Abstract

We report a two-photon double-slit interference experiment. Two correlated photons, generated by spontaneous parametric downconversion and traveling in different directions, are made to pass through a double slit in such a way that the path of one photon can be identified with that of the other photon. No single-photon interference pattern is observed even if the path is not actually identified, but a spatial interference pattern appears in the joint detection counting rates of the photon pairs. It can easily be shown that these nonlocal phenomena cannot be exhibited by any kind of classical field.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  43. J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).
  44. S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
    [CrossRef]
  45. Z. Y. Ou and L. Mandel, “Classical treatment of the Franson two-photon correlation experiment,” J. Opt. Soc. Am. B 7, 2127–2131 (1990).
    [CrossRef]
  46. J. D. Franson, “Violation of a simple inequality for classical fields,” Phys. Rev. Lett. 67, 290–293 (1991).
    [CrossRef] [PubMed]
  47. M. H. Rubin and Y. H. Shih, “Models of a two-photon Einstein–Podolsky–Rosen interference experiment,” Phys. Rev. A 45, 8138–8147 (1992).
    [CrossRef] [PubMed]
  48. J. H. Shapiro and K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
    [CrossRef]

1996 (2)

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

1995 (3)

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the quantum eraser,” Phys. Rev. Lett. 75, 3034–3037 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

1994 (4)

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space–time,” Phys. Rev. A 50, 2564–2568 (1994).
[CrossRef] [PubMed]

P. R. Tapster, J. G. Rarity, and P. C. M. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

J. H. Shapiro and K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

1993 (4)

Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Einstein–Podolsky–Rosen state for space–time variables in a two-photon interference experiment,” Phys. Rev. A 47, 1288–1293 (1993).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[CrossRef]

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46(8), 22–29 (1993).
[CrossRef]

1992 (4)

J. G. Rarity and P. R. Tapster, “Fourth-order interference effects at large distances,” Phys. Rev. A 45, 2052–2056 (1992).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ‘quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A 45, 7729–7739 (1992).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, L. J. Wang, and L. Mandel, “Can an ‘empty’ de Broglie pilot wave induce coherence?” Phys. Rev. Lett. 68, 3667–3669 (1992).
[CrossRef] [PubMed]

M. H. Rubin and Y. H. Shih, “Models of a two-photon Einstein–Podolsky–Rosen interference experiment,” Phys. Rev. A 45, 8138–8147 (1992).
[CrossRef] [PubMed]

1991 (5)

J. D. Franson, “Violation of a simple inequality for classical fields,” Phys. Rev. Lett. 67, 290–293 (1991).
[CrossRef] [PubMed]

J. Brendel, E. Mohler, and W. Martienssen, “Time-resolved dual-beam two-photon interferences with high visibility,” Phys. Rev. Lett. 66, 1142–1145 (1991).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67, 318–321 (1991).
[CrossRef] [PubMed]

J. D. Franson, “Two-photon interferometry over large distances,” Phys. Rev. A 44, 4552–4555 (1991).
[CrossRef] [PubMed]

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[CrossRef]

1990 (8)

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (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, 321–324 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (1990).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A 41, 5139–5146 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495–2498 (1990).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Classical treatment of the Franson two-photon correlation experiment,” J. Opt. Soc. Am. B 7, 2127–2131 (1990).
[CrossRef]

1989 (4)

M. O. Scully and H. Walther, “Quantum optical test of observation and complementarity in quantum mechanics,” Phys. Rev. A 39, 5229–5235 (1989).
[CrossRef] [PubMed]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference in parametric downconversion,” J. Opt. Soc. Am. B 6, 1221–1226 (1989).
[CrossRef]

Z. Y. Ou and L. Mandel, “Further evidence of nonclassical behavior in optical interference,” Phys. Rev. Lett. 62, 2941–2944 (1989).
[CrossRef] [PubMed]

1988 (3)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

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

Y. H. Shih and C. O. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric downconversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

1987 (2)

R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of two photons,” Phys. Rev. Lett. 59, 1903–1905 (1987).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef] [PubMed]

1986 (1)

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

1985 (1)

S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
[CrossRef]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

1964 (1)

J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric downconversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

Barbosa, G. A.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

Bell, J. S.

J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).

Brendel, J.

J. Brendel, E. Mohler, and W. Martienssen, “Time-resolved dual-beam two-photon interferences with high visibility,” Phys. Rev. Lett. 66, 1142–1145 (1991).
[CrossRef] [PubMed]

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Campos, R. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

Chiao, R. Y.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[CrossRef]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ‘quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A 45, 7729–7739 (1992).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Englert, B.-G.

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[CrossRef]

Franson, J. D.

J. D. Franson, “Two-photon interferometry over large distances,” Phys. Rev. A 44, 4552–4555 (1991).
[CrossRef] [PubMed]

J. D. Franson, “Violation of a simple inequality for classical fields,” Phys. Rev. Lett. 67, 290–293 (1991).
[CrossRef] [PubMed]

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
[CrossRef]

Ghosh, R.

R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of two photons,” Phys. Rev. Lett. 59, 1903–1905 (1987).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

Grayson, T.

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, L. J. Wang, and L. Mandel, “Can an ‘empty’ de Broglie pilot wave induce coherence?” Phys. Rev. Lett. 68, 3667–3669 (1992).
[CrossRef] [PubMed]

Greenberger, D. M.

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46(8), 22–29 (1993).
[CrossRef]

Herzog, T. J.

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the quantum eraser,” Phys. Rev. Lett. 75, 3034–3037 (1995).
[CrossRef] [PubMed]

Hong, C. K.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
[CrossRef]

Horne, M. A.

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46(8), 22–29 (1993).
[CrossRef]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[CrossRef] [PubMed]

Jakeman, E.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Kwiat, P. G.

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the quantum eraser,” Phys. Rev. Lett. 75, 3034–3037 (1995).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[CrossRef]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ‘quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A 45, 7729–7739 (1992).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

Larchuk, T.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

Machado da Silva, J. C.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Mandel, L.

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, L. J. Wang, and L. Mandel, “Can an ‘empty’ de Broglie pilot wave induce coherence?” Phys. Rev. Lett. 68, 3667–3669 (1992).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67, 318–321 (1991).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (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, 321–324 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Classical treatment of the Franson two-photon correlation experiment,” J. Opt. Soc. Am. B 7, 2127–2131 (1990).
[CrossRef]

Z. Y. Ou and L. Mandel, “Further evidence of nonclassical behavior in optical interference,” Phys. Rev. Lett. 62, 2941–2944 (1989).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

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

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef] [PubMed]

R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of two photons,” Phys. Rev. Lett. 59, 1903–1905 (1987).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
[CrossRef]

Martienssen, W.

J. Brendel, E. Mohler, and W. Martienssen, “Time-resolved dual-beam two-photon interferences with high visibility,” Phys. Rev. Lett. 66, 1142–1145 (1991).
[CrossRef] [PubMed]

Mattle, K.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Michler, M.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

Migdall, A.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

Mohler, E.

J. Brendel, E. Mohler, and W. Martienssen, “Time-resolved dual-beam two-photon interferences with high visibility,” Phys. Rev. Lett. 66, 1142–1145 (1991).
[CrossRef] [PubMed]

Nathel, H.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

Ou, Z. Y.

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (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, 321–324 (1990).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Classical treatment of the Franson two-photon correlation experiment,” J. Opt. Soc. Am. B 7, 2127–2131 (1990).
[CrossRef]

Z. Y. Ou and L. Mandel, “Further evidence of nonclassical behavior in optical interference,” Phys. Rev. Lett. 62, 2941–2944 (1989).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

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

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

Owens, P. C. M.

P. R. Tapster, J. G. Rarity, and P. C. M. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

Pádua, S.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Pittman, T. B.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Rarity, J. G.

P. R. Tapster, J. G. Rarity, and P. C. M. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference effects at large distances,” Phys. Rev. A 45, 2052–2056 (1992).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A 41, 5139–5146 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495–2498 (1990).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference in parametric downconversion,” J. Opt. Soc. Am. B 6, 1221–1226 (1989).
[CrossRef]

Ribeiro, P. H. S.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Rubin, M. H.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Einstein–Podolsky–Rosen state for space–time variables in a two-photon interference experiment,” Phys. Rev. A 47, 1288–1293 (1993).
[CrossRef] [PubMed]

M. H. Rubin and Y. H. Shih, “Models of a two-photon Einstein–Podolsky–Rosen interference experiment,” Phys. Rev. A 45, 8138–8147 (1992).
[CrossRef] [PubMed]

Saleh, B. E. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

Scully, M. O.

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[CrossRef]

M. O. Scully and H. Walther, “Quantum optical test of observation and complementarity in quantum mechanics,” Phys. Rev. A 39, 5229–5235 (1989).
[CrossRef] [PubMed]

Sergienko, A. V.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space–time,” Phys. Rev. A 50, 2564–2568 (1994).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Einstein–Podolsky–Rosen state for space–time variables in a two-photon interference experiment,” Phys. Rev. A 47, 1288–1293 (1993).
[CrossRef] [PubMed]

Shapiro, J. H.

Shih, Y. H.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space–time,” Phys. Rev. A 50, 2564–2568 (1994).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Einstein–Podolsky–Rosen state for space–time variables in a two-photon interference experiment,” Phys. Rev. A 47, 1288–1293 (1993).
[CrossRef] [PubMed]

M. H. Rubin and Y. H. Shih, “Models of a two-photon Einstein–Podolsky–Rosen interference experiment,” Phys. Rev. A 45, 8138–8147 (1992).
[CrossRef] [PubMed]

Y. H. Shih and C. O. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric downconversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

Shimony, A.

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[CrossRef] [PubMed]

Steinberg, A. M.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[CrossRef]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ‘quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A 45, 7729–7739 (1992).
[CrossRef] [PubMed]

Strekalov, D. V.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can two-photon interference be considered the interference of two photons?” Phys. Rev. Lett. 77, 1917–1920 (1996).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Sun, K.-X.

Tapster, P. R.

P. R. Tapster, J. G. Rarity, and P. C. M. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference effects at large distances,” Phys. Rev. A 45, 2052–2056 (1992).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495–2498 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A 41, 5139–5146 (1990).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference in parametric downconversion,” J. Opt. Soc. Am. B 6, 1221–1226 (1989).
[CrossRef]

Teich, M. C.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[CrossRef] [PubMed]

Vareka, W. A.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

Walther, H.

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[CrossRef]

M. O. Scully and H. Walther, “Quantum optical test of observation and complementarity in quantum mechanics,” Phys. Rev. A 39, 5229–5235 (1989).
[CrossRef] [PubMed]

Wang, L. J.

X. Y. Zou, T. Grayson, L. J. Wang, and L. Mandel, “Can an ‘empty’ de Broglie pilot wave induce coherence?” Phys. Rev. Lett. 68, 3667–3669 (1992).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67, 318–321 (1991).
[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, 321–324 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[CrossRef] [PubMed]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Weinfurter, H.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the quantum eraser,” Phys. Rev. Lett. 75, 3034–3037 (1995).
[CrossRef] [PubMed]

Zeilinger, A.

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the quantum eraser,” Phys. Rev. Lett. 75, 3034–3037 (1995).
[CrossRef] [PubMed]

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46(8), 22–29 (1993).
[CrossRef]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[CrossRef] [PubMed]

Zou, X. Y.

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, L. J. Wang, and L. Mandel, “Can an ‘empty’ de Broglie pilot wave induce coherence?” Phys. Rev. Lett. 68, 3667–3669 (1992).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67, 318–321 (1991).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (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, 321–324 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (3)

Nature (London) (1)

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[CrossRef]

Opt. Commun. (1)

S. Friberg, C. K. Hong, and L. Mandel, “Intensity dependence of the normalized intensity correlation function in parametric down-conversion,” Opt. Commun. 54, 311–316 (1985).
[CrossRef]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Phys. Rev. A (17)

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34, 3962–3968 (1986).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A 51, 3495–3498 (1995).
[CrossRef] [PubMed]

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53, R1209–R1212 (1996).
[CrossRef] [PubMed]

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurement,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

J. D. Franson, “Two-photon interferometry over large distances,” Phys. Rev. A 44, 4552–4555 (1991).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Fourth-order interference effects at large distances,” Phys. Rev. A 45, 2052–2056 (1992).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ‘quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A 45, 7729–7739 (1992).
[CrossRef] [PubMed]

Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Einstein–Podolsky–Rosen state for space–time variables in a two-photon interference experiment,” Phys. Rev. A 47, 1288–1293 (1993).
[CrossRef] [PubMed]

X. Y. Zou, T. Grayson, G. A. Barbosa, and L. Mandel, “Control of visibility in the interference of signal photons by delays imposed on the idler photons,” Phys. Rev. A 47, 2293–2295 (1993).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[CrossRef]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space–time,” Phys. Rev. A 50, 2564–2568 (1994).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, “Evidence for phase memory in two-photon down conversion through entanglement with the vacuum,” Phys. Rev. A 41, 566–568 (1990).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41, 2910–2913 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A 41, 5139–5146 (1990).
[CrossRef] [PubMed]

M. H. Rubin and Y. H. Shih, “Models of a two-photon Einstein–Podolsky–Rosen interference experiment,” Phys. Rev. A 45, 8138–8147 (1992).
[CrossRef] [PubMed]

M. O. Scully and H. Walther, “Quantum optical test of observation and complementarity in quantum mechanics,” Phys. Rev. A 39, 5229–5235 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (19)

J. D. Franson, “Violation of a simple inequality for classical fields,” Phys. Rev. Lett. 67, 290–293 (1991).
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Figures (3)

Fig. 1
Fig. 1

Experimental setup: DS, double slit; L’s, cylindrical lenses; M’s, mirrors; OF’s, optical fibers; IF’s, interference filters; D’s detectors.

Fig. 2
Fig. 2

Measured signal photon-counting (●) and coincidence-counting (■) rates as functions of (y-y). Error bars for the signal photon-counting rate are smaller than the dot size. The solid curve is the least-squares fit to relation (7).

Fig. 3
Fig. 3

Schematics of the experimental setup. (a) Four beams of the signal and the idler fields selected by a double slit. (b) Signal photon-detection scheme with a double slit. The box outlines the arrangement to identify the path of the signal photon with the idler photon. (c) Two-photon joint detection scheme with two identical double slits.

Equations (11)

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|ψ=12(|1, 1s1, i1+|1, 1s2, i2),
E^s(+)(y)=E^s1(+)(y)+E^s2(+)(y)=exp(iks1y)a^s1+exp(iks2y)a^s2,
ps(y)=Kψ|E^s(-)(y)E^s(+)(y)|ψ,=K2[exp(iks1y)|1i1+exp(iks2y)|1i2]2,=K,
ps(id)(y)=K{ψ|a^i1a^i1E^s(-)(y)E^s(+)(y)|ψ+ψ|a^i2a^i2E^s(-)(y)E^s(+)(y)|ψ}=K2[exp(iks1y)|vac2+exp(iks2y)|vac2]=K,
E^i(+)(y)=E^i1(+)(y)+E^i2(+)(y)=exp(iki1y)a^i1+exp(iki2y)a^i2.
psi(y, y)=Kψ|E^s(-)(y)E^i(-)(y)E^i(+)(y)E^s(+)(y)|ψ=K2[exp(iks1y)exp(iki1y)|vac+exp(iks2y)exp(iki2y)|vac]2=K[1+cos(ks1-ks2)(y-y)],
psi(y, y)K1+Vsin Δk(y-y)/2Δk(y-y)/22×cos(k1-k2)(y-y),
I(y)I(y)=|exp(iks1y)vs1+exp(iks2y)vs2|2|exp(iki1y)vi1+exp(iki2y)vi2|2=|vs1|2|vi1|2+|vs1|2|vi2|2+|vs2|2|vi1|2+|vs2|2|vi2|2+exp(-iθi)|vs1|2vi1vi2+exp(iθi)|vs1|2vi1vi2+exp(-iθi)×|vs2|2vi1vi2+exp(iθi)|vs2|2vi1vi2+exp(-iθs)|vi1|2vs1vs2+exp(iθs)×|vi1|2vs1vs2+exp(-iθs)×|vi2|2vs1vs2+exp(iθs)|vi2|2vs1vs2+exp[-i(θs+θi)]vs1vs2vi1vi2+exp[-i(θs-θi)]vs1vs2vi1vi2+exp[i(θs-θi)]vs1vs2vi1vi2+exp[i(θs+θi)]vs1vs2vi1vi2,
|vs1|2|vi2|2=|vs2|2|vi1|2=(|v|2)2,
|uw|2|u|2|w|2,
I(y)I(y)=|vs1|2|vi1|2+|vs2|2|vi2|2+O[(|v|2)2].

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