  We present a theoretical treatment for the dissipative two-resonator circuit
quantum electrodynamics setup referred to as quantum switch. There, switchable
coupling between two superconducting resonators is mediated by a
superconducting qubit operating in the dispersive regime, where the qubit
transition frequency is far detuned from those of the resonators. We derive an
effective Hamiltonian for the quantum switch beyond the rotating wave
approximation and study the dissipative dynamics within a Bloch-Redfield
quantum master equation approach. We derive analytically how the qubit affects
the quantum switch even if the qubit has no dynamics, and we estimate the
strength of this influence. The analytical results are corroborated by
numerical calculations, where coherent oscillations between the resonators, the
decay of coherent and Fock states, and the decay of resonator-resonator
entanglement are studied. Finally, we suggest an experimental protocol for
extracting the damping constants of qubit and resonators by measuring the
quadratures of the resonator fields.
