Quantum Phase Diagrams of Matter-Field Hamiltonians I: Fidelity, Bures Distance, and Entanglement.

A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for $N_a=1$ and $N_a=4$ particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with $2$ modes of electromagnetic field. For $N_a=1$, the reduced density matrix of the matter allows us to determine the phase regions in a $2$-simplex (associated to a general three dimensional density matrix), on the different $3$-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.