Tunable $\pmφ$, $φ_0$ and $φ_0\pmφ$ Josephson junction

We study a 0-$π$ dc superconducting quantum interference device (SQUID) with asymmetric inductances and critical currents of the two Josephson junctions (JJs). By considering such a dc SQUID as a black box with two terminals, we calculate its effective current-phase relation $I_s(ψ)$ and the Josephson energy $U(ψ)$, where $ψ$ is the Josephson phase across the terminals. We show that there is a domain of parameters where the black box has the properties of a $φ$ JJ with degenerate ground state phases $ψ=\pmφ$. The $φ$ domain is rather large, so one can easily construct a $φ$ JJ experimentally. We derive the current phase relation and show that it can be tuned \emph{in situ} by applying an external magnetic flux resulting in a continuous transition between the systems with static solutions $ψ=\pmφ$, $ψ=φ_0$ ($φ_0 \neq 0,π$) and even $ψ=φ_0\pmφ$. The dependence of $φ_0$ on applied magnetic flux is not $2π$ (one flux quantum) periodic.

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