Abstract

A formula for the photoelectron-counting distribution for a two-photon detector is derived quantum mechanically assuming that the ionising transitions in the atoms of the detector take place through the simultaneous absorption of two photons. It is assumed that the incident light is quasimonochromatic. It is shown that the distribution of the photoelectrons is given by the average of a Poisson distribution, the parameter of the distribution being proportional to the time integral of the square of the instantaneous light intensity. Counting distributions for the thermal (gaussian) light and for some models of laser light are obtained for the limiting case when the counting-time interval T is short compared to the coherence time Tc of the light. An approximate formula for arbitrary time intervals for the counting distribution of thermal light is also proposed.

© 1969 Optical Society of America

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References

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  1. M. Goeppert–Mayer, Ann. Physik 9, 273 (1931).
    [CrossRef]
  2. R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964);S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965);E. M. Logothetics and P. L. Hartman,  18, 581 (1967);and some of the references quoted in these papers.
    [CrossRef]
  3. S. Kielich, Acta Phys. Polon. 30, 393 (1966);R. Wallace, Mol. Phys. 11, 457 (1966);and some of the references quoted therein.
    [CrossRef]
  4. A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965);H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
    [CrossRef]
  5. L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).
  6. B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.
    [CrossRef]
  7. G. S. Agarwal, Phys. Rev. 177, 400 (1969).
    [CrossRef]
  8. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  9. L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.
    [CrossRef]
  10. J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1967), p.211, and some of the references quoted in this review article.
    [CrossRef]
  11. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (W. A. Benjamin, Inc., New York, 1968).
  12. L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958);Proc. Phys. Soc. (London) 74, 233 (1959).
    [CrossRef]
  13. The original derivation of Mandel was based on semiclassical arguments.The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches1964,C. Dewitt, A. Blandin, and C. Cohen–Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65.See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
    [CrossRef]
  14. E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964);G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
    [CrossRef]
  15. N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966);Phys. Rev. 151, 1076 (1966);See also R. Wallace, Phys. Rev. Letters 17, 397 (1966);Mol. Phys. 11, 457 (1966).
    [CrossRef]
  16. The contribution of the A2 term is also easily taken into account. It may be shown that the final result is still given by Eq. (30) except that the quantum-efficiency parameter α changes slightly. The new value of α isα=2πN∑{μ}∑jkMj(μ1,μ2)Mk*(μ3,μ4)∊μ1∊μ2∊μ3*∊μ4*(ωj−ω0)(ωk−ω0)ρ(2ω0)+2πN∑{μ}(e22ℏmc2)2|〈f|0〉|2∊μ1*∊μ1*∊μ2∊μ2ρ(2ω0).Here 〈f|0〉 represents the scalar product between the ground-state wavefunction and the final-state (free-electron) wave-function.
  17. E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).A detailed discussion of the diagonal representation is given in Ref. 11.See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.
  19. L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
    [CrossRef]
  20. The representation of V in the form Eq. (24) assumes that the incident light is plane polarized. However, the results are easily generalized to the case of partially polarized light. If the incident light is of thermal origin, then as well known we may consider it as the superposition of two linearly polarized, statistically independent components with average intensities equal to 12(1+P)〈I〉 and 12(1−P)〈I〉, respectively, where P is the degree of polarization. It is also assumed that the condition of cross-spectral purity is satisfied. In this case, the final result (30) remains unchanged. (See also Sec. IIIA.) For the partially polarized light of nonthermal origin, the expression for the photoelectron counting distribution may be shown to bep(n,T)=∫Φ({υks})[W({υks})]nexp{−W({υks})}n!d2({υks}),whereW({υks})=∑{μ}αμ1μ2μ3μ4∫0TdtVμ1(t)Vμ2(t)Vμ3*(t)Vμ4*(t),andαμ1μ2μ3μ4=2πN∑jkMj(μ1,μ2)Mk*(μ3,μ4)(ωj−ω0)(ωk−ω0)ρ(2ω0).
  21. See Ref. 9, Appendix B.Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).
    [CrossRef]
  22. See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.
  23. See, for example, Ref. 8, Eq. (4.41).
  24. A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169;R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and V. Hochuli, Phys. Letters 25A, 272 (1967).
  25. G. Bedard, Phys. Letters 24A, 613 (1967).
  26. H. Risken, Z. Physik 186, 85 (1965).
    [CrossRef]
  27. G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
    [CrossRef]
  28. S. O. Rice, Bell System Tech. J. 24, 46 (1945).
    [CrossRef]

1969 (1)

G. S. Agarwal, Phys. Rev. 177, 400 (1969).
[CrossRef]

1968 (1)

B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.
[CrossRef]

1967 (2)

G. Bedard, Phys. Letters 24A, 613 (1967).

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[CrossRef]

1966 (4)

See Ref. 9, Appendix B.Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).
[CrossRef]

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966);Phys. Rev. 151, 1076 (1966);See also R. Wallace, Phys. Rev. Letters 17, 397 (1966);Mol. Phys. 11, 457 (1966).
[CrossRef]

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).

S. Kielich, Acta Phys. Polon. 30, 393 (1966);R. Wallace, Mol. Phys. 11, 457 (1966);and some of the references quoted therein.
[CrossRef]

1965 (3)

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965);H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

H. Risken, Z. Physik 186, 85 (1965).
[CrossRef]

1964 (3)

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964);G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
[CrossRef]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[CrossRef]

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964);S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965);E. M. Logothetics and P. L. Hartman,  18, 581 (1967);and some of the references quoted in these papers.
[CrossRef]

1963 (1)

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).A detailed discussion of the diagonal representation is given in Ref. 11.See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

1958 (1)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958);Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

1945 (1)

S. O. Rice, Bell System Tech. J. 24, 46 (1945).
[CrossRef]

1931 (1)

M. Goeppert–Mayer, Ann. Physik 9, 273 (1931).
[CrossRef]

1169 (1)

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169;R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and V. Hochuli, Phys. Letters 25A, 272 (1967).

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. 177, 400 (1969).
[CrossRef]

Armstrong, J. A.

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169;R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and V. Hochuli, Phys. Letters 25A, 272 (1967).

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1967), p.211, and some of the references quoted in this review article.
[CrossRef]

Barry Bebb, H.

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965);H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
[CrossRef]

Bedard, G.

G. Bedard, Phys. Letters 24A, 613 (1967).

Bédard, G.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

Braunstein, R.

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964);S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965);E. M. Logothetics and P. L. Hartman,  18, 581 (1967);and some of the references quoted in these papers.
[CrossRef]

Chang, J. C.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[CrossRef]

Cohan, N. V.

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966);Phys. Rev. 151, 1076 (1966);See also R. Wallace, Phys. Rev. Letters 17, 397 (1966);Mol. Phys. 11, 457 (1966).
[CrossRef]

Glauber, R. J.

The original derivation of Mandel was based on semiclassical arguments.The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches1964,C. Dewitt, A. Blandin, and C. Cohen–Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65.See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Goeppert–Mayer, M.

M. Goeppert–Mayer, Ann. Physik 9, 273 (1931).
[CrossRef]

Gold, A.

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965);H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
[CrossRef]

Gradshteyn, I. S.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

Hameka, H. V.

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966);Phys. Rev. 151, 1076 (1966);See also R. Wallace, Phys. Rev. Letters 17, 397 (1966);Mol. Phys. 11, 457 (1966).
[CrossRef]

Keldysh, L. V.

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).

Kielich, S.

S. Kielich, Acta Phys. Polon. 30, 393 (1966);R. Wallace, Mol. Phys. 11, 457 (1966);and some of the references quoted therein.
[CrossRef]

Klauder, J. R.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (W. A. Benjamin, Inc., New York, 1968).

Mandel, L.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[CrossRef]

See Ref. 9, Appendix B.Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[CrossRef]

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958);Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.
[CrossRef]

Mehta, C. L.

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964);G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
[CrossRef]

Mollow, B. R.

B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.
[CrossRef]

Ockman, N.

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964);S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965);E. M. Logothetics and P. L. Hartman,  18, 581 (1967);and some of the references quoted in these papers.
[CrossRef]

Rice, S. O.

S. O. Rice, Bell System Tech. J. 24, 46 (1945).
[CrossRef]

Risken, H.

H. Risken, Z. Physik 186, 85 (1965).
[CrossRef]

Ryzhik, I. M.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

Smith, A. W.

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169;R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and V. Hochuli, Phys. Letters 25A, 272 (1967).

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1967), p.211, and some of the references quoted in this review article.
[CrossRef]

Sudarshan, E. C. G.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[CrossRef]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).A detailed discussion of the diagonal representation is given in Ref. 11.See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (W. A. Benjamin, Inc., New York, 1968).

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[CrossRef]

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964);G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

Acta Phys. Polon. (1)

S. Kielich, Acta Phys. Polon. 30, 393 (1966);R. Wallace, Mol. Phys. 11, 457 (1966);and some of the references quoted therein.
[CrossRef]

Ann. Physik (1)

M. Goeppert–Mayer, Ann. Physik 9, 273 (1931).
[CrossRef]

Bell System Tech. J. (1)

S. O. Rice, Bell System Tech. J. 24, 46 (1945).
[CrossRef]

Phys. Letters (1)

G. Bedard, Phys. Letters 24A, 613 (1967).

Phys. Rev. (5)

See Ref. 9, Appendix B.Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).
[CrossRef]

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[CrossRef]

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964);S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965);E. M. Logothetics and P. L. Hartman,  18, 581 (1967);and some of the references quoted in these papers.
[CrossRef]

B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.
[CrossRef]

G. S. Agarwal, Phys. Rev. 177, 400 (1969).
[CrossRef]

Phys. Rev. Letters (5)

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965);H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
[CrossRef]

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964);G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
[CrossRef]

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966);Phys. Rev. 151, 1076 (1966);See also R. Wallace, Phys. Rev. Letters 17, 397 (1966);Mol. Phys. 11, 457 (1966).
[CrossRef]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).A detailed discussion of the diagonal representation is given in Ref. 11.See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169;R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and V. Hochuli, Phys. Letters 25A, 272 (1967).

Proc. Phys. Soc. (London) (2)

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[CrossRef]

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958);Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Sov. Phys. JETP (1)

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).

Z. Physik (1)

H. Risken, Z. Physik 186, 85 (1965).
[CrossRef]

Other (9)

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

See, for example, Ref. 8, Eq. (4.41).

L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.
[CrossRef]

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1967), p.211, and some of the references quoted in this review article.
[CrossRef]

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (W. A. Benjamin, Inc., New York, 1968).

The original derivation of Mandel was based on semiclassical arguments.The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches1964,C. Dewitt, A. Blandin, and C. Cohen–Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65.See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

The contribution of the A2 term is also easily taken into account. It may be shown that the final result is still given by Eq. (30) except that the quantum-efficiency parameter α changes slightly. The new value of α isα=2πN∑{μ}∑jkMj(μ1,μ2)Mk*(μ3,μ4)∊μ1∊μ2∊μ3*∊μ4*(ωj−ω0)(ωk−ω0)ρ(2ω0)+2πN∑{μ}(e22ℏmc2)2|〈f|0〉|2∊μ1*∊μ1*∊μ2∊μ2ρ(2ω0).Here 〈f|0〉 represents the scalar product between the ground-state wavefunction and the final-state (free-electron) wave-function.

The representation of V in the form Eq. (24) assumes that the incident light is plane polarized. However, the results are easily generalized to the case of partially polarized light. If the incident light is of thermal origin, then as well known we may consider it as the superposition of two linearly polarized, statistically independent components with average intensities equal to 12(1+P)〈I〉 and 12(1−P)〈I〉, respectively, where P is the degree of polarization. It is also assumed that the condition of cross-spectral purity is satisfied. In this case, the final result (30) remains unchanged. (See also Sec. IIIA.) For the partially polarized light of nonthermal origin, the expression for the photoelectron counting distribution may be shown to bep(n,T)=∫Φ({υks})[W({υks})]nexp{−W({υks})}n!d2({υks}),whereW({υks})=∑{μ}αμ1μ2μ3μ4∫0TdtVμ1(t)Vμ2(t)Vμ3*(t)Vμ4*(t),andαμ1μ2μ3μ4=2πN∑jkMj(μ1,μ2)Mk*(μ3,μ4)(ωj−ω0)(ωk−ω0)ρ(2ω0).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

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Equations (67)

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p ( n ; t , T ) = W n e W n ! P ( W ) d W ,
W = α t t + T I ( t ) d t ,
Ĥ I ( t ) = e m c p ̂ ( t ) · Â ( x , t ) + e 2 2 m c 2 Â 2 ( x , t ) ,
 ( x , t )  ( t ) =  ( + ) ( t ) +  ( ) ( t ) ,
A ( + ) ( t ) = k , s ( C k L 3 ) 1 2 ɛ ( k , s ) â k , s e i w k t .
P ( t ) = { f } f | f f | a f | Û I ( t , t 0 ) | 0 a | i f | 2 .
Û I ( t , t 0 ) = T ̂ exp { i t 0 t Ĥ I ( t ) d t } ,
P ( 2 ) ( t ) = { f } f | t 0 t 0 + Δ t d t 1 t 0 t 1 d t 2 μ 1 μ 2 j M j ( μ 1 , μ 2 ) × exp { i ( ω j ω f ) t 1 + i ω j t 2 } × f f | Â μ 1 ( + ) ( t 1 ) Â μ 2 ( + ) ( t 2 ) | i f | 2 .
M j ( μ 1 , μ 2 ) ( e m c ) 2 a f | p ̂ μ 1 | j a a j | p ̂ μ 2 | 0 a ,
Π ( t ) = 1 Δ t 0 P ( 2 ) ( t ) ρ ( ω f ) d ω f .
ρ ̂ i f = Φ ( { υ k , s } ) | { υ k , s } { υ k , s } | d 2 ( { υ k , s } ) .
A ( + ) ( x , t ) | { υ k , s } = V ( x , t ) | { υ k , s } ,
V ( x , t ) k , s ( c k L 3 ) 1 2 ɛ ( k , s ) υ k , s e i ω k t .
V ( x , t ) = 0 A ( x , ω ) e i ω t d ω .
P ( 2 ) ( t ) = { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 4 ) t 0 t 0 + Δ t d t 1 t 0 t 1 d t 2 t 0 t 0 + Δ t d t 1 t 0 t 1 d t 2 exp { i ( ω j ω f ) t 1 + i ω j t 2 + i ( ω k ω f ) t 1 i ω k t 2 } V μ 1 ( t 1 ) V μ 2 ( t 2 ) V μ 3 * ( t 1 ) V μ 4 * ( t 2 ) .
P ( 2 ) ( t ) = { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 4 ) 0 d ω 1 0 d ω 2 0 d ω 1 0 d ω 2 A μ 1 ( ω 1 ) A μ 2 ( ω 2 ) A μ 3 * ( ω 1 ) A μ 4 * ( ω 2 ) × t 0 t 0 + Δ t d t 1 t 0 t 1 d t 2 t 0 t 0 + Δ t d t 1 t 0 t 1 d t 2 exp { i t 1 ( ω j ω f + ω 1 ) + i t 2 ( ω j ω 2 ) + i ( ω k ω f + ω 1 ) t 1 i t 2 ( ω k ω 2 ) } .
P ( 2 ) ( t ) = { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 4 ) 0 d ω 1 0 d ω 2 0 d ω 1 0 d ω 2 A μ 1 ( ω 1 ) A μ 2 ( ω 2 ) A μ 3 * ( ω 1 ) A μ 4 * ( ω 2 ) ( t ) 2 × ( Δ t ) 2 { sin [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] sin [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] } × exp { i ( ω 1 + ω 2 ω 1 ω 2 ) ( t 0 + Δ t 2 ) } / { ( ω j ω 2 ) ( ω k ω 2 ) } .
I = 0 d ω f ρ ( ω f ) { sin [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] sin [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω f ) Δ t ] } .
( ω 1 + ω 2 ) ( ω 1 + ω 2 ) 1 Δ t 2 ω 0 .
I = 2 π Δ t { ρ ( ω 1 + ω 2 ) ρ ( ω 1 + ω 2 ) } 1 2 × sin [ 1 2 ( ω 1 + ω 2 ω 1 ω 2 ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω 1 ω 2 ) Δ t ] .
sin [ 1 2 ( ω 1 + ω 2 ω 1 ω 2 ) Δ t ] / [ 1 2 ( ω 1 + ω 2 ω 1 ω 2 ) Δ t ] = 1 Δ t exp { i ( ω 1 + ω 2 ω 1 ω 2 ) Δ t 2 } × 0 Δ t exp { i ( ω 1 + ω 2 ω 1 ω 2 ) τ } d τ ,
Π ( t ) = 2 π Δ t 0 Δ t d τ { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 4 ) 0 d ω 1 0 d ω 2 0 d ω 1 0 d ω 2 A μ 1 ( ω 1 ) A μ 2 ( ω 2 ) A μ 3 * ( ω 1 ) A μ 4 * ( ω 2 ) × { ρ ( ω 1 + ω 2 ) ρ ( ω 1 + ω 2 ) } 1 2 ( ω j ω 2 ) ( ω k ω 2 ) exp { i ( ω 1 + ω 2 ω 1 ω 2 ) ( t + τ ) } .
Π ( t ) = 2 π { { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 4 ) ρ ( 2 ω 0 ) ( ω j ω 0 ) ( ω k ω 0 ) } 1 Δ t 0 Δ t V μ 1 ( t + τ ) V μ 2 ( t + τ ) V μ 3 * ( t + τ ) V μ 4 * ( t + τ ) d τ .
V ( t ) = ɛ υ ( t ) ,
1 Δ t 0 Δ t { υ * ( t + τ ) υ ( t + τ ) } 2 d τ { υ * ( t ) υ ( t ) } .
P ( t ) Δ t = α { υ * ( t ) υ ( t ) } 2 Δ t ,
α = 2 N π { μ } j , k M j ( μ 1 , μ 2 ) M k * ( μ 3 , μ 2 ) μ 1 μ 2 μ 3 * μ 4 * ρ ( 2 ω 0 ) ( ω j ω 0 ) ( ω k ω 0 )
P ( t ) Δ t = α I 2 ( t ) Δ t .
p ( n ; t , T ) = W n e W n ! ,
W = α t t + T I 2 ( t ) d t .
p ( n ; t , T ) = ϕ ( { υ k , s } ) × [ W ( { υ k , s } ) ] n e W ( { υ k , s } ) n ! d 2 ( { υ k , s } ) ,
p ( n ; t , T ) = 0 W n e W n ! P ( W ) d W ,
P ( W ) = ϕ ( { υ k , s } ) δ ( W α t t + T I 2 ( t ) d t ) × d 2 ( { υ k , s } ) .
W = α α t + T I 2 ( t ) d t α T I 2 ( t ) .
p ( n , T ) = ( α I 2 T ) n n ! e α I 2 T P ( I ) d I .
n [ k ] = n n ( n 1 ) ( n k + 1 ) p ( n ; T ) = ( α T ) k I 2 k .
P ( I ) = 1 I e I / I .
p ( n , T ) = 2 n ! n ! 2 n e 1 2 n n 1 2 D ( 2 n + 1 ) ( n 1 2 ) .
D p ( Z ) = 1 Γ ( p ) 0 x p 1 exp { x 2 2 Z x Z 2 4 } d x , Re p < 0 .
P ( I ) = 1 P I { exp { 2 I I ( 1 + P ) } exp { 2 I I ( 1 P ) } } ,
p ( n , T ) = 2 n ! n ! 2 n ( 4 n P 2 3 + P 2 ) 1 2 { D ( 2 n + 1 ) [ 1 ( 1 + P ) ( 3 + P 2 n ) 1 2 ] × exp { 3 + P 2 4 n ( 1 + P ) 2 } D ( 2 n + 1 ) [ 1 ( 1 P ) ( 3 + P 2 n ) 1 2 ] × exp { 3 + P 2 4 n ( 1 P ) 2 } } ,
n = α T I 2 ( 3 + P 2 2 ) .
P ( I ) = δ ( I I ) .
p ( n , T ) = n n e n n ! ,
n = α T I 2 .
P ( I ) = 2 π I 0 [ e ω 2 1 + err ( ω ) ] exp { I 2 π I 0 2 + 2 ω I π I 0 } , I 0 .
p ( n , T ) = 2 n ! n ! 2 n 2 π σ 0 n ( 1 + σ 0 ) n + 1 2 × exp { ω 2 2 ( 1 + 2 σ 0 1 + σ 0 ) } / [ 1 + err ( ω ) ] . × D ( 2 n + 1 ) [ ω / ( 1 + σ 0 2 ) 1 2 ] ,
σ 0 = π n 0 = π α T I 0 2 .
n [ k ] = 2 k ! σ 0 k 2 k 2 π { exp ( ω 2 2 ) / 1 + err ( ω ) } · × D ( 2 k + 1 ) ( ω 2 ) .
P ( W ) = a 2 k 2 Γ ( 2 k ) W k 1 e a W 1 2 ,
W = α T I 2 ,
( Δ W ) 2 = α 2 0 T 0 T d t 1 d t 2 { I 2 ( t 1 ) I 2 ( t 2 ) I 2 ( t 1 ) I 2 ( t 2 ) } .
I 2 = 2 I 2
I 2 ( t 1 ) I 2 ( t 2 ) I 2 ( t 1 ) I 2 ( t 2 ) = I 4 { 16 | γ ( t 1 t 2 ) | 2 + 4 | γ ( t 1 t 2 ) | 4 } ,
( Δ W ) 2 = 4 α 2 T 2 ξ ( T ) T I 4 ,
ξ ( T ) = 1 2 0 T ( 1 τ T ) { 16 | γ ( τ ) | 2 + 4 | γ ( τ ) | 4 } d τ .
k = 1 4 [ ( 4 T ξ ( T ) 1 ) + { 16 ( T ξ ( T ) ) 2 + 16 T ξ ( T ) + 1 } 1 2 ]
a = { 2 k ( 2 k + 1 ) W } 1 2 .
k = 1 2 and a = ( 2 W ) 1 2 .
P ( W ) = ( 2 W W ) 1 2 e ( 2 W / W ) 1 2 ,
p ( n ; T ) = a 2 k Γ ( 2 n + 2 k ) 2 n + k n ! Γ ( 2 k ) e a 2 / 8 D ( 2 n + 2 k ) ( a / 2 ) .
n = W
( Δ n ) 2 = n { 1 + n ξ ( T ) T } ,
α=2πN{μ}jkMj(μ1,μ2)Mk*(μ3,μ4)μ1μ2μ3*μ4*(ωjω0)(ωkω0)ρ(2ω0)+2πN{μ}(e22mc2)2|f|0|2μ1*μ1*μ2μ2ρ(2ω0).
p(n,T)=Φ({υks})[W({υks})]nexp{W({υks})}n!d2({υks}),
W({υks})={μ}αμ1μ2μ3μ40TdtVμ1(t)Vμ2(t)Vμ3*(t)Vμ4*(t),
αμ1μ2μ3μ4=2πNjkMj(μ1,μ2)Mk*(μ3,μ4)(ωjω0)(ωkω0)ρ(2ω0).