Georgios G. Pyrialakos, Julius Beck, Matthias Heinrich, Lukas J. Maczewsky, Nikolaos V. Kantartzis, Mercedeh Khajavikhan, Alexander Szameit, Demetrios N. Christodoulides
Topological theories have established a new set of rules that govern the transport properties in a wide variety of wave-mechanical settings. In a marked departure from the established approaches that induce Floquet topological phases by specifically tailored discrete coupling protocols or helical lattice motions, we introduce a new class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to unlock new topological features in the system. In exploring a 'chain-driven' generalization of the archetypical Floquet honeycomb lattice, we identify a rich phase structure that can host multiple non-trivial topological phases associated simultaneously with both Chern-type and anomalous chiral states. Experiments carried out in photonic waveguide lattices reveal a unique and strongly confined helical edge state that, owing to its origin in bulk flat bands, can be set into motion in a topologically protected fashion, or halted at will, without compromising its adherence to individual lattice sites.
We experimentally realized Floquet topological photonic insulators using a square lattice of direct-coupled octagonal resonators. Unlike previously reported topological insulator systems based on microring lattices, the nontrivial topological behaviors of our system arise directly from the periodic evolution of light around each octagon to emulate a periodically-driven system. By exploiting asynchronism in the evanescent coupling between adjacent octagonal resonators, we could achieve strong and asymmetric couplings in each unit cell, which are necessary for observing Anomalous Floquet Insulat…
Achieving topologically-protected robust transport in optical systems has recently been of great interest. Most topological photonic structures can be understood by solving the eigenvalue problem of Maxwell's equations for a static linear system. Here, we extend topological phases into dynamically driven nonlinear systems and achieve a Floquet Chern insulator of light in nonlinear photonic crystals (PhCs). Specifically, we start by presenting the Floquet eigenvalue problem in driven two-dimensional PhCs and show it is necessarily non-Hermitian. We then define topological invariants associated …
Commonly, a two-dimensional topological insulator is characterized by non-zero Chern numbers associated with its band structure. In our work, we present the experimental demonstration of an anomalous topological insulator, for which the Chern numbers are always zero, and the winding number is the appropriate quantity describing the topological character of the system. We probe the robustness of the chiral edge states in such anomalous topological insulators and analyze the phase transition between topological and trivial phase.
We show the presence of Floquet-Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice. The Weyl points in the three-dimensional Brillouin zone and Fermi-arc surface states are clearly demonstrated in the quasienergy spectrum of the system in the Weyl phase. In addition, chiral surface states coexist in this phase. The Floquet-topological-insulator phase is characterized by the winding number of two in the reflection matrices of the semi-infinite system and resulting two gapless surface states in the quasienergy g ap of the bulk. The phase diagram of th…
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 the trivial-to-topological transition. The quasienergy 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 …
We demonstrate the photonic Floquet topological insulator (PFTI) in an atomic vapor with nonlinear susceptibilities. The interference of three coupling fields splits the energy levels periodically to form a periodic refractive index structure with honeycomb symmetry that can be adjusted by the choice of frequency detunings and intensities of the coupling fields, which all affect the appearance of Dirac cones in the momentum space. When the honeycomb lattice sites are helically ordered along the propagation direction, we obtain a PFTI in the atomic vapor in which an obliquely incident beam move…
We consider a topological Floquet insulator consisting of two honeycomb arrays of identical waveguides having opposite helicities. The interface between the arrays supports two distinct topological edge states, which can be resonantly coupled by additional weak longitudinal refractive index modulation with a period larger than the helix period. In the presence of Kerr nonlinearity, such coupled edge states enable topological Bragg solitons. Theory and examples of such solitons are presented.
One dimensional laser-written modulated photonic lattices are known to be particularly suitable for diffraction management purposes. Here, we address the connection between discrete non-diffracting states and topological properties in such devices through the experimental observation and identification of three classes of non-diffracting state. The first one corresponds to topologically protected edge states, recently predicted in Floquet topological insulators, while the second and third are both bulk modes. One of them testifies of a topological transition, although presenting topological fe…
A recently-proposed class of photonic topological insulators is shown to map onto Chalker-Coddington-type networks, which were originally formulated to study disordered quantum Hall systems. Such network models are equivalent to the Floquet states of periodically-driven lattices. We show that they can exhibit topologically protected edge states even if all bands have zero Chern number, which is a characteristic property of Floquet bandstructures. These edge states can be counted by an adiabatic pumping invariant based on the winding number of the coefficient of reflection from one edge of the …
Edge modes in topological insulators are known to be robust against defects. We investigate if this also holds true when the defect is not static, but varies in time. We study the influence of defects with time-dependent coupling on the robustness of the transport along the edge in a Floquet system of helically curved waveguides. Waveguide arrays are fabricated via direct laser writing in a negative tone photoresist. We find that single dynamic defects do not destroy the chiral edge current, even when the temporal modulation is strong. Quantitative numerical simulation of the intensity in the …
We propose a class of photonic Floquet topological insulators based on staggered helical lattices and an efficient numerical method for calculating their Floquet bandstructure. The lattices support anomalous Floquet topological insulator phases with vanishing Chern number and tunable topological transitions. At the critical point of the topological transition, the bandstructure hosts a single unpaired Dirac cone, which yields a variety of unusual transport effects: a discrete analogue of conical diffraction, weak antilocalization not limited by intervalley scattering, and suppression of Anders…
Observation of excitonic Floquet states in a one-dimensional organic Mott insulator using mid-infrared pump near-infrared probe reflection spectroscopy
Photonic and acoustic topological insulators exhibiting one-way transportation that is robust against defects and impurities are typically realized in coupled arrays of two-dimensional ring resonators. These systems have produced a series of applications, including optical isolators, delay lines, and lasers. However, the structures are complicated because an additional coupler ring between neighboring rings is needed to construct photonic pseudo-spin. In this work, a photonic analogue of the quantum spin Hall effect is proposed and experimentally demonstrated in an anomalous Floquet topologica…
We introduce topological vector edge solitons in a Floquet insulator, consisting of two honeycomb arrays of helical waveguides with opposite directions of rotation in a focusing nonlinear optical medium. Zigzag edges of two arrays placed in contact create a zigzag-zigzag interface between two structures with different topology. A characteristic feature of such a photonic insulator is that, in the linear limit, it simultaneously supports two topologically protected chiral edge states at the interface between the two arrays. In the presence of nonlinearity, either bright or dark scalar Floquet e…
We introduce a class of topological lasers based on the photonic Floquet topological insulator concept. The proposed system is realized as a truncated array of the lasing helical waveguides, where the pseudo-magnetic field arises due to twisting of the waveguides along the propagation direction that breaks the time-reversal symmetry and opens up a topological gap. When sufficient gain is provided in the edge channels of the array then the system lases into the topological edge states. Topological lasing is stable only in certain intervals of the Bloch momenta, that ensure a dynamic, but stable…
The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically-driven waveguides, which serves as a paradigmatic Floquet topological insulator. Here it is shown that discrete topologically protected edge modes do not suffer from the typical slowdown associated with the Peierls-Nabarro barrier. Instead, as a result of their topological nature, the modes al…
In this work, we reported a ubiquitous presence of topological Floquet time crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry (DTS) breaking in our model require necessarily coexistence of anomalous topological invariants (0 modes and $π$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between zero and pi modes can produce the period-doubling (2T, two cycles of the drive…
We describe topological edge solitons in a continuous dislocated Lieb array of helical waveguides. The linear Floquet spectrum of this structure is characterized by the presence of two topological gaps with edge states residing in them. A focusing nonlinearity enables families of topological edge solitons bifurcating from the linear edge states. Such solitons are localized both along and across the edge of the array. Due to the non-monotonic dependence of the propagation constant of the edge states on the Bloch momentum, one can construct topological edge solitons that either propagate in diff…
Topological edge states form at the edges of periodic materials with specific degeneracies in their modal spectra, such as Dirac points, under the action of effects breaking certain symmetries of the system. In particular, in Floquet topological insulators unidirectional edge states appear upon breakup of the effective time-reversal symmetry due to dynamical modulations of the underlying lattice potential. However, such states are usually reported for certain simple lattice terminations, for example, at zigzag or bearded edges in honeycomb lattices. Here we show that unconventional topological…
Spatially uniform excitations can induce Floquet topological bandstructures within insulators which have equal characteristics to those of topological insulators. Going beyond we demonstrate in this article the evolution of Floquet topological quantum states for electromagnetically driven semiconductor bulk matter. We show the direct physical impact of the mathematical precision of the Floquet-Keldysh theory when we solve the driven system of a generalized Hubbard model with our framework of dynamical mean field theory (DMFT) in the non-equilibrium. We explain the physical consequences of the …
Topological insulators are unique physical structures that are insulators in their bulk, but support currents at their edges which can be unidirectional and topologically protected from scattering on disorder and inhomogeneities. Photonic topological insulators can be crafted in materials that exhibit a strong nonlinear response, thus opening the door to the exploration of the interplay between nonlinearity and topological effects. Among the fascinating new phenomena arising from this interplay is the formation of topological edge solitons -- hybrid asymmetric states localized across and along…
Optical control of electrons in a Floquet topological insulator
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；University of Central Florida（中佛罗里达大学）；Technion - Israel Institute of Technology（以色列理工学院）；Max Planck Institute for the Structure and Dynamics of Matter（马普物质结构与动力学研究所）；Ludwig-Maximilians-Universität, München（慕尼黑路德维希-马克西米利安大学）
Topological phases in quantum and classical systems have been of significant recent interest due to their fascinating physical properties. While a range of different mechanisms to induce topological order have been introduced, a quest for the most viable solution for practical systems is still open. Floquet topological insulator represent one of possible venues to engineer topological phases, yet they have been so far largely restricted to temporal modulation of Hermitian potentials. On the other hand, in many physical systems, including acoustic and optical systems, modulating loss or gain ca…
A salient feature of solid-state topological materials in two dimensions is the presence of conducting electronic edge states that are insensitive to scattering by disorder. Such unidirectional edge states have been explored in many experimental settings beyond solid-state electronic systems, including in photonic devices, mechanical and acoustic structures, and others. It is of great interest to understand how topological states behave in the presence of inter-particle interactions and nonlinearity. Here we experimentally demonstrate unidirectional soliton-like nonlinear states on the edge of…
The topological insulator is a fundamentally new phase of matter, with the striking property that the conduction of electrons occurs only on its surface, not within the bulk, and that conduction is topologically protected. Topological protection, the total lack of scattering of electron waves by disorder, is perhaps the most fascinating and technologically important aspect of this material: it provides robustness that is otherwise known only for superconductors. However, unlike superconductivity and the quantum Hall effect, which necessitate low temperatures or magnetic fields, the immunity to…
Akin to zero-mode anomalies, such as the chiral anomaly of edge states in quantum Hall effect, in this work, a pi-mode anomaly is proposed in a 1+1 dimensional periodically-driven topological/normal insulator (TI/NI) heterostructure. Usually, when coupling in a background gauge field, the zero modes on domain walls would provide an anomalous current term that is eventually canceled by additional boundary contributions from the topological bulk, via the Callan-Harvey mechanism. This anomaly cancellation associated with the generalization of bulk-boundary correspondence is called anomaly inflow.…
