  Under optical cooling of nuclei, a strongly correlated nuclear-spin polaron
state can form in semiconductor nanostructures with localized charge carriers
due to the strong hyperfine interaction of the localized electron spin with the
surrounding nuclear spins. Here we develop a kinetic-equation formalism
describing the nuclear-spin polaron formation. We present a derivation of the
kinetic equations for an electron-nuclear spin system coupled to reservoirs of
different electron and nuclear spin temperatures which generate the exact
thermodynamic steady state for equal temperatures independent of the system
size. We illustrate our approach using the analytical solution of the central
spin model in the limit of an Ising form of the hyperfine coupling. For
homogeneous hyperfine coupling constants, i.e., the box model, the model is
reduced to an analytically solvable form. Based on the analysis of the
nuclear-spin distribution function and the electron-nuclear spin correlators,
we derive a relation between the electron and nuclear spin temperatures, where
the correlated nuclear-spin polaron state is formed. In the limit of large
nuclear baths, this temperature line coincides with the critical temperature of
the mean-field theory for polaron formation. The criteria of the polaron
formation in a finite-size system are discussed. We demonstrate that the
system's behavior at the transition temperature does not depend on details of
the hyperfine-coupling distribution function but only on the effective number
of coupled bath spins. In addition, the kinetic equations enable the analysis
of the temporal formation of the nuclear-polaron state, where we find the
build-up process predominated by the nuclear spin-flip dynamics.

Mesoscale and Nanoscale Physics