# Detection Efficiency Characterization for Free-Space Single-Photon
Detectors: Measurement Facility and Wavelength-Dependence Investigation

S. Virzì<EMAIL_ADDRESS>Istituto Nazionale di Ricerca Metrologica, str.
delle Cacce 91, 10135 Turin, Italy  A. Meda<EMAIL_ADDRESS>Istituto
Nazionale di Ricerca Metrologica, str. delle Cacce 91, 10135 Turin, Italy  E.
Redolfi  Istituto Nazionale di Ricerca Metrologica, str. delle Cacce 91, 10135
Turin, Italy  Physis Department, University of Turin, via P. Giuria 1, 10125
Turin, Italy  Istituto Nazionale di Fisica Nucleare, sez. di Torino, via P.
Giuria 1, 10125 Turin, Italy  M. Gramegna  Istituto Nazionale di Ricerca
Metrologica, str. delle Cacce 91, 10135 Turin, Italy  G. Brida  Istituto
Nazionale di Ricerca Metrologica, str. delle Cacce 91, 10135 Turin, Italy  M.
Genovese  Istituto Nazionale di Ricerca Metrologica, str. delle Cacce 91,
10135 Turin, Italy  Istituto Nazionale di Fisica Nucleare, sez. di Torino, via
P. Giuria 1, 10125 Turin, Italy  I. P. Degiovanni  Istituto Nazionale di
Ricerca Metrologica, str. delle Cacce 91, 10135 Turin, Italy  Istituto
Nazionale di Fisica Nucleare, sez. di Torino, via P. Giuria 1, 10125 Turin,
Italy

###### Abstract

In this paper, we present a new experimental apparatus for the measurement of
the detection efficiency of free-space single-photon detectors based on the
substitution method. For the first time, we extend the analysis to account for
the wavelength dependence introduced by the transmissivity of the optical
window in front of the detector’s active area. Our method involves measuring
the detector’s response at different wavelengths and comparing it to a
calibrated reference detector. This allows us to accurately quantify the
efficiency variations due to the optical window’s transmissivity. The results
provide a comprehensive understanding of the wavelength-dependent efficiency,
which is crucial for optimizing the performance of single-photon detectors in
various applications, including quantum communication and photonics research.
This characterization technique offers a significant advancement in the
precision and reliability of single-photon detection efficiency measurements.

††preprint: AIP/123-QED

The detection of single photons is a critical component in a variety of
scientific and technological applications, including quantum
communicationZapatero  _et al._ (2023); Diamanti  _et al._ (2016); Pirandola
_et al._ (2020), quantum computingLadd  _et al._ (2010); Gill  _et al._
(2022); Krantz  _et al._ (2019), quantum imagingGenovese (2016) and quantum
sensingDegen, Reinhard, and Cappellaro (2017); Aslam  _et al._ (2023);
Bakhshandeh (2022); Petrini  _et al._ (2020) with photons. Accurate photon
detection is essential for ensuring the reliability and precision of these
applications. Consequently, the calibration of the detection efficiency of
single-photon detectors (SPDs) is of paramount importanceMigdall  _et al._
(1995, 2002); Ghazi-Bellouati  _et al._ (2005); Polyakov and Migdall (2009a,
b); Chunnilall  _et al._ (2014); López, Hofer, and Kück (2015a); Bae  _et al._
(2019); López  _et al._ (2020); Gerrits  _et al._ (2020); Georgieva  _et al._
(2021); Jin, Gerrits, and Gamouras (2022). Such an efficiency is
definedBienfang  _et al._ (2023) as the probability of a SPD producing a
measurable signal in response to one incident photon, depending on the
wavelength and detection rate, with specific wavelength and count rate
specifications.

However, unlike detectors calibrated using classical radiometric techniques,
there is currently no established standard for calibrating the detection
efficiency of SPDs based on the measured counts. This lack of standardization
presents a significant challenge, as it hinders the ability to ensure
traceability of calibration results to the International System of Units (SI).
For this reason, a pilot studystu  has recently been initiated among various
national metrology institutes (NMIs) worldwide to attempt to define a
characterization standard for free-space silicon single-photon avalanche
diodes (Si-SPADs) detecting photons with a wavelength of $850\mathrm{\,nm}$.
This collaborative effort aims to establish a unified and precise methodology
for assessing the performance and detection efficiency of these detectors,
thereby providing a reliable benchmark for scientific research and
technological applications that depend on accurate single-photon detection.

Si-SPADs are SPDs operating in Geiger modeBronzi  _et al._ (2016); Villa  _et
al._ (2021); Bruschini  _et al._ (2019); Migdall  _et al._ (2013). They are
widely exploited due to their high detection efficiency in the visible range
(up to $80\%$ for wavelengths around $650\mathrm{\,nm}$), their low dark count
rate (tens of counts per second), and their short dead time and jitter
(typically tens of nanoseconds and hundreds of picoseconds, respectively).
They can be exploited for a broad wavelength interval, from approximately
$400\mathrm{\,nm}$ to $1000\mathrm{\,nm}$. In particular, they find huge
application for free-space Quantum Key Distribution (QKD)Nielsen and Chuang
(2001), wherein the employed wavelength is often around $850\mathrm{\,nm}$,
because of the transmissivity of the atmosphere just considering the range
detectable from silicon-based SPDs, that is also the wavelength considered in
the pilot study.

The detection efficiency is inherently dependent on the wavelength of the
incident photons. In addition, in SPDs operating in free-space with an optical
window typically made of glass, there can be an additional (nonlinear)
dependency on wavelength due to the interference effect that occurs between
the two optical surfaces of the window, changing the transmissivity. Glass and
quartz windows, while offering high transparency across a broad spectrum,
still exhibit such a behavior, acting as an optical cavity, that can affect
the overall detection efficiency. These variations must be carefully
characterized and accounted for to ensure accurate and reliable photon
detection across different wavelengths. Furthermore, there is ongoing debate
about interference effects due to a spatial nonuniform detector response, that
could significantly impact the detection efficiency (see, e.g., the study
carried on at NPLArabskyj  _et al._ (2024)).

In this work, we present the INRiM new experimental setup for the measurement
of the detection efficiency for free-space SPDs based on the substitution
methodLópez, Hofer, and Kück (2015b); López  _et al._ (2020). In particular,
we demonstrate such a technique on free-running Si-SPADs at
$850\mathrm{\,nm}$, in the framework of the aforementioned pilot study.
Additionally, we provide a model for the transmissivity of the quartz optical
window to account for its impact on the overall detection efficiency.

Calibration technique. The substitution method consists in a technique for
comparing the signal measured by a SI-traceable detector with respect to the
one measured from a SPD after a proper attenuation. This comparison regards
very different light fluxes, deviating of several orders of magnitude. For
example, a photon flux of about $1000\mathrm{\,counts/s}$ with wavelength
$850\mathrm{\,nm}$ corresponds to an average optical power in the order of
$10^{-16}W$. Hence, the required attenuation between the high-flux regime and
the single-photon level is usually around six or seven orders of magnitude,
and it is of the utmost importance to characterize such an attenuation,
containing its related uncertainty.

Then, the detection efficiency $\eta(\lambda)$ of a SPD can be estimated
comparing the macroscopic optical power $\mathcal{P}$ of a laser source,
measured with a SI-traceable calibrated detector, and the rate $\mathcal{R}$
measured by the SPD after attenuating the same signal down to the single-
photon level:

$\eta(\lambda)=\frac{hc}{\lambda}\frac{\mathcal{R}}{\tau\mathcal{P}}$ (1)

where $h$ is the Planck constant, $c$ is the speed of light in the vacuum and
$\tau$ is the transmissivity due to the introduced attenuation.

In Fig. 1 we show the measurement apparatus.

Figure 1: Experimental setup. The source is a tunable Ti:Sa laser followed by
a variable optical attenuator. After a mechanical shutter, photons are fiber-
coupled into a single-mode optical fiber through a $20\mathrm{\,X}$ microscope
objective, and two 50:50 beamsplitters address part of the photons to a
powermeter and a lambdameter for monitoring the source stability. A double
attenuator system allows introducing the transmissivity $\tau$ addressing the
photons through two unbalanced beam splitters: the former is a 999:1
($30\mathrm{\,dB}$), the latter 9999:1 ($40\mathrm{\,dB}$). The selected path
is controlled by three optical switches. Then, the output photons are directed
through free space into a shielding box, where a motorized stage allows
choosing between two detectors: a silicon photodiode (Si-ph) for high-flux
measurements and the quantum device under test (DUT) consisting in a free-
space Si-SPAD in free-running mode. In front of the latter, photons are
focused by a lens ($f=8\mathrm{\,mm}$). The whole setup is fully automated and
can be controlled through a LabVIEW programmed interface.

The source is a CW Ti:Sa laser with tunable wavelength. The laser light
intensity is controlled by a first variable optical attenuator stage
exploiting polarizers and half waveplates. Then, after a mechanical shutter
the laser light is fiber coupled into a single-mode fiber, optimizing the
emitting spatial profile, and with two 50:50 beam splitters it is addressed to
a monitor stage for checking both the emission wavelength and optical power
stability, and to a second fully-pigtailed attenuator stage, that represents
the novelty of our experimental apparatus. It consists of two unbalanced beam
splitters: the former 999:1 and the latter 9999:1, respectively allowing
introducing a $30\mathrm{\,dB}$ and a $40\mathrm{\,dB}$ attenuation. The light
path is selected thanks to three pc-controlled optical switches. Selecting the
path corresponding to the maximum attenuation (nominal $70\mathrm{\,dB}$), the
double-attenuator stage reproduces the transmissivity $\tau$ required in Eq.
1. Then, the photons will be out-coupled and collimated in free space with a
Gaussian spatial mode to be sent into a shielding box for minimizing
environmental photons.

A motorized translation stage allows selecting the measuring detector
depending on the introduced attenuation: a SI-traceable Silicon photodiode
(Si-ph) for the measurements of $\mathcal{P}$, and the Si-SPAD device under
test (DUT) for the measurement of $\mathcal{R}$, when the attenuation
reproduces $\tau$ and the photon flux goes down to the single-photon level.

Since the usual diameter of a free-space Si-SPAD active area is about hundreds
of micrometers, a lens focuses the photons spatial distribution on it in a
diameter of $40\mathrm{\,\mu m}$. The DUT position is optimized exploiting
three actuators in the $x$,$y$ and $z$ directions. All the devices of the
depicted set-up are connected to a computer via LabVIEW interface, and the
measurement routines are automatized.

The DUT detection efficiency is estimated adapting Eq. 1 to the measurements
realized with the experimental apparatus described in the previous section,
i.e. comparing the SPAD measurements and the Si-ph output. The SPAD produces a
macroscopic pulse for each revealed photon and sends it to the electronics
that collects $N$ counts in a time interval $t$, whereas the Si-ph generates
photoelectrons proportionally to the incident optical power with a sensitivity
$s$, and the resulting current is revealed by a picoamperometer with
calibration factor $C$.

The transmissivity $\tau$ is pc-controlled, and it has been independently
measured (see below). Furthermore, one has to consider also the transmissivity
$T$ of the lens in front of the SPAD (see Fig. 1). Finally, taking into
account the (intrinsic and environmental) noise level for both the SPAD and
the Si-ph measurements ($N^{\mathrm{env}}$ and $A^{\mathrm{env}}$), that
account for dark counts and background photons as well as for the dark
current. For the $i$-th measurement run Eq. 1 becomes:

$\eta_{i}(\lambda)=\frac{hc}{\lambda
t}\frac{s\left(N_{i}-N_{i}^{\mathrm{env}}\right)}{\tau
C\left(A_{i}-A_{i}^{\mathrm{env}}\right)T}$ (2)

The transmittance $\tau$ is characterized exploiting the Si-ph detector. Our
dual-attenuator approach enables us to divide the introduced attenuation,
which is crucial for maintaining the linear response of the detector.
Introducing the entire attenuation at once would compromise this linearity,
increasing the related uncertainty and the accuracy and reliability of the
experimental data. Therefore, $\tau$ is evaluated as
$\tau=\tau_{30\mathrm{\,dB}}\tau_{40\mathrm{\,dB}}$, where:

$\tau_{x_{dB}}=\frac{A_{x\mathrm{\,dB}}-A^{\mathrm{env}}}{A_{0\mathrm{\,dB}}-A^{\mathrm{env}}}$
(3)

with the subscript $x$ indicating the selected attenuation by the optical
switches (see Fig. 1).

Measurement and Results. In our experiment, we characterize the behavior of a
free-running Si-SPAD with a circular active area of $200\mathrm{\,\mu m}$ in
diameter with light at the wavelength of $\lambda=(850.711\pm
0.006)\mathrm{\,nm}$. Since the active area of a commercial free-space
detector is generally not uniform, it is necessary to scan it to find a quite
flat region.

Figure 2: $(300\times 300)\mathrm{\,\mu m^{2}}$ scansion of the Si-SPAD’s
active area obtained by setting a step size of $0.01\mathrm{\,mm}$ in both the
directions transversal to the photons propagation. The distance with respect
to the focal lens was previously optimized and fixed obtaining a focused beam
with waist around $40\mathrm{\,\mu m}$.

Fig. 2 shows that the surface of the active area is relatively uniform except
for two dips located on the left side of the scan.

Once the DUT position is fixed far from the two dips, we start the procedure
for obtaining the DUT detection efficiency. First of all, we characterize
$\tau$ (Eq. 3) averaging over a sequence of $100$ measurements. Then, we
repeat ten times the $\tau$ characterization in different days, i.e.
evaluating the repeatability behavior of our double-attenuator system,
obtaining $\tau=(2.1601\pm 0.0070)\times 10^{-7}$, highlighting a reasonable
repeatability of our system day by day.

We underline that our double-attenuator approach allows good repeatability
since it does not require to disconnect the optical fibers. To monitor the
source stability, we exploit the powermeter measurements at the source monitor
stage (see Fig. 1). This allows us to correct the measured $N_{i}$ and $A_{i}$
in Eq.s 2,3 with respect to the source fluctuations according to:

$\begin{split}N_{i}&\rightarrow N^{\prime}_{i}=N_{i}\rho_{i}^{\mathrm{DUT}}\\\
A_{i}&\rightarrow A^{\prime}_{i}=\varepsilon A_{i}\rho_{i}^{\mathrm{Si-
ph}}\end{split}$ (4)

where $\rho_{i}^{\mathrm{DUT}}=\langle
P_{i}^{\mathrm{DUT}}\rangle/P_{i}^{\mathrm{DUT}}$ represents the correction
with respect to the monitor powermeter measurement $P$ occurred during the
$i$-th run of the DUT measurement $N_{i}$, meaning $\langle X_{i}\rangle$ the
average value of the variable $X$; the same argument holds for
$\rho_{i}^{\mathrm{Si-ph}}$. Furthermore, since an unbalancement may happen
between the average source optical power emission during the DUT and the Si-ph
measurements, we have introduced the parameter $\varepsilon=\langle
P_{i}^{\mathrm{DUT}}\rangle/\langle P_{i}^{\mathrm{Si-ph}}\rangle$.

To remove the arbitrary dependence on the count rate, we estimate the zero-
flux efficiency $\eta_{0}$, i.e. the detection efficiency extrapolated to the
zero-flux level, whose value is not affected by the presence of the SPD dead
time. The behavior of the detection efficiency can be described asCastelletto,
Degiovanni, and Rastello (2000); Brida, Genovese, and Novero (2000); Migdall
_et al._ (2013):

$\eta_{\lambda}(\langle N^{\prime}_{i}\rangle)=\eta_{0}-D\frac{\langle
N^{\prime}_{i}-N_{\mathrm{env}}\rangle}{t}$ (5)

where $D$ is the dead time and $N^{\prime}_{i}$ is defined in Eq. 4. Hence,
estimating $\eta_{\lambda}(\langle N^{\prime}_{i}\rangle)$ for different
photon fluxes, $\eta_{0}$ comes out from a linear regression.

We measured the DUT detection efficiency (accordingly to Eq. 2) at different
photon fluxes acting on the variable attenuator depicted in Fig. 1, obtaining
various count-rate regimes from $5000\mathrm{\,counts/s}$ to $2\times
10^{6}\mathrm{\,counts/s}$. After collecting ten data points, we perform the
linear regression as described in Eq. 5 to estimate $\eta_{0}$. The results
are shown in Fig. 3.

Figure 3: Detection efficiency estimation for different count rates
$\mathcal{R}=\langle N_{i}^{\prime}-N_{i}^{\mathrm{env}}\rangle/t$. The blue
dots represent the calculated detection efficiencies with their related
uncertainties, as determined by Eq. 2, while the red line is the result of the
linear regression according to Eq. 5. $\eta_{0}$ is obtained as the intercept
of the fit with the $y$-axis. All the shown uncertainties consider a coverage
factor $K=1$.

As shown in Fig. 3, the experimental results align well with the fitted
behavior. From the presented measurement run, we obtained $\eta_{0}=0.5526\pm
0.0029$ (coverage factor $K=1$). Uncertainties are propagated from Eq. 2,
considering both statistical and non-statistical contributions. An example of
uncertainty budget for a fixed count rate is reported in Table 1.

Table 1: Uncertainty budget related to the detection efficiency (see Eq. 2) $\langle\eta_{i}(N^{\prime},\lambda)\rangle$, with $N^{\prime}=20655\pm 27$, showed in Fig. 3. Coefficient | Value | Uncertainty |  $\%$ Contribution
---|---|---|---
$N^{\prime}$ | $20655$ | $27$ | $5.47$
$N^{\mathrm{env}}$ | $28$ | $1$ | $0.012$
$A^{\prime}$ | $1.92807\times 10^{-8}\mathrm{\,A}$ | $4.9\times 10^{-12}\mathrm{\,A}$ | $0.06$
$A^{\mathrm{env}}$ | $4.88\times 10^{-14}\mathrm{\,A}$ | $1.3\times 10^{-15}\mathrm{\,A}$ | $1.5\times 10^{-8}$
$\tau$ | $2.1601\times 10^{-7}$ | $7.0\times 10^{-10}$ | $33.83$
$\varepsilon$ | $1.0148$ | $1.4\times 10^{-3}$ | $5.70$
$s$ | $0.4766\mathrm{\,W/A}$ | $1.9\times 10^{-3}\mathrm{\,W/A}$ | $51.55$
$C$ | $1.000023$ | $1.0\times 10^{-5}$ | $3.2\times 10^{-4}$
$T$ | $0.985000$ | $3.0\times 10^{-5}$ | $3.0\times 10^{-3}$
$\lambda$ | $8.50711\times 10^{-7}\mathrm{\,m}$ | $6\times 10^{-12}\mathrm{\,m}$ | $1.6\times 10^{-4}$
$t$ | $1.0000\mathrm{\,s}$ | $1.0\times 10^{-3}\mathrm{\,s}$ | $3.22$
$\langle\eta(N^{\prime},\lambda)\rangle$ | $0.5514$ | $0.0031$ |

Our approach demonstrates the possibility to measure the DUT detection
efficiency in a SI-traceable manner, maintaining a contained uncertainty. To
assess the robustness of our technique, we repeat the entire estimation
process ten times. The results are illustrated in Fig. 4.

Figure 4: Si-SPAD detection efficiencies at zero-photon flux ($\eta_{0}$)
obtained from ten independent estimations. The coverage factor of the
uncertainties is $K=1$.

Once more, the resulting estimations of $\eta_{0}$ exhibit strong agreement.
The average value obtained for the detection efficiency at zero-photon flux is
$\langle\eta_{0}\rangle=0.5510\pm 0.0030$. This consistency across multiple
estimations underlines the robustness and reliability of our measurement
technique.

Finally, we investigate the detection efficiency as a function of photon
wavelength. To accomplish this, we replicate the estimation of
$\langle\eta_{i}(\lambda)\rangle$ (Eq. 2) maintaining a constant
$N^{\prime}\simeq 10^{5}$, while varying the emission wavelength of our
source. The value of $N^{\prime}$ is arbitrarily chosen, and it represents a
reasonable trade-off between reduced distortion effect due to SPD dead time
and efficient data collection. The experimental data present a peculiar
sinusoidal behavior (see Fig. 5), that we interpret as an etaloning effect of
the two surfaces of the optical window of the SPAD packaging. Indeed, for a
window with thickness $L$ and refractive index $n$, the transmissivity depends
on the wavelength $\lambda$ through the parameter $\Gamma$, that isMacleod and
Macleod (2010):

$\Gamma(\lambda,n,L)=\frac{\gamma(1-\exp\left[{-2i\frac{2\pi}{\lambda}nL}\right])}{1-\gamma^{2}\exp\left[{-2i\frac{2\pi}{\lambda}nL}\right]}$
(6)

where $\gamma=(n-n_{a})/(n+n_{a})$ and $n_{a}$ represents the air refractive
index. Then, the overall detection efficiency in Eq. 2 can be generalized as:

$\eta_{i}(\lambda,n,L)=\eta_{i}(\lambda)\bigg{(}1-|\Gamma(\lambda,n,L)|^{2}\bigg{)}$
(7)

The results of this analysis are presented in Fig. 5.

Figure 5: Characterization of detection efficiency as a function of
wavelength. The number of detected photons for each acquisition lasting
$t=1\mathrm{\,s}$ was fixed at approximately $N^{\prime}\simeq 10^{5}$ counts.
The blue dots represent the experimental results with their associated
uncertainties ($K=1$), while the red line corresponds to the fitted model
described by Eq. 7.

Our proposed model aligns closely with the experimental data. Moreover, these
findings highlight the non-negligible impact of transmittance effects caused
by the optical window. Within a range of approximately $3\mathrm{\,nm}$, the
detection efficiency varies by up to $5\%$. Consequently, it is imperative
from a metrological perspective to consider such effects when characterizing
free-space SPADs, rather than only focusing on a single wavelength.

Conclusions. In this work, we presented the new experimental set-up for the
measurement of the detection efficiency of free-space SPDs, exploiting the
substitution method. Specifically, the fully computer-controlled pigtailed
attenuation stage optimizes the detection efficiency measurement time thanks
to an excellent reproducibility, minimizing the uncertainty contribution
associated to the attenuation measurement. Then, we extended our analysis to
the variation of the detection efficiency as a function of wavelength, taking
into account the transmissivity of the optical window positioned in front of
the detector’s sensitive area. This comprehensive characterization is crucial
for optimizing the performance of Si-SPADs in various applications, including
quantum communication and photonics research. By understanding the wavelength
dependence and the influence of the optical window, we can better estimate the
efficiency of these detectors, leading to improved accuracy and reliability in
single-photon detection. This study provides a valuable foundation for future
metrological characterizations of Si-SPAD technology in both scientific and
industrial contexts.

This work was financially supported in the context of the following projects:
Qu-Test project, which has received funding from the European Union’s Horizon
Europe Research and Innovation Programme under grant agreement No. 101113901;
QUID (QUantum Italy Deployment) and EQUO (European QUantum ecOsystems)
projects which are funded by the European Commission in the Digital Europe
Programme under the grant agreements number 101091408 and 101091561; the
project G6026 SPS NATO; the EMPIR 19NRM06 METISQ and 23NRM04 NoQTeS. The
projects 19NRM06 and 23NRM04 NoQTeS have received funding from the European
Partnership on Metrology, co-financed from the European Union’s Horizon Europe
Research and Innovation Programme and by the Participating States.

Conflict of Interest. The authors have no conflicts to disclose.

Author Contributions. All authors participated in the initial investigation on
how to carry out the work under the supervision of IPD, AM, and GB. The
experimental apparatus was built by SV and ER. Data acquisition was performed
by ER under the supervision of SV and AM. Data analysis was conducted by SV,
AM, and ER under the supervision of MGr and MGe. Data validation was carried
out by AM and GB under the supervision of IPD. All authors contributed to the
writing and revision of the article.

Data Availability. The data that support the findings of this study are
available from the corresponding authors upon reasonable request.

## References

## References

  * Zapatero  _et al._ (2023) V. Zapatero, T. van Leent, R. Arnon-Friedman, W.-Z. Liu, Q. Zhang, H. Weinfurter, and M. Curty, “Advances in device-independent quantum key distribution,” npj Quantum Information 9, 10 (2023).
  * Diamanti  _et al._ (2016) E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Information 2, 16025 (2016).
  * Pirandola  _et al._ (2020) S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” Adv. Opt. Photon. 12, 1012–1236 (2020).
  * Ladd  _et al._ (2010) T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45–53 (2010).
  * Gill  _et al._ (2022) S. S. Gill, A. Kumar, H. Singh, M. Singh, K. Kaur, M. Usman, and R. Buyya, “Quantum computing: A taxonomy, systematic review and future directions,” Software: Practice and Experience 52, 66–114 (2022), https://onlinelibrary.wiley.com/doi/pdf/10.1002/spe.3039 .
  * Krantz  _et al._ (2019) P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “A quantum engineer’s guide to superconducting qubits,” Applied Physics Reviews 6, 021318 (2019), https://pubs.aip.org/aip/apr/article-pdf/doi/10.1063/1.5089550/16667201/021318_1_online.pdf .
  * Genovese (2016) M. Genovese, “Real applications of quantum imaging,” Journal of Optics 18, 073002 (2016).
  * Degen, Reinhard, and Cappellaro (2017) C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. 89, 035002 (2017).
  * Aslam  _et al._ (2023) N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, “Quantum sensors for biomedical applications,” Nature Reviews Physics 5, 157–169 (2023).
  * Bakhshandeh (2022) S. Bakhshandeh, “Quantum sensing goes bio,” Nature Reviews Materials 7, 254–254 (2022).
  * Petrini  _et al._ (2020) G. Petrini, E. Moreva, E. Bernardi, P. Traina, G. Tomagra, V. Carabelli, I. P. Degiovanni, and M. Genovese, “Is a quantum biosensing revolution approaching? perspectives in nv-assisted current and thermal biosensing in living cells,” Advanced Quantum Technologies 3, 2000066 (2020).
  * Migdall  _et al._ (1995) A. L. Migdall, R. Datla, A. Sergienko, J. Orszak, and Y. Shih, “Absolute detector quantum-efficiency measurements using correlated photons,” Metrologia 32, 479 (1995).
  * Migdall  _et al._ (2002) A. Migdall, S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Intercomparison of a correlated-photon-based method to measure detector quantum efficiency,” Applied optics 41, 2914–2922 (2002).
  * Ghazi-Bellouati  _et al._ (2005) A. Ghazi-Bellouati, A. Razet, J. Bastie, M. Himbert, I. Degiovanni, S. Castelletto, and M. L. Rastello, “Radiometric reference for weak radiations: comparison of methods,” Metrologia 42, 271 (2005).
  * Polyakov and Migdall (2009a) S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” Journal of Modern Optics 56, 1045–1052 (2009a), https://doi.org/10.1080/09500340902919477 .
  * Polyakov and Migdall (2009b) S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” Journal of Modern Optics 56, 1045–1052 (2009b), https://doi.org/10.1080/09500340902919477 .
  * Chunnilall  _et al._ (2014) C. J. Chunnilall, I. P. Degiovanni, S. Kück, I. Müller, and A. G. Sinclair, “Metrology of single-photon sources and detectors: a review,” Optical Engineering 53, 081910–081910 (2014).
  * López, Hofer, and Kück (2015a) M. López, H. Hofer, and S. Kück, “Detection efficiency calibration of single-photon silicon avalanche photodiodes traceable using double attenuator technique,” Journal of Modern Optics 62, 1732–1738 (2015a).
  * Bae  _et al._ (2019) I.-H. Bae, S. Park, K.-S. Hong, H. S. Park, H. J. Lee, H. S. Moon, J. S. Borbely, and D.-H. Lee, “Detection efficiency measurement of single photon avalanche photodiodes by using a focused monochromatic beam tunable from 250 nm to 1000 nm,” Metrologia 56, 035003 (2019).
  * López  _et al._ (2020) M. López, A. Meda, G. Porrovecchio, R. Starkwood, M. Genovese, G. Brida, M. Šmid, C. Chunnilall, I. Degiovanni, and S. Kück, “A study to develop a robust method for measuring the detection efficiency of free-running ingaas/inp single-photon detectors,” EPJ Quantum Technology 7, 14 (2020).
  * Gerrits  _et al._ (2020) T. Gerrits, A. Migdall, J. C. Bienfang, J. Lehman, S. W. Nam, J. Splett, I. Vayshenker, and J. Wang, “Calibration of free-space and fiber-coupled single-photon detectors,” Metrologia 57, 015002 (2020).
  * Georgieva  _et al._ (2021) H. Georgieva, M. López, H. Hofer, N. Kanold, A. Kaganskiy, S. Rodt, S. Reitzenstein, and S. Kück, “Absolute calibration of a single-photon avalanche detector using a bright triggered single-photon source based on an ingaas quantum dot,” Optics Express 29, 23500–23507 (2021).
  * Jin, Gerrits, and Gamouras (2022) J. Jin, T. Gerrits, and A. Gamouras, “Calibration and comparison of detection efficiency for free-space single-photon avalanche diodes at 850 nm,” Applied Optics 61, 5244–5249 (2022).
  * Bienfang  _et al._ (2023) J. C. Bienfang, J. Bienfang, T. Gerrits, P. Kuo, A. Migdall, S. Polyakov, and O. T. Slattery,  _Single-photon Sources and Detectors Dictionary_ (US Department of Commerce, National Institute of Standards and Technology, 2023).
  * (25) Pilot study website: https://www.bipm.org/en/committees/cc/ccpr/wg/ccpr-wg-sp-tg11.
  * Bronzi  _et al._ (2016) D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sensors Journal 16, 3–12 (2016).
  * Villa  _et al._ (2021) F. Villa, F. Severini, F. Madonini, and F. Zappa, “Spads and sipms arrays for long-range high-speed light detection and ranging (lidar),” Sensors 21, 3839 (2021).
  * Bruschini  _et al._ (2019) C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon avalanche diode imagers in biophotonics: review and outlook,” Light: Science & Applications 8, 87 (2019).
  * Migdall  _et al._ (2013) A. Migdall, S. V. Polyakov, J. Fan, and J. C. Bienfang,  _Single-photon generation and detection: physics and applications_ (Academic Press, 2013).
  * Nielsen and Chuang (2001) M. A. Nielsen and I. L. Chuang,  _Quantum computation and quantum information_ , Vol. 2 (Cambridge university press Cambridge, 2001).
  * Arabskyj  _et al._ (2024) L. Arabskyj, P. R. Dolan, B. Dejen, S. T. S., M. Lucamarini, and C. J. Chunnilall, “Interference effects in commercially-available free-space silicon single-photon avalanche diodes,” Manuscript in preparation (private communication) (2024).
  * López, Hofer, and Kück (2015b) M. López, H. Hofer, and S. Kück, “Detection efficiency calibration of single-photon silicon avalanche photodiodes traceable using double attenuator technique,” Journal of Modern Optics 62, 1732–1738 (2015b).
  * Castelletto, Degiovanni, and Rastello (2000) S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613 (2000).
  * Brida, Genovese, and Novero (2000) G. Brida, M. Genovese, and C. Novero, “An application of two-photon entangled states to quantum metrology,” Journal of modern optics 47, 2099–2104 (2000).
  * Macleod and Macleod (2010) H. A. Macleod and H. A. Macleod,  _Thin-film optical filters_ (CRC press, 2010).
  * Brida, Genovese, and Gramegna (2005) G. Brida, M. Genovese, and M. Gramegna, “Twin-photon techniques for photo-detector calibration,” Laser Physics Letters 3, 115 (2005).