Steering the motion of rotary solitons in radial lattices

Y. J. He,1,2 Boris A. Malomed,3 and H. Z. Wang1,*1State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, China

2School of Electronics and Information Engineering, Guangdong Polytechnic Normal University, Guangzhou 510665, China3Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,

Tel Aviv 69978, Israel�Received 13 June 2007; published 2 November 2007�

We demonstrate that rotary motion of a two-dimensional soliton trapped in a Bessel lattice can be preciselycontrolled by application of a finite-time push to the lattice, due to the transfer of the lattice’s linear momentumto the orbital momentum of the soliton. A simple analytical consideration treating the soliton as a particleprovides for an accurate explanation of numerical findings. Some effects beyond the quasi-particle approxi-mation are explored too, such as destruction of the soliton by a hard push.

DOI: 10.1103/PhysRevA.76.053601 PACS number�s�: 03.75.Lm, 42.65.Jx

I. INTRODUCTION

Techniques which make it possible to manipulate coherentmatter-wave beams by means of specially designed poten-tials are crucial ingredients in many applications of atomoptics, such as inertial sensors �1� and atomic holography�2�. In that context, an important topic is the control of rota-tional motion of Bose-Einstein condensates �BECs�. RotatingBECs support Abrikosov lattices of quantized vortices �3�, acounterpart of the fractional quantum Hall effect �4�, persis-tent currents �5�, and quantum phase transitions �6�. RotatingBECs can be trapped in toroidal magnetic �7� and opticalwaveguides �8�. Recently, the transfer of angular momentumfrom photons to atoms using a two-photon stimulated Ramanprocess with Laguerre-Gaussian beams was demonstrated togenerate a BEC vortex state �9�.

Optical lattice �OL� is another versatile tool for the stud-ies of dynamical properties of cold quantum gases �10�.Many fundamental effects in BECs loaded in OLs have beenreported, including Bloch oscillations �11�, Landau-Zenertunneling �12�, and superfluid-insulator transitions �13�.Matter-wave solitons of the gap type were predicted �14� andcreated �10� in BEC loaded in an OL. It was also shown thatmotion of the OL relative to external trap can be used for thedynamical control of matter-wave solitons �15�.

A promising application of the OLs is the stabilization ofmultidimensional solitons in BEC �16�. Rotating OLs arealso available in the experiment �17�. It was predicted that2D solitons can be stabilized by a rotating lattice in either oftwo forms: as a fully localized soliton trapped by a slowlyrevolving lattice and rotating in sync with it, at some dis-tance from the rotation pivot, or a ring-shaped soliton orvortex, supported by a rapidly revolving OL �18�. Phase tran-sitions of vortex matter in rotating lattices were predicted too�19�. In the limit of very rapid rotation, the potential of therevolving OL becomes equivalent to that of a radial lattice�18�. Previously, it had been demonstrated that such OLs ofthe Bessel type support stable 2D rotary solitons becausethey may perform circular motion in an annular potential

trough �lattice ring� �20�. The same model supports stable 3Dsolitons �21�. If the nonlinearity is repulsive, the Bessel ra-dial potential maintains ring solitons in the form of azimuthaldipoles and quadrupoles �22�, and a potential periodic alongthe radius gives rise to radial gap solitons �23�. Experimen-tally, optical spatial solitons localized at the center or form-ing a ring, as well as rotating ones, were created in acylindrical OL �24�.

An essential issue is steering the motion of rotary solitonsin the Bessel lattice. Here, we aim to demonstrate that 2Dsolitons trapped in a Bessel OL at a distance from the centercan be set in rotation by pushing the lattice in a transversedirection, for a finite time �“jerking”�. In other words, theOL’s linear momentum may be effectively converted into theorbital angular momentum of the rotary soliton. After the OLcomes to a halt, the soliton keeps rotating due to inertia. Thismechanism makes it possible to design a “motor” for accu-rate control of motion and transfer of rotary matter-wavesolitons in radial lattices.

II. THE STATIONARY MODEL

The dynamics of the self-attractive BEC in radial poten-tial V�r� is governed by the 2D Gross-Pitaevskii equation�GPE� for the mean-field wave function q�r , t�. The scaledform of the equation is

i�q/�t = �− �1/2��2 + V�r� − �q�2�q , �1�

where �2=�2 /�x2+�2 /�y2 with r2=x2+y2. To cast the GPEin the form of Eq. �1�, one rescales the wave function andother variable as follows: q→q�g2D with g2D=2�2��as� /aL����� /�L�, where as�0 is the s-wave scattering length, ��

is the transverse trap frequency, the characteristic length isaL=e /� �where e is the lattice period�, and the characteristicfrequency is �L=EL /�. The recoil energy is EL=�2 / �2maL

2��m is the mass of the trapped atoms�. The radial potentialwith the minimum at r=0, corresponding to the Bessel OL,is

V�r� = − pJ0��8�x2 + y2�� , �2�

where p�0 is the strength of the lattice, its intrinsic scalewas fixed using the invariance of Eq. �1�, and J0 is the Bessel*stswhz@mail.sysu.edu.cn

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function �18,20�. In this work, we will consider the OL mov-ing along the y axis at velocity −v; to this end, y will bereplaced by y+vt.

First, axisymmetric solutions to Eq. �1� with the quiescentOL�v=0�, in the form of a soliton trapped in the centralpotential well, is looked for as q�r , t�= f�r�exp�ibt�, where −bis chemical potential, and f�r� is a real function obeying theequation f�+r−1f�−2�V�r�+b�f +2f3=0, that can be solvedby the shooting method.

The norm of the axisymmetric soliton solution, U=2��0

�f2�r�rdr, is found to be a monotonically growingfunction of b �Fig. 1�a��, which implies the stability of thesolitons according to the Vakhitov-Kolokolov criterion �25�.Similar to the situation known in the case of 2D solitonssupported by the 2D or quasi-1D OL �16�, the solitons existnot at all positive values of b, but only at b�bCO�0. Thecutoff value bCO grows with the OL strength �Fig. 1�b��, andthe soliton’s norm vanishes at b→bCO �Fig. 1�a��.

To find stationary solitons trapped in the circular troughcorresponding to the first off-center minimum of radial po-tential �2�, at r=rmin�2.47, a stable stationary soliton foundin the central potential well was placed in the trough, and Eq.�1� was then directly integrated in time, demonstrating stablerelaxation to the corresponding 2D soliton �Fig. 1�c��. Theestablished shape of the latter soliton was found to be iden-tical to that which could be found from Eq. �1� by dint of theimaginary-time integration, while the present method pro-vides for a faster convergence to the numerically accuratesolution.

The stability of the axisymmetric solitons against azi-muthal perturbations, which depend on angular variable �,

was examined in a rigorous form, by looking for a perturbedsolution to Eq. �1� in the form of q= �f�r�+u�r�exp�t+ in�+w*�r�exp�*t− in��exp�ibt�, where u�r� and w�r�are eigenmodes of infinitesimal perturbations with integerazimuthal index n and �complex� instability growth rate .The linearization of Eq. �1� leads an eigenvalue problem,which was solved numerically. The stability conditionRe��=0 was explicitly checked for n=0,1 ,2 ,3 ,4 ,5, an in-stability at still higher values of n is very implausible �as alsosuggested by the VK criterion�. The stability was also veri-fied in direct simulations of the evolution of the solitons withadded random perturbations, whose relative amplitude wasset at the 10% level.

III. THE MOVING LATTICE

Proceeding to the model with the radial OL suddenly setin motion at velocity −v along the y axis, we show in Fig. 2that the static soliton placed in the annular trough �the sameas in Fig. 1�c��, starts rotary motion. In this case, the push,with v=1, was applied perpendicular to the radial directionfrom the center to the initial location of the soliton, and thelattice stopped after having passed a short distance, d=1.5�i.e., the lattice was subjected to a “jerk” during short time�=1.5�. An elementary kinematic consideration treating thesoliton as a quasiparticle shows that, after the lattice comesto a halt at t=�, the residual tangential velocity of the solitonis vres=v�1−cos�v� /rmin���0.18. Accordingly, the period ofthe resulting rotary motion is expected to be T=2�rmin/vres�87, which matches the numerical picture in Fig. 2. Takingtypical values of physical parameters relevant to the experi-ment with the BEC of seven Li atoms, viz., transverse trap-ping frequency ��=2��90 Hz and number of atoms 4�105, and undoing rescaling implied in the derivation of Eq.�1�, we conclude that x=1, t=1, and v=1 correspond, inphysical units, to 15 �m, 1.8 ms, and 15 mm/s, respectively.

In a more general situation, the push is applied to the OLwith the soliton trapped at an azimuthal position � �Fig.

FIG. 1. �Color online� �a� Norm of the stationary axisymmetricsoliton trapped in the central potential well versus the absolutevalue of the chemical potential b, for lattice strength p=10; theinset is the radial profile of the soliton at the point marked by circle�b=5.4�. �b� Chemical-potential cutoff bCO �the lower bound for theexistence of the solitons� versus p; the inset represents the radial-lattice potential. �c� Stable evolution of the soliton from panel �a�which was put, at t=0, in the annular trough corresponding to thefirst potential minimum at finite r; 2D density distributions in theright panel corresponds to particular moments of time, as indicatedby the red lines. Below, figures are arranged in the same way.

FIG. 2. �Color online� Stable rotary motion of the soliton�the same as in Fig. 1�c��, triggered by a jerk applied to the lattice.

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3�a��. It is obvious that quadrant pairs �I, IV� and �II, III� inthe figure are equivalent, hence it is sufficient to confine theconsideration to quadrants I and IV. Repeating the abovekinematic analysis which treats the soliton as a quasiparticle,it is easy to derive the following result for the soliton’s re-sidual tangential velocity after the application of the jerk ofduration �=d /v:

vres = 2v sin��d/2rmin�sin ��cos�� − �d/2rmin�sin �� , �3�

vres�0 corresponding to the counterclockwise direction ofthe rotary motion. Equation �3� provides for an accurate de-scription for numerical results collected in Fig. 3�b�, whichdisplays the outcome of the simulations as a diagram inplane �� ,d� for v=1, viz., the counterclockwise and clock-wise rotation in domains A and C, respectively �examples areshown in panels 3�d� and 3�e��. In narrow domains B and D,the soliton does not set in the progressive rotary motion, butrather features small oscillations. The existence of domain B

is readily explained by Eq. �3�: indeed, in the range consid-ered, d 10 �recall rmin�2.47�, it follows from Eq. �3� thatvres vanishes along curve d / �2rmin�= ��−� /2� / sin � �redcurve in Fig. 3�b��. If the soliton comes to a halt after theapplication of the jerk, the finite shift of the soliton in theazimuthal direction can be found too, ��=−�d /rmin�sin �.

The above results were explained within the framework ofthe quasiparticle �adiabatic� approximation for the soliton. Anonadiabatic effect, which manifests the wave nature of thesoliton, is the existence of a threshold value of the latticedisplacement dthr�0.6: a very short jerk, with d�0.6 doesnot set the soliton in progressive motion, generating onlysmall oscillations about the initial position �domain D in Fig.3�b��. The threshold value depends on �, attaining a mini-mum at �=� /4. Of course, the rotary motion cannot be gen-erated by the application of the push in directions �=0 and�=�, i.e., dthr diverges at these points, that is why the dia-gram in Fig. 3�b� is limited to range � /6���5� /6.

A more dramatic nonadiabatic effect is the destruction ofthe soliton if the velocity suddenly applied to the holdinglattice exceeds a critical value vmax, see Fig. 3�f� �in somecases, a part of the initial soliton’s norm may be kept by asmaller soliton which is pushed, along the radial direction,into the central potential well or the adjacent annular trough�.A numerically found dependence of vmax on � is displayed inFig. 3�c�, the maximum of vmax at �=� /4 correlating withthe above mentioned minimum of dthr observed at the samepoint.

Finally, the application of the jerk can be used to acceler-ate, decelerate, reverse, and stop a rotary soliton which wasoriginally moving at velocity v0. In particular, a straightfor-ward generalization of Eq. �3� makes it possible to predictthe duration of the jerk �with velocity v�, which is necessaryto stop the soliton: �= ��0−sin−1�sin �0−v0 /v�� / �sin �0

FIG. 4. �Color online� Examples of the �a� acceleration, �b�deceleration, �c� reversal, and �d� stoppage of an initially movingrotary soliton by the application of the jerk to the radial lattice, withv=1 and d=1.5.

FIG. 3. �Color online� �a� Orientation of the velocity applied tothe radial lattice, with respect to the initial azimuthal position of thesoliton �. �b� Outcomes of the application of the jerk of durationd /v, with v=1: counterclockwise �A�, clockwise rotation �C�, andsmall oscillations �B and D; the red curve is the analytical predic-tion for B�. �c� The maximum value of the velocity suddenly ap-plied to the radial lattice, beyond which the soliton suffers destruc-tion, versus azimuth angle �, for d=1.5. �d� and �e� Examples of thecounterclockwise and clockwise rotation, with d=5.5 and d=1.5.�f� Destruction of soliton with v=2.5 and d=1.5. In �d�–�f�, theinitial azimuthal position of the solitons is �=2� /3.

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−v0 /v�, where �0 is the azimuthal position of the soliton atthe moment of the application of the jerk. Examples of theabove-mentioned outcomes are displayed in Fig. 4.

A possible generalization is to consider the motion of therotary soliton driven by periodic jitter of the radial lattice,y�t�=a sin��t�. In the adiabatic approximation, the respec-tive equation of motion for the soliton in the reference frame

attached to the lattice is rmin�̈=a�2 sin��t�sin �. It is wellknown that this equation may produce both regular and cha-otic dynamical regimes. The same driving technique may beapplied to predicted matter-wave solitons in toroidal traps�26�.

IV. CONCLUSION

We demonstrate that a 2D matter-wave soliton trapped inthe radial lattice can be set in controlled motion, or its mo-tion can be altered as required, by jerking the holding lattice.The use of this “lattice motor” may avoid heating of the BECimplicated by other soliton-transport techniques, such as

dragging by a focused laser beam. If the radial OL traps twosolitons in the same annular trough, the jerk may be used toinitiate collisions between them. It is also relevant to men-tion that, while the present model considered the BEC withattraction between atoms, in the case of repulsion the radialOL may trap stable multipole annular patterns. Due to theirsymmetry, the patterns cannot be set in motion by jerking.However, a more sophisticated approach may be possible inthis case: first, the symmetry of the pattern can be broken bya sufficiently strong jerk, and then another jerk, in the per-pendicular direction, will initiate the rotary motion of thedeformed pattern.

ACKNOWLEDGMENTS

This work was supported by the National Natural ScienceFoundation of China �Grant No. 10674183� and National 973Project of China �Grant No. 2004CB719804�, and Ph. D.Degrees Foundation of Ministry of Education of China�Grant No. 20060558068�.

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