The dc-Josephson effect with more than four superconducting leads

By definition, the $p$-terminal dc-Josephson current is sensitive to the superconducting phase variables of $p$ terminals. In the paper, we establish protocol for direct detection of the $p$-terminal dc-Josephson effect with $p\ge 3$ in a device containing $N$ superconducting leads $S_1,\,S_2,\,...,\,S_N$ having the phase variables $\varphi_1,\,\varphi_2,\,...,\,\varphi_N$. The calculated signal ${\chi}^{(N)}$ can be probed in microwave experiments, and it corresponds to the higher-order nonlocal inverse inductance obtained from differentiating the current $I_1$ through $S_1$ with respect to the remaining $N-2$ independent phase differences $\varphi_2-\varphi_N,\,\varphi_3-\varphi_N,\, ...,\varphi_{N-1}-\varphi_N$. We find that the values $p\le N-2$ do not contribute to ${\chi}^{(N)}$, and that ${\chi}^{(N)}\ne 0$ implies evidence for the $p=N-1$ or the $p=N$-terminal dc-Josephson currents. For $N=4$ superconducting leads, we demonstrate that ${\chi}^{(4)}\ne 0$ implies evidence for the $p=3$ or $p=4$ dc-Josephson effect, irrespective of the $p=2$-terminal dc-Josephson current. Thus, we provide a way to demonstrate the dc-Josephson effect with more than three terminals (i.e. with $p\ge 3$) in a device containing more than four superconducting leads (i.e. with $N\ge 4$). The predicted ${\chi}^{(4)}$ is "yes or no" answer to the $p\ge 3$ dc-Josephson effect, i.e. for $N=4$, nonvanishingly small $\chi^{(4)}\ne 0$ implies the $p=3$ or $p=4$-terminal dc-Josephson effect and vanishingly small $\chi^{(4)}=0$ implies absence of the $p=3$ and $p=4$-terminal dc-Josephson effect. The paper can be viewed as generalizing the recently considered $\varphi$-junctions in Andreev molecules to arbitrary number $N$ of the superconducting leads, and it relies on basic properties of the dc-Josephson effect that are not directly related to nontrivial topology and Weyl point singularities.