Abstract

We evaluate the commutation relations and quantum-mechanical Langevin equations satisfied by two-level operators in the case of exponential decay of the driving field. We rewrite the dynamical equation in the form of a difference equation by Laplace transformation. From the solution of the difference equation, the instantaneous resonance fluorescence spectrum is calculated.

© 1993 Optical Society of America

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References

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  1. B. R. Mollow, Phys. Rev. 188, 1969 (1969).
    [CrossRef]
  2. X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
    [CrossRef]
  3. J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
    [CrossRef]
  4. K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509 (1984).
    [CrossRef]
  5. J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
    [CrossRef]
  6. M. Lewenstein, J. Zakrzewski, and K. Rzazewski, J. Opt. Soc. Am. B 3, 22 (1986).
    [CrossRef]
  7. A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981).
    [CrossRef]
  8. F. T. Hioc, Phys. Rev. A 30, 2100 (1984).
    [CrossRef]
  9. F. T. Hioc and C. E. Carroll, J. Opt. Soc. Am. B 2, 497 (1985).
    [CrossRef]
  10. M. Lewenstein, J. Zakrzewski, and K. Rzązewski, J. Opt. Soc. Am. B 3, 22 (1986).
    [CrossRef]
  11. M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
    [CrossRef]
  12. J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
    [CrossRef]
  13. W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2042 (1963); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 248.
  14. H. Haken, “Laser theory,” in Light and Matter, Vol. XXV/2c of Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1970), pp. 42–43.
  15. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Science (Springer-Verlag, Berlin, 1983), pp. 16–17.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 736.

1986 (2)

1985 (3)

F. T. Hioc and C. E. Carroll, J. Opt. Soc. Am. B 2, 497 (1985).
[CrossRef]

J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
[CrossRef]

M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
[CrossRef]

1984 (2)

K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509 (1984).
[CrossRef]

F. T. Hioc, Phys. Rev. A 30, 2100 (1984).
[CrossRef]

1983 (1)

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

1982 (1)

X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
[CrossRef]

1981 (1)

A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981).
[CrossRef]

1980 (1)

J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

1969 (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

1963 (1)

W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2042 (1963); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 248.

Bambini, A.

A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981).
[CrossRef]

Berman, P. R.

A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981).
[CrossRef]

Carroll, C. E.

Eberly, J. H.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
[CrossRef]

J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

Florjanczyk, M.

M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
[CrossRef]

K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509 (1984).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Science (Springer-Verlag, Berlin, 1983), pp. 16–17.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 736.

Haken, H.

H. Haken, “Laser theory,” in Light and Matter, Vol. XXV/2c of Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1970), pp. 42–43.

Hioc, F. T.

Huang, X. Y.

X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
[CrossRef]

Kuklinski, R.

J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
[CrossRef]

Kunasz, C. V.

J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

Lengyel, B. A.

W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2042 (1963); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 248.

Lewenstein, M.

Mollow, B. R.

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Narozhny, N. B.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 736.

Rzazewski, K.

M. Lewenstein, J. Zakrzewski, and K. Rzazewski, J. Opt. Soc. Am. B 3, 22 (1986).
[CrossRef]

M. Lewenstein, J. Zakrzewski, and K. Rzązewski, J. Opt. Soc. Am. B 3, 22 (1986).
[CrossRef]

M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
[CrossRef]

K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509 (1984).
[CrossRef]

Sanchez-Mondragon, J. J.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Tanas, R.

X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
[CrossRef]

Wagner, W. G.

W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2042 (1963); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 248.

Wodkiewicz, K.

J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

Zakrzewski, J.

M. Lewenstein, J. Zakrzewski, and K. Rzązewski, J. Opt. Soc. Am. B 3, 22 (1986).
[CrossRef]

M. Lewenstein, J. Zakrzewski, and K. Rzazewski, J. Opt. Soc. Am. B 3, 22 (1986).
[CrossRef]

M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
[CrossRef]

J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
[CrossRef]

J. Appl. Phys. (1)

W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2042 (1963); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 248.

J. Opt. Soc. Am. B (3)

J. Phys. B (3)

J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, J. Phys. B 13, 217 (1980); J. H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509 (1984).
[CrossRef]

J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
[CrossRef]

Phys. Rev. (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Phys. Rev. A (4)

X. Y. Huang, R. Tanas, and J. H. Eberly, Phys. Rev. A 26, 892 (1982).
[CrossRef]

M. Florjanczyk, K. Rzązewski, and J. Zakrzewski, Phys. Rev. A 31, 1558 (1985).
[CrossRef]

A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981).
[CrossRef]

F. T. Hioc, Phys. Rev. A 30, 2100 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Other (3)

H. Haken, “Laser theory,” in Light and Matter, Vol. XXV/2c of Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1970), pp. 42–43.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Science (Springer-Verlag, Berlin, 1983), pp. 16–17.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 736.

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Figures (6)

Fig. 1
Fig. 1

Optical driving pulse: a, η = 0.2, b, η = 0.5; c, η = 1.

Fig. 2
Fig. 2

Variation of IMC and INC as functions of t; γ1 = 1.0, γ2 = 0.5. IMC, Solid curves; INC, dotted–dashed curves.

Fig. 3
Fig. 3

Instantaneous resonance spectra: γ1 = 1; γ2 = 0.5; η = 0.2; Ω = 1, 3, 5.

Fig. 4
Fig. 4

Instantaneous resonance spectra: γ1 = 1; γ2 = 0.5; η = 0.5; Ω = 1, 3, 5.

Fig. 5
Fig. 5

Instantaneous resonance spectra: γ1 = 1; γ2 = 0.5; η = 1.0; Ω = 1, 3, 5.

Fig. 6
Fig. 6

Instantaneous resonance spectra: γ1 = 1; γ2 = 0.5; Ω = 5; η = 0.1, 0.05, 0.

Equations (74)

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d σ Z d t = γ 1 ( σ Z σ ¯ Z ) i λ g λ ( b λ + b λ ) σ + i λ g λ ( b λ + b λ ) σ + + Γ Z , d σ d t = i 2 λ g λ * ( b λ + b λ ) σ Z ( i ω 0 + γ 2 ) σ + Γ , d σ + d t = i 2 λ g λ ( b λ + b λ ) σ Z + ( i ω 0 γ 2 ) σ + + Γ + .
σ Z = 1 2 ( a 2 + a 2 a 1 + a 1 ) , σ = a 1 + a 2 , σ + = a 2 + a 1 .
2 g λ ( b λ + b λ ) = Ω ˜ = Ω [ exp ( i ω t ) + exp ( i ω t ) ] .
d σ Z d t = γ 1 σ Z 1 Ω ˜ 2 σ + i Ω ˜ 2 σ + + γ 1 σ ¯ Z + Γ Z , d σ d t = i Ω ˜ σ Z ( γ 2 + i ω 0 ) σ + Γ , d σ + d t = i Ω ˜ σ Z ( γ 2 i ω 0 ) σ + + Γ + .
σ ± σ ± exp ( ± i ω 0 t ) , σ Z σ Z , Γ ± Γ ± exp ( ± i ω 0 t ) , Γ Z Γ Z ,
d σ d t = M σ + γ σ ¯ + Γ ,
σ = [ σ Z σ σ + ] , γ σ ¯ = [ γ 1 σ ¯ Z 0 0 ] , Γ = [ Γ Z Γ Γ + ] , M = [ γ i i Ω 2 i Ω 2 i Ω γ 2 0 i Ω 0 γ 2 ] .
σ = σ S + L , d σ S d t = M σ S + γ σ ¯ ,
d L d t = M L + Γ .
σ S ( t + τ ) = U ( t , τ ) σ S ( t ) + V ( t , τ ) γ σ ¯ , L ( t + τ ) = U ( t , τ ) L ( t ) + t t + τ U ( t , τ ) Γ ( t ) d t , L ( t ) = U ( t , 0 ) L ( 0 ) + 0 t U ( t , 0 ) Γ ( t ) d t , U = exp ( t t + τ M d t ) , V = t t + τ U ( t , τ ) d t ,
D ( t , τ ) = L ( t ) L r ( t + τ ) ,
D ( t , τ ) = L ( t ) L r ( t ) U r ( t , τ ) + t t + τ L ( t ) Γ r ( t ) U r ( t , τ ) d t .
L ( t ) Γ r ( t ) = U ( t , 0 ) L ( 0 ) + 0 t U ( t , 0 ) Γ ( t ) d t Γ r ( t ) .
D ( t , τ ) = D ( t , 0 ) U r ( t , τ ) ,
σ ( t ) σ r ( t ) = [ σ S ( t ) + L ( t ) ] [ σ S r ( t ) + L t ( t ) ] = σ S ( t ) σ S r ( t ) + D ( t , 0 ) = σ S ( t ) σ S r ( t ) + D ( t , 0 ) ,
σ ( t ) σ r ( t + τ ) = σ S ( t ) σ S r ( t + τ ) + D ( t , τ ) = σ S ( t ) σ S r ( t ) U r ( t , τ ) + σ S ( t ) γ σ ¯ r V r ( t , τ ) + [ σ ( t ) σ r ( t ) σ S ( t ) σ S r ( t ) ] U S r ( t , τ ) = σ ( t ) σ r ( t ) U r ( t , τ ) + σ S ( t ) γ σ ¯ r V r ( t , τ ) .
σ + σ σ σ + = 2 σ Z , σ ± σ Z σ Z σ ± = σ ± , σ 2 = σ + 2 = 0 σ + σ + σ σ + = 1 , σ ± σ Z + σ Z σ ± = 0 , σ Z 2 = 1 / 4 .
σ σ r = [ σ Z σ Z σ Z σ σ Z σ + σ σ Z σ 2 σ σ + σ + σ Z σ + σ σ + 2 ] = [ 1 4 σ 2 σ + 2 σ 2 0 1 2 σ Z σ + 2 1 2 + σ Z 0 ] .
σ σ r = [ 1 4 σ S 2 σ S + 2 σ S 2 0 1 2 σ Z S σ S + 2 1 2 + σ Z S 0 ] .
1 t 0 t σ + ( t + τ ) σ ( t ) d t .
γ ( ω ) = 0 t exp ( i ω t ) χ ( t ) d t ,
S ( ω ) = lim t 1 2 π t | γ ( ω ) | 2 = lim t 1 π t Re S t exp ( s γ ) d τ S t τ χ ( t ) χ ( t + τ ) d t | S = i ω .
G ( ω ) = 1 π t Re 0 t exp ( s τ ) d τ 0 t τ σ ( t ) σ r ( t + τ ) d t | S = i ω = 1 π t Re [ 0 t exp ( s τ ) d τ 0 t τ σ ( t ) σ r ( t ) U r ( t , τ ) d t ] S = i ω + 1 π t Re [ 0 t exp ( s τ ) d τ 0 t τ σ ( t ) γ σ ¯ V r ( t , τ ) d t ] S = i ω ,
G ( ω ) 1 π t Re [ 0 t σ ( t ) σ r ( t ) d t 0 t exp ( s τ ) U r ( t , τ ) d τ ] S = i ω + 1 π t Re { 0 t σ ( t ) γ σ ¯ d τ 0 t exp ( s τ ) V r ( t , τ ) d τ } S = i ω ,
ϕ ( t , ω , Γ ) = 2 Γ Re 0 t d t 2 exp [ Γ ( t t 2 ) ] 0 t t 2 d τ × exp [ ( Γ / 2 i D ) τ ] S ˆ 21 ( t 2 + τ ) S ˆ 12 ( t 2 ) 2 Γ Re 0 t d t 2 0 t t 2 d τ exp ( i D τ ) × S ˆ 21 ( t 2 + τ ) S ˆ 1 2 ( t 2 ) = 2 Γ Re 0 t exp ( i D τ ) d τ 0 t τ d t × S ˆ 21 ( t + τ ) S ˆ 12 ( t ) .
G ( ω ) = 1 t 1 2 Γ ϕ ( t , ω , Γ ) .
d σ d t = M exp ( η t ) σ
σ = exp { M η [ 1 exp ( η t ) ] } σ 0 , = n 1 n ! ( M η ) n [ 1 exp ( η t ) ] n σ 0 .
s σ ˜ ( s ) σ 0 = M σ ˜ ( s + η ) , σ ˜ ( s ) = 0 exp ( s t ) σ d t , σ ˜ ( s ) = σ 0 s M s σ ˜ ( s + η ) = σ 0 s + σ 0 s ( M ) s + η + + σ 0 s ( M ) n ( s + η ) ( s + n η ) +
1 s ( s + η ) ( s + n η ) = 1 η n ( 1 ) k k | ( n k ) | 1 s + k η
σ ˜ ( s ) = n = 0 1 n ! ( M η ) n n = 0 s ( 1 ) k n ! k | ( n k ) | 1 s + k η σ 0
σ ( t ) = n = 0 1 n ! ( M η ) n [ 1 exp ( η t ) ] n σ 0 .
σ = [ σ Z σ σ + ] , M = [ 0 i 2 i 2 i 0 0 i 0 0 ] Ω .
M 2 = [ 1 0 0 0 1 2 1 2 0 1 2 1 2 ] = E ,
M 4 = E , E 2 = E , E M = M ;
σ ( t ) = n = 0 ( Ω 2 ) n ( 2 n ) ! [ 1 exp ( η t ) η ] 2 n E σ 0 + n = 0 ( 1 ) n Ω 2 n + 1 ( 2 n + 1 ) ! [ 1 exp ( η t ) η ] 2 n + 1 M σ 0 .
C ( Ω , η , t ) = n = 0 ( Ω 2 ) n ( 2 n ) ! [ 1 exp ( η t ) η ] 2 n C ( Ω , η , t ) = n = 0 ( 1 ) n Ω 2 n + 1 ( 2 n + 1 ) ! [ 1 exp ( η t ) η ] 2 n + 1 .
C ( Ω , η , t ) cos ( Ω t ) , S ( Ω , η , t ) sin ( Ω t ) .
σ ( t ) = ( C E + S M ) σ 0 ;
[ σ Z σ σ + ] = C [ σ Z 0 1 2 ( σ 0 σ 0 + ) 1 2 ( σ 0 σ 0 + ) ] + S [ 1 2 ( σ 0 σ 0 + ) i σ Z 0 i σ Z 0 ] .
σ ( t + τ ) = ( C E + S M ) σ ( t ) , C = C [ Ω exp ( η t ) , η , τ ] , S = S ( Ω exp ( η t ) , η , τ ) .
d σ d t = γ ( σ σ ¯ ) + Ω M exp ( η t ) σ ,
γ = [ γ 1 γ 2 γ 2 ] , σ ¯ = [ σ ¯ Z 0 0 ] , M = [ 0 i 2 i 2 i 0 0 i 0 0 ] .
( s + γ ) σ ˜ ( s ) σ 0 = 1 s γ σ ¯ + Ω M σ ˜ ( s + η ) , σ ˜ ( s ) = ( 1 s + γ + 1 s + γ M Ω s + η + γ + 1 s + γ M Ω s + η + γ M Ω s + 2 η + γ + ) ( σ 0 σ ¯ ) + ( 1 s + Ω s + γ M 1 s + η + Ω s + γ M Ω s + η + γ M 1 s + 2 η + ) σ ¯ .
D ( s + γ ) = [ 1 s + γ 1 1 s + γ 2 1 s + γ 2 ] , D ˜ ( s + γ ) = [ 1 s + γ 2 1 s + γ 1 1 s + γ 1 ] .
D ( s + γ ) M = M D ˜ ( s + γ ) , D ˜ ( s + γ ) = D ( s + γ ) ,
1 s + γ M Ω s + η + γ M Ω s + n η + γ = D 0 M D 1 M D n ( Ω ) n = { M n Ω n D ˜ 0 D 1 D n ( n odd ) M n Ω n D 0 D ˜ 1 D n ( n even ) .
σ ˜ ( s ) = n = 0 Ω 2 n M 2 n D 0 D ˜ 1 D 2 D 2 n × [ ( σ 0 σ ¯ ) + D 2 n 1 1 s + 2 n η σ ¯ ] + n = 0 Ω 2 n + 1 M 2 n + 1 D ˜ 0 D 1 D ˜ 2 D 2 n + 1 × [ ( σ 0 σ ¯ ) + D 2 n + 1 1 1 s + ( 2 n + 1 ) η σ ¯ ]
D 2 n 1 = s + 2 n η + γ , D 2 n + 1 1 = s + ( 2 n + 1 ) η + γ ,
σ ˜ ( s ) = n = 0 Ω 2 n M n D 0 D ˜ 1 D 2 D 2 n ( σ 0 + γ 1 s + 2 n η σ ¯ ) + n = 0 Ω 2 n + 1 M 2 n + 1 D ˜ 0 D 1 D ˜ 2 D 2 n + 1 × [ σ 0 + γ 1 s + ( 2 n + 1 ) η σ ¯ ] .
M 2 n = ( 1 ) n E , n 0 , M 2 n + 1 = ( 1 ) n M ,
σ ˜ ( s ) = ( 1 E ) D 0 ( σ 0 + γ 1 s σ ¯ ) + E ( I ˜ 1 σ 0 + I ˜ , γ σ ¯ ) + Ω M ( I ˜ 2 σ 0 + I ˜ 4 γ σ ¯ ) ,
I ˜ 1 = n = 0 ( 1 ) n Ω 2 n D 0 D ˜ 1 D 2 D 2 n , I ˜ 2 = n = 0 ( 1 ) n Ω 2 n D ˜ 0 D 1 D ˜ 2 D 2 n + 1 , I ˜ 3 = n = 0 ( 1 ) n Ω 2 n D 0 D ˜ 1 D 2 D 2 n 1 s + 2 n η , I ˜ 4 = n = 0 ( 1 ) n Ω 2 n D ˜ 0 D 1 D ˜ 2 D 2 n + 1 1 2 + ( 2 n + 1 ) η .
I ˜ j = [ i j ι ¯ j ι ¯ j ] ( j = 1 4 ) ,
[ σ ˜ Z σ ˜ σ ˜ + ] = [ i i σ Z 0 1 2 ι ¯ 1 ( σ 0 σ 0 + ) 1 2 i ¯ 1 ( σ 0 σ + ) ] + [ i Ω 2 i ¯ 2 ( σ 0 σ 0 + ) i Ω i 2 σ Z 0 i Ω i 2 σ Z 0 ] . + [ i 3 γ 1 σ ¯ Z 0 0 ] + [ 0 i Ω i 4 γ 1 σ ¯ Z i Ω i 4 γ 1 σ ¯ Z ] + [ 0 1 2 σ 0 + σ 0 + s + γ 2 1 2 σ 0 + σ 0 + s + γ 2 ] .
σ ( τ ) = 1 2 I ¯ 1 ( τ ) ( σ 0 σ 0 + ) i I 2 ( τ ) Ω σ z 0 + 1 2 exp ( γ 2 τ ) ( σ 0 + σ 0 + ) I 4 ( τ ) Ω γ 1 σ ¯ Z
σ ( t + τ ) = 1 2 I ¯ 1 ( τ ) [ σ 0 ( t ) σ 0 + ( t ) ] i I 2 ( τ ) Ω σ z 0 ( t ) + 1 2 exp ( γ 2 τ ) [ σ 0 ( t ) + σ 0 + ( t ) ] i I 4 ( τ ) Ω ( t ) γ 1 σ ¯ Z , Ω ( t ) = Ω exp ( η t ) ,
σ + ( t ) σ ( t + τ ) = 1 2 [ I ¯ 1 ( t ) + exp ( γ 2 τ ) ] σ + ( t ) σ ( t ) i I 2 ( τ ) Ω ( t ) σ + ( t ) σ Z ( t ) i I 4 ( τ ) Ω ( t ) γ 1 σ ¯ Z σ + ( t ) .
G 32 ( ω ) = 1 π t Re [ 0 t exp ( s τ ) d τ × 0 t τ σ + ( t ) σ ( t + τ ) d t ] S = i ω 1 π t Re ( 0 t σ + ( t ) σ ( t ) d t 0 t exp ( s τ ) × { 1 2 [ I ¯ 1 ( τ ) + exp ( γ 2 τ ) ] } d τ i 0 t Ω ( t ) σ + ( t ) σ Z ( t ) d t × 0 t exp ( s τ ) I 2 ( τ ) d τ i 0 t Ω ( t ) σ + ( t ) γ 1 σ ¯ Z d t × 0 t exp ( s τ ) I 4 ( τ ) d τ ) S = i ω ,
[ σ Z S ( t ) σ S ( t ) σ S + ( t ) ] = [ I 1 ( t ) 1 2 I ¯ 1 ( t ) ( σ 0 σ 0 + ) 1 2 I ¯ 1 ( t ) ( σ 0 σ 0 + ) ] + [ i Ω 2 I ¯ 2 ( t ) ( σ 0 σ 0 + ) i Ω I 2 ( t ) σ Z 0 i Ω I 2 ( t ) σ Z 0 ] + [ I 3 ( t ) γ 1 σ ¯ Z 0 0 ] + [ 0 i Ω I 4 ( t ) γ 1 σ ¯ Z i Ω I 4 ( t ) γ 1 σ ¯ Z ] + [ 0 1 2 ( σ 0 + σ 0 + ) exp ( γ 2 τ ) 1 2 ( σ 0 + σ 0 + ) exp ( γ 2 τ ) ] .
IMC = 1 t 0 t σ + ( t ) σ ( t ) d t / σ + ( ) σ ( ) , INC = 1 t 0 t exp ( η t ) σ + ( t ) d t / σ + ( ) , σ + ( ) σ ( ) = Ω 2 / 2 ( γ 1 2 / 2 ) + Ω 2 , σ + ( ) = i ( γ 1 / 2 ) Ω ( γ 1 2 / 2 ) + Ω 2 .
σ 0 + = σ 0 = 0 , σ Z 0 = σ ¯ Z = 1 / 2 .
1 t 0 t σ + ( t ) σ ( t ) d t , i t 0 t Ω ( t ) σ + ( t ) d t ,
1 ( s ( s + η ) ( s + n η ) = 1 η n ( 1 ) k k | ( n k ) | ( s + k η ) 1 n | η n [ 1 exp ( η t ) ] n .
D 0 D ˜ 1 D 2 D 2 n = [ d n 1 d ¯ n 1 d ¯ n 1 ] , d n 1 = 1 s + γ 1 1 s + γ 2 + η 1 s + γ 1 + 2 η 1 s + γ 1 + 2 n η = ( s + γ 2 η ) 1 s + γ 1 1 s + γ 1 + 2 η 1 s + γ 1 + 2 n η × 1 s + γ 2 η 1 s + γ 2 + η 1 s + γ 2 + ( 2 n 1 ) η ( d d t + γ 2 η ) 1 ( n ! ) 2 × 0 t { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } n × exp [ γ 1 τ ( γ 2 η ) ( t τ ) ] d τ ,
i i = ( 1 ) n Ω 2 n d n 1 I 1 , I 1 = ( d d t + γ 2 η ) exp [ ( γ 2 η ) t ] × 0 t exp [ ( γ 1 γ 2 + η ) τ ] × n = 0 1 ( n ! ) 2 { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } n d τ = ( d d t + γ 2 η ) exp [ ( γ 2 η ) t ] × 0 t exp [ ( γ 1 γ 2 + η ) τ ] × J 0 ( 2 Ω { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } 1 / 2 ) d τ .
I 1 ( d d t + γ 2 ) exp ( γ 2 t ) 0 t exp [ ( γ 1 γ 2 ) τ ] × J 0 { 2 Ω [ τ ( t τ ) ] 1 / 2 } d τ = ( d d t + γ 2 ) exp ( γ 1 + γ 2 2 t ) × 0 t cosh ( γ 1 γ 2 2 χ ) J 0 [ Ω ( t 2 χ 2 ) 1 / 2 ] d χ = ( d d t + γ 2 ) exp ( γ 1 + γ 2 2 t ) sin ( Ω t ) Ω , Ω = [ Ω 2 ( γ 1 γ 2 2 ) 2 ] 1 / 2 .
D ˜ 0 D 1 D ˜ 2 D 2 n + 1 = [ d n 2 d ¯ n 2 d ¯ n 2 ] ,
d n 2 = 1 s + γ 2 1 s + γ 2 + 2 η 1 s + γ 2 + 2 n η × 1 s + γ 1 + η 1 s + γ 1 + 3 η 1 s + γ 1 + ( 2 n + 1 ) η 1 ( n ! ) 2 0 t { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } n × exp [ ( γ 1 + η ) T γ 2 ( t τ ) ] d τ , i 2 = n = 0 ( 1 ) n Ω 2 n d n 2 I 2 , I 2 = exp ( γ 2 t ) 0 t exp [ ( γ 1 γ 2 + η ) τ ] × J 0 ( 2 Ω { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } 1 / 2 ) d τ ;
I 2 exp ( η 2 t ) 0 t exp [ ( γ 1 γ 2 ) τ ] J 0 { 2 Ω [ τ ( t τ ) ] } 1 / 2 d τ = exp ( γ 1 + γ 2 2 t ) sin ( Ω t ) Ω .
d n 3 = d n 1 1 s + 2 n η ( d d t + γ 2 η ) × 0 t exp [ 2 n η ( t t ) ] d t × 1 ( n ! ) 2 0 t { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } n × exp [ γ 1 τ ( τ 2 η ) ( t τ ) ] d τ , i 3 = ( 1 ) n Ω 2 n d n 3 I 3 , I 3 = ( d d t + γ 2 η ) 0 t d t exp [ ( γ 2 η ) t ] × 0 t exp [ ( γ 1 γ 2 + η ) τ ] × J 0 ( 2 Ω exp [ η ( t t ) ] × { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } 1 / 2 ) d τ ;
I 3 ( d d t + γ 2 ) 0 t d t exp ( γ 2 t ) 0 t exp [ ( γ 1 γ 2 + η ) τ ] × J 0 { 2 Ω [ τ ( t τ ) ] 1 / 2 } d τ = ( d d t + γ 2 ) 0 t d t exp ( γ 1 + γ 2 2 t ) sin ( Ω t ) Ω = exp ( γ 1 γ 2 2 t ) sin ( Ω t ) Ω + γ 2 i 2 Ω [ exp ( λ 1 t ) 1 λ 2 exp ( λ 2 t ) 1 λ 2 ] ,
d n 4 = d n 2 1 s + ( 2 n + 1 ) η 0 t exp [ ( 2 n + 1 ) ( t t ) ] d t × ( 1 n ! ) 2 0 t ( 1 exp ( 2 η τ ) 2 η × 1 exp [ 2 η ( t τ ) ] 2 η ) n × exp [ ( γ 1 + η ) τ γ 2 ( t τ ) ] d τ , i 4 = ( 1 ) n Ω 2 n d n 4 I 4 , I 4 = 0 t d t exp [ η ( t t ) ] d t exp ( γ 2 t ) × 0 t exp [ ( γ 1 γ 2 + η ) τ ] × J 0 ( 2 Ω exp [ η ( t t ) ] × { 1 exp ( 2 η τ ) 2 η 1 exp [ 2 η ( t τ ) ] 2 η } 1 / 2 ) d τ ;
I 4 0 t d t exp ( γ 1 + γ 2 2 t ) sin ( Ω t ) Ω = 1 i 2 Ω ( exp ( λ 1 t ) 1 λ 2 exp ( λ 2 t ) 1 λ 2 ) , λ 1 = γ 1 + γ 2 2 + i Ω , λ 2 = γ 1 + γ 2 2 i Ω .

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