Fig. 1: The equivalent two-level system in the higher-order regime.

a The original spin-1/2 setting describes the spinnings of a quantum particle in a driven magnetic field, where the spin up \((\varrho =+1/2)\) and spin down \((\varrho =-1/2)\) constitute the two-level eigenstates. b In the higher-order optical regime, the eigenstates \(\hat{R}\) and \(\hat{L}\) simultaneously coupling with the SAM and OAM are defined as the spin up \((-\varrho {\mathscr{+}}{\mathscr{l}}{\mathscr{)}}\) and spin down \((+\varrho {\mathscr{-}}{\mathscr{l}}{\mathscr{)}}\) equivalents in a pseudo spin-1/2 system, respectively. These pseudo spin up and spin down are coupled by a synthetic magnetic field. c The higher-order Poincaré sphere is introduced to represent the spin–orbit states, with two poles denoting the eigenstates. d Typical states mapped on the sphere (marked points in c) showing spatial variations of polarization (upper panels) and phase (bottom panels) distributions with polar angle. The arrows in the upper panels denote polarizations; while the black lines in the bottom panels represent the phase contours