Josephson effect of superconductors with $J=3/2$ electrons

The angular momentum of an electron is characterized well by pseudospin with $J=3/2$ in the presence of strong spin-orbit interactions. We study theoretically the Josephson effect of superconductors in which such two $J=3/2$ electrons form a Cooper pair. Within even-parity symmetry class, pseudospin-quintet pairing states with $J=2$ can exist as well as pseudospin-singlet state with $J=0$. We focus especially on the Josephson selection rule among these even-parity superconductors. We find that the selection rule between quintet states is severer than that between spin-triplet states formed by two $S=1/2$ electrons. The effects of a pseudospin-active interface on the selection rule are discussed as well as those of odd-frequency Cooper pairs generated by pseudospin dependent band structures.