  Decoherence of a localized electron spin in a solid state material (the
``central spin'' problem) at low temperature is believed to be dominated by
interactions with nuclear spins in the lattice. This decoherence is partially
suppressed through the application of a large magnetic field that splits the
energy levels of the electron spin and prevents depolarization. However,
dephasing decoherence resulting from a dynamical nuclear spin bath cannot be
removed in this way. Fluctuations of the nuclear field lead to uncertainty of
the electron's precessional frequency in a process known as spectral diffusion.
This article considers the effect of the electron's wavefunction shape upon
spectral diffusion and provides wavefunction dependent decoherence time
formulas for free induction decay as well as spin echoes and concatenated
dynamical decoupling schemes for enhancing coherence. We also discuss dephasing
of a qubit encoded in singlet-triplet states of a double quantum dot. A central
theoretical result of this work is the development of a continuum approximation
for the spectral diffusion problem which we have applied to GaAs and InAs
materials specifically.
