Quantum Kicked Dynamics and Classical Diffusion

We consider the quantum counterpart of the kicked harmonic oscillator showing that it undergoes the effect of delocalization in momentum when the classical diffusional threshold is obeyed. For this case the ratio between the oscillator frequency and the frequency of the kick is a rational number, strictly in analogy with the classical case that does not obey the Kolmogorov-Arnold-Moser theorem as the unperturbed motion is degenerate. A tight-binding formulation is derived showing that there is not delocalization in momentum for irrational ratio of the above frequencies. In this way, it is straightforwardly seen that the behavior of the quantum kicked rotator is strictly similar to the one of the quantum kicked harmonic oscillator, although the former, in the classical limit, obeys the Kolmogorov-Arnold-Moser theorem.