  We use theory and first-principles calculations to explore mechanisms for
control of the translational and point group symmetries of crystals in
ultrafast optical experiments. We focus in particular on mechanisms that
exploit anharmonic (biquadratic) lattice couplings between a driven
infrared-active phonon mode and other modes at arbitrary wave vector, which are
always allowed by symmetry in any space group. We use Floquet theory to develop
a general phase diagram depicting the various dynamical regimes accessible to
materials, with simulated dynamics to illustrate how the biquadratic coupling
changes materials structure depending on both extrinsic factors (light pulse
characteristics) and intrinsic materials parameters (phonon frequencies, phonon
coupling strengths). We use our phase diagram, in conjunction with density
functional theory calculations, both to suggest experiments to reveal hidden
structural order in perovskite KTaO$_3$, and to provide additional insights
into recently reported experiments on SrTiO$_3$ and LiNbO$_3$.
