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Studies of the dynamics of a quantum system coupled to baths are typically performed by utilizing the Nakajima-Zwanzig memory kernel (${\mathcal{K}}$) or the influence functions ($\mathbf{{I}}$), especially when the dynamics exhibit memory effects (i.e., non-Markovian). Despite their significance, the formal connection between the memory kernel and the influence functions has not been explicitly made. We reveal their relation by inspecting the system propagator for a broad class of problems where an $N$-level system is linearly coupled to Gaussian baths (bosonic, fermionic, and spin.) With this connection, we also show how approximate path integral methods can be understood in terms of approximate memory kernels. For a certain class of open quantum system problems, we devised a non-perturbative, diagrammatic approach to construct ${\mathcal{K}}$ from $\mathbf{{I}}$ for (driven) systems interacting with Gaussian baths without the use of any projection-free dynamics inputs required by standard approaches. Lastly, we demonstrate a Hamiltonian learning procedure to extract the bath spectral density from a set of reduced system trajectories obtained experimentally or by numerically exact methods, opening new avenues in quantum sensing and engineering. The insights we provide in this work will significantly advance the understanding of non-Markovian dynamics, and they will be an important stepping stone for theoretical and experimental developments in this area. |
