Multimode Ultrastrong Coupling in Three-Dimensional Photonic-Crystal Cavities

Recent theoretical studies have highlighted the role of spatially varying cavity electromagnetic fields in exploring novel cavity quantum electrodynamics (cQED) phenomena, such as the potential realization of the elusive Dicke superradiant phase transition. One-dimensional photonic-crystal cavities (PCCs), widely used for studying solid-state cQED systems, have uniform spatial profiles in the lateral plane. Three-dimensional (3D) PCCs, which exhibit discrete in-plane translational symmetry, overcome this limitation, but fabrication challenges have hindered the achievement of strong coupling in 3D-PCCs. Here, we report the realization of multimode ultrastrong coupling in a 3D-PCC at terahertz frequencies. The multimode coupling between the 3D-PCC's cavity modes and the cyclotron resonance of a Landau-quantized two-dimensional electron gas in GaAs is significantly influenced by the spatial profiles of the cavity modes, leading to distinct coupling scenarios depending on the probe polarization. Our experimental results are in excellent agreement with a multimode extended Hopfield model that accounts for the spatial inhomogeneity of the cavity field. Guided by the model, we discuss the possible strong ground-state correlations between different cavity modes and introduce relevant figures of merit for the multimode ultrastrong coupling regime. Our findings emphasize the importance of spatially nonuniform cavity mode profiles in probing nonintuitive quantum phenomena expected for the ground states of cQED systems in the ultrastrong coupling regime.