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Hawking spectrum for a fiber-optical analog of the event horizon David Bermudez∗ Departamento de F´ısica. Cinvestav, A.P. 14-740, 07000 Ciudad de México, Mexico Ulf Leonhardt† Department of Physics of Complex Systems. Weizmann Institute of Science, Rehovot 76100, Israel Hawkingradiationhasbeenregardedasamoregeneralphenomenonthaningravitationalphysics, in particular in laboratory analogs of the event horizon. Here we consider the fiber-optical analog oftheeventhorizon,whereintenselightpulsesinfibersestablishhorizonsforprobelight. Then,we calculatetheHawkingspectruminanexperimentallyrealizablesystem. WefoundthattheHawking radiationispeakedaroundgroup-velocityhorizonsinwhichthespeedofthepulsematchesthegroup velocity of the probe light. The radiation nearly vanishes at the phase horizon where the speed of the pulse matches the phase velocity of light. 6 PACSnumbers: 42.50.-pQuantum optics. 42.81.-iFiberoptics. 04.70.DyQuantumaspectsof blackholes, 1 evaporation,thermodynamics. 0 2 I. INTRODUCTION vacuum: particles; also, astrongelectricfieldcandetach n u electron-positronpairsfromthevacuum,whichisknown J More than forty years ago Hawking [1, 2] predicted as the Schwinger effect [6]; and finally, the dynamical 7 thatthehorizonofablackholeisnotblackafterall,but Casimir effect [7, 8], where the change in the boundary emits thermal radiation with a characteristic tempera- conditions of an electromagnetic field creates photons. ] ture consistent with Bekenstein’s black-hole thermody- In this work we will deal exclusively with Hawking c q namics [3]. Since then, Hawking’s radiation and Beken- radiation, which is considered one of the most secure - stein’s entropy have been the crucial tests for potential hypotheses of a future quantum theory of gravity. In gr quantum theories of gravity, although these tests have fact, it is used to check the viability of new theories [ remained unproven themselves—there is no experimen- [9]. Nevertheless, it rests on two questionable assump- tal evidence for Hawking radiation in astrophysics, and tions: first, the derivation of Hawking radiation needs 4 this is likely to remain for the foreseeable future. How- wavelengths shorter than the Planck length, where we v 6 ever, a new approach to Hawking radiation has become expect the known physics to fail, which is known as the 1 experimentally accessible: analogs of gravity. Studying trans-Planckian problem; second, assuming that we ex- 8 such analogs, we have already gained insights into the pectnonewphysicsinthoseregimeswouldimplythatwe 6 trans-Planckian problem that arises due to the infinite cannot use Hawking radiation as a test of new theories. 0 frequency shifts at horizons. In analog systems, the fre- Therefore, the challenge of a theory of quantum grav- 1. quencyshiftislimitedduetothefrequencyorwavelength ity is not to reproduce the Hawking radiation hypothe- 0 dependence of the wave velocity, i.e., due to dispersion. sis, but rather to explain what happens to the quantum 6 However, one of the unavoidable consequences of disper- fields around an event horizon, the boundary that limits 1 sive systems is the loss of strict thermality in the spec- a black hole. : v trumofHawkingradiation. Whatexactlyistheexpected Anyone studying Hawking radiation should accept i Hawkingspectrumforexperimentallyrealizablesystems? these issues and, if possible, strive to explain them. Yet, X This is the question we answer here for fiber-optical sys- Hawking radiation is a good starting point to study r a tems [4]. the connection between gravity and quantum physics. Quantum field theory (QFT) tells us that there is a Its study combines naturally different research areas— physical state that fills the entire universe; it is the state gravity,quantumtheory,andthermodynamics—butitis of absolute darkness, the quantum vacuum. Quantum still simple enough to be addressed theoretically. There vacuum is predicted to have physical consequences: the is one problem though: the nearest black hole is thou- strong gravitational field around a black hole produces sands of light years away from Earth and, even more, it Hawking radiation [1, 2]; a related phenomenon is the seemsthatintheforeseeablefutureitwillnotbepossible Unruh effect [5], where an accelerated detector in the to measure radiation coming from it due to cosmic noise Minkowski vacuum measures something more than just from the cosmic microwave background radiation. Thereareseveralphenomenathatsurpassourpresent, and sometimes foreseeable, observational capacities, yet we believe in them due to their strong theoretical sup- port. One way scientists have come up to study these ∗ dbermudez@fis.cinvestav.mx; http://www.fis.cinvestav.mx/∼dbermudez/ phenomena is via analog systems, where a part of the † [email protected]; actualsystemisreplicatedwithadifferentonesuchthat http://ulfleonhardt.weizmann.ac.il itsequationsaresimilar. Moreover, analogshavethead- 2 vantage that they can be designed to be more efficient only have mass M (no charge Q nor angular momentum than the original systems, thus enhancing our capacity L). TheseblackholesaredescribedbytheSchwarzschild to learn from them through the understanding of their metricds,whichisthemetricofaspherically-symmetric similarities and differences. space with a mass M at the origin. This metric is given The first realization that Hawking radiation could be in Painlevé [27], Gullstrand [28] and Lemaˆıtre [29] coor- a more general effect was given by Unruh [5], although dinates by: it was largely unrecognized at the time. This work included an analog model based on fluid flow. Over time, (cid:18) (cid:114)r (cid:19)2 ds2 =c2dt2− dr+ Scdt −r2dΩ2, (1) Physicistshavestudiedtheconsequencesofthisworkand r concludedthatHawkingradiationhasnothingtodowith generalrelativityper se, butthatitisamorefundamen- where dΩ2 = dθ+sin2θdφ2 is the solid-angle element, talphenomenonderiveddirectlyfromcurved-spaceQFT (r,θ,φ) are the spherical coordinates, c is the speed of and is present wherever there is a horizon. light in vacuum, and r =2GM/c2 is the Schwarzschild S Several analog systems have appeared since then, in- radius. As we are interested in the Hawking effect and cluding water waves [10, 11], Bose-Einstein condensates the fluid analog we will only consider a 1+1 metric by [12], chains of superconducting quantum interference setting dΩ=0. devices (SQUIDs) [13, 14], excito-polariton superfluids Thelightcones(thatfulfillds2 =0)ofthismetrichave [15, 16] and ultra-short laser pulses [4, 17]. These sys- the following trajectories: tems have something in common: they mimic quantum effects in such a way that it may be possible to mea- (cid:18) (cid:114) (cid:19)−1 dt 1 r sure them in a laboratory. In this work we will focus on =± 1∓ S . (2) dr c r quantum-optical analogs using optical fibers [4]. The first optical-analog proposed the use of slow light Consider the behavior of light rays in two different [18], although it was later realized that superluminal ve- regimes: forr (cid:29)r ,theirspeedapproachesc;forr →r locities [19] are an essential for particle creation [20]. S S itgoestozero. Then,lightraystravelingtowardsr can Nevertheless, it inspired another optical analog, using S never reach it (it would take them an infinite time) light pulses in fibers [4], which fulfills the conditions for The plus sign describes light falling in, the minus sign particle creation: superluminal velocities and negative light traveling out, or trying to do so. For r (cid:29) r the norm in the group-velocity horizon. S speed of both rays approaches c. For r → r the speed After, dispersive systems were studied [21–25]. These S oftheoutgoingwavesapproacheszero, forr <r isneg- systemskeepthecreationofparticlesduetotheHawking S ative: evenoutgoingwavesaredrawninandfalltowards effect,butitsspectrumwouldnotbethermalanymore,it the central singularity. Therefore, the surface r =r de- will rather be strongly dependent on dispersion. In the S fines the event horizon [30]. The behavior of these two present work, we employ a recently developed numeri- geodesics is shown in Fig. 1, where the analogy with a cal method [26] to solve a scattering problem entirely in moving fluid starts to be useful: light rays moving with Fourier space. This method can be applied to the event the space-time fluid pass the horizon without problem, horizon of the optical analog with a realistic dispersion i.e.,thereisnothingspecialforthemthere. Ontheother of the fiber to obtain its scattering spectrum, which we hand, light rays moving against the fluid are not regular calledtheHawkingspectrum,whichwillbestronglynon- at the horizon, as we just saw, their velocity is exactly thermal. zero there. Here we define exactly what we mean by an analog of (cid:112) the event horizon. If we replace the term −c r /r with II. THE OPTICAL ANALOG OF THE EVENT S a general velocity profile u(r) in the (1+1)-dimensional HORIZON version of Eq. (1) we obtain the metric: Alltheanalogsoftheeventhorizonconsidertheblack- ds2 =c2dt2−[dr−u(r)dt]2, (3) holespacetimeasamovingmedium,i.e.,asafluidwhose movement is caused by gravity. For the optical case the where c is now the speed of rays with respect to the analogy goes one step further: the waves are light waves medium. The point here is not only that the general and the moving fluid is replaced by propagation inside velocity profile can be different from ∝ r−1/2 as in the a dielectric material. We will summarize and compare black-holemetric,butalsothatthisprofiledoesnotneed both analogies. to be caused by gravity. Hence, this is an analog system in which we generalize both shape and origin of the effects. This is the common proposal in the analog gravity A. Space time as a moving fluid community: someeffectscausedbygravityaremoregen- eral and could also have different origins. Specifically, Asweareinterestedinthemostbasicfeaturesofblack Hawking radiation is a phenomenon from curved-space holes,wechoosetostudythesimplestofthem: thosethat QFT originated by the event horizon, independently of 3 � �� -�� -�� )� � ω /� -�� (� �� -�� -�� -�� -� � � �� (�/� ) � �� FIG. 1. (Color online). Space-time diagram of light-ray trajectories near the horizon given by Eq. (2). Straight lines (dashed orange) represent rays traveling with the fluid (co- �� propagating waves), for which there is nothing special in the horizon. Curved lines (solid blue) represent rays traveling �� against the fluid (counterpropagating waves) and they are ω)� split into two at the horizon. The center black-line is the ( horizon. The radius r is written in terms of r−1 units. �� S �� what causes the horizon (gravity or otherwise). To dif- ferentiate this point of view, the new systems are called analogsoftheoriginalones. Weshouldkeepinmindthis -�� -� � � �� difference(forareviewofworksinanaloggravityseeRef. �(�/� ) � [31]). Let us describe the analogy of Hawking radiation in FIG.2. (Coloronline). Space-timediagramoflightraysnear detail. In 1974, Hawking [1] proposed that the event the horizon for the velocity profile in Eq. (7) for two cases: horizon of a black hole emits thermal radiation consis- dispersionless (up) and dispersion c(k) from Eq. (8) (down). tent with Bekenstein’s black hole thermodynamics [3]. Inthefirstcasethehorizoniswelldefined,butinthesecond Hawking radiation has effective temperature T given by oneis“fuzzy”asitdependsonk. Weshowthreepairsofrays that conserve ω−uk, one with positive (blue) and the other [1, 2]: with negative (green) frequency. (cid:126)κ k T = , (4) B 2π change of speed in the velocity profile and thus, improve where kB is Boltzmann’s constant and κ is the surface the production of Hawking radiation [4]. gravity. For the astrophysical case κ is given by Equation(3)isobtainedwithadispersionlessfluidand theanalogyleadstothesameequationsastheastrophys- c3 κ= , (5) ical case. For example, for a velocity profile u(z) whose 4GM geodesics can still be solved analytically is: where M is the mass of the black hole. The same equa- u +u u −u (cid:16)z(cid:17) tion(4)can be obtained[32,33]for theanalogcasewith u(z)= R L + R L tanh , (7) 2 2 a the general velocity profile u(r), and with a = 1/k , u = −1.2c , u = −0.8c , k defines (cid:12) 0 L 0 R 0 0 κ= ∂u(cid:12)(cid:12) . (6) thecharacteristiclengthofthedispersion,uL anduR are ∂r(cid:12) the fluid speed at the far left and right of the horizon, horizon and c is the speed of waves in the fluid (−c defines the 0 0 IfweusethevelocityprofilefromtheSchwarzschildmet- horizons.) Thetrajectoriesareillustratedinthetoppart (cid:112) ric u(r) = −c r /r we obtain Hawking’s original for- of Fig. 2 S mula (5). However, dispersion is unavoidable in any experimen- In the astrophysical case, the smaller the mass, the talrealizationaswaveswilltravelinsiderealmaterials— higherthetemperatureoftheHawkingradiationandthe such as silica or a BEC—and when we include it in the stronger its emission [30], but the Chandrasekhar limit theoreticaltreatment, welosetheexactanalogy. Forthe is a lower bound for the mass of a star to form a black fluidcase,thedispersionisgivenintermsofc(k),thede- hole. For an analog system, we can try to increase the pendenceofitsvelocitywithrespecttothewavenumber 4 k. The simplest superluminal dispersion is given by: Fluid Optics t ζ (cid:115) k2 z τ c(k)=c 1− . (8) 0 k2 k −ω 0 ω ω(cid:48) In this case the geodesics have to be solved numerically c(k) n(ω) and they are shown in the bottom part of Fig. 2. By comparingthesetwoplots, weseethatinthedispersion- TABLE I. Relationship between variables in two analog sys- less case there is a defined z that separates light rays in tems for event horizons: the fluid model and the optical one. two,butinthecasewithdispersion,thehorizonbecomes an extended region that depends on the initial k value. We say that the horizon becomes “fuzzy”. where we have defined If we include dispersion, light rays scattered by the (cid:16) v (cid:17) horizon will no longer have a thermal spectrum. In this ω(cid:48) =ω−v k = 1−n(ω) 0 ω, (15) 0 c work we investigate the effect of dispersion on the emission and we obtain its Hawking spectrum. usingthedispersionrelationk(ω)=n(ω)ω/c[infiberop- tics [34] k(ω) is usually written as β(ω)]. Therefore, the role of time is played by ζ, the propagation distance di- B. Waves in a moving fluid vs light in fibers videdbyv ,theretardedtimeτ playstheroleofdistance 0 and,accordingtothephase,wealsohavek andω played Weconsiderlightwavespropagatinginadielectricma- by−ω andω(cid:48),respectively. InTableIwesummarizethe terial, usuallyafiber, indirectionz andtimetmeasured relation between each quantity in the two frames. in the laboratory frame. Furthermore, these waves can In the comoving frame the pulse is at rest and the be described by their frequency ω or their wave number medium (fiber) travels with speed −v , effectively cre- 0 k. The dispersion of a fluid is defined by c(k) but for ating a moving medium and opening the possibility to theopticalcaseweareinterestedin,thedispersionisde- establish horizons. Furthermore, due to this change of fined by n(ω), the dependence of its refractive index on frame, the speed v defines the direction of propagation 0 the frequency of light. in the comoving frame: waves traveling slower than v 0 Consideraframemovingwithvelocityv0 followingthe appear to be traveling in the direction of the moving direction of the waves: the comoving frame. There are medium, as copropagating, while pulses traveling faster two different options to do this. First: than v continue to be counterpropagating. 0 z(cid:48) =z−v t, t(cid:48) =t, (9) 0 from which we can obtain III. THE SCATTERING PROCESS ∂ =∂ −v ∂ . (10) t t(cid:48) 0 z(cid:48) One simple way of studying a scattering process is to consideritsconservationlaws. Foranon-relativisticpro- Thus, the frequency ω =i∂ is not invariant. This leads t cessinastationarybackground,theconservedquantities to a complicated functional form for n(ω). The second are the frequency and the number of particles. For a rel- option is ativistic process, the second one is replaced by a more z z general condition: the conservation of norm. If there are τ =t− , ζ = , (11) v v waves with opposite-sign norms, this condition is more 0 0 like a conservation of charge, e.g., when a neutral par- which leads to ticle decays into two particles, one having positive and the other negative charge. The conservation of norm ∂ =∂ (12) t τ appears in all the pair production processes in particle and ω is invariant. For this simplicity, (τ,ζ) are the co- physics, e.g., when a high-energy photon decays into an ordinates used by the fiber-optics community [34]. Let electron and a positron, effectively creating two parti- us study how the phase φ transforms between the labo- cles, one with positive norm and the other with negative ratory and the comoving frame. In the laboratory frame norm. we have The quantum vacuum state is defined by the absence (cid:90) of quanta; mathematically, it is the state that fulfils: φ= (kdz−ωdt), (13) aˆ|0(cid:105)=0, ∀aˆ, (16) and in the comoving frame given by Eq. (11) we obtain where aˆ is the annihilation operator and |0(cid:105) the zero- (cid:90) eigenvalue eigenstate. Both the annihilation operator φ=− (ω(cid:48)dζ+ωdτ), (14) and the quantum vacuum depend on the choice of the 5 basis for the modes, in a scattering process we can de- considering that the vectors A(cid:126)out, A(cid:126)in, and the matrix S fine it with in- and out-modes. Moreover, in QFT when are normalized. The elements of S−1 are the coefficients waveschangethesignoftheirnormsafterscattering,the α and β of Eq. (17). The coefficients of S are the same annihilation operators for the in-modes contain creation with the labels “in” and “out” exchanged. operators for the out-modes and vice versa. The nonequivalence of incoming and outgoing modes is present in all scattering processes; it is not an un- IV. THE CALCULATION METHOD usual phenomenon by itself, but usually this results in a conversion from incoming waves to outgoing ones such There are several ways to implement the analogy of that the norm of each is conserved. However, if the con- the event horizon. From an experimental point of view, verted waves have norms with opposite signs, then this the optical analogs are very attractive, because light is a process could be an amplification [35] for both positive- simple quantum object and efficient methods and mate- andnegative-normwaves. Ifapositive-normwavegener- rial are available in quantum optics. Additionally, since ates a negative-norm component, norm conservation im- its inception quantum optics has offered a reliable test- pliesthatthepositive-normcomponentmustgrowlarger: ing ground for new theories. In fact, some of the most the wave is amplified. strikingpredictionsofquantummechanicshavebeenver- Furthermore, this process also occurs when the ampli- ifiedinquantumoptics,e.g.,entanglementandnonlocal- tude of the initial wave is small, even if the state is the ity. In this section we describe the method to calculate quantum vacuum. When the vacuum is scattered by a the Hawking spectrum from a fiber-optical analog of the horizon, there is an amplification of this quantum noise event horizon [4]. andparticlescanbecreated[35]. Thisisoneofthemain physical points of Hawking radiation. Thein-andtheout-modesformtwodifferentsetsofor- A. The soliton pulse and its half Fourier transforms thonormal modes and the transformation between them satisfies the following equation: In the optical case, the vacuum state will be scattered (cid:88) (cid:88) by a pulse. It is useful to consider the shortest possi- φout,+ = α(ω;k ,k )φin,+ + β(ω;k ,k )φin,−, ω,ki i j ω,kj i j ω,kj ble pulses [4]. Currently, there are commercially avail- j j able short-pulse lasers that produce light in the optical (17) range of ∼6fs full-width half-maximum (FWHM) dura- where the modes φ are all normalized to ±1 accord- ω,ki tion,i.e.,veryclosetothesingle-cycleregime. Ingeneral ing to their superscripts. Since the out-modes are given these pulses have a bell shape, which we will model as by both positive- and negative-norm modes, they com- sech,becausethesepulseshaveastablesolutionthatbal- bine creation and annihilation operators. On the other ances the opposite effects of dispersion and nonlinearity hand, amplitudes α and β fulfill the following norm- when traveling inside a dielectric material, the funda- conservation equation: mentalsoliton. Thisallowsthemtotravellongdistances (cid:88) (cid:88) inside dielectrics without losing their shapes. To form |α(ω;k ,k )|2− |β(ω;k ,k )|2 =1. (18) i j i j a soliton, the pulse duration and its amplitude cannot j j be chosen independently, but one is fixed by the other Equation (17) can be seen as a Bogoliubov transforma- and some fiber parameters. Usually, lasers have a de- tion, and it is known that its coefficients have interpre- fineddurationsowehavetotunetheintensitytogetthe tation as scattering amplitudes. In general, they have fundamental soliton. Their shape is given by: to be calculated numerically. The radiation spectrum of (cid:18)τ (cid:19)2 an outgoing mode is the sum of squared amplitudes of χ(τ)=χ sech , (21) the opposite-norm ingoing waves β(ω;ki,kj). Therefore, 0 τ0 thenumberofemittedparticlesN onthek branchper ki i where χ is the nonlinear susceptibility, τ is the pulse unit time per unit frequency is [36]: 0 duration(usuallygivenintermsoftheFWHMtime)and ∂2Nki = 1 (cid:88)|β(ω;k ,k )|2. (19) χ0 is fixed by τ0 and some fiber parameters [34]. In the ∂ω∂t 2π i j Appendixwepresentthefullintegralmethodtocalculate j the scattering matrix. To apply this method we need the Fourier transform and two half Fourier transforms The radiation coming from this branch of the scattering (left and right) for the pulse. For Eq. (21), the Fourier is the Hawking radiation. Hence, to obtain the Hawking transform is given by: spectrum we need to calculate the scattering amplitudes fortheingoingmodesA(cid:126)in andfortheoutgoingonesA(cid:126)out (cid:16)πτ ω(cid:17) χ(ω)=χ πτ2ωcsch 0 , (22) for a process that mixes waves of opposite norm, i.e., we (cid:101) 0 0 2 must find the scattering matrix S that fulfills: where the limit ω → 0 should be calculated with care A(cid:126)out =SA(cid:126)in, (20) anditisequalto2χ τ . ThetwohalfFouriertransforms 0 0 6 �� χ) � (� � � � � -�� -� � � �� ω(��) FIG. 4. (Color online). Dispersion relation β(ω) for coun- FIG. 3. (Color online). The Fourier transform of the sech(t) terpropagating waves using the model in Eq. (26) with two pulse(solidblue),therealpartofthehalfFouriertransforms terms(solidblue). Wealsoshowthenegativeofthisfunction (dashed orange), the imaginary parts of left- (dot-dashed (dashedred)thatisusefultomatchthenegativefrequencies. green) and right- (dotted red) Fourier transforms. are where the last equality is just the Taylor expansion of χL(ω)=χ τ ω(cid:104)1−iτ0ω(H −H )(cid:105), β around a given frequency ω0. This is a very common (cid:101) 0 0 2 iτ0ω/4 −1/2+iτ0ω/4 wayofstudyingitseffectanditgivessomemathematical (23a) advantages. Also,thephysicalmeaningofthefirstterms χR(ω)=χ τ ω(cid:104)1+iτ0ω(H −H )(cid:105), of the expansion is known: β1 is the inverse of the group (cid:101) 0 0 2 −iτ0ω/4 −1/2−iτ0ω/4 velocity, β is the group-velocity dispersion (GVD), and 2 (23b) β is the third-order dispersion (TOD) [34]. 3 where H is the harmonic number function or, more ex- In this work we are looking for a model that includes n actly, its generalization for continuous complex values, the essential properties for modeling the dispersion of which is usually defined through the (cid:122) function Ψ and light in a fiber and that creates the analog of the event 0 the Euler-Mascheroni constant γ as (see Ref. [37] for horizon in the optical regime. To do that, we can ap- more details): proximate β2(ω) by the following equation H =γ+Ψ (n+1). (24) n 0 ω2 β2(ω)= (b +b ω2), (26) InFig. 3weshowtheFouriertransformandthetwohalf c2 1 2 Fourier transforms given by Eqs. (22) and (23). where b and b are parameters. The dispersionless case 1 2 is obtained with b = 1 and b = 0. Also, the sign of 1 2 B. Modeling the dispersion relation b determines if we have a subluminal or superluminal 2 dispersion. This is a simple approximation, but it con- When electromagnetic waves travel inside a medium, tains enough detail for modeling the main part of the they interact with the bound electrons such that their optical-fiber dispersion, which will be an inflexion point. velocity depends on the optical frequency ω of the wave. In Fig. 4 there is a plot of the usual shape of a sublu- This is modeled by the material dispersion and, far from minal dispersion β(ω) for optical fibers from Eq. (26). resonances, it is usually well approximated by any of theemployedmodels: Lorentzian,Drude,Debye,orSell- Wewouldliketoexpressparametersb andb interms 1 2 meier (the most commonly used). For a list of Sellmeier of others more physically meaningful. For this, we have coefficients of different materials see Ref. [38]. In our twospecialpointstochoosefrom(seeFig. 5.) Oneisthe case, we consider optical fibers because their geometry phasehorizonorzero-frequencypoint,whereω(cid:48) =0[39]; modifies further the material dispersion, giving us also the other fulfills dω(cid:48)/dω = 0 and we will call it group- some geometrical parameters to be modeled for conve- velocity horizon or simply the horizon for reasons that nience. will become clear soon. We describe the points in the In the study of optical fibers one usually works with a dispersion by their frequency in the laboratory frame ω dispersion relation of a related function β, which is the asitissingle-valued,whilethefrequencyinthecomoving wave number k when it depends on ω. It is defined by: frame is not. β(ω)=n(ω)ω =(cid:88)∞ βj(ω−ω )j, (25) weTchaenreobatraeintwaosyusntkemnowofntwpoareaqmueatteiorsns(bb1yatnhde tbw2)o,caonnd- c j! 0 ditions for the values of ω(cid:48) = 0 and dω(cid:48)/dω = 0 at the j=0 7 �� C. Kerr effect inside an optical fiber ω � � Weneedapropertythatchangesthespeedofthewaves �� in the medium to obtain the analog. To do this we use a fairly common effect in nonlinear optics [34, 41]: the )� � Kerr effect, which is a nonlinear phenomenon and, as � �� ω ω ( � � such, needs relatively high pulse-intensities. The change ω� of the refractive index of a fiber due to Kerr effect is -�� n2 (ω,t)=n2(ω)+χ(ω,t), (31) eff where n is the refractive index of the fiber at rest and -�� χ isthenonlinearsusceptibility, whichisproportionalto -� -� -� � � � � the pulse intensity with the constants of proportional- ω(��) ity given by the material response. This is why in Eq. (21) we already wrote the pulse in terms of χ. Initially, FIG. 5. (Color online). Dispersion relation in the comoving frameω(cid:48) =ω(cid:48)(ω)fromEq. (15)andβ(ω)fromEq. (26). We n = n(ω) depends on the frequency but it is constant showthefunctionforthecopropagatingwaves(solidblue),its along the fiber. We obtain n(ω) from β(ω) through Eq. negative (dot-dashed red) to match with negative-frequency (25)withthemodelwedescribedintheprevioussection. waves, and the counterpropagating waves (dashed orange). Due to the Kerr effect, the refractive index also depends We also show the two points of interest: the phase horizon ontimeasinEq. (31),becausepulsestravelingalongthe (ωz,0) and the group horizon (ωh,ωh(cid:48)). fiberchangethedispersion. Thiseffectisusuallynegligi- ble, but in the case where the phase or group velocity of thewavesareveryclosetotheonesofthepulsetheymay points of interest. For the phase horizon ω we have z changesignificantly. Ifweapproximatetofirstorder,Eq. ω(cid:48)(ωz)=0, ddωω(cid:48)(cid:12)(cid:12)(cid:12)(cid:12) =−(cid:15), (27) (31) becomes w=wz n (ω,t)(cid:39)n(ω)+ χ(ω,t) =n(ω)+δn(ω,t). (32) where (cid:15) is the new parameter that now has physical eff 2n(ω) meaning: it is the negative derivative of ω(cid:48) evaluated at the phase horizon. The solution of the system is: D. Dispersion relation in the comoving frame c2 c2 (cid:15) b = (1−(cid:15)), b = . (28) 1 u2 2 u2w2 The final step to achieve the analogy with the mov- z ing fluid is to write down these equations in the comov- For the horizon ω , the system is given by h ing frame; the differential equation that appears in the ω(cid:48)(ωh)=ωh(cid:48), ddωω(cid:48)(cid:12)(cid:12)(cid:12)(cid:12) =0, (29) p[3r4o,p4a2g–a4ti4o]n,,isthuesunaolnlylinseoalvreSdchtrhöidsiwngaeyr. eWqueatcioonnsi(dNeLrStEhe) w=wh Doppler effect in the change of frame. Thus, the fre- where ωh(cid:48) is the new physical parameter, which is the quency ω(cid:48) in the comoving frame is frequencyofthehorizoninthecomovingframe(theother (cid:18) (cid:19) u n (ω) is ωh itself). The solution of the new system is: ω(cid:48)(ω)=ω∓uβeff(ω)=ω∓ωcneff(ω)=ω 1∓ neff(ω ) b = c2 (cid:18)1− ωh(cid:48) (cid:19)(cid:18)1−2ωh(cid:48) (cid:19), (30a) (cid:18) n(ω)+δn(ω)(cid:19) g 0 1 u2 ω ω (cid:39)ω 1∓ , (33) h h n (ω ) c2 ω(cid:48) (cid:18) ω(cid:48) (cid:19) g 0 b2 = u2ωh3 1− ωh , (30b) where we drop the explicit dependence on t in Eq. (32) h h as it only appears as a parameter, ω is the laboratory so the new parametrization is given in terms of ω and frequency and n (ω ) = c/u is the group velocity of the h g 0 ω(cid:48). This parametrization is not only physical, but also pulse. The sign of the dispersion corresponds to waves h much closer to the analog event horizon in optical fibers. traveling with the pulse (copropagating, negative sign) InFig. 5weshowthesamedispersionasinFig. 4but or against it (counterpropagating, positive sign). In the now in the comoving frame. The two points ω and ω(cid:48) comovingframe,thephasevelocityisω(cid:48)/kandthegroup h h are also shown, as well as the negative of the dispersion velocity is given by dω(cid:48)/dk. The shape of the dispersion function,asitisusefultoobtainthematchingconditions forcounter-andcopropagatingwavescanbeseeninFig. for negative frequencies by only looking at positive ones 5. [40]. We choose the parameters w = 2.62645 PHz and Wewillseethatifthesolitonisstrongenough,aprobe h w(cid:48) = 0.91108 PHz, which gives not only the right order pulsethatinteractswithitcansurpassthegroupvelocity h of magnitude but also a good agreement with possible of the fiber for certain frequencies and create an event experimental materials. horizon for those frequencies. 8 E. Analogue of the event horizon �� ω ω ω �� Inthissectionwestudymorecloselytheω frequency, �� h which fulfills v (ω ) = 0, establishing what is known as g h a group-velocity horizon. )�� A scattering process combines different frequencies in � � the laboratory frame, but in the comoving frame ω(cid:48) is ( ω�� conserved. So,allthescatteredwaveshavethefrequency ω(cid:48) in the comoving frame. For our model of dispersion, we will have three modes in ω (and four different values) �� wherethewavescouldbescatteredto[45,46], whichare shown in Fig. 6. Here we define counter- and copropa- �� ω� � � � � � � � gating according to the direction in the comoving frame, ω(��) whichopposesthedirectionofthepulseinthelaboratory frame. This is to be in accord with the fluid model. FIG. 6. (Color online). The dispersion relation for counterpropagating(solidblue)andcopropagatingwaves(dashed • Mode 1 describes the negative-frequency waves, orange); the negative of the counterpropagating (dot-dashed which we obtained by naturally extending the dis- red) dispersion is also plotted as it indicates the matching persion relation to negative values, taking advan- with the negative frequency. The horizontal line (dotted tage of the fact that β(ω) is an odd function. green)representstheconservationofω(cid:48) andthepointsshown satisfy the matching conditions for the unperturbed system. • Mode2containsthecopropagatingwaves,withthe The labels 1, 2, 3, and 3(cid:48) correspond to the identification of dispersion given by Eq. (33) with a negative sign. the modes for the numerical solution. • Mode 3 describes the counterpropagating waves, withapositivesigninEq. (33). Itmustbesplitin Thefrequenciesthatareclosertothehorizononlyneeda two due to the existence of the horizon, 3 and 3(cid:48), little help from the pulse, while the limit frequency ω(cid:48) in order to conserve the uniqueness in the matrix min issetbythepeakintensityofthepulseδn . FromEq. S from Eq. (20). max (33) we have: Figure 6 illustrates the modes. There we draw a horizontal line in ω(cid:48) = ω(cid:48), which marks the conservation of ω(cid:48) =ω(cid:48) −ω δn , (34) h min h h max ω(cid:48) and allows us to see that a scattering process from ω(cid:48) h could lead from ω to two other accessible frequencies h which is represented by the diagonal line in Fig. 7. in the laboratory frame: ω , which is the negative- max1 frequency matching and ω , which is the copropagat- The other modes also have a new range of frequencies max2 ing one. Usually, the amount of scattering to this last that are able to reach the horizon, not only the points mode is very small because it travels in the opposite di- ωmax1 andωmax2. Thesefrequenciesreachthehorizonin rection of the initial waves, but it is included in the cal- the comoving frame but their laboratory frequencies are culation for completeness. From the figure we can also still very different and these are the ones that would be see that any other value of ω(cid:48) <ω(cid:48) leads to two possible measured with a detector in a laboratory. We see in Fig. valuesformode3(whichwecalledh3and3(cid:48)). Thetransi- 7 that even though in the comoving frame ωm(cid:48) in is very tion from one to the other allows the input modes to be close to ωh(cid:48), they have very different frequencies in the converted into outgoing modes and we will see that the laboratory frame ω (see the scales for ω(cid:48) and ω in Fig. consequenceisthecreationofparticles(fortheblack-hole 7), thus facilitating their detection with the usual tools andwhite-holehorizons). Therefore,anessentialingredi- of optics laboratories. For our model, the frequencies ent for Hawking radiation is the group-velocity horizon. that can get to the horizon are all in the optical range On the other hand, as the pulse travels through the of the spectrum where there are commercially available fiber, it creates an effective moving medium due to the detectors, which makes this experiment feasible. In this change of refractive index given by δn(ω). Then, while case, the frequencies are inside [ωmin2, ωmax2], [ωmin3, the pulse passes through a certain fixed point in the co- ωmin3’] and [ωmin1, ωmax1], as marked in the Fig. 7. moving frame [according to Eq. (11), this point is mov- Wemustremarkthatweareconsideringapulse,which ing with speed v with respect to the laboratory frame], meansthatδnchangestwice: 0→δn andδn →0, 0 max max δn(ω) varies from zero to δn , defined by the peak in- both can create an horizon: in front of the pulse it is max tensity of the pulse, i.e., where the susceptibility reaches a black-hole horizon and in the back a white-hole hori- its maximum, χ(ω,t)=χ ; and then from δn to zero zon (the time-reversed equivalent of a black hole). Our 0 max when the pulses completes its passing. Due to the Kerr methodconsiderstheradiationspectrumfromtheblack- effect, the pulse is able to “push” frequencies in the co- hole–white-holepairthatisthemostlikelyconfiguration movingframethatrangefromω(cid:48) toω(cid:48) tothehorizon. of the optical analogs to be measured in the laboratory. h min 9 ω ω ω ω �� ω ω ω �� )�� )�� /� � �� (�� ω� (� � �� ω�� ω�� ω�� ω�� ω(��) � � � � � � � ω(��) FIG. 7. (Color online). Close up on ω(cid:48) from Fig. 6. The h diagonal purple line shows the maximum slope reached by FIG. 8. (Color online). Full Hawking spectrum for the dis- thepulsewithδnmax anddefinestheedgeoffrequenciesωm(cid:48) in persion. The peaks are in ωh and ωmax1. The vertical gray from the three modes that are able to reach the horizon due linesmarktheshownfrequencies,thelinesonthesidesofωh totheKerreffectwhenweconsiderthesoliton. Thisrangeis areωmin3 andωmin3’. Also,totherightofωmax2 isωmin2 and shown in green shadow. totheleftofωmax1 isωmin1,bothindistinguishablefromthis scale. These lines limit the region for creation of Hawking radiation. V. HAWKING SPECTRUM istheessentialquantityfortheproductionrateofHawk- In this section we obtain the Hawking spectrum for a ing radiation in an analog system for the optical case, fiber using the numerical method described in the Ap- whilethechangeofspeeduisgivenbytheKerreffectδn pendix. Then we analyze the numerical errors of the in Eq. (32). For the soliton, the derivative starts from method by checking the norm-conservation. zero away from the pulse and increases slowly up to a The algorithm gives the number of photons per unit t per unit ω(cid:48) per pulse. The experimentally available maximum point and then decreases rapidly to zero only atthehorizon,whichexplainsthedipinthespectrumin quantity is actually the number of photons per unit of Fig. 8. We show a close up to the spectrum around the time around a certain ω. In order to do that we need to consider the interaction distance, the change between ω(cid:48) horizon in the top part of Fig. 9. and ω andthe repetition rateof thelaser. Basically, this TheHawkingradiationintheUVregionpresentssim- last one is the most important figure, as all the others ilar features going to zero in ωh, as seen in Fig. 8. In will keep the order of magnitude of the result for typical this case we do not have the dip, because mode 1 only values of an experiment. Therefore, to simplify matters has one branch. These effect can be seen in the close up we just multiply our result by the repetition rate, which to mode 1 in the bottom part of Fig. 9. we considered to be 80 MHz. Of course this Hawking radiation will have the usual properties expected from its production of quantum origin, i.e., it will be quantized and the Hawking spectrum A. Numerical results willeventuallybebuildupfromanaccumulationofpho- tons (like in the classic double-slit experiment with sin- ThefullHawkingspectrumforthedispersionfromSec. gle photons). Also, we must remember that ω(cid:48) is the IV and a soliton pulse given by χ = 10−3 and FWHM conserved quantity, therefore all particles created by the 0 time of 2fs is shown in Fig. 8. This spectrum has two Hawking mechanism at a given time will have the same main features. First, even though in a first approxima- value in ω(cid:48). This presents as an entanglement between tion the only regions where Hawking radiation could be all particles with the same ω(cid:48) created at the same time. producedaretheonesshownattheendofSec. IVEand Thus,onesimplewayofverifyingiftheemittedradiation inFig. 7, thisspectrumcontainsHawkingradiationout- isreallyHawkingradiationistomeasureitscorrelations. side those regions. This result is somehow expected, as Particles from mode 1 ranging from ω to ω will min1 max1 theHawkingproductionshouldbecontinuous: theemis- be correlated with those from mode 3 from ω to ω min3 h sion rate is much lower outside the expected region, but and those from mode 3(cid:48) from ω to ω . For example, min3’ h not zero. On the other hand, the Hawking spectrum is the orange lines in both sides of Fig. 9 show the cor- higher around the horizon in ω and it presents a clear responding ω in the modes 3, 3(cid:48), and 1 for the same ω(cid:48) h dip exactly there, i.e., the production of Hawking radi- that lies inside the region of creation of Hawking radia- ation is exactly zero at the horizon. At first, this seems tion. In theory, the same happens for mode 2, but as we contradictory but it also follows from the theory in Sec. saw,itisverysmallbecauseofitspropagationintheop- IIA. As can be seen from Eq. (6), the relative derivative positedirection. Astrongtestofthequantumnessofthe 10 �� ω ω ω �� )�� α+(ω�)� /� �� ( �� -β-(ω�)� � -�� ω�(��) ω(��) ω�� ω�� FIG. 10. (Color online). Norm conservation. The positive- �� norm (solid blue) is the sum of the norm of modes 2, 3, and 3(cid:48), the negative norm (dashed green) is the norm of mode 1. )� �� Wealsoshowthesumofbothofthemandwecheckthenorm /� conservation(dottedred). Thenormisnormalizedaccording �� to the maximum (close to the horizon). � � ( � �� negative-norm waves, this process is more like a charge �� conservation. So, from the physical point of view, the conservation of norm gives us a very strong test of our �� method, specially in the region close to the horizon. We �� obtain the sum of positive-norm modes α (ω(cid:48)) (norms + ω(��) 2, 3, and 3(cid:48) from Fig. 6) from the negative-norm mode (mode 1) β (ω(cid:48)): − FIG. 9. (Color online). Close up of the Hawking spectrum around the horizon in ωh and around the negative frequency |α (ω(cid:48))|2 =|α(ω(cid:48);ω )|2+|α(ω(cid:48);ω )|2+|α(ω(cid:48);ω )|2, horizon in ω . In both cases we observe that the particle + 2 3 3(cid:48) max1 (35a) production dips to zero exactly at the horizon. The dashed orangelinescorrespondtothesamearbitraryvalueofω(cid:48) and |β (ω(cid:48))|2 =|β(ω(cid:48);ω )|2. (35b) − 1 they will be correlated. The notation in these equations is slightly different from that in Eq. (18), but remember that, according to Table radiation is to check if photons at those frequencies are I, in the optical analog the role of k and ω is played correlated and entangled, which is something routinely by −ω and ω(cid:48). In Fig. 10 we show the results of this performed in quantum-optics laboratories. analysis. The norm is plotted in terms of ω(cid:48), from zero to ω(cid:48). We also show the sum of all the norms and see h thatitstaysveryclosetozeroandonlyincreasesnearthe horizon (right-hand side of the plot) although it is still B. Analysis of numerical errors relatively small. All the norms are normalized according to the maximum absolute value of the positive norm (it Aswepointedoutearlier,wheneverweuseanumerical is almost the same as the negative maximum absolute method to solve a differential equation the question of value). We can also see that the emission of Hawking stability becomes important. We checked the numerical radiation outside the horizon is possible although much method in two ways: we examined the stability of the smaller than inside. results while changing the grid spacing and the norm Anotherimportantfigureofmeritistherelativediffer- conservation of scattered waves. ence between the positive and negative norms [the rela- In the first case, the main parameter that governs the tive error E (ω(cid:48))], because the spectrum has most of the r stability of the algorithm is the number of points in the radiation inside the horizon region. Given the normal- ω(cid:48) region (or conversely, the grid spacing combined with ization in the previous plot we have: the limits of the grid). We tried several number of grid points until the point where adding or subtracting some |α (ω(cid:48))|2−|β (ω(cid:48))|2 E (ω(cid:48))= + − . (36) of them will make no difference on the results. r |β (ω(cid:48))|2 − For the second point, as we mentioned in Sec. III and in Eq. (18), the norm should be conserved during the InFig. 11weplotE (ω(cid:48))andwefindacontinuousline r scattering process. Furthermore, in cases when there are thatincreasesoutsidethehorizonanditstaysclosetothe