  A remarkable property of quantum mechanics in two-dimensional (2D) space is
its ability to support "anyons," particles that are neither fermions nor
bosons. Theory predicts that these exotic excitations can be realized as bound
states confined near topological defects, like Majorana zero modes trapped in
vortices in topological superconductors. Intriguingly, in the simplest cases
the nontrivial phase that arises when such defects are "braided" around one
another is not intrinsically quantum mechanical; rather, it can be viewed as a
manifestation of the geometric (Pancharatnam-Berry) phase in wave mechanics,
enabling the simulation of such phenomena in classical systems. Here we report
the first experimental measurement in any system, quantum or classical, of the
geometric phase due to such a braiding process. These measurements are obtained
using an interferometer constructed from highly tunable 2D arrays of photonic
waveguides. Our results introduce photonic lattices as a versatile playground
for the experimental study of topological defects and their braiding,
complementing ongoing efforts in solid-state systems and cold atomic gases.
