  We present a theory of magnon-polaritons in quantum Ising materials, and
develop a formalism describing the coupling between light and matter as an
Ising system is tuned through its quantum critical point. The theory is applied
to Ising materials having multilevel single-site Hamiltonians, in which
multiple magnon modes are present, such as the insulating Ising magnet
LiHoF$_4$ . We find that the magnon-photon coupling strengths may be tuned by
the applied transverse field, with the coupling between the soft mode present
in the quantum Ising material and a photonic resonator mode diverging at the
quantum critical point of the material. A fixed system of spins will not
exhibit the diamagnetic response expected when light is coupled to mobile spins
or atoms. Without the diamagnetic response, one expects a divergent
magnon-photon coupling strength to lead to a superradiant quantum phase
transition. However, this neglects the effects of damping and decoherence
present in any real system. We show that damping and decoherence may block the
superradiant quantum phase transition, and lead to weak coupling between the
soft magnon mode and the resonator mode. The results of the theory are applied
to experimental data on the model system LiHoF$_4$ in a microwave resonator.
