Universal Response of Quantum Systems with Chaotic Dynamics

The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.

Figure 1

Width $\sigma$ of the LDOS as a function of the scaled perturbation strength $\chi =N\delta k$, for a local perturbation. Solid symbols correspond to the quantum case and solid lines to the semiclassical calculation ${\sigma }_{\mathrm{sc}}^{(p)}$. We use $N=800$ and $q_0=0.01$. In panel (a) we use a width $\beta =0.2$ (circles) and $\beta =0.4$ (squares). In panel (b) $\beta =0.7$. Left inset in (b): Schematic figure showing the used local perturbation. The scaled shear [$f(q)\equiv 2\pi \epsilon (q,k)/k$ (with $k$ the strength of the perturbation)] is plotted as a function of $q$. The limits of the perturbed region are indicated with $q_0$ and $q_1$, with $\beta =q_1-q_0$ its width. Right inset in (b): The LDOS $\rho$ for $\beta =0.7$, with $\chi =78.4$ and 89.6. In solid (red) lines we plot periodized Lorentzian functions with the corresponding widths.

Figure 2

Idem Fig. 1 for global perturbations. The perturbation in the main plot is the same as in Fig. 1. For the inset, the perturbation is a shear in momentum and position (see text for details).

Figure 3

$\sigma$ as a function of the perturbation strength $\delta x$ for the stadium billiard in the region of wave number around 200 (solid circles). The semiclassical approximation ${\sigma }_{\mathrm{sc}}$ using Eq. (9) is plotted with a solid line. Inset: Two Bunimovich stadium billiards with differently shaped parameters; the perturbed system is indicated with a dashed line.