  We show that the optical spin conductivity being a small AC response of a
bulk spin current and elusive in condensed matter systems can be measured in
ultracold atoms. We demonstrate that this conductivity contains rich
information on quantum states by analyzing experimentally achievable systems
such as a spin-1/2 superfluid Fermi gas, a spin-1 Bose-Einstein condensate, and
a Tomonaga-Luttinger liquid. The obtained conductivity spectra being absent in
the Drude conductivity reflect quasiparticle excitations and non-Fermi liquid
properties. Accessible physical quantities include the superfluid gap and the
contact for the superfluid Fermi gas, gapped and gapless spin excitations as
well as quantum depletion for the Bose-Einstein condensate, and the spin part
of the Tomonaga-Luttinger liquid parameter elusive in cold-atom experiments.
Unlike its mass transport counterpart, the spin conductivity serves as a probe
applicable to clean atomic gases without disorder and lattice potentials. Our
formalism can be generalized to various systems such as spin-orbit coupled and
nonequilibrium systems.
