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Active optical media leading to interaction Hamiltonians of the form $ H =\tilde{\lambda}\, (a + a^{\dagger})^{\zeta}$ represent a crucial resource forquantum optical technology. In this paper, we address the characterization ofthose nonlinear media using quantum probes, as opposed to semiclassical ones.In particular, we investigate how squeezed probes may improve individual andjoint estimation of the nonlinear coupling $\tilde{\lambda}$ and of thenonlinearity order $\zeta$. Upon using tools from quantum estimation, we showthat: i) the two parameters are compatible, i.e. the may be jointly estimatedwithout additional quantum noise; ii) the use of squeezed probes improvesprecision at fixed overall energy of the probe; iii) for low energy probes,squeezed vacuum represent the most convenient choice, whereas for increasingenergy an optimal squeezing fraction may be determined; iv) using optimizedquantum probes, the scaling of the corresponding precision with energyimproves, both for individual and joint estimation of the two parameters,compared to semiclassical coherent probes. We conclude that quantum probesrepresent a resource to enhance precision in the characterization of nonlinearmedia, and foresee potential applications with current technology.

Quantum probes for the characterization of nonlinear media