Abstract

We investigate the feasibility and performance of photon-number-resolved photodetection employing single-photon avalanche photodiodes (SPADs) with low dark counts. While the main idea, to split n photons into m detection modes with a vanishing probability of more than one photon per mode, is not new, we investigate here a important variant of this situation where SPADs are side-coupled to the same waveguide rather than terminally coupled to a propagation tree. This prevents the nonideal SPAD quantum efficiency from contributing to photon loss. We propose a concrete SPAD segmented waveguide detector based on a vertical directional coupler design, and characterize its performance by evaluating the purities of Positive-Operator-Valued Measures (POVMs) in terms of number of SPADs, photon loss, dark counts, and electrical cross-talk.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (4)

K. L. Nicolich, C. Cahall, N. T. Islam, G. P. Lafyatis, J. Kim, A. J. Miller, and D. J. Gauthier, “Universal model for the turn-on dynamics of superconducting nanowire single-photon detectors,” Phys. Rev. Appl. 12(3), 034020 (2019).
[Crossref]

J. P. Höpker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated transition edge sensors on titanium in-diffused lithium niobate waveguides,” APL Photonics 4(5), 056103 (2019).
[Crossref]

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

R. Nehra, A. Win, M. Eaton, R. Shahrokhshahi, N. Sridhar, T. Gerrits, A. Lita, S. W. Nam, and O. Pfister, “State-independent quantum state tomography by photon-number-resolving measurements,” Optica 6(10), 1356–1360 (2019).
[Crossref]

2018 (3)

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26(10), 12499–12505 (2018).
[Crossref]

F. M. Miatto, A. Safari, and R. W. Boyd, “Explicit formulas for photon number discrimination with on/off detectors,” Appl. Opt. 57(23), 6750–6754 (2018).
[Crossref]

Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
[Crossref]

2017 (3)

S. J. van Enk, “Time-dependent spectrum of a single photon and its positive-operator-valued measure,” Phys. Rev. A 96(3), 033834 (2017).
[Crossref]

S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Commun. 1(4), 045001 (2017).
[Crossref]

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

2016 (3)

C. Veerappan and E. Charbon, “A low dark count pin diode based spad in cmos technology,” IEEE Trans. Electron Devices 63(1), 65–71 (2016).
[Crossref]

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

C. Lee, S. Ferrari, W. H. P. Pernice, and C. Rockstuhl, “Sub-poisson-binomial light,” Phys. Rev. A 94(5), 053844 (2016).
[Crossref]

2015 (5)

D. D’Agostino, G. Carnicella, C. Ciminelli, P. Thijs, P. J. Veldhoven, H. Ambrosius, and M. Smit, “Low-loss passive waveguides in a generic inp foundry process via local diffusion of zinc,” Opt. Express 23(19), 25143–25157 (2015).
[Crossref]

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
[Crossref]

X. Jiang, M. Itzler, K. O’Donnell, M. Entwistle, M. Owens, K. Slomkowski, and S. Rangwala, “Inp-based single-photon detectors and geiger-mode apd arrays for quantum communications applications,” IEEE J. Sel. Top. Quantum Electron. 21(3), 5–16 (2015).
[Crossref]

F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40(7), 1548–1551 (2015).
[Crossref]

F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting nanowire detectors,” Supercond. Sci. Technol. 28(10), 104001 (2015).
[Crossref]

2013 (1)

2012 (2)

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109(9), 093601 (2012).
[Crossref]

2010 (2)

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

M. Ramilli, A. Allevi, V. Chmill, M. Bondani, M. Caccia, and A. Andreoni, “Photon-number statistics with silicon photomultipliers,” J. Opt. Soc. Am. B 27(5), 852–862 (2010).
[Crossref]

2009 (1)

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

2008 (3)

I. Rech, A. Ingargiola, R. Spinelli, I. Labanca, S. Marangoni, M. Ghioni, and S. Cova, “Optical crosstalk in single photon avalanche diode arrays: a new complete model,” Opt. Express 16(12), 8381–8394 (2008).
[Crossref]

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Levy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
[Crossref]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16(5), 3032–3040 (2008).
[Crossref]

2007 (1)

B. E. Kardynał, S. S. Hees, A. J. Shields, C. Nicoll, I. Farrer, and D. A. Ritchie, “Photon number resolving detector based on a quantum dot field effect transistor,” Appl. Phys. Lett. 90(18), 181114 (2007).
[Crossref]

2005 (1)

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71(6), 061803 (2005).
[Crossref]

2003 (2)

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[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]

2001 (1)

P. Kok and S. L. Braunstein, “Detection devices in entanglement-based optical state preparation,” Phys. Rev. A 63(3), 033812 (2001).
[Crossref]

2000 (1)

A. J. Shields, M. P. O’Sullivan, I. Farrer, D. A. Ritchie, R. A. Hogg, M. L. Leadbeater, C. E. Norman, and M. Pepper, “Detection of single photons using a field-effect transistor gated by a layer of quantum dots,” Appl. Phys. Lett. 76(25), 3673–3675 (2000).
[Crossref]

1996 (2)

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53(6), 4528–4533 (1996).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76(23), 4344–4347 (1996).
[Crossref]

1969 (1)

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. iii. theory of photoelectron counting statistics,” Phys. Rev. 179(2), 368–374 (1969).
[Crossref]

Achilles, D.

Agarwal, G. S.

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109(9), 093601 (2012).
[Crossref]

Allevi, A.

Ambrosius, H.

Andreoni, A.

Avella, A.

Banaszek, K.

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76(23), 4344–4347 (1996).
[Crossref]

Baños, R.

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Barkhofen, S.

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

Bartley, T. J.

J. P. Höpker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated transition edge sensors on titanium in-diffused lithium niobate waveguides,” APL Photonics 4(5), 056103 (2019).
[Crossref]

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Bartnick, M.

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Barton, J. S.

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “An ultra-low-loss (<0.1 dB/m) planar silica waveguide platform,” IEEE Photon. Soc. Newslett. p. 4 (2011).

Bauters, J. F.

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “An ultra-low-loss (<0.1 dB/m) planar silica waveguide platform,” IEEE Photon. Soc. Newslett. p. 4 (2011).

Beling, A.

Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
[Crossref]

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P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
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J. P. Höpker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated transition edge sensors on titanium in-diffused lithium niobate waveguides,” APL Photonics 4(5), 056103 (2019).
[Crossref]

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P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
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Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
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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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F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting nanowire detectors,” Supercond. Sci. Technol. 28(10), 104001 (2015).
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B. E. Kardynał, S. S. Hees, A. J. Shields, C. Nicoll, I. Farrer, and D. A. Ritchie, “Photon number resolving detector based on a quantum dot field effect transistor,” Appl. Phys. Lett. 90(18), 181114 (2007).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Levy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
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K. L. Nicolich, C. Cahall, N. T. Islam, G. P. Lafyatis, J. Kim, A. J. Miller, and D. J. Gauthier, “Universal model for the turn-on dynamics of superconducting nanowire single-photon detectors,” Phys. Rev. Appl. 12(3), 034020 (2019).
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B. E. Kardynał, S. S. Hees, A. J. Shields, C. Nicoll, I. Farrer, and D. A. Ritchie, “Photon number resolving detector based on a quantum dot field effect transistor,” Appl. Phys. Lett. 90(18), 181114 (2007).
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A. J. Shields, M. P. O’Sullivan, I. Farrer, D. A. Ritchie, R. A. Hogg, M. L. Leadbeater, C. E. Norman, and M. Pepper, “Detection of single photons using a field-effect transistor gated by a layer of quantum dots,” Appl. Phys. Lett. 76(25), 3673–3675 (2000).
[Crossref]

Rockstuhl, C.

C. Lee, S. Ferrari, W. H. P. Pernice, and C. Rockstuhl, “Sub-poisson-binomial light,” Phys. Rev. A 94(5), 053844 (2016).
[Crossref]

Roeloffzen, C. G. H.

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Rosenberg, D.

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71(6), 061803 (2005).
[Crossref]

Safari, A.

Sahin, D.

F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting nanowire detectors,” Supercond. Sci. Technol. 28(10), 104001 (2015).
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Santandrea, M.

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Scully, M. O.

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. iii. theory of photoelectron counting statistics,” Phys. Rev. 179(2), 368–374 (1969).
[Crossref]

Seleznev, V.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Levy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
[Crossref]

Shahrokhshahi, R.

Shields, A. J.

B. E. Kardynał, S. S. Hees, A. J. Shields, C. Nicoll, I. Farrer, and D. A. Ritchie, “Photon number resolving detector based on a quantum dot field effect transistor,” Appl. Phys. Lett. 90(18), 181114 (2007).
[Crossref]

A. J. Shields, M. P. O’Sullivan, I. Farrer, D. A. Ritchie, R. A. Hogg, M. L. Leadbeater, C. E. Norman, and M. Pepper, “Detection of single photons using a field-effect transistor gated by a layer of quantum dots,” Appl. Phys. Lett. 76(25), 3673–3675 (2000).
[Crossref]

Silberhorn, C.

J. P. Höpker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated transition edge sensors on titanium in-diffused lithium niobate waveguides,” APL Photonics 4(5), 056103 (2019).
[Crossref]

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref]

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Sliwa, C.

Slomkowski, K.

X. Jiang, M. Itzler, K. O’Donnell, M. Entwistle, M. Owens, K. Slomkowski, and S. Rangwala, “Inp-based single-photon detectors and geiger-mode apd arrays for quantum communications applications,” IEEE J. Sel. Top. Quantum Electron. 21(3), 5–16 (2015).
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Sperling, J.

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109(9), 093601 (2012).
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J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012).
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Sridhar, N.

Srinivasan, K.

Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Stassen, A.

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Sun, K.

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26(10), 12499–12505 (2018).
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Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Thiele, F.

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Thijs, P.

Tiedau, J.

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

Tisa, S.

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

Tosi, A.

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

Ulm, G.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
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van Dijk, P. W.

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

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S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Commun. 1(4), 045001 (2017).
[Crossref]

S. J. van Enk, “Time-dependent spectrum of a single photon and its positive-operator-valued measure,” Phys. Rev. A 96(3), 033834 (2017).
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C. Veerappan and E. Charbon, “A low dark count pin diode based spad in cmos technology,” IEEE Trans. Electron Devices 63(1), 65–71 (2016).
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Verma, V.

J. P. Höpker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Towards integrated superconducting detectors on lithium niobate waveguides,” in Quantum Photonic Devices, vol. 10358C. Soci, M. Agio, and K. Srinivasan, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 21–27.

Villa, F.

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

Vogel, W.

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109(9), 093601 (2012).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012).
[Crossref]

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53(6), 4528–4533 (1996).
[Crossref]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53(6), 4528–4533 (1996).
[Crossref]

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P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
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J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
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Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
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Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
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Wódkiewicz, K.

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76(23), 4344–4347 (1996).
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Xie, L.

Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Xie, X.

Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
[Crossref]

Ye, N.

Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Yu, Q.

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26(10), 12499–12505 (2018).
[Crossref]

Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
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Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Zappa, F.

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

Zervas, M.

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Q. Yu, N. Ye, J. Gao, K. Sun, L. Xie, K. Srinivasan, M. Zervas, G. Navickaite, M. Geiselmann, and A. Beling, “High-responsivity photodiodes heterogeneously integrated on silicon nitride waveguides,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optical Society of America, 2019), p. IW3A.5.

Zhang, L.

Zhou, Z.

F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting nanowire detectors,” Supercond. Sci. Technol. 28(10), 104001 (2015).
[Crossref]

Zwiller, V.

Zwinkels, J. C.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

APL Photonics (1)

J. P. Höpker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated transition edge sensors on titanium in-diffused lithium niobate waveguides,” APL Photonics 4(5), 056103 (2019).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

B. E. Kardynał, S. S. Hees, A. J. Shields, C. Nicoll, I. Farrer, and D. A. Ritchie, “Photon number resolving detector based on a quantum dot field effect transistor,” Appl. Phys. Lett. 90(18), 181114 (2007).
[Crossref]

A. J. Shields, M. P. O’Sullivan, I. Farrer, D. A. Ritchie, R. A. Hogg, M. L. Leadbeater, C. E. Norman, and M. Pepper, “Detection of single photons using a field-effect transistor gated by a layer of quantum dots,” Appl. Phys. Lett. 76(25), 3673–3675 (2000).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (3)

X. Jiang, M. Itzler, K. O’Donnell, M. Entwistle, M. Owens, K. Slomkowski, and S. Rangwala, “Inp-based single-photon detectors and geiger-mode apd arrays for quantum communications applications,” IEEE J. Sel. Top. Quantum Electron. 21(3), 5–16 (2015).
[Crossref]

Y. Wang, Z. Wang, Q. Yu, X. Xie, T. Posavitz, M. Jacob-Mitos, A. Ramaswamy, E. J. Norberg, G. A. Fish, and A. Beling, “High-power photodiodes with 65 GHz bandwidth heterogeneously integrated onto silicon-on-insulator nano-waveguides,” IEEE J. Sel. Top. Quantum Electron. 24(2), 1–6 (2018).
[Crossref]

P. Munoz, P. W. van Dijk, D. Geuzebroek, M. Geiselmann, C. Dominguez, A. Stassen, J. D. Doménech, M. Zervas, A. Leinse, C. G. H. Roeloffzen, B. Gargallo, R. Baños, J. Fernández, G. M. Cabanes, L. A. Bru, and D. Pastor, “Foundry developments toward silicon nitride photonics from visible to the mid-infrared,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

IEEE Sens. J. (1)

D. Bronzi, F. Villa, S. Tisa, A. Tosi, and F. Zappa, “Spad figures of merit for photon-counting, photon-timing, and imaging applications: A review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

IEEE Trans. Electron Devices (1)

C. Veerappan and E. Charbon, “A low dark count pin diode based spad in cmos technology,” IEEE Trans. Electron Devices 63(1), 65–71 (2016).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Commun. (1)

S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Commun. 1(4), 045001 (2017).
[Crossref]

Metrologia (1)

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47(5), R15–R32 (2010).
[Crossref]

Nat. Photonics (1)

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Levy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
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Nat. Phys. (1)

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
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New J. Phys. (1)

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
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Opt. Express (5)

Opt. Lett. (2)

Optica (1)

Phys. Rev. (1)

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R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
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J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71(6), 061803 (2005).
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C. Lee, S. Ferrari, W. H. P. Pernice, and C. Rockstuhl, “Sub-poisson-binomial light,” Phys. Rev. A 94(5), 053844 (2016).
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S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53(6), 4528–4533 (1996).
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S. J. van Enk, “Time-dependent spectrum of a single photon and its positive-operator-valued measure,” Phys. Rev. A 96(3), 033834 (2017).
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K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76(23), 4344–4347 (1996).
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F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting nanowire detectors,” Supercond. Sci. Technol. 28(10), 104001 (2015).
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Figures (13)

Fig. 1.
Fig. 1. Principle sketch of a segmented detector. Guided photons are detected alongside propagation by SPADs which frustrate total internal reflection. The quantum efficiency (QE) of SPAD #$j$ is $\alpha _{j}^{2}$. The design goal is to eschew detection losses, which are distinct from the nonunity of $\alpha _{j}^{2}$, and keep all undetected photons in the waveguide for further detection.
Fig. 2.
Fig. 2. Model for detection alongside propagation. If this is the $\rm j^{th}$ SPAD, then $t_{j}=tt' + rr'\sqrt {1-\alpha ^{2}}$, where $t_{j}=T_{j}^{1/2}$ in Fig. 3.
Fig. 3.
Fig. 3. Model of a PNR segmented photodetector with $R_{j}+T_{j}\equiv r_{j}^{2}+t_{j}^{2}=1, \forall j \in [1,m]$.
Fig. 4.
Fig. 4. (a) Cross-section of waveguide photodiode [24]; (b) Side view schematic of light propagating in the segmented waveguide photodetector. (c) Normalized optical power and QE of PD1 to PD6 in the segmented photodetector. The inset shows the optical intensity in PD1 with a PD length of 32$\mu$m.
Fig. 5.
Fig. 5. (a) Cross-section of of new waveguide photodiode design with thin absorber; (b) Side view schematic of light propagating in the segmented waveguide photodetector with cladding layer.
Fig. 6.
Fig. 6. (a) Simulated optical intensity in PDs with various lengths; (b) Normalized optical power with (black) and without (red) absorption in the 50-element segmented detector with 8 mm total length.
Fig. 7.
Fig. 7. Conditional probabilities $P_{2000}(k|n)$ versus $n$, for $\eta = 1$.
Fig. 8.
Fig. 8. POVM element $\textrm {Purity}(\prod _{k})$ versus click number $k$, for different $m$.
Fig. 9.
Fig. 9. Conditional probabilities $P_{50}(k|n)$ versus $n$, for $\eta = 0.9$.
Fig. 10.
Fig. 10. POVM element $\textrm {Purity}(\prod _{k})$ versus click number $k$, for several values of $\eta$ at $m = 50$
Fig. 11.
Fig. 11. POVM element $\textrm {Purity}(\prod _{k})$ versus click number $k$, for several values of dark count probabilities, $\delta$ at $m = 16$
Fig. 12.
Fig. 12. Conditional probabilities $P_{16}(k|n)$ versus $n$, for $\eta = 0.9$, $\delta = 0.1$, and $\epsilon = 0.01$.
Fig. 13.
Fig. 13. POVM element $\textrm {Purity}(\Pi _{k})$ versus click number $k$, for several values of $\delta$ and $\epsilon$ at $m = 16$.

Equations (27)

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t r r t 1 α 2 = 0.
p k = Tr ( ρ Π k ) .
Π k = n = 0 P ( k | n ) | n n | ,
Π k Π k δ k k Π k .
Purity ( Π k ) = Tr ( Π k 2 ) Tr ( Π k ) 2 .
1 D Purity ( Π k ) 1 ,
| n = a 1 n n ! | 0 m .
a 1 = r 1 a 1 + k = 2 m 1 [ l = 1 m 1 t l ] r k 1 a k + l = 1 m 1 t l a m ,
| ψ o u t = 1 n ! ( r 1 a 1 + k = 2 m 1 l = 1 m 1 t l r k 1 a k + l = 1 m 1 t l a m ) n | 0 m
| ψ o u t = n 1 = 0 n n m = 0 n j = 1 m n j = n n ! n 1 ! n 2 ! n m ! r 1 n 1 k = 2 m 1 τ 1 , k 1 n k r k n k τ m 1 , 1 n m i = 1 m a i n i | 0 m ,
P m ( k | n ) = n 1 = 0 n n m = 0 n ( ) i = 1 k n i = n n ! i = 1 m n i !   X ,
where  X = ( r 1 n 1 k = 2 m 1 τ 1 , k 1 n k r k n k τ m 1 , 1 n m ) 2
= R 1 n 1 T 1 n n 1 × R 2 n 2 T 2 n n 1 n 2 × × R m 1 n m 1 T m 1 n j = 1 m 1 n j .
X = 1 m n ,
P m ( k | n ) = n ! m n ( m k ) n 1 = 1 n n k = 1 n j = 1 k n j = n 1 i = 1 k n i ! .
P 1 ( 0 | n , η ) = ( 1 η ) n .
P 1 ( 1 | n , η ) = k = 1 n ( n k ) η k ( 1 η ) n k = 1 ( 1 η ) n .
P m ( k | n , η ) = n ! ( 1 η m ) n ( m k ) n 1 = 0 n n m = 0 n j = 1 m n j = n 1 j = 1 m n j ! l = n 1 n k [ ( 1 1 η ) l 1 ] ,
P 0 d = ( 1 δ ) ( 1 η ) n ,
P 1 d = 1 P 0 d = 1 ( 1 δ ) ( 1 η ) n ,
Π 1 d = n = 0 n = [ 1 ( 1 δ ) ( 1 η ) n ] | n n | .
P m ( k | n , η ) = n ! ( 1 δ ) m ( 1 η m ) n ( m k ) n 1 = 0 n n m = 0 n j = 1 m n j = n 1 j = 1 m n j ! l = n 1 n k [ 1 1 δ ( 1 1 η ) l 1 ] .
P 0 d , ϵ = ( 1 2 ) n n 1 = 0 n n 2 = 0 n j = 1 2 n j = n n ! n 1 ! n 2 ! ( 1 δ ) ( 1 η ) n 1 ( 1 ϵ ) n 2 * ( 1 δ ) ( 1 η ) n 2 ( 1 ϵ ) n 1 #
= ( 1 δ ) 2 ( 1 η ) n ( 1 ϵ ) n ,
P 1 d , ϵ = ( 1 2 ) n n 1 = 0 n n 2 = 0 n j = 1 2 n j = n n ! n 1 ! n 2 ! { [ 1 ( 1 δ ) ( 1 η ) n 1 ( 1 ϵ ) n 2 ] First SPAD clicked [ ( 1 δ ) ( 1 η ) n 2 ( 1 ϵ ) n 1 ] No click from second SPAD + [ 1 ( 1 δ ) ( 1 η ) n 2 ( 1 ϵ ) n 1 ] [ ( 1 δ ) ( 1 η ) n 1 ( 1 ϵ ) 2 n ] The other way around } ,
P 2 d , ϵ = ( 1 2 ) n n 1 = 0 n n 2 = 0 n j = 1 2 n j = n n ! n 1 ! n 2 ! [ 1 ( 1 δ ) ( 1 η ) n 1 ( 1 ϵ ) n 2 ] [ ( 1 ( 1 δ ) ( 1 η ) n 2 ( 1 ϵ ) n 1 ] .
P m ( k | n , η , δ , ϵ ) = C n ! ( 1 η m ) n ( m k ) n 1 = 0 n n m = 0 n j = 1 m n j = n { 1 j = 1 m n j ! l = n 1 n k [ 1 ( 1 δ ) ( 1 ϵ ) n ( 1 ϵ 1 η ) l 1 ] } ,

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