Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

Optical control of topological end states via soliton formation in a 1D lattice

Discrete spatial solitons are self-consistent solutions of the discrete nonlinear Schrödinger equation that maintain their shape during propagation. Here we show, using a pump-probe technique, that soliton formation can be used to optically induce and control a linear topological end state in the bulk of a Su-Schrieffer-Heeger lattice, using evanescently-coupled waveguide arrays. Specifically, we observe an abrupt nonlinearly-induced transition above a certain power threshold due to an inversion symmetry-breaking nonlinear bifurcation. Our results demonstrate all-optical active control of topological states.

Nonlinear tuning of multiple topological edge states in photovoltaic photonic lattices

The fusion of topology and nonlinearity has led to groundbreaking advancements in complex systems, paving the way for new discoveries and innovative device development. However, the interaction between topological states and self-defocusing nonlinearities in complex systems with multiple topological gaps has not yet been explored. Here we demonstrate two distinct topological edge solitons tuned by photovoltaic nonlinearity in Fe-doped lithium niobate waveguide arrays. By establishing a photonic nontrivial decorated Su-Schrieffer-Heeger lattice with two topological gaps, we reveal the emergence of multiple topological edge solitons derived from linear square-root edge states within these gaps. Interestingly, the bulk photovoltaic effect in Fe-doped lithium niobate crystals can generate an internal electric field that drives the electro-optical effect, enabling real-time dynamic manipulation of topological states. As a result, we experimentally observe the complete self-defocusing nonlinear tuning process of topological states within a system with multiple topological gaps, demonstrating the transitions between localization and delocalization. Our research establishes a novel platform for exploring nonlinear topology and sets the stage for further investigation into other intriguing nonlinear phenomena, offering both theoretical insights and practical applications.

Transition from the topological to the chaotic in the nonlinear Su-Schrieffer-Heeger model

Recent studies on topological materials are expanding into the nonlinear regime, while the central principle, namely the bulk-edge correspondence, is yet to be elucidated in the strongly nonlinear regime. Here, we reveal that nonlinear topological edge modes can exhibit the transition to spatial chaos by increasing nonlinearity, which can be a universal mechanism of the breakdown of the bulk-edge correspondence. Specifically, we unveil the underlying dynamical system describing the spatial distribution of zero modes and show the emergence of chaos. We also propose the correspondence between the absolute value of the topological invariant and the dimension of the stable manifold under sufficiently weak nonlinearity. Our results provide a general guiding principle to investigate the nonlinear bulk-edge correspondence that can potentially be extended to arbitrary dimensions.

All-optical switching in nonlinear topological waveguide arrays

Photonic topological states are prospective in integrated optical devices due to their robustness to perturbations and defects. When taking into account the nonlinear effects of the system, the functionality of topological photonics can be further enhanced. Here, we investigated the interplay between topological edge states and nonlinear effects based on the Su-Schrieffer-Heeger (SSH) model. Relying on the theory prediction that topological edge states would shift upward under the action of nonlinearity, two types of optical switching are designed and experimentally realized in femtosecond laser direct-write waveguide arrays. This work provides a new, to the best of our knowledge, approach to preparing all-optical switches and offers a new perspective on the application of nonlinearity in topological optical devices.

Observation of nonlinear fractal higher order topological insulator

Higher-order topological insulators (HOTIs) are unique materials hosting topologically protected states, whose dimensionality is at least by 2 lower than that of the bulk. Topological states in such insulators may be strongly confined in their corners which leads to considerable enhancement of nonlinear processes involving such states. However, all nonlinear HOTIs demonstrated so far were built on periodic bulk lattice materials. Here, we demonstrate the first nonlinear photonic HOTI with the fractal origin. Despite their fractional effective dimensionality, the HOTIs constructed here on two different types of the Sierpiński gasket waveguide arrays, may support topological corner states for unexpectedly wide range of coupling strengths, even in parameter regions where conventional HOTIs become trivial. We demonstrate thresholdless spatial solitons bifurcating from corner states in nonlinear fractal HOTIs and show that their localization can be efficiently controlled by the input beam power. We observe sharp differences in nonlinear light localization on outer and multiple inner corners and edges representative for these fractal materials. Our findings not only represent a new paradigm for nonlinear topological insulators, but also open new avenues for potential applications of fractal materials to control the light flow.

Arbitrarily Configurable Nonlinear Topological Modes

Topological modes (TMs) are typically localized at boundaries, interfaces and dislocations, and exponentially decay into the bulk of a large enough lattice. Recently, the non-Hermitian skin effect has been leveraged to delocalize the wave functions of TMs from the boundary and thus to increase the capacity of TMs dramatically. Here, we explore the capability of nonlinearity in designing and configuring the wave functions of TMs. With growing intensity, wave functions of these in-gap nonlinear TMs undergo an initial deviation from exponential decay, gradually merge into arbitrarily designable plateaus, then encompass the entire nonlinear domain, and eventually concentrate at the nonlinear boundary. Intriguingly, such extended nonlinear TMs are still robust against defects and disorders, and stable in dynamics under external excitation. Advancing the conceptual understanding of the nonlinear TMs, our results open new avenues for increasing the capacity of TMs and developing compact and configurable topological devices.

Vortex solitons in topological disclination lattices

The existence of thresholdless vortex solitons trapped at the core of disclination lattices that realize higher-order topological insulators is reported. The study demonstrates the interplay between nonlinearity and higher-order topology in these systems, as the vortex state in the disclination lattice bifurcates from its linear topological counterpart, while the position of its propagation constant within the bandgap and localization can be controlled by its power. It is shown that vortex solitons are characterized by strong field confinement at the disclination core due to their topological nature, leading to enhanced stability. Simultaneously, the global discrete rotational symmetry of the disclination lattice imposes restrictions on the maximal possible topological charge of such vortex solitons. The results illustrate the strong stabilizing action that topologically nontrivial structures may exert on excited soliton states, opening new prospects for soliton-related applications.

Topological solitons in coupled Su-Schrieffer-Heeger waveguide arrays

We investigate the formation of multipole topological solitons at the edges of two and three coupled parallel Su-Schrieffer-Heeger (SSH) waveguide arrays. We show that independent variations of waveguide spacing in the unit cells (dimers) in coupled waveguide arrays result in the emergence at their edges of several topological edge states with different internal symmetries. The number of emerging edge states is determined by how many arrays are in topologically nontrivial phase. In the presence of nonlinearity, such edge states give rise to families of multipole topological edge solitons with distinct stability properties. Our results illustrate that coupling between quasi-one-dimensional topological structures substantially enriches the variety of stable topological edge solitons existing in them.

Observation of π solitons in oscillating waveguide arrays

Floquet systems with periodically varying in time parameters enable realization of unconventional topological phases that do not exist in static systems with constant parameters and that are frequently accompanied by appearance of novel types of the topological states. Among such Floquet systems are the Su-Schrieffer-Heeger lattices with periodically-modulated couplings that can support at their edges anomalous π modes of topological origin despite the fact that the lattice spends only half of the evolution period in topologically nontrivial phase, while during other half-period it is topologically trivial. Here, using Su-Schrieffer-Heeger arrays composed from periodically oscillating waveguides inscribed in transparent nonlinear optical medium, we report experimental observation of photonic anomalous π modes residing at the edge or in the corner of the one- or two-dimensional arrays, respectively, and demonstrate a new class of topological π solitons bifurcating from such modes in the topological gap of the Floquet spectrum at high powers. π solitons reported here are strongly oscillating nonlinear Floquet states exactly reproducing their profiles after each longitudinal period of the structure. They can be dynamically stable in both one- and two-dimensional oscillating waveguide arrays, the latter ones representing the first realization of the Floquet photonic higher-order topological insulator, while localization properties of such π solitons are determined by their power.

Theory of nonlinear corner states in photonic fractal lattices

We study linear and nonlinear higher-order topological insulators (HOTIs) based on waveguide arrays arranged into Sierpiński gasket and Sierpiński carpet structures, both of which have non-integer effective Hausdorff dimensionality. Such fractal structures possess different discrete rotational symmetries, but both lack transverse periodicity. Their characteristic feature is the existence of multiple internal edges and corners in their optical potential landscape, and the formal absence of an insulating bulk. Nevertheless, we show that a systematic geometric shift of the waveguides in the first generation of such fractal arrays, which affects the coupling strengths between sites of this building block as well as in subsequent structure generations, enables the formation of corner states of topological origin at the outer corners of the array. We find that, in contrast to HOTIs based on periodic arrays, Sierpiński gasket arrays always support topological corner states, irrespective of the direction of the shift of the waveguides, while in Sierpiński carpet structures, corner states emerge only for one direction of the waveguide shift. We also find families of corner solitons bifurcating from linear corner states of fractal structures that remain stable practically in the entire gap in which they form. These corner states can be efficiently excited by injecting Gaussian beams into the outer corner sites of the fractal arrays. Our results pave the way toward the investigation of nonlinear effects in topological insulators with non-integer dimensionality and enrich the variety of higher-order topological states.

Rotating topological edge solitons

We address the formation of topological edge solitons in rotating Su-Schrieffer-Heeger waveguide arrays. The linear spectrum of the non-rotating topological array is characterized by the presence of a topological gap with two edge states residing in it. Rotation of the array significantly modifies the spectrum and may move these edge states out of the topological gap. Defocusing nonlinearity counteracts this tendency and shifts such modes back into the topological gap, where they acquire the structure of tails typical of topological edge states. We present rich bifurcation structure for rotating topological solitons and show that they can be stable. Rotation of the topologically trivial array, without edge states in its spectrum, also leads to the appearance of localized edge states, but in a trivial semi-infinite gap. Families of rotating edge solitons bifurcating from the trivial linear edge states exist too, and sufficiently strong defocusing nonlinearity can also drive them into the topological gap, qualitatively modifying the structure of their tails.

Topological edge solitons in χ(2) waveguide arrays

We address the formation of χ⁽²⁾ topological edge solitons emerging in a topologically nontrivial phase in Su-Schrieffer-Heeger (SSH) waveguide arrays. We consider edge solitons, whose fundamental frequency (FF) component belongs to the topological gap, while the phase mismatch determines whether the second harmonic (SH) component falls into topological or trivial forbidden gaps of the spectrum for the SH wave. Two representative types of edge solitons are found, one of which is thresholdless and bifurcates from the topological edge state in the FF component, while the other exists above a power threshold and emanates from the topological edge state in the SH wave. Both types of soliton can be stable. Their stability, localization degree, and internal structure strongly depend on the phase mismatch between the FF and SH waves. Our results open up new prospects for the control of topologically nontrivial states by parametric wave interactions.

Observation of nonlinearity-controlled switching of topological edge states

We report the experimental observation of the periodic switching of topological edge states between two dimerized fs-laser written waveguide arrays. Switching occurs due to the overlap of the modal fields of the edge states from topological forbidden gap, when they are simultaneously present in two arrays brought into close proximity. We found that the phenomenon occurs for both strongly and weakly localized edge states and that switching rate increases with decreasing spacing between the topological arrays. When topological arrays are brought in contact with nontopological ones, switching in topological gap does not occur, while one observes either the formation of nearly stationary topological interface mode or strongly asymmetric diffraction into the nontopological array depending on the position of the initial excitation. Switching between topological arrays can be controlled and even completely arrested by increasing the peak power of the input signal, as we observed with different array spacings.

Topologically tuned terahertz confinement in a nonlinear photonic chip

Compact terahertz (THz) functional devices are greatly sought after for high-speed wireless communication, biochemical sensing, and non-destructive inspection. However, controlled THz generation, along with transport and detection, has remained a challenge especially for chip-scale devices due to low-coupling efficiency and unavoidable absorption losses. Here, based on the topological protection of electromagnetic waves, we demonstrate nonlinear generation and topologically tuned confinement of THz waves in an engineered lithium niobate chip forming a wedge-shaped Su-Schrieffer-Heeger lattice. Experimentally measured band structures provide direct visualization of the THz localization in the momentum space, while robustness of the confined mode against chiral perturbations is also analyzed and compared for both topologically trivial and nontrivial regimes. Such topological control of THz waves may bring about new possibilities in the realization of THz integrated circuits, promising for advanced photonic applications.

Observation of Edge Solitons in Topological Trimer Arrays

We report the experimental observation of nonlinear light localization and edge soliton formation at the edges of fs-laser written trimer waveguide arrays, where transition from nontopological to topological phases is controlled by the spacing between neighboring trimers. We found that, in the former regime, edge solitons occur only above a considerable power threshold, whereas in the latter one they bifurcate from linear states. Edge solitons are observed in a broad power range where their propagation constant falls into one of the topological gaps of the system, while partial delocalization is observed when considerable nonlinearity drives the propagation constant into an allowed band, causing coupling with bulk modes. Our results provide direct experimental evidence of the coexistence and selective excitation in the same or in different topological gaps of two types of topological edge solitons with different internal structures, which can rarely be observed even in nontopological systems. This also constitutes the first experimental evidence of formation of topological solitons in a nonlinear system with more than one topological gap.

Valley Hall edge solitons in a photonic graphene

We predict the existence and study properties of the valley Hall edge solitons in a composite photonic graphene with a domain wall between two honeycomb lattices with broken inversion symmetry. Inversion symmetry in our system is broken due to detuning introduced into constituent sublattices of the honeycomb structure. We show that nonlinear valley Hall edge states with sufficiently high amplitude bifurcating from the linear valley Hall edge state supported by the domain wall, can split into sets of bright spots due to development of the modulational instability, and that such an instability is a precursor for the formation of topological bright valley Hall edge solitons localized due to nonlinear self-action and travelling along the domain wall over large distances. Topological protection of the valley Hall edge solitons is demonstrated by modeling their passage through sharp corners of the Ω-shaped domain wall.

Dynamically Emerging Topological Phase Transitions in Nonlinear Interacting Soliton Lattices

We demonstrate dynamical topological phase transitions in evolving Su-Schrieffer-Heeger lattices made of interacting soliton arrays, which are entirely driven by nonlinearity and thereby exemplify an emergent nonlinear topological phenomenon. The phase transitions occur from the topologically trivial-to-nontrivial phase in periodic succession with crossovers from the topologically nontrivial-to-trivial regime. The signature of phase transition is the gap-closing and reopening point, where two extended states are pulled from the bands into the gap to become localized topological edge states. Crossovers occur via decoupling of the edge states from the bulk of the lattice.

Nonlinear control of photonic higher-order topological bound states in the continuum

Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

Quantized nonlinear Thouless pumping

The topological protection of wave transport, originally observed in the context of the quantum Hall effect in two-dimensional electron gases¹, has been shown to apply broadly to a range of physical platforms, including photonics^2-5, ultracold atoms in optical lattices^6-8 and others^9-12. That said, the behaviour of such systems can be very different from the electronic case, particularly when interparticle interactions or nonlinearity play a major role^13-22. A Thouless pump²³ is a one-dimensional model that captures the topological quantization of transport in the quantum Hall effect using the notion of dimensional reduction: an adiabatically, time-varying potential mathematically maps onto a momentum coordinate in a conceptual second dimension^24-34. Importantly, quantization assumes uniformly filled electron bands below a Fermi energy, or an equivalent occupation for non-equilibrium bosonic systems. Here we theoretically propose and experimentally demonstrate quantized nonlinear Thouless pumping of photons with a band that is decidedly not uniformly occupied. In our system, nonlinearity acts to quantize transport via soliton formation and spontaneous symmetry-breaking bifurcations. Quantization follows from the fact that the instantaneous soliton solutions centred upon a given unit cell are identical after each pump cycle, up to translation invariance; this is an entirely different mechanism from traditional Thouless pumping. This result shows that nonlinearity and interparticle interactions can induce quantized transport and topological behaviour without a linear counterpart.

Optically Reconfigurable Spin-Valley Hall Effect of Light in Coupled Nonlinear Ring Resonator Lattice

Scattering immune propagation of light in topological photonic systems may revolutionize the design of integrated photonic circuits for information processing and communications. In optics, various photonic topological circuits have been developed, which were based on classical emulation of either quantum spin Hall effect or quantum valley Hall effect. On the other hand, the combination of both the valley and spin degrees of freedom can lead to a new kind of topological transport phenomenon, dubbed spin-valley Hall effect (SVHE), which can further expand the number of topologically protected edge channels and would be useful for information multiplexing. However, it is challenging to realize SVHE in most known material platforms, due to the requirement of breaking both the (pseudo)fermionic time-reversal (T) and parity symmetries (P) individually, but leaving the combined symmetry S≡TP intact. Here, we propose an experimentally feasible platform to realize SVHE for light, based on coupled ring resonators mediated by optical Kerr nonlinearity. Thanks to the inherent flexibility of cross-mode modulation, the coupling between the probe light can be engineered in a controllable way such that spin-dependent staggered sublattice potential emerges in the effective Hamiltonian. With delicate yet experimentally feasible pump conditions, we show the existence of spin-valley Hall-induced topological edge states. We further demonstrate that both degrees of freedom, i.e., spin and valley, can be manipulated simultaneously in a reconfigurable manner to realize spin-valley photonics, doubling the degrees of freedom for enhancing the information capacity in optical communication systems.

Nonlinear tuning of PT symmetry and non-Hermitian topological states

Topology, parity-time (PT) symmetry, and nonlinearity are at the origin of many fundamental phenomena in complex systems across the natural sciences, but their mutual interplay remains unexplored. We established a nonlinear non-Hermitian topological platform for active tuning of PT symmetry and topological states. We found that the loss in a topological defect potential in a non-Hermitian photonic lattice can be tuned solely by nonlinearity, enabling the transition between PT-symmetric and non-PT-symmetric regimes and the maneuvering of topological zero modes. The interaction between two apparently antagonistic effects is revealed: the sensitivity close to exceptional points and the robustness of non-Hermitian topological states. Our scheme using single-channel control of global PT symmetry and topology via local nonlinearity may provide opportunities for unconventional light manipulation and device applications.

Weakly nonlinear topological gap solitons in Su-Schrieffer-Heeger photonic lattices

We study both theoretically and experimentally the effect of nonlinearity on topologically protected linear interface modes in a photonic Su-Schrieffer-Heeger (SSH) lattice. It is shown that under either focusing or defocusing nonlinearity, this linear topological mode of the SSH lattice turns into a family of topological gap solitons. These solitons are stable. However, they exhibit only a low amplitude and power and are thus weakly nonlinear, even when the bandgap of the SSH lattice is wide. As a consequence, if the initial beam has modest or high power, it will either delocalize, or evolve into a soliton not belonging to the family of topological gap solitons. These theoretical predictions are observed in our experiments with optically induced SSH-type photorefractive lattices.