Long-time signatures of short-time dynamics in decaying quantum-chaotic systems

We analyze the decay of classically chaotic quantum systems in the presence of fast ballistic escape routes on the Ehrenfest time scale. For a continuous excitation process, the form factor of the decay cross section deviates from the universal random-matrix result on the Heisenberg time scale, i.e., for times much larger than the time for ballistic escape. We derive an exact analytical description and compare our results with numerical simulations for a dynamical model.

Figure 1

(Color online) Form factor $⟨\tilde{C}\left(t\right)⟩$ times $e^{Nt}$ for the case of time-reversal-invariant decay (real initial coherent wave packet in the open kicked rotor) and for different amounts of overlap of the initial state with the regions of fast decay (solid lines). Here, $M=4096$ and $N=50$. The corresponding theoretical results [Eq. (7), with $\mid {\alpha }^{+}⟩=\mid {\alpha }^{-}⟩$] are shown with dashed lines. From top to bottom ${\parallel {\alpha }^{\pm }\parallel }^2=0,0.25,0.417,0.5$.

Figure 2

Numerical data and theory for $Y\left(t\right)$, for $N=20$ channels (left panels) and $N=50$ channels (right panels). The overlap of the initial state with the regions of fast decay is given by ${\parallel {\alpha }^{-}\parallel }^2=0$ (a), 0.595 (b), 0.907 (c), 0 (d), 0.571 (e), and 0.906 (f). Areas in gray represent the range of the standard deviation of $Y\left(t\right)$, while areas in black represent the standard deviation for $⟨Y\left(t\right)⟩$ when the average is taken over 100 samples. The dashed lines show the theoretical curves for the different values for ${\parallel {\alpha }^{-}\parallel }^2$; the respective theoretical curve corresponding to the numerical result is plotted in light gray.