  Two-color spin-noise spectroscopy of interacting electron spins in singly
charged semiconductor quantum dots provides information on the inter quantum
dot interactions. We investigate the spin cross-correlation function in a
quantum dot ensemble using a modified semiclassical approach. Spin-correlation
functions are calculated using a Hamilton quaternion approach maintaining local
quantum mechanical properties of the spins. This method takes into account the
effects of the nuclear-electric quadrupolar interactions, the randomness of the
coupling constants, and the electron g factor on the spin-noise power-spectra.
We demonstrate that the quantum dot ensemble can be mapped on an effective
two-quantum dot problem and discuss how the characteristic length scale of the
inter-dot interaction modifies the low-frequency cross-correlation spectrum. We
argue that details on the interaction strength distribution can be extracted
from the cross-correlation spectrum when applying a longitudinal or a
transversal external magnetic field.
