  We establish, within the second quantization method, the general
dipole-dipole Hamiltonian interaction of a system of $n$-level atoms. The
variational energy surface of the $n$-level atoms interacting with $\ell$-mode
fields and under the Van Der Waals forces is calculated with respect the
tensorial product of matter and electromagnetic field coherent states. This is
used to determine the quantum phase diagram associated to the ground state of
the system and quantify the effect of the dipole-dipole Hamiltonian
interaction. By considering real induced electric dipole moments, we find the
quantum phase transitions for $2$- and $3$-level atomic systems interacting
with $1$- and $2$- modes of the electromagnetic field, respectively. The
corresponding order of the transitions is established by means of Ehrenfest
classification; for some undetermined cases, we propose two procedures: the
difference of the expectation value of the Casimir operators of the $2$-level
subsystems, and by maximizing the Bures distance between neighbor variational
solutions.
