Observation of optical solitons in PT-symmetric lattices

Lossless and stable propagation of surface plasmon polaritons in quasi-P T potential

We propose a scheme to study the propagation properties of surface plasmon polaritons (SPPs) in a dielectric/metal/dielectric waveguide with a refraction index of the dielectric modulated as quasi-P T-symmetric potentials. By treating the propagation loss of SPPs as the absorption background, we prove that a total gain-loss balance system which allows SPPs losslessly and stably propagating can be achieved. In addition, the propagation robustness of single peak and dipole peak transverse structured SPPs in quasi-P T-symmetric potentials under noise perturbations is discussed. The results may have certain significance for expanding applications of non-Hermitian optics in micro-/nano-optical information processing.

Non-Hermitian dynamical topological winding in photonic mesh lattices

Topological winding in non-Hermitian systems is generally associated to the Bloch band properties of lattice Hamiltonians. However, in certain non-Hermitian models, topological winding naturally arises from the dynamical evolution of the system and is related to a new form of geometric phase. Here we investigate dynamical topological winding in non-Hermitian photonic mesh lattices, where the mean survival time of an optical pulse circulating in coupled fiber loops is quantized and robust against Hamiltonian deformations. The suggested photonic model could provide an experimentally accessible platform for the observation of non-Hermitian dynamical topological windings.

Dephasing-Induced Mobility Edges in Quasicrystals

Mobility edges (ME), separating Anderson-localized states from extended states, are known to arise in the single-particle energy spectrum of certain one-dimensional lattices with aperiodic order. Dephasing and decoherence effects are widely acknowledged to spoil Anderson localization and to enhance transport, suggesting that ME and localization are unlikely to be observable in the presence of dephasing. Here it is shown that, contrary to such a wisdom, ME can be created by pure dephasing effects in quasicrystals in which all states are delocalized under coherent dynamics. Since the lifetimes of localized states induced by dephasing effects can be extremely long, rather counterintuitively decoherence can enhance localization of excitation in the lattice. The results are illustrated by considering photonic quantum walks in synthetic mesh lattices.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Anderson localization without eigenstates in photonic quantum walks

Anderson localization is ubiquitous in wavy systems with strong static and uncorrelated disorder. The delicate destructive interference underlying Anderson localization is usually washed out in the presence of temporal fluctuations or aperiodic drives in the Hamiltonian, leading to delocalization and restoring transport. However, in one-dimensional lattices with off diagonal disorder, Anderson localization can persist for arbitrary time-dependent drivings that do not break a hidden conservation law originating from the chiral symmetry, leading to the dubbed "localization without eigenstates." Here it is shown that such an intriguing phenomenon can be observed in discrete-time photonic quantum walks with static disorder applied to the coin operator and can be extended to non-Hermitian dynamics as well.

Transient non-Hermitian skin effect

The discovery of non-Hermitian skin effect (NHSE) has opened an exciting direction for unveiling unusual physics and phenomena in non-Hermitian system. Despite notable theoretical breakthroughs, actual observation of NHSE's whole evolvement, however, relies mainly on gain medium to provide amplified mode. It typically impedes the development of simple, robust system. Here, we show that a passive system is fully capable of supporting the observation of the complete evolution picture of NHSE, without the need of any gain medium. With a simple lattice model and acoustic ring resonators, we use complex-frequency excitation to create virtual gain effect, and experimentally demonstrate that exact NHSE can persist in a totally passive system during a quasi-stationary stage. This results in the transient NHSE: passive construction of NHSE in a short time window. Despite the general energy decay, the localization character of skin modes can still be clearly witnessed and successfully exploited. Our findings unveil the importance of excitation in realizing NHSE and paves the way towards studying the peculiar features of non-Hermitian physics with diverse passive platforms.

Thermalization Dynamics of Nonlinear Non-Hermitian Optical Lattices

We develop a rigorous theoretical framework based on principles from statistical mechanics that allows one to predict the equilibrium response of classical non-Hermitian arrangements in the weakly nonlinear regime. In this respect, we demonstrate that a pseudo-Hermitian configuration can always be driven into thermal equilibrium when a proper nonlinear operator is paired with the linear Hamiltonian of the system. We show that, in this case, the system will thermodynamically settle into an irregular pattern that does not resemble any known statistical distribution. Interestingly, this stable equilibrium response is associated with a Rayleigh-Jeans law when viewed within an appropriately transformed space that displays unitary dynamics. By considering a non-Hermitian Su-Schrieffer-Heeger chain, our results indicate that the thermodynamic equilibrium will always favor the edge modes instead of the ground state, in stark contrast to conventional nonlinear Hermitian configurations. Moreover, non-Hermitian lattices are shown to exhibit unusually high heat capacities, potentially acting as optical heat reservoirs to other Hermitian systems, by employing only a small number of sites and low power levels.

Observation of Anderson localization beyond the spectrum of the disorder

Anderson localization predicts that transport in one-dimensional uncorrelated disordered systems comes to a complete halt, experiencing no transport whatsoever. However, in reality, a disordered physical system is always correlated because it must have a finite spectrum. Common wisdom in the field states that localization is dominant only for wave packets whose spectral extent resides within the region of the wave number span of the disorder. Here, we show experimentally that Anderson localization can occur and even be dominant for wave packets residing entirely outside the spectral extent of the disorder. We study the evolution of wave packets in synthetic photonic lattices containing bandwidth-limited (correlated) disorder and observe strong localization for wave packets centered at twice the mean wave number of the disorder spectral extent and at low wave numbers, both far beyond the spectrum of the disorder. Our results shed light on fundamental aspects of disordered systems and offer avenues for using spectrally shaped disorder for controlling transport.

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.

Superfluidity of Light and Its Breakdown in Optical Mesh Lattices

Hydrodynamic phenomena can be observed with light thanks to the analogy between quantum gases and nonlinear optics. In this Letter, we report an experimental study of the superfluid-like properties of light in a (1+1)-dimensional nonlinear optical mesh lattice, where the arrival time of optical pulses plays the role of a synthetic spatial dimension. A spatially narrow defect at rest is used to excite sound waves in the fluid of light and measure the sound speed. The critical velocity for superfluidity is probed by looking at the threshold in the deposited energy by a moving defect, above which the apparent superfluid behavior breaks down. Our observations establish optical mesh lattices as a promising platform to study fluids of light in novel regimes of interdisciplinary interest, including non-Hermitian and/or topological physics.

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.

Phase diagram and quantum criticality of a non-HermitianXYmodel with a complex transverse field

We study a one dimensional non-Hermitian quantumXYmodel with a complex transverse field, where the imaginary transverse field can be generated by three-level atoms with spontaneous decay. Many-body spectrum can be obtained analytically. The phase diagram in thermodynamical limit indicates that the imaginary transverse field induces non-Hermitian degeneracy and shrinks the ferromagnetic phase for anisotropic interaction. We also analyze the effect of imaginary field on quantum criticality through ground state geometry phase. It causes a shift of transition point but does not change the order of phase transition in anisotropic interaction region. However, the phase transition is related to the original Hermitian degeneracy in anisotropic interaction region, which indicates that the imaginary transverse field cannot change phase boundary and critical behavior.

Nonlinear scattering by non-Hermitian multilayers with saturation effects

We theoretically investigate the optical properties of a one-dimensional non-Hermitian dispersive layered system with saturable gain and loss. We solve the nonhomogeneous Helmholtz equation perturbatively by applying the modified transfer matrix method and we obtain closed-form expressions for the reflection or transmission coefficients for TM incident waves. The nonreciprocity of the scattering process can be directly inferred from the analysis of the obtained expressions. It is shown that by tuning the parameters of the layers we can effectively control the impact of nonlinearity on the scattering characteristics of the non-Hermitian layered structure. In particular, we investigate the asymmetric and nonreciprocal characteristics of the reflectance and transmittance of multilayered parity-time (PT)-symmetric slab. We demonstrate that incident electromagnetic wave may effectively tunnel through the PT-symmetric multilayered structures with zero reflection. The effect of nonlinearity to the scattering matrix eigenvalues is systematically examined.

Nonlinear tuning of PT symmetry and non-Hermitian topological states

Topology, parity-time (PT) symmetry, and nonlinearity are at the origin of many fundamental phenomena in complex systems across the natural sciences, but their mutual interplay remains unexplored. We established a nonlinear non-Hermitian topological platform for active tuning of PT symmetry and topological states. We found that the loss in a topological defect potential in a non-Hermitian photonic lattice can be tuned solely by nonlinearity, enabling the transition between PT-symmetric and non-PT-symmetric regimes and the maneuvering of topological zero modes. The interaction between two apparently antagonistic effects is revealed: the sensitivity close to exceptional points and the robustness of non-Hermitian topological states. Our scheme using single-channel control of global PT symmetry and topology via local nonlinearity may provide opportunities for unconventional light manipulation and device applications.

Real-time observation of frequency Bloch oscillations with fibre loop modulation

Bloch oscillations (BOs) were initially predicted for electrons in a solid lattice to which a static electric field is applied. The observation of BOs in solids remains challenging due to the collision scattering and barrier tunnelling of electrons. Nevertheless, analogies of electron BOs for photons, acoustic phonons and cold atoms have been experimentally demonstrated in various lattice systems. Recently, BOs in the frequency dimension have been proposed and studied by using an optical micro-resonator, which provides a unique approach to controlling the light frequency. However, the finite resonator lifetime and intrinsic loss hinder the effect from being observed practically. Here, we experimentally demonstrate BOs in a synthetic frequency lattice by employing a fibre-loop circuit with detuned phase modulation. We show that a detuning between the modulation period and the fibre-loop roundtrip time acts as an effective vector potential and hence a constant effective force that can yield BOs in the modulation-induced frequency lattices. With a dispersive Fourier transformation, the pulse spectrum can be mapped into the time dimension, and its transient evolution can be precisely measured. This study offers a promising approach to realising BOs in synthetic dimensions and may find applications in frequency manipulations in optical fibre communication systems.

Exceptional points enhance sum sideband generation in a mechanical PT-symmetric system

Systems exhibiting parity-time (PT) symmetry are, in general, non-Hermitian systems, in which exceptional points (EPs) emerge when the system transits from the PT-symmetric phase to the broken-PT-symmetric phase. Based on the abnormal exponential amplification effect in EPs, it is often used to generate, control and transmit light in non-Hermitian systems. In this paper, we theoretically analyze the generation of the frequency components at the sum sideband by considering the nonlinear terms of the optomechanical dynamics in a double-probe-field-driven mechanical PT-symmetric system. Using experimentally achievable parameters, we demonstrate that the efficiency of sum sideband generation (SSG) can be significantly enhanced in EPs, even that the efficiency of SSG can be raised by three orders of magnitude compared to the general optomechanical system by adjusting the appropriate system parameters. These results are beneficial to explore the transmission and conversion of light in chip-scale optical communications.

Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities

We demonstrate that the fractional cubic-quintic nonlinear Schrödinger equation, characterized by its Lévy index, maintains ring-shaped soliton clusters ("necklaces") carrying orbital angular momentum. They can be built, in the respective optical setting, as circular chains of fundamental solitons linked by a vortical phase field. We predict semi-analytically that the metastable necklace-shaped clusters persist, corresponding to a local minimum of an effective potential of interaction between adjacent solitons in the cluster. Systematic simulations corroborate that the clusters stay robust over extremely large propagation distances, even in the presence of strong random perturbations.

Breaking Anti-PT Symmetry by Spinning a Resonator

Non-Hermitian systems, with symmetric or antisymmetric Hamiltonians under the parity-time (PT) operations, can have entirely real or imaginary eigenvalues. This fact has led to surprising discoveries such as loss-induced lasing and topological energy transfer. A merit of anti-PT systems is free of gain, but in recent efforts on making anti-PT devices, nonlinearity is still required. Here, counterintuitively, we show how to achieve anti-PT symmetry and its spontaneous breaking in a linear device by spinning a lossy resonator. Compared with a Hermitian spinning device, significantly enhanced optical isolation and ultrasensitive nanoparticle sensing are achievable in the anti-PT-broken phase. In a broader view, our work provides a new tool to study anti-PT physics, with such a wide range of applications as anti-PT lasers, anti-PT gyroscopes, and anti-PT topological photonics or optomechanics.

Energy Band Attraction Effect in Non-Hermitian Systems

The energy band attraction (EBA) caused by the nonorthogonal eigenvectors is a unique phenomenon in the non-Hermitian (NH) system. However, restricted by the required tight-binding approximation and meticulously engineered complex potentials, such an effect has never been experimentally demonstrated before. Here by a suitable design of all-dielectric Mie resonators in a parallel-plate transmission line, we for the first time verify the photonic analog of the EBA effects both theoretically and experimentally. The evolution of the EBA effect in a two-level NH system from gapped bands to gapless bands to flat bands is observed by precisely tuning the loss of the Mie resonators. The transmission spectra can be theoretically connected to the eigenvalues and eigenvectors of the NH Hamiltonian. Furthermore we extend our methods to a graphenelike two-dimensional NH system. Our works show a metamaterial approach toward NH topological photonics and foster a deeper understanding of band theory in open systems.

Parity-Time Symmetry in Non-Hermitian Complex Optical Media

The exploration of quantum-inspired symmetries in optical and photonic systems has witnessed immense research interest both fundamentally and technologically in a wide range of subject areas in physics and engineering. One of the principal emerging fields in this context is non-Hermitian physics based on parity-time symmetry, originally proposed in the studies pertaining to quantum mechanics and quantum field theory and recently ramified into a diverse set of areas, particularly in optics and photonics. The intriguing physical effects enabled by non-Hermitian physics and PT symmetry have enhanced significant application prospects and engineering of novel materials. In addition, there has been increasing research interest in many emerging directions beyond optics and photonics. Here, the state-of-the art developments in the field of complex non-Hermitian physics based on PT symmetry in various physical settings are brought together, and key concepts, a background, and a detailed perspective on new emerging directions are described. It can be anticipated that this trendy field of interest will be indispensable in providing new perspectives in maneuvering the flow of light in the diverse physical platforms in optics, photonics, condensed matter, optoelectronics, and beyond, and will offer distinctive application prospects in novel functional materials.

Asymmetric localization and symmetric diffraction-free transmission in synthetic photonic lattice with anti-parity-time symmetry

We study, to the best of our knowledge, the first observations of light propagation in synthetic photonic lattice with anti-parity-time symmetry by tuning the gain or loss of two coupled fiber rings alternatively and corresponding phase distribution periodically. By tuning the phase φ and the wave number Q in the lattice, asymmetric transmission of the light field can be achieved for both long and short loops when φ≠nπ/2 (n is an integer). Further investigations demonstrate that asymmetric localization of the light field in the long loop and symmetric diffraction-free transmission in two loops can both be realized by changing these two parameters. Our work provides a new method to obtain anti-parity-time symmetry in synthetic photonic lattice and paves a broad way to achieve novel optical manipulation in photonic devices.

Soliton Maxwell demons and long-tailed statistics in fluctuating optical fields

We demonstrate experimentally in biased photorefractive crystals that collisions between random-amplitude optical spatial solitons produce long-tailed statistics from input Gaussian fluctuations. The effect is mediated by Raman nonlocal corrections to Kerr self-focusing that turn soliton-soliton interaction into a Maxwell demon for the output wave amplitude.

Multipole gap solitons in fractional Schrödinger equation with parity-time-symmetric optical lattices

We investigate the existence and stability of in-phase three-pole and four-pole gap solitons in the fractional Schrödinger equation supported by one-dimensional parity-time-symmetric periodic potentials (optical lattices) with defocusing Kerr nonlinearity. These solitons exist in the first finite gap and are stable in the moderate power region. When the Lévy index decreases, the stable regions of these in-phase multipole gap solitons shrink. Below a Lévy index threshold, the effective multipole soliton widths decrease as the Lévy index increases. Above the threshold, these solitons become less localized as the Lévy index increases. The Lévy index cannot change the phase transition point of the PT-symmetric optical lattices. We also study transverse power flow in these multipole gap solitons.

Microwave Tunneling and Robust Information Transfer Based on Parity-Time-Symmetric Absorber-Emitter Pairs

Robust signal transfer in the form of electromagnetic waves is of fundamental importance in modern technology, yet its operation is often challenged by unwanted modifications of the channel connecting transmitter and receiver. Parity-time- (PT-) symmetric systems, combining active and passive elements in a balanced form, provide an interesting route in this context. Here, we demonstrate a PT-symmetric microwave system operating in the extreme case in which the channel is shorted through a small reactance, which acts as a nearly impenetrable obstacle, and it is therefore expected to induce large reflections and poor transmission. After placing a gain element behind the obstacle, and a balanced lossy element in front of it, we observe full restoration of information and overall transparency to an external observer, despite the presence of the obstacle. Our theory, simulations, and experiments unambiguously demonstrate stable and robust wave tunneling and information transfer supported by PT symmetry, opening opportunities for efficient communication through channels with dynamic changes, active filtering, and active metamaterial technology.

Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

It was recently shown that the nonlinear Schrodinger equation with a simplified dissipative perturbation features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield’s associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields the complex Ginzburg–Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation, define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show, in a similar way to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case, the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.

2D Solitons in PT-Symmetric Photonic Lattices

Over the last few years, parity-time (PT) symmetry has been the focus of considerable attention. Ever since, pseudo-Hermitian notions have permeated a number of fields ranging from optics to atomic and topological physics, as well as optomechanics, to mention a few. Unlike their Hermitian counterparts, nonconservative systems do not exhibit a priori real eigenvalues and hence unitary evolution. However, once PT symmetry is introduced, such dissipative systems can surprisingly display a real eigenspectrum, thus ensuring energy conservation during evolution. In optics, PT symmetry can be readily established by incorporating, in a balanced way, regions having an equal amount of optical gain and loss. However, thus far, all optical realizations of such PT symmetry have been restricted to a single transverse dimension (1D), such as arrays of optical waveguides or active coupled cavity arrangements. In most cases, only the loss function was modulated-a restrictive aspect that is only appropriate for linear systems. Here, we present an experimental platform for investigating the interplay between PT symmetry and nonlinearity in two-dimensional (2D) environments, where nonlinear localization and soliton formation can be observed. In contrast to typical dissipative solitons, we demonstrate a one-parameter family of soliton solutions that are capable of displaying attributes similar to those encountered in nonlinear conservative arrangements. For high optical powers, this new family of PT solitons tends to collapse on a discrete network-thus giving rise to an amplified, self-accelerating structure.

Kapitza light guiding in photonic mesh lattice

We experimentally demonstrate the transverse confinement of light in the presence of a longitudinally periodic photonic potential with vanishing average. In agreement with Kapitza's original findings in classical mechanics, we confirm that light undergoes a transverse localization due to the action of an effective potential proportional to the square of the first derivative of the potential. Experiments are performed based on (1+1) D synthetic dimensions realized in a fiber loop system, allowing for complete control of the transverse and longitudinal distributions of the potential.

Parity–Time Symmetry in Bidirectionally Coupled Semiconductor Lasers

We report on the numerical analysis of intensity dynamics of a pair of mutually coupled, single-mode semiconductor lasers that are operated in a configuration that leads to features reminiscent of parity–time symmetry. Starting from the rate equations for the intracavity electric fields of the two lasers and the rate equations for carrier inversions, we show how these equations reduce to a simple 2 × 2 effective Hamiltonian that is identical to that of a typical parity–time (PT)-symmetric dimer. After establishing that a pair of coupled semiconductor lasers could be PT-symmetric, we solve the full set of rate equations and show that despite complicating factors like gain saturation and nonlinearities, the rate equation model predicts intensity dynamics that are akin to those in a PT-symmetric system. The article describes some of the advantages of using semiconductor lasers to realize a PT-symmetric system and concludes with some possible directions for future work on this system.

Tuning the Effective PT Phase of Plasmonic Eigenmodes

We experimentally observe an effective PT-phase transition through the exceptional point in a hybrid plasmonic-dielectric waveguide system. Transmission experiments reveal fundamental changes in the underlying eigenmode interactions as the environmental refractive index is tuned, which can be unambiguously attributed to a crossing through the plasmonic exceptional point. These results extend the design opportunities for tunable non-Hermitian physics to plasmonic systems.

Anti-Parity-Time Symmetric Optical Four-Wave Mixing in Cold Atoms

Non-Hermitian optical systems with parity-time (PT) symmetry have recently revealed many intriguing prospects that outperform conservative structures. The previous works are mostly rooted in complex arrangements with controlled gain-loss interplay. Here, we demonstrate anti-PT symmetry inherent in the nonlinear optical interaction based upon forward optical four-wave mixing in a laser-cooled atomic ensemble with negligible linear gain and loss. We observe that the pair of frequency modes undergo a nontrivial anti-PT phase transition between coherent power oscillation and optical parametric amplification in presence of a large phase mismatch.

Experimental Realization of PT-Symmetric Flat Bands

The capability to temporarily arrest the propagation of optical signals is one of the main challenges hampering the ever more widespread use of light in rapid long-distance transmission as well as all-optical on-chip signal processing or computations. To this end, flat-band structures are of particular interest, since their hallmark compact eigenstates not only allow for the localization of wave packets, but importantly, also protect their transverse profile from deterioration without the need for additional diffraction management. In this work, we experimentally demonstrate that, far from being a nuisance to be compensated, judiciously tailored loss distributions can, in fact, be the key ingredient in synthesizing such flat bands in non-Hermitian environments. We probe their emergence in the vicinity of an exceptional point and directly observe the associated compact localized modes that can be excited at arbitrary positions of the periodic lattice.

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.

Collapse on the line - how synthetic dimensions influence nonlinear effects

Power induced wave collapse is one of the most fascinating phenomena in optics as it provides extremely high intensities, thus stimulating a range of nonlinear processes. For low power levels, propagation of beams in bulk media is dominated by diffraction, while above a certain threshold self-focusing is steadily enhanced by the action of a positive nonlinearity. An autocatalytic blow-up occurs, which is only stopped by saturation of the nonlinearity, material damage or the inherent medium discreteness. In the latter case, this leads to energy localization on a single site. It is commonly believed that for cubic nonlinearities, this intriguing effect requires at least two transverse dimensions to occur and is thus out of reach in fiber optics. Following the concept of synthetic dimensions, we demonstrate that mixing short and long-range interaction resembles a two-dimensional mesh lattice and features wave collapse at mW-power levels in a genuine 1D system formed by coupled fiber loops.

How to Compute Spectra with Error Control

Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics.

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

We introduce a model based on the one-dimensional nonlinear Schrödinger equation with critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

Transport properties of the non-Hermitian T-shaped quantum router

In this study, we design a T-shaped quantum router that comprises two-level systems (TLSs), an infinite coupled resonator waveguide (CRW), and a semi-infinite CRW. The loss (absorption) and gain (amplification) of the energy levels of the TLSs can be considered as energy exchange between the system and its environment. Considering loss in the ground state and gain in the excited state of the TLSs and loss of cavities, the system is non-energy-conserving and non-Hermitian. Loss in the system consists of loss of cavities and TLSs. The total transmission probabilities (TPs) of photons in the system are equal to 1 or lower when the system has loss only. Loss causes a bounce-back phenomenon in the TPs. The TPs have a divergent point when the TLSs have gain, and we obtain this divergent condition. The reflection probability has a minimal point only when photons are incident from the semi-infinite CRW and the system has loss. The TPs of the non-Hermitian router are increased by gain, decreased by loss, and conserved under certain conditions.

Robust [Formula: see text] symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

The real spectrum of bound states produced by [Formula: see text]-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of [Formula: see text]-symmetric systems for various applications. On the other hand, it is known that the [Formula: see text] symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) [Formula: see text] symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose [Formula: see text] symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a [Formula: see text]-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the [Formula: see text] symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable [Formula: see text] symmetry, while generic soliton families are found in a numerical form. The [Formula: see text]-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.

Observation of localized modes at effective gauge field interface in synthetic mesh lattice

We predict a generic mechanism of wave localization at an interface between uniform artificial gauge fields, arising due to propagation-dependent phase accumulation similar to Aharonov-Bohm phenomenon. We realize experimentally a synthetic mesh lattice with real-time control over the vector gauge field, and observe robust localization under a broad variation of gauge strength and direction, as well as structural lattice parameters. This suggests new possibilities for confining and guiding waves in diverse physical systems through the synthetic gauge fields.

Parity-time symmetry in coherent asymmetric double quantum wells

A coherently prepared asymmetric double semiconductor quantum well (QW) is proposed to realize parity-time (PT) symmetry. By appropriately tuning the laser fields and the pertinent QW parameters, PT-symmetric optical potentials are obtained by three different methods. Such a coherent QW system is reconfigurable and controllable, and it can generate new approaches of theoretically and experimentally studying PT-symmetric phenomena.

Anderson localization in synthetic photonic lattice with random coupling

A synthetic photonic lattice (SPL) is a re-configurable test-bed for studying the dynamics of one-dimensional mesh lattices including the photonic implementations of discrete time quantum walks. Unlike other realizations of photonic lattices, SPL possesses easy and fast control of lattice parameters. Here we consider disordered SPL where the coupling ratio between the two fiber loops realizing the lattice is random but does not change between the round trips. We obtain a new analytical result for the localization length (inverse Lyapunov exponent) for a practical case of weak coupling disorder. We also numerically study the dynamics of the pulse train circulating within the loops and observe that despite delocalization transition at the band center the pulse spreading is arrested even at small values of the disorder.

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.

Vector solitons in nonparity-time-symmetric complex potentials

The existence and stability of vector solitons in non-parity-time (PT)-symmetric complex potentials are investigated. We study the vector soliton family, in which the propagation constants of the two components are different. It is found that vector solitons can be stable below and above the phase transition of the non-PT-symmetric complex potentials. Below the phase transition, vector solitons are stable in the low power region. Above the phase transition, there are two continuous stable intervals in the existence region. The profiles of two components of these vector solitons show the asymmetry and we also study the transverse power flow in the two components of these vector solitons in the non-PT-symmetric complex potentials.

Asymmetric localization induced by non-Hermitian perturbations with PT symmetry in photonic lattice

We study both theoretically and numerically the asymmetric localization of lightwave in a three-layered photonic lattice with non-Hermitian perturbations. The results indicate that the gauge potential for photons can arise from the non-Hermitian perturbations, once the perturbations satisfy parity-time symmetry. Further study shows that the Peierls phase between adjacent waveguides has an important impact on the shapes of the band structures, which result in asymmetric localization of a lightwave in such a system when the wave number and Peierls phase satisfy k=ϕ=±π/2. This Letter provides a new way to control the light transmission and a feasible method to realize gauge potential for photons.

𝒫𝒯-symmetric and antisymmetric nonlinear states in a split potential box

We introduce a one-dimensional [Formula: see text]-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height ε, and constant linear gain and loss, γ, in each half-box. The system may be readily realized in microwave photonics. Using numerical methods, we construct [Formula: see text]-symmetric and antisymmetric modes, which represent, respectively, the system's ground state and first excited state, and identify their stability. Their instability mainly leads to blowup, except for the case of ε=0, when an unstable symmetric mode transforms into a weakly oscillating breather, and an unstable antisymmetric mode relaxes into a stable symmetric one. At ε>0, the stability area is much larger for the [Formula: see text]-antisymmetric state than for its symmetric counterpart. The stability areas shrink with increase of the total power, P In the linear limit, which corresponds to [Formula: see text], the stability boundary is found in an analytical form. The stability area of the antisymmetric state originally expands with the growth of γ, and then disappears at a critical value of γThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Optical gradient forces in PT-symmetric coupled-waveguide structures

Optical gradient force in a parity-time (PT)-symmetric coupled-waveguide system is theoretically studied. We find that when the system evolves from PT-symmetric region to broken-PT-symmetric region, the normalized optical forces of the two eigenmodes decrease first and become the same when the exceptional point is reached. Besides, the optical force induced PT phase transition is demonstrated. It is worth noting that, when the system is in the broken-PT-symmetric region and the length of the waveguide is much longer than the propagation length of the lossy eigenmode, the total optical gradient force acting on the two waveguides will decrease with the decreasing of the gap. This work gives us a new understanding of integrated optomechanics by combining with PT symmetry.

Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media

We study the three-wave interaction that couples an electromagnetic pump wave to two frequency down-converted daughter waves in a quadratic optical crystal and PT-symmetric potentials. PT symmetric potentials are shown to modulate stably nonlinear modes in two kinds of three-wave interaction models. The first one is a spatially extended three-wave interaction system with odd gain-and-loss distribution in the channel. Modulated by the PT-symmetric single-well or multi-well Scarf-II potentials, the system is numerically shown to possess stable soliton solutions. Via adiabatical change of system parameters, numerical simulations for the excitation and evolution of nonlinear modes are also performed. The second one is a combination of PT-symmetric models which are coupled via three-wave interactions. Families of nonlinear modes are found with some particular choices of parameters. Stable and unstable nonlinear modes are shown in distinct families by means of numerical simulations. These results will be useful to further investigate nonlinear modes in three-wave interaction models.

Observation of Time Reversed Light Propagation by an Exchange of Eigenstates

As time flow dictates all evolution, its effective reversal is a topic of active research in a broad range of disciplines, including acoustics, hydrodynamics and optics. This multifarious set of environments is reflected by a great diversity of approaches to observe various echoes of wave functions. Here, we experimentally demonstrate time reversal of a pulse sequence propagating through a photonic mesh lattice realized by two coupled loops of telecommunication fibres. Our system features a symmetric band structure, which allows for almost perfect reversal of its evolution by exchanging the population between two opposing bands. The protocol applied is based on a non-adiabatic and instantaneous exchange of eigenstates resulting in highly efficient time reversal of a pulse chain.

Integrated Optical Modulator Based on Transition between Photonic Bands

An area efficient novel optical modulator with low operation voltage is designed based on integrated Mach-Zehnder Interferometer with a photonic crystal slab structure as the phase shifter. Plasma dispersion effect is utilized so that photonic band-to-band transition occurs at the operating frequency leading to a high index change (Δn = ~4) for π-phase shift on the modulator. This approach reduces the phase shifter length to a few micrometers (~5 µm) in a silicon on insulator platform and operating voltage required is around 1 V. Low voltage together with short optical interaction length decrease optical losses and power consumption during modulation process providing a great opportunity for size and system cost optimization.

Parity–time symmetric photonics

The establishment of non-Hermitian quantum mechanics (such as parity–time (PT) symmetry) stimulates a paradigmatic shift for studying symmetries of complex potentials. Owing to the convenient manipulation of optical gain and loss in analogy to complex quantum potentials, photonics provides an ideal platform for the visualization of many conceptually striking predictions from non-Hermitian quantum theory. A rapidly developing field has emerged, namely, PT-symmetric photonics, demonstrating intriguing optical phenomena including eigenstate coalescence and spontaneous PT-symmetry breaking. The advance of quantum physics, as the feedback, provides photonics with brand-new paradigms to explore the entire complex permittivity plane for novel optical functionalities. Here, we review recent exciting breakthroughs in PT-symmetric photonics while systematically presenting their underlying principles guided by non-Hermitian symmetries. The potential device applications for optical communication and computing, biochemical sensing and healthcare are also discussed.