Classical transients and the support of open quantum maps

The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that the classical repeller—the set of nonescaping orbits in the future and past evolution—is the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case. For this purpose we study recently introduced families of tribaker maps [L. Ermann, G. G. Carlo, J. M. Pedrosa, and M. Saraceno, Phys. Rev. E 85, 066204 (2012)], which share the same asymptotic properties but differ in their short-time behavior. We have found that although the eigenvalue distribution of the evolution operator of these maps follows the fractal Weyl law prediction, the theory of short periodic orbits for open maps fails to describe the resonance eigenfunctions of some of them. This is a strong indication that new elements must be included in the semiclassical description of open quantum systems. We provide an interpretation of the results in order to have hints about them.

Figure 1

Iteration of the classical open baker map. The unit square divided into nine regions by means of three different gray intensities (three different colors) is shown at the top left. One iteration of the colored unit square with the open baker map is shown for $k=1$ (shift or intersection family) at the top right and for $k=2$ for the shift (bottom left) and intersection (bottom right) cases.

Figure 2

Ordered decay factor ${\mu }_n$ of the usual open tribaker map as a function of $n/N$. Hilbert dimensions $N=3^l$ with $l=4$ [open (green) squares], $l=5$ [filled (blue) squares], $l=6$ [open (turquoise) circles], $l=7$ [filled (red) circles], $l=8$ [open (gray) triangles], and $l=9$ [filled (black) triangles]. Inset: Fraction of states $N_{\mu }$ with decay rate $\mu >0.1$ as a function of the Hilbert-space dimension. The solid (red) line corresponds to the prediction of the FWL ($N_{\mu }\sim N^{\ln \left(2\right)/\ln \left(3\right)}$).

Figure 3

Ordered decay factor ${\mu }_n$ of the open map ${\tilde{B}}_3^s$ as a function of $n/N$. The Hilbert-space dimension is $N=3^l$ with $l=4$ [open (green) squares], $l=5$ [filled (blue) squares], $l=6$ [open (turquoise) circles], $l=7$ [filled (red) circles], $l=8$ [open (gray) triangles], and $l=9$ [filled (black) triangles]. Inset: Fraction of states $N_{\mu }$ with decay rate $\mu >0.1$ as a function of the Hilbert-space dimension. The solid (red) line corresponds to the prediction of the FWL ($N_{\mu }\sim N^{\ln \left(2\right)/\left(\ln \left(3\right)\right)}$).

Figure 4

Ordered decay factor ${\mu }_n$ of the open map ${\tilde{B}}_4^s$ as a function of $n/N$. The Hilbert-space dimension is $N=3^l$ with $l=4$ [open (green) squares], $l=5$ [filled (blue) squares], $l=6$ [open (turquoise) circles], $l=7$ [filled (red) circles], $l=8$ [open (gray) triangles], and $l=9$ [filled (black) triangles]. Inset: Fraction of states $N_{\mu }$ with decay rate $\mu >0.1$ as a function of the Hilbert-space dimension. The solid (red) line corresponds to the prediction of the FWL ($N_{\mu }\sim N^{\ln \left(2\right)/\left(\ln \left(3\right)\right)}$). Filled squares plot $N=3^l$ with $l=3$, 4, 5, 6, 7, 8, and 9; dotted lines, $N=81\times i$, with $i=2,3,\cdots$.

Figure 5

Ordered decay factor ${\mu }_n$ of the open map ${\tilde{B}}_2^i$ as a function of $n/N$. The Hilbert-space dimension is $N=3^l$ with $l=4$ [open (green) squares], $l=5$ [filled (blue) squares], $l=6$ [open (turquoise) circles], $l=7$ [filled (red) circles], $l=8$ [open (gray) triangles], and $l=9$ [filled (black) triangles]. Inset: Fraction of states $N_{\mu }$ with decay rate $\mu >0.1$ as a function of the Hilbert-space dimension. The solid (red) line corresponds to the prediction of the FWL ($N_{\mu }\sim N^{\ln \left(2\right)/\left(\ln \left(3\right)\right)}$).

Figure 6

Ordered decay factor ${\mu }_n$ of the open map ${\tilde{B}}_3^i$ as a function of $n/N$. The Hilbert-space dimension is $N=3^l$ with $l=4$ [open (green) squares], $l=5$ [filled (blue) squares], $l=6$ [open (turquoise) circles], $l=7$ [filled (red) circles], $l=8$ [open (gray) triangles], and $l=9$ [filled (black) triangles]. Inset: Fraction of states $N_{\mu }$ with decay rate $\mu >0.1$ as a function of the Hilbert-space dimension. The solid (red) line corresponds to the prediction of the FWL ($N_{\mu }\sim N^{\ln \left(2\right)/\left(\ln \left(3\right)\right)}$).

Figure 7

Performance $P$ of the short POs approach for the case of the shift family: fraction of long-lived resonances found as a function of the number of POs used in the calculations $N_{POs}$. Hilbert space dimension is $N=3^l$ with $l=5$, all possible $k$ are shown. (Color) gray lines with squares correspond to $k=1$, circles to $k=2$, triangles up to $k=3$, triangles down to $k=4$, and diamonds to $k=5$.

Figure 8

Performance $P$ of the short POs approach for the case of the intersection family: fraction of long-lived resonances found as a function of the number of POs used in the calculations $N_{POs}$. Hilbert space dimension is $N=3^l$ with $l=5$, all possible $k$ are shown. We use the same colors and symbols as in Fig. 7.

Figure 9

Performance $P$ of the short PO approach in the case of the shift family: fraction of long-lived resonances found as a function of the number of POs used in the calculations $N_{\mathrm{POs}}$. The Hilbert-space dimension is $N=3^l$ with $l=6$; all possible $k$ values are shown. Line with squares corresponds to $k=1$; line with circles, to $k=2$; line with upward-pointing triangles, $k=3$; line with downward-pointing triangles, $k=4$; line with diamonds, to $k=5$; and line with pentagons, to $k=6$.

Figure 10

Performance $P$ of the short PO approach in the case of the intersection family: fraction of long-lived resonances found as a function of the number of POs used in the calculations $N_{\mathrm{POs}}$. The Hilbert-space dimension is $N=3^l$ with $l=6$; all possible $k$ values are shown. Colors and symbols are the same as in Fig. 9.

Figure 11

Phase-space representation of $Q_{32}$ for the exact and short PO approach with $l=5$ and $N_{\mathrm{POs}}=32$. A map with $k=1$ (both shift and intersection families) is shown (a) for exact eigenstates and (b) for the short PO approach. For cases of the shift family a map with $k=3,4$ shows (c, e) the exact eigenstates and (d, f) the short PO approach, respectively. The intensities of the phase-space representation are white for a 0 value and dark gray (red) for the maximum.