Perturbations and chaos in quantum maps

The local density of states (LDOS) is a distribution that characterizes the effects of perturbations on quantum systems. Recently, a semiclassical theory was proposed for the LDOS of chaotic billiards and maps. This theory predicts that the LDOS is a Breit-Wigner distribution independent of the perturbation strength and also gives a semiclassical expression for the LDOS width. Here, we test the validity of such an approximation in quantum maps by varying the degree of chaoticity, the region in phase space where the perturbation is applied, and the intensity of the perturbation. We show that for highly chaotic maps or strong perturbations the semiclassical theory of the LDOS is accurate to describe the quantum distribution. Moreover, the width of the LDOS is also well represented for its semiclassical expression in the case of mixed classical dynamics.

Figure 1

Classical phase space of the Harper map for $k=0.3$ (left) and $k=200$ (right). See text for details.

Figure 2

Width $\sigma$ vs. $\Gamma$ for a periodized Lorentzian function [Eq. (20)]. The limit of $\sigma$ for $\Gamma \to \infty$ which corresponds to a constant LDOS is plotted with a red dotted line.

Figure 3

$\Gamma$ and the phase $\varphi$ as a function of the scaled perturbation $\chi$ for the perturbed cat map.

Figure 4

Width $\sigma$ of the LDOS as a function of the scaled perturbation strength $\chi =(k-k_0)N$ for the cat map with $G_1$ ($\circ$) and $G_2$ ($\times$). The red solid line is the semiclassical approximation of $\sigma$. The number of states of the Hilbert space is $N=2000$ and $k_0=0.01$. We indicate with arrows the perturbation strength of the LDOS displayed in Fig. 5.

Figure 5

LDOS $\rho$ (points) and its semiclassical approximation (solid line). (a) Cat map of Eq. (3) with $G=G_2$ for a scaled perturbation strength $\chi =159.6$ (main plot), $\chi =14.4$ (left inset), and $\chi =18.0$ (right inset). (b) Cat map of Eq. (3) with $G=G_1$ for $\chi =159.6$, $\chi =14.4$ (left inset), and $\chi =18.0$ (right inset).

Figure 6

Mean value of the amplitude fidelity $\overline{O\left(n\right)}$ for the cat map with $G_1$ ($\circ$) and $G_2$ ($\times$) with perturbation strength $\chi =14.4$. The exponential decay given by $\exp (-\Gamma n)$ is plotted with a solid blue line. We also plot with a dotted red line the exponential decay given by $\exp (-\lambda n/2)$, with $\lambda$ the Lyapunov exponent for the cat map with $G_1$.

Figure 7

$\Gamma$ and the phase $\varphi$ as a function of the scaled perturbation $\chi$ for the cat map when the perturbation is applied in a $q$ strip from $q_0=0.25$ to $q_1=0.46$. The mean value of the quantum LDOS is also plotted ($\square$).

Figure 8

LDOS $\rho$ for local perturbations. The cat map with $G_1$ is perturbed from $q_0=0.25$ to $q_1=0.46$. The scaled perturbation strength is $\chi =8$ ($△$) and $\chi =28$ ($\square$). The semiclassical approximation of the LDOS is plotted with a red solid line. Inset: Width $\sigma$ of the LDOS as a function of the scaled perturbation strength $\chi$ ($\circ$) and, plotted with a red solid line, the semiclassical approximation.

Figure 9

$\Gamma$ as a function of the scaled perturbation strength $\chi$ for the Harper map.

Figure 10

Width $\sigma$ of the LDOS as a function of the scaled perturbation strength $\chi$ for the Harper map with $k_0=0.3$ ($\circ$) and $k_0=200$ ($\times$). The semiclassical approximation ${\sigma }_{\text{sc}}$ is plotted with a red line. We indicate with arrows the perturbation strengths of the LDOS displayed in Fig. 11.

Figure 11

$\rho$ (points) and its semiclassical approximation ${\rho }_{\text{sc}}$ (solid line) for the Harper map. (a) $k_0=200$. In the main plot $\chi =17$ and in the inset $\chi =1.8$. In both plots it is seen that the semiclassical LDOS describes the full quantum result. (b) $k_0=0.3$ (mixed dynamics). In the main plot $\chi =17$ and in the inset $\chi =1.8$.