insteinondensatesate

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Physics Letters A 332 (2004) 291–297

www.elsevier.com/locate/pla

Collective oscillations in two-dimensional Bose–Einsteincondensate

Arup Banerjee

Laser Physics Division, Centre for Advanced Technology, Indore 452013, India

Received 25 August 2004; accepted 30 August 2004

Available online 6 October 2004

Communicated by V.M. Agranovich

Abstract

We study the effect of lower-dimensional geometry on the frequencies of the collective oscillations of a Bose–Econdensate confined in a trap. To study the effect of two-dimensional geometry we consider a pancake-shaped cconfined in a harmonic trap and employ various models for the coupling constant depending on the thickness of the condenrelative to the value of the scattering length. These models correspond to different scattering regimes ranging from qudimensional to strictly two-dimensional regimes. Using these models for the coupling parameter and sum rule approamany-body response theory we derive analytical expressions for the frequencies of the monopole and the quadrupole mshow that the frequencies of monopole mode of the collective oscillations are significantly altered by the reduced dimensionalitand also study the evolution of the frequencies as the system makes transition from one regime to another. 2004 Elsevier B.V. All rights reserved.

tud-thehar-

calfer-un-on-

trap-

on-cter-k-nityen-con-

wo-in

rent

ingcon-

Recently, several theoretical and experimental sies devoted to the influence of dimensionality onproperties of a Bose–Einstein condensate in amonic trap have been reported in the literature[1–16].The reduction of dimensionality affects the physiproperties of the condensates resulting in very difent features from their three-dimensional (3D) coterparts. In current experiments on Bose–Einstein cdensates with alkali atoms confined in a magneticthe anisotropy parameterλ (defined as the ratio be

E-mail address: banerjee@cat.ernet.in(A. Banerjee).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.08.065

tween the frequencies of the trap in thez- and thetransverse directions) may be varied to achieve cdensates with special properties which are charaistic of the low dimensionality. For example, by maing the anisotropy parameter much larger than u(λ 1) a flatter and flatter (pancake-shaped) condsate can be produced. Such a pancake-shapeddensate is expected to exhibit special features of tdimensional (2D) condensate. It is well known thatthe 2D case the scattering properties are very diffeas compared to the 3D case and this in turn leads to asignificant modification of the boson–boson couplconstant. For example, the boson–boson coupling

.

292 A. Banerjee / Physics Letters A 332 (2004) 291–297

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stant in 2D limit becomes density dependent evenlow density and zero temperature. In contrast tofor 3D case, to the lowest order in the density theteractions are described by constant coupling streand deviations occur only when quantum depletand finite temperature effects are taken into consiations[17].

In a pancake-shaped 3D condensate as the anispy parameter is increased the physical properties ocondensate first change due to the modified shapthe confinement[12] and then also due to the alteratiof the scattering properties. With respect to the stering properties of a flat pancake-shaped condenthree regimes can be identified which are characterizeby different expressions for the coupling constant[10].When the linear dimension of the condensate alz-axis given byaz = (h/mωz)

1/2 (whereωz is thez-component of the angular frequency of the trapppotential andm is the mass of the trapped atoms)still much larger than the 3D scattering lengtha (az a), the collisions still take place in three dimensioand this is referred to as quasi-3D (Q3D) regime.further increasing the anisotropy parameter the cdensate gets more tightly confined along thez-axis butthe assumption that the scattering is unaffected beto break down whena ≈ az and the condensate is sato be in quasi-2D (Q2D) regime. A fully 2D condesate is achieved whenaz a so that the collisions arrestricted only in the transversex, y plane. It is nat-ural to expect that as the condensate evolves frofully 3D to a fully 2D regime its static and dynamproperties undergo dramatic changes. In Ref.[10] theevolution of the density profile of a condensate hbeen investigated as the system crosses from the 3the 2D regime. They have found that the widths ofdensity distribution crucially depend on the collisionproperties or the boson–boson interaction paramein the different regimes.

The main aim of this Letter is to study the effeof lower dimensionality on the frequencies of the clective oscillations of a condensate. In particularcalculate the collective oscillation frequencies ofquadrupole and monopole modes of a flat pancashaped condensate and study how these frequeevolve as the system undergo transitions fromQ3D to the Q2D and to a strictly 2D regime. Fthis purpose we make use of the sum rule approof many-body response theory[19,20]. By employ-

-

s

ing this method we derive analytical expressionsthe frequencies of the quadrupole and the monomodes. In the following we first derive these exprsions and then discuss the results.

We consider a dilute condensate withN bosonsconfined in an anisotropic (pancake-shaped) harmtrap characterized by the frequenciesω⊥ and ωz =λω⊥ with the anisotropy parameterλ being muchlarger than unity. Within the density functional theothe ground properties of a condensate can be cpletely described by the ground state condensatesity ρ(r) in x, y plane. The ground state densitythe condensate can be determined by minimizinglocal density energy functional

E[ρ] =∫

d2r[

h2

2m

∣∣∣∇√ρ(r)

∣∣∣2

(1)+ vext(r)ρ(r) + ε(ρ)ρ(r)],

wherevext(r) is the external harmonic potential in thtransverse direction given by

(2)vext(r) = 1

2mω2⊥

(x2 + y2).

In the above equation(1) the first and the third termrepresent the kinetic energy and the energy due tointeratomic interaction within local density approxmation (LDA), respectively. Within this approximation the interaction energy per particleε(ρ) is givenby

(3)ε(ρ) = g

2ρ(r).

Whereg is the coupling constant whose form depenon the collisional properties of the condensate. Forample,g is independent of the density for the 3D caon the other hand in the purely 2D regime it depelogarithmically on the density. In the following wbriefly describe the models for the coupling constg valid in the different collisional regimes.

For the 3D system the coupling parameterg whichis a constant completely determined by the s-wscattering lengtha and it is given by

(4)g = 4πh2

ma.

When the linear dimensionaz of the condensate alonthe z-direction is much larger then 3D scatteri

A. Banerjee / Physics Letters A 332 (2004) 291–297 293

ivef

e inis

n-by

r-ni-

. Itthensity

oreda

ime

n-the

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u-forthe

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ingIn

-a-

thistorhis

t of-ct

x-

ap-

ergy

nht-n-

length a (az a), the collisions still take place inthree dimensions. Under this condition the effectcoupling constantgQ3D which includes the effects oreduced dimensionality only is given by[4,12]

(5)gQ3D = 2√

2πh2a

maz

.

On further increasing the anisotropy andaz becomingcomparable toa (az ≈ a), the collisions start gettingaffected by the tight confinement along thez-direction.Under such a condition the condensate is said to bQ2D regime. The coupling constant in this regimegiven by

(6)gQ2D =2√

2π h2amaz

1+ a√2πaz

| ln(2(2π)3/2ρ(r)aaz)| .

Hereρ(r) is the ground state density of the condesate. The above expression was originally derivedPetrov et al.[5,6] by studying the scattering propeties of a bosonic system which is trapped harmocally in thez-direction and uniform in thex, y plane.A similar expression was later derived by Lee et al.[9]also by employing many-body T-matrix approachis important to note here that in the Q2D regimecoupling constant becomes dependent on the dein accordance with the behaviour of collisions in twdimensions. Finally, whenaz becomes much smallethana (az a), the collisions can be safely assumto be taking place in two dimensions resulting in2D condensate. The coupling constant in 2D regis given by

(7)g2D = 4πh2

m

1

| lnρ(r)a2| .

Notice that the expression forg2D also depends othe densityρ(r) but the information about the confinement direction is absent as it corresponds topurely 2D case. The expression forg2D was derivedin Ref. [18] for a homogeneous Bose gas of hard dThis form ofg2D has been employed to study the proerties confined bosons in two dimensions[3,7] and therigorous justification for this use was provided by Liet al.[8].

Having described the different models for the copling constant, we now briefly discuss the methodobtaining the frequencies of the monopole and

quadrupole modes of collective oscillations. As mtioned earlier for this purpose we employ the surule approach of the many-body response theory.most important advantage of this method is thatcalculation of frequencies requires the knowledgethe ground-state wave function (or the correspondground-state density) of many-body system only.accordance to the basic results of the sum-rule approach[19,20] the upper bound of the lowest excittion energy is given by

(8)hΩex =√

m3

m1,

where

(9)mp =∑n

∣∣〈0|F |n〉∣∣2(hω0n)p

is thepth order moment of the excitation energyhω0n

associated with the excitation operatorF andΩex isthe frequency excitation. Herehωn0 = En − E0 is theexcitation energy of eigenstate|n〉 of the HamiltonianH of the system. The upper bound given by Eq.(8)is close to the exact lowest excited state whenstate is highly collective, that is, when the oscillastrength is almost exhausted by a single mode. Tcondition is satisfied by the trapped bosons in mosthe cases. Moreover, Eq.(8) can be used for computation of the excitation energies by exploiting the fathat the momentsm1 andm3 can be expressed as epectation values of the commutators betweenF andH

in the ground state|0〉 [19,20]:

m1 = 1

2〈0|[F †, [H,F ]]|0〉,

(10)m3 = 1

2〈0|[[F †,H

],[H, [H,F ]]]|0〉.

For the purpose of calculation ofm1 andm3 as givenby the above equation we need to first choose anpropriate excitation operatorF . Following Ref. [13]the excitation operatorF is written as

(11)F = x2 + αy2,

with α = 1 and α = −1 for the monopole and thquadrupole modes, respectively. By using the enefunctional given by Eq.(1) along with the expressiofor ε(ρ) we find after some tedious although straigforward algebra following expressions for the freque

294 A. Banerjee / Physics Letters A 332 (2004) 291–297

they

-

u-

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a-n-

iesthe

pylsote

s of

ein-es

hatcyon-r

,ki-cted

eticof

hen-no-ionoleu-in-weree

theole

cies of the quadrupole

(12)Ωq = √2

(1+ T

U

)1/2

,

and the monopole

(13)Ωm = √2

(1+ T

U+ Yint

U

)1/2

,

modes. In the above equationsT andU denote the ki-netic and the harmonic confinement energies andare given by

T = h2

2m

∫d2r

∣∣∣∇√ρ(r)

∣∣∣2,(14)U = 1

2mω2⊥

∫d2r

(x2 + y2)ρ(r).

On the other handYint arises from the interaction energy (third term) term of Eq.(1). As a result of thisthe values ofYint depend on the model of the copling constant. The expressions forYint (in the unit ofNhω⊥) corresponding to the three regimes are givby

YQ3Dint = E

Q3Dint ,

YQ2Dint = (

EQ2Dint + 2kI

Q2D1 + 2k2I

Q2D2

),

(15)Y 2Dint = (

E2Dint + 2I2D

1 − 2I2D2

),

with k = a/√

π and the dimensionless interaction prametera = a/az. The general expression for the iteraction energyE i

int (i = Q3D,Q2D,2D) is given by

(16)E iint =

∫d2r

gi

2ρ2(r).

By using coupling constant given by Eqs.(5)–(7) inthe above integral we obtain the interaction energin the corresponding regime. On the other hand,expressions for other terms appearing in Eq.(15) canbe written as

IQ2D1 = √

2πNa

∫dr2 ρ2(r)

1+ k| ln(2(2π)3/2Nρ(r) aλ)|2 ,

IQ2D2 = √

2πNa

∫dr2 ρ2(r)

1+ k| ln(2(2π)3/2Nρ(r) aλ)|3 ,

I2D1 = 2πN

∫dr2 ρ2(r)

| ln(Nρ(r)a2)|2 ,

(17)I2D2 = 2πN

∫dr2 ρ2(r)

2 3 .

| ln(Nρ(r)a )|

The density appearing in the above equation (Eq.(17))is normalized to unity and notice that the anisotroparameterλ is explicitly appearing in the integrafor the Q2D case. Before proceeding further we nthat the expressions given by Eqs.(12) and (13)forthe Q3D case match with the corresponding resultRef. [13].

It is evident from Eq.(12) that the frequency of thquadrupole mode is not explicitly dependent on theteraction energy and it is true in all the three regimconsidered in this Letter. We wish to note here tEq.(12)is identical to the expression for the frequenof the quadrupole mode of a 3D condensate with cstant coupling parameter[21]. In the absence of inteparticle interactions we haveT = U and this leads toharmonic oscillator resultΩq = 2. On the other handwhen interaction energy is much larger than thenetic energy so that the kinetic energy can be negle(Thomas–Fermi approximation) we getΩq = √

2. Inthe general case one needs to know the value of kinenergy of the ground state for accurate estimationthe frequency of quadrupole mode.

Now we focus our attention on the frequency of tmonopole mode of the collective oscillations. In cotrast to the quadrupole case the frequencies of mopole mode are explicitly dependent on the interactenergy. Therefore, the frequencies of the monopmode will be affected by different models of the copling constant. To illustrate the dependence of theteraction energy on the frequencies more clearlysubstitute the virial relations associated with the thmodels

T − U + EQ3Dint = 0,

T − U + EQ2Dint + kI

Q2D1 = 0,

(18)T − U + E2Dint + I2D

1 = 0

in the respective expressions in Eq.(13) to obtain

ΩQ3Dm = 2,

ΩQ2Dm = 2

(1+ k

IQ2D1

2U+ k2I

Q2D2

U

)1/2

,

(19)Ω2Dm = 2

(1+ I

Q2D1

2U− I

Q2D2

U

)1/2

.

The first of the above equations shows that unlikeQ2D and the 2D cases the frequency of the monop

A. Banerjee / Physics Letters A 332 (2004) 291–297 295

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al-fe-

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mode in the Q3D regime is independent of the cpling constant. The frequency of the monopole moin the Q3D regime is given by the frequency of a 2harmonic oscillator. It is important to point out thatcontrast to the Q3D case the frequency of the mopole mode of a 3D condensate is explicitly dependon the interaction strength[21].

Now we turn to the detail study of the frequencof the monopole mode for the Q2D and the 2D moels of the coupling constant. To this end we first neto evaluate the integrals given in Eq.(17). For this pur-pose we employ the ground state densitiesρ(r) of thecondensates which are obtained within the ThomFermi (TF) approximation. Furthermore, to obtain tdensities within the TF approximation the spatial dpendence of the coupling constants is also negleby using results of the homogeneous system to rethe density to the chemical potential. It has been shin Refs.[9,10] that for largeN the TF approximationand spatially independent form of the coupling costant yield sufficiently accurate results.

We begin the discussion of the results with the vues of relevant parameters from the experiments oGorlitz et al. [15] with 23Na atoms. These paramters areN = 105, λ = 26.33 anda = 3.8× 10−3. Wenote here that this value ofN is consistent with theTF approximation. These parameters indicate thatcondensate produced in the experiment falls witthe Q3D regime. The numbers obtained by usthe monopole frequencies of this system areΩQ2D =2.001 andΩ2D = 2.084. These results clearly shothat for the above system the Q2D result is very clto the corresponding Q3D number. However, themodel overestimates the frequency of the monopmode and this is anticipated as this model isapplicable to the condensate achieved in the abmentioned experiment.

Now to study the effect of different models of thcoupling constant and their applicability we choothree different values of the parametera: a = 3.8 ×10−3, 0.33 and 2.68. These values are chosen sucway that the first, the second and the third numbfall in the Q3D, the Q2D and the 2D regimes, resptively. In addition to this we also chooseλ = 2 × 105

so that the condensate has negligible length in thz-direction and the motion along this axis is completfrozen. The results with these parameters are presentin Table 1. Again we can see fromTable 1that for

Table 1Frequencies of the monopole mode in the units ofω⊥ for threedifferent values of the dimensionless interaction parametera cal-culated using Eq.(19) for N = 5× 105 andλ = 2× 105

a ΩQ3D ΩQ2D Ω2D

3.8× 10−3 2.00 2.001 2.0370.33 2.00 2.049 2.0842.68 2.00 2.292 2.182

a = 3.8×10−3 the numbers predicted by the Q3D athe Q2D models are very close and the corresponresult from the 2D model is quite higher than thetwo numbers. Fora = 0.33 the numbers obtained bthe both Q2D and 2D models are markedly differfrom the result of Q3D model. As has been discusbefore fora = 0.33 along with the large value ofλ thescattering properties start to get influenced by the cfinement. In this situation it is expected that the Qand 2D models will give significantly different numbers in comparison to the corresponding Q3D resIn the light of our earlier discussion we expect thata = 0.33 the coupling constant is better describedthe Q2D model. On the other hand, for this valuea the frequency of the monopole mode obtained wthe 2D model is higher than that of Q2D model.the interaction parametera is further increased tovaluea = 2.68 the scattering properties become trtwo-dimensional, consequently for this value ofa the2D model should be able to predict the frequencythe monopole mode of a two-dimensional condensIn contrast to the case ofa = 0.33, for a = 2.68 thevalue of the frequency obtained by the 2D modelower than the corresponding number from the Qmodel.

Finally, for the sake of completeness we plotFig. 1the frequencies of the monopole mode obtainwith three different models as a function of theteraction parametera. The curves are drawn with thanisotropy parameterλ = 5 × 105 and the number oatomsN = 105. It can be clearly seen fromFig. 1that the frequencies of the monopole mode obtaiwith the three models of the coupling constant exhdifferent trend with the increase in the interaction parmetera. For example, in contrast to the constant vaof the frequency for the Q3D case the frequencobtained by employing the Q2D and the 2D modincrease asa is increased. As mentioned before it

296 A. Banerjee / Physics Letters A 332 (2004) 291–297

fith

re-layed

bythe

her

ingentore,re-iourys-2Dlso

heri-

s ofn-

con-

en-the

-be

ibedac-lec-els.h ofnd-on.d-in

eady.inncy

en-ted

ndal

ev.

1)

ev.

th.

v.

tt.

02)

Fig. 1. Frequencies (in units ofω⊥) of the monopole mode o5 × 105 23Na atoms confined in a highly deformed trap wλ = 2× 105 as a function of the interaction parametera. The solidline represents results for Q2D case while the corresponding 2Dsults are shown by the dashed line and the Q3D results are dispby the horizontal line.

only for a 1 the values of frequencies obtainedthe Q3D and the Q2D models are very close and2D model gives quite different numbers. On the othand, whena exceeds the value 10−2, the effects ofreduced dimensionality start affecting the scatterproperties and both Q2D and 2D models give differresults as compared to the Q3D numbers. Therefwe conclude from our results that the collective fquencies of the monopole mode and their behavcan be used to identify the dimensionality of the stems as the system evolves from Q3D to a strictlyregime. The results of theoretical calculations can abe used to test the validity of different models of tcoupling constant by comparing them with the expemental values.

In summary, we have calculated the frequenciecollective oscillations of the Bose–Einstein condesate confined in a flat pancake-shaped trap. Thedensate is tightly trapped alongz-axis such that themotion along this axis is frozen. For such a condsate depending on the value of the ratio between

size of the condensate along thez-direction and the swave scattering length, three different regimes canidentified. These three different regimes are descrby three different models for the boson–boson intertions. We have calculated the frequencies of the coltive oscillations corresponding to these three modFor this purpose we have used sum-rule approacmany-body response theory along with the groustate density obtained within the TF approximatiThe main result of this Letter is that the different moels for the coupling constant are clearly manifestedthe frequency of the monopole mode and they lto distinct results in the region of their applicabilitThe effect of modification of the collision propertiestwo dimension start changing the monopole frequewhen the value ofa becomes more than 10−2. On theother hand, quadrupole mode is not explicitly depdent on the coupling parameter and thus not affecby the boson–boson interactions.

Acknowledgement

It is a pleasure to thank Dr. S.C. Mehendale aDr. M.P. Singh for helpful discussions and criticreading of the manuscript.

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