  We 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 titraversing 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.
