Experimental evidence of crystal-field, Zeeman splitting, and spin-phonon excitations in the quantum supersolid Na2BaCo(PO4)2

Drawing inspiration from the recent breakthroughs in the \ce{Na_{2}BaCo(PO_{4})_{2}} quantum magnet, renowned for its spin supersolidity phase and its potential for revolutionary cooling applications, our study delves into the intricate interplay among lattice, spin, and orbital degrees of freedom within this intriguing compound. Using meticulous temperature, field, and pressure-dependent Raman scattering techniques, we present compelling experimental evidence revealing pronounced crystal-electric field (CEF) excitations, alongside the interplay of CEF-phonon interactions. Notably, our experiments elucidate all electronic transitions from $j_{1 / 2}$ to $j_{3 / 2}$ and from $j_{1 / 2}$ to $j_{5 / 2}$, with energy level patterns closely aligned with theoretical predictions based on point-charge models. Furthermore, the application of a magnetic field and pressure reveals Zeeman splittings characterized by Land\'e-g factors as well as the CEF-phonon resonances. The anomalous shift in coupled peak at low temperatures originates from the hybridization of CEF and phonon excitations due to their close energy proximity. These findings constitute a significant step towards unraveling the fundamental properties of this exotic quantum material for future research in fundamental physics or engineering application.