Abstract

We discuss a system with two discrete levels embedded in one continuum, which, after Fano diagonalization, gives a double Fano profile with two zeros. The second zero, which is located between the two levels, does not disappear even when the asymmetry parameter q goes to infinity. The spectrum of photoelectrons from such a system is calculated for any strength of the exciting field.

© 1987 Optical Society of America

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References

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  1. K. Rzążewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981), and references cited therein.
    [Crossref]
  2. L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
    [Crossref]
  3. Yu. I. Heller and A. K. Popov, Opt. Commun. 18, 449 (1976); Yu. I. Heller, V. F. Lukinykh, A. K. Popov, and V. V. Slabko, Phys. Lett. A 82, 4 (1981).
    [Crossref]
  4. A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
    [Crossref]
  5. P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 (1981).
    [Crossref]
  6. K. Rzążewski and J. H. Eberly, Phys. Rev. A 27, 2026 (1983).
    [Crossref]
  7. J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
    [Crossref]
  8. G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
    [Crossref]
  9. G. S. Agarwal and D. Agassi, Phys. Rev. A 27, 2254 (1983).
    [Crossref]
  10. A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
    [Crossref]
  11. L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
    [Crossref]
  12. M. Crance and L. Armstrong, J. Phys. B 15, 3199 (1982).
    [Crossref]
  13. K. Rzążewski, Phys. Rev. A 28, 2565 (1983).
    [Crossref]
  14. P. E. Coleman and P. L. Knight, J. Phys. B 14, 2139 (1981).
    [Crossref]
  15. Z. Deng and J. H. Eberly, J. Opt. Soc. Am. B 1, 102 (1984).
    [Crossref]
  16. G. Alber and P. Zoller, Phys. Rev. A 29, 2290 (1984).
    [Crossref]
  17. Y. S. Kim and P. Lambropoulos, Phys. Rev. A 29, 3159 (1984).
    [Crossref]
  18. U. Fano, Phys. Rev. 124, 1866 (1961).
    [Crossref]
  19. S. Kielich, Acta Phys. Polon. 30, 393 (1966), and references cited therein.
  20. See, for instance L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  21. A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

1985 (1)

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

1984 (3)

Z. Deng and J. H. Eberly, J. Opt. Soc. Am. B 1, 102 (1984).
[Crossref]

G. Alber and P. Zoller, Phys. Rev. A 29, 2290 (1984).
[Crossref]

Y. S. Kim and P. Lambropoulos, Phys. Rev. A 29, 3159 (1984).
[Crossref]

1983 (4)

K. Rzążewski, Phys. Rev. A 28, 2565 (1983).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. A 27, 2026 (1983).
[Crossref]

G. S. Agarwal and D. Agassi, Phys. Rev. A 27, 2254 (1983).
[Crossref]

A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
[Crossref]

1982 (4)

J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
[Crossref]

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
[Crossref]

M. Crance and L. Armstrong, J. Phys. B 15, 3199 (1982).
[Crossref]

1981 (3)

P. E. Coleman and P. L. Knight, J. Phys. B 14, 2139 (1981).
[Crossref]

P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 (1981).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981), and references cited therein.
[Crossref]

1978 (1)

L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
[Crossref]

1976 (1)

Yu. I. Heller and A. K. Popov, Opt. Commun. 18, 449 (1976); Yu. I. Heller, V. F. Lukinykh, A. K. Popov, and V. V. Slabko, Phys. Lett. A 82, 4 (1981).
[Crossref]

1975 (1)

L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
[Crossref]

1966 (1)

S. Kielich, Acta Phys. Polon. 30, 393 (1966), and references cited therein.

1961 (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and D. Agassi, Phys. Rev. A 27, 2254 (1983).
[Crossref]

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

Agassi, D.

G. S. Agarwal and D. Agassi, Phys. Rev. A 27, 2254 (1983).
[Crossref]

J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
[Crossref]

Alber, G.

G. Alber and P. Zoller, Phys. Rev. A 29, 2290 (1984).
[Crossref]

Allen, L.

See, for instance L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Andryushin, A. I.

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
[Crossref]

Armstrong, L.

M. Crance and L. Armstrong, J. Phys. B 15, 3199 (1982).
[Crossref]

L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
[Crossref]

L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
[Crossref]

Beers, B. L.

L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
[Crossref]

Burnett, K.

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

Coleman, P. E.

P. E. Coleman and P. L. Knight, J. Phys. B 14, 2139 (1981).
[Crossref]

Cooper, J.

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

Crance, M.

M. Crance and L. Armstrong, J. Phys. B 15, 3199 (1982).
[Crossref]

Deng, Z.

Eberly, J. H.

Z. Deng and J. H. Eberly, J. Opt. Soc. Am. B 1, 102 (1984).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. A 27, 2026 (1983).
[Crossref]

J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981), and references cited therein.
[Crossref]

See, for instance L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Fano, U.

U. Fano, Phys. Rev. 124, 1866 (1961).
[Crossref]

Fedorov, M. V.

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
[Crossref]

Feneuille, S.

L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
[Crossref]

Haan, S. L.

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

Haus, J. W.

A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
[Crossref]

Heller, Yu. I.

Yu. I. Heller and A. K. Popov, Opt. Commun. 18, 449 (1976); Yu. I. Heller, V. F. Lukinykh, A. K. Popov, and V. V. Slabko, Phys. Lett. A 82, 4 (1981).
[Crossref]

Kazakov, A. E.

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
[Crossref]

Kielich, S.

S. Kielich, Acta Phys. Polon. 30, 393 (1966), and references cited therein.

Kim, Y. S.

Y. S. Kim and P. Lambropoulos, Phys. Rev. A 29, 3159 (1984).
[Crossref]

Knight, P. L.

P. E. Coleman and P. L. Knight, J. Phys. B 14, 2139 (1981).
[Crossref]

Lambropoulos, P.

Y. S. Kim and P. Lambropoulos, Phys. Rev. A 29, 3159 (1984).
[Crossref]

P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 (1981).
[Crossref]

Lewenstein, A. M.

A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
[Crossref]

Popov, A. K.

Yu. I. Heller and A. K. Popov, Opt. Commun. 18, 449 (1976); Yu. I. Heller, V. F. Lukinykh, A. K. Popov, and V. V. Slabko, Phys. Lett. A 82, 4 (1981).
[Crossref]

Rzazewski, K.

K. Rzążewski and J. H. Eberly, Phys. Rev. A 27, 2026 (1983).
[Crossref]

A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
[Crossref]

K. Rzążewski, Phys. Rev. A 28, 2565 (1983).
[Crossref]

J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981), and references cited therein.
[Crossref]

Theodosiou, C. E.

L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
[Crossref]

Wall, M. J.

L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
[Crossref]

Zoller, P.

G. Alber and P. Zoller, Phys. Rev. A 29, 2290 (1984).
[Crossref]

P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 (1981).
[Crossref]

Acta Phys. Polon. (1)

S. Kielich, Acta Phys. Polon. 30, 393 (1966), and references cited therein.

J. Opt. Soc. Am. B (1)

J. Phys. B (2)

P. E. Coleman and P. L. Knight, J. Phys. B 14, 2139 (1981).
[Crossref]

M. Crance and L. Armstrong, J. Phys. B 15, 3199 (1982).
[Crossref]

Opt. Commun. (1)

Yu. I. Heller and A. K. Popov, Opt. Commun. 18, 449 (1976); Yu. I. Heller, V. F. Lukinykh, A. K. Popov, and V. V. Slabko, Phys. Lett. A 82, 4 (1981).
[Crossref]

Phys. Rev. (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[Crossref]

Phys. Rev. A (8)

L. Armstrong, B. L. Beers, and S. Feneuille, Phys. Rev. A 12, 1903 (1975).
[Crossref]

P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 (1981).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. A 27, 2026 (1983).
[Crossref]

K. Rzążewski, Phys. Rev. A 28, 2565 (1983).
[Crossref]

G. S. Agarwal and D. Agassi, Phys. Rev. A 27, 2254 (1983).
[Crossref]

L. Armstrong, C. E. Theodosiou, and M. J. Wall, Phys. Rev. A 18, 2538 (1978).
[Crossref]

G. Alber and P. Zoller, Phys. Rev. A 29, 2290 (1984).
[Crossref]

Y. S. Kim and P. Lambropoulos, Phys. Rev. A 29, 3159 (1984).
[Crossref]

Phys. Rev. Lett. (4)

A. M. Lewenstein, J. W. Haus, and K. Rzążewski, Phys. Rev. Lett. 50, 417 (1983); J. W. Haus, M. Lewenstein, and K. Rzążewski, Phys. Rev. A 28, 2269 (1983); J. Opt. Soc. Am. B 1, 641 (1984).
[Crossref]

J. H. Eberly, K. Rzążewski, and D. Agassi, Phys. Rev. Lett. 49, 693 (1982); D. Agassi, K. Rzążewski, and J. H. Eberly, Phys. Rev. A 28, 3648 (1983).
[Crossref]

G. S. Agarwal, S. L. Haan, K. Burnett, and J. Cooper, Phys. Rev. Lett. 48, 1164 (1982); Phys. Rev. A 26, 2277 (1982); G. S. Agarwal, S. L. Haan, and J. Cooper, Phys. Rev. A 29, 2552, 2565 (1984).
[Crossref]

K. Rzążewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981), and references cited therein.
[Crossref]

Zh. Eksp. Teor. Fiz. (2)

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 82, 91 (1982) [Sov. Phys. JETP 55, 53 (1982)]; A. I. Andryushin, M. V. Fedorov, and A. E. Kazakov, J. Phys. B 15, 2851 (1982).
[Crossref]

A. I. Andryushin, A. E. Kazakov, and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 88, 1153 (1985) [Sov. Phys. JETP 61, 678 (1985)].

Other (1)

See, for instance L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Cited By

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Figures (8)

Fig. 1
Fig. 1

Simplified atomic-level scheme. Configuration-interaction coupling of levels |1〉 and |2〉 to |ω〉 leads to the double Fano continuum |ω〉. This continuum is coupled to discrete state |0〉 by a laser of frequency ωL.

Fig. 2
Fig. 2

Double Fano profile for various sets of parameters. Solid line: γ1 = γ2 = 0.5, q1 = 2, q2 = 3. Dotted line: γ1 = γ2 = 0.5, q1 = 2, q2 = 1. Dashed–dotted line: γ1 = γ2 = 0.5, q1 = q2 = 2. Dashed line: γ1 = 0.1, γ2 = 0.9, q1 = q2 = 2. The separation of the two levels ω21 = 2.

Fig. 3
Fig. 3

Long-time photoelectron spectrum in the degenerate case ω21 = 0 for two strengths of the field, Ω0 = 1, 3. The dotted and dashed–dotted lines are for the same asymmetry parameters q1 = q2 = 100, the solid and dashed lines for different q’s: q1 = 90, q2 = 100. Autoionization widths are γ1 = γ2 = 0.5, and the laser is tuned to the lower level (δ = 0).

Fig. 4
Fig. 4

Same as Fig. 3 but for small q values. Dashed–dotted and dashed lines, q1 = q2 = 2; solid and dotted lines, q1 = 2, q2 = 3.

Fig. 5
Fig. 5

Strong-field (Ω0 = 3) photoelectron spectrum for the nondegenerate case ω21 = 2, γ1 = γ2 = 0.5, δ = 0. Dashed line, q1 = q2 = 2; dashed–dotted line, q1 = 2, q2 = 10; solid line q1 = 10, q2 = 2.

Fig. 6
Fig. 6

Same as Fig. 5 but for different widths γ1 = 0.1, γ2 = 0.9. Solid line, q1 = q2 = 2; dashed line, q1 = 10, q2 = 2; dashed–dotted line, q1 = 2, q2 = 10.

Fig. 7
Fig. 7

Strong-field photoelectron spectrum for large asymmetry parameters q1 = q2 = 100 and various separations of the levels; γ1 = γ2 = 0.5, δ = 0, Ω0 = 3.

Fig. 8
Fig. 8

Same as Fig. 7 but for other level separations. Solid line, ω21 = 1.01; dashed line, ω21 = 2; dashed–dotted line, ω21 = 3.

Equations (26)

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

H ˆ = H ˆ 0 + H ˆ 1 + H ˆ 2 + H ˆ ω + V ˆ Coul + V ˆ rad ,
H ˆ i = ω i | i i | , i = 0 , 1 , 2 ,
H ˆ ω = d ω ω | ω ω | .
V 1 ( ω ) = 1 | V ˆ Coul | ω ,
V 2 ( ω ) = 2 | V ˆ Coul | ω ,
V 01 = 0 | V ˆ rad | 1 ,
V 02 = 0 | V ˆ rad | 2 ,
V 0 ( ω ) = 0 | V ˆ rad | ω .
H ˆ L | ω ) = ω | ω )
V ˆ rad = d ω Ω ( ω ) | 0 ( ω | + H L .
Ω ( ω ) = V 0 ( ω ) e i φ × ( ω ω 1 ) ( ω ω 2 ) + ω ( q 1 γ 1 + q 2 γ 2 ) ( ω 1 q 2 γ 2 + ω 2 q 1 γ 1 ) ( ω ω 1 ) ( ω ω 2 ) i ω ( γ 1 + γ 2 ) + i ( ω 1 γ 2 + ω 2 γ 1 ) ,
ω = ω 1 q 1 γ 1 + ω 2 q 2 γ 2 2 ± 1 2 [ ω 21 2 + ( q 1 γ 1 + q 2 γ 2 ) 2 2 ω 21 ( q 2 γ 2 q 1 γ 1 ) ] 1 / 2 ,
Ω 0 4 π Γ ( Q + i ) V 0 ( ω ) e i φ ,
Ω ( ω ) = Ω 0 4 π Γ ( A + ω ω + + A ω ω + 1 Q + i ) ,
ω ± = ω 1 + ω 2 ± ν 2 + i Γ ± η 2 ,
η = 1 2 { [ ( ω 21 2 Γ 2 ) 2 + 4 ω 21 2 ( γ 2 γ 1 ) 2 ] 1 / 2 ω 21 2 + Γ 2 } 1 / 2 , ν = 1 2 { [ ( ω 21 2 Γ 2 ) 2 + 4 ω 21 2 ( γ 2 γ 1 ) 2 ] 1 / 2 + ω 21 2 Γ 2 } 1 / 2 .
A ± = Γ 2 ( 1 ± ω 21 κ + i Γ ν + i η ) ,
κ = q 2 γ 2 q 1 γ 1 + i ( γ 2 γ 1 ) Γ ( Q + i ) .
| ψ ( t ) = α ( t ) | 0 + d ω β ω ( t ) | ω ) ,
α ˙ ( t ) = i d ω Ω ( ω ) β ω ( t ) ,
β ˙ ω ( t ) = i ( ω ω L ) β ω ( t ) i Ω * ( ω ) α ( t ) .
W ( ω ) = lim t | β ω ( t ) | 2 .
W ( ω ) = | Ω ( ω ) G [ z = i ( ω ω L ) ] | 2 ,
G ( z ) = 1 α ¯ ( z ) = z + d ω | Ω ( ω ) | 2 z + i ( ω ω L ) .
G [ z = i ( ω ω L ) ] = i ( ω ω L ) Ω 0 2 4 Γ ( B + ω ω + + B ω ω + 1 Q 2 + 1 ) ,
B ± = 2 A ± [ A ± * i ( Γ ± η ) + A * i Γ ± ν + 1 Q i ] .

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