Abstract

Control of non-circular and non-spreading wave packet states by a resonant radiation field is predicted and numerically confirmed for hydrogen.

© 1998 Optical Society of America

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References

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  1. I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
    [CrossRef] [PubMed]
  2. D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
    [CrossRef]
  3. J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
    [CrossRef] [PubMed]
  4. J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
    [CrossRef]
  5. H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. 211, 249 (1991).
    [CrossRef]
  6. J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A 45, 1978 (1992).
    [CrossRef] [PubMed]
  7. M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals 5, 1143 (1995).
    [CrossRef]
  8. D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys 35, 85 (1994).
    [CrossRef]
  9. M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
    [CrossRef] [PubMed]
  10. M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A 53, 1715 (1996).
    [CrossRef] [PubMed]
  11. B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263 (1979).
    [CrossRef]
  12. G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. 162, 95 (1992).
  13. K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55,  568 (1997).
  14. e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A 46, 364 (1992) and references therein.
    [CrossRef]
  15. D. Brouwer and G. Clemence, Methods of celestial mechanics (Academic Press, New York and London, 1961).
  16. e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
    [CrossRef]
  17. M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys.,  120, 402 (1979).
    [CrossRef]
  18. R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys.,  55, 261 (1986).
    [CrossRef]
  19. B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A 27, 6579 (1994).
    [CrossRef]
  20. R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. 54, 3065 (1971).
    [CrossRef]
  21. T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
    [CrossRef]
  22. N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press (1947).
  23. X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
    [CrossRef] [PubMed]

1997 (3)

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
[CrossRef]

K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55,  568 (1997).

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

1996 (1)

M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A 53, 1715 (1996).
[CrossRef] [PubMed]

1995 (3)

D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
[CrossRef]

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
[CrossRef] [PubMed]

M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals 5, 1143 (1995).
[CrossRef]

1994 (3)

D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys 35, 85 (1994).
[CrossRef]

I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
[CrossRef] [PubMed]

B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A 27, 6579 (1994).
[CrossRef]

1992 (3)

J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A 45, 1978 (1992).
[CrossRef] [PubMed]

G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. 162, 95 (1992).

e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A 46, 364 (1992) and references therein.
[CrossRef]

1991 (2)

H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. 211, 249 (1991).
[CrossRef]

X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
[CrossRef] [PubMed]

1986 (1)

R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys.,  55, 261 (1986).
[CrossRef]

1983 (1)

T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
[CrossRef]

1979 (3)

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263 (1979).
[CrossRef]

e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
[CrossRef]

M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys.,  120, 402 (1979).
[CrossRef]

1971 (1)

R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. 54, 3065 (1971).
[CrossRef]

Augustin, S. D.

e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
[CrossRef]

Berman, G. P.

G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. 162, 95 (1992).

Bialynicki-Birula, I.

I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
[CrossRef] [PubMed]

Breuer, H. P.

H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. 211, 249 (1991).
[CrossRef]

Brouwer, D.

D. Brouwer and G. Clemence, Methods of celestial mechanics (Academic Press, New York and London, 1961).

Buchleitner, A.

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
[CrossRef]

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
[CrossRef] [PubMed]

D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys 35, 85 (1994).
[CrossRef]

Chan, F. T.

X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
[CrossRef] [PubMed]

Chirikov, B. V.

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263 (1979).
[CrossRef]

Clemence, G.

D. Brouwer and G. Clemence, Methods of celestial mechanics (Academic Press, New York and London, 1961).

Delande, D.

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
[CrossRef]

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
[CrossRef] [PubMed]

D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys 35, 85 (1994).
[CrossRef]

Eberly, J. H.

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A 53, 1715 (1996).
[CrossRef] [PubMed]

I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
[CrossRef] [PubMed]

Farrelly, D.

D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
[CrossRef]

Guo, S. H.

X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
[CrossRef] [PubMed]

Henkel, J.

J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A 45, 1978 (1992).
[CrossRef] [PubMed]

Holthaus, M.

M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals 5, 1143 (1995).
[CrossRef]

J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A 45, 1978 (1992).
[CrossRef] [PubMed]

H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. 211, 249 (1991).
[CrossRef]

Howard, J. E.

e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A 46, 364 (1992) and references therein.
[CrossRef]

Kalinski, M.

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear.
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A 53, 1715 (1996).
[CrossRef] [PubMed]

I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
[CrossRef] [PubMed]

Kolovsky, A. R.

G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. 162, 95 (1992).

Korsch, H. J.

B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A 27, 6579 (1994).
[CrossRef]

Leacock, R. A.

R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys.,  55, 261 (1986).
[CrossRef]

Lee, E.

D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
[CrossRef]

Marcus, R. A.

T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
[CrossRef]

R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. 54, 3065 (1971).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press (1947).

Mirbach, B.

B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A 27, 6579 (1994).
[CrossRef]

Moshinsky, M.

M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys.,  120, 402 (1979).
[CrossRef]

Noid, D. W.

T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
[CrossRef]

Pladgett, M. J.

R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys.,  55, 261 (1986).
[CrossRef]

Rabitz, H.

e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
[CrossRef]

Sacha, K.

K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55,  568 (1997).

See, e.g.

e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A 46, 364 (1992) and references therein.
[CrossRef]

e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
[CrossRef]

Seligman, T. H.

M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys.,  120, 402 (1979).
[CrossRef]

Shapiro, E. A.

Uzer, T.

D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
[CrossRef]

T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
[CrossRef]

Yang, X. L.

X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
[CrossRef] [PubMed]

Zakrzewski, J.

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
[CrossRef]

K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55,  568 (1997).

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
[CrossRef] [PubMed]

Adv. At. Mol. Opt. Phys (1)

D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys 35, 85 (1994).
[CrossRef]

Am. J. Phys. (1)

R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys.,  55, 261 (1986).
[CrossRef]

Ann. Phys. (2)

H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. 211, 249 (1991).
[CrossRef]

M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys.,  120, 402 (1979).
[CrossRef]

Chaos, Solitons and Fractals (1)

M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals 5, 1143 (1995).
[CrossRef]

J. Chem. Phys. (3)

e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. 71, 4956 (1979).
[CrossRef]

R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. 54, 3065 (1971).
[CrossRef]

T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. 79, 4412 (1983).
[CrossRef]

J. Phys. A (1)

B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A 27, 6579 (1994).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A 204, 359–372 (1995).
[CrossRef]

Phys. Rep. (1)

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263 (1979).
[CrossRef]

Phys. Rev. A (4)

e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A 46, 364 (1992) and references therein.
[CrossRef]

J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A 45, 1978 (1992).
[CrossRef] [PubMed]

M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A 53, 1715 (1996).
[CrossRef] [PubMed]

X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A 43, 1186–1205 (1991).
[CrossRef] [PubMed]

Phys. Rev. A 55 (1)

K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55,  568 (1997).

Phys. Rev. Lett. (2)

I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. 73, 1777 (1994).
[CrossRef] [PubMed]

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. 75, 4015 (1995).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. 162, 95 (1992).

Z. Phys B (1)

J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B 103, 115 (1997).
[CrossRef]

Other (2)

D. Brouwer and G. Clemence, Methods of celestial mechanics (Academic Press, New York and London, 1961).

N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press (1947).

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Figures (3)

Fig. 1.
Fig. 1.

(left) Coefficient η 1 + ζ 1 in equation (9). (right) Energy spectrum in the rotating frame. Color dashed lines are the predictions of the Mathieu theory.

Fig. 2.
Fig. 2.

(a) Probability distribution of the field-free 2D state n = 20, l = 14. (b) Probability distribution of the field-dressed state obtained with the help of matrix elements of 3D aligned states. The gauge in the left lower corner of left plot indicates the distance equal to 100 atomic units.

Fig. 3.
Fig. 3.

Two snapshots of time evolution of the dressed state n = 20, l = 14 obtained by ab initio calculation. The gauge in the left lower corner of left plot indicates the distance equal to 100 atomic units.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

n ˙ = 0 , i = 0 , θ ˙ = 1 / n 3 , φ ˙ = 0
𝛨 = 𝛨 0 + cos ωt + sin ω t
l ˜ = l n , n ˜ = n φ ˜ = φ , θ ˜ = θ + φ ωt
𝛨 = 1 2 n ˜ 2 ω n ˜ + ε k = [ η k n l + ζ k n l ] cos ( k θ ˜ ( k 1 ) ( φ ˜ ωt ) )
1 2 n ˜ 2 = 1 2 n 0 2 + ( n ˜ n 0 ) n 0 3 3 ( n ˜ n 0 ) 2 2 n 0 4 .
𝛨 = 1 2 n 0 2 ωn 0 3 2 n 0 4 ( n ˜ n 0 ) 2 + ε [ η 1 + ζ 1 ] cos θ ˜
[ 3 2 n 0 4 ( i θ ˜ n 0 + 1 / 2 ) 2 + ( 1 2 n 0 2 ωn 0 E + ε [ η 1 + ζ 1 ] cos θ ˜ ) ψ θ ˜ φ ˜ = 0 . ]
3 2 n 0 4 g ′′ + ( E N 0 E dr + ε [ η 1 + ζ 1 ] cos θ ˜ ) g = 0 ,
ψ NL dr θ ˜ φ ˜ = C * exp [ iN θ ˜ ] exp [ iL φ ˜ ] M ( θ ˜ ) ,
ψ nl dr = const * e ilφ e i ( n 0 1 / 2 ) θ e 0 [ ( θ + φ ωt π ) 2 ]
ψ n 0 l dr const * e A ( ϕ ωt ) 2 e AB 2 sin 2 θ e 2 AB sin θ ( ϕ ω t ) * ψ n 0 l 0
A = [ η 1 + ζ 1 ] ε n 4 2 3 n B = 2 e 1 4 e 3 +
E n 0 l dr E n 0 3 ε [ η 1 + ζ 1 ] 2 n 0 2 + ε [ η 1 + ζ 1 ] .
𝛨 = 𝛨 0 ωl + ε x

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