Four-terminal Josephson junctions: diode effects, anomalous currents and transverse currents

We study a multi-terminal Josephson junction consisting of a central spin-orbit-coupled (SOC) region with an in-plane Zeeman field connected to four superconducting terminals. This setup allows for the simultaneous measurement of both longitudinal and transverse Josephson currents in response to a phase bias and provides a platform to probe the planar Hall effect in superconducting transport. We find that the system exhibits anomalous Josephson effect (AJE) and Josephson diode effect (JDE) when the symmetry between opposite momentum modes is broken in SOC region. Specifically, breaking the symmetry between $k_x$ and $-k_x$ results in JDE and AJE in the longitudinal Josephson current, while breaking the symmetry between $k_y$ and $-k_y$ leads to a finite transverse Josephson current that also exhibits JDE and AJE. Transverse Josephson diode effect coefficient attains values as large as $500\%$ for realistic set of parameters. Furthermore, for specific parameter choices, the current-phase relation in the transverse direction supports unidirectional transport, highlighting its potential for superconducting circuit applications. Our setup offers a new route to engineering nonreciprocal superconducting transport.