Relaxation of the Superconducting Order Parameter

We investigate the nonequilibrium behavior of a pure type-I superconductor and take into account the effects of the electron-phonon interaction. The formulation is based on Kadanoff-Baym transport equations suitably generalized to the case of superconductors. In order to investigate the relaxation behavior of the order parameter due to real phonon transitions, we solve the transport equations in the spatially homogeneous case when there is a slow time variation of the energy-gap function. We solve the transport equations treating the real phonon transitions in the weakest possible way, much in the spirit of the Bardeen-Rickayzen-Tewordt (BRT) transport equation used in the treatment of thermal conductivity. We discuss in detail, however, why the BRT equation fails in this case, and in general when there is a time variation of the order parameter. For weak-coupling materials, we find that for $0.9\le \frac{T}{T_c}\le 0.99$, the gap relaxation rate (${10}^8$-${10}^9$ ${\sec }^{-1\mathrm{}}$) is about an order of magnitude slower than the quasiparticle decay rate, but that very close to $T_c$ it disappears as $T-T_c$.