This document summarizes research on controlling the rotational motion of solitons trapped in a radial optical lattice (Bessel lattice) by applying a finite push to the lattice. Numerical simulations show that a push applied perpendicular to the radial direction can set the initially stationary soliton into circular rotation. The residual rotational velocity after the push can be accurately predicted by treating the soliton as a quasiparticle. More generally, applying a push at different azimuthal positions results in either clockwise or counterclockwise rotation, with the direction determined by the push parameters. Non-adiabatic effects like a threshold push amplitude for initiating rotation and soliton destruction by too strong a push are also examined.

1. PHYSICAL REVIEW A 76, 053601 2007
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, China
3Department 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 precisely
controlled by application of a finite-time push to the lattice, due to the transfer of the lattice’s linear momentum
to the orbital momentum of the soliton. A simple analytical consideration treating the soliton as a particle
provides 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 numbers: 03.75.Lm, 42.65.Jx
I. INTRODUCTION
Techniques which make it possible to manipulate coherent
matter-wave beams by means of specially designed poten-tials
are crucial ingredients in many applications of atom
optics, 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. Rotating
BECs support Abrikosov lattices of quantized vortices 3, a
counterpart of the fractional quantum Hall effect 4, persis-tent
currents 5, and quantum phase transitions 6. Rotating
BECs can be trapped in toroidal magnetic 7 and optical
waveguides 8. Recently, the transfer of angular momentum
from photons to atoms using a two-photon stimulated Raman
process with Laguerre-Gaussian beams was demonstrated to
generate 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 been
reported, including Bloch oscillations 11, Landau-Zener
tunneling 12, and superfluid-insulator transitions 13.
Matter-wave solitons of the gap type were predicted 14 and
created 10 in BEC loaded in an OL. It was also shown that
motion of the OL relative to external trap can be used for the
dynamical control of matter-wave solitons 15.
A promising application of the OLs is the stabilization of
multidimensional solitons in BEC 16. Rotating OLs are
also available in the experiment 17. It was predicted that
2D solitons can be stabilized by a rotating lattice in either of
two forms: as a fully localized soliton trapped by a slowly
revolving lattice and rotating in sync with it, at some dis-tance
from the rotation pivot, or a ring-shaped soliton or
vortex, 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 the
revolving OL becomes equivalent to that of a radial lattice
18. Previously, it had been demonstrated that such OLs of
the Bessel type support stable 2D rotary solitons because
they may perform circular motion in an annular potential
trough lattice ring 20. The same model supports stable 3D
solitons 21. If the nonlinearity is repulsive, the Bessel ra-dial
potential maintains ring solitons in the form of azimuthal
dipoles and quadrupoles 22, and a potential periodic along
the 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 a
cylindrical OL 24.
An essential issue is steering the motion of rotary solitons
in the Bessel lattice. Here, we aim to demonstrate that 2D
solitons trapped in a Bessel OL at a distance from the center
can be set in rotation by pushing the lattice in a transverse
direction, for a finite time “jerking”. In other words, the
OL’s linear momentum may be effectively converted into the
orbital angular momentum of the rotary soliton. After the OL
comes to a halt, the soliton keeps rotating due to inertia. This
mechanism makes it possible to design a “motor” for accu-rate
control of motion and transfer of rotary matter-wave
solitons in radial lattices.
II. THE STATIONARY MODEL
The dynamics of the self-attractive BEC in radial poten-tial
Vr is governed by the 2D Gross-Pitaevskii equation
GPE for the mean-field wave function qr , t. The scaled
form of the equation is
iq/t = − 1/22 + Vr − q2q, 1
L 2
where 2=2 /x2+2 /y2 with r2=x2+y2. To cast the GPE
in the form of Eq. 1, one rescales the wave function and
other variable as follows: q→qg2D with g2D=22as /aL
/L, where as0 is the s-wave scattering length,
is the transverse trap frequency, the characteristic length is
aL=e/ where e is the lattice period, and the characteristic
frequency is L=EL /. The recoil energy is EL=2 / 2ma
m is the mass of the trapped atoms. The radial potential
with the minimum at r=0, corresponding to the Bessel OL,
is
Vr = − pJ08x2 + y2, 2
where p0 is the strength of the lattice, its intrinsic scale
*stswhz@mail.sysu.edu.cn was fixed using the invariance of Eq. 1, and J0 is the Bessel
1050-2947/2007/765/0536015 053601-1 ©2007 The American Physical Society

2. HE, MALOMED, AND WANG PHYSICAL REVIEW A 76, 053601 2007
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 be
replaced by y+vt.
First, axisymmetric solutions to Eq. 1 with the quiescent
OLv=0, in the form of a soliton trapped in the central
potential well, is looked for as qr , t= frexpibt, where −b
is chemical potential, and fr is a real function obeying the
equation f+r−1f−2Vr+bf +2f3=0, that can be solved
by the shooting method.
The norm of the axisymmetric soliton solution, U
f2rrdr, is found to be a monotonically growing
=20
function of b Fig. 1a, which implies the stability of the
solitons according to the Vakhitov-Kolokolov criterion 25.
Similar to the situation known in the case of 2D solitons
supported by the 2D or quasi-1D OL 16, the solitons exist
not at all positive values of b, but only at bbCO0. The
cutoff value bCO grows with the OL strength Fig. 1b, and
the soliton’s norm vanishes at b→bCO Fig. 1a.
To find stationary solitons trapped in the circular trough
corresponding to the first off-center minimum of radial po-tential
2, at r=rmin2.47, a stable stationary soliton found
in the central potential well was placed in the trough, and Eq.
1 was then directly integrated in time, demonstrating stable
relaxation to the corresponding 2D soliton Fig. 1c. The
established shape of the latter soliton was found to be iden-tical
to that which could be found from Eq. 1 by dint of the
imaginary-time integration, while the present method pro-vides
for a faster convergence to the numerically accurate
solution.
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 perturbed
solution to Eq. 1 in the form of q=fr+urexp t
+in
+w*rexp *t−in
expibt, where ur and wr
are eigenmodes of infinitesimal perturbations with integer
azimuthal index n and complex instability growth rate .
The linearization of Eq. 1 leads an eigenvalue problem,
which was solved numerically. The stability condition
Re =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 also
suggested by the VK criterion. The stability was also veri-fied
in direct simulations of the evolution of the solitons with
added random perturbations, whose relative amplitude was
set at the 10% level.
III. THE MOVING LATTICE
Proceeding to the model with the radial OL suddenly set
in motion at velocity −v along the y axis, we show in Fig. 2
that the static soliton placed in the annular trough the same
as in Fig. 1c, starts rotary motion. In this case, the push,
with v=1, was applied perpendicular to the radial direction
from the center to the initial location of the soliton, and the
lattice 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 the
soliton as a quasiparticle shows that, after the lattice comes
to a halt at t=, the residual tangential velocity of the soliton
is vres=v1−cosv / rmin0.18. Accordingly, the period of
the resulting rotary motion is expected to be T=2rmin/vres
87, which matches the numerical picture in Fig. 2. Taking
typical values of physical parameters relevant to the experi-ment
with the BEC of seven Li atoms, viz., transverse trap-ping
frequency =290 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, in
physical units, to 15

3. m, 1.8 ms, and 15 mm/ s, respectively.
In a more general situation, the push is applied to the OL
with the soliton trapped at an azimuthal position Fig.
FIG. 1. Color online a Norm of the stationary axisymmetric
soliton trapped in the central potential well versus the absolute
value of the chemical potential b, for lattice strength p=10; the
inset 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 the
existence 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 the
first potential minimum at finite r; 2D density distributions in the
right panel corresponds to particular moments of time, as indicated
by 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. 1c, triggered by a jerk applied to the lattice.
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4. STEERING THE MOTION OF ROTARY SOLITONS IN… PHYSICAL REVIEW A 76, 053601 2007
3a. It is obvious that quadrant pairs I, IV and II, III in
the figure are equivalent, hence it is sufficient to confine the
consideration to quadrants I and IV. Repeating the above
kinematic 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 of
duration =d/v:
vres = 2v sind/2rminsin cos − d/2rminsin , 3
vres0 corresponding to the counterclockwise direction of
the rotary motion. Equation 3 provides for an accurate de-scription
for numerical results collected in Fig. 3b, which
displays the outcome of the simulations as a diagram in
plane ,d for v=1, viz., the counterclockwise and clock-wise
rotation in domains A and C, respectively examples are
shown in panels 3d and 3e. In narrow domains B and D,
the soliton does not set in the progressive rotary motion, but
rather features small oscillations. The existence of domain B
FIG. 4. Color online Examples of the a acceleration, b
deceleration, c reversal, and d stoppage of an initially moving
rotary soliton by the application of the jerk to the radial lattice, with
v=1 and d=1.5.
is readily explained by Eq. 3: indeed, in the range consid-ered,
d
10 recall rmin2.47, it follows from Eq. 3 that
vres vanishes along curve d/ 2rmin= −/2 / sin red
curve in Fig. 3b. If the soliton comes to a halt after the
application of the jerk, the finite shift of the soliton in the
azimuthal direction can be found too, =−d/ rminsin .
The above results were explained within the framework of
the quasiparticle adiabatic approximation for the soliton. A
nonadiabatic effect, which manifests the wave nature of the
soliton, is the existence of a threshold value of the lattice
displacement dthr0.6: a very short jerk, with d0.6 does
not set the soliton in progressive motion, generating only
small oscillations about the initial position domain D in Fig.
3b. 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. 3b is limited to range /65/6.
A more dramatic nonadiabatic effect is the destruction of
the soliton if the velocity suddenly applied to the holding
lattice exceeds a critical value vmax, see Fig. 3f in some
cases, a part of the initial soliton’s norm may be kept by a
smaller 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 in
Fig. 3c, the maximum of vmax at =/4 correlating with
the above mentioned minimum of dthr observed at the same
point.
Finally, the application of the jerk can be used to acceler-ate,
decelerate, reverse, and stop a rotary soliton which was
originally moving at velocity v0. In particular, a straightfor-ward
generalization of Eq. 3 makes it possible to predict
the duration of the jerk with velocity v, which is necessary
to stop the soliton: =0−sin−1sin 0−v0 /v / sin 0
FIG. 3. Color online a Orientation of the velocity applied to
the radial lattice, with respect to the initial azimuthal position of the
soliton . b Outcomes of the application of the jerk of duration
d/v, with v=1: counterclockwise A, clockwise rotation C, and
small 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 the
counterclockwise 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, the
initial azimuthal position of the solitons is =2/3.
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5. HE, MALOMED, AND WANG PHYSICAL REVIEW A 76, 053601 2007
−v0 /v, where 0 is the azimuthal position of the soliton at
the moment of the application of the jerk. Examples of the
above-mentioned outcomes are displayed in Fig. 4.
A possible generalization is to consider the motion of the
rotary soliton driven by periodic jitter of the radial lattice,
yt=a sint. In the adiabatic approximation, the respec-tive
equation of motion for the soliton in the reference frame
attached to the lattice is rmin ¨
=a2 sintsin . It is well
known that this equation may produce both regular and cha-otic
dynamical regimes. The same driving technique may be
applied to predicted matter-wave solitons in toroidal traps
26.
IV. CONCLUSION
We demonstrate that a 2D matter-wave soliton trapped in
the 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 BEC
implicated by other soliton-transport techniques, such as
dragging by a focused laser beam. If the radial OL traps two
solitons in the same annular trough, the jerk may be used to
initiate collisions between them. It is also relevant to men-tion
that, while the present model considered the BEC with
attraction between atoms, in the case of repulsion the radial
OL may trap stable multipole annular patterns. Due to their
symmetry, the patterns cannot be set in motion by jerking.
However, a more sophisticated approach may be possible in
this case: first, the symmetry of the pattern can be broken by
a sufficiently strong jerk, and then another jerk, in the per-pendicular
direction, will initiate the rotary motion of the
deformed pattern.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China Grant No. 10674183 and National 973
Project of China Grant No. 2004CB719804, and Ph. D.
Degrees Foundation of Ministry of Education of China
Grant No. 20060558068.
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