Demonstration of site-selective angular-resolved absorption spectroscopy of the $^{4}I_{15/2} \rightarrow $$^{4}I_{13/2}$ erbium transition in the monoclinic crystal Y$_2$SiO$_5$

We study the angular dependence in polarized light of the optical absorption for the \er transition $^{4}I_{15/2} \rightarrow $$^{4}I_{13/2}$ in Y$_2$SiO$_5$, revealing thus the associated anisotropy and the orientation of the related absorption principal directions in the dielectric plan perpendicular to the monoclinic axis b. The measurements are performed at low temperature. This allows us to isolate the lowest crystal field levels in the ground and excited states. We spectrally resolve and independently characterize the two yttrium substitution sites in the Y$_2$SiO$_5$ matrix. The absorption tensor components cannot be unambiguously determined yet while only considering the investigated dielectric plane. Still, measurements remarkably demonstrate that this transition of interest for quantum optical memories is not only a magnetic-dipole allowed transition but indeed a hybrid electric-magnetic transition.