  A defining property of particles is their behavior under exchange. In two
dimensions anyons can exist which, opposed to fermions and bosons, gain
arbitrary relative phase factors or even undergo a change of their type. In the
latter case one speaks of non-Abelian anyons - a particularly simple and
aesthetic example of which are Fibonacci anyons. They have been studied in the
context of fractional quantum Hall physics where they occur as quasiparticles
in the $k=3$ Read-Rezayi state, which is conjectured to describe a fractional
quantum Hall state at filling fraction $\nu=12/5$. Here we show that the
physics of interacting Fibonacci anyons can be studied with strongly
interacting Rydberg atoms in a lattice, when due to the dipole blockade the
simultaneous laser excitation of adjacent atoms is forbidden. The Hilbert space
maps then directly on the fusion space of Fibonacci anyons and a proper tuning
of the laser parameters renders the system into an interacting topological
liquid of non-Abelian anyons. We discuss the low-energy properties of this
system and show how to experimentally measure anyonic observables.
