Extending Phenomenological Crystal-Field Methods to $C_1$ Point-Group Symmetry: Characterization of the Optically-Excited Hyperfine Structure of $^{167}$Er$^{3+}$:Y$_2$SiO$_5$

TL;DR

This paper extends phenomenological crystal-field methods to $C_1$ symmetry, enabling accurate modeling of hyperfine structures in rare-earth doped crystals for quantum information applications.

Contribution

It demonstrates the first successful crystal-field calculations for $C_1$ symmetry, including hyperfine structures, with high accuracy for $^{167}$Er$^{3+}$:Y$_2$SiO$_5$.

Findings

Achieved better than 50 MHz accuracy in hyperfine transition predictions.

Included comprehensive spectroscopic data up to 20,000 cm$^{-1}$.

Enabled systematic evaluation of coherence properties in rare-earth doped crystals.

Abstract

We show that crystal-field calculations for $C_1$ point-group symmetry are possible, and that such calculations can be performed with sufficient accuracy to have substantial utility for rare-earth based quantum information applications. In particular, we perform crystal-field fitting for a C$_1$-symmetry site in $^{167}$Er$^{3+}$:Y$_2$SiO$_5$. The calculation simultaneously includes site-selective spectroscopic data up to 20,000 cm$^{-1}$, rotational Zeeman data, and ground- and excited-state hyperfine structure determined from high-resolution Raman-heterodyne spectroscopy on the 1.5 $\mu$m telecom transition. We achieve an agreement of better than 50 MHz for assigned hyperfine transitions. The success of this analysis opens the possibility of systematically evaluating the coherence properties, as well as transition energies and intensities, of any rare-earth ion doped into Y$_2$SiO$_5$ .