Excitation Gap of a Confined Chiral Mode Measures Critical Zero-Point Fluctuations

Abstract

Where uniform and chiral superradiant boundaries meet in a quantum Rabi ring, the quadratic theory leaves a degenerate chiral ladder with finite radial width. We show that adding one chiral quantum widens the ladder's radial wave function; the quartic coupling then raises the uniform quadratic coefficient. The resulting gap is proportional to the uniform zero-point variance divided by the atomic-to-cavity frequency ratio $\eta$: it closes as $\eta^{-2/3}$ while the variance grows as $\eta^{1/3}$. A quartic Schr\"odinger equation fixes both observables and the first correction to their ratio, as confirmed by full spin-photon diagonalization. A triangle of collective spins with Dzyaloshinskii-Moriya exchange obeys the same relation, with a different coefficient and a correction of opposite sign. The ring's three-coordinate spectrum connects this gap to the neighboring $\eta^{-1}$ and $\eta^{-1/3}$ laws and to photon-number saturation.