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Physical review. B./Physical review. B

Benchmarking quantum master equations beyond ultraweak coupling

Camilo Santiago Tello Breuer, Tobias Becker, André Eckardt

Published Aug 21, 2024• 5 citations
Publisher

Abstract

Open quantum systems are ubiquitous, however, their description beyond ultraweak system-bath coupling is challenging. The standard Redfield master equation can violate positivity. Recently, Nathan and Rudner proposed an alternative equation, not suffering from this problem. The authors compare here both equations to the exact solution of a simple model. The Redfield equation performs noticeably better for transient dynamics. For the steady state it is slightly better for high temperatures, but fails at low temperatures, where the Nathan-Rudner equation is still accurate.

DOI
10.1103/physrevb.110.064319

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Benchmarking quantum master equations beyond ultraweak coupling

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Abstract Quantum master equations are widely used to describe the dynamics of open quantum systems. All these different master equations rely on specific approximations that may or may not be justified. Starting from a microscopic model, applying the standard Born and Markov approximations results in the Redfield equation that does not guarantee to preserve positivity. The latter is typically achieved by additionally applying the secular approximation resulting in a quantum master equation in Lindblad form. There are other ways to obtain an equation in Lindblad form, one of which is the recently proposed universal Lindblad equation (ULE). It has been shown that it is in the same equivalence class of approximations as the Redfield master equation although avoiding the heuristic secular approximation (Nathan and Rudner 2020 Phys. Rev. B 102 115109). In this work, we prove that the solutions of the master equation obtained by applying the secular approximation are also obtained by an approximation of the same order as the one performed to obtain the Redfield equation. We hereby provide a mathematical justification for the secular approximation. We show that the result of applying the secular approximation is obtained naturally by applying a self-consistency argument. This shows that the resulting master equation is also in the same equivalence class of approximations as the Redfield master equation and the ULE. We furthermore give numerical considerations about the accuracy of the secular master equation compared to the ULE.

arXiv:2505.06786•Journal of Physics A Mathematical and Theoretical•Aug 24, 2026•0 citations
Open quantum systems are ubiquitous, however, their description beyond ultraweak system-bath coupling is challenging. The standard Redfield master equation can violate positivity. Recently, Nathan and Rudner proposed an alternative equation, not suffering from this problem. The authors compare here both equations to the exact solution of a simple model. The Redfield equation performs noticeably better for transient dynamics. For the steady state it is slightly better for high temperatures, but fails at low temperatures, where the Nathan-Rudner equation is still accurate.
Recently, Nathan and Rudner derived a Gorini-Kossakowski-Sudarshan-Lindblad master equation from the Redfield equation. The claim is that the level of approximation is equal to that of the Redfield equation. Here we benchmark the Nathan-Rudner equation (NRE) against the exact solution of a damped harmonic oscillator and compare its performance to that of the time-dependent Redfield equation (RE). We find that which of the equations performs better depends on the regime considered. It turns out that the short-time dynamics is generally much better captured by the RE, whereas the NRE delivers results comparable to those of the rotating-wave approximation. For the steady state, in the high-temperature limit the RE again performs better and its solution approaches the exact result for ultrahigh temperatures. Nevertheless, here also the NR equation constitutes a good approximation. In the low-temperature limit, in turn, the NRE provides a better approximation than the RE. For too strong coupling, here the RE might even fail completely by predicting unphysical behaviour.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
Quantum master equations are an invaluable and highly successful tool for the approximate, yet accurate, description of undriven open quantum systems weakly coupled to their environment. In contrast, results on the description of the equally important case of driven open quantum systems using quantum master equations remain scarce. Existing approaches typically impose restrictions on the driving protocol, including, for instance, Floquet master equations for periodic driving and adiabatic master equations for slow driving. However, no generally accepted formulation for arbitrary driving has yet been established. A promising candidate from which all existing approaches can be derived by approximations based on protocol constraints is the driven Redfield master equation. Strikingly, it has received very little attention so far. A main reason is that, in analogy to its undriven counterpart, the driven Redfield master equation notoriously violates complete positivity. In the undriven case, this problem can be successfully addressed using the rotating wave approximation (RWA). For approaches derived from the driven Redfield master equation for specific protocols, such as Floquet master equations and adiabatic master equations, a generalized form of the RWA can still be applied. However, a full generalization directly applicable to the driven Redfield master equation under arbitrary driving has not yet been developed. To address this issue, we propose a straightforward and natural generalization of the RWA based on the version of the RWA used for adiabatic master equations. We then apply the driven Redfield master equation to the paradigmatic, exactly solvable model of a driven quantum harmonic oscillator coupled to a bosonic bath and benchmark it against the exact solution. In particular, we compare the original driven Redfield master equation without an RWA to its generalized RWA version. We find that the driven Redfield master equation without an RWA exhibits excellent agreement with the exact solution across all driving speeds, while the violation of complete positivity shows no notable effect. By contrast, the RWA-based version of the driven Redfield master equation fails to reproduce the dynamics accurately beyond slow driving at short times. Additionally, we analyze known adiabatic master equations, both without and with an RWA. Remarkably, the non-RWA versions agree with the exact dynamics far beyond the strictly adiabatic regime, while the RWA-based versions already show noticeable deviations even for slow driving, contrasting previous observations in the literature. Taken together, these results indicate that the driven Redfield master equation and adiabatic master equations without RWA have been significantly underestimated in their practical usefulness, whereas RWA-based approaches in driven settings may have been overestimated. This strongly suggests that future work should focus on alternative approximation methods applied to the driven Redfield master equation that preserve its high numerical accuracy while restoring complete positivity.
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system toward a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, we use the exact mean-force Gibbs state to correct the Redfield quantum master equation. By benchmarking it with the exact solution of the damped harmonic oscillator, we show that our method also helps correct the long-standing issue of positivity violation, albeit without complete positivity. Our method of a canonically consistent quantum master equation opens a new perspective in the theory of open quantum systems leading to a reduced density matrix accurate beyond the commonly used Redfield and Lindblad equations, while retaining the same conceptual and numerical complexity.
Abstract Quantum master equations are widely used to describe the dynamics of open quantum systems. All these different master equations rely on specific approximations that may or may not be justified. Starting from a microscopic model, applying the standard Born and Markov approximations results in the Redfield equation that does not guarantee to preserve positivity. The latter is typically achieved by additionally applying the secular approximation resulting in a quantum master equation in Lindblad form. There are other ways to obtain an equation in Lindblad form, one of which is the recently proposed universal Lindblad equation (ULE). It has been shown that it is in the same equivalence class of approximations as the Redfield master equation although avoiding the heuristic secular approximation (Nathan and Rudner 2020 Phys. Rev. B 102 115109). In this work, we prove that the solutions of the master equation obtained by applying the secular approximation are also obtained by an approximation of the same order as the one performed to obtain the Redfield equation. We hereby provide a mathematical justification for the secular approximation. We show that the result of applying the secular approximation is obtained naturally by applying a self-consistency argument. This shows that the resulting master equation is also in the same equivalence class of approximations as the Redfield master equation and the ULE. We furthermore give numerical considerations about the accuracy of the secular master equation compared to the ULE.