Abstract Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.
Abstract Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.
Master equations play a pivotal role in investigating open quantum systems. In particular, the Bloch-Redfield equation stands out due to its relation to a concrete physical environment. However, without further approximations it does not lead to a Lindblad master equation that guarantees that the density matrix stays completely positive, which has raised some concerns regarding its use. This study builds on previous efforts to transform the Bloch-Redfield framework into a mathematically robust Lindblad equation while fully preserving the effects that are lost within the secular approximation that is commonly used to guarantee positivity. These previous approaches introduced two potential deficiencies: The environment-induced energy shift can be non-Hermitian and some decay rates can be negative, violating the assumptions of Lindblad's theorem. Here we propose and evaluate straightforward solutions to both problems. Our approach offers an effective and general procedure for obtaining a Lindblad equation, derived from a concrete physical environment, while mitigating the unphysical dynamics present in the Bloch-Redfield equation.
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.
The Markovian dynamics of open quantum many-body systems are typically governed by the Lindblad master equation, yet obtaining the reduced density matrix for bosonic and fermionic systems remains a formidable numerical challenge due to the exponential growth of the Hilbert space. Here, we introduce a computationally efficient framework for constructing the reduced density matrix by modeling the environment as a copy of the primary system with a monotonically decaying coupling that enforces unidirectional energy flow. Our method directly yields exact solutions that rigorously satisfy the Lindblad master equation for both bosonic and fermionic cases, bypassing the need for costly Liouvillian diagonalization. We validate our approach through applications to paradigmatic models, demonstrating accurate reproduction of dissipative dynamics across a broad parameter regime. This work provides a straightforward and powerful tool for simulating Markovian open-system evolution, with immediate applicability to quantum transport and control problems in many-body physics.
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.
It is very common in the literature to write a Markovian quantum master equation in Lindblad form to describe a system with multiple degrees of freedom and weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in a system-bath coupling is of the so-called Redfield form, which is known to not preserve complete positivity in most cases. Additional approximations to the Redfield equation are required to obtain a Lindblad form. We lay down some fundamental requirements for any further approximations to the Redfield equation, which, if violated, leads to physical inconsistencies such as inaccuracies in the leading order populations and coherences in the energy eigenbasis, violation of thermalization, and violation of local conservation laws at the nonequilibrium steady state. We argue that one or more of these conditions will generically be violated in all the weak system-bath-coupling Lindblad descriptions existing in the literature to our knowledge. As an example, we study the recently derived universal Lindblad equation and use these conditions to show the violation of local conservation laws due to inaccurate coherences but accurate populations in the energy eigenbasis. Finally, we exemplify our analytical results numerically in an interacting open quantum spin system.