  Topological quantum phases cannot be characterized by Ginzburg-Landau type
order parameters, and are instead described by non-local topological
invariants. Experimental platforms capable of realizing such exotic states now
include "synthetic" many-body systems such as ultracold atoms or photons.
Unique tools available in these systems enable a new characterization of
strongly correlated many-body states. Here we propose a general scheme for
detecting topological order using interferometric measurements of elementary
excitations. The key ingredient is the use of mobile impurities which bind to
quasiparticles of a host many-body system. Specifically we show how fractional
charges can be probed in the bulk of fractional quantum Hall systems. We
demonstrate that combining Ramsey interference with Bloch oscillations can be
used to measure Chern numbers of individual quasiparticles, which gives a
direct probe of their fractional charges. We discuss possible extensions of our
method to other topological many-body systems, such as spin liquids.
