  We study the influence of a strong off-resonant driving signal to the energy
levels of a superconducting flux qubit both experimentally and theoretically.
In the experiment, we carry out a three-tone spectroscopy. This allows us to
directly observe the modification of the qubit's energy levels by the dynamical
Stark shift caused by the driving signal. A theoretical treatment including
corrections from both, rotating and counter-rotating frame, allowed us to
completely explain the observed experimental results and to reconstruct the
influence of the strong driving to the dissipative dynamics as well as to the
coupling constants of the qubit. As one potential application, the tunability
of the minimal energy-level splitting of a superconducting qubit by a microwave
induced dynamical Stark shift can help to overcome the parameter spread induced
by the micro fabrication of superconducting artificial quantum circuits.
