Resonance-assisted tunneling in 4D normal-form Hamiltonians

A significant enhancement of tunneling may be caused by the presence of nonlinear resonances in the corresponding classical phase space. We describe this resonance-assisted tunneling using a 4D normal-form Hamiltonian as it captures all fundamental features. We consider single as well as double resonances and study the morphology of quantum states. Quantitatively we describe tunneling in terms of the weight of these states in classically disconnected regions. By applying perturbative methods we reveal the underlying mechanism and obtain excellent qualitative and quantitative agreement with numerical data. Using a minimal $4 \times 4$ matrix model, an intuitive understanding of the basic features of resonance-assisted tunneling in the normal-form Hamiltonians is obtained.