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

TL;DR

This paper investigates the nonlinear dynamics of coupled oscillators near an exceptional point degeneracy, revealing how nonlinear saturation affects sensitivity and stability, and demonstrating potential for enhanced sensor applications.

Contribution

It extends the analysis of exceptional point degeneracies to nonlinear, saturable systems, providing a comprehensive understanding of steady-state regimes and sensitivity behavior.

Findings

Sensitivity near third-order SS-EPD exhibits cubic-root behavior.

Stable and bistable regions depend on saturated gain values.

Operation within the weakly coupled regime optimizes sensitivity.

Abstract

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.