  We address the quantum estimation of the diamagnetic, or $A^2$, term in an
effective model of light-matter interaction featuring two coupled oscillators.
First, we calculate the quantum Fisher information of the diamagnetic parameter
in the interacting ground state. Then, we find that typical measurements on the
transverse radiation field, such as homodyne detection or photon counting,
permit to estimate the diamagnetic coupling constant with near-optimal
efficiency in a wide range of model parameters. Should the model admit a
critical point, we also find that both measurements would become asymptotically
optimal in its vicinity. Finally, we discuss binary discrimination strategies
between the two most debated hypotheses involving the diamagnetic term in
circuit QED. While we adopt a terminology appropriate to the Coulomb gauge, our
results are also relevant for the electric dipole gauge. In that case, our
calculations would describe the estimation of the so-called transverse $P^2$
term. The derived metrological benchmarks are general and relevant to any
implementation of the model, cavity and circuit QED being two relevant
examples.
