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Measuring the Lyapunov exponent using quantum mechanics.

F M Cucchietti1, C H Lewenkopf, E R Mucciolo

  • 1Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 15, 2002
PubMed
Summary

This study reveals how chaotic systems exhibit exponential decay in wave packet overlap. Weak perturbations follow Fermi

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Area of Science:

  • Quantum mechanics
  • Chaos theory
  • Statistical physics

Background:

  • Understanding wave packet dynamics is crucial in quantum mechanics.
  • Chaotic systems exhibit sensitive dependence on initial conditions.
  • The overlap function quantifies the similarity between quantum states over time.

Purpose of the Study:

  • To investigate the time evolution of wave packet overlap in chaotic systems.
  • To determine the conditions leading to exponential decay of the overlap function.
  • To connect quantum decay rates to classical chaos measures.

Main Methods:

  • Theoretical analysis of wave packet dynamics under slightly different Hamiltonians.
  • Application of Fermi's golden rule for weak perturbations.

Related Experiment Videos

  • Comparison with classical Lyapunov exponents for larger perturbations.
  • Numerical simulations of a two-dimensional dynamical system.

    • Main Results:

    • Established conditions for exponential decay of wave packet overlap in chaotic systems.
    • Demonstrated that weak perturbations lead to decay governed by Fermi's golden rule.
    • Showed that stronger perturbations result in decay times related to the system's Lyapunov exponent.
    • Confirmed theoretical predictions through numerical investigation of overlap decay.

    Conclusions:

    • The study provides a unified framework for understanding wave packet decay in chaotic systems.
    • Links quantum mechanical wave packet dynamics to classical chaos indicators.
    • Offers insights into decoherence and information loss in quantum chaotic systems.