## Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes |

Optics Express, Vol. 19, Issue 14, pp. 13268-13276 (2011)

http://dx.doi.org/10.1364/OE.19.013268

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### Abstract

We demonstrate the use of two high speed avalanche photodiodes in exploring higher order photon correlations. By employing the photon number resolving capability of the photodiodes the response to higher order photon coincidences can be measured. As an example we show experimentally the sensitivity to higher order correlations for three types of photon sources with distinct photon statistics. This higher order correlation technique could be used as a low cost and compact tool for quantifying the degree of correlation of photon sources employed in quantum information science.

© 2011 OSA

## 1. Introduction

1. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963). [CrossRef]

2. D. F. Walls and G. J. Milburn, “Coherence properties of the electromagnetic field,” in *Quantum Optics* (Springer-Verlag, 2008), pp. 29–55. [CrossRef]

3. K. Usami, Y. Nambu, B. S. Shi, A. Tomita, and K. Nakamura, “Observation of antinormally ordered Hanbury Brown-Twiss correlations,” Phys. Rev. Lett. **92**(11), 113601 (2004). [CrossRef] [PubMed]

4. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**(4497), 27–29 (1956). [CrossRef]

5. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature **445**(7126), 402–405 (2007). [CrossRef] [PubMed]

6. H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. **39**(11), 691–695 (1977). [CrossRef]

7. G. S. Agarwal, “Field–correlation effects in multiphoton absorption processes,” Phys. Rev. A **1**(5), 1445–1459 (1970). [CrossRef]

8. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**(1), 145–195 (2002). [CrossRef]

9. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single–photon turnstile device,” Science **290**(5500), 2282–2285 (2000). [CrossRef] [PubMed]

11. Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single–photon source,” Science **295**(5552), 102–105 (2002). [CrossRef]

12. Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Boosting up quantum key distribution by learning statistics of practical single-photon sources,” New J. Phys. **11**(11), 113033 (2009). [CrossRef]

7. G. S. Agarwal, “Field–correlation effects in multiphoton absorption processes,” Phys. Rev. A **1**(5), 1445–1459 (1970). [CrossRef]

13. Y. Qu, S. Singh, and C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. **76**(8), 1236–1239 (1996). [CrossRef] [PubMed]

14. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High–order temporal coherences of chaotic and laser light,” Opt. Express **18**(2), 1430–1437 (2010), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-18-2-1430. [CrossRef] [PubMed]

15. M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, “Accessing higher order correlations in quantum optical states by time multiplexing,” Phys. Rev. Lett. **104**(6), 063602 (2010). [CrossRef] [PubMed]

16. D. A. Kalashikov, S. H. Tan, M. V. Chekhova, and L. A. Krivitsky, “Accessing photon bunching with photon number resolving multi-pixel detector,” Opt. Express **19**(10), 9352–9363 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-10-9352. [CrossRef]

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

4. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**(4497), 27–29 (1956). [CrossRef]

5. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature **445**(7126), 402–405 (2007). [CrossRef] [PubMed]

*g*(2) defined as

*â*

^{†}and

*â*are the usual photon creation and annihilation operators. The form of

*g*(2) presented above is the most salient in terms of categorizing photon states. However, we demonstrate that due to the PNR capability of the detectors, two-photon, three photon and up to

*n*-photon events can be selected using photon counting discriminators, where

*n*is the PNR capability of the detector. Coupled with the joint photon detection of using both PNR detectors together, the setup is sensitive to higher order correlations. As we show below for a bunched photon source, the sensitivity to photon bunching is significantly improved over the standard

*g*(2) measurement.

## 2. Theory

1. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963). [CrossRef]

*n*th order, normally ordered, normalized correlation function is given by: When

*n*= 2,

*g*(2) can be measured using the usual Hanbury-Brown Twiss experiment with a double single photon detector arrangement. Such a setup is depicted in Fig. 1(a) with two InGaAs APDs as the detectors. In this case, both detector discriminators are set so 0-photons are rejected while 1-photon and multi-photon events are sampled. This is readily achieved using most non-photon number resolving single photon detectors. As an example, if the source under investigation were purely thermal with a coherence time greater than the temporal resolution of the detector, a standard HBT experiment would yield

*g*(2) = 2. Specifically, the measurement involves sampling the coincident counts at times smaller than the coherence time of the source and dividing by the accidental coincidences (obtained at times greater than the source coherence time).

*γ*(4):

*γ*(4) clearly contains information regarding higher order correlation functions as we can rewrite

*γ*(4) in terms of the usual normalized correlation functions: In the example given above with the thermal source, the higher order coincidence rate would be higher than

*g*(2) with

*γ*(4) = 6.

*n*indicates that the window discrimination levels of detector

_{i}*i*is placed around the

*n*photon number states. Eq. (4) can also be re-written in terms of normalized correlation functions: If the photon number resolution of both detectors is

_{i}*n*, then the higher order coincidence can be measured up to 2

*n*. Note that this arrangement circumvents the non-unity detector efficiency of the detectors and any inherent optical losses in the system.

*n*is the maximum photon number the detector can measure and

_{max}*η*is the

_{i}*i*system single photon detection efficiency. The photon sources appropriate for use with this technique have detected mean photon fluxes much less than 2

*n*. Note that Eq. (6) approximates Eq. (4) when

_{max}*η*is relatively small. This is due to the weighting of the higher order terms in the sums of Eq. (6) scaling as

_{i}## 3. Photon number resolution with InGaAs avalanche photodiodes

*n*in the incident light field. The avalanche voltage distribution can be satisfactorily modeled assuming the photon number distribution of the source is Poissonian with a mean detected photon flux,

*μ*∼ 2.6, black line in Fig. 2. Each photon peak,

*n*≥ 1 is assumed to be Gaussian and to reflect the statistical broadening due to avalanche noise. The widths of the photon peaks are scaled as √

*n*relative to the 1-photon peak width. Such a technique has successfully been employed to model photon number distributions for both InGaAs and Silicon APDs [19, 20

20. O. Thomas, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Efficient photon number detection with silicon avalanche photodiodes,” Appl. Phys. Lett. **97**(3), 031102 (2010). [CrossRef]

*μ*∼ 2.8.

22. A. R. Dixon, J. F. Dynes, Z. L. Yuan, A. W. Sharpe, A. J. Bennett, and A. J. Shields, “Ultrashort dead time of photon-counting InGaAs avalanche photodiodes,” Appl. Phys. Lett. **94**(23), 231113 (2009). [CrossRef]

*μ*and a chaotic field with mean photon number

_{s}*μ*[23

_{n}23. A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. **16**(25), 1169–1172 (1966). [CrossRef]

24. G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. **138**(4B), B1012–B1016 (1965). [CrossRef]

*μ*&

_{s}*μ*from experimentally measurable quantities, namely average detected photon flux,

_{n}*μ*=

*μ*+

_{s}*μ*and

_{n}*g*(2) =

*μ*(

_{n}*μ*+ 2

_{n}*μ*)/

_{s}*μ*

^{2}+1 [23

23. A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. **16**(25), 1169–1172 (1966). [CrossRef]

24. G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. **138**(4B), B1012–B1016 (1965). [CrossRef]

*μ*= 2.8 and

*g*(2) = 1.2 [25], the resulting modeled avalanche distribution is shown in Fig. 2 (red line).

*n*> 3 to be reduced in size (the linear model assumes a linear avalanche voltage dependence on photon number). The quenching effect can be somewhat quantified by comparing the experimental curves crossover point with the theoretical crossover point. These occur at 0.18 and 0.26V respectively, as indicated by the dashed lines in Fig. 2. Hence there is a 30% reduction in avalanche height at an avalanche voltage of 0.26V compared to the simple linear model prediction.

## 4. Higher order correlations of the photon field

*t*= 0) a prominent correlation peak is observed due to photon bunching. The correlation here is given by Eq. (4),

*γ*. The correlation value is measured as the ratio of peak height at t=0 to the average height at Δ

*t*≠ 0.

*γ*is much higher than the the equivalent

*g*(2) value of

*g*(2) = 1.2, as shown by the histogram in the inset of Fig. 3(a).

*γ*evaluated. The detector discriminator level of detector SD-APD 1 was kept at 300mV. Figure 3(b) plots the resulting correlations,

*γ*as a function of SD-APD 2 discrimination level for the three sources. All three sources show

*γ*> 1 even at the lowest discrimination level used. This can be readily understood in terms of elevated photon bunching through higher order correlations. The

*n*order correlation is the expectation of the joint detections of

^{th}*n*photons correlations. For photons that are indistinguishable and statistically dependent, there is a factorial increase of the available permutations of photon amplitudes as

*n*rises [26

26. M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science **325**(5938), 297–300 (2009). [CrossRef] [PubMed]

2. D. F. Walls and G. J. Milburn, “Coherence properties of the electromagnetic field,” in *Quantum Optics* (Springer-Verlag, 2008), pp. 29–55. [CrossRef]

*γ*for this source attains ∼ 1.3 at the very highest D2 discrimination level employed, Fig. 3(b), inset. Operating the same laser near to threshold (LNT) shows a marked increase in correlation which is due to increased intensity fluctuations leading to increased photon bunching. Source FML displays the highest correlation values; attaining

*γ*∼ 47.1 ±4.5 for these measurements. Intensity fluctuations for this source are by far the highest of all sources under test.

*μ*&

_{s}*μ*. These mean photon numbers are derived from the total average photon number

_{n}*μ*and

*g*(2). The second order correlation function

*g*(2) was measured separately for each of the sources FML, LNT & LAT and the values

*g*(2) = 1.2, 1.075 & 1.01 respectively were used in the calculations (the inset of Fig. 3(a) shows the measured

*g*(2) photon coincidence histogram for source FML).

*n*in the model for the detectors. Figure 4 shows the simulated results based on Eq. (6) for

_{max}*γ*plotted with up to

*n*= 7 photon detection events. Each point corresponds to selecting exactly

_{max}*n*avalanche detections based on a linear dependence of the avalanche peaks (as shown by the arrows in Fig. 3(b)). The x-axis is plotted reciprocally to emphasize the avalanche self-limiting.

*γ*for all sources shows a similar growth and dependence to that observed experimentally, qualitatively confirming our underlying model.

28. E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. **8**(1), 4 (2006). [CrossRef]

*n*and

*n*± 1 photon number avalanche peaks.

## 5. Conclusion

*g*(2) measurement. We believe the technique will be of use and importance in characterizing photon sources involved in the rapidly evolving field of quantum information science.

## Acknowledgments

## References and links

1. | R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. |

2. | D. F. Walls and G. J. Milburn, “Coherence properties of the electromagnetic field,” in |

3. | K. Usami, Y. Nambu, B. S. Shi, A. Tomita, and K. Nakamura, “Observation of antinormally ordered Hanbury Brown-Twiss correlations,” Phys. Rev. Lett. |

4. | R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature |

5. | T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature |

6. | H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. |

7. | G. S. Agarwal, “Field–correlation effects in multiphoton absorption processes,” Phys. Rev. A |

8. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

9. | P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single–photon turnstile device,” Science |

10. | C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. |

11. | Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single–photon source,” Science |

12. | Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Boosting up quantum key distribution by learning statistics of practical single-photon sources,” New J. Phys. |

13. | Y. Qu, S. Singh, and C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. |

14. | M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High–order temporal coherences of chaotic and laser light,” Opt. Express |

15. | M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, “Accessing higher order correlations in quantum optical states by time multiplexing,” Phys. Rev. Lett. |

16. | D. A. Kalashikov, S. H. Tan, M. V. Chekhova, and L. A. Krivitsky, “Accessing photon bunching with photon number resolving multi-pixel detector,” Opt. Express |

17. | R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics |

18. | Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. |

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

20. | O. Thomas, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Efficient photon number detection with silicon avalanche photodiodes,” Appl. Phys. Lett. |

21. | L. Mandel and E. Wolf, “Quantum theory of photoelectric light detection,” in |

22. | A. R. Dixon, J. F. Dynes, Z. L. Yuan, A. W. Sharpe, A. J. Bennett, and A. J. Shields, “Ultrashort dead time of photon-counting InGaAs avalanche photodiodes,” Appl. Phys. Lett. |

23. | A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. |

24. | G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. |

25. | The value of |

26. | M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science |

27. | P. Meystre and M. Sargent, “Field quantization,” in |

28. | E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. |

**OCIS Codes**

(030.5290) Coherence and statistical optics : Photon statistics

(270.5290) Quantum optics : Photon statistics

(270.5570) Quantum optics : Quantum detectors

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 30, 2011

Revised Manuscript: May 9, 2011

Manuscript Accepted: May 10, 2011

Published: June 24, 2011

**Citation**

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, and A. J. Shields, "Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes," Opt. Express **19**, 13268-13276 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13268

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### References

- R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130(6), 2529–2539 (1963). [CrossRef]
- D. F. Walls and G. J. Milburn, “Coherence properties of the electromagnetic field,” in Quantum Optics (Springer-Verlag, 2008), pp. 29–55. [CrossRef]
- K. Usami, Y. Nambu, B. S. Shi, A. Tomita, and K. Nakamura, “Observation of antinormally ordered Hanbury Brown-Twiss correlations,” Phys. Rev. Lett. 92(11), 113601 (2004). [CrossRef] [PubMed]
- R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]
- T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445(7126), 402–405 (2007). [CrossRef] [PubMed]
- H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39(11), 691–695 (1977). [CrossRef]
- G. S. Agarwal, “Field–correlation effects in multiphoton absorption processes,” Phys. Rev. A 1(5), 1445–1459 (1970). [CrossRef]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
- P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single–photon turnstile device,” Science 290(5500), 2282–2285 (2000). [CrossRef] [PubMed]
- C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. 86(8), 1502–1505 (2001). [CrossRef] [PubMed]
- Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single–photon source,” Science 295(5552), 102–105 (2002). [CrossRef]
- Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Boosting up quantum key distribution by learning statistics of practical single-photon sources,” New J. Phys. 11(11), 113033 (2009). [CrossRef]
- Y. Qu, S. Singh, and C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996). [CrossRef] [PubMed]
- M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High–order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-18-2-1430 . [CrossRef] [PubMed]
- M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, “Accessing higher order correlations in quantum optical states by time multiplexing,” Phys. Rev. Lett. 104(6), 063602 (2010). [CrossRef] [PubMed]
- D. A. Kalashikov, S. H. Tan, M. V. Chekhova, and L. A. Krivitsky, “Accessing photon bunching with photon number resolving multi-pixel detector,” Opt. Express 19(10), 9352–9363 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-10-9352 . [CrossRef]
- R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009). [CrossRef]
- Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91(4), 041114 (2007). [CrossRef]
- B. E. Kardynal, Z. L. Yuan, and A. J. Shields, “An avalanche–photodiode–based photon–number–resolving detector,” Nat. Photonics 2(7), 425–428 (2008).
- O. Thomas, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Efficient photon number detection with silicon avalanche photodiodes,” Appl. Phys. Lett. 97(3), 031102 (2010). [CrossRef]
- L. Mandel and E. Wolf, “Quantum theory of photoelectric light detection,” in Optical Coherence and Quantum Optics (Cambridge University Press, 1995), pp. 683–740.
- A. R. Dixon, J. F. Dynes, Z. L. Yuan, A. W. Sharpe, A. J. Bennett, and A. J. Shields, “Ultrashort dead time of photon-counting InGaAs avalanche photodiodes,” Appl. Phys. Lett. 94(23), 231113 (2009). [CrossRef]
- A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. 16(25), 1169–1172 (1966). [CrossRef]
- G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. 138(4B), B1012–B1016 (1965). [CrossRef]
- The value of g(2) ∼ 1.2 is corroborated experimentally by an independent measurement of g(2).
- M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009). [CrossRef] [PubMed]
- P. Meystre and M. Sargent, “Field quantization,” in Elements of Quantum Optics (Springer–Verlag, 1998), pp. 263–285.
- E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8(1), 4 (2006). [CrossRef]

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