Robust Phase Estimation of Squeezed State ShibdasRoy*,IanR.Petersen,ElanorH.Huntington SchoolofEngineeringandInformationTechnology,UniversityofNewSouthWales,Canberra,Australia. *[email protected] Abstract: Optimal phase estimation of a phase-squeezed quantum state of light has been recently shown to beat the coherent-state limit. Here, the estimation is made robust to 3 uncertaintiesinunderlyingparametersusingarobustfixed-intervalsmoother. 1 0 © 2013 OpticalSocietyofAmerica 2 OCIScodes: 120.0120,270.0270. n a J 1. Introduction 0 3 Preciseestimationandtrackingofarandomlyvaryingopticalphaseiskeytoapplicationslikecommunication[1],and metrology[2].OptimalsmoothedestimationofawidelyvaryingphaseundertheinfluenceofastochasticOrnstein- ] Uhlenbeck(OU)processforaphase-squeezedbeamnoticeablysurpassesthemaximumaccuracythatcanbeobtained h p for coherentlight by 15 4% [3]. But, the estimation is very sensitive to fluctuations in the parametersunderlying ± - the phasenoiseorsqueezing,andis desiredto bemaderobust to suchuncertainties.We havealreadydemonstrated t n in Ref. [4],superiorphase estimationascomparedto the optimalcase fora coherentstate [5],usinga robustfixed- a intervalsmootherwheretheuncertainsystemsatisfiesacertainintegralquadraticconstraint(IQC)[6].Here,therobust u smoothingtechniqueisappliedtothecaseofsqueezedstateincomparisontotheoptimalcaseconsideredinRef.[3]. q [ 2. ModelofAdaptivePhaseEstimationinRef.[3] 1 v 2.1. ProcessModel 4 Thephasef (t)ofthecontinuousopticalphase-squeezedbeamismodulatedwithanOUnoiseprocess,suchthat 4 1 f˙(t)= lf (t)+√kn (t), (1) − 7 . wherel −1 isthecorrelationtimeoff (t),k determinesthemagnitudeofthephasevariationandn (t)isazero-mean 1 whiteGaussiannoisewithunityamplitude. 0 3 2.2. MeasurementModel 1 v: Thephase-modulatedbeamismeasuredbyhomodynedetectionusingalocaloscillator,thephaseofwhichisadapted i withthefilteredestimatef (t),therebyyieldinganormalizedhomodyneoutputcurrentI(t), X f ar I(t)dt≃2|a |[f (t)−f f(t)]dt+qR¯sqdW(t), R¯sq=s 2fe2rp+(1−s 2f)e−2rm, (2) where a istheamplitudeoftheinputphase-squeezedbeam,anddW(t)isWienernoisearisingfromsqueezedvacuum fluctua|tio|ns.TheparameterR¯ isdeterminedbythedegreeofsqueezing(r 0)andanti-squeezing(r r )and sq m p m bys 2= [f (t) f (t)]2 .Weusethemeasurementappropriatelyscaledasour≥measurementmodel, ≥ f h − f i q (t):=1/ R¯ [I(t)+2a f (t)]=2a / R¯ f (t)+w (t), (3) sq f sq q | | | | q wherew (t)= dW(t) isanotherzero-meanwhiteGaussiannoisewithunityamplitude. dt 3. RobustModel 3.1. UncertainSystem Weemploytherobustfixed-intervalsmoothingtechniquefromRef.[6]asusedinRef.[4].Weintroduceuncertainties d andd ,respectively,intheparametersl and2a / R¯ ,suchthatEq.(2.5)inRef.[6]takestheform: 1 2 sq | | Process: f˙=( l +√k D 1K)f +√kn , pMeasurement: q = 2a / R¯sq+D 2K f +w , (4) − (cid:18) | | q (cid:19) ml /√k where D 1 = d 1 0 ,D 2 = 0 d 2 ,|d 1|≤1,|d 2|≤1,K =(cid:20) 2m−a / R¯sq (cid:21). 0≤m <1 is the levelof uncer- (cid:2) (cid:3) (cid:2) (cid:3) | | tainty.D 1andD 2satisfy:||[ D ′1Q21 D ′2R12 ]||≤1,suchthatQ=R=1.ThepIQCofEq.(2.4)inRef.[6]forourmodel is: T T (w2+v2)dt 1+ z 2dt, (5) Z0 ≤ Z0 || || wherez=Kf istheuncertaintyoutput,andw=D Kf +n andv=D Kf +w aretheuncertaintyinputs. 1 2 3.2. Robustvs.OptimalSmoothersfortheUncertainSystem WeusethemethodfromRef.[4]tocomputeandcomparetheerrorsoftherobustandoptimalsmoothersasafunction ofd andd ,forvariousvaluesofm andwithk =1.9 104rad/s,l =5.9 104rad/s, a 2=1.00 106s 1,r =0.36, 1 2 − m andr =0.59asusedinRef.[3].Duetotheimplicitd×ependenceofR¯ an×ds 2,wecom| p|utetheer×rors 2uponrunning p sq f s severaliterationsuntils 2 matchesuptothe6th decimalplacetothatoftheprioriterationineachcase.Fig.1shows f thecomparisonfor20%and40%uncertainties.Clearly,the robustsmootherperformsmuchbetterthantheoptimal smootherasd and/ord approach 1foralllevelsofuncertainty. 1 2 − Optimal Smoother Optimal Smoother Robust Smoother Robust Smoother 0.04 0.08 0.07 0.035 0.06 2σs0.03 2σs0.05 0.04 0.025 0.03 0.02 0.02 1 1 0.5 1 0.5 1 0 0.5 0 0.5 δ1 −0.5 −1 −1 −0.5 0 δ2 δ1 −0.5 −1 −1 −0.5 0δ2 (a) (b) Fig.1.ComparisonofSmoothedErrorsforUncertainSystem: (a)m =0.2,(b)m =0.4. 4. Conclusion Thisworkextendsthetreatmentofrobustsmoothingtothecaseofsqueezedstate. ForR¯ =1,itboilsdowntothe sq coherentstatecaseofRef.[4]whenD =0.Theseresultsmaybedemonstratedexperimentallyaspartoffurtherwork. 2 References 1. J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates belowthestandardquantumlimitusingaconditionalnullingreceiver,”NaturePhotonics6,374(2012). 2. V.Giovannetti,S.Lloyd,andL.Maccone,“Advancesinquantummetrology,”NaturePhotonics5,222(2011). 3. H.Yonezawa,D.Nakane,T.A.Wheatley,K.Iwasawa,S.Takeda,H.Arao,K.Ohki,K.Tsumura,D.W.Berry, T.C.Ralph,H.M.Wiseman,E.H.Huntington,andA.Furusawa,“Quantum-enhancedoptical-phasetracking,” Science337,1514(2012). 4. S. Roy, I. R. Petersen, and E. H. Huntington,“Adaptive continuoushomodynephase estimation using robust fixed-intervalsmoothing,”ToappearintheProceedingsoftheAmericanControlConference,2013. 5. 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