Polychromatic phase diagram for $n$-level atoms interacting with $\ell$ modes of electromagnetic field

A system of $N_a$ atoms of $n$-levels interacting dipolarly with $\ell$ modes of electromagnetic field is considered. The energy surface of the system is constructed from the direct product of the coherent states of U$(n)$ in the totally symmetric representation for the matter times the $\ell$ coherent states of the electromagnetic field. A variational analysis shows that the collective region is divided into $\ell$ zones, inside each of which only one mode of the electromagnetic field contributes to the ground state. In consequence, the polychromatic phase diagram for the ground state naturally divides itself into monochromatic regions. For the case of $3$-level atoms in the $\Xi$-configuration in the presence of $2$ modes, the variational calculation is compared with the exact quantum solution showing that both are in agreement.