  Spin-noise measurements can serve as direct probe for the microscopic
decoherence mechanism of an electronic spin in semiconductor quantum dots
(QD).We have calculated the spin-noise spectrum in the anisotropic central spin
model using a Chebyshev expansion technique which exactly accounts for the
dynamics up to an arbitrary long but fixed time in a finite size system. In the
isotropic case, describing QD charged with a single electron, the short-time
dynamics is in good agreement with a quasi-static approximation for the
thermodynamic limit. The spin-noise spectrum, however, shows strong deviations
at low frequencies with a power-law behavior. In the Ising limit, applicable to
QDs with heavy-hole spins, the spin-noise spectrum exhibits a threshold
behavior above the Larmor frequency. In the generic anisotropic central spin
model we have found a crossover from a Gaussian type of spin-noise spectrum to
a more Ising-type spectrum with increasing anisotropy in a finite magnetic
field. In order to make contact with experiments, we present ensemble averaged
spin-noise spectra for QD ensembles charged with single electrons or holes. The
Gaussian-type noise spectrum evolves to a more Lorentzian shape spectrum with
increasing spread of characteristic time-scales and g-factors of the individual
QDs.
