  Reciprocal and nonreciprocal effects in dielectric and magnetic materials
provide crucial information about the microscopic properties of electrons.
However, experimentally distinguishing the two has proven to be challenging,
especially when the associated effects are extremely small. To this end, we
propose a contact-less detection using a cross-cavity device where a material
of interest is placed at its centre. We show that the optical properties of the
material, such as Kerr and Faraday rotation, or, birefringence, manifest in the
coupling between the cavities' electromagnetic modes and in the shift of their
resonant frequencies. By calculating the dynamics of a geometrical photonic
state, we formulate a measurement protocol based on the quantum metric and
quantum process tomography that isolates the individual components of the
material's complex refractive index and minimizes the quantum mechanical
Cram\'er-Rao bound on the variance of the associated parameter estimation. Our
approach is expected to be applicable across a broad spectrum of experimental
platforms including Fock states in optical cavities, or, coherent states in
microwave and THz resonators.
