Classical Analog of Linear and Quasi-Linear Quantum Tunneling

  • Leonid I. ManevitchEmail author
  • Agnessa Kovaleva
  • Valeri Smirnov
  • Yuli Starosvetsky
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)


In this part of the report, we develop an analytical framework to investigate irreversible energy transfer in a system of two unforced weakly coupled oscillators with slowly time-varying frequencies. In the system under consideration, one of the oscillators is initially excited by an initial impulse, while the second one is initially at rest. As shown in the previous sections, these initial conditions provide motion along the LPT with a maximum possible energy transfer from the excited oscillator to the second one.


  1. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA (2000)zbMATHGoogle Scholar
  2. Ishkhanyan, A., Joulakian B., Suominen, K.-A.: Variational ansatz for the nonlinear Landau–Zener problem for cold atom association. J. Phys. B. 42, 221002–221006 (2009)Google Scholar
  3. Itin, A.P., Törmä, P.: Dynamics of a many-particle Landau-Zener model: inverse sweep. Phys. Rev. A. 79, 055602(1–3) (2009)Google Scholar
  4. Itin, A.P., Törmä, P.: Dynamics of quantum phase transitions in Dicke and Lipkin-Meshkov-Glick models, arXiv:0901.4778v1 (2010)
  5. Itin, A.P., Watanabe, S.: Universality in nonadiabatic behavior of classical actions in nonlinear models with separatrix crossings. Phys. Rev. E. 76, 026218(1–16) (2007)Google Scholar
  6. Khomeriki, R.: Multiple Landau-Zener tunnelling in two weakly coupled waveguide arrays. Euro. Phys. J. D: Atom. Mole. Opt. Phys 61, 193–197 (2011)CrossRefGoogle Scholar
  7. Kosevich, YuA, Manevitch, L.I., Manevitch, E.L.: Vibrational analogue of nonadiabatic Landau-Zener tunneling and a possibility for the creation of a new type of energy traps. Phys. Uspekhi 53, 1281–1287 (2010)CrossRefGoogle Scholar
  8. Kovaleva, A., Manevitch L.I.: Classical analog of quasilinear Landau-Zener tunneling. Phys. Rev. E. 85, 016202(1–8) (2012)Google Scholar
  9. Kovaleva, A., Manevitch, L.I., Kosevich, Y.A.: Fresnel integrals and irreversible energy transfer in an oscillatory system with time-dependent parameters. Phys. Rev. E 83, 026602(1–12) (2011)Google Scholar
  10. Landau, L.D.: Zur Theorie der Energieubertragung. II, Phys. Z. Sowjetunion 2, 46–51 (1932)Google Scholar
  11. de Lima Jr. M.M., Kosevich, Y.A., Santos, P.V., Cantarero, A.: Surface acoustic Bloch oscillations, the Wannier-Stark ladder, and Landau-Zener tunneling in a solid. Phys. Rev. Lett. 104, 165502(1–4) (2010)Google Scholar
  12. Liu, J., Fu, L., Ou, B.-Y., Chen, S.-G., Choi, D.-I., Wu, B., Niu, Q.: Theory of nonlinear Landau-Zener tunneling. Phys. Rev. A 66, 023404(1–7) (2002)Google Scholar
  13. Manevitch, L.I., Kovaleva, A.: Nonlinear energy transfer in classical and quantum systems. Phys. Rev. E 87, 022904(1–12) (2013)Google Scholar
  14. Manevitch, L.I., Kosevich, Y.A., Mane, M., Sigalov, G., Bergman, L.A., Vakakis, A.F.: Towards a new type of energy trap: classical analog of quantum Landau-Zener tunneling. Int. J. Nonlinear Mech 46, 247–252 (2011)CrossRefGoogle Scholar
  15. Nakamura, H.: Nonadiabatic Transitions: Concepts, Basic Theories and Applications. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  16. Rosam, B., Leo, K., Glück, M., Keck, F., Korsch, H.J., Zimmer, F., Köhler, K.: Lifetime of Wannier-Stark states in semiconductor superlattices under strong Zener tunneling to above-barrier bands. Phys. Rev. B 68, 125301(1–7) (2003)Google Scholar
  17. Sahakyan, N., Azizbekyan, H., Ishkhanyan, H., Sokhoyan, R., Ishkhanyan, A.: Weak coupling regime of the Landau-Zener transition for association of an atomic Bose-Einstein condensate. Laser Phys 20, 291–297 (2010)CrossRefGoogle Scholar
  18. Saito, K., Wubs, M., Kohler, S., Hanggi, P., Kayanuma, Y.: Quantum state preparation in circuit QED via Landau-Zener tunneling. Europhys. Lett 76, 22–28 (2006)CrossRefGoogle Scholar
  19. Sanchis-Alepuz, H., Kosevich, Y.A., Sanchez-Dehesa, J.: Acoustic analogue of Bloch oscillations and Zener tunneling in ultrasonic superlattices. Phys. Rev. Lett. 98, 134301(1–4) (2007)Google Scholar
  20. Trimborn, F., Witthaut, D., Kegel, V., Korsch, H.J.: Nonlinear Landau–Zener tunneling in quantum phase space. New J. Phys. 12, 05310(1–20) (2010)Google Scholar
  21. Trompeter, H., Pertsch, T., Lederer, F., Michaelis, D., Streppel, U., Bräuer, A., Peschel, U.: Visual observation of Zener tunneling. Phys. Rev. Lett. 96, 023901(1–4) (2006)Google Scholar
  22. Zener, C.: Non-adiabatic crossing of energy levels. Proc. R. Soc. London A 137, 696–702 (1932)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Leonid I. Manevitch
    • 1
    Email author
  • Agnessa Kovaleva
    • 2
  • Valeri Smirnov
    • 1
  • Yuli Starosvetsky
    • 3
  1. 1.Institute of Chemical PhysicsRussian Academy of ScienceMoscowRussia
  2. 2.Space Research InstituteRussian Academy of ScienceMoscowRussia
  3. 3.Technion—Israel Institute of TechnologyFaculty of Mechanical EngineeringHaifaIsrael

Personalised recommendations