Heisenberg-Langevin versus quantum master equation

Abstract

The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the \emph{exact} solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the \emph{exact} correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation \emph{with and without} the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the \emph{exact} asymptotic limit under the factorization approximation \emph{only}. We find discrepancies that could be significant when the bandwidth of the bath $\Lambda$ is much larger than the typical scales of the system. We study the \emph{exact} interaction energy as a \emph{proxy} for the correlations missed by the Born approximation and find that its dependence on $\Lambda$ is similar to the \emph{discrepancy} between the exact solution and that of the quantum master equation in the Born approximation. We quantify the regime of validity of the quantum master equation in the Born approximation with or without the Markov approximation in terms of the system's relaxation rate $\gamma$, its \emph{unrenormalized} natural frequency $\Omega$ and $\Lambda$: $\gamma/\Omega \ll 1$ and \emph{also} $\gamma \Lambda/\Omega^2 \ll 1$. The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.