Russian Journal of Mathematical Physics

, Volume 25, Issue 4, pp 500–508 | Cite as

Bi-Orbital States in Hyperbolic Traps

  • M. KarasevEmail author
  • E. Vybornyi


We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is the sum of a hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under the resonance 1 : (−1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, which define distinct classically allowed regions in the space of motion constants of the ideal trap. The corresponding stationary states of the trapped charge can form bi-orbital states, i.e., a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.


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  1. 1.
    M. Vogel, Particle Confinement in Penning Traps (Springer, 2018).CrossRefGoogle Scholar
  2. 2.
    Trapped Charged Particles and Fundamental Interactions, Ed. by K. Blaum and F. Herfurth (Springer-Verlag, 2008).Google Scholar
  3. 3.
    F. G. Major, V. Gheorghe, and G. Werth, Charged Particle Traps (Springer, 2002).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).ADSCrossRefGoogle Scholar
  5. 5.
    F. Galve, P. Fernandez, and G. Werth, “Operation of a Planar Penning Trap,” Eur. Phys. J. D 40, 201–204 (2006).ADSCrossRefGoogle Scholar
  6. 6.
    F. Galve and G. Werth, “Motional Frequencies in a Planar Penning Trap,” Hyperfine Interact. 174, 397–402 (2007).CrossRefGoogle Scholar
  7. 7.
    J. Goldman and G. Gabrielse, “Optimized Planar Penning Traps for Quantum Information Studies,” Hyperfine Interact. 199, 279–289 (2011).ADSCrossRefGoogle Scholar
  8. 8.
    O. Blagodyreva, M. Karasev, and E. Novikova, “Cubic Algebra and Averaged Hamiltonian for the Resonance 3: (-1) Penning–Ioffe Trap,” Russ. J. Math. Phys. 10 (4), 441–450 (2012).MathSciNetzbMATHGoogle Scholar
  9. 9.
    M. V. Karasev and E. M. Novikova, “Secondary Resonances in Penning Traps. Non-Lie Symmetry Algebras and Quantum States,” Russ. J. Math. Phys. 20 (3), 283–294 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. V. Karasev and E. M. Novikova, “Planar Penning Trap with Combined Resonance and Top Dynamics on Quadratic Algebra,” Russ. J. Math. Phys. 22, 463–468 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. V. Karasev, E. M. Novikova, and E. V. Vybornyi, “Non-Lie Top Tunneling and Quantum Bilocalization in Planar Penning Trap,” Math. Notes 100 (5–6), 807–819 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Karasev, E. Novikova, and E. Vybornyi, “Bi-States and 2-Level Systems in Rectangular Penning Traps,” Russ. J. Math. Phys. 22 (4), (2017).Google Scholar
  13. 13.
    M. Karasev, E. Novikova, and E. Vybornyi, “Instantons Via Breaking Geometric Symmetry in Hyperbolic Traps,” Math. Notes 102 (5–6), 776–786 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Razavy, Quantum Theory of Tunneling (World Scientific, 2003).CrossRefzbMATHGoogle Scholar
  15. 15.
    E. V. Vybornyi, “Tunnel Splitting of the Spectrum and Bilocalization of Eigenfunctions in an Asymmetric Double Well,” Theoret. and Math. Phys. 178 (1), 93–114 (2014).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. D. Landau and E. M. Lifshitz. Quantum Mechanics, Non-Relativistic Theory: Vol. 3 of Course of Theoretical Physics (Gos. izd. RSFSR, Leningrad (1948);) (English transl., Pergamon, Oxford, 1958).zbMATHGoogle Scholar
  17. 17.
    M. J. Davis and E. J. Heller, “Quantum Dynamical Tunneling in Bound States,” J. Chemical Phys. 75 (1), 246–254 (1981).ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Keshavamurthy and P. Schlagheck, Dynamical Tunneling: Theory and Experiment (CRC Press, 2011).CrossRefGoogle Scholar
  19. 19.
    E. V. Vybornyi, “Energy Splitting in Dynamical Tunneling,” Theoret. and Math. Phys. 181 (2), 1418–1427 (2014).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. Y. Dobrokhotov and A. Shafarevich, “Momentum Tunneling between Tori and the Splitting of Eigenvalues of the Laplace–Beltrami Operator on Liouville Surfaces,” Math. Phys. Anal. Geom. 2 (2), 141–177 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Le Deunff and A. Mouchet, “Instantons Re-Examined: Dynamical Tunneling and Resonant Tunneling,” Phys. Rev. E 81 (4), 046205 (2010).ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    E. V. Vybornyi, “On the WKB Method for Difference Equations: Weyl Symbol and the Phase Geometry,” Nanostructures. Math. Phys. and Modelling 2 (15), 5–20 (2016) [in Russian].Google Scholar
  23. 23.
    P. Braun, “WKB Method for Three-Term Recursion Relations and Quasienergies of an Anharmonic Oscillator,” Theoret. and Math. Phys. 37 (3), 1070–1081 (1978).ADSCrossRefGoogle Scholar
  24. 24.
    H. Harada, A. Mouchet, and A. Shudo, “Riemann Surfaces of Complex Classical Trajectories and Tunnelling Splitting in One-Dimensional Systems,” J. Phys. A: Math. Theor. 50 (43), 435204 (2017). arXiv preprint arXiv:1709.10144.ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.National Research University Higher School of Economics Laboratory for Mathematical Methods in Natural SciencesMoscowRussia

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