Measuring the photon distribution by ON/OFF photodectors

0.44 MB·English

by  M. Genovese

Measuring thephoton distribution by ON/OFFphotodectors
Marco Genovese, Marco Gramegna, Giorgio Brida
Istituto Elettrotecnico Nazionale, IEN, Galileo Ferraris, Torino, Italia
Maria Bondani, Guido Zambra, Alessandra Andreoni
INFM-CNRandDipartimentodiFisicaeMatematica, Universita` degliStudidell’Insubria, Como, Italia
Andrea R. Rossi, Matteo G. A. Paris
Dipartimento di Fisica dell’ Universita` di Milano, Italia.
(Dated:February1,2008)
Reconstruction of photon statisticsof optical statesprovide fundamental information on the nature of any
opticalfieldandfindvariousrelevantapplications. Nevertheless,nodetectorthatcanreliablydiscriminatethe
6
number of incident photons is available. On the other hand the alternative of reconstructing density matrix
0
by quantum tomography leads tovarious technical difficultiesthat areparticular severe inthe pulsed regime
0
(wheremodematchingbetweensignalanlocaloscillatorisverychallenging).Evenifon/offdetectors,asusual
2
avalanchePhotoDiodesoperatinginGeigermode,seemuselessasphotocounters,recentlyitwasshownhow

n reconstructionofphotonstatisticsispossiblebyconsideringavariablequantumefficiency.Herewepresentex-
a perimentalreconstructionsofphotonnumberdistributionsofbothcontinuous-waveandpulsedlightbeamsina
J schemebasedonon/offavalanchephotodetectionassistedbymaximum-likelihoodestimation.Reconstructions

6 ofthedistributionforbothsemiclassicalandquantumstatesoflight(assinglephoton,coherent,pseudothermal
2 andmultithermalstates)arereportedforsingle-modeaswellasformultimodebeams. Thestabilityandgood
accuracyobtainedinthereconstructionofthesestatesclearlydemonstratetheinterestingpotentialitiesofthis

1 simpletechnique.
v
5
PACSnumbers:42.50.Ar,42.50.Dv,03.65.Wj
7 Keywords:Reconstructionofquantumopticalstates,photonstatistics
1
1
0 I. INTRODUCTION challengingtaskinthecaseofpulsedopticalfields.
6
Ontheotherhand,thephotodetectorsusuallyemployedin
0
quantumoptics,suchasAvalanchePhotoDiodes(APD’s)op-
/ The evaluation of diagonalelements of the density matrix
h eratinginthe Geigermode[6, 11](thathaverelativelylarge
p forquantumopticalstates,i.e. ofthestatisticaldistributionof quantumefficiencies),arenotsuitedfordistinguishingdiffer-
- thenumberofphotons,providesfundamentalinformationon
t entnumberof incidentphotons, since they have the obvious
n thenatureofanyopticalfieldandfindsvariousrelevantappli-
drawback that the breakdown current is independent of the
a cations,rangingfromstudiesonfoundationsofquantumme-
number of detected photons. The outcome of these APD’s
u chanics[1]toquantuminformation[2],andquantummetrol-
q is either ”off” (no photons detected) or ”on” i.e. a ”click”,
ogy. Despite the importance of photon distribution, photon
: indicating the detection of one or more photons. Actually,
v detectors allowing an effective discrimination among differ-
suchanoutcomecanbeprovidedbyanyphotodetector(PMT,
i entnumberofincidentphotonsarenotyetavailable. Among
X hybrid photodetector, cryogenic thermal detector) for which
the possible candidatesto numberresolvingphoto-detectors,
r thechargecontainedindarkpulsesisdefinitelybelowthatof
a PhotoMultiplierTubes(PMT’s)[3]andhybridphotodetectors theoutputcurrentpulsescorrespondingtothedetectionofat
[4]havethedrawbackofalowquantumefficiency,sincethe
least one photon. Notice that for most high-gain PMT’s the
detectionstartswiththeemissionofanelectronfromthepho-
anodicpulsescorrespondingtothe”no-photons”(”no-click”)
tocathode. Ontheotherhand,solidstatedetectorswithinter-
event can be easily discriminated by a threshold from those
nalgain,inwhichthenatureoftheprimarydetectionprocess
corresponding to the detection of one or more photons. On
ensureshigherefficiency,arestillunderdevelopment.Highly
the other hand, , as we will describe in the next paragraph,
efficientthermalphotoncountershavealsobeenused. How-
these detectors can be used for reconstructing photon statis-
ever, since they operate in cryogenic conditions they are far
tic when measurementsat differentquantum efficiencies are
to allow commonuse [5, 6]. Furthermore,theirefficiencyis
performed.
limitedbytheopticalwindowforenteringthecryostat.
In this paper we present in some details (see [8] for a
Quantum tomography provides an alternative method to shorter summary of these results) experimental reconstruc-
measure photon number distributions [9]. However, the to- tionsofphotonnumberdistributionsofbothcontinuous-wave
mographyofastate,whichhasbeenappliedtoseveralquan- andpulsedlightbeamsinaschemebasedonon/offavalanche
tum states [10], needs the implementation of homodyne de- photodetection assisted by maximum-likelihood estimation.
tection,whichinturnrequirestheappropriatemodematching Reconstructionsofthedistributionforbothsemiclassicaland
ofthesignalwithasuitablelocaloscillatoratabeamsplitter. quantum states of light (as single photon, coherent, pseu-
This technique is therefore, in general, not of simple imple- dothermal and multithermal states) are reported for single-
mentation and, in particular, such mode matching is a very modeaswellasformultimodebeams.
2
II. THEORETICALMETHOD
100 FCoochkerent
Thestatistics ofthe”no-click”and”click”eventsfroman Thermal
Multithermal
on/offdetector,assumingnodarkcounts,isgivenby 10-1
p (η) = (1 η)n̺ , (1) ()i 10-2
0 n
−
Xn 10-3
adnisdtrpib>u0t(ioηn)o=ft1he−qpu0a(nηtu),mwshtaetree̺̺nan=dηhni|s̺t|hneiqisuathnetupmhoetfofin- 10-40 200 400 600 800 1000
Iteration number, i
ciencyof thedetector, i.e. the probabilityof a single photon
toberevealed.Atfirstsightthestatisticsofanon/offdetector
FIG.1: Plotofthetotalerrorε(i)asafunctionoftheiterationnum-
appearsto providequite a scarce piece of informationabout
berforthevariousstateofthefield(correspondingtothestatesex-
the state underinvestigation. However,if the statistics about
perimentally investigated, see next Sections). For the thermal dis-
p (η)iscollectedforasuitablylargesetofefficiencyvalues
0 tributionthevalueofthetotalerrorisplottedateachiterationstep
thentheinformationisenoughtoreconstructthewholepho- (valueonthexaxis),whilefortheotherstatesthevalueofε(i)isthe
tondistribution̺n ofthesignal,uponasuitabletruncationat averageoveransuitablesetofiterations.
n¯ oftheHilbertspace.
The reconstruction of photon distribution through on/off
detectionatdifferentefficiencieshasbeenanalyzed[12]and performedseveralnumericalsimulationsinordertocheckthe
its statistical reliability investigated in some details [13]. In accuracy and reliability of this method by varying the dif-
addition,thecaseoffewandsmallvaluesofη [14]hasbeen ferent parameters. Since the solution of the ML estimation
addressed. However, whilst these theoretical studies found isobtainediteratively,the mostimportantaspecttokeepun-
an application to realize a multichannel fiber loop detector dercontrolis its convergence. Of course, the degreeof con-
[15,16],anexperimentalimplementationofthistechniquefor vergence at a given step can be checked evaluating the log-
reconstructingphotondistributionofa free-propagatingfield likelihoodfunction =logL,
L
isstillmissing.Inviewoftherelevanceofphotondistribution
for applications in quantum information and foundations of K
L= pn0ν (1 p )nν−n0ν .
quantummechanics,ourpurposeistoshowthatareconstruc- ν − ν
tionofthephotondistributionbyusingthistechniquecanbe νY=1
effectivelyrealizedgatheringresultsobtainedfrommeasure- However,a moresuitable parameteris givenbythe totalab-
ment at different quantum efficiencies. As we will see this soluteerroratthei-thiterationi.e.
method leads to excellent results both for free-propagating
continuous-wave(cw)andpulsedlightbeams,forbothsingle- K
mmooddeessteamteisc.lassicalandquantumstates, aswellasformulti- ε(i) =νX=0(cid:12)(cid:12)fν −pν[{̺n(i)}](cid:12)(cid:12) . (3)
(cid:12) (cid:12)
The procedure consists in measuring a given signal by
Indeed,thetotalerrormeasuresthedistanceoftheprobabili-
on/off detection using different values ην (ν = 1,...,K) of tiesp [ ̺(i) ],ascalculatedatthei-thiteration,fromtheac-
the quantum efficiency. The information provided by ex- ν { n }
tualexperimentalfrequenciesandthus,besidesconvergence,
perimental data is contained in the collection of frequencies
itquantifieshowtheestimateddistributionreproducestheex-
f = f (η ) = n /n where n is the number of ”no
ν 0 ν 0ν ν 0ν perimentaldata. Thetotaldistanceisadecreasingfunctionof
click” events and n the total number of runs with quantum
ν the number of iterations. Its stationary value is proportional
efficiency η . Then we consider expression (1) as a statisti-
ν totheaccuracyoftheexperimentalfrequencies f . Forfi-
cal model for the parameters ̺n to be solved by maximum- nitedata samplethisvalueis oforder1/√n fo{reνa}chvalue
likelihood (ML) estimation. Upon defining p p (η ) ν
and A = (1 η )n we rewrite expressionν(1)≡as p0 ν= ofην,givingusaroughestimateofn¯/√nν forthetotalerror
PtonbeAeννsnnti̺mna.teSsian−rceepνtohseitimveod(LelINisPOlinSeaprroabnldemth),ethpeanratmheeνtseors- fvoarluteheofreεc(oi)nisstroufcttehdisporrodbearbwileityhapvνe[{d̺on(ui)b}le].cIhfetchkeedstathtieoncoarny-
lution can be well approximated by using the Expectation- vergence of the whole method. Notice that these properties
Maximizationalgorithm(EM)[17]. Byimposingtherestric- arenotsharedbythelog-likelihood: itsstationaritycertainly
tion ̺ =1,weobtaintheiterativesolution reveals convergence, but the value depends on the statistics
n n
to be retrieved and so can not even be estimated by a priori
P
K considerations.
A f
̺(i+1) =̺(i) νn ν (2) Inordertochecktheconvergenceoftheiterationsin(2),we
n n Xν=1 λAλnpν[{̺(ni)}] run simulated experiments to reconstruct some of the states
P
subsequently investigated experimentally (see next Sections
wherepν[ ̺n(i) ]aretheprobabilitiespν,ascalculatedbyus- andFigs.3-7).TheresultsareshowninFig.1,wherethetotal
{ }
ingthereconstructeddistribution ̺(i) atthei-thiteration. error(ratioitsstationaryvalue)isreportedasafunctionofthe
n
{ }
Before going to the experimental implementation we have numberofiterations. Asitisapparentfromtheplotthetotal
3
errorshowsatransientbehaviourandthenquicklyconverges based on the specific propertiesof PDC emission. PDC is a
toitsstationaryvalue.Therateofconvergencedependsonthe quantumeffectwithoutclassicalcounterpartsandconsistsofa
signalunderinvestigation. Ourresultsshowthattheiterative spontaneousdecay,insideanon-linearcrystal,ofonephoton
algorithmalways convergesand the asymptotic value of ε(i) fromapumpbeam(usuallygeneratedbyalaser)intoacouple
isoftheexpectedorderonmagnitude. ofphotonsconventionallycalledsignalandidler. Thisdecay
Anestimateoftheconfidenceintervalonthedetermination processobeys(phasematchinglaws)toenergyconservation
oftheelement̺ canbegivenintermsofthevariance
n
ω =ω +ω (7)
0 i s
σ =( F )−1/2, (4)
n n
N andmomentumconservation
being the total number of measurements, and F the
N n ~k =~k +~k (8)
Fisher’sinformation[7] 0 i s
1 ∂q 2 where ω0,ωi,ωs are the frequencies and~k0,~ki,~ks the wave
F = ν , (5) vectors of pump, idler and signal photon respectively. Fur-
n
q (cid:18)∂̺ (cid:19)
Xν ν n thermore,thetwophotonsareproduced,withinfewtensfem-
toseconds,atthesametime. Theprobabilityofaspontaneous
where
decayintoapairofcorrelatedphotonsisusuallyverylow,of
p A ̺ theorderof10−9orlower;thereforewithtypicalpumppower
q = ν = n νn n , (6)
ν oftheorderofsomemilliwatts,thefluorescenceemissionlies
p P A ̺
ν ν νn νn n at the levels of photon counting regime. Since the photons
P P
representstherenormalizedprobabilitiesoftheno-clickevent areproducedinpairsandbecauseoftheenergyandmomen-
atquantumefficiencyη . tum conservationrestrictions, the detectionof one photonin
ν
a certain direction and with a given energy indicates the ex-
istence of the pair correlated one, with definite energy in a
III. EXPERIMENTALDATAINCWREGIME well defined direction. This property allows to ”heralding”
thesecondphotonofthepaironcethefirstoneisdetectedin
In the following we present various different applications aprecisedirection(andtemporalandspectralwindow). This
of this method both to cw and pulsed regime, with the pur- ”heralded photon”was the state to be measured. In our set-
poseofdemonstratingthepotentialitiesofthistechnique.For up,afterhavingeliminatedthepumplaserbeamwithafilter,
what concerns cw regime we have studied reconstruction of the two photonsof the pair are separated by means of a po-
diagonalelementofthedensitymatrixforsinglephotonFock larizingbeamsplitter. Thedetectionofonephotoninoneof
statesandaweakcoherentone. the two conjugated directions was then used to open a win-
dowof∆t = 4.9nsfordetectioninarm2. Thiswasrealised
byaddressingthefirstdetectionsignalasastarttoaTimeto
Argon
laser AmplitudeConverter,thesignalfromseconddetector(aftera
delayline)wasthenaddressedtothesameTACasastopand
MCA
countedonlyifarrivinginawindowof∆t.
351 nmType II BBOUPeVmD fCiilste scrioonllinear PBS 7 0 I2F xxxxxxxxxxxnxxxxxxxxxxxxxxxxxxxxxxmxxxxxxxxxxx L pah(vSDtoaritilgloaicgdnoeiconrhd)eestart TstAopC/SCACcooinucindetnecre
1.0
7 0 I2F Lnmxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 1.0 f0.9
pahvSoatilloaicdnoiconhdee 0.8 0.8
0.00 0.05 0.10 0.15 0.20
FIG.2: SetupforheraldedsinglePDCphotonreconstruction. An n0.6
Ar+ laser beam pumps a type II BBO crystal generating collinear
0.4
degenerate PDC. After an anti-UV filter, pairs are split on a PBS.
ObservationofaphotonontriggerdetectorD(precededbyalens,L,
0.2
andaninterferencefilterIF)startsaTACrampwhichiseventually
closedbyastopsignalderivingfromtheobservationofaphotonin 0.0
0 1 2 3 4
theseconddetector.
n
FIG.3: Reconstruction ofthephotondistributionfortheheralded
The single photon states have been generated by produc- single-photonstateproducedinspontaneoustypeIIPDC.Inset:Ex-
ingParametricDownConversion(PDC)heraldedphotons.In perimental frequencies fν of no-click events as a function of the
somemoredetail(seeFig.2),pairsofcorrelatedphotonshave quantum efficiency ην for a PDC heralded photon state compared
beengeneratedbypumpingincollineardegenerategeometry withthetheoreticalcurvepν =1−ην .
a5mmlongβ-Barium-Borate(BBO) crystal,cutfortypeII
phasematching(i.ethetwophotonsofthepairhaveorthogo- The photodetection apparatuses were constituted by a sil-
nalpolarizations),witha0.3WcwArgonionlaserbeamwith icon avalanche photodiode detector preceded by an iris and
wavelengthof351nm. Theheraldedsinglephotonschemeis an interference filter (IF) at 702 nm, 4 nm FWHM, inserted
4
with the purposeof reducingthe stray light. Both thedetec-
torsweresiliconavalanchephotodiodeones(SPCM-AQR-15, 1
PerkinElmer). Thequantumefficiencyofthe”heraldedpho- 1.0
f
ton”detectionapparatus(includingIFandiris)wasmeasured
to be 20% by using the standard calibration scheme based 0.1 0.90.00.10.20.30.40.50.60.7
n
oncorrelationpropertiesofPDCemission(see[18]). Lower
quantum efficiencies, needed for the reconstruction scheme, 0.01
were simulated by inserting calibrated neutral density filters
on the optical path. A comparison of the observed frequen-
ciesfν withthetheoreticalcurve(1 ην)ispresentedinthe 0 1 2 3
− n
insetofFig.3.Thephotondistributionhasbeenreconstructed
usingK =34differentvaluesofthequantumefficiencyfrom
FIG.4: Reconstructionofthephotondistributionforaweakcoher-
η 0.01%to η 20%with n = 106 runsforeach η .
Rνes≃ultsatiteratioνni≃= 106areshoνwninFig. 3. Asexpecteνd eevnetnsttsataes.aIfnuntchteioinnsoefttthheeqeuxapneturimmeenffitacliefnrecqyuηeνncfoiersafPνDoCfhneor-aclldiecdk
the PDC heralded photon state largely agrees with a single photonstatearecomparedwithatypicalcurveforaweakcoherent
photonFockstate. However, also a small two photonscom- beampν ≃1−ην|α|2.
ponentandavacuumoneareobserved.Theρ contributionis
2
expected, by estimating the probabilitythat a second photon
randomlyentersthedetectionwindow,tobe1.85%ofρ , in the measurements, the light was delivered to a photo multi-
1
agreementwithwhatobserved.Anonzeroρ isalsoexpected pliertube(PMT,Burle8850)throughamultimodefiber(100
0
due to background. This quantity can be evaluated by mea- µm core diameter). Althoughthe PMT has the capability of
suringthecountswhenthepolarizationofthepumpbeamis counting the number of photoelectrons produced by one or
rotatedbyaλ/2waveplatetoavoidgenerationofparametric morephotons[3], for the presentapplicationwe used it in a
fluorescence.Inthiscaseaswell,ourestimate,(2.7 0.2)%, Geiger-like configuration, by setting a threshold to discrimi-
isingoodagreementwiththereconstructedρ . ± nateon/offevents. Furthermorewetakeadvantageofthelin-
0
Asasecondexamplewehavereconstructedthestatisticof earity of the mean anodic charge, Ac, see Inset in Fig. 6, as
astronglyattenuatedcoherentstate,whichhasbeenproduced a function of the mean energy of the measured light [20] to
byaHe-Nelaserbeamattenuatedtophoton-countingregime obtainthe valuesof ην. In fact, if we set Ac = 0 forη = 0
byinsertionofneutralfilters.Thesamesiliconavalanchepho- andAc = (Ac)max forη = ηP (nominalquantumefficiency
todiode detector of the previouscase was used here as well. ofthePMT),foralltheintermediatequantumefficienciesob-
Thecountsweremeasuredinabout400nswindowobtained tained by attenuating the light with neutral filters, we have
by gating the photo-detector with a periodic signal (10 kHz η = (ηP/(Ac)max)Ac. Thisprocedureallowustovarycon-
rate).Itmustbenoticedthatinthiscasewedonothaveinter- tinuouslythequantumefficiency.
ferencefiltersoririsesinfrontofthedetectorandalltheother
attenuationscanbeincludedinthegenerationofthestate(i.e. a) 349nm
theycontributetotheabsorbtiontogetherwithneutralfilters):
thusthehighestquantumefficiencyisassumedtobe66%as UV b) D 349nm F MF PMT
Nd:YLF
declaredbythemanufacturerdata-sheetforthephotodetector.
Thereconstructeddistribution,withK = 15differentvalues 420nm GI
c) BBO
of the quantum efficiency from ην 0.1% to ην 66% Stop MCA
with n = 106 runs for each η , ag≃rees well with w≃hat ex-
ν ν
pected for a coherent state with average number of photons FIG.5: Setupforthegeneration of pulsedoptical states. Starting
α2 0.02. In Fig. 4 are shown both the frequencies fν withthethird-harmonicsofaNd:YLFlaser,weobtain: a)gaussian
| | ≃
as a function of η compared with the theoretical prediction state;b)thermalstate;c)multithermalstate.D,rotatingground-glass
ν
p =exp η α2 1 η α2andthereconstructedpho- diffuser; BBO,typeInonlinear crystal; F,variablefilter; MF,mul-
ν ν ν
ton statist{ic−. Fi|na|ll}y,≃we−briefl|y|acknowledge that a compa- timode fiber; PMT, photomultiplier detector; GI, gated integrator;
rableresultwasalsoobtainedwithaverystronglyattenuated MCA,multichannelanalyzer.
thermalstate,i.e.lightemittedbyathermalsource,atungsten
lampattenuatedbyneutraldensityfilters. Thefirstmeasurementwasperformedonthepulseemerg-
ing from the laser source. Due to the pulsed nature of the
source, we do notexpect to recovera true Poissonian statis-
tics as in the cw measurement described above. Rather, we
IV. EXPERIMENTALDATAINPULSEDREGIME
expectaGaussiandistributionoftheform[21]
Inthepulseddomain,wehavemeasuredthreedifferentop- 1 (n N)2
̺ = exp − , (9)
tical states generated starting from the third harmonics (349 n,G 2π(N +σ2) (cid:20)−2(N +σ2)(cid:21)
nm, 4.45 ps) of a cw mode-locked Nd:YLF laser regenera-
p
tivelyamplifiedatarepetitionrateof500Hz(HighQLaser). whichtakesintoaccountthepresenceofnoise;N isthepho-
ThegeneralexperimentalsetupissketchedinFig.5. Forall ton mean value and σ2/N can be taken as a measure of the
5
deviationfromPoissonianstatistics. InFig. 7a)weshowthe where µ is the number of temporal modes. The photon dis-
photondistribution̺ , reconstructedatthei = 50000itera- tribution reconstructed at the i = 1500 iteration, is shown
n
tionoftheMLalgorithm,alongwiththebestfitobtainedwith in Fig. 7 c) along with the best fit of the data using (11)
themodel(9)(fittingparametersN = 4.88andσ2 = 0.63). (N = 6.17 and µ = 5); the fidelity of the reconstruction is
The inset of the figure comparesthe experimentalfrequency given by G 1. In the inset of the figure we show the ex-
f data(K = 37valuesofη,n = 104 runsforeachη)asa perimentalfr≃equencyf data(K =18valuesofη,n =104
ν ν ν ν
functionof η with the theoreticalvaluescalculated through runsforeachη)andtheirtheoreticalvaluesascalculatedac-
ν
(1)andtheparametersgivenbythefitofthephotondistribu- cordingto (1). As a commentto the experimentalresults in
tion.Boththereconstructeddistributionandtheexperimental thepulsedregime,wenotethatthebestreconstructionofthe
frequenciesagreesverywellwiththeaboveGaussianmodel. photondistributionisachievedatadifferentnumberofitera-
The fidelity of the reconstruction is G 0.998. Using the tions for the three differentmeasured opticalstates, and that
estimated value of σ2 a deviation of abo≃ut 13% of the laser the absolute error ε doesnot approachthe same value. This
photonnumberdistributionfromthePoissonianstatisticscan is due to the presence of excess noise in our measurements,
bederived. since the stability and the repetition rate of our source (500
Hz) limits to n 104 the number of runs for each value
ν
∼
of the quantumefficiency[13]. The choiceof the best itera-
6000 1.2 tiontostopthealgorithmisdrivenbythepossibilitytofitthe
1.0
45000000 A(V)c000...468 dniostari-bpurtiioorni dweicthisiaonsuiintacbhleoomsiondgela. GWaeussstiraenssdtihsatrtibthuetiroenwfoasr
unts3000 00..020.0 0.2 0.4 0.6 0.8 1.0 case a) or of a multithermal distribution for case c), but, on
co T thecontrary,wefollowedthea-posterioriobservationthatno
2000 Threshold otherdistributioncouldfitequallywellthereconstructeddata.
1000
0 1 2 3
Ac(V)
1.0
FIG.6: HistogramofthedetectedanodicchargeofthePMT(data 0.16 f0.8
corresponding tothemeasurement of agaussian fieldhavingN = 0.12 0.6
4.88withη=0.20).Thearrowindicatesthethresholdvalueusedto n 0.40.00 0.05 0.10 0.15 0.20
0.08
discriminateon/offevents.Inset:linearitytestforthePMTdetector:
meananodiccurrentasafunctionofthetransmittanceofcalibrated 0.04
filters.
0.00
0 5 10 15 20
n
A second measurement was performed on the laser pulse
diffused by a moving ground glass. When the photons are 0.16 1.0
collected from within an area of spatial coherence, the sys- f0.8
0.12
tem acts as a pseudo-thermalsource, whose photon number 0.6
n
distributionsatisfies 0.08 0.00 0.05 0.10 0.15 0.20
Nn 0.04
̺ = . (10)
n,T (N +1)n+1
0.00
0 5 10 15 20 25 30
Figure7b)showsthephotondistribution̺ ,asreconstructed n
n
atthe i = 400iterationand thebestfit of thedata with (10)
(N = 5.33);thefidelityisgivenbyG 0.995. Theinsetof 0.12 1.0
thefigurecontainstheexperimentalfre≃quencyfν data(K = 0.10 f00..68
24valuesofη,n =104runsforeachη)andtheirtheoretical 0.08 0.4
ν n
valuesascalculatedfrom(1). 0.06 0.00 0.05 0.10 0.15 0.20 0.25
0.04
The last measurement was performed on the blue portion
0.02
(420nm)ofthedownconversionfluorescenceproducedbya
0.00
typeIBBO crystal(10mmdepth,cutat34deg)pumpedby 0 5 10 15 20 25 30
n
thelaserpulse.Thepump,incidentorthogonallytothecrystal
face, had an intensity 60 GW/cm2. In this experimental FIG.7: Reconstructedphotondistribution(blackbars)andbestthe-
∼
condition we expect a coherence time of the generated field oreticalfit(greybars)forthreedifferentstatesinthepulsedregime:
of 1 ps, that corresponds to measuring a convolution of a)laser pulse, b) diffusedlaser pulse, c)multimode stateproduced
∼
4-5temporalmodes[19]. Thephotonnumberdistributionis intypeIPDC.Insets:Experimentalfrequencyfν datainfunctionof
expectedtobea”multithermal”distributionoftheform thequantumefficiencyην andtheoreticalmodelforeachoneofthe
states.
(n+µ 1)!
̺ = − , (11)
n,M n!(µ 1)!(1+N/µ)n(1+µ/N)µ
−
6
V. CONCLUSIONS ics, quantuminformationandquantummetrology. Someap-
plicationsinthesefieldsarenowunderrealisationinourlab-
In summary, we have presented experimental results on oratories(reconstructionoffurtherquantumopticalstates,of
the reconstructionof the photondistribution based on on/off entangledstates[22],characterizationofahighspectralselec-
detection at different quantum efficiency followed by a tionheraldedphotonsource[23],etc.).
maximum-likelihooditerativealgorithmbasedonthetheoret-
icalanalysispresentedin[13].
Our results concern single-photon (PDC heralded) and
VI. ACKNOWLEDGEMENT
weak coherent states in the cw regime, as well as to coher-
ent, thermal and multi-thermal states in the pulsed regime.
Thestabilityandthegoodaccuracyshowninthereconstruc- This work has been supported by MIUR (FIRB
tionofthesestates,togetherwiththesimplicityofthemethod, RBAU01L5AZ-002 and RBAU014CLC-002), by INFM
clearlydemonstratetheinterestingpotentialitiesofthistech- (PRA-CLON), by ”Regione Piemonte” and by Fondazione
nique, suggesting relevant future applications ranging from SanPaolo. ARandMGAPthanksStefanoOlivaresformany
studieson quantumoptics, foundationsof quantummechan- fruitfuldiscussions.
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