We present a microscopic theory of the optical properties of self-assembled quantum dots doped with a single magnetic manganese (Mn) impurity and containing a controlled number of electrons. The single-particle electron and heavy-hole electronic shells are described by two-dimensional harmonic oscillators. The electron-electron, electron-hole Coulomb as well as the short-range electron spin-Mn spin and hole spin-Mn spin contact exchange interactions are included. The electronic states of the photo-excited electron-hole-Mn complex and of the final electron-Mn complex are expanded in a finite number of configurations and the full interacting Hamiltonian is diagonalized numerically. The emission spectrum is predicted as a function of photon energy for a given number of electrons and different number of confined electronic quantum dot shells. We show how emission spectra allow to identify the number of electronic shells, the number of electrons populating these shells and, most importantly, their spin. We show that electrons not interacting directly with the spin of Mn ion do so via electron-electron interactions. This indirect interaction is a strong effect even when Mn impurity is away from the quantum dot center.
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