  We study the properties of superconducting electrical circuits, realizing
cavity QED. In particular we explore the limit of strong coupling, low
dissipation, and elevated temperatures relevant for current and future
experiments. We concentrate on the cavity susceptibility as it can be directly
experimentally addressed, i.e., as the impedance or the reflection coefficient
of the cavity. To this end we investigate the dissipative Jaynes-Cummings model
in the strong coupling regime at high temperatures. The dynamics is
investigated within the Bloch-Redfield formalism. At low temperatures, when
only the few lowest levels are occupied the susceptibility can be presented as
a sum of contributions from independent level-to-level transitions. This
corresponds to the secular (random phase) approximation in the Bloch-Redfield
formalism. At temperatures comparable to and higher than the oscillator
frequency, many transitions become important and a multiple-peak structure
appears. We show that in this regime the secular approximation breaks down, as
soon as the peaks start to overlap. In other words, the susceptibility is no
longer a sum of contributions from independent transitions. We treat the
dynamics of the system numerically by exact diagonalization of the Hamiltonian
of the qubit plus up to 200 states of the oscillator. We compare the results
obtained with and without the secular approximation and find a qualitative
discrepancy already at moderate temperatures.
