Localization of a dipolar Bose–Einstein condensate in a bichromatic optical lattice

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 (8pp) doi:10.1088/0953-4075/43/20/205305

Localization of a dipolar Bose–Einsteincondensate in a bichromatic optical latticeP Muruganandam1,2, R Kishor Kumar2 and S K Adhikari1

1 Instituto de Fısica Teorica, UNESP-Universidade Estadual Paulista, 01.140-070 Sao Paulo,Sao Paulo, Brazil2 School of Physics, Bharathidasan University, Palkalaiperur Campus, Tiruchirappalli 620024,Tamilnadu, India

E-mail: anand@cnld.bdu.ac.in and adhikari@ift.unesp.br

Received 17 June 2010, in final form 29 July 2010Published 4 October 2010Online at stacks.iop.org/JPhysB/43/205305

AbstractBy numerical simulation and variational analysis of the Gross–Pitaevskii equation we studythe localization, with an exponential tail, of a dipolar Bose–Einstein condensate (DBEC) of52Cr atoms in a three-dimensional bichromatic optical lattice (OL) generated by twomonochromatic OLs of incommensurate wavelengths along three orthogonal directions. For afixed dipole–dipole interaction, a localized state of a small number of atoms (∼1000) could beobtained when the short-range interaction is not too attractive or not too repulsive. A phasediagram showing the region of stability of a DBEC with a short-range interaction anddipole–dipole interaction is given.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The localization of a non-interacting wave form in a disorderedpotential, predicted by Anderson [1], has been the topic ofvigorous research in different areas. Localization has beenobserved experimentally in diverse contexts [2] includingBose–Einstein condensates (BEC) [3, 4]. Billy et al [3]demonstrated the localization of a cigar-shaped interacting87Rb BEC released into a one-dimensional (1D) waveguidewith controlled disorder created by a laser speckle. Roatiet al [4] observed the localization of a non-interacting 39K BECin a bichromatic quasi-periodic optical-lattice (OL) potentialcreated by superposing of two standing-wave polarized laserbeams of incommensurate wavelengths. The non-interactingBEC was created [4] by tuning the atomic scattering lengtha to zero near a Feshbach resonance [5]. The disorder in aquasi-periodic OL potential is deterministic, in contrast to thecomplete disorder in an optical speckle potential. Localizationin such a quasi-periodic potential is a special case of Andersonlocalization in a fully disordered potential and is well describedby the Aubry–Andre model [6]. Such a localization isoften termed Aubry–Andre localization. However, these twomechanisms of localization are distinct. While the Andersonlocalization of wavefunctions with exponential tails is a purequantum effect, the Aubry–Andre localization may occur in

a classical phase space [7]. The localization of a BEC in adisordered potential has been the subject matter of varioustheoretical [8–12] and experimental [2–4] studies.

In the presence of strong disorder one has strong Andersonlocalization [3, 4], where the localized state could be quitesimilar to a localized state of the Gaussian shape in aninfinite potential or a potential of very high barriers. Thenthe quantum state cannot escape the strong barriers of thedisordered potential. However, the more interesting caseof localization is in the presence of a weak disorder whenthe system is localized due to the quasi-periodic (disordered)nature of the potential [3, 4] and not due to the strength ofthe lattice. Localization takes place due to cancellation ofwaves coming after multiple scattering from many barriersof the random potential. When this happens, the localizedstate acquires a pronounced exponential tail.

The usual dilute BEC with negligible dipole momentinteracting via short-range interaction is described by themean-field Gross–Pitaevskii (GP) equation. More recently,it has been possible [13] to obtain a BEC of 52Cr atomswith large dipole moment leading to a long-range dipole–dipole (dipolar) interaction superposed on the usual short-range interaction, which can be varied using a Feshbachresonance [5]. This allows us to study the dipolar BEC

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

(DBEC) of 52Cr atoms with a variable short-range interaction[13]. Because of the anisotropic dipolar interaction, the DBECpossesses many distinct features [13–17], which are underactive investigation by different research groups [18, 19]. Forexample, the stability of a DBEC depends not only on the valueof the scattering length, but also strongly on the geometry ofthe trapping potential [13–15]. A pancake-shaped trappingpotential gives a repulsive mean field of the dipolar interactionand thus the dipolar condensate is more stable. In contrast,a cigar-shaped trapping potential yields an attractive meanfield of the dipolar interaction and hence leading to a dipolarcollapse [13, 20–23]. Peculiarities in the collective low-energyshape oscillations of DBEC have been studied by Yi and You[21]. Analogue of roton–maxon instability and the appearanceof roton minimum in the Bogoliubov spectrum [24, 25] andangular collapse of DBEC [26] have been studied. Numericalstudies on the stability show certain unusual structure of theHartree ground state of the dipolar condensate in an anisotropictrap [15, 16].

After the experiment [4] on the localization of a 1DBEC in a quasi-periodic trap, a natural extension of thisphenomenon would be to achieve localization in two and threedimensions (3D) [12, 27], both for a BEC and a DBEC. Thetheoretical description of a dilute weakly interacting DBECcan be formulated by including a dipolar interaction term in theGP equation maintaining the formal simplicity, nevertheless,increasing vastly the numerical complexity. Using thenumerical and variational solutions of the GP equation, westudy the localization of a BEC and a DBEC in 3D, in thepresence of bichromatic OL potentials along the orthogonaldirections. Although the localized states have an exponentialtail in a weak quasi-periodic potential, the central part ofsuch localized states, responsible for the major contributionto the total density, may have a Gaussian distribution. In thispaper we consider such localized states with an exponentialtail and a Gaussian central part, which can also be studied bythe variational approximation. The variational approximationprovides an analytical understanding of the localizationand also yields an interesting result when the numericalprocedure is difficult to implement. Such a variationalGaussian approximation has successfully been applied to thelocalization of a BEC without dipolar interaction in one [10],two and three dimensions [12] in a bichromatic OL potentialas well as a speckle potential [11] in one dimension.

Due to the angle dependence of the long-range dipolarinteraction, the localization of a DBEC is more interestingthan that of a BEC with only a short-range interaction.The dipolar interaction is weak in the spherically symmetricshape, attractive in the cigar shape (aligned dipoles arrangedlinearly attract each other) and repulsive in the pancake shape(aligned dipoles arranged side-by-side repeal each other) [19].Because of this, the trap configuration plays an essential rolein the localization of a DBEC. The controllable short-rangeinteraction together with the exotic dipolar interaction makesthe DBEC an attractive system for experimental localizationin a bichromatic OL potential and a challenging system fortheoretical investigation allowing us to study the interplaybetween the dipolar interaction and the short-range interactionin dipolar atoms.

The cigar- and pancake-shaped localized DBEC areobtained by considering bichromatic OL potentials of differentstrengths along the orthogonal directions. A localizedDBEC, without short-range interaction, can be achieved in abichromatic OL trap for a small number of atoms by tuning [13]the atomic scattering length to zero near a Feshbach resonance[5]. For a cigar-shaped DBEC, without short-range interaction,as the number of atoms increases it becomes highly attractiveand suffers from collapse instability. In the presence of arepulsive short-range interaction, the effect of the attractivedipolar interaction in the cigar shape can be compensatedleading to a localized DBEC for a small number of atoms;delocalization may take place due to excess of repulsion fora large number of atoms. For a pancake-shaped DBEC,the dipolar interaction is repulsive and a localized DBEC isobtained for a small number of atoms. For a large numberof atoms excess of repulsion should lead to delocalization.From a variational analysis of the localization of a DBEC,a phase diagram illustrating its stability for different short-range interactions, number of atoms and the geometry of thebichromatic OL is given. To obtain localization, the short-range interaction should be small. As in 1D [9, 10, 28], a largerepulsive short-range interaction destroys the localization andthe localized state escapes to infinity in all cases. A largeattractive short-range interaction leads to collapse instabilityand destroys the localization.

In section 2 we present a brief account of the modified GPequation with the bichromatic OL potential in the presenceof a dipolar interaction together with a Gaussian variationalanalysis. In section 3 we present numerical and variationalstudies of the localization of a dipolar interaction in thepresence and absence of a short-range interaction. In section 4we present a brief summary of the present investigation.

2. Analytical formulation

We study the localization of the DBEC of N atoms, each ofmass m, using the following mean-field GP equation with thebichromatic OL potential V (r) [13]:

m

h2 μφ(r) =[−1

2∇2 + V (r) + 4πaN |φ(r)|2

+∫

Udd(r − r′)|φ(r′)|2 dr′]

φ(r), (1)

with

Udd(r) = 3Nadd(1 − 3 cos2 θ)/r3 (2)

and

V (r) =2∑

l=1

Al[νρ{sin2(klx) + sin2(kly)} + νz sin2(klz)], (3)

where φ(r) is the DBEC wavefunction of the normalization∫φ(r)2dr = 1, μ is the chemical potential, a is the atomic

scattering length, θ is the angle between r and the direction ofpolarization, here taken along the z axis, Al = k2

l sl/2, λl arethe wavelengths, kl = 2π/λl are the wave numbers and sl are

2

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

the strengths of the OL potentials. The parameters νρ and νz

control the relative strengths of the OLs in different directionsand can be varied to achieve the pancake- (νz > 1, νρ = 1)and cigar-shaped (νρ > 1, νz = 1) DBEC. The constantadd = μ0μ

2m/(12πh2) in the dipolar interaction Udd is alength characterizing the strength of the dipolar interactionand its experimental value for 52Cr atoms is 15a0 [13], with a0

being the Bohr radius, μ the (magnetic) dipole moment of asingle atom and μ0 the permeability of free space.

The GP equation (1) can be solved variationally byminimizing the energy functional [13]

m

h2 E[φ] =∫ [

1

2|∇φ|2 + V (r)φ2 + 2πaNφ4

+φ2

2

∫Udd(r − r′)|φ(r′)|2 dr′

]dr (4)

with the following Gaussian ansatz for φ(r) [29]:

φ(r) =(

π−3/2

w2ρwz

)1/2

exp

(− ρ2

2w2ρ

− z2

2w2z

), (5)

where wρ is the width in the radial direction ρ and wz isthe width in the axial direction z. We have assumed circularsymmetry in the x–y plane. Equations (4) and (5) lead to [29]

m

h2 E = 1

4

[2

w2ρ

+1

w2z

]+

N√2π

1

w2ρwz

[a − addf (κ)]

+1

2

2∑l=1

Al[2νρ + νz − 2νρEρ,l − νzEz,l], (6)

f (κ) = 1 + 2κ2

1 − κ2− 3κ2atanh

√1 − κ2

(1 − κ2)3/2, κ ≡ wρ

wz

, (7)

where Eρ,l = exp(−w2

ρk2l

)and Ez,l = exp

(−w2z k

2l

). For a

stationary state the energy mE/h2 should have a minimum asa function of wρ and wz: ∂E/∂wρ = ∂E/∂wz = 0, togetherwith the condition(

∂2E

∂wρ∂wz

)2

− ∂2E

∂w2ρ

∂2E

∂w2z

< 0. (8)

The minima conditions ∂E/∂wρ = ∂E/∂wz = 0 become inexplicit notation

2νρw4ρ

2∑l=1

Alk2l Eρ,l − N√

2πwz

[2a − addg(κ)] = 1, (9)

2νzw4z

2∑l=1

Alk2l Ez,l − 2Nwz√

2πw2ρ

[a − addh(κ)] = 1, (10)

where

g(κ) = 2 − 7κ2 − 4κ4

(1 − κ2)2+

9κ4atanh√

1 − κ2

(1 − κ2)5/2, (11)

h(κ) = 1 + 10κ2 − 2κ4

(1 − κ2)2− 9κ2atanh

√1 − κ2

(1 − κ2)5/2. (12)

The solution of (9)–(12) determines the widths wz and wρ .For a certain set of parameters, localization is possible if thereis a minimum of the energy. Hence from (6) and (9)–(12)we can determine if a BEC or DBEC would be localizedor not.

3. Numerical study

We perform a full 3D numerical simulation in Cartesian x, y, z

variables using the imaginary- and real-time propagations withthe Crank–Nicolson discretization [30] employing small space(∼0.025) and time (∼0.0001) steps necessary for obtainingconverged results. For this purpose we use the FORTRANprograms provided in [31] after transforming (1) into a time-dependent form by replacing mμ/h2 by i∂/∂t , where t isthe time. The dipolar interaction term is evaluated bythe usual fast Fourier transformation technique [14]. Theimaginary- and real-time propagations lead essentially to thesame localized states. This not only ensures the correctnessof the calculational scheme, but also guarantees that thelocalized states are stationary and not a result of dynamicalself-trapping [32]. (The imaginary-time propagation can onlyfind the stationary localized states and cannot obtain thedynamical self-trapped states.) The stability of the localizedstate was tested by the real-time propagation allowing smallperturbations of potential or interaction parameters. (In theabsence of the dipolar interaction, the localized states havebeen demonstrated explicitly to be stable in one [10] and two[12] dimensions.) The accuracy of the numerical simulationwas tested by varying the size of space and time steps and thetotal number of space and time steps.

To compare the numerical results with variationalanalysis, we only consider localized states mostly occupyinga single site of the bichromatic OL. Throughout this study thestrength parameters of the two components of the bichromaticOL are taken as s1 = s2 = 4 and the correspondingwavelengths are taken as λ1 = 5 μm, λ2 = 0.862λ1. Therelative strengths of the radial and axial bichromatic OLs arevaried by adjusting the parameters νρ and νz in equation (3).To demonstrate that with these sets of parameters we are inthe limit of localization with an exponential tail in a weakpotential, we solve the 1D linear Schrodinger equation indimensionless variables [31]

− 1

2φ′′(x) +

2∑l=1

2π2sl

λ2l

sin2

(2π

λl

x

)φ(x) = Eφ(x), (13)

with the above sets of parameters, where the prime denotesthe x-derivative and E denotes the energy. In the limitof zero nonlinearity (a = 0) and zero dipolar interaction(add = 0), (1) decouples into three equations like (13).In figures 1(a) and (b) we plot the density φ2(x) of (13)versus x in linear and log scales together with its Gaussianvariational counterpart φ2(x) = exp(−x2/w2)/(π1/2w), withw = 0.4865 being the variational width, and the exponentialfit φ2(x) = exp[−abs(x)/ lloc]/(0.4865π1/2), with lloc = 0.75being the localization length [3], providing a measure ofthe exponential tail. For a strong localization with a strongbichromatic potential lloc → 0 and the localized state has a

3

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

0

0.4

0.8

1.2

-3 -2 -1 0 1 2 3

φ2 (x)

x

(a) λ1 = 5λ2/λ1 = 0.862s1 = s2 = 4

Num

0

0.4

0.8

1.2

-3 -2 -1 0 1 2 3

φ2 (x)

x

λ1 = 5λ2/λ1 = 0.862s1 = s2 = 4

NumVar

10-910-810-710-610-510-410-310-210-1100

-8 -4 0 4 8

φ2 (x)

x

λ1 = 5λ2/λ1 = 0.862s1 = s2 = 4

Num

10-910-810-710-610-510-410-310-210-1100

-8 -4 0 4 8

φ2 (x)

x

λ1 = 5λ2/λ1 = 0.862s1 = s2 = 4

(b)NumVar

10-910-810-710-610-510-410-310-210-1100

-8 -4 0 4 8

φ2 (x)

x

λ1 = 5λ2/λ1 = 0.862s1 = s2 = 4

NumVarExp

Figure 1. (a) Normalized numerical (Num) and variational (Var) densities φ2(x) of (13) versus x for the 1D localized state withbichromatic OL wavelengths λ1 = 5, λ2 = 0.862λ1 and strengths s1 = s2 = 4. (b) The same densities together with the exponentialfit φ2(x) = exp[−2abs(x)/ lloc]/(0.4865

√π), lloc = 0.75 on a log scale.

pure Gaussian tail. However, when lloc > xrms with xrms beingthe root mean square size, the exponential tail is pronouncedand the limit of localization in a weak quasi-periodic potentialis attained [3, 4]. In the present example, xrms = 0.53 andlloc = 0.75, and hence we are in the limit of localization ina weak potential. This is clear from figure 1(b), where thecentral part of the localized state is fitted to the variationalGaussian solution, whereas the large-x parts are fitted to anexponential function with a large localization length over aboutnine orders of magnitude. In figure 1(b), one can identifyseveral minor peaks in successive wells of the bichromaticOL potential. The state of figure 1 is localized by theweak bichromatic lattice and if we substantially reduce thestrength of the potential, no localized state will emerge. In theexperiments of [3, 4] and in some other studies on localization[9] in a weak potential, localized states with a pronouncedundulating tail over many wells of the localization potentialwere considered. However, a weak limit of localization withan exponential tail can also be achieved in the absence of apronounced undulating tail, as we have shown here. In thispaper we consider localization in the presence of a pronouncedGaussian peak and a weak undulating tail as can be seenfrom figure 1(b). The existence of a pronounced Gaussianpeak will be turned to good advantage in predicting accurateanalytical variational results for the localized state and for ananalytical understanding of the localization. We also considerthe localization in the presence of a short-range interaction anddipolar interaction. The inclusion of a repulsive short-rangeinteraction will in general destroy localization by increasingthe localization length and thus creating a more pronouncedexponential tail [9]. The inclusion of the dipolar interactionwill have an effect on localization, which we study here,and such inclusion should not destroy the exponential tail oflocalization as illustrated in figure 1(b).

A variational analysis is useful for a qualitativeunderstanding of the problem and we present the same beforeconsidering a numerical solution of (1). In figure 2 we plotthe variational widths wρ and wz for a DBEC of 170 52Cratoms in the cigar and pancake shapes. This figure showsthe evolution of the widths from the pancake to cigar shapeswhile the radial width wρ reduces and the axial width wz

increases as expected. For a = 0, the axial width wz does not

1 10 100νz

νρ = 1νz = 1pancakecigar

1 10 100νz

νρ = 1νz = 1pancakecigar

1 10 100νz

νρ = 1νz = 1pancakecigar

1 10 100νz

νρ = 1νz = 1pancakecigar

0

0.2

0.4

0.6

0.8

1

1 10

wρ,

wz

(μm

)

νρ

0

0.2

0.4

0.6

0.8

1

1 10

wρ,

wz

(μm

)

νρ

0

0.2

0.4

0.6

0.8

1

1 10

wρ,

wz

(μm

)

νρ

0

0.2

0.4

0.6

0.8

1

1 10

wρ,

wz

(μm

)

νρ

Figure 2. Variational widths wρ and wz of the DBEC versusbichromatic OL relative strength parameters νρ and νz (namelyequation (3)) for 170 52Cr atoms for scattering lengths a = 0and 20a0.

increase with the increase of νρ , as in the cigar shape the dipolarinteraction becomes attractive and does not permit the increaseof wz.

Next we study, using (6), the set of values of theparameters for which the energy can have a minimum andallows a stable localized DBEC. The region of stability forthe dipole strengths add = 0 and 15a0 is shown in figure 3for (a) 170 and (b) 500 52Cr atoms as a phase plot of a/a0

versus the relative strengths of the bichromatic OL νρ andνz in the radial and axial directions. The stability of aDBEC in a harmonic trap has also been studied [14]. Fora fixed add and N the stability region lies between the twocorresponding lines in figure 3. Above the upper line thesystem becomes too repulsive (positive Na/a0) to be confinedby the weak bichromatic OL. Below the lower line the systembecomes too attractive (negative Na/a0) and suffers fromcollapse instability. The localization of a BEC (without dipolarinteraction) is controlled by the nonlinearity 4πaN alone of(1) and the effect of the dipolar interaction on the stability ofthe DBEC is clearly exhibited in figure 3. In these figureswe also plot numerical results for collapse instability withnegative (attractive) scattering length (the lower limit of thestability in this figure), which agree well with the variational

4

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

(a) (b)

Figure 3. Stability region in the a/a0 versus νρ and νz (relative strengths of the bichromatic OL in the radial and axial directions)phase plots for (a) 170 and (b) 500 atoms for add = 15a0 (52Cr atoms) and add = 0 variational—var (lines), numerical—num (points).Localization is possible between the upper and lower lines of the same data set.

(a) (b)

Figure 4. Numerical (num) and variational (var) energies E versus relative strengths of the bichromatic OLs νρ and νz for (a) 170 and(b) 500 52Cr atoms.

results. Starting from a stable localized state, the stabilitylines (points) are obtained by slowly changing the scatteringlength a at a fixed trap symmetry (by fixing the parameters νρ

and νz) until no localized state is obtained (by numerical orvariational means). On the pancake side the dipolar interactionis repulsive and the localization is destroyed for a smallervalue of the repulsive scattering length compared to the casewhere the dipolar interaction is absent, as can be seen infigure 3. The opposite happens on the cigar side where thedipolar interaction is attractive. On the cigar side the dipolarinteraction is attractive and the localization is destroyed for alarger value of the repulsive scattering length compared to thecase where the dipolar interaction is absent, as can be seen infigure 3(b) for 500 atoms. This effect is smaller in figure 3(a)for 170 atoms.

From (7) we find that f (κ) is positive for the cigar shapeleading to an attractive contribution to energy (6), whereas itis negative in the pancake shape leading to a repulsive termin energy. For N = 170 there is a symmetric state withwz = wρ and f (κ) = 0 near νρ = 1, νz ≈ 1.3 where thedipolar interaction does not contribute and where the DBECis most stable at the maxima as shown in figure 3(b). At thispoint the DBEC acts like a ‘normal BEC’ and the Na/a0 valueat the maximum is independent of N. In the pancake shape,

the localization of the DBEC can be easily destroyed due toa large repulsive dipolar interaction in the weak bichromaticOL. In the cigar shape, the large attractive dipolar interactionmay lead to the collapse of the localized DBEC more easilythan in the absence of the dipolar interaction. These aspectsare clearly illustrated in figure 3(b). For N = 500 theinteractions are stronger than for N = 170 and the domain ofthe allowed localization in terms of the number of atoms hasreduced.

In figures 4(a) and (b) we compare the numerical andvariational energies of the 52Cr DBEC of 170 atoms witha = 0 and 20a0 and of 500 atoms with a = 0 and 10a0,respectively. A smaller value of a is required for the stabilityof 500 atoms (see figure 3(b)). The agreement betweenvariational and numerical results is good in general exceptfor νz → 100, νρ = 1. In this limit the localized DBECoccupies two sites of the bichromatic OL and does not have aGaussian shape. This justifies a small disagreement betweenthe numerical and variational results in this limit. Thevariational result for a = 10a0 in figure 4(b) only exists up toνz ≈ 35 (see figure 3(b)).

Next we study how the DBEC solely under the effectof the dipolar interaction (a = 0) changes its shape as wemove from the cigar- to pancake-shaped configuration. In

5

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

(a)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

z (μ

m)

0

0.2

0.4

0.6

0.8

1 (b)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

z ( μ

m)

0

0.2

0.4

0.6

0.8

1

(c)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

y (μ

m)

0

0.2

0.4

0.6

0.8

1 (d)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

y (μ

m)

0

0.2

0.4

0.6

0.8

1

Figure 5. 2D Contour plot of density φ2(x, 0, z) in the y = 0 plane for the cigar-shaped DBEC with νρ = 10, νz = 1 of N = 500 52Cratoms with a = 0: (a) numerical, (b) variational. 2D Contour plot of the density φ2(x, y, 0) in the z = 0 plane for the same DBEC:(c) numerical, (d) variational.

figures 5(a) and (b) we show the 2D contour plot from thenumerical and variational analysis, respectively, for the densityφ2(x, 0, z) in the y = 0 plane for a cigar-shaped DBEC withνρ = 10, νz = 1 for 500 52Cr atoms for a = 0. The 2D contourplot from the numerical and variational analysis, respectively,for the density φ2(x, y, 0) in the z = 0 plane for the sameDBEC is shown in figures 5(c) and (d), respectively. Thislocalized numerical DBEC state is of small size, comparedto its variational counterpart, due to the attractive dipolarinteraction in the cigar-shaped DBEC. The attraction due tothe dipolar interaction has caused the DBEC to contract froman average Gaussian shape. The corresponding numericaland variational energies of the localized states of figure 5 are15.16 and 15.53, respectively. The localized DBEC statesin figure 5 (and also in figure 6) occupy practically a singlebichromatic OL site at the origin along x, y and z: −1.25 μm< x, y, z < 1.25 μm. However, these localized states haveexponential tails.

The effect of the dipolar interaction is small in thesymmetric case with νz = νρ = 1 and N = 500.Nevertheless, due to the complicated dipolar interaction, thedensity φ2(x, 0, z) does not have a pure circular shape in thiscase (not explicitly shown). This density distribution willbe circular for a BEC without dipolar interaction. (A nearlycircular shape for this density is obtained for a DBEC with

νρ = 1 and νz = 1.6.) The numerical and variational energiesfor this state are 6.97 and 7.06, respectively.

We next consider the localization for νz = 10, νρ = 1and a = 0. In this pancake shape, the dipolar interaction isrepulsive and for 500 52Cr atoms a localized DBEC is obtained.In figures 6(a) and (b) we show the numerical and variational2D contour plots of the density φ2(x, 0, z) in the y = 0 plane,respectively, for this DBEC. The numerical and variational2D contour plots for the density φ2(x, y, 0) in the z = 0plane for the same DBEC is shown in figures 6(c) and (d),respectively. In this case, the dipolar interaction is repulsivein nature and hence the size of the DBEC is larger than theGaussian variational shape (see figure 5 where the oppositehappens for a DBEC with an attractive dipolar interaction.).The numerical and variational energies for this state are 14.16and 14.26, respectively. The change from a cigar shape toa pancake shape of the localized DBEC as we move fromνρ = 10 and νz = 1 to νρ = 1 and νz = 10 is obvious fromthe density distribution in figures 5(a) and 6(a). Althoughthe variational and numerical energies (studied in figures 5and 6) are quite close to each other, the numerically obtainedmatter density should have some peculiarities not obtainablefrom the variational calculation due to the anisotropic dipolarinteraction. (The variational calculation is based on an axiallysymmetric Gaussian distribution.)

6

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 205305 P Muruganandam et al

(a)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

z (μ

m)

0

0.2

0.4

0.6

0.8

1 (b)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

z (μ

m)

0

0.2

0.4

0.6

0.8

1

(c)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

y (μ

m)

0

0.2

0.4

0.6

0.8

1 (d)

-1 -0.5 0 0.5 1

x (μm)

-1

-0.5

0

0.5

1

y (μ

m)

0

0.2

0.4

0.6

0.8

1

Figure 6. Same as in figure 5 for the DBEC in the pancake trap with νρ = 1, νz = 10.

4. Summary and conclusion

We investigated the localization of a 52Cr DBEC in a weakbichromatic OL trap in the presence and absence of a short-range interaction using the numerical and variational solutionsof the 3D GP equation (1). Of the two solutions, the numericalsolution is the most precise and should be used in the caseof disagreement with the variational solution. Although thedensity of the central part of the localized states has a near-Gaussian distribution, the density distribution also has a longexponential tail [3, 4]. The Gaussian distribution near thecenter permits a variational analysis of localization, which isused for an analytical understanding of the problem. A DBECof a small number of atoms with a weak short-range interactioncould be localized by a relatively weak bichromatic OL trap.From the variational solution we obtain a phase diagram(figure 3(b)) illustrating the effect of the dipolar interaction onthe localization as a function of the strengths of the traps νz andνρ in the axial and radial directions, respectively. We find thatfor 52Cr atoms, the dipolar interaction has a moderate effect onlocalization. (A larger effect will certainly appear for dipolarmolecules where the dipole moment could be larger by an orderof magnitude compared to the dipole moment of 52Cr atoms.)The numerical and variational energies of the DBEC, as well asthe corresponding densities, are in reasonable agreement witheach other. In the absence of a short-range interaction, thelocalized DBEC can accommodate the largest number of 52Cr

atoms (∼1000) in the spherical configuration and this numberreduces for both cigar and pancake shapes due to the attractiveand repulsive dipolar interaction, respectively. The attractivedipolar interaction leads to collapse and the repulsive dipolarinteraction leads to leakage to infinity. We hope that this studywill motivate experiments on the localization of a 52Cr DBECin a bichromatic OL trap. The estimate of the number oflocalized 52Cr atoms, their radial and axial sizes and shapes,etc as predicted in the present study can be verified in theexperiment.

Acknowledgments

FAPESP and CNPq (Brazil) and DST (India) provided partialsupport.

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