Disorder in two-level atom array chirally coupled via waveguiding mode
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Disorder in two-level atom array chirally coupled via waveguiding mode
G. Fedorovich, ∗ D. Kornovan, and M. Petrov
ITMO University, Birzhevaya liniya 14, 199034 St.-Petersburg, Russia
In this paper we studied a one-dimensional array of quantum emitters asymmetrically coupleddue to chiral interaction through a waveguiding mode. We have showed that disorder and couplingasymmetry compete with each other in forming and destroying Anderson localized states. Wefound that for wide range of the disorder strength there exists certain asymmetry parameter, whichdestroys the localization of the states. We have also numerically obtained the dependence of thecritical asymmetry strength on the amplitude of the disorder. We believe that our findings areimportant for rapidly developing field of waveguide quantum electrodynamics, where the chiralinteractions and disorder play a critical role.
Keywords: Anderson localization, chiral interaction, two-level systems, polaritonic states, waveguiding mode
I. INTRODUCTION
The coupling of quantum ensembles with nanopho-tonics systems such as microcavities [1], photonic crys-tal waveguides, [2] and optical nanofibers [3, 4]provideunique opportunities to enhance the light-matter inter-actions. At the same time, nanophotonic systems canalso drastically modify the nature of this interactions, forexample, by tailoring the polarization states of the pho-tons and fields. This phenomenon has recently led to ap-pearance of chiral coupling between nanophotonic modesand quantum emitters [5], i.e., coupling that depends onthe propagation direction (forwards or backwards) of thelight. This field has attracted even bigger interest withthe emergence of topological photonics [6], where the uni-directional propagation of protected edge states becameone of the central topics. Recently the quantum emittersinteractions with the topological modes has been sug-gested as a new quantum optics platform [7–9].In this prospective, unidirectional emitter-emitter cou-pling [10] and absence of back-reflection significantlymodifies the character of light-matter interactions suchas super- and sub-radiance [11, 12], quantum coherence[13] and disorder induced photon localization. The lattereffect has vital strong importance in one-dimensional chi-ral quantum systems based on waveguides or topologicaledge states, where, on the one hand, the disorder is in-evitably present due to technological imperfections, andon the other has strong impact due to low-dimensionalityof the system.The localization of excitation in disordered media staysone of the central universal concepts in modern physics[14] since its invention [15]. One-dimensional systemsstand aside of others as, according to classical scalingtheory [16], all the states appear to be localized. How-ever, the situation changes drastically in non-Hermitiandisordered quantum systems, where states can be eitherlocalized or delocalized [17–20]. In this sense, the chi-ral photonic system are strongly non-Hermitian owing to ∗ [email protected] (a) Bloch vector, qa/ π (b) l i g h t l i n e F r e q u e n c y , Δ ω / γ ω - ω + ξ =1 ξ =0 ξ =0.5 ξ =0.1 ξ =0.01
00 0.50.5 -0.5-0.51-1 z ...... γ L γ R γ L γ R γ L γ R ω ω ω N a a FIG. 1. a) The geometry of an array of regularly spaced quan-tum emitters separated with distance a and coupled througha waveguiding mode. (b) The dispersion of polaritonic modeswith account for chiral interactions shown for different valuesof asymmetry parameter ξ . optical losses typical for any optical system and to uni-directional character of interactions, which destroy theinternal symmetry of the problem.There have been a number of studies, which addressthe optical properties of chiral quantum optic systems.The interaction of guided light with disordered atomicarrays coupled to a waveguide was studied in detailsin Ref. [21] in the absence of chiral interactions. Al-ternatively, the effects of disorder on spectral proper-ties of semiconductor polaritonic lattices have been alsostudied [22–24]. While light transmission through chi-ral disordered atomic arrays was considered recently inRef. [25–27]. Namely, a complex two-parameter scalingtheory of localization in chiral systems has been sug-gested [28]. Nevertheless, the localization properties ofthe eigen states in chiraly coupled quantum systems havenot been discovered yet and are addressed in this paper.We believe that the results obtained in our work are im-portant both for understanding fundamental propertiesof disordered quantum systems and for future develop-ment of chiral quantum optics.The paper is organized as follows: in Sec. II we providethe general formulation of the problem; in Sec. III wefocus on the optical properties of symmetric and chirallycoupled arrays of two-level quantum emitters; in Sec. IVwe study effects of diagonal disorder and the localizationproperties of the eigenstates depending on asymmetry ofinteraction and disorder amplitude. II. FORMULATION OF THE PROBLEM
In our work, we consider a one-dimensional (1D) arrayof N two-level quantum emitters located at the coor-dinates z n , and coupled through a single guided modewhich is schematically shown in Fig. 1 (a).In the case of a finite system, the effective Hamiltonianof the considered system can be represented as: b H = b H + b V , (1) b H = ~ (cid:16) ω m − i γ m (cid:17) N X m =1 b σ + m b σ − m , b V = ~ N X m,n =1 m = n g n,m b σ + n b σ − m , here ω m , and γ m are the transition frequency, and theradiative emission rate of the m -th emitter, respec-tively, and g m,n are the coupling constants. Whilewe assume that the transition frequency may fluctu-ate from one emitter to another, the emission ratesare fixed to be constant γ m = γ . The coupling con-stants between two emitters can be defined throughthe electromagnetic Green’s function [29, 30] g m,n = − πk d ∗ m G ( r m , r n , ω ) d n , where d n is the transitiondipole moment of the n -th emitter, and k = ω /c isthe wavenumber. Thus, the coupling constants are de-fined by the polarization properties of both the guidingmode, and the transition dipole moments, and takes thefollowing form: g m,n = − i γ R e iϕ mn for m > n, − i γ L e iϕ mn for m < n, (2)where γ R and γ L are the photon emission rates to theright and left directions, correspondingly, and the pa-rameter ϕ mn = k | z m − z n | is the phase that acquiresthe photon from a guided mode while travelling betweenthe emitters at positions z n , and z m . Indeed, the leftand right emission rates can be different: for circularly -50-40-30-20-1001020304050 (a) F r equen cy , Δ ω / γ F r equen cy , Δ ω / γ ξ=1 R ad i a t i v e l o ss e s , γ / ( γ / ) -2 -4 -6 -8 -10 -12 -14 Participation ratioN N/2 N/4
Subradiative -1 -0.5 0 0.5 1
Superradiative
Subradiative (b) -1 -0.5 0 0.5 1-15-10-5051015 ξ=10 - R ad i a t i v e l o ss e s , γ / ( γ / ) Participation ratioN N/2 N/4
Subradiative
Superradiative
Wavenumber, qa/ π Subradiative
FIG. 2. (a) The resonant states of N = 400 symmetricallycoupled ( ξ = 1) array of quantum emitters separated with ϕ = π/
2. The dispersion of the infinite system is shown withsolid grey line. The color of labelling point denotes the nor-malized radiation loss rate for each state. The diameter of thelabelling point corresponds to normalized participation ratio.The typical
P R values are shown for eye guidance in the in-set. The inset figures of mode profiles are plotted for N = 50for clearness. (b) The resonant states of chirally coupled ar-ray with asymmetry parameter ξ = 10 − . The computationalparameters are the same as in (a). polarized dipole transitions d n = d / √ e x − i e z ) andcircular polarization of the guided mode travelling in+ z direction E + ( z ) = E / √ e x + i e z ) e ik z the emis-sion rate γ R = 0, while the guided mode propagating inreverse − z direction will have the opposite polarization E − ( z ) = E / √ e x − i e z ) e − ik z , and γ L ∼ | d ∗ n · E − | = 0.Thus, one can vary the asymmetry in coupling betweenthe emitters either by controlling the polarization of thetransition dipole moment [31] or through controlling thepolarization state of the guided mode [10], which pro-vides the ground for a chiral quantum optical platform.For further convenience, we introduced the asymmetryparameter ξ = γ L /γ R , which is assumed to be freely var-ied from ξ = 1 for symmetric interaction to ξ = 0 forfully asymmetric interaction.In the following sections, we will study the propertiesof the eigenstates of the Hamiltonian (1) describing thepolaritonic modes in the ensemble of coupled emitters.We will assume that all of the emitters are ordered in anarray having an equal spacing a providing a single phaseparameter ϕ n = ϕ = k | z n +1 − z n | = k a . We will startwith revising the polaritonic states in a regular array inthe following section, while the system with fluctuationswill be considered in Sec. IV. III. REGULAR ARRAY OF TWO-LEVELEMITTERSA. Infinite periodically ordered system
First of all, it is very illustrative for further analysis torecall a photon dispersion in an infinite system of two-level quantum emitters with equal transition frequencies ω n = ω with a separation distance between the neigh-bors a . The interaction of emitters through the guidedmode leads to the formation of polaritonic states [32–35],which dispersion ω ( k ) depends on the asymmetry param-eter ξ and takes the form:∆ ω ( q ) = − iγ − ξ )+ (3) γ (cid:18) cot (cid:18) ϕ − qa (cid:19) + ξ cot (cid:18) ϕ + qa (cid:19)(cid:19) , where ∆ ω = ω ( q ) − ω , q ∈ [ − π/a, π/a ) is quasi-momentum. In the limiting cases of symmetric coupling ξ = 1 and ideal unidirectional coupling ξ = 0, the disper-sion takes a simple form of∆ ω ( q ) = γ ϕ )cos( qa ) − cos( ϕ ) , for ξ = 1 , ∆ ω ( q ) = − iγ γ (cid:18) ϕ − qa (cid:19) , for ξ = 0 . (4)The dispersion curve of the polaritonic states is shown inFig. 1 (b) for a fixed period qa = π/
2, and several val-ues of the asymmetry parameter ξ . The black solid linerepresents the case of the symmetric coupling, clearlydemonstrating the avoided crossing behaviour close tothe light line. At the same time, in the case of an idealchiral coupling, the asymmetry of the dispersion curveindicates the directional transport of the quantum exci-tation in the positive direction across the whole Brillouinzone. One can also notice, that the gradual change of theasymmetry parameter from ξ = 1 to 0 leads to closing ofthe band gap. B. Finite regular chain with symmetric coupling
Once the system becomes finite, the non-zero radiativelosses appear due to photon scattering at the edge of the -6 -1 -6 -1 -2 -4 -6 -8
10 100 1000 ξξ=1 γ –› γ /4 Superradiaitve ~NSubradiaitve ~N -3 Number of atoms, N R a d i a i t i v e l o ss e s , γ / ( γ / ) Subradiant stateSuperradiant state
Assymetry parameter ξ P a r t i c i pa t i on r a t i o , P R / N -1 -2 -3 -4 -5 -6 -7 FIG. 3. (a)The radiative losses of the states with largest(superradiaitve) and smallest (subradiative) radiation lossessas a function emitter’s number for different asymmetry pa-rameter ξ and for ϕ = π/
2. (b) The participation ratio of thecorresponding super- and subradiative states as a function ofasymmetry parameter for N = 400 and ϕ = π/ array, and the eigenfrequencies of the collective statesacquire the imaginary parts: b H | ψ k i = ~ Ω k | ψ k i , Ω k = ω k − iγ k / . In the complex plane the real and imaginary parts ofeigenfrequencies from a circular structure [33] typical forToeplitz-type matrices [36]. In order to plot the disper-sion of a finite system, one can map the obtained eigen-frequencies of collective states to the first Brillouin zoneof an infinite structure. The quasi-momentum q k asso-ciated with the mode k in this case can be extractedfrom the corresponding modal profile | ψ k i , based on themodal profile [37, 38]. The eigenfrequencies for an ar-ray of N = 100 emitters, and phase parameter ϕ = π/ | ψ k i = P n c nk | n i for sub- P a r t i c i p a t i o n r a t i o , P R / N D e n s i t y o f s t a t e s -1-2-3 1 2 300.40.80.60.20 Frequency, Δω / γ Frequency, Δω / γ D e n s i t y o f s t a t e s Symmetric ξ =1 Asymmetric ξ =0.01 P a r t i c i p a t i o n r a t i o , P R / N Band gapExtended states (a) (c) (e)
Band gapLocalized states Extended statesExtended statesExtended states Localized states -1 -2 -3 -4 Frequency, Δω / γ Scaling factor, β A s y mm e t r y p a r a m e t e r , ξ Extended states region Extended states region
Localized statesregion
Asymmetric ξ =0.01 Critical asymmetrySymmetric ξ =1 (b) (d) DisorderedDisordered Ordered Disordered OrderedOrdered DisorderedOrdered
FIG. 4. (a) Density of states of the ordered ( δ = 0) and disordered ( δ = 0 .
01) symmetric system. (c) The dependence of
P R for the systems shown in (a): blue squares correspond to ordered system, and red circles to disordered. (b) Density ofstates of the ordered ( δ = 0) and disordered ( δ = 0 .
01) chiral system ( ξ = 0 . P R for the systemsshown in (c): blue squares correspond to ordered system, and red circles to disordered. (e) The spectral map of the scalingfactor β as function of frequency detuning ∆ ω and asymmetry parameter ξ for fixed disorder amplitude δ = 0 .
01. The blueregion corresponds to localized states, while the green one to delocalized. The dashed lines denote ξ = 1 and ξ = 0 .
01 regioncorresponding to the cases plotted in (a)-(d). The critical asymmetry dash-dot line denote the critical value of ξ after whichall the states become delocalized. the simulations parameter for (a)-(e) are N = 400 and ϕ = π/
2. The number of randomrealizations for averaging was equal to 1000 for (a)-(d), and 500 for (e). radiant, and superradiant states, where | n i state corre-sponds to n -th emitter being excited, while all the othersare in the ground state: | n i = | e n i| g i ⊗ ( N − . The modestructure of these states has an envelope and a Bloch-type phase factor and partially inherits their structurefrom the nearest neighbour model [37, 39]. The radiativelosses of subradiant and superradiant states scale withthe size of the system as γ sub ∝ N − [39, 40], while theemission rates of superradiant states γ sup ∝ N [21, 41].This can be clearly seen from Fig. 3 (a), where the ra-diative losses of superradiant and subradiant states areplotted in double logarithmic scale as functions of theemitter number N for various values of the asymmetryparameter ξ .Finally, the last parameter, which characterizes theeigenstates, and has a critical importance for identifica-tion of localization effects is the participation ratio ( P R )[42], that quantifies the effective number of the occupiedsites by a single excitation, and which can be expressedas:
P R k = N P i =1 | c ik | N P i =1 | c ik | , (5)for the k -th state. Since in the case of the ordered struc-ture, the eigenmodes of the system are constructed fromthe Bloch waves, the excitation occupies almost all of thelattice sites P R ∼ N . We have depicted the normalizedparticipation ratio P R/N for each mode in Fig. 2 (a) with the diameter of the circle labelling the PR value foreach state. One can see, that the superradiative stateshave the smallest PR, while the subradiant states, onthe contrary, are the most distributed but still having
P R ≈ N . Moreover, all the states in the ordered arrayscale linearly with the system size, so P R ∝ N , which isa sign of their truly extended nature.One also needs to mention a special case of ϕ =0, which corresponds to discrete Bardin-Cooper-Schiffermodel [43, 44] and is proposed for description of super-conducting states in lattice models. In this case, thereappear N − γ N = N γ and constant mode profile within-phase amplitudes | ψ i = 1 / √ N P n | n i . C. Finite regular chain with asymmetric coupling
Once the chiral coupling is introduced for a finite sys-tem, discrete resonant states follow the dispersion behav-ior of the infinite structure as shown with a solid grey linein Fig. 2 (b). One can see that the anticrossing at qa = ϕ vanishes in the negative region of the Brillouin zone, andthe resonant states close to this point posses the lowestradiative losses among all of the states that are compa-rable to γ in value. Thus, the absence of backscatteringin an array of asymmetrically coupled emitters destroysthe subradiance effect, which bases on the destructiveinterference of radiation coming from emiiters. At thesame time, the radiative losses of superradiant state inthe case of chiral coupling stay of the same order as in thesystem with the symmetric coupling. These effects canbe clearly seen from Fig. 3 (a), where radiative losses ofsubradiant, and superradiant states are shown for a widerange of ξ parameter. While losses of subradiant statesbecome constant with the increasing emitter number N ,and beahve like γ sub → γ / − ξ ) (see Eq. (3)) for large N [12], the superradiant state losses tend to γ sup → N γ for ξ →
0, and large N .Subradiant states have the largest P R value close to N , therefore, the excitation occupies most of the arraysites as shown by the label diameter Fig. 2 (b). However,now the modes become localized at the edge of the chainas it is shown in the insets of Fig. 2 (b). If ξ becomessmall enough the excitation in the system noticeably pre-vails on the right side of the chain. This is explained bythe fact that each emitter radiates to the left side muchweaker than to the right in the asymmetric coupling case.In the limit ξ → | ψ i = | N i .The participation ratio strongly depends on the asym-metry parameter, which is reflected in Fig. 3 (b), wherethe value of P R is plotted for the states with the highestand lowest radiative losses. One can see that as ξ → P R decreases reaching value of
P R ≈ P R forchirally coupled emitters becomes much smaller than N ,our calculations show that the states are still extendedas scaling behaviour P R ( N, ξ → ∝ N is observed. IV. DISORDER IN THE ARRAY
Finally, in this section we will focus on the effectsof localization induced by diagonal disorder due to thefluctuations of transition frequencies of k -th emitter: ω k = ω + ∆ ω k , where ∆ ω k is normally distributed ran-dom value with standard deviation equal to δ · γ in theabsence of correlations between the emitters. A. Disorder in an array with achiral coupling
The effects of disorder in long-range interacting sys-tems have been studied in a number of previous workswith focus on cold atoms coupled to through a waveguid-ing mode [21], and exciton-polaritons in one-dimensionalphotonic crystal structures [23, 34]. Most of them wereaddressing the effects of Anderson localization due to di-agonal [22], and positional (off-diagonal) [21, 23] disor-der, and related spectral properties of the system suchas inhomogeneous broadening of reflection or transmis-sion coefficients. In this section, will overview the effectsof diagonal disorder on the localization properties of thesystem eigenmodes.We start with the density-of-states (DOS) function asa proper measure of disordered structures. The DOS profiles shown in Fig. 4 (a) have typical structure that canbe understood from the dispersion curves shown in Fig. 1:they have a band gap width equal to γ in case of ϕ = π/
2, and possess van Hove singularities typical for one-dimensional systems. One can notice, that introductionof disorder leads to the smearing of the band edges, andformation of the Urbach tails [42]. The normalized valueof
P R drastically drops in the are close to the band gapas shown in Fig. 4 (b). At the the same time, the statesfar from the band gap edge are spread over the largenumber of sites that is close to N .The large relative value of P R does not immediatelyprovides the delocalization of the eigenstates. In orderto check that, we have traced the exponent parameter β in the scaling behaviour of the P R ( N ) ∝ N β . The spec-tral dependence of the scaling factor β (∆ ω ) is shown inFig. 4 (e) for different values of the asymmetry param-eter ξ . One can see that for the symmetric case ξ = 1the states close to the band edge appear to be localizedas β is close to 0 providing that the localization lengthdoes not scale up with the system size. Meanwhile, farfrom the band gap region the states are clearly delocal-ized, despite the 1D nature of the system. The absenceof localization for most of the states is provided by anon-Hermitian character of the problem [17]. Note thatHaakh et al. [21] have related the delocalization of statesto a quasi-3D nature of the interaction as in their reportthe vacuum dipole-dipole interaction was also considered,and the coupling efficiency to the guided mode was lessthan 1. Clearly, that is not the case for the system con-sidered in this work as there is no free space coupling,and yet not all of the states are localized.With the decrease of the asymmetry parameter ξ theband gap gets narrower as can be seen in DOS spectrain Fig. 4 (c) for ξ = 0 .
01, and the localization region alsobecomes narrower (see Fig. 4 (d)). The spectral width ofthe localized states region reduces with the further de-crease of the asymmetry parameter, and once the asym-metry parameter reaches the critical value all states be-come delocalized (see Fig. 4 (e) below the dash-dot line).This occurs once the band gap width becomes compara-ble to the energy smearing due to disorder, when, roughlyspeaking, Urbach tails are greater or equal to the bandgap size. We believe that the map shown in Fig. 4 (e)is one of the most important results of the paper, clearlydemonstrating the delocalization transition driven by chi-ral interactions in the disordered system.In order to illustrate the competing effects of disorderand asymmetry on Anderson localization, we have ana-lyzed the delocalization transition for different values ofthe disorder amplitude δ , while the results shown in Fig. 4were obtained for a fixed value of disorder amplitude δ .Fig. 5 (a) shows the dependence of the normalized P R on the disorder amplitude δ , and asymmetry parameter ξ for states in a band gap close to its edge as shown inFig. 4 (e). One can see that for symmetric case, and forsmall disorder the states extend over the whole lengthof the system. With the increase of disorder amplitude δ -4 -3 -2 -1 Participation Ratio, PR/N Scaling factor, β -4 -3 -2 -1 Asymmetry parameter, ξ D i s o r de r pa r a m e t e r , δ Asymmetry parameter, ξ D i s o r de r pa r a m e t e r , δ -4 -3 -2 -1 -4 -3 -2 -1 -0.4-0.200.20.40.60.811.2 (b)(a) Delocalized:symmetricDelocalized: asymmetric Localized LocalizedDelocalized: asymmetric T r an s i t i on t h r e s ho l d FIG. 5. (a) Dependence of the
P R of the state close to band-edge on the asymmetry parameter ξ and disorder amplitude δ forarray of N = 100 emitter, averaged over 500 realizations. (b) The spectral map of the scaling factor β as function of asymmetryparameter ξ and disorder amplitude δ . The blue region corresponds to localized states, while the yellow one to delocalized.The calculations were provided for an array of N = 400 emitters with averaging over 100 realizations. the states become localized and P R/N . .
1, which is asa sign of localization. However, at small enough asym-metry parameter ξ ∼ − the state occupies a smallfraction of the system comparable to localized state, butthat is due to chiral coupling rather than disorder. Asdiscussed in Sec. III, in case of strong chiral couplingthe excitation is located at the right edge of the system,and its normalized P R tends to 0 as ξ →
0. Despite ofthat, it stays delocalized in the sense of scaling of the
P R parameter with the system size N . To prove that,we have plotted a map of a scaling factor β shown inFig. 5 (b) in parallel to P R map in (a). One can see,that β ∼ β -factoris close to β ∼
1, which is a clear sign of a delocalizedcharacter of the state. We have marked the transitionregion separating the localized and of delocalized areaswith a dashed line for eye guidance.
V. CONCLUSION
Concluding, we have considered a one-dimensional ar-ray of two-level emitters coupled through a waveguiding modes a with account for chiral interaction. Introductionof the disorder in such system results in partial localiza-tion of the eigenstates, which can be destroyed by chiralnature of the interaction. We show that for particularstrength of the interaction asymmetry all the states be-come delocalized, forced by the unidirectional propaga-tion of excitation in chiral system. There exists certaincompetition between the disorder and coupling asymme-try strengths and the critical values of the asymmetrystrength have been identified numerically.We believe that our findings will be important forrapidly developing field of waveguide-QED, where thechiral interactions and disorder play a critical role. More-over, extension of the obtained results to multiphotondomain will be of significant interest due to rapid devel-opment of theoretical[45–47] and experimental studies inthis area[13, 48].
VI. ACKNOWLEDGEMENT
The authors are thankful to Diedrik Wiersma, IvanIorsh, and Vladimir Yudson for fruitful discussions. [1] M. Scheucher, A. Hilico, E. Will, J. Volz, andA. Rauschenbeutel, Science , 1577 (2016),arXiv:1609.02492. [2] J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu,S. P. Yu, D. E. Chang, and H. J. Kimble,Proceedings of the National Academy of Sciences of the United States of America , 10507 (2016),arXiv:1603.02771. [3] E. Vetsch, D. Reitz, G. Sague, R. Schmidt,S. T. Dawkins, and A. Rauschenbeutel,Physical Review Letters , 203603 (2010).[4] S. T. Dawkins, R. Mitsch, D. Re-itz, E. Vetsch, and a. Rauschenbeutel,Physical Review Letters , 243601 (2011).[5] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeu-tel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller,Nature , 473 (2017).[6] L. Lu, J. D. Joannopoulos, and M. Soljaˇci´c,Nature Photonics , 821 (2014), arXiv:1408.6730.[7] S. Barik, A. Karasahin, S. Mittal, E. Waks,and M. Hafezi, Physical Review B , 1 (2020),arXiv:1906.11263.[8] M. Jalali Mehrabad, A. P. Foster, R. Dost,A. M. Fox, M. S. Skolnick, and L. R. Wil-son, arXiv (2019), 10.1364/optica.393035,arXiv:1912.09943.[9] M. Jalali Mehrabad, A. P. Foster, R. Dost,E. Clarke, P. K. Patil, I. Farrer, J. Hef-fernan, M. S. Skolnick, and L. R. Wilson,Applied Physics Letters (2020), 10.1063/1.5131846,arXiv:1910.07448.[10] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs,Nature Photonics , 789 (2015).[11] F. Le Kien and A. Rauschenbeu-tel, Physical Review A , 1 (2017),arXiv:arXiv:1612.04516v1.[12] D. Kornovan, M. Petrov, and I. Iorsh,Physical Review B , 115162 (2017),arXiv:1701.06311.[13] A. S. Prasad, J. Hinney, S. Mahmoodian, K. Hammerer,S. Rind, P. Schneeweiss, A. S. Sørensen, J. Volz, andA. Rauschenbeutel, Nature Photonics , 719 (2020),arXiv:1911.09701.[14] A. Lagendijk, B. Van Tiggelen, and D. S. Wiersma,Physics Today , 24 (2009).[15] P. W. Anderson, Physical Review , 1492 (1958).[16] E. Abrahams, P. W. Anderson, D. C.Licciardello, and T. V. Ramakrishnan,Physical Review Letters (1979), 10.1103/PhysRevLett.42.673.[17] N. Hatano and D. R. Nelson,Physical Review Letters , 570 (1996).[18] P. Brouwer, P. Silvestrov, and C. Beenakker,Physical Review B - Condensed Matter and Materials Physics , R4333 (1997).[19] P. W. Brouwer, C. Mudry, B. D. Simons, and A. Altland,Physical Review Letters , 862 (1998).[20] F. H´ebert, M. Schram, R. T. Scalettar, W. B. Chen, andZ. Bai, European Physical Journal B , 465 (2011).[21] H. R. Haakh, S. Faez, and V. Sandoghdar,Physical Review A , 1 (2016).[22] G. Malpuech and A. Kavokin,Semiconductor Science and Technology , 1031 (1999).[23] V. A. Kosobukin, Physics of the Solid State , 1145 (2003).[24] V. A. Kosobukin and A. N. Poddubny˘ı,Physics of the Solid State , 1977 (2007).[25] I. M. Mirza, J. G. Hoskins, and J. C. Schotland,Physical Review A , 30 (2017), arXiv:1708.00902. [26] I. M. Mirza and J. C. Schotland,Journal of the Optical Society of America B , 1149 (2018),arXiv:1709.04641.[27] H. H. Jen, Physical Review A , 1 (2020),arXiv:2005.09855.[28] K. Kawabata and S. Ryu, , 1 (2020), arXiv:2005.00604.[29] T. Gruner and D.-G. Welsch,Physical Review A , 1818 (1996).[30] A. Asenjo-Garcia, J. D. Hood, D. E. Chang, and H. J.Kimble, Physical Review A , 1 (2017).[31] R. J. Coles, D. M. Price, J. E. Dixon, B. Royall,E. Clarke, P. Kok, M. S. Skolnick, A. M. Fox, and M. N.Makhonin, Nature Communications , 1 (2016).[32] E. Ivchenko, A. Nesvizhskii, and S. Jorda, Superlatticesand Microstructures , 1156 (1994).[33] M. R. Vladimirova, E. L. Ivchenko, and A. V. Kavokin,Semiconductors , 90 (1998).[34] G. Angelatos and S. Hughes, Optica , 370 (2016),arXiv:arXiv:1509.01613v1.[35] D. F. Kornovan, A. S. Sheremet, and M. I.Petrov, Physical Review B , 245416 (2016),arXiv:arXiv:1608.03202v1.[36] R. Movassagh and L. P. Kadanoff, Journal of Statistical Physics , Vol. 167 (SpringerUS, 2017) pp. 959–996, arXiv:1604.08295.[37] W. H. Weber and G. W. Ford,Physical Review B , 125429 (2004).[38] M. Petrov, Physical Review A , 023821 (2015).[39] A. Asenjo-Garcia, M. Moreno-Cardoner, A. Al-brecht, H. J. Kimble, and D. E. Chang,Physical Review X , 1 (2017), arXiv:1703.03382.[40] Y.-X. Zhang, C. Yu, and K. Mølmer,Physical Review Research , 1 (2020).[41] R. H. Dicke, Physical Review , 99 (1954).[42] B. Van Tiggelen, Localization of waves (1999).[43] R. Modak, S. Mukerjee, E. A.Yuzbashyan, and B. S. Shastry,New Journal of Physics (2016), 10.1088/1367-2630/18/3/033010,arXiv:1503.07019.[44] G. L. Celardo, R. Kaiser, and F. Borgonovi,Physical Review B , 1 (2016), arXiv:1604.07868.[45] A. N. Poddubny, arXiv (2019),arXiv:1908.01818.[46] S. Mahmoodian, M. ˇCepulkovskis, S. Das,P. Lodahl, K. Hammerer, and A. S.Sørensen, Physical Review Letters , 1 (2018),arXiv:1803.02428.[47] S. Mahmoodian, G. Calaj´o, D. E. Chang, K. Ham-merer, and A. S. Sørensen, arXiv1910.05828