Long time evolution of a spin interacting with a spin bath in arbitrary magnetic field

We introduce a completely different method to calculate the evolution of a spin interacting with a sufficient large spin bath, especially suitable for treating the central spin model in a quantum dot (QD). With only an approximation on the envelope of central spin, the symmetry can be exploited to reduce a huge Hilbert space which cannot be calculated with computers to many small ones which can be solved exactly. This method can be used to calculate spin-bath evolution for a spin bath containing many (say, 1000) spins, without a perturbative limit such as strong magnetic field condition, and works for long-time regime with sufficient accuracy. As the spin-bath evolution can be calculated for a wide range of time and magnetic field, an optimal dynamic of spin flip-flop can be found, and more sophisticated approaches to achieve extremely high polarization of nuclear spins in a QD could be developed.