  We propose an approach to the study of open quantum systems based on a
parametric representation of the principal system. The representation is
obtained introducing generalized coherent states for the environment, and is
such that the evolution from a fully quantum environment to a classical one can
be inherently followed. The method is applied to the specific case where the
principal system is a central qubit and the environment is a surrounding ring
of interacting spins, with the overall Hamiltonian being that of the $s=1/2$
Heisenberg star with frustration. In this case the quantum character of the
environment, embodied by the total spin $S$ of the ring, can be physically
varied by acting on the ratio between the couplings entering the Hamiltonian,
and formally followed by increasing $S$. We find that when the star is in any
of its eigenstates, the qubit behaves as if it were under the effect of an
external magnetic field, whose direction in real space is set by the variables
defining the environmental coherent states, and is hence broadened according to
their quantum probability distribution. When the quantum character of the
environment is reduced, such distribution becomes narrower, finally collapsing
into a Dirac-$\delta$ as $S\to\infty$. The entanglement between the central
qubit and its environment remains well defined even in this limit, where it
comes across as the binary entropy of the $2\pi$-normalized Berry phase
characterizing the dynamics of a qubit in an adiabatically precessing magnetic
field.
