  Simulation of the interaction of light with matter, including at the
few-photon level, is important for understanding the optical and optoelectronic
properties of materials, and for modeling next-generation non-linear
spectroscopies that use entangled light. At the few-photon level the quantum
properties of the electromagnetic field must be accounted for with a quantized
treatment of the field, and then such simulations quickly become intractable,
especially if the matter subsystem must be modeled with a large number of
degrees of freedom, as can be required to accurately capture many-body effects
and quantum noise sources. Motivated by this we develop a quantum simulation
framework for simulating such light-matter interactions on platforms with
controllable bosonic degrees of freedom, such as vibrational modes in the
trapped ion platform. The key innovation in our work is a scheme for simulating
interactions with a continuum field using only a few discrete bosonic modes,
which is enabled by a Green's function (response function) formalism. We
develop the simulation approach, sketch how the simulation can be performed
using trapped ions, and then illustrate the method with numerical examples. Our
work expands the reach of quantum simulation to important light-matter
interaction models and illustrates the advantages of extracting dynamical
quantities such as response functions from quantum simulations.
