Single-photon sources: Approaching the ideal through multiplexing.

We review the rapid recent progress in single-photon sources based on multiplexing multiple probabilistic photon-creation events. Such multiplexing allows higher single-photon probabilities and lower contamination from higher-order photon states. We study the requirements for multiplexed sources and compare various approaches to multiplexing using different degrees of freedom.


I. INTRODUCTION
Single-photon sources, inherently nonclassical in their nature, are quite distinct from the light sources of a century ago. And since the first efforts toward building nonclassical sources of light a half century ago 1,2 , significant progress has been made. Now, sources that produce photons in pairs, allowing for the heralding of a single photon, are the workhorse for a wide array of applications, from tests of fundamental physics 3,4 to metrology 5,6 , to even entanglement-enhanced microscopy 7 . Systems built from sources of photon pairs rely on either spontaneous parametric down-conversion (PDC) or spontaneous four-wave mixing (FWM) and can now reach production rates of millions of heralded single photons per second in controlled states 8 , with tailored spectral properties 9,10 and near-perfect spatial modes 11 . However because these nonlinear optical processes are probabilistic, they cannot simultaneously achieve a high probability of producing a photon and a high single-photon fidelity 12 (See Appendix for detailed definitions). This inherent tradeoff can be a severe constraint in many applications.
The multiplexing of many of these probabilistic single-photon sources offers a path to overcoming this tradeoff. By having many low-probability, but high-fidelity, heralded singlephoton sources, it is possible to create a system that boosts the probability of successfully generating an output, while retaining high single-photon fidelity. Multiplexing in such sources is achieved through the use of time, space, and/or frequency degrees of freedom to parallelize the spontaneous photon creation in a number of different modes, then actively switch the photons into a single output mode based on feedback from heralding detection events.
We will review the history and recent rapid progress in this exciting field. From a few theoretical proposals near the year 2000, the field has sharply grown: numerous distinct multiplexing schemes have been proposed, with ≈ 9 experiments realized in the last three years alone (plus many more relying on similar ideas or technologies), and that rate is increasing. It seems likely that through the use of source multiplexing, one can expect states with ten photons at rates of ≈ 10 3 /s, and states of 50 photons in some finite time, are no longer a pipe dream. This should be enough for a conclusive quantum advantage over classical computers 13 .
The paper is organized as follows: first we discuss photon source issues in general, and provide definitions for single-photon and related metrological parameters. Then we give a brief history of the field of source multiplexing and show how and why multiplexing helps photon sources; next, we discuss requirements for multiplexed single-photon sources, including the required performance of the heralded sources, the switches, and the heralding detectors. Then we compare the different degrees of freedom that allow multiplexing. Finally, we provide an outlook and possible future directions.

A. Single-photon sources
An ideal single-photon source is one that produces a single photon in a known single mode each and every time one is called for, and that mode must be identical each time. That is what is meant by an on-demand single-photon source. While this ideal source can never be achieved in the real world due to inevitable losses and nonzero multiphoton rates, there have been efforts at making better and better approximations to such an ideal source. There are two basic approaches to constructing single-photon sources (beyond the original approximation to single photons of strongly attenuated laser light 14 , which is limited to ≈ 37 % fidelity due to the photon statistics of coherent states). Current single-photon sources ( Fig.  1) are based on either isolated single quantum systems that can only emit one photon at a time, or on sources that emit photons in pairs so that the detection of one photon heralds the existence of the second photon. While isolated quantum systems are often called deterministic sources (because they can, in principle, produce a single photon on demand, with no other noise or higher-order photon terms) in practice there is always some level of residual noise and the efficiency of photon collection is less than 100%. The difference between deterministic and on-demand sources is that the former produce photons at certain fixed times, whereas the latter can produce them at arbitrary times. This distinction can be blurred by reconfigurable delay lines, or by pumping schemes or storage cavities that are necessarily periodic, e.g. modelocked lasers. By contrast, the pair sources used to herald single photons are called probabilistic sources, since one cannot know even in principle when a heralding signal will come, but when and only when it does come, the existence of the output photon is guaranteed. In reality, the line between deterministic and probabilistic is rather a continuum. Deterministic sources become more probabilistic as extraction and coupling efficiencies to other systems (such as fibers) decrease, and probabilistic sources can become more deterministic by multiplexing.
Each of these approaches has advantages and drawbacks. With nonlinear-optical pair sources (and faint laser sources), the probability of one-and two-photon pair excitations are strongly linked, making optimizing both of them simultaneously impossible. Thus these sources are usually operated in the regime of low excitation probability, meaning the state generated is mostly vacuum, so when a photon is actually produced is unpredictable. However unlike faint laser sources, the fidelity (after heralding to eliminate the vacuum contribution) of pair sources can be very near unity, as the multiphoton component can be made arbitrarily small 15 , and the indistinguishability can be made high through source engineering 16 . Of deterministic sources, those based on quantum dots suffer from lower indistinguishability and fidelity due to charge fluctuations in their local environment 17 . However, this problem is being addressed by techniques such as resonant pumping [18][19][20][21] , and efficiencies continue to increase 22,23 , leading closer to the ideal. There are also efforts to enable deterministic growth and/or placement within an integrated platform 24 . Nitrogen-vacancy and other color centres in crystals are also emerging as reliable quantum emitters 25,26 , though spectral drift in these systems still limits indistinguishability 27 . Single atoms and molecules have also shown promise 28,29 , but are arguably more difficult to engineer and integrate than quantum dot-type emitters.

B. Single photons: definitions
Single photons, that is, single excitations of modes of the electromagnetic field as solutions to Maxwell's equations, are useful in metrology 5,30 , quantum computing 31,32 , imaging 33,34 , quantum communication 35,36 , and randomness generation 37 . (Single-photon detectors are used in far wider applications as low-noise low-intensity detectors, but here we focus on applications requiring also single-photon sources.) An ideal single photon is in the state |1〉 k , where k defines the field mode (spatial mode, continuous-wave or pulsed temporal mode) in which the photon resides.
While that is the ideal, often more than one photon will be produced by the source, and once produced, the state always encounters some nonzero optical loss before reaching the application, a transformation that results in a state that contains both vacuum and multiphoton components, along with the desired single photon. Furthermore, the photon can also be spread over multiple modes in a superposition or mixture, requiring a sum or integral over k.
As the engineering of single-photon sources improves, the metrology of their performance characteristics becomes more challenging and more critical. While the need for good metrology obviously impacts comparisons of devices and schemes, its use as a diagnostic tool is also key to the advance of these sources.
In surveying the state of development of these devices, it clear that complete reporting of performance data is needed and something we strongly encourage. With that goal in mind, we provide a list of definitions (Appendix Tables V to VII) and later present a table with performance results as best as can be determined from available literature (Table III). Unfortunately, published results are often incomplete and ill defined. Hopefully this compendium of results will help clarify the progress in the field and suggest standards of reporting of results. Even the missing table entries serve a purpose, highlighting the need for better reporting of results.
For a wider look at single photons and their applications, we refer the reader to many excellent reviews 38-48 . 1. Single-photon fidelity and indistinguishability-One way to assess the quality of the output of a single-photon source is to measure its absolute fidelity to a single photon in a single mode, which is defined as the overlap of the reconstructed output density matrix with |1〉 k . For the most general case, this is an unheralded fidelity (i.e. without postselection), which requires quantum state tomography by homodyning 49 . This can be difficult and time-consuming, so other single-photon metrics have been developed. For example, the normalized Glauber second-order correlation g 2 0 = a † a † aa a † a 2 is particularly useful, as it quantifies, independent of losses, the multiphoton component of the state with respect to the single-photon component i.e. |n > 1〉 versus |1〉. We also make use of the heralded fidelity F h , which is the fidelity of the photon to |1〉 after heralding, but before any losses. This is assessed via the g (2) (0) as it is intrinsically insensitive to losses, and as such, is the fidelity in the subspace excluding the vacuum.
Another characteristic that needs consideration is indistinguishability, which is defined as the modal similarity of the photons (spectrum, time, polarization and space). A subtle point is that this includes the purity of the photon in each of these degrees of freedom, as impure states imply the need for purifications that in principle provide distinguishing information.
(Indistinguishability is sometimes also used to quantify the interference between two photons from the same pair, but we do not consider that here.) The indistinguishability between photons from different sources or between photons from the same source, but generated at different times, is critical for interference experiments. For single emitters it is often much harder to achieve indistinguishability between different sources due to narrow spectra and differing electronic/magnetic environments, which can cause spectral variations and wandering. Indistinguishability is quantified using the visibility of the Hong-Ou-Mandel interference of the photons 50,51 . Single-photon sources should also not emit noise photons at unwanted or unheralded times. This can be quantified by the output noise factor, the ratio of unheralded or untriggered photons to the total photons emitted photon by the single-photon source 52,53 .
Regarding photon modes, ideal single photons are in a well-defined spatial mode (e.g. of a single-mode fiber) and nicely behaved spectral-temporal modes (e.g. Fourier-transformlimited Gaussian modes). Unfortunately, photons are often emitted from sources that are spatially multimode, violating the ideal requirement and making coupling to fiber difficult. Photons are often also spectrally multimode, either due to spectral correlations between the two photons of a pair, or due to electronic or other noise near isolated quantum emitters. For the photon pair case, once one is detected, the other is left in a mixture of modes, whose natural basis is given by the Schmidt decomposition of the two-photon joint spectral amplitude 54 . The effective number of contributing modes is then given by the Schmidt number. For the rest of the paper we assume the photon has been engineered (meaning the source is constructed in such a way as to produce intrinsically pure photons without subsequent filtering and the additional loss that comes with it), or filtered (meaning the photons are strongly filtered to a single mode, decreasing brightness and heralding efficiency) to be spectrally and spatially single-mode.
We can also gain some information looking at the bandwidth and temporal duration of the photons: it is generally desirable to produce an output with a Fourier-transform-limited timebandwidth product. For heralded sources, a photon that is transform-limited is also spectrally single mode 55 . For isolated quantum emitters, transform-limited photons indicate the long-term noise near the emitter is controlled or eliminated, meaning the emission frequency is not wandering with time 18 . In both cases, transform-limited photons enable interference applications.

Source Brightness-
The brightness of a single-photon source has taken different definitions in different communities; for example, for quantum dots it is often the probability of receiving a photon per excitation event, while for photon pair sources it is normally the number of photons per pump pulse or per second, normalized to pump power and sometimes to photon bandwidth. Here we define it as the probability to have exactly one photon in a single-mode fiber at a given clock cycle, irrespective of whether a heralding detection is received or not. Thus for a pulsed pair source, brightness can be written as B unheralded = p h p 1 h η coupl (1) where p h , the heralding probability, is the probability of a heralding event per pulse, and p (1|h) is the conditional probability of producing a single photon given a heralding event. The coupling from the source to single-mode fiber is η coupl . For an isolated quantum emitter the brightness is B Q = p 1 η coupl (2) as heralding is not relevant, so here we use simply p 1 as the single-photon probability. These two definitions, both defined in terms of a single spatial mode, allow for clear comparison between these two types of sources, at least in most applications. We note that for quantum dot emitters, η coupl is often defined as the collection by the first lens, a much less useful definition than our single-mode-based definition.
The brightness can be assessed by measuring the detection rate in combination with the multiphoton component via g (2) (0). To find brightness from non-photon-number-resolvingdetector count rates, we neglect terms beyond two photons, such that the click probability in the heralded mode given a heralding signal is where S det is the rate of heralded single-photon detections, S h is the heralding rate (or repetition rate for quantum emitters), η tot = η coupl η det is the total detection efficiency, composed of coupling and detector efficiencies, and p (2|h) is the conditional probability of generating two photons given a heralding event. Also, the (heralded) second-order correlation can be measured with a 50:50 beamsplitter and two detectors as g h Here the heralded coincidence rate is C 12 and heralded singles rates are S 1h(2h) . This directly gives p 2 h ≈ g h 2 (0)/2, since a † a † aa ≈ 2p 2 h η tot 2 and 〈a † a〉 ≈ η tot . Then the brightness in terms of easily measurable quantities is which can be seen as the heralded click rate, corrected for detector efficiency, minus the fraction of the click rate that is made of two-photon events.
Of course, most applications require not just a single photon one time, but rather a stream of single photons. Thus the source repetition rate (the rate of attempts to extract a single photon from the source, R src ) is an important factor in source design, with higher source repetition rates allowing, for example, faster experiments and better signal-to-noise ratios. The total rate of usable single photons out of the source is then R src B.
3. Efficiencies-In experiments, there are a number of efficiencies at play. For pair sources, the efficiency of transforming pump photons into photon pairs (conversion or generation efficiency) and the coupling efficiency η coupl (i.e. loss due to coupling) for both the heralding photons and heralded photons into single-mode fiber are of critical importance. The total efficiency of detecting a photon given a herald event (comprising coupling efficiency η coupl and detector efficiency η det ) is called the Klyshko, heralding, or total efficiency η tot , given in the low power regime by the ratio of coincidence counts to heralding counts 56 . For single emitters, the excitation efficiency, the quantum efficiency of emitting a photon given the dot is in the excited state, losses due to non-radiative decay 57 , and the coupling efficiency from the dot to the fiber largely determine the brightness 45 . The various sources of non-unit efficiency differ in origin between pair sources and quantum emitters, making comparisons difficult. Thus we use the brightness to best compare the ultimate performance. To allow for fair comparisons, a delineation of where the source ends and where the transfer to the application begins needs to be clearly made,

II. THE ADVANTAGE OF MULTIPLEXING
In this work we focus on transforming probabilistic sources (namely, sources based on photon-pair emission) into deterministic via multiplexing. It has been shown in theory 12 and experiment 58 that a single ideal photon-pair source cannot be used to herald single photons with heralded fidelity (fidelity to |1〉 after heralding but before any losses) F h = 1 with greater than p h = 25 % heralding probability (i.e. unheralded fidelity F < 25%), and this bound is only achieved if the heralding detector can perfectly resolve photon number with perfect efficiency. For single-photon detectors that cannot resolve photon number (bucket, click, or so-called photon-counting detectors, that merely declare the presence of one or more photons) the fidelity and heralding probability are bounded 12 as F h + p h = 1. Only source multiplexing can break these bounds for heralded photon sources.
Photon source multiplexing allows independent control of the amplitudes of the emitted photon numbers of the component sources. By allowing many low-probability chances to produce a single photon, the single-photon term for the multiplexed system can be increased without increasing the higher-order terms. To first order, the output photons maintain the high fidelity of a single, weakly-pumped heralded source, while increasing the brightness linearly with the number of sources employed. We now examine these scaling arguments in detail.
The general structure of a multiplexed spontaneous source is seen in Fig. 1c. Multiple pair sources are pumped simultaneously, and each one has its own heralding detector. If one heralding detector fires, the corresponding partner mode is actively routed to the output. If multiple detectors fire, just one of the partner modes is routed to the output, perhaps based on which path is the least lossy. It is key that the other modes, those not selected to route to the output, are removed and not allowed to propagate with the true output mode, so that any photons produced in these modes do not pollute the single-photon state.

A. History of multiplexing
The first ideas for multiplexing many down-conversion events to engineer quantum states came as early as 2001 59 . The next year, multiplexing for single-photon generation was independently formalized by Pittman and colleagues for the temporal degree of freedom 60 , and by Migdall and colleagues for the spatial degree of freedom 61

B. Theory of multiplexing
The two-mode state emerging from a photon pair source is well-described by the squeezed vacuum state 76 with the squeezing parameter r contained in the squeezing strength 76 λ = tanh r, and n being the number of signal and idler photons are each in single modes. It is immediately clear that the probability of generating two photon pairs is directly linked to the probability of generating one, namely its square. The heralding event by a click detector is represented by the positive operator-valued measure (POVM) element Π = n 0 ∞ 1 1 η h n n n , (6) where η h is the detection efficiency of the herald including all optical coupling efficiencies and the heralding detector efficiency. Applying those efficiencies to one mode of the PDC state and tracing over the other gives the heralding probability and the heralded single-photon state 12 where in each second line we keep only two orders in λ.
Before we look at the low-power approximation, we can check the fundamental limits of brightness and heralded fidelity for a standard single PDC source. Without optical or detector losses the heralding probability for click detectors is p h = λ 2 , then the brightness is B = p h p (1|h) = (1 − λ 2 ) λ 2 and the heralded fidelity is F h = p (1|h) = (1 − λ 2 ), satisfying the limit F h + p h = 1, and allowing us to plot the single PDC source limit curves in Fig. 2. The heralded fidelity is lowered only due to multiphoton components and does not include losses on the heralded mode. These losses are instead captured in the brightness through η coupl .
In the low power regime, in order to approach commonly measured quantities, we can directly extract the heralded probability of producing one and two photons as p (1|h) ≈ η h λ 2 / p h and p (2|h) ≈ (2 − η h )η h λ 4 /p h . The brightness is B = p h p (1|h) η coupl ≈ η h λ 2 η coupl , and to second order, the heralded fidelity is The heralded g h 2 0 can also be related to the coincidences-to-accidentals ratio CAR = (C − A)/A, where C is the total number of measured coincidences, and A is the measured accidentals, as Again, for perfect heralding efficiency, p h = λ 2 , so p (1|h) = 1 − λ 2 , giving F h + p h = p (1|h) + p h = 1. Then the bound F h + p h < 1 applies for imperfect efficiency. Multiplexing can break this bound.
Instead of a single |ψ〉, multiplexing uses many, and routes to the output one whose corresponding heralding detector clicks. This K-source state is described by the tensor Now we have also K heralding detectors, and since we want to actually switch the heralded photon to the output mode, we can accept any number > 0 of detectors firing. From the geometric distribution, the heralding probability now is which can be interpreted as one minus the probability that none of the heralding detectors fires.
The state after heralding and switching is identical to Eq. (8) and thus has the same heralded fidelity, but the brightness is directly improved with increasing number of sources. Again for lossless detectors, F h + p h ≈ 1 − λ 2 + Kλ 2 > 1, breaking the single-source bound. In fact, for K → ∞ and λ 2 → 0, F h and p h both approach 1. For a realistic efficiency of η h = 80%, to reach p h = F h = 99% with non-photon-number-resolving detectors requires λ 2 = 0.0083 and K = 692 sources, before any other losses are considered. These and some other cases are summarized in Table I. The large number of sources can also be reduced by using photon number-resolving detectors in the heralding arm, allowing stronger pumping while preserving the fidelity; for details see Christ et al. 12 Next we describe the physical requirements for multiplexed photon sources and the effects of imperfections.

A. Probabilistic single-photon sources
A good multiplexed single-photon source starts with a good photon pair source. As shown above, the fidelity of the multiplexed single photon is the same as the heralded single photon from a single source. Thus, photon pair sources used for multiplexing should be low-noise, i.e. their g h 2 0 should not exceed 2λ 2 , and they should be spatially and spectrally singlemode to allow multiphoton interference. Achieving single-mode operation remains a significant effort in source engineering, and many common sources do not fulfill this condition. Furthermore, the coupling efficiency from the source to the heralding detector and to the multiplexer should be high, as both significantly degrade the achievable improvement. Whether the sources must be bright depends on the application: for optimized single-photon fidelity, many sources of low squeezing strength are needed (λ 2 < 0.01), such that the multiphoton component of the output state is minimized. By contrast, for the highest rates the emission probability of each source should also be large enough (λ 2 ≈ 0.1) such that the probability to have a heralding event is high, or the number of sources increased. The tradeoff is that for high emission probability the multiphoton component will be large, decreasing the output single-photon fidelity.
Very many sources can satisfy these requirements, notably three-wave mixing sources based on bulk 10,77,78 and waveguided 9,79 KTP (unlike PPLN, which does not produce spectrally single-mode states directly), and bulk KDP 16 crystals. Resonator sources that restrict the photon pairs to single spectral modes are also good candidates 80,81 . Some integrated sources also show reasonably good performance, especially if photons would not have to be coupled off-chip for multiplexing, for example four-wave mixing in silicon wire waveguides 82 and microdisks 83 , photonic crystals 84 and birefringent 85 fibers, and laser-written silica waveguides 86 .

B. Optical switches
Equally critical to source multiplexing is the switch or switches, that receive a signal from the heralding detector to switch the heralded photon into the output mode. Multiplexing schemes have used integrated switches, such as fast opto-ceramic switches 65,87,88 and electrooptic switches 66,67 , bulk electrooptic polarization rotating switches, with 62,69 and without 60,70,73 polarization-independent interferometers. Spectral multiplexing requires frequency shifts rather than path switching, which has been accomplished with electrooptic modulators 68 and four-wave mixing 89 . In all cases the heralded photons must be delayed to allow time to process the heralding signals and activate the switch. Most switching times are short enough that the needed optical delay is dominated by electronic processing time. Delay (or latency) times in source-multiplexing experiments range from 200 ns to 1000 ns, but none of these has been strongly optimized. If this required delay can be met by sending the heralded photons through optical fiber, and if the photons are at telecom wavelengths, not much loss is encountered. However in other cases and for scaling to applications, this delay time should be reduced significantly from its current average.
For spatial and temporal multiplexing, the photons pass through switches multiple times, making the insertion loss of the switch a critical parameter. Most integrated switches mentioned above have ≈ 1 dB loss, while the bulk polarization rotators can reach 0.03 dB 74 . This is one of the contributing factors why the integrated implementations in Fig. 2 remain on the left part of the graph, as switching loss enters exponentially in the photon rate. However, integrated switches can be faster than bulk, allowing the source(s) to be pumped faster, creating a linear speedup in photon rate. In current implementations the speed advantage for integrated switches is not yet large (e.g. 1 MHz 88 maximum repetition rate for integrated switches versus 0.5 MHz 74 for bulk), but this gap should grow with continued research. The contrast of the switches should also be high, such that only the desired heralded mode is coupled to the output, suppressing unwanted counts from all other modes. This high contrast has been shown in a "multiplexing of one" scheme, wherein the output of a single source is physically gated based on the heralding signal, providing an extremely low output noise factor 52,53,90 . Switch parameters for a number of multiplexing experiments are shown in Table II. A promising new direction is the use of four-wave mixing in interferometers as a switch 91 , which promises rates up to 500 MHz (currently 5 MHz) with losses below 1 dB.

C. Heralding detector
The final important component in source multiplexing is the heralding detector(s), which must be efficient enough to detect a significant fraction of the herald photons, fast enough for high-rate pumping, and have low enough dark counts to avoid heralding the vacuum. In recent years it is very common to use superconducting nanowire single-photon detectors 92 , as they have the highest overall figure of merit 38 H = η det /(d r Δt) (defined as the detector efficiency divided by the dark count rate and the timing resolution), and satisfy the requirements above, albeit at the cost of cryogenic operation.
As seen above, the heralding probability for a single source is proportional to the efficiency of detecting the heralding photon. For a fixed pump power, the effect of herald detection efficiency on the heralding probability and heralded fidelity for K = 20 and K = 50 multiplexed sources is shown in Fig. 3. From Eqs. (9) and (12) we can approximate the number of sources required to surpass the single-source threshold of F h + p h = 1 given herald detection efficiency η h as which shows that for lower herald detection efficiency, more sources are required. Assuming weak pumping, this threshold is independent of the pumping strength. Or equivalently, for K sources, a herald detection efficiency of η h ≈ 2/ K + 1 (14) is needed to break the single-source bound.
The dark counts of the heralding detector also play a role. Adding dark counts to the detector POVM 93 Eq. (6) where d is the dark count probability during the heralding coincidence window, and in the limits λ ≪ 1 and d ≪ λ. Now the heralded fidelity including the vacuum is Reaching the same 99 % fidelity now with d = 10 −5 instead of a noiseless detector requires decreasing the pump power (lowering the multiphoton contributions to compensate for the vacuum contribution) to λ 2 = 0.0068, increasing the number of sources required to reach heralding probability 99% from K = 692 to K = 845 (Table I). However modern superconducting nanowire detectors have d < 10 −7 in a 1 ns window, making dark counts largely irrelevant 92 .
Finally, heralding detector dead time can decrease the heralding rate, or require adopting detector multiplexing strategies 94,95 Dead time arises from the need for suppressing afterpulses after the detector fires, or in the case of gated detectors from the need to reset to a ready state even after a gate with no detection events 95 . Dead time is a bigger problem for temporally-multiplexed sources, since just one detector is responsible for detecting all modes, whereas in other multiplexing methods, different detectors are coupled to different source modes.

D. Integrated versus bulk optics
Photon sources, switches, and detectors all exist in both bulk-optic and integrated forms. Integrated optics promise bright sources due to strong confinement and long interaction lengths (however, source engineering is essential to produce spatially and spectrally singlemode photons), high-speed switching enabled by low required voltages, and high-efficiency low-noise detectors due to small active areas and strong interactions. Integrated optics devices are also compact and robust, leading to easier adoptability in applications. However, so far the problems of coupling losses from the chips to external detectors and noise filtering have not been sufficiently solved to permit performance competitive with bulk optics, with the notable exception of detection. Here, superconducting nanowire detectors outperform other types of detectors, especially in the telecommunications wavelength bands. They normally require coupling via optical fiber, but the integration of detectors on waveguide structures is showing promise as the need for off-chip coupling is eliminated 96,97 .
Integrated sources, however, suffer from losses coupling between disparate elements, ≈ 1 dB (20 % loss) per interface 98 , as well as waveguide losses in the source itself (e.g. ≈ 1 dB over the 196 nm long source waveguide in ref. 87). And not only are integrated filters lossy, they tend to have low extinction (e.g. 2.8 dB loss for 40 dB extinction in ref. 99), requiring cascading filters to achieve the required pump and noise suppression 66 . Nonetheless, strong gains are being made all the time in integrated quantum optics, which should allow surpassing bulk-optic performance in a few years, in particular in scaling to many sources and many photons 86 .

IV. DEGREES OF FREEDOM AND MULTIPLEXING SCHEMES
Now we present a survey of the various degrees of freedom that have been proposed for multiplexing, and the implementations using them, with experimental results summarized in Table III.

A. Spatial source multiplexing
Spatial multiplexing requires multiple down-conversion events to occur in separate spatial modes, either in the same crystal e.g. in different directions, or in physically separate crystals (see Fig. 4). First proposed in 2002 61 , it was shown theoretically that it is possible to decouple the probability of producing one and more than one photon by pumping multiple sources simultaneously and placing a heralding detector on each one. When a detector fires, its corresponding partner photon is switched into the output. The first experiment using multiplexed spatial modes came nearly ten years later: Ma et al. 69 pumped two BBO crystals to produce photon pairs, and additionally polarization entanglement, effectively giving four PDC sources (two spatially separated sources and two polarizations for each source). A fast router based on Pockels cells directed the heralded photons to the output mode.
After this implementation, there were many theoretical refinements in spatial schemes, adding realistic experimental imperfections 102,103 . With an eye toward scalability, modularity was introduced, allowing the connection of many identical components into massive multiplexing setups 64 , particularly to reduce the average number of switches traversed [104][105][106] , or to switch the pump instead to eliminate switching of the single photons entirely 107 and the associated switch loss. This last scheme is sometimes referred to as "dump the pump." A number of schemes combined spatial and temporal multiplexing, increasing the number of sources available to multiplex 67,108,109 . Applications of such sources include BosonSampling 110,111 and quantum key distribution 112 .
In parallel, partially integrated spatial multiplexing took off, pioneered in the Eggleton group, including discrete but waveguided sources and switches 65,66,87 , and fully fiber-based experiments 99 . However, due to high losses these experiments have not yet approached the single-source limits. A great engineering push to reduce losses is ongoing across the integrated quantum optics community, and this will greatly benefit these multiplexed sources.
One new direction is the spatial multiplexing of atom-photon entanglement in a quantum memory 113 . This allows multiple excitations to be stored in parallel, heralded by the detection of a single photon correlated to each excitation on a single-photon camera. Then the multiple stored excitations can be emitted simultaneously as a multiphoton state. Switching the arbitrary heralded spatial modes into desired modes remains a challenge.

B. Temporal source multiplexing
A proposal resembling temporal source multiplexing was published in 2001 59 , but the field really began with the experiment of Pittman, Jacobs, and Franson ( Fig. 5(a)) 60 , published just a month before the first spatial multiplexing proposal 61 . They showed the storage of heralded photons in a free-space loop conditioned on heralding events, with the purpose of providing photons on pseudo-demand. Pseudo-demand here means that the photons can come at any time, but then be stored for an integer number of the roundtrip times of the memory loop and released at a selected, albeit constrained, time. This was followed by analyses by Kwiat and colleagues to put the idea on solid theoretical footing comparable with the spatial case 62 , and to introduce the production of Fock states with more than one photon by repeated down-conversion 63 .
Compared to spatial multiplexing, temporal multiplexing can claim a big advantage in the scaling of physical resources (as well as the experimental effort in alignment). Since the down-conversion events happen at different times rather than different locations, only one PDC crystal and a single pair of output ports are needed for an arbitrary number of multiplexed modes. This makes scaling to large multiplexing considerably easier experimentally. However, temporal schemes generally have to run at lower repetition rates than spatial ones, both for the fundamental reason of needing many temporal bins to generate photons, and for the practical limitation of slow switching speeds. In the spatial case, any one switch has to operate on average less often than the rate at which photons are produced, but the temporal switch has to switch at least once for each photon.
Relative multiplexing 106,114 of photons, another multiplexing variant, allows multiple photons to synchronize not at a fixed clock time, but at some convenient common time bin, or in the spatial mode which requires least switching. This can improve both temporal and spatial schemes.
Temporal multiplexing can be divided into two main types: those based on networks of delay lines (Fig. 5(b)) and those based on storage cavities or loops (Fig. 5(a)). Delay lines use a fixed number of fixed-length relative delays to align the heralded photon to an output clock, while storage loops use an arbitrary number of roundtrips in a single path. Examples of delay lines include both on-chip delays 115 , and proposals 116 and implementations 67,100,114 using specific fiber lengths. Storage loops have been proposed 117,118 and implemented 88 using fiber loops, and proposed with simple free-space linear cavities 62 , and implemented with free-space Herriot cells 73 , providing about the same delays but allowing the use of freespace Pockels cells which have much lower losses than integrated switches. The best performance so far has come from short free-space ring cavities 74 , with output single-photon probabilities of up to 66.7%, and additionally spectrally-engineered pure photons, giving up to F h + p h ≈ 1.83, strongly outperforming the single-source limit. Values of F h + p h for many multiplexing experiments are given in Table IV. Related to storage loops are delay cavities, which use resonance of the photons for storage and release, rather than polarization or physical switches. These have found use in the continuous-variable community for narrowband photons [119][120][121] , and also in a fullyintegrated proposal in silicon photonics 122 . Using true quantum memories based on Raman scattering or rare-earth ions has also been proposed to enable time multiplexing 123 . Finally, time multiplexing of spin-wave excitations has been shown to increase atom-photon entanglement rates in the field of quantum repeaters 124,125 .
There is an important point in the bookkeeping of the source repetition rate (the rate of attempts to extract a single photon from the source) of spatial versus temporal source multiplexing. In temporal multiplexing the source repetition rate is chosen to be compatible with the maximum switching rate. (Note that the switching rate is not to be confused with the switching transition time, which is typically much faster.) In this case the pump rate is many times faster than the maximum switching rate, which provides for the multiplexing of many pump pulses to one single-photon extraction attempt. In spatial multiplexing, the pump repetition rate equals the source repetition rate (since the multiplexing happens in space rather than time). However, in current spatial multiplexing experiments, the source is pumped much faster than the maximum switching rate, and the maximum switching rate is enforced after the fact by a holdoff time programmed in switching logic after a heralding event. This holdoff time reduces the single-photon rate, but not the source repetition rate, which remains at the full laser repetition rate. This can be likened to making the photons available on pseudo-deomand 60 because they come at random laser pulse times, rather than at pre-chosen ones as in the temporal case. The way around this for spatial multiplexing is to select specific pulses (or pulse-pick) from the pump laser to a rate compatible with the switches. This will increase the brightness at the specified clock times, at the cost of lowering the overall per-second photon rate, which may not be a worthwhile tradeoff until losses are reduced.

C. Spectral source multiplexing
The newest form of multiplexing uses the frequency degree of freedom, spectrally-correlated photon pairs, and spectrally-resolved detection of herald photons. For this form, frequency shifting of the heralded photons is employed to switch photons between modes as in Fig. 6. Spectral multiplexing was pioneered using a linear phase ramp from an integrated phase modulator to shift the frequency of the heralded photon depending on the frequency of the herald, for three discrete frequency bins 68 . The method has been extended to continuous frequency resolution 126 , using a time-of-flight spectrometer 10,127 .
A different approached soon followed: frequency conversion of the heralded photon using four-wave mixing with different pump frequencies, again for three frequency bins of the herald photon 89 . Both approaches to spectral multiplexing will face technical challenges in expanding to many sources, the phase ramp approach due to limited bandwidth and depth of the modulators, and the four-wave mixing approach due to the need for a different pump laser for each frequency bin.

D. Related experiments
One of the main reasons for desiring single photons on demand is the production of multiple photons for interference-based applications 75,101,107,[128][129][130][131] . The fast switching inherent in source multiplexing is additionally useful for synchronizing multiple sources, or demultiplexing a photon stream from a single source to multiple spatial modes. The former has been accomplished with two temporally multiplexed sources, relatively multiplexed to show an increase in the two-photon interference rate of ≈30 times over non-multiplexed sources using bulk components 132 , and ≈2 times compared to non-relative temporal multiplexing using fiber-integrated components 114 . Both the previous experiments also demonstrated high indistinguishability of the multiplexed photons. As proposed in 2014 116 , de-multiplexing has been employed to take subsequently produced photons from a single quantum dot and route them to different spatial modes, using integrated switches 133 , passive (probabilistic) bulk optics 130 , and multiple free-space switches 75 . Finally, higher-order Fock states can be prepared using the ideas of multiplexing through repeated down-conversion 63 , or simply storing higher-order heralded states until required 134 .

V. SUMMARY
It is clear that from the beginning, what were offered as ideal single-photon sources were woefully deficient in many characteristics, particularly for any application requiring more than two photons. And it is also clear that there has been a great flowering of multiplexing methods in terms of the degrees of freedom used and how to best use those degrees given limitations of existing components. One benefit of these efforts to implement multiplexed systems is that it has drawn attention to those component limitations and encouraged improvements. Advances in detector efficiency, dark count rate, and switch loss have all contributed to improved operation of heralded sources and their multiplexing. Also the careful engineering of pair sources to generate pure photon states has taken great strides. All these advances have greatly improved single-photon rates and brightnesses, to the point where experiments requiring handfuls of photons are possible and experiments requiring 15 or more are imaginable.
What is still clear is that while multiplexing is the only path to scalability, components of multiplexed photonic quantum systems also need to be scalable. This can be seen as heralded photon and quantum dot sources both use multiplexing to improve to achieve higher rates in multiphoton experiments. And as a result both must move to integrated components as system size increases. With the emergence of numerous "quantum vendors," the development and advance of efficient integrated components will be interesting to watch. While the current leader in the highest photon experiments uses a single time-multiplexed quantum dot 101 , it remains to be seen if that lead will hold. What ultimately may tip the balance is the relative ease with which indistinguishable photons can made from separate heralded-photon sources versus the individual manipulation required to make two photons from separate dots indistinguishable. This will be a key factor in determining which approach succeeds and in which applications. Another important factor is the convenience of noncryogenic operation.
We have noted the continuing difficulty in comparing results and the often lack of clarity in defining and stating performance parameters. That was apparent to the authors in assembling results into Table III. To help, we have created a table of definitions and measurement protocols that we hope will be a convenience to subsequent authors and help promote clarity. Definitions of terms relevant for single-photon sources.

Deterministic photon source
A photon source that can in principle produce a single photon with high probability at fixed, regular times. Isolated quantum emitters and multiplexed heralded sources seek to reach this goal.
Heralded singlephoton source A source of single photons based on heralding one photon from a photon pair source by detecting its partner photon. This is a probabilistic source that can nevertheless produce highquality photons, albeit at arbitrary times.

Isolated quantum system
A source based on a single isolated quantum system, like a single atom, color centre, or quantum dot, which emits single photons after optical or electronic excitation.

On-demand photon source
A photon source that can in principle produce a single photon with high probability at arbitrary times. Isolated quantum emitters and multiplexed heralded sources seek to reach this goal.

Photon pair source
A source of photons based on nonlinear optics, either three-wave (parametric downconversion) or four-wave mixing, where one or two pump photons decay to a pair of correlated photons. One photon can be directly detected to herald the presence of the other.

Photon-counting (click) detector
A single-photon-sensitive detector that clicks when one or more photons are detected, thus "photon counting" is a misnomer.

Photon number resolving detector
A single-photon-sensitive detector that reports the number of photons detected.

Probabilistic photon source
A source of single photons that emits photons at various times with some probability, typically much lower than unity. Often these are pair sources, where detecting one photon at an arbitrary time heralds the presence of a coincident partner photon. Isolated quantum systems also become probabilistic under realistic losses.

Transform limited
A photon source emits transform limited photons if they are in a single spectral-temporal mode without drift of the central frequency (for isolated quantum systems) or mixedness across multiple Schmidt modes (for pair sources). Such photons are highly indistinguishable and are suitable for multiphoton interference. The efficiency to collect an emitted photon in singlemode fiber from the photon source, including filtering and other losses. In this work we include loss from the multiplexing process here.
For pair sources, measure the Klyshko (heralding) efficiency and divide out the detector efficiency and downstream losses.
Detector efficiency η det The efficiency of a detector to produce an electrical output click when one photon is incident on the detector.
Detector efficiency can be measured using calibrated attenuation of laser light.
Klyshko efficiency 56 , heralding efficiency, total efficiency η tot For pair sources, the raw probability to detect a heralded photon given a herald detection, composed of coupling and detection efficiency, η tot = η coupl η det .
For the signal's Klyshko/ heralding/total efficiency, divide the coincidence rate by the idler singles rate, and vice versa.
Total herald detection efficiency For pair sources, the total detection efficiency (Klyshko efficiency) of the herald photon.
Coincidence rate divided by singles rate in the other channel.

Emission efficiency
The probability (normally for a single-emitter source) that a photon is emitted at the desired time, including the excitation probability.
Measure the count rate then divide out all losses.

Generation efficiency
The number of emitted photon pairs (for pair sources) per second per milliwatt of input pump power, sometimes also normalized per photon bandwidth.
Measure the coincidence (C) and singles rates (S 1 , S 2 ), then find the rate of emitted pairs as before any losses S 1 S 2 /C, then divide by pump power and possibly photon bandwidth.  Given the maximum and minimum coincidence rates (C max , C min ) across the dip, V HOM = (C max − C min )/C max .

Indistinguishability I
The indistinguishability of two photons from separate sources or from the same source at different times quantifies their ability to interfere. Perfect Measure a Hong-Ou-Mandel dip 50 between the two modes. The indistinguishability is equal to the visibility of the dip; I = V HOM .

Symbol Definition
How to measure indistinguishability (I = 1) implies perfect interference visibility.

Multiphoton component
The fraction of the total photon state made up of Fock states of more than one photon. A heuristic rather than precise quantity.
A sense of the multiphoton component can be obtained from the g (2) (0) and similar correlation functions.
Number of multiplexed sources K The number of sources whose outputs are combined through switching in a multiplexed singlephoton source.
Count the number of sources you've built.
Output noise factor 52 The ratio in the output mode of a photon source of the background counts to the total output counts (sum of background counts and true counts).
Use a coincidence timing histogram to identify the true, narrow, photon peak, and the broad background counts. Divide these background counts by the sum of true and background counts.

Schmidt number K S
The effective number of optical modes (spatial and/or temporalspectral) into which the photon pairs are emitted. Sources with Schmidt number greater than 1 produce non-transform-limited photons.
Measure the unheralded g unh 2 0 for either photon of the pair, and find the Schmidt Second-order correlation function g (2) (0) g h 2 0 g unh 2 0 In a single mode, gives information on the multiphoton componenet. Variants specific to pair sources are the heralded g h 2 0 , measured conditioned on a heralding event, and the unheralded g unh 2 0 , measured unconditionally.
For low multiphoton component, the g (2) (0) can be approximated as follows: split the mode on a 50:50 beamsplitter, and place single-photon detectors at each output, recording the singles on each detector S 1 and S 2 and the coincidences C. Then g (2) (0) ≈ S 1 S 2 /C.

Single-photon rate S det
The rate of single-photon detections that have been heralded by a herald event.
Count the number of heralded single-photon detection events per unit time (i.e. coincidences between signal and idler).

Source repetition rate R src
The rate of attempts to extract a single photon from the source.
For spatially-multiplexed sources, generally the same as the pump laser repetition rate. For other cases, this source repetition rate is often much lower than the pump laser repetition rate, so take care in counting the number of attempts made.
Squeezing strength The overall strength of the squeezing interaction or parametric down-conversion; depends on pump power, nonlinearity, crystal geometry, phasematching.
Can be inferred from single-photon count rate in the low-squeezing regime. Related to the squeezing parameter r as λ = tanh r.

FIG. 1.
Types of single-photon sources: (a) isolated quantum systems (e.g. a single particle in an optical cavity, with ground g and excited e states), (b) heralded single-photon sources from photon pairs, (c) multiplexed source (as one example, spatial multiplexing is shown).

FIG. 2.
Experimental source multiplexing (MUX) performance for state-of-the-art single-photon parametric down-conversion (PDC) sources. For comparison quantum dot performance is also shown. Left: The heralded fidelity to a single photon versus the brightness (i.e. the probability of finding just one photon per pump laser pulse). Single-source PDC brightness is bounded by the black line and limited to the grey region. Optimal sources are toward the upper right, and PDC source multiplexing has outperformed the best quantum dots in brightness. Right: The heralded g (2) (0) (= 0 for ideal single photons) versus the heralding probability. For standard PDC, g (2) (0) is bounded by the black line and limited to the grey region. Currently only bulk time multiplexing and quantum dots have achieved g (2) (0) better than this limit.   Spectral source multiplexing. The idler photon, which shares a correlated joint spectrum with the signal photon, is detected with spectral resolution (ν i+ , ν i0 , or ν i− ), then the corresponding signal photon's frequency is shifted to the output spectral band ν s0 . Reprinted  Heralded fidelity and heralding probability for given source and detection parameters. The first row is the base case of a weakly-pumped single source providing 99 % heralded fidelity. The next two rows are for a very reachable number of sources of 10, the latter showing the effect of increasing the squeezing strength by a factor 10. The next row shows what could be achieved with 100 sources. The last two rows show the number of multiplexed sources needed to reach simultaneously 99 % fidelity and heralding probability, without and with dark counts in the heralding detector respectively.  Parameters and metrics for source-multiplexing experiments. In the type of multiplexing, "int." denotes integrated sources, switches, or both. Values that could not be extracted or estimated from published data are marked with '-'. For papers with multiple data points, the values with the lowest and highest pump powers are given. The data from Eggleton13a 87 are for the dual-input device.  Best values for the sum of heralding probability and heralded fidelity from multiplexed sources. All values are inferred from published data, the fidelity from the heralded g (2) (0) or CAR, and the probability from the heralding rate and experiment clock rate. Papers where one of these parameters is not provided are not included in the