Cross-tunneling and phonon bottleneck effects in the relaxation pheomena of the XY pyrochlore antiferromagnet Er$_{\mathbf{2}}$\textbf{Ti}$_{\mathbf{2}}$\textbf{O}$_{\mathbf{7}}$\textbf{.}*

Multiple time-scale relaxation dynamics are revealed by alternating-current (ac) susceptibility measurements of a single crystal of Er$_{2}$Ti$_{2}$O$_{7}$ studied at high temperatures ($k_{B}T$ \textgreater \textgreater $J/k_{B})$ and in a wide-range of static magnetic fields. The analysis of the frequency dependence of the ac susceptibility revealed the existence of two relaxation mechanisms identified as an Orbach process with a pronounced effect of phonon bottleneck and cross-tunneling. The origin of the phonon bottleneck is attributed to a resonant phonon trapping. The relevance of the obtained results for relaxation phenomena found in other rare--earth pyrochlores, studied under similar conditions, is discussed.