The Accuracy of Perturbative Master Equations

TL;DR

This paper reveals that perturbative master equations in open quantum systems require higher-order corrections for accurate solutions, impacting the validity of many commonly used second-order approaches.

Contribution

It demonstrates that second-order master equations are insufficient for accurate solutions, showing the need for higher-order equations to avoid inaccuracies and positivity violations.

Findings

Order-2n solutions are inaccurate without order-(2n+2) equations.

Steady state and positivity violations occur at order-2n.

Exact solutions illustrate the inaccuracies in common approximations.

Abstract

We consider open quantum systems with dynamics described by master equations that have perturbative expansions in the system-environment interaction. We show that, contrary to intuition, full-time solutions of order-2n accuracy require an order-(2n+2) master equation. We give two examples of such inaccuracies in the solutions to an order-2n master equation: order-2n inaccuracies in the steady state of the system and order-2n positivity violations, and we show how these arise in a specific example for which exact solutions are available. This result has a wide-ranging impact on the validity of coupling (or friction) sensitive results derived from second-order convolutionless, Nakajima-Zwanzig, Redfield, and Born-Markov master equations.