With a terahertz time-domain spectrometer (0.3 - 3.0 THz) we have measured the optical conductivity of the type-I clathrate Ba$_8$Ga$_{16}$Sn$_{30}$ at temperatures from 300 K down to 7 K. Independent six spectra superimposed on the Drude conductivity are identified to infrared active vibrational modes of guest Ba ions and the cages. While the spectra of five higher-frequency modes depend hardly on temperature, the lowest-lying spectrum with a peak at 0.72 THz due to the Ba(2) ions off-centering vibration in the oversized cage changes with temperature characteristically. With lowering temperature, the spectral shape of this so-called rattling phonon continues to become so broad that the line-width amounts to be comparable to the peak frequency. Furthermore, below about 100 K, the single broad peak tends to split into two subpeaks. While this splitting can be explained by assuming a multi-well anharmonic potential, the strong enhancement of the line-width broadening toward low temperature, cannot be understood, since the Boltzmann factor generally sharpens the low-temperature spectra.
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