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The Theory of Optical Black Hole Lasers Jose´ L.Gaona-Reyes∗andDavidBermudez† DepartmentofPhysics,Cinvestav,A.P.14-740,07000MexicoCity,Mexico Abstract 7 Theeventhorizonofblackholesandwhiteholescanbeachievedinthecontextofanaloguegravity. 1 Itwasprovenforasoniccasethatifthesetwohorizonsareclosetoeachothertheirdynamicsresemble 0 alaser,ablackholelaser,wheretheanalogueofHawkingradiationistrappedandamplified. Optical 2 analoguesarealsoverysuccessfulandasimilarsystemcanbeachievedthere.Inthisworkwedevelop r thetheoryofopticalblackholelasersandprovethattheamplificationisalsopossible.Then,westudy p theopticalsystembydeterminingtheforwardpropagationofmodes,obtaininganapproximationfor A the phase difference which governs the amplification, and performing numerical simulations of the pulsepropagationofoursystem. 3 ] Keywords: black hole laser; analogue gravity; Hawking radiation; pulse dynamics; negative fre- c quency. q - r g 1 Introduction [ 2 The emerging field of analogue gravity [1–3] has been successful in relating gravity with analogue v systems taken from different fields of physics such as water waves [4,5], Bose-Einstein condensates 5 (BECs)[6],opticalpulsesinfibers[7],andmorerecently,Type-IIWeylFermions[8]andmagnetization 5 dynamics [9]. Furthermore, advances in both gravitational and analogue systems have been possible 6 5 through the study of these analogies. Part of their success is due to the interplay between theoretical 0 modelsandexperimentalresults[7,10,11]. Oneofthemaintopicsofinterestisthecreationofparticles . attheeventhorizonofablackhole,i.e.,Hawkingradiation. 1 0 S. Hawking predicted in 1974 that black holes emit particles with a thermal spectrum [12] and W. 7 Unruh established in 1981 that the motion of sound waves in a convergent fluid flow corresponds to 1 themodelofthebehaviorofaquantumfieldinaclassicalgravitationalfield[13]. Sincethoseseminal : v works,Hawkingradiationhasbeenconsideredafundamentalphenomenonofquantumfieldtheoryin i curvedspaceandhasbeenusedasanacidtestforquantumtheoriesofgravity[14]. X Furthermore,in1999,S.CorleyandT.Jacobsonprovedforsonichorizons(suchasthoseachievedin r a BECs)thatiftheanaloguesofablackhole(BH)andwhitehole(WH)horizonsareclosetoeachother, the Hawking process is self-amplifying under certain conditions. As the mechanism of amplification resembles that of a laser, this phenomenon was called black hole laser (BHL) [15]. The conditions for amplificationareabosonicfield,adefinedorderofthehorizonsthatwecallWH–BHorder,ananoma- lousdispersion,andachangeofthevelocityprofileofthefluidflow. Inparticular,modelsinwater,BECsandopticsareamongthemostsuccessfulinstudyinganalogue Hawkingradiation. Intheseareas,thetheoryfromwhichthecorrespondingdispersionrelationsisde- rivediswell-known. Also,thestudyofobjectssuchaswhiteholesisjustified,evenwhenthereareno knownmechanismsofformationinthegravitationalcase[16]. Anumericalstudyofthehydrodynam- icalBHLhasbeenrecentlypublished[17]Inaddition,experimentstoverifythetheoreticalmodelsare ∗email:jgaona@fis.cinvestav.mx †email:dbermudez@fis.cinvestav.mx;web:http://www.fis.cinvestav.mx/∼dbermudez 1 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez feasible. Forexample,in2014J.Steinhauerclaimedtohavemeasuredradiationfromablackholelaser formedinaBEC[18]. Ontheotherhand,opticalanaloguesarealsoverysuccessfulinprovingthequantumpropertiesof Hawking radiation. An optical black hole laser (OBHL) was presented by D. Faccio et al. in 2012 and numerical simulations were shown as evidence of the phenomenon [19]. However, the conditions in theopticalcasearedifferentthanthoseinthesonicone,andtherefore,theproofofamplificationofthe lattercannotbeusedintheformer. Inthiswork,wepresentatheoreticaldescriptionoftheOBHLfollowingtheapproachofCorleyand Jacobson,thatis,byprovidingaWKBdescriptionoftheevolutionoffrequencymodesthroughacavity andallowingmodeconversionprocessesatthehorizons. Intheopticalcontext,thecavityisformedby apairoflightpulses. Inparticular,westudytheHawkingprocessalsoforabosonicfield,inthiscase photons,butinthenormaldispersionregime. Thisforcesaninverseorderofthehorizons(aBH–WH order)togettheproperkinematicbehavior. Moreover,thechangeofvelocityisnowduetodispersion, andnotasaconsequenceofmodificationsinthefluidflow. Hence,isHawkingradiationamplifiedin theopticalanalogue? Andifso,underwhatconditions? Hereweanswerthesequestions. In addition, we derive the forward propagation of modes and, in this way, the heuristic argument fortheamplificationiseasiertofollow. Furthermore,itisknownthattheamplificationdependsmainly onthephasedifferenceofthemodesinbothhorizons[19,20]. Here, wedevelopamethodtoapprox- imate this phase difference and study its behavior. Finally, we present some numerical simulations of theOBHLbasedonthenonlinearSchro¨dingerequation(NLSE)includingnegativefrequencies[21,22], which are usually not considered in this kind of simulations but that are necessary to obtain the cor- rect modes of the black hole laser and its amplification [23]. Besides, they have been measured in the laboratory[24]. This work is organized as follows. The concept of the analogue of the event horizon in an optical contextisreviewedinSection2. InSection3wedescribethefundamentalideasbehindtheconstruction ofanOBHLandprovidethenecessarytheorytodefinethevaluesofthefrequencymodesthatinteractin thecavityfollowingtheWKBdescription.Then,thetheoreticalanalysisofthepropagationoffrequency modes through the cavity and the proof of the amplification of the Hawking process in the OBHL are bothdetailedinSection4. Numericalsimulationsofthepropagationofmodesinthecavityareshown inSection5. Finally,wepresentourconclusionsinSection6. 2 The optical analogue of the event horizon Alltheanaloguesoftheeventhorizonconsidertheblackholespacetimeasamovingmedium, i.e., as a fluid whose movement is caused by gravity, and consider light as waves moving in this fluid. For the optical case the analogy goes one step further: the waves are light waves and the moving fluid is replaced by propagation inside a dielectric material [7]. We will now summarize and compare both analogies. 2.1 Spacetimeasamovingfluid Asweareinterestedinthemostbasicfeaturesofblackholes,wechoosetostudythesimplestofthem: the one that only has mass M (no charge Q nor angular momentum L). This black hole is described bythe Schwarzschildmetric, whichcharacterizes aspherically-symmetric spacewith amass M atthe origin. ThecorrespondingmetricisgiveninPainleve´-Gullstrand-Lemaˆıtrecoordinates[25–27]by: (cid:18) (cid:114) (cid:19)2 r ds2 = c2dt2− dr+ Scdt −r2dΩ2, (1) r wheredΩ2 = dθ2+sin2θdφ2 isthesolid-angleelement, (r,θ,φ)arethesphericalcoordinates, cisthe speedoflightinvacuum, and r = 2GM/c2 istheSchwarzschildradius. FortheHawkingeffectand S the fluid analogue it is only necess√ary to consider a 1+1 dimensional metric, which can be done by settingdΩ = 0. Thetermv(r) = c r /rofEq. (1)canbeinterpretedasthevelocityprofileofspaceif S itisthoughtofasafluid[28]. Inthispicture,spaceemergesoutofthewhiteholeorfallsintotheblack 2 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez r S v rS v Figure1:(Coloronline). Flowingriverofspaceinanastrophysicalwhitehole(left)andblackhole(right). At thehorizons(yellowlines),thespaceflow(blackarrows)equalsc. Counter-propagatingmodes(redarrows) cannotescapefromtheregioninsidetheeventhorizonintheblackholeorpenetratetothesameregioninthe whitehole. holeasshowninFig. 1. AttheSchwarzschildradiusr = r ,theflowvelocityequalscsothespeedof S light is exceeded by the flow beyond that point. Therefore, it is impossible to escape out of the black holeonceinsidetheSchwarzschildradius. Inthetimereversalcase,correspondingtothewhitehole,it isimpossibletopenetratebeyondr . S 2.2 Opticalanalogyofahorizon Therewereattemptstocreateaflowinanopticalcontextinordertoinduceakinematichorizonusing slowlight[29]. Itwaslaterproposedthatalocalizedpulsethatpropagatesthroughanopticalfiberand achangeofreferenceframearesufficienttorecreatethehorizon[7]. In terms of the laboratory coordinates (z,t), a retarded time τ and a propagation time ζ can be definedas z z τ = t− , ζ = , (2) u u where u is the group velocity of the localized pulse, which acts as a perturbation. The coordinates (ζ,τ)definetheco-movingframe. Wewillconsiderafixed-shapepulsepropagatinginzwithconstant velocity. Then,thepropertiesoftheeffectivemediumintheco-movingframedependonlyonτ,which playstheroleofspace(whereasζ playstheroleoftime). Thiscanbeachievedexactlybyusingsolitons orasanapproximationifthehorizondynamicsisfasterthanthepulsedynamicsinthefiber. Oneofthemainadvantagesofworkingintheco-movingframeisthatthefrequencyinthisframe ω(cid:48) is invariant [30]. Actually, it can be shown that frequency conservation in the co-moving frame is equivalent to momentum conservation in the laboratory frame, and in turn, to a phase-matching condition[31]. Thefrequencyω(cid:48) isexplicitlygivenbytheDopplerrelation u ω(cid:48)(ω) = ω−ω n(ω). (3) c Theeffectofthelocalizedpulseistochangetheoriginalphaseindexofthefibern(ω)byanadditional contributionδn,whichdependsontheintensityofthepulse,i.e.,δn ∝ I(τ),andmovesinsidethefiber withaconstantvelocityu. ThisiscalledtheopticalKerreffect[7,31]. Theeffectivephaseindexn is eff thengivenby n (ω,τ) = n(ω)+δn(τ). (4) eff 2.3 Interactionofaprobepulsewithalightperturbation Given a localized perturbation moving through an optical fiber with velocity u, we consider a probe pulsewithvelocity v inthefiber. Twosituationsmayoccuraccordingtothevaluesof u and v. Inthe case where u < v, the perturbation will be caught up by the probe pulse approaching from behind, as viewed in the laboratory frame. As the probe pulse reaches the trailing edge of the perturbation, it willbesloweddownbythehighergroupindex,anditwilleventuallybeblockedatapointz inthe WH 3 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez u u �n �n max max zBH zWH ⌧WH ⌧BH z ⌧ Figure2: (Coloronline). (Left)Cavityformedbytwosolitonsinthelaboratoryframe. Thewhitehole(WH) andblackhole(BH)horizonsareinthetrailingandleadingedgeofthepulses,respectively. (Right)Location ofthehorizonsintheco-movingframe,withanapparentflipofthehorizons.Theprobepulseisblockedatthe WHandisunabletoescapefromtheBH(magentaarrows)duetotheflowvelocity(orangearrowsindicate themagnitudeintheco-movingframe). laboratoryframeorτ intheco-movingone. Thefollowingrelationisthensatisfied WH c u = , (5) n +δn(τ ) g WH wheren isthegroupindexofthefiber,whichisinverselyproportionaltothegroupvelocityv . This g g isreminiscentofawhiteholehorizon. Ontheotherhand,ifu > v,ananalogueofablackholehorizon isobtainedfortheleadingedgeoftheperturbation. Therefore,undertheseconditions,asinglepertur- bation recreates the analogue of both a white hole (trailing edge) and a black hole (leading edge). In addition, it should be noted that there is an apparent flip between left and right when we transform betweenthelaboratoryandtheco-movingframes,asshowninFig. 2. Theconditionofoccurrenceoftheeventhorizonsisgivenbytherelation c c ≤ u ≤ , (6) n +δn n g max g whereδn isthemaximumchangeinthephaseindex,whichisreachedatthepeakoftheperturbation. max TherearetwopossiblewaysofinterpretingEq. (6). Ifthebackgroundgroupindexn andperturbation g amplitude δn aregiven, onlyperturbationsthatmoveatavelocity u intheintervaldefinedbyEq. max (6) give rise to both analogue horizons. On the other hand, if u and δn are given, the effect of the max horizonswilloccurforpulseswithfrequenciesωsuchthatn (ω)satisfiesEq. (6)[16]. g Ifweconsidertheeffectoftheperturbation,ω(cid:48) ismodifiedbytheeffectivephaseindexas u u ω(cid:48) (ω) = ω−ω [n(ω)+δn] = ω(cid:48)(ω)−ω δn, (7) eff c c where ω(cid:48)(ω) is given by Eq. (3). Notably, ω(cid:48) is the conserved quantity in the system formed by the eff probeandtheperturbation. Wewillusethisfactinthederivationofthenextsection. 3 The optical black hole laser Thebasicideaofanopticalblackholelaseristotrapaprobepulseinacavityformedbylightpertur- bations. As previously mentioned, this probe will be confined because it will be slowed down when it reaches any of the perturbations due to the Kerr effect. A cavity can be constructed by using two positive-valuedperturbationsgeneratedbytwoindependentlaserpulsesthatareplacedacertaindis- tancez(ortimeτ)apartinsidethefiber[19],asshowninFig. 2. In order to construct a cavity that does not modify its shape during ζ-propagation (so as to isolate theeffectofthelocaldecreaseinthegroupvelocity),weusetwosolitonscenteredatthesamefrequency ω and separated a fixed time τ (c stands for cavity). Usually, the dispersion of the fiber is in the so- c c callednormaldispersionregime,i.e.,n (ω)increaseswithincreasingω.Nevertheless,solitonscanonly g existintheanomalousregime,inwhichn (ω)decreaseswithincreasingω. Thus,theremustbesome g frequency interval in the dispersion of the fiber that is anomalous. This situation is very common for exampleinphotoniccrystalfibers[31]. 4 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez �� !0(!h) !h �� ) !e0↵(!h-max) !h-max � ! ! �� min max � (�� ω� �� ω (��) Figure3: (Coloronline). ThefrequenciesforwhichthehorizonsoccurarefoundbysolvingEq. (8)forδn = 0 (orange) and δn = δnmax (green), yielding the frequencies ωh and ωh-max, respectively. The laboratory frequenciesintheinterval(ωmin,ωmax)experiencetheeffectofthehorizonsasitscorrespondingvalueofthe co-movingfrequencyisintheinterval(cid:0)ω(cid:48) (ω ),ω(cid:48)(ω )(cid:1). eff h-max h 3.1 Determinationoffrequencymodes The range of frequencies that experience the horizons is found by solving Eq. (6). Alternatively, the frequenciescanbeobtainedbydeterminingthenullvaluesofthederivativeofω(cid:48) (ω)inEq. (7),that eff is, dω(cid:48) dω(cid:48) u eff = − δn =0. (8) dω dω c Then,thefrequenciesthatcorrespondtotheanalogueoftheeventhorizonsdependonthevalueofδn. ThelimitingvaluesofEq.(6)canbeobtainedbysettingδn =0andδn = δn .Inparticular,forregions max ofnormaldispersionwithpositivevaluesofω(cid:48) ,thesolutionscorrespondtomaximaofω(cid:48) (ω),soifω eff eff h isthesolutionofEq. (8)correspondingtoδn =0andω correspondstoδn = δn ,thentherange h-max max offrequenciesthatexperiencethehorizonsissuchthatthecorrespondingfrequenciesintheco-moving framearecontainedintheinterval(cid:0)ω(cid:48) (ω ),ω(cid:48)(ω )(cid:1),asshowninFig. 3. Thefrequencyintervalin eff h-max h thelaboratoryframeisthusgivenbythesolutionsof ω(cid:48)(ω) = ω(cid:48) (ω ). InFig. 3, wecalled ω eff h-max min and ω the frequencies that satisfy this condition, so the laboratory frequencies that experience the max horizonsarecontainedintheinterval(ω ,ω ). min max The dynamical evolution of pulses in this interval corresponds to a region of normal dispersion. Giventhisregion,asimplifiedmodelofthedispersionrelation(seeRef.[30])canbeusedtostudythe effectoftheperturbation.Thismodelincludestheessentialpropertiestodescribethedispersionoflight in a fiber, and is able to create the analogue of the event horizons. In the absence of perturbation, the propagationconstantβ(ω)canbeapproximatedbytherelation ω2 (cid:16) (cid:17) β2(ω) = b2+b ω2 , (9) c2 1 2 where the parameters b and b are found by imposing that ω(cid:48)(ω ) is a maximum of the dispersion 1 2 h relation. Thefrequencymodesthatconserve ω(cid:48) aretheonesthatcanexistafteranonlinearinteractionwith thecavitythatcausesthemodeconversionprocesses. WecanfindthesemodesbysolvingEq. (7)using the approximated model for β(ω) given by Eq. (9). For a given input frequency ω , the co-moving IN frequencyisω(cid:48) ≡ ω(cid:48)(ω ),thatis 0 IN ω (cid:18)(cid:113) (cid:19) ω(cid:48) = ω−u b2+b ω2+δn . (10) 0 c 1 2 AswementionedinSection2.2,ω(cid:48) isaconservedquantity. Ontheotherhand,thisequationshowsthe 0 twomaindifferencesbetweentheblackholelaserintheopticalandsoniccontexts(seeEqs. (2.6)and 5 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez � �� !N !T !00 !IN !P � !P �� )�� !00 !IN )�� BH ( ( ω ω�-�� IN -� !T -�� T P !N -�� N -� -�� -�� -� -� -� � � � � β(�/μ�) ω(��) Figure4:(Coloronline).Modescorrespondingtoafixedvalueoftheco-movingfrequencyω(cid:48) inthelaboratory 0 frame(left)andintheco-movingframe(right). ThethreemodesIN, P, andNarefoundforthedispersion relationwithδn=0(orangeline). Forthedispersionwithδn=δnmax(greenline),onlytheTmodeisfound. TheinsetshowsmodepropagationinthevicinityofaBH. (2.7)inRef.[15]). First,thedispersionrelationfortheOBHLisnormal,incontrastwiththeanomalous dispersion considered for the sonic case. In addition, in the sonic case there is a velocity profile v(x) that yields the solutions of Eq. (10) by fixing a value of v(x) that corresponds to a superluminal or a subluminalvelocity. Incontrast,intheopticalcasethereisnovelocityprofilebecausethevelocityuof theframeco-movingwiththeperturbationisfixed. Thecorrespondingmodesarefoundbytheeffectof theperturbationwhichischangingthedispersionrelationbyanadditionalcontributionδntothephase index. Inadditiontotheinputmode(IN)withfrequencyω ,twoothersolutionsofEq. (10)arefoundfor IN δn =0. TheywillbedenotedasP(positivelaboratoryfrequencymode)andN(negativelaboratoryfre- quencymode). Thesethreemodesaretrappedinthecavityformedbytheperturbationsand,according to Eq. (6), correspond to a superluminal velocity. On the other hand, the effect of the contribution δn canbethoughtofasaclockwiserotationofthecurveω(cid:48)(ω). Therefore,foralargeenoughδn = δn max thereisonlyoneadditionalsolutionthatshallbedenotedasT(transmittedmode)thatexistsoutsidethe cavityandcorrespondstoasubluminalvelocity. ThefourpossiblesolutionsareshowninFig. 4. The numberofsolutionsforthesuperluminalandsubluminalvelocitiesshowsanotherdifferencebetween theopticalandsoniccases. IfthepossiblesolutionsofthefrequencymodesintheOBHLwerefoundby fixingthevaluesofavelocityprofile,asinthesoniccase,thenthenumberofsolutionsinthesubluminal andsuperluminalwouldbeinverted. TheapproachofCorleyandJacobson[15]canbefollowedbynoticingthatarearrangementofEq. (10)yields W4 [−W +Wv(τ)]2 = W2+ , (11) 0 Ω2 0 whereW = b ω,theconstantsW andΩ aregivenbytherelationsW = ω(cid:48)c/u,Ω2 = b4/b andwe 1 0 0 0 0 0 1 2 definethequantity c (cid:104) u (cid:105) v(τ) = 1− δn(τ) . (12) b u c 1 TheimportanceoftherearrangementshowninEq. (11)isthatitsstructureisofthesameformasthat usedinthesoniccase. So,evenifthefrequencymodesarenotgivenbythepossiblevaluesofavelocity profile,theycanbefoundaccordingtothevaluesofthequantityv(τ),thusestablishinganequivalence between the velocity profile in the sonic case and the modification of the dispersion relation by the additionalcontributionδnintheopticalcase. 3.2 Propagationoffrequencymodes ThegroupvelocityofeachmodecanbefoundusingEq. (8). Thedirectioncaneasilybeobtainedwith the sign of the derivative in the ω(cid:48)(ω) diagram (see Fig. 4). The modes that correspond to δn = 0 6 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez WH BH WH BH T IN2 T2 N N 1 2 P1 P2 T 1 T IN IN 1 1 1 N2 ⇣ N1 P P 2 1 T 2 IN IN 2 ⌧ Figure5:(Coloronline).SpacetimeevolutionofanoutgoingTmodebackwardsinζ(left)andofanincoming IN mode forwards in ζ (right). Destruction of wave modes by mode conversion is indicated by the end of lines.Solidlinesrefertopositivefrequency(andnorm)modesanddashedlinestonegativeones. propagate through the interior of the cavity in such way that the IN mode travels in the BH→WH direction, whereasPandNpropagateintheoppositedirection. Ontheotherhand, theTmodeexists only outside the cavity, when δn = δn . This mode propagates away from the BH to the right or max approaches the WH from the left, as shown in the inset of Fig. 4. Strictly speaking, after this mode moves away from the localized perturbation it transforms into an N mode (which also has negative- frequencyandthatmovesinthesamedirection)andleavesthesystem. Inourstudywewillcontinue labelingthismodeasTtodistinguishitfromtheNmodeinsidethecavity. In particular, the evolution of a final T mode backwards in ζ can be described based on a WKB frameworkcomplementedbymodeconversionprocessesatthehorizons. Inthispicture, asshownin thediagramofFig. 5(left), thefinalTmodeapproachestheBH,whereittransformsintoanNmode. In addition, mode conversion allows the existence of a P mode that is not obvious based only on the dispersioncurve,butthatispossiblebecausethismodeconservesω(cid:48) (seeFig. 4). EventhoughtheIN 0 mode also conserves the frequency ω(cid:48), there is no mode conversion to this mode because it moves in 0 theoppositedirection.Then,thepairofPandNmodespropagatetowardstheWH.Whentheyreachit, thetwomodesgiveorigintoaTmodethatpropagatesfurthertotheleftofthecavity,andanINmode that propagates again to the BH. As the IN mode approaches the BH, a pair of P and N modes arises throughmodeconversionandthesubsequentprocessesareaspreviouslydescribed. On the other hand, let us consider the evolution of an IN mode inside the cavity forwards in ζ, as seeninFig. 5(right). ThismodetravelstowardstheWH,whereitisconvertedintoapairofPandN modesbymodeconversion. ConversiontotheNmodewithnegativefrequencyispossiblebecausethis modealsoconservesω(cid:48) (seeFig. 4),andmovesintheproperdirectionunliketheTmode. Then,these 0 modestraveltowardstheBH,wheretheygiveorigintoaTandanINmodes. Atthisstage,theTmode escapesthecavityandtheINmoderepeatstheprocess. 3.3 Approximationoffrequencymodes Westartbyproposingthefollowingactionforthebosonicfieldinthe(τ,ζ)coordinates S = (cid:90) dζL = (cid:101)0 (cid:90) dζdτ(cid:110)(cid:2)(cid:0)∂ −v(τ)∂ (cid:1)Φ(cid:3)2+ΦFˆ(∂ )Φ(cid:111), (13) ζ τ τ 2 wheretheoperatorFˆ(∂ )isdefinedas τ ∂4 Fˆ = ∂2 − τ (14) τ Ω2 0 Giventhisaction,theEuler-Lagrangeequationsyieldthefollowingequationofmotion (cid:2)∂ −∂ v(τ)(cid:3)(cid:2)∂ −v(τ)∂ (cid:3)Φ = Fˆ(∂ )Φ. (15) ζ τ ζ τ τ 7 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez Proposingasolutionoftheform (cid:18) (cid:90) (cid:19) Φ(τ,ζ) =exp iW ζ+i dτW(τ) , (16) 0 andneglectingderivativesofbothW(τ)andv(τ),wereproducethedispersionrelationofEq. (11). On theotherhand,allowingamoregeneralsolutionoftheformΦ(τ,ζ) =exp(iW ζ)φ(τ),substitutionin 0 Eq. (15)yieldsthefollowingrelation − 1 φ(iv)(τ)+(cid:104)1−v2(τ)(cid:105)φ(cid:48)(cid:48)(τ)+2v(τ)(cid:2)iW −v(cid:48)(τ)(cid:3)φ(cid:48)(τ)−iW (cid:2)iW −v(cid:48)(τ)(cid:3)φ(τ) =0. (17) Ω2 0 0 0 0 FollowingtheWKBmethod,wechooseφ(τ)astheτ-terminEq. (16)andobtainthefollowingexpres- sionforW(τ) 1 (cid:104) (cid:105) − W4− 1−v2(τ) W2−2v(τ)W W +W2 Ω2 0 0 0 (cid:40) (cid:41) = −i d 2 W3+(cid:104)1−v2(τ)(cid:105)W2+W v(τ) − 1 (cid:104)4WW(cid:48)(cid:48)+3(cid:0)W(cid:48)(cid:48)(cid:1)2(cid:105)+ i W(3). dτ Ω2 0 Ω2 Ω2 0 0 0 (18) Afterlettingτ → ατandassumingthatW(τ)maybeexpandedininversepowersofα,(whereαis just an auxiliary parameter that will be set as α = 1 at the end), the condition that each power of 1/α vanishes separately gives an infinite set of equations, where the lowest orders produce the following frequenciesofthemodesinconsideration W W =− 0 , (19) IN,T 1−v(τ) (cid:113) W v(τ) 3 d (cid:104) (cid:105) W =±Ω v2(τ)−1+ 0 +i ln v2(τ)−1 . (20) P,N 0 1−v2(τ) 4dτ Thesesolutionsshouldbeevaluatedwith δn = 0fortheIN,P,andNmodesandwith δn = δn for max theTmode. 4 Hawking Amplification in the OBHL Inthissectionwewillobtaintheconnectionformulas, whichdescribetherelationbetweenthemodes at both sides of the horizons. In addition, we will derive the equations for the evolution of packets in theOBHLandwewillprovetheamplificationoftheHawkingradiation. Finally,wewilluseasimple modeltostudytheconditionsofmaximalamplification. 4.1 Connectionformulas Wecansettheoriginofour τ-coordinatesothat τ = 0, andworkwith b = 1inasetofunitssuch BH 1 thatv(0) = 1inEq. (12). ThisarrangementcanbeseeninFig. 6. Underthiscondition,weexpandv(τ) uptofirst-order,whichisusefultodefineaparameterκthatplaystheroleofthesurfacegravityasitis commoninanaloguesystems[7,15]. Theparameterκisexplicitlygivenby (cid:12) (cid:12) dv(cid:12) 1 dδn(τ)(cid:12) κ = (cid:12) = − (cid:12) , (21) dτ(cid:12) b dτ (cid:12) τ=0 1 τ=0 and is a negative quantity for the leading edge where the BH is located. Following the approach of Corley [32], the Laplace transform method can be used to solve Eq. (17). Under the assumptions that |κ/τ| (cid:28)1and|κ2/κ| (cid:28)1(whereκ isthesecondtermintheexpansionofv(τ)aroundτ =0),Eq. (17) 1 1 takestheform 8 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez v(⌧) ⌧˜ ⌧˜ WH BH v max ⌧ ⌧ WH BH 1 ⌧ 0 Figure6: (Coloronline). Equivalentvelocityprofilev(τ)inthecaseofacavityformedbytwosolitons. The pointsτ andτ aredefinedasthesolutionsofv(τ) = 1. Notethatτ˜ andτ˜ areimportanttimesin WH WH BH WH theapproximationthephasedifferenceinSection4.5. −Ω−2φ(iv)(τ)−2κτφ(cid:48)(cid:48)(τ)+2(iW −κ)φ(cid:48)(τ)−iW (iW −κ) =0 (22) 0 0 0 0 ThesolutionsderivedfromtheLaplacemethodcanbeexpressedintermsofthoseobtainedbytheWKB approach(seeEqs. (18-20)). Fromthere,thefollowingconnectionformulashold (cid:16) (cid:17) K e−πW0/(2|κ|)φ +eπW0/(2|κ|)φ ←→ φ , (23) P N T (cid:16) (cid:17) −φ +K eπW0/(2|κ|)φ +e−πW0/(2|κ|)φ ←→0, (24) IN P N wheretheparameterKisdefinedas (cid:115) W (cid:18)πW (cid:19) K = 0 sinh−1 0 , (25) 2Ω |κ| 0 and the arrows indicate a connection between the solutions at both sides of the horizon. It can be shown[15]thatthesolutionsofEq. (15)aroundtheWH,locatedatτ ,arethecomplexconjugatesof WH thoseattheBHforasymmetriccavity,sothecorrespondingconnectionformulasfortheWHhavethe samestructureasthoseshowninEqs. (23)and(24). Moreimportantly, thesolutionsatbothhorizons differinaphaseonly,sotheycanbeexpressedwithoutlossofgeneralityas φ˜ (τ −τ) =eiθIN(W0)φ (τ), (26) IN WH IN φ˜ (τ −τ) =eiθP,N(W0)φ (τ). (27) P,N WH P,N 4.2 Normofthefrequencymodes AsthegeneralizedLagrangiandensityLoftheaction(13)isinvariantwithrespecttothetransformation φ(cid:48) = eiλφ of the complex field φ, an associated current is conserved such that its ζ-component can be usedtodefinethefollowinginnerproduct (φ ,φ ) = i(cid:101)0c2 (cid:90) dτ(cid:2)φ∗(cid:0)∂ −v(τ)∂ (cid:1)φ −φ (cid:0)∂ −v(τ)∂ (cid:1)φ∗(cid:3), (28) 1 2 uh¯ 1 ζ τ 2 2 ζ τ 1 whichsatisfies ∂ (φ,φ) = 0, thatis, thenormisconserved. Thisinnerproductcoincideswiththatde- ζ finedinRef.[7]exceptforafactorof−1.Thissignisusuallychosensothatthesignsofthenormandthe laboratoryfrequencycoincide. Normconservationwillproveessentialtounderstandtheamplification oftheanalogueofHawkingradiationintheOBHL.Wecanconstructwavepacketsfromthesolutions ofthefrequencymodesas (cid:90) dW ψ = √W0 GW0eiW0ζφ[W(W0)], (29) 0 9 TheTheoryofOpticalBlackHoleLasers Gaona-Reyes,Bermudez whereGW istheamplitudeobtainedfrommatchingtheconnectionformulasinEqs. (23)and(24)and 0 theevolutionformulas. Thelatterdescribethebehaviorofwavepacketsastheyevolvewithrespectto ζ andwillbeobtainedinthefollowingsection. Wewillalsoshowthatthesignofthenormofthewave packetagreeswiththesignofthefrequencyinthelaboratoryframe.Wecanchoosetoevaluatethenorm of each packet in a region where v(τ) is approximately constant, as it is conserved in ζ. For the three frequencymodescorrespondingtoδn =0(IN,P,N),thepreviousapproximationisjustifiedifthecavity islongcomparedwiththedurationoftheperturbations. Ontheotherhand,ifwewanttodescribethe processoccurringinthesystemformedbythecavityandtheprobepulse,thebestapproximationthat canbemadeistoevaluatethenormoftheTpacketwithδn = δn . max ItispossibletoverifythatthesolutionsofEq. (17)canbeexpressedintheform φW (τ) =CW eiW(W0)τ, (30) 0 0 where (cid:12)(cid:12)CW0(cid:12)(cid:12)IN,T =1, (cid:12)(cid:12)CW0(cid:12)(cid:12)P,N = (cid:104)v2(τ)−1(cid:105)−3/4. (31) These relations hold if we redefine the modes P and N by only considering the first two terms of Eq. (20),asthelasttermisalreadyincludedintheexpressionfor|CW0|P,N. Ontheotherhand,theexplicit calculationoftheinnerproductyieldsthefollowingrelations 4π(cid:101) c2 (cid:90) (ψIN,T,ψIN,T) (cid:39) ± uh¯0 dW0|GW0|2, (32) (ψ ,ψ ) (cid:39) ±4π(cid:101)0c2Ω0 (cid:90) dW |GW0|2. (33) P,N P,N uh¯ 0 W 0 Havingobtainedtheseapproximatedexpressionsforthenormsofthepackets,wecangiveaqualita- tiveexplanationoftheamplificationoftheanalogueofHawkingradiationintheOBHL.Letusconsider theevolutionforwardsinζ ofaninitialINpacketinsidethecavity,asstudiedinSection3.2andshown inFig. 5(right). Duetonormconservation,thefollowingrelationshold (IN ,IN ) = (IN ,IN )+(T ,T ), j ≥0, (34) j j j+1 j+1 j+1 j+1 where IN is just the initial IN packet mentioned before. According to Eqs. (32) and (33), the IN (T ) 0 j j packets have positive (negative) norm, therefore Eq. (34) indicates that (IN ,IN ) > (IN ,IN ), j+1 j+1 j j where j ≥ 0. Foreachcycleofmodeconversionprocessesatthehorizons,thenormoftheIN packets j grows by a fixed multiple. By rearranging Eq. (34) in order to express the norm of the T packets in j termsoftheIN ones,itcanbeseenthat (T ,T ) > (T ,T ),andmoreover,thatthenormoftheT j j+1 j+1 j j j packets grows (in magnitude) by the same multiple as the IN packets. This results in an increase of j emittedradiationwhenviewedforwardsinζ,becausethenumberofparticlesemittedisrelatedtothe expectationvalueofthenumberoperatordefinedintermsoftheTpackets,aswewillseeinSection4.4. Ontheotherhand,iftheevolutionbackwardsinζ oftheoutgoingTpacketwithnegativenormof Fig. 5(left)isconsidered,thenthenormconservationyieldsthesamerelationasinEq. (34),sothatthe normofbothpacketsgrowsbackwardsinζ,butwhenviewedforwardsinζ,thisfactisconsistentwith theconclusionpreviouslystatedthattheHawkingprocessisself-amplifying[15]. 4.3 Evolutionformulas In this section we will derive the evolution formulas that describe quantitatively the propagation of modesasweoutlinedinSection3.2.First,wechecktheresults(seeRef.[15])fortheevolutionbackwards inζ ofafinaloutgoingpacketT,asshowninFig. 5(left). Thebackwardevolutionformulasarebased onthe(a),(b),(c),and(d)diagramsofFig. 7(left). Thefinalevolutionformuladescribestheevolution ofapairofTpacketsemittedfromtheBHatcyclesnandmofamplification,wheren > m,intermsof allthepacketsinsidetheWHinvolvedintheprocess. Thefollowingexpressionisobtained n−m ∑ ψ(cid:96),n,T+ χ(cid:96),n−j,T−ψ(cid:96),m,T −→ ψn,T−ψm,T, (35) j=1 10