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We study the collective effects that emerge in waveguide quantumelectrodynamics where several (artificial) atoms are coupled to aone-dimensional superconducting transmission line. Since single microwavephotons can travel without loss for a long distance along the line, real andvirtual photons emitted by one atom can be reabsorbed or scattered by a secondatom. Depending on the distance between the atoms, this collective effect canlead to super- and subradiance or to a coherent exchange-type interactionbetween the atoms. Changing the artificial atoms transition frequencies,something which can be easily done with superconducting qubits (two levelsartificial atoms), is equivalent to changing the atom-atom separation andthereby opens the possibility to study the characteristics of these collectiveeffects. To study this waveguide quantum electrodynamics system, we extendprevious work and present an effective master equation valid for an ensemble ofinhomogeneous atoms driven by a coherent state. Using input-output theory, wecompute analytically and numerically the elastic and inelastic scattering andshow how these quantities reveal information about collective effects. Thesetheoretical results are compatible with recent experimental results usingtransmon qubits coupled to a superconducting one-dimensional transmission line[A.~F.~van Loo {\it et al.}].

Input-output theory for waveguide QED with an ensemble of inhomogeneous atoms