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We study decoherence effects in qubits coupled to environments that exhibitresonant frequencies in their spectral function. We model the coupling of thequbit to its environment via the Caldeira-Leggett formulation of quantumdissipation/coherence, and study the simplest example of decoherence effects incircuits with resonances such as a dc SQUID phase qubit in the presence of anisolation circuit, which is designed to enhance the coherence time. Weemphasize that the spectral density of the environment is strongly dependent onthe circuit design, and can be engineered to produce longer decoherence times.We begin with a general discussion of superconducting qubits such as the fluxqubit, the Cooper pair box and the phase qubit and show that in these kinds ofsystems appropriate circuit design can greatly modify the spectral density ofthe environment and lead to enhancement of decoherence times. In the particularcase of the phase qubit, for instance, we show that when the frequency of thequbit is at least two times larger than the resonance frequency of theenvironmental spectral density, the decoherence time of the qubit is a feworders of magnitude larger than that of the typical ohmic regime, where thefrequency of the qubit is much smaller than the resonance frequency of thespectral density. In addition, we demonstrate that the environment does notonly affect the decoherence time, but also the frequency of the transitionitself, which is shifted from its environment-free value. Second, we show thatwhen the qubit frequency is nearly the same as the resonant frequency of theenvironmental spectral density, an oscillatory non-Markovian decay emerges, asthe qubit and its environment self-generate Rabi oscillations of characteristictime scales shorter than the decoherence time.

Enhancing decoherence times in superconducting qubits via circuit design