  We explore theoretically electromagnetically-induced transparency (EIT) in a
superconducting quantum circuit (SQC). The system is a persistent-current flux
qubit biased in a $\Lambda$ configuration. Previously [Phys. Rev. Lett. 93,
087003 (2004)], we showed that an ideally-prepared EIT system provides a
sensitive means to probe decoherence. Here, we extend this work by exploring
the effects of imperfect dark-state preparation and specific decoherence
mechanisms (population loss via tunneling, pure dephasing, and incoherent
population exchange). We find an initial, rapid population loss from the
$\Lambda$ system for an imperfectly prepared dark state. This is followed by a
slower population loss due to both the detuning of the microwave fields from
the EIT resonance and the existing decoherence mechanisms. We find analytic
expressions for the slow loss rate, with coefficients that depend on the
particular decoherence mechanisms, thereby providing a means to probe,
identify, and quantify various sources of decoherence with EIT. We go beyond
the rotating wave approximation to consider how strong microwave fields can
induce additional off-resonant transitions in the SQC, and we show how these
effects can be mitigated by compensation of the resulting AC Stark shifts.
