Fig. 1: Unification of open quantum dynamics framework for Class 1.
From: Unified framework for open quantum dynamics with memory
a An open quantum system, where the environment is characterized by the spectral density J(ω), can be described with the generalized quantum master equation (GQME) and the influence functional path integral (INFPI). The former distills environmental correlations through the memory kernels \({{{\mathcal{K}}}}\) while the latter through the influence functionals \({{{\mathcal{I}}}}\). In this work, we show both are related through Dyck Paths, and that, furthermore, we can use the Dyck construction for extracting J(ω) by simply knowing how the quantum system evolves. b Cumulant expansion of memory kernel. Examples through Eq. (6) for N  =  2 and N  = 3. Solid arcs of diameter k filled with all possible arcs of diameters smaller than k denote propagator Uk. c Dyck path diagrams. Examples for N = 2 and N  = 3 and their corresponding influence function diagrams, which composes \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{2}\) and \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{3}\), respectively. Solid lines denote influence functions I and dashed lines denote \(\tilde{{{{\bf{I}}}}}\).
