Initial correlation in a system of a spin coupled to a spin bath through an intermediate spin

The strong system-bath correlation is a typical initial condition in many condensed matter and some quantum optical systems. Here, the dynamics of a spin interacting with a spin bath through an intermediate spin are studied. Initial correlations between the spin and the intermediate spin are taken into account. The exact analytical expression for the evolution operator of the spin is found. Furthermore, correlated projection superoperator techniques are applied to the model and a time-convolutionless master equation to second order is derived. It is shown that the time-convolutionless master equation to second order reproduces the exact dynamics for time-scales of the order 1/{\gamma}, where {\gamma} is the coupling of the central spin to the intermediate spin. It is found that there is a strong dependence on the initial system-bath correlations in the dynamics of the reduced system, which cannot be neglected.