Fidelity decay for local perturbations: Microwave evidence for oscillating decay exponents

We study fidelity decay in classically chaotic microwave billiards for a local, pistonlike boundary perturbation. We experimentally verify a predicted nonmonotonic crossover from the Fermi golden rule to the escape-rate regime of the Loschmidt echo decay with increasing local boundary perturbation. In particular, we observe pronounced oscillations of the decay rate as a function of the piston position which quantitatively agree with corresponding theoretical results based on a refined semiclassical approach for local boundary perturbations.

Figure 1

Geometry of the chaotic Sinai-shaped billiard (length of 472 mm, width of 200 mm, and a quarter-circle of radius 70 mm) with a variable pistonlike local boundary deformation. The piston position can be changed from a displacement $h=45\hspace{0.5em}$mm to $h=0\hspace{0.5em}$mm for four different piston widths $w=20,\hspace{0.5em}40,\hspace{0.5em}70,\hspace{0.5em}\text{and}98\hspace{0.5em}$mm. At position $a$ the measuring antenna is introduced. The additional elements were inserted to perform ensemble averages (rotatable ellipse) and to reduce the influence of bouncing balls.

Figure 2

Measured scattering fidelity decay $F(t)$, Eq. (2) (solid lines with symbols) for three different piston displacements $h_1=1\hspace{0.5em}$mm (blue triangles), $h_2=5\hspace{0.5em}$mm (green circles), $h_3=10\hspace{0.5em}$mm (red squares), for a frequency range $17–18$ GHz corresponding to a mean de Broglie wavelength $\mathrm{\lambda ̄}\approx 17\hspace{0.5em}$mm. The dashed lines show the corresponding semiclassical prediction, Eq. (4), for the LE decay, with $\kappa$ chosen as free parameter: ${\kappa }_1=0.26$, ${\kappa }_2=2.78$, and ${\kappa }_3=1.09$, respectively. The time is given in units of the dwell time $1/\gamma$, with $\gamma$ determined from experimental parameters via Eq. (8) with $w=40\hspace{0.5em}$mm.

Figure 3

Decay rate $\kappa$ as a function of piston displacement $h$ for a piston of width $w=40\hspace{0.5em}$mm in a frequency range $17–18\hspace{0.5em}$GHz corresponding to a mean de Broglie wavelength $\mathrm{\lambda ̄}\approx 17\hspace{0.5em}$mm. The asterisks represent the data points obtained from fitting the decay exponent of the measured scattering fidelity. The three cases discussed in Fig. 2 are marked by correspondingly colored symbols. The dashed curve shows the theoretical approximation (7) (valid for $h⩽w$), and the solid curve is a result of the numerical evaluation of the full semiclassical expression (6).

Figure 4

Decay rate $\kappa$ as a function of $\chi =\sqrt{8/3}h\hspace{0.5em}2\pi /\mathrm{\lambda ̄}$. The fuzzy trace depicts the overlayed experimental data and the dashed curve the theoretical prediction (7).

Figure 5

Decay rate $\kappa$ as a function of the displacement $h$ for a thin piston of width $w=20\hspace{0.5em}$mm in a frequency range $17–18\hspace{0.5em}$GHz corresponding to a mean de Broglie wavelength $\mathrm{\lambda ̄}\approx 17\hspace{0.5em}$mm. The asterisks show the data points extracted again from the fit exponent of the exponential decay of the observed scattering fidelity. The dashed and solid curves show the theoretical predictions based on Eq. (7) and the numerical evaluation of Eq. (6).

Figure 6

Examples of correlated trajectory pairs, unperturbed (blue dash-dotted line) and perturbed (red solid line), belonging to sets ${\Omega }_1$ (a), ${\Omega }_3$ (b), and ${\Omega }_5$ (c) (see text in the Appendix).

Figure 7

“Unfolded” representation of correlated trajectory pairs belonging to sets ${\Omega }_1$ (a), ${\Omega }_3$ (b), and ${\Omega }_n$ (c) with an odd integer $n$.

Figure 8

Schematic representation of the regions ${\Omega }_1$, ${\Omega }_3$, and ${\Omega }_5$ [see Eq. (A4)]. Further regions, ${\Omega }_{2k+1}$ with $k⩾3$, contributing to the sum on the right-hand side of Eq. (6) are not shown in the figure; they cluster as narrow stripes “to the right” of ${\Omega }_5$ and approach $\theta =\pi /2$ in the limit $k\to \infty$.