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We show that crystal-field calculations for $C_1$ point-group symmetry arepossible, and that such calculations can be performed with sufficient accuracyto have substantial utility for rare-earth based quantum informationapplications. In particular, we perform crystal-field fitting for aC$_1$-symmetry site in $^{167}$Er$^{3+}$:Y$_2$SiO$_5$. The calculationsimultaneously includes site-selective spectroscopic data up to 20,000cm$^{-1}$, rotational Zeeman data, and ground- and excited-state hyperfinestructure determined from high-resolution Raman-heterodyne spectroscopy on the1.5 $\mu$m telecom transition. We achieve an agreement of better than 50 MHzfor assigned hyperfine transitions. The success of this analysis opens thepossibility of systematically evaluating the coherence properties, as well astransition energies and intensities, of any rare-earth ion doped intoY$_2$SiO$_5$ .

physical review letters

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$_5$