## Abstract

An experimental scheme and data processing approaches are proposed for measuring by analog photo detectors the normalized second-order correlation function of the biphoton field generated under spontaneous parametric down-conversion. Obtained results are especially important for quantum SPDC-based technologies in the long-wave spectral ranges, where it is difficult to use the single-photon detector at least in one of the two biphoton channels. The methods of discrimination of analog detection samples are developed to eliminate the negative influence of the detection noises and get quantitatively true values of both the correlation function and the detector quantum efficiency. The methods are demonstrated depending on whether two single-photon avalanche photo detectors are used in both SPDC channels, or at least one single-photon detector is replaced by a photo-multiplier tube which cannot operate in the photon counting mode.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum-correlated pairs of optical photons (biphotons) are widely used in modern quantum technologies, from quantum communication [1], computing [2], metrology [3], to various types of quantum spectroscopy, imaging and sensing [4–11]. Spontaneous parametric down-conversion (SPDC) process provides a remarkable possibility to generate biphotons with the highest possible level of correlation. The normalized second-order correlation function *g ^{(2)}* is usually considered as a quantitative measure of this level. Today the single-photon detectors and fast coincidence circuits are the necessary key elements in a vast majority of quantum optical schemes. They are successfully used in monitoring quantum properties of biphotons generated in the X-ray [12] and optical ranges up to near-IR [13]. However, if a frequency of at least one photon from the biphoton pair goes down from mid-IR down to the terahertz (THz) frequency range, application of the coincidence technique becomes impossible. The reasons are in great difficulties in creating THz receivers that can operate in the single-photon or photon-number resolving detection modes [14,15]. This greatly constrains the advancement of quantum-optical technologies to terahertz range, and up to now all the experimental progress within this direction is concerned with detection of optical signal radiation which is assumed to be correlated with its THz idler counterpart [16–22].

In principle, one can measure the correlation function applying the oldest approach [23] based on measuring the correlation between photocurrents of analog detectors instead of coincidences of single-photon detectors. This possibility was studied theoretically for the optical [24] and optical-terahertz biphotons [25]. It was shown that if the noise contribution to samples of analog detectors is negligible, the both methods should give the same result. However, the quantum contribution to a biphoton correlation function is significant at low photon fluxes, where, unfortunately, the effect of noise on the detector samples is usually quite large. Experiments show that this drastically decreases the results of *g ^{(2)}* measuring if it is done directly by taking into account all the photocurrents from the initially obtained raw set of statistical data. Problems with noises are resolved and the biphoton correlations are successfully used in calibrating quantum efficiencies of that analog detectors that can operate in single-photon detection modes in principle, from the photon number resolving detectors [26] up to ECCD cameras [27]. But the challenges in exact quantitative measuring of

*g*still remain when the outputs of the analog detectors cannot be mapped onto integer photoelectron counts. The main goal of the present work was to develop a relevant experimental procedure for processing the analog samples and determining

^{(2)}*g*using analog detectors of this type. Since the true

^{(2)}*g*value must be known in advance, the study was carried out completely in the optical range, in which the exact

^{(2)}*g*value could be pre-measured using the conventional photon counting detection technique. A number of intuitively meaningful processing procedures were applied to analog samples. Some of them were shown to give an artificially increased

^{(2)}*g*value. Finally, after quantitative comparing the results with the expected true

^{(2)}*g*values in each case, the optimal approaches were selected. The proposed methods open the possibility of quantum optical measurements with a wide range of sensitive analog detectors, which, however, cannot operate in the photon counting modes. They will be especially useful for developing quantum technologies at THz and mid-IR frequencies.

^{(2)}In the final Section 6 of this paper, the best of proposed procedures is described and realized experimentally for all-optical biphotons by two photo detectors, one of which (a PMT module) cannot in any way operate in the single-photon detection mode. The procedure validity is verified by comparing the obtained values of *g ^{(2)}* with the results of measuring

*g*in the photon counting mode using two single-photon avalanche photo diodes (APDs). Apart from measuring the values of

^{(2)}*g*, we calibrate quantum efficiency of one of APDs, first in a standard photon-counting scheme, secondly in an analog scheme with the same APDs, and finally in an analog scheme with PMT. In all the cases, the same biphoton field was characterized and used; its generation is described in Sec.2. In Sec. 3 we remind some general aspects of the Klyshko method for SPDC-based calibration of quantum efficiency of single-photon detectors [28–31], and consider modification of this method when analog samples of these detectors are measured. Results of application of the standard photon counting approach for measuring

^{(2)}*g*and efficiencies of APDs are presented in Sec.4. In the next Sec.5, we switch the electronic circuit to an analog detection mode and show that the same results are obtained for

^{(2)}*g*and APD efficiency in the modified experimental scheme. Finally, in Sec.6, one of APDs is substituted by PMT module, and we present a method of data processing which enables to get the same (true) values of the biphoton correlation function

^{(2)}*g*. As an additional marker of validity of the proposed procedure, we also provide here the results of SPDC calibration of the quantum efficiency of APD when the analog PMT stands in the idler channel. It is shown, that our method of

^{(2)}*g*characterization gives the same value of APD quantum efficiency as in case when there is a single-photon detector in the idler channel.

^{(2)}## 2. Generation of biphotons

We generate orthogonally polarized collinear frequency-degenerate biphotons by using type-II SPDC process in 41.2°-cut beta barium borate (BBO) crystal pumped at 405 nm. Schematics of the optical part of an experimental setup is shown in Fig. 1(a).

As a pump source, we used a spatially single-mode diode laser with a tunable power. A Faraday isolator was installed to prevent reflections from different optical elements from entering the laser diode. A half-wave plate was used to control the laser polarization. In order to get rid of radiation with longer wavelengths, the laser beam was passed through the filter F1 blocking the wavelengths longer than 450 nm. Then the radiation was focused by a 40 mm-focus quartz lens L1 onto 1 mm thick BBO crystal. The filter F2 was placed in front of the second lens L2 (F = 45 mm) after the crystal to cut off the pumping radiation with a wavelength of less than 600 nm. The transmitted part of SPDC radiation was collimated by the lens L2 and incident on a polarizing beam splitter, which made it possible to distribute generated photons into the “signal” or “idler” channels depending on their orthogonal polarizations.

To apply the Klyshko method for quantum efficiency calibration with high accuracy [28–32], unequal sets of spatio-temporal modes were chosen for detection in the two channels. The signal channel was wider in space and frequency ranges; it collected *e*-polarized SPDC photons. Signal radiation passed through an interference filter *F _{s}* with a center wavelength 800 nm and full width at half maximum 40 nm. Then it was focused by the 11 mm focal lens

*L*onto a multimode optical fiber of 62.5 μm transverse section, connected to the photodetector

_{s}*D*. In the idler channel, the

_{s}*o*-polarized SPDC radiation passed through an interference filter

*F*with a center wavelength 810 nm and a full width at a half maximum 10 nm. Then it was focused by 11 mm focal lens

_{i}*L*onto a multimode optical fiber of 50 μm transverse section, connected to the photodetector

_{i}*D*.

_{i}In the process of measuring the radiation at the output of signal and idler channels, three different photodetectors were changed at positions *D _{s}* and

*D*. Two of them were single-photon avalanche Si photo detectors, APD #1 (Laser Components COUNT NIR module with a dead time 45 ns) and APD#2 (a homemade module with a dead time 220 ns), providing electrical pulses in the standard TTL format. The third one was H7422-50 Hamamatsu photosensor module with a sensitive photomultiplier tube (PMT). PMT could operate in an analog mode only.

_{i}An intensity transmission coefficient $K_s^{}$ of optical elements standing between BBO crystal and the detector *D _{s}* in the signal channel was measured using radiation of CW diode laser at 808 nm. It was found that $K_s^{} = 0.39$ for e-polarized photons. Remarkably, the intensity transmission coefficient measured by the same method for the idler channel is not actual for our further consideration, since it does not account the losses of that idler photons which are correlated with detected signal photons but do not pass through the narrower spectral and spatial selective elements in the idler channel.

## 3. Klyshko method of SPDC-based absolute quantum efficiency calibration

According to its general definition, the normalized biphoton second-order correlation function ${g^{(2)}}$ describes correlation of electric fields in idler and signal channels. In case of single-mode detection, it is connected with averages of the photon number operators in signal and idler modes as [33]

*j*denotes the corresponding pair of single modes, and ${N_j}$ scales with the parametric gain coefficient ${G_{is}}$ as ${N_j}\sim {sh ^2}\sqrt {{G_{is}}} $ for phase-matched signal and idler modes. Equation (1) follows from the specific quantum nature of SPDC process, where the numbers of generated signal and idler photons are the same in exactly correlated signal and idler modes, so that $\left\langle {\hat{N}{{_{}^j}_i}} \right\rangle = \left\langle {\hat{N}{{_{}^j}_s}} \right\rangle = {N_j}$. In our case, the multimode fields with total numbers of photons $\left\langle {{N_a}} \right\rangle = \sum\limits_j {\left\langle {\hat{N}_a^j} \right\rangle } \approx M\left\langle {\hat{N}_a^j} \right\rangle$ (here and below, $a = s,i$) are detected in the signal and idler channels, and the biphoton correlation function ${g^{(2)}}$ decreases [34,35] according to

A mean number of photo-counts $\left\langle {{n_a}} \right\rangle$ detected by each photo-detector within ${t_{det}}$ should be proportional to the mean number of generated SPDC photons, optical transmission of the channel $K_a^{}$, and quantum efficiency of the detector ${\eta _a}$: $\left\langle {{n_a}} \right\rangle = {K_a}{\eta _a}\left\langle {{N_a}} \right\rangle$. Quantum efficiency of each detection channel does not affect the measured value of ${g^{(2)}}$. However, according to Klyshko method [28–30], ${\eta _a}{K_a}$ can be determined after multiplying $({{g^{(2)}} - 1} )$ by the mean number of photo-counts detected in this channel. Following this approach, absolute quantum efficiency of the detector placed in the signal channel is determined as

The number $\left\langle {{n_s}} \right\rangle$ can be easily determined if single-photon or photon-number resolving detectors are used, but has a more complex relation to samples of other type detectors.Generally, any event of photon absorption by an intensity-sensitive detector leads to formation of a short-time photocurrent surge $i_0^m$ at the detector’s output. The number of such events $n_{}^{}$ during the time ${t_{det}}$ is directly determined by detection in a photon counting mode. But an analog detector issues only the total current $i$ detected over ${t_{det}}$, that is equal to

Here, $z_0^m \equiv i_0^m\tau _0^m$ denotes a charge carried by each elementary current pulse, and $\tau _0^m$ corresponds to its duration. Statistical distribution of elementary charges could be very different in different detector types [24,36], so that it is impossible to give a general receipt for estimating $\left\langle n \right\rangle$ according to a measured value of $\left\langle i \right\rangle$. This makes it difficult to apply the Klyshko method using the analog data in the most part of cases. However, if the statistical distribution of the elementary charges $P(z_0^m)$ is narrow (i.e. its dispersion is small in comparison with the average value of an elementary single-photon charge $\left\langle {{z_0}} \right\rangle$), it does not affect significantly the distribution of currents $P(i)$. Then the relation between a photo-current and a number of photo-counts can be described by mean parameters of the elementary current pulses, so thatHere, $\left\langle {{i_0}} \right\rangle$ is an average single-photon current issued when only one photon is detected during ${t_{det}}$. We will show in the next sections that approximation (4a) is valid at least when a single-photon detector is used in the analog detection mode. In this case the mean number of photo counts $\left\langle n \right\rangle$ is simply determined after measuring $\left\langle i \right\rangle$ as $\left\langle n \right\rangle = \left\langle i \right\rangle /\left\langle {{i_0}} \right\rangle$, and the modified Klyshko approach can be applied for calibrating the quantum efficiency using analog samples of such a detector.

## 4. Characterization of the biphoton correlation function and quantum efficiencies of APDs in the photon-counting mode

First, we applied the standard approach for characterizing both the biphoton normalized second-order correlation function ${g^{(2)}}$ and quantum efficiencies of single-photon APDs. During a properly extended data acquisition time (*T*∼1s), we determined the total numbers of single-photon counts from each detector placed in signal (${m_s}$) and idler (${m_i}$) channels, and the number ${m_{cc}}$ of the single-photon count coincidences. Measurements were done using the electronic circuit shown in Fig. 1(b). The time window of the coincidence scheme ${t_{cc}}$ corresponded to the detection time in general theory, ${t_{det}} = {t_{cc}} = 8\,\textrm{ns}$. To account the dark noise counts of each detector, we measured the corresponding samples ${m_{s0}}$ and ${m_{i0}}$ when the pump radiation was blocked. The mean numbers of photo-counts per detection time (${t_{cc}}$) was calculated as $\left\langle {{n_a}} \right\rangle = ({{m_a} - {m_{a0}}} ){t_{cc}}/T$ ($a = i,s$). The mean number of coincidences per detection time was calculated as $\left\langle {{n_{cc}}} \right\rangle = {m_{cc}}{t_{cc}}/T$, without taking into account a negligibly small number of dark noise coincidences.

Applying the standard procedure with single-photon detectors, we determined ${g^{(2)}}$ experimentally, using digital samples of single-photon APDs as

Quantum efficiencies of both APDs were measured when the detector under calibration was placed in the signal channel. They were calculated in accordance with Eqs. (3) and (5):

The results obtained at different pump powers for APD#1 are shown in Fig. 3 by red circles. As it should, within the experiment uncertainties the values of quantum efficiencies do not demonstrate any dependence on the pump power. Noticeably, the experimental accuracy was less for the quantum efficiency of APD#1 than for APD#2. It takes place due to a higher dark noise level of APD#2, which detected lower photon fluxes in the narrow idler channel when the quantum efficiency of APD#1 was measured. In both cases, the values of ${\eta _s}$ were less than the passport data of the devices. It indicates that not all the losses in the signal channel were accounted in the applied value of ${K_s}$. For example, there could be possible losses in connecting the multimode fiber with the detector input window. However, the full compliance with the passport data was not the direct goal of this measurement. For our study, it was important, without any changes in the optical scheme, to obtain the same values of the quantum efficiency when using the analog circuit.## 5. Switching to analog detection mode with the same APDs

At the next step we changed an electronic circuit to detect “almost instantaneous” photocurrents from the both detectors, ${i_s}$ and ${i_i}$. They were measured by two gated integrator and boxcar averager modules SR250 (Fig. 1(c)) and further converted into the digital form by an analog-to-digital converter (ADC). An external pulse generator was used to trigger equal gates at each module with 4 kHz frequency; an optimal delay time between two triggers was adjusted for the maximum of the measured biphoton correlation function. The detection time in this setup corresponded to the gate duration, ${t_{det}} = {t_{gate}}$, and could be changed from 2 ns to 15 µs. We selected different ${t_{gate}}$ exceeding the dead times of our detectors (from 220 ns and above). The number of detected modes changed correspondingly, as $M\sim {t_{gate}}$, and it was checked that the values of the measured biphoton correlation function vary in accordance with Eq. (2). The data on statistical distributions of the values of ${i_s}$, ${i_i}$, and their product ${i_s}{i_i}$, detected within the same gate intervals, was collected within 20-30 min. and analyzed by PC.

Figure 4 shows examples of non-normalized histograms, which correspond to statistical distributions of “instantaneous” signal currents $P({i_s})$, recorded by APD#1 at the same detection time ${t_{gate}}$=500 ns and different values of the laser pump power ${P_{pump}}$, 17.2 mW, 4.5 mW, 0.15 mW, and 0 mW. In each histogram one can see some noise peak in the region of the lowest currents. Only this peak remains when the pump radiation is blocked and no SPDC photons are incident on the detectors. When the pump power increases, the noise peak amplitudes and widths typically grow up. The peaks at higher currents appear at nonzero ${P_{pump}}$. Evidently, they correspond to the cases of detecting one, two or more elementary photo counts during the gate time. When the pump power decreases, each peak position remains stable, but a number of observed peaks decreases together with the amplitude of each peak. The gentle character of the left slope and the sharp right slope of each peak can be easily explained by considering the cases when only a part of an asymmetric single-photon current pulse falls at the beginning or end of the strobe time interval. Up to some possible constant shift along the horizontal axis, the photo count maxima have coordinates equal to $p\left\langle {{i_{ph}}} \right\rangle$, where *p* is an ordinal number of the peak. An average current $\left\langle {{i_{ph}}} \right\rangle$ carried by a single detector’s photo count is easily determined as a distance between two neighboring photo count peaks. Finally, we observe only the first peak in the lowest part of ${P_{pump}}$ scale. Amplitude of this peak gradually “drowns” in noise when the photon flux decreases. If the detector is placed in the low-transmission idler channel, this effect comes earlier, at higher pump powers. However, consistently placing APD#1 and APD#2 into the signal channel we determined the values of $\left\langle {{i_{ph}}} \right\rangle$ for each APD.

To measure the true correlation between the samples of the detectors, first we excluded from statistical distributions the data on that current intervals, where ${i_s}$ or ${i_i}$ were not within the areas of the first and other (if applicable) photon peaks. These areas are shaded in examples presented in Fig. 4. The detector samples outside these areas were taken as equal to 0. By this discrimination we could eliminate not only the main part of noise samples of each detector, but also to decrease substantially the noise part in their correlation ${i_i}{i_s}$. After that, the averages $\left\langle {{i_a}} \right\rangle$, $\left\langle {{i_i}{i_s}} \right\rangle$ were determined for the entire sample. Also, to exclude a possible residual noise contribution to the mean SPDC currents, we measured $\left\langle {{i_{a0}}} \right\rangle$ by the same treatment of the histograms detected when the pump radiation was blocked. After that the biphoton correlation function was calculated as

To compare the biphoton correlation functions detected by analog and photon-counting experimental set-ups, we recalculated $g_M^{(2)}$ into $g_{}^{(2)}$, which corresponds to the same number of modes as in the previous experiment. It was done using the direct consequence from (2)

Since the number of spatial modes, as well as the spectral intervals of detection did not change, we accounted here only an effect of different detection times. The results on this reduced $g_{}^{(2)}$, obtained at different values of the pump power, are shown in Fig. 2 by the black square dots. Here and below, for all measurements in the analog regime, the vertical error bars indicate possible deviations calculated by taking dispersions of the current samples in each statistical data set, including the set of current products ${i_i}{i_s}$. In the experiment, the gate time was taken ${t_{gate}}$=500 ns, longer than the dead times (220 ns and 45 ns) and the single-photon pulse times (18 ns and 16 ns) of APDs. Thus, the scale factor required for comparison with the photon-counting results was 500/8 = 62.5. It can be seen that the values of correlation function $g_{}^{(2)}$ obtained by different detection methods with two APDs coincide within the experimental errors. The both sets of experimental values of the correlation function were determined when APD#1 was placed in the signal channel and APD#2 was in the idler channel.We have also calculated the quantum efficiencies of APD#1 and APD#2. It was done using a modified relation (6) in the form

The results obtained using (9) for APD#1 are shown by black open squares in Fig. 3. It is seen that the experimental error grows up when the pump power and, correspondingly, the SPDC photon fluxes scale down. The accuracy of the results can be further increased by collecting a more representative set of statistical data. However, quite good agreement is already noticeable between the values of the quantum efficiency obtained using the analog and the photon counting calibration procedures. This testifies a general applicability of the modified calibration method to measuring quantum efficiency of single-photon APDs when they operate in an analog detection mode. But fortunately, at this stage we could isolate the main part of the analog detection noise by selecting the regions of the photon count peaks in the histograms due to the single-photon response of our APDs.

## 6. Analog detection with PMT

At the final stage of the study we removed APD#2 from the analog electronic detection scheme, while APD#1 remained in the signal channel. Since we excluded APD # 2 with the longest dead time, we could take a smaller gate time ${t_{gate}}$=100 ns. In the idler channel, APD#2 was replaced by H7422-50 Hamamatsu module with PMT (Fig. 1(c)). The module could produce samples in analog mode only. Its sensitive element is unable to operate in a photon-counting regime, as clearly demonstrate histograms in Fig. 5. Figure 5(a) presents histograms of PMT samples in the idler channel, taken at the same pump powers as the histograms of APD#1 in the signal channel in Fig. 4. In contrary to the case of single-photon APDs, in the PMT histograms it is impossible to separate unambiguously the samples related to absorption of certain number of photons and the pure noise samples. To study how large is the noise contribution, we considered the conditional distributions of the PMT samples. Figure 5(b) shows example of such distributions, created from the PMT samples detected when

All the conditional histograms were obtained from the samples recorded at ${P_{pump}}$ = 17.2 mW and present the corresponding parts of the total histogram shown in Fig. 5(a) by the purple color. One can see in Fig. 5(b) that the overwhelming majority of PMT samples is recorded when no photons were detected in the idler channel during the same strobe time. Of course, it does not mean that there were no photons at all in the signal channel, since the quantum efficiency of APD#1 is below 100%. But the signal channel is wider. Thus, some rather significant part of the samples on the yellow histogram is still purely noise in nature. And these noise samples can have the amplitudes within the whole range of detected PMT samples. Most probably, it takes place due to a wide spread of possible elementary single-photon currents ${i_{ph}}$ [36] generated in PMT.

The presence of noise contribution can decrease noticeably the experimentally measured correlation function. For example, a simple subtraction of the mean noise current $\left\langle {{i_{i0}}} \right\rangle$, measured when the pump radiation is blocked (the black histogram in Fig. 6(a)), from the mean current $\left\langle {{i_i}} \right\rangle$ (measured through the total histograms such as the red histogram in Fig. 6(a) recorded at 17.2 mW pump power) is not enough to get the true values of the biphoton correlation function. The noise contribution to the values of $\left\langle {{i_s}{i_i}} \right\rangle$ remains incompatible in this case and the recorded correlation will be lower than it actually is. This demonstrates Fig. 6(b), where the ${g^{(2)}}$ values obtained by this way are presented together with the results of ${g^{(2)}}$ characterization using the single photon detection technique.

Evidently, some PMT samples should be eliminated (i.e. replacing by zero) before processing the whole PMT statistical set. This procedure decreases the effective value of the PMT quantum efficiency, but, in contrast to the noise contribution, can have no influence on the measured value of ${g^{(2)}}$. We studied different ways of such discrimination of the PMT samples. The simplest one was to eliminate the low-level PMT samples. Indeed, the higher is a single sample ${i_i}$, the lower should be a relative contribution of the noise to this sample. Thus, by eliminating (i.e. replacing by zero) sufficiently low values of ${i_i}$, one can obtain higher values of ${g^{(2)}}$, up to its true level measured at both previous stages. This method enabled to obtain the larger and larger ${g^{(2)}}$ values when the level of discrimination ${i_{i,thr}}$ was increased, finally approaching the true value of ${g^{(2)}}$ obtained by the photon counting technique for the same SPDC field. As for example, Fig. 7 shows this dependency calculated using the data set recorded at ${P_{pump}}$ = 17.2 mW; the red line corresponds to the true level of ${g^{(2)}}$. It is seen that, at the same time, uncertainty of the obtained ${g^{(2)}}$ values increases gradually when the statistical set of samples is depleted so much. The uncertainties can be too high especially at low pump powers, and very long expositions will be necessary to decrease them. Anyway, this discrimination method could be very important when the both detectors, the idler and the signal ones, do not possess the single-photon response property.

In our case there is a single-photon APD in an opposite SPDC channel. The regions of pure noise samples and the samples with the photon-induced contributions are clearly spaced in its histogram. This provides an opportunity to make optimal discrimination of the PMT samples based on the samples of APD#1 in the signal channel. However, leaving only that PMT samples, which were obtained when APD#1 detected photon counts, one can artificially increase the level of correlations. This illustrates Fig. 8. Here, as for example, we provide at the bottom graph the histograms of APD#1 recorded at 17.2 mW pump power and when the pump was blocked. Before calculating ${g^{(2)}}$, we discriminated (replaced by 0) those PMT samples, that were obtained simultaneously with the APD#1 samples below a certain threshold level ${i_{s,thr}}$. After that the APD#1 samples were discriminated by the same way as in previous section. The processed data including all the zero values was used for calculation all the distribution moments. Discrimination using the same principle was done when the pure noise data from PMT was processed to get $\left\langle {{i_{i0}}} \right\rangle$ for substituting into Eq. (7). The upper graph shows results of ${g^{(2)}}$ evaluation using this algorithm with different ${i_{s,thr}}$ values. It is seen, that the obtained ${g^{(2)}}$ values grow when the threshold level ${i_{s,thr}}$ is increased. This growth in the initial range of ${i_{s,thr}}$ from 0 up to 30-50 finally gives the true ${g^{(2)}}$ value detected with two single photon detectors (a horizontal line in the upper graph). Remarkably, namely this range corresponds to that APD#1 histogram which was detected in the absence of any incident photons (red color in the bottom graph). Further increase of ${i_{s,thr}}$ gives an artificial increase of the measured ${g^{(2)}}$ above its true value. This can be explained by the growing presence of photon count contributions in the corresponding ranges of the APD#1 histogram.

Thus, our analysis shows that the discrimination of PMT samples according to the samples of single-photon APD#1 in the opposite channel is optimal from the point of minimal statistical errors. The threshold level ${i_{s,thr}}$ should be taken at the end of the pure noise (recorded without any incident photons) APD histogram to avoid the systematic errors. Only a small part of all PMT samples is discriminated by this way (see the conditional distribution of PMT samples marked by blue color in Fig. 6(a)), but it enables to eliminate the influence of analog detector noise correctly. Other case, the obtained ${g^{(2)}}$ can be less (due to non-eliminated noises) or higher (due to artificial procedure-induced correlations) than its true value. Apparently, using this method, we can eliminate the effect of pure electrical noise, which has the same effect on both APD and PMT samples, and, nevertheless, not artificially impose a superfluous photon correlation. The same values of the biphoton correlation function as in previous cases of single-photon APDs in the both SPDC channels are obtained with a fairly small uncertainty for all the considered pump power levels. Results of this method are presented in Fig. 2 by blue triangles. Evidently, the applied method of discrimination can be especially useful for detecting true biphoton correlations in strongly non-degenerate SPDC processes, when one of the biphoton frequencies falls into the range where there are no single photon detectors.

By substituting obtained with PMT $g_M^{(2)}$ values into Eq. (9), we have determined the quantum efficiency of APD#1. Obtained by this way values of quantum efficiency of APD#1 are shown in Fig. 3 together with results of other calibration procedures. Within the experimental error, they coincide with the results of calibration by the two previous methods. As it is seen also from Fig. 3, the uncertainties increase for the data of all three types when the pump power scales down. Much more pronounced at pump powers below 3 mW, this effect takes place mostly due to increase of statistical errors in detecting the photon fluxes which become weaker when the pump power decreases. Obviously, the statistical uncertainties can be decreased by increasing the total data acquisition time, as it was observed in our experiments also. However, the pump power dependences of vertical errors in Fig. 3 tend to saturate at the high pump powers. One can estimate the contribution of possible systematic errors to the total uncertainty budget by taking as an upper level the uncertainties detected at the highest pump power, 17.2 mW in our case. Reasoning in this way, we obtain the relative systematic error for measuring the quantum efficiency of APD # 1 in all three detection regimes to be no more than the same level of 9%. It is noteworthy that application of the discrimination procedure proposed for PMT gives the value of quantum efficiency, which coincides with much greater accuracy with the result of direct application of Klyshko method in the photon counting regime.

## 7. Summary

We have developed an experimental scheme and proposed the data processing approaches for measuring by analog photo detectors the normalized second-order correlation function of the biphoton SPDC fields ${g^{(2)}}$. The approaches are alternative to the traditional method based on the use of single-photon detectors in the photon-counting scheme, and can be very important in experimental conditions when the detectors which can operate in the photon counting mode are not available. At all steps of our study we compared the determined values of ${g^{(2)}}$ and, in parallel, of the quantum efficiency of one of the single-photon detectors, with the corresponding results of the conventional photon-counting technique.

The experimental scheme operates with two gated integrator boxcar modules which simultaneously determine photo currents from detectors of signal and idler SPDC radiation. These “almost instantaneous” currents (samples) are actually results of averaging over a small gate time. It is shown that this time corresponds to the time window of the coincidence scheme in the photon-counting approach. The gate time scales the number of detected spectral modes and, correspondingly, the value of the detected correlation function ${g^{(2)}}$ of the multi-mode SPDC field. However, statistical distributions of the samples should be specially processed for calculating the exact ${g^{(2)}}$ values.

It is shown that high noise contributions to the analog samples can drastically decrease the determined ${g^{(2)}}$ below its true value. To avoid this effect it is necessary to apply a proper discrimination procedure to the raw sets of samples before calculating the moments of the sample distributions. During the discrimination procedure some noisy samples are left equal to zero in the set. In case of APD with single-photon response, the statistical distribution of the samples contains several well-resolved peaks corresponding to pure noise samples, and the samples with one photon count during the gate time, two photons, et cet. We found that the true results on ${g^{(2)}}$ value with low uncertainty can be obtained if all the APD samples beyond the photon count peaks are substituted by 0. For the analog PMT detector the direct discrimination of low-amplitude samples and the heralding discrimination procedures with the cut-off levels corresponding to APD samples of different origin are studied. It is found that some of them can lead to artificially increased ${g^{(2)}}$ values. The direct discrimination approach needs longer measurement times to get the final result with a satisfactory uncertainty, but is applicable regardless of which type detector is in the opposite SPDC channel. As the best method we recognize the heralding-type approach which requires a detector with a single-photon property in the opposite channel. In this case, the PMT samples are discriminated or not, depending on whether the amplitudes of the analog samples, simultaneously recorded by the opposite detector, are below or above their characteristic threshold value. The correct choice of the threshold value is shown to be an important point for successful application of the heralding approach. The threshold value is determined according to the pure noise histogram of the single-photon samples recorded in complete darkness.

In general, discrimination can lead to underestimated results on quantum efficiencies of the detectors if the method of quantum calibration based on correlation between signal and idler photons is applied. Fortunately, this does not concern the discrimination procedure proposed for the single-photon detectors inserted in the analog circuit. This is demonstrated for both cases, when an analog PMT, and when another single-photon APD operate in the opposite SPDC channel.

The obtained results will be most relevant in applications of quantum SPDC technologies in the long-wave spectral ranges, where it is difficult to use the single-photon detectors at least in one channel of the quantum-correlated biphoton field.

## Funding

Russian Science Foundation (17-12-01134).

## Acknowledgements

We are grateful to Maria Chekhova, Pavel Prudkovskii for stimulating discussions and to Anton Konovalov for the technical help.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The main part of the data underlying the results presented in this paper is available within the article, the not publicly available at this time and more detailed part may be obtained from the authors upon reasonable request.

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