Gap solitons in rocking optical lattices and waveguides with undulating gratings
Thawatchai Mayteevarunyoo
1
and Boris A. Malomed
2
1
Department of Telecommunication Engineering, Mahanakorn University of Technology, Bangkok 10530, Thailand
2
Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,
Tel Aviv 69978, Israel
Received 4 May 2009; published 27 July 2009
We report results of a systematic analysis of the stability of one-dimensional solitons in a model including
the self-repulsive or attractive cubic nonlinearity and a linear potential represented by a periodically shaking
lattice, which was recently implemented in experiments with Bose-Einstein condensates. In optics, the same
model applies to undulated waveguiding arrays, which are also available to the experiment. In the case of the
repulsive nonlinearity, stability regions are presented, in relevant parameter planes, for fundamental gap soli-
tons and their two-peak and three-peak bound complexes, in the first and second finite band gaps. In the model
with the attractive nonlinearity, stability regions are produced for fundamental solitons and their bound states
populating the semi-infinite gap. In the first finite and semi-infinite gaps, unstable solitons gradually decay into
radiation, while, in the second finite band gap, they are transformed into more complex states, which may
represent new species of solitons. For a large amplitude of the rocking-lattice drive, the model is tantamount to
that with a “flashing” lattice potential, which is controlled by periodic sequences of instantaneous kicks. Using
this correspondence, we explain generic features of the stability diagrams for the solitons. We also derive a
limit case of the latter system, in the form of coupled-mode equations with a “flashing” linear coupling.
DOI: 10.1103/PhysRevA.80.013827 PACS numbers: 42.65.Tg, 03.75.Lm, 42.82.Et, 05.45.Yv
I. INTRODUCTION
Optical lattices OLs provide a highly efficient tool for
the control of dynamics of collective excitations in Bose-
Einstein condensates BECs1. In the experiment, OLs are
created as interference patterns by coherent laser beams illu-
minating the condensate from opposite directions or by a set
of parallel beams shone through an effectively one- or two-
dimensional 1D or 2D condensate. In photonics, the trans-
mission of light beams may be controlled by counterparts of
OLs in the form of gratings representing a periodic modula-
tion of the refractive index in the direction transverse to the
propagation axis of the probe beam. In particular, in photo-
refractive crystals, gratings may be induced by the interfer-
ence of transverse beams with the ordinary polarization if the
probe beam is launched in the extraordinary polarization 2.
In bulk silica, permanent material gratings can be written by
means of a different optical technique 3. The transverse
structure of photonic-crystal fibers may also be considered,
in a crude approximation, as a pattern of the periodic
refractive-index modulation 4.
In experimental and theoretical studies of BEC, OLs were
found to be especially efficient in supporting matter-wave
solitons. It has been predicted that 2D and three-dimensional
3D OLs can arrest the collapse and thus stabilize solitons,
in the space of the same dimension, in the case of attractive
interactions between atoms 5. Low-dimensional 1D and 2D
lattices, created, respectively, in the 2D 6 or 3D 6,7 space,
may also stabilize multidimensional solitons, allowing them
to move in the unrestricted direction 6. Similarly, a cylin-
drical OL may lend the stability to 2D 8,9 and 3D 10
solitons. The stabilization of matter-wave solitons against the
collapse was also analyzed in the framework of the 1D equa-
tion which combines the OL potential and nonpolynomial
nonlinearity resulting from the tight confinement in the trans-
verse plane 11.
In photonics, lattice structures have been used as a me-
dium for the creation of spatial solitons which would not be
possible otherwise. Notable results are 2+1D fundamental
12, vortical 13, and necklace 14 solitons in photorefrac-
tive crystals equipped with the square-shaped lattice, as well
as solitons supported by a photoinduced circular lattice 15.
Dipole-mode lattice solitons in a photorefractive medium
featuring the self-defocusing nonlinearity were reported too
16. The creation of solitons was also reported in bundled
arrays of parallel waveguides inscribed in bulk silica 17.
Theoretically, various types of spatial solitons were investi-
gated in models of 1D 18 and 2D 4 photonic crystals. The
stability of localized vortices in photoinduced lattices 19,
and of 2+1D solitons in low-dimensional 1D lattices of the
same type 20, was studied too.
For the case of repulsive interactions between atoms in
BEC, it was predicted that stable gap solitons GSs could be
supported by OL potentials, in both 1D 21 and multidimen-
sional 22–26 geometries. Localized modes in the form of
radial gap solitons were predicted in the 2D condensate
trapped in an axisymmetric potential represented by a peri-
odic function of the radial coordinate 9. In the experiment,
a GS was created in the condensate of
87
Rb atoms trapped in
a quasi-1D configuration equipped with the longitudinal OL
1,27. Then, extended confined states were discovered in the
strong OL 28 and explained as segments of a nonlinear
Bloch wave trapped in the OL potential 29.
Another theoretically studied versatile tool for steering
the dynamics of nonlinear excitations in BEC is based on the
periodic time modulation management 30 of various pa-
rameters affecting the condensate, such as the trap’s strength
in the cases of the self-repulsion 31 or attraction 32, and
the periodic modulation of the nonlinearity strength, through
the Feshbach resonance, by a low-frequency ac magnetic
field Feshbach-resonance management, FRM. It was pre-
dicted that FRM may stabilize 2D solitons 33, and also 3D
ones, if combined with the one-dimensional OL potential
34. In the 1D geometry proper, the FRM may support
stable second-order soliton states 35 and multistability
PHYSICAL REVIEW A 80, 013827 2009
1050-2947/2009/801/01382713 ©2009 The American Physical Society 013827-1