Nodes and Spin Windings for Topological Transitions in Light-Matter Interactions

The anisotropic quantum Rabi model (QRM) is the fundamental model of light-matter interactions with indispensable counter-rotating terms in ultra-strong couplings. By extracting different levels of topological information a new light is shed on the energy spectrum of the anisotropic QRM. Besides conventional topological transitions (TTs) at gap closing, abundant unconventional TTs are unveiled underlying level anticrossings without gap closing by tracking the wave-function nodes, including a particular unconventional TT which turns out to be universal for different energy levels. On the other hand, it is found that the nodes have a correspondence to spin windings, which not only endows the nodes a more explicit topological character in supporting single-qubit TTs but also turns the topological information physically detectable. Furthermore, hidden small-spin-knot transitions are exposed for the ground state, while more kinds of spin-knot transitions emerge in excited states including unmatched node numbers and spin winding numbers. Based on node sorting, algebraic formulation is established to decode the topological information encoded geometrically in the wave functions and spin windings.