Analytical and Numerical Study of Lindblad Equations

Yu Hua Cao, Jianfeng Lu

Jan Rais, Adrian Koenigstein, Niklas Zorbach +1 more

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.

Tuure Orell

Axel U. J. Lode

Michael Riesch, Alek Pikl, Christian Jirauschek

The Lindblad master equation is a valuable tool in quantum mechanics, which describes the dynamics of open systems. In the scope of our research, it is combined with the one-dimensional Maxwell's equations to form the generalized Maxwell-Bloch equations. Since analytical solutions are not available in the general case, numerical methods have to be employed to solve the Lindblad equation. In this work, we focus on methods that are completely positive trace preserving (CPTP), i.e., that guarantee to preserve the properties of the density matrix. We review existing approaches and compare the most promising candidates in terms of computational performance.

Yu Hua Cao, Jianfeng Lu

For multi-level open quantum systems, the interaction between different levels could pose a challenge to understand the quantum system both analytically and numerically. In this work, we study the approximation of the dynamics of the Anderson-Holstein model, as a model of the multi-level open quantum system, by Redfield and Lindblad equations. Both equations have a desirable property that if the density operators for different levels are diagonal initially, they remain to be diagonal for any time. Thanks to this nice property, the semiclassical limit of both Redfield and Lindblad equations could be derived explicitly; the resulting classical master equations share similar structures of transport and hopping terms. The Redfield and Lindblad equations are also compared from the angle of time dependent perturbation theory.