  We develop a "chain-boson model" master equation, within the Born-Markov
approximation, for a few superconducting quantum interference devices (SQUIDs)
coupled into a chain and exchanging their angular momenta with a low
temperature phonon bath. Our master equation has four generators; we
concentrate on the damping and diffusion and use them to study the relaxation
and decoherence of a Heisenberg SQUID chain whose spectrum exhibits critical
point energy-level crossings, entangled states, and pairs of resonant
transitions. We note that at an energy-level crossing the relevant bath
wavelengths are so long that even well-spaced large SQUIDs can partially
exhibit collective coupling to the bath, dramatically reducing certain
relaxation and decoherence rates. Also, transitions into entangled states can
occur even in the case of an independent coupling of each SQUID to the bath.
Finally, the pairs of resonant transitions can cause decaying oscillations to
emerge in a lower energy subspace.
