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Quadratic Zeeman effect and spin-lattice relaxation of Tm$^{3+}$:YAG at high magnetic fields

Anisotropy of the quadratic Zeeman effect for the $^3{\rm H}_6 \rightarrow \, ^3{\rm H}_4$ transition at 793 nm wavelength in $^{169}$Tm$^{3+}$-doped Y$_3$Al$_5$O$_{12}$ is studied, revealing shifts ranging from near zero up to + 4.69 GHz/T$^2$ for ions in magnetically inequivalent sites. This large range of shifts is used to spectrally resolve different subsets of ions and study nuclear spin relaxation as a function of temperature, magnetic field strength, and orientation in a site-selective manner. A rapid decrease in spin lifetime is found at large magnetic fields, revealing the weak contribution of direct-phonon absorption and emission to the nuclear spin-lattice relaxation rate. We furthermore confirm theoretical predictions for the phonon coupling strength, finding much smaller values than those estimated in the limited number of past studies of thulium in similar crystals. Finally, we observe a significant -- and unexpected -- magnetic field dependence of the two-phonon Orbach spin relaxation process at higher field strengths, which we explain through changes in the electronic energy-level splitting arising from the quadratic Zeeman effect.