Zeeman split Kramers doublets in spin-supersolid candidate Na$_{2}$BaCo(PO$_{4}$)$_{2}$

Na$_{2}$BaCo(PO$_{4}$)$_{2}$ is a triangular antiferromagnet that displays highly efficient adiabatic demagnetization cooling (J. Xiang $\textit{et al.}$ Nature ${\bf{625}}$, 270 (2024)) near a quantum critical point at $\mu_{0}H_{c}\sim 1.6$ T, separating a low-field magnetically disordered from a high-field fully polarized ferromagnetic phase. We apply high resolution backscattering neutron spectroscopy in an applied field to study the magnetic excitations near $\mu_{0}H_{c}$. At large fields we observe ferromagnetic fluctuations that gradually transition to being overdamped in energy below $\mu_{0}H_{c}$ where the magnetism is spatially disordered. We parameterize the excitations in the high field polarized phase in terms of coupled Zeeman split Kramers doublets originating from the presence of spin-orbit coupling. On reducing the field, the splitting between the Kramers doublets is reduced and if done adiabatically, provides a mechanism for reducing temperature. On lowering the applied field through the $\mu_{0}H_{c}$ the excitations characterize a textured phase that we suggest is inefficient for cooling. Low temperature disordered frustrated magnets built on Kramers doublets with nearby quantum critical points provide a route for efficient magnetocalorics.