  Experiments with ultracold atoms in optical lattices usually involve a weak
parabolic trapping potential which merely serves to confine the atoms, but
otherwise remains negligible. In contrast, we suggest a different class of
experiments in which the presence of a stronger trap is an essential part of
the set-up. Because the trap-modified on-site energies exhibit a slowly varying
level spacing, similar to that of an anharmonic oscillator, an additional
time-periodic trap modulation with judiciously chosen parameters creates
nonlinear resonances which enable efficient Floquet engineering. We employ a
Mathieu approximation for constructing the near-resonant Floquet states in an
accurate manner and demonstrate the emergence of effective ground states from
the resonant trap eigenstates. Moreover, we show that the population of the
Floquet states is strongly affected by the phase of a sudden turn-on of the
trap modulation, which leads to significantly modified and rich dynamics. As a
guideline for further studies, we argue that the deliberate population of only
the resonance-induced effective ground states will allow one to realize Floquet
condensates which follow classical periodic orbits, thus providing challenging
future perspectives for the investigation of the quantum-classical
correspondence.
