Photon Correlations in Spectroscopy and Microscopy

: Measurements of photon temporal correlations have been the mainstay of experiments in quantum optics. Over the past several decades, advancements in detector technologies have supported further extending photon correlation techniques to give rise to novel spectroscopy and imaging methods. This Perspective reviews the evolution of these techniques from temporal autocorrelations through multidimensional photon correlations to photon correlation imaging. State-of-the-art single-photon detector technologies are discussed, highlighting the main challenges and the unique current perspective of photon correlations to usher in a new generation of spectroscopy and imaging modalities.

T he world is undergoing a new quantum revolution, with the rapid maturation of quantum computing, quantum communication, and quantum sensing technologies over the past decade. The holy grail in all of these is to obtain a "quantum advantage", that is, to harness the power of quantum mechanics to outperform classical capabilities. Many recent advances in quantum sensing rely heavily on quantum entanglement or squeezing. Examples can be seen in magnetic sensing using superconducting qubits 1 and enhanced phase sensitivity in laser interferometry (with applications ranging from gravitational interferometry 2 to optical microscopy 3 ). However, these intricate quantum states are not prerequisites for quantum advantage, as manifested in the extensive field of nonclassical photon statistical correlations. Such photon correlation experiments have been at the forefront of quantum optics for half a century and are currently experiencing a renewed interest, as they expand to new regimes and unexpected applications. The advances and outlook of this field are the topics of the present Perspective.
The use of photon correlations as a metrology or spectroscopy tool has its roots in the work of Hanbury Brown and Twiss, who showed for the first time how the temporal correlation (there still of classical origin) between the signal observed from two single-photon detectors (SPDs) can facilitate the measurement of the angle subtended by a star. 4 This idea was rapidly taken up to demonstrate some of the most fundamental properties of both light and of quantum emitters of light, such as the indivisibility of single photons. 5 Yet, only in the past two decades, applications of photon correlations to more pragmatic spectroscopy and imaging applications, especially aiming toward bioimaging and materials spectroscopy, have sprung up. This is to a great extent due to the tremendous advances in detector and correlator technology, which now enable to massively parallelize measurements that would have taken a prohibitively long time not too long ago. This evolution is schematically described in Figure 1. Whereas, initially, correlation experiments used a pair of single pixel detectors to generate a one-dimensional temporal autocorrelation trace (Figure 1, left), linear arrays of SPDs now allow parallelize detection along another dimension, such as energy (as exemplified in Figure 1, middle). In fact, this has also dramatically simplified the optical setup since the use of diffraction or dispersion can obviate the need for the use of a beamsplitter and the concomitant signal loss when both photons arrive at the same channel. Finally, as 2D imaging SPDs with rapid readout emerge, multidimensional correlations (involving time and space or time and spatial frequency dimensions, as shown in Figure 1, right) enable augmentation of classical imaging modalities in terms of spatial resolution, noise insensitivity, or information content.
The aim of this Perspective is to highlight the evolution of the use of photon correlations from "simple" temporal autocorrelations, through multidimensional correlations to imaging and to tie these developments to technological advances. Finally, we present an outlook on promising new avenues for the exploitation of photon correlations and on the requisite advances in detector technology to achieve them.
While this Perspective focuses on correlations emerging from the photons' quantized nature, we note that classical intensity correlations constitute a related and rich scientific field. Fluorescence correlation spectroscopy (FCS) 8 is a powerful and diverse family of techniques, particularly useful in chemical and biological contexts where fluctuations occur due to, for example, diffusion, rotation, and chemical reactions. 9−12 The interested reader is referred to a recently published review by Sarkar et al., 13 depicting the relations between classical and quantum fluctuation and their analysis in FCS studies. Classical fluorescence intensity correlations are also useful for microscopy and serve as the imaging contrast of super-resolution optical fluctuation imaging (SOFI). 14−16

■ PHOTON CORRELATIONS IN A SINGLE DIMENSION
The basic Hanbury Brown and Twiss (HBT) setup as applied to measure autocorrelations at the single-photon level is shown in Figure 2a. Light from a single-photon emitter impinges on a beam splitter evenly distributing the intensity between two low-noise SPDs. For the purpose of this perspective article, we define the results of this measurement, the number of detected photon pairs versus the delay time between them, as a onedimensional photon correlation analysis. Until recent years, setups beyond the basic HBT, including more than two channels, required a high level of technical expertise. As a result, photon correlation experiments heavily relied on the elementary two-identical-channel setup, observing the temporal autocorrelation of the impinging light. Despite the specificity of the method, these measurements have become a cornerstone of quantum optics experiments and found multiple spectroscopic uses. This section explores the nature of the extra information provided by one-dimensional photon correlation experiments and its uses.
Early Studies of Quantum Light Sources. As mentioned above, initial applications of HBT measurements were aimed at sources of astronomical length scales. Making an impact on the quantum world required observing physical systems on the opposite end of the size scale, nanosized light emitters. For these systems, acquiring correlations became possible only once the technology of efficient photomultiplier tube (PMT) detectors and precise timing circuitry had matured, toward the end of the 1970s. Initial experiments set off to explore the very nature of quantum light, proving 5,21,22 what were already decades-old fundamental theories. For example, the very existence (and indivisibility) of the photon 23,24 was validated by measuring photon antibunching in the emission of quantized two-level systems. 5 Figure 2a illustrates this concept: light emerging from an emitter is evenly split between two detectors in order to measure the light's second-order autocorrelation function, g (2) (τ) . If the emission occurs one photon at a time, simultaneous detections will not occur and the detectors' outputs are completely anticorrelated, as manifested by g (2) (0) ∼ 0 (see Figure 2a, bottom).
Building up on early demonstrations that examined the interaction of light with isolated atoms, the following wave of research studied systems of growing complexity, that is, those containing more degrees of freedom. Photon antibunching was measured from various nanosized objects, including single dye molecules, 25−27 single semiconductor quantum dots, 17,28 and color centers in diamond. 29 While the observation of antibunching in the resonant fluorescence of single atoms, an isolated two-level system, was expected according to theoretical arguments, its observation in more complex systems was hardly trivial. That is, one could not anticipate that antibunching would persist in systems with up to thousands of atoms under off-resonant excitation and even under ambient environmental conditions. These measurements were thus considered as breakthrough discoveries and unlocked the potential for low-cost, room-temperature sources of quantum light.
Photon correlations also played a pivotal role in the discovery and implementation of three-and four-wave mixing techniques to generate entangled photon pairs and postselected single photons. 30,31 Under the three-wave mixing scheme, spontaneous parametric down conversion (SPDC), one high-energy photon is converted into two lower-energy photons in a nonlinear medium. To date, SPDC light is the most common light source in quantum-optics experiments, in general, and a frequent one for quantum-based imaging and sensing schemes, in particular (see the section on imaging below).
Multiple Emitters or Multiple Emission Centers. A natural step forward is measuring photon correlations from a group of quantum emitters. When multiple localized excitations only weakly interact, one-dimensional photon correlation analysis can provide an accurate estimate for the number of active emitting centers. One practically important application is quantitative microscopy, providing an absolute reading for the density of labels in a fluorescent microscope image. 32 An interesting technique that achieves this goal, based on measuring photon correlations, is termed counting by photon statistics (CoPS). 33−36 Its foundation is a back-of-theenvelope calculation: for n equal-intensity and single-photon emitters (light blue circles in Figure 2b), the zero-time-delay autocorrelation becomes = g (0) 1 n (2) 1 . Developed mainly by the Herten group, CoPS is capable, for example, to rather accurately count up to 30 fluorescent molecules within a 0.5 s acquisition time 18 (see Figure 2b). The same principle has been successfully utilized to monitor the fluctuating number of active emitters inside a diffraction limited spot in order to enhance the capabilities of super-resolution localization microscopy. 37 Excitation in a complex emitter with multiple emission centers can hop from one center to the next and interact with one another (see Figure 2c). For such complex emitters, including synthesized organic molecules, 19,38,39 polymers, 40,41 large biomolecules, 42 and sophisticated inorganic nanostructures, 43 photon correlations can provide valuable input regarding the excited electron dynamics. For the simplest case, two identical emitters connected by a molecular bridge (Figure 2c), measuring g (2) provides the probability for the singlet−singlet annihilation, a nonradiative recombination process that involves two nearby excitations. 19,44 Indeed, with increasing inter-emitter distance, g (2) (0) grows from zero toward one-half (Figure 2c), reflecting a reduction in the Coulombic interaction between excitations.
Pursuing the above-mentioned complex systems, an intriguing question is "How localized are the excited charge carriers?", that is, to what extent excitations in different centers within the quantum system sense each other? Strikingly, signatures of antibunching are observed in polymer chains made of hundreds of units 41 and even polymer aggregates. 45 This is an indication that such excitations sample a large volume during their lifetime and are thus able to interact with one another.
On the other hand, even when excitations are delocalized and occupy the same confined space, g (2) (0) does not necessarily vanish. In colloidal semiconductor nanocrystals, for example, the extent of antibunching can be controlled according to size and shape of the particle. Extending at least one of the particle's dimensions beyond the exciton Bohr radius, the average eletron-hole separation, strongly reduces the inter-exciton interaction strength ( Figure 2d). As a result, nonradiative Auger recombination becomes less efficient, and the occurrence of pairs (or triplets, etc.) of photons is more frequent. 46,47 Even when the antibunching dip drops to nearly zero, its temporal width can be used as a spectroscopic indicator for molecular dynamics. Although often it reflects only the excited state lifetime (which can be readily measured otherwise), in principle it also monitors the dynamics of the ground state. This principle has been applied to monitor the rates of chemical reactions at the single molecule level. 48,49 Spectroscopic Input from Higher-Order Correlations. Noting the multiple types of information extracted via secondorder photon correlations, it is natural to consider the potential input of higher-order correlations. Technically, this requires extending the number of channels in an HBT setup by adding more beamsplitters and detectors 50 or using a small pixelated detector array. 51 Just as measuring g (2) targets the doubly excited state of the system, probing g (n) exclusively observes its n-times excited state. 52,53 Amgar et al. provide a notable example of the extra information contained in higher-order analyses. 20 In twodimensional colloidal nanocrystals (nanoplatelets), 54 the emission of multiple photons after one excitation pulse is only partially quenched and critically depends on the lateral size of the nanocrystal. 47 Remarkably, despite the wide distribution of g (2) (0) for different nanoplatelets, the values of third-and fourth-order antibunching are precisely determined by that of the second-order. The measured dependence of g (3) (0) versus g (2) (0), for example, enables the observation of a small deviation from the commonly used two-body interaction model. While higher-order photon correlation measurements require relatively high detection rates and long-term stability, they hold significant potential for the study of the multi-emitter systems introduced above.
Photon Correlations in Electrically Pumped Systems. Optically active nanosized materials have already greatly contributed to the fabrication of low-cost and energy-efficient electro-optic devices. 55 Characterizing the dynamics of the charge carriers in the process of absorption and emission is crucial for these applications. This information is typically obtained by time-resolved spectroscopy -measuring timeresolved photon emission after a pulsed excitation. Conversely, as part of a device, the emitters are continuously pumped, most commonly by electrical, rather than optical, pumping. In the past few years, several experimental demonstrations have shown that measuring photon correlations under electrical pumping can test the dynamics of carriers even without pulsed excitation. 56,57 Furthermore, within an electron microscope (EM) 58 or a scanning-tunneling microscope (STM) 56 excitation can be done in a localized manner, well beyond the resolution provided by an optical microscope. While these methods can provide unprecedented simultaneous temporal and spatial resolution, they currently rely on highly sophisticated experimental setups and make use of long acquisition times. Their general potential for spectroscopy of different quantum systems will unfold in the coming years.

■ MULTIDIMENSIONAL PHOTON CORRELATIONS
The experiments discussed thus far all employ "one-dimensional" photon correlation, performing the correlation operation along a single temporal coordinate, the time delay between photon detections. However, photons generally carry much more information, and extending the correlation operation to additional photon properties introduces new opportunities to extract useful spectroscopic information. Correlating photons across their time delay and an additional temporal property is perhaps the most straightforward twodimensional correlation to apply, as it essentially relies on the same experimental setup and instrumentation (Figure 3a, top). Correlating time and a different property (e.g., photon polarization or energy) requires adaptation of the experimental apparatus and can be performed either in a series of measurements (Figure 3b, top) or using parallelized detection along the nontemporal dimension (Figure 3c, top). These three classes of multidimensional photon correlations are explored through this section.
2D Temporal Correlations. A temporal property of particular interest in single-emitter photoluminescence experiments is the delay between excitation and photon arrival time (t d ). The time scale of this delay is extensively used to probe the rate of excited state relaxation and is often implicitly measured in antibunching measurements. A two-dimensional correlation of t d and antibunching can be realized by postselecting detected photons arriving within a temporal gate following each excitation pulse (t d > t gate ) and then applying a standard g (2) (τ) analysis to the filtered signal to extract the gated-antibunching, ĝ( 2) (τ = 0|t d > t gate ). Multiply excited states are the exclusive source of zero-delay detection pairs in fluorescent single-emitters, but typically decay significantly faster than the singly excited state. Hence, ĝ( 2) (τ = 0|t d > t gate ) rapidly diminishes with increasing t gate . Magnum et al. demonstrate that this trend can be utilized to identify single emitters, even when their emission is not significantly antibunched 61 (significant antibunching is implicitly assumed in the counting methods described in Figure 2b). The dominant cause for the rapid decay of multiply excited states in single emitters is efficient nonradiative relaxation mechanisms. Benjamin et al. extract the rate of these dark transitions from the rate ĝ( 2) decays with t gate (Figure 3a, bottom). 59 They show that for semiconductor nanoplatelets (Figure 2d, top) the nonradiative decay rate is temperature-independent.
Hedley et al. have recently demonstrated that twodimensional correlations are useful also to investigate multicenter emitters. 62 They realized a slightly modified gatedantibunching scheme that allows both counting the number of emission centers (Figure 2b) and investigating the interaction between them (Figure 2c) within a single measurement. The modified ĝ( 2) (τ = 0|t d ) can be used to calculate the number of excited emission centers at time t d . While immediately after an excitation pulse ĝ( 2) (τ = 0|t d = 0) reflects the total number of excited emission centers, tracing its evolution with t d reveals the dynamics of the interaction between excitations. Naturally, this elegant solution is sufficient only when the interaction dynamics are slow enough to be resolved by the detection's temporal resolution. Another noteworthy application of twodimensional temporal photon correlations has been recently demonstrated by Wein et al. to characterize a multiphoton quantum light source. 63 There, a two-dimensional correlation of photon delays from excitation is used to investigate an entangled photon-number state generated from a quantumdot-micropillar source.
Sequential Measurement of 2D Correlations. Extending multidimensional photon correlations to nontemporal properties entails some added complexity to the optical setups. An early example is the analysis of polarization correlation of photons emitted from an atomic cascade by Kocher and Commins (Figure 3b). 60 Photons emitted from excited calcium atoms are passed through a polarization-sensitive HBT setup, realized by placing a linear polarizer before each of the two SPDs. Sequential one-dimensional photon correlation experiments are conducted where the two polarizers are either parallel or perpendicular to each other. The observation of a photon bunching peak at zero-delay only for the parallel polarization arrangement indicates that the pairs of photons emitted in the cascade exhibit correlated polarizations.
Another photon property that carries valuable information is photon energy. Vonk et al. have recently realized a temporalspectral photon correlation apparatus by introducing a scanning monochromator in one arm of an HBT setup, and applied it to directly probe doubly excited states of semiconductor quantum dots. 64 In this scheme, one-dimensional temporal photon correlation analysis is used to isolate photon pairs originating in emission cascades (doubly excited → singly excited → ground state) from an overwhelming single-photon background. The monochromator position at the moment of detection indicates the energy of one of the photons. Over repeated scans, the spectrum of both transitions in the cascade can be extracted.
Parallel Measurement of 2D Correlations. Parallel twodimensional correlation measurements have emerged to address two limitations of sequential methods: they supply only projections of the full correlation information, and they often dictate very long acquisition times (∼day and ∼  In the case of polarization, parallelization is conceptually simple due to the low dimensionality of this property. Aspect et al. realized a parallelized version of Kocher and Commins's polarization correlation experiment by replacing the polarizers with polarizing beam splitters, and placing a detector at each port of the beam splitters. 65 They thus conducted a true dichotomic measurement of the linear polarization at each arm of the HBT setup. The parallel approach not only allowed dramatically shorter acquisition times (∼minutes), but could be utilized to demonstrate a direct violation of Bell's inequalities, significantly larger than any violation demonstrated before.
The same principle can be applied to discrete photon energy correlations by placing dichroic mirrors or spectral filters in the arms of an HBT setup. In this case, photon energy (light color) is discretized to a pair of bands and the correlation operation is applied between multiplexed color channels. For a pair of spectrally distinguishable emitters, g (2) (τ) can be measured for each emission color separately (autocorrelation) and between the two colors (crosscorrelation). With this analysis, the rate of the nonradiative Forster resonance energy transfer (FRET) between the two emitters can be unambiguously analyzed. 39 Similar schemes have been applied to investigate other twocolor emitter scenarios, such as two-color-emitting nanocrystal systems, 43 and emission from highly excited states in nanocrystals. 53 Recent advances in detector technology allow adapting this approach to less-discretized properties, and specifically to continuous photon energy. Namely, single-photon-sensitive, fast, pixelated detectors allow building photon correlation spectrometers, as demonstrated by Lubin et al. (Figure 3c). 6,66 The traditional HBT setup is replaced in this experiment by a dispersive element (a diffraction grating), mapping photon energies to different pixels of the pixelated detector. This allows parallel characterization of both the time-of-arrival and energy of each detected photon. Applying this method to semiconductor quantum dots enables measuring the doubly excited state spectrum in a few minutes, a fraction of the time needed for sequential measurements. Moreover, acquiring the full 2D correlation gives access to new insights, such as the spectral diffusion dynamics of multiply excited states.
The examples above harness temporal correlations to isolate signal (two-photon) events from the single-photon background. Correlations in the second measured dimension, such as polarization or energy, typically constitute the quantity of interest. Alternatively, both correlation dimensions can be used to identify the relevant detection pairs. Zhang et al. demonstrate this for enhanced isolation of SPDC photon pairs. 67 Both the temporal and the spectral correlations are used to enhance the background rejection beyond the reach of temporal correlations alone. The underlying SPDC temporalspectral correlations used in this work were also studied by other groups applying similar photon correlation approaches. 68,69 These works map the photon energy onto a single-photon-sensitive imaging detector's lateral dimension and study the emerging two-dimensional photon correlations.

■ IMAGING WITH PHOTON CORRELATIONS
In parallel to the progress in utilization of temporal photon correlations, the past decade has seen significant advances in the use of spatial photon correlations in imaging applications. Essentially, imaging is a particular realization of a multidimensional correlation measurement using two spatial (or spatial frequency) coordinates and often also the temporal one. Thus, applications in imaging and microscopy generally rely on higher dimensionality of correlations, compared to spectroscopic applications, discussed earlier. Within this context, various imaging protocols that utilize illumination either with classically correlated or with entangled photon pairs generated by SPDC, have been developed. 70−72 Taking advantage of the extra information encapsulated in the correlation of photon pairs has facilitated spatial super resolution, enhanced phase sensitivity and an improved signal-to-noise ratio (SNR) over classical imaging approaches. Another strategy is to use classical uncorrelated illumination and take advantage of the naturally occurring phenomenon of photon antibunching in the emission of fluorescent markers as the resource for improvement of the image quality or resolution. 7 At the heart of nearly all the imaging schemes that exploit photon pairs is the parallelized measurement of coincidence events between many detector pixels. Therefore, the advent of high quality, multipixel sensitive detectors, such as electron multiplying charged coupled-device cameras (EMCCDs) and single-photon avalanche diode (SPAD) arrays, has propelled developments and more efficient implementations in this arena. In what follows, we review some of the main classes and current capabilities of quantum imaging methods.
Super-Resolution Microscopy. Although the below mentioned methods for super-resolution appear very different from one another, they all rely on a single principle: photons that arrive at different times stem from multiple emission events and can pertain to different emission centers, whereas simultaneous photons are associated with a single emission event and thus to a single position. Therefore, the ability to tell whether photons arrived simultaneously or not holds additional spatial information, which gives rise to super-resolution capabilities. One such approach to extract spatial information relies on the centroid estimation of pairs of photons, which are correlated both in time and in real or momentum space. Evaluation of the average position of simultaneous photon detections, which are associated with the same point in the object, facilitates an up to 2-fold resolution improvement. 74 Experimental realizations employing entangled photons as illumination, such as the one shown in Figure 4a, demonstrate resolution-enhanced full-field images in bright-field imaging. To perform centroid estimation one must acquire multiple frames, with each sufficiently sparse, so that photons from different pairs are unlikely to spatially overlap. 70,75 The use of quantum correlations in microscopy is not restricted to transmission-based approaches, but appears also in fluorescence imaging. Obtaining a quantum advantage in fluorescence microscopy seems, at first, less intuitive due to the incoherent nature of fluorescent emission. Yet, one can take advantage of the fact that practically all fluorescent labels used in imaging applications are single-photon emitters. As such, they exhibit photon antibunching (see Figure 2a), the absence of simultaneously emitted photon pairs. Thus, every emitter can be considered as a "source" of missing detection pairs in the image. Therefore, measurement of the antibunching anticorrelations using a single-photon-sensitive imaging device enables spatial resolution enhancement beyond the diffraction limit, similar to that obtainable by centroid estimation. Interestingly, this can, in principle, be extended to higher correlation orders (that is, every emitter is also a source of missing photon triplets), an effect that has been experimentally demonstrated. 76 To date, antibunching-based super-resolution ACS Photonics pubs.acs.org/journal/apchd5 Perspective has been applied with either nitrogen-vacancy centers 77 or quantum dots 7,51,76,78 as fluorescent markers. Yet, similar principles are readily applicable to molecular dyes. Notably, fluorescence antibunching imaging has achieved superresolution in confocal scans of fixed biological samples (see Figure 4b). 7,78 Enhanced Phase Imaging Using Entangled Photons. Quantum illumination protocols along with photon coincidence measurements can surpass the standard quantum limit of sensitivity in phase sensing. In principle, with these methods one can achieve the much lower Heisenberg limit. The underlying idea is that phase information can be probed by measuring photon coincidence rates. As depicted in Figure 4c, the use of polarization NOON states, entangled states in which N photons are in a superposition where all photons are linearly polarized along either one polarization axis or the other, yielded supersensitive phase images. Basically, phase variations in the object give rise to modulation of the photon coincidence rate. The modulation becomes more sensitive to a phase difference with a higher number of entangled photons. The outcome is an enhanced SNR for the phase estimate relative to equivalent classical measurements consisting of the same number of photons. 72,79 Whereas early demonstrations employed a scanning microscope configuration, recently, supersensitive quantum holography was shown in a nonscan configuration over a large field of view, although acquisition times were in the order of several hours. 80 Another work demonstrated a wide-field holographic reconstruction scheme, where an entangled state is split to a signal and reference beams and the phase information is encoded into the polarization degree of freedom. 81 The quantum advantage in this case is manifested in a few manners. In addition to a reduced sensitivity to classical noise and an improved spatial resolution compared with classical coherent holographic systems, the method is robust with respect to dynamic phase disorder. The latter is achieved since the interference between the two-photon states is insensitive to path differences between the two beams, but only to polarization-dependent disorder. Therefore, phase information encoded onto the polarization degree of freedom of the entangled photons can be preserved, even though classical coherence between the arms is not maintained. 81 Another phase imaging technique uses the bunching of indistinguishable photon pairs at the output of an interferometer, namely, the Hong-Ou-Mandel (HOM) effect, over many spatial modes (many pixels). 82,83 Practically, coincidence measurements at the output ports of a beamsplitter serve to reconstruct a sample's depth profile. 82 The underlying concept is that a signal photon that passes through a sample experiences a group velocity delay compared to the idler photon, which shifts the characteristic HOM dip and changes the photon coincidence rate. Compared to classical interferometric or phase imaging approaches, the axial field of view of HOM interference can be significantly larger than the optical wavelength. It is determined by the half-width of the HOM dip and can reach a few tens of microns.
Noise and Background Reduction. Another benefit of quantum correlations is the reduction of noise and uncorrelated background. Consider entangled-photons illumination schemes, where one beam interacts with the sample and its twin serves as a reference. Measuring coincidence events between spatially correlated pixels again holds the potential for enhancement of the image, since the quantum nature of the illumination is intrinsically different from that of classical light. Coincidences of the correlated entangled photons serve as a discriminating agent over coincidences induced by uncorrelated classical light and, thus, can be used to separate between them, as illustrated in Figure 4d. We bring a few examples that successfully accomplished noise and background suppression using the quantum advantage. Sub-shot-noise quantum imaging was achieved by measuring the intensity pattern of the signal photons that passed through an object and then subtracting the locally correlated noise pattern measured in the twin idler beam, which did not interact with the object. 84,85 Alternatively, instead of subtracting correlated intensity images, utilization of coincidence events of entangled pairs as the actual imaging contrast has led to background reduction of classical uncorrelated light and sensor noise, 73,86 as well as to resolution-enhanced imaging. 71,81 Furthermore, the isolation of the correlation signal between entangled photons from classical light interference has been shown to have a potential application in LiDAR systems. 87 Notably, some of the works revolving around image distillation have succeeded to benefit from quantum correlations even with a large number of photons per pixel during the exposure time of a single frame. Essentially, instead of measuring actual pairs, the joint probability distribution of entangled pairs (i.e., statistics of genuine coincidences) is estimated by intensity correlations which are affected by photon statistics. 86,88 Thus, measuring quantum correlations is not exclusive to single-photonsensitive detectors and a photon-sparse regime, and the quantum edge can be exploited even with conventional detectors, although less efficiently. This is opposed to a standard quantum optics measurement that relies on the single-photon counting regime and is typically very vulnerable to sources of classical noise, such as background illumination, and spurious reflections. 71,84−86 ■ DETECTOR TECHNOLOGIES FOR PHOTON CORRELATIONS Advancing many of the above-described ideas to practical technologies depends on the availability of suitable technologies for the rapid and efficient measurement of photon correlations. SPDs are at the heart of photon correlation experiments and are the subject of several comprehensive reviews. 89−92 Notably, the past few years have seen tremendous progress in fast, single-photon-sensitive array detector technology, which has already been instrumental in demonstrating several new techniques featured in this text. We therefore dedicate the following section to a brief overview and outlook of single-photon detector technologies in the specific context of photon correlation experiments.
Detector Characteristics. Considering the wide variety of available photon counting detector technologies, it is useful to define several performance parameters crucial for photoncorrelation measurements. The detection efficiency is defined as the probability that a photon incident on the detector is detected. The nth order correlation signal scales with the nth power of the detection efficiency, making it a particularly important parameter when measuring photon correlations. A second parameter is the timing jitter, defined as the stochastic variation in the delay between the arrival of the incident photon at the detector and the generation of an output electrical pulse. A smaller timing jitter translates into a better temporal resolution, enabling the study of faster physical processes. In addition, most detectors are rendered inactive ACS Photonics pubs.acs.org/journal/apchd5 Perspective after the detection of a photon for a time period known as dead time. Detectors also exhibit a finite probability of registering a signal in the absence of an incident photon. The number of such false detections per unit time is termed the dark count rate (DCR) . Multipixel detectors require the optimization of additional parameters. First, the fill factor, which is the ratio of the photosensitive area to the total area of the detector. Second, the crosstalk, the probability that a detection in one pixel will also register a false detection in another pixel. While crosstalk is typically a negligible factor in intensity measurements, it introduces artificial zero-delay correlations that can easily overwhelm photon correlation signals. Therefore, when present, crosstalk severely hampers measurements of correlation functions, and needs to be corrected carefully. 51 Third, correlation measurements can be considered a series of frames, each consisting of a binary matrix of detection or no detection in each pixel. The frequency at which such frames can be acquired, termed the frame rate, fundamentally limits the data acquisiton rate in correlation measurements.
Single Pixel Detectors. For a significant period of time, photomultipliers (PMTs) have been the dominant technology employed in measurements of temporal photon correlations. 21 PMTs are based on vacuum tube technology and offer a large detection area (∼cm 2 ) and timing jitters of the order of 100 ps. However, PMTs have now been largely superseded by singlephoton avalanche diodes (SPADs) which offer better quantum efficiency for visible light. Silicon SPADs typically operate at wavelengths between 400 and 1000 nm. Commercial devices based on custom SPAD architectures routinely achieve efficiencies of >60% at 600−700 nm and timing jitters of hundreds of ps. By reducing the thickness of the photon absorption layer, sub-100 ps jitter has been demonstrated, but at the price of reduced efficiency at longer wavelengths (>600 nm), although efforts are underway to mitigate the latter problem. 93 Si SPADs can also be fabricated via a standard complementary metal-oxide semiconductor (CMOS) process. While CMOS-SPADs exhibit slightly lower efficiencies than their custom counterparts, they can be engineered for better timing jitter, with recent demonstrations reaching 20 ps. 94 Due to their efficiency and timing performance, Si SPADs are currently the dominant photon detection technology for correlation experiments using visible light.
While Si-based technologies are essentially limited to the detection of visible light, infrared photon detection can be achieved with SPADs based on an InGaAs/InP architecture. However, their efficiency is currently limited to ≈30% and their performance is further hampered by the high dark count rate (∼10 4 s −1 at room temperature). Superconducting nanowire SPDs (SNSPDs) are an emerging class of detectors that offer a much higher detection efficiency in the infrared. When a single photon is absorbed by the narrow strip of superconductor in an SNSPD, it breaks superconductivity in a section of the strip. The change in current that results from the absorption is then registered by the electronics. Since these detectors can operate with high efficiency (≈80%) at telecom wavelengths (≈1550 nm), their development is driven by the need for efficient SPDs for quantum key distribution and related technologies. Recent demonstrations include near-unity efficiency, 95 negligible dark counts, 96 and <10 ps jitter, 97 but not all at the same time. 98 Unlike the other discussed technologies, SNSPDs require cooling to cryogenic temperatures (<4 K), and their architecture hinders efficient coupling to multimode fibers or free-space. These aspects, together with their high price, currently impede the widespread use of SNSPDs in applications that require a large number of detection channels operating in parallel.
Multipixel Detectors. All the one-dimensional correlation experiments and several multidimensional correlation experiments were performed using single-pixel detectors. However, these technologies offer limited scalability to measurements that require correlating a large number of SPDs. Such experiments would benefit significantly from multipixel detector technologies. Indeed, the emergence of high-performance multipixel SPDs has supported and inspired many of the works described in the second half of this perspective, from parallel temporal-spectral correlations to photon correlation imaging.
Single-photon detection can be achieved in the pixels of Sibased CCDs with inbuilt physical light amplification. In an electron multiplying CCD (EMCCD), electrons are multiplied within the Si CCD during the readout process. An intensified CCD (ICCD) camera incorporates a microchannel plate together with a phosphor screen that amplifies light before it is detected by a Si CCD. Both can resolve photon detections and perform correlations. 76 Photon detection in ICCDs can be time-gated, with recent demonstrations achieving gate times below 20 ps. 99,100 EMCCDs do not possess a time-gating ability, but can reach detection efficiencies of 90%, much higher than ICCDs. 101 Both EMCCDs and ICCDs have been employed in recent demonstrations of correlation-based imaging. 70,86,100 However, both technologies are limited by the frame rate (∼kHz) and readout noise typical of CCDs. The more recent TimePix cameras combine a pixelated camera, an image intensifier and a data-driven readout scheme to increase the frame rate. 102,103 The latest variant, the Tpx3Cam, is capable of detecting photons with an efficiency of ≈90% and a timing resolution of ≈1.5 ns and has recently been utilized in demonstrating better background rejection in correlation imaging via spatiotemporal correlation measurements. 67 The temporal limitations of EMCCDs have also stimulated the scale-up of several photon-counting technologies to array detectors with pixel-wise timing circuitry. Of the technologies mentioned in the previous section, CMOS SPADs, which are based on industry standard fabrication processes, are the most readily scalable to arrays. 104 Smaller monolithic CMOS SPAD arrays (tens of pixels) incorporate time-digital converters (TDCs) to perform time stamping and already feature a performance comparable to single-pixel SPADs. 51 While it is possible to integrate a TDC with each pixel, this reduces the fill factor of the detector significantly, and it is important to optimize this trade-off. 105 Implementing the entire timing circuit on an FPGA enables greater flexibility, such as in case of the LinoSPAD detector for spectroscopy, where 64 TDCs are shared by all the pixels in a linear array. 106 Such an architecture is ideal for applications where only a few detectors register photons at any instant, such as spectroscopy of few-photon emitters. 6,66 The recent inception of megapixel-scale CMOS SPAD arrays, featuring frame rates 2 orders of magnitude faster than EMCCDs, promises further progress in this field. 107 However, the large arrays are typically equipped with only time-gating capabilities and have primarily been used in fluorescence lifetime imaging (FLIM) demonstrations. 108,109 The efficiency of CMOS SPADs is significantly lower at longer wavelengths (>600 nm). 104 This has inspired efforts to engineer custom fabricated Si SPAD arrays. 110 While not as ACS Photonics pubs.acs.org/journal/apchd5 Perspective readily scalable to large arrays as CMOS SPADs, custom Si SPAD arrays achieve a higher efficiency at longer wavelengths, with recent demonstrations reaching 33% at 800 nm. 111,112 SNSPDs have also been successfully scaled up into arrays. 113 The number of pixels in these arrays is typically limited by the heat generated by pixel-wise readout cables that places a significant load on the cryogenic cooler. In order to overcome this limitation, alternate readout mechanisms have been implemented, such as superconducting signal-processing circuits, 114 addressing pixels via frequency multiplexing, 115 and row-column readout schemes that reduce the number of readout lines. 116 Other superconducting detectors such as kinetic inductance detectors (MKIDs) are designed to operate with frequency multiplexing and are inherently scalable. 117 InGaAs/InP SPAD arrays have also been explored for photon detection in the IR, but current implementations suffer from a poor crosstalk of several tens of percent. 118,119 IR photon counting array detectors are key to extending quantum enhanced and photon-correlation-based spectroscopy and imaging to the infrared regime; however, such arrays have not reached widespread availability.

■ CONCLUSIONS AND OUTLOOK
In the decades following the early experiments of Hanbury Brown and Twiss, the opportunities for the use of photon correlations were limited to those afforded by the limitations of the experimental setup comprising two SPDs and a correlator. Hence, while photon correlation measurements served as a cornerstone in developing a basic understanding of quantum phenomena, their use in a broader context of imaging and spectroscopy was limited. The growing interest in recent years in the photophysics of nanometric objects and, particularly, the growing need for the development of nonclassical light sources have significantly increased the importance of photoncorrelation measurements.
The rising scientific interest was fueled by significant technological advances, enabling highly multiplexed singlephoton detection, leading to a dramatic expansion of the application of photon correlations. The advent of singlephoton-sensitive imaging detectors, first in the form of ICCDs and EMCCDs, later in the development of SPAD and SNSPD arrays, along with an increase in detection efficiency, enabled to rapidly perform measurements that were considered to take prohibitively long times or were simply impossible. This opened a broad range of spectroscopy and imaging applications, leading to many "proof of concept" experiments highlighting the utility of information obtained from photon correlations in contexts far removed from fundamental quantum mechanics. Moreover, the technological advances in detector technology are rapidly democratizing the use of photon correlations by introducing systems that may soon obviate the need for a strong expertise in optics to perform these measurements. Two fields of study where we particularly expect the impact of photon-correlation measurements to grow significantly are the study of quantum emitters of light and bioimaging, in both of which multidimensional correlation can significantly contribute to the extraction of information from the photon stream.
Quantum sources of light are expected to become a significant component in computation and cryptography in the near future. The large variety of such potential sources, especially nanoscale ones, semiconductor quantum dots (epitaxially and colloidally grown), defects in crystals, or in 2D materials and dopants in crystals, to name a few, requires simple tools for their characterization, which can be employed by people whose expertise is in materials science and not necessarily in optics. Especially considering the tendency to move from ensemble measurements to single-emitter measurements in spectroscopy, so as to avoid artifacts involving heterogeneity within an ensemble, photon correlations are expected to become a handy tool in the materials characterization toolbox.
Bioimaging applications increasingly take advantage of highend imaging detectors, especially since those can provide access to previously inaccessible properties of the samples, such as fluorescence lifetime, quantitative information, an improved spatial resolution, or access to a broader range of infrared frequencies. The mere fact that some of these technologies will enable the extraction of photon correlation data will likely lead to the exploitation of this information in various imaging scenarios.
As was evident in recent years, further progress in utilizing photon correlations for quantum advantage will heavily rely on technological advances. For SPAD arrays, the quantum efficiency and the fill factor of single-photon array detectors still has to improve, especially in the near-infrared, to be more competitive with conventional silicon-based imagers. Further into the infrared, both InGaAs SPAD arrays and SNSPDs present significant opportunities that are yet to be fulfilled. Both have already reached about 1000 pixel arrays, but the overall detection efficiency is still relatively low. The increasing number of pixels also puts a significant burden on the data link from the detector to the data collection system, which would likely necessitate a higher degree of on-chip processing so as to reduce the volume of data to be transferred from the detector. Processes such as spatial correlation and crosstalk reduction are examples of operations, which can be performed on-chip, especially due to their local nature.
All in all, the applicability of photon correlations beyond the rather exotic quantum optics experiments can be perceived as a pleasant surprise. The above-described applications came to life despite the fact that photon-correlation measurements still require the independent development of a dedicated lab setup and acquiring signal from dim light emitters. Naturally, both of these obstacles become simpler to surmount as detector technology and material synthesis improve. Given the intensifying drive to harvest technological benefits from devices based on single nanosized quantum objects, we foresee that the role of quantum photon correlations, as that of other quantum optics protocols, will significantly grow. ■ REFERENCES