Time-dependent Bogoliubov-de Gennes equations -: Mean-field and density-functional theory

An inhomogeneous superconductor in electromagnetic fields is considered as a system in contact with a particle and energy reservoir of chemical potential mu and temperature T. The reservoir contains the source of the electromagnetic fields which is switched on at the time t = t(0). From the Heisenberg equations of motion of the fermionic particle-field operators the Time-dependent Bogoliubov-de Gennes Equations (TdBdGE) are derived. They are wave equations of the electron- and hole-components u(l)((r) over right arrow, t) and v(l)((r) over right arrow, t) of a BCS-like quasiparticle (I) characterized in the stationary situation at times t less than or equal to to by the set I of quantum numbers. The Schrodinger-like wave equations are coupled by the superconducting pair potential Delta((r) over right arrow, t). The mean field each quasiparticle feels at temperature T takes into account the definite occupancy of the considered quasiparticle state (I), which results in a phase shift S-l((r) over right arrow, t) of all quasiparticle wavefunctions. The continuity equation for charge and current density of the many-body system is satisfied only, if there is a groundstate current density proportional to <(del)over right arrow> S-l with a divergence equal to 4(e/(h) over bar)Im(Delta*v(l)*v(l)). This is non-zero when in electron-hole (Andreev) scattering processes from spatial inhomgeneities of the pair potential Cooper pairs are created or destroyed. Considering strongly correlated inhomogeneous superconductors with repulsive and attractive interactions and time-dependent external scalar, vector, and pair potentials one can derive a Hohenberg-Kohn-type theorem for gauge invariant current and anomalous densities. As a consequence it is possible to compute the densities of the interacting system as densities of a noninteracting system with appropriate single-particle potentials. These densities include exchange-correlation efffects. They are built-up from wavefunctions u(n)((r) over right arrow, t) and v(n)((r) over right arrow, t) which satisfy a set of Time-dependent Density-Functional Bogoliubov-de Gennes Equations.