Closures of the functional expansion hierarchy in the non-Markovian quantum state diffusion approach

To find a practical scheme to numerically solve the non-Markovian Quantum State Diffusion equation (NMQSD), one often uses a functional expansion of the functional derivative that appears in the general NMQSD equation. This expansion leads to a hierarchy of coupled operators. It turned out that if one takes only the zeroth order term into account, one has a very efficient method that agrees remarkably well with the exact results for many cases of interest. We denote this approach as zeroth order functional expansion (ZOFE). In the present work, we investigate two extensions of ZOFE. Firstly, we investigate how the hierarchy converges when taking higher orders into account (which, however, leads to a fast increase in numerical size). Secondly, we demonstrate that by using a terminator that approximates the higher order contributions, one can obtain significant improvement, at hardly any additional computational cost. We carry out our investigations for the case of absorption spectra of molecular aggregates.