# Recent Progress in Multiplexed Single-Photon Sources
## 4 Theory of Multiplexed Single-Photon Sources
Assuming that such a set allows for using an optional detection strategy that cannot be realized without a photon-number-resolving detector, the elements of the set \(S\) are positive integers less than or equal to a value \(J_{b}\) characterizing the resolving capability of the photon-number-resolving detector: this number is the highest number of photons that can still be distinguished by the detector. Such a detector can also be used as a threshold detector by assuming the set \(S\) to contain all positive integers. There are two terms in Equation (4). The first one quantifies the likelihood of detecting an unexpected number of photons, that is, the number of photons registered in the multiplexed unit is outside the set \(S\). Considering that for an unexpected number of photons detected in a multiplexed unit the corresponding input port of the multiplexer does not open, this first term contributes to the probability \(P_{0}^{(S)}\) of obtaining no outgoing photons from the multiplexer. Consequently, all events corresponding to the detection of an expected number of photons in a multiplexed unit are described by the second term. Here, it is assumed that \(l\) photons enter the multiplexer from the \(n\)th unit subsequent to the heralding, but the output is reached by only \(i\) of these photons owing to the losses present in the multiplexer. In this term, the first factor expresses the fact that no expected number of photons is detected in the first \(n-1\) multiplexed units. The theory can also be used to characterize the multiphoton contribution at the output of the multiplexed system. This is generally quantified by the normalized second-order autocorrelation function [23]  
\[g^{(2)}(0)=\frac{\sum\limits_{i=2}^{\infty}P_{i}^{(S)}i(i-1)}{\left(\sum\limits_ {i=1}^{\infty}P_{i}^{(S)}i\right)^{2}} \tag{10}\]  
that measures the multiphoton components of the output state relative to the single-photon component. In this formula, the probabilities \(P_{i}^{(S)}\) (\(i=1,2,\dots\)) of obtaining \(i\) photons at the output can be calculated by using Equation (4). Next, we summarize how the theoretical description can be used to optimize a given multiplexed system. As it was stated before, Equation (4) can be used to determine the probability of getting a single signal photon at the output of the multiplexer. After fixing the values of the various losses characterizing the multiplexed SPS such as the transmission coefficients \(V_{r}\) and \(V_{t}\) of the individual routers building up the spatial multiplexer or, in the case of temporal multiplexing, the corresponding quantity \(V_{S}\) characterizing the switch, the detector efficiency \(V_{D}\), and the general transmission coefficient \(V_{b}\), Equation (4) still contains two free parameters, or variables, namely, the number of multiplexed units \(N\) and the input mean photon number \(\lambda\). Note that the formula allows for the use of different input mean photon numbers \(\lambda_{n}\) in the various multiplexed units, which means that the number of variables in Equation (4) is \(N+1\), namely, the input mean photon numbers \(\lambda_{n}\), \(n=1,\dots,N\) and the number of multiplexed units \(N\). Then, the optimization of a given multiplexed system means determining the optimal values of these variables, that is, the optimal number of multiplexed units \(N_{\text{opt}}\) and the optimal input mean photon number or numbers \(\lambda_{\text{opt}}\) or \(\lambda_{n,\text{opt}}\), respectively, for which the value of the single-photon probability \(P_{1,\text{max}}\) is maximal. This can be accomplished in two steps. First, the number of multiplexed units \(N\) is fixed, and the input mean photon number \(\lambda\) or, generally, photon numbers \(\lambda_{n}\) have to be determined for which the single-photon probability \(P_{1}\) is maximal.