Exactly solvable spin dynamics of an electron coupled to large number of nuclei and the electron-nuclear spin echo in a quantum dot

The model considered in the paper is used nowadays to describe spin dynamics of quantum dots after optical excitation. Based on the exact diagonalization of a model Hamiltonian, we solve the problems of the electron spin polarization decay and magnetic field dependence of the steady state polarization. The important role of the nuclear state is shown and methods of its calculation for different regimes of optical excitation are proposed. The effect of spin echo observed after application of the magnetic field $\pi$-pulse is predicted.