Adiabatic Elimination and Sub-space Evolution of Open Quantum Systems

TL;DR

This paper develops a method for effectively describing open quantum systems by eliminating fast degrees of freedom, providing corrections for the trace of the density matrix, and illustrating the approach with models involving continuous and discrete states.

Contribution

It introduces a frequency space construction of effective operators with trace corrections, applicable to systems with continuous or discrete fast subspaces, and analyzes their convergence at high dissipation.

Findings

Effective operators in frequency space improve system descriptions.

Trace correction accounts for non trace-preserving evolution.

Models converge at high dissipation and coherent population trapping.

Abstract

Efficient descriptions of open quantum systems can be obtained by performing an adiabatic elimination of the fast degrees of freedom and formulating effective operators for the slow degrees of freedom in reduced dimensions. Here, we perform the construction of effective operators in frequency space, and using the final value theorem or alternatively the Keldysh theorem, we provide a correction for the trace of the density matrix which takes into account the non trace-preserving character of the evolution. We illustrate our results with two different systems, ones where the eliminated fast subspace is constituted by a continuous set of states and ones with discrete states. Furthermore, we show that the two models converge for very large dissipation and at coherent population trapping points. Our results also provide an intuitive picture of the correction to the trace of the density matrix as a detailed balance equation.