  We present a quantum Monte-Carlo simulation for a pumped atom in a strong
coupling cavity with dissipation, where two ordered spatial modes are formed
for the atomic probability density, with the peaks distributed either only in
the odd sites or only in the even ones of the lattice formed by the cavity
field. A mixed state density operator model, which describes the coupling
between different atomic spatial modes and the corresponding cavity field
components, is proposed, which goes beyond the pure state interpretation. We
develop a new decomposition treatment to derive the atomic spatial modes as
well as the cavity field statistics from the simulation results for the steady
state. With this treatment, we also investigate the dynamical process for the
probabilities of the atomic spatial modes in the adiabatic limit. According to
the analysis of the fitting error between the simulation results and the
density operator model, the latter is a good description for the system.
