Abstract

We describe how a two-mode, above-threshold, optical parametric oscillator can generate nonclassical states of light with a large average number of photons: the fluctuation spectrum of the various fields is calculated, and several quantities are shown to have squeezed fluctuations. In particular, a perfect quantum noise suppression is predicted on the difference between the intensities of the two generated beams. We describe an experiment designed to demonstrate such an effect and show how it can be used to generate high-intensity amplitude-squeezed states.

© 1987 Optical Society of America

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  1. D. Burnham and D. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
    [Crossref]
  2. S. Friberg, C. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
    [Crossref] [PubMed]
  3. B. Mollow and R. Glauber, Phys. Rev. 160, 1097 (1967).
    [Crossref]
  4. B. Mollow, Phys. Rev. A 8, 2684 (1973).
    [Crossref]
  5. C. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).
    [Crossref] [PubMed]
  6. L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [Crossref] [PubMed]
  7. E. Jakeman and J. Walker, Opt. Commun. 55, 219 (1985).
    [Crossref]
  8. R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
    [Crossref]
  9. J. Giordmaine and R. Miller, Phys. Rev. Lett. 14, 973 (1965).
    [Crossref]
  10. S. Harris, Proc. IEEE 57, 2096 (1969).
    [Crossref]
  11. R. Smith, “Optical parametric oscillators,” in Laser Handbook I, T. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1973).
  12. R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
    [Crossref]
  13. More precisely, α0, α1, and α2are the classical amplitudes associated with the annihilation operators a0, a1, and a2corresponding to the three-cavity eigenmodes coupled by the parametric interaction.
  14. M. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984).
    [Crossref]
  15. B. Yurke, Phys. Rev. A 29, 408 (1984); Phys. Rev. A 32, 300 (1985).
    [Crossref]
  16. M. Collett and D. Walls, Phys. Rev. A 32, 2887 (1985).
    [Crossref] [PubMed]
  17. M. Reid and D. Walls, Phys. Rev. A 31, 1622 (1985).
    [Crossref] [PubMed]
  18. This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).
  19. A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
    [Crossref]
  20. N. C. Wong and J. Hall, J. Opt. Soc. Am. B 2, 1527 (1985).
    [Crossref]
  21. M. Gehrtz, G. Bjorklund, and E. Whittaker, J. Opt. Soc. Am. B 2, 1510 (1985).
    [Crossref]
  22. Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
    [Crossref] [PubMed]

1987 (1)

A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
[Crossref]

1986 (3)

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
[Crossref] [PubMed]

1985 (8)

C. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

E. Jakeman and J. Walker, Opt. Commun. 55, 219 (1985).
[Crossref]

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

N. C. Wong and J. Hall, J. Opt. Soc. Am. B 2, 1527 (1985).
[Crossref]

M. Gehrtz, G. Bjorklund, and E. Whittaker, J. Opt. Soc. Am. B 2, 1510 (1985).
[Crossref]

M. Collett and D. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

M. Reid and D. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

1984 (2)

M. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

B. Yurke, Phys. Rev. A 29, 408 (1984); Phys. Rev. A 32, 300 (1985).
[Crossref]

1973 (1)

B. Mollow, Phys. Rev. A 8, 2684 (1973).
[Crossref]

1970 (1)

D. Burnham and D. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[Crossref]

1969 (1)

S. Harris, Proc. IEEE 57, 2096 (1969).
[Crossref]

1968 (1)

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

1967 (1)

B. Mollow and R. Glauber, Phys. Rev. 160, 1097 (1967).
[Crossref]

1965 (1)

J. Giordmaine and R. Miller, Phys. Rev. Lett. 14, 973 (1965).
[Crossref]

Bjorklund, G.

Brown, R.

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

Burnham, D.

D. Burnham and D. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[Crossref]

Collett, M.

M. Collett and D. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

M. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

Friberg, S.

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

Gardiner, C.

M. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

Gehrtz, M.

Gensic, J.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Giordmaine, J.

J. Giordmaine and R. Miller, Phys. Rev. Lett. 14, 973 (1965).
[Crossref]

Glauber, R.

B. Mollow and R. Glauber, Phys. Rev. 160, 1097 (1967).
[Crossref]

Hall, J.

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

N. C. Wong and J. Hall, J. Opt. Soc. Am. B 2, 1527 (1985).
[Crossref]

Harris, S.

S. Harris, Proc. IEEE 57, 2096 (1969).
[Crossref]

Heidmann, A.

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

Hong, C.

C. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

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

Horowicz, R.

A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
[Crossref]

Imoto, N.

Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
[Crossref] [PubMed]

Jakeman, E.

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

E. Jakeman and J. Walker, Opt. Commun. 55, 219 (1985).
[Crossref]

Kimble, H.

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Levinstein, H.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Machida, S.

Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
[Crossref] [PubMed]

Mandel, L.

C. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

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

Miller, R.

J. Giordmaine and R. Miller, Phys. Rev. Lett. 14, 973 (1965).
[Crossref]

Mollow, B.

B. Mollow, Phys. Rev. A 8, 2684 (1973).
[Crossref]

B. Mollow and R. Glauber, Phys. Rev. 160, 1097 (1967).
[Crossref]

Pike, E.

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

Pinard, M.

A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
[Crossref]

Raimond, J. M.

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

Rarity, J.

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

Reid, M.

M. Reid and D. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

Reynaud, S.

A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
[Crossref]

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

Rubin, J.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Singh, S.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Smith, R.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

R. Smith, “Optical parametric oscillators,” in Laser Handbook I, T. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1973).

Tapster, P.

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

Van Uitert, L.

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Walker, J.

E. Jakeman and J. Walker, Opt. Commun. 55, 219 (1985).
[Crossref]

Walls, D.

M. Collett and D. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

M. Reid and D. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

Weinberg, D.

D. Burnham and D. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[Crossref]

Whittaker, E.

Wong, N. C.

Wu, H.

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Wu, L.

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Yamamoto, Y.

Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
[Crossref] [PubMed]

Yurke, B.

B. Yurke, Phys. Rev. A 29, 408 (1984); Phys. Rev. A 32, 300 (1985).
[Crossref]

Zagury, N.

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

Appl. Phys. Lett. (1)

R. Smith, J. Gensic, H. Levinstein, J. Rubin, S. Singh, and L. Van Uitert, Appl. Phys. Lett. 12, 308 (1968).
[Crossref]

Europhys. Lett. (1)

R. Brown, E. Jakeman, E. Pike, J. Rarity, and P. Tapster, Europhys. Lett. 2, 279 (1986).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

This can be shown by a treatment analogous to the one given in A. Heidmann, J. M. Raimond, S. Reynaud, and N. Zagury, Opt. Commun. 61, 142 (1985).

A related derivation, but applied to the case of four-wave mixing, can be found in R. Horowicz, M. Pinard, and S. Reynaud, Opt. Commun. 61, 142 (1987).
[Crossref]

E. Jakeman and J. Walker, Opt. Commun. 55, 219 (1985).
[Crossref]

Phys. Rev. (1)

B. Mollow and R. Glauber, Phys. Rev. 160, 1097 (1967).
[Crossref]

Phys. Rev. A (7)

B. Mollow, Phys. Rev. A 8, 2684 (1973).
[Crossref]

C. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

M. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

B. Yurke, Phys. Rev. A 29, 408 (1984); Phys. Rev. A 32, 300 (1985).
[Crossref]

M. Collett and D. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

M. Reid and D. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

D. Burnham and D. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[Crossref]

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

L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

J. Giordmaine and R. Miller, Phys. Rev. Lett. 14, 973 (1965).
[Crossref]

Proc. IEEE (1)

S. Harris, Proc. IEEE 57, 2096 (1969).
[Crossref]

Other (2)

R. Smith, “Optical parametric oscillators,” in Laser Handbook I, T. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1973).

More precisely, α0, α1, and α2are the classical amplitudes associated with the annihilation operators a0, a1, and a2corresponding to the three-cavity eigenmodes coupled by the parametric interaction.

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Figures (3)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Theoretical predictions for the fluctuations V of the components r, q, and q0 calculated for ω = 0 as functions of the reduced pump parameter σ = (I/I0)1/2 (I, pump intensity; I0, threshold). The variance V = 1 corresponds to the vacuum fluctuations.

Fig. 3
Fig. 3

Theoretical prediction for the noise spectrum SI(ω) of the intensity difference between the two signal beams as a function of the frequency.

Equations (39)

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α ˙ 1 = 2 χ α 2 * α 0 - λ α 1 ,
α ˙ 2 = 2 χ α 1 * α 0 - λ α 2 ,
α ˙ 0 = - 2 χ α 1 α 2 - λ 0 α 0 + 1 2 λ 0 λ χ σ .
p j = α j + α j * ,
q j = - i ( α j - α j * ) .
p = 1 2 ( p 1 + p 2 ) , q = 1 2 ( q 1 + q 2 ) ; r = 1 2 ( p 1 - p 2 ) , s = 1 2 ( q 1 - q 2 ) .
p ˙ = χ ( p 0 p + q 0 q ) - λ p , q ˙ = χ ( q 0 p - p 0 q ) - λ q , r ˙ = - χ ( p 0 r + q 0 s ) - λ r , s ˙ = - χ ( q 0 r - p 0 s ) - λ s , p ˙ 0 = - 1 2 χ ( p 2 - r 2 - q 2 + s 2 ) - λ 0 p 0 + 1 χ λ 0 λ σ , q ˙ 0 = - χ ( p q - r s ) - λ 0 q 0 .
χ p ¯ 0 p ¯ = λ p ¯ ,
1 2 χ p ¯ 2 + λ 0 p ¯ 0 - 1 χ λ 0 λ σ = 0 ,
p ¯ = 0 ,             p ¯ 0 = λ / χ σ ,
p ¯ 0 = λ χ ,
p ¯ 2 = [ ( 2 λ 0 λ ) / χ 2 ] ( σ - 1 ) ,
σ = I / I 0 ,
δ p ˙ = χ ( p ¯ 0 δ p + p ¯ δ p 0 ) - λ δ p + t P in ,
δ q ˙ = χ ( p ¯ δ q 0 - p ¯ 0 δ q ) - λ δ q + t Q in ,
δ r ˙ = - χ p ¯ 0 δ r - λ δ r + t R in ,
δ s ˙ = χ p ¯ 0 δ s - λ δ s + t S in ,
δ p ˙ = - χ p ¯ δ p - λ 0 δ p 0 + t 0 P 0 in ,
δ q ˙ 0 = - χ p ¯ δ q - λ 0 δ q 0 + t 0 Q 0 in ,
t 2 = 2 λ τ ,             t 0 2 = 2 λ 0 τ ,
σ p ˙ = χ ( p ¯ 0 δ p + p ¯ δ p 0 ) + λ δ p - t P out ,
δ q ˙ = χ ( p ¯ δ q 0 - p ¯ 0 δ q ) + λ δ q - t Q out ,
δ r ˙ = - χ p ¯ 0 δ r + λ δ r - t R out ,
δ s ˙ = χ p ¯ 0 δ s + λ δ s - t S out ,
δ p ˙ 0 = - χ p ¯ δ p + λ 0 δ p 0 - t 0 P 0 out ,
δ q ˙ 0 = - χ p ¯ δ q + λ 0 δ q 0 - t 0 Q 0 out ,
( i ω + χ p ¯ 0 + λ ) δ r ˜ ( ω ) = t R in ( ω ) ,
( i ω + χ p ¯ 0 - λ ) δ r ˜ ( ω ) = - t R ˜ out ( ω ) ,
R ˜ out ( ω ) = - i ω + χ p ¯ 0 - λ i ω + χ p ¯ 0 + λ R in ( ω ) .
R ˜ in ( ω ) R ˜ in ( - ω ) = 1.
V R = R ˜ out ( ω ) R ˜ out ( - ω ) = ω 2 + ( χ p ¯ 0 - λ ) 2 ω 2 + ( χ p ¯ 0 + λ ) 2 .
V R ( ω = 0 ) = ( 1 - σ 1 + σ ) 2             for σ < 1.
V R ( ω = 0 ) = 0             for σ < 1.
V S = ω 2 + ( χ p ¯ 0 + λ ) 2 ω 2 + ( χ p ¯ 0 - λ ) 2 = 1 V R .
V P = ( 1 + σ 1 - σ ) 2 ,             V Q = ( 1 - σ 1 + σ ) 2 ,             V P 0 = V Q 0 = 1
V P = 1 + 1 ( σ - 1 ) 2 , V Q = σ 2 - 1 σ 2 , V P 0 = σ + 1 σ - 1 . V Q 0 = 1 - 2 ( σ - 1 ) σ 2 .
I = I 1 - I 2 p 1 2 + q 1 2 - p 2 2 - q 2 2 .
δ I p ¯ 1 δ p 1 - p ¯ 2 δ p 2 = 2 p ¯ 1 δ r .
S I ( ω ) = S 0 ω 2 ω 2 + 4 λ 2 ,

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