Pairing dynamics in particle transport

TL;DR

This paper investigates the effects of pairing on particle transport using time-dependent Hartree-Fock-Bogoliubov (HFB) theory, highlighting its advantages over BCS approximations and comparing numerical implementations with exact solutions.

Contribution

The study clarifies the unique equations of motion in HFB, critiques BCS-based methods, and evaluates different numerical approaches against exact two-particle Schrödinger solutions.

Findings

HFB equations conserve physical quantities and are more reliable than BCS.

TDHF+BCS can violate particle number conservation and show unphysical oscillations.

HFB solutions match short-time particle emission rates from exact solutions, but differ at long times.

Abstract

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 TDHF+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, but there are other problems with the BCS that make it doubtful for reaction dynamics. 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 time-dependent HFB solutions with numerically exact solutions of the two-particle Schrodinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schrodinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory.