Abstract

We extend the quantum theory of nondegenerate four-wave mixing by including the effects of the finite bandwidth of the driving-pump field. The interaction of a beam of two-level atoms with the two opposite driving-pump fields that have finite bandwidth inside a bimodal cavity is considered. The master equation for the cavity-field modes, averaged over the stochastic process, is derived. We use our theory to study the effects of phase fluctuations, associated with the driving-pump field, on the generation of two-mode squeezing inside the cavity. The steady-state squeezing is achieved with the same driving-pump field as a local oscillator in the balanced homodyne-detection system. Our results show that, in spite of instantaneous phase locking between the cavity field and the local oscillator, the time-delay effects associated with the exponential decay of atomic coherence relate the steady-state squeezing to the diffusion constant of the driving-pump field.

© 1997 Optical Society of America

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References

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  1. R. W. Hellwarth, “Generation of time reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [CrossRef]
  2. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [CrossRef] [PubMed]
  3. H. P. Yuen and J. H. Shapiro, “Generation and detection of two-photon coherent state in degenerate four-wave mixing,” Opt. Lett. 4, 334–336 (1979).
    [CrossRef] [PubMed]
  4. For a review, see K. Zaheer and M. S. Zubairy, “Squeezed states of the radiation field,” in Advances in Atomic, Molecular, and Optical Physics, D. R. Bates and B. Bederson, eds. (Academic, New York, 1991), Vol. 28, pp. 143–235.
  5. R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
    [CrossRef]
  6. M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
    [CrossRef] [PubMed]
  7. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [CrossRef] [PubMed]
  8. B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408–410 (1984).
    [CrossRef]
  9. M. D. Reid and D. F. Walls, “Quantum theory of nondegenerate four-wave mixing,” Phys. Rev. A 34, 4929–4955 (1986).
    [CrossRef] [PubMed]
  10. D. A. Holm, M. Sargent III, and B. A. Capron, “Generation of squeezed states by nondegenerate multiwave mixing in two-level media,” Opt. Lett. 11, 443–445 (1986).
    [CrossRef] [PubMed]
  11. D. A. Holm and M. Sargent III, “Quantum theory of multiwave mixing. VIII. Squeezed states,” Phys. Rev. A 35, 2150–2163 (1987).
    [CrossRef] [PubMed]
  12. S.-T. Ho, P. Kumar, and J. H. Shapiro, “Quantum theory of nondegenerate multiwave mixing. General formulation,” Phys. Rev. A 37, 2017–2032 (1988); S.-T. Ho, P. Kumar, and J. H. Shapiro, “Quantum theory of nondegenerate multiwave mixing. III. Application to single-beam squeezed state generation,” J. Opt. Soc. Am. B 8, 37–57 (1991).
    [CrossRef] [PubMed]
  13. M. Sargent III, D. A. Holm, and M. S. Zubairy, “Quantum theory of multiwave mixing. I. General formulation,” Phys. Rev. A 31, 3112–3123 (1985).
    [CrossRef]
  14. B. Wilhelmi, “Distortion of phase-conjugation in degenerate four-wave mixing with phase fluctuations of pump and signal,” Opt. Commun. 82, 89–93 (1991).
    [CrossRef]
  15. D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
    [CrossRef] [PubMed]
  16. K. Wodkiewicz and M. S. Zubairy, “Effects of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 27, 2003–2007 (1983).
    [CrossRef]
  17. J. Gea-Banacloche and M. S. Zubairy, “Influence of pump-phase fluctuations on the squeezing in a degenerate parametric oscillator,” Phys. Rev. A 42, 1742–1751 (1990).
    [CrossRef] [PubMed]
  18. A. H. Toor and M. S. Zubairy, “Effects of finite bandwidth on the resonance fluorescence spectrum inside of a cavity,” Phys. Rev. A 49, 449–460 (1994).
    [CrossRef] [PubMed]
  19. D. A. Holm, M. Sargent III, and S. Stenholm, “Quantum theory of multiwave mixing. IV. Effects of cavities on the spectrum of resonance fluorescence,” J. Opt. Soc. Am. B 2, 1456–1463 (1985).
    [CrossRef]
  20. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
    [CrossRef] [PubMed]
  21. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  22. C. M. Caves and D. D. Crouch, “Quantum wideband traveling-wave analysis of degenerate parametric amplifier,” J. Opt. Soc. Am. B 4, 1535–1545 (1987).
    [CrossRef]

1994 (1)

A. H. Toor and M. S. Zubairy, “Effects of finite bandwidth on the resonance fluorescence spectrum inside of a cavity,” Phys. Rev. A 49, 449–460 (1994).
[CrossRef] [PubMed]

1992 (1)

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

1991 (1)

B. Wilhelmi, “Distortion of phase-conjugation in degenerate four-wave mixing with phase fluctuations of pump and signal,” Opt. Commun. 82, 89–93 (1991).
[CrossRef]

1990 (1)

J. Gea-Banacloche and M. S. Zubairy, “Influence of pump-phase fluctuations on the squeezing in a degenerate parametric oscillator,” Phys. Rev. A 42, 1742–1751 (1990).
[CrossRef] [PubMed]

1987 (2)

C. M. Caves and D. D. Crouch, “Quantum wideband traveling-wave analysis of degenerate parametric amplifier,” J. Opt. Soc. Am. B 4, 1535–1545 (1987).
[CrossRef]

D. A. Holm and M. Sargent III, “Quantum theory of multiwave mixing. VIII. Squeezed states,” Phys. Rev. A 35, 2150–2163 (1987).
[CrossRef] [PubMed]

1986 (2)

1985 (4)

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

M. Sargent III, D. A. Holm, and M. S. Zubairy, “Quantum theory of multiwave mixing. I. General formulation,” Phys. Rev. A 31, 3112–3123 (1985).
[CrossRef]

D. A. Holm, M. Sargent III, and S. Stenholm, “Quantum theory of multiwave mixing. IV. Effects of cavities on the spectrum of resonance fluorescence,” J. Opt. Soc. Am. B 2, 1456–1463 (1985).
[CrossRef]

1984 (2)

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408–410 (1984).
[CrossRef]

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

1983 (2)

H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
[CrossRef] [PubMed]

K. Wodkiewicz and M. S. Zubairy, “Effects of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 27, 2003–2007 (1983).
[CrossRef]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1979 (1)

1977 (2)

Aspect, A.

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

Bondurant, R. S.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

Capron, B. A.

Caves, C. M.

Chan, V. W. S.

Cooper, J.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Crouch, D. D.

Ewart, P.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Gea-Banacloche, J.

J. Gea-Banacloche and M. S. Zubairy, “Influence of pump-phase fluctuations on the squeezing in a degenerate parametric oscillator,” Phys. Rev. A 42, 1742–1751 (1990).
[CrossRef] [PubMed]

Hellwarth, R. W.

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Holm, D. A.

Kumar, P.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

Levenson, M. D.

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

Maeda, M.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

Meacher, D. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Pepper, D. M.

Reid, M. D.

M. D. Reid and D. F. Walls, “Quantum theory of nondegenerate four-wave mixing,” Phys. Rev. A 34, 4929–4955 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

Sargent III, M.

Shapiro, J. H.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

H. P. Yuen and J. H. Shapiro, “Generation and detection of two-photon coherent state in degenerate four-wave mixing,” Opt. Lett. 4, 334–336 (1979).
[CrossRef] [PubMed]

Shelby, R. M.

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Smith, P. G. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Stenholm, S.

Toor, A. H.

A. H. Toor and M. S. Zubairy, “Effects of finite bandwidth on the resonance fluorescence spectrum inside of a cavity,” Phys. Rev. A 49, 449–460 (1994).
[CrossRef] [PubMed]

Valley, F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Walls, D. F.

M. D. Reid and D. F. Walls, “Quantum theory of nondegenerate four-wave mixing,” Phys. Rev. A 34, 4929–4955 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

Wilhelmi, B.

B. Wilhelmi, “Distortion of phase-conjugation in degenerate four-wave mixing with phase fluctuations of pump and signal,” Opt. Commun. 82, 89–93 (1991).
[CrossRef]

Wodkiewicz, K.

K. Wodkiewicz and M. S. Zubairy, “Effects of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 27, 2003–2007 (1983).
[CrossRef]

Yariv, A.

Yuen, H. P.

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408–410 (1984).
[CrossRef]

Zubairy, M. S.

A. H. Toor and M. S. Zubairy, “Effects of finite bandwidth on the resonance fluorescence spectrum inside of a cavity,” Phys. Rev. A 49, 449–460 (1994).
[CrossRef] [PubMed]

J. Gea-Banacloche and M. S. Zubairy, “Influence of pump-phase fluctuations on the squeezing in a degenerate parametric oscillator,” Phys. Rev. A 42, 1742–1751 (1990).
[CrossRef] [PubMed]

M. Sargent III, D. A. Holm, and M. S. Zubairy, “Quantum theory of multiwave mixing. I. General formulation,” Phys. Rev. A 31, 3112–3123 (1985).
[CrossRef]

K. Wodkiewicz and M. S. Zubairy, “Effects of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 27, 2003–2007 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

B. Wilhelmi, “Distortion of phase-conjugation in degenerate four-wave mixing with phase fluctuations of pump and signal,” Opt. Commun. 82, 89–93 (1991).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (10)

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source of squeezed-state light,” Phys. Rev. A 30, 343–353 (1984).
[CrossRef]

M. D. Levenson, R. M. Shelby, A. Aspect, M. D. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by non-degenerate four-wave mixing in an optical fiber,” Phys. Rev. A 32, 1550–1562 (1985).
[CrossRef] [PubMed]

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408–410 (1984).
[CrossRef]

M. D. Reid and D. F. Walls, “Quantum theory of nondegenerate four-wave mixing,” Phys. Rev. A 34, 4929–4955 (1986).
[CrossRef] [PubMed]

D. A. Holm and M. Sargent III, “Quantum theory of multiwave mixing. VIII. Squeezed states,” Phys. Rev. A 35, 2150–2163 (1987).
[CrossRef] [PubMed]

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

K. Wodkiewicz and M. S. Zubairy, “Effects of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 27, 2003–2007 (1983).
[CrossRef]

J. Gea-Banacloche and M. S. Zubairy, “Influence of pump-phase fluctuations on the squeezing in a degenerate parametric oscillator,” Phys. Rev. A 42, 1742–1751 (1990).
[CrossRef] [PubMed]

A. H. Toor and M. S. Zubairy, “Effects of finite bandwidth on the resonance fluorescence spectrum inside of a cavity,” Phys. Rev. A 49, 449–460 (1994).
[CrossRef] [PubMed]

M. Sargent III, D. A. Holm, and M. S. Zubairy, “Quantum theory of multiwave mixing. I. General formulation,” Phys. Rev. A 31, 3112–3123 (1985).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Other (2)

For a review, see K. Zaheer and M. S. Zubairy, “Squeezed states of the radiation field,” in Advances in Atomic, Molecular, and Optical Physics, D. R. Bates and B. Bederson, eds. (Academic, New York, 1991), Vol. 28, pp. 143–235.

S.-T. Ho, P. Kumar, and J. H. Shapiro, “Quantum theory of nondegenerate multiwave mixing. General formulation,” Phys. Rev. A 37, 2017–2032 (1988); S.-T. Ho, P. Kumar, and J. H. Shapiro, “Quantum theory of nondegenerate multiwave mixing. III. Application to single-beam squeezed state generation,” J. Opt. Soc. Am. B 8, 37–57 (1991).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram for the interaction of a beam of two-level atoms with two strong driving-pump fields inside a bimodal cavity.

Fig. 2
Fig. 2

Schematic diagram for the generation of squeezing in four-wave mixing and detection with the BHDS.

Fig. 3
Fig. 3

Minimum squared quadrature variance (Δd1)2 versus dimensionless beat frequency Δ1/γ for: Γ=2γ; D/γ equal to 0, 0.01, 0.02, and 0.03; ν/(2Q)=0.07αo; Δ=9γ; and Ω/γ =(50)1/2.

Fig. 4
Fig. 4

Minimum squared quadrature variance (Δd1)2 versus dimensionless beat frequency Δ1/γ for Ω/γ=10; other parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Diagram for the phase-space representation for the observed squared quadrature variances, i.e., (Δd1)2 and (Δd2)2, through the BHDS: (a) D/γ=0 and (b) D/γ0.

Fig. 6
Fig. 6

(Δd2)2 versus dimensionless beat frequency Δ1/γ for the same parameters as in Fig. 3.

Equations (67)

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Vint=Δ2σz+j=1,3gjUjaj exp(-iΔjt)σ+-Ω2exp[-iϕ(t)]σ++H.c.,
ϕ(t)=ϕ0+ϕ1(t),
F(t)=ϕ˙1(t),
F(t)F(t)=2Dδ(t-t).
ϕ1(t)ϕ1(t)=D(t+t-|t-t|),
ρ˙AF=-i[Vint,ρAF].
ρ˙F=-i([Vab, ρba]+[Vba, ρab]).
Vab=j=1,3gjUj*aj exp(iΔjt)+Ω2exp[iϕ(t)].
ρab(1)(t)=-j=1,3igjUjλj-(D-iΔj)(κ1-iΔj)(γ-iΔ+4D-iΔj)+Ω22[(1-Λ)ajρF-ΛρFaj]+μ(j)[(κ2+iΔj)ajρF-(κ1-iΔj)ρFaj]exp(-iΔjt)+igjUj*λj+×ξ(j)[(κ2-iΔj)ρFaj-(κ1-iΔj)×ajρF]+Ω22(D+iΔj)[(β-1)×ρFaj+βajρF]exp(iΔjt).
κ1=Γ+D,
κ2=Γ-D,
Λ=Ω22α1(γ+D),
μ(j)=Ω24α1Γ(γ-iΔ+4D-iΔj)(γ+D+iΔ),
β=Ω22α2(γ+5D),
ξ(j)=Ω24α2(Γ+4D)(γ+9D-iΔ)×(γ+iΔj+4D-iΔ),
α1=Γ(γ+D-iΔ)(γ+D+iΔ)+(γ+D)Ω2,
α2=(Γ+4D)(γ+9D-iΔ)(γ+D+iΔ)+(γ+5D)Ω2,
λj±=(D±iΔj)[Ω2(γ+2D±iΔj)+(D+Γ±iΔj)×(γ±iΔj-iΔ+4D)(γ+iΔ±iΔj)].
ρ˙F=-A1(ρFa1a1-a1ρFa1)-B1+ν2Q×(a1a1ρF-a1ρFa1)+C1(a1a3ρF-a3ρFa1)+D1(ρFa3a1-a1ρFa3)+(samewith13)+H.c.,
A1=g12λ1-(D-iΔ1)(κ1-iΔ1)(γ-iΔ+4D-iΔ1)+Ω22Λ+μ(1)(Γ+D-iΔ1),
B1=g12λ1-(D-iΔ1)(κ1-iΔ1)(γ-iΔ+4D-iΔ1)+Ω22(1-Λ)+μ(1)(Γ-D+iΔ1),
C1=g1g3U1*U3*λ3+ξ(3)(κ1+iΔ3)-Ω22(D+iΔ3)β,
D1=g1g3U1*U3*λ3+ξ(3)(κ2-iΔ3)-Ω22(D+iΔ3)×(1-β).
d=12(a1 exp{i[θ+ϕ(t)]}+a3 exp{i[θ+ϕ(t)]}),
d1=12(d+d),
d2=12i(d-d).
(Δdi)2=14+14(a1a1+a3a3±{a1a3×exp[-2iϕ(t)+2iθ]+c.c.}).
ρF=ρF exp[-2iϕ(t)].
ddt(ρF)=ρ˙F exp[-2iϕ(t)]-2iϕ˙(t)ρF.
ρ˙F=-i({Vab, ρba exp[-2iϕ(t)]}+{Vba, ρab exp[-2iϕ(t)]})-2iϕ˙(t)ρF exp[-2iϕ(t)].
ρba(1)(t)exp[-2iϕ(t)]=-j=1,3igjUj*δ1, j+(D+iΔj)(κ1+iΔj)(γ+iΔ-iΔj)+Ω22(Λ1ajρF-(1-Λ1)ρFaj)+η1(j)[(κ1+iΔj)ajρF-(κ2-iΔj)ρFaj]×exp(iΔjt)+igjUjδ1j-ζ1(j)[(κ1+D-iΔj)ρFaj-(κ2+iΔj)ajρF]+Ω22(D-iΔj)[(1-Λ2)×ajρF-Λ2ρFaj]exp(-iΔjt),
ρab(1)(t)exp[-2iϕ(t)]=-j=1,3igjUjδ2, j-(9D-iΔj)(κ3-iΔj)(γ-iΔ-iΔj+16D)+Ω22[(1-Λ1)ajρF-Λ1ρFaj]+η2(j)[(κ4+iΔj)ajρF-(κ3-iΔj)ρFaj]exp(-iΔjt)+igjUj*δ2j+×ζ2(j)[(κ3+iΔj)ajρF-(κ4-iΔj)×ρFaj]+Ω22(9D+iΔj)[Λ3ajρF-(1-Λ3)ρFaj]exp(iΔjt).
κ3=Γ+9D,
κ4=Γ-9D,
Λ1=Ω22α1(γ+5D),
Λ2=Ω22α2(γ+D),
Λ3=Ω22α3(γ+17D),
η1(j)=Ω24α1(Γ+4D)(γ-iΔ+9D)×(γ+iΔ+iΔj),
η2(j)=Ω24α1(Γ+4D)(γ+iΔ+D)×(γ+16D-iΔ-iΔj),
ζ1(j)=Ω24α2Γ(γ+D+iΔ)(γ+iΔ-iΔj),
ζ2(j)=Ω24α3(Γ+16D)(γ-iΔ+25D)×(γ+16D-iΔ+iΔj),
α1=(Γ+4D)(γ+iΔ+D)(γ+9D-iΔ)+Ω2(γ+5D),
α2=Γ(γ+iΔ+D)(γ+D-iΔ)+Ω2(γ+D),
α3=(Γ+16D)(γ-iΔ+25D)(γ+9D+iΔ)+Ω2(γ+17D),
δ1, j(±)=(D±iΔj)(γ+4D±iΔj-iΔ)(Γ+D±iΔj)×(γ+iΔ±iΔj)+Ω2(γ+2D±iΔj),
δ2, j(±)=(9D±iΔj)[(γ+4D±iΔj+iΔ)(Γ+9D±iΔj)×(γ+16D-iΔ±iΔj)+Ω2(γ+10D±iΔj)].
ρ˙F=-E1(ρFa1a1-a1ρFa1)-F1+ν2Q×(a1a1ρF-a1ρFa1)-E1(a1a1ρF-a1ρFa1)-F1+ν2Q(ρFa1a1-a1ρFa1)+G1(a1a3ρF-a3ρFa1)+H1(ρFa3a1-a1ρFa3)+G1(ρFa1a3-a1ρFa3)+H1(a1a3ρF-a3ρFa1)+(samewith13).
E1=g12δ2,1-(9D-iΔ1)(κ3-iΔ1)(γ-iΔ+16D-iΔ1)+Ω22Λ1+η2(1)(κ3-iΔ1),
F1=g12δ2,1-(9D-iΔ1)(κ3-iΔ1)(γ-iΔ+16D-iΔ1)+Ω22(1-Λ1)+η2(1)(κ4-iΔ1),
E1=g12δ1,1+(D+iΔ1)(κ1+iΔ1)(γ+iΔ+iΔ1)+Ω22Λ1+η1(1)(κ1+iΔ1),
F1=g12δ1,1+(D+iΔ1)(κ1+iΔ1)(γ+iΔ+iΔ1)+Ω22×(1-Λ1)+η1(1)(κ2-iΔ1),
G1=g1g3U1*U3*δ2,3+ζ2(3)(κ3+iΔ3)-Ω22(9D+iΔ3)Λ3,
H1=g1g3U1*U3*δ2,3+ζ2(3)(κ4-iΔ3)-Ω22(9D+iΔ3)×(1-Λ3),
G1=g1g3U1U3δ1,3-ζ1(3)(κ1-iΔ3)-Ω22(D-iΔ3)Λ2,
H1=g1g3U1U3δ1,3-ζ1(3)(κ2+iΔ3)-Ω22(D-iΔ3)×(1-Λ2).
(Δd1)2=14+14{a1a1+a3a3-2|a1a3 exp[-2iϕ(t)]|}.
ddta1a1=A1+A1*-B1-B1*-νQa1a1+(C1-D1)a1a3 exp[2iϕ(t)]+(C1*-D1*)×a1a3 exp[-2iϕ(t)]+A1+A1*,
ddta1a3 exp[-2iϕ(t)]=E1*+E3*-F1*-F3*-νQ+4D×a1a3 exp[-2iϕ(t)]+(G1*-H1*)a3a3+(G3*-H3*)a1a1+G1*+G3*,
ddta3a3=A3+A3*-B3-B3*-νQa3a3+(C3-D3)a1a3 exp[2iϕ(t)]+(C3*-D3*)a1a3 exp[-2iϕ(t)]+A3+A3*,
ddta1a3 exp[2iϕ(t)]=E1+E3-F1-F3-νQ+4D×a1a3 exp[2iϕ(t)]+(G1-H1)a3a3+(G3-H3)a1a1+G1+G3.
a1a1=1χ{[M(MN3-L1*P3*)-L1P3M*](A1+A1*)-[P1*(MN3-L1P3)+L1P1P3*](G1*+G3*)+(MP1*L1*+L1P1M*)(A3+A3*)-[P1(M*N3-L1*P3*)+L1*P1*P3]×(G1+G3)},
a1a3 exp[-2iϕ(t)]=1χ{[L3*(MN3-L1P3)+L1*P3L3](A1+A1*)+[N1(MN3-L1P3)-L3P1N3](G1*+G3*)-[MN1L1*+P1(L1L3*-L3L1*)](A3+A3*)+(N1L1*P3+L3*P1N3)(G1+G3)},
χ=N1[M*(N3M-L1P3)-L1*P3*M]-P1*[L3*(N3M-L1P3)+L1*P3*L3]-P1[L3(N3M*-L1*P3*)+L3*P3*L1],
M=F1+F3-E1-E3+νQ+4D,
Nj=Bj+Bj*-Aj-Aj*+νQ,
Lj=Hj-Gj,
Pj=Dj-Cj.

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