  We study a superconducting charge qubit coupled to an intensive
electromagnetic field and probe changes in the resonance frequency of the
formed dressed states. At large driving strengths, exceeding the qubit
energy-level splitting, this reveals the well known Landau-Zener-Stuckelberg
(LZS) interference structure of a longitudinally driven two-level system. For
even stronger drives we observe a significant change in the LZS pattern and
contrast. We attribute this to photon-assisted quasiparticle tunneling in the
qubit. This results in the recovery of the qubit parity, eliminating effects of
quasiparticle poisoning and leads to an enhanced interferometric response. The
interference pattern becomes robust to quasiparticle poisoning and has a good
potential for accurate charge sensing.
