Two-band parallel conductivity at terahertz frequencies in the   superconducting state of MgB$_2$.
  The optical response of the two-band superconductor MgB$_2$ has been studied in the 0.7-4 THz range on films with very low impurity level. The effect of the high-energy $\sigma$-gap is observed in the ratio $R_S/R_N$ between the normal and superconducting state reflectance, while in a neutron irradiated film with a slightly higher impurity level mainly the effect of the $\pi$-gap is evident as reported in previous experiments. At terahertz frequencies, the electrodynamic of MgB$_2$ can be well described by the two-band parallel conductivity model and is dominated by the $\pi$-bands when the impurity level is only slightly higher than that of an ultra-clean sample. 
M. Ortolani, P. Dore, D. Di Castro, A. Perucchi, S. Lupi, V. Ferrando,   M. Putti, I. Pallecchi, C. Ferdeghini and X. X. Xi