  We study decoherence effects in qubits coupled to environments that exhibit
resonant frequencies in their spectral function. We model the coupling of the
qubit to its environment via the Caldeira-Leggett formulation of quantum
dissipation/coherence, and study the simplest example of decoherence effects in
circuits with resonances such as a dc SQUID phase qubit in the presence of an
isolation circuit, which is designed to enhance the coherence time. We
emphasize that the spectral density of the environment is strongly dependent on
the circuit design, and can be engineered to produce longer decoherence times.
We begin with a general discussion of superconducting qubits such as the flux
qubit, the Cooper pair box and the phase qubit and show that in these kinds of
systems appropriate circuit design can greatly modify the spectral density of
the environment and lead to enhancement of decoherence times. In the particular
case of the phase qubit, for instance, we show that when the frequency of the
qubit is at least two times larger than the resonance frequency of the
environmental spectral density, the decoherence time of the qubit is a few
orders of magnitude larger than that of the typical ohmic regime, where the
frequency of the qubit is much smaller than the resonance frequency of the
spectral density. In addition, we demonstrate that the environment does not
only affect the decoherence time, but also the frequency of the transition
itself, which is shifted from its environment-free value. Second, we show that
when the qubit frequency is nearly the same as the resonant frequency of the
environmental spectral density, an oscillatory non-Markovian decay emerges, as
the qubit and its environment self-generate Rabi oscillations of characteristic
time scales shorter than the decoherence time.
