Two Toy Spin Chain Models of Decoherence

We solve for the decoherence dynamics of two models in which a simple qubit or Central Spin couples to a bath of spins; the bath is made from a chain of spins. In model 1, the bath spins are Ising spins; in Model 2, they are coupled by transverse spin-spin interactions, and the chain supports spin waves. We look at (i) the case where the Hamiltonian is static, with a constant system/bath coupling, and (ii) where this coupling varies in time.