**Abstract.**
The generation, measurement, and manipulation of light at the single- and few-photon levels underpin a rapidly expanding range of applications. These range from applications moving into the few-photon regime in order to achieve improved sensitivity and/or energy efficiency, as well as new applications that operate solely in this regime, such as quantum key distribution and physical quantum random number generation. There is intensive research to develop new quantum optical technologies, for example, quantum sensing, simulation, and computing. These applications rely on the performance of the single-photon sources and detectors they employ; this review article gives an overview of the methods, both conventional and recently developed, that are available for measuring the performance of these devices, with traceability to the SI system.

# Metrology of single-photon sources and detectors: a review

**Christopher J. Chunnilall**

National Physical Laboratory (NPL), Hampton Road, Teddington, TW11 0LW, United Kingdom

**Ivo Pietro Degiovanni**

Istituto Nazionale di Ricerca Metrologica (INRIM), Strada delle Cacce 91, 10135 Torino, Italy

**Stefan Kück**

Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany

**Ingmar Müller**

Physikalisch-Technische Bundesanstalt (PTB), Abbestraße 2-12, 10587 Berlin, Germany

**Alastair G. Sinclair**

National Physical Laboratory (NPL), Hampton Road, Teddington, TW11 0LW, United Kingdom

*Opt. Eng*. 53(8), 081910 (Jul 10, 2014). doi:10.1117/1.OE.53.8.081910

#### Open Access

Applications that use single-photon sources and detectors cover an expanding range of topics, as demonstrated by the contents of these proceedings. They can be considered as falling under overlapping categories. The first, which is primarily the domain of single-photon detectors, is concerned with detection at the lowest of light levels, and the discrete nature of light is a phenomenon that has to be accommodated. Examples of these are sensing applications, such as low ambient light sensing and surveillance, medical imaging, and astronomy. The second is that where the quantized energy structure of matter is the important factor. Examples of these are photoelectric and thermal detection, single-photon generation, and the majority of spectroscopic applications, which may range from solid-state physics to a multiplicity of biological applications. The final category covers applications where the quantum nature of light is the key factor. This is relevant in the fields of quantum information processing and quantum metrology where industrial technologies based on the production, manipulation, and detection of single photons are emerging, and this is the field that is driving the need for new metrology. Within this category lies the application of nonclassical correlations, such as entanglement. Quantum key distribution (QKD) and quantum random number generators (QRNGs) are two of the most commercially advanced technologies and among the first to directly harness the peculiar laws of quantum physics. Figure 1 shows some of the main application areas.

This review is concerned with the traceable methods that are available for reliable characterization of single-photon detectors and sources. Traceable means that the result of a measurement, no matter where it is made, can be related to a national or international measurement standard and that this relationship is documented. The concept of traceability is important because it makes possible the comparison of the accuracy of measurements worldwide according to a standardized procedure for estimating measurement uncertainty.^{1} The chosen system of units is the International System of Units (SI). SI is not static but evolves to match the world’s increasingly demanding requirements for measurements at all levels of precision and in all areas of science and technology. However, any changes in the SI system are designed to ensure that any step-changes in units are minimized. Hence, traceability to the SI ensures comparison of measurements not only worldwide, but also between one year and another.

Current optical power scales are based on measurements in the 0.1- to 1-mW regime, which are suitable for traditional requirements.^{2} The rapid development of single-photon sources and detectors, and the growth of associated technologies, such as QKD and quantum computing, require measurements in the single- and few-photon regime. The international acceptance of these new quantum-based technologies requires improved traceability and reliability of measurements at the few-photon level. This has led to a proposal to expand the formulation of candela, the SI unit of optical power, to include one based on photon number. The reader is directed to Zwinkels et al.^{3} for further details.

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 $g(2)(\tau )$ determines how antibunched the emitted photons are; for an ideal source, $g(2)(0)=0$. 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 $lc$) 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.

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:

- photon-number resolution (the ability to distinguish the number of photons in each detected pulse)
- detection efficiency, $\eta $ (the probability that a photon incident at the optical input of the detector within a detection gate will be detected and produce an output signal)
- dark count probability (the probability that a detector registers a detection event within a detection gate in the absence of incident photons)
- after-pulse probability (the probability that a detector registers a false detection event in the absence of illumination, conditional on a true photon detection event in a preceding detection gate)
- dead-time (the time interval after a detection event when the detector is unable to provide a response to an incoming photon)
- recovery time (the shortest time duration after a photon detection event for the detection efficiency to return to its steady-state value)
- jitter (the temporal variation in the output signal produced by the detector upon registering an event)
- linearity of response (the property that the detector response is unchanged as the number of incident photons per pulse varies over a specified range)
- maximum exposure level (the photon flux above which the detector may undergo a temporary or permanent change in characteristics).

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.

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}

Correlated pairs of photons produced in a three- or four-wave nonlinear process can be used as a source of single photons. Detection of one of the created photons heralds the existence of its single-photon twin. The creation process is random in time, and multiphoton events can still arise. Until true deterministic sources are realized, these probabilistic processes are the pragmatic option for producing single- and entangled-photon states and exploring the potential offered by entanglement and nonclassical correlations.

Spontaneous parametric downconversion (SPDC) is a three-wave mixing process where a photon in a medium (usually a crystal) with $\chi (2)$ nonlinearity can be downconverted into two photons under the constraint of energy and momentum conservation, commonly referred to as phase matching. SPDC can produce correlations^{19}^{,}^{20} (in the pair of downconverted particles) in time, energy, momentum, polarization, and angular momentum. In addition to producing heralded single photons, SPDC can also be used to produce pairs of photons that are entangled in one or more of their observables.^{21}^{,}^{22} Periodic poling has been applied to $\chi (2)$ crystals in order to achieve phase-matching conditions not otherwise possible,^{23} and has been applied to waveguide structures^{24}^{,}^{25} where the downconverted photons are constrained to the single (or few) spatial mode(s) of the waveguide. We highlight a few implementations—entanglement-based QKD over 144 km through free space,^{26} a 1.55-*μ*m source using a fast shutter to suppress background counts,^{27} and multiplexing of four sources to provide a high output rate while suppressing multiphoton states.^{28}

Spontaneous four-wave mixing (SFWM) is a $\chi (3)$ nonlinear process, where two pump photons create the pair of downconverted photons. This process is weaker than the $\chi (2)$ process, so a longer interaction path is required. Initial work focused on silica optical fiber,^{29}^{,}^{30} and recent efforts have applied the SFWM process to on-chip waveguides in silica.^{31}^{–}^{33} In silica, a particular problem to overcome is the Raman background.

Atomic particles can be trapped and cooled almost to rest, thus providing a reproducible single quantum emitter. A single trapped ion will exhibit antibunching,^{34} and such systems have been used successfully in demonstrating probabilistic atom-photon entanglement,^{35}^{,}^{36} atom-atom entanglement,^{37} and quantum teleportation between remote atomic particles.^{38} A necessary requirement for this work is that the photons from separate sources exhibit quantum interference.^{39} Cavity quantum electrodynamics offers the route to a deterministic source; the ion is confined in an optical cavity which is near resonant and strongly coupled to one of two transitions in a Λ-configuration of the atomic-level structure.^{40} The technique of stimulated Raman scattering via adiabatic passage can generate single photons deterministically with high efficiency, although at limited rate. An analogous approach using single atoms confined in an optical dipole trap has yielded deterministic sources,^{41} which exhibit quantum interference.^{42} These atom-cavity systems have been used as nodes^{43}^{,}^{44} in the demonstration of an elementary quantum network.^{45}

Using advanced micro- and nanofabrication techniques, III-V semiconductor heterostructures can be engineered to create SPSs.^{14}^{,}^{17} For example, InAs quantum dots can be grown within a structure based on GaAs; these can be excited by optical^{46} or electrical^{47} means. The Purcell effect can be used to enhance the rate and directionality of the emission by embedding the quantum dot in a pillar microcavity;^{48}^{,}^{49} layers of a semiconductor with an alternating refractive index form Bragg mirrors, and the structure is etched to a micron-scale diameter to minimize the cavity’s mode volume and maximize enhancement. Optically pumped structures can generate indistinguishable photons,^{46} which have been used to demonstrate entanglement^{50} and teleportation.^{51} Optically pumped semiconductor sources have also been used to generate polarization-entangled photons.^{52}^{,}^{53} Indistinguishability of single photons emitted by an electrically pumped source has also been demonstrated.^{54} Quantum dot emission is inhomogeneous; this requires that emission from separate sources is tuned to enable photon indistinguishability. As an alternative to pillar microcavities, locating quantum dots in a photonic crystal cavity has also been investigated.^{55} Cryogenic temperatures are necessary for all of these semiconductor approaches.

Color center defects in diamond is another approach to generating single photons,^{15} the nitrogen-vacancy (NV) defect^{56} being the most widely studied. Recent work has shown coupling of individual NV centers to a microring resonator^{57} and a photonic crystal cavity^{58} in single-crystal diamond, as well as coupling of a single NV center to a fiber microcavity.^{59} These approaches will be used in quantum information applications in conjunction with the internal energy levels of the NV defect. Other diamond defects can also be used for generating single photons, one example being that due to chromium.^{60}

Molecules have also been investigated.^{61} Their rich vibrational structure leads to spectral broadening at ambient temperatures. At very low temperatures, the transition connecting the ground vibrational states of the ground and excited electronic states is a very narrow line, the zero-phonon line (ZPL). ZPL line-widths are often lifetime-limited at low temperatures. Photostability is a serious issue.^{62} Recent work has demonstrated quantum interference from separate molecules^{63} and improved the spontaneous emission rate.^{64}

Finally, we highlight novel structures that have recently been used to enhance the coupling efficiency from solid-state single emitters.^{65}^{–}^{68}

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}

The photomultiplier tube (PMT) was the first established photon-counting technology^{70} and consists of a vacuum tube with a light absorbing photocathode from which electrons are liberated through the photoelectric effect. The few-electron photocurrent is multiplied by a cascade of secondary electron emissions from a series of electrodes positively biased with respect to the previous one (dynodes) in order to obtain a macroscopic current pulse. Different photocathode materials can be chosen in order to optimize the spectral response, ranging from UV to telecom wavelengths. An evolution of the PMT idea is the microchannel plate PMT, where glass capillaries are fused in parallel and coated with secondary electron-emitting materials to obtain a single continuous biased dynode^{71} with an improved temporal resolution with respect to the original PMT (tens of picoseconds versus hundreds of picoseconds).

Single-photon avalanche photodiodes (SPADs) operating in Geiger mode are the most common and commercially successful solution for single-photon counting, having replaced PMTs in many applications. SPADs are based on an avalanche photodiode structure reversely biased above the breakdown voltage (known as Geiger mode operation), so that electron-hole pairs generated by photon absorption are multiplied in an avalanche gain process. To control this effect, the avalanche must be stopped and the device reset by a passive or active quenching circuit.^{72}^{–}^{75} Silicon-based SPADs operating in Geiger mode achieve single-photon sensitivity in the VIS-NIR, with low dark counts and timing jitter reduced to tens of picoseconds.

SPADs for the 1.3- and 1.55-*μ*m telecommunication bands use lower-band-gap semiconductor materials, such as Ge and InGaAs/InP.^{76}^{–}^{80} These devices suffer from dark count rates that are orders of magnitude higher than that for their Si counterparts and are typically operated in gated Geiger mode, although free-running operation has recently been achieved.^{81}^{,}^{82} Much of the ongoing effort to improve InGaAs SPAD performance is targeted at the commercial development of fiber-based QKD systems.^{83} Novel biasing and gating schemes, employing a dc bias just below the avalanche breakdown voltage on top of which a high-speed periodic low amplitude bias signal is added,^{84}^{,}^{85} enable increased device clock rates,^{86} a feature that is particularly important for QKD.^{87}

Frequency upconversion schemes are used to convert photons in the 1.3- and 1.55-*μ*m telecommunication bands to a shorter wavelength that can be detected by Si-SPADs, which typically have a higher detection efficiency than infrared SPADs.^{88}^{–}^{91} Upconversion exploits sum-frequency phenomena in nonlinear optical media, where a weak signal (the single-/few-photon state) is combined with a strong pump signal to upconvert the weak signal to the frequency that is the sum of the frequency of the two incoming signals. Despite what seems a very simple and beautiful solution, there are several technical challenges and drawbacks to frequency upconversion detectors related to the stability of, and fluorescence and other optical losses within, the nonlinear medium.

Superconducting nanowire single-photon detectors (SNSPDs) exhibit low dark counts ($\u223c1\u2009\u2009Hz$), short recovery times ($\u223c10\u2009\u2009ns$), and low timing jitter ($<100\u2009\u2009ps$). The detection element is an $\u223c100-nm-wide$ nanowire of superconducting material in a meander structure. The superconducting wire is biased just below its critical current, and a localized resistive hot-spot is created when a photon strikes the wire, triggering the voltage-pulse that signifies the detection of the photon.^{92}^{–}^{94} In contrast to SPADs that operate at room temperature, or temperatures achievable using thermoelectric cooling, these devices operate at a few kelvin. The first devices used NbN nanowires, but recent improvements in detection efficiency have been achieved by exploiting optical cavities^{95} and amorphous W-Si nanowires.^{96}

PMTs, SPADs, and, more recently, SNSPDs are available commercially and are reasonably straightforward to use, but they do not have a photon-number resolving (PNR) ability. On the contrary, detectors with some kind of PNR ability are often research prototypes.

One approach is to spatially or temporally multiplex non-PNR detectors.^{69}^{,}^{97}^{–}^{101} In addition to single chip spatial multiplexing of SNSPDs,^{102} there have been several efforts to fabricate SPAD arrays on a single chip—one example is a silicon photomultiplier device consisting of an array of SPAD pixels that are read in parallel.^{103} Other examples are arrays where each SPAD of the array is integrated directly with quenching circuitry and millimeter-scale SPAD arrays.^{104}^{–}^{108}

In addition to these extrinsic PNR detectors based on multiplexed non-PNR detectors, there are laboratory-prototype intrinsic PNR detectors, e.g., visible-light photon counters (VLPCs) and superconducting transition-edge sensors (TESs).

VLPCs are low-temperature ($\u223c6\u2009\u2009K$) semiconductor-based high-efficiency PNR detectors.^{109}^{–}^{111} In these devices, the absorption of a photon produces an electron–hole pair that, in the low-voltage gain region, produces a multiplication process close to the theoretical minimum. This gives rise to a signal that is proportional to the photon number and has been proven successful for counting up to five photons.

Superconducting TESs provide almost ideal detection efficiency and intrinsic PNR ability.^{112}^{–}^{114} A major limitation is that they must be operated in the tens of millikelvin regime since they are essentially very sensitive bolometers. The sensor is a superconducting film maintained at the superconducting transition; thus, any change in temperature will cause a change in resistance, which is detected using a SQUID amplifier. Another drawback of the TES is that the energy resolution and recovery time are, respectively, directly and inversely proportional to the superconducting transition temperature. This currently limits the maximum repetition rate to around $1\u2009\u2009MHz$.^{115}^{,}^{116}

QKD^{83}^{,}^{117} is arguably the first commercialized quantum optical technology and uses single photons to establish a secret key (cipher) between two parties over an open optical channel, such as free space^{26} or an optical fiber.^{118} If a hacker intercepts these photons, (s)he will disturb their encoding in a way that can be detected. QKD does not prevent hacking, but reveals whether a hacker has been able to compromise the key. The simplest configuration, known as “prepare-and-measure” or “one-way,” comprises a transmitter (Alice), which encodes bits on single photons, and a receiver (Bob), which decodes these bits. In the “plug-and-play” or “go-and-return” configuration, Bob transmits photons, Alice encodes information on them, and resends them to Bob for decoding. The most commercially advanced QKD systems operate over an optical fiber, use attenuated laser pulses (faint pulses) as an approximation to true single photons, and encode information on the phase of the photons.^{119} Photons are distributed in attenuated laser pulses according to Poissonian statistics; hence, some pulses will contain two or more photons. In order to reduce the power of photon-number-splitting^{120} attacks on these multiphoton pulses, the Scarani-Acin-Ribordy-Gisin^{121} and decoy state^{122} protocols were developed.

QKD is a physical, as opposed to an algorithmic, process. The laws of physics prove the security of QKD if faithfully implemented. The physical performance of the QKD system at the time of creating the secret key is, therefore, essential to its security. As such, it is one of the drivers for single-photon source and detector metrology.

Measurement of the performance of the optical components of a QKD system can be used to establish (1) whether they satisfy the assumptions and requirements of security proofs,^{123}^{,}^{124} (2) the performance of the system in terms of expected secure bit-rate and range, (3) immunity from side-channel attacks,^{125}^{,}^{126} and (4) whether component performance has changed, either from natural ageing or from device manipulation,^{127}^{–}^{129} i.e., hacking attacks via the open optical channel.

For in-fiber faint-pulse QKD systems, the most important properties of the emitted photons are the mean photon number(s), timing jitter, and any means of distinguishing the photons apart from their phase (e.g., from their spectral or temporal characteristics). For the photon receiver, the relevant properties are photon detection probability, dark count and after-pulse probability, dead time, recovery time, and spectral and temporal distinguishability.

Random number generators are also essential components of QKD systems, since the encoding, as well as the intensity for decoy protocol systems, must be varied in a truly random way. Optical quantum random number generators operate at the single-photon level and depend on the performance of their constituent single-photon sources and detectors.^{130}

Characterization of an SPS is achieved through the estimation of its relevant parameters, using dedicated measurement techniques. A provisional list of these parameters, and the associated measurement techniques, is presented below.

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.

$\alpha (\tau )=Q(2)/[Q(r)(1)Q(t)(1)]$, 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 $g(2)(0)$, but we prefer to refer to $\alpha $ since $g(2)(0)$ 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.

**F3 :**

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 $\alpha (t1,t2)$, where $t1$ and $t2$ 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}

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).

**F4 :**

The first-order field-field correlation $g(1)(\tau )$ 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 $Nc$ as the time delay $\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}

**F5 :**

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 ^{140} with permission).

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., $FSR\u226b\Delta \nu source$, to enable an unambiguous measurement of the source spectrum, yet there must be a line-width $\Delta \nu cavity\u226a\Delta \nu source$ 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.

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.

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}

Photon polarization is the quantum mechanical equivalent of the classical electromagnetic light polarization. The quantum polarization state vector for a single photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Thus, quantum state tomography is equivalent to the estimation of Stokes parameters for classical light, the only difference being that instead of measuring light power, what is experimentally observed are the relative detection frequencies of single-photon detection, i.e., conditional probabilities of detection of single photons. Tomographic reconstruction of the polarization state at the single-photon level is more affected by detection imperfections (e.g., dark counts, afterpulses, etc.) than conventional polarimetry operating in the macroscopic optical regime. For this reason, proper reconstruction algorithms, such as maximum likelihood algorithms, are employed to reconstruct the physically meaningful polarization state of the single photons.^{142}^{,}^{149} This technique also allows one to reconstruct the evolution of the qubit states traveling and/or interacting in photonic devices through the formalism of quantum process tomography.^{163}

The characterization of detectors is fundamental to characterizing single-photon devices, since current traceability of optical scales, i.e., the SI system, is based on cryogenic radiometers, which are detectors of optical power [Fig. 6(a)]. Free-space monochromatic radiation at the 100-*μ*W power level can be measured with this technique, with an uncertainty around 0.005% ($k=2$).^{164}^{,}^{165}

**F6 :**

(a) Principle of the cryogenic radiometer. The temperature change caused by incident optical radiation is compared with the electrical power required to change the temperature of the cavity by the same amount. (b) A prototype picowatt fiber-coupled radiometer made from a superconducting transition-edge sensor [(b) is reproduced from ^{166} with permission].

However, a cryogenic radiometer is simply a well-characterized instrument and, unless linked to a more fundamental concept, has the potential for unknown systematic errors or drifts that can then propagate into all other radiometric quantities. Comparisons with other traceability routes have been carried out, at least to uncertainty levels around 0.02%, confirming the underlying principle of cryogenic radiometry.^{3} In the discussion below on detection efficiency, we describe various alternatives to cryogenic radiometry that may, in time, achieve the necessary accuracy to enable such a test of cryogenic radiometry, although our focus in this review is their relevance to calibrating devices operating in the photon-counting regime.

An interesting development is a prototype microscale picowatt cryogenic radiometer for electrical substitution optical fiber power measurements [Fig. 6(b)].^{166} The absorber is a superconducting TES, and it, the electrical heater, and the thermometer are on a micromachined membrane of $<1\u2009\u2009mm$ square. Initial measurements at 1550 nm with input powers from 50 fW to 20 nW ($\u223c4.106$ to $5.108\u2009\u2009photons\u2009s\u22121$) show a response inequivalence between electrical and optical power of 8%. A comparison of the response to electrical and optical input powers between 15 and 70 pW yields a repeatability better than $\xb10.3%$ ($k=2$). The system has a noise equivalent power of $\u223c5\xd710\u221215\u2009\u2009W\u2009Hz\u22121/2$.

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.

The traditional approach is the substitution method based on the comparison of a traceably calibrated reference device (detector) with the detector under test (DUT). The power of incident radiation (CW or pulsed and in the appropriate power regime) is measured with the reference detector. This power can then be further attenuated by a measurable amount (either through the use of a monitor detector operating at high power^{167} or calibrated attenuators) to the single-photon regime and used to calibrate the DUT.

The low power limit of a reference detector is determined by the noise floor of the detector and any amplification used to obtain a measureable signal. The large transimpedance value ($>1011\u2009VA\u22121$) required to convert sub-picoampere-level photocurrents poses a series of challenges to the traditional current-to-voltage converter with a feedback resistor, such as noise amplification,^{168} long time constant, and a not negligible I/V conversion factor uncertainty ($>0.1%$). The switched integrator amplifier that employs a capacitor in place of the feedback resistor can offer a shorter time constant, overall better noise performance, and I/V conversion uncertainty better than 0.01%.^{169}^{,}^{170}

Using an attenuation chain, a reference detector can be calibrated at the 100-pW level ($\u223c108$ to $109$ photons per second) with an uncertainty of $\u223c0.1%$ (visible, free space) to 1% (1550 nm, fiber-coupled) ($k=2$).

A synchrotron can function as a variable attenuator since the radiant intensity of the synchrotron radiation can be adjusted by many decades in a controlled manner with low uncertainties. A traceably calibrated reference detector is used to measure the photon flux at high power (high current). The ring current is then reduced and the count rate of the DUT measured. The ratio of the ring currents yields the photon flux on the DUT.

The Schwinger equation ^{171} describes the spectral energy irradiated per solid angle by one electron moving on a circular arc, i.e., moving in a homogeneous magnetic field. Adapting this to electron storage rings, where the electron revolves, the Schwinger equation is multiplied by the revolution frequency ν, yielding the spectral radiant intensity for one stored electron. The spectral radiant power $\Phi Schwinger(\lambda ,1e\u2212)$ is given by integration over the appropriate solid angle and can be calculated from theory. Synchrotron radiation can, therefore, be considered an absolute source. Its use in such a mode requires accurate knowledge of all the storage ring parameters entering into the calculation.^{172} Its application to absolute calibrations of detectors would require the theoretical consideration of the coupling losses into the DUT to be evaluated, which is not feasible for low uncertainty calibrations.

If $N$ electrons are stored, which is equivalent to a stored electron beam current $I=Ne\nu $, the spectral radiant power is written as $\Phi \lambda (\lambda ,N)=N\Phi Schwinger$$(\lambda ,1e\u2212)[1+\epsilon (\lambda )]$. The parameter $\epsilon (\lambda )$ accounts for the influence of the finite vertical source size and vertical divergence of the stored electron beam. At the Metrology Light Source (MLS), the value of $\epsilon (\lambda )$, which is dependent on the wavelength and the vertical acceptance angle, is well below $10\u22124$ for wavelengths longer than $\u223c100\u2009\u2009nm$.^{172} The direct proportionality between radiant power and the number of stored electrons holds not only for bending magnet radiation as described above, but also for the radiation of devices, such as undulators, installed in the storage ring, which can produce radiation of a much higher power compared to bending magnet radiation. Therefore, the linearity of the reference detection system has to be known only in the microwatt power regime. At the MLS, the stored electron beam current can be varied by more than 11 decades from a maximum current of $\u223c200\u2009\u2009mA$ down to one stored electron (1 pA).^{173}

Uncertainties ($k=1$) of 0.17% and 0.16% for the measurement of the detection efficiency of two Perkin-Elmer single-photon counting modules at 651 nm were achieved^{174} using a reference trap detector calibrated traceably to the Physikalisch-Technische Bundesanstalt cryogenic radiometer. The absolute photon rate per stored electron in the focus was determined in the high ring current regime, where the current can be measured with a relative standard uncertainty of better than $5.10\u22124$. The ring current was then reduced to several hundred picoamperes, i.e., several hundred stored electrons, and the count rate of the DUT per stored electron was measured. Optical filters can adjust the calibration wavelength to any desired value covered by the synchrotron radiation spectrum. Recent work^{175} has extended this technique to fiber-coupled SNSPDs.

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 *μ*W.^{177} This device could, therefore, provide traceability around the 0.02% uncertainty level for visible wavelength gigahertz photon fluxes.

Another alternative to cryogenic radiometry is a method for absolutely measuring the radiance of a fiber-coupled source at $\u223c1542\u2009\u2009nm$.^{179}^{,}^{180} This compares the spontaneous emission of an erbium amplifier to the emission stimulated by the source. Using spontaneous emission as a standard was originally proposed in 1970s^{181}^{,}^{182} and was implemented using SPDC in bulk crystals,^{183}^{,}^{184} but the free-space nature of the setup made it challenging to accurately define the number of spatial modes involved. The method operates best at the one photon/mode level, which is $\u223c7\u2009\u2009nW$ at 1550 nm, and has so far demonstrated uncertainties around the 1% level. A detector of known relative spectral response can be used to transfer this to measurements at other wavelengths and the power level attenuated to the single photon level for calibrating a photon counter.

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 $pn$ is to always indicate a detection event except for the case in which zero photons are in a pulse. Hence, the probability $pdet$ for a real detector to detect a photon event is given by

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)

**F8 :**

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

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

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

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-*μ*W 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.

The extension of Klyshko’s technique (Sec. 5.1.5) to other kinds of single-photon detector is quite straightforward, with careful consideration of any nonidealities associated with the detection model. For example, for a PNR detector, a generalized version of Klyshko’s technique accommodating the photon number resolving ability of the detector has been implemented.^{192}

Another calibration technique for a PNR detector inspired by Klyshko’s technique, but explicitly taking into account multiple twin-photon events produced in the SPDC process, utilizes the PNR detector’s capability to measure the photon-number distribution of an optical mode.^{193} Using two PNR detectors, the joint photon-number statistics between the two electromagnetic field modes of the PDC source, including photon-number correlations and individual photon-number distributions, can be determined. For each element of the resulting joint photon statistics, one can find a formula giving the quantum efficiencies of the two PNR detectors. Optimization techniques are necessary to estimate the detector efficiencies from the increased photon number of measurements. A drawback of this technique is that it is strongly dependent on the probabilistic detection model assumed (e.g., the Bernoulli model), which should be correct in its entirety not just in terms of mean values (as is the case for Klyshko’s technique). Inadequacies in the assumptions immediately propagate to the accuracy of the estimation of photon detection efficiency. Another drawback is the use of an optimization algorithm that, in general, does not yield a provable uncertainty estimation.

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.