Multiple Landau polariton branches in the ultrastrong coupling regime in a multi-mode three-dimensional photonic-crystal cavity

One-dimensional (1D) photonic-crystal cavities (PCCs) have been widely used to study the strong coupling and ultrastrong coupling (USC) of matter with high Q-factor cavity photons. Three-dimensional (3D) PCCs can possess several unique features as compared to 1D PCCs, such as the existence of: (i) a complete photonic band gap, and (ii) multiple cavity modes within a narrow frequency range in the band gap. However, a 3D-PCC coupled system operating beyond the weak coupling regime has never been reported due to challenges in fabrication. Here we have fabricated a terahertz 3D-PCC through deep reactive ion etching and studied Landau polariton dispersions reflecting the complex interactions between multiple cavity modes in the 3D-PCC and the cyclotron resonance of a high-mobility two-dimensional electron gas. Our data showed good agreement with calculations based on a microscopic model that takes into account multiple cavity modes. We show that the observed Landau polariton branches can be well understood in terms of photonic modes that are either decoupled or coupled together via inter-Landau-level excitations, depending on the polarization of the probe. Our findings highlight features that are unique to 3D-PCC and will enable further explorations of the USC regime in novel geometries where multiple photonic modes in a narrow frequency range contribute to the light-matter coupling.