Collisionless dynamics of the pairing amplitude in disordered superconductors

I will discuss relaxation of the pairing amplitude in a disordered s-wave superconductor in the absence of the two-particle collisions.

My main assumption is that disorder scattering rate is much smaller than the value of the superconducting energy gap. I will present a system of nonlinear equations which describe the collisionless relaxation of the pairing amplitude following a quench of the pairing strength. After analysing these equations I find that in a superconductor contaminated with paramagnetic impurities, the pairing amplitude in a steady state varies periodically with time even for small deviations from equilibrium. It is shown that such a steady state emerges due to scattering on paramagnetic impurities which pushes the frequency of the amplitude mode below the the edge of the continuum spectrum of states.