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 exact solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the exact correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation with and without the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the exact asymptotic limit under the factorization approximation only. We find discrepancies that could be significant when the bandwidth of the bath Λ is much larger than the typical scales of the system. We study the exact interaction energy as a proxy for the correlations missed by the Born approximation and find that its dependence on Λ is similar to the 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 γ, its unrenormalized natural frequency Ω and Λ: γ/Ω ≪ 1 and also γΛ/Ω 2 ≪ 1. The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.

Testing the Validity of the ‘Local’ and ‘Global’ GKLS Master Equations on an Exactly Solvable Model

“…The steady state for an all-linear model can also be found exactly by solving the corresponding quantum Langevin equations [30][31][32]. In this section, we will limit ourselves to outline the procedure to calculate the stationary covariances for our particular problem, while full details on its application to similar settings can be found in e.g.…”

Quantum dissipation of planar harmonic systems: Maxwell-Chern-Simons theory

“…We show first the derivation forG αβ ij , and an identical procedure can be carried out to obtainF αβ ij . Substituting the corresponding expressions (17), (20) and (25) in (D2), we find 2G αβ ij (ω, ω , t 0 ) (2π) 2 e it0(ω−ω ) = e 4 κ 2 4(2π) 4 L 4 ωω n l,m=1 k,k e −2σ(|k| 2 +|k | 2 ) e i(k ·(qj +qm)−k·(qi+q l )) × κ 2 k α k β ιν νγ ι ν ν γ k ι k ι k γ k γ e i κ |κ| (θ(k )−θ(k)) |k | 2 |k| 2 ω(k)ω(k )…”

Partition of energy for a dissipative quantum oscillator

“…where the spectral density J(ω) is defined in Eq. (5). For an even function f (t), we define the pair of Fourier cosine transforms by the relations…”

“…As an example, it has been lately applied in the problem of quantum-to-classical transition, formation of dynamical spectrum broadcast structures and classical objectivity as a property of quantum states [4]. Finally, we subjectively cite only a few papers [5][6][7][8][9] published in the last two years to confirm that it is still the topic of active research. We also wish to revisit the dissipative quantum oscillator and discuss a quite different aspect, namely, the quantum counterpart of the theorem of energy equipartition (TEE) in classical statistical physics.…”