Abstract

A detailed analytical description of a Doppler-free probe response (absorption and dispersion) is presented for a coherently prepared four-level atomic tripod driven by two coupling fields. Under the density matrix formalism the field-dependent and field-independent coherence are introduced through the off-diagonal matrix elements. Field-dependent coherence appears due to the interference between the probability amplitudes of finding the atom in energy levels associated with the allowed transition, while field-independent phases measure the atomic coherence for a coherently prepared lower state triplet. The double-control coherent effects on the probe response are investigated by controlling the relative driving contribution (coherence) of the two coupling fields, whereas the driving contribution can be controlled by the use of field-independent coherence, along with two coupling parameters like Rabi frequency and detuning of the respective coupling field. We report on the possibility of modifying the Rabi splitting into a perfect electromagnetically induced transparent window at probe resonance by a coherent double-control mechanism in an on-resonance coupling situation. Moreover, for off-resonance coupling, we exhibit the enhancement of multiphoton (two- or three-photon) absorption via coherent control of multiresonance processes.

© 2012 Optical Society of America

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  1. S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
    [CrossRef]
  2. K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
    [CrossRef]
  3. A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
    [CrossRef]
  4. G. S. Agarwal, “Inhibition of spontaneous emission noise in laser without inversion,” Phys. Rev. Lett. 67, 980–982 (1991).
    [CrossRef]
  5. S. Wielandy and A. L. Gaeta, “Coherent control of the polarization of an optical field,” Phys. Rev. Lett. 81, 3359–3362 (1998).
    [CrossRef]
  6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [CrossRef]
  7. E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
    [CrossRef]
  8. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
    [CrossRef]
  9. G. W. Chantry, ed., “Microwave–microwave double resonance,” in Modern Aspects of Microwave Spectroscopy (Academic, 1979), pp. 65–122.
  10. P. Meystre and M. Sargent, Elements of Quantum Optics, 3rd ed. (Springer, 2006).
  11. S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate,” Phys. Rev. A 47, 4934–4945 (1993).
    [CrossRef]
  12. S. Ghosh and S. Mandal, “Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects,” J. Phys. B 42, 145403 (2009).
    [CrossRef]
  13. S. Ghosh and S. Mandal, “A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ-type five level system,” Opt. Commun. 284, 376–387 (2011).
    [CrossRef]
  14. S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B 43, 245505 (2010).
    [CrossRef]
  15. G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996).
    [CrossRef]
  16. N. Mulchan, D. G. Ducreay, R. Pina, M. Yan, and Y. F. Zhu, “Nonlinear excitation by quantum interference in a Doppler-broadened rubidium atomic system,” J. Opt. Soc. Am. B 17, 820–826 (2000).
    [CrossRef]
  17. J. Q. Shen and P. Zhang, “Double-control quantum interferences in a four-level atomic system,” Opt. Express 15, 6484–6493 (2007).
    [CrossRef]
  18. Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
    [CrossRef]
  19. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
    [CrossRef]
  20. S. Stenholm, Foundations of Laser Spectroscopy (Wiley, 1983).
  21. J. Miynek and W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
    [CrossRef]

2011 (2)

S. Ghosh and S. Mandal, “A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ-type five level system,” Opt. Commun. 284, 376–387 (2011).
[CrossRef]

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

2010 (1)

S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B 43, 245505 (2010).
[CrossRef]

2009 (1)

S. Ghosh and S. Mandal, “Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects,” J. Phys. B 42, 145403 (2009).
[CrossRef]

2007 (1)

2005 (1)

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

2002 (1)

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[CrossRef]

2000 (1)

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1998 (1)

S. Wielandy and A. L. Gaeta, “Coherent control of the polarization of an optical field,” Phys. Rev. Lett. 81, 3359–3362 (1998).
[CrossRef]

1996 (1)

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996).
[CrossRef]

1993 (1)

S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate,” Phys. Rev. A 47, 4934–4945 (1993).
[CrossRef]

1991 (3)

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[CrossRef]

G. S. Agarwal, “Inhibition of spontaneous emission noise in laser without inversion,” Phys. Rev. Lett. 67, 980–982 (1991).
[CrossRef]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

1990 (1)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

1989 (1)

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[CrossRef]

1979 (1)

J. Miynek and W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996).
[CrossRef]

G. S. Agarwal, “Inhibition of spontaneous emission noise in laser without inversion,” Phys. Rev. Lett. 67, 980–982 (1991).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

Ducreay, D. G.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Field, J. E.

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

Gaeta, A. L.

S. Wielandy and A. L. Gaeta, “Coherent control of the polarization of an optical field,” Phys. Rev. Lett. 81, 3359–3362 (1998).
[CrossRef]

Ghosh, P. N.

S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate,” Phys. Rev. A 47, 4934–4945 (1993).
[CrossRef]

Ghosh, S.

S. Ghosh and S. Mandal, “A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ-type five level system,” Opt. Commun. 284, 376–387 (2011).
[CrossRef]

S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B 43, 245505 (2010).
[CrossRef]

S. Ghosh and S. Mandal, “Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects,” J. Phys. B 42, 145403 (2009).
[CrossRef]

Gong, S.

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[CrossRef]

Harshawardhan, W.

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

Knight, P. L.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[CrossRef]

Krishna, A.

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

Lange, W.

J. Miynek and W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Mandal, S.

S. Ghosh and S. Mandal, “A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ-type five level system,” Opt. Commun. 284, 376–387 (2011).
[CrossRef]

S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B 43, 245505 (2010).
[CrossRef]

S. Ghosh and S. Mandal, “Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects,” J. Phys. B 42, 145403 (2009).
[CrossRef]

S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate,” Phys. Rev. A 47, 4934–4945 (1993).
[CrossRef]

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics, 3rd ed. (Springer, 2006).

Miynek, J.

J. Miynek and W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Mulchan, N.

Natarajan, V.

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

Niu1, Y.

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

Pandey, K.

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

Paspalakis, E.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[CrossRef]

Peng, Y.

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

Pina, R.

Qi, Y.

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

Sargent, M.

P. Meystre and M. Sargent, Elements of Quantum Optics, 3rd ed. (Springer, 2006).

Scully, M. O.

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[CrossRef]

Shen, J. Q.

Stenholm, S.

S. Stenholm, Foundations of Laser Spectroscopy (Wiley, 1983).

Wasan, A.

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

Wielandy, S.

S. Wielandy and A. L. Gaeta, “Coherent control of the polarization of an optical field,” Phys. Rev. Lett. 81, 3359–3362 (1998).
[CrossRef]

Yan, M.

Zhang, P.

Zhou, F.

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

Zhu, Y. F.

Europhys. Lett. (1)

A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using electromagnetically induced transparency,” Europhys. Lett. 72, 221–227 (2005).
[CrossRef]

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

J. Phys. B (3)

Y. Qi, Y. Niu1, F. Zhou, Y. Peng, and S. Gong, “Phase control of coherent pulse propagation and switching based on electromagnetically induced transparency in a four-level atomic system,” J. Phys. B 44, 085502 (2011).
[CrossRef]

S. Ghosh and S. Mandal, “Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects,” J. Phys. B 42, 145403 (2009).
[CrossRef]

S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B 43, 245505 (2010).
[CrossRef]

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meter per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Opt. Commun. (2)

S. Ghosh and S. Mandal, “A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ-type five level system,” Opt. Commun. 284, 376–387 (2011).
[CrossRef]

J. Miynek and W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (2)

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[CrossRef]

S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate,” Phys. Rev. A 47, 4934–4945 (1993).
[CrossRef]

Phys. Rev. Lett. (7)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

G. S. Agarwal, “Inhibition of spontaneous emission noise in laser without inversion,” Phys. Rev. Lett. 67, 980–982 (1991).
[CrossRef]

S. Wielandy and A. L. Gaeta, “Coherent control of the polarization of an optical field,” Phys. Rev. Lett. 81, 3359–3362 (1998).
[CrossRef]

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[CrossRef]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Obsevation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[CrossRef]

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996).
[CrossRef]

Other (3)

S. Stenholm, Foundations of Laser Spectroscopy (Wiley, 1983).

G. W. Chantry, ed., “Microwave–microwave double resonance,” in Modern Aspects of Microwave Spectroscopy (Academic, 1979), pp. 65–122.

P. Meystre and M. Sargent, Elements of Quantum Optics, 3rd ed. (Springer, 2006).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of a driven four-level atomic tripod. The transitions |1|4 and |3|4 are driven by two coupling fields with Rabi frequencies Ω1and Ω2 respectively, while the |2|4 transition is probed by a relatively weak field with Rabi frequency Ωp. The detuning of the two coupling fields are Δ1 and Δ2, while the detuning and effective linewidth for probe transition are Δp and Γp, respectively.

Fig. 2.
Fig. 2.

Diagrammatic representation of transitions. Dipole-allowed transitions between the levels |1|4, |2|4, and |3|4 depend on Ω12, Ωp2, and Ω22, respectively, whereas the paths |1|2, |2|3, and |1|3 depend on Ω12Ωp2, Ω22Ωp2, and Ω12Ω22 respectively.

Fig. 3.
Fig. 3.

Absorption is plotted against probe detuning (Δp) for different values of the Rabi frequencies of the two coupling fields, with ϕ12=ϕ23=0 and Δ1=Δ2=0. (a) Ω1=Ω2=0, (b) Ω1=Ω2=0.01GHz, (c) Ω1=Ω2=0.02GHz.

Fig. 4.
Fig. 4.

Absorption and dispersion (dotted curve) are plotted against probe detuning (Δp) for different values of the Rabi frequencies of the two coupling fields, with ϕ12=π, ϕ23=0, and Δ1=Δ2=0. (a) Ω1=Ω2=0.02GHz, (b) Ω1=0.021GHz, Ω2=0.02GHz, (b) dispersion corresponding to absorption curve (b), (c) Ω1=Ω2=0.

Fig. 5.
Fig. 5.

Absorption (solid curve) and dispersion (dotted curve) are plotted against probe detuning (Δp) for ϕ12=ϕ23=0, Δ1=0.25GHz, Δ2=0.35GHz, and Ω1=Ω2=0.02GHz.

Fig. 6.
Fig. 6.

Absorption is plotted against probe detuning (Δp) for different values of the phases (ϕij), with Δ1=0.3GHz, Δ2=0.3GHz, and Ω1=Ω2=0.02GHz. (a) ϕ12=ϕ23=0, (b) ϕ12=ϕ23=π6, (c) ϕ12=ϕ23=π4, (d) ϕ12=ϕ23=π2.

Fig. 7.
Fig. 7.

Absorption is plotted against probe detuning (Δp) for different values of the phases (ϕij), with Δ1=0.6GHz, Δ2=0.3GHz, and Ω1=Ω2=0.02GHz. (a) ϕ12=ϕ23=0, (b) ϕ12=ϕ23=π6, (c) ϕ12=ϕ23=π4.

Equations (15)

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H=H02με1cos(ω1t)2με2cos(ω2t)2μεpcos(ωpt),
ρ11˙=γ1ρ112iΩ1cosω1t(ρ41ρ14),ρ22˙=γ2ρ222iΩpcosωpt(ρ42ρ24),ρ33˙=γ3ρ332iΩ2cosω2t(ρ43ρ34),ρ44˙=γ4ρ442iΩpcosωpt(ρ24ρ42)2iΩ1cosω1t(ρ14ρ41)2iΩ2cosω2t(ρ34ρ43),ρ12˙=ρ21*˙=(γ12+iω12)ρ122iΩ1cosω1tρ42+2iΩpcosωptρ14,ρ13˙=ρ31*˙=(γ13+iω13)ρ132iΩ1cosω1tρ43+2iΩ2cosω2tρ14,ρ23˙=ρ32*˙=(γ23+iω23)ρ23+2iΩ2cosω2tρ242iΩpcosωptρ43,ρ14˙=ρ41*˙=(γ14+iω14)ρ14+2iΩ1cosω1t(ρ11ρ44)+2iΩ2cosω2tρ13+2iΩpcosωptρ12,ρ24˙=ρ42*˙=(γ24+iω24)ρ24+2iΩpcosωpt(ρ22ρ44)+2iΩ2cosω2tρ23+2iΩ1cosω1tρ21,ρ34˙=ρ43*˙=(γ34+iω34)ρ34+2iΩ2cosω2t(ρ33ρ44)+2iΩpcosωptρ32+2iΩ1cosω1tρ31.
ρ14=ρ˜14exp(iω1t),ρ24=ρ˜24exp(iωpt),ρ34=ρ˜34exp(iω2t),
2cos(ωit)exp(iωit)=exp(2iωit)+11.
ρ11(t)=ρ11(0)eγ1tiΩ1γ1(ρ˜41ρ˜14)(1eγ1t),ρ22(t)=ρ22(0)eγ2tiΩpγ2(ρ˜42ρ˜24)(1eγ2t),ρ33(t)=ρ33(0)eγ3tiΩ2γ3(ρ˜43ρ˜34)(1eγ3t),ρ44(t)=ρ44(0)eγ4t+[iΩ1γ4(ρ˜41ρ˜14)+iΩpγ4(ρ˜42ρ˜24)+iΩ2γ4(ρ˜43ρ˜34)](1eγ4t).
ρ12(t)=ρ12(0)e(γ12+iω12)t+i[eiδ1te(γ12+iω12)tγ12+i(ω12δ1)t](Ωpρ˜14Ω1ρ˜42),ρ23(t)=ρ23(0)e(γ23+iω23)t+i[eiδ2te(γ23+iω23)tγ23+i(ω23δ2)t](Ω2ρ˜24Ωpρ˜43),ρ13(t)=ρ13(0)e(γ13+iω13)t+i[eiδte(γ13+iω13)tγ13+i(ω13δ)t](Ω2ρ˜14Ω1ρ˜43).
(a11a12a13a21a22a23a31a32a33)(ρ˜41ρ˜42ρ˜43)=(l1l2l3),
ρ˜42=iΩp[ρ22(0)eγ2tρ44(0)eγ4t]iΩ1ρ12(0)e[γ12+i(ΔpΔ1)]tiΩ2ρ32(0)e[γ23+i(ΔpΔ2)]tγ24+iΔp+Ω12γ12+i(ΔpΔ1)+Ω22γ23+i(ΔpΔ2),
P=r0T(μ24ρ42(t))dt,
ρ12(0)γ12+i(ΔpΔ1)=ρ12(0)γ122+(ΔpΔ1)2exp(iϕ12).
ρ23(0)γ23i(ΔpΔ2)=ρ23(0)γ232+(ΔpΔ2)2exp(iϕ23).
Re(P˜)=K×[Γp{(Ω1Ωp)|ρ12(0)|sinϕ12γ122+(ΔpΔ1)2+(Ω2Ωp)|ρ23(0)|sinϕ23γ232+(ΔpΔ2)2}+Δp{ρ22(0)γ2ρ44(0)γ4+(Ω1Ωp)|ρ12(0)|cosϕ12γ122+(ΔpΔ1)2+(Ω2Ωp)|ρ23(0)|cosϕ23γ232+(ΔpΔ2)2}Γp2+(Δp)2],
Im(P˜)=K×[Γp{ρ22(0)γ2ρ44(0)γ4+(Ω1Ωp)|ρ23(0)|cosρ23γ122+(ΔpΔ1)2+(Ω2Ωp)|ρ23(0)|cosϕ23γ232+(ΔpΔ2)2}+Δp{(Ω1Ωp)|ρ12(0)|sinϕ12γ122+(ΔpΔ1)2(Ω2Ωp)|ρ23(0)|sinϕ23γ232+(ΔpΔ2)2}Γp2+(Δp)2],
Γp=γ24+Ω12γ12γ122+(ΔpΔ1)2+Ω22γ23γ232+(ΔpΔ2)2,
Δp=ΔpΩ12(ΔpΔ1)γ122+(ΔpΔ1)2+Ω22(ΔpΔ2)γ232+(ΔpΔ2)2.

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