Making non-Markovian master equations accessible with approximate environments

Accurate and efficient simulation of open quantum systems remains a significant challenge, particularly for non-Markovian dynamics. We demonstrate the profound utility of expressing the environmental correlation function as a sum of damped sinusoidals within master equations. While not strictly required, this decomposition offers substantial benefits, crucially reducing the cost of Lamb-shift and decay-rate calculations without sacrificing accuracy. Furthermore, this approach enables straightforward calculation of Lamb-shift corrections, bypassing the need for complex principal value integration. We show that these Lamb-shift effects are demonstrably non-negligible in heat transport scenarios and are needed for an accurate description. Unlike in the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation, the noncommuting nature of the Lamb shift with the Hamiltonian in non-Markovian descriptions, coupled with GKLS's inaccuracies at early times, brings the necessity of non-Markovian descriptions for finite-time thermodynamics. In the weak-coupling regime, our master equation formulations with exponential decomposition achieve accuracy comparable to numerically exact methods. This methodology significantly simplifies and accelerates the simulation of non-Markovian dynamics in open quantum systems, offering a more reliable and computationally tractable alternative akin to a global master equation.