The internal friction $Q^{-1\mathrm{}}$ and the sound velocity $\delta v/v$ of vitreous silica were measured at very low temperatures using mechanical double paddle resonators operated at frequencies ranging from 0.33 to 14 kHz. Below $\sim 40\mathrm{mK}$ the internal friction showed an unexpected temperature and frequency dependence, with absolute values of $Q^{-1\mathrm{}}$ clearly exceeding those predicted by the standard tunneling model. Even though the most plausible origin of the observed excess internal friction appears to be the mutual interaction between tunneling states, the results are difficult to reconcile quantitatively with present theories taking into account this interaction.

• Received 11 November 1999

DOI:https://doi.org/10.1103/PhysRevLett.84.2176