Localization of light in a three-dimensional disordered crystal of atoms
LLocalization of light in a three-dimensional disordered crystal of atoms
S.E. Skipetrov ∗ Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France (Dated: September 28, 2020)We demonstrate that a weak disorder in atomic positions introduces spatially localized opticalmodes in a dense three-dimensional ensemble of immobile two-level atoms arranged in a diamondlattice and coupled by the electromagnetic field. The frequencies of the localized modes concentratenear band edges of the unperturbed lattice. Finite-size scaling analysis of the percentiles of Thoulessconductance reveals two mobility edges and yields an estimation ν = 0 . I. INTRODUCTION
Photonic crystals are periodic arrangements of scat-tering units (typically, dielectric spheres or rods) thatexhibit frequency ranges (band gaps) for which no opti-cal modes exist in the infinite structure and light prop-agation is forbidden [1, 2]. Thus, photonic crystals playthe same role for light as semiconductor crystals do forelectrons. They have numerous promising prospects forapplications in optical technologies and, in particular, forguiding of light [3, 4], lasing [5, 6] and quantum optics[7, 8].Photonic crystals exist in nature [9] (e.g., natural opals[10] or wings of some butterflies [11, 12]) or can be fabri-cated using modern nanofabrication techniques [13–16].However, neither nature nor humans do a perfect joband real-life photonic crystals always have some degreeof imperfection: fluctuating sizes or positions of elemen-tary building units, vacancies, interstitial or substitutionimpurities, cracks [17, 18]. Whereas these imperfectionsdo not destroy the band gap provided that they are nottoo strong, they introduce an interesting new feature inthe spectrum: spatially localized optical modes appearin the band gap, especially near its edges [19]. Localiza-tion of eigenmodes of wave equations or of eigenstates ofthe Schr¨odinger equation by disorder is a ubiquitous phe-nomenon discovered by Philip Anderson [20] and bearinghis name [21, 22]. Anderson localization of electromag-netic waves in general and of light in particular has beenpredicted by Anderson himself [23] and by Sajeev John[24]. Later on, it has been observed in fully disorderedone- [25, 26], quasi-one- [27, 28] and two-dimensional[29, 30] disordered media whereas observing it in threedimensions (3D) turned out to be difficult [31, 32]. Eventhough Sajeev John proposed a way to facilitate localiza-tion of light in 3D by using disordered photonic crystalsinstead of fully disordered suspensions or powders a longtime ago [19, 33], no clear experimental realization of thisidea has been reported up to date. Some signatures ofAnderson localization have been observed in reflection ofshort optical pulses from a disordered photonic crystal[34] although the authors did not claim the observationof Anderson localization.The idea of facilitating localization of light in 3D by us- ing a photonic structure with a band gap arises from thelocalization criterion following from the scaling [35] andthe self-consistent [36, 37] theories of localization [38]: N EM ( ω ) D ( ω ) (cid:96) ∗ ( ω ) (cid:46) const ∼ , (1)where N EM ( ω ) is the density of electromagnetic modes(states), D ( ω ) = v E (cid:96) ∗ ( ω ) / v E isthe energy transport velocity [39, 40], and (cid:96) ∗ ( ω ) is thetransport mean free path in the absence of localizationeffects. In a fully disordered isotropic medium withoutany short- or long-range order, N EM ( ω ) ∼ k ( ω ) /v E andwe obtain the standard Ioffe-Regel criterion of localiza-tion: k(cid:96) ∗ ∼ k(cid:96) (cid:46) const ∼
1, where k ( ω ) is the effectivewave number, (cid:96) is the scattering mean free path, andwe made use of the fact that (cid:96) ∗ and (cid:96) are of the sameorder. This criterion corresponds to a very strong scat-tering with (cid:96) shorter than the wavelength of light. If,however, the density of states N EM ( ω ) is suppressed withrespect to its value in the fully disordered medium, thecriterion (1) becomes easier to obey. In a photonic crys-tal, N EM ( ω ) → a r X i v : . [ c ond - m a t . d i s - nn ] S e p is a possible way to circumvent this obstacle [51, 52] butstrong fields are required [53]. An easier way towardslight localization by cold atoms may be to arrange atomsin a periodic 3D lattice and enjoy the relaxation of thelocalization criterion (1) near an edge of a photonic bandgap.In this paper, we investigate spatially localized quasi-modes that are introduced in an open 3D diamond atomiclattice of finite size by a randomness in atomic positions.Randomly displacing the atoms from their positions inthe lattice is different from introducing disorder by ran-domly removing the atoms—a situation studied in Ref.54—and allows for varying the strength of disorder whilekeeping the atom number constant. Thus, we can followa transition from the perfect photonic crystal for vanish-ing disorder to a fully disordered system for strong dis-order. After discussing the impact of boundary states,we establish that for a moderate amount of disorder W ,two localization transitions exist near edges of a photonicband gap that the diamond lattice exhibits. A finite-sizescaling analysis of one of these transitions yields the pre-cise position of the mobility edge and an estimation of thecritical exponent ν of the localization length. Increasing W eventually leads to the closing of the band gap and thedisappearance of localized states. A relation between theband gap formation, Anderson localization, and the near-field dipole-dipole coupling between the atoms is conjec-tured. Finally, implications of our results to experimentswith cold atoms are discussed. II. THE MODEL
We consider N identical two-level atoms arranged ina diamond lattice. The lattice is a superposition oftwo face-centered cubic lattices (lattice constant a ) withbasis vectors e = (0 , a/ , a/ e = ( a/ , , a/ e = ( a/ , a/ ,
0) and e + e , e + e , e + e , where e = ( a/ , a/ , a/ L and volume V = ( π/ L centered at the origin (see the inset of Fig. 1 for a 3D ren-dering of the resulting sample). Disorder is introduced bydisplacing each atom by a random distance ∈ [0 , W a ] in arandom direction, with W being a dimensionless param-eter characterizing the strength of disorder. The atomshave resonance frequencies ω and resonance widths Γ ;their ground states have the total angular momentum J g = 0 while their excited states have J e = 1 and are thusthree-fold degenerate, with the three excited states hav-ing the same energies but different projections J z = m ( m = 0, ±
1) of J e on the quantization axis z . We havealready used such a model of resonant two-level atomscoupled via the electromagnetic field to study randomensembles of atoms in our previous work [49] where theHamiltonian of the system was given. The model wasgeneralized to include external dc magnetic [51, 52] orelectric [55] fields. It has been also used to study pho- tonic crystals that we consider here [54, 56]. Followingthese previous works, we will study localization proper-ties of quasimodes ψ m of the atomic system found aseigenvectors of a 3 N × N Green’s matrix ˆ G :ˆ G ψ m = Λ m ψ m , m = 1 , . . . , N. (2)The matrix ˆ G describes the coupling between the atomsvia the electromagnetic waves (light) and is composedof N × N blocks of size 3 ×
3. A block ˆ G jn gives theelectric field created at a position r n of the atom n by anoscillating point dipole at a position r j of the atom j ( j , n = 1 , . . . , N ). It has elements G µνjn = iδ jn δ µν + (1 − δ jn ) 32 e ik r jn k r jn × (cid:34) P ( ik r jn ) δ µν + Q ( ik r jn ) r µjn r νjn ( r jn ) (cid:35) , (3)where P ( x ) = 1 − /x + 1 /x , Q ( x ) = − /x − /x , r jn = r n − r j , and the indices µ , ν = x, y, z denote theprojections of r jn on the axes x , y , z of the Cartesiancoordinate system: r xjn = x jn , r yjn = y jn , r zjn = z jn . Theinverse of the resonant wave number of an isolated atom k = ω /c provides a convenient length scale by whichwe will normalize all other length scales. Here c is thespeed of light in the free space.An eigenvector ψ m = ( ψ m , . . . , ψ Nm ) T of the matrixˆ G describes the spatial structure of the m -th quasi-mode: ψ j − µm gives the µ -th component of the elec-tric field on the atom j . The corresponding eigenvalueΛ m yields the eigenfrequency ω m and the decay rateΓ m / ω m = ω − (Γ / m andΓ m / / m . Spatial localization of quasimodescan be quantified by the so-called inverse participationratio (IPR):IPR m = N (cid:88) j =1 (cid:40) (cid:88) µ =1 (cid:12)(cid:12)(cid:12) ψ j − µm (cid:12)(cid:12)(cid:12) (cid:41) , (4)where we assume that the eigenvectors ψ m are normal-ized: N (cid:88) j =1 3 (cid:88) µ =1 (cid:12)(cid:12)(cid:12) ψ j − µm (cid:12)(cid:12)(cid:12) = 1 . (5)It is easy to see that IPR m = 1 for a state localized on asingle atom and IPR m = 1 /N for a state that is uniformlydelocalized over all N atoms of the system. Generally,IPR m ∼ /M for a state localized on M atoms.The spectral distribution of quasimodes can be char-acterized by the density of states (DOS) N ( ω ) defined inan open system as [56, 57] N ( ω ) = 13 N π N (cid:88) m =1 (Γ m / ω − ω m ) + (Γ m / . (6) N ( ω ) is normalized such that the number of states in-side an infinitely narrow frequency interval dω is dN = FIG. 1. (a) Density of states of perfect ( W = 0, black) anddisordered (red, green, blue) photonic crystals for differentdisorder strengths: W = 0 . W = 0 .
1, 0.2, and the fully random case, respectively. Verti-cal dashed lines show band edges. Inset: A 3D rendering ofa perfect diamond lattice of atoms. (b) Zoom on the bandgap. Yellow shading shows frequency ranges in which we findlocalized quasimodes for W = 0 . N N ( ω ) dω . Thanks to such a normalization, N ( ω )converges to a limiting shape corresponding to the infi-nite crystal as the size of the crystal increases [56]. Notethat in our formalism, the number of quasimodes is equalto the size 3 N of the matrix ˆ G and hence increases with N for all frequencies, including those inside the bandgap. However, as discussed elsewhere [56], the quasi-modes corresponding to the frequencies inside the bandgap are confined near the crystal boundary and hencetheir number grows proportionally to the crystal surface πL ∝ N / . This growth is slower than the growth ofthe total number of modes and hence the relative weightof these quasimodes tends to zero in the thermodynamiclimit N → ∞ and N ( ω ) ∝ /L [56, 58].In this paper, we will present results for crystals offour different sizes k L = 30, 40, 50 and 60 composed of N = 2869, 6851, 13331 and 22929 atoms, respectively.These numbers of atoms have been adjusted to maintainthe same lattice constant k a = 3 . ρ/k = 0 .
2. The lattice constantis chosen small enough for a band gap to open in thespectrum of the ideal lattice [59] as we illustrate by DOScalculations shown in Fig. 1 for the perfect ( W = 0)and disordered crystals of size k L = 50. For disor-dered lattices, DOS has been averaged over many inde-pendent random atomic configurations using the MonteCarlo method [60]. DOS inside the band gap is differentfrom zero due to the finite size of the considered sample[56, 58]. We observe that the band gap narrows whendisorder in atomic positions is introduced ( W = 0 .
1) andcloses for strong enough disorder ( W = 0 . r j are chosen randomly insidea sphere without any reference to the periodic diamondstructure. Therefore, it turns out that our disorderedphotonic crystal preserves a band gap only for relativelyweak disorder W < . N ( ω ) reflects onlythe atomic component of elementary excitations of thesystem comprising the atoms and the electromagneticfield. Thus, low N ( ω ) does not necessarily correspondto a small number of excitations at a given frequency ω but can simply mean that the atomic subsystem isweakly involved and the excitations look very much likefreely propagating photons. This typically happens farfrom the atomic resonance, for | ω − ω | (cid:29) Γ , where thecoupling of light with atoms is inefficient. The absenceof free-field solutions that have no atomic component forfrequencies inside the band gap has been demonstratedpreviously [59, 61]. A gap in N ( ω ) thus corresponds to agap in the total density of states and a gap in the densityof electromagnetic modes N EM ( ω ) entering the localiza-tion criterion (1), even though N ( ω ) (cid:54) = N EM ( ω ).In addition to DOS N ( ω ), another interesting quan-tity is the local density of states (LDOS) N ( ω, r ). In aphotonic crystal of finite size, LDOS exhibits rapid spa-tial variations within each unit cell of the crystal andslow overall evolution with the distance to the boundaries[62, 63]. Disorder introduces fluctuations of LDOS andthe statistics of the latter may serve as a criterion for An-derson localization [64]. However, calculation of LDOSfor our model would require finding eigenvectors ψ m ofthe matrix ˆ G which is a much more time-consuming com-putational task than finding the eigenvalues Λ m that areneeded to calculate N ( ω ) [see Eq. (6)]. Even though wepresent some results for ψ m in Figs. 2–4 below, their sta-tistical analysis including the calculation of the averageLDOS (cid:104)N ( ω, r ) (cid:105) is beyond the scope of this work. III. LOCALIZED STATES INSIDE THE BANDGAP
It follows from Fig. 1(b) that some quasimodes crossover the edges of the band gap when disorder is intro-duced in the photonic crystal (compare DOS correspond-ing to W = 0 and W = 0 . FIG. 2. Eigenvalues of a single realization of the Green’s matrix for perfect [left column, panels (a) and (c)] and disordered[right column, panels (b) and (d)] diamond crystals of two different sizes k L = 40 (upper row) and 60 (lower row). Each pointin the graph corresponds to an eigenvalue and its grey scale represents the IPR of the corresponding eigenvector, from lightgrey for IPR = 0 (extended eigenvectors) to black for the maximum IPR (different for each panel, most localized eigenvectors).Vertical dashed lines show band edges. Only a part of the eigenvalue spectrum ( ω − ω ) / Γ ∈ [ − ,
2] is shown. localization properties of these modes, we show quasi-mode eigenfrequencies ω and decay rates Γ together withtheir IPR for the perfect diamond crystal and a single re-alization of the disordered crystal in Fig. 2. For the per-fect crystal [left column of Fig. 2, panels (a) and (c)], thevast majority of the modes both inside and outside theband gap are extended and have IPR ∼ /N . The distri-bution of quasimodes on the frequency-decay rate planechanges only slightly upon increasing the size of the sys-tem from k L = 40 to k L = 60 [compare panels (a) and(c) of Fig. 2]. In contrast, the disordered photonic crys-tal exhibits some localized modes with appreciable IPRnear band edges and in particular near the upper bandedge, see Fig. 2(b) and (d). These modes have decayrates (life times) that are significantly smaller (longer)than the decay rates (life times) of any modes of the per-fect crystal. In addition, the number of such localizedmodes increases and their decay rates decrease signifi-cantly when the disordered crystal gets bigger [comparepanels (b) and (d) of Fig. 2]. Such a combination ofspatial localization with small decay rates and the scal-ing with the sample size is typical for disorder-inducedquasimode localization [49, 65].In addition to extended modes everywhere in the spec- trum, isolated localized modes appear in the middle ofthe band gap of the perfect crystal [see Fig. 2(a) and(c)]. Their IPR ∼ × − is small but still considerablylarger than 1 /N ∼ − expected for extended modes.Such modes do not disappear and become even more nu-merous in the disordered crystal [see Fig. 2(b) and (d)].They differ from the modes near band edges by theirmuch larger decay rates that are virtually independentof the crystal size. Our previous work suggests that allmodes in the middle of the band gap of a photonic crys-tal are confined near the crystal boundary, which mayexplain their IPR ∝ /N / (cid:29) /N [56]. In the presenceof disorder, some of these modes may, in addition, be re-stricted to a small part of sample surface [66], which mayexplain their larger IPR. To confirm this explanation, wecompute the center of mass of a mode ψ m as r ( m )CM = N (cid:88) j =1 r j (cid:34) (cid:88) µ =1 | ψ j − µm | (cid:35) . (7)Figure 3 shows that the modes in the middle of the bandgap, including those having large IPR, tend to have theabsolute value of their center of mass r CM to be of orderof the radius L/ FIG. 3. Eigenvalues of a single realization of the Green’smatrix for a disordered diamond crystal of size k L = 30.Each point in the graph corresponds to an eigenvalue and itsgrey scale represents the IPR (a) or the center of mass r CM (b) of the corresponding eigenvector. The spatial structure ofthe eigenvectors corresponding to the two eigevalues indicatedby arrows in panel (a) is illustrated in Fig. 4. Vertical dashedlines show band edges. Only a part of the eigenvalue spectrum( ω − ω ) / Γ ∈ [ − ,
2] is shown. therefore confined at the sample boundary as we have an-ticipated. The confinement at the boundary explains therelatively large decay rates of these modes and the weakdependence of decay rates on the sample size. Althoughthe role of surface modes discussed above may appear tobe important in the calculations presented in this work,this is due to the relatively small sizes k L = 30–60 ofconsidered atomic samples limited by the computationalconstraints to which our numerical calculations are sub-jected. In the limit of k L → ∞ relevant for the analysisof modes localized by disorder in the bulk, surface modesplay no role. In finite samples accessible to numericalcalculations, the impact of surface modes can be mini-mized by using a scaling analysis presented in the nextsection. The need for a scaling analysis is also due to theabsence of a univocal relation between the decay rate Γof a quasimode and its localization properties. Indeed,some of the black points in Fig. 3(a) correspond to muchlarger Γ than some of the grey points, showing that the FIG. 4. Visualization of eigenvectors (quasimodes) corre-sponding to the eigenvalues indicated by arrows in Fig. 3(a).A quasimodes ψ m is represented by N red spheres centeredat the locations r j ( j = 1 , . . . , N ) of the N atoms and havingradii proportional to the intensities I jm = (cid:80) µ =1 | ψ j − µm | of the quasimode on the atom. The quasimode (a) is spatiallylocalized and has a relatively high IPR whereas the quasimode(b) is spatially extended. Grey spheres in both panels visual-ize the spherical region occupied by the disordered photoniccrystal. IPR and Γ are not directly related. However, a relationcan be established between the scaling of (normalized)Γ with the sample size L and the spatial localization ofquasimodes at a given frequency. Surface states do notfollow the same scaling with the sample size as the modeslocalized in the bulk, which provides a way of discrimi-nating between these two types of modes.Similarly to the panels (b) and (d) of Fig. 2, Fig. 3(a)shows that quasimodes with large IPR appear inside theband gap of the photonic crystal due to disorder. Thespatial structure of these spatially localized quasimodesis very different from that of the extended quasimodeswith frequencies outside the band gap, as we illustrate inFig. 4. IV. FINITE-SIZE SCALING
The finite-size scaling analysis is a way to access the be-havior of an infinitely large system from the experimentalor numerical data available for finite-size systems only. Itis a common approach for analysis of phase transitions[60, 67] and has been widely used to characterize Ander-son localization transitions in electronic [68–71], optical[52, 72] and mechanical [73, 74] systems. Very gener-ally, one chooses a quantity (let’s denote it by Ω) thatis supposed to take two very different values (say, 0 and ∞ ) for the infinitely large system in the two differentphases. The behavior of the quantity Ω is then studiedas a function of sample size L and regions of the parame-ter space are identified in which Ω increases or decreaseswith L . A point (for 1D parameter space), a line (2D), ora surface (3D) separating these regions is identified as aboundary between the two phases at which Ω is indepen-dent of L . Moreover, it often turns out that even whenthe parameter space of the physical system under con-sideration is multidimensional, all the parameters can becombined into a single one that is the only relevant nearthe phase transition point. In this situation known asthe ‘single-parameter scaling’ [68], the critical exponentsof the transition can be estimated from the behavior ofΩ with L for finite L .In the context of Anderson localization, the (dimen-sionless) electrical conductance g of a sample of size L was identified as the most relevant quantity to consider:Ω = g [35]. Obviously, the conductance of a 3D metalliccube of side L in which all the electronic eigenstates areextended, g ∝ L , grows with L whereas one expects adecreasing conductance g ∝ exp( − L/ξ ) if the electroniceigenstates are localized at the scale of localization length ξ and the sample is an (Anderson) insulator. We thus seethat in the limit of L → ∞ , g → ∞ if the eigenstates areextended and g → g to be independent of L at thecritical point [35]. This is, by the way, the essence of theThouless criterion of Anderson localization g ∼ const[35, 75], where ‘const’ is a number of order unity.The conceptual picture described above needs some ad-justments when it comes to its application to the phys-ical reality. Indeed, in a disordered system, g is a ran-dom quantity and it is not clear how exactly its scalingwith the sample size should be understood [76]. Thesimplest option of analyzing its average value (cid:104) g (cid:105) turnedout to be not always appropriate because (cid:104) g (cid:105) may bedominated by rare realizations of disorder with large g [76, 77]. Another, more intelligent guess is to use theaverage of the logarithm of g , (cid:104) ln g (cid:105) . This indeed allowsto obtain reasonable results [70] but has the weaknessof being somewhat arbitrary as a choice: why (cid:104) ln g (cid:105) andnot (cid:104) (ln g ) (cid:105) , (cid:104) (ln g ) (cid:105) or the mean value of some otherfunction of g ? Although averaging different functions of g may yield identical results for the critical propertiesof the localization transition in some models [70], it isnot so for the model of point scatterers considered here [65]. This is why studying the full probability distri-bution function P ( g ) instead of statistical moments of g or ln g is necessary [76, 77]. Conductance g and itsprobability distribution function P ( g ) are not the onlyquantities that can be used for the scaling analysis of theAnderson transition. Alternatives include the distribu-tion of eigenvalue (level) spacings [69] or the multifractalspectrum [78] as the most prominent examples. Notethat although initially proposed for Hermitian systems[69], the finite-size scaling of spacings between eigenval-ues has been recently extended to the non-Hermitian case[79, 80] and thus can, in principle, be applied to analyzeopen disordered systems as the one considered in thiswork. However, g and P ( g ) still remain the most simpleand computationally accessible quantities to analyze.Conductance as a ratio of the electric current to thevoltage that causes it, is a notion that is proper to elec-tronics and seems to be impossible to generalize to light.However, Thouless has noticed that if one divides the typ-ical decay rate Γ / ω , the resulting ‘Thouless conductance’ isequal to the electrical conductance g for a metal wire:(Γ / / ∆ ω = g [75]. The advantage of the Thouless defi-nition is that it can be readily generalized to any wavesindependent of any electrical currents or potential differ-ences in the considered physical system. In our open,finite-size photonic crystal we define g m = Γ m / (cid:104)| ω m − ω m − |(cid:105) = ImΛ m (cid:104)| ReΛ m − ReΛ m − |(cid:105) , (8)where the eigenfrequencies ω m are assumed to be or-dered. We note that in a closed system the matrix ˆ G would be Hermitian and its eigenvalues real. Then thedenominator of Eq. (8) would be equal to 1 / [3 N N ( ω )].However, in the open system that we consider, the rela-tion between the average spacing between eigenfrequen-cies ω m and DOS is only approximate because the defi-nition of DOS (6) involves decay rates of quasimodes aswell. In practice, we can still approximately write g m (cid:39) Γ m N ( ω m )3 N. (9)Using this definition instead of Eq. (8) would barely mod-ify the results following from the finite-size scaling anal-ysis below because neither (cid:104)| ω m − ω m − |(cid:105) − nor N ( ω )exhibit singularities at the localization transition points.In a disordered photonic crystal, the Thouless conduc-tance defined by Eq. (8) is a random quantity and atfixed scatterer density ρ and disorder strength W , itsstatistical properties can be characterized by a proba-bility density function P (ln g ; ω, L ). Here we choose towork with ln g instead of g because g varies in a ratherwide range. The probability density is parameterized bythe frequency ω of the quasimodes and the sample size L .We estimate P (ln g ; ω, L ) for different ω around the upperedge of the band gap observed in Fig. 1 by numericallydiagonalizing many independent random realizations ofthe matrix ˆ G for different sizes L of the disordered pho-tonic crystal. Figure 5 shows the results for W = 0 . ω = ω − . for whichthe so-called Harald Cram´er’s distance between probabil-ity density functions corresponding to the smallest andlargest L is minimized (see the inset of Fig. 5). The Har-ald Cram´er’s distance is D ( ω ) = ∞ (cid:90) −∞ d (ln g ) (cid:12)(cid:12) P (ln g ; ω, L = 30 k − ) − P (ln g ; ω, L = 60 k − ) (cid:12)(cid:12) . (10)Interestingly, the frequency ω for which D ( ω ) is min-imal also corresponds to the frequency for which distri-butions P (ln g ; ω, L ) corresponding to different L tend tocoincide for small g , see the main panel of Fig. 5. Fol-lowing our previous work [65], we identify this relative L -independence of P (ln g ; ω, L ) as a signature of a criticalpoint of a localization transition (also called a mobilityedge). The probability density of conductance near thetransition from extended to localized states has been ex-tensively studied in the past for both quasi-1D [81, 82]and 3D [83–85] disordered systems without band gaps.For small g , our P (ln g ; ω, L ) exhibits a tail decreasingto zero as g → P (ln g ; ω, L ) does not have a smooth, size-independent shape for large g . We attribute this fact tothe following reason. The realistic physical model of two-level atoms arranged in a diamond lattice that we con-sider, may exhibit other physical phenomena in additionto the eigenmode localization near band edges. Thesephenomena may be due, for example, to the collectiveinteraction between atoms (sub- [86–88] and superrradi-ance [89, 90]) or to the specific structure of their spatialarrangement (potential topological phenomena [91, 92]).Without having any relation to quasimode localization,these phenomena may cause some particular features of P (ln g ; ω, L ) and exhibit some L -dependence. Some ofthese features may disappear in the limit of k L → ∞ but it is impossible to claim such a disappearance fromour calculations performed for finite k L = 30–60, whichare likely to be insufficient to clearly observe the behaviorexpected in the limit of k L → ∞ . For example, we seefrom Fig. 5 that P (ln g ; ω, L ) exhibits a pronounced peakat ln g (cid:38)
5. The peak shifts to larger g and reduces inmagnitude as L increases. This peak corresponds to su-perradiant states with short lifetimes which always existin a finite-size system but which have a statistical weightdecreasing with L . It is likely that the peak would vanishin the limit of L → ∞ which is, however, inaccessible forour calculations.We will use the small- g part of P (ln g ; ω, L ) that be-comes L -independent at ω (cid:39) ω − . (see Fig. 5),to quantify the localization transition. The finite-sizescaling analysis of P (ln g ; ω, L ) can be conveniently per-formed by analyzing its percentiles ln g q [93]. The q -th FIG. 5. Probability density of the logarithm of the Thoulessconductance g at the critical point of the localization tran-sition for different sizes of the disordered crystal: k L = 30(black), 40 (red), 50 (green), 60 (blue). The numbers of ran-dom realizations of the matrix ˆ G used for different sizes are2200, 900, 461 and 180, respectively. All eigenvalues within afrequency interval of width 0 . around ω − ω = − . are used to estimate P (ln g ; ω, L ). Probability densities cor-responding to different sizes coincide for small g ; the greyshaded area below P (ln g ; ω, L ) illustrates the notion of q -thpercentile ln g q for the firth percentile q = 0 .
05. Inset: thedistance D ( ω ) between probability densities corresponding to k L = 30 and 60 attains a minimum at the critical point( ω − ω ) / Γ (cid:39) − .
44. The step of frequency discretization is0 . for this figure. percentile ln g q is defined by a relation: q = ln g q (cid:90) −∞ P (ln g ; ω, L ) d (ln g ) (11)illustrated in Fig. 5 for q = 0 .
05 (firth percentile). Inde-pendence of the small- g part of P (ln g ; ω, L ) of L impliesthat ln g q should be L -independent for small q as well. Vi-sual inspection of Fig. 5 suggests that q = 0 .
05 is more orless the maximal value of q for which the L -independenceof P (ln g ; ω, L ) can be assumed. For larger q , the dashedvertical line in Fig. 5 would shift to the right and enterinto the range of ln g in which P (ln g ; ω, L ) correspond-ing to different L are clearly different. The grey shadedarea q on the left from the dashed line would then beill-defined.We have computed and analyzed the percentiles ln g q for q = 0 . q = 0 .
05 inFig. 6. The results for smaller q are similar but exhibitstronger fluctuations and larger error bars due to poorerstatistics. As discussed above, crossings between ln g q corresponding to different L are potential signatures oflocalization transitions. Figure 6(a) suggests that thereare two pairs of such crossings, a pair near the lower edgeof the band gap and another pair near the upper edge.Panels (b) and (c) zoom on the corresponding frequencyranges. Let us discuss the behavior with increasing thefrequency ω . First, a transition to localized states can FIG. 6. (a) Firth percentile ln g q =0 . of the Thouless conductance as a function of frequency ω for four different sizes k L ofthe disordered photonic crystal. Very large error bars in the range ( ω − ω ) / Γ ∈ ( − . , − .
54) are not shown. Vertical dashedlines show the band edges. Panels (b) and (c) zoom on the spectral ranges in which ln g q =0 . drops near the lower and upperband edges, respectively. (d) Finite-size scaling analysis of the localization transition taking place at ω = ω c (cid:39) ω − . where curves corresponding to different crystal sizes cross in a single point { ( ω c − ω ) / Γ , ln g ( c ) q } . Solid lines represent a jointpolynomial fit of Eq. (15) with m = n = 3 to the numerical data, dashed lines show their extrapolation beyond the range ofdata ln g q ∈ [ln g ( c ) q − δ (ln g q ) , ln g ( c ) q + δ (ln g q )] used for the fit. δ (ln g q ) = 2 for this figure. The inset shows the best-fit valuesof the critical exponent ν for q = 0 . .
05 with errors bars corresponding to the standard deviation, the grey area showingthe 95% confidence interval, and the dashed horizontal line indicating the average of ν over q . be identified around ( ω − ω ) / Γ (cid:39) − .
015 where a com-mon crossing of ln g q corresponding to k L = 40, 50 and60 takes place. The line corresponding to k L = 30 doesnot pass through this common crossing point, most prob-ably because this sample size is insufficient to observe theexpected large-sample behavior. ln g q remains a decreas-ing function of L for ( ω − ω ) / Γ (cid:38) − .
015 and up to( ω − ω ) / Γ (cid:39) − .
97. This is consistent with the ap-pearance of states localized in the bulk of the disorderedcrystal at frequencies near a band egde (see Figs. 2 and3). The states with frequencies in the middle of the bandgap, − . (cid:46) ( ω − ω ) / Γ (cid:46) − .
57 in Fig. 6(a), appear asrelatively localized according to their IPR in Figs. 2 and3 but show a scaling behavior that identify them as ex-tended (i.e., ln g q grows with L ). This is consistent withtheir surface nature: indeed, surface states are restrictedto the boundary of the sample and hence the number ofatoms on which they have significant amplitudes growsas L instead of L for extended states in the bulk. Thus,they have larger IPR as compared to the extended states in the bulk, but this IPR still decreases with L (as IPR ∝ /L instead of 1 /L ). This decrease is reflected inthe growth of ln g q shown in Fig. 6(a). A second bandof localized states arises near the upper edge of the bandgap, for − . (cid:46) ( ω − ω ) / Γ (cid:46) − . g q ( ω, L ) around ( ω − ω ) / Γ (cid:39) − . q percentiles of g in the frame-work of the single-parameter scaling hypothesis [93]. Thelatter postulates that in the vicinity of the localizationtransition point, ln g q is a function of a single parameter L/ξ ( ω ), where | ξ ( ω ) | is the localization length on the oneside from the mobility edge ω c and the correlation lengthon the other side: ln g q ( ω, L ) = F q [ L/ξ ( ω )]. Assumingthat the divergence of ξ ( ω ) at the transition is power-law,we represent ξ ( ω ) as ξ ( ω ) = m (cid:88) j =1 A j w j − ν (12)near w = ( ω − ω c ) /ω c = 0. Here A j are constants and ν is the critical (localization length) exponent. We thuscan writeln g q ( ω, L ) = F q [ L/ξ ( ω )] = F q [ ψ ( ω, L )] (13)with a scaling variable ψ ( ω, L ) = (cid:20) Lξ ( ω ) (cid:21) /ν = L /ν m (cid:88) j =1 A j w j . (14)Finally, the scaling function F q ( ψ ) is expanded in Taylorseries: F q ( ψ ) = ln g ( c ) q + n (cid:88) j =1 B j ψ j , (15)where ln g ( c ) q is the critical value of ln g q independent of L .Fits of Eq. (15) to the numerical data are performedwith with ω c , ln g ( c ) q , ν , A j ( j = 1 , . . . , m ), and B j ( j =1 , . . . , n ) as free fit parameters. The orders m and n ofthe expansions (14) and (15) are chosen large enough toensure that the χ statistics χ = 1 N data N data (cid:88) j =1 {F q [ ψ ( ω j , L )] − ln g ( j ) q } σ j (16)is of the order 1. Here N data is the number of data points { ω j , ln g ( j ) q } and σ j are statistical errors of ln g ( j ) q shownby error bars in Fig. 6. We only fit the numerical datain the range ln g q ∈ [ln g ( c ) q − δ (ln g q ) , ln g ( c ) q + δ (ln g q )]around the critical value ln g ( c ) q estimated in advance bylooking for the minimum of the sum of squares of differ-ences between points corresponding to different L .A joint fit to the numerical data corresponding to fourdifferent values of L and q = 0 .
05 is shown in Fig. 6(d). Ityields ω c = − . ± . ν = 0 . ± .
02 as thebest fit parameters. We repeated the fits for other valuesof q in the range from 0.01 to 0.05 with the same fre-quency resolution 0 . as in Fig. 6(d) [see the inset ofFig. 6(d) for the best-fit ν ] and with a twice finer resolu-tion and δ (ln g q ) = 1 instead of δ (ln g q ) = 2 in Fig. 6(d).In addition, we varied the orders m and n of the seriesexpansions (14) and (15) from 1 to 3 and introduced anadditional, irrelevant scaling variable [71]. All fits yieldconsistent values of ( ω c − ω ) / Γ in the range [ − . − . ν = 0 . δ (ln g q ) = 1. V. DISCUSSION
Whereas the position of the mobility edge found fromthe finite-size scaling analysis agrees well with the esti-mation following from the analysis of P (ln g ; ω, L ) (seeFig. 5), the value of the critical exponent ν turns out tobe well below ν AM (cid:39) .
57 found numerically for the An-derson model (AM) in the 3D orthogonal symmetry classand believed to be universal for disorder-induced local-ization transitions in 3D systems in the absence of anyparticular symmetry breaking mechanisms [71]. Cold-atom experiments mimicking the so-called quasiperiodickicked rotor model indeed yielded values of ν compatiblewith ν AM [94], but ν (cid:46) ν AM were reported in low-temperature electron transport ex-periments in doped semiconductors [95, 96]. Recently,values of ν (cid:46) W (cid:54) = 0) is not clearlyseparated from the band of propagating modes (i.e., themodes in the bands of the perfect crystal) either (see Fig.1). This may be a reason for the value of the critical ex-ponent ν different from ν AM . Other possible reasons mayinclude a strong anisotropy of optical properties of a pho-tonic crystal near a band edge due to the fact that thefirst modes that become allowed upon crossing a bandedge propagate only in certain directions, and, of course,the vector nature of light of which the full impact onAnderson localization still remains to be understood.To determine the precise value of ν and to obtain abetter estimation of its uncertainty, more accurate calcu-lations are required. Unfortunately, such calculations aredifficult to perform using our approach. Indeed, the ap-proach is based on the diagonalization of large 3 N × N non-Hermitian matrices ˆ G and has an advantage of yield-ing the whole spectrum of a single realization of an opendisordered system at once. The downsides of this are that(i) the approach does not allow for focusing on a partic-ular frequency range at a lower computational cost and(ii) studying large systems ( N (cid:38) ) is computation-ally expensive. Because the localization transition takesplace in a narrow frequency range, only a small fractionof eigenvalues obtained by the numerical diagonalizationof ˆ G is actually used for the estimation of ν . Indeed,in Fig. 6(d) we have chosen to analyze the behavior ofln g q within an interval ln g ( c ) q ±
2, which restricts thenumber of eigenvalues of ˆ G used in the calculations of ω c and ν to less than 1% of the total number of eigen-values. Narrowing the interval of considered ln g q onlydecreases the fraction of useful eigenvalues whereas ex-panding this interval and using more eigenvalues wouldcorrespond to leaving the critical regime and hence isnot desirable. Thus, significantly increasing the statis-0tical accuracy of calculations requires large amounts ofcomputations. Although this drawback of our approachis general and complicates the analysis of fully randomensembles of atoms as well [52, 65], its impact is ampli-fied here by the particular narrowness of the frequencyrange in which the localization transition takes place andthe low DOS in this range. Indeed, for scalar waves ina random ensemble of point scatterers studied in Ref.65, ln g q =0 . grows from ln g ( c ) q =0 . − g ( c ) q =0 . + 1in a frequency range δω/ Γ (cid:39) .
08 whereas in the pho-tonic crystal studied here the same growth takes placewithin δω/ Γ (cid:39) .
02 [see Fig. 6(d)]. In addition, DOSof the fully random system has no particular features inthe transition region, whereas in the photonic crystal,the localization transition takes place near a band edgewhere DOS is quite low [see Fig. 1(b)]. These factorslimit the statistical accuracy of our numerical data andmake the high-precision estimation of ν a heavy compu-tational task.The frequency range in which the quasimodes are lo-calized can be broadened and DOS in this range can beraised by increasing the strength of disorder W . How-ever, the space for increase of W without closing the gapand loosing localization altogether is rather limited. Aswe show in Fig. 1, the gap closes already for W = 0 . W = 0 . L of the disordered photonic crystal as in Fig. 6. Contraryto the latter figure, no crossings between lines ln g q ( ω, L )occur in Fig. 7(a), signaling the absence of localizationtransitions. Moreover, the values of ln g q ( ω, L ) in Fig.7(a) are rather high: ln g q ( ω, L ) > ω . Thismeans that at any frequency, less than 5% of g valuesobtained for different atomic configurations are smallerthan exp(2) ≈
7, which is incompatible even with the“weakest” form of the Thouless localization criterion re-quiring that some typical value of g ( (cid:104) g (cid:105) , exp( (cid:104) g (cid:105) ), me-dian g , etc.) is less than 1. Finally, another signature ofthe absence of localization is the monotonous growth ofln g q with L at all frequencies indicating that most prob-ably, ln g q → ∞ in the limit of L → ∞ , as it should befor spatially extended modes.Further increase of the strength of disorder W beyond W = 0 . g q and its monotonous growth with L ) and hence confirmsthe previously discovered absence of the localization oflight in the fully random system [49].The presence of localization only at weak disorderhighlights the important differences between localiza-tion phenomena in disordered crystals and fully randommedia. As it has been largely discussed in the litera- FIG. 7. Firth percentile ln g q =0 . of the Thouless conduc-tance as a function of frequency ω for different diameters L of a disordered crystal with disorder strength W = 0 . L confirms theabsence of localization transitions in these systems. ture starting from the pioneering works of Sajeev John[19, 33, 38], the localization in a photonic crystal takesplace due to an interplay of order and disorder in con-trast to the localization in a fully random medium thatis due to disorder only. Whereas localized states may ap-pear in a 3D random medium only when the strength ofdisorder exceeds some critical value, even a weak disor-der introduces spatially localized modes in the band gapof a disordered photonic crystal and the notion of criticaldisorder does not exist. However, the possibility of reach-ing localization at arbitrary weak disorder is counterbal-anced by the narrowness of frequency ranges inside theband gap in which the density of states is large enoughto allow for observation of the localization of light in anexperiment or a numerical simulation. Increasing disor-der widens the relevant frequency ranges but also tends toclose the band gap and hence to suppress the ‘order’ partof the interplay between order and disorder. A compro-mise is reached at some intermediate disorder strength1that is sufficient to significantly affect wave propagationat frequencies near band edges but not large enough toclose the band gap. For the atomic crystal considered inthis work, such a compromise seems to be reached around W = 0 . W allows for an additionalinsight about the reasons behind the absence of Ander-son localization of light in a completely random 3D en-semble of point scatterers. Indeed, recent work [55, 98]has confirmed the initial suggestion [49] that the reso-nant dipole-dipole coupling between scatterers impedesthe formation of spatially localized optical modes in 3D.This explanation seems to be supported by the fact thatlocalized modes do arise in a photonic crystal wherethe distance ∆ r between any two scatterers (atoms) isalways larger than a certain minimal distance ( a √ / a consid-ered in this work) and hence the strength of the dipole-dipole coupling between scatterers that scales as 1 / ∆ r ,is bounded. The increase of W enhances chances for twoatoms to be closer, the minimum possible distance be-tween atoms being equal to ( √ / − W ) a in our model.The probability for two neighboring atoms to get in-finitely close because of disorder becomes different fromzero for W ≥ √ / (cid:39) .
22. This estimation of disor-der strength for which dipole-dipole interactions shouldbecome particularly strong, is reasonably close to theapproximate value W (cid:39) . ω < ω ) in contrast to the blue-shifted localized modesthat arise in a fully disordered ensemble of atoms in astrong magnetic field [51, 52].A final remark concerns the spatially extended quasi-modes in the middle of the band gap, corresponding tolarge ln g q =0 . (cid:38) ω − ω ) / Γ (cid:39) − .
97 and − .
57 in Fig. 6. As we have illustrated already in Fig. 3,most of these quasimodes are bound to the surface of thecrystal. Their statistical weight is thus expected to de- crease with L roughly as the surface-to-volume ratio 1 /L ,which tends to zero when L → ∞ but remains significantin our calculations restricted to rather small L . Neverthe-less, we clearly see from Figs. 6(a–c) that the frequencyrange in the middle of the bandgap where ln g q =0 . takeslarge values ln g q =0 . (cid:38) L , shrinks as L increases. No transi-tion point where curves ln g q corresponding to different L cross can be identified around ( ω − ω ) / Γ (cid:39) − .
97 or − .
57, which is especially clear in Fig. 6(b) whereas lessobvious in Fig. 6(c) due to much stronger fluctuations ofthe numerical data. We note that the above picture ofsurface modes playing less and less important role as L increases is certainly only a rough approximation to thecomplete explanation of the evolution of the spectrumin the middle of the band gap. Nontrivial features thatare already seen from our results and call for explana-tion include the nonmonotonous behavior of ln g q with L near the high-frequency end of the interval − . (cid:46) ( ω − ω ) / Γ (cid:46) − .
57 [note the red line that crosses thegreen line around ( ω − ω ) / Γ (cid:39) − . ω − ω ) / Γ (cid:39) − . ω − ω ) / Γ (cid:39) − .
97 [compare Figs. 6(c)and (b)]. Unfortunately, a study of these puzzling fea-tures is difficult to perform using our numerical methodbecause it mobilizes significant computational power toobtain the full spectrum of the system of which only avery small fraction [i.e., a small number of eigenvaluesΛ m of the matrix (3)] fall in the band gap where thedensity of states is low. VI. CONCLUSIONS
We performed a thorough theoretical study of the lo-calization of light in a 3D disordered photonic crystalmade of two-level atoms. The atoms are first arrangedin a diamond lattice with a lattice constant a and arethen slightly displaced in random directions by randomdistances up to W a . We show that spatially localizedquasimodes appear near edges of the band gap of theideal crystal when the disorder strength is W = 0 . W = 0 . ν is in the interval 0.8–1.1, which is different from ν AM (cid:39) .
57 corresponding tothe Anderson transition of the 3D orthogonal universal-ity class to which the investigated transition might beexpected to belong because of the absence of any partic-ular symmetry breaking mechanisms and, in particular,the preserved time-reversal symmetry.From the practical standpoint, arranging atoms in adiamond lattice may be a realistic alternative to sub-jecting them to strong magnetic fields in order to reachthe localization of light in cold-atom systems. Indeed,2atomic lattices can be readily designed by loading atomsin optical potentials created by interfering laser beamswith carefully adjusted phases and propagation direc-tions [99, 100]. Some degree of disorder may arise insuch lattices due to experimental imperfections; ways tocreate additional, controlled disorder have been largelyexplored in recent years [45]. Calculations presentedin this work provide quantitative estimates for disorderstrengths and frequency ranges for which localized quasi-modes should appear in lattices of cold atoms featuringa J g = 0 → J e = 1 transition. Examples of appropriatechemical elements for vapors of which laser cooling tech-nologies are readily available include strontium (Sr) orytterbium (Yb). Multiple scattering of light in large en-sembles of Sr atoms has been already reported [101] and high atomic number densities have been reached in ex-periments with Yb [102]. In addition, some of our conclu-sions may hold for atomic species with more complicatedlevel structure, which may be easier to manipulate andcontrol in an experiment (e.g., rubidium). This opens away towards the experimental observation of phenomenareported in this work. ACKNOWLEDGMENTS
All the computations presented in this paper were per-formed using the Froggy platform of the CIMENT infras-tructure ( https://ciment.ujf-grenoble.fr ), which issupported by the Rhone-Alpes region (grant CPER07 _ Investissements d’Avenir supervised by the
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