Are Dicke Superradiant Phase Transitions Possible in Zeeman Polaritons in the Ultrastrong Coupling Regime?

The interaction between an ensemble of two-level atoms and a coherent light mode, described by the Dicke Hamiltonian, is a well-known topic in quantum optics. However, experimental efforts examining condensed matter in cavities typically consist of bosonic matter modes, which behave as simple harmonic oscillators. Zeeman polaritons, a hybridized state consisting of an ensemble of paramagnetic spins strongly coupled to a coherent magnetic field, demonstrate temperature-dependent vacuum Rabi splitting originating from the population difference between the finite energy states. Thus, Zeeman polaritons, as a finite-level spin system strongly coupled with a cavity magnetic field, are compatible with the Dicke model and well-suited to pursue experimental realizations of Dicke phenomena in condensed matter. Of particular interest is the pursuit of the Dicke superradiant phase transition, at which point the atomic polarization and photonic field become finite without an external driving field. While this phase transition is forbidden in the case of electric dipole coupling due to the diamagnetic term in the Hamiltonian, it has not been ruled out for interactions based on a magnetic dipole. Here, we examine the conditions in which an ensemble of paramagnetic spins may undergo the superradiant phase transition and present spectroscopic evidence to interrogate the behavior of paramagnets ultrastrongly coupled with cavity photons.