Understanding the fundamental collective excitations of an emergent many-body interacting system has been a crosscutting theme throughout many branches of physics. Key questions about the dynamics of these excitations in the presence of both many-body physics and topology are currently driving numerous major research efforts in quantum materials. One such material is (TaSe_4)_2I - a Weyl semimetal that undergoes charge density wave (CDW) ordering below 260 K. I will discuss two of our recent experiments using nonlinear light-matter interaction in the terahertz (THz) range to directly probe the dynamics of the collective excitations of (TaSe_4)_2I. I will first show how upon transient photoexcitation at low temperatures (TaSe_4)_2I strikingly emits coherent, narrow-band THz radiation. The frequency, polarization and temperature dependence of the emitted radiation imply the existence of a phason that acquires mass by coupling to long-range Coulomb interaction, giving a direct measurement of the Anderson-Higgs mechanism. Second, I will show our recent results using THz pump, IR Kerr probe spectroscopy on (TaSe_4)_2I to highlight how THz-driven phonons can induce a sizeable dynamic optical Kerr rotation in this otherwise time-reversal invariant system. I will briefly discuss both these results in the context of the predicted axion electrodynamics in (TaSe_4)_2I.