Anderson Localization in Nonlocal Nonlinear Media

0.21 MB·

by  Viola Folli

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