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Here, we consider \u039b\u0302(\u03c4) arising from suddenly coupling a system to one or more thermal baths with a strength that is neither weak nor strong. Given no clear separation of characteristic system\/bath time scales, \u039b\u0302(\u03c4) is generically expected to be non-Markovian; however, we do assume the ensuing dynamics has a unique steady state, implying the baths possess a finite memory time \u03c4m. By combining several techniques within a tensor network framework, we directly and accurately extract \u039b\u0302(\u03c4) for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. First, we use an orthogonal polynomial mapping and thermofield doubling to arrive at a purified chain representation of the baths whose length directly equates to a time over which the dynamics of the infinite baths is faithfully captured. Second, we employ the Choi\u2013Jamiolkowski isomorphism so that \u039b\u0302(\u03c4) can be fully reconstructed from a single pure state calculation of the unitary dynamics of the system, bath and their replica auxiliary modes up to time \u03c4. From \u039b\u0302(\u03c4), we also compute the time local propagator L\u0302(\u03c4). By examining the convergence with \u03c4 of the instantaneous fixed points of these objects, we establish their respective memory times \u03c4m\u039b and \u03c4mL. Beyond these times, the propagator L\u0302(\u03c4) and dynamical map \u039b\u0302(\u03c4) accurately describe all the subsequent long-time relaxation dynamics up to stationarity. These timescales form a hierarchy \u03c4mL\u2264\u03c4m\u039b\u2264\u03c4re, where \u03c4re is a characteristic relaxation time of the dynamics. Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where \u03c4re \u226b \u03c4m, where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit. Our results also show that having access to \u039b\u0302(\u03c4) affords a number of insightful analyses of the open system thus far not commonly exploited.<\/jats:p>","DOI":"10.1063\/5.0228428","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T15:19:31Z","timestamp":1729264771000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Extracting dynamical maps of non-Markovian open quantum systems"],"prefix":"10.1063","volume":"161","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9500-6373","authenticated-orcid":false,"given":"David J.","family":"Strachan","sequence":"first","affiliation":[{"name":"H. H. 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