Temporal flooding of regular islands by chaotic wave packets

We investigate the time evolution of wave packets in systems with a mixed phase space where regular islands and chaotic motion coexist. For wave packets started in the chaotic sea on average the weight on a quantized torus of the regular island increases due to dynamical tunneling. This flooding weight initially increases linearly and saturates to a value which varies from torus to torus. We demonstrate for the asymptotic flooding weight universal scaling with an effective tunneling coupling for quantum maps and the mushroom billiard. This universality is reproduced by a suitable random matrix model.

Figure 1

Flooding in a system with a mixed phase space. A wave packet started at $t=0$ in the chaotic sea (dots) floods the regular island (closed lines) for large times.

Figure 2

(a) Classical phase space of the designed map with chaotic dynamics (blue dots) and regular tori (red lines). (b) Husimi representation of all regular basis states $m=0,1,2,3$ for $h_{\text{eff}}=\frac{1}{20}$. The border of the island is indicated by a solid line.

Figure 3

(a) The weight $p_{m,{\theta }_q,|{\phi }_0\rangle }^{\mathrm{reg}}$ according to Eq. (8) on the regular torus $m=0$ of the designed map with the initial state $|{\phi }_0\rangle$ being a momentum eigenstate with $p\approx -0.25$ and under variation of the Bloch phase ${\theta }_q$ ($0.2,0.3,0.55,0.75$, blue, red, green, cyan) for $h_{\text{eff}}=\frac{1}{20}$. The black line shows the regular weight $p_m^{\mathrm{reg}}$ according to Eq. (9) averaged over $20000$ pairs ${\theta }_q,|{\phi }_0\rangle$. Here ${\theta }_q\in [0,1)$ is chosen equidistantly and $|{\phi }_0\rangle$ is a momentum eigenstate with $p\in [-0.45,-0.05]$ chosen randomly. (b) Double-logarithmic representation of the same data.

Figure 4

(a) Flooding weights for a wave packet started in the chaotic sea of the designed map with $h_{\text{eff}}=\frac{1}{20}$ for all regular basis states $m=0,1,2,3$. (b) Double-logarithmic representation of the same data together with the predictions of Eq. (20) (solid lines). The Heisenberg time ${\tau }_{\text{H,ch}}$ is indicated by a dashed line. Inset: Phase space for $M=1$.

Figure 5

(a) Flooding weights for a wave packet started in the chaotic sea of the designed map with $h_{\text{eff}}=\frac{144}{2825}\approx \frac{1}{20}$ and $M=144$ for all regular basis states $m=0,1,2,3$. (b) Double-logarithmic representation of the same data together with the predictions of Eq. (20) (solid lines). The Heisenberg time ${\tau }_{\text{H,ch}}$ is indicated by a dashed line.

Figure 6

Asymptotic flooding weights $f^{\infty }$ vs effective coupling $v_{\mathrm{eff}}$ of the designed map for various parameters $h_{\text{eff}}=\frac{1}{10}$, $\frac{1}{20}$, $\frac{1}{30}$, $\frac{1}{40}$, $M=1$, 13,144, 1597, 17711, and $m=1,\cdots ,8$ (black crosses), random matrix prediction (blue solid line), and Eq. (42) (red dashed line).

Figure 7

The regular spectrum of the $m$th torus of a transporting island is equidistant with spacing ${\Delta }_{\text{reg}}$ and the chaotic spectrum is modeled by the circular orthogonal ensemble with mean spacing ${\Delta }_{\text{ch}}$. The typical regular to chaotic coupling is $v_{\mathrm{eff}}$.

Figure 8

Ratio of saturation time $t_{\mathrm{sat}}$ to Heisenberg time ${\tau }_{\text{H,ch}}$ vs. effective coupling $v_{\mathrm{eff}}$ of the designed map for various parameters $h_{\text{eff}}=\frac{1}{10},\frac{1}{20},\frac{1}{30},\frac{1}{40}$, $M=1,13,144,1597,17711$, and $m=1,\cdots ,8$ (black crosses) and Eq. (49) (red dashed line).

Figure 9

Flooding weights $f_m\left(t\right)$ of the standard map for $K=2.9$, $N=29$, and $M=1$, for the regular tori $m=0,1,2$ (symbols). They are compared to the prediction of Eqs. (20) and (42) (solid lines).

Figure 10

Asymptotic flooding weights $f^{\infty }$ (symbols) for the mushroom billiard (inset) with $R=1$, $a=0.7$ vs $v_{\mathrm{eff}}$ compared to the prediction Eq. (42) (red dashed line). As explained in the text, $v_{\mathrm{eff}}$ is either determined analytically (diamonds) or numerically (crosses).

Figure 11

(a) Regular weights $p_m^{\mathrm{reg}}\left(t\right)$ of the designed map for the case of complete flooding, $N=3853335$, $M=196418$, $h_{\text{eff}}=\frac{1}{20}$, for $m=0,1,2,3$ (symbols) compared to Eq. (B5) (dotted lines). (b) Corresponding flooding weights $f_m\left(t\right)$ and the analytical result of Eq. (50) (solid lines, almost indistinguishable from the dotted lines). (c) Double-logarithmic representation of the same data as in (b).