Spawning rings of exceptional points out of Dirac cones

Abstract

The Dirac cone underlies many unique electronic properties of graphene 1 and topological insulators 2 , and its band structure-two conical bands touching at a single point-has also been realized for photons in waveguide arrays 3 , atoms in optical lattices 4 , and through accidental degeneracy 5,6 . Deformations of the Dirac cone often reveal intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels 7 . A seemingly unrelated phenomenon is the exceptional point [8][9][10][11] , also known as the parity-time symmetry breaking point [12][13][14][15] , where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency 16 , unidirectional transmission or reflection [17][18][19][20][21][22][23] , and lasers with reversed pump dependence [24][25][26] or single-mode operation 27, 28 . These two fields of research are in fact connected: here we discover the ability of a Dirac cone to evolve into a ring of exceptional points, which we call an "exceptional ring." We experimentally demonstrate this concept in a photonic crystal slab. Angle-resolved reflection measurements of the photonic crystal slab reveal that the peaks of reflectivity follow the conical band structure of a Dirac cone from accidental degeneracy, whereas the complex eigenvalues of the system are deformed into a two-dimensional flat band enclosed by an exceptional ring. This deformation arises from the dissimilar radiation rates of dipole and quadrupole resonances, which play a role analogous to the loss and gain in parity-time symmetric systems. Our results indicate that the radiation that exists in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain.Closed and lossless physical systems are described by Hermitian operators, which guarantee realness of the eigenvalues and a complete set of eigenfunctions that are orthogonal to each 2 other. On the other hand, systems with open boundaries 10, 29 or with material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] are non-Hermitian 8 and have non-orthogonal eigenfunctions with complex eigenvalues where the imaginary part corresponds to decay or growth. The most drastic difference between Hermitian and non-Hermitian systems is that the latter exhibit exceptional points (EPs) where both the real and the imaginary parts of the eigenvalues coalesce. At an EP, two (or more) eigenfunctions collapse into one so the eigenspace no longer forms a complete basis, and this eigenfunction becomes orthogonal to itself under the unconjugated inner product [8][9][10][11] . To date, most studies of EP and its intriguing consequences concern parity-time symmetric systems that rely on material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] , but EP is a general property that requires only non-Hermiticity. Here, we...

Coalescence of exceptional points and phase diagrams for one-dimensionalPT-symmetric photonic crystals

“…EPs can also close band gaps in PT -symmetry honeycomb PCs [29,30]. A ring of EPs can exist near a Dirac-like cone in two-dimensional (2D) PCs [33]. However, if the whole Brillouin zone is considered, the coalescence of two states occurs in the continuous spectrum and is called a spectral singularity in the literature [34,35].…”

“…For systems with continuous spectra, various PT -symmetric photonic crystals (PCs) have been considered by studying their complex band structures. For each Bloch * Corresponding author: phchan@ust.hk k, the coalescence of two eigenstates in a discrete spectrum is normally called an EP [28][29][30][31][32][33]. For example, in the PT -symmetric plasmonic waveguides, EPs can close the gap formed by two branches of surface plasmon polaritons [28].…”

Large Sagnac frequency splitting in a ring resonator operating at an exceptional point

“…In systems with periodic structures, such as photonic crystals, there are EPs near a point (κ = 0) of the Bloch wave-number space [37,54]. For κa 1 and | D | 1, the eigenmode is determined by the wave equation…”

“…An EP is a non-Hermitian degeneracy, where two or more eigenvalues and the corresponding eigenmodes coalesce [32][33][34], and it is distinct from Hermitian degeneracy (normal degeneracy), where the eigenmodes of the degenerate eigenvalues are linearly independent. The occurrences of EPs have been experimentally and numerically demonstrated in a variety of systems [35][36][37][38][39][40][41][42]. If the system with two coalescing eigenvalues at an EP is subjected to a perturbation of strength , the magnitude of the resulting eigenvalue splitting is typically proportional to 1/2 [27,32,[43][44][45][46].…”

Eigenvalue dynamics in the presence of nonuniform gain and loss

“…The conditions presented here vastly expand the design space for observing these effects. We also show that a similarly broad class of systems exhibit a loss-induced narrowing of the density of states.Recently, the study of parity-time (PT ) symmetric optical systems has highlighted the importance of exploring non-Hermitian systems with patterned gain and loss [1][2][3][4][5][6][7][8][9][10][11], and has led to the discovery of a remarkable array of phenomena, such as loss-induced transmission in waveguides [12], unidirectional transport behavior [13][14][15][16][17], reversed pump dependence in lasers [18][19][20], and band flattening in periodic structures [4,[21][22][23][24]. These effects are leading to new possibilities for constructing on-chip integrated photonic circuits for the manipulation of light.…”