  We solve the time evolution of a nonlinear optomechanical Hamiltonian with
arbitrary time-dependent mechanical displacement, mechanical single-mode
squeezing and a time-dependent optomechanical coupling up to the solution of
two second-order differential equations. The solution is based on identifying a
minimal and finite Lie algebra that generates the time-evolution of the system.
This reduces the problem to considering a finite set of coupled ordinary
differential equations of real functions. To demonstrate the applicability of
our method, we compute the degree of non-Gaussianity of the time-evolved state
of the system by means of a measure based on the relative entropy of the
non-Gaussian state and its closest Gaussian reference state. We find that the
addition of a constant mechanical squeezing term to the standard optomechanical
Hamiltonian generally decreases the overall non-Gaussian character of the
state. For sinusoidally modulated squeezing, the two second-order differential
equations mentioned above take the form of the Mathieu equation. We derive
perturbative solutions for a small squeezing amplitude at parametric resonance
and show that they correspond to the rotating-wave approximation at times
larger than the scale set by the mechanical frequency. We find that the
non-Gaussianity of the state increases with both time and the squeezing
parameter in this specific regime.
