Bragg condition for scattering into a guided optical mode
BBragg condition for scattering into a guided optical mode
B. Olmos,
1, 2
C. Liedl, I. Lesanovsky,
1, 2 and P. Schneeweiss School of Physics and Astronomy and Centre for the Mathematicsand Theoretical Physics of Quantum Non-Equilibrium Systems,The University of Nottingham, Nottingham, NG7 2RD, United Kingdom Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Department of Physics, Humboldt-Universit¨at zu Berlin, 10099 Berlin, Germany (Dated: March 1, 2021)We theoretically investigate light scattering from an array of atoms into the guided modes ofa waveguide. We show that the scattering of a plane wave laser field into the waveguide modesis dramatically enhanced for angles that deviate from the geometric Bragg angle. We derive amodified Bragg condition, and show that it arises from the dispersive interactions between theguided light and the atoms. Moreover, we identify various parameter regimes in which the scatteringrate features a qualitatively different dependence on the atom number, such as linear, quadratic,oscillatory or constant behavior. We show that our findings are robust against voids in the atomicarray, facilitating their experimental observation and potential applications. Our work sheds newlight on collective light scattering and the interplay between geometry and interaction effects, withimplications reaching beyond the optical domain.
Introduction.
Bragg diffraction was originally discov-ered when investigating crystalline solids using X-rays.However, Bragg scattering is based on the constructiveinterference of partial waves that originate from periodi-cally arranged scatterers and is, thus, a very general phe-nomenon that plays a central role in many branches ofphysics, most notably in optics [1]. One well-known andtechnologically relevant application of Bragg scatteringare dielectric mirrors, which enable the reflection of lightwithout almost any losses. More recently, Bragg scatter-ing phenomena that occur when laser-cooled atoms areused as scatterers for light have been the matter of nu-merous theoretical and experimental studies [2–10].While the resonances of the materials the dielectricmirror is made of are far-detuned with respect to thewavelength of the reflected light, this can be distinctlydifferent in the case of atomic scatterers. When the lightis resonant or near-resonant with an atomic transition,the light can be absorbed by the atom, with the scat-tered light acquiring a phase shift relative to the incidentlight. Close to resonance, the scattering cross section issignificantly enhanced, such that multiple scattering be-tween different atoms becomes relevant [11, 12]. More-over, single atoms can scatter only one photon at a time,giving rise to non-linear optical effects [13–15]. The in-terplay between Bragg scattering and cooperative effectsstemming from coherent scattering of light between emit-ters gives rise to surprising phenomena, such as photonicband gaps [16], sub-radiant atomic mirrors [17, 18], im-proved optical quantum memories [19, 20], guided lightin atomic chains [21, 22], collective enhancement of chiralphoton emission into a waveguide [23], or the modifica-tion of Bragg scattering from atoms in an optical lattice[24].In this work we theoretically investigate the scatter- ing of light from an atomic emitter array into the guidedoptical modes of a waveguide. The emitters are coher-ently driven by an external plane wave light field suchthat the scattered light from the different emitters caninterfere constructively. We demonstrate that the disper-sive waveguide-mediated atom-atom interactions lead toa modified Bragg condition, i.e., the maximum scatteringrate into the guided mode is reached at laser incidenceangles different from the one determined by the geomet-ric Bragg relation. Here the maximum scattering rate isshown to be dramatically enhanced and to grow linearlywith the number of emitters. This is in stark contrast toother incidence angles for which a saturation is observed.We also identify situations in which the scattering ratescales quadratically and even oscillates as a function ofthe atom number. Strikingly, all these qualitatively dif-ferent scalings are shown to be largely independent ofthe asymmetry (or “chirality”) of the emitter-waveguidecoupling [25] and also robust against voids in the atomicarray.
System.
We consider a one-dimensional array of N atomic emitters with nearest neighbor distance a situatedparallel to an optical waveguide (here a silica nanofiber),as sketched in Figure 1(a). Each emitter is modelled as atwo-level system with ground and excited states denotedby | g (cid:105) and | e (cid:105) , respectively. The atoms are externallydriven by a plane wave monochromatic light field withRabi frequency Ω, detuning ∆, and wave vector k thatencloses an angle θ with the array. When an atom isexcited, it can decay back into its ground state emittinga photon with wavelength λ = 2 π/k with k = | k | .Due to the proximity of the nanofiber, the photon can beemitted into one of the two counter-propagating guidedmodes supported by the nanofiber (at a rate γ R and γ L for the right- and left-propagating mode, respectively). It a r X i v : . [ qu a n t - ph ] F e b FIG. 1.
Scattering into waveguide mode. (a): An arrayof emitters coupled to a waveguide is driven by a laser withwave vector k (forming an angle θ with the array), Rabi fre-quency Ω, and detuning ∆. The rate of photons emitted intothe guided mode propagating to the right, Γ R ( θ, ∆), can bewell approximated by considering the interference of the scat-tering processes indicated with the dashed, colored arrows.(b): Γ R ( θ, ∆) / ˜Γ , where ˜Γ is the single-atom scattering rateinto the waveguide on resonance. The maxima of the scatter-ing rate occur at an angle θ MB (red line) that deviates fromthe geometric Bragg angle θ GB , and ∆ (cid:54) = 0. The cuts on theright show qualitatively different spectra depending on thechoice of θ . Here, N = 144 and D = 1. also can be emitted into the unguided modes (at a rate γ u ), whose modification due to the presence of the fiber istaken into account [26, 27]. The efficiency of the couplinginto the guided modes is quantified by the so-called betafactor β = ( γ R + γ L ) / Γ, where Γ = γ R + γ L + γ u is thetotal single-atom decay rate. Moreover, depending on theorientation of the dipole moment of the atomic transition,an asymmetry of the emission into the guided modes canbe present, such that γ R (cid:54) = γ L [25]. We will quantify thisasymmetry via the parameter D = ( γ R − γ L ) / ( γ R + γ L ).Under the Born-Markov approximation, the dynamicsand stationary state of the system are determined by themaster equation [26, 27]˙ ρ = − i (cid:126) [ H L , ρ ] − i (cid:88) j (cid:54) = l (cid:104) V jl σ † j σ l , ρ (cid:105) (1)+ (cid:88) jl Γ jl (cid:18) σ j ρσ † l − (cid:110) σ † j σ l , ρ (cid:111)(cid:19) , where σ j = | g j (cid:105) (cid:104) e j | for j = 1 , . . . N . Here, the first term describes the action of the laser field: H L = (cid:126) N (cid:88) j =1 (cid:104) Ω (cid:16) e i k aj cos θ σ † j + h . c . (cid:17) + ∆ σ † j σ j (cid:105) . (2)Note, that in the following we will assume that the laserdriving is weak, such that the saturation parameter issmall, i.e. Ω (cid:28) Γ. The second term in Eq. (1) repre-sents dipole-dipole interactions induced by the exchangeof virtual photons between the j -th and l -th atom at arate V jl . Finally, the last term describes the incoherentemission of photons. The decay rates of the entire systemof coupled atoms, γ c , are given by the eigenvalues of thedissipation coefficient matrix Γ jl . For a single atom thedecay rate is simply Γ, i.e., the atom’s excited state pop-ulation decay rate including possible modification due tothe proximity of the nanofiber [28, 29]. For several atoms,the decay becomes collective, and the corresponding de-cay rates γ c can be either superradiant ( γ c (cid:29) Γ), orsubradiant ( γ c (cid:28) Γ) [23, 30, 31].It is convenient to separate the contribution of theguided and unguided modes in both the coherent andincoherent interaction matrix coefficients as V jl = V Rjl + V Ljl + V u jl and Γ jl = Γ Rjl + Γ
Ljl + Γ u jl , respectively. Thecharacter of the interactions mediated by the unguidedmodes is fundamentally different from that of the guidedones: while the unguided modes give rise to interactionsthat decay with the distance between the atoms, the in-teractions mediated by guided modes are infinite-ranged[26, 27, 32, 33].We are here interested in the photon emission rate intothe guided modes. In particular, we will analyze thescattering rate into the right-propagating guided modein the stationary state, defined asΓ R ( θ, ∆) = (cid:88) jl Γ Rjl (cid:68) σ † j σ l (cid:69) ss . (3)As one can observe in Fig. 1(b), the resonances of the flu-orescence excitation spectrum Γ R ( θ, ∆) are qualitativelymodified when the angle θ is close to the one given bythe conventional, geometric Bragg condition. The latteris given by cos θ GB = 2 πm/ ( ak ) − k f /k , with m ∈ Z and k f being the propagation constant inside the nanofiber.For most angles the spectrum is well approximated by aLorentzian centered at ∆ = 0. As θ GB is approached,the scattering rate increases rapidly. The spectrum thenstarts to display a maximum that is off resonance, in par-ticular for an excitation under the modified Bragg angle θ MB . Exactly at θ = θ GB , the spectrum splits symmetri-cally around ∆ = 0 into two peaks [9]. Scattering into the waveguide for unidirectional cou-pling.
In order to understand the origin of the intricaciesof the spectrum and to investigate the scaling with thesystem parameters, we make use of a simplified modelwhich reproduces the main features found with the fullone described by Eq. (1). In this model, we only accountfor the waveguide-mediated interactions. The decay intothe unguided modes is considered to be diagonal: eachatom decays with rate γ u into the unguided modes andno interactions are induced between the atoms via thisdissipative channel. Moreover, for simplicity we considerthat the coupling into the waveguide is fully directional,i.e., γ L = 0 and D = 1. Here, the scattering rate (3) isfound to beΓ R ( θ, ∆) = ˜Γ ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) m =0 t m e − i mk eff a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4)where we have defined the effective wave number k eff = k cos θ + k f , the single-atom scattering rate into theguided modes ˜Γ ∆ = 4Ω β Γ / (4∆ + Γ ), and thecomplex-valued amplitude transmission coefficient t =1 − β Γ / (2∆ + iΓ).Expression (4) can be alternatively obtained by coher-ently summing the contributions of the light scatteredinto the waveguide by all atoms, as illustrated in Fig.1(a). The light that each atom scatters into the waveg-uide exhibits a phase difference with respect to the oneemitted by an atom that is one lattice constant to itsright: The contribution related to the plane wave excita-tion is given by k a cos θ and the one due to propagationin the fiber is k f a . Moreover, t describes the transmissionof the light when it passes an atom in the chain, yieldinga phase shift given by arg t and an amplitude reduction | t | . The total scattering rate Γ R is then obtained as theabsolute value squared of the sum of all amplitudes, mul-tiplied by ˜Γ ∆ .In order to analyze the dependence of the scatteringrate with the number of atoms N , we perform the sumin (4) formally such thatΓ R ( θ, ∆) = ˜Γ ∆ | t | N − | t | N cos( bN )1 + | t | − | t | cos( b ) , (5)where b = arg t − k eff a . The numerator contains a termdue to which the scattering rate oscillates as a function of N , see Fig. 2(a) for an example. Since N can only takeinteger values, the oscillations are sampled with a fre-quency f s = 1, and one observes oscillations at an angu-lar aliasing frequency b alias = min || b |− πkf s | , for k ∈ N .From expression (5), one can also see that the oscilla-tions are damped via the term | t | N describing the fielddecay, such that for large enough N a saturation value isreached. Conversely, for small values of N , the scatter-ing rate grows proportionally to N for all | N ln | t || (cid:28) N is smaller than the period of theoscillations, N < N p = 2 π/b alias .We now discuss the case when the atoms are drivenunder the geometric Bragg condition [23], i.e. θ = θ GB .Here, b = arg t − mπ , and, for large enough atom num-bers N , the spectrum splits into two peaks [see Fig. 1(b)]. FIG. 2.
Scattering rate for unidirectional coupling. (a):Normalized scattering rate into the right-propagating modefor θ − θ GB = − .
05 rad and ∆ = Γ as a function of N . (b):Detuning and (c): maximum scattering rate as a function of N as the laser drives the chain at θ GB (red) and θ MB (blue).In all cases, (a)-(c), there is a good agreement between thepredictions from the simplified model (lines) and the full mas-ter equation (markers). In the limit ∆ (cid:29)
Γ, and for large N , we find that thedetunings at which the two maxima occur are approxi-mately given by ∆ max θ GB ≈ ± Γ βNπ , (6)with corresponding maximum scattering rateΓ R ( θ GB , ∆ max θ GB ) ≈ Ω β Γ (cid:18) − π − β )2 βN (cid:19) . (7)Hence, the maximum scattering rate approaches a sat-uration value 4Ω / ( β Γ) when N → ∞ . Note that,rather counterintuitively, this saturation value is largerthe weaker the coupling β .Eq. (5) allows to infer a modified Bragg conditioncos θ MB = cos θ GB + arg tk a . (8)Here b is a multiple of 2 π , the scattering rate reaches amaximum and it does not oscillate with N . In agree-ment with the numerical results shown in Fig. 1(b), themaximum guided scattering rate is therefore not assumedwhen the emitter array is driven at the geometric Braggangle, but rather slightly away from it. As arg t dependson the detuning, so does θ MB , as depicted by the red solidline in Fig. 1(b). As can also be seen, the maxima of thespectrum Γ R ( θ MB , ∆) are shifted away from resonance.However, comparing to the geometric Bragg conditioncase, we find different scalings of the optimal detuningsand the maximum scattering rate with the number ofatoms N , given approximately by∆ max θ MB ∝ ± (cid:112) N (1 − β ) β Γ , (9)and Γ max R, MB ∝ Ω N (1 − β )Γ , (10)respectively. Notably, now the maximum scattering ratedoes not saturate for large values of N , but rather growslinearly with N , eventually diverging as N → ∞ . As aconsequence, while there is a collective enhancement ofthe total scattering for excitation under θ GB , the scat-tering rates are dramatically further enhanced at θ MB .For example, in the case shown in Fig. 2(c), 150 atomscan scatter as much as ∼
600 independent atoms into thewaveguide. Finally, note that all the discussed scalingsare confirmed by the numerical simulation of the full mas-ter equation (1), cf. Fig. 2.
Asymmetric and symmetric coupling.
Up to now, wehave assumed the special situation where the emissioninto the guided modes is completely unidirectional, i.e. D = 1. While this allowed us to obtain analytic results,this is usually not the situation found in realistic experi-mental settings, where | D | < D = 0, i.e. thereis symmetric emitter-waveguide coupling. We have in-vestigated this situation numerically and found that thescaling with N is independent of the value of D . This isexemplified in Fig. 3(a) and (b), where we compare thescaling of ∆ max θ and Γ R ( θ, ∆ max θ ) obtained for θ = θ MB and θ GB for different values of D . Robustness against voids.
In experiments, laser-cooledatoms can be trapped next to an optical waveguide ina periodic array of trapping sites [34]. Here, while theresidual motion of the atoms is small enough to observeBragg scattering phenomena [7, 8], it is challenging toobtain atomic arrays where indeed every trapping site isoccupied. We investigate the robustness of our findingsagainst voids in the atomic array using our simplifiedmodel. For this purpose, we simulate an array of N sites = N/η sites, with η being the filling factor.In Fig. 3(c), we show the average scattering rate over1000 randomly chosen configurations for N = 50 atomsdistributed over N sites = 100 sites ( η = 0 .
5, as e.g. inRef. [35]). For this filling factor, the average spacing be-tween two atoms is 2 a . Hence, here we use the modifiedBragg angle (8) corresponding to a lattice constant of 2 a in order to maximize the scattering rate, evaluated at∆ = ∆ max θ MB . We compare this scattering rate with themaximum value that is obtained for a completely filledarray with 50 atoms and nearest neighbor distance a asa function of the atom–waveguide coupling. For small β , the scattering rate for arrays with η = 0 . FIG. 3.
Asymmetry and voids. (a): Detuning ∆ max θ and (b): maximum scattering rate, as a function of N as the laser drives the chain at the geometric Bragg an-gle (GB) and the modified (MB) one, by numerical evalu-ation of the full master equation (1). Data is shown for D = 0 , .
85 and 1, and γ R / Γ = 0 . R ( θ MB , ∆ max θ MB ) = Γ voids R ( θ MB , ∆ max θ MB ) / Γ R ( θ MB , ∆ max θ MB ) againstrandom voids with a filling factor of η = 0 .
5. We averageover 1000 randomly chosen void configurations and displaythe standard deviation of the scattering rate as the shadedarea. (d): Oscillations of the scattering rate with N for θ = θ GB + 0 .
004 rad and ∆ = − as high as for perfect filling, confirming the robustnessagainst voids. However, as β → θ MB , eachvoid leads to a phase difference of − k eff a = 2 mπ − arg t compared to the perfect chain. For small β , arg t is alsosmall, such that the phase shift due to a void is close to amultiple of 2 π and therefore the scattering properties arenot significantly altered. For β → t → π and thus the voids inhibit the build-up of constructiveinterference along the chain.In Fig. 3(d), we study the influence of imperfect fillingon the oscillations of the scattering rate with N at anangle θ slightly away from the Bragg resonance θ GB . Onecan see that, despite the imperfect filling, the oscillationsare still visible, although they feature a smaller amplitudeand a larger oscillation frequency. For an arbitrary fillingfactor η , this frequency is simply given by b voids = arg t − k eff aη . These findings indicate that an observation of theeffects presented in this work are within reach of currentexperimental capabilities. Conclusion and outlook.
We have studied the collec-tive emission of an array of atoms into a single guidedoptical mode upon excitation with a plane wave. Weshow that waveguide-mediated atom-atom interactionslead to a qualitative modification of the Bragg scatter-ing condition. We find simple analytical expressions forthe scattering rate into the waveguide and reveal fourregimes, each one exhibiting a different scaling with thenumber of emitters. These findings are shown to be ro-bust against changes in the asymmetry of the couplingand also against voids in the emitter array.We have first indications that not only the scatter-ing into the guided mode studied here but also the totalscattering of a waveguide-coupled array shows collectiveeffects, leading for example to a stronger extinction ofthe excitation light field compared to a free-space atomicarray. Moreover, we noticed that the emission spectrumof the coupled emitters into unguided modes can be per-fectly spectrally flat over a large range of detunings, de-spite the fact that each individual emitter has a Loren-zian line shape [16]. In addition to further investigat-ing these observations, future work will include studyingnon-linear effects [13, 22], the generalization of these re-sults for other scatterers such as plasmonic nanostruc-tures [36], and the exploitation of the described effectsfor quantum information transfer.We thank A. Rauschenbeutel and J. Volz for insight-ful comments and discussions. Financial support fromthe European Union’s Horizon 2020 research and in-novation program under grant agreement No. 800942(ErBeStA) is gratefully acknowledged. We also acknowl-edge funding by the Alexander von Humboldt Founda-tion in the framework of the Alexander von HumboldtProfessorship endowed by the Federal Ministry of Educa-tion and Research. 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