Tunable $\pm\varphi$, $\varphi_0$ and $\varphi_0\pm\varphi$ Josephson junction

TL;DR

This paper analyzes a tunable superconducting quantum interference device (SQUID) with asymmetric Josephson junctions, demonstrating the ability to realize and control a variety of phase states, including the $ extit{ extbf{ extit{ extphi}}}$ Josephson junction, through external magnetic flux.

Contribution

It introduces a method to realize and tune a $ extit{ extbf{ extphi}}$ Josephson junction with degenerate ground states using a 0-$ extit{ extbf{ extpi}}$ SQUID with asymmetric parameters.

Findings

Identification of parameter domains for $ extit{ extbf{ extphi}}$ JJ behavior.

Demonstration of in situ tunability of the current-phase relation.

Discovery of non-$2 extpi$ periodic dependence of $ extit{ extbf{ extphi}}_0$ on magnetic flux.

Abstract

We study a 0-$\pi$ 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 $I_s(\psi)$ and the Josephson energy $U(\psi)$, where $\psi$ 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 $\varphi$ JJ with degenerate ground state phases $\psi=\pm\varphi$. The $\varphi$ domain is rather large, so one can easily construct a $\varphi$ 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 $\psi=\pm\varphi$, $\psi=\varphi_0$ ($\varphi_0 \neq 0,\pi$) and even $\psi=\varphi_0\pm\varphi$. The dependence of $\varphi_0$ on applied magnetic flux is not $2\pi$ (one flux quantum) periodic.