Experimental Verification of Fidelity Decay: From Perturbative to Fermi Golden Rule Regime

The scattering matrix was measured for a flat microwave cavity with classically chaotic dynamics. The system can be perturbed by small changes of the geometry. We define the “scattering fidelity” in terms of parametric correlation functions of scattering matrix elements. In chaotic systems and for weak coupling, the scattering fidelity approaches the fidelity of the closed system. Without free parameters, the experimental results agree with random matrix theory in a wide range of perturbation strengths, reaching from the perturbative to the Fermi golden rule regime.

Figure 1

Logarithmic plot of the correlation function $\widehat{C}[S_{11}^{*},S_{11}^{\prime }]$ for $\nu =5–6\mathrm{GHz}$, $l=1\mathrm{mm}$, and $\lambda =0.047$. The experimental results for the autocorrelation are shown in black, while the correlation of perturbed and unperturbed system are shown in gray/orange. The smooth solid curve corresponds to the theoretical autocorrelation function, and the dashed curve to the product of autocorrelation function and fidelity amplitude. The inset shows the billiard geometry used. Moveable parts are marked with an arrow. See text for dimensions.

Figure 2

Logarithmic plot of the scattering fidelity amplitude $f_{11}(t)$ for $\nu =5–6\mathrm{GHz}$, $l=1\mathrm{mm}$, and $\lambda =0.047$. The smooth curve shows the linear-response result for the fidelity amplitude $f(t)$, where the perturbation strength $\lambda$ was obtained from the variance of the level velocities.

Figure 3

Logarithmic plot of the fidelity amplitude $f(t)$ for three different perturbation strengths. Linear-response result (smooth solid gray line), exact theoretical result (dark dashed line), and experimental result (solid dark line). The perturbation parameter $\lambda$ has been fitted to each experimental curve. The parameters were $\nu =3–4\mathrm{GHz}$, $l=0.4\mathrm{mm}$, $\lambda =0.01$ for (a), $\nu =13–14\mathrm{GHz}$, $l=0.6\mathrm{mm}$, $\lambda =0.13$ for (b), and $\nu =16–17\mathrm{GHz}$, $l=0.8\mathrm{mm}$, $\lambda =0.21$ for (c).

Figure 4

(a) Perturbation strength ${\lambda }^2$ as a function of the shift $l$ of the billiard wall for the frequency windows $\nu =3–4$ (stars), 9–10 (diamonds), and 16–17 GHz (triangles). The slope of the straight lines is 2. (b) ${\lambda }^2$ as a function of the frequency $\nu$ for $l=0.2$ (stars), 0.6 (diamonds), and 2.0 mm (triangles). The slope of the straight lines is 3.