Abstract

The excitation of Rydberg atom transitions by submillimeter-wavelength radiation in high-Q cavities forms the basis of the micromaser. The excitation dynamics of a micromaser is known to be dependent on the detailed photon statistics in the interaction cavity and can exhibit well-known collapses and revivals of the atomic inversion, dipole moment, and photon number. We study these effects in a two-photon model in which the time evolution is exactly periodic. We study the field entropy in two two-photon cases and link the fluctuations in the field phase to the changes in the field entropy. We also calculate the statistical Q function of the field and show how the periodicity of the two-photon dynamics is linked to a periodic splitting of the Q function in phase space. Finally this periodicity is linked to the nature of the atom-field dressed states involved in two-photon resonance.

© 1990 Optical Society of America

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References

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  1. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [Crossref]
  2. J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
    [Crossref]
  3. N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
    [Crossref]
  4. For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).
  5. P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982);Phys. Lett. 90A, 342 (1982).
    [Crossref]
  6. G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
    [Crossref] [PubMed]
  7. J. Eiselt, Opt. Commun. 72, 351 (1989).
    [Crossref]
  8. B. Buck and C. V. Sukumar, Phys. Lett. 81A, 132 (1981);C. V. Sukumar and B. Buck, Phys. Lett. 83A, 211 (1981).
  9. V. Buzek, Phys. Rev. A 39, 3196 (1989).
    [Crossref] [PubMed]
  10. P. L. Knight, Phys. Scr. T12, 51 (1986).
    [Crossref]
  11. A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
    [Crossref] [PubMed]
  12. S. Singh, Phys. Rev. A 25, 3206 (1982);P. Alsing and M. S. Zubairy, J. Opt. Soc. Am. B 4, 177 (1987);R. R. Puri and R. K. Bullough, J. Opt. Soc. Am. B 5, 2021 (1988);H. I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 215 (1989);R. R. Puri and G. S. Agarwal, Phys. Rev. A 37, 3879 (1988);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 557 (1989).
    [Crossref] [PubMed]
  13. S. J. D. Phoenix and P. L. Knight, Ann. Phys. (NY) 186, 381 (1988).
    [Crossref]
  14. M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
    [Crossref] [PubMed]
  15. H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).
    [Crossref]
  16. S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986).
    [Crossref]
  17. R. Loudon, The Quantum Theory of Light, 1st. ed. (Oxford U. Press, Oxford, 1973), p. 143.
  18. S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989);D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
    [Crossref] [PubMed]
  19. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
    [Crossref]
  20. See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  21. B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985).
    [Crossref] [PubMed]
  22. D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
    [Crossref] [PubMed]
  23. S. Stenholm, Phys. Rep. C6, 1(1973).
    [Crossref]

1989 (3)

J. Eiselt, Opt. Commun. 72, 351 (1989).
[Crossref]

V. Buzek, Phys. Rev. A 39, 3196 (1989).
[Crossref] [PubMed]

S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989);D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
[Crossref] [PubMed]

1988 (1)

S. J. D. Phoenix and P. L. Knight, Ann. Phys. (NY) 186, 381 (1988).
[Crossref]

1987 (4)

M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
[Crossref] [PubMed]

A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
[Crossref] [PubMed]

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[Crossref]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[Crossref] [PubMed]

1986 (2)

S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986).
[Crossref]

P. L. Knight, Phys. Scr. T12, 51 (1986).
[Crossref]

1985 (2)

B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985).
[Crossref] [PubMed]

D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[Crossref] [PubMed]

1982 (2)

S. Singh, Phys. Rev. A 25, 3206 (1982);P. Alsing and M. S. Zubairy, J. Opt. Soc. Am. B 4, 177 (1987);R. R. Puri and R. K. Bullough, J. Opt. Soc. Am. B 5, 2021 (1988);H. I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 215 (1989);R. R. Puri and G. S. Agarwal, Phys. Rev. A 37, 3879 (1988);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 557 (1989).
[Crossref] [PubMed]

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982);Phys. Lett. 90A, 342 (1982).
[Crossref]

1981 (2)

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
[Crossref]

B. Buck and C. V. Sukumar, Phys. Lett. 81A, 132 (1981);C. V. Sukumar and B. Buck, Phys. Lett. 83A, 211 (1981).

1980 (1)

J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

1973 (1)

S. Stenholm, Phys. Rep. C6, 1(1973).
[Crossref]

1970 (1)

H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).
[Crossref]

1963 (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Aliskenderov, E. I.

A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
[Crossref] [PubMed]

Araki, H.

H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).
[Crossref]

Barnett, S. M.

S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989);D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
[Crossref] [PubMed]

S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986).
[Crossref]

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

Brune, M.

M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
[Crossref] [PubMed]

Buck, B.

B. Buck and C. V. Sukumar, Phys. Lett. 81A, 132 (1981);C. V. Sukumar and B. Buck, Phys. Lett. 83A, 211 (1981).

Buzek, V.

V. Buzek, Phys. Rev. A 39, 3196 (1989).
[Crossref] [PubMed]

Caves, C. M.

B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985).
[Crossref] [PubMed]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Eberly, J. H.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
[Crossref]

J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Eiselt, J.

J. Eiselt, Opt. Commun. 72, 351 (1989).
[Crossref]

Filipowicz, P.

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

Haroche, S.

M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
[Crossref] [PubMed]

Javanainen, J.

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Kien, F. L.

A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
[Crossref] [PubMed]

Klein, N.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[Crossref] [PubMed]

Knight, P. L.

S. J. D. Phoenix and P. L. Knight, Ann. Phys. (NY) 186, 381 (1988).
[Crossref]

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[Crossref]

P. L. Knight, Phys. Scr. T12, 51 (1986).
[Crossref]

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982);Phys. Lett. 90A, 342 (1982).
[Crossref]

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

Lieb, E.

H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).
[Crossref]

Loudon, R.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[Crossref]

R. Loudon, The Quantum Theory of Light, 1st. ed. (Oxford U. Press, Oxford, 1973), p. 143.

Louisell, W. H.

See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Meschede, D.

D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[Crossref] [PubMed]

Meystre, P.

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

Muller, G.

D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[Crossref] [PubMed]

Narozhny, N. B.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
[Crossref]

J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Pegg, D. T.

S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989);D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
[Crossref] [PubMed]

S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986).
[Crossref]

Phoenix, S. J. D.

S. J. D. Phoenix and P. L. Knight, Ann. Phys. (NY) 186, 381 (1988).
[Crossref]

Radmore, P. M.

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982);Phys. Lett. 90A, 342 (1982).
[Crossref]

Raimond, J. M.

M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
[Crossref] [PubMed]

Rempe, G.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[Crossref] [PubMed]

Sanchez-Mondragon, J. J.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
[Crossref]

J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Schumaker, B. L.

B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985).
[Crossref] [PubMed]

Schumovsky, A. S.

A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
[Crossref] [PubMed]

Singh, S.

S. Singh, Phys. Rev. A 25, 3206 (1982);P. Alsing and M. S. Zubairy, J. Opt. Soc. Am. B 4, 177 (1987);R. R. Puri and R. K. Bullough, J. Opt. Soc. Am. B 5, 2021 (1988);H. I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 215 (1989);R. R. Puri and G. S. Agarwal, Phys. Rev. A 37, 3879 (1988);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 557 (1989).
[Crossref] [PubMed]

Stenholm, S.

S. Stenholm, Phys. Rep. C6, 1(1973).
[Crossref]

Sukumar, C. V.

B. Buck and C. V. Sukumar, Phys. Lett. 81A, 132 (1981);C. V. Sukumar and B. Buck, Phys. Lett. 83A, 211 (1981).

Walther, H.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[Crossref] [PubMed]

D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[Crossref] [PubMed]

Ann. Phys. (NY) (1)

S. J. D. Phoenix and P. L. Knight, Ann. Phys. (NY) 186, 381 (1988).
[Crossref]

Commun. Math. Phys. (1)

H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).
[Crossref]

J. Mod. Opt. (2)

S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989);D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
[Crossref] [PubMed]

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[Crossref]

J. Phys. A (1)

S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986).
[Crossref]

Opt. Commun. (1)

J. Eiselt, Opt. Commun. 72, 351 (1989).
[Crossref]

Phys. Lett. (2)

B. Buck and C. V. Sukumar, Phys. Lett. 81A, 132 (1981);C. V. Sukumar and B. Buck, Phys. Lett. 83A, 211 (1981).

A. S. Schumovsky, F. L. Kien, and E. I. Aliskenderov, Phys. Lett. 124A, 351 (1987);N. N. Bogolubov, F. L. Kien, and A. S. Schumovsky, Phys. Lett. 107A, 436 (1985);S. Y. Zhu, J. Mod. Opt. 36, 499 (1989);X. S. Li and N. Y. Bei, Phys. Lett. 101A, 169 (1984);X. S. Li and Y. N. Peng, Phys. Rev. A 32, 1501 (1985);X. S. Li and C. D. Gong, Phys. Rev. A 33, 2807 (1986).
[Crossref] [PubMed]

Phys. Rep. (1)

S. Stenholm, Phys. Rep. C6, 1(1973).
[Crossref]

Phys. Rev. A (6)

B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985).
[Crossref] [PubMed]

S. Singh, Phys. Rev. A 25, 3206 (1982);P. Alsing and M. S. Zubairy, J. Opt. Soc. Am. B 4, 177 (1987);R. R. Puri and R. K. Bullough, J. Opt. Soc. Am. B 5, 2021 (1988);H. I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 215 (1989);R. R. Puri and G. S. Agarwal, Phys. Rev. A 37, 3879 (1988);A. Joshi and R. R. Puri, J. Mod. Opt. 36, 557 (1989).
[Crossref] [PubMed]

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982);Phys. Lett. 90A, 342 (1982).
[Crossref]

M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 35, 154 (1987);L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. A 36, 3771 (1987);M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, Phys. Rev. Lett. 59, 1899 (1987).
[Crossref] [PubMed]

V. Buzek, Phys. Rev. A 39, 3196 (1989).
[Crossref] [PubMed]

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981).
[Crossref]

Phys. Rev. Lett. (3)

J. H. Eberly, J. J. Sanchez-Mondragon, and N. B. Narozhny, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[Crossref] [PubMed]

D. Meschede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985);P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[Crossref] [PubMed]

Phys. Scr. (1)

P. L. Knight, Phys. Scr. T12, 51 (1986).
[Crossref]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Other (3)

For a review of some recent developments in the JCM, see S. M. Barnett, P. Filipowicz, J. Javanainen, P. L. Knight, and P. Meystre, in Frontiers of Quantum Optics, E. R. Pike and S. Sarkar, eds.(Hilger, Bristol, UK, 1986).

R. Loudon, The Quantum Theory of Light, 1st. ed. (Oxford U. Press, Oxford, 1973), p. 143.

See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

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Figures (9)

Fig. 1
Fig. 1

Level scheme for the Raman-coupled JCM. Levels |e〉 and |g〉 are degenerate.

Fig. 2
Fig. 2

Atomic inversion, labeled W here, in the RCM. The atom is initially in the state |e〉 and the field in a coherent state with an average photon number of 10, and we have set λ = 1.

Fig. 3
Fig. 3

Field entropy, Sf, in the RCM. The atom is initially in the state |e〉 and the field in a coherent state with an average photon number of 10, and we have set λ = 1.

Fig. 4
Fig. 4

Evolution of the Q function in the RCM with an initially coherent state field and ψ = 0.

Fig. 5
Fig. 5

Evolution of the Q function in the RCM with an initially squeezed vacuum state field and θ = 0.

Fig. 6
Fig. 6

Average photon number, in the cavity for the two-photon JCM with zero detuning. The atom is initially inverted and the field in a coherent state with an average photon number of 2, and we have set = 1.

Fig. 7
Fig. 7

Same as Fig. 6 but with a detuning equal to the coupling constant.

Fig. 8
Fig. 8

Field entropy, Sf, in the two-photon JCM with zero detuning. The atom is initially inverted and the field in a coherent state with an average photon number of 2, and we have set λ = 1.

Fig. 9
Fig. 9

Same as Fig. 8 but with a detuning equal to the coupling constant.

Equations (89)

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H = ω a a + λ a a ( σ + + σ ) ,
U ( t ) = [ cos [ λ t ( a a ) ] i sin [ λ t ( a a ) ] i sin [ λ t ( a a ) ] cos [ λ t ( a a ) ] ] .
ρ F ( t ) = C ρ F ( 0 ) C + S ρ F ( 0 ) S ,
C = cos [ λ t ( a a ) ] , S = sin [ λ t ( a a ) ] .
ρ F ( t ) = n = 0 m = 0 [ ρ F ( 0 ) ] n m cos [ λ t ( n m ) ] | n m | .
a ˙ = i [ ω + λ ( σ + + σ ) ] a ,
σ ˙ = i λ a a σ 3 ,
σ ˙ 3 = i 2 λ a a ( σ + σ ) .
a ( t ) = e i ω t cos λ t a ( 0 ) .
X = ½ a + a = | α | cos ( ω t ϕ ) cos λ t ,
σ ¨ 3 + 4 λ 2 ( a a ) 2 σ 3 = 0 .
σ 3 ( t ) = 2 Re ( α α e i 2 λ t ) ,
σ 3 ( t ) = exp ( 2 n ¯ sin 2 λ t ) cos ( n ¯ sin 2 λ t ) ,
σ 3 ( t ) n = cos ( 2 n λ t ) ,
ρ ( t ) = C | ψ ψ | C + S | ψ ψ | S ,
π ± = e i ϕ / 2 e ± θ / 2 C | ψ ± e i ϕ / 2 e θ / 2 S | ψ ,
C S = | C S | e i ϕ ,
θ = sinh 1 ( δ / 2 ) ,
δ = ( C C S S ) | C S | .
C C = n = 0 p ( n ) cos 2 ( n λ t ) ,
S S = n = 0 p ( n ) sin 2 ( n λ t ) ,
C S = n = 0 p ( n ) sin ( n λ t ) cos ( n λ t ) .
π ± = C C ± | C S | e θ ,
p ( n ) = exp ( n ¯ ) n ¯ n / n ! ,
π ± = ½ [ 1 ± exp ( 2 n ¯ sin 2 λ t ) ] .
S f = ( π + ln π + + π ln π ) .
π ± = A ± ( t ) cos [ λ t ( a a ) ] α ± B ± ( t ) sin [ λ t ( a a ) ] α .
π + = ½ [ A ( t ) i B ( t ) ] α e i λ t + ½ [ A ( t ) + i B ( t ) ] α e i λ t ,
ξ ± = A ± ( t ) e ± B ± ( t ) g ,
θ j = lim s ( s + 1 ) 1 / 2 n = 0 s exp ( i n θ j ) n .
ρ F ( t ) = ½ [ R ( λ t ) ρ F ( 0 ) R ( λ t ) + R ( λ t ) ρ F ( 0 ) R ( λ t ) ] ,
R ( ξ ) = exp ( i ξ a a ) .
R ( ξ ) θ j = θ j + ξ .
ρ F ( t ) = ½ ( θ j λ t θ j λ t + θ j + λ t θ j + λ t ) .
P ( θ , t ) = ½ P 1 ( θ , t ) + ½ P 2 ( θ , t ) ,
S θ = 0 2 π d θ P ( θ ) ln P ( θ ) .
S θ 0 π ½ P 1 ln ½ P 1 d θ π 2 π ½ P 2 ln ½ P 2 d θ ,
S θ ln 2 + S i .
Q ( α , α * ) = α | ρ | α .
exp ( λ a a ) β = exp [ ½ | β | 2 ( 1 e 2 Re λ ) ] β e λ ,
ρ F ( t ) = ½ ( β e i λ t β e i λ t | + | β e i λ t β e i λ t | ) .
Q ( α , α * ) = ½ [ exp ( | α β e i λ t | 2 ) + exp ( | α β e i λ t | 2 ) ] .
Q ( α , α * ) = Q + ( α , α * ) + Q ( α , α * ) ,
Q ± ( x , y ) = ½ exp [ ( x ± 2 + y ± 2 ) ] ,
x ± = x | β | cos ( λ t ± ψ ) , y ± = y | β | sin ( λ t ± ψ ) .
S ( r , θ ) 0 = ( cosh r ) 1 / 2 exp ( ½ a 2 e i θ tanh r ) 0 .
α | S ( r , θ ) | 0 = ( cosh r ) 1 / 2 exp ( ½ α * 2 e i θ tanh r ) α 0 .
Q = ( cosh r ) 1 exp [ x 2 ( 1 + ω cos θ ) y 2 ( 1 ω cos θ ) 2 x y ω sin θ ] .
r , θ = n = 0 f ( n , r ) e i n θ 2 n ,
ρ F ( t ) = ½ ( | r , θ 2 λ t r , θ 2 λ t + r , θ + 2 λ t r , θ + 2 λ t ) .
( Δ X ) SQ 2 = ¼ + ½ [ sinh 2 r cos 2 λ t cos ( θ 2 ω t ) sinh r cosh r ] .
( Δ X ) SQ 2 = ¼ + ¼ [ ( e 2 r 1 ) + 2 sin 2 λ t sinh 2 r ] .
( Δ X ) COH 2 = ¼ + | α | 2 sin 2 ( ω t ϕ ) sin 2 λ t .
H = ω a a + ½ ω 0 σ 3 + λ ( a 2 σ + a 2 σ + ) ,
H = H 0 + V ,
H 0 = ω ( a a + σ 3 ) ,
V = Δ 2 σ 3 + λ ( a 2 σ + a 2 σ + ) ,
exp ( iVt / ) = n = 0 ( i λ t ) n n ! [ Δ / 2 λ ã ã Δ / 2 λ ] n ,
[ Δ / 2 λ ã ã Δ / 2 λ ] 2 m = [ ( Δ 2 / 4 λ 2 + ã ã ) m 0 0 ( Δ 2 / 4 λ 2 + ã ã ) m ] ,
[ Δ / 2 λ ã ã Δ / 2 λ ] 2 m + 1 = [ Δ 2 λ ( Δ 2 / 4 λ 2 + ã ã ) m ( Δ 2 / 4 λ 2 + ã ã ) m ã ( Δ 2 / 4 λ 2 + ã ã ) m ã Δ 2 λ ( Δ 2 / 4 λ 2 + ã ã ) m ]
exp ( iVt / ) = [ cos ( λ t δ ) + i Δ 2 λ sin ( λ t δ ) δ i sin ( λ t δ ) δ ã i sin ( λ t ) ã cos ( λ t ) i Δ 2 λ sin ( λ t ) ] ,
δ 2 = Δ 2 4 λ 2 + ã ã ,
2 = Δ 2 4 λ 2 + ã ã .
ρ F ( t ) = C ρ F ( 0 ) C + S ( 0 ) ρ F S ,
C = cos ( λ t δ ) i Δ 2 λ sin ( λ t δ ) δ ,
S = ã sin ( λ t δ ) δ .
δ 2 n = [ Δ 2 4 λ 2 + j = 1 m ( n + j ) ] n = δ n 2 n ,
ρ F ( t ) = [ cos 2 ( λ t δ n ) + Δ 2 4 λ 2 sin 2 ( λ t δ n ) δ n 2 ] | n n | + ( n + 1 ) ( n + 2 ) δ n 2 sin 2 ( λ t δ n ) | n + 2 n + 2 | .
a a n = n + 2 [ 1 + Δ 2 4 λ 2 ( n + 1 ) ( n + 2 ) ] 1 sin 2 ( λ t δ n ) .
a a = n = 0 p ( n ) a a n .
δ n 2 = Δ 2 4 λ 2 + ( n + 1 ) ( n + 2 ) = n 2 + 3 n + γ 2 ,
C C = 1 n = 0 p ( n ) [ 1 Δ 2 4 λ 2 δ n 2 ] sin 2 ( λ t δ n ) ,
S S = 1 C C ,
C S = α * 2 n = 0 p ( n ) sin ( λ t δ n ) δ n × [ cos ( λ t δ n + 2 ) + i Δ 2 λ sin ( λ t δ n + 2 ) δ n + 2 ] ,
H 0 { n , e n + 2 , g } = ω ( n + 1 ) { n , e n + 2 , g } ,
V n , e = ½ Δ n , e + λ ( n + 1 ) 1 / 2 × ( n + 2 ) 1 / 2 n + 2 , g ,
V n + 2 , g = ½ Δ n + 2 , g + λ ( n + 1 ) 1 / 2 × ( n + 2 ) 1 / 2 n , e ,
H = [ E n + Δ / 2 ( Ω n 2 Δ 2 ) 1 / 2 / 2 ( Ω n 2 Δ 2 ) 1 / 2 / 2 E n Δ / 2 ] ,
E n = ω ( n + 1 )
Ω n = 2 λ [ Δ 2 4 λ 2 + ( n + 1 ) ( n + 2 ) ] 1 / 2 .
μ ± ( n ) = E n ± Ω n / 2 .
ψ ± ( n ) = 1 2 ( 1 ± Δ Ω n ) 1 / 2 n , e ± 1 2 ( 1 Δ Ω n ) 1 / 2 n + 2 , g ,
ψ ( n ) ( t = 0 ) = C + ( n ) ψ + ( n ) + C ( n ) ψ ( n ) ,
ψ ( n ) ( t ) = exp ( i E n t / ) [ C + ( n ) exp ( i Ω n t / 2 ) ψ + ( n ) + C ( n ) exp ( i Ω n t / 2 ) ψ ( n ) ] .
τ R ( n ) = 2 π / Ω n .
a a n = | C + ( n ) | 2 ψ + ( n ) | a a | ψ + ( n ) + | C ( n ) | 2 ψ ( n ) | a a | ψ ( n ) + C + ( n ) C ¯ ( n ) exp ( i Ω n t ) ψ ( n ) | a a | ψ + ( n ) + C ¯ + ( n ) C ( n ) exp ( i Ω n t ) ψ + ( n ) | a a | ψ ( n ) ,
τ R ( 0 ) = 2 π α ( n ) / Ω n ,
α ( n ) = Ω n / Ω 0 .
α ( n ) = ( n + 3 / 2 ) .

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