  In this work we study the validity of the rotating wave approximation of an
ideal system composed of two harmonic oscillators evolving with a quadratic
Hamiltonian and arbitrarily strong interaction. We solve the dynamics
analytically by employing tools from symplectic geometry. We focus on systems
with initial Gaussian states and quantify exactly the deviation between the
state obtained through the rotating approximation and the state obtained
through the full evolution, therefore providing an answer for all values of the
coupling. We find that the squeezing present in the full Hamiltonian and in the
initial state governs the deviation from the approximated evolution.
Furthermore, we also show that the rotating wave approximation is recovered for
resonant frequencies and vanishing coupling to frequency ratio. Finally, we
give a general proof of the rotating wave approximation and estimate its
convergence on Fock states. Applications and potential physical implementations
are also discussed.
