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Article

Non-Lie Top tunneling and Quantum bilocalization in Planar Penning Trap

Mathematical notes. 2016. Vol. 100. No. 6. P. 807-819.

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under breaking its axial symmetry (inclination of the magnetic field) and turning 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 the avoided crossing effect and 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 the tunneling of quantum states of the electron between 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting interms of symplectic area of membranes bounded by invariantly defined instantones.