  We study the coupling of light and the structural order parameter in the
charge density wave (CDW) state of the layered transition-metal dichalcogenide,
Tantalum Disulfide ($1T-\mathrm{TaS_2}$). Using time-dependent density
functional theory calculations of the dielectric properties along the
distortions coordinates, we show that $1T-\mathrm{TaS_2}$ displays very large
change in its dielectric function along the amplitude (Higgs) mode due to the
coupling of the periodic lattice distortion with an in-plane metal-insulator
transition, leading to optomechanical coupling coefficients two orders of
magnitude larger than the ones of diamond and ErFeO$_3$. In addition, we derive
an effective model of the light-induced dynamics, which is in quantitative
agreement with experimental observations in $1T-\mathrm{TaS_2}$. We show that
light-induced dynamics of the structural order parameter in $1T-\mathrm{TaS_2}$
can be deterministically controlled to engineer large third-order non-linear
optical susceptibilities. Our findings suggest that CDW materials are promising
active materials for non-linear optics.
