  A cavity quantum electrodynamical (QED) system beyond the strong-coupling
regime is expected to exhibit intriguing quantum phenomena. Here we report a
direct measurement of the photon-dressed qubit transition frequencies up to
four photons by harnessing the same type of state transitions in an
ultrastrongly coupled circuit-QED system realized by inductively coupling a
superconducting flux qubit to a coplanar-waveguide resonator. This demonstrates
a convincing observation of the photon-dressed Bloch-Siegert shift in the
ultrastrongly coupled quantum system. Moreover, our results show that the
photon-dressed Bloch-Siegert shift becomes more pronounced as the photon number
increases, which is a characteristic of the quantum Rabi model.
