  Active optical media leading to interaction Hamiltonians of the form $ H =
\tilde{\lambda}\, (a + a^{\dagger})^{\zeta}$ represent a crucial resource for
quantum optical technology. In this paper, we address the characterization of
those nonlinear media using quantum probes, as opposed to semiclassical ones.
In particular, we investigate how squeezed probes may improve individual and
joint estimation of the nonlinear coupling $\tilde{\lambda}$ and of the
nonlinearity order $\zeta$. Upon using tools from quantum estimation, we show
that: i) the two parameters are compatible, i.e. the may be jointly estimated
without additional quantum noise; ii) the use of squeezed probes improves
precision at fixed overall energy of the probe; iii) for low energy probes,
squeezed vacuum represent the most convenient choice, whereas for increasing
energy an optimal squeezing fraction may be determined; iv) using optimized
quantum probes, the scaling of the corresponding precision with energy
improves, both for individual and joint estimation of the two parameters,
compared to semiclassical coherent probes. We conclude that quantum probes
represent a resource to enhance precision in the characterization of nonlinear
media, and foresee potential applications with current technology.
