Mappings of open quantum systems onto chain representations and Markovian embeddings

TL;DR

This paper introduces a method to map open quantum systems onto chain models with nearest neighbor interactions, enabling better spectral density analysis and convergence understanding as environmental degrees of freedom increase.

Contribution

It presents a novel mapping technique using measures and Jacobi matrices to represent open quantum systems as semi-infinite chains, with new convergence theorems for spectral densities.

Findings

Derived measures with special properties for chain mapping

Expressed spectral densities for embedded environments

Proved convergence theorems for residual spectral densities

Abstract

We derive a sequence of measures whose corresponding Jacobi matrices have special properties and a general mapping of an open quantum system onto 1D semi infinite chains with only nearest neighbour interactions. Then we proceed to use the sequence of measures and the properties of the Jacobi matrices to derive an expression for the spectral density describing the open quantum system when an increasing number of degrees of freedom in the environment have been embedded into the system. Finally, we derive convergence theorems for these residual spectral densities.