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Theory of a Slow-Light Catastrophe Ulf Leonhardt School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland In diffraction catastrophes such as the rainbow the wave nature of light resolves ray singularities and draws delicate interference patterns. In quantum catastrophes such as the black hole the quantumnatureoflightresolveswavesingularitiesandcreatescharacteristicquantumeffectsrelated to Hawking radiation. The paper describes the theory behind a recent proposal [U. Leonhardt, arXiv:physics/0111058, Nature (in press)] to generate a quantumcatastrophe of slow light. 2 0 42.50.Gy, 03.70.+k, 04.70.Dy 0 2 n a I. INTRODUCTION terms [8], a monochromatic light wave of frequency ω J oscillates as Θ(r r )(r r )iµ when the radius r ap- 9 Catastrophes [1] are at the heart of many fascinating proaches the hori−zonsrs, −withs µ = 2rsω/c. A logarith- opticalphenomena. Themostprominentexampleofsuch mic phase singularity will develop. Potential quantum ] s a catastrophe is the rainbow. Light rays from the Sun effects of such a wave singularity are effects of the quan- c i enter waterdroplets floatingin the air. After two refrac- tum vacuum. The gravitational collapse [7] of the star pt tionsandonereflectioninsideeachdroptheraysreachan intotheblackholehassweptalongthevacuum. Thevac- o observer. Above a critical observation angle no rays ar- uum thus shares the fate of an inward-falling observer. . rive,whereasbelowtheangletworaysstriketheobserver. Yet such an observer would not notice anything unusual s c A brightbow, the rainbow,appearsat the criticalangle, at the event horizon. In mathematical terms, the vac- si because here the cross section of light rays diverges [2]. uum modes are analytic across the horizon[8,9]. On the y (The critical angle depends on the refractive index that other hand, the modes perceived by an outside observer h varieswiththefrequencyoflightindispersivemediasuch are essentially non-analytic, because they vanish beyond p aswater,givingrisetotherainbowcolors.) Thedirection the horizon where the observer has no access to. Con- [ ofalightrayisproportionaltothegradientofthephase. sequently, the observer does not see the electromagnetic 2 The rainbow thus represents a singularity of a gradient field in the vacuum state. Instead, the observer notices v map,a catastrophein the senseofThom[3]andArnol’d the quanta of Hawking radiation [10] with the Planck 0 7 [4]. Structurallystablesingularitiesofgradientmapsfall spectrum (e2πµ−1)−1. The quantum vacuum does not 1 into distinct classes,depending onthe number ofcontrol assumecatastrophicwavesofthetypeΘ(r rs)(r rs)iµ, − − 1 parameters involved [3,4]. Structural stability is the key henceresolvingsotheassociatedwavesingularityand,si- 1 to Nature’s way of focusing light [5] in the caustics cre- multaneously,generatingquantumradiationwithachar- 1 ated by ray catastrophes. Yet the wave nature of light acteristic spectrum. At the heart of such a catastrophe 0 / smooths the harsh singularities of rays. Simultaneously, lies a time-dependent process, for example the gravita- s characteristic interference effects appear. For example, tional collapse in the case of the black hole [7]. The c i thepairsoflightraysbelowtherainbowcreateadelicate processhasdisconnectedthespatialregionswherewaves s y pattern of supernumerary arcs [1] that are visible under canpropagateandhascreatedalogarithmicphasesingu- h favorable weather conditions (when the floating droplets larityatthe interface. Anytime-dependent phenomenon p are nearly uniform in size [2]). Every class of diffraction willgeneratesomeradiation,aslongastheprocesslasts. v: catastrophes generates its distinct interference structure In remarkable contrast, a quantum catastrophe creates i [1]. quanta continuously. X Catastropheopticsdescribesthewavepropertiesofray This paper describes the theory behind a recent idea ar singularities. In the hierarchyof physicalconcepts, wave [11] to generate a quantum catastrophe of slow light optics refines and embraces rayoptics, and quantum op- [12–15,17–21]. An experiment is proposed based on tics rules above wave optics. So, what would be the Electromagnetically-Induced Transparency (EIT) [17]. quantum effects of wave catastrophes [6]? First, what In EIT a controlbeam determines the optical properties are quantum catastrophes? It might be a good idea to of slow light in a suitable medium. Changing the inten- begin with an example, the black hole [7]. When a star sity of the control light from a uniform to a parabolic collapsesto a black hole aneventhorizonis formed,cut- profile creates s slow-light catastrophe [11], see Fig. 1. ting space into two disconnected regions. Seen from an This catastrophe resembles the event horizon of a black outside observer, time stands still at the horizon, freez- hole but also shows some characteristic differences. In ing all motion. A light wave wouldfreeze as well, propa- Section II we put forward a rather general phenomeno- gating with ever-shrinking wavelength. In mathematical logical quantum field theory of slow light. Appendix A 1 justifies the theoryforthe mostcommonmethodtoslow modifies the optical properties of the atom. In partic- down light using EIT [17]. In Section III we address the ular, the coupling of the excited states affects the tran- specific theory of the slow-light catastrophe [11]. Ap- sition from the atomic ground state to one of the up- pendix B contains some estimations that are relevant to per states, i.e., the ability of the atom to absorb probe the experimental aspects involved. Section IV summa- photonswithmatchingtransitionfrequency. Destructive rizes the results and draws a further vision of quantum quantum interference between the paths of the transi- catastrophes. tion process turns out to eliminate absorption at exact resonance [17]. A medium composed of such optically- manipulatedatomsistransparentataspectrallinewhere it wouldotherwisebe completely opaque. Inthe vicinity of the transparency frequency ω the medium is highly 0 dispersive,i.e. therefractiveindexchangeswithinanar- row frequency interval. In turn, probe light pulses with a carrier frequency at ω travel with a very low group I 0 velocityv [18]. The intensity I of the controlbeamde- g c termines the group velocity of the probe pulse, as long asI isstrongerthantheprobe. Paradoxically,thelower c I is the slower moves the pulse, which, however, is only c possiblewhentheelectronicstatesoftheatomsfollowdy- namically the control field [19], causing the probe-light Z intensity to fall accordingly [19]. In this regime light freezes when I vanishes [13–16]. c The theoryofEIT[17]oftenemploysanatomic three- level scheme: Two levels account for the excited states coupled by the control field and one level represents the groundstate,seeAppendix A.Inreality,atomsaremore complicated and, when details matter, an accurate de- scriptioninvolvesthe fullatomicsub-levelstructure[20]. Hereweputforwardaphenomenologicalquantumtheory ofslowlightthatisratherindependentofthemicroscopic mechanisms used in practice. We assume only that the FIG.1. Schematicdiagram oftheproposedexperiment. A beamofcontrollight withintensityIc generatesElectromag- slow-lightmedium is transparentwith a realsusceptibil- netically-Induced Transparency [15] in a medium, strongly ity that depends linearly on the detuning from ω0. In modifyingitsopticalpropertiesforasecondfieldofslowlight. EIT our model is restricted to the narrow transparency Whenaninitially uniformcontrolintensityisturnedintothe windowaroundω thatcanmaximallyreachthe natural 0 parabolic profile shown in the figure, the slow-light field suf- line width of the atomic transition. Appendix A shows fersaquantumcatastrophe. Toslow-lightwaves,theinterface thatourmodelagreeswiththethree-leveltheoryofEIT. Z of zero control intensity cuts space into two disconnected Our theory is simple enough to be treated analytically regionsandcreatesalogarithmicphasesingularity,inanalogy and yet sufficiently complex to capture the essence of totheeffect[7]ofaneventhorizon[6]. Thequantumvacuum slow-light quanta. We postulate an effective Lagrangian ofslowlightcannotassumesuchcatastrophicwaves. Inturn, L, show that L is consistent with the known dynamics pairs of slow-light quanta,propagating in opposite directions awayfromZ,areemittedwithacharacteristicspectrum. The of EIT within the validity range of our model, calculate waves shown below the intensity profile refer to the emitted theenergyandquantizethefieldofslow-lightpolaritons. light with themodes wR and wL of Eq. (50). Our approach has the additional advantage that it may be applicable to other mechanisms [21] for creating slow light that do not rely on EIT. II. THE MODEL A. Lagrangian Electromagnetically-Induced Transparency (EIT) [17] has served as a method to slow down light significantly Wecharacterizeslowlightbyarealscalarfieldϕignor- [12] or, ultimately, to freeze light completely [13–16]. ingthe polarization. Theopticalfieldϕshallbe givenin Likeothersuccessfultechniques,EITisbasedonasimple unitsofthevacuumnoise[22]. Forsimplicity,weassume idea[17]: Acontrolbeamoflaserlightcouplestheupper uniformity in two spatial directions,x and y, and regard levels of an atom, and, in this way, the beam strongly the optical field as a function of time t and position z. Throughoutthispaperwedenotepartialtimederivatives 2 by dots, by differential operators ∂ or simply by suf- At a laterstage we need to considernegative frequencies t fixest, whateverhappens tobe mostconvenient. Spatial as well. To verify that the dispersion relation (6) is also derivatives are denoted by dashes, differential operators validinthecorrespondingnegativetransparencywindow ∂ or suffixes z. We postulate the effective Lagrangian weutilizethegeneralpropertyofaspectralsusceptibility z L = ¯h2 (1+α)ϕ˙2−c2ϕ′2−αω02ϕ2 . (1) χ(−ω)=χ∗(ω) (9) (cid:0) (cid:1) which implies near ω 0 We see in the next subsection that the real parameter − function α determines the group velocity, and hence we 2α χ= (ω+ω ). (10) call α group index. In EIT [17] the parameter α is in- −ω 0 0 versely proportional to the control-field intensity, We see that as long as the frequency of the probe light κ α(t,z)= , (2) lies within the transparency windows of EIT, the La- Ic(t,z) grangian (1) reproduces the typical linear slope of the spectral susceptibility. In fact, up to a trivial prefactor, with a coupling strength κ that is proportional to the L is the only Lagrangian that is quadratic in the field modulus squared of the atomic dipole-transition matrix and its derivatives and that is consistent with the spec- element and to the number of atoms per unit volume. Without the EIT medium present, L is the Lagrangian tralsusceptibilities (8)and(10). ThereforeweregardL as a suitable Lagrangianfor slow light. of a free electromagnetic field, ε0(E2 c2B2) with fixed 2 − polarization. We seethatϕisrelatedtotheelectricfield strength E in SI units by B. Dynamics 1/2 ¯h E = ωϕ (3) The ability to freeze light by turning off the control (cid:18)ε (cid:19) 0 field depends crucially on the dynamics of the process. whereω denotesthe frequencyoflightandε isthe elec- Consider a time-dependent group index α without sig- 0 tric permittivity of the vacuum. nificantspatialvariations. In this caseslow lightis dom- In order to motivate the Lagrangian (1) we consider inated by oscillations within an optical wave length and the corresponding Euler-Lagrangeequation [23], an optical cycle. We express the wave as ∂tδδLϕ˙ +∂zδδLϕ′ = δδLϕ , (4) (cid:18)ε¯h(cid:19)1/2ωϕ=Eeikz−iωt+c.c., k = ωc , (11) 0 that leads to the wave equation with the slowly varying electric-field amplitude in SI E units. We approximate ∂ (1+α)∂ c2∂2+αω2 ϕ=0. (5) t t− z 0 (cid:0) (cid:1) 1/2 This is the propagation equation of slow light based on ¯h ωe−ikz+iωt ϕ¨ ω2 2iω∂t , the traditional three-level model, see Appendix A, but (cid:18)ε0(cid:19) ≈(cid:0)− − (cid:1)E the equation holds on more general grounds: Equation ¯h 1/2 ikz+iωt (5)describeslightinmediawithlinearspectralsuscepti- ωe− ϕ˙ iω , (cid:18)ε (cid:19) ≈− E 0 bility. Assuming that the optical field oscillates at much 1/2 shorter time and length scales than any variations of α, ¯h ωe−ikz+iωtϕ′′ k2+2ik∂t , (12) we replace i∂t by the frequency ω and −i∂z by the wave (cid:18)ε0(cid:19) ≈(cid:0)− (cid:1)E number k. We arrive at the dispersion relation and get from the wave equation (5) ω2 (ω+ω )(ω ω ) k2 α 0 − 0 =0 (6) 2iω(1+α)˙ − c2 − c2 − E (1+α)ω2+iωα˙ c2k2+2ikc2∂ αω2 which, in the positive transparency window near ω , ≈ − z − 0 E 0 =(cid:0)2iωc∂ +iωα˙ +α(ω2 ω2) (cid:1) agrees with z − 0 E (cid:0) α˙ (cid:1) ω2 =2iω c ′+ (13) k2 (1+χ)=0 (7) (cid:18) E 2E(cid:19) − c2 whenthecarrierfrequencyωisequaltothetransparency and the linear spectral susceptibility [18,24] resonance ω . We apply the relation (2) between the 0 2α control-fieldintensity Ic and the group index α, write Ic χ= (ω ω ). (8) ω0 − 0 as the square ofthe field strengthEc, and obtain,finally, 3 α˙ κ ˙+c ′ = α˙ = ∂ E . (14) In the experiment [13], slow light enters the EIT sam- t E E − E − 2E − c c ple and is then frozen inside by turning off the control E E field. Switching on the control releases the stored light. This is exactly the propagation equation of slow light Thepulseemergeswithanintensitythatdepends onthe in the adiabatic and perturbative limit (Eq. (9) of Ref. control field and that may exceed the initial intensity, [19]withtheRabifrequencyΩbeingproportionalto ). Ec in agreement with Eq. (15). Clearly, this phenomenon Consequently, the Lagrangian (1) has codified naturally is only possible if energy is indeed transferred from the the correctdynamic regimeincluding the 1α˙ termthat 2 E control beam to the probe light. describes reversible stimulated Raman scattering [19]. In order to understand the principal behavior of or- dinary slow-light pulses, consider the case of a spatially D. Polaritons uniform yet time-dependent group index. Equation (14) has the solution Finally, we realize the full potential of the Lagrangian (1) in setting up an effective quantum theory of slow (t,z)= z v dt v /c (15) E E0(cid:16) −Z g (cid:17)q g light. The classical canonical momentum density of the field ϕ is [23,25] in terms of the velocity [24] δL c =h¯(1+α)ϕ˙. (20) vg = . (16) δϕ˙ 1+α Wequantizethe fieldbyregardingϕandδL/δϕ˙ asHer- We see that the pulse envelope propagates with the E mitian operators ϕˆ and πˆ, respectively, with the canoni- speed v called group velocity. When the group velocity g changes in time, the intensity 2 reacts proportionally. cal commutation relations [23,25] |E| Theratiobetweenthecontrol(2)andthe pulseintensity [ϕˆ(t,z),ϕˆ(t,z′)]=[πˆ(t,z),πˆ(t,z′)]=0, (15), (I +κ)/ 2, remains large, even in the limit of a c 0 vanishingcontr|oElfi|eldwhenv vanishesaswell,aslongas [ϕˆ(t,z),πˆ(t,z′)]=i¯hδ(z z′). (21) g − κislarge(forasufficientlydensemedium). Thespectral Letusdecomposethefieldϕîntomodeswithdimension- spreadofapulseisreducedbyv /candastandingpulse g less mode indices q oscillates just with the carrier frequency. In this regime slowlightdoesnotleavethetransparencywindowofEIT [14]. One can freeze light without losing control [13,14]. ϕˆ(t,z)=Z aˆquq(t,z)+aˆ†qu∗q(t,z) dq. (22) (cid:0) (cid:1) In order to guarantee that ϕˆ satisfies the wave equation C. Energy (5) the mode functions u are required to obey Eq. (5) q as well. The u shall be normalized according to q After having gained confidence in our field-theoretical approach,we use the Lagrangian(1) to calculate the en- uq,uq′ =δ(q−q′), uq,u∗q′ =0, (23) ergy balance of slow light. According to Noether’s theo- (cid:0) (cid:1) (cid:0) (cid:1) with the Klein-Gordon-type scalar product [23,25] rem [23] we obtain the energy density δL ¯h +∞ I = δϕ˙ ϕ˙ −L = 2 (cid:0)(1+α)ϕ˙2+c2ϕ′2+αω02ϕ2(cid:1) (17) (ϕ1,ϕ2)=iZ−∞ (cid:16)ϕ∗1ϕ˙2−ϕ˙∗1ϕ2(cid:17)(1+α)dz. (24) and the energy flux (Poynting vector) The scalar product is chosen such that it remains con- stant during the propagation of ϕ and ϕ , δL 1 2 P = ϕ˙ = ¯hc2ϕ˙ϕ′. (18) δϕ′ − +∞ ∂ (ϕ ,ϕ ) =i ϕ∗∂ (1+α)ϕ˙ ϕ ∂ (1+α)ϕ˙∗ dz The energy balance I +P is then, as a consequence of t 1 2 Z−∞ (cid:16) 1 t 2− 2 t 1(cid:17) t z +∞ the wave equation (5), =ic2 ∂ ϕ∗ϕ′ ϕ ϕ∗′ dz Z z 1 2− 2 1 ¯hα˙ −∞ (cid:0) (cid:1) I +P = ϕ˙2+ω2ϕ2 . (19) =0. (25) t z 2 0 (cid:0) (cid:1) Using these postulates and definitions we calculate the Temporal changes in the control field, modifying the commutation relation of the mode operators group index (2), do not conserve energy. In fact, the experiment[13]indicatesthatthecontrolbeamcanamplify light stored in an EIT medium with zero group velocity. 4 [aˆ ,aˆ† ]= (u ,ϕˆ) u∗ ,ϕˆ + u∗ ,ϕˆ (u ,ϕˆ) One canuse similar argumentsasin the subsectionon q q′ − q q′ q′ q +∞ (cid:0) (cid:1) (cid:0) (cid:1) thedynamicsofslowlighttoprovethattheproposedpo- = u∗ϕˆ ϕû˙∗ (1+α)dz laritontheoryis consistentwiththe adiabaticthree-level Z q t− q −∞ (cid:0) (cid:1) model[19]. Yetourapproachisnotrestrictedtoaregime +∞ dominatedbyspatialoscillationsoftheformexp(iωz/c), (uq′ϕˆt ϕû˙q′)(1+α)dz′ ×Z − see Eq. (11), which is in essence the regime of geomet- −∞ +∞ rical optics [27]. One can easily relax this unnecessary (uq′ϕˆt ϕû˙q′)(1+α)dz′ constraint of the three-level theory, because the electro- −Z − −∞ magneticresponseofanatomislocal,aslongasthewave +∞ u∗ϕˆ ϕû˙∗ (1+α)dz lengthoflightislargecomparedwiththesizeoftheatom. ×Z q t− q The adiabatic polaritontheory[19]is, asours,restricted −∞ (cid:0) (cid:1) 1 +∞ +∞ to a narrow frequency range with respect to time, i.e., = ¯hZ−∞ Z−∞ (cid:16)u∗qu˙q′[ϕˆ(z′),πˆ(z)](1+α(z′)) to the transparency window of the electromagnetically- manipulated medium. Yet this restriction in frequency − u˙∗quq′[ϕˆ(z),πˆ(z′)](1+α(z))(cid:17)dzdz′ does not exclude rapid spatial oscillations beyond the scale of the wave length in vacuum, 2πc/ω. We will see +∞ =iZ u∗qu˙q′ −u˙∗quq′ (1+α)dz in the next section that such oscillations occur near a −∞ (cid:0) (cid:1) spatial singularity of the group index. In this situation =δ(q q′). (26) our quantum field theory of slow-light polaritons turns − into a perfect tool for analyzing the quantum physics of In a similar way we prove that a wave catastrophe. [aˆq,aˆq′]=0. (27) Consequently, and in agreement with the spin-statistics III. THE CATASTROPHE theorem [23], slow light consists of bosons. Let us express the total energy (18) in terms of the annihilation Imagine that the control beam illuminates the EIT and creation operators aˆ and aˆ†. Consider the case of q q mediumfromabove. Initially,thecontrolintensityisuni- a stationarygroupindex α whenthe totalenergy is con- form,butthenthecontrollightdevelopsadarknodethat served. We obtain after partialintegration, via the wave continuesasaninterface ofzerointensity I througha c equation (5) for the field operators, Z part of the medium. One could use computer-generated +∞ hologramsto achievethis situation, similar to the gener- Tˆ00dz ationofLaguerre-Gaussianbeams[28]. Supposethatthe Z −∞ interfaceissufficientlyflatandcutsdeepenoughintothe ¯h +∞ = (1+α)(∂ ϕˆ)2 c2ϕˆ∂2ϕˆ+αω2ϕˆ2 dz medium to justify our model. (We have assumed unifor- 2 Z−∞ (cid:0) t − z 0 (cid:1) mity in the two spatial directions x and y that are par- ¯h +∞ allel to the interface. Most probably, uniformity over a = (∂ ϕˆ)2 ϕˆ∂2ϕˆ (1+α)dz. (28) 2 Z t − t fewwavelengthswouldsuffice.) Nearazerointheinten- −∞ (cid:0) (cid:1) sity, the control field strength must grow linearly. Con- We employ the mode expansion(22) with the norm (23) sequently, I depends quadratically on z and, according c with respect to the scalar product (24), use the commu- to the relation (2), the group index α forms a quadratic tation relation (26), and find, finally, singularity where the group velocity (16) vanishes, +∞ 1 Tˆ00dz = ¯hω aˆ†aˆ + dq. (29) a2 Z Z (cid:18) q q 2(cid:19) α= . (30) −∞ z2 Consequently,theannihilationandcreationoperatorsaˆ q The parameter a sets the scale of the group-index pro- and aˆ† refer indeed to the energy quanta of slow light. q file. We assume that the spatialprofile (30)ofthe group The quasiparticles of light in a dielectric medium are index extends overa sufficiently long range. For simplic- called polaritons in analogy to the optical excitations in ity, we consider a one-dimensionalmodel where the slow a solid [26]. They combine characteristic features of free lightpropagatesinz directiononly. AppendixBgeneral- photons with the properties of the dipole oscillations of izesour resultsto arealistic three-dimensionalsituation. theatomsconstitutingthemedium. Whenlightisslowed Apartfromthese idealizationsandfromthe physicscap- downphotonsturnintoatomicexcitationsthat,afterac- turedinourLagrangian(1),we makeno furtherapprox- celeration, may emerge as photons again [13,14]. The imations. polariton picture contains implicitly the correct book- keeping of the photonic and atomic features of light in linear media. 5 A. Horizon ν =iµ. (37) This regime can be reached by adjusting the gradient of At a node ofthe controlfieldthe groupvelocityof the thecontrolfieldthatplaysadecisiveroleinEqs.(2)and probe light vanishes. Therefore we would expect that (30). The smaller the gradientis the largeris a2. There- no wave packet of slow light can pass the interface . Z fore, a sufficiently small control-field gradient gives rise Considerslowlightsubjecttothewaveequation(5)with to an imaginary Bessel index ν, even within the narrow the group-indexprofile(30). We representawaveϕ(t,z) transparencywindow in frequency ω =kc. On the other as hand, as we will see at a later stage, the gradientshould ϕ=√zφ (31) beaslargeaspossibleforproducingamaximalquantum effect. When the Bessel index ν is imaginary the inci- and find dent and the reflected waves are not balanced anymore, as we see from the asymptotics (36) of the Bessel func- z2∂2+a2(∂2+ω2) c2 z∂ z∂ 1 φ=0. (32) tions. Only a fraction of the incident wave is reflected t t 0 − z z− 4 (cid:2) (cid:0) (cid:1)(cid:3) and the rest must be transmitted somewhere. To find We could multiply φ by step functions Θ( z) and still the transmitted wave, we focus on the behavior of the ± get solutions of the wave equation (32), because Bessel functions for small arguments. We use the first term in the power series [29], z∂ Θ( z)φ(z)=Θ( z)z∂ φ(z) φ(z)zδ(z) z z ± ± ± iµ =Θ( z)z∂ φ(z). (33) 1 ζ ± z J (ζ) , ζ =kz, ζ 0. (38) iµ ∼ (iµ)!(cid:18)2(cid:19) → Consequently, waves on different sides of are indepen- Z dentfromeachother. As longas slowlightis concerned, Therefore, near the interface of zero group velocity the theinterface hascutspaceintotwodisconnectedparts. light waves are proportional to Z Consider monochromatic probe light oscillating with ζiµ+1/2 = ζ exp(iµlnζ)= ζ exp(iS). (39) frequency ω. Inthis case the waveequation(32)reduces p p to Bessel’s differential equation [29] with the index The logarithmic phase S reduces dramatically the wave length near z =0, because ν = 1 a2(k2 k2). (34) q4 − − 0 2π 2π λ= = z. (40) Here k abbreviates ω/c and k0 refers to ω0/c. So, in Sz µ mathematical terms, the monochromatic waves of slow Thetransmittedwavethusfreezesinfrontoftheinterface light are products of a square root with Bessel functions of zero group velocity. [29], We regard a process that creates an interface where waves separate and develop a logarithmic phase singu- iωt ϕ=√zJ±ν(kz)e− . (35) larity as a wave catastrophe. The gravitational collapse of a star into a black hole has created an horizon that Depending on the detuning of the frequency ω with re- cuts space into two disconnected parts as well [7]. Close spect to the exact transparency resonance ω , two cases 0 to the event horizon, an outside observer would see a emerge. First,when4a2(k2 k2)isbelowunitytheBessel − 0 similarbehaviorofwaves[8]. Allmotionfreezesnearthe index ν is real. We apply the asymptotics of the Bessel horizonoftheholewhich,therefore,hasalsobeentermed functions for large and positive arguments ρ [29], frozenstar[31]. Inviewofthisanalogyweregardthein- 1 π π terface of zero group velocity as the optical analog of J (ρ) exp iρ iν i Z ν ∼ √2πρh (cid:16) − 2 − 4(cid:17) a horizon. π π +exp iρ+iν +i . (36) (cid:16)− 2 4(cid:17)i B. Modes We see that in the far field the light waves with real Bessel indices are in a perfectly balanced superposition Consider a superposition of sufficiently blue-detuned ofincidentandemergingplanewaves. Inotherwords,the slow-lightwaves(35)with imaginaryBesselindices (37). light is totally reflected away from the interface of zero To analyze the quantum effects of the horizonon polari- groupvelocity, similar to the reflection of radio waves at tons, we must turn the waves into modes (22), i.e., we the Earth’s ionosphere [30]. mustnormalizethewavefunctionsaccordingtothescalar Amoreinterestingscenarioappearsinthesecondcase product (24). Waves (35) with different wave numbers when the light is sufficiently blue-detuned to evoke an k are orthogonal to each other and possess a continu- imaginary Bessel index ous spectrum. Hence they shouldbe normalizedto delta 6 functions. Weemploytheratiok/k asthedimensionless (ϕ ,ϕ )=2ceπµδ(k k ). (45) 0 1 2 1 2 − modeindexq thatoccursinthescalarproduct(24). The normalizationfactorisentirelydeterminedbythewayin Consequently,inthespatialregionrightfromthehorizon which the norm (24) diverges to reach the δ singularity (z >0) the normalized wave functions are [32]. Consequently[32],we canignoreallfinite, converg- Θ(z) ing contributions to the normalization integral given by u± = e−πµ/2 k zJ (kz)e−iωt. (46) Eqs. (24), (30), and (35), R √2c p 0 ±iµ ∞ Left from the horizon (z <0) we chose for convenience, (ϕ ,ϕ ) =(ω +ω ) J∗ (k z)J (k z)z 1 2 1 2 Z ±iµ1 1 ±iµ2 2 0 u±(z)=u∓( z). (47) a2 L R − 1+ dz. (41) ×(cid:18) z2(cid:19) The step function Θ in the definition (46) guarantees that the u modes ondifferent sides of the horizondo not The integral (41) diverges both at z = 0 and at z = overlapand,therefore,theyareautomaticallyorthogonal . We accountforthe twodivergencesseperately inthe ∞ to eachother. However,the degeneratedwaveson the integrals I and I , ± 0 ∞ same side are not orthogonal. In fact, we obtain along similar lines as in the normalization procedure, (ϕ ,ϕ )=2ω(I +I ) . (42) 1 2 0 ∞ Ignoring any convergent contributions to I0, we cut off u+R,u−R =e−πµδ(q1−q2). (48) theintegralatsomesmallz ,regardz(1+a2/z2)asa2/z, (cid:0) (cid:1) 0 Yet we construct easily the orthogonal partners w± to and use the asymptotics (38) near the origin, the u∓ waves, z0 a2 I0 =Z J±∗iµ1(k1z)J±iµ2(k2z) z dz w± = 1 u± e−πµu∓ , 0 R √1 e−2πµ R− R a2 ζ0 dζ − (cid:0) (cid:1) = ζ±i(µ2−µ1) w±(z)=w∓( z). (49) (iµ)!2 Z ζ L R − | | 0 a2 lnζ0 Wechoseastheorthonormalbasisinthemodeexpansion =sinh(πµ) exp[ i(µ µ )ξ] dξ µπ Z ± 2− 1 (22) −∞ =sinh(πµ)aµ2 δ(µ2−µ1) wR ≡wR+, uR ≡u−R, wL ≡wL−, uL ≡u+L. (50) =k−1sinh(πµ)δ(k k ). (43) Finally, to find an interpretation of the w modes, we use 1 2 − the asymptotics (36) of the Bessel functions and get for In the last step we have utilized that µ(∂µ/∂k) equals z on the appropriate sides of the horizon, a2k. Furthermore, we have used Eq. 1.2.(6) of Ref. [33] | |→∞ for the gamma function x! Γ(x+1). Let us address 1 e−2πµ 1/2 ω π the other integral, ≡ wR − exp +i z iωt i , ∼(cid:18)4πcω/ω0 (cid:19) (cid:16) c − − 4(cid:17) I = ∞J∗ (k z)J (k z)zdz 1 e−2πµ 1/2 ω π ∞ Zz∞ ±iµ1 1 ±iµ2 2 wL ∼(cid:18)4π−cω/ω0 (cid:19) exp(cid:16)−ic z−iωt−i4(cid:17) . (51) 1 ∞ π π = exp ik1z µ1 +i The asymptotics (51) shows that the w modes turn into 2πk Zz∞ h (cid:16)− ± 2 4(cid:17) planewavespropagatingawayfromthehorizon. Inother π π +exp +ik1z µ1 i words, the w modes are the ones that reach an external (cid:16) ∓ 2 − 4(cid:17)i π π photon detector. exp +ik z µ i 2 2 ×h (cid:16) ± 2 − 4(cid:17) π π +exp ik2z µ2 +i dz C. Analyticity (cid:16)− ∓ 2 4(cid:17)i 1 ∞ π = exp i(k k )z (µ +µ ) 2 1 1 2 The stationary modes (50) of catastrophic slow light 2πk Zz∞ h (cid:16) − ± 2(cid:17) are severely non-analytic. The modes vanish on either π +exp i(k1 k2)z (µ1+µ2) dz the left or the right side of the horizon with a charac- (cid:16) − ∓ 2(cid:17)i teristic essential singularity as a precursor. Waves near =k−1cosh(πµ)δ(k k ). (44) 1− 2 the event horizonof a black hole suffer a similar fate [8]. Combining the two integrals (43) and (44) gives in total Seen from an outside observer,the wavesfreeze near the (42) Schwarzschild radius rs with an essential singularity of 7 the type (r r )iµ where µ = 2r ω/c [8]. Yet an ob- s s − server falling into the hole would see little difference in wavesnearthehorizonandcouldpassthepointofnore- turn without noticing. Like the inward-falling observer, the quantum vacuum flows towards the central singular- t Z ity of the hole and, similarly, the horizon should not be a special place for the vacuum either. In mathematical terms, the wave function of the vacuum is analytic [8,9]. Consequently, the modes seen by the outside observer must not be in their vacuum states. In fact, they carry the quantaofHawkingradiation[10]. The historyofthe hole formation during a gravitational collapse turns out to be responsible for the analyticity of the vacuum [8,9]. Inspired by the analogy between a black hole and our slow-light catastrophe, let as consider the history of our horizon. z Suppose that the group index α was initially a largely uniform α . Then, by tuning the controlfield, the group 0 index develops a quadratic singularity, for example as FIG. 2. Space-time diagram of a slow-light catastrophe. α=a2(t)/(z2+b2(t))witha2( ) ,b2( ) , Thefigureillustratesthefateofawavepacketϕ(t,z)thatex- a2( )/b2( ) α , and −fin∞ally→b2∞(+ )−∞0.→T∞he periencestheformationofthehorizonZ. Initially,thepacket −∞ −∞ → 0 ∞ → oscillates with positive frequencies in time t and propagates details of the process do not matter. Immediately af- fromthelefttotherightinspacez. Thehorizoncannotgen- ter a time t the group index will possess the quadratic c erate negative frequencies in the reflected light, apart from a singularity that we are studying, creating a slow-light brief burst that we neglect. Onthe left side of Z we thusre- catastrophe. Given a uniform group index as the initial gardϕ(t,z)asanalyticintonthelowerhalfofthecomplext condition and no slow-light injected, the polariton vac- plane. Furthermore, ϕ(t,z) isanalytic in z on theupperhalf uumoccupiesinitiallypacketsofplanewavesthatwecan planethroughoutthehistoryofthewavepacket,becausethe sort into right- or left-traveling waves, process (5) conserves analyticity. Yet ϕ(t,z) is not analytic intontheothersideofthehorizon,asthesolution(54)indi- z cates. Here waves with negative frequencies are continuously ϕ = A (ω) exp i ω2+α (ω2 ω2) iωt ± Z ± (cid:18)± c q 0 − 0 − (cid:19) peelingawayfrom thehorizon,correspondingtoastationary creation of slow-light quanta. dω. (52) × Regarded as a function of complex z, the ϕ± waves are Consider the analytic properties of the vacuum waves analyticintheupper(+)orlower( )halfplane,respec- with respect to time. Picture a wave with positive fre- − tively, because here the integral (52) converges. When quencies incident from the left, see Fig. 2. After the the control field creates a horizon the vacuum modes catastrophea partofthe wavemay freeze atthe horizon must follow the wave equation (5). We assume that α and the rest is reflected. If the wave happens to arrive is analytic apart from poles. Consider closed contour in- during the formationof the horizona brief burst of light tegrals in either one of the half planes. We obtain from with negative frequencies may be generated. However, the wave equation (5), in the stationary regime we are interested in, the reflected light contains always positive frequencies. There- ∂ (1+α)ϕ dz =c2 ϕ dz ω2 αϕdz. (53) foreweregardthewavefunctionϕ(t,z)ontheleftsideof tI t I zz − 0I the horizon as analytic in t on the lower complex plane. Given an analytic signal ϕ(t) at some z we can propa- Due to the analyticity of the initial wave packets (52) gateit in spaceaccordingto the waveequation(5) with- ϕdz and ϕ dz were initially zero. Equation (53) in- t out loss in analyticity, but we cannot pass the horizon, Hdicates thatHboth integrals remain zero, as long as α is because here α is singular. It is therefore conceivable analytic. At single poles of α we get ϕ = ω2ϕ, which t − 0 that beyond the horizon ϕ(t) is not analytic anymore. cannot generate a singularity. Higher poles of α do not In other words, ϕ(t) may contain negative frequencies. contributetothe closedcontourintegrals. Consequently, In fact, we show in the next subsection that negative the vacuum wave functions are always analytic in z. frequencies in time are unavoidable for not running into conflict with the analyticity in the spatial coordinate z. Waves propagating to the right are analytic in z on the upper half plane and they must have originated from a wave incident from the left. The analytic properties of 8 the vacuumwavesinspacearethus connectedwiththeir u+ ie+πµu−, u+ ie+πµu+∗e−2iωt0, R− L R− L analyticpropertiesintime. Analyticityintheupperhalf u− ie−πµu+, u− ie−πµu−∗e−2iωt0 (57) of the z plane is linked to analyticity in t onthe left side R− L R− L and, using similar arguments, analyticity in the lower are analytic on the lower half plane. The analyticity of half of the z plane goes hand in hand with analyticity the modes (54) follows from the analytic properties of in t on the right side. We utilize the analytic properties the combinations (56) and (57). Here it is sufficient to of the vacuum waves in space and time as a marker for focusonthevicinityoftheoriginwheretheleftandright distinguishing the vacuum modes. modesareconnected. AsaconsequenceofEqs.(38),(46) and (47), we get in terms of ζ =kz, D. Combinations Θ(z) 2∓iµ u± e−µπ/2 e−iωtζ±iµ+1/2, R ∼ 2cω/ω ( iµ)! The vacuum states of slow-light polaritons are char- 0 ± u±(ζ)=up∓( ζ). (58) acterized by analytic wave functions in z. Therefore, to L R − describe the vacuum after the formation of the horizon, Consider the analytic properties of ζiµ+1/2 for an arbi- we should construct orthonormal combinations v of the traryrealµ. Weindicatewithasuffix whetherζiµ+1/2 non-analytic u and w modes that are analytic on either ± should be regarded as analytic on the upper (+) or on the upper or the lower half of the complex z plane at thelower( )halfplane,respectively. Withthisnotation some arbitrary time t0. The solution is we get − v = 1sech(πµ)w iw ie−2iωt01sech(πµ)w∗ , R 2 L− R− 2 R ζiµ+1/2 =Θ(ζ)ζiµ+1/2+Θ( ζ)ζiµ+1/2 v⊥ = 1 (u +ieπµu ) , ± − ± R √1+e2πµ R L =Θ(ζ)ζiµ+1/2+Θ( ζ)( 1)iµ+1/2( ζ)iµ+1/2 − − ± − vL = 21sech(πµ)wR−iwL−ie−2iωt012sech(πµ)wL∗ , =Θ(ζ)ζiµ+1/2+Θ(−ζ)e±iπ(iµ+1/2)(−ζ)iµ+1/2 v⊥ = 1 (u +ieπµu ) . (54) =Θ(ζ)ζiµ+1/2 ie∓πµΘ( ζ)( ζ)iµ+1/2. (59) L √1+e2πµ L R ± − − This relation proves the analyticity of the combinations Becausetheuandwmodesareorthonormalwithrespect (56) and (57) and, as a result, the analyticity of the to the scalar product (24), one can easily verify that the modes (54). v modes form an orthonormalset as well. However, is the set of modes (54) unique? In princi- The modes (54) are given on the real z axis and are ple, we could perform linear canonical transformations subject to analytic continuation. We prove that the v R that convert the modes (54) into a new set, yet such and v⊥ modes are analytic on the upper half plane and R transformations are severely restricted by the required that the v and v⊥ modes are analytic on the lower half L L analyticity in space and time. For example, we can plane. First, we use the definitions (49) and (50), and construct superpositions of the v and v⊥ modes to write v and v in the form R R R L formthe new modes v cosθ+eiγv⊥sinθ andv⊥cosθ R R R − 2cosh(πµ) 1 e−2πµvR e−iγvRsinθ, analogous to the mode transformation of =u−L −ie+pπµ−(cid:16)(1+e−2πµ)u+R−e−2πµu−R∗e−2iωt0(cid:17) ianbtheaemBosgpoliltiutebrov[3t4r]a.nsOforrmwaetimonayvRcocmosbhinξe+veRiγvwL∗itshinhvL∗ξ e−πµ u+ ie−πµ (1+e+2πµ)u− and vR⊥coshξ +eiγvL∗ sinhξ, analogous to a parametric − L − R amplifier [35]. However, only transformations of this e+2(cid:2)πµu+∗e−2iωt0(cid:0) type maintain the analyticity on one of the half planes − R (cid:17)i of complex z. We are not allowed to combine v with R 2=cous+h(πµie)+pπ1µ−(e1−+2πeµ−v2Lπµ)u− e−2πµu+∗e−2iωt0 vtiR⊥on∗soarrevRfurwtihtehrvrLes.trYicettedt:heapBoosgsoibliluebmovodtreantrsafonrsmfoartmioan- R− (cid:16) L − L (cid:17) v coshξ+eiφv∗ sinhξ wouldgeneratenegative frequen- − e−πµ u−R−ie−πµ (1+e+2πµ)u+L ciRes on the leftLside of the horizon. We have argued e+2(cid:2)πµu−∗e−2iωt0(cid:0) . (55) that this must not happen. Similar arguments apply − L (cid:17)i to all other Bogoliubov transformations. Consequently, Then we show that the combinations ourmodesareuniquelydefineduptosuperpositions,but such transformations do not change the vacuum state u− ie+πµu+, u− ie+πµu−∗e−2iωt0, L − R L − R [34]. Therefore, the v modes are indeed the vacuum u+ ie−πµu−, u+ ie−πµu+∗e−2iωt0 (56) modes. L − R L − R are analytic on the upper half plane and that the corresponding combinations 9 E. Radiation The slow-lightcatastrophe generatesa maximalparti- clenumberpermode (63)of1/4,whichisquitesubstan- As a consequence of a slow-light catastrophe, the po- tial,consideringthefactthatbrightsunlightwitharadi- lariton field is decomposed into two different sets of ationtemperatureof6 103Kcarriesamere0.01photons · modes. The v modes contain the polariton vacuum, per mode in the optical range of the Planck spectrum. whereasthe uandw modesguidethe detectablequanta, However, the photon number (63) is sharply peaked as a function of µ and, in any case, our pair-production mechanismis restrictedto the narrowfrequency window ϕˆ= aˆ w +aˆ u +aˆ w +aˆ u +H.c. dq, R R R⊥ R L L L⊥ L Z (cid:16) (cid:17) of EIT [17]. To maximize the generated quantum radiation, one should create a situation where µ is near zero = ˆb v +ˆb v⊥+ˆb v +ˆb v⊥+H.c. dq. (60) Z (cid:16) R R R⊥ R L L L⊥ L (cid:17) over an as large as possible spectral range. In terms of the experimentalparameters,wegetinthe transparency The aˆ operatorsarethe annihilationoperatorsofthe de- window near ω , 0 tector modes and the ˆb operators refer to the vacuum modes. H.c. denotes the Hermitian conjugate and the 1 δ 1/2 ω ω 0 µ= 1 , δ = − , q are the mode indices k/k0. Notice that the vacuum 2(cid:18)δ0 − (cid:19) ω0 modescontainbothpositiveandnegativefrequencycom- c2 1 λ 2 ponents,becausetheset(54)involvescomplexconjugate δ = = 0 . (64) 0 8a2ω2 32π2 (cid:18) a (cid:19) w modes. Therefore we expect that the corresponding 0 operator transformations combine annihilation with cre- Pair production occurs on the blue side of the critical ationoperators. Thisisthedecisivesignofpaircreation, detuning δ (for δ > δ ) where δ also determines the 0 0 0 similar to the production of photon pairs in parametric width of the spectrum (63). The smaller the scale a of downconversion [22]. We represent aˆR as (wR,ϕˆ), aˆR⊥ the group-index profile (30) is, the larger is the criti- as (uR,ϕˆ) et cetera, use the normalization (23) and the cal detuning (64) and the wider is the spectrum (63) properties of the scalar product (24), and arrive at the of particle production. Equations (2) and (16) indicate Bogoliubov transformations [25] thatthe scaleais smallfor asteepgroup-velocityprofile created by a large control-field gradient. Gravitational aˆ = 1sech(πµ)ˆb iˆb +ie2iωt01sech(πµ)ˆb† , R 2 L− R 2 R blackholesshowasimilarbehavior[10]. Thesmallerthe aˆ = 1 ˆb +ieπµˆb (61) hole is, the larger is the gravity gradient at the horizon R⊥ √1+e2πµ (cid:16) R⊥ L⊥(cid:17) and the stronger is the Hawking radiation [10] generated. Returning to our case, the parameter-dependence and at analogous relations for the aˆ and aˆ . With- L L⊥ ofthe particle-numberspectrum (63)underlines the cru- out initial probe light injected, the dynamically formed cial role of a spatially varying group velocity in creating slow-light catastrophe will cause spontaneous radiation a quantum catastrophe. A mere zero of the groupveloc- ofprobepolaritonsataconstantrate,becauseweobtain ity would not suffice to produce a measurable radiation. for v-mode vacua The control-field gradient matters. Close to the horizon the susceptibility of slow light aˆ†(q )aˆ (q ) = aˆ†(q )aˆ (q ) =n¯δ(q q ) (62) h R 1 R 2 i h L 1 L 2 i 1− 2 diverges. Yet Nature tends to prevent infinite suscepti- bilities: Instead of responding infinitely strongly, opti- with the average particle number cal media become absorptive or non-linear. Considering 1 gravitationalblack holes, Nature could prevent the exis- n¯ = . (63) (eπµ+e−πµ)2 tence oftrue eventhorizonsas well. Here wavesaresup- posed to shrink in wave lengths beyond the Planck scale Theradiationenergywillbetakenfromthecontrolbeam. where the physics is unknown. Hawking radiation seems The initial formation of the horizon is a time-dependent to stem from these extremely shortened waves. This process that, therefore, transfers energy to the polariton trans-Planckianproblem[36]wasanalyzedintheoretical field. Yet, in addition to an initial brief burst of energy, toy models of sonic black holes in moving fluids [37,38]. the control beam creates a wave catastrophe that would Here the inter-atomic distance provides a naturalcut-off force polaritons into a state they cannot occupy. The for extremely shortened sound waves. Nevertheless [38], frustrated polariton field reacts in attempting to alter the mere threat of a horizon seems to be sufficient for theparabolicprofileofthecontrolintensity. Thisprocess generating Hawking sound. takes energy awayfrom the controlbeam and allows the Returning to slow light, an EIT medium in linear re- creationofpolaritionpairs. Pairproductioncontinuesas sponse is transparent within a narrow spectral window long as the control beam is not significantly depleted. A around the resonance frequency ω . The spectral width 0 running wave of control light will produce a steady flow of the transparencywindow is proportionalto the inten- of slow-light quanta. sityofthe controlbeam[17]. Therefore,slow-lightwaves 10