Cooperative effects for qubits in a semi-infinite one-dimensional transmission line

In a one-dimensional (1D) transmission line, it is possible to achieve nearly ideal spatial mode-matching for a microwave field. Taking advantage of this, experiments have demonstrated strong interaction between resonant propagating microwave photons and superconducting qubits in such a system. When the system contains more than one qubit, photon-mediated interactions between the qubits gives rise to many interesting phenomena. In this work, we embedded two transmon qubits, separated by wavelength distances, in a 1D transmission line terminated by a mirror. By tuning the qubits into resonance, we observe collective effects. In particular, both the relaxation rates and the Lamb shifts of the qubits are modified as a function of the inter-qubit and qubit-mirror distances.