Abstract

We present an analytical expression for the response of a two-level atom to a frequency-modulated optically coherent pulse train. The optical beam has sinusoidal frequency modulation and is chopped to have a square-wave envelope. We assume that the laser pulses are short compared with the atomic-decay time, the pulse-repetition time, and the modulation period. With this short-pulse assumption we are able to use a method similar to Temkin’s [J. Opt. Soc. Am. B 10, 830–839 (1993)] and solve the optical Bloch equations in closed form.

© 1998 Optical Society of America

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  1. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorption and dispersion,” Opt. Lett. 5, 15–17 (1980);G. C. Bjorklund, “Method and device for detecting a specific spectral feature,” U.S. patent4,297,035 (October27, 1981).
    [Crossref]
  2. J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994), and references therein.
    [Crossref] [PubMed]
  3. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B: Photophys. Laser Chem. 32, 145–152 (1983).
    [Crossref]
  4. J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
    [Crossref] [PubMed]
  5. Xiang Zhu and D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B 14, 1945–1950 (1997).
    [Crossref]
  6. G. S. Agarwal, “Frequency-modulated spectra of coherently driven systems,” Phys. Rev. A 23, 1375–1381 (1981).
    [Crossref]
  7. N. Nayak and G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
    [Crossref] [PubMed]
  8. H.-R. Xia, J. I. Cirac, S. Swartz, B. Kohler, D. S. Elliott, J. L. Hall, and P. Zoller, “Phase shifts and intensity dependence in frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 11, 721–730 (1994).
    [Crossref]
  9. W. M. Ruyten, “Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach,” Phys. Rev. A 42, 4226–4245 (1990).
    [Crossref] [PubMed]
  10. M. A. Kramer, R. W. Boyd, L. W. Hillman, and C. R. Stroud, “Propagation of modulated optical fields through saturable-absorbing media: a general theory of modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1444–1455 (1985).
    [Crossref]
  11. A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
    [Crossref]
  12. W. M. Ruyten, “Comment on absorption and fluorescence in strong frequency–modulated and amplitude-modulated fields,” Phys. Rev. A 39, 442–444 (1989).
    [Crossref] [PubMed]
  13. A. V. Alekseev and N. V. Sushilov, “Analytic solutions of Bloch and Maxwell–Bloch equations in the case of arbitrary field amplitude and phase modulation,” Phys. Rev. A 46, 351–355 (1992).
    [Crossref] [PubMed]
  14. Ping Koy Lam and C. M. Savage, “Complete atomic population inversion using correlated sidebands,” Phys. Rev. A 50, 3500–3504 (1994).
    [Crossref] [PubMed]
  15. S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
    [Crossref]
  16. Jun Ye, Long-Sheng Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6–15 (1998).
    [Crossref]
  17. K. Namjou, S. Cai, E. A. Whittaker, J. Faist, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Sensitive absorption spectroscopy with a room-temperature distributed-feedback quantum-cascade laser,” Opt. Lett. 23, 219–221 (1998).
    [Crossref]
  18. J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Appl. Phys. Lett. 70, 2670–2672 (1997).
    [Crossref]
  19. T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, and W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28–30 (1982).
    [Crossref] [PubMed]
  20. N. H. Tran, R. Kachru, T. F. Gallagher, J. P. Watjen, and G. C. Bjorklund, “Pulsed frequency-modulation spectroscopy at 3302 Å,” Opt. Lett. 8, 157–159 (1983).
    [Crossref] [PubMed]
  21. N. H. Tran, R. Kachru, P. Pillet, H. B. van Linden van den Heuvell, T. F. Gallagher, and J. P. Watjen, “Frequency-modulation spectroscopy with a pulsed dye laser: experimental investigations of sensitivity and useful features,” Appl. Opt. 23, 1353–1360 (1984).
    [Crossref]
  22. E. E. Eyler, S. Gangopadhyay, N. Melikechi, J. C. Bloch, and R. W. Field, “Frequency-modulation spectroscopy with transform-limited nanosecond laser pulses,” Opt. Lett. 21, 225–227 (1996).
    [Crossref] [PubMed]
  23. P. Peterson and A. Gavrielides, “Periodic response of the Bloch equations to a phase modulated pulse train; an application to mesospheric sodium,” Opt. Commun. 104, 53–56 (1993).
    [Crossref]
  24. N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
    [Crossref] [PubMed]
  25. F. T. Hioe, “Solution of Bloch equation involving amplitude and frequency modulations,” Phys. Rev. A 30, 2100–2103 (1984).
    [Crossref]
  26. L. C. Bradley, “Pulse-train excitation of sodium for use as a synthetic beacon,” J. Opt. Soc. Am. B 9, 1931–1944 (1992).
    [Crossref]
  27. G. F. Thomas, “Excitation of a multilevel system by a train of identical phase-coherent Gaussian-shaped laser pulses,” Phys. Rev. A 41, 1645–1652 (1990).
    [Crossref] [PubMed]
  28. P. W. Milonni and L. E. Thode, “Theory of mesospheric sodium fluorescence excited by pulse trains,” Appl. Opt. 31, 785–800 (1992).
    [Crossref] [PubMed]
  29. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
    [Crossref]
  30. P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), especially Chaps. 6 and 8.
  31. P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, New York, 1990).
  32. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
  33. B. W. Shore, The Theory of Coherent Atomic Excitation: Simple Atoms and Fields (Wiley, New York, 1990), Vol. 1, especially Chap. 8.
  34. R. J. Temkin, “Excitation of an atom by a train of short pulses,” J. Opt. Soc. Am. B 10, 830–839 (1993).
    [Crossref]

1998 (2)

1997 (2)

1996 (1)

1995 (1)

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref] [PubMed]

1994 (3)

1993 (2)

R. J. Temkin, “Excitation of an atom by a train of short pulses,” J. Opt. Soc. Am. B 10, 830–839 (1993).
[Crossref]

P. Peterson and A. Gavrielides, “Periodic response of the Bloch equations to a phase modulated pulse train; an application to mesospheric sodium,” Opt. Commun. 104, 53–56 (1993).
[Crossref]

1992 (4)

1990 (2)

G. F. Thomas, “Excitation of a multilevel system by a train of identical phase-coherent Gaussian-shaped laser pulses,” Phys. Rev. A 41, 1645–1652 (1990).
[Crossref] [PubMed]

W. M. Ruyten, “Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach,” Phys. Rev. A 42, 4226–4245 (1990).
[Crossref] [PubMed]

1989 (1)

W. M. Ruyten, “Comment on absorption and fluorescence in strong frequency–modulated and amplitude-modulated fields,” Phys. Rev. A 39, 442–444 (1989).
[Crossref] [PubMed]

1985 (2)

N. Nayak and G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[Crossref] [PubMed]

M. A. Kramer, R. W. Boyd, L. W. Hillman, and C. R. Stroud, “Propagation of modulated optical fields through saturable-absorbing media: a general theory of modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1444–1455 (1985).
[Crossref]

1984 (2)

1983 (2)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B: Photophys. Laser Chem. 32, 145–152 (1983).
[Crossref]

N. H. Tran, R. Kachru, T. F. Gallagher, J. P. Watjen, and G. C. Bjorklund, “Pulsed frequency-modulation spectroscopy at 3302 Å,” Opt. Lett. 8, 157–159 (1983).
[Crossref] [PubMed]

1982 (2)

1981 (1)

G. S. Agarwal, “Frequency-modulated spectra of coherently driven systems,” Phys. Rev. A 23, 1375–1381 (1981).
[Crossref]

1980 (1)

1976 (1)

S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
[Crossref]

1957 (1)

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[Crossref]

Agarwal, G. S.

N. Nayak and G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[Crossref] [PubMed]

G. S. Agarwal, “Frequency-modulated spectra of coherently driven systems,” Phys. Rev. A 23, 1375–1381 (1981).
[Crossref]

Alekseev, A. V.

A. V. Alekseev and N. V. Sushilov, “Analytic solutions of Bloch and Maxwell–Bloch equations in the case of arbitrary field amplitude and phase modulation,” Phys. Rev. A 46, 351–355 (1992).
[Crossref] [PubMed]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Bjorklund, G. C.

Bloch, J. C.

Boyd, R. W.

Bradley, L. C.

Brewer, R. G.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[Crossref]

Cai, S.

Capasso, F.

Cassidy, D. T.

Cho, A. Y.

Cirac, J. I.

DeVoe, R. G.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[Crossref]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), especially Chaps. 6 and 8.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Elliott, D. S.

Eyler, E. E.

Faist, J.

Feneuille, S.

S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
[Crossref]

Feynman, R. P.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[Crossref]

Field, R. W.

Gallagher, T. F.

Gangopadhyay, S.

Gavrielides, A.

P. Peterson and A. Gavrielides, “Periodic response of the Bloch equations to a phase modulated pulse train; an application to mesospheric sodium,” Opt. Commun. 104, 53–56 (1993).
[Crossref]

Gmachl, C.

Gounand, F.

Hall, J. L.

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[Crossref]

Hillman, L. W.

Hioe, F. T.

F. T. Hioe, “Solution of Bloch equation involving amplitude and frequency modulations,” Phys. Rev. A 30, 2100–2103 (1984).
[Crossref]

Hutchinson, A. L.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Appl. Phys. Lett. 70, 2670–2672 (1997).
[Crossref]

Kachru, R.

Knight, P. L.

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref] [PubMed]

Kohler, B.

Kramer, M. A.

Lam, Ping Koy

Ping Koy Lam and C. M. Savage, “Complete atomic population inversion using correlated sidebands,” Phys. Rev. A 50, 3500–3504 (1994).
[Crossref] [PubMed]

Lenth, W.

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B: Photophys. Laser Chem. 32, 145–152 (1983).
[Crossref]

Ma, Long-Sheng

Melikechi, N.

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, New York, 1990).

Milonni, P. W.

Namjou, K.

Nayak, N.

N. Nayak and G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[Crossref] [PubMed]

Oliver, G.

S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
[Crossref]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B: Photophys. Laser Chem. 32, 145–152 (1983).
[Crossref]

Peterson, P.

P. Peterson and A. Gavrielides, “Periodic response of the Bloch equations to a phase modulated pulse train; an application to mesospheric sodium,” Opt. Commun. 104, 53–56 (1993).
[Crossref]

Pillet, P.

Ruyten, W. M.

W. M. Ruyten, “Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach,” Phys. Rev. A 42, 4226–4245 (1990).
[Crossref] [PubMed]

W. M. Ruyten, “Comment on absorption and fluorescence in strong frequency–modulated and amplitude-modulated fields,” Phys. Rev. A 39, 442–444 (1989).
[Crossref] [PubMed]

Sargent, M.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, New York, 1990).

Savage, C. M.

Ping Koy Lam and C. M. Savage, “Complete atomic population inversion using correlated sidebands,” Phys. Rev. A 50, 3500–3504 (1994).
[Crossref] [PubMed]

Schenzle, A.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[Crossref]

Schweighofer, M.-G.

S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
[Crossref]

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation: Simple Atoms and Fields (Wiley, New York, 1990), Vol. 1, especially Chap. 8.

Silver, J. A.

Sirtori, C.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Appl. Phys. Lett. 70, 2670–2672 (1997).
[Crossref]

Sivco, D. L.

Stroud, C. R.

Supplee, J. M.

Sushilov, N. V.

A. V. Alekseev and N. V. Sushilov, “Analytic solutions of Bloch and Maxwell–Bloch equations in the case of arbitrary field amplitude and phase modulation,” Phys. Rev. A 46, 351–355 (1992).
[Crossref] [PubMed]

Swartz, S.

Temkin, R. J.

Thode, L. E.

Thomas, G. F.

G. F. Thomas, “Excitation of a multilevel system by a train of identical phase-coherent Gaussian-shaped laser pulses,” Phys. Rev. A 41, 1645–1652 (1990).
[Crossref] [PubMed]

Tran, N. H.

van Linden van den Heuvell, H. B.

Vernon, F. L.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[Crossref]

Vitanov, N. V.

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref] [PubMed]

Watjen, J. P.

Whittaker, E. A.

Xia, H.-R.

Ye, Jun

Zhu, Xiang

Zoller, P.

Appl. Opt. (4)

Appl. Phys. B: Photophys. Laser Chem. (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B: Photophys. Laser Chem. 32, 145–152 (1983).
[Crossref]

Appl. Phys. Lett. (1)

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Appl. Phys. Lett. 70, 2670–2672 (1997).
[Crossref]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[Crossref]

J. Opt. Soc. Am. B (6)

J. Phys. B (1)

S. Feneuille, M.-G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 2003–2009 (1976).
[Crossref]

Opt. Commun. (1)

P. Peterson and A. Gavrielides, “Periodic response of the Bloch equations to a phase modulated pulse train; an application to mesospheric sodium,” Opt. Commun. 104, 53–56 (1993).
[Crossref]

Opt. Lett. (5)

Phys. Rev. A (10)

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[Crossref]

W. M. Ruyten, “Comment on absorption and fluorescence in strong frequency–modulated and amplitude-modulated fields,” Phys. Rev. A 39, 442–444 (1989).
[Crossref] [PubMed]

A. V. Alekseev and N. V. Sushilov, “Analytic solutions of Bloch and Maxwell–Bloch equations in the case of arbitrary field amplitude and phase modulation,” Phys. Rev. A 46, 351–355 (1992).
[Crossref] [PubMed]

Ping Koy Lam and C. M. Savage, “Complete atomic population inversion using correlated sidebands,” Phys. Rev. A 50, 3500–3504 (1994).
[Crossref] [PubMed]

G. S. Agarwal, “Frequency-modulated spectra of coherently driven systems,” Phys. Rev. A 23, 1375–1381 (1981).
[Crossref]

N. Nayak and G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[Crossref] [PubMed]

W. M. Ruyten, “Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach,” Phys. Rev. A 42, 4226–4245 (1990).
[Crossref] [PubMed]

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref] [PubMed]

F. T. Hioe, “Solution of Bloch equation involving amplitude and frequency modulations,” Phys. Rev. A 30, 2100–2103 (1984).
[Crossref]

G. F. Thomas, “Excitation of a multilevel system by a train of identical phase-coherent Gaussian-shaped laser pulses,” Phys. Rev. A 41, 1645–1652 (1990).
[Crossref] [PubMed]

Other (4)

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), especially Chaps. 6 and 8.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, New York, 1990).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

B. W. Shore, The Theory of Coherent Atomic Excitation: Simple Atoms and Fields (Wiley, New York, 1990), Vol. 1, especially Chap. 8.

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Figures (3)

Fig. 1
Fig. 1

Inversion versus detuning. M=0.25, ωm=0.15 GHz, γ=0.02 GHz, tr=10 ns, tp=0.2 ns, pulse area is π/8. The major peaks are at zero detuning and the first optical Ramsey fringe (trΔ=2π). The modulation sidebands are clearly visible. The solid curve shows w after 30 complete cycles (the instant the 31st pulse turns on). The dashed curve shows w after 50 complete cycles.

Fig. 2
Fig. 2

Inversion (after 30 complete cycles) versus modulation frequency. M=0.25, Δ=0.314 GHz, γ=0.02 GHz, tr=10 ns, tp=0.2 ns, pulse area is π/8. w remains near -1 until the modulation frequency is sufficient to make one sideband nearly resonant with the atom. This curve is very sensitive to the number of pulses.

Fig. 3
Fig. 3

Inversion versus time. M=0.25, Δ=0.314 GHz, ωm=0.307 GHz, γ=0.02 GHz, tr=10 ns, tp=0.2 ns, pulse area is π/8. (Parameters correspond to a peak in Fig. 2.) The main oscillation, period 439 ns, is the beating of the pulse-repetition frequency with the second harmonic of the modulation frequency.

Equations (26)

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

E(t)=E0 exp[i(ω0t+M sin ωmt)]=E0 exp(iM sin ωmt)exp(iω0t).
E(t)=E(t)exp(iω0t),
E(t)=E0 exp(iM sin ωmt).
Re(E )=E0 cos(M sin ωmt),
Im(E )=E0 sin(M sin ωmt).
u˙=-βu-Δv+κ Im(E )w,
v˙=Δu-βv+κ Re(E )w,
w˙=-κ Im(E )u-κ Re(E )v-γ(w+1).
Δωatom-ω0,
δ(M sin ωmt)2π.
tp2πMωm.
tp1γ.
u˙=-Δv+bnw,
v˙=Δu+anw,
w˙=-bnu-anv,
ddt uvw=0-ΔbnΔ0an-bn-an0 uvw.
anκ Re[E(t)]=χ cos(M sin ωmtn),
bnκ Im[E(t)]=χ sin(M sin ωmtn),
u(τ)v(τ)w(τ)1=An(τ)unvnwn1,
An(τ)=1Ω2an2+(Δ2+bn2)cos Ωτ-anbn(1-cos Ωτ)-ΔΩ sin Ωτ-Δan(1-cos Ωτ)+Ωbn sin Ωτ0-anbn(1-cos Ωτ)+ΔΩ sin Ωτbn2+(Δ2+an2)cos ΩτΔbn(1-cos Ωτ)+Ωan sin Ωτ0-Δan(1-cos Ωτ)-Ωbn sin ΩτΔbn(1-cos Ωτ)-Ωan sin ΩτΔ2+χ2 cos Ωτ0000Ω2.
u˙=-γ2 u-Δv,
v˙=Δu-γ2 v,
w˙=-γ(w+1).
u(τ)v(τ)w(τ)1=B(τ)unvnwn1,
B(τ)=x cos Δτ-x sin Δτ00x sin Δτx cos Δτ0000x2x2-10001.
uvw1=B(τ)Aj(tp)n=j-11B(tr-tp)An(tp) 00-11.

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