Optical pumping simulations and optical Rabi frequency measurements in $^{151}\mathrm{Eu}^{3+}\!:\mathrm{Y}_2\mathrm{SiO}_5$ under magnetic field

Europium-doped crystal $\mathrm{Y}_2\mathrm{SiO}_5$ is an interesting platform for optical quantum memories, due to its long optical and spin coherence times. In this work, we investigate $^{151}\mathrm{Eu}^{3+}\!:\mathrm{Y}_2\mathrm{SiO}_5$ under the application of magnetic field, a complex system with 36 different optical-hyperfine transitions. We present a simple numerical simulator for calculating the effect of optical pumping schemes that allow the isolation of a single frequency class of ions, applicable to any multi-level atom with inhomogeneous broadening. Experimentally, we develop methodologies for estimating the magnetic field vector, demonstrating high precision in predicting spectral features based on the known spin Hamiltonians. We measured the optical Rabi frequency of 21 transitions among the 36 possible optical-hyperfine transitions, allowing for the construction of a 6x6 branching ratio matrix of relative transition strengths. From the Rabi frequency measurements, we derive the optical dipole moment for the $^7\!F_0 \leftrightarrow {}^5\!D_0$ transition, yielding a value of $(6.94 \pm 0.09) \times 10^{-33}~\mathrm{C} \cdot \mathrm{m}$. Our results provide a critical test of the spin Hamiltonian models, demonstrating high precision in predicting energy levels and relative transition strengths, which are key abilities for quantum applications with $^{151}\mathrm{Eu}^{3+}\!:\mathrm{Y}_2\mathrm{SiO}_5$ under magnetic field.