Double resonances in the dynamics of quantum systems with a dense discrete spectrum

  • V. A. Benderskii
  • L. N. Gak
  • E. I. Kats
Statistical, Nonlinear, and Soft Matter Physics


The quantum dynamics of a simple model of a nanoparticle has been investigated. This model suggests that the initially prepared state (system) interacts with the other states (reservoir) forming a dense discrete spectrum. In contrast to our previous papers concerned with this problem [1, 2], in which only the dynamics of the system has been studied, the present paper is devoted to the description of the evolution of reservoir states. In the initial recurrence cycles, the reverse transitions from the reservoir to the system generate a double resonance (an echo at frequencies of the reservoir states transitions). Since different states of the reservoir are depleted at different instants of time, the Loschmidt echo in the system is inhomogeneously broadened, whereas the double resonances remain narrower and more intense. Apart from the main resonances, there arise satellites due to the redistribution of the populations between the reservoir states during the cycle. In mixing cycles regime, the regular evolution of the reservoir states (like the system state) transforms into a stochastic-like evolution. It is noted that the predicted double resonances can be experimentally detected and used in analyzing vibrational relaxation of large molecules and nanoparticles.

PACS numbers

03.65-w 82.20.-w 82.20.Bc 


  1. 1.
    V. A. Benderskii, L. A. Falkovsky, and E. I. Kats, Pis’ma Zh. Éksp. Teor. Fiz. 86(3), 249 (2007) [JETP Lett. 86 (3), 221 (2007)].Google Scholar
  2. 2.
    V. A. Benderskii, L. N. Gak, and E. I. Kats, Zh. Éksp. Teor. Fiz. 135(1), 176 (2009) [JETP 108 (1), 159 (2009)].Google Scholar
  3. 3.
    R. Zwanzig, Lect. Theor. Phys. 3, 106 (1960).Google Scholar
  4. 4.
    S. Luzzatto, arXiv:math.DS/0409085.v1.Google Scholar
  5. 5.
    M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).zbMATHGoogle Scholar
  6. 6.
    F. Haake, Quantum Signature of Chaos (Springer, Berlin, 1991).Google Scholar
  7. 7.
    K. Nakamura, Quantum versus Chaos: Questions Emerging from Mesoscopic Cosmos (Kluwer, Dordrecht, 1997).zbMATHGoogle Scholar
  8. 8.
    T. User and W. H. Miller, Phys. Rep. 199, 73 (1991).CrossRefADSGoogle Scholar
  9. 9.
    Time-Resolved Vibrational Spectroscopy, Ed. by S. Takahashi (Springer, Berlin, 1992).Google Scholar
  10. 10.
    V. A. Benderskii, P. A. Stunzhas, E. A. Sokolov, and L. A. Blumenfeld, Nature (London) 220, 365 (1968).CrossRefADSGoogle Scholar
  11. 11.
    A. D. Milov, M. D. Shchirov, V. E. Khmelinskiĭ, and Yu. D. Tsvetkov, Dokl. Akad. Nauk SSSR 218, 878 (1974).Google Scholar
  12. 12.
    K. M. Salikhov, A. G. Semenov, and Yu. D. Tsvetkov, Electron Spin Echo and Its Applications (Nauka, Novosibirsk, 1976) [in Russian].Google Scholar
  13. 13.
    Ya. B. Zeldovich, Zh. Éksp. Teor. Fiz. 39, 776 (1961) [Sov. Phys. JETP 12, 542 (1961)].Google Scholar
  14. 14.
    V. A. Benderskii and E. I. Kats, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 65, 036217 (2002).ADSGoogle Scholar
  15. 15.
    V. I. Arnold, Ordinary Differential Equations (Nauka, Moscow, 1984; Massachusetts Inst. of Technology Press, Cambridge, MA, United States, 1978).Google Scholar
  16. 16.
    G. M. Zaslavsky, Chaos in Dynamical Systems (Nauka, Moscow, 1984; Harwood, New York, 1985).Google Scholar
  17. 17.
    V. Ya. Demikhovskii, F. M. Izrailev, and A. I. Malyshev, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 66, 036211 (2002).MathSciNetADSGoogle Scholar
  18. 18.
    F. M. Izrailev, Phys. Rep. 196, 299 (1990).CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    M. E. Flatte and M. Holthaus, Ann. Phys. (San Diego, CA, United States) 245, 299 (1996).Google Scholar
  20. 20.
    A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983).zbMATHGoogle Scholar
  21. 21.
    G. M. Zaslavsky, Phys. Rep. 80, 159 (1981).CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    U. Weiss, Quantum Dissipative Systems (World Sci., Singapore, 1999).zbMATHGoogle Scholar
  23. 23.
    L. H. Yu and C.-P. Sun, Phys. Rev. A: At., Mol., Opt. Phys. 49, 592 (1994).ADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of Problems of Chemical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia
  2. 2.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia
  3. 3.Institut Laue-LangevinGrenoble, Cedex 9France

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