  We investigate quantum dynamics of a generic model of light-matter
interaction in the context of high impedance waveguides, focusing on the
behavior of the emitted photonic states, in the framework of the spin-boson
model Quantum quenches as well as scattering of an incident coherent pulse are
studied using two complementary methods. First, we develop an approximate
ansatz for the electromagnetic waves based on a single multimode coherent state
wavefunction; formally, this approach combines ideas from adiabatic
renormalization, the Born-Markov approximation, and input-output theory.
Second, we present numerically exact results for scattering of a weak intensity
pulse by using NRG calculations. NRG provides a benchmark for any linear
response property throughout the ultra-strong coupling regime. We find that in
a sudden quantum quench, the coherent state approach produces physical
artifacts, such as improper relaxation to the steady state. These previously
unnoticed problems are related to the simplified form of the ansatz that
generates spurious correlations within the bath. In the scattering problem, NRG
is used to find the transmission and reflection of a single photon, as well as
the inelastic scattering of that single photon. Simple analytical formulas are
established and tested against the NRG data that predict quantitatively the
transport coefficients for up to moderate environmental impedance. These
formulas resolve pending issues regarding the presence of inelastic losses in
the spin-boson model near absorption resonances, and could be used for
comparison to experiments in Josephson waveguide QED. Finally, the scattering
results using the coherent state wavefunction approach are compared favorably
to the NRG results for very weak incident intensity. We end our study by
presenting results at higher power where the response of the system is
nonlinear.
