Analytical and Numerical Study of Lindblad Equations

Abstract

Lindblad equations, since introduced in 1976 by Lindblad, and by Gorini, Kossakowski, and Sudarshan, have received much attention in many areas of scientific research. Around the past fifty years, many properties and structures of Lindblad equations have been discovered and identified. In this dissertation, we study Lindblad equations from three aspects: (I) physical perspective; (II) numerical perspective; and (III) information theory perspective.In Chp. 2, we study Lindblad equations from the physical perspective. More specifically, we derive a Lindblad equation for a simplified Anderson-Holstein model arising from quantum chemistry. Though we consider the classical approach (i.e., the weak coupling limit), we provide more explicit scaling for parameters when the approximations are made. Moreover, we derive a classical master equation based on the Lindbladian formalism.In Chp. 3, we consider numerical aspects of Lindblad equations. Motivated by the dynamical low-rank approximation method for matrix ODEs and stochastic unraveling for Lindblad equations, we are curious about the relation between the action of dynamical low-rank approximation and the action of stochastic unraveling. To address this, we propose a stochastic dynamical low-rank approximation method. In the context of Lindblad equations, we illustrate a commuting relation between the dynamical low-rank approximation and the stochastic unraveling.In Chp. 4, we investigate Lindblad equations from the information theory perspective. We consider a particular family of Lindblad equations: primitive Lindblad equations with GNS-detailed balance. We identify Riemannian manifolds in which these Lindblad equations are gradient flow dynamics of sandwiched Rényi divergences. The necessary condition for such a geometric structure is also studied. Moreover, we study the exponential convergence behavior of these Lindblad equations to their equilibria, quantified by the whole family of sandwiched Rényi divergences.