Abstract

The optical properties of a four-level double-V-type closed-loop system is studied in the multiphoton resonance condition. It is shown that the phase-dependent electromagnetically induced transparency (EIT) window is established and by changing the relative phase of the applied fields, EIT switches to electromagnetically induced absorption. Moreover, the optical bistability (OB) behavior of the system is studied, and it is demonstrated that the cross-Kerr-effect-induced OB depends on the relative phase of the applied fields, and it can be controlled by either intensity or relative phase of laser fields. By applying a microwave field to drive the hyperfine transition between two lower states, the optical properties change and OB switches to optical multistability. This can be used for detection of longer wavelengths than visible light.

© 2014 Optical Society of America

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References

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  1. See for example, a review by L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984), Vol. 21, pp. 69–216.
  2. H. M. Gibbs and D. Sarid, “Optical bistability: controlling light by light,” Phys. Today 40(8), 71 (1987).
    [CrossRef]
  3. J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
    [CrossRef]
  4. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).
  5. J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
    [CrossRef]
  6. S.-T. Chen and M. R. Chatterjee, “Dual-input hybrid acousto-optic set-reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147–3154 (1997).
  7. M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
    [CrossRef]
  8. F. S. Almehmadi and M. R. Chatterjee, “Numerical examination of the nonlinear dynamics of a hybrid acousto-optic Bragg cell with positive feedback under profiled beam propagation,” J. Opt. Soc. Am. B 31, 833–841 (2014).
    [CrossRef]
  9. A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).
  10. E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
    [CrossRef]
  11. M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multi-photon resonance condition,” Phys. Rev. A 74, 063827 (2006).
    [CrossRef]
  12. E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
    [CrossRef]
  13. N. Heidari and M. Mahmoudi, “Phase and amplitude control of optical bistability in the three-coupled quantum wells,” Physica E 44, 1288–1294 (2012).
  14. S. H. Asadpour, M. Jaberi, and H. R. Soleimani, “Phase control of optical bistability and multistability via spin coherence in a quantum well waveguide,” J. Opt. Soc. Am. B 30, 1815–1820 (2013).
    [CrossRef]
  15. A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
    [CrossRef]
  16. Z. Wang, A. X. Chen, Y. Bai, W. X. Yang, and R. K. Lee, “Coherent control of optical bistability in an open Λ-type three-level atomic system,” J. Opt. Soc. Am. B 29, 2891–2896 (2012).
    [CrossRef]
  17. H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
    [CrossRef]
  18. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Radio-frequency-induced optical gain in Pr3+:Y2SiO5,” J. Opt. Soc. Am. B 15, 1541–1544 (1998).
    [CrossRef]

2014

2013

S. H. Asadpour, M. Jaberi, and H. R. Soleimani, “Phase control of optical bistability and multistability via spin coherence in a quantum well waveguide,” J. Opt. Soc. Am. B 30, 1815–1820 (2013).
[CrossRef]

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

2012

Z. Wang, A. X. Chen, Y. Bai, W. X. Yang, and R. K. Lee, “Coherent control of optical bistability in an open Λ-type three-level atomic system,” J. Opt. Soc. Am. B 29, 2891–2896 (2012).
[CrossRef]

N. Heidari and M. Mahmoudi, “Phase and amplitude control of optical bistability in the three-coupled quantum wells,” Physica E 44, 1288–1294 (2012).

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
[CrossRef]

2009

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
[CrossRef]

2007

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).

2006

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multi-photon resonance condition,” Phys. Rev. A 74, 063827 (2006).
[CrossRef]

1999

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

1998

1997

1987

H. M. Gibbs and D. Sarid, “Optical bistability: controlling light by light,” Phys. Today 40(8), 71 (1987).
[CrossRef]

Almehmadi, F. S.

Al-Saedi, M.

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Al-Saedi, M. A.

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

Asadpour, S. H.

Bai, Y.

Baluschev, S.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

Chatterjee, M. R.

F. S. Almehmadi and M. R. Chatterjee, “Numerical examination of the nonlinear dynamics of a hybrid acousto-optic Bragg cell with positive feedback under profiled beam propagation,” J. Opt. Soc. Am. B 31, 833–841 (2014).
[CrossRef]

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

S.-T. Chen and M. R. Chatterjee, “Dual-input hybrid acousto-optic set-reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147–3154 (1997).

Chen, A. X.

Chen, S.-T.

Cheng, S.

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Deng, L.

H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
[CrossRef]

Evers, J.

M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multi-photon resonance condition,” Phys. Rev. A 74, 063827 (2006).
[CrossRef]

Ghosh, A. K.

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Gibbs, H. M.

H. M. Gibbs and D. Sarid, “Optical bistability: controlling light by light,” Phys. Today 40(8), 71 (1987).
[CrossRef]

Goharshenasan, S.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

Ham, B. S.

Heidari, N.

N. Heidari and M. Mahmoudi, “Phase and amplitude control of optical bistability in the three-coupled quantum wells,” Physica E 44, 1288–1294 (2012).

Hemmer, P. R.

Huang, G.

H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
[CrossRef]

Huang, Q. J.

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

Huck, R. C.

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Huss, A.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

Jaberi, M.

Korsunsky, E. A.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

Kosachiov, D. V.

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

Lee, R. K.

Leinfellner, N.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

Li, H. J.

H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
[CrossRef]

Li, J. H.

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

Lu, X. Y.

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

Lugiato, L. A.

See for example, a review by L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984), Vol. 21, pp. 69–216.

Luo, J. M.

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

MacDonald, K. F.

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
[CrossRef]

Mahmoudi, M.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

N. Heidari and M. Mahmoudi, “Phase and amplitude control of optical bistability in the three-coupled quantum wells,” Physica E 44, 1288–1294 (2012).

M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multi-photon resonance condition,” Phys. Rev. A 74, 063827 (2006).
[CrossRef]

Mortezapour, A.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

Nozari, N.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

Sarid, D.

H. M. Gibbs and D. Sarid, “Optical bistability: controlling light by light,” Phys. Today 40(8), 71 (1987).
[CrossRef]

Shahriar, M. S.

Soleimani, H. R.

Vafafard, A.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

Verma, P.

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Wang, Z.

Windholz, L.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

Yang, W. X.

Zhang, J.

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
[CrossRef]

Zheludev, N. I.

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
[CrossRef]

Appl. Opt.

Eur. Phys. J. D

H. J. Li, L. Deng, and G. Huang, “Dynamics of coupled ultraslow optical solitons in a coherent four-state double-Λ system,” Eur. Phys. J. D 55, 99–109 (2009).
[CrossRef]

J. Lumin.

A. Vafafard, S. Goharshenasan, N. Nozari, A. Mortezapour, and M. Mahmoudi, “Phase-dependent optical bistability in the quantum dot nanostructure molecules via inter-dot tunneling,” J. Lumin. 134, 900–905 (2013).
[CrossRef]

J. Opt. Soc. Am. B

Light Sci. Appl.

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light Sci. Appl. 1, e18 (2012).
[CrossRef]

Opt. Eng.

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

Phys. Rev. A

J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[CrossRef]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multi-photon resonance condition,” Phys. Rev. A 74, 063827 (2006).
[CrossRef]

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302–2305 (1999).
[CrossRef]

Phys. Rev. B

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).

Phys. Today

H. M. Gibbs and D. Sarid, “Optical bistability: controlling light by light,” Phys. Today 40(8), 71 (1987).
[CrossRef]

Physica E

N. Heidari and M. Mahmoudi, “Phase and amplitude control of optical bistability in the three-coupled quantum wells,” Physica E 44, 1288–1294 (2012).

Proc. SPIE

A. K. Ghosh, P. Verma, S. Cheng, R. C. Huck, M. R. Chatterjee, and M. Al-Saedi, “Design of acousto-optic chaos based secure free-space optical communication links,” Proc. SPIE 7464, 74640L (2009).

Other

See for example, a review by L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984), Vol. 21, pp. 69–216.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic diagram of a four-level double-V-type closed-loop system. The lower states are the two hyperfine levels |0=5S1/2 (F=1, MF=1) and |3=5S1/2 (F=2, MF=1), separated by 6.83 GHz. The excited levels are |1=5P1/2 (F=1, MF=0) and |2=5P1/2 (F=2, MF=0), separated by 816.65 MHz. Here, F and F are the total atomic angular momentum quantum numbers. MF stands for magnetic quantum number of the corresponding state. (b) Unidirectional ring cavity with double-V sample of length L. EpI and EpT are the incident and transmitted fields, respectively. Ec is the pumping field which does not circulate in the cavity. For mirror 1 and 2 it is assumed that R+T=1, and mirrors 3 and 4 have perfect reflectivity.

Fig. 2.
Fig. 2.

(a) Dispersion and (b) absorption of the probe field for different relative phases of the applied fields versus Δp. Parameters used are γ13=γ23=0.2γ, γ10=γ20=0.2γ/3, Δc1=Δc2=0, Δp1=Δp2=Δp, Ωc1=Ωc2=0.1γ, and Ωp1=Ωp2=0.001γ.

Fig. 3.
Fig. 3.

Different paths to create the probe coherences ρ10 and ρ20. (a) The scattering of the two pumping fields and one of the probe fields into the other probe field mode via wave mixing. (b) The cross-phase modulation of one of the coupling fields. (c) The direct scattering of the probe field.

Fig. 4.
Fig. 4.

OB of the system for ϕ=0 (solid line), π (dashed line), Δp1=Δp2=Δc1=Δc2=0, and C=200. Other parameters are the same as in Fig. 2.

Fig. 5.
Fig. 5.

OB for different values of pumping fields. Parameters used are Δp1=Δp2=Δc1=Δc2=0, C=200, and ϕ=π. The other parameters are the same as in Fig. 2.

Fig. 6.
Fig. 6.

OM of a system in the presence of a microwave field for different values of Ωr=0.0 (solid line), 0.15γ(dashed line) for Ωc1=0.15γ and Ωc2=0.25γ. The other parameters used are the same as in Fig. 2.

Equations (13)

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

Hint=Ωp1eiΔp1tiϕp1|01|Ωp2eiΔp2tiϕp2|02|Ωc1eiΔc1tiϕc1|31|Ωc2eiΔc2tiϕc2|32|ΩreiΔrtiϕr|03|+H.c.,
ρ˙00=(iΩp1ρ10+iΩp2ρ20+iΩrρ30+c.c)+γ10ρ11+γ20ρ22,ρ˙11=(iΩp1ρ10iΩc1eiϕρ13+c.c)γ13ρ11γ10ρ11,ρ˙22=(iΩc2eiϕρ23iΩp2ρ20+c.c)γ23ρ22γ20ρ22,ρ˙23=(iΔc2Γ23)ρ23iΩc1*ei((ΔΔ)tϕ)ρ21iΩrρ20eiΔtiΩc2*eiϕ(ρ22ρ33)+iΩp2*eiΔtρ03,ρ˙13=(iΔc1Γ13)ρ13iΩrρ10eiΔtiΩc1*(ρ11ρ33)eiϕiΩc2*ei((ΔΔ)tϕ)ρ12+iΩp1*eiΔtρ03,ρ˙30=(iΔrΓ30)ρ30+iΩr*(ρ00ρ33)+iΩc1ei(Δtϕ)ρ10+iΩc2ei(Δtϕ)ρ20iΩp1*eiΔtρ31iΩp2*eiΔtρ32,ρ˙12=(i(Δp1Δp2)Γ12)ρ12iΩp2ρ10iΩc2ei((ΔΔ)t+ϕ)ρ13+iΩp1*ρ02+iΩc1*ei((ΔΔ)tϕ)ρ32,ρ˙20=(iΔp2Γ20)ρ20+iΩp2*(ρ00ρ22)+iΩc2*ei(Δtϕ)ρ30iΩp1*ρ21iΩr*eiΔtρ23,ρ˙10=(iΔp1Γ10)ρ10+iΩp1*(ρ00ρ11)+iΩc1*ei(Δtϕ)ρ30iΩp2*ρ12iΩr*eiΔtρ13,
χ(ωp)=2α0Γkpρp(ωp)Ωp.
ρ10=Ωpeiϕp1[|Ωc|2eiϕ|Ωc|2+Δp(iΓ2+Δp)]A,
ρ20=Ωpeiϕp2[|Ωc|2eiϕ|Ωc|2+Δp(iΓ1+Δp)]A,
|D1=1Ωc2+Ωp2(Ωc|0+Ωp|3),|D2=12(|1+|2),|D3=Ωp2Ωc2+Ωp2|012|112|2+Ωc2Ωc2+Ωp2|3,|D4=Ωp2Ωc2+Ωp2|0+12|1+12|2+Ωc2Ωc2+Ωp2|3,
Λ1=0,Λ2=0,Λ3=2Ωc2+Ωp2,Λ4=2Ωc2+Ωp2.
|d1=12(|1|2+2|3),|d2=12(|1+|2+2|3),|d3=12(2|0+|1+|2),|d4=12(2|0+|1+|2),
λ1=2Ωc,λ2=2Ωc,λ3=2Ωp,λ4=2Ωp.
E=E⃗p1ei(ωp1t+ϕp1)+E⃗p2ei(ωp2t+ϕp2)+E⃗c1ei(ωc1t+ϕc1)+E⃗c2ei(ωc2t+ϕc2)+c.c.,
Ept+cEpz=iωp2ε0P(ωp),
Ep(0)=TEpI+REp(L),Ep(L)=EpT/T.
y=xiC(ρ01+γ20γ10ρ02),

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