Fractal Weyl laws for quantum decay in dynamical systems with a mixed phase space

Weyl’s law approximates the number of states in a quantum system by partitioning the energetically accessible phase-space volume into Planck cells. Here, we show that resonances in open quantum systems can follow a modified fractal Weyl law, even when their classical dynamics is not globally chaotic but also contains domains of regular motion. Using an appropriate phase-space representation for open quantum systems, we connect this behavior to emerging quantum-to-classical correspondence.

Figure 1

(Color online) Quantum versus classical escape in a kicked rotator (kicking strength $k=2.0$) with opening at $0<q<0.2$. (a) Classical escape zones in phase space $\left(q,p\right)$, color coded according to escape after one (red), two (blue), three (green), or four (magenta) iterations (for gray scale see the markers in the figure). The black dots are trajectory segments in the closed system, whose phase space is a mixture of regular and chaotic motion. (b) Husimi representation of short-lived quantum resonances with decay time ${\tau }_n<1/\left(2\text{ln}10\right)$ (decay factor $\left|{\mu }_n\right|<0.1$), for Hilbert-space dimension $M=1280$. The other panels show the fast [panels (c) and (d), $\left|{\mu }_n\right|<0.1$] and slow [panels (e) and (f), $\left|{\mu }_n\right|>0.98$] Husimi-Schur representation, introduced in this work, for [(c) and (e)] $M=160$ and [(d) and (f)] $M=1280$.

Figure 2

(Color online) Fractal Weyl law for resonances in the open kicked rotator with a mixed classical phase space (kicking strength $k=2.0$, opening at $0<q<0.2$). (a) Fraction $P\left(\mu \right)$ of resonances with decay factor $\left|{\mu }_n\right|>\mu$, corresponding to a lifetime ${\tau }_n>-1/\left(2\text{ln}\mu \right)$. (b) Fraction $P_{\text{int}}$ of resonances in a band of intermediate lifetimes $\left(0.1<\left|{\mu }_n\right|<0.98\right)$, as a function of dimensionless system size $M$. In this double-logarithmic representation, the fractal Weyl law results in a linear dependence with a negative slope.

Figure 3

(Color online) Same as Fig. 2, but with opening shifted to $0.2<q<0.4$.

Figure 4

(Color online) (a) Classical escape zones and (b) fast Husimi-Schur representation of short-lived quantum resonances $\left(M=1280\right)$ for the kicked rotator in Fig. 1, with opening shifted to $0.2<q<0.4$.

Figure 5

(Color online) Fractal Weyl law for resonances in the open kicked rotator with $k=2.5$ and various positions of the opening: (a) opening at $0<q<0.2$; (b) opening at $0.2<q<0.4$; (c) opening at $0.4<q<0.6$. First column: fraction $P\left(\mu \right)$ of resonances with decay factor $\left|{\mu }_n\right|>\mu$, corresponding to a lifetime ${\tau }_n>-1/\left(2\text{ln}\mu \right)$. Second column: fraction $P_{\text{int}}$ of resonances in a band of intermediate lifetimes $\left(0.1<\left|{\mu }_n\right|<0.98\right)$, as a function of dimensionless system size $M$. In this double-logarithmic representation, the fractal Weyl law results in a linear dependence with a negative slope. Third column: classical escape zones. Last column: fast Husimi-Schur representation of short-lived quantum resonances $\left(M=1280\right)$.

Figure 6

Phase-space portraits of the kicked rotator with kicking strengths (a) $k=2.5$, (b) $k=5.5$, and (c) $k=7.5$.

Figure 7

(Color online) Same as Fig. 5, but for kicking strength $k=5.5$.

Figure 8

(Color online) Same as Fig. 5, but for kicking strength $k=7.5$, for which the system is strongly chaotic (since the phase space is globally chaotic, results are shown for only one position of the opening $\left(0.4<q<0.6\right)$; see also Ref. [4]).