  We propose a method of two-beam spin noise spectroscopy to test the spin
transport at equilibrium via analysis of correlations between time-shifted spin
fluctuations at different space locations. This method allows one to determine
the strength of spin-orbit interaction and spin relaxation time and separate
spin noise of conducting electrons from the background noise of localized
electrons. We formulate a theory of two-beam spin noise spectroscopy in
semiconductor wires with Bychkov-Rashba spin-orbit interaction taking into
account several possible spin relaxation channels and finite size of laser
beams. Our theory predicts a peak shift with respect to the Larmor frequency to
higher or lower frequencies depending on the strength of spin orbit interaction
and distance between the beams. The two-beam spin noise spectroscopy could find
applications in experimental studies of semiconductors, emergent materials and
many other systems.
