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A system of $N_a$ atoms of $n$-levels interacting dipolarly with $\ell$ modesof electromagnetic field is considered. The energy surface of the system isconstructed from the direct product of the coherent states of U$(n)$ in thetotally symmetric representation for the matter times the $\ell$ coherentstates of the electromagnetic field. A variational analysis shows that thecollective region is divided into $\ell$ zones, inside each of which only onemode of the electromagnetic field contributes to the ground state. Inconsequence, the polychromatic phase diagram for the ground state naturallydivides itself into monochromatic regions. For the case of $3$-level atoms inthe $\Xi$-configuration in the presence of $2$ modes, the variationalcalculation is compared with the exact quantum solution showing that both arein agreement.

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