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


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.


symmetry breaking double-resonance symplectic instanton Kirillov form tunnel splitting 


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  1. 1.
    M. Kretzschmar, “Single particle motion in a Penning trap: Description in the classical canonical formalism,” Phys. Scripta 46, 544–554 (1992).CrossRefGoogle Scholar
  2. 2.
    F. G. Major, V. Gheorghe, and G. Werth, Charged Particle Traps (Springer, 2002).Google Scholar
  3. 3.
    Trapped Charged Particles and Fundamental Interactions, Ed. by K. Blaum and F. Herfurth (Springer-Verlag, 2008).Google Scholar
  4. 4.
    S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint, T. Valenzuela, J. Verdu, M. Vogel, and G. Werth, “A planar Penning trap,” Eur. Phys. J. D 32, 139–146 (2005).CrossRefGoogle Scholar
  5. 5.
    F. Galve and G. Werth, “Motional frequencies in a planar Penning trap,” Hyperfine Interact. 174, 41–46 (2007).CrossRefGoogle Scholar
  6. 6.
    J. Goldman and G. Gabrielse, “Optimized planar Penning traps for quantum information studies,” Hyperfine Interact. 199, 279–289 (2011).CrossRefGoogle Scholar
  7. 7.
    M. V. Karasev and E. M. Novikova, “Secondary resonances in Penning traps. Non-Lie symmetry algebras and quantum states,” Russian J. Math. Phys. 20 (3), 283–294 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. V. Karasev and E. M. Novikova, “Inserted perturbations generating asymptotical integrability,” Math. Notes 96 (5–6), 965–970 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. V. Karasev and E. M. Novikova, “Planar Penning trap with combined resonance and top dynamics on quadratic algebra,” Russian J. Math. Phys. 22, 463–468 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. V. Karasev and E. M. Novikova, “Stable two-dimensional tori in Penning trap under a combined frequency resonance,” Nanostructures. Math. Phys. and Modelling 13 (2), 55–92 (2015).Google Scholar
  11. 11.
    M. V. Karasev, “Quantization due to breaking the commutativity of symmetries. Wobbling oscillator and anharmonic Penning trap,” Russian J. Math. Phys. 24 (4), 483–489 (2016).MathSciNetGoogle Scholar
  12. 12.
    E. V. Vybornyi, “Tunnel splitting of the spectrumand bilocalization of eigenfunctions in an asymmetric double well,” Teoret. Mat. Fiz. 178 (1), 107–130 (2014) [Theoret. and Math. Phys. 178 (1), 93–114 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. F. Pankratova, “Quasimodes and exponential splitting of a hammock,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 195, 103–112 (1991) [J. SovietMath. 62 (6), 3117–3122 (1992)].MathSciNetGoogle Scholar
  14. 14.
    V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Nauka, Moscow, 1976; Springer Science & BusinessMedia, 2001).Google Scholar
  15. 15.
    L. D. Landau and E. M. Lifshitz, Theoretical Physics, Vol. 3: QuantumMechanics. Nonrelativistic Theory (Gos. Izdat. RSFSR, Leningrad, 1948; Pergamon, Oxford, 1958).Google Scholar
  16. 16.
    J. von Neumann and E. P. Wigner, “Uber merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen,” Zhurnal Physik 30, 467–470 (1929).zbMATHGoogle Scholar
  17. 17.
    V. I. Arnol’d, “Modes and quasimodes,” Funktsional. Anal. Prilozhen. 6 (2), 12–20 (1972) [Functional Anal. Appl. 6 (2), 94–101 (1972). ]MathSciNetzbMATHGoogle Scholar
  18. 18.
    S. Keshavamurthy and P. Schlagheck, Dynamical Tunneling: Theory and Experiment (CRC Press, 2011).zbMATHGoogle Scholar
  19. 19.
    M. Razavy, Quantum Theory of Tunneling (World Scientific, 2003).CrossRefzbMATHGoogle Scholar
  20. 20.
    J. LeDeunff and A. Mouchet, “Instantons re-examined: Dynamical tunneling and resonant tunneling,” Phys. Rev. E 81 (4), 046205 (2010).MathSciNetCrossRefGoogle Scholar
  21. 21.
    S. Y. Dobrokhotov and A. Shafarevich, “Momentum tunneling between tori and the splitting of eigenvalues of the Laplace–Beltrami operator on Liouville surfaces,” Mathematical Physics, Analysis and Geometry 2 (2), 141–177 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    V. P. Maslov, “Global exponential asymptotics of solutions of tunnel equations and problems concerning large deviations,” Tr. Mat. Inst. Steklova 163, 150–180 (1984) [Proc. Steklov Inst. Math. 163, 177–209 (1985)].zbMATHGoogle Scholar
  23. 23.
    S. Yu. Dobrokhotov, V. N. Kolokol’tsov, and V. P. Maslov, “Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation hu t = h 2Δu/2 -V (x)u,” Teoret. Mat. Fiz. 87 (3), 561–599 (1991) [Theoret. and Math. Phys. 87 (3), 561–599 (1991)].MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J. Q. Liang and H. J. W. Müller-Kirsten, “Periodic instantons and quantum mechanical tunneling at high energy,” Phys. Rev. D 46 (10), 4685–4690 (1992).CrossRefGoogle Scholar
  25. 25.
    H. J. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (World Scientific, 2006).CrossRefzbMATHGoogle Scholar
  26. 26.
    E. V. Vybornyi, “On the WKB method for difference equations: Weyl symbol and the phase geometry,” Nanostructures. Math. Phys. and Modelling 15 (2), 5–22 (2016).Google Scholar
  27. 27.
    P. Braun, “Discrete semiclassicalmethods in the theory of Rydberg atoms in external fields,” Rev. Mod. Phys. 65 (1), 115–161 (1993).CrossRefGoogle Scholar
  28. 28.
    P. Braun, “WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator,” Teoret. Mat. Fiz. 37 (3), 355–370 (1978) [Theoret. and Math. Phys. 37 (3), 1070–1081 (1978)].MathSciNetGoogle Scholar
  29. 29.
    O. Costin and R. Costin, “Rigorous WKB for finite-order linear recurrence relations with smooth coefficients,” SIAM J. Math. Anal. 27 (1), 110–134 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    E. V. Vybornyi, “Energy splitting in dynamical tunneling,” Teoret. Mat. Fiz. 181 (2), 337–348 (2014) [Theoret. and Math. Phys. 181 (2), 1418–1427 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    A. Garg, “Application of the discreteWentzel–Kramers–Brillouinmethod to spin tunneling,” J. Math. Phys. 39 (10), 5166–5179 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    A. Garg, “Quenched spin tunneling and diabolical points in magnetic molecules. I: Symmetric configurations,” Phys. Rev. B 64 (9), 094413 (2001).CrossRefGoogle Scholar
  33. 33.
    A. Garg, “Quenched spin tunneling and diabolical points in magnetic molecules. II: Asymmetric configurations,” Phys. Rev. B 64 (9), 094414 (2001).CrossRefGoogle Scholar

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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