The dynamics of magnetic moments coupled to phonons is of great interest for understanding spin transport in solids as well as for our ability to control magnetism via tailored phonon modes. For spin $S>\frac{1}{2}$, spin-orbit coupling permits an unusual linear coupling of phonons to quadrupolar moments, so that phonons act as a dynamical transverse field for the spins. Here, we develop a generalized $\text{SU}\left(N\right)$ spin-phonon Monte Carlo and molecular dynamics technique to simulate the equilibrium and nonequilibrium properties of such spin-orbital-phonon coupled Mott insulators, and apply it to a spin-1 model with competing XY antiferromagnet (AFM) and quadrupolar paramagnet (QPM) phases. We uncover a rich set of nonequilibrium phenomena from driven phonons, including the generation of a uniform magnetization in the QPM and AFM, strengthening of Néel order, gapping of the AFM Nambu-Goldstone mode by Floquet-Ising anisotropy, and creating Floquet copies of transverse and longitudinal spin waves. Our work is relevant for driven spin-1 magnets, such as ${\mathrm{Ba}}_2{\mathrm{FeSi}}_2{\mathrm{O}}_7$, and we highlight broader implications for nonequilibrium multipolar magnetism.

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• Magnetism
• Multipole order
• Phonons
• Spin dynamics
• Floquet systems
• Molecular dynamics
• Monte Carlo methods