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

TL;DR

This study uses advanced relativistic methods to calculate transition matrix elements in cadmium, supporting the feasibility of vacuum-ultraviolet light generation for thorium-229 nuclear clock applications.

Contribution

It provides detailed relativistic calculations of transition energies and matrix elements in cadmium, comparing two methods and applying results to VUV generation for nuclear clock spectroscopy.

Findings

Good agreement between methods for low-lying states

Large discrepancies for high-lying states

Supports VUV generation via four-wave mixing in Cd vapor

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.