Exact Solution of the Thomas-Fermi Equation for a Trapped Bose-Einstein Condensate with Dipole-Dipole Interactions

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provided by Sussex Research Online PHYSICAL REVIEW A 71, 033618 ͑2005͒

Exact solution of the Thomas-Fermi equation for a trapped Bose-Einstein condensate with dipole-dipole interactions

Claudia Eberlein,1 Stefano Giovanazzi,2 and Duncan H. J. O’Dell1 1Department of Physics & Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, England 2School of Physics & Astronomy, University of St Andrews, North Haugh, St. Andrews KY16 9SS, Scotland ͑Received 20 August 2004; published 15 March 2005͒

We derive an exact solution to the Thomas-Fermi equation for a Bose-Einstein condensate ͑BEC͒ which has dipole-dipole interactions as well as the usual s-wave contact interaction, in a harmonic trap. Remarkably, despite the nonlocal anisotropic nature of the dipolar interaction the solution is an inverted parabola, as in the pure s-wave case, but with a different aspect ratio. We explain in detail the mathematical tools necessary to describe dipolar BECs with or without cylindrical symmetry. Various properties such as electrostriction and stability are discussed.

DOI: 10.1103/PhysRevA.71.033618 PACS number͑s͒: 03.75.Hh, 34.20.Cf, 32.80.Qk, 75.80.ϩq

I. INTRODUCTION can be expected to exhibit novel behavior including unusual stability properties ͓4–6͔, exotic ground states such as super- Usually the dominant interatomic interactions in an solid ͓7,8͔ and checkerboard phases ͓8͔, and modified exci- atomic Bose-Einstein condensate ͑BEC͒ are asymptotically tation spectra ͓9,10͔, even to the extent of a roton minimum of the van der Waals ͑vdW͒ type, which falls off as r−6 and is ͓11–13͔ in the dispersion relation. We have recently reported short range in comparison to the de Broglie wavelength of on the exact dynamics of a dipolar BEC in the Thomas- the atoms. These interactions can be incorporated into a Fermi limit ͓14͔, including the quadrupole and monopole mean-field description of the condensate via a ␦-function shape oscillation frequencies. Here we give the full deriva- pseudopotential ͓1,2͔ tion of the static solution, which was only stated in ͓14͔, and ͑ ͒ ␲ 2 ␦͑ ͒ ϵ ␦͑ ͒ ͑ ͒ ͑ U r =4 q as r /m g r , 1 investigate stability and electrostriction change in volume due to an applied electric field͒. We also calculate the dipole- involving just the s-wave scattering length as and atomic dipole potential outside the boundary of the condensate, mass m. The interactions then appear as a cubic nonlinearity which has a bearing on the distribution of thermal atoms and in the Gross-Pitaevskii equation ͓3͔ for the order parameter on the stability of a dipolar BEC, but also on a lattice array ␺͑r͒ of a trapped zero-temperature BEC: of dipolar BECs, since the effective giant dipole on each site ͓ ͔ ប2 is coupled to its neighbors by dipole-dipole interactions 15 . -␮␺͑ ͒ ͭ ٌ2 ͑ ͒ ͉␺͑ ͉͒2ͮ␺͑ ͒ ͑ ͒ The long-range part of the interaction between two di r = − + Vtrap r + g r r , 2 2m poles separated by r and aligned by an external field along a where ␮ is the chemical potential. The trapping potential unit vector eˆ is given by Vtrap due to a magnetic or optical trap is typically harmonic, C ͑␦ −3rˆ rˆ ͒ 2 2 2 2 2 2 ͑ ͒ dd ij i j ͑ ͒ V =͑m/2͓͒␻ x +␻ y +␻ z ͔. Udd r = eîeˆ j , 4 trap x y z 4␲ r3 The Thomas-Fermi regime for a trapped BEC is reached ͑ ␲͒ when the zero-point kinetic energy is vanishingly small in where the coupling Cdd/ 4 depends on the specific realiza- comparison to both the potential energy due to the trap and tion. Marinescu and You ͓16͔ investigated the low-energy the interaction energy between atoms ͓2͔. Many of the cur- scattering of two atoms with dipole-dipole interactions in- ͓ ͔ 2␣2 ⑀ rent atomic BEC experiments 1 satisfy these conditions, duced by a static electric field, E=Eeˆ, so that Cdd=E / 0. which tend to hold for condensates containing a large num- Yi and You ͓17͔ then went on to consider a BEC composed ber of atoms. When the kinetic energy is neglected the time- of such atoms. Another possible scenario is permanent mag- ͑ ͒ independent Gross-Pitaevskii equation 2 can be trivially netic dipoles dm aligned by an external magnetic field, B ␮ 2 solved for the static condensate density, =Beˆ, leading to a coupling Cdd= 0dm. A BEC with magnetic dipole-dipole interactions was first discussed by Góral, n͑r͒ϵ͉␺͑r͉͒2 = ͓␮ − V ͑r͔͒/g for n͑r͒ ജ 0, ͑3͒ trap Rzążewski, and Pfau ͓4͔. A measure of the strength of the and n͑r͒=0 elsewhere. Thus the density profile is completely long-range dipole-dipole interaction relative to the s-wave determined by the trapping potential, and in a harmonic trap scattering energy is provided by the dimensionless quantity ͑ ͒ ͑ ͒ n r has an inverted parabolic profile and the same aspect C ␧ ϵ dd ͑ ͒ ratio as the trap. dd . 5 3g Our aim here is to obtain similar exact results for a BEC in which dipole-dipole interactions play an important role. This definition arises naturally from an analysis of the fre- Compared to the vdW interaction the dipolar interaction is quencies of collective excitations ͑Bogoliubov spectrum͒ in a long range and anisotropic, and consequently these systems homogeneous dipolar BEC ͓4,18͔. As we shall see later on,

ഛ␧ Ͻ the BEC is stable as long as 0 dd 1, but loses that sta- this external field is rotated in resonance with a collective ␧ Ͼ bility in the Thomas-Fermi limit when dd 1. Note that in excitation frequency of the system, then the presence of di- the presence of a strong electric field the s-wave scattering polar interactions could be demonstrated even in a Rb or Na ͓ ͔ ␧ ͓ ͔ length can be modified 9,16 , and therefore g and hence dd condensate 18 . An alternative method to enhance the role should be treated as effective quantities when dealing with played by dipolar interactions is of course to reduce the electrically induced dipoles. s-wave interaction using a Feshbach resonance ͓19͔. ͓ ͔ The plan of this paper is as follows. However, as pointed out in 4 , one atom that has an in- Section II briefly outlines some of the experimental pos- trinsically large dipole moment is chromium: it has dm ␮ 52 =6 B. The more common bosonic isotope Cr has as sibilities for realizing a dipolar BEC. ͑ ͒ ͓ ͔ ␧ Ϸ Sections III and IV set up the mathematical framework for = 170±40 a0 20 , yielding dd 0.089. The less common 50 ͑ ͒ ͓ ͔ describing a harmonically trapped BEC with both short- bosonic isotope Cr has as = 40±15 a0 20 , which leads to ␧ Ϸ ͓ ͔ range s-wave and long-range dipole-dipole interactions. The the much higher dd 0.36. So far two experiments 20,21 key point is that the problem of interacting dipoles can be have achieved promising results towards cooling chromium to degeneracy. In a crossed optical trap with ␻ =␻ =2␲ reduced to the problem of static charges with Coulomb in- x y ϫ170 s−1 and ␻ =2␲ϫ240 s−1, atom numbers in a 52Cr teractions and thus the well-known machinery of electrostat- z BEC on the order of N=104 appear to be within reach ͓22͔. ics brought to bear. For oblate or prolate BECs with cylin- ␥ ␻ ␻ The trap anisotropy = z / x =1.41 has been chosen because drical symmetry one can work in spheroidal coordinates it promises to be optimal for enhancing condensate shape where expressions for the Coulomb potential are readily deformations induced by dipolar interactions upon ballistic available. expansion ͓23͔. In a harmonically trapped BEC with just Section V gives the mean-field potential inside the BEC s-wave scattering, the criterion for the Thomas-Fermi limit is due to dipole-dipole interactions, as calculated from the elec- ͓ ͔ that the parameter Nas /aho must be large 2 , where aho trostatic analogy. The radii of the condensate as a function of =ͱប/m␻ is the harmonic oscillator length of the trap and N the dipolar interaction strength are calculated. is the total number of atoms. For an experiment with 52Cr Ϸ 2 Section VI calculates the dipolar potential outside the under the conditions listed above, one finds Nas /aho 10 ,so condensate. The existence of this potential is unique to BECs that an analysis within the Thomas-Fermi regime is appro- with long-range interactions; it does not occur in the usual priate. case of short-range s-wave interactions. Other suggestions to realize BECs with strong dipolar in- Section VII discusses three instabilities expected to be teractions include polar molecules ͓4͔, a variety of which present in a dipolar BEC: local density perturbations, scaling have recently been cooled and trapped ͓24,25͔, and Rydberg deformations, and a “Saturn-ring” instability, which is due to atoms ͓5͔. Generally speaking, molecules can have much a local minimum in the potential outside the condensate and larger electric polarizabilities than atoms, so the formation peculiar to the case of long-range interactions. by photoassociation of ultracold heteronuclear molecules ͓ ͔ Section VIII calculates the electrostriction ͑change in vol- 26 and, in particular, the remarkable realization of molecu- ͓ ͔ ume͒ of a dipolar BEC as a function of the dipolar interac- lar BECs 27 are very significant steps on the road to making a dipolar superfluid. tion strength. It also works out the effect of dipolar interac- ͑ ͒ tions upon the release energy of the BEC. The release energy Laser-induced dynamic dipole-dipole interactions differ from the static case of Eq. ͑4͒ by extra retarded terms, in- allows a straight-forward experimental measurement of dipo- cluding an r−1 term which is always attractive; they are dis- lar interactions. cussed in Refs. ͓28,29͔. In certain situations this very-long- Section IX gives a summary of the most important results range part of the interaction is important and can be and key formulas. responsible for unique features such as self-binding and plas- The Appendix explains the mathematics that one needs if monlike collective excitations. Here, however, we confine the BEC does not have cylindrical symmetry—i.e., if all ourselves to the static case, which could be realized by static three condensate axes are different. fields, but also by using a laser provided the atomic separation is considerably smaller than the laser wavelength.

II. EXPERIMENTAL REALIZATION III. THOMAS-FERMI EQUATION FOR A DIPOLAR BEC The magnetic dipole-dipole interaction between two at- Proceeding from the Thomas-Fermi equation for a static oms is determined by their magnetic moment, which for al- BEC—i.e., the time-independent Gross-Pitaevskii equation ␮ ͑ ͒ kali metals can be up to dm =1 B one Bohr magneton . without the kinetic energy term—we seek an exact solution However, the magnetic dipole-dipole interaction between for the density n͑r͒ of a condensate with both dipole-dipole alkali-metal atoms is usually masked by a stronger s-wave interactions and the usual short-range s-wave scattering in a 87 harmonic trap, which for simplicity ͑though not necessity͒ interaction. Rb has an s-wave scattering length of as Ϸ ͑ ͒ we take to be cylindrically symmetric ͑␻ =␻ ͒. The 103a0 Bohr radii , and so the ratio between the dipole- x y dipole interaction strength and the s-wave scattering Thomas-Fermi equation then reads ͑ ͒ ␧ Ϸ strength, as defined by Eq. 5 ,is dd 0.007. For Na one 1 ␮ ͑␻2␳2 ␻2 2͒ ͑ ͒ ⌽ ͑ ͒ ͑ ͒ finds ␧ Ϸ0.004. Thus, magnetic dipolar effects in alkali- = m + z + gn r + dd r , 6 dd 2 x z metal BECs are small, at least in the stationary case. ␳2 2 2 ⌽ ͑ ͒ Throughout this paper we imagine an experimental setup where =x +y and dd r is the mean-field potential due where the atomic dipoles are aligned by an external field. If to dipole-dipole interactions:

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in the pure s-wave case, in the presence of dipole-dipole ⌽ ͑ ͒ϵ͵ 3 Ј ͑ Ј͒ ͑ Ј͒ ͑ ͒ dd r d r Udd r − r n r . 7 interactions a parabolic density profile is also an exact solution of the Thomas-Fermi problem in a harmonic trap, al- The intuitive form of Eq. ͑7͒, which corresponds to the Born though this time we should expect that the condensate aspect approximation for two-body scattering, has been shown by ratio differs from that of the trap. It remains to determine the Yi and You ͓9͔ to be accurate for dipole moments of the coefficients appearing in Eqs. ͑11͒ and ͑13͒ and adjust them order of a Bohr magneton and collisions away from any in such a way that the Thomas-Fermi equation is satisfied. To shape resonances. Recently Derevianko ͓30͔ proposed a this end we shall evaluate the integral ͑10͒ for a density of more sophisticated approach to the dipolar scattering prob- the form ͑11͒. This is an arduous task because the domain of lem which suggests that dipole-dipole interactions can be integration is bounded by and has the symmetry of a spher- substantially larger than previously estimated ͓31͔. However, oid or, in the general case, even of an ellipsoid. Calculating it appears that, provided the condensate is not too strongly the integral is possible only if one takes explicit account of deformed, the basic form of Eq. ͑7͒ with the bare interaction this symmetry, and we shall demonstrate two independent ͑4͒ remains valid, albeit with a renormalized coupling ways of doing that. One is to transform into spheroidal co- ͓ ͔ Cdd 31 . ordinates, use the known Green’s function of Poisson’s equa- The presence of the nonlocal dipolar mean-field potential tion in these coordinates ͓32͔, and subsequently transform ⌽ ͑ ͒ ͑ ͒ dd r means that the Thomas-Fermi equation 6 is an inte- back into Cartesian coordinates. The other is to start from gral equation and so less trivial to solve than in the purely basics and integrate over successive thin ellipsoidal shells. local ␦-function pseudopotential case. However, it is straight- While the former approach is also quite involved, it is sim- forward to demonstrate that this equation also admits an in- pler than the latter. However, if we were to drop our simpli- verted parabola as a self-consistent solution. We begin our fying assumption of cylindrical symmetry, the second ap- analysis with a suggestive recasting of the dipole-dipole term proach is the only workable as the general solution of using the mathematical identity Poisson’s equation in general ellipsoidal coordinates is un- manageably complicated because the separation constants do ͑␦ −3rˆ rˆ ͒ 1 4␲ ͓͑ ͔ ͒ ij i j =−ٌ ٌ − ␦ ␦͑r͒. ͑8͒ not separate in these coordinates 32 ,p.757. The second r3 i j r 3 ij approach is presented in Appendix A. We can then write ␦ IV. GREEN’S FUNCTION IN SPHEROIDAL ⌽ ͑r͒ =−C eˆ eˆ ٌͩ ٌ ␾͑r͒ + ijn͑r͒ͪ, ͑9͒ dd dd i j i j 3 COORDINATES with We now demonstrate the Green’s function approach. For prolate spheroidal coordinates ͑␰,␩,␸͒ we have x 1 d3rЈn͑rЈ͒ ͱ͑␰2 ͒͑ ␩2͒ ␸ ͱ͑␰2 ͒͑ ␩2͒ ␸ ␰␩ ␾͑r͒ϵ ͵ . ͑10͒ =q −1 1− cos ,y=q −1 1− sin ,z=q . 4␲ ͉r − rЈ͉ Surfaces of constant ␰ are confocal spheroids whose eccentricity is 1/␰, and ␰ runs between 1 and ϱ. Surfaces of con- The problem thereby reduces to an analogy with electrostat- stant ␩ are confocal two-sheet hyperboloids of revolution, ics, and we need only calculate the “potential” ␾͑r͒ arising ␩ Ͼ ͑ ͒ ␾͑ ͒ and runs between −1 and 1. For Rz Rx the boundary of from the “static charge” distribution n r . In particular, r the density profile ͑11͒ is a prolate ͑cigarlike͒ spheroid with =given by Eq. ͑10͒ must obey Poisson’s equation ٌ2␾ semimajor axis Rz, semiminor axis Rx, and eccentricity −n͑r͒. We adopt the following inverted parabola as an ansatz ͱ 2 2 1−Rx /Rz . To make the spheroidal coordinate system con- for the density profile of an N-atom condensate: focal to that boundary we need to choose the scaling constant ͱ 2 2 ␳2 z2 q= R −R . Then we can use the Green’s function in prolate ͑ ͒ ͫ ͬ ͑ ͒ ജ ͑ ͒ z x n r = n0 1− 2 − 2 for n r 0, 11 spheroidal coordinates ͓32͔ to write the potential ͑10͒ as Rx Rz 2 2 ␰ 1 with radii R =R and R and where the central density n is R − R x y z 0 ␾͑␰,␩,␸͒ = z x ͫ͵ d␰Ј͵ d␩Ј͑␰Ј2 − ␩Ј2͒n͑␰Ј,␩Ј͒ constrained by normalization to be 2 1 −1 ͑ ␲ 2 ͒ ͑ ͒ ϱ n0 =15N/ 8 RxRz . 12 ϫ ͚ ͑2ᐉ +1͒Pᐉ͑␩͒Pᐉ͑␩Ј͒Qᐉ͑␰͒Pᐉ͑␰Ј͒ Then Poisson’s equation is satisfied by an “electrostatic po- ᐉ=0 tential” of the form 1 1/ͱ1−R2/R2 ␾͑ ͒ ␳2 2 ␳4 4 ␳2 2 ͑ ͒ ͵ x z ␰Ј͵ ␩Ј͑␰Ј2 ␩Ј2͒ ͑␰Ј ␩Ј͒ r = a0 + a1 + a2z + a3 + a4z + a5 z . 13 + d d − n , ␰ −1 ͑ ͒ ⌽ ͑ ͒ However, by Eq. 9 , the physical dipolar contribution dd r ϱ to the mean-field potential inside the inverted-parabola BEC ϫ ͚ ͑ ᐉ ͒ ͑␩͒ ͑␩Ј͒ ͑␰͒ ͑␰Ј͒ͬ 2 +1 Pᐉ Pᐉ Pᐉ Qᐉ , ͑11͒ will now itself also be parabolic, just like the potentials ᐉ=0 due to the harmonic trap and the local s-wave scattering interaction. Thus the Thomas-Fermi equation ͑6͒ contains where Pᐉ are Legendre functions of the first and Qᐉ of the only parabolic and constant terms and so, remarkably, just as second kind. Since n͑r͒ is quadratic in x,y, and z, it is qua-

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boundary of n͑r͒. Using the Green’s function in oblate spheroidal coordinates ͓32,33͔ we ﬁnd for the potential

R2 − R2 ␰ 1 ␾͑␰,␩,␸͒ = x z ͫ͵ d␰Ј͵ d␩Ј͑␰Ј2 + ␩Ј2͒n͑␰Ј,␩Ј͒ 2 0 −1 ϱ

ϫ i͚ ͑2ᐉ +1͒Pᐉ͑␩͒Pᐉ͑␩Ј͒Qᐉ͑i␰͒Pᐉ͑i␰Ј͒ ᐉ=0 1 1/ͱR2/R2−1 + ͵ x z d␰Ј͵ d␩Ј͑␰Ј2 + ␩Ј2͒n͑␰Ј,␩Ј͒ ␰ −1 ϱ ϫ ͚ ͑ ᐉ ͒ ͑␩͒ ͑␩Ј͒ ͑ ␰͒ ͑ ␰Ј͒ͬ i 2 +1 Pᐉ Pᐉ Pᐉ i Qᐉ i . ᐉ=0 To return to Cartesian coordinates we need to make the substitutions ␰=͑ͱx2 +y2 +͑z+iq͒2 +ͱx2 +y2 +͑z−iq͒2͒/͑2q͒ and ␩ ͑ͱ 2 2 ͑ ͒2 ͱ 2 2 ͑ ͒2͒ ͑ ͒ ͑ ͒ = x +y + z+iq − x +y + z−iq / 2iq . Then we FIG. 1. Color online An illustration of oblate spheroidal coor- ͑ ͒ dinates. Surfaces of constant ␰ are confocal spheroids, with ␰=0 find that the result for the potential is the same as in Eq. 14 being a flat disk of radius q and ␰→ϱ an infinitely large sphere. but with ͉␩͉ ͑ Surfaces of constant are one-sheet hyperboloids narrow-waisted 2 reels͒; ␩ changes sign with z. The surface described by ␩=0 is the ⌶ ϵ arctanͱ␬2 − 1 for ␬ Ͼ 1 ͑oblate͒. ͑16͒ ͱ 2 xy plane with a circular hole of radius q, and ͉␩͉=1 is the z axis. ␬ −1 The prolate and oblate cases are of course connected by ana- dratic in ␰ and ␩, and all integrals in the above expression lytic continuation, which, however, cannot be used to deter- are elementary. Performing the ␩Ј integration first we see mine one from the other because of the ambiguity of sheets that the only contributing ᐉ are 0, 2, and 4. To reexpress the in the complex plane. result for ␾͑␰,␩,␸͒ in Cartesian coordinates we need to In order to simplify the expressions that will follow and in ␰ ͑ ͒ ͑ ͒ ␩ ͑ make the substitutions = r1 +r2 / 2q and = r1 order to conform with existing notation in the literature ͒ ͑ ͒ ͓ 2 2 ͑ ͒2͔1/2 ͓ 2 2 ͑ −r2 / 2q with r1 = x +y + z+q and r2 = x +y + z ͓9,23͔, rather than working with the function ⌶͑␬͒, we shall −q͒2͔1/2. We thereby obtain a “potential” of the form pre- work instead with f͑␬͒: dicted by Eq. ͑13͒: 2+␬2͓4−3⌶͑␬͔͒ ͑␬͒ϵ ͑ ͒ f 2 . 17 n R2 2͑1−␬ ͒ ␾͑r͒ = 0 x ͭ24⌶͑1−␬2͒2 +48͑1−␬2͒͑2−⌶͒ 192͑1−␬2͒2 f͑␬͒ is a monotonically decreasing function of ␬, having val- ജ ͑␬͒ജ ␬ z 2 ␳ 2 ues in the range 1 f −2, passing through zero at =1. ϫͩ ͪ −24͑1−␬2͒͑2−␬2⌶͒ͩ ͪ Rz Rx z 4 V. SOLUTION OF THE THOMAS-FERMI EQUATION +8͑2␬2 −8+3⌶͒ͩ ͪ +3͓2͑2−5␬2͒ +3␬4⌶͔ Rz In this paper we shall take the external field to be along ␳ 4 ␳ 2 z 2 the z axis. Then the result ͑14͒ for the “electrostatic poten- ϫͩ ͪ +24͑2+4␬2 −3␬2⌶͒ͩ ͪ ͩ ͪ ͮ, ͑14͒ ␾͑ ͒ ͑ ͒ R R R tial” r yields, by virtue of Eq. 9 , a parabolic dipolar x x z potential ␬ϵ where Rx /Rz is the aspect ratio of the BEC and n C ␳2 2z2 3 ␳2 −2z2 ⌽ = 0 ddͫ − − f͑␬͒ͩ1− ͪͬ, ͑18͒ dd 3 R2 R2 2 R2 − R2 1 1+ͱ1−␬2 x z x z ⌶ ϵ ln for ␬ Ͻ 1 ͑prolate͒. ͑15͒ ͑ ͱ1−␬2 1−ͱ1−␬2 which is valid inside the condensate the potential outside the condensate boundary will be discussed in the next section͒. Ͼ ͑ ͒ ⌽ ͑ ͒ ͑ ͒ If Rx Rz, then the boundary of the density profile 11 is Substituting dd r into the Thomas-Fermi equation 6 and an oblate ͑pancakelike͒ spheroid, and we have to use oblate comparing the coefficients of ␳2 ,z2, and 1 yields three spheroidal coordinates x=qͱ͑␰2 +1͒͑1−␩2͒cos ␸,y coupled equations. The first equation, due to the constant =qͱ͑␰2 +1͒͑1−␩2͒sin ␸,z=q␰␩. Surfaces of constant ␰ are terms, gives the chemical potential again confocal spheroids but now with eccentricity 1/ͱ␰+1, ␮ = gn ͓1−␧ f͑␬͔͒. ͑19͒ and ␰ running between 0 and ϱ. An illustration of oblate 0 dd spheroidal coordinates is given in Fig. 1. We have to choose This equation indicates that, all other things being equal, the ͱ 2 2 q= Rx −Rz to make the coordinate system confocal to the effect of dipole-dipole interactions is to lower the chemical

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FIG. 2. Aspect ratio of the condensate as a function of the FIG. 3. Aspect ratio of the condensate as a function of ␧ . This ␧ dd dipole-dipole to s-wave coupling ratio dd. Each line is for a differ- is an expanded version of Fig. 2, illustrating the surprising result ␥ ␬͑␧ ent trap aspect ratio , which can be read off by noting that dd that beyond a certain critical oblateness of the trap ͑␥Ͼ␥ ͒ ␥ Ͻ␬Ͻ ␬Ͼ crit =0 = . When 0 1 the condensate is prolate; for 1itis =5.1701͒ the system is metastable to scaling deformations, even for Ͻ␥Ͻ ␥ oblate. Likewise, when 0 1 the trap is prolate, and when arbitrarily high values of the dipolar interaction strength ͑but is not Ͼ 1 the trap is oblate. Dashed lines indicate unstable branches. necessarily stable to other types of perturbations such as phonons— ͒ ␥ ␥ ␬→ see later . At the boundary value = crit, one finds 2.5501 for ␧ →ϱ potential ͑which is proportional to the mean-field energy per dd . particle͒ of a prolate ͑␬Ͻ1͒ condensate, while raising that of ͑␬Ͼ ͒ ͑ ͒ an oblate 1 condensate. The radii Rx =Ry and Rz of =0, the condensate aspect ratio matches that of the trap, ␬ the exact parabolic solution ͑11͒ are obtained from the coef- =␥. When the dipole-dipole interactions are switched on the ficients of ␳2 and z2. We find condensate becomes more prolate than the trap and one al- ␬Ͻ␥ ഛ␧ Ͻ 15gN␬ 3 ␬2f͑␬͒ 1/5 ways has . As long as 0 dd 1, the transcendental ͫ ͭ ␧ ͩ ͪͮͬ ͑ ͒ equation ͑21͒ has a single solution ␬ for any choice of trap ␥. Rx = Ry = 1+ dd −1 20 4␲m␻2 2 1−␬2 ␧ Ͼ x The behavior for dd 1 is more complicated and requires an ␬ ␬ analysis of the stability properties of a dipolar BEC, which and Rz =Rx / . The aspect ratio is determined by solving a transcendental equation we give in Sec. VII. ␥2 f͑␬͒ VI. DIPOLAR POTENTIAL OUTSIDE THE BEC 3␬2␧ ͫͩ +1ͪ −1ͬ + ͑␧ −1͒͑␬2 − ␥2͒ =0, dd 2 1−␬2 dd For a variety of applications it is very useful to know the ͑21͒ potential outside a dipolar condensate. For example, in an array of dipolar BECs on a lattice the condensate at each ␥ ␻ ␻ where = z / x is the ratio of the harmonic trapping fre- lattice site can behave as a single mesoscopic spin ͓15͔.In quencies. In fact, a property such as the aspect ratio is insen- order to determine the spin-spin coupling between sites one sitive to the details of the density profile and Eq. ͑21͒ has needs to know the external potential generated by each con- been obtained previously from a Gaussian variational ansatz densate. In Sec. VII C below we shall see that knowledge of for the density ͓9,23͔. Figures 2 and 3 show examples of the the outside potential also gives insight into the problem of ␬ ␧ dependence of upon dd for oblate, spherical, and prolate the stability of a single dipolar condensate. ⌽ ͑ ͒ ͑ ͒ traps. The effect of dipole-dipole forces polarized along the z In order to calculate dd r in Eq. 7 outside the conden- axis is to make the condensate more cigar shaped along z. sate, we again use relation ͑8͒ and write the outside dipole- For an oblate trap ͑␥Ͼ1͒ the BEC becomes exactly spherical dipole potential in the same way as in Eq. ͑9͒, except that the ␧ ͑ ͒͑␥2 ͒ ͑␥2 ͒ ␦ ͑ ͒ when dd= 5/2 −1 / +2 . term with ij does not arise because n r is obviously zero In order to illustrate the static properties of the Thomas- outside the condensate. Using the Green’s function in prolate Fermi solution for a dipolar BEC we imagine an experiment spheroidal coordinates we find for the potential ͑10͒ outside with a large fixed number of atoms, N, in a trap set to a ͱ 2 2 1/ 1−␬2 1 particular aspect ratio ␥ where the value of ␧ is adiabati- Rz − Rx dd ␾͑␰,␩,␸͒ = ͫ͵ d␰Ј͵ d␩Ј͑␰Ј2 − ␩Ј2͒ cally increased from zero. For electrically induced dipoles 2 1 −1 this would involve increasing the electric field, whereas for ϱ magnetic dipoles one could either rotate the external mag- ϫ ͑␰Ј ␩Ј͒ ͑ ᐉ ͒ netic field, gradually changing the angle of rotation ͓18͔,or n , ͚ 2 +1 ᐉ reduce the s-wave scattering length using a Feshbach reso- =0 nance. The system then follows one of the curves shown in ␧ ϫ ͑␩͒ ͑␩Ј͒ ͑␰͒ ͑␰Ј͒ͬ Fig. 2 and 3. In the absence of the external field, when dd Pᐉ Pᐉ Qᐉ Pᐉ ,

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where, as before, only ᐉ=0, 2, 4 actually contribute to the ␬→0—since the system lowers its energy by arranging the sum. In the oblate case a similar formula applies, with just ␰ dipoles end to end. However, in reality a transition to another and ␰Ј replaced by i␰ and i␰Ј and the ␰Ј integration running ͑more structured?͒ state ͑see͓4,7,8,11,12͔͒ presumably occurs from0to1/ͱ␬2 −1. We obtain in preference to the system becoming truly singular. Bearing

2 this in mind we shall consider below three nominally differ- ͑ ͒ 3g␧ n ␬ ⌽ outside ͑r͒ =− dd 0 ͭ6␰͑1−3␩2͒ ent types of instability: local density perturbations, “scaling” dd 4͑1−␬2͒3/2 deformations, and the “Saturn-ring” instability. The latter occurs due to a peculiarity in the potential that would be seen ␰ +1 + ͓9␰2␩2 −3͑␰2 + ␩2͒ +1͔ln ͮ ͑22͒ by an atom located outside the boundary of the BEC and ␰ −1 may result in a previously unforeseen type of instability. All ␧ ജ in the prolate case and of them predict the onset of instability when dd 1. Never- ␧ theless, we have included in the figures values of dd exceed- 2 ͑ ͒ 3g␧ n ␬ ing unity, our justification being partly mathematical curios- ⌽ outside ͑r͒ =− dd 0 ͕6␰͑1−3␩2͒ dd 4͑␬2 −1͒3/2 ity and partly the fact that the inclusion of kinetic energy would extend the stability of a dipolar BEC beyond that of ͓ ␰2␩2 ͑␰2 ␩2͒ ͔͑␲ ␰͖͒ + 9 −3 − −1 − 2 arctan the strict Thomas-Fermi limit considered here. ͑23͒ in the oblate case. These expressions can easily be converted A. Local density perturbations back from prolate or oblate spheroidal coordinates ͑␰,␩͒ Phonons have already been predicted ͓4,18͔ to cause in- ␧ Ͼ into Cartesian coordinates, as was described above, stabilities in a homogeneous dipolar BEC when dd 1. This by substituting ␰=͑ͱx2 +y2 +͑z+q͒2 +ͱx2 +y2 +͑z−q͒2͒/͑2q͒ can be seen directly from the Bogoliubov dispersion relation 2 2 2 2 2 2 and ␩=͑ͱx +y +͑z+q͒ −ͱx +y +͑z−q͒ ͒/͑2q͒ for between the energy EB and momentum p for phonons in the prolate spheroidal coordinates and ␰=͑ͱx2 +y2 +͑z+iq͒2 gas: +ͱx2 +y2 +͑z−iq͒2͒/͑2q͒ and ␩=͑ͱx2 +y2 +͑z+iq͒2 p2 2 p2 ͱ 2 2 ͑ ͒2͒ ͑ ͒ ͱͩ ͪ ͕ ␧ ͑ 2␪ ͖͒ ͑ ͒ − x +y + z−iq / 2iq for oblate spheroidal coordinates. EB = +2gn 1+ dd 3 cos −1 , 25 ⌽͑outside͒͑ ͒ 2m 2m While easy to obtain, the expression for dd r is ␧ Ͼ rather lengthy and unwieldy in Cartesian coordinates. Thus, which can become imaginary when dd 1, indicating an if one has to work in Cartesian coordinates, one may prefer instability. This dispersion relation has an angular depen- the asymptotic expression for the dipole-dipole interaction dence ͑␪ is the angle between the momentum of the phonon potential at large distances r=͑␳2 +z2͒1/2, which is approxi- and the external polarizing field͒ which further illustrates the mately richness of dipolar systems in comparison to the usual non- dipolar case. Equation ͑25͒ is derived by adding to the Fou- N 3z2 R2 − R2 9 45 z2 ⌽͑outside͒͑ ͒Ӎ ͫͩ ͪ x z ͩ rier transform g of the contact interaction g␦͑r͒ that appears r Cdd 1− + − dd 4␲r3 r2 r2 14 7 r2 in the usual Bogoliubov dispersion relation, the Fourier 4 4 transform C ͑kˆ kˆ −␦ /3͒ of the dipole-dipole interaction 15 z Rx,Rz dd i j ij + ͪ + Oͩ ͪ ͬ, ͑24͒ ͑4͒. This is an approximation that assumes, as we do 2 r4 r throughout this paper, that there is no screening of two di- and holds for both prolate and oblate condensates. It turns poles by the other dipoles lying between them and also that out that this asymptotic expression serves remarkably well the scattering of these two particles by the dipole-dipole in- even quite close to the condensate. We give a derivation from teraction takes place within the Born approximation, as men- integration over thin ellipsoidal shells in Appendix B. Note tioned above in Sec. III. that Eq. ͑24͒ says that to a first approximation, when seen In a trapped BEC with negligible kinetic energy, as con- from outside, the dipolar condensate behaves like a single sidered here, we should expect an analogous instability due giant dipole of N times the single-atom dipole magnitude. to local density perturbations having a wavelength much The higher multipoles depend on the shape of the BEC. smaller than the dimensions of the condensate. For example, Santos et al. ͓12͔ recently showed that an infinite-pancake dipolar BEC, homogenous in two directions and parabolic in ␧ Ͼ VII. STABILITY OF A DIPOLAR BEC the third, is unstable when dd 1 for a density exceeding a critical value. The partially attractive nature of the dipole-dipole interaction ͑4͒ has been widely predicted ͓4–6,9͔ to lead to a collapse of the BEC when the dipolar interaction strength B. Scaling deformations ␧crit exceeds a certain critical value, dd . In the Thomas-Fermi We use the term “scaling” deformation to describe pertur- ␧crit ␥ ͑ ͒ limit dd depends only upon the trap aspect ratio . Both the bations that merely rescale the parabolic solution 11 —i.e. ͑ ͒ parabolic solution presented here and the Gaussian varia- that change Rx =Ry and Rz from their equilibrium values 20 , ␧crit tional ansatz indicate that above dd the system is liable to but leave the basic form of the parabolic solution the same. collapse towards an infinitely thin and long prolate “pencil” Since the equilibrium values of the radii are determined by oriented along the field polarization direction—i.e., the transcendental equation ͑21͒, which also occurs in the

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͓ ͔ ␥Ͼ␥ Santos et al. 5 , when crit there exists a solution metastable to scaling perturbations at a ﬁnite value of ␬ for all ␧ ␧ ϱ ͑ values of dd, strictly speaking even for dd= . Note, how- ␥ ͓ ͔ ever, that our value for crit disagrees with that of Ref. 5 but is that same as the one given in Ref. ͓9͔.͒ For completeness we would like to mention that prima ͑ ͒ ␧ Ͼ facie the transcendental equation 21 for dd 1 has solutions also for ␬Ͼ␥. These come in pairs, one corresponding to a maximum and the other to a saddle point in the energy landscape. However, inspection reveals that these solutions have no physical relevance as for them the radius Rx of the condensate, Eq. ͑20͒, comes out imaginary.

C. Saturn-ring instability

␧crit FIG. 4. The critical value of the dipolar coupling, dd , above An examination of the dipolar potential outside the con- which the condensate becomes strictly unstable—even the meta- densate, which is seen, for example by a single test atom stable state ͑local minimum in the energy landscape͒ no longer placed beyond the boundary ␳2 /R2 +z2 /R2 =1, reveals a new ␥ x z exists. However, above crit=5.1701, there is always a solution type of possible instability. Like the local density perturba- ␧ metastable to scaling deformations even for arbitrarily large dd tions, it also does not preserve the parabolic form of the ͑ although not necessarily stable to local density perturbations—see density profile. It turns out that for ␧ Ͼ1 the potential seen ͒ dd text . by atoms just outside the condensate exhibits a local minimum i.e., the sum of trap and dipole-dipole potentials is context of a Gaussian variational solution, much of what we locally lower than the chemical potential, which causes at- shall say below has already been described by other authors, oms to spill out from the condensate and fill this dip in the including Santos et al. ͓5͔ and Yi and You ͓9͔. potential. Such an effect is peculiar to condensates with in- Information on the stability of the parabolic solution can duced dipole-dipole interactions because these are long range be gained from analyzing the behavior of the energy func- and thus give rise to a potential even outside the condensate, tional Etot=Etrap+Es-wave+Edd evaluated over a general para- whereas the potential due to s-wave scattering is short range bolic density profile ͑11͒, and thus zero outside the condensate. To investigate the dip in the outside potential we need to use the dipolar potential N ␥2 15 N2g ͑ ͒ ͑ ͒ E = m␻2R2ͩ2+ ͪ + ͓1−␧ f͑␬͔͒, 22 and 23 that we calculated in Sec. VI. As the expres- tot x x ␬2 ␲ 2 dd 14 28 RxRz sions are rather awkward in Cartesian coordinates, we shall analyze them in spheroidal coordinates. ͑26͒ The outside potential V is the sum of the trap potential ⌽͑outside͒͑ ͒ in the vicinity of the solution ͑20͒ and ͑21͒ and across the and the dipole-dipole interaction potential dd r . The ͑␬ ␧ ␥͒ trap potential is positive and monotonically rising from the whole parameter space , dd, . One finds that for 0 ഛ␧ Ͻ center. The dipole-dipole interaction potential outside the dd 1 the solution given by the transcendental equation ͑ ͒ condensate is a solution of the ͑homogeneous͒ Laplace equa- 21 is always stable, in the sense that it corresponds to a ͑ ͒ ␧ ⌽ outside ͑ ͒ global minimum of the energy functional. As dd is increased tion; i.e., dd r is a harmonic function. Thus the maxi- ⌽͑outside͒͑ ͒ and passes through unity, the solution matches smoothly onto mum principle applies, and dd r must assume its one that is only metastable; i.e., it is only a local minimum in maximum and minimum on the boundaries of the domain, the energy landscape, and the global minimum is then a ͑pro- either at infinity or on the surface of the condensate. At in- late͒ collapsed state with ␬→0. Simultaneously with the finity the dipole-dipole potential vanishes, which means that turning of the stable into a metastable solution, a second at large distances the total outside potential is positive and branch of solutions appears at a smaller value of ␬, which dominated by the trap potential. To ascertain whether the correspond to unstable saddle points that separate the local outer potential V has a local minimum one only needs to minimum of the metastable solution from the global mini- check whether the sum of trap and dipole-dipole potentials mum of the collapsed state at ␬=0 in the energy landscape. has a negative first derivative at the surface of the condensate ␧ If one continues to increase dd, then one of two things hap- in some outward direction. It is easy to see that local minima pens, depending on the value of ␥:if␥ is less than a critical of V can occur only for ␩=0: at a local minimum of V its ␥Ͻ␥ ␧ value, crit=5.1701, then, as dd increases, eventually the first derivative with respect to ␰ and ␩ must vanish, but both ␧ ␧crit metastable and unstable solutions coalesce at dd= dd , above the trap potential and the dipole-dipole potential are qua- ␩ is proportionalץ/Vץ which there are no solutions, not even metastable ones, and dratic in ␩, so that the first derivative the energy landscape is just a continuous slope down towards to ␩ and thus vanishes only for ␩=0 unless it vanishes for ␬ ␧crit ␩ a collapsed state with =0. This critical value dd as a func- all . ␰ץ Vץ ␥Ͼ␥ ␥ tion of is plotted in Fig. 4. If, however, crit, then Therefore we only need to examine the derivative / something rather surprising happens. As first remarked by at ␩=0;wefind

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FIG. 5. Contour plot of the potential V ͑in units of the chemical FIG. 6. The volume V of the condensate for traps with aspect ␮͒ ␬ ␧ potential outside a condensate with =2 at dd=1.5. The closed ratios ␥=͕0.1,0.2,0.5,1,2,3,4,5,8͖ as a function of the dipole- V ␮Ͻ ␧ ␧ contours drawn as dotted lines have / 1; the lowest in the dipole coupling strength dd.At dd=0—i.e., without dipole-dipole middle is at V/␮=0.992 and the difference to the next higher is interactions—the volume is proportional to ␥−2/5, so that on the left roughly 0.0017. The cutoff contours outside the condensate go from edge of the graph higher volume corresponds to lower ␥. That re- V ␮ ␧ / =1.002 to 1.04 in steps of approximately 0.0034. verses with increasing dd.

͓ ͑ ␲ ␻2͔͒3/5 ␧ V 2͑1−␧ ͒ of 15gN/ 4 m as a function of dd for various trap ץ ͯ ͩ ͪͯ = dd , ͑27͒ x ␰ ␮ ␰ ␬2͓ ␧ ͑␬͔͒ aspect ratios ␥. The figure shows that condensates in prolateץ ␰ ␰ ␩ B 1− ddf = B, =0 traps and slightly oblate traps ͑with ␥Ͻ1.6630͒ get com- ␰ ␰ where B is the value of the spheroidal variable on the pressed by increasing dipolar interactions, while condensates ␰ ͱ ␬2 ␥ surface of the condensate—i.e., B =1/ 1− for a prolate in traps with aspect ratios above crit=5.1701 are being ␰ ͱ␬2 ␧ ␥ ␥ BEC and B =1/ −1 for an oblate BEC. For oblate BECs pulled apart as dd rises. Between =1.6630 and crit is a the denominator in Eq. ͑27͒ is always positive, and thus range of trap aspect ratios for which the condensate is pulled ␰ ␧ Ͼץ Vץ / is negative if and only if dd 1. For prolate BECs the apart at first but eventually compressed into collapse by denominator is always positive in the stable and metastable higher values of the dipolar interaction strength. If, during an ␰ is negative experiment, the condensate was imaged in the trap, then theץ/Vץ regions of Fig. 2, and thus in these regions ␰ץ Vץ ␧ Ͼ if and only if dd 1. Whenever / is negative on the volume could be measured either directly from the radii or condensate surface, a local minimum of the potential lies by the central density which is inversely proportional to the ͑ ͒ somewhere outside the surface. The fact that this happens at volume, V= 5/2 N/n0. ␩=0 means that the local dip in the potential occurs along a If the trap is turned off, the condensate expands ballisti- ring at z=0 around the condensate, and the flowing out of cally and the s-wave and dipole-dipole interaction energies atoms from the main condensate into the dip causes the con- are converted into kinetic energy, the so-called release en- densate to take on a Saturn-like appearance, which then leads ergy, which can be measured in an experiment. For the exact to further instability. Figure 5 gives an illustration of a typi- parabolic solution the release energy is given by ␳ cal potential, plotted as contours in the -z plane. The flat 2͓ ␧ ͑␬͔͒ ͑ ␲ 2 ͒ ͑ ͒ part to the left of the center is the constant chemical potential Erel =15gN 1− ddf / 28 RxRz . 30 inside the condensate. The release energy can also be expressed in terms of the ͑ ͒ ͑ ͒ ␮ chemical potential 19 as Erel= 2/7 N , which is the stan- ͓ ͔ VIII. ELECTROSTRICTION AND RELEASE ENERGY dard expression for the Thomas-Fermi limit 2 . The volume V of the spheroidal BEC can be expressed as IX. CONCLUSIONS AND SUMMARY OF RESULTS ␲ ␲ 3 4 4 Rx V = R2R = . ͑28͒ The long-range anisotropic nature of dipole-dipole inter- 3 x z 3 ␬ actions gives condensates composed of dipolar atoms or ͑ ͒ Substituting Rx from Eq. 20 we can write it as molecules novel properties compared to those with only short-range interactions. Furthermore, the dipole-dipole in- 4␲ 15gN 3/5 3 ␬2f͑␬͒ 3/5 V = ͫ ͬ ͫ1+␧ ͩ −1ͪͬ , teractions can be controlled in sign, magnitude, and direc- ␬2/5 ␲ ␻2 dd ␬2 ␧ ͑ ͒ 3 4 m x 2 1− tion. The parameter dd defined in Eq. 5 gives a measure ͑29͒ for the strength of the dipole-dipole interactions relative to the usual s-wave interactions. and then we can use the transcendental equation ͑21͒ to The main point of our paper has been to show that, just ␬ ␥ ␧ eliminate in favor of and dd. In Fig. 6 we plot V in units like for a BEC with pure s-wave interactions, a simple in-

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verted parabola remains an exact solution for the density mean that degenerate quantum gases with dipole-dipole in- profile of a harmonically trapped dipolar BEC in the teractions can now be explored experimentally. Thomas-Fermi limit. We explain the necessary mathematics for describing a dipolar BEC with cylindrical symmetry in ACKNOWLEDGMENTS Sec. IV and those for a general ellipsoidal BEC, whose three axes are all different, in the Appendix. The results for the It is a pleasure to thank Gabriel Barton and Tilman Pfau cylindrical case are the density profile given by Eq. ͑11͒ and for discussions. We would like to acknowledge financial sup- the axis lengths ͑20͒, with substitutions from Eqs. ͑17͒ and port from the UK Engineering and Physical Sciences Re- ͑ ͒ ͑ ͒ ␬ 15 or 16 . The condensate aspect ratio is obtained from search Council ͑D.O’D.͒, the European Union ͑S.G.͒, and the ͑ ͒ the solution of the transcendental equation 21 , and in Figs. Royal Society ͑C.E.͒. ␧ 2 and 3 we have plotted it as a function of dd for various trap aspect ratios ␥. We have calculated the dipolar potential both inside and APPENDIX A: INTEGRATING OVER HOMOEOID outside the condensate region. The exact expressions for the SHELLS potential outside a prolate or oblate dipolar BEC are given Here we demonstrate how the potential ͑14͒ can be ob- by Eqs. ͑22͒ or ͑23͒, respectively, though for most practical tained by integrating over successive thin shells, a method applications the asymptotic form ͑24͒ should suffice. This is the potential that would be felt by an isolated atom outside that works even in the general case of an ellipsoid. The prob- the condensate—for example, by an atom belonging to the lem of the electrostatic/gravitational field within and outside thermal cloud or, if one dealt with an array of condensates, a charged/massive ellipsoid is associated with some of the great names of nineteenth century mathematical physics. In by the atoms of another condensate at a different lattice site. ͓ ͔ The parabolic solution is stable, in the strict Thomas- his 1892 treatise on statics, Routh 34 notes that Chasles, ␧ Ͻ ␧ Ͼ Dirichlet, Jacobi, and Poisson all made contributions. Rod- Fermi limit, only for dd 1. For dd 1 it is unstable against scaling perturbations as seen in the energy functional, against rigues is credited with being the first to evaluate the potential of a solid ellipsoid of constant density, seemingly in 1815, perturbations of different symmetry, such as phonons of ͓ ͔ small wavelengths, and also due to a local minimum appear- and in 1833 Green 35 “treated the subject in a very general ing in the potential outside the condensate, as indicated in manner,” giving solutions for various cases of inhomoge- Fig. 5. neous density, as in the current situation. Our derivation is The effect of the dipole-dipole interactions upon the BEC adapted from Routh. Although we take a cylindrically sym- is to change both its aspect ratio and its volume. The manner metric density as input, the derivation holds for the general of this electrostriction depends on the aspect ratio of the ellipsoidal case. ͑ ͒2 ͑ ͒2 external trap; as a function of the strength of the dipole- Consider an ellipsoidal surface x/Rx + y/Ry ͑ ͒2 ͑ ͒ dipole interactions ͑relative to the s-wave scattering͒ the vol- + z/Rz =1, having semiaxes Rx ,Ry ,Rz . A continuous fam- ume of the BEC can either increase or decrease or even ily of concentric similar ellipsoids lying inside this outer ͑ ͒ initially increase and then decrease at higher values of the surface can then be defined via the semiaxes sRx ,sRy ,sRz , coupling. An exact expression for the volume is given by Eq. where 0ഛsഛ1. Note that by similar we mean that the mem- ͑29͒, but since it contains the condensate aspect ratio ␬ and bers of this family all share the same aspect ratios among not the externally controllable trap aspect ratio ␥, one may their semiaxes, but are consequently not confocal. A surface 2 ͑ 2 ␭͒ gain more information by looking at Fig. 6 where we have which is confocal to the original surface obeys x / Rx + ␧ 2 ͑ 2 ␭͒ 2 ͑ 2 ␭͒ ␭ plotted the volume as a function of dd for various trap aspect +y / Ry + +z / Rz + =1, so that =0 is obviously the ratios. The release energy, which is usually a very important original surface, and surfaces with ␭Ͼ0 lie outside the origi- experimental quantity to measure, is given by Eq. ͑30͒, with nal. Considered in terms of the parametrization specified by ͑ substitutions from Eqs. ͑20͒, ͑17͒, and ͑15͒ or ͑16͒. s, the parabolic density profile can be written as n=n0 1 In a companion paper ͓14͔ we have used the exact- −s2͒. Since the equidensity surfaces within the density pro- equilibrium Thomas-Fermi solution derived here to calculate file are similar, when computing the integral ͑10͒ it makes the low-energy collective excitation frequencies of a dipolar sense to take as our basic volume element a thin “homoeoid” BEC. Like previous studies ͓9,10͔ we find dynamical insta- ͓34͔, defined as the shell bounded by two similar ellipsoidal bilities ͑imaginary frequencies͒ above a threshold magnitude surfaces, parametrized by s and s+ds, respectively. The vol- ␧ ␲ 2 of dd which is consistent with the picture developed here. ume of the thin homoeoid is dV=4 RxRyRzs ds. When com- Furthermore, calculations for dipolar BECs which are homo- puting the potential at a point P͑x,y,z͒ within a charged geneous in one ͓11,13͔ and two ͓12͔ dimensions lead to a ellipsoid the contribution from the homoeoids interior to that Bogoliubov spectrum which has a “roton” minimum. Refer- point is of a different nature to that from those exterior to it, ence ͓13͔ speculates that the roton minimum indicates an so we consider the two parts separately. instability towards the formation of a density wave. Taken with the results obtained here for the equilibrium case, it 1. Potential ␾in due to the “charge” interior to P seems that quantum gases with dipole-dipole interactions are rich systems which would reward further study. Let d␾ denote the contribution to the total potential ␾ Note added in proof. Two recent achievements, the ex- from a thin homoeoid shell. In this section we require the perimental and theoretical investigation of Feshbach reso- potential d␾in outside a thin homoeoid, labeled by its inner 52 52 ͑ 2͒ nances in Cr ͓38͔ and the Bose condensation of Cr ͓39͔, surface s, whose “charge” density is n=n0 1−s . The equi-

033618-9 EBERLEIN, GIOVANAZZI, AND O’DELL PHYSICAL REVIEW A 71, 033618 ͑2005͒

ϱ potential surfaces outside a charged thin homoeoid s are con- n R R R ˜s2 ˜s4 dv ͓ ͔ ␾in = 0 x y z ͭͩ − ͪ͵ focal ellipsoids 34 2 2 4 ͱ͑ 2 ͒͑ 2 ͒͑ 2 ͒ 0 Rx + v Ry + v Rz + v ϱ 2 2 2 1 x2 y2 z2 x y z − ͵ ͫ ͩ + + ͪ + + =1. ͑A1͒ 2 ␴ 2 ␴ 2 ␴ 2 2 ␭ 2 2 ␭ 2 2 ␭ 0 2 R + R + R + s Rx + s Ry + s Rz + x y z 1 x2 y2 z2 2 ͑ ⑀ ͒ − ͩ + + ͪ ͬ In our analogy in which the dielectric constant 0 =1 the 4 R2 + ␴ R2 + ␴ R2 + ␴ “potential” on a confocal surface ␭ outside a thin homoeoid x y z labeled by s, of density n and volume dV is d␴ ϫ ͮ, ͑A5͒ ͱ͑ 2 ␴͒͑ 2 ␴͒͑ 2 ␴͒ Rx + Ry + Rz + ϱ ndV du ␾in͑ ͒ ͵ d ␭ s = . where in the second term we have used Leibniz’s formula for 8␲ ͱ͑ 2 2 ͒͑ 2 2 ͒͑ 2 2 ͒ ␭ s Rx + u s Ry + u s Rz + u differentiating an integral and used Eq. ͑A1͒ to replace s2 ͑A2͒ and s4.

This potential is most easily obtained by noticing that the 2. Potential ␾ex due to the “charge” exterior to P “charge” distribution on the homoeoid is identical to that of a solid ellipsoidal conductor with the same external dimen- The potential inside a homoeoid is much simpler to cal- sions and with the same total “charge” ͑ndV͒ distributed over culate since, just as in the case of the spherical shell, it is a constant, independent of position ͓37͔. Returning to the solid its surface. A homoeoid shell does not have uniform thick- ellipsoidal conductor model, the potential throughout a con- ness dh, being slightly thicker at the points farthest from the ductor is the same as on the surface defined by ␭=0. Thus center: ϱ ndV du d␾ex͑s͒ = ͵ . sds 8␲ ͱ͑ 2 2 ͒͑ 2 2 ͒͑ 2 2 ͒ dh = . ͑A3͒ 0 s Rx + u s Ry + u s Rz + u x2 y2 z2 ͱ + + 4 4 4 Note that because the lower limit of the integral is this time Rx Ry Rz independent of s, the ensuing treatment is simple. Integrating from the surface ˜s, which includes P, out the boundary s If rather than a shell of thickness dh we consider the =1, we require ␾ex=͐1 d␾ex͑s͒. Making once again the sub- “charge” it contains to in fact be surface charge, of variable s=˜s stitution u=s2v one immediately finds surface density ␻͑x,y,z͒—i.e., so that ␻=ndh—we obtain exactly the same ␻͑x,y,z͒ as in the well-known problem of n R R R 1 1 the charged ellipsoid conductor ͑see, e.g. ͓36͔͒, and thus the ␾ex = 0 x y z ͫ ͑1−˜s2͒ − ͑1−˜s4͒ͬ resulting potentials are also the same. 2 2 4 ϱ Having established the potential due to a homoeoid shell dv we must integrate over all the shells s lying inside P, which ϫ͵ . ͑A6͒ ͱ͑ 2 ͒͑ 2 ͒͑ 2 ͒ is located on the confocal surface ␭. While P is of course 0 Rx + v Ry + v Rz + v fixed in space, the surface ␭ passing through it is a different Combining Eqs. ͑A5͒ and ͑A6͒ we see that the parameter surface ͑i.e., different aspect ratio͒ for each homoeoid—i.e., ˜s drops from the sum ␾tot=␾in+␾ex. In the general ellipsoi- ␭=␭͑s͒—becoming at the last instant a similar surface to the dal case the remaining integrals in ␾tot can be expressed in final homoeoid ͑upon which P, itself sits͒. To simplify the ͑ ͒ terms of elliptic integrals. In the spheroidal case Rx =Ry 2 algebra we put ␭=s ␴ in the equation for the ellipsoidal they can be written in terms of more elementary functions. surface, Eq. ͑A1͒. We see that if s=0, then ␴=ϱ. We choose There are two cases to distinguish, depending upon whether s=˜s to describe the similar ellipsoidal surface upon which P ͑ ͒ Ͼ ͑ the density profile is prolate a cigar , Rz Rx, or oblate a lies, and thus we have ␴=0 for s=˜s. To obtain the potential ͒ Ͼ pancake , Rx Rz. If we define due to the charge interior to P we must evaluate ␾in ͐˜s ␾in͑ ͒ 2 ͑ ͒ ϱ = s=0d ␭ s . Setting u=s v in the integral A2 ,wefind dv J͑a,c͒ϵ͵ ͑ 2 ͒ͱ͑ 2 ͒ 0 Rx + v Rz + v ˜s n0RxRyRz ␾in = ͵ ds͑1−s2͒s 1 1+ͱ1−R2/R2 2 ͩ x z ͪ 0 ln prolate, ͱ 2 2 ͱ 2 2 ϱ Rz − Rx 1− 1−Rx/Rz dv = ϫ͵ , ͑A4͒ Ά 2 ͱ 2 2 · ͱ͑ 2 ͒͑ 2 ͒͑ 2 ͒ arctan͑ R /R −1͒ oblate, ␴͑s͒ Rx + v Ry + v Rz + v ͱ 2 2 x z Rx − Rz which can be integrated by parts to give then the total potential at P is most compactly expressed as

033618-10 EXACT SOLUTION OF THE THOMAS-FERMI EQUATION… PHYSICAL REVIEW A 71, 033618 ͑2005͒

2J ary of the condensate, upon which the point P sits, and soץ J ␳4ץ Jץ J ␳2 2 tot n0RxRz 2 ␾ = ͩ + + z + obeys the quadratic equation ␳2 /͑R2 +␭͒+z2 /͑R2 +␭͒=1. R2͒2 x z͑ץ R2͒ 8͑ץ R2͒͑ץ 2 4 2 x z x When solving for ␭ one should take the positive solution 2Jץ 2Jץ 4 z 2 2 + + ␳ z ͪ ͑A7͒ ␳2 2 2 2 Rx + z − Rz − 2͒ ͑ ץ 2͒ ͑ץ 2͒2 ͑ץ 3 Rz Rx Rz ␭ = 2 and is identical to the Green’s function result ͑14͒. The rela- tionship between the function ⌶ defined in the Green’s func- ͱ͑␳2 − R2 + z2 − R2͒2 +4͑␳2R2 + z2R2 − R2R2͒ J J ⌶ + x z z x x z . tion approach and defined here is simply = /Rz. 2 APPENDIX B: THE POTENTIAL OUTSIDE ␾ THE CONDENSATE Note that expressing the solution for the potential in the form ͑A7͒ requires that when taking the derivatives with re- The calculation of the potential at a point P outside the ␭ spect to Rx and Rz, then is to be treated as a constant and is boundary of the condensate follows very similar lines to that not to be differentiated. On the other hand, when going on to presented above for a point inside. The main difference is ⌽ calculate the dipolar potential outside the BEC, dd= that one no longer has to consider “charge” located exterior ٌ ٌ ␾͑ ͒ ␭ −Cddeîeˆ j i j r , then is to be treated as a function of to P. The result is identical to Eq. ͑A7͒ except that the inte- ͑␳,z͒ and should be differentiated. This makes the exact ex- gral J is now given by ⌽ pression for dd outside the condensate complicated. How- ϱ ␳ӷ ӷ dv ever, in the limit Rx ,z Rz, one may use the asymptotic J͑a,c,␭͒ϵ͵ result ͑ 2 ͒ͱ͑ 2 ͒ ␭ Rx + v Rz + v ͑ 2 2͒͑␳2 2͒ ͑␳2 2͒2 ͱ␭ 2 ͱ 2 2 N Rx − Rz −2z +14 + z 1 + Rz + Rz − Rx ␾outside ϳ ͑ ͒ ͩ ͪ 2 2 5/2 , B1 ln prolate, 8␲ 7͑␳ + z ͒ ͱR2 − R2 ͱ␭ + R2 − ͱR2 − R2 = z a z z x 2 2 which turns out in practice to be remarkably accurate, even Ά 2 ͩͱRx − Rz ͪ · arctan 2 oblate, right up to the condensate surface, providing the condensate ͱR2 − R2 ␭ + R x z z is not too aspherical ͑in which case the next terms in the where ␭ parametrizes an ellipse, confocal to the outer bound- multipole expansion should included͒.

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