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We show how the use of variational states to approximate the ground state ofa system can be employed to study a multi-mode Dicke model. One of the maincontributions of this work is the introduction of a not very commonly usedquantity, the cooperation number, and the study of its influence on thebehavior of the system, paying particular attention to the quantum phasetransitions and the accuracy of the used approximations. We also show how thesephase transitions affect the dependence of the expectation values of some ofthe observables relevant to the system and the entropy of entanglement withrespect to the energy difference between atomic states and the couplingstrength between matter and radiation, thus characterizing the transitions indifferent ways.

Characterization of the quantum phase transition in a two-mode Dicke model for different cooperation numbers