Non-Hermitian photonics promises exceptional topology of light

The band degeneracy, either the exceptional point of a non-Hermitian system or the Dirac point associated with a topological system, can feature distinct symmetry and topology. Their synergy will further produce more exotic topological effects in synthetic matter.

leads to a robust asymmetric jump to a final state that is different for a clockwise and a counterclockwise loop. The resulting asymmetric transmission is a result of the breakdown of the adiabaticity. Topological operations close to an EP-that is, performed in a closed circuit irrespective of its precise geometry -thus achieve certain goals depending solely on whether the encircling is adiabatic or non-adiabatic in nature. This topological energy transport around an EP has been experimentally verified in a microwave cavity experiment 4 , a designed waveguide structure 5 and an optomechanical setup 6 (among others) and may find applications in quantum control and switching.
Dirac points in topological photonics. Hermitian photonics crystals or waveguide arrays can be engineered to possess a different type of degeneracy: a Dirac point, that is, a linear band crossing point in the Brillouin zone where the conduction and valence bands touch each other (Fig. 1b). The Dirac point can separate distinct topological phases and mark the creation or annihilation of gapless edge modes 1,7 , but trivial Dirac points without a topological transition also exist.
In a topological insulator, gapless edge modes are protected by time-reversal and crystal symmetry, and the band structure of a topological insulator is topologically distinct from that of a conventional insulator. The Hamiltonians describing this band structures cannot be deformed smoothly into each other without closing the energy gap at the degeneracy, in the same topological manner as a Möbius strip cannot be deformed into a hollow cylinder without cutting the surface. The topological phases can be characterized by the Chern number C, which is a topological quantum number. It is defined by the Berry phase equal to 2πC, which can be calculated by considering the integral of the Berry curvature around a band degeneracy over a closed manifold in the Brillion zone. Because the Chern number is insensitive to the local geometry of the manifold, states at an interface of two topologically distinct materials (characterized by different Chern numbers), can propagate undisturbed, even in the presence of disorder.
Despite the striking similarities between non-Hermitian and topological photonics, for instance-both have degeneracies and non-vanishing Berry phase-the two have a few characteristics that set them apart: firstly, the modes become parallel at an EP, whereas they remain linearly orthogonal at a Dirac point; secondly, the Dirac point manifests robust interface states-a topological consequence very different from the adiabatic state flipping at an EP. Nevertheless, the similarity between the two types of degeneracies has recently been revealed by a complex perturbation, in terms of radiation loss, to an otherwise conservative photonic crystal slab. The perturbation forces the Dirac point to bifurcate in a continuous ring of EPs, forming an exceptional ring in reciprocal space 8 .
Combining non-Hermitian and topological photonics. The dichotomy and similarity between topological characters of a non-Hermitian EP and a Hermitian band degeneracy leads to a natural question: what benefits can be harvested from the interplay between optical non-Hermiticity and topological physics? A straightforward answer is that the efficient manipulation of gain and loss using non-Hermiticity can selectively enhance the desired topological interface state 9 , making it stand out from other topologically trivial bulk states, even in a structure of the same topological order 10 . Another example is the recent experimental demonstration of topological insulator lasers 11 , which is energy-efficient because of its non-Hermiticity, and at the same time its topological nature makes it robust against defects. Beyond the non-Hermitian topological coupling in mode selection, non-Hermitian photonics promises an extra flavour to expand the parameter space of topological photonic structures into the full complex domain, paving the way towards unexplored mechanisms of bulk-edge correspondence in open optical systems 12,13 . To conceptually visualize the prospect, we have shown in Fig. 2a and b a scheme to induce a topological phase transition solely by non-Hermitian engineering, where the onsite gain and loss effectively transform into non-Hermiticitycontrolled coupling. This new degree of freedom enables the couplings to be tuned, which is useful for lifting the degeneracy and results in topological states from the band inversion. For example, when the non-Hermitian coupling between the two elements of the dimers continuously increased by reducing the magnitude of the gain/loss coefficient, the trivially gapped bands reach the degeneracy and, upon further reduction, open a topological non-trivial gap. This convenient manipulation of gain/loss via pump-controlled or gate-controlled optical absorption helps reconfigure the topological order that is otherwise as good as unchangeable in the Hermitian limit. The intimate interplay between non-Hermitian physics and topological photonics therefore offers novel solutions to control the path and number of topological states, promising robust switches and routers for next-generation optical communications.
Additionally, non-Hermiticity can be incorporated to attain unique topological features that have no counterpart in Hermitian systems. For example, the π-Berry phase associated with the EP in non-Hermitian systems has been shown to possess a half-integer topological charge 14 . This may open the door to versatile generation of half-twisted vector beams. Furthermore, the conceptual hybridization of topology and non-Hermiticity promises a new class of topological phase transitions: when two oppositely charged EPs merge, the topological charge can disappear without opening a gap 12 . Moreover, in contrast to the synthetic (real) gauge field with a hopping phase imbalance in Hermitian configurations, strategic non-Hermitian engineering can produce an imaginary gauge field 15 . Photons subject to such an imaginary vector potential accumulate an imaginary phase along their trajectory, becoming amplified or attenuated depending on the direction in which they travel. As shown in Fig. 2c-e, such an imaginary gauge field and directional coupling can be implemented by linking two resonators with a non-Hermitianengineered anti-resonance ring. For sufficiently large values of the gauge field perfect transmission, even in the presence of disorder, can be achieved because back-scattered waves are evanescent (rather than propagating) when an imaginary gauge field is applied. An intriguing feature of the imaginary gauge field is its ability to transform a protected interface state into an extended state, thus preventing localization and restoring mobility.
Despite the demonstrated transformative potential for fundamental and applied photonics, research efforts on the synergy between quantum symmetries and topology are still in their infancy. One largely unexplored area is the consideration of symmetry paradigms other than non-Hermiticity. For example, supersymmetry, which regulates the relation between fermions and bosons in quantum field theory, is proven to be a faithful tool for spectral engineering of certain physical systems. In general, symmetry engineering of photonics offers a powerful toolbox for topological photonics research by providing a platform to design materials with topological properties on demand. Novel functionalities from the interplay between optical non-Hermiticity and topological photonics. a shows the unit cell of a coupled resonator array, with hopping rate κ and onsite gain/loss coefficient γ, which is equivalent to its Hermitian counterpart with a modified coupling κ NH ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The non-Hermiticity controlled coupling accomplishes the topological phase transition from the trivial to non-trivial band structures via the degeneracy by tuning the gain/loss coefficient as shown in (b). Here C represents the Chern number related to the bulk band. c shows a typical unit cell of a coupled resonator array subjected to a pseudo-spin-dependent imaginary gauge field engineered by an anti-resonance link ring with gain on the lower half and loss on the upper half. This coupling scheme is equivalent to two site resonators with asymmetric hopping rates: the clockwise circulating light hops from left to right with a rate of exp(h), while it hops from right to left with a much smaller rate of exp(−h), where h is the single-pass amplification/attenuation of the lower/upper halves of the link ring. d shows the topological pseudo-spin subbands, subjected to the imaginary gauge field, containing the amplifying propagation state with forward group velocity v þ g (orange curve) and the attenuating propagation state with backward group velocity v À g (blue curve). Owing to the imaginary gauge field, only the forward propagation can be observed in real space while the backscattering is absorbed in the link rings, resulting in robust one-way propagation, as shown in (e)