Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems

Abstract

Recent progress on nonlinear properties of parity-time ($\mathcal{P}\mathcal{T}$-) symmetric systems is comprehensively reviewed in this article. $\mathcal{P}\mathcal{T}$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\mathcal{P}\mathcal{T}$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\mathcal{P}\mathcal{T}$-symmetric system. The natural inclusion of nonlinearity into these $\mathcal{P}\mathcal{T}$ systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above $\mathcal{P}\mathcal{T}$-symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear $\mathcal{P}\mathcal{T}$-symmetric systems arising from various physical disciplines are presented; nonlinear properties of these systems are thoroughly elucidated; and relevant experimental results are described. In addition, emerging applications of $\mathcal{P}\mathcal{T}$ symmetry are pointed out.

Review paper; to appear in Reviews of Modern Physics. 65 two-column pages, 46 figures. Link to the abstract online at the RMP site: http://journals.aps.org/rmp/accepted/dd07cE7bE181340ae03857487ac98892c5c69e322

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• Interaction effects on $\mathcal{P}\mathcal{T}$-symmetry breaking transition in atomic gases
• Dissipation-facilitated molecules in a Fermi gas with non-Hermitian spin-orbit coupling
• $\mathcal{P}\mathcal{T}$-symmetry-breaking enhanced cavity optomechanical magnetometry
• Non-Hermitian Topology in Hermitian Topological Matter
• Impact of near-PT symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model
• Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT-symmetric Su-Schrieffer-Heeger model
• Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media
• Entanglement Dynamics in Anti-$\mathcal{P}\mathcal{T}$-Symmetric Systems
• Non-Hermitian Skin Effect in Periodically-Driven Dissipative Ultracold Atoms
• Non-Hermitian impurities in Dirac systems
• General criterion for non-Hermitian skin effects and Application: Fock space skin effects in many body systems
• High-order series expansion of non-Hermitian quantum spin models
• Parity-time-symmetric vector rational rogue wave solutions in any n-component nonlinear Schrödinger models
• Spin-orbit-coupled spinor gap solitons in Bose-Einstein condensates
• Energy-level-attraction and heating-resistant-cooling of mechanical resonators with exceptional points
• Dissipative optical solitons in asymmetric Rosen-Morse potential
• Non-normal Hamiltonian dynamics in quantum systems and its realization on quantum computers
• Spinor solitons and their $\mathcal{P}\mathcal{T}$-symmetric offspring
• Designing lasing and perfectly absorbing potentials
• Stationary through-flows in a Bose-Einstein condensate with a PT-symmetric impurity
• Weak Ergodicity Breaking in Non-Hermitian Many-body Systems
• Asymmetric transfer matrix analysis of Lyapunov exponents in one-dimensional non-reciprocal quasicrystals
• Asymmetric scattering by non-hermitian potentials
• Anti-parity-time symmetry hidden in a damping linear resonator
• Snakes and ghosts in a parity-time-symmetric chain of dimers
• Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition
• Anisotropic Topological Anderson Transitions in Chiral Symmetry Classes
• Spectral singularities of odd-$PT$-symmetric potential
• Non-Hermitian non-Abelian topological insulators with $PT$ symmetry
• Super-exponential diffusion in nonlinear non-Hermitian systems
• Real spectra, Anderson localization, and topological phases in one-dimensional quasireciprocal systems
• Asymmetric Acoustic Energy Transport in Non-Hermitian Metamaterials
• Coupling-induced nonunitary and unitary scattering in anti-PT-symmetric non-Hermitian systems
• Symmetry breaking with opposite stability between bifurcated asymmetric solitons in parity-time-symmetric potentials
• Rational solutions of the defocusing nonlocal nonlinear Schrodinger equation: Asymptotic analysis and soliton interactions
• Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach
• Exceptional points in oligomer chains
• Experimental Observation of Partial Parity-Time Symmetry and Its Phase Transition with a Laser-Driven Cesium Atomic Gas
• Solitons in a $\mathcal{P}\mathcal{T}$- symmetric ${\chi }^{(2)}$ coupler
• $S$-matrix pole symmetries for non-Hermitian scattering Hamiltonians
• Field theory of non-Hermitian disordered systems
• Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions
• Nonlinear wave dynamics near phase transition in $\mathcal{P}\mathcal{T}$-symmetric localized potentials
• Nonlinear anti-directional couplers with gain and loss
• Extension of Noether's theorem in PT-symmetric systems and its experimental demonstration in an optical setup
• Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension
• Cavity solitons in a microring dimer with gain and loss
• Non-Hermitian heterostructure for two-parameter sensing
• Optomechanical devices based on traveling-wave microresonators
• PT-symmetric quantum oscillator in an optical cavity
• Nonlinear currents in a ring-shaped waveguide with balanced gain and dissipation
• Floquet control of global $PT$ symmetry in 2D arrays of quadrimer waveguides
• Stationary and traveling solitons via local dissipations in Bose-Einstein condensates in ring optical lattices
• Quasinormal mode analysis of chiral power flow from linearly polarized dipole emitters coupled to index-modulated microring resonators close to an exceptional point
• PT-symmetric quantum graphs
• Characterizing and Tuning Exceptional Points Using Newton Polygons
• Parameter estimation and quantum entanglement in PT symmetrical cavity magnonics system
• Laser Mode Bifurcations Induced by $\mathcal{P}\mathcal{T}$-Breaking Exceptional Points
• Cross-symmetric dipolar-matter-wave solitons in double-well chains
• Localization-Delocalization Transitions in Non-Hermitian Aharonov-Bohm Cages
• Edge and bulk dissipative solitons in modulated PT-symmetric waveguide arrays
• $\mathcal{P}\mathcal{T}$ symmetric evolution, coherence and violation of Leggett-Garg inequalities
• Non-Hermitian Origin of Detachable Boundary States in Topological Insulators
• Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices
• Fundamental and vortex dissipative quadratic solitons supported by spatially localized gain
• Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States
• On the spontaneous time-reversal symmetry breaking in synchronously-pumped passive Kerr resonators
• Non-Bloch PT symmetry breaking: Universal threshold and dimensional surprise
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Unitary Scattering Protected by Pseudo-Hermiticity
• Solitons in $\mathcal{P}\mathcal{T}$-symmetric systems with spin-orbit coupling and critical nonlinearity
• On the K-theoretic classification of dynamically stable systems
• Hopf Bifurcation of Nonlinear Non-Hermitian Skin Effect
• Quantum fluctuations on top of a $\mathcal{P}\mathcal{T}$-symmetric Bose-Einstein Condensate
• Anomalous Faraday effect in a $\mathcal{P}\mathcal{T}$-symmetric dielectric slab
• Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation
• Bistable soliton switching dynamics in a $\mathcal{P}\mathcal{T}$-symmetric coupler with saturable nonlinearity
• Resonant interaction of ${\varphi }^4$ kink with spatially periodic $\mathcal{P}\mathcal{T}$-symmetric perturbation
• Imaginary gap-closed points and dynamics in a class of dissipative systems
• Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models
• Interplay of nonreciprocity and nonlinearity on mean-field energy and dynamics of a Bose-Einstein condensate in a double-well potential
• Dynamical control of solitons in a parity-time-symmetric coupler by periodic management
• Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons
• Integrable nonlinear Klein-Gordon systems with $\mathcal{P}\mathcal{T}$ nonlocality and/or space-time exchange nonlocality
• $CPT$-symmetric coupler with intermodal dispersion
• Non-Hermitian boost deformation
• Integrable Local and Non-local Vector Non-linear Schrodinger Equation with Balanced loss and Gain
• Metastable two-component solitons near an exceptional point
• Non-Hermitian skin effect enforced by nonsymmorphic symmetries
• Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials
• Modulational instability in $\mathcal{P}\mathcal{T}$-symmetric Bragg grating structures with saturable nonlinearity
• Non-Hermitian dynamics of a two-spin system with PT symmetry
• Extended imaginary gauge transformation in a general nonreciprocal lattice
• Analysis of Dirac exceptional points and their isospectral Hermitian counterparts
• Transformations between nonlocal and local integrable equations
• Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation
• Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations
• No-signaling principle and quantum brachistochrone problem in $PT$-symmetric fermionic two- and four-dimensional models
• Hybridization of topological surface states with a flat band
• Dynamical localization transition in the non-Hermitian lattice gauge theory
• Correlation versus dissipation in a non-Hermitian Anderson impurity model
• Quantum Squeezing of Slow-Light Dark Solitons via Electromagnetically Induced Transparency
• Standing and Traveling Waves in a Model of Periodically Modulated One-dimensional Waveguide Arrays
• Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation
• Parity-time symmetric systems with memory
• Wireless energy transfer in non-Hermitian quantum battery
• Boundary-dependent dynamical instability of bosonic Green's function: Dissipative Bogoliubov-de Gennes Hamiltonian and its application to non-Hermitian skin effect
• Controllable high-speed polariton waves in a PT-symmetric lattice
• Solitons in a chain of charge-parity-symmetric dimers
• Asymmetric soliton mobility in competing linear-nonlinear PT-symmetric lattices
• Non-Markovian anti-parity-time symmetric systems: theory and experiment
• Non-Hermitian gauged laser arrays with localized excitations: Anomalous threshold and generalized principle of selective pumping
• Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems
• Time-dependent spectral renormalization method
• Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schrödinger equation with balanced loss-gain
• Overcoming the diffraction limit on the size of dielectric resonators using an amplifying medium
• Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a PT-symmetric potential
• PT-symmetric gain and loss in a rotating Bose-Einstein condensate
• Kink-inhomogeneity interaction in the sine-Gordon model
• Wannier-Stark localization in one-dimensional amplitude-chirped lattices
• Anyonic $\mathcal{P}\mathcal{T}$ symmetry, drifting potentials and non-Hermitian delocalization
• Pseudo-conservative dynamics of coupled polariton condensates
• Solitons and nonlinear dynamics in dual-core optical fibers
• Stable solitons in a nearly PT-symmetric ferromagnet with spin-transfer torque
• Coherent scattering from semi-infinite non-Hermitian potentials
• $\mathcal{P}\mathcal{T}$ symmetry of a square-wave modulated two-level system
• Non-Hermitian matter wave mixing in Bose Einstein condensates: dissipation induced amplification
• Universal hard-edge statistics of non-Hermitian random matrices
• Unbalanced gain and loss in a quantum photonic system
• Solitons in a Hamiltonian $PT$-symmetric coupler
• Coexistence of non-Hermitian skin effect and extended states in one-dimensional nonreciprocal lattices
• Low-frequency scattering defined by the Helmholtz equation in one dimension
• Tunable nonlinear spectra of anti-directional couplers
• Quantization of the mean decay time for non-Hermitian quantum systems
• Self-trapping and skin solitons in two-dimensional non-Hermitian lattices
• Nonequilibrium steady states of Bose-Einstein condensates with a local particle loss in double potential barriers
• Effects of the third-order dispersion on continuous waves in complex potentials
• Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials
• Linear localization of zero modes in weakly coupled non-Hermitian reservoirs
• Quantum-optical implementation of non-Hermitian potentials for asymmetric scattering
• Unidirectional Reflection and Invisibility in Nonlinear Media with an Incoherent Nonlinearity
• The Quantum Skin Hall Effect
• Non-Hermitian tuned topological band gap
• Complex entanglement entropy for complex conformal field theory
• $\mathcal{P}\mathcal{T}$-symmetric quantum Rabi model: Solutions and exceptional points
• Chiral phase modulation and tunable broadband perfect absorber using the coherent cold atomic ensemble
• Topological gap solitons in a 1D non-Hermitian lattice
• Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
• Nonlinear transition between PT-symmetric and PT-broken modes in coupled fiber lasers
• Spacing gain and absorption in a simple $\mathcal{P}\mathcal{T}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances
• Interplay between nonclassicality and $\mathcal{P}\mathcal{T}$ symmetry in an effective two level system with open system effects
• Loss compensation symmetry in dimers made of gain and lossy nanoparticles
• Small-amplitude nonlinear modes under the combined effect of the parabolic potential, nonlocality and $\mathcal{P}\mathcal{T}$ symmetry
• Integrability and trajectory confinement in $\mathcal{P}\mathcal{T}$-symmetric waveguide arrays
• Analytic eigenvalue structure of a coupled oscillator system beyond the ground state
• Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions
• Parity-Time Symmetry and Exceptional points: A Tutorial
• Symmetry breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials
• Origin of robust exceptional points: a restricted bulk zero mode
• Quantum dynamics of PT-symmetrically kicked particle confined in a 1D box
• New families of non-parity-time-symmetric complex potentials with all-real spectra
• Exceptional second-order topological insulators
• Krein signature for instability of $\mathcal{P}\mathcal{T}$-symmetric states
• $\mathcal{P}\mathcal{T}$-symmetric photonic lattices with type-II Dirac cones
• Optical solitons in periodically managed PT-symmetric media
• Antichiral and trap-skin dynamics in a nonreciprocal bosonic two-leg ladder with artificial magnetic flux
• Non-Hermitian off-diagonal disordered optical lattices
• Modes and exceptional points in waveguides with impedance boundary conditions
• Composite Quantum Phases in Non-Hermitian Systems
• Impurity in a Fermi gas under non-Hermitian spin-orbit coupling
• Analytical Study of the Non-Hermitian Semiclassical Rabi Model
• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• Continuous families of non-Hermitian surface solitons
• Simultaneous Quantum Squeezing of Light Polarizations and Atomic Spins in a Cold Atomic Gas
• Non-Hermitian physics and engineering in silicon photonics
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Polarization dynamics of trapped polariton condensates with $\mathcal{P}\mathcal{T}$-symmetry
• Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity
• Nonlinear behaviors of parity-time-symmetric lasers
• Exceptional Points in Gyrator-Based Circuit and Nonlinear High-Sensitivity Oscillator
• Exact States in Waveguides With Periodically Modulated Nonlinearity
• Adjusting exceptional points using saturable nonlinearities
• Gain engineering and atom lasing in a topological edge state in synthetic dimensions
• Controllable suppression of Non-Hermitian skin effects
• Manipulating Quantum Spins by a Spin-Polarized Current: An Approach Based Upon PT-Symmetric Quantum Mechanics
• Hybrid patterns and solitonic frequency combs in non-Hermitian Kerr Cavities
• Quantum phases transition revealed by the exceptional point in Hopfield-Bogoliubov matrix
• Resonant scattering of two-dimensional honeycomb PT dipole structure
• Mechanochemical feedback drives complex inertial dynamics in active solids
• Anderson-skin dualism: A boundary-dependent effect in non-Hermitian disordered coupled systems
• Quantumness and its hierarchies in PT-symmetric down-conversion models
• Defect-driven incoherent skin localization
• Raising the $\mathcal{P}\mathcal{T}$ transition threshold by strong coupling to neutral chains
• Non-linear Schr$\ddot{o}$dinger equation with time-dependent balanced loss-gain and space-time modulated non-linear interaction
• Making the PT symmetry unbreakable
• Unidirectional segregation of Bright-Bright soliton through a $\mathcal{P}\mathcal{T}$-symmetric potential
• Recent advances in the combination of nonlinearity and exceptional points
• Dissolution of the non-Hermitian skin effect in one-dimensional lattices with linearly varying nonreciprocal hopping
• Stability restoration by asymmetric nonlinear states in non-Hermitian double-well potentials
• Nonlinear Schrödinger equations with amplitude-dependent Wadati potentials
• A systematic construction of parity-time ($\mathcal{P}\mathcal{T}$)-symmetric and non-$\mathcal{P}\mathcal{T}$-symmetric complex potentials from the solutions of various real nonlinear evolution equations
• Non-adiabatic transitions in parabolic and super-parabolic $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian systems
• PT-symmetric and antisymmetric nonlinear states in a split potential box
• Non-Hermitian Hopf insulators
• Negative Gilbert damping in cavity optomagnonics
• Interactions-disorder duality and critical phenomena in nodal semimetals, dilute gases and other systems
• Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations
• Existence and Stability of Dissipative Solitons in a Dual-Waveguide Lattice with Linear Gain and Nonlinear Losses
• Dynamic transition of the density-matrix topology under parity-time symmetry
• Fundamental issues with light propagation through $\mathcal{P}\mathcal{T}$-symmetric systems
• Can a photonic thermalization gap arise in disordered non-Hermitian Hamiltonian systems?
• Dual opposing quadrature-PT symmetry
• Creating moving gap solitons in spin-orbit-coupled Bose-Einstein condensates
• On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system
• Gaussian eigenstate pinning in non-Hermitian quantum mechanics
• Continuous time crystals as a PT symmetric state and the emergence of critical exceptional points
• Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I Symmetry preserving and symmetry breaking solutions
• Real spectra and phase transition of skin effect in nonreciprocal systems
• Bosonic Holes in Quadratic Bosonic Systems
• Quantum transport in nonlinear Rudner-Levitov models
• Beyond the Petermann limit: can exceptional points increase sensor precision?
• Long time asymptotic behavior for the nonlocal nonlinear Schrödinger equation with weighted Sobolev initial data
• Higher-dimensional soliton generation, stability and excitations of the PT-symmetric nonlinear Schrödinger equations
• Classical Hamiltonian Systems with Balanced loss and gain
• Spontaneous symmetry emergence in a Hermitian system of coupled oscillators without symmetry
• Using mixed many-body particle states to generate exact $\mathcal{P}\mathcal{T}$-symmetry in a time-dependent four-well system
• A Hermitian bypass to the non-Hermitian quantum theory
• On general rogue waves in the parity-time-symmetric nonlinear Schrodinger equation
• Propagation of light through amplifying honeycomb photonic lattice
• Bright solitons in a PT-symmetric chain of dimers
• Self-trapped spatially localized states in combined linear-nonlinear periodic potentials
• Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential
• Topological phase dynamics described by overtone-synthesized classical and quantum Adler equations
• Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity
• Synthetic high-order PT symmetry in a single coil resonator
• On non-existence of continuous families of stationary nonlinear modes for a class of complex potentials
• Superexponential amplification, power blowup, and solitons sustained by non-Hermitian gauge potentials
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Broadband quasi-PT Symmetry Sustained by Inhomogeneous Broadening of the Spectral Line
• Characteristics of rogue waves in the scalar and vector nonlocal nonlinear Schrödinger equations
• Non-Hermitian superintegrable systems
• A unified inverse scattering transform and soliton solutions of the nonlocal modified KdV equation with non-zero boundary conditions
• A class of 2x2 correlated random-matrix models with Brody spacing distribution
• Localization with non-Hermitian off-diagonal disorder
• Interferometric Geometric Phases of $\mathcal{P}\mathcal{T}$-symmetric Quantum Mechanics
• Gain-loss-engineering: a new platform for extreme anisotropic thermal photon tunneling
• Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI${}^{†}$ and AII${}^{†}$
• Stability band structure for periodic states in periodic potentials
• Geometric origin of self-intersection points in non-Hermitian energy spectra
• State Diagram of the Non-Reciprocal Cahn-Hilliard Model and the Effects of Symmetry
• Floquet multiple exceptional points with higher-order skin effect
• Molecular resonance identification in complex absorbing potentials via integrated quantum computing and high-throughput computing
• Topological superconducting phase in a non-Hermitian Kitaev chain with staggered pairing imbalance
• Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations
• Nonreciprocity Induced Fractional Nonlinear Thouless Pumping
• Coexistence of topological Anderson insulator and multifractal critical phase in a non-Hermitian quasicrystal
• Tunable photon blockade in a whispering-gallery-mode microresonator coupled with two nanoparticles
• Characteristic influence of exceptional points in quantum dynamics
• Analytical construction of soliton families in one- and two-dimensional nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials
• Darboux transformations and solutions of nonlocal Hirota and Maxwell-Bloch equations
• Reductions of nonlocal nonlinear Schrödinger equations to Painlevé type functions
• Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry
• Linear and integrable nonlinear evolution of the qutrit
• Theory of light diffusion through amplifying photonic lattice
• Nonreciprocal Coupling Induced Non-Hermitian Entanglement in Optomechanical Systems
• Large Chern numbers in a dissipative dice model
• Non-linear and weak-coupling expansion in Quantum Field Theory
• Pursuing complex optical energy through Exceptional Points of Degeneracy (EPDs)
• Solitary waves in the resonant nonlinear Schrödinger equation: stability and dynamical properties
• Construction of non-PT-symmetric complex potentials with all-real spectra
• Editorial (a preface to a Special Issue of journal Applied Sciences on "Guided-Wave Optics")