  We study the relaxation mechanism of a highly excited carrier propagating in
the antiferromagnetic background modeled by the $t$-$J$ Hamiltonian on a square
lattice. We show that the relaxation consists of two distinct stages. The
initial ultrafast stage with the relaxation time $\tau\sim
(\hbar/t_0)(J/t_0)^{-2/3}$ (where $t_0$ is the hopping integral and $J$ is the
exchange interaction) is based on generation of string states in the close
proximity of the carrier. This unusual scaling of $\tau$ is obtained by means
of comparison of numerical results with a simplified $t$-$J_z$ model on a Bethe
lattice. In the subsequent (much slower) stage local spin excitations are
carried away by magnons. The relaxation time on the two-leg ladder system is an
order of magnitude longer due to the lack of string excitations. This further
reinforces the importance of string excitations for the ultrafast relaxation in
the two-dimensional system.
