Single-Mode Emission by Phase-Delayed Coupling Between Nanolasers

Near-field coupling between nanolasers enables collective high-power lasing but leads to complex spectral reshaping and multimode operation, limiting the emission brightness, spatial coherence, and temporal stability. Many lasing architectures have been proposed to circumvent this limitation based on symmetries, topology, or interference. We show that a much simpler and robust method exploiting phase-delayed coupling, where light exchanged by the lasers carries a phase, can enable stable single-mode operation. Phase-delayed coupling changes the modal amplification: for pump powers close to the anyonic parity-time (PT) symmetric exceptional point, a high phase delay completely separates the mode thresholds, leading to single-mode operation. This is shown by stability analysis with nonlinear coupled mode theory and stochastic differential equations for two coupled nanolasers and confirmed by a realistic semianalytical treatment of a dimer of lasing nanospheres. Finally, we extend the mode control to large arrays of nanolasers featuring lowered thresholds and higher power. Our work promises a novel solution to engineer bright and stable single-mode lasing from nanolaser arrays with important applications in photonic chips for communication and LIDAR.

Quantum and Classical Exceptional Points at the Nanoscale: Properties and Applications

Exceptional points (EPs) are the spectral singularities and one of the central concepts of non-Hermitian physics, originating from the inevitable energy exchange with the surrounding environment. EPs exist in diverse physical systems and give rise to many counterintuitive effects, offering rich opportunities to control the dynamics and alter the properties of optical, electronic, acoustic, and mechanical states. The last two decades have witnessed the flourishing of non-Hermitian physics and associated applications related to coalesced eigenstates at EPs in a plethora of classical systems. While stemming from the quantum mechanism, the implementation of EPs in real quantum systems still faces challenges of tuning and stabilizing the systems at EPs, as well as the additional noises that hinder the observation of relevant phenomena. This review mainly focuses on summarizing the current efforts and opportunities offered by quantum EPs that result from or produce observable quantum effects. We introduce the concepts of Hamiltonian and Liouvillian EPs in the quantum regime and focus on their different properties in connection with quantum jumps and decoherence. We then provide a comprehensive discussion covering the theoretical and experimental advances in accessing EPs in diverse quantum systems and platforms. Special attention is paid to EP-based quantum-optics applications with state-of-art technologies. Finally, we present a discussion on the existing challenges of constructing quantum EPs at the nanoscale and an outlook on the fundamental science and applied technologies of quantum EPs, aiming to provide valuable insights for future research and building quantum devices with high performance and advanced functionalities.

Exceptional lines and higher-order exceptional points enabled by uniform loss

This study theoretically investigates the realization of exceptional points (EPs) in space-time invariant Lorentz dispersive media with uniform loss, which contrasts sharply with conventional approaches that rely on spatial or temporal differential losses. Using the derived full and reduced Hamiltonians, we reveal that uniform loss in Lorentz dispersive media not only introduces attenuation to the eigenmodes of the lossless medium but also enables two distinct types of non-Hermitian couplings: reciprocal and non-reciprocal. Both coupling mechanisms independently contribute to the emergence of EPs. The EPs manifest as exceptional lines (ELs) in the parameter space, and a fourth-order EP (EP4) is formed when three ELs intersect at a single point. Remarkably, at the EP4, all eigenmodes exhibit maximum optical chirality density, presenting potential applications such as chiral sorting. Our findings provide valuable insights into the mechanisms underpinning the formation of EPs.

Unified One-Parameter Scaling Function for Anderson Localization Transitions in Nonreciprocal Non-Hermitian Systems

Using dimensionless conductances as scaling variables, the conventional one-parameter scaling theory of localization fails for nonreciprocal non-Hermitian systems such as the Hatano-Nelson model. Here, we propose a one-parameter scaling function using the participation ratio as the scaling variable. Employing a highly accurate numerical procedure based on exact diagonalization, we demonstrate that this one-parameter scaling function can describe Anderson localization transitions of nonreciprocal non-Hermitian systems in one and two dimensions of symmetry classes AI and A. The critical exponents of correlation lengths depend on symmetries and dimensionality only, a typical feature of universality. Then, we derive a complex-gap equation based on the self-consistent Born approximation to determine the critical disorder at which the point gap closes. The obtained critical disorders perfectly match those given by the one-parameter scaling function. Moreover, we propose a one-parameter β function that can describe the critical properties of such Anderson localization transitions. Finally, we show that the one-parameter scaling function is also valid for Anderson localization transitions in reciprocal non-Hermitian systems such as the two-dimensional class AII^{†} and can, thus, serve as a unified scaling function for disordered non-Hermitian systems.

Flatband solitons in the nonlinear media with parity-time symmetric sawtooth lattices

This paper demonstrates solitons in the nonlinear media with parity-time (PT) symmetric sawtooth lattices. These PT lattices exhibit several uncommon features, of which the most striking are the flat Bloch bands, in which the local modes are stable and the depths of imaginary parts are large when the PT symmetry breaks up. Furthermore, we investigate the formation, properties, and dynamics of fundamental, multipole, and two kinds of vortex-carrying solitons that all reside in the semi-infinite bandgaps of the underlying linear Bloch wave spectrum. The stability of these localized gap modes is inspected by direct dynamical simulations of the perturbed solitons, and the results reveal that the lattice parameters significantly influence their stability. Our findings provide valuable insights into soliton physics within the versatile platform of PT-symmetric systems.

Experimental Observation of Dirac Exceptional Points

The energy-level degeneracies, also known as exceptional points (EPs), are crucial for comprehending emerging phenomena in materials and enabling innovative functionalities for devices. Since EPs were proposed over half a century ago, only two types of EPs have been experimentally discovered, revealing intriguing phases of materials such as Dirac and Weyl semimetals. These discoveries have showcased numerous exotic topological properties and novel applications, such as unidirectional energy transfer. Here, we report the observation of a novel type of EP, named the Dirac EP, utilizing a nitrogen-vacancy center in diamond. Two of the eigenvalues are measured to be degenerate at the Dirac EP and remain real in its vicinity. This exotic band topology associated with the Dirac EP enables the preservation of the symmetry when passing through, and makes it possible to achieve adiabatic evolution in non-Hermitian systems. We examined the degeneracy between the two eigenstates by quantum state tomography, confirming that the degenerate point is a Dirac EP rather than a Hermitian degeneracy. Our research of the distinct type of EP contributes a fresh perspective on dynamics in non-Hermitian systems and is potentially valuable for applications in quantum control in non-Hermitian systems and the study of the topological properties of EPs.

Exceptional points in a passive strip waveguide

Exceptional points (EPs) in non-Hermitian systems have attracted significant interest due to their unique behaviors, including novel wave propagation and radiation. While EPs have been explored in various photonic systems, their integration into standard photonic platforms can expand their applicability to broader technological domains. In this work, we propose and experimentally demonstrate EPs in an integrated photonic strip waveguide configuration, exhibiting unique deep wave penetration and uniform-intensity radiation profiles. By introducing the second-order grating on one side of the waveguide, forward and backward propagating modes are coupled both directly through second-order coupling and indirectly through first-order coupling via a radiative intermediate mode. To describe the EP behavior in a strip configuration, we introduce modified coupled-mode equations that account for both transverse and longitudinal components. These coupled-mode formulas reveal the formation of EPs in bandgap closure, achieved by numerically optimizing the grating's duty cycle to manipulate the first- and second-order couplings simultaneously. Experimental observations, consistent with simulations, confirm the EP behavior, with symmetric transmission spectra and constant radiation profiles at the EP wavelength, in contrast to conventional exponential decay observed at detuned wavelengths. These results demonstrate the realization of EPs in a widely applicable strip waveguide configuration, paving the way for advanced EP applications in nonlinear and ultrafast photonics, as well as advanced sensing technologies.

Selective filtering of photonic quantum entanglement via anti-parity-time symmetry

Entanglement is a key resource for quantum computing, sensing, and communication, but it is susceptible to decoherence. To address this, research in quantum optics has explored filtering techniques such as photon ancillas and Rydberg atom blockade to restore entangled states. We introduce an approach to entanglement retrieval that exploits the features of non-Hermitian systems. By designing an anti-parity-time two-state guiding configuration, we demonstrate efficient extraction of entanglement from any input state. This filter is implemented on a lossless waveguide network and achieves near-unity fidelity under single- and two-photon excitation and is scalable to higher photon levels, remaining robust against decoherence during propagation. Our results offer an approach to using non-Hermitian symmetries to address central challenges in quantum technologies.

On-chip reconfigurable transmission in spatially chirped Floquet parity-time symmetric photonics

Phase transition in parity-time (PT) symmetry is one of the most intriguing discoveries in non-Hermitian physics, giving rise to plenty of physical phenomena and strategies to develop advanced devices and systems, such as unconventional lasers, nonreciprocal transmission, and enhanced sensitivity. Floquet PT-symmetric systems are characterized by time-periodic Hamiltonians, in which the gain or loss is modulated to steer the PT phase, providing an additional dimension for realizing phase transitions. In this study, we introduce frequency-varying modulation, specifically spatially chirped modulation, into on-chip Floquet PT-symmetric photonic waveguides to explore their unique properties. The waveguides exhibit distinct forward and backward transmissions when the system dynamically evolves around phase transition points, i.e., exceptional points. Furthermore, reconfigurable asymmetric transmission systems are developed by integrating tunable mode switches. Combining non-Hermitian physics with the advanced technologies of photonic integrated circuits holds great potential to create devices and systems with improved functionalities and enhanced performance.

Polarization-controlled chiral transport

Handedness-selective chiral transport is an intriguing phenomenon that not only holds significant importance for fundamental research but also carries application prospects in fields such as optical communications and sensing. Currently, on-chip chiral transport devices are static, unable to modulate the output modes based on the input modes. This limits both device functionality reconfiguration and information transmission capacity. Here, we propose to use the incident polarization diversity to control the Hamiltonian evolution path, achieving polarization-dependent chiral transport. By mapping the evolution path of TE and TM polarizations onto elaborately engineered double-coupled waveguides, we experimentally demonstrate that different polarizations yield controllable modal outputs. This work combines Multiple-Input, Multiple-Output, and polarization diversity concepts with chiral transport and challenges the prevailing notion that the modal outputs are fixed to specific modes in chiral transport, thereby opening pathways for the development of on-chip reconfigurable and high-capacity handedness-selective devices.

Critical Relaxation in the Quantum Yang-Lee Edge Singularity

We study the relaxation dynamics near the critical points of the Yang-Lee edge singularities (YLESs) in the quantum Ising chain in an imaginary longitudinal field with a polarized initial state. We find that scaling behaviors are manifested in the relaxation process after a non-universal transient time. We show that for the paramagnetic Hamiltonian, the magnetization oscillates periodically with the period being inversely proportional to the gap between the lowest energy level; for the ferromagnetic Hamiltonian, the magnetization decays to a saturated value; while for the critical Hamiltonian, the magnetization increases linearly. A scaling theory is developed to describe these scaling properties. In this theory, we show that for a small- and medium-sized system, the scaling behavior is described by the (0+1)-dimensional YLES.

Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra

The non-Hermitian skin effect is an iconic phenomenon characterized by the aggregation of eigenstates near the system boundaries in non-Hermitian systems. While extensively studied in one dimension, understanding the skin effect and extending the non-Bloch band theory to higher dimensions encounter a formidable challenge, primarily due to infinite lattice geometries or open boundary conditions. This work adopts a point-gap perspective and unveils that non-Hermitian skin effect in all spatial dimensions originates from point gaps. We introduce the concept of uniform spectra and reveal that regardless of lattice geometry, their energy spectra are universally given by the uniform spectra, even though their manifestations of skin modes may differ. Building on the uniform spectra, we demonstrate how to account for the skin effect with generic lattice cuts and establish the connections of skin modes across different geometric shapes via momentum-basis transformations. Our findings highlight the pivotal roles point gaps play, offering a unified understanding of the topological origin of non-Hermitian skin effect in all dimensions.

Crossing exceptional points in non-Hermitian quantum systems

Exceptional points facilitate peculiar dynamics in non-Hermitian systems. Yet, in photonics, they have mainly been studied in the classical realm. In this work, we reveal the behavior of two-photon quantum states in non-Hermitian systems across the exceptional point. We probe the lossy directional coupler with an indistinguishable two-photon input state and observe distinct changes of the quantum correlations at the output as the system undergoes spontaneous breaking of parity-time symmetry. Moreover, we demonstrate a switching in the quantum interference of photons directly at the exceptional point, where Hong-Ou-Mandel dips are transformed into peaks by a change of basis. These results show that quantum interference and exceptional points are linked in curious ways that can now be further explored.

Quantum and thermal noise in coupled non-Hermitian waveguide systems with different models of gain and loss

Non-Hermitian (NH) photonic systems leverage gain and loss to open new directions for nanophotonic technologies. However, the quantum and thermal noise intrinsically associated with gain/loss affects the eigenvalue/eigenvector structure of NH systems, and thus the existence of exceptional points, as well as the practical noise performance of these systems. Here, we present a comparative analysis of the impact of different gain and loss mechanisms on the noise generated in gain-loss compensated NH waveguide systems. Our results highlight important differences in the eigenvalue/eigenvector structure, noise power, photon statistics and squeezing. At the same time, we identify some universal properties such as the occurrence of phase-transition points in parameter space and intriguing phenomena related to them, including coalescence of pairs of eigenvectors, gain-loss compensation, and linear scaling of the noise with the length of the waveguide. We believe that these results contribute to a better understanding of the impact of the gain/loss mechanism on the noise generated in NH systems.

Observation of non-Hermitian boundary induced hybrid skin-topological effect excited by synthetic complex frequencies

The hybrid skin-topological effect (HSTE) has recently been proposed as a mechanism where topological edge states collapse into corner states under the influence of the non-Hermitian skin effect (NHSE). However, directly observing this effect is challenging due to the complex frequencies of eigenmodes. In this study, we experimentally observe HSTE corner states using synthetic complex frequency excitations in a transmission line network. We demonstrate that HSTE induces asymmetric transmission along a specific direction within the topological band gap. Besides HSTE, we identify corner states originating from non-chiral edge states, which are caused by the unbalanced effective onsite energy shifts at the boundaries of the network. Furthermore, our results suggest that whether the bulk interior is Hermitian or non-Hermitian is not a key factor for HSTE. Instead, the HSTE states can be realized and relocated simply by adjusting the non-Hermitian distribution at the boundaries. Our research has deepened the understanding of a range of issues regarding HSTE, paving the way for advancements in the design of non-Hermitian topological devices.

Hydrodynamic moiré superlattice

The structural periodicity in photonic crystals guarantees the crystal's effective energy band structure, which is the fundamental cornerstone of topological and moiré physics. However, the shear modulus in most fluids is close to zero, which makes it challenging for fluids to maintain spatial periodicity akin to photonic crystals. We realized periodic vortices in hydrodynamic metamaterials and created a bilayer moiré superlattice by stacking and twisting two such vortex fluids. We observed energy delocalization and localization when the twist angles, respectively, result in the Pythagorean and non-Pythagorean triples in the fluidic moiré superlattice. Anomalous localization was found even in commensurate moiré fluids with large lattice constants that satisfy Pythagorean triples. Our work reports the moiré phenomena in fluids and opens an unexpected door to controlling the energy transfer, mass transport, and particle navigation through the elaborate dynamics of vortices in fluidic moiré superlattices.

Non-Hermitian Theory of Valley Excitons in Two-Dimensional Semiconductors

Electron-hole exchange interaction in two-dimensional transition metal dichalcogenides is extremely strong due to the dimension reduction, which promises valley-superposed excitonic states with linearly polarized optical emissions. However, strong circular polarization reflecting valley-polarized excitonic states is commonly observed in helicity-resolved optical experiments. Here, we present a non-Hermitian theory of valley excitons by incorporating optical pumping and intrinsic decay, which unveils an anomalous valley-polarized excitonic state with elliptically polarized optical emission. This novel state arises from the non-Hermiticity induced parity-time (PT) symmetry breaking, which impedes the experimental observation of intervalley excitonic coherence effect. At large excitonic center-of-mass momenta, the PT symmetry is restored and the excitonic states recover their valley coherence. Interestingly, the linear polarization directions in optical emissions from these valley-superposed excitonic states are nonorthogonal and even become parallel at exceptional points. Our non-Hermitian theory also predicts a nonzero Berry curvature for valley excitons, which admits a topological excitonic Hall transport beyond the Hermitian predictions.

Experimental realization of on-chip few-photon control around exceptional points

Non-Hermitian physical systems have attracted considerable attention in recent years for their unique properties around exceptional points (EPs), where the eigenvalues and eigenstates of the system coalesce. Phase transitions near exceptional points can lead to various interesting phenomena, such as unidirectional wave transmission. However, most of those studies are in the classical regime and whether these properties can be maintained in the quantum regime is still a subject of ongoing studies. Using a non-Hermitian on-chip superconducting quantum circuit, here we observe a phase transition and the corresponding exceptional point between the two phases. Furthermore, we demonstrate that unidirectional microwave transmission can be achieved even in the few-photon regime within the broken symmetry phase. This result holds some potential applications, such as on-chip few-photon microwave isolators. Our study reveals the possibility of exploring the fundamental physics and practical quantum devices with non-Hermitian systems based on superconducting quantum circuits.

Time-variant parity-time symmetry in frequency-scanning systems

Parity-time (PT) symmetry is an active research area that provides a variety of new opportunities for different systems with novel functionalities. For instance, PT symmetry has been used in lasers and optoelectronic oscillators to achieve single-frequency lasing or oscillation. A single-frequency system is essentially a static PT-symmetric system, whose frequency is time-invariant. Here we investigate time-variant PT symmetry in frequency-scanning systems. Time-variant PT symmetry equations and eigenfrequencies for frequency-scanning systems are developed. We show that time-variant PT symmetry can dynamically narrow the instantaneous linewidth of frequency-scanning systems. The instantaneous linewidth of a produced frequency-modulated continuous-wave (FMCW) waveform is narrowed by a factor of 14 in the experiment. De-chirping and radar imaging results also show that the time-variant PT-symmetric system outperforms a conventional frequency-scanning one. Our study paves the way for a new class of time-variant PT-symmetric systems and shows great promise for applications including FMCW radar and lidar systems.

Dynamic gain and frequency comb formation in exceptional-point lasers

Exceptional points (EPs)-singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce-feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell-Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an "EP comb" has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell-Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

Parity-Time symmetry helps breaking a new limit

Parity-Time (PT) symmetry is an emerging concept in quantum mechanics where non-Hermitian Hamiltonians can exhibit real eigenvalues. Now, PT symmetric optical microresonators have been demonstrated to break the bandwidth-efficiency limit for nonlinear optical signal processing.

Clearing a path for light through non-Hermitian media

The performance of all active photonic devices today is greatly limited by loss. Here, we show that one can engineer a low loss path in a metal-clad lossy multi-mode waveguide while simultaneously achieving high-performance active photonic devices. We leverage non-Hermitian systems operating beyond the exceptional point to enable the redistribution of losses in a multi-mode photonic waveguide. Consequently, our multi-mode waveguide offers low propagation losses for fundamental mode while other higher order modes experience prohibitively high losses. Furthermore, we show an application of this non-Hermitian waveguide platform in designing power-efficient thermo-optic phase shifters with significantly faster response times than conventional silicon-based thermo-optic phase shifters. Our device achieves a propagation loss of less than 0.02 dB μm-1 for our non-Hermitian waveguide-based phase shifters with high performance efficiency of P π ⋅ τ = 19.1 mW μs. In addition, our phase shifters have significantly faster response time (rise/fall time), τ ≈ 1.4 μs, compared to traditional silicon based thermo-optic phase shifters.

Parity-time and anti-parity-time symmetries in heat transfer

This perspective briefly reviews the recent developments of non-Hermitian parity-time and anti-parity-time physics in dissipative heat transfer systems, highlighting their potentials in novel functional thermal devices in the future.

Hybrid Patterns and Solitonic Frequency Combs in Non-Hermitian Kerr Cavities

We unveil a new scenario for the formation of dissipative localized structures in nonlinear systems. Commonly, the formation of such structures arises from the connection of a homogeneous steady state with either another homogeneous solution or a pattern. Both scenarios, typically found in cavities with normal and anomalous dispersion, respectively, exhibit unique fingerprints and particular features that characterize their behavior. However, we show that the introduction of a periodic non-Hermitian modulation in Kerr cavities hybridizes the two established soliton formation mechanisms, embodying the particular fingerprints of both. In the resulting novel scenario, the stationary states acquire a dual behavior, playing the role that was unambiguously attributed to either homogeneous states or patterns. These fundamental findings have profound practical implications for frequency comb generation, introducing unprecedented reversible mechanisms for real-time manipulation.

Giant strong coupling in a Q-BICs' tetramer metasurface

Due to their ultrahigh Q-factor and small mode volume, bound states in the continuum (BICs) are intriguing for the fundamental study of the strong coupling regime. However, the strong coupling generated by BICs in one metasurface is not always strong enough, which highly limits its efficiency in applications. In this work, we realize a giant strong coupling of at most 60 meV in a quasi-BICs' (Q-BICs) tetramer metasurface composed of four Si cylinders with two different sets of diagonal lengths. The Q-BICs are induced from two types of electric quadrupole (EQ), for which detuning can be flexibly controlled by manipulating the C4v symmetry breaking Δd. The giant Rabi splitting in our proposed metasurface performs more than 15 times of the previous works, which provides more possibilities for important nonlinear and quantum applications, such as nanolaser and quantum optics.

Topologically protected entanglement switching around exceptional points

The robust operation of quantum entanglement states is crucial for applications in quantum information, computing, and communications1-3. However, it has always been a great challenge to complete such a task because of decoherence and disorder. Here, we propose theoretically and demonstrate experimentally an effective scheme to realize robust operation of quantum entanglement states by designing quadruple degeneracy exceptional points. By encircling the exceptional points on two overlapping Riemann energy surfaces, we have realized a chiral switch for entangled states with high fidelity. Owing to the topological protection conferred by the Riemann surface structure, this switching of chirality exhibits strong robustness against perturbations in the encircling path. Furthermore, we have experimentally validated such a scheme on a quantum walk platform. Our work opens up a new way for the application of non-Hermitian physics in the field of quantum information.

Design and simulation of a tunable parity-time symmetric optoelectronic oscillator utilizing integrated components

Non-Hermitian photonics, relaying on parity-time (PT) symmetry, have shown promise in achieving mode selection for optical or microwave single-mode oscillation. Typically, a PT-symmetric system is constructed using two coupled loops with identical geometry. This article utilizes the PT-symmetry property to select a single frequency mode in an optoelectronic oscillator (OEO). However, traditional OEO implementations often involve discrete components, limiting widespread adoption due to factors such as size, weight, power consumption, and cost. Our aim in this paper is to leverage integrated components within the OEO loop. The proposed structure incorporates an integrated micro-ring resonator (MRR) with a high-quality factor (Q-factor) that serves both as a modulator and a resonator. Additionally, we suggest employing an adjustable integrated power splitter utilizing a micro heater to balance the gain and loss of two mutually coupled OEO loops. In this configuration, two integrated photo detectors (PD) are also utilized. In this setup, the single-frequency mode can be easily identified by simultaneously utilizing the properties of PT-symmetry and an integrated high-Q-factor resonator, obviating the need for a narrowband microwave filter. By adjusting the center frequency of the microwave photonic filter (MPF), the frequency of the generated signal can be tuned over a wide range. For instance, setting the generated frequency of the microwave signal to 11.5 GHz results in a measured phase noise of - 76.5 dBc/Hz at a 10-kHz offset frequency, with a side mode suppression ratio (SMSR) of 40 dB.

Creating Anti-Chiral Exceptional Points in Non-Hermitian Metasurfaces for Efficient Terahertz Switching

Non-Hermitian degeneracies, also known as exceptional points (EPs), have presented remarkable singular characteristics such as the degeneracy of eigenvalues and eigenstates and enable limitless opportunities for achieving fascinating phenomena in EP photonic systems. Here, the general theoretical framework and experimental verification of a non-Hermitian metasurface that holds a pair of anti-chiral EPs are proposed as a novel approach for efficient terahertz (THz) switching. First, based on the Pancharatnam-Berry (PB) phase and unitary transformation, it is discovered that the coupling variation of ±1 spin eigenstates will lead to asymmetric modulation in two orthogonal linear polarizations (LP). Through loss-induced merging of a pair of anti-chiral EPs, the decoupling of ±1 spin eigenstates are then successfully realized in a non-Hermitian metasurface. Final, the efficient THz modulation is experimentally demonstrated, which exhibits modulation depth exceeding 70% and Off-On-Off switching cycle less than 9 ps in one LP while remains unaffected in another one. Compared with conventional THz modulation devices, the metadevice shows several figures of merits, such as a single frequency operation, high modulation depth, and ultrafast switching speed. The proposed theory and loss-induced non-Hermitian device are general and can be extended to numerous photonic systems varying from microwave, THz, infrared, to visible light.

Free-space laser emission from Nd:YAG elliptical microdisks

The utilization of deformed microcavities, such as elliptical microdisks, has been widely acknowledged as an effective solution for achieving free-space emission in microcavity lasers. However, the deformations introduced in the microcavity structure tend to decrease the quality factor (Q factor), resulting in weakened output intensity. To address this issue, one potential approach is to employ highly efficient laser gain media that can compensate for the negative impact of the structure on the output intensity. In this study, we employed the exceptional laser crystal material Nd:YAG as the laser gain medium and successfully fabricated an elliptical microdisk laser with a major semiaxis of 15 µm and an eccentricity ratio of 0.15. By utilizing an 808 nm laser for pumping, we were able to achieve free-space laser emission with a slope efficiency of 1.7% and a remarkable maximum output power of 58 µW. This work contributes toward the advancement of the application of deformation microcavity lasers.

Dynamical encircling of multiple exceptional points in anti-PT symmetry system

Exceptional points (EPs) in non-Hermitian systems have turned out to be at the origin of many intriguing effects with no counterparts in Hermitian cases. A typically interesting behavior is the chiral mode switching by dynamically winding the EP. Most encircling protocols focus on the two-state or parity-time (PT) symmetry systems. Here, we propose and investigate the dynamical encircling of multiple EPs in an anti-PT-symmetric system, which is constructed based on a one-dimensional lattice with staggered lossy modulation. We reveal that dynamically encircling the multiple EPs results in the chiral dynamics via multiple non-Hermiticity-induced nonadiabatic transitions, where the output state is always on the lowest-loss energy sheet. Compared with the PT-symmetric systems that require complicated variation of the gain/loss rate or on-site potentials, our system only requires modulations of the couplings which can be readily realized in various experimental platforms. Our scheme provides a route to study non-Hermitian physics by engineering the EPs and implement novel photonic devices with unconventional functions.

Band flipping and bandgap closing in a photonic crystal ring and its applications

The size of the bandgap in a photonic crystal ring is typically intuitively considered to monotonically grow as the modulation amplitude of the grating increases, causing increasingly large frequency splittings between the "dielectric" and "air" bands. In contrast, here we report that as the modulation amplitude in a photonic crystal ring increases, the bandgap does not simply increase monotonically. Instead, after the initial increase, the bandgap closes and then reopens again with the two bands flipped in energy. The air and dielectric band edges are degenerate at the bandgap closing point. We demonstrate this behavior experimentally in silicon nitride photonic crystal microrings, where we show that the bandgap is closed to within the linewidth of the optical cavity mode, whose intrinsic quality factor remains unperturbed with a value ≈ 1×106. Moreover, through finite-element simulations, we show that such bandgap closing and band flipping phenomena exist in a variety of photonic crystal rings with varying unit cell geometries and cladding layers. At the bandgap closing point, the two standing wave modes with a degenerate frequency are particularly promising for single-frequency lasing applications. Along this line, we propose a compact self-injection locking scheme that integrates many core functionalities in one photonic crystal ring. Additionally, the single-frequency lasing might be applicable to distributed-feedback (DFB) lasers to increase their manufacturing yield.

Toward Heisenberg scaling in non-Hermitian metrology at the quantum regime

Non-Hermitian quantum metrology, an emerging field at the intersection of quantum estimation and non-Hermitian physics, holds promise for revolutionizing precision measurement. Here, we present a comprehensive investigation of non-Hermitian quantum parameter estimation in the quantum regime, with a special focus on achieving Heisenberg scaling. We introduce a concise expression for the quantum Fisher information (QFI) that applies to general non-Hermitian Hamiltonians, enabling the analysis of estimation precision in these systems. Our findings unveil the remarkable potential of non-Hermitian systems to attain the Heisenberg scaling of 1/t, where t represents time. Moreover, we derive optimal measurement conditions based on the proposed QFI expression, demonstrating the attainment of the quantum Cramér-Rao bound. By constructing non-unitary evolutions governed by two non-Hermitian Hamiltonians, one with parity-time symmetry and the other without specific symmetries, we experimentally validate our theoretical analysis. The experimental results affirm the realization of Heisenberg scaling in estimation precision, marking a substantial milestone in non-Hermitian quantum metrology.

Spatial Information Lasing Enabled by Full-k-Space Bound States in the Continuum

Optical amplification and massive information transfer in modern physics depend on stimulated radiation. However, regardless of traditional macroscopic lasers or emerging micro- and nanolasers, the information modulations are generally outside the lasing cavities. On the other hand, bound states in the continuum (BICs) with inherently enormous Q factors are limited to zero-dimensional singularities in momentum space. Here, we propose the concept of spatial information lasing, whose lasing information entropy can be correspondingly controlled by near-field Bragg coupling of guided modes. This concept is verified in gain-loss metamaterials supporting full-k-space BICs with both flexible manipulations and strong confinement of light fields. The counterintuitive high-dimensional BICs exist in a continuous energy band, which provide a versatile platform to precisely control each lasing Fourier component and, thus, can directly convey rich spatial information on the compact size. Single-mode operation achieved in our scheme ensures consistent and stable lasing information. Our findings can be expanded to different wave systems and open new scenarios in informational coherent amplification and high-Q physical frameworks for both classical and quantum applications.

Spatio-spectral control of coherent nanophotonics

Fast modulation of optical signals that carry multidimensional information in the form of wavelength, phase or polarization has fueled an explosion of interest in integrated photonics. This interest however masks a significant challenge which is that independent modulation of multi-wavelength carrier signals in a single waveguide is not trivial. Such challenge is attributed to the longitudinal direction of guided-mode propagation, limiting the spatial separation and modulation of electric-field. Here, we overcome this using a single photonic element that utilizes active coherent (near) perfect absorption. We make use of standing wave patterns to exploit the spatial-degrees-of-freedom of in-plane modes and individually address elements according to their mode number. By combining the concept of coherent absorption in spatio-spectral domain with active phase-change nanoantennas, we engineer and test an integrated, reconfigurable and multi-spectral modulator operating within a single element. Our approach demonstrates for the first time, a non-volatile, wavelength-addressable element, providing a pathway for exploring the tunable capabilities in both spatial and spectral domains of coherent nanophotonics.

Experimental Demonstration of Controllable PT and Anti-PT Coupling in a Non-Hermitian Metamaterial

Non-Hermiticity has recently emerged as a rapidly developing field due to its exotic characteristics related to open systems, where the dissipation plays a critical role. In the presence of balanced energy gain and loss with environment, the system exhibits parity-time (PT) symmetry, meanwhile as the conjugate counterpart, anti-PT symmetry can be achieved with dissipative coupling within the system. Here, we demonstrate the coherence of complex dissipative coupling can control the transition between PT and anti-PT symmetry in an electromagnetic metamaterial. Notably, the achievement of the anti-PT symmetric phase is independent of variations in dissipation. Furthermore, we observe phase transitions as the system crosses exceptional points in both anti-PT and PT symmetric metamaterial configurations, achieved by manipulating the frequency and dissipation of resonators. This work provides a promising metamaterial design for broader exploration of non-Hermitian physics and practical application with a controllable Hamiltonian.

Exceptional-point-enhanced phase sensing

Optical sensors, crucial in diverse fields like gravitational wave detection, biomedical imaging, and structural health monitoring, rely on optical phase to convey valuable information. Enhancing sensitivity is important for detecting weak signals. Exceptional points (EPs), identified in non-Hermitian systems, offer great potential for advanced sensors, given their marked response to perturbations. However, strict physical requirements for operating a sensor at EPs limit broader applications. Here, we introduce an EP-enhanced sensing platform featuring plug-in external sensors separated from an EP control unit. EPs are achieved without modifying the sensor, solely through control-unit adjustments. This configuration converts and amplifies optical phase changes into quantifiable spectral features. By separating sensing and control functions, we expand the applicability of EP enhancement to various conventional sensors. As a proof-of-concept, we demonstrate a sixfold reduction in the detection limit of fiber-optic strain sensing using this configuration. This work establishes a universal platform for applying EP enhancement to diverse phase-dependent structures, promising ultrahigh-sensitivity sensing across various applications.

Parity-time symmetry enabled ultra-efficient nonlinear optical signal processing

Nonlinear optical signal processing (NOSP) has the potential to significantly improve the throughput, flexibility, and cost-efficiency of optical communication networks by exploiting the intrinsically ultrafast optical nonlinear wave mixing. It can support digital signal processing speeds of up to terabits per second, far exceeding the line rate of the electronic counterpart. In NOSP, high-intensity light fields are used to generate nonlinear optical responses, which can be used to process optical signals. Great efforts have been devoted to developing new materials and structures for NOSP. However, one of the challenges in implementing NOSP is the requirement of high-intensity light fields, which is difficult to generate and maintain. This has been a major roadblock to realize practical NOSP systems for high-speed, high-capacity optical communications. Here, we propose using a parity-time (PT) symmetric microresonator system to significantly enhance the light intensity and support high-speed operation by relieving the bandwidth-efficiency limit imposed on conventional single resonator systems. The design concept is the co-existence of a PT symmetry broken regime for a narrow-linewidth pump wave and near-exceptional point operation for broadband signal and idler waves. This enables us to achieve a new NOSP system with two orders of magnitude improvement in efficiency compared to a single resonator. With a highly nonlinear AlGaAs-on-Insulator platform, we demonstrate an NOSP at a data rate approaching 40 gigabits per second with a record low pump power of one milliwatt. These findings pave the way for the development of fully chip-scale NOSP devices with pump light sources integrated together, potentially leading to a wide range of applications in optical communication networks and classical or quantum computation. The combination of PT symmetry and NOSP may also open up opportunities for amplification, detection, and sensing, where response speed and efficiency are equally important.

Supplementary Information

The online version contains supplementary material available at 10.1186/s43593-024-00062-w.

Advances in Semiconductor Lasers Based on Parity-Time Symmetry

Semiconductor lasers, characterized by their high efficiency, small size, low weight, rich wavelength options, and direct electrical drive, have found widespread application in many fields, including military defense, medical aesthetics, industrial processing, and aerospace. The mode characteristics of lasers directly affect their output performance, including output power, beam quality, and spectral linewidth. Therefore, semiconductor lasers with high output power and beam quality are at the forefront of international research in semiconductor laser science. The novel parity-time (PT) symmetry mode-control method provides the ability to selectively modulate longitudinal modes to improve the spectral characteristics of lasers. Recently, it has gathered much attention for transverse modulation, enabling the output of fundamental transverse modes and improving the beam quality of lasers. This study begins with the basic principles of PT symmetry and provides a detailed introduction to the technical solutions and recent developments in single-mode semiconductor lasers based on PT symmetry. We categorize the different modulation methods, analyze their structures, and highlight their performance characteristics. Finally, this paper summarizes the research progress in PT-symmetric lasers and provides prospects for future development.

Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems

We present a pedagogical review of the periodically driven non-Hermitian systems, particularly on the rich interplay between the non-Hermitian skin effect and the topology. We start by reviewing the non-Bloch band theory of the static non-Hermitian systems and discuss the establishment of its generalized bulk-boundary correspondence (BBC). Ultimately, we focus on the non-Bloch band theory of two typical periodically driven non-Hermitian systems: harmonically driven non-Hermitian system and periodically quenched non-Hermitian system. The non-Bloch topological invariants were defined on the generalized Brillouin zone and the real space wave functions to characterize the Floquet non-Hermtian topological phases. Then, the generalized BBC was established for the two typical periodically driven non-Hermitian systems. Additionally, we review novel phenomena in the higher-dimensional periodically driven non-Hermitian systems, including Floquet non-Hermitian higher-order topological phases and Floquet hybrid skin-topological modes. The experimental realizations and recent advances have also been surveyed. Finally, we end with a summarization and hope this pedagogical review can motivate further research on Floquet non-Hermtian topological physics.

Single-Mode Laser in the Telecom Range by Deterministic Amplification of the Topological Interface Mode

Photonic integrated circuits are paving the way for novel on-chip functionalities with diverse applications in communication, computing, and beyond. The integration of on-chip light sources, especially single-mode lasers, is crucial for advancing those photonic chips to their full potential. Recently, novel concepts involving topological designs introduced a variety of options for tuning device properties, such as the desired single-mode emission. Here, we introduce a novel cavity design that allows amplification of the topological interface mode by deterministic placement of gain material within a topological lattice. The proposed design is experimentally implemented by a selective epitaxy process to achieve closely spaced Si and InGaAs nanorods embedded within the same layer. This results in the first demonstration of a single-mode laser in the telecom band using the concept of amplified topological modes without introducing artificial losses.

Exponentially Enhanced Non-Hermitian Cooling

Certain non-Hermitian systems exhibit the skin effect, whereby the wave functions become exponentially localized at one edge of the system. Such exponential amplification of wavefunction has received significant attention due to its potential applications in, e.g., classical and quantum sensing. However, the opposite edge of the system, featured by exponentially suppressed wave functions, remains largely unexplored. Leveraging this phenomenon, we introduce a non-Hermitian cooling mechanism, which is fundamentally distinct from traditional refrigeration or laser cooling techniques. Notably, non-Hermiticity will not amplify thermal excitations, but rather redistribute them. Hence, thermal excitations can be cooled down at one edge of the system, and the cooling effect can be exponentially enhanced by the number of auxiliary modes, albeit with a lower bound that depends on the dissipative interaction with the environment. Non-Hermitian cooling does not rely on intricate properties such as exceptional points or nontrivial topology, and it can apply to a wide range of excitations.

The second-order coherence analysis of number state propagation through dispersive non-Hermitian multilayered structures

To examine the second-order coherence of light propagation of quantum states in arbitrary directions through dispersive non-Hermitian optical media, we considered two sets of non-Hermitian periodic structures that consist of gain/loss unit cells. We show that each batch can satisfy the parity-time symmetry conditions at a distinct frequency. We then varied the gain/loss strength in the stable electromagnetic regime to evaluate the transmittance of N-photon number states through each structure. The results show both sets preserve their antibunching characteristics under specific incident light conditions. Furthermore, s(p)-polarized light exhibits higher (lower) second-order coherence at larger incident angles. In addition, the antibunching features of the transmitted states degrade with an increase in the number of unit cells in multilayered structures for both polarizations.

Chiral transmission by an open evolution trajectory in a non-Hermitian system

Exceptional points (EPs), at which two or more eigenvalues and eigenstates of a resonant system coalesce, are associated with non-Hermitian Hamiltonians with gain and/or loss elements. Dynamic encircling of EPs has received significant interest in recent years, as it has been shown to lead to highly nontrivial phenomena, such as chiral transmission in which the final state of the system depends on the encircling handedness. Previously, chiral transmission for a pair of eigenmodes has been realized by establishing a closed dynamical trajectory in parity-time- (PT-) or anti-PT-symmetric systems. Although chiral transmission of symmetry-broken modes, more accessible in practical photonic integrated circuits, has been realized by establishing a closed trajectory encircling EPs in anti-PT-symmetric systems, the demonstrated transmission efficiency is very low due to path-dependent losses. Here, we demonstrate chiral dynamics in a coupled waveguide system that does not require a closed trajectory. Specifically, we explore an open trajectory linking two infinite points having the same asymptotic eigenmodes (not modes in PT- and anti-PT-symmetric systems), demonstrating that this platform enables high-efficiency chiral transmission, with each eigenmode localized in a single waveguide. This concept is experimentally implemented in a coupled silicon waveguide system at telecommunication wavelengths. Our work provides a new evolution strategy for chiral dynamics with superior performance, laying the foundation for the development of practical chiral-transmission devices.

Macroscopic Noise Amplification by Asymmetric Dyads in Non-Hermitian Optical Systems for Generative Diffusion Models

We study noise amplification by asymmetric dyads in freely expanding non-Hermitian optical systems. We show that modifications of the pumping strengths can counteract bias from natural imperfections of the system's hardware while couplings between dyads lead to systems with nonuniform statistical distributions. Our results suggest that asymmetric non-Hermitian dyads are promising candidates for efficient sensors and ultrafast random number generators. We propose that the integrated light emission from such asymmetric dyads can be efficiently used for analog all-optical degenerative diffusion models of machine learning to overcome the digital limitations of such models in processing speed and energy consumption.

Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics

The exotic physics associated with exceptional points (EPs) is always under the scrutiny of theoretical and experimental science. Recently, considerable effort has been invested in the combination of nonlinearity and non-Hermiticity. The concept of nonlinear EPs (NEPs) has been introduced, which can avoid the loss of completeness of the eigenbasis in dynamics while retaining the key features of linear EPs. Here, we present the first direct experimental demonstration of a NEP based on two non-Hermition coupled circuit resonators combined with a nonlinear saturable gain. At the NEP, the response of the eigenfrequency to perturbations demonstrates a third-order root law and the eigenbasis of the Hamiltonian governing the system dynamics is still complete. Our results bring this counterintuitive aspect of the NEP to light and possibly open new avenues for applications.

Nonlocal Metasurface with Chiral Exceptional Points in the Telecom-Band

The exceptional point (EP) is the critical phase transition point in parity-time (PT) symmetry systems, offering many unique physical phenomena, such as a chiral response. Achieving chiral EP in practical applications has been challenging due to the delicate balance required between gain and loss and complicated fabrication, limiting both working band and device miniaturization. Here, we proposed a nonlocal metasurface featuring orthogonal gold nanorods, where loss modulation is achieved through rod size and lattice pitch. By tuning the coupling strength, we experimentally observed the PT symmetry phase transition and chiral EP in the telecom-band. The experimental and simulated circular conversion dichroism at EP reach 0.79 and 0.99, respectively. We also demonstrated an abrupt phase flip of a specific component near EP theoretically. This work provides a feasible scheme for exploring EP in polarized space within the telecom-band, which may find applications in polarization control, wavelength division multiplexing, ultrasensitive sensing, imaging, etc.

Tunable single frequency Hz-magnitude narrow linewidth Brillouin fiber laser based on parity-time symmetry

An Hz-magnitude ultra-narrow linewidth single-frequency Brillouin fiber laser (BFL) is proposed and experimentally demonstrated. The single frequency of the laser is selected by parity-time (PT) symmetry, which consists of a stimulated Brillouin scatter (SBS) gain path excited by a 24 km single-mode fiber (SMF) and an approximately equal length loss path tuned with a variable optical attenuator (VOA). These paths are coupled through a fiber Bragg grating (FBG) into a wavelength space. Accomplishing single-frequency oscillation involves the precise adjustment of polarization control (PC) and VOA to attain the PT broken phase. In the experiment, the linewidth of the proposed BFL is 9.58 Hz. The optical signal-to-noise ratio (OSNR) reached 78.89 dB, with wavelength and power fluctuations of less than 1pm and 0.02 dB within one hour. Furthermore, the wavelength can be tuned from 1549.9321 nm to 1550.2575 nm, with a linewidth fluctuation of 1.81 Hz. The relative intensity noise (RIN) is below -74 dB/Hz. The proposed ultra-narrow single-frequency BFL offers advantages such as cost-effectiveness, ease of control, high stability and excellent output characteristics, making it highly promising for the applications in the coherent detection.

Photonic Floquet Skin-Topological Effect

Non-Hermitian skin effect and photonic topological edge states are of great interest in non-Hermitian physics and optics. However, the interplay between them is largely unexplored. Here, we propose and demonstrate experimentally the non-Hermitian skin effect constructed from the nonreciprocal flow of Floquet topological edge states, which can be dubbed "Floquet skin-topological effect." We first show the non-Hermitian skin effect can be induced by structured loss when the one-dimensional (1D) system is periodically driven. Next, based on a two-dimensional (2D) Floquet topological photonic lattice with structured loss, we investigate the interaction between the non-Hermiticity and the topological edge states. We observe that all the one-way edge states are imposed onto specific corners, featuring both the non-Hermitian skin effect and topological edge states. Furthermore, a topological switch for the skin-topological effect is presented by utilizing the phase-transition mechanism. Our experiment paves the way for realizing non-Hermitian topological effects in nonlinear and quantum regimes.

Geometric Origin of Non-Bloch PT Symmetry Breaking

The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to real (complex) energy spectrum when the non-Hermiticity is below (above) a threshold. Recently, it has been demonstrated that the non-Hermitian skin effect generates a new type of PT symmetry, dubbed the non-Bloch PT symmetry, featuring unique properties such as high sensitivity to the boundary condition. Despite its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for this generic phenomenon even in one spatial dimension. Here, we uncover the geometric mechanism of non-Bloch PT symmetry and its breaking. We find that non-Bloch PT symmetry breaking occurs by the formation of cusps in the generalized Brillouin zone (GBZ). Based on this geometric understanding, we propose an exact formula that efficiently determines the breaking threshold. Moreover, we predict a new type of spectral singularities associated with the symmetry breaking, dubbed non-Bloch van Hove singularity, whose physical mechanism fundamentally differs from their Hermitian counterparts. This singularity is experimentally observable in linear responses.

Third-order exceptional line in a nitrogen-vacancy spin system

Exceptional points (EPs) are singularities in non-Hermitian systems, where k (k ≥ 2) eigenvalues and eigenstates coalesce. High-order EPs exhibit richer topological characteristics and better sensing performance than second-order EPs. Theory predicts even richer non-Hermitian topological phases for high-order EP geometries, such as lines or rings formed entirely by high-order EPs. However, experimental exploration of high-order EP geometries has hitherto proved difficult due to the demand for more degrees of freedom in the Hamiltonian's parameter space or a higher level of symmetries. Here we observe a third-order exceptional line in an atomic-scale system. To this end, we use a nitrogen-vacancy spin in diamond and introduce multiple symmetries in the non-Hermitian Hamiltonian realized with the system. Furthermore, we show that the symmetries play an essential role in the occurrence of high-order EP geometries. Our approach can in future be further applied to explore high-order EP-related topological physics at the atomic scale and, potentially, for applications of high-order EPs in quantum technologies.