Complete positivity of non-Markovian quantum dynamics

Abstract

The hierarchy equations of motion provide an elegant formalism for the description of non-Markovian quantum dynamics, which has successfully been applied to a broad variety of physical problems ranging from light-harvesting complexes to resonant tunnelling junctions. When these equations are derived from microscopic models of system and environment, then they typically expand to an infinite series of coupled differential equations. The truncation of this series is accompanied by an approximation of the dynamics and can give rise to a violation of complete positivity.

The goal of this thesis is to investigate under what conditions a set of hierarchy equations of motion induces completely positive dynamics. Such conditions are in particular crucial for a phenomenological understanding of the equations of motion. Several approaches to this problem are proposed and compared for classical systems before a generalisation to quantum mechanical problems is carried out. The main result is a concise set of operator inequalities that are sufficient for complete positivity and that can be implemented efficiently in terms of a semi-definite program. The applicability of this technique is demonstrated by various explicit examples.

For truncated hierarchy equations for which a violation of complete positivity is inevitable the framework is extended such that the degree of this violation can be estimated and the validity of the truncation can be gauged. In particular in the presence of initial system-environment correlations a dynamical process can also be considered physically reasonable, when the time evolution is not completely positive and it is shown how hierarchy equations that describe such scenarios can be identified. Eventually, the framework is altered such that it gives rise to conditions on hierarchy equations that are sufficient not only for completely positive but even Markovian dynamics.