# Recent Progress in Multiplexed Single-Photon Sources
## 3 Multiplexed Single-Photon Sources
### Temporal Multiplexing  
Temporal multiplexing was first proposed and demonstrated experimentally in 2002 [37]. In this type of multiplexing, a non-deterministic photon generation process of a heralded source is repeated in time with period \(T\). The planned time period \(T_{S}\) of the multiplexed source must be chosen as an integer multiple of the time period \(T\), that is, \(T_{S}=NT\). Within a period \(T_{S}\), if the heralding event occurs at the \(n\)th time bin (\(n=1,\ldots,N\)), the corresponding signal photon is delayed so that it reaches the output of the multiplexer at the end of the period \(T_{S}\). Accordingly, in the case of temporal multiplexing, the temporal modes of the same heralded source are multiplexed. The signal photons potentially arriving from the heralded source during the observation time \(T_{S}\) can be reasonably delayed by at least \(T_{S}\), which makes it possible to choose the photon from the time bin closer to the end of the predefined time period \(T_{S}\). At this choice, the signal photon experiences lower loss during time multiplexing. Note, however, that the switching realized by the control unit requiresadditional time; therefore, the realized time period \(T_{R}\) of the temporally multiplexed SPS is greater than the planned time period, that is, \(T_{R}>T_{\rm S}\). The time bins and the delay lines together play the role of multiplexed unit in the case of temporally multiplexed SPSs. Two setups realizing temporal multiplexing were proposed in the literature. One setup utilizes storage loop (or cavity) delays [37; 39; 46; 47; 49] with a fixed delay time \(T\). The scheme of an SPS based on such a setup is shown in Figure 2a. In this system, a heralded photon goes around the loop as many times as needed to be delayed to arrive at the end of the time period \(T_{\rm S}\). This system uses a single \(2\times 2\) switch that routes the photon into or out of the delay loop, or it simply forwards the photon to its output if no delay is needed. The various experimental realizations of this type of multiplexer are summarized in Ref. [9]. The transmission coefficient \(V_{n}\) assigned to the \(n\)th time bin can be written as  
\[V_{n}=V_{b}V_{d}^{N-n}V_{s}^{N+1-n}, \tag{2}\]  
where \(V_{b}\) is a general transmission coefficient, \(V_{s}\) denotes the loss characterizing the switch, \(V_{d}\) is the transmission coefficient of one-time-bin delay (that is, the propagation loss), and \(N\) is the total number of time bins. The other temporal multiplexer proposed in the literature is based on binary-divided delays [42; 51]. The scheme of an SPS based on this type of multiplexer is presented in Figure 2b. It is based on several delay lines the lengths of which are power-of-two multiples of that of a unit-length delay line that realizes the delay \(T\). Assume that the amount of delay required for the chosen signal photon to arrive at the end of the predefined period \(T_{\rm S}\) is \(nT\). The integer multiplicative factor \(n\) can be represented by a binary number. The lowest (rightmost) digit of the binary number corresponds to the shortest delay line realizing \(T\), while the highest (leftmost) digit belongs to the delay line realizing the longest delay. The signal photon chosen to be routed through the multiplexer is directed into those delay lines of the multiplexer for which the corresponding digit in the binary number \(n\) is \(1\). In this case, the signal photon goes through all switches exactly once. Then, the transmission coefficient \(V_{n}\) assigned to the \(n\)th time bin can be written as  
\[V_{n}=V_{b}V_{d}^{N-n}V_{s}^{m+1}, \tag{3}\]  
where \(m=\lceil\log_{2}N\rceil\) and \(\lceil x\rceil\) represent the ceiling function that maps its argument \(x\) to the smallest integer larger than or equal to \(x\). It is worth mentioning here that the combination of spatial and temporal multiplexing was also proposed in the literature [23; 48; 65; 122; 123]. SPSs based on combined multiplexing can outperform those based on any of the multiplexing schemes [48].