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Structure of Vortices in Two-component Bose-Einstein Condensates D. M. Jezek1,2, P. Capuzzi1, and H. M. Cataldo 1,2. 1Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, RA-1428 Buenos Aires, Argentina 2Consejo Nacional de Investigaciones Cient´ıficas y Técnicas, Argentina 2 little is said about the structure of the stationary states 0 0 andnocomments aremade aboutsegregationofspecies. 2 We develop a three-dimensional analysis of the phase sep- In a recent work, Chui et al. [7] computed the energy aration of two-species Bose-Einstein condensates in the pres- n ence of vorticity within the Thomas-Fermi approximation. of different configurations within the Thomas-FermiAp- a proximation(TFA). They varied the number of particles We find different segregation features according to whether J and interchanged the species in order to derive conclu- themorerepulsivecomponentisinavortexorinavortex-free 0 state. Anapplicationofthisstudyisaimedatdescribingsys- sions about the relative stability. But, although they 3 temsformedbytwoalmostimmisciblespeciesofrubidium-87 used a 3D model and considered phase separationin the thatarecommonlyusedinBose-Einsteincondensationexper- mixture,ahypothesiswasmaderegardingtheformofthe ] t iments. In particular, in this work we calculate the density interface between species that we find is not consistent f o profiles of condensates for the same conditions as the states with our results and apparently also with new available s prepared in the experiments performed at JILA [ Matthews, experimental data [8]. We shall comment on this later. t. et al., Phys. Rev. Lett. 83, 2498 (1999)]. In 1996, in a pioneering work, Ho and Shenoy [9] con- a m structed an algorithm to determine the density profiles 03.75. Fi, 67.57.Fg, 67.90.+z of binary mixtures of alkali atoms within the TFA. We - d develop here a different procedure which allows one to n Vortices in Bose-Einstein condensates (BECs) have obtainconclusionsabout the generalfeaturesof segrega- o thoroughlybeen studiedfroma theoreticalpointof view tion through simple calculations. c [ since the first experiments on condensates were carried The aim of the present article is to analyze and per- out,duetotheirdirectconnectiontosuperfluidity. Fora form a classificationof the structure of condensates near 1 reviewoftheseissuesseeforexample[1],andforthemost phaseseparation,whentwoalmostimmisciblespeciesare v 3 recent works references in [2]. But until the experiment involved. Our starting point for calculating the station- 6 of Matthews et al. [3] was performed, no evidence about arystates are the Gross-Pitaevskiiequationsin the TFA 5 their existence had ever been reported. This experiment [1]: 1 was based on a method proposed by Williams and Hol- 0 µ Ψ =(V +N G Ψ 2+N G Ψ 2)Ψ , 2 land[4]tocreatevorticesintwo-componentBECs. They 1 1 1 1 1,1| 1| 2 1,2| 2| 1 0 trapped atoms of 87Rb in two different hyperfine states µ2Ψ2 =(V2+N1G1,2 Ψ1 2+N2G2,2 Ψ2 2)Ψ2, (1) | | | | / F = 1,m = 1 and F = 2,m = 1 , henceforth t | f − i | f i where N denotes the number of atoms of species i, and a denoted as 1 and 2 , respectively, and created vortices i m eitherinthe| i1 or 2| icomponent,whiletheotherspecies Gk,l =uk,lU, with uk,l the relative interaction strengths | i | i between species k and l. We set the most repulsive com- - remained without rotation. Because of the relation be- d ponent 1 in the Ψ state, and fix u = 1 (> u ), on tswtaeteens hinatvrea-thaenpdroinpteerrt-ypaorftbiceleinsgcaaltmteorsintgimlemnigstchibs,let.hWesee so U =|4πiMh¯2a, a bei1ng the scattering 1le,1ngth of the2|,12i- c shall see that this fact has important consequences on species and M the atom mass. v: the structure of the vortices. For simplicity, we consider the system in a spherically symmetric trap and make a change of variables accord- i In connection with this experiment interesting theo- X retical work has recently been developed [5–7] in order ingto Mw ~r ~r,withw thetrapangularfrequency. ar to describe different properties of many types of two- Thus tqhe2eff0ect→ive potentia0ls Vi written in cylindrical componentcondensatesystems. Garc´ıa-RipollandPérez variables read Garc´ıa,bymeansofarobustformalism,analyzedthesta- κ bility and dynamics of a variety of two-species conden- Vi =r2+z2+ r2i. (2) sates,takingintoaccountthe effectofthe interchangeof species in the system, and also the variationof the num- The first two terms correspond to the trapping poten- ber of particles. In particular, in Ref. [5] they set the tial, while the third includes the centrifugal term [9], relative number of particles equal in both species and κ = m2ih¯2w02, with m the number of quanta of circu- i 4 i varied the total number of particles, and in Ref. [6] they lation associated with the wave function for species i. pursuedthe researchby also varyingthe relativecompo- LetusfirstanalyzewhathappenswhenD u u 1,1 2,2 ≡ − sition using a 2D model. However, in these works [5,6] u2 = 0. Physically the system starts undergoing phase 1,2 1 separation. Mathematically, for both Ψ 2 >0, Eqs. (1) shell around it. In the presence of vorticity (B =0), the i | | 6 become linearly dependent. So to have a compatible set location of each species can be predicted from energetic of equations the following condition must hold: considerations. The state with greater vorticity should be located in the region of space that excludes r = 0, µ V µ V 1 1 = 2 2 . (3) because of the centrifugal term. G − G G − G 1,1 1,1 1,2 1,2 From this simple analysis we may state that close to phase separation the presence of vorticity dramatically If we define β =u /u , the above equation can be 2 1,2 1,1 changes the segregationstructure. rewritten as Now,letusconsiderthegeneralcaseD =0. Thesolu- 6 µ2−β2µ1 r2 z2+ β2κ1−κ2 =0. (4) tion of Eqs. (1 ) can be easily obtained and has the fol- 1 β − − (1 β )r2 lowing different expressions depending on whether there 2 2 − − exists any overlap between the wave functions of both This equation defines different families of surfaces that species. a)Inthe regionwhere only one wavefunction is we shall call i, where i labels the family. As was men- S0 non-vanishing ( Ψi 2 =0 and Ψk 2 =0, for i=k) Eqs. tioned above,this is the only regionof space where both | | 6 | | 6 (1) are decoupled and the solution is Ψ may be non-zero, so these surfaces necessarily deter- i mine the interfaces between species. Such families are Ψi 2 = µi r2 z2 κi/r2 /(Gi,iNi). (6) | | − − − characterized by the values of (cid:2) (cid:3) b) In the region where both wave functions are non- A= µ2−β2µ1, B = β2κ1−κ2, (5) vanishing, |Ψi|2 >0, one obtains 1 β (1 β ) 2 2 µ β µ κ β κ − − Ψ 2 = 1− 1 2 r2 z2+ 2 1− 1 A u , (note that D =0 implies β2 <1). | 1| (cid:20) (1−β1) − − r2(1−β1)(cid:21) 1 2,2 Depending on the values of the above parameters, µ β µ κ β κ Ψ 2 = 2− 2 1 r2 z2+ 1 2− 2 A u , (7) three topologically different kinds of surfaces exist, | 2| (cid:20) (1 β ) − − r2(1 β )(cid:21) 2 1,1 2 2 − − namely, spheres, cylinder-like surfaces, and toroids. In Fig. 1weshowapartitionoftheparameterspace(A,B) where β1 = uu12,,22 and Ai = (U1N−βiDi). indicating the domains corresponding to each family. In We call ( ) the surface defined by equating the 1 2 S S ordertoclassifythesesurfacesletusstudythez(r)curves expression inside the square bracket in Ψ 2 (Ψ 2) to 1 2 | | | | definedbyEq. (4). Ifz(r)intersectsthe z axis,theasso- zero. Itisinterestingtonoticethatthe expressionsmen- ciatedsurfaces a arespheres,and this occursfor B =0 tionedaboveareofthesametype astheonethatdefines S0 and A > 0. If z(r) has only one root the corresponding (cf. Eq. (4)), and so a similar classification can be 0 S surfaces b are cylinder-like, and this takes place when applied to these surfaces. The surfaces and define S0 S1 S2 B >0. Finally, if z(r) has two roots the related surfaces the boundary of the coexisting region of the 1 and 2 | i | i c aretoroids,anditiseasytoverifythattheconditions species. Depending on the value of the interparticle in- S0 to be fulfilled are A2+4B > 0, A >0, and B < 0. The teraction strength, the regions where the overlapping of remaining region of the parameter space has no associ- the waves function occurs exhibit different features. For ated surfaces. D > 0 and u small enough, the coexisting region can 1,2 To perform a displacement of a point within the pa- beverylarge,asmaybeobservedinthefiguresshownin rameter space two mechanisms are possible. On the one Ref. [9]. When increasing u this region becomes nar- 1,2 hand,theAvaluemaybechangedbyvaryingthenumber rower. At D = 0, it is easy to verify that = = 1 2 S S of particles in each species, which will cause a change in ,andhencethecoexistingregionisreducedtoasingle 0 S thechemicalpotentials. Avariationofthistypedoesnot surface. When u increases making D <0, this surface 1,2 change the family, but the shape of the surface is mod- splits into two distinct surfaces,forming thus a new nar- ified. For example, for B > 0 one can get a straighter row 3D overlapping region, due to mutual repulsion of “cylinder”bymovingfrompositivetonegativeAvalues, the species. and,intermsofthenumberofparticles,thiscorresponds To havethe problemcompletely solvedone has to cal- to a decrease in the N /N ratio. On the other hand, B culatethechemicalpotentials. Inordertoobtainµ and 2 1 1 canonlytakediscretevaluesanditismodifiedbychang- µ the normalization condition for the wave functions 2 ing the vorticity. given by Eqs. (6) and (7) must be used. This leads to Itisworthwhilementioningthattheonlypossiblereal- a system of two coupled integral equations that must be ization for B =0, (and hence for a), within reasonable solved numerically. We have accomplished this in order S0 valuesofm ,isavorticityfreecondensatem =m =0. to describe the states obtained in the JILA experiment i 1 2 In such a case, regarding the location of each species, it [3], choosing parameters equal to those of the experi- is easyto conclude thatthe energyminimum is obtained ment. Namely, the trapping potential has an angular when the less repulsive component lies inside the sphere frequencyw =2πν ,beingν =7.8Hz. Theinter- 0 trap trap of radius √A, while the other species forms a spherical actionstrengthsrelativetothe 1 componentofthe87Rb | i 2 species are u = 1,u = .97 and u = .94. And the With respect to the hypothesis made by Chui et al. 1,1 1,2 2,2 number of particles of each species is N =N =4 105. [7],theyassumedforcondensatesincludingvorticityina 1 2 × Note that for these values of the interaction strengths, symmetric trap that one component is confined inside a D = 9 10−4 is close to the phase separation value, so ball and the other one in a sphericalshell aroundit; and − × the region between interfaces is very narrow, and hence thatthespecieswithvorticityhaveasmallhealinglength . Thus hereafter, in order to identify the sizedvortexcore. Inthecontextofourarticlethismeans 1 2 0 S ∼ S ∼ S type of segregationof species encountered, we will make that they worked as if segregation of species, including use of the classification we performed for . vortices, always evolves as a, which is not consistent S0 ∼ S0 For completeness, we also include calculations for the with our results. In a very recent experimental work [8] two species without vorticity, although the experimen- Anderson et al. haveincluded a figure (Fig. 3)in which tal realization consisted only of one vortex and one non- atwo-componentvortexisviewedalongitsaxisdirection vortex state in each component. and along an orthogonal one. Views of a vorticity free In Fig. 2 we show the density contours of each wave condensatearealsoincluded. Thein-trappicturesofRef. function in the (r,z) plane for the following three con- [8] cannot rule out a healing length core because such a figurations. (i) A vortex-free system (m ,m ) = (0,0) feature would be too small ( 0.7µm ) to resolve. So 1 2 ∼ yields chemical potentials in units of h¯w as µ = 24.50 in this experiment Chui’s condensates should look like a 0 1 and µ = 24.06. It may be seen that in this case the vortex free condensate. As it may be seen in the above 2 segregationevolvesas a. The more repulsivecompo- mentionedfiguretheimagesofasystemwithandwithout ∼ S0 nentisonasphericalshellaroundtheother,asexpected. vorticity are quite different. In particular, we want to (ii) For (m ,m ) = (1,0) we obtained µ = 24.54 and note that in the vortex case there is a visible evidence 1 2 1 µ =24.04. The vortex segregates in a ring surrounding of a cylinder type segregation. However to be conclusive 2 the non vortex state. It is easy to verify that the species aboutthesestatementsmoreexperimentalinformationis become separated by an interface of the type b. The required. S0 coresizeis about9µmandthewholevortexhasaradius In Refs. [5,6] the authors employed, for describing the of 26µm. This reproduces very well the experimental stationary states, wave functions expanded on a finite data. (iii) When the vortex is in the less repulsive com- subset of the harmonic oscillator basis. This means that ponent (m ,m ) = (0,1), we obtained µ = 24.50 and they did not work with exact solutions; unfortunately in 1 2 1 µ = 24.15. In this case the vortex is segregated inside these articles there is no precise information about the 2 thenon-vortexstate. Thecorrespondingsurfacethatde- structure of the states. However, looking at both the scribes such segregation is the toroid c. The inner and domains of the wave function (Fig. 2) and the form of S0 outer radii of the torus are 5µm and 18µm, respectively. thedensityitself(Fig. 3),itseemsthattheyshouldhave However,this does notcorrespondto whatwas obtained taken into account a large number of elements of this at the first stage of the experiment, which looks quite basis to describe similar configurations. similar to the pattern described in (ii). Mainly, in the It is well known that the TFA turns out to be very experiment, the radius of the vortex was approximately accurate when systems with a large number of particles 26µm, while according to our calculations it should be areconsidered,sothemainfeaturesofouranalysiscould 18µm. Taking into account that after vortex creation, be regardedas a guide to check the accuracy of different the vortex immediately shrank, a possible and likely ex- approximations. planation for the discrepancy is that the system was not Finally, we want to mention that very interesting ex- createdneartheequilibriumconfigurationbutmovedto- perimental [12] and theoretical [13] work dealing with wards it. The case with a vorticity of (m ,m ) = (1,0) off-centeredvorticesin two-speciescondensates, is forth- 1 2 is created much closer to equilibrium [10]. It is impor- coming. tant to notice that in every case, even when the vortex In conclusion, we think that our analysis of states configurationisoutofequilibrium,the wholecondensate within the TFA could be helpful both from the exper- is almost spherical with the same radius value of 26µm. imental and theoretical viewpoints. In the first case, to In Fig. 3 we display the density profiles as a function prepare a configuration as close as possible to a station- of the radial coordinate at z = 0, for the same three ary state, and in the second, to make a suitable choice configurations described in Fig. 2. It may be seen in forthesetofwavefunctionsinwhichsuchkindsofstates all of the cases that the shape of the total density is are to be expanded. like a parabola, a characteristic that also holds for any We gratefully acknowledge useful conversations with other direction considered. Regarding the densities of B. P. Anderson. each one of the components separately, they abruptly go to zero in the narrow region between species. It is worthwhiletomentionthatthisbehaviorwouldnotbeso sharp for real systems and some interpenetrating region would exist [11]. 3 [1] F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari Rev.Mod. Phys. 71, 463 (1999). 30 30 30 [2] B. Jackson, J. F. McCann, and C. S. Adams, Phys. |ψ |2 |ψ |2 |ψ |2 20 1 20 1 20 1 Rev. Lett. 80, 3903 (1998), S. Stringari, Phys. Rev. z z z Lett.82,4371(1999),B.Caradoc-Davis,R.Ballaguand, 10 10 10 and K. Burnett, Phys. Rev. Lett. 83, 895 (1999), D. Feder,C.Clark,B.Schneider,Phys.Rev.Lett.82,4956 00 10 20 30 00 10 20 30 00 10 20 30 (1999), and A. L. Fetter and A. A. Svidzinsky, cond- r r r mat/0102003. 30 30 30 [3] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. |ψ |2 |ψ |2 |ψ |2 Hall,C.E.Wieman,andE.A.Cornell, Phys.Rev.Lett. 20 2 20 2 20 2 z z z 83, 2498 (1999). 10 10 10 [4] J. Williams and M. Holland, Nature (London) 401, 568 (1999). 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 [5] J. J. Garc´ıa-Ripoll and V. M. Pérez-Garc´ıa, Phys. Rev. r r r Lett.84, 4264 (2000). FIG. 2. Density contours in the (r,z) plane for compo- [6] V. M. Pérez-Garc´ıa and J. J. Garc´ıa-Ripoll, Phys. Rev. nent |1i (|2i) in the first (second) row of subfigures. The A.62, 033601 (2000). columns correspond to the following situations: (left) no [7] S. T. Chui, V. N. Ryzhov, and E. E. Tareyeva, Phys. vortex (m1,m2) = (0,0), (middle) a vortex in |1i species Rev.A.63, 023605 (2001). (m1,m2) = (1,0), and (right) a vortex in the less repulsive [8] B. P. Anderson, C. A. Regal, D. L. Feder, L. A. Collins, component(m1,m2)=(0,1). Thedensitycontourspacingis C. W. Clark, and E. A. Cornell, Phys. Rev. Lett. 86, 10−5 µm−3. 2926 (2001). [9] Tin-LunHoandV.BShenoy,Phys.Rev.Lett.77,3276 (1996). [10] B. P. Anderson,privatecommunication. [11] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev.Lett. 81, 1543 (1998). [12] B. P. Anderson, P. C. Haljan, C. E. Wieman, and E. A. Cornell, Phys. Rev.Lett. 85, 2857 (2000). 1 1 1 |ψ|2 [13] S.A. McGee and M. J. Holland, cond-mat/0007143. i 0.5 0.5 0.5 B 0 0 0 z z 0 20 40 0 20 40 0 20 40 Sa r (µm) r (µm) r (µm) 0 r r Sb 0 FIG. 3. Density profiles |Ψ1|2 (solid line) and |Ψ2|2 (dashedline)asfunctionofr atz=0forthesamethreedis- z A tributions of vorticity described in the previous figure. The No surfaces density is given in units of 10−4 µm−3. rSc 0 i FIG. 1. Domains of the different families of surfaces S0 inparameterspace(A,B). Ineachregion weindicatethere- latedz(r)curve. ThethicklineB=0andA>0corresponds tospheres. TheB>0regioncorrespondstocylinder-likesur- faces. AndfortheconditionsA>0,B<0,andforB>−A2 4 thesurfaces aretoroids. Thedashed linedeterminesthesep- aration between these families of surfaces and emptysets. 4