Metamaterials are artificial optical media composed of sub-wavelength metallic and dielectric building blocks that feature optical phenomena not present in naturally occurring materials. Although they can serve as the basis for unique optical devices that mould the flow of light in unconventional ways, three-dimensional metamaterials suffer from extreme propagation losses. Two-dimensional metamaterials (metasurfaces) such as hyperbolic metasurfaces for propagating surface plasmon polaritons have the potential to alleviate this problem. Because the surface plasmon polaritons are guided at a metal-dielectric interface (rather than passing through metallic components), these hyperbolic metasurfaces have been predicted to suffer much lower propagation loss while still exhibiting optical phenomena akin to those in three-dimensional metamaterials. Moreover, because of their planar nature, these devices enable the construction of integrated metamaterial circuits as well as easy coupling with other optoelectronic elements. Here we report the experimental realization of a visible-frequency hyperbolic metasurface using single-crystal silver nanostructures defined by lithographic and etching techniques. The resulting devices display the characteristic properties of metamaterials, such as negative refraction and diffraction-free propagation, with device performance greatly exceeding those of previous demonstrations. Moreover, hyperbolic metasurfaces exhibit strong, dispersion-dependent spin-orbit coupling, enabling polarization- and wavelength-dependent routeing of surface plasmon polaritons and two-dimensional chiral optical components. These results open the door to realizing integrated optical meta-circuits, with wide-ranging applications in areas from imaging and sensing to quantum optics and quantum information science.

We demonstrate that two-dimensional atomic emitter arrays with subwavelength spacing constitute topologically protected quantum optical systems where the photon propagation is robust against large imperfections while losses associated with free space emission are strongly suppressed. Breaking timereversal symmetry with a magnetic field results in gapped photonic bands with nontrivial Chern numbers and topologically protected, long-lived edge states. Due to the inherent nonlinearity of constituent emitters, such systems provide a platform for exploring quantum optical analogs of interacting topological systems. DOI: 10.1103/PhysRevLett.119.023603 Charged particles in two-dimensional systems exhibit exotic macroscopic behavior in the presence of magnetic fields and interactions. These include the integer [1], fractional [2], and spin [3] quantum Hall effects. Such systems support topologically protected edge states [4,5] that are robust against defects and disorder. There is a significant interest in realizing topologically protected photonic systems. Photonic analogs of quantum Hall behavior have been studied in gyromagnetic photonic crystals [6][7][8][9][10][11], helical waveguides [12], two-dimensional lattices of optical resonators [13][14][15] and in polaritons coupled to optical cavities [16]. An outstanding challenge is to realize optical systems which are robust not only to some specific backscattering processes but to all loss processes, including scattering into unconfined modes and spontaneous emission. Another challenge is to extend these effects into a nonlinear quantum domain with strong interactions between individual excitations. These considerations motivate the search for new approaches to topological photonics.In this Letter, we introduce and analyze a novel platform for engineering topological states in the optical domain. It is based on atomic or atomlike quantum optical systems [17], where time-reversal symmetry can be broken by applying magnetic fields and the constituent emitters are inherently nonlinear. Specifically, we focus on optical excitations in a two-dimensional honeycomb array of closely spaced emitters. We show that such systems maintain topologically protected confined optical modes that are immune to large imperfections as well as to the most common loss processes such as scattering into free-space modes. Such modes can be used to control individual atom emission, and to create quantum nonlinearity at a single photon level.The key idea is illustrated in Fig. 1(a). We envision an array with interatomic spacing a and quantization axisẑ

Two-dimensional atomic arrays exhibit a number of intriguing quantum optical phenomena, including subradiance, nearly perfect reflection of radiation, and long-lived topological edge states. Studies of emission and scattering of photons in such lattices require complete treatment of the radiation pattern from individual atoms, including long-range interactions. We describe a systematic approach to perform the calculations of collective energy shifts and decay rates in the presence of such long-range interactions for arbitrary two-dimensional atomic lattices. As applications of our method, we investigate the topological properties of atomic lattices both in free space and near plasmonic surfaces.