Exact solutions and stability of rotating dipolar Bose-Einsteincondensates in the Thomas-Fermi limitCitation for published version (APA):Bijnen, van, R. M. W., Dow, A. J., O'Dell, D. H. J., Parker, N. G., & Martin, A. M. (2009). Exact solutions andstability of rotating dipolar Bose-Einstein condensates in the Thomas-Fermi limit. Physical Review A : Atomic,Molecular and Optical Physics, 80(3), 033617. [033617]. https://doi.org/10.1103/PhysRevA.80.033617

DOI:10.1103/PhysRevA.80.033617

Document status and date:Published: 01/01/2009

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Exact solutions and stability of rotating dipolar Bose-Einstein condensatesin the Thomas-Fermi limit

R. M. W. van Bijnen,1,2,3 A. J. Dow,1 D. H. J. O’Dell,3 N. G. Parker,3,4 and A. M. Martin1

1School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

3Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M14School of Food Science and Nutrition, University of Leeds, Leeds LS2 9JT, United Kingdom

Received 14 May 2009; published 17 September 2009

We present a theoretical analysis of dilute gas Bose-Einstein condensates with dipolar atomic interactionsunder rotation in elliptical traps. Working in the Thomas-Fermi limit, we employ the classical hydrodynamicequations to first derive the rotating condensate solutions and then consider their response to perturbations. Wethereby map out the regimes of stability and instability for rotating dipolar Bose-Einstein condensates and, inthe latter case, discuss the possibility of vortex lattice formation. We employ our results to propose severalroutes to induce vortex lattice formation in a dipolar condensate.

DOI: 10.1103/PhysRevA.80.033617 PACS numbers: 03.75.Kk, 34.20.Cf, 47.20.k

I. INTRODUCTION

The successful Bose-Einstein condensation of 52Cr atoms1–3 enables the realization of Bose-Einstein condensatesBECs with significant dipole-dipole interactions. Theselong-range and anisotropic interactions introduce rich physi-cal effects, as well as opportunities to control BECs. A basicexample is how dipole-dipole interactions modify the shapeof a trapped BEC. In a prolate elongated dipolar gas withthe dipoles polarized along the long axis the net dipolar in-teraction is attractive, whereas for an oblate flattened con-figuration with the dipoles aligned along the short axis thenet dipolar interaction is repulsive. As a result, in comparisonto s-wave BECs which we define as systems in which atom-atom scattering is dominated by the s-wave channel, a di-polar BEC elongates along the direction of an applied polar-izing field 4,5.

A full theoretical treatment of a trapped BEC involvessolving the Gross-Pitaevskii equation GPE for the conden-sate wave function 6,7. The nonlocal nature of the mean-field potential describing dipole-dipole interactions meansthat this task is significantly harder for dipolar BECs than fors-wave ones. However, in the limit where the BEC containsa large number of atoms the problem of finding the ground-state density profile and low-energy dynamics simplifies. In a

harmonic trap with oscillator length aho= / m, a BECcontaining N atoms of mass m which have repulsive s-waveinteractions characterized by scattering length a enters theThomas-Fermi TF regime for large values of the parameterNa /aho 6,7. In the TF regime the zero-point kinetic energycan be ignored in comparison to the interaction and trappingenergies and the Gross-Pitaevskii equation reduces to theequations of superfluid hydrodynamics at T=0 6–8. Whenapplied to a trapped s-wave BEC these equations are knownto admit a large class of exact analytical solutions 9. TheTF approximation can also be applied to dipolar BECs 10.Although the resulting superfluid hydrodynamic equationsfor a dipolar BEC contain the nonlocal dipolar potential, ex-act solutions can still be found 11,12 and we make an ex-tensive use of them here. The calculations in this paper areall made within the TF regime.

Condensates are quantum fluids described by a macro-scopic wave function = expiS, where is the conden-sate density and S is the condensate phase. This constrainsthe velocity field v = /m S to be curl free v =0 . In anexperiment the rotation of the condensate can be accom-plished by applying a rotating elliptical deformation to thetrapping potential 13,14. At low rotation frequencies theelliptical deformation excites low-lying collective modesquadrupole, etc. with a quantized angular momentumwhich may be viewed as surface waves and which obey v =0 . Above a certain critical rotation frequency vorticesare seen to enter the condensate and these satisfy the v=0 condition by having a quantized circulation. The hydro-dynamic equations for a BEC provide a simple and accuratedescription of the low-lying collective modes. Furthermore,they predict that these modes become unstable for certainranges of rotation frequency 15,16. Comparison with ex-periments 13,14 and full numerical simulations of the GPE17–19 have clearly shown that the instabilities are the firststep in the entry of vortices into the condensate and the for-mation of a vortex lattice. Crucially, the hydrodynamic equa-tions give a clear explanation of why vortex lattice formationin s-wave BECs was only observed to occur at a muchgreater rotation frequency than that at which they becomeenergetically favorable. It is only at these higher frequenciesthat the vortex-free condensate becomes dynamicallyunstable.

Individual vortices 20–23 and vortex lattices 24–26 indipolar condensates have already been studied theoretically.However, a key question that remains is how to make suchstates in the first place. In this paper we extend the TF ap-proximation for rotating trapped condensates to include di-polar interactions, building on our previous work 27,28.Specifically, starting from the hydrodynamic equations ofmotion, we obtain the stationary solutions for a condensatein a rotating elliptical trap and find when they become dy-namically unstable to perturbations. This enables us to pre-dict the regimes of stable and unstable motions of a rotatingdipolar condensate. For a nondipolar BEC in the TF limitthe transition between stable and unstable motions is inde-

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pendent of the interaction strength and depends only on therotation frequency and trap ellipticity in the plane perpen-dicular to the rotation vector 15,16. We show that for adipolar BEC it is additionally dependent on the strength ofthe dipolar interactions and also the axial trapping strength.All of these quantities are experimentally tunable and thisextends the routes that can be employed to induce instability.Meanwhile, the critical rotation frequency at which vorticesbecome energetically favorable v is also sensitive to thetrap geometry and dipolar interactions 21 and means thatthe formation of a vortex lattice following the instability can-not be assumed. Using a simple prediction for this frequency,we indicate the regimes in which we expect vortex latticeformation to occur. By considering all of the key and experi-mentally tunable quantities in the system, we outline severalaccessible routes to generate instability and vortex lattices indipolar condensates.

This paper is structured as follows. In Sec. II we introducethe mean-field theory and the TF approximation for dipolarBECs, in Sec. III we derive the hydrodynamic equations fora trapped dipolar BEC in the rotating frame, and in Sec. IVwe obtain the corresponding stationary states and discusstheir behavior. In Sec. V we show how to obtain the dynami-cal stability of these states to perturbations and in Sec. VI weemploy the results of the previous sections to discuss pos-sible pathways to induce instability in the motion of the BECand discuss the possibility that such instability leads to theformation of a vortex lattice. Finally in Sec. VII we concludeour findings and suggest directions for future work.

II. MEAN-FIELD THEORY OF A DIPOLAR BEC

We consider a BEC with long-range dipolar atomic inter-actions, with the dipoles aligned in the z direction by anexternal field. The condensate wave function mean-field or-der parameter for the condensate r , t satisfies theGPE which is given by 4,29,30

i

t= −

2

2m2 + Vr,t +ddr,t + g2 , 1

where m is the atomic mass. The 2 term arises from kineticenergy and Vr , t is the external confining potential. BECstypically feature s-wave atomic interactions which gives riseto a local cubic nonlinearity with coefficient g=42a /m,where a is the s-wave scattering length. Note that a, andtherefore g, can be experimentally tuned between positiverepulsive interactions and negative attractive interactionsvalues by means of a Feshbach resonance 2,3. The dipolarinteractions lead to a nonlocal mean-field potential ddr , twhich is given by 5

ddr,t = d3r Uddr − rr,t , 2

where r , t= r , t2 is the condensate density and

Uddr =Cdd

4

1 − 3 cos2

r33

is the interaction potential of two dipoles separated by a vec-tor r, where is the angle between r and the polarization

direction, which we take to be the z axis. The dipolar BECsmade to date have featured permanent magnetic dipoles.Then, assuming the dipoles to have moment dm and be

aligned in an external magnetic field B = kB, the dipolar cou-pling is Cdd=0dm

2 29, where 0 is the permeability of freespace. Alternatively, for dipoles induced by a static electric

field E = kE, the coupling constant Cdd=E22 / 0 30,31,where is the static polarizability and 0 is the permittivityof free space. In both cases, the sign and the magnitude ofCdd can be tuned through the application of a fast-rotatingexternal field 32.

We will specify the interaction strengths through theparameter

dd =Cdd

3g, 4

which is the ratio of the dipolar interactions to the s-waveinteractions 32. We take the s-wave interactions to be re-pulsive, g0, and so where we discuss negative values ofdd, this corresponds to Cdd0. We will also limit our analy-sis to the regime of −0.5dd1, where the Thomas-Fermiapproach predicts that nonrotating stationary solutions arerobustly stable 11. Outside of this regime the situation be-comes more complicated since the nonrotating system be-comes prone to collapse 33.

We are concerned with a BEC confined by an ellipticalharmonic trapping potential of the form

Vr = 12m

2 1 − x2 + 1 + y2 + 2z2 . 5

In the x-y plane the trap has mean trap frequency andellipticity . The trap strength in the axial direction, and in-deed the geometry of the trap itself, is specified by the trapratio =z /. When 1 the BEC shape will typically beoblate flattened while for 1 it will typically be prolateelongated, although for strong enough dipolar interactionsthe electrostrictive or magnetostrictive effect can cause aBEC in an oblate trap to become prolate itself.

The time-dependent GPE 1 can be reduced to its time-independent form by making the substitution r , t=r expit /, where is the chemical potential of thesystem. We employ the TF approximation whereby the ki-netic energy of static solutions is taken to be negligible incomparison to the potential and interaction energies. The va-lidity of this approximation in dipolar BECs has been dis-cussed elsewhere 10. Then, the time-independent GPE re-duces to

Vr +ddr + gr = . 6

For ease of calculation the dipolar potential ddr can beexpressed as

ddr = − 3gdd 2

z2r +1

3r , 7

where r is a fictitious “electrostatic” potential defined by11,12

VAN BIJNEN et al. PHYSICAL REVIEW A 80, 033617 2009

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r =1

4 d3r r

r − r. 8

This effectively reduces the problem of calculating the dipo-lar potential 2 to the calculation of an electrostatic potentialof form 8, for which a much larger theoretical body ofliterature exists. Exact solutions of Eq. 6 for r, r, andhence ddr can be obtained for any general parabolic trap,as proven in Appendix A of Ref. 12. In particular, thesolutions of r take the form

r = 01 −x2

Rx2 −

y2

Ry2 −

z2

Rz2 for r 0, 9

where 0=15N / 8RxRyRz is the central density. Remark-ably, this is the general inverted parabola density profile fa-miliar from the TF limit of nondipolar BECs. An importantdistinction, however, is that for the dipolar BEC the aspectratio of the parabolic solution differs from the trap aspectratio.

III. HYDRODYNAMIC EQUATIONS INTHE ROTATING FRAME

Having introduced the TF model of a dipolar BEC wenow extend this to include rotation and derive hydrodynamicequations for the rotating system. We consider the rotation toact about the z axis, described by the rotation vector ,where = is the rotation frequency and the Hamiltonianin the rotating frame is given by

Heff = H0 − · L , 10

where H0 is the Hamiltonian in the absence of the rotation

and L =−ir is the quantum-mechanical angular mo-mentum operator. Using this result with the Hamiltonian H0from Eq. 1, we obtain 34,35

ir,t

t= −

2

2m2 + Vr +ddr,t + gr,t2

−

ix

y− y

xr,t . 11

Note that all space coordinates r are those of the rotatingframe and the time-independent trapping potential Vr,given by Eq. 5, is stationary in this frame. Momentum co-ordinates, however, are expressed in the laboratory frame34–36.

We can express the condensate mean field in terms of adensity r , t and phase Sr , t as r , t=r , texpiSr , t, so that the condensate velocity is v = /m S.Substituting into the time-dependent GPE 11 and equatingimaginary and real terms leads to the following equations ofmotion:

t= − · v − r , 12

mvt

= − 1

2mv · v + Vr +ddr + g − mv · r .

13

In the absence of dipolar interactions dd=0 Eqs. 12 and13 are commonly known as the superfluid hydrodynamicequations 6–8 since they resemble the equation of continu-ity and the Euler equation of motion from dissipationlessfluid dynamics. Here, we have extended them to includedipolar interactions.

Note that the form of condensate velocity leads to therelation

v =

m S = 0 , 14

which immediately reveals that the condensate is irrotational.The exceptional case is when the velocity potential /mS issingular, which arises when a quantized vortex occurs in thesystem.

IV. STATIONARY SOLUTION OF THE HYDRODYNAMICEQUATIONS

We now search for stationary solutions of the hydrody-namic equations 12 and 13. These states satisfy the equi-librium conditions

t= 0,

vt

= 0. 15

Following the approach of Recati et al. 15 we assume thevelocity field ansatz

v = xy . 16

Here, is a velocity field amplitude that will provide us witha key parameter to parametrize our rotating solutions. Notethat this is the velocity field in the laboratory frame ex-pressed in the coordinates of the rotating frame, and also thatit satisfies the irrotationality condition 14. Actually, the ve-locity field amplitude can be given even more physicalmeaning by noting that, according to the continuity Eq. 12,it can be written as 7

= − D 17

where D is the deformation of the BEC in the x-y plane

D = y2 − x2 y2 + x2

=y

2 − x2

y2 + x

2 . 18

In the first term on the right-hand side of this expression ¯ signifies the expectation value in the stationary state,and in the second term on the right-hand side we have intro-duced the condensate aspect ratios defined as x=Rx /Rz andy =Ry /Rz.

Combining Eqs. 13 and 16 we obtain the relation

=m

2x

2x2 + y2y2 + z

2z2 + gr +ddr , 19

where the effective trap frequencies x and y are given by

EXACT SOLUTIONS AND STABILITY OF ROTATING… PHYSICAL REVIEW A 80, 033617 2009

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x2 =

2 1 − + 2 − 2 , 20

y2 =

2 1 + + 2 + 2 . 21

The dipolar potential inside an inverted parabola density pro-file 9 has been found in Refs. 12,27 to be

dd

3gdd=0xy

2001 −

x2101 + y2011 + 3z2002

Rz2 −

3,

22

where the coefficients ijk are given by

ijk = 0

ds

x2 + si+1/2y

2 + s j+1/21 + sk+1/2 , 23

where i, j, and k are integers. Note that for the cylindricallysymmetric case, where x=y =, the integrals ijk evaluateto 37

ijk = 22F1k +

1

2,1;i + j + k +

3

2;1 − 2

1 + 2i + 2j + 2k2i+j , 24

where 2F1 denotes the Gauss hypergeometric function 38.Thus, we can rearrange Eq. 19 to obtain an expression forthe density profile,

=

−m

2x

2x2 + y2y2 + z

2z2

g1 − dd

+

3gddn0xy

2Rz2 x2101 + y2011 + 3z2002 − Rz

2001

g1 − dd.

25

Comparing the x2, y2, and z2 terms in Eqs. 9 and 25 wefind three self-consistency relations that define the size andthe shape of the condensate:

x2 = z

x21 + dd3

2x

3y101 − 1

, 26

y2 = z

y21 + dd3

2y

3x011 − 1

, 27

Rz2 =

2g0

mz2 , 28

where =1−dd1− 9xy /2002. Furthermore, by insert-ing Eq. 25 into Eq. 12 we find that stationary solutionssatisfy the condition

0 = +x2 −

3

2dd

2 xy

2

101

+ −y2 −

3

2dd

2 xy

2

011 . 29

We can now solve Eq. 29 to give the velocity field ampli-tude for a given dd, , and trap geometry. In the limitdd=0 this amplitude is independent of the s-wave interac-tion strength g and the trap ratio . However, in the presenceof dipolar interactions the velocity field amplitude becomesdependent on both g and . For fixed dd and trap geometry,Eq. 29 leads to branches of as a function of rotationfrequency . These branches are significantly different be-tween traps that are circular =0 or elliptical 0 in thex-y plane, and so we will consider each case in turn. Notethat we restrict our analysis to the range : for the static solutions can disappear, with the condensatebecoming unstable to a center-of-mass instability 15.

A. Circular trapping in the x-y plane: =0

We first consider the case of a trap with no ellipticity inthe x-y plane =0. In Fig. 1a we plot the solutions of Eq.29 as a function of rotation frequency for a sphericallysymmetric trap =1 and for various values of dd. Beforediscussing the specific cases, let us first point out that foreach dd the solutions have the same qualitative structure. Upto some critical rotation frequency, only one solution existscorresponding to =0. At this critical point the solution bi-furcates, giving two additional solutions for 0 and

0.5 0.6 1.0

0.2

0.6

1.0

-0.2

-0.6

-1.0

(a)

0.7 0.8 0.9

x

x

y

y

Ω/ω

α/ω

εdd

1γ

Ω/ω

b

(b)

102 10410-210-40.55

0.6

0.65

0.7

0.75

0.8

εdd

FIG. 1. a Irrotational velocity field amplitude of the staticcondensate solutions as a function of the trap rotation frequency in a spherically symmetric trap =1 and =0. Various values ofdd are presented: dd=−0.49, 0, 0.5, and 0.99. Insets illustrate thegeometry of the condensate in the x-y plane. b The bifurcationfrequency b the point at which the solutions of in a bifurcateaccording to Eq. 32 versus the trap ratio . Plotted are the resultsfor dd=−0.49, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 0.9, and 0.99. In aand b dd increases in the direction of the arrow.

VAN BIJNEN et al. PHYSICAL REVIEW A 80, 033617 2009

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0 on top of the original =0 solution. We term this criticalfrequency the bifurcation frequency b.

For dd=0 we regain the results of Refs. 15,16 with abifurcation point at b= /2 and, for b, nonzerosolutions given by =22−

2 / 15. The physicalsignificance of the bifurcation frequency has been estab-lished for the nondipolar case and is related to the fact thatthe system becomes energetically unstable to the spontane-ous excitation of quadrupole modes for /2. In theTF limit, a general surface excitation with angular momen-tum l=qlR, where R is the TF radius and ql is the quan-tized wave number, obeys the classical dispersion relationl

2= ql /mRV involving the local harmonic potential V=m

2 R2 /2 evaluated at R see p. 183 of 7. Consequently,for the nonrotating and nondipolar BEC, l=l. Mean-while, inclusion of the rotational term in the Hamiltonian10 shifts the mode frequency by −l. Then, in the rotatingframe, the frequency of the l=2 quadrupole surface excita-tion becomes 2=2−2 7. The bifurcation fre-quency thus coincides with the vanishing of the energy of thequadrupolar mode in the rotating frame, and the two addi-tional solutions arise from excitation of the quadrupole modefor /2.

For the nondipolar BEC it is noteworthy that b does notdepend on the interactions. This feature arises because themode frequencies l themselves are independent of g. How-ever, in the case of long-range dipolar interactions the poten-tial dd of Eq. 7 gives nonlocal contributions, breaking thesimple dependence of the force − V upon R 11. Thus, weexpect the resonant condition for exciting the quadrupolarmode, i.e., b=l / l with l=2, to change with dd. In Fig.1a we see that this is the case: as dipole interactions areintroduced, our solutions change and the bifurcation point bmoves to lower higher frequencies for dd0 dd0.Note that the parabolic solution still satisfies the hydrody-namic equations providing −0.5dd1. Outside this rangethe parabolic solution may still exist but it is no longer guar-anteed to be stable against perturbations.

Density profiles for =0 have zero ellipticity in the x-yplane. By contrast, the 0 solutions have an ellipticaldensity profile, even though the trap itself has zero ellipticity.This remarkable feature arises due to a spontaneous breakingof the axial rotational symmetry at the bifurcation point. For0 the condensate is elongated in x while for 0 it iselongated in y, as can be seen from Eq. 17 and as illustratedin the insets in Fig. 1a. In the absence of dipolar interac-tions the 0 solutions can be interpreted solely in termsof the effective trapping frequencies x and y given by Eqs.20 and 21. The introduction of dipolar interactions con-siderably complicates this picture since they also modify theshape of the solutions. Notably, for dd0 the dipolar inter-actions make the BEC more prolate, i.e., reduce x and y,while for dd0 they make the BEC more oblate, i.e., in-crease x and y.

In Fig. 1a we see that as the dipole interactions areincreased the bifurcation point b moves to lower frequen-cies. The bifurcation point can be calculated analytically asfollows. First, we note that for =0 the condensate is cylin-drically symmetric and x=y =. In this case the aspect

ratio is determined by the transcendental equation11,12,30

2

2+ 1 f

1 − 2 − 1 +dd − 12 − 2

32dd= 0, 30

where

f =2 + 24 − 3000

21 − 231

with 000= 1 /1−2ln1+1−2 / 1−1−2 for theprolate case 1, and 000= 2 /2−1arctan2−1 forthe oblate case 1. For small →0+, we can calculatethe first-order corrections to x and y with respect to fromEqs. 26 and 27. We can then insert these values in Eq.29 and solve for , noting that in the limit →0 we have→b. Thus, we find

b

=1

2+

3

42dd

2 2201 − 101

1 − dd1 −9

22002 . 32

In Fig. 1b we plot b Eq. 32 as a function of forvarious values of dd. For dd=0 we find that the bifurcationpoint remains unaltered at b=x /2 as =z /x ischanged 15,16. As dd is increased the value of for whichb is a minimum changes from a trap shape which is oblate1 to prolate 1. Note that for dd=0.99 the mini-mum bifurcation frequency occurs at b0.55, which isover a 20% deviation from the nondipolar value. For moreextreme values of dd we can expect b to deviate evenfurther, although the validity of the inverted parabola TFsolution does not necessary hold. For a fixed we also findthat as dd increases the bifurcation frequency decreasesmonotonically.

B. Elliptical trapping in the x-y plane: 0

Consider now the effect of finite ellipticity in the x-yplane 0. Rotating elliptical traps have been created ex-perimentally with laser and magnetic fields 13,14. Follow-ing the experiment of Madison et al. 13, we will employ aweak trap ellipticity of =0.025. In Fig. 2a we have plottedthe solutions to Eq. 29 for various values of dd in a =1trap. As predicted for nondipolar interactions 15,16, thesolutions become heavily modified for 0. There exists anupper branch of 0 solutions which exists over the wholerange of and a lower branch of 0 solutions whichback-bends and is double valued. We term the frequency atwhich the lower branch back-bends to be the back-bendingfrequency b. The bifurcation frequency in nonellipticaltraps can be regarded as the limiting case of the back-bending frequency, with the differing nomenclature em-ployed to emphasize the different structures of the solutionsat this point. However, for convenience we will employ thesame parameter for both, b. No =0 solution exists forany nonzero . In the absence of dipolar interactions theeffect of increasing the trap ellipticity is to increase the back-bending frequency b. Turning on the dipolar interactions,

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as in the case of =0, reduces b for dd0, and increasesb for dd0. This is more clearly seen in Fig. 2b whereb is plotted versus dd for various values of the trap ratio .

Importantly, the back-bending of the lower branch canintroduce an instability. Consider the BEC to be on the lowerbranch at some fixed rotation frequency . Now consider adecreasing dd. The back-bending frequency b increasesand at some point can exceed . In other words, the staticsolution of the BEC suddenly disappears and the BEC findsitself in an unstable state. We will see in Sec. VI that thistype of instability can also be induced by variations in and .

As in the =0 case, increasing dd decreases both x andy, i.e., the BEC becomes more prolate. As explained in theIntroduction, this distortion is expected because of the aniso-tropy of dipolar interactions. However, because the dipolarinteractions are isotropic in the x-y plane, it is perhaps sur-prising to find that they increase the deformation of the BECin that plane. This can be clearly seen in Fig. 2a where wesee that the magnitude of increases as dd is increased forany fixed value of . See Eq. 17 for the relationship be-tween and the deformation of the BEC in the x-y plane.

V. DYNAMICAL STABILITY OF STATIONARYSOLUTIONS

Although the solutions derived above are static solutionsin the rotating frame, they are not necessarily stable, and soin this section we analyze their dynamical stability. Considersmall perturbations in the BEC density and phase of the form=0+ and S=S0+S. Then, by linearizing the hydrody-namic equations 12 and 13, the dynamics of such pertur-bations can be described as

tS

= − v c · g1 + ddK/m

· 0 · v + v c · S

, 33

where v c=v − r and the integral operator K is defined as

Kr = − 32

z2 rdr

4r − r− r . 34

The integral in the above expression is carried out over thedomain where 00, that is, the general ellipsoidal domainwith radii Rx , Ry , Rz of the unperturbed condensate. Ex-tending the integration domain to the region where 0+0 adds higher-order effects since it is exactly in this do-main that 0=O. To investigate the stability of the BEC,we look for eigenfunctions and eigenvalues of operator 33:dynamical instability arises when one or more eigenvalues possess a positive real part. The size of the real eigenvaluesdictates the rate at which the instability grows. Note that theimaginary eigenvalues of Eq. 33 relate to stable collectivemodes of the system 39, e.g., sloshing and breathing, andhave been analyzed elsewhere for dipolar BECs 40. In or-der to find such eigenfunctions we follow Refs. 16,27 andconsider a polynomial ansatz for the perturbations in the co-ordinates x, y, and z of total degree N. All operators in Eq.33, acting on polynomials of degree N, result in polynomi-als of at most the same degree, including the operator K.This latter fact was known to 19th century astrophysicistswho calculated the gravitational potential of a heterogeneousellipsoid with a polynomial density 41,42. The integral ap-pearing in Eq. 34 is exactly equivalent to such a potential.A more recent paper by Levin and Muratov summarizesthese results and presents a more manageable expression forthe resulting potential 43. Hence, using these results theoperator K can be evaluated for a general polynomial densityperturbation =xpyqzr, with p, q, and r being non-negativeintegers and p+q+rN. Therefore, the perturbation evolu-tion operator 33 can be rewritten as a scalar matrix opera-tor, acting on vectors of polynomial coefficients, for whichfinding eigenvectors and eigenvalues is a trivial computa-tional task.

Using the above approach we determine the real positiveeigenvalues of Eq. 33 and thereby predict the regions ofdynamical instability of the static solutions. We focus on thecase of an elliptical trap since this is the experimentally rel-evant case. Recall the general form of the branch diagram forthis case, i.e., Fig. 2a. In the 0 half plane, the staticsolutions nearest the =0 axis never become dynamicallyunstable, except for a small region , due to a center-of-mass instability of the condensate 44. The other lower-branch solutions are always dynamically unstable and there-fore expected to be irrelevant to experiment. Thus, we onlyconsider dynamical instability for the upper-branch solu-tions, i.e., the branch in the upper half plane where 0.In Fig. 3 we plot the maximum positive real eigenvalues ofthe upper-branch solutions as a function of for a fixedellipticity =0.025. The maximum polynomial perturbationwas set at N=3 since for this ellipticity it was found thathigher-order perturbations did not alter the region of insta-

x

y

x

y

0.2

0.6

1.0

-0.2

-0.6

-1.00 0.2 1.00.4 0.6 0.8

(a)

Ω/ω

α/ω

Ω/ω

εdd

0.6

0.65

0.7

0.75

0.8

0.85

-0.5 0 0.5 1.0εdd

b

(b)

FIG. 2. a Irrotational velocity field amplitude as a functionof the trap rotation frequency for a trap ratio =1 and ellipticity =0.025. Various values of dd are presented, dd=−0.49, 0, 0.5,and 0.99, with dd increasing in the direction of the arrow. Insetsillustrate the geometry of the condensate in the x-y plane. b Back-bending point b versus dd for =0.025 and =0.5 solid curve,1.0 long dashed curve, and 2.0 short dashed curve.

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bility, and such modes are therefore not displayed but seethe end of this section for further comment on the higher-order perturbations.

For a given dd and there exists a dynamically unstableregion in the - plane. An illustrative example is shown inFig. 3 inset for dd=0 and =1. The instability regionshaded consists of a series of crescents 16. Each crescentcorresponds to a single value of the polynomial degree N,where higher values of N add extra crescents from above. Atthe high-frequency end these crescents merge to form a mainregion of instability, characterized by large eigenvalues. Atthe low-frequency end the crescents become vanishingly thinand are characterized by very small eigenvalues which are atleast one order of magnitude smaller than in the main insta-bility region 19. As such these regions will only induceinstability in the condensate if they are traversed very slowly.This was confirmed by numerical simulations in Ref. 19where it was shown that the narrow instability regions havenegligible effect when ramping at rates greater thand /dt=210−4

2 . It is unlikely that an experiment couldbe sufficiently long lived for these narrow instability regionsto play a role. For this reason we will subsequently ignorethe narrow regions of instability and define our instabilityregion to be the main region, as bounded by the dashed linein Fig. 3 inset. For the experimentally relevant trap ellip-ticities 0.1 the unstable region is defined solely by theN=3 perturbations.

We define the lower bound of the instability region to bei this corresponds to the dashed line in the inset. This isthe key parameter to characterize the dynamical instability.As dd is increased i decreases and, accordingly, the un-stable range of widens. Note that the upper bound of theinstability region is defined by the end point of the upperbranch at .

Finally, we consider the higher perturbations N3. Wefind that as we increase the size of matrix 33 to N=3,4 ,5 , . . . the higher-lying modes that are thereby de-scribed also develop real eigenvalues as is increased, butthese lie within the region of instability already shown inFig. 3 for N=3 and so do not alter the region of instability, as

mentioned above. Significantly, we find that the modes be-come unstable in order, i.e., the perturbations contained inN=4 that are not present in N=3 become unstable at highervalues of than those in N=3, and similarly for N=5 incomparison to N=4. We take this as circumstantial evidencethat there is no “roton” minimum in the energy spectrum forthe parameters we have considered. The possibility of a rotonminimum in the Bogoliubov energy spectrum of a dipolarBEC has been widely discussed in recent literature see, e.g.,45–47. In analogy to the celebrated dispersion relation ofliquid 4He, the roton minimum refers to a minimum in theenergy spectrum at a finite value of the eigenvalue e.g.,momentum p labeling the excitation. This means that, coun-terintuitvely, some higher-lying modes can have lower en-ergy than lower-lying modes and this causes important ef-fects in flowing systems. Pitaevskii 48 discussed the caseof superfluid 4He flowing through a pipe at velocity v. In thelaboratory frame the energy spectrum is Galilean shiftedsuch that E→E− pv and this leads, for large enough v, to theroton mode pr being brought down to zero energy first.Crudely speaking, this is expected to trigger an instability tothe formation of a density wave with wavelength pr

−1. Inthe present case we have rotational flow and the Galileanshifted energy E→E−L can presumably result in angularroton modes 46 becoming unstable as is increased. How-ever, our empirical observation that the modes become un-stable in order as is increased seems to rule out the pres-ence of an angular roton minimum at finite angularmomentum Lr for our parameters, at least up to N=5. This isnot surprising because roton minima in dipolar BECs have sofar only been predicted to occur outside of the range −0.5dd1 where the system is stable against dipolar collapse.A proper treatment outside this stable range requires goingbeyond the TF approximation since the zero-point energymust be included. The interplay between rotational instabili-ties and dipolar collapse instabilities remains a fascinatingtopic for future exploration, although very relevant theoreti-cal work has been performed 46.

VI. ROUTES TO INSTABILITY AND VORTEX LATTICEFORMATION

A. Procedures to induce instability

For a nondipolar BEC the static solutions and their stabil-ity in the rotating frame depend only on rotation frequency and trap ellipticity . Adiabatic changes in and can beemployed to evolve the condensate through the static solu-tions and reach a point of instability. Indeed, this has beenrealized both experimentally 13,14 and numerically17,18, with an excellent agreement with the hydrodynamicpredictions. For the case of a dipolar BEC we have shown inSecs. IV and V that the static solutions and their instabilitydepend additionally on the trap ratio and the interactionparameter dd. Since all of these parameters can be experi-mentally tuned in time, one can realistically consider eachparameter as a distinct route to traverse the parameter spaceof solutions and induce instability in the system.

Examples of these routes are presented in Fig. 4. Specifi-cally, Fig. 4 shows the static solutions of Eq. 29 as a

FIG. 3. The maximum positive real eigenvalues of Eq. 33solid curves for the upper-branch solutions of as a function of. We assume =0.025, =1, and N=3 and present various dipolarstrengths dd=−0.49, 0, 0.5, and 0.99, with dd increasing in thedirection of the arrow. The inset shows the full region of dynamicalinstability in the - plane for dd=0. The narrow regions havenegligible effect and so we only consider the main instability regionbounded by the dashed line.

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function of Fig. 4a, Fig. 4b, dd Fig. 4c, and Fig. 4d. In each case the remaining three parameters arefixed at =0.025, =1, =0.7, and dd=0.99. Dynami-cally unstable solutions are indicated with red circles. Grayarrows mark routes toward instability the point of onset ofinstability being marked by an asterisk, where the free pa-rameter , , dd, or is varied adiabatically until either adynamical instability is reached or the solution branch back-bends and so ceases to exist. For solutions with 0, theinstability is always due to the system becoming dynamicallyunstable dashed arrows, whereas for 0 the instability isalways due to the solution branch back-bending on itselfsolid arrows and so ceasing to exist. Numerical studies 18indicate that these two types of instability involve differentdynamics and possibly have distinct experimental signatures.

Below we describe the adiabatic variation of each param-eter in more general detail, beginning with the establishedroutes toward instability in which i and ii are varied,and then routes, which are unique to dipolar BECs, based onadiabatic changes in iii dd and iv . In each case it iscrucial to consider the behavior of the points of instability,namely, the back-bending point b and the onset of dynami-cal instability of the upper branch i.

i Adiabatic introduction of . The relevant parameterspace of and is presented in Fig. 5a, with the instabilityfrequencies b solid curves and i dashed curvesindicated. For a BEC initially confined to a nonrotating trapwith finite ellipticity , as the rotation frequency is in-creased adiabatically the BEC follows the upper-branch so-lution Fig. 4a dashed arrow. This particular route tracesout a horizontal path in Fig. 5a until it reaches i , where

the stationary solution becomes dynamically unstable. Forthe specific parameters of Fig. 4a the system becomes un-stable at =i 0.65. More generally, Fig. 5ashows that as dd is increased, i is decreased and as suchinstabilities in the stationary solutions will occur at lowerrotation frequencies. At 0.1, the curve for i displays asharp kink, arising from the shape of the dynamically un-stable region, as shown in Fig. 3 inset.

ii Adiabatic introduction of . Here, we begin with acylindrically symmetric =0 trap, rotating at a fixed fre-quency . The trap ellipticity is then increased adiabati-cally, and in the phase diagram of Fig. 5a the BEC tracesout a vertical path starting at =0. The ensuing dynamicsdepend on the trap rotation speed relative to b =0.

a For b =0 the condensate follows the upperbranch of the static solutions shown in Fig. 4a. This branchmoves progressively to larger . For i the BEC re-mains stable but as is increased further the condensateeventually becomes dynamically unstable. Figure 5a showsthat as dd is increased i is decreased and as such thedynamical instability of the stationary solutions occurs at alower trap ellipticity.

b For b =0 the condensate accesses the lower-branch solutions nearest the =0 axis. These solutions arealways dynamically stable and the criteria for instability are

0.4

α

Ω/ω0.6 0.8 1.0

0

0.5

1.0

-0.5

-1.0

(a)

α 0

0.5

-0.5

0 0.4

(c)

0.2 0.6 0.8 1.0εdd

α 0

0.5

-0.5

0 0.05 0.1 0.15ε

(b)

α 0

0.5

-0.5

0.1 1.0 10γ

(d)

FIG. 4. Color online Stationary states in the rotating trap char-acterized by the velocity field amplitude , determined from Eq.29. Dynamically unstable solutions are marked with red circles. Ineach of the figures the a trap rotation frequency , b trap ellip-ticity , c dipolar interaction strength dd, and d axial trappingstrength are varied adiabatically, while the remaining parametersremain fixed at =0.7, =0.025, dd=0.99, and =1. The adia-batic pathways to instability onset marked by red asterisk areschematically shown by the dashed and solid arrows. Dashed ar-rows indicate a route toward dynamical instability, whereas solidarrows indicate an instability due to disappearance of the stationarystate.

ε

0

0.05

0.1

0.5 0.6 0.7Ω/ω

(i)(ii)(iii)

(a)

0.8 0.9

Ω/ω0.80.70.6

-0.5

0

0.5

1.0

0.1

1.0

10

εdd

(i)

(ii)(iii)

(iv)

(i)

(iii)

(ii)

(b)

(c)

γ

FIG. 5. Color online a Phase diagram of versus b solidcurves and i dashed curves for =1 and i dd=−0.49, ii 0.5,and iii 0.99. b Phase diagram of dd versus b solid curves andi dashed curves for =0.025 and i =0.5, ii 1, and iii 2. cPhase diagram of versus b solid curves and i dashed curvesfor =0.025 and i dd=−0.49, ii 0, iii 0.5, and iv 0.99. Ineach case the solid dashed arrows depict the routes to instabilityshown in Fig. 4.

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instead determined by whether the solution exists. As isincreased the back-bending frequency b increases.Therefore, when exceeds some critical value, the lower-branch solutions disappear for the chosen value of rotationfrequency . This occurs when b . Figure 5ashows that as dd is increased b is decreased and as suchinstabilities in the system will occur at a higher trap elliptic-ity. At this point the parabolic condensate density profile nolonger represents a stable solution. The particular route indi-cated in Fig. 4b is included in Fig. 5a as a vertical solidgray arrow.

iii Adiabatic change in dd. The relevant parameterspace of dd and is shown in Fig. 5b for several differenttrap ratios. Consider that we begin from an initial BEC in atrap with finite ellipticity =0.025 and rotation frequency .This can be achieved, for example, by increasing fromzero at fixed . Then, by changing dd adiabatically an in-stability can be induced in two ways:

a For bdd the condensate follows the upper-branch solutions until they become unstable. This route toinstability is shown in Fig. 4c by the dashed arrow, with thecorresponding path in Fig. 5b shown by the vertical dashedarrow. Thus, for idd the motion remains stable.However, for idd the upper branch becomes dy-namically unstable. In Fig. 5b, idd dashed curves isplotted for different trap ratios. As can be seen, the stableregion of the upper branch becomes smaller as dd isincreased.

b For bdd the condensate follows the lower-branch solutions nearest the =0 axis. These solutions arealways stable and hence an instability can only be inducedwhen this solution no longer exists, i.e., bdd. Figure5b shows bdd solid curves for various trap aspectratios. As can be seen the back-bending frequency b de-creases as dd is increased. Thus, if dd is increased the sys-tem will remain stable. However, if dd is decreased then thesystem will become unstable when =bdd.

iv Adiabatic change in . Figure 5c shows the param-eter space of and . Consider, again, an initial stable con-densate with finite trap rotation frequency and ellipticity =0.025. Then through adiabatic changes in the conden-sate can traverse the parameter space and, depending on theinitial conditions, the instability can arise in two ways:

a For b the condensate exists on the upperbranch. It is then relevant to consider the onset of dynamicalinstability i dashed curves in Fig. 5c. Providing i the solution remains dynamically stable. However,once i the upper-branch solutions become unstable.

b For b the condensate exists on the lowerbranch nearest the =0 axis. These solutions are always dy-namically stable and instability can only occur when the mo-tion of the back-bending point causes the solution to disap-pear. This occurs when b, with b shown in Fig.5c by solid curves for various dipolar interaction strengths.These two paths to instability are shown in Fig. 4d and arealso indicated in Fig. 5c as vertical gray arrows, where thedashed solid arrow corresponds to the 0 0 path.

B. Is the final state of the system a vortex lattice?

Having revealed the points at which a rotating dipolarcondensate becomes unstable, we will now address the ques-

tion of whether this instability leads to a vortex lattice. First,let us review the situation for a nondipolar BEC. The pres-ence of vortices in the system becomes energetically favor-able when the rotation frequency exceeds a critical frequencyv. Working in the TF limit, with the background densitytaking the parabolic form 9, v can be approximated as49

v =5

2

mR2 ln0.67R

!s. 35

Here, the condensate is assumed to be circularly symmetricwith radius R and !s= /2m0g is the healing length thatcharacterizes the size of the vortex core. For typical conden-sate parameters v0.4. It is observed experimentally,however, that vortex lattice formation occurs at considerablyhigher frequencies, typically 0.7. This differencearises because above v, the vortex-free solutions remainremarkably stable. It is only once a hydrodynamic instabilityoccurs which occurs in the locality of 0.7 that thecondensate has a mechanism to deviate from the vortex-freesolution and relax into a vortex lattice. Another way of visu-alizing this is as follows. Above v the vortex-free conden-sate resides in some local energy minimum, while the globalminimum represents a vortex or vortex lattice state. Since thevortex is a topological defect, there typically exists a consid-erable energy barrier for a vortex to enter the system. How-ever, the hydrodynamic instabilities offer a route to navigatethe BEC out of the vortex-free local energy minimum towardthe vortex lattice state.

Note that vortex lattice formation occurs via nontrivialdynamics. The initial hydrodynamic instability in the vortex-free state that we have discussed in this paper is only the firststep 18. For example, if the condensate is on the upperbranch of hydrodynamic solutions e.g., under adiabatic in-troduction of and undergoes a dynamical instability, thisleads to the exponential growth of surface ripples in the con-densate 13,18. Alternatively, if the condensate is on thelower branch and the static solutions disappear e.g., follow-ing the introduction of , the condensate undergoes largeand dramatic shape oscillations. In both cases the destabili-zation of the vortex-free condensate leads to the nucleationof vortices into the system. A transient turbulent state ofvortices and density perturbations then forms, which subse-quently relaxes into a vortex lattice configuration 18,50.

In the presence of dipolar interactions, however, the criti-cal frequency for a vortex depends crucially on the trap ge-ometry and the strength of the dipolar interactions dd.Following Ref. 21 we will make a simple and approxi-mated extension of Eq. 35 to a dipolar BEC. We will con-sider a circularly symmetric dipolar condensate with radiusR=Rx=Ry that satisfies Eqs. 26–28 and insert this intoEq. 35 for the condensate radius. This method still assumesthat the size of the vortex is characterized by the s-wavehealing length !s. Although one does expect the dipolar in-teractions to modify the size of the vortex core, it should benoted that Eq. 35 only has a logarithmic accuracy and isrelatively insensitive to the choice of vortex core lengthscale. The dominant effect of the dipolar interactions in Eq.

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35 comes from the radial size and is accounted for. Notealso that this expression is for a circularly symmetric systemwhile we are largely concerned with elliptical traps. How-ever we will employ a very weak ellipticity =0.025 forwhich we expect the correction to the critical frequency to becorrespondingly small.

As an example, we take the parameter space of rotationfrequency and dipolar interactions dd. We first considerthe behavior in a quite oblate trap with =10. In Fig. 6a weplot the instability frequencies i and b for this system as afunction of the dipolar interactions dd. Depending on thespecifics of how this parameter space is traversed, either byadiabatic changes in vertical path or in dd horizontalpath, the condensate will become unstable when it reachesone of the instability lines short and long dashed lines.These points of instability decrease weakly with dipolar in-teractions and have the approximate value ib0.75. On the same plot we present the critical rotationfrequency v according to Eq. 35. In order to calculate thiswe have assumed a BEC of 150 000 52Cr atoms confinedwithin a trap with =2200 Hz. In this oblate systemwe see that the dipolar interactions lead to a decrease in v,as noted in 21. This dependence is very weak at this valueof , and throughout the range of dd presented it maintainsthe approximate value v0.1. Importantly these resultsshow that when the condensate becomes unstable a vortex orvortex lattice state is energetically favored. As such, we ex-pect that in an oblate dipolar BEC a vortex lattice will ulti-mately form when these instabilities are reached.

In Fig. 6b, we make a similar plot but for a prolate trapwith =0.1. The instability frequencies show a somewhatsimilar behavior to the oblate case. However, v is drasti-cally different, increasing significantly with dd. We find that

this qualitative behavior occurs consistently in prolate sys-tems, as noted in 21. This introduces two regimes depend-ing on the dipolar interactions. For dd0.8, i,bv, andso we expect a vortex or vortex lattice state to form follow-ing the instability. However, for dd"0.8 we find an intrigu-ing regime in which i,bv. In other words, while theinstability in the vortex-free parabolic density profile stilloccurs, a vortex state is not energetically favorable. The finalstate of the system is therefore not clear. Given that a prolatedipolar BEC is dominated by attractive interactions sincethe dipoles lie predominantly in an attractive end-to-end con-figuration one might expect similar behavior to the case ofconventional BECs with attractive interactions g0 wherethe formation of a vortex lattice can also be energeticallyunfavorable. Suggestions for final state of the condensate inthis case include center-of-mass motion and collective oscil-lations, such as quadrupole modes or higher angular-momentum-carrying shape excitations 51–53. However, thenature of the true final state in this case is beyond the scopeof this work and warrants further investigation.

VII. CONCLUSIONS

By calculating the static hydrodynamic solutions of a ro-tating dipolar BEC and studying their stability, we have pre-dicted the regimes of stable and unstable motions. In generalwe find that the back-bending or the bifurcation frequencyb decreases with increasing dipolar interactions. In addi-tion, the onset of dynamical instability in the upper-branchsolutions, i, decreases with increasing dipolar interactions.Furthermore these frequencies depend on the aspect ratio ofthe trap.

By utilizing the features of dipolar condensates we detailseveral routes to traverse the parameter space of static solu-tions and reach a point of instability. This can be achievedthrough adiabatic changes in trap rotation frequency , trapellipticity , dipolar interactions dd, and trap aspect ratio ,all of which are experimentally tunable quantities. While theformer two methods have been employed for nondipolarBECs, the latter two methods are unique to dipolar BECs. Inan experiment the latter instabilities would therefore demon-strate the special role played by dipolar interactions. Further-more, unlike for conventional BECs with repulsive interac-tions, the formation of a vortex lattice following ahydrodynamic instability is not always favored and dependssensitively on the shape of the system. For a prolate BECwith strong dipolar interactions, there exists a regime inwhich the rotating spheroidal parabolic Thomas-Fermi den-sity profile is unstable and yet it is energetically unfavorableto form a lattice. Other outcomes may then develop, such asa center-of-mass motion of the system or collective modeswith angular momentum. However, for oblate dipolar con-densates, as well as prolate condensates with weak dipolarinteractions, the presence of vortices is energetically favoredat the point of instability and we expect the instability to leadto the formation of a vortex lattice.

We acknowledge financial support from the AustralianResearch Council A.M.M., the Government of CanadaN.G.P., and the Natural Sciences and Engineering ResearchCouncil of Canada D.H.J.O.D..

Ω/ω

εdd

(a)

(b)

0

0.2

0.4

0.6

0.8

1.00

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1.0

Ω/ω

FIG. 6. Color online The relation between the instability fre-quencies, b long dashed red curve and i short dashed curve,and the critical rotation frequency for vorticity v solid curve fora an oblate trap =10 and b a prolate trap =0.1. The instabilityfrequencies are based on a trap with ellipticity =0.025 while v isobtained from Eq. 35 under the assumption of a 52Cr BEC with150 000 atoms and scattering length as=5.1 nm in a circularlysymmetric trap with =2200 Hz.

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