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.

Long-lived polariton condensates can propagate well beyond the area of their initial excitation while still maintaining spatial coherence. This enables direct and controllable manipulation of the condensate wavefunction. Cavity exciton-polaritons1,2 (polaritons) are bosonic quasi-particles offering a unique solid-state system for investigating interacting condensates3,4,5,6,7,8,9,10. Up to now, disorder-induced localization and short lifetimes4,6,11 have prevented the establishment of long-range off-diagonal order12 needed for any quantum manipulation of the condensate wavefunction. In this work, using a wire microcavity with polariton lifetimes much longer than in previous samples, we show that polariton condensates can propagate over macroscopic distances outside the excitation area, while preserving their spontaneous spatial coherence. An extended condensate wavefunction builds up with a degree of spatial coherence larger than 50% over distances 50 times the polariton de Broglie wavelength. The expansion of the condensate is shown to be governed by the repulsive potential induced by photogenerated excitons within the excitation area. The control of this local potential offers a new and versatile method to manipulate extended polariton condensates. As an illustration, we demonstrate synchronization of extended condensates by controlled tunnel coupling13,14 and localization of condensates in a trap with optically controlled dimensions.

Polariton lasing is demonstrated on the zero-dimensional states of single $\mathrm{GaAs}/\mathrm{GaAlAs}$ micropillar cavities. Under nonresonant excitation, the measured polariton ground-state occupancy is found as large as ${10}^4$. Changing the spatial excitation conditions, competition between several polariton lasing modes is observed, ruling out Bose-Einstein condensation. When the polariton state occupancy increases, the emission blueshift is the signature of self-interaction within the half-light half-matter polariton lasing mode.