Fig. 3

Genuine non-Abelian AB effect for light. a Sketch of the non-Abelian AB system with two optical paths \(\gamma _{\mathrm{I}}\), \(\gamma _{{\mathrm{II}}}\) interfering on the screen, where the background light blue (red) arrows denote the \(\hat \sigma _1\) (\(\hat \sigma _2\)) component \({\cal{A}}^1\) \(({\cal{A}}^2)\) of the non-Abelian vector potential. b \(\gamma _{\mathrm{I}}\) (\(\gamma _{{\mathrm{II}}}\)) can be divided into a closed loop \(c_{\mathrm{I}}\) (\(c_{{\mathrm{II}}}\)) and a common path γ₀. c, d c_I and c_II can, respectively, deform continuously into a closed path that winds around the two vortices successively but in opposite sequences. e Snapshot of the simulated field intensity for the proposed non-Abelian optical interferometer with incident spinor \((1,{\mathrm{i}}1/5)^ \top\) for both beams and the vortex fluxes \({\mathrm{\Phi }}_1 = - 2\pi /3\), \({\mathrm{\Phi }}_2 = - \pi /3\). f Spin evolution on the Bloch sphere along two beams \(\gamma _{\mathrm{I}}\), \(\gamma _{{\mathrm{II}}}\), which share the same initial spin \(\vec s_0\) but achieve different final spins \(\vec s_{\mathrm{I}}\) and \(\vec s_{{\mathrm{II}}}\). g Spin density interference corresponding to e, where each arrow denotes the local pseudo-spin density \(|\psi |^2\vec s\) at a point on the screen. All of the local spins \(\vec s(y)\) are perpendicular to \(\Delta \vec s = \vec s_{\mathrm{I}} - \vec s_{{\mathrm{II}}}\), and thus fall on the green circle in f. The corresponding intensity interference \(|\psi |^2(y)\) and the two Euler angles α, β of the local spins \(\vec s(y)\) on the screen are shown in h–j, where blue circles and red curves indicate simulated and theoretical results, respectively, and \(\delta \theta\), b are the phase shift and relative amplitude relative to the case of \(\hat {\cal{A}} = 0\) (L₀ is the period of \(\Delta \theta (y)\ {\mathrm{{mod}}}\ 2\pi\)). The green lines correspond to the “control experiment”