Quantum Corrections to Fidelity Decay in Chaotic Systems

Abstract

By considering correlations between classical orbits we derive semiclassical expressions for the
decay of the quantum fidelity amplitude for classically chaotic quantum systems, as well as for its
squared modulus, the fidelity or Loschmidt echo. Our semiclassical results for the fidelity amplitude
agree with random matrix theory (RMT) and supersymmetry predictions in the universal Fermi
golden rule regime. The calculated quantum corrections can be viewed as arising from a static ran-
dom perturbation acting on nearly self-retracing interfering paths, and hence will be suppressed for
time-varying perturbations. Moreover, using trajectory-based methods we show a relation, recently
obtained in RMT, between the fidelity amplitude and the cross-form factor for parametric level cor-
relations. Beyond RMT, we compute Ehrenfest-time effects on the fidelity amplitude. Furthermore
our semiclassical approach allows for a unified treatment of the fidelity, both in the Fermi golden
rule and Lyapunov regimes, demonstrating that quantum corrections are suppressed in the latter.