  We study the cavity field's and atomic asymptotic mean excitation numbers due
to anti-rotating term (AMENDART) in the circuit Quantum Electrodynamics
(circuit QED) system, composed of a two-level atom and a single cavity field
mode, subject to Markovian damping and dephasing mechanisms. We show that the
AMENDART are above the thermal values, and their behavior is analyzed
analytically and numerically for typical parameters in circuit QED
implementations described by the Rabi Hamiltonian. We point out that "parasitic
elements", such as other cavity modes or eventual off-resonant atoms, also
contribute substantially to AMENDART.
