Abstract

Interactions of two two-level atoms with a single-mode quantized radiation field are studied, and exact expressions for the atom-field probability amplitudes in the rotating-wave approximation are presented. These solutions are employed in the study of the quantum collapse and revivals of the atomic coherence.

© 1988 Optical Society of America

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References

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  1. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [CrossRef]
  2. F. W. Cummings, Phys. Rev. 140, A1051 (1965).
    [CrossRef]
  3. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
    [CrossRef]
  4. N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
    [CrossRef]
  5. H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).
  6. Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
    [CrossRef]
  7. D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
    [CrossRef]
  8. G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
    [CrossRef] [PubMed]
  9. M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
    [CrossRef]
  10. F. W. Cummings and A. Dorri, Phys. Lett. 95A, 263 (1983).
  11. S. M. Barnett and P. L. Knight, Opt. Acta 31, 435, 1203 (1984).
    [CrossRef]
  12. S. Mahmood and M. S. Zubairy, Phys. Rev. A 35, 425 (1987).
    [CrossRef] [PubMed]
  13. Z. Deng, Opt. Commun. 54, 222 (1985).
    [CrossRef]

1987 (2)

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

S. Mahmood and M. S. Zubairy, Phys. Rev. A 35, 425 (1987).
[CrossRef] [PubMed]

1985 (2)

Z. Deng, Opt. Commun. 54, 222 (1985).
[CrossRef]

D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef]

1984 (1)

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435, 1203 (1984).
[CrossRef]

1983 (2)

F. W. Cummings and A. Dorri, Phys. Lett. 95A, 263 (1983).

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

1981 (1)

H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).

1980 (2)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
[CrossRef]

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
[CrossRef]

1968 (1)

M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
[CrossRef]

1965 (1)

F. W. Cummings, Phys. Rev. 140, A1051 (1965).
[CrossRef]

1963 (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Barnett, S. M.

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435, 1203 (1984).
[CrossRef]

Cummings, F. W.

F. W. Cummings and A. Dorri, Phys. Lett. 95A, 263 (1983).

M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
[CrossRef]

F. W. Cummings, Phys. Rev. 140, A1051 (1965).
[CrossRef]

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Deng, Z.

Z. Deng, Opt. Commun. 54, 222 (1985).
[CrossRef]

Dorri, A.

F. W. Cummings and A. Dorri, Phys. Lett. 95A, 263 (1983).

Eberly, J. H.

H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
[CrossRef]

Goy, P.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Gross, M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Haroche, S.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Kaluzny, Y.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Klein, N.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Knight, P. L.

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435, 1203 (1984).
[CrossRef]

Mahmood, S.

S. Mahmood and M. S. Zubairy, Phys. Rev. A 35, 425 (1987).
[CrossRef] [PubMed]

Mesehede, D.

D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef]

Muller, G.

D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef]

Narozhny, N. B.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
[CrossRef]

Raimond, J. M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Rempe, G.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
[CrossRef]

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
[CrossRef]

Tavis, M.

M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
[CrossRef]

Walther, H.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef]

Yoo, H. Y.

H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).

Zubairy, M. S.

S. Mahmood and M. S. Zubairy, Phys. Rev. A 35, 425 (1987).
[CrossRef] [PubMed]

J. Phys. B (1)

H. Y. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. B 14, 1383 (1981).

Opt. Acta (1)

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435, 1203 (1984).
[CrossRef]

Opt. Commun. (1)

Z. Deng, Opt. Commun. 54, 222 (1985).
[CrossRef]

Phys. Lett. (1)

F. W. Cummings and A. Dorri, Phys. Lett. 95A, 263 (1983).

Phys. Rev. (2)

M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
[CrossRef]

F. W. Cummings, Phys. Rev. 140, A1051 (1965).
[CrossRef]

Phys. Rev. A (2)

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1980).
[CrossRef]

S. Mahmood and M. S. Zubairy, Phys. Rev. A 35, 425 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980);Sov. J. Quantum Electron. 10, 1261 (1981).
[CrossRef]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

D. Mesehede, H. Walther, and G. Muller, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Single-atom population inversion W versus dimensionless time τ = g1t with n ¯ = 10.

Fig. 2
Fig. 2

(a) W1 versus τ = (g12 + g22)½t for Dg2/g1 = 0.5 with n ¯ = 10. (b) W1 = WT versus τ for D = 1.0 with n ¯ = 10.

Fig. 3
Fig. 3

(a) Probability for both atoms in upper state Paa versus τ for D = 0.5 with n ¯ = 10. (b) Probability for both atoms in ground state Pbb versus τ for D = 0.5 with n ¯ = 10.

Fig. 4
Fig. 4

(a) Paa versus τ for D = 1.0 with n ¯ = 10. (b) Pbb versus τ for D = 1.0 with n ¯ = 10.

Fig. 5
Fig. 5

WT versus τ for D = 0.5 with n ¯ = 10.

Tables (1)

Tables Icon

Table 1 Coefficients Aij in Eq. (10)

Equations (26)

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H I = i = 1 2 g i ( a σ i + a σ i ) ,
| Ψ ( t ) = C a a n ( t ) | a , a , n + C a b n + 1 ( t ) | a , b , n + 1 + C b a n + 1 ( t ) | b , a , n + 1 + C b b n + 2 ( t ) | b , b , n + 2 .
| Ψ ˙ ( t ) = i H I | Ψ ( t ) .
C ˙ = i M C ,
C = [ C a n n C a b n + 1 C b a n + 1 C b b n + 2 ] ,
M = [ 0 g 2 ( n + 1 ) 1 / 2 g 1 ( n + 1 ) 1 / 2 0 g 2 ( n + 1 ) 1 / 2 0 0 g 1 ( n + 2 ) 1 / 2 g 1 ( n + 1 ) 1 / 2 0 0 g 2 ( n + 2 ) 1 / 2 0 g 1 ( n + 2 ) 1 / 2 g 2 ( n + 2 ) 1 / 2 0 ] .
C i ( t ) = j l α i l exp ( i λ l t ) α j * l C j ( 0 ) ,
λ l = ± 1 2 [ ( g 1 2 + g 2 2 ) ( 2 n + 3 ) ± β ] 1 / 2 ,
β = [ ( 2 n + 3 ) 2 ( g 1 2 + g 2 2 ) 2 4 ( n + 1 ) ( n + 2 ) ( g 1 2 g 2 2 ) 2 ] 1 / 2 .
α 1 l = ± λ l 2 ( g 1 2 + g 2 2 ) ( n + 2 ) [ β 2 β ( g 1 2 + g 2 2 ) ] 1 / 2 ,
α 2 l = ± g 2 ( n + 1 ) 1 / 2 [ λ l 2 + ( g 1 2 g 2 2 ) ( n + 2 ) ] λ l [ β 2 β ( g 1 2 + g 2 2 ) ] 1 / 2 ,
α 3 l = ± g 1 ( n + 1 ) 1 / 2 [ λ l 2 ( g 1 2 g 2 2 ) ( n + 2 ) ] λ l [ β 2 β ( g 1 2 + g 2 2 ) ] 1 / 2 ,
α 4 l = ± 2 g 1 g 2 [ ( n + 1 ) ( n + 2 ) ] 1 / 2 [ β 2 β ( g 1 2 + g 2 2 ) ] 1 / 2 .
C i ( t ) = A i j ( t ) C j ( 0 ) .
W 1 ( t ) = n [ | C 1 ( t ) | 2 + | C 2 ( t ) | 2 | C 3 ( t ) | 2 | C 4 ( t ) | 2 ] .
W 2 ( t ) = n [ | C 1 ( t ) | 2 + | C 3 ( t ) | 2 | C 2 ( t ) | 2 | C 4 ( t ) | 2 ] .
W T ( t ) = ½ ( W 1 + W 2 ) = n [ | C 1 ( t ) | 2 | C 4 ( t ) | 2 ] .
W 1 ( t ) = n [ | A 11 | 2 p ( n ) + ( | A 21 | 2 | A 31 | 2 ) p ( n 1 ) + | A 41 | 2 p ( n 2 ) ] ,
W T ( t ) = n [ | A 11 | 2 p ( n ) | A 41 | 2 p ( n 2 ) ] .
W 1 ( t ) = n ē n ¯ n ¯ n n ! [ | A 11 | 2 n n ¯ ( | A 21 | 2 | A 31 | 2 ) n ( n 1 ) n ¯ 2 | A 41 | 2 ] ,
W T ( t ) = n ē n ¯ n ¯ n n ! [ | A 11 | 2 n ( n 1 ) n ¯ 2 | A 41 | 2 ] ,
W 1 = 1 8 n ē n ¯ n ¯ n n ! [ 1 β 2 ( 2 [ β 2 + ( g 1 2 + g 2 2 ) 2 ] + [ β ( g 1 2 + g 2 2 ) ] 2 × cos ( 2 λ 1 t ) + [ β + ( g 1 2 + g 2 2 ) ] 2 cos ( 2 λ 3 t ) + 2 [ β 2 ( g 1 2 + g 2 2 ) 2 ] { cos [ ( λ 1 + λ 3 ) t ] + cos [ ( λ 1 λ 3 ) t ] } ) + n 2 n ¯ ( β ) 2 × ( [ 1 cos ( 2 λ 1 t ) ] ( g 1 2 X 1 2 g 1 2 X 3 2 ) / λ 1 2 + [ 1 cos ( 2 λ 3 t ) ] ( g 2 2 X 2 2 g 1 2 X 4 2 ) / λ 3 2 + 2 { cos [ ( λ 1 + λ 3 ) t ] cos [ ( λ 1 λ 3 ) t ] } × ( g 2 2 X 1 X 2 g 1 2 X 3 X 4 ) / λ 1 λ 3 ) 16 n 2 ( n 1 ) 2 g 1 2 g 2 2 n ¯ 2 β 2 × { 2 + cos ( 2 λ 1 t ) + cos ( 2 λ 3 t ) 2 cos [ ( λ 1 + λ 3 ) t ] 2 cos [ ( λ 1 λ 3 ) t ] } ] ,
W T = 1 8 n ē n ¯ n ¯ n n ! [ 1 β 2 ( 2 [ β 2 + ( g 1 2 + g 2 2 ) 2 ] + [ β ( g 1 2 + g 2 2 ) ] 2 × cos ( 2 λ 1 t ) + [ β + ( g 1 2 + g 2 2 ) ] 2 cos ( 2 λ 3 t ) + 2 [ β 2 ( g 1 2 + g 2 2 ) 2 ] { cos [ ( λ 1 + λ 3 ) t ] + cos [ ( λ 1 λ 3 ) t ] } ) 16 n 2 ( n 1 ) 2 g 1 2 g 2 2 n ¯ 2 β 2 × { 2 + cos ( 2 λ 1 t ) + cos ( 2 λ 3 t ) 2 cos [ ( λ 1 λ 3 ) t ] 2 cos [ ( λ 1 λ 3 ) t ] } ] ,
X 1 , 2 = ( 4 n + 3 ) g 1 2 g 2 2 ± β ,
X 3 , 4 = ( 4 n + 3 ) g 2 2 g 1 2 ± β .
A 11 ( t ) = β ( g 1 2 + g 2 2 ) 2 β cos λ 1 t + β + ( g 1 2 + g 2 2 ) 2 β cos λ 3 t A 12 ( t ) = A 21 ( t ) = i g 2 ( n + 1 ) 1 / 2 2 β [ ( 4 n + 7 ) g 1 2 g 2 2 + β λ 1 sin λ 1 t ( 4 n + 7 ) g 1 2 g 2 2 β λ 3 sin λ 3 t ] A 13 ( t ) = A 31 ( t ) = i g 1 ( n + 1 ) 1 / 2 2 β [ ( 4 n + 7 ) g 2 2 g 1 2 + β λ 1 sin λ 1 t ( 4 n + 7 ) g 2 2 g 1 2 β λ 3 sin λ 3 t ] A 14 ( t ) = A 41 ( t ) = 2 g 1 g 2 [ ( n + 1 ) ( n + 2 ) ] 1 / 2 β [ cos λ 1 t cos λ 3 t ] A 22 ( t ) = 2 g 2 2 ( n + 1 ) β { [ λ 1 2 + ( g 1 2 g 2 2 ) ( n + 2 ) ] 2 λ 1 2 [ β ( g 1 2 + g 2 2 ) ] cos λ 1 t + [ λ 3 2 + ( g 1 2 g 2 2 ) ( n + 2 ) ] 2 λ 3 2 [ β ( g 1 2 + g 2 2 ) ] cos λ 3 t } A 23 ( t ) = A 32 ( t ) = 2 g 1 g 2 ( n + 1 ) β { λ 1 4 ( g 1 2 g 2 2 ) 2 ( n + 2 ) 2 λ 1 2 [ β ( g 1 2 + g 2 2 ) ] cos λ 1 t + λ 1 4 ( g 1 2 g 2 2 ) 2 ( n + 2 ) 2 λ 3 2 [ β + ( g 1 2 + g 2 2 ) ] cos λ 3 t } A 24 ( t ) = A 42 ( t ) = 4 i g 1 g 2 2 ( n + 1 ) ( n + 2 ) 1 / 2 β { λ 1 2 + ( g 1 2 g 2 2 ) ( n + 2 ) λ 1 [ β ( g 1 2 + g 2 2 ) ] sin λ 1 t λ 3 2 + ( g 1 2 g 2 2 ) ( n + 2 ) λ 3 [ β + ( g 1 2 + g 2 2 ) ] sin λ 3 t } A 34 ( t ) = A 43 ( t ) = 4 i g 1 2 g 2 ( n + 1 ) 1 / 2 ( n + 2 ) β { λ 1 2 ( g 1 2 g 2 2 ) ( n + 2 ) λ 1 [ β ( g 1 2 + g 2 2 ) ] sin λ 1 t λ 3 2 ( g 1 2 g 2 2 ) ( n + 2 ) λ 3 [ β + ( g 1 2 + g 2 2 ) ] sin λ 3 t } A 44 ( t ) = 8 g 1 2 g 2 2 ( n + 1 ) ( n + 2 ) β [ cos λ 1 t β ( g 1 2 + g 2 2 ) + cos λ 3 t β + ( g 1 2 + g 2 2 ) ]

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