## Abstract

Heralded single-photon sources (HSPS) are widely used in experimental quantum science because they have negligibly small jitter and can therefore achieve high visibility for quantum interference. However, it is necessary to decrease the photon generation rate to suppress multi-photon components. To address this problem, two methods have been proposed and discussed: spatial (or temporal) source multiplexing and photon-pair number discrimination. Here, we report the experimental realization of a HSPS combining these two methods that can suppress the two-photon probability to 44.2 ± 0.7% of that of a normal HSPS. We also provide a theoretical analysis and a discussion of the effect of combining the two methods, considering a detector cascade as a practical photon number discriminating detector. The experimental results agreed well with the theoretical predictions.

© 2016 Optical Society of America

## 1. Introduction

Single-photon sources are indispensable for photonic quantum technologies [1–5]. Various kinds of single-photon sources have been developed, using various technologies such as spontaneous parametric down conversion (SPDC) [6, 7], spontaneous four-wave mixing [8, 9], quantum dots [10, 11] and color centers in diamond [12]. Heralded single photon sources (HSPS) using SPDC have been widely used for generating photons with negligibly small jitter, which makes it possible to achieve high visibility for quantum interference [13]. However, a problem with HSPS is that occasionally events that generate more than two photons occur, which may cause errors in the operation of quantum circuits. This is a crucial problem for multi-photon interference experiments when there are many input photons.

To solve this problem, two methods have been proposed. The first method is called the multiplexed heralded single-photon source (Multiplexed-HSPS). A Multiplexed-HSPS consists of multiple SPDC sources, and upon the detection of an idler photon from one of the sources, the corresponding signal photon is selected and emitted [14–23]. The second method uses a photon-number-resolving detector (PND) to monitor the number of generated photon pairs from a SPDC source and select cases for which only one pair is created [19, 24–27]. To evaluate the performance of these methods, Shapiro and Wong conducted a theoretical analysis assuming a Poisson distribution for the generated photons from the SPDC source [16], and D. Bonneau et al. conducted an analysis assuming a thermal distribution of the generated SPDC [19]. However, in these theoretical analyses, the PNDs are assumed to be able to distinguish a number of photons up to infinity. Although the development of PNDs is under investigation, there are still limitations in the detection efficiencies and response times. Furthermore, the realization of multiplexing of heralded single photon sources using photon number resolving detectors (Multiplexed-HSPS with PND) has not yet been reported.

In this paper, we report the experimental realization of Multiplexed-HSPS with PND. For this, we adopted a PND using cascaded detectors. We also developed the theory to describe such Multiplexed-HSPS with PND considering a limited number of cascaded detectors. We found that even a small number of detector cascade is very effective in reducing excess photon emission in certain circumstances. With the realized Multiplexed-HSPS with PND equipped with two SPDC sources and a PND consisting of two on–off detectors, we found that the two-photon components are reduced to 44.2 ± 0.7% compared to a normal HSPS.

## 2. Model

Figure 1(a) shows a conventional HSPS. A nonlinear crystal (NLC) is pumped by a pulsed laser and generates a photon pair via an SPDC process. One of the photons in a pair is detected by an on–off detector, which generates a heralding signal upon detection. In this paper, we call the photon detected to generate heralding signals ‘idler photon’ and the photon to be selected and emitted ‘signal photon’. However, an on–off detector generates only one detection signal, even when more than one photon hit the detector. Therefore, in some cases where multiple pairs are generated in one pulse duration time, more than two photon signals will be produced for one heralding signal because the on-off detector cannot distinguish such events.

To prevent multiple photon signals from being produced, a HSPS using a photon number resolving detector (PND) [Fig. 1(b)] generates a heralding signal only when the PND detects only one photon. The excess-photon emission can thus be suppressed.

Although considerable effort has been put into realizing PNDs, the current devices are well short of being an ideal detector that can distinguish a number of incident photon up to infinity. As a more practical approach, we consider a PND using cascaded on–off detectors. As shown in Fig. 1(c), an incoming mode is divided equally into 2* ^{D}* paths using 50:50 beam splitters (BS) and directs them into 2

*on–off detectors, where*

^{D}*D*is the number of detector cascade. Figure 1(c) illustrates an example of such a cascaded detector [

*D*= 2]. Note that recently developed PNDs consisting of multiple superconducting nanowire single-photon detectors can be considered such a type of cascaded detector, with

*D*= 6 [28]. If

*S*→ ∞, an ideal cascaded detector is able to distinguish a number of photons up to infinity.

Multiplexed-HSPS is an alternative approach to reducing the two or more photon probability. A Multiplexed-HSPS consists of multiple SPDC sources. Figure 1(d) shows a schematic of a Multiplexed-HSPS for two SPDC sources, *S* = 2, where *S* is the number of SPDC sources per Multiplexed-HSPS. The pump laser is split equally by a BS, and injected into each NLC. The heralding signals to detect idler photons are used to identify which source has created a photon pair. The signal photon of the generated pair is guided to an output port by an optical switch. Even if each idler photon is detected by an on–off detector, the excess photons are suppressed. By replacing the on–off detectors with PNDs, the performance of Multiplexed-HSPS with PND is improved.

## 3. Theory

Here, we derive the photon number distribution of a Multiplexed-HSPS using cascaded detectors *P*_{h}(*n*) under non-ideal experimental conditions. For such conditions, we consider that a cascaded detector has a non-unity detection efficiency (*η*_{h}), and optical switches exhibit losses (*t*).

A SPDC source is characterized by the mean number of photon pairs in a pulse (
$\overline{N}$), the probability that a heralding signal is produced *P*_{herald} and the probability that *n* photons are detected at the output when the heralding signal is obtained *P*_{C}(*n*). We derive *P*_{herald} and *P*_{C}(*n*) using a cascaded detector. The output state *P*(*n*) of a single-mode SPDC follows a thermal distribution *P*_{th}(*n*):

*P*(

*n*) follows a Poisson distribution

*P*

_{Po}(

*n*):

When only one of the on–off detectors is activated, a cascaded detector produces a heralding signal with a probability of

where*P*

_{D=0}(1|

*m*) is a probability of one detector triggered by one or more photons and generate a detection signal when

*m*photons are incident to the detector. Note that for Eq. (4), the number of detector cascade

*D*is 0, meaning that there is only one detector. Similarly,

*P*

_{D}(1|

*m*) denotes the probability where only one detector in the detector cascade generates the detection signal when

*m*photons are incident to the detector cascade.

*η*

_{h}is detection efficiency of a cascaded detector.

*C*

_{n}*is used for the combination selecting*

_{a}*a*items from

*n*distinct elements. In the case of

*D*→ ∞, the cascaded detector can distinguish a number of photons up to infinity with

*P*

_{D}_{=∞}(1|

*m*) ≈

*mη*

_{h}(1 −

*η*

_{h})

^{m}^{−1}.

The coincidence probability *P*_{C}(*n*) that *n* photons are detected at the output when the heralding signal is obtained is given as

*t*is the transmittance between the SPDC sources and the output.

*P*_{h}(*n*) is the *n*-photon output probability for a heralding signal for a Multiplexed-HSPS using cascaded detectors, and can be derived using Eqs. (3) and (6):

*i*th SPDC source, and ${P}_{\mathrm{C}}^{i}\left(n\right)$ is the coincidence probability for the cascaded detector for the

*i*th SPDC source. Here, we used the mean number of photon pairs per one SPDC source, which is given by $\overline{N}/S$.

If *P*(*n*) follows a Poisson distribution and *D* → ∞, our model is consistent with the theory in [16]. If *P*(*n*) are output follows a thermal distribution and *D* → ∞, our model is also consistent with the theory in [19].

Depending on the experimental setup, an produced photon from an SPDC source contains just a few modes, and the produced photon is neither single nor multi-mode [29, 30]. To analyze this condition, we use the following photon number distribution consisting of the thermal and Poisson distribution connected with a parameter *γ* (0 ≤ *γ* ≤ 1):

When the mean number of photon pairs is small (
$\overline{N}\ll 1$), the two photon probability *P*_{h}(2) in Eq. (7) can be approximated to the following linear function of
$\overline{N}$:

## 4. Numerical analysis

Here, using our theory in section 3, we numerically analyze the suppression of the two-photon probability [*P*_{h}(2)] for an HSPS using a cascaded detector, a Multiplexed-HSPS and a Multiplexed-HSPS with PND. For this calculation, we assumed a thermal distribution *P*_{th}(*n*) for *P*(*n*). Note that a smaller *P*_{h}(2) indicates a higher quality single-photon source.

Figure 2(a) shows *P*_{h}(2) versus the mean photon pair number
$\overline{N}$, which corresponds to the generation rate of the HSPS. We assumed a detection efficiency of the detector cascade of *η*_{h} = 0.5 and an overall transmittance between the SPDC sources and the output of *t* = 0.5. The solid black lines for *S* = 1 and *D* = 0 in Fig. 2(a) correspond to the conventional HSPS. The solid lines colored red, blue and purple indicate *P*_{h}(2) for *S* = 1 (no source multiplexing) and *D* = 1, 2 and ∞, respectively. Note that two and four detectors are used for *D* = 1 and *D* = 2 at each SPDC sources. These lines show that *P*_{h}(2) decreases with increasing *D*. The results also show that *P*_{h}(2) rapidly approaches the value for *D* = ∞ when we increase *D*, suggesting that we can effectively decrease *P*_{h}(2) even with a relatively small *D*.

By multiplexing two SPDCs [*S* = 2], we could halve *P*_{h}(2) for *D* = 0 (black dashed lines in Fig. 2(a)) compared with the conventional HSPS (solid black line). The dashed lines colored red, blue and purple indicate *P*_{h}(2) for *S* = 2 and *D* = 1, 2 and ∞, respectively. These lines show *P*_{h}(2) at the output of a Multiplexed-HSPS with PND. A Multiplexed-HSPS with PND suppresses *P*_{h}(2) more efficiently than a Multiplexed-HSPS. However, since *P*_{h}(2) for *S* = 2 and *D* = 0 is lower than that for *S* = 1 and *D* = ∞, the effect of source multiplexing is higher than that of detector cascading in the case of *η*_{h} = 0.5.

If we can achieve a higher detection efficiency, *η*_{h} = 0.9 (*t* = 0.5), the situation changes as shown in Fig. 2(b). The solid lines correspond to no source multiplexing (black) and detector cascade numbers of *D* = 2 (blue) and ∞ (purple). The blue and purple solid lines achieve lower *P*_{h}(2) than the black dashed line, corresponding to source multiplexing [*S* = 2] and no detector cascading [*D* = 0]. Thus, the effect of the suppression using the detector cascading method increases with *η*_{h}. On the other hand, the effect of source multiplexing does not depend on *η*_{h}.

Figure 3 shows how the suppression ratio depends on the detection efficiency *η*_{h}, where we assume the case of
$\overline{N}=0.05$, *t* = 0.5. The suppression ratio *R* is defined as the ratio of the two photon probability of a Multiplexed-HSPS with PND to that of a normal HSPS, and smaller *R* indicates better performance as a heralded single photon source. *R* can be derived using Eq. (7):

*P*

_{h}(2)[

*S*,

*D*] is the two-photon probability of Multiplexed-HSPS with PND. The results show that Multiplexed-HSPS [

*D*= 0] gives a suppression ratio of about 1/

*S*regardless of

*η*

_{h}. On the other hand, for HSPS using a PND [

*S*= 1,

*D*= 1], the suppression ratio decreases with increasing detection efficiency. Without source multiplexing, to reduce the two-photon probability by 0.51 (the ratio of Multiplexed-HSPS [

*S*= 2]), detector cascading for

*D*= 1 and ∞ requires

*η*

_{h}= 0.99 and 0.65, respectively. With

*η*

_{h}= 1.0, Multiplexed-HSPS with PND [

*S*,

*D*] gives a suppression ratio of about 1/(

*S*× 2

*).*

^{D}In the theoretical analysis, the detection efficiency is an important parameter determining the ability of a cascaded detector. On the other hand, it is found that a source multiplexing system does not depend on the detection efficiency.

## 5. Experimental setup

A schematic illustration of the experimental setup is shown in Fig. 4. We implemented a Multiplexed-HSPS with PND consisting of two SPDC sources (SPDC-A and SPDC-B), each of which is equipped with two cascaded detectors [*S* = 2, *D* = 1]. A femtosecond pulse laser (repetition rate of 82 MHz and central wavelength of 390 nm, Spectra-Physics, Tsunami) is used to pump a BBO crystal. The injected pump pulses generate photon pairs in the forward direction (SPDC-A). After passing through the crystal, the pump pulse is reflected back by a dichroic mirror (DM) and generates photon pairs in the backward direction (SPDC-B). For each of the sources (SPDC-A and SPDC-B), idler photons are detected by cascaded two single-photon counting modules (SPCMs, Excelitas Technologies, SPCM-AQR), and the signal photons are sent to the optical switch via long polarization maintaining fibers. A long fiber (65 m) is necessary to compensate the delay caused by the electric signal processing. A heralding signal-A or signal-B is generated when only one of the two SPCMs monitoring SPDC-A or SPDC-B, respectively, detects the idler photon. The final heralding signal is generated when either both heralding signal-A or heralding signal-B is generated.

According to whether heralding signal-A or heralding signal-B from the SPDC sources is generated, the signal photons are appropriately routed and gated by the optical switch. The optical switch consists of an EOM and two PBSs. The optical switch passes the signal photons from SPDC-A (SPDC-B) when the heralding signal-A from SPDC-A exists (non exists). Note that when signal-A and signal-B are generated, only the signal photon generated by SPDC-A is produced and the photon from SPDC-B is cut.

In the next section, we evaluate the two-photon probability at the output *P*_{h}(2) to check the performance of our single-photon source while changing the mean number of photon pairs. The conditional two-photon probability for the final heralding signal was estimated using three fold coincidence events among the detectors (D1, D2) and the final heralding signal. Note that the detection efficiencies of the detectors (D1, D2) and the effect of the 50:50 FBS are compensated, and we assumed that all of the three-fold coincidences are due to *P*_{h}(2).

## 6. Experimental results and discussion

Figure 5 shows the two-photon probability *P*_{h}(2) at the output of our Multiplexed-HSPS with PND versus the coincidence rate. The horizontal axis shows the number of events where only one of the two detectors (D1, D2) detects a photon when the final heralding signal is produced. The two-photon probability was measured with various pump powers. The blue and red circles indicate HSPS [*S* = 1, *D* = 0] (SPDC-A) and HSPS using a cascaded detector [*S* = 1, *D* = 1] (SPDC-A), respectively. The blue and red squares indicate Multiplexed-HSPS [*S* = 2, *D* = 0] and Multiplexed-HSPS with PND [*S* = 2, *D* = 1], respectively. These data points show that *P*_{h}(2) increases with the coincidence rate. For constant coincidence rate, Multiplexed-HSPS has a two times lower two-photon probability than HSPS. We achieved further suppression with the Multiplexed-HSPS with PND.

The dashed lines in Fig. 5 show the theoretical curves calculated with Eq. (7). For *η*_{h} and transmittance *t*, we used the separately measured values (Table 1). Since we used the BPFs which have a relatively broad bandwidth of 4 nm in order to increase the detection efficiency of the cascaded detectors, the output state of the SPDC follows neither the thermal distribution nor the Poisson distribution. Thus, for the photon number distribution *P*(*n*), we adopted the intermediate distribution in Eq. (8). The dashed lines are the theoretical curve when *γ* is 0.59, 0.72. The theoretical lines are in good agreement with the experimental results. Note that plotted curves are derived from Eq. (7), not from the approximated one shown in Eq. (9). Note also that we compensated a small effective loss in the optical switch which appeared when we used the multiplexing method. The effective loss was caused by the deadtime of the electrical amplifiers used for driving the EOMs.

Note also that, in our experiment, the detector darkcounts (∼ 200 counts/s) does not impact the results since it is negligibly small when compared to the average count rates of each of the detectors in the cascade (∼ 1.5 × 10^{5} counts/s). Note also that the effect of the deadtime of the detectors is also negligible. In more detail, the average count rate at the heralding arm was about 1.5 ×10^{5} counts/s and is much smaller than 2.5 ×10^{8}, which is the inverse of the deadtime (40 ns) of our detector.

To confirm the suppression clearly, we calculated the suppression ratio *R* in Eq. (10) defined as the two-photon probability ratio between a normal HSPS (SPDC-A) [*S* = 1, *D* = 0] and the other HSPS. The suppression ratios of HSPS using PND, Multiplexed-HSPS and Multiplexed-HSPS with PND were found to be 91.1 ± 0.7%, 47.6 ± 0.7% and 44.2 ± 0.6%, respectively. Thus, a Multiplexed-HSPS with PND exhibits a significant improvement, compared to the normal HSPS.

Due to the limited effective detection efficiencies (about 0.14 for SPDC-A, about 0.19 for SPDC-B) in our experiment, the suppressions of two photon probability with the help of cascaded detectors were not as significant as shown in Fig. 2 where the detection efficiencies were assumed to be 0.5 and 0.9. However, we would like to note that the experimental result is consistent with Fig. 3 for the given detection efficiencies. It is important to improve the effective detection efficiencies by adopting highly efficient single photon detectors [28, 32] and highly efficient coupling of down converted photons into single mode optical fibers [33, 34].

## 7. Conclusion

In conclusion, we realized a Multiplexed-HSPS with PND using cascaded detectors. We also developed the theory for describing such a Multiplexed-HSPS with PND considering a limited number of cascaded detectors. We found that even a small number of detector cascade is very effective in reducing the excess photon emission in certain circumstances. With the realized a source equipped with two SPDC sources and a PND consisting of two on-off detectors, we found that the two-photon components were reduced to 44.2 ± 0.6% of that of a HSPS.

## Funding

JSPS-KAKENHI (No. 26220712, 26610052, 25707034); JST-CREST; JSPS-FIRST; the Project for Developing Innovation Systems of MEXT; the Research Foundation for Opto-Science and Technology.

## Acknowledgments

We wish to thank K. Tsujino for helpful comments.

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