Quantum-to-Classical Crossover of Quasibound States in Open Quantum Systems

In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.

Figure 1

(a) Regions of escape after one to four iterations (as indicated in the figure) in the open standard map (9) with $K=7.5$, and escape for $x\in (0.2,0.4)$ (${\tau }_{\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=5$). The other panels show Husimi representations of quasibound states for Hilbert space dimension $M=160$. (b) and (c) Short-lived states with instantaneous ballistic decay ($|\mu |<{10}^{-8}$), localized on the classical preimage of the opening. (d) and (e) Long-lived states with random-wave characteristics. (f) Trapped long-lived state (only one state of this kind exists for $M=160$; it has the smallest decay rate).

Figure 2

Ordered decay factors $|{\mu }_n|=\exp (-{\gamma }_n/2)$ of the open kicked rotator with $M=2^m\times 80$, $m=0,\dots ,8$, and the other parameters as in Fig. 1a, plotted as a function of the relative index $n/M$ [panel (a)] or of the renormalized relative index $n/\overline{M}$, where $\overline{M}$ is obtained from Eq. (7) [panel (b)]. The solid curve is the RMT result [22]. The dashed curve is this result with $\overline{M}$ fitted to the data.