Effect of loss on multiplexed single-photon sources

All the MUX sources we consider are composed of a core component called the HSPS (figure 1(a)). A HSPS is a non-deterministic source with a logic output set to 1 when it emits a photon and set to 0 otherwise. We consider a HSPS composed of (1) a photon pair generation stage in which photons are produced at non-degenerate wavelengths ${{\lambda }_{s}}$ (signal) and ${{\lambda }_{i}}$ (idler), (2) a filter which removes the pump and separates the signal from the idler into two different paths, and (3) a detector on the idler arm which heralds the emission of the signal photon. For now, we assume the photon pairs are generated as a biphoton two-mode squeezed vacuum state such that their joint spectrum is disentangled [28]. Furthermore, we assume that all signal photons produced from different HSPSs are perfectly indistinguishable in all degrees of freedom except for the spatial mode in which they are generated.

The source is characterized by a squeezing parameter ξ, determined by the pump intensity and strength of the nonlinearity, a trigger probability p_trig, the probability for the heralded state to be a single photon p_single, also called the state fidelity or heralding efficiency, and the probability for the heralded state to be contaminated with multiple photons, p_multi. Quantifying multi-photon contamination separately from vacuum emissions is especially important; vacuum emissions can be treated as effective loss in a linear optical quantum circuit while multi-photon events can result in other types of errors, and linear optical quantum circuits have been shown to have a much higher tolerance to loss than to other errors [10].

We aim at deriving expressions for p_trig, p_single, and p_multi for the case in which the heralding detector is number-resolving and the case in which the detector is non-number resolving (also called a threshold detector). The state produced by the pair source, assuming spectral disentanglement, is of the form [29]:

$$\begin{eqnarray}&&|\psi \rangle =\sqrt{1-|\xi {{|}^{2}}}\left( |0{{\rangle }_{i}}|0{{\rangle }_{s}}+\xi |1{{\rangle }_{i}}|1{{\rangle }_{s}}+\mathop{\sum }\limits_{n=2}^{\infty }{{\xi }^{n}}|n{{\rangle }_{i}}|n{{\rangle }_{s}} \right),\end{eqnarray} \tag{ 1 }$$

where i and s are the idler and signal modes. In practice, sources and filters have losses, and the heralding detector does not have a unit efficiency detection. We call ${{\eta }_{i}}$ the global collection efficiency on the idler arm accounting for all these effects, and ${{\eta }_{s}}$ is the overall transmission on the signal arm accounting for losses in the sources and filters. Each lossy component is modelled as an ideal component preceded by an ideal beamsplitter with a non-unit transmission probability. The full state, after accounting for losses and tracing over loss modes, can therefore be written:

$$\begin{eqnarray}&&\hat{\rho }=\left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}{{{\hat{\rho }}}_{p,k}} \right),\end{eqnarray} \tag{ 2 }$$

where ${{\hat{\rho }}_{p,k}}=|p{{\rangle }_{i}}|k{{\rangle }_{s}}\langle p{{|}_{i}}\langle k{{|}_{s}}$ and C_n^k is the binomial coefficient.

This expression will be the starting point to derive ${{p}_{{\rm trig}\,D}}$, ${{p}_{{\rm single}\,D}}$, and ${{p}_{{\rm multi}\,D}}$ for the two detection schemes. In this paper we use the subscript D on expressions to indicate the type of detector considered: NRD for number-resolving detector and TD for threshold detector. Detailed derivations can be found in appendix A.

Threshold detector

Starting with the reduced state (equation (2)) and tracing out the signal mode, the probability for the detector on the idler arm to trigger is given by summing all the contribution of the states having at least one photon and is given by:

$$\begin{eqnarray}&&{{p}_{{\rm trig}\,{\rm TD}}}=|\xi {{|}^{2}}{{\eta }_{i}}1-|\xi {{|}^{2}}1\eta i.\end{eqnarray} \tag{ 3 }$$

The heralded state in the signal arm is expressed, renormalizing by dividing by ${{p}_{{\rm trig}\,{\rm TD}}}$, as:

$$\begin{eqnarray}&&{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}=\frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=1}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}|k{{\rangle }_{s}}\langle k{{|}_{s}} \right).\end{eqnarray} \tag{ 4 }$$

We can compute the probability that the heralded state is a single-photon using ${{p}_{{\rm single}\,{\rm TD}}}={{\langle 1{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|1\rangle }_{s}}$:

$$\begin{eqnarray}&&{{p}_{{\rm single}\,{\rm TD}}}=\left( 1-|\xi {{|}^{2}} \right){{\eta }_{s}}\frac{\left[ 1-{{\left( |\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right) \right)}^{2}}\left( 1-{{\eta }_{i}} \right) \right]\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right]}{{{\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right) \right]}^{2}}{{\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right) \right]}^{2}}}.\end{eqnarray} \tag{ 5 }$$

The probability that the heralded state contains multi-photon contamination is given by ${{p}_{{\rm multi}\,{\rm TD}}}=\sum _{m=2}^{\infty }{{\langle m{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|m\rangle }_{s}}$:

$$\begin{eqnarray}&&{{p}_{{\rm multi}\,{\rm TD}}}=\frac{{{Z}_{{\rm TD}}}}{{{p}_{{\rm trig}\,{\rm TD}}}}-{{p}_{{\rm single}\,{\rm TD}}},\end{eqnarray} \tag{ 6 }$$

with

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{Z}_{{\rm TD}}} & = & \left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}\left( \frac{1}{1-|\xi {{|}^{2}}}+\frac{\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right)} \right. \\ {} & {} & -\left. \frac{\left( 1-{{\eta }_{i}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)}-\frac{\left( 1-{{\eta }_{s}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)} \right). \\ \end{array}\end{eqnarray} \tag{ 7 }$$

Note that we can also similarly derive an expression for the probability that the heralded state is any Fock number state using ${{\langle n{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|n\rangle }_{s}}$.

Number-resolving detector

We now consider the case in which the detector is number resolving. In practice, we only need to discriminate the single-photon state from the vacuum and from states having two or more photons. Therefore, the detector does not need to be able to distinguish between two and $N\gt 2$ photon states. Starting again from equation (2), tracing out the signal mode and calculating the probability to get only one photon gives:

$$\begin{eqnarray}&&{{p}_{{\rm trig}\,{\rm NRD}}}=\frac{\left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}{{\eta }_{i}}}{{{\left( 1-\left( 1-{{\eta }_{i}} \right)|\xi {{|}^{2}} \right)}^{2}}}.\end{eqnarray} \tag{ 8 }$$

The corresponding heralded state is:

$$\begin{eqnarray}&&{{\hat{\rho }}_{{\rm heralded}\,{\rm NRD}}}=\frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm NRD}}}}{{\eta }_{i}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}n{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}|k{{\rangle }_{s}}\langle k{{|}_{s}} \right).\end{eqnarray} \tag{ 9 }$$

As in the previous case, we can compute the probability the heralded state contains one photon ${{p}_{{\rm single}\,{\rm NRD}}}={{\langle 1{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm NRD}}}|1\rangle }_{s}}$:

$$\begin{eqnarray}&&{{p}_{{\rm single}\,{\rm NRD}}}={{\left( 1-\left( 1-{{\eta }_{i}} \right)|\xi {{|}^{2}} \right)}^{2}}{{\eta }_{s}}\left( \frac{\left( 1+\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right)|\xi {{|}^{2}} \right)}{{{\left( 1-\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right)|\xi {{|}^{2}} \right)}^{3}}} \right).\end{eqnarray} \tag{ 10 }$$

Not surprisingly, contrary to the previous case, number-resolving detectors enable unit fidelity in the absence of losses (${{p}_{{\rm single}\,{\rm NRD}}}=1$ when ${{\eta }_{i}}={{\eta }_{s}}=1$). The probability for the heralded state to contain multi-photon contamination is given by:

$$\begin{eqnarray}&&{{p}_{{\rm multi}\,{\rm NRD}}}={{\eta }_{s}}\frac{1-\left( 1-{{\eta }_{s}} \right){{\left( |\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right)}^{2}}}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right) \right)}^{2}}}-{{p}_{{\rm single}\,{\rm NRD}}}.\end{eqnarray} \tag{ 11 }$$

We can again similarly derive an expression for the probability that the heralded state is any Fock number state using ${{\langle n{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm NRD}}}|n\rangle }_{s}}$.

We now have a complete framework for characterizing any HSPS. We assumed no joint spectral entanglement between photons in a generated photon pair. However, we can account for joint spectral entanglement by redefining the fidelity as $p_{{\rm single}\,D}^{\prime }={{p}_{{\rm single}\,D}}\times P$ where P is the purity of the heralded single photon. Recall that, in the ideal case of lossless components, the probability ${{p}_{{\rm trig}\,{\rm NRD}}}$ of heralding a unit fidelity single photon is bounded by 0.25 (achieved when $|\xi {{|}^{2}}=0.5$ maximizes the probability $|\xi {{|}^{2}}\left( 1-|\xi {{|}^{2}} \right)$). Clearly, HSPS do not suffice on their own to function as a near-deterministic single-photon source, providing in the ideal scenario one photon every four pulses on average.

As we will see in section 3, the probability of multi-photon contamination from individual HSPSs approximates the probability of multi-photon contamination from MUX sources. We therefore graph ${{p}_{{\rm multi}\,D}}$ as a function of idler transmission ${{\eta }_{i}}$ for several squeezing parameters $|\xi {{|}^{2}}$ in figures 2(a) and (b). To focus on the effect of ${{\eta }_{i}}$, we take ${{\eta }_{s}}=1$; the plots therefore serve as an upper bound on ${{p}_{{\rm multi}\,D}}$. We see that threshold detectors only reliably herald a single-photon state for very low levels of squeezing, while number-resolving detectors achieve a much lower level of contamination with the same squeezing parameter. With number-resolving detectors, 10% loss in the idler arm (n_i = 0.90), and pumping with ${{p}_{{\rm pair}}}=0.09$ $(|\xi {{|}^{2}}=0.1)$, the multi-photon contamination level is almost an order of magnitude lower ($\sim 0.02)$ compared with that of threshold detectors with the same squeezing parameter ($\sim 0.1)$. Despite their enhanced performance compared to threshold detectors, number-resolving detectors with the highest pumping parameters are not immune to loss; even 10% loss in the idler arm with ${{p}_{{\rm pair}}}=0.25$ $(|\xi {{|}^{2}}=0.5)$ results in a probability of multi-photon contamination of $\sim 0.1.$ Achieving a lower level of contamination requires a reduction in either the squeezing parameter or the loss in the idler arm.

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In section 3 we will show that a practical multiplexed source will require operation in the strong pumping regime, thus showing that threshold detectors can not be used for multiplexed sources with the highest efficiency and low levels of multi-photon contamination. For this reason, and due to space constraints, we will only consider number-resolving detectors in the remainder of the paper. However, the derived expression for threshold detectors can still be used in the framework for multiplexed sources.

A general MUX source can be characterized in a similar way to the HSPS. The probability per clock-cycle that at least one HSPS in an array of N HSPSs triggers is given by:

$$\begin{eqnarray}&&p_{{\rm trig}}^{{\rm MUX}}=1-{{\left( 1-{{p}_{{\rm trig}}} \right)}^{N}},\end{eqnarray} \tag{ 12 }$$

and the probability per clock-cycle that at least one source emits a triggered single-photon is:

$$\begin{eqnarray}&&{{p}_{1}}={{p}_{{\rm single}}}\left( 1-{{\left( 1-{{p}_{{\rm trig}}} \right)}^{N}} \right).\end{eqnarray} \tag{ 13 }$$

p₁ can in principle be made arbitrarily close to one, such that a near-deterministic single-photon source (>99% emission probability) can be made out of 17 HSPSs using a lossless switching network to route the photon from the HSPS which triggered to the output [13]. However, in any implementation, the switching network will have loss due to the optical delay lines—required for allowing enough time to reconfigure the switching network upon trigger from the HSPS—and the intrinsic loss of the switch components, for example 2 × 2 couplers and phase modulators for MZI-type switches (we will assume all 2 × 2 switches are MZI-type switches). The network loss, ${{\eta }_{{\rm network}}}(N)$, is proportional to the number of sources used in the MUX source, since additional sources require a larger switching network for routing. Provided the losses are equally distributed in the network (this assumption holds for two of the three schemes presented in the further sections), the network loss applies the same amount of loss to every HSPS.

For switching networks with balanced loss, the probability per clock-cycle for a MUX source to emit a single photon, conditioned on the MUX source triggering, is given by p_single^MUX, and is calculated from equation (5) or (10) (depending on the type of detector) by replacing ${{\eta }_{s}}$ in the expression with the full transmission ${{\eta }_{s}}{{\eta }_{{\rm network}}}(N)$. Similarly, the probability for a MUX source to emit multi-photon contamination, conditioned on the MUX source triggering, is given by p_multi^MUX, and is calculated from either equation (6) or (11) (depending on the type of detector) by replacing ${{\eta }_{s}}$ in the expression with the full transmission ${{\eta }_{s}}{{\eta }_{{\rm network}}}(N)$. Using this definition of p_single^MUX, it then follows that the probability per clock-cycle for a multiplexed source to emit a triggered single-photon can be written:

$$\begin{eqnarray}&&{{q}_{{\rm MUX}}}=p_{{\rm single}}^{{\rm MUX}}\times p_{{\rm trig}}^{{\rm MUX}}.\end{eqnarray} \tag{ 14 }$$

For switching networks with balanced loss, a convenient lower bound for this probability is given by:

$$\begin{eqnarray}&&q_{{\rm MUX}}^{*}={{p}_{{\rm single}}}\times p_{{\rm trig}}^{{\rm MUX}}\times {{\eta }_{{\rm network}}}(N),\end{eqnarray} \tag{ 15 }$$

which uses only the single-photon emission probability from the HSPSs and neglects cases in which multi-photon contamination from the HSPS reduces to a single-photon due to loss from the switching network.

For a multiplexed source pumped by a pulsed laser with repetition rate R, the emission rate of triggered single-photons is:

$$\begin{eqnarray*}&&{{R}_{{\rm MUX}}}=R\times {{q}_{{\rm MUX}}}.\end{eqnarray*}$$

Large-scale linear optical experiments require M multiplexed sources in parallel for generating M single photons. The M-photon state is heralded by the simultaneous triggering of all M MUX sources (at least one HSPS per MUX source). For a source pumped by a pulse laser with repetition rate R, the M-photon generation rate is:

$$\begin{eqnarray}&&R_{{\rm MUX}}^{M}=R\times {{\left( {{q}_{{\rm MUX}}} \right)}^{M}}.\end{eqnarray} \tag{ 16 }$$

The multi-photon contamination probability conditioned on heralding the M-photon state is:

$$\begin{eqnarray}&&p_{{\rm multi}}^{{\rm MUX}\,M}=1-{{\left( 1-p_{{\rm multi}}^{{\rm MUX}} \right)}^{M}}.\end{eqnarray} \tag{ 17 }$$

The analysis so far has been kept general, and can be applied to any MUX source with balanced network loss. In the next sections we focus on three specific architectures.

While photon loss and inefficient detectors are likely to be the dominant source of error for the near-term implementation of MUX sources, we note that there are other sources of error which will also have an effect on MUX performance. These include dark counts from single-photon detectors [30], mode-mismatch [31], and circuit faults. These effects are left to a future analysis. We also assume that the detector deadtimes are smaller than the system clock period, and therefore loss effects due to detector deadtimes are not considered in our current work.

The first MUX implementation we consider is called the log-tree source. In this scheme, the output optical ports of N HSPS sources are connected to an $N\;\times$ 1 reconfigurable switch, which is a logarithmic tree composed of 2 × 2 switches (figure 3(a)). Using N sources requires a log tree with a depth of $\left\lceil {\frac{{\rm ln} N}{{\rm ln} 2}} \right\rceil$ 2 × 2 switches in order to route any of the N sources to the output. The signal photons are stored in delay lines as the electrical trigger output is sent to a logic circuit and the network configuration is determined and set.

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The network loss is given by ${{\eta }_{{\rm network}}}(N)={{\eta }^\left\lceil {{\frac{{\rm ln} N}{{\rm ln} 2}}} \right\rceil }{{\eta }_{{\rm delay}}}$ where η is the transmission of a switch in the log tree network and ${{\eta }_{{\rm delay}}}$ is the loss of the delay line. From equation (14), the probability per clock-cycle for the log-tree multiplexed source to emit a triggered single-photon is given by:

$$\begin{eqnarray}&&{{q}_{{\rm tree}}}=p_{{\rm single}}^{{\rm tree}}\left( 1-{{\left( 1-{{p}_{{\rm trig}}} \right)}^{N}} \right).\end{eqnarray} \tag{ 18 }$$

and has the lower bound (equation (15)):

$$\begin{eqnarray}&&q_{{\rm tree}}^{*}={{p}_{{\rm single}}}\left( 1-{{\left( 1-{{p}_{{\rm trig}}} \right)}^{N}} \right){{\eta }^\left\lceil {{\frac{{\rm ln} N}{{\rm ln} 2}}} \right\rceil }{{\eta }_{{\rm delay}}}.\end{eqnarray} \tag{ 19 }$$

The optimal number of sources for a given switch loss is found by numerically finding the N which maximizes equation (19). To study the effect of switching loss in isolation from other sources of loss, the optimal N and $q_{{\rm tree}}^{*}$ with ${{\eta }_{{\rm delay}}}=1$ and ${{p}_{{\rm single}}}=1$ are plotted as a function of the 2 × 2 switch loss in figures 3(b) and (c). As expected, we see that the probability of triggered single-photon emission tends towards 1 in the limit of low switching loss for all trigger probabilities. The number of HSPSs required in the weak pump regime (p_trig < 0.01) is likely to be impractical: obtaining a probability of single photon emission $q_{{\rm tree}}^{*}\gt 0.9$ with ${{p}_{{\rm trig}}}=0.01$ requires 512 sources in parallel and switches with 0.05 dB loss (∼0.989 transmission). Obtaining $q_{{\rm tree}}^{*}\gt 0.9$ with ${{p}_{{\rm trig}}}=0.1$ requires 64 sources in parallel and switches with 0.07 dB loss (∼0.984 transmission). With the maximum trigger probability ${{p}_{{\rm trig}}}=0.25$, only 16 sources and switches with 0.1 dB loss (∼0.977 transmission) are required.

Because the switching network is balanced, the probability for a triggered log-tree MUX source to emit multi-photon contamination p_multi^MUX is calculated as explained in section 3.1 using ${{\eta }_{{\rm network}}}(N)={{\eta }^\left\lceil {{\frac{{\rm ln} N}{{\rm ln} 2}}} \right\rceil }{{\eta }_{{\rm delay}}}$. Since the loss from the switching network can only decrease p_multi^MUX, the multi-photon contamination probabilities for the HSPSs in figures 2(a) and (b) serve as valid upper bounds for p_multi^MUX. We will consider these expressions further when we consider M-photon state generation in section 4.

Like the log-tree source, this scheme also uses N HSPS sources connected to a $N\times 1$ reconfigurable switch (figure 4(a)). However, here the $N\times 1$ switch is a generalized Mach–Zehnder interferometer (GMZ)- composed of two N × N balanced splitters enclosing N phase modulators. N phase modulators are sufficient to route any input to a given fixed output port. This is achieved by setting half of the phases to π and setting the other half to 0, or by applying 0 to all the phases to obtain a full swap. The N × N passive splitter can either be a N × N MMI or built from of 2 × 2 couplers. Fabricating large N × N balanced MMIs with low loss is challenging, so we propose using cascaded couplers (having a reflectivity of 0.5) and crossings as shown in figures 4(b) and (c).

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The lower bound on the probability to emit a triggered single photon $q_{{\rm GMZ}}^{*}$ is, from equation (15):

$$\begin{eqnarray}&&q_{{\rm GMZ}}^{*}={{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}\left( 1-{{\left( 1-{{p}_{{\rm trig}}} \right)}^{N}} \right){{\eta }_{{\rm modulator}}}\eta _{N\times N}^{2},\end{eqnarray} \tag{ 20 }$$

where ${{\eta }_{{\rm modulator}}}$ is the transmission of the modulator section and ${{\eta }_{N\times N}}$ is the loss induced by the balanced N×N switch. If implemented with couplers with transmission ${{\eta }_{{\rm coupler}}}$, then ${{\eta }_{N\times N}}=\eta _{{\rm coupler}}^{N-1}$. To show the effect of coupler loss, the optimal number of sources to achieve a probability of triggered single photon emission $q_{{\rm GMZ}}^{*}$ with ${{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}{{\eta }_{{\rm modulator}}}=1$ is plotted in figure 5.

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The choice between the log tree and the GMZ depends on the dominant component loss. In the GMZ, the photon must pass only a single phase modulator, instead of a logarithmic number as in the log-tree scheme. However, the GMZ requires a linear scaling in the number of directional couplers the photon passes through, instead of the logarithmic scaling given by the log tree scheme. Also, the GMZ requires $O({{n}^{2}})$ couplers while the total number of components for the log tree scales linearly. Table 1 shows a summary of the different resource scalings for the two architectures.

Table 1.  Comparison of the requirements and component depth between the log tree and generalized MZI architectures as a function of the number of HSPSs N. The first two columns give the total number of component of the given type required for building the switch. The last two columns provide the depth of the circuit for each component—or the number of components of each type that each photon has to go through.

Source type	Total number of phase modulators	Total number of directional couplers	Phase modulator depth	Coupler depth
Log tree	$N-1$	2 $(N-1)$	${{{\rm log} }_{2}}(N)$	2 ${{{\rm log} }_{2}}(N)$
GMZ	N	$\frac{N\left( N+{{{\rm log} }_{2}}(N)-1 \right)}{4}$	1	2 $(N-1)$

Since the switching network is balanced, the exact probability for emitting a triggered single-photon q_GMZ and the output state multi-photon contamination probability p_multi^GMZ are calculated as explained in section (3.1) using ${{\eta }_{{\rm network}}}={{\eta }_{{\rm delay}}}{{\eta }_{{\rm modulator}}}\eta _{N\times N}^{2}$. We will consider these expressions further in section 3.

In this scheme, unit cells—each composed of a HSPS, a delay line, and a 2 × 2 switch—are cascaded to build a chain of sources (figure 6(a)). The electrical output trigger of the HSPS directly drives a switch which routes the emitted photon to the output port while the input port is routed towards a blocked path. As in the other schemes, an optical delay line allows sufficient time for the detection of the idler photon and the configuration of the switch.

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The chained scheme benefits from very simple control logic requirements, since each switch is only driven by one HSPS. For switches driven by voltage, for example, the logic consists only of amplifying the trigger signal from the HSPS to the required switching voltage. This switching logic privileges the HSPS emitting closest to the output of the chain, and hence the photon from the HSPS suffering the least from the losses. As an example, if two cells are cascaded and cell 2 triggers before cell 1, the switch corresponding to cell 1 will remove the photon from the HSPS of cell 2 and replace it with a new photon from the HSPS of cell 1.

Since the switching network applies different amounts of loss to the different HSPSs, we cannot derive the lower bound for the single-photon emission probability $q_{{\rm chain}}^{*}$ using the method given in section 3.1. We therefore show a different derivation here. Given a chain of N unit cells, assuming HSPSs with a probability of triggering p_trig, a probability for the heralded state to be a single photon p_single, and switches with transmission η, $q_{{\rm chain}}^{*}$ is given by summing the probabilities for each source to fire given that the subsequent ones do not, and accounting for switching losses, which gives:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} q_{{\rm chain}}^{*} & = & {{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}\mathop{\sum }\limits_{n=1}^{N}{{p}_{{\rm trig}}}{{\eta }^{n}}{{\left( 1-{{p}_{{\rm trig}}} \right)}^{n-1}} \\ {} & = & {{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}{{p}_{{\rm trig}}}\eta \frac{1-{{\left[ \left( 1-{{p}_{{\rm trig}}} \right)\eta \right]}^{N}}}{1-\left( 1-{{p}_{{\rm trig}}} \right)\eta }. \\ \end{array}\end{eqnarray} \tag{ 21 }$$

The maximal probability for emitting a single photon is obtained in the limit of an infinite chain ${{q}_{{\rm max} }}={{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}\frac{{{p}_{{\rm trig}}}\eta }{1-\left( 1-{{p}_{{\rm trig}}} \right)\eta }$. In practice, to operate at a fraction f of the maximal probability $q_{{\rm max} }^{*}$, the length of the chain is given by solving $fq_{{\rm max} }^{*}={{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}{{p}_{{\rm trig}}}\eta \frac{1-{{\left[ \left( 1-{{p}_{{\rm trig}}} \right)\eta \right]}^{N}}}{1-\left( 1-{{p}_{{\rm trig}}} \right)\eta }$ which gives:

$$\begin{eqnarray}&&N=\left\lceil {\frac{{\rm ln} (1-x)}{{\rm ln} \left[ \left( 1-{{p}_{{\rm trig}}} \right)\eta \right]}} \right\rceil. \end{eqnarray} \tag{ 22 }$$

We show the single-photon emission probability $q_{{\rm chain}}^{*}$ with ${{p}_{{\rm single}}}{{\eta }_{{\rm delay}}}=1$ and the number of cells required N as a function of the switching loss for f = 0.9 in figure 6(b). As with the log tree architecture, the required number of sources becomes impractical for small p_trig, but can be kept below 8 when operating at ${{p}_{{\rm trig}}}=0.25$. The performance of the chained scheme compares well with the performance of the log tree, as shown in figure 6(c). It has better performances for ${{p}_{{\rm trig}}}=0.1$ (for a switch loss less than 1.6 dB) and is close to the log-tree performances for ${{p}_{{\rm trig}}}=0.25$.

The derivation for the exact probability of triggered single photon emission q_chain and the output state multi-photon contamination probability p_multi^chain are shown in appendix B.

The source output state after accounting for losses is given by:

$$\begin{eqnarray}&&\hat{\rho }=\left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}{{{\hat{\rho }}}_{p,k}} \right),\end{eqnarray} \tag{ 23 }$$

where ${{\hat{\rho }}_{p,k}}=|p{{\rangle }_{i}}|k{{\rangle }_{s}}\langle p{{|}_{i}}\langle k{{|}_{s}}$ and C_n^k is the binomial coefficient.

The probability for the detector placed on the idler arm to trigger is given by summing all the contribution of states having at least one photon. Starting with the reduced state after tracing out the signal mode:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{p}_{{\rm trig}\,{\rm TD}}} & = & \mathop{\sum }\limits_{m=1}^{\infty }{{\langle m{{|}_{i}}\left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}|p{{\rangle }_{i}}\langle p{{|}_{i}} \right)|m\rangle }_{i}} \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=1}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}} \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\left( \mathop{\sum }\limits_{p=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}-{{\left( 1-{{\eta }_{i}} \right)}^{n}} \right) \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\left( 1-{{\left( 1-{{\eta }_{i}} \right)}^{n}} \right) \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \frac{1}{1-|\xi {{|}^{2}}}-\frac{1}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)} \right). \\ {{p}_{{\rm trig}\,{\rm TD}}} & = & \frac{|\xi {{|}^{2}}{{\eta }_{i}}}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)}. \\ \end{array}\end{eqnarray} \tag{ 24 }$$

We can check the validity of the expression in the two extreme cases. If ${{\eta }_{i}}=1$, the probability to trigger is maximal, ${{p}_{{\rm trig}}}=|\xi {{|}^{2}}$. If ${{\eta }_{i}}=0$ then ${{p}_{{\rm trig}}}=0$.

The heralded state in the signal arm is expressed, renormalizing by dividing by p_trig, as:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{{\hat{\rho }}}_{H\,{\rm TD}}} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}\mathop{\sum }\limits_{m=1}^{\infty }{{\langle m{{|}_{i}}\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}{{{\hat{\rho }}}_{p,k}} \right)|m\rangle }_{i}} \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=1}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}|k{{\rangle }_{s}}\langle k{{|}_{s}} \right). \\ \end{array}\end{eqnarray*}$$

We can compute the probability that the heralded state is a single photon using ${{p}_{{\rm single}\,{\rm TD}}}={{\langle 1{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|1\rangle }_{s}}$:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{p}_{{\rm single}\,{\rm TD}}} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}C_{n}^{1}{{\eta }_{s}}{{\left( 1-{{\eta }_{s}} \right)}^{n-1}}\mathop{\sum }\limits_{p=1}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}} \right) \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}C_{n}^{1}{{\eta }_{s}}{{\left( 1-{{\eta }_{s}} \right)}^{n-1}}\left( 1-{{\left( 1-{{\eta }_{i}} \right)}^{n}} \right) \right) \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}|\xi {{|}^{2}}\left( {{\eta }_{s}}\mathop{\sum }\limits_{n=1}^{\infty }n|\xi {{|}^{2(n-1)}}{{\left( 1-{{\eta }_{s}} \right)}^{n-1}} \right. \\ {} & {} & -\left. \left( 1-{{\eta }_{i}} \right){{\eta }_{s}}\mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2(n-1)}}n{{\left( 1-{{\eta }_{s}} \right)}^{n-1}}{{\left( 1-{{\eta }_{i}} \right)}^{n-1}} \right). \\ \end{array}\end{eqnarray*}$$

Using the identity (obtained from taking the derivative of the geometric series) $\sum _{n=0}^{\infty }n{{x}^{n-1}}=1/{{(1-x)}^{2}}$:

$$\begin{eqnarray*}&&{{p}_{{\rm single}\,{\rm TD}}}=\frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm TD}}}}|\xi {{|}^{2}}{{\eta }_{s}}\left( \frac{1}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right) \right)}^{2}}}-\frac{\left( 1-{{\eta }_{i}} \right)}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right) \right)}^{2}}} \right).\end{eqnarray*}$$

$$\begin{eqnarray}&&{{p}_{{\rm single}\,{\rm TD}}}=\left( 1-|\xi {{|}^{2}} \right){{\eta }_{s}}\frac{\left[ 1-{{\left( |\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right) \right)}^{2}}\left( 1-{{\eta }_{i}} \right) \right]\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right]}{{{\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right) \right]}^{2}}{{\left[ 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right) \right]}^{2.}}}\end{eqnarray} \tag{ 25 }$$

In the lossless case, we obtain ${{p}_{{\rm single}\,{\rm TD}}}=\left( 1-|\xi {{|}^{2}} \right)$, which, with a triggering probability ${{p}_{{\rm trig}}}=|\xi {{|}^{2}}$ (from equation (24)), is consistent with a probability of getting one photon of $|\xi {{|}^{2}}\left( 1-|\xi {{|}^{2}} \right)$ (from equation (1)).

The probability that the heralded state contains multi-photon contamination is derived as follows. First, we define:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{Z}_{{\rm TD}}} & = & \mathop{\sum }\limits_{q=1}^{\infty }\mathop{\sum }\limits_{m=1}^{\infty }\langle q,m|\hat{\rho }|q,m\rangle \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=1}^{n}\mathop{\sum }\limits_{k=1}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}} \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\left( 1-{{\left( 1-{{\eta }_{i}} \right)}^{n}} \right)\left( 1-{{\left( 1-{{\eta }_{s}} \right)}^{n}} \right) \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\left( 1+{{\left( 1-{{\eta }_{s}} \right)}^{n}}{{\left( 1-{{\eta }_{i}} \right)}^{n}}-{{\left( 1-{{\eta }_{i}} \right)}^{n}}-{{\left( 1-{{\eta }_{s}} \right)}^{n}} \right) \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}\left( \frac{1}{1-|\xi {{|}^{2}}}+\frac{\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)\left( 1-{{\eta }_{i}} \right)}-\frac{\left( 1-{{\eta }_{i}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)}-\frac{\left( 1-{{\eta }_{s}} \right)}{1-|\xi {{|}^{2}}\left( 1-{{\eta }_{s}} \right)} \right). \\ \end{array}\end{eqnarray*}$$

Then it follows that ${{p}_{{\rm multi}\,{\rm TD}}}=\sum _{m=2}^{\infty }{{\langle m{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|m\rangle }_{s}}$:

$$\begin{eqnarray}&&{{p}_{{\rm multi}\,{\rm TD}}}=\frac{{{Z}_{{\rm TD}}}}{{{p}_{{\rm trig}\,{\rm TD}}}}-{{p}_{{\rm single}\,{\rm TD}}}.\end{eqnarray} \tag{ 26 }$$

Now we consider the number-resolving detector. Starting with equation (2), tracing out the signal mode and calculating the probability to get only one photon:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{p}_{{\rm trig}\,{\rm NRD}}} & = & {{\langle 1{{|}_{i}}\left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}|p{{\rangle }_{i}}\langle p{{|}_{i}} \right)|1\rangle }_{i}} \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}C_{n}^{1}{{\eta }_{i}}{{\left( 1-{{\eta }_{i}} \right)}^{n-1}} \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}{{\eta }_{i}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2(n-1)}}n{{\left( 1-{{\eta }_{i}} \right)}^{n-1}} \right). \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}&&{{p}_{{\rm trig}\,{\rm NRD}}}=\frac{\left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}{{\eta }_{i}}}{{{\left( 1-\left( 1-{{\eta }_{i}} \right)|\xi {{|}^{2}} \right)}^{2}}}.\end{eqnarray} \tag{ 27 }$$

We can check the validity of the expression in the two extreme cases. If ${{\eta }_{i}}=1$, the probability to trigger is maximal, ${{p}_{{\rm trig}}}=\left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}$. If ${{\eta }_{i}}$=0 then ${{p}_{{\rm trig}}}=0$.

The corresponding heralded state is:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{{\hat{\rho }}}_{H\,{\rm NRD}}} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm NRD}}}}{{\langle 1{{|}_{i}}\left( \mathop{\sum }\limits_{n=0}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{p=0}^{n}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{p}\eta _{i}^{p}{{\left( 1-{{\eta }_{i}} \right)}^{n-p}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}{{{\hat{\rho }}}_{p,k}} \right)|1\rangle }_{i}} \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm NRD}}}}{{\eta }_{i}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}n{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}\mathop{\sum }\limits_{k=0}^{n}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}}|k{{\rangle }_{s}}\langle k{{|}_{s}} \right). \\ \end{array}\end{eqnarray*}$$

As in the previous case, We can compute the probability that the heralded state is a single photon using ${{p}_{{\rm single}\,{\rm NRD}}}={{\langle 1{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm TD}}}|1\rangle }_{s}}$:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{p}_{{\rm single}\,{\rm NRD}}} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm NRD}}}}{{\eta }_{i}}\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}n{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}n\eta _{s}^{1}{{\left( 1-{{\eta }_{s}} \right)}^{n-1}} \right) \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)}{{{p}_{{\rm trig}\,{\rm NRD}}}}{{\eta }_{i}}{{\eta }_{s}}|\xi {{|}^{2}}\left( \mathop{\sum }\limits_{n=1}^{\infty }{{n}^{2}}{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}{{\left( 1-{{\eta }_{s}} \right)}^{n-1}}|\xi {{|}^{2(n-1)}} \right). \\ \end{array}\end{eqnarray*}$$

Using $\sum _{n=1}^{\infty }{{n}^{2}}{{x}^{n-1}}=\sum _{n=1}^{\infty }n(n-1){{x}^{n-1}}+\sum _{n=1}^{\infty }n{{x}^{n-1}}=\frac{2x}{{{(1-x)}^{3}}}+\frac{1}{{{(1-x)}^{2}}}:$

$$\begin{eqnarray*}&&{{p}_{{\rm single}\,{\rm NRD}}}={{\left( 1-\left( 1-{{\eta }_{i}} \right)|\xi {{|}^{2}} \right)}^{2}}{{\eta }_{s}}\left( \frac{\left( 1+\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right)|\xi {{|}^{2}} \right)}{{{\left( 1-\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right)|\xi {{|}^{2}} \right)}^{3}}} \right).\end{eqnarray*}$$

The probability that the heralded state contains multi-photon contamination is derived as follows. First, we define:

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{Z}_{{\rm NRD}}} & = & \mathop{\sum }\limits_{m=1}^{\infty }\langle 1,m|\hat{\rho }|1,m\rangle \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)\left( \mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2n}}\mathop{\sum }\limits_{k=1}^{n}C_{n}^{1}{{\eta }_{i}}{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}C_{n}^{k}\eta _{s}^{k}{{\left( 1-{{\eta }_{s}} \right)}^{n-k}} \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}\left( {{\eta }_{i}}\mathop{\sum }\limits_{n=1}^{\infty }|\xi {{|}^{2(n-1)}}n{{\left( 1-{{\eta }_{i}} \right)}^{n-1}}\left( 1-{{\left( 1-{{\eta }_{s}} \right)}^{n}} \right) \right) \\ {} & = & \left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}{{\eta }_{i}}\left[ \frac{1}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right)}^{2}}}-\frac{\left( 1-{{\eta }_{s}} \right)}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right) \right)}^{2}}} \right] \\ {} & = & \frac{\left( 1-|\xi {{|}^{2}} \right)|\xi {{|}^{2}}{{\eta }_{i}}{{\eta }_{s}}}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right)}^{2}}}\left( \frac{1-\left( 1-{{\eta }_{s}} \right){{\left( |\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right)}^{2}}}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right) \right)}^{2}}} \right). \\ \end{array}\end{eqnarray*}$$

Then it follows that the probability that the heralded state contains multi-photon contamination is given by ${{p}_{{\rm multi}\,{\rm NRD}}}=\sum _{m=2}^{\infty }{{\langle m{{|}_{s}}{{\hat{\rho }}_{{\rm heralded}\,{\rm NRD}}}|m\rangle }_{s}}$:

$$\begin{eqnarray}&&{{p}_{{\rm multi}\,{\rm NRD}}}=\frac{{{Z}_{{\rm NRD}}}}{{{p}_{{\rm trig}\,{\rm NRD}}}}-{{p}_{{\rm single}\,{\rm NRD}}}.\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray*}&&{{p}_{{\rm multi}\,{\rm NRD}}}={{\eta }_{s}}\frac{1-\left( 1-{{\eta }_{s}} \right){{\left( |\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right) \right)}^{2}}}{{{\left( 1-|\xi {{|}^{2}}\left( 1-{{\eta }_{i}} \right)\left( 1-{{\eta }_{s}} \right) \right)}^{2}}}-{{p}_{{\rm single}\,{\rm NRD}}}.\end{eqnarray*}$$