Proposal for detecting the $\pi-$shifted Cooper quartet supercurrent

The multiterminal Josephson effect aroused considerable interest recently, in connection with theoretical and experimental evidence for correlations among Cooper pairs, that is, the so-called Cooper quartets. It was further predicted that the spectrum of Andreev bound states in such devices could host Weyl-point singularities. However, the relative phase between the Cooper pair and quartet supercurrents has not yet been addressed experimentally. Here, we propose an experiment involving four-terminal Josephson junctions with two independent orthogonal supercurrents, and calculate the critical current contours (CCCs) from a multiterminal Josephson junction circuit theory. We predict nonconvex shapes for the CCCs in zero magnetic field, where the dissipative state reenters into the superconducting one. This observation will allow providing firm experimental evidence of the $\pi$-shifted Cooper quartet current-phase relation.