  We study the collective effects that emerge in waveguide quantum
electrodynamics where several (artificial) atoms are coupled to a
one-dimensional superconducting transmission line. Since single microwave
photons can travel without loss for a long distance along the line, real and
virtual photons emitted by one atom can be reabsorbed or scattered by a second
atom. Depending on the distance between the atoms, this collective effect can
lead to super- and subradiance or to a coherent exchange-type interaction
between the atoms. Changing the artificial atoms transition frequencies,
something which can be easily done with superconducting qubits (two levels
artificial atoms), is equivalent to changing the atom-atom separation and
thereby opens the possibility to study the characteristics of these collective
effects. To study this waveguide quantum electrodynamics system, we extend
previous work and present an effective master equation valid for an ensemble of
inhomogeneous atoms driven by a coherent state. Using input-output theory, we
compute analytically and numerically the elastic and inelastic scattering and
show how these quantities reveal information about collective effects. These
theoretical results are compatible with recent experimental results using
transmon qubits coupled to a superconducting one-dimensional transmission line
[A.~F.~van Loo {\it et al.}].

Quantum Physics