Fig. 3: Intensity oscillations fit and the Bloch metric maps.

a Photonic oscillations at a specific point (experimental data) are fitted by the two interfering LGs model (see “Methods”). The procedure allows to retrieve the intensities of the two normal modes in time (solid purple and green lines for the UP and LP modes, respectively) and their relative phase at each point and moment of time. Repeating the procedure all along the spatial domain allows to plot their 2D profiles. Inset: the polariton Bloch sphere with indicated UP, LP states at the poles, photon (C) and exciton (X) states at the equator, and the sphere metric consisting in the azimuthal angle φ_LU (relative phase) and \(s=\cos \theta\) (local UP-LP content imbalance). An example of area item \({{{{{{{\rm{d}}}}}}}}{A}^{{{{{{{{\rm{sphere}}}}}}}}}={l}_{{{{{{{{\rm{p}}}}}}}}}{l}_{{{{{{{{\rm{m}}}}}}}}}=\sin \theta {{{{{{{\rm{d}}}}}}}}\varphi {{{{{{{\rm{d}}}}}}}}\theta\) is marked by the orange patch. b, c Relative phase map φ_LU(x, y) and local content imbalance s(x, y) (at t = 2.7 ps). The superimposed lines (red, blue) represent the experimental orbits of photonic vortex cores along the second Rabi cycle after the pulse B arrival (time range t = 2.7–3.5 ps). One clearly observes their rotation around the two relative phase singularities corresponding to the two LP-mode vortex cores. The asymmetry manifests itself both in the fact that the UP mode cores are more distant from each other than the two LP cores, and that the displacement lines of sight are mutually oblique. The superimposed orange patches in (b) mark the real-space area elements dA^real corresponding to a given element on the sphere. d Module of the total density ∣Ψ_tot(x, y)∣ (with ∣Ψ_tot∣² = ∣ψ_L∣² + ∣ψ_U∣²). The shape is a footprint mainly of the LP mode. e Normalized Berry curvature map in log scale, retrieved from the maps in (b, c).