Purcell effect in a circuit-QED implementation of a universal two-qubit gate

We study spontaneous emission processes of two capacitively coupled superconducting qubits embedded in transmission-line resonators in a circuit-quantum electrodynamics architecture. We find that under proper conditions of the entanglement generating operation, a Purcell protection from spontaneous emission occurs. We discuss the dependence of this effect on the coupling energy between the two qubits and comment on the possibility of finding a trade-off between optimization of the gate with respect to low-frequency noise and relaxation effects.