  We introduce a lattice model of interacting spins and bosons that leads to
Luttinger-liquid physics, and allows for quantitative tests of the theory of
bosonization by means of trapped-ion or superconducting-circuit experiments. By
using a variational bosonization ansatz, we calculate the power-law decay of
spin and boson correlation functions, and study their dependence on a single
tunable parameter, namely a bosonic driving. For small drivings,
Matrix-Product-States (MPS) numerical methods are shown to be efficient and
validate our ansatz. Conversely, even static MPS become inefficient for
large-driving regimes, such that the experiment can potentially outperform
classical numerics, achieving one of the goals of quantum simulations.
