Abstract

A theoretical construct is presented for fourth-order interference between the signal and the idler beams of a parametric downconverter. Previous quantum treatments of fourth-order interference have employed correlated single-photon wave packets. The introduced approach, however, relies on Gaussian-state field correlations, which were previously used to characterize quadrature-noise squeezing produced by an optical parametric amplifier and nonclassical twin-beam generation in an optical parametric oscillator. Three principal benefits accrue from the correlation-function formalism. First, the quantum theory of fourth-order interference is unified with that for the other nonclassical effects of χ(2) interactions, i.e., squeezing and twin-beam production. Second, the semiclassical photodetection limit on Gaussian-state fourth-order interference is established; a purely quantum effect can be claimed at fringe visibilities substantially below the 50% level. Finally, both photon-coincidence counting (within the low-photon-flux regime) and intensity interferometry (in the high-photon-flux limit) are easily analyzed within a common framework.

© 1994 Optical Society of America

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  1. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric downconversion," Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  2. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
    [CrossRef] [PubMed]
  3. A. Aspect, P. Grangier, and G. Roger, "Dualité onde-particule pour un photon unique," J. Opt. (Paris) 20, 119 (1989).
    [CrossRef]
  4. R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
    [CrossRef] [PubMed]
  5. L.-A. Wu, M. Xiao, and H. J. Kimble, "Squeezed states of light from an optical parametric oscillator," J. Opt. Soc. Am. B 4, 1465 (1987).
    [CrossRef]
  6. S. Reynaud, C. Fabre, and E. Giacobino, "Quantum fluctuations in a two-mode parametric oscillator," J. Opt. Soc. Am. B 4, 1520 (1987).
    [CrossRef]
  7. N. C. Wong, K. W. Leong, and J. H. Shapiro, "Quantum correlation and absorption spectroscopy in an optical parametric oscillator in the presence of pump noise," Opt. Lett. 15, 891 (1990).
    [CrossRef] [PubMed]
  8. R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
    [CrossRef] [PubMed]
  9. R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Fourth-order interference of joint single-photon wave packets in lossless optical systems," Phys. Rev. A 42, 4127 (1990).
    [CrossRef] [PubMed]
  10. H. P. Yuen and J. H. Shapiro, "Optical communication with two-photon coherent states’Part III: Quantum measurements realizable with photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78 (1980).
    [CrossRef]
  11. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  12. M. H. Rubin and Y. H. Shih, "Models of a two-photon Ein-stein-Podolsky-Rosen interference experiment," Phys. Rev. A 45, 8138 (1992).
    [CrossRef] [PubMed]
  13. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation in a measurement of the single-photon propagation velocity," Phys. Rev. Lett. 68, 2421 (1992).
    [CrossRef] [PubMed]
  14. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer," Phys. Rev. A 45, 6659 (1992).
    [CrossRef] [PubMed]
  15. Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.
  16. J. H. Shapiro, "Quantum noise and excess noise in optical ho-modyne and heterodyne receivers," IEEE J. Quantum Electron. QE-21, 237 (1985).
    [CrossRef]
  17. Because we have suppressed the spatial and polarization characteristics of these fields, we say that our parametric interaction is degenerate if ωs = ωI = ωP/2 and nonde-generate if |Ωs - ωI| » Δω, where Δω is the common bandwidth of the signal and the idler emissions. Our development implicitly assumes that the signal and the idler beams are nondegenerate in either space or in polarization when they are degenerate in frequency. Spatial nondegen-eracy is ordinarily the case in parametric downconverters, and type-II phase-matched OPA’s and OPO’s produce orthogonally polarized signal and idler beams, so little generality is lost through this implicit assumption.
  18. L. G. Joneckis and J. H. Shapiro, "Quantum propagation in a Kerr medium: lossless dispersionless fiber," J. Opt. Soc. Am. B 10, 1102 (1993).
    [CrossRef]
  19. H. P. Yuen, "Two-photon coherent states of the radiation field," Phys. Rev. A 13, 2226 (1976).
    [CrossRef]
  20. Because well-known procedures are available for accounting for subunity quantum efficiency in both quantum and somi-classical photodetection (see, e.g., Ref. 16), we treat only the ideal case of unity quantum efficiency. Furthermore, because our principal goal is to study the semiclassical and quantum fringe visibilities in fourth-order interference ex-periments, it is worth noting that these visibilities are independent of detector quantum efficiency; see the quantum efficiency discussion in Section 4.
  21. J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, "Theory of light detection in the presence of feedback," J. Opt. Soc. Am. B 4, 1604 (1987).
    [CrossRef]
  22. It is well known that fourth moments of zero-mean, jointly Gaussian random variables factor into sums of products of their second moments; see, e.g., Ref. 24. For an arbitrary Gaussian state this classical result can be combined with the quantum theory of heterodyne detection10 to produce a quantum moment-factoring theorem for anti-normally ordered field-operator moments. The normally ordered fourth-moment result that we employ is then found by repeated use of the delta-function commutator.
  23. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 205.
  24. This result does not depend on assuming that P (ω) is an even function.
  25. C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044 (1987).
  26. This quantum preservation of joint Gaussian behavior is just like the well-known result for classical Gaussian random processes (see, e.g., Ref. 24, Chap. 3). It can be proved by performance of the state transformation implied by Eq. (39) with antinormally ordered characteristic functions.28
    [CrossRef] [PubMed]
  27. H. P. Yuen and J. H. Shapiro, "Optical communication withtwo-photon coherent states’Part I: Quantum state propagation and quantum-noise reduction," IEEE Transf. Inf. Theory IT-24, 657 (1978).
  28. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980) Chap. 11.
    [CrossRef]
  29. Strictly speaking, we are concerned with a coincidence-rate dip at T = 0, not with a white-light fringe. Thus an experimentalist might prefer to gauge the depth of the destructive interference that occurs at T = 0 by computing {maxT[C(T; τg)] - minT[C(T; τg)]}/maxT[C(T; τg)] rather than γ.
  30. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Observation of nonlocal interference in separated photon channels," Phys. Rev. Lett. 65, 321 (1990).
  31. Z. Y. Ou and L. Mandel, "Classical treatment of the Franson two-photon correlation experiment," J. Opt. Soc. Am. B 7, 2127 (1990).
    [CrossRef] [PubMed]
  32. J. D. Franson, "Violation of a simple inequality for classical fields," Phys. Rev. Lett. 67, 290 (1991).
    [CrossRef]
  33. The configuration considered by Ou and Mandel32 is not that of the dispersion-cancellation experiment. Their principal assumption, however, is configuration independent; they require the coincidence-gate duration to be much longer than both the field- and the intensity-correlation times of the signal and the idler beams. Thus the Ou-Mandel proof can be adapted to the Fig. 1 configuration.
    [CrossRef] [PubMed]
  34. Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).
  35. Our concluding analysis from Subsection 3.A can be adapted to show that the nonergodic classical-field model from Eqs. (65) and (66) does not predict high-visibility coincidence-rate fringes. The Ou-Mandel theory32 can be used to disqualify any ergodic model from producing high-visibility fringes.

1993 (1)

1992 (3)

M. H. Rubin and Y. H. Shih, "Models of a two-photon Ein-stein-Podolsky-Rosen interference experiment," Phys. Rev. A 45, 8138 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation in a measurement of the single-photon propagation velocity," Phys. Rev. Lett. 68, 2421 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer," Phys. Rev. A 45, 6659 (1992).
[CrossRef] [PubMed]

1991 (1)

J. D. Franson, "Violation of a simple inequality for classical fields," Phys. Rev. Lett. 67, 290 (1991).
[CrossRef]

1990 (4)

N. C. Wong, K. W. Leong, and J. H. Shapiro, "Quantum correlation and absorption spectroscopy in an optical parametric oscillator in the presence of pump noise," Opt. Lett. 15, 891 (1990).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, "Classical treatment of the Franson two-photon correlation experiment," J. Opt. Soc. Am. B 7, 2127 (1990).
[CrossRef] [PubMed]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Fourth-order interference of joint single-photon wave packets in lossless optical systems," Phys. Rev. A 42, 4127 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Observation of nonlocal interference in separated photon channels," Phys. Rev. Lett. 65, 321 (1990).

1989 (1)

A. Aspect, P. Grangier, and G. Roger, "Dualité onde-particule pour un photon unique," J. Opt. (Paris) 20, 119 (1989).
[CrossRef]

1987 (6)

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044 (1987).

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

S. Reynaud, C. Fabre, and E. Giacobino, "Quantum fluctuations in a two-mode parametric oscillator," J. Opt. Soc. Am. B 4, 1520 (1987).
[CrossRef]

L.-A. Wu, M. Xiao, and H. J. Kimble, "Squeezed states of light from an optical parametric oscillator," J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef]

J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, "Theory of light detection in the presence of feedback," J. Opt. Soc. Am. B 4, 1604 (1987).
[CrossRef]

1986 (2)

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric downconversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
[CrossRef] [PubMed]

1985 (1)

J. H. Shapiro, "Quantum noise and excess noise in optical ho-modyne and heterodyne receivers," IEEE J. Quantum Electron. QE-21, 237 (1985).
[CrossRef]

1981 (1)

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
[CrossRef]

1980 (1)

H. P. Yuen and J. H. Shapiro, "Optical communication with two-photon coherent states’Part III: Quantum measurements realizable with photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

1978 (1)

H. P. Yuen and J. H. Shapiro, "Optical communication withtwo-photon coherent states’Part I: Quantum state propagation and quantum-noise reduction," IEEE Transf. Inf. Theory IT-24, 657 (1978).

1976 (1)

H. P. Yuen, "Two-photon coherent states of the radiation field," Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Alley, C. O.

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, "Dualité onde-particule pour un photon unique," J. Opt. (Paris) 20, 119 (1989).
[CrossRef]

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Fourth-order interference of joint single-photon wave packets in lossless optical systems," Phys. Rev. A 42, 4127 (1990).
[CrossRef] [PubMed]

Camy, G.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Chiao, R. Y.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation in a measurement of the single-photon propagation velocity," Phys. Rev. Lett. 68, 2421 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer," Phys. Rev. A 45, 6659 (1992).
[CrossRef] [PubMed]

Fabre, C.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

S. Reynaud, C. Fabre, and E. Giacobino, "Quantum fluctuations in a two-mode parametric oscillator," J. Opt. Soc. Am. B 4, 1520 (1987).
[CrossRef]

Franson, J. D.

J. D. Franson, "Violation of a simple inequality for classical fields," Phys. Rev. Lett. 67, 290 (1991).
[CrossRef]

Ghosh, R.

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
[CrossRef] [PubMed]

Giacobino, E.

S. Reynaud, C. Fabre, and E. Giacobino, "Quantum fluctuations in a two-mode parametric oscillator," J. Opt. Soc. Am. B 4, 1520 (1987).
[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Grangier, P.

A. Aspect, P. Grangier, and G. Roger, "Dualité onde-particule pour un photon unique," J. Opt. (Paris) 20, 119 (1989).
[CrossRef]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric downconversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Heidmann, A.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Ho, S.-T.

Hong, C. K

R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
[CrossRef] [PubMed]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044 (1987).

Horowicz, R. J.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Jacobs, I. M.

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 205.

Joneckis, L. G.

Kiess, T. E.

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Kimble, H. J.

L.-A. Wu, M. Xiao, and H. J. Kimble, "Squeezed states of light from an optical parametric oscillator," J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric downconversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Kumar, P.

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer," Phys. Rev. A 45, 6659 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation in a measurement of the single-photon propagation velocity," Phys. Rev. Lett. 68, 2421 (1992).
[CrossRef] [PubMed]

Leong, K. W.

Mandel, L.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Observation of nonlocal interference in separated photon channels," Phys. Rev. Lett. 65, 321 (1990).

Z. Y. Ou and L. Mandel, "Classical treatment of the Franson two-photon correlation experiment," J. Opt. Soc. Am. B 7, 2127 (1990).
[CrossRef] [PubMed]

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044 (1987).

R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
[CrossRef] [PubMed]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, "Classical treatment of the Franson two-photon correlation experiment," J. Opt. Soc. Am. B 7, 2127 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Observation of nonlocal interference in separated photon channels," Phys. Rev. Lett. 65, 321 (1990).

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044 (1987).

R. Ghosh, C. K Hong, Z. Y. Ou, and L. Mandel, "Interference of two photons in parametric downconversion," Phys. Rev. A 34, 3962 (1986).
[CrossRef] [PubMed]

Reynaud, S.

S. Reynaud, C. Fabre, and E. Giacobino, "Quantum fluctuations in a two-mode parametric oscillator," J. Opt. Soc. Am. B 4, 1520 (1987).
[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Roger, G.

A. Aspect, P. Grangier, and G. Roger, "Dualité onde-particule pour un photon unique," J. Opt. (Paris) 20, 119 (1989).
[CrossRef]

Rubin, M. H.

M. H. Rubin and Y. H. Shih, "Models of a two-photon Ein-stein-Podolsky-Rosen interference experiment," Phys. Rev. A 45, 8138 (1992).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Fourth-order interference of joint single-photon wave packets in lossless optical systems," Phys. Rev. A 42, 4127 (1990).
[CrossRef] [PubMed]

J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, "Theory of light detection in the presence of feedback," J. Opt. Soc. Am. B 4, 1604 (1987).
[CrossRef]

Saplakoglu, G.

Sergienko, A. V.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

Shapiro, J. H.

L. G. Joneckis and J. H. Shapiro, "Quantum propagation in a Kerr medium: lossless dispersionless fiber," J. Opt. Soc. Am. B 10, 1102 (1993).
[CrossRef]

N. C. Wong, K. W. Leong, and J. H. Shapiro, "Quantum correlation and absorption spectroscopy in an optical parametric oscillator in the presence of pump noise," Opt. Lett. 15, 891 (1990).
[CrossRef] [PubMed]

J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, "Theory of light detection in the presence of feedback," J. Opt. Soc. Am. B 4, 1604 (1987).
[CrossRef]

J. H. Shapiro, "Quantum noise and excess noise in optical ho-modyne and heterodyne receivers," IEEE J. Quantum Electron. QE-21, 237 (1985).
[CrossRef]

H. P. Yuen and J. H. Shapiro, "Optical communication with two-photon coherent states’Part III: Quantum measurements realizable with photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

H. P. Yuen and J. H. Shapiro, "Optical communication withtwo-photon coherent states’Part I: Quantum state propagation and quantum-noise reduction," IEEE Transf. Inf. Theory IT-24, 657 (1978).

Shih, Y. H.

M. H. Rubin and Y. H. Shih, "Models of a two-photon Ein-stein-Podolsky-Rosen interference experiment," Phys. Rev. A 45, 8138 (1992).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980) Chap. 11.
[CrossRef]

Steinberg, A. M.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer," Phys. Rev. A 45, 6659 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Dispersion cancellation in a measurement of the single-photon propagation velocity," Phys. Rev. Lett. 68, 2421 (1992).
[CrossRef] [PubMed]

Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Fourth-order interference of joint single-photon wave packets in lossless optical systems," Phys. Rev. A 42, 4127 (1990).
[CrossRef] [PubMed]

J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, "Theory of light detection in the presence of feedback," J. Opt. Soc. Am. B 4, 1604 (1987).
[CrossRef]

Wang, L. J.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Observation of nonlocal interference in separated photon channels," Phys. Rev. Lett. 65, 321 (1990).

Wong, N. C.

Wozencraft, J. M.

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 205.

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric downconversion," Phys. Rev. Lett. 57, 2520 (1986).
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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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Other (12)

The configuration considered by Ou and Mandel32 is not that of the dispersion-cancellation experiment. Their principal assumption, however, is configuration independent; they require the coincidence-gate duration to be much longer than both the field- and the intensity-correlation times of the signal and the idler beams. Thus the Ou-Mandel proof can be adapted to the Fig. 1 configuration.
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," Phys. Rev. A (to be published).

Our concluding analysis from Subsection 3.A can be adapted to show that the nonergodic classical-field model from Eqs. (65) and (66) does not predict high-visibility coincidence-rate fringes. The Ou-Mandel theory32 can be used to disqualify any ergodic model from producing high-visibility fringes.

Because well-known procedures are available for accounting for subunity quantum efficiency in both quantum and somi-classical photodetection (see, e.g., Ref. 16), we treat only the ideal case of unity quantum efficiency. Furthermore, because our principal goal is to study the semiclassical and quantum fringe visibilities in fourth-order interference ex-periments, it is worth noting that these visibilities are independent of detector quantum efficiency; see the quantum efficiency discussion in Section 4.

This quantum preservation of joint Gaussian behavior is just like the well-known result for classical Gaussian random processes (see, e.g., Ref. 24, Chap. 3). It can be proved by performance of the state transformation implied by Eq. (39) with antinormally ordered characteristic functions.28
[CrossRef] [PubMed]

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980) Chap. 11.
[CrossRef]

Strictly speaking, we are concerned with a coincidence-rate dip at T = 0, not with a white-light fringe. Thus an experimentalist might prefer to gauge the depth of the destructive interference that occurs at T = 0 by computing {maxT[C(T; τg)] - minT[C(T; τg)]}/maxT[C(T; τg)] rather than γ.

Because we have suppressed the spatial and polarization characteristics of these fields, we say that our parametric interaction is degenerate if ωs = ωI = ωP/2 and nonde-generate if |Ωs - ωI| » Δω, where Δω is the common bandwidth of the signal and the idler emissions. Our development implicitly assumes that the signal and the idler beams are nondegenerate in either space or in polarization when they are degenerate in frequency. Spatial nondegen-eracy is ordinarily the case in parametric downconverters, and type-II phase-matched OPA’s and OPO’s produce orthogonally polarized signal and idler beams, so little generality is lost through this implicit assumption.

It is well known that fourth moments of zero-mean, jointly Gaussian random variables factor into sums of products of their second moments; see, e.g., Ref. 24. For an arbitrary Gaussian state this classical result can be combined with the quantum theory of heterodyne detection10 to produce a quantum moment-factoring theorem for anti-normally ordered field-operator moments. The normally ordered fourth-moment result that we employ is then found by repeated use of the delta-function commutator.

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 205.

This result does not depend on assuming that P (ω) is an even function.

Y. H. Shih, A. V. Sergienko, T. E. Kiess, M. H. Rubin, and C. O. Alley, "Two-photon interference in a standard Mach-Zehnder interferometer," in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 273.

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Figures (2)

Fig. 1
Fig. 1

Schematic for the dispersion-cancellation experiment of Steinberg et al.13,14 PD, degenerate parametric downconverter; DE, dispersive element; BS, 50/50 beam splitter whose position is changed to vary the delay between the reflected signal and the idler beams reaching detectors D1 and D2, respectively; SP, signal processor used to measure the singles- and the coincidence-counting rates.

Fig. 2
Fig. 2

Schematic for the fourth-order interference Mach–Zehnder interferometer of Shih et al.15 PD, nondegenerate parametric downconverter; VD, variable delay between the interferometer’s two arms; SF and IF, respectively, the signal-beam and the idler-beam passband optical filters; DS and DI, respectively, the signal-beam and the idler-beam detectors; SP, signal processor used to measure the singles- and the coincidence-counting rates.

Equations (89)

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[ E ^ j ( t ) , E ^ j ( u ) ] = δ ( t - u ) ,             for j = S , I .
E ^ j ( t + τ ) E ^ k ( t ) = δ j k exp ( i ω j τ ) d ω 2 π P ( ω ) exp ( i ω τ ) ,
E ^ j ( t + τ ) E ^ k ( t ) = ( 1 - δ j k ) exp [ - i ( ω P t + ω j τ ) ] × d ω 2 π { P ( ω ) [ P ( ω ) + 1 ] } 1 / 2 × exp ( - i ω τ ) ,
E ^ j IN ( ω ) d t E ^ j IN ( t ) exp ( i ω t )             for j = S , I ,
E ^ j ( ω ) d t E ^ j ( t ) exp ( i ω t )             for j = S , I ,
E ^ S ( ω S + ω ) = [ P ( ω ) + 1 ] 1 / 2 E ^ S IN ( ω S + ω ) + [ P ( ω ) ] 1 / 2 E ^ I IN ( ω I - ω ) ,
E ^ I ( ω I - ω ) = [ P ( ω ) + 1 ] 1 / 2 E ^ I IN ( ω I - ω ) + [ P ( ω ) ] 1 / 2 E ^ S IN ( ω S + ω ) .
E ^ S ( ω S + ω ) E ^ S ( ω S + ω ) - E ^ I ( ω I - ω ) E ^ I ( ω I - ω ) = E ^ S IN ( ω S + ω ) E ^ S IN ( ω S + ω ) - E ^ I IN ( ω I - ω ) E ^ I IN ( ω I - ω )             for all ω .
P ( ω ) = ( 2 π P / Δ ω ) exp ( - ω 2 / 2 Δ ω 2 ) ,
E ^ S ( t + τ ) E ^ S ( t ) exp ( - i ω S τ ) = E ^ I ( t + τ ) E ^ I ( t ) × exp ( - i ω I τ ) = P exp ( - τ 2 Δ ω 2 / 2 ) .
E ^ S ( t + τ ) E ^ I ( t ) exp ( i ω S τ ) = E ^ I ( t + τ ) E ^ S ( t ) exp ( i ω I τ ) ( 2 / π ) 1 / 4 P Δ ω exp ( - i ω P t - τ 2 Δ ω 2 ) .
E ^ S ( t + τ ) E ^ I ( t ) exp ( i ω S τ ) = E ^ I ( t + τ ) E ^ S ( t ) exp ( i ω I τ ) P exp ( - i ω P t - τ 2 Δ ω 2 / 2 ) .
i ^ θ ( t ) q Re { 2 P LO exp [ i ( ω P t / 2 - θ ) ] [ E ^ S ( t ) + E ^ I ( t ) ] } .
E j * ( t + τ ) E k ( t ) = δ j k exp ( - i ω j τ ) d ω 2 π P ( ω ) exp ( i ω τ ) ,
E j ( t + τ ) E k ( t ) = ( 1 - δ j k ) exp [ - i ( ω P t + ω j τ ) ] × d ω 2 π P ( ω ) exp ( - i ω τ ) ,
E S * ( t + τ ) E S ( t ) exp ( - i ω S τ ) = E I * ( t + τ ) E I ( t ) exp ( - i ω I τ ) = P exp ( - τ 2 Δ ω 2 / 2 ) ,
E S ( t + τ ) E I ( t ) exp ( i ω S τ ) = E I ( t + τ ) E S ( t ) exp ( i ω I τ ) = P exp ( - i ω P t - τ 2 Δ ω 2 / 2 ) ,
i ( t ) = q E ( t ) 2 + i n ( t ) ,
i n ( t ) i n ( u ) = q 2 E ( t ) 2 δ ( t - u )
S shot ( ω ) d τ i n ( t + τ ) i n ( t ) exp ( - i ω τ ) = q 2 E 2 .
i θ ( t ) = q Re { 2 P LO exp [ i ( ω P t / 2 - θ ) ] [ E S ( t ) + E I ( t ) ] } + i LO ( t ) .
S θ ( ω ) d τ i θ ( t + τ ) i θ ( t ) exp ( - i ω τ ) = q 2 P LO [ 1 + P ( ω ) ] 1 / 2 + exp ( - i 2 θ ) [ P ( ω ) ] 1 / 2 2
S θ ( ω ) = q 2 P LO [ 1 + P ( ω ) 1 + exp ( - i 2 θ ) 2 ] .
S min ( ω ) = { q 2 P LO { [ 1 + P ( ω ) ] 1 / 2 - [ P ( ω ) ] 1 / 2 } 2 quantum theory q 2 P LO semiclassical theory .
( 1 + x - x ) 2 = 1 / ( 1 + x + x ) 2 < 1             for x > 0 ,
Δ N T q - 1 - T / 2 T / 2 d t Δ i ( t ) ,
S Δ i ( ω ) d τ Δ i ( t + τ ) Δ i ( t ) exp ( - i ω τ ) ,
Δ i ( t + τ ) Δ i ( t ) = q 2 { δ ( τ ) [ E ^ S ( t ) E ^ S ( t ) + E ^ I ( t ) E ^ I ( t ) ] + E ^ S ( t + τ ) E ^ S ( t ) E ^ S ( t + τ ) E ^ S ( t ) + E ^ I ( t + τ ) E ^ I ( t ) E ^ I ( t + τ ) E ^ I ( t ) - E ^ S ( t + τ ) E ^ I ( t ) E ^ S ( t + τ ) E ^ I ( t ) - E ^ I ( t + τ ) E ^ S ( t ) E ^ I ( t + τ ) E ^ S ( t ) } .
Δ i ( t + τ ) Δ i ( t ) = q 2 { δ ( τ ) [ E ^ S ( t ) E ^ S ( t ) + E ^ I ( t ) E ^ I ( t ) ] + E ^ S ( t + τ ) E ^ S ( t ) E ^ S ( t ) E ^ S ( t + τ ) + E ^ I ( t + τ ) E ^ I ( t ) E ^ I ( t ) E ^ I ( t + τ ) - E ^ S ( t + τ ) E ^ I ( t ) E ^ S ( t + τ ) E ^ I ( t ) - E ^ I ( t + τ ) E ^ S ( t ) E ^ I ( t + τ ) E ^ S ( t ) } .
S Δ i ( ω ) = 2 q 2 d ω 2 π { P ( ω ) + P ( ω ) P ( ω - ω ) - [ P ( ω ) [ P ( ω ) + 1 ] P ( ω - ω ) [ P ( ω - ω ) + 1 ] ] 1 / 2 } .
lim ω S Δ i ( ω ) = 2 q 2 d ω 2 π P ( ω ) .
Δ i ( t ) = q [ E S ( t ) 2 - E I ( t ) 2 ] + Δ i n ( t ) ,
Δ i ( t + τ ) Δ i ( t ) = q 2 { δ ( τ ) [ E S ( t ) 2 + E I ( t ) 2 ] + E S * ( t + τ ) E S * ( t ) E S ( t + τ ) E S ( t ) + E I * ( t + τ ) E I * ( t ) E I ( t + τ ) E I ( t ) - E S * ( t + τ ) E I * ( t ) E S ( t + τ ) E I ( t ) - E I * ( t + τ ) E S * ( t ) E I ( t + τ ) E S ( t ) } .
Δ i ( t + τ ) Δ i ( t ) = q 2 δ ( τ ) [ E S ( t ) 2 + E I ( t ) 2 ] ,
S Δ i ( ω ) = 2 q 2 d ω 2 π P ( ω ) = 2 q 2 P ,
Δ N T 2 / T = d ω 2 π T [ sin ( ω T / 2 ) ω T / 2 ] 2 S Δ i ( ω ) .
Δ N T 2 / T = 2 P             for all T
Δ N T 2 / T { 2 P as Δ ω T 0 0 as Δ ω T .
E ^ SD ( t ) d τ E ^ S ( τ ) h ( t - τ ) ,
h ( t ) exp ( - i ω S t ) d ω 2 π H ( ω ) exp ( i ω t )
H ( ω ) exp ( i ω 2 ϕ ¨ / 2 )
Δ t = ( 1 + ϕ ¨ 2 Δ ω 4 ) 1 / 2 / Δ ω .
[ E ^ SD ( t ) , E ^ SD ( u ) ] = δ ( t - u ) ,
E ^ j ( t + τ ) E ^ k ( t ) = δ j k exp ( i ω j τ ) d ω 2 π P ( ω ) exp ( i ω τ ) ,
E ^ j ( t + τ ) E ^ k ( t ) = ( 1 - δ j k ) exp [ - i ( ω P t + ω j τ ) ] × d ω 2 π { P ( ω ) [ P ( ω ) + 1 ] } 1 / 2 × H ( ω ) exp ( - i ω τ ) ,
E ^ SD ( t + τ ) E ^ SD ( t ) = E ^ I ( t + τ ) E ^ I ( t ) = P exp ( i ω P τ / 2 - τ 2 Δ ω 2 / 2 ) ,
E ^ SD ( t + τ ) E ^ I ( t ) = E ^ I ( t + τ ) E ^ SD ( t ) ( 2 π ) 1 / 4 ( P Δ ω 1 - 2 i ϕ ¨ Δ ω 2 ) 1 / 2 × exp [ - i ω P ( t + τ / 2 ) - τ 2 Δ ω 2 1 - 2 i ϕ ¨ Δ ω 2 ] ,
E ^ 1 ( t ) [ E ^ SD ( t - T / 2 ) + E ^ I ( t ) ] / 2 ,
E ^ 2 ( t ) [ E ^ SD ( t ) - E ^ I ( t + T / 2 ) ] / 2
S j ( T ) i j ( t ) / q             for j = 1 , 2.
C ( T ; τ g ) q - 2 d τ i 1 ( t + τ ) i 2 ( t ) exp ( - τ 2 / τ g 2 ) ,
E j * ( t + τ ) E k ( t ) = δ j k exp ( i ω j τ ) d ω 2 π P ( ω ) exp ( i ω τ ) ,
E j ( t + τ ) E k ( t ) = ( 1 - δ j k ) exp [ - i ( ω P t + ω j τ ) ] × d ω 2 π P ( ω ) H ( ω ) exp ( - i ω τ ) ,
E SD * ( t + τ ) E SD ( t ) = E I * ( t + τ ) E I ( t ) = P exp ( i ω P τ / 2 - τ 2 Δ ω 2 / 2 ) ,
E SD ( t + τ ) E I ( t ) = E I ( t + τ ) E SD ( t ) = P ( 1 - i ϕ ¨ Δ ω 2 ) 1 / 2 × exp [ - i ω P ( t + τ / 2 ) - τ 2 Δ ω 2 / 2 1 - i ϕ ¨ Δ ω 2 ] .
S j ( T ) = P             for j = 1 , 2 ,
q - 2 i 1 ( t + τ ) i 2 ( t ) = P 2 + 4 - 1 E ^ SD ( t + τ - T / 2 ) E ^ I ( t + T / 2 ) - E ^ I ( t + τ ) E ^ SD ( t ) 2
q - 2 i 1 ( t + τ ) i 2 ( t ) = P 2 + 4 - 1 E SD ( t + τ - T / 2 ) E I ( t + T / 2 ) - E I ( t + τ ) E SD ( t ) 2
C ( T ; τ g ) = P 2 π τ g + 4 - 1 d τ 2 P Δ ω [ π ( 1 + 4 ϕ ¨ 2 Δ ω 4 ) ] 1 / 2 × exp ( - τ 2 / τ g 2 ) ( exp [ - 2 ( τ - T ) 2 Δ ω 2 1 + 4 ϕ ¨ 2 Δ ω 4 ] + exp ( - 2 τ 2 Δ ω 2 1 + 4 ϕ ¨ 2 Δ ω 4 ) - 2 Re { exp [ - ( τ - T ) 2 Δ ω 2 1 - 2 i ϕ ¨ Δ ω 2 ] × exp ( - τ 2 Δ ω 2 1 + 2 i ϕ ¨ Δ ω 2 ) } ) .
C ( T ; τ g ) = P { π P τ g + 2 - 1 [ 1 - exp ( - Δ ω 2 T 2 / 2 ) ] } .
C ( T ; τ g ) = π P 2 τ g { 1 + ( 2 Δ ω τ g ) - 1 [ 1 - exp ( - Δ ω 2 T 2 / 4 ) ] } .
γ max T [ C ( T ; τ g ) ] - min T [ C ( T ; τ g ) ] max T [ C ( T ; τ g ) ] + min T [ C ( T ; τ g ) ]
= 1 / ( 1 + 4 π P τ g ) 1             for P τ g 1.
γ = 1 / ( 1 + 4 Δ ω τ g ) 1             for Δ ω τ g 1.
E S ( t ) = P exp [ - i ( ω S + ω ˜ ) t - i θ ˜ ] ,
E I ( t ) = P exp [ - i ( ω I - ω ˜ ) t + i θ ˜ ] ,
E SD ( t ) = P H ( ω ˜ ) exp [ - i ( ω S + ω ˜ ) t - i θ ˜ ] .
q - 2 i 1 ( t + τ ) i 2 ( t ) = E 1 ( t + τ ) 2 E 2 ( t ) 2
= P 2 ( 1 - 2 - 1 Re { exp [ i ω ˜ ( 2 τ - T ) ] } )
= P 2 { 1 - 2 - 1 exp [ - 2 ( τ - T / 2 ) 2 Δ ω 2 ] } .
C ( T ; τ g ) = P 2 π τ g [ 1 - ( 8 Δ ω τ g ) - 1 exp ( - T 2 / 4 τ g 2 ) ] .
E S out ( t ) [ E S ( t - T ) + E S ( t ) ] / 2 ,
E I out ( t ) [ E I ( t ) - E I ( t - T ) ] / 2
S j ( T ) = i j ( t ) / q = { ( P / 2 ) [ 1 + exp ( - Δ ω 2 T 2 / 2 ) cos ( ω S T ) ] for j = S ( P / 2 ) [ 1 - exp ( - Δ ω 2 T 2 / 2 ) cos ( ω I T ) ] for j = I ,
C ( T ; τ g ) = q - 2 d τ i S ( t + τ ) i I ( t ) exp ( - τ 2 / τ g 2 ) ,
q - 2 i S ( t + τ ) i I ( t ) = S S ( T ) S I ( T ) + ( 1 / 16 ) [ E ^ S ( t + τ - T ) + E ^ S ( t + τ ) ] × [ E ^ I ( t ) - E ^ I ( t - T ) ] 2 ,
q - 2 i S ( t + τ ) i I ( t ) = S S ( T ) S I ( T ) + ( 1 / 16 ) [ E S ( t + τ - T ) + E S ( t + τ ) ] × [ E I ( t ) - E I ( t - T ) ] 2 ,
C ( T ; τ g ) = 4 - 1 ( π P 2 τ g + K ) { 1 + cos ( ω S T ) - cos ( ω I T ) - 2 - 1 cos [ ( ω S - ω I ) T ] - 2 - 1 cos ( ω P T ) } .
C ( T ; τ g ) = 4 - 1 { π P 2 τ g + K [ 1 - 2 - 1 cos ( ω P T ) ] } .
C ( T ; τ g ) = 4 - 1 { π P 2 τ g + ( K / 2 ) [ 1 - cos ( ω P T ) ] } .
K = d ω 2 π P ( ω ) [ P ( ω ) + 1 ] = π P 2 / Δ ω + P
K = d ω 2 π P 2 ( ω ) = π P 2 / Δ ω
γ = { 1 + ( Δ ω / π P ) 1 + 2 Δ ω τ g + ( Δ ω / π P ) quantum mechanically 1 1 + 2 Δ ω τ g semiclassically .
γ { 1 quantum mechanically 1 / 2 Δ ω τ g 1 semiclassically ,
γ Q γ SC + 1 / 2 π P τ g 1
γ Q / γ SC 1 + Δ ω / π P 1
S min ( ω ) = { q 2 η P LO ( ( 1 - η ) + η { [ 1 + P ( ω ) ] 1 / 2 - [ P ( ω ) ] 1 / 2 } 2 ) quantum theory q 2 η P LO semiclassical theory ,
Δ N T 2 / T = 2 η P             for all T ,
Δ N T 2 / T { 2 η P as Δ ω T 0 2 η ( 1 - η ) P as Δ ω T ,

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