NIPG-DG schemes for transformed master equations modeling open quantum systems

TL;DR

This paper develops a Discontinuous Galerkin numerical scheme for a transformed master equation in open quantum systems, demonstrating reduced computational costs and validating results against analytical solutions for harmonic and non-harmonic potentials.

Contribution

It introduces a DG method tailored for the transformed master equation, offering an efficient computational approach for open quantum systems with non-harmonic potentials.

Findings

Reduced computational cost compared to Wigner-Fokker-Planck models.

Accurate numerical solutions for harmonic, linear, and quartic potentials.

Validation against analytical steady-state solutions for harmonic case.

Abstract

This work presents a numerical analysis of a Discontinuous Galerkin (DG) method for a transformed master equation modeling an open quantum system: a quantum sub-system interacting with a noisy environment. It is shown that the presented transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of non-harmonic potentials via DG schemes. Specifics of a Discontinuous Galerkin (DG) numerical scheme adequate for the system of convection-diffusion equations obtained for our Lindblad master equation in position basis are presented. This lets us solve computationally the transformed system of interest modeling our open quantum system problem. The benchmark case of a harmonic potential is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem. Two non-harmonic cases are then presented: the linear and quartic potentials are modeled via our DG framework, for which we show our numerical results.