  A complete analysis of the decoherence properties of a Josephson junction
qubit is presented. The qubit is of the flux type and consists of two large
loops forming a gradiometer and one small loop, and three Josephson junctions.
The contributions to relaxation (T_1) and dephasing (T_\phi) arising from two
different control circuits, one coupled to the small loop and one coupled to a
large loop, is computed. We use a complete, quantitative description of the
inductances and capacitances of the circuit. Including two stray capacitances
makes the quantum mechanical modeling of the system five dimensional. We
develop a general Born-Oppenheimer approximation to reduce the effective
dimensionality in the calculation to one. We explore T_1 and T_\phi along an
optimal line in the space of applied fluxes; along this "S line" we see
significant and rapidly varying contributions to the decoherence parameters,
primarily from the circuit coupling to the large loop.
