  The interaction of qubits with quantized modes of electromagnetic fields has
been largely addressed in the quantum optics literature under the rotating wave
approximation (RWA), where rapid oscillating terms in the qubit-mode
interaction picture Hamiltonian can be neglected. At the same time, it is
generally accepted that provided the interaction is sufficiently strong or for
long times, the RWA tends to describe physical phenomena incorrectly. In this
work, we extend the investigation of the validity of the RWA to a more involved
setup where two qubit-mode subsystems are brought to interaction through their
harmonic coordinates. Our treatment is all analytic thanks to a sequence of
carefully chosen unitary transformations which allows us to diagonlize the
Hamiltonian within and without the RWA. By also considering qubit dephasing, we
find that the purity of the two-qubit state presents non-Markovian features
which become more pronounced as the coupling between the modes gets stronger
and the RWA loses its validity. In the same regime, there occurs fast
generation of entanglement between the qubits which is also not correctly
described under the RWA. The setup and results presented here clearly show the
limitations of the RWA in a scenario amenable to exact description and free
from numerical uncertainties. Consequently, it may be of interest for the
community working with cavity or circuit quantum electrodynamic systems in the
strong coupling regime.
