Mechanism of Ultrafast Relaxation of a Photo-Carrier in Antiferromagnetic Spin Background

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.