Unbounded quantum Fisher information in two-path interferometry with finite photon number

0.36 MB·English

by  Y. R. Zhang

Unbounded quantum Fisherinformationintwo-path interferometry withfinite photon number
Y. R. Zhang1,∗ G. R. Jin2, J. P. Cao1, W. M. Liu1, and H. Fan1†
1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Department of Physics, Beijing Jiaotong University, Beijing 100044, China
(Dated:January17,2013)
The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in
accordancetotheCrame´r-Raobound. WeindicatethatonlysuperposedNOONstatesbysimultaneousmea-
surementscanachievethemaximumquantumFisherinformationwithformhNˆ2iforagivenphotonnumber
distributionbya POVM inlineartwo-path interferometerphase measurement. Wepresent a seriesof speci-
fiedsuperposed stateswithinfinitequantum Fisherinformationbutwithfiniteaveragephotonnumbers. The
advantage ofthisunbounded quantumFisherinformationwillbebeneficial tomanyapplicationsinquantum
3 technology.
1
0 PACSnumbers:42.50.St,42.50.Dv,42.50.Ex,42.50.Lc
2

n
I. INTRODUCTION thatthe optimalmeasurementscheme can be expressedbya
a
group of compatible observables. We confirm that the max-
J
6 Precise interferometric measurement plays a key role in imum QFI for a definite photon number probability distri-
1 manyscientificandtechnologicalapplications,suchasquan- bution, written as Nˆ2 , is saturated with a superposition of
h i
tummetrology,imaging,sensingandinformationprocessing NOONstatesinMMZI.Inparticular,wepresentasuperposi-

] [1]. The fundamental sensitivity bounds of phase measure- tionofNOONstateswithaspecifiedprobabilitydistribution
h
mentintheMach-Zehnderinterferometer(MZI),asshownin andafiniteaveragephotonnumber. WefindthatQFIforthis
p
- Fig.1(a), areofbroadinterestforthoseareas. For N identi- state can be infinite, i.e. an arbitrary high phase sensitivity
nt caluncorrelatedparticles,theerroraboutthephasemeasured is obtained by Crame´r-Rao inequality while it still has a fi-
a in MZI on the average decreases as 1/ Nˆ 1/2 which is the niteaveragephotonnumber.Apparently,thisphasesensitivity
u shot-noise limit (SNL). In 1980s, it washpoiinted out that by doesnotviolateHeisenberguncertaintyrelation,butseemsto
q usingcoherentlighttogetherwithsqueezedvacuumwecould breakHL.
[
beatSNL[2]. ItisalsoshownthatusingNOONstate[3]and

5 quantumentanglementallowsinterferometricsensitivity that
Nevertheless, CRB is only asymptotically tight for in-
v also surpasses this limit. Instead of SNL, the ultimate limit
finitely many trials of measurements under the unbiased es-
0
imposedbyquantummechanicsistheHeisenberglimit(HL)
9 timateassumption[14,15],anditincludesnopriorinforma-
withagenerallyacceptedform1/ Nˆ [4,5].Experimentsex-
9 tionofthephaseprobabilitydistribution[16]. Therefore,HL
h i
2 ploringthose topicshave beenperformedin varioussystems canmerelybeviolatedundersomeveryspecialsituation,for
5. withphotons[6,7],ions[8],cold-atoms[9]andBose-Einstein example the phase is in a restricted neighborhoodnear zero
condensates [10, 11]. However, it seems to be indicated in
0 asthecaseofdistinguishingstates, andthelimitobtainedby
some worksthat the HL with form 1/ Nˆ could be violated
1 CRB isoflittleusewhenweconsidertheresourcesrequired
h i
1 whiletheexperimentsareperformedwithafluctuatingnum- forthepriorinformationoftheproblem[17–19].Similarthe-
: ber of particles [12, 13]. Then, HL for an unfixed number
v oreticalresultsbasedonthequantumspeedlimit[20]arealso
of particles has become a focus of great attention as well as
i obtainedinRef.[21]. AlthoughCRB cansometimesgrossly
X contentioninalltheparameterestimationschemes.
underestimatethe achievableerror, QFI andCRB are still of
r In this Letter, we investigatethe quantumFisher informa- great use for many applications in quantum techniques such
a
tion(QFI)intwokindsoftwo-pathinterferometersforafluc-
as parameterestimation fornoisy systems [22, 23], limits of
tuatingnumberofphotonssincethewell-knownCrame´r-Rao
imaging [24], distinguishability of states [25] and quantum
bound (CRB) whose leading role is played by QFI is often
Zenoeffect[26]. Relatedtopicsandrecentdevelopmentscan
usedtoestimatesensitivityofphasemeasurement.Toachieve
befoundinRefs.[27–36].
themaximumQFIandthelowestCRB,weshouldusetheop-
timalstates, andimplementthe optimalmeasurements(opti-
malPOVM).We investigatethemeasurementappliedintwo Thispaperis organizedas follows. In Sec. II, we demon-
kindsof linear two-pathinterferometers, as shown in Fig. 1. stratetheoptimalmeasurementschemebysimultaneousmea-
Note that Fig. 1(a) is for MZI, and Fig. 1(b) is a modified surementsin lineartwo-pathinterferometers. In Sec. III, we
Mach-Zehnderinterferometer(MMZI)where thefirst beam- obtainthemaximumQFIintheoptimalmeasurementscheme.
splitterisreplacedbyan entangledphotonsource. We show In Sec. IV, we discuss the measurement scheme written as
oneoperator.InSec.V,wepresentasuperpositionofNOON
stateswithinfiniteQFIandfiniteaveragephotonnumber.Su-
perpositionsof NOON states in MMZI and dual Fock states
∗[email protected] inMZIarecomparedinSec.VI.Finally,aconclusionisgiven
†[email protected] inSec.VII.
2
cial example in which phase transformation repeats n times
duringeachmeasurement.
(a)
FIG.2.(coloronline).Generalphaseestimationprocedureinvolving
statespreparation,phasetransformationandmeasurement.
In this section, our discussion is focused on the measure-
ment part. We perform photon number counting measure-
mentsattwoportsoflineartwo-pathinterferometerstomea-
sure the output state Ψ , respectively, written as nˆ and
(b) | iout a
nˆ . It is obviousthat nˆ and nˆ are compatibleobservables,
b a b
[nˆ ,nˆ ] = 0. Thus, measurements nˆ and nˆ can be per-
FIG.1. (coloronline). (a) Setupof Mach-Zehnder Interferometer. a b a b
formedsimultaneously.Thispairofmeasurementscannotbe
|Ψi istheinput state. Bothbeam-splittersare 50 : 50. Theun-
in expressedasnˆ nˆ becauseweobtaintwoindependentre-
known phase shiftφ istobeestimated. Acontrollable phase θ, in a b
⊗
theotherarm,isusedintheexperimenttoachievethebestprecision. sultsthatcanbewrittenasacolumnmatrix, (na,nb)T,from
Uˆ isthephase operator and Uˆ |Ψi istheintermediate state. simultaneousmeasurementsofanytwo-modeFockstate,
φ−θ φ−θ
Passingthroughthesecondbeam-splitteristheoutputstate|Ψi .
out
j
Weperformphotonnumbercountingmeasurementsattwoports of nˆ
a C j+m j m =
tMivaeclyh.-Z(be)hSndeteurpIonftemrfoerdoifimeedteMr atochm-ZeeahsnudreerthIneteorufetproumt settaetre.,Trheespfiercs-t (cid:18) nˆb (cid:19)Xj mX=−j j,m| ia| − ib
beam-splitterisreplacedbyanentangledphotonsource.Weassume m=j
j+m
thattheentangledphotonsourcecanproduceNOONstatesandsu- C j+m j m , (3)
perposedNOONstates. Xj mX=−j j,m(cid:18)j−m(cid:19)| ia| − ib
where j C 2 = 1. However, the operator
j m=−j| j,m|
II. SIMULTANEOUSMEASUREMENTS nˆa nˆPb onPly obtains a result na nb, and is useless to de-
⊗ ×
scribea pairofsimultaneousmeasurements. Herewe define
Two-pathopticalinterferometersarewidelystudiedtoesti- simultaneous measurements as using several compatible ob-
mateacompletelyunknownopticalphaseφfromthephoton servables to performa united measurementfor a state at the
numbersobservedintwoportsoftheoutput.Hereweexpress same time. In fact, it is sufficient and necessary for them to
the quantummechanicsof two-pathinterferometersin terms bemeasurableconsecutively,suchthatthejointprobabilityof
of the spin-N/2 algebra of the Schwinger representation, as the resultsdoesnotdependonthe orderof themeasurement
showninRef.[37], [1]. Moreover,weshowthesepairsofsimultaneousmeasure-
mentsareequivalent,
Jˆ =(aˆ†ˆb+ˆb†aˆ)/2, Jˆ =(aˆ†ˆb ˆb†aˆ)/2i,
1 Jˆ3 =(aˆ†aˆ 2ˆb†ˆb)/2,− (1) nˆa nˆb Nˆ ∆ˆ , (4)
− (cid:18) nˆb (cid:19)⇔(cid:18)nˆa (cid:19)⇔(cid:18) ∆ˆ (cid:19)⇔(cid:18)Nˆ (cid:19)
whereaˆandˆbaretheannihilationoperatorsofthetwopaths.
EigenstatesofJˆ canbedefinedasusualbyFockspacerep- whereNˆ = nˆa+nˆb istotaloutputphotonnumber,and∆ˆ =
3
nˆ nˆ isthedifferencebetweenthetwoports.
resentationas, b a
−
Then, we show that the simultaneous measurements in
j,m = j+m j m . (2) two-pathinterferometerscan be describedby two equivalent
a b
| i | i | − i
POVMs,
The phase shift of φ between the arms of the interferom-
eter can be expressed by a unitary transformation Uˆφ = Eˆ(N,∆)⇔Eˆ′(na,nb) (5)
exp( iφJˆ3). The unitary transformation of a beam-splitter N2,∆2 N2,∆2 ⇔ N−2∆ a N+2∆ b N+2∆ b N−2∆ a.
isgiv−enbyBˆ =exp(iπJˆ1/2). Eˆ(cid:12)(cid:12)(n ,n(cid:11)(cid:10)) and(cid:12)(cid:12)Eˆ(N(cid:12)(cid:12) ,∆)(cid:11)ar(cid:12)(cid:12)ecom(cid:11)ple(cid:10)teinth(cid:12)(cid:12)e(cid:10)senset(cid:12)(cid:12)hat,
Then, we consider that a linear phase estimation scheme { a b } { }
in two-path interferometers can be divided to three parts:
N
states preparation, phase transformation and measurement Eˆ(n ,n )= Eˆ(N,∆)=Iˆ, (6)
a b
[4, 38, 39], see Fig. 2. The procedure in Ref. [40] is a spe- nXa,nb XN ∆X=−N
3
whereIˆisaunitoperator. Insubsequentsections,weusethe the situation that the photon number is fixed, FI of ψ is
N
equivalentsimultaneousmeasurements Eˆ(N,∆) forsimple- FQ[ψN ] = N2 ifandonlyif ψN isaNOONstate,|andiFI
| i | i
ness. ofanyotherstateislessthanN2. Therefore,wehave,
F [Ψ ] maxF [Ψ ]= Nˆ2 , (12)
III. FISHERINFORMATIONANDCRAME´R-RAOBOUND Eˆ(N,∆) | i ≤ |Ψi Q | i h i
where the equality can only be achieved when Ψ is a su-
| i
Toestimatetheparameterφ,aPOVM Eˆ(ξ) isperformed perposition of NOON states. Moreover this is QFI, and
ontheoutputstate,andφisinferredfrom{theme}asurementre- Eˆ(N,∆) istheoptimalPOVM.AccordingtotheCrame´r-
{ }
sultsξ. ThefinalmeasurementresultsarerelatedwithFisher Raoinequality,weobtainCRBas
information(FI)whichtakestheform,
∆φ 1/ Nˆ2 1/2. (13)
∂lnP(ξ φ) 2 ≥ h i
FEˆ(ξ)[|Ψφi]= P(ξ|φ)(cid:18) ∂φ | (cid:19) , (7) Therefore, we find the optimal measurement and optimal
Xξ states for the maximum QFI in linear two-path interferome-
ters. WeremarkthatwecancalculateFIbyusingeitherinter-
whΨhφer|Eeˆ(|Ψξ)φ|Ψi φ=i isUˆφth|Ψeip,roabnadbilliikteyliohfoomdeafusunrcetimonenPt r(eξs|uφl)ts=ξ mnoeddiiaffteersetnacteeUfˆoφr|ΨthieorresfiunlatlwohuetpnuPtO|ΨViMoutiEnˆ(Fξi)g.1is. Iotpmtiamkaels,
given that the true system phase is φ. CRB places a limit { }
aspointedoutinRef.[38].
onthemeanofthesquareoftheunbiasedphaseerrorviathe
HereweemphasizethatCRBisasymptoticallytightforun-
Crame´r-Raoinequality,
biasedestimation with infinitetrials ofmeasurements, and it
does not provide a rigorous basis for HL when considering
∆φ 1/ F [Ψ ]. (8)
≥ Eˆ(ξ) | φi thepriorprobabilitydistributionofoverallphaseshift. Even
q
so, QFI is still an extremely important quantity in quantum
We note that FI is additive, and for M repeated trials of
physics. When the system interacts with the environment,
the same measurement, the FI is M F [Ψ ], which
× Eˆ(ξ) | φi such as quantum noise and photon losses, QFI changes and
leads to CRB on the phase uncertainty, ∆φ 1/(M
F [Ψ ])1/2. QFIforstate Ψ isthemaxim≥um, × theprecisionofphasesensitivityisaffected. Itisshownthat
Eˆ(ξ) | φi | φi QFIflow,writtenas∂tFQ[Ψ ],directlycharacterizesthenon-
| i
Markovianityofthequantumdynamicsofopensystems[42].
F [Ψ ]:= max F [Ψ ], (9)
Q | φi {Eˆ(ξ)} Eˆ(ξ) | φi Therefore,QFIcanalsobeevaluatedbythephasesensitivity
in theexperimenttomeasurethe non-Markovianityofquan-
whichissaturatedbyaparticularPOVM[25,41]. Inthefol-
tumopensystems. Moreover,anenvironment-assistedpreci-
lowingdiscussion, weshowthatthemeasurementwithform
sionmeasurementisalsoavailable[43].
Eˆ(N,∆)inEq.(5)saturatestheQFIwhenstate Ψ isasu-
| i
perposedNOONstate.
When total photon number fluctuates, we consider a pure IV. MEASUREMENTASONEOBSERVABLEESTIMATOR
stategeneratedfromtheentangledphotonsourceinFig.1(b),
If a phase shift φ is measured by the outcomes of an ob-
Ψ = P(N)ψ , (10)
| i | Ni servable estimator Aˆ in linear two-path interferometers, the
XN p
estimatorshouldbeexpressedasabinaryfunctionof nˆ and
a
whereP(N)referstotheprobabilitydistributionoftotalpho- nˆ ,
b
ton number N, and ψ = N/2 p(m)N/2,m .
| Ni m=−N/2 | i Aˆ=f(nˆ ,nˆ ), (14)
When the phase is φ, we writePthe probapbility of any out- a b
put result (N,∆)T as P(N,∆φ) = ΨEˆ(N,∆)Ψ , whichisthecaseconsideredinRefs.[44–46].Becausesimul-
out out
| h | | i
in which Ψ = BˆUˆ Ψ is the output state. For the taneous measurements are complete for the two-mode Fock
out φ
| i | i
output result N = n + n , the probability is P(N φ) = space,andaresufficientforphaseφ,weobtain,
a b
|
N ΨEˆ(N,∆)Ψ . Theconditionalprobability
iPs P∆=(∆−NNo,uφth) =| P(N,∆| φi)o/uPt (N φ). Because [Bˆ,Nˆ] = FAˆ[|Ψφi]≤FEˆ′(nˆa,nˆb)[|Ψφi], (15)
| | |
[Uˆφ,Nˆ] = 0, the probability distribution of photon num- wheretheequalholdsifthemeasurementAˆofthestate Ψφ
berdoesnotchangeafterbeam-splitterorphase, P(N φ) = is a sufficient statistic for underlying parameter φ [47]|; foir
|
P(N)and∂P(N φ)/∂φ=0. Hence,FIcanbewrittenas, example,anobservableAˆ=nˆ +√2nˆ issufficientforQFI
| a b
foranystateinthetwo-modeFockspace.
F [Ψ ]= P(N)F [ψ ]. (11)
Eˆ(N,∆) | i Eˆ(∆) | Ni NotethatoneobservableestimatorthatissufficientforQFI
XN whenconsideringafixedphotonnumbermaybeinsufficient
This means that the FI of the state with a fluctuating pho- when the photon number is fluctuating. For example, if the
ton number equals to the probability summation of each FI totalphotonnumberisfixedandknowntobeN,momentum
ofthesuperposedstatewhichhasafixedphotonnumber. As operator Jˆ [48] is sufficient to achieve the maximum QFI
3
4
when state Ψ generated from the entangled photon source
| i
inMMZIisaNOONstate; butweconfirmthatJˆ isnotthe
3
optimalmeasurementforQFIwhentotalphotonnumberfluc-
tuates.
Weshouldalsonoticethatsomeoperators,forexamplethe
parity operator Πˆ = exp(iπnˆ ) [12, 49], are still usefulin
a a
thequantummetrologyforhighpreciseparameterestimation,
althoughtheycan notsaturate QFI forstates with a fluctuat-
ingphotonnumber. Moreover,whenderivingtheappropriate
formofHL,itisreasonabletoconsiderthesituationthatthe
parameterisestimatedfromtheresultsofquantummeasure-
mentwhichisasufficientstatisticfortheparameter.
(a)
V. ASUPERPOSITIONOFNOONSTATESWITH
ARBITRARYHIGHPHASESENSITIVITYANDFINITE
AVERAGEPHOTONNUMBER
Aswehavealreadyshown,asuperpositionofNOONstates
willalwayssaturatethemaximumQFIinEq.(12). Herewe
present an interesting example of such a state which has an
infinite QFI when the average photon number is finite. We
considerasuperpositionofNOONstatesasthefollowing,
∞
1 1 N 0 + 0 N
Ψ(x) = | ia| ib | ia| ib,(16) (b)
| i ζ(x) √Nx √2
NX=1
where x (1,+p ), ζ(x) = ∞ 1/Nx is Riemann Zeta FIG.3.(coloronline).(a)Phaseuncertainty∆φagainstaveragepho-
∈ ∞ N=1 tonnumberhNˆi.Greendash-dotlineisfortheSNL1/hNˆi1/2,black
function, and here it is the noPrmalization factor. The prob-
ability of each NOON state, (N 0 + 0 N )/√2, is dash-dotlineisforconventionalHL1/hNˆi,purpledash-dotlinede-
1/(ζ(x) Nx)forphotonnum| beira|Ni.bWh|eniax| ib2 the av- notes CRB of TMSV dual Fock state 1/(hNˆ2i+ 2hNˆi)1/2, blue
× ≤ dashlineindicatestheminimumCRBinTMSVstates1/(2hNˆ2i+
erageofphotonnumberisinfinite. Wenextconsiderthecase
2hNˆi)1/2,andredsolidlineisforCRBofsuperposedNOONstates
x=3,thephotonnumberonaveragecanbecalculatedas,
inEq.(16). WhenhNˆi ≥ 1.369, stateinEq.(16)obtainsaninfi-
∞
1 N ζ(2) niteQFIandazeroCRB.(b)Phaseuncertainty∆φagainstaverage
hNˆi= ζ(3) N3 = ζ(3) ≈1.369. (17) photonnumberhNˆi. RedsolidlineisfortheCRBobtainedbythe
NX=1 superposition of NOON states similar as Eq. (16), and black solid
This means that a superposed state, Ψ(3) in Eq. (16), has lineisforthesuperpositionofdualFockstates.WhenhNˆi<2.737,
afiniteaveragephotonnumber,while|theaiverageofsquared QFI for NOON state is larger than dual Fock state, while when
photon numbers, Nˆ2 , can be infinite, as calculated in the hNˆi≥2.737,bothQFIsareinfinite.
h i
following,
1 ∞ N2 ζ(1)
Nˆ2 = = , (18) VI. SUPERPOSITIONSOFNOONSTATESANDDUAL
h i ζ(3) N3 ζ(3) →∞ FOCKSTATES
NX=1
where we have used the fact, ζ(1) . For the general
case,both Nˆ and Nˆ2 1/2 aredepe→nde∞ntonx,therelation DualFockstate, |Nia|Nib, iscloselyrelatedwithNOON
h i h i state and is also widely used in the optical interferometric
betweenthephotonnumberonaverageandCRBofphasees-
quantum measurement. For example, when input is a dual
timationforstateEq.(16),∆φ=1/ Nˆ2 1/2,isshowninFig.
3(a). WecanseefromFig.3(a)thathCRiBreacheszerowhen Fockstate|1ia|1ib inMZI,theintermediatestateisaNOON
state withphotonnumberN = 2, see [6]. We nextcompare
Nˆ approachesabout1.369,andas Nˆ increases,CRB re-
h i h i QFIof superpositionofdualFockstateswith thatofNOON
mainsaszero.
statesintwoexamples. WeemphasizethatNOONstatesand
NotethatinRefs.[4,50],afollowingstatewasproposed,
dual Fock states are compared in different interferometers.
M NOON states are generated by the entangled photon source
1
Φ =A m ,(M 1,A 6/π2).(19) inFig.1(b),whiledualFockstatesaretheinputstatesofMZI
ssw
| i m+1| i ≫ ≃
mX=0 p inFig.1(a). ThereasonwhyweconsiderdualFockstatesis
It is shown thatQFI of this state can be arbitraryhigh when thatgeneratingdualFockstates is easierthan generatingthe
the average photon number is infinite, while, in our case as NOONstateswhenphotonnumbergrowshigh.
Eq.(16),theaveragephotonnumbercanbefinite. The first example: we considerthe inputstate as the two-
5
mode squeezed vacuum (TMSV), which is a superposition as the effects of entanglement as well as superposition. A
of dual Fock states, Ψ = ∞ p ( Nˆ )N N , similar discussion can also be found in Ref. [13]. Another
| iin N=0q N h i | ia| ib interestingresultisthatthestateswithinfiniteQFImightnot
wherepN(hNˆi)=(1−thNˆi)tNhNˆPiwiththNˆi =1/(1+2/hNˆi), have anyZeno dynamicssince the quantumZeno time scale
see [12]. When a dualFockstate Ψ in = N a N b passes expressedintermsofQFI,writtenas τQZ = 2/ mFQ [26],
| i | i | i
throughthefirstbeam-splitter,theintermediatestateis, approacheszero. p
N Ck (2N 2k)!(2k)!
Ψ = N −
| i pN!2N+1/2 × VII. CONCLUSIONANDDISCUSSION
kX=0
[( 1)N 2N 2k 2k + 2k 2N 2k ]. (20)
− | − ia| ib | ia| − ib We propose that the maximum QFI of states with a fluc-
QFIofeachsuperposedstateis(2N 4k)2.Hence,QFIofthe tuating or fixed number of particles takes the form, Nˆ2 in
− h i
intermediatestateEq.(20)canbecalculatedtobe2N2+2N. two-path interferometer. Our result shows the superposed
Then QFI of the TMSV is ∞ p ( Nˆ )(2N2 +2N) = NOONstatesbysimultaneousmeasurementsachievethisop-
Nˆ 2 +2 Nˆ . IncomparisoPn,Nf=or0aNsupherpiositionofNOON timalQFIinMMZI.Wealsopresentaspecifiedsuperposition
h i h i ofNOONstateswithprobabilitydistributionrelatedwithRie-
states with a same probability distribution as TMSV, Ψ =
| i mannZetafunction. ThisstatehasaninfiniteQFIbutwitha
∞N=0qpN(hNˆi)(|2Nia|0ib +|0ia|2Nib)/√2, the QFI is finiteaveragephotonnumbervialineartwo-pathinterferom-
P∞ p ( Nˆ )4N2 = 2 Nˆ 2 +2 Nˆ . Thisstate saturates eter measurements. We also compare the case of dual Fock
tPheNm=0axiNmuhmiQFI for thihs pirobabihlityi distribution of total state in MZI with that of NOON state in MMZI. Our work
photonnumber. CRBs of these two states are shown in Fig. presentsthattheadvantageofthisunboundedquantumFisher
3(a). informationcanbeobtainedbyonlyafewsuperposedterms,
Second example: similar as the state in Eq. (16), we con- whichwillbebeneficialtothefuturedevelopmentofquantum
siderasuperpositionofdualFockstateswiththesameprob- technology.
ability distribution of the photonnumber, Ψ(x) = We assume that the entangled photon source in Fig. 1(b)
dualFock
∞ N N / ζ(x) Nx. When x|= 3,ithe average could produce superposed NOON states. One area for fu-
PphoNt=on1|numiab|eriibs pNˆ =×2×ζ(2) 2.737, and QFI can be ture research is to prepare superposed NOON states when
h i ζ(3) ≈ someachievementshavebeenmadefortheformationofhigh-
calculated as, F (φ) = ∞ 2×ζ(1) + 2×ζ(2) . NOONstates[51,52].
Q N=1h ζ(3) ζ(3) i → ∞
Subsequently,itiswithazPeroCRBsimilarasthestateinEq.
(16). The results of these two states are plotted in Fig. 3(b).
Notethatforprobabilitydistribution1/(ζ(x) Nx),thepho- ACKNOWLEDGMENTS
×
ton numberof NOON state is 2N in comparingwith that of
dualFockstate. Wecanseethatwhen Nˆ < 2.737,QFIfor WethankLorenzoMaccone,SethLloydandMichaelHall
h i
NOONstatecaseislargerthancaseofdualFockstate;when for useful discussions. The authors are also grateful to
Nˆ 2.737,CRBsofbothstatesarezeroes. Tian-Si Han and Si-Wen Li for discussions. We also ac-
h i≥
Generally, for any pure intermediate state written as Eq. knowledge Referees for their useful suggestions which help
(10), the form of superposed state ψ and the distribution us a lot to improve our work. This work is supported by
N
| i
ofphotonnumberP(N)arebothimportanttoachieveahigh NSFC (10934010, 10974233, 10974247), “973” program
QFIinlineartwo-pathinterferometers,whichcanberegarded (2010CB922904,2011CB921500).
[1] H.M.Wiseman,andG.J.Milburn,QuantumMeasurementand Lett.104,073604(2010).
Control(CambraidgeUniv.Press,NewYork,2010). [10] G.B.Jo,Y.Shin,S.Will,T.A.Pasquini,M.Saba,W.Ketterle,
[2] C.M.Caves,Phys.Rev.D23,1693(1981). andD.E.Pritchard,Phys.Rev.Lett.98,030407(2007).
[3] B.C.Sanders,Phys.Rev.A40,2417(1989). [11] A.Sørensen,L.M.Duan,J.I.Cirac,andP.Zoller,Nature409,
[4] Z.Y.Ou,Phys.Rev.A55,2598(1997). 63(2001).
[5] V.Giovannetti, S.Lloyd, andL.Maccone, Science306, 1330 [12] P.M. Anisimov, G.M. Raterman, A.Chiruvelli, W.N. Plick,
(2004). S.D.Huver, H.Lee, andJ.P.Dowling, Phys.Rev. Lett.104,
[6] G.Y.Xiang,B.L.Higgins,D.W.Berry,H.M.Wiseman,and 103602(2010).
G.J.Pryde,Nat.Photon.5,43(2011). [13] A´.Rivas,andA.Luis,arXiv:1105.6310v2.
[7] T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. [14] V.Giovannetti,S.Lloyd,andL.Maccone,Nat.Photon.5,222
Takeuchi,Science316,726(2007). (2011).
[8] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. [15] D. W.Berry, M. J. W.Hall, M. Zwierz, and H.M. Wiseman,
Itano,C.Monroe,andD.J.Wineland,Phys.Rev.Lett.86,5870 arXiv:1209.3547v1.
(2001). [16] M.Tsang,Phys.Rev.Lett.108,230401(2012).
[9] M.H.Schleier-Smith,I.D.Leroux,andV.Vuleti´c,Phys.Rev. [17] V.Giovannetti andL.Maccone, Phys.Rev.Lett.108, 210404
6
(2012). [36] U.Dorner,R.Demkowicz-Dobrzanski, B.J.Smith,J.S.Lun-
[18] M. J.W. Hall, D.W. Berry, M. Zwierz, and H.M. Wiseman, deen, W.Wasilewski,K.Banaszek, andI.A.Walmsley,Phys.
Phys.Rev.A85,041802(R)(2012). Rev.Lett.102,040403(2009).
[19] M. J. W. Hall, and H.M Wiseman, New J. Phys.14, 033040 [37] B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944
(2012). (1995).
[20] N.Margolus,andL.B.Levitin,PhysicaD120,188(1998). [38] V.Giovannetti,S.Lloyd,andL.Maccone,Phys.Rev.Lett.96,
[21] V.Giovannetti,S.Lloyd,andL.Maccone,Phys.Rev.Lett.108, 010401(2006).
260405(2012). [39] M.Zwierz,C.A.P´erez-Delgado,andP.Kok,Phys.Rev.Lett.
[22] B.M. Escher, R. L. deMatos Filho, and L.Davidovich, Nat. 105,180402(2010).
Phys.7,406(2011). [40] B.L.Higgins,D.W.Berry,S.D.Bartlett,H.M.Wiseman,and
[23] B.M. Escher, L.Davidovich, N. Zagury, and R. L. de Matos G.J.Pryde,Nature450,393(2007).
Filho,Phys.Rev.Lett.109,190404(2012). [41] P. Hyllus, L. Pezz´e, and A. Smerzi, Phys. Rev. Lett. 105,
[24] C.A.Pe´rez-Delgado,M.E.PearceandP.Kok,Phys.Rev.Lett. 120501(2010).
109,123601(2012). [42] X.M.Lu,X.G.Wang,andC.P.Sun,Phys.Rev.A82,042103
[25] S.L.Braunstein, and C. M. Caves, Phys. Rev. Lett.72, 3439 (2010).
(1994). [43] G.Goldstein,P.Cappellaro,J.R.Maze,J.S.Hodges,L.Jiang,
[26] A.Smerzi,Phys.Rev.Lett.109,150410(2012). A.S.Sørensen,andM.D.Lukin,Phys.Rev.Lett.106,140502
[27] A. Politi, J. C. F. Matthews, M. G. Thompson, and J. L. (2011).
O’Brien,IEEEJ.SelTop.Quant.Electron15,1673(2009). [44] H.F.Hofmann,Phys.Rev.A79,033822(2009).
[28] H. Cable, and G. A. Durkin, Phys. Rev. Lett. 105, 013603 [45] H.Lee,P.Kok,andJ.P.Dowling,J.Mod.Opt.49,2325(2002).
(2010). [46] J.P.Dowling,Phys.Rev.A57,4736(1998).
[29] L.K.Shalm,R.B.A.Adamson, andA.M.Steinberg,Nature [47] I.A.Ibragimov,andR.Z.Has’minskii,StatisticalEstimation:
457,67(2009). Asymptotictheory(Springer,NewYork,1981).
[30] L.Pezz´e,andA.Smerzi,Phys.Rev.Lett.102,100401(2009). [48] M.Hillery,andL.Mlodinow,Phys.Rev.A48,1548(1993).
[31] D.Braun,andJ.Martin,Nat.Comm.2,223(2011). [49] R.A.Campos, C.C.Gerry,andA.Benmoussa, Phys.Rev.A
[32] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, 68,023810(2003).
R. Ramponi, and R. Osellame, Phys. Rev. Lett.105, 200503 [50] J.H.Shapiro,S.R.Shepard,andN.C.Wong,Phys.Rev.Lett.
(2010). 62,2377(1989).
[33] H.F.Hofmann,andT.Ono,Phys.Rev.A76,031806(2007). [51] ItaiAfek,OronAmbar,andYaronSilberberg,Science328,879
[34] M.Napolitano,M.Koschorreck,B.Dubost,N.Behbood,R.J. (2010).
Sewell,andM.W.Mitchell,Nature471,486(2011). [52] H.Cable,F.Laloe¨,andW.J.Mullin,Phys.Rev.A83,053626
[35] A´.Rivas,andA.Luis,Phys.Rev.Lett.105,010403(2010). (2011).