Balanced Gain Research Articles

Non-Hermitian topological systems with eigenvalues that are always real

The effect of non-Hermiticity in band topology has sparked many discussions on non-Hermitian topological physics. It has long been known that non-Hermitian Hamiltonians can exhibit real energy spectra under the condition of parity-time ($PT$) symmetry---commonly implemented with balanced loss and gain---but only when non-Hermiticity is relatively weak. Sufficiently strong non-Hermiticity, on the other hand, will destroy the reality of energy spectra, a situation known as spontaneous $PT$-symmetry breaking. Here, based on nonreciprocal coupling, we show a systematic strategy to construct non-Hermitian topological systems exhibiting bulk and boundary energy spectra that are always real, regardless of weak or strong non-Hermiticity. Such nonreciprocal-coupling-based non-Hermiticity can directly drive a topological phase transition and determine the band topology, as demonstrated in a few non-Hermitian systems from one dimensional to two dimensional. Our work develops a theory that can guarantee the reality of energy spectra for non-Hermitian Hamiltonians, and offers an avenue to explore non-Hermitian topological physics.

Fully integrated parity\u2013time-symmetric electronics

Harnessing parity–time symmetry with balanced gain and loss profiles has created a variety of opportunities in electronics from wireless energy transfer to telemetry sensing and topological defect engineering. However, existing implementations often employ ad hoc approaches at low operating frequencies and are unable to accommodate large-scale integration. Here we report a fully integrated realization of parity–time symmetry in a standard complementary metal–oxide–semiconductor process technology. Our work demonstrates salient parity–time symmetry features such as phase transition as well as the ability to manipulate broadband microwave generation and propagation beyond the limitations encountered by existing schemes. The system shows 2.1 times the bandwidth and 30% noise reduction compared to conventional microwave generation in the oscillatory mode, and displays large non-reciprocal microwave transport from 2.75 to 3.10 GHz in the non-oscillatory mode due to enhanced nonlinearities. This approach could enrich integrated circuit design methodology beyond well-established performance limits and enable the use of scalable integrated circuit technology to study topological effects in high-dimensional non-Hermitian systems.

PT phase transition in open quantum systems with Lindblad dynamics

We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed recently by Huber \textit{et al}. [J. Huber \textit{et al}., SciPost Phys. $\textbf{9}$, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$ symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a $\mathcal{PT}$ unbroken phase, some eigenvalues are pure imaginary numbers while in a $\mathcal{PT}$ broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed $\mathcal{PT}$ symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber \textit{et al}. of Liouvillian $\mathcal{PT}$ symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian $\mathcal{PT}$ symmetry.

Edge states in a non-Hermitian topological crystalline insulator

Breaking Hermiticity in topological systems gives rise to intriguing phenomena, such as the exceptional topology and the non-Hermitian skin effect. In this work, we study a non-Hermitian topological crystalline insulator sitting on the Kekul\'{e}-modulated honeycomb lattice with balanced gain and loss. We find that the gaplessness of the topological edge states in the non-Hermitian system is insensitive to edge geometries under moderate strength of gain and loss, unlike the cases of Hermitian topological crystalline insulators that depend on edge geometries crucially. We focus on two types of gain and loss configurations, which are $PT$-symmetric and $PT$-asymmetric, respectively. For the $PT$-symmetric configuration, the Dirac point of the topological edge states in the Hermitian molecular-zigzag-terminated ribbons splits into a pair of exceptional points. The edge gap in the Hermitian armchair-terminated ribbons vanishes and a Dirac point forms as far as moderate gain and loss is induced. The band gaps of edge and bulk states in the Hermitian armchair-terminated ribbons close simultaneously for the $PT$-asymmetric configuration.

Optical isolator based on backward Brillouin scattering

We propose to realize an optical isolator based on backward Brillouin scattering in a whispering-gallery-mode (WGM) cavity that supports both traveling optical and acoustic waves. When the WGM cavity is optically driven in one direction, the time-reversal symmetry for light transmission is broken due to the unidirectional photon–phonon coupling. We show that the signal light is either transmitted or completely absorbed depending on whether it propagates in the same or opposite direction as the strong pump laser. The perfect absorption can be interpreted as a phase transition in a parity–time–symmetric Hamiltonian with the balanced gain and loss. The present work paves a way for the realization of an on-chip optical isolator with the high degree of isolation, large bandwidth, and low insertion loss simultaneously.

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.

Parity-time symmetry in monolithically integrated graphene-assisted microresonators.

Recently, optical systems with parity-time (PT) symmetry have attracted considerable attention due to its remarkable properties and promising applications. However, these systems usually require separate photonic devices or active semiconductor materials. Here, we investigate PT symmetry and exceptional points (EPs) in monolithically integrated graphene-assisted coupled microresonators. Raman effect and graphene cladding are utilized to introduce the balanced gain and loss. We show that PT-symmetry breaking and EPs can be achieved by changing the pump power and the chemical potential. In addition, the intracavity field intensities experience suppression and revival as the graphene-induced loss increases. Due to the unique distribution of optical field, tunable nonreciprocal light transmission is theoretically demonstrated when introducing the gain saturation nonlinearity. The maximum isolation ratio can reach 26 dB through optimizing the relevant parameters. Our proposed scheme is monolithically integrated, CMOS compatible, and exhibits remarkable properties for microscale light field manipulation. These superior features make our scheme has promising applications in optical communication, computing and sensing.

Relevant‐Based Feature Ranking (RBFR) Method for Text Classification Based on Machine Learning Algorithm

High dimensionality of the feature space is one of the problems in the field of text classification. Identification of optimal subset of features can optimize text classification process in terms of processing time and performance. In this paper, we propose a novel Relevant‐Based Feature Ranking (RBFR) algorithm which identifies and selects smaller subsets of more relevant features in the feature space. We compared the performance of the RBFR against other existing feature selection methods such as balanced accuracy measure, information gain, Gini index, and odds ratio on 3 datasets, namely, 20 newsgroup, Reuters, and WAP datasets. We have used 5 machine learning models (SVM, NB, kNN, RF, and LR) to test and evaluate the proposed feature selection method. We found that the performance of the proposed feature selection method is 25.4305% times more effective than the existing feature selection methods in terms of accuracy.

Unidirectional transparency in epsilon-near-zero based rectangular waveguides induced by parity-time symmetry

Epsilon-near-zero (ENZ) metamaterials present exciting features such as an almost infinite effective wavelength and phase velocity. However, a major challenge when using such artificial media is to achieve impedance matching due to their vanishing permittivity. To address this, it has recently been shown that wave propagation within ENZ media combined with parity-time (PT) symmetry concepts can exhibit asymmetric reflection and enhanced transmission. Here, we exploit this technique by studying a PT-symmetric ENZ structure consisting of a rectangular waveguide operating near the cutoff frequency of the dominant TE10 mode (emulating an ENZ metamaterial) and placed between balanced thin loss and gain blocks. Theoretical studies based on the transmission line theory, eigenvalue problems, and full-wave numerical simulations are performed. It is shown how unbroken and broken PT-symmetric phases can exist with exceptional points in between, demonstrating asymmetric reflection and enhanced transmission while preserving the ENZ behavior of the system.

Classification of exceptional nodal topologies protected by PT symmetry

Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and parity-time ($\mathcal{PT}$) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in non-Hermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by $\mathcal{PT}$ symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as second-order knotted surfaces of arbitrary genus, third-order knots and fourth-order points.

Nontrivial amplitude death in coupled parity-time-symmetric Liénard oscillators.

We unravel the collective dynamics exhibited by two coupled nonlinearly damped Liénard oscillators exhibiting parity and time symmetry, which is a classical example of the position-dependent damped systems. The coupled system facilitates the onset of limit-cycle and aperiodic oscillations in addition to large-amplitude oscillations. In particular, a nontrivial amplitude death state emerges as a consequence of balanced linear loss and gain of the coupled PT-symmetric systems, where gain in the amplitude of oscillation in one oscillator is exactly balanced by the loss in the other. Further, quasiperiodic attractors exist in the parameter space of a neutrally stable trivial steady state. We deduce analytical critical curves enclosing the stable regions of a nontrivial fixed point, leading to the manifestation of nontrivial amplitude death state, and neutrally stable trivial steady state. The latter loses its stability leading to the emergence of the former. The analytical critical curves exactly match with the simulation boundaries. There is also a reemergence of dynamical states as a function of the coupling strength and multistability among the observed dynamical states. The basin of attraction provides an explanation for the observed probability of dynamical states.

Multimode -symmetry thresholds and third-order exceptional points in coupled dielectric waveguides with loss and gain

A pair of coupled dielectric waveguides with balanced loss and gain is the basic system for observing the phenomena of parity-time () symmetry and exceptional points (EPs). We use the multimode analytical approach to study the peculiarities of interaction between the modes excited in such an archetypal structure. With this approach, the conditions for symmetry breaking are obtained in the special case of the hybrid HE and EH modes resulting in the ordinary second-order EPs (or -symmetry thresholds). The coupling of modes with the different azimuthal indices (the TM01 and HE21 modes in our study) generates the uncommon types of EPs. First, there are the so-called multimode -symmetry thresholds due to interaction of hybrid and transverse modes. Second, the third-order EPs appear as a result of three-modes degeneracy of real eigenvalues. It is noteworthy that an increase in the EPs order is achieved without both breaking the symmetry and introducing additional elements into the waveguide structure. The results obtained with the multimode analytical approach are in perfect agreement with the independent full-wave simulations.

Supermetal-insulator transition in a non-Hermitian network model

We study a non-Hermitian and non-unitary version of the two-dimensional Chalker-Coddington network model with balanced gain and loss. This model belongs to the class D^dagger with particle-hole symmetry^dagger and hosts both the non-Hermitian skin effect as well as exceptional points. By calculating its two-terminal transmission, we find a novel contact effect induced by the skin effect, which results in a non-quantized transmission for chiral edge states. In addition, the model exhibits an insulator to 'supermetal' transition, across which the transmission changes from exponentially decaying with system size to exponentially growing with system size. In the clean system, the critical point separating insulator from supermetal is characterized by a non-Hermitian Dirac point that produces a quantized critical transmission of 4, instead of the value of 1 expected in Hermitian systems. This change in critical transmission is a consequence of the balanced gain and loss. When adding disorder to the system, we find a critical exponent for the divergence of the localization length \nu \approx 1, which is the same as that characterizing the universality class of two-dimensional Hermitian systems in class D. Our work provides a novel way of exploring the localization behavior of non-Hermitian systems, by using network models, which in the past proved versatile tools to describe Hermitian physics.

Classical Hamiltonian Systems with balanced loss and gain

Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be removed completely through appropriate co-ordinate transformations with its effect manifested in modifying the strength of the velocity-mediated coupling. The effect of the Lorentz interaction in improving the stability of classical solutions as well as allowing a possibility of defining the corresponding quantum problem consistently on the real line, instead of within Stokes wedges, is also discussed. Several exactly solvable models based on translational and rotational symmetry are discussed which include coupled cubic oscillators, Landau Hamiltonian etc. The role of -symmetry on the existence of periodic solution in systems with balanced loss and gain is critically analyzed. A few non--symmetric Hamiltonian as well as non-Hamiltonian systems with balanced loss and gain, which include mechanical as well as extended system, are shown to admit periodic solutions. An example of Hamiltonian chaos within the framework of a non--symmetric system of coupled Duffing oscillator with balanced loss-gain and/or positional non-conservative forces is discussed. It is conjectured that a non--symmetric system with balanced loss-gain and without any velocity mediated interaction may admit periodic solution if the linear part of the equations of motion is necessarily symmetric —the nonlinear interaction may or may not be -symmetric. Further, systems with velocity mediated interaction need not be -symmetric at all in order to admit periodic solutions. Results related to nonlinear Schrödinger and Dirac equations with balanced loss and gain are mentioned briefly. A class of solvable models of oligomers with balanced loss and gain is presented for the first time along with the previously known results.

Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems

Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general -symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.

PT symmetry and the evolution speed in open quantum systems 1

The dynamics of an open quantum system with balanced gain and loss is not described by a PT-symmetric Hamiltonian but rather by Lindblad operators. Nevertheless the phenomenon of PT-symmetry breaking and the impact of exceptional points can be observed in the Lindbladean dynamics. Here we briefly review the development of PT symmetry in quantum mechanics, and the characterisation of PT-symmetry breaking in open quantum systems in terms of the behaviour of the speed of evolution of the state.

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.

Low-frequency scattering defined by the Helmholtz equation in one dimension

The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.

Significant optical force enhancements of nanostructure with balanced gain and loss.

In this paper, we theoretically analyze the optical force between a pair of active and passive plasmonic core-shell nanoparticles (NPs). The optical force between the NPs can be either attractive or repulsive near the critical point while the passive dimer provides only attractive force. We reveal that the reversal of attractive or repulsive force is determined by the relative phase of electric dipole (ED) modes, which can be strongly affected by the gain and loss coefficient κ. Compared with the passive dimer with the same size, the active-passive dimer can exhibit a very high repulsive force (about two orders of magnitude) while remaining the same order magnitude attractive force when the value of coefficient is 0.345. Interestingly, we find that the position of the maximum repulsive force occurs near the critical point. We also investigate the influence of variations in geometrical parameters of the dimer and polarization angle on the force. Finally, the numerical results demonstrate that when the dimer is illuminated by a laser beam, the attractive and repulsive forces can also be achieved. The manipulation of optical force can find potential in optical sorting and transport of NPs.

Photonic spin Hall effect in a parity-time symmetric cavity and its sensing application

By inserting an air gap between two epsilon-near-zero dielectric slabs of balanced gain and loss, the parity-time (PT) symmetric cavity is established. In this contribution, the scattering behaviors, including reflection and transmission, as well as the photonic spin Hall effect (SHE) in the PT symmetric cavity are investigated. This cavity is found to exhibit a multiple PT symmetry characterized by alternating symmetric and broken phases. This multiple PT symmetry allows the photonic SHE of transmitted light to be easily manipulated in the ranges of broad angles. Near the resonance angle of coherent perfect absorption (CPA)-laser mode, the enhanced photonic SHE of both transmitted and reflected light can reach its upper limitation (i.e. half of the beam waist), which are much larger than that of the PT-symmetric structure without air gap. Finally, a novel sensing scheme based on the photonic SHE enhanced by the excitation of CPA-laser mode in this PT-symmetric cavity is proposed. The results show that this refractive index sensor has a superior intensity sensitivity of 8762 um/RIU, which outperforms the photonic SHE sensors enhanced by surface plasmon resonance. These findings provide an effective method to enhance the photonic SHE and therefore open the opportunity for developing the optical sensors based on the photonic SHE.