Abstract

Random point processes are often classified as bunched, unbunched, or antibunched, depending on the value assumed by their variance-to-mean ratio (the Fano factor); in the low-degeneracy limit, the Fano factor of a Bose–Einstein process approaches the Poisson value of unity. This fact probably added strength to the conjecture that in the limit of zero degeneracy a Bose–Einstein process reduces to a memoryless Poisson process. After considering some statistics related both to counting and to arrival times, we have experimentally verified that a Bose–Einstein process retains its correlation properties down to values of the degeneracy parameter of the order of 0.01. It also follows that the two most common definitions of photon bunching, the one based on the Fano factor and the one based on the correlation function, are not equivalent at low degeneracy levels.

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References

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  1. B. A. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1999).
  2. B. A. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  3. D. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).
  4. L. Mandel, "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233-243 (1959), p. 236.
    [CrossRef]
  5. M. C. Teich and B. A. A. Saleh, "Effects of random deletion and additive noise on bunched and antibunched photon-counting statistics," Opt. Lett. 7, 365-367 (1982).
    [CrossRef] [PubMed]
  6. G. Vannucci and M. C. Teich, "Computer simulation of superposed coherent and chaotic radiation," Appl. Opt. 19, 548-553 (1980).
    [CrossRef] [PubMed]
  7. J. Goodman, Statistical Optics (Wiley, New York, 1985), Sect. 9.2.2.
  8. P. Holgate, "The use of distance methods for the analysis of spatial distribution of points," in Stochastic Point Processes, P.A. W.Lewis, ed. (Wiley-Interscience, New York, 1972), p. 122.
  9. L. Mandel, "Fluctuations of light beams," in Progress in Optics Vol. 2, E.Wolf ed. (North-Holland, Amsterdam, 1963), p. 230.
  10. B. A. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle and photon detection," Proc. IEEE 70, 229-245 (1982).
    [CrossRef]
  11. Ref. , pp. 168-169.
  12. K. Matsuo, M. C. Teich, and B. A. A. Saleh, "Thomas point process in pulse, particle, and photon detection," Appl. Opt. 22, 1898-1908 (1983).
    [CrossRef] [PubMed]
  13. W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
    [CrossRef] [PubMed]
  14. D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).
    [CrossRef]
  15. L. Mandel, "Coherence properties of photons," in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, New York, 1967), pp. 143-165.
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
    [CrossRef]
  17. L. Basano and P. Ottonello, "Direct memory access digital events analyzer," Rev. Sci. Instrum. 60, 1184-1188 (1989).
    [CrossRef]

1999 (1)

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

1989 (1)

L. Basano and P. Ottonello, "Direct memory access digital events analyzer," Rev. Sci. Instrum. 60, 1184-1188 (1989).
[CrossRef]

1983 (1)

1982 (2)

B. A. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle and photon detection," Proc. IEEE 70, 229-245 (1982).
[CrossRef]

M. C. Teich and B. A. A. Saleh, "Effects of random deletion and additive noise on bunched and antibunched photon-counting statistics," Opt. Lett. 7, 365-367 (1982).
[CrossRef] [PubMed]

1980 (1)

1959 (1)

L. Mandel, "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233-243 (1959), p. 236.
[CrossRef]

Basano, L.

L. Basano and P. Ottonello, "Direct memory access digital events analyzer," Rev. Sci. Instrum. 60, 1184-1188 (1989).
[CrossRef]

Cox, D. R.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985), Sect. 9.2.2.

Holgate, P.

P. Holgate, "The use of distance methods for the analysis of spatial distribution of points," in Stochastic Point Processes, P.A. W.Lewis, ed. (Wiley-Interscience, New York, 1972), p. 122.

Kim, J.

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

Lewis, P. A. W.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).
[CrossRef]

Liu, R. C.

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel, "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233-243 (1959), p. 236.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

L. Mandel, "Coherence properties of photons," in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, New York, 1967), pp. 143-165.

L. Mandel, "Fluctuations of light beams," in Progress in Optics Vol. 2, E.Wolf ed. (North-Holland, Amsterdam, 1963), p. 230.

Matsuo, K.

Oliver, W. D.

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

Ottonello, P.

L. Basano and P. Ottonello, "Direct memory access digital events analyzer," Rev. Sci. Instrum. 60, 1184-1188 (1989).
[CrossRef]

Saleh, B. A. A.

K. Matsuo, M. C. Teich, and B. A. A. Saleh, "Thomas point process in pulse, particle, and photon detection," Appl. Opt. 22, 1898-1908 (1983).
[CrossRef] [PubMed]

B. A. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle and photon detection," Proc. IEEE 70, 229-245 (1982).
[CrossRef]

M. C. Teich and B. A. A. Saleh, "Effects of random deletion and additive noise on bunched and antibunched photon-counting statistics," Opt. Lett. 7, 365-367 (1982).
[CrossRef] [PubMed]

B. A. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1999).

B. A. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

Snyder, D.

D. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

Teich, M. C.

Vannucci, G.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

Yamamoto, Y.

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

Appl. Opt. (2)

Opt. Lett. (1)

Proc. IEEE (1)

B. A. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle and photon detection," Proc. IEEE 70, 229-245 (1982).
[CrossRef]

Proc. Phys. Soc. London (1)

L. Mandel, "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233-243 (1959), p. 236.
[CrossRef]

Rev. Sci. Instrum. (1)

L. Basano and P. Ottonello, "Direct memory access digital events analyzer," Rev. Sci. Instrum. 60, 1184-1188 (1989).
[CrossRef]

Science (1)

W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, "Hanbury Brown and Twiss-type experiments with electrons," Science 284, 299-301 (1999).
[CrossRef] [PubMed]

Other (10)

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).
[CrossRef]

L. Mandel, "Coherence properties of photons," in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, New York, 1967), pp. 143-165.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

J. Goodman, Statistical Optics (Wiley, New York, 1985), Sect. 9.2.2.

P. Holgate, "The use of distance methods for the analysis of spatial distribution of points," in Stochastic Point Processes, P.A. W.Lewis, ed. (Wiley-Interscience, New York, 1972), p. 122.

L. Mandel, "Fluctuations of light beams," in Progress in Optics Vol. 2, E.Wolf ed. (North-Holland, Amsterdam, 1963), p. 230.

Ref. , pp. 168-169.

B. A. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1999).

B. A. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

D. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

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Figures (4)

Fig. 1
Fig. 1

Plots of counting probabilities for Bose–Einstein (asterisks) and Poisson (open circles) distributions, when (a) n = 5 ; (b) n = 0.05 . Plot of term-by-term ratios, Eqs. (11, 12), for (c) n = 5 , (d) n = 0.05 .

Fig. 2
Fig. 2

Block diagram of the experimental apparatus (see text). The input–output Board is an ADLink 7300A digital DMA acquisition board.

Fig. 3
Fig. 3

Experimental plot of the conditional probability density of detecting a pulse at delay t given that a pulse has been detected at delay zero for (a) motionless disk, (b) with rotating disk. The average interevent time is approximately 40 times larger than the coherence time.

Fig. 4
Fig. 4

Plot of experimental results (marks connected by thick lines) of term-by-term ratios for the rotating disk case (the average interevent time is approximately 40 times larger than the coherence time). For comparison, the theoretical expectations for Bose–Einstein and Poisson processes are also shown.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

R S C = sampling time coherence time 1 .
R C I = coherence time average interevent time .
P ( n , T ) = 1 n ! [ 0 T I ( t ) d t ] n exp [ 0 T I ( t ) d t ] .
P ( n , T ) = 1 n ! exp ( n ) n n ,
P ( n , T ) = n n ( 1 + n ) n + 1 , R S C 1 .
Q M ( s ) = exp ( s M ) .
Q M ( s ) = 1 1 + n M n M exp ( s ) ,
Q X ( s ) = 1 p + exp ( s ) ,
Q N ( s ) = Q M ln Q X ( s ) .
Q N ( s ) = 1 1 + n N n N exp ( s ) ,
F n = Var ( n ) n = n + n 2 n = 1 + n .
F = 1 + Q ,
R BE = P ( n + 1 , T ) P ( n , T ) = n 1 + n .
R n P = P ( n + 1 , T ) P ( n , T ) = n 1 + n ,
R BE R n P = 1 + n 1 + n ,
sampling time coherence time average interevent time .
P c ( photon at τ photon at 0 ) N ( photon at τ and photon at 0 ) N ( photon at 0 ) .

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