Master equation approach to the central spin decoherence problem: Uniform coupling model and role of projection operators

The generalized master equation of Nakajima-Zwanzig (NZ) type has been used extensively to investigate the coherence dynamics of the central spin model with a nuclear bath in a narrowed state characterized by a well-defined value of the Overhauser field. We reconsider 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 time scales during which most of the coherence is lost. Our key finding is that, in order to extend the time scale 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 time scales and magnetic fields.