  We investigate the influence of temperature and dissipation on the
Landau-Zener transition probability in circuit QED. Dissipation is modelled by
coupling the transmission line to a bath of harmonic oscillators. The reduced
description for the density operator is treated within Bloch-Redfield theory. A
phase-space representation allows an efficient numerical implementation of the
resulting master equation. It provides reliable results which are valid even
for rather low temperatures. We find that the spin-flip probability as a
function of temperature and dissipation strength exhibits a non-monotonic
behaviour. Our numerical results are complemented by analytical solutions for
zero temperature and for vanishing dissipation strength.
