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We have utilized time-domain magneto-terahertz spectroscopy to investigatethe low frequency optical response of topological insulatorCu$_{0.02}$Bi$_2$Se$_3$ and Bi$_2$Se$_3$ films. With both field and frequencydependence, such experiments give sufficient information to measure themobility and carrier density of multiple conduction channels simultaneously. Weobserve sharp cyclotron resonances (CRs) in both materials. The small amount ofCu incorporated into the Cu$_{0.02}$Bi$_2$Se$_3$ induces a true bulk insulatorwith only a \textit{single} type of conduction with total sheet carrier density$\sim4.9\times10^{12}/$cm$^{2}$ and mobility as high as 4000cm$^{2}/$V$\cdot$s. This is consistent with conduction from two virtuallyidentical topological surface states (TSSs) on top and bottom of the film witha chemical potential $\sim$145 meV above the Dirac point and in the bulk gap.The CR broadens at high fields, an effect that we attribute to anelectron-phonon interaction. This assignment is supported by an extended Drudemodel analysis of the zero field Drude conductance. In contrast, in normalBi$_2$Se$_3$ films two conduction channels were observed and we developed aself-consistent analysis method to distinguish the dominant TSSs and coexistingtrivial bulk/2DEG states. Our high-resolution Faraday rotation spectroscopy onCu$_{0.02}$Bi$_2$Se$_3$ paves the way for the observation of quantized Faradayrotation under experimentally achievable conditions to push chemical potentialin the lowest Landau Level.

High-Resolution Faraday Rotation and Electron-Phonon Coupling in Surface States of the Bulk-Insulating Topological Insulator Cu$_{0.02}$Bi$_2$Se$_3$