  Higher-order transitions can occur in the ultrastrong-coupling regime of
circuit QED through virtual processes governed by the counter-rotating
interactions. We propose a feasible way to probe higher-order transitions
through the scattering of propagating microwave photons incident on the hybrid
qubit-cavity system. The lineshapes in the scattering spectra can indicate the
coherent interaction between the qubits and the cavity, and the higher-order
transitions can be identified in the population spectra. We further find that
if the coupling strengths between the two qubits and the cavity are tuned to be
asymmetric, the dark antisymmetric state with the Fano-lineshape can also be
detected from the variations in the scattering spectra.

Quantum Physics