  In connection with the recent Harvard group experiment on graphene-based
four-terminal Josephson junctions containing a grounded loop, we consider
biasing at opposite voltages on the quartet line and establish lowest-order
perturbation theory in the tunnel amplitudes between a two-dimensional (2D)
metal and four superconducting leads in the dirty limit. We present in addition
general nonperturbative and nonadiabatic results. The critical current on the
quartet line $I_c(\Phi/\Phi_0)$ depends on the reduced flux $\Phi/\Phi_0$ via
interference between the three-terminal quartets (3TQ) and the nonstandard
four-terminal split quartets (4TSQ). The 4TSQ result from synchronizing two
Josephson junctions by exchange of two quasiparticles "surfing" on the 2D
quantum 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 linear
dimension which is large compared to the elastic mean free path. We establish
the 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 the
quartet line and arbitrary interface transparencies. This statement relies on
physically-motivated approximations leading to the Ambegaokar-Baratoff-type
formula for the quartet critical current-flux relation. It is concluded that
the recent experiment mentioned above finds evidence for the four-terminal
4TSQ.
