This dissertation demonstrates the usefulness of exactly solvable quantum models in the investigation of light-matter interaction phenomena associated with the propagation of ultrashort laser pulses through gaseous media. This work fits into the larger research effort towards remedying the weaker portions of the standard set of medium modeling equations commonly used in simulations. The ultimate goal is to provide a self-consistent quantum mechanical description that can integrate Maxwell and Schrödinger systems and provide a means to realistically simulate nonlinear optical experiments on relevant scales. The study of exactly solvable models begins with one of the simplest quantum systems available, one with a 1D Dirac-delta function potential plus interaction with the light field. This model contains, in the simplest form, the most important "ingredients" that control optical filamentation, i.e. discrete and continuum electronic states. The importance of both states is emphasized in the optical intensity regime in which filaments form, where both kinds of electronic states simultaneously play a role and may not even be distinguishable. For this model atom, an analytical solution for the time-dependent light-induced atomic response from an arbitrary excitation waveform is obtained. Although this system is well-known and has been studied for decades, this result is probably the most practically useful and general one obtained thus far. Numerical implementation details of the result are also given as the task is far from trivial. Given an efficient implementation, the model is used in light-matter interaction simulations and from these it is apparent that even this toy model can qualitatively reproduce many of the nonlinear phenomena seen in experiments. Not only does this model capture the basic physics of optical filamentation, but it is also well-suited for high harmonic generation simulations. Next, a theoretical framework for using Stark resonant states (or metastable states) to represent the medium's polarization response is presented. Researchers have recognized long ago the utility of Gamow resonant states as a description of various decay processes. Even though a bound electron experiences a similar decay-like process as it transitions into the continuum upon ionization, it was unclear whether field-induced Stark resonant states carry physically relevant information. It is found that they do, and in particular it is possible to use them to capture a medium's polarization response. To this end, two quantum systems with potentials represented by a 1D Dirac-delta function and a 1D square well are solved, and all the necessary quantities for their use as medium models are presented. From these results it is possible to conjecture some general properties that hold for all resonance systems, including systems that reside in higher than one dimensional space. Finally, as a practical application of this theory, the Metastable Electronic State Approach (MESA) is presented as a quantum-based replacement for the standard medium modeling equations.
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