  The generalized Master equation of the Nakajima-Zwanzig (NZ) type has been
used extensively to investigate the coherence dynamics of the central spin
model with the nuclear bath in a narrowed state characterized by a well defined
value of the Overhauser field. We revisit the perturbative NZ approach and
apply it to the exactly solvable case of a system with uniform hyperfine
couplings. This is motivated by the fact that the effective Hamiltonian-based
theory suggests that the dynamics of the realistic system at low magnetic
fields and short times can be mapped onto the uniform coupling model. We show
that the standard NZ approach fails to reproduce the exact solution of this
model beyond very short times, while the effective Hamiltonian calculation
agrees very well with the exact result on timescales during which most of the
coherence is lost. Our key finding is that in order to extend the timescale of
applicability of the NZ approach in this case, instead of using a single
projection operator one has to use a set of correlated projection operators
which properly reflect the symmetries of the problem and greatly improve the
convergence of the theory. This suggests that the correlated projection
operators are crucial for a proper description of narrowed state free induction
decay at short times and low magnetic fields. Our results thus provide
important insights toward the development of a more complete theory of central
spin decoherence applicable in a broader regime of timescales and magnetic
fields.
