Observation of higher-order exceptional points using frequency-dependent gain

Non-Hermitian systems exhibit exponentially amplified spectral sensitivity at higher-order exceptional points (HOEPs), where the eigenvalue response grows with the EP order. Conventional characterization relies on identifying reflection minima, implicitly imposing a fixed-gain constraint. Here, we demonstrate that the attainable order of the EP can be substantially elevated by replacing the fixed gain with a frequency-dependent gain, thereby obviating the necessity of nonlinear-gain mechanisms traditionally deemed indispensable. Capitalizing on the fact that the characteristic polynomial is encoded in the input impedance, we introduce an impedance observer that simultaneously serves as the frequency-dependent gain. By directly interrogating the zero-crossing analysis of the imaginary part of the input impedance, the eigenfrequencies of systems harboring HOEPs can be deterministically retrieved. Under the purely real-gain condition, a PT-symmetric dimer experimentally confirms a third-order EP, whereas an anti-PT-symmetric trimer verifies a fifth-order EP via the proposed observation. Our work offers a versatile and experimentally accessible route for realizing HOEPs, thereby enriching the landscape of non-Hermitian physics. Higher-order exceptional points in non-Hermitian systems enable extraordinarily sensitive detection, but require complex nonlinear gain. The authors show that frequency-dependent gain from simple impedance observers achieves a third-order exceptional point in a parity-time-symmetric dimer and a fifth-order exceptional point in an anti-parity-time-symmetric trimer, offering a practical route to enhanced sensing without nonlinear complications