Strongly correlated quantum systems can exhibit exotic behaviour called topological order which is characterized by non-local correlations that depend on the system topology. Such systems can exhibit remarkable phenomena such as quasiparticles with anyonic statistics and have been proposed as candidates for naturally error-free quantum computation. However, anyons have never been observed in nature directly. Here, we describe how to unambiguously detect and characterize such states in recently proposed spin-lattice realizations using ultracold atoms or molecules trapped in an optical lattice. We propose an experimentally feasible technique to access non-local degrees of freedom by carrying out global operations on trapped spins mediated by an optical cavity mode. We show how to reliably read and write topologically protected quantum memory using an atomic or photonic qubit. Furthermore, our technique can be used to probe statistics and dynamics of anyonic excitations.