Energy relaxation of superconducting charge qubit via Andreev processes

We study fundamental limitations on the energy relaxation rate of a superconducting charge qubit with a large-gap Cooper-pair box, $\Delta_b > \Delta_r$. At a sufficiently large mismatch between the gap energies in the box $\Delta_b$ and in the reservoir $\Delta_r$,"quasiparticle poisoning" becomes ineffective even in the presence of nonequilibrium quasiparticles in the reservoir. The qubit relaxation still may occur due to higher-order (Andreev) processes. In this paper we evaluate the qubit energy relaxation rate due to Andreev processes.