Solving the Lindblad equation with methods from computational fluid dynamics

Liouvillian dynamics describes the evolution of a density operator in closed quantum systems. One extension toward 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 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 analytical 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 into the Lindblad equation from its reformulation in terms of an advection-diffusion equation with source and sink terms.