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Quantum interferometry withthree-dimensional geometry Nicolo` Spagnolo,1 Lorenzo Aparo,1 Chiara Vitelli,2,1 Andrea Crespi,3,4 Roberta Ramponi,3,4 Roberto Osellame,3,4 Paolo Mataloni,1,5 and Fabio Sciarrino1,5 1Dipartimentodi Fisica,Sapienza Universita` diRoma, PiazzaleAldoMoro5, I-00185Roma, Italy 2CenterofLifeNanoScience@LaSapienza,IstitutoItalianodiTecnologia,VialeReginaElena,255,I-00185Roma,Italy 3IstitutodiFotonicaeNanotecnologie,ConsiglioNazionaledelleRicerche(IFN-CNR),PiazzaLeonardodaVinci32,I-20133Milano,Italy 4DipartimentodiFisica, PolitecnicodiMilano, PiazzaLeonardo daVinci32, I-20133 Milano, Italy 5Istituto Nazionale di Ottica (INO-CNR), Largo E. Fermi 6, I-50125 Firenze, Italy Quantuminterferometryusesquantumresourcestoim- proaches increases with the number of particles involved in 3 prove phase estimationwith respect to classical methods. thecorrelatedprobestate [5]. Two-photonNOONstatescan 1 Hereweproposeandtheoreticallyinvestigateanewquan- be produced deterministically simply by quantum interfer- 0 tum interferometric scheme based on three-dimensional enceoftwoidenticalphotonsonabalancedtwo-modebeam- 2 waveguide devices. These can be implemented by fem- splitter. However, two-mode NOON states with more than n tosecond laser waveguide writing, recently adopted for two particlesare difficultto produce. Increasingthe number a quantum applications. In particular, multiarm interfer- ofmodestomorethantworepresentsaninterestingpossibil- J ometersinclude“tritter”and“quarter”asbasicelements, ity to extend the concept of multiparticle interferometry, as 8 corresponding to the generalizationof a beam splitter to pointedoutbyGreenbergeretal.[8].Thisrequiresmulti-port ] a 3- and 4-port splitter, respectively. By injecting Fock devices, instead of simple two-modebeam-splitters, to build h states in the input ports of such interferometers, fringe multi-arm interferometers. Multi-port beam splitters can be p patterns characterized by nonclassical visibilities are ex- potentially realized by properly combining several balanced - t pected. This enables outperforming the quantum Fisher two-port beam-splitters and phase shifters [9, 10]: such im- n informationobtainedwithclassicalfieldsinphaseestima- plementationhasanywaytightrequirementsoninterferomet- a u tion.Wealsodiscussthepossibilityofachievingthesimul- q taneous estimation of more than one optical phase. This [ approachisexpectedtoopennewperspectivestoquantum 1 enhancedsensing andmetrologyperformedinintegrated v photonic. 7 7 5 1 INTRODUCTION . 1 0 Quantummetrologyisoneofthemostfascinatingfrontiers 3 of the science of measurement: the counterintuitive laws of 1 quantummechanicsareexploitedtomaximizetheamountof : v information extracted from an unknown sample, beating the i limitsimposedbyclassicalphysics. Thelowdecoherenceof X photons, enabling the observation of quantum effects in an r a easier way, makes optical interferometrya promising candi- datefordemonstratingquantumenhancedsensitivity.Thees- timationofanopticalphaseφthroughinterferometricexper- imentsis indeedan ubiquitoustechniquein physics, ranging from the investigation of fragile biological samples, such as tissues [1] or blood proteins in aqueous buffer solution [2], to gravitational wave measurements [3, 4]. Whereas optical interferometry relying on classical interference is intrinsically a single-particle process, quantum advantages arise FIG. 1: (a) 3D structure of tritter. (b) 3-dimensional structure of whenquantum-correlatedstatesofmorethanoneparticleare quarter.(c)Geometryshowingthedirectcouplingbetweenthethree modesof thetritter. (d) Geometryshowingthedirectcoupling be- employed [5], such as Greenberger-Horne-Zeilinger (GHZ) tweenthefourmodesofthequarter. (e)Indirectcouplingbetween [6] and NOON [7] states. NOON states, in particular, allow thefourmodesof thequarterbymeansof oneancillarymode. (f) to saturate the Heisenberg limit of sensitivity: the ultimate 3-mode interferometer built by using two cascaded tritters. (g) 4- limittothe precisionofa measurementimposedbythelaws modeinterferometerbuiltbycascadingtwoquarterdevices. (f)-(g) ofphysics. φ: phase to be measured. ψ: additional phase for adaptive phase Generally, the quantum advantage over classical ap- estimation. 2 ricstability, makingitseffectiverealizationchallengingwith foracertaininteractionlengthandcouplebyevanescentfield. bulkoptics. Infact,thefewexperimentalrealizationsofsuch The feasibility of such structures, in the case of three-ports devices are reported on fiber-based [11] or integrated-optics (tritter),hasalreadybeendemonstratedbyfemtosecondlaser [12] multi-mode devices; in both cases the characterization writing [25, 27], albeit characterization has only been per- wasperformedwithonlytwo-photonstates. Multi-photonin- formedin aclassicalframework. Notethatin thesecompact terferometryis,indeed,stillawidelyunexploredfield. multi-waveguidestructurestheinteractionbetweenthediffer- Times are mature to demonstrate multi-photonand multi- entarmshappenssimultaneously,withoutthedecomposition portdevices. Infact,ontheonehandinthelastfewyearsthe intocascadedtwo-modebeamsplitters. Thisismadepossible efficiencyofquantummulti-photonsourceshavedramatically bythe3Dcapabilitiesoffemtosecondlasermicromachining, improvedleadingtoseveralexperimentswithupto8photons thusrelaxingthestrictrequirementsonpath-lengthcontrolof [13]. On the other hand, the advent of integrated quantum alternativeapproaches. photonics have opened exciting perspectives for the realiza- The symmetric configurationof a tritter can be easily ob- tion of scalable, miniaturized and intrinsically stable optical tainedbyadoptingatriangulargeometry,asshowninFig. 1 setups[14]. Inparticular,theultrafastlaser-writingtechnique (c).Inthisconfigurationitispossibletoobtainequalcoupling [15–18] has proved to be a powerful tool for demonstrat- coefficients,sothatasinglephotonenteringinoneinputport ing new quantum integrated-optics devices, able to perform hasthesameprobabilityofexitingfromoneofthethreeout- quantumlogicoperations[19]aswellastwo-photonquantum put ports. The symmetric configuration of a quarter can be walks[20, 21]. Thistechniqueexploitsnonlinearabsorption achievedbyadoptingtwopossiblesolutions. Inthefirstone, offocusedfemtosecondlaserpulsestoinducepermanentand thefourmodesaredirectlycoupled,andthe symmetriccon- localizedincreaseoftherefractiveindexintransparentmateri- ditioncanbeobtainedbyappropriatelytuningtheinteraction als. Waveguidesaredirectlyfabricatedinthematerialbulkby length [Fig. 1 (d)]. Alternatively, an indirect geometrymay translating the sample at constant velocity along the desired beexploitedbycouplingwaveguides1-4throughanancillary pathwithuniquethree-dimensional(3D)capabilities. mode,asshowninFig. 1(e),andwithoutanydirectcoupling In the field of quantum metrology, recent results have betweenthem. demonstrated the possibility of using two-mode path- entangledstates in integratedstructures forphase estimation belowtheStandardQuantumLimit(SQL)[2,22–24]. Com- Outputstates binationofalltheaboveelementswouldenablesteppinginto multi-photon/multi-port quantum metrology. However, the -Westartbyconsideringtheactionofasingletritterandof potentialsofthisapproachhavenotyetbeeninvestigatedthe- asinglequarter. Theactionofthesedevicesonaninputstate oretically. ψ is expressed by a unitary matrix (III) [ (IV)], which | i U U Inthisworkwe introducethe conceptof3D multi-photon mapstheinputfieldoperatorsa† totheoutputfieldoperators i interferometry. First, we propose novel geometries for in- b† accordingto b† = P (k)a†,withk = III,IV [9,10, tegrated multi-arm interferometers based on three-port (trit- 2i8] (see Supplemientaryi,MjUatiejrialj). Let’s consider the Fock ter)andfour-port(quarter)devices. Second,wetheoretically state 1,1,1 ,where i,j,l = i k j k l k istheinputstate study possible measurement protocols, based on the injec- inthe|tritter.iByappl|yingi(II|I)iw1e| oibt2a|ini t3heoutputstate: tion of multi-photon Fock states in this kind of multi-port U 1,1,1 c 1,1,1 +c 3,0,0 . (1) devices, demonstrating relevant metrological advantages in {1,1,1} {3,0,0} | i→ | i |{ }i phase-estimationtasks. Ourresultsarenotmerelyspeculative Here c = eı2π/3/√3, c = eı4π/3p2/3, and 1,1,1 {3,0,0} sinceboththerealizationofsuchintegratedmulti-portdevices − i,j,l is the symmetric superposition of three-photon [25] and the generationof multi-photonquantum states [26] |{ }i stateswhere(i,j,l)photonsexitinthethreeoutputports. A appearto be within reach of presentstate-of-the-arttechnol- similarresultisobtainedforaquarterfedwitha 1,1,1,1 = ogy. | i 1 k 1 k 1 k 1 k state: | i 1| i 2| i 3| i 4 1,1,1,1 c 1,1,1,1 +c 2,2,0,0 + {1,1,1,1} {2,2,0,0} | i→ | i |{ }i RESULTS +c 4,0,0,0 , {4,0,0,0} |{ }i (2) Multi-arminterferometricschemes where c = 1/2, c = √6/4, c = 1,1,1,1 {2,2,0,0} {4,0,0,0} √6/4.Inbothcases,weobservethatsometermsoftheout- − A multi-arm interferometer can be realized by cascading putstates are suppresseddue to quantuminterference. They twomulti-portbeamsplitters. Inparticular,athree-arminter- correspond in particular to the contributions 2,1,0 in the { } ferometerisbuildbythecombinationoftwotrittersandafour tritter case, 3,1,0,0 and 2,1,0,0 in the quarter one. { } { } arminterferometerbythecombinationoftwoquarters.Inour This feature is a N-mode analogue of the Hong-Ou-Mandel approachtheintegrated-opticsmulti-portdeviceisdevisedas bosoniccoalescence effect[28–30]. The two devicescan be a 3D multi-waveguidedirectional coupler[Fig. 1 (a)-(b)], a exploited to generate maximally-entangledN00N states in a structureinwhichthewaveguidesarebroughtclosetogether post-selectedconfiguration[31]. 3 FIG.2:(a)-(c)OutputfringepatternsPF ofthe3-modesinterferometerfedwitha|1,1,1iinputstate.(d)DiagramoftheN-foldvisibilities m,n,q VF oftheoutputfringepatternsPF foraN =3interferometerwitha|1,1,1iinputstate,comparedwiththeclassicalboundΓcl . m,n,q m,n,q m,n,q (e)Correspondingdiagramforthe4-modescase. N-modesinterferometry Nonclassicalitycriterionforsub-Rayleighfringepatterns - The 3D, N-port structure of tritter and of quarter de- - In order to formally address the nonclassicality of this vices can be exploited to implement an integrated N-modes setup we have extended the criterion proposed by Afek et interferometer. Thiscanberealizedbyachainoftwosubse- al. in Ref. [33] from a 2-modes to a 3-modes interferome- quentmultiportbeam-splitter,leadingtoageneralizedMach- ter. This criterion sets an upper bound Γcml,n,q for classical Zehnder structure [see Fig. 1 (f)-(g)]. The phase shifting statesontheN-foldvisibilitesoftheFourierexpansionofthe could be introduced for instance by adopting a microfluidic fringepatterndistributionsPm,n,q(φ) = P∞k=0Akcos(kφ− channel as reported in Ref. [2, 32]. Let us now consider δk), where N = m + n + q. The visibilities are defined the3-modessystemobtainedwithtwotritterdevices,andthe as the ratio between the Fourier coefficient Am,n,q of the action of a relative phase shift φ in the optical mode k3 in- term oscillating as Nφ and the constant coefficient A0, i.e. side the interferometer, which is described by the operator Vm,n,q = Am,n,q/A0 . The classical bound for Vm,n,q is | | =exp( ın φ),beingn thephotonnumberoperatorfor obtainedbycalculatingtheprobabilitydistributionofhaving φ 3 3 Umodek . T−heoutputprobabilitydistributionsP (φ)cor- (m,n,q) photons in the output modes (1,2,3) respectively, 3 m,n,q respondingto the detection of (m,n,q) photonsin the three by feedingthe interferometerwith a classical coherentstate: outputports, are obtained by the overallevolution of the in- α1,α2,α3 , where αi = αi eıθi. Then, the N-foldvisibil- | i | | tuesrfceoronmsideeterraUga(IinIIa)UnφinUp(uIItIF)oaccktinstgatoen1t,h1e,1inp;utthestaotbet.ainLeedt ittoymVmax,nV,qmi,sn,mq a=ximΓicmzle,nd,qw.itIhnrgeespneecratlt,oit|αcai|nabnedsθhio,wlenadtihnagt fringe patterns PmF,n,q(φ), symmetric f|or an iindex exchange for any classical state Vm,n,q ≤ Γcml,n,q, since any classical (m,n,q),arereportedinFigs. 2(a)-(c).Weobservethepres- statecanbeexpandedinthecoherentstatesbasisaccordingto enceofinterferometricpatternspresentingthesumofdiffer- ρC =R d2αP(α)α α withawellbehavedP(α). Thecor- | ih | ent harmonics up to cos(3φ). Furthermore we note that the respondingN-foldvisibilitiesVF fortheinputFockstate m,n,q PF (φ) term presents a sub-Rayleigh λ/3 behavior with a 1,1,1 areobtainedfromtheprobabilitydistributionsPF 2,1,0 | i m,n,q unitaryvisibility.Theseresultssuggestthattheoutputstateof showninFigs. 2(a)-(c),leadingtohighervisibilitiesthanthe theinterferometer,fora 1,1,1 inputstate,presentsnonclas- classical limits, except for the PF case [Fig. 2 (d)]. The | i 1,1,1 sicalfeatures.Similarresultscanbeobtainedwhena4-modes samecriterioncanbeextendedtoaN-modesinterferometer. interferometerisfedwitha 1,1,1,1 inputstate[Fig. 1(g)] WethenrepeatedthesameanalysisfortheN =4casewhen | i (seeSupplementaryMaterial). the interferometerisbuiltby two subsequentquarterdevices 4 FIG.3:(a)ComparisonoftheQFIsfortheN =2,N =3,N =4interferometersfedwithaFockstateHF,acoherentstatewithanexternal φ phase reference beam HC,(i), and acoherent statewithout anexternal phase reference beamHC,(ii). (b)-(c)QFI HF and FIIF for(b) a φ φ φ φ 3-modesinterferometerwitha|1,1,1iinputstate,and(c)fora4-modesinterferometerwithaninput|1,1,1,1istate. Theoptimalworking pointsareobtainedwhenIF=HF. φ φ [Fig. 1 (g)], showing the presenceof nonclassicalbehaviors ber of photons impinging onto the phase shifter. Further- inthefringepatterns[seeFig. 2(e)]. more, for a coherent state input two different cases can be identified. (i) If an external reference beam, providing an absolute reference frame for the optical phase, is available Phaseestimation at the measurement stage, the QFI is calculated for a pure α ,α ,α input state: HC,(i). (ii) If no reference frame | 1 2 3i φ -Thepresentinterferometricconfigurationcanbeadopted is available at the measurement stage (such as for photon- to perform a phase estimation protocol. In this context, the counting detection), one needs to average the input state on aim is to measure an unknown phase shift φ introduced in a random phase shift θ common to all input modes leading aninterferometerwiththebestpossibleprecisionbyprobing to HC,(ii). In absence of an externalreferencebeam, an in- φ the system with a N-photon state, and by measuring the re- put state ̺ has to be replaced with the phase-averaged state sulting output state. The classical limit is provided by the ̺′ = (2π)−1R2πdθ 1 2 3̺ 1† 2† 3†, where i is the SQL, which sets a lower bound to the minimum uncertainty phaseshiftope0ratorfUorθmUθoUdeθ kU.θTUhθetUwθoconditionUsθ(i)and i δφSQL >1/√MN whichcanbeobtainedonφbyexploiting (ii) are equivalent for the 1,1,1 probe, since this state has classicalN-photonstatesontwomodesandM repeatedmea- a fixednumberofphotons|[39]. iWe thenevaluatedthethree surements[34]. Recently,ithasbeenshownthattheadoption quantitiesHC,(i),HC,(ii),HF,andobtainedthattheadoption φ φ φ ofquantumstatescanleadtoabetterscalingwithN,setting of a 1,1,1 probe state leads to quantum improved perfor- the ultimate precision to δφ > 1/(√MN), correspond- | i HL mances. The same result is found for the 4-modes case, as ing to the Heisenberg limit [5, 35, 36]. Hereafter, we show shown in Fig. 3 (a). We note that, while HC,(i) is fixed, φ thatthepresentintegratedtechnologycanleadtoa sub-SQL theQFIachievablewithaFockstateinputincreaseswiththe performanceintheestimationofanopticalphase,exploiting numberofmodes,leadingprogressivelytoagreateradvantage multi-modeinterferometry. inphaseestimationprotocols. Furthermore,nopost-selection isneededtogeneratetherequiredprobestate[40]. QuantumFisherinformation Achievingtheoptimalbound - In orderto characterize the present 3-modesinterferometer fed with a 1,1,1 input state, and analogously the 4- - We now show thatthe QCR boundprovidedby HF can | i φ modescase, we needto determinethequantumFisherinfor- beachievedbyadoptingafeasibleandpracticalchoiceofthe mation (QFI) HF of the outputstate [37, 38]. This quantity measurementsetup,consistinginaphoton-countingapparatus φ sets the maximum amount of information which can be ex- recordingthe numberof outputphotonson each mode. The tractedonthephaseφfromastate̺ accordingtothequan- detectionapparatuscanbeimplementedbysplittingeachout- φ tumCrame´r-Rao(QCR)bound:δφ (MH )−1/2[37]. The putmodeinthreepartsbymeansofachainofbeam-splitters, φ ≥ classical limit is provided by the quantum Fisher informa- andbyplacingasingle-photondetectoroneachpart. Theoc- tion HC when a coherent state α ,α ,α is injected into currenceof 1, 2 or3 simultaneousclicks of the detectorson φ | 1 2 3i theinterferometer.Notethatthecomparisonbetweentheper- the same mode corresponds to the detection of 1, 2, 3 pho- formancesachievablewithaninputcoherentstate andanin- tons.Forafixedchoiceofthemeasurementsetup,theamount put 1,1,1 Fockstatemustbeperformedforthesamenum- ofinformationwhichcanbeextractedonφisprovidedbythe | i 5 Crame´r-Rao(CR)boundδφ (MIφ)−1/2 [37],whereIφ is 0.0036 0.01 ≥ the Fisher information of the output probability distribution t of the measurement outcomes. The results for IF with the es φ 0.003 φ 0 1,1,1 inputstateandphoton-countingmeasurementsarere- − | i portedin Fig. 3 (b). By comparingthe trendof IF with the φ φ correspondingQFIHF,weobservethattheultimateprecision -0.01 φ φ0.0024 −π/3 π/6 2π/3 7π/6 5π/3 givenbytheQCR boundcanbeachievedwiththischoiceof δ φ themeasurementapparatusfor φ = 0,2π/3,4π/3. Ananal- ogousresultisfoundforthe N = 4 case, wherethe optimal 0.0018 workingpointsarenowφ=0,π[Fig. 3(c)]. 0.0012 Adaptiveprotocol −π/3 π/6 2π/3 7π/6 5π/3 φ -Theobtainedφ-dependenceoftheFisherinformationI φ suggeststhatanadaptiveprotocol[41]isnecessarytoobtain FIG. 4: Results for the phase estimation error δφ of a numerical optimalperformancesinthefullphaserange,thatis,tosatu- simulation with M = 105 repeated measurements with the three rate the QCR boundforall valuesof φ. To thispurpose, we modesinterferometerofFig. 1(f). Bluecircularpoints: δφforthe considertheadoptionofathree-stepadaptivestrategywhere adaptive protocol as a function of φ. Blue solid line: QCR bound given by HF. Redcross points: δφfor thenon-adaptive protocol. the first two steps of the protocol are performed to obtain a φ Red dashed line: Fisher information IF, providing the bound for roughestimateφ ofthephaseφ(moredetailsontheprotocol φ r the non-adaptive strategy. Shaded region corresponds to sub-SQL canbefoundintheSupplementaryinformation).Theamount performancesachievablewithquantumresources. Inset: plotofthe ofmeasurementsperformedinthesestepsisa smallfraction differencebetweenthetruevalueφandtheestimatedvalueφ ob- est of the overall resources M, namely M1 = M2 = √M. In tainedwiththeadaptiveprotocol. thefirst step, a roughestimate ofthephaseis obtainedupto a two-fold degeneracydue to the symmetry of the interfero- metricfringes(φ,4π/3 φ). Thesecondstepisexploitedto removethisdegeneracy−fortheestimateφ .Finally,thethird by the optical mode k1, and the phases to be measured, φ2 est andφ , correspondrespectivelyto the opticalmodes k and stepconsistingofM =M M M measurementsisper- 3 2 formedby sendingt3he 1,1−,1 i1np−uts2tate, andthe system is k3 [seeFig. 5(a)]. Inordertoevaluatethemaximumpreci- | i sionachievableinthetwoparameterproblem,itisnecessary tuned to operate in the optimal regime (φ+ψ 2π/3) by ≃ to extend the concept of quantum Fisher information to the meansofanadditionalphaseshiftψ. Ateachstepofthepro- multiparameter case [44]. Indeed, it is possible to define a tocol,themeasurementoutcomesareanalyzedbyaBayesian quantumFisherinformationmatrix(QFIM)H correspond- approachandassumingnoa-prioriknowledgeonφ[42]. The µν ing to the set of parameter λ = (λ ,...,λ ), which in the resultsofthenumericalsimulationforM = 105arereported 1 n in Fig. 4, together with the results for a numerical simula- poufrtheestsaetteocfagseen|eψraλtior=sGe−=ıP(GµG,µ.λ.µ.|,ψG0i)isfodretfihneepdarianmteertmerss tion of a one step non-adaptivestrategy. We observe that in 1 n λ. Theerroron the single parameter λ fora fixed valueof the latter case, the uncertainty δφ associated to the estima- µ the other parameters λ , with ν = µ is bounded by the in- tionprocessresemblescloselytheCRboundprovidedbythe equality: δλ [(H−1ν) /M]1/62.Whenperformingthesi- Fisher information IφF. A better result is obtained with the multaneouseµst≥imationofµtµhesetofparametersλ,thesumof adaptive strategy, showing that the quantum Fisher informa- the variancesis boundedby the multiparameterCramer-Rao tionisachieved,leadingtosub-SQLperformancesinthefull phaseinterval. ThechoiceoftheBayesianapproachleadsto inequality:PµVar(λµ)≥Tr[H−1]/M. anunbiasedestimationprocess,i.e.,theestimatedphaseφ We now consider as input state a coherent state α with est | i converges to the true value φ (see inset of Fig. 4), and the n = 3, which defines the standard quantum limit for a h i errorontheestimationprocesscanbedirectlyretrievedfrom N = 3photonprobe. Asforthesingleparameterscenario,it theoutputdistributionforφ[43]. isnecessarytoevaluatethequantumFisherinformationmatri- cesboth(i)inpresenceofanexternalphasereferenceHC,(i) φ2,φ3 and(ii)inabsenceofanexternalphasereferenceHC,(ii). The Twoparameterphaseestimation φ2,φ3 correspondingbounds for the sensitivities after M measure- mentsinthetwocasesaregivenbyδφC,(i) (MH˜C,(i))−1/2 -Letusnowconsideradifferentscenario.The3-modesin- µ ≥ φµ terferometerbuiltwith two cascadedtrittersproposedin this and δφCµ,(ii) ≥ (MH˜φCµ,(ii))−1/2. When a threephotonstate paper may be adopted to perform a two parameters estima- 1,1,1 isinjectedintotheinterferometer,theboundsforthe | i tion process, consisting of the simultaneousmeasurementof parametersφ andφ arethesameforbothcases(i)and(ii) 2 3 twoopticalphases. Inthiscase,thereferenceφ isprovided leadingto δφF (MH˜F )−1/2. Thesame analysismaybe ref µ ≥ φµ 6 FIG.5:(a)-(b)Schemesfortwoparametersphaseestimationwiththe3-and4-modesintegratedinterferometers.(c)Comparisonbetweenthe effectivequantumFisherinformations(H˜C,(i),H˜C,(ii),H˜F )forthetwo-parametersproblemwhentheinterferometerisfedwithFockstates φµ φµ φµ orcoherentstates. performedin the case of a 4-modes interferometer, fed with formphaseestimationprotocolsleadingtoquantum-enhanced a coherent state of n = 4 photons or with a Fock state performances. We providedandsimulatedafullprotocolfor h i 1,1,1,1 , where the two parametersare now the phases φ sub-SQLphasemeasurements,byexploitingFockinputstates 3 a|ndφ oni modesk andk [seeFig. 5(b)]. Theresultsare and photon-counting detection, thus not requiring any post- 4 3 4 showninFig. 5(c),showingthatinabsenceofaphaserefer- selection for the generation of the probe state. We also dis- ence the adoption of Fock probe states can lead to quantum cussed the application of the same multimode structure for enhanced performances in the measurement of two optical multiparameterestimationpurposes. Thepresenttechnology phases. Furthermore,in fullanalogywith the oneparameter is expectedto lead to the developmentof new phase estima- case,agreateradvantangewithrespecttotheclassicalstrate- tionprotocolsabletoreachHeisenberg-limitedperformances giesmaybeprogressivelyachievedbyincreasingthenumber [31] and to open a new scenario for the simultaneous mea- ofmodes. surementofmorethanoneopticalphase. Indeed,thepresent In the single parameter case, the quantum Cramer-Rao approachcanbeadoptedasanaccessibletestbenchtoinvesti- bound can always be asymptotically achieved [41] perform- gatetheoreticallyandexperimentallythestillunexploredsce- ingasuitablemeasurementandchoosingtherightestimator. narioofmultiparameterestimation. Furtherperspectivesmay Onthecontrary,inthemultiparametercasetheboundforthe leadtotheapplicationofthismultiportsplittersinothercon- statistical errors defined by the quantum Fisher information texts, such as quantum simulations [48], linear-optical com- matrix is not in general achievable [45, 46]. This depends puting[49]andnonlocalitytests[50]. on the fact that the optimal measurements for the individual parametersmaynotbe compatibleobservables. A necessary conditionfortheachievabilityofthemultiparameterquantum Cramer-Raoboundis then givenbythe weak commutativity ACKNOWLEDGEMENTS condition: Tr[ρλ[Lµ,Lν]] = 0. Here, the Lµ operators are thesymmetriclogarithmicderivatives(SLD),whichdefinethe optimalmeasurementandestimatorsfortheindividualparam- This work was supported by FIRB-Futuro in Ricerca eters[47]. 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Characterizationof symmetric[3x3]directional couplers Supplementary material: Quantum interferometry withthree-dimensional geometry Nicolo` Spagnolo,1 Lorenzo Aparo,1 Chiara Vitelli,2,1 Andrea Crespi,3,4 Roberta Ramponi,3,4 Roberto Osellame,3,4 Paolo Mataloni,1,5 and Fabio Sciarrino1,5 1Dipartimentodi Fisica,Sapienza Universita` diRoma, PiazzaleAldoMoro5, I-00185Roma, Italy 2CenterofLifeNanoScience@LaSapienza,IstitutoItalianodiTecnologia,VialeReginaElena,255,I-00185Roma,Italy 3IstitutodiFotonicaeNanotecnologie,ConsiglioNazionaledelleRicerche(IFN-CNR),PiazzaLeonardodaVinci,32,I-20133Milano,Italy 4DipartimentodiFisica,PolitecnicodiMilano, PiazzaLeonardo daVinci, 32, I-20133Milano, Italy 5Istituto Nazionale di Ottica (INO-CNR), Largo E. Fermi 6, I-50125 Firenze, Italy InthisSupplementarymaterialweprovidemoredetailsontheresultspresentedinthepaper. Wefirstreport theunitaryevolution matriceswhich describetheaction of tritterand quarterdevices. Then, weanalyze the multiarminterferometersobtainedbycascadingtwotritter(quarter)devicesandderivingtheprobabilitydistri- 3 butionsassociatedtotheoutput photonnumberstatisticsfortheinputstate|1,1,1i(|1,1,1,1i). Wediscuss 1 moreindetailtheanalyzedphaseestimationprotocolbasedonthe3-dimensionalinterferometricarchitecture 0 obtainedwithmultiportdevices.Finally,weperformafirstanalysisonthetwoparameterestimationproblem. 2 n a TRITTERANDQUARTERUNITARYEVOLUTION J 8 The basic element of a tritter (quarter)is a directional coupler(DC) in which 3 (4) waveguidesare broughtclose together ] and coupled by evanescent field. A DC is described by transmission coefficients Tmn = |Tmn|eıφmn, giving the probability h amplitudeofthephotonexitingin thesameopticalmode(m = n), andthecouplingbetweendifferentwaveguides(m = n). 6 p In the symmetric case we have T = 1/√N for any (m,n), where N is the numberof waveguidesin the DC. The time mn t- evolutionofthefieldoperatorsis|obta|inedasb† = (k)a†,wherek = III,IV, b† aretheoutputmodesoperators,and n i i,jUij j { i} a {a†i}aretheinputmodesoperators. Thematrix U(PIII) inthesymmetriccaseisobtainedbyimposingtheunitarietycondition u onthetransformationinducedbythelinearcouplingcoefficientsT andR,leadingto: q 1 [ (III) = 1 eı123π eı123π eeıı2233ππ . (1) v U √3eı23π eı23π 1  7   7 Followingthesameproceduretheunitarymatrixwhichdescribestheevolutioninthesymmetricquarterdeviceisgivenby: 5 1 1 1 1 1 . − − − 1 1 1 1 1 1 0 (IV) = − − − . (2) U 2 1 1 1 1 3 − − −  1 1 1 1  1 − − −  :   v AsshowninRef.[1],thereisnouniquechoiceforthematrices (III)and (IV). Inthetrittercase,thereisonlyoneequivalence i U U X classforthematrices (III),thatis,alltheallowedtransformationsareequivalentup tophaseshiftsinsertedintheinputorin U theoutputmodes. Inthequartercasethereisasetofequivalenceclassesforthematrices (IV),parametrizedbyacontinuous r a parameterθ [0,2π). U ∈ INTERFERENCEFRINGES A 3-photon Mach-Zehnder interferometer is obtained by cascading two balanced tritters, and considering the action of a phaseshiftφonmodek describedbytheunitaryevolution = exp( ın φ). Theoutputstate ψ oftheinterferometer 3 φ 3 out U − | i correspondingtoaninputstate ψ isobtainedas ψ = ψ ,where = (III) (III)representstheoverallevolution in out in φ | i | i U| i U U U U of the system. By feeding the interferometer with the input state 1,1,1 the output probability distributions of obtaining | i (m,n,q)photonsonthethreeoutputmodestaketheform: 29 24 12 16 PF (φ)= cos(φ+π/3) cos(2φ π/3)+ cos(3φ), (3) 1,1,1 81 − 81 − 81 − 81 4 PF (φ)= [1 cos(3φ)], (4) 2,1,0 81 − 2 28 24 12 8 PF (φ)= cos(φ 2π/3)+ cos(2φ π/3)+ cos(3φ). (5) 3,0,0 243 − 243 − 243 − 243 AlltheprobabilitydistributionsPF aresymmetricwithrespecttoanexchangeoftheoutputindices(m,n,q). m,n,q By an analogousprocedure, the output probability distributions for the 4-modes interferometer, obtained by two cascaded quarterdevicesandfedwiththeinputstate 1,1,1,1 ,read: | i 167 168 108 24 45 PF (φ)= + cosφ+ cos(2φ)+ cos(3φ)+ cos(4φ), (6) 1,1,1,1 512 512 512 512 512 47+60cosφ+21cos(2φ)sin4(φ/2) PF (φ)= , (7) 2,2,0,0 128 3sin4(φ) PF (φ)= , (8) 3,1,0,0 128 3(5+3cosφ)sin6(φ/2) PF (φ)= , (9) 4,0,0,0 64 [17+15cos(2φ)]sin2φ PF (φ)= . (10) 2,0,1,1 256 Asforthe3-modeinterferometerfedwiththe 1,1,1 state,theoutputprobabilitydistributionsPF ofthe4-modecaseare | i m,n,q,p symmetricwithrespecttoanexchangeoftheoutputindices(m,n,q,p). PHASEESTIMATIONWITH3-MODEAND4-MODEINTERFEROMETERS Letusconsiderthe3-modeandthe4-modeinterferometersobtainedbycascadingtwotritter(quarter)devices,fedwith the input state 1,1,1 (1,1,1,1 ). We calculate the quantum Fisher information associated to the adopted probe state, and the | i | i classical Fisher information obtained with photon-counting detection. We then discuss an adaptive 3-step phase estimation protocolabletoachieveoptimalperformanceforanunknownphaseshiftlyinginthefullphaseinterval,thatis,abletosaturate thequantumFisherinformationofthescheme. QuantumFisherinformation In the context of parameter estimation, the quantum Fisher information H represents the maximum amount of infor- φ mation which can be extracted on the phase φ from a family of states ̺ , according to the quantum Crame´r-Rao bound: φ δφ (MH )−1/2. When the family of states ̺ is obtained from unitary evolution = exp( ıGφ) of an input pure φ φ φ ≥ U − state ψ as̺ = ψ ψ †,thequantumFisherinformationisevaluatedasH =4 ψ (∆G)2 ψ [2]. Inthecaseofthe | 0i φ Uφ| 0ih 0|Uφ φ h 0| | 0i 3-modeinterferometerfedwitha 1,1,1 inputstate,thequantumFisherinformationreads: | i 16 HF =4 1,1,1 (III)†(∆n )2 (III) 1,1,1 = , (11) φ h |U 3 U | i 3 where n = G isthe generatorofthe phaseshiftonmode k . Inthecase ofthe 4-modeinterferometerfed witha 1,1,1,1 3 3 | i inputstate,thequantumFisherinformationreads: HF =4 1,1,1,1 (IV)†(∆n )2 (IV) 1,1,1,1 =6, (12) φ h |U 4 U | i wheren =Gisthegeneratorofthephaseshiftonmodek . 4 4 3 ClassicalFisherinformation WhenfixingthedetectionapparatustothesetofPOVMoperators Π ,themaximumamountofinformationwhichcanbe x { } extractedonthephaseφfromafamilyofstates̺ measuredby Π isquantifiedbytheCrame´r-Raoboundδφ (MI )−1/2, φ x φ { } ≥ whereI istheclassicalFisherinformation: φ I = p(xφ)[∂ logp(xφ)]2, (13) φ φ | | x X wherep(xφ)istheconditionalprobabilitydistributionofthemeasurementoutcomes. Inthecaseofthe3-modeinterferometer | fedwitha 1,1,1 inputstate,whenwechoosephoton-countingΠ = m,n,q m,n,q asthemeasurementapparatus,we m,n,q | i | ih | obtain: 2 2sin2 3φ √3sin φ +cos φ +2cos 3φ IF = 16 3cos2 3φ + 2 2 2 2 φ 9  2 6√3sin(φ)+(cid:16)3co(cid:17)s((cid:16)2φ)+4(cid:16)cos(cid:17)(3φ)+6(cid:16) √(cid:17)3sin(φ)(cid:16)+1(cid:17)c(cid:17)os(φ)+14 (cid:18) (cid:19) − (14)   (cid:0)2 (cid:1) sin(φ)+sin(2φ) 4sin(3φ)+√3cos(φ) √3cos(2φ) + − − . 12√3(cid:0)sin(φ) 6√3sin(2φ) 12cos(φ) 6cos(2φ)+16cos(3(cid:1)φ)+29! − − − Ananalogousresultisobtainedforthe4-modeinterferometer. Three-stepbayesianadaptiveprotocol We have also developedand simulated an adaptive phase estimation protocoltailored to reach asymptoticallythe quantum Crame´r-Rao bound of the input state for all values of the phase φ. We adopt a Bayesian approach [3], which is based on invertingtheprobabilitydistributionofthemeasurementoutcomesaccordingtotheBayesformula. Bayesiananalysispresent therelevantpropertiesofbeingasymptoticallyoptimalandunbiased,thatis,theestimatedvalueconvergestothetruevalueof theparameterwithuncertaintyreachingthequantumlimitforlargemeasurementsM. Furthermore,theoutputdistributionfor the phaseφ presentsa fast convergenceto a Gaussian shape, and the erroron the estimationprocesscanbe directlyretrieved fromtheoutputdistributionforφ[4]. Inthecaseofthe3-modeinterferometerunderanalysis,theprobabilitydistributionofM repeatedphoton-countingmeasurementoutcomesP( N(i),N(i),N(i) M φ)isinvertedtoobtainaconditionaldistributionfor { 1 2 3 }i=1| thephaseφ: P(φ) P(φ N(i),N(i),N(i) M )= P( N(i),N(i),N(i) M φ), (15) |{ 1 2 3 }i=1 { 1 2 3 }i=1| N where isanormalizationconstant,evaluatedas: N 2π = dφP( N(i),N(i),N(i) M φ)P(φ). (16) N { 1 2 3 }i=1| Z0 Here, P(φ) is the phase probability distribution which represents the a-priori knowledge on the value of the phase φ. Fi- nally, the estimated value of the phase φ and the associated variance σ2 are obtained from the conditional distribution est est P(φ N(i),N(i),N(i) M )as: |{ 1 2 3 }i=1 φ = dφφP(φ N(i),N(i),N(i) M ), (17) est |{ 1 2 3 }i=1 ZΩ σ2 = dφ(φ φ )2P(φ N(i),N(i),N(i) M ), (18) est − est |{ 1 2 3 }i=1 ZΩ whereΩisthephaseinterval. Morespecifically,weconsideredthecaseinwhichthereismaximumignoranceonthevalueof thephaseφ,correspondingauniformdistributionP(φ)intheintervalΩ=[ π/3,5π/3). − Thestrategyanalyzedinthepaperisanadaptiveprotocol,sinceitisbasedondrivinganadditionalphaseψaccordingtothe resultsofeachstepoftheexperiment.Theprotocolisdividedinthreemainsteps: