  We propose a scheme to realize the fractional quantum Hall system with atoms
confined in a two-dimensional array of coupled cavities. Our scheme is based on
simple optical manipulation of atomic internal states and inter-cavity hopping
of virtually excited photons. It is shown that as well as the fractional
quantum Hall system, any system of hard-core bosons on a lattice in the
presence of an arbitrary Abelian vector potential can be simulated solely by
controlling the phases of constantly applied lasers. The scheme, for the first
time, exploits the core advantage of coupled cavity simulations, namely the
individual addressability of the components and also brings the gauge potential
into such simulations as well as the simple optical creation of particles.
