Abstract

We have measured the probability distribution of the phase difference ϕ2ϕ1 between two signal fields produced by parametric downconversion from two cascaded nonlinear crystals for several different values of the degree of mutual coherence. The experimental results are compared with theory, based on our operational approach to phase operators for quantum fields, and good agreement is obtained. These results are of interest because they constitute what we believe is the first experimental test of our phase theory for nonclassical states of light.

© 1994 Optical Society of America

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  1. See, for example, the special issue on quantum phase and phase dependent measurement,W. P. Schleich, S. M. Barnett, eds., Phys. Scr. T48,5–142 (1993).
  2. J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
    [CrossRef] [PubMed]
  3. J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
    [CrossRef] [PubMed]
  4. J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
    [CrossRef]
  5. D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
    [CrossRef] [PubMed]
  6. D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
    [CrossRef] [PubMed]
  7. D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
    [CrossRef]
  8. M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
    [CrossRef] [PubMed]
  9. D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
    [CrossRef] [PubMed]
  10. J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
    [CrossRef] [PubMed]
  11. X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
    [CrossRef] [PubMed]
  12. L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
    [CrossRef] [PubMed]
  13. J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
    [CrossRef] [PubMed]
  14. D. T. Pegg, S. M. Barnett, Phys. Rev. A 39, 1665 (1984).
    [CrossRef]
  15. S. M. Barnett, D. T. Pegg, J. Mod. Opt. 36, 7 (1989).
    [CrossRef]
  16. S. M. Barnett, D. T. Pegg, Phys. Rev. A 42, 6713 (1990).
    [CrossRef] [PubMed]

1993

J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
[CrossRef]

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
[CrossRef] [PubMed]

1992

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
[CrossRef] [PubMed]

1991

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
[CrossRef] [PubMed]

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
[CrossRef] [PubMed]

L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
[CrossRef] [PubMed]

1990

S. M. Barnett, D. T. Pegg, Phys. Rev. A 42, 6713 (1990).
[CrossRef] [PubMed]

1989

S. M. Barnett, D. T. Pegg, J. Mod. Opt. 36, 7 (1989).
[CrossRef]

1984

D. T. Pegg, S. M. Barnett, Phys. Rev. A 39, 1665 (1984).
[CrossRef]

Barnett, S. M.

S. M. Barnett, D. T. Pegg, Phys. Rev. A 42, 6713 (1990).
[CrossRef] [PubMed]

S. M. Barnett, D. T. Pegg, J. Mod. Opt. 36, 7 (1989).
[CrossRef]

D. T. Pegg, S. M. Barnett, Phys. Rev. A 39, 1665 (1984).
[CrossRef]

Beck, M.

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
[CrossRef] [PubMed]

Belsley, M.

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

Cooper, J.

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
[CrossRef] [PubMed]

Faridani, A.

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

Fougères, A.

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
[CrossRef]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
[CrossRef] [PubMed]

Mandel, L.

J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
[CrossRef]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
[CrossRef] [PubMed]

X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
[CrossRef] [PubMed]

L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
[CrossRef] [PubMed]

Noh, J. W.

J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
[CrossRef]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
[CrossRef] [PubMed]

Pegg, D. T.

S. M. Barnett, D. T. Pegg, Phys. Rev. A 42, 6713 (1990).
[CrossRef] [PubMed]

S. M. Barnett, D. T. Pegg, J. Mod. Opt. 36, 7 (1989).
[CrossRef]

D. T. Pegg, S. M. Barnett, Phys. Rev. A 39, 1665 (1984).
[CrossRef]

Raymer, M. G.

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

Smithey, D. T.

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

Wang, L. J.

X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
[CrossRef] [PubMed]

L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
[CrossRef] [PubMed]

Wedding, K.

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

Zou, X. Y.

X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
[CrossRef] [PubMed]

L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
[CrossRef] [PubMed]

J. Mod. Opt.

S. M. Barnett, D. T. Pegg, J. Mod. Opt. 36, 7 (1989).
[CrossRef]

Phys. Rev. A

S. M. Barnett, D. T. Pegg, Phys. Rev. A 42, 6713 (1990).
[CrossRef] [PubMed]

D. T. Pegg, S. M. Barnett, Phys. Rev. A 39, 1665 (1984).
[CrossRef]

L. J. Wang, X. Y. Zou, L. Mandel, Phys. Rev. A 44, 4614 (1991).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 46, 2840 (1992).
[CrossRef] [PubMed]

M. G. Raymer, J. Cooper, M. Beck, Phys. Rev. A 48, 4617 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. A 45, 424 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

X. Y. Zou, L. J. Wang, L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 67, 1426 (1991).
[CrossRef] [PubMed]

J. W. Noh, A. Fougères, L. Mandel, Phys. Rev. Lett. 71, 2579 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Belsley, K. Wedding, M. G. Raymer, Phys. Rev. Lett. 67, 2446 (1991).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

Phys. Scr. T

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, A. Faridani, Phys. Scr. T 48, 35 (1993).
[CrossRef]

J. W. Noh, A. Fougères, L. Mandel, Phys. Scr. T 48, 29 (1993).
[CrossRef]

Other

See, for example, the special issue on quantum phase and phase dependent measurement,W. P. Schleich, S. M. Barnett, eds., Phys. Scr. T48,5–142 (1993).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the experimental setup.

Fig. 2
Fig. 2

Experimental results for the probability distribution of the phase difference ϕ2ϕ1 superimposed upon the theoretical curve given by Eq. (9) with γ 12 ( P ) ( τ 0 ) = 0.32 for (a) T = 1 , (b) T = 0.43 , (c) T = 0 . The error bar above each bin corresponds to one standard deviation.

Equations (9)

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{ ( n ^ 4 - n ^ 3 ) + i ( n ^ 6 - n ^ 5 ) [ ( n ^ 4 - n ^ 3 ) 2 + ( n ^ 6 - n ^ 5 ) 2 ] 1 / 2 } C ( x )
P m ( ϕ 2 - ϕ 1 θ ) = 1 2 π - C ( x θ ) × exp [ - i x ( ϕ 2 - ϕ 1 - θ ) ] d x .
ρ ^ signal = M vac s 1 s 2 s 1 s 2 vac + f 1 2 I 1 1 s 1 0 s 2 s 2 0 s 1 1 + f 2 2 I 2 0 s 1 1 s 2 s 2 1 s 1 0 + T * f 1 * f 2 I 1 I 2 γ 12 ( P ) ( τ 0 ) × exp ( i θ 0 ) 1 s 2 0 s 1 s 1 1 s 2 0 + h . c .
C ( x θ ) = ¼ [ exp ( i π x ) + exp ( i 2 π x ) + exp ( - i π x / 2 ) + exp ( i π x / 2 ) ] + D exp ( - i θ ) + D * exp ( i θ ) 4 ( A + B ) × [ - exp ( i π x ) + exp ( i 2 π x ) + exp ( - i π x / 2 ) - i exp ( i π x / 2 ) ] ,
A = f 1 2 I 1 , B = f 2 2 I 2 , D = T * f 1 * f 2 ( I 1 I 2 ) 1 / 2 γ 12 ( P ) ( τ 0 ) exp ( i θ 0 ) .
P ( ϕ 2 - ϕ 1 ) = 1 2 π [ 1 + 2 D A + B cos ( ϕ 2 - ϕ 1 - arg D ) ] .
P PB ( ϕ 1 , ϕ 2 ) = ( s + 1 ) 2 ( 2 π ) 2 ϕ 2 ϕ 1 ρ ^ ϕ 1 ϕ 2 ,
P PB ( ϕ 1 , ϕ 2 ) = ( s + 1 ) 2 ( 2 π ) 2 × n 1 s n 1 s n 2 s n 2 s exp [ i ( n 1 - n 1 ) ϕ 1 ] exp [ i ( n 2 - n 2 ) ϕ 2 ] ( s + 1 ) 2 × n 2 n 1 ρ ^ n 1 n 2 = 1 ( 2 π ) 2 { M + A + B + D × exp [ - i ( ϕ 2 - ϕ 1 ) ] + D * exp [ i ( ϕ 2 - ϕ 1 ) ] } .
P PB ( ϕ 2 - ϕ 1 ) = 1 2 π [ 1 + 2 D cos ( ϕ 2 - ϕ 1 - arg D ) ] ,

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