Algebraic Fidelity Decay for Local Perturbations

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 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.

Figure 1

Part of the reflection spectrum for two perturber positions differing by $|\Delta r|=4\mathrm{mm}$. The inset shows a sketch of the used microwave billiard (scatterers are shown 3 times larger). The antenna position is marked by a cross.

Figure 2

Scattering fidelity for different shifts of the perturber $|\Delta r|=1\mathrm{mm}$ (open circles, black), 2 mm (crosses, blue), and 4 mm (closed squares, red). The solid line corresponds to the theoretical prediction of Eq. (10). The results were obtained for the frequency range from 3.5 to 6 GHz. The inset shows the same results in a double-logarithmic plot.

Figure 3

Same as Fig. 2, but plotted on a rescaled time axis using the scaling property of Eq. (10).

Figure 4

Ordinary fidelity determined as obtained from the resonance position and depths, see text. The meaning of the symbols is the same as in Fig. 2.

Figure 5

Scattering fidelity, obtained from a superposition of the results from all perturber shifts (blue crosses) and ordinary fidelity (red squares) on a rescaled time axis. The solid line corresponds to the theoretical prediction.