Superluminal k-Gap Solitons in Nonlinear Photonic Time Crystals.

We propose superluminal solitons residing in the momentum gap (k gap) of nonlinear photonic time crystals. These gap solitons are structured as plane waves in space while being periodically self-reconstructing wave packets in time. The solitons emerge from modes with infinite group velocity causing superluminal evolution, which is the opposite of the stationary nature of the analogous Bragg gap soliton residing at the edge of an energy gap (or a spatial gap) with zero group velocity. We explore the faster-than-light pulsed propagation of these k-gap solitons in view of Einstein's causality by introducing a truncated input seed as a precursor of a signal velocity forerunner, and find that the superluminal propagation of k-gap solitons does not break causality.

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.

Observation of Anderson localization beyond the spectrum of the disorder.

Anderson localization predicts that transport in one-dimensional uncorrelated disordered systems comes to a complete halt, experiencing no transport whatsoever. However, in reality, a disordered physical system is always correlated because it must have a finite spectrum. Common wisdom in the field states that localization is dominant only for wave packets whose spectral extent resides within the region of the wave number span of the disorder. Here, we show experimentally that Anderson localization can occur and even be dominant for wave packets residing entirely outside the spectral extent of the disorder. We study the evolution of wave packets in synthetic photonic lattices containing bandwidth-limited (correlated) disorder and observe strong localization for wave packets centered at twice the mean wave number of the disorder spectral extent and at low wave numbers, both far beyond the spectrum of the disorder. Our results shed light on fundamental aspects of disordered systems and offer avenues for using spectrally shaped disorder for controlling transport.

Fractal photonic topological insulators.

Topological insulators constitute a newly characterized state of matter that contains scatter-free edge states surrounding an insulating bulk. Conventional wisdom regards the insulating bulk as essential, because the invariants that describe the topological properties of the system are defined therein. Here, we study fractal topological insulators based on exact fractals composed exclusively of edge sites. We present experimental proof that, despite the lack of bulk bands, photonic lattices of helical waveguides support topologically protected chiral edge states. We show that light transport in our topological fractal system features increased velocities compared with the corresponding honeycomb lattice. By going beyond the confines of the bulk-boundary correspondence, our findings pave the way toward an expanded perception of topological insulators and open a new chapter of topological fractals.

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.

Imprinting the quantum statistics of photons on free electrons.

The interaction between free electrons and light stands at the base of both classical and quantum physics, with applications in free-electron acceleration, radiation sources, and electron microscopy. Yet to this day, all experiments involving free-electron­light interactions are fully explained by describing the light as a classical wave. We observed quantum statistics effects of photons on free-electron­light interactions. We demonstrate interactions that pass continuously from Poissonian to super-Poissonian and up to thermal statistics, revealing a transition from quantum walk to classical random walk on the free-electron energy ladder. The electron walker serves as the probe in nondestructive quantum detection, measuring the second-order photon-correlation g(2)(0) and higher-orders g(n)(0). Unlike conventional quantum-optical detectors, the electron can perform both quantum weak measurements and projective measurements by evolving into an entangled joint state with the photons. These findings inspire hitherto inaccessible concepts in quantum optics, including free-electron­based ultrafast quantum tomography of light.

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.

Generalized laws of refraction and reflection at interfaces between different photonic artificial gauge fields.

Artificial gauge fields the control over the dynamics of uncharged particles by engineering the potential landscape such that the particles behave as if effective external fields are acting on them. Recent years have witnessed a growing interest in artificial gauge fields generated either by the geometry or by time-dependent modulation, as they have been enablers of topological phenomena and synthetic dimensions in many physical settings, e.g., photonics, cold atoms, and acoustic waves. Here, we formulate and experimentally demonstrate the generalized laws of refraction and reflection at an interface between two regions with different artificial gauge fields. We use the symmetries in the system to obtain the generalized Snell law for such a gauge interface and solve for reflection and transmission. We identify total internal reflection (TIR) and complete transmission and demonstrate the concept in experiments. In addition, we calculate the artificial magnetic flux at the interface of two regions with different artificial gauge fields and present a method to concatenate several gauge interfaces. As an example, we propose a scheme to make a gauge imaging system-a device that can reconstruct (image) the shape of an arbitrary wavepacket launched from a certain position to a predesigned location.

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.

Observation of branched flow of light.

When waves propagate through a weak disordered potential with correlation length larger than the wavelength, they form channels (branches) of enhanced intensity that keep dividing as the waves propagate1. This fundamental wave phenomenon is known as branched flow. It was first observed for electrons1-6 and for microwave cavities7,8, and it is generally expected for waves with vastly different wavelengths, for example, branched flow has been suggested as a focusing mechanism for ocean waves9-11, and was suggested to occur also in sound waves12 and ultrarelativistic electrons in graphene13. Branched flow may act as a trigger for the formation of extreme nonlinear events14-17 and as a channel through which energy is transmitted in a scattering medium18. Here we present the experimental observation of the branched flow of light. We show that, as light propagates inside a thin soap membrane, smooth thickness variations in the film act as a correlated disordered potential, focusing the light into filaments that display the features of branched flow: scaling of the distance to the first branching point and the probability distribution of the intensity. We find that, counterintuitively, despite the random variations in the medium and the linear nature of the effect, the filaments remain collimated throughout their paths. Bringing branched flow to the field of optics, with its full arsenal of tools, opens the door to the investigation of a plethora of new ideas such as branched flow in nonlinear media, in curved space or in active systems with gain. Furthermore, the labile nature of soap films leads to a regime in which the branched flow of light interacts and affects the underlying disorder through radiation pressure and gradient force.

Deep learning reconstruction of ultrashort pulses from 2D spatial intensity patterns recorded by an all-in-line system in a single-shot.

We propose a simple all-in-line single-shot scheme for diagnostics of ultrashort laser pulses, consisting of a multi-mode fiber, a nonlinear crystal and a camera. The system records a 2D spatial intensity pattern, from which the pulse shape (amplitude and phase) are recovered, through a fast Deep Learning algorithm. We explore this scheme in simulations and demonstrate the recovery of ultrashort pulses, robustness to noise in measurements and to inaccuracies in the parameters of the system components. Our technique mitigates the need for commonly used iterative optimization reconstruction methods, which are usually slow and hampered by the presence of noise. These features make our concept system advantageous for real time probing of ultrafast processes and noisy conditions. Moreover, this work exemplifies that using deep learning we can unlock new types of systems for pulse recovery.

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.

Self-Induced Diffusion in Disordered Nonlinear Photonic Media.

We find that waves propagating in a 1D medium that is homogeneous in its linear properties but spatially disordered in its nonlinear coefficients undergo diffusive transport, instead of being Anderson localized as always occurs for linear disordered media. Specifically, electromagnetic waves in a multilayer structure with random nonlinear coefficients exhibit diffusion with features fundamentally different from the traditional diffusion in linear noninteracting systems. This unique transport, which stems from the nonlinear interaction between the waves and the disordered medium, displays anomalous statistical behavior where the fields in multiple different realizations converge to the same intensity value as they penetrate deeper into the medium.

Topological protection of biphoton states.

The robust generation and propagation of multiphoton quantum states are crucial for applications in quantum information, computing, and communications. Although photons are intrinsically well isolated from the thermal environment, scaling to large quantum optical devices is still limited by scattering loss and other errors arising from random fabrication imperfections. The recent discoveries regarding topological phases have introduced avenues to construct quantum systems that are protected against scattering and imperfections. We experimentally demonstrate topological protection of biphoton states, the building block for quantum information systems. We provide clear evidence of the robustness of the spatial features and the propagation constant of biphoton states generated within a nanophotonics lattice with nontrivial topology and propose a concrete path to build robust entangled states for quantum gates.