Absorbing boundary conditions for dynamical many-body quantum systems

In numerical studies of the dynamics of unbound quantum mechanical systems, absorbing boundary conditions are frequently applied. Although this certainly provides a useful tool in facilitating the description of the system, its applications to systems consisting of more than one particle is problematic. This is due to the fact that all information about the system is lost upon absorption of one particle; a formalism based solely on the Scrh{\"o}dinger equation is not able to describe the remainder of the system as particles are lost. Here we demonstrate how the dynamics of a quantum system with a given number of identical fermions may be described in a manner which allows for particle loss. A consistent formalism which incorporates the evolution of sub-systems with a reduced number of particles is constructed through the Lindblad equation. Specifically, the transition from an $N$-particle system to an $(N-1)$-particle system due to a complex absorbing potential is achieved by relating the Lindblad operators to annihilation operators. The method allows for a straight forward interpretation of how many constituent particles have left the system after interaction. We illustrate the formalism using one-dimensional two-particle model problems.