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Radiometric characterization of a triggered narrow-bandwidth single-photon source and its use for the calibration of silicon single-photon avalanche detectors

The traceability of measurements of the parameters characterizing single-photon sources, such as photon flux and optical power, paves the way towards their reliable comparison and quantitative evaluation. In this paper, we present an absolute measurement of the optical power of a single-photon source based on an InGaAs quantum dot under pulsed excitation with a calibrated single-photon avalanche diode (SPAD) detector. For this purpose, a single excitonic line of the quantum dot emission with a bandwidth below 0.1 nm was spectrally filtered by using two tilted interference filters. Since high count rates are essential for many metrological applications, we optimized the setup efficiency by combining high overall transmission of the optical components with a geometrical enhancement of the extraction efficiency of a single quantum dot by a monolithic microlens to reach photon fluxes up to 3.7⋅105 photons per second at the SPADs. A relative calibration of two SPAD detectors with a relative standard uncertainty of 0.7% was carried out and verified by the standard calibration method using an attenuated laser. Finally, an Allan deviation analysis was performed giving an optimal averaging time of 92 s for the photon flux.


Introduction
The range of possible implementations of single-photon sources and detectors is rapidly growing. They find application in many quantum optical technologies such as quantum key distribution, quantum-enhanced imaging, quantum computing, quantum metrology and quantum radiometry [1]. Therefore, there is an increasing need to ensure high accuracy and metrological traceability of measurements in these fields, also for metrology itself.
In quantum metrology, the identity principle guarantees universality and reproducibility of the observed quantum phenomena, whereas the existence of discrete energy states opens the possibility of reaching high resolution in energy [2]. Moreover, energy conservation for transitions between these states enables a direct connection between macroscopic physical units and the fundamental constants by counting single quanta [3]. For example, single-electron transport pumps [4] could be utilized to realize the unit of Ampere by counting the number of transferred elementary charges per time. Similarly, one can make use of the quantization of the electromagnetic radiation to define the optical power by counting photons. The optical power depends only on the photon flux and on the wavelength of the electromagnetic radiation. A possible choice for a detection system are transition edge sensors, whose main advantage is the number-resolving capability, but they suffer from a reduced temporal resolution [5]. The most widely used photon counting devices nowadays, the SPAD detectors, are unable to resolve the photon number. Consequently, for calibration purposes, highest precision can be reached only by ensuring single-photon purity. This means that there has to be exactly one photon per time interval, while every photon should have the same energy. Laser light fulfills the requirement of monochromatic light, but the nature of the Poisson statistics sets a theoretical limit because of the non-zero probability for multiphoton events. This probability can be reduced by decreasing the mean photon number, but at the cost of a substantial vacuum component. On the other hand, single-photon sources based on quantum emitters and on parametric down conversion can in principle reach simultaneously high efficiency and high single-photon purity [6][7][8][9].
Different types of quantum emitters have already been investigated, ranging from trapped atoms or ions to color centers in diamond, quantum dots and organic molecules [10,11], but yet only a few have been metrologically characterized to allow for a quantitative comparison and evaluation in the field of quantum radiometry. For instance, a nitrogen vacancy center in a nanodiamond was found to have a total photon flux of approx. 2.6 · 10 5 photons per second and a g (2) (0) value of 0.1 at room temperature; however, its spectral radiant flux was below 350 aW nm −1 due to the broad emission spectrum [12]. The first direct detection efficiency calibration of a SPAD detector at a specific wavelength was achieved by using a single organic dye molecule as a light source with a tunable optical radiant flux in the range from 36.5 fW to 334 fW, measured with an analog reference low-noise detector, and a g (2) (0)value below 0.1 [13]. The bright narrow-bandwidth emission at (785.6 ± 0.1) nm was obtained at 3 K via continuous wave pumping. Therefore, the smallest time interval between two consecutive photons ∆t min is given by the inverse decay rate of the excited state. A more sophisticated approach would be to use pulsed excitation, so that the time interval is determined by ∆t min = 1/f, where f is the laser repetition frequency-an external parameter that could be used to tune the temporal properties of the single-photon source, as it was carried out for a silicon vacancy center based single-photon source [14].
In this paper, we present the absolute characterization of semiconductor quantum dots (QDs) under pulsed excitation and their direct application for the relative calibration of two SPAD detectors. The exciton recombination lines of InGaAs QDs under non-resonant excitation have typical linewidths of less than 100 µeV [15,16] (corresponding to 70 pm at 930 nm) and even reach the Fourier transform limited homogenous linewidth under resonant excitation [17], and hence fulfill the condition of nearly monochromatic radiation. Moreover, their mechanical robustness and conformity with other semiconductor devices, enabling on-chip operation, give them an important advantage over molecule-and nanoparticle-based sources. Furthermore, InGaAs QDs are an important platform for the field of quantum key distribution [18] and allow for the generation of polarization-entangled photons via the biexciton-exciton radiative cascade [19].

Single-photon source
The single-photon emitter consists of an InGaAs QD deterministically embedded into a monolithic microlens, positioned on top of a distributed Bragg reflector (see figure 1(a)). Numerical simulations of a similar geometrical configuration [20] predict an enhancement of the extraction efficiency of up to 23%, when using a collecting lens with a numerical aperture (NA) of 0.7. An additional antireflection layer on top of the microlens surface further decreases the coupling losses. The fabrication process is described in detail in reference [21].
The confocal setup, shown in figure 1(b), has a simple and compact design, optimized to reach high transmission at 930 nm. It consists of a self-made confocal microscope with implemented spectral filtering. Bearing in mind that only the actual signal arriving at the detector area is of any relevance for calibrations, we treat the whole system comprised of a quantum emitter and optics for spatial and spectral filtering as a self-contained single-photon source. A diode laser (PicoQuant, LDH-D-C-850), which either operates in continuous wave mode or in pulsed mode with repetition frequencies between 1 kHz and 80 MHz and a pulse width of about 64 ps (full width at half maximum (FWHM)) is used for nonresonant excitation of the QD. The laser beam is reflected at a dichroic beam splitter (Semrock, 875 nm edge Bright-Line) and focused on the sample, which is placed in a helium flow cryostat to maintain a constant temperature of approximately 10 K. The QD fluorescence is collected by an infinity corrected objective (Mitutoyo, 100x Plan Apo NIR HR, NA = 0.7). The fluorescent light passes through one long-pass and two bandpass filters and is then coupled into a single mode (9 µm core diameter) or a multimode (62.5 µm core diameter) optical fiber serving as a pinhole for confocal imaging. The optical fiber can be connected either to a spectrometer, a SPAD detector or a fiber-based Hanbury Brown and Twiss interferometer (HBTI) to analyze the spectral characteristics, measure the photon flux or determine the single-photon purity. For micro-photoluminescence scans, the position of objective 1 is varied in steps of 0.1 µm via an XYZ piezo scanner and only emission above 900 nm reaches the SPAD detector (Perkin Elmer, SPCM-AQRH-13-FC) due to the long-pass filtering.
In the case of non-resonant excitation, the selection of a spectral line corresponding to the recombination of a specific excitonic state is conventionally carried out by a monochromator, which guarantees high spectral resolution at the cost of polarization dependent transmission and rather high optical losses. Instead, we are using two identical narrow bandpass filters (Alluxa, 935.0-0.45 OD5), based on thin film interference with a transmission of about 90% each (see figures 2(c) and (d)). The FWHM of the transmission window was determined to be 0.5 nm by using a spectrophotometer with a spectral bandwidth of 0.05 nm. For a required wavelength λ c , defined as the central wavelength of the filter's transmission window, the incident angle θ has to be [22]: where n is the effective refractive index and λ 0 = λ c (θ = 0). Both filters are mounted on rotation stages enabling a precise rotation in steps of 100 µrad. The stages are rotated by about the same angle θ in opposite directions to compensate for the angle dependent beam shift (figure 2(a)). A rotation up to 20 • completely covers the relevant sample fluorescence range from 915 nm to 935 nm, as shown in figure 2(b). The overall setup transmission was measured by coupling the monochromatic emission (FWHM = 0.1 nm) of a tunable laser to the optical fiber next to objective 3 and replacing the cryostat by a silicon photodiode. The output power of the laser was monitored to correct for any temporal variations. The transmission of the cryostat window was subsequently added to the overall result, yielding T setup = (25.9 ± 0.4) % at (923.1 ± 0.1) nm (θ = 15 • ).
The total efficiency depends not only on the wavelength dependent transmission of all optical components, but also on further dependencies which we currently cannot investigate independently with our setup. For example, the unknown complex intensity profile of the emitter has a strong influence on the extraction efficiency into the first lens, as well as on the coupling efficiency into the optical fiber. A non-zero beam divergence could also lead to optical losses due the high directional sensitivity of the bandpass filters. Furthermore, a non-ideal internal quantum efficiency of the QD and the excitation probability for a specific excitonic transition also play an important role.
For a precise and reliable photon flux determination, the detection efficiency η detection of a SPAD is calibrated against a reference analog detector by means of the double attenuator technique [23]. Therefore, the absolute photon flux φ and thus the absolute optical power P are traceable to the primary standard for optical power, the cryogenic radiometer.
With the redefinition of the SI units from May 2019, the Planck constant h and the speed of light c have exact values with no uncertainties. The narrow bandwidth of the quantum dot spectral lines leads to a small uncertainty for the wavelength λ (about 0.01%). Hence, the main uncertainty contribution comes from the measured photon fluxṄ measured . An Allan deviation analysis is conducted to determine the optimal averaging time leading to lowest standard uncertainty.

Metrological characterization of the single-photon source
The emission of the QDs embedded into individual monolithic microlenses is spatially and spectrally filtered with the confocal setup shown in figure 1(b) (see previous section). A 2D micro-photoluminescence scan of the QD layer showing the sample emission above 900 nm is used to precisely determine the position of each deterministically fabricated QD-microlens. For the reminder of this work, we choose the QD-microlens marked in figure 3(a), since it simultaneously fulfills the selection criteria of having high brightness as well as high spectral and single-photon purity. The very low signal intensity inside the square region is due to the removal of all QDs and the associated wetting layer between the microlenses by reactive ion etching as part of the fabrication process. Furthermore, the large distance between neighboring QD-microlenses on the order of 5 µm facilitates the convenient spatial selection of a single QD for spectral analysis. A typical spectrum for continuous wave excitation below saturation is presented in figure 3  is determined to be 42 pm and is limited by the spectrometer's resolution.
For a weak excitation, the linear dependence of the normalized peak area on the pump power in figure 4 indicates the recombination of a neutral or a charged exciton. The solution of a system of rate equations [24,25] describing the photon emission rate of the exciton ground state is given by The fit parameters k 1 and k 2 are proportional to the exciton and biexciton radiative lifetimes τ X and τ XX , whereas k 3 is a normalization constant. The variable x represents the exciton creation rate and is thus proportional to the optical power of the excitation laser. Equation (3) is fitted to the data set under the assumption k 1 /k 2 = τ X /τ XX = 1.8 [26] and by using the fit parameters k 2 = (0.78 ± 0.07) µW −1 and k 3 = 2.6 ± 0.1.
Next, the second-order correlation function g (2) a,b was measured under pulsed excitation with a repetition frequency of 40 MHz using an HBTI with an overall temporal resolution of 1 ns. The raw data is shown in figure 5(a). This measurement is a result of the detection of two uncorrelated sources with intensities a and b, where a arises from the investigated source and b from the dark counts of the detection system, i.e. when the excitation laser is switched off. The dark counts are assumed to follow Poisson statistics and are completely uncorrelated from the sample emission, i.e. g 2 b = 1. The function g can be expressed as [27,28] g (2) a,b = g (2) and g (2) a is the dark-count corrected intensity correlation function of the source. The value for g (2) a,b is calculated by binning the coincidences in a time window of 10.5 ns (corresponding to 41 bins), centered at τ = n/f rep , where n is an integer, and dividing the area of the central peak at τ = 0 by the mean area of the six nearest peaks. This yields g (2) a,b (0) = 0.35 ± 0.02 and g (2) a (0) = 0.24 ± 0.06. There is a constant offset of about 0.1 in the normalized coincidence histogram, considerably affecting the g (2) (0) value. Its origin can be evaluated by a decay rate measurement, presented in figure 5(b). It shows a histogram of the exciton decay for the same excitation repetition frequency of 40 MHz. After 12.5 ns, corresponding to half the repetition period, the signal drops to 3.8%. The decay can be best described by a double exponential function with decay constants of (1.11 ± 0.02) ns and (8.2 ± 0.7) ns and a constant offset of 0.8%. The uncertainties are determined from a comparison of measurements at laser repetition frequencies of 20 MHz, 40 MHz and 80 MHz. A decay time on the order of 1 ns is typical for this type of QDs [29], whereas the second slow decay component creates a peak overlap, which can partially explain the observed offset in figure 5(a). This slow component could be caused by additional dark exciton transitions [30] or by a re-excitation due to a hole recapture process taking place at crystal defects or fluctuations of the wetting layer potential [31].
Another background contribution, that could be observed mainly in the intensity correlation histogram because of the significantly smaller coincidence rate, is fluorescent emission from the sample matrix, e.g. from the carrier-overloaded wetting layer. It causes an increasing g (2) (0) value at higher excitation intensities. Since this is an inherent property of the nonresonant excitation, exploiting other excitation schemes (e.g. p-shell and s-shell resonant excitation) is expected to further reduce the value. The last step of the source characterization is to determine the photon flux and the optical power according to equation (2) (see table 1). The highest optical power amounts to (79.4 ± 1.2) fW. Considering the spectral width of 42 pm, one can calculate a maximum spectral radiant flux of about 1.9 pW nm −1 for an excitation rate of 80 MHz.

Detection efficiency calibration of Si-SPAD detectors-method
The ratio of detection efficiencies of two commercial silicon SPAD detectors (Perkin Elmer, SPCM-AQRH-13-FC) is determined by using the emission of a single InGaAs QD as a light source. The applied fiber exchange technique is wellknown in the field of optical telecommunications [32] and has been successfully implemented for a relative SPAD calibration using the spectrally filtered emission of a nitrogen vacancy center in nanodiamond; however, with count rates below 3000 counts per second and a spectral bandwidth of 19 nm [33]. The model presented here is extended to take into account further systematic deviations.
The spatially and spectrally filtered QD emission is split by a 50:50 fiber-based beam splitter, as shown in figure 6. For the first measurement (I), fibers a and b are connected via FC/PC couplers to the standard detector (S) and the device under test (T), respectively. The electrical signal from both detectors is integrated over a period of one second by two counters, synchronized by an external trigger with a frequency of 0.5 Hz. The dark-count corrected counts per secondṄ =Ṅ measured − N dark for each detector can be expressed as a product of the incoming photon flux φ, the fiber transmission κ and the detection efficiency η: For the second measurement (II), the positions of the detectors are switched, so thatṄ II The following formula for the ratio r of the detection efficiencies can be derived from these four equations: Consequently, the value of r is independent of the incident photon flux and of the splitting ratio SR itself, but it depends in general on the variation of this ratio over the course of several measurement series, taken into account by the factor f SR , with f SR = 1 ± u (f SR ). The uncertainty u (f SR ) is set equal to the measured standard deviation of the splitting ratio (SR meas ). However, the variations of SR cannot be directly decoupled from those of the incoming photon flux, since SR meas = φ I /φ II · SR, but the temporal stability of φ can be estimated from two consecutive measurement series in each of the two detector configurations.

Detection efficiency calibration of Si-SPAD detectors-result
The detection efficiency ratio r was determined from ten independent measurement series (see figure 7). Each of them is conducted with a photon flux of approx. 1.2 · 10 5 photons per second (about 2 · 10 4 counts per second at each detector), originating from a single spectral line of the QD emission spectrum. This photon flux corresponds to an optical power of 26 fW, being well in the linear regime of the silicon SPAD detectors. Therefore, no attenuation is required, allowing us to achieve a lower overall uncertainty.
The black error bars in figure 7 display the type A uncertainties, which vary between 0.12% and 0.17%. They originate from the statistical variations of 30 measurements per data point. These ten data points are combined into a single value of r = 1.059 by taking the weighted mean, whereas their uncertainties are combined by calculating the external consistency, also known as the Birge ratio [34], resulting in a total type A uncertainty of 0.24%.
The discrepancy of the data points is clearly due to an additional uncertainty contribution, most probably due to variations of the splitting ratio, which are accounted for by the factor f SR in equation (5). Changes of the splitting ratio can be caused by insufficient reproducibility of the coupling losses for both connector pairs upon each detector exchange. The relative mean temporal deviation of the incoming photon flux was measured to be 0.30%. Although this contribution has no effect on the uncertainty budget, its value shows a possible overestimation of the type B uncertainty, which amounts to u (f SR ) = 0.67%. All values in figure 7 lie within the total expanded uncertainty (k = 2) denoted by the blue area.
One can estimate the expected impact of multiphoton emission from the quantum dot on the detection efficiency calibration. Let p n be the probability of n photons arriving simultaneously at the detection area. In the limit p 2 ≫ p 3 , p 4 , p 5 , . . . the g (2) (0) value can be expessed as g (2) (0) ≈ 2p2 n 2 [35], where ⟨n⟩ ≈ 0 · p 0 + 1 · p 1 + 2 · p 2 is the mean photon number, which differs from the measured value ⟨n measured ⟩ = φ/f by the factor ⟨n measured ⟩ = ⟨n⟩ · (1 − ϵ) due to multiphoton events. The relative correction ϵ when using a non-photon number resolving detector would be given by the number of not counted photons per pulse divided by the mean photon number: The only physically reasonable solution gives an estimated correction of ϵ = 0.009%, which is significantly smaller than the relative standard uncertainty of 0.7% for the calibration. Additionally, an absolute detection efficiency calibration method [23] using attenuated laser light (PicoQuant, LDH-D-C-930) at 930 nm was applied to verify the measurement results yielding η S = 0.327 and η T = 0.307 with a relative standard uncertainty of 1%. To ensure comparability, both calibration methods are applied for the same pulse repetition frequency of 80 MHz, count rate of approx. 2 · 10 4 counts per second and a fiber core diameter of 62.5 µm. The ratios r QD = 1.059 ± 0.008 and r laser = 1.063 ± 0.015 from both independent methods are in very good agreement within the stated uncertainty, see figure 7. It should be noted that the spectral mismatch of 7.6 nm between laser emission and QD emission leads to an estimated correction by 0.002, which is negligible compared to the standard uncertainties.

Count rate stability analysis
The Allan deviation was found to be a very useful tool not only for quantifying the frequency stability of oscillators, but also for stability analysis of an arbitrary signal in the time domain. For example, it has been used for the characterization of gyroscopes [36], microwave radiometers [37], direct current nanovoltmeters [38], SPAD detectors [39] and others. Here, it is used to investigate the temporal stability of the photon flux. The optical signal is split by a 1:2 coupler in order to investigate the dependence of the noise amplitude on the light intensity. The count rate stemming from the QD emission was monitored over a period of seven hours (see figure 8). An increase of signal fluctuations is observed with increasing count rate. The lack of any significant change of the count rates during the whole measurement period demonstrates the stability of the quantum emitter and of the experimental setup.
We are applying the Overlapping Allan deviation (OADEV) [40] because of the improved confidence interval compared to the standard method due to the use of overlapping time intervals. The OADEV σ y (τ ) is a function of the averaging time τ = m · τ 0 , where τ 0 is the basic measurement interval of one second and m is an averaging factor. The calculation for M measurements with a count rate y i is done according to the following equation [41]: The relative OADEV expressed as a percentage of the mean count rate is presented in figure 9. The counts measured from both SPAD detectors have almost identical values for the relative deviation for averaging times between 1 s and 1500 s (blue and red triangles in figure 9), which means that all dominating noise contributions within this time interval scale with the mean value of the count rate. The main noise source is expected to stem from the quantum nature of light; fluctuations of the photon number in a light beam with constant intensity follow the Poisson law [42] and manifest as a shot noise of the electric current upon the electrical conversion of the optical signal by the detection system [43]. Given the fact that the signal-to-noise ratio of the Poisson distribution goes as the square root of the mean photon number, the measurement uncertainty could be further reduced by using a brighter single-photon source.
There is additional intrinsic shot noise originating from the fluctuations of the dark count rate, whose mean value can be determined precisely and subtracted, but its corresponding noise contribution still must be taken into account. For this measurement, it is expected to comprise about 10% of the total shot noise, since the dark counts amount to 207 s −1 for SPAD 1 and 333 s −1 for SPAD 2.
Another possible noise type arises from the thermal energy of the current carriers and is known as Johnson noise [42]. Both the shot and Johnson noise exhibit no frequency dependence and have a slope of −0.5 [38] in the double logarithmic Allan deviation plot (see figure 9). These seem to be the dominating noise contributions for integration times below 20 s, where the slope has a measured value of −0.44. Then the curve starts to flatten out and reaches its minimum at 92 s. Therefore, this is the optimal averaging time with the smallest achievable uncertainty of 0.58%. The increase of the deviation towards even longer integration times with a slope of 0.40 can be partially explained by a random walk with an expected slope of 0.5 [37]. The low frequency quasi-random fluctuations can have many origins, e.g. a drift in the optical power of the pump laser or in the cryostat temperature. Therefore, the laser power and the Figure 9. Double logarithmic plot of the relative overlapping Allan deviation for the measured count rate from SPAD 1 (red triangles pointing up), SPAD 2 (blue triangles pointing down) and for the ratio of both count rates (green diamond symbol). The black lines represent fitted slopes to the data points to estimate the dominating noise sources at different averaging times. A minimum deviation of the count rate of 0.6% is reached for a 92 s averaging interval. The count rate ratio can be more precisely determined with a minimum deviation of only 0.1%. cryostat temperature were monitored during the measurement, yielding correlation coefficients of −0.08 and −0.10 correspondingly. The lack of significant correlation implies that the laser power and temperature shifts of 13 nW and 0.1 K within 7 h have a negligible effect on the count rate stability.
Another possible explanation for the fluctuations on a time scale above 90 s are mechanical instabilities of the optical setup, as slight changes in the position of the objective relative to the quantum emitter would lead to a substantial decrease of the collected light intensity. Additionally, the slight temperature gradient of 0.014 K per hour may induce a mechanical strain in the cryostat, causing a non-linear displacement affecting the collected light intensity. Such a non-linear dependence would not be reflected in the corresponding correlation coefficient.
Of particular interest for the signal stability of the relative calibration method is the OADEV of the ratio of the simultaneously measured count rates of both SPAD detectors (data points marked with diamonds in figure 9). The slope of-0.50 for averaging times below 32 s implies the presence of white noise. A minimum deviation of 0.11% is reached at an averaging time of 128 s. Consequently, the ratio of count rates can be determined with a 5 times better statistical uncertainty compared to the count rate from a single SPAD, since the noise contributions of the simultaneously measured count rates partially cancel out. Responsible for this behavior is their strong correlation (correlation coefficient of 0.95) due to the direct linear dependence of the count rates on the source intensity.

Conclusion
A traceable single-photon source based on a non-resonantly driven InGaAs QD with a spectral bandwidth of 42 pm (FWHM) and a single-photon purity corresponding to a g (2) (0) value of 0.24 ± 0.06 was presented. The highest optical power under non-resonant pulsed excitation with a repetition frequency of 80 MHz amounts to (79.4 ± 1.2) fW for an emission wavelength of 922.4 nm. High optical power is achieved by extraction efficiency enhancement through monolithic microlenses and by low loss bandpass filtering, leading to a high setup transmission of (25.9 ± 0.4) %. The photon flux stability of the source was analysed using the overlapping Allan deviation. Here, white noise dominates for short sampling times and a minimum deviation of 0.58% is reached for averaging 92 values, each with an integration period of 1 s. This is the first absolutely characterized quantum dot based single-photon source emitting photons under pulsed excitation. The triggered emission enables full control over the time interval between two consecutive photons, which can be used to tune the photon flux over a wide range without changing the saturation properties. Furthermore, we have demonstrated the feasibility of detector calibrations in the near infrared. The ratio of detection efficiencies of two similar SPAD detectors has been determined with a standard uncertainty of 0.7%. This result has been confirmed with the standard double attenuator technique.
The simultaneous count rate measurement as part of the relative calibration allows for reaching statistical uncertainties as low as 0.11% due to the lack of dependence on the temporal stability of the photon flux. Moreover, the count rate ratio does not depend on the brightness of the single-photon source, as long as a sufficient signal-to-noise ratio can be reached.