A superatom picture of collective nonclassical light emission and dipole blockade in atom arrays
AA superatom picture of collective nonclassical light emission and dipole blockade inatom arrays
L. A. Williamson, M. O. Borgh, and J. Ruostekoski Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Faculty of Science, University of East Anglia, Norwich NR4 7TJ, United Kingdom (Dated: July 22, 2020)We show that two-time, second-order correlations of scattered photons from planar arrays andchains of atoms display nonclassical features that can be described by a superatom picture of thecanonical single-atom g ( τ ) resonance fluorescence result. For the superatom, the single-atomlinewidth is replaced by the linewidth of the underlying collective low light-intensity eigenmode.Strong light-induced dipole-dipole interactions lead to a correlated response, suppressed joint pho-ton detection events, and dipole blockade that inhibits multiple excitations of the collective atomicstate. For targeted subradiant modes, nonclassical nature of emitted light can be dramaticallyenhanced even compared with that of a single atom. The first direct evidence for the quantum natureof light was observed in resonance fluorescence of anatom [1–5], defining a significant historical milestone inquantum optics. Such quantum correlations can be iden-tified by measuring the second-order correlation functionfor the emitted field that represents a joint probabilityof two photon detection events appearing a time τ apartand can be defined as g ( τ ) ≡ lim t →∞ h : ˆ n ( t + τ )ˆ n ( t ) : ih ˆ n ( t ) i , (1)where : : denotes normal ordering and ˆ n ( t ) is the numberoperator for detected photons. Classically, g (0) ≥ g ( τ );hence g (0) < g ( τ ) implies quantum correlations in thephoton emission, and also defines antibunched photonemission [6, 7].Going beyond a single atom, in a noninteracting en-semble atoms will emit photons independently, leadingto an adulteration of the single-atom photon antibunch-ing that (neglecting interferences) scales as 1 − N − , withthe atom number N [3, 8, 9]. Correlated excitations foratomic ensembles have been observed for highly-excitedRydberg atoms in the microwave regime. The correlatedresponse is generated by dipolar interactions that inhibittransitions into all but singly-excited states, represent-ing dipole blockade [10–16], with applications to scalablequantum logic gates.In dense ensembles of cold atoms, also light-mediatedinteractions between the atoms can lead to drastic andunexpected phenomena [17–20] as multiple resonant scat-tering events give rise to a correlated response. Corre-lations can emerge even for the classical optical regimein the limit of low light intensity (LLI) of an incidentlaser [21, 22], and the quest for observing the effects ofstrong light-mediated interactions is attracting consid-erable attention [23–34]. Regular arrays of atoms areparticularly interesting for the exploration and manipula-tion of collective optical responses, as more recently stud-ied also in the quantum regime [35–47]. Transmission-resonance narrowing due to collective subradiance in the classical limit in a planar optical lattice was already ob-served [48] and other related experiments are rapidlyemerging [49].Here we show that photon emission events from pla-nar arrays and chains of atoms can still be described bythe single isolated atom picture, representing a collec-tive response of the entire atomic ensemble as one super-atom . By resonantly targeting LLI collective excitationeigenmodes, we show that even at high light intensitiesthe many-atom joint photon emission g ( τ ) displays thesame functional form as the single isolated atom g ( τ )of Eq. (1), but with the single atom linewidth replacedby the linewidth of the targeted LLI collective mode.We find that for sufficiently small lattice spacings stronglight-induced interactions can increase antibunching byestablishing correlations between the atoms that repre-sent inhibited multiple excitations of the collective stateof the atoms, or dipole blockade . Remarkably, for under-lying LLI eigenmodes for which the resonance linewidthis much narrower than the one for an isolated atom (sub-radiance), the nonclassical nature of emitted light can bedramatically enhanced to much longer time scales evencompared with those of a single atom.We consider two-level atoms with the dipole matrixelement d , coupled by light-mediated interactions andsubject to an incident laser field. The atom dynamics inthe rotating-wave approximation follows from the many-body quantum master equation (QME) for the reduceddensity matrix [50–52], dρdt = − i ~ X j [ H j , ρ ] + i X j‘ ( ‘ = j ) ∆ j‘ [ˆ σ + j ˆ σ − ‘ , ρ ]+ X j‘ γ j‘ (cid:0) σ − j ρσ + ‘ − σ + ‘ σ − j ρ − ρσ + ‘ σ − j (cid:1) (2)with the atomic operators ˆ σ + j = (ˆ σ − j ) † = | e i jj h g | , ˆ σ eej =ˆ σ + j ˆ σ − j , for ground | g i j and excited | e i j states of atom j located at r j and H j ≡ − ~ δ ˆ σ eej − d · E + ( r j )ˆ σ + j − d ∗ · E − ( r j )ˆ σ − j . (3) a r X i v : . [ phy s i c s . a t o m - ph ] J u l We take the positive-frequency component E + ( r ) = E e i k · r ˆ e of the laser field to be a monochromatic planewave of frequency ω = kc = 2 πc/λ and wavevector k ,detuned from the single-atom transition frequency ω by δ ≡ ω − ω . The light and atomic field amplitudesare here defined as slowly varying with the rapid oscil-lations at the laser frequency factored out. The light-mediated interactions between the atoms have both co-herent ∆ j‘ and dissipative γ j‘ contributions [ γ jj = γ ≡| d | k / (6 π ~ (cid:15) ) is the single atom linewidth]. These arethe real and imaginary parts, respectively, of d ∗ · G ( r j − r ‘ ) d / ~ (cid:15) , with G ( r ) the dipole radiation kernel of a pointdipole at the origin [52, 53].In the limit of LLI the dynamics reduces to that ofclassical coupled dipoles [54, 55]. In this regime we maydescribe [52] the optical response using LLI collective ra-diative excitation eigenmodes u m of H j‘ = ∆ j‘ + iγ j‘ (with ∆ jj ≡ ζ m + iυ m representing the collective linewidth υ m and line shift ζ m from the single-atom resonance. The linewidths can spanmany orders of magnitude, from extremely subradiant tosuperradiant [29, 56, 57].To calculate the rate of the detected photons for thesecond-order correlation function g ( τ ) of Eq. (1) weassume all the scattered photons are detected and in-tegrate ˆ n ( t ) = (2 (cid:15) c/ ~ ω ) R S dS ˆ E − sc ( r , t ) · ˆ E +sc ( r , t ) overa closed surface enclosing the atoms to give ˆ n =2 P j‘ γ j‘ σ + j σ − ‘ [52], where (cid:15) ˆ E +sc ( r , t ) = P j G ( r − r j ) d ˆ σ − j ( t ) denotes the scattered electric field summedover all the atoms. For a single isolated atom, a closedexpression for g ( τ ) can be derived analytically and isgiven by [1, 58], g ( γ,κ )2 ( τ ) ≡ − e − γτ/ (cid:18) cosh κγτ + 32 sinh κγτκ (cid:19) , (4)where κ ≡ p − I in /I s , and I in ≡ (cid:15) c |E ˆ e · ˆ d | / I s ≡ ~ ck γ/ π are the incident light and satu-ration intensities, respectively. For g ( γ,κ )2 (0) = 0 andlim τ →∞ g ( γ,κ )2 ( τ ) = 1; a single isolated atom thereforeshows photon antibunching, a manifestation of the factthat an atomic energy level can contain at most a singleexcitation.For the many-body system, g ( τ ) [Eq. (1)] in gen-eral needs to be evaluated by first solving the QME(2) numerically. The existence of nonclassical effectsfor a many-atom ensemble is less obvious than in thesingle-atom case. This can be illustrated by a simplecounting example of N independently emitting, non-interacting atoms: Neglecting interferences then yields g ( τ ) = 1 + N − [ g ( γ,κ )2 ( τ ) − FIG. 1. Superatom picture and nonclassical light scatteringfor a 3 × a = 0 . λ ) with a drivefield resonant with (a) the uniform superradiant ( υ ≈ . γ , NI in = 2 I s ) and (b) a subradiant ( υ ≈ . γ , NI in = 0 . I s )LLI collective eigenmode; g ( τ ) for the full quantum solution(blue solid line), superatom (black dashed line), and singleisolated atom (black dotted line). The red star marks thenoninteracting, interfering result of g (0), showing that inter-actions substantially enhance photon antibunching. For sub-radiant mode the nonclassical emission is enhanced comparedwith a single atom. Insets show the corresponding photondetection rates. many-body system with long-range dipole-dipole inter-actions. While we have also numerically calculated g for such situations, our key observation is that for sev-eral strongly correlated regimes of interest, Eq. (4) re-markably can still provide a qualitative description foremitted photon correlations that also exhibit nonclassicalscattered light and inhibited multiple excitations (dipoleblockade) even for increasing atom numbers. This is be-cause atoms collectively respond as one giant superatom ,where effectively the single-particle resonance linewidthis replaced by the resonance linewidth of the dominantunderlying LLI collective excitation eigenmode.The dominant eigenmode in a regular array is deter-mined by the resonance frequency and phase-matchingprofile with the incident field. We find then that themany-body g ( τ ) obeys a functional form analogous toEq. (4), g ( τ ) ≈ b h g ( υ,κ )2 ( τ ) − i , (5)where υ = υ ‘ is the linewidth of the resonant LLI eigen-mode u ‘ (found by diagonalising H j‘ [52]) and κ ≡ p − I I in /I s , with I s ≡ ~ ck υ/ π . The overlap of thedrive field with u ‘ , I = | P j e − i k · r j u ‘ ( r j ) | , representsthe sum of the coupling strengths of light over all theatoms and can for uniform targeted modes with perfectphase-matching be replaced by N , reflecting the collec-tive N -enhancement of the response. There is an overallnormalization in Eq. (5) by b ≈ − g (0) that accountsfor nonclassical light emission at zero delay due to many-body correlations. When b > N − , these are enhancedcompared to the noninteracting, noninterfering case.In the numerics, we consider 2D square arrays of atomsin the xy plane and 1D chains along the x axis, with theincident light direction ˆ k = ˆ z , polarized along the atomicdipoles ˆ d = ˆ x . We solve the QME by directly integratingEq. (2) or by unraveling the evolution into stochasticquantum trajectories of state vectors [52, 59–62].We demonstrate nonclassically scattered light from astrongly interacting 3 × enhanced due to interactions. This corresponds to inhibited multi-ple excitations of the collective atomic state due to light-mediated dipole-dipole interactions, representing dipoleblockade of optical transitions, analogous to collectivesuppression of microwave Rydberg excitations [11]. Thedrive, which is uniform across the plane, couples moststrongly to the most superradiant LLI eigenmode withno phase variation across the atoms. We show that thesuperatom picture (SAP) [Eq. (5)] provides an excellentdescription of g ( τ ) for light resonant with this mode( υ ≈ . γ , I ≈ . N ) [Fig. 1(a)]. The antibunchingdelay time is much shorter than that of a single atom.The incident light can also be tuned to target a subra-diant eigenmode. Here we consider the eigenmode withthe fourth broadest resonance, with υ ≈ . γ and I ≈ . N . We find that the SAP again accuratelydescribes the dynamics [Fig. 1(b)]. The mode is approx-imately u ‘ ( r j ) ≈ . π ˆ x · r j / a ) − .
12 with the con-stant giving rise to nonorthogonality of the eigenmodes.The linewidths of the superradiant and subradiant eigen-modes differ by two orders of magnitude, resulting invery different responses, and in both cases radically de-parting from the single-atom result. The substantiallylarger values of 1 − g (0) compared to those of noninter-acting atoms show enhanced antibunching due to interac-tions. In the subradiant case nonclassical effects are en-hanced compared even with those of a single atom, withthe nonclassical delay time of g approximately 10 timeslarger than that of a single atom. Subradiant excitationscan therefore provide much extended antibunching timescales compared with Rydberg atom based vapor cell de-vices [16], also avoiding two-photon excitations and theinvolvement of highly excited Rydberg states that aresensitive to electric and magnetic field gradients.The SAP also provides an excellent description of thetransient photon scattering rate h ˆ n ( t ) i (insets to Fig. 1and Fig. S1 in [52]). The SAP for the photon scat-tering rate is h ˆ n ( t ) i ≈ n ( υ,κ ) ( t ), where n ( γ,κ ) ( t ) ≡ [ I in / ( I in + I s )] g ( γ,κ )2 ( t ) is the photon scattering rate for asingle, isolated atom [58, 63].The suppressed short-delay joint photon detectionevents in g represent dipole blockade that inhibits mul-tiple excitations of the collective atomic state, as illus-trated in the excited-state atom number distributions(Fig. 2). Already for a 2 × a = 0 . λ , the two-excitation weight is . − at N I in = 2 I s , but rapidly increases to 0.1for a = 0 . λ , as the antibunching is reduced and the -6 -3 n = 1 n = 22 . I s . I s a/ λ P ( n ) (a) n = 1 n = 20 . I s a/ λ (b) FIG. 2. Dipole blockade by the occupation weights P ( n ) ofstates with n = 1 and 2 excited atoms as a function of thelattice spacing a in a 2 × a , n = 2 statesare suppressed by the blockade, regardless of intensity, but theblockade is weakened for larger a and the occupation increasesdramatically while the weight of n = 1 states changes little incomparison. Drive intensity NI in given as a multiple of I s . -20 0 2000.51 10 -3 FIG. 3. Validity of the superatom picture and the effectof position fluctuations of the atoms for a field resonant withthe uniform superradiant LLI mode. (a) Relative error η ofthe SAP as a function of lattice spacing (bottom axis) for a3 × NI in = 0 . I s (blue circles) and NI in =2 I s (red diamonds), and as a function of atom number (topaxis, crosses) for a chain at a = 0 . λ , NI in = 0 . I s ; (b)position fluctuations of the atoms improve the accuracy ofSAP ( a = 0 . λ , 2 × . a ateach lattice site (yellow dashed-dotted line), superatom (blackdashed line), and single atom (black dotted line). The red starmarks the noninteracting, interfering result of g (0). Inset:Power spectrum for a 2 × I in = 2 I s , a = 0 . λ )showing a superradiant central peak (SAP result: dashed line)with additional small excitations far off resonance. dipole blockade removed. The origin of the blockade canbe understood also in the excitation spectrum P (Ω) ∝ R dτ e i Ω τ P j‘ γ j‘ h ˆ σ + j ( t + τ )ˆ σ − ‘ ( t ) i [inset to Fig. 3(b)] thatshows how the second photon excitation is shifted due tothe dipole-dipole interactions.The accuracy and the regimes of validity of theSAP in both planar arrays and chains are analyzed inFig. 3. The uniform phase profile of the drive acrossthe atoms most strongly couples to the superradiant,uniform eigenmode, and we show the relative devia-tions η ≡ max τ<τ | g ( τ ) /b − g ( υ,κ )2 ( τ ) | (calculated un-til τ , such that for all τ . τ , g ( τ ) <
1; see alsoFig. S2 [52]). The SAP describes the behavior of g ( τ )very well for a . . λ and remains qualitatively accurateup to a ∼ . λ (a 9 atom chain gives similar results). Theonset of the plateau around a ≈ . λ , irrespective oflight intensity, coincides with LLI eigenmode resonancesoverlapping with the superradiant mode. For a & . λ ,the SAP deviates from g ( τ ). The deviations as a func-tion of N in Fig. 3(a) show how the accuracy of the SAPdecreases gradually in larger systems.Increasing deviations for large values of τ for a & . λ are due to the presence of a persistent oscillation [aweak oscillation is also visible in Fig. 1(a)]. To un-derstand this behavior, we look at the steady-state oc-cupations of the LLI modes for h σ − j i , defined as [64] L m ≡ P j | u m ( r j ) h σ − j i| / P j‘ | u ‘ ( r j ) h σ − j i| . The pres-ence of the persistent oscillation coincides with a simulta-neous nonnegligible occupation of two eigenmodes. Onecan then qualitatively understand the effect of the two-mode interference from the linear combination g ( τ ) ≈ b h Cg ( υ ,κ )2 ( τ ) + (1 − C ) g ( υ ,κ )2 ( τ ) − i , (6)where the increasing contribution from the less radiantmode with increasing lattice spacing leads to deviationsfrom the simple SAP at large τ . Although we consideronly chains and arrays, systems with higher symmetrysuch as rings [65] or configurations that optimize inter-actions offer the potential to more effectively target in-dividual superatom resonances and enhance the photonblockade.For atoms in optical lattices, proposals exist to producea tight atom confinement [66], but generally the atomicpositions fluctuate. We can take into account the positionfluctuations in the numerics by ensemble-averaging overmany stochastic realizations of randomly sampled atompositions in each lattice site [56]. We find in Fig. 3(b)that the accuracy of the superatom picture increases dueto the fluctuations, as the oscillations resulting from thesecond eigenmode contribution are washed out. However,increasing position fluctuations eventually also start in-creasing g (0).The normalization of the SAP two-time correlationfunction at zero delay g (0) in Eq. (5) represents thestrength of nonclassical and correlated light emission ofthe atoms. For noninteracting atoms in the absence ofmultiple scattering, interference effects only slightly mod-ify the result g (0) = 1 − N − . Strong light-mediatedcorrelations, however, can substantially shift the value of g (0), directly reflected in the antibunching of the emit-ted photons. In Fig. 4(a) we show g (0) as a function oflattice spacing and atom number, with the drive tuned tothe uniform LLI eigenmode. We find that light-mediatedinteractions enhance the nonclassical nature of light forsmall lattice spacing (up to a . . λ ), which coincideswith the regime where the SAP shows good accuracyover all values of τ . For chains with large lattice spacing( a & . λ ), light-mediated interactions between atomsare no longer sufficient to establish collective correlation FIG. 4. Enhanced antibunching due to quantum correlationsof light-induced dipole-dipole interactions in a 9-atom chainand 3 × g (0) for a chain as a function of latticespacing (inset: array) compared with noninteracting atoms(dashed line); (b) g (0) as a function of atom number forchains with different lattice spacing a/λ compared with non-interacting, noninterfering atoms (dashed line). Solid linesare guides for the eye. effects, and g (0) follows the noninteracting, noninter-fering scaling g (0) = 1 − N − [Fig. 4(b)], with smallor absent antibunching. In denser arrays, however, wefind that nonclassical collective effects persist also as theatom number increases. For example, g (0) ≈ .
08 for a9-atom chain with a = 0 . λ .In Rydberg atoms, dipole blockade inhibits multipleexcitations within the blockade radius R [67]. Due to thelong-range interactions present in our system, R is in gen-eral not well defined. However, power-law-fit estimatesof the dependence of g (0) on the system size can be ob-tained from Fig. 4(b), resulting in R of the order of λ ,with a small roughly linear increase in R with decreasinglattice spacing [68]. Correlations can be suppressed witha sufficiently broad laser [69] with increasing contribu-tions from multiple modes [Eq. (6)] when the bandwidthnotably exceeds γ .The time-honoured two-time correlation function (1)for joint photon emission events from a single atom re-veals nonclassical resonance fluorescence of light [1, 58].Here we showed that the same functional form also de-scribes emission from strongly coupled arrays of atoms,representing a superatom picture of correlated many-atom resonance fluorescence. For a single atom the sup-pression of joint photon emission events is a direct conse-quence of the fermionic statistics with (ˆ σ ± ) = 0 for thesingle excitation; after the photon emission the electronis in the ground state and cannot re-emit before being ex-cited again. For a many-atom system, the antibunchingwith g (0) ’ g ( τ ) correlations illustrates how rel-atively simple and intuitive representations could possi-bly more generally be extended to understand stronglycorrelated many-body phenomena in quantum optics farbeyond linearly responding coupled classical dipoles.We have become aware of a related parallel the-oretical work on the calculation of dipolar blockadein atom chains in Ref. [71]. We acknowledge finan-cial support from the Engineering and Physical Sci-ences Research Council (Grants Nos. EP/S002952/1 andEP/P026133/1) and discussions with L. F. dos Santos. [1] H J Carmichael and D F Walls, “Proposal for the mea-surement of the resonant Stark effect by photon corre-lation techniques,” Journal of Physics B: Atomic andMolecular Physics , L43–L46 (1976).[2] H. J. Kimble, M. Dagenais, and L. Mandel, “Photonantibunching in resonance fluorescence,” Phys. Rev. Lett. , 691–695 (1977).[3] H. J. Kimble, M. Dagenais, and L. Mandel, “Multiatomand transit-time effects on photon-correlation measure-ments in resonance fluorescence,” Phys. Rev. A , 201–207 (1978).[4] M. Dagenais and L. Mandel, “Investigation of two-timecorrelations in photon emissions from a single atom,”Phys. Rev. A , 2217–2228 (1978).[5] D. F. Walls, “Evidence for the quantum nature of light,”Nature , 451–454 (1979).[6] X. T. Zou and L. Mandel, “Photon-antibunching andsub-poissonian photon statistics,” Phys. Rev. A , 475–476 (1990).[7] H. Paul, “Photon antibunching,” Rev. Mod. Phys. ,1061–1102 (1982).[8] E Jakeman, ER Pike, PN Pusey, and JM Vaughan,“The effect of atomic number fluctuations on photon an-tibunching in resonance fluorescence,” J. Phys. A: Math.Gen. , L257 (1977).[9] HJ Carmichael, P Drummond, P Meystre, andDF Walls, “Intensity correlations in resonance fluores-cence with atomic number fluctuations,” J. Phys. A:Math. Gen. , L121 (1978).[10] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cˆot´e,and M. D. Lukin, “Fast quantum gates for neutralatoms,” Phys. Rev. Lett. , 2208–2211 (2000).[11] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan,D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole block-ade and quantum information processing in mesoscopicatomic ensembles,” Phys. Rev. Lett. , 037901 (2001).[12] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D.Yavuz, T. G. Walker, and M. Saffman, “Observation ofrydberg blockade between two atoms,” Nature Physics ,110–114 (2009).[13] Alpha Ga¨etan, Yevhen Miroshnychenko, Tatjana Wilk,Amodsen Chotia, Matthieu Viteau, Daniel Comparat,Pierre Pillet, Antoine Browaeys, and Philippe Grang- ier, “Observation of collective excitation of two individualatoms in the rydberg blockade regime,” Nature Physics , 115–118 (2009).[14] M. Saffman, T. G. Walker, and K. Mølmer, “Quantuminformation with rydberg atoms,” Rev. Mod. Phys. ,2313–2363 (2010).[15] Peter Schauß, Marc Cheneau, Manuel Endres, TakeshiFukuhara, Sebastian Hild, Ahmed Omran, Thomas Pohl,Christian Gross, Stefan Kuhr, and Immanuel Bloch,“Observation of spatially ordered structures in a two-dimensional rydberg gas,” Nature , 87–91 (2012).[16] Fabian Ripka, Harald K¨ubler, Robert L¨ow, and TilmanPfau, “A room-temperature single-photon source basedon strongly interacting Rydberg atoms,” Science ,446–449 (2018).[17] Juha Javanainen, Janne Ruostekoski, Yi Li, and Sung-Mi Yoo, “Shifts of a resonance line in a dense atomicsample,” Phys. Rev. Lett. , 113603 (2014).[18] S. E. Skipetrov and I. M. Sokolov, “Absence of ander-son localization of light in a random ensemble of pointscatterers,” Phys. Rev. Lett. , 023905 (2014).[19] W. Guerin, M.T. Rouabah, and R. Kaiser, “Light in-teracting with atomic ensembles: collective, cooperativeand mesoscopic effects,” Journal of Modern Optics ,895–907 (2017).[20] Juha Javanainen, Janne Ruostekoski, Yi Li, and Sung-Mi Yoo, “Exact electrodynamics versus standard opticsfor a slab of cold dense gas,” Phys. Rev. A , 033835(2017).[21] Olivier Morice, Yvan Castin, and Jean Dalibard, “Re-fractive index of a dilute Bose gas,” Phys. Rev. A ,3896–3901 (1995).[22] Janne Ruostekoski and Juha Javanainen, “Quantum fieldtheory of cooperative atom response: Low light inten-sity,” Phys. Rev. A , 513–526 (1997).[23] S. Balik, A. L. Win, M. D. Havey, I. M. Sokolov, andD. V. Kupriyanov, “Near-resonance light scattering froma high-density ultracold atomic Rb gas,” Phys. Rev. A , 053817 (2013).[24] Julien Chab´e, Mohamed-Taha Rouabah, Louis Bellando,Tom Bienaim´e, Nicola Piovella, Romain Bachelard, andRobin Kaiser, “Coherent and incoherent multiple scat-tering,” Phys. Rev. A , 043833 (2014).[25] J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sor-tais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski,“Observation of suppression of light scattering inducedby dipole-dipole interactions in a cold-atom ensemble,”Phys. Rev. Lett. , 133602 (2014).[26] C. C. Kwong, T. Yang, M. S. Pramod, K. Pandey, D. De-lande, R. Pierrat, and D. Wilkowski, “Cooperative emis-sion of a coherent superflash of light,” Phys. Rev. Lett. , 223601 (2014).[27] S. Jennewein, M. Besbes, N. J. Schilder, S. D. Jenkins,C. Sauvan, J. Ruostekoski, J.-J. Greffet, Y. R. P. Sortais,and A. Browaeys, “Coherent scattering of near-resonantlight by a dense microscopic cold atomic cloud,” Phys.Rev. Lett. , 233601 (2016).[28] S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Both-well, J. Schachenmayer, T. L. Nicholson, R. Kaiser, S. F.Yelin, M. D. Lukin, A. M. Rey, and J. Ye, “Collectiveatomic scattering and motional effects in a dense coher-ent medium,” Nat Commun , 11039 (2016).[29] S. D. Jenkins, J. Ruostekoski, J. Javanainen, R. Bour-gain, S. Jennewein, Y. R. P. Sortais, and A. Browaeys, “Optical resonance shifts in the fluorescence of thermaland cold atomic gases,” Phys. Rev. Lett. , 183601(2016).[30] P. C. Bons, R. de Haas, D. de Jong, A. Groot, andP. van der Straten, “Quantum enhancement of the indexof refraction in a Bose-Einstein condensate,” Phys. Rev.Lett. , 173602 (2016).[31] William Guerin, Michelle O. Ara´ujo, and Robin Kaiser,“Subradiance in a large cloud of cold atoms,” Phys. Rev.Lett. , 083601 (2016).[32] Shimon Machluf, Julian B. Naber, Maarten L. Soudijn,Janne Ruostekoski, and Robert J. C. Spreeuw, “Col-lective suppression of optical hyperfine pumping in denseclouds of atoms in microtraps,” Phys. Rev. A , 051801(2019).[33] L. Corman, J. L. Ville, R. Saint-Jalm, M. Aidels-burger, T. Bienaim´e, S. Nascimb`ene, J. Dalibard, andJ. Beugnon, “Transmission of near-resonant light througha dense slab of cold atoms,” Phys. Rev. A , 053629(2017).[34] R. J. Bettles, T. Ilieva, H. Busche, P. Huillery, S. W.Ball, N. L. R. Spong, and C. S. Adams, “Collectivemode interferences in light-matter interactions,” (2018),arXiv:1808.08415.[35] Martin Hebenstreit, Barbara Kraus, Laurin Ostermann,and Helmut Ritsch, “Subradiance via entanglement inatoms with several independent decay channels,” Phys.Rev. Lett. , 143602 (2017).[36] Ryan Jones, Reece Saint, and Beatriz Olmos, “Far-fieldresonance fluorescence from a dipole-interacting laser-driven cold atomic gas,” Journal of Physics B: Atomic,Molecular and Optical Physics , 014004 (2017).[37] Yu-Xiang Zhang and Klaus Mølmer, “Theory of subra-diant states of a one-dimensional two-level atom chain,”Phys. Rev. Lett. , 203605 (2019).[38] A. Grankin, P. O. Guimond, D. V. Vasilyev, B. Vermer-sch, and P. Zoller, “Free-space photonic quantum linkand chiral quantum optics,” Phys. Rev. A , 043825(2018).[39] P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Verm-ersch, and P. Zoller, “Subradiant bell states in distantatomic arrays,” Phys. Rev. Lett. , 093601 (2019).[40] Robert J Bettles, Mark D Lee, Simon A Gardiner, andJanne Ruostekoski, “Quantum and Nonlinear Effectsin Light Transmitted through Planar Atomic Arrays,”(2019), arXiv:1907.07030.[41] K. E. Ballantine and J. Ruostekoski, “Subradiance-protected excitation spreading in the generation of col-limated photon emission from an atomic array,” Phys.Rev. Research , 023086 (2020).[42] Jemma A Needham, Igor Lesanovsky, and Beatriz Ol-mos, “Subradiance-protected excitation transport,” NewJournal of Physics , 073061 (2019).[43] Chunlei Qu and Ana M. Rey, “Spin squeezing and many-body dipolar dynamics in optical lattice clocks,” Phys.Rev. A , 041602 (2019).[44] L. A. Williamson and J. Ruostekoski, “Optical responseof atom chains beyond the limit of low light intensity:The validity of the linear classical oscillator model,”Phys. Rev. Research , 023273 (2020).[45] Yu-Xiang Zhang, Chuan Yu, and Klaus Mølmer, “Subra-diant bound dimer excited states of emitter chains cou-pled to a one dimensional waveguide,” Phys. Rev. Re-search , 013173 (2020). [46] Lo¨ıc Henriet, James S. Douglas, Darrick E. Chang, andAndreas Albrecht, “Critical open-system dynamics in aone-dimensional optical-lattice clock,” Phys. Rev. A ,023802 (2019).[47] Ana Asenjo-Garcia, H. J. Kimble, and Darrick E. Chang,“Optical waveguiding by atomic entanglement in multi-level atom arrays,” Proceedings of the National Academyof Sciences , 25503–25511 (2019).[48] Jun Rui, David Wei, Antonio Rubio-Abadal, Simon Hol-lerith, Johannes Zeiher, Dan M. Stamper-Kurn, Chris-tian Gross, and Immanuel Bloch, “A subradiant opti-cal mirror formed by a single structured atomic layer,”(2020), arXiv:2001.00795.[49] A. Glicenstein, Ferioli, Sibalic G., N.L. Brossard,I. Ferrier-Barbut, and A. Browaeys, “Collective shiftof resonant light scattering by a one-dimensional atomicchain, eprint arxiv:2004.05395,” (2020).[50] R. H. Lehmberg, “Radiation from an N -Atom System. I.General Formalism,” Phys. Rev. A , 883–888 (1970).[51] G. S. Agarwal, “Master-equation approach to sponta-neous emission,” Phys. Rev. A , 2038–2046 (1970).[52] See Supplemental Material for technical details, whichincludes Refs. [1, 22, 50, 51, 53, 58–61, 63, 64, 72–78].[53] John David Jackson, Classical Electrodynamics , 3rd ed.(Wiley, New York, 1999).[54] Juha Javanainen, Janne Ruostekoski, Bjarne Vester-gaard, and Matthew R. Francis, “One-dimensional mod-eling of light propagation in dense and degenerate sam-ples,” Phys. Rev. A , 649–666 (1999).[55] Mark D. Lee, Stewart D. Jenkins, and Janne Ru-ostekoski, “Stochastic methods for light propagationand recurrent scattering in saturated and nonsaturatedatomic ensembles,” Phys. Rev. A , 063803 (2016).[56] Stewart D. Jenkins and Janne Ruostekoski, “Controlledmanipulation of light by cooperative response of atomsin an optical lattice,” Phys. Rev. A , 031602 (2012).[57] R. T. Sutherland and F. Robicheaux, “Collective dipole-dipole interactions in an atomic array,” Phys. Rev. A ,013847 (2016).[58] HJ Carmichael and DF Walls, “A quantum-mechanicalmaster equation treatment of the dynamical Stark ef-fect,” J. Phys. B: At. Mol. Phys. , 1199 (1976).[59] Jean Dalibard, Yvan Castin, and Klaus Mølmer, “Wave-function approach to dissipative processes in quantumoptics,” Phys. Rev. Lett. , 580–583 (1992).[60] L. Tian and H. J. Carmichael, “Quantum trajectorysimulations of two-state behavior in an optical cavitycontaining one atom,” Phys. Rev. A , R6801–R6804(1992).[61] R. Dum, P. Zoller, and H. Ritsch, “Monte carlo sim-ulation of the atomic master equation for spontaneousemission,” Phys. Rev. A , 4879–4887 (1992).[62] Howard Carmichael, An open systems approach to quan-tum optics (Springer-Verlag, 1993).[63] B. R. Mollow and M. M. Miller, “The damped driventwo-level atom,” Ann. Phys. , 464–478 (1969).[64] G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, “Stor-ing light with subradiant correlations in arrays of atoms,”Phys. Rev. Lett. , 243601 (2016).[65] H. Dong, S.-W. Li, Z. Yi, G. S. Agarwal, and M. O.Scully, “Photon-blockade induced photon anti-bunchingin photosynthetic antennas with cyclic structures, eprintarxiv:1608.04364,” (2016).[66] Y. Wang, S. Subhankar, P. Bienias, M. Lkacki, T-C. Tsui, M. A. Baranov, A. V. Gorshkov, P. Zoller, J. V.Porto, and S. L. Rolston, “Dark state optical lattice witha subwavelength spatial structure,” Phys. Rev. Lett. ,083601 (2018).[67] D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P.Zhang, R. Cˆot´e, E. E. Eyler, and P. L. Gould, “Localblockade of rydberg excitation in an ultracold gas,” Phys.Rev. Lett. , 063001 (2004).[68] In Fig. 4(b) the 1 /r interaction is cancelled due to ˆ d · ˆ x = 1, but different dipole orientations for which the 1 /r interaction is nonzero show analogous behavior.[69] Nobuyuki Takei, Christian Sommer, Claudiu Genes,Guido Pupillo, Haruka Goto, Kuniaki Koyasu, HisashiChiba, Matthias Weidem¨uller, and Kenji Ohmori, “Di-rect observation of ultrafast many-body electron dynam-ics in an ultracold Rydberg gas,” Nat. Commun. , 13449(2016).[70] G. Facchinetti and J. Ruostekoski, “Interaction of lightwith planar lattices of atoms: Reflection, transmis-sion, and cooperative magnetometry,” Phys. Rev. A ,023833 (2018).[71] A. Cidrim, T. S. do Espirito Santo, J. Schachenmayer,R. Kaiser, and R. Bachelard, “Photon blockade with ground-state neutral atoms, eprint arxiv:2004.14720,”(2020).[72] MR Andrews, M-O Mewes, NJ Van Druten, DS Dur-fee, DM Kurn, and W Ketterle, “Direct, nondestructiveobservation of a Bose condensate,” Science , 84–87(1996).[73] HJ Carmichael and Kisik Kim, “A quantum trajectoryunraveling of the superradiance master equation,” Opt.Commun. , 417–427 (2000).[74] H. J. Kimble and L. Mandel, “Theory of resonance fluo-rescence,” Phys. Rev. A , 2123–2144 (1976).[75] B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. , 1969–1975 (1969).[76] Klaus Mølmer, Yvan Castin, and Jean Dalibard, “MonteCarlo wave-function method in quantum optics,” J. Opt.Soc. Am. B , 524–538 (1993).[77] J. P. Clemens, L. Horvath, B. C. Sanders, and H. J.Carmichael, “Collective spontaneous emission from a lineof atoms,” Phys. Rev. A , 023809 (2003).[78] Janne Ruostekoski, M. J. Collett, Robert Graham, andDan F. Walls, “Macroscopic superpositions of bose-einstein condensates,” Phys. Rev. A , 511–517 (1998). upplemental Material to“A superatom picture of collective nonclassical light emission and dipole blockade in atom arrays” L. A. Williamson, M. O. Borgh, and J. Ruostekoski Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Faculty of Science, University of East Anglia, Norwich NR4 7TJ, United Kingdom (Dated: July 22, 2020)
S.A. Formalism
The dynamics in the interaction picture for an array of N two-level atoms driven by a coherent laser field is describedby the many-body quantum master equation (QME) for thereduced density matrix ρ [S1, S2], d ρ dt = − i ~ X j [ H j , ρ ] + i X j ‘ ( ‘ , j ) ∆ j ‘ [ ˆ σ + j ˆ σ − ‘ , ρ ] + X j ‘ γ j ‘ (cid:16) σ − j ρσ + ‘ − σ + ‘ σ − j ρ − ρσ + ‘ σ − j (cid:17) . (S1)Here ˆ σ + j = ( ˆ σ − j ) † = | e i j j h g | , ˆ σ eej = ˆ σ + j ˆ σ − j are the atomicraising (lowering) and excited state population operators, withground | g i j and excited | e i j states of atom j . The Hamiltonianoperator H j ≡ − ~ δ ˆ σ eej − d · E + ( r j ) ˆ σ + j − d ∗ · E − ( r j ) ˆ σ − j . (S2)describes the dynamics of a single atom at position r j with thedipole moment d ≡ D ˆ d . Here D is the reduced dipole ma-trix element that we assume is real without loss of generality.The atoms are driven by a plane-wave drive with positive fre-quency component E + ( r ) = E e i k · r ˆ e = [ E − ( r )] ∗ . The drivefield frequency ω is detuned from the single-atom transitionfrequency ω by δ ≡ ω − ω . Here the atomic and light fieldsare slowly varying, such that the rapidly rotating phase fac-tors e ± i ω t are removed by moving into an interaction pictureand making the rotating wave approximation (by omitting thefast co-rotating terms ˆ σ − m e i ω t , ˆ σ + m e − i ω t ). The single-atom dy-namics is thus described by H j together with the decay terms γ (2 σ − j ρσ + j − σ + j σ − j ρ − ρσ + j σ − j ), where γ ≡ D k / (6 π ~ (cid:15) ) isthe single atom Wigner-Weisskopf linewidth.The scattered light is given as a sum of the scattered lightfrom all the atoms (cid:15) ˆ E + sc ( r , t ) = X j G ( r − r j ) d ˆ σ − j ( t ) (S3)where the dipole radiation kernel [S3], G ( r ) d = − d δ ( r )3 + k π ( (ˆ r × d ) × ˆ r e ikr kr − [3ˆ r (ˆ r · d ) − d ] " i ( kr ) − kr ) e ikr ) , (S4)represents the monochromatic positive frequency componentof the scattered light at r from the dipole d located at the ori-gin. The interaction terms in Eq. (S1) arise from each atom j being driven by the light scattered from all other atoms ‘ , j .These radiative dipole-dipole couplings have coherent ∆ j ‘ anddissipative γ j ‘ contributions given by the real and imaginaryparts of ∆ j ‘ + i γ j ‘ = ~ (cid:15) d ∗ · G ( r j − r ‘ ) d . (S5)Note that Eq. (S5) gives γ j j = γ . A proper calculation of ∆ j j would involve evaluation of the Lamb shift, and we assumethis is incorporated to the single-atom detuning δ .The total field at position r is given as a sum of the inci-dent field E + ( r ) and the scattered light ˆ E + sc ( r , t ). We assumethat the incident field has been blocked before detection, forexample by a thin wire as in the dark-ground imaging tech-nique of [S4]. Hence only the scattered field is detected, withintensity I sc ( r , t ) = (cid:15) c h ˆ E − sc ( r , t ) · ˆ E + sc ( r , t ) i . (S6)Integrating the scattered intensity over the detector surface S gives the total count rate, which is the expectation value ofthe operatorˆ n ( t ) = (cid:15) c ~ ω Z S dS ˆ E − sc ( r , t ) · ˆ E + sc ( r , t ) = X j ,‘ I j ‘ ˆ σ + j ( t ) ˆ σ − ‘ ( t ) . (S7)with interference integrals I j ‘ ≡ c ~ (cid:15) ω Z S dS h G ( r − r j ) d i ∗ G ( r − r ‘ ) d (S8)We assume the detector lies in the radiation zone kr (cid:29)
1, hence we can expand the dipole radiation kernels to ob-tain [S5] I j ‘ = γ π Z S d θ d φ sin θ (cid:16) − | ˆ r · ˆ d | (cid:17) e ik ˆ r · ( r j − r ‘ ) = γ j ‘ . (S9)Hence we arrive atˆ n ( t ) = X j ‘ γ j ‘ ˆ σ + j ( t ) ˆ σ − ‘ ( t ) (S10)for the photon-number operator. S.B. Single-atom physics
For a single isolated atom, both the photon detection rate h ˆ n ( t ) i and the second-order correlation function g ( τ ) can be a r X i v : . [ phy s i c s . a t o m - ph ] J u l evaluated analytically to yield [S6–S9] g ( τ ) = − e − γτ/ cosh κγτ +
32 sinh κγτκ ! h ˆ n ( t ) i = I in I in + I s g ( τ ) (S11)The parameter κ = √ − I in / I s depends on the ratio of theincident intensity to the single atom saturation intensity, anddetermines the spectral properties of the atom [S10]. For lowincident light intensity I in (cid:28) I s , Eqs. (S11) are dominated bya term proportional to the single decay e − γ t . Here the sin-gle atom linewidth exceeds the single atom Rabi frequency γ √ I in / I s and Rabi oscillations are suppressed. Conversely,when I in & I s , the parameter κ is imaginary and hence boththe photon scattering rate and g ( τ ) display decaying Rabi os-cillations. S.C. Limit of low light intensity
A consistent low light intensity (LLI) theory of Eq. (S1),can be obtained [S11] from the equations of motions by re-taining terms containing at most one of either σ ± j or the inci-dent field amplitude. The only remaining equations of motionfor the expectation values of atomic operators from Eq. (S1)are those for h σ ± j i , which in the LLI are, d h σ − j i dt = i δ h σ − j i + i X ‘ H j ‘ h σ − ‘ i + i d · E + ( r j ) ~ (S12)with H j ‘ ≡ ∆ j ‘ + i γ j ‘ (with ∆ j j ≡
0; recall that γ j j = γ ). Hencethe atom dynamics evolves linearly in terms of the drive. Herewe expand the complex symmetric matrix H j ‘ in a completebasis of eigenstates u m , m = , ..., N , which are the LLI col-lective eigenmodes, X ‘ H j ‘ u m ( r ‘ ) = ( ζ m + i υ m ) u m ( r j ) , (S13)where the imaginary part, υ m , of the eigenvalue gives the col-lective linewidth of the eigenmode u m and the real part the lineshift ζ m from the single-atom resonance. Note that the eigen-states u m are not necessarily orthogonal, however, they do sat-isfy the biorthogonality condition P j u m ( r j ) u n ( r j ) = δ mn (afterappropriate normalization of the u m ) apart from possible rarecases when P j u m ( r j ) u m ( r j ) = h σ − j i , a measure ofthe occupation of the LLI collective mode u m is given by [S12] L m ≡ P j | u m ( r j ) h σ − j i| P j ‘ | u ‘ ( r j ) h σ − j i| . (S14) S.D. Quantum trajectories
A direct way to solve the QME (S1) is via matrix expo-nentiation of the density-matrix evolution operator. This is convenient for small atom numbers. For larger systems, how-ever, the size of the density matrix becomes prohibitively large( ∼ N ). A more profitable scaling is to employ the MonteCarlo wavefunction method of quantum trajectories [S13–S16]. The evolution of the density matrix is then representedas the ensemble average of many individual realizations ofthe evolution of a many-body wavefunction ψ ( t ), whose sizescales as ∼ N , under a non-Hermitian Hamiltonian operator H S − i ~ X j ˆ J † j ˆ J j , (S15)where ˆ J j are jump operators derived from the dissipative termsof QME and H S represents Hermitian Hamiltonian evolution.Incoherent evolution is incorporated via stochastic quantumjumps that happen with a probability proportional to the lossof norm of the wavefunction as it evolves under (S15). Onecan show that this formalism is exactly equivalent to QME forthe operator expectation values [S16].A many-body system supports multiple decay channels andunraveling of the QME into an explicit mixture of pure statessubject to stochastic evolution can be done in several di ff er-ent ways, corresponding to di ff erent constructions of the jumpoperators, as long as the full incoherent evolution in Eq. (S1)is accounted for. For the driven array of two-level atoms ofthe QME (S1) we follow here the “source-mode” quantumtrajectory formalism [S5, S17]. In the single-excitation limit,these jumps correspond to the emission of photons, while theirphysical interpretation is more convoluted at su ffi ciently highlight intensities to cause multiple excitations when the jumpoperators become formal constructions that do not necessarilycorrespond to any specific measurement record. They, how-ever, provide a straightforward mapping of Eq. (S1) to the evo-lution of quantum trajectories of state vectors.To formulate the source-mode jump operators, the matrix γ j ‘ is diagonalized to find its eigenvalues λ j and the corre-sponding eigenvectors b j = ( b j , . . . , b N j ) T . The jump opera-tors are then defined asˆ J j = p λ j b Tj ˆ Σ , ˆ J † j = p λ j ˆ Σ † b j , (S16)where ˆ Σ = ˆ σ − ... ˆ σ − N , ˆ Σ † = (cid:16) ˆ σ + , . . . , ˆ σ + N (cid:17) . (S17)Then defining H S = X j H j + ~ X j ‘ ( ‘ , j ) ∆ j ‘ ˆ σ + j ˆ σ − ‘ , (S18)the problem has been cast in the form of quantum trajecto-ries and the corresponding non-Hermitian Hamiltonian for thewavefunction evolution follows from Eq. (S15). The quantumtrajectory evolution can then be evaluated as described in, e.g.,Ref. [S18]. Thanks to the source-mode unraveling, the dissi-pative component of Eq. (S1) is now diagonal in the jumpoperators ˆ J j , which is computationally expedient. Further, the FIG. S1. Transient dynamics of the photon detection rate for a 3 × υ ≈ . γ , I ≈ . N ) and (b) a subradiant ( υ ≈ . γ , I ≈ . N ) LLI collective eigenmode, with NI in = I s , a = . λ .The full quantum solution (blue solid line) agrees very well with thesuperatom (black dashed line); black dotted line shows the singleisolated atom solution. The gray shading gives the standard errorfrom ∼ quantum trajectories. Interestingly, examining just theincoherent contribution to the scattering rates in (b) gives even betteragreement between the SAM and full quantum solution. FIG. S2. Relative error η of the superatom picture as a functionof lattice spacing for a field resonant with the uniform superradi-ant LLI mode for a 3 × NI in = . I s (blue circles)and NI in = I s (red diamonds). Unfilled markers show results for η ≡ max | g ( τ ) / [1 − g (0)] − g ( υ,κ )2 ( τ ) | τ<τ , where the deviation iscalculated until τ , such that for all τ . τ , g ( τ ) <
0, while filledmarkers for η ≡ max | g ( τ ) / [1 − g (0)] − g ( υ,κ )2 ( τ ) | all τ . The two deviateat a ∼ . λ due to a persistent oscillation arising from a second modeat larger τ . Inset: Example g ( τ ) for a = . λ (blue dotted curve)and a = . λ (red solid curve) compared to the SAP results (blackdashed curves), with the larger lattice spacing showing an oscillation. stochastic wavefunction evolution requires exponentiating amatrix of size 2 N , as opposed to the 2 N matrix governing thedensity matrix evolution, providing a significant numerical ad-vantage as the system size increases beyond a few atoms. [S1] R. H. Lehmberg, “Radiation from an N -Atom System. I. Gen-eral Formalism,” Phys. Rev. A , 883–888 (1970).[S2] G. S. Agarwal, “Master-equation approach to spontaneousemission,” Phys. Rev. A , 2038–2046 (1970).[S3] John David Jackson, Classical Electrodynamics , 3rd ed. (Wi-ley, New York, 1999).[S4] MR Andrews, M-O Mewes, NJ Van Druten, DS Durfee,DM Kurn, and W Ketterle, “Direct, nondestructive observa-tion of a Bose condensate,” Science , 84–87 (1996).[S5] HJ Carmichael and Kisik Kim, “A quantum trajectory unrav-eling of the superradiance master equation,” Opt. Commun. , 417–427 (2000). [S6] B. R. Mollow and M. M. Miller, “The damped driven two-level atom,” Ann. Phys. , 464–478 (1969).[S7] H J Carmichael and D F Walls, “Proposal for the measurementof the resonant Stark e ff ect by photon correlation techniques,”Journal of Physics B: Atomic and Molecular Physics , L43–L46 (1976).[S8] HJ Carmichael and DF Walls, “A quantum-mechanical masterequation treatment of the dynamical Stark e ff ect,” J. Phys. B:At. Mol. Phys. , 1199 (1976).[S9] H. J. Kimble and L. Mandel, “Theory of resonance fluores-cence,” Phys. Rev. A , 2123–2144 (1976).[S10] B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. , 1969–1975 (1969).[S11] Janne Ruostekoski and Juha Javanainen, “Quantum field the-ory of cooperative atom response: Low light intensity,” Phys.Rev. A , 513–526 (1997).[S12] G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, “Storinglight with subradiant correlations in arrays of atoms,” Phys.Rev. Lett. , 243601 (2016).[S13] Jean Dalibard, Yvan Castin, and Klaus Mølmer, “Wave-function approach to dissipative processes in quantum optics,”Phys. Rev. Lett. , 580–583 (1992).[S14] L. Tian and H. J. Carmichael, “Quantum trajectory simula-tions of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A , R6801–R6804 (1992).[S15] R. Dum, P. Zoller, and H. Ritsch, “Monte carlo simulation ofthe atomic master equation for spontaneous emission,” Phys.Rev. A , 4879–4887 (1992).[S16] Klaus Mølmer, Yvan Castin, and Jean Dalibard, “MonteCarlo wave-function method in quantum optics,” J. Opt. Soc.Am. B , 524–538 (1993).[S17] J. P. Clemens, L. Horvath, B. C. Sanders, and H. J.Carmichael, “Collective spontaneous emission from a line ofatoms,” Phys. Rev. A , 023809 (2003).[S18] Janne Ruostekoski, M. J. Collett, Robert Graham, and Dan F.Walls, “Macroscopic superpositions of bose-einstein conden-sates,” Phys. Rev. A57