  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 $\nu = 7/3$ FQH state through detection of
the predicted statistical phase angle of $2\pi/3$, consistent with a change of
the anyonic particle number by one. The $\nu = 5/2$ 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.
