Absence of Anderson localization of light in a random ensemble of point scatterers

S.E. Skipetrov1, ∗ and I.M. Sokolov2, †

1Universite Grenoble 1/CNRS, LPMMC UMR 5493, B.P. 166, 38042 Grenoble, France2Department of Theoretical Physics, State Polytechnic University, 195251 St. Petersburg, Russia

(Dated: January 14, 2014)

As discovered by Philip Anderson in 1958, strong disorder can block propagation of waves andlead to the localization of wave-like excitations in space. Anderson localization of light is particularlyexciting in view of its possible applications for random lasing or quantum information processing.We show that, surprisingly, Anderson localization of light cannot be achieved in a random three-dimensional ensemble of point scattering centers that is the simplest and widespread model to studythe multiple scattering of waves. Localization is recovered if the vector character of light is neglected.This shows that, at least for point scatterers, the polarization of light plays an important role inthe Anderson localization problem.

Anderson localization—the appearance and dominanceof localized states in strongly disordered systems—is be-lieved to be a universal phenomenon for all quantum andclassical waves [1–3]. In particular, three-dimensional(3D) disordered systems are expected to exhibit a tran-sition from the “metallic” phase with extended states tothe “insulating” one with localized states, upon increas-ing the disorder [4]. This transition was observed forelectrons in disordered solids [5], ultrasound [6], and coldatoms [7–9]. Reports of Anderson localization of lightin 3D also exist [10–12]. Here we present a theoreticalstudy of light scattering in a 3D ensemble of resonantpoint scatterers (atoms) at random positions. We showthat Anderson localization takes place only in the scalarapproximation and disappears when the vector characterof light is taken into account. Our results raise the issueof the role that polarization effects play in the problem ofAnderson localization of light in general. They suggestthat it might be important to better understand theseeffects in more complex photonic media used in experi-ments: semiconductor [10, 13] or dielectric [11, 12] pow-ders, porous semiconductors [14], or disordered photoniccrystals [15].

The point-scatterer model is useful to understand thegeneric behavior of waves in disordered media [16, 17].In addition, this model is excellent for ensembles of coldatoms which therefore provide a fantastic and practi-cally realizable playground for testing the theory [18].Let us apply the point-scatterer model to study Ander-son localization of light and try to go as far as possiblewithout additional approximations. For concreteness, weassume that the point scatterers are immobile two-levelatoms each having a non-degenerate ground state |gi〉with energy Eg and the total angular momentum Jg = 0and an excited state |ei〉 with Ee = Eg + hω0, Je = 1,and lifetime 1/Γ0 (h is the Planck’s constant and the in-dex i = 1, . . . , N denotes quantities corresponding to theatom i among N atoms). The excited state is thus triplydegenerate and splits in 3 sub-states |eim〉 with differentprojections m = −1, 0, 1 of the angular momentum Jeon the quantization axis z. The system is described by a

standard Hamiltonian [19, 20]

H =

N∑i=1

1∑m=−1

hω0|eim〉〈eim|+∑s⊥k

hck

(a†ksaks +

1

2

)

−N∑i=1

Di · E(ri) +1

2ε0

N∑i6=j

Di · Djδ(ri − rj), (1)

where the first two terms correspond to noninteractingatoms and the free electromagnetic field, respectively, thethird term describes the interaction between the atomsand the field in the dipole approximation, and the last,contact term ensures correct description of the electro-magnetic field radiated by the atoms [19]. Here a†ks andaks are operators of creation and annihilation of a photonhaving the wave vector k and the polarization s, c is thespeed of light in free space, Di is the dipole operator ofthe atom i, and E(ri) is the electric displacement vectordivided by the vacuum permittivity ε0 at the position riof the atom i. Formally solving Heisenberg equations ofmotion for aks, substituting the solution into equationsfor atomic operators, and applying the so-called polarapproximation (i.e. neglecting retardation effects [21]),one obtains a system of equations for the latter opera-tors only, with the coupling between atoms described bythe so-called “Green’s matrix” G [22–24]. It is essentiallybuilt up of Green’s functions of Maxwell equations, de-scribing propagation of light from one atom to another.G is a 3N × 3N random matrix of which a particularrealization is determined by the ensemble of random po-sitions {ri} of N atoms in 3D Euclidean space [22, 23]:

Geimejm′ = iδeimejm′ −2

Γ0(1− δeimejm′ )

×∑µ,ν

dµeimgidνgjejm′

eik0rij

hr3ij

×

{δµν

[1− ik0rij − (k0rij)

2]

−rµijr

νij

r2ij

[3− 3ik0rij − (k0rij)

2]}

. (2)

arX

iv:1

303.

4655

v3 [

phys

ics.

optic

s] 1

3 Ja

n 20

14

2

Here deimgi = 〈Jem|Di|Jg0〉 is the matrix element of the

dipole moment operator Di, rij = ri−rj , and k0 = ω0/c.The superscripts µ and ν denote projections of vectorson axes of the reference frame. Note that Eq. (2) exhibitsa 1/r3ij singularity for rij → 0 which can be related tothe transverse nature of electromagnetic waves.

Any excitation of the ensemble of N atoms coupledthrough the electromagnetic field can be expanded overeigenvectors ψn of the matrix G. The real and imaginaryparts of its eigenvalues Λn yield the frequencies ωn =ω0− (Γ0/2)ReΛn and decay rates Γn/2 = (Γ0/2)ImΛn ofthe corresponding eigenstates. G is therefore the funda-mental object to study in order to understand the behav-ior of collective excitations in the atomic ensemble. Wewill be interested in the spatial localization of ψn andwill compare the properties of the matrix (2) that takesinto account the vector character of light with those ofits scalar approximation

Geiej = iδeiej + (1− δeiej )eik0rij

k0rij(3)

that is often used to further simplify the problem [25, 26].Note that Eq. (3) has a weaker singularity for rij →0 than Eq. (2). Matrices similar to (2) and (3) werepreviously studied in Refs. 27–29.

The vector character of light is considered to be irrele-vant for the Anderson localization problem in a mediumwith a fluctuating dielectric constant [30–32]. How-ever, our model (1) is fundamentally different becauseit does not reduce to the macroscopic Maxwell equationsin the interesting regime of intermediate atomic densityρ ∼ k30. Therefore, significant differences between vectorand scalar cases cannot be excluded beforehand.

We first analyze the density of eigenvalues Λ of theGreen’s matrices (2) and (3) with a particular atten-tion paid to the part of the spectrum corresponding tolong-lived states with ImΛ < 1. Random realizationsof Green’s matrices (2) and (3) were generated by ran-domly choosing N = (2 ÷ 8) × 103 points in a sphereof radius R and volume V [see the inset of Fig. 1(d)].Their eigenvalues Λn and eigenvectors ψn = {ψneim}obeying Gψn = Λnψn were computed for a sufficientnumber of random realizations. The density of eigen-values Λ for N = 4 × 103 is shown in Fig. 1. At lowdensities ρ = N/V , the results obtained for scalar andvector models are similar, most of the eigenvalues beingrestricted to a region delimited by a line following fromthe diffusion theory of light scattering [24, 25]. At den-sities exceeding ρ/k30 ≈ 0.1, however, we observe thatin the scalar model, a significant fraction of eigenvaluescross this line and acquire very small decay rates ImΛ.No such long-lived states appear in the vector model.

To test the intuitive conjecture that the long-livedstates corresponding to eigenvalues with small imaginaryparts may be localized in space, we show in Fig. 2 mapsof the average inverse participation ratio (IPR) for the

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FIG. 1. Density of eigenvalues of the random Green’s matrix.Grayscale density plots of the probability density p(Λ) for Λ’scorresponding to long-lived states (ImΛ < 1). Dashed linesshow the border of the eigenvalue domain following from thediffusion theory and the spiral branches along which eigen-values corresponding to subradiant states are concentrated[24, 25]. Panels (a) and (b) correspond to a low densityof atoms at which the majority of eigenvalues are containedwithin the boundary imposed by the diffusion theory. Panels(c) and (d) correspond to a high density, for which states withvery small decay rates ImΛ appear in the scalar model, butnot in the vector one. The smallest ImΛ of the vector modelis even larger than the prediction of the diffusion theory. Theinset of panel (d) shows N atoms (black dots) randomly dis-tributed in a sphere of radius R.

same parameters as in Fig. 1. IPRn =∑Ni=1 |ψnei |4/

(∑Ni=1 |ψnei |2)2 quantifies the degree of spatial localiza-

tion of the eigenvector ψn. It is of order 1/M for aneigenvector localized on M atoms. In the vector model,each ψnei is a vector with 3 components ψneim and |ψnei |should be understood as its length. As we see from Fig.2, states localized on a small number of atoms exist evenat small densities. They are typically localized on pairsof very closely located atoms and are due to the phe-nomenon of subradiance that does not require multiplescattering and therefore has nothing to do with Andersonlocalization [24–26]. Their eigenvalues are concentratedalong the dashed lines that depict the evolution of thesmallest eigenvalue of a 2× 2 Green’s matrix as the dis-tance between the two atoms is varied. In the scalarmodel, however, localized states of a different type ap-pear at densities larger than ρ/k30 ≈ 0.1. These stateshave very small decay rates, in agreement with Fig. 1.

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FIG. 2. Inverse participation ratio of eigenvectors. Grayscaledensity plot of the average inverse participation ratio (IPR) asa function of the eigenvalue Λ of the corresponding eigenvec-tor. Dashed lines are the same as in Fig. 1. At low density,subradiant states localized on pairs of closely located scat-terers exist in both scalar (a) and vector (b) models. Thesestates have IPR ' 1/2. At high density, Anderson-localizedstates with large IPR appear in the scalar model (c), but notin the vector one (d).

Once again, no such localized states are seen in the vec-tor case.

To convince ourselves that the localized states ap-pearing at large densities in the scalar model are dueto Anderson localization, we perform the scaling anal-ysis [33]. We compute the average dimensionless life-time of eigenstates 〈1/ImΛ〉 = δω−1 and the averagespacing of nearest dimensionless eigenfrequencies ∆ω =〈ReΛn − ReΛn−1〉 for eigenvalues Λn in a strip of unitwidth around ReΛ = −1 where, according to Figs. 1and 2, the localization effects are important in the scalarmodel. The localization transition for light at frequencyω = ω0 + Γ0/2 is expected to take place when theThouless number (also called dimensionless conductance)g = δω/∆ω becomes of order unity [33, 34]. In Fig. 3 weshow g as a function of the bare Ioffe-Regel parameterk0`0, with the on-resonance mean free path `0 calculatedin the independent-scattering approximation (ISA) [16].In the scalar case, the curves g(k0`0) corresponding todifferent N cross at g ≈ 1, k0`0 ≈ 1, as expected fromthe Thouless and Ioffe-Regel criteria of localization [4].A second crossing takes place at much smaller k0`0 (cor-responding to a very large density ρ at which ISA is not

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FIG. 3. Scaling in scalar and vector models. (a) Thoulessnumber g as a function of the bare Ioffe-Regel parameter k0`0for the scalar model at frequency ω = ω0 + Γ0/2 and fordifferent N . The curves cross at g ≈ k0`0 ≈ 1 and then againat k0`0 � 1 and g ≈ 1. Localization transitions take placeat these points, as confirmed by the analysis of the scalingfunction β(g) that changes sign at g ≈ 1 (inset). (b) Thesame for the vector model. Solid lines in the insets are guidesfor the eye.

a good approximation for the mean free path `) and sig-nals the disappearance of localization; the system startsto approach the effective medium regime. The closenessof the latter is manifest in the tendency of eigenvalues Λwith large imaginary parts to concentrate around pointson the complex plane that correspond to quasi-modes of ahomogeneous sphere with some effective refractive index[25]; such a tendency is observed in both the scalar andvector models. A finite-difference estimate of the scalingfunction β(g) = ∂ ln g/∂ ln k0R obtained from all possiblepairs of curves of the main plot is shown in the inset ofFig. 3(a). As expected, β(g) changes sign at g ≈ 1, con-firming Anderson transition in the scalar model. How-ever, none of the above signatures of Anderson localiza-tion is seen in Fig. 3(b) where we present the resultsfor the vector model. g(k0`0) corresponding to differ-

4

ent N do not cross, g always remains larger than 1, andβ(g) > 0 does not change sign, suggesting no localizationtransition.

Let us now elucidate the reasons that prevent Ander-son localization in the vector model. In the scalar ap-proximation, the behavior of our system is analogous tothat of a system of spinless fermions and β(g) exhibitsthe behavior expected for the orthogonal symmetry class(see Ref. [4] for a summary of the symmetry classifica-tion of disordered Hamiltonians). However, because thepolarization of electromagnetic waves does not play ex-actly the same role as the spin of electrons, no directanalogy can be drawn between vector electromagneticwaves and the well-studied case of disordered fermionicsystems. In the system described by Eq. (1), the propaga-tion of elementary excitations from one atom to anothercan be mediated not only by the transverse electromag-netic waves but also by the direct interaction of atomicdipole moments which is accounted for by the longitu-dinal component of the electromagnetic field [16]. Thelatter phenomenon becomes more and more efficient asthe typical distance between neighboring atoms decreaseswith increasing the number density of atoms. The possi-ble importance of resonant dipole-dipole interactions inthe context of Anderson localization was pointed out bySajeev John in the reply [35] to a comment on his paper[32]. Later on, Nieuwenhuizen et al. developed a pertur-bational approach to show that the dipole-dipole interac-tions between atoms in a dilute cloud of two-level atomsyield a small positive correction to the photon diffusioncoefficient D and thus compete with the weak localiza-tion phenomenon that tends to decrease D [36]. How-ever, being limited to low densities of atoms ρ/k30 � 1,this result does not allow one to draw any conclusionsconcerning the fate of energy transport in the interestingregime of high atomic densities. Our calculations go be-yond the perturbation theory of Ref. [36] and show thatat high densities ρ/k30

>∼ 1, the resonant dipole-dipoleinteractions become sufficiently strong to overcome thesuppression of transport due to Anderson localization ef-fects and thereby prevent spatial localization of elemen-tary excitations in the system described by Eq. (1). Thedipole-dipole interactions are discarded in the scalar ap-proximation (3) which explains essential differences be-tween vector and scalar models. It is interesting to notethat the vector character of a wave does not suppress An-derson localization if it is not accompanied by significantmodifications of the near-field behavior. To demonstratethis, we repeated the calculations presented above forelastic waves which, in contrast to the electromagneticcase, can also have a propagating longitudinal compo-nent. The elastic Green’s function exhibits the same1/rij divergence for rij → 0 as the scalar one (3) andour calculations show clear signatures of Anderson local-ization transition, similar to the scalar case [37].

At low densities ρ/k30 � 1, the photon-mediated trans-

port dominates and the dimensionless conductance ofa disordered system of size R is g ∝ M`/R, whereM ∝ (k0R)2 is the number of transport channels [16].Assuming ` = `0 and noticing that R ∝ (N/ρ)1/3, weobtain g ∝ (k0`0)4/3 at a constant N . As can be seen inFig. 3, this scaling is indeed obeyed at k0`0 > 1 for bothscalar and vector models, confirming the transport of en-ergy via the multiple scattering of photons. At higherdensities, corresponding to k0`0 < 1, Anderson localiza-tion suppresses transport in the scalar model, leading tovery small values of g, whereas the non-radiative trans-port channel takes over in the vector model. As followsfrom the approximate scaling g ∝ (k0`0)1/3 observed inFig. 3(b), the mean free path ` is essentially independentof density ρ in this regime.

Our discovery of the absence of Anderson localiza-tion of light in a 3D random ensemble of point scat-terers shows that clouds of randomly distributed coldatoms—for which the Hamiltonian (1) applies providedthat the dipole approximation for light-matter interac-tion is acceptable,— are not suitable for observation ofthis phenomenon. We demonstrated the importance ofthe vector character of electromagnetic waves in the con-text of the Anderson localization problem and elucidatedthe role of resonant dipole-dipole interactions in multiplelight scattering. In addition, our results suggest that thesimple point-scatterer model might not be suitable for de-scription of multiple light scattering in complex photonicmedia like, for example, the media used in recent exper-iments [10–15]. However, the latter can be modeled bygrouping many point scattering centers in clusters repre-senting large dielectric particles which, in their turn, canbe distributed in space randomly or with certain spatialcorrelations. The role of order [38] and long- or short-range correlations [39] in scatterer positions can also bestudied in the framework of the approach developed inthis work.

SES thanks A. Goetschy and B.A. van Tiggelen forfruitful discussions. This work was supported by theFederal Program for Scientific and Scientific-PedagogicalPersonnel of Innovative Russia for 2009–2013 (contractNo. 14.B37.21.1938).

∗ Sergey.Skipetrov@grenoble.cnrs.fr† ims@is12093.spb.edu

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