Algebraic fidelity decay for local perturbations

TL;DR

This paper investigates how local perturbations, specifically the movement of a scatterer in a microwave billiard, cause fidelity decay, deriving an algebraic decay law and confirming it through experiments.

Contribution

It provides an analytic expression for fidelity decay due to local scatterer shifts using the random-plane-wave conjecture, validated by experiments.

Findings

Fidelity decay follows an algebraic 1/t law for long times.

Experimental results agree with theoretical predictions.

Scaling behavior depends on the scatterer shift distance.

Abstract

From a reflection measurement in a rectangular microwave billiard with randomly distributed scatterers the scattering and the ordinary fidelity was studied. The position of one of the scatterers is the perturbation parameter. Such perturbations can be considered as {\em local} since wave functions are influenced only locally, in contrast to, e. g., the situation where the fidelity decay is caused by the shift of one billiard wall. Using the random-plane-wave conjecture, an analytic expression for the fidelity decay due to the shift of one scatterer has been obtained, yielding an algebraic $1/t$ decay for long times. A perfect agreement between experiment and theory has been found, including a predicted scaling behavior concerning the dependence of the fidelity decay on the shift distance. The only free parameter has been determined independently from the variance of the level velocities.