Transient features of quantum open maps

We study families of open chaotic maps that classically share the same asymptotic properties—forward and backward trapped sets, repeller dimensions, and escape rate—but differ in their short time behavior. When these maps are quantized we find that the fine details of the distribution of resonances and the corresponding eigenfunctions are sensitive to the initial shape and size of the openings. We study phase space localization of the resonances with respect to the repeller and find strong delocalization effects when the area of the openings is smaller than $\hslash$.

Figure 1

Phase space representation of the openings for shift and intersection families with $k⩽3$, in top and bottom panels, respectively. The allowed regions are shown in white, while the corresponding openings are in light blue (gray).

Figure 2

Spectrum in the complex plane for three different BVS quantum open baker maps with dimension $D=3^5=243$. Black circles represent the eigenvalues for the first member of both the shift and intersection families ($k=1$), red (gray) squares stand for the shift family with $k=5$, and blue (dark gray) diamonds stand for intersection with $k=5$.

Figure 3

Distribution of resonances for shift (top panel) and intersection (bottom panel) quantum open baker maps with dimension $D=3^7=2187$. Moduli of eigenvalues are represented by black circles for $k=1,3,5,7$ from top to bottom, respectively. Histograms of these moduli for 100 intervals from 0 to 1 are shown by means of red (gray) lines.

Figure 4

Norm ratio ($\eta$) for ${\widehat{h}}_j$ operators of the shift quantum open baker maps with dimension $D=3^5=243$. Black circles, red (gray) squares, and green (light gray) diamonds represent $k=1$, $k=2,$ and $k=5$ members, respectively. $\eta$ values for ${\widehat{Q}}_j$ with $k=1$, $k=2,$ and $k=5$ are shown in black, red (gray), and green (light gray) dashed lines, respectively. Three Husimi functions $(h_j(q,p)\equiv |\langle q,p|{\widehat{h}}_j|q,p\rangle |)$ for $k=5$ and $j=1$, $j=13,$ and $j=28$ are shown going from zero [blue (dark gray)] to maximum value [red (gray)] on top panels (from left to right). The horizontal dot-dashed blue (dark gray) line is the norm ratio value obtained for the uniform quantum distribution on the repeller $I_{\text{rep}}$.

Figure 5

Norm ratio ($\eta$) for ${\widehat{h}}_j$ operators of the intersection quantum open baker maps with dimension $D=3^5=243$. Black circles, red (gray) squares, and green (light gray) diamonds represent $k=1$, $k=2,$ and $k=5$ members, respectively. $\eta$ values for ${\widehat{Q}}_j$ with $k=1$, $k=2,$ and $k=5$ are shown in black, red (gray), and green (light gray) dashed lines, respectively. Four Husimi functions $(h_j(q,p)\equiv |\langle q,p|{\widehat{h}}_j|q,p\rangle |)$ for $k=1$ and $k=5$ are shown going form zero [blue (dark gray)] to maximum [red (gray)]. The most localized cases corresponding to one periodic orbit scar ($k=1$ and $j=2$) and two periodic orbit scars ($k=5$ and $j=16$) are shown on the right and left bottom panels, respectively. Some cases of uniformly distributed functions on the repeller, for $k=5$ and $j=9$ and $k=1$ and $j=18$, are illustrated on left and right top panels, respectively. The horizontal dot-dashed blue (dark gray) line is the norm ratio value obtained for the uniform quantum distribution on the repeller $I_{\text{rep}}$.

Figure 6

Phase space representation of $\langle q,p\left|I_{\text{rep}}\right|q,p\rangle$ (left top panel) and $Q_{32}$ operators for all members of the shift family with $l=5$. $Q$ functions are shown for $k=1,2,3,4,5$ and $j=2^5=32$ in a $D$-dimensional Hilbert space with $D=3^5=243$. The values of the functions go from zero in blue (dark gray) to the maximum value in red (gray).

Figure 7

Average norm ratio $\langle \eta \rangle$, for both families of quantum open tri-baker maps for $D=3^6=729$. Top and bottom panels show shift and intersection families, respectively. The average norm ratio for the first 20 and 64 eigenstates are shown in red (gray) and black lines, respectively. Black and red (gray) circles represent individual values of norm ratios for the first 20 eigenstates and for the next 42 eigenstates, respectively. The horizontal dashed blue (dark gray) line is the norm ratio value obtained for the uniform quantum distribution on the repeller $I_{\text{rep}}$.