Abstract

An atomic model is discussed that contains one structured continuum interacting with an electromagnetic field of quantum nature. It is assumed that the field is initially in a squeezed state. Three kinds of squeezed state are discussed: (1) the squeezed vacuum state, (2) the squeezed state with a strong coherent component, and (3) the Holstein–Primakoff SU(1, 1) coherent state. A derivation is given of a fully analytical formula for the probability of finding the system in the continuum in a long-time regime. This result is obtained by a nonperturbative method and hence is valid for arbitrary field strengths. The result is discussed for various values of the parameters describing the electromagnetic field and its interaction with the atomic model and exhibits a strong dependence of the probability on the type and the squeezing parameters of the squeezed state.

© 1993 Optical Society of America

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References

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  1. R. Laudon, Rep. Prog. Phys. 43, 913 (1980).
    [CrossRef]
  2. D. F. Walls, Nature (London) 306, 141 (1983).
    [CrossRef]
  3. R. Laudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
    [CrossRef]
  4. A. Ekert and K. Rza̧żewski, Opt. Commun. 65, 225 (1988).
    [CrossRef]
  5. C. C. Gerry, Phys. Rev. A 37, 2683 (1988).
    [CrossRef] [PubMed]
  6. K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985).
    [CrossRef]
  7. V. Buzek, Phys. Rev. A 39, 3196 (1989), and references therein.
    [CrossRef] [PubMed]
  8. A. Orłowski and K. Wódkiewicz, J. Mod. Opt. 37, 295 (1990).
    [CrossRef]
  9. U. Fano, Phys. Rev. 124, 1866 (1961).
    [CrossRef]
  10. P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
    [CrossRef]
  11. K. Rza̧żewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981).
    [CrossRef]
  12. W. Leoński, R. Tanaś, and S. Kielich, J. Opt. Soc. Am. B 4, 72 (1987).
    [CrossRef]
  13. J. Zakrzewski, J. Phys. B 17, 719 (1984).
    [CrossRef]
  14. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [CrossRef]

1990 (2)

A. Orłowski and K. Wódkiewicz, J. Mod. Opt. 37, 295 (1990).
[CrossRef]

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

1989 (1)

V. Buzek, Phys. Rev. A 39, 3196 (1989), and references therein.
[CrossRef] [PubMed]

1988 (2)

A. Ekert and K. Rza̧żewski, Opt. Commun. 65, 225 (1988).
[CrossRef]

C. C. Gerry, Phys. Rev. A 37, 2683 (1988).
[CrossRef] [PubMed]

1987 (2)

R. Laudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[CrossRef]

W. Leoński, R. Tanaś, and S. Kielich, J. Opt. Soc. Am. B 4, 72 (1987).
[CrossRef]

1985 (1)

1984 (1)

J. Zakrzewski, J. Phys. B 17, 719 (1984).
[CrossRef]

1983 (1)

D. F. Walls, Nature (London) 306, 141 (1983).
[CrossRef]

1981 (1)

K. Rza̧żewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981).
[CrossRef]

1980 (1)

R. Laudon, Rep. Prog. Phys. 43, 913 (1980).
[CrossRef]

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1961 (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Buzek, V.

V. Buzek, Phys. Rev. A 39, 3196 (1989), and references therein.
[CrossRef] [PubMed]

Dalton, B. J.

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Eberly, J. H.

K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985).
[CrossRef]

K. Rza̧żewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981).
[CrossRef]

Ekert, A.

A. Ekert and K. Rza̧żewski, Opt. Commun. 65, 225 (1988).
[CrossRef]

Fano, U.

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Gerry, C. C.

C. C. Gerry, Phys. Rev. A 37, 2683 (1988).
[CrossRef] [PubMed]

Kielich, S.

Knight, P. L.

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

R. Laudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[CrossRef]

Lauder, M. A.

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Laudon, R.

R. Laudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[CrossRef]

R. Laudon, Rep. Prog. Phys. 43, 913 (1980).
[CrossRef]

Leonski, W.

Orlowski, A.

A. Orłowski and K. Wódkiewicz, J. Mod. Opt. 37, 295 (1990).
[CrossRef]

Rza¸zewski, K.

A. Ekert and K. Rza̧żewski, Opt. Commun. 65, 225 (1988).
[CrossRef]

K. Rza̧żewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981).
[CrossRef]

Tanas, R.

Walls, D. F.

D. F. Walls, Nature (London) 306, 141 (1983).
[CrossRef]

Wódkiewicz, K.

A. Orłowski and K. Wódkiewicz, J. Mod. Opt. 37, 295 (1990).
[CrossRef]

K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985).
[CrossRef]

Yuen, H. P.

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Zakrzewski, J.

J. Zakrzewski, J. Phys. B 17, 719 (1984).
[CrossRef]

J. Mod. Opt. (2)

R. Laudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[CrossRef]

A. Orłowski and K. Wódkiewicz, J. Mod. Opt. 37, 295 (1990).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

J. Zakrzewski, J. Phys. B 17, 719 (1984).
[CrossRef]

Nature (London) (1)

D. F. Walls, Nature (London) 306, 141 (1983).
[CrossRef]

Opt. Commun. (1)

A. Ekert and K. Rza̧żewski, Opt. Commun. 65, 225 (1988).
[CrossRef]

Phys. Rep. (1)

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Phys. Rev. (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Phys. Rev. A (3)

V. Buzek, Phys. Rev. A 39, 3196 (1989), and references therein.
[CrossRef] [PubMed]

C. C. Gerry, Phys. Rev. A 37, 2683 (1988).
[CrossRef] [PubMed]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

K. Rza̧żewski and J. H. Eberly, Phys. Rev. Lett. 47, 408 (1981).
[CrossRef]

Rep. Prog. Phys. (1)

R. Laudon, Rep. Prog. Phys. 43, 913 (1980).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Squeezed vacuum-induced spectra V(E) = πW(E) for weak-field coupling (Ω = 1) and a small q parameter (q = 2). The energies are E1 = EL = 1. All the energies are measured in units of the continuum width.

Fig. 2
Fig. 2

Same as Fig. 1 but for Ω = 3.

Fig. 3
Fig. 3

Spectra for the same parameters as those for Fig. 1, except for a high q parameter (q = 100).

Fig. 4
Fig. 4

Same as Fig. 3 but for strong-field coupling (Ω = 3).

Fig. 5
Fig. 5

Same as Fig. 4 but for various photon energies EL. The squeezing parameter is s = 0.7.

Fig. 6
Fig. 6

Long-time spectra V(E) induced by a squeezed state with a strong coherent component. The field coupling is strong (Ω = 3), and the q parameter is high (q = 100). The energies are EL = E1 = 1, and the phase difference is D = 0.

Fig. 7
Fig. 7

Same as Fig. 6 but for various values of D. The modulus of the squeezing parameter is assumed to be equal to 0.5.

Fig. 8
Fig. 8

Spectra induced by an SU(1, 1) coherent state for weak-field coupling (Ω = 1) and a high q parameter (q = 100). The energies are E1 = EL = 1.

Fig. 9
Fig. 9

Same as Fig. 8 but for q = 2.

Fig. 10
Fig. 10

Same as Fig. 9 but for various photon energies. The parameter |ξ| is equal to 0.85.

Fig. 11
Fig. 11

Strong-field spectra (Ω = 3). The energies are E1 = EL = 1, and q = 2.

Fig. 12
Fig. 12

Same as Fig. 11 but for a high asymmetry q parameter (q = 100).

Equations (23)

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H = E o O O + d E E c c + a + a E L + [ a d E S ( E ) c O + H . c . ] .
S ( E ) = ( q + i ) O V R C E - E 1 + Γ q E - E 1 - i Γ exp ( i f ) ,
Φ ( t ) = exp [ - i E L ( n - 1 ) t ] × n = 0 [ a n ( t ) O + d E b n ( E , t ) c ] ,
O O n , c c n - 1 .
Φ G ( t = 0 ) = n = 0 Q n O .
d a n ( t ) d t = ( E O + E L ) a n ( t ) + d E b n ( E , t ) S ( E ) n + i δ ( t ) Q n ,
d b n ( E , t ) d t = S ( E ) n a n ( t ) + E b n ( E , t ) ,
a n ( t ) = d exp ( - i t ) A n ( ) ,
b n ( E , t ) = d exp ( - i t ) B n ( E , ) ,
( - E O - E L ) A n ( ) - d E B n ( E , ) S ( E ) n = Q n ,
S ( E ) n A n ( ) - ( - E ) B n ( E , ) = 0.
W ( E ) = lim t n = 0 b n ( E , t ) 2 .
Q n A n ( ) = ( - E O - E L ) - n d E S ( E ) 2 - E .
π W ( E ) Γ 0 = n = 0 | n ( E - E 1 ) Q n ( E - E 1 + i Γ ) ( E - E 0 - E L + i n Γ 0 ) - n Γ 0 Γ ( q - i ) 2 | 2 ,
Γ 0 = π O V R C 2 .
Ω = ( 4 π Γ ) 1 / 2 ( q + i ) O V R C exp ( i θ ) ,
P n = Q n 2 = 1 n ! cosh s ( tanh s 2 ) n H n ( 0 ) 2 ,
P 0 = 1 cosh s ,             P i = 0 , P n = P n - 2 n - 1 n ( tanh s ) 2 .
P n = 1 [ 2 π ( Δ n ) 2 ] 1 / 2 exp [ - ( n - α 2 ) 2 2 ( Δ n ) 2 ] ,
( Δ n ) 2 = α 2 [ exp ( - 2 s ) cos 2 ( σ - ½ β ) + exp ( 2 s ) sin 2 ( σ - ½ β ) ] .
ξ = ( 1 - ξ 2 ) k m = 0 Γ ( m + 2 k ) 1 / 2 [ m ! Γ ( 2 k ) ] 1 / 2 ξ μ m , k ,
K 0 = ( a + a + ½ ) / 2 ,             K + = ( a + ) 2 / 2 ,             K - = a 2 / 2.
ξ = n = 0 Q n n = n = 0 ξ n ( 1 - ξ 2 ) 1 / 2 n ,

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