  We examine the decoherence properties of a quantum open system as modeled by
a quantum optical system in the Markov regime. We look for decoherence in both
the Environment Induced Superselection (EIS) and Consistent Histories (CH)
frameworks. We propose a general measure of the coherence of the reduced
density matrix and find that EIS decoherence occurs in a number of bases for
this model. The degree of ``diagonality'' achieved increases with bath
temperature. We evaluate the Decoherence Functional of Consistent Histories for
coarse grained phase space two-time projected histories. Using the measures
proposed by Dowker and Halliwell we find that the consistency of the histories
improves with increasing bath temperature, time and final grain size and
decreases with initial grain size. The peaking increases with increasing grain
size and decreases with increasing bath temperature. Adopting the above
proposed measure of ``coherence'' to the Decoherence Functional gives similar
results. The results agree in general with expectations while the anomalous
dependence of the consistency on the initial grain size is discussed.
