  Decoherence in superconducting qubits is known to arise because of a variety
of environmental degrees of freedom. In this article, we focus on the influence
of thermal fluctuations in a weakly damped circuit resonance coupled to the
qubit. Because of the coupling, the qubit frequency is shifted by an amount $n
\delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations
induce temporal variations $n(t)$ and thus dephasing. We give an approximate
formula for the qubit dephasing time as a function of $\delta \nu_0$. We
discuss the specific case of a flux-qubit coupled to the plasma mode of its
DC-SQUID detector. We first derive a plasma mode-qubit interaction hamiltonian
which, in addition to the usual Jaynes-Cummings term, has a coupling term
quadratic in the oscillator variables coming from the flux-dependence of the
SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in
certain non-trivial bias conditions for which dephasing due to thermal
fluctuations should be suppressed.
