  In this work we study numerically and analytically the interaction of two
qubits in a one-dimensional waveguide, as mediated by the photons that
propagate through the guide. We develop strategies to assert the Markovianity
of the problem, the effective qubit-qubit interactions and their individual and
collective spontaneous emission. We prove the existence of collective
Lamb-shifts that affect the qubit-qubit interactions and the dependency of
coherent and incoherent interactions on the qubit separation. We also develop
the scattering theory associated to these models and prove single photon
spectroscopy does probes the renormalized resonances of the single- and
multi-qubit models, in sharp contrast with earlier toy models where individual
and collective Lamb shifts cancel.
