Nonlinear Exceptional Points with a Complete Basis in Dynamics

Exceptional points (EPs) are special spectral singularities at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become identical. In conventional wisdom, the coalescence of eigenvectors inevitably leads to the loss of completeness of the eigenbasis. Here, we show that this scenario breaks down in general at nonlinear EPs (NEPs). As an example, we realize a fifth-order NEP (${\mathrm{NEP}}_{5}$) within only three coupled resonators with both a theoretical model and simulations in circuits. One stable and another four auxiliary steady eigenstates of the nonlinear Hamiltonian coalesce at the ${\mathrm{NEP}}_{5}$, and the response of eigenfrequency to perturbations demonstrates a fifth-order root law. Intriguingly, the biorthogonal eigenbasis of the Hamiltonian governing the system dynamics is still complete, and this fact is corroborated by a finite Petermann factor instead of a divergent one at conventional EPs. Consequently, the amplification of noise, which diverges at other EPs, converges at our ${\mathrm{NEP}}_{5}$. Our finding transforms the understanding of EPs and shows potential for miniaturizing various key applications operating near EPs.