  The role of spontaneous emission in the interaction between a two-level atom
and a pumped micro-cavity in the strong coupling regime is discussed in this
paper. Especially, using a quantum Monte-Carlo simulation, we investigate
atomic spatial coherence. It is found that atomic spontaneous emission destroys
the coherence between neighboring lattice sites, while the cavity decay does
not. Furthermore, our computation of the spatial coherence function shows that
the in-site locality is little affected by the cavity decay, but greatly
depends on the cavity pump amplitude.
