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It is well known that a current proportional to $J\propto\sin(\phi_1-\phi_2)$ flows when two superconductor with phases $\phi_1$ and $\phi_2$ are connected. Also, a current which depends on the phases according to $\sin[(\phi_1-\phi_2)/2]$ is claimed to be fractional Josephson effect (which is believed to be a signature of Majorana fermions). But I see that even with two s-wave superconductors connected through a barrier, the current could be proportional to $\sin[(\phi_1-\phi_2)/2]$ (see for instance - when the barrier is zero. I do not understand this well. Does it mean that $J\propto\sin(\phi_1-\phi_2)$ is true only in the large barrier limit ? Then, what is holy about fractional Josephson current?

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In the paper you cited, the formula (Eq.13) for Josephson current in the low barrier limit has another phase-dependent factor $\tanh(\alpha\cos((\phi_1-\phi_2)/2))$, where $\alpha$ is some dimensionless constant. Therefore, the periodicity of the phase dependence is still $2\pi$, not $4\pi$. – Isidore Seville Jan 5 '14 at 7:55
Thank you Isidore! – FasInfLovCom Jan 5 '14 at 14:59

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