Theoretical study of transition matrix elements in cadmium for vacuum-ultraviolet generation in ${}^{229}$Th nuclear clock applications

Abstract

The relativistic Fock-space coupled-cluster methods are applied to the cadmium atom. A large number of transition energies and matrix elements are calculated for the $5s^2$${}^{1}S\to 5snp{}^{1,3}{P}^o$, $5s6s{}^{1}S\to 5snp{}^{1,3}{P}^o$ and $5s5d{}^{1}D\to 5snp{}^{1,3}{P}^o$ transitions for a wide range of $p$ states accounting for relativistic and electron-correlation effects. The results obtained within two different approaches (Fock-space coupled cluster and configuration interaction) are compared with available experimental and theoretical data. Good agreement is found between the two methods for transitions involving low-lying excited $p$-states, whereas for high-lying states, the discrepancy becomes large. The calculated values are used to determine the third-order nonlinear susceptibility of cadmium vapor, with an agreement within 5\% between the different methods. The results of the present computations support the feasibility of generating vacuum ultraviolet light in Cd vapor via a four-wave mixing process for the spectroscopy of ${}^{229}$Th isomer transition.