Journal of Russian Laser Research

, Volume 39, Issue 4, pp 389–400 | Cite as

Squeezing of Relative and Center-of-Orbit Coordinates of a Charged Particle by Step-Wise Variations of a Uniform Magnetic Field with an Arbitrary Linear Vector Potential

  • Viktor V. DodonovEmail author
  • M. B. Horovits


We consider a quantum charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the linear vector potential A = H(t) [−y(1 + β), x(1 − β)] /2 with a fixed parameter β. The systems with different values of β are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are ellipses for |β| < 1 and hyperbolas for |β| > 1. Using the approximation of the stepwise variation of the magnetic field H(t), we obtain explicit formulas describing the evolution of the principal squeezing in two pairs of noncommuting observables: the coordinates of the center of orbit and relative coordinates with respect to this center. Analysis of these formulas shows that no squeezing can arise for the circular gauge (β = 0). On the other hand, for any nonzero value of β, one can find the regimes of excitations resulting in some degree of squeezing in the both pairs. The maximum degree of squeezing can be obtained for the Landau gauge (|β| = 1) if the magnetic field is switched off and returns to the initial value after some time T, in the limit T → ∞.


circular gauge Landau gauge arbitrary linear gauge stepwise variation center-of-orbit coordinates relative coordinates principal squeezing elliptic and hyperbolic solenoids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Landau, Z. Phys., 64, 629 (1930).ADSCrossRefGoogle Scholar
  2. 2.
    M. H. Johnson and B. A. Lippmann, Phys. Rev., 76, 828 (1949).ADSCrossRefGoogle Scholar
  3. 3.
    J. E. Avron, I. W. Herbst, and B. Simon, Ann. Phys. (NY), 114, 431 (1978).ADSCrossRefGoogle Scholar
  4. 4.
    B. R. Johnson, J. O. Hirschfelder, and K.-H. Yang, Rev. Mod. Phys., 55, 109 (1983).ADSCrossRefGoogle Scholar
  5. 5.
    R. von Baltz, Phys. Lett. A, 105, 371 (1984).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Mielnik and A. Ramírez, Phys. Scr., 84, 045008 (2011).ADSCrossRefGoogle Scholar
  7. 7.
    I. A. Malkin and V. I. Man’ko, Zh. Éksp. Teor. Fiz., 55, 1014 (1968) [English translation: Sov. Phys. – JETP, 28, 527 (1969)].Google Scholar
  8. 8.
    A. Feldman and A. H. Kahn, Phys. Rev. B, 1, 4584 (1970).ADSCrossRefGoogle Scholar
  9. 9.
    W. G. Tam, Physica, 54, 557 (1971).ADSCrossRefGoogle Scholar
  10. 10.
    V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, Physica, 59, 241 (1972).ADSCrossRefGoogle Scholar
  11. 11.
    M. H. Boon, Helv. Phys. Acta, 48, 551 (1975).Google Scholar
  12. 12.
    S. Varró, J. Phys. A: Math. Gen., 17, 1631 (1984).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    K. Kowalski and J. Rembieliński, J. Phys. A: Math. Gen., 38, 8247 (2005).ADSCrossRefGoogle Scholar
  14. 14.
    V. V. Dodonov, E. V. Kurmyshev, and V. I. Man’ko, “Correlated coherent states,” in: A. A. Komar (Ed.), Classical and Quantum Effects in Electrodynamics, Proceedings of the P. N. Lebedev Physical Institute, Nauka, Moscow (1986), Vol. 176, p. 128 [English translation by Nova Science, Commack, New York (1988), p. 169].Google Scholar
  15. 15.
    C. Aragone, Phys. Lett. A, 175, 377 (1993).ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    V. V. Dodonov, V. I. Man’ko, and P. G. Polynkin, Phys. Lett. A, 188, 232 (1994).ADSCrossRefGoogle Scholar
  17. 17.
    M. Ozana and A. L. Shelankov, Fiz. Tverd. Tela, 40, 1405 (1998) [English translation: Phys. Solid State, 40, 1276 (1998)].Google Scholar
  18. 18.
    V. Fock, Z. Phys., 47, 446 (1928).ADSCrossRefGoogle Scholar
  19. 19.
    L. Page, Phys. Rev., 36, 444 (1930).ADSCrossRefGoogle Scholar
  20. 20.
    C. G. Darwin, Math. Proc. Cambridge Philos. Soc., 27, 86 (1931).ADSCrossRefGoogle Scholar
  21. 21.
    I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, Phys. Rev. D, 2, 1371 (1970).ADSCrossRefGoogle Scholar
  22. 22.
    A. Bechler, Phys. Lett. A, 130, 481 (1988).ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    A. Jannussis, E. Vlahos, D. Skaltsas, et al., Nuovo Cimento B, 104, 53 (1989).ADSCrossRefGoogle Scholar
  24. 24.
    M. S. Abdalla, Phys. Rev. A, 44, 2040 (1991).ADSCrossRefGoogle Scholar
  25. 25.
    B. Baseia, Phys. Lett. A, 170, 311 (1992).ADSCrossRefGoogle Scholar
  26. 26.
    B. Baseia, S. S. Mizrahi, and M. H. Y. Moussa, Phys. Rev. A, 46, 5885 (1992).ADSCrossRefGoogle Scholar
  27. 27.
    F. C. Delgado and B. Mielnik, J. Phys. A: Math. Gen., 31, 309 (1998).ADSCrossRefGoogle Scholar
  28. 28.
    J. E. Santos, N. M. R. Peres, and J. M. B. Lopes dos Santos, Phys. Rev. A, 80, 053401 (2009).ADSCrossRefGoogle Scholar
  29. 29.
    H. Takahasi, “Information theory of quantum mechanical channels,” in: A. V. Balakrishnan (Ed.), Advances in Communication Systems. Theory and Applications, Academic Press, New York (1965), Vol. 1, p. 227.Google Scholar
  30. 30.
    V. V. Dodonov and V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, in: M. A. Markov (Ed.), Proceedings of the P. N. Lebedev Physical Institute, Nauka, Moscow (1987), Vol. 183 [English translation by Nova Science, Commack, New York (1989)].Google Scholar
  31. 31.
    V. V. Dodonov and V. I. Man’ko, “Correlated and squeezed coherent states of time-dependent quantum systems,” in: M. Evans and S. Kielich (Eds.), Advances in Chemical Physics, Vol. LXXXV. Modern Nonlinear Optics, pt. 3, Wiley, New York (1994), p. 499.Google Scholar
  32. 32.
    V. V. Dodonov, “Parametric excitation and generation of nonclassical states in linear media,” in: V. V. Dodonov and V. I. Man’ko (Eds.), Theory of Nonclassical States of Light, Taylor & Francis, London (2003), p. 153.Google Scholar
  33. 33.
    J. Janszky and Y. Y. Yushin, Opt. Commun., 59, 151 (1986).ADSCrossRefGoogle Scholar
  34. 34.
    T. Kiss, J. Janszky, and P. Adam, Phys. Rev. A, 49, 4935 (1994).ADSCrossRefGoogle Scholar
  35. 35.
    M. Kira, I. Tittonen, W. K. Lai, and S. Stenholm, Phys. Rev. A, 51, 2826 (1995).ADSCrossRefGoogle Scholar
  36. 36.
    I. Tittonen, S. Stenholm, and I. Jex, Opt. Commun., 124, 271 (1996).ADSCrossRefGoogle Scholar
  37. 37.
    R. Graham, J. Mod. Opt., 34, 873 (1987).ADSCrossRefGoogle Scholar
  38. 38.
    X. Ma and W. Rhodes, Phys. Rev. A, 39, 1941 (1989).ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    A. Lukš, V. Peřinová, and Z. Hradil, Acta Phys. Polon. A, 74, 713 (1988).Google Scholar
  40. 40.
    V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt., 4, R1 (2002).ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    V. V. Dodonov, J. Phys. A: Math. Gen., 33, 7721 (2000).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Physics and International Center for PhysicsUniversity of BrasiliaBrasiliaBrazil

Personalised recommendations