Pump-induced exceptional points in lasers

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.

Strong Laser Emission Modulation by Coherent Perfect Absorption Inside Complex-Coupled Distributed Feedback Laser Diodes

The proof-of-concept of the exploitation of Coherent Perfect Absorption (CPA) in electrically-injected distributed-feedback laser sources is reported. Capitalizing on the essence of CPA as "light extinction by light", an integrated laser-modulator scheme emerges. The key ingredient compared to conventional single-frequency laser diodes is a careful periodic in-phase modulation of both real and imaginary parts of the complex grating index profile that enables both single-frequency operation and 40 dB line purity at the Bragg frequency. It is shown that this combination is most apt for the operation of CPA as a modulation mechanism that respects the laser spectral purity. The specific proof-of-concept is based on an ultra-short external cavity formed by a metallic micro-mirror, whose role is to generate the second beam of more conventional CPA interferometric approaches. The implemented complex-coupled grating is compatible with existing industrial technologies and promising for real-life laser source applications. Furthermore, the concept can be directly transferred to other material platforms and other wavelengths ranging from terahertz to ultraviolet.

Enhanced Frequency Conversion in Parity-Time Symmetry Line

Non-Hermitian degeneracies reveal intriguing and nontrivial behaviors in open physical systems. Examples like parity-time (PT) symmetry breaking, topological encircling chirality, and enhanced sensing near an exceptional point (EP) are often associated with the abrupt nature of the phase transition around these degeneracies. Here we experimentally observe a cavity-enhanced second-harmonic frequency (SHG) conversion on a PT symmetry line, i.e., a set consisting of open-ended isofrequency or isoloss lines, both terminated at EPs on the Riemann surface in parameter space. The enhancement factor can reach as high as 300, depending on the crossing point whether in the symmetry or the broken phase of the PT line. Moreover, such enhancement of SHG enables sensitive distance sensing with a nanometer resolution. Our works may pave the way for practical applications in sensing, frequency conversion, and coherent wave control.

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.

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.

Tracking exceptional points above the lasing threshold

Recent studies on exceptional points (EPs) in non-Hermitian optical systems have revealed unique traits, including unidirectional invisibility, chiral mode switching and laser self-termination. In systems featuring gain/loss components, EPs are commonly accessed below the lasing threshold, i.e., in the linear regime. In this work, we experimentally demonstrate that EP singularities in coupled semiconductor nanolasers can be accessed above the lasing threshold, where they become branch points of a nonlinear dynamical system. Contrary to the common belief that unavoidable cavity detuning impedes the formation of EPs, here we demonstrate that such detuning is necessary for compensating the carrier-induced frequency shift, hence restoring the EP. Furthermore, we find that the pump imbalance at lasing EPs varies with the total pump power, enabling their continuous tracking. This work uncovers the unstable nature of EPs above laser threshold in coupled semiconductor lasers, offering promising opportunities for the realization of self-pulsing nanolaser devices and frequency combs.

Normal mode analysis in multi-coupled non-Hermitian optical nanocavities

Coupled optical cavities are an attractive on-chip optical platform for realizing quantum mechanical concepts in electrodynamics and further developing non-Hermitian photonics. In such systems, an intercavity interaction is often considered as a key parameter to understand the system's behaviors but its estimation/calculation is typically limited for some simplified systems owing to extended complexities. For example, multi-coupled photonic crystal (PhC) nanocavities exhibiting strong resonances with a large free spectral range can serve as an excellent test-bed to study non-Hermitian optical properties when spatially non-uniform gain is introduced. However, the detailed quantitative analysis such as spectral tracing of cavity normal modes is often limited in commercially available numerical tools because of the required massive computation resources. Herein, we report on a concept of spatial overlap integrals (SOIs) between the eigenmodes in non-coupled PhC nanocavities and utilize them to obtain the intercavity interactions in passively coupled PhC nanocavity systems. With the help of coupling strength factors calculated from SOIs, we were able to fully exploit the coupled mode theory (CMT) and readily trace the detailed spectral behaviors of normal modes in various multi-coupled PhC nanocavities. Full-wave numerical simulation results verified the proposed method, revealing that the characteristics of original eigenmodes from non-coupled PhC nanocavities can act as key building blocks for analyzing the normal modes of multi-coupled PhC nanocavities. We further applied this SOI method to various multi-coupled PhC nanocavities with non-symmetric optical gain/loss distributions and successfully observed the unusual spectral evolution of normal modes and the correspondingly occurring unique non-Hermitian behaviors.

Topological Microlaser with a Non-Hermitian Topological Bulk

Bulk-edge correspondence, with quantized bulk topology leading to protected edge states, is a hallmark of topological states of matter and has been experimentally observed in electronic, atomic, photonic, and many other systems. While bulk-edge correspondence has been extensively studied in Hermitian systems, a non-Hermitian bulk could drastically modify the Hermitian topological band theory due to the interplay between non-Hermiticity and topology, and its effect on bulk-edge correspondence is still an ongoing pursuit. Importantly, including non-Hermicity can significantly expand the horizon of topological states of matter and lead to a plethora of unique properties and device applications, an example of which is a topological laser. However, the bulk topology, and thereby the bulk-edge correspondence, in existing topological edge-mode lasers is not well defined. Here, we propose and experimentally probe topological edge-mode lasing with a well-defined non-Hermitian bulk topology in a one-dimensional (1D) array of coupled ring resonators. By modeling the Hamiltonian with an additional degree of freedom (referred to as synthetic dimension), our 1D structure is equivalent to a 2D non-Hermitian Chern insulator with precise mapping. Our Letter may open a new pathway for probing non-Hermitian topological effects and exploring non-Hermitian topological device applications.

Bound chiral magnonic polariton states for ideal microwave isolation

Bound states in the continuum (BICs) present a unique solution for eliminating radiation loss. So far, most reported BICs are observed in transmission spectra, with only a few exceptions being in reflection spectra. The correlation between reflection BICs (r-BICs) and transmission BICs (t-BICs) remains unclear. Here, we report the presence of both r-BICs and t-BICs in a three-mode cavity magnonics. We develop a generalized framework of non-Hermitian scattering Hamiltonians to explain the observed bidirectional r-BICs and unidirectional t-BICs. In addition, we find the emergence of an ideal isolation point in the complex frequency plane, where the isolation direction can be switched by fine frequency detuning, thanks to chiral symmetry protection. Our results demonstrate the potential of cavity magnonics and also extend the conventional BICs theory through the application of a more generalized effective Hamiltonians theory. This work offers an alternative idea for designing functional devices in general wave optics.

Non-Hermitian Anharmonicity Induces Single-Photon Emission

Single-photon sources are in high demand for quantum information applications. A paradigmatic way to achieve single-photon emission is through anharmonicity in the energy levels, such that the absorption of a single photon from a coherent drive shifts the system out of resonance and prevents absorption of a second one. We identify a novel mechanism for single-photon emission through non-Hermitian anharmonicity, i.e., anharmonicity in the losses instead of in the energy levels. We demonstrate the mechanism in two types of systems, including a feasible setup consisting of a hybrid metallodielectric cavity weakly coupled to a two-level emitter, and show that it induces high-purity single-photon emission at high repetition rates.

Generalized PT Symmetry in Non-Hermitian Wireless Power Transfer Systems

We show that, by using a saturable gain g_{sat}, generalized PT (GPT) symmetry can be achieved in the intrinsically unbalanced (non-PT-symmetric) high-order wireless power transfer systems. A topology decomposition approach is implemented to analyze the parity of the high-order wireless power transfer systems. In the coupling parametric space, a global GPT-symmetric eigenstate is observed along with the spontaneous phase transition of the local GPT-symmetric eigenstates on the exceptional contour. GPT symmetry guarantees a highly efficient and stable power transfer across the distinct coupling regions, which introduces a new paradigm for a broad range of application scenarios involving asymmetric coupling.

Overcoming the Diffraction Limit on the Size of Dielectric Resonators Using an Amplifying Medium

Existing methods for the creation of subwavelength resonators use either structures with negative permittivity, by exploiting subwavelength plasmonic resonances, or dielectric structures with a high refractive index, which reduce the wavelength. Here, we provide an alternative to these two methods based on a modification of the modes of dielectric resonators by means of an active medium. On the example of the dielectric active layer of size substantially smaller than a half-wavelength of light, we demonstrate that there is a gain at exceeding of which the change in phase due to the reflection at the layer boundaries compensates the change in phase due to propagation over the layer. Above this value of the gain, an unconventional mode forms, in which the phase shift after a round-trip of the light is zero. We show that this mode can be exploited to create a laser, the size of which is much smaller than the wavelength of the generated light and scales inversely with the square of absolute value of the refractive index in the active medium. Our results pave the way to creation of dielectric lasers of subwavelength size.

Unveiling the Enhancement of Spontaneous Emission at Exceptional Points

Exceptional points (EPs), singularities of non-Hermitian physics where complex spectral resonances degenerate, are one of the most exotic features of nonequilibrium open systems with unique properties. For instance, the emission rate of quantum emitters placed near resonators with EPs is enhanced (compared to the free-space emission rate) by a factor that scales quadratically with the resonance quality factor. Here, we verify the theory of spontaneous emission at EPs by measuring photoluminescence from photonic-crystal slabs that are embedded with a high-quantum-yield active material. While our experimental results verify the theoretically predicted enhancement, they also highlight the practical limitations on the enhancement due to material loss. Our designed structures can be used in applications that require enhanced and controlled emission, such as quantum sensing and imaging.

Far-Field Correlations Verifying Non-Hermitian Degeneracy of Optical Modes

An experimental verification of an exceptional point (EP) in a stand-alone chaotic microcavity is a tough issue because as deformation parameters are fixed the traditional frequency analysis methods cannot be applied any more. Through numerical investigations with an asymmetric Reuleaux triangle microcavity (ARTM), we find that the eigenvalue difference of paired modes can approach near-zero regardless of nonorthogonality of the modes. In this case, for a definite verification of EPs in experiments, wave function coalescence should be confirmed. For this, we suggest the method of exploiting correlation of far-field patterns (FFPs), which is directly related to spatial mode patterns. In an ARTM, we demonstrate that the FFP correlation of paired modes can be used to confirm wave function coalescence when an eigenvalue difference approaches near zero.

Anomalous Single-Mode Lasing Induced by Nonlinearity and the Non-Hermitian Skin Effect

Single-mode operation is a desirable but elusive property for lasers operating at high pump powers. Typically, single-mode lasing is attainable close to threshold, but increasing the pump power gives rise to multiple lasing peaks due to inter-modal gain competition. We propose a laser with the opposite behavior: multimode lasing occurs at low output powers, but pumping beyond a certain value produces a single lasing mode, with all other candidate modes experiencing negative effective gain. This phenomenon arises in a lattice of coupled optical resonators with non-fine-tuned asymmetric couplings, and is caused by an interaction between nonlinear gain saturation and the non-Hermitian skin effect. The single-mode lasing is observed in both frequency domain and time domain simulations. It is robust against on-site disorder, and scales up to large lattice sizes. This finding might be useful for implementing high-power laser arrays.

Exceptional Precision of a Nonlinear Optical Sensor at a Square-Root Singularity

Exceptional points (EPs)-spectral singularities of non-Hermitian linear systems-have recently attracted interest for sensing. While initial proposals and experiments focused on enhanced sensitivities neglecting noise, subsequent studies revealed issues with EP sensors in noisy environments. Here we propose a single-mode Kerr-nonlinear resonator for exceptional sensing in noisy environments. Based on the resonator's dynamic hysteresis, we define a signal that displays a square-root singularity reminiscent of an EP. However, our sensor has crucial fundamental and practical advantages over EP sensors: the signal-to-noise ratio increases with the measurement speed, the square-root singularity is easily detected through intensity measurements, and both sensing precision and information content of the signal are enhanced around the singularity. Our sensor also overcomes the fundamental trade-off between precision and averaging time characterizing all linear sensors. All these unconventional features open up new opportunities for fast and precise sensing using hysteretic resonators.

Observation of photonic constant-intensity waves and induced transparency in tailored non-Hermitian lattices

Light propagation is strongly affected by scattering due to imperfections in the complex medium. It has been recently theoretically predicted that a scattering-free transport through an inhomogeneous medium is achievable by non-Hermitian tailoring of the complex refractive index. Here, we implement photonic constant-intensity waves in an inhomogeneous, linear, discrete mesh lattice. By extending the existing theoretical framework, we experimentally show that a driven non-Hermitian tailoring allows us to control the propagation and diffraction of light even in highly disordered systems. In this vein, we demonstrate the transmission of shape-preserving beams and the seemingly undistorted propagation of light excitations across a strongly inhomogeneous non-Hermitian photonic lattice that can be realized by coupled optical fiber loops. Our results lead to a deeper understanding of non-Hermitian wave control and further contribute to the development of non-Hermitian photonics.

Coalescence of Anderson-localized modes at an exceptional point in 2D random media

In non-Hermitian settings, the particular position at which two eigenstates coalesce in the complex plane under a variation of a physical parameter is called an exceptional point. An open disordered system is a special class of non-Hermitian system, where the degree of scattering directly controls the confinement of the modes. Herein a non-perturbative theory is proposed which describes the evolution of modes when the permittivity distribution of a 2D open dielectric system is modified, thereby facilitating to steer individual eigenstates to such a non-Hermitian degeneracy. The method is used to predict the position of such an exceptional point between two Anderson-localized states in a disordered scattering medium. We observe that the accuracy of the prediction depends on the number of localized states accounted for. Such an exceptional point is experimentally accessible in practically relevant disordered photonic systems.

A new type of non-Hermitian phase transition in open systems far from thermal equilibrium

We demonstrate a new type of non-Hermitian phase transition in open systems far from thermal equilibrium, which can have place in the absence of an exceptional point. This transition takes place in coupled systems interacting with reservoirs at different temperatures. We show that the spectrum of energy flow through the system caused by the temperature gradient is determined by the [Formula: see text]-potential. Meanwhile, the frequency of the maximum in the spectrum plays the role of the order parameter. The phase transition manifests itself in the frequency splitting of the spectrum of energy flow at a critical point, the value of which is determined by the relaxation rates and the coupling strength. Near the critical point, fluctuations of the order parameter diverge according to a power law with the critical exponent that depends only on the ratio of reservoirs temperatures. The phase transition at the critical point has the non-equilibrium nature and leads to the change in the energy flow between the reservoirs. Our results pave the way to manipulate the heat energy transfer in the coupled out-of-equilibrium systems.

Continuous wave vertical emission from terahertz microcavity lasers with a dual injection scheme

Quantum cascade lasers (QCLs) represent a most promising compact source at terahertz (THz) frequencies, but efficiency of their continuous wave (CW) operation still needs to be improved to achieve large-scale exploitation. Here, we demonstrate highly efficient operation of a subwavelength microcavity laser consisting of two evanescently coupled whispering gallery microdisk resonators. Exploiting a dual injection scheme for the laser cavity, single mode CW vertical emission at 3.3 THz is obtained at 10 K with 6.4 mA threshold current and 145 mW/A slope efficiency up to 320 μW emitted power measured in quasi-CW mode. The tuning of the laser emission directionality is also obtained by independently varying the pumping strength between the microdisks. By connecting the resonators through a suspended gold bridge, the laser out-coupling efficiency in the vertical direction is strongly enhanced. Owing to the high brightness, low-power consumption and CW operation, the proposed microcavity laser design could allow the realization of high-performance CW THz QCLs ready for massive parallelization.

Suppression and revival of single-cavity lasing induced by polarization-dependent loss

For most photonics devices and systems, loss is desperately averted, since it will increase the power consumption and degrade the performance. However, in some non-Hermitian systems, loss can induce a modal gain when the parity-time symmetry is broken, which offers a new way to manipulate the lasing of active cavities. Here we experimentally observe the counterintuitive phenomenon in a single laser cavity assisted by the polarization-dependent loss. A parity-time symmetric system is constituted by the two orthogonally polarized photonic loops in a single laser cavity, which can guarantee the consistency of two coupling loops. The measured output power of the cavity depends on the cross-polarization loss, which reveals virtually opposite relationships before and after the critical point. It provides a novel, to the best of our knowledge, understanding of polarization loss and shows great potential for lasing manipulation in a single cavity with polarization control.

Rayleigh scatterer-induced steady exceptional points of stable-island modes in a deformed optical microdisk

A formation of second-order non-Hermitian degeneracies, called exceptional points (EPs), in a chaotic oval-shaped dielectric microdisk is studied. Different symmetric optical modes localized on a stable period-3 orbit coalesce to form chiral EPs. Unlike a circular microdisk perturbed by two scatterers (CTS), our proposed system requires only one scatterer to build chiral EPs. The scatterer positions for counterpropagating EP modes are far distant from one another and almost steady against varying scatterer sizes in contrast to the CTS case. Our results can contribute to establishing a more solid platform for EP-based-device applications with flexibility and easy feasibility in obtaining EPs.

Perturbation theory in the complex plane: exceptional points and where to find them

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.

Recent advances in nanocavities and their applications

High quality factor and small mode volume in nanocavities enable the demonstration of efficient nanophotonic devices with low power consumption, strong nonlinearity, and high modulation speed, due to the strong light-matter interaction. In this review, we focus on recent state-of-the-art nanocavities and their applications. We introduce single nanocavities including semiconductor nanowires, plasmonic cavities, and nanostructures based on quasi-bound states in the continuum (quasi-BIC), for laser, photovoltaic, and nonlinear applications. In addition, nanocavity arrays with unique feedback mechanisms, including BIC cavities, parity-time symmetry coupled cavities, and photonic topological cavities, are introduced for laser applications. These various cavity designs and underlying physics in single and array nanocavities are useful for the practical implementation of promising nanophotonic devices.

One-Way Zero Reflection in an Insulator-Metal-Insulator Structure Using the Transfer Matrix Method

We numerically demonstrate one-way zero reflection using the transfer matrix method. Using simulations, we adjusted the thickness of SiO2 layers in a simple SiO2-Au-SiO2 layer structure. We found two solutions, 47 nm-10 nm-32 nm and 71 nm-10 nm-60 nm, which are the thicknesses for one-way zero reflection at a wavelength of 560 nm. We confirmed it with reflection spectra, where reflectance is zero for forwardly incident light and 2.5% for backwardly incident light at the wavelength 560 nm, and thickness 47 nm-10 nm-32 nm.

Experimental Investigation of Quantum PT-Enhanced Sensor

PT-symmetric theory is developed to extend quantum mechanics to a complex region, but it wins its great success first in classical systems, for example, optical waveguides and electric circuits, etc., because there are so many counterintuitive phenomena and striking applications, including unidirectional light transport, PT-enhanced sensors (one kind of exceptional-point-based sensor), and wireless power transfer. However, these phenomena and applications are mostly based on the ability to approach a PT-symmetric broken region, which makes it difficult to transfer them to the quantum regime, since the broken quantum PT-symmetric system has not been constructed effectively, until recently several methods have been raised. Here, we construct a quantum PT-symmetric system assisted by weak measurement, which can effectively transit from the unbroken region to the broken region. The full energy spectrum including the real and imaginary parts is directly measured using weak values. Furthermore, based on the ability of approaching a broken region, we for the first time translate the previously mentioned PT-enhanced sensor into the quantum version, and investigate its various features that are associated to the optimal conditions for sensitivity enhancement. In this experiment, we obtain an enhancement of 8.856 times over the conventional Hermitian sensor. Moreover, by separately detecting the real and imaginary parts of energy splitting, we can derive the additional information of the direction of perturbations. Our work paves the way of leading classical interesting PT phenomena and applications to their quantum counterparts. More generally, since the PT system is a subset of non-Hermitian systems, our work will be also helpful in the studies of general exception point in the quantum regime.

Laser Control of Resonance Tunneling via an Exceptional Point

According to the familiar Breit-Wigner formula, tunneling through a potential barrier is strongly enhanced when the energy of the projectile is equal to the resonance energy. Here we show how a weak continuous wave laser can qualitatively change the character of resonance tunneling, and enforce a sudden and total suppression of the transmission by inducing an exceptional point (EP, special non-Hermitian degeneracy). Our findings are relevant not only for laser control of transmission in the resonance tunneling diodes, but also in the context of electron scattering through any type of metastable (e.g., autoionization, Auger, intermolecular Coulombic decay) atomic or molecular states, and even in the case of transmission of light or sound waves in active systems with gain and loss.

Equivalence of Effective Non-Hermitian Hamiltonians in the Context of Open Quantum Systems and Strongly Correlated Electron Systems

Recently, it has become clear that non-Hermitian phenomena can be observed not only in open quantum systems experiencing gain and loss but also in equilibrium single-particle properties of strongly correlated systems. However, the circumstances and requirements for the emergence of non-Hermitian phenomena in each field are entirely different. While the implementation of postselection is a significant obstacle to observe the dynamics governed by a non-Hermitian Hamiltonian in open quantum systems, it is unnecessary in strongly correlated systems. Until now, a relation between both descriptions of non-Hermitian phenomena has not been revealed. In this Letter, we close this gap and demonstrate that the non-Hermitian Hamiltonians emerging in both fields are identical, and we clarify the conditions for the emergence of a non-Hermitian Hamiltonian in strongly correlated materials. Using this knowledge, we propose a method to analyze non-Hermitian properties without the necessity of postselection by studying specific response functions of open quantum systems and strongly correlated systems.

Passage through exceptional point: case study

The description of unitary evolution using non-Hermitian but ‘hermitizable’ Hamiltonians H is feasible via an ad hoc metric Θ  =  Θ ( H ) and a (non-unique) amendment 〈 ψ 1 | ψ 2 〉 → 〈 ψ 1 | Θ | ψ 2 〉 of the inner product in Hilbert space. Via a proper fine-tuning of Θ ( H ) this opens the possibility of reaching the boundaries of stability (i.e. exceptional points) in many quantum systems sampled here by the fairly realistic Bose–Hubbard (BH) and discrete anharmonic oscillator (AO) models. In such a setting, it is conjectured that the EP singularity can play the role of a quantum phase-transition interface between different dynamical regimes. Three alternative ‘AO ↔ BH’ implementations of such an EP-mediated dynamical transmutation scenario are proposed and shown, at an arbitrary finite Hilbert-space dimension N , exact and non-numerical.

Vector Exceptional Points with Strong Superchiral Fields

Exceptional points (EPs), branch points of complex energy surfaces at which eigenvalues and eigenvectors coalesce, are ubiquitous in non-Hermitian systems. Many novel properties and applications have been proposed around the EPs. One of the important applications is to enhance the detection sensitivity. However, due to the lack of single-handed superchiral fields, all of the proposed EP-based sensing mechanisms are only useful for the nonchiral discrimination. Here, we propose theoretically and demonstrate experimentally a new type of EP, which is called a radiation vector EP, to fulfill the homogeneous superchiral fields for chiral sensing. This type of EP is realized by suitably tuning the coupling strength and radiation losses for a pair of orthogonal polarization modes in the photonic crystal slab. Based on the unique modal-coupling property at the vector EP, we demonstrate that the uniform superchiral fields can be generated with two beams of lights illuminating the photonic crystal slab from opposite directions. Thus, the designed photonic crystal slab, which supports the vector EP, can be used to perform surface-enhanced chiral detection. Our findings provide a new strategy for ultrasensitive characterization and quantification of molecular chirality, a key aspect for various bioscience and biomedicine applications.

Synthetic Anti-PT Symmetry in a Single Microcavity

Non-Hermitian systems based on parity-time (PT) and anti-PT symmetry reveal rich physics beyond the Hermitian regime. So far, realizations of such symmetric systems have been limited to the spatial domain. Here we theoretically and experimentally demonstrate synthetic anti-PT symmetry in a spectral dimension induced by nonlinear Brillouin scattering in a single optical microcavity, where Brillouin scattering induced transparency or absorption in two spectral resonances provides the optical gain and loss to observe a phase transition between two symmetry regimes. This scheme provides a new paradigm towards the investigation of non-Hermitian physics in a synthetic photonic dimension for all-optical signal processing and quantum information science.

Exceptional cones in 4D parameter space

The notion of synthetic dimensions has expanded the realm of topological physics to four dimensional (4D) space lately. In this work, non-Hermiticity is used as a synthetic parameter in PT-symmetric photonic crystals to study the topological physics in 4D non-Hermitian synthetic parameter space. We realize a 3D exceptional hypersurface (EHS) in such 4D parameter space, and the degeneracy points emerge due to the symmetry of synthetic parameters. We further demonstrate the existence of exceptional degenerate points (EDPs) on the EHS that originates from the chirality of exceptional points (EPs), and the exceptional surface near EDPs behaves like a Dirac cone. We further show that a very narrow reflection plateau can be found near these EDPs, and their sensitivity towards the PT-symmetry breaking environmental perturbation can make these degeneracy points useful in optical sensing and many other nonlinear and quantum optical applications.

Observation of Critical Phenomena in Parity-Time-Symmetric Quantum Dynamics

We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- (PT-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the PT-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the PT-unbroken and PT-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in PT-symmetric nonunitary quantum dynamics.

Experimental Observation of PT Symmetry Breaking near Divergent Exceptional Points

Standard exceptional points (EPs) are non-Hermitian degeneracies that occur in open systems. At an EP, the Taylor series expansion becomes singular and fails to converge-a feature that was exploited for several applications. Here, we theoretically introduce and experimentally demonstrate a new class of parity-time symmetric systems [implemented using radio frequency (rf) circuits] that combine EPs with another type of mathematical singularity associated with the poles of complex functions. These nearly divergent exceptional points can exhibit an unprecedentedly large eigenvalue bifurcation beyond those obtained by standard EPs. Our results pave the way for building a new generation of telemetering and sensing devices with superior performance.

New topological invariants in non-Hermitian systems

Both theoretical and experimental studies of topological phases in non-Hermitian systems have made a remarkable progress in the last few years of research. In this article, we review the key concepts pertaining to topological phases in non-Hermitian Hamiltonians with relevant examples and realistic model setups. Discussions are devoted to both the adaptations of topological invariants from Hermitian to non-Hermitian systems, as well as origins of new topological invariants in the latter setup. Unique properties such as exceptional points and complex energy landscapes lead to new topological invariants including winding number/vorticity defined solely in the complex energy plane, and half-integer winding/Chern numbers. New forms of Kramers degeneracy appear here rendering distinct topological invariants. Modifications of adiabatic theory, time-evolution operator, biorthogonal bulk-boundary correspondence lead to unique features such as topological displacement of particles, 'skin-effect', and edge-selective attenuated and amplified topological polarizations without chiral symmetry. Extension and realization of topological ideas in photonic systems are mentioned. We conclude with discussions on relevant future directions, and highlight potential applications of some of these unique topological features of the non-Hermitian Hamiltonians.

Sensing with Exceptional Surfaces in Order to Combine Sensitivity with Robustness

Exceptional points (EPs) are singularities that arise in non-Hermitian physics. Current research efforts focus only on systems supporting isolated EPs characterized by increased sensitivity to external perturbations, which makes them potential candidates for building next generation optical sensors. On the downside, this feature is also the Achilles heel of these devices: they are very sensitive to fabrication errors and experimental uncertainties. To overcome this problem, we introduce a new design concept for implementing photonic EPs that combine the robustness required for practical use together with their hallmark sensitivity. Particularly, our proposed structure exhibits a hypersurface of Jordan EPs embedded in a larger space, and having the following peculiar features: (1) A large class of undesired perturbations shift the operating point along the exceptional surface (ES), thus, leaving the system at another EP which explains the robustness; (2) Perturbations due to back reflection or backscattering force the operating point out of the ES, leading to enhanced sensitivity. Importantly, our proposed geometry is relatively easy to implement using standard photonics components and the design concept can be extended to other physical platforms such as microwave or acoustics.

Perfectly Absorbing Exceptional Points and Chiral Absorbers

We identify a new kind of physically realizable exceptional point (EP) corresponding to degenerate coherent perfect absorption, in which two purely incoming solutions of the wave operator for electromagnetic or acoustic waves coalesce to a single state. Such non-Hermitian degeneracies can occur at a real-valued frequency without any associated noise or nonlinearity, in contrast to EPs in lasers. The absorption line shape for the eigenchannel near the EP is quartic in frequency around its maximum in any dimension. In general, for the parameters at which an operator EP occurs, the associated scattering matrix does not have an EP. However, in one dimension, when the S matrix does have a perfectly absorbing EP, it takes on a universal one-parameter form with degenerate values for all scattering coefficients. For absorbing disk resonators, these EPs give rise to chiral absorption: perfect absorption for only one sense of rotation of the input wave.

Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point

Evanescently coupled pairs of microdisk lasers have emerged as a useful platform for studying the non-Hermitian physics of exceptional points. It remains an open question how scalable and versatile such phenomena can be when carried over to other designs. Here we have studied the effect of gain/loss modulation in an evanescently coupled pair of microdisk optical resonators fabricated from solution-processed colloidal quantum dots. The emission spectra of these structures are sensitive to small imperfections, which cause frequency-splitting of the whispering gallery modes. Despite this inherent disorder, we found that when spatially modulating the optical pump to vary the gain differential between the coupled microdisks, the coupling drives the split parasitic intra-cavity modes into coalescence at an exceptional point of the resulting three-mode system. This unusual behavior is rationalized via a Hamiltonian that incorporates the intra-cavity coupling as well as the anisotropic inter-cavity coupling between modes in the microdisk pair.

Pairs of microdisk lasers are one of the most common systems for studying optical PT-symmetry. Here, Lafalce, Zeng et al. study the influence of fabrication imperfections in a disk pair made from colloidal quantum dots and show that the resulting three modes also coalesce at an exceptional point.

Dynamics for encircling an exceptional point in a nonlinear non-Hermitian system

We study the dynamics near an exceptional point in a nonlinear non-Hermitian system consisting of a pair of resonators. One of the resonators has a linear loss, and the other resonator has a saturable gain. We show that the system dynamics exhibit chiral characteristics. And moreover, unique to the nonlinear system, such dynamics allow one to adiabatically switch between bistable states at the same system parameter. Such bistable switching is potentially interesting in optical memory based on coupled laser systems.

Exceptional points in optics and photonics

Exceptional points are branch point singularities in the parameter space of a system at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become degenerate. Such peculiar degeneracies are distinct features of non-Hermitian systems, which do not obey conservation laws because they exchange energy with the surrounding environment. Non-Hermiticity has been of great interest in recent years, particularly in connection with the quantum mechanical notion of parity-time symmetry, after the realization that Hamiltonians satisfying this special symmetry can exhibit entirely real spectra. These concepts have become of particular interest in photonics because optical gain and loss can be integrated and controlled with high resolution in nanoscale structures, realizing an ideal playground for non-Hermitian physics, parity-time symmetry, and exceptional points. As we control dissipation and amplification in a nanophotonic system, the emergence of exceptional point singularities dramatically alters their overall response, leading to a range of exotic optical functionalities associated with abrupt phase transitions in the eigenvalue spectrum. These concepts enable ultrasensitive measurements, superior manipulation of the modal content of multimode lasers, and adiabatic control of topological energy transfer for mode and polarization conversion. Non-Hermitian degeneracies have also been exploited in exotic laser systems, new nonlinear optics schemes, and exotic scattering features in open systems. Here we review the opportunities offered by exceptional point physics in photonics, discuss recent developments in theoretical and experimental research based on photonic exceptional points, and examine future opportunities in this area from basic science to applied technology.

White light cavity formation and superluminal lasing near exceptional points

We study theoretically a superluminal laser system comprising active and passive (lossy) coupled micro-resonators with equal gain/loss. It is shown that when the system satisfies the white light cavity (WLC) condition, corresponding to zero group index, it also forms a PT-symmetric system (PTSS) at its exceptional point (EP). Slightly above lasing threshold, in the broken symmetry regime near the EP, the system exhibits "superluminal" lasing - a unique lasing condition which is highly attractive for sensing and precision metrology applications. It is also shown that some of the latest experimental studies involving PTSSs have indirectly demonstrated such superluminal lasing.

Non-Hermitian Chern Bands

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

Edge States and Topological Invariants of Non-Hermitian Systems

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Incident Direction Independent Wave Propagation and Unidirectional Lasing

We propose an incident direction independent wave propagation generated by properly assembling different unidirectional destructive interferences (UDIs), which is a consequence of the appropriate match between synthetic magnetic fluxes and the incident wave vector. Single-direction lasing at spectral singularity is feasible without introducing nonlinearity. UDI allows unidirectional lasing and unidirectional perfect absorption; when they are combined in a parity-time-symmetric manner, the spectral singularities vanish with bounded reflections and transmissions. Furthermore, the simultaneous unidirectional lasing and perfect absorption for incidences from opposite directions is created. Our findings provide insights into light control and may shed light on the explorations of desirable functionality in fundamental research and practical applications.

Observation of parity-time symmetry in microwave photonics

Symmetry plays a crucial role in explorations of the laws of nature. Parity-time (PT) symmetry phenomena can lead to entirely real spectra in non-Hermitian systems, which attracts considerable attention in the fields of optics and electronics because these phenomena provide a new tool for the manipulation of oscillation modes and non-reciprocal signal transmission. A potential new field of application is microwave photonics, an interdisciplinary field in which the interaction between microwaves and optical signals is exploited. In this article, we report the experimental use of PT symmetry in an optoelectronic oscillator (OEO), a key microwave photonics system that can generate single-frequency sinusoidal signals with high spectral purity. PT symmetry is theoretically analyzed and experimentally observed in an OEO with two mutually coupled active oscillation cavities via a precise manipulation of the interplay between gain and loss in the two oscillation cavities. Stable single-frequency microwave oscillation is achieved without using any optical/electrical filters for oscillation mode selection, which is an indispensable requirement in traditional OEOs. This observation opens new avenues for signal generation and processing based on the PT symmetry principle in microwave photonics.

Ring quantum cascade lasers with twisted wavefronts

We demonstrate the on-chip generation of twisted light beams from ring quantum cascade lasers. A monolithic gradient index metamaterial is fabricated directly into the substrate side of the semiconductor chip and induces a twist of the light's wavefront. This significantly influences the obtained beam pattern, which changes from a central intensity minimum to a maximum depending on the discontinuity count of the metamaterial. Our design principle provides an interesting alternative to recent implementations of microlasers operating at an exceptional point.