Abstract

Observations of the oscillatory exchange of excitation between N two-state atoms and a single mode of a high-finesse optical cavity are reported in a regime of weak-field excitation and of comparable atomic and cavity damping rates. The observed frequencies of oscillation, approximately given by gN, where g is the single-photon Rabi frequency, are in reasonable agreement with theoretical predictions.

© 1995 Optical Society of America

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  1. P. Meystre, "Cavity quantum optics and the quantum measurement process," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1992), Vol. XXX, pp. 261–355.
    [CrossRef]
  2. E. A. Hinds, "Cavity quantum electrodynamics," in Advances in Atomic and Molecular Physics, B. Bederson, ed. (Academic, New York, 1991), Vol. 28, pp. 271–289.
  3. S. Haroche, "Cavity quantum electrodynamics," in Fundamental Systems in Quantum Optics, Les Houches Session LIII, J. Dalibard, J. M. Raimond and J. Zinn-Justin, eds. (Elsevier, New York, 1991), pp. 767–940.
  4. P. Berman, ed., Cavity Quantum Electrodynamics, Supplement 2 of Advances in Atomic, Molecular and Optical Physics series (Academic, San Diego, Calif., 1994).
  5. Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
    [CrossRef]
  6. M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
    [CrossRef] [PubMed]
  7. D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
    [CrossRef]
  8. G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
    [CrossRef] [PubMed]
  9. G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
    [CrossRef] [PubMed]
  10. R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).
  11. L. A. Orozco, "Optical bistability with two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1987).
  12. R. J. Brecha, "Nonclassical photon correlations from two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).
  13. H. J. Kimble, "Structure and dynamics in cavity-quantum electrodynamics" in Ref. 4, pp. 203–265.
  14. M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
    [CrossRef] [PubMed]
  15. M. G. Raizen, "Squeezing and spectroscopy of two-level atoms in a cavity," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).
  16. R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
    [CrossRef] [PubMed]
  17. M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
    [CrossRef] [PubMed]
  18. L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
    [CrossRef]
  19. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
    [CrossRef] [PubMed]
  20. G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
    [CrossRef] [PubMed]
  21. L. A. Lugiato, "Theory of optical bistability," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
    [CrossRef]
  22. H. J. Carmichael, "Theory of quantum fluctuations in optical bistability," in Frontiers in Quantum Optics, E. R. Pike, ed. (Hilger, Bristol, 1986), pp. 120–203.
  23. P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
    [CrossRef]
  24. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
  25. M. Xiao, "Quantum fluctuations in nonlinear optics," Ph.D. dissertation (University of Texas, Austin, Tex., 1988).
  26. D. Z. Anderson, J. C. Frisch, and C. S. Masser, "Mirror reflectometer based on optical cavity decay time," Appl. Opt. 23, 1238–1245 (1984).
    [CrossRef] [PubMed]
  27. D. J. Heinzen and M. S. Feld, "Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator," Phys. Rev. Lett. 59, 2623–2626 (1987).
    [CrossRef] [PubMed]
  28. M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
    [CrossRef] [PubMed]
  29. A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
    [CrossRef] [PubMed]
  30. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
    [CrossRef] [PubMed]
  31. J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.
  32. J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

1994

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

1992

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
[CrossRef] [PubMed]

1991

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

1990

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef] [PubMed]

1989

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

1987

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
[CrossRef]

D. J. Heinzen and M. S. Feld, "Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator," Phys. Rev. Lett. 59, 2623–2626 (1987).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

1986

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

1984

1983

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

1981

P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

1946

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Anderson, D. Z.

Arakawa, Y.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

Bernardot, F.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

Björk, G.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

Boyd, T. L.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

Brecha, R. J.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

R. J. Brecha, "Nonclassical photon correlations from two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).

Brune, M.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Cao, H.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

Carmichael, H. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

H. J. Carmichael, "Theory of quantum fluctuations in optical bistability," in Frontiers in Quantum Optics, E. R. Pike, ed. (Hilger, Bristol, 1986), pp. 120–203.

Davidovich, L.

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Drummond, P. D.

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Feld, M. S.

D. J. Heinzen and M. S. Feld, "Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator," Phys. Rev. Lett. 59, 2623–2626 (1987).
[CrossRef] [PubMed]

Frisch, J. C.

Gardiner, C. W.

P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Goy, P.

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

Gross, M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

Haroche, S.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

S. Haroche, "Cavity quantum electrodynamics," in Fundamental Systems in Quantum Optics, Les Houches Session LIII, J. Dalibard, J. M. Raimond and J. Zinn-Justin, eds. (Elsevier, New York, 1991), pp. 767–940.

Heinzen, D. J.

D. J. Heinzen and M. S. Feld, "Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator," Phys. Rev. Lett. 59, 2623–2626 (1987).
[CrossRef] [PubMed]

Hinds, E. A.

E. A. Hinds, "Cavity quantum electrodynamics," in Advances in Atomic and Molecular Physics, B. Bederson, ed. (Academic, New York, 1991), Vol. 28, pp. 271–289.

Ishikawa, A.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

Jacobson, J. M.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

Kaluzny, Y.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

Kimble, H. J.

R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

H. J. Kimble, "Structure and dynamics in cavity-quantum electrodynamics" in Ref. 4, pp. 203–265.

Klein, N.

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

Lee, W. D.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

Lugiato, L. A.

L. A. Lugiato, "Theory of optical bistability," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
[CrossRef]

Maali, A.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

Masser, C. S.

Meschede, D.

D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
[CrossRef]

Meystre, P.

P. Meystre, "Cavity quantum optics and the quantum measurement process," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1992), Vol. XXX, pp. 261–355.
[CrossRef]

Morin, S. E.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Mossberg, T. W.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Müller, G.

D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
[CrossRef]

Nishioka, M.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

Norris, T. B.

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

Nussenzveig, P.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

Orozco, L. A.

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

L. A. Orozco, "Optical bistability with two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1987).

Pau, S.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

Purcell, E. M.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Raimond, J. M.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

Raizen, M. G.

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

M. G. Raizen, "Squeezing and spectroscopy of two-level atoms in a cavity," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).

Rempe, G.

R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

Rhee, J.-K.

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

Rosenberger, A. T.

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

Schmidt-Kaler, F.

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef] [PubMed]

Thompson, R. J.

R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

Walls, D. F.

P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Walther, H.

G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
[CrossRef]

Weisbuch, C.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

Wu, Q.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Xiao, M.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

M. Xiao, "Quantum fluctuations in nonlinear optics," Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

Yamamoto, Y.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

Zhu, Y.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. B

L. A. Orozco, M. G. Raizen, M. Xiao, R. J. Brecha, and H. J. Kimble, "Squeezed-state generation in optical bistability," J. Opt. Soc. Am. B 4, 1490–1500 (1987).
[CrossRef]

R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, "Mode splitting for two-level atoms inside an optical resonator," J. Opt. Soc. Am. B 3, 238 (1986); R. J. Brecha, L. A. Orozco, M. G. Raizen, M. Xiao, and H. J. Kimble, Bull. Am. Phys. Soc. 31, 1973 (1986).

Phys. Rev.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Phys. Rev. A

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond, "Absorptive optical bistability in two-state atoms," Phys. Rev. A 43, 6284–6302 (1991).
[CrossRef] [PubMed]

P. D. Drummond, C. W. Gardiner, and D. F. Walls, "Quasiprobability methods for nonlinear chemical and optical systems," Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Phys. Rev. Lett.

D. J. Heinzen and M. S. Feld, "Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator," Phys. Rev. Lett. 59, 2623–2626 (1987).
[CrossRef] [PubMed]

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, "From Lamb shifts to light shifts: vacuum and subphoton cavity fields measured by atomic phase-sensitive detection," Phys. Rev. Lett. 72, 3339–3342 (1994).
[CrossRef] [PubMed]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-polariton mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314–3317 (1992).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240–243 (1989).
[CrossRef] [PubMed]

R. J. Thompson, G. Rempe, and H. J. Kimble, "Observation of normal-mode splitting for an atom in an optical cavity," Phys. Rev. Lett. 68, 1132–1135 (1992).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, "Squeezed-state generation by normal modes of a coupled system," Phys. Rev. Lett. 59, 198–201 (1987).
[CrossRef] [PubMed]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear dispersion theory: analysis and experimental observations," Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, "Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727–1730 (1991).
[CrossRef] [PubMed]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, "Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance," Phys. Rev. Lett. 51, 1175–1178 (1983).
[CrossRef]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, "Realization of a two-photon maser oscillator," Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Müller, "One-atom maser," Phys. Rev. Lett. 54, 551–554 (1987).
[CrossRef]

G. Rempe, F. Schmidt-Kaler, and H. Walther, "Observation of sub-Poissonian photon statistics in a micromaser," Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

Other

P. Meystre, "Cavity quantum optics and the quantum measurement process," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1992), Vol. XXX, pp. 261–355.
[CrossRef]

E. A. Hinds, "Cavity quantum electrodynamics," in Advances in Atomic and Molecular Physics, B. Bederson, ed. (Academic, New York, 1991), Vol. 28, pp. 271–289.

S. Haroche, "Cavity quantum electrodynamics," in Fundamental Systems in Quantum Optics, Les Houches Session LIII, J. Dalibard, J. M. Raimond and J. Zinn-Justin, eds. (Elsevier, New York, 1991), pp. 767–940.

P. Berman, ed., Cavity Quantum Electrodynamics, Supplement 2 of Advances in Atomic, Molecular and Optical Physics series (Academic, San Diego, Calif., 1994).

L. A. Lugiato, "Theory of optical bistability," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
[CrossRef]

H. J. Carmichael, "Theory of quantum fluctuations in optical bistability," in Frontiers in Quantum Optics, E. R. Pike, ed. (Hilger, Bristol, 1986), pp. 120–203.

L. A. Orozco, "Optical bistability with two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1987).

R. J. Brecha, "Nonclassical photon correlations from two-level atoms," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).

H. J. Kimble, "Structure and dynamics in cavity-quantum electrodynamics" in Ref. 4, pp. 203–265.

M. G. Raizen, "Squeezing and spectroscopy of two-level atoms in a cavity," Ph.D. dissertation (University of Texas, Austin, Tex., 1990).

M. Xiao, "Quantum fluctuations in nonlinear optics," Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

J.-K. Rhee, T. B. Norris, Y. Arakawa, M. Nishioka, and C. Weisbuch, "Time-resolved study of vacuum-Rabi oscillations in a semiconductor microcavity," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QThE2.

J. M. Jacobson, H. Cao, S. Pau, G. Björk, and Y. Yamamoto, "Direct time-domain observation of microcavity exciton-polariton emission," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper QPD21.

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Figures (11)

Fig. 1
Fig. 1

Dependence of the oscillatory decay on the number of atoms in the cavity, which is proportional to the parameter C (see text). For (a)–(d), μ = 1, the detunings are Δ = Θ = 0, α = 0 (input field turns off completely), and a = 0 (input field turns off instantaneously). (a) C = 1, (b) C = 4, (c) C = 20, (d) C = 100.

Fig. 2
Fig. 2

Dependence of the oscillatory decay on a, the turnoff rate of the input field (see text). For both (a) and (b), C = 50, μ = 1, and the detunings are Δ = Θ = 0, α = 0 (input field turns off completely). (a) a = 2, (b) a = 0.2.

Fig. 3
Fig. 3

Dependence of the oscillatory decay on α, the switching ratio of the input field (see text). Parameters are C = 50, μ = 1, and the detunings are Δ = Θ = 0, a = ∞ (input field turns off infinitely fast). (a) α = 0.2, (b) α = 0.8.

Fig. 4
Fig. 4

Dependence of the oscillatory decay on Θ, the cavity detuning (see text). For both (a) and (b), C = 20, μ = 1, and the detunings are Δ = 0, a = 0, α ~ ∞ (input field turns off infinitely fast). (a) Θ = 0, (b) Θ = 10.

Fig. 5
Fig. 5

Dependence of the oscillatory decay on Δ, the atomic detuning (see text). For both (a) and (b), C = 100, μ = 1, and the detunings are Θ = 0, α = 0, a ~ ∞ (input field turns off infinitely fast). (a) Δ = 0, (b) Δ = 2.

Fig. 6
Fig. 6

Schematic of the experiment. The main components for the experiments described in the text are the electro-optic modulator (EOM), which is used to turn the input field from one level to another. It is driven by a pulse generator, which also triggers the photon-counting system. The time-to-digital converter (TDC) collects the data in the form of standardized pulses originating at the photodetector and stores it in the histogramming memory unit, from which it can later be read by the PC. The atomic beams cross the cavity waist perpendicularly as shown.

Fig. 7
Fig. 7

Data taken by the use of the setup described in the text: (a) the input laser pulse, as recorded with neither atoms nor cavity present, (b) the transmission of the empty cavity, tuned to resonance. τ = 0, the point at which the input field was switched, is marked on each of the figures by the arrow at ~45 ns.

Fig. 8
Fig. 8

Raw data for the number of counts observed per time interval of 0.625 ns, as recorded by the photon-counting electronics, with both atoms and cavity present. Common parameters for all traces are μ = 0.75, a = 2.7, and α = 0.81. The detunings are held close to 0 as well. Measured values of C are (a) C ~ 10, (b) C = 51 ± 11, (c) C = 73 ± 12, (d) C = 130 ± 20.

Fig. 9
Fig. 9

Observed oscillation frequency Ωexp versus theoretical prediction Ωth. The frequency is the inverse of the period determined from the data of Fig. 8. The theoretical value is calculated from the value of the parameter C, which in turn was from a calibrated fluorescence signal.

Fig. 10
Fig. 10

Data from the second experimental configuration for which the input field was turned off more completely. Common parameters for all traces are μ = 0.75, a = 4.5, and α = 0.2. The detunings are held close to 0 as well. Measured values of the cooperativity parameter C are (a) C = 29 ± 7, (b) C = 50 ± 23, (c) C = 101 ± 23, (d) C = 175 ± 37. The input field switching (indicated by the arrow) occurs at ~37 ns.

Fig. 11
Fig. 11

Plot of the peak height versus the square of the cooperativity parameter, with data from the second experimental configuration. Experimental parameters are μ = 0.75, a = 4.5, and α = 0.2.

Equations (10)

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

H ^ = H ^ 0 + H ^ a + H ^ c
= i g [ J ^ - a ^ + a ^ J ^ + ] + ( ω a / 2 ) J ^ z + ω c a ^ a ^ .
d x d τ = - μ 2 ( 1 + i Θ ) x + μ C Γ v + μ 2 y ( τ ) , d v d τ = - 1 2 Γ ( 1 + i Δ ) v - 1 2 Γ x , d m d τ = - [ m ( τ ) + 1 ] .
y ( τ ) = y 1 ,             τ < 0 = ( y 1 - y 2 ) exp ( - a τ ) + y 2 ,             τ > 0 ,
y j = [ ( 1 + 2 C 1 + Δ 2 ) + i ( Θ - 2 C Δ 1 + Δ 2 ) ] x j ,             j = 1 , 2 ,
| x ( τ ) x ( 0 ) | = | A 1 exp ( - b 2 τ ) sin Ω τ + A 2 exp ( - b 2 τ ) × cos Ω τ + A 3 exp ( - a τ ) + A 4 | ,
A 1 - μ C ( 1 - i Δ ) ( 1 + Δ 2 ) Ω - 1 Ω [ 1 4 ( μ - 1 Γ ) + i 4 ( μ Θ - Δ Γ ) ] + d α 2 Ω × [ μ - ( 1 + i Δ ) b ( 1 + 2 C ) - Δ Θ + i ( Δ + Θ ) ] + μ d ( 1 - α ) ( b - 2 a ) 2 + 4 Ω 2 × { 2 Ω + ( b - 2 a ) Ω [ 1 4 ( μ - 1 Γ ) + i 4 ( μ Θ - Δ Γ ) ] } , A 2 1 - α d ( 1 + i Δ ) ( 1 + 2 C ) - Δ Θ + i ( Δ + Θ ) - μ d ( 1 - α ) ( b - 2 a ) 2 + 4 Ω 2 [ ( 1 + i Δ ) Γ - 2 a ] , A 3 μ d ( 1 - α ) ( b - 2 a ) 2 + 4 Ω 2 [ ( 1 + i Δ ) Γ - 2 a ] , A 4 α d ( 1 + i Δ ) ( 1 + 2 C ) - Δ Θ + i ( Δ + Θ ) ,
b 1 2 ( μ + 1 Γ ) + i 2 ( μ Θ + Δ Γ ) , d ( 1 + 2 C 1 + Δ 2 ) + i ( Θ - 2 C Δ 1 + Δ 2 ) , Ω [ μ C 2 Γ - 1 16 ( μ - 1 Γ ) 2 + 1 16 ( μ Θ - Δ Γ ) 2 + i 8 ( μ - 1 Γ ) ( Δ Γ - μ Θ ) ] 1 / 2 .
x ( τ ) = exp ( - b 2 τ ) × [ 1 4 Ω ( 1 Γ - μ - 4 μ C ) sin Ω τ + cos Ω τ ] .
x ( τ ) = x 2 + ( x 1 - x 2 ) exp [ - 2 ( 1 + i Θ ) μ τ ] ,

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