# Recent Progress in Multiplexed Single-Photon Sources
## 2 Heralded Single-Photon Sources  
A heralded single-photon source (heralded SPS) is composed of a source of photon pairs based on some nonlinear processes such as spontaneous parametric down-conversion [12; 13; 14; 16; 17; 18; 19; 20; 21; 22; 23; 24; 26; 27; 29; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76] or spontaneous four-wave mixing [15; 25; 28; 30; 31; 32; 36], and a heralding detector. The twin photons generated in the process are termed idler and signal. The presence of the signal photon is heralded by the detection of the idler photon; hence, they are also called heralded and heralding photons, respectively. Note that most applications require the periodicity of the SPS. This can be ensured by applying pulsed pumping of the photon pair source. Photon pair generation is an inherently probabilistic process; hence, the possible values of the number of generated photon pairs can be described by a probability distribution. In the case of multimode nonlinear processes having weaker spectral filtering, a Poissonian distribution can be assumed. For single-mode processes characterized by stronger filtering, the distribution describing the number of photon pairs is thermal [59; 64; 77; 78; 79; 80; 81]. In the latter case, the heralded SPS can yield almost identical single photons required in various experiments and applications. Assuming that the mean number of generated photon pairs is \(\lambda=1\), in the case of thermal distribution, the probability of having one generated photon pair is \(P_{1}=0.25\), while for a Poissonian distribution, this probability is \(P_{1}\approx 0.367\). These are the highest probabilities of having a single-photon pair from a photon pair generation process. Hence, even applying an ideal detector for heralding, it is not possible to obtain higher single-photon probabilities than these values from a heralded SPS [82]. Note that the single-photon probability, that is, the single-photon yield per pump pulse (i.e., at a clock cycle), is generally termed as brightness for SPSs [9]. Another quantity characterizing the performance of heralded SPSs is the heralded fidelity \(F_{h}\), which is the fidelity of the heralded photon to the one-photon state \(|1\rangle\)[9; 82]. The multiphoton components of the output of the heralded SPS can also be quantified by the normalized second-order correlation \(g^{(2)}(0)\). Applying photon-number-resolving detectors for heralding can reduce the multiphoton noise, that is, it can increase the fidelity of the heralded photons compared to the case when a single-click (that is, threshold) detector is applied [14; 22; 34; 35; 82]. One possible realization of photon-number-resolving detectors is based on the use of click detectors in combination with either a temporal or spatial multiplexing scheme [83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97]. The performance and limitations of such schemes in the presence of various deficiencies and losses have been studied in Refs. [90; 92; 93; 95; 96]. Note that superconducting nanowire detectors widely used in quantum optical experiments are generally single-click ones without photon number resolution. However, the multi-pixel ones show this capability [98; 99]. Spatial multiplexing of multiple nanowires can also result in a photon-number-resolving capability [100; 101; 102; 103; 104; 105]. Superconducting nanowire detectors are noted for their high quantum efficiency, excellent timing resolution, low dark counts, and sensitivity across a broad spectral range. A lot of effort has also been paid to develop high-efficiency inherent photon-number-resolving detectors. The best known realizations of such devices are fast-gated avalanche photodiodes [106; 107], quantum dot field-effect transistors [108; 109], superconducting nanowire detectors [110; 111], and transition edge sensors [111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121].