  Combining the Josephson effect with magnetism, or spin dependence in general,
creates novel physical phenomena. The spin-asymmetric Josephson effect is a
predicted phenomenon where a spin-dependent potential applied across a
Josephson junction induces a spin-polarized Josephson current. Here, we propose
an approach to observe the spin-asymmetric Josephson effect with spin-dependent
superlattices, realizable, e.g., in ultracold atomic gases. We show that
observing this effect is feasible by studying numerically the quantum dynamics
of the system in one dimension. Furthermore, we show that the enhancement, or
tunability, of the critical supercurrent in ferromagnetic Josephson junctions
[F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 86, 3140
(2001)] can be explained by the spin-asymmetric Josephson effect.
