Abstract

New analytic solutions to the problem of a three-state system driven simultaneously by resonant optical pulses of different shapes are presented. The solutions are useful for prescribing the conditions for complete population transfer from one state to another or for complete population return.

© 1988 Optical Society of America

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References

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  1. See, e.g., L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  2. F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
    [Crossref] [PubMed]
  3. J. Zakrzewski, M. Lewenstein, and R. Kuklinski, J. Phys. B 18, 4631 (1985).
    [Crossref]
  4. C. E. Carroll and F. T. Hioe, J. Phys. A 19, 3579 (1986).
    [Crossref]
  5. References 1–4 contain many other references to the two-state model.
  6. E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
    [Crossref]
  7. R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
    [Crossref]
  8. H. R. Gray, R. M. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
    [Crossref] [PubMed]
  9. J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).
  10. R. J. Cook and B. W. Shore, Phys. Rev. A 20, 539 (1979).
    [Crossref]
  11. J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
    [Crossref]
  12. F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 (1981);Phys. Rev. A 25, 2168 (1982).
    [Crossref]
  13. H. P. W. Gottlieb, Phys. Rev. A 26, 3713 (1982);Phys. Rev. A 32, 653 (1985).
    [Crossref]
  14. F. T. Hioe, Phys. Rev. A 29, 3434 (1984).
    [Crossref]
  15. D. T. Pegg, J. Phys. B 18, 415 (1985).
    [Crossref]
  16. F. T. Hioe, Phys. Rev. A 28, 879 (1983);Phys. Rev. A 30, 3097 (1984);Phys. Rev. A 32, 2824 (1985);J. Opt. Soc. Am. B 5, 859 (1988).
    [Crossref]
  17. S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
    [Crossref]
  18. F. T. Hioe, J. Opt. Soc. Am B 4, 1327 (1987).The (3, 1) element for j= 1 in Eq. (16) of this reference should be +b*2.
    [Crossref]
  19. C. E. Carroll and F. T. Hioe, Phys. Rev. A 36, 724 (1987).
    [Crossref] [PubMed]
  20. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.
  21. Ref. 20, p. 182.
  22. N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).
    [Crossref]

1987 (2)

F. T. Hioe, J. Opt. Soc. Am B 4, 1327 (1987).The (3, 1) element for j= 1 in Eq. (16) of this reference should be +b*2.
[Crossref]

C. E. Carroll and F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[Crossref] [PubMed]

1986 (2)

C. E. Carroll and F. T. Hioe, J. Phys. A 19, 3579 (1986).
[Crossref]

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

1985 (3)

D. T. Pegg, J. Phys. B 18, 415 (1985).
[Crossref]

F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
[Crossref] [PubMed]

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

1984 (1)

F. T. Hioe, Phys. Rev. A 29, 3434 (1984).
[Crossref]

1983 (1)

F. T. Hioe, Phys. Rev. A 28, 879 (1983);Phys. Rev. A 30, 3097 (1984);Phys. Rev. A 32, 2824 (1985);J. Opt. Soc. Am. B 5, 859 (1988).
[Crossref]

1982 (1)

H. P. W. Gottlieb, Phys. Rev. A 26, 3713 (1982);Phys. Rev. A 32, 653 (1985).
[Crossref]

1981 (1)

F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 (1981);Phys. Rev. A 25, 2168 (1982).
[Crossref]

1979 (2)

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

R. J. Cook and B. W. Shore, Phys. Rev. A 20, 539 (1979).
[Crossref]

1978 (1)

1977 (1)

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

1976 (1)

E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
[Crossref]

1975 (1)

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[Crossref]

1932 (1)

N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).
[Crossref]

Allen, L.

See, e.g., L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Arimondo, E.

E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
[Crossref]

Barnett, S. M.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

Bialynicka-Birula, I.

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

Bialynicka-Birula, Z.

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

Bowden, C. M.

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

Brewer, R. G.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[Crossref]

Buckle, S. J.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

Carroll, C. E.

C. E. Carroll and F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[Crossref] [PubMed]

C. E. Carroll and F. T. Hioe, J. Phys. A 19, 3579 (1986).
[Crossref]

F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
[Crossref] [PubMed]

Cook, R. J.

R. J. Cook and B. W. Shore, Phys. Rev. A 20, 539 (1979).
[Crossref]

Eberly, J. H.

F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 (1981);Phys. Rev. A 25, 2168 (1982).
[Crossref]

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

See, e.g., L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.

Gottlieb, H. P. W.

H. P. W. Gottlieb, Phys. Rev. A 26, 3713 (1982);Phys. Rev. A 32, 653 (1985).
[Crossref]

Gray, H. R.

Hahn, E. L.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[Crossref]

Hioe, F. T.

C. E. Carroll and F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[Crossref] [PubMed]

F. T. Hioe, J. Opt. Soc. Am B 4, 1327 (1987).The (3, 1) element for j= 1 in Eq. (16) of this reference should be +b*2.
[Crossref]

C. E. Carroll and F. T. Hioe, J. Phys. A 19, 3579 (1986).
[Crossref]

F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
[Crossref] [PubMed]

F. T. Hioe, Phys. Rev. A 29, 3434 (1984).
[Crossref]

F. T. Hioe, Phys. Rev. A 28, 879 (1983);Phys. Rev. A 30, 3097 (1984);Phys. Rev. A 32, 2824 (1985);J. Opt. Soc. Am. B 5, 859 (1988).
[Crossref]

F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 (1981);Phys. Rev. A 25, 2168 (1982).
[Crossref]

Knight, P. L.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

Kuklinski, R.

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

Lauder, M. A.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

Lewenstein, M.

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

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.

Orriols, G.

E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
[Crossref]

Pegg, D. T.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

D. T. Pegg, J. Phys. B 18, 415 (1985).
[Crossref]

Rosen, N.

N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).
[Crossref]

Shore, B. W.

R. J. Cook and B. W. Shore, Phys. Rev. A 20, 539 (1979).
[Crossref]

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

Stettler, J. D.

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

Stroud, C. R.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.

Whitley, R. M.

Witriol, N. M.

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

Zakrzewski, J.

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

Zener, C.

N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).
[Crossref]

J. Opt. Soc. Am B (1)

F. T. Hioe, J. Opt. Soc. Am B 4, 1327 (1987).The (3, 1) element for j= 1 in Eq. (16) of this reference should be +b*2.
[Crossref]

J. Phys. A (1)

C. E. Carroll and F. T. Hioe, J. Phys. A 19, 3579 (1986).
[Crossref]

J. Phys. B (2)

D. T. Pegg, J. Phys. B 18, 415 (1985).
[Crossref]

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

Nuovo Cimento Lett. (1)

E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
[Crossref]

Opt. Acta (1)

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
[Crossref]

Opt. Lett. (1)

Phys. Lett. (1)

J. D. Stettler, C. M. Bowden, N. M. Witriol, and J. H. Eberly, Phys. Lett. 73A, 171 (1979).

Phys. Rev. (1)

N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).
[Crossref]

Phys. Rev. A (8)

C. E. Carroll and F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[Crossref] [PubMed]

R. J. Cook and B. W. Shore, Phys. Rev. A 20, 539 (1979).
[Crossref]

J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, and I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
[Crossref]

F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
[Crossref] [PubMed]

F. T. Hioe, Phys. Rev. A 28, 879 (1983);Phys. Rev. A 30, 3097 (1984);Phys. Rev. A 32, 2824 (1985);J. Opt. Soc. Am. B 5, 859 (1988).
[Crossref]

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[Crossref]

H. P. W. Gottlieb, Phys. Rev. A 26, 3713 (1982);Phys. Rev. A 32, 653 (1985).
[Crossref]

F. T. Hioe, Phys. Rev. A 29, 3434 (1984).
[Crossref]

Phys. Rev. Lett. (1)

F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 (1981);Phys. Rev. A 25, 2168 (1982).
[Crossref]

Other (4)

See, e.g., L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

References 1–4 contain many other references to the two-state model.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.

Ref. 20, p. 182.

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Figures (3)

Fig. 1
Fig. 1

Optical pulse shapes given by Eqs. (4.2) and (4.5) with τ = 1. This implies that z(t) runs from zero to one and that Ω1(t) has a hyperbolic-secant shape. Also we use α1 = 2π and α2 = 2π which gives complete transfer of the occupation probability from state 1 to state 3 as t increases from −∞ to +∞.

Fig. 2
Fig. 2

Time-dependent occupation probabilities corresponding to the first line of Table 1; we used Eq. (4.5) with τ = 1.

Fig. 3
Fig. 3

Time-dependent occupation probabilities corresponding to the last line of Table 1; we used Eq. (4.5) with τ = 1.

Tables (1)

Tables Icon

Table 1 Cases of Complete Transfer of the Occupation Probability from State 1 to State 3a

Equations (68)

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i Ψ t = Ĥ ( t ) Ψ
Ĥ ( t ) = [ 0 α 12 ( t ) i α 13 ( t ) α 12 ( t ) 0 α 23 ( t ) i α 13 ( t ) α 23 ( t ) 0 ] ,
α 12 ( t ) = ½ Ω 1 ( t ) , α 23 ( t ) = ½ Ω 2 ( t ) , α 13 ( t ) = ½ Ω 3 ( t ) ,
Ĥ ( t ) = [ 0 ½ Ω 1 ( t ) i ½ Ω 3 ( t ) ½ Ω 1 ( t ) 0 ½ Ω 2 ( t ) i ½ Ω 3 ( t ) ½ Ω 2 ( t ) 0 ] .
Ĥ ( t ) = α 12 ( t ) Ĵ 1 + α 23 ( t ) Ĵ 2 + α 13 ( t ) Ĵ 3 ,
Ĵ 1 = [ 0 1 0 1 0 0 0 0 0 ] , Ĵ 2 = [ 0 0 0 0 0 1 0 1 0 ] , Ĵ 3 = [ 0 1 i 0 0 0 i 0 0 ] .
[ Ĵ 1 , Ĵ 2 ] = i Ĵ 3 , [ Ĵ 2 , Ĵ 3 ] = i Ĵ 1 , [ Ĵ 3 , Ĵ 1 ] = i Ĵ 2
Û = [ 1 / 2 0 1 / 2 0 1 0 i / 2 0 i / 2 ] , Û = [ 1 / 2 0 i / 2 0 1 0 1 / 2 0 i / 2 ] ,
Ĵ k = Û Ĵ k Û , k = 1 , 2 , 3 ,
Ĵ 1 = [ 0 1 0 1 0 1 0 1 0 ] , Ĵ 2 = 1 2 [ 0 i 0 i 0 i 0 i 0 ] , Ĵ 3 = [ 1 0 0 0 0 0 0 0 1 ] .
i Ψ t = Ĥ ( t ) Ψ ,
Ψ = Û Ψ
Ĥ ( t ) = Û Ĥ ( t ) Û
Ĥ ( t ) = α 12 ( t ) Ĵ 1 + α 23 ( t ) Ĵ 2 + α 13 ( t ) Ĵ 3 .
i ϕ t = ̂ ( t ) ϕ ,
̂ ( t ) = α 12 ( t ) ŝ 1 + α 23 ( t ) ŝ 2 + α 13 ( t ) ŝ 3 ,
̂ ( t ) = [ ½ Δ 0 ( t ) ½ Ω 0 ( t ) ½ Ω 0 * ( t ) ½ Δ 0 ( t ) ] ,
Ω 0 ( t ) = α 12 ( t ) + α 23 ( t ) = ½ [ Ω 1 ( t ) i Ω 2 ( t ) ]
Δ 0 ( t ) = α 13 ( t ) = ½ Ω 3 ( t ) .
ϕ 1 = ϕ 1 exp [ i 1 2 t Δ 0 ( t ) d t ] ,
ϕ 2 = ϕ 2 exp [ i 1 2 t Δ 0 ( t ) d t ] .
i ϕ t = ̂ ( t ) ϕ ,
̂ ( t ) = [ 0 ½ A ˙ ( t ) e i B ( t ) ½ A ˙ ( t ) e i B ( t ) 0 ] ,
A ˙ ( t ) = | Ω 0 | = ½ [ Ω 1 2 ( t ) + Ω 2 2 ( t ) ] 1 / 2 ,
( t ) = Ω 1 2 ( t ) Ω 1 2 ( t ) + Ω 2 2 ( t ) d d t [ Ω 2 ( t ) Ω 1 ( t ) ] + ½ Ω 3 ( t ) .
[ ϕ 1 ϕ 2 ] = [ a b b * a * ] [ ϕ 1 ( 0 ) ϕ 2 ( 0 ) ] .
Ψ ( t ) = D ̂ ( 1 ) ( a , b ) Ψ ( 0 ) ,
[ Ψ 1 Ψ 2 Ψ 3 ] = [ a 2 2 a b b 2 2 a b * | a | 2 | b | 2 2 a * b b * 2 2 a * b * a * 2 ] [ Ψ 1 ( 0 ) Ψ 2 ( 0 ) Ψ 3 ( 0 ) ] ,
Ψ ( t ) = Û D ̂ ( 1 ) ( a , b ) Û Ψ ( 0 ) ,
| Ψ 1 ( 0 ) | = 1 , Ψ 2 ( 0 ) = Ψ 3 ( 0 ) = 0 ,
Ψ 1 ( t ) = ½ [ a 2 + a * 2 + b 2 + b * 2 ] Ψ 1 ( 0 ) ,
Ψ 2 ( t ) = ( a * b a b * ) Ψ 1 ( 0 ) ,
Ψ 3 ( t ) = ½ i [ a 2 a * 2 + b 2 b * 2 ] Ψ 1 ( 0 ) .
Ω 1 ( t ) = [ f ( z ) cos θ ( z ) ] ż , Ω 2 ( t ) = [ f ( z ) sin θ ( z ) ] ż , Ω 3 ( t ) = g ( z ) ż ,
A ˙ ( t ) = ½ f ( z ) ż ,
( t ) = [ d θ ( z ) d z + 1 2 g ( z ) ] ż .
f ( z ) = 2 α π [ z ( 1 z ) ] 1 / 2 , g ( z ) = 2 γ π z ( 1 z ) ,
θ ( z ) = θ 0 β π ln ( 1 z ) ,
A ˙ ( t ) = α π 1 [ z ( 1 z ) ] 1 / 2 ż
( t ) = β z + γ π z ( 1 z ) ż .
a = F 2 1 [ R i β / ( 2 π ) , R i β / ( 2 π ) ; ½ + i γ / π ; z ] ,
b = α ( 2 γ + i π ) z [ 1 / 2 ( i γ / π ) ] F 2 1 [ 1 2 + R i ( β + 2 γ ) 2 π , 1 2 R i ( β + 2 γ ) 2 π ; 3 2 i γ π ; z ] ,
R = ( 2 π ) 1 ( α 2 β 2 ) 1 / 2 .
f ( z ) = 2 α π [ z ( 1 z ) ] 1 / 2 , g ( z ) = 2 β π ( 1 z ) ,
θ ( z ) = θ 0 2 γ π tanh 1 ( 1 2 z ) ,
f ( z ) = 2 α π ( z 2 + 1 ) , g ( z ) = 2 γ z π ( z 2 + 1 ) ,
θ ( z ) = θ 0 + β π tan 1 z ,
A ˙ ( t ) = α π 1 z 2 + 1 ż
( t ) = β + γ z π ( z 2 + 1 ) ż .
a ( + ) = ( cos r + i β 2 r sin r ) exp ( i β / 2 ) ,
b ( + ) = i β 2 r sin r ,
r = ½ ( α 2 + β 2 ) 1 / 2 .
| Ψ 1 ( + ) | 2 = [ ( cos 2 r β 2 4 r 2 sin 2 r ) cos β + β 2 r sin ( 2 r ) sin β α 2 4 r 2 sin 2 r ] 2 ,
| Ψ 2 ( + ) | 2 = ( α r sin r ) 2 ( cos r cos 1 2 β + β 2 r sin r sin 1 2 β ) 2 ,
| Ψ 3 ( + ) | 2 = [ ( cos 2 r β 2 4 r 2 sin 2 r ) sin β β 2 r sin ( 2 r ) cos β ] 2 ,
1 + a 1 ! β δ γ z + α ( α + 1 ) 2 ! β ( β + 1 ) δ ( δ + 1 ) γ ( γ + 1 ) ( + 1 ) z 2 + .
Ω 1 ( t ) = α 1 π [ z ( 1 z ) ] 1 / 2 ż , Ω 2 ( t ) = α 2 π z ( 1 z ) 1 / 2 ż , Ω 3 ( t ) = 0 ,
Ψ 1 = 3 F 2 ( 1 2 , α 1 2 π , α 1 2 π ; 1 2 + i α 2 2 π , 1 2 i α 2 2 π ; z ) ,
Ψ 2 = i α 1 / π 1 + α 2 2 / π 2 [ z ( 1 z ) ] 1 / 2 F 3 2 ( 3 2 , 1 + α 1 2 π , 1 α 1 2 π ; 3 2 + i α 2 2 π , 3 2 i α 2 2 π ; z ) ,
Ψ 3 = α 1 α 2 / π 2 1 + α 2 2 / π 2 z 1 / 2 F 3 2 ( 1 2 , 1 + α 1 2 π , 1 α 1 2 π ; 3 2 + i α 2 2 π , 3 2 i α 2 2 π ; z ) .
F 3 2 ( 1 2 , α 1 2 π , α 1 2 π ; 1 2 + i α 2 2 π , 1 2 i α 2 2 π ; 1 ) = 0 .
Ω 1 ( t ) d t ,
z ( t ) = ½ [ 1 + tanh ( π t / τ ) ] ,
Ω 1 ( t ) = [ f ( z ) cos θ ( z ) ] ż , Ω 2 ( t ) = [ f ( z ) sin θ ( z ) ] ż , Ω 3 ( t ) = 0 ,
f ( z ) = α π z ( 1 z ) 1 / 2 , θ ( z ) = β π sin 1 ( 1 z ) + π β 2 ,
Ψ 1 = ( sin θ ) 3 F 2 ( 1 2 , β π , β π ; 1 2 + i α 2 π , 1 2 i α 2 π ; z ) + 2 β cos θ π ( 1 + α 2 / π 2 ) [ z ( 1 z ) ] 1 / 2 F 3 2 ( 3 2 , 1 + β π , 1 β π ; 3 2 + i α 2 π , 3 2 i α 2 π ; z ) ,
Ψ 2 = 2 i α β / π 2 1 + α 2 / π 2 z 1 / 2 F 3 2 ( 1 2 , 1 + β π , 1 β π ; 3 2 + i α 2 π , 3 2 i α 2 π ; z ) ,
Ψ 3 = 2 β sin θ π ( 1 + α 2 / π 2 ) [ z ( 1 z ) ] 1 / 2 F 3 2 ( 3 2 , 1 + β π , 1 β π ; 3 2 + i α 2 π , 3 2 i α 2 π ; z ) ( cos θ ) F 3 2 ( 1 2 , β π , β π ; 1 2 + i α 2 π , 1 2 i α 2 π ; z ) .

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