Complete crystal field calculation of Zeeman-hyperfine splittings in europium

TL;DR

This paper develops a comprehensive computational model including electric quadrupole and nuclear Zeeman interactions to accurately predict Zeeman-hyperfine structures in Eu$^{3+}$, aiding quantum information applications.

Contribution

The study introduces a novel crystal field model that incorporates typically omitted interactions, enabling precise fitting of Eu$^{3+}$ hyperfine structures across multiple sites.

Findings

Achieved close fits for Eu$^{3+}$ sites in CaF$_2$ and EuCl$_3$.6H$_2$O.

Quantified large crystal-field-induced state mixing in C$_{3v}$ site.

Determined signs of Zeeman-hyperfine parameters for all studied sites.

Abstract

Computational crystal-field models have provided consistent models of both electronic and Zeeman-hyperfine structure for several rare earth ions. These techniques have not yet been applied to the Zeeman-hyperfine structure of Eu$^{3+}$ because modeling the structure of the $J=0$ singlet levels in Eu$^{3+}$ requires inclusion of the commonly omitted lattice electric quadrupole and nuclear Zeeman interactions. Here, we include these terms in a computational model to fit the crystal field levels and the Zeeman-hyperfine structure of the $^7F_0$ and $^5D_0$ states in three Eu$^{3+}$ sites: the C$_{4v}$ and C$_{3v}$ sites in CaF$_2$ and the C$_2$ site in EuCl$_3$.6H$_2$O. Close fits are obtained for all three sites which are used to resolve ambiguities in previously published parameters, including quantifying the anomalously large crystal-field-induced state mixing in the C$_{3v}$ site and determining the signs of Zeeman-hyperfine parameters in all three sites. We show that this model allows accurate prediction of properties for Eu$^{3+}$ important for quantum information applications of these ions, such as relative transition strengths. The model could be used to improve crystal field calculations for other non-Kramers singlet states. We also present a spin Hamiltonian formalism without the normal assumption of no $J$ mixing, suitable for other rare earth ion energy levels where this effect is important.