Abstract

We analyze the transient response induced by a resonant, weak probe laser in a three-level V-type system driven by a strong, resonant coupling laser. The Rabi oscillation of the probe transition contains two frequencies determined by the coupling laser, and the ratio of the two-frequency components can be controlled by a time delay between the probe and the coupling laser. In particular, when both lasers are turned on simultaneously, the optical nutation with the frequency approximately equal to the Rabi frequency of the coupling laser is dominant; when the probe laser is turned on well after the steady state is reached for the coupling transition, the Rabi oscillation of the probe transition still occurs, and the oscillatory frequency is approximately equal to half of the Rabi frequency of the coupling laser. When both the probe and the coupling lasers are strong, the mutual interaction of the two fields and the atomic system may revive the optical nutation with each other.

© 1997 Optical Society of America

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1996

G. Vemuri, G. S. Agarwal, and B. D. N. Rao, Phys. Rev. A 53, 2842 (1996).
[CrossRef] [PubMed]

Y. Zhu and J. Lin, Phys. Rev. A 53, 1767 (1996).
[CrossRef] [PubMed]

G. G. Padmabandu, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 76, 2053 (1996).
[CrossRef] [PubMed]

Y. Zhu, Phys. Rev. A 53, 2742 (1996).
[CrossRef] [PubMed]

J. L. Cohen and P. R. Berman, Phys. Rev. A 54, 5262 (1996).
[CrossRef] [PubMed]

1995

N. Wang and H. Rabitz, Phys. Rev. A 52, R17 (1995).
[CrossRef]

Y. Li and M. Xiao, Opt. Lett. 20, 1489 (1995).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

1994

A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleischhauer, Phys. Rev. Lett. 73, 1789 (1994).
[CrossRef] [PubMed]

J. A. Kleinfeld and A. D. Streater, Phys. Rev. A 49, R4301 (1994).
[CrossRef]

J. Eberly, M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 66, 2593 (1994).

M. Fleischhauer, Phys. Rev. Lett. 72, 989 (1994).
[CrossRef] [PubMed]

1993

K. K. Meduri, G. A. Wilson, P. B. Sellin, and T. W. Mossberg, Phys. Rev. Lett. 77, 4311 (1993).
[CrossRef]

A. Nottelman, C. Peters, and W. Lange, Phys. Rev. Lett. 70, 1783 (1993).
[CrossRef]

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

W. van der Veer, R. J. van Diest, A. Donszelmann, and H. B. Van Linden Van den Henrell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

1992

O. Kocharovskaya, Phys. Rep. 219, 175 (1992).
[CrossRef]

M. O. Scully, Phys. Rep. 219, 191 (1992).
[CrossRef]

M. O. Scully and M. Fleischhauer, Phys. Rev. Lett. 69, 1360 (1992).
[CrossRef] [PubMed]

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Y. Zhu, Phys. Rev. A 45, R6149 (1992).
[CrossRef]

1991

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, C. H. Keitel, S. Y. Zhu, and H. M. Doss, Opt. Commun. 86, 324 (1991).
[CrossRef]

G. B. Prasad and G. S. Agarwal, Opt. Commun. 86, 409 (1991).
[CrossRef]

A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev. Lett. 66, 1154 (1991).
[CrossRef]

1990

G. S. Agarwal, S. Ravi, and J. Cooper, Phys. Rev. A 41, 4721 (1990).
[CrossRef] [PubMed]

S. E. Harris, J. E. Feld, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

1989

A. Lyras, X. Tang, P. Lambropoulos, and J. Zhang, Phys. Rev. A 40, 4131 (1989).
[CrossRef] [PubMed]

1987

H. W. H. Lee and J. E. Wessel, Phys. Rev. Lett. 59, 1416 (1987).
[CrossRef] [PubMed]

1986

N. Lu, P. R. Berman, A. G. Yodh, Y. S. Bai, and T. W. Mossberg, Phys. Rev. A 33, 3956 (1986), and references therein.
[CrossRef] [PubMed]

1977

P. F. Liao, J. E. Bjorkholm, and J. P. Gordon, Phys. Rev. Lett. 39, 15 (1977).
[CrossRef]

1975

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

1972

R. L. Shoemaker and R. G. Brewer, Phys. Rev. Lett. 28, 1430 (1972).
[CrossRef]

Agarwal, G. S.

G. Vemuri, G. S. Agarwal, and B. D. N. Rao, Phys. Rev. A 53, 2842 (1996).
[CrossRef] [PubMed]

G. B. Prasad and G. S. Agarwal, Opt. Commun. 86, 409 (1991).
[CrossRef]

G. S. Agarwal, S. Ravi, and J. Cooper, Phys. Rev. A 41, 4721 (1990).
[CrossRef] [PubMed]

Bai, Y. S.

N. Lu, P. R. Berman, A. G. Yodh, Y. S. Bai, and T. W. Mossberg, Phys. Rev. A 33, 3956 (1986), and references therein.
[CrossRef] [PubMed]

Berman, P. R.

J. L. Cohen and P. R. Berman, Phys. Rev. A 54, 5262 (1996).
[CrossRef] [PubMed]

N. Lu, P. R. Berman, A. G. Yodh, Y. S. Bai, and T. W. Mossberg, Phys. Rev. A 33, 3956 (1986), and references therein.
[CrossRef] [PubMed]

Bjorkholm, J. E.

P. F. Liao, J. E. Bjorkholm, and J. P. Gordon, Phys. Rev. Lett. 39, 15 (1977).
[CrossRef]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

Bowden, C. M.

A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleischhauer, Phys. Rev. Lett. 73, 1789 (1994).
[CrossRef] [PubMed]

Brewer, R. G.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

R. L. Shoemaker and R. G. Brewer, Phys. Rev. Lett. 28, 1430 (1972).
[CrossRef]

Cohen, J. L.

J. L. Cohen and P. R. Berman, Phys. Rev. A 54, 5262 (1996).
[CrossRef] [PubMed]

Cooper, J.

G. S. Agarwal, S. Ravi, and J. Cooper, Phys. Rev. A 41, 4721 (1990).
[CrossRef] [PubMed]

Donszelmann, A.

W. van der Veer, R. J. van Diest, A. Donszelmann, and H. B. Van Linden Van den Henrell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

Doss, H. M.

L. M. Narducci, M. O. Scully, C. H. Keitel, S. Y. Zhu, and H. M. Doss, Opt. Commun. 86, 324 (1991).
[CrossRef]

Dowling, J. P.

A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleischhauer, Phys. Rev. Lett. 73, 1789 (1994).
[CrossRef] [PubMed]

Eberly, J.

J. Eberly, M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 66, 2593 (1994).

Feld, J. E.

S. E. Harris, J. E. Feld, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

Field, J. E.

A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev. Lett. 66, 1154 (1991).
[CrossRef]

Fleischhauer, M.

A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleischhauer, Phys. Rev. Lett. 73, 1789 (1994).
[CrossRef] [PubMed]

M. Fleischhauer, Phys. Rev. Lett. 72, 989 (1994).
[CrossRef] [PubMed]

M. O. Scully and M. Fleischhauer, Phys. Rev. Lett. 69, 1360 (1992).
[CrossRef] [PubMed]

Fry, E. S.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Gordon, J. P.

P. F. Liao, J. E. Bjorkholm, and J. P. Gordon, Phys. Rev. Lett. 39, 15 (1977).
[CrossRef]

Guo, C.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Guo, J. Y.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Guo, X.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Hahn, E. L.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Haq, H. R.

J. Eberly, M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 66, 2593 (1994).

Harris, S. E.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev. Lett. 66, 1154 (1991).
[CrossRef]

S. E. Harris, J. E. Feld, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

Hollberg, L.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev. Lett. 66, 1154 (1991).
[CrossRef]

S. E. Harris, J. E. Feld, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

Jiang, D.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Jiang, Y.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Jin, G.

J. Y. Guo, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, and D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Keitel, C. H.

L. M. Narducci, M. O. Scully, C. H. Keitel, S. Y. Zhu, and H. M. Doss, Opt. Commun. 86, 324 (1991).
[CrossRef]

Kleinfeld, J. A.

J. A. Kleinfeld and A. D. Streater, Phys. Rev. A 49, R4301 (1994).
[CrossRef]

Kocharovskaya, O.

O. Kocharovskaya, Phys. Rep. 219, 175 (1992).
[CrossRef]

Lambropoulos, P.

A. Lyras, X. Tang, P. Lambropoulos, and J. Zhang, Phys. Rev. A 40, 4131 (1989).
[CrossRef] [PubMed]

Lange, W.

A. Nottelman, C. Peters, and W. Lange, Phys. Rev. Lett. 70, 1783 (1993).
[CrossRef]

Lee, H. W. H.

H. W. H. Lee and J. E. Wessel, Phys. Rev. Lett. 59, 1416 (1987).
[CrossRef] [PubMed]

Li, X.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Li, Y.

Liao, P. F.

P. F. Liao, J. E. Bjorkholm, and J. P. Gordon, Phys. Rev. Lett. 39, 15 (1977).
[CrossRef]

Lin, J.

Y. Zhu and J. Lin, Phys. Rev. A 53, 1767 (1996).
[CrossRef] [PubMed]

Lu, N.

N. Lu, P. R. Berman, A. G. Yodh, Y. S. Bai, and T. W. Mossberg, Phys. Rev. A 33, 3956 (1986), and references therein.
[CrossRef] [PubMed]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

Lyras, A.

A. Lyras, X. Tang, P. Lambropoulos, and J. Zhang, Phys. Rev. A 40, 4131 (1989).
[CrossRef] [PubMed]

Manka, A. S.

A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleischhauer, Phys. Rev. Lett. 73, 1789 (1994).
[CrossRef] [PubMed]

Meduri, K. K.

K. K. Meduri, G. A. Wilson, P. B. Sellin, and T. W. Mossberg, Phys. Rev. Lett. 77, 4311 (1993).
[CrossRef]

Mossberg, T. W.

K. K. Meduri, G. A. Wilson, P. B. Sellin, and T. W. Mossberg, Phys. Rev. Lett. 77, 4311 (1993).
[CrossRef]

N. Lu, P. R. Berman, A. G. Yodh, Y. S. Bai, and T. W. Mossberg, Phys. Rev. A 33, 3956 (1986), and references therein.
[CrossRef] [PubMed]

Narducci, L. M.

L. M. Narducci, M. O. Scully, C. H. Keitel, S. Y. Zhu, and H. M. Doss, Opt. Commun. 86, 324 (1991).
[CrossRef]

Nikonov, D.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Nikonov, D. E.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

Nottelman, A.

A. Nottelman, C. Peters, and W. Lange, Phys. Rev. Lett. 70, 1783 (1993).
[CrossRef]

Padmabandu, G. G.

G. G. Padmabandu, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 76, 2053 (1996).
[CrossRef] [PubMed]

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Peters, C.

A. Nottelman, C. Peters, and W. Lange, Phys. Rev. Lett. 70, 1783 (1993).
[CrossRef]

Pons, M. L.

J. Eberly, M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 66, 2593 (1994).

Prasad, G. B.

G. B. Prasad and G. S. Agarwal, Opt. Commun. 86, 409 (1991).
[CrossRef]

Rabitz, H.

N. Wang and H. Rabitz, Phys. Rev. A 52, R17 (1995).
[CrossRef]

Rao, B. D. N.

G. Vemuri, G. S. Agarwal, and B. D. N. Rao, Phys. Rev. A 53, 2842 (1996).
[CrossRef] [PubMed]

Ravi, S.

G. S. Agarwal, S. Ravi, and J. Cooper, Phys. Rev. A 41, 4721 (1990).
[CrossRef] [PubMed]

Robinson, H. G.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

Scully, M. O.

G. G. Padmabandu, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 76, 2053 (1996).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 1499 (1995).
[CrossRef] [PubMed]

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

M. O. Scully, Phys. Rep. 219, 191 (1992).
[CrossRef]

M. O. Scully and M. Fleischhauer, Phys. Rev. Lett. 69, 1360 (1992).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, C. H. Keitel, S. Y. Zhu, and H. M. Doss, Opt. Commun. 86, 324 (1991).
[CrossRef]

Sellin, P. B.

K. K. Meduri, G. A. Wilson, P. B. Sellin, and T. W. Mossberg, Phys. Rev. Lett. 77, 4311 (1993).
[CrossRef]

Shoemaker, R. L.

R. L. Shoemaker and R. G. Brewer, Phys. Rev. Lett. 28, 1430 (1972).
[CrossRef]

Smith, A. V.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Streater, A. D.

J. A. Kleinfeld and A. D. Streater, Phys. Rev. A 49, R4301 (1994).
[CrossRef]

Tang, X.

A. Lyras, X. Tang, P. Lambropoulos, and J. Zhang, Phys. Rev. A 40, 4131 (1989).
[CrossRef] [PubMed]

Tittel, F. K.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

van der Veer, W.

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Figures (7)

Fig. 1
Fig. 1

(a) Coherently coupled V-type three-level atomic system. (b) Sequential turn-on scheme of the coupling laser (Ω) and the probe laser (g). t0 is the variable delay time.

Fig. 2
Fig. 2

Calculated time evolution of the atomic responses in the three-level V system when the delay time t0=0. The dashed curve represents Im(ρ13)/g (proportional to the probe gain–absorption coefficient) versus the normalized time τγ31, and the solid curve represents Im(ρ12)/Ω (proportional to the coupling-laser gain–absorption coefficient) versus the normalized time τγ31. For comparison, Im(ρ12) [Im(ρ13)] is normalized by Ω (g). The chosen parameters are γ21=2γ31, Ω=20γ31, g=0.1γ31, Δ1=0, and Δ=0. The initial conditions are ρ11(0)=1 and ρij(0)=0 (i, j=13).

Fig. 3
Fig. 3

Calculated time evolution of the atomic responses in the three-level V system. The coupling laser is turned on at t=0, whereas the probe laser is turned on at t0=3/γ31. The dashed curve is Im(ρ13)/g versus the normalized time τγ31; the solid curve is Im(ρ12)/Ω versus the normalized time τγ31. Note that the nutation frequency for Im(ρ13) is about half of the nutation frequency for Im(ρ12). The chosen parameters are the same as those in Fig. 2.

Fig. 4
Fig. 4

Two-photon atomic coherence Re(ρ23) (dashed curve) and the population probability ρ33 (solid curve) versus the normalized time τγ31. The delay time of the probe laser is t0 =3/γ31. The nutation frequency is the same as that for Im(ρ13), shown in Fig. 3. The parameters are the same as those in Fig. 2.

Fig. 5
Fig. 5

Calculated time evolution of the atomic responses in the three-level V system. The coupling laser is turned on at τ=0, while the probe laser is turned on at the second maximum of the optical nutation of Im(ρ12). The dashed curve is Im(ρ13)/g versus the normalized time τγ31, and the solid curve is Im(ρ12)/Ω versus the normalized time τγ31. Note that the damped optical nutation of Im(ρ13) is anharmonic, indicating the effect of two different oscillation frequencies. The chosen parameters are the same as that in Fig. 2.

Fig. 6
Fig. 6

Calculated time evolution of the atomic responses in the three-level V system interacting with two strong laser fields. The coupling laser is turned on at t=0, while the probe laser is turned on at t0=3/γ31. The dashed curve is Im(ρ13) versus the normalized time τγ31, and the solid curve, Im(ρ12) versus the normalized time τγ31. Note that the optical nutation of Im(ρ12) is revived by the strong probe laser. The chosen Rabi frequency of the probe laser is 2g=40γ31, and the other parameters are the same as those in Fig. 2.

Fig. 7
Fig. 7

(a) Atomic population distribution in the V system interacting with two strong laser fields. The coupling laser is turned on at t=0, while the probe laser is turned on at t0=3/γ31. Both laser fields have the same Rabi frequency 2Ω=2g=40γ31. The dashed curve is ρ33 versus τγ31; the solid curve is ρ11 versus τγ31. Note that ρ22=1-ρ11-ρ33. The parameters are the same as those in Fig. 6.

Equations (25)

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dρ11dt=γ21ρ22+γ31ρ33+iΩρ21-iΩ*ρ12+igρ31-ig*ρ13,
dρ22dt=-γ21ρ22+iΩ*ρ12-iΩρ21,
dρ33dt=-γ31ρ33+ig*ρ13-igρ31,
dρ12dt=-γ212+iΔ1ρ12+igρ32+iΩ(ρ22-ρ11),
dρ13dt=-γ312+iΔρ13+iΩρ23+ig(ρ33-ρ11),
dρ23dt=-γ21+γ312+i(Δ-Δ1)ρ23-igρ21+iΩ*ρ13.
ρ12=-i Ωγ214|Ω2|+γ212/21-cos λt+3γ214λ-4|Ω2|+γ212/2λγ21sin λtexp-3γ21t4,
ρ22=2|Ω|24|Ω2|+γ212/21-cos λt+3γ214λsin λt×exp-3γ21t4,
ρ11=1-ρ22,
dρ13dt=-γ312ρ13+iΩρ23-igρ11,
dρ23dt=-γ31+γ212ρ23+iΩ*ρ13-igρ21.
d2ρ23dt2+2γ31+γ212dρ13dt+|Ω|2+γ31(γ31+γ21)4ρ13
+ig(γ31+γ21)2ρ11-Ωgρ21+ig dρ11dt=0.
ρ13=-i g4Ω2γ31-[A1 cos λt+A2 sin λt]×exp-3γ21t4+i[B1 cos βt+B2 sin βt]×exp-2γ31+γ214t.
β=|Ω|2+γ31(γ31+γ21)2-(2γ31+γ21)2161/2.
A1=28γ21-9γ318,A2=-2Ω,
B1=g 9γ31-14γ2118|Ω|2,
B2=g 56γ212-65γ31γ21+36γ312144|Ω|3.
ρ12=-i Ωγ214|Ω|2+γ2122,
ρ22=2|Ω|24|Ω|2+γ2122,
ρ11=2|Ω|2+γ21224|Ω|2+γ2122.
ρ13=A+{a+ exp[iβ(t-t0)]+a- exp[-iβ(t-t0)]}×exp-2γ31+γ214(t-t0).
A=-i g|Ω|2γ31(4|Ω2|+γ212/2)(|Ω2|+(γ31+γ21)γ31/2),
a+=ig|Ω|2γ31-gβ2Ω4+Ω2γ3122+3γ31γ214+γ2122+γ31γ212(γ31+γ21)22(4|Ω2|+γ212/2)(|Ω2|+(γ31+γ21)γ31/2),
a-=ig|Ω|2γ31+gβ2Ω4+Ω2γ3122+3γ31γ214+γ2122+γ31γ212(γ31+γ31)22(4|Ω2|+γ212/2)(|Ω2|+(γ31+γ21)γ31/2).

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