  We study the low temperature thermodynamic properties of a superconducting
double-island qubit. For an odd number of electrons in the device, the ground
state corresponds to the intrinsic quasiparticle bound to the tunneling
contact. The ground state is separated from the continuum of excited states by
a finite gap of order of the Josephson energy E_J. The presence of the bound
quasiparticle state results in a nonmonotonic temperature dependence of the
width of the transition region between Coulomb blockade plateaus. The minimum
width occurs at the ionization temperature of the bound state, T_i\sim
E_J/\ln(\sqrt{\Delta E_J}/\delta), with \Delta and \delta being respectively
the superconducting gap and single particle mean level spacing in the island.
For an even number of electrons in the system, we show that the Coulomb
enhancement of the Josephson energy can be significantly stronger than that in
the case of a single grain coupled to a superconducting lead. If the
electrostatic energy favors a single broken Cooper pair, the resulting
quasiparticles are bound to the contact with an energy that is exponentially
small in the inverse dimensionless conductance.
