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Nonperturbative coupling of light with condensed matter in an optical cavityis expected to reveal a host of coherent many-body phenomena and states. Inaddition, strong coherent light-matter interaction in a solid-state environmentis of great interest to emerging quantum-based technologies. However, creatinga system that combines a long electronic coherence time, a large dipole moment,and a high cavity quality ($Q$) factor has been a challenging goal. Here, wereport collective ultrastrong light-matter coupling in an ultrahigh-mobilitytwo-dimensional electron gas in a high-$Q$ terahertz photonic-crystal cavity ina quantizing magnetic field, demonstrating a cooperativity of $\sim$360. Thesplitting of cyclotron resonance (CR) into the lower and upper polaritonbranches exhibited a $\sqrt{n_\mathrm{e}}$-dependence on the electron density($n_\mathrm{e}$), a hallmark of collective vacuum Rabi splitting. Furthermore,a small but definite blue shift was observed for the polariton frequencies dueto the normally negligible $A^2$ term in the light-matter interactionHamiltonian. Finally, the high-$Q$ cavity suppressed the superradiant decay ofcoherent CR, which resulted in an unprecedentedly narrow intrinsic CR linewidthof 5.6 GHz at 2 K. These results open up a variety of new possibilities tocombine the traditional disciplines of many-body condensed matter physics andcavity-based quantum optics.

Collective, Coherent, and Ultrastrong Coupling of 2D Electrons with Terahertz Cavity Photons