No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Controllable topological edge mode in an optically excited exciton-polariton lattice
Abstract
We propose an all-optical scheme of topological lasing and switching based on the Aubry-André-Harper (AAH) model of an exciton-polariton chain. We theoretically show that the phase parameter of the optical potential, with a tunable effective quasimomentum, allows the system to exhibit nontrivial topological properties which are attributed to higher dimensions. The topological modes emerging within the bulk band gaps are spatially localized at the edges of the polariton lattice, and their topological properties are characterized by the nonzero Chern numbers of the bulk bands. Polariton lasing in topological edge modes exhibits a higher efficiency and better robustness than in bulk modes, and can be switched between two opposite edges of the lattice by nonresonant excitation, which paves a way for topologically protected optical circuits.
... Indeed, nonlinearities of polaritons governed by polaritonic interactions have become one of the most studied topics in the field of polariton physics. A variety of nonlinear phenomena, such as inter-and intra-band parametric scattering, 2,[8][9][10][11] polariton blockade, [12][13][14] topological edge mode, [15][16][17] and evaporative cooling 18) have been reported in recent years. Understanding and controlling the nonlinearities of polaritons is critical for both fundamental polariton physics and their potential optoelectronic applications. ...
In this work, we report the experimental observation of the ballistic transport of condensed exciton-polaritons at room temperature in a one-dimensional whispering-gallery microcavity. Such coherent transport, initiated by the pronounced inter-particle interactions of polaritons, leads to the generation of two symmetric emission beams in the momentum (angular) space. By means of spatially filtered angle-resolved photoluminescence imaging spectroscopy, we were able to identify their origin and successfully rationalize these observations using the potential energy-to-kinetic energy conversion picture. The energy-dependent emission linewidth, as well as TM to TE inter-mode scattering, have also been discussed.
... A less intensely studied but not less promising route is the Aubry-André-Harper (AAH) model [54] which translates into a Chern class in synthetic space [55]. Such kind of model has recently been proposed in polariton systems in 1D by using optically induced potentials [56]. ...
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.
... 261 There is another paradigmatic quasiperiodic model called Aubry-Andr e (AA) model, with a cosine modulation incommensurate with the underlying periodic lattice spacing. 262 The two models belong to the same topological class, and can be continuously deformed to an interpolating Aubry-Andr e-Fibonacci (IAAF) model. A recent investigation of IAAF model in polaritonic 1D wires discovered that the eigenmodes underwent a cascade of band-selective localization/delocalization transitions from AA to Fibonacci limits. ...
The past 30 years have witnessed remarkable developments of microcavity exciton–polaritons, which have made a great impact on photonics and optoelectronics from fundamental physics to device applications. New materials and optical structures have been developed for novel polariton lasers for the sake of room temperature operation, flexible mode engineering, and high power efficiency. More powerful spectroscopic techniques have also promoted the understanding of polariton dynamics, coherence, nonlinearity, and topology. In this review, we start with a brief introduction to the picture of polaritons, and various polariton systems based on different microcavity structures and semiconductor materials. Then, we present several important spectroscopic techniques and numerical tools for characterizing polaritons experimentally and theoretically. Next, we address the macroscopic quantum phenomena observed in the polariton systems and review the physics and applications of polariton nonlinearity. Moreover, we highlight the new emerging fields of topological and non-Hermitian polaritons. In the end, we conclude with the future perspectives of microcavity exciton–polaritons.
We propose an approach for the excitation of polariton Tamm states with a controllable nontrivial topology. The Tamm polaritons emerge at the interface of two binary one-dimensional photonic crystals belonging to a C3v point group, with an exciton resonance within their matching band gaps. The external magnetic field applied in the Faraday geometry endows the dispersion of the Tamm polaritons with the nonreciprocity: it lifts the degeneracy between opposite propagation directions in the interface plane. The phenomenology of Tamm polariton currents closely resembles one of a Z2 topological insulator. The proposed structure acts as an optical spin splitter controlled by the magnetic field magnitude.
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.
The rise of quantum science and technologies motivates photonics research to seek new platforms with strong light-matter interactions to facilitate quantum behaviors at moderate light intensities. Topological polaritons (TPs) offer an ideal platform in this context, with unique properties stemming from resilient topological states of light strongly coupled with matter. Here we explore polaritonic metasurfaces based on 2D transition metal dichalcogenides (TMDs) as a promising platform for topological polaritonics. We show that the strong coupling between topological photonic modes of the metasurface and excitons in TMDs yields a topological polaritonic Z2 phase. We experimentally confirm the emergence of one-way spin-polarized edge TPs in metasurfaces integrating MoSe2 and WSe2. Combined with the valley polarization in TMD monolayers, the proposed system enables an approach to engage the photonic angular momentum and valley and spin of excitons, offering a promising platform for photonic/solid-state interfaces for valleytronics and spintronics.
Topological insulators are a class of electronic materials exhibiting robust edge states immune to perturbations and disorder. This concept has been successfully adapted in photonics, where topologically nontrivial waveguides and topological lasers were developed. However, the exploration of topological properties in a given photonic system is limited to a fabricated sample, without the flexibility to reconfigure the structure in situ. Here, we demonstrate an all-optical realization of the orbital Su–Schrieffer–Heeger model in a microcavity exciton-polariton system, whereby a cavity photon is hybridized with an exciton in a GaAs quantum well. We induce a zigzag potential for exciton polaritons all-optically by shaping the nonresonant laser excitation, and measure directly the eigenspectrum and topological edge states of a polariton lattice in a nonlinear regime of bosonic condensation. Furthermore, taking advantage of the tunability of the optically induced lattice, we modify the intersite tunneling to realize a topological phase transition to a trivial state. Our results open the way to study topological phase transitions on-demand in fully reconfigurable hybrid photonic systems that do not require sophisticated sample engineering.
We consider exciton-polaritons in a honeycomb lattice of micropillars subjected to circularly polarized (σ±) incoherent pumps, which are arranged to form two domains in the lattice. We predict that the nonlinear interaction between the polaritons and the reservoir excitons gives rise to the topological valley Hall effect where in each valley two counterpropagating helical edge modes appear. Under a resonant pump, σ± polaritons propagate in different directions without being reflected around bends. The polaritons propagating along the interface have extremely high effective lifetimes and show fair robustness against disorder. This paves the way for robust exciton-polariton spin separating and transporting channels in which polaritons attain and maintain high degrees of spin polarization, even in the presence of spin relaxation.
Strong light-matter interaction enriches topological photonics by dressing light with matter, which provides the possibility to realize active nonlinear topological devices with immunity to defects. Topological exciton polaritons-half-light, half-matter quasiparticles with giant optical nonlinearity-represent a unique platform for active topological photonics. Previous demonstrations of exciton polariton topological insulators demand cryogenic temperatures, and their topological properties are usually fixed. Here, we experimentally demonstrate a room temperature exciton polariton topological insulator in a perovskite zigzag lattice. Polarization serves as a degree of freedom to switch between distinct topological phases, and the topologically nontrivial polariton edge states persist in the presence of onsite energy perturbations, showing strong immunity to disorder. We further demonstrate exciton polariton condensation into the topological edge states under optical pumping. These results provide an ideal platform for realizing active topological polaritonic devices working at ambient conditions, which can find important applications in topological lasers, optical modulation, and switching.
Microcavity polaritons are light-matter quasiparticles that arise from the strong coupling between excitons and photons confined in a semiconductor microcavity. They are typically studied at visible or near visible wavelengths. They combine the properties of confined electromagnetic fields, including a sizeable spin-orbit coupling, and the sensitivity to external magnetic fields and particle interactions inherited from their partly matter nature. These features make polaritons an excellent platform to study topological phases in photonics in one and two-dimensional lattices, whose band properties can be directly accessed using standard optical tools. In this review, we describe the main properties of microcavity polaritons and the main observations in the field of topological photonics, which include, among others, lasing in topological edge states, the implementation of a polariton Chern insulator under an external magnetic field, and the direct measurement of fundamental quantities, such as the quantum geometric tensor and winding numbers in one- and two-dimensional lattices. Polariton interactions open exciting perspectives for the study of nonlinear topological phases.
We present a theoretical study of the temporal and spatial coherence properties of a topological laser device built by including saturable gain on the edge sites of a Harper-Hofstadter lattice for photons. For small enough lattices, the Bogoliubov analysis applies, and the coherence time is almost determined by the total number of photons in the device in agreement with the standard Schawlow-Townes phase diffusion. In larger lattices, looking at the lasing edge mode in the comoving frame of its chiral motion, the spatiotemporal correlations of long-wavelength fluctuations display a Kardar-Parisi-Zhang (KPZ) scaling. Still, at very long times, when the finite size of the device starts to matter, the functional form of the temporal decay of coherence changes from the KPZ stretched exponential to a Schawlow-Townes-like exponential, while the nonlinear dynamics of KPZ fluctuations remains visible as a broadened linewidth as compared to the Bogoliubov-Schawlow-Townes prediction. While we establish the above behaviors also for nontopological 1D laser arrays, the crucial role of topology in protecting the coherence from static disorder is finally highlighted: Our numerical calculations suggest the dramatically reinforced coherence properties of topological lasers compared to corresponding nontopological devices. These results open exciting possibilities for both fundamental studies of nonequilibrium statistical mechanics and concrete applications to laser devices.
We consider exciton polaritons in a zigzag chain of coupled elliptical micropillars subjected to incoherent excitation. The driven-dissipative nature of the system along with the naturally present polarization splitting inside the pillars gives rise to nonreciprocal dynamics, which eventually leads to the non-Hermitian skin effect, where all the modes of the system collapse to one edge. As a result, the polaritons propagate only in one direction along the chain, independent of the excitation position, and the propagation in the opposite direction is suppressed. The system shows robustness against disorder and, using the bistable nature of polaritons to encode information, we show one-way information transfer. This paves the way for compact and robust feedback-free one-dimensional polariton transmission channels without the need for external magnetic field, which are compatible with proposals for polaritonic circuits.
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.
Rapidly growing demands for fast information processing have launched a race for creating compact and highly efficient optical devices that can reliably transmit signals without losses. Recently discovered topological phases of light provide novel opportunities for photonic devices robust against scattering losses and disorder. Combining these topological photonic structures with nonlinear effects will unlock advanced functionalities such as magnet-free nonreciprocity and active tunability. Here, we introduce the emerging field of nonlinear topological photonics and highlight the recent developments in bridging the physics of topological phases with nonlinear optics. This includes the design of novel photonic platforms which combine topological phases of light with appreciable nonlinear response, self-interaction effects leading to edge solitons in topological photonic lattices, frequency conversion, active photonic structures exhibiting lasing from topologically protected modes, and many-body quantum topological phases of light. We also chart future research directions discussing device applications such as mode stabilization in lasers, parametric amplifiers protected against feedback, and ultrafast optical switches employing topological waveguides.
Topological phases have recently witnessed rapid progress in non-Hermitian systems. Here we study a one-dimensional non-Hermitian Aubry-André-Harper (AAH) model with imaginary periodic or quasiperiodic modulations. We demonstrate that the non-Hermitian off-diagonal AAH models can host zero-energy modes at the edges. In contrast to the Hermitian case, the zero-energy mode can be localized only at one edge. Such a topological phase corresponds to the existence of a quarter winding number defined by eigenenergy in momentum space. We further find the coexistence of a zero-energy mode located only at one edge and topological nonzero-energy edge modes characterized by a generalized Bott index. In the incommensurate case, a topological non-Hermitian quasicrystal is predicted where all bulk states and two topological edge states are localized at one edge. Such topological edge modes are protected by the generalized Bott index. Finally, we propose an experimental scheme to realize these non-Hermitian models in electric circuits. Our findings add another direction for exploring topological properties in Aubry-André-Harper models.
We theoretically study topological laser operation in a bosonic Harper-Hofstadter model featuring a saturable optical gain. Crucial consequences of the chirality of the lasing edge modes are highlighted, such as a sharp dependence of the lasing threshold on the geometrical shape of the amplifying region and the possibility of ultraslow relaxation times and of convectively unstable regimes. The different unstable regimes are characterized in terms of spatiotemporal structures sustained by noise and a strong amplification of a propagating probe beam is anticipated to occur in between the convective and the absolute (lasing) thresholds. The robustness of topological laser operation against static disorder is assessed.
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 energy splitting between the 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.
Topological insulators—materials that are insulating in the bulk but allow electrons to flow on their surface—are striking examples of materials in which topological invariants are manifested in robustness against perturbations such as defects and disorder¹. Their most prominent feature is the emergence of edge states at the boundary between areas with different topological properties. The observable physical effect is unidirectional robust transport of these edge states. Topological insulators were originally observed in the integer quantum Hall effect² (in which conductance is quantized in a strong magnetic field) and subsequently suggested3–5 and observed⁶ to exist without a magnetic field, by virtue of other effects such as strong spin–orbit interaction. These were systems of correlated electrons. During the past decade, the concepts of topological physics have been introduced into other fields, including microwaves7,8, photonic systems9,10, cold atoms11,12, acoustics13,14 and even mechanics¹⁵. Recently, topological insulators were suggested to be possible in exciton-polariton systems16–18 organized as honeycomb (graphene-like) lattices, under the influence of a magnetic field. Exciton-polaritons are part-light, part-matter quasiparticles that emerge from strong coupling of quantum-well excitons and cavity photons¹⁹. Accordingly, the predicted topological effects differ from all those demonstrated thus far. Here we demonstrate experimentally an exciton-polariton topological insulator. Our lattice of coupled semiconductor microcavities is excited non-resonantly by a laser, and an applied magnetic field leads to the unidirectional flow of a polariton wavepacket around the edge of the array. This chiral edge mode is populated by a polariton condensation mechanism. We use scanning imaging techniques in real space and Fourier space to measure photoluminescence and thus visualize the mode as it propagates. We demonstrate that the topological edge mode goes around defects, and that its propagation direction can be reversed by inverting the applied magnetic field. Our exciton-polariton topological insulator paves the way for topological phenomena that involve light–matter interaction, amplification and the interaction of exciton-polaritons as a nonlinear many-body system.
We present a scheme to generate an artificial gauge field for the system of neutral bosons, represented by polaritons in micropillars arranged into a square lattice. The splitting between the two polarizations of the micropillars breaks the time-reversal symmetry (TRS) and results in the effective phase-dependent hopping between cavities. This can allow for engineering a nonzero flux on the plaquette, corresponding to an artificial magnetic field. Changing the phase, we observe a characteristic Hofstadter's butterfly pattern and the appearance of chiral edge states for a finite-size structure. For long-lived polaritons, we show that the propagation of wave packets at the edge is robust against disorder. Moreover, given the inherent driven-dissipative nature of polariton lattices, we find that the system can exhibit topological lasing, recently discovered for active ring cavity arrays. The results point to a static way to realize artificial magnetic field in neutral spinful systems, avoiding the periodic modulation of the parameters or strong spin-orbit interaction. Ultimately, the described system can allow for high-power topological single-mode lasing which is robust to imperfections.
Non-perturbative coupling of photons and excitons produces hybrid particles, exciton–polaritons, which have exhibited a variety of many-body phenomena in various microcavity systems. However, the vacuum Rabi splitting (VRS), which defines the strength of photon–exciton coupling, is usually a single constant for a given system. Here, we have developed a unique architecture in which excitons in an aligned single-chirality carbon nanotube film interact with cavity photons in polarization-dependent manners. The system reveals ultrastrong coupling (VRS up to 329 meV or a coupling-strength-to-transition-energy ratio of 13.3%) for polarization parallel to the nanotube axis, whereas VRS is absent for perpendicular polarization. Between these two extremes, VRS is continuously tunable through polarization rotation with exceptional points separating crossing and anticrossing. The points between exceptional points form equienergy arcs onto which the upper and lower polaritons coalesce. The demonstrated on-demand ultrastrong coupling provides ways to explore topological properties of polaritons and quantum technology applications.
Topology describes properties that remain unaffected by smooth distortions. Its main hallmark is the emergence of edge states localized at the boundary between regions characterized by distinct topological invariants. Because their properties are inherited from the topology of the bulk, these edge states present a strong immunity to distortions of the underlying architecture. This feature offers new opportunities for robust trapping of light in nano- and micrometre-scale systems subject to fabrication imperfections and environmentally induced deformations. Here, we report lasing in such topological edge states of a one-dimensional lattice of polariton micropillars that implements an orbital version of the Su-Schrieffer-Heeger Hamiltonian. We further demonstrate that lasing in these states persists under local deformations of the lattice. These results open the way to the implementation of chiral lasers in systems with broken time-reversal symmetry and, when combined with polariton interactions, to the study of nonlinear phenomena in topological photonics.
The vast majority of real-life optimization problems with a large number of degrees of freedom are intractable by classical computers, since their complexity grows exponentially fast with the number of variables. Many of these problems can be mapped into classical spin models, such as the Ising, the XY or the Heisenberg models, so that optimization problems are reduced to finding the global minimum of spin models. Here, we propose and investigate the potential of polariton graphs as an efficient analogue simulator for finding the global minimum of the XY model. By imprinting polariton condensate lattices of bespoke geometries we show that we can engineer various coupling strengths between the lattice sites and read out the result of the global minimization through the relative phases. Besides solving optimization problems, polariton graphs can simulate a large variety of systems undergoing the U(1) symmetry-breaking transition. We realize various magnetic phases, such as ferromagnetic, anti-ferromagnetic, and frustrated spin configurations on a linear chain, the unit cells of square and triangular lattices, a disordered graph, and demonstrate the potential for size scalability on an extended square lattice of 45 coherently coupled polariton condensates. Our results provide a route to study unconventional superfluids, spin liquids, Berezinskii–Kosterlitz–Thouless phase transition, and classical magnetism, among the many systems that are described by the XY Hamiltonian.
Polariton lasing is the coherent emission arising from a macroscopic polariton condensate first proposed in 1996. Over the past two decades, polariton lasing has been demonstrated in a few inorganic and organic semiconductors in both low and room temperatures. Polariton lasing in inorganic materials significantly relies on sophisticated epitaxial growth of crystalline gain medium layers sandwiched by two distributed Bragg reflectors in which combating the built-in strain and mismatched thermal properties is nontrivial. On the other hand, organic active media usually suffer from large threshold density and weak nonlinearity due to the Frenkel exciton nature. Further development of polariton lasing towards technologically significant applications demand more accessible materials, ease of device fabrication and broadly tunable emission at room temperature. Herein, we report the experimental realization of room-temperature polariton lasing based on an epitaxy-free all-inorganic cesium lead chloride perovskite microcavity. Polariton lasing is unambiguously evidenced by a superlinear power dependence, macroscopic ground state occupation, blueshift of ground state emission, narrowing of the linewidth and the build-up of long-range spatial coherence. Our work suggests considerable promise of lead halide perovskites towards large-area, low-cost, high performance room temperature polariton devices and coherent light sources extending from the ultraviolet to near infrared range.
The interaction between light and matter can give rise to novel topological states. This principle was recently exemplified in Floquet topological insulators, where classical light was used to induce a topological electronic band structure. Here, in contrast, we show that mixing single photons with excitons can result in new topological polaritonic states—or “topolaritons.” Taken separately, the underlying photons and excitons are topologically trivial. Combined appropriately, however, they give rise to nontrivial polaritonic bands with chiral edge modes allowing for unidirectional polariton propagation. The main ingredient in our construction is an exciton-photon coupling with a phase that winds in momentum space. We demonstrate how this winding emerges from the finite-momentum mixing between s-type and p-type bands in the electronic system and an applied Zeeman field. We discuss the requirements for obtaining a sizable topological gap in the polariton spectrum and propose practical ways to realize topolaritons in semiconductor quantum wells and monolayer transition metal dichalcogenides.
A gas of electrons in a one-dimensional periodic potential can be transported
even in the absence of a voltage bias if the potential is modulated slowly and
periodically in time. Remarkably, the transferred charge per cycle is only
sensitive to the topology of the path in parameter space. Although this
so-called Thouless charge pump has first been proposed more than thirty years
ago, it has not yet been realized. Here we report the first demonstration of
topological Thouless pumping using ultracold atoms in a dynamically controlled
optical superlattice. We observe a shift of the atomic cloud as a result of
pumping and extract the topological invariance of the pumping process from this
shift. We demonstrate the topological nature of the Thouless pump by varying
the topology of the pumping path and verify that the topological pump indeed
works in the quantum region by varying speed and temperature.
More than 30 years ago, Thouless introduced the concept of a topological
charge pump that would enable the robust transport of charge through an
adiabatic cyclic evolution of the underlying Hamiltonian. In contrast to
classical transport, the transported charge was shown to be quantized and
purely determined by the topology of the pump cycle, making it robust to
perturbations. On a fundamental level, the quantized charge transport can be
connected to a topological invariant, the Chern number, first introduced in the
context of the integer quantum Hall effect. A Thouless quantum pump may
therefore be regarded as a 'dynamical' version of the integer quantum Hall
effect. Here, we report on the realization of such a topological charge pump
using ultracold bosonic atoms that form a Mott insulator in a dynamically
controlled optical superlattice potential. By taking in-situ images of the atom
cloud, we observe a quantized deflection per pump cycle. We reveal the genuine
quantum nature of the pump by showing that, in contrast to ground state
particles, a counterintuitive reversed deflection occurs when particles are
prepared in the first excited band. Furthermore, we were able to directly
demonstrate that the system undergoes a controlled topological phase transition
in higher bands when tuning the superlattice parameters.
We study the topological structure of mixed matter-light particles, so called
polaritons, in a quantum spin Hall insulator coupled to photonic cavity modes.
We identify a topological invariant in the presence of time reversal symmetry,
and find protected helical edge states with energies below the lower polariton
branch. Applying a Zeeman field allows us to relate our topological index to a
different index defined for an effective pseudospin model for polaritons. A
nontrivial phase is characterized by a pseudospin which covers the entire Bloch
sphere twice. Both, helical edge states and the topologically nontrivial
polariton pseudospin structure can be probed directly by optical techniques.
Recent search for optical analogues of topological phenomena mainly focuses
on mimicking the key feature of quantum Hall and quantum spin Hall effects (QHE
and QSHE): edge currents protected from disorder. QHE relies on time-reversal
symmetry breaking, which can be realised in photonic gyromagnetic crystals. In
the optical range, the weak magneto-optical activity may be replaced with
helical design of coupled waveguides, converting light propagation into a
time-dependent perturbation. Finally, optical QHE due to artificial gauge
fields was predicted in microcavity lattices. Here, we consider honeycomb
arrays of microcavity pillars as an alternative optical-frequency 2D
topological insulator. We show that the interplay between the photonic
spin-orbit coupling natively present in this system and the Zeeman splitting of
exciton-polaritons in external magnetic fields leads to the opening of a
non-trivial gap characterised by set of band Chern numbers and to the
formation of topologically protected one-way edge states.
A generalization of the Aubry-Andr\'e-Harper (AAH) model is developed,
containing a tunable phase shift between on-site and off-diagonal modulations.
By varying only this phase, keeping all other model parameters constant, a bulk
localization transition can be induced. The model, and its localization phase
diagram, can be derived from a quantum Hall system with spatially non-uniform
magnetic flux. Families of these generalized AAH models form bandstructures
with either topologically trivial gaps, or non-trivial gaps spanned by boundary
states. By slowly varying the phase shift, the boundary states can be pumped
across the system; the breakdown of adiabatic pumping due to the onset of bulk
localization can also be observed.
We report exciton-polariton condensation in a new family of fully hybrid ZnO-based microcavity demonstrating the best-quality ZnO material available (a bulk substrate), a large quality factor (∼4000) and large Rabi splittings (∼240 meV). Condensation is achieved between 4 and 300 K and for excitonic fractions ranging between 17% and 96%, which corresponds to a tuning of the exciton-polariton mass, lifetime, and interaction constant by 1 order of magnitude. We demonstrate mode switching between polariton branches allowing, just by controlling the pumping power, to tune the photonic fraction by a factor of 4.
In the past decade, a two-dimensional matter-light system called the microcavity exciton-polariton has emerged as a new promising candidate of Bose-Einstein condensation (BEC) in solids. Many pieces of important evidence of polariton BEC have been established recently in GaAs and CdTe microcavities at the liquid helium temperature, opening a door to rich many-body physics inaccessible in experiments before. Technological progress also made polariton BEC at room temperatures promising. In parallel with experimental progresses, theoretical frameworks and numerical simulations are developed, and our understanding of the system has greatly advanced. In this article, recent experiments and corresponding theoretical pictures based on the Gross-Pitaevskii equations and the Boltzmann kinetic simulations for a finite-size BEC of polaritons are reviewed.
In this paper, we begin with the nonlinear Schrodinger/Gross-Pitaevskii
equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear
optics as well as other applications, and discuss their dynamical properties
ranging from time reversible, time transverse invariant, mass and energy
conservation, dispersion relation to soliton solutions. Then, we review and
compare different numerical methods for solving the NLSE/GPE including finite
difference time domain methods and time-splitting spectral method, and discuss
different absorbing boundary conditions. In addition, these numerical methods
are extended to the NLSE/GPE with damping terms and/or an angular momentum
rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate
a quantized vortex lattice dynamics in a rotating BEC are reported.
The Aubry-Andr\'e or Harper (AAH) model has been the subject of extensive
theoretical research in the context of quantum localization. Recently, it was
shown that one-dimensional quasicrystals described by the incommensurate AAH
model has a nontrivial topology. In this Letter, we show that the commensurate
off-diagonal AAH model is topologically nontrivial in the gapless regime and
supports zero-energy edge modes. Unlike the incommensurate case, the nontrivial
topology in the off-diagonal AAH model is attributed to the topological
properties of the one-dimensional Majorana chain. We discuss the feasibility of
experimental observability of our predicted topological phase in the
commensurate AAH model.
The unrelated discoveries of quasicrystals and topological insulators have in
turn challenged prevailing paradigms in condensed-matter physics. We find a
surprising connection between quasicrystals and topological phases of matter:
(i) quasicrystals exhibit nontrivial topological properties and (ii) these
properties are attributed to dimensions higher than that of the quasicrystal.
Specifically, we show, both theoretically and experimentally, that
one-dimensional quasicrystals are assigned two-dimensional Chern numbers and,
respectively, exhibit topologically protected boundary states equivalent to the
edge states of a two-dimensional quantum Hall system.We harness the topological
nature of these states to adiabatically pump light across the quasicrystal. We
generalize our results to higher-dimensional systems and other topological
indices. Hence, quasicrystals offer a new platform for the study of topological
phases while their topology may better explain their surface properties.
Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behavior. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyze the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry.
By using rational and irrational on-site modulation in graphene waveguide arrays, we construct both commensurate and incommensurate Aubry-André-Harper (AAH) models and achieve the topological edge states and localization transition for graphene surface plasmon polaritons (SPPs). We show that when the commensurate AAH graphene array is truncated, topological edge states emerge, which are immune to the random perturbation of modulation depth in each individual graphene sheet. While for an incommensurate AAH array, localization transition for SPPs modes have also been achieved when the modulation depth reaches a critical value. Our work points out that graphene waveguide array provides a promising platform for robust light transport and compact localization within a fully deep subwavelength scale.
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.
Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the K=m11n matrix (even m for boson and odd n for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological (mn−1)-fold ground-state degeneracy equivalent to the determinant of the K matrix; (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the K matrix; and (iii) two parallel-propagating chiral edge branches with level counting 1,2,5,10 consistent with the conformal field theory description.
Polaritons with a twist
The ability to design and fabricate optical systems with tunable topological features makes them especially attractive for developing analogs of topological condensed matter systems, which by themselves tend to be fixed or limited in their tunability. Liu et al. now show that the combination of a two-dimensional material with a photonic crystal can be used to develop an analogous quantum spin Hall system. The strong coupling between the monolayer tungsten disulfide excitons with a nontrivial hexagonal photonic crystal gives rise to helical topological polaritons observed at up to 200 kelvin. The topological polaritons can be actively tuned by temperature and may further be manipulated with electric or magnetic fields, thereby providing a flexible platform with which to explore exotic topological phenomena and new phases of quantum matter.
Science , this issue p. 600
A large number of optimization problems of extremely complex systems in the real world remain highly challenging for conventional digital computers. Some such problems can be mapped into the Ising model, and then efficiently solved by searching for the global minimum of the Ising Hamiltonian. Finding an appropriate physical system for efficient simulation of the Ising model is a promising way of addressing such optimization problems that has recently emerged. Here we report on the realization of analog-spin chains of exciton-polariton condensates that mimic a chain of classical spins. This is done with a room-temperature system based on a one-dimensional polariton lattice that is induced by exciting a ZnO microrod with a controllable periodic laser pattern. Depending on the lattice constant chosen, the spontaneously phase-locked condensates show either an antiphase (π) or an in-phase (zero) ordering in the steady state, which mimics the antiferromagnetic or ferromagnetic state in the one-dimensional classical Ising model. In addition, when the external excitation power is increased, a chain of coupled condensate pairs, characterized by a small phase shift between the neighboring condensates that is induced by the tunneling effect, arises at a lower energy. These observations pave the way to the realization at room temperature of analog-spin simulators based on periodic condensates of exciton polaritons.
In this paper we study the formation of topological Tamm states at the interface between a semi-infinite one-dimensional (1D) photonic crystal and a metal. We show that when the system is topologically nontrivial there is a single Tamm state in each of the band gaps, whereas if it is topologically trivial the band gaps host no Tamm states. We connect the disappearance of the Tamm states with a topological transition from a topologically nontrivial system to a topologically trivial one. This topological transition is driven by the modification of the dielectric functions in the unit cell. Our interpretation is further supported by an exact mapping between the solutions of Maxwell's equations and the existence of a tight-binding representation of those solutions. We show that the tight-binding representation of the 1D photonic crystal, based on Maxwell's equations, corresponds to a Su-Schrieffer-Heeger–type model (SSH model) for each set of pairs of bands. By expanding this representation near the band edge we show that the system can be described by a Dirac-like Hamiltonian. It allows one to characterize the topology associated with the solution of Maxwell's equations via the winding number. In addition, for the infinite system, we provide an analytical expression for the photonic bands from which the band gaps can be computed.
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.
There have been significant recent advances in realizing bandstructures with geometrical and topological features in experiments on cold atomic gases. We provide an overview of these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. We describe the different methods that have been used to generate these novel bandstructures for cold atoms, as well as the physical observables that have allowed their characterization. We focus on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. However, we also describe how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, we describe the forms of inter-particle interactions that emerge when atoms are subjected to these energy bands, and some of the many-body phases that may be sought in future experiments.
Topological photonics is a rapidly-emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realise exotic topological models and to probe and exploit topological effects in new ways. In this article, we review experimental and theoretical developments in topological photonics across a wide-range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics and circuit-QED. We discuss how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases, and we review progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading towards new collective phenomena and novel strongly-correlated states of light, such as an analogue of the fractional quantum Hall effect.
Topological protection for lasers
Ideas based on topology, initially developed in mathematics to describe the properties of geometric space under deformations, are now finding application in materials, electronics, and optics. The main driver is topological protection, a property that provides stability to a system even in the presence of defects. Harari et al. outline a theoretical proposal that carries such ideas over to geometrically designed laser cavities. The lasing mode is confined to the topological edge state of the cavity structure. Bandres et al. implemented those ideas to fabricate a topological insulator laser with an array of ring resonators. The results demonstrate a powerful platform for developing new laser systems.
Science , this issue p. eaar4003 , p. eaar4005
Topological lasing
Resonant cavities that confine light are crucial components of lasers. Typically, these cavities are designed to high specification to get the best possible output. That, however, can limit their integration into photonic devices and optical circuits. Bahari et al. fabricated resonant cavities of arbitrary shape within a hybrid photonic crystal structure. The confinement of light to topologically protected edge states resulted in lasing at communication wavelengths. Relaxing the resonant cavity design criteria should be useful in designing photonic devices.
Science , this issue p. 636
We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow one to investigate generalized edge states forming in the gaps of the fractal energy spectrum. We employ these generalized edge states to determine the topological invariants of the quasicrystal. When varying a structural degree of freedom (phason) of the Fibonacci sequence, the edge states spectrally traverse the gaps, while their spatial symmetry switches: The periodicity of this spectral and spatial evolution yields direct measurements of the gap topological numbers. The topological invariants that we determine coincide with those assigned by the gap-labeling theorem, illustrating the direct connection between the fractal and topological properties of Fibonacci quasicrystals.
Superfluidity---the suppression of scattering in a quantum fluid at velocities below a critical value---is one of the most striking manifestations of the collective behaviour typical of Bose-Einstein condensates. This phenomenon, akin to superconductivity in metals, has until now only been observed at prohibitively low cryogenic temperatures. For atoms, this limit is imposed by the small thermal de Broglie wavelength, which is inversely related to the particle mass. Even in the case of ultralight quasiparticles such as exciton-polaritons, superfluidity has only been demonstrated at liquid helium temperatures. In this case, the limit is not imposed by the mass, but instead by the small exciton binding energy of Wannier-Mott excitons, which places the upper temperature limit. Here we demonstrate a transition from normal to superfluid flow in an organic microcavity supporting stable Frenkel exciton-polaritons at room temperature. This result paves the way not only to table-top studies of quantum hydrodynamics, but also to room-temperature polariton devices that can be robustly protected from scattering.
Photonic lattices provide an excellent platform for simulating conventional topological systems, and they can also be explored for the study of novel topological phases. However, a direct measurement of bulk topological invariants remains a great challenge. Here we study topological features of generalized commensurate Aubry-Andre-Harper (AAH) photonic lattices and construct a topological phase diagram by calculating all bulk Chern numbers, and then explore the bulk-edge correspondence by analyzing the topological edge states and their winding numbers. In contrast to incommensurate AAH models, we find that diagonal and off-diagonal commensurate AAH models are not topologically equivalent. In particular, nontrivial topological phases with large Chern numbers and topological phase transitions are possible. By implementing Thouless pumping of light in photonic lattices, we propose a simple scheme to measure the bulk Chern numbers.
Topology is revolutionizing photonics, bringing with it new theoretical
discoveries and a wealth of potential applications. This field was inspired by
the discovery of topological insulators, in which interfacial electrons
transport without dissipation even in the presence of impurities. Similarly,
new optical mirrors of di?fferent wave-vector space topologies have been
constructed to support new states of light propagating at their interfaces.
These novel waveguides allow light to flow around large imperfections without
back-reflection. The present review explains the underlying principles and
highlights the major findings in photonic crystals, coupled resonators,
metamaterials and quasicrystals.
We demonstrate a novel way to realize room-temperature polariton parametric scattering in a one-dimensional ZnO microcavity. The polariton parametric scattering is driven by a polariton condensate, with a balanced polariton pair generated at the adjacent polariton mode. This parametric scattering is experimentally investigated by the angle-resolved photoluminescence spectroscopy technique under different pump powers and it is well described by the rate equation of interacting bosons. The direct relation between the intensity of the scattered polariton signal and that of the polariton reservoir is acquired under nonresonant excitation, exhibiting the explicit nonlinear characteristic of this room-temperature polariton parametric process.