Regular-to-Chaotic Tunneling Rates Using a Fictitious Integrable System

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.

Figure 1

Dynamical tunneling rates from a regular island to the chaotic sea for the kicked system [27]: Numerical results (dots) and prediction following from Eq. (3) (lines) vs $1/{\hslash }_{\mathrm{eff}}$ for quantum numbers $m\le 5$. The insets show Husimi representations of the regular states $m=0$ and $m=5$ at $1/h_{\mathrm{eff}}=50$. The prediction of Ref. 14 for $m=0$ with a fitted prefactor is shown (dotted line).

Figure 2

(a)–(c) The classical phase space corresponding to some quantum maps $U$, $U_{\mathrm{reg}}$, and $U_{\mathrm{ch}}$. (d)–(f) Husimi representation of eigenstates of such maps. Eigenstates of $U$ have a regular and a chaotic component, as illustrated in the strongest form of a hybrid state (d). Eigenstates $|{\psi }_{\mathrm{reg}}⟩$ ($|{\psi }_{\mathrm{ch}}⟩$) of $U_{\mathrm{reg}}$ ($U_{\mathrm{ch}}$) are purely regular (chaotic).

Figure 3

Predicted tunneling rate, Eq. (3), normalized by the numerical value for $m=0$, $h_{\mathrm{eff}}=1/32$ vs order $N$ of $H_{\mathrm{reg}}$ corresponding to (a) Fig. 1 ($N=10$) and (b) Fig. 4 ($N=4$).

Figure 4

Tunneling rates for the standard map ($K=2.9$) for $m\le 2$. Prediction of Eq. (3) (lines) and numerical results (dots), obtained using an absorbing boundary at $q=\pm 1/4$ (gray-shaded area of the inset). Prediction of Ref. 14 for $m=0$ with a fitted prefactor (dotted line).