N. Y. Kim, K. Kusudo, A. Löffler, S. Höfling, A. Forchel, Y. Yamamoto

• Dirac particles, massless relativistic entities, obey linear energy dispersions and hold important implications in particle physics. The recent discovery of Dirac fermions in condensed matter systems including graphene and topological insulators has generated a great deal of interest in exploring the relativistic properties associated with Dirac physics in solid-state materials. In addition, there are stimulating research activities to engineer Dirac particles, elucidating their exotic physical properties in a controllable setting. One of theDirac particles, massless relativistic entities, obey linear energy dispersions and hold important implications in particle physics. The recent discovery of Dirac fermions in condensed matter systems including graphene and topological insulators has generated a great deal of interest in exploring the relativistic properties associated with Dirac physics in solid-state materials. In addition, there are stimulating research activities to engineer Dirac particles, elucidating their exotic physical properties in a controllable setting. One of the successful platforms is the ultracold atom-optical lattice system, whose dynamics can be manipulated and probed in a clean environment. A microcavity exciton-polariton-lattice system offers the advantage of forming high-orbital condensation in non-equilibrium conditions, which enables one to explore novel quantum orbital order in two dimensions. In this paper, we experimentally construct the band structures near Dirac points, the vertices of the first hexagonal Brillouin zone with exciton-polariton condensates trapped in a triangular lattice. Due to the finite spectral linewidth, the direct map of band structures at Dirac points is elusive; however, we identify the linear part above Dirac points and its associated velocity value is similar to ~0.9-2 x \(10^8 cm s^{-1}\), consistent with the theoretical estimate \(1 x 10^8 cm s^{-1}\) with a \(2 \mu m\) lattice constant. We envision that the exciton-polariton condensates in lattices would be a promising solid-state platform, where the system order parameter can be accessed in both real and momentum spaces.…