We report the experimental observation of high-order sideband transitions at the single-photon level in a quantum circuit system of a flux qubit ultrastrongly coupled to a coplanar waveguide resonator. With the coupling strength reaching 10% of the resonator's fundamental frequency, we obtain clear signatures of higher-order redand first-order blue-sideband transitions, which are mainly due to the ultrastrong Rabi coupling. Our observation advances the understanding of ultrastrongly-coupled systems and paves the way to study high-order processes in the quantum Rabi model at the single-photon level.

We study cross-Kerr (CK) effect on an optomechanical system driven by two-tone fields. We show that in the presence of the CK effect, a bistable behavior of the mean photon number in the cavity becomes more robust against the fluctuations of the frequency detuning between the cavity mode and the control field. The bistability can also be turned into a tri-stability within the experimentally accessible range of the system parameters. Also, we find that the symmetric profile of the optomechanically induced transparency is broken and the zero-absorption point is shifted in the presence of the CK effect. This shift can be used to measure the strength of the CK effect, and the asymmetric absorption profiles can be employed to engineer a high quality factor of the cavity.Comment: 7 pages, 4 figure