  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.
