  We theoretically and numerically investigate the spin fluctuations induced in
a thermal atomic ensemble by an external fluctuating uniaxial magnetic field,
in the context of a standard spin noise spectroscopy (SNS) experiment. We show
that additional spin noise is excited, which dramatically depends on the
magnetic noise variance and bandwidth, as well as on the power of the probe
light and its polarization direction. We develop an analytical perturbative
model proving that this spin noise first emerges from the residual optical
pumping in the medium, which is then converted into spin fluctuations by the
magnetic noise and eventually detected using SNS. The system studied is a
spin-1 system, which thus shows both Faraday rotation and ellipticity noises
induced by the random magnetic fluctuations. The analytical model gives results
in perfect agreement with the numerical simulations, with potential
applications in future experimental characterization of stray field properties
and their influence on spin dynamics.
