Distribution of resonances in the quantum open baker map

We study relevant features of the spectrum of the quantum open baker map. The opening consists of a cut along the momentum $p$ direction of the 2-torus phase space, modeling an open chaotic cavity. We study briefly the classical forward trapped set and analyze the corresponding quantum nonunitary evolution operator. The distribution of eigenvalues depends strongly on the location of the escape region with respect to the central discontinuity of this map. This introduces new ingredients to the association among the classical escape and quantum decay rates. Finally, we could verify that the validity of the fractal Weyl law holds in all cases.

Figure 1

(Color online) Area of the forward trapped set for the tenth iteration of the open baker map $A_{fw}\left(10\right)$ as a function of the center of the opening $q_c$ for different widths, with $\Delta q=0.2$ (black solid line), $\Delta q=0.1$ (red dotted line), and $\Delta q=0.05$ (green dashed line). The insets show the forward trapped sets (in black) for $t=10$ and $\Delta q=0.05$, for $q_c=0.3$ (left-hand side) and $q_c=0.5$ (right-hand side).

Figure 2

Eigenvalues of the open baker map in the complex plane. In the upper panels we show the case for $N=602$ and in the lower ones for $N=2048$. In the left-hand panels the opening is centered at $q_c=0.3$ and in the right-hand ones at $q_c=0.5$; $\Delta q=0.1$ in all cases.

Figure 3

(Color online) Cumulative number of resonances $n$ as a function of $\nu$. The green short dashed line corresponds to $N=2048$ and the black dotted-dashed line corresponds to $N=602$, both for $q_c=0.5$; the blue long dashed line corresponds to $N=2048$ and the red dotted line corresponds to $N=602$, both for $q_c=0.3$.

Figure 4

(Color online) Histograms corresponding to the eigenvalue distribution $W$ as a function of $\nu$ for $q_c=0.3$ (red dotted lines) and $q_c=0.5$ (black dashed lines). In the upper panel $N=2048$, in the middle panel $N=602$, and in the lower panel $N=1782$.

Figure 5

(Color online) Width $\sigma$ of the eigenvalue distributions $W$ as a function of the dimension of the Hilbert space. The red dotted line corresponds to $q_c=0.5$, and the black solid line corresponds to $q_c=0.3$, being $\Delta q=0.1$ in both cases. Only values for $\nu >0.7$ were considered.

Figure 6

(Color online) Same distributions as in Fig. 4, shown as a function of the decay rate and rescaled with ${\gamma }_{\mathit{cl}}$. Thin red lines correspond to $q_c=0.3$ and thick black lines correspond to $q_c=0.5$. Solid lines correspond to $N=2048$, dashed lines to $N=602$, and dotted-dashed lines to $N=1782$.

Figure 7

(Color online) Logarithmic plot of the fraction of eigenstates $N_{\nu }$ for $\nu >0.3$, as a function of the dimension of the Hilbert space $N$. Lines correspond to the prediction given by the fractal Weyl law. Symbols correspond to the numerically calculated values. In the upper panel $q_c=0.3$ and in the lower panel $q_c=0.5$. In both cases black solid lines and circles correspond to $\Delta q=0.05$, red dotted lines and squares correspond to $\Delta q=0.1$, and green dashed lines and triangles correspond to $\Delta q=0.2$.