  The two-leg ladder system consisting of the Kitaev chains is known to exhibit
a richer phase diagram than that of the single chain. We theoretically
investigate the variety of the Josephson effects between the ladder systems. We
consider the Josephson phase difference $\theta$ between these two ladder
systems as well as the phase difference $\phi$ between the parallel chains in
each ladder system. The total energy of the junction at $T = 0$ is calculated
by a numerical diagonalization method as functions of $\theta$, $\phi$, and
also a transverse hopping $t_{\perp}$ in the ladders. We find that, by
controlling $t_{\perp}$ and $\phi$, the junction exhibits not only the
fractional Josephson effect for the phase difference $\theta$, but also the
usual 0-junction and even $\pi$-junction properties.
