Intrinsic limitations to the quality of pulsed spontaneousparametric downconversion sources for quantum
information applications
Jean-Loup Smirr,1 Robert Frey,1,2 Eleni Diamanti,1 Romain Alléaume,1 and Isabelle Zaquine1,*1Institut Télécom/Télécom ParisTech, CNRS/LTCI, 46 rue Barrault, 75013 Paris, France
2Laboratoire Charles Fabry de l’Institut d’Optique, Institut d’Optique Graduate School, CNRS—Univ. Paris Sud, CampusPolytechnique RD 128, 91127 Palaiseau Cedex, France
1. INTRODUCTIONQuantum communications and quantum information proces-sing applications have rapidly progressed in the last years.In particular, quantum key distribution [1,2] has allowedthe exchange of unconditionally secure secret messages overtens of kilometers, while the first quantum cryptography net-work deployed on installed optical fibers was recently imple-mented [3]. The next challenge will be the implementation ofadvanced quantum networks, which will employ quantumrepeaters [4] and will allow very long distance quantum com-munications. The quantum repeater technology involves thegeneration and distribution of entangled photon states [5,6],the realization of quantum memories [7–11], and the imple-mentation of an appropriate interface between these elements[12]. The generation of entangled photon states is also re-quired in several other applications in quantum informationprocessing [13].
Sources of entangled photon pairs are therefore an essen-tial element of systems used in applications in this field. Inparticular, such sources must emit very high-quality entangledstates with a near-unity fidelity and a maximum repetition ratefor the photon pair generation. In the past few years, severalentangled photon pair sources have been implemented usingspontaneous parametric downconversion (SPDC) [14–30].Most of them offer high fidelity and reasonable repetition
rates, at least for spectral bandwidths larger than 0:1 nm. How-ever, at lower bandwidths, high fidelity is always associatedwith low repetition rates [15–22].
High fidelities are often obtained after correcting imperfec-tions due to unbalanced probabilities of the two-photon com-ponents of the entangled state as well as spectral, temporal,and spatial distinguishabilities. However, even after a carefulcorrection of these defects, the fidelity of the entangled stateis still altered by accidental coincidences occurring betweenphotons from two different pairs since there are no longerquantum correlations in this case.
This intrinsic limitation has been addressed in a number ofcases [16,24,31], using either the second-order correlationfunction for nondegenerate SPDC or the true-to-accidental co-incidence rate ratio, better suited to pulsed sources (moreconvenient for potential applications to quantum communica-tions networking and quantum information processing). A sys-tematic investigation of the effects encountered in practicalsystems is lacking, however. Therefore, it is very importantto derive general rules about the best expected performanceof any pulsed SPDC source, and apply these rules to the de-sign of an optimized source: through the impact of multiplepair generation, the beamsplitting, spectral bandwidth ofthe filter used on each photon path, and associated losses im-pose limitations on the global performance of the system. We
832 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Smirr et al.
provide here a comprehensive evaluation of these limitations,using true coincidence rate and visibility as key quality param-eters of the pulsed source. These results can be used to opti-mize the source design and derive the best possible trade-offin terms of performance for any given pulsed SPDC source ofentangled photons when a specific application scenario is tar-geted. Although our analysis is valid for any pulsed SPDCsource, the results presented hereafter are especially interest-ing to help increasing the useful pair rate of low bandwidthhigh-fidelity SPDC sources.
The paper is organized as follows. The problem and the re-levant parameters are described in Section 2, while the countand coincidence probability calculations are detailed in Sec-tion 3. In Section 4, the results are analyzed to understand theinfluence of every parameter (spectral bandwidth and trans-mission) on source quality. Section 5 presents the maximumoutput photon pair probabilities that can be reached for agiven visibility and defines acceptable ranges of spectral band-width, transmission loss and optical noise for an optimalsource. Comparison of results with previously reported ex-perimental data is also performed in this section. Finally, inSection 6 we investigate the influence of the imperfectionsof the system (pump-induced noise and detector dark counts)on the results of the preceding sections before concluding inSection 7.
2. MODEL DESCRIPTIONMost of the present SPDC sources operate in the continuouswave regime and count and coincidence rates are measured.These free running sources can be safely used in point to pointquantum communications. However, synchronizable pulsedSPDC sources are probably the most suitable devices forfuture applications in quantum information processing, in-cluding quantum communication networking. Our analysisis performed in the framework of these pulsed SPDC sourceswith the count and coincidence probabilities during a pumppulse as the key parameters. The measured rates are thenthe product of these probabilities by the pump repetition rate.
The investigated setup is shown in Fig. 1. The collinear geo-metry has been chosen in the figure for the sake of simplicitybut the analysis can also be applied to the case of noncollinearSPDC as is justified hereafter. As we are only interested incoincidences, the results reported here do not depend onthe phase-matching type of the source either. In most ofthe subsequent analysis, we suppose that the source producesonly photon pairs through spontaneous parametric down-conversion. The specific cases of pump-induced parasitic lightand detector noise are addressed in Sections 6.A and 6.B,respectively.
The frequencies of the generated photons are νs and νi, re-spectively, and the pulsed pump beam can be described by itsfrequency νp ¼ νs þ νi, its pulse durationΔt, and its repetitionrate f 0. The device S splits the pairs towards paths A and B.
The splitting is deterministic in experimental situations suchas noncollinear SPDC (spatial splitting of the two photons ofpairs emitted in different directions), nondegenerate down-conversion (frequency splitting by a dichroic mirror) or polar-ization splitting for downconversion using type II phasematching, even in the case of quasi-degenerate downconver-sion (νs ≃ νi ≃ νp=2).
However, collinear quasi-degenerate downconversionusing type I phase matching requires a statistical splittingof the photon pairs, using a beam splitter defined by its powerreflection and transmission coefficients RA and RB
(RA þ RB ¼ 1). It should be noted that this case is of greatinterest as it is the way to implement SPDC sources basedon periodically poled lithium niobate (PPLN), which hasthe highest presently available nonlinear coefficient.
All subsequent calculations have been performed for bothdeterministic and statistical splittings of the pairs produced byspontaneous parametric downconversion.
The filters FA and FB used on paths A and B have an idealrectangular transmission spectrum, as shown in Fig. 2. Theirtransmission is zero outside of their respective spectral band-widthsΔνA andΔνB, and we assumeΔνA ≤ ΔνB. Their centerfrequencies νA and νB are related by νA þ νB ¼ νp. Note thatthis spectral shape corresponds to an ideal filtering situation.Although such ideal rectangular filters are not available com-mercially (the apodized filters used in telecommunicationswould provide the closest spectral shape), we havedeliberately chosen this ideal case, in order to derive the bestpossible performance of the SPDC source. For convenience,the power transmission coefficients XA and XB take into ac-count not only the filter losses but also all optical losses of thecorresponding path (including beamsplitter, propagation andfiber coupling losses, and detector efficiency).
As we are interested in SPDC sources for quantum informa-tion processing based on Bell inequalities, “Bell” splitters BSAand BSB are used in channel A and B, respectively. For in-stance, the Bell splitters are rotating polarization analyzersin the case of polarization entanglement [14] with the relativeangle between these analyzers as the control parameter. In thecase of time-bin entanglement [32], the Bell splitters areFranson interferometers with the relative phase betweenthem as the control parameter. Two detectors DI1 and DI2are placed on each subchannel of channel I (I ¼ A;B). For thesake of simplicity, the detector quantum efficiencies ηA1;2
andηB1;2
are chosen such that total losses are identical in channelsI1 and I2. (This can always be done by adjusting the gain of theavalanche photodiodes). All detectors are gated with a gatewidth T > Δt so that all signal and idler photons emitted
Fig. 1. (Color online) Schematic of the setup. SPDC, spontaneousparametric downconversion; S, splitting device; FA, FB, filters;DA1;2
, DB1;2, single-photon detectors.
Fig. 2. (Color online) Transmission spectrum of the ideal filter usedin this model, where νI is the center frequency,ΔνI the spectral band-width, XI the transmission (I ¼ A;B). Note that XI includes lossesdue to the splitting device, propagation, fiber coupling, and detectorefficiency.
Smirr et al. Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 833
during the pump pulse can be detected by the avalanchephotodiodes.
For the setup under study, we calculate the probabilitiesPAi
and PBi(i ¼ 1; 2) of having a detectable photon on the
corresponding detector during a pump pulse on paths Ai
and Bj , respectively, and the coincidence probability betweenchannels Ai and Bj (i; j ¼ 1; 2) corresponding either to twinphotons PTCij
(true coincidences) or to photons of indepen-dent pairs emitted during the same pump pulse PACij
(acciden-tal coincidences). The source coherence arises from the highdegree of correlation between the photons of a pair and it isdegraded because of multiple pairs.
Several techniques can be used to evaluate the fidelity ofentangled states [33]. It is usually deduced from the measure-ment of coincidence rates as a function of the specific controlparameter used in the Bell-type experiment considered. Thisevaluation is made quantitatively through the measurement ofthe visibility (or contrast) of the two-photon interferencefringes vij ¼ ðPmax
CAiBj− Pmin
CAiBjÞ=ðPmax
CAiBjþ Pmin
CAiBjÞ where Pmin
CAiBj
and PmaxCAiBj
are, respectively, the minimum and maximum val-
ues of the coincidence probabilities PCAiBj(also valid with
rates in this case) between detectors DAiand DBj
(i; j ¼ 1; 2) that can be obtained when varying the controlparameter. In our idealized setup, PCAiBj
¼ PTCAiBjþ PACAiBj
only results from true and accidental coincidences. The prob-ability PACAiBj
of accidental coincidences between detectable
photons on the detectorsDAiandDBj
is independent of i and j,
regardless of the control parameter, since there is no quantumcoherence between them.
For Bell splitters, the probabilities of true coincidencesbetween channels Ai and Bj satisfy the relations:PTCA1B1
¼ PTCA2B2; PTCA1B2
¼ PTCA2B1; PTCAiB1
þ PTCAiB2¼
PmaxCAiB1
− PminCAiB1
¼ PmaxCAiB2
− PminCAiB2
ði ¼ 1; 2Þ.Let PI be the probability of getting a detectable photon on
channel I, and PAC and PTC, the probabilities of accidental andtrue coincidences when the contribution of the two subchan-nels are added, i.e., counts on detectors DA1
and DA2on the
one hand, and DB1and DB2
, on the other hand. In this case,PI ¼ PI1 þ PI2 with PIi (i ¼ 1; 2) the probability of getting adetectable photon on detector DIi . In the same man-ner, PACAiBj
¼ PminCAiBj
¼ PAC=4 and PTCAiB1þ PTCAiB2
¼ PTC=2.Therefore,
vij ¼ V ¼ 11þ PAC=PTC
: ð1Þ
It follows from this analysis that the degradation of visibilitydue to multiple pairs can be measured independently of thecontrol parameter, i.e., by adding counts from the two detec-tors on each channel (or by using only one detector on eachchannel), which is much easier than determining Pmax
Cijand
PminCij
by varying the control parameter. This feature can bequite useful when setting up a SPDC source for quantum in-formation processing. for this reason, the simpler quantitiesPI , PAC, and PTC are used in the following sections.
3. PROBABILITY CALCULATIONIn this section, we derive the expressions of the probabilitiesof detecting one photon on channel A or B as well as the prob-
ability of true coincidences due to the two photons of one pairand the probability of accidental coincidences due to twophotons of different pairs. This is performed in both casesof deterministic and statistical splittings, taking into accountin both cases the degree of coherence of the photons pairs.
Let us first discuss the photon pair coherence issue. Theprobability spectral density of pair generation p0 is consideredto be uniform in all spectral intervals considered, so as to getan insight into the best achievable performance for the SPDCsource. The statistics of double pairs depend on the coherenceof the four-photon state: the probability spectral density ofdouble pair generation is given by [34]
p2 ¼12p20½J2
2 þ J4�; ð2Þ
where J2, J22, and J4 are normalized spectral distributions of
pair, two independent pair state, and four-photon state, re-spectively. J4 ¼ χJ2
2, where χ ∈ ½0; 1�, represents the probabil-ity ratio of the four-photon state to the two independent pairgeneration. In our approximation of uniform spectral distribu-tion of the photon pairs, J2 ¼ 1 and the total spectral densityof double pair probability is then
p2 ¼12p20½1þ χ�: ð3Þ
For two independent pair states, the statistics is Poissonian(χ ¼ 0) and the probability spectral density of double pair gen-eration is p2P ¼ p20=2. This is the case when pump pulses arelong compared to the coherence time of the signal and idlerphotons. If the coherence time of signal and idler photons islarge compared to the duration of pump pulses, a coherentcontribution is added to the incoherent one: the four-photonstate where the emission of the “second pair” is stimulated bythe presence of the first one, and the probability spectral den-sity of double pair generation is then p2S ¼ p20 (χ ¼ 1), whichdiffers from p2P by a factor of 2.
Taking the filters into account, we write the double pairgeneration probability as [34]
P2 ¼12p20½J2AJ2B þ J4AB �; ð4Þ
where the first term represents the product of the probabil-ities of measuring one photon of one pair through the filterFA and one photon of the other pair through filter FB and thelast one is the probability of measuring one photon of the four-photon state passing through the filter FA and another onethrough filter FB. Using Eq. (3), (4) in our simplified casebecomes
P2 ¼12p20½ΔνAΔνB þ χΔν2A�: ð5Þ
From Eq. (5), it is possible to derive the probabilities of trueand accidental coincidences in both cases of deterministicand statistical splitting. Only the case of deterministic splittingwas reported in [35].
In both cases, the schematic setup of Fig. 1 can be idealizedas a pulsed SPDC source emitting photon pairs with a spectraldensity p0 and two channels exhibiting losses due to the
834 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Smirr et al.
filters, propagation, and detector efficiency in each channel.The total transmissions of channels A and B are denoted byXA and XB, respectively. The only difference between the twocases is the statistics to be used for the photon splitting andthe maximum value of XI (XImax
¼ 1 in the deterministic caseand XImax
¼ RI in the statistical one).
A. Deterministic SplittingIn such a setup, the signal and idler photons are separatedfrom each other and the probability PðDÞ
N ðnA; nBÞ of gettingnA and nB detectable photons in channels A and B, respec-tively, for N pairs generated by SPDC in the crystal duringa pump pulse, is simply the product of the probabilities of get-ting nA photons in channel A and nB photons in channel B:
PðDÞN ðnA; nBÞ ¼ ½CN−nA
N ð1 − XAÞN−nAXnAA �½CN−nB
N ð1 − XBÞN−nBXnBB �ð6Þ
In Eq. (6), the factors CN−nIN (I ¼ A;B) (where CP
N ¼ N !=½ðN −
PÞ!P!� is the binomial coefficient, i.e., the number of P-elementsubsets of a N -element set) give the number of possible casesfor ðN − nIÞ lost photons in channel I. The probability of get-ting a photon detected by DI (I ¼ A;B) during a pump pulse isthen
PðDÞA ¼ p0ΔνA½PðDÞ
1 ð1; 1Þ þ PðDÞ1 ð1; 0Þ� ¼ p0ΔνAXA; ð7Þ
PðDÞB ¼ p0ΔνB½PðDÞ
1 ð1; 1Þ þ PðDÞ1 ð0; 1Þ� ¼ p0ΔνBXB: ð8Þ
As we have assumedΔνB ≥ ΔνA, true coincidences can be ob-tained only in the spectral bandwidthΔνA with the probability
PðDÞTC ¼ p0ΔνAPðDÞ
1 ð1; 1Þ ¼ p0ΔνAXAXB: ð9Þ
Coincidences are also obtained when two pairs are generatedduring the same pump pulse. However, the four-photon stateproduces only true coincidences, while incoherent doublepairs give rise to true coincidences in half of the cases. Theprobability of getting an accidental coincidence during onepump pulse is then simply derived from Eqs. (5) and (6):
PðDÞAC ¼ 1
4p20ΔνAΔνB½PðDÞ
2 ð2; 2Þ þ PðDÞ2 ð2; 1Þ þ PðDÞ
2 ð1; 2Þ
þ PðDÞ2 ð1; 1Þ�
¼ 14p20ΔνAΔνBXAXBð2 − XAÞð2 − XBÞ ð10Þ
Note that PðDÞAC does not depend on the coherence of the pairs,
as already mentioned in [35]. The visibility defined by Eq. (1)is thus given by
V ðDÞ ¼ 11þ p0ΔνBð2 − XAÞð2 − XBÞ=4
: ð11Þ
The visibility is only slightly degraded in the case of lowtransmission coefficients, but it can be significantly reducedif the spectral bandwidth on path B is much larger than onpath A. Let us note that it is because of our arbitrary hypoth-esisΔνB ≥ ΔνA that the visibility only depends onΔνB, so thatthe best possible visibility in the deterministic splitting setupcorresponds to ΔνB ¼ ΔνA.
B. Statistical SplittingIn such a setup, photons are statistically split between the twochannels. The probability PðSÞ
N ðnA; nBÞ of getting nA and nB de-tectable photons in channels A and B, respectively, for Nphoton pairs generated by SPDC in the crystal during a pumppulse is then given by
PðSÞN ðnA; nBÞ ¼ C2N−nA−nB
2N ð1 − XA − XBÞ2N−nA−nBCnAnAþnB
XnAA XnB
B
ð12Þ
Indeed, in such a case (2N − nA − nB) photons are lost, eachwith a probability 1 − XA − XB and the detectable photons aredistributed between channels A and B with probabilities XnA
Aand XnB
B , respectively. The factors C2N−nA−nB2N and CnA
nAþnBeval-
uate the number of possible cases corresponding to theð2N − nA − nBÞ lost photons and nA detectable photons onchannel A, respectively.
The probability of detection of a photon by detector I dur-ing a pump pulse is then given by
PðSÞA ¼ p0ΔνA½PðSÞ
1 ð2; 0Þ þ PðSÞ1 ð1; 1Þ þ PðSÞ
1 ð1; 0Þ�¼ p0ΔνAXAð2 − XAÞ ð13Þ
PðSÞB ¼ p0ΔνB½PðSÞ
1 ð0; 2Þ þ PðSÞ1 ð1; 1Þ þ PðSÞ
1 ð0; 1Þ�¼ p0ΔνBXBð2 − XBÞ ð14Þ
True coincidences (only obtained in the spectral bandwidthΔνA) occur with probability
PðSÞTC ¼ p0ΔνAPðSÞ
1 ð1; 1Þ ¼ 2p0ΔνAXAXB: ð15Þ
When two pairs are generated during the same pump pulse,the four-photon state produces true coincidences in two-thirds of the cases. The probability of an accidental coinci-dence during a pump pulse is then simply derived fromEqs. (5)and (12):
PðSÞAC ¼
2þ χ ΔνAΔνB
6p20ΔνAΔνB
× ½PðSÞ2 ð3; 1Þ þ PðSÞ
2 ð1; 3Þ þ PðSÞ2 ð2; 2Þ þ PðSÞ
2 ð2; 1Þþ PðSÞ
2 ð1; 2Þ þ PðSÞ2 ð1; 1Þ�
¼ 2p20ΔνAΔνBXAXBK; ð16Þ
where K ¼ r3½6 − 6ðXA þ XBÞ
þ 2ðX2A þ X2
BÞ þ 3XAXB�; ð17Þ
and r ¼ 1þ χ2ΔνAΔνB
: ð18Þ
Contrary to the case of deterministic splitting, the value ofPðSÞAC depends on the coherence of the photon pairs, expressed
by χ. However, the expression of r shows the limited influenceof the statistics of double pair emission on our analysis: r ¼ 1when ΔνB ≫ ΔνA, as in the case of Poissonian pair distribu-tion, independently of the value of χ, and ifΔνB ¼ ΔνA, r onlyvaries between 1 and 3=2 when χ varies from 0 to 1.
Smirr et al. Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 835
Using Eqs. (1), (15), and (16), the visibility is given by
V ðSÞ ¼ 11þ p0ΔνBK
: ð19Þ
From Eq. (17), we can see, e.g., that for incoherent pairs, Konly varies from its minimum value 1.17 obtained for the max-imum value 0.5 of XA and XB (obtained when RA ¼ RB ¼ 0:5)to its maximum value 4 obtained for XA ¼ XB ¼ 0. It is note-worthy, then, that the largest spectral bandwidth of the twopaths (here ΔνB) is the key parameter for visibility whereasthe transmission factors XA and XB are the relevant param-eters concerning the true coincidence probability [Eq. (15)].
4. RESULT ANALYSISThe preceding results can now be analyzed in terms of theinfluence of spectral bandwidth and global transmission onthe visibility, as a function of pair generation probability. Hereit is assumed that XA ¼ XB. As we have shown that the ratio offour-photon states to the total double pair number has nosignificant effect on the visibility, we will only consider herethe case of purely Poissonian double pair statistics (r ¼ 1).
A. Influence of Relative Spectral Bandwidthon the VisibilityFigure 3 shows the evolution of the SPDC visibility as a func-tion of the probability of photon pair generation in the spectralbandwidth ΔνA for various values of the relative spectralbandwidth of the two paths ΔνB=ΔνA. To focus on the effectof this bandwidth ratio, our calculations were performed for alossless system, using Eqs. (11) (with XA ¼ XB ¼ 1) and (19)(with XA ¼ XB ¼ 0:5) for deterministic and statistical split-ting, respectively.
The visibility is higher than 95% for pair generation prob-ability of the order of 0.1 whenΔνA ¼ ΔνB. But if the spectralbandwidth on path B is significantly higher (by a factor of 10to 100), then the visibility drops significantly at such high gen-eration rate. Indeed, acceptable values (V > 80%) can be ob-tained only for very low pair generation probabilities (of theorder of 5 × 10−3). This important drawback is due to thephotons on path B that do not have a twin on path A any long-er but strongly contribute to the accidental coincidences.
It should be stressed that the high visibility V ≃ 98% re-ported in [25], obtained in a highly asymmetric bandwidthSPDC source, was measured after substracting the accidentalcoincidences. When taking them into account (using Fig. 2 of[25]), the visibility goes down to ≃72%. This value is in qua-litative agreement with our analysis considering the very lowprobability of pair generation in the 22MHz bandwidth(p0ΔνA ≃ 10−5) which can be inferred (at least qualitatively)from the experimental results (see Section 6 for the estimationof probabilities in the case of CW sources).
The comparison between Figs. 3(a) and 3(b) also showsthat for a given generation rate and relative spectral band-width of the two paths, the visibility is significantly lowerin the statistical splitting configuration. This fact becomesall the more significant as the spectral bandwidth ratio be-tween paths A and B becomes larger. For instance, forp0ΔνA ¼ 0:02, the normalized visibility difference
ΔV ¼V ðDÞ
�ΔνBΔνA
�− V ðSÞ
�ΔνBΔνA
�
V ðDÞ�
ΔνBΔνA
�þ V ðSÞ
�ΔνBΔνA
� ð20Þ
is almost negligible for ΔνBΔνA ¼ 1 and reaches 18% for ΔνB
ΔνA ¼ 100.
B. Influence of Transmission Losses on the VisibilityFigure 4 shows the evolution of the visibility as a function ofthe pair generation probability for various transmission val-ues. These results were obtained for a symmetric system(ΔνA ¼ ΔνB and XA ¼ XB), using Eqs. (11) and (19) for deter-ministic and statistical splitting, respectively. When comparedto the relative spectral bandwidth of the two paths, transmis-sion losses have a very small effect on the visibility, thoughthey necessarily reduce the measured coincidence probabil-ity. The next section focuses on the compromise betweenthe expected visibility and the measurable true coincidenceprobability.
5. BEST EXPECTED PERFORMANCEThe most important figures of merit for a SPDC source are theuseful photon pair (i.e., true coincidence) probability andthe visibility. These are the relevant parameters for possible
Fig. 3. (Color online) Visibility V plotted as a function of the prob-ability p0ΔνA for various relative spectral widths of paths A and B,using (a) deterministic or (b) statistical splitting.
836 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Smirr et al.
applications to quantum communication networks and quan-tum information processing and they are given by Eqs. (9) and(11) in the case of deterministic splitting and Eqs. (15) and(19) in the case of statistical splitting of the photon pairs. Thissection is devoted to the investigation of the best achievableprobability of getting a useful photon pair generated by a fil-tered pulsed source, for a desired visibility as a function of thespectral bandwidth ratio between the two paths and theirtransmission losses. We also compare the results of our anal-ysis to reported experimental data and discuss the impact ofthese results on the design of a SPDC source for quantumcommunications.
A. Influence of Relative Spectral Bandwidth on theUseful Pair ProbabilityFigure 5 shows the probability of a useful photon pair PTC as afunction of the desired visibility for various values of the re-lative spectral bandwidths ΔνB=ΔνA of the two paths. Theseresults are obtained for a lossless system, using Eqs. (9) and(11) in the case of deterministic splitting and Eqs. (15) and(19) in the case of statistical splitting. It is noteworthy thatthe useful pair probability can be large for any relative spec-tral bandwidth, but with a strongly decreasing visibility with
increasing bandwidth asymmetry. However the useful pairprobability decreases strongly with increasing relative spec-tral bandwidth (by a factor of 10 or 100 when ΔνB=ΔνA ¼10 or 100).
Unfortunately, this result dismisses the use of asymmetricfiltering, although it could have been interesting, for instance,in quantum communication networking where photons ofchannel A would have been stored in a quantum memory(with a small bandwidth ΔνA < 1GHz) while their twinswould have propagated with a larger bandwidth, thus relaxingthe precision on the frequency stabilization of the narrowbandwidth filter.
Moreover, comparison of Figs. 5(a) and 5(b) shows thatthe useful pair probability is about 5 times smaller in thestatistical splitting case, because of the beamsplitter(RA ¼ RB ¼ 0:5), thus making deterministic splitting muchmore advantageous.
For applications in quantum information processing thatuse quantum memories, small bandwidths (Δν < 1GHz) arerequired. As a consequence, nondegenerate SPDC seems im-practicable due to the high difficulty arising from the stabili-zation of two independent filters on two given frequencies,while keeping their sum constant. Nevertheless, as already
Fig. 4. (Color online) Visibility V plotted as a function of the prob-ability p0ΔνA for symmetric paths (ΔνA ¼ ΔνB) and for various valuesof the transmissions XA ¼ XB, using (a) deterministic or (b) statisticalsplitting. In the latter case, note that the maximum transmision valueis XA ¼ XB ¼ 0:5.
Fig. 5. (Color online) Probability of a useful photon pair PTC plottedas a function of the desired visibility V for various values of the spec-tral bandwidth ratioΔνB=ΔνA using (a) deterministic or (b) statisticalsplitting.
Smirr et al. Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 837
pointed out, deterministic splitting is possible for quasi-degenerate SPDC when using noncollinear phase-matchingin type I or II nonlinear crystals, or polarization splitting ofthe photon pair in collinear SPDC in type II nonlinear crystals(PPKTP, for instance). These two techniques, however, sufferfrom lower efficiency than collinear SPDC in PPLN, due to thesmall crystal length that can be used for the former, and to thelower nonlinear coefficient for the latter. These drawbackscan make the statistical splitting of collinear quasi-degenerateSPDC in type I PPLN interesting in the case of low requiredbandwidths. However, the various possibilities must be anal-yzed thoroughly before choosing a given type of SPDC sourcefor a given application. We believe that the present analysisprovides some help in this direction.
B. Influence of Transmission Loss on the Useful PairProbabilityFigure 6 shows the probability of a useful pair as a function ofthe desired visibility for various transmission values. Theseresults are obtained for a symmetric system (ΔνA ¼ ΔνBand XA ¼ XB ¼ X), using Eqs. (9), (11), (15), and (19), for de-terministic and statistical splitting, respectively. The decrease
of the useful pair probability becomes steep at high values ofthe visibility. It is important to point out the harmful effect oflosses on the performance of SPDC sources: if a visibility V ¼90% is required, for instance, almost an order of magnitude islost on the useful pair probability for X ¼ 0:5 and 2 orders ofmagnitude for X ¼ 0:2.
The useful pair probability is significantly lower in the caseof statistical splitting than in the case of deterministic split-ting, as can be seen by comparing Figs. 6(a) and 6(b). Hereagain, losses caused by the beamsplitter are detrimental tothe source performance.
Our analysis can be directly compared to the few experi-mental results dealing with pulsed SPDC sources. Scaraniet al. [35] provided the visibility as a function of the probabilityper pump pulse p0ΔνA of creating a pair in the filter band-width. For instance, a visibility of 90% is reported forp0ΔνA ∼ 0:07. Although the comparison cannot go further be-cause the authors did not take losses into account, this resultis consistent with the ones plotted in Fig. 4(a). The compar-ison can be made more quantitative in the case of [27], wherethe authors report total losses in each channel XA ¼ XB ¼ 0:1and an emission probability of p0ΔνA ∼ 1:1 × 10−2, a probabil-ity PTC ≃ 10−4 of detected coincidences and a visibility of 97%–98%. In our approach, the maximum visibility of the system,calculated using Eq. (11), is slightly higher (V ¼ 99%), dueto the ideal square filter considered in our analysis.
Continuous wave SPDC sources cannot be quantitativelyanalyzed using the present work, since count and coincidencerates are the only relevant parameters in that case and the the-ory remains to be developped for such sources. Nevertheless,a qualitative comparison can be performed by relating therates to probabilities, using a typical frequency f 0. This fre-quency can be defined as the largest value of the filter band-width ΔνA and the inverse of the time resolution of the de-tection system. In the following, we use this definition to anal-yze the experimental results reported in Refs. [26,30].
In Ref. [26], a visibility V ¼ 97% is obtained with a pumppower of 4mW, corresponding to an approximate rate of230photonpairs=s in the small ΔνA ¼ 96MHz bandwidth. Asthe bandwidth limits the maximum frequency, this corre-sponds to a very low probability of pair generation (p0ΔνA∼2:4 × 10−5). Even considering high losses (XA ¼ XB ¼ 0:1 isprobably too pessimistic in this experiment at 780nm), ourmodel predicts a visibility of almost 1, showing that multiplepairs are not causing any limitation to this experiment.
Reference [30] reports a very high visibility (V ¼ 99%) overa wide spectral region (∼25nm). For this value, obtained at apump power of 0:25mW, the generation rate is 2:05 ×104 photon pairs =s in the ΔνA ¼ ΔνB ¼ 0:3 nm bandwidth.In this case, the maximum frequency (f 0 ¼ 2:3 × 108 Hz) is lim-ited by the coincidence time gates of 4:4 ns. The equivalentprobability of generating pairs in the useful bandwidth isp0ΔνA ∼ 9 × 10−5. Our model predicts V ¼ 99:99%, in goodagreement with the experimental results. Unfortunately, notmany experimental results are available in the SPDC sourcenarrow bandwidth regime. Nevertheless, a comparison is pos-sible with the experimental results of Ref. [26]. The authorsreport a linewidth of 9:6MHz (which corresponds to the pumprepetition rate) and a pair rate of 1500 s−1 (for a 27mW pumppower), which leads to a probability of 1:6 × 10−4 of detectinga pair per pump pulse. This value is in agreement with the
Fig. 6. (Color online) Probability of a useful photon pair PTC plottedas a function of the desired visibility V for various values of the trans-missions XA ¼ XB in the cases of (a) deterministic or (b) statisticalsplitting. In the latter case, note that the maximum transmision valueis XA ¼ XB ¼ 0:5.
838 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Smirr et al.
reported visibility of 97% with a transmission of X ≃ 0:1,which would be a quite reasonable value with the reportedexperimental setup.
In the present analysis, we used probabilities of getting sin-gle counts or coincidences during a pump pulse. In experi-ments, count or coincidence rates (numbers per second)are the relevant quantities because they give a direct knowl-edge about the quantum device rate. As already mentioned inthe introduction, in a pulsed SPDC source, the rate is simplythe probability multiplied by the pulse repetition rate, so that,for a given probability calculated in Section 3, the maximumrate is obtained for the maximum possible pump repetitionrate f M . Note, by the way, that due to this limitation of therepetition rate of the SPDC source, the spectral power densityof pairs (number of pairs per unit pump power and unit sourcebandwidth) is totally irrelevant at very high pump power orvery small bandwidth. For a pump pulse duration Δt,f M ¼ 1=ðNΔtÞ, with N large enough to generate independentpump pulses. It should also be noted that Δt ≫ 1=ΔνA in ouranalysis of the best possible quality of the SPDC source. Tak-ing ΔνA ¼ 40GHz (Δλ ¼ 0:3 nm at λ ¼ 1:5 μm), Δt ¼3=ΔνA ¼ 75ps, and N ¼ 3, one gets f M ¼ 4:8GHz. For a highvisibility V ¼ 95% of the two-photon state generated by aSPDC source using statistical splitting with XA ¼ XB ¼ 0:02(a low transmission value, due to the low detector efficiencyat λ ¼ 1:5 μm), the maximum achievable probability of de-tected pairs is PTC ¼ 2:1 × 10−5 [see Fig. 6(b)], which providesa rate of the order of 100KHz for the detected pairs. Note how-ever that these rates could be smaller if the gate duration Twas larger than the pulse duration Δt. If small bandwidths(Δν ∼ 1GHz for instance [36]) are required, as for currentquantum memories, the maximum rate of detected pairs isthen only 2KHz, which is most probably insufficient for futureapplications. Larger quantum memory bandwidth acceptancewill, therefore, be necessary in order to increase these rates.However, the present rates are sufficient to test quantumoptical circuits including quantum memories.
6. INFLUENCE OF NOISEThe coincidence probabilities derived in the preceding sec-tions do not include the effect of noise. This section succes-sively deals with the influence of pump-induced stray light andconstant noise (such as detector dark counts).
A. Influence of Pump-Induced Stray LightThe pump-induced stray light considered here is generatedwith a constant spectral probability density pON proportionalto the pump power. In the model proposed here, the opticalnoise can only get through the spectral bandwidth allowed bythe filter on the considered path I with the global transmissionXI . We can then use this approximation to estimate the prob-ability of a coincidence caused by a noise photon in a simpleway. The probability of such a coincidence has a spectral den-sity p0pON. The spectral probability density pON is consideredsmall enough for coincidences between two noise photons tobe negligible.
We investigate hereafter in what way the source quality canbe affected by the parasitic coincidences resulting from thenoise. Indeed, their contribution increases the accidental co-incidences number and degrades the visibility as can be seenin the expression
VON ¼ 1
1þ PACþPONCPTC
; ð21Þ
where PONC is the probability of an optical noise-inducedcoincidence. This probability is given by
PðDÞONC ¼ 2p0pONΔνAΔνBXAXB ð22Þ
in the case of deterministic splitting and
PðSÞONC ¼ p0pONΔνAΔνBXAXB ð23Þ
in the case of statistical splitting. As an example of the influ-ence of pumped induced stray light, we consider a symmetricsystem: ΔνB ¼ ΔνA and XA ¼ XB ¼ X in the case of determi-nistic splitting. The visibility is derived from Eq. (21) usingEqs. (9), (10), and (22), for the true, accidental, and opticalnoise-induced coincidences. Figures 7(a) and 7(b) show thegeneration probability of a useful pair PTC as a function ofthe desired visibility VON for various pON=p0 values with mod-erate losses X ¼ 0:1 and high losses X ¼ 0:02, respectively.
As expected, for a given visibility, the probability of a use-ful pair decreases when noise increases. This effect is
Fig. 7. (Color online) Probability of a useful photon pair plotted as afunction of the desired visibility VON using deterministic splitting andfor various values of the relative optical noise probability spectral den-sity pON=p0 in the cases of (a) moderate losses (XA ¼ XB ¼ 0:1) and(b) high losses (XA ¼ XB ¼ 0:02).
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nevertheless quite moderate: for instance for a visibility of95%, the probability is only divided by two when the probabil-ity ratio of noise to photon pair emission is equal to 0.3. Thismeans that even if it is clearly always better to suppress op-tical noise, its effect on source quality is not as important asthe asymmetry of spectral filtering or the transmission losses.
B. Influence of Dark CountsIf PDNA;B
is the noise count probability in the chosen detectionwindow on detector DA or DB, the consequent coincidenceprobability is
PDNC ¼ PDNAPB þ PDNB
PA þ PDNAPDNB
ð24Þ
using the expressions of PðDÞI and PðSÞ
I given by Eqs. (7) and(13), in the respective cases of deterministic and statisticalsplitting. The corresponding visibility becomes
VDN ¼ 1
1þ PACþPDNCPTC
: ð25Þ
Figures 8(a) and 8(b) show an example of the influence ofdetector noise. The useful pair probability is plotted as a func-
tion of visibility for various detector noise coefficients in thecase of deterministic splitting with symmetric paths for thetwo cases of moderate losses (X ¼ 0:1) and high losses(X ¼ 0:02), respectively.
In both cases, the visibility is limited to a given maximumdepending on the experimental situation. The visibility de-creases both at low and high true coincidence probabilitiesdue to dark counts and high pair generation rates, respec-tively. It should be noted that dark counts do not influencestrongly the maximum value of the true coincidence probabil-ity, even at moderate losses [see Fig. 8(a)]. Their effect is onlysignificant at high losses and high detector noise probability[see Fig. 8(b)]. This situation can actually occur for SPDCsources at telecom wavelengths exhibiting the low linewidthspresently necessary for applications in quantum communica-tion networking.
7. CONCLUSIONWhen quantum communication networking and quantum in-formation processing applications are considered, the visibi-lity and the coincidence rates or probabilities per pump pulseare essential qualities of SPDC sources. We have derived theoptimum probabilities of true coincidences as a function ofvisibility, taking into account source parameters such as spec-tral bandwidth and transmission losses on each path. We haveshown that even with an idealized model of uniform pair gen-eration and rectangular filtering spectra and when all spectral,spatial or temporal distinguishabilities have been taken careof, multiple pair generation still limits the performance ofSPDC sources.
This analysis demonstrates the advantage of working withidentical spectral bandwidths on the two paths, as the opti-mum probability of true coincidences decreases proportion-ally to their bandwidth ratio. These results then clearly leadto a preferred choice of a symmetric setup, for which the filterbandwidths of the two paths are identical. Transmissionlosses can also be harmful as the probability of true coinci-dences decreases by a factor of 6 to 8 when the transmissiondecreases by a factor of 2 for a given value of the visibility.One of the important issues of source design is hence photoncollection, and therefore low loss filters and optics shouldbe used.
The performance reduction of statistical compared to de-terministic splitting is larger than the mere beamsplitter lossfigure by a factor of 3. However, this advantage of determinis-tic splitting is obvious only when the symmetric spectral band-widths are large enough. New problems arise when smallspectral bandwidths are required, as, for instance, when cou-pling to a quantum memory is considered. If type I PPLN crys-tals are required for obtaining sufficiently high coincidencerates, it is then a real challenge to stabilize the two filters in-dependently while keeping constant the sum of their centerfrequencies. Statistical splitting of nearly identical signaland idler frequencies is much easier to deal with in that case.
We have shown that the optical noise that is proportional topump power does not have a significant effect if it is kept suf-ficiently lower than the pair generation rate, although itshould, of course, be minimized to achieve optimal perfor-mance. On the contrary, a threshold is observed, due to detec-tor dark counts, beyond which the desired visibility can nolonger be obtained, with this threshold being lower at high
Fig. 8. (Color online) Probability of a useful pair PTC plotted as afunction of visibility VDN using deterministic splitting with symmetricpaths (XA ¼ XB, ΔνA ¼ ΔνB) and for various detector noise coeffi-cients for the two cases of (a) moderate losses (XA ¼ XB ¼ 0:1)and (b) high losses (XA ¼ XB ¼ 0:02).
840 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Smirr et al.
losses. This clearly confirms that much care must be devotedto the system design, and highlights the importance of usinglow-noise detectors.
It is important to note that this work provides the means ofevaluating the best possible performance of any pulsed SPDCsource, taking the entire quantum link into account throughthe overall losses XA, XB, and points out the key parametersof its design.
This analysis can be applied to any pulsed SPDC source foruse in quantum communication applications. It is even of par-ticular interest applied to quantum optical devices includingquantum memories which, at least presently, require smallbandwidth SPDC sources.
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