Resonance- and Chaos-Assisted Tunneling in Mixed Regular-Chaotic Systems

We present evidence that nonlinear resonances govern the tunneling process between symmetry-related islands of regular motion in mixed regular-chaotic systems. In a similar way as for near-integrable tunneling, such resonances induce couplings between regular states within the islands and states that are supported by the chaotic sea. On the basis of this mechanism, we derive a semiclassical expression for the average tunneling rate, which yields good agreement in comparison with the exact quantum tunneling rates calculated for the kicked rotor and the kicked Harper.

Figure 1

Chaos-assisted tunneling in the kicked rotor at (a) $K=2$ and (b) $K=3$. Plotted are, as a function of $N=2\pi /\hslash$, the splittings $\Delta \phi$ between the eigenphases of the ground states at the Bloch phases $\xi =0$ and $\pi$. The associated regular islands are shown in the lower panels. The steplike curve is the semiclassical prediction $\overline{\Delta \phi }$ for the eigenphase splittings, taking into account the $10:2$ resonance at $K=2$ and the $10:3$ resonance at $K=3$.

Figure 2

Chaos-assisted tunneling in the kicked Harper at (a) $\tau =2$ and (b) $\tau =3$. (a) The splittings between the eigenphases of the ground state at $\xi =0$ and $\pi$. (b) The eigenphase splittings between the symmetric and the antisymmetric states that are localized on the center of the small, bifurcated islands (at fixed $\xi =0$). The steplike curves represent the semiclassical predictions, based on the $8:2$ resonance at $\tau =2$ and on the $9:1$ resonance at $\tau =3$.