Kinetics of the superconducting charge qubit in the presence of a quasiparticle

We investigate the energy and phase relaxation of a superconducting qubit caused by a single quasiparticle. In our model, the qubit is an isolated system consisting of a small island (Cooper-pair box) and a larger superconductor (reservoir) connected with each other by a tunable Josephson junction. If such a system contains an odd number of electrons, then even at lowest temperatures a single quasiparticle is present in the qubit. Tunneling of a quasiparticle between the reservoir and the Cooper-pair box results in the relaxation of the qubit. We derive master equations governing the evolution of the qubit coherences and populations. We find that the kinetics of the qubit can be characterized by two time scales—quasiparticle escape time from the reservoir to the box Γ_in^-1 and quasiparticle relaxation time τ . The former is determined by the dimensionless normal-state conductance g_T of the Josephson junction and one-electron level spacing δ_r in the reservoir (Γ_in∼g_Tδ_r) , and the latter is due to the electron-phonon interaction. We find that phase coherence is damped on the time scale of Γ_in^-1 . The qubit energy relaxation depends on the ratio of the two characteristic times τ and Γ_in^-1 and also on the ratio of temperature T to the Josephson energy E_J .