Anderson Localization in Nonlocal Nonlinear Media Viola Folli1,2, Claudio Conti1,2 1Department of Physics, University Sapienza, Piazzale Aldo Moro, 5, 00185, Rome (IT) 2ISC-CNR, Dep. Physics, Univ. Sapienza, Piazzale Aldo Moro 5, 00185, Rome (IT) The effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media is theoretically and numerically investigated. A perturbativeapproach is developed to solve the nonlocal nonlinear Schroedinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states. 2 Disorder and nonlinearity are two leading mechanisms lossofgenerality,we assumethat the horizontalaxis has 1 0 promoting wave localization. On one hand, a sufficient been shifted such that x0 = 0. From H0, one roughly 2 strengthofdisorderfostersthetransitionfromadiffusive finds the link between the eigenvalue andthe averagelo- n regime to a wave-function exponentially decaying over a calization lenght l by a taking the average wavefunction Ja tcohaarsacAtenrdisetriscondislotacanlciezalt;ioans[c1e–n1a0r].ioOcnomthmeonotlyherrefhearnredd, ψha0s: htβhe0ih=ighh(eψst0,dHeg0rψe0e)iof∼=lo−ca1l/izl2a;ttiohne.lowestenergystate 8 in a nonlinear medium, diffraction, or dispersion, can be Highly Nonlocal Limit —Thenonlocalityisdescribedby 1 compensatedbythenonlinearproperties;whenthechar- χ(x), which is typically bell-shaped with a characteris- acteristic lengths of these phenomena (nonlinearity and tic length σ. For an average localization length l much ] s diffraction) are comparable, localized solitary waves, or shorterthanthenonlocalitydegreeσ ofthe medium,the c solitons,emerge. It is wellknownthatnonlocalityin the response function χ(x x) can be expanded aroundthe ti nonlinear response can largely affect the localized wave- localization center, x ′=−0, and one has in (1) p ′ o forms, depending on the degree of nonlocality σ [12–15]. + s. However the interplay between the disorder induced lo- ψ ∞χ(x′ x)ψ(x′)2dx′ =Pχ(x). (2) c calization length l and the characteristic length of non- Z − | | si locality σ has never been considered before. Here we −∞ In this highly nonlocal limit (HNL), the nonlinearity y analyze the effect of a nonlocal nonlinearity on Ander- h son localization; we show that in the framework of the can be treated as an interaction Hamiltonian Hint = p sχ(x)P, where P = +∞ ψ(x)2dx is overal energy, [ highly nonlocal approximation, it is possible to derive −or beam power, which iRs−c∞on|served| during evolution af- closed form expressions to describe the role of nonlin- 1 earity on Anderson localizations, and that these states ter Eq.(1). This limit is valid in the regime of a wave v dominated by a single localization, which, without loss become more stable when the degree of nonlocality in- 3 ofgenerality,istakencenteredatx =0. Thisalsoholds creases, meaning that the power needed to destabilize 0 2 true as far as during the dynamics, additional localiza- 9 them increases with σ/l; this result unveils a fundamen- tion are generated among those located in proximity of 3 tal connection between nonlocality and disorder. 1. Model — The nonlocal nonlinear Schroedinger equation x = 0. Conversely, if two distant localizations are ex- cited, χ(x) will be composed by two nonlocal responses 0 reads as centered in the two Anderson states, this case will be 2 :1 iψt+ψxx =V(x)ψ−sψZ +∞χ(x′−x)|ψ(x′)|2dx′ (1) isntavnesdtaigrdatpederteulsrebwahtieorne.thTeohrey HofNqLuaanltlouwmsmtoecahpapnliycstfhoer v −∞ deriving closed form expressions for the effect of non- i X where ψ = ψ(x,t), V(x) is a random potential and linearity on the Anderson states. We write the field as r χ(x) is the response function of the nonlocal medium anexpansionin P, ψ =√P ψ +Pψ(1)+P2ψ(2)+... , a 0 normalized such that χ(x)dx = 1; s 1 corresponds where we take at the leadin(cid:0)g order ψ = √Pψ to focu(cid:1)s ± 0 to a focusing (s = 1)Ror defocusing (s = 1) nonlin- ontheeffectofnonlocalnonlinearityonthefundamental − earity. Eq. (1) applies to a variety of physical prob- state. We obtain the correction to the Anderson ground lems, including nonlinear optics and Bose Einstein con- state (at second order in P) eigenvalue: densation [16]. We define the “unperturbed” Hamilto- nian as H = ∂2 + V(x); its eigenstates are written χ 2 as H0ψn =0 βn−ψnxwith (ψn,ψm) = δnm. H0 sustains β0(P)=β0−sPχ00+P2X β|0 n0β|n, (3) n=0 − exponentially localized states, corresponding to negative 6 eigenvalues βn. The fundamental state can be approx- with the matrix elements of the nonlocality given by imated by ψ0(x) = √1le−|x−x0|/l, where the average lo- χnm = χ(x)ψn(x)ψm(x)dx. As the degree of nonlocal- calization length l is determined by the strength of the ity increRases, χ(x) can be treated as a constant in χ , nm randompotential V , and x is the location of the eigen- such that χ = χ(0)δ . This shows that the pertur- 0 0 mn nm function with eigenvalue β . In the following, without bation to the Hamiltonian is diagonal, therefore, in the 0 2 HNL, the effect of the nonlocality is to shift the eigen- valuesuchthatβ (P)=β sχ(0)P,whereχ(0)depends 0 0 − onthespecificχ(x), andexplicitlycontainsthedegreeof nittdzhooiaaenbtgilneooononcenaasxlllttiotriucnyera.mntlihTetoelyhuy.setarTtoomobvheeuebrssseaettlaelwlitogHeicetsaahnomlsifrtzieaaHlstttpoe0iosne,ncihaatslenstnaoocHrieetnhAh=teehnnnHedcoep0enrrsl+eeionxsnHeepnaleiocrncicetttaeyoild.s-f Critical Power1122305050 000...01155δβsidual Coupling 5 Re Instability of Anderson states — The effect of nonlinear- 0 5 10 15 20 25 30 35 40 ity on Anderson states becomes relevant when the term Degree of nonlocality linear in P is comparable with β (higher order correc- 0 FIG. 1: (Color online) Left axis: critical power for ex- tions vanish in the HNL); this allows to define through Eq. (3), the critical power Pc : psioannen(tPiacl=(Pcl|β=0|e|xβp0(|(−2σσ2/+l2l))/,Edrafsch(eσd/-ld),otctoedntilninueo)u,sGlianues)-, β quadratic (Pc = |β0|/|χ(0)|(1−l2/2σ2), dots) and rectangu- Pc = +∞ ψ0|(x0)|2χ(x)dx. (4) lfaorr(aPcsi=ng2leσ|rβe0a|l/iz(a1t−ione−o2fσ/tlh),eddiaismorodnedrs;)rriegshpto,nrseesifduunaclticoonus-, | | R−∞ pling δβ Vs degree of nonlocality, averaged over 10 disorder For a defocusing medium (s = 1), this is the power realizations (V0=1). − needed to change the sign of the eigenvalue β (P), from 0 negative to positive; such that the localization is de- stroyed. Conversely, for a focusing medium (s=1), this the system even if nonlinearity is present (however, its can be interpreted as the power were the averagedegree degree of localization may be largely affected). oflocalizationisstronglyaffectedbynonlinearity,indeed Nonlocal Responses — We analyzed a few specific case as l(P)=1/ β(P) , one has of response function χ(x) [12] (rectangular χ = 1/(2σ) ∼ h i p for x < σ, χ = 0 elsewhere; exponential χ(x) = l(P)∼= l (5) e−|x||/σ|/(2σ); Gaussian χ(x) = e−x2/σ2/√πσ2; and 1+sP/hPci quadratic χ(x) = χ(0)+χ2x2). In Fig. 1, we report the p behavior of the critical power as a function of the non- such that at critical power, the localization length is re- locality degree σ for the analyzed response functions. In duced by a factor √2 for the focusing case s = 1, and allofthese cases,the calculatedcriticalpoweris linearly divergesforthe defocusingcases= 1. Thisshowsthat − dependent on the unperturbed eigenvalue of the state. forP >P nolocalizedstatesareexpectedfors= 1,as c − So, the higher the strength of the disorder V0, the lower the corresponding eigenvalue changes sign. This trends β , the higher the power needed to affect the localiza- 0 applies as far as additional effects, like the excitation of tion. Furthermore,weemphasizethatthecriticalpower, further localizations,occur. From Eq.(4), we obtain the depending on the Anderson eigenvalue, has a statistical expression for P in the HNL, P Pˆ = β /χ(0) with Pˆ = σ/2lc2 [χ(0) = 1/2σ focr a≡n excpone|nt0i|al|non-| distribution depending on the disorder configuration. h ci ∼ Numerical Results — We numerically solved Eq.(1) for locality]. Because of the Cauchy-Schwarzinequality, one fixed disorder configurations: we consider a Gaussianly has +∞ ψ0(x)2χ(x)dx < 1, and one readily sees that distributedV(x) with zeromeanandstandarddeviation | | for aR−fi∞nite nonlocality P < Pˆ ; as the nonlocality in- V , first obtain the eigenstates, then by using a pseudo- c c 0 creases the power needed to destabilize the Anderson spectral Runge-Kutta algorithm, we evolve the Ander- states grows. sonlocalizationsinanonlocalmediumwithagivenχ(x) Ausefulmeasuretoquantifytheeffectofanonlocalnon- [anexponentialresponsehereafter,similarresultsareob- linearityonAndersonstatesistheresidualvalueofβ(P) tained for other χ(x)]. Fig. 2 shows the dynamics of the at the critical power P , which can be written as ground-state intensity: for P < P the state remains al- c c most unperturbed (Fig. 2(a), attractive case; Fig. 2(c), β (P ) χ 2 0 c n0 repulsive case). By increasing the power beyond the δβ(σ)= = | | . (6) β0 X χ00 2(1 βn/β0) critical threshold, we observe two different phenomena. n=0| | − 6 In the focusing case, the state becomes more localized As it is determined by the off-diagonal elements χ , δβ Fig. 2(b). At higher powers, a temporal beating pattern n0 can be taken as the “residual coupling” due to the non- can be observed. This is mainly due to the coupling of local nonlinearity, which vanishes in the HNL limit. In the Anderson ground state with other localizations. In Fig.1,wenumericallyshow[foranexponentialχ(x)]that fact, for σ =10 the highly nonlocallimit is not satisfied, δβ goesto zerowhen increasingσ: as the nonlocalityin- and a residual nonlinear coupling between the Anderson creases the nonlinear coupling of ψ with other states is ground state and additional localized modes is present 0 moderated, hence it tends to behave as an eigenstate of (see Fig. 1, 2nd y-axis), and causes the observed oscil- 3 newexperimentalinvestigations,canbeextendedtosev- eral related problems, as quantum phase diffusion and coherence [17], ultrashort pulses in fibers [18], second harmonicgeneration[19]andBose-Einsteincondensation [6, 7], and furnish a novel theoretical framework for the interplay of nonlinearity and disorder [20]. We acknowledge CINECA-ISCRA and the Humboldt foundation. The researchleading to these results has re- ceived funding from the European Research Council un- dertheEuropeanCommunity’sSeventhFrameworkPro- FIG.2: (Coloronline)Evolutionoftheground-stateintensity gram (FP7/2007-2013)/ERCgrant agreement n.201766. for a fixed disorder realization, for σ =10, V0 =10; focusing (attractive) case s = 1, for P = 0.04P (a) and for P = 4P c c (b); defocusing (repulsive) case s = −1, for P = 0.04P (c) c and for P = 4P (d). The superimposed dashed line in the c panels represents theground-state at theinitial time. [1] P. W. Anderson,Phys. Rev.109, 1492 (1958). [2] P. Sheng, ed., Scattering and Localization of Classical 0.5 Waves in Random Media (World Scientific, Singapore, 1990). (a) (b) st [3] D.S.Wiersma, P.Bartolini, A.Lagendijk, and R.Righ- Wai0.4 ini, Nature390, 671 (1997). [4] S. John, Phys. Rev.Lett. 58, 2486 (1987). 0.3 [5] C. ContiandA.Fratalocchi, Nat.Physics4,794(2008). 0 0.5 1 0 0.5 1 [6] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, Evolution Coordinate Evolution Coordinate P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A.Aspect, Nature453, 891 (2008). FIG. 3: (Color online) Wavefunction waist for focusing (a), [7] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, and defocusing (b) cases, for P = 0.04P (dash-dotted line), c M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus- P =0.64P (continuousline),P =2P (dottedline),P =4P c c c cio, Nature 453, 895 (2008). (dashed line), (σ = 10, V0 = 10). Results averaged over 10 [8] I.V.Shadrivov,K.Y.Bliokh,Y.P.Bliokh,V.Freilikher, disorder realizations. andY.S.Kivshar,Phys.Rev.Lett.104, 123902 (2010). [9] Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008). lations. In the defocusing case, we observe the break- [10] T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Na- ingofthe Andersonlocalization,asexpected(Fig.2(d)). ture 446, 52 (2007). Fig. 3 shows the localization length of the ground state [11] V. Folli and C. Conti, Opt.Lett. 36, 2830 (2011). for various powers. In the focusing case, the localization [12] O. Bang, W. Krolikowski, J. Wyller, and J. J. Ras- lengthdecreasesandthebeatingpatternisobserved. For mussen, Phys. Rev.E 66, 046619 (2002). s = 1, the perturbation delocalizes the eigenmode as [13] P. Rasmussen, O. Bang, and W. Kr´olikowski, Physical P >−P . Review E 72, 066611 (2005). c [14] F. Maucher, W. Krolikowski, and S. Skupin, Conclusions — We reported on a theoretical analysis of arXiv:1008.1891 (2010). the effect of a nonlocal nonlinearity on disorder induced [15] C.Conti,M.Peccianti,andG.Assanto,Phys.Rev.Lett. localization. We derive explicit formulas to predict the 91, 073901 (2003). critical power for destabilizing the Anderson states, in [16] V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 quantitative agreement with numerical simulations. We (2010). have shownthat an increasingdegree of nonlocality pro- [17] S.BatzandU.Peschel,Phys.Rev.A83,033826(2011). ducesasubstantialgrowthofthe powerneededtodesta- [18] C. Conti, M. A. Schmidt, P. S. J. Russell, and F. Bian- calana, Phys.Rev.Lett. 105, 263902 (2010). bilize Anderson states, which turn out to be very robust [19] C.Conti,E.D’Asaro,S.Stivala,A.Busacca,andG.As- with respect to nonlinear effects. This can also be ex- santo, Opt.Lett. 35, 3760 (2010). plainedby the fact thatnonlocalityreducesthe coupling [20] C.ContiandL.Leuzzi,Phys.Rev.B83,134204(2011). between Anderson states. These results may stimulate