Yang, YikaiBabkevich, PeterGaal, RichardŽivković, IvicaRønnow, Henrik M.2025-01-252025-01-252025-01-252024-12-0110.1038/s41598-024-75978-x2-s2.0-85207345587https://infoscience.epfl.ch/handle/20.500.14299/24409139448735We first present a formalism that incorporates the input-output formalism and the linear response theory to employ cavity-magnon-polariton coupling as a spectroscopic tool for investigating strongly hybridized electro-nuclear spin excitations. A microscopic relation between the generalized susceptibility and the scattering parameter |S11| in strongly hybridized cavity-magnon-polariton systems has been derived without resorting to semi-classical approximations. The formalism is then applied to both analyze and simulate a specific systems comprising a model quantum Ising magnet (LiHoF4) and a high-finesse 3D re-entrant cavity resonator. Quantitative information on the electro-nuclear spin states in LiHoF4 is extracted, and the experimental observations across a broad parameter range were numerically reproduced, including an external magnetic field traversing a quantum critical point. The method potentially opens a new avenue not only for further studies on the quantum phase transition in LiHoF4 but also for a wide range of complex magnetic systems.entruetext::journal::journal article::research article