Abstract

The mutual influence of second-harmonic generation and nondegenerate downconversion inside the same cavity is studied. The threshold behavior and the stability of a semiclassical solution with all four modes oscillating resonantly is investigated. It is shown that under certain conditions the self-pulsing instability that occurs in second-harmonic generation is absent in the combined system. The squeezing spectra for the quadrature-phase amplitudes in the second harmonic and the fundamental, as well as the photon statistics in the downconverted modes, are derived analytically. As one of the various nonclassical features of this system, the second harmonic is found to exhibit excellent noise suppression at very large frequencies, with the frequency component of best squeezing being tunable by the pump parameter. The intricate interaction of the two nonlinear processes can give rise to twin beams of correlated photons, with sub-Poissonian statistics in each beam. These results are contrasted with the results for each nonlinear process by itself.

© 1995 Optical Society of America

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  1. S. Schiller and R. L. Byer, J. Opt. Soc. Am. B 10, 1696 (1993).
    [Crossref]
  2. G. T. Moore, K. Koch, and E. C. Cheung, Opt. Commun. 113, 463 (1995).
    [Crossref]
  3. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanheerzeele, Opt. Lett. 17, 28 (1992); R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, Opt. Lett. 18, 194 (1993).
    [Crossref] [PubMed]
  4. D. Y. Kim, W. E. Torruellas, J. Kang, C. Bosshard, G. I. Stegeman, P. Vidakovic, J. Zyss, W. E. Moerner, R. Twieg, and G. Bjorklund, Opt. Lett. 19, 868 (1994); S. Nitti, H. M. Tan, G. P. Banfi, and V. Degorgio, Opt. Commun. 106, 263 (1994).
    [Crossref] [PubMed]
  5. C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 650 (1993).
    [Crossref]
  6. M. A. Marte, Phys. Rev. A 49, 3166 (1994).
    [Crossref]
  7. P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980).
    [Crossref]
  8. P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 28, 211 (1981).
    [Crossref]
  9. P. Mandel and T. Erneux, Opt. Acta 29, 7 (1982).
    [Crossref]
  10. M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
    [Crossref] [PubMed]
  11. K. Fiedlesr, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, Opt. Lett. 18, 1786 (1993).
    [Crossref]
  12. R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
    [Crossref] [PubMed]
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  14. K. J. McNeil and C. W. Gardiner, Phys. Rev. A 28, 1560 (1983).
    [Crossref]
  15. S. Reynaud, C. Fabre, and E. Giacobino, J. Opt. Soc. Am. B 4, 1520 (1987).
    [Crossref]
  16. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
    [Crossref] [PubMed]
  17. M. J. Collett and R. Loudon, J. Opt. Soc. Am. B 4, 1525 (1987).
    [Crossref]
  18. S. F. Pereira, M. Xiao, H. J. Kimble, and J. Hall, Phys. Rev. A 38, 4931 (1988).
    [Crossref] [PubMed]
  19. A. S. Lane, M. D. Reid, and D. F. Walls, Phys. Rev. A 38, 788 (1988).
    [Crossref] [PubMed]
  20. J. J. Slosser and G. J. Milburn, Phys. Rev. A 50, 793 (1994).
    [Crossref] [PubMed]
  21. G. Björk and Y. Yamamoto, Phys. Rev. A 37, 125 (1988).
    [Crossref]
  22. J. Martinez-Linares and P. Garcia-Fernandes, J. Mod. Opt. 41, 67 (1994).
    [Crossref]
  23. A. Eschmann and M. D. Reid, Phys. Rev. A 49, 2881 (1994).
    [Crossref] [PubMed]
  24. Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
    [Crossref] [PubMed]
  25. S. Friberg and L. Mandel, Opt. Commun. 48, 439 (1984).
    [Crossref]
  26. C. Cabrillo and F. J. Bermejo, Phys. Lett. A 170, 300 (1992); C. Cabrillo and F. J. Bermejo, Phys. Rev. A 48, 2433 (1993).
    [Crossref] [PubMed]
  27. K. M. Gheri, D. F. Walls, and M. A. M. Marte, Phys. Rev. A 46, 6002 (1992).
    [Crossref] [PubMed]
  28. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
  29. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983); C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1992).
    [Crossref]
  30. N. P. Pettiaux, P. Mandel, and C. Fabre, Phys. Rev. Lett. 66, 1838 (1991).
    [Crossref] [PubMed]
  31. M. A. M. Marte, Phys. Rev. Lett. 76, 4815 (1995).
    [Crossref]
  32. X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991).
    [Crossref] [PubMed]
  33. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
    [Crossref] [PubMed]
  34. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1994).
    [Crossref]
  35. W. J. Munro and M. D. Reid, Phys. Rev. A 50, 3661 (1994).
    [Crossref] [PubMed]
  36. P. Kwiat, H. Weinfurter, T. Herzog, and A. Zeilinger, Phys. Rev. Lett. 72, 624 (1994).
  37. P. D. Townsend, J. G. Rarity, and P. R. Tapster, Electron. Lett. 29, 1291 (1993).
    [Crossref]
  38. A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992).
    [Crossref] [PubMed]
  39. P. D. Drummond and M. D. Reid, Phys. Rev. A 37, 1806 (1988).
    [Crossref] [PubMed]
  40. C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
    [Crossref]

1995 (2)

G. T. Moore, K. Koch, and E. C. Cheung, Opt. Commun. 113, 463 (1995).
[Crossref]

M. A. M. Marte, Phys. Rev. Lett. 76, 4815 (1995).
[Crossref]

1994 (10)

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1994).
[Crossref]

W. J. Munro and M. D. Reid, Phys. Rev. A 50, 3661 (1994).
[Crossref] [PubMed]

P. Kwiat, H. Weinfurter, T. Herzog, and A. Zeilinger, Phys. Rev. Lett. 72, 624 (1994).

J. Martinez-Linares and P. Garcia-Fernandes, J. Mod. Opt. 41, 67 (1994).
[Crossref]

A. Eschmann and M. D. Reid, Phys. Rev. A 49, 2881 (1994).
[Crossref] [PubMed]

Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
[Crossref] [PubMed]

J. J. Slosser and G. J. Milburn, Phys. Rev. A 50, 793 (1994).
[Crossref] [PubMed]

D. Y. Kim, W. E. Torruellas, J. Kang, C. Bosshard, G. I. Stegeman, P. Vidakovic, J. Zyss, W. E. Moerner, R. Twieg, and G. Bjorklund, Opt. Lett. 19, 868 (1994); S. Nitti, H. M. Tan, G. P. Banfi, and V. Degorgio, Opt. Commun. 106, 263 (1994).
[Crossref] [PubMed]

M. A. Marte, Phys. Rev. A 49, 3166 (1994).
[Crossref]

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

1993 (4)

S. Schiller and R. L. Byer, J. Opt. Soc. Am. B 10, 1696 (1993).
[Crossref]

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 650 (1993).
[Crossref]

P. D. Townsend, J. G. Rarity, and P. R. Tapster, Electron. Lett. 29, 1291 (1993).
[Crossref]

K. Fiedlesr, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, Opt. Lett. 18, 1786 (1993).
[Crossref]

1992 (5)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992).
[Crossref] [PubMed]

C. Cabrillo and F. J. Bermejo, Phys. Lett. A 170, 300 (1992); C. Cabrillo and F. J. Bermejo, Phys. Rev. A 48, 2433 (1993).
[Crossref] [PubMed]

K. M. Gheri, D. F. Walls, and M. A. M. Marte, Phys. Rev. A 46, 6002 (1992).
[Crossref] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanheerzeele, Opt. Lett. 17, 28 (1992); R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, Opt. Lett. 18, 194 (1993).
[Crossref] [PubMed]

1991 (2)

N. P. Pettiaux, P. Mandel, and C. Fabre, Phys. Rev. Lett. 66, 1838 (1991).
[Crossref] [PubMed]

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

1989 (1)

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
[Crossref]

1988 (4)

P. D. Drummond and M. D. Reid, Phys. Rev. A 37, 1806 (1988).
[Crossref] [PubMed]

G. Björk and Y. Yamamoto, Phys. Rev. A 37, 125 (1988).
[Crossref]

S. F. Pereira, M. Xiao, H. J. Kimble, and J. Hall, Phys. Rev. A 38, 4931 (1988).
[Crossref] [PubMed]

A. S. Lane, M. D. Reid, and D. F. Walls, Phys. Rev. A 38, 788 (1988).
[Crossref] [PubMed]

1987 (3)

1985 (1)

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

1984 (1)

S. Friberg and L. Mandel, Opt. Commun. 48, 439 (1984).
[Crossref]

1983 (1)

K. J. McNeil and C. W. Gardiner, Phys. Rev. A 28, 1560 (1983).
[Crossref]

1982 (1)

P. Mandel and T. Erneux, Opt. Acta 29, 7 (1982).
[Crossref]

1981 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 28, 211 (1981).
[Crossref]

1980 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980).
[Crossref]

Aitchison, J. S.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 650 (1993).
[Crossref]

Arnold, J. M.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 650 (1993).
[Crossref]

Bachor, H. A.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

Bermejo, F. J.

C. Cabrillo and F. J. Bermejo, Phys. Lett. A 170, 300 (1992); C. Cabrillo and F. J. Bermejo, Phys. Rev. A 48, 2433 (1993).
[Crossref] [PubMed]

Björk, G.

G. Björk and Y. Yamamoto, Phys. Rev. A 37, 125 (1988).
[Crossref]

Bjorklund, G.

Bosshard, C.

Byer, R. L.

Cabrillo, C.

C. Cabrillo and F. J. Bermejo, Phys. Lett. A 170, 300 (1992); C. Cabrillo and F. J. Bermejo, Phys. Rev. A 48, 2433 (1993).
[Crossref] [PubMed]

Camy, G.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Cheung, E. C.

G. T. Moore, K. Koch, and E. C. Cheung, Opt. Commun. 113, 463 (1995).
[Crossref]

Collett, M.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

Collett, M. J.

M. J. Collett and R. Loudon, J. Opt. Soc. Am. B 4, 1525 (1987).
[Crossref]

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

DeSalvo, R.

Drummond, P. D.

P. D. Drummond and M. D. Reid, Phys. Rev. A 37, 1806 (1988).
[Crossref] [PubMed]

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 28, 211 (1981).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980).
[Crossref]

Ekert, A. K.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1994).
[Crossref]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992).
[Crossref] [PubMed]

Erneux, T.

P. Mandel and T. Erneux, Opt. Acta 29, 7 (1982).
[Crossref]

Eschmann, A.

A. Eschmann and M. D. Reid, Phys. Rev. A 49, 2881 (1994).
[Crossref] [PubMed]

Fabre, C.

N. P. Pettiaux, P. Mandel, and C. Fabre, Phys. Rev. Lett. 66, 1838 (1991).
[Crossref] [PubMed]

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
[Crossref]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

S. Reynaud, C. Fabre, and E. Giacobino, J. Opt. Soc. Am. B 4, 1520 (1987).
[Crossref]

Fiedler, K.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

Fiedlesr, K.

Friberg, S.

S. Friberg and L. Mandel, Opt. Commun. 48, 439 (1984).
[Crossref]

Garcia-Fernandes, P.

J. Martinez-Linares and P. Garcia-Fernandes, J. Mod. Opt. 41, 67 (1994).
[Crossref]

Gardiner, C. W.

K. J. McNeil and C. W. Gardiner, Phys. Rev. A 28, 1560 (1983).
[Crossref]

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983); C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1992).
[Crossref]

Gheri, K. M.

K. M. Gheri, D. F. Walls, and M. A. M. Marte, Phys. Rev. A 46, 6002 (1992).
[Crossref] [PubMed]

Giacobino, E.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
[Crossref]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

S. Reynaud, C. Fabre, and E. Giacobino, J. Opt. Soc. Am. B 4, 1520 (1987).
[Crossref]

Hagan, D. J.

Hall, J.

S. F. Pereira, M. Xiao, H. J. Kimble, and J. Hall, Phys. Rev. A 38, 4931 (1988).
[Crossref] [PubMed]

Heidmann, A.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
[Crossref]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Herzog, T.

P. Kwiat, H. Weinfurter, T. Herzog, and A. Zeilinger, Phys. Rev. Lett. 72, 624 (1994).

Horne, M. A.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1994).
[Crossref]

Horowicz, R. J.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Ironside, C. N.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 650 (1993).
[Crossref]

Kang, J.

Kim, D. Y.

Kimble, H. J.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

S. F. Pereira, M. Xiao, H. J. Kimble, and J. Hall, Phys. Rev. A 38, 4931 (1988).
[Crossref] [PubMed]

H. J. Kimble, “Quantum fluctuations in quantum optics, squeezing and related phenomena,”C. Fabre and S. Reynaud, “Quantum noise in optical systems: a semiclassical approach,” in Les Houches Lecture Notes, Session LIII, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992).

Koch, K.

G. T. Moore, K. Koch, and E. C. Cheung, Opt. Commun. 113, 463 (1995).
[Crossref]

Kürz, P.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

K. Fiedlesr, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, Opt. Lett. 18, 1786 (1993).
[Crossref]

Kwiat, P.

P. Kwiat, H. Weinfurter, T. Herzog, and A. Zeilinger, Phys. Rev. Lett. 72, 624 (1994).

Lane, A. S.

A. S. Lane, M. D. Reid, and D. F. Walls, Phys. Rev. A 38, 788 (1988).
[Crossref] [PubMed]

Loudon, R.

Mandel, L.

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

S. Friberg and L. Mandel, Opt. Commun. 48, 439 (1984).
[Crossref]

Mandel, P.

N. P. Pettiaux, P. Mandel, and C. Fabre, Phys. Rev. Lett. 66, 1838 (1991).
[Crossref] [PubMed]

P. Mandel and T. Erneux, Opt. Acta 29, 7 (1982).
[Crossref]

Marte, M. A.

M. A. Marte, Phys. Rev. A 49, 3166 (1994).
[Crossref]

Marte, M. A. M.

M. A. M. Marte, Phys. Rev. Lett. 76, 4815 (1995).
[Crossref]

K. M. Gheri, D. F. Walls, and M. A. M. Marte, Phys. Rev. A 46, 6002 (1992).
[Crossref] [PubMed]

Martinez-Linares, J.

J. Martinez-Linares and P. Garcia-Fernandes, J. Mod. Opt. 41, 67 (1994).
[Crossref]

McNeil, K. J.

K. J. McNeil and C. W. Gardiner, Phys. Rev. A 28, 1560 (1983).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 28, 211 (1981).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980).
[Crossref]

Milburn, G. J.

J. J. Slosser and G. J. Milburn, Phys. Rev. A 50, 793 (1994).
[Crossref] [PubMed]

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

Mlynek, J.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

K. Fiedlesr, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, Opt. Lett. 18, 1786 (1993).
[Crossref]

Moerner, W. E.

Moore, G. T.

G. T. Moore, K. Koch, and E. C. Cheung, Opt. Commun. 113, 463 (1995).
[Crossref]

Munro, W. J.

W. J. Munro and M. D. Reid, Phys. Rev. A 50, 3661 (1994).
[Crossref] [PubMed]

Ou, Z. Y.

Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
[Crossref] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

Palma, G. M.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992).
[Crossref] [PubMed]

Paschotta, R.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H. A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[Crossref] [PubMed]

K. Fiedlesr, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, Opt. Lett. 18, 1786 (1993).
[Crossref]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

S. F. Pereira, M. Xiao, H. J. Kimble, and J. Hall, Phys. Rev. A 38, 4931 (1988).
[Crossref] [PubMed]

Pettiaux, N. P.

N. P. Pettiaux, P. Mandel, and C. Fabre, Phys. Rev. Lett. 66, 1838 (1991).
[Crossref] [PubMed]

Rarity, J. G.

P. D. Townsend, J. G. Rarity, and P. R. Tapster, Electron. Lett. 29, 1291 (1993).
[Crossref]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992).
[Crossref] [PubMed]

Reid, M. D.

W. J. Munro and M. D. Reid, Phys. Rev. A 50, 3661 (1994).
[Crossref] [PubMed]

A. Eschmann and M. D. Reid, Phys. Rev. A 49, 2881 (1994).
[Crossref] [PubMed]

A. S. Lane, M. D. Reid, and D. F. Walls, Phys. Rev. A 38, 788 (1988).
[Crossref] [PubMed]

P. D. Drummond and M. D. Reid, Phys. Rev. A 37, 1806 (1988).
[Crossref] [PubMed]

Reynaud, S.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, J. Phys. (Paris) 50, 1209 (1989).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the nonlinear χ(2) cascade in the cavity. The pump field 1 of frequency ω1 is frequency doubled, the second harmonic is downconverted into a pair of unequal photons of frequencies ω± = ω1 ± Δ, and vice versa.

Fig. 2
Fig. 2

Stability region for quadruply resonant above-threshold oscillation. The region of stability is bounded on the left by the vertical line 1/thresh = 1 corresponding to the threshold condition. The solution is stable, provided that the relative ratio r of the nonlinear couplings is smaller than a critical value rmax; for r > rmax stability regions may extend into the upper right-hand quadrant, with the exact boundary depending on rmax. The top, middle, and bottom curves correspond to γ2/γ1 = 5, 10, 100, respectively.

Fig. 3
Fig. 3

Time-evolution sequence of nonlinear dynamics. A numerical hysteresis loop for the self-pulsing behavior is illustrated by variation of the parameters, as indicated in the stability diagram (with points 3 and 5 and points 2 and 6 being identical).

Fig. 4
Fig. 4

Squeezing variance for the amplitude noise quadrature of the second harmonic. Good squeezing is seen to occur in satellite peaks that move to larger frequencies with increasing e1 = 1/thresh. The parameters are γ2 = 20, γ1 = 5 (in units of χ), s = γ ¯/γ1 = 1, r = rmax = 5.

Fig. 5
Fig. 5

Comparison of the single-beam intensity fluctuation spectra Vsingle for the cascaded system (solid curve), for the NDOPO (dotted curve), and for the NDOPO with adiabatic elimination of the pump field (dashed curve), with all the intensities assumed to be the same. (a) γ2 = 10, γ1 = 2 (in units of χ), s = 1, r = rmax = 6; (b) γ2 = 2, γ1 = 10 (in units of χ), s = 1, r = rmax = 1.1.

Fig. 6
Fig. 6

Influence of the coupling to the SHG nonlinearity on the intensity fluctuations in the NDOPO modes of the cascaded system: Vsingle (solid curve) and Va2 (dotted curve) for e1 = 8, γ2 = 10, γ1 = 2 (in units of χ), s = 3, with increasing coupling to the SHG modes from left to right. The high-frequency noise Vsingle is seen to follow Va2 and to drop below shot noise under the influence of the second nonlinearity.

Equations (23)

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H = ω 1 α 1 a 1 + ω 2 a 2 a 2 + ω + α + a + + ω - a - a - + i ( κ / 2 ) ( a 1 2 a 2 - a 1 2 a 2 ) + i χ ( a 2 a + a - - a 2 a + a - ) + i [ E 1 ( t ) a 1 - E 1 * ( t ) a 1 ] .
α ˙ 1 = - γ 1 α 1 + κ α 1 * α 2 + E 1 , α ˙ 2 = - γ 2 α 2 - ( κ / 2 ) α 1 2 - χ α + α - , α ˙ + = - γ + α + + χ α 2 α - * , α ˙ - = - γ - α - + χ α 2 α + * .
α ˙ 1 = - γ 1 α 1 + κ α 1 * α 2 + E 1 , α ˙ 2 = - γ 2 α 2 - ( κ / 2 ) α 1 2 - χ U , U ˙ = - ( γ + + γ - ) U + χ α 2 ( I + + I - ) , I ˙ ± = - 2 γ ± I ± + χ ( α 2 U * + α 2 * U ) .
α 1 0 = e 1 ( 2 γ ¯ γ 2 χ 2 r ) 1 / 2             α 2 0 = - γ ¯ χ , U 0 = γ ¯ γ 2 χ 2 ( 1 - e 1 2 ) ,             I ± 0 = ( - U 0 ) γ γ ± .
E thresh = γ 1 χ ( 1 + r s ) ( 2 γ 2 γ ¯ r ) 1 / 2 ,
r κ SHG / κ NDOPO = κ / χ , γ ¯ = γ + γ - , s = γ ¯ / γ 1 .
r r max 1 s ( 1 + γ 2 γ 1 ) .
S X Y ( ω ) = - + d t exp ( - i ω t ) : X ^ ( t ) , Y ^ ( 0 ) : s ,
S ( ω ) = ( i ω + A 0 ) - 1 D 0 [ - i ω + ( A 0 ) T ] - 1
D 0 = χ 2 [ D κ 0 0 D χ ] ,
D κ = r 2 [ a 2 0 b 2 0 0 0 b 2 0 - a 2 0 0 0 0 0 0 0 0 0 0 0 ] , D χ = [ u 0 a 2 0 - v 0 b 2 0 v 0 a 2 0 + u 0 b 2 0 a 2 0 I + 0 a 2 0 I - 0 v 0 a 2 0 + u 0 b 2 0 - u 0 a 2 0 + v 0 b 2 0 b 2 0 I + 0 b 2 0 I - 0 I + 0 a 2 0 I + 0 b 2 0 0 u 2 a 2 0 + v 0 b 2 0 I - 0 a 2 0 I - 0 b 2 0 u 0 a 2 0 + v 0 b 2 0 0 ] ,
V i j coh ( ω ) = 1 + ( 2 γ i 2 γ j ) 1 / 2 S i j ( ω ) 0             i , j { a 1 , b 1 , a 2 , b 2 } .
V i j incoh ( ω ) = 1 + ( 2 γ i 2 γ j I i I j ) 1 / 2 S i j ( ω ) 0             i , j { I + , I - }
V a 1 , a 1 ( ω ) = 1 - 4 γ 1 2 ω 2 ( ω 2 + γ 2 2 ) - 2 0 2 ( ω 2 + γ 2 γ ¯ ) ω 2 [ ω 2 ( γ 1 + γ 2 + 2 ) 2 + N + 2 ( ω ) ] + 0 2 M + ( ω ) , V b 1 , b 1 ( ω ) = 1 + 4 γ 1 2 ( ω 2 + 4 γ ¯ 2 ) ( ω 2 + γ 2 2 ) - 2 0 2 ( ω 2 - γ 2 γ ¯ ) ( ω 2 + 4 γ ¯ 2 ) [ ω 2 ( γ 1 + γ 2 - 2 ) 2 + N - 2 ( ω ) ] + 0 2 M - ( ω ) , V a 2 , a 2 ( ω ) = 1 - 4 γ 2 2 ω 2 1 2 - 0 2 Γ + 2 ( ω ) ( 1 / r ) ω 2 [ ω 2 ( γ 1 + γ 2 + 2 ) 2 + N + 2 ( ω ) ] + 0 2 M + ( ω ) , V b 2 , b 2 ( ω ) = 1 + 4 γ 2 2 ( ω 2 + 4 γ ¯ 2 ) 1 2 - 0 2 Γ - 2 ( ω ) ( 1 / r ) ( ω 2 + 4 γ ¯ 2 ) [ ω 2 ( γ 1 + γ 2 - 2 2 ) 2 + N - 2 ( ω ) ] + 0 2 M - ( ω )
Γ ± 2 ( ω ) = ω 2 + ( γ 1 ± 2 ) 2 , N ± ( ω ) = ω 2 - γ 2 ( γ 1 ± 2 ) - 1 2 , M + ( ω ) = - 2 ω 2 [ Γ + 2 ( ω ) - 1 2 ] + 0 2 Γ + 2 ( ω ) , M - ( ω ) = - 2 ω 2 [ Γ - 2 ( ω ) - 1 2 ] + 0 2 Γ - 2 ( ω ) + 4 γ ¯ [ γ 2 Γ - 2 ( ω ) + 1 2 ( γ 1 - 2 ) ] .
1 = κ α 1 0 ,             2 = κ α 2 0 , 0 = χ I 0
λ SHG = 1 / 2 { - ( γ 1 + γ 2 ) - 2 ± [ ( 2 + γ 1 + γ 2 ) 2 + 4 ( - γ 1 γ 2 - γ 2 2 - 1 2 ) ] 1 / 2 } = - ( γ 1 + γ 2 ) + r s γ 1 2 ± i { [ γ 1 γ 2 ( 1 - r s + 2 e 1 r s ) - ( γ 1 + γ 2 - r s γ 1 ) 2 / 4 ] 1 / 2 }
V diff ( ω ) = 1 + 2 γ ¯ I ± 0 [ S single ( ω ) - S cross ( ω ) ] = ω 2 ω 2 + 4 γ ¯ 2 .
V single ( ω ) = 1 + ( 2 γ ¯ / I ± 0 ) S single ( ω ) = 1 + 2 γ ¯ 2 ω 2 + 4 γ ¯ 2 × 4 γ ¯ 2 R ( ω ) + 0 2 T - ( ω ) - 0 4 Γ + 2 ( ω ) ω 2 R ( ω ) - 2 ω 2 0 2 [ Γ + 2 ( ω ) - 1 2 ] + 0 4 Γ + 2 ( ω ) ,
V sum ( ω ) = 1 + ( 2 γ ¯ / I ± 0 ) [ S I ± I ± ( ω ) + S I ± I ( ω ) ] = 1 + 4 γ ¯ 2 × R ( ω ) - r 0 2 1 2 ω 2 R ( ω ) - 2 ω 2 0 2 [ Γ + 2 ( ω ) - 1 2 ] + 0 4 Γ + 2 ( ω ) ,
R ( ω ) = ( ω 2 + γ 2 2 ) Γ + 2 ( ω ) - 2 1 2 N + ( ω ) - 1 4 , T ± ( ω ) = 2 ω 2 [ Γ + 2 ( ω ) - 1 2 ] ± r 1 2 ( ω 2 + 4 γ ¯ 2 ) .
V single adiab ( ω ) = 1 - 4 γ ¯ 2 ω 2 + 4 γ ¯ 2 2 γ ¯ 2 e 1 2 ( e 1 2 - 2 ) ω 2 - 4 γ ¯ 2 ( e 1 2 - 1 ) 2 ,
V single ( ω = 0 ) = 1 2 + 4 e 1 2 r s + 1 + 2 r s ( 1 + r s ) + 1 2 ( e 1 2 - 1 ) 2 1 2 ,

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