  Time-varying optical materials have attracted recent interest for their
potential to enable frequency conversion, nonreciprocal physics, photonic
time-crystals, and more. However, the description of time-varying materials has
been primarily limited to regimes where material resonances (i.e., dispersion)
can be neglected. In this work, we describe how the optics of these dispersive
time-varying materials emerges from microscopic quantum mechanical models of
time-driven systems. Our results are based on a framework for describing the
optics of dispersive time-varying materials through quantum mechanical linear
response theory. Importantly, we clarify how response functions for
time-varying materials are connected to energy transfer. We provide three
examples of our framework applied to systems which can be used to model a wide
variety of experiments: few level models that can describe atoms, spins, or
superconducting qubits, oscillator models which can describe the strong
response of polar insulators, and strongly driven atom models which can
describe the highly nonperturbative optical response of materials undergoing
high harmonic generation. We anticipate that our results will be broadly
applicable to electromagnetic phenomena in strongly time-varying systems.
