# Recent Progress in Multiplexed Single-Photon Sources
## 3 Multiplexed Single-Photon Sources
### Spatial Multiplexing  
The idea of spatial multiplexing proposed first in Ref. [38] relies on the parallel use of a number of pulsed heralded SPSs. Photon pairs needed in the individual heralded sources can be created in separate nonlinear processes or in physically separate spatial modes of a single-photon pair source. Following the successful heralding of a heralding photon in one of the heralded sources, a spatial multiplexer, that is, an \(N\times 1\) switch, routes the twin of the photon to a single output. Such a switch can be realized by using binary photon routers. A binary photon router is actually a \(2\times 2\) switch, that is, it has two input and two output ports. One of the output ports is used as the actual output of the router, while the other output port, the so-called noise port of the router, is used to remove unnecessary photons from the multiplexer. Photons arriving at any of the input ports of the router experience losses while propagating from the input port to the output port of the router. The losses assigned to the two input ports of the router are described by transmission coefficients denoted by \(V_{r}\) and \(V_{t}\), as were introduced in Refs. [48; 53]. In general, \(V_{r}\
eq V_{t}\), meaning the routers are asymmetric. There are various types of optical switching devices that can act as photon routers. In multiplexed single-photon experiments, bulk electro-optic polarization rotating switches [37; 39; 58; 62; 64], integrated opto-ceramic switches [59; 60; 67] or electro-optic switches [61; 65] were applied. Photon routers can be used to build binary tree spatial multiplexers with various structures. The operation of the multiplexer is governed by a control unit, the operation of which is triggered by the heralding detectors. To guarantee the time needed to realize the switching, the signal photons are generally retarded by delay lines to delay their arrival to the input of the multiplexer. To facilitate the unified discussion of multiplexed SPSs, it is useful to apply the concept of multiplexed units comprising a source of heralded photons and an optional delay line. In the case of multiple heralding events in different multiplexed units, the control unit decides which heralded photons are to be forwarded to the output. Though the structure of the multiplexer can be arbitrary, special attention has been given to certain systematic building logics. The following terms can be used for describing a multiplexer. The arms of the multiplexer are formed by the connections between the photon routers. A branch is a subtree of the multiplexer. A level of a multiplexer is formed by photon routers at an equal distance from the output of the multiplexer. As real multiplexers always have losses, a photon entering the multiplexer vanishes with a certain probability. The losses of the arms of the multiplexer can be described by transmission coefficients denoted by \(V_{n}\). Note that in the case of multiple detection events in different multiplexedunits, the control unit should choose the arm with the smallest loss, that is, with the highest transmission coefficient \(V_{n}\). One type of spatial multiplexer is called symmetric, log-tree, or complete binary-tree multiplexer. Such a multiplexer is characterized by a power-of-two number of inputs (that is equal to the number of multiplexed units) and photon routers arranged so that all the levels of the binary tree are complete. Such multiplexers are discussed in Refs. [58; 59; 61; 63; 64]. The formula describing the transmission coefficients of a symmetric multiplexer can be found in Ref. [48]. Another remarkable group of spatial multiplexers are asymmetric multiplexers [41; 45; 52]. Figure 1a shows the schematic figure of an SPS based on an asymmetric multiplexer. During the building of such a multiplexer, the output of the next photon router is connected to one of the inputs of the previous router; thus, the consecutive photon routers form a chain-like structure.