  We report terahertz time-domain spectroscopy (TDTS) experiments demonstrating
strong light-matter coupling in a terahertz (THz) LC-metamaterial in which the
phonon resonance of a topological insulator (TI) thin film is coupled to the
photonic modes of an array of electronic split-ring resonators. As we tune the
metamaterial resonance frequency through the frequency of the low frequency
$\alpha$ mode of (Bi$_x$Sb$_{1-x}$)$_2$Te$_3$ (BST), we observe strong mixing
and level repulsion between phonon and metamaterial resonance. This hybrid
resonance is a phonon polariton. We observe a normalized coupling strength,
$\eta$ = $\Omega_R$/$\omega_c$ $\approx$ 0.09, using the measured vacuum Rabi
frequency and cavity resonance. Our results demonstrate that one can tune the
mechanical properties of materials by changing their electromagnetic
environment and therefore modify their magnetic and topological degrees of
freedom via coupling to the lattice in this fashion.
