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.
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