  We use an erbium doped CaWO$_4$ crystal as a resonant transducer between the
RF and optical domains at 12 GHz and 1532 nm respectively. We employ a RF
resonator to enhance the spin coupling but keep a single-pass (non-resonant)
setup in optics. The overall efficiency is low but we carefully characterize
the transduction process and show that the performance can be described by two
different metrics that we define and distinguish: the electro-optics and the
quantum efficiencies. We reach an electro-optics efficiency of -84 dB for 15.7
dBm RF power. The corresponding quantum efficiency is -142 dB for 0.4 dBm
optical power. We develop the Schr\"odinger-Maxwell formalism, well-known to
describe light-matter interactions in atomic systems, in order to model the
conversion process. We explicitly make the connection with the cavity quantum
electrodynamics (cavity QED) approach that are generally used to describe
quantum transduction.

Quantum Physics;Optics