Abstract

The incoherent part of the Mollow spectrum for resonance fluorescence has a subnatural linewidth in the weak-field limit. We show that this is due to squeezing of the fluctuations in the induced atomic dipole. The reduced linewidth persists for driving field intensities of approximately 12% of the saturation intensity, where Rabi sidebands begin to appear. We find a similar linewidth narrowing in the transmitted and fluorescent spectra for a single atom in a weakly driven resonant cavity. In this system single-quantum frequency splitting can produce a two-peaked spectrum. Both peaks have a narrowed linewidth. The spectrum of the transmitted light shows a second nonclassical effect that is due to squeezing. A spectral hole may appear at line center, giving a two-peaked spectrum even when there is no frequency splitting.

© 1988 Optical Society of America

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References

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  1. R. Loudon, Rep. Prog. Phys.43, 913 (1980).
    [CrossRef]
  2. J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
    [CrossRef]
  3. Accounts of the current status of research on squeezed states may be found in J. Opt. Soc. Am. B 4(10) (1987)and J. Mod. Opt. 34(6/7) (1987).
  4. H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
    [CrossRef] [PubMed]
  5. H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
    [CrossRef]
  6. H. J. Carmichael, Phys. Rev. Lett. 55, 2790 (1985).
    [CrossRef] [PubMed]
  7. P. R. Rice and H. J. Carmichael, “Single-atom cavity-enhanced absorption. I: Photon statistics in the bad-cavity limit,” IEEE J. Quantum Electron. (to be published).
  8. J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
    [CrossRef]
  9. G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
    [CrossRef]
  10. Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
    [CrossRef]
  11. H. J. Carmichael, Phys. Rev. A 33, 3262 (1986).
    [CrossRef] [PubMed]
  12. R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).
  13. L. A. Lugiato, Nuovo Cimento 50B, 89 (1979).
  14. B. R. Mollow, Phys. Rev. 188, 1969 (1969).
    [CrossRef]
  15. H. J. Carmichael and D. F. Walls, J. Phys. B 9, L43 (1976).
    [CrossRef]
  16. H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
    [CrossRef]
  17. D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
    [CrossRef]
  18. F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
    [CrossRef]
  19. F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
    [CrossRef]
  20. W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
    [CrossRef]
  21. R. I. Sokolovskii, Sov. Phys. JETP 32, 438 (1971).
  22. H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
    [CrossRef]
  23. B. R. Mollow, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 1–43.
    [CrossRef]
  24. M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
    [CrossRef]
  25. H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987).
    [CrossRef]
  26. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
    [CrossRef] [PubMed]
  27. M. J. Collett and R. Loudon, J. Opt. Soc. Am. B 4, 1525 (1987).
    [CrossRef]
  28. C. M. Savage and H. J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. (to be published).

1987 (5)

Accounts of the current status of research on squeezed states may be found in J. Opt. Soc. Am. B 4(10) (1987)and J. Mod. Opt. 34(6/7) (1987).

H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
[CrossRef]

H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987).
[CrossRef]

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

1986 (3)

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1986).
[CrossRef] [PubMed]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

1985 (1)

H. J. Carmichael, Phys. Rev. Lett. 55, 2790 (1985).
[CrossRef] [PubMed]

1984 (2)

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
[CrossRef]

1983 (2)

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

1981 (1)

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[CrossRef]

1979 (1)

L. A. Lugiato, Nuovo Cimento 50B, 89 (1979).

1977 (1)

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
[CrossRef]

1976 (3)

H. J. Carmichael and D. F. Walls, J. Phys. B 9, L43 (1976).
[CrossRef]

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[CrossRef]

1975 (1)

F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
[CrossRef]

1974 (1)

F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
[CrossRef]

1971 (1)

R. I. Sokolovskii, Sov. Phys. JETP 32, 438 (1971).

1969 (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

Brecha, R. J.

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

Carmichael, H. J.

H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
[CrossRef]

H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987).
[CrossRef]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1986).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. Lett. 55, 2790 (1985).
[CrossRef] [PubMed]

H. J. Carmichael and D. F. Walls, J. Phys. B 9, L43 (1976).
[CrossRef]

P. R. Rice and H. J. Carmichael, “Single-atom cavity-enhanced absorption. I: Photon statistics in the bad-cavity limit,” IEEE J. Quantum Electron. (to be published).

C. M. Savage and H. J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. (to be published).

Collett, M. J.

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

M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
[CrossRef]

Cresser, J. D.

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

Dagenais, M.

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
[CrossRef]

Eberly, J. H.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Ezekiel, S.

F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
[CrossRef]

Goy, P.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Gross, M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Grove, R. E.

F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
[CrossRef]

Häger, J.

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

Haroche, S.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Hartig, W.

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

Hercher, M.

F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
[CrossRef]

Kaluzny, Y.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Kimble, H. J.

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[CrossRef]

Lane, A. S.

H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
[CrossRef]

H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
[CrossRef] [PubMed]

Leuchs, G.

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

Loudon, R.

Lugiato, L. A.

L. A. Lugiato, Nuovo Cimento 50B, 89 (1979).

Mandel, L.

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[CrossRef]

Mollow, B. R.

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

B. R. Mollow, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 1–43.
[CrossRef]

Narozhny, N. B.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Orozco, L. A.

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

Raimond, J. M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Raizen, M. G.

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

Rasmussen, W.

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

Rateike, M.

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

Rice, P. R.

P. R. Rice and H. J. Carmichael, “Single-atom cavity-enhanced absorption. I: Photon statistics in the bad-cavity limit,” IEEE J. Quantum Electron. (to be published).

Sanchez-Mondragon, J. J.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Savage, C. M.

C. M. Savage and H. J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. (to be published).

Schieder, R.

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

Schuda, F. Y.

F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
[CrossRef]

Sokolovskii, R. I.

R. I. Sokolovskii, Sov. Phys. JETP 32, 438 (1971).

Stroud, C. R.

F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
[CrossRef]

Walls, D. F.

H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
[CrossRef]

M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
[CrossRef]

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[CrossRef]

H. J. Carmichael and D. F. Walls, J. Phys. B 9, L43 (1976).
[CrossRef]

Walther, H.

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

Wu, F. Y.

F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
[CrossRef]

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

Xiao, M.

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

Zoller, P.

M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
[CrossRef]

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[CrossRef]

J. Mod. Opt. (1)

H. J. Carmichael, A. S. Lane, and D. F. Walls, J. Mod. Opt. 34, 821 (1987).
[CrossRef]

J. Opt. Soc. Am. B (4)

Accounts of the current status of research on squeezed states may be found in J. Opt. Soc. Am. B 4(10) (1987)and J. Mod. Opt. 34(6/7) (1987).

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 3(13), P238 (1986).

H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987).
[CrossRef]

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

J. Phys. B (2)

H. J. Carmichael and D. F. Walls, J. Phys. B 9, L43 (1976).
[CrossRef]

F. Y. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B 7, L198 (1974).
[CrossRef]

Nuovo Cimento (1)

L. A. Lugiato, Nuovo Cimento 50B, 89 (1979).

Opt. Commun. (1)

M. J. Collett, D. F. Walls, and P. Zoller, Opt. Commun. 52, 145 (1984).
[CrossRef]

Phys. Rev. (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Phys. Rev. A (2)

H. J. Carmichael, Phys. Rev. A 33, 3262 (1986).
[CrossRef] [PubMed]

H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[CrossRef]

Phys. Rev. Lett. (9)

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986);L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).
[CrossRef]

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).
[CrossRef]

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[CrossRef]

H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett. 58, 2539 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. Lett. 55, 2790 (1985).
[CrossRef] [PubMed]

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Sov. Phys. JETP (1)

R. I. Sokolovskii, Sov. Phys. JETP 32, 438 (1971).

Z. Phys. A (1)

W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
[CrossRef]

Other (5)

P. R. Rice and H. J. Carmichael, “Single-atom cavity-enhanced absorption. I: Photon statistics in the bad-cavity limit,” IEEE J. Quantum Electron. (to be published).

R. Loudon, Rep. Prog. Phys.43, 913 (1980).
[CrossRef]

J. D. Cresser, J. Häger, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum Optics, R. Bonifacio, ed. (Springer-Verlag, Berlin, 1982), pp. 21–59.
[CrossRef]

B. R. Mollow, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 1–43.
[CrossRef]

C. M. Savage and H. J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. (to be published).

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Figures (6)

Fig. 1
Fig. 1

Squeezing-induced linewidth narrowing in the incoherent spectrum of resonance fluorescence. S inc ( ω ¯ ) is compared with a Lorentzian of natural linewidth: Y2 ≪ 1 [Eq. (3)] (——), Y2 = 1/8 [Eq. (5)] (– – –), the Lorentzian ( 1 / π ) [ 1 + ( ω ¯ ω ¯ 0 ) 2 ] 1 (— · — ·).

Fig. 2
Fig. 2

Subtraction of two Lorentzians to obtain a reduced linewidth. The solid curve is the difference between the dashed and dashed–dotted curves, which are plotted from Eq. (6) with Y2 = 0.04.

Fig. 3
Fig. 3

Single-quantum level splitting for a two-level atom coupled to a single mode of the radiation field. ∣0〉 and ∣1〉 are Fock states for the field mode, and ∣+〉 and ∣−〉 are the upper and lower states, respectively, of the resonant atomic transition.

Fig. 4
Fig. 4

Squeezing-induced linewidth narrowing for a coupled atom and cavity mode where the spectrum shows single-quantum frequency splitting: spectrum of squeezing (– – –) and incoherent part of the optical spectrum for the transmitted light (——). C = 4, μ = 1 (ns = 1/16).

Fig. 5
Fig. 5

Squeezing-induced spectral holes for a coupled atom and cavity mode: spectrum of squeezing (– – –) and incoherent part of the optical spectrum for the transmitted light (——). (a) C = 1, μ = 1/40; (b) C = 5, μ = 1/200 (both curves have ns = 10).

Fig. 6
Fig. 6

Subtraction of two Lorentzians to obtain a spectral hole. The solid curve is the difference between the dashed and dashed–dotted curves, which are plotted from the formula for the optical spectrum in the good-cavity limit [Eq. (9.18) of Ref. 13].

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

π S inc ( ω ¯ ) = 1 2 ( Y 2 1 + Y 2 ) 1 1 1 + ( ω ¯ ω ¯ 0 ) 2 ¼ [ 1 / Y 2 1 + ( 1 / Y 2 5 ) ( 1 / 2 δ ) ] × 3 / 2 + δ ( 3 / 2 + δ ) 2 + ( ω ¯ ω ¯ 0 ) 2 ¼ [ 1 / Y 2 1 ( 1 / Y 2 5 ) ( 1 / 2 δ ) ] × 3 / 2 + δ ( 3 / 2 + δ ) 2 + ( ω ¯ ω ¯ 0 ) 2 ,
δ = ½ 1 8 Y 2 ,
π S inc ( ω ¯ ) = 2 [ 1 + ( ω ¯ ω ¯ 0 ) 2 ] 2 .
Δ ω = ( 2 1 ) 1 / 2 ( γ / 2 ) 0.64 ( γ / 2 ) ,
π S inc ( ω ¯ ) = 9 2 { 1 1 + ( ω ¯ ω ¯ 0 ) 2 3 + ( ω ¯ ω ¯ 0 ) 2 [ 9 / 4 + ( ω ¯ ω ¯ 0 ) 2 ] 2 } .
π S inc ( ω ¯ ) = ( 1 / 2 Y 2 ) [ 1 1 + ( ω ¯ ω ¯ 0 ) 2 ( 1 Y 2 ) 1 + 2 Y 2 ( 1 + 2 Y 2 ) 2 + ( ω ¯ ω ¯ 0 ) 2 ] .
π S inc ( ω ¯ ) = ( Δ σ + Δ σ ss ) 1 × { 0 d τ ¯ exp [ i ( ω ¯ ω ¯ 0 ) τ ¯ Δ σ + ( 0 ) Δ σ ( τ ¯ ) ss } ,
Δ σ = σ σ ss , Δ σ + = σ + σ + ss ;
S ( ω ¯ , θ ) = 8 η 0 d τ ¯ cos ω ¯ τ ¯ [ Δ σ + ( 0 ) Δ σ ( τ ¯ ) ss + e 2 i θ Δ σ + ( 0 ) Δ σ + ( τ ¯ ) ss ] ,
π S inc ( ω ¯ + ω ¯ 0 ) = ( 16 η Δ σ + Δ σ ss ) 1 [ S ( ω ¯ , θ ) + S ( ω ¯ , θ + π / 2 ) ] .
S ( ω ¯ , 0 ) = 4 η Y 2 1 + Y 2 ω ¯ 2 + 4 ( 1 2 Y 2 ) ω ¯ 4 + ω ¯ 2 ( 5 4 Y 2 ) + 4 ( 1 + Y 2 ) 2 .
S ( ω ¯ , π / 2 ) = 4 η Y 2 1 + Y 2 1 1 + ω ¯ 2 .
π S inc ( ω ¯ + ω ¯ 0 ) = ( 16 η Δ Δ ss ) 1 [ S ( ω ¯ , θ ) + S ( ω ¯ , θ + π / 2 ) ] .
π S inc ( ω ¯ ) = 2 ( 1 λ 2 ) [ ( 1 + λ ) 2 + ( ω ¯ ω ¯ 0 ) 2 ] [ ( 1 λ ) 2 + ( ω ¯ ω ¯ 0 ) 2 ] ,
ρ = [ , ρ ] + g [ σ σ + , ρ ] + γ 2 ( 2 σ ρ σ + σ + σ ρ ρ σ + σ ) + κ ( 2 ρ ρ ρ ) ,
Y = / κ n s , C = g 2 / κ γ , n s = γ 2 / 8 g 2 .
κ / γ , n s 0 , / κ 0 ,
Δ ω 0.64 [ γ ( 1 + 2 C ) / 2 ] .
λ ¯ ± = 2 λ ± / γ = ½ ( μ + 1 ) ± ½ [ ( μ 1 ) 2 8 μ C ] 1 / 2 ,
μ = 2 κ / γ .
S ( ω ¯ , θ ) = cos 2 θ 4 η 2 C 1 + 2 C Y 2 ( 1 + 2 C ) 2 1 + μ 1 + μ + 2 C × μ 2 ( 1 + 2 C ) 2 | λ ¯ + + i ω ¯ | 2 | λ ¯ + i ω ¯ | 2 = cos 2 θ 4 η 2 C 1 + 2 C Y 2 ( 1 + 2 C ) 2 1 + μ 1 + μ + 2 C × μ 2 [ μ ω ¯ 2 / ( 1 + 2 C ) ] 2 + ( 1 + μ ) 2 ω ¯ 2 / ( 1 + 2 C ) 2 .
S ( ω ¯ , θ ) = cos 2 θ 4 η 2 C 1 + 2 C Y 2 ( 1 + 2 C ) 2 1 1 + ω ¯ 2 / ( 1 + 2 C ) 2 .
S ( ω ¯ , θ ) = cos 2 θ 4 η 2 C 1 + 2 C Y 2 ( 1 + 2 C ) 3 1 1 + ω ¯ 2 / μ 2 ( 1 + 2 C ) 2 .
π S inc ( ω ¯ ) = 2 z + z [ i ( z + + z ) ] 3 × ( ω ¯ ω ¯ 0 ) 2 [ ( ω ¯ ω ¯ 0 ) 4 + ( 1 1 / n s ) ( ω ¯ ω ¯ 0 ) 2 + 1 / 4 n s 2 ] 2 ,
z ± = ½ ( 1 1 / n s ) ± ½ 1 2 / n s .
λ ¯ ± = ½ ± ½ 1 2 / n s .
π S inc ( ω ¯ ) = ( 1 / C μ ) ( ω ¯ ω ¯ 0 ) 2 / ( 2 C μ ) 2 [ 1 + ( ω ¯ ω ¯ 0 ) 2 / ( 2 C μ ) 2 ] 2 ,
d d t ( x y ) = ( | a | c 0 | b | ) ( x y ) .
d d t ( x ( 0 ) x ( τ ) ss x ( 0 ) y ( τ ) ss ) = ( | a | c 0 | b | ) [ x ( 0 ) x ( τ ) ss x ( 0 ) y ( τ ) ss ] .
1 2 π d τ e i ω τ x ( 0 ) x ( τ ) ss = ( x 2 ss c | a | | b | x y ss ) × | a | | a | 2 + ω 2 + c | a | | b | x y ss | b | | b | 2 + ω 2 .

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