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In connection with the recent Harvard group experiment on graphene-basedfour-terminal Josephson junctions containing a grounded loop, we considerbiasing at opposite voltages on the quartet line and establish lowest-orderperturbation theory in the tunnel amplitudes between a two-dimensional (2D)metal and four superconducting leads in the dirty limit. We present in additiongeneral nonperturbative and nonadiabatic results. The critical current on thequartet line $I_c(\Phi/\Phi_0)$ depends on the reduced flux $\Phi/\Phi_0$ viainterference between the three-terminal quartets (3TQ) and the nonstandardfour-terminal split quartets (4TSQ). The 4TSQ result from synchronizing twoJosephson junctions by exchange of two quasiparticles "surfing" on the 2Dquantum wake, and this mechanism is already operational at equilibrium.Perturbation theory in the tunnel amplitudes shows that the 3TQ are$\pi$-shifted but the 4TSQ are $0$-shifted if the contacts have lineardimension which is large compared to the elastic mean free path. We establishthe gate voltage dependence of the quartet critical current oscillations$I_c(\Phi/\Phi_0)$. It is argued that "Observation of $I_c(0)\ne I_c(1/2)$"implies "Evidence for the four-terminal 4TSQ" for finite bias voltage on thequartet line and arbitrary interface transparencies. This statement relies onphysically-motivated approximations leading to the Ambegaokar-Baratoff-typeformula for the quartet critical current-flux relation. It is concluded thatthe recent experiment mentioned above finds evidence for the four-terminal4TSQ.

Inversion in a four-terminal superconducting device on the quartet line: I. Two-dimensional metal and the quartet beam splitter