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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...

We formulate and exploit a computational inverse-design method based on topology optimization to demonstrate photonic crystal structures supporting complex spectral degeneracies. In particular, we discover photonic crystals exhibiting third-order Dirac points formed by the accidental degeneracy of monopolar, dipolar, and quadrupolar modes. We show that, under suitable conditions, these modes can coalesce and form a third-order exceptional point, leading to strong modifications in the spontaneous emission (SE) of emitters, related to the local density of states. We find that SE can be enhanced by a factor of 8 in passive structures, with larger enhancements ∼ ffiffiffiffiffi n 3 p possible at exceptional points of higher order n. [3,4], and as precursors to nontrivial topological effects [5][6][7]. Recent work also showed that Dirac-point degeneracies can give rise to rings of exceptional points [8]. An exceptional point (EP) is a singularity in a non-Hermitian system where two or more eigenvectors and their corresponding complex eigenvalues coalesce, leading to a nondiagonalizable, defective Hamiltonian [9,10]. EPs have been studied in various physical contexts, most notably lasers and atomic as well as molecular systems [11,12]. In recent decades, interest in EPs has been reignited in connection with non-Hermitian parity-time symmetric systems [13], especially optical media involving carefully designed gain and loss profiles [14][15][16][17][18][19][20], where they can lead to intriguing phenomena such as excess noise [21,22], chiral modes [23], directional transport [24,25], and anomalous lasing behavior [26][27][28]. Also recently, it became possible to directly observe EPs in photonic crystals (PhCs) [8] and optoelectronic microcavities [29]. Thus far, however, the main focus of these works has been the effect of second-order exceptional points (EP2s) realized through photonic radiations, where only two modes coalesce; apart from a few mathematical analyses [30][31][32] or works focused on acoustic systems [33], there has been little or no investigation into the design and consequences of EPs of higher order (where more than two modes collapse).In this Letter, we formulate and exploit a powerful inverse-design method, based on topology optimization (TO), to develop complex photonic crystals supporting Dirac points formed out of the accidental degeneracy [34] of modes belonging to different symmetry representations. We show that such higher-order Dirac points can be exploited to create third-order exceptional points (EP3s) along with complex contours of EP2s. Furthermore, we consider possible enhancements and spectral modifications in the spontaneous emission (SE) rate of emitters, showing that the local density of states (LDOS) at an EP3 (14) can be enhanced eightfold (in passive systems) and can exhibit a cubic Lorentzian spectrum under special conditions. More generally, we find enhancement factors ∼ ffiffiffiffiffi n 3 p with increasing EP order n. Although the area of photonic inverse design is not new ...

Abstract:We present a general theory of spontaneous emission at exceptional points (EPs)-exotic degeneracies in non-Hermitian systems. Our theory extends beyond spontaneous emission to any light-matter interaction described by the local density of states (e.g., absorption, thermal emission, and nonlinear frequency conversion). Whereas traditional spontaneous-emission theories imply infinite enhancement factors at EPs, we derive finite bounds on the enhancement, proving maximum enhancement of 4 in passive systems with second-order EPs and significantly larger enhancements (exceeding 400×) in gain-aided and higher-order EP systems. In contrast to non-degenerate resonances, which are typically associated with Lorentzian emission curves in systems with low losses, EPs are associated with non-Lorentzian lineshapes, leading to enhancements that scale nonlinearly with the resonance quality factor. Our theory can be applied to dispersive media, with proper normalization of the resonant modes. References and links1. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681--681 (1946). (CRC Press, 1995, vol. X). 3. S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010 H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities

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 8-11 , 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 17-23 , 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 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 8-11 . 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 show the existence of EPs in a photonic cryst...

The immune system is essential to body defense and maintenance. Specific antibodies to foreign invaders function in body defense, and it has been suggested that autoantibodies binding to self molecules are important in body maintenance. Recently, the autoantibody repertoires in the bloods of healthy mothers and their newborns were studied using an antigen microarray containing hundreds of self molecules. It was found that the mothers expressed diverse repertoires for both IgG and IgM autoantibodies. Each newborn shares with its mother a similar repertoire of IgG antibodies, which cross the placental but its IgM repertoire is more similar to those of other newborns. Here, we took a system-level approach and analyzed the correlations between autoantibody reactivities of the previous data and extended the study to new data from newborns at birth and a week later, and from healthy young women. For the young women, we found modular organization of both IgG and IgM isotypes into antigen cliquessubgroups of highly correlated antigen reactivities. In contrast, the newborns were found to share a universal congenital IgM profile with no modular organization. Moreover, the IgG autoantibodies of the newborns manifested buds of the mothers' antigen cliques, but they were noticeably less structured. These findings suggest that the natural autoantibody repertoire of humans shows relatively little organization at birth, but, by young adulthood, it becomes sorted out into a modular organization of subgroups (cliques) of correlated antigens. These features revealed by antigen microarrays can be used to define personal states of autoantibody organizational motifs.immune development ͉ immune holography ͉ immune network ͉ network cliques

We present a multimode laser-linewidth theory for arbitrary cavity structures and geometries that contains nearly all previously known effects and also finds new nonlinear and multimode corrections, e.g., a correction to the α factor due to openness of the cavity and a multimode Schawlow-Townes relation (each linewidth is proportional to a sum of inverse powers of all lasing modes). Our theory produces a quantitatively accurate formula for the linewidth, with no free parameters, including the full spatial degrees of freedom of the system. Starting with the Maxwell-Bloch equations, we handle quantum and thermal noise by introducing random currents whose correlations are given by the fluctuation-dissipation theorem. We derive coupled-mode equations for the lasing-mode amplitudes and obtain a formula for the linewidths in terms of simple integrals over the steady-state lasing modes.

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 8-11 , 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 17-23 , 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 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 8-11 . 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 show the existence of EPs in a photonic cryst...

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