Flooding of Chaotic Eigenstates into Regular Phase Space Islands

We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect is observed for the example of island chains with just ten islands.

Figure 1

(a) Husimi representation of a chaotic state flooding the regular islands. Shown are three phase-space cells out of $M=1597$. The full curves are classical tori close to the border of the regular islands and the black dots are iterates of a chaotic orbit. This eigenstate extends well into the islands, while having no weight in their central region. (b) Average of the eigenstate over all $M$ cells. (c) For $M=1$ eigenstates concentrate either on the chaotic component (left) or over the $m$th quantized regular torus. For all plots $h_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 1/30$ is used.

Figure 2

Distribution $P(W)$ of the weights $W$ of the eigenstates in the regular islands for $h_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 1/10$. (a) $M=1$: the main peak near $W=0$ is due to chaotic states and two further peaks are due to regular states $m=0,1$ (see the insets). (b) For increasing $M$ the main peak shifts to larger values of $W$ (white line) and the two peaks from the regular states disappear sequentially. (c) $M=10946$: a narrow peak remains around $W=A_{\mathrm{r}\mathrm{e}\mathrm{g}}$. Three phase-space cells of an eigenstate show a complete flooding of the islands.

Figure 3

Fraction $f_{\mathrm{r}\mathrm{e}\mathrm{g}}$ of regular states vs system size $M$ for $h_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 1/5$ (triangles), $1/10$ (circles), and $1/20$ (squares). An approximately linear decrease with $\ln M$ to $f_{\mathrm{r}\mathrm{e}\mathrm{g}}=0$ can be seen (lines are a guide to the eye). Already for small system sizes $M\approx 10$ and small $h_{\mathrm{e}\mathrm{f}\mathrm{f}}$ a significant reduction of regular states is observed.

Figure 4

Weight $W$ in the island vs time $t$ for a wave packet initially started as a coherent state in the chaotic sea at $(x,p)=(0.4,-0.2)$ for system sizes $M=1,13,144,1597$ and $h_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 1/30$. The thick (green) line guides the eye to the linear increase with $\ln t$ until $W=A_{\mathrm{r}\mathrm{e}\mathrm{g}}$. The insets show the time-evolved wave packet averaged over all cells for the case $M=1597$, demonstrating the progressive flooding of the island.