A tropical geometric approach to exceptional points

<sec> We theoretically and experimentally demonstrate the anisotropic exceptional points (EPs) in the cavity magnonics system where magnons in a one millimeter-diameter yttrium iron garnet (YIG) sphere are coherently coupled with the microwave photons in a three-dimensional microwave cavity. The damping nature makes the cavity magnonics system inherently non-Hermitian. By solving the eigenvalues and eigenvectors of non-Hermitian Hamiltonian, a series of interesting and essential characteristics of the system can be obtained. Therefore, non-Hermitian physics has received more and more attention in both theory and experiment communities. Among them, exceptional points correspond to the non-Hermitian system’s degenerate states where the eigenvalues of the non-Hermitian system are identical, and the eigenvectors are parallel. The coupled cavity photon-magnon system has high tunability of coupling strength and cavity external damping rate, which is very suitable for studying EPs -related physics.</sec><sec> Exceptional points (EPs) are crucial in all kinds of non-Hermitian physical systems, which have both fundamental and applicational importance. For instance, it can be used for sensitive detection by monitoring spectrum splitting of degenerate modes when a perturbation to be sensed occurs. The EPs can be anisotropic, which means that it has a different function relation when the system approaches the EPs along different parameter paths of the system. In this paper, by carefully designing the parameter space, we observe the anisotropic exceptional point in the coupled cavity photon-magnon system. It shows the linear and square-root behavior when the EPs are approached from different directions in the parameter space. One of the parameters is the position of the YIG sphere in the cavity, which determines the coherent coupling strength between the cavity mode and the magnon mode. Another parameter is the number of the gasket between the cavity signal loading port and the cavity external surface, which determines the external damping rate of the cavity mode. Both of these parameters can be easily and accurately adjusted experimentally.</sec><sec> Our study paves the way for exploring anisotropic EPs based sensing technologies and more non-Hermitian related physics in the cavity magnonics system.</sec>

Bound states in the continuum (BICs) and exceptional points (EPs), as two distinct physical singularities represented by complex frequencies in non-Hermitian systems, have garnered significant attention and clear definitions in their respective fields in recent years. They share overlapping applications in areas such as high-sensitivity sensing and laser emission. However, the transition between the two, inspired by these intersections, remains largely unexplored. In this work, we reveal the transition process in a non-Hermitian two-mode system, evolving from one bound singularity to a two-dimensional exceptional ring, where the EP is the coalescent state of the quasi-Friedrich-Wintgen (FW)-BIC. This phenomenon is experimentally validated through pored dielectric metasurfaces in terahertz band. Furthermore, external pumping induced photocarriers as the dissipative perturbation, facilitates the breaking of degeneracy in the complex eigenfrequency and enables dynamic EP switching. Finally, we experimentally demonstrate a switchable terahertz beam deflection driven by the phase singularities of the EP. These findings are instrumental in advancing the development of compact devices for sensing and wavefront control within non-Hermitian systems.

Engineered non-Hermitian systems featuring exceptional points (EPs) can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, and electronics to atomic physics. In optics, non-Hermitian dynamics are typically realized using dissipation and phase-insensitive gain accompanied by unavoidable fluctuations. Here, we introduce non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sensitive amplification, and rich quantum and classical nonlinear dynamics. We show that two coupled OPOs can exhibit spectral anti-parity-time (anti-PT) symmetry and a EP between its degenerate and nondegenerate operation regimes. To demonstrate the distinct potentials of the coupled OPO system compared to conventional non-Hermitian systems, we present higher-order EPs with two OPOs, tunable Floquet EPs in a reconfigurable dynamic non-Hermitian system, and the generation of a squeezed vacuum around EPs, all of which are not easy to realize in other non-Hermitian platforms. We believe our results show that coupled OPOs are an outstanding non-Hermitian setting with unprecedented opportunities to realize nonlinear dynamical systems for enhanced sensing and quantum information processing.

As a kind of phase singularity in physical systems, exceptional points (EPs) give rise to a plethora of counterintuitive phenomena, such as the asymmetric state transfer and unidirectional reflection. Especially, the eigenvectors of EP are self-orthogonal, and an inherent chirality can be determined because of the “missing dimension.” Beyond traditional wisdom, in this work, we proposed theoretically an effective anti-parity-time symmetry in a non-Hermitian system under the external excitation, and the chirality reversal from resonance EP to absorption EP is demonstrated experimentally. The essence of this phenomenon is that the external input is involved in the establishment of the eigenstates of the system. Moreover, the critical condition of chirality reversal between two types of EPs in the non-Hermitian system is analytically given. This external excitation-controlled EP not only paves an effective way to study the fundamental non-Hermitian physics in an open system but also holds great potential in the applied photonic devices such as the chiral antenna, polarization converter, and wireless communication.

Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we have investigated entropic uncertainty relation (EUR) in a non-Hermitian system and revealed a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitarity dynamics determined by a Hermitian Hamiltonian, the behaviors of EUR can be well defined in two different ways depending on whether the system is located in unbroken phase or broken phase regimes. In unbroken phase regime, EUR undergoes an oscillatory behavior while in broken phase regime where the oscillation of EUR breaks down. The exceptional points mark the oscillatory and non-oscillatory behaviors of the EUR. In the dynamical limit, we have identified the witness of critical behavior of non-Hermitian systems in terms of the EUR. Our results reveal that the witness can detect exactly the critical points of non-Hermitian systems beyond (anti-) PT-symmetric systems. Our results may have potential applications to witness and detect phase transition in non-Hermitian systems.

Quantum phase transitions occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal critical behaviors for quantum phase transitions and quantum entanglement in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems. First, it bridges the non-analytic behavior of physical observable and no-analytic behavior of steady state energy, which explains why the quantum phase transitions in non-Hermitian systems occur for finite systems. Second, it predicts universal scaling behaviors of any physical observable at non-Hermitian phase transition point with scaling exponent being (1 − 1/p) with p being the number of coalesced states at the exceptional point. Third, it reveals that quantum entanglement in non-Hermitian phase transition point presents universal scaling behaviors with critical exponents being (1 − 1/p). These results uncover universal critical behaviors in non-Hermitian phase transitions and provide profound connections between entanglement and phase transition in non-Hermitian quantum many-body physics.

Exceptional points (EPs) in non-Hermitian systems offer a remarkably strong response to weak perturbations, but the nonorthogonal nature of the corresponding eigenvectors causes noise to diverge, hindering EPs practical application. Here, we report a twelve-fold enhancement of signal-to-noise ratio (SNR) in magnetic field sensing enabled by a third-order EP of coherent perfect absorption (CPA EP3) in a passive cavity magnonic system. This non-Hermitian magnonic platform comprises two identical yttrium iron garnet (YIG) spheres coherently coupled to a cavity mode, in which the CPA EP3 is realized by engineering the three-mode loss to form a pseudo-Hermitian absorption Hamiltonian. By independently tailoring the absorption EP apart from the resonance EP, the system circumvents the noise divergence caused by eigenbasis collapse. Notably, we harness the sensitivity of the minimum output intensity near CPA to perturbations, yielding a seventy-fold SNR improvement and a 400-fold increase in responsivity compared with non-CPA system. A comprehensive noise analysis over one hundred repeated measurements confirms the suppression of frequency noise near the CPA EP3. This demonstrates that our scheme not only avoids the noise divergence plaguing conventional higher-order EP sensors but also provides a general strategy to exploit both CPA and EP for SNR enhancement in passive non-Hermitian systems.

Topological insulators and bound states in the continuum represent two fascinating topics in the optical and photonic domain. The exploration of their interconnection and potential applications has emerged as a current research focus. Here, we investigated non-Hermitian photonics based on a parallel cascaded-resonator system, where both direct and indirect coupling between adjacent resonators can be independently manipulated. We observed the emergence of topological Fabry-Pérot bound states in the continuum in this non-Hermitian system, and theoretically validated its robustness. We also observed topological phase transitions and exceptional points in the same system. By elucidating the relationship between topological insulators and bound states in the continuum, this work will enable various applications that harness the advantages of bound states in the continuum, exceptional points, and topology. These applications may include optical delay and storage, highly robust optical devices, high-sensitivity sensing, and chiral mode switching.

Radiative emission experienced by a subwavelength particle near a resonant cavity is typically characterized by the well-known Purcell factor figure of merit. In recent work, we presented a generalization of Purcell enhancement that applies to situations involving exceptional points (EP)---spectral singularities in non-Hermitian systems where two or more eigenvectors and their corresponding complex eigenvalues coalesce, leading to a non-diagonalizable, defective Hamiltonian. EPs are attended by many intriguing physical effects and have been studied in various contexts, including lasers, atomic and molecular systems, photonic crystals, parity-time symmetric lattices, and optomechanical resonators. Thus far, the main focus of these works has been on analyzing the impact of second-order exceptional points on scattering from eternally incident light, e.g. for unidirectional transmission. An important but little explored property of EPs related to light-matter interactions is their ability to modify and enhance the local density of states (LDOS). Recently, we showed that EPs can modify the spontaneous emission rate or Purcell factor of narrow-band emitters embedded in resonant cavities. In this talk, we show that EPs can have an even greater impact on nonlinear optical processes like frequency conversion. In particular, we derive a general formula quantifying radiative emission from a subwavelength emitter in the vicinity of a triply resonant χ(2) cavity that supports an EP near the emission frequency and a bright mode at the second harmonic. We show that the resulting frequency up-conversion process can be enhanced by up to two orders of magnitude compared to nondegenerate scenarios and that, in contrast to the recently predicted spontaneous-emission enhancements, nonlinear EP enhancements can persist even when considering spatial distributions of broadband emitters, provided that the cavity satisfies special nonlinear selection rules. This is demonstrated via a two-dimensional proof-of-concept PhC designed to partially fulfill the various criteria needed to approach the derived bounds on the maximum achievable up-conversion efficiencies. Along these lines, we show that similar enhancements can arise in quantum systems consisting of single and multi-level atoms embedded in photonic cavities. Our predictions suggest an indirect but practically relevant route to experimentally observe the impact of EPs on spontaneous emission and related light–matter interactions, with implications to quantum information science.

The physics of topological singularities, namely exceptional points (EPs), has been a key to wide range of intriguing and unique physical effects in non-Hermitian systems. In this context, the mutual interactions among four coupled states around fourth-order EPs (EP4s) are yet to be explored. Here we report a four-level parameter-dependent perturbed non-Hermitian Hamiltonian, mimicking quantum or wave-based systems, to explore the physical aspects of an EP4 analytically as well as numerically. The proposed Hamiltonian exhibit different orders of interaction schemes with the simultaneous presence of different higher-order EPs. Here an EP4 has been realized by mutual interaction between four coupled states with proper parameter manipulation. We comprehensively investigate the dynamics of corresponding coupled eigenvalues with stroboscopic parametric variation in the vicinity of the embedded EP4 to establish a new successive state-switching phenomenon among them; which proves to be robust even in the presence of different order of EPs. Implementing the relation of the perturbation parameters with the coupling control parameters, we exclusively report a region to host multiple EP4 in a specific system. The chiral behaviour of successive state-exchange has also been established near EP4. Proposed scheme enriched with physical aspects of EP4s should provide a new light manipulation tool in any anisotropic multi-state integrated system.

Non-Hermitian systems display remarkable response effects that directly reflect a variety of distinct spectral scenarios, such as exceptional points where the eigensystem becomes defective. However, present frameworks treat the different scenarios as separate cases, following the singular mathematical change between different spectral decompositions from one scenario to another. This not only complicates the coherent description near the spectral singularities where the response qualitatively changes, but also impedes the application to practical systems, as the determination of these decompositions is manifestly ill conditioned. Here we develop a general response theory of non-Hermitian systems that uniformly applies across all spectral scenarios. We unravel this response by formulating a uniform expansion of the spectral quantization condition, as well as a uniform expansion of the Green's function, where both expansions exclusively involve directly calculable data from the Hamiltonian. These data smoothly vary with external parameters and energy as spectral singularities are approached and attained, and nevertheless capture the qualitative differences of the response in these scenarios. We furthermore present two direct applications of this framework. First, in the context of the quantization condition, we determine the precise conditions for spectral degeneracies of geometric multiplicity greater than unity, as well as the perturbative behavior around these cases. Second, in the context of the Green's function, we formulate a hierarchy of spectral response strengths that varies continuously across all parameter space, and thereby also reliably determines the response strength of exceptional points. Finally, we join both themes and demonstrate, both generally and in concrete examples, that the previously inaccessible scenarios of higher geometric multiplicity result in unique variants of the super-Lorentzian response. Our approach widens the scope of non-Hermitian response theory to capture all spectral scenarios on an equal and uniform footing, identifies the exact mechanisms that lead to the qualitative changes of physical signatures, and renders non-Hermitian response theory fully applicable to numerical descriptions of practical systems. Published by the American Physical Society 2025

Biosensors have important implications for medical research and disease prevention. High-quality optical microcavities with whispering-gallery modes (WGMs) can enhance light-matter interactions in small volumes, making them ideal for making biosensors. However, the sensitivity of the optical microcavity biosensors can further improve. The sensing performance of the biosensor based on a microtube or microbubble can be further improved by exceptional points (EPs) in non-Hermitian physics. Non-Hermitian systems are open systems, exchanging energy with the external environment. EPs as one representative feature of non-Hermitian systems are a branch point singularity, where eigenvalues and their corresponding eigenvectors coalesce in the parameter space of the system. In the vicinity of EPs, the eigenvalues exhibit non-trivial topology, which can be exploited for highly sensitive detections. Based on this principle, a novel configuration with multiple layers is proposed, where the coupling of modes can be tailored for EPs by choosing the thickness of layers. Simulated results demonstrated that the design has strong biosensing performance. The sensitivity can be enhanced by 1000 times more than that of conventional methods. The ability to prevent environmental interference has also been improved. This is significant for highly sensitive low-concentration biomolecule detection in microtubes in the future.

The adiabatic theorem, a corollary of the Schrödinger equation, manifests itself in a profoundly different way in non-Hermitian arrangements, resulting in counterintuitive state transfer schemes that have no counterpart in closed quantum systems. In particular, the dynamical encirclement of exceptional points (EPs) in parameter space has been shown to lead to a chiral phase accumulation, non-adiabatic jumps and topological mode conversion1-8. Recent theoretical studies, however, have shown that contrary to previously established demonstrations, this behaviour is not strictly a result of winding around a non-Hermitian degeneracy9. Instead, it seems to be mostly attributed to the non-trivial landscape of the Riemann surfaces, sometimes because of the presence of an EP in the vicinity9-11. Here, in an effort to bring this counterintuitive aspect of non-Hermitian systems to light and confirm this hypothesis, we provide a set of experiments to directly observe the field evolution and chiral state conversion in an EP-excluding cycle in a slowly varying non-Hermitian system. To do so, a versatile yet unique fibre-based photonic emulator is realized that utilizes the polarization degrees of freedom in a quasi-common-path single-ring arrangement. Our observations may open up new avenues for light manipulation and state conversion, as well as providing a foundation for understanding the intricacies of the adiabatic theorem in non-Hermitian systems.

In non-Hermitian systems, enhancing sensitivity under exceptional point (EP) conditions offers an ideal solution for reconciling the trade-off between sensitivity and size constraints in sensing applications. However, practical application is limited by undesired sensitivity to external fluctuations, noise, and errors in signal amplification synchronization. This paper presents a precisely controlled EP tracking and detection system (EPTDS) that achieves long-term rapid tracking and locking near the EP by constructing a second-order non-Hermitian optical sensing unit, employing an optical power adaptive control method, and utilizing a combinatorial demodulation-based dual-loop cascaded control (CDCC) technique to selectively suppress traditional noise at different frequencies. The system locking time is 10 ms, and in room temperature conditions, the output frequency error over 1 hour is reduced by more than 30 times compared to before locking. To assess its sensing capabilities, the EPTDS undergoes testing in a rotational experiment based on the Sagnac effect, with the output bias instability based on Allan deviation measured at 0.036 °/h. This is the best result for EP-enhanced angular rate sensing that we are aware of that has been reported. The EPTDS method can be extended to various sensing fields, providing a new path for transitioning non-Hermitian sensing from the laboratory to practical applications.

Recently, a Dirac exceptional point (EP) was reported in a non-Hermitian system. Unlike a Dirac point in Hermitian systems, this Dirac EP has coalesced eigenstates in addition to the degenerate energy. Also different from a typical EP, the two energy levels connected at this Dirac EP remain real in its vicinity and display a linear instead of square root dispersion, forming a tilted Dirac cone in the hybrid space consisting of a momentum dimension and a synthetic dimension for the strength of non-Hermiticity. In this report, we first present simple three-band and two-band matrix models with a Dirac EP, where the linear dispersion of the tilted Dirac cone can be expressed analytically. Importantly, our analysis also reveals that there exist Hermitian and non-Hermitian systems that have the same (real-valued) energy spectrum in their entire parameter space, with the exception that one or more degeneracies in the former are replaced by Dirac EPs in the later. Finally, we show the existence of an imaginary Dirac cone with an EP at its center.