# Recent Progress in Multiplexed Single-Photon Sources
## 4 Theory of Multiplexed Single-Photon Sources
In this case,  
\[P_{n}^{(\lambda_{n})}(l)=\frac{\lambda_{n}^{l}e^{-\lambda_{n}}}{l!}, \tag{5}\]  
where \(\lambda_{n}\) represents the mean number of photon pairs originating from the nonlinear source in the multiplexed units, so it is the input of the heralding process. In the following, the term input mean photon number is used for this quantity. The other distribution assumed in the literature is thermal distribution that is valid for the single mode, i.e., spectrally narrow-filtered spontaneous parametric down-conversion or spontaneous four-wave mixing processes. The probability of having \(n\) photon pairs in the case of thermal distribution can be expressed as  
\[P_{n}^{(\lambda_{n})}(l)=\frac{\lambda_{n}^{l}}{(1+\lambda_{n})^{1+l}}. \tag{6}\]  
Multiplexed SPSs in which thermal distribution is assumed can yield highly indistinguishable single photons required for the realization of various optical quantum information experiments. We note that the entangled two-mode output state of these systems determines the actual photon distribution of the photon pair generation [78]. The conditional probability \(P^{(D)}(j|l)\) describes the event that \(j\) photons are registered provided that \(l\) photons fall onto the detector (\(j\leq l\)). Characterizing the detector by the detector efficiency \(V_{D}\), this probability can be written as  
\[P^{(D)}(j|l)=\binom{l}{j}V_{D}^{j}(1-V_{D})^{l-j}. \tag{7}\]Another detection-related quantity is the probability \(P_{n}^{(D)}(j)\) corresponding to the event that exactly \(j\) photons are detected in the \(n\)th multiplexed unit. Using the above two probabilities \(P^{(D)}(j|l)\) and \(P_{n}^{(\lambda_{n})}(l)\), one can write  
\[P_{n}^{(D)}(j)=\sum_{l=j}^{\infty}P^{(D)}(j|l)P_{n}^{(\lambda_{n})}(l). \tag{8}\]  
Refs. [42; 50; 52] introduce the three versions of the statistical description of multiplexed SPSs, where no other detector imperfections were taken into consideration but the finite detector efficiency described by \(V_{D}\). It was also stated that omitting other imperfections poses no significant limitation against the realistic nature of the model [50]. We note that the effect of the dark counts on the performance of temporally multiplexed SPSs was analyzed in Ref. [127]. Lastly, \(V_{n}(i|l)\) is the conditional probability quantifying the likelihood of the event that \(i\) photons leave the multiplexer provided that \(l\) signal photons arrive to the system from the \(n\)th multiplexed unit. It can be calculated as  
\[V_{n}(i|l)=\binom{l}{i}V_{n}^{i}(1-V_{n})^{l-i}, \tag{9}\]  
where \(V_{n}\) denotes the total transmission coefficient characterizing the losses of the \(n\)th arm of the multiplexer. In the case of heralding events occurring in multiple multiplexed units, the preferred unit in Equation (4) is the one the sequential number \(n\) of which is the smallest. The unit controlling the operation of the switches of the multiplexer should implement this priority logic. Hence, the total transmission coefficients \(V_{n}\) describing the losses experienced by the photons in the arms of the multiplexer are to be sorted into a descending order for a given set of the loss parameters. Such a renumbering ensures that the multiplexer arm characterized by the highest \(V_{n}\) that corresponds to the smallest loss is favored by the priority logic in the case of multiple heralding events occurring in the system [50; 53]. In this way, the probability that the input signal photon vanishes in the multiplexer is reduced; hence, the output single-photon probability is higher. The set \(S\) contains the number of photons detected in a multiplexed unit that triggers the opening of the corresponding input port of the multiplexer to let the generated signal photons into it.