Highly nonlocal optical nonlinearities in atoms trapped near a waveguide
Ephraim Shahmoon, Pjotrs Grisins, Hans Peter Stimming, Igor Mazets, Gershon Kurizki
HHighly nonlocal optical nonlinearities in atoms trapped near a waveguide
Ephraim Shahmoon,
1, 2, 3
Pjotrs Griˇsins, Hans Peter Stimming,
5, 6
Igor Mazets,
4, 6, 7 and Gershon Kurizki Department of Physics, Harvard University, Cambridge MA 2138, USA Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 7610001, Israel Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 7610001, Israel Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria Fakult¨at f¨ur Mathematik, Universit¨at Wien, 1090 Vienna, Austria Wolfgang Pauli Institute, Universit¨at Wien, 1090 Vienna, Austria Ioffe Physico-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia (Dated: October 3, 2018)Nonlinear optical phenomena are typically local. Here we predict the possibility of highly nonlo-cal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide,where long-range interactions between the atoms can be tailored. When the atoms are in anelectromagnetically-induced transparency configuration, the atomic interactions are translated tolong-range interactions between photons and thus to highly nonlocal optical nonlinearities. Wederive and analyze the governing nonlinear propagation equation, finding a roton-like excitationspectrum for light and the emergence of order in its output intensity. These predictions open thedoor to studies of unexplored wave dynamics and many-body physics with highly-nonlocal interac-tions of optical fields in one dimension.
I. INTRODUCTION
Optical nonlinearities are commonly described by lo-cal nonlinear response of the material to the optical field,resulting in the dependence of the refractive index atpoint z on the field at the same point, E ( z ) [1]. Re-cently however there has been a growing interest in non-local nonlinear optics, namely, in mechanisms wherebythe refractive index at z depends on the field intensity atdifferent points z (cid:48) in the material [2]. Mechanisms thatgive rise to such nonlocal nonlinearities include e.g. heatdiffusion [3, 4], molecular reorientation in liquid crystals[5] and atomic diffusion [6, 7]. This paper discusses anew regime of extremely nonlocal nonlinearities affectingboth the frequency and the quadratic dispersion of opti-cal waves [see Eq. (1) below]. The physical mechanismthat leads to this new regime is very different than thoseexplored previously [2–7]: it relies on an atomic mediumprepared in an electromagnetically-induced transparency(EIT) configuration, whose optical nonlinearity is con-trolled by shaping the nonlocal dipolar interactions be-tween the atoms.EIT is associated with the lossless and slow propaga-tion of light pulses in resonant atomic medium subjectto coherent driving of an auxiliary atomic transition [8].Since the early days of EIT, it has been explored as ameans of enhancing optical nonlinearities [8–11]. A par-ticulary effective mechanism for giant optical nonlinear-ities is provided by dipolar interactions between atomsthat form the medium: Since EIT can be described bythe propagation of the so-called dark-state polariton [12],which is a superposition of the light field and an atomicspin wave, the inherently nonlocal dipolar interactionsbetween atoms are translated to nonlocal nonlinearitiesin polariton propagation. In the case of dipolar interac-tions between Rydberg atoms in free space, most theo-retical [13–21] and experimental [22–26] studies have fo- cused on their remarkable strength, a useful feature inquantum information, whereas their nonlocal aspect hasreceived less attention [27].The present work rests on two recently explored mech-anisms: that of EIT polaritons and that of modified long-range dipolar interactions in confining geometries, suchas fibers, waveguides, photonic band structures or trans-mission lines, which currently attracts considerable in-terest [28–37]. Yet, we show that the combined effect ofthese mechanisms may allow for new and unfamiliar pos-sibilities of nonlocal nonlinear optics. More specifically,we show how dispersive laser-induced dipolar interactionsbetween atoms coupled to a nano-waveguide with a grat-ing, which can be designed to extend over hundreds ofoptical wavelengths [31, 32], are translated via EIT intoextremely nonlocal optical nonlinearities. We then ana-lyze light propagation in this medium along the waveg-uide and find a roton-like excitation spectrum for lightand the emergence of spatial self-order in its output in-tensity. II. SUMMARY AND SCOPE OF THE RESULTS
The main general result of this work is the derivation ofa nonlinear propagation equation for the (possibly quan-tum) EIT polariton field ˆΨ comprised of light guided bythe waveguide, and an atomic medium tightly trappedalong the waveguide axis z with an effectively 1d inter-atomic potential U ( z − z (cid:48) ),( ∂ t + v∂ z ) ˆΨ( z ) = − i ˆ∆ c α ˆΨ( z ) − i ˆ∆ c Cv ∂ z ˆΨ( z ) , ˆ∆ c = δ c + ˆ δ NL , ˆ δ NL = α (cid:90) L dz (cid:48) U ( z − z (cid:48) ) ˆΨ † ( z (cid:48) ) ˆΨ( z (cid:48) ) . (1) a r X i v : . [ qu a n t - ph ] M a r 𝑒 𝑔 𝑑𝛿 𝑐 𝛿 𝑁𝐿 Ωℰ (b)(a) 𝜃 𝐿 Ω 𝐿 ℰ, Ω 𝑧 𝑟 𝑎 𝑑 𝑠𝑑𝑠 Ω 𝐿 𝛿 𝐿 𝑈(𝑧) (c)
FIG. 1: (a) Setup: atoms (black dots), illuminated by theEIT fields [see (b)] ˆ E and Ω (thin blue arrow), are trapped ata distance r a from a nano-waveguide (gray cylinder) along z ,from z = 0 to z = L . A far-detuned laser Ω L (thick orange ar-row), tilted by an angle θ L from the z axis, induces long-rangeinteractions between the atoms, mediated by the waveguidemodes [see (c)]. (b) EIT atomic configuration: the probe fieldˆ E is resonantly coupled to the | g (cid:105) → | e (cid:105) transition whereas thecoupling field Ω is coupled to the | d (cid:105) → | e (cid:105) transition with de-tuning δ c . Interaction between the atoms in the | d (cid:105) -level [see(c)] induces its energy shift δ NL which is effectively added tothe detuning δ c . (c) Laser-induced dipolar interactions: thelaser with Rabi frequency Ω L and detuning δ L operates onthe | d (cid:105) → | s (cid:105) transition of all atoms, | s (cid:105) being an additionallevel, thus inducing a dipolar potential U ( z ) between pairs ofatoms ( z apart) populating the state | d (cid:105) [57]. The left-hand side of the equation describes an envelopeof a wave travelling with a group velocity v , whereas thefirst and second terms in the right-hand side present itsfrequency shift ˆ∆ c (multiplied by a coefficient α ) andits quadratic dispersion with a coefficient proportionalto the detuning ˆ∆ c (and to a constant C ), respectively.Nonlinearity comes about by noting the detuning ˆ∆ c : itcontains a linear component δ c which is controlled bythe EIT configuration (the so-called coupling-field de-tuning, see Fig. 1b), and a nonlocal nonlinear detuningˆ δ NL which depends on field intensities integrated over themedium with the interaction kernel U ( z ). The appear-ance of a nonlocal nonlinearity not only in the frequencyshift but also in the dispersion coefficient gives rise toa new regime of nonlocal nonlinear optics. The physicalsystem and reasoning that lead to Eq. (1), as well as itsderivation, are discussed in Sec. III below. In principle,we assume sufficient conditions for a lossless EIT prop-agation (coefficients v , α and C are real), whereas lossand decoherence mechanisms due to imperfections andscattering are analyzed in Sec. VI.The second part of this work (Sec. IV and V) is dedi-cated to study some first implications on wave propaga-tion in such a medium, and specifically to the analysisof wave excitations on top of a continuous wave (CW)background. General results for the excitation spectrum(dispersion relation) and the intensity correlations at theoutput are presented in Sec. IV, Eqs. (8) and (11),respectively. These are followed in Sec. V by specificresults for a medium of atoms trapped near a waveguide- grating which supports extremely long-range interactions U ( z ), as per Eq. (12). The results for the excitation spec-trum exhibit roton-like narrow-band shape, which maybe probed by a homodyne detection scheme (Fig. 3).The roton-like spectrum signifies the tendency of light inthis regime to exhibit spatial self-order, namely, crystal-like correlations; these can be revealed by measuring thephoton intensity at the waveguide’s output (Fig. 4).The predicted self-order of light constitutes a new,hitherto unexplored, optical ”phase”, analogous to thespatial structure of cold atomic media subject to light-induced dipolar interactions [31, 38–43]. We discuss im-portant aspects and prospects of this work in Sec. VII. III. EIT POLARITONS WITH NONLOCALINTERACTIONSA. The system
Consider a medium of identical atoms in an EIT con-figuration as in Fig. 1: The atoms are trapped at adistance r a from a nano-waveguide along its longitudinal z axis [30, 44–52] (Fig. 1a). A strong (external) cou-pling field with a constant and uniform Rabi frequencyΩ drives the | d (cid:105) → | e (cid:105) atomic transition with detuning δ c and wavenumber k c , whereas a weak (possibly quan-tized) probe field with carrier frequency ω , wavenumber k and envelope ˆ E is resonantly coupled to the | g (cid:105) → | e (cid:105) transition ( ω = ω eg ) [Fig. 1b]. Under tight transversetrapping (around r a ) with respect to the transition wave-length, the atomic positions can be characterized by theirlongitudinal component z (Appendix A1) [31, 44]. We as-sume the existence of a dipolar interaction U ( z ) betweenatoms that occupy the state | d (cid:105) (Fig. 1c and Sec. V Abelow). Then, the energy of level | d (cid:105) of an atom at z isshifted by δ NL ∼ n a (cid:82) dz (cid:48) U ( z − z (cid:48) ) P d ( z (cid:48) ), n a being theatomic density (per unit length) and P d ( z (cid:48) ) the occupa-tion of state | d (cid:105) in an atom at z (cid:48) .We may now explain the physical reasoning that leadsto Eq. (1). In Fig. 2 we plot the complex linear suscep-tibility χ of the EIT medium to the probe field ˆ E as afunction of its detuning ∆ p , in the presence of a couplingfield detuned by ∆ c [8], which in our case is given by∆ c = δ c + δ NL (Fig. 1b). When ∆ c = 0 (Fig. 2a), theabsorption coefficient Im χ is symmetric with respect to∆ p whereas the dispersion Re χ is antisymmetric, so thatno (real) quadratic dispersion exists for the probe enve-lope centered around ∆ p = 0. By contrast, when ∆ c (cid:54) = 0is introduced (Fig. 2b), Re χ is no longer antisymmet-ric and quadratic dispersion exists, which explains theterm ∆ c Cv ∂ z in Eq. (1). However, this comes at theprice of non-vanishing losses at ∆ p = 0. For this rea-son we choose to work in the so-called Autler-Townesregime, Ω (cid:29) γ, ∆ c [8], where γ is the width of the level | e (cid:105) . Then, for ∆ c smaller than the single-atom trans-parency window, ∆ c (cid:28) Ω /γ, Ω, but still larger than γ ,the absorption per atom can become negligible while dis- (cid:45) (cid:45) (cid:68) p (cid:45) (cid:45) Χ (cid:64) a.u (cid:68)(cid:72) a (cid:76) : (cid:68) c (cid:61) (cid:45) (cid:45) (cid:68) p (cid:45) (cid:45) Χ (cid:64) a.u (cid:68)(cid:72) b (cid:76) : (cid:68) c (cid:61) Γ FIG. 2: Linear susceptibility of the EIT atomic medium tothe probe field as a function of its detuning ∆ p [8]. (a) Fortotal detuning of the coupling field ∆ c = 0, the absorptionIm χ (red dashed line) and dispersion Re χ (blue solid line) aresymmetric and antisymmetric, respectively, with respect to∆ p , so that no (real) quadratic dispersion exists for the probeenvelope centered around ∆ p = 0. (b) For ∆ c (cid:54) = 0, Re χ is notantisymmetric, so that quadratic dispersion exists, giving riseto the term ∆ c Cv ∂ z in Eq. (1), with ∆ c = δ c + δ NL (Fig.1b). persion is still significant, as illustrated in Fig. 2b (seealso Appendix A). This explains the lossless propagationdescribed by Eq. (1) with real parameters α, v, C . Aslong as the absorption, associated with dissipation due tospontaneous emission at rate γ , is negligible, so are thenoise effects of vacuum fluctuations; Eq. (1) then holdsin operator form without additional Langevin quantumnoise operators. B. Derivation of Eq. (1)
The formal derivation of Eq. (1) goes as follows(more details in Appendix A). The field envelope ˆ E ( z ) = (cid:80) k ˆ a k e ikz / √ L , with commutation relations [ˆ a k , ˆ a k (cid:48) ] = δ kk (cid:48) and hence [ ˆ E ( z ) , ˆ E † ( z (cid:48) )] = δ ( z − z (cid:48) ), is assumed tobe spectrally narrow and is guided by a transverse modeof the waveguide (later taken to be the HE mode of afiber) with effective area A at the atomic position r a andpolarization vector e . The Hamiltonian in the interac-tion picture is (see also Appendix A1) H AF = − (cid:126) n a (cid:90) dz (cid:104) ig ˆ E ( z ) e ik z ˆ σ eg ( z ) + h . c . (cid:105) ,H AC = − (cid:126) n a (cid:90) dz (cid:2) i Ω e − iδ c t e ik c z ˆ σ ed ( z ) + h . c . (cid:3) ,H DD = 12 n a (cid:126) (cid:90) dz (cid:90) dz (cid:48) U ( z − z (cid:48) )ˆ σ dd ( z )ˆ σ dd ( z (cid:48) ) , (2)and H F = (cid:80) k (cid:126) ck ˆ a † k ˆ a k , where g = (cid:112) ω / (2 (cid:15) (cid:126) A ) d · e , d being the dipole matrix element of the | g (cid:105) → | e (cid:105) tran-sition, and ˆ σ ij ( z ) = | i (cid:105)(cid:104) j | for an atom at z , with i, j rep-resenting the states { g, d, e } . By writing the Heisenbergequations for the atom and field operators and assum-ing a sufficiently weak probe field such that the atomic | g (cid:105) → | e (cid:105) transition is far from saturation, we obtain cou-pled equations for the | g (cid:105) ↔ | d (cid:105) spin-wave and the field (Appendix A2),¯ σ gd ( z ) = − g Ω ˆ E ( z ) − (cid:20) ( ∂ t + γ ) (cid:18) ∗ (cid:19) × (cid:16) ∂ t + iδ c + i ˆ S ( z ) (cid:17) ¯ σ gd ( z ) − ˆ F (cid:105) , ( ∂ t + c∂ z ) ˆ E ( z ) = n a g ∗ Ω ∗ (cid:104) ∂ t + iδ c + i ˆ S ( z ) (cid:105) ¯ σ gd ( z ) , (3)where ¯ σ gd ( z ) = ˆ σ gd ( z ) e i ( k c − k ) z e − iδ c t . Here γ and ˆ F are the spontaneous emission rate and correspondingLangevin noise operator, respectively due to the couplingof the | g (cid:105) → | e (cid:105) transition to the reservoir formed by pho-ton modes other than those guided by the waveguide.The effect of interaction is a nonlinear detuning for thecoupling field,ˆ S ( z ) = n a (cid:90) L dz (cid:48) U ( z − z (cid:48) )¯ σ † gd ( z (cid:48) )¯ σ gd ( z (cid:48) ) . (4)Moving to the polariton picture of EIT [12], we definethe dark and bright polaritons, ˆΨ and ˆΦ respectively, (cid:18) ˆΨ( z )ˆΦ( z ) (cid:19) = (cid:18) cos θ −√ N sin θ sin θ √ N cos θ (cid:19) (cid:18) ˆ E ( z )¯ σ gd ( z ) / √ L (cid:19) , (5)with tan θ = n a g / | Ω | , and transform Eqs. (3) intoequations of motion for the polaritons, (cid:0) ∂ t + c cos θ∂ z (cid:1) ˆΨ( z ) = − sin θ cos θc∂ z ˆΦ( z ) − i sin θ (cid:16) δ c + ˆ S ( z ) (cid:17) (cid:104) sin θ ˆΨ( z ) − cos θ ˆΦ( z ) (cid:105) , ˆΦ( z ) = cos θ | Ω | ( ∂ t + γ ) (cid:16) ∂ t + iδ c + i ˆ S ( z ) (cid:17) × (cid:104) sin θ ˆΨ( z ) − cos θ ˆΦ( z ) (cid:105) + √ n a cos θ Ω ˆ
F . (6)In the adiabatic regime, where the probe field is aCW and in the absence of detunings ( δ c , U = 0), thebright polariton ˆΦ vanishes [12]. Here, we take thefirst non-adiabatic correction (Appendix A3) by insert-ing the equation for ˆΦ into that of ˆΨ, and assume alldetunings to be smaller than the EIT transparency win-dow δ tr = Ω / ( γ √ OD ), with OD = ( n a /A ) Lσ a and σ a the cross section of the | g (cid:105) → | e (cid:105) transition, finallyarriving at Eq. (1) with α = sin θ , v = c cos θ , C = sin θ (2 − θ ) / | Ω | . IV. COLLECTIVE EXCITATIONS ANDPROPAGATION IN A CW BACKGROUNDA. Excitation spectrum of polariton waves
With Eq. (1) in hand, we now turn to the analysis ofthe polariotn wave propagation it predicts. Specifically,let us assume a CW polariton, and find the dispersion re-lation of wave excitations around this CW background,in analogy to the Bogoliubov spectrum of excitations in aBose-Einstein condensate (BEC) [53, 54]. The CW solu-tion of (1) is ψ ( t ) = ψ e − i ( αδ c + n p U ) t , with ψ = | ψ | e iφ , n p = α | ψ | being the effective photon density per unitlength and U the k = 0 component of the spatial Fouriertransform of the potential U k = (cid:82) ∞−∞ dzU ( z ) e − ikz . Herewe have neglected edge effects by assuming l < z < L − l , l being the range of the potential U ( z ). The dispersionrelation of small fluctuations ϕ ( z, t ) around the large av-erage CW field (cid:104) Ψ (cid:105) = ψ ( t ) are found as usual upon in-serting Ψ = ψ ( t ) + ϕ ( z, t ) into Eq. (1) and linearizingit by keeping the fluctuations ϕ to linear order. Then,introducing the ansatz [53], ϕ ( z, t ) = e i [ φ − ( αδ c + n p U ) t ] × (cid:104) u k e ikz e − i ( ω k + kv ) t − v ∗ k e − ikz e i ( ω k + kv ) t (cid:105) , (7)into the linearized equation for ϕ , u k and v k being c-number (Bogoliubov) coefficients, and using standardprocedures, we find the modified Bogoliubov spectrum(Appendix B1) ω k = (cid:113) ω k ( ω k + 2 n p U k ) , ω k = ( n p U /α + δ c ) Cv k . (8)This means that a polariton wave distortion (about theCW solution) with a wavenumber k relative to the car-rier wavenumber (inside the EIT medium), oscillates ata frequency (relative to the carrier frequency ω ) ω ( k ) = αδ c + n p U ± ( vk + ω k ) , (9)where the ± sign is for positive/negative k , respectively.The dispersion relation (9) is composed of the detuning αδ c due to the coupling field, the self-phase modulationof the CW component n p U (analogous to the chemicalpotential in a BEC [53]), the linear dispersion vk , and themodified Bogoliubov excitation spectrum ω k . The spec-trum ω k is determined by the interplay between the in-teraction Fourier transform U k and the ”kinetic-energy”quadratic dispersion ω k , which is affected by both thedetuning δ c and by the k = 0 component of U k . This in-terplay is further discussed below for U ( z ) resulting fromlaser-induced interactions near a waveguide grating. B. Generation of two-mode correlations
The parametric process described by the foregoingmodified Bogoliubov theory also entails the dynamic gen-eration of two-mode squeezing, i.e. pairs of entangled po-laritons with wavenumbers ± k . The analysis is similar tothat of propagation in fibers with local Kerr nonlinearity[55, 56]. Upon inserting the expansion of small quan-tum fluctuations in the longitudinal wavenumber modes k , ˆ ϕ ( z ) = (cid:80) k ˆ a k e ikz / √ L , into the linearized equationfor ϕ ( z, t ), we obtain coupled first-order differential equa-tions (in time) for ˆ a k ( t ) and ˆ a †− k ( t ), whose solution is a dynamic Bogoliubov transformation (Appendix B2),ˆ a k ( t ) = e − i ( n p U + αδ c + kv ) t (cid:104) µ k ( t )ˆ a k (0) + e i φ ν k ( t )ˆ a †− k (0) (cid:105) ,µ k ( t ) = cos( ω k t ) − i n p U k + ω k ω k sin( ω k t ) ,ν k ( t ) = − i n p U k ω k sin( ω k t ) . (10)The number of entangled pairs generated after prop-agation time t at wavenumbers ± k can be quantifiedby the so-called squeezing spectrum, whose opti-mum is given by G k = ( | µ k | − | ν k | ) [55] ( G k < g (2) ( z, z (cid:48) , t ) =[ (cid:104) ˆΨ † ( z ) ˆΨ † ( z (cid:48) ) ˆΨ( z (cid:48) ) ˆΨ( z ) (cid:105) ] / [ (cid:104) ˆΨ † ( z ) ˆΨ( z ) (cid:105)(cid:104) ˆΨ † ( z (cid:48) ) ˆΨ( z (cid:48) ) (cid:105) ](all fields measured at the waveguide’s output afterpropagation time t = L/v ), where the averaging isperformed with respect to the initial probability distri-bution, e.g. the quantum state, of the polariton (probe)field. For initial zero-mean fluctuations (around the CWsolution) with average polariton occupation at mode k , (cid:104) ˆ a † k (0)ˆ a k (cid:48) (0) (cid:105) = N k δ kk (cid:48) , and vanishing anomalouscorrelations (cid:104) ˆ a k (0)ˆ a k (cid:48) (0) (cid:105) = 0, we find (Appendix B3) g (2) ( z, z (cid:48) ) ≈ α πn p (cid:90) ∞ dk (cid:2) | µ k | N k + | ν k | ( N k + 1)+(2 N k + 1) | µ k || ν k | cos φ k ] cos[ k ( z − z (cid:48) )] , (11)where φ k = arg( µ k ν k ) and we used (1 /L ) (cid:80) k → (cid:82) dk/ (2 π ). As we shall see below, these correlations mayreveal the ordering of a nonlocal system caused by pair-generation at preferred k -values. V. HIGHLY NONLOCAL LASER-INDUCEDINTERACTION VIA WAVEGUIDE GRATING
Our analysis up to this point was kept general, with-out specifying the interaction potential U ( z ). Let us nowturn to a particulary interesting case of an extremelylong-range interaction, where novel nonlinear optical ef-fects can be illustrated. A. Shaping the interaction potential
The illumination of atoms by an off-resonant laser vir-tually excites the atoms and allows them to resonantlyinteract via virtual photons. The spatial dependence ofthe resulting interaction potential U ( z ) then follows thespatial structure of the mediating photon modes. This isthe essence of the laser-induced interaction potential wewish to employ [57]. Specifically, consider another laserwith Rabi frequency Ω L which is detuned by | δ L | (cid:29) Ω L from the transition | d (cid:105) → | s (cid:105) , | s (cid:105) being a fourth atomiclevel (Fig. 1c), and assume that this transition is dis-tinct and separated from the transitions used for EIT(Fig. 1b) either spectrally or by polarization. Then, theextended waveguide modes can mediate long-range inter-actions between the trapped atoms (Fig. 1a) (AppendixC). As a specific example, we consider a nano-waveguidethat incorporates a grating, i.e. periodic perturbation ofthe refractive index with period Λ ≡ π/k B , so that thephoton modes exhibit a bandgap (see e.g. Refs. [47, 58]).Then, for a laser frequency ω L inside the gap and closeto its edge (the probe field’s carrier frequency ω beingoutside the gap), the laser-induced interaction potentialbecomes [31] (Appendix C) U ( z ) = − U L
12 cos( k zL z ) cos( k B z ) e −| z | /l , (12)where l can extend over hundreds of wavelengths [31, 32].Here k zL = k L cos θ L with k L the laser wavenumber and θ L its orientation with respect to the waveguide axis z (Fig. 1c), and U L depends on the laser parameters,atomic transition and effective area at r a (see AppendixD). The resulting spatial Fourier transform U k then con-sists of four Lorentzian peaks of width ∼ /l , centeredaround the spatial beating frequencies ± ( k zL − k B ) and ± ( k zL + k B ). We note that the laser Ω L and the grat-ing are unrelated to the linear propagation of the probefield and their sole role is to induce the long-range dipo-lar interaction Eq. (12) between atoms, which is in turntranslated via EIT to interaction between polariotns asin Eq. (1) (see also Discussion, Sec. VII B). B. Roton and Anti-Roton spectra
Let us focus on the peak of U k around k R ≡ k zL − k B and its effect on the dispersion relation (spectrum), Eq.(8). We first consider the case of anomalous dispersion,where the signs of ω k and U k are opposite. In analogyto BEC, this describes the case of an attractive potential U k that competes with the ”kinetic energy” ω k . Then,for k -values satisfying | ω k | > n p | U k | , ω k is real and ex-hibits a dip around k R , in contrast to the case of a localpotential for which U k is independent of k (standard Bo-goliubov spectrum) and this feature is absent. This isseen in Fig. 3a, where both the analytical results of Eq.(8) and numerical simulations of the nonlinear equation(1) (Appendix E), are plotted and shown to agree verywell. The narrow-band ”dip” of this ω k spectrum is inanalogy with the roton minimum in He II [59]. It reflectsthe fact that wave distortions about the CW field withspatial frequencies around k R cost less energy and arehence favorable. This feature implies that the intensityof the polariton field in its ground state would tend toself-order with typical wavenumber k R [59].Turning to the case of normal dispersion, where thesigns of ω k and U k are identical, ω k exhibits an ”anti-roton” peak around k R (Fig. 3b). This means that dis-tortions of spatial frequencies around k R are costly, so
500 1000 1500 k (cid:64) (cid:144) m (cid:68) Ω k (cid:64) (cid:144) s (cid:68) (cid:72) a (cid:76)
500 1000 1500 k (cid:64) (cid:144) m (cid:68) Ω k (cid:64) (cid:144) s (cid:68) (cid:72) b (cid:76) (c) ℰ 𝑖𝑛 ℰ 𝑜𝑢𝑡 𝐷 𝐿𝑂 𝑧 = 0 𝑧 = 𝐿 Ω, Ω 𝐿 FIG. 3: Dispersion relations for EIT polaritons withwaveguide-grating mediated atomic interactions ( k values pre-sented in all plots are within the EIT transparency window;values of physical parameters used here are given in AppendixD). (a) Roton-like excitation spectrum (dispersion relation) ω k for the potential of Eq. (12) in the anomalous dispersioncase (opposite signs of ω k and U k ). The analytical resultsfrom Eq. (8) (blue solid line) agree well with those of di-rect numerical simulations of Eq. (1) (gray dots). Comparedwith the spectrum of a local interaction ( U k independent of k , dashed red line), the roton-like spectrum exhibits a dip ina narrow band of k -values around k R = k zL − k B . (b) Anti-roton peak of the spectrum ω k in a narrow band around k R inthe normal dispersion case (identical signs of ω k and U k ). (c)Possible homodyne detection scheme: the input probe fieldconsists of a CW field + perturbation at wavenumber k andfrequency ω (0) ( k ). The field is split before entering the EITmedium ( z = 0), so that a local oscillator of ω (0) ( k ) is formed(lower arm) by e.g. filtering out the CW component. Then,mixing the output signal ( z = L ) with the local oscillatorreveals their phase difference, from which ω k can be inferred(see text). that the system prefers to avoid these spatial variations.This behavior again manifests the tendency of the sys-tem to order, since it indicates the spatial distortions thatthe system is unlikely to be found in, should it be in itsground state.In order to measure the roton and anti-roton spec-tra, we first recall the meaning of the dispersion rela-tion ω ( k ) from Eq. (9): without an interaction, the fre-quency associated with a wave envelope at wavenumber k traveling inside the EIT medium is given by ω (0) ( k ) = αδ c + vk + δ c Cv k = ω ( k ) − n p U − ( ω k − δ c Cv k ), sothat ω k (together with n p U ) expresses a frequency, orphase velocity, shift due to the nonlinearity. Namely, thewavenumber k describes a spatial eigenmode of propaga-tion, both with and without interaction, with an eigen-frequency ω ( k ) and ω (0) ( k ), respectively. Now, supposewe let a weak quasi-CW pulse of length L p < L andfrequency ω p = ω (0) ( k ) enter the medium (on top of thestrong CW), when the laser and hence the interaction U k are turned off. Since upon entering the medium the fielddoes not change its frequency ω p , we deduce from thedispersion relation in the absence of interaction, ω (0) ( k )that the field inside the medium exhibits a perturbation ϕ ( z ) at spatial frequency k on top of the strong CW.Subsequently, when the entire pulse is in the medium,we immediately (non-adiabatically) turn on the laser Ω L ,and hence the interaction U k , so that the temporal fre-quency of the perturbation ϕ ( z ) at wavenumber k be-comes ω ( k ) = ω (0) ( k ) + n p U + ( ω k − δ c Cv k ). Thefrequency shift n p U + ( ω k − δ c Cv k ) can be thoughtof as an extra energy acquired by the mode k due tothe interaction energy. Therefore, the spectrum ω k canbe inferred from the frequency shift, measurable by ho-modyne detection of the pulse ϕ ( z ) that exits the EITmedium (Fig. 3c). A similar procedure was proposedfor measuring the tachyon-like spectrum of polaritons ininverted media [60]. C. Dynamical instability: pair generation
In the anomalous dispersion case, consider now a suf-ficiently strong interaction such that for a narrow bandof k -values around k R , where U k is peaked, the condition2 n p | U k | > | ω k | is satisfied and ω k becomes imaginary.Then, field perturbations around k R become exponen-tially unstable (Fig. 4a), resulting in parametric ampli-fication and generation of entangled photon pairs in thisnarrow band of unstable k -values. The strength of theamplified perturbation and generated field is character-ized by the magnitude of the coefficients µ k ( t ) and ν k ( t )from Eq. (10), which grow exponentially with propaga-tion time t and are largest for the narrow peak around k R . The resulting squeezing spectrum G k at the out-put t = L/v (Fig. 4a) may be measured by homodynedetection [55].
D. Dynamical instability: emergence of self-order
An interesting implication of the extremely nonlocalpotential of Eq. (12) is the dynamic formation of or-der in the system. Consider that at t = 0 there ex-ist fluctuations around the CW, with a spatial spec-trum N k = N e − ( k/q tr ) , i.e. ” δ -correlated” noise limitedby the EIT transparency window of width δ tr = vq tr .Then, fluctuations at k -values around the peak k R willbe parametrically amplified as they propagate throughthe medium, as verified in Fig. 4b, where the spatialspectrum of the polariton field, N k ( t ) = (cid:104) ˆ a † k ( t )ˆ a k ( t ) (cid:105) , atdifferent propagation times t ( t = L/v describing the out-put field) is calculated both analytically and via direct(classical) numerical simulations of Eq. (1). This sug-gests that the system becomes spatially ordered with aperiod ∼ π/k R , which may be revealed by measuringthe correlation function g (2) between the intensities of (cid:64) (cid:144) m (cid:68) Γ k (cid:64) Π(cid:42) (cid:144) s (cid:68) , G k (cid:72) a (cid:76) (cid:64) (cid:144) m (cid:68) N k (cid:72) t (cid:76) (cid:72) b (cid:76) t (cid:61) (cid:144) v0.5L (cid:144) v0.75L (cid:144) vL (cid:144) v (cid:72) z (cid:45) z' (cid:76) (cid:64) mm (cid:68) g (cid:72) (cid:76) (cid:72) c (cid:76) t (cid:61) (cid:61) (cid:144) vt (cid:61) (cid:144) vt (cid:61) L (cid:144) v FIG. 4: Instability and self-ordering (values of physical pa-rameters are given in Appendix D). (a) γ k = Im ω k is theexponential growth rate of unstable perturbations at spatialfrequency k (anomalous dispersion case). It exhibits a narrowpeak around k R (blue dotted line), compared to the broad-band instability of the local-interaction case (red dotted line).The instability is accompanied by generation of quantum en-tanglement, characterized by a narrow-band squeezing spec-trum G k (blue solid line; G k < ± k photon modes), in contrast to the broadbandspectrum for a local interaction (horizontal red solid line).(b) Dynamics of spatial spectrum N k ( t ) of the field intensityfor different propagation times t inside the medium ( t = L/v at the output). The emergence of a large peak around k R (and − k R ) out of the initial Gaussian perturbation is clearly seen.Results of both the linearized theory (solid lines) and numeri-cal simulations of Eq. (1) (dots) are shown: slight differencesat later t -values are attributed to nonlinear corrections. (c)Self-ordering of the field intensity: considering the narrowpeaks of N k ( t ) around ± k R , fluctuations at these k -valuesbecome dominant, resulting in ordered intensity correlations g (2) which grow with propagation time t ( t = L/v at theoutput). Excellent agreement between the theory, Eq. (11),(solid lines) and numerical simulations of Eq. (1) (dots) is ob-served. The correlations oscillate with a period 2 π/k R ∼ . l ≈ . L = 2 .
68 cmand v = 4340 ms − ), determined by the long-range interac-tion U ( z ). This is in contrast to the local-interaction case,where the correlations vanish ( g (2) = 1) after a short distance π/q tr ≈ .
75 mm, which is determined by the bandwidth q tr of the initial fluctuations [see inset: local (red thin line) andnonlocal (blue thick line) cases for the output field at t = L/v ]. the output field that arrive at a detector at the waveg-uide’s end ( z = L ) and time difference ( z − z (cid:48) ) /v (schemefrom Fig. 3c without the lower local-oscillator arm). Weobtain the corresponding g (2) by numerically integrat-ing over k in the classical limit of Eq. (11), where thevacuum-fluctuations contributions are neglected. Thiscalculation is compared to g (2) measured via direct nu-merical simulations of Eq. (1), yielding excellent agree-ment. Fig. 4c reveals the emergence of order by pre-senting the intensity-correlations g (2) at different prop-agation times t through the medium ( t = L/v at theoutput): The resulting g (2) exhibits oscillations with aperiod z − z (cid:48) ∼ π/ ( k R ), which persist over a few l . Con-sidering that the interaction range l can reach hundredsor even thousands of optical wavelengths ( l ≈ λ L and L ∼ l ∼ .
026 m in our example, see AppendixD), the light intensity clearly becomes ordered due tothe long-range interaction U ( z ), as can also be seen bythe comparison to the local-interaction case (inset of Fig.4c). VI. SCATTERING AND IMPERFECTIONS
So far we have considered a purely coherent evolutionof the polariton field. In Appendix F we address threemain sources of scattering and loss of field excitations;we estimate the decoherence rate each of them imposeson the polariton field and its possible effect on the ob-servability of the self-ordering effects discussed above.First, since the interaction U ( z ) from Fig. 1b is in-duced by the illumination of the atoms by an off-resonantlaser, it is accompanied by an incoherent process of scat-tering of laser photons Ω L from the | d (cid:105) → | s (cid:105) transitionto non-guided modes at rate R fs [57]. This process limitsthe coherence time of the ¯ σ gd spin wave and hence thatof the polariton to be below ∼ R − fs , which in the exam-ple of Figs. 3 and 4 is nevertheless much longer than theexperiment time L/v (Appendix D).Second, consider the losses due to propagation in theEIT medium with a non-vanishing detuning of the cou-pling field, ∆ c = δ c + δ NL , where here δ NL = n p U /α isthe mean-field value of ˆ δ NL . The transmission throughthe EIT medium of length L is given by e − (1 / k χ (cid:48)(cid:48) L ,leading to a decoherence rate R EIT ∼ (1 / k vχ (cid:48)(cid:48) for thepolariton, where χ (cid:48)(cid:48) = Im χ (∆ p = 0 , ∆ c ) is the imaginarypart of the EIT susceptibility [8] evaluated for simplicityat the center of the probe pulse (∆ p = 0). In AppendixF2 we show that the effect of the EIT losses on the observ-ability of ω k can be significantly decreased by reducingboth n p and δ c , resulting in a loss rate smaller than themain features of the ω k (roton and anti-roton cases) and γ k = Im ω k (instability case) spectra.Finally, we consider material imperfections in thewaveguide grating structure (e.g. defects and surfaceroughness) which give rise to scattering of photons offthe guided modes. This leads to a decay rate (width) κ for each of the longitudinal (guided) photon-modes k (seeAppendix F3 and [32, 48]). Then, since atoms at level | d (cid:105) are coupled via Ω L to these lossy waveguide-gratingmodes, the ¯ σ gd spin wave and hence the polariton are de-cohered at a rate R (1) im ∼ ( κ/δ u ) U L , δ u being the detuningof the laser Ω L from the upper bandedge of the grating. However, the effect of R (1) im on the spectra ω k and the ob-servables mentioned above can in principle be made arbi-trary small, by noting that ω k depends on n p U L whereas R (1) im depends solely on U L , reflecting the fact that R (1) im isa single-polariton loss mechanism whereas ω k is a coop-erative effect. Then, decreasing | U L | while keeping n p U L constant (by increasing the CW power n p ) reduces R (1) im but keeps ω k unchanged.Nevertheless, a cooperative decoherence can in fact re-sult from κ and U ( z ); namely, a pair of atoms, at adistance z apart, can jointly scatter photons at a rate ∼ ( κ/δ u ) U ( z ), since the virtual photons that mediatetheir interaction U are now lossy and enable excitationdecay to non-guided modes. Then, for a single polaritonat z , the scattering induced by the entire atomic mediumbecomes R (2) im ∼ n p (cid:82) dz (cid:48) ( κ/δ u ) U ( z (cid:48) ) ∼ ( κ/δ u ) n p U . InAppendix F3 we show that the limitation imposed bythis dephasing channel on the observability of ω k , γ k canbe reduced by increasing the EIT coupling laser Ω.In principle, all loss terms discussed above shouldbe accompanied by corresponding quantum noise terms,which are not considered here, restricting this discussionto the classical-field case (for which all of the above re-sults apply, excluding the squeezing spectrum G k ). VII. DISCUSSION
This study predicts a new and hitherto unexploredregime of nonlinear optics; namely, that of highly nonlo-cal interactions between photons in one dimension (1d).These nonlocal optical nonlinearities arise for light prop-agation inside driven atomic media in the vicinity of awaveguide, and affect both the frequency and quadraticdispersion of the light field. We have derived the non-linear equation that governs light propagation in thisregime, and have analyzed it around its CW solution,finding a narrow roton-like dispersion relation (Figs.3a,b) and squeezing (entanglement) spectrum (Fig. 4a),and the emergence of order in the field-intensity (Figs.4b,c), all of which reflect the tendency of the system toself-organize, which in turn results from the long-rangeinteractions between photons. In the following we wishto discuss some important aspects of this work.
A. Structure and generality of Eq. (1)
It is important to note the nonlocal and nonlinear dis-persion term appearing in Eq. (1), absent in previousworks on nonlocal nonlinear optics [2, 4–7] and EIT-based nonlinear optics [13–17, 19, 21, 27]. We identifythe conditions under which this term becomes significantand may lead to new phenomena, namely, in the AutlerTownes regime of EIT and for sufficiently large nonlineardetuning.Another interesting point concerns the generality ofour work in relation to different confining geometries .The derivation of Eq. (1) does not depend on the spe-cific form of the potential U ( z ), which naturally opensthe way to the exploration of nonlocal nonlinear opticsdue to laser-induced dipolar potentials mediated by con-fined photon modes of geometries other than the waveg-uide grating considered here. For example, the laser-induced potential in a cavity along z is in general nottranslational-invariant and depends on the z -positions ofboth interacting atoms (rather than just their difference),a case which is qualitatively different from the one ana-lyzed here. B. Origin and length-scale of self ordering
We stress that the length-scale associated with order, k R = k L cos θ L − k B originates from the interaction po-tential of the light field with itself, Eq. (12), hence order-ing spontaneously occurs in this optical system much likein other condensed-mater systems and crystals. This isin contrast to e.g. order of atoms in a deep-potential op-tical lattice, where the atoms are situated at lattice sitesdetermined by the potential imposed by an external laserrather than by their mutual interactions. In our case, therole of the grating is not to trap the propagating polari-ton but merely to create dispersive and long-range dipo-lar interaction between atoms U ( z ) (induced by an ex-ternal laser Ω L that is unrelated to the light-componentof the polariotn), which underlies the nonlinearity in thepolariton propagation.Moreover, it is important to note that the length-scale k R is a signature of the specific spatial shape of the po-tential U ( z ) [sinusoidal in the case of Eq. (12)], as alsorevealed by the excitation and instability spectra of Figs.3 and 4(a). This is in contrast to e.g. the polariton crys-tallization process in a short-range potential describedin Ref. [18] using Luttinger-liquid theory, where the spe-cific shape of the potential is irrelevant. In this respect,our results are more related to the modulational instabil-ity discussed in Ref. [27] for Rydberg-atom EIT with apower-law potential, though there the considered systemis three-dimensional and the quadratic dispersion is lin-ear and emerges simply due to diffraction. Other recentworks where photon spatial correlations follow those ofthe interaction potential include Refs. [23] and [61, 62],which however treat a probe field with only a few photons(typically two), in contrast with the largely collective be-havior discussed in the present work. C. Prospects
To conclude, this work opens the way to experimentaland theoretical investigations of new nonlinear wave phe-nomena, especially by venturing beyond the linearizedregime and exploring the role of the nonlocal and non-linear dispersion term ˆ δ NL Cv ∂ z . Specific directions of further research may include: Nonlocal nonlinear opticsin 1d , concerning the study of solitons, where the lensingeffect created by the nonlinear refractive index changeis now highly nonlocal, following U ( z ); Thermalizationin 1d , which has been considered both experimentally[63] and theoretically [64] for isolated BEC in 1d, maybecome qualitatively different here due to the nonlocalcharacter of the designed interactions U ( z ) and the non-linear dispersion (”mass”); Effects of non-additivity ofsystems with long-range interactions [65] may be studiedhere for quantum/classical optical fields.
Acknowledgments
We acknowledge discussions with Ofer Firstenberg,Arno Rauschenbeutel, Philipp Schneeweiß, DarrickChang, Nir Davidson, Tommaso Caneva and Boris Mal-omed and the support of ISF, BSF and FWF [Projectnumbers P25329-N27, SFB F41 (VICOM) and I830-N13(LODIQUAS)].
Appendix A: EIT with nonlocal interactions
Here we elaborate on the derivation of Eq. (1). Sec. 1and 2 below review rather standard results, whereas theadiabatic expansion leading to nonlocal and nonlineardispersion is discussed in Sec. 3 and 4.
1. Effective 1d description
The system from Fig. 1 is modeled by the Hamiltonian H A = (cid:90) d r ρ ( r ) [ (cid:126) ω eg ˆ σ ee ( r ) + (cid:126) ( ω eg − ω ed )ˆ σ dd ( r )] ,H F = (cid:88) k (cid:126) ( ω + ck )ˆ a † k ˆ a k ,H AF = − (cid:126) (cid:90) d r ρ ( r ) (cid:88) k (cid:2) ig k ( r )ˆ a k e ik z ˆ σ eg ( r ) + h . c . (cid:3) ,H AC = − (cid:126) (cid:90) d r ρ ( r ) (cid:2) i Ω e − iω c t e ik c z ˆ σ ed ( z ) + h . c . (cid:3) ,H DD = 12 (cid:90) d r ρ ( r ) (cid:90) d r (cid:48) ρ ( r (cid:48) ) U ( r − r (cid:48) )ˆ σ dd ( r )ˆ σ dd ( r (cid:48) ) , (A1)where ˆ σ ij ( r ) = | i (cid:105)(cid:104) j | are the continuum (coarse-grained)atomic operators around r [12], and ρ ( r ) is the atomicdensity. Here { k } denote longitudinal modes guidedby the waveguide with mode function f k ( x, y ) e ikz / √ L and frequency ω k , f k being the profile of the fundamen-tal transverse mode of the waveguide at wavenumber k ,so that g k ( r ) = (cid:112) ω k / (2 (cid:15) (cid:126) ) d · f k e ikz / √ L . Assumingthat the relevant k wavenumbers form a relatively nar-row band around k = ω /c , we take a linear dispersionrelation ω k = ω + kc and ignore the dependence of thetransverse profile on k , f k ( x, y ) = e f ( x, y ).The atoms are trapped along the waveguide between z = 0 and z = L with a constant density (per unit length)along z , n a , and a transverse profile p ( x, y ) (per unitarea) of width w around some point r a = ( x a , y a ). Con-sidering the trapping width w to be much smaller thanthe spatial variation of f ( x, y ) (e.g. ∼ ρ ( r ) = n a p ( x, y ) ≈ n a δ ( x − x a ) δ ( y − y a ). Then, identifying the effective area f ( x a , y a ) = 1 / √ A and the collective atomic operatorsˆ σ ij ( z ) = (cid:82) dx (cid:82) dyp ( x, y )ˆ σ ij ( r ), and moving also to aninteraction picture with respect to H A + (cid:80) k (cid:126) ω ˆ a † k ˆ a k ,we obtain the effectively 1d Hamiltonian, Eq. (2) in themain text.
2. Weak probe approximation
Using the Hamiltonian (2), we derive the followingHeisenberg equations of motion,( ∂ t + c∂ z ) ˆ E ( z ) = n a g ∗ ˜ σ ge ( z ) ,∂ t ˜ σ ge ( z ) = − Ω − iδ c t ˜ σ gd ( z ) + g ˆ E ( z ) [ˆ σ ee ( z ) − ˆ σ gg ( z )] − γ ˜ σ ge + ˆ F ,∂ t ˜ σ gd ( z ) = Ω ∗ e iδ c t ˜ σ ge ( z ) + g ˆ E ( z )ˆ σ ed ( z ) e ik c z − in a (cid:90) dz (cid:48) U ( z − z (cid:48) )ˆ σ dd ( z (cid:48) )˜ σ gd ( z ) , (A2)with ˜ σ ge = ˆ σ ge e − ik z and ˜ σ gd = ˆ σ ge e i ( k c − k ) z . The sec-ond line of the equation for ∂ t ˜ σ ge describes the effect ofthe reservoir formed by all photonic modes not includedin the Hamiltonian (non-guided modes), namely, sponta-neous emission from the level | e (cid:105) and its correspondingLangevin noise operator. We now assume that all atomsare initially in the state | g (cid:105) and that the probe field ˆ E isweak enough so that the photon density in the medium ismuch smaller than the atom density hence the individual atoms respond linearly to the probe (i.e. the transition | g (cid:105) → | e (cid:105) is far from saturation): ˆ σ ij = O ( ˆ E ) for i (cid:54) = j ,and ˆ σ gg ∼
1. Applying such linearization to Eqs. (A2)while retaining the nonlinearity due to the interaction U between different atoms, we can eliminate ˜ σ ge and obtainthe coupled equations for ˜ σ gd and ˆ E , Eqs. (3) from themain text.
3. Adiabatic expansion
The equations of motion for the polaritons, Eqs. (6),were obtained directly from the ones for the atoms andweak probe field, Eq. (3) without further approxima-tions. In order to arrive at our central equation, Eq. (1),we then perform an adiabatic expansion in the spirit ofRef. [12]. The main idea is to consider an overall EITconfiguration which is very close to both single-photon and two-photon resonances, namely, a slowly varyingprobe envelope ˆ E ( z ) and small detunings for the couplingfield. In turn, this implies small temporal derivatives ofboth polariton fields on the one hand, and small linearand nonlinear detunings on the other hand, respectively.Namely, our small parameter in the expansion, denoted T − , scales as T − (cid:38) ∂ t ˆΨ , ∂ t ˆΦ , δ c , ˆ S. (A3)For example, in the adiabatic limit T − → ∂ t + c cos θ∂ z ) ˆΨ = − i sin θ [ δ c + ˆ S ( z )] ˆΨ , ˆΦ = 0 . (A4)In order to find the lowest order corrections to this limit,we first go back to Eqs. (6) and insert the equation forˆΦ into that of ∂ t ˆΨ, obtaining an equation of the form( ∂ t + c cos θ∂ z ) ˆΨ = D (cid:48) + D (cid:48)(cid:48) , (A5)where D (cid:48) are terms associated with dispersion effectsand D (cid:48)(cid:48) with dissipative effects. In the following wefocus on each of these terms separately. Dissipative terms . We find, D (cid:48)(cid:48) = − sin θ cos θ | Ω | γ [ c∂ z − i ( δ c + ˆ S )][ ∂ t + i ( δ c + ˆ S )] ˆΨ+ sin θ cos θ | Ω | γ [ c∂ z − i ( δ c + ˆ S )][ ∂ t + i ( δ c + ˆ S )] ˆΦ − sin θ cos θ Ω √ n a [ c∂ z − i ( δ c + ˆ S )] ˆ F . (A6)For the lowest order correction to the adiabatic limit, weinsert ˆΨ and ˆΦ from the adiabatic solution, Eqs. (A4)into D (cid:48)(cid:48) . Then, the second line in Eq. (A6) vanishes,whereas the first line scales as sin θ ( γ/ | Ω | ) T − ˆΨ [notingfrom Eqs. (A4) and (A3) that c cos θ∂ z ∼ T − ]. Then,this term of D (cid:48)(cid:48) is negligible with respect to the leadingcontribution of the left hand side of Eq. (A5) (scaling as T − ˆΨ) if T − (cid:28) | Ω | γ θ . (A7)For typical EIT conditions with v (cid:28) c , we have sin θ ≈ T − , should be within the single-atom EITtransparency bandwidth | Ω | /γ . It should be noted how-ever that the usefulness of the single-atom transparencywindow in predicting the losses in our system relies onthe expansion in T − , i.e. on very narrow detunings, andthat in the Autler-Townes regime for example, consider-able losses are expected for T − ∼ Ω (cid:28) | Ω | /γ . Never-theless, the real condition for lossless propagation has totake into account the entire medium and not just a singleatom. It can be found by noting that the first line in D (cid:48)(cid:48) L are negligible for T − (cid:28) | Ω | γ √ OD ≡ δ tr , (A8)where δ tr is the transparency window of the entiremedium, as usual [12], which for large OD can becomemuch narrower than both Ω | /γ and Ω.Finally, the quantum fluctuations induced by thereservoir formed by the vacuum of all non-guided modes,captured by the ˆ F term, are related to the losses by thefluctuation-dissipation theorem. Therefore, when lossesare negligible, so is the effect of these fluctuations, uponcomparison to the reservoir-free quantum fluctuations ofˆΨ. We have verified this, using the equation for ˆΦ in Eq.(6), by showing that the variance which ˆ F adds to ˆΦ isnegligible with respect to the vacuum noise of ˆΦ. Dispersive terms . We find, D (cid:48) = − sin θ cos θ | Ω | [ c∂ z − i ( δ c + ˆ S )] ∂ t [ ∂ t + i ( δ c + ˆ S )] ˆΨ+ sin θ cos θ | Ω | [ c∂ z − i ( δ c + ˆ S )] ∂ t [ ∂ t + i ( δ c + ˆ S )] ˆΦ − i sin θ ( δ c + ˆ S ) ˆΨ . (A9)Inserting the adiabatic solutions, Eqs. (A4), into theabove expression, we obtain D (cid:48) up to order T − ˆΨ. Then,for approximately lossless propagation as in Eq. (A7), wetake D (cid:48)(cid:48) ≈
0, and Eq. (A5) takes the form( ∂ t + c cos θ∂ z ) ˆΨ = ( i ˆ A + ˆ B∂ z + i ˆ C∂ z + ˆ D∂ z ) ˆΨ , (A10)with hermitian (real) coefficients ˆ A, ˆ B, ˆ C and ˆ D ex-panded up to order T − , T − , T − and T , respectively.Taking the lowest non-vanishing order for each coeffi-cient, we find ˆ S ≈ ˆ δ NL and end up with Eq. (1) fromthe main text, plus an extra ˆ D -coefficient term withˆ D = − (sin θ/ | Ω | ) v . Nevertheless, for the regime ex-plored in this work, namely, that of a probe-pulse band-width smaller than the combined detuning δ c + ˆ δ NL , suchcubic dispersion term is negligible (this was also verifiedby repeating calculations while including this term).Finally, it should be noted that our derivation of Eq.(1) can be shown to be consistent with a similar adiabaticexpansion in Ref. [12], upon relating the linear and non-linear detunings δ c + ˆ S in Eqs. (6) to the time-dependenceof the EIT mixing angle in [12] (Sec. V therein).
4. Significant dispersion in the lossless regime
The question is now under which conditions the above dispersive corrections to adiabaticity, giving rise to the ∂ z term in Eq. (1), are significant while losses are nev-ertheless negligible. By comparing the first line of D (cid:48) to that of D (cid:48)(cid:48) in Eqs. (A9) and (A6), respectively, weobserve that dispersive effects are dominant over dissipa-tive ones when T − (cid:29) γ . On the other hand, for losslesspropagation we have from (A7), T (cid:29) | Ω | /γ , leading tothe Autler-Townes condition Ω (cid:29) γ , as also explainedIn sec. II A and Fig. 2. More realistically, we considerlossless propagation with the condition (A8), leading tothe requirement γ (cid:28) T − (cid:28) | Ω | γ √ OD , (A11)and hence to a modified Autler-Townes-like condition γOD / (cid:28) Ω. In this work, we consider a case wheredeviations from adiabaticity are mainly due to the over-all detuning of the coupling field, ∆ c = δ c + δ NL (inanalogy to the intuitive explanation of sec. II A), so thatthe condition (A11) has to be satisfied for T − ∼ ∆ c . Appendix B: Bogoliubov theory for a nonlocalinteraction1. Excitation spectrum
The polariton wave excitation spectrum (dispersion re-lation) around the CW solution, Eq. (8), is found asfollows. We first write the polariton field as a sum ofan average CW (cid:104) ˆΨ( z, t ) (cid:105) = ψ ( t ) and a small fluctuationˆ ϕ ( z, t ), ˆΨ( z, t ) = ψ ( t ) + ˆ ϕ ( z, t ) , (B1)where the average CW solution is that of self-phase mod-ulation, ψ ( t ) = ψ e − i ( αδ c + n p U ) t with ψ = (cid:112) n p /α e iφ , U = U k =0 , U k being the spatial Fourier transform of thepotential U ( z ), U k = (cid:90) ∞−∞ dzU ( z ) e − ikz . (B2)The integral for U that arises in the above CW solution ψ ( t ) runs over z (cid:48) = 0 to z (cid:48) = L around some point z , (cid:82) L dz (cid:48) U ( z − z (cid:48) ), and not from z (cid:48) = −∞ to z (cid:48) = ∞ around z = 0 as in the definition (B2). However, for a symmetricpotential [ U ( z ) = U ( − z )] with range l , and points z wellwithin the medium, i.e. l < z < L − l , the edges z = 0and z = L have no effect, so that U can be approximatedas (cid:82) L dz (cid:48) U ( z − z (cid:48) ) ≈ (cid:82) ∞−∞ dz (cid:48) U ( − z (cid:48) ) for all such points z . Thus, we may neglect the edge effects of a sufficientlylong medium L (cid:29) l .Inserting Eq. (B1) into the nonlinear equation (1) inthe main text and keeping terms only to linear order inthe perturbation ˆ ϕ , we obtain( ∂ t + v∂ z ) ˆ ϕ ( z ) = − i ( αδ c + n p U ) ˆ ϕ ( z )+ i ( n p U /α + δ c ) Cv ∂ z ˆ ϕ ( z ) − in p (cid:90) L dz (cid:48) U ( z − z (cid:48) ) ˆ ϕ ( z (cid:48) ) − in p e i φ − ( αδ c + n p U ) t ] (cid:90) L dz (cid:48) U ( z − z (cid:48) ) ˆ ϕ † ( z (cid:48) ) . (B3)1This linearized equation is valid as long as the fluctua-tions ˆ ϕ around the mean ψ are small, i.e. for | ψ | (cid:29) |(cid:104) ˆ ϕ (cid:105)| , (cid:113) (cid:104) ˆ ϕ † ˆ ϕ (cid:105) , (cid:112) (cid:104) ˆ ϕ (cid:105) . (B4)In order to find the dispersion relation of the fluctuationsit is enough to consider the case of a classical field ˆ ϕ → ϕ .We then insert the ansatz for ϕ ( z, t ) from Eq. (7) into thelinearized equation (B3), obtaining coupled equations for u k and v k (cid:18) ω k + n p U k − ω k − n p U k − n p U k ω k + n p U k + ω k (cid:19) (cid:18) u k v k (cid:19) = 0 . (B5)In order to arrive at the equations (B5), we used (cid:90) L dz (cid:48) U ( z − z (cid:48) ) e ikz (cid:48) ≈ (cid:90) ∞−∞ U ( ξ ) e ikξ e ikz = U k e ikz , (B6)which again amounts to neglecting edge effects, andwhere in the last equality we assumed that U ( z ) is sym-metric and real so that U − k = U k . Eq. (B5) has anontrivial solution only if the determinant of the matrixon the left-hand side is zero, yielding an extra equationwhose solution is the spectrum ω k from Eq. (8).
2. Dynamic Bogoliubov theory
Here we address the quantum description of the dy-namics of the fluctuations ˆ ϕ around the CW solution andarrive at the dynamical Bogoliubov transformation withthe coefficients of Eq. (10) in the main text. We expandthe quantum field ˆ ϕ ( z, t ) in spatial Fourier modes ˆ a k ( t ),ˆ ϕ ( z ) = (cid:80) k (1 / √ L ) e ikz ˆ a k , and the transformed modesˆ c k ( t ) as,ˆ ϕ ( z, t ) = (cid:88) k √ L e ik ( z − vt ) e i [ φ − ( αδ c + n p U ) t ] ˆ c k ( t ) , ˆ a k ( t ) = e iφ e − i ( αδ c + n p U + kv ) ˆ c k ( t ) , (B7)where the operators ˆ a k ( t ) satisfy the equal-time commu-tation relations [ˆ a k ( t ) , ˆ a † k (cid:48) ( t )] = δ kk (cid:48) and hence so do theoperators ˆ c k ( t ). Inserting Eq. (B7) into the linearizedequation (B3) and neglecting edge effects as in Eq. (B6),we obtain an equation for ∂ t ˆ c k . Then, upon taking itsHermitian conjugate, we end up with coupled equationsof motion for ˆ c k and ˆ c †− k , ∂ t (cid:18) ˆ c k ˆ c †− k (cid:19) = − i (cid:18) ω k + n p U k n p U k − n p U k − ω k − n p U k (cid:19) (cid:18) ˆ c k ˆ c †− k (cid:19) . (B8)By diagonalizing the matrix, we find the solution for ˆ c k ( t )from which we obtain the dynamics of ˆ a k ( t ) as the Bo-goliubov transformation from Eq. (10) in the main text.
3. Intensity correlations
Intensity correlations are characterized by the normal-ized second order coherence function defined by [66] g (2) ( z, z (cid:48) ) = G (2) ( z, z (cid:48) ) G (1) ( z ) G (1) ( z (cid:48) ) (B9)with G (2) ( z, z (cid:48) ) = (cid:104) ˆΨ † ( z, t ) ˆΨ † ( z (cid:48) , t ) ˆΨ( z (cid:48) , t ) ˆΨ( z, t ) (cid:105) ,G (1) ( z ) = (cid:104) ˆΨ † ( z, t ) ˆΨ( z, t ) (cid:105) . (B10)Here we assume that all points of the field ˆΨ( z ) haveexperienced the same duration of interaction t upon ar-rival at the detector. The detector measures at differenttimes different points of the field ˆΨ( z ), so that the cor-relations between detector-signals at different times arein fact correlations of the field in space as in g (2) ( z, z (cid:48) )which quantifies the autocorrelation of the field between z and z (cid:48) .Upon expanding the field in its longitudinal Fouriermodes, we recall that the k = 0 mode can be approxi-mated by the strong average CW solution,ˆΨ( z, t ) = 1 √ L (cid:88) k e ikz ˆ a k ( t ) ≈ a ( t ) √ L + 1 √ L (cid:88) k (cid:54) =0 e ikz ˆ a k ( t ) ,a ( t ) / √ L = (cid:113) n p /α e iφ e − i ( αδ c + n p U ) t , (B11)where ˆ a k ( t ) is given by Eq. (10) in the main text. In-serting Eq. (B11) into G (1) ( z ) from Eq. (B10), we find G (1) ( z ) = n p α + 1 L (cid:88) k (cid:54) =0 (cid:2) | µ k | N k + | ν k | ( N k + 1) (cid:3) , (B12)where we have assumed the following statistics of theinitial fluctuations ˆ a k (cid:54) =0 : (cid:104) ˆ a k (0) (cid:105) = 0, (cid:104) ˆ a † k (0)ˆ a k (cid:48) (0) (cid:105) = N k δ kk (cid:48) with N k = N − k , and (cid:104) ˆ a k (0)ˆ a k (cid:48) (0) (cid:105) = 0. Mov-ing to the second-order coherence, we insert Eq. (B11)into G (2) from Eq. (B10), keeping terms only to secondorder in the fluctuations ˆ a k (cid:54) =0 , in accordance with theassumptions (B4), and obtain G (2) ( z, z (cid:48) ) = n p α + 2 n p α × L (cid:88) k (cid:54) =0 (cid:8)(cid:2) | µ k | N k + | ν k | ( N k + 1) (cid:3) (1 + cos[ k ( z − z (cid:48) )])+ | ν k || µ k | (1 + 2 N k ) cos[ k ( z − z (cid:48) ) − φ k ] } , (B13)where φ k = arg( µ k ν k ). Inserting G (1) and G (2) from Eqs.(B12) and (B13) in Eq. (B9), we note that to lowestorder in the fluctuations we can take G (1) ≈ n p /α suchthat g (2) ∝ G (2) . Then, upon taking the continuum limit(1 /L ) (cid:80) k → (1 / π ) (cid:82) ∞−∞ dk , we split the integral into itspositive and negative k -values and obtain Eq. (11) fromthe main text.2 Appendix C: Laser-induced dipole-dipole interaction
The laser-induced interaction potential between a pairof atoms, e.g. at positions z and z along the z -axis, isgiven by [57] U ( z , z ) = − | Ω L | δ L (cid:126) ∆ dd ( z , z ) cos[ k zL ( z − z )] , (C1)with the same notations from Sec. V (and Fig. 1c)and where ∆ dd ( z , z ) is the so-called resonant dipole-dipole interaction evaluated at the laser frequency ω L .∆ dd ( z , z ) is responsible for dispersive interactions (ex-citation exchange) between the atoms mediated by vir-tual excitations of the photon modes that couple to theatoms. Therefore, the spatial dependence of ∆ dd ( z , z ),which in turn determines the potential U ( z , z ), directlyfollows that of the photon modes. And in a confined ge-ometry, these can alter dramatically and lead to modifiedand long range dipolar interactions ∆ dd and potentials U [28, 31, 32, 34, 57]. The case considered here, of adipolar interaction at a frequency inside the bandgap ofwaveguide grating, leads to ∆ dd ( z , z ) ∝ η cos[ k B ( z − z )] e −| z − z | /l , with η, l ∝ / √ ω u − ω L , where ω u is thefrequency of the upper bandedge [28, 31, 32]. Therefore,for ω L not too far from the bandedge, strong and long-range interaction potentials can be achieved. Appendix D: Quantitative illustration
Let us specify the system parameters used in the il-lustration in Figs. 3 and 4. Motivated by the experi-ment in Ref. [44], we take the wavelength, dipolar ma-trix element and radiative decay rate of the D1 line ofCs atoms [67] as the typical parameters of all dipolaratomic transitions in Fig. 1, and consider atoms trappedat a distance r a ∼
500 nm from the center of a ta-pered fiber with radius a = 250 nm and refractive in-dex n = 1 . transverse mode A = 4 . µ m [31]. Taking anatom density of n a ∼ × m − , length L = 2 . × s − (2 × s − and 1 . × s − in Figs. 3b and4, respectively) and detuning δ c = − . × s − inFig. 3a (4 . × s − and − . × s − and inFigs. 3b and 4), we obtain α = 0 . . . v = 48216 ms − (12055ms − in 4340 ms − in Figs. 3b and 4, respectively) and δ tr = 8 . × s − (2 . × s − and 7 . × s − in Figs. 3b and 4), so that Figs. 3a,b and 4a,b are ef-fectively cut at k = q tr = δ tr /v = 1795 m − . For theBragg grating imprinted on the fiber/waveguide [47, 58],we assume periodic perturbations ∆ n = 0 .
02 of the re-fractive index about n with period of length Λ = 396nm. For the laser induced interaction we take a detuning δ L = − π × .
65 GHz and intensity I = 2 × Wm − (0 . × Wm − and 10 Wm − in Figs. 3b and 4), yielding U L = 2 ηR fs = 1 . × s − (2 . × s − and 5696 s − in Figs. 3b and 4), where η ∼
12 is the ratiobetween emission to the fiber grating modes and to free-space, and R fs = γ | Ω L | / (2 δ L ) = 47443 s − (11860 s − and 237 s − in Figs. 3b and 4) is the resulting scatteringrate to free space form the | d (cid:105) → | s (cid:105) transition, leadingto a potential range of l ≈ λ L ≈ . L ≈ . θ L = 0 . k R = k L cos θ L − k B = 1019 m − . We assume a CWbackground with a power 2 × − W in Fig. 3a (10 − W and 4 × − W in Figs. 3b and 4) , giving n p ≈ − (37347 m − and 414981 m − in Figs. 3b and 4). Forthe N k ( t ) and g (2) calculations in Figs. 4b,c, we consideran initial Gaussian spectrum of intensity fluctuations, N k = N e − ( k/q tr ) , with N = 5. Finally, wherever acomparison with the local-interaction case is made, thelocal potential is taken as U ( z ) = (1 / U L lδ ( z ). Appendix E: Numerical simulations of Eq. (1)
In order to obtain the ω k spectrum, we perform numer-ical simulations of the full nonlinear equation (1). We usea three-term splitting method, capable of dealing withthe nonlocal and nonlinear dispersion coefficient that ap-pears in Eq. (1), unlike the more common (two-term)split-step method. We study the dynamics of the field,initially comprised of the CW solution ψ and weak per-turbations ϕ at a spatial frequency k , and extract thefield’s temporal oscillation frequency ω ( k ) leading to thespectrum ω k . For weak enough perturbations where theBogoliubov theory applies, the results of the simulationof Eq. (1) agree very well with those of the simpler split-step method simulations, where the nonlinear dispersionterm ˆ δ NL Cv ∂ z is approximated by its mean-field value, δ NL = n p U /α , so that the equation becomes of semilin-ear type. For the case of instability (Fig. 4b,c), we beginwith the initial intensity fluctuations with a spectrum N k and run split-step simulations with the mean-fieldnonlinear dispersion term δ NL = n p U /α . Then, afterpropagation time t , we measure N k ( t ) = (cid:104) a ∗ k ( t ) a k ( t ) (cid:105) and g (2) = (cid:104) I ( z ) I ( z (cid:48) ) (cid:105) / ( (cid:104) I ( z ) (cid:105)(cid:104) I ( z (cid:48) ) (cid:105) ), with I ( z ) = | Ψ( z ) | for a classical field Ψ. Appendix F: Scattering and imperfections
In the following we address the main loss and dephas-ing mechanisms which affect the polariton evolution. Foreach of them we estimate the rate at which they decoherethe polariton field; then, by comparing the analytical ex-pression of this decoherence rate with that of the spectra ω k (or γ k = Im ω k in the case of instability), we studyhow the limitations it imposes on the observability of thespectra can be relaxed.3
1. Scattering of off-resonant laser photons
Beginning with the illumination of the | d (cid:105) → | s (cid:105) atomictransition by the off-resonant laser Ω L , it leads to aninteraction potential U ( z ), but is also accompanied byscattering of photons to non-guided modes at a rate R fs = γ | Ω L | / (2 δ L ) [57], which decoheres the spin wave¯ σ gd , and hence the polariotn, at rate R fs . This rate hasto be compared to the frequency scale U L = 2 ηR fs of thecoherent interaction-related effects such as ω L , where η is a geometrical factor related to the mode-density of theguided modes. For the case of waveguide-grating modes, η can be larger than 1 ( η ∼
12 in our numerical exam-ple, see Appendix D) so coherent effects prevail [31]. Infact, a deeper reason for the dominance of the coher-ent effects discussed here (the spectra ω k and γ k ) resultsfrom the fact that these are all cooperative effects, pro-portional to the density of excitations (EIT polaritons) n p in the system, whereas the scattering R fs is essen-tially a single-atom effect since photons are scattered tonon-guided, non-confined modes (which mediate muchweaker cooperative effects) [31]. Therefore, ω k scales as n p U L ∝ n p ηR fs [Eq. (8))] so that even for small η , itcan be made much larger than R fs by e.g. increasing n p (increasing the laser power of the background CWprobe field). Indeed, for the specific numerical exam-ple taken here (Appendix D), we obtain R fs = 47443s − , − ,
237 s − for the roton, anti-roton and in-stability cases, respectively, yielding decoherence rates R fs much smaller than the typical frequency resolutionof the corresponding spectra ω k of the roton and anti-roton (Figs. 3a and 3b), and γ k of the instability (Fig.5a).
2. EIT propagation losses
Next, we recall that lossless propagation of EIT po-laritons is only exact for CW polaritons and vanishingcoupling-field detuning ∆ c = 0. As seen in Fig. 2, theEIT susceptibility in fact exhibits a finite spectral ”trans-parency window”, leading to a loss rate [8] R EIT = (1 / k v Im χ (∆ p , ∆ c )= v ODL p − ∆ c ) γ γ (∆ p − ∆ c ) + [Ω − p (∆ p − ∆ c )] , (F1) γ being the width of the atomic level | e (cid:105) and OD the opti-cal depth of the medium (see main text). For simplicity,we evaluate R EIT at the center of the pulse (∆ p = 0)and for ∆ c = δ c + δ NL , where δ NL = n p U /α is themean-field value of the nonlinear detuning. Then, theeffect of this EIT-related decoherence rate on ω k can bemade much smaller by the following: consider decreas-ing both n p and δ c by a factor f , resulting in a decreaseof ω k by the same factor [see Eq. (8)]. On the otherhand, Eq. (F1) reveals that R EIT scales approximately
500 1000 1500 k (cid:64) (cid:144) m (cid:68) Γ k (cid:64) (cid:144) s (cid:68) (cid:72) a (cid:76)
500 1000 1500 k (cid:64) (cid:144) m (cid:68) Ω k (cid:64) (cid:144) s (cid:68) (cid:72) b (cid:76)
500 1000 1500 k (cid:64) (cid:144) m (cid:68) Γ k (cid:64) (cid:144) s (cid:68) (cid:72) c (cid:76) FIG. 5: (a) Instability case. Spectrum of amplification γ k =Im ω k for the example considered in the main text (as in Fig.4a therein): nonlocal potential (blue solid curve) and localpotential (red dashed curve). (b) Overcoming cooperativedecoherence R (2) im : anti-roton case. Same parameters as in themain text (Fig. 3b therein) apart from Ω which is increasedby a factor 12. (c) Same as (b), however this time for the caseof instability γ k and where Ω is increased by a factor 8.5. as ∆ c = ( δ c + n p U /α ) and hence decreases by f , sothat it can become much smaller than ω k and its relatedeffects (the scaling R EIT ∼ ∆ c holds for large Ω in cor-respondence with the Autler-Townes regime of EIT thatwe consider).Referring back to our numerical example we obtain R EIT ∼ . × s − for the roton case, so that this deco-herence becomes comparable to the frequency resolutionof the main features in ω k [e.g. the difference between theroton and usual (local-interaction) Bogoliubov spectra atthe roton ”dip” around k R is about 0 . × s − , see Fig.3a]. Then, decreasing (multiplying) both n p and δ c by afactor f = 0 . R EIT ∼ . × s − whereasthe typical frequency resolution for the roton spectrumbecomes 0 . × s − , following the expected f and f scalings, respectively, and yielding a much suppressed ef-fect of the losses. For the anti-roton case, again with thenumbers from Appendix D, we have R EIT ∼ . × s − , which is already much smaller than the typical reso-lution of the anti-roton feature, ∼ × s − (Fig. 3b).Finally, in the case of instability we find R fs ≈ . × s − , which is smaller than but still rather close to thepeak value of the amplification rate γ k in Fig. 5a. Thiscan be improved by again decreasing n p and δ c by f = 0 . R fs ≈ . × s − with γ k and its fea-tures reduced only by f = 0 .
3. Scattering by imperfections in the waveguidestructure
The effect of imperfections on the guided modes k istypically described by adding an imaginary part − iκ totheir mode frequencies. This can be consistently shownby e.g. considering pure modes which are dipole-coupledto scatterers modelled as lossy and localized harmonicoscillators far-detuned from the modes k [68]. Recallingthat the interaction U ( z ) between the atoms is mediatedby the photon modes k , and taking their width κ into4account, U now becomes complex, U ( z ) = U (cid:48) ( z ) − iU (cid:48)(cid:48) ( z ) ,U (cid:48) ( z ) = − cos( k zL z ) Ω L δ L (cid:90) ∞ dω G ( ω, z )( ω − ω L )( κ/ + ( ω − ω L ) ,U (cid:48)(cid:48) ( z ) = − cos( k zL z ) Ω L δ L (cid:90) ∞ dω G ( ω, z ) κ/ κ/ + ( ω − ω L ) , (F2)where G ( ω, z ) (real function) is the autocorrelation spec-trum of the photon-modes reservoir coupled to a pair ofatoms z -apart [28, 32, 57, 68]. The existence of the imagi-nary part − iU (cid:48)(cid:48) reflects the fact that now the modes k notonly mediate a dispersive interaction between the atoms( U (cid:48) ), but also mediate scattering of laser photons Ω L tonon-guided modes via the atoms and the imperfectionsin the waveguide.For the modes k of a waveguide with a bandgap, as inour example, we have G ( ω, z ) ∝ / √ ω − ω u , with ω u thefrequency of the upper bandedge [28, 31], so the integralsin Eq. (F2) get most of their contribution around ω ∼ ω u . Then, as a simple estimation, we consider the lowestorder in κ , so that U (cid:48) ( z ) ≈ U ( z ) , U (cid:48)(cid:48) ( z ) ≈ κ δ u U ( z ) , δ u ≡ ω u − ω L , (F3)with U ( z ) from Eq. (12) in the main text. We nowturn to the estimation of three distinct decoherencerates associated with this imperfection-induced lossmechanism. Single-atom scattering.–
The resulting decoherencerate on a single atom is obtained by taking z → R (1) im ∼ (1 / | U L | ( κ/δ u ). We note thatthis is a single-polariotn decoherence rate and hencecan be made much smaller than the cooperative effectsrepresented by ω k , by increasing n p (as explained for the R fs case above). Cooperative scattering.–
Atoms located at a dis-tance z apart can also scatter photons in a cooperativemanner, leading to a mutual decoherence rate U (cid:48)(cid:48) ( z ).The resulting cooperative decoherence per atom isthen obtained by summing over interfering mutualscattering from all | d (cid:105) -populated atoms in the medium, R (2) im ∼ . n p (cid:82) L dz (cid:48) ( κ/δ u ) U ( z (cid:48) ) ∼ . κ/δ u ) n p U ,where n p is the density of polariton excitations in the medium (which is roughly the density of | d (cid:105) -populatedatoms). We now need to compare R (2) im ∼ . κ/δ u ) n p U to ω k = (cid:112) ω k ( ω k + 2 n p U k ). It is easy to show that ω k ∝ Cv ∝ Ω , so that near the peak of ω k , where2 n p U k is dominant over ω k , we have ω k ∝ Ω. On theother hand, R (2) im is independent of Ω, so that in principle ω k can always be made dominant over R (2) im by increasingΩ.For our numerical example we first need to estimate κ :current state-of-the-art nano-waveguide-based cavitiescan have an imperfection-induced Q -factor of up to ∼ × for a cavity resonance at ∼
800 nm, so that κ ∼ × s − . Going back to our numerical examplefrom the main text, we typically have δ u ∼ s − ,yielding R (2) im ∼ . n p U . Then, for the anti-rotoncase, we find R (2) im ∼ × s − , much larger thanthe anti-roton peak of a few 10 s − (Fig. 3b). Thiscan be improved by increasing Ω by e.g. a factor of f = 12, yielding the pronounced anti-roton spectrum ofFig. 5b. Similarly, for the case of instability, we find R (2) im ∼ . × s − , much larger than the amplificationrate γ k which is peaked at a value of about 1 . × s − . Again, this can be remedied upon increasing Ω by afactor f = 8 . R (2) im may pose an important limitation on the its observability. Polariton propagation losses.–
Eq. (5) defines thepolariotn ˆΨ as a superposition of an atom (spin wave)and a propagating photon. The origin of both R (1) im and R (2) im discussed above are in its atomic component,whereas its photonic component, essentially comprisedof the modes k , also decays at a rate κ due to photonpropagation in the presence of imperfections. Thismeans that the polariton field suffers from an additionaldecoherence given by κ times the fraction of its photoncomponent cos θ (likewise, R (1 , im should be multipliedby sin θ , however as typical of EIT, we have sin θ ≈ R fs , R EIT and R (1 , im ), this loss mechanism is independent of the existence of the interaction U ( z ) (itdoes not depend on the laser Ω L ), and hence can be firstmeasured in the absence of U and then corrected for inmeasurements where U is switched on. [1] R. B. Boyd, Nonlinear Optics (Academic Press, Orlando2008).[2] W. Kr´olikowski, O. Bang, N. I. Nikolov, D. Neshev, J.Wyller, J. J. Rasmussen and D. 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