Effect of quasiparticles on the parameters of a gap-engineered transmon

We evaluate the quasiparticle contribution to the frequency shift and relaxation rates of a transmon with the Josephson junctions connecting superconductors that have unequal energy gaps. The gap difference substantially affects the transmon characteristics. We investigate their dependence on the density and effective temperature of the quasiparticles, and on the nominal (unperturbed by the quasiparticles) transmon frequency. At temperatures low compared to the qubit frequency, the gap difference can induce an anomalous positive frequency shift, resulting in a non-monotonic temperature dependence of the transmon frequency. The qubit relaxation rate exhibits a resonance when the qubit frequency matches the gap difference; the shape of the resonance is strongly temperature-dependent. We propose to use these effects to access the details of the quasiparticle energy distribution.