Characterizing the non-classicality of mesoscopic optical twin-beam states
We present a robust tool to analyze nonclassical properties of multimode twin-beam states in the mesoscopic photon-number domain. The measurements are performed by direct detection. The analysis exploits three different non-classicality criteria for detected photons exhibiting complementary behavior in the explored intensity regime. Joint signal-idler photon-number distributions and quasi-distributions of integrated intensities are determined and compared with the corresponding distributions of detected photons. Experimental conditions optimal for nonclassical properties of twin-beam states are identified.
pacs:42.50.Dv, 42.50.Ar, 42.65.Lm, 85.60.Gz
Quantum nature of an optical state is mandatory
for exploiting the state in many useful applications including
those in quantum information and metrology
(1); (2); (3); (4); (5). By definition,
a state is nonclassical whenever it cannot be written as a
positive superposition of coherent states. Using the
Glauber-Sudarshan representation of a statistical operator
(6); (7), nonclassical states are described by
negative or even singular probability -functions
(quasi-distributions). However, as -functions introduced in
this representation cannot be directly observed, also other
non-classicality criteria based on measurable quantities have
been derived (8); (9); (10); (11); (12).
For instance, the negativity of the Wigner function of a state
available experimentally is commonly used as a non-classicality
indicator (13); (14); (15). Unfortunately,
this function is defined only for single-mode states and so it
cannot be used to describe the usual spectrally and spatially
multimode fields (16); (17). Moreover, the retrieval of Wigner function,
typically obtained through optical homodyne tomography, is in
general challenging as it requires optimal spatio-temporal
matching between the state under investigation and a local
oscillator (18); (19); (20).
An alternative approach to investigate the quantum properties of a state is provided by the direct detection of the number of photons in the state. Direct detection offers the possibility to reconstruct the photon-number distribution and evaluate possible correlations between the components of a bipartite state (21); (22); (23). The non-classicality of a photon-number distribution can be indicated by the values of its Fano factor ( and stand for variance and mean value, respectively): means nonclassical sub-Poissonian statistics (24); (25). On the other hand, when a bipartite state exhibits photon-number correlations, a noise reduction factor ( and are the signal and idler photon numbers) having values lower than 1 indicates non-classicality (26); (27); (28); (29); (30).
As one has no direct access to photons, it is of paramount importance to define non-classicality criteria in terms of detected photons. In fact, the introduction and exploitation of non-classicality conditions for measurable quantities give the possibility to avoid the use of photon-number reconstruction methods that are in general complex. In this paper, we experimentally investigate optical multimode twin-beam (TWB) states containing sizeable numbers of photon pairs. We report on the characterization of their quantumness by means of a direct detection scheme involving two photon-counting detectors that are able to operate in the mesoscopic photon-number domain, in which more than one pair of photons is produced at each laser shot. In particular, we compare three different non-classicality criteria based on detected photon-number correlations and discuss the conditions suitable for their application. Moreover, we compare these criteria with the genuine definition of non-classicality using both the measured joint signal-idler detected-photon distributions and reconstructed joint signal-idler photon-number distributions and the corresponding quasi-distributions of integrated intensities (31).
Even if the overall detection efficiency of our apparatus is relatively low, we demonstrate that quantities determined for detected photons are sufficient to reveal the quantum features of the generated TWB states. The presented comprehensive approach can thus be considered as a robust tool for discriminating nonclassical TWB states in different experimental regimes.
The paper is organized as follows. Experimental setup is described in Sec. II. Nonclassical characteristics of twin-beams derived for detected photons are analyzed in Sec. III. Sec. IV is devoted to the reconstruction of joint signal-idler photon-number distributions, the determination of quasi-distributions of integrated intensities and their nonclassical features. Conclusions are drawn in Sec. V.
Ii Experimental implementation of multimode TWB states
According to the experimental setup shown in Fig. 1, mesoscopic TWB states were obtained in spontaneous parametric down-conversion (SPDC) in a nonlinear crystal with susceptibility. In particular, we sent the third harmonics (at 266 nm) of a cavity-dumped Kerr-lens mode-locked Ti:Sapphire laser (Mira 900, Coherent Inc. and PulseSwitch, A.P.E.) to a type I -BaBO crystal (BBO hereafter, 8x8x5 mm, cut angle ) tuned for slightly non-collinear interaction geometry. 100-fs long pump-beam pulses were delivered at frequency 11 kHz.
The TWB states generated by the apparatus are intrinsically multimode, both in spatial and spectral domains. By assuming that the output energy is equally distributed among modes in each beam, the overall multimode state can be written as a tensor product of identical single-mode twin-beam states (32); (33); (34); (35), i.e.,
where represents an -photon state coming from equally-populated modes that impinge on the detector and
is a multimode thermal photon-number distribution having mean photons (36). The TWB state in Eq. (1) exhibits photon-number correlations that are provided by pairwise character of SPDC. To investigate the nature of such correlations and describe their properties, we collected two frequency-degenerate (at 532 nm) parties of the TWB state using two symmetric cage systems. The light in each arm was spectrally filtered by a bandpass filter at high transmissivity, spatially selected by an iris with variable aperture, focused by a lens ( mm) into a multimode fiber (600-m-core diameter) and delivered to the photodetector. In particular, we used a pair of hybrid photodetectors (HPD, mod. R10467U-40, Hamamatsu, Japan). These detectors are composed by a photocathode, whose quantum efficiency is about in the investigated spectral region (37); (38), followed by an avalanche diode operated below breakdown threshold. The internal amplification has a gain profile narrow enough to allow photon-number resolution. The output of each HPD was amplified (preamplifier A250 plus amplifier A275, Amptek), synchronously integrated (SGI, SR250, Stanford) and digitized (ADC, PCI-6251, National Instruments). To perform a systematic characterization of the generated TWB states, each experimental run was repeated 200,000 times for fixed choices of pump mean power and iris sizes.
Iii Nonclassical characteristics of detected photons
By exploiting the self-consistent analysis method extensively described in (37); (39), we processed the output of each detection chain, obtained detected-photon-number distributions and evaluated shot-by-shot photon-number correlations. In accordance with Eq. (2) and by taking into account invariance of the functional form of statistics under Bernoullian detection (40), the detected photon-number distributions are described by multimode thermal distributions, in which the number of modes can be determined as (34); (38), where stands for the number of detected photons, denotes the number of photons and is the quantum detection efficiency. In Fig. 2 we plot the experimental detected-photon-number distributions in the signal arm for three different values of the pump-beam power keeping fixed the value of iris size (dots). Lines are the expected theoretical curves obtained from Eq. (2) by replacing by the measured mean number of photons. The mean detected-photon numbers presented in Fig. 2 demonstrate the capability of the detection apparatus to capture TWB states in different intensity regimes. Nevertheless, it is worth noting that the SPDC gain is linear in the whole investigated photon-number domain. This is evident in Fig. 3, where we show the mean values of photons detected in the signal arm as functions of the pump mean power for different values of iris sizes.
The observed detected-photon-number correlations were quantified by means of the correlation coefficient
that is plotted in Fig. 4 as a function of the value of iris sizes
(41). However, as already demonstrated in
(42), the existence of correlations is not sufficient
to discriminate between quantum and classical states. For example,
bipartite states obtained by dividing classical super-Poissonian states at a beam splitter also display photon-number correlations (43); (44).
The noise reduction factor mentioned above is an explicit marker of non-classicality originating in photon-number correlations. For detected photons it is determined along the formula
It has been shown (45) that whenever the value of lies in between and 1 (41), the detected state is nonclassical. In this case, we have sub-shot-noise correlations since the fluctuations in the detected photon-number correlations are below the shot-noise level (46); (47); (48). The behavior of as a function of the value of iris sizes is quantified in Fig. 4, in which the nonclassical character of all obtained data is confirmed (49). To produce the theoretical values shown in Fig. 4, we inserted in Eqs. (3) and (4) the experimental values of , , and obtained in a self-consistent way (33) for each considered value of the iris sizes. This results in the irregular behavior of the curve connecting the obtained points in the graphs in Fig. 4. Comparison of the curves in Figs. 4 and 4 reveals complementary behavior of values of the correlation coefficient and noise reduction factor . Moreover, it follows from the curves in Fig. 4 that the noise reduction factor attains its minimum for a certain value of iris sizes.
This occurs when the irises are -mm wide and select the largest possible portions of the twin-beam cones (50). This explanation is confirmed by the behavior of mean detected-photon numbers in the signal arm depending on the iris sizes. As shown in Fig. 5 the mean detected-photon numbers stop increasing linearly with the iris size at the same value. Also the maximum extension of emission cones beyond the filters was reached in the horizontal plane at this value. Further increase in mean detected-photon numbers is caused only by additional contributions in the vertical plane.
The values of and plotted in Fig. 4 may be
divided into three groups depending on different values of iris
sizes. For small values of the iris sizes, and get smaller
and higher values, respectively, as only a small portion of the
twin beam is collected. For moderate values of the iris sizes,
and reach their highest and smallest values,
respectively, due to optimum collection conditions. For large
values of the iris sizes, smaller values of together with
greater values of are observed because the irises exceed
the width of the cone.
We discuss advantages and limitations of the noise reduction factor as nonclassicality quantifier in comparison with other two quantities. In particular, we consider a ratio derived from the Schwarz inequality (51) for detected photons:
If the state is nonclassical. The second analyzed quantity is determined from a more recent criterion based on higher-order detected-photon-number correlations (33):
where is the th-order correlation function and represents its symmetrized version. If the state is nonclassical.
In Fig. 6, we show the results obtained by applying the above non-classicality criteria to the experimental data. The three quantities are plotted as functions of the mean number of photons detected in one of the two arms: good quality of our data is confirmed by the fact that all criteria are satisfied simultaneously. For each criterion the data are distributed into three groups differing in iris sizes, as already mentioned in the description of Fig. 4. It is also interesting to note that all the experimental points (except a very few of them) obtained for different values of pump mean powers and iris sizes are in good agreement with the corresponding theoretical predictions calculated for the actual values of experimental parameters. In particular, the theoretical curve of noise reduction factor was drawn along the formula
that represents a generalization of the expression derived in (42) to the multimode case. In Eq. (7), gives the average of the signal and idler mode numbers, and are the experimental mean signal and idler detected-photon numbers and a common quantum detection efficiency was determined from the formula valid for an ideal twin beam (33). As the curves in Fig. 6 document, the values of noise reduction factor are practically independent of the mean detected-photon numbers. On the other hand, quantities related to the other two non-classicality criteria depend strongly on the mean detected-photon numbers. Whereas the Schwarz inequality is more suitable for detecting non-classicality for small mean detected-photon numbers, the inequality based on higher-order moments is preferred for larger mean detected-photon numbers. In fact, this criterion is more sensitive to noise with respect to the other two criteria because of the presence of higher-order moments. As a consequence, when the mean numbers of photons are very low, a lot of acquisitions is required for successful application of this criterion.
Iv Nonclassical characteristics of the reconstructed photon fields
The generated TWB states are highly nonclassical as they are
composed of photon pairs. The amount of their non-classicality
decreases during their propagation towards the detectors as some
of photons lose their twins. However, by far the largest loss of
non-classicality occurs during the detection by hybrid
photodetectors as their actual overall detection efficiencies lie
around 17, as confirmed by the minimum value achieved by .
Despite this and in accordance with the results of the previous
Section, even the detected photons exhibit strong pairwise
correlations that guarantee nonclassical behavior of the
detected-photon fields. Nevertheless, the amount of
non-classicality found in the detected-photon fields is
considerably lower compared to that of the original TWB containing
For this reason, it is important to reconstruct the original TWB in terms of photon numbers starting from the experimental detected-photon distributions in order to reveal the quantum nature of state emitted in the nonlinear process. The reconstructed joint signal-idler photon-number distributions can be obtained either by applying the maximum-likelihood approach (52); (53); (54); (55) or by fitting the experimental detected-photon distributions using a special analytical form of the photon-number distribution (35); (54). The second approach is more convenient as it allows us to determine also quantum detection efficiencies and of the signal and idler beams, respectively (55). The method only assumes that the detected non-ideal TWB can be decomposed into three statistically independent parts, namely the paired part, the signal noise part and the idler noise part, which are all described by multimode thermal fields. According to this model, the joint signal-idler photon-number distribution (34) can be written as
in which the Mandel-Rice distributions are written as and denotes the -function. In Eq. (8), mean photon (photon-pair) numbers
per mode and numbers of independent modes for
the paired part (), noise signal part () and noise
idler part () as suitable characteristics of the analyzed
TWBs have been introduced. As the Mandel-Rice distributions in
Eq. (8) are defined for arbitrary nonnegative real numbers of modes, the same applies also to the distribution in Eq. (8). This allows to consider a broader
class of analytic distributions when fitting the experimental
data. We note that the formula (2) has been derived
for an integer number of modes, but its generalization to real
nonnegative is straightforward (34).
The photon-number distribution is related to the theoretical detected-photon distribution by quantum detection efficiencies and (53). Since detection by hybrid photodetectors is characterized by the Bernoulli distribution, we can express this relation as
using the Bernoulli coefficients ,
A fitting procedure that minimizes the declination between the
experimental histogram and theoretical
detected-photon distribution under the
assumption of equality of the first and second experimental and
theoretical detected photon-number moments (for details, see
(31)) allows us to determine both quantum detection
efficiencies , , and parameters and , , of the analyzed TWB. To give a typical
example, we consider the experimental data obtained for pump mean
power 49.2 W and iris sizes’ area 46 mm (see the
marginal distribution plotted as black dots in
Fig. 2). The fitting procedure assigned
the following parameters to the experimental distribution :
, , , , , , , and .
First of all, we note that the values of quantum efficiencies obtained by
the reconstruction method are comparable with the value obtained from the noise reduction
factor for the same set of data (see points at 46 mm in Fig. 4(b)) (56).
Second, we remark that the paired part of TWB
representing more than 98% of the entire field is
described by a multi-thermal field with 31 independent modes. We
note that the mean number of photons in paired fields equals 8, whereas the means
of noisy signal and idler photon numbers lay below 0.1. On the
other hand, the noise signal and idler parts have numbers
of modes much less than one which means that their probability
densities have appreciated values only very close to the zero
photon number. This is a consequence of very low noise signal and
idler intensities observed in the experiment. We attribute the
found numbers of modes much less than one to distortions
of electronic signals inside the detection chains
Finally, we point out that whereas the joint signal-idler experimental detected-photon histogram provided covariance equal to 0.16, covariance of photon numbers in the reconstructed photon-number distribution is equal to 0.85. The reconstruction also decreased the value of noise reduction factor to 0.2. This dramatic increase of correlations between the signal and idler fields in a TWB after the reconstruction also changes the shape of the corresponding joint signal-idler (detected) photon-number distributions (see Fig. 7). In fact, the presence of nonzero off-diagonal elements in the detected photon-number distribution in Fig. 7 makes its nonclassical character less evident compared to the reconstructed photon-number distribution plotted in Fig. 7 and clearly showing the prevailing pairwise character of the TWB (the off-diagonal elements attain values lower than 1% of those of diagonal elements). Also, the sum of diagonal elements gives 98.2% of the entire joint signal-idler photon-number distribution. This is in accord with the relative weights of paired, noise signal and noise idler parts of the TWB expressed in mean pair/photon numbers.
A substantial difference in the nonclassical behavior of detected-photon-number and photon-number distributions can be observed in the corresponding distributions of the sum and difference of the signal and idler detected-photon and photon numbers, respectively. The resulting distributions are compared with those obtained by the combination of two independent classical fields with Poissonian statistics. This comparison applied to the experimental detected-photon distribution reveals only weak signatures of non-classicality in the distributions and of the sum and difference of the signal and idler detected-photon numbers defined as:
where denotes the Kronecker symbol. As shown in Fig. 8, the experimental distribution of the difference is slightly narrower than the reference distribution. On the other hand, a slightly broader experimental distribution of the sum with respect to the reference distribution is drawn in Fig. 9. The reconstruction of joint photon-number distribution clearly reveals non-classicality of TWBs, as documented by the photon-number distributions and plotted in Figs. 8 and 9. The distribution of photon-number difference plotted in Fig. 8 demonstrates the prevailing pairwise character of TWBs that is also confirmed by a ’teeth-like’ character of the photon-number distribution of photon-number sum depicted in Fig. 9.
An ultimate criterion for discriminating quantum and classical multimode fields is related to the properties of quasi-distribution of integrated intensities, i.e. electric-field intensities integrated over the detection interval, related to normal ordering of field operators (for more details, see, e.g., (34); (24); (57)). The reason is that integrated intensities describe the fields before detection that may conceal nonclassical features of these fields. The relation between integrated intensities and detected photons is provided by Mandel’s detection formula (24). This formula can be inverted (34) and then used for the determination of quasi-distributions of integrated intensities from the photon-number distributions obtained from experimental data. According to quantum theory of radiation (58); (34) if the quasi-distribution attains negative values or is even singular, the field is nonclassical. The quasi-distribution of signal () and idler () integrated intensities can be written in the form of two-fold convolution, which is a consequence of Eq. (8) for the photon-number distribution (31):
Quasi-distributions of integrated intensities introduced
in Eq. (12) describe the paired (), signal noise () and idler noise () parts of the TWB. More details
can be found in (35); (31).
As we have demonstrated, many non-classicality criteria indicate quantum behavior of even experimental distributions written in terms of detected photons. Following the genuine definition of non-classicality, we can define a quasi-distribution of ‘detected-photon intensities’ following the approach developed for photons and assuming perfect quantum detection efficiencies () (29). Of course, the obtained quasi-distribution characterizes a fictitious ‘detected-photon’ boson field, as it contains only those photons that are captured by the detectors. As in the case of quasi-distribution of integrated intensities, the existence of negative regions in the quasi-distribution for detected photons confirms the nonclassical character of the state. The quasi-distribution of ‘detected-photon intensities’ determined from the analyzed experimental distribution is shown in Fig. 10. In order to see a detained behavior of this quasi-distribution and in particular to investigate in which regions it attains values close to zero, we plot only a part of the function in Fig. 10 and remark that the maximum of the peak in the origin reaches the value 7. The smallest negative values, equal to -0.2, are found close to the and axes. The highly prevailing positive part of quasi-distribution indicates that the measured state is close to a classical one. However, the presence of a negative part (even small) shows that the low detection efficiency has preserved the pairwise character of TWB. The comparison of the quasi-distribution of ‘detected-photon intensities’ with the genuine quasi-distribution of photon intensities [see Fig. 10] reveals much stronger non-classicality in the case of photons. We note that the peak value of in Fig. 10 equals 0.99 which is considerably lower than the peak value of quasi-distribution shown in Fig. 10. Nevertheless, both quasi-distributions attain negative values and so both describe a nonclassical field. The contour plots of both quasi-distributions depicted in Figs. 10 and 10 reveal that negative values of these distributions are localized in parallel strips whose orientation originates in the pairwise character of TWBs.
Using spontaneous parametric down-conversion in the linear gain regime, we generated multimode twin-beam states in the mesoscopic photon-number regime. We studied nonclassical properties of the twin beams by applying three different non-classicality criteria written in terms of detected photons. Whereas the noise reduction factor is a suitable indicator of non-classicality independent of the twin-beam intensity, the Schwarz inequality is useful for weak twin beams and the criterion derived from higher-order detected-photon-number moments finds its application for intense twin beams. To compare these criteria with the genuine definition of non-classicality we also determined quasi-distributions of detected-photon and photon integrated intensities for normally ordered field operators. Despite the low detection efficiency (around ) negative values of these quasi-distributions found in typical strips were observed both for photons and detected photons, confirming non-classicality of the generated twin beams. The set of criteria we presented can thus be considered as a robust tool for quantifying non-classicality of multimode twin beams used in many schemes, including that for conditional generation of nonclassical and non-Gaussian states.
The research leading to these results has been supported by MIUR (FIRB ÒLiCHISÓ - RBFR10YQ3H). Support by projects P205/12/0382 of GA ČR, Operational Program Research and Development for Innovations - European Regional Development Fund project CZ.1.05/2.1.00/03.0058 and Operational Program Education for Competitiveness - European Social Fund project CZ.1.07/2.3.00/20.0058 of MŠMT ČR are acknowledged.
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