1.1.

Single-Photon Sources

A true single-photon source (SPS) will emit individual photons at periodic intervals. To guarantee its operation, the core of the source will be a solitary quantum emitter, such as an atomic particle, molecule, or quantum dot. Deterministic emission is triggered by a periodic electronic or optical excitation of the source. An ideal source will exhibit highly efficient polarized emission into a well-defined spatial optical mode; there will be negligible temporal jitter of the photon emission with respect to the clock signal triggering the emission.

Various measurements exist to quantify the operating parameters of a practical SPS. The degree of second order coherence ${\mathrm{g}}^{(2)}(\tau )$ determines how antibunched the emitted photons are; for an ideal source, ${\mathrm{g}}^{(2)}(0)=0$. This parameter is a measure of the probability of multiphoton emission events, which is greatly reduced in comparison to a coherent light source. The coherence time ${\tau }_c$ (measured via the coherence length $l_c$) and the source emitter’s lifetime ${\tau }_s$ are important for evaluating the quantity ${2\tau }_s{/\tau }_c$. For an ideal source, no dephasing of the emitted photons exists and the ratio is ${2\tau }_s{/\tau }_c=1$, meaning that the photons are wholly indistinguishable. This can be quantified further through the observation of two-photon quantum interference with perfect visibility.

In the context of quantum-photonic technologies based on entanglement, the antibunched and indistinguishable nature of photons is paramount. However, real (imperfect) sources will deviate from the ideal case, and the emission will not be perfectly antibunched or indistinguishable or free of jitter. The parameters described above can be used as metrics to determine the utility of a practical SPS in such applications.

1.2.

Single-Photon Detectors

Single-photon detectors, also referred to as photon counters, operate in either gated or nongated (i.e., continuously gated) modes, and are only able to detect an incoming optical pulse during these gates. Single-photon detectors can be characterized by the following properties:

These properties may be wavelength and temperature dependent, as well as varying across the spatial dimensions of the detector.⁴ A property that is important in QKD is the photon counter indistinguishability, i.e., the extent to which the detector output voltage pulses of detectors can be distinguished in the time domain.

An ideal detector would be photon number resolving for all $n$, where $n$ is the number of photons in a pulse, as well as having unity detection efficiency, and zero dark count and after-pulse probabilities with no dead time or recovery time. Various single-photon detection technologies exist, and a particular technology may only exhibit a subset of the characteristics listed above.

2.1.

Sources

The simplest approximation to an SPS would take thermal radiation (from a hot filament or discharge lamp) and attenuate it down to the photon-counting level. While this form of light may be relevant in ambient low-light sensing applications, the most common SPS is an attenuated laser, where the photons in the attenuated output are distributed within time intervals or pulses [continuous wave (CW) or pulsed radiation, respectively] according to Poissonian statistics. Thermal light also gives rise to a distribution of photon number states, and both types of light can have less (zero) or more than one photon per time-interval/pulse.

Practical SPSs can be divided into two classes. The first comprises heralded SPSs, which are based on correlated pairs of photons produced in a nonlinear medium. The second class uses a single quantum emitter, for example, as in demonstrations with trapped atoms and ions, as well as with molecules, color centers, and quantum dots in a solid-state host.

The drive to develop true, deterministic, SPSs was provided by the realization that such states, together with interference and detection, could be used to achieve quantum computation and other photonic quantum technologies,^5–7 including applications to metrology.^8–11 Deterministic sources would provide scalability (i.e., being able to manipulate large numbers of photons using finite resources), which is considered to be problematic for heralded photons created in spontaneous nonlinear processes. Reviews covering the topic in more detail have been published.^12–18

2.2.

Detectors

Several types of single-photon detectors have been developed, and a brief overview of the most commonly used types is given below. For detailed reviews, see Hadfield,⁶⁹ Eisaman et al.,¹⁶ and Migdall et al.¹⁸

4.1.

Single-Photon Emission

The most important measurement is evaluating the probability of having more than one photon emitted by the source within a prescribed time interval. This is commonly performed with a Hanbury Brown-Twiss (HBT) interferometer operating at single-photon level. It is usually implemented using two threshold (click/no-click) detectors placed at the output ports of a 50:50 beam splitter [Fig. 2(a)].^16,18,131 How efficiently an SPS emits a single photon can be quantified by means of the α parameter proposed by Grangier et al.¹³¹ This is essentially an anticorrelation criterion based on the parameter.

Fig. 2

Single-photon measurements: (a) $\alpha$-parameter measured with a Hanbury Brown-Twiss interferometer. (b) Coherence time ${\tau }_c$ measured with a Michelson interferometer. (c) Indistinguishability V measured with a Hong-Ou-Mandel interferometer.

$\alpha (\tau )=Q(2)/[Q^{(r)}(1)Q^{(t)}(1)]$, where Q(1) is the probability of a count in the reflection (r) or transmission (t) port of the beam splitter, Q(2) is the probability of a coincidence in counts, and τ is the time delay between separate detection events at the beam splitter output ports. In the single-photon community, the parameter $\alpha$ is often called the second-order correlation function ${\mathrm{g}}^{(2)}(0)$, but we prefer to refer to $\alpha$ since ${\mathrm{g}}^{(2)}(0)$ has a different definition,¹³² despite the fact that in the few-photon regime the two definitions are asymptotically equivalent. Idealized plots of $\alpha (\tau )$ for different sources are illustrated in Fig. 3.

Fig. 3

Characteristic plot of $\alpha$ as a function of time τ between coincidence clicks (each detector being equidistant from the beam splitter). For a Poissonian source, such as an attenuated laser, $\alpha =1$ for all $\tau$ (blue, continuous); a thermal source exhibits bunching, i.e., $\alpha (0)>1$ (red, dotted) and is said to be super-Poissonian; a single-photon source exhibits antibunching (green, dashed) and is said to be sub-Poissonian. For an ideal single-photon source, $\alpha (0)=0$.

Each pulse from the SPS is expected to contain $n$ photons ($n=1$ in the ideal case). Thus, by considering the proper detection model for the click/no-click detectors, the probability of the detector firing due to an optical pulse containing $n$ photons, as well as the probability of observing a coincidence between the two detectors of the HBT due to a single pulse from the SPS, can be properly evaluated.¹⁸

It is worth noting that with typical click/no-click detectors the parameter $\alpha$ is almost independent of the detection efficiency of the detectors (once they are fairly similar), while it can be strongly affected by the presence of dark counts or counts due to stray light. For this reason, time-correlated single-photon counting (TCSPC) measurement techniques can be helpful in providing proper estimation of the background counts. Furthermore, the detector dead times and imbalance of the HBT interferometer may bias the estimation of the $\alpha$ parameter. Proper estimation of these nonidealities is necessary to implement the needed corrections in order to have a faithful estimation for $\alpha$. Examples of proper detection models in HBT interferometers can be found in Brida et al.¹³³ and Migdall et al.¹⁸

A two-time correlation function $\alpha (t_1,t_2)$, where $t_1$ and $t_2$ are the delay times between the excitation pulse and photon detection on the two detectors, can be used to analyze the dynamics of the single-photon emission.¹³⁴

4.2.

Coherence Time, Emission Lifetime, and Indistinguishability

These measurements are important for characterizing SPSs designed for entanglement-based applications (see Sec. 1.1).

The coherence time ${\tau }_c$ can be measured using a Michelson or Mach-Zehnder interferometer to produce single-photon self-interference [Fig 2(b)].¹³⁵ The coherence length is the 1/e decay point of the interference envelope when measured in optical path difference units, and the coherence (decay) time is obtained by dividing this distance by the speed of light (Fig. 4).

Fig. 4

The first-order field-field correlation ${\mathrm{g}}^{(1)}(\tau )$ of the total emission from a single nitrogen-valency defect center in diamond. The decay time is found to be 13 fs {reprinted [Fig. 2(c)] with permission from F. Jelezko et al., Phys. Rev. A 67, 041802(R), 2003. Copyright (2003) by the American Physical Society}.

The lifetime ${\tau }_s$ of a single emitter, which will generally be a combination of the intrinsic radiation lifetime, the dephasing time, and jitter (due to nonradiative transitions), can be measured by using pulsed excitation. Exploiting TCSPC, and performing coincidence measurements between the triggering signal and the detected photon, one obtains a temporal profile corresponding to the convolution of the different components of the experiment, i.e., the radiation from the emitter, the detector, and the TCSPC instrumentation (the latter is usually negligible). The ideal situation is when both TCSPC electronics as well as the single-photon detector have negligible jitter with respect to the source. When this is not the case, a proper characterization of the detector and TCSPC electronics should be performed in order to deconvolve the profile of interest. A single-photon detector with the lowest possible jitter is needed for this measurement. Superconducting nanowire detectors, which can have a jitter of just tens of picoseconds,^16,18,69,136 appear to be the best for this purpose.

Indistinguishability is measured using Hong-Ou-Mandel (HOM) interference.^137,138 If two photons are perfectly indistinguishable, i.e., they are in exactly the same mode and are each incident at the same time at the separate input ports of a 50/50 beam splitter, they will bunch or coalesce, i.e., both will exit together from one of the exit ports [Fig. 2(c)]. The interference curve is usually measured by placing a photon-counting detector at each output port of the beam splitter and measuring the detection coincidences $N_c$ as the time delay $\mathrm{\Delta }\tau$ between the photons being incident at the input ports is varied. The detection coincidences will be a minimum for zero time delay; fully destructive interference will occur only if the two photons are completely indistinguishable. The widely accepted definition for the measured Hong-Ou-Mandel visibility is given by

Data fitting is usually applied to the interference curve, whose form depends on the spectrum of the interfering photons,¹³⁹ as well as imperfections in the experimental setup, in order to extract a reliable value for the interference visibility and, hence, the indistinguishability (Fig. 5).¹⁴⁰

Fig. 5

Hong-Ou-Mandel interference obtained from degenerate photons produced by spontaneous parametric downconversion in beta-barium borate. The abscissa axis is in units of optical path difference. The measured visibility of 0.79, after correction for instrumentation effects, yielded a photon indistinguishability of 0.95 (reproduced from Ref. 140 with permission).

4.3.

Wavelength and Spectral Line-Width

It is possible to measure the wavelength and spectral line-width of an SPS with a monochromator or a wavemeter coupled to a detector operating at the single-photon level.

An interesting solution for measuring the spectral line-width exploits a stable, tunable Fabry-Perot resonator.¹⁴¹ The technique requires that the cavity free spectral range (FSR) is greater than the line-width of the SPS, i.e., ${\mathrm{FSR}\gg \mathrm{\Delta }\nu }_{\text{source}}$, to enable an unambiguous measurement of the source spectrum, yet there must be a line-width ${\mathrm{\Delta }\nu }_{\text{cavity}}{\ll \mathrm{\Delta }\nu }_{\text{source}}$ to adequately resolve any spectral structure. When used in transmission mode, the Fabry-Perot cavity can be tuned to resonance with the SPS spectral profile using a photon-counting detector.

4.4.

Mean Photon Number and the Variation in Mean Photon Number

A measurement that is important, particularly for a pseudo SPS based on an attenuated laser, is related to the estimation of the mean photon number and its variance. A solution suitable for any kind of SPS exploits a calibrated single-photon detector. The mean number of photons and its variance will be estimated on the basis of assumptions about the statistical model of the detection process and the photon statistics of the source (see Sec. 5.1.4). If the latter is not available, quantum tomographic techniques should be employed.

4.5.

Quantum Tomography and State Reconstruction

Knowledge of the density matrix of a quantum optical state is fundamental for several applications, and considerable effort has been devoted to finding reliable methods to fully, or partially, reconstruct it (see Refs. 142 and 143 and references therein).

Quantum tomography is an experimental procedure to reconstruct the density matrix of an unknown quantum state when many identical copies are available in the same state, so that a different measurement can be performed on each copy of the state. Balanced homodyne detection is able to measure all possible linear combinations of position and momentum operators (the quadratures) of a quantum optical field. The probability distribution of the quadrature operators is demonstrated to be just the Radon transformation of the Wigner function of the quantum optical state.^142,143

In principle, quantum tomography allows perfect reconstruction of the state in the limit of an infinite number of measurements. However, in the practical finite-measurement case, statistical errors affect the quality of the reconstruction. Data analysis strategies and optimization algorithms, e.g., adaptive tomography or maximum-likelihood strategies, have been investigated in order to obtain a physical and unbiased reconstructed density matrix.^144–148

Direct reconstruction of the density matrix of a specific degree of freedom of a quantum optical field instead of the Wigner function has been routinely performed in finite-dimension Hilbert spaces, e.g., in the case of photon polarization (including qubits and qutrits)^{144,149–153} and in the case of photon optical angular momentum.^154,155 This is achieved by making a quorum of direct projective measurements (with a single-photon detector) on the different copies of the quantum optical system (typically a single photon). Optimization algorithms have been developed to obtain the reconstructed physical density matrix that most likely corresponds to the measured data.¹⁴²

It is often not necessary to reconstruct the full density matrix, as the experimentalist is interested only in reconstructing the diagonal elements in the photon number basis, i.e., the photon statistics. The most direct way to measure a photon number distribution is by using photon number resolving detectors. It is possible to deconvolve the photon statistics of the incoming light field by knowing the mode of operation of the PNR detector (e.g., linear detection in the case of TES detectors^156–158 and nonlinear detection in the case of temporally or spatially multiplexed detectors^97–100,103) as well as its inefficiencies (e.g., quantum efficiency, dark counts, reliability of the photon number discrimination, pixel cross-talk, etc.).

PNR detection has also been used to reconstruct just the underlying mode structure of multimode classical and nonclassical light fields instead of the full density matrix. Full characterization of the mode structure involves a series of separate measurements in spatial, temporal, frequency, and polarization domains, requiring a range of instrumentation. This method uses only the measurement of the photon number distribution of a field and exploits an optimization algorithm together with some hypotheses about the field modes.¹⁵⁹

Another approach uses a non-PNR detector, which can only distinguish between when photons are absorbed by the detector (producing a “click” or an “on” signal) and when no photons are absorbed (producing an “off” signal, i.e., no “click” signal). The data used for the reconstruction of the probability distribution are the probability of “no-click” for different values of the quantum efficiency of the “on/off” detector. This quantum efficiency variation is obtained by using a calibrated attenuator in front of the detector and applying specific optimization algorithms in order to obtain the most likely photon number distribution.^160,161 A minor modification of this technique has been proven to be able to also reconstruct some off-diagonal elements of the density matrix.¹⁶²

5.1.

Detection Efficiency

Detection efficiency can be measured by sending single photons onto the detector at a known repetition rate and recording the number of detection events. The detection efficiency is the ratio of detection events to incident photon events. An ideal SPS that emits only one photon within a predetermined temporal window at a known (and variable) repetition rate does not yet exist.

5.1.2.

Predictable quantum efficient detector

A potential alternative to cryogenic radiometry for obtaining traceability to the SI is the predictable quantum efficient detector. It comprises two custom-made induced-junction silicon photodiodes operated under reverse bias voltage and arranged in a wedge light-trap configuration (Fig. 7).^176–178 Its spectral response can, in principle, be calculated from measurements of the specular and diffuse reflectances of the photodiodes, together with calculation of the intrinsic quantum deficiency. Comparison with cryogenic radiometry has shown agreement at the 0.01% level, and its nonlinearity of response has been measured to be $<0.02\%$ from 100 pW to 400 μW.¹⁷⁷ This device could, therefore, provide traceability around the 0.02% uncertainty level for visible wavelength gigahertz photon fluxes.

Fig. 7

The arrangement of photodiodes in the predictable quantum efficient detector. The incident beam undergoes seven reflections before the residual beam exits, leading to an overall reflectance of $<3$ parts in ${10}^4$.

5.1.4.

Photon statistics

The methods described in Secs. 5.1.1 to 5.1.3 will, at some point, include a comparison between measuring incident optical power with an analog detector, whose response is linear (in the appropriate power regime) with respect to the incoming photon flux, and measuring the response to this or an attenuated flux with a photon-counting detector. Laser light (Poissonian light) and thermal light sources have photon statistics which both give rise to multiple photon events. These have to be taken into account when calibrating non-PNR detectors.^101,185,186. A convenient way of analyzing this is to model a detector with finite detection efficiency $\eta$ by an ideal detector ($\eta =1$) placed behind a beam splitter with transmittance $\tau =\eta$. The ideal detector’s response to a train of pulses with photon statistics $p_n$ is to always indicate a detection event except for the case in which zero photons are in a pulse. Hence, the probability $p^{\text{det}}$ for a real detector to detect a photon event is given by

Eq. (2)

$$p^{\det }=1-p_0^{\eta },$$

where $p_n^{\eta }$ is given by the Bernouilli transformation of the incoming photon statistics $p_m$.

Eq. (3)

$$p_n^{\eta }=\sum _{m=n}^{\infty }\left(\begin{array}{c}m\\ n\end{array}\right){\eta }^n{(1-\eta )}^{m-n}{p}_m.$$

The case of incoming Poissonian light is easy to analyze, since the Bernouilli transformation leads to another Poissonian distribution, with the mean photon number reduced from $\mu$ to $\eta$ $\mu$; hence (Fig. 8)

Eq. (4)

$$p^{\text{det}}{=1-\mathrm{e}}^{-\eta \mu }.$$

Fig. 8

Analysis for measurements involving a Poissonian distribution incident on a non-photon-number resolving detector of detection efficiency $\eta$. The input Poissonian distribution ${\text{P}}_{\text{A}}$ (mean photon number $\mu$) is transformed into another Poissonian distribution ${\text{P}}_{\text{B}}$ (mean photon number $\eta \mu$) after the beam splitter.

In order to obtain $\eta$ from a measured $p_{\det }$ and known $\mu$, we rearrange Eq. (4) as follows:

Eq. (5)

$$\eta =-\frac{1}{\mu }\ln (1-p_{\det }).$$

Conversely (Sec. 4.4), if we wish to measure μ with a detector of known $\eta$, we rearrange Eq. (4) as

Eq. (6)

$$\mu =-\frac{1}{\eta }\ln (1-p_{\det }).$$

Figure 9 illustrates the effect of different photon probability distributions on $p^{\text{det}}$.

Fig. 9

Ratio $Q$ of the count rates between a SPAD detector (detection efficiency, $\eta =0.6$) and an ideal detector, as a function of the mean photon number.

5.1.5.

Heralded single photons

An alternative technique to the traditional one of radiometric substitution is based on the use of parametric downconversion (Sec. 2.1.1) to produce a heralded SPS.^19,20 Detection of one of the downconverted photons (by a single-photon detector) heralds the existence of its twin, which can be directed to the DUT (Fig. 10). This approach still suffers from multiple photon events, and various experiments^187–191 have been carried out to demonstrate the equivalence of the two methods at the photon-counting level. However, optical scales remain based on cryogenic radiometry since the lowest uncertainty so far achieved with the heralded single-photon approach (0.18%) (at the single-photon level) is over an order of magnitude less accurate that that based on cryogenic radiometry (0.005%) (at the 100-μW level). This is mainly due to the need to estimate the absorption in the path the heralded photon takes from the point of creation within the nonlinear medium until it is incident on the detector, which may include geometrical or absorptive spectral filtering. The method suffers from limited spectral tunability at high accuracy and is limited to detectors that are either free-running or can be randomly gated. At present, the importance of this technique lies in the fact that it establishes an absolute means of measuring detection efficiency which is independent of cryogenic radiometry, and operates in the single-/few-photon regime.

Fig. 10

Schematic of heralded single-photon setup. All optics for spectral filtering is placed in the heralding arm. ${\eta }_A={\text{N}}_{\text{C}}/{\text{N}}_{\text{B}}$; the detection efficiency of the herald detector does not need to be known, but does affect the time required to obtain good statistics.

5.2.

Dark Count Probability

The dark count probability of a detector can be measured by recording detection events per gate or per unit time in the absence of photon flux illuminating the detector’s sensitive area. To perform the measurements, a counting device records the detector’s output signal. In order to count only detection events during gates, a time-correlated photon-counting device can be used to record the detector output signal.

5.3.

After-Pulse Probability

In SPAD detectors, charge carriers created during the avalanche process become trapped at atomic defect sites in the multiplication region. The subsequent detrapping of these carriers at a later time can trigger spurious additional avalanches known as after-pulses. After-pulses are a type of dark count, but unlike other dark count mechanisms—such as thermal excitations or tunneling effects—that occur randomly in time, after-pulses are strongly correlated to previous avalanches during which trap sites were populated.¹⁹⁴

The preferred measurement sequence for a detector that exhibits after-pulsing is to measure the dark count probability, followed by the after-pulse probability^194,195 and then the detection efficiency, as described by Yuan et al.⁸⁵

The detector is illuminated by a pulsed laser source attenuated to the single-photon level. The laser and detector are triggered by a pulse generator, where the laser pulse frequency is stepped down by an integer factor R compared to the detector gate rate. The arrival of the laser pulses at the detector is synchronized to occur during the detector gates. A time-correlated photon counting device is used to record a time histogram of laser triggers and detections. At zero time delay with respect to the laser pulse, the histogram peak is composed of detection events observed under laser light illumination (plus dark counts). Peaks at a time delay in this histogram not corresponding to an illuminated gate are generated by photon events caused by the after-pulse effect (and dark counts). By normalizing the detected count rate to the total number of applied gates, the total after-pulse probability can be calculated using Eq. (7), where ${\mathrm{C}}_i$ and ${\mathrm{C}}_{n-i}$ are the average number of counts per illuminated and nonilluminated gate, and $C_d$ is the number of dark counts, calculated from the previously established dark count probability.

With knowledge of the dark count probability $p_d$ and the after-pulse probability $p_{\text{after}}$, the photon detection probability $p_{\text{det}}$ can be obtained from Eq. (8). $p_i$ is the probability to detect a photon at each illuminated gate and is given by $C_i/N_i$, where $N_i$ is the total number of illuminated gates

The detection efficiency can then be calculated from Eq. (5). The mean number of photons per laser pulse, $\mu$, can be obtained by calibrating the attenuated laser source against a traceable standard. The latter is currently not available at the single-photon level. A practical solution is to use a calibrated detector to measure the power for a given pulse repetition rate and then use a calibrated attenuator to reduce the pulse photon number to the single-photon level. Figure 11 shows an example of data collected using this technique. The after-pulse probability can also be analysed as a function of time after a ‘true’ detection, and this is illustrated in Fig. 12.

Fig. 11

Example of data collected using the method described in Sec. 5.3. Thick red lines indicate laser pulse triggers. Narrow blue lines correspond to detections. The illuminated gate is to the immediate right of the corresponding laser pulse trigger.

Fig. 12

After-pulse plus dark count probability of a gated InGaAs detector, measured as a function of time after an incident optical pulse. The detector was gated at 4 MHz, and the laser at $4\mathrm{MHz}/256=15.625\mathrm{kHz}$. The maximum time after a pulse of 63.75 ms corresponds to 255 gates. At long times, the probability approaches the dark count level. [Data & figure courtesy G. Lepert, NPL, 2014]

5.4.

Dead Time, Reset Time, and Recovery Times

These parameters limit the maximum count rate of a single-photon detector. There are differing definitions of these terms in the literature, and we shall adapt those given by Migdall et al.¹⁸ After a detection event, there will be a time interval when the detector as a whole is unable to provide an output in response to incoming photons at the single-photon level, which may be due to intrinsic processes within the detector or its control electronics. We shall call this the dead time, $t_{\text{dead}}$. After the dead time has elapsed, the detector is able to detect incident photons; however, it may take some further time before its detection efficiency recovers to its steady-state value. We shall call this the reset time, $t_{\text{reset}}$. We shall define the sum of these times as the recovery time, $t_{\text{recovery}}$, i.e., $t_{\text{recovery}}=t_{\text{dead}}+t_{\text{reset}}$. If the detector recovers to its normal value slowly, it may be useful to specify a shorter recovery time where the detection efficiency is some fraction (e.g., 90%) of the final value. We shall call this the partial recovery time, $t_{\text{partialrecovery}}$. We note that in the literature, $t_{\text{recovery}}$ has sometimes been defined as the dead time.¹⁹⁶

The dead time, reset time, and recovery times can be measured using the two-pulse method.^196–198 A train of double pulses of equal intensity, separated by a tunable time $\mathrm{\Delta }t$ and attenuated to the single-photon level, are sent to the detector. In the case of gated detectors, the photons will be synchronized to the detector gates and their time separation incremented in steps of a gating period. The probabilities of detecting the first photon, $p_1$, the second photon, $p_2$, and both photons, $p_{12}$, will be recorded as a function of their time separation by recording detections for several incident pairs of pulses at each time separation. The time between pairs of pulses should be able to be made large enough to exceed the expected recovery time and, in the case of SPAD detectors, ensure a negligible after-pulse probability. We note that

From Fig. 13, $p_{12}$ will be zero for $\mathrm{\Delta }t=0$, and then will become nonzero at some point $\mathrm{\Delta }t=t_{\text{dead}}$. In order to estimate $t_{\text{fullrecovery}}$, i.e., the point at which, by our definition, $p_2=p_1$, we find the value of $\mathrm{\Delta }t$ for which

Fig. 13

Detection efficiency as a function of time (adapted from Ref. 18).

Similarly, estimation of $t_{\text{partialrecovery}}$ at the 90% level requires finding the value of $\mathrm{\Delta }t$ for which

A check can be made that $p_1=p_2$ for $\mathrm{\Delta }t\ge t_{\text{recovery}}$, i.e., where the effect of after-pulses is negligible.

5.5.

Maximum Count Rate

To measure the achievable maximum count rate, the detector is illuminated by pulsed laser light at the same frequency of the detector gating rate, corresponding to an illumination pulse every detector gate. By measuring the detector count rate as a function of the photon flux, the number of detection events per gate will saturate at the detection rate limit of the SPAD. The results can be compared with the prediction of the maximum count rate as a function of the detection efficiency and dead time of the detector.

5.6.

Timing Jitter

To ensure good timing resolution of a single-photon detector, the time interval between the absorption of a photon and the generation of an output electrical signal should be stable, corresponding to a small timing jitter. A common technique to determine this parameter is to measure the full-width half-maximum (FWHM) of the detector’s instrument response function. For that purpose, the FWHM of the laser pulses illuminating the detector should be less than the timing jitter (typically $>100\mathrm{ps}$) of the detector. By correlating many detection events with the trigger signal of the laser, a time delay histogram can be observed by a time-correlated counter, from which the detector’s response function can be calculated. Many detectors have non-Gaussian and asymmetric response functions which can be taken into account in a detailed analysis.

5.7.

Positive-Operator Valued Measure Reconstruction

Detector characterization is normally carried out by measuring the parameters of a trusted model describing the detector operation. Where characterization of the mode of operation of a detector without preliminary assumptions is needed, quantum detector tomography may present the ideal solution. It consists of determining the positive-operator valued measure (POVM) corresponding to the detection process, i.e., given an input quantum optical state, POVM is the operator that determines the probability of having a certain macroscopic output signal from the detector.

Measurements with a quorum of probe states enable a complete determination of the POVM of a detector to be achieved. Experiments have been performed on phase-insensitive^199–201 and phase-sensitive²⁰² PNR detectors using coherent states as probes. As usual with tomographic techniques, experimental errors and statistical fluctuations may lead to unphysical POVM elements, and specific optimization algorithms constraining POVM elements to be physical should be employed.

POVM extraction with a large quorum of probe states suffers from slow convergence,^199,201 and it was shown²⁰³ that taking advantage of quantum resources (e.g., entanglement) can increase the speed of convergence. Taking advantage of quantum correlations with an ancillary state, a first experimental POVM reconstruction was carried out.²⁰⁴ Here, it was assumed that the POVM was diagonal in the Fock (photon number) basis, as was the case in most of the previous POVM reconstructions discussed,^199,201 while only in the case of Natarajan et al.²⁰² were nondiagonal POVM elements considered.