  Motivated by ongoing work on pump-probe experiments and Floquet driven
magnets, we theoretically study the impact of pumped and driven Einstein
phonons with weak dissipation in multipolar Mott insulators. For Mott
insulators with pseudospin-$1/2$ non-Kramers doublets hosting octupolar and
quadrupolar moments, we show that pumping the ${\cal E}_g$ phonon modes which
couple linearly to the quadrupoles leads to characteristic chiral phonon
dynamics below the octupolar ordering transition. We also explore steady states
in the presence of continuously driven phonons, showing that a coherent
two-phonon drive with a suitable phase choice can be used to 'train' octupolar
order or even switch its sign. For spin-$1$ magnets, which host dipolar and
quadrupolar moments, we show that one-phonon and two-phonon drives can generate
Floquet flavor-wave excitations and induce dipolar magnetization. Our results
are obtained using a generalized $SU(N)$ Monte Carlo for the multipolar degrees
of freedom with numerical integration of the coupled spin-phonon equations of
motion. Our work opens up pathways to explore phonon-driven dynamics in
multipolar magnets.
