Abstract

The resonance fluorescence system of a coherently driven two-level atom in the presence of a broadband squeezed vacuum field whose central frequency is not in resonance with either the driving field frequency or the atomic transition frequency is analyzed. Solutions of the Bloch equations of the system are presented in terms of continued fractions for arbitrary strength of the coherent driving field. Numerical results are presented for the first-harmonic quadrature components of the absorption spectrum and the fluorescent intensity. The effects of the squeezed vacuum field detuning on the absorption spectrum result in hole burning or a dip structure in the weak-field case and two absorption–amplification peaks in the strong-field case. Results are sensitive to the relative phase of the squeezed vacuum.

© 1995 Optical Society of America

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  1. H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
    [CrossRef] [PubMed]
  2. H. Ritsch and P. Zoller, "Absorption spectrum of a two-level system in a squeezed vacuum," Opt. Commun. 64, 523–528 (1987).
    [CrossRef]
  3. S. Smart and S. Swain, "Dispersive profiles in resonance fluorescence of a two-level atom in a squeezed vacuum," Phys. Rev. A 48, R50–R53 (1993).
    [CrossRef] [PubMed]
  4. M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
    [CrossRef]
  5. A. S. Shumovsky and T. Quang, "Spectral and statistical properties of collective resonance fluorescence in a squeezed vacuum," J. Phys. B 22, 131–142 (1989); T. Quang, M. Kozierowski, and L. H. Lan, "Collective resonance fluorescence in a squeezed vacuum," Phys. Rev. A 39, 644–646 (1989).
    [CrossRef] [PubMed]
  6. Z. Ficek, "Effect of a broadband squeezed vacuum on twoatom spontaneous emission," Opt. Commun. 82, 130–136 (1991).
    [CrossRef]
  7. M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).
  8. Z. Ficek and B. J. Dalton, "Squeezing-induced transparency in a two-level atom," Opt. Commun. 102, 231–237 (1993).
    [CrossRef]
  9. S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
    [CrossRef]
  10. Z. Ficek and B. C. Sanders, "Resonance fluorescence of a two-level atom in an off-resonance squeezed vacuum," J. Phys. B 27, 809–824 (1994).
    [CrossRef]
  11. N. Nayak and G. S. Agarwal, "Absorption and fluorescence in frequency-modulated fields under conditions of strong modulations and saturation," Phys. Rev. A 31, 3175–3182 (1988).
    [CrossRef]
  12. G. S. Agarwal and N. Nayak, "Saturation of optical susceptibilities in strongly amplitude-modulated fields," J. Phys. B 19, 3385–3400 (1986).
    [CrossRef]
  13. G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, "Spectrum of radiation from two-level atoms under intense bichromatic excitation," J. Opt. Soc. Am. B 8, 1163–1167 (1991).
    [CrossRef]
  14. S. Papademetriou, S. Chakmakjian, and C. R. Stroud, Jr., "Optical subharmonic Rabi resonances," J. Opt. Soc. Am. B 9, 1182–1188 (1992).
    [CrossRef]
  15. R. Kubo, "Statistical mechanical theory of irreversible processes: I—General theory and simple applications to magnetic and conduction problems," J. Phys. Soc. Jpn. 12, 570–586 (1957).
    [CrossRef]
  16. B. R. Mollow, "Stimulated emission and absorption near resonance for driven systems," Phys. Rev. A 5, 2217–2222 (1972).
    [CrossRef]

1995 (1)

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

1994 (1)

Z. Ficek and B. C. Sanders, "Resonance fluorescence of a two-level atom in an off-resonance squeezed vacuum," J. Phys. B 27, 809–824 (1994).
[CrossRef]

1993 (2)

S. Smart and S. Swain, "Dispersive profiles in resonance fluorescence of a two-level atom in a squeezed vacuum," Phys. Rev. A 48, R50–R53 (1993).
[CrossRef] [PubMed]

Z. Ficek and B. J. Dalton, "Squeezing-induced transparency in a two-level atom," Opt. Commun. 102, 231–237 (1993).
[CrossRef]

1992 (1)

1991 (2)

1989 (1)

A. S. Shumovsky and T. Quang, "Spectral and statistical properties of collective resonance fluorescence in a squeezed vacuum," J. Phys. B 22, 131–142 (1989); T. Quang, M. Kozierowski, and L. H. Lan, "Collective resonance fluorescence in a squeezed vacuum," Phys. Rev. A 39, 644–646 (1989).
[CrossRef] [PubMed]

1988 (1)

N. Nayak and G. S. Agarwal, "Absorption and fluorescence in frequency-modulated fields under conditions of strong modulations and saturation," Phys. Rev. A 31, 3175–3182 (1988).
[CrossRef]

1987 (2)

H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
[CrossRef] [PubMed]

H. Ritsch and P. Zoller, "Absorption spectrum of a two-level system in a squeezed vacuum," Opt. Commun. 64, 523–528 (1987).
[CrossRef]

1986 (1)

G. S. Agarwal and N. Nayak, "Saturation of optical susceptibilities in strongly amplitude-modulated fields," J. Phys. B 19, 3385–3400 (1986).
[CrossRef]

1972 (1)

B. R. Mollow, "Stimulated emission and absorption near resonance for driven systems," Phys. Rev. A 5, 2217–2222 (1972).
[CrossRef]

1957 (1)

R. Kubo, "Statistical mechanical theory of irreversible processes: I—General theory and simple applications to magnetic and conduction problems," J. Phys. Soc. Jpn. 12, 570–586 (1957).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, "Spectrum of radiation from two-level atoms under intense bichromatic excitation," J. Opt. Soc. Am. B 8, 1163–1167 (1991).
[CrossRef]

N. Nayak and G. S. Agarwal, "Absorption and fluorescence in frequency-modulated fields under conditions of strong modulations and saturation," Phys. Rev. A 31, 3175–3182 (1988).
[CrossRef]

G. S. Agarwal and N. Nayak, "Saturation of optical susceptibilities in strongly amplitude-modulated fields," J. Phys. B 19, 3385–3400 (1986).
[CrossRef]

Bullough, R. K.

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
[CrossRef] [PubMed]

Chakmakjian, S.

Dalton, B. J.

Z. Ficek and B. J. Dalton, "Squeezing-induced transparency in a two-level atom," Opt. Commun. 102, 231–237 (1993).
[CrossRef]

Ficek, Z.

Z. Ficek and B. C. Sanders, "Resonance fluorescence of a two-level atom in an off-resonance squeezed vacuum," J. Phys. B 27, 809–824 (1994).
[CrossRef]

Z. Ficek and B. J. Dalton, "Squeezing-induced transparency in a two-level atom," Opt. Commun. 102, 231–237 (1993).
[CrossRef]

Z. Ficek, "Effect of a broadband squeezed vacuum on twoatom spontaneous emission," Opt. Commun. 82, 130–136 (1991).
[CrossRef]

Gauthier, D. J.

Hassan, S. S.

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
[CrossRef]

Kubo, R.

R. Kubo, "Statistical mechanical theory of irreversible processes: I—General theory and simple applications to magnetic and conduction problems," J. Phys. Soc. Jpn. 12, 570–586 (1957).
[CrossRef]

Lane, A. S.

H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
[CrossRef] [PubMed]

Mollow, B. R.

B. R. Mollow, "Stimulated emission and absorption near resonance for driven systems," Phys. Rev. A 5, 2217–2222 (1972).
[CrossRef]

Mossberg, T. W.

Nayak, N.

N. Nayak and G. S. Agarwal, "Absorption and fluorescence in frequency-modulated fields under conditions of strong modulations and saturation," Phys. Rev. A 31, 3175–3182 (1988).
[CrossRef]

G. S. Agarwal and N. Nayak, "Saturation of optical susceptibilities in strongly amplitude-modulated fields," J. Phys. B 19, 3385–3400 (1986).
[CrossRef]

Papademetriou, S.

Puri, R. R.

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

Quang, T.

A. S. Shumovsky and T. Quang, "Spectral and statistical properties of collective resonance fluorescence in a squeezed vacuum," J. Phys. B 22, 131–142 (1989); T. Quang, M. Kozierowski, and L. H. Lan, "Collective resonance fluorescence in a squeezed vacuum," Phys. Rev. A 39, 644–646 (1989).
[CrossRef] [PubMed]

Ritsch, H.

H. Ritsch and P. Zoller, "Absorption spectrum of a two-level system in a squeezed vacuum," Opt. Commun. 64, 523–528 (1987).
[CrossRef]

Sanders, B. C.

Z. Ficek and B. C. Sanders, "Resonance fluorescence of a two-level atom in an off-resonance squeezed vacuum," J. Phys. B 27, 809–824 (1994).
[CrossRef]

Saunders, R.

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

Shumovsky, A. S.

A. S. Shumovsky and T. Quang, "Spectral and statistical properties of collective resonance fluorescence in a squeezed vacuum," J. Phys. B 22, 131–142 (1989); T. Quang, M. Kozierowski, and L. H. Lan, "Collective resonance fluorescence in a squeezed vacuum," Phys. Rev. A 39, 644–646 (1989).
[CrossRef] [PubMed]

Smart, S.

S. Smart and S. Swain, "Dispersive profiles in resonance fluorescence of a two-level atom in a squeezed vacuum," Phys. Rev. A 48, R50–R53 (1993).
[CrossRef] [PubMed]

Stroud, C. R.

Swain, S.

S. Smart and S. Swain, "Dispersive profiles in resonance fluorescence of a two-level atom in a squeezed vacuum," Phys. Rev. A 48, R50–R53 (1993).
[CrossRef] [PubMed]

Timonen, J.

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
[CrossRef]

Wahiddin, M. R.

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
[CrossRef]

Walls, D. F.

H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
[CrossRef] [PubMed]

Zhu, Y.

Zoller, P.

H. Ritsch and P. Zoller, "Absorption spectrum of a two-level system in a squeezed vacuum," Opt. Commun. 64, 523–528 (1987).
[CrossRef]

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

J. Phys. B (3)

G. S. Agarwal and N. Nayak, "Saturation of optical susceptibilities in strongly amplitude-modulated fields," J. Phys. B 19, 3385–3400 (1986).
[CrossRef]

A. S. Shumovsky and T. Quang, "Spectral and statistical properties of collective resonance fluorescence in a squeezed vacuum," J. Phys. B 22, 131–142 (1989); T. Quang, M. Kozierowski, and L. H. Lan, "Collective resonance fluorescence in a squeezed vacuum," Phys. Rev. A 39, 644–646 (1989).
[CrossRef] [PubMed]

Z. Ficek and B. C. Sanders, "Resonance fluorescence of a two-level atom in an off-resonance squeezed vacuum," J. Phys. B 27, 809–824 (1994).
[CrossRef]

J. Phys. Soc. Jpn. (1)

R. Kubo, "Statistical mechanical theory of irreversible processes: I—General theory and simple applications to magnetic and conduction problems," J. Phys. Soc. Jpn. 12, 570–586 (1957).
[CrossRef]

Opt. Commun. (3)

Z. Ficek and B. J. Dalton, "Squeezing-induced transparency in a two-level atom," Opt. Commun. 102, 231–237 (1993).
[CrossRef]

Z. Ficek, "Effect of a broadband squeezed vacuum on twoatom spontaneous emission," Opt. Commun. 82, 130–136 (1991).
[CrossRef]

H. Ritsch and P. Zoller, "Absorption spectrum of a two-level system in a squeezed vacuum," Opt. Commun. 64, 523–528 (1987).
[CrossRef]

Phys. A (1)

S. S. Hassan, M. R. Wahiddin, R. Saunders, and R. K. Bullough, "Analysis of the master equation for the Dicke model in squeezed vacua," Phys. A 215, 556–576 (1995).
[CrossRef]

Phys. Rev. A (3)

N. Nayak and G. S. Agarwal, "Absorption and fluorescence in frequency-modulated fields under conditions of strong modulations and saturation," Phys. Rev. A 31, 3175–3182 (1988).
[CrossRef]

S. Smart and S. Swain, "Dispersive profiles in resonance fluorescence of a two-level atom in a squeezed vacuum," Phys. Rev. A 48, R50–R53 (1993).
[CrossRef] [PubMed]

B. R. Mollow, "Stimulated emission and absorption near resonance for driven systems," Phys. Rev. A 5, 2217–2222 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

H. J. Carmichael, A. S. Lane, and D. F. Walls, "Resonance fluorescence from an atom in a squeezed vacuum," Phys. Rev. Lett. 58, 2539–2542 (1987).
[CrossRef] [PubMed]

Other (2)

M. R. Wahiddin, S. S. Hassan, J. Timonen, and R. K. Bullough, "Transient and spectral properties of several Rydberg atoms in a squeezed vacuum," in Coherence and Quantum Optics VI, H. J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990), pp. 1195–1200.
[CrossRef]

M. R. Wahiddin, S. S. Hassan, R. R. Puri, J. Timonen, and R. K. Bullough, "Collective optical phenomena for several or many Rydberg atoms in a squeezed vacuum," in Proceedings of the European Conference on Optics, Optical Systems and Applications, Inst. Phys. Conf. Ser. 115, 85–90 (1991).

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Figures (9)

Fig. 1
Fig. 1

Two quadrature components Ac and As of the absorption spectrum as functions of the atomic detuning Δ in the weak coherent field case (Ω/γ = 0.1) and for squeezed vacuum parameters N =1, M = N ( N + 1 ), and a, ϕ = 0 and b, ϕ = −(π/2) with fixed squeezed field detuning parameter q/γ ≡ 2δ/γ = 0.1.

Fig. 2
Fig. 2

Same as Fig. 1 but with q/γ = 2: a, b, ϕ = 0 and ϕ = π/2, respectively.

Fig. 3
Fig. 3

Components Ac and As as functions of the squeezed field detuning parameter q at exact resonance (Δ = 0) and for Ω/γ = 0.1, N = 0.1, and a, ϕ = 0 and b, ϕ = −(π/2).

Fig. 4
Fig. 4

Same as Fig. 3 but for Δ/γ = 5.

Fig. 5
Fig. 5

Same as Fig. 2 but for a strong coherent field (Ω/γ = 25) and a, ϕ = 0 and b, ϕ = −(π/2).

Fig. 6
Fig. 6

Components Ac and As as functions of the squeezed field detuning parameter q for a strong coherent field (Ω/γ = 25) and for N = 1, ϕ = 0. a, b, Δ = 0 and Δ/γ = 5, respectively.

Fig. 7
Fig. 7

Two quadrature components Ic and Is of the fluorescent intensity as functions of the detuning parameter q in the weak coherent field case (data as in Fig. 3).

Fig. 8
Fig. 8

Same as Fig. 7 but for Δ/γ = 5.

Fig. 9
Fig. 9

Components Ic and Is as functions of the detuning parameter q in the strong coherent field case (data as in Fig. 6).

Equations (52)

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S ˙ + = - ( Γ + i Δ ) S + - γ M exp ( 2 i δ t ) S - + S z = S ˙ - * , S ˙ z = - γ 2 - 2 Γ S z - 1 2 ( S - + * S + ) ,
[ S + ( t ) S - ( t ) S z ( t ) ] = n = - [ U n ( t ) V n ( t ) W n ( t ) ] exp ( - i n q t ) .
U ˙ n = - ( Γ + i Δ - i n q ) U n + W n - γ M V n + 1 ,
V ˙ n = - ( Γ - i Δ - i n q ) V n + * W n - γ M * U n - 1 ,
W ˙ n = - ( 2 Γ - i n q ) W n - ½ ( V n + * U n ) - γ 2 δ n , 0 .
W n = - ( V n + * U n ) 2 ( 2 Γ - i n q ) - γ 2 ( 2 Γ - i n q ) δ n , 0 .
a n V n - b n V n + 1 - c n V n - 1 = d n δ n , 0 + e n δ n , 1 ,
a n = - ( Γ - i Δ - i n q + k n ) + k n 2 ( Γ + i Δ - i n q + k n ) + γ 2 M 2 [ Γ + i Δ - i ( n - 1 ) q + k n - 1 ] , b n = - γ k n M ( * / ) ( Γ + i Δ - i n q + k n ) , c n = - γ k n - 1 M * ( * / ) [ Γ + i Δ - i ( n - 1 ) q + k n - 1 ] , d n = γ k n + γ k n 2 ( Γ + i Δ - i n q + k n ) , e n = - γ 2 M * k n - 1 * [ Γ + i Δ - i ( n - 1 ) q + k n - 1 ] , k n = 1 2 2 ( 2 Γ - i n q ) .
X n = V n / V n - 1 ,             V n 0             for all n .
X n = c n ( a n - b n X n + 1 ) - 1 ,             n { 0 , 1 } .
Y - n = b - n - 1 ( a - n - 1 - c - n - 1 Y - n - 1 ) - 1 .
X 0 - 1 Y 0 = b - 1 ( a - 1 - c - 1 Y - 1 ) - 1 .
V 0 = d 0 ( a 0 - b 0 X 1 - c 0 X 0 - 1 ) - 1 , V 1 = e 1 ( a 1 - b 1 X 2 - c 1 X 1 - 1 ) - 1 .
X 1 = [ d 0 c 1 + e 1 ( a 0 - c 0 X 0 - 1 ) ] [ d 0 ( a 1 - b 1 X 2 ) + b 0 e 1 ] - 1 .
d W d t = E · d P d t ,
P ( t ) = μ [ S + ( t ) exp ( i ω l t ) + c . c . ]
= μ n = - { U n ( t ) exp [ i ( ω l - n q ) t ] + V n ( t ) exp [ - i ( ω l + n q ) t ] } ,
E = [ exp ( - i ω l t ) + c . c . ] .
( d W d t ) S S = i μ · ( ω l ( U 0 - V 0 ) + n = 1 { [ ( ω l + n q ) U - n - ( ω l - n q ) V - n ] × exp ( i n q t ) + [ ( ω l - n q ) U n - ( ω l + n q ) V n ] × exp ( - i n q t ) } ) .
n = - V n exp ( - i n q t ) = n = - U n * exp ( i n q t ) .
V - n = U n * ,             V n * = U - n
U 0 * = V 0 .
( d W d t ) S S = h n = 1 Im { [ ( ω l - n q ) U n - ( ω l + n q ) V n ] × exp ( - i n q t ) } + ( d w ( 0 ) d t ) S S ,
( d W ( 0 ) d t ) S S = - h ω l Im ( U 0 )
( d W ( 1 ) d t ) S S = - h ω l ( A c cos q t + A s sin q t ) ,
A c = Im ( U 1 - V 1 ) , A s = Re ( V 1 - U 1 )
A c = [ γ Ω M ( 1 + 2 N ) ] { [ Γ 2 - ( q 2 4 - r ) ] cos ϕ + Γ q sin ϕ } [ Γ 2 - ( q 2 - r ) 2 ] [ Γ 2 + ( q 2 + r ) 2 ] ,
A s = [ γ Ω M ( 1 + 2 N ) ] { Γ q cos ϕ - [ Γ 2 - ( q 2 4 - r ) ] sin ϕ } [ Γ 2 + ( q 2 - r ) 2 ] [ Γ 2 + ( q 2 + r ) 2 ] .
Γ 2 - ( q 2 4 - r ) = 0 ,
Δ = - q 2 ± 1 2 q 2 - γ 2 .
( - q 2 - 1 2 q 2 - γ 2 ,             - q 2 + 1 2 q 2 - γ 2 ) ;
A s q [ Γ 2 + ( q + Δ ) 2 ] - 1 ,
A c [ ¼ + Δ ( q + Δ ) ] [ Γ 2 + ( q + Δ ) 2 ] - 1 ,
I ( t ) = ½ + S z ( t ) = ½ + n = - w n exp ( - i n q t )
= ½ + w 0 + 2 n = 1 [ Re ( ω n ) cos n q t + Im ( w n ) sin n q t ] ,
I c = - Re ( W 1 ) ,             I s = - Im ( W 1 ) ,
S z ( ) W 0 - 1 2 ( 1 + 2 N ) - 1 2 k .
S ˙ + = - ( Γ + i Δ ) S + + i Ω k - γ M exp ( 2 i δ t ) S - = S ˙ - * .
S ± ( t ) = R ± ( t ) exp ( ± i δ t ) ,
R ˙ + ( t ) = - [ Γ + i ( Δ + δ ) ] R + ( t ) - γ M R - ( t ) + ( i Ω / k ) exp ( - i δ t ) = R ˙ - ( t ) * .
R ¯ + ( s ) = { R + ( 0 ) [ S + Γ - i ( Δ + δ ) ] - γ M R - ( 0 ) } × [ p ( s ) ] - 1 + i Ω k [ S + Γ - i ( Δ + δ ) ( S + i δ ) + γ M S - i δ ] [ p ( s ) ] - 1 ,
p ( s ) = ( s + Γ ) 2 + ( Δ + δ ) 2 - γ 2 M 2 .
s 1 , 2 = - Γ ± i r ,             r = ( Δ + δ ) 2 - γ 2 M 2 > 0.
s + ( t ) = exp ( + i δ t ) R + ( t ) = exp ( + i δ t ) exp ( - Γ t ) × ( exp ( i r t ) { r - ( Δ + δ ) 2 r S + ( 0 ) - γ M 2 i r S - ( 0 ) + Ω k i [ r - ( Δ + δ ) ] 2 r [ i ( r + δ ) - Γ ] + Ω k γ M 2 r [ i ( r - δ ) - Γ ] } + exp ( - i r t ) { ( r + Δ + δ ) 2 r S + ( 0 ) + γ M 2 i r S - ( 0 ) + Ω k i ( r + Δ + δ ) 2 r [ i ( δ - r ) - Γ ] + Ω k γ M 2 r [ i ( δ + r ) + Γ ] } ) + i Ω k [ Γ - i ( Δ + 2 δ ) ] [ Γ - i ( r + δ ) ] [ Γ + i ( r - δ ) ] + exp ( 2 i δ t ) i Ω k γ M [ Γ + i ( r + δ ) ] [ Γ - i ( r - δ ) ] = S - ( t ) * .
S + ( ) = i Ω k [ Γ - i ( Δ + 2 δ ) ] [ Γ - i ( r + δ ) ] [ Γ + i ( r - δ ) ] + exp ( 2 i δ t ) i Ω k γ M [ Γ + i ( r + δ ) ] [ Γ - i ( r - δ ) ] S - ( ) * .
S + ( ) = U 0 + U - 1 exp ( - i q t ) ,
U 1 = 0 , U - 1 * = V 1 = - i Ω γ M * k 1 [ Γ - i ( r + δ ) ] [ Γ + i ( r - δ ) ] .
A c = - Im ( V 1 ) ,             A s = Re ( V 1 ) .
W 1 = - ( V 1 + * U 1 ) 2 ( 2 Γ - i q ) .
I c = - Re ( W 1 ) = γ M Ω 2 D ( C 1 cos ϕ + S 1 sin ϕ ) , I s = - Im ( W 1 ) = γ M Ω 2 D ( S 1 cos ϕ - C 1 sin ϕ ) ,
C 1 = γ k ( Γ 2 + r 2 - δ 2 ) - γ k δ q , S 1 = q ( Γ 2 + r - δ 2 ) + γ 2 k 2 δ , D = k ( Γ 2 + q 2 ) [ Γ 2 + ( q 2 - r ) 2 ] × [ Γ 2 + ( q 2 + r ) 2 ] .
[ Γ 2 + ( q 2 - r ) 2 ] [ Γ 2 + ( q 2 + r ) 2 ] [ ( Γ 2 - r 1 - q 2 4 ) 2 + q 2 Γ 2 ] .

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