Abstract

The dynamical behavior of a two-level atom under the influence of two intense pump beams is examined. Exact equations for the nonlinear susceptibilities to all orders in the field strengths are derived. Numerical results for the nonlinear susceptibilities giving the absorption of the radiation from either of the pump beams and the generation of the four-wave mixing signal are given. The susceptibility for four-wave mixing, in addition to showing various saturation peaks, shows a peak at ω1 = ω2 even in the case of radiative relaxation. Higher-order multiphoton peaks in the absorption spectra are found to be extremely sensitive to the temporal fluctuations in the pumps.

© 1984 Optical Society of America

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  1. B. R. Mollow, Phys. Rev. 188, 1969–1975 (1969); R. P. Hackel and S. Ezekiel, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 88.
    [CrossRef]
  2. B. R. Mollow, Phys. Rev. A5, 2217–2222 (1972).
    [CrossRef]
  3. F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
    [CrossRef]
  4. A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).
  5. R. G. Gush and H. P. Gush, Phys. Rev. A10, 1474–1487 (1974).
    [CrossRef]
  6. N. Tsukada and T. Nakayama, Phys. Rev. A25, 964–997 (1982).
    [CrossRef]
  7. A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).
  8. A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).
  9. G. S. Agarwal, Phys. Rev. A18, 1490–1506 (1978).
    [CrossRef]
  10. G. S. Agarwal, Z. Physik B33, 111 (1979).
  11. G. I. Toptygina and E. E. Fradkin, Sov. Phys. JETP 55, 246–251 (1982).
  12. Note that, since both the fields can be strong, there is really no distinction between probe and pump.
  13. N. Bloembergen, in Laser Spectroscopy, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.
  14. For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
    [CrossRef]
  15. N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
    [CrossRef]
  16. Substitution rules in the context of optical resonance have been emphasized by J. H. Eberly, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 80.
    [CrossRef]
  17. Complications that arise because of the temporal fluctuations are discussed in G. S. Agarwal and C. V. Kunasz, Phys. Rev. A27, 996–1012 (1983).
    [CrossRef]
  18. In the framework of higher-order perturbation theory, H. Friedmann and A. D. Wilson-Gordon [Phys. Rev. A26, 2768–2777 (1982)] have also found the presence of the extra resonance, even in the case of radiative relaxation.
    [CrossRef]
  19. The values at the peak ω1 = ω2 calculated from the analytic formulas (4.4) and (4.5) are in agreement with the numerical computations.

1983

Complications that arise because of the temporal fluctuations are discussed in G. S. Agarwal and C. V. Kunasz, Phys. Rev. A27, 996–1012 (1983).
[CrossRef]

1982

In the framework of higher-order perturbation theory, H. Friedmann and A. D. Wilson-Gordon [Phys. Rev. A26, 2768–2777 (1982)] have also found the presence of the extra resonance, even in the case of radiative relaxation.
[CrossRef]

G. I. Toptygina and E. E. Fradkin, Sov. Phys. JETP 55, 246–251 (1982).

N. Tsukada and T. Nakayama, Phys. Rev. A25, 964–997 (1982).
[CrossRef]

1981

A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
[CrossRef]

1979

Substitution rules in the context of optical resonance have been emphasized by J. H. Eberly, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 80.
[CrossRef]

N. Bloembergen, in Laser Spectroscopy, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.

G. S. Agarwal, Z. Physik B33, 111 (1979).

A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).

1978

G. S. Agarwal, Phys. Rev. A18, 1490–1506 (1978).
[CrossRef]

For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
[CrossRef]

1977

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

1976

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

1974

R. G. Gush and H. P. Gush, Phys. Rev. A10, 1474–1487 (1974).
[CrossRef]

1972

B. R. Mollow, Phys. Rev. A5, 2217–2222 (1972).
[CrossRef]

1969

B. R. Mollow, Phys. Rev. 188, 1969–1975 (1969); R. P. Hackel and S. Ezekiel, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 88.
[CrossRef]

Agarwal, G. S.

Complications that arise because of the temporal fluctuations are discussed in G. S. Agarwal and C. V. Kunasz, Phys. Rev. A27, 996–1012 (1983).
[CrossRef]

G. S. Agarwal, Z. Physik B33, 111 (1979).

G. S. Agarwal, Phys. Rev. A18, 1490–1506 (1978).
[CrossRef]

Bloembergen, N.

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
[CrossRef]

N. Bloembergen, in Laser Spectroscopy, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.

Bogdan, A. R.

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
[CrossRef]

Bonch-Bruevich, A. M.

A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).

A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

Chandra, S.

For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
[CrossRef]

Chigir, N. A.

A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).

A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

Compaan, A.

For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
[CrossRef]

Downer, M. W.

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
[CrossRef]

Ducloy, M.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Eberly, J. H.

Substitution rules in the context of optical resonance have been emphasized by J. H. Eberly, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 80.
[CrossRef]

Ezekiel, S.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Fradkin, E. E.

G. I. Toptygina and E. E. Fradkin, Sov. Phys. JETP 55, 246–251 (1982).

Friedmann, H.

In the framework of higher-order perturbation theory, H. Friedmann and A. D. Wilson-Gordon [Phys. Rev. A26, 2768–2777 (1982)] have also found the presence of the extra resonance, even in the case of radiative relaxation.
[CrossRef]

Gush, H. P.

R. G. Gush and H. P. Gush, Phys. Rev. A10, 1474–1487 (1974).
[CrossRef]

Gush, R. G.

R. G. Gush and H. P. Gush, Phys. Rev. A10, 1474–1487 (1974).
[CrossRef]

Khodovoi, V. A.

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

Kunasz, C. V.

Complications that arise because of the temporal fluctuations are discussed in G. S. Agarwal and C. V. Kunasz, Phys. Rev. A27, 996–1012 (1983).
[CrossRef]

Mollow, B. R.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

B. R. Mollow, Phys. Rev. A5, 2217–2222 (1972).
[CrossRef]

B. R. Mollow, Phys. Rev. 188, 1969–1975 (1969); R. P. Hackel and S. Ezekiel, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 88.
[CrossRef]

Nakayama, T.

N. Tsukada and T. Nakayama, Phys. Rev. A25, 964–997 (1982).
[CrossRef]

Przhibelskii, S. G.

A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

Toptygina, G. I.

G. I. Toptygina and E. E. Fradkin, Sov. Phys. JETP 55, 246–251 (1982).

Tsukada, N.

N. Tsukada and T. Nakayama, Phys. Rev. A25, 964–997 (1982).
[CrossRef]

Vartanyan, T. A.

A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).

Wiener-Avnear, E.

For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
[CrossRef]

Wilson-Gordon, A. D.

In the framework of higher-order perturbation theory, H. Friedmann and A. D. Wilson-Gordon [Phys. Rev. A26, 2768–2777 (1982)] have also found the presence of the extra resonance, even in the case of radiative relaxation.
[CrossRef]

Wu, F. Y.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Phys. Rev.

B. R. Mollow, Phys. Rev. 188, 1969–1975 (1969); R. P. Hackel and S. Ezekiel, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 88.
[CrossRef]

B. R. Mollow, Phys. Rev. A5, 2217–2222 (1972).
[CrossRef]

R. G. Gush and H. P. Gush, Phys. Rev. A10, 1474–1487 (1974).
[CrossRef]

N. Tsukada and T. Nakayama, Phys. Rev. A25, 964–997 (1982).
[CrossRef]

G. S. Agarwal, Phys. Rev. A18, 1490–1506 (1978).
[CrossRef]

Phys. Rev. Lett.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Sov. Phys. JETP

A. M. Bonch-Bruevich, S. G. Przhibelskii, V. A. Khodovoi, and N. A. Chigir, Sov. Phys. JETP 43, 230–236 (1976).

A. M. Bonch-Bruevich, S. G. Przhibelskii, and N. A. Chigir, Sov. Phys. JETP 53, 285–291 (1981).

A. M. Bonch-Bruevich, T. A. Vartanyan, and N. A. Chigir, Sov. Phys. JETP 50, 901–906 (1979).

G. I. Toptygina and E. E. Fradkin, Sov. Phys. JETP 55, 246–251 (1982).

Z. Physik

G. S. Agarwal, Z. Physik B33, 111 (1979).

Other

Note that, since both the fields can be strong, there is really no distinction between probe and pump.

N. Bloembergen, in Laser Spectroscopy, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.

For example, the saturation effects in the susceptibility ψ(5) (−3ω2 + 2ω1, ω2, ω22, −ω1, −ω1), which describes higher-order coherent anti-Stokes Raman Spectroscopy [A. Compaan, E. Wiener-Avnear, and S. Chandra, Phys. Rev. A17, 1083 (1978)] can be studied.
[CrossRef]

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
[CrossRef]

Substitution rules in the context of optical resonance have been emphasized by J. H. Eberly, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 80.
[CrossRef]

Complications that arise because of the temporal fluctuations are discussed in G. S. Agarwal and C. V. Kunasz, Phys. Rev. A27, 996–1012 (1983).
[CrossRef]

In the framework of higher-order perturbation theory, H. Friedmann and A. D. Wilson-Gordon [Phys. Rev. A26, 2768–2777 (1982)] have also found the presence of the extra resonance, even in the case of radiative relaxation.
[CrossRef]

The values at the peak ω1 = ω2 calculated from the analytic formulas (4.4) and (4.5) are in agreement with the numerical computations.

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Figures (7)

Fig. 1
Fig. 1

A, rate of absorption W1 [∝ 〈ψ1[−1]〉] as a function of Ω for g1 = 30, g2 = 20, Δ = 0, Γ1 = Γ2 = 0, and T1 = T2 = 1. B, W1 for various values of Γ1 and Γ2 as a function of Ω for g1 = 30, g2 = 20, Δ = 0, and T1 = T2 = 1. Curve a, Γ1 = 0.4, Γ2 = 0.0. Curve b, Γ1 = Γ2 = 0.4. Curve c, Γ1 = 0.0, Γ2 = 0.8. Curve d, Γ1 = Γ2 = 0.8. Note that the spectrum for Γ1 = 0.0 and Γ2 = 0.4 is similar to that for curve a. For convenience of illustration, curves c and d have been shifted.

Fig. 2
Fig. 2

Absorption spectrum W2 [∝ 〈ψ1(0)〉] as a function of Ω for g1 = 30, g2 = 20, Δ = 0, and T1 = T2 = 1. Curve a, Γ1 = Γ2 = 0.0. Curve b, Γ1 = 0.2, Γ2 = 0.0. Curve c, Γ1 = 0.8, Γ2 = 0.0.

Fig. 3
Fig. 3

Multiphoton absorption region of the spectrum W1 (solid line) and W2 (dashed line) for g1 = 30, g2 = 20, Δ = 0, and T1 = T2 = 1. Curve a, Γ1 = Γ2 = 0.0. Curve b, Γ1 = 0.05, Γ2 = 0.0.

Fig. 4
Fig. 4

Absorption spectrum W1 (curves a) and W2 (curves b) for high values of Γ1 and Γ2. g1 = 30, g2 = 20, Δ = 0, and T1 = T2 = 1. (Solid line, Γ1 = 10.0, Γ2 = 0.0; dashed line, Γ1 = 0.0, Γ2 = 10.0; and dashed–dotted line, Γ1 = Γ2 = 10.0.)

Fig. 5
Fig. 5

Rate of absorption W1 for g2 > g1. The numerical values are g1 = 9.52, g2 = 11.9, Δ = 0, and T1 = T2 = 1. The various values for Γ1 and Γ2 are curve a, Γ1 = Γ2 = 0.0; curve b, Γ1 = 0.0, Γ2 = 0.05; curve c, Γ1 = 0.0, Γ2 = 0.2.

Fig. 6
Fig. 6

Four-wave mixing signal S(α|ψ1[1]|2) as a function of (ω1ω2) for curve a, inset: Δ = 200, T1 = T2 = 1, g1 = 4, and g2 = 15 and for T1 = 0.5, T2 = 1.0, Δ = 0; curve b, g1 = 1, g2 = 15; curve c, g1 = 4, g2 = 15.

Fig. 7
Fig. 7

Multiphoton absorption region of S as a function of (ω1ω2) for g1 = 30, g2 = 20, T1 = T2 = 1, and Δ = 0.

Equations (30)

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

E ( t ) = 1 exp [ - i ω 1 t - i Φ 1 ( t ) ] + 2 exp [ - i ω 2 t - i Φ 2 ( t ) ] + c . c . , = ( t ) + c . c . ,
Φ ˙ i ( t ) = μ i ( t ) ,             i = 1 , 2.
μ i ( t ) = 0 ,             μ i ( t ) μ j ( t ) = 2 Γ i j δ ( t - t ) .
d d t [ ρ 21 ρ 12 ρ 11 - ρ 12 ] = [ i ω 0 - 1 T 2 0 + i d · * ( t ) ρ 21 0 - i ω 0 - 1 T 2 - i d · ( t ) ρ 12 2 i d · ( t ) - 2 i d · * ( t ) - 1 T 1 ρ 11 - ρ 22 ] + [ 0 0 ρ 11 ( 0 ) - ρ 22 ( 0 ) T 1 ] ,
ρ 21 = exp [ i ω 2 t + i Φ 2 ( t ) ] ψ 1 , ρ 12 = exp [ - i ω 2 t - i Φ 2 ( t ) ] ψ 2 , ρ 11 - ρ 22 = 2 ψ 3 ,
ψ ˙ = A ψ + i μ 2 ( t ) f ψ + I + B + exp { i [ Φ ( t ) + Ω t ] } ψ + B - exp { - i [ Φ ( t ) + Ω t ] } ψ , Ω = ω 1 - ω 2 ,             Φ = Φ 1 - Φ 2 ,
f 11 = - 1 ,             f 22 = + 1 ,             I 3 = η ( ρ 11 ( 0 ) - ρ 22 ( 0 ) ) / T 1 , ( B + ) 13 = 2 i d · 1 * ,             ( B + ) 32 = - i d · 1 * , ( B - ) 23 = - 2 i d · 1 ,             ( B - ) 31 = i d · 1 ,
A = [ i Δ - 1 T 2 0 2 i d · 2 * 0 - i Δ - 1 T 2 - 2 i d · 2 i d · 2 - i d · 2 * - 1 T 1 ] , Δ = ω 0 - ω 2 .
ψ [ n ] = exp { i [ Ω t + Φ ( t ) ] n } ψ ,
ψ ˙ [ n ] = A ψ [ n ] + i μ 2 ( t ) f ψ [ n ] + ( i Ω n + i μ 1 n - i μ 2 n ) ψ [ n ] + B + ψ [ n + 1 ] + B - ψ [ n - 1 ] + I exp { i n [ Ω t + Φ ( t ) ] } .
ψ ˙ [ n ] = A ψ [ n ] + B + ψ [ n + 1 ] + B - ψ [ n - 1 + δ n 0 I + [ i n Ω - Γ 1 n 2 - Γ 2 ( f - n ) 2 - 2 Γ 12 n ( f - n ) ] ψ n .
ψ 1 [ n ] = ( 2 i g 2 ψ 3 [ n ] + 2 i g 1 ψ 3 [ n + 1 ] ) / D n ,
ψ 2 [ n ] = ( - 2 i g 2 ψ ˙ 3 [ n ] - 2 i g 1 ψ 3 [ n - 1 ] / D - n * ,
D n = 1 T 2 = i Δ 2 - i n Ω + Γ 1 n 2 + Γ 2 ( n + 1 ) 2 + 2 Γ 12 n ( n + 1 ) .
a n X n + b n X n + 1 + c n X n - 1 = η δ n 0 ,
a n = 1 T 1 + n 2 ( Γ 1 + Γ 2 - 2 Γ 12 ) - i n Ω + 2 g 2 2 D - 1 * + 2 g 1 2 D n - 1 + 2 g 1 2 D - n - 1 * + 2 g 2 2 D n , b n = 2 g 1 g 2 [ 1 D n + 1 D - n - 1 * ] , c n = 2 g 1 g 2 [ 1 D n - 1 + 1 D - 1 * ] .
X 0 = η [ a 0 + 2 Re ( b 0 Y 1 ) ] - 1 ,
Y n = - c n [ a n + b n Y n + 1 ] - 1 ,             Y n X n / X n - 1 ,             n 0.
W 1 2 ω 1 g 1 Im ψ 1 [ - 1 ] ,
W 2 2 ω 2 g 2 Im ψ 1 [ 0 ] .
ψ ¨ = A ψ - Γ 2 f 2 ψ + I + B + exp ( i Ω t ) ψ + B - exp ( - i Ω t ) ψ ,
ψ = ψ [ n ] exp ( - i n Ω t ) ,
A ψ [ n ] + B + ψ [ n + 1 ] + B - ψ [ n - 1 ] + δ n 0 I + ( i n Ω - Γ 2 f 2 ) ψ [ n ] = 0.
ψ 1 [ - 1 ] = n = 0 g 1 2 n + 1 α 2 n + 1 ,
1 T 2 1 T 2 + Γ .
ρ 12 ( t ) = n ψ 2 [ n ] exp { - i t [ ω 2 ( 1 - n ) + n ω 1 ] }
= ψ 2 [ - 1 ] exp [ - i t ( 2 ω 2 - ω 1 ) ] + other terms .
ψ 1 [ 1 ] = 2 i ( g 2 + g 1 Y 2 ) Y 1 X 0 D 1 - 1 .
Y 1 = - a 2 b [ 1 - ( 1 - 4 c b a 2 ) 1 / 2 ] ,
a = 1 T 1 + 4 ( g 1 2 + g 2 2 ) T 2 [ Δ 2 2 + ( 1 T 2 ) 2 ] ,             b = c = 4 g 1 g 2 T 2 [ ( 1 T 2 ) 2 + Δ 2 2 ] .

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