Spectral diffusion, phonon echoes, and saturation recovery in glasses at low temperatures

Many features of glasses below 1 K are explicable in terms of localized tunneling levels, for which a spin-$\frac{1}{2}$ analogy exists. Here we show that spectral diffusion, resulting from fluctuations in resonant frequency, is essential to our understanding of recent ultrasonic experiments. Our model involves a coupling among the levels of the form $J_{\mathrm{ij}}{S}_z^i{S}_z^j$, which acquires a time dependence when a spin-flipping rate $T_1^{-1\mathrm{}}$ is introduced. For two- and three-pulse phonon-echo experiments near $T=20\mathrm{}$ mK, we predict phase-memory times which agree qualitatively with the experimental results of Golding and Graebner. For saturation recovery, we predict a linewidth whose time dependence should be observable near $T=100\mathrm{}$ mK. Estimates of tunneling-model parameters and comparison with specific-heat experiments suggest that glasses may contain two types of tunneling levels.