Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling

Electrical Control of Polariton Josephson Junctions via Exciton Stark Effect

We propose harnessing the tools of modern nanofabrication to provide electrical control of exciton-polariton (EP) condensates. We develop the theory of a device based on the Josephson effect in which electric fields can be used to both switch between and monitor various dynamical modes. In particular, both the bias potential and the Josephson energy can be tuned electrically via the exciton component. We model the device by a Gross-Pitaevskii equation assuming that ideal EP condensates are established with well-balanced pumping and dissipation. We find that the EP condensates can be manipulated through degrees of freedom not easily accessible in other coherent quantum systems, and the dynamics of EP Josephson junctions are richer than that of the conventional superconducting junctions. The ability to control and monitor the condensate by both optical and electrical means allows new ways to study its physics not possible by either, alone.

Flat-Band Lasing in Silicon Waveguide-Integrated Metasurfaces

Photonic flat bands are crucial for enabling strong localization of light and enhancing light-matter interactions, as well as tailoring the angular distribution of emission from photonic structures. These unique properties open pathways for developing robust photonic devices, efficient nonlinear optical processes, and novel platforms for exploring topological and quantum phenomena. So far, experimental realizations of lasing in photonic flat bands have been limited to structures that emulate geometrically frustrated lattices in the tight-binding, i.e., short-range coupling, regime. Here, we consider a periodic metasurface with long-range couplings combined with guided modes and report experimental observation of lasing in photonic nearly flat modes. By carefully tuning the thickness of the guiding layer and periodicities, the observed flat lasing spectrum extends up to approximately k y = 2 μm-1 in reciprocal space. Simulations show that the observed modes exhibit localization in both the waveguiding and active layers. In addition, we observe accidental bound states in the continuum (BICs) at the lasing frequencies, manifesting through polarization vortices with a topological charge |q| = 1.

Polariton lattices as binarized neuromorphic networks

We introduce a novel neuromorphic network architecture based on a lattice of exciton-polariton condensates, intricately interconnected and energized through nonresonant optical pumping. The network employs a binary framework, where each neuron, facilitated by the spatial coherence of pairwise coupled condensates, performs binary operations. This coherence, emerging from the ballistic propagation of polaritons, ensures efficient, network-wide communication. The binary neuron switching mechanism, driven by the nonlinear repulsion through the excitonic component of polaritons, offers computational efficiency and scalability advantages over continuous weight neural networks. Our network enables parallel processing, enhancing computational speed compared to sequential or pulse-coded binary systems. The system's performance was evaluated using diverse datasets, including the MNIST dataset for image recognition and the Speech Commands dataset for voice recognition tasks. In both scenarios, the proposed system demonstrates the potential to outperform existing polaritonic neuromorphic systems. For image recognition, this is evidenced by an impressive predicted classification accuracy of up to 97.5%. In voice recognition, the system achieved a classification accuracy of about 68% for the ten-class subset, surpassing the performance of conventional benchmark, the Hidden Markov Model with Gaussian Mixture Model.

Topological Moiré Polaritons

The combination of an in-plane honeycomb potential and of a photonic spin-orbit coupling (SOC) emulates a photonic or polaritonic analog of bilayer graphene. We show that modulating the SOC magnitude allows us to change the overall lattice periodicity, emulating any type of moiré-arranged bilayer graphene with unique all-optical access to the moiré band topology. We show that breaking the time-reversal symmetry by an effective exciton-polariton Zeeman splitting opens a large topological gap in the array of moiré flat bands. This gap contains one-way topological edge states whose constant group velocity makes an increasingly sharp contrast with the flattening moiré bands.

Observation of perovskite topological valley exciton-polaritons at room temperature

Topological exciton-polaritons are a burgeoning class of topological photonic systems distinguished by their hybrid nature as part-light, part-matter quasiparticles. Their further control over novel valley degree of freedom (DOF) has offered considerable potential for developing active topological optical devices towards information processing. Here, employing a two-dimensional (2D) valley-Hall perovskite lattice, we report the experimental observation of valley-polarized topological exciton-polaritons and their valley-dependent propagations at room temperature. The 2D valley-Hall perovskite lattice consists of two mutually inverted honeycomb lattices with broken inversion symmetry. By measuring their band structure with angle-resolved photoluminescence spectra, we experimentally verify the existence of valley-polarized polaritonic topological kink states with a large gap opening of ~ 9 meV in the bearded interface at room temperature. Moreover, these valley-polarized states exhibit counter-propagating behaviors under a resonant excitation at room temperature. Our results not only expand the landscape of realizing topological exciton-polaritons, but also pave the way for the development of topological valleytronic devices employing exciton-polaritons with valley DOF at room temperature.

Dirac Cones and Room Temperature Polariton Lasing Evidenced in an Organic Honeycomb Lattice

Artificial 1D and 2D lattices have emerged as a powerful platform for the emulation of lattice Hamiltonians, the fundamental study of collective many-body effects, and phenomena arising from non-trivial topology. Exciton-polaritons, bosonic part-light and part-matter quasiparticles, combine pronounced nonlinearities with the possibility of on-chip implementation. In this context, organic semiconductors embedded in microcavities have proven to be versatile candidates to study nonlinear many-body physics and bosonic condensation, and in contrast to most inorganic systems, they allow the use at ambient conditions since they host ultra-stable Frenkel excitons. A well-controlled, high-quality optical lattice is implemented that accommodates light-matter quasiparticles. The realized polariton graphene presents with excellent cavity quality factors, showing distinct signatures of Dirac cone and flatband dispersions as well as polariton lasing at room temperature. This is realized by filling coupled dielectric microcavities with the fluorescent protein mCherry. The emergence of a coherent polariton condensate at ambient conditions are demonstrated, taking advantage of coupling conditions as precise and controllable as in state-of-the-art inorganic semiconductor-based systems, without the limitations of e.g. lattice matching in epitaxial growth. This progress allows straightforward extension to more complex systems, such as the study of topological phenomena in 2D lattices including topological lasers and non-Hermitian optics.

Topological edge and corner states in coupled wave lattices in nonlinear polariton condensates

Topological states have been widely investigated in different types of systems and lattices. In the present work, we report on topological edge states in double-wave (DW) chains, which can be described by a generalized Aubry-André-Harper (AAH) model. For the specific system of a driven-dissipative exciton polariton system we show that in such potential chains, different types of edge states can form. For resonant optical excitation, we further find that the optical nonlinearity leads to a multistability of different edge states. This includes topologically protected edge states evolved directly from individual linear eigenstates as well as additional edge states that originate from nonlinearity-induced localization of bulk states. Extending the system into two dimensions (2D) by stacking horizontal DW chains in the vertical direction, we also create 2D multi-wave lattices. In such 2D lattices multiple Su-Schrieffer-Heeger (SSH) chains appear along the vertical direction. The combination of DW chains in the horizonal and SSH chains in the vertical direction then results in the formation of higher-order topological insulator corner states. Multistable corner states emerge in the nonlinear regime.

Photonic Flatband Resonances in Multiple Light Scattering

We introduce the concept of photonic flatband resonances and demonstrate it for an array of high-index dielectric particles. We employ the multiple Mie scattering theory and demonstrate that both short- and long-range interactions between the resonators are crucial for the emerging collective resonances and their associated photonic flatbands. By examining both near- and far-field characteristics, we uncover how the flatbands emerge due to a fine tuning of resonators' radiation fields, and predict that hybridization of a flatband resonance with an electric hot spot can lead to giant values of the Purcell factor for the electric dipolar emitters.

Manipulating nonlinear exciton polaritons in an atomically-thin semiconductor with artificial potential landscapes

Exciton polaritons in atomically thin transition-metal dichalcogenide microcavities provide a versatile platform for advancing optoelectronic devices and studying the interacting Bosonic physics at ambient conditions. Rationally engineering the favorable properties of polaritons is critically required for the rapidly growing research. Here, we demonstrate the manipulation of nonlinear polaritons with the lithographically defined potential landscapes in monolayer WS2 microcavities. The discretization of photoluminescence dispersions and spatially confined patterns indicate the deterministic on-site localization of polaritons by the artificial mesa cavities. Varying the trapping sizes, the polariton-reservoir interaction strength is enhanced by about six times through managing the polariton-exciton spatial overlap. Meanwhile, the coherence of trapped polaritons is significantly improved due to the spectral narrowing and tailored in a picosecond range. Therefore, our work not only offers a convenient approach to manipulating the nonlinearity and coherence of polaritons but also opens up possibilities for exploring many-body phenomena and developing novel polaritonic devices based on 2D materials.

Observation of Zitterbewegung in photonic microcavities

We present and experimentally study the effects of the photonic spin-orbit coupling on the real space propagation of polariton wavepackets in planar semiconductor microcavities and polaritonic analogues of graphene. In particular, we demonstrate the appearance of an analogue Zitterbewegung effect, a term which translates as 'trembling motion' in English, which was originally proposed for relativistic Dirac electrons and consisted of the oscillations of the centre of mass of a wavepacket in the direction perpendicular to its propagation. For a planar microcavity, we observe regular Zitterbewegung oscillations whose amplitude and period depend on the wavevector of the polaritons. We then extend these results to a honeycomb lattice of coupled microcavity resonators. Compared to the planar cavity, such lattices are inherently more tuneable and versatile, allowing simulation of the Hamiltonians of a wide range of important physical systems. We observe an oscillation pattern related to the presence of the spin-split Dirac cones in the dispersion. In both cases, the experimentally observed oscillations are in good agreement with theoretical modelling and independently measured bandstructure parameters, providing strong evidence for the observation of Zitterbewegung.

A machine learning route between band mapping and band structure

The electronic band structure and crystal structure are the two complementary identifiers of solid-state materials. Although convenient instruments and reconstruction algorithms have made large, empirical, crystal structure databases possible, extracting the quasiparticle dispersion (closely related to band structure) from photoemission band mapping data is currently limited by the available computational methods. To cope with the growing size and scale of photoemission data, here we develop a pipeline including probabilistic machine learning and the associated data processing, optimization and evaluation methods for band-structure reconstruction, leveraging theoretical calculations. The pipeline reconstructs all 14 valence bands of a semiconductor and shows excellent performance on benchmarks and other materials datasets. The reconstruction uncovers previously inaccessible momentum-space structural information on both global and local scales, while realizing a path towards integration with materials science databases. Our approach illustrates the potential of combining machine learning and domain knowledge for scalable feature extraction in multidimensional data.

Enhanced Cavity Optomechanics with Quantum-Well Exciton Polaritons

Semiconductor microresonators embedding quantum wells can host tightly confined and mutually interacting excitonic, optical, and mechanical modes at once. We theoretically investigate the case where the system operates in the strong exciton-photon coupling regime, while the optical and excitonic resonances are parametrically modulated by the interaction with a mechanical mode. Owing to the large exciton-phonon coupling at play in semiconductors, we predict an enhancement of polariton-phonon interactions by 2 orders of magnitude with respect to mere optomechanical coupling: a near-unity single-polariton quantum cooperativity is within reach for current semiconductor resonator platforms. We further analyze how polariton nonlinearities affect dynamical backaction, modifying the capability to cool or amplify the mechanical motion.

Domain-Wall Topology Induced by Spontaneous Symmetry Breaking in Polariton Graphene

We present a numerical study of exciton-polariton (polariton) condensation in a staggered polariton graphene showing a gapped s band. The condensation occurs at the kinetically favorable negative mass extrema (K and K^{'} valleys) of the valence band. Considering attractive polariton-polariton interaction allows us to generate a spatially extended condensate. The symmetry breaking occurring during the condensate buildup leads to the formation of valley-polarized domains. This process can either be spontaneous, following the Kibble-Zurek scenario, or triggered, leading to a controlled spatial distribution of valley-polarized domains. The selection of a single valley breaks time-reversal symmetry, and the walls separating domains exhibit a reconfigurable topologically protected chiral current. This current emerges as a result of the interplay between the nontrivial valley topology and the condensation-induced symmetry breaking.

Halide perovskites enable polaritonic XY spin Hamiltonian at room temperature

Exciton polaritons, the part-light and part-matter quasiparticles in semiconductor optical cavities, are promising for exploring Bose-Einstein condensation, non-equilibrium many-body physics and analogue simulation at elevated temperatures. However, a room-temperature polaritonic platform on par with the GaAs quantum wells grown by molecular beam epitaxy at low temperatures remains elusive. The operation of such a platform calls for long-lifetime, strongly interacting excitons in a stringent material system with large yet nanoscale-thin geometry and homogeneous properties. Here, we address this challenge by adopting a method based on the solution synthesis of excitonic halide perovskites grown under nanoconfinement. Such nanoconfinement growth facilitates the synthesis of smooth and homogeneous single-crystalline large crystals enabling the demonstration of XY Hamiltonian lattices with sizes up to 10 × 10. With this demonstration, we further establish perovskites as a promising platform for room temperature polaritonic physics and pave the way for the realization of robust mode-disorder-free polaritonic devices at room temperature.

Electronic Quantum Materials Simulated with Artificial Model Lattices

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin–orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two- and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin–orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekulé and “breathing” kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.

Spin-orbit-coupled exciton-polariton condensates in lead halide perovskites

Spin-orbit coupling (SOC) is responsible for a range of spintronic and topological processes in condensed matter. Here, we show photonic analogs of SOCs in exciton-polaritons and their condensates in microcavities composed of birefringent lead halide perovskite single crystals. The presence of crystalline anisotropy coupled with splitting in the optical cavity of the transverse electric and transverse magnetic modes gives rise to a non-Abelian gauge field, which can be described by the Rashba-Dresselhaus Hamiltonian near the degenerate points of the two polarization modes. With increasing density, the exciton-polaritons with pseudospin textures undergo phase transitions to competing condensates with orthogonal polarizations. Unlike their pure photonic counterparts, these exciton-polaritons and condensates inherit nonlinearity from their excitonic components and may serve as quantum simulators of many-body SOC processes.

Geometric characterization of anomalous Landau levels of isolated flat bands

According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.

Landau levels are normally bounded by upper and lower band edges at zero magnetic field. Here, the authors predict a system with isolated flat bands, where anomalous Landau level spreading appears outside the zero-field energy bounds.

Quantum Geometry and Flat Band Bose-Einstein Condensation

We study the properties of a weakly interacting Bose-Einstein condensate (BEC) in a flat band lattice system by using the multiband Bogoliubov theory and discover fundamental connections to the underlying quantum geometry. In a flat band, the speed of sound and the quantum depletion of the condensate are dictated by the quantum geometry, and a finite quantum distance between the condensed and other states guarantees stability of the BEC. Our results reveal that a suitable quantum geometry allows one to reach the strong quantum correlation regime even with weak interactions.

Quantum fluids of light in all-optical scatterer lattices

One of the recently established paradigms in condensed matter physics is examining a system's behaviour in artificial potentials, giving insight into phenomena of quantum fluids in hard-to-reach settings. A prominent example is the matter-wave scatterer lattice, where high energy matter waves undergo transmission and reflection through narrow width barriers leading to stringent phase matching conditions with lattice band formation. In contrast to evanescently coupled lattice sites, the realisation of a scatterer lattice for macroscopic matter-wave fluids has remained elusive. Here, we implement a system of exciton-polariton condensates in a non-Hermitian Lieb lattice of scatterer potentials. By fine tuning the lattice parameters, we reveal a nonequilibrium phase transition between distinct regimes of polariton condensation: a scatterer lattice of gain guided polaritons condensing on the lattice potential maxima, and trapped polaritons condensing in the potential minima. Our results pave the way towards unexplored physics of non-Hermitian fluids in non-stationary mixtures of confined and freely expanding waves.

Exciton-Photonics: From Fundamental Science to Applications

Semiconductors in all dimensionalities ranging from 0D quantum dots and molecules to 3D bulk crystals support bound electron-hole pair quasiparticles termed excitons. Over the past two decades, the emergence of a variety of low-dimensional semiconductors that support excitons combined with advances in nano-optics and photonics has burgeoned an advanced area of research that focuses on engineering, imaging, and modulating the coupling between excitons and photons, resulting in the formation of hybrid quasiparticles termed exciton-polaritons. This advanced area has the potential to bring about a paradigm shift in quantum optics, as well as classical optoelectronic devices. Here, we present a review on the coupling of light in excitonic semiconductors and previous investigations of the optical properties of these hybrid quasiparticles via both far-field and near-field imaging and spectroscopy techniques. Special emphasis is given to recent advances with critical evaluation of the bottlenecks that plague various materials toward practical device implementations including quantum light sources. Our review highlights a growing need for excitonic material development together with optical engineering and imaging techniques to harness the utility of excitons and their host materials for a variety of applications.

Room-Temperature Topological Polariton Laser in an Organic Lattice

Interacting bosonic particles in artificial lattices have proven to be a powerful tool for the investigation of exotic phases of matter as well as phenomena resulting from nontrivial topology. Exciton-polaritons, bosonic quasi-particles of light and matter, have been shown to combine the on-chip benefits of optical systems with strong interactions, inherited from their matter character. Technologically significant semiconductor platforms strictly require cryogenic temperatures. In this communication, we demonstrate exciton-polariton lasing for topological defects emerging from the imprinted lattice structure at room temperature. We utilize red fluorescent protein derived from DsRed of Discosoma sea anemones, hosting highly stable Frenkel excitons. Using a patterned mirror cavity, we tune the lattice potential landscape of a linear Su-Schrieffer-Heeger chain to design topological defects at domain boundaries and at the edge. We unequivocally demonstrate polariton lasing from these topological defects. This progress has paved the road to interacting boson many-body physics under ambient conditions.

Enhanced Scattering between Electrons and Exciton-Polaritons in a Microcavity

The interplay between strong light-matter interactions and charge doping represents an important frontier in the pursuit of exotic many-body physics and optoelectronics. Here, we consider a simplified model of a two-dimensional semiconductor embedded in a microcavity, where the interactions between electrons and holes are strongly screened, allowing us to develop a diagrammatic formalism for this system with an analytic expression for the exciton-polariton propagator. We apply this to the scattering of spin-polarized polaritons and electrons, and show that this is strongly enhanced compared with exciton-electron interactions. As we argue, this counterintuitive result is a consequence of the shift of the collision energy due to the strong light-matter coupling, and hence this is a generic feature that applies also for more realistic electron-hole and electron-electron interactions. We furthermore demonstrate that the lack of Galilean invariance inherent in the light-matter coupled system can lead to a narrow resonancelike feature for polariton-electron interactions close to the polariton inflection point. Our results are potentially important for realizing tunable light-mediated interactions between charged particles.

Flat-Band Localization in Creutz Superradiance Lattices

Flat bands play an important role in diffraction-free photonics and attract fundamental interest in many-body physics. Here we report the engineering of flat-band localization of collective excited states of atoms in Creutz superradiance lattices with tunable synthetic gauge fields. Magnitudes and phases of the lattice hopping coefficients can be independently tuned to control the state components of the flat band and the Aharonov-Bohm phases. We can selectively excite the flat band and control the flat-band localization with the synthetic gauge field. Our study provides a room-temperature platform for flat bands of atoms and holds promising applications in exploring correlated topological materials.

Spontaneously coherent orbital coupling of counterrotating exciton polaritons in annular perovskite microcavities

Exciton-polariton condensation is regarded as a spontaneous macroscopic quantum phenomenon with phase ordering and collective coherence. By engineering artificial annular potential landscapes in halide perovskite semiconductor microcavities, we experimentally and theoretically demonstrate the room-temperature spontaneous formation of a coherent superposition of exciton-polariton orbital states with symmetric petal-shaped patterns in real space, resulting from symmetry breaking due to the anisotropic effective potential of the birefringent perovskite crystals. The lobe numbers of such petal-shaped polariton condensates can be precisely controlled by tuning the annular potential geometry. These petal-shaped condensates form in multiple orbital states, carrying locked alternating π phase shifts and vortex-antivortex superposition cores, arising from the coupling of counterrotating exciton-polaritons in the confined circular waveguide. Our geometrically patterned microcavity exhibits promise for realizing room-temperature topological polaritonic devices and optical polaritonic switches based on periodic annular potentials.

Coherent transfer of topological interface states

We demonstrate the controlled coherent transfer of topological interface states in a one-dimensional non-Hermitian chain of interacting Bose-Einstein condensates. The topological protection stems from a spatially patterned pump in an open-dissipative system. As a test bed setup of the proposed phenomenon, we consider a chain of coupled micropillars with embedded quantum wells, possessing exciton-polariton resonances. The transfer of an interface state is driven by spatially localised, adiabatic pump modulation in the vicinity of the interface state. The stochastic calculations prove the coherent nature of the interface state transfer. For appropriate system parameters the coherence degree is preserved after multiple transitions, paving the way towards long-range transfer of a coherent quantum state.

Topological edge states of nonequilibrium polaritons in hollow honeycomb arrays

We address topological currents in polariton condensates excited by uniform resonant pumps in finite honeycomb arrays of microcavity pillars with a hole in the center. Such currents arise under combined action of the spin-orbit coupling and Zeeman splitting, which breaks the time-reversal symmetry and opens a topological gap in the spectrum of the structure. The most representative feature of this structure is the presence of two interfaces, inner and outer ones, where the directions of topological currents are opposite. Due to the finite size of the structure, polariton-polariton interactions lead to coupling of the edge states at the inner and outer interfaces, which depends on the size of the hollow region. Moreover, switching between currents can be realized by tuning the pump frequency. We illustrate that currents in this finite structure can be stable and study bistability effects arising due to the resonant character of the pump.

Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases

Synthetic crystal lattices provide ideal environments for simulating and exploring the band structure of solid-state materials in clean and controlled experimental settings. Physical realisations have, so far, dominantly focused on implementing irreversible patterning of the system, or interference techniques such as optical lattices of cold atoms. Here, we realise reprogrammable synthetic band-structure engineering in an all optical exciton-polariton lattice. We demonstrate polariton condensation into excited states of linear one-dimensional lattices, periodic rings, dimerised non-trivial topological phases, and defect modes utilising malleable optically imprinted non-Hermitian potential landscapes. The stable excited nature of the condensate lattice with strong interactions between sites results in an actively tuneable non-Hermitian analogue of the Su-Schrieffer-Heeger system.

To simulate band structures of solid state materials synthetic lattices are usually generated by optical lattices or by irreversible patterning the system. Here, the authors present reconfigurable synthetic band-structures in optical exciton-polariton lattices and generate non-Hermitian topological phases.

The non-Hermitian geometrical property of 1D Lieb lattice under Majorana's stellar representation

The topological properties of non-Hermitian Hamiltonian is a hot topic, and the theoretical studies along this research line are usually based on the two-level non-Hermitian Hamiltonian (or, equivalently, a spin-1/2 non-Hermitian Hamiltonian). We are motivated to study the geometrical phases of a three-level Lieb lattice model (or, equivalently, a spin-1 non-Hermitian Hamiltonian) with the flat band in the context of a polariton condensate. The topological invariants are calculated by both winding numbers in the Brillouin zone and the geometrical phase of Majorana stars on the Bloch sphere. Besides, we provide an intuitive way to study the topological phase transformation with the higher spin, and the flat band offers a platform to define the topological phase transition on the Bloch sphere. According to the trajectories of the Majorana stars, we calculate the geometrical phases of the Majorana stars. We study the Lieb lattice with a complex hopping and find their phases have a jump when the parameters change from the trivial phase to the topological phase. The correlation phase of Majorana stars will rise along with the increase of the imaginary parts of the hopping energy. Besides, we also study the Lieb lattice with different intracell hopping and calculate the geometrical phases of the model using non-Bloch factor under the Majorana's stellar representation. In this case, the correlation phases will always be zero because of the normalized coefficient is always a purely real number and the phase transition is vividly shown with the geometrical phases of the Majorana stars calculated by the mean values of the total phases of both right and the joint left eigenstates.

Multi-orbital tight binding model for cavity-polariton lattices

In this work we present a tight-binding model that allows to describe with a minimal amount of parameters the band structure of exciton-polariton lattices. This model based on s and p non-orthogonal photonic orbitals faithfully reproduces experimental results reported for polariton graphene ribbons. We analyze in particular the influence of the non-orthogonality, the inter-orbitals interaction and the photonic spin-orbit coupling on the polarization and dispersion of bulk bands and edge states.

Partial quantum revivals of localized condensates in distorted lattices

We report on a peculiar propagation of bosons loaded by a short Laguerre-Gaussian pulse in a nearly flat band of a lattice potential. Taking a system of exciton polaritons in a kagome lattice as an example, we show that an initially localized condensate propagates in a specific direction in space, if anisotropy is taken into account. This propagation consists of quantum jumps, collapses, and revivals of the whole compact states, and it persists given any direction of anisotropy. This property reveals its signatures in the tight-binding model, and, surprisingly, it is much more pronounced in a continuous model. Quantum revivals are robust to the repulsive interaction and finite lifetime of the particles. Since no magnetic field or spin-orbit interaction is required, this system provides a new kind of easily implementable optical logic.

Nonlinear Polariton Fluids in a Flatband Reveal Discrete Gap Solitons

Phase frustration in periodic lattices is responsible for the formation of dispersionless flatbands. The absence of any kinetic energy scale makes flatband physics critically sensitive to perturbations and interactions. We report on the experimental investigation of the nonlinear response of cavity polaritons in the gapped flatband of a one-dimensional Lieb lattice. We observe the formation of gap solitons with quantized size and abrupt edges, a signature of the frozen propagation of switching fronts. This type of gap soliton belongs to the class of truncated Bloch waves, and has only been observed in closed systems up to now. Here, the driven-dissipative character of the system gives rise to a complex multistability of the flatband nonlinear domains. These results open up an interesting perspective regarding more complex 2D lattices and the generation of correlated photon phases.

Topological phase transition with p orbitals in the exciton-polariton honeycomb lattice

We study the topological phase transition with the TE-TM splitting in the p-orbital exciton-polariton honeycomb lattice. We find that some Dirac points survive at the high-symmetry points with space-inversion symmetry breaking, which reflects the characteristic of p orbitals. A phase diagram is obtained by the gap Chern number, from which the topological phase transition takes place in the intermediate gap. There is no topological phase transition in the bottom or top gap, and its edge state has the potential application for transporting signals in optoelectronic devices. When taking into account the non-degenerate p orbitals, we find that the bottom gap arises owing to the competition between the Zeeman energy and rotating angular velocity, and topological phase transition also appears in the complete gaps. These results can facilitate the experimental investigations of the topological properties of p-orbital exciton-polariton lattice structure.

Two-Dimensional Topological Polariton Laser

We provide proof-of-principle illustration of lasing in a two-dimensional polariton topological insulator. Topological edge states may arise in a structured polariton microcavity under the combined action of spin-orbit coupling and Zeeman splitting in the magnetic field. Their properties and lifetime are strongly affected by gain. Thus, gain concentrated along the edge of the insulator can counteract intrinsic losses in such a selective way that the topologically protected edge states become amplified, while bulk modes remain damped. When gain is compensated by nonlinear absorption the metastable nonlinear edge states are formed. Taking a triangular structure instead of an infinite edge we observed persistent topological currents accompanied by the time-periodic oscillations of the polariton density.

Unconventional Flatband Line States in Photonic Lieb Lattices

Flatband systems typically host "compact localized states" (CLS) due to destructive interference and macroscopic degeneracy of Bloch wave functions associated with a dispersionless energy band. Using a photonic Lieb lattice (LL), such conventional localized flatband states are found to be inherently incomplete, with the missing modes manifested as extended line states that form noncontractible loops winding around the entire lattice. Experimentally, we develop a continuous-wave laser writing technique to establish a finite-sized photonic LL with specially tailored boundaries and, thereby, directly observe the unusually extended flatband line states. Such unconventional line states cannot be expressed as a linear combination of the previously observed boundary-independent bulk CLS but rather arise from the nontrivial real-space topology. The robustness of the line states to imperfect excitation conditions is discussed, and their potential applications are illustrated.

Platform for Electrically Pumped Polariton Simulators and Topological Lasers

Two-dimensional electronic materials such as graphene and transition metal dichalgenides feature unique electrical and optical properties due to the conspirative effect of band structure, orbital coupling, and crystal symmetry. Synthetic matter, as accomplished by artificial lattice arrangements of cold atoms, molecules, electron patterning, and optical cavities, has emerged to provide manifold intriguing frameworks to likewise realize such scenarios. Exciton polaritons have recently been added to the list of promising candidates for the emulation of system Hamiltonians on a semiconductor platform, offering versatile tools to engineer the potential landscape and to access the nonlinear electro-optical regime. In this work, we introduce an electronically driven square and honeycomb lattice of exciton polaritons, paving the way towards real world devices based on polariton lattices for on-chip applications. Our platform exhibits laserlike emission from high-symmetry points under direct current injection, hinting at the prospect of electrically driven polariton lasers with possibly topologically nontrivial properties.

Edge-emitting polariton laser and amplifier based on a ZnO waveguide

We demonstrate edge-emitting exciton-polariton (polariton) laser operation from 5 to 300 K and polariton amplifiers based on polariton modes within ZnO waveguides. The guided mode dispersion below and above the lasing threshold is directly measured using gratings placed on top of the sample, fully demonstrating the polaritonic nature of the lasing modes. The threshold is found to be smaller than that expected for radiative polaritons in planar ZnO microcavities below 150 K and comparable above. These results open up broad perspectives for guided polaritonics by enabling easier and more straightforward implementation of polariton integrated circuits that exploit fast propagating polaritons, and, possibly, topological protection.

Nonlinear Control of Electromagnetic Topological Edge States

Topological photonics has emerged recently as a smart approach for realizing robust optical circuitry, and the study of nonlinear effects is expected to open the door for tunability of photonic topological states. Here we realize experimentally nonlinearity-induced spectral tuning of electromagnetic topological edge states in arrays of coupled nonlinear resonators in the pump-probe regime. When nonlinearity is weak, we observe that the frequencies of the resonators exhibit spectral shifts concentrated mainly at the edge mode and affecting only weakly the bulk modes. For a strong pumping, we describe several scenarios of the transformation of the edge states and their hybridization with bulk modes, and also predict a parametrically driven transition from topological stationary to unstable dynamic regimes.

Z2 topological edge state in honeycomb lattice of coupled resonant optical waveguides with a flat band

Two-dimensional (2D) coupled resonant optical waveguide (CROW), exhibiting topological edge states, provides an efficient platform for designing integrated topological photonic devices. In this paper, we propose an experimentally feasible design of 2D honeycomb CROW photonic structure. The characteristic optical system possesses two-fold and three-fold Dirac points at different positions in the Brillouin zone. The effective gauge fields implemented by the intrinsic pseudo-spin-orbit interaction open up topologically nontrivial bandgaps through the Dirac points. Spatial lattice geometries allow destructive wave interference, leading to a dispersionless, near-flat energy band in the vicinity of the three-fold Dirac point in the telecommunication frequency regime. This nontrivial structure with a near-flat band yields topologically protected edge states. These characteristics underpin the fundamental importance as well as the potential applications in various optical devices. Based on the honeycomb CROW lattice, we design the shape-independent topological cavity and the beam splitter, which demonstrate the relevance for a wide range of photonic applications.

Topological dynamics and excitations in lasers and condensates with saturable gain or loss

We classify symmetry-protected and symmetry-breaking dynamical solutions for nonlinear saturable bosonic systems that display a non-hermitian charge-conjugation symmetry, as realized in a series of recent groundbreaking experiments with lasers and exciton polaritons. In particular, we show that these systems support stable symmetry-protected modes that mirror the concept of zero-modes in topological quantum systems, as well as symmetry-protected power-oscillations with no counterpart in the linear case. In analogy to topological phases in linear systems, the number and nature of symmetry-protected solutions can change. The spectral degeneracies signalling phase transitions in linear counterparts extend to bifurcations in the nonlinear context. As bifurcations relate to qualitative changes in the linear stability against changes of the initial conditions, the symmetry-protected solutions and phase transitions can also be characterized by topological excitations, which set them apart from symmetry-breaking solutions. The stipulated symmetry appears naturally when one introduces nonlinear gain or loss into spectrally symmetric bosonic systems, as we illustrate for one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.

Conservation of Quantum Correlations in Multimode Systems with U(1) Symmetry

We present a theoretical investigation of the properties of quantum correlation functions in a multimode system. We define a total mth order equal-time correlation function, summed over all modes, which is shown to be conserved if the Hamiltonian possesses U(1) symmetry. It is also conserved in the presence of dissipation, provided the loss rate is the same for all modes of the system. As examples, we demonstrate this conservation using numerical simulations of a coupled cavity system and the Jaynes-Cummings model.