  We study the non-Gaussian character of quantum optomechanical systems
evolving under the fully nonlinear optomechanical Hamiltonian. By using a
measure of non-Gaussianity based on the relative entropy of an initially
Gaussian state, we quantify the amount of non-Gaussianity induced by both a
constant and time-dependent cubic light-matter coupling and study its general
and asymptotic behaviour. We find analytical approximate expressions for the
measure of non-Gaussianity and show that initial thermal phonon occupation of
the mechanical element does not significantly impact the non-Gaussianity. More
importantly, we also show that it is possible to continuously increase the
amount of non-Gaussianity of the state by driving the light-matter coupling at
the frequency of mechanical resonance, suggesting a viable mechanism for
increasing the non-Gaussianity of optomechanical systems even in the presence
of noise.
