  The Josephson effect is generally described as Cooper pair tunneling, but it
can also be understood in a more general context. The DC Josephson effect is
the pseudo-Goldstone boson of two coupled systems with a broken continuous
abelian U(1) symmetry. Hence, an analog should exist for systems with broken
continuous non-Abelian symmetries. To exhibit the generality of the phenomenon
and make predictions from a realistic model, we study tunneling between
antiferromagnets and also between antiferromagnetic d-wave superconductors.
Performing a calculation analogous to that of Ambegaokar and Baratoff for the
Josephson junction, we find an equilibrium current of the staggered
magnetization through the junction that, in antiferromagnets, is proportional
to s_L X s_R where s_L and s_R are the Neel vectors on either sides of the
junction. Microscopically, this effect exists because of the coherent tunneling
of spin-one particle-hole pairs. In the presence of a magnetic field which is
different on either sides of the junction, we find an analog of the AC
Josephson effect where the angle between Neel vectors depends on time. In the
case of antiferromagnetic d-wave superconductors we predict that there is a
contribution to the critical current that depends on the antiferromagnetic
order and a contribution to the spin-critical current that depends on
superconducting order. The latter contributions come from tunneling of the
triplet Cooper pair that is necessarily present in the ground state of an
antiferromagnetic d-wave superconductor. All these effects appear to leading
order in the square of the tunneling matrix elements.
