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We propose an approach to the study of open quantum systems based on aparametric representation of the principal system. The representation isobtained introducing generalized coherent states for the environment, and issuch that the evolution from a fully quantum environment to a classical one canbe inherently followed. The method is applied to the specific case where theprincipal system is a central qubit and the environment is a surrounding ringof interacting spins, with the overall Hamiltonian being that of the $s=1/2$Heisenberg star with frustration. In this case the quantum character of theenvironment, embodied by the total spin $S$ of the ring, can be physicallyvaried 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 anyof its eigenstates, the qubit behaves as if it were under the effect of anexternal magnetic field, whose direction in real space is set by the variablesdefining the environmental coherent states, and is hence broadened according totheir quantum probability distribution. When the quantum character of theenvironment is reduced, such distribution becomes narrower, finally collapsinginto a Dirac-$\delta$ as $S\to\infty$. The entanglement between the centralqubit and its environment remains well defined even in this limit, where itcomes across as the binary entropy of the $2\pi$-normalized Berry phasecharacterizing the dynamics of a qubit in an adiabatically precessing magneticfield.

Parametric representation of open quantum systems and crossover from quantum to classical environment