Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes

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.