  Decoherence of quantum systems from entanglement with an unmonitored
environment is to date the most compelling explanation of the emergence of a
classical picture from a quantum world. While it is well understood for a
single Lindblad operator, the role in the einselection process of a complex
system-environment interaction remains to be clarified. In this paper, we
analyze an open quantum dynamics inspired by CQED experiments with two
non-commuting Lindblad operators modeling decoherence in the number basis and
dissipative decoherence in the coherent state basis. We study and solve exactly
the problem using quantum trajectories and phase-space techniques. The
einselection optimization problem, which we consider to be about finding states
minimizing the variation of some entanglement witness at a given energy, is
studied numerically. We show that Fock states remain the most robust states to
decoherence up to a critical coupling.
