# Recent Progress in Multiplexed Single-Photon Sources
## 5 Theoretical Results and Experiments
### Theoretical Results
[57], a method was proposed to reduce the multiphoton contribution of a multiplexed SPS with a moderate reduction in the single-photon probability compared to the maximal one obtained from the optimization above. In that paper, it was shown that the relation between the normalized second-order autocorrelation function \(g^{(2)}\) and the input mean photon number \(\lambda\) is practically linear. Hence, the reduction of \(\lambda\) results in a decrease in \(g^{(2)}\), but this also leads to a reduction in the single-photon probability \(P_{1}\). The solution to this problem is the reoptimization of the system, that is, determining the optimal number of multiplexed units and optimal input mean photon numbers for the prescribed lower value of the normalized second-order autocorrelation function for which the reduction of the single-photon probability is minimal. The theoretical analyses also showed that the optimal system sizes, that is, the optimal numbers of multiplexed units, are higher, and the optimal input mean photon numbers are lower for lower losses of the multiplexer [52; 53]. In contrast, for higher detector efficiencies \(V_{\text{D}}\), the values of the optimal input mean photon number \(\lambda_{\text{opt}}\) are higher, while the corresponding values of the optimal number of multiplexed units \(N_{\text{opt}}\) are lower. Obviously, in the case of higher losses of the multiplexer, keeping the system sizes at high values increases the probability that the photon vanishes in the multiplexer. Also, applying a better photon-number-resolving detector for heralding can better suppress the multiphoton contribution in the signal entering the multiplexer. The performance of multiplexed SPSs can be further improved by using different input mean photon numbers in the different multiplexed units [52; 56; 41]. This method takes into account that different arms of the multiplexer generally introduce different losses. In Ref. [41], a definite functional dependence for the different input mean photon numbers was proposed. The idea of using different input mean photon numbers was inserted into the general theory of multiplexed sources in Ref. [52]. The theory presented in the previous section includes this possibility. The optimal values of the different input mean photon numbers can be determined by using standard multivariate optimization methods such as the genetic algorithm. The theory has been applied for SPSs based on asymmetric multiplexers [52] and for those based on OMAXVs [56]. It was shown that the method applying a full optimization for all the individual input mean photon numbers surpasses the one assuming a functional dependence for the different input mean photon numbers. The application of photon-number-resolving detectors in the multiplexed units has opened the possibility to optimize multiplexed SPSs in which a heralding strategy can be chosen based on the actual number of detected photons [50; 52; 53]. The theory shown in Section 4 also includes this possibility. The analyses showed that for multiplexers with higher losses, it can be advantageous to choose a strategy in which two or more photons also induce a heralding event [50; 52; 53]. The theoretical models described above and proposed in the literature were also used to analyze multiplexed SPSs with suboptimal system sizes, that is, for the number of multiplexed units less than what would be optimal for a given set of loss parameters [52; 53; 54; 55]. The relevance of analyzing SPSs with suboptimal system sizes is that using relatively low, suboptimal number of optical elements is the typical situation in experiments realized so far [58; 59; 61; 63; 64]. Next, we present results that show the typical behavior of multiplexed SPSs for suboptimal system sizes. Figure 5 shows the characteristics of SPSs based on the multiplexers presented in Table 1 for suboptimal values of the number of multiplexed units.