Brownian motion of a parametric oscillator: A model for ion confinement in radio frequency traps

  • R. Blatt
  • P. Zoller
  • G. Holzmüller
  • I. Siemers


The distribution function for Brownian motion of a parametric oscillator is calculated exactly with the help of continued fraction expansions in the long time limit. We derive expressions for the energy and the widths of the spatial and velocity distribution. Our results are relevant to understand confinement of particles in radio frequency ion traps.


32.70. -n 32.80.Pj 05.40.+j 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ince, E.L.: Ordinary differential equations. New York: Longmans 1977Google Scholar
  2. 1a.
    Mc Lachlan, N.W.: Theory and application of Mathieu functions. Oxford: Clarendon Press 1947Google Scholar
  3. 2.
    Wang, M.C., Uhlenbeck, G.E.: Rev. Mod. Phys.17, 323 (1945)Google Scholar
  4. 3.
    Risken, H.: The Fokker Planck equation: methods of solution and applications. In: Springer Series in synergetics. Vol. 18. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag 1984Google Scholar
  5. 4.
    Kampen, N.G. van: Stochastic processes in physics and chemistry. Amsterdam: North-Holland 1981Google Scholar
  6. 5.
    Dehmelt, H.G.: Adv. At. Mol. Phys.3, 53 (1967)Google Scholar
  7. 6.
    Wineland, D.J., Itano, W.M., Dyck, R.S. van, jr.: Adv. At. Mol. Phys.19, 135 (1983)Google Scholar
  8. 7.
    Meixner, T., Schäffke, F.W.: Mathieu'sche Funktionen und Sphäroidfunktionen. Berlin, Göttingen, Heidelberg: Springer-Verlag 1954Google Scholar
  9. 8.
    Knight, R.D., Prior, M.H.: J. Appl. Phys.50, 3044 (1979)Google Scholar
  10. 9.
    Schaaf, H., Werth, G.: Appl. Phys.25, 249 (1981)Google Scholar
  11. 10.
    Toschek, P.E.: Les Houches XXXVIII. 383 (1984). Amsterdam, Oxford, New York, Tokyo: North-Holland 1984Google Scholar
  12. 11.
    For a review on laser cooling see Stenholm, S.: Rev. Mod. Phys. (to be published)Google Scholar
  13. 12.
    Cook, R., Shankland, D.G., Wells, A.L.: Phys. Rev. A31, 564 (1985)Google Scholar
  14. 13.
    Haken, H.: Laser theory. In: Encyclopedia of Physics. Vol. XXV/2c. Berlin, Heidelberg, New York: 1970Google Scholar
  15. 14.
    Ichimaru, S.: Basic principles of plasma physics: a statistical approach. In: Frontiers in Physics Lecture Note Series. Reading, Ma.: Benjamin/Cummings 1973Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • R. Blatt
    • 1
  • P. Zoller
    • 2
  • G. Holzmüller
    • 2
  • I. Siemers
    • 1
  1. 1.I. Institut für ExperimentalphysikUniversität HamburgFederal Republic of Germany
  2. 2.Institut für Theoretische PhysikUniversität InnsbruckAustria

Personalised recommendations