Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 8 / c

Let ${\delta }_1$ and ${\delta }_2$ be gauge-invariant phase drops across the two identical junctions, oriented between the same terminal phases. Their currents obey the DC Josephson effect: $I_1=I_C\sin {\delta }_1$ and $I_2=I_C\sin {\delta }_2$. For negligible loop self-inductance and no additional phase drops in the arms, fluxoid consistency fixes their difference modulo $2\pi$:Define the mean terminal phase $\delta =({\delta }_1+{\delta }_2)/2$ and $\varphi =\pi \Phi /{\Phi }_0$. Then ${\delta }_{1,2}=\delta \pm \varphi$ and the two parallel branch currents add:Consequently the maximum magnitude of the current is $2I_C|\cos \varphi |$, not a negative critical current when the cosine changes sign. The phase shifts by $\pi$ between such stable branches. Junction asymmetry or appreciable self-generated flux modifies the simplest interference pattern, so the stated idealization is required before treating $\Phi$ as an externally prescribed flux.