Quantization of the Kicked Rotator With Dissipation

  • T. Dittrich
  • R. Graham
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 294)


The effect of dissipation on a quantum system exhibiting chaos in its classical limit is studied by coupling the kicked quantum rotator to a reservoir with angular momentum exchange. A master equation is derived which maps the density matrix from one kick to the subsequent one. The limits of h→0 and of vanishing dissipation reproduce the classical kicked damped rotator and the kicked quantum rotator, respectively. In the semi-classical limit the quantum map reduces to a classical map with quantum mechanically determined classical noise terms. For sufficiently small dissipation quantum mechanical interference effects render the Wigner distribution negative in some parts of phase space and prevent its interpretation in classical terms.


Angular Momentum Master Equation Classical Limit Wigner Function Wigner Distribution 
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.
    R. Shaw, Z. Naturforsch. 36a, 80 (1981)ADSMATHMathSciNetGoogle Scholar
  2. 2.
    M.V. Berry, N.L. Balazs, M. Tabor, A. Voros, Ann. Phys. (N.Y.) 122, 26 (1979)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Casati, B.V. Chirikov, F.M. Izraelev, J. Ford, Lecture Notes in Physics, Vol. 93, 334, Springer, Berlin 1979Google Scholar
  4. 4.
    F.M. Izraelev, D.L. Shepelyanski, Teor. Mat. Fiz. 49,117 (1980) (Teor. Math. Phys. 43, 553 (1980)Google Scholar
  5. 5.
    H. Hannay, M.V. Berry, Physica 1D, 267 (1980)MATHMathSciNetGoogle Scholar
  6. 6.
    J.S. Hutchinson, R.E. Wyatt, Chem. Phys. Lett. 72,378 (1980)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    H.J. Korsch, M.V. Berry, Physica 3D,627 (1981)MATHMathSciNetGoogle Scholar
  8. 8.
    G.M. Zaslavsky, Phys. Rep. 80, 157 (1981)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    T. Hogg, B.V. Huberman, Phys. Rev. Lett. 48, 711 (1982)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    T. Hogg, B.V. Huberman, Phys. Rev. A28,22 (1983)ADSCrossRefGoogle Scholar
  11. 11.
    S. Fishman, D.R. Grempel, R.E. Prange, Phys. Rev. Lett. 49,509 (1982)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    D.L. Shepelyansky, Physica 8D,208 (1983)Google Scholar
  13. 13.
    D.R. Grempel, R.E. Prange, S. Fishman, Phys. Rev. A29,1639 (1984)ADSCrossRefGoogle Scholar
  14. 14.
    D.R. Grempel, S. Fishman, R.E. Prange, Phys. Rev. Lett. 53,1212 (1984)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Casati, I. Guarneri, Comm. Math. Phys. 95,121 (1984)ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    E. Ott, T.M. Antonsen, J.D. Hanson, Phys. Rev. Lett. 53,2187 (1984)ADSCrossRefGoogle Scholar
  17. 17.
    R. Graham, Z. Phys. B59,75 (1985)CrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Graham, T. Tel, Z. Phys. B60,127 (1985)CrossRefMathSciNetGoogle Scholar
  19. 19.
    B.V. Chirikov, Phys. Rep. 52,263 (1979)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    G.M. Zaslavski, Phys. Lett. 69A,145 (1978)CrossRefMathSciNetGoogle Scholar
  21. 20a.
    G.M. Zaslavski, Kh.-R.Ya. Rachko, Zh. Eksp. Teor. Fiz. 76,2052 (1979) (Sov. Phys. JETP 49,1039 (1979))ADSGoogle Scholar

Copyright information

© Springer-Verlag Wien 1987

Authors and Affiliations

  • T. Dittrich
    • 1
  • R. Graham
    • 1
  1. 1.Universität EssenFederal Republic of Germany

Personalised recommendations