Semi-Dirac Transport and Anisotropic Localization in Polariton Honeycomb Lattices
B. Real, O. Jamadi, M. Milićević, N. Pernet, P. St-Jean, T. Ozawa, G. Montambaux, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, S. Ravets, J. Bloch, A. Amo
Compression dramatically changes the transport and localization properties of graphene. This is intimately related to the change of symmetry of the Dirac cone when the particle hopping is different along different directions of the lattice. In particular, for a critical compression, a semi-Dirac cone is formed with massless and massive dispersions along perpendicular directions. Here we show direct evidence of the highly anisotropic transport of polaritons in a honeycomb lattice of coupled micropillars implementing a semi-Dirac cone. If we optically induce a vacancy-like defect in the lattice, we observe an anisotropically localized polariton distribution in a single sublattice, a consequence of the semi-Dirac dispersion. Our work opens up new horizons for the study of transport and localization in lattices with chiral symmetry and exotic Dirac dispersions.
Spin-momentum locking of polariton edge states in honeycomb lattices
Universidad Autónoma de Madrid(马德里自治大学);Università di Padova(帕多瓦大学);Indian Institute of Technology Bombay(孟买印度理工学院);Indian Institute of Technology Dharwad(达尔瓦德印度理工学院);Julius-Maximilians-Universität Würzburg(维尔茨堡朱利叶斯-马克西米利安大学)
Pseudorelativistic Dirac quasiparticles have emerged in a plethora of artificial graphene systems that mimic the underlying honeycomb symmetry of graphene. However, it is notoriously difficult to manipulate their properties without modifying the lattice structure. Here we theoretically investigate polaritons supported by honeycomb metasurfaces and, despite the trivial nature of the resonant elements, we unveil rich Dirac physics stemming from a non-trivial winding in the light-matter interaction. The metasurfaces simultaneously exhibit two distinct species of massless Dirac polaritons, namely …
The experimental study of edge states in atomically-thin layered materials remains a challenge due to the difficult control of the geometry of the sample terminations, the stability of dangling bonds and the need to measure local properties. In the case of graphene, localised edge modes have been predicted in zig-zag and bearded edges, characterised by flat dispersions connecting the Dirac points. Polaritons in semiconductor microcavities have recently emerged as an extraordinary photonic platform to emulate 1D and 2D Hamiltonians, allowing the direct visualization of the wavefunctions in both…
We theoretically explore nonresonantly pumped polaritonic graphene, a system consisting of a honeycomb lattice of micropillars in the regime of strong light-matter coupling. We demonstrate that, depending on the parameters of the structure, such as intensity of the pump and coupling strength between the pillars, the system shows rich variety of macroscopic ordering, including analogs of ferromagnetic, antiferromagnetic, and resonant valence bond phases. Transitions between these phases are associated with dramatic reshaping of the spectrum of the system connected with spontaneous appearance of…
Two-dimensional lattices of coupled micropillars etched in a planar semiconductor microcavity offer a workbench to engineer the band structure of polaritons. We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene. Using energy-resolved photoluminescence we directly observe Dirac cones, around which the dynamics of polaritons is described by the Dirac equation for massless particles. At higher energies, we observe p orbital bands, one of them with the nondispersive character of a flatband. The realization of t…
We report the realization of a synthetic magnetic field for photons and polaritons in a honeycomb lattice of coupled semiconductor micropillars. A strong synthetic field is induced in both the s and p orbital bands by engineering a uniaxial hopping gradient in the lattice, giving rise to the formation of Landau levels at the Dirac points. We provide direct evidence of the sublattice symmetry breaking of the lowest-order Landau level wavefunction, a distinctive feature of synthetic magnetic fields. Our realization implements helical edge states in the gap between n=0 and n=1 Landau levels, expe…
The electronic bandstructure of a solid is a collection of allowed bands separated by forbidden bands, revealing the geometric symmetry of the crystal structures. Comprehensive knowledge of the bandstructure with band parameters explains intrinsic physical, chemical and mechanical properties of the solid. Here we report the artificial polaritonic bandstructures of two-dimensional honeycomb lattices for microcavity exciton-polaritons using GaAs semiconductors in the wide-range detuning values, from cavity-photon-like (red-detuned) to exciton-like (blue-detuned) regimes. In order to understand t…
Light-matter coupling in van der Waal's materials holds significant promise in realizing Bosonic condensation and superfluidity. The underlying semiconductor's crystal asymmetry, if any, can be utilized to form anisotropic half-light half-matter quasiparticles. We demonstrate generation of such highly anisotropic exciton-polaritons at the interface of a biaxial layered semiconductor, stacked on top of a distributed Bragg reflector. The spatially confined photonic mode in this geometry couples with polarized excitons and their Rydberg states, creating a system of highly anisotropic polariton ma…
Exciton polaritons in semiconductor microcavities exhibit many fundamental physical effects, with some of them amenable to being controlled by external fields. The polariton transport is affected by the polaritonic spin-orbit interaction, which is caused by the splitting of transverse-electric and transverse-magnetic (TE-TM) modes. This is the basis for a polaritonic Hall effect, called optical spin Hall effect (OSHE), which is related to the formation of spin/polarization textures in momentum space, determining anisotropic ballistic transport, as well as related textures in real space. Owing …
As one of the most fundamental optical phenomena, refraction between isotropic media is characterized by light bending towards the normal to the boundary when passing from a low- to a high-refractive-index medium. However, in anisotropic media, refraction is a much more exotic phenomenon. The most general case of refraction between two anisotropic media remains unexplored, particularly in natural media and at the nanoscale. Here, we visualize and comprehensively study refraction of electromagnetic waves between two strongly anisotropic (hyperbolic) media, and, importantly, we do it with the us…
We derive the low-energy Hamiltonian for a honeycomb lattice with anisotropy in the hopping parameters. Taking the reported Dirac Hamiltonian for the anisotropic honeycomb lattice, we obtain its optical conductivity tensor and its transmittance for normal incidence of linearly polarized light. Also, we characterize its dichroic character due to the anisotropic optical absorption. As an application of our general findings, which reproduce the case of uniformly strained graphene, we study the optical properties of graphene under a nonmechanical distortion.
In recent years, novel materials supporting in-plane anisotropic polaritons have attracted a lot of research interest due to their capability of shaping nanoscale field distributions and controlling nanophotonic energy flows. Here we report a nano-optical imaging study of waveguide exciton polaritons (EPs) in tin sulfide (SnS) in the near-infrared (IR) region using the scattering-type scanning near-field optical microscopy (s-SNOM). With s-SNOM, we mapped in real space the propagative EPs in SnS, which show sensitive dependence on the excitation energy and sample thickness. Moreover, we found …
We determine experimentally the spinor interaction constants of lower-polaritons $α_1$ and $α_2$ using resonant pump-probe spectroscopy with a spectrally narrow pump pulse. Our experimental findings are analyzed with the Bogoliubov-like theory and mean-field two channel model based on the lower-polariton and biexciton basis. We find an enhancement of the attractive interaction and dissipative nonlinearity of lower-polaritons with anti-parallel spins in the vicinity of the biexciton resonance when the energy of two lower-polaritons approaches energetically that of the biexciton. These observati…
Polaritons in polar biaxial crystals with extreme anisotropy offer a promising route to manipulate nanoscale light-matter interactions. The dynamical modulation of their dispersion is great significance for future integrated nano-optics but remains challenging. Here, we report a momentum-directed strategy, a coupling between the modes with extra momentum supported by the interface and in-plane hyperbolic polaritons, to tailor topological transitions of anisotropic polaritons in biaxial crystals. We experimentally demonstrate such tailored polaritons at the interface of heterostructures between…
In this paper we reveal the physics behind the formation of tri- and tetra-hyperbolic phases in anisotropic metamaterials without magnetoelectric coupling and describe the anti-crossing splitting phenomenon in the hyperbolic dispersion which arises due to the hybridization of the plasmonic and magnetic Bloch high-k polaritons. This considerably deepens the understanding of the high-k polaritons and the topology of the optical iso-frequency surfaces in k-space and will find applications in optical nano-resolution imaging and emission rate and directivity control. To accomplish this, we develop …
We explore the exciton-polariton condensation in the two degenerate orbital states. In the honeycomb lattice potential, at the third band we have two degenerate vortex-antivortex lattice states at the inequivalent K and K'-points. We have observed energetically degenerate condensates within the linewidth ~ 0.3 meV, and directly measured the vortex-antivortex lattice phase order of the order parameter. We have also observed the intensity anticorrelation between polariton condensates at the K- and K'-points. We relate this intensity anticorrelation to the dynamical feature of polariton condensat…
We consider exciton-polaritons in a honeycomb lattice of micropillars subjected to circularly polarized (${σ_\pm}$) 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, ${σ_\pm}$ polaritons propagate in different directions without being reflected around bends. The polaritons propagating along the interface have extremely high effective lifeti…
Theory anticipates that the in-plane px, py orbitals in a honeycomb lattice lead to new and potentially useful quantum electronic phases. So far, p orbital bands were only realized for cold atoms in optical lattices and for light and exciton-polaritons in photonic crystals. For electrons, in-plane p orbital physics is difficult to access since natural electronic honeycomb lattices, such as graphene and silicene, show strong s-p hybridization. Here, we report on electronic honeycomb lattices prepared on a Cu(111) surface in a scanning tunneling microscope that, by design, show (nearly) pure orb…
We study the repulsive Hubbard model on an anisotropic honeycomb lattice within a mean-field and a slave-rotor treatment. In addition to the known semi-metallic and band-insulating phases, obtained for very weak interactions, and the anti-ferromagnetic phase at large couplings, various insulating spin-liquid phases develop at intermediate couplings. Whereas some of these spin liquids have gapless spinon excitations, a gapped one occupies a large region of the phase diagram and becomes the predominant phase for large hopping anisotropies. This phase can be understood in terms of weakly-coupled …
Natural hyperbolic materials (HMs) in two dimensions (2D) have an extraordinarily high anisotropy and a hyperbolic dispersion relation. Some of them can even sustain hyperbolic polaritons with great directional propagation and light compression to deeply sub-wavelength scales due to their inherent anisotropy. Herein, the anisotropic optical features of 2D natural HMs are reviewed. Four hyperbolic polaritons (i.e., phonon polaritons, plasmon polaritons, exciton-polaritons, and shear polaritons) as well as their generation mechanism are discussed in detail. The natural merits of 2D HMs hold prom…
Terahertz (THz) electromagnetic radiation is key to optically access collective excitations such as magnons (spins), plasmons (electrons), or phonons (atomic vibrations), thus bridging between optics and solid-state physics. Confinement of THz light to the nanometer length scale is desirable for local probing of such excitations in low dimensional systems, thereby inherently circumventing the large footprint and low spectral density of far-field THz optics. For that purpose, phonon polaritons (PhPs, i.e., light coupled to lattice vibrations in polar crystals) in anisotropic van der Waals (vdW)…
We present systematic study of zero modes and gaps by introducing effects of anisotropy of hopping integrals for a tight-binding model on the honeycomb lattice in a magnetic field. The condition for the existence of zero modes is analytically derived. From the condition, it is found that a tiny anisotropy for graphene is sufficient to open a gap around zero energy in a magnetic field. This gap behaves as a non-perturbative and exponential form as a function of the magnetic field. The non-analytic behavior with respect to the magnetic field can be understood as tunneling effects between energy …
Propagating, directionally dependent, polarized spin-currents are created in an anisotropic planar semiconductor microcavity, via Rayleigh scattering of optically injected polaritons in the optical spin Hall regime. The influence of anisotropy results in the suppression or enhancement of the pseudospin precession of polaritons scattered into different directions. This is exploited to create intense spin currents by excitation on top of localized defects. A theoretical model considering the influence of the total effective magnetic field on the polariton pseudospin quantitatively reproduces the…
In this paper, we investigate the band properties of 2D honeycomb plasmonic lattices consisting of metallic nanoparticles. By means of the coupled dipole method and quasi-static approximation, we theoretically analyze the band structures stemming from near-field interaction of localized surface plasmon polaritons for both the infinite lattice and ribbons. Naturally, the interaction of point dipoles decouples into independent out-of-plane and in-plane polarizations. For the out-of-plane modes, both the bulk spectrum and the range of the momentum $k_{\parallel}$ where edge states exist in ribbon…
We study the formation of topological defects (quantum vortices) during the formation of a 2D polariton condensate at the $Γ$ point of a honeycomb lattice via the Kibble-Zurek mechanism. The lattice modifies the single-particle dispersion. The typical interaction energies at the quench time correspond to the linear part of the dispersion. The resulting scaling exponent for the density of topological defects is numerically found as $0.95\pm0.05$. This value differs from the one expected for 2D massive particles (1/2), but is indeed compatible with the one expected for a linear dispersion. We mo…
Modification of interatomic distances due to high pressure leads to exotic phenomena, including metallicity, superconductivity and magnetism, observed in materials not showing such properties in normal conditions. In two-dimensional crystals, such as graphene, atomic bond lengths can be modified by more that 10 percent by applying in-plane strain, i.e., without generating high pressure in the bulk. In this work, we study the strain-induced Mott transition on a honeycomb lattice by using computationally inexpensive techniques, including Gutzwiller Wave Function (GWF) and different variants of G…
Transport properties of exciton-polaritons in anisotropic quantum microcavities are considered theoretically. Microscopic symmetry of the structure is taken into account by allowing for both the longitudinal-transverse (TE-TM) and anisotropic splitting of polariton states. The splitting is equivalent to an effective magnetic field acting on polariton pseudospin, and polarization conversion in microcavities is shown to be caused by an interplay of exciton-polariton spin precession and elastic scattering. In addition, we considered the spin-dependent interference of polaritons leading to weak lo…
The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.
We report polariton lasing in localised gap states in a honeycomb lattice of coupled micropillars. Localisation of the modes is induced by the optical potential created by the excitation beam, requiring no additional engineering of the otherwise homogeneous polariton lattice. The spatial shape of the gap states arises from the interplay of the orbital angular momentum eigenmodes of the cylindrical potential created by the excitation beam and the hexagonal symmetry of the underlying lattice. Our results provide insights into the engineering of defect states in two-dimensional lattices.
By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation.…
A convenient scheme is presented for calculating potential energy of van der Waals interacting bilayer graphene and other similar 2D compounds. It is based on the notion of the existence of two types of local symmetry of carbon atoms ordering, a 3- and 6-fold one. Potential energy of an atom is expressed as a sum of contributions from rings of equidistant atoms on neighboring layer. Methods are described to compute the radius of rings of equidistant atoms and number of atoms they contain. Exact positions of atoms are found as well, allowing to apply the introduced method in modelling of anisot…
We study the formation of Majorana fermions in honeycomb-lattice structures in the presence of a Zeeman field, Rashba spin-orbit coupling, and in the proximity of an s-wave superconductor. We show that an exact mapping exists between an anisotropic hexagonal-lattice nanoribbon at k = 0 and a one-dimensional chain, for which the existence of Majorana fermions has been extensively discussed. Consequently we can predict the conditions for the emergence of Majorana fermions at the edges of such ribbon, and relate the existence of Majoranas to a band inversion in the bulk band structure. Moreover w…
We propose a family of free fermion lattice models that have "Dirac loops", closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply-connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility…