# Recent Progress in Multiplexed Single-Photon Sources
## 5 Theoretical Results and Experiments
### Theoretical Results
To obtain the data presented in the table for SPSs based on temporal multiplexers, we have used Equations (2) and (3). The table shows that the differences between maximal single-photon probabilities \(P_{1,\text{max}}\) that can be obtained with SPSs based on these multiplexers are moderate for the \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & \multicolumn{4}{c}{**State-of-the-Art**} & \multicolumn{4}{c}{**Realistic**} \\
**Type** & \(\mathbf{P_{1,max}}\) & \(g^{(2)}\) & \(\mathbf{N_{\text{opt}}}\) & \(\mathbf{\lambda_{\text{opt}}}\) & \(\mathbf{P_{1,max}}\) & \(g^{(2)}\) & \(\mathbf{N_{\text{opt}}}\) & \(\mathbf{\lambda_{\text{opt}}}\) \\ \hline Spatial ASYM & 0.905 & 0.061 & 31 & 0.453 & 0.739 & 0.166 & 23 & 0.757 \\ Spatial SOBTM & 0.916 & 0.041 & 42 & 0.259 & 0.763 & 0.128 & 29 & 0.500 \\ Spatial OMAXV & 0.913 & 0.039 & 39 & 0.249 & 0.749 & 0.128 & 21 & 0.502 \\ Temporal binary-divided & 0.867 & 0.063 & 32 & 0.472 & 0.679 & 0.156 & 16 & 0.685 \\ Temporal loop & 0.881 & 0.076 & 24 & 0.637 & 0.722 & 0.174 & 20 & 0.809 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the performances of SPSs based on various temporal and spatial multiplexer structures for two sets of loss parameters. State-of-the-art parameters: transmission coefficients \(V_{r}=0.99\), \(V_{t}=0.985\), detector efficiency \(V_{D}=0.95\), general transmission coefficient \(V_{b}=0.98\). Realistic parameters: transmission coefficients \(V_{r}=0.95\), \(V_{t}=0.95\), detector efficiency \(V_{D}=0.9\), general transmission coefficient \(V_{b}=0.95\). In both cases for temporal multiplexers, the transmission coefficient of one-time-bin delay is \(V_{d}=0.99\), and the loss values characterizing a switch is in the two cases are \(V_{s}=0.99\) and \(V_{s}=0.95\), respectively. Figure 4: Single-photon probability \(P_{1}\) as a function of the input mean photon number \(\lambda\) for SPSs based on five different types of multiplexers: asymmetric multiplexer (ASYM), minimum-based, maximum-logic output-extended incomplete binary tree multiplexers (OMAXVs), stepwise-optimized binary tree multiplexers (SOBTMs), temporal multiplexers based on binary-divided delays (binary-divided) and on storage loops (loop), for the number of multiplexed units \(N=10\), the transmission coefficients \(V_{r}=0.95\) and \(V_{t}=0.95\), the general transmission coefficient \(V_{b}=0.95\), the detector efficiency \(V_{D}=0.9\), and, in the case of temporal multiplexers, the transmission coefficient of one-time-bin delay \(V_{d}=0.99\), as well as when the loss characterizing a switch is \(V_{s}=0.95\). parameter sets used. However, the required optimal numbers of multiplexed units \(N_{\text{opt}}\) and the corresponding values of the normalized second-order autocorrelation function \(g^{(2)}\), as well as the optimal input mean photon numbers \(\lambda_{\text{opt}}\), are considerably different. The highest single-photon probability that can be achieved both for state-of-the-art parameters and for realistic parameter sets can be obtained with SPSs based on SOBTMs. Generally, the relation between the performances of SPSs based on different multiplexing methods depends on the chosen sets of the loss parameters. A detailed comparison of SPSs based on OMAXVs and asymmetric multiplexers in this respect can be found in Ref. [54]. That analysis shows that, though SPSs based on OMAXVs surpass those based on asymmetric multiplexers in the present examples, in the case of multiplexers built of highly asymmetric photon routers, SPSs based on asymmetric multiplexers usually show better performance than those based on OMAXVs. The optimization of SPSs usually means maximizing the single-photon probability. Note, however, that the highest single-photon probabilities obtained by the optimization are not necessarily accompanied by low values of the second-order autocorrelation function. In Ref.