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Loschmidt echo and Lyapunov exponent in a quantum disordered system.

Y Adamov1, I V Gornyi, A D Mirlin

  • 1Institut für Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 6, 2003
PubMed
Summary

We studied how disordered systems respond to perturbations. The fidelity, or return probability, decays exponentially then by a power law for short-range disorder, and exponentially with a Lyapunov exponent for long-range disorder.

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Area of Science:

  • Condensed matter physics
  • Quantum chaos
  • Disordered systems

Background:

  • Disordered systems exhibit complex behavior under perturbation.
  • Fidelity, or the Loschmidt echo, quantifies the sensitivity of quantum systems to Hamiltonian changes.

Purpose of the Study:

  • To investigate the sensitivity of disordered systems to weak external perturbations.
  • To analyze the fidelity decay dynamics for different types of disorder (short-range vs. long-range).

Main Methods:

  • Diagrammatic calculations for short-range scatterers.
  • Path-integral technique and kinetic equation derivation for long-range disorder.

Main Results:

  • For short-range disorder, fidelity shows initial exponential decay (golden rule) followed by power-law decay (diffusive dynamics).

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  • For long-range disorder, an intermediate regime appears, and fidelity decays exponentially, governed by the classical Lyapunov exponent.

    • Conclusions:

    • The fidelity decay in disordered systems depends significantly on the nature of the scatterers (range of disorder).
    • The study provides insights into quantum dynamics and chaos in disordered media.