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Studies of the dynamics of a quantum system coupled to a bath are typicallyperformed by utilizing the Nakajima-Zwanzig memory kernel (${\mathcal{K}}$) orthe influence functions ($\mathbf{{I}}$), especially when the dynamics exhibitmemory effects (i.e., non-Markovian). Despite their significance, the formalconnection between the memory kernel and the influence functions has not beenexplicitly made. We reveal their relation through the observation of adiagrammatic structure underlying the system propagator, $\mathbf{{I}}$, and${\mathcal{K}}$. Based on this, we propose a non-perturbative, diagrammaticapproach to construct ${\mathcal{K}}$ from $\mathbf{{I}}$ for (driven) systemsinteracting with harmonic baths without the use of any projection-free dynamicsinputs required by standard approaches. With this construction, we also showhow approximate path integral methods can be understood in terms of approximatememory kernels. Furthermore, we demonstrate a Hamiltonian learning procedure toextract the bath spectral density from a set of reduced system trajectoriesobtained experimentally or by numerically exact methods, opening new avenues inquantum sensing and engineering. The insights we provide in this work willsignificantly advance the understanding of non-Markovian dynamics, and theywill be an important stepping stone for theoretical and experimentaldevelopments in this area.

Unified Framework for Open Quantum Dynamics with Memory