Abstract

Detection of photons from electromagnetic radiation can be considered as the appearance of random events on the time axis. When an attenuator is placed in front of the detector, which attenuates the intensity by a factor of α, the statistical properties of the detected photons are altered. We show that simple relations exist between the statistical functions of the photons detected from the attenuated field and the same functions for the photons that would be detected from the unattenuated field. We also derive several recurrence relations for the statistical functions involving their dependence on the parameter α. For photon detection from resonance fluorescence, the parameter α appears naturally as the probability that an emitted photon is detected. In this case, there is no attenuator, but the parameter α appears in the same way. We show that the probability for the emission (α=1) of n photons in a given time interval can easily be computed, and with the general theory we can then obtain the result for the detection of n photons (α<1).

© 2013 Optical Society of America

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References

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  1. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [CrossRef]
  2. R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. Lond. A 242, 300–324 (1957).
  3. R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. Lond. A 243, 291–319 (1958).
  4. F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
    [CrossRef]
  5. H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
    [CrossRef]
  6. M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Phys. Rev. A 18, 2217–2228 (1978).
    [CrossRef]
  7. R. Short and L. Mandel, “Observation of sub-poissonian photon statistics,” Phys. Rev. Lett. 51, 384–387 (1983).
    [CrossRef]
  8. G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
    [CrossRef]
  9. R. L. Stratonovich, Topics in the Theory of Random Noise(Gordon and Breach, 1963), Vol. 1, Chap. 6.
  10. P. L. Kelly and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
    [CrossRef]
  11. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, 1965), pp. 65–185.
  12. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, 2007), Chap. 2.
  13. H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” J. Mod. Opt. 59, 1002–1015 (2012).
    [CrossRef]
  14. H. F. Arnoldus and G. Nienhuis, “Conditions for sub-poissonian photon statistics and squeezed states in resonance fluorescence,” Optica Acta 30, 1573–1585 (1983).
    [CrossRef]
  15. G. S. Agarwal, “Time factorization of the higher-order intensity correlation functions in the theory of resonance fluorescence,” Phys. Rev. A 15, 814–816 (1977).
    [CrossRef]
  16. D. Lenstra, “Photon-number statistics in resonance fluorescence,” Phys. Rev. A 26, 3369–3377 (1982).
    [CrossRef]
  17. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), p. 221.
  18. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 908.
  19. H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Phys. Lett. A 376, 2584–2587 (2012).
    [CrossRef]

2012 (2)

H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” J. Mod. Opt. 59, 1002–1015 (2012).
[CrossRef]

H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Phys. Lett. A 376, 2584–2587 (2012).
[CrossRef]

1990 (1)

G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef]

1983 (2)

H. F. Arnoldus and G. Nienhuis, “Conditions for sub-poissonian photon statistics and squeezed states in resonance fluorescence,” Optica Acta 30, 1573–1585 (1983).
[CrossRef]

R. Short and L. Mandel, “Observation of sub-poissonian photon statistics,” Phys. Rev. Lett. 51, 384–387 (1983).
[CrossRef]

1982 (1)

D. Lenstra, “Photon-number statistics in resonance fluorescence,” Phys. Rev. A 26, 3369–3377 (1982).
[CrossRef]

1978 (1)

M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Phys. Rev. A 18, 2217–2228 (1978).
[CrossRef]

1977 (2)

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
[CrossRef]

G. S. Agarwal, “Time factorization of the higher-order intensity correlation functions in the theory of resonance fluorescence,” Phys. Rev. A 15, 814–816 (1977).
[CrossRef]

1966 (1)

F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
[CrossRef]

1964 (1)

P. L. Kelly and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[CrossRef]

1958 (1)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. Lond. A 243, 291–319 (1958).

1957 (1)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. Lond. A 242, 300–324 (1957).

1956 (1)

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Time factorization of the higher-order intensity correlation functions in the theory of resonance fluorescence,” Phys. Rev. A 15, 814–816 (1977).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 908.

Arnoldus, H. F.

H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Phys. Lett. A 376, 2584–2587 (2012).
[CrossRef]

H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” J. Mod. Opt. 59, 1002–1015 (2012).
[CrossRef]

H. F. Arnoldus and G. Nienhuis, “Conditions for sub-poissonian photon statistics and squeezed states in resonance fluorescence,” Optica Acta 30, 1573–1585 (1983).
[CrossRef]

Dagenais, M.

M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Phys. Rev. A 18, 2217–2228 (1978).
[CrossRef]

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
[CrossRef]

Gatti, E.

F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, 1965), pp. 65–185.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. Lond. A 243, 291–319 (1958).

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. Lond. A 242, 300–324 (1957).

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Kelly, P. L.

P. L. Kelly and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[CrossRef]

Kimble, H. J.

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
[CrossRef]

Kleiner, W. H.

P. L. Kelly and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[CrossRef]

Lenstra, D.

D. Lenstra, “Photon-number statistics in resonance fluorescence,” Phys. Rev. A 26, 3369–3377 (1982).
[CrossRef]

Mandel, L.

R. Short and L. Mandel, “Observation of sub-poissonian photon statistics,” Phys. Rev. Lett. 51, 384–387 (1983).
[CrossRef]

M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Phys. Rev. A 18, 2217–2228 (1978).
[CrossRef]

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
[CrossRef]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), p. 221.

Nienhuis, G.

H. F. Arnoldus and G. Nienhuis, “Conditions for sub-poissonian photon statistics and squeezed states in resonance fluorescence,” Optica Acta 30, 1573–1585 (1983).
[CrossRef]

Rempe, G.

G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef]

Riehle, R. A.

H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Phys. Lett. A 376, 2584–2587 (2012).
[CrossRef]

H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” J. Mod. Opt. 59, 1002–1015 (2012).
[CrossRef]

Schmidt-Kaler, F.

G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef]

Short, R.

R. Short and L. Mandel, “Observation of sub-poissonian photon statistics,” Phys. Rev. Lett. 51, 384–387 (1983).
[CrossRef]

Sona, A.

F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
[CrossRef]

Stratonovich, R. L.

R. L. Stratonovich, Topics in the Theory of Random Noise(Gordon and Breach, 1963), Vol. 1, Chap. 6.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. Lond. A 243, 291–319 (1958).

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. Lond. A 242, 300–324 (1957).

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

van Kampen, N. G.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, 2007), Chap. 2.

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), p. 221.

Walther, H.

G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 908.

J. Mod. Opt. (1)

H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” J. Mod. Opt. 59, 1002–1015 (2012).
[CrossRef]

Nature (1)

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Optica Acta (1)

H. F. Arnoldus and G. Nienhuis, “Conditions for sub-poissonian photon statistics and squeezed states in resonance fluorescence,” Optica Acta 30, 1573–1585 (1983).
[CrossRef]

Phys. Lett. (1)

F. T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. 20, 27–29 (1966).
[CrossRef]

Phys. Lett. A (1)

H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Phys. Lett. A 376, 2584–2587 (2012).
[CrossRef]

Phys. Rev. (1)

P. L. Kelly and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[CrossRef]

Phys. Rev. A (3)

G. S. Agarwal, “Time factorization of the higher-order intensity correlation functions in the theory of resonance fluorescence,” Phys. Rev. A 15, 814–816 (1977).
[CrossRef]

D. Lenstra, “Photon-number statistics in resonance fluorescence,” Phys. Rev. A 26, 3369–3377 (1982).
[CrossRef]

M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Phys. Rev. A 18, 2217–2228 (1978).
[CrossRef]

Phys. Rev. Lett. (3)

R. Short and L. Mandel, “Observation of sub-poissonian photon statistics,” Phys. Rev. Lett. 51, 384–387 (1983).
[CrossRef]

G. Rempe, F. Schmidt-Kaler, and H. Walther, “Observation of sub-poissonian photon statistics in a micromaser,” Phys. Rev. Lett. 64, 2783–2786 (1990).
[CrossRef]

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695(1977).
[CrossRef]

Proc. R. Soc. Lond. A (2)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. Lond. A 242, 300–324 (1957).

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. Lond. A 243, 291–319 (1958).

Other (5)

R. L. Stratonovich, Topics in the Theory of Random Noise(Gordon and Breach, 1963), Vol. 1, Chap. 6.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), p. 221.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 908.

R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, 1965), pp. 65–185.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, 2007), Chap. 2.

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Figures (5)

Fig. 1.
Fig. 1.

Function f(t) for various values of the parameter Ω^. Curves a, b, and c correspond to Ω^=0.4, 2, and 8, respectively.

Fig. 2.
Fig. 2.

Solid curve is the probability for the detection of zero photons in [0,T] from resonance fluorescence radiation, for Ω^=0.4 and α=1. The dashed curve is the corresponding function for an independent event process with the same intensity.

Fig. 3.
Fig. 3.

Graph compares the probability for the detection of one photon from resonance fluorescence (solid curve), with Ω^=0.4, to the same probability for a Poisson process with the same intensity.

Fig. 4.
Fig. 4.

Solid curve is the probability to detect one photon in [0,T] for resonance fluorescence, with Ω^=2 and α=1. The dashed curve is the corresponding probability for a Poisson process.

Fig. 5.
Fig. 5.

Graph illustrates that for high laser power (Ω^ large, and Ω^=8 for the graph), photon detection from resonance fluorescence becomes indistinguishable from a Poisson process.

Tables (2)

Tables Icon

Table 1. Several Values of Ank

Tables Icon

Table 2. Several Values of Bnk

Equations (98)

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Ik(t1,,tk)dt1dtk=probability for anevent in[t1,t1+dt1],andand anevent in[tk,tk+dtk],irrespective of events at othertimes,and witht1<<tk.
Ik(t1,,tk)=I(t1)I(tk),
Ik(t1,,tk)=ζkE(t1)()E(tk)()E(tk)(+)E(t1)(+)
Sk(ta,tb)=n=kn!(nk)!Pn(ta,tb),k=0,1,2,,
G(x,ta,tb)=n=0xnPn(ta,tb).
G(x,ta,tb)=k=0(x1)kk!Sk(ta,tb).
Pn(ta,tb)=1n!nxnG(x,ta,tb)|x=0.
Pn(ta,tb)=(1)nn!k=n(1)k(kn)!Sk(ta,tb),
S0(ta,tb)=1,
S1(ta,tb)=tatbdtI(t),
Sk(ta,tb)=k!tatbdtktatkdtk1tat2dt1Ik(t1,,tk),k=2,3,.
μ(ta,tb)=n=0nPn(ta,tb)=S1(ta,tb)
I(t)=tμ(ta,t).
wn(ta,t)dt=probability that thenth event occurs in[t,t+dt],=probability forn1events in[ta,t]and an eventin[t,t+dt].
wn(ta,t)=tm=0n1Pm(ta,t).
Pn(ta,tb|ta)=probability fornevents in[ta,tb],after anevent in[tadta,ta].
Pn(ta,tb|ta)=1I(ta)tam=0nPm(ta,tb),
wn(ta,t|ta)dt=probability that thenth even to occurs in[t,t+dt],after an event in[tadta,ta].
wn(ta,t|ta)=1I(ta)tam=1nwm(ta,t).
n=0Pn(ta,tb)=S0(ta,tb)=1.
k=0(1)kk!Sk(ta,tb)=P0(ta,tb).
wn(ta,t)=tm=nPm(ta,t).
n=1wn(ta,t)=tm=0mPm(ta,t).
n=1wn(ta,t)=I(t),
n=0Pn(ta,tb|ta)=1.
n=1wn(ta,t|ta)=I2(ta,t)I(ta).
Ik(t1,,tk;α)=αkIk(t1,,tk;1).
Sk(ta,tb;α)=αkSk(ta,tb;1).
G(x,ta,tb;α)=G(α(x1)+1,ta,tb;1).
Pn(ta,tb;α)=m=n(mn)αn(1α)mnPm(ta,tb;1).
n=0Pn(ta,tb;α)=m=0Pm(ta,tb;1).
wn(ta,t;α)=tm=nPm(ta,t;α).
wn(ta,t;α)=tm=nan,m(α)Pm(ta,t;1).
an,m(α)=k=nm(mk)αk(1α)mk,mn0.
an,m(1)=1.
tPm(ta,t;1)=wm(ta,t;1)wm+1(ta,t;1),m1,
wn(ta,t;α)=αnwn(ta,t;1)+m=n+1[an,m(α)an,m1(α)]wm(ta,t;1),
an,n(α)=αn.
gn(y;α)=m=nan,m(α)ym,n0.
m=km!(mk)!um=k!uk(1u)k+1,
gn(y;α)=11y(αy1y+αy)n.
an,n+1(α)=[1+(1α)n]an,n(α),
(mn)an,m(α)+[(α1)(m1)+nm]an,m1(α)+(1α)(m1)an,m2=0,mn+2.
(mn)[an,m(α)an,m1(α)]=(1α)(m1)[an,m1(α)an,m2(α)],mn+2,
an,m(α)an,m1(α)=(1α)mn1(m1)!n!(mn)![an,n+1(α)an,n(α)],
an,n+1(α)an,n(α)=n(1α)αn.
an,m(α)an,m1(α)=(m1n1)αn(1α)mn,mn+1.
wn(ta,t;α)=m=n(m1n1)αn(1α)mnwm(ta,t;1),
n=1wn(ta,t;α)=αm=1wm(ta,t;1).
I(t;α)=αI(t;1),
Pn(ta,tb|ta;α)=1I(ta;α)tam=n+1Pm(ta,tb;α),
Pm(ta,tb;α)=k=m(km)αm(1α)kmPk(ta,tb;1).
Pn(ta,tb|ta;α)=1αI(ta;1)tam=n+1an+1,m(α)Pm(ta,tb;1),
1I(ta;1)taPm(ta,tb;1)=Pm(ta,tb|ta;1)Pm1(ta,tb|ta;1),m1,
Pn(ta,tb|ta;α)=αnPn(ta,tb|ta;1)+1αm=n+1[an+1,m+1(α)an+1,m(α)]Pm(ta,tb|ta;1).
Pn(ta,tb|ta;α)=m=n(mn)αn(1α)mnPn(ta,tb|ta;1).
wn(ta,t|ta;α)=1I(ta;α)ta[I(t;α)m=n+1wm(ta,t;α)],
wn(ta,t|ta;α)=1αI(ta;1)tam=n+1wm(ta,t;α).
wn(ta,t|ta;α)=1I(ta;1)tam=n+1an,m1(α)wm(ta,t;α).
1I(ta;1)tawm(ta,t;1)=wm(ta,t|ta;1)wm1(ta,t|ta;1),m2.
wn(ta,t|ta;α)=m=n(m1n1)αn(1α)mnwm(ta,t|ta;1)
P0(ta,tb;α)=m=0(1α)mPm(ta,tb;1).
nαnP0(ta,tb;α)=(1)nm=nm!(mn)!(1α)mnPm(ta,tb;1),
Pn(ta,tb;α)=(α)nn!nαnP0(ta,tb;α).
Pn+1(ta,tb;α)=αn+1n+1α(Pn(ta,tb;α)αn)
(n+1)Pn+1(ta,tb;α)=nPn(ta,tb;α)ααPn(ta,tb;α).
1αwn(ta,t;α)=(α)n1(n1)!n1αn1(1αw1(ta,t;α)).
wn+1(ta,t;α)=αn+1nα(wn(ta,t;α)αn),
nwn+1(ta,t;α)=nwn(ta,t;α)ααwn(ta,t;α),
I(α)=αAne,
Ik(t1,,tk;α)=(αA)kf(tktk1)f(t2t1)ne.
f(0)=0,
ne=Ω^21+2Ω^2,
f(t)=ne{1e34t^[34ρsinh(ρt^)+cosh(ρt^)]},
ρ=116Ω^2.
S˜k(0,s;α)=0esTSk(0,T;α)dT,
S˜0(0,s;α)=1s,
S˜k(0,s;α)=I(α)k!s2[αAf˜(s)]k1,k=1,2,,
P˜0(0,s;α)=1sI(α)s211+αAf˜(s),
P˜n(0,s;α)=I(α)s2[αAf˜(s)]n1[1+αAf˜(s)]n+1,n=1,2,.
f˜(s)=12sΩ2(s+A)(s+12A)+Ω2,
P˜0(0,s;α)=1sαAnes(s+A)(s+12A)+Ω2s(s+A)(s+12A)+Ω2(s+12αA).
P˜0(0,s;1)=1sAnes1s+12A(s+A)(s+12A)+Ω2(s+12A)2(γA)2,
γ=14Ω^2.
P˜n(0,s;1)=Ane(12AΩ2)n1[(s+A)(s+12A)+Ω2]2(s+12A)n+1[(s+12A)2(γA)2]n+1,n=1,2,.
P0(0,T;1)=18γ211+2Ω^2e12T^[16Ω^4+4γsinh(γT^)+(1+4γ2)cosh(γT^)],
Pn(0,T;1)=Ane(12AΩ2)n1e12T^1{[s(s+12A)+Ω2]2sn+1[s2(γA)2]n+1}.
Pn(0,T;1)=Ω^2n1+2Ω^28(2γ)3n+2e12T^k=0nr=0nk(γT^)nkr(nkr)!×{12[(1)kzr(γ)eγT^+zr(γ)eγT^]An,kzr(0)Bn,k},n=1,2,
z0(x)=[x(x+12)+Ω^2]2,
z1(x)=γ(4x+1)[x(x+12)+Ω^2],
z2(x)=γ2[6x2+3x+14+2Ω^2],
z3(x)=γ3(4x+1),
z4(x)=γ4,
zr(x)=0,r>4.
An,k=m=0k(n+mn)(n+kmn)2nm,n,k=0,1,2,,
Bn,k=0,k=1,3,,
Bn,2m=(1)n22n(n+mn),m=0,1,.
Pn(0,T;α)=(IT)nn!eIT.

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