Parity-time Symmetry Breaking Research Articles

Homotopy, symmetry, and non-Hermitian band topology

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ( PT ) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in PT -symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by Chern–Euler and Chern–Stiefel–Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

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.

Fundamental and dipole gap solitons and their dynamics in the cubic–quintic fractional nonlinear Schrödinger model with a [formula omitted]-symmetric lattice

The interplay of two linear controlled terms – fractional diffraction and parity-time (PT) symmetric lattice – gives rise to unique and interesting linear Bloch gap structures within where the nonlinear localized gap modes may exist. In this study, we explore the formation and dynamics of one-dimensional gap solitons in the cubic–quintic physical model combining the fractional diffraction and PT symmetric lattice. Two classes of gap solitons, which we name the fundamental gap solitons and dipole ones, are constructed and their stability regions within the first finite gap of the associated linear Bloch spectrum are identified by means of linear-stability analysis and direct perturbed numerical simulations. We stress that the gap solitons are always unstable under the condition of PT symmetry breaking (the imaginary part of which is above 0.5). The excitations of the stable two classes of gap solitons are also investigated by using the adiabatic variation of the system’s parameters. The results predicted here shed some light on soliton physics in physical systems with combined fractional diffraction and PT symmetric lattice and the competing nonlinearities.

Observation of energy localization in nonlinear non-Hermitian systems

Energy localization in Hermitian systems has been utilized to generate ultra-sensitivity. Here, we report the interplay between non-Hermitian parity-time (PT) symmetry breaking and the mode localization transition. In our scheme, a PT-symmetric system consists of two coupled LC (inductor–capacitor) resonators: one has a linear loss and the other has a saturated gain described by a nonlinear model. The nonlinear gain is initially set to be slightly higher than the loss, and the system is operated at the exact PT-symmetric phase close to an exceptional point. The capacitance variation applied on the loss resonator, i.e., perturbation, causes PT-symmetry to break, generating complex frequencies. As a result of nonlinear gain, the resonator will grow to reach its steady state and saturate out the gain. This stable oscillation eliminates the complex frequencies, and the mode is ultimately localized at the gain side. We have observed that the voltage amplitude of the gain resonator due to the perturbation has experienced drastic changes. The amplitude ratio before and after the perturbation is sensitive to the perturbation. Our results provide an approach to study perturbation-driven localization phenomena in a PT-symmetric system and pave the way for sensors with ultrahigh sensitivity.

Topological phases and non-Hermitian topology in tunable nonreciprocal cyclic three-mode optical systems

We propose a method for simulating a 1D non-Hermitian Su-Schrieffer-Heeger model with modulated nonreciprocal hopping using a cyclic three-mode optical system. The current system exhibits different localization of topologically nontrivial phases, which can be characterized by the winding number. We find that the eigenenergies of such a system undergo a real-complex transition as the nonreciprocal hopping changes, accompanied by a non-Bloch parity-time symmetry breaking. We explain this phase transition by considering the evolution of saddle points on the complex energy plan and the ratio of complex eigenenergies. Additionally, we demonstrate that the skin states resulting from the non-Hermitian skin effect possess higher-order exceptional points under the critical point of the non-Bloch parity-time phase transition. Furthermore, we investigate the non-Hermitian skin phase transition by the directional mean inverse participation ratio and the generalized Brillouin zone. This work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.

Activating non-Hermitian skin modes by parity-time symmetry breaking

Parity-time (PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{PT}}}}}}}}$$\\end{document}) symmetry is a cornerstone of non-Hermitian physics as it ensures real energies for stable experimental realization of non-Hermitian phenomena. In this work, we propose PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{PT}}}}}}}}$$\\end{document} symmetry as a paradigm for designing rich families of higher-dimensional non-Hermitian states with unique bulk, surface, hinge or corner dynamics. Through systematically breaking or restoring PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{PT}}}}}}}}$$\\end{document} symmetry in different sectors of a system, we can selectively activate or manipulate the non-Hermitian skin effect (NHSE) in both the bulk and topological boundary states. Some fascinating phenomena include the directional toggling of the NHSE, and the flow of boundary states without chiral or dynamical pumping, developed from selective boundary NHSE. Our results extend richly into 3D or higher, with more sophisticated interplay with selective bulk and boundary NHSE and charge-parity (CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{CP}}}}}}}}$$\\end{document}) symmetry. Based on non-interacting lattices, PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{PT}}}}}}}}$$\\end{document}-activated NHSEs can be observed in various optical, photonic, electric and quantum platforms that admit gain/loss and non-reciprocity.

Geometric Origin of Non-Bloch PT Symmetry Breaking.

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

PT symmetry characterization and dynamics of periodically modulated four-channel optical waveguides

The control of parity-time (PT) symmetry in cosmic-time PT symmetry systems holds paramount significance, yet experimental realization of such optical configurations using current technologies poses formidable challenges. Conversely, the approach of periodic modulation emerges as a more viable alternative. Notably, periodic modulation in optical systems is predominantly executed through the cyclic alteration of complex refractive index materials. Distinct from conventional approaches that align periodically modulated waveguides in parallel with gain-dissipative counterparts to satisfy PT symmetry, this paper innovatively introduces a physical model featuring the cross-placement of these waveguides, marking the first instance of leveraging this configuration to manipulate PT symmetry. This paper examines the effect of periodic modulation on the energy spectrum of the system within the high-frequency approximation, elucidating the dynamical evolution of light in a non-Hermitian four-channel optical waveguide through a synergistic approach that combines analytical and numerical methods. Adjusting the modulation parameter <i>A/ω</i> reveals a dual capability: it modulates the extent of the real energy spectrum and precisely controls the PT symmetry of the system. Notably, at <i>A/ω</i>=0, the structure exhibits a fully real energy spectrum, diverging from conventional parallel four-channel waveguide configurations. Furthermore, as <i>A/ω</i> varies from 0 to 2.4, the relative intensity and optical periodicity within each waveguide exhibit enhanced stability compared to their conventionally arranged counterparts. Furthermore, our examination of PT symmetry's effect on light tunneling dynamics within individual waveguides reveals that in the unbroken PT symmetry phase, light oscillates periodically between waveguides, whereas in the broken PT symmetry phase, light propagation within each waveguide becomes stable. In the presence of waveguide coupling, it is observed that each waveguide within the system attains steady-state light regardless of the initial light injection point. Furthermore, under weak coupling between the gain-dissipative two-channel waveguide and the neutral waveguide, light, regardless of its entry point, becomes localized in the gain waveguide with propagation distance, vanishing from other waveguides, ultimately reaching a steady-state configuration. The findings reveal that, in contrast to traditional four-channel optical waveguide systems, periodic modulation not only narrows the range of existence for the fully real energy spectrum but also enables its earlier observation. Furthermore, the relative light intensity and optical periodicity within the four-channel waveguide exhibit greater stability against variations in modulation parameters. Hence, this theoretical inquiry not only profoundly encapsulates the ubiquitous principle of PT-symmetric tetramers, elucidating that spontaneous PT symmetry breaking drastically alters optical transmission properties, transforming periodic light propagation into steady-state illumination, but also presents an enhanced, more robust configuration for the manipulation of PT symmetry.

Time-reversal and parity-time symmetry breaking in non-Hermitian field theories.

We study time-reversal symmetry breaking in non-Hermitian fluctuating field theories with conserved dynamics, comprising the mesoscopic descriptions of a wide range of nonequilibrium phenomena. They exhibit continuous parity-time (PT) symmetry-breaking phase transitions to dynamical phases. For two concrete transition scenarios, exclusive to non-Hermitian dynamics, namely, oscillatory instabilities and critical exceptional points, a low-noise expansion exposes a pretransitional surge of the mesoscale (informatic) entropy production rate, inside the static phases. Its scaling in the susceptibility contrasts conventional critical points (such as second-order phase transitions), where the susceptibility also diverges, but the entropy production generally remains finite. The difference can be attributed to active fluctuations in the wavelengths that become unstable. For critical exceptional points, we identify the coupling of eigenmodes as the entropy-generating mechanism, causing a drastic noise amplification in the Goldstone mode.

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases.

We study fluctuating field models with spontaneously emerging dynamical phases. We consider two typical transition scenarios associated with parity-time symmetry breaking: oscillatory instabilities and critical exceptional points. An analytical investigation of the low-noise regime reveals a drastic increase of the mesoscopic entropy production toward the transitions. For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields, we find physical interpretations in terms of actively propelled interfaces and a coupling of eigenmodes of the linearized dynamics near the critical exceptional point.

A Symmetric Parity–Time Coupled Optoelectronic Oscillator Using a Polarization–Dependent Spatial Structure

We propose and experimentally investigate a symmetric parity-time (PT) coupled optoelectronic oscillator (COEO) based on a polarization-dependent spatial structure. In such a COEO system, the gain/loss and coupling coefficients of two orthogonal polarization optical waves can be controlled by adjusting the polarization controller (PC) and the bias voltage of a Mach-Zehnder modulator (MZM). The single-mode selection of a microwave signal can be implemented by the PT symmetry breaking of a special mode. The performance of the proposed COEO is experimentally examined, and a 10.0 GHz microwave signal with a phase noise of −109.1 dBc/Hz @ 10 kHz and a side mode suppression ratio of 51.4 dB is generated. Moreover, an optical frequency comb with a comb tooth spacing of 10.0 GHz and a bandwidth of 100 GHz within a 10 dB amplitude variation can be simultaneously generated.

Multiple exceptional points and phase transitions of a one-dimensional PT-symmetric Lieb photonic lattice

Exceptional points (EPs) in non-Hermitian systems have attracted enormous attention and spawned intriguing prospects for the manipulation of waves. Despite many efforts focusing on the exotic behaviors about EPs, there are only a few studies of phase transitions involving multiple EPs. Here, by employing staggered couplings as well as two pairs of on-site gain/loss, we propose a one-dimensional parity-time (PT)-symmetric Lieb photonic lattice and demonstrate diverse phase transitions of such a multiband structure. Owing to the non-Hermitian chiral symmetry, symmetry-protected higher-order EPs are constructed, and the system exhibits PT symmetry breaking beyond a certain threshold. More importantly, both the relative couplings and the on-site gain/loss can be flexibly reconfigured on demand, which yields the degeneracy of different bands, i.e., the emergence of multiple EPs. We also unveil that the EPs will no longer exist in the presence of a non-Hermitian diagonal disorder. In contrast, the spectrum remains symmetric and the EPs, along with the flatband, are robust against the off diagonal disorder due to the preserved non-Hermitian particle-hole symmetry. Our work not only provides a controllable platform for studying EPs but also sheds light on the exciting non-Hermitian physics based on exceptional degeneracies.

Nonreciprocal Dicke Model.

We investigate the physics of an open two-component Dicke model, where the light field mediates nonreciprocal interactions between two spin species. We show that the model, which we dub nonreciprocal Dicke model, exhibits a discrete parity-time (PT) symmetry and we characterize the emergence of a nonstationary phase, so far explained in terms of dissipation-induced instability, as spontaneous breaking of PT symmetry. We further show that such PT symmetry breaking embodies an instance of a nonreciprocal phase transition, a concept recently introduced by Fruchart etal. [Nature (London) 592, 363 (2021)NATUAS0028-083610.1038/s41586-021-03375-9]. Remarkably, the phase transition in our model does not necessitate the presence of any underlying broken symmetry or exceptional points in the spectrum, both believed to be essential requirements for nonreciprocal phase transitions. Our results establish driven-dissipative light-matter systems as a new avenue for exploring nonreciprocal phase transitions and contribute to the theory of nonreciprocal collective phenomena.

Tunable strong plasmon–exciton coupling based modulator employing borophene and deep subwavelength perovskite grating

Two-dimensional materials support deeply confined and tunable plasmonic modes, which have great potential for achieving device miniaturization and flexible manipulation. In this paper, we propose a diffraction-unlimited system (period ≈ λ/20) composed of borophene layer and perovskite grating to investigate the strong coupling between the borophene guiding plasmon (BGP) and perovskite exciton (PE) modes. The resonant energy of the BGP mode could be electrically tuned to match the energy of the PE mode, and a remarkable Rabi splitting is attained under zero-detuning conditions. The splitting energy could reach 230 meV due to the strong field enhancement provided by the BGP mode. Taking advantage of the proposed electrically tunable hybrid system, not only is the reflective amplitude modulation depth up to 99.9%, but the 1.76π phase range modulation is achieved. Furthermore, by increasing the distance between the borophene layer and perovskite grating, a passive parity-time symmetry breaking could be observed with the vanished energy splitting. Our results deepen our understanding of light–matter interactions at the sub-wavelength scale and provide a guideline for designing active plasmonic devices.

A multifunctional layered photonic structure with NOR logical operation and multiscale detection based on the PT-symmetry breaking

In this paper, a multifunctional layered photonic structure (LPS) with NOR logical operation and multiscale detection based on the parity-time (PT) symmetry breaking composed of magnetized yttrium iron garnet (YIG) is proposed in theory. The accurate NOR logical operation is completed by properly modulating the target resonant absorption peak (AP) by the external magnetic field and using the peak to ascertain the logical operation. Given the high-quality factor of the resulting APs, the proposed structure can be used to detect four important physical quantities, which are angle, magnetic field, the thickness of ferrite1, and refractive index (RI). YIG is a typical representative of ferrite. Due to the magneto-optical effect, the PT-symmetry is broken, resulting in a nonreciprocal phenomenon, thereby further realizing NOR logic operation and multi-physical quantity detection on the front and rear scales. Since the RI of YIG is modulated by the magnetic field, the change of the magnetic field can cause the AP generated by the resonance to shift, so that the magnetic field can be detected.

Unification of quantum Zeno–anti Zeno effects and parity-time symmetry breaking transitions

Unification of quantum Zeno–anti Zeno effects and parity-time symmetry breaking transitions

Experimentally Realizable PT Phase Transitions in Reflectionless Quantum Scattering.

Above-barrier quantum scattering with truncated real potentials V(x)=-|x|^{p} provides an experimentally accessible platform that exhibits spontaneous parity-time symmetry breaking as p is varied. The unbroken phase has reflectionless states that correspond to bound states in the continuum of the nontruncated potentials at arbitrarily high discrete real energies. In the fully broken phase there are no bound states. There is a mixed phase in which exceptional points occur at specific energies and values of p. These effects should be observable in cold-atom scattering experiments.

Non-reciprocal transmission of coupled LC resonators through parity-time symmetry breaking

Non-reciprocal devices that allow a signal to be transmitted only in one direction are important for full-duplex communications. Due to the requirements of miniaturized systems, there has been an increase interest in non-magnetic non-reciprocal devices in recent years. Based on parity-time (PT) symmetric inductors-capacitors (LC) resonators, this paper has proposed non-reciprocal transmission configurations by PT-symmetry breaking. In the configuration, the coupled capacitance between the two coupled LC resonators can be adjusted so that the transmission frequency is tunable. At the same time, the resonant frequency and transmission frequency have been discriminated to optimize the non-reciprocal transmission. The configuration has been implemented by utilizing discrete components on a printed circuit board (PCB). It demonstrates that the center operation frequency of 14.05 MHz with the bandwidth 4 MHz, the insertion loss 0.32 dB, and the isolation 11 dB is adjusted to the center operation frequency of 14.95 MHz with the bandwidth 4.6 MHz, the insertion loss 0.716 dB, and the isolation 14.5 dB.

Generation of an ultrashort pulse train through ultrafast parity-time symmetry switching.

We propose a scheme for the direct generation of an ultrashort pulse train as well as the further compression of pulsed lasers based on the nonlinearity inherent to parity-time (PT) symmetric optical systems. Implementation of optical parametric amplification in a directional coupler of χ(2) waveguides enables ultrafast gain switching through pump-controlled breaking of PT symmetry. We theoretically demonstrate that pumping such a PT symmetric optical system with a periodically amplitude-modulated laser enables periodic gain switching, which can directly convert a continuous-wave signal laser into a train of ultrashort pulses. We further demonstrate that by engineering the PT symmetry threshold, an apodized gain switching that enables the production of ultrashort pulses without side lobes. This work suggests a new approach for exploring the non-linearity inherent to various PT symmetric optical structures to extend optical manipulation capabilities.

Parity-time symmetry breaking optical nanocircuit.

Gain and loss balanced parity-time (PT) inversion symmetry has been achieved across multiple platforms including acoustics, electronics, and photonics. Tunable subwavelength asymmetric transmission based on PT symmetry breaking has attracted great interest. However, due to the diffraction limit, the geometric size of an optical PT symmetric system is much larger than the resonant wavelength, which limits the device miniaturization. Here, we theoretically studied a subwavelength optical PT symmetry breaking nanocircuit based on the similarity between a plasmonic system and an RLC circuit. Firstly, the asymmetric coupling of an input signal is observed by varying the coupling strength and gain-loss ratio between the nanocircuits. Furthermore, a subwavelength modulator is proposed by modulating the gain of the amplified nanocircuit. Notably, the modulation effect near the exceptional point is remarkable. Finally, we introduce a four-level atomic model modified by the Pauli exclusion principle to simulate the nonlinear dynamics of a PT symmetry broken laser. The asymmetric emission of a coherent laser is realized by full-wave simulation with a contrast of about 50. This subwavelength optical nanocircuit with broken PT symmetry is of great significance for realizing directional guided light, modulator and asymmetric-emission laser at subwavelength scales.