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Mathematical Notes

, Volume 100, Issue 5–6, pp 807–819 | Cite as

Non-Lie top tunneling and quantum bilocalization in planar Penning trap

  • M. V. KarasevEmail author
  • E. M. Novikova
  • E. V. Vybornyi
Article

Abstract

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under the breaking of its axial symmetry (inclination of the magnetic field) and tuning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation of the voltage on the trap electrode, there exists an avoided crossing effect and a corresponding effect of bilocalization of quantum states on pairs of closed trajectories belonging to common energy levels. This quantum tunneling happens on the symplectic leaves of the symmetry algebra, and hence it generates a tunneling of quantum states of the electron between the 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting in terms of symplectic area of membranes bounded by invariantly defined instantons.

Keywords

symmetry breaking double-resonance symplectic instanton Kirillov form tunnel splitting 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • M. V. Karasev
    • 1
    Email author
  • E. M. Novikova
    • 1
  • E. V. Vybornyi
    • 1
  1. 1.Laboratory for Mathematical Methods in Natural ScienceNational Research University Higher School of EconomicsMoscowRussia

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