© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1303.6580 · Physical Review A

Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations

Christian Majenz, Tameem Albash, Heinz‐Peter Breuer, Daniel A. Lidar

Published Jul 8, 2013• 77 citations

Abstract

We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.

DOI: 10.1103/physreva.88.012103

Related papers

Coarse-graining in the derivation of Markovian master equations and its significance in quantum thermodynamics

James D. Cresser, Christopher Facer

The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.

arXiv:1710.09939•Oct 26, 2017•15 citations

Markovian and non-Markovian master equations versus an exactly solvable model of a qubit in a cavity

Zihan Xia, Juan Garcia-Nila, Daniel A. Lidar

Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.

arXiv:2403.09944•Physical Review Applied•Jul 11, 2024•14 citations

On derivation of Markovian master equation with time coarse graining

Yasusada Nambu, Shingo Kukita

We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.

•Apr 13, 2015•0 citations

Comparison between non-Markovian dynamics with and without rotating wave approximation

Ning Tang, Tian-Tian Xu, Hao‐Sheng Zeng

In the limit of weak coupling between a system and its reservoir, we derive the time-convolutionless (TCL) non-Markovian master equation for a two-level system interacting with a zero-temperature structured environment with no rotating wave approximation (NRWA). By comparing the dynamics with RWA, we demonstrate the impact of RWA on the system dynamics, as well as the effects of non-Markovianity on the preservation of atomic coherence, squeezing, and entanglement.

Chinese Physics B•Mar 1, 2013•14 citations

Time-Uniform Error Bound for Temporal Coarse Graining in Markovian Open Quantum Systems

Teruhiro Ikeuchi, Takashi Mori

Several approximation procedures, such as the full or partial rotating-wave, time-averaging, and geometric-arithmetic approximations, have been proposed to derive Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators from the Born-Markov quantum master equation (e.g., the Redfield equation). Establishing rigorous error bounds for these approximations is of fundamental and practical importance. However, existing bounds face two major limitations: they are highly specific to individual methods, and, more critically, they diverge in the long-time limit, ensuring the accuracy of the derived GKSL generator only in short-time regimes. In this Letter, we resolve both issues by deriving a unified, rigorous error bound for a general class of approximation methods -- termed temporal coarse graining -- that encompasses all aforementioned schemes. Crucially, our error bound is time-uniform. This guarantees that GKSL generators obtained via temporal coarse graining remain accurate for arbitrarily long times, provided the dissipation timescale is significantly longer than the bath correlation timescale.

cond-mat.stat-mechCross-listedarXiv:2604.21366•Apr 24, 2026•0 citations

Comparison between non-Markovian dynamics with and without rotating wave approximation

唐宁, 徐甜甜, 曾浩生

In the limit of weak coupling between a system and its reservoir,we derive the time-convolutionless(TCL) nonMarkovian master equation for a two-level system interacting with a zero-temperature structured environment with no rotating wave approximation(NRWA).By comparing the dynamics with RWA,we demonstrate the impact of RWA on the system dynamics,as well as the effects of non-Markovianity on the preservation of atomic coherence,squeezing,and entanglement.

中国物理B：英文版•Jan 1, 2013•0 citations

We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.

The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.

Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.

We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.

Several approximation procedures, such as the full or partial rotating-wave, time-averaging, and geometric-arithmetic approximations, have been proposed to derive Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators from the Born-Markov quantum master equation (e.g., the Redfield equation). Establishing rigorous error bounds for these approximations is of fundamental and practical importance. However, existing bounds face two major limitations: they are highly specific to individual methods, and, more critically, they diverge in the long-time limit, ensuring the accuracy of the derived GKSL generator only in short-time regimes. In this Letter, we resolve both issues by deriving a unified, rigorous error bound for a general class of approximation methods -- termed temporal coarse graining -- that encompasses all aforementioned schemes. Crucially, our error bound is time-uniform. This guarantees that GKSL generators obtained via temporal coarse graining remain accurate for arbitrarily long times, provided the dissipation timescale is significantly longer than the bath correlation timescale.