Abstract

We investigate classically and quantum-mechanically the coherent interaction between a single mode of the electromagnetic field and a stream of two-level atoms under the assumptions that only one atom at a time is coupled to the field and that the interaction times are all equal. This is a quantum-optics analog of a coherently kicked harmonic oscillator. We find that the classical system always evolves toward a marginally stable steady state at the threshold of type-1 intermittency, independently of the initial state of inversion of the atoms. But there are infinitely many such steady states, and which one is reached may depend sensitively on the initial conditions. In contrast, in the case of inverted atoms the quantized system usually does not reach a steady state: The intrinsic quantum fluctuations of the field almost always force it eventually to grow past the classical fixed points. A notable exception occurs under conditions such that the sequence of inverted atoms injected into the cavity leads to the preparation of a highly excited Fock state of the cavity mode.

© 1986 Optical Society of America

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  1. D. Kleppner, in Atomic Physics and Astrophysics, M. Chretien and E. Lipworth, eds. (Gordon and Breach, New York, 1971), p. 5; K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165; D. Kleppner, Phys. Rev. Lett. 47, 233 (1981); G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67 (1985); for a recent review, see P. Filipowicz, P. Meystre, and H. Walther, Opt. Acta 32, 1115 (1985).
    [CrossRef] [PubMed]
  2. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [CrossRef]
  3. F. W. Cummings, Phys. Rev. 140, 1051 (1965); P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento 25, 521 (1975); T. von Foerster, J. Phys. A: Gen. Phys. 8, 95 (1975); S. Stenholm, Phys. Rep. 6, 1 (1973).
    [CrossRef]
  4. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
    [CrossRef]
  5. D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
    [CrossRef] [PubMed]
  6. See e.g. H. G. Schuster, Deterministic Chaos: An Introduction (Physik-Verlag, Weinheim, 1984).
  7. See, e.g., M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), in particular Chap. 17.
  8. See, e.g., L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  9. G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
    [CrossRef]
  10. G. Casati, in Trends and Developments of the Eighties, Proceedings of the Bielefeld Symposium 1983, S. Albeverio and Ph. Blanchard, eds. (World, New York, to be published).
  11. G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
    [CrossRef] [PubMed]
  12. We use “diffusive” in a loose sense here, since the average growth of the field energy does not appear to be linear in time.
  13. The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
    [CrossRef]

1985 (1)

D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
[CrossRef] [PubMed]

1984 (2)

G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
[CrossRef] [PubMed]

The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
[CrossRef]

1980 (1)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
[CrossRef]

1965 (1)

F. W. Cummings, Phys. Rev. 140, 1051 (1965); P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento 25, 521 (1975); T. von Foerster, J. Phys. A: Gen. Phys. 8, 95 (1975); S. Stenholm, Phys. Rep. 6, 1 (1973).
[CrossRef]

1963 (1)

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

Ackerhalt, J. R.

The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
[CrossRef]

Allen, L.

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

Casati, G.

G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
[CrossRef] [PubMed]

G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
[CrossRef]

G. Casati, in Trends and Developments of the Eighties, Proceedings of the Bielefeld Symposium 1983, S. Albeverio and Ph. Blanchard, eds. (World, New York, to be published).

Chirikov, B. V.

G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
[CrossRef] [PubMed]

G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
[CrossRef]

Cummings, F. W.

F. W. Cummings, Phys. Rev. 140, 1051 (1965); P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento 25, 521 (1975); T. von Foerster, J. Phys. A: Gen. Phys. 8, 95 (1975); S. Stenholm, Phys. Rep. 6, 1 (1973).
[CrossRef]

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

Eberly, J. H.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
[CrossRef]

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

Ford, J.

G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
[CrossRef]

Galbraith, H. W.

The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
[CrossRef]

Israilev, F. M.

G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
[CrossRef]

Jaynes, E. T.

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

Kleppner, D.

D. Kleppner, in Atomic Physics and Astrophysics, M. Chretien and E. Lipworth, eds. (Gordon and Breach, New York, 1971), p. 5; K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165; D. Kleppner, Phys. Rev. Lett. 47, 233 (1981); G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67 (1985); for a recent review, see P. Filipowicz, P. Meystre, and H. Walther, Opt. Acta 32, 1115 (1985).
[CrossRef] [PubMed]

Lamb, W. E.

See, e.g., M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), in particular Chap. 17.

Meschede, D.

D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
[CrossRef] [PubMed]

Milonni, P. W.

The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
[CrossRef]

Müller, G.

D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
[CrossRef] [PubMed]

Narozhny, N. B.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
[CrossRef]

Sanchez-Mondragon, J. J.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
[CrossRef]

Sargent, M.

See, e.g., M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), in particular Chap. 17.

Schuster, H. G.

See e.g. H. G. Schuster, Deterministic Chaos: An Introduction (Physik-Verlag, Weinheim, 1984).

Scully, M. O.

See, e.g., M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), in particular Chap. 17.

Shepelyanski, D. L.

G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
[CrossRef] [PubMed]

Walther, H.

D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
[CrossRef] [PubMed]

Phys. Rev. (1)

F. W. Cummings, Phys. Rev. 140, 1051 (1965); P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento 25, 521 (1975); T. von Foerster, J. Phys. A: Gen. Phys. 8, 95 (1975); S. Stenholm, Phys. Rep. 6, 1 (1973).
[CrossRef]

Phys. Rev. Lett. (4)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984).
[CrossRef]

D. Meschede, H. Walther, and G. Müller, Phys. Rev. Lett. 54, 551 (1985); P. Goy, J. D. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 (1983).
[CrossRef] [PubMed]

G. Casati, B. V. Chirikov, and D. L. Shepelyanski, Phys. Rev. Lett. 53, 2525 (1984); see also J. E. Bayfield and A. Pinnaduwage, Phys. Rev. Lett. 54, 313 (1985).
[CrossRef] [PubMed]

The study of chaos in the Jaynes–Cummings model without rotating-wave approximation has been performed by P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50, 966 (1984) in the semiclassical case and by R. Graham and M. Höhnerbach, Z. Phys. B 57, 233 (1984) for both quantized and semiclassical theories.
[CrossRef]

Proc. IEEE (1)

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

Other (7)

D. Kleppner, in Atomic Physics and Astrophysics, M. Chretien and E. Lipworth, eds. (Gordon and Breach, New York, 1971), p. 5; K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165; D. Kleppner, Phys. Rev. Lett. 47, 233 (1981); G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67 (1985); for a recent review, see P. Filipowicz, P. Meystre, and H. Walther, Opt. Acta 32, 1115 (1985).
[CrossRef] [PubMed]

We use “diffusive” in a loose sense here, since the average growth of the field energy does not appear to be linear in time.

See e.g. H. G. Schuster, Deterministic Chaos: An Introduction (Physik-Verlag, Weinheim, 1984).

See, e.g., M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), in particular Chap. 17.

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

G. Casati, B. V. Chirikov, F. M. Israilev, and J. Ford, in Stochastic Behavior in Classical and Quantum Systems, G. Casati and J. Ford, eds., Vol. 93 of Lecture Notes in Physics (Springer-Verlag, New York, 1979); G. Casati, in Proceedings of the Second Workshop on Quantum Probability and Applications (Springer-Verlag, Berlin, to be published).
[CrossRef]

G. Casati, in Trends and Developments of the Eighties, Proceedings of the Bielefeld Symposium 1983, S. Albeverio and Ph. Blanchard, eds. (World, New York, to be published).

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Figures (5)

Fig. 1
Fig. 1

Average (solid line) and rms spread (dashed line) of the photon distribution evolving from an initial Poisson distribution with 〈n〉 = 8, as a function of the number of atoms that have traversed the cavity for the interaction time κ t int = 2 π 2 / 199.

Fig. 2
Fig. 2

Classical return map n+1 = f(n) for initially inverted atoms (η = 1) for the dimensionless interaction time τint = 5. Also shown are the straight line n+1 = n and the iteration of the field toward 3* ≃ 3.5 from the initial value 0 ≃ 0.2.

Fig. 3
Fig. 3

Number of orbits converging to the fixed points * for an ensemble of 200 orbits starting from initial values evenly distributed in the interval [0, 1]. The interaction time is τint = 20.

Fig. 4
Fig. 4

The return map for absorbing atoms (η = −1) is plotted for τint = 5.

Fig. 5
Fig. 5

Comparison of the classical and quantized models. The dimensionless intensity 2/4 of the classical model (solid line) and its direct counterpart 〈n〉 for the quantum case (dashed line) are presented as a function of the number of atoms that have passed through the cavity. The interaction time is τ int = π 2 / 199, R = 1, and initially 2/4 = 〈n〉 = 8. The initial quantum field is taken to have Poissonian photon statistics.

Equations (20)

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H = ( ћ ω 0 / 2 ) S 3 + ћ ω a a + ћ κ / 2 ( S + a + a S ) ,
σ ˙ = i Δ σ + i ( κ / 2 ) σ 3 E , σ ˙ 3 = i κ ( σ + E E * σ ) , E ˙ = i ( κ / 2 ) σ ,
E = a exp ( i ω t ) , σ = S exp ( i ω t ) , σ 3 = S 3 .
ρ F ( t i + t int ) = Tr A [ U ( t int ) ρ ( t i ) ] ,
ρ n ( t i + t int ) = 1 e x + 1 [ ( α n + 1 + α n e x ) ρ n ( t i ) + β n + 1 e x ρ n + 1 ( t i ) + β n ρ n 1 ( t i ) ] .
β n = 1 α n = ( κ 2 n / Ω n 2 ) sin 2 ( Ω n t int / 2 ) ,
ρ a a = e x ρ b b ,
B n β n ( ρ n 1 e x ρ n ) = 0
ρ n = e x ρ n 1 .
1 1 + e x [ α k 1 + e x α k 2 β k 1 e x 0 0 β k 1 α k + e x α k 1 0 0 0 0 α k 1 + e x α k β k + 1 e x 0 0 β k + 1 α k + 2 + e x α k + 1 ] .
κ k t int = 2 q π ,
i ( σ + σ ) = R sin θ , σ 3 = R cos θ ,
R / ξ = 0 , θ / ξ = E , E / ξ = ( R / 4 ) sin θ .
τ = ξ | R | / 2 , = 2 E / | R | .
θ / τ = ,
/ τ = η sin θ ,
n + 1 = f ( n ) .
θ 2 2 + η cos θ = n 2 2 + η ,
n f ( n ) d [ 1 ( n 2 + 2 2 + η ) 2 ] 1 / 2 = η τ int .
n + 1 n + ( 2 η / n ) sin 2 ( n τ int / 2 ) .

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