  Recalling that the rotating wave approximation (RWA) is only valid in the
weak coupling regimes, the purpose of this paper is to study the Hamiltonian
dynamics describing the full quantum mechanical approach of the interaction
between various configurations of three-level atoms Lambda, V and ladder
distinctly with a two-mode radiation field, while the RWA is not considered;
the counter-rotating terms (CRTs) are taken into account. Generally, the
presence of CRTs in the Hamiltonian prevents one to achieve an analytical
solution. Moreover, as we will show in the present work, using the perturbation
theory, analytical solvable Hamiltonians can be successfully obtained.
According to our calculations, the contribution of CRTs within the ordinary
Hamiltonian is equivalent to the replacement of the constant detuning with a
specific intensity dependent detuning in the first order, and the constant atom
field coupling with a particular intensity dependent (f deformed) coupling in
the second order of the associated perturbation parameter. Moreover, noticing
that according to the initial expression of the Hamiltonian, each mode of the
field interacts only with a specific pair of the allowed transitions of each
type of the three-level atom, it is surprisingly observed that via applying the
mentioned approach, the obtained intensity dependent coupling functions depend
on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are
removed with the price of arriving at some intermixed intensity dependent atom
field coupling functions of the two modes of the field. In this way, the
obtained final Hamiltonians are analytically solvable. At last, by determining
the time evolution of the atom field wave function, we study the effects of
CRTs on a few nonclassical properties of the state of the system, including the
atomic population inversion and photon statistics.
