Few-mode models have been a cornerstone of the theoretical work in quantum optics, with the famous single-mode Jaynes-Cummings model being only the most prominent example. In this work, we develop ab initio few-mode theory, a framework connecting few-mode system-bath models to ab initio theory. We first present a method to derive exact few-mode Hamiltonians for non-interacting quantum potential scattering problems and demonstrate how to rigorously reconstruct the scattering matrix from such few-mode Hamiltonians. We show that upon inclusion of a background scattering contribution, an ab initio version of the well known input-output formalism is equivalent to standard scattering theory. On the basis of these exact results for non-interacting systems, we construct an effective few-mode expansion scheme for interacting theories, which allows to extract the relevant degrees of freedom from a continuum in an open quantum system. As a whole, our results demonstrate that few-mode as well as input-output models can be extended to a general class of problems, and open up the associated toolbox to be applied to various platforms and extreme regimes. We outline differences of the ab initio results to standard model assumptions, which may lead to qualitatively different effects in certain regimes. The formalism is exemplified in various simple physical scenarios. In the process we provide proof-of-concept of the method, demonstrate important properties of the expansion scheme, and exemplify new features in extreme regimes.
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