Multimode ultrastrong coupling in three-dimensional photonic-crystal cavities

Recent theoretical studies have highlighted how spatially varying cavity electromagnetic fields enable novel cavity quantum electrodynamics phenomena, such as the Dicke superradiant phase transition. Three-dimensional photonic-crystal cavities, which exhibit discrete in-plane translational symmetry, overcome this limitation, but fabrication challenges have hindered the achievement of strong coupling. Here, we demonstrate multimode ultrastrong coupling between cavity modes of a three-dimensional photonic-crystal cavity at terahertz frequencies and the cyclotron resonance of a Landau-quantized two-dimensional electron gas in gallium arsenide. The multimode coupling depends on the spatial profiles of the cavity modes, resulting in distinct coupling scenarios based on probe polarization. Our results align with an extended multimode Hopfield model that accounts for spatial field variations. Guided by the model, we discuss possible strong ground-state correlations between cavity modes and introduce relevant figures of merit for multimode ultrastrong coupling. Our findings highlight the crucial role of spatial inhomogeneity in multimode ultrastrong coupling.