1. Introduction

The development of single-photon sources (SPSs) is still in the focus of research due to their substantial role in the effective realization of a number of experiments in the fields of quantum information processing and photonic quantum technology [1,2]. Ideal SPSs yield indistinguishable single photons in near-perfect spatial modes with known polarization on demand. Multiplexed single-photon sources are promising candidates to meet all these criteria. Such sources are based on heralded SPSs [3–8] in which the detection of one member of a correlated photon pair generated in spontaneous four-wave mixing or spontaneous parametric down-conversion heralds the presence of its twin photon. Using pulsed pumping in these nonlinear processes can ensure the emission of photon pairs to be on demand. As the aforementioned nonlinear processes possess an inherent probabilistic nature, beside the generation of single pairs of photons the occurrence of multipair events has a nonzero probability. This multiphoton noise can be reduced by using single-photon detectors with photon-number-resolving capabilities for heralding [6,9–14] or by decreasing the mean photon number of the generated photon pairs that can be easily realized by adjusting appropriately the power of the pumping beam of the nonlinear process. However, the reduction of the mean photon number implies the decrease of the probability of successful heralding that can be compensated by multiplexing several sources of heralded photons. Multiplexing can be accomplished by a suitable switching system that reroutes heralded photons generated in particular multiplexed units realized in space or in time to a single output mode. Various schemes have been proposed for spatial [9,10,13,15–23] and temporal multiplexing [21,24–35], and some of them have been successfully implemented in experiments [10,17,18,20,22,28,30,31,33].

In an ideal lossless multiplexed system the decrease of the mean photon number of the photon pairs and the simultaneous increase of the number of multiplexed units in principle can lead to a perfect SPS in which the single-photon probability approaches one asymptotically. However, the presence of various losses in real multiplexed systems results in the degradation of the performance of the multiplexed SPSs [9,19]. In real systems the output single-photon probability can be maximized by determining the optimal number of multiplexed units and the mean number of photon pairs generated in the units. Such an optimization can be performed by applying the full statistical frameworks developed for the description of any kind of multiplexed SPSs [11–13,21,23]. These frameworks are capable of taking into account all relevant loss mechanisms and the application of various photon detectors at the heralding stage. Note that these theories can be also applied to optimize multiplexed SPSs with suboptimal system sizes that is a typical situation in current experiments.

The analysis of multiplexed SPSs showed that the achievable single-photon probabilities are high while the multiphoton contribution can be kept low when these systems are built by using state-of-the-art optical elements with minimal losses [11–13]. The reported single-photon probabilities are considerably higher than the ones communicated for quantum dot-based SPSs which seems to be the most promising isolated quantum emitter-based systems [36,37]. Another inherent advantage of multiplexed SPSs is that indistinguishability of the generated photons can be ensured more easily.

One important goal of the researches on multiplexed SPSs is finding novel multiplexing schemes that can further improve the performance of SPSs. In the current research corresponding to this line we address spatial multiplexing. The idea of this kind of multiplexing is based on using several individual pulsed heralded SPSs in parallel. Photon pairs needed in these heralded sources can be generated in physically separate nonlinear processes or in separate spatial modes of a single source of photon pairs. After a successful heralding event in one of the heralded sources, a set of binary photon routers forming a spatial multiplexer is used to reroute the corresponding heralded signal photon to a single output. Binary photon routers can be realized either in bulk optics using e.g. polarization beam splitters and Pockels cells, or in integrated optics. Routers are generally asymmetric, that is, they can be characterized with two transmission coefficients assigned to their two input ports. Photon routers can be used to build spatial multiplexers with various geometries. The three main types of multiplexer geometries considered thus far in the literature are symmetric (complete binary-tree) multiplexers, asymmetric (chain-like) structures, and incomplete binary-tree multiplexers. While the number of multiplexed units for symmetric multiplexers is always a power of two, this number can be arbitrary both for asymmetric multiplexers and for incomplete binary-tree multiplexers. Experimental realizations of spatial multiplexing have been reported up to four multiplexed units by using spontaneous parametric down-conversion in bulk crystals [10,17] and waveguides [20], and by using spontaneous four-wave mixing up to two multiplexed units in photonic crystal fibers [18,22]. In all these experiments symmetric multiplexers were applied.

In Ref. [13] we showed that using incomplete binary-tree multiplexers can lead to higher single-photon probabilities than that can be achieved with symmetric multiplexers. The analysis also revealed that the application of incomplete binary-tree multiplexers extended on their outputs (termed as output-extended ones) leads to higher performances than applying incomplete binary-tree multiplexers extended on their inputs (termed as input-extended ones). In Ref. [13] the structure of the output-extended incomplete binary-tree multiplexers (OIBTMs) was defined following a simple geometric logic for the construction. Obviously, the performance of any spatial multiplexing system is strongly affected by the losses along the various arms of the multiplexer. Therefore, it is worth taking the losses into account in the process of constructing the multiplexer.

In the present paper we consider four novel types of output-extended incomplete binary-tree multiplexers built with asymmetric routers. The rule of the construction of these multiplexers is determined by the losses of those multiplexer arms where the next photon router can be added to. We select the multiplexer the application of which yields a SPS with the highest single-photon probability in the considered parameter range. We analyze this system in detail by using the statistical theory introduced in Ref. [11]. We compare the chosen system with SPSs based on asymmetric multiplexers and we identify the parameter range for which SPSs based on the selected multiplexer outperform those based on asymmetric multiplexers.

2. Output-extended incomplete binary-tree multiplexers constructed following transmission-based logics

Spatial multiplexing schemes are based on multiplexing devices formed by a set of binary photon routers (PRs) that convey photons generated in a set of multiplexed units (MUs) to a single output. Each multiplexed unit contains a nonlinear photon pair source, a detector for detecting the idler photons of the photon pairs, and an optional delay line placed into the path of the signal photons. As most applications pose the requirement of periodicity on the resulting single-photon output, one can use pulsed pumping of the source generating the photon pairs to ensure this property. The detection of the idler photons of the photon pairs triggers the coupling of the corresponding signal photons into the multiplexer. In this paper we assume that the multiplexed units contain single-photon detectors with photon-number-resolving capabilities [38–44]. The delay lines are used to enable the operation of the logic controlling the routers by introducing a sufficiently long delay into the arrival time of the signal photon. Binary PRs have two input and one output ports. Such routers are generally asymmetric, that is, they can be characterized by two different photon losses belonging to the two input ports. In Refs. [13,23] we presented a possible bulk-optical realization of such an asymmetric PR. In those papers, the efficiencies characterizing the two input ports of the PR are termed as transmission and reflection efficiencies and they are denoted by $V_t$ and $V_r$, respectively. We use these terms to denote the general inputs of the PRs in the present paper.

In OIBTMs, an initially $m$-level symmetric multiplexer is extended step-by-step toward another, $m+1$-level symmetric multiplexer by adding new PRs and MUs to the system at the outputs of the initial symmetric system. In Ref. [13] novel routers were added to the multiplexers following a special geometric rule. However, the performance of the multiplexer strongly depends on the losses of the particular arms of the multiplexer characterized by the total transmission coefficients $V_n$. Hence, in the present paper we keep the systematic way of building OIBTMs but we apply a transmission-based logic for the step-by-step construction instead of the geometric rule. We propose four novel types of OIBTM where the rule of placement of the novel PRs is determined by the magnitudes of the total transmission coefficients of the optional connection points in the level under construction.

Figure 1 shows the schematic figure of the proposed OIBTMs. We assume that all PRs in the multiplexer are identical. All these schemes are based on initially complete $m$-level binary-tree multiplexers (CBTMs) indicated by light red background in the figure (in this case, $m=4$). The output of this multiplexer is coupled into one of the inputs with transmission coefficient $V_{\mathrm {1B}}$ of a newly added router hereinafter referred to as base router. The transmission coefficient of the other input of the base router is denoted by $V_{\mathrm {2B}}$. In Fig. 1 such a base router is denoted by PR$_{16}$, but for lower numbers of constituent routers PR$_8$, PR$_4$, and PR$_2$ play the same role. The left and right inputs of the other routers are denoted by $V_1$ and $V_2$, respectively. The output of the next router to be added to the multiplexer is conveyed into the other input of the base router. In our figure such a router is denoted by PR$_{17}$, and the part of the multiplexer containing this novel router and those built below that is called the incomplete branch. Then the subsequent routers can be added to the incomplete branch of the multiplexer one by one until the given level is completed. The numbering of the PRs reflect the order in which they are added to the multiplexer. As the efficiencies $V_r$ and $V_t$ characterizing the two inputs of any router are different in general, choosing the higher or lower efficiency for the left or right input of the routers can lead to considerably different results. Therefore, instead of simply assigning $V_t$ and $V_r$ to the left and right inputs of a router, we introduce the notations for the larger and smaller of these quantities as $V_{\max }=\max (V_r,V_t)$ and $V_{\min }=\min (V_r,V_t)$. The cases when the output of the initial CBTM is coupled into the input of the novel base router with the higher or lower transmission efficiency will be referred to as maximum-based and minimum-based setups, respectively. Also, the next router on the level under construction in the incomplete branch can be added to the arm with the highest or lowest total transmission coefficient $V_n$ of the optional connection points in the level under construction in the incomplete branch. These cases will be referred to as maximum-logic or minimum-logic building strategies, respectively. Accordingly, four novel OIBTMs can be distinguished: maximum-based, maximum-logic OIBTM (OMAX multiplexers); minimum-based, minimum-logic OIBTM (OMIN multiplexers); minimum-based, maximum-logic OIBTM (OMAXV multiplexers); maximum-based, minimum-logic OIBTM (OMINV multiplexers). Following the building strategy presented in Fig. 1 it can be found that for maximum-logic OIBTMs the left and right inputs of the PRs are $V_1=V_{\max }$ and $V_2=V_{\min }$, respectively, while for minimum-logic OIBTMs these two quantities are opposite, that is, $V_1=V_{\min }$ and $V_2=V_{\max }$. Also, in the arrangement shown in Fig. 1 for the maximum-based setups the losses of the base router are $V_{\mathrm {1B}}=V_{\max }$ and $V_{\mathrm {2B}}=V_{\min }$, while for the minimum-based setups these quantities are opposite, that is, $V_{\mathrm {1B}}=V_{\min }$ and $V_{\mathrm {2B}}=V_{\max }$. Therefore, the above four novel OIBTMs can be characterized by the following combinations of the router losses: for OMAX multiplexers $V_1=V_{\mathrm {1B}}=V_{\max }$ and $V_2=V_{\mathrm {2B}}=V_{\min }$; for OMIN multiplexers $V_1=V_{\mathrm {1B}}=V_{\min }$ and $V_2=V_{\mathrm {2B}}=V_{\max }$; for OMAXV multiplexers $V_1=V_{\mathrm {2B}}=V_{\max }$ and $V_2=V_{\mathrm {1B}}=V_{\min }$; finally, for OMINV multiplexers $V_1=V_{\mathrm {2B}}=V_{\min }$, $V_2=V_{\mathrm {1B}}=V_{\max }$.

 

Fig. 1. Schematic figure of a SPS based on the proposed OIBTM schemes. MU$_i$s denote multiplexed units and PR$_i$s denote 2-to-1 photon routers. The numbering of the PRs reflect the order in which they are added to the multiplexer.

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From these definitions one would conclude that OMIN multiplexers for the regions of the transmissions $V_r\leq V_t$ and OMAX multiplexers for the regions of the transmissions $V_r\geq V_t$ coincide with the OIBTMs presented in Ref. [13]. However, the building strategy of the transmission coefficient-based logics differs from that of the geometric logic. Therefore, the sequences of the PRs in the proposed multiplexers are generally different from that in the OIBTM presented in Ref. [13]. In Fig. 1 we denote the first deviation in the sequence of the PRs; see PR$_{27}$ and PR$_{28}$ distinguished by red frames. Note that for further levels of the multiplexer more complex deviation patterns can be observed. We also note that the four proposed OIBTMs together with the OIBTM built by following a geometric rule introduced in Ref. [13] cover all reasonable configurations corresponding to the expectation included in the original definition of incomplete binary-tree multiplexers that they are systematic extensions of symmetric multiplexers. Note that there can be general incomplete binary-tree multiplexers without the restriction of systematic building of levels, hence our analysis will yield the optimal multiplexer only in this context.

The total transmission coefficients $V_n$ of the proposed OIBTMs can be characterized by a single formula as follows. Assume that the overall number of inputs of the multiplexer is $N$. In Fig. 1 this number, that is, the number of multiplexed units is $N=28$. Let us denote the number of inputs of the initial CBTM by $N_1$, while the number of inputs on the incomplete branch is $N_2$; accordingly, $N=N_1+N_2$. In Fig. 1 $N_1=16$, and consequently $N_2=12$. Then the total transmission coefficients $V_n$ characterizing the arms of the proposed four OIBTMs read

We note that the sequential numbers of the total transmission coefficients $V_n$ in Eq. (1) follow a mathematical logic and they do not coincide with the left-to-right numbering of the sequential numbers $i$ of the multiplexed units MU$_i$ in Fig. 1. As the statistical description presented in the next Section includes the rearrangement of the total transmission coefficients into a descending order, the physical results are not influenced by the indexing applied in Eq. (1).

For showing an example that the proposed four OIBTM schemes are generally characterized by different total transmission coefficients, in Table 1 we present the total transmission coefficients $V_i$ of the four schemes for $N=13$ multiplexed units. In the table, the sequential numbers $i$ correspond to the numbering of the multiplexed units in Fig. 1, and the total transmission coefficients are expressed through the general notations $V_{\max }$ and $V_{\min }$. It is easy to check that, though these lists contain identical elements, the lists are different for the four schemes. For example, in the lists corresponding to the OMAX, OMIN, and OMINV geometries there are five, three, and two different elements, respectively, from the list corresponding to the OMAXV scheme. Naturally, this difference depends on the number of multiplexed units. For obvious reasons, the systems coincide only when the numbers of multiplexed units are powers of two, that is, when the binary-tree multiplexer is complete and, evidently, for $V_{\max }=V_{\min }$.

Table 1. Total transmission coefficients $V_i$ of the four proposed OIBTMs for $N=13$ multiplexed units. The sequential number $i$ corresponds to the numbering of the multiplexed units in Fig. 1.

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In subsequent sections of this paper we compare results achieved for the proposed OIBTMs with those for asymmetric multiplexers (ASYM multiplexers) [9,12,19]. Therefore, we also present the formula for describing the total transmission coefficients $V_n$ of the $n$th arm of these asymmetric spatial multiplexers built from bulk optical asymmetric routers (Fig. 2). However, the formula below takes into account the fact that for such an asymmetric structure the resulting maximal single-photon probability $P_{1,\max }$ can strongly depend on the actual values of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively, of the asymmetric routers. It can be easily seen that the achievable single-photon probability of such systems is highest when the multiplexer is extended on the arms with the largest transmission coefficient. Accordingly, the formula for asymmetric routers can be written as follows:

 

Fig. 2. Schematic figure of a SPS based on asymmetric spatial multiplexing. MU$_i$s denote multiplexed units and PR$_i$s denote 2-to-1 photon routers.

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3. Statistical theory

The statistical description of any periodic SPSs based on either spatial or temporal multiplexing equipped with photon-number-resolving detectors realizing any detection strategy was presented in Refs. [11,13]. As our aim is to find the spatial multiplexing scheme that yields the highest performance SPS, we consider systems characterized by the best loss parameters for which single-photon detection is certainly the optimal detection strategy in view of the previous results on similar systems [11,13]. Accordingly, below we summarize the applied theoretical description restricted to such type of detection.

The output $i$-photon probability of multiplexed SPSs equipped with single-photon detectors can be written as

In this formula $P^{(\lambda )}(l)$ is the probability of generating $l$ photon pairs in a multiplexed unit that is characterized by the input mean photon number $\lambda$, that is, by the mean photon number of the generated photon pairs. We assume that the probability distribution of the photon pair generation $P^{(\lambda )}(l)$ is Poissonian, that is,

The conditional probability $P^{(D)}(1|l)$ of registering a single photon provided that $l$ photons arrive at the detector can be expressed as

The probability $P^{(D)}_1$ describing the event that a single photon is detected can be written as

In our calculations we assumed that the probabilities $P^{(D)}_1$, $P^{(D)}(1|l)$, $P^{(\lambda )}(l)$, and the input mean photon number $\lambda$ are independent of the sequential number $n$ of the multiplexed unit.

In Eq. (5) the conditional probability $V_n(i|l)$ of the event that $i$ photons reach the output of the multiplexer provided that $l$ signal photons arrive from the $n$th multiplexed unit into the system is expressed as

Note that Eq. (5) assumes a priority logic that favors the MU with the smallest sequential number $n$ in the case of multiple heralding events in different MUs. As the MUs can be numbered arbitrarily, we can choose a numbering for which the associated total transmission coefficients $V_n$ are arranged into a decreasing order, that is, $V_1\ge V_2\ge \dots \ge V_N$. Obviously, the numbering of the multiplexer arms having identical total transmission coefficients is arbitrary. As such a numbering implies that the multiplexer prefers the arm with the highest $V_n$, that is, the smallest loss caused by the optical elements, applying this numbering decreases the probability of photon loss in the multiplexer and so the resulting single-photon probabilities can be higher.

The contribution of multiphoton components in the output state compared to that of the single-photon component can be quantified by the normalized second-order autocorrelation function

Using the above formulas it is possible to determine the optimal number of multiplexed units $N_{\text {opt}}$ and the optimal input mean photon number $\lambda _{\text {opt}}$ corresponding to the maximal value of the output single-photon probability $P_{1,\max }$. For the proposed OIBTM schemes the function describing the single-photon probability against the number of multiplexed units and the input mean photon number, $P_1(N,\lambda )$ has a well-defined global maximum, therefore one possible way to find the optimal values of $N$ and $\lambda$ corresponding to the maximal value of $P_1$ is using e.g. a single objective function genetic algorithm. However, for asymmetric (chain-like) multiplexers the same function $P_1(N,\lambda )$ monotonically increases with the number of multiplexed units and it eventually saturates [9,12,19]. Therefore seeking a maximum, if possible at all, would result in inconveniently high optimal values of $N$. Accordingly, for asymmetric multiplexers we first maximized the single-variable function $P_{1,\max }(N=100,\lambda )$ (considering $N=100$ to be a high-enough number of multiplexed units) and then we determined the maximal single-photon probabilities $P_{1,\max }$ and the corresponding optimal input mean photon numbers $\lambda _{\text {opt}}$ for values of the number of multiplexed units $N$ increasing from 2. The number of multiplexed unit $N$ for which the absolute difference $\Delta _P=\left |P_{1,\max }(N,\lambda _{\text {opt}})-P_{1,\max }(N=100,\lambda _{\text {opt}})\right |$ between the corresponding maximal single-photon probability $P_{1,\max }(N,\lambda _{\text {opt}})$ and the single-photon probability $P_{1,\max }(N=100,\lambda _{\text {opt}})$ determined for a high-enough number of multiplexed units, respectively, is $\Delta _P<=0.001$ is considered to be the optimal number of multiplexed units.

4. Results

In this section we present our results for the optimization of SPSs based on output-extended incomplete binary-tree multiplexers constructed following transmission-based logics and composed of general asymmetric routers. As our goal is to find the construction logic from the proposed ones that yields the spatially multiplexed SPS having the best performance we restrict our calculations for high transmission and detector efficiencies that can be realized experimentally using state-of-the-art devices. Therefore, in this section we fixed the detector efficiency to $V_D=0.95$, the highest value reported in [40], and the general transmission coefficient to $V_b=0.98$, hence we do not indicate these values in the following. The effect of choosing lower values for these parameters is analyzed at the end of this section where the proposed schemes are considered for suboptimal system sizes having relevance in practical realizations. The highest transmission efficiencies $V_r=0.99$ and $V_t=0.985$ were reported for routers built of bulk-optical elements [33,46]. These values are used in our calculations whenever individual parameter sets or sweeps for other parameters are analyzed. However, due to symmetry considerations discussed earlier, in parameter sweeps for $V_r$ and $V_t$ the upper boundaries of the ranges of these efficiencies are chosen to be $V_r=V_t=0.99$, while the lower boundaries are chosen so that the maximal single-photon probability for single-photon detection strategy is higher than that for any other detection strategy for the whole parameter range, that is, $V_r=V_t=0.85$ (see e.g. [12]).

We calculated the maximal single-photon probabilities $P_{1,\max }$ that can be achieved by using optimal number of multiplexed units $N_{\text {opt}}$ and optimal input mean photon numbers $\lambda _{\text {opt}}$ for SPSs based on OIBTMs constructed by following the four proposed transmission-based logics for the considered ranges of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively. Figure 3 shows the differences of these probabilities for two relevant cases. Figure 3(a) presents the difference $\Delta _P^{\text {omaxv}-\text {omin}}=P_{1,\max }^{\text {omaxv}}-P_{1,\max }^{\text {omin}}$ between the maximal single-photon probabilities for SPSs based on OMAXV and OMIN multiplexers, respectively. The figure shows that using SPSs based on OMAXV multiplexers leads to higher single-photon probabilities for the considered domain. We also found that the single-photon probabilities obtained with SPSs based on OMINV multiplexers are always higher than those achieved for SPSs based on OMAX multiplexers, and the characteristics of the difference between the maximal single-photon probabilities for SPSs based on these multiplexers are qualitatively similar to the corresponding difference for systems based on OMAXV and OMIN multiplexers. Figure 3(b) shows the difference $\Delta _P^{\text {omaxv}-\text {ominv}}=P_{1,\max }^{\text {omaxv}}-P_{1,\max }^{\text {ominv}}$ between the maximal single-photon probabilities for SPSs based on OMAXV and OMINV multiplexers, respectively. The figure shows that SPSs based on OMAXV multiplexers yield higher single-photon probabilities for most part of the considered domain than those based on OMINV multiplexers. The largest difference $\Delta _P^{\text {omaxv}-\text {ominv}}$ is close to 0.01. SPSs based on OMINV multiplexers can yield higher single-photon probabilities only for a small region of notably different transmission and reflection efficiencies, that is, for very asymmetric routers. Note that the advantage of the OMINV multiplexer is small, that is, $\left |\Delta _P^{\text {omaxv}-\text {ominv}}\right |\lessapprox 0.001$. We have also found that SPSs based on OMAXV multiplexers outperform those based on OIBTMs constructed by a pure geometric logic. From these results we concluded that SPSs based on OMAXV multiplexers can generally lead to the highest maximal single-photon probabilities. A plausible physical explanation of this conclusion is the following. Recall that the priority logic favors the MU with the smallest sequential number, and the numbering of the MUs is chosen so that the associated total transmission coefficients $V_n$ are arranged into a decreasing order. In the case of OIBTMs the arms in the incomplete branch generally have higher $V_n$s than the arms in the complete branch. Obviously, the building logic of OMAXV multiplexers leads to higher $V_n$s, that is, smaller losses in these arms at a given $N$.

 

Fig. 3. (a) The difference $\Delta _P^{\text {omaxv}-\text {omin}}$ between the maximal single-photon probabilities for SPSs based on OMAXV and OMIN multiplexers, respectively. (b) The difference $\Delta _P^{\text {omaxv}-\text {ominv}}$ between the maximal single-photon probabilities for SPSs based on OMAXV and OMINV multiplexers, respectively. The differences are plotted against the transmission efficiency $V_t$ and reflection efficiency $V_r$. The black continuous line indicates the zero-level $\Delta _P^{\text {omaxv}-\text {ominv}}=0$ wherever the difference changes sign.

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Next, we analyze the performance of SPSs based on OMAXV multiplexers in detail. We calculated the maximal single-photon probabilities $P_{1,\max }^{\text {omaxv}}$ and the optimal number of multiplexed units $N_{\text {opt}}^{\text {omaxv}}$ as functions of the transmission and reflection efficiencies $V_t$ and $V_r$. The results of these calculations are presented in Fig. 4. Obviously, for increasing values of either $V_t$ or $V_r$ the resulting maximal single-photon probabilities $P_{1,\max }$ and the corresponding optimal number of multiplexed units $N_{\text {opt}}$ are also increased. We found that for the highest reported values of the transmission efficiency $V_t=0.985$ and the reflection efficiency $V_r=0.99$ the maximal single-photon probability is $P_{1,\max }=0.9222$ and the optimal number of multiplexed units is $N_{\text {opt}}=38$. We note that the unique values of the optimal number of multiplexed units in Fig. 4(b) are 6, 10, 11, 19, 20, 21, 37, and 38. As the possible number of multiplexed units in CBTMs are powers of two, the unique numbers in Fig. 4(b) clearly show that optimized OMAXV multiplexers always contain short arms having lower losses in their incomplete branches. The presence of these low-loss arms can improve the performance of SPSs based on OMAXV multiplexers.

 

Fig. 4. (a) The maximal single-photon probability $P_{1,\max }^{\text {omaxv}}$ and (b) the optimal number of multiplexed units $N_{\text {opt}}^{\text {omaxv}}$ for SPSs based on OMAXV multiplexers as functions of the transmission efficiency $V_t$ and the reflection efficiency $V_r$.

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In Ref. [13] we found that SPSs based on OIBTM multiplexers always yield higher single-photon probabilities than SPSs based on CBTM multiplexers. In the following we compare SPSs based on OMAXV multiplexers with SPSs based on the other known type of spatial multiplexers, that is, asymmetric ones. Fig. 5 shows the difference $\Delta _P^{\text {omaxv}-\text {asym}}=P_{1,\max }^{\text {omaxv}}-P_{1,\max }^{\text {asym}}$ between the maximal single-photon probabilities and the difference $\Delta _N^{\text {omaxv}-\text {asym}}=N_{\text {opt}}^{\text {omaxv}}-N_{\text {opt}}^{\text {asym}}$ between the optimal number of multiplexed units for SPSs based on OMAXV and ASYM multiplexers, respectively, as a function of the transmission and reflection efficiencies $V_t$ and $V_r$. Figure 5(a) shows that for highly asymmetric routers $V_r \gg V_t$ or $V_r \ll V_t$ SPSs based on OMAXV multiplexers can yield higher single-photon probabilities than those based on ASYM multiplexers. The largest difference is $\Delta _P^{\text {omaxv}-\text {asym}}=0.0143$ for the transmission efficiency $V_t=0.85$ and the reflection efficiency $V_r=0.9814$. For simultaneously high values of the router efficiencies $V_t\approx V_r\approx 1$ the maximal single-photon probability calculated for SPSs based on OMAXV multiplexers is also higher than that of the asymmetric scheme, though the maximal difference is smaller, $\Delta _P^{\text {omaxv}-\text {asym}}=0.0027$.

 

Fig. 5. (a) The difference $\Delta _P^{\text {omaxv}-\text {asym}}$ between the maximal single-photon probabilities and (b) the difference $\Delta _N^{\text {omaxv}-\text {asym}}$ between the optimal number of multiplexed units for SPSs based on OMAXV and ASYM multiplexers, respectively, as functions of the transmission efficiency $V_t$ and the reflection efficiency $V_r$. The black continuous lines indicates the zero-level $\Delta _P^{\text {asym}-\text {omaxv}}=0$ or $\Delta _N^{\text {asym}-\text {omaxv}}=0$ wherever the difference changes sign.

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The characteristics of Fig. 5(b) are inherited basically from those of Fig. 4(b) as, for plausible reasons, the optimal numbers of multiplexed units for asymmetric multiplexers (the corresponding figures not presented here) do not exhibit bigger jumps when the transmission coefficients are changed. For high values of the efficiencies $V_r$ and $V_t$, the large jump in the difference between the optimal numbers of multiplexed units for OMAXV multiplexers and for ASYM multiplexers ($\Delta _N^{\rm omax-asym}>15$) corresponds to the jump that can be observed in Fig. 4(b) for the same values of these efficiencies. This large difference can be explained by the argument that the total transmission coefficients $V_n$, on average, are higher for OMAXV multiplexers than for asymmetric multiplexers for this range of parameters, this can be deduced from Eqs. (1) and (4). Generally, lower losses in the multiplexer can result in higher optimal number of multiplexed units. It is worth noting that generally a larger multiplexer containing higher number of multiplexed units can exploit better the idea of multiplexing, that is, multiphoton events are avoided by increasing the number of multiplexed units in parallel with decreasing the input mean photon number. Hence, though the performance of a spatially multiplexed single-photon source is influenced by the actual values of the total transmission coefficients characterizing the multiplexer and not simply by the number of the multiplexed units, the characteristics of the difference $\Delta _N^{\text {omaxv}-\text {asym}}$ between the optimal numbers of multiplexed units presented in Fig. 5(b) can support the observations for the difference $\Delta _P^{\text {omaxv}-\text {asym}}$ between the maximal single-photon probabilities shown in Fig. 5(a). In Fig. 6 we present the second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ for SPSs based on OMAXV multiplexers and the difference $\Delta _{g^{(2)}}^{\text {asym}-\text {omaxv}}=g^{(2)}_{\text {asym}}-g^{(2)}_{\text {omaxv}}$ between the second-order autocorrelation functions $g^{(2)}$ for SPSs based on ASYM multiplexers and OMAXV multiplexers, respectively, as functions of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively. The normalized second-order autocorrelation function $g^{(2)}$ is calculated at optimal values of the number of multiplexed units $N_\text {opt}$ and the input mean photon number $\lambda _\text {opt}$ corresponding to the maximal single-photon probabilities $P_{1,\max }$ for SPSs based on various multiplexing schemes. Figure 6(a) shows the normalized second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ for SPSs based on OMAXV multiplexers. For smaller values of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively, the autocorrelation function $g^{(2)}$ shows higher values. This finding is obvious because for smaller values of $V_t$ and $V_r$ the calculated maximal single-photon probabilities are also smaller, consequently, the multiphoton contribution in the output state is higher. Figure 6(b) shows that for a wide region around the line $V_t=V_r$ the difference $\Delta _{g^{(2)}}^{\text {asym}-\text {omaxv}}$ is positive and so, in this respect, OMAXV multiplexers outperform asymmetric multiplexers. We also note that for simultaneously high values of $V_r\approx V_t\approx 1$ the maximal single-photon probability $P_{1,\max }$ is higher while the normalized second-order autocorrelation function $g^{(2)}$ is lower for SPSs based on OMAXV multiplexers than that for SPSs based on ASYM multiplexers, therefore for such router efficiencies $V_t$ and $V_r$ SPSs based on OMAXV multiplexers outperform the ones based on ASYM multiplexers in every relevant characteristics, albeit at the cost of the application of higher optimal numbers of multiplexed units $N_\text {opt}$. We found that for the highest reported values of the transmission efficiency $V_t=0.985$ and the reflection efficiency $V_r=0.99$ the value of the normalized second-order autocorrelation function for SPSs based on OMAXV multiplexers is $g^{(2)}_{\text {omaxv}}=0.0264$ and the difference between the normalized second-order autocorrelation functions for SPSs based on ASYM and OMAXV multiplexers is $\Delta _{g^{(2)}}^{\text {asym}-\text {omaxv}}=0.0202$.

 

Fig. 6. (a) The second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ for SPSs based on OMAXV multiplexers, and (b) the difference $\Delta _{g^{(2)}}^{\text {asym}-\text {omaxv}}=g^{(2)}_{\text {asym}}-g^{(2)}_{\text {omaxv}}$ between the second order autocorrelation functions $g^{(2)}$ for SPSs based on ASYM multiplexers and OMAXV multiplexers, respectively, as a function of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively. The black continuous line indicates the zero-level $\Delta _{g^{(2)}}^{\text {asym}-\text {omaxv}}=0$ wherever the difference changes sign.

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We have also performed calculations for SPSs based on OMAXV multiplexers assuming thermal distribution for the input photon pairs. In Fig. 7(a) we present the difference $\Delta _P^{\text {omaxv}{},P-T}=P_{1,\max }^{\text {omaxv}{},P}-P_{1,\max }^{\text {omaxv}{},T}$ between the maximal single-photon probabilities for these systems obtained by assuming Poisson (P) and thermal (T) distributions, respectively, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. We found that, in accordance with the expectations, assuming Poisson distribution always leads to higher maximal single-photon probabilities than assuming thermal distribution. For simultaneously high values of $V_t$ and $V_r$ the difference is relatively small, $\Delta _P^{\text {omaxv}{},P-T}\approx 0.01$ but for simultaneously low values of these efficiencies $V_t$ and $V_r$ the difference is larger, it can be as high as $\Delta _P^{\text {omaxv}{},P-T}\approx 0.055$. Figure 7(b) shows the difference $\Delta _N^{\text {omaxv}{},P-T}=N_{\text {opt}}^{\text {omaxv}{},P}-N_{\text {opt}}^{\text {omaxv}{},T}$ between the optimal number of multiplexed units for SPSs based on OMAXV multiplexers obtained by assuming Poisson and thermal distributions, respectively, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. The results show that the optimal number of multiplexed units $N_\text {opt}$ obtained by assuming thermal distribution is always higher than that for Poisson distribution, and the difference is higher for high values of the efficiencies $V_t$ and $V_r$. Figure 7(c) shows the difference $\Delta _{g^{(2)}}^{\text {omaxv}{},P-T}=g^{(2)}_{\text {omaxv}{},P}-g^{(2)}_{\text {omaxv}{},T}$ between the normalized second-order autocorrelation functions for SPSs based on OMAXV multiplexers obtained by assuming Poisson and thermal distributions, respectively, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. The difference is negative on the whole analyzed domain, that is, assuming Poisson distribution leads to smaller values for $g^{(2)}$ than assuming thermal distribution.

 

Fig. 7. (a) The difference $\Delta _P^{\text {omaxv}{},P-T}=P_{1,\max }^{\text {omaxv}{},P}-P_{1,\max }^{\text {omaxv}{},T}$ between the maximal single-photon probabilities, (b) the difference $\Delta _N^{\text {omaxv}{},P-T}=N_{\text {opt}}^{\text {omaxv}{},P}-N_{\text {opt}}^{\text {omaxv}{},T}$ between the optimal number of multiplexed units, and (c) the difference $\Delta _{g^{(2)}}^{\text {omaxv}{},P-T}=g^{(2)}_{\text {omaxv}{},P}-g^{(2)}_{\text {omaxv}{},T}$ between the normalized second-order autocorrelation functions for SPSs based on OMAXV multiplexers obtained by assuming Poisson and thermal distributions, as a function of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively.

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We have also calculated the differences $\Delta _P^{\text {omaxv-asym}{}{},T}$ and $\Delta _{g^{(2)}}^{\text {asym}{}}\hbox{-}{\text {omaxv}{},T}$ between the results for SPSs based on OMAXV and ASYM multiplexers obtained by assuming thermal distribution. They are considerably different both qualitatively and quantitatively from those obtained by assuming Poisson distribution. Figure 8(a) shows the difference $\Delta _P^{\text {omaxv-asym},T}=P_{1,\max }^{\text {omaxv},T}-P_{1,\max }^{\text {asym},T}$ between the maximal single-photon probabilities for SPSs based on OMAXV and ASYM multiplexers, respectively, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. Comparing it with Fig. 5(a) it can be seen that in the case of thermal distribution SPSs based on OMAXV multiplexers outperform those based on ASYM multiplexers for a considerably larger range of the parameters $V_r$ and $V_t$ than in the case of Poisson distribution, and this range contains the highest values of these parameters. The advantage of OMAXV multiplexers can be as high as $\Delta _P^{\text {omaxv-asym},T}\approx 0.01$. For those regions of the parameter values where SPSs based on ASYM multiplexers give higher maximal single-photon probabilities $P_{1,\max }$ the difference $\Delta _P^{\text {omaxv}-\text {asym}}$ for thermal distribution is generally smaller than for Poisson distribution (c.f. Figure 5(a)), that is, $\left |\Delta _P^{\text {omaxv-asym},T}\right |\lessapprox 0.005$.

 

Fig. 8. (a) The difference $\Delta _P^{\text {omaxv-asym},T}=P_{1,\max }^{\text {omaxv},T}-P_{1,\max }^{\text {asym},T}$ between the maximal single-photon probabilities for SPSs based on OMAXV and ASYM multiplexers, respectively, obtained by assuming thermal distribution, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. The black continuous line indicates the zero-level $\Delta _{P}^{\text {asym-omaxv},T}=0$. (b) The difference $\Delta _{g^{(2)}}^{\text {asym-omaxv},T}=g^{(2)}_{\text {asym},T}-g^{(2)}_{\text {omaxv},T}$ between the normalized second order autocorrelation functions $g^{(2)}$ corresponding to the maximal single-photon probabilities for SPSs based on ASYM multiplexers and OMAXV multiplexers, respectively, obtained by assuming thermal distribution, as a function of the transmission and reflection efficiencies $V_t$ and $V_r$, respectively. The black continuous line indicates the zero-level $\Delta _{g^{(2)}}^{\text {asym-omaxv},T}=0$ wherever the difference changes sign.

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Figure 8(b) shows the difference $\Delta _{g^{(2)}}^{\text {asym-omaxv},T}=g^{(2)}_{\text {asym},T}-g^{(2)}_{\text {omaxv},T}$ between the normalized second order autocorrelation functions $g^{(2)}$ corresponding to the maximal single-photon probabilities for SPSs based on ASYM multiplexers and OMAXV multiplexers, respectively, obtained by assuming thermal distribution, as a function of the $V_t$ transmission and $V_r$ reflection efficiencies. The figure shows that in the case of thermal distribution SPSs based on OMAXV multiplexers outperform those based on ASYM multiplexers in a larger range of the parameters $V_t$ and $V_r$. Comparing this figure with Fig. 6(b) it can be seen that this range is larger than that for Poisson distribution. The differences for thermal distribution can be as high as $\Delta _{g^{(2)}}^{\text {asym-omaxV},T}\approx 0.035$ while for Poisson distribution it is only $\Delta _{g^{(2)}}^{\text {asym-omaxv},P}\approx 0.022$. Summarizing the findings related to Fig. 8 it can be deduced that in the case of thermal distribution for high values of the efficiencies $V_t$ and $V_r$ SPSs based on OMAXV multiplexers outperform in every respect the ones based on ASYM multiplexers. The advantage of OMAXV multiplexers is significant in the suppression of the multiphoton noise shown by the relevantly lower values of the second order autocorrelation function $g^{(2)}$.

A plausible explanation of the lower values of $g^{(2)}_\text {omaxv}$ in the OMAXV scheme for the parameter ranges shown in Figs. 6(b) and 8(b) can be the following. Obviously, the occurrence of the multiphoton events can be suppressed most efficiently by decreasing the input mean photon number in the multiplexed units. As a consequence, for spatially multiplexed single-photon sources, the characteristics of the optimal input mean photon number $\lambda _{\rm opt}$ qualitatively coincides with those of the function $g^{(2)}$ for a given set of loss parameters. For smaller values of the optimal input mean photon number $\lambda _{\rm opt}$ the corresponding values of the function $g^{(2)}$ are also smaller. The optimization of spatially multiplexed single-photon sources is a complex process that is influenced by the actual values of all the total transmission coefficients $V_n$ characterizing the system, but it is quite plausible that generally, higher transmission coefficients representing smaller losses can provide smaller optimal input mean photon number $\lambda _{\rm opt}$. Comparing the total transmission coefficients $V_n$ of the OMAXV and asymmetric multiplexers (see Eqs. (1) and (4)) one can conclude that, for a given number of multiplexed units $N$, these coefficients contain the powers of $V_{\max }$ scaling from 1 to $N-1$ for asymmetric multiplexers while the corresponding coefficients for OMAXV multiplexers contain the products of the powers of $V_{\max }$ and $V_{\min }$ and the sum of the exponents are lower than or equal to $\log _2N$. This implies that the total transmission coefficients $V_n$ can be, on average, larger for OMAXV multiplexers for approximately equal or similar values of the transmission coefficients ($V_{\max }\approx V_{\min }$). Accordingly, for such coefficients the optimized values of $\lambda _{\rm opt}$ and $g^{(2)}$ can be smaller for OMAXV multiplexers. The significant advantage of OMAXV multiplexers in the performance of the second order autocorrelation function $g^{(2)}$ around the best values of the transmission coefficients $V_{\max }$ and $V_{\min }$ (i.e., $V_r$ and $V_t$) can be explained by the fact that the optimal numbers of multiplexed units $N_{\rm opt}$ in such multiplexers are relevantly higher compared to those in asymmetric multiplexers (see Figs. 5(b) and 7(b)) that entails the further decrease in the optimal input mean photon number $\lambda _{\rm opt}$.

The calculations thus far focused on determining the maximal single-photon probabilities $P_{1,\max }$ that can be obtained by the considered systems and finding the optimal numbers of multiplexed units $N_{\text {opt}}$ and optimal input mean photon numbers $\lambda _{\text {opt}}$ for which these probabilities can be achieved. However, the calculated $N_{\text {opt}}$ values are generally higher than what is reasonable to realize in experiments. Consequently, we performed calculations for suboptimal system sizes for SPSs based on OMAXV multiplexers. Figure 9 shows the achievable single-photon probabilities $P_1^{\text {omaxv}}$ and the normalized second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ for SPSs based on OMAXV multiplexers obtained by assuming Poisson distribution as functions of the number of multiplexed units $N$ for the best reported values of the transmission efficiency $V_t=0.985$ and the reflection efficiency $V_r=0.99$, for the general transmission coefficient $V_b=0.98$, and for various values of the detector efficiency $V_D$. The figures show that by increasing the value of $N$ the achievable single-photon probabilities also increase while the corresponding values of the normalized second-order autocorrelation function $g^{(2)}$ decrease. Similarly, increasing the detector efficiency $V_D$ leads to an increase in $P_1$ and a decrease in $g^{(2)}$, and $g^{(2)}$ is more sensitive to a change in $V_D$ than $P_1$. The calculations showed that the resulting achievable single-photon probabilities $P_{1,\max }^{\text {omaxv}}$ are reasonably high even for lower numbers of the applied multiplexed units, and the corresponding values of the normalized second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ are relatively low. The achievable single-photon probability obtained for the considered loss parameters and for the detector efficiency $V_D=0.95$, and assuming $N=6$ multiplexed units is as high as $P_{1,\max }^{\text {omaxv}}=0.854$ with $g^{(2)}_{\text {omaxv}}=0.089$. By further increasing the number of multiplexed units to $N=10$ the resulting achievable single-photon probability reaches $P_{1,\max }^{\text {omaxv}}=0.904$ with even smaller value of the normalized second-order autocorrelation function $g^{(2)}_{\text {omaxv}}=0.067$. Assuming thermal distribution for the input photon pairs the corresponding values for $N=6$ are $P_{1,\max }^{\text {omaxv}}=0.747$, $g^{(2)}_{\text {omaxv}}=0.114$, and for $N=10$ these are $P_{1,\max }^{\text {omaxv}}=0.853$, $g^{(2)}_{\text {omaxv}}=0.094$, that is, the single-photon probability achieved for $N=6$ for Poisson distribution can be reached for $N=10$ with thermal distribution.

 

Fig. 9. (a) The achievable single-photon probabilities $P_1^{\text {omaxv}}$ and (b) the normalized second-order autocorrelation function $g^{(2)}_{\text {omaxv}}$ for SPSs based on OMAXV multiplexers as functions of the number of multiplexed units $N$ for the transmission efficiency $V_t=0.985$, the reflection efficiency $V_r=0.99$, the general transmission coefficient $V_b=0.98$, and for various values of the detector efficiency $V_D$.

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5. Conclusion

We have considered four novel types of spatially multiplexed single-photon sources based on output-extended incomplete binary-tree multiplexers containing general asymmetric routers where the rule of placement of the novel photon routers during the construction is determined by the magnitudes of the losses of the optional connection points. We have found that the multiplexers termed as minimum-based, maximum-logic OIBTMs outperform the others. We have analyzed and optimized single-photon sources based on such multiplexers and realized with photon-number-resolving detectors using a general statistical theory that includes all relevant loss mechanisms. We have determined the ranges of the loss parameters for which single-photon sources based on such systems yield higher single-photon probabilities and lower values of the second-order autocorrelation function than that can be achieved by using asymmetric multiplexers. The advantage of the proposed multiplexers appears for high values of the transmission coefficients characterizing single-photon sources having the best performance. We have shown that these multiplexers can be used especially beneficially in single-mode single-photon sources characterized by thermal statistics of the input photon pairs and the application of them yields high performance single-photon sources even for suboptimal system sizes that is a typical situation in current experiments.