Solving the Lindblad equation with methods from computational fluid dynamics

TL;DR

This paper introduces a novel application of a computational fluid dynamics method, the Kurganov-Tadmor scheme, to numerically solve the Lindblad equation for open quantum systems, offering potential improvements in efficiency and stability.

Contribution

It demonstrates the effectiveness of CFD techniques, specifically the Kurganov-Tadmor scheme, for solving the Lindblad equation in quantum physics, with benchmark validation and new qualitative insights.

Findings

The CFD method provides efficient and stable solutions for the Lindblad equation.

Benchmark tests show good agreement with analytical solutions.

Reformulation reveals new qualitative understanding of open quantum system dynamics.

Abstract

Liouvillian dynamics describes the evolution of a density operator in closed quantum systems. One extension towards open quantum systems is provided by the Lindblad equation. It is applied to various systems and energy regimes in solid state physics as well as also in nuclear physics. A main challenge is that analytical solutions for the Lindblad equation are only obtained for harmonic system potentials or two-level systems. For other setups one has to rely on numerical methods. In this work, we propose to use a method from computational fluid dynamics, the Kurganov-Tadmor central (finite volume) scheme, to numerically solve the Lindblad equation in position-space representation. We will argue, that this method is advantageous in terms of the efficiency concerning initial conditions, discretization, and stability. On the one hand, we study, the applicability of this scheme by performing benchmark tests. Thereby we compare numerical results to analytic solutions and discuss aspects like boundary conditions, initial values, conserved quantities, and computational efficiency. On the other hand, we also comment on new qualitative insights to the Lindblad equation from its reformulation in terms of an advection-diffusion equation with source/sink terms.