Solitary vortices and gap solitons in rotating optical lattices
Hidetsugu Sakaguchi
1
and Boris A. Malomed
2
1
Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences,
Kyushu University, Kasuga, Fukuoka 816-8580, Japan
2
Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,
Tel Aviv 69978, Israel
Received 13 January 2009; revised manuscript received 18 February 2009; published 10 April 2009
We report on results of a systematic analysis of two-dimensional solitons and localized vortices in models
including a rotating periodic potential and the cubic nonlinearity, with the latter being both self-attractive and
self-repulsive. The models apply to Bose-Einstein condensates stirred by rotating optical lattices and to twisted
photonic-crystal fibers, or bundled arrays of waveguides, in nonlinear optics. In the case of the attractive
nonlinearity, we construct compound states in the form of vortices, quadrupoles, and supervortices, all trapped
in the slowly rotating lattice, and identify their stability limits fundamental solitons in this setting were studied
previously. In rapidly rotating potentials, vortices decouple from the lattice in the azimuthal direction and
assume an annular shape. In the model with the repulsive nonlinearity, which was not previously explored in
this setting, gap solitons and vortices are found in both cases of the slow and rapid rotations. It is again
concluded that the increase in the rotation frequency leads to the transition from fully trapped corotating
vortices to ring-shaped ones. We also study “crater-shaped” vortices in the attraction model, which, unlike their
compound counterparts, are trapped, essentially, in one cell of the lattice. Previously, only unstable vortices of
this type were reported. We demonstrate that they have a certain stability region. Solitons and vortices are
found here in the numerical form, and, in parallel, by means of the variational approximation.
DOI: 10.1103/PhysRevA.79.043606 PACS numbers: 03.75.Lm, 05.45.Yv, 42.65.Tg
I. INTRODUCTION
Spatially periodic potentials provide for a universal set-
ting which can support two-dimensional 2D solitons and
solitary vortices vortex solitons in nonlinear media where,
in the absence of an external support, localized modes are
either unstable or nonexistent. Two classes of such media in
physics, which are most significant for the ongoing studies,
are represented by Bose-Einstein condensates BECs and
optical waveguides of the photonic-crystal type. In BEC, ef-
fective periodic potentials can be induced by optical lattices
OLs, which are created either by beam arrays shone
through a “pancake-shaped” condensate, or by the interfer-
ence of counterpropagating beams illuminating the pancake
in lateral directions 1. It was predicted that OLs can stabi-
lize 2D as well as three-dimensional 3D matter-wave soli-
tons against collapse in the BEC with attraction between
atoms 2,3. The OL also suppresses the splitting instability
of vortex solitons, which destroys them in the free space,
even if the nonlinearity is too weak to cause the collapse 4.
It is relevant to stress that, although the OL potential breaks
the 2D rotation invariance, topological charge S vorticity,
which determines vortex states, can be defined in the pres-
ence of the lattice. In addition to vortices with S =1, the OL
readily stabilizes higher-order localized vortical modes, with
S 2 S = 2 actually corresponds to quadrupoles, which are
represented by real stationary wave functions, unlike com-
plex solutions for other values of S, see below, and “super-
vortices” ring-shaped chains of compact vortices, each car-
rying individual topological charge s =1, with global
vorticity S = 1 imprinted onto the chain; supervortices with
|S| 1 can exist too5,6.
Quasi-one-dimensional 1D and quasi-2D periodic po-
tentials, i.e., those which do not depend on one coordinate,
are sufficient to support stable 2D and 3D solitons, respec-
tively 7. 2D solitons and localized vortices in anisotropic
OLs, with different strengths of 1D sublattices, were recently
studied too 8. Axisymmetric Bessel potentials 9,10, as
well as circular lattices periodic in the radial direction 11,
stabilize their own species of 2D solitons, as well as their 3D
counterparts 12.
In optics, 2D spatial solitons 13 and vortices 14 were
predicted in photonic-crystal fibers PCFs. Fundamental
spatial solitons were experimentally created in bundled
waveguiding arrays written in bulk silica 15. A powerful
experimental technique, which makes it possible to create 2D
spatial solitons in a different optical setting, is based on the
use of laser-induced lattices in photorefractive crystals 16.
This technique has produced fundamental solitons 17, vor-
tices 18, dipoles 19, necklace arrays 20, and solitons in
a radial photonic lattice 21.
In the case of the self-repulsive cubic nonlinearity 22,
which is more typical to BEC, OLs give rise to gap solitons
GSs12 in any dimension and the corresponding 2D soli-
tary vortices 23. Effectively one-dimensional GSs were
created experimentally in the condensate of
87
Rb atoms 24.
As a tool for the formation and stabilization of dynamical
patterns in the setting featuring additional physics induced
by the Coriolis force, one can use rotating optical lattices
ROLs. In BEC, they have been created by means of a re-
volving mask through which a broad laser beam illuminates
the pancake-shaped condensate 25. In optics, a similar
model applies to twisted PCFs, which are also available to
the experiment 26. The above-mentioned bundled
waveguiding array written in bulk silica 15 may also be
subjected to the twist, thus offering an alternative realization
of the model in optics.
PHYSICAL REVIEW A 79, 043606 2009
1050-2947/2009/794/04360611 ©2009 The American Physical Society 043606-1