Coarse-grained controllability of wavepackets by free evolution and phase shifts

We describe an approach to controlling wavepacket dynamics and a criterion of wavepacket controllability based on discretized properties of the wavepacket's localization on the orbit. The notion of "coarse-grained control" and the coarse-grained description of the controllability in infinite-dimensional Hilbert spaces are introduced and studied using the mathematical apparatus of loop groups. We prove that 2D rotational wavepackets are controllable by only free evolution and phase kicks by AC Stark shift implemented at fractional revivals. This scheme works even if the AC Stark shifts can have only a smooth coordinate dependence, correspondent to the action of a linearly polarized laser field.