  Anyons are exotic quasi-particles with fractional charge that can emerge as
fundamental excitations of strongly interacting topological quantum phases of
matter. Unlike ordinary fermions and bosons, they may obey non-abelian
statistics--a property that would help realize fault tolerant quantum
computation. Non-abelian anyons have long been predicted to occur in the
fractional quantum Hall (FQH) phases that form in two-dimensional electron
gases (2DEG) in the presence of a large magnetic field, su ch as the
$\nu=\tfrac{5}{2}$ FQH state. However, direct experimental evidence of anyons
and tests that can distinguish between abelian and non-abelian quantum ground
states with such excitations have remained elusive. Here we propose a new
experimental approach to directly visualize the structure of interacting
electronic states of FQH states with the scanning tunneling microscope (STM).
Our theoretical calculations show how spectroscopy mapping with the STM near
individual impurity defects can be used to image fractional statistics in FQH
states, identifying unique signatures in such measurements that can distinguish
different proposed ground states. The presence of locally trapped anyons should
leave distinct signatures in STM spectroscopic maps, and enables a new approach
to directly detect - and perhaps ultimately manipulate - these exotic
quasi-particles.
