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