Abstract

Einstein–Podolsky–Rosen (EPR) correlations produced for two-photon entangled wave packets are analyzed by use of optical Wigner functions. In particular, type-II optical parametric downconversion systems are used, and the effects of the length of the crystal and the filter bandwidth on the Wigner functions are studied. The relation between the optical Wigner functions and the interference experiment with a beam splitter is studied. We find that, whereas in first-order correlation experiments (typified, for example, by Young’s interference experiment) the “photon interferes with itself” [P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, UK, 1958), p. 9], in the EPR correlations the two-photon amplitude function interferes with itself. Such interference is analyzed by use of the optical Wigner functions.

© 1999 Optical Society of America

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  3. K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); K. E. Cahil and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
    [CrossRef]
  4. Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).
  5. B. R. Mollow and R. J. Glauber, “Quantum theory of parametic amplification. II,” Phys. Rev. 160, 1097–1108 (1967).
    [CrossRef]
  6. U. Leonhardt, “Quantum statistics of a two-mode SU(1, 1) interferometer,” Phys. Rev. A 49, 1231–1242 (1994).
    [CrossRef] [PubMed]
  7. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U. Press, Cambridge, 1988), pp. 196–200.
  8. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolski–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992); Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of Einstein–Podolski–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
    [CrossRef] [PubMed]
  9. U. Leonhardt, “On Bell correlations for the phase space of two entangled light modes,” Phys. Lett. A 182, 195–200 (1993).
    [CrossRef]
  10. A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
    [CrossRef]
  11. K. Banaszek and K. Wodkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
    [CrossRef]
  12. A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
    [CrossRef]
  13. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [CrossRef]
  14. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  15. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
  16. Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
    [CrossRef] [PubMed]
  17. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
    [CrossRef] [PubMed]
  18. B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
    [CrossRef]
  19. J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
    [CrossRef]
  20. W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
    [CrossRef]
  21. M. H. Rubin, “Transverse correlation in optical spontaneous parametric down conversion,” Phys. Rev. A 54, 5349–5360 (1996).
    [CrossRef] [PubMed]
  22. Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
    [CrossRef] [PubMed]
  23. H. M. Wiseman, F. E. Harrison, M. J. Collet, D. F. Walls, S. M. Tan, and R. B. Killip, “Nonlocal Momentum Transfer in ‘Welcher Weg’ Schemes” in Quantum Interferometry, Proceedings of an Adriatico Conference at the International Center for Theoretical Physics, Trieste, 1996, F. De Martini, G. Denardo, and Y. H. Shih, eds. (VCH, Weinheim, 1996), pp. 267–278.
  24. P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, UK, 1958), p. 9.

1999

K. Banaszek and K. Wodkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
[CrossRef]

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

1998

A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
[CrossRef]

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
[CrossRef]

1997

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[CrossRef]

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

1996

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down conversion,” Phys. Rev. A 54, 5349–5360 (1996).
[CrossRef] [PubMed]

1994

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

U. Leonhardt, “Quantum statistics of a two-mode SU(1, 1) interferometer,” Phys. Rev. A 49, 1231–1242 (1994).
[CrossRef] [PubMed]

1993

U. Leonhardt, “On Bell correlations for the phase space of two entangled light modes,” Phys. Lett. A 182, 195–200 (1993).
[CrossRef]

1988

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[CrossRef] [PubMed]

1984

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

1967

B. R. Mollow and R. J. Glauber, “Quantum theory of parametic amplification. II,” Phys. Rev. 160, 1097–1108 (1967).
[CrossRef]

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Alley, C. O.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

Banaszek, K.

K. Banaszek and K. Wodkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
[CrossRef]

Casado, A.

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

Deuar, P.

A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
[CrossRef]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Fernandez-Rueda, A.

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

Gilchrist, A.

A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
[CrossRef]

Glauber, R. J.

B. R. Mollow and R. J. Glauber, “Quantum theory of parametic amplification. II,” Phys. Rev. 160, 1097–1108 (1967).
[CrossRef]

Grice, W. P.

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[CrossRef]

Hillery, M.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Joobeur, A.

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
[CrossRef]

Jost, B. M.

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

Kiess, T. E.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Quantum statistics of a two-mode SU(1, 1) interferometer,” Phys. Rev. A 49, 1231–1242 (1994).
[CrossRef] [PubMed]

U. Leonhardt, “On Bell correlations for the phase space of two entangled light modes,” Phys. Lett. A 182, 195–200 (1993).
[CrossRef]

Mandel, L.

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[CrossRef] [PubMed]

Marshal, T.

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

Mollow, B. R.

B. R. Mollow and R. J. Glauber, “Quantum theory of parametic amplification. II,” Phys. Rev. 160, 1097–1108 (1967).
[CrossRef]

O’Connel, R. F.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[CrossRef] [PubMed]

Perina Jr., J.

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

Reid, M. D.

A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
[CrossRef]

Risco-Delgado, R.

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

Rubin, M. H.

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down conversion,” Phys. Rev. A 54, 5349–5360 (1996).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Saleh, B. E. A.

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
[CrossRef]

Santos, E.

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Sergienko, A. V.

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

Shih, Y. H.

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Teich, M. C.

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
[CrossRef]

Walmsley, I. A.

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wigner, E. P.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Wodkiewicz, K.

K. Banaszek and K. Wodkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
[CrossRef]

Phys. Lett. A

U. Leonhardt, “On Bell correlations for the phase space of two entangled light modes,” Phys. Lett. A 182, 195–200 (1993).
[CrossRef]

Phys. Rep.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Phys. Rev.

B. R. Mollow and R. J. Glauber, “Quantum theory of parametic amplification. II,” Phys. Rev. 160, 1097–1108 (1967).
[CrossRef]

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A

A. Casado, A. Fernandez-Rueda, T. Marshal, R. Risco-Delgado, and E. Santos, “Fourth-order interference in the Wigner representation for parametric down-conversion experiments,” Phys. Rev. A 55, 3879–3890 (1997).
[CrossRef]

U. Leonhardt, “Quantum statistics of a two-mode SU(1, 1) interferometer,” Phys. Rev. A 49, 1231–1242 (1994).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon entanglement in type-II parametric down conversion,” Phys. Rev. A 50, 23–28 (1994).
[CrossRef] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998).
[CrossRef]

J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, “Dispersion in femtosecond entangled two-photon interference,” Phys. Rev. A 59, 2359–2368 (1999).
[CrossRef]

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[CrossRef]

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down conversion,” Phys. Rev. A 54, 5349–5360 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[CrossRef] [PubMed]

A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. 80, 3169–3172 (1998).
[CrossRef]

K. Banaszek and K. Wodkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
[CrossRef]

Prog. Opt.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Other

L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); K. E. Cahil and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U. Press, Cambridge, 1988), pp. 196–200.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolski–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992); Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of Einstein–Podolski–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef] [PubMed]

H. M. Wiseman, F. E. Harrison, M. J. Collet, D. F. Walls, S. M. Tan, and R. B. Killip, “Nonlocal Momentum Transfer in ‘Welcher Weg’ Schemes” in Quantum Interferometry, Proceedings of an Adriatico Conference at the International Center for Theoretical Physics, Trieste, 1996, F. De Martini, G. Denardo, and Y. H. Shih, eds. (VCH, Weinheim, 1996), pp. 267–278.

P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, UK, 1958), p. 9.

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Equations (36)

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ωe=Ωe+ν,ωo=Ωo-ν,Ωd=12(Ωe-Ωo),
ωo+ωe=Ωo+Ωe=ωP,
ϕ(Δkz)=exp(iΔkzL)-1i(ΔkzL).
Δkz=ν1uo-1ueνD,
f(t1, t2)=Af1(t1-t2)f2(t1+t2),
f1(t1-t2)=-dν1-exp(-iνDL)iνDLexp-ν22σ2exp[-i(ν+Ωd)(t1-t2)],
f2(t1+t2)=exp-iωPt1+t22.
q-=q1-q2,q+=q1+q2,
T-=T1-T2,T+=T1+T2.
PW(q-, p-; q+, p+)=PW(q-, p-)PW(q+, p+),
p-=p1-p22,p+=p1+p22.
Dˆ-(ξ-, η-)=exp[-i(ξ-qˆ-+η-pˆ-)]
Dˆ+(ξ+, η+)=exp[-i(ξ+qˆ++η+pˆ+)]
C(ξ-, η-)=-f1*(q-)Dˆ-(ξ-, η-)f1(q-)dq-.
Dˆ-(ξ-, η-)f1=expiη-ξ-2exp(-iξ-q-)×exp-η-q-f1.
exp-ηq-f1(q-)=-dν11-exp(-iν1DL)iν1DL×exp-ν122σ2×exp-i(ν1+Ωd)T--q-c×exp-i(Ωd+ν1)η-c.
- exp(-iξ-q-) expi(ν1-ν2)q-cdq-
=2πδξ-+ν2-ν1c.
C(ξ-, η-)=2πc-ϕ*[(ν1-ξ-c)DL]ϕ(ν1DL)×exp-ν122σ2exp-(ν1-ξ-c)22σ2×exp-iη-Ωd+ν1c-ξ-2×exp(-iT-ξ-c)dν1.
ϕ(x)=1-exp(-ix)ix,x=νDL.
PW(q-, p-)=(1/2π)2-dξ--dη-×exp[i(ξ-q-+η-p-)]C(ξ-, η-),
ν1=ν1-ξ-c2.
PW(q-, p-)=B-dξ- exp[iξ-(q--T-c)]×exp-ξ-c2σ2ϕ*p-c2-Ωd-ξ-c2DLϕp-c-Ωd+ξ-c2DL×exp-p-c-Ωdσ2,
--PW(q-, p-)dq-dp-=1.
PW(q-, p-)=1πexp-(q--T-c)2σ2c2×exp-p--Ωdc2c2σ2.
PW(q-, p-)dq-=PW(p-)=G exp-(p-c-Ωd)2σ2×sin c2(p-c-Ωd)DL2,
PW(q-, p-)=-×θτ+DL4-θτ-DL4θt-τ+DL4-θt-τ-DL4×exp(iΩτ)exp[-iΩ(t-τ)]dτ.
t=T--q-c+DL2;Ω=p-c-Ωd.
PW(q-, p-)=RexpiΩDL2exp(-iΩt)-exp-iΩDL2exp(iΩt)i2Ω
f2(t1+t2)=A+ exp-iωPt1+t22×exp[-σ+2(t1+t2)2].
PW(q+, p+)=1πexp-(q+-T+c)2σ+2c2×exp-(ωe+ω0)2σ+2.
RC=1T0TdT-f1T--q-c2×0TdT+f2T+-q+c2.
exp[i(ν+Ωd)(Δl/c)]+exp[-i(ν+Ωd)(Δl/c)]
PW(q-, p-)=G-+ exp[iξ-(q--T-c)]×exp-(p-c-Ωd)2σ2×exp-(ξ-c)24σ2{exp(iξ-Δl)+exp(-iξ-Δl)+exp(ip-2Δl)+exp(-ip-2Δl)}dξ-.
Rc-+d(p-c-Ωd)exp-(p-c-Ωd)2σ2×{2+exp(ip-2Δl)+exp(-ip-2Δl)}.
Rc1+exp-σΔlc2cosΩd2Δlc.

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