Oriented or signed Cayley graphs with all eigenvalues integer multiples of $\sqrt{\Delta}$

Let $G$ be a finite abelian group. Bridges and Mena characterized the Cayley graphs of $G$ that have only integer eigenvalues. Here we consider the $(0,1,-1)$ adjacency matrix of an oriented Cayley graph or of a signed Cayley graph $X$ on $G$. We give a characterization of when all the eigenvalues of $X$ are integer multiples of $\sqrt{\Delta}$ for some square-free integer $\Delta$. These are exactly the oriented or signed Cayley graphs on which the continuous quantum walks are periodic, a necessary condition for walks on such graphs to admit perfect state transfer. This also has applications in the study of uniform mixing on oriented Cayley graphs, as the occurrence of local uniform mixing at vertex $a$ in an oriented graph $X$ implies periodicity of the walk at $a$. We give examples of oriented Cayley graphs which admit uniform mixing or multiple state transfer.