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In systems with a real Bloch Hamiltonian band nodes can be characterised by anon-Abelian frame-rotation charge. The ability of these band nodes toannihilate pairwise is path dependent, since by braiding nodes in adjacent gapsthe sign of their charges can be changed. Here, we theoretically construct andnumerically confirm two concrete methods to experimentally probe thesenon-Abelian braiding processes and charges in ultracold atomic systems. Weconsider a coherent superposition of two bands that can be created by movingatoms through the band singularities at some angle in momentum space. Analyzingthe dependency on the frame charges, we demonstrate an interferometry schemepassing through two band nodes, which reveals the relative frame charges andallows for measuring the multi-gap topological invariant. The second methodrelies on a single wavepacket probing two nodes sequentially, where the framecharges can be determined from the band populations. Our results present afeasible avenue for measuring non-Abelian charges of band nodes and theexperimental verification of braiding procedures directly, which can be appliedin a variety of settings including the recently discovered anomalousnon-Abelian phases arising under periodic driving.

Interferometry of non-Abelian band singularities and Euler class topology