Exploring Non-Linearity In Exceptional Singularities Of Non-Hermitian Systems

The exotic physics at and around singularities has always been in the scrutiny of theoretical and experimental investigations.In non-Hermitian systems,a type of unique singularity known as exceptional point（EP）is defined as the point at which two or more eigenvalues,and their corresponding eigenvectors,coalesce and become identical.Af-ter being experimentally demonstrated in microwave cavities,EPs were subsequently experimentally observed in various classical and quantum systems.An array of excit-ing physics and novel phenomena related to EPs have been elucidated.For example,amplification of a detected signal in microtoroid cavities,mode discrimination in multi-mode laser cavities,topological energy transfer in optomechanical systems,polarization states conversion in an optical wave guide,on-chip optical devices such as optical iso-lators,and directional lasing.However,the implementation of EP-related technologies in various schemes faces several potential challenges,including the stringent parameter requirements and noise.For example,the possibility of implementing EPs to enhance the signal-to-noise ratio of sensors has triggered an ongoing debate over the past few years.Specifically,the response of eigenfrequencies can demonstrate?^1/M-dependence when M degenerate eigenmodes are lifted by the perturbation?,which is highly desir-able in a realistic sensing protocol.Current schemes for achieving exceptional points require a considerable number of design parameters.And the enhanced noise due to the coalescence of eigenstates compensates for the improved responsivity and does not lead to an improvement in the signal-to-noise ratio.Previous treatments near an EP mainly focused on linear systems,where the gain and loss have fixed values independent of the fields’amplitudes.In principle,a proper description of a natural physical system requires nonlinear elements.For example,practical gain media saturate as the signal intensity reaches a certain value.So exploring the nonlinear gain in the non-Hermitian systems is a more realistic scenario.Meanwhile,the nonlinearity can also lead to beneficial applications such as wireless energy transfer and topological insulator lasers.Here,we investigate the contribution of nonlinear saturation gain to the forma-tion and applications of exceptional points.By introducing nonlinear saturation gain in a coupled dual-cavity system and conducting circuit experiments,we discover that the exceptional nexuses（EXs）previously believed to only exist in systems with three or more cavities,can actually exist in a two-cavity system,the multi-valued characteristics of nonlinearity can effectively relieve the stringent parameter requirements.Addition-ally,the feedback mechanism of saturation gain effectively suppresses the system noise enhanced by the coalescence of eigenvectors during the sensing process,thereby sig-nificantly improving the signal-to-noise ratio.However,the impact of noise on other applications based on EP points remains unclear.Detailed analysis specific to each ap-plication is evidently not wise.Therefore,we pose raise a fundamental question:Can the completeness of the eigenbasis be revived while the key features of EPs are still preserved?Thus,the issue associated with noise can be solved once and for all.To answer this question,we construct a combination of a resonator with a nonlinear saturation gain and another resonator with a linear gain,and achieve a nonlinear fifth-order exceptional point（NEP₅）in a three-cavity coupled resonator system.One stable and another four auxiliary steady eigenstates of the nonlinear Hamiltonian coalesce at the NEP₅.In the vicinity of NEP₅,the slope of the responsiveness of eigenvalues in logarithmic coordinates is 1/5.Interestingly,at the NEP₅,the instantaneous eigenbasis of the Hamiltonian governing the system dynamics is complete.A finite Petermann factor at the NEP₅confirms our conclusion once again.Additionally,we verify the chiral state conversion characteristic around the NEP₅.In a noisy environment,compared to the counterparts in linear systems,the chiral characteristics around the NEP₅can be maintained with smaller parameter trajectories.Finally,we perform simulations that verify the existence of a NEP₅in a circuit system.Although our experiments and simulations are based on circuit systems,our find-ings are universal and applicable to various non-Hermitian classical wave systems and semi-classical quantum systems.On the one hand,our research provides new insights into nonlinear and non-Hermitian EPs,further promoting interdisciplinary research in nonlinear and non-Hermitian domains.On the other hand,our results contribute to re-ducing the stringent parameter requirements for practical applications of non-Hermitian singular points and significantly reducing the negative impact of noise.Our works,pro-vide guidance for the development and realization of numerous emerging technologies based on EPs（NEPs）.