Nature Communications (2016) 7

DOI: 10.1038/ncomms13731

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Weyl points, as a signature of 3D topological states, have been extensively studied in condensed matter systems. Recently, the physics of Weyl points has also been explored in electromagnetic structures such as photonic crystals and metamaterials. These structures typically have complex three-dimensional geometries, which limits the potential for exploring Weyl point physics in on-chip integrated systems. Here we show that Weyl point physics emerges in a system of two-dimensional arrays of resonators undergoing dynamic modulation of refractive index. In addition, the phase of modulation can be controlled to explore Weyl points under different symmetries. Furthermore, unlike static structures, in this system the non-trivial topology of the Weyl point manifests in terms of surface state arcs in the synthetic space that exhibit one-way frequency conversion. Our system therefore provides a versatile platform to explore and exploit Weyl point physics on chip. A Weyl point is a point degeneracy between two bands in a three-dimensional (3D) band structure, with linear dispersion in all three dimensions in its vicinity 1–5 . The simplest Hamiltonian in the wavevector space (k-space) that supports a Weyl point is H ¼ v x k x s x þ v y k y s y þ v z k z s z . As s x,y,z together with the identity matrix form a complete basis for 2 Â 2 Hermitian matrices, any perturbation on H that preserves the translational symmetry can be written as a linear superposition of these four matrices. Thus, any small perturbation in k-space can only shift the Weyl point without destroying the degeneracy and opening a gap 5,6 . Weyl points are 3D topological states: they are monopoles of Berry curvature in the wavevector space 7 . Any closed two-dimensional (2D) surface surrounding the Weyl point has a unit Chern number. This implies the existence of topological surface states, in the form of a Fermi arc connecting two Weyl points of opposite charges for a finite system with its bulk described by H 8–10 . A Weyl point is a 3D object. Therefore, previous works on Weyl points in photonics 11–15 , plasmonics 16,17 and acoustics 18,19 have complex 3D geometries, which limits the potential for exploring Weyl point physics in on-chip integrated systems. To explore a Weyl point in a planar 2D geometry, one may use a synthetic dimension 20,21 to simulate the third spatial dimension. The notion of synthetic dimension was previously proposed for superconducting qubits 22 , cold atoms 23 and optics 24 based on the idea of increasing local mode connectivity. One can also form the synthetic dimension using the modes of a ring resonator at different frequencies 25–27 . The size of the synthetic dimension, which corresponds to the number of modes in each individual ring, can be rather large without increasing the system complexity. Here we create a synthetic 3D space by dynamically modulating a 2D array of on-chip ring resonators. Each resonator supports a set of discrete modes equally spaced in resonant frequency. These discrete modes thus form a periodic lattice in the third, synthetic frequency dimension. The two spatial dimensions and one synthetic frequency dimension together form a 3D space. Dynamic modulation of the refractive index leads to effective coupling of modes in the synthetic dimension 25–29 . We show that proper design of the modulation leads to Weyl points in the synthetic space. Our proposed approach is specifically designed for implementation using an existing on-chip integrated photonic platform. Compared with the complex 3D electromagnetic or acoustic structure previously used to demonstrate Weyl point physics 11–19 , our approach provides a far more flexible platform to explore a wide range of phase space. For example, by changing the dynamic modulation phases, the same device can be tuned to exhibit line nodes and Weyl points under inversion or/and time-reversal symmetry breaking. This system also provides a novel manifestation of Weyl point physics in terms of a surface state in the synthetic space that exhibits one-way frequency conversion. More generally, in the context of topological photonics 25–39 , this work points to the significant richness in using dynamic refractive index modulation to achieve novel topological effects. Results Model Hamiltonian system and Weyl points. Our exemplary system consists of a 2D honeycomb array of identical ring resonators as shown in Fig. 1a. In the vicinity of a resonant frequency o 0 , each ring resonator supports a discrete set of resonant modes at frequencies described by [b(o m) À b(o 0)] Â L ¼ 2mp (m ¼ 0, ±1, ±2...), where L is the circumference of the ring and b is the effective wavevector. In the absence of group velocity dispersion, these modes are equally spaced in frequency by the free spectral range O ¼ 2p

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Lin, Q., Xiao, M., Yuan, L., & Fan, S. (2016). Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension. *Nature Communications*, *7*. https://doi.org/10.1038/ncomms13731

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