We investigate the scattering properties of an anti-parity-symmetric non-Hermitian system. The anti-parity-symmetric scattering center possesses imaginary nearest-neighbor hoppings and real onsite potentials, it has been experimentally realized through dissipative coupling and frequency detuning between atomic spin waves. We find that such anti-parity-symmetric system displays three salient features: Firstly, the reflection and transmission are both reciprocal. Secondly, the reflection and transmission probabilities satisfy R+T=1 or R-T=1, which depends on the parity of the scattering center size. Thirdly, the scattering matrix satisfies (S_z)(S_z)*=I for scattering center with even-site; for scattering center with odd-site, the dynamics exhibits Hermitian scattering behavior, possessing unitary scattering matrix SS†=I.
In the past decade, the concept of parity-time ($\mathcal{PT}$) symmetry, originally introduced in non-Hermitian extensions of quantum mechanical theories, has come into thinking of photonics, providing a fertile ground for studying, observing, and utilizing some of the peculiar aspects of $\mathcal{PT}$ symmetry in optics. Together with related concepts of non-Hermitian physics of open quantum systems, such as non-Hermitian degeneracies (exceptional points) and spectral singularities, $\mathcal{PT}$ symmetry represents one among the most fruitful ideas introduced in optics in the past few yea…
We study a topological band degeneracy in non-Hermitian systems with parity-time ($PT$) and parity-particle-hole ($CP$) symmetries. In $d$-dimensional non-Hermitian systems, it is shown that $(d-1)$-dimensional exceptional surfaces can appear from band touching thanks to $PT$ or $CP$ symmetry. We investigate the topological stability and zero-gap quasiparticles for the exceptional surfaces due to the band degeneracy. We also demonstrate the band degeneracy by using lattice models of a topological semimetal and a superconductor.
Non-Hermitian skin effect denotes the exponential localization of a large number of eigen-states in a non-Hermitian lattice under open boundary conditions. Such a non-Hermiticity-induced skin effect can offset the penetration depth of in-gap edge states, leading to counterintuitive delocalized edge modes, which have not been studied in a realistic photonic system such as photonic crystals. Here, we analytically reveal the non-Hermitian skin effect and the delocalized edge states in Maxwell's equations for non-Hermitian chiral photonic crystals with anomalous parity-time symmetry. Remarkably, w…
Light propagation in systems with anti-Hermitian coupling, described by a spinor-like wave equation, provides a general route for the observation of anti parity-time ($\mathcal{PT}$ ) symmetry in optics. Remarkably, under a different definition of parity operator, a $\mathcal{PT}$ symmetry can be found as well in such systems. Such symmetries are ubiquitous in nonlinear optical interactions and are exemplified by considering modulation instability in optical fibers and optical parametric amplification.
A theoretical model based on two-point scatterers is suggested to investigate scattering of partially coherent radiation by a non-Hermitian localized structure, invariant under the simultaneous symmetry operations of parity inversion and time reversal. Within the first-order Born approximation, and the formalism of classical coherence theory, the spectral density of the scattered field in the far zone is obtained analytically. We find that the unidirectional character of the scattered radiation, which is one of the principal effects present in non-Hermitian structures, may be suppressed due to…
We investigate the effects of disorder on single particle time-evolution and two-particle correlations in an array of evanescently coupled waveguides with position-dependent tunneling rates. In the clean limit, the energy spectrum of such an array is widely tunable. In the presence of a Hermitian on-site or tunneling disorder, we find that the localization of a wave packet is highly sensitive to this energy spectrum. In particular, for an input confined to a single waveguide, we show that the fraction of light localized to the original waveguide depends on the tunneling profile. We compare the…
Parity-time ($\mathcal{PT}$) symmetry, originally conceived for non-Hermitian open quantum systems, has opened an excitingly new avenue for the coherent control of light. By tailoring optical gain and loss in integrated photonic structures, $\mathcal{PT}$ symmetric non-Hermitian photonics has found applications in many fields ranging from single mode lasing to novel topological matters. Here we propose a new paradigm towards non-Hermitian photonics based on the charge-parity ($\mathcal{CP}$) symmetry that has the potential to control the flow of light in an unprecedented way. In particular, we…
Parity-time (PT) symmetry in non-Hermitian optical systems promises distinct optical effects and applications not found in conservative optics. Its counterpart, anti-PT symmetry, subscribes another class of intriguing optical phenomena and implies complementary techniques for exotic light manipulation. Despite exciting progress, so far anti-PT symmetry has only been realized in bulky systems or with optical gain. Here, we report an on-chip realization of non-Hermitian optics with anti-PT symmetry, by using a fully-passive, nanophotonic platform consisting of three evanescently coupled waveguid…
We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian $\mathcal{H}$ satisfying the generalized (anyonic) parity-time symmetry $\mathcal{PT H}= \exp(2 i φ) \mathcal{HPT}$, where $\mathcal{P}$ and $ \mathcal{T}$ are the parity and time-reversal operators. For a stationary potential, the anyonic phase $φ$ just rotates the energy spectrum of $\mathcal{H}$ in complex plane, however for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilea…
We have studied a three-level Λ-type atomic system with all the energy levels exhibiting decay. The system is described by a pseudo-Hermitian Hamiltonian and subject to certain conditions, the Hamiltonian shows parity-time (PT) symmetry. The probability amplitudes of various atomic levels both below and above the PT-theshold is worked out.
Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we …
The explorations of the quantum-inspired symmetries in optical and photonic systems have witnessed immense research interests both fundamentally and technologically in a wide range of subjects of 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, recently ramified into diverse set of areas, particularly in optics and photonics. The intriguing physical effects enabled by non-Hermitian physics and PT symmetry have enhanced…
Non-Hermitian systems based on parity-time (PT) symmetry reveal rich physics beyond the Hermitian regime. So far, realizations of PT-symmetric systems have been limited to the spatial domain. Here we theoretically and experimentally demonstrate PT symmetry in a synthetic spectral dimension induced by nonlinear Brillouin scattering in a single optical microcavity, where electromagnetically induced transparency or absorption in two spectral resonances provides the optical gain and loss to observe a phase transition between two symmetry regimes. This scheme provides a new paradigm towards the inv…
We investigate the properties of multidimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system's periodicity. This creates a natural set of degeneracies which can undergo thresholdless $\mathcal{PT}$ transitions. We derive a $\mathbf{k} \cdot \mathbf{p}$ perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless $\mathcal{PT}$ transitions are shown to yield significant control over the band structure…
We show that topologically protected defect states can exist in open (leaky or lossy) systems even when these systems are topologically trivial in the closed limit. The states appear from within the continuum, thus in absence of a band gap, and are generated via exceptional points (a spectral transition that occurs in open wave and quantum systems with a generalized time-reversal symmetry), or via a degeneracy induced by charge-conjugation-symmetry (which is related to the pole transition of Majorana zero modes). We demonstrate these findings for a leaking passive coupled-resonator optical wav…
Temporal interfaces introduced by abrupt switching of the constitutive parameters of unbounded media enable unusual wave phenomena. So far, their explorations have been mostly limited to lossless media. Yet, non-Hermitian phenomena leveraging material loss and gain, and their balanced combination in parity-time (PT)-symmetric systems, have been opening new vistas in photonics. Here, we unveil the role that temporal interfaces offer in non-Hermitian physics, introducing the dual of PT symmetry for temporal boundaries. Our findings reveal unexplored interference mechanisms enabling extreme energ…
The physical aspects of partially coherent radiation interacting with deterministic non-Hermitian periodic materials remain largely unexplored in the statistical optics literature. Here, we consider the scattering of partially coherent radiation by a deterministic periodic medium, symmetric under the simultaneous transformations of parity inversion and time reversal, that is, a parity-time (PT)-symmetric periodic medium. Taking into account light fluctuations, one is able to describe the spectrum changes on propagation and the influence of the coherence-driven angular divergence effect. The fa…
The physics of systems that cannot be described by a Hermitian Hamiltonian, has been attracting a great deal of attention in recent years, motivated by their nontrivial responses and by a plethora of applications for sensing, lasing, energy transfer/harvesting, topology and quantum networks. Electromagnetics is an inherently non-Hermitian research area because all materials are lossy, loss and gain distributions can be controlled with various mechanisms, and the underlying systems are open to radiation. Therefore, the recent developments in non-Hermitian physics offer exciting opportunities fo…
The quantum harmonic oscillator with parity-time ($\mathcal{PT}$) symmetry, obtained from the ordinary (Hermitian) quantum harmonic oscillator by an imaginary displacement of the spatial coordinate, provides an important and exactly-solvable model to investigate non-Hermitian extension of the Ehrenfest theorem. Here it is shown that transverse light dynamics in an optical resonator with off-axis longitudinal pumping can emulate a $\mathcal{PT}$-symmetric quantum harmonic oscillator, providing an experimentally accessible system to investigate non-Hermitian coherent state propagation.
Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal (C and K) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity (Q symmetry) or the inversion (P) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symm…
We introduce a one dimensional non-Hermitian four band tight binding lattice system. We find stable topological edge states protected by particle-hole and parity-time symmetries. We show that topological phase appears in the system. We discuss that the system can also be used as a topological laser if the gain and loss positions in a unit cell are judiciously arranged.
Recently, the study of non-Hermitian physics has attracted considerable attention. The modified bulk-boundary correspondence has been proposed to understand topological edge states in non-Hermitian static systems. Here we report a new experimental observation of edge states in non-Hermitian periodically driven systems. Some unconventional edge states are found not to be satisfied with the bulk-boundary correspondence when the system belongs to the broken parity-time (PT) symmetric phase. The experiments are performed in our constructed non-Hermitian light quantum walk platform with left and ri…
Open quantum systems consisting of coupled bound and continuum states have been studied in a variety of physical systems, particularly within the scope of nuclear, atomic and molecular physics. In the open systems, the effects of the continuum decay channels are accounted for by indirect non-Hermitian couplings among the quasi-bound states. Here we explore anti-Hermitian coupling in a plasmonic system for spatially manipulating light on the nano-scale. We show that by utilizing the anti-Hermitian coupling, plasmonic antennas closely packed within only λ/15 separations can be individually excit…
A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices the bulk properties are largely affected by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time ($\mathcal{PT}$) symmetry breaking phase transitions, and suggest a bulk probing method to detect such boundary-d…
The discovery of topological phases in non-Hermitian open classical and quantum systems challenges our current understanding of topological order. Non-Hermitian systems exhibit unique features with no counterparts in topological Hermitian models, such as failure of the conventional bulk-boundary correspondence and non-Hermitian skin effect. Advances in the understanding of the topological properties of non-Hermitian lattices with translational invariance have been reported in several recent studies, however little is known about non-Hermitian quasicrystals. Here we disclose topological phases …
Non-Hermitian Hamiltonians, and particularly parity-time (PT) and anti-PT symmetric Hamiltonians, play an important role in many branches of physics, from quantum mechanics to optical systems and acoustics. Both the PT and anti-PT symmetries are specific instances of a broader class known as anyonic-PT symmetry, where the Hamiltonian and the PT operator satisfy a generalized commutation relation. Here, we study theoretically these novel symmetries and demonstrate them experimentally in coupled lasers systems. We resort to complex coupling of mixed dispersive and dissipative nature, which allow…
Optical instabilities in moving media are linked to a spontaneous parity-time symmetry breaking of the system. It is shown that in general the time evolution of the electromagnetic waves in moving media is determined by a non-Hermitian parity-time symmetric operator. For lossless systems the frequency spectrum of the time evolution operator may be complex valued, and has a mirror symmetry with respect to the real-frequency axis. The possibility of optical amplification of a light pulse in the broken parity-time symmetry regime is demonstrated.
The emergence of parity-time ($\mathcal{PT}$) symmetry has greatly enriched our study of symmetry-enabled non-Hermitian physics, but the realization of quantum $\mathcal{PT}$-symmetry faces an intrinsic issue of unavoidable symmetry-breaking Langevin noises. Here we construct a quantum pseudo-anti-$\mathcal{PT}$ (pseudo-$\mathcal{APT}$) symmetry in a two-mode bosonic system without involving Langevin noises. We show that the spontaneous pseudo-$\mathcal{APT}$ symmetry breaking leads to an exceptional point, across which there is a transition between different types of quantum squeezing dynamic…
A non-Hermitian coupled waveguide system with periodically varying parameters, in which the Berry curvature is analogous to a hyperbolic magnetic monopole or antimonopole, is investigated. It is shown to have a purely imaginary Berry connection, and is consequently influenced by a geometric multiplier. It is possible for this multiplier to induce net gain or loss in the system, corresponding to the existence of the antimonopole or monopole in parameter space, respectively. For the right choice of parameters, the system will display an apparent non-adiabatic change in behaviour, which implies a…
The exotic physics emerging in non-Hermitian systems with balanced distributions of gain and loss has drawn a great deal of attention in recent years. These systems exhibit phase transitions and exceptional point singularities in their spectra, at which eigen-values and eigen-modes coalesce and the overall dimensionality is reduced. Among several peculiar phenomena observed at exceptional points, an especially intriguing property, with relevant practical potential, consists in the inherently enhanced sensitivity to small-scale perturbations. So far, however, these principles have been implemen…
Parity-time ($\mathcal{PT}$) symmetry plays an important role both in non-Hermitian and topological systems. In non-Hermitian systems $\mathcal{PT}$ symmetry can lead to an entirely real energy spectrum, while in topological systems $\mathcal{PT}$ symmetry gives rise to stable and protected Dirac points. Here, we study a $\mathcal{PT}$-symmetric system which is both non-Hermitian and topological, namely a $\mathcal{PT}$-symmetric Dirac semimetal with non-Hermitian perturbations in three dimensions. We find that, in general, there are only two types of symmetry allowed non-Hermitian perturbatio…
Realization and manipulation of parity-time (PT) symmetry in multidimensional systems are highly desirable for exploring nontrivial physics and uncovering exotic phenomena in non-Hermitian systems. Here, we report the first experimental observation of partial PT (pPT) symmetry in a cesium atomic gas coupled with laser fields, where a two-dimensional pPT-symmetric optical potential for probe laser beam is created. A transition of the pPT symmetry from an unbroken phase to a broken one is observed through changing the beam-waist ratio of the control and probe laser beams, and the domains of unbr…
In the presence of loss and gain, the coupled mode equation on describing the mode hybridization of various waveguides or cavities, or cavities coupled to waveguides becomes intrinsically non-Hermitian. In such non-Hermitian waveguides, the standard coupled coupled mode theory fails. We generalize the coupled mode theory with a properly defined inner product based on reaction conservation. We apply our theory to the non-Hermitian parity-time symmetric waveguides, and obtain excellent agreement with results obtained by finite element fullwave simulations. The theory presented here is typically …
Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic topologically stable exceptional points according to two types of complex-energy gaps and fundamental symmetries of charge conjugation, parity, and time reversal. This classification reveals unique non-Hermitian gapless structures with no Hermitian analogs and systematically predicts unknown non-Hermitian semimetals and nodal superconductors; a topological dumbbe…
We analyze the stability of a non-Hermitian coupler with respect to modulational inhomogeneous perturbations in the presence of unbalanced gain and loss. At the parity-time (PT) symmetry point the coupler is unstable. Suitable symmetry breakings lead to an asymmetric coupler, which hosts nonlinear supermodes. A subset of those broken symmetry cases finally yields nonlinear supermodes which are stable against modulational perturbations. The lack of symmetry requirements will facilitate experimental implementations and relevant photonics applications.
We describe the process of parametric amplification in a directional coupler of quadratically nonlinear and lossy waveguides, which belong to a class of optical systems with spatial parity-time (PT) symmetry in the linear regime. We identify a distinct spectral parity-time anti-symmetry associated with optical parametric interactions, and show that pump-controlled symmetry breaking can facilitate spectrally selective mode amplification in analogy with PT lasers. We also establish a connection between breaking of spectral and spatial mode symmetries, revealing the potential to implement unconve…
Non-Hermitian topological systems simultaneously posses two antagonistic features: ultra sensitivity due to exceptional points and robustness of topological zero energy modes, and it is unclear which one prevails under different perturbations. We study that question by applying the pseudospectra theory on the prototypical non-Hermitian SSH (NHSSH) lattice. Topological modes are robust with respect to chiral perturbations and sensitive to parity-time (PT) symmetry preserving perturbations. In fact, the chiral symmetry exactly at the exceptional point leads to the suppression of its sensitivity,…
Non-Hermitian optical systems with parity-time (PT) symmetry have recently revealed many intriguing prospects that outperform conservative structures. The prevous works are mostly rooted in complex arrangements with controlled gain-loss interplay. Here, we demonstrate anti-PT symmetry inherent in nonlinear optical interactions based upon forward optical four-wave mixing in a laser-cooled atomic ensemble with negligible linear gain and loss. We observe the pair of frequency modes undergo a nontrivial anti-PT phase transition between coherent power oscillation and optical parametric amplificatio…
Detecting the size of single nanoparticle with high precision is crucial to understanding the characteristic of the nanoparticle. In this paper, we research the single particle detection based on the Anti-parity time symmetric (APT) indirectly coupled WGMs. The results show that the Anti-parity time symmetric WGM nanoparticle sensor exhibits giant enhancement in frequency splitting compared with single WGM sensor, when the system operating at exceptional point (EP). With respect to the parity-time symmetric nanoparticle sensor, our research exhibits a real eigenfrequency splitting, which can b…
We study state conversion in parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circli…
The Chern topological numbers of a material platform are usually written in terms of the Berry curvature, which depends on the normal modes of the system. Here, we use a gauge invariant Green's function method to determine from first principles the topological invariants of photonic crystals. The proposed formalism does not require the calculation of the photonic band-structure, and can be easily implemented using the operators obtained with a standard plane-wave expansion. Furthermore, it is shown that the theory can be readily applied to the classification of topological phases of non-Hermit…
