  Experimental realizations of topologically ordered states of matter, such as
fractional quantum Hall states, with cold atoms are now within reach. In
particular, optical lattices provide a promising platform for the realization
and characterization of such states, where novel detection schemes enable an
unprecedented microscopic understanding. Here we show that the central charge
can be directly measured in current cold atom experiments using the number
entropy as a proxy for the entanglement entropy. We perform density-matrix
renormalization-group simulations of Hubbard-interacting bosons on coupled
chains subject to a magnetic field with $\alpha=\frac{1}{4}$ flux quanta per
plaquette. Tuning the inter-chain hopping, we find a transition from a trivial
quasi-one dimensional phase to the topologically ordered Laughlin state at
magnetic filling factor $\nu=\frac{1}{2}$ for systems of three or more chains.
We resolve the transition using the central charge, on-site correlations,
momentum distributions and the many-body Chern number. Additionally, we propose
a scheme to experimentally estimate the central charge from Fock basis
snapshots. The model studied here is experimentally realizable with existing
cold atom techniques and the proposed observables pave the way for the
detection and classification of a larger class of interacting topological
states of matter.
