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Related Experiment Videos

Quantum resonances and regularity islands in quantum maps

Sokolov1, Zhirov, Alonso

  • 1Budker Institute of Nuclear Physics, Novosibirsk, Russia. vsokolov@inp.nsk.su

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|October 14, 2000
PubMed
Summary

This study analyzes quantum maps near resonances, revealing that specific quantum resonances, particularly those with low orders, dominate dynamics for extended periods. Analytical and numerical methods explain resonant energy growth and saturation effects.

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Area of Science:

  • Quantum dynamics
  • Quantum chaos
  • Mathematical physics

Background:

  • Quantum maps are discrete-time quantum dynamical systems.
  • Quantum resonances occur when a system's energy levels align with external driving frequencies.
  • The unitary unimodular group SU(q) describes symmetries in quantum systems with q degrees of freedom.

Purpose of the Study:

  • To analytically and numerically investigate the dynamics of a quantum map near a quantum resonance of order q.
  • To understand the role of the quasi-Hamiltonian and gauge transformations in resonant energy growth.
  • To explore the connection between quantum motion and classical nonlinear resonance phenomena.

Main Methods:

  • Analytical and numerical study of quantum map dynamics.

Related Experiment Videos

  • Embedding the quantum map into a continuous unitary transformation generated by a quasi-Hamiltonian.
  • Power expansion of the quasi-Hamiltonian near resonance.
  • Relating the problem to the motion of a quantum particle in a magnetic field.

    • Main Results:

    • Resonant energy growth is linked to zero Liouville eigenmodes of the generator in the SU(q) group's adjoint representation.
    • Nonzero modes contribute a time-saturating effect.
    • Resonances with orders much smaller than the localization length (q << l) significantly influence motion for long durations.
    • Explicit analytical solutions are found for the strongest, lowest-order resonances.

    Conclusions:

    • Low-order quantum resonances play a critical role in controlling quantum map dynamics over extended time scales.
    • The study provides a framework for understanding quantum chaotic behavior near resonances using group theory and classical analogies.
    • Analytical solutions for specific resonances offer insights into the long-term evolution of quantum systems.