Abstract

Improving photon-number resolution of single-photon sensitive detectors is important for many applications, as is extending the range of such detectors. Here we seek improved resolution for a particular superconducting transition-edge sensor (TES) through better processing of the TES output waveforms. With that aim, two algorithms to extract number resolution from TES output waveforms are compared. The comparison is done by processing waveform data sets from a TES illuminated at nine illumination levels by a pulsed laser at 1550 nm. The algorithms are used to sort the individual output waveforms and then create clusters associated with individual photon numbers. The first uses a dot product with the waveform mean (for each illumination level), while the second uses K-means clustering modified to include knowledge of the Poisson distribution. The first algorithm is shown to distinguish adjacent peaks associated with photon numbers up to 19, whereas the second algorithm distinguishes photon numbers up to 23, using the same data.

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References

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  1. A. Migdall, “Differences explained in correlated-photon metrology techniques,” Phys. Today 52, 41–46 (1999).
    [CrossRef]
  2. A. P. Worsley, H. B. Coldenstrodt-Ronge, J. S. Lundeen, P. J. Mosley, B. J. Smith, G. Puentes, N. Thomas-Peter, and I. A. Walmsley, “Absolute efficiency estimation of photon-number-resolving detectors using twin beams,” Opt. Express 17, 4397–4411 (2009).
    [CrossRef]
  3. A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
    [CrossRef]
  4. 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, R15–R32 (2010).
    [CrossRef]
  5. R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
    [CrossRef]
  6. H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
    [CrossRef]
  7. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [CrossRef]
  8. S. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” Massachusetts Institute of Technology Research Laboratory of Electronics Quarterly Progress Report, Vol. 111, 115–120 (1973).
  9. C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
    [CrossRef]
  10. F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
    [CrossRef]
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  12. While always keeping in mind that this correspondence is never 100% assured, we refer to these as photon-number peaks.
  13. M. Mehmet, A. Ast, T. Eberle, S. Steinlechner, H. Vahlbruch, and R. Schnabel, “Squeezed light at 1550 nm with a quantum noise reduction of 12.3 dB,” Opt. Express 19, 25763–25772 (2011).
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  14. A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
    [CrossRef]
  15. M. Fujiwara and M. Sasaki, “Direct measurement of photon number statistics at telecom wavelengths using a charge integration photon detector,” Appl. Opt. 46, 3069–3074 (2007).
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    [CrossRef]
  17. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photon. 3, 696–705 (2009).
    [CrossRef]
  18. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
    [CrossRef]
  19. D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H. Fujino, H. Ishii, T. Itatani, S. Inoue, and T. Zama, “Titanium-based transition-edge photon number resolving detector with 98 percent detection efficiency with index-matched small-gap fiber coupling,” Opt. Express 19, 870–875 (2011).
    [CrossRef]
  20. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
    [CrossRef]
  21. C. Ding and X. He, “K-means clustering via principal component analysis,” in ICML Proceedings 21st International Conference on Machine Learning (Assoc. Comp. Mach., 2004), pp. 1–9.
  22. G. Hamerly and C. Elkan, “Alternatives to the k-means algorithm that find better clusterings,” in Proceedings of the Eleventh International Conference on Information and Knowledge Management (ACM, 2002), CIKM ’02, pp. 600–607.
  23. W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.
  24. T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).
  25. L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
    [CrossRef]
  26. P. E. Black, “Greedy algorithm,” in Dictionary of Algorithms and Data Structures [online], P. E. Black, ed., U. S. National Institute of Standards and Technology (2February2005) (accessed March 5, 2012). Available from http://www.nist.gov/dads/HTML/greedyalgo.html .
  27. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef]
  28. D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
    [CrossRef]
  29. Curves were classified as containing background photons if the initial or final voltage was above 16.8 mV, or if there was a peak of at least 16.8 mV after a time of 3.4 μs. The number of curves removed was consistent with the background rate given above.
  30. The mention of commercial products does not imply endorsement by the authors’ institutions nor does it imply that they are the best available for the purpose.

2011 (4)

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, R15–R32 (2010).
[CrossRef]

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[CrossRef]

2009 (4)

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

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

A. P. Worsley, H. B. Coldenstrodt-Ronge, J. S. Lundeen, P. J. Mosley, B. J. Smith, G. Puentes, N. Thomas-Peter, and I. A. Walmsley, “Absolute efficiency estimation of photon-number-resolving detectors using twin beams,” Opt. Express 17, 4397–4411 (2009).
[CrossRef]

2008 (2)

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

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

2007 (1)

2004 (2)

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

2003 (1)

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[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]

1999 (1)

A. Migdall, “Differences explained in correlated-photon metrology techniques,” Phys. Today 52, 41–46 (1999).
[CrossRef]

1998 (1)

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

1973 (1)

S. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” Massachusetts Institute of Technology Research Laboratory of Electronics Quarterly Progress Report, Vol. 111, 115–120 (1973).

Amemiya, K.

Andersen, U. L.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

Ast, A.

Baumgartner, G.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

Beaumont, A. R.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

Becerra, F. E.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

Black, P. E.

P. E. Black, “Greedy algorithm,” in Dictionary of Algorithms and Data Structures [online], P. E. Black, ed., U. S. National Institute of Standards and Technology (2February2005) (accessed March 5, 2012). Available from http://www.nist.gov/dads/HTML/greedyalgo.html .

Blackshaw, S.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Briegel, H. J.

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Buzhan, P.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Cabrera, B.

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

Cai, L.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Calkins, B.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Cassemiro, K. N.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

Cepko, C.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Cheung, J. Y.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

Chunnilall, C. J.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

Cirac, J. I.

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Coldenstrodt-Ronge, H. B.

Ding, C.

C. Ding and X. He, “K-means clustering via principal component analysis,” in ICML Proceedings 21st International Conference on Machine Learning (Assoc. Comp. Mach., 2004), pp. 1–9.

Dolgoshein, B.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Dolinar, S.

S. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” Massachusetts Institute of Technology Research Laboratory of Electronics Quarterly Progress Report, Vol. 111, 115–120 (1973).

Dur, W.

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Eberle, T.

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

Elkan, C.

G. Hamerly and C. Elkan, “Alternatives to the k-means algorithm that find better clusterings,” in Proceedings of the Eleventh International Conference on Information and Knowledge Management (ACM, 2002), CIKM ’02, pp. 600–607.

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

Figueroa-Feliciano, E.

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

Fixsen, D. J.

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.

Fox, N. P.

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, R15–R32 (2010).
[CrossRef]

Fujii, G.

Fujino, H.

Fujiwara, M.

Fukuda, D.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Gerrits, T.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Goldhar, J.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

Hadfield, R. H.

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

Hamerly, G.

G. Hamerly and C. Elkan, “Alternatives to the k-means algorithm that find better clusterings,” in Proceedings of the Eleventh International Conference on Information and Knowledge Management (ACM, 2002), CIKM ’02, pp. 600–607.

He, X.

C. Ding and X. He, “K-means clustering via principal component analysis,” in ICML Proceedings 21st International Conference on Machine Learning (Assoc. Comp. Mach., 2004), pp. 1–9.

Huang, H. Y.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Ikonen, E.

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, R15–R32 (2010).
[CrossRef]

Ilyin, A.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Inoue, S.

Ireland, J.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

Ishii, H.

Itatani, T.

Kaplin, V.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Klein, R.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[CrossRef]

Klemin, S.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[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]

Kosloski, J. T.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[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]

Leuchs, G.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

Lita, A. E.

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

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Liu, J. S.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Lundeen, J. S.

Martin, J. M.

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[CrossRef]

Mehmet, M.

Migdall, A.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

A. Migdall, “Differences explained in correlated-photon metrology techniques,” Phys. Today 52, 41–46 (1999).
[CrossRef]

S. V. Polyakov, A. Migdall, and S. W. Nam, “Real-time data-acquisition platform for pulsed measurements,” in Advances in Quantum Theory, AIP Conference Proceedings, Vol. 1327 (Växjö, 2010), pp. 505–519.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

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]

Miller, A. J.

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

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[CrossRef]

Mirin, R.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Mirzoyan, R.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Moseley, S. H.

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

Mosley, P. J.

Nam, S. W.

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

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[CrossRef]

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

S. V. Polyakov, A. Migdall, and S. W. Nam, “Real-time data-acquisition platform for pulsed measurements,” in Advances in Quantum Theory, AIP Conference Proceedings, Vol. 1327 (Växjö, 2010), pp. 505–519.

Numata, T.

Polyakov, S. V.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

S. V. Polyakov, A. Migdall, and S. W. Nam, “Real-time data-acquisition platform for pulsed measurements,” in Advances in Quantum Theory, AIP Conference Proceedings, Vol. 1327 (Växjö, 2010), pp. 505–519.

Popova, E.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

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W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.

Puentes, G.

Rastello, M. L.

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, R15–R32 (2010).
[CrossRef]

Sasaki, M.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
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M. Fujiwara and M. Sasaki, “Direct measurement of photon number statistics at telecom wavelengths using a charge integration photon detector,” Appl. Opt. 46, 3069–3074 (2007).
[CrossRef]

Schnabel, R.

Sergienko, A. V.

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[CrossRef]

Smith, B. J.

Steinlechner, S.

Takeoka, M.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

Teshima, M.

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.

Thomas-Peter, N.

Thornagel, R.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[CrossRef]

Tomlin, N.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Tsuchida, H.

Ulm, G.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
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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, R15–R32 (2010).
[CrossRef]

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Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

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W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.

Walmsley, I. A.

White, M. G.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

Wittmann, C.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
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Wong, W. H.

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

Worsley, A. P.

Yoshizawa, A.

Zama, T.

Zoller, P.

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

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, R15–R32 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. J. Miller, S. W. Nam, J. M. Martin, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83, 791–793 (2003).
[CrossRef]

Genome Biol. (1)

L. Cai, H. Y. Huang, S. Blackshaw, J. S. Liu, C. Cepko, and W. H. Wong, “Clustering analysis of SAGE data using a Poisson approach,” Genome Biol. 5, R51.1 (2004).
[CrossRef]

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S. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” Massachusetts Institute of Technology Research Laboratory of Electronics Quarterly Progress Report, Vol. 111, 115–120 (1973).

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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, R15–R32 (2010).
[CrossRef]

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[CrossRef]

Nat. Photon. (1)

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

Nature (1)

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

Nucl. Instrum. Methods Phys. Res. A (3)

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Instrum. Methods Phys. Res. A 610, 183–187 (2009).
[CrossRef]

P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kaplin, S. Klemin, R. Mirzoyan, E. Popova, and M. Teshima, “The cross-talk problem in SiPMs and their use as light sensors for imaging atmospheric Cherenkov telescopes,” Nucl. Instrum. Methods Phys. Res. A 610, 131–134 (2009).
[CrossRef]

D. J. Fixsen, S. H. Moseley, B. Cabrera, and E. Figueroa-Feliciano, “Pulse estimation in nonlinear detectors with nonstationary noise,” Nucl. Instrum. Methods Phys. Res. A 520, 555–558 (2004).
[CrossRef]

Opt. Express (4)

Phys. Rev. A (1)

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[CrossRef]

Phys. Today (1)

A. Migdall, “Differences explained in correlated-photon metrology techniques,” Phys. Today 52, 41–46 (1999).
[CrossRef]

Rev. Sci. Instrum. (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon source and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Other (9)

C. Ding and X. He, “K-means clustering via principal component analysis,” in ICML Proceedings 21st International Conference on Machine Learning (Assoc. Comp. Mach., 2004), pp. 1–9.

G. Hamerly and C. Elkan, “Alternatives to the k-means algorithm that find better clusterings,” in Proceedings of the Eleventh International Conference on Information and Knowledge Management (ACM, 2002), CIKM ’02, pp. 600–607.

W. H. Press, S. A. Teukolsky, W. T. Vettenberg, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007), Sections 13.3 and 16.1.

T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, S. W. Nam, and R. Mirin, “Extending single-photon optimized superconducting transition edge sensors beyond the single-photon counting regime,” presented at the Quantum Electronics and Laser Science Conference (2012).

Curves were classified as containing background photons if the initial or final voltage was above 16.8 mV, or if there was a peak of at least 16.8 mV after a time of 3.4 μs. The number of curves removed was consistent with the background rate given above.

The mention of commercial products does not imply endorsement by the authors’ institutions nor does it imply that they are the best available for the purpose.

P. E. Black, “Greedy algorithm,” in Dictionary of Algorithms and Data Structures [online], P. E. Black, ed., U. S. National Institute of Standards and Technology (2February2005) (accessed March 5, 2012). Available from http://www.nist.gov/dads/HTML/greedyalgo.html .

S. V. Polyakov, A. Migdall, and S. W. Nam, “Real-time data-acquisition platform for pulsed measurements,” in Advances in Quantum Theory, AIP Conference Proceedings, Vol. 1327 (Växjö, 2010), pp. 505–519.

While always keeping in mind that this correspondence is never 100% assured, we refer to these as photon-number peaks.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of the ordering of waveforms in the two algorithms. A waveform may be viewed as a point in a high-dimensional space. Specifically, a waveform discretized at times t1,,tNt may be considered to be the point (V(t1),,V(tNt))=(V1,,VNt). Just two dimensions of this high-dimensional space are shown here. The evolution of the waveform with photon number is seen for the dot-product algorithm in the straight line resulting from Eq. (2) and for the Poisson-influenced K-means algorithm (PIKA) in the curved line representing Eq. (13). The values of n=3, 4, or 5 are shown for both cases (small dots). Each large dot represents a particular waveform Vi(t) to be ordered. These waveforms are projected onto the line and the curve to show how they would be ordered by the two algorithms. One feature of the actual curves represented here is the uniform spacing of n for the dot-product method and the decreasing spacing for PIKA.

Fig. 2.
Fig. 2.

(a) Cluster waveform means for 0n12 derived from the two lowest illumination data sets n=2.00 (solid red) and n=2.83 (dotted blue). The n=0 response is approximately zero and is not very visible in the plot. (b) Waveform cluster means are given as derived from the n=22.6 (solid red) and n=31.6 (dashed blue) with n=6,9,,42 and n=12,15,,54, respectively. The solid red curves nearly obscure the dashed blue curves for 0n9 in (a) and for many n’s in (b), showing that the cluster means derived from independent data sets are nearly identical except where the number of waveforms in the data set is very few.

Fig. 3.
Fig. 3.

Counts of waveforms per bin (of width 0.05 photon) for 19 091 pulses (derived from 20 480 pulses after rejection of observations with blackbody photons) with an average of n=2.00 photons per pulse (blue points). The histogram was fit to a series of Gaussians centered on the integers leaving only the amplitude and width of each Gaussian as parameters (red line). Effective photon numbers n(eff) are found for each waveform using the Poisson-influenced K-means algorithm. Results for the dot-product algorithm (not shown) are very similar for this case.

Fig. 4.
Fig. 4.

Histogram of waveforms versus (a) effective photon number found using the dot-product algorithm or (b) interpolated photon number found using the Poisson-influenced K-means algorithm, all for 20 480 pulses, an average of n=22.6 photons per pulse, and a bin width of 0.05 of a photon. The histograms were fit as in Fig. 3 (red line).

Fig. 5.
Fig. 5.

Peak visibility of the dot-product algorithm (blue diamonds) and PIKA (red circles) fits of the counts in Figs. 3 and 4 (along with others not shown), calculated using (maxmin)/(max+min) with minima taken at half-integer photon numbers and the maxima the average of the two surrounding values. The visibility uncertainties were obtained by assuming Poisson statistics for the counts using a Monte Carlo technique to sample variations in the fitted curve. Uncertainties are two standard deviations, and are purely statistical.

Fig. 6.
Fig. 6.

Full width at half maximum (a) and the peak voltage (b) of the waveform cluster means as a function of photon number. Curves are derived from the data sets with n=2.00, 2.84, 4.02, 5.68, 7.99, 11.3, 16.0, 22.6, and 31.6 (shown as different color points).

Fig. 7.
Fig. 7.

Waveform cluster mean corresponding to a photon number of n=37 (red, solid) is compared to the n=25 cluster mean (green, dotted) with the latter curve shifted right by 3.2 μs (taken from the n=31.6 data set). This common falling edge shape being independent of photon number is typical behavior for the cluster means.

Equations (21)

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Pn(n)=ennnn!.
V^n(t)=1Mi=1MVi(t),
ni(eff)=ntVi(t)V^n(t)t[V^n(t)]2.
OK=n=n0n0+K1iCn1Ntt[Vi(t)V¯n(t)]2.
V¯n(t)=1mniCnVi(t),
OKPC=12σ2OK+OPC,
OPC=lnL(mn0,,mn0+K1;n).
L=LPLC.
LP=n=n0n0+K1(ennnn!)mn.
LC=M!(n=n0n0+K1mn!)1,
Φ(ν)=|S(ν)|2|S(ν)|2+|N(ν)|2.
Vi(t)αiV¯n(t)+(1αi)V¯n(t),
ni(eff)=αin+(1αi)n.
OK=n=n0n0+K1Jn,
Jn=iCnt[Vi(t)V¯n(t)]2.
Ja+=Ja+mama+1t[Vj(t)V¯n(t)]2.
Jb=Jbmb1mbt[Vj(t)V¯n(t)]2.
V¯n(t)=mnV¯n(t)±Vj(t)mn±1
lnLP(mn0,,ma,,mb,,mn0+K1;μ)=μM+n=n0n0+K1mn[nlnμln(mn!)].
lnLP(mn0,,ma+1,,mb1,,mn0+K1;μ)lnLP(mn0,,ma,,mb,,mn0+K1;μ)=(ba)lnμ+(ma+1)ln(ma+1)+ln(ma!)(mb)ln(mb)ln[(mb1)!].
ΔlnLC=mbma+1,

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