  We study theoretically the formation of the nuclear-spin polaron state in
semiconductor nanosystems within the Lindblad equation approach. To this end,
we derive a general Lindblad equation for the density operator that complies
with the symmetry of the system Hamiltonian and address the nuclear-spin
polaron formation for localized charge carriers subject to an arbitrarily
anisotropic hyperfine interaction when optically cooling the nuclei. The
steady-state solution of the density matrix for an anisotropic central spin
model is presented as a function of the electron and nuclear spin bath
temperature. Results for the electron-nuclear spin correlator as well as data
for the nuclear spin distribution function serve as a measure of
spin-entanglement. The features in both of them clearly indicate the formation
of the nuclear polaron state at low temperatures where the crossover regime
coincides with an enhancement of quantum fluctuations and agrees with the
mean-field prediction of the critical temperature line. We can identify two
distinct polaron states dependent upon the hyperfine anisotropy which are
separated by a quantum phase transition at the isotropic point. These states
are reflected in the temporal spin auto-correlation functions accessible in
experiment via spin-noise measurements.
