Effective model for a short Josephson junction with a phase discontinuity

We consider a short Josephson junction with a phase discontinuity $kappa$ created, e.g., by a pair of tiny current injectors, at some point $x_0$ along the length of the junction. We derive the effective current-phase relation (CPR) for the system as a whole, i.e., reduce it to an effective point-like junction. From the effective CPR we obtain the ground state of the system and predict the dependence of its critical current on $kappa$. We show that in a large range of $kappa$ values the effective junction behaves as a $varphi_0$ Josephson junction, i.e., has a unique ground state phase $varphi_0$ within each $2pi$ interval. For $kappaapproxpi$ and $x_0$ near the middle of the junction one obtains a $varphi_0pmvarphi$ junction, i.e., the Josephson junction with degenerate ground state phase $varphi_0pmvarphi$ within each $2pi$ interval. Further, in view of possible escape experiments especially in the quantum domain, we investigate the scaling of the energy barrier and eigenfrequency close to the critical currents and predict the behavior of the escape histogram width $sigma(kappa)$ in the regime of the macroscopic quantum tunneling.