effect of time reversal symmetry on the current-phase relation in josephson junctions including two-gap superconductors

From a general viewpoint, the josephson effect is a quantum phenomenon that the tunnelling of cooper electron pairs can describe. this junction involves two or more superconductors separated by a thin layer of non-superconducting material. a supercurrent can flow without dissipation through this junction without applying voltage. the dependence of the josephson current on the phase difference between the two superconductors is of significant importance and is investigated in this study. this research considers a josephson junction consisting of two gap superconductors separated by a thin insulator. additionally, we consider a spin-singlet s-wave pair potential in each conduction band. we study the effect of time-reversal symmetry on the phase difference and, consequently, on the josephson current in two cases. the first case is when only one of the superconductors has time-reversal symmetry, and the second is when neither superconductor has time-reversal symmetry. these states are compared with the condition where both superconductors have time-reversal symmetry.