Spectral densities of superconducting qubits with environmental resonances

In this paper we derive the environmental spectral density for flux, phase and charge qubits when each of them is coupled to an environment with a resonance. From the spectral density we obtain the characteristic spontaneous emission (relaxation) lifetimes $T_1$ for each of these qubits, and show that there is a substantial enhancement of $T_1$ beyond the resonant frequency of the environment.