Scattering of two-dimensional solitons in dipolar Bose-Einst(2)

We analyze the scattering of bright solitons in dipolar Bose-Einstein condensates placed in unconnected layers. Whereas for short-range interactions unconnected layers are independent, a remarkable consequence of the dipole interaction is the appearance of

tons,allowingforacomplexscatteringphysicsbetweenthem.Thisnovelphysicsincludesinelasticfusionintoexcitedsoliton-moleculesforsu cientlyslowsolitons,aswellasstrongsymmetricandasymmetricinelasticreso-nancesforintermediatevelocities.Inaddition,wediscussafundamentallynew2D-scatteringscenarioinmatter-waves,showingthatinelasticsolitonspiralingsimilartothatobservedinphotorrefractivematerials[27,28]ispos-sibleindipolarBEC.Finallyweconsiderthescatteringof1Ddipolarsolitonsinunconnectedwires,andcommentaboutobservabilityinon-goingexperiments.

Inthefollowing,weconsideradipolarBECtransver-sallycon nedinthez-directionbyatwo-wellpotential,withwellslocatedatz=±z0,andseparatedbyasuf- cientlylargepotentialbarrierthatpreventstunnelingbetweenthem.Ateachwellthez-con nementisap-proximatedbyanharmonicpotentialoffrequencyωz,whereasthereisnocon nementonthexy-plane.WeconsiderineachwellaBECofNparticles(amoregen-eralcasewillbeanalyzedlateron)withelectricdipoled(theresultsareequallyvalidformagneticdipoles)ori-entedinthez-directionbyasu cientlylargeexternal eld,andthathenceinteractviaadipole-dipolepoten-tial:Vd( r)=gd(1 3cos2θ)/r3,wheregd=αNd2/4π 0,with 0thevacuumpermittivity,θtheangleformedbythevectorjoiningtheinteractingparticlesandthedipoledirection,and 1/2≤α≤1atunableparameterbymeansofrotatingorienting elds[29].Atsu cientlylowtemperaturesoursystemisdescribedbythefollowingtwocoupledNLSEwithnonlocalnonlinearity[4]:

i

2+Uj(z)+g|Ψj( r)|2+

2m

d r′Vd( r r′)(|Ψ1( r′)|2+|Ψ 1( r′)|2

) Ψj( r),(1)wherej=±1isthelayer-index,Ψjaretheateachwell,Uj(z)=mω2z(z+jz0)2

/2, wavefunctions|Ψj( r,t)|2d r=

1,andg=4π 2

aN/mcharacterizesthecontactinterac-tion,withathes-wavescatteringlength.Inthefollowingweconsidera>0,i.e.repulsiveshort-rangeinteractions.Weassumea2Ddynamicsineachwell.Thisapproxi-mationdemandsthatthecorrespondingchemicalpoten-tialµ ωz.Inthatcase,Ψj( r) ψj(ρ ) j(z)with j(z)theground-statewave-functionoftheharmonicoscillatorinthelayerj.Employingthisfactorization,theconvolutiontheorem,dipole-dipolepotential,V theFouriertransformofthe

d(k)=(4π/3)(3k2integratingoverthez-direction,wearrivez/k2atasystem 1),and

oftwocoupled2DNLSE:

i 2g|ψj|2+4√

where 2m ρ+2πlz3√ (2π)2eikρρn j( kρ)F(kρlz,0)+n j( kρ)F(kρlz,2z0/lz)l2

ψj(2)

z= /mωzistheharmonic-oscillatorlength,n j

2

istheFouriertransformof|ψj(ρ )|2andF(

√

2λ)=

2e λ2

(3

√2π ωzl3

z,andβ=gd/g.Henceβmustbesu cientlylargeandnegative,whichdemandstunability(α<0).ForNa/lz0.12.Asdiscussed 1,stableinRef.solutions[23],theappearDDIformay|β|destabilize>3/8π thesolitonsiftheybecome3D.Inthefollowing,weshallrestrictourselvestothe2Dregime[23].

Weintroduceavariationalformalismwhichallowsustostudytheequilibriumpropertiesanddynamicsofthetwosolitons.WeconsideraGaussianAnsatz[30]:

jη0)2

ψj(ρ ,t)=A

η=

e

(η x,y

qU,(4)

i

whereq{i}=x0,y0,wx,wyarethedynamicalvariables,and

U=

2

w2+

1

x

√

12π2

2

dk3kz

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