Abstract

Single Photon Detectors are integral to quantum optics and quantum information. Superconducting Nanowire based detectors exhibit new levels of performance, but have no accepted quantum optical model that is valid for multiple input photons. By performing Detector Tomography, we improve the recently proposed model [M.K. Akhlaghi and A.H. Majedi, IEEE Trans. Appl. Supercond. 19, 361 (2009)] and also investigate the manner in which these detectors respond nonlinearly to light, a valuable feature for some applications. We develop a device independent model for Single Photon Detectors that incorporates this nonlinearity.

© 2011 OSA

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    [CrossRef] [PubMed]
  3. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
    [CrossRef]
  4. O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
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  5. R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for raman lidar,” Appl. Opt. 48, 3903–3914 (2009).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

2009

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[CrossRef]

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

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, 27–30 (2009).
[CrossRef]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond. 19, 341–345 (2009).
[CrossRef]

Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. 56, 380–384 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 361–366 (2009).
[CrossRef]

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for raman lidar,” Appl. Opt. 48, 3903–3914 (2009).
[CrossRef] [PubMed]

2008

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

2007

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

2004

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[CrossRef] [PubMed]

2003

2002

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176–S183 (2002).
[CrossRef]

C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. 38, 1578–1579 (2002).
[CrossRef]

2001

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C 351, 349–356 (2001).
[CrossRef]

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A 63, 020102 (2001).
[CrossRef]

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

1996

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Achilles, D.

Akhlaghi, M. K.

Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. 56, 380–384 (2009).
[CrossRef]

M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 361–366 (2009).
[CrossRef]

Banaszek, K.

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Blais-Ouellette, S.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, 2004).

Brambilla, E.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176–S183 (2002).
[CrossRef]

Branderhorst, M. P. A.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

Brida, G.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Calkins, B.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Carignan, C.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Chulkova, G.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Ciavarella, L.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Clayton, M.

Clement, T. S.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Coldenstrodt-Ronge, H.

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, 27–30 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

Coldenstrodt-Ronge, H. B.

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

Cross, A.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Daigle, O.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Degiovanni, I. P.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Dzardanov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Eisert, J.

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[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, 27–30 (2009).
[CrossRef]

Feito, A.

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, 27–30 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

Ferrare, R.

Fortin, C. A.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Gach, J. L.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Gatti, A.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176–S183 (2002).
[CrossRef]

Genovese, M.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Gerrits, T.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Glancy, S.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Gol’tsman, G. N.

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C 351, 349–356 (2001).
[CrossRef]

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Goltsman, G.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Grangier, P.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

Guillaume, C.

O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. 121, 866–884 (2009).
[CrossRef]

Hadfield, R. H.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[CrossRef]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Jeong, H.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

Knill, E.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Korneev, A.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Korneev, A. A.

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C 351, 349–356 (2001).
[CrossRef]

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J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
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E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Lipatov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Lita, A. E.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

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C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. 38, 1578–1579 (2002).
[CrossRef]

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G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

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[CrossRef] [PubMed]

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, 27–30 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
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M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
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K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A 63, 020102 (2001).
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M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 361–366 (2009).
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Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. 56, 380–384 (2009).
[CrossRef]

J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond. 19, 341–345 (2009).
[CrossRef]

Mattle, K.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Mauerer, W.

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
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M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Mielke, B.

Migdall, A. L.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Miller, A. J.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Milostnaya, I.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Minaeva, O.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Mingolla, M. G.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Mirin, R. P.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Mitchell, M. W.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[CrossRef] [PubMed]

Mollenauer, L.

C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. 38, 1578–1579 (2002).
[CrossRef]

Nam, S. W.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

Newsom, R. K.

Okunev, O.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Orgiazzi, J. L.

Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. 56, 380–384 (2009).
[CrossRef]

Orgiazzi, J. L. F. X.

J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond. 19, 341–345 (2009).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

Paris, M. G. A.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Pearlman, A.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Piacentini, F.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Plenio, M. B.

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, 27–30 (2009).
[CrossRef]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

Pregnell, K. L.

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[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, 27–30 (2009).
[CrossRef]

Puentes, G.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

Rajteri, M.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Ralph, T. C.

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, 27–30 (2009).
[CrossRef]

Resch, K. J.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A 63, 020102 (2001).
[CrossRef]

Rubtsova, I.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Semenov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Semenov, A. D.

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C 351, 349–356 (2001).
[CrossRef]

Silberhorn, C.

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[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, 27–30 (2009).
[CrossRef]

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

Sivaraman, C.

Sliwa, C.

Slysz, W.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Smirnov, K.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Smith, B. J.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

Sobolewski, R.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006).
[CrossRef] [PubMed]

Steinberg, A. M.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[CrossRef] [PubMed]

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A 63, 020102 (2001).
[CrossRef]

Taralli, E.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

Tualle-Brouri, R.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

Turner, D. D.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, 2004).

Verevkin, A.

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Voronov, B.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).

Walmsley, I. A.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. 102 (2009).
[CrossRef] [PubMed]

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, 27–30 (2009).
[CrossRef]

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. 56, 432–441 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. 11 (2009).
[CrossRef]

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

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

Weinfurter, H.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Williams, C.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Xu, C.

C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. 38, 1578–1579 (2002).
[CrossRef]

Yan, Z.

Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. 56, 380–384 (2009).
[CrossRef]

Zeilinger, A.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001).
[CrossRef]

Electron. Lett.

C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. 38, 1578–1579 (2002).
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Figures (3)

Fig. 1
Fig. 1

(a) Schematics of the experimental setup. (b) Click Probability at different bias currents. The critical current was measured to be 26.0±0.5μA. The blue circles are measured Click probabilities (not all points are shown). The red lines are calculated using the linear model in Eq. (1) by calculating the efficiency parameter P1 when the click probability is equal to 0.1. P1 equals 2.44e-3, 7.3e-5, 4.87e-6 for 25, 20, and 16μA, respectively. The black dotted lines are from the nonlinear SPD Model in Eq. (5). {Pn} equals {7.30e-4, 2.49e-3}, {9.72e-6, 7.15e-5, 8.14e-9}, {0, 7.33e-8, 2.87e-10, 2.81e-14} for 25, 20, and 16μA, respectively. Zero values indicate they change the minimum of Eq. (6) by less than 1%.

Fig. 2
Fig. 2

(a) The Q-function (i.e. click probability for coherent state inputs) of the scaled detector at different bias currents. The scaling factor (k) equals 0.025, 0.0015, 0.00037 for 25, 20, and 16μA, respectively. The raw scaled data (black circles) agrees well with the tomographic POVM (blue line) and the AM Model (red dashed line). (b) The corresponding Click POVM operator. The operator found from tomography (blue circles) disagrees with that from the AM Model (red dashed line) but agrees with that from the nonlinear SPD Model (black dotted line). In the latter, {Pn} equals {7.29e-4, 9.95e-2}, {1.08e-5, 4.76e-2, 3.74e-3, 1.13e-4}, {0, 1.97e-4, 2.01e-3, 4.87e-4, 5.07e-5} for 25, 20, and 16μA, respectively. Zero values indicate they change the minimum of Eq. (6) by less than 1%.

Fig. 3
Fig. 3

(a) The Nonlinear Single Photon Detector model. Each element represents an n-Photon Detector. A broadly applicable model is created by logically ORing these elements. (b) {Pn} at 16μA found under different scaling factors, k. ◇, ○, □, ▽, △, and ✩ represent Pn=0 to Pn=5 respectively. The corresponding dashed lines have slopes equal to n on a log-log plot. The plot shows a linear optical loss of η scales PnηnPn.

Equations (6)

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π 0 SPD = n = 0 ( 1 P 1 ) n | n n | ,
π 0 NPD = m = 0 ( 1 P n ) ( m n ) | m m | ,
π 1 N L = 1 m = 0 n = 0 M 1 ( 1 P n ) ( m n ) | m m | ,
P Click ( α ) = 1 m = 0 e | α | 2 | α | 2 m m ! n = 0 M 1 ( 1 P n ) ( m n ) .
C = F Π = F ( E exp ( GH ) ) ,
min C FE + F exp ( G H ) C 2 ,

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