Time Refraction and Time Reflection above Critical Angle for Total Internal Reflection.

We study the time reflection and time refraction of waves caused by a spatial interface with a medium undergoing a sudden temporal change in permittivity. We show that monochromatic waves are transformed into a pulse by the permittivity change, and that time reflection is enhanced at the vicinity of the critical angle for total internal reflection. In this regime, we find that the evanescent field is transformed into a propagating pulse by the sudden change in permittivity. These effects display enhancement of the time reflection and high sensitivity near the critical angle, paving the way to experiments on time reflection and photonic time crystals at optical frequencies.

Multi-dimensional band structure spectroscopy in the synthetic frequency dimension.

The concept of synthetic dimensions in photonics provides a versatile platform in exploring multi-dimensional physics. Many of these physics are characterized by band structures in more than one dimensions. Existing efforts on band structure measurements in the photonic synthetic frequency dimension however are limited to either one-dimensional Brillouin zones or one-dimensional subsets of multi-dimensional Brillouin zones. Here we theoretically propose and experimentally demonstrate a method to fully measure multi-dimensional band structures in the synthetic frequency dimension. We use a single photonic resonator under dynamical modulation to create a multi-dimensional synthetic frequency lattice. We show that the band structure of such a lattice over the entire multi-dimensional Brillouin zone can be measured by introducing a gauge potential into the lattice Hamiltonian. Using this method, we perform experimental measurements of two-dimensional band structures of a Hermitian and a non-Hermitian Hamiltonian. The measurements reveal some of the general properties of point-gap topology of the non-Hermitian Hamiltonian in more than one dimensions. Our results demonstrate experimental capabilities to fully characterize high-dimensional physical phenomena in the photonic synthetic frequency dimension.

Photonic time-crystals - fundamental concepts [Invited].

Photonic Time-Crystals (PTCs) are materials in which the refractive index varies periodically and abruptly in time. This medium exhibits unusual properties such as momentum bands separated by gaps within which waves can be amplified exponentially, extracting energy from the modulation. This article provides a brief review on the concepts underlying PTCs, formulates the vision and discusses the challenges.

Photonic time crystals: a materials perspective [Invited].

Recent advances in ultrafast, large-modulation photonic materials have opened the door to many new areas of research. One specific example is the exciting prospect of photonic time crystals. In this perspective, we outline the most recent material advances that are promising candidates for photonic time crystals. We discuss their merit in terms of modulation speed and depth. We also investigate the challenges yet to be faced and provide our estimation on possible roads to success.

Photonic topological insulator induced by a dislocation in three dimensions.

The hallmark of topological insulators (TIs) is the scatter-free propagation of waves in topologically protected edge channels1. This transport is strictly chiral on the outer edge of the medium and therefore capable of bypassing sharp corners and imperfections, even in the presence of substantial disorder. In photonics, two-dimensional (2D) topological edge states have been demonstrated on several different platforms2-4 and are emerging as a promising tool for robust lasers5, quantum devices6-8 and other applications. More recently, 3D TIs were demonstrated in microwaves9 and  acoustic waves10-13, where the topological protection in the latter  is induced by dislocations. However, at optical frequencies, 3D photonic TIs have so far remained out of experimental reach. Here we demonstrate a photonic TI with protected topological surface states in three dimensions. The topological protection is enabled by a screw dislocation. For this purpose, we use the concept of synthetic dimensions14-17 in a 2D photonic waveguide array18 by introducing a further modal dimension to transform the system into a 3D topological system. The lattice dislocation endows the system with edge states propagating along 3D trajectories, with topological protection akin to strong photonic TIs19,20. Our work paves the way for utilizing 3D topology in photonic science and technology.

Amplified emission and lasing in photonic time crystals.

Photonic time crystals (PTCs), materials with a dielectric permittivity that is modulated periodically in time, offer new concepts in light manipulation. We study theoretically the emission of light from a radiation source placed inside a PTC and find that radiation corresponding to the momentum bandgap is exponentially amplified, whether initiated by a macroscopic source, an atom, or vacuum fluctuations, drawing the amplification energy from the modulation. The radiation linewidth becomes narrower with time, eventually becoming monochromatic in the middle of the bandgap, which enables us to propose the concept of nonresonant tunable PTC laser. Finally, we find that the spontaneous decay rate of an atom embedded in a PTC vanishes at the band edge because of the low density of photonic states.

Light emission by free electrons in photonic time-crystals.

Photonic time-crystals (PTCs) are spatially homogeneous media whose electromagnetic susceptibility varies periodically in time, causing temporal reflections and refractions for any wave propagating within the medium. The time-reflected and time-refracted waves interfere, giving rise to Floquet modes with momentum bands separated by momentum gaps (rather than energy bands and energy gaps, as in photonic crystals). Here, we present a study on the emission of radiation by free electrons in PTCs. We show that a free electron moving in a PTC spontaneously emits radiation, and when associated with momentum-gap modes, the electron emission process is exponentially amplified by the modulation of the refractive index. Moreover, under strong electron-photon coupling, the quantum formulation reveals that the spontaneous emission into the PTC bandgap experiences destructive quantum interference with the emission of the electron into the PTC band modes, leading to suppression of the interdependent emission. Free-electron physics in PTCs offers a platform for studying a plethora of exciting phenomena, such as radiating dipoles moving at relativistic speeds and highly efficient quantum interactions with free electrons.

Topological insulator vertical-cavity laser array.

Topological insulator lasers are arrays of semiconductor lasers that exploit fundamental features of topology to force all emitters to act as a single coherent laser. In this study, we demonstrate a topological insulator vertical-cavity surface-emitting laser (VCSEL) array. Each VCSEL emits vertically, but the in-plane coupling between emitters in the topological-crystalline platform facilitates coherent emission of the whole array. Our topological VCSEL array emits at a single frequency and displays interference, highlighting that the emitters are mutually coherent. Our experiments exemplify the power of topological transport of light: The light spends most of its time oscillating vertically, but the small in-plane coupling is sufficient to force the array of individual emitters to act as a single laser.

Synthetic-Space Photonic Topological Insulators Utilizing Dynamically Invariant Structure.

Synthetic-space topological insulators are topological systems with at least one spatial dimension replaced by a periodic arrangement of modes, in the form of a ladder of energy levels, cavity modes, or some other sequence of modes. Such systems can significantly enrich the physics of topological insulators, in facilitating higher dimensions, nonlocal coupling, and more. Thus far, all synthetic-space topological insulators relied on active modulation to facilitate transport in the synthetic dimensions. Here, we propose dynamically invariant synthetic-space photonic topological insulators: a two-dimensional evolution-invariant photonic structure exhibiting topological properties in synthetic dimensions. This nonmagnetic structure is static, lacking any kind of modulation in the evolution coordinate, yet it displays an effective magnetic field in synthetic space, characterized by a Chern number of one. We study the evolution of topological states along the edge, and on the interface between two such structures with opposite synthetic-space chirality, and demonstrate their robust unidirectional propagation in the presence of defects and disorder. Such topological structures can be realized in photonics and cold atoms and provide a fundamentally new mechanism for topological insulators.

Disordered Photonic Time Crystals.

We study the propagation of electromagnetic waves in disordered photonic time crystals: spatially homogenous media whose refractive index changes randomly in time. We find that the group velocity of a pulse propagating in such media decreases exponentially, eventually coming to a complete stop, while experiencing exponential growth in intensity. These effects greatly depend on the Floquet band structure of the photonic time crystal, with the strongest sensitivity to disorder occurring in superluminal modes. Finally, we analyze the ensemble statistics and find them to coincide with those of Anderson localization, exhibiting single parameter scaling.

Identifying Topological Phase Transitions in Experiments Using Manifold Learning.

We demonstrate the identification of topological phase transitions from experimental data using diffusion maps: a nonlocal unsupervised machine learning method. We analyze experimental data from an optical system undergoing a topological phase transition and demonstrate the ability of this approach to identify topological phase transitions even when the data originates from a small part of the system, and does not even include edge states.

Photonic Floquet topological insulators in a fractal lattice.

We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics.

Photonic topological insulator in synthetic dimensions.

Topological phases enable protected transport along the edges of materials, offering immunity against scattering from disorder and imperfections. These phases have been demonstrated for electronic systems, electromagnetic waves1-5, cold atoms6,7, acoustics8 and even mechanics9, and their potential applications include spintronics, quantum computing and highly efficient lasers10-12. Typically, the model describing topological insulators is a spatial lattice in two or three dimensions. However, topological edge states have also been observed in a lattice with one spatial dimension and one synthetic dimension (corresponding to the spin modes of an ultracold atom13-15), and atomic modes have been used as synthetic dimensions to demonstrate lattice models and physical phenomena that are not accessible to experiments in spatial lattices13,16,17. In photonics, topological lattices with synthetic dimensions have been proposed for the study of physical phenomena in high dimensions and interacting photons18-22, but so far photonic topological insulators in synthetic dimensions have not been observed. Here we demonstrate experimentally a photonic topological insulator in synthetic dimensions. We fabricate a photonic lattice in which photons are subjected to an effective magnetic field in a space with one spatial dimension and one synthetic modal dimension. Our scheme supports topological edge states in this spatial-modal lattice, resulting in a robust topological state that extends over the bulk of a two-dimensional real-space lattice. Our system can be used to increase the dimensionality of a photonic lattice and induce long-range coupling by design, leading to lattice models that can be used to study unexplored physical phenomena.