Abstract

The approximate analytical solution of equations for the density matrix of three-level atoms moving in a strong bichromatic pumping field and in a weak probe-wave field (V system) is obtained. It is demonstrated that two series of resonances with different parameters appear in the gain spectrum of the probe wave. When certain conditions are met, some resonances are found to be free from Doppler broadening. The relations describing the resonance broadening and shifts are derived.

© 1992 Optical Society of America

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  1. Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
    [CrossRef]
  2. N. C. Kothari, G. S. Agarwal, “Effects of collisional mixing on subharmonic Raman resonances in nonlinear spectroscopy,” Opt. Commun. 74, 342 (1990).
    [CrossRef]
  3. G. S. Agarwal, S. S. Jha, “Theory of resonant Raman scattering of intense optical waves,” J. Phys. B 12, 2655 (1979).
    [CrossRef]
  4. H. R. Schlossberg, A. Javan, “Saturation behavior of a Doppler-broadened transition involving levels with closely spaced structure,” Phys. Rev. 150, 267 (1966).
    [CrossRef]
  5. S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).
  6. S. Stenholm, Foundations of Laser Spectroscopy (Wiley, New York, 1987).
  7. G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53, 1 (1979).
    [CrossRef]
  8. M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).
  9. J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
    [CrossRef]
  10. B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
    [CrossRef]
  11. W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
    [CrossRef]
  12. E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
    [CrossRef]
  13. S. Stenholm, W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618 (1969).
    [CrossRef]
  14. B. J. Feldman, M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375 (1970).
    [CrossRef]
  15. A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).
  16. I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).
  17. L. S. Gaida, S. A. Pulkin, “Three-level atomic system in the strong resonance bichromatic optical field and the weak probe field on consequent transition,” Opt. Spectrosc. (USSR) 67, 761 (1989).
  18. Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
    [CrossRef] [PubMed]
  19. S. Chakmakjian, K. Koch, C. R. Stroud, “Observation of resonances at subharmonics of the Rabi frequency in the saturated absorption of a 100% amplitude-modulated laser beam,” J. Opt. Soc. Am. B 5, 2015 (1988).
    [CrossRef]
  20. B. R. Mollow, “Power spectrum of light scattered by two-level system,” Phys. Rev. 188, 1969 (1969).
    [CrossRef]
  21. B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
    [CrossRef]
  22. L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
    [CrossRef] [PubMed]
  23. T.-S. Ho, S.-I. Chu, “Semiclassical many-mode Floquet theory: II. Non-linear multiphoton dynamics of a two-level system in a strong bichromatic field,” J. Phys. B 17, 2101 (1984).
    [CrossRef]
  24. M. Ya. Agre, L. P. Rappoport, “Generalized quasienergy states and the atom spectrum in the intense multimode radiation field,” Sov. Phys. JETP 90, 1154 (1986).
  25. A. K. Popov, Introduction to Nonlinear Spectroscopy (Nauka, Novosibirsk, USSR, 1983).
  26. A. Ya. Kazakov, “Two-level atom in a strong polychromatic electromagnetic field,” Sov. Phys. JETP 99, 705 (1991).
  27. H. Freedhoff, Z. Chen, “Resonance fluorescence of a two-level atom in a strong bichromatic field,” Phys. Rev. A 41, 6013 (1990).
    [CrossRef] [PubMed]
  28. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).
  29. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  30. L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
    [CrossRef]
  31. L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operations,” J. Opt. Soc. Am. B 2, 211 (1985).
    [CrossRef]
  32. T. W. Mossberg, M. Lewenstein, “Stability and phase-dependent dynamics of an atom driven by strong resonant field and a tunable moderate-strength perturber field,” Phys. Rev. A 39, 163 (1989).
    [CrossRef] [PubMed]

1991 (1)

A. Ya. Kazakov, “Two-level atom in a strong polychromatic electromagnetic field,” Sov. Phys. JETP 99, 705 (1991).

1990 (4)

H. Freedhoff, Z. Chen, “Resonance fluorescence of a two-level atom in a strong bichromatic field,” Phys. Rev. A 41, 6013 (1990).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

N. C. Kothari, G. S. Agarwal, “Effects of collisional mixing on subharmonic Raman resonances in nonlinear spectroscopy,” Opt. Commun. 74, 342 (1990).
[CrossRef]

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

1989 (3)

L. S. Gaida, S. A. Pulkin, “Three-level atomic system in the strong resonance bichromatic optical field and the weak probe field on consequent transition,” Opt. Spectrosc. (USSR) 67, 761 (1989).

M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).

T. W. Mossberg, M. Lewenstein, “Stability and phase-dependent dynamics of an atom driven by strong resonant field and a tunable moderate-strength perturber field,” Phys. Rev. A 39, 163 (1989).
[CrossRef] [PubMed]

1988 (3)

B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
[CrossRef]

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

S. Chakmakjian, K. Koch, C. R. Stroud, “Observation of resonances at subharmonics of the Rabi frequency in the saturated absorption of a 100% amplitude-modulated laser beam,” J. Opt. Soc. Am. B 5, 2015 (1988).
[CrossRef]

1986 (1)

M. Ya. Agre, L. P. Rappoport, “Generalized quasienergy states and the atom spectrum in the intense multimode radiation field,” Sov. Phys. JETP 90, 1154 (1986).

1985 (1)

1984 (3)

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[CrossRef]

T.-S. Ho, S.-I. Chu, “Semiclassical many-mode Floquet theory: II. Non-linear multiphoton dynamics of a two-level system in a strong bichromatic field,” J. Phys. B 17, 2101 (1984).
[CrossRef]

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

1982 (2)

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

1979 (2)

G. S. Agarwal, S. S. Jha, “Theory of resonant Raman scattering of intense optical waves,” J. Phys. B 12, 2655 (1979).
[CrossRef]

G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53, 1 (1979).
[CrossRef]

1978 (1)

A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).

1976 (1)

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

1972 (1)

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

1970 (1)

B. J. Feldman, M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375 (1970).
[CrossRef]

1969 (2)

S. Stenholm, W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618 (1969).
[CrossRef]

B. R. Mollow, “Power spectrum of light scattered by two-level system,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

1966 (1)

H. R. Schlossberg, A. Javan, “Saturation behavior of a Doppler-broadened transition involving levels with closely spaced structure,” Phys. Rev. 150, 267 (1966).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Agarwal, G. S.

N. C. Kothari, G. S. Agarwal, “Effects of collisional mixing on subharmonic Raman resonances in nonlinear spectroscopy,” Opt. Commun. 74, 342 (1990).
[CrossRef]

G. S. Agarwal, S. S. Jha, “Theory of resonant Raman scattering of intense optical waves,” J. Phys. B 12, 2655 (1979).
[CrossRef]

Agre, M. Ya.

M. Ya. Agre, L. P. Rappoport, “Generalized quasienergy states and the atom spectrum in the intense multimode radiation field,” Sov. Phys. JETP 90, 1154 (1986).

Avila, G.

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

Baklanov, Ye. V.

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

Beterov, I. M.

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

Bonch-Bruevich, A. M.

A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).

Borisovsky, S. P.

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

Boyd, R. W.

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[CrossRef]

Cerez, P.

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

Chakmakjian, S.

Chebotayev, V. P.

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

Chen, Z.

H. Freedhoff, Z. Chen, “Resonance fluorescence of a two-level atom in a strong bichromatic field,” Phys. Rev. A 41, 6013 (1990).
[CrossRef] [PubMed]

Chigir, N. A.

A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).

Chu, S.-I.

T.-S. Ho, S.-I. Chu, “Semiclassical many-mode Floquet theory: II. Non-linear multiphoton dynamics of a two-level system in a strong bichromatic field,” J. Phys. B 17, 2101 (1984).
[CrossRef]

de Clercq, E.

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

de Labachellerie, M.

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

Dubetsky, B. Ya.

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

Ezekiel, S.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Feld, M. S.

B. J. Feldman, M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375 (1970).
[CrossRef]

Feldman, B. J.

B. J. Feldman, M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375 (1970).
[CrossRef]

Fellman, T.

B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
[CrossRef]

Freedhoff, H.

H. Freedhoff, Z. Chen, “Resonance fluorescence of a two-level atom in a strong bichromatic field,” Phys. Rev. A 41, 6013 (1990).
[CrossRef] [PubMed]

Gaida, L. S.

L. S. Gaida, S. A. Pulkin, “Three-level atomic system in the strong resonance bichromatic optical field and the weak probe field on consequent transition,” Opt. Spectrosc. (USSR) 67, 761 (1989).

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

Gauthier, D. J.

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

Gawlik, W.

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

Gorny, M. B.

M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).

Hemmer, P. R.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Hillman, L. W.

L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operations,” J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[CrossRef]

Ho, T.-S.

T.-S. Ho, S.-I. Chu, “Semiclassical many-mode Floquet theory: II. Non-linear multiphoton dynamics of a two-level system in a strong bichromatic field,” J. Phys. B 17, 2101 (1984).
[CrossRef]

Javan, A.

H. R. Schlossberg, A. Javan, “Saturation behavior of a Doppler-broadened transition involving levels with closely spaced structure,” Phys. Rev. 150, 267 (1966).
[CrossRef]

Jha, S. S.

G. S. Agarwal, S. S. Jha, “Theory of resonant Raman scattering of intense optical waves,” J. Phys. B 12, 2655 (1979).
[CrossRef]

Junger, P.

B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
[CrossRef]

Kazakov, A. Ya.

A. Ya. Kazakov, “Two-level atom in a strong polychromatic electromagnetic field,” Sov. Phys. JETP 99, 705 (1991).

Khodovoy, V. A.

A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).

Koch, K.

Komar, V. N.

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

Korbetsky, R. V.

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

Kothari, N. C.

N. C. Kothari, G. S. Agarwal, “Effects of collisional mixing on subharmonic Raman resonances in nonlinear spectroscopy,” Opt. Commun. 74, 342 (1990).
[CrossRef]

Kowalski, J.

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

Krasinski, J.

L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operations,” J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[CrossRef]

Lamb, W. E.

S. Stenholm, W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618 (1969).
[CrossRef]

Leiby, C. C.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Lewenstein, M.

T. W. Mossberg, M. Lewenstein, “Stability and phase-dependent dynamics of an atom driven by strong resonant field and a tunable moderate-strength perturber field,” Phys. Rev. A 39, 163 (1989).
[CrossRef] [PubMed]

Lezama, A.

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

Matisov, B. G.

M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).

Mollow, B. R.

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

B. R. Mollow, “Power spectrum of light scattered by two-level system,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

Mossberg, T. W.

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

T. W. Mossberg, M. Lewenstein, “Stability and phase-dependent dynamics of an atom driven by strong resonant field and a tunable moderate-strength perturber field,” Phys. Rev. A 39, 163 (1989).
[CrossRef] [PubMed]

Narducci, L. M.

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

Oppo, G.-L.

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

Orriols, G.

G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53, 1 (1979).
[CrossRef]

Perebyakin, V. A.

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

Picard, P. H.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Popov, A. K.

A. K. Popov, Introduction to Nonlinear Spectroscopy (Nauka, Novosibirsk, USSR, 1983).

Pulkin, S. A.

L. S. Gaida, S. A. Pulkin, “Three-level atomic system in the strong resonance bichromatic optical field and the weak probe field on consequent transition,” Opt. Spectrosc. (USSR) 67, 761 (1989).

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

Rappoport, L. P.

M. Ya. Agre, L. P. Rappoport, “Generalized quasienergy states and the atom spectrum in the intense multimode radiation field,” Sov. Phys. JETP 90, 1154 (1986).

Rozhdestvensky, Yu. V.

M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).

Ru, P.

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

Schlossberg, H. R.

H. R. Schlossberg, A. Javan, “Saturation behavior of a Doppler-broadened transition involving levels with closely spaced structure,” Phys. Rev. 150, 267 (1966).
[CrossRef]

Scully, M. O.

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

Smirnov, M. Z.

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

Stahlberg, B.

B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
[CrossRef]

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Stenholm, S.

S. Stenholm, W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618 (1969).
[CrossRef]

S. Stenholm, Foundations of Laser Spectroscopy (Wiley, New York, 1987).

Stroud, C. R.

Tetu, M.

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

Thomas, J. E.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Trager, F.

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

Tredicce, J. R.

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

Vollmer, M.

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

Willis, C. R.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Wu, Q.

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

Zakharenko, Yu. G.

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

Zeilikovich, I. S.

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

Zhu, Y.

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

Appl. Phys. (1)

Ye. V. Baklanov, I. M. Beterov, V. P. Chebotayev, B. Ya. Dubetsky, “Power resonances in a Raman gas laser,” Appl. Phys. 11, 75 (1976).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. (Paris) (1)

E. de Clercq, M. de Labachellerie, G. Avila, P. Cerez, M. Tetu, “Laser diode optically pumped caesium beam,” J. Phys. (Paris) 45, 239 (1984).
[CrossRef]

J. Phys. B (2)

G. S. Agarwal, S. S. Jha, “Theory of resonant Raman scattering of intense optical waves,” J. Phys. B 12, 2655 (1979).
[CrossRef]

T.-S. Ho, S.-I. Chu, “Semiclassical many-mode Floquet theory: II. Non-linear multiphoton dynamics of a two-level system in a strong bichromatic field,” J. Phys. B 17, 2101 (1984).
[CrossRef]

Nuovo Cimento (1)

G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53, 1 (1979).
[CrossRef]

Opt. Commun. (2)

N. C. Kothari, G. S. Agarwal, “Effects of collisional mixing on subharmonic Raman resonances in nonlinear spectroscopy,” Opt. Commun. 74, 342 (1990).
[CrossRef]

B. Stahlberg, P. Junger, T. Fellman, “Stabilization of a single mode dye laser to a He–Ne laser using magnetic modulation of a three-level neon system,” Opt. Commun. 68, 209 (1988).
[CrossRef]

Opt. Spectrosc. (USSR) (2)

A. M. Bonch-Bruevich, V. A. Khodovoy, N. A. Chigir, “Investigation of the absorption spectrum of atoms in the non-resonant bichromatic radiation field,” Opt. Spectrosc. (USSR) 44, 228 (1978).

L. S. Gaida, S. A. Pulkin, “Three-level atomic system in the strong resonance bichromatic optical field and the weak probe field on consequent transition,” Opt. Spectrosc. (USSR) 67, 761 (1989).

Phys. Rev. (3)

S. Stenholm, W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618 (1969).
[CrossRef]

H. R. Schlossberg, A. Javan, “Saturation behavior of a Doppler-broadened transition involving levels with closely spaced structure,” Phys. Rev. 150, 267 (1966).
[CrossRef]

B. R. Mollow, “Power spectrum of light scattered by two-level system,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

Phys. Rev. A (6)

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

L. M. Narducci, M. O. Scully, G.-L. Oppo, P. Ru, J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42, 1630 (1990).
[CrossRef] [PubMed]

T. W. Mossberg, M. Lewenstein, “Stability and phase-dependent dynamics of an atom driven by strong resonant field and a tunable moderate-strength perturber field,” Phys. Rev. A 39, 163 (1989).
[CrossRef] [PubMed]

H. Freedhoff, Z. Chen, “Resonance fluorescence of a two-level atom in a strong bichromatic field,” Phys. Rev. A 41, 6013 (1990).
[CrossRef] [PubMed]

B. J. Feldman, M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375 (1970).
[CrossRef]

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, T. W. Mossberg, “Resonance fluorescence of two-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, 6574 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

W. Gawlik, J. Kowalski, F. Trager, M. Vollmer, “New mechanism for the production of optical resonances with sub-natural linewidths,” Phys. Rev. Lett. 48, 871 (1982).
[CrossRef]

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[CrossRef]

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, P. H. Picard, C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Sov. Phys. JETP (4)

A. Ya. Kazakov, “Two-level atom in a strong polychromatic electromagnetic field,” Sov. Phys. JETP 99, 705 (1991).

M. Ya. Agre, L. P. Rappoport, “Generalized quasienergy states and the atom spectrum in the intense multimode radiation field,” Sov. Phys. JETP 90, 1154 (1986).

I. S. Zeilikovich, S. A. Pulkin, L. S. Gaida, V. N. Komar, “Susceptibility spectrum and parametric generation in atomic vapor in a resonant light field,” Sov. Phys. JETP 94, 76 (1988).

M. B. Gorny, B. G. Matisov, Yu. V. Rozhdestvensky, “Coherent radiation trapping in the medium with high optical density,” Sov. Phys. JETP 95, 1263 (1989).

Other (5)

S. P. Borisovsky, Yu. G. Zakharenko, R. V. Korbetsky, V. A. Perebyakin, M. Z. Smirnov, “Cw visible He–Ne lasers and their application” (Central Scientific and Research Institute Electronica, Moscow, 1990);Rev. Electron. Instrum. Ser. 11 4, 1538 (1990).

S. Stenholm, Foundations of Laser Spectroscopy (Wiley, New York, 1987).

A. K. Popov, Introduction to Nonlinear Spectroscopy (Nauka, Novosibirsk, USSR, 1983).

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the three-wave interaction. Ω1 ± p are the frequencies of the strong bichromatic field components, and Ω2 is the weak probe-wave frequency. If Ω1 is held fixed near tho 1–2 transition frequency ω1 and Ω2 is scanned over the 2–3 transition, the gain (absorption) spectrum of the probe wave with two series of resonances is obtained.

Fig. 2
Fig. 2

Regions of amplification and absorption of the probe wave for γ12/γ = 1.5; N2 > N1. Region I, amplification occurs for both the even and the odd resonances; region II, amplification occurs for all the even resonances and absorption for all the odd ones; regions III, absorption occurs for all the even resonances and amplification for all the odd ones; region IV, absorption occurs for both the even and the odd resonances.

Fig. 3
Fig. 3

Absorption spectrum of the probe wave (the case of near-resonant bichromatic pumping) for N1 = N3 = 0, γ13/γ23 = 0.3, γ12/γ = 1.5, and p/γ23 = 8. a, σ = 0.5, Δ1 = 0; b, σ = 1, Δ1 = γ23; c, σ = 1.5, Δ1 = −2.5 γ23.

Fig. 4
Fig. 4

Absorption spectrum of the probe wave (the case of non-resonant bichromatic pumping) in the vicinity of zero detuning (Δ2 = λ0γ23) for I = 0.1. The solid curve corresponds to the case Δ1 > 0 and the dashed one to Δ1 < 0.

Equations (61)

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E ( z , t ) = E 1 ( z , t ) exp [ i ( Ω 1 t k 1 z ) ] + E 2 exp [ i ( Ω 2 t k 2 z ) ] + h . c . ,
| Δ 1 | , | Δ 2 | ω 1 , ω 2 , | ω 1 ω 2 | ,
E 1 = E 0 exp ( i φ 0 ) { exp [ i p ( t z / c ) ] + a exp ( i φ ) exp [ i p ( t z / c ) ] } ,
W = [ Ω 1 ( n 1 + n 2 n 3 n 4 ) + p ( n 1 n 2 n 3 + n 4 ) Δ 2 ] .
Δ 3 = p [ 2 ( n 1 n 3 ) 1 ] , Δ 2 = 2 ( n 1 n 2 ) p ,
( t + υ z ) ρ = Λ i [ H , ρ ] + A ( ρ ) .
μ 12 = μ 21 * μ 1 , μ 23 = μ 32 * μ 2 .
ρ 12 = exp ( i Ω 1 t ) [ ρ 1 exp ( i k 1 z ) + r 1 exp ( i k 1 z ) ] ,
ρ 23 = exp ( i Ω 2 t ) [ ρ 2 exp ( i k 2 z ) + r 2 exp ( i k 2 z ) ] ,
ρ 13 = exp ( i Ω 3 t ) [ ρ 3 exp ( i k 3 z ) + r 3 exp ( i k 3 z ) + ρ 4 exp ( i k 4 z ) + r 4 exp ( i k 4 z ) ] ,
ρ ˙ 2 = C 1 ρ 2 + i d 1 ρ 4 , ρ ˙ 4 = C 2 ρ 4 + i d 1 * ρ 2 ,
2 = C 3 r 2 + i d 1 r 3 i d 2 ( ρ 22 ρ 33 ) , 3 = C 4 r 3 + i d 1 * r 2 i d 2 ρ 1 ,
C 1 = [ γ 23 + i ( Δ 2 + k 2 υ ) ] , C 2 = [ γ 13 + i ( Δ 3 + k 4 υ ) ] , C 3 = [ γ 23 + i ( Δ 2 k 2 υ ) ] , C 4 = [ γ 13 + i ( Δ 3 k 3 υ ) ] , Δ 3 Ω 3 ω 3 , ω 3 ω 2 ω 1 , d 1 μ 1 * E 1 / , d 2 μ 2 E 2 / .
ρ ˙ 11 = Λ 1 γ 1 ρ 11 i ( d 1 ρ 1 d 1 * ρ 1 * ) , ρ ˙ 22 = Λ 2 γ 2 ρ 22 + i ( d 1 ρ 1 d 1 * ρ 1 * ) , ρ ˙ 33 = Λ 3 γ 3 ρ 33 , ρ ˙ 1 = C 5 ρ 1 i d 1 * ( ρ 11 ρ 22 ) ,
t = t φ / ( 2 p ) z / c
d 1 ( t ) = 2 α d 0 cos ( p t ) ,
α = exp [ i φ 0 ( z ) ] , d 0 = μ 1 * E 0 / .
ρ 33 = N 3 , ρ 11 + ρ 22 = N ,
x = i ( α ρ 1 α * ρ 1 * ) , y = α ρ 1 + α * ρ 1 * ,
ρ ˙ 11 = R ( N 1 ρ 11 ) 2 σ ( cos τ ) x ,
= A x B y + 4 σ ( cos τ ) ( 2 ρ 11 N ) ,
= A y + B x ,
σ = d 0 / p , R = γ / p , A = γ 12 / p , B = ( Δ 1 k 1 υ ) / p .
σ 1 ρ ˙ 11 = ( N 1 ρ 11 ) 2 ( cos τ ) x ,
σ 1 = S 1 x S 2 y + 4 ( cos τ ) ( 2 ρ 11 N ) ,
σ 1 = S 1 y + S 2 x .
ρ 11 = Z 0 ( τ ) + Z 1 ( τ ) + , x = x 0 ( τ ) + x 1 ( τ ) + , y = y 0 ( τ ) + y 1 ( τ ) .
Z 0 ( τ ) = { N α 0 [ sin u ( τ ) ] + β 0 [ cos u ( τ ) ] } / 2 , x 0 ( τ ) = α 0 [ cos u ( τ ) ] + β 0 [ sin u ( τ ) ] , y 0 = const ( τ ) ,
y ( τ ) = ϕ 0 ( τ ) [ y 0 + 0 τ ϕ 0 1 ( ξ ) b ( ξ ) d ξ ] ,
y 1 = σ S 1 y 0 τ + σ S 2 0 τ x 0 ( ξ ) d ξ + ν 1 ,
z = [ Z 1 x 1 ] , ϕ 0 ( τ ) = [ cos u sin u 2 ( sin u ) 2 ( cos u ) ] , y 0 = [ α 1 β 1 ] , b = σ [ b 1 b 2 ] .
b 1 = [ α 0 ( sin u ) β 0 ( cos u ) N 21 ] / 2 , b 2 = S 1 [ α 0 ( cos u ) + β 0 ( sin u ) ] S 2 y 0 ,
β 0 = 2 N 21 J 0 ( 4 σ ) / [ S 1 1 ( S 1 + 1 ) J 0 ( 8 σ ) ] , y 0 = α 0 = 0 ,
2 = θ r 2 + 2 i α σ ( cos τ ) r 3 i σ g [ N N 3 Z 0 ( τ ) ] exp ( i δ τ ) ,
3 = ψ r 3 + 2 i α * σ ( cos τ ) r 2 σ g ( α * / 2 ) x 0 ( τ ) exp ( i δ τ ) ,
r 2 , 3 = r 2 , 3 exp ( i δ τ ) , δ = Δ 2 / p , g = d 2 / γ , θ = σ ( γ 23 i k 2 υ ) / γ , ψ = σ [ γ 13 i ( Δ 1 + k 3 υ ) ] / γ .
r 2 = f [ cos ( u / 2 ) ] + h [ sin ( u / 2 ) ] , r 3 = i α * { f [ sin ( u / 2 ) ] h [ cos ( u / 2 ) ] } ,
f = i g σ 2 N 3 N + β 0 2 i δ 0 + ( θ + ψ ) + ( θ ψ ) J 0 ( 4 σ ) × J 2 n ( 2 σ ) exp ( i δ 0 τ ) , h = 0 ,
h = g σ 2 N 3 N β 0 2 i δ 0 + ( θ + ψ ) ( θ ψ ) J 0 ( 4 σ ) × J 2 n + 1 ( 2 σ ) exp ( i δ 0 τ ) , f = 0 .
r 2 ( t ) υ = d 2 K 1 u ( 2 N 3 N + β 0 ) J 2 n ( 2 σ ) Z ( D 1 + Γ 1 K 1 u ) × exp ( impt ) cos [ u ( t ) 2 ]
r 2 ( t ) υ = ± i d 2 K 2 u ( 2 N 3 N β 0 ) J 2 n + 1 ( 2 σ ) Z ( D 2 + i Γ 2 K 2 u ) × exp ( impt ) sin [ u ( t ) 2 ]
Γ j = γ 23 + γ 13 ( 1 ) j ( γ 23 γ 13 ) J 0 ( 4 σ ) ,
D j = Δ 1 [ 1 + ( 1 ) j J 0 ( 4 σ ) ] 2 Δ 2 ,
K j = 2 k 2 k 1 [ 1 + ( 1 ) j J 0 ( 4 σ ) ] ,
E 2 ( z , t ) = E 2 ( 0 , t ) exp [ G ( t ) z ] , I 2 ( z , t ) = I 2 ( 0 , t ) exp [ G I ( t ) z ] ,
G = i 2 c μ 0 Ω 2 | μ 2 | 2 d 2 r 2 ( t )
= 3 π c 2 W 2 K j u ω 2 2 [ 2 N 3 N + ( 1 ) j β 0 ] × J | m | 2 ( 2 σ ) Z i ( D j + i Γ j K j u ) ,
m > 1 2 m 12 | m 12 | [ 1 + ( 1 ) j 2 J 0 ( 4 σ ) ( Γ + 1 ) ( Γ 1 ) J 0 ( 8 σ ) ] ,
Γ γ 12 / γ , m ( N 3 N 2 ) / | N 2 N 1 | .
I = G 0 { 2 m 32 m 12 [ 1 + ( 1 ) j 2 J 0 ( 4 σ ) ( Γ + 1 ) ( Γ 1 ) J 0 ( 8 σ ) ] } × 1 Γ j J | m | 2 ( 2 σ ) 1 1 + ( D j / Γ j ) 2 ,
m i j = ( N i N j ) / N 2 , i , j = 1 , 2 , 3 , i j ; D j = D j / γ 23 , Γ j = Γ j / γ 23 .
| Δ 1 | γ , p , d 0 , k 1 , 2 , 3 u
ρ 11 = Z 0 + σ Z 1 + , x = x 0 + σ x 1 + , y = y 0 + σ y 1 + ,
2 = ( C 3 / p ) r 2 + 2 i α σ ( cos τ ) r 3 i ( d 2 / p ) ( N N 3 ρ 11 ) ,
3 = ( C 4 / p ) r 3 + 2 i α * σ ( cos τ ) r 2 i ( d 2 / p ) ρ 1 .
r 2 = r 20 + σ r 21 + , r 3 = r 30 + σ r 31 + ,
I = G 0 m 23 L ( λ 0 ) [ 1 4 I λ 0 L ( λ 0 ) 4 I 2 ( 1 3 λ 0 2 ) L 2 ( λ 0 ) ] ,
I = G 0 γ 23 | Δ 1 | { J m 13 L 1 ( λ 1 ) 2 J 2 L ( λ 1 ) × [ 4 Γ + 1 2 γ 13 Δ 1 m 12 2 m 13 λ 1 L ( λ 1 ) ] } ,
I = 1 4 G 0 ( γ 23 p ) 2 m 23 I 2 L ( λ 2 ) .
2 k 2 / k 1 = 1 J 0 ( 4 σ ) [ 2 k 2 / k 1 = 1 + J 0 ( 4 σ ) ]
Z 0 = 1 2 [ N 2 γ N 21 J 0 2 ( 4 σ ) γ 12 + γ ( γ 12 γ ) J 0 ( 8 σ ) ] .

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