Approaches to the standard-free calibration of the quantum efficiency of a wide class of analog detectors based on the measurement of statistical characteristics of fields generated by parametric down-conversion are analyzed. General expressions are obtained for the noise reduction factor of the difference photocurrent and for the covariance of photocurrents in the signal and idler channels that take into account the possibility of strong fluctuations in the amplitudes of single-photon response functions of the detectors used. It is shown that the measurement of the noise reduction factor of the difference photocurrent using detectors that cannot operate in the photon counting mode is on its own insufficient to directly characterize the level of two-mode squeezing in the down-conversion field and to determine the quantum efficiency of photosensitive elements without additional calibration procedures. A method to determine the quantum efficiency of such detectors based on measuring the dependence of the normalized covariance of photocurrents on the parametric gain is proposed.

## INTRODUCTION

In recent years, there has been growing interest in the studies of the statistical parameters of terahertz (THz) radiation at the photon level and the generation of THz fields with quantum properties [1–10]. Some of the first works in this direction dealt with the investigation and application of quantum-correlated pairs of photons of the optical and terahertz ranges, the so-called optical–terahertz biphotons. Biphotons of this kind are produced by parametric down-conversion in the strongly frequency-nondegenerate regime [5, 11, 12]. In the course of this work, the spectra of the optical (“signal”) component of the generated radiation were investigated and the fundamental possibility of determining the dispersion properties of different media in the THz range without direct detection of THz waves was shown [4, 7, 13–15]. The first measurements of “idler” THz radiation were also carried out [16, 17]. However, the possibility of measuring the quantum characteristics of optical–terahertz biphoton fields has so far been directly demonstrated only by the example of the measurement of the normalized correlation function \({{g}^{{(2)}}}\) under conditions close to spontaneous parametric down-conversion [10]. As far as the observation of a weak flux of THz photons characteristic of the spontaneous parametric down-conversion regime is very difficult because of the low sensitivity and large noise level of THz detectors, it was possible to demonstrate only a weak difference (on the order of tenths of percent) of \({{g}^{{(2)}}}\) from its classical value in these first experiments.

Obviously, the high-gain regime of parametric down-conversion would be more convenient to investigate the quantum properties of optical–terahertz fields. In the studies of twin beams generated in this regime in the case of frequency-degenerate parametric down-conversion, covariance, noise reduction factor (NRF), and other related characteristics that are calculated on the basis of current readings from optical detectors in the signal and idler parametric down-conversion channels are parameters typically measured to distinguish between classical and nonclassical radiation [18, 19]. Modern methods of quantum sensorics and standard-free quantum photometry are based on the measurement of these parameters [20–22]. However, these methods cannot simply be transferred to the THz range for one simple reason: only single-photon detectors are invariably used in the optical signal and idler channels of parametric down-conversion. Then, regardless of whether the detection is carried out in the analog or photon-counting mode, the statistical correlations of detector readings unambiguously determine the correlation of photons in the parametric down-conversion field to be characterized [23]. Meanwhile, single-photon detectors working in the THz range are still practically unavailable.

Here, we consider the NRF and the relative covariance of the currents measured in the signal and idler channels of parametric down-conversion with conventional analog photodetectors that do not necessarily have the functionality of single-photon detection. The derived general expressions make it possible to estimate the impact of the statistical spread in the single-photon response of such detectors on the characterization of the quantum properties of the initial parametric down-conversion field and, furthermore, to develop approaches to the standard-free measurement of the quantum efficiency of the sensitive elements of a wide class of conventional analog detectors.

## SINGLE-PHOTON RESPONSE, MODES, AND AVERAGE CURRENTS IN THE PARAMETRIC DOWN-CONVERSION CHANNELS

The response of both optical and THz detectors measuring the intensity of incident radiation features an initial stage where a single pulse of photocurrent is formed as a result of the absorption of a single photon in the photosensitive layer. On average, the number \(\langle m\rangle \) of these pulses that occur over the time interval allocated for taking an individual reading is certainly different from the number \(\langle n\rangle \) of photons incident over the same time and equals \(\langle m\rangle = \eta \langle n\rangle \), where η is the quantum efficiency of the detector. Each elementary event results in the formation of a photocurrent pulse at the output of the detector described by

Here, \({{q}_{k}}\) is the charge carried by the *k*th single-photon pulse and \({{f}_{k}}(t)\) describes its temporal shape (it is implied that \(\int {{f}_{k}}(t)dt = 1\)). The shape \({{f}_{k}}(t)\) of elementary pulses and their duration are determined by the parameters of the built-in electronic circuit that amplifies and conditions the signal coming from the photosensitive layer. In most cases, fluctuations in the pulse shape are negligible and it can be assumed that \({{f}_{k}}(t)\) is the same in all elementary acts of photon detection. This description is valid not only for so-called “single-photon” detectors, or photon counters, where a deliberately standardized output current pulse is formed upon every instant of photon absorption, but also for those detectors that cannot operate in the photon-counting mode. In contrast, fluctuations in the value of \({{q}_{k}}\) can be large in detectors of the latter type (referred to as analog detectors).

When the detector operates in the analog mode, i.e., when the full output current is fed to the signal-processing circuit, elementary photocurrent pulses from the detector sensitive element are superimposed on each other, and the net time-varying output current is

where *m* is the number of elementary single-photon pulses that appear within the detection time. For the sake of simplicity, hereinafter, we consider insignificant and disregard possible additional noise contributions of electronic nature to the total current. The detection time is determined by the speed of the signal processing circuit; if the statistics of the current readings is measured using a gated integrator, this time is the strobe duration [24]. In the general case, the photocurrent readout is a random variable whose average value is proportional to the average intensity of radiation incident on the detector. In terms of the photon numbers *N* per input radiation mode and taking into account the total number \(M = {{M}_{\parallel }}{{M}_{ \bot }}\) of radiation modes received by the detector (where \({{M}_{\parallel }}\) and \({{M}_{ \bot }}\) are the numbers of longitudinal and transverse modes, respectively), we can write [25]

Such a description is applicable to detectors that record photon pairs generated upon parametric down-conversion in a nonlinear crystal under the action of a monochromatic laser pump. Each photon from the generated pair is detected by its own detector: the photon of the higher frequency (optical) by the detector in the signal channel with quantum efficiency η_{s}, while the photon of the lower frequency (THz or also optical) by the detector in the idler channel with quantum efficiency η_{i}. In the simplest idealized case, photons are detected by two detectors whose angular and linear input apertures are matched so that the number of radiation modes \({{M}_{\parallel }}{{M}_{ \bot }} = {{M}_{{\text{c}}}}\) in the mutually correlated photon fluxes received by each detector is the same. In a real experiment, there may always be some mismatch between the spectral intervals and/or spatial apertures of detection. Because of such a mismatch, one or both detectors also receive uncorrelated radiation in their respective frequency ranges. If *M*_{uj} is the number of modes of uncorrelated radiation received by the signal or idler detector (here, *j* = i or s), Eq. (3) for the average photocurrent recorded by each detector can be rewritten as

The average number \(\langle N\rangle \) of photons per mode of parametric down-conversion radiation depends on the parametric gain [25]

and the phase mismatch. Here, \({{\chi }^{{(2)}}}\) and *L* are the effective nonlinear susceptibility and length of the crystal, respectively; *E*_{p} is the pump field amplitude; and *k*_{sz} and *k*_{iz} are the components of the signal and idler wave vectors along the pump direction, respectively. In the case of a large number of received modes, the total number of detected photons in each channel increases with the parametric gain coefficient as [26]

## TWO-MODE SQUEEZE FACTOR AND DIFFERENCE-PHOTOCURRENT NOISE REDUCTION FACTOR

The signal and idler beams generated in the parametric down-conversion process feature equal fluctuations in the photon numbers for any value of the parametric gain. The commonly accepted quantum-optical characteristic of two-mode photon-number squeezing is the variance of the difference in the numbers of photocounts in the signal and idler channels (*m*_{s} and *m*_{i}, respectively) divided by the sum of the corresponding average values [18, 21, 22, 25, 27–29]:

Let us call this quantity the photon NRF. In the case of ideal detectors with \({{\eta }_{j}} = 1\), noise of the difference photocurrent is suppressed completely. The lower the quantum efficiency of the detector, the closer the photon NRF to the maximum value of 1, characteristic of classical uncorrelated photon fluxes.

The measurement of the two-mode squeeze factor using single-photon detectors is the basis of the standard-free calibration of the quantum efficiency of photon-counting detectors operating either directly in the photon counting mode or, for the detection of high-power optical signals, in the analog mode [18, 21, 22, 25, 27–29]. In the case of analog detection, the quantity determined from experimental data is the variance of the difference between current readings of the detectors in the signal and idler channels divided by the sum of the corresponding average currents:

To reduce the number of variables, the “reduced” current in one of the channels (e.g., the signal channel) is frequently introduced as \(\langle i_{s}^{'}\rangle = \alpha \langle {{i}_{s}}\rangle \), where the coefficient \(\alpha \equiv \langle {{i}_{i}}\rangle {\text{/}}\langle {{i}_{s}}\rangle \) is easily measured experimentally [25]. Calculating the reduced “current” NRF

and normalizing it to the single-photon charge \(\langle {{q}_{i}}\rangle \), one can find the two-mode photon-number squeeze factor \(\sigma _{{{\text{red}}}}^{{({\text{phot}})}}\) and the quantum efficiency of the counting detector in the signal channel.

Let us derive the general expression for the current NRF \(\sigma _{{{\text{red}}}}^{{{\text{(anal)}}}}\) without assuming any constraints on the amplitude fluctuations of elementary pulses from the detectors. According to the definition (6c),

Using the results obtained in [25], let us write expressions for the variance of photocurrents from each detector

and for the correlation between these currents

Then, we obtain for the current NRF

Here, the parameters \({{\varepsilon }_{j}} \equiv {\text{Var}}({{q}_{j}}){\text{/}}{{\langle {{q}_{j}}\rangle }^{2}}\) (*j* = i or s) characterize the magnitude of fluctuations in the amplitude of single-photon response of each of the detectors.

## MEASUREMENT OF THE QUANTUM EFFICIENCY OF THE DETECTOR USING CURRENT NRF DATA

Fluctuations in the amplitude of the single-photon response are nearly absent in detectors capable of operating in the photon counting mode; i.e., in this case, the variance of the charge \({{q}_{j}}\) is zero and \({{\varepsilon }_{j}} = 0\). The “instantaneous” current readings are related to the number of photocounts simply as \({{m}_{j}} = {{i}_{j}}{\text{/}}{{q}_{j}}\). Then, the ratio of the average readings of the detectors in the idler and signal channels (see Eq. (3a)) is

Here, coefficients κ_{j} ≡ *M*_{uj} /*M*_{c} represent the fraction of uncorrelated modes received by each detector, which is determined by their respective apertures and sp-ectral bands. The reduced two-mode squeeze factor \(\sigma _{{{\text{red}}}}^{{({\text{phot}})}} \equiv \frac{{{\text{Var}}({{\alpha }^{{({\text{phot}})}}}{{m}_{s}} - {{m}_{i}})}}{{{{\alpha }^{{({\text{phot}})}}}\langle {{m}_{s}}\rangle + \langle {{m}_{i}}\rangle }}\) (where \({{\alpha }^{{({\text{phot}})}}} \equiv \) \(\langle {{m}_{i}}\rangle {\text{/}}\langle {{m}_{s}}\rangle = \alpha {{q}_{s}}{\text{/}}{{q}_{i}}\)) coincides with the normalized value of the current NRF; i.e., \(\sigma _{{{\text{red}}}}^{{({\text{phot}})}} = \sigma _{{{\text{red}}}}^{{({\text{anal}})}}{\text{/}}{{q}_{i}}\). The general relation described by Eq. (10) then leads to the following expression for the normalized current NRF in the case where single-photon detectors are used:

In the case of exact matching between the modes received by the two detectors, *M*_{ui} = *M*_{us} = 0 and this expression assumes the simple form

Relations (12) and (12a) underlie the known method of standard-free calibration of photon-counting detectors [21, 22, 28, 29]. Certainly, it is most convenient to perform measurements directly in the counting mode, because there will be no need to know the value of the single-photon charge. By measuring the NRF \(\sigma _{{{\text{red}}}}^{{({\text{phot}})}}\) and the ratio \({{\alpha }^{{({\text{phot}})}}}\) of the average reading of the two detectors, one can determine the quantum efficiency η_{i} of the detector in the idler channel using Eq. (12a), and the quantum efficiency of the detector in the signal channel is then found as \({{\eta }_{s}} = {{\eta }_{i}}{\text{/}}{{\alpha }^{{({\text{phot}})}}}\). In the more complicated case where the presence of unmatched modes cannot be disregarded, Eq. (12) has to be used. If the photon fluxes are so high that single-photon current pulses overlap, adaptation to the analog detection mode is also straightforward: according to the above expressions, we only need to additionally measure the value of the single-photon charge.

Now, on the basis of the general expression (10), let us derive a relation similar to Eq. (12) that is not restricted to the case of photon-counting detectors. In the current detection mode implemented with arbitrary detectors, the relation between the current NRF and the quantum efficiency η_{i} assumes the form

The simple linear relation of the current NRF with the two-mode squeeze factor no longer holds, and its linear relation with the quantum efficiency of the idler detector becomes quite indirect. One can see that, if \({{\varepsilon }_{j}} \ne 0\) (the detectors are not single-photon), the relation between the current NRF and the quantum efficiency η_{i} will depend on the number of photons \(\langle N\rangle \) per parametric down-conversion mode even in the absence of uncorrelated modes:

Even in the case where both analog detectors have the maximum quantum efficiency, fluctuations of the difference photocurrent cannot be suppressed completely. The current NRF has a nonzero value of \(\frac{{{{\varepsilon }_{s}} + {{\varepsilon }_{i}}}}{2}\langle {{q}_{i}}\rangle \) for zero photon number \(\langle N\rangle \) per parametric down-conversion mode and increases with \(\langle N\rangle \) as \(\sigma _{{{\text{red}}}}^{{({\text{anal}})}} = \frac{{{{\varepsilon }_{s}} + {{\varepsilon }_{i}}}}{2}(1 + \langle N\rangle )\langle {{q}_{i}}\rangle \).

In order to absolutely determine the quantum efficiency of an analog detector from the corresponding current NRF, it is necessary to know the characteristics of the single-photon response of the detector, i.e., its average charge \(\langle {{q}_{j}}\rangle \) and variance Var\(({{q}_{j}})\). These can be obtained by fitting the statistical distributions of “instantaneous” current readings of each detector. Analysis of the histograms yielding the average value and variance of the single-photon charge has been successfully carried out for analog detectors working in different spectral ranges [29–31]. However, this is insufficient. In most experiments, especially in the case of strongly frequency-nondegenerate parametric down-conversion, where the detection schemes of the signal and idler channels are fundamentally different, one cannot be sure that the contributions from uncorrelated modes are negligible. The coefficients κ_{i} in Eq. (13) can be rather large (for at least one of the detectors), and their values must also be determined in advance.

Let us consider one of the possible approaches to the measurement of the quantum efficiency of an analog THz detector in the THz parametric down-conversion channel by the current NRF. Suppose a single-photon photodetector with ε_{s} = 0 is placed in the signal optical channel. Owing to the large difference in wavelengths, the idler THz radiation has a much greater angular divergence [12] and is more difficult to focus onto the corresponding detector than the optical component of parametric down-conversion radiation. Under these conditions, it can be assumed that, if the input apertures and the detection band in the signal channel are maximized, the THz detector receives only correlated modes, while the optical detector can capture a much larger fraction of radiation in the signal channel. Therefore, κ_{i} = 0, but κ_{s} has to be determined experimentally. Let a filter with variable transmission coefficient *T* be placed in front of the THz detector in the idler channel of the parametric down-conversion setup. Consider the linear dependences of the normalized current NRF measured in such a setup on the number \(\langle N\rangle \) of photons per parametric down-conversion mode and the transmission coefficient *T* of the filter:

By varying the laser pump power at a constant *T*, we vary the value of \(\langle N\rangle \) according to Eqs. (4) and (5). A linear fit for the resulting dependence then yields the parameters

Performing these fits for different values of the filter transmittance *T*, we can find the coefficient in the linear dependences \({{A}_{1}}(T) = {{A}_{2}} + {{B}_{2}}T\) and \({{B}_{1}}(T) = {{B}_{3}}T\):

With these three quantities known, we can find the quantum efficiency η_{i} and the relative variance ε_{i} of the single-photon charge of the THz detector and, in addition, the mode filling factor κ_{s} of the signal channel. The values \(\langle {{q}_{s}}\rangle \) and \(\langle {{q}_{i}}\rangle \) of the average single-photon charge for each detector should be determined beforehand from, e.g., the analysis of statistical distributions of the corresponding current readings. The accuracy of the outlined method for the measurement of the quantum efficiency is to a large degree determined by the accuracy of the fitting procedures involved.

Therefore, although the measurement of the current NRF using analog detectors cannot be directly used to characterize the two-photon squeeze factor of the parametric down-conversion field, it is still possible to determine the quantum efficiency of such detectors from the NRF. However, in addition to direct measurements of the current NRF, several fitting procedures will be required for this purpose. The accuracy of the result obtained will obviously be lower than that in the case of similar calibration of single-photon detectors.

## CALIBRATION OF THE QUANTUM EFFICIENCY USING CURRENT COVARIANCE DATA

Let us now consider the normalized covariance of the idler and reduced signal currents \(C \equiv \) \(\langle \delta i_{s}^{'}\delta {{i}_{i}}\rangle {\text{/}}(\langle {{i}_{i}}\rangle + \langle i_{s}^{'}\rangle )\). Taking into account Eq. (9), we can write for any type of detectors used

Analyzing the dependence of this correlation parameter on the laser power, we can directly determine the ratio \(\frac{{{{\eta }_{i}}}}{{(1 + {{\kappa }_{s}})}}\); no additional information on the mode matching parameter κ_{i} in the idler channel and the variance of single-photon response in both channels is required. This ratio can also be measured for a fixed known number of photons \(\langle N\rangle \) by varying the transmission coefficient *T* of the filter in the idler channel. However, in order to determine the quantum efficiency, the parameter 1 + κ_{s} is required. Therefore, the quantum efficiency can be determined on the basis of covariance measurements only when the absence of uncorrelated modes in the signal channel can be ensured and, thus, κ_{s} = 0. We note that the additional measurement of the average charge \(\langle {{q}_{i}}\rangle \) and the normalization of covariance by this value can be discarded if, instead of the quantum efficiency of the detector η_{i}, we need its current sensitivity, equal to \(\langle {{q}_{i}}\rangle {{\eta }_{i}}{\text{/}}(\hbar {{\omega }_{i}})\).

Thus, the measurement of the quantum efficiency of an analog detector on the basis of covariance data is much less demanding: there is no need for a single-photon detector in the signal channel, and uncorrelated modes can occur in the idler channel. One just needs to narrow the set of modes in the signal channel so that only modes matched with at least some of the parametric down-conversion modes received by the idler detector are present.

With both of the considered methods, the accuracy of the measurement of η_{i} will depend on the accuracy of current readings, on the accuracy of determining the amplitude \(\langle {{q}_{i}}\rangle \) of the single-photon response of the detector to be calibrated, and on the accuracy of determining the average number \(\langle N\rangle \) of photons in the field mode. The amplitude \(\langle {{q}_{i}}\rangle \) is determined by fitting the histograms of statistical distributions of current readings with an accuracy that depends significantly on the approach taken to model the noise of each specific type of detector [29–31]. The corresponding contribution to the relative error can reach 1–5%. The value of \(\langle N\rangle \) is determined to an accuracy limited by that of the measurement of β_{si}, which is carried out by fitting the experimental dependences of the power of the detected parametric down-conversion radiation on the pump power in the high-gain mode. Although the corresponding relative error in \(\langle N\rangle \) increases with β_{si}, a regime with a moderate value of β_{si} < 5 can be selected for the measurement of the quantum efficiency. Then, the contribution of this type of error will also be around a few percent. As can be seen from Eq. (17), only these three factors determine the ultimate error of measuring the quantum efficiency from covariance data, which can also be brought to levels below 10%. In the case of measurements based on current NRF data, additional linear approximation procedures will be required to determine the coefficients \({{A}_{2}}\), \({{B}_{2}}\), and \({{B}_{3}}\) (see Eq. (16)). Without conducting direct experiments, it is difficult to predict the error introduced by these procedures and to understand how well they will compensate the NRF measurement error associated with the inaccuracy of individual current readings.

In summary, we have considered general relations describing the effect of the characteristics of the fluctuating single-photon responses of the detectors in the signal and idler channels on the difference-photocurrent noise reduction factor and on the normalized covariance of the currents from the signal and idler detectors in the multimode parametric down-conversion setup. The analysis of these relations demonstrates that the measurement of the current NRF using analog detectors does not allow direct characterization of the two-photon squeeze factor in the parametric down-conversion field. To determine the quantum efficiency of the detector from NRF data, one needs, in addition to direct measurements of the current NRF, to measure the statistical distributions of current readings of the detectors and the NRF dependences on the number of photons per parametric down-conversion mode and on the channel transmission coefficient and to carry out fitting procedures with these dependences. Another proposed method to determine the quantum efficiency of analog detectors, which uses the data on the covariance of signal and idler currents, may be much simpler and more accurate. The results obtained will be important in the development of the experimental foundations of standard-free methods for calibrating quantum efficiency and measuring the current sensitivity of a wide range of analog detectors, including THz detectors, as well as detectors of other spectral ranges that are incapable of operating directly in the photon-counting mode.

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## Funding

This study was supported by the Russian Science Foundation, project no. 22-12-00055 (the development of methods to determine the quantum efficiency of analog detectors), and by the Russian Foundation for Basic Research, project nos. 20-32-90232 (calculations of the noise reduction factor of the difference photocurrent) and 20-02-00621 (calculations of covariance).

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Translated by M. Skorikov

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Novikova, T.I., Leontyev, A.A. & Kitaeva, G.K. Measurement of the Quantum Efficiency of Analog Detectors in the Parametric Down-Conversion Field.
*Jetp Lett.* **116**, 353–359 (2022). https://doi.org/10.1134/S0021364022601701

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DOI: https://doi.org/10.1134/S0021364022601701