Possible Observation of Quadrupole Waves in Spin Nematics

Discovery of new states of matter is a key objective in modern condensed matter physics, which often leads to revolutionary technological advancements such as superconductivity. Quantum spin nematic, a "hidden order" that evades conventional magnetic probes, is one such state. Na_{2}BaNi(PO_{4})_{2} is a potential spin nematic material, suggested by the observation of a two-magnon Bose-Einstein condensation from above the saturation field. However, direct confirmation of the spin nematicity remains elusive. This Letter presents inelastic neutron scattering spectra from the putative spin nematic phases of Na_{2}BaNi(PO_{4})_{2}, revealing low-energy quadrupole waves that are absent in the neighboring conventional magnetic phases. A spin-one model quantitatively captures the full details of the spin excitation spectra across all low-temperature phases, providing direct evidence of the spin nematic orders. Additionally, we show evidence of the three-magnon continuum and two-magnon bound states in the 1/3-magnetization plateau, revealing condensation of the two-magnon bound state as the origin of the low-field spin nematic supersolid phase.