DENSITY-MATRIX METHODS AND TIME DEPENDENCE OF ORDER PARAMETER IN SUPERCONDUCTORS

The density-matrix equations for a superconductor are used to obtain corrections to the order parameter by means of a time-dependent perturbation technique. In particular, under certain restrictions a time-dependent differential equation for the order parameter in pure and dirty superconductors near the critical temperature is derived. Compared with the various equations obtained by the Green's function method, our equation is similar to those showing a diffusive character. Above T cfluctuations of the order parameter relax according to the same equation. The density cannot be written in a simple differential form; however an explicit expression for the time derivative of the density is given. From this we are able to show that within the scheme of the approximation used, the continuity equation is satisfied. The diffusion term, together with the derivative of the density with respect to time are shown to vanish whenever Josephson's equation is satisfied. Time-dependent perturbative solutions of the Hartree-Bogoliubov equations are given at any temperature and for any order in the pair potential. The first-order term is explicitly calculated. © 1969 Società Italiana di Fisica.