  A model system consisting of a mesoscopic superconducting grain coupled by
Josephson junctions to two macroscopic superconducting electrodes is studied.
We focus on the effects of ohmic dissipation caused by resistive shunts and
superconducting-normal charge relaxation within the grain. As the temperature
is lowered, the behavior crosses over from uncoupled Josephson junctions,
similar to situations analyzed previously, to strongly interacting junctions.
The crossover temperature is related to the energy-level spacing of the grain
and is of the order of the inverse escape time from the grain. In the limit of
zero temperature, the two-junction system exhibits five distinct quantum
phases, including a novel superconducting state with localized Cooper pairs on
the grain but phase coherence between the leads due to Cooper pair cotunneling
processes. In contrast to a single junction, the transition from the fully
superconducting to fully normal phases is found to be controlled by an
intermediate coupling fixed point whose critical exponents vary continuously as
the resistances are changed. The model is analyzed via two component
sine-Gordon models and related Coulomb gases that provide effective low
temperature descriptions in both the weak and the strong Josephson coupling
limits. The complicated phase diagram is consistent with symmetries of the two
component sine-Gordon models, which include weak to strong coupling duality and
permutation triality. Experimental consequences of the results and potential
implications for superconductor to normal transitions in thin wires and films
are discussed briefly.
