In this paper, we report on the study of Abelian and non-Abelian statistics through Fabry-Perot interferometry of fractional quantum Hall (FQH) systems. Our detection of phase slips in quantum interference experiments demonstrates a powerful, new way of detecting braiding of anyons. We confirm the Abelian anyonic braiding statistics in the $ u = 7/3$ FQH state through detection of the predicted statistical phase angle of $2pi/3$, consistent with a change of the anyonic particle number by one. The $ u = 5/2$ FQH state is theoretically believed to harbor non-Abelian anyons which are Majorana, meaning that each pair of quasiparticles contain a neutral fermion orbital which can be occupied or unoccupied and hence can act as a qubit. In this case our observed statistical phase slips agree with a theoretical model where the Majoranas are strongly coupled to each other, and strongly coupled to the edge modes of the interferometer. In particular, an observed phase slip of approximately $pi$ is interpreted as a sudden flip of a qubit, or entry of a neutral fermion into the interferometer. Our results provide compelling support for the existence of non-Abelian anyons.
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