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typeset using JPSJ.sty <ver.1.0b> Topological Excitations in Spinor Bose-Einstein Condensates Shunji Tuchiya and Susumu Kurihara ∗ Department of Physics, Waseda University,Tokyo169-8555 1 (Received February1,2008) 0 0 2 We investigate the properties of skyrmion in the ferromagnetic state of spin-1 Bose-Einstein condensates by means of the mean-field theory and show that the size of skyrmion is fixed to n the order of the healing length. It is shown that the interaction between two skyrmions with a oppositely rotating spin texturesis attractive when their separation is large, following a unique J power-law behavior with a power of -7/2. 0 1 KEYWORDS: Bose-Einstein condensation, spindegree offreedom,ferromagnetic state,skyrmion,spintexture ] t f o s Recent experimental success1) in obtaining Bose- the field annihilation operator of hyperfine state 1,α , . | i t Einsteincondensation(BEC)of23Naatomsinanoptical n(r) = Ψ (r)2, ζ (r) is a normalized spinor i.e. a α α | | m trap has opened up new directions in the study of BEC. Xα Although confining atoms in a magnetic traps has been ζ† ζ =1. This determines the average local spin - · d a successfulmethod, it alignsthe orientationofspindue F ζ (r)F ζ (r). (2) n totheexistanceofstrongmagneticfield. Forthisreason, h i≡ α∗ αβ β o spin of the trapped atoms cannot be regarded as a de- Ho2)showedthatinthecaseofspin-1bosonsconsider- c [ greeoffreedom. Incontrast,opticaltrapisbasedonthe ationoftwogroundstatesisnecessarysincetheeffective optical dipole force. Therefore atoms are confined in all interaction between two spins can be either ferromag- 2 hyperfine states. This offers the possibility of studying neticorantiferromagnetic. Thegroundstatesofthe for- v condensates with a spin degree of freedom, the spinor mer is minimized for F 2 = 1, and the latter, which 9 h i 0 condensates. is known for the polar state, is minimized for F = 0. h i 1 A variety of novel phenomena has been predicted for Thepropertiesofvorticesaredifferentbetweenthesetwo 1 spinor condensates, such as changes in ground state states, and skyrmions are possible only in the ferromag- 0 structures with interaction parameters, propagation of netic state. In this paper, we shall discuss their shape, 1 spin waves and internal vortex structure.2,3,4) In par- size and interactions in the ferromagnetic state. 0 / ticular, the possibility of creating vortex states with- The energy of the condensate is t a out core, or skyrmions, in spinor condensates has been ¯h2 ¯h2 m pointed out.2,3) Skyrmions, which do not have an or- K = d3r ( √n)2+ ( ζ) ( ζ)n † Z (cid:26)2M ∇ 2M ∇ ∇ - dinary vortex core due to the spin degree of freedom, d offer us many new physical phenomena beyond those 1 n +(U µ)n+ g n2 , (3) o presented by other vortex states. One interesting fea- − 2 2 (cid:27) c ture is the reduction of kinetic energy associated with : the rotation by transferring this energy to the spin and where µ is the chemical potential, g2 = 4π¯h2a2/M, a2 v is the s-wave scattering length for two colliding atoms i eliminating energetically unfavorable normal core. This X with total spin 2, and U is the trapping potential. reducestheenergyofskyrmiontoavaluelowerthanthat r of an ordinary singular vortex. All spinors in the ferromagnetic state are related each a other by gauge transformation eiθ and spin rotations In this paper we investigate the properties of skyrmions in spin-1 Bose condensates. First we discuss R(α,β,τ)=e−iFzαe−iFyβe−iFzτ, where (α,β,τ) are the Euler angles. The spinor ζ is given by2) the basic propertiesof a single skyrmion,then we exam- inethe interactionbetweentwoskyrmionswithopposite 1 phases. ζ(r)=eiθR(α,β,τ)0 We shall consider bosons with hyperfine spin F = 1. 0   This includes alkali atoms with nuclear spin I = 3/2 1e i(γ+α)(1+cosβ) such as 23Na, 39K, and 87Rb. The order parameter of 2 − the Bose condensate is characterizedby = √12e−iγsinβ , (4) 1e i(γ α)(1 cosβ) ψ (r) =Ψ (r) (1) 2 − − −  h α i α whereγ =τ θ. Thespintextureandsuperfluidvelocity − = n(r)ζ (r) (α=1,0, 1) of the ferromagnetic state are given by α − p where α indicates three hyperfine components, ψ (r) is F =(sinβcosα)xˆ+(sinβsinα)yˆ +(cosβ)zˆ, (5) α h i ∗E-mail: [email protected] 2 ShunjiTuchiyaandSusumuKurihara ¯h v = ( γ+cosβ α). (6) s −M ∇ ∇ 2p Equation (5) shows that α, β characterizes the spin ori- entation. Moreovereq.(6)showsthatspatialvariationof spin in general leads to superflows. Note that the spin ) r texture with superfluid velocity takes the same form as ( b the angular momentum texture with superfluid velocity p in 3He-A.6) To illustrate the skyrmion, we use cyrindrical co- ordinates (r,φ,z) and assume that the spin texture and superfluid velocity of the condensate are both cylindri- (cid:0) cally symmetric. Taking α = φ, γ = φ, β = β(r), − where β(r) is a function increasing from β =0 at r =0. (cid:0) (cid:1)(cid:0) (cid:2)(cid:0) (cid:3)(cid:0) r/l The spin texture and superfluid velocity field are F =(sinβcosφ)xˆ +(sinβsinφ)yˆ+(cosβ)zˆ. (7) h i Fig. 1. The variational function β(r) when the spatial depen- denceofdensityisneglected. β(r)isamonotonicallyincreasing ¯h functionobeyingβ(0)=0andβ(∞)=2π. v = (1 cosβ)e , (8) s φ Mr − Equation(8) showsthat the superfluid velocity vanishes continuously as r 0, and energetically unfavourable whereRisthecharacteristicdimensionofthecondensate. → normal core is avoided.2) For β(r)=π/2, the superfluid velocity of the skyrmionreduces to an ordinary singular vortex. To obtain a precise order parameter of the skyrmion, (cid:7)(cid:4) we have to determine n(r) and β(r) that minimize the energy. To investigate only properties of the skyrmion ) weneglectthe trappingpotential. Fromvariationalcon- M ditions δK/δn = 0, δK/δβ = 0, we can derive a pair of n /(cid:6)(cid:4) 2 coupled equations for n(r) and β(r) (cid:9) ( d2f 1 df 1 dβ 2 1 E/s (cid:5)(cid:4) + + (3 cosβ)(1 cosβ) f dx2 x dx −(cid:26)2(cid:18)dx(cid:19) 2x2 − − (cid:27) (f2 1)f =0, (9) − − (cid:4) d2β 1 2df dβ 1 (cid:4) (cid:8) (cid:5)(cid:4) (cid:5)(cid:8) (cid:6)(cid:4) + + (2 cosβ)sinβ =0, (10) R/l dx2 (cid:18)x f dx(cid:19)dx − x2 − where x = r/ξ and f = n/n¯. Here we define ξ = Fig. 2. The energy of the skyrmion in case of neglecting spatial (h¯2/2Mg2n¯)1/2 as the healping length of the condensate dependence of the density. For R ≫ λ, it approaches the con- 2 and n¯ = µ/g as the density of the condensate without stant(1+π/2)4π¯h n¯/M. 2 flow. In view of the absence of the core, we shall neglect the spatial dependence of density for the moment, and Itisclearfromeq.(12)thattheskyrmionhaslessenergy replaceitwithanaveragen¯. Inthis case,the solutionof than that of a singular vortex which is logarithmically eq.(10) is dependent on the size of system due to the existence of singularityat the core. The R dependence of the energy 4kx β(x)=2tan−1 . (11) of the skyrmion is shown in Fig. 2. (cid:18)8 k2x2(cid:19) − Figure 2 shows that the energy of the skyrmion is We plot this function in Fig. 1. Here we put β (0)=k. minimum when λ is infinite. This result is in fact not ′ λ=ξ/k expresses the typical size of the skyrmion. unreasonable physically. Our previous assumption has The energy of skyrmion per unit length along the z- neglected the spatial dependence of density, and only axis is considerdspatial dependence ofζ. Since ζ tends to vary E = 4π¯h2n¯ Rλ 4+ Rλ 4+64 tan−1 81 Rλ 2 aersgsyl,owtheassipzeososifbtleheinskoyrrdmeriotnobdeeccormeaesseitnhfieniktienlyetilcaregne-. s M (cid:0) (cid:1) (cid:0)(cid:0) (cid:1) R 4+(cid:1)64 (cid:0) (cid:0) (cid:1) (cid:1) However as the size of the skyrmion increase, the re- λ π 4π¯h2n¯ (cid:0) (cid:1) gion in which n(r) deviate from n¯ becomes large. This 1+ (R λ) (12) increases the interaction energy. Therefore considering ∼(cid:16) 2(cid:17) M ≫ spatial dependence of density, the balance between the TopologicalExcitations inSpinorBose-EinsteinCondensates 3 kinetic energy and the interactionenergy of the condensate determine the typical size of the skyrmion. (a) We can assume without loss of generality that devi- r < F > ation of n(r) from the average density n¯ is small i.e. n(r) = n¯(1 ǫ(r)), ǫ(r) 1 for r . Linearizing − ≪ → ∞ eq.(9) with ǫ(r) and using eq.(11), we obtain the following equation R (b) 0 d2ǫ 1 dǫ + 2ǫ dx2 xdx − 8k3x2+43k 2 1 = 2 1 ǫ . (13) − (cid:18) k4x4+43 (cid:19) (cid:18) − 2 (cid:19) R 0 The solution of the above equation has the following asymtotic behavior (c) C e √2x (x 1) 1 − ǫ(x)=C2I0(√k2+2x)+ k22k+22 (x≫≪1), (14) r r + where I0 isa modified Bessel function. Equation (14) r - shows that the variation of the density is spreaded over a distance ξ. Therefore the size of skyrmion is of the f f f orderofthehealinglengthofcondensatei.e. λ ξ. The 2 o 1 ∼ skyrmionisthusstabilizedbythefactthatasitbecomes R large the gradientof spin texture on one hand decreases 0 the kinetic energy and the variation of density increases the interaction energy on the other. Fig. 3. (a) Schematic representation oftwo skyrmions. Bold ar- Let us consider now two skyrmions with spin textures rows represent magnetic moment. The direction of circulation oppositely rotating of each skyrmion is represented by a circular arrow. (b) Inter- 1(1+cosβ(r)) sectionofthespintexture. (c)Viewfromthe z axis. 2 ζs(r,φ)= √12eiφsinβ(r)  1e2iφ(1 cosβ(r)) 2 −  1(1+cosβ(r)) 2 (cid:6)(cid:4) ζ¯s(r,φ)= −√12eiφsinβ(r) , (15) 1e2iφ(1 cosβ(r)) 2 −  whereβ(r)=2tan 1( 4r/λ ). Theminus signinfront (cid:3)(cid:2) − 8 (r/λ)2 ) ofthesecondcomponen−tofζ representsoppositephase. M ¯s 2 Thetextureinthepresenceoftwoskyrmionsisschemat- / n ically represented in Fig. 3. 2 (cid:5) Weshalldeterminetwo-skyrmionenergyasafunction /( (cid:2) of their separation R . Here we neglect the spatial de- wo (cid:4) pendence of density. 0The difference between the energy Et (cid:4) (cid:2) (cid:1)(cid:4) (cid:1)(cid:2) (cid:3)(cid:4) at a given separation R and the energy of two isolated 0 skyrmions per unit length is (cid:0)(cid:1)(cid:2) ¯h2 R /l Etwo(R0)= n¯ d2r( ζtwo)†( ζtwo) 2Es.(16) 0 2M Z ∇ ∇ − For R λ we approximately express the order param- 0 Fig. 4. Theenergyoftwoskyrmionsasafunctionofthedistance ≫ eter in the presence of two skyrmions as a sum of each between thecenter oftwoskyrmions. order parameter. ζ (r,φ)=ζ (r ,φ )+ζ (r ,φ ). (17) two s − 1 ¯s + 2 thekineticenergy. Sincetexturetendstovarysmoothly, skyrmions with large separation tends to come closer to Our result is plotted in Fig. 3. The curve has a mini- each other, thus interaction becomes attractive. In con- mumatR =r wherer isapproximately5λ. Itcanbe trast, if they are too closer together, the region where 0 0 0 seenthat for R <r the interactionbetweenskyrmions velocity increases become large and the energy of two 0 0 is repulsive, while R > r the interaction is attractive. skyrmion increases. Thus with sufficiently close separa- 0 0 As twoskyrmionsarebroughttogetherthe regionwhere tion the interaction energy is repulsive. spin texture varies gradually overlap, thereby changes We have investigatednumerically the attractive inter- 4 ShunjiTuchiyaandSusumuKurihara (cid:2) ) M) (cid:5)(cid:3)(cid:4) 2 n/ 2 (cid:5) /((cid:9) (cid:5)(cid:3)(cid:6) (cid:2)(cid:3)(cid:2) (cid:2)(cid:3)(cid:7) (cid:2)(cid:3)(cid:4) (cid:2)(cid:3)(cid:8) o Etw(cid:0)(cid:5)(cid:3)(cid:4) - ( 0 g1 (cid:0)(cid:2) o l (cid:0)(cid:2)(cid:3)(cid:4) (cid:0)(cid:1) ( ) log R /l 10 0 Fig. 5. Logarithmic plot of energy of two skyrmions. The solid line,withaslopeof-7/2isthebestfittothedata. action between skyrmions. Figure 5 shows logarithmic plotofenergyoftwoskyrmions. Itshowsthatattractive interaction between skyrmions is approximately propor- tional to R0−7/2. Inconclusion,wehaveinvestigatedtheenergyandthe size of a skyrmionin ferromagnetic state of spin-1 Bose- Einstein condensates and have shown that the size is of theorderofthehealinglength. Wealsohaveshownthat the skyrmion has lower excitation energy than that of a singular vortex. Therefore skyrmions can be excited more easily than singular vortices in scalar superfluids. This means that skyrmions are important in determin- ing thermodynamic properties of the condensate at low temperatures. It also means a reduction of the critical superfluid velocity. We have shown that the interaction between two skyrmions with opposite phases is attractive when their separationislargerthanr andrepulsivewhentheirsep- 0 arationissmallerthanr . Physicaloriginandthepower 0 of attractive force between skyrmions is still unknown, and interactions between skyrmions with other textures is also an interesting subject to invetigate. Acknowledgement The authorswouldliketo thank M. Nishida foruseful discussions. [1] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H,-J. Miesner, J. Stenger, and W. Ketterle: Phys. Rev.Lett.80(1998) 2027. [2] T.-L.Ho: Phys.Rev.Lett.81(1998) 742. [3] T.Ohmi,K.Machida: J.Phys.Soc.Jpn.67(1998)1823. [4] C. K. Law, H. Pu, and N. P. Bigelow: Phys. Rev. Lett. 81 (1999) 5257. [5] S.-K.Yip: Phys.Rev.Lett. 83(1999)4677. [6] D. Vollhardt and P. Wo¨lfle, The Superfluid Phases of He3 (Tailor&Francis,London, 1990)