4 0 0 Depletion of a Bose-Einstein condensate by laser-induced 2 n dipole-dipole interactions a J 8 I.E. Mazets1,3, D.H.J. O’Dell2, G. Kurizki3, N. Davidson3 and W.P. Schleich4 2 1Ioffe Physico-Technical Institute, St.Petersburg 194021, Russia ] 2Department of Physics & Astronomy, University of Sussex, Brighton t f BN1 9QH, United Kingdom o s 3Weizmann Institute of Science, 76100 Rehovot, Israel t. 4Abteilung fu¨r Quantenphysik, Universit¨at Ulm, Ulm D-89069, Germany. a m - d Abstract n WestudyagaseousatomicBose-Einsteincondensatewithlaser-induceddipole-dipoleinteractions o c using theHartee-Fock-Bogoliubov theory within thePopov approximation. The dipolar interactions [ introduce long-range atom-atom correlations which manifest themselves as increased depletion at momenta similar to that of thelaser wavelength, as well as a ‘roton’ dip in theexcitation spectrum. 2 Suprisingly, the roton dip and the corresponding peak in the depletion are enhanced by raising the v temperature above absolute zero. 2 PACS: 03.75.Fi, 05.30.Jp 7 6 0 1 Introduction 1 3 0 One of the novel features of gaseous Bose-Einstein condensates (BECs), when considered from a many- / t body physics perspective, is the ability to directly manipulate the interatomic interaction using external a electromagnetic fields [1]. In the BECs realized thus far the interatomic interactions can be described m by a single parameter, the s-wavescattering length, a, and are short-range in comparison to the average - interatomic distance. The experiment by Inouyeet al [2] demonstratedhow the s-wavescatteringlength d n canbe changedbymagneticfields viaaFeshbachresonance. Hereweconsiderenhancingandcontrolling o the interatomic interactions using laser-induced dipole-dipole forces. These interactions are intrinsically c long-range and so affect the gas in a way profoundly different from a Feshbach resonance. In particular, : v dipole-dipole interactions, both laser-induced [3] and static [4], have been predicted to be capable of i introducing a ‘roton’ dip into the Bogoliubov excitation spectrum of the BEC. This is behaviour remi- X niscent of that of the strongly correlated quantum liquid helium II, and can be understood in terms of r a Feynman’s relation which provides an upper bound for the spectrum E(k) of helium II ¯h2k2 E(k) (1) ≤ 2mS(k) whereS(k),knownasthestaticstructurefactor,istheFouriertransformofthepairdistributionfunction. S(k) is a measure of 2nd order correlation in the system. Feynman’s formula interprets the roton dip in 1 heliumIIasbeingduetoapeakinS(k)andhenceasbeingduetostrongcorrelations,whichareatvalues ofkcorrespondingtophononshavingwavelengthsontheorderoftheinteratomicspacing(intuitively: the excitations at these wavelengthsrequire less energyto excite in comparisonto neighbouringwavelengths which cost more compressional energy). InagaseousBECatzerotemperaturethe BogoliubovtheoryofaweaklyinteractingdegenerateBose gas [5] is valid and shows that 2nd order correlationarises predominantly from pairs of atoms scattering out of the condensate to form the so-called quantum depletion, or ‘above condensate’ fraction. In a homogenous system the atoms in these pairs are prefectly correlated, in the sense of perfect two mode- squeezing [6, 7, 8]. However, in gaseous BECs the quantum depletion, and hence 2nd order correlation, is typically very small on account of their diluteness, as expressed through the gas parameter na3 1, ≪ wherenistheatomicdensity. Indeed,withintheBogoliubovtheorytheFeynmanrelation(1)isanexact equality, yet in a measurement of the bulk excitation spectrum of a regular gaseous BEC interacting via s-wave scattering by Steinhauer et al [9] no roton was seen. Laser-induceddipole-dipole interactions, on the otherhand,introducecontrollable (vialaserintensity etc.) long-rangecorrelationsonthescaleofthe laser wavelength, and one might expect some significant depletion at the corresponding momentum. In our previous investigations into laser-induced dipole-dipole interactions in a BEC [3, 10] (and references therein) we considered the zero-temperature case, and the calculations were limited to the basic Bogoliubov method [5] which does not treat the depletion self-consistently. Our purpose here is to usetheso-calledPopovversionoftheHartree-Fock-Bogoliubovmethod[11,12],whichtreatsthedepletion asaself-consistentmean-fieldinordertoincludetheeffectsofthetheback-actionofthedepletedfraction upon the condensate. This method was primarily developed to treat the finite temperature case where depletionisimportant,butourhopehereistoaccountforsignificantquantumdepletionduetoenhanced interactions and also for thermal depletion due to non-zero temperature. The most significant results we find here are that, perhaps contrary to expectation, the depletion can increase the effects of the dipole-dipole interactions, deepening the roton dip in the spectrum, so that the roton is not diminished by finite-temperature effects but can actually be enhanced. 2 Laser induced dipole-dipole interactions The idea of exploiting dipole-dipole forces to modify the properties of atomic BECs first arose in the context of static dipolar interactions. A BEC of atoms with a large permanent magnetic moment (e.g. chromium) ordered by an external magnetic field, or equivalently, polarizable atoms in a static electric field,hasbeenconsideredbyanumberofauthors[13,14,15,16,17,18]. Here,however,weareinterested in fully retarded dynamic dipole-dipole interactions, such as those induced by an electromagnetic wave (e.g. laserbeam). The dynamicdipole-dipoleinteractionisdistinguishedfromthestaticcasebyalonger range (the retardation gives it an attractive r−1 component which can be used in certain geometries to mimic gravity [19]) and a huge enhancement of atomic polarizabilty close to a resonance. We consider a cigar-shaped BEC tightly confined in the radial (x,y) plane, irradiated by a plane- wave laser as in [3] (see Fig. 1). The laser polarization is along the long z-axis of the condensate to suppress collective (“superradiant”) Rayleigh scattering [20] or coherent atomic recoil lasing [21] that areforbiddenalongthe directionofpolarization. The tightconfinementalongthe propagationdirection, together with the far off-resonance condition, enables us to treat the electromagnetic field within the Born approximation. The dipole-dipole potential induced by far off-resonance electromagnetic radiation 2 of intensity I, wave-vectorkL =kLyˆ (along the y-axis), and polarizationeˆ=zˆ (along the z-axis) is [22] Iα2(ω) 1 Udd(r)= 4πcε2 r3 1−3cos2(θ) cos(kLr)+kLrsin(kLr) −sin2(θ)kL2r2cos(kLr) cos(kLy). (2) 0 h(cid:0) (cid:1)(cid:0) (cid:1) i Here r is the interatomic axis, and θ is the angle between r and the z-axis. The far-zone (kLr 1) behavior of (2) along the z-axis is proportional to sin(kLr)/(kLr)2. In terms of the condensa≫te density n(r) = ψ(r)2, the mean-field energy functional−accounting for atom-atom interactions is taken to be the sum|Hdd |+ Hs [14, 13, 19], where Hdd = (1/2) n(r)Udd(r r′)n(r′)d3rd3r′, and Hs = (1/2)(4πa¯h2/m) n2(r)d3r is due to short-range (r−6) van der Waals inte−ractions, which are described, R as is usual, by a delta function pseudo-potential Us(r)=(4πa¯h2/m)δ(r). In momentum space Hdd takes R the form, Hdd = (1/2)(2π)−3 Udd(k)n˜(k)n˜( k)d3k, where Udd(k) = exp[ ik r]Udd(r)d3r is the − − · Fourier transformof the dipole-dipole potential (2), the explicit form of which, for the laser propagation R R and polarization as in Fig. 1, weehave given in Eq. (4) of [3]. e When the radial trapping is sufficiently tight so that the BEC is in its radial ground state we may adopt a cylindrically symmetric gaussian ansatz for the density profile whose width, w , is r n3D(r) N(πw2)−1n(z)exp (x2+y2)/w2 , (3) ≡ r − r where N is the total number of atoms and n(z), the 1D(cid:2)axial density, (cid:3)is normalized to 1, but is un- specified for the time being. Denoting by n˜3D(k) the Fourier transform of n3D(r), we have n˜3D(k) = Nn(k )exp w2(k2+k2)/4 , in which n(k ) is the Fourier transform of the axial density n(z). This z − r x y z ansatz allows the principal value of the radialintegrationin Hdd to be evaluated analyticallyso that the (cid:2) (cid:3) dipole-dipole energy reduces to a one dimensional functional along the axial (zˆ) direction e e Hdd =(N2/2) n(z)n(z′)Udzd(z−z′)dzdz′ =(N2/4π) n(kz)n(−kz)Udzd(kz)dkz. (4) Z Z Defining the variables e e g ζ =k2w2/2 , ξ =(k2 k2)w2/2, (5) L r z − L r the one-dimensional (1D) axial potential that appears in Hdd is Iα2k2 Uz (k )= L Q[ξ(w ,k ),ζ(w )], (6) dd z 4πǫ2 c r z r 0 where Q[ξ(w ,k ),ζ(w )]= 1 g2 +F(ξ,ζ) , and r z r 2ζ −3 (cid:2) (cid:3) ∞ ζj F(ξ,ζ)=2ξ exp(ξ ζ) Ej+1(ξ) . (7) − j! ℜ{ } j=0 X E [z] is the realpartofthe generalizedexponentialintegral[23]. However,the suminthe expression j ℜ{ } (6) is rather unwieldy to use in calculations, so instead we will substitute it by the following asymptotic representation for the function F: 2ξ 1 exp (ζ+ξ) 1 ζ , ξ <0 F(ξ,ζ) ζ+ξ − − −ξ . (8) ≈( 2ξ ,n h (cid:16) q (cid:17)io ξ 0 ζ+ξ ≥ 3 This representation is very good for ξ > 0 and gives the correct asymptotics for ξ 0 (k k ) and z L → → ξ ζ (k 0). For ζ <ξ <0 the agreement is reasonably good (see Figure 2). z →− → − Usingtheaboveexpression,theFourierTransformofthetotal(s-waveplusdipole-dipole)1Dreduced interatomic potential is Utzot(kz)=4ERa (kLwr)−2+ Q(wr,kz) (9) I where ER =h¯2kL2/2m is the phgoton recoil energ(cid:0)y of an atom and I is th(cid:1)e dimensionless laser ‘intensity’ parameter Iα2(ω)m = . (10) I 8πε2c¯h2a 0 It is emphasized that the radial degree of freedom is contained in (9) via the radius w . Note that the r Fourier Transform of the total potential is an even function: Uz (k )=Uz ( k ). tot z tot z − g g 3 Hartree-Fock-Bogoliubov-Popov method TheHartree-Fock-Bogoliubov(HFB)methodisconciselydescribedin[12]. Areasonableapproachwithin theHFBmethodisthePopovapproximation[11],whichneglectsthecontributionoftheabove-condensate fractiontotheanomalouscorrelationfunction. TheadvantageofthePopovapproximationisthatitgives agaplessspectrumofelementaryexcitationsofadegeneratebosonicsysteminawidetemperaturerange (up to the critical temperature). In the present paper we apply the Popov approximation to a quasi-1D BEC with the laser-induced dipole-dipole interactions at finite temperature. Of course, for an infinitely long 1D gas there is no condensate, even if this gas is non-ideal. This is because the depletion diverges at infinitely long wavelength. However, taking into account the finite length ℓ of the sample, we set an effective cutoff at a wavelength ℓ and thus removethe divergence. Hence, a macroscopicpopulation of ∼ the ground state becomes possible (in the case of an interacting gas). The 1D ansatz for the atomic field operator is Ψˆ(x,y,z,t) = ψˆ(z,t)exp[ (x2+y2)/(2w2)]/(√πw ). − r r Then the 1D Hamiltonian reads as ¯h2 ∂2 1 Hˆ = dzψˆ†(z) ψˆ(z)+ dz dz′ψˆ†(z′)ψˆ†(z)Uz (z z′)ψˆ(z)ψˆ(z′), (11) −2m∂z2 2 tot − Z (cid:18) (cid:19) Z Z where Uz (z) is the inverse Fourier Transform of Uz (k ) defined by Eq. (9). Decomposing the atomic tot tot z field operator in plane waves ψˆ(z)= ℓ−1/2exp(ikz)aˆ we rewrite Eq. (11) in the momentum space: k k g ¯h2Pk2 1 Hˆ = 2m aˆ†kaˆk+ 2ℓ Utzot(q)aˆ†k+qaˆ†k′−qaˆk′aˆk. (12) k k,k′,q X X g The Hamiltonian of Eq. (12) applies to a system with a fixed number of particles. For simplicity we will work within the grand canonical ensemble and use the chemical potential, µ, to fix the mean particle number N. Then the Heisenberg equation of motion for the atomic annihilation operators aˆ , is given k by the commutator of aˆ with Hˆ µ aˆ†aˆ , and reads k − k k k ∂ ¯hP2k2 1 ih¯∂taˆk = 2m −µ aˆk+ ℓ Utzot(q)aˆ†k′+qaˆk′aˆk+q. (13) (cid:18) (cid:19) k′,q X g 4 Sincethestatewithzeromomentumismacroscopicallypopulated,weinvokebrokensymmetryarguments [5] and replace aˆ0 by a c-number √Ncexp(iϕ), where Nc is the number of atoms in the condensate. The phase ϕ is not an observable quantity, and its choice is arbitrary, thus we set ϕ = 0. The linear (1D) condensate density is defined as n = N /ℓ. Atoms with k = 0 comprise the above-condensate fraction c c 6 (i.e., depletion) with the linear density 1 n = aˆ†aˆ , (14) a ℓ k k kX6=0D E which, together with n , yields the total linear density c n=n +n N/ℓ. c a ≡ Finally, in the Popov approximationthe chemical potential is given by the expression µ=Uz (0)(n +n )+W(0), (15) tot c a where g f 1 W(k)= ℓ Utzot(k′) aˆ†k′+kaˆk′+k . (16) k′X6=−k D E f g For the modes with k =0 we obtain, from Eq. (13) 6 ∂ ¯h2k2 ih¯ aˆ = µ aˆ + Uz (0)+Uz (k) n aˆ + Uz (0)n +W(k) aˆ +Uz (k)n aˆ† . (17) ∂t k 2m − k tot tot c k tot a k tot c −k (cid:18) (cid:19) h i h i g g g f g WhenderivingEq.(17),wetakeintoaccountthat aˆ†kaˆk′ = aˆ†kaˆk δk′k. Alsonotethattheanomalous correlation functions for the above-condensate opeDrators,EsuchDas aˆEkaˆk′ , are neglected in Eqs. (15, 17), h i as required by the Popov approximation. We now implement the standard Bogoliubov transformation [5] aˆ =u exp( iω t)ˆb v exp(iω t)ˆb† , k k − k k− k k −k wherethenew(quasiparticle)operatorsˆb†,ˆb obeytheusualbosoniccommutationrelations. Thesolution k k of the generalized eigenvalue problem gives the dispersion relation ¯hω = T(k) T(k)+2Uz (k)n , (18) k tot c r h i where g ¯h2k2 T(k)= +W(k) W(0). (19) 2m − The transformationcoefficients are f f T(k)+Uz (k) n 1 T(k)+Uz (k) n 1 u = tot a + , v = tot a . (20) k k s 2h¯ω 2 s 2h¯ω − 2 k k g g 5 Note that if Uz (k) were k-independent (as it is in the case of absence of the external laser radiation, tot = 0) then T(k) would coincide with the free atom kinetic energy, and the well-known dispersion I relation for agBEC with short-range interactions in the Popov approximation [12, 11] would hold. At a finite temperature Θ (measured in energy units) we have [12] aˆ†aˆ =u2[exp(h¯ω /Θ) 1]−1+v2. (21) k k k k − k D E The Hartree-Fock-Bogoliubov-Popov equations should be solved self-consistently to obtain the con- densate and above-condensate occupation numbers, under the constraint that the total density is fixed, n=n +n . This involves an iterative procedure [12]: c a 1) As a starting point we choose the occupation numbers given by the u ,v coefficients found from k k { } the basic Bogoliubov scheme [5] (which does not treat the depletion self-consistently). 2) The occupation numbers are used to calculate W(k) [Eq. (16)] and hence the dispersion relation h¯ω k [Eq. (18)]. 3)Theoccupationnumbersaresummedtogivethefnewtotaldensityn′,whichisusedtorenormalizethe occupation numbers of the condensate and above condensate modes, in order to keep the total density fixed. 4) The dispersion relation, incorporating the new occupation numbers, is then used to recalculate the u ,v coefficients [Eq. (20)]. k k { } 5) Repeat above steps 2–4 until convergence is achieved. 4 Results and Discussion WehavesolvedtheHFBequationswiththePopovapproximation,asdescribedaboveforthebasicsetup shown in Figure 1. In Figures 3–6 we present the results for two sets of parameters: Case 1: kLwr =2.4, na=0.66, ℓ/wr =74, Θ/ER =0.68, =1.12; I Case 2: kLwr =8.0, na=4.7, ℓ/wr =28, Θ/ER =2.3, =0.28. For example, for 87Rb these sets of parameters mean expIlicitly: Case 1: The condensate radius w is set at 0.3 µm, the 3D density at the trap centre is equal to r 4 1014 cm−3, ℓ = 22 µm, Θ = 120 nK. The laser intensity can be found from the relation I/∆2 = · 2 0.26 W/(cm GHz) , where ∆ is the laser detuning from resonance. · Case 2: w = 1.0 µm, the peak 3D density is the same as in case 1, ℓ = 28 µm, Θ = 410 nK, r I/∆2 =0.064 W/(cm GHz)2. · Case1isrepresentativeofaBECatthecrossoverbetweentheideal-gasandThomas-Fermilimitswith respecttotheradialmotion,whileincase2theradialmotionsatisfiestheconditionsoftheThomas-Fermi limit. The parameters in cases 1 and 2 are chosen so that a significant ‘roton’ dip appears only in the moresophisticatedHartree-Fock-Bogoliubov-Popovmethod, but notinthe simpleBogoliubovapproach. In each of Figures 3–6 we have plotted three curves. The solid line gives the results of the Hartee- Fock-Bogoliubov-Popovcalculation described in section 3. At convergencewe found that in case 1 55 % of atoms remain in the condensate and in case 2 the condensate fraction is 82 %. The two other curves in the figures give the results obtained within the simple Bogoliubov approach and differ in the density thatappearsinthe Bogoliubovdispersionrelation. Thevariant(a)usesasthe densityintheBogoliubov dispersionrelationthecondensatedensityn obtainedinthenumericalcalculations. Thevariant(b)uses c in the same context as the density the total density of the system, n = n +n . Similarly, we calculate c a 6 the depletion within the simple Bogoliubov approach for the variants (a) and (b). The position of the dip centre remains unchanged with temperature. Figures 3 and 5 show that when the depletion is taken into account self-consistently, the ‘roton’ dip in the excitation spectrum can actually become deeper. This means that the ‘roton’ dip persists at finite temperatures (of course, below the condensation temperature). Thus the sample heating due to spontaneous scattering of laser photons need not prevent observation of the ‘roton’ dip in the spectrum of the elementary excitations. Contrary to what one might expect, finite temperature effects not only do not wash out or diminish these intermode correlations, but rather serve to enhance them. This is surprising,considering the noisy character of finite-temperature density fluctuations. This might in part bebecausethedipole-dipolepotentialhasamomentumdependencesuchthatitismoreeffectiveatfinite interatomic momenta (see Fig. 2). Onemayask: whathappensifthelaserintensityisincreasedsothattherotondippassesthroughthe zero-energy axis, so that we obtain imaginary excitation frequencies? Two scenarios are then possible. The first one is that this would create a dynamical instability similar to that found in a BEC of atoms with negative scattering length [24]. The instability would grow exponentially, finally leaving us with a totally depleted BEC. Another scenario is the following: when the laser intensity reaches the value at which the roton minimum touches the zero-energy axis we may access a new ground state that is periodically modulated along the z-axis: a supersolid. An analogous behaviour had been predicted for superfluid liquid helium passing through a pipe [25]. The formation of a supersolid state in a BEC with laser-induced dipole-dipole interactions has recently been studied [26] by numerically integrating the time-dependent generalized Gross-Pitaevskii equation describing a cigar-shaped BEC irradiated by a circularly polarized laser (in contrast to the linear case considered in the present paper), whose wave vector is along the major axis, z, of the cigar. This new supersolid state has to be stable against small perturbations. One can choose between these two scenariosby numerical investigationof the BEC dynamics in the case of gradually-increasinglaser intensity, based on a time-dependent generalization of the Popov approximation. Such an investigation is beyond the scope of the present paper. Finally, we would like to briefly mention experimentally detectable signatures of the roton minimum. Whilst the most convincing method would certainly be to map out the dispersion formula using Bragg spectroscopy, as in the measurement by Steinhauer et al [9], Bragg spectroscopy is technically difficult andtime consuming. Asimplermethodwouldbe toperformatime-of-flightmeasurementafterreleasing the BEC from the trap in order to detect the enhanced number of atoms having momenta around the roton minimum. To summarize, we have presented a method allowingself-consistent calculationof the effects of laser- induced dipole-dipole interactions upon a BEC in the presence of significant condensate depletion. This is especially relevant to the experimental situation where the temperature cannot be significantly lower thanthe chemicalpotential. This may help us cope with the adverseeffects ofthe Rayleighscattering of laser light [3], which tend to heat up the sample. 5 Acknowledgements We are grateful to the Engineering and Physical Sciences Research Council UK (EPSRC), the German- Israeli Foundation (GIF), and the EU QUACS and CQG networks for funding. I.E.M. also thanks the Russian funding sources (RFBR 02–02–17686,UR.01.01.040,E02–3.2–287). 7 References [1] K. Burnett, Nature 392, 125 (1998). [2] S. Inouye, M.R. Andrews, J. Stenger, H.J. Miesner, D.M. Stamper-Kurn and W. Ketterle, Nature 392, 151 (1998). [3] D.H.J. O’Dell, S. Giovanazzi, and G. Kurizki, Phys. Rev. Lett. 90, 110402 (2003). [4] L. Santos, G.V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 90, 250403 (2003). [5] E.M.LifshitzandL.P.Pitaevskii,StatisticalPhysicsPart2 (Butterworth-Heinemann,Oxford,1998). [6] T. Gasenzer, D.C. Roberts, and K. Burnett, Phys. Rev. A 65, 021605 (2002). [7] J. Rogel-Salazar,G.H.C. New, S. Choi and K. Burnett, Phys. Rev. A 65, 023601(2002). [8] S.M.BarnettandP.M.Radmore,Methods inTheoretical QuantumOptics (Clarendonpress,Oxford, 1997). [9] J. Steinhauer, R. Ozeri, N. Katz and N. Davidson, Phys. Rev. Lett. 88, 120407 (2002). [10] G. Kurizki, S. Giovanazzi, D.H.J. O’Dell, and A.I. Artemiev in Dynamics and Thermodynamics of Systems with Long-Range Interactions, edited by T. Dauxois et al, Lecture Notes in Physics 602, p382 (Springer, Berlin, 2002). [11] V.N. Popov, Functional Integrals and Collective Modes (Cambridge University Press, NY, 1987). [12] A. Griffin, Phys. Rev. B 53, 9341 (1996). [13] S. Yi and L. You, Phys. Rev. A 61, 041604(2000); ibid. 63, 053607 (2001). [14] K. G´oral, K. Rza¸z˙ewski, and T. Pfau, Phys. Rev. A 61, 051601 (2000). [15] J.-P. Martikainen, Matt Mackie, and K.-A. Suominen, Phys. Rev. A 64, 037601 (2001). [16] L. Santos, G.V. Shlyapnikov, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. 85, 1791 (2000). [17] K. G´oral, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 88, 170406 (2002). [18] K. G´oral and L. Santos, Phys. Rev. A 66, 023613 (2002). [19] D. O’Dell, S. Giovanazzi, G. Kurizki and V.M. Akulin, Phys. Rev. Lett. 84, 5687 (2000). [20] S.Inouye,A.P.Chikkatur,D.M.Stamper-Kurn,J.Stenger,D.E.Pritchard,andW.Ketterle,Science 285, 571 (1999); M.G. Moore and P. Meystre, Phys. Rev. Lett., 83, 5202 (1999). [21] N. Piovella,R. Bonifacio, B.W.J. McNeil, and G.R.M. Robb, Opt. Commun. 187, 165 (2001). [22] T. Thirunamachandran,Mol. Phys. 40, 393 (1980); D.P. Craig and T. Thirunamachandran,Molec- ular Quantum Electrodynamics (Academic Press, London, 1984). [23] M.AbramowitzandI.Stegun,Handbook of Mathematical Functions (NationalBureauofStandards, Washington, 1964). 8 [24] C.C. Bradley, C.A. Sackett, J.J. Tollet, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997); C.A. Sackett, H.T.C. Stoof, and R.G. Hulet, Phys. Rev. Lett. 80, 2031 (1998). [25] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 39, 423 (1984). [26] M. Kalinski, I.E. Mazets, G. Kurizki, B.A. Malomed, K. Vogel, and W.P. Schleich, “Dynamics of laser-induced supersolid formation in Bose-Einsten condensates” cond-mat/0310480. 9 POLARIZATION Z Y Figure 1: The laser beam and condensate geometry. 2 0 1 L Ζ , Ξ -2 H F -4 2 -1 -0.5 0 0.5 1 Ξ(cid:144)Ζ Figure 2: Comparisonof the exact(dashed line) and approximate (solid line) expressionsfor F(ξ,ζ) for (1)ζ =2.9,thesolidanddashedlinesarepracticallyindistinguishable,(2)ζ =32,thedifferencebetween the two lines is small but visible. 10