# Recent Progress in Multiplexed Single-Photon Sources
## 4 Theory of Multiplexed Single-Photon Sources  
In principle, in an ideal multiplexed system free of losses, increasing the number of multiplexed units and at the same time decreasing the mean number of the generated photon pairs yields single-photon probabilities that tend to one asymptotically. However, in real multiplexed systems, all the optical constituents applied in the multiplexing system and the heralding stage have losses. In order to analyze multiplexed SPSs, theoretical descriptions are required that are capable of taking into account these losses. In Refs. [39; 46; 47; 49], mathematical descriptions of temporally multiplexed SPSs based on storage loops were presented. Theoretical analyses of temporally multiplexed SPSs based on binary-divided delays were outlined in Refs. [40; 44; 51]. Theoretical models describing spatially multiplexed SPSs were developed in Refs. [41; 43; 45]. A full statistical theory describing any type of multiplexed SPSs was introduced in Ref. [42]. That theory can be used to determine the output probability distribution of the photon numbers of the multiplexed system for any type of input photon statistics of the photon pairs generated in the multiplexed units. It assumes threshold, that is, click detectors for heralding in the units and includes all relevant loss mechanisms. Recently, this theory has been extended to include the application of photon-number-resolving detectors capable of realizing any type of detection strategy and the possibility of using different mean numbers of the generated photon pairs \(\lambda_{n}\) in different multiplexed units [50; 52]. Photon-number-resolving detectors were used in recent multiplexed SPS experiments [59; 62; 63; 64; 66; 67; 70]. The significance of the theory is that it can be used to optimize any multiplexed SPS characterized by a certain set of the loss parameters of the system, thus determining the  
Figure 3: Schematic figure of spectral multiplexing. The demultiplexer (DM) filters the idler photons into distinct spectral modes. The frequency shift \(\Delta\
u\) on the signal photons is realized depending on the detected frequency \(\
u_{n}^{\text{det}}\). optimal number of multiplexed units and the optimal mean number of the photon pairs generated in the multiplexed units for which the single-photon probability is maximal. The theory can also be applied to determine the normalized second-order autocorrelation \(g^{(2)}(0)\) characterizing the multiphoton contribution at the output of the multiplexed system. In the following, we summarize this extended statistical theory. Assume that a certain type of multiplexer contains \(N\) multiplexed units and that \(l\) photon pairs are generated by the nonlinear source in the \(n\)th multiplexed unit. Furthermore, assume that only when a photon-number-resolving detector detects a predefined number of photons \(j\) (\(j\leq l\)) will the corresponding input ports of the multiplexer be open. Then, the probability of the arrival of \(i\) signal photons at the output of the multiplexer takes the form  
\[ \begin{split} P_{l}^{(S)}&=\prod_{k=1}^{N}\big{(}1- \sum_{j\in S}P_{k}^{(D)}(j)\big{)}\delta_{i,0}\\ &+\sum_{n=1}^{N}\left[\prod_{k=1}^{n-1}\big{(}1-\sum_{j\in S}P_{k} ^{(D)}(j)\big{)}^{(1-\delta_{1,n})}\sum_{l=i}^{\infty}\sum_{j\in S}P^{(D)}(j|l )P_{n}^{(\lambda_{n})}(l)V_{n}(i|l)\right].\end{split}  \tag{4}\]  
The formula contains various probabilities corresponding to different events. The probability \(P_{n}^{(\lambda_{n})}(l)\) corresponds to the event that the nonlinear source in the \(n\)th multiplexed unit generates \(l\) photon pairs, with the mean number of the generated photon pairs set to \(\lambda_{n}\) in the chosen unit. There are two probability distributions that can be assumed to be related to the SPSs. In the case of multimode spontaneous parametric down-conversion or spontaneous four-wave mixing processes, that is, for weaker spectral filtering [59, 64, 77, 78, 79, 80, 81], Poissonian distribution can be assumed.