  The phonon inverse Faraday effect describes the emergence of a DC
magnetization due to circularly polarized phonons. In this work we present a
microscopic formalism for the phonon inverse Faraday effect. The formalism is
based on time-dependent second order perturbation theory and electron phonon
coupling. While our final equation is general and material independent, we
provide estimates for the effective magnetic field expected for the
ferroelectric soft mode in the oxide perovskite SrTiO$_3$. Our estimates are
consistent with recent experiments showing a huge magnetization after a
coherent excitation of circularly polarized phonons with THz laser light.
Hence, the theoretical approach presented here is promising for shedding light
into the microscopic mechanism of angular momentum transfer between ionic and
electronic angular momentum, which is expected to play a central role in the
phononic manipulation of magnetism.
