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.

 

Fig. 1 (a) Scheme of a conventional HSPS [S = 1]. (b) Scheme of HSPS using a PND [S = 1, D = 1]. (c) Scheme of a cascaded detector [D = 2]. The detector consists of 50:50 beam splitters (BS) and on–off detectors. (d) Scheme of Multiplexed-HSPS [S = 2] with on-off detectors/Multiplexed-HSPS with PND [S = 2] with PNDs.

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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 2D paths using 50:50 beam splitters (BS) and directs them into 2D on–off detectors, where 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 Ph(n) under non-ideal experimental conditions. For such conditions, we consider that a cascaded detector has a non-unity detection efficiency (ηh), and optical switches exhibit losses (t).

A SPDC source is characterized by the mean number of photon pairs in a pulse ( N¯), the probability that a heralding signal is produced Pherald and the probability that n photons are detected at the output when the heralding signal is obtained PC(n). We derive Pherald and PC(n) using a cascaded detector. The output state P(n) of a single-mode SPDC follows a thermal distribution Pth(n):

P(n)=Pth(n)N¯n(1+N¯)1+n,
where N¯ is the mean number of photon pairs in a pulse. For a multi-mode SPDC, P(n) follows a Poisson distribution PPo(n):
P(n)=PPo(n)N¯neN¯n!.

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

Pherald=m=1P(m)PD(1|m),
where
PD=0(1|m)=1(1ηh)m,
PD(1|m)=12m1a=0m mCa{PD1(1|a)(1ηh)ma},
where PD=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, PD(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. ηh is detection efficiency of a cascaded detector. nCa is used for the combination selecting a items from n distinct elements. In the case of D → ∞, the cascaded detector can distinguish a number of photons up to infinity with PD=∞(1|m) ≈ h(1 − ηh)m−1.

The coincidence probability PC(n) that n photons are detected at the output when the heralding signal is obtained is given as

PC(n)=m=nP(m)PD(1|m)mCntn(1t)mn,
where t is the transmittance between the SPDC sources and the output.

Ph(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):

Ph(n)=(PC1(n)+(1Pherald1)(PC2(n)+(1Pherald2)(PC3(n)+))1(1Pherald1)(1Pherald2)(1Pherald3),
where Pheraldi is the probability that the idler photon is detected with the cascaded detector for the ith SPDC source, and PCi(n) is the coincidence probability for the cascaded detector for the ith SPDC source. Here, we used the mean number of photon pairs per one SPDC source, which is given by 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):

P(n)=γPth(n)+(1γ)PPo(n).

When the mean number of photon pairs is small ( N¯1), the two photon probability Ph(2) in Eq. (7) can be approximated to the following linear function of N¯:

Ph(2)~(1+γ)t2PD(1|2)2ηhS×N¯.

4. Numerical analysis

Here, using our theory in section 3, we numerically analyze the suppression of the two-photon probability [Ph(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 Pth(n) for P(n). Note that a smaller Ph(2) indicates a higher quality single-photon source.

Figure 2(a) shows Ph(2) versus the mean photon pair number N¯, which corresponds to the generation rate of the HSPS. We assumed a detection efficiency of the detector cascade of ηh = 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 Ph(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 Ph(2) decreases with increasing D. The results also show that Ph(2) rapidly approaches the value for D = ∞ when we increase D, suggesting that we can effectively decrease Ph(2) even with a relatively small D.

 

Fig. 2 Two-photon probability from the Multiplexed-HSPS with PND versus the mean number of photons pairs for (a) ηh=0.5, t=0.5, (b) ηh=0.9, t=0.5. Black line: S = 1, D = 0; red line: S = 1, D = 1; blue line: S = 1, D = 2; purple line: S = 1, D = ∞; black dashed line: S = 2, D = 0; red dashed line: S = 2, D = 1; blue dashed line: S = 2, D = 2; purple dashed line: S = 2, D = ∞:

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By multiplexing two SPDCs [S = 2], we could halve Ph(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 Ph(2) for S = 2 and D = 1, 2 and ∞, respectively. These lines show Ph(2) at the output of a Multiplexed-HSPS with PND. A Multiplexed-HSPS with PND suppresses Ph(2) more efficiently than a Multiplexed-HSPS. However, since Ph(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 ηh = 0.5.

If we can achieve a higher detection efficiency, ηh = 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 Ph(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 ηh. On the other hand, the effect of source multiplexing does not depend on ηh.

Figure 3 shows how the suppression ratio depends on the detection efficiency ηh, where we assume the case of 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):

R=Ph(2)[S,D]Ph(2)[S=1,D=0],
where Ph(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 ηh. 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 ηh = 0.99 and 0.65, respectively. With ηh = 1.0, Multiplexed-HSPS with PND [S, D] gives a suppression ratio of about 1/(S × 2D).

 

Fig. 3 Suppression ratio versus detection efficiency. Ratio between HSPS [S = 1, D = 0] and Multiplexed-HSPS with PND [S, D] for N¯=0.05, t = 0.5. Black curve: S = 1, D = 0; blue curve: S = 2, D = 0; red curve: S = 4, D = 0; black dashed curve: S = 1, D = 1; blue dashed curve: S = 2, D = 1; black long dashed/short dashed curve: S = 1, D = ∞; blue long dashed/short dashed curve: S = 2, D = ∞.

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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.

 

Fig. 4 Schematic diagram of experimental set up. BBO: beta-barium-borate crystal (Type-I); DM: dichroic mirror; FC: fiber coupler; SPCM: single-photon counting module; XOR: XOR gate; AND: AND gate; EOM: electro-optic modulator; PBS: polarizing beam splitter; HWP: half wave plate; BPF: band pass filter (∆λ = 4 nm); PMF : polarization maintaining fiber; 50:50 FBS: 50:50 fiber beam splitter; AMP: high-voltage pulsed amplifier; fiber delay is 65m.

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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 Ph(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 Ph(2).

6. Experimental results and discussion

Figure 5 shows the two-photon probability Ph(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 Ph(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.

 

Fig. 5 Experimental results: The vertical axis is the two photon probability at the output. The horizontal axis shows the number of events where one of the two detectors (D1, D2) detects a photon when the final heralding signal is produced. HSPS (SPDC-A) [S=1, D=0] (blue circles), HSPS (SPDC-A) using PND [S=1, D=1] (red circles), Multiplexed-HSPS [S=2, D=0] (blue squares), a Multiplexed-HSPS with PND [S=2, D=1] (red square), where all contain the optical switch. The dashed lines are based on our model.

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The dashed lines in Fig. 5 show the theoretical curves calculated with Eq. (7). For ηh 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.

Tables Icon

Table 1. Separately measured parameters. ηh,A/B: detection efficiency of the detector in SPDC-A/B. tA/B: transmittance of the single photon between the SPDC-A/B and the output.

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 × 105 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 ×105 counts/s and is much smaller than 2.5 ×108, 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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References

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  1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [Crossref]
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  3. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
    [Crossref]
  4. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
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  10. C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
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  11. M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
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  12. C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
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  13. M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express 20(14), 15275–15285 (2012).
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  14. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
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  15. A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
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  16. J. H. Shapiro and F. N. Wong, “On-demand single-photon generation using a modular array of parametric downconverters with electro-optic polarization controls,” Opt. Lett. 32(18), 2698–2700 (2007).
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  18. M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).
  19. D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompsom, “Effect of loss on multiplexed single-photon sources,” New. J. Phys. 17(4), 043057 (2015).
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  20. G. J. Mendoza, R. Santagati, J. Munns, E. Hemsley, M. Piekarek, E. Martin-Lopez, G. D. Marshall, D. Bonneau, M. G. Thompson, and J. L. O’Brien, “Active temporal and spatial multiplexing of photon,” Optica 3(2), 127–132 (2016).
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  21. F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P. G. Kwiat, “Time-multiplexed heralded single-photon source,” Optica 2(12), 1010–1013 (2015).
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  23. C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
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  24. K. Tsujino, S. Takeuchi, and K. Sasaki, “Detailed analysis of the fidelity of quantum teleportation using photons: Considering real experimental parameters,” Phys. Rev. A 66(4), 042314 (2002).
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    [Crossref]
  26. M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
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  27. P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61(4), 042304 (2000).
    [Crossref]
  28. S. Miki, T. Yamashita, Z. Wang, and H. Terai, “A 64-pixel NbTiN superconducting nanowire single-photon detector array for spatially resolved photon detection,” Opt. Express 22(7), 7811–7820 (2011).
    [Crossref]
  29. E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light,” J. Phys. A 1, 128 (1968).
    [Crossref]
  30. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
    [Crossref]
  31. S. Ramelow, A. Mech, M. Giustina, S. Groblacher, W. Wieczorek, J. Beyer, B. Calkins, T. Gerrits, S. W. Nam, A. Zeilinger, and R. Ursin, “Highly efficient heralding of entangled single photons,” Opt. Express 21(6), 6707–6717 (2013).
    [Crossref]
  32. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
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  33. X. L. Niu, Y. F. Huang, G. Y. Xiang, G. C. Guo, and Z. Y. Ou, “Beamlike high-brightness source of polarization-entangled photon pairs,” Opt. Lett. 33(9), 968–970 (2008).
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  34. O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
    [Crossref]

2016 (2)

G. J. Mendoza, R. Santagati, J. Munns, E. Hemsley, M. Piekarek, E. Martin-Lopez, G. D. Marshall, D. Bonneau, M. G. Thompson, and J. L. O’Brien, “Active temporal and spatial multiplexing of photon,” Optica 3(2), 127–132 (2016).
[Crossref]

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

2015 (3)

2013 (3)

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

S. Ramelow, A. Mech, M. Giustina, S. Groblacher, W. Wieczorek, J. Beyer, B. Calkins, T. Gerrits, S. W. Nam, A. Zeilinger, and R. Ursin, “Highly efficient heralding of entangled single photons,” Opt. Express 21(6), 6707–6717 (2013).
[Crossref]

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

2012 (3)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

A. Aspuru-guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express 20(14), 15275–15285 (2012).
[Crossref]

2011 (3)

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

S. Miki, T. Yamashita, Z. Wang, and H. Terai, “A 64-pixel NbTiN superconducting nanowire single-photon detector array for spatially resolved photon detection,” Opt. Express 22(7), 7811–7820 (2011).
[Crossref]

2009 (2)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17(19), 16558–16570 (2009).
[Crossref]

2008 (3)

H. J. Kimble, “Quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

X. L. Niu, Y. F. Huang, G. Y. Xiang, G. C. Guo, and Z. Y. Ou, “Beamlike high-brightness source of polarization-entangled photon pairs,” Opt. Lett. 33(9), 968–970 (2008).
[Crossref]

O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
[Crossref]

2007 (3)

2006 (1)

2003 (1)

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

2002 (5)

K. Tsujino, S. Takeuchi, and K. Sasaki, “Detailed analysis of the fidelity of quantum teleportation using photons: Considering real experimental parameters,” Phys. Rev. A 66(4), 042314 (2002).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
[Crossref]

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[Crossref]

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref]

2000 (2)

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61(4), 042304 (2000).
[Crossref]

1998 (1)

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[Crossref]

1968 (1)

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light,” J. Phys. A 1, 128 (1968).
[Crossref]

Aspuru-guzik, A.

A. Aspuru-guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Baek, B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Baets, R. G.

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Beyer, J.

Bogaerts, W.

Bonneau, D.

Branning, D.

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[Crossref]

Braunstein, S. L.

P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61(4), 042304 (2000).
[Crossref]

Calkins, B.

Castelletto, S.

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Chae, C. J.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

Chen, Z.-B.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Cho, Y. W.

O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
[Crossref]

Choi, D.-Y.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

Christensen, B. G.

Clark, A. S.

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Clemmen, S.

Collins, M. J.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

Collons, M. J.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Dusek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Eggleton, B. J.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

Emplit, P.

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

Fattal, D.

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

Fitch, M. J.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

Foster, M. A.

Franson, J. D.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
[Crossref]

Gaeta, A. L.

Gerrits, T.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

S. Ramelow, A. Mech, M. Giustina, S. Groblacher, W. Wieczorek, J. Beyer, B. Calkins, T. Gerrits, S. W. Nam, A. Zeilinger, and R. Ursin, “Highly efficient heralding of entangled single photons,” Opt. Express 21(6), 6707–6717 (2013).
[Crossref]

Giustina, M.

Groblacher, S.

Guo, G. C.

Harrington, S.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Hasegawa, T.

He, J.

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Helt, L. G.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

Hemsley, E.

Horokiri, T.

T. Horokiri, Y. Takeno, A. Yabushita, and T. Kobayashi, “Photon-number-resolved heralded-photon source for improved quantum key distribution,” Phys. Rev. A 76(1), 012306 (2007).
[Crossref]

Huang, Y. F.

Huy, K. P.

Jacobs, B. C.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
[Crossref]

Jakeman, E.

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light,” J. Phys. A 1, 128 (1968).
[Crossref]

Jennewein, T.

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

Jizan, I.

Kaneda, F.

Kim, Y. H.

O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
[Crossref]

Kimble, H. J.

H. J. Kimble, “Quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref]

Kobayashi, T.

T. Horokiri, Y. Takeno, A. Yabushita, and T. Kobayashi, “Photon-number-resolved heralded-photon source for improved quantum key distribution,” Phys. Rev. A 76(1), 012306 (2007).
[Crossref]

Kofler, J.

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

Kok, P.

P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61(4), 042304 (2000).
[Crossref]

Krauss, T. F.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Kumar, P.

Kurtsiefer, C.

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

Kwiat, P. G.

Kwon, O.

O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
[Crossref]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref]

Lee, K. F.

Leong, P. H. W

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

Lita, A. E.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Liu, Z.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

Lopson, M.

Lu, C.-Y.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Lutkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Ma, X. S.

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

Mahendra, A.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

Marshall, G. D.

Marsili, F.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Martin-Lopez, E.

Massar, S.

Matsui, M.

Mayer, S.

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

McCusker, K. T.

Mech, A.

Mendoza, G. J.

Migdall, A.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

Migdall, A. L.

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[Crossref]

Miki, S.

Milburn, G. J.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref]

Mirin, R. P.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Munns, J.

Nam, S. W.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

S. Ramelow, A. Mech, M. Giustina, S. Groblacher, W. Wieczorek, J. Beyer, B. Calkins, T. Gerrits, S. W. Nam, A. Zeilinger, and R. Ursin, “Highly efficient heralding of entangled single photons,” Opt. Express 21(6), 6707–6717 (2013).
[Crossref]

Nishioka, T.

Niu, X. L.

O’Brien, J. L.

Okamoto, R.

Ou, Z. Y.

Pan, J.-W.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Park, H. S.

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Pelton, M.

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

Piekarek, M.

Pike, E. R.

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light,” J. Phys. A 1, 128 (1968).
[Crossref]

Pittman, T. B.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
[Crossref]

Polyakov, S. V.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

Ramelow, S.

Rarity, J. G.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[Crossref]

Reardon, C.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Rey, I. H.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Santagati, R.

Santori, C.

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

Sasaki, K.

A. Soujaeff, T. Nishioka, T. Hasegawa, S. Takeuchi, T. Tsurumaru, K. Sasaki, and M. Matsui, “Quantum key distribution at 1550 nm using a pulse heralded single photon source,” Opt. Express 15(2), 726–734 (2007).
[Crossref]

K. Tsujino, S. Takeuchi, and K. Sasaki, “Detailed analysis of the fidelity of quantum teleportation using photons: Considering real experimental parameters,” Phys. Rev. A 66(4), 042314 (2002).
[Crossref]

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Schmidt, B. S.

Shahnia, S.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Shapiro, J. H.

Sharping, J. E.

Shaw, M. D.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Solomon, G. S.

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

Soujaeff, A.

Steel, M. J.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Stern, J. A.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Takeno, Y.

T. Horokiri, Y. Takeno, A. Yabushita, and T. Kobayashi, “Photon-number-resolved heralded-photon source for improved quantum key distribution,” Phys. Rev. A 76(1), 012306 (2007).
[Crossref]

Takeuchi, S.

Tanida, M.

Tapster, P. R.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[Crossref]

Terai, H.

Thompsom, M. G.

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompsom, “Effect of loss on multiplexed single-photon sources,” New. J. Phys. 17(4), 043057 (2015).
[Crossref]

Thompson, M. G.

Tsujino, K.

K. Tsujino, S. Takeuchi, and K. Sasaki, “Detailed analysis of the fidelity of quantum teleportation using photons: Considering real experimental parameters,” Phys. Rev. A 66(4), 042314 (2002).
[Crossref]

Tsurumaru, T.

Turner, A. C.

Ursin, R.

Vayshenker, I.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Verma, V. B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Vo, T. D.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Vuckovic, J.

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

Walther, P.

A. Aspuru-guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Wang, Z.

Weinfurter, H.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

Wieczorek, W.

Wong, F. N.

Wong, J. J.

Xiang, G. Y.

Xiong, C.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

Xoing, C.

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Yabushita, A.

T. Horokiri, Y. Takeno, A. Yabushita, and T. Kobayashi, “Photon-number-resolved heralded-photon source for improved quantum key distribution,” Phys. Rev. A 76(1), 012306 (2007).
[Crossref]

Yamamoto, Y.

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

Yamashita, T.

Zarda, P.

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

Zeilinger, A.

S. Ramelow, A. Mech, M. Giustina, S. Groblacher, W. Wieczorek, J. Beyer, B. Calkins, T. Gerrits, S. W. Nam, A. Zeilinger, and R. Ursin, “Highly efficient heralding of entangled single photons,” Opt. Express 21(6), 6707–6717 (2013).
[Crossref]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

Zhang, B.

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

Zhang, X.

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

X. Zhang, I. Jizan, J. He, A. S. Clark, D.-Y. Choi, C. J. Chae, B. J. Eggleton, and C. Xiong, “Enhancing the heralded single-photon rate from a silicon nanowire by time and wavelength division multiplexing pump pulses,” Opt. Lett. 40(11), 2489–2492 (2015).
[Crossref]

Zotter, S.

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

Zukowski, M.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

J. Mod. Opt. (1)

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[Crossref]

J. Phys. A (1)

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light,” J. Phys. A 1, 128 (1968).
[Crossref]

Nat. Comm. (2)

C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, D.-Y. Choi, C. J. Chae, P. H. W Leong, and B. J. Eggleton, “Active temporal multiplexing of indistinguishable heralded single photons,” Nat. Comm. 7, 10853 (2016).
[Crossref]

M. J. Collons, C. Xoing, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat. Comm. 4, 2582 (2013).

Nat. Photon. (1)

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photon. 7, 210–214 (2013).
[Crossref]

Nat. Phys. (1)

A. Aspuru-guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Nature (2)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref]

H. J. Kimble, “Quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

Nature (London) (1)

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature (London) 419, 594 (2002).
[Crossref]

New. J. Phys. (1)

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompsom, “Effect of loss on multiplexed single-photon sources,” New. J. Phys. 17(4), 043057 (2015).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Optica (2)

Phys. Rev. A (8)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudodemand from stored parametric down-conversion,” Phys. Rev. A 66(4), 042303 (2002).
[Crossref]

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[Crossref]

O. Kwon, Y. W. Cho, and Y. H. Kim, “Single-mode coupling efficiencies of Type-II spontaneous parametric down -conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78(5), 053825 (2008).
[Crossref]

X. S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83(4), 043814 (2011).
[Crossref]

K. Tsujino, S. Takeuchi, and K. Sasaki, “Detailed analysis of the fidelity of quantum teleportation using photons: Considering real experimental parameters,” Phys. Rev. A 66(4), 042314 (2002).
[Crossref]

T. Horokiri, Y. Takeno, A. Yabushita, and T. Kobayashi, “Photon-number-resolved heralded-photon source for improved quantum key distribution,” Phys. Rev. A 76(1), 012306 (2007).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61(4), 042304 (2000).
[Crossref]

Phys. Rev. Lett. (2)

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, and G. S. Solomon, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref]

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[Crossref]

Rev. Mod. Phys. (2)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Rev. Sci. Instrum. (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Scheme of a conventional HSPS [S = 1]. (b) Scheme of HSPS using a PND [S = 1, D = 1]. (c) Scheme of a cascaded detector [D = 2]. The detector consists of 50:50 beam splitters (BS) and on–off detectors. (d) Scheme of Multiplexed-HSPS [S = 2] with on-off detectors/Multiplexed-HSPS with PND [S = 2] with PNDs.
Fig. 2
Fig. 2 Two-photon probability from the Multiplexed-HSPS with PND versus the mean number of photons pairs for (a) ηh=0.5, t=0.5, (b) ηh=0.9, t=0.5. Black line: S = 1, D = 0; red line: S = 1, D = 1; blue line: S = 1, D = 2; purple line: S = 1, D = ∞; black dashed line: S = 2, D = 0; red dashed line: S = 2, D = 1; blue dashed line: S = 2, D = 2; purple dashed line: S = 2, D = ∞:
Fig. 3
Fig. 3 Suppression ratio versus detection efficiency. Ratio between HSPS [S = 1, D = 0] and Multiplexed-HSPS with PND [S, D] for N ¯ = 0.05 , t = 0.5. Black curve: S = 1, D = 0; blue curve: S = 2, D = 0; red curve: S = 4, D = 0; black dashed curve: S = 1, D = 1; blue dashed curve: S = 2, D = 1; black long dashed/short dashed curve: S = 1, D = ∞; blue long dashed/short dashed curve: S = 2, D = ∞.
Fig. 4
Fig. 4 Schematic diagram of experimental set up. BBO: beta-barium-borate crystal (Type-I); DM: dichroic mirror; FC: fiber coupler; SPCM: single-photon counting module; XOR: XOR gate; AND: AND gate; EOM: electro-optic modulator; PBS: polarizing beam splitter; HWP: half wave plate; BPF: band pass filter (∆λ = 4 nm); PMF : polarization maintaining fiber; 50:50 FBS: 50:50 fiber beam splitter; AMP: high-voltage pulsed amplifier; fiber delay is 65m.
Fig. 5
Fig. 5 Experimental results: The vertical axis is the two photon probability at the output. The horizontal axis shows the number of events where one of the two detectors (D1, D2) detects a photon when the final heralding signal is produced. HSPS (SPDC-A) [S=1, D=0] (blue circles), HSPS (SPDC-A) using PND [S=1, D=1] (red circles), Multiplexed-HSPS [S=2, D=0] (blue squares), a Multiplexed-HSPS with PND [S=2, D=1] (red square), where all contain the optical switch. The dashed lines are based on our model.

Tables (1)

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Table 1 Separately measured parameters. ηh,A/B: detection efficiency of the detector in SPDC-A/B. tA/ B : transmittance of the single photon between the SPDC-A/B and the output.

Equations (10)

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P ( n ) = P th ( n ) N ¯ n ( 1 + N ¯ ) 1 + n ,
P ( n ) = P Po ( n ) N ¯ n e N ¯ n ! .
P herald = m = 1 P ( m ) P D ( 1 | m ) ,
P D = 0 ( 1 | m ) = 1 ( 1 η h ) m ,
P D ( 1 | m ) = 1 2 m 1 a = 0 m   m C a { P D 1 ( 1 | a ) ( 1 η h ) m a } ,
P C ( n ) = m = n P ( m ) P D ( 1 | m ) m C n t n ( 1 t ) m n ,
P h ( n ) = ( P C 1 ( n ) + ( 1 P herald 1 ) ( P C 2 ( n ) + ( 1 P herald 2 ) ( P C 3 ( n ) + ) ) 1 ( 1 P herald 1 ) ( 1 P herald 2 ) ( 1 P herald 3 ) ,
P ( n ) = γ P th ( n ) + ( 1 γ ) P Po ( n ) .
P h ( 2 ) ~ ( 1 + γ ) t 2 P D ( 1 | 2 ) 2 η h S × N ¯ .
R = P h ( 2 ) [ S , D ] P h ( 2 ) [ S = 1 , D = 0 ] ,

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