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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*[email protected]
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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�
063846-6