  Individual quantum systems may be interacting with surrounding environments
having a small number of degrees of freedom. It is therefore relevant to
understand the extent to which such small (but uncontrollable) environments
could affect the quantum properties of the system of interest. Here we discuss
a simple system-environment toy model, constituted by a two-level atom (atom 1)
interacting with a single mode cavity field. The field is also assumed to be
(weakly) coupled to an external noisy subsystem, the small environment, modeled
as a second two-level atom (atom 2). We investigate the action of the minimal
environment on the dynamics of the linear entropy (state purity) and the atomic
dipole squeezing of atom 1, as well as the entanglement between atom 1 and the
field. We also obtain the full analytical solution of the two atom
Tavis-Cummings model for both arbitrary coupling strengths and frequency
detunings, necessary to analyze the influence of the field-environment detuning
on the evolution of the above mentioned quantum properties. For
complementarity, we discuss the role of the degree of mixedness of the
environment by analyzing the time-averaged linear entropy of atom 1.
