Steady-State Exceptional Point Degeneracy and Sensitivity of Nonlinear Saturable Coupled Oscillators

A coupled oscillator system displays enhanced sensitivity of its saturated steady-state (SS) oscillation frequency to small parameter perturbations near an exceptional point degeneracy (EPD), a property that can be used to realize EPD-based sensors. Linear $\mathcal{PT}$-symmetric systems, consisting of two coupled resonators, exhibit EPDs around which square-root sensitivity is observed. However, linear models are insufficient for realistic systems that rely on nonlinear, saturable gain elements, particularly when $\mathcal{PT}$-symmetry is broken. Thus, we study the SS of a general system of two coupled oscillators featuring EPDs and saturable nonlinear gain, using coupled-mode theory. We do this by synthesizing and extending prior SS analyses of the system's stability, and its square-root and cubic-root oscillation frequency sensitivity at a unique third-order SS-EPD. We include an SS analysis of the saturated gain values, energy, and the oscillation frequency's sensitivity in the vicinity of the third-order SS-EPD, providing a comprehensive analysis of the system's various SS regimes. We determine that the stable and bistable regions in parameter space directly depend on the saturated gain values; that the dynamic range of high sensitivity around degenerate conditions is extended by increasing losses, consequently reducing the system's stored energy; and that, to exploit the cubic-root-like sensitivity associated to the third-order SS-EPD, the suggested working regime is best confined to operation within the weakly coupled regime and not exactly at the third order SS-EPD. Finally, we apply the model to two electronic circuits that exhibit cubic-root sensitivity, demonstrating the application and limitations of this analysis.