  The Josephson effect is a kind of macroscopic quantum phenomenon that
supercurrent flows through a Josephson junction without any voltage applied. We
predict a novel vortex-state-mediated Josephson effect in an SNS Josephson
junction supporting vortices. The vortex-state-mediated supercurrent is
enhanced or reduced significantly in magnitude depending on the junction
length, and exhibits several steps with the number of effective propagating
channels in current-phase evolution at zero temperature. At finite
temperatures, these supercurrent steps persist in the short junction limit, and
develop into sawtooth oscillations if the junction length becomes comparable to
the coherence length $\xi=\hbar v_F/\Delta$ of the superconductor, and in later
case a supercurrent reversal can be observed. These findings may provide a
smoking-gun signature of vortex bound states in superconductors and promise
possible applications in future Josephson devices.
