Abstract

This paper reviews our current understanding of the dynamics of a laser-cooled trapped particle in the Lamb–Dicke regime, where the quantum structure of the energy levels cannot be ignored. The derivation and validity of a master equation are surveyed, and its physical interpretation is discussed in some detail. The structure and physical nature of the ultimate steady state are discussed. Using a generating function method, we can solve for both the complete eigenvalue spectrum and the general time-dependent solution of the master equation. These results are derived and interpreted physically. They have earlier been scattered in our various publications and are presented here in a coherent way for the first time. Also included are some new results and a physical discussion of the situation. The paper concludes with a discussion of the validity and limitations of the model as treated so far.

© 1985 Optical Society of America

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  1. D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
    [CrossRef]
  2. W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
    [CrossRef]
  3. P. E. Toschek, W. Neuhauser, in Atomic Physics, D. Kleppner, F. M. Pipkin, eds. (Plenum, New York, 1981), Vol. 7, p. 529.
  4. H. G. Dehmelt, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 353.
    [CrossRef]
  5. W. M. Itano, D. J. Wineland, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 360.
    [CrossRef]
  6. J. Javanainen, S. Stenholm, Appl. Phys. 21, 283 (1980).
    [CrossRef]
  7. J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [CrossRef]
  8. S. Stenholm, Phys. Rep. 43, 152 (1978).
    [CrossRef]
  9. V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 2 (1981).
    [CrossRef]
  10. J. Javanainen, S. Stenholm, Appl. Phys. 24, 71 (1981).
    [CrossRef]
  11. J. Javanainen, S. Stenholm, Appl. Phys. 24, 151 (1981).
    [CrossRef]
  12. J. Javanainen, J. Phys. B 14, 2519 (1981).
    [CrossRef]
  13. J. Javanainen, J. Phys. B 14, 4191 (1981).
    [CrossRef]
  14. J. Javanainen, J. Phys. B 18, 1549 (1985).
    [CrossRef]
  15. M. Lindberg, J. Phys. B 17, 2129 (1984).
    [CrossRef]
  16. M. Lindberg, S. Stenholm, J. Phys. B 17, 3375 (1984).
    [CrossRef]
  17. J. Javanainen, M. Lindberg, S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984).
    [CrossRef]
  18. D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
    [CrossRef]
  19. W. M. Itano, D. J. Wineland, Phys. Rev. A 25, 35 (1982).
    [CrossRef]
  20. J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
    [CrossRef]
  21. A. Yu. Pusep, Sov. Phys. JETP 43, 441 (1976).
  22. J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
    [CrossRef]
  23. L. Mandel, J. Opt. (Paris) 10, 51 (1979).
    [CrossRef]
  24. M. O. Scully, W. E. Lamb, Phys. Rev. 159, 208 (1967).
    [CrossRef]
  25. S. Stenholm, Phys. Rev. A 27, 2513 (1983).
    [CrossRef]
  26. J. Javanainen, Appl. Phys. 23, 175 (1980).
    [CrossRef]
  27. J. Javanainen, Opt. Commun. 34, 375 (1980).
    [CrossRef]
  28. R. J. Cook, Opt. Commun. 35, 437 (1980).
    [CrossRef]

1985

J. Javanainen, J. Phys. B 18, 1549 (1985).
[CrossRef]

1984

M. Lindberg, J. Phys. B 17, 2129 (1984).
[CrossRef]

M. Lindberg, S. Stenholm, J. Phys. B 17, 3375 (1984).
[CrossRef]

J. Javanainen, M. Lindberg, S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984).
[CrossRef]

1983

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

1982

W. M. Itano, D. J. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

1981

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 2 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 71 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 151 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 14, 2519 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 14, 4191 (1981).
[CrossRef]

1980

J. Javanainen, S. Stenholm, Appl. Phys. 21, 283 (1980).
[CrossRef]

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

J. Javanainen, Appl. Phys. 23, 175 (1980).
[CrossRef]

J. Javanainen, Opt. Commun. 34, 375 (1980).
[CrossRef]

R. J. Cook, Opt. Commun. 35, 437 (1980).
[CrossRef]

1979

L. Mandel, J. Opt. (Paris) 10, 51 (1979).
[CrossRef]

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

1978

S. Stenholm, Phys. Rep. 43, 152 (1978).
[CrossRef]

D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

1976

A. Yu. Pusep, Sov. Phys. JETP 43, 441 (1976).

1967

M. O. Scully, W. E. Lamb, Phys. Rev. 159, 208 (1967).
[CrossRef]

Ashkin, A.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

Cook, R. J.

R. J. Cook, Opt. Commun. 35, 437 (1980).
[CrossRef]

Dehmelt, H. G.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

H. G. Dehmelt, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 353.
[CrossRef]

Drullinger, R. E.

D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

Gordon, J. P.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

Hohenstatt, M.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Itano, W. M.

W. M. Itano, D. J. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

W. M. Itano, D. J. Wineland, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 360.
[CrossRef]

Javanainen, J.

J. Javanainen, J. Phys. B 18, 1549 (1985).
[CrossRef]

J. Javanainen, M. Lindberg, S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 71 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 151 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 14, 2519 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 14, 4191 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 283 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
[CrossRef]

J. Javanainen, Appl. Phys. 23, 175 (1980).
[CrossRef]

J. Javanainen, Opt. Commun. 34, 375 (1980).
[CrossRef]

Lamb, W. E.

M. O. Scully, W. E. Lamb, Phys. Rev. 159, 208 (1967).
[CrossRef]

Letokhov, V. S.

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 2 (1981).
[CrossRef]

Lindberg, M.

J. Javanainen, M. Lindberg, S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984).
[CrossRef]

M. Lindberg, J. Phys. B 17, 2129 (1984).
[CrossRef]

M. Lindberg, S. Stenholm, J. Phys. B 17, 3375 (1984).
[CrossRef]

Mandel, L.

L. Mandel, J. Opt. (Paris) 10, 51 (1979).
[CrossRef]

Minogin, V. G.

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 2 (1981).
[CrossRef]

Neuhauser, W.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

P. E. Toschek, W. Neuhauser, in Atomic Physics, D. Kleppner, F. M. Pipkin, eds. (Plenum, New York, 1981), Vol. 7, p. 529.

Pusep, A. Yu.

A. Yu. Pusep, Sov. Phys. JETP 43, 441 (1976).

Scully, M. O.

M. O. Scully, W. E. Lamb, Phys. Rev. 159, 208 (1967).
[CrossRef]

Stenholm, S.

M. Lindberg, S. Stenholm, J. Phys. B 17, 3375 (1984).
[CrossRef]

J. Javanainen, M. Lindberg, S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984).
[CrossRef]

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 71 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 151 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 283 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
[CrossRef]

S. Stenholm, Phys. Rep. 43, 152 (1978).
[CrossRef]

Toschek, P.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Toschek, P. E.

P. E. Toschek, W. Neuhauser, in Atomic Physics, D. Kleppner, F. M. Pipkin, eds. (Plenum, New York, 1981), Vol. 7, p. 529.

Walls, F. L.

D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

Wineland, D. J.

W. M. Itano, D. J. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

W. M. Itano, D. J. Wineland, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 360.
[CrossRef]

Appl. Phys.

J. Javanainen, S. Stenholm, Appl. Phys. 21, 283 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 71 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 24, 151 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

J. Javanainen, Appl. Phys. 23, 175 (1980).
[CrossRef]

J. Opt. (Paris)

L. Mandel, J. Opt. (Paris) 10, 51 (1979).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

J. Javanainen, J. Phys. B 14, 2519 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 14, 4191 (1981).
[CrossRef]

J. Javanainen, J. Phys. B 18, 1549 (1985).
[CrossRef]

M. Lindberg, J. Phys. B 17, 2129 (1984).
[CrossRef]

M. Lindberg, S. Stenholm, J. Phys. B 17, 3375 (1984).
[CrossRef]

Opt. Commun.

J. Javanainen, Opt. Commun. 34, 375 (1980).
[CrossRef]

R. J. Cook, Opt. Commun. 35, 437 (1980).
[CrossRef]

Phys. Rep.

S. Stenholm, Phys. Rep. 43, 152 (1978).
[CrossRef]

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 2 (1981).
[CrossRef]

Phys. Rev.

M. O. Scully, W. E. Lamb, Phys. Rev. 159, 208 (1967).
[CrossRef]

Phys. Rev. A

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

W. M. Itano, D. J. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

Phys. Rev. Lett.

D. J. Wineland, R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. Toschek, H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Sov. Phys. JETP

A. Yu. Pusep, Sov. Phys. JETP 43, 441 (1976).

Other

P. E. Toschek, W. Neuhauser, in Atomic Physics, D. Kleppner, F. M. Pipkin, eds. (Plenum, New York, 1981), Vol. 7, p. 529.

H. G. Dehmelt, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 353.
[CrossRef]

W. M. Itano, D. J. Wineland, in Laser Spectroscopy V, A. R. W. McKellar, T. Oka, B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 360.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

To lowest order in the laser intensity we have only two types of processes transferring population from the level n to levels n ± 1. (a) This is a direct induced process followed by spontaneous decaying. The process acts as a diffusive spreading. (b) This process contains an induced process utilizing the vibrational quanta from the trap motion followed by spontaneous emission. Depending on the detuning Δ this is either a heating (nn + 1) or a cooling (nn − 1) process.

Fig. 2
Fig. 2

The rates A± are the quantum rates for harmonic oscillator transitions up or down the energy-level ladder.

Fig. 3
Fig. 3

The time evolution of an initial Planck-type distribution relaxing toward a final one without changing its form. The initial average excitation energy <n> = 10, and the final one is 1.5.

Fig. 4
Fig. 4

The time evolution of an initial Poisson distribution with the initial excitation level <n> = 10 toward the final value 1.5 as in Fig. 3.

Fig. 5
Fig. 5

Relaxing from the initial pure oscillator state with N = 10, the oscillator cools through stages such as the Poissonian case of Fig. 4 toward the same final state with <n> = 1.5 as in Figs. 3 and 4.

Equations (81)

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η 2 = q 2 / 2 M ν = 2 π 2 ( a 0 λ ) 2 ,
γ 1 2 Γ .
P ( Δ ) 2 π ( μ E / 2 ) 2 δ ( Δ ) Γ I γ 2 Δ 2 + γ 2 ,
I = 1 2 ( μ 2 E 2 2 γ Γ ) .
d d t n ρ n + k = i k ν n ρ n + k + η 2 { [ ( n + 1 ) ( n + 1 + k ) ] 1 / 2 A - n + 1 ρ n + 1 + k - ( n + 1 + 1 2 k ) A + n ρ n + k - ( n + 1 2 k ) A - n ρ n + k + [ n ( n + k ) ] 1 / 2 A + n - 1 ρ n - 1 + k } .
A - ( ν ) = A + ( - ν ) = Γ P ( Δ ) Γ + 2 P ( Δ ) ( α - 1 ) - γ Γ I Γ + 2 P ( Δ ) Im { 1 1 + 2 γ Γ I Δ 2 - ( ν + i γ ) 2 × [ Γ ( 1 Δ + i γ - 1 Δ - ν - i γ ) - 2 P ( Δ ) ν ( ν + i Γ ) ( ν + i γ ) Δ 2 - ( ν + i γ ) 2 ] } .
A ( ν ) = α P ( Δ ) + P ( Δ ν ) .
A - > A + ,
γ ν = Δ
Γ I ˜ = P ( ν ) ( γ / ν ) 2 I Γ 4 Γ P ( Δ + ν ) .
A + = 1 4 I ˜ Γ + α I ˜ Γ ,
A - = I ˜ Γ + α I ˜ Γ .
W = η 2 ( A - - A + ) = η 2 Γ I + O ( I ˜ ) ,
s A + A - = ( γ ν ) 2 ( 1 4 + α ) + 1.
P ( n ) = n ρ n
P ( n ) = η 2 { ( n + 1 ) A - P ( n + 1 ) - [ ( n + 1 ) A + + n A - ] P ( n ) + n A + P ( n - 1 ) } .
A - P ( n + 1 ) = A + P ( n ) .
P 0 ( n ) = ( 1 - s ) s n ,
E fin = ν ( n + 1 2 ) = ν n = 0 P ( n ) ( n + 1 2 ) = ν ( s 1 - s + 1 2 ) .
s = exp ( - ν / k B T f ) ,
E fin = ν { P ( Δ + ν ) + α P ( Δ ) P ( Δ - ν ) - P ( Δ + ν ) + 1 2 } ,
γ ν ,
P ( Δ P ( Δ + ν ) Γ I γ 2 γ 2 + Δ 2 ,
P ( Δ - ν ) - P ( Δ + ν ) = 4 Γ I γ 2 Δ ν ( γ 2 + Δ 2 ) 2 ,
E fin = γ ( 1 + α 4 ) ( γ Δ + Δ γ ) ;
E fin = 1 2 γ ( 1 + α ) .
E f ν ( s + 1 2 ) = ν [ ( γ ν ) 2 ( α + 1 4 ) + 1 2 ] .
E ex = ( α + 1 4 ) ( γ ν ) γ γ ,
E 0 = 1 2 ν γ
k B T f = ν log ( s - 1 )
k B T f ν 2 1 log ( ν / γ ) > γ 2 .
E f = ν γ Δ φ ( ν , γ , Δ , I ) ,
θ ( n , k ) = e - i v k t [ ( n + 1 ) ! n ! ] 1 / 2 n ρ n + k ,
d d t θ ( n , k ) = η 2 { ( n + 1 ) A - θ ( n + 1 , k ) - ( n + 1 + 1 2 k ) A + θ ( n , k ) - ( n + 1 2 k ) A - θ ( n , k ) + ( n + k ) A + θ ( n - 1 , k ) } ,
G k ( z , t ) = n = 0 z n θ ( n , k ) ,
η 2 A - { ( 1 - s z ) ( 1 - z ) G k z - [ k 2 ( 1 - s ) + s ( 1 + k ) ( 1 - z ) ] } G k = G k t ,
s = A + A - < 1.
n P ¯ = n = 0 n P P ( n ) = [ ( z d d z ) P G 0 ( z , t ) ] z = 1 .
G 0 ( 1 , t ) = 1.
n ρ n + k e i k ν t ,
G k ( z , t ) = exp ( - n 2 A - t ) g k ( z )
( 1 - s z ) ( 1 - z ) g k z - [ k 2 ( 1 - s ) + s ( 1 + k ) ( 1 - z ) - ] g k = 0
g k ( z ) = C ( 1 - s z ) - ( k + 1 + σ ) ( 1 - z ) σ ,
σ = - 1 2 k + 1 - s .
= ( 1 - s ) ( 1 2 k + N ) .
η 2 A - ( 1 - s ) = W ,
n ρ n + k exp ( i k ν t - 1 2 W k t ) +
P ( n ) ~ P 0 ( n ) + O ( e - W t ) ,
g t ( z ) = C 1 - s z ,
C = ( 1 - s ) ,
η 2 A - d t = d z ( 1 - z ) ( s z - 1 ) = d G k [ 1 2 k ( s - 1 ) + s ( 1 + k ) ( z - 1 ) ] G k .
C 1 = 1 - z s z - 1 e - W t ,
C 2 = G k ( 1 - z ) 1 2 k ( s z - 1 ) 1 + 1 2 k .
G k ( z , t ) = 1 ( 1 - z ) 1 2 k ( s z - 1 ) 1 + 1 2 k Φ [ ( 1 - z s z - 1 ) e - W t ] .
Φ ( x ) = x 1 2 k N = 0 C N x N ,
G k ( z , t ) = e - W k t / 2 ( s z - 1 ) 1 + k N = 0 C N ( 1 - z s z - 1 ) N e - N W t .
n ( t ) ¯ = - s ( s - 1 ) 2 Φ [ 0 ] - 1 ( s - 1 ) 2 Φ [ 0 ] e - W t .
n ( ) ¯ = s ( 1 - s ) ( s - 1 ) Φ ( 0 ) ,
Φ ( 0 ) = ( s - 1 )
n ¯ ( ) - n ¯ ( 0 ) = 1 ( s - 1 ) 2 Φ ( 0 ) ,
n ¯ ( t ) = n ¯ ( 0 ) e - W t + n ¯ ( ) ( 1 - e - W t ) .
s ( 0 ) ( A + / A - ) = s .
G 0 ( z , 0 ) = 1 - s ( 0 ) 1 - z s ( 0 ) .
1 - s ( 0 ) 1 - z s ( 0 ) = 1 s z - 1 Φ ( 1 - z s z - 1 ) ,
Φ I ( y ) = [ 1 - s ( 0 ) ] ( s - 1 ) [ s - s ( 0 ) ] y + [ 1 - s ( 0 ) ] .
G 0 ( z , t ) = 1 - s ( t ) 1 - s ( t ) z ,
s ( t ) = [ s - s ( 0 ) ] e - W t + [ s ( 0 ) - 1 ] s [ s - s ( 0 ) ] e - W t + [ s ( 0 ) - 1 ] .
P n = [ 1 - s ( t ) ] [ s ( t ) ] N
P n ( 0 ) = e - N n ! N n ,
G 0 ( z , 0 ) = e - N n = 0 N n z n n ! = e N ( z - 1 ) .
Φ II ( y ) = s - 1 s y + 1 exp [ N ( 1 - s ) 1 + s y y ] .
G 0 ( z , t ) = ( 1 - s ) ( 1 - s e - W t ) exp [ N ( s - 1 ) e - W t 1 - s e - W t ] exp ( - ρ x 1 - x ) 1 - x ,
x = s ( 1 - e - W t ) 1 - s e - W t z
ρ = N ( 1 - s ) 2 e - W t s ( s e - W t - 1 ) ( 1 - e - W t ) .
P ( n , t ) = ( 1 - s ) ( 1 - s e - W t ) exp [ N ( s - 1 ) e - W t 1 - s e - W t ] × 1 n ! L n [ N ( 1 - s 2 ) e - W t s ( s e - W t - 1 ) ( 1 - e - W t ) ] [ s ( 1 - e - W t ) ( 1 - s e - W t ) ] n .
lim L n t 0 [ N ( 1 - s ) 2 e - W t s ( s e - W t - 1 ) ( 1 - e - W t ) ] , [ s ( 1 - e - W t ) 1 - s e - W t ] n = N n ,
G 0 ( z , 0 ) = z N ,
Φ I I I ( y ) = ( s - 1 ) ( 1 + y ) N ( s y + 1 ) N + 1 ,
G 0 ( z , t ) = ( s - 1 ) [ ( e - W t - 1 ) + ( s - e - W t ) z ] N [ ( s e - W t - 1 ) + s ( 1 - e - W t ) z ] N + 1 = ( s - 1 ) l = 0 N k = 0 ( N l ) ( - N - 1 k ) z k z l ( e - W t - 1 ) N - l × ( s - e - W t ) l ( s e - W t - 1 ) - N - 1 - k ( s - s e - W t ) k .
P n N ( t ) = ( 1 - s ) k = Max { 0 , n - N } n ( - 1 ) k ( N n - k ) ( - N - 1 k ) × s k ( 1 - e - W t ) N - n + 2 k ( e - W t - s ) n - k ( 1 - s e - W t ) - N - k - 1 .
P n ( t ) = N = 0 P n N ( t ) P 0 ( N ) .

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