  Trapped ions driven by electromagnetic radiation constitute one of the most
developed quantum technologies to date. The scenarios range from
proof-of-principle experiments to on-chip integration for quantum information
units. In most cases, these systems have operated in a regime where the
magnitude of the ion-radiation coupling constant is much smaller than the trap
and electronic transition frequencies. This regime allows the use of simple
effective Hamiltonians based on the validity of the rotating wave
approximation. However, novel trap and cavity designs now permit regimes in
which the trap frequency and the ion-radiation coupling constant are
commensurate. This opens up new venues for faster quantum gates and state
transfers from the ion to a photon, and other quantum operations. From the
theoretical side, however, there is not yet much known in terms of models and
applications that go beyond the weak driving scenario. In this work, we will
present two main results in the scenario of stronger drivings. First, we
revisit a known protocol to reconstruct the motional Wigner function and expand
it to stronger driving lasers. This extension is not trivial because the
original protocol makes use of effective Hamiltonians valid only for weak
drivings. The use of stronger fields or faster operations is desirable since
experimental reconstruction methods of that kind are usually hindered by
decoherence. We then present a model that allows the analytical treatment of
stronger drivings and that works well for non-resonant interactions, which are
generally out of the reach of the previous models.
