The $(e_g \otimes e_u) \otimes E_g$ product Jahn-Teller effect in the neutral group-IV--vacancy quantum bits in diamond

The product Jahn-Teller (pJT) effect may occur for such coupled electron-phonon systems in solids where single electrons occupy double degenerate orbitals. We propose that the excited state of the neutral $X$V split-vacancy complex in diamond, where $X$ and V labels a group-IV impurity atom of $X$=Si, Ge, Sn, Pb and the vacancy, respectively, is such a system with $e_g$ and $e_u$ double degenerate orbitals and $E_g$ quasi-localized phonons. We develop and apply \emph{ab initio} theory to quantify the strength of electron-phonon coupling for neutral $X$V complexes in diamond, and find a significant impact on the corresponding optical properties of these centers. Our results show good agreement with recent experimental data on the prospective SiV($0$) quantum bit, and reveals the complex nature of the excited states of neutral $X$V color centers in diamond.