Loschmidt-echo decay from local boundary perturbations

We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensional chaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analytically and numerically, the time decay of the Loschmidt echo (LE). We find the LE to decay exponentially in time, with the rate equal to the classical escape rate from an open billiard obtained from the original one by removing the perturbation-affected region of its boundary. Finally, we propose a principal scheme for the experimental observation of the LE decay.

Figure 1

An unperturbed, chaotic billiard (solid line), together with the perturbation (dashed line). The boundary of the unperturbed billiard consists of two segments, ${\mathcal{B}}_0$ and ${\mathcal{B}}_1$. The perturbation replaces the latter segment by ${\tilde{\mathcal{B}}}_1$, rendering the perturbed billiard to be bounded by ${\mathcal{B}}_0$ and ${\tilde{\mathcal{B}}}_1$. The initial Gaussian wave packet is centered at ${\mathbf{r}}_0$. Three possible types of trajectories, $s_0$, $s_1$, and ${\tilde{s}}_1$, leading from ${\mathbf{r}}_0$ to another point $\mathbf{r}$, are shown.

Figure 2

Forward-time wave packet evolution in the unperturbed DD billiard, followed by the reversed-time evolution in the perturbed billiard. The initial Gaussian wave packet is characterized by the size $\sigma =12$ and de Broglie wavelength $ƛ=15∕\pi$; the arrow shows the momentum direction of the initial wave packet. The DD billiard is characterized by $L=400$ and $R=200\sqrt{10}$. The perturbation is defined by $w=60$ and $r=30$. The propagation time corresponds to approximately ten collisions of the classical particle.

Figure 3

(Color online) The Loschmidt-echo (LE) decay in the DD billiard for four different values of the curvature radius $r$ of the arc perturbation. The width of the perturbation region is fixed, $w=60$. The other system parameters are $L=400$, $R=200\sqrt{10}$, $\sigma =3$, and $ƛ=5∕\pi$. The solid straight line gives the trend of the $\exp (-2\gamma t)$ decay, with $\gamma$ given by Eq. (9). The inset presents the decay of the average LE, with averaging performed over individual LE curves corresponding to different values of $r$.