Magnetic $g$ tensors for the ${}^4\mathrm{I}_{15∕2}$ and ${}^4\mathrm{I}_{13∕2}$ states of ${\mathrm{Er}}^{3+}:{\mathrm{Y}}_2\mathrm{Si}{\mathrm{O}}_5$

We present the complete Zeeman $g$ tensors for the lowest-energy ${}^4\mathrm{I}_{15∕2}$ and ${}^4\mathrm{I}_{13∕2}$ states of ${\mathrm{Er}}^{3+}$ doped into ${\mathrm{Y}}_2\mathrm{Si}{\mathrm{O}}_5$ for both crystallographic sites deduced from orientation-dependent optical Zeeman spectroscopy over three orthogonal crystal planes. From these data, principal axes of the $g$ tensors were determined for each crystallographic site. Along axes with maximum values, the effective $g$ factors are 14.65 (site 1) and 15.46 (site 2) for the ground state, and 12.97 (site 1) and 13.77 (site 2) for the excited state. To minimize optical decoherence and spectral diffusion in device applications and high resolution spectroscopy, special directions for applying an external magnetic field have been found for each site, for which the ground- and excited-state $g$ factors are equal. Among those directions, choices are presented that also maximize the ground-state splittings for all four magnetically inequivalent sites, thus optimizing the prospects for freezing out electron spin fluctuations and reducing decoherence and spectral diffusion significantly.

Figure 1

(Color online) Sense of rotation adopted for the magnetic field $B$ in the $D_1-D_2$ plane. The $b$ axis points out of the page. Note that clockwise and counterclockwise rotations are not equivalent for this case.

Figure 2

(Color online) Zeeman laser absorption spectra for site 2 with $B=1\mathrm{T}$ at $\theta =16.5°$ in the $b\text{−}{D}_1$ plane for $T=1.5\mathrm{K}$ and $T=5\mathrm{K}$, showing thermal depopulation of the upper Zeeman components of the ground state in the lower temperature spectra; the two magnetic subclasses have been labeled with indices I and II, respectively. Transition labels are shown in the inset.

Figure 3

Experimental Zeeman data for site 1 in three orthogonal planes taken in a field of $0.484\mathrm{T}$. (a) shows the $D_1-D_2$ plane where the angle zero denotes the $D_1$ axis, (b) the $D_2\text{−}b$ plane, and (c) the $b\text{−}{D}_1$ plane with angle zero denoting the $b$ axis. In (c), the field maps out a slightly conic surface in the crystal. Line types are not coordinated for each site.

Figure 4

Experimental Zeeman data for site 2 in three orthogonal planes. The $D_1-D_2$ data were taken in a field of $0.290\mathrm{T}$ due to larger $g$ factors in that plane, and data from the other two planes were taken in a field of $0.484\mathrm{T}$. Subplot designations and angle labeling conventions are the same as in Fig. 3. In (b), the field maps out a slightly conic surface in the crystal.

Figure 5

Effective $g$-factor variations for site 1 in all three planes for the lowest-energy ${}^4\mathrm{I}_{15∕2}$ and ${}^4\mathrm{I}_{13∕2}$ states of ${\mathrm{Er}}^{3+}:{\mathrm{Y}}_2\mathrm{Si}{\mathrm{O}}_5$. Solid lines are least-squares fits to the data showing excellent agreement. Labels I and II are for magnetic subclasses; $g$ and $e$ for ground and excited states.

Figure 6

Effective $g$-factor variations for site 2 in all three planes for the lowest-energy ${}^4\mathrm{I}_{15∕2}$ and ${}^4\mathrm{I}_{13∕2}$ states of ${\mathrm{Er}}^{3+}:{\mathrm{Y}}_2\mathrm{Si}{\mathrm{O}}_5$. Solid lines are least-squares fits to the data showing excellent agreement. Labels I and II are for magnetic subclasses; $g$ and $e$ for ground and excited states.

Figure 7

(Color online) Calculated equal $g$-factor directions for subclass I of site 1 are shown in the upper plot. Those for subclass II can be determined from a $C_2$ rotation. In the bottom plot, all of the ground-state $g$ factors are plotted for these directions.

Figure 8

(Color online) Calculated equal $g$-factor directions for subclass I of site 2. Those for subclass II can be determined from a $C_2$ rotation. In the bottom plot, all of the ground-state $g$ factors are plotted for these directions.