Topological triple phase transition in non-Hermitian Floquet quasicrystals.

Phase transitions connect different states of matter and are often concomitant with the spontaneous breaking of symmetries. An important category of phase transitions is mobility transitions, among which is the well known Anderson localization1, where increasing the randomness induces a metal-insulator transition. The introduction of topology in condensed-matter physics2-4 lead to the discovery of topological phase transitions and materials as topological insulators5. Phase transitions in the symmetry of non-Hermitian systems describe the transition to on-average conserved energy6 and new topological phases7-9. Bulk conductivity, topology and non-Hermitian symmetry breaking seemingly emerge from different physics and, thus, may appear as separable phenomena. However, in non-Hermitian quasicrystals, such transitions can be mutually interlinked by forming a triple phase transition10. Here we report the experimental observation of a triple phase transition, where changing a single parameter simultaneously gives rise to a localization (metal-insulator), a topological and parity-time symmetry-breaking (energy) phase transition. The physics is manifested in a temporally driven (Floquet) dissipative quasicrystal. We implement our ideas via photonic quantum walks in coupled optical fibre loops11. Our study highlights the intertwinement of topology, symmetry breaking and mobility phase transitions in non-Hermitian quasicrystalline synthetic matter. Our results may be applied in phase-change devices, in which the bulk and edge transport and the energy or particle exchange with the environment can be predicted and controlled.

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.

Parity-time-symmetric photonic topological insulator.

Topological insulators are a concept that originally stems from condensed matter physics. As a corollary to their hallmark protected edge transport, the conventional understanding of such systems holds that they are intrinsically closed, that is, that they are assumed to be entirely isolated from the surrounding world. Here, by demonstrating a parity-time-symmetric topological insulator, we show that topological transport exists beyond these constraints. Implemented on a photonic platform, our non-Hermitian topological system harnesses the complex interplay between a discrete coupling protocol and judiciously placed losses and, as such, inherently constitutes an open system. Nevertheless, even though energy conservation is violated, our system exhibits an entirely real eigenvalue spectrum as well as chiral edge transport. Along these lines, this work enables the study of the dynamical properties of topological matter in open systems without the instability arising from complex spectra. Thus, it may inspire the development of compact active devices that harness topological features on-demand.

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.

Photonic topological Anderson insulators.

The hallmark property of two-dimensional topological insulators is robustness of quantized electronic transport of charge and energy against disorder in the underlying lattice1. That robustness arises from the fact that, in the topological bandgap, such transport can occur only along the edge states, which are immune to backscattering owing to topological protection. However, for sufficiently strong disorder, this bandgap closes and the system as a whole becomes topologically trivial: all states are localized and all transport vanishes in accordance with Anderson localization2,3. The recent suggestion4 that the reverse transition can occur was therefore surprising. In so-called topological Anderson insulators, it has been predicted4 that the emergence of protected edge states and quantized transport can be induced, rather than inhibited, by the addition of sufficient disorder to a topologically trivial insulator. Here we report the experimental demonstration of a photonic topological Anderson insulator. Our experiments are carried out in an array of helical evanescently coupled waveguides in a honeycomb geometry with detuned sublattices. Adding on-site disorder in the form of random variations in the refractive index of the waveguides drives the system from a trivial phase into a topological one. This manifestation of topological Anderson insulator physics shows experimentally that disorder can enhance transport rather than arrest it.

Bimorphic Floquet topological insulators.

Topological theories have established a unique 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 class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to reveal rich 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 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.

Fermionic time-reversal symmetry in a photonic topological insulator.

Much of the recent attention directed towards topological insulators is motivated by their hallmark feature of protected chiral edge states. In electronic (or fermionic) topological insulators, these states originate from time-reversal symmetry and allow carriers with opposite spin-polarization to propagate in opposite directions at the edge of an insulating bulk. By contrast, photonic (or bosonic) systems are generally assumed to be precluded from supporting edge states that are intrinsically protected by time-reversal symmetry. Here, we experimentally demonstrate counter-propagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is the design of a Floquet driving protocol that incorporates effective fermionic time-reversal symmetry, enabling the realization of the photonic version of an electronic topological insulator. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby offering alternative opportunities for photonics as well as acoustics, mechanical waves and cold atoms.

Topological Defect Engineering and PT Symmetry in Non-Hermitian Electrical Circuits.

We employ electric circuit networks to study topological states of matter in non-Hermitian systems enriched by parity-time symmetry PT and chiral symmetry anti-PT (APT). The topological structure manifests itself in the complex admittance bands which yields excellent measurability and signal to noise ratio. We analyze the impact of PT-symmetric gain and loss on localized edge and defect states in a non-Hermitian Su-Schrieffer-Heeger (SSH) circuit. We realize all three symmetry phases of the system, including the APT-symmetric regime that occurs at large gain and loss. We measure the admittance spectrum and eigenstates for arbitrary boundary conditions, which allows us to resolve not only topological edge states, but also a novel PT-symmetric Z_{2} invariant of the bulk. We discover the distinct properties of topological edge states and defect states in the phase diagram. In the regime that is not PT symmetric, the topological defect state disappears and only reemerges when APT symmetry is reached, while the topological edge states always prevail and only experience a shift in eigenvalue. Our findings unveil a future route for topological defect engineering and tuning in non-Hermitian systems of arbitrary dimension.

Edge solitons in Lieb topological Floquet insulator.

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 nonmonotonic dependence of the propagation constant of the edge states on the Bloch momentum, one can construct topological edge solitons that either propagate in different directions along the same boundary or do not move. This allows us to study collisions of edge solitons moving in opposite directions. Such solitons always interpenetrate each other without noticeable radiative losses; however, they exhibit a spatial shift that depends on the initial phase difference.

Experimental Realization of PT-Symmetric Flat Bands.

The capability to temporarily arrest the propagation of optical signals is one of the main challenges hampering the ever more widespread use of light in rapid long-distance transmission as well as all-optical on-chip signal processing or computations. To this end, flat-band structures are of particular interest, since their hallmark compact eigenstates not only allow for the localization of wave packets, but importantly, also protect their transverse profile from deterioration without the need for additional diffraction management. In this work, we experimentally demonstrate that, far from being a nuisance to be compensated, judiciously tailored loss distributions can, in fact, be the key ingredient in synthesizing such flat bands in non-Hermitian environments. We probe their emergence in the vicinity of an exceptional point and directly observe the associated compact localized modes that can be excited at arbitrary positions of the periodic lattice.

Photonic Floquet topological insulators.

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism-one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport-a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate (z) acts as 'time'. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

Tapering of femtosecond laser-written waveguides.

The vast development of integrated quantum photonic technology enables the implementation of compact and stable interferometric networks. In particular, laser-written waveguide structures allow for complex 3D circuits and polarization-encoded qubit manipulation. However, the main limitation in the scaling up of integrated quantum devices is the single-photon loss due to mode-profile mismatch when coupling to standard fibers or other optical platforms. Here we demonstrate tapered waveguide structures realized by an adapted femtosecond laser writing technique. We show that coupling to standard single-mode fibers can be enhanced up to 77% while keeping the fabrication effort negligible. This improvement provides an important step for processing multiphoton states on chip.

Real and virtual propagation dynamics of angular accelerating white light beams.

Accelerating waves have received significant attention of late, first in the optical domain and later in the form of electron matter waves, and have found numerous applications in non-linear optics, material processing, microscopy, particle manipulation and laser plasma interactions. Here we create angular accelerating light beams with a potentially unlimited acceleration rate. By employing wavelength independent digital holograms for the creation and propagation of white light beams, we are able to study the resulting propagation in real and virtual space. We find that dephasing occurs for real propagation and that this can be compensated for in a virtual propagation scheme when single plane dynamics are important. Our work offers new insights into the propagation dynamics of such beams and provides a versatile tool for further investigations into propagating structured light fields.

Measuring the Aharonov-Anandan phase in multiport photonic systems.

Beyond the adiabatic limit, the Aharonov-Anandan phase is a generalized description of Berry's phase. In this regime, systems with time-independent Hamiltonians may also acquire observable geometric phases. Here we report on a measurement of the Aharonov-Anandan phase in photonics. Different from previous optical experiments on geometric phases, the implementation is based on light modes confined in evanescently coupled waveguides rather than polarization-like systems, thereby physical models in more than two-dimensional Hilbert spaces are achievable. In a tailored photonic lattice, we realize time-independent quantum-driven harmonic oscillators initially prepared in the vacuum state and achieve a measurement of the Aharonov-Anandan phase via integrated interferometry.

Transport in Sawtooth photonic lattices.

We investigate, theoretically and experimentally, a photonic realization of a Sawtooth lattice. This special lattice exhibits two spectral bands, with one of them experiencing a complete collapse to a highly degenerate flat band for a special set of inter-site coupling constants. We report the observation of different transport regimes, including strong transport inhibition due to the appearance of the non-diffractive flat band. Moreover, we excite localized Shockley surface states residing in the gap between the two linear bands.

Universal Sign Control of Coupling in Tight-Binding Lattices.

We present a method of locally inverting the sign of the coupling term in tight-binding systems, by means of inserting a judiciously designed ancillary site and eigenmode matching of the resulting vertex triplet. Our technique can be universally applied to all lattice configurations, as long as the individual sites can be detuned. We experimentally verify this method in laser-written photonic lattices and confirm both the magnitude and the sign of the coupling by interferometric measurements. Based on these findings, we demonstrate how such universal sign-flipped coupling links can be embedded into extended lattice structures to impose a Z_{2}-gauge transformation. This opens a new avenue for investigations on topological effects arising from magnetic fields with aperiodic flux patterns or in disordered systems.

Observation of unconventional edge states in 'photonic graphene'.

Graphene, a two-dimensional honeycomb lattice of carbon atoms, has been attracting much interest in recent years. Electrons therein behave as massless relativistic particles, giving rise to strikingly unconventional phenomena. Graphene edge states are essential for understanding the electronic properties of this material. However, the coarse or impure nature of the graphene edges hampers the ability to directly probe the edge states. Perhaps the best example is given by the edge states on the bearded edge that have never been observed-because such an edge is unstable in graphene. Here, we use the optical equivalent of graphene-a photonic honeycomb lattice-to study the edge states and their properties. We directly image the edge states on both the zigzag and bearded edges of this photonic graphene, measure their dispersion properties, and most importantly, find a new type of edge state: one residing on the bearded edge that has never been predicted or observed. This edge state lies near the Van Hove singularity in the edge band structure and can be classified as a Tamm-like state lacking any surface defect. The mechanism underlying its formation may counterintuitively appear in other crystalline systems.

Goos-Hänchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces.

We present a theoretical study of the Goos-Hänchen and Imbert-Fedorov shifts for a fundamental Gaussian beam impinging on a surface coated with a single layer of graphene. We show that the graphene surface conductivity σ(ω) is responsible for the appearance of a giant and negative spatial Goos-Hänchen shift.

Effect of Orbital Angular Momentum on Nondiffracting Ultrashort Optical Pulses.

We introduce a new class of nondiffracting optical pulses possessing orbital angular momentum. By generalizing the X-wave solution of the Maxwell equation, we discover the coupling between angular momentum and the temporal degrees of freedom of ultrashort pulses. The spatial twist of propagation invariant light pulse turns out to be directly related to the number of optical cycles. Our results may trigger the development of novel multilevel classical and quantum transmission channels free of dispersion and diffraction. They may also find application in the manipulation of nanostructured objects by ultrashort pulses and for novel approaches to the spatiotemporal measurements in ultrafast photonics.