Abstract

Optical parametric down-conversion (PDC) is a central tool in quantum optics experiments. The number of collected down-converted modes greatly affects the quality of the produced photon state. We use Silicon Photomultiplier (SiPM) number-resolving detectors in order to observe the photon-number distribution of a PDC source, and show its dependence on the number of collected modes. Additionally, we show how the stimulated emission of photons and the partition of photons into several modes determine the overall photon number. We present a novel analytical model for the optical crosstalk effect in SiPM detectors, and use it to analyze the results.

© 2012 OSA

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2011 (1)

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef] [PubMed]

2010 (1)

2009 (4)

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

M. Akiba, K. Tsujino, K. Sato, and M. Sasaki, “Multipixel silicon avalanche photodiode with ultralow dark count rate at liquid nitrogen temperature,” Opt. Express 17, 16885–16897 (2009).
[CrossRef] [PubMed]

2008 (4)

B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425 (2008).
[CrossRef]

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
[CrossRef] [PubMed]

W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A 78, 033831 (2008).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (3)

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73, 033814 (2006).
[CrossRef]

O. A. Ivanova, T. S. Iskhakov, A. N. Penin, and M. V. Chekhova, “Multiphoton correlations in parametric down-conversion and their measurement in the pulsed regime,” Quantum Electron. 36, 951 (2006).
[CrossRef]

2005 (2)

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

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

2004 (5)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

F. Paleari, A. Andreoni, G. Zambra, and M. Bondani, “Thermal photon statistics in spontaneous parametric downconversion,” Opt. Express 12, 2816–2824 (2004).
[CrossRef] [PubMed]

2002 (3)

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

G. A. Durkin, C. Simon, and D. Bouwmeester, “Multiphoton entanglement concentration and quantum cryptography,” Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

2001 (1)

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

2000 (2)

D. Lincoln, “A large statistics study of the performance and yields of generation-6 vlpcs (histe-vi),” Nucl. Instrum. Methods Phys. Res. A 453, 177–181 (2000).
[CrossRef]

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

1998 (2)

G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
[CrossRef]

M. Vasilyev, S.-K. Choi, P. Kumar, and G. M. D’Ariano, “Investigation of the photon statistics of parametric fluorescence in a traveling-wave parametric amplifier by means of self-homodyne tomography,” Opt. Lett. 23, 1393–1395 (1998).
[CrossRef]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

1990 (1)

C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608–1616 (1990).
[CrossRef] [PubMed]

1959 (1)

L. Mandel, “Fluctuations of photon beams: The distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233 (1959).
[CrossRef]

Achilles, D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

Adami, C.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

Afek, I.

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

Akiba, M.

Ambar, O.

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

Andreoni, A.

Anisimov, P. M.

Avenhaus, M.

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
[CrossRef] [PubMed]

Bakstein, M.

L. Dovrat, M. Bakstein, D. Istrati, and H. Eisenberg, “Simulations of the detection process in silicon photomultiplier detectors,” arXiv:1109.0698v1 (2011).

Banaszek, K.

W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A 78, 033831 (2008).
[CrossRef]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

Bartlett, S. D.

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

Berggren, K.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

Bondani, M.

Bondarenko, G.

G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
[CrossRef]

Bouwmeester, D.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

G. A. Durkin, C. Simon, and D. Bouwmeester, “Multiphoton entanglement concentration and quantum cryptography,” Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

Braunstein, S. L.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

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

Buzhan, P.

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

Chekhova, M. V.

O. A. Ivanova, T. S. Iskhakov, A. N. Penin, and M. V. Chekhova, “Multiphoton correlations in parametric down-conversion and their measurement in the pulsed regime,” Quantum Electron. 36, 951 (2006).
[CrossRef]

Choi, S.-K.

Christ, A.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef] [PubMed]

Coldenstrodt-Ronge, H. B.

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
[CrossRef] [PubMed]

D’Ariano, G. M.

Dauler, E.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

Diamanti, E.

E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73, 033814 (2006).
[CrossRef]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

Dolgoshein, B.

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
[CrossRef]

Dovrat, L.

L. Dovrat, M. Bakstein, D. Istrati, and H. Eisenberg, “Simulations of the detection process in silicon photomultiplier detectors,” arXiv:1109.0698v1 (2011).

Dowling, J. P.

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
[CrossRef]

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

Durkin, G. A.

H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

G. A. Durkin, C. Simon, and D. Bouwmeester, “Multiphoton entanglement concentration and quantum cryptography,” Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

Eckstein, A.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef] [PubMed]

Eisenberg, H.

L. Dovrat, M. Bakstein, D. Istrati, and H. Eisenberg, “Simulations of the detection process in silicon photomultiplier detectors,” arXiv:1109.0698v1 (2011).

Eisenberg, H. S.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Eraerds, P.

Filatov, L.

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

Fitch, M. J.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

Frankowski, R.

W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A 78, 033831 (2008).
[CrossRef]

Franson, J. D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

Gao, Y.

Gisin, N.

Golovin, V.

G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
[CrossRef]

Goltsman, G.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
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Hamiltom, S. A.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
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H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

Hodelin, J. F.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
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R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
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G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
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O. A. Ivanova, T. S. Iskhakov, A. N. Penin, and M. V. Chekhova, “Multiphoton correlations in parametric down-conversion and their measurement in the pulsed regime,” Quantum Electron. 36, 951 (2006).
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O. A. Ivanova, T. S. Iskhakov, A. N. Penin, and M. V. Chekhova, “Multiphoton correlations in parametric down-conversion and their measurement in the pulsed regime,” Quantum Electron. 36, 951 (2006).
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D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
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B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425 (2008).
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E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
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Khoury, G.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
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H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
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G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
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E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
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H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
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P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A 61, 042304 (2000).
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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
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E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
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M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
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C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608–1616 (1990).
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Lee, H.

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
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H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
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Legré, M.

Lincoln, D.

D. Lincoln, “A large statistics study of the performance and yields of generation-6 vlpcs (histe-vi),” Nucl. Instrum. Methods Phys. Res. A 453, 177–181 (2000).
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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, 061803 (2005).
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Luine, J.

Lundeen, J. S.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Mauerer, W.

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
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Milburn, G.

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

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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, 061803 (2005).
[CrossRef]

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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

Mosley, P. J.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

Nam, S. W.

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

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I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

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R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

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R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

Paleari, F.

Penin, A. N.

O. A. Ivanova, T. S. Iskhakov, A. N. Penin, and M. V. Chekhova, “Multiphoton correlations in parametric down-conversion and their measurement in the pulsed regime,” Quantum Electron. 36, 951 (2006).
[CrossRef]

Pittman, T. B.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

Popova, E.

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, “Limited geiger-mode silicon photodiode with very high gain,” Nucl. Phys. B. (Proc. Suppl) 61, 347 (1998).
[CrossRef]

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W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A 78, 033831 (2008).
[CrossRef]

Robinson, B.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

Rochas, A.

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, 061803 (2005).
[CrossRef]

Sanders, B. C.

E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73, 033814 (2006).
[CrossRef]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

Sasaki, K.

R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

Sasaki, M.

Sato, K.

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Shields, A. J.

B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425 (2008).
[CrossRef]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

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I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

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A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef] [PubMed]

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

Simon, C.

H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

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D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

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P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

Takeuchi, S.

R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science 323, 483–485 (2009).
[CrossRef] [PubMed]

Teshima, M.

P. Buzhan, B. Dolgoshein, L. Filatov, A. Ilyin, V. Kaplin, A. Karakash, S. Klemin, R. Mirzoyan, A. Otte, E. Popova, V. Sosnovtsev, and M. Teshima, “Large area silicon photonmultipliers: Performance and appli,” Nucl. Instrum. Methods Phys. Res. A 567, 78 (2006).
[CrossRef]

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U’Ren, A. B.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

Vasilyev, M.

Voronov, B.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

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E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73, 033814 (2006).
[CrossRef]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

Walmsley, I. A.

M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101, 053601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

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W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A 78, 033831 (2008).
[CrossRef]

Wasylczyk, P.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

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Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73, 033814 (2006).
[CrossRef]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

Yang, J.

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
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A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1991).

Yuan, Z.

B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425 (2008).
[CrossRef]

Yurtsever, U.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

Zambra, G.

Zbinden, H.

Zeilinger, A.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

J. Mod. Opt. (3)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).

E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov, G. Goltsman, S. A. Hamiltom, and K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Opt. 56, 365 (2009).
[CrossRef]

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).

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

Nat. Photonics (1)

B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425 (2008).
[CrossRef]

Nature (1)

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

Nucl. Instrum. Methods Phys. Res. A (2)

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Figures (5)

Fig. 1
Fig. 1

(a) The experimental setup. A type-II collinear BBO crystal is pumped by 390 nm amplified pulses at a repetition rate of 250 kHz. The signal and idler photons are split at a Polarizing Beam Splitter (PBS) according to their polarization and detected by two SiPM detectors. The number of spatial and spectral modes which are collected is varied using interference filters (IF) with different bandwidths and different optical fibers. (b) The detection configuration. An SiPM detector produces a signal whose intensity is proportional to the number of impinging photons. This signal is amplified, digitized using analog-to-digital converters (ADC) and analyzed using FPGA electronics in real-time. The data is continuously transmitted to a computer, which displays the photon-number distribution.

Fig. 2
Fig. 2

A histogram of the electrical output signal level for a coherent input state. The data was accumulated over a period of 30 seconds. Good peak separation is maintained up to 20 photons.

Fig. 3
Fig. 3

Measurements of a thermal state conducted on a polarization mode of a type-II collinear PDC source, spatially and spectrally filtered using a single-mode fiber and a 3 nm bandpass filter. The experimental data was fitted to the obtained statistics using the model presented in this work (solid lines). The reconstruction of the original photon statistics from the measured data is also shown. The data was obtained at a temperature of −10°C. The bias voltage values and the fit parameters are: Vbias = 68.3 V, η = 6 · 10−3 ± 1 · 10−3, λdk = 2.4 · 10−3 ± 6 · 10−4, = 8.9 ± 0.5, ɛ = 0.280 ± 0.005 (green circles), Vbias = 67.7 V, η = 4.9 · 10−3 ± 3 · 10−4, λdk = 1.0 · 10−3 ± 3 · 10−4, = 8.8 ± 0.5, ɛ = 0.140 ± 0.006 (red upward triangles), and Vbias = 67.4 V, η = 4.2 · 10−3 ± 5 · 10−4, λdk = 2.5 · 10−4 ± 1 · 10−4, = 7.8 ± 2.7, ɛ = 0.040 ± 0.006 (blue downward triangles). The corresponding crosstalk values obtained with the numerical model of Ref. [28] are ɛnn = 0.078 ± 0.001, ɛnn = 0.038 ± 0.001, and ɛnn = 0.010 ± 0.001, respectively. Fits using the numerical model are presented by black dashed lines.

Fig. 4
Fig. 4

(a) Representative photon-number distribution measurements and their fits. The results are presented for distributions which range between thermal and Poissonian statistics. Experimental errors are smaller than their respective symbol size. The average photon number for each data set is reduced as less modes are collected. (b–e) A close-up of each of the measured distributions along with distributions calculated for the same fit parameters while changing the mode-number value between a single, 1.4 and 5.5 modes, and a Poisson distribution. The measurements were taken with the following parameters: η = 0.013, λdk = 1.95 · 10−3, ɛ = 0.26, 3 nm bandwidth filter, single mode fiber for 780 nm, integration time of 780 minutes (green circles), η = 0.019, λdk = 2.1 · 10−3, ɛ = 0.23, 3 nm bandwidth filter, single mode fiber for 1500 nm, integration time of 100 minutes (blue squares), η = 0.013, λdk = 1.69 · 10−3, ɛ = 0.27, 10 nm bandwidth filter, single mode fiber for 1500 nm, integration time of 100 minutes (orange triangles), and η = 0.020, λdk = 3 · 10−3, ɛ = 0.22, 3 nm bandwidth filter, multimode fiber for 780 nm, integration time of 115 minutes (red diamonds).

Fig. 5
Fig. 5

The average number of photons collected from multiple modes as a function of the pump intensity. The solid lines are fits to Eq. 8 with the following parameters: (a) s = 1.1 ± 0.7, α = 0.08 ± 0.02 (green circles) (b) s = 1.4 ± 0.5, α = 0.10 ± 0.02 (blue squares) (c) s = 6.3 ± 0.9, α = 0.08 ± 0.01 (orange triangles). (d) s = 23 ± 8, α = 0.06 ± 0.01 (red diamonds).

Tables (1)

Tables Icon

Table 1 The number of modes as obtained from the photon-number statistics. The table shows the number of modes for interference filters with different bandwidths and for optical fibers with different mode field diameters (MFD) and numerical apertures (NA). Measurements in the first row were taken with a single-mode fiber for 780 nm, the second row with a single-mode fiber for 1550 nm and the third row with a standard graded-index multimode fiber. Distributions are considered Poissonian for s > 10.

Equations (8)

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p m = M p o .
M loss ( n | m ) = { ( m n ) η n ( 1 η ) m n n m 0 n > m ,
M d k ( n | m ) = { 0 n < m λ d k n m exp ( λ d k ) ( n m ) ! n m ,
M c t ( n | m ) = { n c t = 0 min ( m , n m ) ( m n c t ) ɛ n c t ( 1 ɛ ) m n c t M c t ( n m | n c t ) n m > 0 1 n = m = 0 0 otherwise ,
p th ( n ) = 1 ( 1 + n ¯ ) ( 1 + 1 n ¯ ) n ,
p s ( n ) = ( s + n 1 s 1 ) i = 1 s p ( n i ) ,
p s ( n ) = ( s + n 1 s 1 ) 1 ( 1 + n ¯ s ) s ( 1 + s n ¯ ) n ,
n ¯ = s n ¯ i = s sinh 2 ( τ i ) = s sinh 2 ( α I ) ,

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