Russian Journal of Mathematical Physics

, Volume 24, Issue 4, pp 454–464 | Cite as

Bi-states and 2-level systems in rectangular Penning traps

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


We introduce a notion of semiclassical bi-states. They arise from pairs of eigenstates corresponding to tunnel-split eigenlevels and generate 2-level subsystems in a given quantum system. As an example, we consider the planar Penning trap with rectangular electrodes assuming the 3: (−1) resonance regime of the charge dynamics. We demonstrate that, under a small deviation of the rectangular shape of electrodes from the square shape (symmetry breaking), there appear instanton pseudoparticles, semiclassical bi-states, and 2-level subsystems in such a quantum trap.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Feynman, “Simulating Physics with Computers,” Internat. J. Theoret. Phys. 6 (21), 467–488 (1982).ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    A. S. Holevo, Quantum Systems, Channels, Information. A Mathematical Introduction (De Gruyter, Berlin–Boston, 2013).zbMATHGoogle Scholar
  3. 3.
    M. Karasev and V. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization (Amer. Math. Soc. Transl. Math. Monogr. 119, 1993).zbMATHGoogle Scholar
  4. 4.
    B. Helffer and J. Sjöstrand, “Multiple Wells in the Semi-Classical Limit. I,” Comm. Partial Differential Equations 9 (4), 337–408 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 4a.
    B. Helffer and J. Sjöstrand, “Puits Multiples en Limite Semi-Classique. II. Interaction Moléculaire. Symétries. Perturbation,” Ann. Inst. H. Poincaré (A) Physique Théorique 42 (2), 127–212 (1985).zbMATHGoogle Scholar
  6. 5.
    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
  7. 5a.
    E. V. Vybornyi, “Coordinate and Momentum Tunneling in One-Dimensional Quantum Systems with Discrete Spectrum,” Nanostructures. Math. Phys. Model. 1 (12), 5–84 (2015) [Russian].Google Scholar
  8. 6.
    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
  9. 6a.
    F. Galve and G. Werth, “Motional Frequencies in a Planar Penning Trap,” Hyperfine Interact. 174, 397–402 (2007).CrossRefGoogle Scholar
  10. 7.
    J. Goldman and G. Gabrielse, “Optimized Planar Penning Traps for Quantum Information Studies,” Hyperfine Interact. 199, 279–289 (2011).ADSCrossRefGoogle Scholar
  11. 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
  12. 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
  13. 10.
    M. V. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances. I,” in “Quantum Algebras and Poisson Geometry in Mathematical Physics,” Ed. by M. Karasev, Amer. Math. Soc. Transl. 216 (2) (Providence, 2005), 1–18; arXiv: math.QA/0412542Google Scholar
  14. 10a.
    M. V. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances. II,” Adv. Stud. Contemp. Math. 11, 33–56 (2005)MathSciNetzbMATHGoogle Scholar
  15. 10b.
    M. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances. III,” Russ. J. Math. Phys. 13 (2), 131–150 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 11.
    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
  17. 12.
    J. S. Geronimo, O. Bruno, and W. Van Assche, “WKB and Turning Point Theory for Second-Order Difference Equations,” Spectral Methods for Operators of Mathematical Physics, Ed. by J. Janas, P. Kurasov, and S. Naboko (Birkhäuser, Basel, 2004), pp. 101–138.CrossRefGoogle Scholar
  18. 13.
    E. V. Vybornyi, “On the WKB Method for Difference Equations: Weyl Symbol and the Phase Geometry,” Nanostructures. Math. Phys. Model. 2 (15), 5–20 (2016) [Russian].Google Scholar
  19. 14.
    P. Braun, “Discrete Semiclassical Methods in the Theory of Rydberg Atoms in External Fields,” Rev. Modern Phys. 65 (1), 115–161 (1993).ADSCrossRefGoogle Scholar
  20. 15.
    A. Garg, “Application of the Discrete Wentzel–Kramers–Brillouin Method to Spin Tunneling,” J. Math. Phys. 39 (10), 5166–5179 (1998).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 16.
    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
  22. 17.
    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
  23. 18.
    E. M. Chudnovsky and J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
  24. 19.
    T. Pankratova, “Quasimodes and Exponential Splitting of a Hammock,” J. Soviet Math. 62 (6), 3117–3122 (1992).MathSciNetCrossRefGoogle Scholar
  25. 20.
    J. Le Deunff and A. Mouchet, “Instantons Re-Examined: Dynamical Tunneling and Resonant Tunneling,” Phys. Rev. E 81 (4), 046205 (2010).ADSMathSciNetCrossRefGoogle Scholar
  26. 21.
    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
  27. 22.
    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.ADSCrossRefzbMATHGoogle Scholar
  28. 23.
    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).ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Laboratory for Mathematical Methods in Natural Sciences at National Research University “Higher School of Economics”MoscowRussia

Personalised recommendations