Abstract

In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed.

© 2020 Optical Society of America

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References

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  1. A. Siegman, Lasers (University Science, 1986).
  2. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).
  3. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, 1985).
  4. D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997).
    [Crossref]
  5. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).
  6. L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).
  7. C. J. McKinstrie, “Gain, loss and the shot-noise rule,” J. Lightwave Technol., doi:10.1109/JLT.2020.2968720 (to be published).
  8. B. van der Pol, “On relaxation oscillations,” Philos. Mag. 2, 978–992 (1926).
    [Crossref]
  9. R. Dunsmuir, “Theory of relaxation oscillations in optical masers,” J. Electron. Control 10, 453–458 (1961).
    [Crossref]
  10. W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918).
    [Crossref]
  11. S. O. Rice, “Mathematical analysis of random noise,” Bell Sys. Tech. J. 23, 282–332 (1944).
    [Crossref]
  12. P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1977).
  13. T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967).
    [Crossref]
  14. K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
    [Crossref]
  15. D. E. McCumber, “Intensity fluctuations in the output of CW laser oscillators,” Phys. Rev. 141, 306–322 (1966).
    [Crossref]
  16. M. Lax, “Quantum noise IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
    [Crossref]
  17. D. E. McCumber, “Intensity fluctuations in the output of laser oscillators,” J. Quantum Electron. 2, 219–221 (1966).
    [Crossref]
  18. H. Haken, “Theory of intensity and phase fluctuations of a homogeneously broadened laser,” Z. Phys. 190, 327–356 (1966).
    [Crossref]
  19. M. Lax, “Quantum noise VII: The rate equations and amplitude noise in lasers,” J. Quantum Electron. 3, 37–46 (1967).
    [Crossref]
  20. M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966).
    [Crossref]
  21. M. Lax, “Quantum noise X: Density-matrix treatment of field and population-difference fluctuations,” Phys. Rev. 157, 213–231 (1967).
    [Crossref]
  22. M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967).
    [Crossref]
  23. J. W. Goodman, Statistical Optics (Wiley, 1985).
  24. C. H. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum Electron. 19, 1391–1397 (1983).
    [Crossref]
  25. C. J. McKinstrie and T. I. Lakoba, “Probability-density function for energy perturbations of isolated optical pulses,” Opt. Express 11, 3628–3648 (2003).
    [Crossref]

2003 (1)

1997 (1)

D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997).
[Crossref]

1983 (1)

C. H. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum Electron. 19, 1391–1397 (1983).
[Crossref]

1967 (4)

M. Lax, “Quantum noise X: Density-matrix treatment of field and population-difference fluctuations,” Phys. Rev. 157, 213–231 (1967).
[Crossref]

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967).
[Crossref]

T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967).
[Crossref]

M. Lax, “Quantum noise VII: The rate equations and amplitude noise in lasers,” J. Quantum Electron. 3, 37–46 (1967).
[Crossref]

1966 (5)

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966).
[Crossref]

D. E. McCumber, “Intensity fluctuations in the output of CW laser oscillators,” Phys. Rev. 141, 306–322 (1966).
[Crossref]

M. Lax, “Quantum noise IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
[Crossref]

D. E. McCumber, “Intensity fluctuations in the output of laser oscillators,” J. Quantum Electron. 2, 219–221 (1966).
[Crossref]

H. Haken, “Theory of intensity and phase fluctuations of a homogeneously broadened laser,” Z. Phys. 190, 327–356 (1966).
[Crossref]

1961 (1)

R. Dunsmuir, “Theory of relaxation oscillations in optical masers,” J. Electron. Control 10, 453–458 (1961).
[Crossref]

1957 (1)

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[Crossref]

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Sys. Tech. J. 23, 282–332 (1944).
[Crossref]

1926 (1)

B. van der Pol, “On relaxation oscillations,” Philos. Mag. 2, 978–992 (1926).
[Crossref]

1918 (1)

W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

Coldren, L. C.

L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).

Corzine, S. W.

L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).

Dunsmuir, R.

R. Dunsmuir, “Theory of relaxation oscillations in optical masers,” J. Electron. Control 10, 453–458 (1961).
[Crossref]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, 1985).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gythiel, A.

D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997).
[Crossref]

Haken, H.

H. Haken, “Theory of intensity and phase fluctuations of a homogeneously broadened laser,” Z. Phys. 190, 327–356 (1966).
[Crossref]

Henry, C. H.

C. H. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum Electron. 19, 1391–1397 (1983).
[Crossref]

Ikegami, T.

T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967).
[Crossref]

Lakoba, T. I.

Lamb, W. E.

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967).
[Crossref]

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966).
[Crossref]

Lax, M.

M. Lax, “Quantum noise X: Density-matrix treatment of field and population-difference fluctuations,” Phys. Rev. 157, 213–231 (1967).
[Crossref]

M. Lax, “Quantum noise VII: The rate equations and amplitude noise in lasers,” J. Quantum Electron. 3, 37–46 (1967).
[Crossref]

M. Lax, “Quantum noise IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
[Crossref]

Lemons, D. S.

D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997).
[Crossref]

Masanovic, M. L.

L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).

McCumber, D. E.

D. E. McCumber, “Intensity fluctuations in the output of laser oscillators,” J. Quantum Electron. 2, 219–221 (1966).
[Crossref]

D. E. McCumber, “Intensity fluctuations in the output of CW laser oscillators,” Phys. Rev. 141, 306–322 (1966).
[Crossref]

McKinstrie, C. J.

C. J. McKinstrie and T. I. Lakoba, “Probability-density function for energy perturbations of isolated optical pulses,” Opt. Express 11, 3628–3648 (2003).
[Crossref]

C. J. McKinstrie, “Gain, loss and the shot-noise rule,” J. Lightwave Technol., doi:10.1109/JLT.2020.2968720 (to be published).

Meyer, P. L.

P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1977).

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Sys. Tech. J. 23, 282–332 (1944).
[Crossref]

Schottky, W.

W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918).
[Crossref]

Scully, M. O.

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967).
[Crossref]

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966).
[Crossref]

Shimoda, K.

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[Crossref]

Siegman, A.

A. Siegman, Lasers (University Science, 1986).

Suematsu, Y.

T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967).
[Crossref]

Takahasi, H.

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[Crossref]

Townes, C. H.

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[Crossref]

van der Pol, B.

B. van der Pol, “On relaxation oscillations,” Philos. Mag. 2, 978–992 (1926).
[Crossref]

Am. J. Phys. (1)

D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997).
[Crossref]

Ann. Phys. (1)

W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918).
[Crossref]

Bell Sys. Tech. J. (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Sys. Tech. J. 23, 282–332 (1944).
[Crossref]

J. Electron. Control (1)

R. Dunsmuir, “Theory of relaxation oscillations in optical masers,” J. Electron. Control 10, 453–458 (1961).
[Crossref]

J. Phys. Soc. Jpn. (1)

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[Crossref]

J. Quantum Electron. (3)

D. E. McCumber, “Intensity fluctuations in the output of laser oscillators,” J. Quantum Electron. 2, 219–221 (1966).
[Crossref]

M. Lax, “Quantum noise VII: The rate equations and amplitude noise in lasers,” J. Quantum Electron. 3, 37–46 (1967).
[Crossref]

C. H. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum Electron. 19, 1391–1397 (1983).
[Crossref]

Opt. Express (1)

Philos. Mag. (1)

B. van der Pol, “On relaxation oscillations,” Philos. Mag. 2, 978–992 (1926).
[Crossref]

Phys. Rev. (4)

D. E. McCumber, “Intensity fluctuations in the output of CW laser oscillators,” Phys. Rev. 141, 306–322 (1966).
[Crossref]

M. Lax, “Quantum noise IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
[Crossref]

M. Lax, “Quantum noise X: Density-matrix treatment of field and population-difference fluctuations,” Phys. Rev. 157, 213–231 (1967).
[Crossref]

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967).
[Crossref]

Phys. Rev. Lett. (1)

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966).
[Crossref]

Proc. IEEE (1)

T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967).
[Crossref]

Z. Phys. (1)

H. Haken, “Theory of intensity and phase fluctuations of a homogeneously broadened laser,” Z. Phys. 190, 327–356 (1966).
[Crossref]

Other (8)

P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1977).

A. Siegman, Lasers (University Science, 1986).

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, 1985).

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).

C. J. McKinstrie, “Gain, loss and the shot-noise rule,” J. Lightwave Technol., doi:10.1109/JLT.2020.2968720 (to be published).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (8)

Fig. 1.
Fig. 1. Levels and transitions for a four-level laser.
Fig. 2.
Fig. 2. Levels and transitions for a type A three-level laser.
Fig. 3.
Fig. 3. Levels and transitions for a type B three-level laser.
Fig. 4.
Fig. 4. Normalized electron and photon numbers $ N/{N_t} $ (blue) and $ S/{N_t} $ (red) plotted as functions of time ($ bt $) for a four-level laser with the gain parameter $ a{N_t}/b = 10 $. The pump parameter $ c/b = 0.01 $ (left) and 0.10 (right).
Fig. 5.
Fig. 5. Normalized electron and photon numbers $ N/{N_t} $ (blue) and $ S/{N_t} $ (red) plotted as functions of time ($ bt $) for a type B three-level laser with the gain parameter $ a{N_t}/b = 10 $. The pump parameter $ c/b = 0.01 $ (left) and 0.10 (right).
Fig. 6.
Fig. 6. Normalized RIN $ Q/Q(0) $ plotted as a function of frequency ($ \omega /b $) for the pump parameters $ c/b = 0.01 $ (blue), 0.03 (green), and 0.10 (red). Left: Four-level laser with the gain parameter $ a{N_t}/b = 10 $. Right: Type B three-level laser with the same gain.
Fig. 7.
Fig. 7. Normalized autocorrelation $ C/C(0) $ plotted as a function of time delay ($ b\tau $) for the pump parameters $ c/b = 0.01 $ (blue), 0.03 (green), and 0.10 (red). Left: Four-level laser with the gain parameter $ a{N_t}/b = 10 $. Right: Type B three-level laser with the same gain. The three- and four-level curves are identical because the normalized frequencies $ {\omega _0}/b $ and $ {\nu _0}/b $ are identical.
Fig. 8.
Fig. 8. Normalized deviation moments $ \left\langle {N_1^2} \right\rangle /{N_t} $ (blue), $ \left\langle {{N_1}{S_1}} \right\rangle /{N_t} $ (green), and $ \left\langle {S_1^2} \right\rangle /{N_t} $ (red) plotted as functions of time for a four-level laser with the gain parameter $ a{N_t}/b = 10 $. The pump parameter $ c/b = 0.01 $ (left) and 0.10 (right).

Tables (3)

Tables Icon

Table 1. Rate Coefficients for Three Types of Lasera

Tables Icon

Table 2. Coupling Coefficients for Three Types of Lasera

Tables Icon

Table 3. Source Strengths for Three Types of Lasera

Equations (192)

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

d t N 3 = a 30 P N 0 a 30 ( P + 1 ) N 3 a 32 N 3 ,
d t N 2 = a 32 N 3 + a 21 S N 1 a 21 ( S + 1 ) N 2 ,
d t N 1 = a 21 S N 1 + a 21 ( S + 1 ) N 2 a 10 N 1 ,
d t N 0 = a 30 P N 0 + a 30 ( P + 1 ) N 3 + a 10 N 1 ,
d t ( N 0 + N 1 + N 2 + N 3 ) = 0.
N 3 = a 30 P N 0 / [ a 32 + a 30 ( P + 1 ) ] a 30 P N 0 / a 32 ,
N 1 = a 21 ( S + 1 ) N 2 / ( a 10 + a 21 S ) a 21 ( S + 1 ) N 2 / a 10 ,
d t N 2 a 30 P N 0 a 21 ( S + 1 ) N 2 ,
d t N 0 a 30 P N 0 + a 21 ( S + 1 ) N 2 .
d t ( N 0 + N 2 ) 0.
d t S = a 21 N 2 ( S + 1 ) a 21 N 1 S b S a 21 N 2 ( S + 1 ) b S ,
d t N 2 = a 20 P N 0 a 20 ( P + 1 ) N 2 + a 21 S N 1 a 21 ( S + 1 ) N 2 a n l N 2 ,
d t N 1 = a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 a 10 N 1 ,
d t N 0 = a 20 P N 0 + a 20 ( P + 1 ) N 2 + a 10 N 1 .
d t ( N 0 + N 1 + N 2 ) = 0.
d t N 2 a 20 P N 0 a 20 ( P + 1 ) N 2 a 21 ( S + 1 ) N 2 a n l N 2 ,
d t N 0 a 20 P N 0 + a 20 ( P + 1 ) N 2 + a 21 ( S + 1 ) N 2 + a n l N 2 .
d t ( N 0 + N 2 ) = 0.
d t S = a 21 N 2 ( S + 1 ) a 21 N 1 S b S , a 21 N 2 ( S + 1 ) b S .
d t N 3 = a 31 P N 1 a 31 ( P + 1 ) N 3 a 32 N 3 ,
d t N 2 = a 32 N 3 + a 21 S N 1 a 21 ( S + 1 ) N 2 a n l N 2 ,
d t N 1 = a 31 P N 1 + a 31 ( P + 1 ) N 3 a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 .
d t ( N 1 + N 2 + N 3 ) = 0.
d t N 2 a 31 P N 1 + a 21 S N 1 a 21 ( S + 1 ) N 2 a n l N 2 ,
d t N 1 a 31 P N 1 a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 .
d t ( N 1 + N 2 ) 0.
d t S = a 21 N 2 ( S + 1 ) a 21 N 1 S b S .
d t S = a M S + a N ( S + 1 ) b S ,
d t N = a M S a N ( S + 1 ) + c M c N d N ,
d t M = a M S + a N ( S + 1 ) c M + c N + d N ,
d t S = a M S + a N ( S + 1 ) b S R a + R a R b ,
d t N = a M S a N ( S + 1 ) + c M c N d N + R a R a + R c R c R d ,
d t M = a M S + a N ( S + 1 ) c M + c N + d N R a + R a R c + R c + R d .
r j ( t ) = 0 , r j ( t ) r k ( t ) = δ j k δ ( t t ) ,
S a = a M S , S a = a N ( S + 1 ) , S b = b S , S c = c M , S c = c N , S d = d N
a N 0 a M 0 = b ,
( a N 0 a M 0 ) S 0 = b S 0 = c M 0 d N 0 .
d t S 1 = ( a N 0 a M 0 b ) S 1 + ( a S 0 ) N 1 ( a S 0 ) M 1 R a + R a R b ,
d t N 1 = ( a M 0 a N 0 ) S 1 ( a S 0 + d ) N 1 + ( a S 0 + c ) M 1 + R a R a + R c R d ,
d t M 1 = ( a N 0 a M 0 ) S 1 + ( a S 0 + d ) N 1 ( a S 0 + c ) M 1 R a + R a R c + R d ,
d t ( M 1 + N 1 ) = 0 ,
d t S 1 = ( a N 0 a M 0 b ) S 1 + ( a S 0 + a S 0 ) N 1 R a + R a R b ,
d t N 1 = ( a M 0 a N 0 ) S 1 ( a S 0 + a S 0 + c + d ) N 1 + R a R a + R c R d .
d t S 1 = γ s s S 1 + γ s n N 1 + R s ,
d t N 1 = γ n s S 1 γ n n N 1 + R n .
R j ( t ) = 0 , R j ( t ) R k ( t ) = R j k δ ( t t ) ,
R s s = S a + S a + S b , R n s = S a S a , R n n = S a + S a + S c + S d .
γ 2 + 2 ν 0 γ + ω 0 2 = 0 ,
ω 0 2 = γ n s γ s n γ n n γ s s , ν 0 = ( γ n n γ s s ) / 2 ,
γ = ( γ s s γ n n ) / 2 ± i [ γ n s γ s n ( γ n n + γ s s ) 2 / 4 ] 1 / 2 .
S 1 ( t ) = G s s ( t ) S 1 ( 0 ) + G s n ( t ) N 1 ( 0 ) ,
N 1 ( t ) = G n s ( t ) S 1 ( 0 ) + G n n ( t ) N 1 ( 0 ) ,
G s s ( t ) = [ cos ( ω r t ) + γ a sin ( ω r t ) / ω r ] exp ( ν 0 t ) ,
G s n ( t ) = [ γ s n sin ( ω r t ) / ω r ] exp ( ν 0 t ) ,
G n s ( t ) = [ γ n s sin ( ω r t ) / ω r ] exp ( ν 0 t ) ,
G n n ( t ) = [ cos ( ω r t ) γ a sin ( ω r t ) / ω r ] exp ( ν 0 t ) ,
[ 0 δ t r j ( t ) d t ] 2 = 0 δ t 0 δ t r j ( t ) r j ( t ) d t d t = δ t .
S 1 ( δ t ) ( 1 + γ s s δ t ) S 1 + ( γ s n δ t ) N 1 + 0 δ t R s ( t ) d t ,
N 1 ( δ t ) ( γ n s δ t ) S 1 + ( 1 γ n n δ t ) N 1 + 0 δ t R n ( t ) d t .
d t S 1 2 = 2 γ s s S 1 2 + 2 γ s n N 1 S 1 + R s s ,
d t N 1 S 1 = γ n s S 1 2 + ( γ s s γ n n ) N 1 S 1 + γ s n N 1 2 + R n s ,
d t N 1 2 = 2 γ n s N 1 S 1 2 γ n n N 1 2 + R n n .
d t F ( m , n , s ) = a m s F ( m , n , s ) + a ( m + 1 ) ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a n ( s + 1 ) F ( m , n , s ) + a ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) b s F ( m , n , s ) + b ( s + 1 ) F ( m , n , s + 1 ) c m F ( m , n , s ) + c ( m + 1 ) F ( m + 1 , n 1 , s ) c n F ( m , n , s ) + c ( n + 1 ) F ( m 1 , n + 1 , s ) d n F ( m , n , s ) + d ( n + 1 ) F ( m 1 , n + 1 , s ) ,
d t T = a 0 0 0 m s F ( m , n , s ) + a 0 1 0 ( m + 1 ) ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n ( s + 1 ) F ( m , n , s ) + a 1 0 1 ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) b 0 0 0 s F ( m , n , s ) + b 0 0 0 ( s + 1 ) F ( m , n , s + 1 ) c 0 0 0 m F ( m , n , s ) + c 0 1 0 ( m + 1 ) F ( m + 1 , n 1 , s ) d 0 0 0 n F ( m , n , s ) + d 1 0 0 ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m s F ( m , n , s ) + a 1 0 1 m s F ( m , n , s ) a 0 1 0 n ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( s + 1 ) F ( m , n , s ) b 0 0 1 s F ( m , n , s ) + b 0 0 1 s F ( m , n , s ) c 1 0 0 m F ( m , n , s ) + c 1 0 0 m F ( m , n , s ) d 0 1 0 n F ( m , n , s ) + d 0 1 0 n F ( m , n , s ) = 0.
d t s = a 0 0 0 m s 2 F ( m , n , s ) + a 0 1 0 ( m + 1 ) s ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n s ( s + 1 ) F ( m , n , s ) + a 1 0 1 ( n + 1 ) s 2 F ( m 1 , n + 1 , s 1 ) b 0 0 0 s 2 F ( m , n , s ) + b 0 0 0 s ( s + 1 ) F ( m , n , s + 1 ) = a 1 0 1 m s 2 F ( m , n , s ) + a 1 0 1 m s ( s 1 ) F ( m , n , s ) a 0 1 0 n s ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( s + 1 ) 2 F ( m , n , s ) b 0 0 1 s 2 F ( m , n , s ) + b 0 0 1 s ( s 1 ) F ( m , n , s ) .
d t s = a m s + a n ( s + 1 ) b s ,
d t n = a 0 0 0 m n s F ( m , n , s ) + a 0 1 0 ( m + 1 ) n ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n 2 ( s + 1 ) F ( m , n , s ) + a 1 0 1 n ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) c 0 0 0 m n F ( m , n , s ) + c 0 1 0 ( m + 1 ) n F ( m + 1 , n 1 , s ) d 0 0 0 n 2 F ( m , n , s ) + d 1 0 0 n ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m n s F ( m , n , s ) + a 1 0 1 m ( n + 1 ) s F ( m , n , s ) a 0 1 0 n 2 ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( n 1 ) ( s + 1 ) F ( m , n , s ) c 1 0 0 m n F ( m , n , s ) + c 1 0 0 m ( n + 1 ) F ( m , n , s ) d 0 1 0 n 2 F ( m , n , s ) + d 0 1 0 n ( n 1 ) F ( m , n , s ) .
d t n = a m s a n ( s + 1 ) + c m d n ,
d t m = a 0 0 0 m 2 s F ( m , n , s ) + a 0 1 0 m ( m + 1 ) ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 m n ( s + 1 ) F ( m , n , s ) + a 1 0 1 m ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) c 0 0 0 m 2 F ( m , n , s ) + c 0 1 0 m ( m + 1 ) F ( m + 1 , n 1 , s ) d 0 0 0 m n F ( m , n , s ) + d 1 0 0 m ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m 2 s F ( m , n , s ) + a 1 0 1 m ( m 1 ) s F ( m , n , s ) a 0 1 0 m n ( s + 1 ) F ( m , n , s ) + a 0 1 0 ( m + 1 ) n ( s + 1 ) F ( m , n , s ) c 1 0 0 m 2 F ( m , n , s ) + c 1 0 0 m ( m 1 ) F ( m , n , s ) d 0 1 0 m n F ( m , n , s ) + d 0 1 0 ( m + 1 ) n F ( m , n , s ) .
d t m = a m s + a n ( s + 1 ) c m + d n .
d t s 2 = a 0 0 0 m s 3 F ( m , n , s ) + a 0 1 0 ( m + 1 ) s 2 ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n s 2 ( s + 1 ) F ( m , n , s ) + a 1 0 1 ( n + 1 ) s 3 F ( m 1 , n + 1 , s 1 ) b 0 0 0 s 3 F ( m , n , s ) + b 0 0 0 s 2 ( s + 1 ) F ( m , n , s + 1 ) = a 1 0 1 m s 3 F ( m , n , s ) + a 1 0 1 m s ( s 1 ) 2 F ( m , n , s ) a 0 1 0 n s 2 ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( s + 1 ) 3 F ( m , n , s ) b 0 0 1 s 3 F ( m , n , s ) + b 0 0 1 s ( s 1 ) 2 F ( m , n , s ) .
d t s 2 = a ( 2 m s 2 + m s ) + a ( 2 n s 2 + 3 n s + n ) + b ( 2 s 2 + s ) .
d t n 2 = a 0 0 0 m n 2 s F ( m , n , s ) + a 0 1 0 ( m + 1 ) n 2 ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n 3 ( s + 1 ) F ( m , n , s ) + a 1 0 1 n 2 ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) c 0 0 0 m n 2 F ( m , n , s ) + c 0 1 0 ( m + 1 ) n 2 F ( m + 1 , n 1 , s ) d 0 0 0 n 3 F ( m , n , s ) + d 1 0 0 n 2 ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m n 2 s F ( m , n , s ) + a 1 0 1 m ( n + 1 ) 2 s F ( m , n , s ) a 0 1 0 n 3 ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( n 1 ) 2 ( s + 1 ) F ( m , n , s ) c 1 0 0 m n 2 F ( m , n , s ) + c 1 0 0 m ( n + 1 ) 2 F ( m , n , s ) d 0 1 0 n 3 F ( m , n , s ) + d 0 1 0 n ( n 1 ) 2 F ( m , n , s ) .
d t n 2 = a ( 2 m n s + m s ) + a ( 2 n 2 s 2 n 2 + n s + n ) + c ( 2 m n + m ) + d ( 2 n 2 + n ) .
d t m 2 = a 0 0 0 m 3 s F ( m , n , s ) + a 0 1 0 m 2 ( m + 1 ) ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 m 2 n ( s + 1 ) F ( m , n , s ) + a 1 0 1 m 2 ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) c 0 0 0 m 3 F ( m , n , s ) + c 0 1 0 m 2 ( m + 1 ) F ( m + 1 , n 1 , s ) d 0 0 0 m 2 n F ( m , n , s ) + d 1 0 0 m 2 ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m 3 s F ( m , n , s ) + a 1 0 1 m ( m 1 ) 2 s F ( m , n , s ) a 0 1 0 m 2 n ( s + 1 ) F ( m , n , s ) + a 0 1 0 ( m + 1 ) 2 n ( s + 1 ) F ( m , n , s ) c 1 0 0 m 3 F ( m , n , s ) + c 1 0 0 m ( m 1 ) 2 F ( m , n , s ) d 0 1 0 m 2 n F ( m , n , s ) + d 0 1 0 ( m + 1 ) 2 n F ( m , n , s ) .
d t m 2 = a ( 2 m 2 s + m s ) + a ( 2 m n s + 2 m n + n s + n ) + c ( 2 m 2 + m ) + d ( 2 m n + n ) .
d t δ s 2 = 2 a δ m s δ s + 2 a δ n s δ s + 2 a δ n δ s 2 b δ s 2 + a m s + a n ( s + 1 ) + b s ,
d t δ n 2 = 2 a δ m s δ n 2 a δ n s δ n 2 a δ n 2 + 2 c δ m δ n 2 d δ n 2 + a m s + a n ( s + 1 ) + c m + d n ,
d t δ m 2 = 2 a δ m s δ m + 2 a δ n s δ m + 2 a δ m δ n 2 c δ m 2 + 2 d δ m δ n + a m s + a n ( s + 1 ) + c m + d n .
d t n s = a 0 0 0 m n s 2 F ( m , n , s ) + a 0 1 0 ( m + 1 ) n s ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 n 2 s ( s + 1 ) F ( m , n , s ) + a 1 0 1 n ( n + 1 ) s 2 F ( m 1 , n + 1 , s 1 ) b 0 0 0 n s 2 F ( m , n , s ) + b 0 0 0 n s ( s + 1 ) F ( m , n , s + 1 ) c 0 0 0 m n s F ( m , n , s ) + c 0 1 0 ( m + 1 ) n s F ( m + 1 , n 1 , s ) d 0 0 0 n 2 s F ( m , n , s ) + d 1 0 0 n ( n + 1 ) s F ( m 1 , n + 1 , s ) = a 1 0 1 m n s 2 F ( m , n , s ) + a 1 0 1 m ( n + 1 ) s ( s 1 ) F ( m , n , s ) a 0 1 0 n 2 s ( s + 1 ) F ( m , n , s ) + a 0 1 0 n ( n 1 ) ( s + 1 ) 2 F ( m , n , s ) b 0 0 1 n s 2 F ( m , n , s ) + b 0 0 1 n s ( s 1 ) F ( m , n , s ) c 1 0 0 m n s F ( m , n , s ) + c 1 0 0 m ( n + 1 ) s F ( m , n , s ) d 0 1 0 n 2 s F ( m , n , s ) + d 0 1 0 n ( n 1 ) s F ( m , n , s ) .
d t n s = a ( m n s + m s 2 m s ) + a ( n 2 s n s 2 + n 2 2 n s n ) b n s + c m s d n s .
d t m n = a 0 0 0 m 2 n s F ( m , n , s ) + a 0 1 0 m ( m + 1 ) n ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 m n 2 ( s + 1 ) F ( m , n , s ) + a 1 0 1 m n ( n + 1 ) s F ( m 1 , n + 1 , s 1 ) c 0 0 0 m 2 n F ( m , n , s ) + c 0 1 0 m ( m + 1 ) n F ( m + 1 , n 1 , s ) d 0 0 0 m n 2 F ( m , n , s ) + d 1 0 0 m n ( n + 1 ) F ( m 1 , n + 1 , s ) = a 1 0 1 m 2 n s F ( m , n , s ) + a 1 0 1 m ( m 1 ) ( n + 1 ) s F ( m , n , s ) a 0 1 0 m n 2 ( s + 1 ) F ( m , n , s ) + a 0 1 0 ( m + 1 ) n ( n 1 ) ( s + 1 ) F ( m , n , s ) c 1 0 0 m 2 n F ( m , n , s ) + c 1 0 0 m ( m 1 ) ( n + 1 ) F ( m , n , s ) d 0 1 0 m n 2 F ( m , n , s ) + d 0 1 0 ( m + 1 ) n ( n 1 ) F ( m , n , s ) .
d t m n = a ( m 2 s m n s m s ) + a ( m n s + n 2 s m n + n 2 n s n ) + c ( m 2 m n m ) + d ( m n + n 2 n ) .
d t m s = a 0 0 0 m 2 s 2 F ( m , n , s ) + a 0 1 0 m ( m + 1 ) s ( s + 1 ) F ( m + 1 , n 1 , s + 1 ) a 0 0 0 m n s ( s + 1 ) F ( m , n , s ) + a 1 0 1 m ( n + 1 ) s 2 F ( m 1 , n + 1 , s 1 ) b 0 0 0 m s 2 F ( m , n , s ) + b 0 0 0 m s ( s + 1 ) F ( m , n , s + 1 ) c 0 0 0 m 2 s F ( m , n , s ) + c 0 1 0 m ( m + 1 ) s F ( m + 1 , n 1 , s ) d 0 0 0 m n s F ( m , n , s ) + d 1 0 0 m ( n + 1 ) s F ( m 1 , n + 1 , s ) = a 1 0 1 m 2 s 2 F ( m , n , s ) + a 1 0 1 m ( m 1 ) s ( s 1 ) F ( m , n , s ) a 0 1 0 m n s ( s + 1 ) F ( m , n , s ) + a 0 1 0 ( m + 1 ) n ( s + 1 ) 2 F ( m , n , s ) b 0 0 1 m s 2 F ( m , n , s ) + b 0 0 1 m s ( s 1 ) F ( m , n , s ) c 1 0 0 m 2 s F ( m , n , s ) + c 1 0 0 m ( m 1 ) s F ( m , n , s ) d 0 1 0 m n s F ( m , n , s ) + d 0 1 0 ( m + 1 ) n s F ( m , n , s ) .
d t m s = a ( m 2 s m s 2 + m s ) + a ( m n s + n s 2 + m n + 2 n s + n ) b m s c m s + d n s .
d t δ n δ s = a ( δ m s δ s δ m s δ n ) + a ( δ n s δ n δ n s δ s ) + a δ n 2 ( a + b + d ) δ n δ s + c δ m δ s a m s a n ( s + 1 ) ,
d t δ m δ n = a ( δ m s δ m δ m s δ n ) + a ( δ n s δ n δ n s δ m ) ( a + c + d ) δ m δ n + ( a + d ) δ n 2 + c δ m 2 a m s a n ( s + 1 ) c m d n ,
d t δ m δ s = a ( δ m s δ m + δ m s δ s ) + a ( δ n s δ m + δ n s δ s ) + a δ m δ n + a δ n δ s ( b + c ) δ m δ s + d δ n δ s + a m s + a n ( s + 1 ) .
d t ( δ m + δ n ) 2 = 0.
d t ( δ n + δ s ) 2 = 0 = d t ( δ m δ s ) 2 .
d t s a m s + a n ( s + 1 ) b s ,
d t n a m s a n ( s + 1 ) + c m d n ,
d t m a m s + a n ( s + 1 ) c m + d n .
d t δ s 2 = 2 a δ m s δ s + 2 a δ n s δ s 2 b δ s 2 + a m s + a n s + b s ,
d t δ n δ s = a δ m s δ n + a δ m s δ s + a δ n s δ n a δ n s δ s ( b + c + d ) δ n δ s a m s a n s ,
d t δ n 2 = 2 a δ m s δ n 2 a δ n s δ n 2 ( c + d ) δ n 2 + a m s + a n s + c m + d n .
d t δ s 2 = 2 ( a s + a s ) δ n δ s + 2 ( a n a m b ) δ s 2 + a m s + a n s + b s ,
d t δ n δ s = ( a s + a s ) δ n 2 + ( a n a m b ) δ n δ s ( a s + a s + c + d ) δ n δ s ( a n a m ) δ s 2 a m s a n s ,
d t δ n 2 = 2 ( a s + a s + c + d ) δ n 2 2 ( a n a m ) δ n δ s + a m s + a n s + c m + d n .
F ( ω ) = T / 2 T / 2 F ( t ) exp ( i ω t ) d t , F ( t ) = F ( ω ) exp ( i ω t ) d ω / 2 π ,
S 1 ( ω ) = ( γ n n i ω ) R s ( ω ) + γ s n R n ( ω ) ω 0 2 2 i ν 0 ω ω 2 ,
N 1 ( ω ) = γ n s R s ( ω ) ( γ s s + i ω ) R n ( ω ) ω 0 2 2 i ν 0 ω ω 2 ,
R j ( ω ) = 0 , R j ( ω ) R k ( ω ) = R j k T .
Q ( ω ) = | S 1 ( ω ) | 2 S 0 2 T = A + B ω 2 S 0 2 [ ( ω 0 2 ω 2 ) 2 + 4 ν 0 2 ω 2 ] ,
A = γ n n 2 R s s + 2 γ n n γ s n R n s + γ s n 2 R n n , B = R s s .
( b S 0 ) Q ( 0 ) = ( c ) 2 + ( a S 0 ) 2 + ( a S 0 + c ) 2 ( a S 0 ) 2 ,
( b S 0 ) Q ( 0 ) = a ( c ) 2 + ( 2 a S 0 ) 2 + ( 2 a S 0 + c ) 2 ( 2 a S 0 ) 2 ,
( b S 0 ) Q ( ω 0 ) = ( c ) 2 + ( a S 0 ) 2 + ( a S 0 + c ) 2 + 2 ( a S 0 ) ( a S 0 ) ( a S 0 + c ) 2 ,
( b S 0 ) Q ( ω 0 ) = a ( c ) 2 + ( 2 a S 0 ) 2 + ( 2 a S 0 + c ) 2 + ( a + 1 ) ( 2 a S 0 ) ( 2 a S 0 ) ( 2 a S 0 + c ) 2 .
C ( τ ) = exp ( ν 0 τ ) 4 ν 0 ω r S 0 2 R e [ A + B ( ω r i ν 0 ) 2 ω r i ν 0 exp ( i ω r τ ) ] .
C ( 0 ) = A + B ω 0 2 4 ν 0 ω 0 2 S 0 2 .
[ 2 γ s s 2 γ s n 0 γ n s γ n n γ s s γ s n 0 2 γ n s 2 γ n n ] [ S 1 2 N 1 S 1 N 1 2 ] = [ R s s R n s R n n ] .
[ S 1 2 N 1 S 1 N 1 2 ] = 1 Δ [ γ n s γ s n + γ n n ( γ n n γ s s ) 2 γ n n γ s n γ s n 2 γ n n γ n s 2 γ n n γ s s γ s n γ s s γ n s 2 2 γ n s γ s s γ n s γ s n γ s s ( γ n n γ s s ) ] [ R s s R n s R n n ] ,
Δ = 2 ( γ n n γ s s ) ( γ n s γ s n γ n n γ s s ) = 4 ν 0 ω 0 2 .
S 1 2 S 0 2 = ( γ n n 2 + γ n s γ s n γ n n γ s s ) R s s + 2 γ n n γ s n R n s + γ s n 2 R n n 2 ( γ n n γ s s ) ( γ n s γ s n γ n n γ s s ) S 0 2 .
F ( ω ) = T / 2 T / 2 F ( t ) exp ( i ω t ) d t , F ( t ) = F ( ω ) exp ( i ω t ) d ω / 2 π ,
T / 2 T / 2 | F ( t ) | 2 d t = | F ( ω ) | 2 d ω / 2 π .
F ( t ) e = F ( t ) t = T / 2 T / 2 F ( t ) d t / T ,
C j k ( τ ) = F j ( t ) F k ( t + τ ) e = F j ( t ) F k ( t + τ ) t .
C j k ( ω ) = F j ( t ) exp ( i ω t ) F k ( ω ) t = F j ( ω ) F k ( ω ) / T .
F j ( t ) e = 0 , F j ( t ) F k ( t ) e = S j k δ ( t t ) ,
F j ( ω ) F k ( ω ) e = F j ( t ) F k ( t ) exp [ i ω ( t t ) ] d t d t e = S j k δ ( t t ) exp [ i ω ( t t ) ] d t d t = S j k T .
F j ( ω ) F k ( ω ) e = S j k T s i n c [ ( ω ω ) T / 2 ] 2 π S j k δ ( ω ω ) .
F ( ω ) F ( ω ) e = S f ( ω ) δ ( ω ω )
T / 2 T / 2 F ˙ ( t ) exp ( i ω t ) d t = F ( t ) exp ( i ω t ) | T / 2 T / 2 i ω T / 2 T / 2 F ( t ) exp ( i ω t ) d t .
d t X i = a i ( X ) + k b i k ( X ) r k ( t ) ,
r k ( t ) = 0 , r k ( t ) r l ( t ) = δ k l δ ( t t ) .
δ w k = 0 , δ w k 2 = δ t .
δ X i = 0 δ t a i [ X ( t ) ] d t + k 0 δ t b i k [ X ( t ) ] r k ( t ) d t a i ( X ) δ t + k 0 δ t [ b i k ( X ) + j j b i k ( X ) l b j l ( X ) 0 t r l ( t ) d t ] r k ( t ) d t = a i ( X ) δ t + k b i k ( X ) 0 δ t r k ( t ) d t + k j l j b i k ( X ) b j l ( X ) 0 δ t 0 t r k ( t ) r l ( t ) d t d t ,
j k b j k ( X ) j b i k ( X ) 0 δ t 0 t r k ( t ) r k ( t ) d t d t .
δ X i a i ( X ) δ t .
δ X i a i ( X ) δ t + j k b j k ( X ) j b i k ( X ) δ t / 2.
δ X i 2 k b i k 2 ( X ) δ t .
X i ( 0 ) r l ( 0 ) { X i ( δ t ) + k b i k [ X ( δ t ) ] δ t 0 r k ( t ) d t } r l ( 0 ) .
X i r l = 0.
X i r l = b i l ( X ) / 2.
R i k ( X , t ) = 0 , R i k ( X , t ) R j l ( X , t ) = b i k ( X ) b j k ( X ) δ k l δ ( t t ) .
δ X i a i ( X ) δ t + k b i k ( X ) 0 δ t r k ( t ) d t .
d t X i = a i ( X ) .
δ f ( X ) i f i ( X ) δ X i + i j f i j ( X ) δ X i δ X j / 2.
δ X i δ X j k l b i k ( X ) b j l ( X ) 0 δ t 0 δ t r k ( t ) r l ( t ) d t d t = k b i k ( X ) b j k ( X ) δ t .
d t f ( X ) = i f i ( X ) d t X i + i j k f i j ( X ) b i k ( X ) b j k ( X ) / 2.
d t X i 2 = 2 X i a i ( X ) + k b i k 2 ( X ) ,
d t X i X j = X j a i ( X ) + X i a j ( X ) + k b i k ( X ) b j k ( X ) .
d t δ X i 2 = 2 δ X i δ a i ( X ) + k b i k 2 ( X ) ,
d t δ X i δ X j = δ X j δ a i ( X ) + δ X i δ a j ( X ) + k b i k ( X ) b j k ( X ) ,
d t X = A X + B R ( t ) ,
R ( t ) = 0 , R ( t ) R T ( t ) = I δ ( t t ) ,
d t Y = e A t B R ( t ) .
X ( t ) = e A t X ( 0 ) + 0 t e A ( t t ) B R ( t ) d t .
X ( t ) = e A t X ( 0 ) .
d t X = A X .
X ( t ) X T ( t ) = [ e A t X + 0 t e A ( t t ) B R ( t ) d t ] × [ X T e A T t + 0 t R T ( t ) B T e A T ( t t ) d t ] ,
X ( t ) X T ( t ) = e A t X X T e A T t + 0 t e A ( t t ) B B T e A T ( t t ) d t .
d t X X T = A X X T + X X T A T + B B T .
d t X i 2 = 2 k a i k X i X k + k b i k 2 ,
d t X i X j = k a i k X j X k + k a j k X i X k + k b i k b j k ,
d t M = a M S + a N ( S + 1 ) c M + d N ,
d t N = a M S a N ( S + 1 ) + c M d N ,
d t S = a M S + a N ( S + 1 ) b S .
d t δ M 2 = 2 a δ M S δ M + 2 a δ N S δ M + 2 a δ M δ N 2 c δ M 2 + 2 d δ M δ N + a M S + a N ( S + 1 ) + c M + d N ,
d t δ N 2 = 2 a δ M S δ N 2 a δ N S δ N 2 a δ N 2 + 2 c δ M δ N 2 d δ N 2 + a M S + a N ( S + 1 ) + c M + d N ,
d t δ S 2 = 2 a δ M S δ S + 2 a δ N S δ S + 2 a δ N δ S 2 b δ S 2 + a M S + a N ( S + 1 ) + b S ,
d t δ M δ N = a δ M S δ N + a δ N S δ N + a δ N 2 c δ M δ N + d δ N 2 + a δ M S δ M a δ N S δ M a δ M δ N + c δ M 2 d δ M δ N a M S a N ( S + 1 ) c M d N ,
d t δ M δ S = a δ M S δ S + a δ N S δ S + a δ N δ S c δ M δ S + d δ N δ S a δ M S δ M + a δ N S δ M + a δ M δ N b δ M δ S + a M S + a N ( S + 1 ) ,
d t δ N δ S = a δ M S δ S a δ N S δ S a δ N δ S + c δ M δ S d δ N δ S a δ M S δ N + a δ N S δ N + a δ N 2 b δ N δ S a M S a N ( S + 1 ) .
d t ( δ M + δ N ) 2 = 2 δ M ( δ A m + δ A n ) + 2 δ N ( δ A m + δ A n ) = 0 ,
d t N = a ( N , S ) + c ( N ) ,
d t S = a ( N , S ) b ( S ) ,
d t F ( n , s ) = a ( n , s ) F ( n , s ) + a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) b ( s ) F ( n , s ) + b ( s + 1 ) F ( n , s + 1 ) c ( n ) F ( n , s ) + c ( n 1 ) F ( n 1 , s ) ,
d t T = 0 0 a ( n , s ) F ( n , s ) + 0 1 a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 b ( s ) F ( n , s ) + 0 0 b ( s + 1 ) F ( n , s + 1 ) 0 0 c ( n ) F ( n , s ) + 1 0 c ( n 1 ) F ( n 1 , s ) = 1 0 a ( n , s ) F ( n , s ) + 1 0 a ( n , s ) F ( n , s ) 0 1 b ( s ) F ( n , s ) + 0 1 b ( s ) F ( n , s ) 0 0 c ( n ) F ( n , s ) + 0 0 c ( n ) F ( n , s ) = 0 ,
d t n = 0 0 n a ( n , s ) F ( n , s ) + 0 1 n a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 n c ( n ) F ( n , s ) + 1 0 n c ( n 1 ) F ( n 1 , s ) = 1 0 n a ( n , s ) F ( n , s ) + 1 0 ( n 1 ) a ( n , s ) F ( n , s ) 0 0 n c ( n ) F ( n , s ) + 0 0 ( n + 1 ) c ( n ) F ( n , s ) .
d t n = a ( n , s ) + c ( n ) ,
d t s = 0 0 s a ( n , s ) F ( n , s ) + 0 1 s a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 s b ( s ) F ( n , s ) + 0 0 s b ( s + 1 ) F ( n , s + 1 ) = 1 0 s a ( n , s ) F ( n , s ) + 1 0 ( s + 1 ) a ( n , s ) F ( n , s ) 0 1 s b ( s ) F ( n , s ) + 0 1 ( s 1 ) b ( s ) F ( n , s ) .
d t s = a ( n , s ) b ( s ) ,
d t n 2 = 0 0 n 2 a ( n , s ) F ( n , s ) + 0 1 n 2 a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 n 2 c ( n ) F ( n , s ) + 1 0 n 2 c ( n 1 ) F ( n 1 , s ) = 1 0 n 2 a ( n , s ) F ( n , s ) + 1 0 ( n 1 ) 2 a ( n , s ) F ( n , s ) 0 0 n 2 c ( n ) F ( n , s ) + 0 0 ( n + 1 ) 2 c ( n ) F ( n , s ) = 2 n a ( n , s ) + a ( n , s ) + 2 n c ( n ) + c ( n ) ,
d t n s = 0 0 n s a ( n , s ) F ( n , s ) + 0 1 n s a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 n s b ( s ) F ( n , s ) + 0 0 n s b ( s + 1 ) F ( n , s + 1 ) 0 0 n s c ( n ) F ( n , s ) + 1 0 n s c ( n 1 ) F ( n 1 , s ) = 1 0 n s a ( n , s ) F ( n , s ) + 1 0 ( n 1 ) ( s + 1 ) a ( n , s ) F ( n , s ) 0 1 n s b ( s ) F ( n , s ) + 0 1 n ( s 1 ) b ( s ) F ( n , s ) 0 0 n s c ( n ) F ( n , s ) + 0 0 ( n + 1 ) s c ( n ) F ( n , s ) = n a ( n , s ) s a ( n , s ) a ( n , s ) n b ( s ) + s c ( n ) ,
d t s 2 = 0 0 s 2 a ( n , s ) F ( n , s ) + 0 1 s 2 a ( n + 1 , s 1 ) F ( n + 1 , s 1 ) 0 0 s 2 b ( s ) F ( n , s ) + 0 0 s 2 b ( s + 1 ) F ( n , s + 1 ) = 1 0 s 2 a ( n , s ) F ( n , s ) + 1 0 ( s + 1 ) 2 a ( n , s ) F ( n , s ) 0 1 s 2 b ( s ) F ( n , s ) + 0 1 ( s 1 ) 2 b ( s ) F ( n , s ) = 2 s a ( n , s ) + a ( n , s ) 2 s b ( s ) + b ( s ) .
d t δ n 2 = 2 δ n δ a ( n , s ) + 2 δ n δ c ( n ) + a ( n , s ) + c ( n ) ,
d t δ n δ s = δ n δ a ( n , s ) δ s δ a ( n , s ) δ n δ b ( s ) + δ s δ c ( n ) a ( n , s ) ,
d t δ s 2 = 2 δ s δ a ( n , s ) 2 δ s δ b ( s ) + a ( n , s ) + b ( s ) ,
C ( τ ) = ( A + B ω 2 ) exp ( i ω τ ) [ ( ω 0 2 ω 2 ) 2 + 4 ν 0 2 ω 2 ] d ω 2 π ,
Γ F ( z ) d z = 2 π i j r e s j ,
( ω ω r i ν 0 ) ( ω ω r + i ν 0 ) ( ω + ω r i ν 0 ) ( ω + ω r + i ν 0 ) .
( 2 i ν 0 ) 2 ( ω r i ν 0 ) ( 2 ω r ) .
A + B ( ω r i ν 0 ) 2 8 ν 0 ω r ( ω r i ν 0 ) exp ( i ω r τ ν 0 τ ) .
A + B ( ω r + i ν 0 ) 2 8 ν 0 ω r ( ω r + i ν 0 ) exp ( i ω r τ ν 0 τ ) .
C ( τ ) = exp ( ν 0 τ ) 4 ν 0 ω r R e [ A + B ( ω r i ν 0 ) 2 ω r i ν 0 exp ( i ω r τ ) ] .
I 1 ( t ) = exp ( i ω t ) d ω ω 0 2 2 i ν 0 ω ω 2 , I 2 ( t ) = i ω exp ( i ω t ) d ω ω 0 2 2 i ν 0 ω ω 2 .
( ω ω r + i ν 0 ) ( ω + ω r + i ν 0 ) .
I 1 ( t ) = exp ( i ω r t ν 0 t ) / 2 i ω r + exp ( i ω r t ν 0 t ) / 2 i ω r = sin ( ω r t ) exp ( ν 0 t ) / ω r ,
I 2 ( t ) = ( i ω r + ν 0 ) exp ( i ω r t ν 0 t ) / 2 i ω r + ( i ω r ν 0 ) exp ( i ω r t ν 0 t ) / 2 i ω r = [ cos ( ω r t ) ν 0 sin ( ω r t ) / ω r ] exp ( ν 0 t ) .

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