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Abstract
We propose a spatially multiplexed single-photon source where the structure of the applied binary-tree multiplexer is optimized systematically during its construction. Along the building procedure of this type of multiplexer, the position of a binary photon router appended to the tree in a step of the expansion is determined by taking into account the current achievable single-photon probability of the source. The method chooses the position where this probability is maximal. We determine the stepwise optimized binary-tree multiplexers for experimentally realizable values of the loss parameters, and for a fixed number of routers. The method is scalable, that is, it is possible to determine the multiplexer with an optimal structure for any number of photon routers. We show that single-photon sources based on stepwise optimized binary-tree multiplexers yield higher single-photon probabilities than single-photon sources based on any spatial multiplexer types discussed in the literature thus far in the considered ranges of the loss parameters.
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1. Introduction
Several applications in the fields of quantum information processing and photonic quantum technology [1,2] require indistinguishable single photons on demand with known polarization and in near-perfect spatial modes. Heralded single-photon sources [3–10] combined with multiplexing, that is, multiplexed single-photon sources (SPSs) can aspire for being an effective source generating photons with these properties. In heralded SPSs, spontaneous parametric down-conversion or spontaneous four-wave mixing is used to generate correlated photon pairs and the detection of one member of the pairs indicates the presence of the other photon. Unfortunately, these nonlinear optical processes have an inherent probabilistic nature that leads to a nonzero probability of generating multiple photons. Possible solutions to diminish the multiphoton noise are using single-photon detectors with photon-number-resolving capabilities for heralding [10–12], or reducing the mean number of the photon pairs generated in the nonlinear process. However, the reduction of the mean photon number also decreases the probability of successful heralding. This detrimental effect can be compensated by multiplexing several heralded sources realized in space or time. The solutions proposed for realizing multiplexed SPSs include a couple of temporal [13–25] and numerous spatial multiplexing schemes [16,26–37] with various successful experimental implementations [18,20,21,23,28,29,31,33,34]. Multiplexed SPSs can be analyzed by applying the full statistical theories developed for the description of such systems [11,16,35,36,38] that include all relevant losses characterizing the specific schemes. These theories can be applied to optimize the systems, that is, to determine the optimal number of multiplexed units, i.e., heralded sources, and the optimal mean number of the photon pairs generated in the units yielding the highest single-photon probability in a particular arrangement. The analyses showed that multiplexed SPSs realized with state-of-the-art optical elements can yield high achievable single-photon probabilities accompanied with low multiphoton noise [11,36–38].
In spatially multiplexed single-photon sources, the individual heralded photon sources can be realized by using physically separate nonlinear processes or in separate spatial modes of a single process. Heralded single photons generated in one of the sources are rerouted to a single output by a spatial multiplexer composed of a set of binary photon routers. According to the theoretical analysis, the structure of the binary tree strongly influences the performance of the multiplexed single-photon source: a particular multiplexer structure applied in a SPS can yield higher single-photon probability than the other for a certain range of the loss parameters characterizing the photon routers [36,37]. In the literature, symmetric (complete binary tree), asymmetric, and incomplete binary-tree multiplexers have been considered and analyzed in detail [11,16,30,36–38].
In the present paper, we propose a spatially multiplexed single-photon source where the structure of the applied binary-tree multiplexer is optimized systematically during its construction in order to further improve the performance of multiplexed SPSs. We show that single-photon sources based on such stepwise optimized binary-tree multiplexers can yield higher single-photon probability than those based on multiplexers proposed earlier in the literature.
2. General binary-tree multiplexers built by following a step-by-step optimization
Single-photon sources based on spatial multiplexing are essentially composed of two main parts: several multiplexed units and a multiplexer built of photon routers (PRs). A multiplexed unit contains a nonlinear photon pair source, a photon detector, and optionally a delay line. The nonlinear process in which photon pairs are generated can be, e.g., spontaneous parametric down-conversion or spontaneous four-wave mixing. The detection of one member of the generated photon pairs (the idler photon) heralds the presence of its twin photon (the signal photon) which can enter the multiplexer. In recent multiplexed single-photon source experiments, single-photon detectors with number resolving capability were generally used [18,20,21,23,29,33,34]. Note that photon-number-resolving detectors with high efficiencies are already available for experiments [39–43]. In the multiplexed unit, an optional delay line can also be used to introduce a sufficient delay to the arrival time of the signal photon so that the logic controlling the PRs has sufficient time for its operation. Spatial multiplexers are usually built of binary photon routers that have two input ports and a single output. The two input ports are generally characterized by two different transmission efficiencies, hence such routers are said to be asymmetric. In Refs. [35,36] the two efficiencies characterizing the input ports are termed as transmission and reflection efficiencies and they are denoted by $V_t$ and $V_r$, respectively. In the present paper we keep the notation of these quantities but we use the term transmission coefficients to refer to both of them.
The building strategy of the proposed stepwise optimized binary-tree multiplexer (SOBTM) containing $N$ multiplexed units, that is, $N-1$ PRs, is as follows. The positioning of the PRs in a binary tree is assumed to be fixed, that is, the inputs of all routers characterized by given transmission efficiencies are in the same geometric position in the tree (e.g., $V_t$ and $V_r$ characterizes the upper and lower inputs, respectively, for all routers in Fig. 1). We start with a single photon router PR$_1$ and append the second photon router PR$_2$ to the first, then to the second input port of the first PR$_1$. We apply these two multiplexers one by one in a SPS and then we determine the achievable output single-photon probabilities for both cases with the aid of the theory presented in the subsequent paragraphs. Then we keep the structure for which the single-photon probability is the highest. The third photon router PR$_3$ is appended to the possible input ports of the selected structure in succession, the achievable single-photon probability is determined for all three cases and the structure yielding the highest single-photon probability is kept. This procedure is repeated for all subsequent photon routers up to the sequential number $N-1$. In the end, we obtain a general binary-tree multiplexer having $N$ inputs in which the structure has been optimized step-by-step by the described method. During this procedure, the input ports of the multiplexers arising in given steps are renumbered following the same geometric logic to guarantee the systematic application of the method. Figure 1 illustrates the method of choosing the positions of photon routers up to PR$_5$. Squares with dashed frames together with dashed double arrows represent the positions of the photon routers in the binary tree not chosen in a given step of the construction, while the boxes PR$_1$, PR$_2$, PR$_3$, PR$_4$, and PR$_5$ represent the optimal positions of these routers in the multiplexer determined by the proposed procedure. The number of iterations required to determine the optimal structure containing $N-1$ PRs using this procedure is $1+2+\cdots +(N-1)=(N-1)N/2$. One iteration consists of determining the achievable single-photon probability of a SPS based on a given structure. This means that the required number of iterations is reasonable even for high numbers of PRs. We note that the proposed method does not consider all possible structures and, accordingly, the globally optimal configuration of the routers might be different to the one determined by this technique.
Fig. 1. Building strategy of a stepwise optimized binary-tree multiplexer containing five photon routers. The construction steps for finding the optimal positions of the photon routers are illustrated for (a) PR$_2$, (b) PR$_3$, (c) PR$_4$, and (d) PR$_5$. Squares with dashed frames together with dashed double arrows represent the positions of the photon routers in the binary tree not chosen in a given step of the construction, while the boxes PR$_1$, PR$_2$, PR$_3$, PR$_4$, and PR$_5$ represent the optimal positions of these routers. These positions are chosen so that the single-photon probability that can be achieved by applying the given multiplexer structures in SPSs are highest compared to the achievable single-photon probabilities of SPSs based on any of the other optional multiplexer structures indicated in the figure.
In the next Section, we will compare the performance of SPSs based on SOBTMs to those of SPSs based on asymmetric (ASYM) or minimum-based, maximum-logic output-extended incomplete binary-tree (OMAXV) multiplexers. In ASYM multiplexers the constituent routers are arranged into a chain-like structure [30,36–38]. In output-extended incomplete binary-tree multiplexers, an initially $m$-level symmetric multiplexer is extended step-by-step toward another, $m+1$-level symmetric multiplexer [36]. In the first step, a PR termed as base router is added with one of its inputs to the output of the initial symmetric system. Next, a PR is appended to the other input of the base router, thus starting an incomplete branch. Then the subsequent new PRs are added to the incomplete branch of the multiplexer one by one on the next levels, starting a new level only after the previous level is completed. The placement of the PRs is based on a chosen building logic, and the same building logic is applied for all levels. This process is repeated until an $m+1$-level symmetric multiplexer is formed. Figure 2 presents the scheme of a single-photon source based on a 4-level output-extended incomplete binary-tree multiplexer containing ten binary photon routers. Note that the numbering of the routers in the figure does not coincide with the order of the construction. In Fig. 2 photon routers PR$_2$ and PR$_{10}$ are on the incomplete branch of the multiplexer connected to the base router PR$_1$, while photon routers PR$_3$ to PR$_9$ form a complete 3-level binary-tree multiplexer. OMAXV multiplexers are a subtype of output-extended incomplete binary-tree multiplexers in which the output of the initial symmetric multiplexer is coupled into the input of the novel base router with the lower transmission efficiency, and any novel router on the level under construction in the incomplete branch are added to the arm with the highest total transmission coefficient $V_n$ of the optional connection points in the level under construction in the incomplete branch [37]. We use these multiplexer types for the comparison with the proposed multiplexer because they were proved to outperform any other spatial multiplexer types reported in the literature [37,38].
Fig. 2. Scheme of a single-photon source based on a 4-level output-extended incomplete binary-tree multiplexer containing ten binary photon routers. Photon routers PR$_2$ and PR$_{10}$ are on the incomplete branch of the multiplexer connected to the base router PR$_1$, while photon routers PR$_3$ to PR$_9$ form a complete 3-level binary-tree multiplexer. MU$_i$s denote the multiplexed units.
Next, we briefly summarize the theory that can be used for the construction of the stepwise optimized binary-tree multiplexers and for the analysis of SPSs based on such multiplexers. The full statistical theory of periodic SPSs based on either temporal or spatial multiplexing operated with photon-number-resolving detectors capable of realizing any detection strategy was presented in Refs. [11,36]. In our calculations, we consider ranges of the loss parameters for which single-photon detection realized by single-photon detectors with photon-number-resolving capability is certainly the optimal detection strategy [11,36]. Therefore, the applied theoretical description outlined below is restricted to such type of detection.
The probability that a multiplexed SPS containing $N$ multiplexed units equipped with single-photon detectors with photon-number-resolving capability emits $i$ photons can be written as
The first term in Eq. (1) describes the case when none of the detectors have detected a single photon in a multiplexed unit. This term contributes only to the probability $P_0$, that is, to the case when no photon is obtained at the output. The first factor in the second term is the probability that none of the detectors have detected a single photon in the first $n-1$ multiplexed units. In this case the detection event occurs in the $n$th unit. We assume that no further photons are allowed into the multiplexer from the subsequent multiplexed units. The second factor in this term, that is, the inner sum, describes the case when, even though there are $l$ photons entering the multiplexer from the $n$th multiplexed unit after a detection event, only $i$ of these reach the output due to the losses of the multiplexer. In this inner sum, the first factor $P^{(D)}(1|l)$ is the probability that the detector registers a single photon out of $l$ incoming photons. The second factor $P^{\lambda }(l)$ is the probability that $l$ photons arrive at the detector. The third factor $V_n(i|l)$ is the probability that $i$ photons leave the multiplexer provided that $l$ signal photons enter the system from the $n$th multiplexed unit. The summation over $n$ in the second term takes into consideration all the possible contributions to the probability $P_i$.
In Eq. (1), the probability $P^{(\lambda )}(l)$ corresponds to the generation of $l$ photon pairs in a multiplexed unit characterized by the input mean photon number $\lambda$. The probability distribution of the photon pair generation $P^{(\lambda )}(l)$ is assumed to be thermal, that is,
which is characteristic to single-mode nonlinear processes with stronger spectral filtering. Assuming thermal distribution leads to multiplexed SPSs generating highly indistinguishable single photons that is required in many experiments and applications [29,34,44]. We note that as the photon probabilities $P_i$ in Eq. (1) depend on $P^{(\lambda )}(l)$, all the results and even the structures of SOBTMs depends on this probability distribution of input photons. Assuming different input statistics, e.g., Poissonian distribution, would result in different optimal multiplexed structures. The probability that a detector registers a single photon out of $l$ incoming photons can be expressed as where $V_D$ denotes the detector efficiency. The term $P^{(D)}_1$ is the total probability of the event of single-photon detection that can be written asAlthough the only imperfection included in Eqs. (3) and 4 is the finite detector efficiency, the realistic nature of the model is not significantly affected by neglecting other detector imperfections, as it was justified in detail in Ref. [11]. In our calculations, the probabilities $P^{(D)}_1$, $P^{(D)}(1|l)$, $P^{(\lambda )}(l)$, and the input mean photon number $\lambda$ were assumed to be independent of the sequential number $n$ of the multiplexed unit. Finally, the conditional probability $V_n(i|l)$ corresponds to the event that $i$ photons leave the multiplexer provided that $l$ signal photons enter the system from the $n$th multiplexed unit. This probability can be written as
where $V_n$ is the total transmission coefficient characterizing the $n$th arm of the particular multiplexer. The total transmission coefficients $V_n$ of the multiplexers proposed earlier can be described by mathematical formulas [11,16,35–38]. For example, the total transmission coefficients characterizing ASYM and OMAVX multiplexers can be found in Ref. [37]. On the other hand, in the case of SOBTM no such formula can be found. Therefore, the total transmission coefficients must be determined separately for each binary tree considered during the construction of the multiplexer. Evidently, all total transmission coefficients can be written as products of the transmission coefficients of the PRs. Accordingly, the symbolic form of any total transmission coefficient of a multiplexer having $N$ inputs can be written as where the multiplicative factor $V_b$ termed as general transmission coefficient is present to characterize all other losses experienced by the photons while propagating to the input of the multiplexer after their heralding.We note that in the case of multiple heralding events in different multiplexed units the priority logic assumed in Eq. (1) favors the multiplexed unit with the smallest sequential number $n$. As the numbering of the multiplexed units in SPSs based on any type of spatial binary-tree multiplexer is arbitrary, one can apply a renumbering of the multiplexed units so that the associated total transmission coefficients $V_n$ of the multiplexer are arranged into a decreasing order, that is, $V_i\ge V_j$ if $i<j$ ($i,j=1,\dots,N$). The multiplexer arms characterized with identical total transmission coefficients can be numbered arbitrarily. The application of such a numbering increases the single-photon probability since it entails that the multiplexer arm with the highest $V_n$ corresponding to the smallest loss is preferred by the logic implying that the probability of photon loss is decreased in the multiplexer [11,36]. Naturally, this renumbering must be applied for all the structures considered during the proposed procedure of the construction of the stepwise optimized binary-tree multiplexers.
Beside single-photon probability, one can use the normalized second-order autocorrelation function defined as
In the following, we describe the way this theory can be used to find the SPS based on a stepwise optimized binary-tree multiplexer for which the single-photon probability is maximal. As it was discussed above, if the positions of the first $N_R-1$ PRs is fixed, the $N_R$th PR can be connected to any of the $N_R$ input ports of the multiplexer, therefore $N_R$ different multiplexer structures must be considered. At each structure $S_m$ ($m=1,\dots,N_R$), the single-photon probability $P_{1}^{S_m}(\lambda )$ of the SPS based on the given multiplexer as a function of the input mean photon number $\lambda$ can be derived from Eq. (1). As it was pointed out in previous papers, such a function always has a maximum [11,16]. Comparing the maximum values of the $P_{1}^{S_m}(\lambda )$ functions for different structures containing $N_R$ PRs, the structure with the highest single-photon probability is said to be optimal and denoted by $S_{\text {opt},N_R}$. Then the highest single-photon probability of the SPS based on the optimal structure containing $N_R$ PRs is termed as achievable single-photon probability and denoted by $P_{1,N}$, while the corresponding optimal input mean photon number is denoted by $\lambda _{\text {opt},N}$. Recall that the number $N$ of multiplexed units is one more than the number $N_R$ of PRs, that is, $N=N_R+1$. By repeating this procedure for various consecutive numbers of multiplexed units from $N=2$, we get the achievable single-photon probability as a (discrete) function of the number of multiplexed units $N$, that is, the function $P_{1,N}(N)$. Earlier it was shown [36,37] that for SPSs based on incomplete binary-tree multiplexers (including those based on complete binary-tree multiplexers as a special case) the function $P_{1,N}(N)$ has a well-defined maximum, therefore there exists a pair of unique optimal values both for the input mean photon number and the number of multiplexed units, $\lambda _{\text {opt}}$ and $N_{\text {opt}}$, respectively, that gives the maximal single-photon probability $P_{1,\max }$ for a given set of the loss parameters. On the other hand, for asymmetric multiplexers, it was shown that the function $P_{1,N}(N)$ saturates [30,38]. As it will be shown in the next section, a similar behavior of $P_{1,N}(N)$ can be observed for SPSs based on SOBTMs. Accordingly, for ASYM multiplexers or SOBTMs the optimal number of multiplexed units $N_{\text {opt}}$ must be defined as a viable value for which the corresponding single-photon probability is close to the saturated value. To this aim, we sequentially determine the differences $\Delta _P(N)=P_{1,N}-P_{1,N-1}$ between the achievable single-photon probabilities for neighboring values of the numbers of multiplexed units, and choose the largest number of multiplexed units $N$ as optimal for which this difference first falls below $10^{-4}$.
3. Results
In this section, we present our results on the optimization of SPSs based on SOBTMs built of general asymmetric routers. Our goal is to prove that using the proposed construction logic generally yields spatially multiplexed SPSs having better performance than SPSs based on the construction logics proposed earlier in the literature. The most relevant question concerns the highest single-photon probability that can be achieved with such a system. Therefore, our calculations are limited to high transmission and detector efficiencies that are experimentally realizable using state-of-the-art devices. The highest reported values of the transmission coefficients of asymmetric photon routers realized with bulk optical elements are $V_t=0.985$ and $V_r=0.99$ [23,45], while the highest detector efficiencies reported in the literature are $V_D=0.98$ achieved by a titanium-based transition-edge sensor [40] and $V_D=0.995$ for a superconducting nanowire single-photon detector [46]. Accordingly, we present results for the ranges of the transmission coefficients $0.85\leq V_r\leq 0.99$ and $0.85\leq V_t\leq 0.99$, and for five discrete values of the detector efficiency $V_D$ chosen from the set $\{0.6,0.8,0.9,0.95,0.98\}$. We use the above highest values of the router transmission coefficients and the detector efficiency $V_D=0.95$ whenever fixed single values of these quantities are assumed. The general transmission coefficient is always fixed at $V_b=0.98$, therefore this value is not specified in the rest of this section.
Figure 3 shows the relationship between the number of multiplexed units $N$ and the two main characteristic quantities, that is, the achievable single-photon probability $P_{1,N}(N)$ (Fig. 3(a)) and the second-order autocorrelation function $g^{(2)}_{N}(N)$ (Fig. 3(b)) for SPSs based on SOBTMs for the transmission coefficients $V_t=0.985$ and $V_r=0.99$, and for various values of the detector efficiency $V_D$. Though the functions $P_{1,N}(N)$ are only plotted up to $N=40$, by performing calculations for even higher values of $N$ and other values of the loss parameters we found that these functions asymptotically approach well-defined values. This asymptotic behavior of the function $P_{1,N}(N)$ justifies the optimization process described in Sec. 2. For the detector efficiencies $V_D=\{0.6,0.8,0.9,0.95,0.98\}$, the maximal single-photon probabilities are $P_{1,\max }=\{0.871, 0.890, 0.904, 0.916, 0.928\}$. The corresponding values of the second-order autocorrelation function are $g^{(2)}_N=\{0.085,0.063,0.050,0.041,0.026\}$. The figures also show that assuming higher values of the detector efficiency leads to higher achievable single-photon probabilities and, simultaneously, lower values of the second-order autocorrelation function. For the detector efficiency $V_D=0.8$, single-photon probabilities exceeding 0.8 can be achieved with $N\geq 15$ multiplexed units, while using $N\geq 26$ multiplexed units can yield single-photon probabilities $P_1>0.85$. Assuming higher detector efficiency $V_D=0.95$, the latter single-photon probability $P_1=0.85$ can be attained with only $N=10$ multiplexed units, and $N=18$ multiplexed units is enough to get an output probability $P_1>0.9$. This finding means that high single-photon probabilities can be achieved with feasibly low number of multiplexed units.
Fig. 3. (a) The achievable single-photon probabilities $P_{1,N}$ and (b) the values of the normalized second-order autocorrelation function $g^{(2)}_{N}$ for SPSs based on SOBTMs as functions of the number of multiplexed units $N$ for the transmission coefficients $V_t=0.985$ and $V_r=0.99$, and for various values of the detector efficiency $V_D$.
In Fig. 4 we present the optimal input mean photon numbers $\lambda _{\text {opt},N}$ for SPSs based on SOBTMs as functions of the number of multiplexed units $N$ for the transmission coefficients (a) $V_t=0.985$, $V_r=0.99$, and (b) $V_t=0.9$, $V_r=0.95$, and for various values of the detector efficiency $V_D$. The figures show that increasing the number of multiplexed units $N$ leads to a decrease in the value of $\lambda _{\text {opt},N}$. This behavior reflects the idea of multiplexing, that is, the input mean photon number is decreased to decrease multiphoton noise while the number of multiplexed units is increased to maintain a high probability of a successful heralding of a signal photon in the system [1,2,11,16]. The function $\lambda _{\text {opt},N}(N)$, similarly to the achievable single-photon probability $P_{1,N}(N)$ and the second-order autocorrelation function $g^{(2)}_N(N)$, have an asymptotic behavior. Note that for the detector efficiencies $V_D=\{0.6,0.8,0.9,0.95,0.98\}$ the corresponding optimal numbers of multiplexed units are $N_{\text {opt}}=\{170, 99, 63, 42, 31\}$ for subfigure (a) and $N_{\text {opt}}=\{48,31,25,24,24\}$ for subfigure (b). We have checked that above these numbers the values $\lambda _{\text {opt},N}$ are practically equal to their asymptotic values. From the figures it can be seen that for higher detector efficiencies $V_D$ the values of $\lambda _{\text {opt},N}$ are higher. It can be explained by the fact that a better photon number resolving detector can better suppress the multiphoton contribution in the signal entering the multiplexer. Accordingly, the value of the $\lambda _{\text {opt},N}$ can be increased to increase the probability of successful heralding. Note that for higher detector efficiencies $V_D$ the values of the optimal input mean photon number $\lambda _{\text {opt}}$ are higher while the corresponding number of multiplexed units $N_{\text {opt}}$ becomes smaller, as it can be seen from the data for $N_{\text {opt}}$ above. Comparing subfigures (a) and (b) one can see that for a fixed value of the detector efficiency $V_D$ and for lower transmission coefficients $V_r$ and $V_t$ the values of $\lambda _{\text {opt},N}$ are higher. In this case the presence of higher losses in the multiplexer has been compensated by increasing the value of the input mean photon number.
Fig. 4. The optimal input mean photon numbers $\lambda _{\text {opt},N}$ for SPSs based on SOBTMs as functions of the number of multiplexed units $N$ for the transmission coefficients (a) $V_t=0.985$, $V_r=0.99$, and (b) $V_t=0.9$, $V_r=0.95$, and for various values of the detector efficiency $V_D$.
Next, we present results on the optimization of SPSs based on SOBTMs for various sets of the loss parameters. In Fig. 5 quantities characterizing the system are presented as functions of the transmission coefficients $V_t$ and $V_r$ for the detector efficiency $V_D=0.95$. Figures 5(a) and 5(b) present the maximal single-photon probability $P_{1,\max }$ and the normalized second-order autocorrelation function $g^{(2)}$, respectively. Figures 5(c) and 5(d) show the difference $\Delta _P^{\text {sobtm}-\text {asym}}= P_{1,\max }^{\text {sobtm}}-P_{1,\max }^{\text {asym}}$ between the maximal single-photon probabilities and the difference $\Delta_{g^{(2)}}^{\text {asym-sobtm }}=g_{\text {asym }}^{(2)}-g_{\text {sobtm }}^{(2)}$ between the second-order autocorrelation functions for optimized SPSs based on SOBTM and ASYM multiplexers, respectively. Lastly, Figs. 5(e) and 5(f) present the difference $\Delta _P^{\text {sobtm}-{\rm omaxv}}= P_{1,\max }^{\text {sobtm}}-P_{1,\max }^{\rm omaxv}$ between the maximal single-photon probabilities and the difference $\Delta_{g^{(2)}}^{\text {omaxv-sobtm }}=g_{\text {omaxv }}^{(2)}-g_{\text {sobtm }}^{(2)}$ between the second-order autocorrelation functions for optimized SPSs based on SOBTM and OMAXV multiplexers, respectively. Recall that maximal single-photon probabilities $P_{1,\max }$ can be achieved for the optimal numbers of multiplexed units $N_{\text {opt}}$ not presented here. The values of the second-order autocorrelation function $g^{(2)}$ are also calculated at the optimal numbers $N_{\text {opt}}$. According to the results presented in Fig. 5, the highest maximal single-photon probability that can be achieved by using SPSs based on SOBTMs for $V_r=V_t=0.99$ is $P_{1,\max }=0.924$ which is $\Delta _P^{\text {sobtm}-\text {asym}}=0.015$ higher than that can be achieved with ASYM multiplexers, and $\Delta _P^{\text {sobtm}-{\rm omaxv}}=0.004$ higher than that can be achieved with OMAXV multiplexers. Figures 5(c) and 5(e) also show that applying SPSs based on SOBTMs gives higher maximal single-photon probabilities than those based on ASYM or OMAXV multiplexers on the whole considered parameter range, and the highest differences exceeding 0.025 are for symmetric or almost symmetric PRs, that is, $V_r\approx V_t$, for both ASYM and OMAXV multiplexers. This result will be explained in the discussion of Fig. 7 where examples of SOBTM structures are shown. We note that applying SPSs based on SOBTMs can also be advantageous for very asymmetric PRs, that is, for the transmission coefficients $V_r\gg V_t$ or $V_r\ll V_t$. However, high values of the maximal single-photon probabilities $P_{1,\max }$ is only one requirement toward SPSs, low values of the second-order autocorrelation function $g^{(2)}$, that is, low multiphoton contribution, is also important. Figure 5(d) presents the difference $\Delta _{g^{(2)}}^{\text {asym}-\text {sobtm}}=g^{(2)}_{\text {asym}}-g^{(2)}_{\text {sobtm}}$ between the second-order autocorrelation functions for SPSs based on ASYM multiplexers and SOBTMs, respectively, while Fig. 5(f) shows the difference $\Delta_{g^{(2)}}^{\text {asym-sobtm }}=g_{\text {asym }}^{(2)}-g_{\text {sobtm }}^{(2)}$ between the second-order autocorrelation functions for SPSs based on OMAXV multiplexers and SOBTMs, respectively. Obviously, positive values in these figures show that SPSs based on SOBTMs have smaller $g^{(2)}$ values, that is, the multiphoton contribution in these points are smaller. The value of the second-order autocorrelation function that can be achieved by using SPSs based on SOBTMs for $V_r=V_t=0.99$ is $g^{(2)}=0.0369$, the difference between the values of this quantity for SPSs based on ASYM multiplexers and SOBTMs for $V_r=V_t=0.99$ is $\Delta _{g^{(2)}}^{\text {asym}-\text {sobtm}}=0.0236$, while the difference between the values of this quantity for SPSs based on OMAXV multiplexers and SOBTMs for $V_r=V_t=0.99$ is $\Delta_{g^{(2)}}^{\mathrm{omaxv}-\mathrm{sobtm}}=-0.0118$. According to Fig. 5(d), SPSs based on SOBTMs have smaller $g^{(2)}$ values than those based on ASYM multiplexers for not very asymmetric PRs, and the highest differences can be observed for symmetric PRs for the highest values of the transmission coefficients $V_t$ and $V_r$. As for the results presented in Fig. 5(f), one can deduce that the difference $\Delta_{g^{(2)}}^{\text {omaxv-sobtm }}$ shows a strong dependence on the values of the transmission coefficients $V_r$ and $V_t$, although the difference is considerably smaller than that observed in Fig. 5(d) for $\Delta _{g^{(2)}}^{\text {asym}-\text {sobtm}}$. These observations can be explained as follows. A SPS based on a multiplexer with higher optimal number of multiplexed units $N_{\text {opt}}$ and, correspondingly, operating with lower optimal input mean photon numbers $\lambda _{\text {opt}}$ has generally lower multiphoton noise, that is, the values of the second-order autocorrelation function $g^{(2)}$ are lower. In Ref. [37] it was shown that the optimal numbers of multiplexed units $N_{\text {opt}}$ for SPSs based on OMAXV multiplexers are higher than those for SPSs based on ASYM multiplexers. Consequently, the values of the second-order autocorrelation function $g^{(2)}$ are lower, that is, better, for SPSs based on OMAXV multiplexers than those for SPSs based on ASYM multiplexers. In the case of SPSs based on SOBTMs the optimal numbers of multiplexed units $N_{\text {opt}}$ are close to those obtained for SPSs based on OMAXV multiplexers. Though detailed data for $N_{\text {opt}}$ are not presented here, some examples of this quantity for SPSs based on SOBTMs are presented in the previous paragraph. These characteristics explain why the values of the second-order autocorrelation functions $g^{(2)}$ for SPSs based on SOBTMs and for those based on OMAXV multiplexers are in the same range; only small differences occur due to the specific differences between the two systems. Also, the properties of the differences $\Delta_g^{\text {asym-sobtm }}$ between the values of the second-order autocorrelation functions $g^{(2)}$ for SPSs based on ASYM multiplexers and for those based on SOBTMs are very similar to the differences $\Delta_{g^{(2)}}^{\text {asym-omaxv }}$ between the values of $g^{(2)}$ for SPSs based on ASYM multiplexers and for those based on OMAXV multiplexers presented in Ref. [37].
Fig. 5. (a) The maximal single-photon probability $P_{1,\max }$ and (b) the normalized second-order autocorrelation function $g^{(2)}$ for SPSs based on SOBTMs, the differences (c) $\Delta _P^{\text {sobtm}-\text {asym}}$ and (e) $\Delta _P^{\text {sobtm}-{\rm omaxv}}$ between the maximal single-photon probabilities, and the differences (d) $\Delta _{g^{(2)}}^{\text {asym}-\text {sobtm}}$ and (f) $\Delta_{g^{(2)}}^{\text {omaxv-sobtm }}$ between the second-order autocorrelation functions for SPSs based on SOBTM and ASYM multiplexers and SOBTM and OMAXV multiplexers, respectively, as functions of the transmission coefficients $V_t$ and $V_r$ for the detector efficiency $V_D=0.95$.
An important output of our calculations is that it is possible to determine the SOBTM for any set of loss parameters and numbers of multiplexed units. In the following, we present results for the optimal structures for a fixed number $N$ of the multiplexed units. Figure 6 presents the occurrence $O_{S_\text {opt}}$ of the various optimal multiplexer structures $S_\text {opt}$ for SPSs based on SOBTMs for $V_r\geq V_t$ and for various values of the detector efficiency for $N=10$ multiplexed units. Each color in a subfigure of Fig. 6 corresponds to a particular multiplexer structure, hence a color shows the values of the transmission coefficients $V_t$ and $V_r$ for which the corresponding structure is optimal. Each sequential number of the occurrence $O_{S_\text {opt}}$ shown in the color bars in Fig. 6 is related to the size of the area occupied by the color assigned to the sequential number in the $V_r$-$V_t$ plane. Sequential number 1 corresponds to the color occupying the largest area, and colors indicated by larger sequential numbers cover smaller areas in the figure. Hence, the multiplexer structure denoted by the color corresponding to the sequential number 1 is the most frequent, that is, it is the optimal multiplexer structure for most of the transmission coefficient pairs $V_r$, $V_t$.
Fig. 6. Occurrence $O_{S_\text {opt}}$ of the various optimal multiplexer structures $S_\text {opt}$ for SPSs based on SOBTMs for $V_r\geq V_t$ and the detector efficiencies (a) $V_D=0.95$, (b) $V_D=0.9$, (c) $V_D=0.85$, and (d) $V_D=0.8$ for the number of multiplexed units $N=10$. A particular color denotes a given structure. Increasing sequential numbers of $O_{S_\text {opt}}$ in the color bar represent decreasing occurrence of a specific structure. Color white represents all the structures in the opposite regions and those with even smaller occurrence.
In Fig. 6 we present the occurrence of the first 20 most frequent structures only, although the actual numbers of occurring structures in the regions $V_r\geq V_t$ are 20 for $V_D=0.95$ (a), 24 for $V_D=0.9$ (b), 37 for $V_D=0.85$ (c), and 40 for $V_D=0.8$ (d). Accordingly, the color white in the figure represents not only all the structures in the opposite region, that is, in the region $V_r<V_t$, but also those that are less frequent than the first 20. Note that a given color in the subfigures of Fig. 6 can indicate different structures for different detector efficiencies $V_D$. We have compared the structures obtained for the detector efficiencies $V_D=\{0.8,0.85,0.9\}$ and $V_D=0.95$. We have found that the numbers of unique structures present for $V_D=0.8$, $V_D=0.85$ and $V_D=0.9$ but absent for $V_D=0.95$ are 33, 26, and 12, respectively, while there are 13, 9, and 8 unique structures present for $V_D=0.95$ and absent for $V_D=0.8$ $V_D=0.85$, and $V_D=0.9$. These results show that the detection efficiency affects the optimal structure obtained for a given pair of transmission coefficients $V_t$ and $V_r$ and for a given number of multiplexed units $N$. Also, as it can be seen above, the decrease of the detector efficiency $V_D$ leads to an increasing number of occurring structures. This observation can be explained by the fact that the decrease in the detector efficiency $V_D$ increases the multiphoton contribution in the signal entering the multiplexer. Different levels of the multiphoton noise can be efficiently handled in our approach by different optimal structures even for identical loss parameters $V_t$ and $V_r$ characterizing the PRs.
In the following, we show examples to the optimal structures of SOBTMs. Figure 7 presents the six most frequent optimal structures of SOBTMs in the region $V_r\geq V_t$ for the detector efficiency $V_D=0.95$ and the number of multiplexed units $N=10$. The order of the labels from (a) to (f) of the subfigures follows the order of the sequential numbers from 1 to 6 of the occurrences $O_{S_{\text {opt}}}$ of the structures. Accordingly, the structure in Fig. 7(a) corresponds to the yellow region in the $V_t-V_r$ plane in Fig. 4(a), that is, this structure is optimal for symmetric or nearly symmetric PRs. The upper and lower inputs of all the individual PRs correspond to the router transmission coefficients $V_t$ and $V_r$, respectively. The optimal structures obtained by using the proposed method are different from the structures reported in the literature, with the exception of the structure shown in Fig. 7(d) that is equivalent to the asymmetric (chain-like) structure. It can be seen in Fig. 7 that SOBTMs are basically chain-like structures but they generally contain additional arms built of low numbers of routers. The losses along such arms are generally lower than those along longer arms. As a consequence, the number of low-loss arms needed for achieving high single-photon probabilities is generally higher in SOBTMs compared to ASYM and OMAXV multiplexers leading to higher performance when they are applied in SPSs. This property is valid for structures containing symmetric or almost symmetric routers (see Fig. 7(a)). On the other hand, this advantage of SOBTMs diminishes when asymmetric routers are applied as in this case low-loss arms can appear in ASYM and OMAXV multiplexers, too [30,37,38]. These characteristics explain the observations in Figs. 5(c) and 5(e). Note that, due to symmetry considerations, the structures that can be obtained for the region $V_r \leq V_t$ are generally the same but mirrored to a horizontal axis and, accordingly, the corresponding figures representing the occurrences $O_{S_\text {opt}}$ of the various optimal multiplexer structures are also the same as those in Fig. 6 but mirrored to the $V_r=V_t$ line. However, the actual structures obtained by the construction logic can be different for a specific pair of loss parameters in the regions $V_r\geq V_t$ and $V_r\leq V_t$. As an example, Fig. 8 shows the fifth most frequent optimal structures of SOBTMs in the two regions for the detector efficiency $V_D=0.95$ and $N=10$ multiplexed units. Obviously, these two structures are different, although the set $\{V_n\}$ of the total transmission coefficients in Fig. 8(a) is the same as the corresponding set in Fig. 8(b). These sets are (a) $\{V_t^3, V_rV_t^2, V_rV_t^2, V_r^2V_t^2, V_r^3V_t^2, V_r^4V_t, V_rV_t, V_r^2V_t, V_r^3V_t, V_r^4\}$ and (b) $\{V_t^4, V_rV_t^4, V_r^2V_t^3, V_rV_t^3, V_r^2V_t^2, V_rV_t^2, V_r^2V_t, V_rV_t, V_r^2V_t, V_r^3\}$. After the reordering of the total transmission coefficients in these sets and swapping the roles of the transmission coefficients $V_r$ and $V_t$ one can see that the resulting ordered sets indeed coincide. Accordingly, the two different structures are physically equivalent. The appearance of different structures in the two regions is due to the applied logic in the implementation of the proposed optimization method. Recall that, after fixing the optimal structure for $N_R$ PRs, the output of the next router is fitted systematically to all possible inputs of that optimal structure, in the geometry of the figures from top to bottom. If the sets $\{V_n\}$ of total transmission coefficients contain the same values for two (or more) of those structures formed in this way then the first structure is kept and the latter ones are not considered at all. This can lead to different structures in the regions $V_r>V_t$ and $V_r<V_t$.
Fig. 7. The six most frequent optimal structures of SOBTMs in the region $V_r\geq V_t$ for the detector efficiency $V_D=0.95$ and the number of multiplexed units $N=10$. $V_t$ and $V_r$ correspond to the upper and lower inputs, respectively, of the individual PRs. The order of the labels from (a) to (f) of the subfigures follows the order of the sequential numbers from 1 to 6 of the occurrences $O_{S_{\text {opt}}}$ of the structures shown in Fig. 6. MU$_i$s denote the multiplexed units.
Fig. 8. The fifth most frequent optimal structures of SOBTM in the regions (a) $V_r\geq V_t$ and (b) $V_r\leq V_t$ for the detector efficiency $V_D=0.95$ and the number of multiplexed units $N=10$. $V_t$ and $V_r$ correspond to the upper and lower inputs, respectively, of the individual PRs. MU$_i$s denote the multiplexed units.
Finally, we compare our results to the ones obtained in recent multiplexed single-photon source experiments. To our knowledge, the highest single-photon probability obtained experimentally is $P_{1}=0.667$ [23]. In this experiment, a time-multiplexed source was realized where an adjustable delay line, that is, a storage loop was used for multiplexing 40 time bin modes. A theoretical analysis of this experimental system was presented in Ref. [11] for the loss parameters of the particular arrangement using the same statistical framework as the one applied here. According to the analysis, the behavior of this system is very similar to that of single-photon sources based on asymmetric multiplexers as the expressions describing the total transmission coefficients of the two systems are mathematically very similar. This foreshadows that SPSs based on SOBTMs can outperform the given time-multiplexed systems in the same way as they outperform SPSs based on ASYM multiplexers. For the proposed system, the single-photon probabilities $P_{1,N}=0.683$ and $P_{1,N}=0.694$ can be achieved by using $N=7$ multiplexed units for the detector efficiency $V_D=0.8$, and $N=5$ units for $V_D=0.95$, respectively, for the experimentally realizable loss parameters applied in Fig. 3. Hence, applying SOBTMs in single-photon sources, single-photon probabilities higher than those obtained in the considered time-multiplexed experiments can be achieved with low numbers of multiplexed units that seem to be experimentally realizable.
Recently, a frequency-multiplexed single-photon source with three frequency modes has been demonstrated [47]. In this experiment, four-wave mixing with different pump frequencies is used for frequency conversion of the heralded photon in the multiplexer realizing the switching between modes. The advantage of this multiplexing scheme is that the loss of the multiplexer is independent of the number of multiplexed frequency modes, though the realization of higher number of modes can be technically challenging. However, this loss is relevantly higher than the losses of the photon routers considered here, e.g., the value of the transmission of the frequency multiplexer is $V_{\rm fm}=0.85$ in the cited experiment. It can be easily checked by applying our theoretical approach and using this value of the transmission that the achievable single-photon probabilities $P_{1,N}$ for such frequency-multiplexed systems are lower for each number of multiplexed units $N$ than those presented in Fig. 3(a) for the proposed system for experimentally realizable loss parameters.
4. Conclusion
We have proposed a novel spatially multiplexed single-photon source based on stepwise optimized binary-tree multiplexer. Along the building procedure of this type of multiplexer, the structure of a binary-tree multiplexer containing $N_R$ photon routers is optimized in each step. That is, starting from an optimized binary-tree multiplexer containing $N_R-1$ photon routers, all possible binary-tree multiplexers are considered that can be created by appending the $N_R$th photon router in succession to all possible inputs of the initial multiplexer. Then, we choose the multiplexer with $N_R$ routers as the optimized multiplexer for an optional next step that yields the highest single-photon probability when it is applied in a single-photon source. This method is scalable, that is, for any number of photon routers it is possible to determine the multiplexer with an optimal structure.
We have shown that single-photon sources based on stepwise optimized binary-tree multiplexers yield higher single-photon probabilities than single-photon sources based on any spatial multiplexer types reported in the literature thus far in the considered ranges of the loss parameters. The multiphoton noise characterized by the second-order autocorrelation function of single-photon sources based on the proposed multiplexer is generally lower than that of sources based on asymmetric multiplexer, while it is in the same range when compared to that of single-photon sources based on minimum-based, maximum-logic output-extended incomplete binary-tree multiplexers.
Funding
National Research, Development and Innovation Office (KKP133827, TKP2021-EGA-17, TKP2021-NVA-04).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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