Long-time saturation of the Loschmidt echo in quantum chaotic billiards

The Loschmidt echo (LE) (or fidelity) quantifies the sensitivity of the time evolution of a quantum system with respect to a perturbation of the Hamiltonian. In a typical chaotic system the LE has been previously argued to exhibit a long-time saturation at a value inversely proportional to the effective size of the Hilbert space of the system. However, until now no quantitative results have been known and, in particular, no explicit expression for the proportionality constant has been proposed. In this paper we perform a quantitative analysis of the phenomenon of the LE saturation and provide the analytical expression for its long-time saturation value for a semiclassical particle in a two-dimensional chaotic billiard. We further perform extensive (fully quantum mechanical) numerical calculations of the LE saturation value and find the numerical results to support the semiclassical theory.

Figure 1

(Color online) Time decay of the Loschmidt echo in the desymmetrized diamond billiard with a boundary deformation for initial wave packets of de Broglie wavelength $\lambda =4\pi$ and dispersion $\sigma =9$. Time is given in units of the free flight time $t_f$ of the corresponding classical particle. The thin (blue) line shows an individual LE decay curve resulted from a single numerical experiment. The thick (red) line represents the result of an averaging over three individual decay curves obtained for different positions ${\mathbf{r}}_0$ of the initial wave packet.

Figure 2

(Color online) Top figure: The Loschmidt echo saturation value $M_{\infty }$ as a function of the dispersion $\sigma$ of the initial wave packet for a fixed de Broglie wavelength $\lambda =4\pi$. Bottom figure: $M_{\infty }$ as a function of $\lambda$ for $\sigma =9$. In both figures the billiard area $A\approx 1.51\times {10}^5$ and the blue dashed line represents the LE saturation value as predicted by Eq. (18).