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, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  2. F. T. Hioe, C. E. Carroll, Phys. Rev. A 32, 1541 (1985).
    [CrossRef] [PubMed]
  3. J. Zakrzewski, M. Lewenstein, R. Kuklinski, J. Phys. B 18, 4631 (1985).
    [CrossRef]
  4. C. E. Carroll, 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, G. Orriols, Nuovo Cimento Lett. 17, 333 (1976).
    [CrossRef]
  7. R. G. Brewer, E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
    [CrossRef]
  8. H. R. Gray, R. M. Whitley, C. R. Stroud, Opt. Lett. 3, 218 (1978).
    [CrossRef] [PubMed]
  9. J. D. Stettler, C. M. Bowden, N. M. Witriol, J. H. Eberly, Phys. Lett. 73A, 171 (1979).
  10. R. J. Cook, B. W. Shore, Phys. Rev. A 20, 539 (1979).
    [CrossRef]
  11. J. H. Eberly, B. W. Shore, Z. Bialynicka-Birula, I. Bialynicka-Birula, Phys. Rev. A 16, 2038 (1977).
    [CrossRef]
  12. F. T. Hioe, 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, 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, F. T. Hioe, Phys. Rev. A 36, 724 (1987).
    [CrossRef] [PubMed]
  20. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, p. 56.
  21. Ref. 20, p. 182.
  22. N. Rosen, C. Zener, Phys. Rev. 40, 502 (1932).
    [CrossRef]

1987

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, F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[CrossRef] [PubMed]

1986

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

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

1985

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

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

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

1984

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

1983

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

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

1981

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

1979

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

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

1978

1977

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

1976

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

1975

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

1932

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

Allen, L.

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

Arimondo, E.

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

Barnett, S. M.

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

Bialynicka-Birula, I.

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

Bialynicka-Birula, Z.

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

Bowden, C. M.

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

Brewer, R. G.

R. G. Brewer, 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, D. T. Pegg, Opt. Acta 33, 1129 (1986).
[CrossRef]

Carroll, C. E.

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

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

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

Cook, R. J.

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

Eberly, J. H.

F. T. Hioe, 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, J. H. Eberly, Phys. Lett. 73A, 171 (1979).

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

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

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, 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, E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Hioe, F. T.

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, F. T. Hioe, Phys. Rev. A 36, 724 (1987).
[CrossRef] [PubMed]

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

F. T. Hioe, 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, 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, D. T. Pegg, Opt. Acta 33, 1129 (1986).
[CrossRef]

Kuklinski, R.

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

Lauder, M. A.

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

Lewenstein, M.

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

Magnus, W.

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

Oberhettinger, F.

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

Orriols, G.

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

Pegg, D. T.

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

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

Rosen, N.

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

Shore, B. W.

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

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

Stettler, J. D.

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

Stroud, C. R.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, 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, J. H. Eberly, Phys. Lett. 73A, 171 (1979).

Zakrzewski, J.

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

Zener, C.

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

J. Opt. Soc. Am B

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

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

J. Phys. B

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

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

Nuovo Cimento Lett.

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

Opt. Acta

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

Opt. Lett.

Phys. Lett.

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

Phys. Rev.

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

Phys. Rev. A

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

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

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

F. T. Hioe, 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, 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.

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

Other

See, e.g., L. Allen, 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, 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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