Exceptional points and quantum dynamics in a non-Hermitian two-qubit system

We study the exceptional-point(EP) structure and the associated quantum dynamics in a system consisting of a non-Hermitian qubit and a Hermitian qubit. We find that the system possesses two sets of EPs, which divide the systemparameter space into PT-symmetry unbroken, partially broken and fully broken regimes, each with distinct quantumdynamics characteristics. Particularly, in the partially broken regime, while the PT-symmetry is generally broken in the whole four-dimensional Hilbert space, it is preserved in a two-dimensional subspace such that the quantum dynamics in the subspace are similar to those in the PT-symmetry unbroken regime. In addition, we reveal that the competition between the inter-qubit coupling and the intra-qubit driving gives rise to a complex pattern in the EP variation with system parameters.

Quantum exceptional points of non-Hermitian Hamiltonian and Liouvillian in dissipative quantum Rabi model

The open quantum system can be described by either a Lindblad master equation or a non-Hermitian Hamiltonian(NHH).However,these two descriptions usually have different exceptional points(EPs),associated with the degeneracies in the open quantum system.Here,considering a dissipative quantum Rabi model,we study the spectral features of EPs in these two descriptions and explore their connections.We find that,although the EPs in these two descriptions are usually different,the EPs of NHH will be consistent with the EPs of master equation in the weak coupling regime.Further,we find that the quantum Fisher information(QFI),which measures the statistical distance between quantum states,can be used as a signature for the appearance of EPs.Our study may give a theoretical guidance for exploring the properties of EPs in open quantum systems.

Fabry–Prot resonance coupling associated exceptional points in a composite grating structure

The exceptional point（EP） is a significant and attractive phenomenon in an open quantum system. The scattering properties of light are similar to those in the open quantum system, which makes it possible to achieve EP in the optic system. Here we investigate the EP in a Fabry–P′erot（F–P） resonant coupling structure. The coupling between different types of F–P resonances leads to a near zero reflection, which results in a degeneration of eigenstates and thus the appearing of EP. Furthermore, the multi-wavelength EPs and unidirectional invisibility can be achieved which may be used in integrated photonics systems.

Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors

Non-orthogonality in non-Hermitian quantum systems gives rise to tremendous exotic quantum phenomena,which can be fundamentally traced back to non-unitarity.In this paper,we introduce an interesting quantity(denoted asη)as a new variant of the Petermann factor to directly and efficiently measure non-unitarity and the associated non-Hermitian physics.By tuning the model parameters of underlying non-Hermitian systems,we find that the discontinuity of bothηand its first-order derivative(denoted as■η)pronouncedly captures rich physics that is fundamentally caused by non-unitarity.More concretely,in the 1D non-Hermitian topological systems,two mutually orthogonal edge states that are respectively localized on two boundaries become non-orthogonal in the vicinity of discontinuity ofηas a function of the model parameter,which is dubbed"edge state transition".Through theoretical analysis,we identify that the appearance of edge state transition indicates the existence of exceptional points(EPs)in topological edge states.Regarding the discontinuity of■η,we investigate a two-level non-Hermitian model and establish a connection between the points of discontinuity of■ηand EPs of bulk states.By studying this connection in more general lattice models,we find that some models have discontinuity of■η,implying the existence of EPs in bulk states.

On-chip single-photon chirality encircling exceptional points

Exceptional points(EPs),which are typically defined as the degener-acy points of a non-Hermitian Hamiltonian,have been investigated in various physical systems such as photonic systems.In particular,the intriguing topological structures around EPs have given rise to novel strategies for manipulating photons and the underlying mechanism is especially useful for on-chip photonic applications.Although some on-chip experiments with the adoption of lasers have been reported,EP-based photonic chips working in the quantum regime largely re-main elusive.In the current work,a single-photon experiment was proposed to dynamically encircle an EP in on-chip photonic waveg-uides possessing passive anti-parity-time symmetry.Photon coinci-dences measurement reveals a chiral feature of transporting single photons,which can act as a building block for on-chip quantum de-vices that require asymmetric transmissions.The findings in the cur-rent work pave the way for on-chip experimental study on the physics of EPs as well as inspiring applications for on-chip non-Hermitian quantum devices.

Nonlinear perturbation of a high-order exceptional point:Skin discrete breathers and the hierarchical power-law scaling

We study the nonlinear perturbation of a high-order exceptional point(EP)of the order equal to the system site number L in a Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity.Notably,we find a class of discrete breathers that aggregate to one boundary,here named as skin discrete breathers(SDBs).The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP.Specifically,the response of nonlinear energy to the perturbation is given by E_(m)∝Γ~(α_(m)),whereα_(m)=3^(m-1)is the power with m=1,...,L labeling the nonlinear energy bands.This is in sharp contrast to the L-th root of a linear perturbation in general.These SDBs decay in a double-exponential manner,unlike the edge states or skin modes in linear systems,which decay exponentially.Furthermore,these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity.They are also stable,as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis.As nonreciprocal nonlinear models may be experimentally realized in various platforms,such as the classical platform of optical waveguides,where Kerr nonlinearity is naturally present,and the quantum platform of optical lattices with Bose-Einstein condensates,our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity,particularly on high-order EPs,and benchmark the relevant simulations.

Reconfigurable metamaterial for asymmetric and symmetric elastic wave absorption based on exceptional point in resonant bandgap

Elastic wave absorption at subwavelength scale is of significance in many engineering applications.Non-Hermitian metamaterials show the ability in high-efficiency wave absorption.However,the single functionality of metamaterials is an important limitation on their practical applications for lack of tunability and reconfigurability.Here,we propose a tunable and reconfigurable non-Hermitian piezoelectric metamaterial bar,in which piezoelectric bars connect with resonant circuits,to achieve asymmetric unidirectional perfect absorption(UPA)and symmetric bidirectional perfect absorption(PA)at low frequencies.The two functions can be arbitrarily switched by rearranging shunted circuits.Based on the reverberation-ray matrix(RRM)method,an approach is provided to achieve UPA by setting an exceptional point(EP)in the coupled resonant bandgap.By using the transfer matrix method(TMM)and the finite element method(FEM),it is observed that a non-Hermitian pseudo-band forms between two resonant bandgaps,and the EP appears at the bottom of the pseudo-band.In addition,the genetic algorithm is used to accurately and efficiently design the shunted circuits for desired low-frequency UPA and PA.The present work may provide new strategies for vibration suppression and guided waves manipulation in wide potential applications.

Majorana tunneling in a one-dimensional wire with non-Hermitian double quantum dots

The combination of non-Hermitian physics and Majorana fermions can give rise to new effects in quantum transport systems. In this work, we investigate the interplay of PT-symmetric complex potentials, Majorana tunneling and interdot tunneling in a non-Hermitian double quantum dots system. It is found that in the weak-coupling regime the Majorana tunneling has pronounced effects on the transport properties of such a system, manifested as splitting of the single peak into three and a reduced 1/4 peak in the transmission function. In the presence of the PT-symmetric complex potentials and interdot tunneling, the 1/4 central peak is robust against them, while the two side peaks are tuned by them. The interdot tunneling only induces asymmetry, instead of moving the conductance peak, due to the robustness of the Majorana modes. There is an exceptional point induced by the union of Majorana tunneling and interdot tunneling. With increased PT-symmetric complex potentials, the two side peaks will move towards each other. When the exceptional point is passed through, these two side peaks will disappear. In the strong-coupling regime, the Majorana fermion induces a 1/4 conductance dip instead of the three-peak structure. PT-symmetric complex potentials induce two conductance dips pinned at the exceptional point. These effects should be accessible in experiments.

Quasi-anti-parity–time-symmetric single-resonator micro-optical gyroscope with Kerr nonlinearity

Parity–time(PT) and quasi-anti-parity–time(quasi-APT) symmetric optical gyroscopes have been proposed recently which enhance Sagnac frequency splitting. However, the operation of gyroscopes at the exceptional point(EP) is challenging due to strict fabrication requirements and experimental uncertainties. We propose a new quasi-APT-symmetric micro-optical gyroscope which can be operated at the EP by easily shifting the Kerr nonlinearity. A single resonator is used as the core sensitive component of the quasi-APT-symmetric optical gyroscope to reduce the size, overcome the strict structural requirements and detect small rotation rates. Moreover, the proposed scheme also has an easy readout method for the frequency splitting. As a result, the device achieves a frequency splitting 10~5 times higher than that of a classical resonant optical gyroscope with the Earth's rotation. This proposal paves the way for a new and valuable method for the engineering of micro-optical gyroscopes.

Observation of the exceptional point in superconducting qubit with dissipation controlled by parametric modulation

Open physical systems described by the non-Hermitian Hamiltonian with parity-time-reversal(PT)symmetry show peculiar phenomena,such as the presence of an exceptional point(EP)at which the PT symmetry is broken and two resonant modes of the Hamiltonian become degenerate.Near the EP,the system could be more sensitive to external perturbations and this may lead to enhanced sensing.In this paper,we present experimental results on the observation of PT symmetry broken transition and the EP using a tunable superconducting qubit.The quantum system of investigation is formed by the two levels of the qubit and the energy loss of the system to the environment is controlled by a method of parametric modulation of the qubit frequency.This method is simple with no requirements for additional elements or qubit device modifications.We believe it can be easily implemented on multi-qubit devices that would be suitable for further exploration of non-Hermitian physics in more complex and diverse systems.

Theory of complex-coordinate transformation acoustics for non-Hermitian metamaterials

Transformation acoustics(TA)has emerged as a powerful tool for designing several intriguing conceptual devices,which can manipulate acoustic waves in a flexible manner,yet their applications are limited in Hermitian materials.In this work,we propose the theory of complex-coordinate transformation acoustics(CCTA)and verify the effectiveness in realizing acoustic non-Hermitian metamaterials.Especially,we apply this theory for the first time to the design of acoustic parity-time(PT)and antisymmetric parity-time(APT)metamaterials and demonstrate two distinctive examples.First,we use this method to obtain the exceptional points(EPs)of the PT/APT system and observe the spontaneous phase transition of the scattering matrix in the transformation parameter space.Second,by selecting the Jacobian matrix's constitutive parameters,the PT/APT-symmetric system can also be configured to approach the zero and pole of the scattering matrix,behaving as an acoustic coherent perfect absorber and equivalent laser.We envision our proposed CCTAbased paradigm to open the way for exploring the non-Hermitian physics and finding application in the design of acoustic functional devices such as absorbers and amplifiers whose material parameters are hard to realize by using the conventional transformation method.

Coherent Perfect Absorption in Higher Order Systems

We investigate the phenomenon of coherent perfect absorption in a high-order system with three passive resonators coupled to a super-surface to form this three-state coherent perfect absorber. The effective parity time (PT) symmetry in the passive system has received much attention, and in this open three-state PT symmetric system, the incident wave is used as the effective gain instead of balancing the material gain and loss. We analyze the variation of coherent perfect absorption of this system with the coupling coefficient of the system by simulation.

Design of a high sensitivity and wide range angular rate sensor based on exceptional surface

It is found that when the parity–time symmetry phenomenon is introduced into the resonant optical gyro system and it works near the exceptional point,the sensitivity can in theory be significantly amplified at low angular rate.However,in fact,the exceptional point is easily disturbed by external environmental variables,which means that it depends on harsh experimental environment and strong control ability,so it is difficult to move towards practical application.Here,we propose a new angular rate sensor structure based on exceptional surface,which has the advantages of high sensitivity and high robustness.The system consists of two fiber-optic ring resonators and two optical loop mirrors,and one of the resonators contains a variable ratio coupler and a variable optical attenuator.We theoretically analyze the system response,and the effects of phase and coupling ratio on the system response.Finally,compared with the conventional resonant gyro,the sensitivity of this exceptional surface angular rate sensor can be improved by about 300 times at low speed.In addition,by changing the loss coefficient in the ring resonator,we can achieve a wide range of 600 rad/s.This scheme provides a new approach for the development of ultra-high sensitivity and wide range angular rate sensors in the future.

Microscale vortex laser with controlled topological charge

A microscale vortex laser is a new type of coherent light source with small footprint that can directly generate vector vortex beams. However, a microscale laser with controlled topological charge, which is crucial for virtually any of its application, is still unrevealed. Here we present a microscale vortex laser with controlled topological charge. The vortex laser eigenmode was synthesized in a metamaterial engineered non-Hermitian micro-ring cavity system at exceptional point. We also show that the vortex laser cavity can operate at exceptional point stably to lase under optical pumping. The microscale vortex laser with controlled topological charge can serve as a unique and general building block for next- generation photonic integrated circuits and coherent vortex beam sources. The method we used here can be employed to generate lasing eigenmode with other complex functionalities.

Anti-PT-symmetric Kerr gyroscope

Non-Hermitian systems can exhibit unconventional spectral singularities called exceptional points(EPs).Various EP sensors have been fabricated in recent years,showing strong spectral responses to external signals.Here we propose how to achieve a nonlinear anti-parity-time(PT)gyroscope by spinning an optical resonator.We show that,in the absence of any nonlinearity,the sensitivity or optical mode splitting of the linear device can be magnified up to 3 orders compared to that of the conventional device without EPs.Remarkably,the PT symmetry can be broken when including the Kerr nonlinearity of the materials and,as a result,the detection threshold can be significantly lowered,i.e.,much weaker rotations which are well beyond the ability of a linear gyroscope can now be detected with the nonlinear device.Our work shows the powerful ability of PT gyroscopes in practice to achieve ultrasensitive rotation measurement.

Anti-parity-time symmetric phase transition in diffusive systems

Parity-time (PT) symmetry/anti-parity-time (APT) symmetry in non-Hermitian systems reveal profound physics andspawn intriguing effects. Recently, it has been introduced into diffusive systems together with the concept of exceptionalpoints (EPs) from quantum mechanics and the wave systems. With the aid of convection, we can generate complex thermalconductivity and imitate various wavelike dynamics in heat transfer, where heat flow can be “stopped” or moving against thebackground motion. Non-Hermitian diffusive systems offer us a new platform to investigate the heat wave manipulation.In this review, we first introduce the construction of APT symmetry in a simple double-channel toy model. Then we showthe phase transition around the EP. Finally, we extend the double-channel model to the four-channel one for showing thehigh-order EP and the associated phase transition. In a general conclusion, the phase difference of adjacent channels isalways static in the APT symmetric phase, while it dynamically evolves or oscillates when the APT symmetry is broken.

Parity-time symmetric acoustic system constructed by piezoelectric composite plates with active external circuits

Researches on parity-time(PT)symmetry in acoustic field can provide an efficient platform for controlling the travelling acoustic waves with balanced loss and gain.Here,we report a feasible design of PT-symmetric system constructed by piezoelectric composite plates with two different active external circuits.By judiciously adjusting the resistances and inductances in the external circuits,we obtain the exceptional point due to the spontaneous breaking of PT symmetry at the desired frequencies and can observe the unidirectional invisibility.Moreover,the system can be at PT exact phase or broken phase at the same frequency in the same structure by merely adjusting the external circuits,which represents the active control that makes the acoustic manipulation more convenient.Our study may provide a feasible way for manipulating acoustic waves and inspire the application of piezoelectric composite materials in acoustic structures.

The Bianisotropic Formulation of the Plane Wave Method from Faraday’s and Ampere’s Laws

The plane wave numerical technique is recast from Ampere’s and Faraday’s laws for materials that are characterized with a bianisotropic form of the constitutive relations. The populating expressions are provided for the eigenvalue matrix system that can be directly solved for the angular frequencies and field profiles when bianisotropy is included. To demonstrate the computation process and expected state diagrams and field profiles, numerical computation examples are provided for a bianisotropic Bragg Array with central defect. It is shown that the location of the magnetoelectric tensor elements has a significant effect on the eigenstates of an equivalent isotropic (anisotropic) structure. One form of the magnetoelectric tensor (diagonal elements only) leads to the observation of merging states and the formation of exceptional points. The numerical approach presented can be implemented as an add-on to the familiar plane wave numerical technique.

Compact asymmetric sound absorber at the exceptional point

We present an asymmetric absorber at an exceptional point(EP) with a compact configuration and deep-subwavelength thickness.Unlike conventional side-branched sound absorbers in dual-port systems, the proposed asymmetric absorber exhibits a compact shape that is coaxial with the waveguide. By tuning the loss and geometric parameters of the non-Hermitian system to reach an EP, we observe extreme asymmetric absorption. This phenomenon is theoretically and experimentally validated by observing a quasi-perfect absorption and a near-total reflection for opposite incidences at the ultra-thin(1/28 th of the operating wavelength)neck-embedded tube employed in this study. Furthermore, we demonstrate an EP-induced tunable asymmetric absorption. Our study proposes novel approaches to manipulate the EP-induced wave phenomena, paving the way for the development of novel acoustic absorbers, sensors, isolators, and directional devices.

Unidirectional lasing in semiconductor microring lasers at an exceptional point [Invited]

Recent experiments demonstrated that chiral symmetry breaking at an exceptional point(EP) is a viable route to achieve unidirectional laser emission in microring lasers. By a detailed semiconductor laser rate equation model,we show here that unidirectional laser emission at an EP is a robust regime. Slight deviations from the EP condition can break preferential unidirectional lasing near threshold via a Hopf instability. However, abovea "second" laser threshold, unidirectional emission is restored.