Strongly coupled photonic molecules as doubly-coupled oscillators

In this work, we present a field-theoretic model of strongly coupled photonic molecules composed of interacting dielectric cavities in a closed, perfect-electric-conductor domain. Within this setting, we treat the resulting inter-mode couplings non-perturbatively. We demonstrate the predictive power of this framework by showing that supermode eigenfrequencies, field profiles, and mode volumes can be obtained directly from the isolated-cavity modes and dielectric environment, without electromagnetic simulations of the composite structure or numerical fitting. While our model affirms the phenomenological approach of modeling coupled cavity modes as simple coordinate-coupled oscillators in the weak coupling regime, we show that this intuition remarkably breaks down for strong coupling. Instead, we demonstrate that strongly coupled cavity modes are analogous to harmonic oscillators we term as \emph{doubly} coupled, with interactions via electric and magnetic fields appearing as independent coordinate-coordinate and momentum-momentum couplings, respectively. We show that this distinction is not merely cosmetic, but gives rise to observable properties while providing deep insights into the physical mechanism behind previously observed phenomena, such as coupling induced frequency shifts. Finally, we illustrate that the complex interplay of these dual couplings suggests the possibility to realize exotic phenomena that typically only occur in the ultrastrong coupling regime, here predicted to emerge for comparably modest mode splittings within a regime we term pseudo-ultrastrong coupling.