Abstract

Density matrix formalism is used to model two-photon free-induction decay from a ladder-type, three-level scheme with a resonant intermediate state. Counterpropagating light beams are assumed. Only the upper transition is frequency shifted out of resonance with the injected field. Results indicate departure from single-photon behavior; in particular, there is a π phase shift in the oscillations when the Rabi frequencies associated with each transition are comparable in magnitude. The theory is applied to the 32S1/2 (F = 1) − 32P1/2 − 42D3/2 excitation scheme of sodium.

© 1986 Optical Society of America

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References

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  1. M. S. Feld, A. Javan, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions,” Phys. Rev. 177, 540–562 (1969).
    [CrossRef]
  2. Th. Hänsch, P. Toschek, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
    [CrossRef]
  3. B. J. Feldman, M. S. Feld, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions. II. Standing-wave features,” Phys. Rev. A 5, 899–918 (1972).
    [CrossRef]
  4. F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
    [CrossRef]
  5. M. D. Levenson, N. Bloembergen, “Observation of two-photon absorption without Doppler broadening on the 3S–5S transition in sodium vapor,” Phys. Rev. Lett. 32, 645–648 (1974).
    [CrossRef]
  6. T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
    [CrossRef]
  7. J. E. Bjorkholm, P. F. Liao, “Resonant enhancement of two-photon absorption in sodium vapor,” Phys. Rev. Lett. 33, 128–131 (1974).
    [CrossRef]
  8. R. L. Shoemaker, R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
    [CrossRef]
  9. M. M. T. Loy, “Observation of two-photon optical nutation and free-induction decay,” Phys. Rev. Lett. 36, 1454–1457 (1976).
    [CrossRef]
  10. D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
    [CrossRef]
  11. R. G. Brewer, E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
    [CrossRef]
  12. M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
    [CrossRef]
  13. P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
    [CrossRef]
  14. R. G. Brewer, R. L. Shoemaker, “Photon echo and optical nutation in molecules,” Phys. Rev. Lett. 27, 631–634 (1971).
    [CrossRef]
  15. R. M. Whitley, C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976).
    [CrossRef]
  16. P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
    [CrossRef]
  17. P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
    [CrossRef]
  18. C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
    [CrossRef]
  19. R. G. Brewer, R. L. Shoemaker, “Optical free induction decay,” Phys. Rev. A 6, 2001–2007 (1972).
    [CrossRef]
  20. I. V. Hertel, W. Stoll, “Collision experiments with laser excited atoms in crossed beams,” in Advances in Atomic and Molecular Physics, D. R. Bates, B. Bederson, eds. (Academic, New York, 1977), pp. 113–228.
  21. W. L. Wiese, M. W. Smith, B. M. Miles, Atomic Transition Probabilities, Vol. II (National Bureau of Standards, Washington, D.C., 1969).

1984 (1)

C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
[CrossRef]

1980 (1)

P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

1979 (1)

P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
[CrossRef]

1977 (2)

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
[CrossRef]

1976 (2)

R. M. Whitley, C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976).
[CrossRef]

M. M. T. Loy, “Observation of two-photon optical nutation and free-induction decay,” Phys. Rev. Lett. 36, 1454–1457 (1976).
[CrossRef]

1975 (2)

D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

R. G. Brewer, E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[CrossRef]

1974 (4)

F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
[CrossRef]

M. D. Levenson, N. Bloembergen, “Observation of two-photon absorption without Doppler broadening on the 3S–5S transition in sodium vapor,” Phys. Rev. Lett. 32, 645–648 (1974).
[CrossRef]

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

J. E. Bjorkholm, P. F. Liao, “Resonant enhancement of two-photon absorption in sodium vapor,” Phys. Rev. Lett. 33, 128–131 (1974).
[CrossRef]

1972 (3)

R. L. Shoemaker, R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

B. J. Feldman, M. S. Feld, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions. II. Standing-wave features,” Phys. Rev. A 5, 899–918 (1972).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Optical free induction decay,” Phys. Rev. A 6, 2001–2007 (1972).
[CrossRef]

1971 (1)

R. G. Brewer, R. L. Shoemaker, “Photon echo and optical nutation in molecules,” Phys. Rev. Lett. 27, 631–634 (1971).
[CrossRef]

1970 (1)

Th. Hänsch, P. Toschek, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

1969 (1)

M. S. Feld, A. Javan, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions,” Phys. Rev. 177, 540–562 (1969).
[CrossRef]

Bassini, M.

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

Biraben, F.

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
[CrossRef]

Bjorkholm, J. E.

P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
[CrossRef]

J. E. Bjorkholm, P. F. Liao, “Resonant enhancement of two-photon absorption in sodium vapor,” Phys. Rev. Lett. 33, 128–131 (1974).
[CrossRef]

Bloembergen, N.

M. D. Levenson, N. Bloembergen, “Observation of two-photon absorption without Doppler broadening on the 3S–5S transition in sodium vapor,” Phys. Rev. Lett. 32, 645–648 (1974).
[CrossRef]

Brewer, R. G.

R. G. Brewer, E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[CrossRef]

R. L. Shoemaker, R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Optical free induction decay,” Phys. Rev. A 6, 2001–2007 (1972).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Photon echo and optical nutation in molecules,” Phys. Rev. Lett. 27, 631–634 (1971).
[CrossRef]

Cagnac, B.

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
[CrossRef]

Feld, M. S.

B. J. Feldman, M. S. Feld, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions. II. Standing-wave features,” Phys. Rev. A 5, 899–918 (1972).
[CrossRef]

M. S. Feld, A. Javan, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions,” Phys. Rev. 177, 540–562 (1969).
[CrossRef]

Feldman, B. J.

B. J. Feldman, M. S. Feld, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions. II. Standing-wave features,” Phys. Rev. A 5, 899–918 (1972).
[CrossRef]

Gordon, J. P.

P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
[CrossRef]

Grischkowsky, D.

D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

Grynberg, G.

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
[CrossRef]

Hahn, E. L.

R. G. Brewer, E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[CrossRef]

Hannaford, P.

P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
[CrossRef]

Hänsch, T. W.

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

Hänsch, Th.

Th. Hänsch, P. Toschek, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

Harvey, K. C.

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

Hertel, I. V.

I. V. Hertel, W. Stoll, “Collision experiments with laser excited atoms in crossed beams,” in Advances in Atomic and Molecular Physics, D. R. Bates, B. Bederson, eds. (Academic, New York, 1977), pp. 113–228.

Javan, A.

M. S. Feld, A. Javan, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions,” Phys. Rev. 177, 540–562 (1969).
[CrossRef]

Knight, P. L.

P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. Bloembergen, “Observation of two-photon absorption without Doppler broadening on the 3S–5S transition in sodium vapor,” Phys. Rev. Lett. 32, 645–648 (1974).
[CrossRef]

Liao, P. F.

D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

J. E. Bjorkholm, P. F. Liao, “Resonant enhancement of two-photon absorption in sodium vapor,” Phys. Rev. Lett. 33, 128–131 (1974).
[CrossRef]

Liao, P. L.

P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
[CrossRef]

Loy, M. M. T.

M. M. T. Loy, “Observation of two-photon optical nutation and free-induction decay,” Phys. Rev. Lett. 36, 1454–1457 (1976).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

MacGillivray, W. R.

C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
[CrossRef]

P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
[CrossRef]

Meisel, G.

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

Miles, B. M.

W. L. Wiese, M. W. Smith, B. M. Miles, Atomic Transition Probabilities, Vol. II (National Bureau of Standards, Washington, D.C., 1969).

Milonni, P. W.

P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Schawlow, A. L.

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

Shoemaker, R. L.

R. L. Shoemaker, R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Optical free induction decay,” Phys. Rev. A 6, 2001–2007 (1972).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Photon echo and optical nutation in molecules,” Phys. Rev. Lett. 27, 631–634 (1971).
[CrossRef]

Smith, M. W.

W. L. Wiese, M. W. Smith, B. M. Miles, Atomic Transition Probabilities, Vol. II (National Bureau of Standards, Washington, D.C., 1969).

Standage, M. C.

C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
[CrossRef]

P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
[CrossRef]

Stoll, W.

I. V. Hertel, W. Stoll, “Collision experiments with laser excited atoms in crossed beams,” in Advances in Atomic and Molecular Physics, D. R. Bates, B. Bederson, eds. (Academic, New York, 1977), pp. 113–228.

Stroud, C. R.

R. M. Whitley, C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976).
[CrossRef]

Toschek, P.

Th. Hänsch, P. Toschek, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

Webb, C. J.

C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
[CrossRef]

Whitley, R. M.

R. M. Whitley, C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976).
[CrossRef]

Wiese, W. L.

W. L. Wiese, M. W. Smith, B. M. Miles, Atomic Transition Probabilities, Vol. II (National Bureau of Standards, Washington, D.C., 1969).

J. Phys. B (1)

C. J. Webb, W. R. MacGillivray, M. C. Standage, “The theory of stepwise electron and laser excitation of atoms: II. Strong optical excitation case,” J. Phys. B 17, 2577–2589 (1984).
[CrossRef]

J. Phys. B. (1)

P. Hannaford, W. R. MacGillivray, M. C. Standage, “Coherent optical transient measurements of absolute and differential Stark shifts in atomic sodium,” J. Phys. B. 12, 4033–4044 (1979).
[CrossRef]

Opt. Commun. (2)

M. Bassini, F. Biraben, B. Cagnac, G. Grynberg, “Raman transients observed in Doppler free two-photon excitation,” Opt. Commun. 21, 263–266 (1977).
[CrossRef]

T. W. Hänsch, K. C. Harvey, G. Meisel, A. L. Schawlow, “Two-photon spectroscopy of Na 3S–4D without Doppler broadening using a cw dye laser,” Opt. Commun. 11, 50–53 (1974).
[CrossRef]

Phys. Rep. (1)

P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Phys. Rev. (1)

M. S. Feld, A. Javan, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions,” Phys. Rev. 177, 540–562 (1969).
[CrossRef]

Phys. Rev. A (5)

B. J. Feldman, M. S. Feld, “Laser-induced line-narrowing effects in coupled Doppler-broadened transitions. II. Standing-wave features,” Phys. Rev. A 5, 899–918 (1972).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

R. G. Brewer, E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Optical free induction decay,” Phys. Rev. A 6, 2001–2007 (1972).
[CrossRef]

R. M. Whitley, C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976).
[CrossRef]

Phys. Rev. Lett. (7)

P. L. Liao, J. E. Bjorkholm, J. P. Gordon, “Observation of two-photon optical free-induction decay in atomic sodium vapor,” Phys. Rev. Lett. 39, 15–18 (1977).
[CrossRef]

R. G. Brewer, R. L. Shoemaker, “Photon echo and optical nutation in molecules,” Phys. Rev. Lett. 27, 631–634 (1971).
[CrossRef]

F. Biraben, B. Cagnac, G. Grynberg, “Experimental evidence of two-photon transition without Doppler broadening,” Phys. Rev. Lett. 32, 643–645 (1974).
[CrossRef]

M. D. Levenson, N. Bloembergen, “Observation of two-photon absorption without Doppler broadening on the 3S–5S transition in sodium vapor,” Phys. Rev. Lett. 32, 645–648 (1974).
[CrossRef]

J. E. Bjorkholm, P. F. Liao, “Resonant enhancement of two-photon absorption in sodium vapor,” Phys. Rev. Lett. 33, 128–131 (1974).
[CrossRef]

R. L. Shoemaker, R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

M. M. T. Loy, “Observation of two-photon optical nutation and free-induction decay,” Phys. Rev. Lett. 36, 1454–1457 (1976).
[CrossRef]

Z. Phys. (1)

Th. Hänsch, P. Toschek, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

Other (2)

I. V. Hertel, W. Stoll, “Collision experiments with laser excited atoms in crossed beams,” in Advances in Atomic and Molecular Physics, D. R. Bates, B. Bederson, eds. (Academic, New York, 1977), pp. 113–228.

W. L. Wiese, M. W. Smith, B. M. Miles, Atomic Transition Probabilities, Vol. II (National Bureau of Standards, Washington, D.C., 1969).

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Figures (6)

Fig. 1
Fig. 1

Atomic energy-level diagram illustrating the counterpropagating light-beam interactions. Stimulated processes between levels are represented by the double-headed arrows, and spontaneous decays by the single-headed arrows.

Fig. 2
Fig. 2

Numerical FID curves for ν2 = 40 MHz, δ = δ′ = 0, δs = 100 MHz, and (a) —, ν1 = 400 MHz; (b) - - - -, ν1 = 200 MHz; (c) -·-·-, ν1 = 40 MHz.

Fig. 3
Fig. 3

FID curves calculated for δ = δ′ = 0, δs = 200 MHz, and (a) —, ν1 = 40 MHz, ν2 = 80 MHz; (b) - - - -, ν1 = 40 MHz, ν2 = 200 MHz; (c) -·-·-, ν1 = 200 MHz, ν2 = 200 MHz. The scale of the Y axis has been doubled for (c).

Fig. 4
Fig. 4

FID curves for ν1 = 400 MHz, ν2 = 40 MHz, δs = 100 MHz, and (a) —, δ = −100 MHz, δ′ = 100 MHz; (b) - - - -, δ = δ′ = 100 MHz.

Fig. 5
Fig. 5

Excitation and relaxation scheme for the sodium levels 3S1/2(F = 1) → 3P1/2 → 4D3/2. The values inserted on the relaxation arrows are the relaxation rates Γ × 10−8(sec−1).21

Fig. 6
Fig. 6

FID curves after an initial preparation time of 200 nsec for ν1 = 400 MHz, ν2 = 40 MHz, δ = δ′ = 0, δs =100 MHz. (a) —, three-level system; (b) - - - -, four-level system. See text for details.

Equations (72)

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

Ω 1 ω 2 - ω 1 ω 21
Ω 2 ω 3 - ω 2 ω 32 .
α = μ 12 E 1 2 ,
β = μ 23 E 2 2 ,
g ( v ) = exp ( - v 2 / v 0 2 ) / [ v 0 ( π ) 1 / 2 ] ,
ρ ˙ 11 = i α ( ρ ˜ 21 - ρ ˜ 12 ) + ρ 22 / T 1 ( 21 ) ,
ρ ˙ 22 = i β ( ρ ˜ 32 - ρ ˜ 23 ) - i α ( ρ ˜ 21 - ρ ˜ 12 ) - ρ 22 / T 1 ( 21 ) + ρ 33 / T 1 ( 32 ) ,
ρ ˙ 33 = - i β ( ρ ˜ 32 - ρ ˜ 23 ) - ρ 33 / T 1 ( 32 )
ρ ˜ ˙ 12 = - i Δ ρ ˜ 12 + i α ( ρ 22 - ρ 11 ) - i β ρ ˜ 13 - ρ ˜ 12 / T 2 ( 21 ) ,
ρ ˜ ˙ 23 = - i Δ ρ ˜ 23 + i β ( ρ 33 - ρ ˜ 22 ) + i α ρ ˜ 13 - ρ ˜ 23 / T 2 ( 32 ) ,
ρ ˜ ˙ 13 = - i ( Δ + Δ ) ρ ˜ 13 + i α ρ ˜ 23 - i β ρ ˜ 12 - ρ ˜ 13 / T 2 ( 31 ) ,
ρ 12 = ρ ˜ 12 exp [ i ( Ω 1 t - k 1 z ) ] ,
ρ 23 = ρ ˜ 23 exp [ i ( Ω 2 ) t + k 2 z ] ,
ρ 13 = ρ ˜ 13 exp { i [ ( Ω 1 + Ω 2 ) t - ( k 1 - k 2 ) z ] } ,
Δ = Ω 1 - ω 21 - k 1 v z ,
Δ = Ω 2 - ω 32 + k 2 v z .
ρ ˜ j i = ρ ˜ i j *
i ρ i i = 1.
T 2 ( 21 ) = 2 T 1 ( 21 ) ,
T 2 ( 31 ) = 2 T 1 ( 32 ) ,
T 2 ( 32 ) = 2 / [ 1 / T 1 ( 32 ) + 1 / T 1 ( 21 ) ] .
P 2 ( z , t ) = N μ 23 [ ρ 32 ( z , t ) AVE + c . c . ] ,
P 2 ( z , t ) = N μ 23 { ρ ˜ 32 ( z , t ) AVE exp [ - i ( Ω 2 t + k 2 z ) ] + c . c . } ,
P 2 ( z , t ) = P 0 ( z , t ) exp [ - i ( Ω 2 t + k 2 z ) ] + c . c .
E s ( z , t ) = E 0 ( z , t ) exp [ - i ( Ω 2 t + k 2 z ) ] + c . c .
2 E s z 2 - 1 c 2 2 E s t 2 = 1 0 c 2 2 P 2 t 2 .
k 2 E 0 E 0 z ;             Ω 2 E 0 E 0 t ,             Ω 2 P 0 P 0 t .
k = ω / c .
E 0 z - 1 c E 0 t = k 2 i 0 P 0
E 0 ( z , t ) z = k 2 i 0 P 0 ( z , t ) .
E 0 ( t ) = - i k 2 0 L 0 P 0 ( z t ) d z .
E 0 ( t ) = i L k 2 0 P 0 ( t ) .
I α E 2 + E 0 2 E 2 2 + 2 E 2 × Re E 0 ,
Re E 0 = L k N μ 23 2 0 Im ρ ˜ 23 ( v , t ) AVE ,
Re ρ ˜ ˙ 23 = ( Δ + δ ω 23 ) Im ρ ˜ 23 - α Im ρ ˜ 13 - Re ρ 23 / T 2 ( 32 ) ,
Im ρ ˜ ˙ 23 = - ( Δ + δ ω 23 ) Re ρ ˜ 23 + α Re ρ ˜ 13 - Im ρ ˜ 23 / T 2 ( 32 ) ,
Re ρ ˜ ˙ 13 = ( Δ + Δ + δ ω 23 ) Im ρ ˜ 13 - α Im ρ ˜ 23 - Re ρ ˜ 13 / T 2 ( 31 ) ,
Im ρ ˜ ˙ 13 = - ( Δ + Δ + δ ω 23 ) Re ρ ˜ 13 + α Re ρ ˜ 23 - Im ρ ˜ 13 / T 2 ( 31 ) ,
Re E 0 ( t ) = C - Im ρ ˜ 23 ( v , t ) g ( v ) d v .
ν 1 = α / π ,
ν 2 = β / π ,
δ = Δ ( v z = 0 ) / 2 π ,
δ = Δ ( v z = 0 ) / 2 π ,
δ s = δ ω 23 / 2 π ,
T 1 ( 21 ) = 16 nsec ,
T 1 ( 32 ) = 50 nsec .
ρ ˙ 44 = Γ 24 ρ 22 ,
ρ ˙ 11 = i α ( ρ ˜ 21 - ρ ˜ 12 ) + Γ 21 ρ 22 ,
Γ 21 = Γ 24 = 1 2 T 1 ( 21 ) .
ρ ˜ 12 = A + i B ,
ρ ˜ 23 = C + i D ,
ρ ˜ 13 = E + i F .
ρ 11 = 1 - ρ 22 - ρ 33 ,
ρ 22 = - 2 α T 1 ( 21 ) B ,
ρ 33 = - 2 β T 1 ( 32 ) D ,
A = T 2 ( 21 ) ( Δ B + β F ) ,
C = T 2 ( 32 ) ( Δ D - α F ) ,
E = T 2 ( 31 ) [ β B - α D + ( Δ + Δ ) F ] ,
B = ( R D - P F ) / O ,
D = W F / X ,
F = α / ( Z - Y W / X ) ,
Z = P Q / O - O ,
Y = R Q / O + U ,
X = S R / O - T ,
W = S P / O - R ,
U = α β [ 2 T 1 ( 32 ) - T 2 ( 31 ) ] ,
P = T 2 ( 31 ) ( Δ + Δ ) 2 + β 2 T 2 ( 21 ) + α 2 T 2 ( 32 ) + 1 / T 2 ( 31 ) ,
R = α [ T 2 ( 31 ) ( Δ + Δ ) + T 2 ( 32 ) Δ ] ,
T = 2 β 2 T 1 ( 32 ) + α 2 T 2 ( 31 ) + Δ 2 T 2 ( 32 ) + 1 / T 2 ( 32 ) ,
S = α β [ 2 T 1 ( 31 ) + T 2 ( 31 ) ] ,
Q = 4 α 2 T 1 ( 21 ) + β 2 T 2 ( 31 ) + Δ 2 T 2 ( 21 ) + 1 / T 2 ( 21 ) ,
O = β [ T 2 ( 31 ) ( Δ + Δ ) + Δ T 2 ( 21 ) ] .

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