General Rules Governing the Dynamical Encircling of an Arbitrary Number of Exceptional Points

Non-Abelian Holonomy in Degenerate Non-Hermitian Systems

Non-Abelian holonomy, a noncommutative process that measures the parallel transport of non-Abelian gauge fields, has so far been realized in degenerate Hermitian systems with degenerate eigenstates or nondegenerate non-Hermitian systems with exceptional points. Here, we introduce non-Abelian holonomy into degenerate non-Hermitian systems possessing degenerate exceptional points and degenerate energy topologies. The interplay between energy degeneracy and energy topology around exceptional points leads to a non-Abelian holonomy with multiple energy levels and multiple degenerate levels simultaneously, going beyond that in degenerate Hermitian systems with a single energy level, or in nondegenerate non-Hermitian systems with a single degenerate level. We exploit an on-chip photonic platform to experimentally demonstrate the holonomy induced non-Abelian phenomenon, including the switching of eigenstates associated with different degenerate exceptional points and sequence-dependent holonomic outcomes. Our work shifts the paradigm of non-Abelian holonomy and adds new degrees of freedom for non-Abelian applications.

Dynamical encircling of multiple exceptional points in anti-PT symmetry system

Exceptional points (EPs) in non-Hermitian systems have turned out to be at the origin of many intriguing effects with no counterparts in Hermitian cases. A typically interesting behavior is the chiral mode switching by dynamically winding the EP. Most encircling protocols focus on the two-state or parity-time (PT) symmetry systems. Here, we propose and investigate the dynamical encircling of multiple EPs in an anti-PT-symmetric system, which is constructed based on a one-dimensional lattice with staggered lossy modulation. We reveal that dynamically encircling the multiple EPs results in the chiral dynamics via multiple non-Hermiticity-induced nonadiabatic transitions, where the output state is always on the lowest-loss energy sheet. Compared with the PT-symmetric systems that require complicated variation of the gain/loss rate or on-site potentials, our system only requires modulations of the couplings which can be readily realized in various experimental platforms. Our scheme provides a route to study non-Hermitian physics by engineering the EPs and implement novel photonic devices with unconventional functions.

Chiral transmission by an open evolution trajectory in a non-Hermitian system

Exceptional points (EPs), at which two or more eigenvalues and eigenstates of a resonant system coalesce, are associated with non-Hermitian Hamiltonians with gain and/or loss elements. Dynamic encircling of EPs has received significant interest in recent years, as it has been shown to lead to highly nontrivial phenomena, such as chiral transmission in which the final state of the system depends on the encircling handedness. Previously, chiral transmission for a pair of eigenmodes has been realized by establishing a closed dynamical trajectory in parity-time- (PT-) or anti-PT-symmetric systems. Although chiral transmission of symmetry-broken modes, more accessible in practical photonic integrated circuits, has been realized by establishing a closed trajectory encircling EPs in anti-PT-symmetric systems, the demonstrated transmission efficiency is very low due to path-dependent losses. Here, we demonstrate chiral dynamics in a coupled waveguide system that does not require a closed trajectory. Specifically, we explore an open trajectory linking two infinite points having the same asymptotic eigenmodes (not modes in PT- and anti-PT-symmetric systems), demonstrating that this platform enables high-efficiency chiral transmission, with each eigenmode localized in a single waveguide. This concept is experimentally implemented in a coupled silicon waveguide system at telecommunication wavelengths. Our work provides a new evolution strategy for chiral dynamics with superior performance, laying the foundation for the development of practical chiral-transmission devices.

A scheme for realizing nonreciprocal interlayer coupling in bilayer topological systems

Nonreciprocal interlayer coupling is difficult to practically implement in bilayer non-Hermitian topological photonic systems. In this work, we identify a similarity transformation between the Hamiltonians of systems with nonreciprocal interlayer coupling and on-site gain/loss. The similarity transformation is widely applicable, and we show its application in one- and two-dimensional bilayer topological systems as examples. The bilayer non-Hermitian system with nonreciprocal interlayer coupling, whose topological number can be defined using the gauge-smoothed Wilson loop, is topologically equivalent to the bilayer system with on-site gain/loss. We also show that the topological number of bilayer non-Hermitian C6v-typed domain-induced topological interface states can be defined in the same way as in the case of the bilayer non-Hermitian Su-Schrieffer-Heeger model. Our results show the relations between two microscopic provenances of the non-Hermiticity and provide a universal and convenient scheme for constructing and studying nonreciprocal interlayer coupling in bilayer non-Hermitian topological systems. This scheme is useful for observation of non-Hermitian skin effect in three-dimensional systems.

Exceptional points and non-Hermitian photonics at the nanoscale

Exceptional points (EPs) arising in non-Hermitian systems have led to a variety of intriguing wave phenomena, and have been attracting increased interest in various physical platforms. In this Review, we highlight the latest fundamental advances in the context of EPs in various nanoscale systems, and overview the theoretical progress related to EPs, including higher-order EPs, bulk Fermi arcs and Weyl exceptional rings. We peek into EP-associated emerging technologies, in particular focusing on the influence of noise for sensing near EPs, improving the efficiency in asymmetric transmission based on EPs, optical isolators in nonlinear EP systems and novel concepts to implement EPs in topological photonics. We also discuss the constraints and limitations of the applications relying on EPs, and offer parting thoughts about promising ways to tackle them for advanced nanophotonic applications.

Observation of delocalization transition in topological waveguide arrays with long-range interactions

Topological edge states are a generic feature of topological insulators, and the long-range interactions, which break certain properties of topological edge states, are always non-negligible in real physical systems. In this Letter, we investigate the influence of next-nearest-neighbor (NNN) interactions on the topological properties of the Su-Schrieffer-Heeger (SSH) model by extracting the survival probabilities at the boundary of the photonic lattices. By introducing a series of integrated photonic waveguide arrays with different strengths of long-range interactions, we experimentally observe delocalization transition of light in SSH lattices with nontrivial phase, which is in good agreement with our theoretical predictions. The results indicate that the NNN interactions can significantly affect the edge states, and that the localization of these states can be absent in topologically nontrivial phase. Our work provides an alternative way to investigate the interplay between long-range interactions and localized states, which may stimulate further interest in topological properties in relevant structures.

Dynamically crossing diabolic points while encircling exceptional curves: A programmable symmetric-asymmetric multimode switch

Nontrivial spectral properties of non-Hermitian systems can lead to intriguing effects with no counterparts in Hermitian systems. For instance, in a two-mode photonic system, by dynamically winding around an exceptional point (EP) a controlled asymmetric-symmetric mode switching can be realized. That is, the system can either end up in one of its eigenstates, regardless of the initial eigenmode, or it can switch between the two states on demand, by simply controlling the winding direction. However, for multimode systems with higher-order EPs or multiple low-order EPs, the situation can be more involved, and the ability to control asymmetric-symmetric mode switching can be impeded, due to the breakdown of adiabaticity. Here we demonstrate that this difficulty can be overcome by winding around exceptional curves by additionally crossing diabolic points. We consider a four-mode [Formula: see text]-symmetric bosonic system as a platform for experimental realization of such a multimode switch. Our work provides alternative routes for light manipulations in non-Hermitian photonic setups.

Edge State, Localization Length, and Critical Exponent from Survival Probability in Topological Waveguides

Edge states in topological phase transitions have been observed in various platforms. To date, verification of the edge states and the associated topological invariant are mostly studied, and yet a quantitative measurement of topological phase transitions is still lacking. Here, we show the direct measurement of edge states and their localization lengths from survival probability. We employ photonic waveguide arrays to demonstrate the topological phase transitions based on the Su-Schrieffer-Heeger model. By measuring the survival probability at the lattice boundary, we show that in the long-time limit, the survival probability is P=(1-e^{-2/ξ_{loc}})^{2}, where ξ_{loc} is the localization length. This length derived from the survival probability is compared with the distance from the transition point, yielding a critical exponent of ν=0.94±0.04 at the phase boundary. Our experiment provides an alternative route to characterizing topological phase transitions and extracting their key physical quantities.

Riemann-Encircling Exceptional Points for Efficient Asymmetric Polarization-Locked Devices

Dynamically encircling exceptional points (EPs) have unveiled intriguing chiral dynamics in photonics. However, the traditional approach based on an open manifold of Hamiltonian parameter space fails to explore trajectories that pass through an infinite boundary. Here, by mapping the full parameter space onto a closed manifold of the Riemann sphere, we introduce a framework to describe encircling-EP loops. We demonstrate that an encircling trajectory crossing the north vertex can realize near-unity asymmetric transmission. An efficient gain-free, broadband asymmetric polarization-locked device is realized by mapping the encircling path onto L-shaped silicon waveguides.

Exceptional point protected robust on-chip optical logic gates

Optical logic gates are crucial components for information processing and communication using photons. Current optical logic gates typically rely on the light interference principle which requires an accurate manipulation of the dynamical phase of light, making the device quite sensitive to system disturbances such as fabrication errors. Here we introduce non-Hermitian principles into the design of optical logic gates that work in the signal transmission process. We propose an exclusive-or gate for silicon-on-insulator platform by employing the physics in the exceptional point (EP) encirclement process. The EP induced mode switching behavior is applied to manipulate the phase of light which is topologically protected by the energy surface around the EP. As a result, the performance of the device is found to be extremely robust to structural parameter disturbances. The proposed non-Hermitian principle is expected to find applications for other on-chip photonic devices toward high robust performance.