  We demonstrate theoretically that in gyrotropic semiconductors and
semiconductor nanosystems the Brownian motion of electrons results in temporal
fluctuations of the polarization plane of light passing through or reflected
from the structure, i.e., in stochastic Faraday or Kerr rotation effects. The
theory of the effects is developed for a number of prominent gyrotropic systems
such as bulk tellurium, ensembles of chiral carbon nanotubes, and GaAs-based
quantum wells of different crystallographic orientations. We show that the
power spectrum of these fluctuations in thermal equilibrium is proportional to
the ac conductivity of the system. We evaluate contributions resulting from the
fluctuations of the electric current, as well as of spin, valley polarization,
and the spin current to the noise of the Faraday/Kerr rotation. Hence,
all-optical measurements of the Faraday and Kerr rotation noise provide an
access to the transport properties of the semiconductor systems.
