Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given $\hslash$ regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.

Figure 1

Dynamical tunneling rates $\gamma$ from the innermost quantized torus ($m=0$) of a regular island with one dominant $5:1$ resonance. Numerical data (dots) are compared to the prediction of Eq. (1) (solid line) and previous results in the quantum regime due to the direct tunneling process [dashed line, Eq. (2), Ref. 10] and in the semiclassical regime due to resonance-assisted tunneling (gray solid line, Ref. 14). The lower inset shows the phase space of the system [24]. The upper insets illustrate the dominant tunneling steps for three values of $h$.

Figure 2

Dynamical tunneling rates $\gamma$ from the quantized torus $m=2$ of a regular island with three dominant resonances. Numerical data (dots) is compared to the prediction of Eq. (1) (solid line). For comparison the direct tunneling contribution (dashed line, Eq. (2), Ref. [10]) and the result of Eq. (1) without the improvements (iia) and (iib) is presented (dotted line). The inset shows the regular island of the system [24] and the dominant $5:1$, $11:2$, and $6:1$ resonances which cause tunneling steps as indicated by labels.

Figure 3

Dynamical tunneling rates $\gamma$ from the innermost quantized torus $m=0$ of the regular island for the standard map at $K=3.5$. Numerical data (dots) is compared to the prediction of Eq. (1) (solid line). For comparison the direct tunneling contribution [dashed line, Eq. (2), Ref. 10] and the result of Eq. (1) without the improvements (iia) and (iib) is presented (dotted line). The inset shows the island with the dominant $6:2$, $14:5$, and $8:3$ resonances and the arrows indicate the regimes where they start to become relevant.