Pairing dynamics in particle transport

We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations.
The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the  Hamiltonian.  In contrast, the theories based on the BCS approximation are more problematic.  In the usual formulation of HFB+BCS, the equation of continuity is violated and one sees  unphysical oscillations in particle densities.  This can be ameliorated by freezing the  occupation numbers during the evolution in TDHF+BCS.  We also compare different numerical implementations of the time-dependent HFB equations.
The  equations of motion for the U and V Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient.  Finally, we compare the HFB solutions with numerically exact solutions for the two-particle system.  We find that TDHFB provides a good description of the particle emission at short time but might deviates significantly from the exact solution at longer time, certainly due to effects beyond TDHFB like four quasi-particles excitations.