  We study Landau-Zener-St\"uckelberg (LZS) interferometry in a cQED
architecture under effects of dissipation. To be specific, we consider a
superconducting qubit driven by a dc+ac signal and coupled to a transmission
line resonator, but our results are valid for general qubit-resonators devices.
To take the environment into account, we assume that the resonator is coupled
to an ohmic quantum bath. The Floquet-Born-Markov master equation is
numerically solved to obtain the dynamics of the system for arbitrary amplitude
of the drive and different time scales. We unveil important differences in the
resonant patterns between the Strong Coupling and Ultra Strong Coupling regimes
in the qubit-resonator interaction, which are mainly due to the magnitude of
photonic gaps in the energy spectrum of the system. We identify in the LZS
patterns the contribution of the qubit gap and the photonic gaps, showing that
for large driving amplitudes the patterns present a weaving structure due to
the combined intercrossing of the different gaps contributions.
