c.m. savage and j. ruostekoski- energetically stable particlelike skyrmions in a trapped...

8/3/2019 C.M. Savage and J. Ruostekoski- Energetically Stable Particlelike Skyrmions in a Trapped Bose-Einstein Condensate

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parameter field becomes uniform sufficiently far from theparticle. As a result, we may enclose the Skyrmion by asphere on which the field configuration has a constantasymptotic value. This sphere can be mapped to a singlepoint in the order parameter space. Topologically, it in-dicates that t he 3D physical region inside the sphere canbe represented by a compact 3-sphere S3 (a sphere in 4D)[14]. Here we consider a SU(2) order parameter space

which topologically also corresponds t o S3

. Then themappings from the compactified 3D physical region tothe order parameter space are represented by the fieldU r: S3 ! SU2. The crucial point is that having asso-ciated the enclosing sphere with a single point in S3, weplace the elements of the physical space into a correspon-dence with the elements of t he compact order parameterspace SU(2). The mappings from S3 to SU(2) fallinto topological classes, each characterized by an integer-valued winding number W [1,7,9]:

W

24 2 Z d3rTrU @U yU @U yU @U y: (1)

Here denotes the completely antisymmetric tensor,the repeated indices are summed over, and @i representsthe derivative with respect to the spatial coordinate xi.The topological charge describes how many times the setof points in physical space is ‘‘wrapped’’ over the orderparameter space of SU(2) for a given element U r.

With cold atoms an SU(2) order parameter space isafforded most simply by a two-component BEC, whoseinteractions effectively fix the local value of the totaldensity j rj2 j ÿrj2 r of the two complexmacroscopic wave functions and ÿ [7]. We may thenexpress the two-component BEC as

r ÿr

r

q U yr

1

0

: (2)

We search for a topologically nontrivial solution for U rwith a nonvanishing winding number, determined byEq. (1). By minimizing the energy of the correspondingtrapped BECs, we may investigate the energetic stability

of the Skyrmion and find its equilibrium configuration.As an initial state for the numerical simulations we useU r expir ~ r̂r [15], or a similar state displacedfrom the trap center, where i denote the Pauli spinmatrices and r̂r represents the unit radial vector. A directsubstitution into Eq. (1) yields W 1 for a monotonicfunction with 0 0 and at the gas boundary.The corresponding BEC wave functions read

r ÿr

p cosr ÿ i sinr cos

ÿi sinr sin expi: (3)

Here ;; can also be understood as the sphericalangles of the 3-sphere. Then the boundary of the atomcloud corresponds to the pole of the 3-sphere.

Because of the topological stability of the Skyrmion,any continuous deformation of Eq. (3), without alteringthe asymptotic boundary values, is still a Skyrmion withW 1. Here, ÿ (the ‘‘line component’’) forms a vortexline with one unit of circulation around the core orientedalong the z axis (Fig. 1). Moreover, (the ‘‘ring com-ponent’’) forms a vortex ring with one unit of circulationaround its core on the circle z 0 ; r ÿ1

2 [7]. The

line component is confined in a toroidal region inside thering component in the neighborhood of the ring core.

Despite its topological stability, the Skyrmion solutionin several physical systems is energetically unstableagainst shrin king to zero size without additional stabiliz-ing features. This can be understood by means of simplescaling arguments: if the kinetic energy density, or theorder parameter ‘‘bending energy,’’ is quadratic in thegradient of the order para meter, this energy density scalesas 1=R2 with respect to the size R of the Skyrmion.Because the Skyrmion occupies a volume proportionalto R3, the energy E / R monotonically decreases with thesize of the Skyrmion. In the Skyrme model, the stability

is provided by an additional interaction term, with theenergy density scaling as 1=R4 [1].

It was proposed that, in a two-component BEC, theseparate conservation of the species can effectively sta-bilize the Skyrmion in a homogeneous space againstcollapse to zero size [9]. In the regime of phase separa-tion, with the scattering lengths satisfying aaÿÿ &a2

ÿ, the two species can strongly repel each other andthe toroidal filling due to the line component can preventthe vortex ri ng from shrinki ng t o zero radius [7,9]. Con-sequently, a filled vortex ring, as opposed to an emptyvortex ring, can be more stable against collapse. More-over, in the Skyrmion the filling has one unit of angular

momentum about t he z axis resulting in a 1=r2 centrifugalbarrier t hat further prevents the shrin king. This should becontrasted to the Skyrmions in a ferromagnetic spin-1BEC [8], which are closely related to the Skyrmion ex-citations proposed in superfluid liquid 3He [16], and canfreely mix the atoms between the different spin compo-nents. As a result, no stabilizing features due to t he atomnumber conservation exist in that case.

We next study numerically the energetic stability of theSkyrmion using the full 3D mean-field theory of thecoupled GPEs without additional simplifications:

i h _ i ÿh2

2mr2 V r Xkikj kj2! i: (4)

Our simulations are performed for the parameters of t heJILA two-species vortex experiment using perfectlyoverlapping isotropic trapping potentials for both statesV r m!2r2=2, with ! 2 7:8 Hz and harmonicoscillator length xho h=m!1=2 ’ 3:86 m [5]. Inthe interaction coefficients, ij 4 h2aijN i=m, the num-ber of atoms in each species is represented by N i, aiidenotes the intraspecies, and aij (i Þ j) the interspecies,

P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 2003VOLUME 91, NUMBER 1

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scattering length. For the jF 2 ; m f 2i and j1 ; 1i spinstates of 87Rb we have [17] a 5:67 nm, aÿÿ 5:34 nm, aÿ 5:50 nm. The GPEs depend on the di-mensionless interaction coefficients 0

ij 4N iaij=xho.Hence, the dynamics is unchanged in any scaling of length and time, which does not change N 2i!.

We found the ground states by imaginary timeevolution of the GPEs (4), using Runge-Kutta [18] and

split-step algorithms. At every time step, we separatelynormalized both wave functions to fix the atom number ineach component. The numerical simulations were fully3D, allowing for cylindrically asymmetric dynamics. Themost demanding numerics was performed on a parallelmultiprocessor supercomputer, using up t o 32 processors.Spatial grids of 1283, 2563, and 5123 were used. A typicalspatial range and imaginary time step were 30 xho and0:0025=!.

As a cylindrically symmetric initial state we used theSkyrmion (3) with the T homas-Fermi density profile[15], resulting in W 1. The winding number was cal-culated during the imaginary time evolution to determine

the stability of the Skyrmion. For small BECs, with N T 104 atoms (N T N N ÿ), the nonlinear repulsion be-tween the two species was too weak to prevent theSkyrmion from shrinking for any N ÿ=N T . Also the linecomponent ÿ rapidly diffused to the boundary of thefinite-size atomic cloud, altering the boundary conditionsand resulting in a decreasing winding number W < 1.Because the t opological stability of t he Skyrmion neces-sitates a well-defined asymptotic boundary conditionwith no phase variation, the Skyrmion was clearly lost.This decay mechanism is characteristic of trapped BECsand cannot occur in homogeneous systems, which im-plicitly assume large N

=N T .

For large BECs with N T > 106, represented in Fig. 2,the same instability mechanism was observed with a verylarge fraction of line component N ÿ=N T * 0:75. In othercases displayed in Fig. 2, the winding number typicallyevaluated to either 0 or 1 to better than 1%, with well preserved at the boundary. A drop from W 1 toW 0 indicated instability of the Skyrmion againstshrinking (Fig. 3 inset). The Skyrmion collapse viashrinking occurred when the high density central partof the ring component, which passes through the linecomponent vortex core, pinched off. T he total density

varies significantly at the trap center when the vortexring core collapses, emphasizing the importance of density fluctuations in the decay process; see Fig. 3.With sufficiently large BECs, even for the collapsedSkyrmions with no vortex ring, the local energeticminimum can correspond to a line component trappedinside a toroidal region with forming an atom-opticalconfinement.

For sufficiently large total and line component atomnumbers, the nonlinear repulsion between the two specieswas strong enough to inhibit the collapse of the vortexring, stabilizing the Skyrmion against shrinking for cy-lindrically symmetric initial states (Fig. 2). However, dueto the inhomogeneous potential, the Skyrmion was stillunstable with respect to drift towards the edge of theBEC where its energy is lower. This was found usingcylindrically asymmetric initial states, emphasizing theimportance of the full 3D simulation. The drift occurredwith the vortex line moving towards the boundary. Oncethe vortex line drifted to a low-density region, the non-linear repulsion was no longer strong enough to prevent

the shrin king of the Skyrm ion and the collapse occurredas described above; see Fig. 3 (bottom). The drift reducesthe total angular momentum of the atoms indicating anenergetic instability. Physically, this results from dissipa-tion, e.g., due to thermal atoms.

We have investigated how Skyrmions might be stabi-lized against the vortex drift instability. Perhaps thesimplest is to rotate the li ne vortex component about thez axis, above its critical angular velocity, thus stabilizingthe vortex. This should always work for cylindricallysymmetric traps in which the symmetric density preventsthe rotation coupling into the other BEC component. For

0 0.25 0.5 0.75 10

4

8

12

16

N− /N

T

N T

/ 1 0 6

FIG. 2. Skymion stability diagram. Total atom number N T vsthe fraction in the line component N ÿ=N T . indicates insta-bility. indicates stability of the cylindrically symmetric stateagainst shrinkage, which may be stabilized against drift asdiscussed in the text.

FIG. 3 (color online). Decaying Skyrmion’s imaginary t imeevolution. Top: densities (unit s of xÿ3

ho ) along the x axis. j j2:it 20 (dash-dotted line), 27.75 (dashed line), 28 (solid line),50 (dotted line). The dotted curve which is zero at x 0 isj ÿj2 at it 50; for other times it is qualitatively similar.W 0 by it 28:25, and it 50 is the stationary state. N N ÿ 2:3 106. Inset: winding number W versus imaginarytime. Bottom: densities on the z 0 plane, left j ÿj2, rightj j2. Since there is no stabilization, the line vortex core (redcircle) has moved towards the boundary and the ring vortex(surrounding the blue circle) is about to collapse. Color mapand parameters same as for Fig. 1.


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the case of Fig. 1 a line component angular velocity of 0:1! stabilized the Skyrmion.

Rotation of both species introduced a new instabilitymechanism. For sufficiently rapid rotations the line com-ponent again reached the boundary, resulting in W < 1.At the same time the ring vortex accommodated therotation by twisting and finally breaking at the gasboundary. Nevertheless, for the case of Fig. 1, we found

a small range of angular velocities around 0:085! forwhich the Skyrmion was stable against this, and againstline vortex drift. It stabilized off the rotation axis, in-dicating an equilibrium between the lower energy of theline vortex and the higher energy of the r ing componentthreading through it with increased distance from thetrap center. The breaking of cylindrical symmetry impliesa family of degenerate Skyrmions parametrized by thepolar a ngle .

Another method for stabilizing the Skyrmion is toinhibit the drift towards low-density regions by creatinga positive density gradient around the vortex line.This can be implemented with a blue-detuned Gauss-

ian laser beam along the z axis [19], providing a cylin-drically symmetric repulsive Gaussian dipole potentialperpendicular to the z axis. The total potential thenhas the ‘‘Mexican hat’’ form: V m!2r2=2 V 0 expÿ2 x2 y2=w2. Setting w 7 xho and V 0 25 h!, corresponding to about twice the width of thevortex line core, successfully stabilized a Skyrmionwith N N ÿ 8 106.

Proving numerically t hat a Skyrmion is stable is diffi-cult. The essential requirement is that it be stationaryunder imaginary time evolution. This was determinedby plotting the density on 1D sections through the sys-tem, and the phase variation on 2D slice planes. Inunstable cases these would evolve until t he Skyrm iondecayed. When they became asymptotically constantas a function of imaginary time the Skyrmion was judgedto be stable. Convergence was particularly slow, andhence difficult to establish, when both species wererotated.

The Skyrmion can be created using electromagneticfields to imprint topological phase singularities on thematter field while changing the internal state of theatoms, as proposed in Ref. [7]. In particular, a vortexring can be engineered in a controlled way with anappropriate phase-coherent superposition of th ree orthog-

onal standing waves. Because of dissipation, the preparedSkyrmions relax to t he ground states calculated here.Different spin textures are often referred to as

‘‘Skyrmions,’’ even when they are not characterized bythe topological invariant (1). It is important to emphasizethat the Skyrmions studied in this Letter are fundamen-tally different from the nonsingular Anderson-Toulouseor Mermin-Ho vortices [20], frequently also calledSkyrm ions [21].

This research was supported by the EPSRC, by theAustralian Research Council, and by the AustralianPartnership for Advanced Computing.

*Electronic address: [email protected]†Electronic address: [email protected]

[1] T. H. R. Skyrme, P roc. R. Soc. London A 260, 127 (1961);Nucl. Phys. 31, 556 (1962); V. Makhankov et al., The

Skyrme Model (Springer-Verlag, New York, 1993);J. Donoghue et al., Dynamics of the Standard Model

(Cambridge University Press, Cambridge, 1992).[2] E. Witten, Nucl. Phys. B223, 422 (1983); B223, 433

(1983).[3] R. A. Battye a nd P. M. Sutclif fe, Phys. Rev. L ett. 79, 363

(1997); 81, 4798 (1998); 86, 3989 (2001).[4] L. Faddeev and A. J. Niemi, Nature (London) 387, 58

(1997).[5] M. R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999).[6] A. E. Leanhardt et al., Phys. Rev. Lett. 90, 140403

(2003).[7] J. Ruostekoski and J. R. Anglin, Phys. Rev. Lett. 86, 3934

(2001); see also J. Ruostekoski, Phys. Rev. A 61, 041603(2000).

[8] U. Al Khawaja and H. T. C. Stoof, Nature (London) 411,918 (2001); and Phys. Rev. A 64, 043612 (2001).

[9] R. A. Battye et al., Phys. Rev. Lett. 88, 80401 (2002).[10] R. A. Battye et al., in Proceedings of the Workshop on

Integrable Theories, Solitons and Duality, edited byL. Ferreira, J. Gomes, and A. Zimerman, J. HighEnergy Phys. Conf. Proc. (PRHEP_unesp2002/009),http://jhep.sissa.it/.

[11] M. Leadbeater et al., Phys. Rev. Lett. 86, 1410 (2001).[12] J. R. A nglin and W. Ketterle, Nature (London) 416, 211

(2002).[13] R. L. Davis a nd E. P. S. Shellard, Nucl. Phys. B323, 209

(1989); J. M. Cornwall, Phys. Rev. D 56, 6146 (1997).[14] All such configurations can be associated with a class of

mappings that comprises an element of the third homo-topy group 3. Two such configurations can be deformedinto one another in a singularity-free space withoutaltering the asymptotic values, if and only if theyare represented by the same element of 3. Here 3SU2 Z indicates that topologically differentconfigurations can be characterized by an integer-valuedwinding number.

[15] Without the loss of generality, we may use as an in itialstate r, since the spherical symmetry of is notconserved in the imaginary time evolution.

[16] G. E. Volovik and V. P. Mineev, Sov. Phys. JET P 46, 401(1977) [Zh. Eksp. Teor. Fiz. 73, 767 (1977)].

[17] D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998).[18] B.M. Caradoc-Davies, Ph.D. thesis, University of Otago:

http://www.physics.otago.ac.nz/bec2/bmcd/ [19] C. Raman et al., Phys. Rev. Lett. 87, 210402 (2001).[20] P.W. Anderson and G. Toulouse, Phys. Rev. Let t. 38, 508

(1977); N. D. Mermin and T. L. Ho, ibid. 36, 594 (1976).[21] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).


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c.m. savage and j. ruostekoski- energetically stable particlelike skyrmions in a trapped...

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