Tunneling rate fluctuations induced by non-linear resonances: a quantitative treatment based on semiclassical arguments

We investigate the tunneling process between two symmetric stable islands of a forced pendulum Hamiltonian in the weak chaos regime. We show that when the tunneling doublet is quantized over a classical non-linear resonance the tunneling rate strongly deviates from the semiclassical prediction. This mechanism is responsible for the irregular dependence of the tunneling rate on the system parameters. The weak-chaos condition allows us to make a theoretical prediction which agrees very well with the numerical results. This opens up a possible avenue to a general theory on the dependence of quantum tunneling on classical chaos.