# Recent Progress in Multiplexed Single-Photon Sources
## 5 Theoretical Results and Experiments  
### Theoretical Results  
The performance outcomes of various SPSs have been discussed in Refs. [39; 40; 41; 43; 44; 45; 46; 47; 49; 51] applying different theoretical models. Using the general theoretical description outlined in the previous section, the properties and characteristics of SPSs based on different multiplexing methods have been analyzed in Refs. [42; 48; 50; 52; 53; 54; 55; 56; 57]. According to these analyses, it can be stated for all multiplexed SPSs that the performance of the system improves for lower system losses, that is, for higher values of the various transmission coefficients and efficiencies. Also, the single-photon probability of multiplexed sources can be increased by using photon-number-resolving detectors [50]. We assume that higher values of the detector efficiency of such detectors lead to higher single-photon probabilities and, simultaneously, lower values of the multiphoton noise. Assuming thermal statistics for the distribution of the generated photon pairs required for the indistinguishability of the generated photons, the resulting output single-photon probabilities are lower than those for Poissonian statistics [53]. Another common characteristic of multiplexed SPSs is that the single-photon probability plotted against the input mean photon number exhibits a single maximum for any value of the number of multiplexed units \(N\). This is shown in Figure 4 for SPSs based on five different types of multiplexers and for \(N=10\) multiplexed units. The maximum shifts toward smaller values of the input mean photon number \(\lambda\) with increasing values of the number of multiplexed units \(N\)[42; 50; 57]. The physical reason behind these observations is that when the input mean photon number \(\lambda\) is low, the likelihood of not obtaining any photons at the output of the system is high, while high values of \(\lambda\) lead to increased multiphoton contribution. The shifting of the maximum can be explained by the fact that the assumption of higher values of the number of multiplexed units \(N\) allows for the use of lower values of the input mean photon number \(\lambda\) to retain the probability of successful heralding high in the whole system. Knowing the maxima of the \(P_{1}(\lambda)\) functions for all reasonable values of \(N\), one can determine the optimal value \(N_{\text{opt}}\) together with the optimal values of \(\lambda_{\text{opt}}\) for which the single-photon probability is the highest, as described in the previous section. This value is called the maximal single-photon probability \(P_{1,\text{max}}\) that can be achieved with the considered SPS. Table 1 shows examples for the maximal single-photon probability \(P_{1,\text{max}}\), the corresponding values of the normalized second-order autocorrelation function \(g^{(2)}\), and the optimal values of the number of multiplexed units \(N_{\text{opt}}\), as well as the input mean photon number \(\lambda_{\text{opt}}\) for SPSs based on five different types of multiplexers: an asymmetric multiplexer (ASYM) [52], stepwise-optimized binary tree multiplexers (SOBTMs) [55], minimum-based, maximum-logic output-extended incomplete binary tree multiplexers (OMAXVs) [54], and temporal multiplexers based on binary-divided delays (binary-divided) and on storage loops (loop) for two sets of loss parameters. With the aim of comparing the performance of the various spatially and temporally multiplexed SPSs, we have calculated all the values shown in the table for the same sets of parameters. The first parameter set characterizes  
state-of-the-art optical devices and elements [70; 119; 128]. The second set comprises parameters that can expectedly be realized in multiplexed SPS experiments. Exact values of the parameters are listed in the table caption. Note that corresponding calculation results for temporally multiplexed SPSs can be found for different parameter sets in Refs. [42; 50].