Using perturbation theory in the strong coupling regime, that is, the dual
Dyson series, and renormalization group techniques to re-sum secular terms, we
obtain the perturbation series of the two-level system driven by a sinusoidal
field till second order. The third order correction to the energy levels is
obtained proving how this correction does not modify at all the localization
condition for a strong field as arising from the zeros of the zero-th Bessel
function of integer order. A comparison with weak coupling perturbation theory
is done showing how the latter is contained in the strong coupling expansion in
the proper limits. This computation gives an explicit analytical form to
Floquet eigenstates and quasi-energies for this problem, supporting recent
theoretical and experimental findings for quantum devices expected to give a
representation for qubits in quantum computation.