  We investigated the magnetoterahertz response of the Dirac semimetal
Cd$_3$As$_2$ and observed a particularly low frequency optical phonon, as well
as a very prominent and field sensitive cyclotron resonance. As the cyclotron
frequency is tuned with field to pass through the phonon, the phonon become
circularly polarized as shown by a notable splitting in their response to
right- and left-hand polarized light. This splitting can be expressed as an
effective phonon magnetic moment that is approximately 2.7 times the Bohr
magneton, which is almost four orders of magnitude larger than ab initio
calculations predict for phonon magnetic moments in nonmagnetic insulators.
This exceedingly large value is due to the coupling of the phonons to the
cyclotron motion and is controlled directly by the electron-phonon coupling
constant. This field tunable circular-polarization selective coupling provides
new functionality for nonlinear optics to create light-induced topological
phases in Dirac semimetals.
