Evidence of low-energy tunneling excitations in the high-$T_c$ superconductor $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$

We have measured the internal friction $Q^{-1\mathrm{}}$ and the Young's-modulus sound velocity $v_E$ at 1 kHz for $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$. Below 20 K the relative change in the sound velocity and the temperature dependence of the internal friction perfectly resemble the behavior obtained for Suprasil W. Below 1 K we observe a logarithmic temperature dependence for the sound velocity and between 1 and 5 K a well-defined plateau in the internal friction is obtained. We find quantitative agreement with the predictions of the standard tunneling model which is widely used to explain the low-temperature properties of amorphous materials. The implications of the existence of tunneling systems in these materials are discussed.