Effects of quasiparticle tunnelling in a circuit-QED realization of a strongly driven two-level system

We experimentally and theoretically study the frequency shift of a driven cavity coupled to a superconducting charge qubit. In addition to previous studies, here we also consider drive strengths large enough to energetically allow for quasiparticle creation. Quasiparticle tunnelling leads to the inclusion of more than two charge states in the dynamics. To explain the observed effects, we develop a master equation for the microwave dressed charge states, including quasiparticle tunnelling. A bimodal behaviour of the frequency shift as a function of gate voltage can be used for sensitive charge detection. However, at weak drives, the charge sensitivity is significantly reduced by nonequilibrium quasiparticles, which induce transitions to a non-sensitive state. Unexpectedly, at high-enough drives, the quasiparticle tunnelling enables a very fast relaxation channel to the sensitive state. In this regime, the charge sensitivity is thus robust against externally injected quasiparticles and the desired dynamics prevail over a broad range of temperatures. We find very good agreement between the theory and experiment over a wide range of drive strengths and temperatures.