  Parametric gates and processes engineered from the perspective of the static
effective Hamiltonian of a driven system are central to quantum technology.
However, the perturbative expansions used to derive static effective models may
not be able to efficiently capture all the relevant physics of the original
system. In this work, we investigate the conditions for the validity of the
usual low-order static effective Hamiltonian used to describe a Kerr oscillator
under a squeezing drive. This system is of fundamental and technological
interest. In particular, it has been used to stabilize Schr\"odinger cat
states, which have applications for quantum computing. We compare the states
and energies of the effective static Hamiltonian with the exact Floquet states
and quasi-energies of the driven system and determine the parameter regime
where the two descriptions agree. Our work brings to light the physics that is
left out by ordinary static effective treatments and that can be explored by
state-of-the-art experiments.
