Measurement of the quantum geometric tensor and of the anomalous Hall drift.

Topological physics relies on the structure of the eigenstates of the Hamiltonians. The geometry of the eigenstates is encoded in the quantum geometric tensor1-comprising the Berry curvature2 (crucial for topological matter)3 and the quantum metric4, which defines the distance between the eigenstates. Knowledge of the quantum metric is essential for understanding many phenomena, such as superfluidity in flat bands5, orbital magnetic susceptibility6,7, the exciton Lamb shift8 and the non-adiabatic anomalous Hall effect6,9. However, the quantum geometry of energy bands has not been measured. Here we report the direct measurement of both the Berry curvature and the quantum metric in a two-dimensional continuous medium-a high-finesse planar microcavity10-together with the related anomalous Hall drift. The microcavity hosts strongly coupled exciton-photon modes (exciton polaritons) that are subject to photonic spin-orbit coupling11 from which Dirac cones emerge12, and to exciton Zeeman splitting, breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured using polarization-resolved photoluminescence. The associated quantum geometry of the bands is extracted, enabling prediction of the anomalous Hall drift, which we measure independently using high-resolution spatially resolved epifluorescence. Our results unveil the intrinsic chirality of photonic modes, the cornerstone of topological photonics13-15. These results also experimentally validate the semiclassical description of wavepacket motion in geometrically non-trivial bands9,16. The use of exciton polaritons (interacting photons) opens up possibilities for future studies of quantum fluid physics in topological systems.

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.

Particlelike Behavior of Topological Defects in Linear Wave Packets in Photonic Graphene.

Topological defects, such as quantum vortices, determine the properties of quantum fluids. Their study has been at the center of activity in solid state and BEC communities. In parallel, the nontrivial behavior of linear wave packets with complex phase patterns was investigated by singular optics. Here, we study the formation, evolution, and interaction of optical vortices in wave packets at the Dirac point in photonic graphene. We show that while their exact behavior goes beyond the Dirac equation and requires a full account of the lattice properties, it can be still approximately described by an effective theory considering the phase singularities as "particles". These particles are capable of mutual interaction, with their trajectory obeying the laws of dynamics.

Stationary Quantum Vortex Street in a Driven-Dissipative Quantum Fluid of Light.

We investigate the formation of a new class of density-phase defects in a resonantly driven 2D quantum fluid of light. The system bistability allows the formation of low-density regions containing density-phase singularities confined between high-density regions. We show that, in 1D channels, an odd (1 or 3) or even (2 or 4) number of dark solitons form parallel to the channel axis in order to accommodate the phase constraint induced by the pumps in the barriers. These soliton molecules are typically unstable and evolve toward stationary symmetric or antisymmetric arrays of vortex streets straightforwardly observable in cw experiments. The flexibility of this photonic platform allows implementing more complicated potentials such as mazelike channels, with the vortex streets connecting the entrances and thus solving the maze.

Effective Theory of Nonadiabatic Quantum Evolution Based on the Quantum Geometric Tensor.

We study the role of the quantum geometric tensor (QGT) in the evolution of two-band quantum systems. We show that all its components play an important role on the extra phase acquired by a spinor and on the trajectory of an accelerated wave packet in any realistic finite-duration experiment. While the adiabatic phase is determined by the Berry curvature (the imaginary part of the tensor), the nonadiabaticity is determined by the quantum metric (the real part of the tensor). We derive, for geodesic trajectories (corresponding to acceleration from zero initial velocity), the semiclassical equations of motion with nonadiabatic corrections. The particular case of a planar microcavity in the strong coupling regime allows us to extract the QGT components by direct light polarization measurements and to check their effects on the quantum evolution.

Chirality of Topological Gap Solitons in Bosonic Dimer Chains.

We study gap solitons which appear in the topological gap of 1D bosonic dimer chains within the mean-field approximation. We find that such solitons have a nontrivial texture of the sublattice pseudospin. We reveal their chiral nature by demonstrating the anisotropy of their behavior in the presence of a localized energy potential.

Kibble-Zurek Mechanism in Topologically Nontrivial Zigzag Chains of Polariton Micropillars.

We consider a zigzag chain of coupled micropillar cavities, taking into account the polarization of polariton states. We show that the TE-TM splitting of photonic cavity modes yields topologically protected polariton edge states. During the strongly nonadiabatic process of polariton condensation, the Kibble-Zurek mechanism leads to a random choice of polarization, equivalent to the dimerization of polymer chains. We show that dark-bright solitons appear as domain walls between polarization domains, analogous to the Su-Schrieffer-Heeger solitons in polymers. The soliton density scales as a power law with respect to the quenching parameter.

Spin-orbit coupling and the optical spin Hall effect in photonic graphene.

We study the spin-orbit coupling induced by the splitting between TE and TM optical modes in a photonic honeycomb lattice. Using a tight-binding approach, we calculate analytically the band structure. Close to the Dirac point, we derive an effective Hamiltonian. We find that the local reduced symmetry (D_{3h}) transforms the TE-TM effective magnetic field into an emergent field with a Dresselhaus symmetry. As a result, particles become massive, but no gap opens. The emergent field symmetry is revealed by the optical spin Hall effect.

Polariton Z topological insulator.

We demonstrate that honeycomb arrays of microcavity pillars behave as an optical-frequency two-dimensional photonic 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 nontrivial gap characterized by a C=±2 set of band Chern numbers and to the formation of topologically protected one-way edge states.

High-speed DC transport of emergent monopoles in spinor photonic fluids.

We investigate the spin dynamics of half-solitons in quantum fluids of interacting photons (exciton polaritons). Half-solitons, which behave as emergent monopoles, can be accelerated by the presence of effective magnetic fields. We study the generation of dc magnetic currents in a gas of half-solitons. At low densities, the current is suppressed due to the dipolar oscillations. At moderate densities, a magnetic current is recovered as a consequence of the collisions between the carriers. We show a deviation from Ohm's law due to the competition between dipoles and monopoles.

Non-Abelian gauge fields in photonic cavities and photonic superfluids.

We show that the TE-TM modes splitting and the structure anisotropy of a semiconductor microcavity combine into a non-Abelian gauge field for exciton-polaritons or cavity photons. The field texture can be tuned simply by rotating the sample and ranges continuously from a Rashba to a monopolar field. In the noninteracting regime, the latter leads to remarkable focusing and conical diffraction effects. In the interacting regime, the spin-orbit coupling induces a breakdown of superfluidity. The spatially homogeneous flows become unstable and dynamically evolve into spin textured states, such as stripes or domain walls.

Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons.

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 this structure which holds massless, massive, and infinitely massive particles opens the route towards studies of the interplay of dispersion, interactions, and frustration in a novel and controlled environment.

Topological Wigner crystal of half-solitons in a spinor Bose-Einstein condensate.

We consider a one-dimensional gas of half-solitons in a spinor Bose-Einstein condensate. We calculate the topological interaction potential between the half-solitons. Using a kinetic equation of the Vlasov-Boltzmann type, we model the coupled dynamics of the interacting solitons. We show that the dynamics of the system in the gaseous phase is marginally stable and spontaneously evolves toward a Wigner crystal.

Transmutation of skyrmions to half-solitons driven by the nonlinear optical spin Hall effect.

We show that the spin domains, generated in the linear optical spin Hall effect by the analog of spin-orbit interaction for exciton polaritons, are associated with the formation of a Skyrmion lattice. In the nonlinear regime, the spin anisotropy of the polariton-polariton interactions results in a spatial compression of the domains and in a transmutation of the Skyrmions into oblique half-solitons. This phase transition is associated with both the focusing of the spin currents and the emergence of a strongly anisotropic emission pattern.

Skyrmion formation and optical spin-Hall effect in an expanding coherent cloud of indirect excitons.

We provide a theoretical description of the polarization pattern and phase singularities experimentally evidenced recently in a condensate of indirect excitons [H. High et al., Nature 483, 584 (2012)]. We show that the averaging of the electron and hole orbital motion leads to a comparable spin-orbit interaction for both types of carriers. We demonstrate that the interplay between a radial coherent flux of bright indirect excitons and the Dresselhaus spin-orbit interaction results in the formation of spin domains and of topological defects similar to Skyrmions. We reproduce qualitatively all the features of the experimental data and obtain a polarization pattern as in the optical spin-Hall effect despite the different symmetry of the spin-orbit interactions.

Realization of a double-barrier resonant tunneling diode for cavity polaritons.

We report on the realization of a double-barrier resonant tunneling diode for cavity polaritons, by lateral patterning of a one-dimensional cavity. Sharp transmission resonances are demonstrated when sending a polariton flow onto the device. We show that a nonresonant beam can be used as an optical gate and can control the device transmission. Finally, we evidence distortion of the transmission profile when going to the high-density regime, signature of polariton-polariton interactions.

From excitonic to photonic polariton condensate in a ZnO-based microcavity.

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.

Propagation and amplification dynamics of 1D polariton condensates.

The dynamics of propagating polariton condensates in one-dimensional microcavities is investigated through time resolved experiments. We find a strong increase in the condensate intensity when it travels through the nonresonantly excited area. This amplification is shown to come from bosonic stimulated relaxation of reservoir excitons into the polariton condensate, allowing for the repopulation of the condensate through nonresonant pumping. Thus, we experimentally demonstrate a polariton amplifier with a large band width, opening the way towards the transport of polaritons with high densities over macroscopic distances.

Backscattering suppression in supersonic 1D polariton condensates.

We investigate the effect of disorder on the propagation of one-dimensional polariton condensates in semiconductor microcavities. We observe a strong suppression of the backscattering produced by the imperfections of the structure when increasing the condensate density. This suppression occurs in the supersonic regime and is simultaneous to the onset of parametric instabilities which enable the "hopping" of the condensate through the disorder. Our results evidence a new mechanism for the strong scattering reduction of polaritons at high speeds.

Annihilation of exceptional points from different Dirac valleys in a 2D photonic system.

Topological physics relies on Hamiltonian's eigenstate singularities carrying topological charges, such as Dirac points, and - in non-Hermitian systems - exceptional points (EPs), lines or surfaces. So far, the reported non-Hermitian topological transitions were related to the creation of a pair of EPs connected by a Fermi arc out of a single Dirac point by increasing non-Hermiticity. Such EPs can annihilate by reducing non-Hermiticity. Here, we demonstrate experimentally that an increase of non-Hermiticity can lead to the annihilation of EPs issued from different Dirac points (valleys). The studied platform is a liquid crystal microcavity with voltage-controlled birefringence and TE-TM photonic spin-orbit-coupling. Non-Hermiticity is provided by polarization-dependent losses. By increasing the non-Hermiticity degree, we control the position of the EPs. After the intervalley annihilation, the system becomes free of any band singularity. Our results open the field of non-Hermitian valley-physics and illustrate connections between Hermitian topology and non-Hermitian phase transitions.