Predicting slow relaxation timescales in open quantum systems

Molecules in open systems may be modeled using so-called reduced descriptions that keep focus on the molecule while including the effects of the environment. Mathematically, the matrices governing the Markovian equations of motion for the reduced density matrix, such as the Lindblad and Redfield equations, belong to the family of non-normal matrices. Tools for predicting the behavior of normal matrices (e.g., eigenvalue decompositions) are inadequate for describing the qualitative dynamics of systems governed by non-normal matrices. For example, such a system may relax to equilibrium on timescales much longer than expected from the eigenvalues. In this paper we contrast normal and non-normal matrices, expose mathematical tools for analyzing non-normal matrices, and apply these tools to a representative example system. We show how these tools may be used to predict dissipation timescales at both intermediate and asymptotic times, and we compare these tools to the conventional eigenvalue analyses. Interactions between the molecule and the environment, while generally resulting in dissipation on long timescales, can directly induce transient or even amplified behavior on short and intermediate timescales.