Statistics of multiphoton events in spontaneous parametric down-conversion
Wojciech Wasilewski, Czeslaw Radzewicz, Robert Frankowski, Konrad Banaszek
aa r X i v : . [ qu a n t - ph ] M a y Statisti s of multiphoton events in spontaneous parametri down- onversionWoj ie h Wasilewski ∗ and Czesªaw Radzewi zInstitute of Experimental Physi s, Warsaw University, Ho»a 69, 00-681 Warsaw, PolandRobert Frankowski and Konrad BanaszekInstitute of Physi s, Ni olaus Coperni us University,Grudzi¡dzka 5, 87-100 Torun, PolandAbstra tWe present an experimental hara terization of the statisti s of multiple photon pairs produ edby spontaneous parametri down- onversion realized in a nonlinear medium pumped by high-energyultrashort pulses from a regenerative ampli(cid:28)er. The photon number resolved measurement has beenimplemented with the help of a (cid:28)ber loop dete tor. We introdu e an e(cid:27)e tive theoreti al des riptionof the observed statisti s based on parameters that an be assigned dire t physi al interpretation.These parameters, determined for our sour e from the olle ted experimental data, hara terize theusefulness of down- onversion sour es in multiphoton interferen e s hemes that underlie proto olsfor quantum information pro essing and ommuni ation.PACS numbers: 42.50.Ar, 42.65.Lm, 42.50.Dv ∗ Ele troni address: wwasilfuw.edu.pl 1 p abff
FIG. 1: (Color online) A s hemati of a spontaneous parametri down- onversion sour e. Thenonlinear rystal X is pumped with a laser beam p . Generated photons are highly orrelated anduseful modes a and b are typi ally sele ted by narrow spatial and frequen y (cid:28)lters f .I. INTRODUCTIONSpontaneous parametri down- onversion (SPDC) is the basi sour e of non- lassi allight in experimental quantum opti s [1℄, testing foundations of the quantum theory [2℄, andimplementing proto ols for quantum information information pro essing and ommuni ation[3℄. The essential feature of SPDC is the guarantee that the photons are always produ edin pairs, and suitable arrangements allow one to generate various types of lassi al andquantum orrelations within those pairs.The physi s of SPDC depends strongly on opti al properties of nonlinear media in whi hthe pro ess is realized. This leads to an interplay between di(cid:27)erent hara teristi s of thesour e and usually imposes trade-o(cid:27)s on its performan e. For example, many experiments re-quire photon pairs to be prepared in well-de(cid:28)ned single spatio-temporal modes. In ontrast,photons generated in typi al media diverge into large solid angles and are often orrelated inspa e and time, as shown s hemati ally in Fig. 1. Spe i(cid:28) modes an be sele ted afterwardsby oupling the output light into single-mode (cid:28)bers and inserting narrowband spe tral (cid:28)l-ters. However, it is usually not guaranteed that both the photons in a pair will always havethe mat hing modal hara teristi s, and in many ases only one of the twin photons will get oupled in [4℄. This e(cid:27)e t, whi h an be modelled as a loss me hanism for the produ ed light,destroys perfe t orrelations in the numbers of twin photons. These losses ome in additionto imperfe t dete tion, and an be des ribed jointly using overall e(cid:30) ien y parameters.The e(cid:27)e ts of losses be ome more riti al when the SPDC sour e is pumped with powersso high that it is no longer possible to negle t the ontribution of events when multiple pairshave been simultaneously produ ed [5℄. Su h a regime is ne essary to arry out multiphotoninterferen e experiments, it an be also approa hed when in reasing the produ tion rate of2hoton pairs. One is then usually interested in postsele ting through photo ounting thedown- onversion term with a (cid:28)xed number of photon pairs and observing its parti ularquantum statisti al features [6℄. In the presen e of losses the same number of photo ounts an be generated by higher-order terms when some of the photons es ape dete tion. However,the statisti al properties of su h events an be ompletely di(cid:27)erent, thus masking the featuresof interest. Although some quantum properties may persist even in this regime, with anotable example of polarization entanglement [7℄, their extra tion and utilization be omes orrespondingly more di(cid:30) ult.The present paper is an experimental study of multiphoton events in spontaneous para-metri down- onversion with parti ular attention paid to the e(cid:27)e ts of (cid:28)ltering and losses.The multiple-pair regime is a hieved by pumping the nonlinear rystal by the frequen y-doubled output of a 300 kHz titanium-sapphire regenerative ampli(cid:28)er system. The kilohertzrepetition rate has allowed us to ount the number of the photons at the output with the helpof the loop dete tor [8℄. Using a simpli(cid:28)ed theoreti al des ription of the SPDC sour e we in-trodu e e(cid:27)e tive parameters that hara terize its performan e in multiphoton experiments.The obtained results illustrate trade-o(cid:27)s involved in experiments with multiple photon pairsand enable one to sele t the optimal operation regime for spe i(cid:28) appli ations.This paper is organized as follows. First we des ribe a theoreti al model for SPDCstatisti s in Se . II. Se . III introdu es e(cid:27)e tive parameters to hara terize SPDC sour es.The experimental setup and measurement results are presented in Se . IV. Finally, Se . V on ludes the paper.II. SPDC STATISTICSWe will start with a simple illustration of the e(cid:27)e ts of higher-order terms in SPDC.Suppose for simpli ity that the sour e produ es a two-mode squeezed state whi h an bewritten in the perturbative expansion as | , i + r | , i + r | , i + . . . , where r measuressqueezing and is assumed to be real. For two-photon experiments, the relevant term is | , i and the ontribution of the higher photon number terms an be negle ted as long as r ≪ .This enables postsele ting the two-photon term and observing asso iated quantum e(cid:27)e ts,su h as Hong-Ou-Mandel interferen e. Suppose now that ea h of the modes is subje t tolosses hara terized by − η , where η is the overall e(cid:30) ien y. Losses may transform the term3 , i into | , i or | , i , whose presen e will lower the visibility of the Hong-Ou-Mandelinterferen e. The two-photon term now o urs with the probability η r , while the four-photon term e(cid:27)e tively produ es one of the states | , i or | , i with the total probabilityequal to − η ) η r . This onstitutes a fra tion of − η ) r of the events that omefrom single pairs produ ed by the sour e. This fra tion an easily be ome omparable withone, espe ially when the losses are large.Let us now develop a general model of photon statisti s produ ed by an SPDC sour e. Inthe limit of a lassi al undepleted pump the output (cid:28)eld is des ribed by a pure multimodesqueezed state. By a suitable hoi e of spatio-temporal modes, alled hara teristi modes,su h a state an be brought to the normal form [9℄ in whi h modes are squeezed pairwise.Denoting the annihilation operators of the hara teristi modes by ˆ a k and ˆ b k , the non-vanishing se ond-order moments an be written as: h ˆ a † k ˆ a l i = h ˆ b † k ˆ b l i = δ kl r k − h ˆ a k ˆ b l i = h ˆ a † k ˆ b † l i ∗ = δ kl r k (1)where r k is the squeezing parameter for the k th pair of modes. Be ause the state of lightprodu ed in SPDC is gaussian, these equations, ombined with the fa t that (cid:28)rst-ordermoments vanish h ˆ a k i = h ˆ b k i = 0 de(cid:28)ne fully quantum statisti al properties of the output(cid:28)eld.Let us (cid:28)rst onsider the ase when the spatial and spe tral (cid:28)lters pla ed after the sour esele t e(cid:27)e tively single (cid:28)eld modes. The annihilation operators ˆ a and ˆ b are given by linear ombinations of the hara teristi modes: ˆ a = X k t k ˆ a k ˆ b = X k t ′ k ˆ b k (2)where t k and t ′ k des ribe amplitude transmissivities of the (cid:28)lters for the hara teristi modesand P k | t k | = P k | t ′ k | = 1 . Be ause the omplete multimode state is gaussian, the redu edstate of the modes ˆ a and ˆ b is also gaussian, and it is fully hara terized by the averagenumbers of photons ¯ n = h ˆ a † ˆ a i and ¯ n ′ = h ˆ b † ˆ b i , and the moment S = h ˆ a ˆ b i . These quantities an be written in terms of the multimode moments given in Eq. (1), but we will not needhere expli it expressions. It will be onvenient to des ribe the quantum state of the modes ˆ a and ˆ b with the help of the Wigner fun tion: W ˆ ρ ( α, β ) = 1 π √ det C exp (cid:18) − α † C − α (cid:19) (3)4here α = ( α, α ∗ , β, β ∗ ) T and C is the orrelation matrix omposed of symmetri ally orderedse ond order moments: C = ¯ n + S n + S ∗ S ¯ n ′ + S ∗ n ′ + . . (4)Given the Wigner fun tion of the redu ed state for the modes ˆ a and ˆ b in the Gaussianform, the al ulation of the joint ount statisti s ρ n,m is straightforward. It will be useful tointrodu e an operator representing the generating fun tion of the joint ount statisti s ˆΞ( x, y ) = X n,m x n y m | n, m ih n, m | = x ˆ a † ˆ a y ˆ b † ˆ b (5)whose expe tation value over the quantum state expanded into the power series yields thejoint ount statisti s ρ nm : ρ n,m = 1 n ! m ! d n + m dx n dy m h ˆΞ( x, y ) i (cid:12)(cid:12)(cid:12)(cid:12) x = y =0 (6)Be ause the operator ˆΞ( x, y ) is formally equal, up to a normalization onstant, to produ t ofdensity matri es des ribing thermal states of modes ˆ a and ˆ b with average photon numbers x/ (1 − x ) and y/ (1 − y ) respe tively, the orresponding Wigner fun tion has a Gaussianform: W ˆΞ ( α, β ) = 4 π (1 + x )(1 + y ) × exp (cid:18) − | α | − x x (cid:19) exp (cid:18) − | β | − y y (cid:19) . (7)Using the above expression it is easy to evaluate the generating fun tion for the joint ount statisti s by integrating the produ t of the respe tive Wigner fun tions: h ˆΞ( x, y ) i = π Z d α d β W ˆΞ ( α, β ) W ˆ ρ ( α, β )= 1 N + 1 − N ( ηx + 1 − η )( η ′ y + 1 − η ′ ) (8)where we introdu ed the following three parameters: η = 1¯ n ′ ( | S | − ¯ n ¯ n ′ ) η ′ = 1¯ n ( | S | − ¯ n ¯ n ′ ) (9) N = ¯ n ¯ n ′ | S | − ¯ n ¯ n ′ = ¯ nη = ¯ n ′ η ′ |h ˆ a ˆ b i| > h ˆ a † ˆ a ih ˆ b † ˆ b i . Then, it is easy to he k that the generating fun tion given in Eq. (8)des ribes the ount statisti s of a plain two-mode squeezed state whose two modes have beensent through lossy hannels with transmissivities η and η ′ , and the average number of photonsprodu ed in ea h of the modes was equal to N . Note that the inequality |h ˆ a ˆ b i| > h ˆ a † ˆ a ih ˆ b † ˆ b i implies non lassi al orrelations between the modes ˆ a and ˆ b . If the opposite ondition issatis(cid:28)ed, the state an be represented as a statisti al mixture of oherent states in modes ˆ a and ˆ b with orrelated amplitudes. In this ase, it is not possible to arry out an absolutemeasurement of losses.III. PARAMETERSIn a realisti situation, the spe tral (cid:28)lters employed in the setup are never su(cid:30) ientlynarrowband to ensure ompletely oherent (cid:28)ltering. Therefore a sum of ounts originat-ing from multiple modes will be observed. We will model this e(cid:27)e t by assuming that thedete ted light is omposed of a ertain number of M modes with identi al quantum sta-tisti al properties des ribed in the pre eding se tion. The generating fun tion Ξ( x, y ) for ount statisti s is therefore given by the M -fold produ t of the expe tation value h ˆΞ( x, y ) i al ulated in Eq. (8): Ξ( x, y ) = h ˆΞ( x, y ) i M (10)The parameter M , whi h we will all the equivalent number of modes, an be read outfrom the varian e of the ount statisti s in one of the arms hara terized by the generatingfun tion Ξ( x, y ) . For a single mode sour e the varian e is equal to that of a thermal statewith (∆ n ) = h n i + h n i . It is easily seen that for M equally populated modes the varian ebe omes redu ed to the value (∆ n ) = h n i + h n i / M . Solving this relation for M leads usto a measurable parameter that will help us to hara terize the e(cid:27)e tive number of dete tedmodes: M = h n i (∆ n ) − h n i . (11)Note that M is losely related to the inverse of the Mandel parameter [10℄.The se ond parameter we will use to hara terize the SPDC sour e measures the overalllosses experien ed by the produ ed photons. Let us (cid:28)rst note that in the perturbativeregime, when all the squeezing parameters r k ≪ , ea h one of the quantities ¯ n , ¯ n ′ , and | S | η ≈ | S | / ¯ n ′ and η ′ ≈ | S | / ¯ n that are independent of thepump intensity. On the other hand, the average photon numbers h n i and h n ′ i in both thearms are linear in the pump power.Following early works on squeezing [11℄, we will introdu e here a parameter that quanti(cid:28)esthe subpoissonian hara ter of orrelations between the ounts n and n ′ in the two arms ofthe setup. First we de(cid:28)ne a sto hasti variable: δ = n h n i − n ′ h n ′ i ,vuut h n i + 1 h n ′ i . (12)This de(cid:28)nition takes into a ount the possibility of di(cid:27)erent losses in the two arms through asuitable normalization of the ount numbers. The subpoissonian hara ter of the orrelations an be tested by measuring the average h δ i . The semi lassi al theory predi ts h δ i ≥ ,while for two beams with ount statisti s hara terized by the generating fun tion h ˆΞ( x, y ) i M one obtains: h δ i = 1 − . (cid:18) η + 1 η ′ (cid:19) < . (13)In the ase of equal e(cid:30) ien ies η = η ′ this expression redu es simply to h δ i = 1 − η . Wewill use the last relation as a method to measure the average overall e(cid:30) ien y of dete tingthe state produ ed by the SPDC sour e.As dis ussed at the beginning of Se . II, the initial quantum state of light used for manyexperiments should ideally be in a state | , i or | , i . In the perfe t ase of two-modesqueezing and negligible losses, su h states an be isolated through postsele tion of theSPDC output on the appropriate number of ounts. In pra ti e, the postsele ted events willalso in lude other ombinations of input photon numbers. Be ause typi al dete tors havelimited or no photon number resolution, one needs to take into a ount also the deleterious ontribution of higher total photon numbers. This leads us to the following de(cid:28)nition of theparameter measuring the ontamination of photon pairs with other terms that annot be ingeneral removed through postsele tion: ε = 1 − ρ , P k + l ≥ ρ k,l (14)An analogous de(cid:28)nition an be given for quadruples of photons, whi h ideally should be7 η −5 −4 −3 −2 −1 −2 −1 p η −10 −8 −6 −4 −2 −2 −1 FIG. 2: Contour plots of the ontamination parameters for single photon pairs ε (upper plot)and double photon pairs ε (lower plot) as a fun tion of the overall e(cid:30) ien y η and the respe tiveprodu tion rates p , and p , . The al ulations have been arried out in the regime of single sele tedmodes when M = 1 .prepared in a state | , i : ε = 1 − ρ , P k + l ≥ ρ k,l . (15)In Fig. 2 we depi t ontour plots of the ontamination parameters ε and ε as a fun tion ofthe produ tion rates and the overall e(cid:30) ien y. It is learly seen that the non-unit e(cid:30) ien yimposes severe bounds on the produ tion rates that guarantee single or double photon pairevents su(cid:30) iently free from spurious terms. The graphs also imply that strong pumping isnot a su(cid:30) ient ondition to a hieve high produ tion rates, but it needs to be ombined witha high e(cid:30) ien y of olle ting and dete ting photons.8 D1D2 IF FCFCFC DMBG RGDMILFL HWPXSHRegA ND CE FIG. 3: (Color online) Experimental setup. RegA, Regenerative ampli(cid:28)er; FL, IL, lenses; XSH,se ond harmoni rystal; DM, di hroi mirrors; BG, blue glass (cid:28)lter; ND, neutral density (cid:28)lter;HWP, half wave plate; X, down onversion rystal; RG, red glass (cid:28)lters; FC, (cid:28)ber oupling stages;IF, interferen e (cid:28)lter; D1, D2, avalan he photodiodes; CE, oin iden e ele troni s. Thin solid linesindi ate single mode (cid:28)bers, dashed lines (cid:22) multimode (cid:28)bers.IV. EXPERIMENTAL RESULTSThe experimental setup is depi ted in the Fig. 3. The master laser (RegA 9000 fromCoherent) produ es a train of 165 fs FWHM long pulses at a 300 kHz repetition rate enteredat the wavelength 774 nm, with 300 mW average power. The pulses are doubled in the se ondharmoni generator XSH based on a 1 mm thi k beta-barium borate (BBO) rystal ut fora type-I pro ess. Ultraviolet pulses produ ed this way have 1.3 nm bandwidth and 30 mWaverage power. They are (cid:28)ltered out of the fundamental using a pair of di hroi mirrors DMand a olor glass (cid:28)lter BG (S hott BG39), and imaged using a 20 m fo al length lens ILon a down oversion rystal X, where they form a spot measured to be 155 µ m in diameter.The power of the ultraviolet pulses, prepared in the polarization perpendi ular to the planeof the setup, is adjusted using neutral density (cid:28)lters ND and a motorized half waveplate.The type-I down- onversion pro ess takes pla e in a 1 mm thi k BBO rystal X ut at . ◦ to the opti axis, and oriented for the maximum sour e intensity.The down- onverted light emerging at the angle of . ◦ to the pump beam is oupledinto a pair of single mode (cid:28)bers pla ed at the opposite ends of the down- onversion one.The (cid:28)bers and the oupling opti s de(cid:28)ne the spatial modes in whi h the down- onversionis observed [12℄. The oupled photons enter the loop dete tor [8℄ in whi h light from eitherarm an propagate towards one of the dete tors through eight distin t paths. The minimal9elay di(cid:27)eren e between two paths is 100 ns, more than twi e the dead time of the dete -tors. Finally the photons exit the (cid:28)ber ir uit, go through interferen e (cid:28)lters IF and are oupled into multimode (cid:28)bers whi h route them dire tly to single photon ounting modulesSPCM (PerkinElmer SPCM-AQR-14-FC) onne ted to fast oin iden e ounting ele tron-i s ( ustom-programmed Virtex4 protype board ML403 from Xilinx) dete ting events in aproper temporal relation to the master laser pulses. The measurement series are arried outwith pairs of interferen e (cid:28)lters of varying spe tral widths.The measurement pro eeds as follows: for ea h pair of interferen e (cid:28)lters the loop dete toris alibrated using data olle ted at a very low pump light intensities, when the han e ofmore than one photon entering the (cid:28)ber ir uit is negligible ompared to the rate singlephotons appear. This allows one to al ulate the omplete matrix of onditional probabilities P k,n of observing k dete tor li ks with n initial photons [8, 13℄ for one arm, and analogously P ′ l,m for the se ond arm. The losses in the dete tors and in the (cid:28)ber ir uit are assumedto ontribute to the overall e(cid:30) ien ies η and η ′ . Thus P k,n and P ′ l,m des ribe lossless loopdete tors for whi h P , = 1 . After the alibration, the ounts are olle ted for approximately master laser pulses for ea h hosen intensity of the pump, whi h yields the probabilities p k,l of observing k li ks in one arm and l in the other one. These probabilities are related tothe joint ount probability ρ n,m orre ted for ombinatorial ine(cid:30) ien ies of the loop dete torthrough the formula: p k,l = X n,m P k,n P ′ l,m ρ n,m . (16)The probabilities ρ n,m an be retrieved from experimentally measured p k,l using the maxi-mum likelihood estimation te hnique [14℄. An exemplary joint photon number distributionre onstru ted from the experimental data is shown in Fig. 4. The results of the re on-stru tion are subsequently used to al ulate the parameters of the sour e dis ussed in thepre eding se tion.In Fig. 5 we depi t the re onstru ted equivalent number of modes M in single arm fordi(cid:27)erent (cid:28)ltering and pump intensities. This quantity an be also understood as a measureof how many in oherent modes a photon from the sour e o upies. Naturally, M dropswith appli ation of narrowband spe tral (cid:28)ltering sin e it erases the information on the exa ttime the photon pair was born in the nonlinear rystal. It also is seen that M is pra ti allyindependent on the pumping intensity, as predi ted in Se . III.10 −4 −1 nn’ ρ nn ’ FIG. 4: An exemplary joint photon number distribution measured without spe tral (cid:28)lters. Theaverage photon numbers in the two arms are h n i =0.15 and h n ′ i =0.18. −3 −2 −1 〈 n 〉 M FIG. 5: The equivalent number of modes M in a single arm as a fun tion of the average photonnumber h n i , for measurements arried out without interferen e (cid:28)lters ( ir les), with 10 nm FWHM(cid:28)lters (squares) and 5 nm FWHM (cid:28)lters ( rosses).The average overall e(cid:30) ien y of the squeezed state al ulated as η = 1 − h δ i is shown inFig. 6. It is seen that the e(cid:30) ien y de reases with an appli ation of narrowband (cid:28)ltering,whi h again is easily understood. In addition, η exhibits a very weak dependen e on thepump intensity whi h again agrees with theoreti al predi tions for the regime when theaverage number of photons is mu h less than one. A relatively large di(cid:27)eren e in η between10 nm and 5 nm interferen e (cid:28)lters an be explained by the fa t that in the latter ase thesele ted bandwidth be omes narrower than the hara teristi s ale of spe tral orrelations11 −2 −1 〈 n 〉 η [ % ] FIG. 6: The equivalent dete tion e(cid:30) ien y η as a fun tion of the total average photon number h n i , measured without interferen e (cid:28)lters ( ir les), with 10 nm FWHM (cid:28)lters (squares) and 5 nmFWHM (cid:28)lters ( rosses).within a pair. Consequently, the (cid:28)lter in one arm sele ts only a fra tion of photons onjugateto those that have passed through the (cid:28)lter pla ed in the se ond arm. This observation is onsistent with the determination of the parameter M , whi h shows that for 5 nm (cid:28)lterse(cid:27)e tively single spe tral modes are sele ted.Finally, in Figs. 7 and 8 we plot the respe tive ontamination parameters for single ε and double ε photon pairs, that des ribes non-postsele table ontributions of other photonterms to the output. As expe ted, the ontamination grows with de reasing overall e(cid:30) ien y η that orresponds to narrowing the spe tral bandwidth, as well as with the in reasing pairprodu tion rates. It is noteworthy that the measurement results agree well with (cid:28)ttedtheoreti al urves whose parameters mat h those that an be read out from Figs. 5 and 6.It is seen that in the regime the experiment was arried out the ontamination is rathersigni(cid:28) ant, and it would a(cid:27)e t substantially e(cid:27)e ts su h as two-photon interferen e. Thislimits in pra ti e the energy of pump pulses(cid:22)and onsequently the produ tion rates(cid:22)thatresult in photon pairs su(cid:30) iently free from spurious terms. The e(cid:27)e t is parti ularly dramati in the ase of ε , where genuine double pairs appear only in approximately half of all theevents. 12 −4 −3 −2 ε [ % ] FIG. 7: ( olor online) The ontamination parameter ε for single photon pairs as a fun tion ofthe pair produ tion rate, measured without interferen e (cid:28)lters ( ir les), with 10nm FWHM (cid:28)lters(squares) and 5nm FWHM (cid:28)lters ( rosses). The parameters of the (cid:28)tted theoreti al urves are: M = 16 . and η = 4 . (red solid line), M = 3 . and η = 4 . (red dashed line) and M = 1 . and η = 2 . (red dash-dot line). −6 −5 −4 −3 ε [ % ] FIG. 8: ( olor online) The ontamination oe(cid:30) ient ε for double photon pairs as a fun tion ofthe produ tion rate, measured without interferen e (cid:28)lters (dots). The parameters of the (cid:28)ttedtheoreti al urve are: M = 16 and η = 4 .9%