Abstract

A two-photon correlation experiment that was proposed by Franson [ Phys. Rev. Lett. 62, 2205 ( 1989)] and was recently carried out is analyzed in terms of electromagnetic waves in order to see whether the observed fourth-order interference effects can be explained classically. The conclusion is that, although a classical field can give rise to such interference effects, no classical field can actually account for the observations reported.

© 1990 Optical Society of America

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References

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  1. C. O. Alley and T. H. Shih, Phys. Rev. Lett. 61, 2921 (1988).
    [CrossRef]
  2. Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).
    [CrossRef] [PubMed]
  3. Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
    [CrossRef]
  4. Z. Y. Ou and L. Mandel, Quantum Opt. 2, 71 (1990).
    [CrossRef]
  5. P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
    [CrossRef] [PubMed]
  6. M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
    [CrossRef] [PubMed]
  7. Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
    [CrossRef] [PubMed]
  8. J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
    [CrossRef] [PubMed]
  9. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
    [CrossRef] [PubMed]
  10. P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
    [CrossRef] [PubMed]
  11. R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).
    [CrossRef] [PubMed]
  12. Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).
    [CrossRef] [PubMed]
  13. See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.
  14. S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
    [CrossRef] [PubMed]

1990 (3)

Z. Y. Ou and L. Mandel, Quantum Opt. 2, 71 (1990).
[CrossRef]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

1989 (4)

M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
[CrossRef] [PubMed]

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).
[CrossRef] [PubMed]

1988 (4)

P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
[CrossRef] [PubMed]

C. O. Alley and T. H. Shih, Phys. Rev. Lett. 61, 2921 (1988).
[CrossRef]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
[CrossRef]

1987 (1)

R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

1985 (1)

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Alley, C. O.

C. O. Alley and T. H. Shih, Phys. Rev. Lett. 61, 2921 (1988).
[CrossRef]

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

Chiao, R. Y.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Franson, J. D.

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef] [PubMed]

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Ghosh, R.

R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

Grangier, P.

P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
[CrossRef] [PubMed]

Hong, C. K.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
[CrossRef]

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Horne, M. A.

M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
[CrossRef] [PubMed]

Kwiat, P. G.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Mandel, L.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, Quantum Opt. 2, 71 (1990).
[CrossRef]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
[CrossRef]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).
[CrossRef] [PubMed]

R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Nathel, H.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, Quantum Opt. 2, 71 (1990).
[CrossRef]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
[CrossRef]

Potsek, M. J.

P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
[CrossRef] [PubMed]

Shih, T. H.

C. O. Alley and T. H. Shih, Phys. Rev. Lett. 61, 2921 (1988).
[CrossRef]

Shimony, A.

M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
[CrossRef] [PubMed]

Vareka, W. A.

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Wang, L. J.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
[CrossRef] [PubMed]

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

Yurke, B.

P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
[CrossRef] [PubMed]

Zeilinger, A.

M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
[CrossRef] [PubMed]

Zou, X. Y.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

Opt. Commun. (1)

Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 67, 159 (1988).
[CrossRef]

Phys. Rev. A (3)

P. Grangier, M. J. Potsek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989).
[CrossRef] [PubMed]

P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (8)

R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990).
[CrossRef] [PubMed]

M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).
[CrossRef] [PubMed]

C. O. Alley and T. H. Shih, Phys. Rev. Lett. 61, 2921 (1988).
[CrossRef]

Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).
[CrossRef] [PubMed]

Quantum Opt. (1)

Z. Y. Ou and L. Mandel, Quantum Opt. 2, 71 (1990).
[CrossRef]

Other (1)

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

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Figures (1)

Fig. 1
Fig. 1

Outline of the experiment being discussed.

Equations (35)

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W 1 ( t ) = α V 1 ( t ) + β V 1 ( t + T + τ 1 ) , W 2 ( t ) = α V 2 ( t ) + β V 2 ( t + T + τ 2 ) ,
τ 1 , τ 2 T C , T CI
V 1 ( t + τ 1 ) = V 1 ( t ) exp ( i ω 1 τ 1 ) , V 2 ( t + τ 2 ) = V 2 ( t ) exp ( i ω 2 τ 2 ) ,
J 1 ( t ) = | W 1 ( t ) | 2 = | α | 2 I 1 ( t ) + | β | 2 I 1 ( t + T ) + α * β V 1 * ( t ) V 1 ( t + T ) exp ( i ω 1 τ 1 ) + c . c . , J 2 ( t ) = | W 2 ( t ) | 2 = | α | 2 I 2 ( t ) + | β | 2 I 2 ( t + T ) + α * β V 2 * ( t ) V 2 ( t + T ) exp ( i ω 2 τ 2 ) + c . c .
J 1 ( t ) = | α | 2 I 1 ( t ) + | β | 2 I 1 ( t + T ) , J 2 ( t ) = | α | 2 I 2 ( t ) + | β | 2 I 2 ( t + T ) ,
P 12 ( t , τ ) = J 1 ( t ) J 2 ( t + τ ) ,
P 12 ( t , τ ) = [ | α | 2 I 1 ( t ) + | β | 2 I 1 ( t + T ) + α * β V 1 * ( t ) V 1 ( t + T ) exp ( i ω 1 τ 1 ) + c . c . ] × [ | α | 2 I 2 ( t + τ ) + | β | 2 I 2 ( t + τ + T ) + α * β V 2 * ( t + τ ) V 2 ( t + τ + T ) × exp ( i ω 2 τ 2 ) + c . c . ] .
P 12 ( t , τ ) = | α | 4 I 1 ( t ) I 2 ( t + τ ) + | β | 4 I 1 ( t + T ) I 2 ( t + τ + T ) + | α | 2 | β | 2 [ I 1 ( t ) I 2 ( t + τ + T ) + I 1 ( t + T ) I 2 ( t + τ ) ] + α * 2 β 2 V 1 * ( t ) V 1 ( t + T ) V 2 * ( t + τ ) V 2 ( t + τ + T ) × exp [ i ( ω 1 τ 1 + ω 2 τ 2 ) ] + c . c . + | α | 2 | β | 2 V 1 * ( t ) V 1 ( t + T ) V 2 * ( t + τ + T ) V 2 ( t + τ ) × exp [ i ( ω 2 τ 2 ω 1 τ 1 ) ] + c . c .
R 12 ( t ) = T R / 2 T R / 2 P 12 ( t , τ ) d τ ,
R ¯ 12 = 1 T M T M / 2 T M / 2 d t R 12 ( t ) = 1 T M T M / 2 T M / 2 d t T R / 2 T R / 2 d τ P 12 ( t , τ ) .
V i * ( t ) V j * ( t + τ ) = Γ i j ( 2 , 0 ) ( t , τ ) ( i , j = 1 , 2 ) , V i * ( t ) V j ( t + τ ) = Γ i j ( 1 , 1 ) ( t , τ ) ( i , j = 1 , 2 ) .
R ¯ 12 = 1 T M T M / 2 T M / 2 d t T R / 2 T R / 2 d τ | α | 4 I 1 ( t ) I 2 ( t + τ ) + | β | 4 I 1 ( t + T ) I 2 ( t + T + τ ) + | a | 2 | β | 2 [ I 1 ( t ) I 2 ( t + T + τ ) + I 1 ( t + T ) I 2 ( t + τ ) ] + α * 2 β 2 T M T M / 2 T M / 2 d t T / 2 T R / 2 d τ × V 1 * ( t ) V 1 ( t + T ) V 2 * ( t + τ ) V 2 ( t + τ + T ) × exp [ i ( ω 1 τ 1 + ω 2 τ 2 ) ] + c . c . + | α | 2 | β | 2 T M T M / 2 T M / 2 d t T R / 2 T R / 2 d τ × V l * ( t ) V 1 ( t + T ) V 2 ( t + τ ) V 2 * ( t + τ + T ) × exp [ i ( ω 2 τ 2 ω 2 τ 1 ) ] + c . c .
V 1 * ( t ) V 2 * ( t + τ ) V 1 ( t + T ) V 2 ( t + T + τ ) = V 1 * ( t ) V 2 * ( t + T ) V 1 ( t + T ) V 2 ( t + T + τ ) ,
1 T M | T M / 2 T M / 2 d t d τ f 1 * ( t ) f 2 ( t + τ ) | 1 2 T M T M / 2 T M / 2 d t × d τ [ f 1 * ( t ) f 1 ( t + τ ) + f 2 * ( t ) f 2 ( t + τ ) ] .
f 1 ( t ) = 0 = f 2 ( t ) ,
f 1 * ( t ) f 1 ( t + τ ) = V 1 * ( t ) V 1 * ( t + T + τ ) × V 1 ( t + T ) V 1 ( t + τ ) = Γ 11 ( 2 , 0 ) ( t , T + τ ) Γ 11 ( 2 , 0 ) * ( t + T , τ T ) .
Modulation Amplitude 1 = 2 | α | 2 | β | 2 T M | T M / 2 T M / 2 d t T R / 2 T R / 2 d τ × V 1 * ( t ) V 1 ( t + T ) V 2 * ( t + τ ) V 2 ( t + τ + T ) | | α | 2 | β | 2 T M T M / 2 T M / 2 d t T C [ I 1 ( t ) I 1 ( t + T ) + I 2 ( t ) I 2 ( t + T ) ] ,
Modulation Amplitude 1 | α | 2 | β | 2 T C ( I ¯ 1 I ¯ 1 + I ¯ 2 I ¯ 2 ) = | α | 2 | β | 2 T C ( I ¯ 1 2 + I ¯ 2 2 ) .
Modulation Amplitude 2 | α | 2 | β | 2 T C ( I ¯ 1 2 + I ¯ 2 2 ) .
1 T M T M / 2 T M / 2 d t T R / 2 T R / 2 d τ I 1 ( t ) I 2 ( t + τ ) = 1 T M T M / 2 T M / 2 d t [ T R / 2 τ ˜ T CI / 2 d τ I 1 ( t ) I 2 ( t + τ ) + τ ˜ + T CI / 2 T R / 2 d τ I 1 ( t ) I 2 ( t + τ ) + τ ˜ T CI / 2 τ ˜ + T CI / 2 d τ I 1 ( t ) I 2 ( t + τ ) ] ( T R / 2 τ ˜ T CI / 2 d τ + τ ˜ + T CI / 2 T R / 2 d τ ) I 1 ( t ) I 2 ( t + τ ) ¯
I 1 ( t ) I 2 ( t + τ ) ¯ = I ¯ 1 I ¯ 2
1 T M T M / 2 T M / 2 d t T R / 2 T R / 2 d τ I 1 ( t ) I 2 ( t + τ ) ( T R T CI ) I ¯ 1 I ¯ 2 .
Unmodulated Term ( T R T CI ) ( | α | 2 + | β | 2 ) I ¯ 1 I ¯ 2 .
V | α | 2 | β | 2 ( | α | 2 + | β | 2 ) 2 I ¯ 1 2 + I ¯ 2 2 I ¯ 1 I ¯ 2 ( T C T R T CI ) ,
V [ T C / ( T R T CI ) ] .
T / 2 T / 2 d t 1 d t 2 g * ( t 1 ) g ( t 2 ) W * ( t 1 ) W ( t 2 ) 0.
T / 2 T / 2 d t 1 T / 2 t 1 T / 2 t 1 d τ W * ( t 1 ) W ( t 1 + τ ) e i ω τ 0.
d τ W * ( t ) W ( t + τ ) e i ω τ = Φ W ( 1 , 1 ) ( t , ω )
T / 2 T / 2 d t Φ W ( 1 , 1 ) ( t , ω ) 0.
W ( t ) = f 1 ( t ) f 2 ( t ) e i ϕ ,
0 1 T T / 2 T / 2 d t d τ [ f 1 * ( t ) f 1 ( t + τ ) + f 2 * ( t ) f 2 ( t + τ ) f 1 * ( t ) f 2 ( t + τ ) e i ϕ f 2 * ( t ) f 1 ( t + τ ) e ] e i ω τ , 0 1 T T / 2 T / 2 d t { Φ 11 ( t , ω ) + Φ 22 ( t , ω ) 2 Re [ Φ 12 ( t , ω ) e i Φ ] } ,
Φ i j ( t , ω ) f i * ( t ) f j ( t + τ ) e i ω τ d τ ( i , j = 1 , 2 ) .
0 1 T T / 2 T / 2 d t ( Φ 11 ( t , 0 ) + Φ 22 ( t , 0 ) 2 Re { Φ 12 ( t , 0 ) × exp [ i ( ω 1 τ 1 + ω 2 τ 2 ) ] } ) ,
1 T T / 2 T / 2 d t Re d τ f 1 * ( t ) f 2 ( t + τ ) exp [ i ( ω 1 τ 1 + ω 2 τ 2 ) ] 1 2 T T / 2 T / 2 d t d τ [ f 1 * ( t ) f 1 ( t + τ ) + f 2 * ( t ) f 2 ( t + τ ) ] .
1 T | T / 2 T / 2 d t d τ f 1 * ( t ) f 2 ( t + τ ) | 1 2 T T / 2 T / 2 d t d τ [ f 1 * ( t ) f 1 ( t + τ ) + f 2 * ( t ) f 2 ( t + τ ) ] .

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