Pretty good fractional revival on abelian Cayley graphs

Let $\Gamma$ be a graph with the adjacency matrix $A$. The transition matrix of $\Gamma$, denoted $H(t)$, is defined as $H(t) := \exp(-\textbf{i}tA)$, where $\textbf{i} := \sqrt{-1}$ and $t$ is a real variable. The graph $\Gamma$ is said to exhibit fractional revival (FR in short) between the vertices $a$ and $b$ if there exists a positive real number $t$ such that $H(t){\textbf{e}_{a}} = \alpha{\textbf{e}_{a}} + \beta{\textbf{e}_{b}}$, where $\alpha, \beta \in \mathbb{C}$ such that $\beta \neq 0$ and $|\alpha|^2 + |\beta|^2 = 1$. The graph $\Gamma$ is said to exhibit pretty good fractional revival (PGFR in short) between the vertices $a$ and $b$ if there exists a sequence of real numbers $\{t_k\}$ with $\lim_{k\to\infty} H(t_k){\textbf{e}_{a}} = \alpha{\textbf{e}_{a}} + \beta{\textbf{e}_{b}}$, where $\alpha, \beta \in \mathbb{C}$ such that $\beta \neq 0$ and $|\alpha|^2 + |\beta|^2 = 1$. In the definition of PGFR, if $\alpha=0$ then $\Gamma$ is said to exhibit pretty good state transfer (PGST in short) between $a$ and $b$. In this paper, we obtain some sufficient conditions for circulant graphs exhibiting PGFR. We also find some sufficient conditions for non-circulant abelian Cayley graphs exhibiting PGFR. From these sufficient conditions, we find infinite families of circulant graphs and non-circulant abelian Cayley graphs exhibiting PGFR that fail to exhibit FR and PGST. Finally, we obtain some necessary conditions for some families of circulant graphs exhibiting PGFR. Some of our results generalize the results of Chan et al. [Pretty good quantum fractional revival in paths and cycles. \textit {Algebr. Comb.} 4(6) (2021), 989-1004.] for cycles.