Three-Dimensional Nonlinear Lattices: From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes

0.28 MB·English

by  R. Carretero-Gonzalez

Three-Dimensional Nonlinear Lattices: From Oblique Vortices and Octupoles to
Discrete Diamonds and Vortex Cubes
R. Carretero-Gonza´lez1, P.G. Kevrekidis2, B.A. Malomed3 and D.J. Frantzeskakis4
1 Nonlinear Dynamical Systems Group, Department of Mathematics & Statistics,
San Diego State University, San Diego CA, 92182-7720
2 Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA
3 Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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4 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
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We construct a variety of novel localized states with distinct topological structures in the 3D
2
discrete nonlinear Schr¨odinger equation. The states can be created in Bose-Einstein condensates

n trapped in strong optical lattices, and crystals built of microresonators. These new structures,
a most of which haveno counterpartsin lower dimensions, range from purelyreal patternsof dipole,
J
quadrupoleandoctupoletypestovortexsolutions, suchas“diagonal” and“oblique” vortices,with

7 axes oriented along the respective directions (1,1,1) and (1,1,0). Vortex “cubes” (stacks of two
2 quasi-planar vortices with like or opposite polarities) and “diamonds” (discrete skyrmions formed
by two vortices with orthogonal axes) are constructed too. We identify stability regions of these

] 3D solutions and compare them with their 2D counterparts, if any. An explanation for the stabil-
t
f ity/instability of most solutions is proposed. The evolution of unstable states is studied as well.
o
s
.
t Introduction. In the past two decades an explosion of nal(wecallthem,respectively,“straight”,“oblique”,and
a
m activity has been observed in the study of intrinsic lo- “diagonal” dipoles). Next, we construct quadrupole and
calizedmodes(ILMs, aliasdiscretesolitons)innonlinear octupolestates,that,similarlytothedipoles,arerealso-
-
d dynamical lattices, especially due to the ability of such lutions. Moresophisticatedstructuresarealsopresented,
n modes to act as energy “hot spots” [1]. The relevance namely “vortex cubes” (concatenations of two straight
o of ILMs has been demonstrated in problems ranging vortices with the same or opposite charges centered on
c
from arraysof nonlinear-opticalwaveguides [2] and pho- parallelplanes), obliqueanddiagonalvortices,and“vor-
[
tonic crystals [3], to Bose-Einstein condensates (BECs) tex diamonds”,formedby a crossedpair ofvorticeswith

1 trapped in optical lattices (OLs) [4] and Josephson- orthogonal axes. Apart from the straight and oblique
v junction ladders [5]. dipoles and quadrupoles, these ILMs have no counter-
0
Auniversalmodel,whichmayariseasanenvelopeap- parts in 2D lattices.
8
proximation from most of the complex nonlinear equa- To present the results, we first introduce the model,
6
1 tions on the lattice and also as direct physical model for and then report systematic numerical results for the
0 theBECs[4]andopticalwaveguidingarrays[6,7],isthe shape and stability of the new localized states. This is
5 discrete nonlinear Schr¨odinger (DNLS) equation [8]. On followedbyconclusions,whichincludeanexplanationfor
0 topofitssignificancetoapplications,theDNLSequation the stability and instability of the majority of patterns
t/ itselfisafundamentallyinterestingdynamicalmodel. In found in this work.
a the 3D case, its direct physical realization is provided, The Model. We consider the DNLS equation on the
m
asmentionedabove,byBECstrappedin strongOLs [4]. cubic lattice with a coupling constant C [12],
- Waveguidearrays,however,cannotbedescribedbya3D
d iφ˙ +C∆φ + φ 2φ =0, (1)
n discrete model, since the evolution variable in the op- l,m,n l,m,n | l,m,n| l,m,n
o tical media is a spatial coordinate, while the temporal with φ˙ =dφ/dt, and the discrete Laplacianis ∆φ
c variable, which effectively plays the role of an additional l,m,n ≡
iv: qotuhaesri-psphaytsiiaclalonreea,liczaantnioontobfethdeisc3rDetDe.NNLSeveeqrtuhaetlieosns,maany- φφll,+m1,,nm−,n1−+6φφl,lm,m+,n1,n. +Soφlul,tmio,nn+s1a+reφlol−ok1,emd,nfo+r aφsl,φml−,m1,,nn +=
X be provided by a crystal built of microresonators [9]. ul,m,nexp(iΛt)withafrequency−Λ(orthechemicalpo-
tentialinthecontextofBEC),wherethestationaryfunc-
r The study ofthe 3DcontinuumNLSequation,includ-
a tions u obey the equation
l,m,n
inga3D[10]orquasi-2D[11]OL,andoftheDNLSmodel
proper [12], has started recently, becoming increasingly Λu =C∆u + u 2u . (2)
l,m,n l,m,n l,m,n l,m,n
accessible to numerical computations. As a result, the | |
first coherent structures, such as discrete vortices of the Profiles used as an initial guess for the fixed-point itera-
topologicalcharge(vorticity)S =1,2and3,wereidenti- tion convergingto solutions displayedbelow, were based
fied and their stability was investigated. The aim of the on the form of the respective solutions (for the same Λ)
presentworkisto studya largevarietyofnovellocalized inthe anti-continuum(AC)limit, C =0. Oncesolutions
3DstructuresintheDNLSequation,manyofwhichturn to Eq. (2) have been obtained, the linear-stability anal-
outtobestable. Inparticular,weconstructstateswhich ysis is performed for a perturbed solution [12], φl,m,n =
include,first,dipoleswiththeaxisorientedalongalattice (cid:2)ul,m,n+ǫ(cid:0)al,m,neλt+bl,m,neλ∗t(cid:1)(cid:3)eiΛt,whereǫisanin-
bond, or along a planar diagonal, or along a 3D diago- finitesimal amplitude of the perturbation, and λ is its
2
eigenvalue. The Hamiltonian nature of the system dic- (a) (b) (c)
∗
tates that if λ is an eigenvalue, then so are λ, λ and 2
∗ − 1 1
λ (in the stable case, λ is imaginary, hence this sym- 1
− n0 n0 n
metry yields only two different eigenvalues, λ and λ). 0
− −1 −1
The stationary solutionis unstable if at leastone pair of 2 2 −1 2
the eigenvalues features nonvanishing real parts. −1l 0 1 −1 0 m1 −1 l 0 1 2−1 0 m1 −1 l 0 1 2−1 0 m1
(d) (e) (f)
Results. Westartbyconstructingpurelyrealsolutions
2 2
of the dipole type (in lower-dimensionalmodels, solitons 1
1
of this type were considered in Refs. [13, 14]). Figure n0 n0 n
0
1 displays generic examples of “tight” dipoles with ad- −1
2 −2 2 −1 2
jisacden=t e1x,ciitnedthseitecsor(rie.esp.,onthdeinsgepuanriatst)ionanbdettwhreeeenptohsesmi- −1 l 0 1 2−1 0 m1 −2 l 0 2−2 0m −1 l 0 1 2−1 0 m1
(g) (h)
ble orientations relative to the lattice: straight (along a 2
2
lattice’s bond), (a), oblique (along a planar diagonal), D,d) 1.5
(b), and diagonal (along a 3D diagonal), (c). In this fig- n0 (3Ccr 1
ure and below, unless stated otherwise, we show a typ- 0.5
−2 2
Λica=l c2a.seTahdembiottridnegrsstoafbtlheesostluatbiiolintsy,wwiinthdoCws,=00.1Cand −2 l 0 2−2 0m 0 1 2 3 4 d 5 6 7
C(3D,d), for the three types (straight, oblique and≤diago≤- FIG. 1: (Color online) Stable multipoles. The top row de-
dip picts stable “tight” dipoles (with d = 1): (a) straight, (b)
nal) of the dipoles are givenby Cd(3ipD−,1s)tr =0.23013±δC, oblique, and (c) diagonal ones. (d) and (e): Quadrupoles in
C(3D,1) = 0.53666 δC, and C(3D,1) = 0.73084 δC, then=0plane,withtheinternalseparationd=1andd=2,
dip−obl ± dip−dia ± respectively. (f) and (g): Octupoles with d = 1 and d = 2.
wheretheerrormarginisδC =0.00001foralltheresults
presented in this work (unless indicated otherwise). We Panel(h)displaysthestabilitythresholdCc(r3D,d)asafunction
of the internal distance d for (from top to bottom) diagonal,
observethatthediagonaldipoleinFig.1(c)remainssta-
oblique,andstraightdipoles,octupolesandquadrupoles. The
ble in a larger interval than its oblique counterparts in
horizontal dashed line corresponds to the stability threshold
Fig. 1(b) that, in turn, is more stable that the diagonal
for the fundamental discrete soliton, C(3D) ≈ 2. Note that,
one in Fig. 1(c). It is also possible to construct dipole fund
for the quadrupole (the bottom graph), C(3D,d) behaves lin-
solutionscorrespondingtod=2,withexcitedsitessepa- quad
early for small d (see the dashed line with slope 0.325 for
ratedbyasinglenearlyemptysite. Suchdipolesolutions
guidance). In panels (a)-(g), level contour corresponding to
(not depicted here) have larger stability windows than
their “tight” (d = 1) counterparts: Cd(3ipD−,2s)tr = 0.66722, Rgrea(yu,l,imn,nth)e=b±la0ck.5-aanrde-swhhoiwtenvienrsbiolune),arnedspreecdti(vdealyr.kAgrlalythaensde
(3D,2) (3D,2) statesarestable(forthecaseshown,withΛ=2andC =0.1).
C = 1.08024, and C = 1.31356. As seen
dip−obl dip−dia
in Fig. 1(h), C(3D,d) further increases with the sepa-
dip
ration distance d, approaching the stability threshold
hasC(2D,1) =0.1485 0.0005[17]. Inthiscase,aswellas
of the fundamental (single-site-based) discrete soliton, quad ±
(3D,∞) (3D) it waswith the dipoles, the 2Dconfigurationstend to be
C C =2.009 0.001 [12], see the horizontal
dip ≡ fund ± slightlymorestablethantheir3Dsiblings. Theoctupole
dashed line in the figure. It can also be found that the
is shown in panel 1(f); it is based on a set of eight con-
same relations, 0 < Cd(3ipD−,dst)r < Cd(3ipD−,do)bl < Cd(3ipD−,dd)ia, as tinuous sites (therefore it is alsoassignedd=1) forming
found for d = 1, are valid for all d, see Fig. 1(h). An- a cubic cell in the 3D lattice. It is stable in the interval
otherrelevantcomparisoniswithdipolesinthe2DDNLS C < C(3D,1) = 0.10030, which is smaller than the above
oct
model, which were studied in Ref. [15]. The comparison ones for the quadrupoles and dipoles. Similar to the
shows that the dipole soliton of the oblique and straight dipoles, multipoles can also “swell” by inserting unpop-
types, which have their counterparts in the 2D case, are ulated sites between the excited ones. The resulting sta-
less stable, although not drastically, than those counter- bilityintervalsford=2arelargerthantheird=1coun-
parts: Cd(3ipD−,1s)tr = 0.23013 < Cd(2ipD−,1s)tr = 0.245± 0.005 terparts: Cq(3uDad,2) =0.503232andCo(3cDt,2) =0.418411,see
for Λ = 2. The weaker stability of the 3D structures is Figs. 1(e) and (g), respectively. It is relevant to mention
explainedbytheanalogywiththemultidimensionalcon- that all the newly found structures have their stability
tinuum NLSequation,where allthe solitonsaredestabi- limits lower than those for the fundamental (single-site-
lized by collapse, which is, respectively, weak and strong based) discrete soliton, C(3D) = 2.009 0.001 [12], see
in the 2D and 3D cases [16]. fund ±
Fig. 1(h).
Quadrupole and octupole solitons are also shown in Stable dipoles have one pair of imaginary eigenvalues
Fig. 1. The quadrupole is based on four contiguous sites in their perturbation-mode spectrum, that, with the in-
(so that we prescribe d = 1 to this structure too) which creaseofC,collideswiththecontinuousspectrum,which
form a square in the plane, Fig. 1(d). It was found to leads to the destabilization. On the other hand, stable
be stable for C <C(3D,1) =0.13836,while its 2D analog quadrupoles have three such pairs (two of them form a
quad
3
(a) (b) (c) (a) (b) (c)
2 2 2 2 2 2
n0 n0 n0 n0
n0 n0
−−22 l 0 2−2 0m 2 −−22 l 0 2−2 0m 2 −−22 l 0 2−2 0m 2 −−22 l 0 2−2 m0 2 −−22 l 0 2−2 0m2 −−22 l 0 2−2 0m2
FIG. 2: (Color online) Vortex cubes for Λ = 2 and C = 0.1. FIG. 3: (Color online) Diagonal and oblique vortices. Panel
Panel(a)showsastable vortexcube,builtoftwoquasi-planar (a) shows an unstable “diagonal” vortex,with theaxis along
vortices,setintheplanesn=±1,withequalvorticities,S1 = thedirection(1,1,1),forΛ=2andC =0.1. Panel(b)shows
S2 = 1, and a phase shift of π. Panel (b) shows an unstable an unstable oblique vortex with the axis oriented along the
cube formed by vortices with opposite charges, S1 = −S2 direction(1,1,0),andpanel(c)showsastable obliquevortex
and panel (c) shows a snapshot (at t=200) of its evolution, of a modified form (see text) for Λ=2 and C =0.01.
clearlydemonstratingthatthephasecoherencebetweensites
forming the pattern is lost. The real level contours are as
in Fig. 1, and the imaginary ones, Im(ul,m,n) = ±0.5, are (a) (b) 2.8
sbhlaocwkn-abnyd-gwrheeitneavnerdsiyoenl)lohwue(sli,grhetspaencdtivveelryy.light gray, in the 2 2|u| 22..46
n0 4 π
doubletforsmallC),andtheoctupoleshaveseven(sixof −−22 l 0 2−2 0m 2 ∆φ(t) − 22 0 ππ0 50 100 150 200 t 250 300 350 400
whichformtwotripletsforsmallC). Moregenerally,the
FIG. 4: (Color online) The vortex diamond (discrete
number ofpotentially unstable eigenvaluepairsisN 1,
skyrmion) for Λ = 2 and C = 0.1. Panel (a) displays an
−
where N is the number of sites on which the structure is unstable “diamond”, and panel (b) depicts its evolution, in
based [18]. terms of the field’s magnitude (top) and phase (bottom) at
The next novel type of a 3D discrete soliton, with no themainsixlatticesites. Initially(t<300),pairsofsites(op-
lower-dimensionalcounterpart,isa“vortexcube”,which posite vertices of the diamond) cyclically change the phase,
isbuiltasastackoftwoquasi-planarvorticeswithequal and subsequently (t > 350) the phase coherence is lost and
topological charges S =S =1 and a phase shift ∆φ= thefieldmagnitudeoscillateserraticallyaboutitsinitiallevel.
1 2
π, separated by an empty layer, so that it has d = 2.
Figure2(a)showsrealandimaginarypartsofthevortex-
cube lattice field. Such a state is stable for 0 C ( 1,1, 1), (0,0, 1), and (1, 1, 1), with the phases
(3D,2) ≤ ≤ a−t them−S (0,α,π−/2,α+π/2,−π,α−+π,3π/2,α+3π/2),
C =0.56324. Onthecontrarytothis,avortexcube
bucuilbt,πas a stack of two in-phase vortices, with ∆φ = 0, where α · tan−1(1/√2). Figure 3(b) depicts such an
≡
is always unstable, through three real eigenvalue pairs. obliquevortex,which,inthis form,isfoundtobealways
Further,Fig.2(b)showsasimilarstack,butcomposedof unstable,similartothediagonalvortex. Nonetheless,the
two vortices with opposite charges, S = S = 1. This obliquevortexcanbestabilizedinamodifiedform,byin-
1 2
configuration is always unstable as well,−due to a real troducing a signshift at the intermediate edge sites [i.e.,
eigenvalue pair. In this case, the instability manifests (0,0,1),( 1,1,0),(0,0, 1)and(1, 1,0)],seeFig.3(c).
− − −
itself as a symmetry breaking between the two planes This sign change avoids contiguous sites with the same
and, hence, the phase coherence of the entire pattern is phase and does not alter the vortex’ topological charge,
eventually lost, see Fig. 2(c). which remains 1. The modified oblique vortex is stable
Another 3D object, with no lower-dimensional analog in a small interval, C <Cv(3oDr−)obl =0.0104.
either, is a vortex with the axis directed along the di- Finally, motivated by the concept of skyrmions [19],
agonal of the cubic lattice, i.e., the vector (1,1,1). Fig- which are distinguished by two topological charges asso-
ure3(a)showssucha“diagonalvortex”constructedbya ciated with closed contours in two perpendicular planes,
continuationprocedurestarting,intheAClimit,withthe we have constructed one more type of vortex structures
following distribution of local phases: φ = 2πSk/6 in the 3D lattice, viz., a “diamond” shown in Fig. 4(a).
l,m,n
(k =0,1,2,...,5),forthesitesthatlieinaplaneorthogo- It is built as a vortex cross, i.e., a nonlinear superposi-
naltotheaxial(diagonal)direction: (1, 1,0),(0, 1,1), tion of two straight site-centered S = 1 vortices, with
− −
( 1,0,1),( 1,1,0),(0,1, 1),(1,0, 1). Obviously,this their axes directed along two orthogonaldirections. The
− − − −
phase pattern bears the vorticity S (S = 1 in Fig. 3). stability analysis shows that, although the diamond has
However,the diagonalvortex turns out to be always un- threeimaginary(stable)eigenvaluepairs,itisalways un-
stable,duetothreerealeigenvaluepairs,anditeventually stable due to a real eigenvalue pair. In direct simula-
settles to a single-site-based soliton. tions,itsinstabilitymanifestsitselfinaratherintriguing
One more species of discrete vortices that may ex- manner, see Fig. 4(b). At t 100, two pairs of oppo-
≈
ist solely in the 3D lattice is an “oblique” one, shown site verticesofthe diamondchangephases(one by+π/2
in Figs. 3(b)-(c), with the axis’ directed along (1,1,0). and the other by π/2) and suffer a momentary mag-
−
In the AC limit, the solution is carried by the array of nitude change. After this, the diamond remains stable
sites (1, 1,0), (1, 1,1), (0,0,1), ( 1,1,1), ( 1,1,0), until t 200, when the same pairs of sites suffer an-
− − − − ≈
4
other phase shift (in the same direction). This process sible to explain the stability/instability of all the struc-
repeats itself almost periodically until (at t 340) the turesthatmayberealizedasboundstatesoftwosimpler
≈
diamond,after4shifts, returnsbacktoitsoriginalphase objects, viz., dipoles, quadrupoles (bound states of two
distribution. Subsequently, the phase shifts accelerate dipoles with opposite orientations), octupoles (bound
and phase coherence is finally lost. Simultaneously, am- states of two quadrupoles), and vortex cubes. Indeed, a
plitude variations of 10% get accumulated, see the top known general principle is that a bound state pinned by
±
panel of Fig. 4(b). Eventually, the solution degenerates thelatticemaybestableonlyifthecoupledobjectsrepel
into a plain single-site-based soliton. each other [15, 20] (i.e., have a phase difference of π be-
Conclusions and discussion. We have introduced sev- tweentheir building blocks). This explains the existence
eralnovelspeciesoftopologicallystructureddiscretesoli- of stability regions for multipoles of all types. Similarly,
tons in 3D dynamicallattices, using the paradigmof the considering the interaction between constituent quasi-
discrete nonlinear Schr¨odinger equation. The solutions planar vortices, one may understand the stability and
have been constructed starting from properly chosen instability of vortex cubes of the types shown in Figs.
anti-continuum approximations, and their linear stabil- 2(a) and 2(b), respectively. Followingthis principle, it is
ity was studied through the computation of the relevant also possible to predict the stability of more exotic 3D
eigenvalues. Previously,onlythefundamentalsingle-site- patterns such as bound states of two oblique or diago-
based solitons and straight vortices, with the axis di- nal dipoles, or octupoles constructed of two such states.
rected along a lattice bond, were known. We have found In those cases when the 3D structures have 2D counter-
three species of dipoles, which differ by the orientation parts,viz.,straightandobliquedipolesandquadrupoles,
relative to the lattice, quadrupoles and octupoles, vor- their stability regions are narrower than in the 2D case,
tex cubes (stacked dual-vortex patterns), diagonal and whichisexplainedbyastrongertrendtocollapseinthree
oblique vortices, and “diamonds” (vortex crosses or dis- dimensions.
crete skyrmions). Except for the straight and oblique
dipoles and quadrupoles, the patterns obtained are en- Future challengesinvolvesemi-analyticalinvestigation
demic to the 3D lattice setting, having no counterparts ofsuchsolutionsviaLyapunov-Schmidttheory,andiden-
in lower dimensions. tification of their stability by means of methods similar
Apart from the diagonal vortices and “diamonds”, all to those developed for the 1D and 2D cases [18]. On the
the patternsconstructedabovehavestabilityregionsbe- experimental side, it may be interesting to create such
low a critical value of the coupling parameter. It is pos- structures in BEC loaded into a strong optical lattice.
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