Josephson junction with magnetic-field tunable current-phase relation

We consider a 0-$π$ Josephson junction consisting of asymmetric 0 and $π$ regions of different lengths $L_0$ and $L_π$ having different critical current densities $j_{c,0}$ and $j_{c,π}$. If both segments are rather short, the whole junction can be described by an \emph{effective} current-phase relation for the spatially averaged phase $ψ$, which includes the usual term $\propto\sin(ψ)$, a \emph{negative} second harmonic term $\propto\sin(2ψ)$ as well as the unusual term $\propto H \cosψ$ tunable by magnetic field $H$. Thus one obtains an electronically tunable current-phase relation. At H=0 this corresponds to the $φ$ Josephson junction.

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