Topological insulators (TIs) are a recently discovered state of matter characterized by an “inverted” band structure driven by strong spin-orbit coupling. One of their most touted properties is the existence of robust "topologically protected" surface states.   I will discuss what topological protection means for transport experiments and how it can be probed using the technique of time-domain THz spectroscopy applied to thin films of Bi ₂ Se ₃ .   By measuring the low frequency optical response, we can follow their transport lifetimes as we drive these materials via chemical substitution through a quantum phase transition into a topologically trivial regime.  I will then discuss our work following the evolution of the response as a function of magnetic field from the classical transport regime to the quantum regime.   In the highest quality samples, we observe a continuous crossover from a low field regime where the response is given by semi-classical transport and observed in the form of cyclotron resonance to a higher field quantum regime.   In the later case, we find evidence for Faraday and Kerr rotation angles quantized in units of the fine structure constant.   This quantized rotation angle can be seen as evidence for a novel magneto-electric of the TI’s surface e.g. the much heralded axion electrodynamics of topological insulators.  Among other aspects this give a purely solid-state measure of fine structure constant.