Effects of optical intra-gap transitions on superexchange interaction in La$ _2$CuO$ _4$ with nonequilibrium photoexcited centers

We investigated the effects of excited many-electron states in the optical control of the magnetic state in undoped Mott-Hubbard insulator. To derive the spin Hamiltonian in material under optical pumping we have used many-electron approach based on the X-operator representation. Extending the projection operators approach on arbitrary energy spectra of the Mott-Hubbard insulator, we obtained the Hamiltonian of superexchange interaction in analytical form. The Hamiltonian includes the spin-exciton variables which are usually missing in discussion on the magnetic response to optical pumping, and is not additive over contributions from the ground and optically excited states. As a test, a microscopic background for the optical induced superexchange was analyzed in La$_2$CuO$_4$(further La214).