  We present theoretical transmission spectra of a strongly driven, damped,
flux qubit coupled to a dissipative resonator in the ultrastrong coupling
regime. Such a qubit-oscillator system, described within a dissipative Rabi
model, constitutes the building block of superconducting circuit QED platforms.
The addition of a strong drive allows one to characterize the system properties
and study novel phenomena, leading to a better understanding and control of the
qubit-oscillator system. In this work, the calculated transmission of a weak
probe field quantifies the response of the qubit, in frequency domain, under
the influence of the quantized resonator and of the strong microwave drive. We
find distinctive features of the entangled driven qubit-resonator spectrum,
namely resonant features and avoided crossings, modified by the presence of the
dissipative environment. The magnitude, positions, and broadening of these
features are determined by the interplay among qubit-oscillator detuning, the
strength of their coupling, the driving amplitude, and the interaction with the
heat bath. This work establishes the theoretical basis for future experiments
in the driven ultrastrong coupling regime and their impact to develop novel
quantum technologies with superconducting circuits.
