Reduction of optimum light power with Heisenberg-limited photon-counting noise in interfero

interferometricgravitational-wavedetectors

ConstantinBrif∗

LIGOProject,CaliforniaInstituteofTechnology,Pasadena,CA91125

Westudyhowthebehaviorofquantumnoise,presentingthefundamentallimitonthesensitivityofinterferometricgravitational-wavedetectors,dependsonpropertiesofinputstatesoflight.Weanalyzethesituationwithspeciallypreparednonclassicalinputstateswhichreducethephoton-countingnoisetotheHeisenberglimit.Thisresultsinagreatreductionoftheoptimumlightpowerneededtoachievethestandardquantumlimit,comparedtotheusualconfiguration.04.80.Nn,42.50.Dv

999SincethepioneeringworkbyCaves[1,2],itiswell1 understoodthattwosourcesofquantumnoise—therpphoton-countingnoiseandtheradiation-pressurenoise—Aconstitutethefundamentallimitationonthesensitiv-ityofaninterferometricgravitational-wavedetector. 3Theselimitationswillbeofpotentialimportanceinlong-1baselineinterferometricdetectorswhicharecurrentlyun- derconstruction(theLIGOproject[3,4]intheUnited1StatesandtheFrench-ItalianVIRGOproject[5]inEu-v3ropearethelargestones).Forexample,thephoton-5countingshotnoisewilldominateatthegravitational-0wavefrequenciesabove1kHzintheVIRGOdetector4andabove200HzintheinitialLIGOdetector.With0afurtherreductionofthethermalnoise,plannedinthe9advancedLIGOinterferometer,theroleoftheshotnoise9willbeevenmoreimportant.

/hForacoherentlaserbeamoflightpowerP,theshotpnoise-−1/2associatedwithphoton-countingstatisticsscalesastPandtheradiation-pressurenoisescalesasP1/2.nThecontributionsofthesetwosourcesofnoisewillbeauequalforsomeoptimumvaluePoptofthelightpower.Providedthatclassicalsourcesofnoise(suchasthermal:qandseismic)aresufficientlysuppressed,theinterferom-veterwiththeoptimumlightpowerwillworkattheso-iXcalledstandardquantumlimit(SQL).Asimplequantumrcalculation,basedontheuseoftheHeisenberguncer-ataintyprinciple,givestheSQLforthemeasurementoftherelativeshiftz=z2−z1inthepositionsoftwoendmirrors:

(∆z)SQL=

󰀅

ωτ4

,(2)

whereωisthelightangularfrequency.FortheinitialLIGOconfiguration,themirrormassism≃11kg,thecavitylengthisL≃4km,thefinesseisF≃200,andthewavelengthoftheNd:YAGlaserisλ≃1.064µm(ω≃1.77×1015Hz).Thisgivesthecavitystoragetimeτ≃8.5×10−4sandaneffectivenumberofbouncesb=τc/2L≃32.ThecorrespondingoptimumlaserpowerisPopt≃191kWandtheSQLofthepositionshiftmea-surementis(∆z)SQL≃1.24×10−19m.AchievingthisSQLwillmakepossibletomeasuregravitationalwaveswithamplitudeshgreaterthan∼3×10−23.

Presently,theavailablelaserlightpowerisinsufficientforachievingtheSQL(forexample,intheinitialLIGOconfigurationtheinputlaserpoweris6Wandthepowerrecyclinggainisabout30).Therefore,inadvancedLIGOconfigurations,itisplanned[4]toreducetheshotnoisebyusingmorepowerfullasers,inconjunctionwiththepower-recyclingtechnique[6,7].However,forveryhighlaserpower,oneencountersserioustechnicalproblemsre-latedtononuniformheatingofthecavitymirrorscausedbyabsorptionofevenasmallportionofcirculatinglight.Theresultingthermalaberrationscanseriouslydeterio-ratetheperformanceoftheinterferometer[7].Therefore,itwillbeveryinterestingtostudypossibilitiesforachiev-ingtheSQLwithlowlightpower.

Thegravitational-wavedetectioncommunityisquitefamiliarwiththeintriguingideabyCaves[2]tore-ducethephoton-countingnoisebysqueezingthevacuumfluctuationsattheunusedinputport.Duringthelastdecade,otherinterestingideashasbeendevelopedinthefieldoftheoreticalquantumoptics,basedontheuseofnonclassicalphotonstatesforthequantumnoisereduc-tioninidealizedopticalinterferometers[8–13].Themain

theoreticalmotivationofallthosepaperswastoshowthepossibilityofbeatingtheshot-noiselimitandachievingthefundamentalHeisenberglimitforthephoton-countingnoiseinanidealinterferometricmeasurement.

Theaimofthepresentworkistoshowthattheop-timumlightpowerneededfortheSQLoperationofaninterferometricgravitational-wavedetectorwithmovablemirrorscanbegreatlyreducedbytheuseofnonclassi-calstatesoflightwiththeHeisenberg-limitedphoton-countingnoise.ThisresultmeansthatHeisenberg-limitedinterferometryisnotonlyinterestingforademon-strationofthefundamentaluncertaintyprinciple,butcanbealsoimportantfortheexperimentaldetectionofgrav-itationalwaves.

Letusconsideralong-baselineMichelsoninterferom-eterwhosearmsareequippedwithhigh-finesseFabry-Perotcavities,withendmirrorsservingasfreetestmasses.Inthequantumdescription,twomodesofthelightfieldentertheinterferometerthroughthetwoin-putportsofa50-50beamsplitter.Afterbeingmixedinthebeamsplitter,thelightmodesspenttimeτintheFabry-Perotcavities,andthenleavetheinterferom-eter(throughthesamebeamsplitter,butintheoppo-sitedirection).Thephotonsleavingtheinterferometerintheoutputmodesarecountedbytwophotodetectors.Agravitationalwaveincidentontheinterferometerwillcausearelativeshiftz=z2−z1inthepositionsoftwoendmirrors,whichresultsinthephaseshiftφ=(ωτ/L)zbetweenthetwoarms.

Theperformanceofsuchaninterferometercanbean-alyzedintheHeisenbergpicture,usinganicegroup-theoreticdescriptionproposedbyYurkeetal.[8].Usingthebosonannihilationoperatorsa1anda2ofthetwoinputmodes,oneconstructstheoperators

Jx=Jy=Jz=

†

(a†1a2+a2a1)/2,

†

−i(a†1a2−a2a1)/2,

†

(a†1a1−a2a2)/2.

rotationsinthe3-dimensionalspace[8].Thefirstmixing

inthebeamsplitterproducesarotationaroundtheyaxisby−π/2,withthetransformationmatrixRy(−π/2).Thesecondmixingcorrespondstotheoppositerotation,withthetransformationmatrixRy(π/2).Therelativephaseshiftproducesarotationaroundthezaxisbyφ,withthetransformationmatrixRz(φ).TheoveralltransformationperformedonJistherotationbyφaroundthexaxis,

Rx(φ)=Ry(π/2)Rz(φ)Ry(−π/2).

(5)

Theinformationonthephaseshiftφisinferredfromthephotonstatisticsoftheoutputbeams.Usually,onemeasuresthedifferencebetweenthenumberofphotonsinthetwooutputmodes,

qout=2Jzout=2[(sinφ)Jy+(cosφ)Jz].

(6)

Ifweassumethattherearenolossesintheinterferome-terandtheclassicalsourcesofnoisearewellsuppressed,thentheuncertaintyintherelativepositionshiftzoftheendmirrorsisduetotwofactors[1,2].Thefirstoneisthephoton-countingnoise.Indeed,sincetherearequan-tumfluctuationsinqout,aphaseshiftisdetectableonlyifitinducesachangein󰀏qout󰀐whichislargerthantheuncertainty∆qout.Consequently,theuncertaintyinthephaseshiftduetothephoton-countingnoiseis

(∆φ)2pc

=

(∆qout)2

󰀏Jz󰀐2

(3)

,Apc=

󰀆

L

Theseoperatorsformthetwo-bosonrealizationofthe

su(2)Liealgebra,[Jk,Jl]=iǫklmJm.TheCasimiroper-atorisaconstant,J2=j(j+1),foranyunitaryirre-duciblerepresentationoftheSU(2)group;sotherepre-sentationsarelabeledbyasingleindexjthattakesthevaluesj=0,1/2,1,3/2,....TherepresentationHilbertspaceH|isspannedbythecompleteorthonormalbasis|j,m󰀐(m=j,j−1,...,−j).UsingEq.(3),onefinds

󰀄†21

N=a†(4)J=1a1+a2a2,2N+1,whereNisthetotalnumberofphotonsenteringthein-terferometer.WeseethatNisanSU(2)invariant;ifthe

inputstateofthetwo-modelightfieldbelongstoH|,thenN=2j.

Theactionsoftheinterferometerelementsonthecolumn-vectorJ=(Jx,Jy,Jz)Tcanberepresentedas

mL

󰀇2

.(9)

Considerthestandardcasewhenthecoherentlaserbeamofamplitudeαenterstheinterferometer’sonein-putport,whilethevacuumenterstheother.Thisin-putstate|in󰀐=|α󰀐1|0󰀐2(where|0󰀐isthevacuumand|α󰀐=exp(αa†−α∗a)|0󰀐isthecoherentstate),satisfies

󰀏Jx󰀐=󰀏Jy󰀐=0,

󰀏Jz󰀐=|α|2/2,

2

22󰀏Jx󰀐=󰀏Jy󰀐=|α|2/4,

¯=|α|2.󰀏N󰀐≡N

Usingtheseresults,onefinds

2¯−1+ArpN.¯(∆z)2=(∆z)2pc+(∆z)rp=ApcN

(10)

¯,oneobtainsOptimizing(∆z)2asafunctionofN

N¯mL2

opt=

2ξa

†2

−

1

2|󰀏Jz󰀐|,

oneobtains

(∆φ)pc≥(2∆Jx)−12.Sinceforanyinputstate|in󰀐∈Hjtherelation(∆Jx)≤1

theeigenvalueequation

2|󰀏Jz󰀐|.

Thesestatesaredeterminedby

(ηJx−iJy)|λ,η󰀐=λ|λ,η󰀐.

(16)

Thespectrumisdiscrete:

λ=im0

󰀅

Ifthetwo-modelightfieldenteringtheinterferometerispreparedintheJx-Jyintelligentstate,thequantumnoisetakestheform

(∆z)2=Apc(2∆Jx)−2+Arp(2∆Jx)2.

(17)

(∆φ)pc=[2j(j+1)]−1/2.Ofcourse,thisimprovementisonaccountofthecorrespondingincreaseintheradiation-pressurenoise,because(∆Jx)2=1

For|η|<1,theintelligentstatesaresqueezedinJyandanti-squeezedinJx,therebyreducingthephoton-countingnoisebelowtheshot-noiselimit,onaccountofincreasingcontributionoftheradiation-pressurenoise.Forη→0,oneobtains[16]

(2∆Jx)2=2|󰀏Jz󰀐/η|≃2(j2−m20+j),

(18)

andtheHeisenberglimitforthephoton-countingnoiseis

achievedwhenm0=0.Then,forlargephotonnumbers¯=2j≫1),weobtain(N

¯−2+(∆z)2≃2ApcN

1

hω2τ3¯

󰀇1/2

,Popt=

󰀆

2¯hmL2

(∂󰀏S󰀐/∂φ)2

=

tan2φ

4j(j+1)

.(21)

Forφ=0(thiscorrespondstoadarkfringeforthemeasurementofS),theHeisenberglimitisachieved:

4

[1][2][3][4]

[5][6]

[7][8][9][10]

C.M.Caves,Phys.Rev.Lett.45(1980)75.C.M.Caves,Phys.Rev.D23(1981)1693.A.Abramovicietal.,Science256(1992)325.

J.K.Blackburn,in:MathematicsofGravitationPartII,GravitationalWaveDetection(BanachCenterPublica-tions,Warsaw,1997),p.95.

B.Caronetal.,Nucl.Phys.B(Proc.Suppl.)54B(1997)167;Class.QuantumGrav.14(1997)1461.

R.W.P.Drever,in:GravitationalRadiation,ed.N.Deru-elleandT.Piran(North-Holland,Amsterdam,1983),p.321.

P.Hello,in:ProgressinOptics,Vol.38,ed.E.Wolf(North-Holland,Amsterdam,1998),p.85.

B.Yurke,S.L.McCallandJ.R.Klauder,Phys.Rev.A33(1986)4033.

M.J.HollandandK.Burnett,Phys.Rev.Lett.71(1993)1355.

B.C.SandersandG.J.Milburn,Phys.Rev.Lett.75

[11][12][13][14]

[15][16][17][18][19]

(1995)2944.

M.HilleryandL.Mlodinow,Phys.Rev.A48(1993)1548.C.BrifandA.Mann,Phys.Rev.A54(1996)4505.

T.Kim,O.Pfister,M.J.Holland,J.NohandJ.L.Hall,Phys.Rev.A57(1998)4004.

D.J.Wineland,J.J.Bollinger,W.M.Itano,F.L.MooreandD.J.Heinzen,Phys.Rev.A46(1992)R6797;

D.J.Wineland,J.J.Bollinger,W.M.ItanoandD.J.Heinzen,Phys.Rev.A50(1994)67;

G.S.AgarwalandR.R.Puri,Phys.Rev.A49(1994)4968.

A.LuisandJ.Peˇrina,Phys.Rev.A53(1996)1886.C.Brif,Int.J.Theor.Phys.36(1997)1651.J.P.Dowling,Phys.Rev.A57(1998)4736.

P.BouyerandM.A.Kasevich,Phys.Rev.A56(1997)R1083.

J.Preskill,privatecommunication.

5