Integrable and Chaotic Behaviour in the Paul Trap and the Hydrogen Atom in a Generalized van der Waals Potential

  • David Farrelly
  • James E. Howard
  • T. Uzer
Part of the NATO ASI Series book series (NSSB, volume 331)


For almost two decades the problem of Rydberg atoms interacting with external fields has proved to be a unique atomic laboratory for the study of chaos.1–5 While some studies have specialized to interactions with explicitly time dependent microwave fields,1 others have dealt instead with static fields.2–4 Many of the most outstanding cases of interest fall into the second category, and, further, their Hamiltonians turn out to be particular limits of the so-called generalized van der Waals Hamiltonian (GVDW)6–7 (e.g., the notoriously chaotic quadratic Zeeman effect2–5). Studies of atomic Rydberg states have clearly led to substantial progress being made in furthering an understanding of the implications of classical chaos in quantum systems. Quite recently, however, investigations of a different problem, the Paul trap,8–9 suggest that this system might prove of comparable value in the study of chaotic dynamics in atomic systems. In this paper we show that the generalized van der Waals and trap Hamiltonians are special cases of a more general Hamiltonian and, remarkably, they share identical integrable limits. Despite their similitude, important differences also exist; the most significant of them being a double well structure in the effective potential energy surface for the Paul trap. This structure gives rise to a new type of two-ion crystal corresponding to a dynamical rather than a static equilibrium, leading to new resonances and modes of regular and chaotic behavior stemming from the dynamics of the associated separatrix layer.


Chaotic Motion Paul Trap Secular Frequency NATO Advance Study Institute Separable Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.E. Bayfield and P.M. Koch, Phys. Rev. Lett., 33:258 (1974).CrossRefADSGoogle Scholar
  2. 2.
    M.L. Zimerman, M.L. Kash and D. D. Kleppner, Phys. Rev. Lett., 45: 1092 (1980).CrossRefADSGoogle Scholar
  3. 3.
    A. Holle, J. Main, G. Wiebusch, H. Rottke and K.H. Welge, Phys. Rev. Lett., 61:161 (1987).Google Scholar
  4. 4.
    J.C. Gay, in: “Atoms in Unusual Situations” J.P. Briand, ed., Proceedings of the NATO Advanced Study Institute, Series B, Physics, Plenum, New York, (1986); 143: 1073.Google Scholar
  5. 5.
    T. Uzer, D. Farrelly, J.A. Milligan, P.E. Raines, and J.P. Skelton, Science, 242:41 (1991).Google Scholar
  6. 6.
    Y. Alhassid, E. A. Hinds and D. Meschede, Phys. Rev. Lett., 59:1545 (1987).CrossRefADSGoogle Scholar
  7. 7.
    K. Ganesan and M. Lakshmanan, Phys. Rev. Lett., 62 (1989) 232.CrossRefADSGoogle Scholar
  8. K. Ganesan and M. Lakshmanan, Phys. Rev. A, 42:3940 (1990).CrossRefADSGoogle Scholar
  9. K. Ganesan and M. Lakshmanan, Phys. Rev. A, 45:1548 (1992).CrossRefADSGoogle Scholar
  10. 8.
    W. Paul, Rev. Mod. Phys., 62:531 (1990).CrossRefADSGoogle Scholar
  11. 9.
    R. Blümel, C. Kappler, W. Quint, and H. Walther, Phys. Rev. A, 40:808 (1989).CrossRefADSGoogle Scholar
  12. R. Blümel, C. Kappler, W. Quint, and H. Walther, Phys. Rev. A, 46:8034 (1993).CrossRefGoogle Scholar
  13. 10.
    G. Baumann and T. F. Nonnenmacher, Phys. Rev. A, 46: 2682 (1992).CrossRefADSGoogle Scholar
  14. 11.
    R. Blatt, P. Gill, and R.C. Thompson, J. Mod. Opt, 39:193 (1992).CrossRefADSGoogle Scholar
  15. D.J. Bate, K. Dholakia, R.C. Thompson and D.C. Wilson, J. Mod. Opt, 39:305 (1992).CrossRefADSGoogle Scholar
  16. 12.
    D. Farrelly and J.E. Howard, Phys. Rev. A, (in press).Google Scholar
  17. 13.
    J.E. Howard and D. Farrelly, Phys. Lett. A, (in press).Google Scholar
  18. 14.
    D. Farrelly and J.E. Howard, Phys. Rev. Lett., (submitted).Google Scholar
  19. 15.
    R. Blümel, Phys. Rev. A, (in press); M. Moore and R. Blümel, Phys. Rev. A, (preprint): R. Blümel, Phys. Lett. A, 174:174 (1993).CrossRefADSGoogle Scholar
  20. 16.
    D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, and C.H. Manney, Phys. Rev. Lett., 59:2935 (1987).CrossRefADSGoogle Scholar
  21. 17.
    L.S. Brown, Phys. Rev. Lett., 66:527 (1991).CrossRefzbMATHMathSciNetADSGoogle Scholar
  22. S. Stenholm, J. Mod. Opt, 39:279 (1992).CrossRefADSGoogle Scholar
  23. 18.
    W.A. Lin and L.E. Ballentine, Phys. Rev. Lett., 65:2927 (1990): D. Farrelly and J. A. Milligan, Phys. Rev. E (in press).CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • David Farrelly
    • 1
  • James E. Howard
    • 1
  • T. Uzer
    • 2
  1. 1.Department of Chemistry and BiochemistryUtah State UniversityLoganUSA
  2. 2.School of PhysicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations