagonal minima atx1 =x2 = d/2 are now more negative.On the other hand, the second natural geminal,2

2x1 ,x2, isstill well approximated by a symmetrized product of the first

and second natural orbital. The behavior of the natural gemi-nals is qualitatively different from that displayed by the natu-ral orbitals. In contrast to the natural orbitals, the naturalgeminals do change their shape during the fragmentationtransition. They only obtain their final forms, when the frag-mentation transition is completed, see Fig.2 and Sec. IV C.

C. Fully fragmented state

When the central barrier is raised to values A24, thetwo parts of the condensate become truly independent. Thenatural orbital occupations approach n1

1 = n21 =N/2, which

reflects the fact that the energies associated with the first andsecond natural orbitals degenerate at infinite barrier heights.The many-body wave function can then be adequately ap-

proximated by a single permanent of the form N/2 ,N/2,i.e., with equal numbers of particles in the first and the sec-ond natural orbitals. Therefore, we refer to barrier heights

A24 as the fully fragmented regime. The additional en-ergy, necessary for the occupation of the second natural or-bital, is outweighed by a lower interaction energy. Note thatthis final form of the wave function is anticipated by theBMF solution at barrier heightsA14.

The natural geminal occupations approach

n12 =NN/2, n2

2 =n32 =N/2 1N/2 25

in the fully fragmented regime. These are the values thatfollow from the BMF solutionN/2 ,N/2.

It is only at barrier heightsA24 that the natural gemi-nals take on their final shapes, compare the third and fourthrows of Fig.2.If we expand the natural geminals in the basisof natural orbitals at these barrier heights, we find that

12=

1

22,00,2, 2

2=1,1 ,

32=

1

22,0+0,2 26

holds to a very good approximation. The first and third natu-ral geminals have equal contributions coming from the firstand the second natural orbitals. The question arises why theiroccupations are different, about 50% and 25%, respectively.Subtracting the permanents2,0and 0,2from one anotheryields a geminal which is localized on theoff-diagonal, see1

2x1 ,x2 in Fig.2at A =24. Adding the permanents 2,0and0,2yields a geminal which is localized on thediagonal,see 3

2x1 ,x2 in Fig.2 at A =24. It is easy to see from the

shape of the natural geminals in the fourth row of Fig.2thatthe integrals over the one-body part in Eq.8 are approxi-mately the same for each of the natural geminals. Given theoccupations in Eq.25, the first natural geminal contributesabout one half of the one-body energy, whereas the secondand the third natural geminal contribute about a fourth each.The situation is different for the two-body part of the Hamil-tonian. Since2

2x1 ,x2and 32x1 ,x2 are localized on the

diagonal, theydocontribute to the interaction energy. In con-trast, 1

2x1 ,x2 is almost zero at coordinate values x1 x2and, due to the contact interaction in Eq.23, it practicallydoes not contribute to the interaction energy. At high barriersa fragmented state allows the system to lower its energy

through the occupation of a natural geminal which is local-ized on the off-diagonal.

We would like to make a remark on the validity of thepresent MCTDHB computation for high barriers. For highbarriers the whole system can be considered as composed oftwo separate condensates. To describe the depletion of eachcondensate it would be necessary to employ M=4 orbitals.We use only M=2 orbitals in the many-body computationand cannot describe this depletion. We justify the use ofM=2 orbitals by noting first that atA =0 the system is almostfully condensed, and the depletion can be safely neglected,see Fig.1.Therefore, we assume that the depletion of each ofthe two condensates can be neglected when the barrier is

very high. This claim is supported by a computation that wecarried out in the harmonic trap at the same interactionstrength 0 =0.01 for 500 particles. The depletion was foundto be even less than forN=1000 particles.

V. FIRST-ORDER CORRELATIONS

A. General analytical considerations

We now describe the first-order correlations in an analyti-cal mean-field model for the two limiting cases of a con-densed and a fully fragmented system. In these cases mean-field theory has been shown to be well applicable, see Sec.


023615-8


9/17

IV. For our purposes the exact shape of the natural orbitals1

1x1 and 21x1 is unimportant. Consider a normalized

one-particle function,x, which is localized at the origin.xmay vary in shape, but is always assumed to resemblea Gaussian. Similarly, we define translated copies 1x=x+ d/2 and 2x =x d/2 ofx, where the pre-viously defined distancedbetween the minima of the poten-

tial wells is taken to be large enough to set products of theform1x2xto zero. Since is localized in some regionaround the origin,1is localized in a regionLto the left and2 in a regionR to the right of the origin.

1. Condensed state

In the condensed regime, 0A13, only one natural or-bital, 1

1x1, is significantly occupied. Therefore, we ap-proximate the first-order reduced density operator of the sys-tem by that of a condensed stateN, 0

N,01 =N1

111 . 27

It then follows from Eq.13thatgN,0

1 x1,x12 = 1. 28

At zero barrier height, the first natural orbital is a Gaussian,broadened by interactions. Therefore, we write 1

1x1=x1, and hence the one-particle density distribution andthe one-particle momentum distribution are of the form

N,01 x1x1=Nx1

2, 29

N,01 k1k1=Nk1

2. 30

Since x1 is a broadened Gaussian, its Fourier transform

k1 is also close to a Gaussian, but narrower in compari-son to a noninteracting system. The momentum distributionof the repulsively interacting system in the harmonic trap istherefore narrower than that of a noninteracting system.

We now turn to the case corresponding toA 13, wherethe system is still condensed, but the first two natural orbitalsare spread out over the two wells. We model the naturalorbitals by

11x1=

1

21x1+2x1 ,

2

1

x1=

1

2

1x1

2x1 . 31

In this case one obtains57

N,01 x1x1=

N

21x1

2 +N

22x1

2, 32

N,01 k1k1=N1 + cosk1dk1

2 33

for the density and the momentum distribution. We note thatthe one-particle momentum distribution displays an oscilla-tory pattern in momentum space at a period which is deter-mined by the separationdof the centers of the two wells.

2. Fully fragmented state

In the true many-body regime, 13A24, where manypermanents contribute to the wave function, a mean-fieldmodel is bound to fail. However, in the fully fragmentedregime it is possible to consider the whole system as twoseparate condensates, and hence a mean-field description is

again applicable. Therefore, we now turn to the case corre-sponding toA24, where the system is fully fragmented andthe many-body state is given by N/2 ,N/2. The first-orderreduced density operator then reads

N/2,N/21 =

N

21

111+

N

22

121. 34

Since the natural orbitals remain qualitatively unchangedduring the fragmentation transition, we approximate1

1x1and 2

1x1 by Eqs.31 and obtain for the density and themomentum distribution57

N/2,N/21 x1x1=

N

2

1x12 +

N

2

2x12, 35

N/2,N/21 k1k1=Nk1

2 . 36

We note that the one-particle momentum distribution of in-dependent condensates does not contain an oscillatory com-ponent and is identical to the momentum distribution of asingle localized condensate ofNparticles within this model,see Eq.30. For the normalized first-order correlation func-tion one finds

gN/2,N/21 x1,x1

2 = 1 ifx1,x1Lor x1,x1R ,0 otherwise.

37

Whereas the stateN, 0is fully first-order coherent, the frag-mented stateN/2 ,N/2 is only first-order coherent in a re-stricted and generally disconnected region. Each of the twocondensates is first-order coherent, but the mutual coherencewhich is present in the condensed regime is lost.

B. Numerical results

We now turn to the discussion of first-order correlations.In particular, we are interested in effects that are due to thetrue many-body nature of the wave function. Along with ourmany-body results we plot the corresponding results of the

BMF solution. From the discussion in Sec. IV it is clear thatwe expect many-body effects to occur during the fragmenta-tion transition at barrier heights 13A24. In the con-densed and in the fully fragmented regime we expect that themany-body results are well approximated by those of theBMF solution. In these cases we can understand the structureof the correlation functions on the basis of the analyticalmean-field model of Sec. V A.

The first column of Fig.3 shows the one-particle densitydistribution1x1 x1of the many-body solutionblue lineand that of the BMF solutiondashed red lineat the barrierheightsA =0 ,13 , 19, 24, from top to bottom. It is remarkablethat the one-particle densities obtained from either the many-


023615-9


10/17

body wave function or the BMF solution give results thatcannot be distinguished from one another at any barrierheight.

In a purely harmonic trap,A =0, the one-particle densitytakes on the form of an interaction-broadened Gaussian. Athigher barriers, the density splits into two parts that are lo-calized in each of the wells. AtA =13, the one-particle den-sity has developed two separated peaks. Note that the system

is still in the condensed regime at this barrier height andmust be considered a single condensate, despite the spatialseparation between the two peaks.

When the central barrier is raised further to values 13A24 the condensate fragments, see Fig.1.At a barrierheight of A =19 the system is halfway on its way from acondensed to a fully fragmented condensate. Many perma-nents contribute to the many-body wave function and onemay wonder how this transition manifests itself in observ-able quantities. However, apart from a small shift of the cen-ter of the two peaks and a reduction at the center of the trap,

1x1 x1 remains largely unaffected by this transition. Ifthe barrier is raised further toA =24, the fragmentation tran-

sition is essentially completed. Also during the transitionfrom a true many-body state to a fully fragmented state thereis no visible indication of this transition in the one-particledensity.

The second column of Fig.3 shows the one-particle mo-mentum distribution1k1 k1at the same barrier heights asbefore. At A =0, the one-particle momentum distribution is

given by a squeezed Gaussian, in agreement with Eq.30.At A =13 the one-particle momentum distribution has devel-oped an oscillatory pattern, typical of a single condensatespread out over two wells. The structure of1k1 k1is wellreproduced by Eq. 33 of the analytical mean-field model.Up to this barrier height the BMF solution is almost identicalto the many-body wave function, and therefore the respectivemomentum distributions are indistinguishable, see the twoupper panels in the second column of Fig.3.

When the system enters the many-body regime, 13A24, the momentum distribution of the many-body solutiondeforms to a Gaussian-like envelope, modulated by an oscil-latory part. The BMF momentum distribution dashed redline, on the other hand, already takes on the form character-

istic of two separate condensates. It agrees with the predic-tion of Eq.36, which is clearly different from the many-body result. This merely reflects the fact that the many-bodywave function is inaccessible to mean-field methods in themany-body regime.

When the state becomes fully fragmented at A =24, themany-body momentum distribution and the BMF momentumdistribution become indistinguishable again, consistent withan explanation in terms of two independent condensates, seeEq. 36. Compared to 1k1 k1 at A =0, the momentumdistribution is broader atA =24, because the density distribu-tion in each of the two wells is narrower than that in theharmonic trap.

The third column of Fig. 3 shows the absolute valuesquared of the normalized first-order correlation functiong1x1 ,x1

2 of the many-body solution only. Here and in allfollowing graphs of correlation functions we restrict the plot-ted region by a simple rule. To avoid analyzing correlationsin regions of space where the density is essentially zero, weplot the respective correlation function only in regions wherethe density is larger than 1% of the maximum value of thedensity in the entire space. We apply the same rule also inmomentum space.

At zero barrier heightg1x1 ,x12 is very close to one in

the region where the density is localized. The system is first-order coherent to a very good approximation and the mean-

field formula, Eq. 28, applies. As the barrier is raised toA =13 the coherence between the two peaks, e.g., at x1=x1, is slightly decreased, while the coherence within eachof the peaks is preserved. Note that the density at the centerof the trap is already below 1% of the maximal value in thiscase. Despite this separation the system remains largely con-densed, but deviations from Eq.28are visible. If the barrieris raised further toA =19, the coherence of the system on theoff-diagonal decreases quickly. Although the bosons in eachwell remain coherent among each other, the overall system isonly partially coherent. At barrier heightsA24, the coher-ence between the two wells is entirely lost. This is also thescenario that the BMF solution anticipates, see Eq.37.

0

0.2

0.4

(1)(x1|x1)/N

0

0.2

0.4

0

0.2

0.4

0

0.2

0.4

-6 -3 0 3 6 x1

0

0.4

0.8

1.2

(1)

(k1|k1)/N

0

0.4

0.8

1.2

0

0.4

0.8

1.2

0

0.4

0.8

1.2

-2 0 2k1

0

0.2

0.4

0.6

0.8

1|g

(1)(x1,x1)|

2

-6

-3

0

3

6

x1

-6

-3

03

6

x1

-6

-3

0

3

6

x1

-6 -3 0 3 6 x1

-6

-3

0

3

6

x1

FIG. 3. Color onlineDensity distribution, momentum distribu-tion and first-order coherence. The first two columns show the one-particle density 1x1 x1 /N and the one-particle momentum dis-tribution1k1 k1 /Nof the many-body solutionsolid blue linesand of the BMF solutiondashed red line, respectively. From top tobottom the height of the central barrier is A =0,13,19,24. The BMFresult agrees well with the many-body result for a large range ofbarrier heights. Only at A =19, in the many-body regime, deviationsare visible in the momentum distribution. See text for details. Thethird column shows the absolute value squared of the normalized

first-order correlation function g1

x1

,x1

2

at the same barrierheights. An initially coherent condensate splits into two separatecondensates which are no longer mutually coherent. Only the co-herence within each of the wells is preserved. The quantities shownare dimensionless.


023615-10


11/17

It is remarkable that not only the density, but also themomentum distribution obtained within mean-field theoryagree so well with the many-body result, when the system isnot in the true many-body regime. This would not be the caseif we had restricted the mean-field approach to the GP equa-

tion, as we shall show now.Up to barrier heightsA =13 the many-body system is con-densed, and the BMF solution coincides with the GP solu-tion. The BMF, and therefore also the GP solution providegood approximations to the interacting many-body system.Above A =13 the results obtained with the GP mean-fieldbecome qualitatively wrong as the barrier is raised. To illus-trate this point, we plot the GP results corresponding to thoseof Fig.3at barrier heightsA =19topandA =24bottominFig.4.A comparison of the respective one-particle densities,shown in the first column of Fig.3and Fig.4, reveals novisible difference. The GP mean-field reproduces the densitydistribution at all barrier heights correctly. However, the GPsolution fails at the description of the momentum distributionand the normalized first-order correlation function, comparethe second and third columns of Figs.3and4at the samebarrier heights. The reason for the failure of the GP meanfield is the assumption that all bosons occupy the same or-bital. It is by construction incapable to describe fragmentedcondensates.

VI. SECOND-ORDER CORRELATIONS

A. General analytical considerations

In this section we extend the analytical mean-field modelof Sec. V A to describe second-order correlations.

1. Condensed state

We found in Sec. IV A that only one natural geminal issignificantly occupied in the condensed regime, where themany-body state is approximately given by a single perma-nent in which all bosons occupy the same single particlestate. Therefore, we approximate the second-order reduced

density operator in the condensed regime by that of the stateN, 0,

N,02 =NN 11

212, 38

where12x1 ,x2 =1

1x111x2is the permanent in which

two bosons reside in the first natural orbital 11. For the

condensed state N, 0 one finds that up to corrections oforderO1 /N the state is second-order coherent,

gN,02 x1,x2,x1,x2= 1

1

N, 39

gN,0

2

k1,k2,k1,k2= 1

1

N. 40

Thus, there are practically no two-body correlations if N1. At zero barrier height the first natural orbital takes onthe shape of a broadened Gaussian,1

1x1 =x1, wherex is defined in Sec. V A. The first natural geminal thenreads 1

2x1 ,x2 =x1x2. It follows that the two-particle density and the two-particle momentum distributionfactorize up to corrections of orderO1 /N into products ofthe respective one-particle distributions,

N,02 x1,x2x1,x2=NN 1x1

2x22 , 41

N,0

2

k1,k2k1,k2=NN 1

k12

k22

. 42At the barrier height A =13, the system is condensed butspread out over the two wells. Then, using Eqs.31 to ap-proximate1

1, we find

N,02 x1,x2x1,x2=

NN 1

4 1x11x2

2

+1x12x22 +2x11x2

2

+2x12x22 , 43

N,02 k1,k2k1,k2=NN 11 + cosk1d1 + cosk2d

k1k22 44

for the two-particle density and the two-particle momentumdistribution. Apart from a correction of order O1 /N, thetwo-particle density and the two-particle momentum distri-bution are again products of the respective one-particle dis-tributions.

2. Fully fragmented state

In Sec. IV C we found that three natural geminals areoccupied in the fully fragmented regime, see Eq.25. Theoccupations of Eq.25 hold exactly for a state of the formN/2 ,N/2. Therefore, we approximate the second-order re-

0

0.2

0.4

(1)(x1|x1)/N

0

0.2

0.4

-6 -3 0 3 6 x1

0

0.4

0.8

1.2

(1)

(k1|k1)/N

0

0.4

0.81.2

-2 0 2k1

0

0.2

0.4

0.6

0.8

1|g

(1)(x1,x1)|

2

-6

-3

0

3

6

x1

-6 -3 0 3 6 x1

-6

-3

0

3

6

x1

FIG. 4. Color onlineDensity distribution, momentum distribu-tion, and first-order coherence obtained by using the GP equationfor high barriers. The first two columns show the GP one-particledensity1x1 x1 /Nleftand the GP one-particle momentum dis-tribution 1k1 k1 /N middle at barrier heights A =19 and A=24solid green lines. In the first column the trapping potential isalso showndashed-dotted black line. The GP equation models the

density well, but fails at the computation of the momentum distri-bution, compare with Fig.3.The third column shows the absolutevalue squared of the normalized first-order correlation functiong1x1 ,x1

2 computed with the GP equation at the same barrierheights. The normalized first-order correlation function is incor-rectly described by the solution of the GP equation. The quantitiesshown are dimensionless.


023615-11


12/17

duced density operator in the fully fragmented regime by thatof the stateN/2 ,N/2,

N/2,N/22 =N

N

21

212+

N

2N

2 12222

+N

2 N

2

1

3

232, 45

where the natural geminals i2 are given by Eq. 26. In

contrast to the condensed state, the normalized second-ordercorrelation function of the fully fragmented state has a morecomplicated structure due to the different terms contributingto Eq.45. We approximate the natural geminals using Eqs.31and find

N/2,N/22 x1,x2x1,x2=

N

2N

2 11x11x22

+2x12x22

+

N

2

N

2 1x12x22

+2x11x22 , 46

N/2,N/22 k1,k2k1,k2=NN 11 + N

N 1

cosk1k2d

2

k1k22 47

for the two-particle density and the two-particle momentumdistribution. This representation allows us to identify the firsttwo terms in Eq. 46 as contributions coming from twoseparate condensates of N/2 bosons each, with condensatewave functions1x1 and 2x1. The remaining terms in

Eq.46are due to the fact that the bosons in the two sepa-rated condensates are identical particles. For the normalizedsecond-order correlation function one finds

gN/2,N/22 x1,x2,x1,x2= 1

2

Nifx1,x2Lor x1,x2R ,

1 otherwise ,

48

which mimics a high degree of second-order coherence, seeEq.39. However, wheng2 is evaluated on the diagonal inmomentum space, one finds

gN/2,N/22 k1,k2,k1,k2=1 1N1 + NN 1 cosk1k2d2 ,49

which displays an oscillatory behavior and deviates signifi-cantly from a uniform value of 11/N. Hence the system isclearly not coherent, see Sec. II. The fact thatgN/2,N/2

2 k1 , k2 , k1 ,k2 oscillates while gN/2,N/22 x1 ,x2 ,x1 ,x2

is almost constant, illustrates the necessity to examine bothquantities in order to quantify second-order coherence. A de-scription of second-order correlations in terms of

2x1 ,x2 x1 ,x2 and g2x1 ,x2 ,x1 ,x2 alone is incomplete,

and 2k1 , k2 k1 , k2 and g2k1 , k2 ,k1 , k2 have to be taken

into account. Although this may seem obvious in the presentcase of a fully fragmented state, this reduction of coherenceis more intricate in a state which is only partially fragmented,see the following section. Whether g2x1 ,x2 ,x1 ,x2 andg2k1 ,k2 , k1 ,k2 together suffice to characterize second-order coherencepossibly up to a phase factoris a matter offurther study.

B. Numerical results

In this section we discuss the second-order correlations ofthe many-body solution. We compare the results to those ofthe BMF solution. When mean-field theory gives a good ap-proximation to the many-body results, we also compare withthe analytical mean-field model of Sec. VI A.

The first two columns of Fig. 5 show the two-particledensity 2x1 ,x2 x1 ,x2 of the many-body left and BMFright solutions at barrier heightsA =0,13,19,24, from topto bottom. At zero barrier height 2x1 ,x2 x1 ,x2 is local-ized at the center of the trap. The two-particle density factor-

izes approximately into a product of the one-particle densi-ties: 2x1 ,x2 x1 ,x2

1x1 x11x2 x2. This remains

true up to barrier heights A =13, where the condensate isspread out over the two wells. The BMF result approximatesthe many-body result well in the condensed regime, and thestructure of 2x1 ,x2 x1 ,x2is that of Eqs.41and 43atbarrier heightsA =0 andA =13, respectively.

When the barrier is raised further to A =19, the systemfragments. Many permanents contribute to the wave functionin this regime and there is no simple formula that relates theoccupations of the natural orbitals to the two-particle density.Similar to the one-particle density, described in Sec. V B, thetwo-particle density seems to take no notice of the transition

from a single to a fragmented condensate. It remains practi-cally unchanged during the transition, apart from a slightshift of the peaks away from each other as the barrier israised.

At even higher barriers,A24, the many-body state be-comes fully fragmented and the wave function approachesN/2 ,N/2. In this limit it is again possible to describe thetwo-particle density on a mean-field level. Therefore, theanalytical results of Sec. VI A for the fully fragmented stateshould apply. In fact, the structure of2x1 ,x2 x1 ,x2in thefully fragmented regime is that predicted by Eq.46.

The two-particle density of the condensed state just belowthe fragmentation transition and of the fully fragmented state

above the fragmentation transition look very much alike. It iseasily verified that Eqs.43 and 46 give rise to the sametwo-particle density profile up to corrections of orderO1 /N.

In contrast, the fragmentation transition is clearly visiblein the two-particle momentum distribution. In the third andfourth columns of Fig.5 the two-particle momentum distri-bution 2k1 , k2 k1 ,k2 of the many-body left and BMFrightwave function are shown.

In the condensed regime the two-particle momentum dis-tribution is approximately given by the product of one-particle momentum distributions of a single condensate. Thisagrees with the analytical predictions of Eq.42 at barrier


023615-12


13/17

heightA =0 and Eq.44at A =13. The mean-field picture isappropriate here.

In the many-body regime the two-particle momentum dis-tribution 2k1 , k2 k1 , k2 contains contributions from manypermanents. The resulting 2k1 ,k2 k1 ,k2 has a structurethat lies somewhat in between the two results, Eq.44 forthe condensed and Eq. 47 for the fragmented state, ob-tained within the analytical mean-field model. The BMF so-lution is fully fragmented and does not provide an accurateapproximation to the many-body two-particle momentum

distribution in this regime, see Fig.5, third and fourth col-umns in the third row from above.When the barrier is raised toA =24, the many-body state

becomes fully fragmented and the mean-field picture is againapplicable. The pattern of a single coherent condensate hasnow vanished completely in favor of a pattern characteristicof two separate condensates. The pattern agrees well with thestructure predicted by Eq.47.

Similar to our results on first-order correlations, discussedin Sec. V B, the fragmentation transition shows up in thetwo-particle momentum distribution, but not in the two-particle density. While this behavior is predictable in the lim-iting cases of a condensed and a fully fragmented state, it is

necessary to solve the many-body problem to determine thelimits of such mean-field approximations. Particularly thebehavior in between the two mean-field limits is only acces-sible to many-body approaches.

We will now address the second-order coherence of thesystem. The first two columns of Fig.6show the diagonalof the normalized second-order correlation functiong2x1 ,x2 ,x1 ,x2 of the many-body left and the BMFright solutions. Note the scale. Equations39 and 48 ofthe analytical mean-field model of Sec. VI A predict very

small correlations in the two-particle density of the con-densed and the fragmented state. This is confirmed in thefirst column of Fig.6.In the condensed regime at zero bar-rier height the effects of the depletion of the condensate ong2x1 ,x2 ,x1 ,x2 are visible. Almost no two-particle densitycorrelations are present. This is equally true in the case of asingle condensate spread out over the two wells and also inthe many-body regime. Above the fragmentation transition,the present computation of the many-body solution cannotdescribe effects on g2 that are due to the depletion of thecondensate. However, since depletion effects are negligiblein the harmonic trap, we are reassured that they are alsonegligible in the fully fragmented regime, see Sec. IV C. The

FIG. 5. Color onlineTwo-particle density and two-particle momentum distribution at different barrier heights. The first two columnsfrom left to rightshow the two-particle density 2x1 ,x2 x1 ,x2 /NN1of the many-bodyleftand BMFrightwave function for thebarrier heights A =0,13,19,24, from top to bottom. At low barrier heightsA =0,13the system is condensed, and the two-particle densityfactorizes into a product of the one-particle densities. At higher barriers A =19,24, the system fragments and the two-particle density doesnot factorize into a product of the one-particle densities. The fragmentation transition is not visible in the two-particle density. The results ofthe many-body and BMF wave function cannot be distinguished at any barrier height. The third and fourth column show the two-particlemomentum distribution 2k1 , k2 k1 , k2 /NN1 of the many-bodyleft and BMF right solution at the same barrier heights as above.The transition from a condensed state to a fragmented state is clearly visible. At A =19 the BMF solution does not reproduce the many-body

results. The system is in a true many-body state, inaccessible to mean-field methods. At higher barriers A24 the system fully fragments,and a mean-field description is applicable again. The quantities shown are dimensionless.


023615-13


14/17

BMF solution predicts almost identical two-body densitycorrelations, see second column of Fig.6.

On the basis of g2x1 ,x2 ,x1 ,x2 alone, the many-bodystate appears to be second-order coherent at all barrierheights. A high degree of second-order coherence requiresEq.14or Eq.15to hold to a very good approximation forp =1 and p =2. This in turn requires the largest eigenvaluesof the first- and second-order RDM to be n1

1 Nand n12

NN1, respectively. We have already demonstrated inSec. IV that these conditions are only satisfied in the con-

densed regime. Therefore, it is obviously tempting, butwrong to conclude fromg2x1 ,x2 ,x1 ,x2 1 that the systemis second-order coherent. This misconception is due to thefact that g2x1 ,x2 ,x1 ,x2 only samples a small part of thefirst- and second-order RDMs of the system.

So, how does the decrease of coherence manifest itselfin second-order correlation functions? For second-ordercoherence to be present, at least approximately, alsog2k1 , k2 ,k1 , k2has to be close to one. The third and fourthcolumn of Fig.6show g2k1 ,k2 , k1 , k2 of the many-bodyleft and BMF right solution. At zero barrier height thesystem is indeed highly second-order coherent since onlyone natural orbital is significantly occupied. Not only

g2x1 ,x2 ,x1 ,x2, but also g2k1 , k2 , k1 ,k2 is very close to

one here. However, at A =13 when the many-body state isstill condensed, g2k1 ,k2 , k1 , k2 starts to develop a struc-ture.

When the barrier is raised to values above A =13, thestructure becomes more and more pronounced. In the many-body regime at A =19, we find that g2k1 ,k2 , k1 ,k2 has acomplicated behavior and deviates significantly from valuesclose to one, thereby proving that strong correlations arepresent. Note that the interaction between the particles isweak and that the strong correlations are due to the transitionfrom a single to a fragmented condensate. This transition isin turn induced by a change of the shape of external poten-tial. Varying the shape of the external potential therefore pro-vides a means to introduce strong correlations between theparticles. The strongest correlationsblack spots in the thirdpanel of the third row of Fig.6occur at those values wherethe two-body momentum distribution has local minima. Atthe values of k1 and k2, where the strongest correlationsoccur, the one-body and the two-body momentumdistributions areclearlydistinct from zero, see third panel inthe middle column of Fig.3 and the third panel in the thirdrow of Fig.5.Their ratio gives rise to these strong correla-tions in g2k1 , k2 , k1 ,k2. Experiments that measure

FIG. 6. Color onlineSecond-order coherence at different barrier heights. The first two columns from left to rightshow the diagonalof the normalized second-order correlation function in real space g2x1 ,x2 ,x1 ,x2 of the many-body left and BMF right solution atbarrier heights A =0,13,19,24, from top to bottom. g 2x1 ,x2 ,x1 ,x2 is very close to one at all barrier heights. Note the scale. The systemseems to be second-order coherent and the results of the many-body and BMF solution agree well with each other. The third and fourthcolumn depict the diagonal of the normalized second-order correlation function in momentum space g2k1 , k2 , k1 , k2 of the many-bodyleftand BMF rightsolution at the same barrier heights. The fragmentation transition is clearly visible between A =13 and A =24. At A=19 there are strong many-body correlations between the momentalocal maxima in black colorand g 2k1 , k2 , k1 , k2 exhibits a compli-cated pattern, see text for more details. A mean-field description fails here. In the limit of high barriers, A24, the correlations of the

many-body state become somewhat weaker and become again describable by those of the BMF solution. The quantities shown aredimensionless.


023615-14


15/17

g2k1 , k2 ,k1 , k2 in ultracold quantum gases have been car-ried out recently, see, e.g., Ref. 58. An experiment thatmeasures g2k1 , k2 , k1 , k2 would find the strongest two-particle momentum correlations at intermediate barrierheights.

When the system becomes fully fragmented at barrierheights A24 the structure of g2k1 , k2 , k1 ,k2 becomes

more regular again. The amplitude of the correlations issmaller than in the many-body regime, and the correlationsbetween different momenta are modulated by a single oscil-latory structure. This structure can be well understood withinthe analytical mean-field model of Sec. VI A. The oscillatorypart of g2k1 , k2 , k1 ,k2 is determined by the difference ofthe wave vectors multiplied by the distance between thewells, see Eq.49. In contrast, the correlations in the many-body regime cannot be explained by analytical mean-fieldmodels.

Hence, we find that only in the condensed regime thesystem is second-order coherent despite the fact thatg2x1 ,x2 ,x1 ,x2 1 at all barrier heights. This merely re-

flects the fact that g2x1 ,x2 ,x1 ,x2 is only the diagonal ofg2x1 ,x2 ,x1 ,x2. On the other hand, g

2k1 ,k2 , k1 , k2 de-pends on all values of 2x1 ,x2 x1 ,x2 and providescomplementary information about the coherence of the state.A description of second-order coherence in terms ofg2x1 ,x2 ,x1 ,x2 alone is therefore incomplete.

The corresponding results of the BMF solution agree wellwith those of the many-body solution as long as the system isnot in the many-body regime at intermediate barrier heights.In the many-body regime the BMF result is inaccurate, but itanticipates the final form of g2x1 ,x2 ,x1 ,x2 in the frag-mented regime.

VII. CONCLUSIONS

In this work we have investigated first- and second-ordercorrelations of trapped interacting bosons. For illustrationpurposes we have investigated the ground state of N=1000weakly interacting bosons in a one-dimensional double-welltrap geometry at various barrier heights on a many-bodylevel. We have obtained the many-body results by solvingthe many-body Schrdinger equation with the recently devel-oped MCTDHB method. This allowed us to compute fromfirst principles the natural orbitals and the natural geminalsof a large interacting many-body system, together with theiroccupation numbers. To our knowledge this is the first com-

putation of the natural geminals of an interacting many-bodysystem of this size.Depending on the height of the double-well barrier we

found that there are three different parameter regimes. At lowbarriers the ground state is condensed and the many-bodywave function is well approximated by a single permanent ofthe form N, 0. At high barriers the ground state becomesfully fragmented and can be well approximated by a singlepermanent of the form N/2 ,N/2. At intermediate barrierheights, where the transition from a single to a fragmentedcondensate occurs, the ground state becomes a true many-body wave function to which many permanents contribute.We have demonstrated that the transition to a fragmented

state results in the occupation of a natural geminal that con-tributes very little to the interaction energy. The overall en-ergy of the system can be lowered by the occupation of sucha geminal, and the ground state becomes fragmented.

We have shown how the transition from a condensed to afully fragmented ground state manifests itself in the one- andtwo-particle momentum distributions. However, the transi-

tion is notcaptured by the one- and two-particle densitydistributions, not even in the many-body regime.In order to determine the coherence of the state during the

fragmentation transition, we have computed the first- andsecond-order normalized correlation functions g1x1 ,x1,g2x1 ,x2 ,x1 ,x2, andg

2k1 ,k2 , k1 , k2. In the condensed re-gime, a high degree of coherence is indeed present in theground state wave function. First- and second-order correla-tions were found to be negligible at the interaction strengthand particle number chosen for our computation. However,with increasing barrier height correlations between the mo-menta of the particles build up. These correlations werefound to be very strong in the many-body regime at interme-diate barrier heights. The ground state at high barriers was

found to be correlated, but not as strongly as the ground stateat intermediate barrier heights.

While the transition from a virtually uncorrelated state toa correlated one is clearly visible in g1x1 ,x1 andg2k1 ,k2 , k1 ,k2, the transition hardly shows up ing2x1 ,x2 ,x1 ,x2. A description of second-order coherence interms ofg2x1 ,x2 ,x1 ,x2alone is, therefore, incomplete andcan lead to wrong predictions.

For comparison we have computed results based onithebest approximation of the many-body wave function withinmean-field theory, the BMF wave function, and ii theGross-Pitaevskii solution. We found that the GP wave func-tion is identical to the BMF solution up to some barrier

height. However, once the true many-body solution starts tofragment the BMF wave function is no longer given by a GPtype permanentN,0, but rather by a fragmented state of theform N/2 ,N/2. In the true many-body regime neither theGP, nor the BMF solution provide an adequate approxima-tion to the many-body wave function, and the predicted cor-relations are inaccurate.

While the GP mean field is only accurate at low barrierheights, the BMF solution provides a very good approxima-tion to the true many-body wave function at low andhighbarriers. We have shown that the GP mean-field predictsqualitatively wrong results at high barriers. The BMF onlyfails at intermediate barrier heights where the true many-

body wave function becomes a superposition of many per-manents. Such many-body effects can, by construction, notbe captured by mean-field methods.

In the mean-field regimes at high and low barriers wehave provided an analytical mean-field model that allows theunderstanding of the general structure of the computed cor-relation functions.

Our work sheds light on the first- and second-order cor-relation functions of interacting many-body systems. Thevariation of the shape of the trapping potential allows one tochange the physics of the system from mean field to stronglycorrelated many-body physics. Particularly, the many-bodyregime in between the condensed and the fully fragmented


023615-15


16/17

regimes has shown to be very rich and promises excitingresults for experiments to come.

ACKNOWLEDGMENTS

Financial support by the DIP Deutsch-IsraelischeProjektkooperation and the DFG Deutsche Forschungs-

gemeinschaft is gratefully acknowledged.

APPENDIX: p -PARTICLE MOMENTUM DISTRIBUTION

The relation of the p-particle momentum distribution tothepth order RDM is shown. Thep-particle RDM is relatedto thep-particle momentum distribution by

pk1, .. . ,kpk1, .. . ,kp;t

= 1

2Dp dprdpreil=1p klrlrl

pr1, .. . ,rpr1, .. . ,rp;t. A1

The change of variablesR i =ri+ri

2 ,s i = ri rifor i = 1 , . . . ,pinEq.A1leads to

pk1, .. . ,kpk1, .. . ,kp;t= dp seil=1p klslp

s1, .. . ,sp;t , A2

where

ps1, .. . ,sp;t= dpR pR1+s 12, .. . ,Rp+

sp

2R1

s1

2, .. . ,Rp

sp

2; t A3

is the average of the value of pr1 , . . . , rp r1 , . . . , rp ; tat distances si between ri and ri. From Eqs.A2 and A3it is clear that the p-particle momentum distribution atlarge momenta is determined by the behavior of

pr1 , . . . ,rp r1 , . . . ,rp ; t at short distances, whereas atlow momenta the off-diagonal long-range behavior of

pr1 , . . . ,rp r1 , . . . ,rp ; t contributes the major part. Of

course, the same analysis remains valid if the roles ofrandkare exchanged.

1M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman,and E. A. Cornell, Science 269, 1981995.

2C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet,Phys. Rev. Lett. 75, 16871995.

3K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten,D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett.75, 39691995.

4 E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A.Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, 3371997.

5L. Cacciapuoti, D. Hellweg, M. Kottke, T. Schulte, W. Ertmer,J. J. Arlt, K. Sengstock, L. Santos, and M. Lewenstein Phys.Rev. A 68, 0536122003.

6B. Laburthe Tolra, K. M. OHara, J. H. Huckans, W. D. Phil-lips, S. L. Rolston, and J. V. Porto, Phys. Rev. Lett. 92, 1904012004.

7S. Flling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I.Bloch, NatureLondon 434, 4812005.

8M. Schellekens, R. Hoppeler, A. Perrin, J. V. Gomes, D.Boiron, A. Aspect, and C. I. Westbrook, Science 310, 6482005.

9A. ttl, S. Ritter, M. Khl, and T. Esslinger, Phys. Rev. Lett.95, 0904042005.

10L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, andD. M. Stamper-Kurn, Phys. Rev. Lett. 98, 1104012007.

11 S. Ritter, A. ttl, T. Donner, T. Bourdel, M. Khl, and T.Esslinger, Phys. Rev. Lett. 98, 0904022007.

12M. Naraschewski and R. J. Glauber, Phys. Rev. A 59, 45951999.

13J. Viana Gomes, A. Perrin, M. Schellekens, D. Boiron, C. I.Westbrook, and M. Belsley, Phys. Rev. A 74, 0536072006.

14M. Holzmann and Y. Castin, Eur. Phys. J. D 7, 4251999.15K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond, and G.

V. Shlyapnikov, Phys. Rev. Lett. 91, 0404032003.

16 G. E. Astrakharchik and S. Giorgini, Phys. Rev. A 68,031602R 2003.

17 R. Bach and K. Rzaewski, Phys. Rev. A 70, 0636222004.18 R. Pezer and H. Buljan, Phys. Rev. Lett. 98, 2404032007.19 S. Zllner, H.-D. Meyer, and P. Schmelcher, Phys. Rev. A 74,

053612 2006.20 S. Zllner, H.-D. Meyer, and P. Schmelcher, Phys. Rev. A 74,

0636112006.21 S. Zllner, H.-D. Meyer, and P. Schmelcher, Phys. Rev. Lett.

100, 0404012008.22 O. Penrose and L. Onsager, Phys. Rev. 104, 5761956.23 P. Nozires and D. Saint James, J. Phys. Paris 43, 1133

1982; P. Nozires, in Bose-Einstein Condensation, edited byA. Griffin, D. W. Snoke, and S. StringariCambridge Univer-sity Press, Cambridge, England, 1996.

24 R. W. Spekkens and J. E. Sipe, Phys. Rev. A 59, 38681999.25 L. S. Cederbaum and A. I. Streltsov, Phys. Lett. A 318, 564

2003.26 A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev.

A 70, 0536072004.

27 O. E. Alon and L. S. Cederbaum, Phys. Rev. Lett. 95, 1404022005.

28 O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Lett. A347, 882005.

29 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. A73, 0636262006.

30 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev.Lett. 99, 0304022007.

31 O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, J. Chem.Phys. 127, 1541032007.

32 O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Rev. A77, 0336132008.

33 O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Rev.


023615-16


17/17

Lett. 95, 0304052005.34P. O. Lwdin, Phys. Rev. 97, 14741955.35A. J. Coleman and V. I. Yukalov, Reduced Density Matrices:

Coulsons ChallengeSpringer, Berlin, 2000.36 Reduced-Density-Matrix Mechanics: With Application to

Many-Electron Atoms and Molecules, edited by D. A. Mazzi-otti, Advances in Chemical Physics Vol. 134 Wiley, New

York, 2007.37C. N. Yang, Rev. Mod. Phys. 34, 6941962.38F. Sasaki, Phys. Rev. 138, B13381965.39R. J. Glauber, Phys. Rev. 130, 25291963.40U. M. Titulaer and R. J. Glauber, Phys. Rev. 140, B676

1965.41 I. Bloch, T. W. Hnsch, and T. Esslinger, Nature London

403, 1662000.42H. Xiong, S. Liu, and M. Zhan, New J. Phys. 8, 2452006.43L. S. Cederbaum, A. I. Streltsov, Y. B. Band, and O. E. Alon,

e-print arXiv:cond-mat/0607556.44L. S. Cederbaum, A. I. Streltsov, Y. B. Band, and O. E. Alon,

Phys. Rev. Lett. 98, 1104052007.

45M. Girardeau, J. Math. Phys. 1, 5161960.46E. H. Lieb and W. Liniger, Phys. Rev. 130, 16051963.47J. B. McGuire, J. Math. Phys. 5, 6221964.

48 M. D. Girardeau and E. M. Wright, Phys. Rev. Lett. 84, 56912000.

49 V. I. Yukalov and M. D. Girardeau, Laser Phys. Lett. 2, 3752005.

50 K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum,Phys. Rev. A 72, 0336132005.

51 M. D. Girardeau, Phys. Rev. Lett. 97, 1004022006.52 A. G. Sykes, P. D. Drummond, and M. J. Davis, Phys. Rev. A

76, 0636202007.53 H. Buljan, R. Pezer, and T. Gasenzer, Phys. Rev. Lett. 100,

080406 2008.54 E. P. Gross, Nuovo Cimento 20, 4541961.55 L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 1961 Sov.

Phys. JETP 13, 4511961.56 C. J. Pethick and H. Smith, Bose-Einstein Condensation in

Dilute Gases Cambridge University Press, Cambridge, En-gland, 2002.

57 L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 83, 42371999.

58 A. Perrin, H. Chang, V. Krachmalnicoff, M. Schellekens, D.Boiron, A. Aspect, and C. I. Westbrook, Phys. Rev. Lett. 99,150405 2007.


023615-17

RDM of Cold Atom Gas Cederbaum

Documents

Transcript of RDM of Cold Atom Gas Cederbaum