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.

量子系统的现实模型必须包括与热环境的耗散相互作用。对于弱阻尼系统，尽管Lindblad形式的马尔可夫主方程对于此任务非常重要，但仅当系统转换的任何子集的频率退化时，或者它们的差值比转换的线宽大得多时，它才适用。在这些机制之外，唯一可用的有效描述是Bloch-Redfield主方程，由于无法保证密度矩阵的正性，其有效性长期以来一直存在争议。在所有情况下有效模拟弱阻尼系统的能力变得越来越重要，尤其是在量子技术中。在这里，我们通过推导适用于所有状态的弱阻尼系统的Lindblad形式主方程来解决这个长期存在的问题。我们进一步表明，当该主方程分解时，所有与时间无关的马尔可夫方程（包括BR方程）也将分解。因此，我们获得了用于热阻尼的BR方程的替代，该替代的精确度更高，结构更简单，完全为正，可以通过有效的量子轨迹方法进行仿真，并且统一了先前的Lindblad主方程。我们还通过精确的仿真表明，新的主方程可以描述在演化过程中跃变频率相互交叉的系统。在系统工程中开发的系统识别工具在我们的分析中起着重要作用。