## Measurements of the dependence of the photon-number distribution on the number of modes in parametric down-conversion |

Optics Express, Vol. 20, Issue 3, pp. 2266-2276 (2012)

http://dx.doi.org/10.1364/OE.20.002266

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### 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

## 1. Introduction

2. 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]

3. 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]

4. 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]

5. 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]

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

7. 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]

8. 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]

10. 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]

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

12. 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]

13. 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]

14. 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]

15. 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]

16. 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]

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

18. 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]

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

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

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

3. 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]

4. 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]

22. 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]

23. 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]

*g*

^{(2)}parameter [24

24. 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]

25. 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]

26. 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]

27. 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]

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

30. 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]

31. P. Eraerds, M. Legré, A. Rochas, H. Zbinden, and N. Gisin, “Sipm for fast photon-counting and multiphoton detection,” Opt. Express **15**, 14539–14549 (2007). [CrossRef] [PubMed]

## 2. The SiPM detection model

*· M*

_{ct}*· M*

_{dk}*, which represent the loss, dark counts (dk) and crosstalk (*

_{loss}*ct*) effects. The individual matrices are constructed as follows:

### Loss

*η*is the overall detection efficiency.

### Dark counts

*λ*is the average number of dark counts.

_{dk}### Crosstalk

*ɛ*is defined as the overall probability for a crosstalk-avalanche to be generated in any of the four neighboring elements. Each stage has

*m*triggered elements which can trigger up to

*m*new crosstalk events. The probability of generating

*n*new crosstalk events is composed of the probability that

_{ct}*n*elements out of the

_{ct}*m*possible will generate crosstalk, while the remaining

*m*–

*n*elements will not. This probability also includes the combinatorial factor of choosing the

_{ct}*n*crosstalk-generating elements out of the possible

_{ct}*m*. The

*n*crosstalk-triggered elements continue to generate additional crosstalk events in a recursive way, until all of the

_{ct}*n*–

*m*crosstalk elements are triggered.

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

30. 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]

31. P. Eraerds, M. Legré, A. Rochas, H. Zbinden, and N. Gisin, “Sipm for fast photon-counting and multiphoton detection,” Opt. Express **15**, 14539–14549 (2007). [CrossRef] [PubMed]

## 3. The experimental setup

*β*– BaB

_{2}O

_{4}(BBO) nonlinear crystal. The degenerate wavelength signal and idler photons at 780 nm, which are created with orthogonal polarizations (horizontal (H) and vertical (V)), are split at a polarizing beam splitter (PBS) and detected separately using two photon-number resolving detectors. This work focuses on measuring the photon-number distribution of a single polarization mode. Thus, the detection of both polarization modes is only used to evaluate the heralded detection efficiency of the SiPM detector. The results presented for a single polarization mode also apply to the photon pair statistics, as a collinear process produces the same statistics for pairs and for individual photons. Before the photons are coupled to the detectors, they are spectrally and spatially filtered by bandpass filters and by optical fibers, respectively. The number of spatial modes which are collected can be changed by using optical fibers with different core diameters and numerical apertures [33], and the number of collected spectral modes can be changed by using filters of different bandwidths.

*Hamamatsu Photonics*, S10362-11-100U) using optical fibers. The output electrical signal from the detector is amplified using low-noise amplifiers, digitized and then analyzed by FPGA electronics, where the number of detected photons is extracted in real time. This data is continuously transmitted to a computer, which presents the statistics. In order to minimize the effects of dark counts and after-pulsing [27

27. 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]

## 4. Results and discussion

*n*photons is proportional to the area of the

*n*Gaussian. We obtain high photon-number discrimination for as high as 20 photons. The photon-number resolution error, which results from the overlapping between neighboring Gaussians, is smaller than 1% below 12 photons and approaches 12% for 20 photons. The number of resolvable photons is limited due to the decrease of the signal to noise ratio as the number of photons is increased.

^{th}*n̄*is the average number of photons. The measured distribution exhibits a lower value of

*n̄*due to losses and deviates from the power-law dependence of Eq. 5 due to optical crosstalk and dark counts. We measured the same thermal state with three different sets of values for the amount of loss, dark counts and crosstalk, obtained by operating the SiPM detector with different bias voltage values [27

27. 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]

*n̄*due to losses and the deviation from a straight line in the semi-log plot when the number of measured photons exceeds 1. The latter is the effect of the crosstalk process, which falsely increases the probability of measuring high numbers of photons. The crosstalk probabilities of these measurements range from the highest probability value (maximum bias voltage), which creates the highest deviation, to the minimal probability (minimal bias voltage) which creates the lowest deviation.

*p⃗*to the function

_{m}*p⃗*= M

_{m}*M*

_{ct}*M*

_{dk}*, where*

_{loss}p⃗_{th}*p⃗*is the thermal distribution of Eq. 5 and

_{th}*n̄*is the only free parameter. The overall detection efficiency

*η*, the average number of dark counts

*λ*, and the crosstalk probability

_{dk}*ɛ*, that are used for the construction of the distortion matrices, are measured separately. The value of

*η*is defined as the heralded efficiency between the two down-converted polarization modes and is determined by the ratio between two-photon coincidence and single-photon counts in the limit

*n̄*≪ 1. The dark counts and crosstalk parameters are obtained through a separate measurement of the Poissonian dark count statistics which are fit to the function

*p⃗*= M

_{m}*M*

_{ct}*1⃗, where 1⃗ is a vector of zeros with 1 at the first position, and*

_{dk}*λ*and

_{dk}*ɛ*are the free fit parameters.

*ɛ*, defined as the probability of generating crosstalk in one particular neighbor among the four nearest neighbors, rather than the overall probability of generating crosstalk among the nearest neighbors, which we define here as

_{nn}*ɛ*. The values of

*ɛ*obtained from the fit correspond to the values of

_{nn}*ɛ*through the relation

*ɛ*= 1 −(1 −

*ɛ*)

_{nn}^{4}[28]. The agreement between the two models shows that although the analytical model does not account for the geometry of the detector, it provides a good description of the crosstalk process for all crosstalk values in the experimental range.

*n*photons are distributed among

*s*modes, such that

*n*

_{1}+

*n*

_{2}+ ...

*n*=

_{s}*n*, then the overall probability of measuring

*n*photons is given by [20

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

*n*photons in each mode

_{i}*i*, and the combinatorial factor amounts to the number of possible arrangements of

*n*indistinguishable photons into

*s*modes. If the photons are distributed evenly among the modes, the average number of photons in each mode is uniform and equals

*p*(

*n*) ≡

_{i}*p*(

_{th}*n*), and by substituting Eq. 5 into Eq. 6, we obtain the following probability distribution known as the negative binomial distribution. This distribution converges to a Poisson distribution when the number of modes approaches infinity.

_{i}*p⃗*= M

_{m}*M*

_{ct}*M*

_{dk}*, where*

_{loss}p⃗_{s}*p⃗*is defined in Eq. 7 with

_{s}*n̄*and

*s*as the free parameters. We detected up to 14 photons and obtained good fits to the data. Notice that the difference between the distributions becomes more marked as the number of photons increases (see Figs. 4(b)–(e)). The difference is almost undetectable for low photon numbers. In fact, measurements of at least 10 photons are required for our experimental parameter range, in order to properly discriminate between distributions with different mode numbers.

34. 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]

25. 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]

24. 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]

25. 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]

*τ*, which has a linear dependence on the pump field, the crystal length and the nonlinear coefficient of the crystal [35

_{i}35. P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A **61**, 042304 (2000). [CrossRef]

*s*modes, the mean number of photons

*n̄*is given by where

*α*is the coupling constant between the pump field and the nonlinear crystal, and

*I*is the pump intensity.

*n̄*on the number of modes, we measured photon-number distributions as a function of the pump intensity for the different numbers of collected modes. The values of

*n̄*were extracted by fitting the different distributions to Eq. 7. The fit values of

*n̄*as a function of the total pump power are shown in Fig. 5. All measurements show a nonlinear dependence, indicative of a stimulated process. The number of modes

*s*obtained from these fits corresponds well to the number of modes obtained through fits to the photon-number distributions (see Fig. 4). Furthermore, all fits resulted in similar values of the

*α*parameter, in agreement with the model of Eq. 8.

## 5. Conclusions

## References and links

1. | L. Mandel and E. Wolf, |

2. | 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. |

3. | 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. |

4. | E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A |

5. | 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. |

6. | C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A |

7. | H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. |

8. | D. Lincoln, “A large statistics study of the performance and yields of generation-6 vlpcs (histe-vi),” Nucl. Instrum. Methods Phys. Res. A |

9. | 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. |

10. | D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A |

11. | B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics |

12. | 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. |

13. | P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A |

14. | 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 |

15. | H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. |

16. | R. Okamoto, J. L. O’Brien, H. F. Hofmann, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science |

17. | E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

18. | T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. |

19. | G. A. Durkin, C. Simon, and D. Bouwmeester, “Multiphoton entanglement concentration and quantum cryptography,” Phys. Rev. Lett. |

20. | L. Mandel, “Fluctuations of photon beams: The distribution of the photo-electrons,” Proc. Phys. Soc. |

21. | F. Paleari, A. Andreoni, G. Zambra, and M. Bondani, “Thermal photon statistics in spontaneous parametric downconversion,” Opt. Express |

22. | 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. |

23. | 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. |

24. | 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. |

25. | W. Wasilewski, C. Radzewicz, R. Frankowski, and K. Banaszek, “Statistics of multiphoton events in spontaneous parametric down-conversion,” Phys. Rev. A |

26. | 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) |

27. | 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 |

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

29. | I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A |

30. | M. Akiba, K. Tsujino, K. Sato, and M. Sasaki, “Multipixel silicon avalanche photodiode with ultralow dark count rate at liquid nitrogen temperature,” Opt. Express |

31. | P. Eraerds, M. Legré, A. Rochas, H. Zbinden, and N. Gisin, “Sipm for fast photon-counting and multiphoton detection,” Opt. Express |

32. | 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. |

33. | A. Yariv, |

34. | 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. |

35. | P. Kok and S. L. Braunstein, “Postselected versus nonpostselected quantum teleportation using parametric down-conversion,” Phys. Rev. A |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5290) Quantum optics : Photon statistics

(190.4975) Nonlinear optics : Parametric processes

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 5, 2011

Revised Manuscript: December 29, 2011

Manuscript Accepted: December 30, 2011

Published: January 17, 2012

**Citation**

L. Dovrat, M. Bakstein, D. Istrati, A. Shaham, and H. S. Eisenberg, "Measurements of the dependence of the photon-number distribution on the number of modes in parametric down-conversion," Opt. Express **20**, 2266-2276 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2266

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- 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]
- 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]
- E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A73, 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]
- C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A42, 1608–1616 (1990). [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]
- D. Lincoln, “A large statistics study of the performance and yields of generation-6 vlpcs (histe-vi),” Nucl. Instrum. Methods Phys. Res. A453, 177–181 (2000). [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).
- D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A71, 061803 (2005). [CrossRef]
- B. E. Kardynal, Z. Yuan, and A. J. Shields, “An avalanche photodiode-based photon-number-resolving detector,” Nat. Photonics2, 425 (2008). [CrossRef]
- 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]
- P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A65, 052104 (2002). [CrossRef]
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