  The central spin decoherence problem has been researched for over 50 years in
the context of both nuclear magnetic resonance and electron spin resonance.
Until recently, theoretical models have employed phenomenological stochastic
descriptions of the bath-induced noise. During the last few years, cluster
expansion methods have provided a microscopic, quantum theory to study the
spectral diffusion of a central spin. These methods have proven to be very
accurate and efficient for problems of nuclear-induced electron spin
decoherence in which hyperfine interactions with the central electron spin are
much stronger than dipolar interactions among the nuclei. We provide an
in-depth study of central spin decoherence for a canonical scale-invariant
all-dipolar spin system. We show how cluster methods may be adapted to treat
this problem in which central and bath spin interactions are of comparable
strength. Our extensive numerical work shows that a properly modified cluster
theory is convergent for this problem even as simple perturbative arguments
begin to break down. By treating clusters in the presence of energy detunings
due to the long-range (diagonal) dipolar interactions of the surrounding
environment and carefully averaging the effects over different spin states, we
find that the nontrivial flip-flop dynamics among the spins becomes effectively
localized by disorder in the energy splittings of the spins. This localization
effect allows for a robust calculation of the spin echo signal in a dipolarly
coupled bath of spins of the same kind, while considering clusters of no more
than 6 spins. We connect these microscopic calculation results to the existing
stochastic models. We, furthermore, present calculations for a series of
related problems of interest for candidate solid state quantum bits including
donors and quantum dots in silicon as well as nitrogen-vacancy centers in
diamond.
