A Chain-Boson Model for the Decoherence and Relaxation of a Few Coupled SQUIDs in a Phonon Bath

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.