Photon bunching in parametric down-conversion with continuous-wave excitation

Bibiane Blauensteiner,1 Isabelle Herbauts,1 Stefano Bettelli,1 Andreas Poppe,2 and Hannes Hübel1,*1Quantum Optics, Quantum Nanophysics and Quantum Information, Faculty of Physics, University of Vienna, Boltzmanngasse 5,

1090 Vienna, Austria2Austrian Research Centers GmbH-ARC, Donau-City-Strasse 1, 1220 Vienna, Austria

�Received 28 October 2008; revised manuscript received 8 May 2009; published 30 June 2009�

Direct measurement of photon bunching �g�2� correlation function� in one output arm of a spontaneous-parametric-down-conversion source operated with a continuous pump laser in the single-photon regime isdemonstrated. The result is in agreement with the statistics of a thermal field of the same coherence length andshows the feasibility of investigating photon statistics with compact continuous-wave-pumped sources. Impli-cations for entanglement-based quantum cryptography are discussed.

DOI: 10.1103/PhysRevA.79.063846 PACS number�s�: 42.65.Lm, 42.50.Ar, 03.67.Mn

I. INTRODUCTION

Light sources based on spontaneous parametric down-conversion �SPDC� �1�, in which photons from a pump laserare sporadically converted into pairs of photons at lower fre-quency in a nonlinear crystal, have become an essential com-ponent in the toolbox of quantum optics laboratories in thelast decade. These sources allow state-of-the-art experimentson entanglement and quantum information, due to the abilityto supply photons that are highly correlated in time, energy,and polarization.

SPDC-based systems have been exploited as a basis forentangled-photons schemes �2� as well as approximations ofsingle-photon sources in the so-called heralded-photonschemes �3�. Such setups are, however, limited by the possi-bility that additional pairs are created within the windowdetermined by the pump-pulse duration or by the electronicgate width of the single-photon detectors �for continuoussources�. In multiphoton interference experiments, additionalpairs in general affect the purity of the investigated state �4�.In quantum cryptography �5� instead, multiphoton pairs mayconstitute a security threat since their information-carryingdegree of freedom �typically polarization� might be corre-lated �6�.

The topic of SPDC-pair emission and correlation, deeplyrelated to the statistical properties of the down-convertedfield, is therefore not only interesting for fundamental quan-tum optics experiments but also concerns applied quantuminformation. Significant theoretical and experimental investi-gations have already been conducted on this subject, but adirect demonstration that the field of one arm of an SPDCsource is thermal is still missing for the special case of asource in single-photon regime operated with a continuouspump. Such evidence is presented in this paper by measuringthe second-order correlation function with a HanburyBrown-Twiss setup �7� and comparing it to a predictionbased both on the detector jitter and a first-order correlationmeasurement with a Michelson interferometer. This ap-proach relies entirely on experimentally observable quanti-ties and no assumptions are needed.

The intrinsic difficulty of the experiment stems from thefact that resolving times are much larger than the coherence

time �c of the down-converted field, so that all relevant quan-tities are averaged out and the statistics becomes similar to aPoissonian one, with the true �thermal� statistics often over-looked.

The remaining part of the introduction presents a reviewof the theoretical background �Sec. I A� and existing experi-mental evidence �Sec. I B� concerning multipair emission, aswell as a discussion of the potential security risk for quantumcryptography �Sec. I C�. The experimental setup, data acqui-sition �Sec. II�, and quantitative expectation model �Sec. III�are then described followed by a discussion of experimentalfindings �Sec. IV�.

A. Field statistics and theoretical description of SPDC

A direct characterization of the temporal statistical prop-erties of a generic quantum electromagnetic field is providedby the �normalized� correlation functions, defined as ��8�,Chap. 6�

g�1��t,t + �� =�ʆ�t + ���t��

�ʆ�t��t��=

G�1��t,t + ��

G�1��t,t�, �1a�

g�2��t,t + �� =�ʆ�t�ʆ�t + ���t + ���t��

�ʆ�t��t���ʆ�t + ���t + ���, �1b�

=G�2��t,t + ��

G�1��t,t�G�1��t + �,t + ��, �1c�

where Ê are field amplitude operators. These quantities are,respectively, field and intensity correlations at timest and t+� at the same spatial location. Since for large timedelays realistic fields are uncorrelated, g�2����=1. The be-havior for finite delays depends, however, on the actual sta-tistics of the field. If g�2� increases around �=0, the field issaid to be bunched.

A chaotic field in an interval short with respect to itscoherence time �c will show thermal statistics, that is, thedistribution Pn of the number of photons is*hannes.huebel@gmail.com

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Pn =�n

�� + 1��n+1�, �2�

where � is the average number of photons in the interval.Chaotic light, for which g�2��0�=2, is therefore bunched. Incontrast, the output of an ideal monomode laser displaysPoissonian behavior, that is, g�2����=1. Hence, in a short-time interval, the probability of counting two photons forchaotic light is twice as large as that expected for Poissonianstatistics.

With the picture of an SPDC process as a spontaneousemission of pairs from the splitting of pump photons, it couldbe naïvely believed that the statistics of pairs is similar tothat of the pump field. It is, however, known �9� that linearinteraction in a resonant medium affects incident light insuch a way that it becomes a mixture of the initial “signal”field and an additional thermal field from spontaneous emis-sion. If the incident field is absent �or already thermal�, theoutput field is therefore exactly thermal. In analogy, since theSPDC process corresponds to optical amplification withoutinput signal �as both signal and idler fields are initially intheir vacuum states�, the statistics of pairs needs not mimicthat of the pump, and thermal behavior is to be expected.

As for incoherent light sources, the quantum interpreta-tion of SPDC bunching relies on constructive two-photoninterference between probability amplitudes for the paths oftwo indistinguishable photons �10,11�. It has been shown thatthis process can equivalently be interpreted as a pair emis-sion stimulated by another down-converted pair ��11�,Chap. 7�.

That the statistics of one arm of a monomode SPDCsource is thermal was theoretically first proven in 1987 byYurke and Potasek �12�, who derived from the pure state ofthe combined single-mode down-converted fields the densityoperator of the mixed state of one field only by tracing outeither signal or idler field. In this model, the quantum state��� of the combined signal-idler fields in the Fock state ex-pansion reads as

ei��a†b†+��ab�� � � = sech����

n=0

� ���� tanh���

n

�n�si

= 1 − 12

���2� � � + ��1�si + �2�2�si + O����3� , �3�where � is a parameter proportional to the pump amplitude,a†�b†� is the creation operator for the idler �signal� modewith fixed polarization, and �n�si=

1n! �a

†b†�n�� � is a Fockstate with n pairs. By tracing over one field, say i, the domi-nant signal-idler correlation is suppressed, and a mixed den-sity matrix is obtained for the residual field that displaysexactly thermal statistics,

�s = �n=0

�

Pn�n�s�n� = �n=0

��n

�� + 1��n+1��n�s�n� , �4�

where �=sinh2��� and Pn is the probability of finding exactlyn signal photons. The SPDC process is nevertheless inher-ently quantum mechanical, and its “thermality” is not an ef-

fect of the trace operation but originates directly from thenature of the pair-production process �12�.

Since SPDC production is far from being monomode,Yurke and Potasek’s analysis was not realistic; however, thethermal nature of a multimode SPDC process was later con-firmed by Tapster and Rarity �13� and, in a more generalway, by Ou, Rhee and Wang �4,14�, who analyzed the case ofsources with pulsed pumping. In their derivation, what isactually measured is the pulse integral of G�1� and G�2�, i.e.,the probabilities of one or two photons in the pulse. Withtime resolution limited to the whole pulse, one must resort totest the “bunching excess,” which was shown �4� to be es-sentially the ratio of the coherence time �c of the photons inthe inspected arm and the duration �p of the pulse, when�c��p. Shorter pump pulses then increase the excess, but—contrary to intuition—the limit of a very short pulse is notsufficient alone to guarantee a full bunching peak �g�2��0�=2�. In ��11�, Chap. 7�, it is shown that this condition canonly be achieved with the help of narrow spectral filtering.

SPDC sources have also been implemented with continu-ous pumping, using continuous-wave �cw� lasers, which areboth more practical and more affordable than femtosecond�fs� pulsed SPDC sources. The field emitted by cw sources,and its statistical properties, is still relatively unexplored.Theoretically, it is usually assumed that resolving times areso much larger than the coherence time of the down-converted field that all relevant quantities are not accessible.It has been stated �15–18� that the statistics of SPDC as amultimode process, i.e., when the pulse duration of the laseris much longer than the coherence time of the produced pho-tons, will lead to Poissonian statistics. Careful experimentaldesign and data analysis, as shown in this paper, reveal how-ever the inherent thermal statistics hidden by an apparentPoissonian behavior.

B. Current experimental evidence

Photon statistics in SPDC processes can be directly ac-cessed using a Hanbury Brown-Twiss setup by looking atphoton emission in a single output arm only, i.e., by measur-ing the g�2� of the signal or idler arm. In such an investiga-tion, however, jitter and arrival-time discretization smear theresult. Due to the aforementioned disproportion between thetime resolution of available single-photon detectors and theSPDC coherence time, most experimental investigationshave exploited fs-pulsed sources and strong filtering to in-crease artificially the coherence time of the measured field.Positive evidence in single-photon regime was first reportedby Tapster and Rarity �13�, who compared quantitative pre-dictions based on a specific experimental hardware withmeasurements performed with various filters; with the nar-rowest one, a very convincing bunching peak of 1.85 couldbe demonstrated. Six years later, de Riedmatten et al. �19��almost� replicated the experiment, with similar shortcom-ings and results, concluding that the amount of bunchingdepends only on the ratio �c /�p. In both cases, an apparenttransition from thermal to Poissonian statistics was observedwhen �c was varied below �p. Söderholm et al. �17� studiedthe dependence of single- and double-click rates on pump

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power in the two cases �c�1.75�p and �c�0.6�p, and foundgood agreement with a thermal model in the first case, andan intermediate behavior in the second case. On the otherhand, Mori et al. �18� studied a source with very long pumppulses �40 ps� and broad filtering leading to a rather shortcoherence time �c ��120 fs� and failed to detect any signifi-cant bunching. Similarly, in a recent article, Avenhaus et al.�15�, using 60 ps laser pulses, observed Poissonian statisticsfor the SPDC process ��c not stated�.

In an experiment where, contrary to the single-photon re-gime, several photons per pulse are created, Paleari et al.�20� were able to demonstrate—with careful fitting of themeasured distribution—a thermal behavior, although their ra-tio �p /�c was rather unfavorable �in the range 10–20�. Vasi-lyev et al. �21� used a strongly filtered optical homodynetomography to extract the photon-number distribution of aSPDC process and compare it with a thermal one, findingalmost perfect agreement �22�.

SPDC sources have also been implemented using cwpumping; this alternative approach is still relatively unex-plored due to the technical difficulty of measuring very shortcorrelation times. Larchuk et al. �16� tried to investigatesources in the single-photon regime with a single detectorand delay lines but failed to obtain any evidence because thedead time �about 1 �s� makes the apparatus blind to theinteresting region �1 ps. Super-Poissonian behavior wasfound by Zhang et al. �23� by directly observing the photo-currents of orthogonally polarized twin beams from a con-tinuously driven KTP crystal in a cavity.

Measurements of the marginal, i.e., one arm, SPDC fieldare not to be confused with measurements of the betterknown statistics of the whole biphoton state, as well as con-ditioned statistics, which have been experimentally con-firmed, among others, by �24–27�.

C. QKD and SPDC bunching

Sources based on continuously pumped SPDC are oftenemployed in quantum cryptographic devices, generating themultiphoton pair state of Eq. �3�. Correlations between mul-tiphoton pairs constitute a potential threat to the security ofquantum key distribution �QKD� as the exchange is thensusceptible to a photon-number-splitting attack �28�. Thecase of entangled-photon sources �29� is still relatively un-explored, and its solution depends on a thorough understand-ing of the extent to which multiple SPDC pairs are corre-lated.

In entangled-photon sources, the state �to first order� cor-responds to a pair completely entangled in the information-carrying degree of freedom �polarization�; e.g., for type-IISPDC �−�= ��Hs ,Vi�− �Vs ,Hi�� /�2. When the second orderof the expansion �Eq. �3�� is considered �two pairs�, twolimiting cases emerge: sister pairs are in well distinguishabletemporal modes and are therefore completely uncorrelated inpolarization, with all polarization combinations equallylikely,

�sisters� =�2H,2V� + �2V,2H� − �2�HV,VH�

2, �5�

while twin pairs are in the same temporal mode �within �c�,and

�twins� =�2H,2V� + �2V,2H� − �HV,VH�

�3 . �6�

For twin states, it has been shown that the probability thatthe two signal photons have the same polarization is 23 �6�;therefore, whereas no kind of photon-number-splitting attackis possible with sister states, twin states are potentially dan-gerous for quantum key distribution. The situation is analo-gous for type-I SPDC.

The works of Tsujino et al. �30� and Ou �31� showed thatpolarization correlation and one-arm bunching are two sidesof the same phenomenon. The peculiar properties of the statespace of identical particles imply that, for delays shorter than�c, signal photons with the same polarization occur morefrequently than expected for classically uncorrelated objects�as differently polarized photons�.

Summarizing, the study of one-arm g�2� in entangled-photon sources �32� gives information about the maximumpossible extent of the security threat posed to quantum keydistribution by higher orders of SPDC.

II. EXPERIMENTAL SETUP AND DATA ACQUISITION

In our SPDC source �33� �Fig. 1�, a 532 nm cw laserpumps a single 30 mm temperature-stabilized nonlinear crys-tal �periodically poled KTP� for type-I down-conversion�532 nm→810 nm+1550 nm�. The signal and idler pho-tons have fixed polarizations, central wavelengths of, respec-tively, 810 and 1550 nm, and are emitted collinearly. Thesignal coherence time, as measured with a Michelson inter-ferometer �Fig. 3�, is �c�2.8 ps FWHM, corresponding to abandwidth of less than 1 nm.

Signal and idler photons are separated according to theirfrequency and coupled into single-mode optical fibers. Sig-nal photons �at 810 nm� are then recollimated and sent to abalanced and nonpolarizing beam splitter. The resultingbeams are collected into multimode fibers. The coupling ef-ficiency is greater than 90% as the light is coupled fromsingle-mode fibers into multimode ones. The multimode fi-bers direct the beams into a PERKINELMER SPCM-AQ4C single-photon detector array. The parameters of each detector are asfollows: quantum efficiency at 810 nm �50%, dark-countrate �500 Hz, dead time �50 ns, and detector saturation

FIG. 1. �Color online� The experimental setup consists of aSPDC source of photon pairs, a balanced beam splitter, and twosingle-photon silicon detectors combined in a Hanbury Brown andTwiss �7� configuration.

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level above 1 MHz. All detector clicks are recorded by atime-tagging unit �TT8, smart systems division, ARC� with acommon time basis and an intrinsic resolution of �TT=82.2 ps. Time tags are then processed to define coinci-dences.

One of the two fibers is a 100 m spool, so that it operatesas a delay line of approximately 500 ns �this delay is re-moved by software during data analysis�. The purpose of thedelay line is to make photons that happen to come with 1 psdelay to impinge on detectors at sufficiently different timesto avoid electronic cross talk and increase sensitivity.

The timing jitter of the overall detection unit was charac-terized by measuring correlations in arrival times of signal-idler photon pairs from an additional degenerate SPDCsource at 810 nm �405 nm→810 nm+810 nm�. This aux-iliary photon-pair source is based on a Sagnac configuration�34�. For the present measurement of jitter, it was operated toyield �1 MHz detected single rates, below detector satura-tion. The coincidence rate of the photon-pair source was ap-proximately 100 kHz. The combined jitter of the detectionunit is estimated to be � j =640 ps��c. This estimatedFWHM is, however, not used in further experimental analy-sis, and no assumption on the shape of the jitter is needed.Instead, the fully normalized jitter curve, as seen in the crosscorrelation in Fig. 2, is used to characterize the detectionunit; the only extracted parameter being the value of the areaunder the jitter curve �611 ps�. Several measurements of thejitter histogram were taken, and the deviation on the jitterarea is estimated to be less than 3%.

A cross-correlation histogram is obtained from recordeddetector clicks. A coincidence from detector D1 to detectorD2 with delay t= t2− t1 is counted if D1 clicked at time t1and D2 clicked at time t2� t1, irrespective of any other click.Coincidences with the role of detectors reversed are definedaccordingly, and the two functions are joined by arbitrarilydefining the delays t from D2 to D1 as negative.

The coincidence density for uncorrelated photons, e.g.,when t is large with respect to any coherence time, is theproduct of the individual experimental detection rates �1 and�2 of D1 and D2 �which include the effects of dead times anddark counts�. Therefore, every bin of the coincidence histo-gram will contain, on average, and in the case of uncorre-lated photons, C=�1�2�TTT counts, where T is the durationof a data-taking run. Since C contains both single rates �1and �2, it is directly proportional to the square of the pumppower.

III. MODEL OF EXPECTED PHOTON BUNCHING ANDMEASUREMENTS

Since the jitter � j of the detection unit is much larger thanthe coherence time �c of the down-converted light, any g�2�peak will be strongly smeared. A rough estimate for the re-sidual peak height is given by the ratio of the coherence timeto the jitter �c /� j �4�10−3.

For a more accurate calculation, we model how a theoret-ical thermal bunching peak �g�2��0�=2� will be transformeddue to our instruments and then compare it with the experi-mental result. SPDC bunching is characterized by the samerelation between g�2� and the first-order coherence g�1� thatholds for chaotic light �8�, namely,

g�2���� = 1 + �g�1�����2. �7�

The g�1� of the signal beam was measured in a Michelsoninterferometer and can be seen in Fig. 3. The setup for thismeasurement consisted of a Michelson interferometer with amotorized movable mirror and some limiting apertures inboth arms to increase visibilities. The maximum visibility ofthe interferogram was measured to be 90%. The mirror is setto move at a constant speed of 0.002 mm/s over a range of 3

FIG. 2. �Color online� Second-order correlation function be-tween the idler and signal arms of an auxiliary degenerate SPDCsource at 810 nm �405 nm→810 nm+810 nm� rescaled to unity.Since the time correlation between photons of the same pair is tight�subpicoseconds�, the FWHM �approximately 640 ps� is essentiallyan estimate of the combined jitter of the two silicon detectors andthe time-tagging unit.

FIG. 3. �Color online� g�1� of the 810 nm arm of the SPDCsource as measured in a Michelson interferometer vs optical pathdifference. g�1� was calculated from the fringe visibility of the in-terferogram. The experimental visibility at zero optical path differ-ence is slightly less than 1. When the maximum of the visibility isrescaled to one, the total area under the squared envelope is�2.2 ps. The inset shows the visibility fringes around the centralpart of the interferogram.

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mm and detection counts were recorded over integrationwindows of 2 ms. At unequal path lengths, the detector countrate at the output of the interferometer was in the range of200 kHz. The interference fringes in Fig. 3 were obtainedfrom the raw data by subtracting the background rates�around 5 kHz� and are displayed with an averaging of 30data points to reduce fluctuations away from the center of theinterferogram.

The envelope of the interferogram as shown in Fig. 3, thatis, �g�1��, is calculated from the visibility of the interferencefringes in the raw data �see inset in Fig. 3�. An estimate ofthe coherence time �c�2.8 ps is obtained from the graph.After rescaling the data points of the �g�1�� envelope to com-pensate for the reduced visibility �90%� in the interferogram,an area of 2.17 ps was found for the squared envelope �g�1��2.The error on this value can be estimated by considering thearea of several interference scans in identical experimentalconditions; a conservative estimate is a 4% error on the cal-culated area for the squared envelope.

To model the washed-out g�2� peak, we note that the totalnumber of coincidences in the excess peak is preserved de-spite the jitter. Hence we expect to observe a smeared peakwith the same temporal profile as that in Fig. 2 but with anarea equal to that of �g�1��2 �35�. This method does not rely onany a priori knowledge of the actual shapes of g�1� or g�2� butonly on a ratio of areas. The estimated coherence time �c andjitter � j of the detection unit from the FWHM of their respec-tive histogram are not used in the calculation neither is anyassumption on curves shape necessary.

The correlation function g�2� of the field from one arm ofa SPDC source was measured using the Hanbury-Brown-Twiss-like setup introduced in the previous section. A cross-correlation histogram for positive and negative delays wasobtained from detector clicks and the bunching peakemerged in the region t�0 over the plateau of accidentalcoincidences. The average count number of the plateau wasthen used to normalize the histogram. Experimental datawere collected during a 16 h data-taking session with anaverage count rate of �1 MHz in each detector �below satu-ration�, giving a plateau count of N4.4�106 events perbin. The observed rates used for jitter characterization wereset to be identical to the rates in this measurement sincedetector jitter is dependent on actual count rates. The doublepair term in Eq. �3� scales with the square of the pumppower, as does the plateau �see the estimate C at the end ofSec. II�; hence the normalized g�2� function is independent ofpump power. The same argument can be used to prove thepeak to be independent of optical losses in the setup.

In Fig. 4, we show a comparison of the measured data tothe expected g�2� of our model. Data points correspond to theexperimental g�2� histogram of the marginal SPDC field. Thesolid line is the expectation for the bunching peak of a fieldwith thermal statistics. This line was obtained by verticallyrescaling the jitter function shown in Fig. 2 to yield the samearea as �g�1��2. The comparison does not involve any freeparameter to be fitted, as the theoretical model is only basedon the specific and independently measured parameters ofthe setup.

The peak was shifted laterally by �50 ps for better com-parison with the experimental data; this is however less than

the intrinsic resolution �TT. Statistical fluctuations of thenumber of events in a bin of the histogram are on the orderof �N. The cumulative error of �5% on the expected g�2�curve, arising from a conservative error estimation from thejitter and �g�1��2 data, is shown in Fig. 4 as a shaded area.Even this 10% error margin fits the data points and, there-fore, the good match of the predicted peak is not accidental.

The emerging bunching from the SPDC field is in excel-lent agreement with the theoretical estimation of a thermalstatistics. The peak protrudes six standard deviations abovethe Poissonian limit of g�2�=1 and is therefore experimen-tally confirmed.

IV. CONCLUSIONS

We experimentally measured the marginal temporalsecond-order correlation function g�2���� of SPDC light withcontinuous-wave excitation in the single-photon regime. Theresults represent a direct observation of photon bunching forsuch a field and strongly confirm its thermal character. Evenwith an experimental excess peak g�2��0� as low as 10−3, ourexperimental accuracy is sufficient to properly observe theresidual bunching peak, which can be modeled solely on theexperimental timing jitter and the output of a first-order in-terferometric measurement �g�1��. With much higher timingprecision �e.g., lower detector jitter�, it is expected that a fullpeak with the shape given by Eq. �7� is recovered. The rel-evant parameter is the ratio between the coherence length �cand the measurement resolution �the jitter � j in the cw case�.

We therefore conclude that the thermal photon statistics ofthe marginal SPDC field can be observed both in the fspulsed and in the cw regime, the latter being more practicaland more affordable. A multimode excitation does not

FIG. 4. �Color online� Second-order correlation function�signal-signal g�2�� of the 810 nm arm of a SPDC source normalizedover a measured plateau of N counts in a bin. The solid line repre-sents the washed-out bunching peak �original height of 2� obtainedfrom the jitter plot rescaled to an area of 2.17 ps. Error bars, rep-resenting statistical fluctuations of the measured data, are set to �N.The shaded gray area, only visible at the very top of the peak,represents the uncertainty in the model prediction. The inset showsan enlarged section of the plateau.

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change the statistics in SPDC but alters only the experimen-tal conditions, so that without a careful analysis an apparentPoissonian character is observed �as presented in some worksreviewed in Sec. I B�, instead of the inherent thermal statis-tics. We have shown in this paper that even with far-from-ideal detection modules and without any additional spectralfiltering, we could demonstrate as a proof of principle thatthe true nature of the field remains accessible.

It is hoped that the results presented here stimulate furtherinvestigations into the nature of SPDC light and its thermalproperties. For cw pumping, there is no clear consensus onthe actual state of multiphoton pairs nor is its extension toentangled states produced by cw pumping well understood.Similar experiments would provide experimental data to con-firm that bunching implies polarization correlations. In addi-

tion, cw pumping allows a continuous temporal investigationin a time domain not accessible to experiments with pulsedpumping and, hence, with an improved timing resolution,access to the shape of g�2�, and not only to its integral �asingle data point� as in pulsed setups.

ACKNOWLEDGMENTS

We would like to thank Michael Hentschel, ThomasLorünser, and Edwin Querasser for technical support withthe experiment, as well as Miloslav Dusek, Momtchil Peev,Thomas Jennewein, Saverio Pascazio, Bahaa Saleh, and An-ton Zeilinger for fruitful discussions and scientific guidance.Financial support is acknowledged from the Austrian FWF�Grants No. SFB15 and No. TRP-L135�.

�1� The Physics of Quantum Information, edited by D. Bouw-meester, A. K. Ekert, and A. Zeilinger �Springer-Verlag, Ber-lin, 2000�, and references therein.

�2� P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 �1995�.

�3� C. K. Hong and L. Mandel, Phys. Rev. A 31, 2409 �1985�, andreferences therein.

�4� Z.-Y. J. Ou, J.-K. Rhee, and L. J. Wang, Phys. Rev. A 60, 593�1999�.

�5� V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek,N. Lütkenhaus, and M. Peev, e-print arXiv:0802.4155, Rev.Mod. Phys. �to be published�.

�6� M. Dušek and K. Brádler, J. Opt. B: Quantum SemiclassicalOpt. 4, 109 �2002�.

�7� R. H. Brown and R. Q. Twiss, Nature �London� 177, 27�1956�.

�8� R. Loudon, The Quantum Theory of Light, 2nd ed. �OxfordUniversity Press, Oxford, 1983�.

�9� S. Carusotto, Phys. Rev. A 11, 1629 �1975�.�10� U. Fano, Am. J. Phys. 29, 539 �1961�.�11� Z.-Y. J. Ou, Multi-Photon Quantum Interference �Springer,

New York, 2007�.�12� B. Yurke and M. Potasek, Phys. Rev. A 36, 3464 �1987�.�13� P. R. Tapster and J. G. Rarity, J. Mod. Opt. 45, 595 �1998�.�14� Z.-Y. J. Ou, Quantum Semiclassic. Opt. 9, 599 �1997�.�15� M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W.

Mauerer, I. A. Walmsley, and C. Silberhorn, Phys. Rev. Lett.101, 053601 �2008�.

�16� T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, in Fundamen-tal Problems in Quantum Theory: A Conference Held in Honorof Professor John A. Wheeler �New York Academy of Sci-ences, New York, 1995�, pp. 680–686.

�17� J. Söderholm, K. Hirano, S. Mori, S. Inoue, and S. Kurimura,in Proceedings of the 8th International Symposium on Foun-dations of Quantum Mechanics in the Light of New Technol-ogy, edited by S. Ishioka and K. Fujikawa �World Scientific,Singapore, 2006�, pp. 46–49.

�18� S. Mori, J. Söderholm, N. Namekata, and S. Inoue, Opt. Com-mun. 264, 156 �2006�.

�19� H. De Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel,H. Zbinden, and N. Gisin, J. Mod. Opt. 51, 1637 �2004�.

�20� F. Paleari, A. Andreoni, G. Zambra, and M. Bondani, Opt.Express 12, 2816 �2004�.

�21� M. Vasilyev, S.-K. Choi, R. Kumar, and G. M. D’Ariano, Opt.Lett. 23, 1393 �1998�.

�22� Homodyne detection with a 10 MHz bandpass filter, henceextremely tight filtering.

�23� Y. Zhang, K. Kasai, and M. Watanabe, Opt. Lett. 27, 1244�2002�.

�24� O. Haderka, J. Peřina, M. Hamar, and J. Peřina, Phys. Rev. A71, 033815 �2005�.

�25� E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, Phys.Rev. A 73, 033814 �2006�.

�26� D. Achilles, C. Silberhorn, and I. A. Walmsley, Phys. Rev.Lett. 97, 043602 �2006�.

�27� M. Tengner and D. Ljunggren, e-print arXiv:0706.2985.�28� G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, Phys.

Rev. Lett. 85, 1330 �2000�.�29� A. Treiber, A. Poppe, M. Hentschel, D. Ferrini, T. Loruenser,

E. Querasser, T. Matyus, H. Huebel, and A. Zeilinger, New J.Phys. 11, 045013 �2009�.

�30� K. Tsujino, H. F. Hofmann, S. Takeuchi, and K. Sasaki, Phys.Rev. Lett. 92, 153602 �2004�.

�31� Z.-Y. J. Ou, Phys. Rev. A 72, 053814 �2005�.�32� As opposed to studying g�2� in attenuated lasers and heralded-

photon sources.�33� This polarization-entangled source was designed for quantum

cryptography, and will be the subject of a future publication�M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger �unpub-lished��. Here the source was employed in unentangled mode,with down-converted fields having fixed linear polarization.

�34� A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A.Zeilinger, Opt. Express 15, 15377 �2007�.

�35� This analysis does not take into account spatial �transversal�coherence. Due to the use of single-mode fibers, the setup wassufficiently optimized for such considerations to have no majorimplications in the model.

BLAUENSTEINER et al. PHYSICAL REVIEW A 79, 063846 �2009�

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