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.

Figure 1

(a) Schematic of our model. The nonlinear gain $g_A(|{\psi }_A|)$ depends on $|{\psi }_A|$. (b) Eigenvalues versus the external perturbation $\epsilon$, where the solid red line and four dashed (cyan, blue, purple, and green) lines represent the stable and four auxiliary modes, respectively. (c) The critical behavior of the stable mode near the ${\mathrm{NEP}}_5$. Here the circles are from the red line in (b), and the solid line represents a straight line with a slope $1/5$. The parameters used are ${\omega }_B=1.18$, ${\omega }_C=1.53$, $l_B=2.87$, $g_C=1.25$, ${\kappa }_2=2$.

Figure 2

(a) Orthogonality function $\chi$ for a Hermitian system (dot-dashed purple line), a PT-symmetric Hamiltonian in Eq. (S11) (dashed gray line), and our system (solid red line). Here, $\chi$ does not reach 1 as the parameters are not far enough from all the possible EPs in the parameter space. (b) PF of the stable state (solid red line) at the ${\mathrm{NEP}}_5$ is finite, contrasting sharply with a divergent one for the linear system (dashed gray line). (c) Evolution of $\mathrm{Re}({\psi }_A)$ (red line) and $g_A(|{\psi }_A|)$ (dark green line) starting from a small initial state $({10}^{-3},{10}^{-3},{10}^{-3}{)}^T$. In a short time, the nonlinear system will reach a stable state. The $g_A(|{\psi }_A|)$ (dark green) matches $g_s$ obtained from Eq. (4) (dark green circles). (d) The real part of the eigenvalues of $H_s$. (e) The amplitude and phase distribution of the stable state. The parameters for the linear system are provided in the Supplemental Material, Sec. 2 [50]. Parameters of the nonlinear system in (a),(b),(d),(e) are the same as those in Fig. 1. The parameters used in (c) are ${\omega }_A=2.9$, ${\omega }_B=4.18$, ${\omega }_C=4.53$, $l_B=2.87$, $g_C=1.25$, ${\kappa }_2=2$, and $g_A=5/(1+{|{\psi }_A|}^2)-3$.

Figure 3

Steady-state frequencies (a),(b) and the corresponding amplitudes (c),(d) around the ${\mathrm{NEP}}_5$ (marked by the black star). In (c),(d), the solid green, dot-dashed blue, and dashed purple lines represent the cases without noise, and the corresponding light lines represent one typical simulation with noises. The chirality function $\mathrm{\Lambda }$ (e) and the corresponding standard deviations $\sigma$ (f) are shown as a function of the encircling radius $r$. In (e),(f), the open cyan circles and gray aplstars are obtained from simulations of over 200 independent noises, and the red and blue lines are for eye guiding. The parameters are ${\omega }_B=1.25$, ${\omega }_C=1.59$, $l_B=3.06$, $g_C=1.40$, ${\kappa }_2=2.18$, $r=0.01$, $T=20000$, and $g_A=5/(1+|{\psi }_A{|}^2)-3$. The amplitude of the noise is given by $D=0.2$.