Splitting of Andreev levels in a Josephson junction by spin-orbit coupling

We consider the effect of spin-orbit coupling on the energy levels of a single-channel Josephson junction below the superconducting gap. We investigate quantitatively the level splitting arising from the combined effect of spin-orbit coupling and the time-reversal symmetry breaking by the phase difference between the superconductors. Using the scattering matrix approach, we establish a simple connection between the quantum mechanical time delay matrix and the effective Hamiltonian for the level splitting. As an application, we calculate the distribution of level splittings for an ensemble of chaotic Josephson junctions. The distribution falls off as a power law for large splittings, unlike the exponentially decaying splitting distribution given by the Wigner surmise-which applies for normal chaotic quantum dots with spin-orbit coupling in the case that the time-reversal symmetry breaking is due to a magnetic field.