  Motivated by recent experiments on pump-induced polar ordering in the quantum
paraelectric SrTiO$_3$, we study a driven phonon system close to a second order
phase transition. Analyzing its classical dynamics, we find that sufficiently
strong driving leads to transitions into polar phases whose structures,
determined by the light polarization, are not all accessible in equilibrium. In
addition, for certain intensity profiles we demonstrate the possibility of
two-step transitions as a function of fluence. For even stronger field
intensities, the possibility of period-doubling and chaotic behavior is
demonstrated. Finally we develop a generalized formalism that allows us to
consider quantum corrections to the classical dynamics in a systematic fashion.
We predict a shift in the critical pump fluence due to quantum fluctuations
with a characteristic dependence on the fluence increase rate, which can be
observed in experiments.
