  The quantum phase diagram for a finite $3$-level system in the $\Lambda$
configuration, interacting with a two-mode electromagnetic field in a cavity,
is determined by means of information measures such as fidelity, fidelity
susceptibility and entanglement, applied to the reduced density matrix of the
matter sector of the system. The quantum phases are explained by emphasizing
the spontaneous symmetry breaking along the separatrix. Additionally, a
description of the reduced density matrix of one atom in terms of a simplex
allows a geometric representation of the entanglement and purity properties of
the system. These concepts are calculated for both, the symmetry-adapted
variational coherent states and the numerical diagonalisation of the
Hamiltonian, and compared. The differences in purity and entanglement obtained
in both calculations can be explained and visualised by means of this simplex
representation.
